I plan to go climb much much higher in the days and weeks ahead using assumptions of sails made or ordinary periodic table elements.

Now, the CMBR incident on the light sail from behind will generally require either a monolithic light sail of near nanometer thickness or perhaps a grid like sail with a cross-weave for which the lines or fibers are separated by less than 0.25 millimeters in order to reflect the vast majority of the incident CMBR for space craft traveling at mildly relativistic velocities. For grid like sails, the advantage of sail porosity enables much higher mass specific capture areas. Since the Doppler blue shifted light incident from directly in front of the sail or nearly so will be much shorter in wavelength than the backwardly incident light for high gamma factor sails, the forwardly incident light can largely pass through the sail openings providing a means for the backwardly incident light to push the sail efficiently forward for cases where the sail is transmissive from front to back to a suitable degree.

Now radiation pressure is equal to σT⁴/C and {2 [σT⁴/C]} for reflected radiation. However, for a space craft traveling through a black body radiation field, the apparent temperature of the radiation increases in proportion to γ and so the black body power impinging on the space craft from directly in front grows in proportion to T⁴ and thus  to γ⁴.

The temperature of a black body is T = {P/[(A)(σ)(e)]}^1/4 = {P/{(A)[5.670400(40) x 10^-8 W m^-2 K^-4]} ^1/4}. Therefore, the cosmic microwave background radiation pressure on a perfectly reflective flat bow  relativistic space craft is roughly equal to  {2 [σT_cmbr ⁴/C]} γ⁴ = {2 [σT_cmbr ⁴/C]} { {1/{1 – [(v/C)²]}^1/2}  ⁴} =   {2 [σT_cmbr ⁴/C]} {1/{1 – [(v/C)²]}²} where γ is constant and T_cmbr is constant. However, the light pressure incident from directly behind will be approximately equal to {2 [σT_cmbr ⁴/C]}/{1/{1 – [(v/C)²]}²}. In actuality, not all of the light is directly incident from the back and so there will be angular affects that result in loss of driving power. However, we will assume that all of the radiation is absorbed and then re-cycled and released as a perfect backwardly directed laser beam.

Assuming that the velocity of the sail starts out at (Zero) C, the optical pressure will be equal to {2 [σT_cmbr ⁴/C]}/γ⁴ = {2 [σT_cmbr ⁴/C]}/{ {1/{1 – [(v/C)²]}^1/2} ⁴} = {2{[5.670400(40) x 10^-8 W m^-2 K^-4][(2.725 K)⁴]/(300,000,000 m/s)}}/(1²) = 2.0844  x 10^-14 Newtons/m². For a 10,000 km by 10,000 km sail, the drive force will be 2.0844 Newtons.

Now, how are we going to deploy such a sail in a meaningful manner? The solution is obvious my dear Watson! Use a grid.

Consider that a monolithic one nanometer thick sail made of STP H₂O density carbonaceous materials would have a mass of  100,000 thousand metric tons, assuming that the space craft plus her sail had a mass of 208,440 metric tons, the craft would start out with an acceleration of F/M = a = 2.0844 N/208,440,000 kg = 10^-8 meter/s². After traveling 6 x 10¹⁵ seconds or about 200,000,000 years, the velocity of the space craft will be 0.20 C and the relativistic Lorentz transformation factor will be 1.0206 thus permitting the above Newtonian formula to apply.

For such a space craft that deployed a linear series of 1,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000 metric tons could be identically accelerated. The mass of the crew quarters would be 108,440,000 metric tons.

For such a space craft that deployed a linear series of 1,000,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000,000  metric tons could be identically accelerated. The mass of the crew quarters would be 108,440,000,000  metric tons. For cosmic journeys, this is not bad.

Some supermaterials already in laboratory existence such as carbon nanotubes can in theory be used to construct large space elevators that would extend from the surface of the Earth near the Equator to locations significantly father than geosynchronous orbit. Such tethers would perhaps have the equivalent of 0.1 G or 1m/s² acceleration based force pulling on it which would be commensurate with a cable roughly 100,000 km long accelerated at 1 m/s². Thus,  a cable that is 10¹³ km long or one light-year long could in theory withstand 10^-8 m/s² levels of acceleration. A linear series of tethered leading sails numbering 1,000,000 where each sail would have a width of 10,000 kilometers and be serially spaced a distance of 100,000 kilometers would have a length of 10¹¹ kilometers.  Thus, a series of 100 million tethered sails might conceivably pull a sail craft combination having a mass of 20,844,000,000,000 metric tons and a crew quarters having a mass of 10,844,000,000,000 metric tons.

Some high-end carbonaceous super-materials include:

1) carbon nano-tubes;
2) boron-nitride nanotubes;
3) buckyball-sheets;
4) layered sheet arrangements of graphene;
5) graphene-oxide paper;
6) fabrics composed of a weave or knit on carbon atom chains;
7) diamond fiber-based fabric;
8) carbon nitride fiber-based fabric;
9) combinations of two or more of the above, and the like material

Metalization would help in these regards.

The sails could have nanotech self-repair mechanisms. An ideal mechanism would entail sails constructed of metallic hydrogen where the hydrogen would be captured from interstellar space and incorporated into the sail membrane(s) in order to re-supply sail atoms knocked loose by interstellar atom and molecular species.

However, much higher sail velocities are anticipatable with much greater accelerations as will be covered in the next post in this series.

However, we can also deploy gridded sails. For example, consider a sail that is comprised on one nanometer wide fibers in a cross-weave where adjacent parallel  fibers are separated by 200 microns. Also consider situations where the fibers are one side reflective and one side transmissive. A sail comprised of such a material will have a mass specific capture area that is 1/100,000 that of a one nanometer thick monolithic sail.

Considering that such a sail that is gridded with the above  one nanometer thick sail fiber construction made of STP H₂O density carbonaceous materials would have a mass of  one metric ton for a sail area of 10⁸ square kilometers, assuming that the space craft plus her sail had a mass of 208,440 metric tons, the craft would start out with an acceleration of F/M = a = 2.0844 N/208,440,000 kg = 10^-8 meter/s². After traveling 6 x 10¹⁵ seconds or about 200,000,000 years, the velocity of the space craft will be 0.20 C and the relativistic Lorentz transformation factor will be 1.0206 thus permitting the above Newtonian formula to apply.

For such a space craft that deployed a linear series of 1,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000 metric tons could be identically accelerated. The mass of the crew quarters would be 208, 439,000 metric tons.

For such a space craft that deployed a linear series of 1,000,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000,000  metric tons could be identically accelerated. The mass of the crew quarters would be 208, 439,000,000 metric tons. For cosmic journeys, this is not bad.

A series of 100 million tethered sails might conceivably pull a sail craft combination having a mass of 20,844,000,000,000 metric tons and a crew quarters having a mass of 20,843,900,000,000 metric tons.

Consider again that such sails which  are gridded with the above  one nanometer thick sail fiber construction made of STP H₂O density carbonaceous materials would have a mass of  one metric ton each, assuming that the space craft plus her sail had a mass of 208,440 metric tons, the craft would start out with an acceleration of F/M = a = 2084.4 N/208,440,000 kg = 10^-5 meter/s² for cases where the craft would utilized 1,000 tethered driving sails. After traveling 6 x 10¹²  seconds or about 200,000 years, the velocity of the space craft will be 0.20 C.

Now consider a space craft having a mass of 208,440 metric tons driven by 10,000 such one metric tons sails. For such a space craft that deployed a linear series of 10,000 tethered sails where each sail was separated by an efficient 10 sail widths, the space craft having a total mass of 208, 440 metric tons would start out with an acceleration of F/M = a = 2084.4 N/208,440,000 kg = 10^-4  meter/s². After traveling 6 x 10¹¹ seconds or about 20,000 years, the velocity of the space craft will be 0.20 C.

We can consider more robust gridded sails such as those made from 10 nanometer diameter fibers spaced 200 microns apart. Each such sail would have a mass of 100 metric  tons. Thus, a space craft having a total mass of 208,440 metric tons that is driven by 1,000 such sails would too start out with an acceleration of F/M = a = 2084.4 N/208,440,000 kg = 10^-5 meter/s² and achieve a velocity of 0.20 C after 200,000 years.

Cnsider a sail that is comprised on 316.2  nanometer wide fibers in a cross-weave where adjacent parallel  fibers are separated by 200 microns. Also consider situations where the fibers are one side reflective and one side transmissive. A sail comprised of such a material will have a mass specific capture area that is equal to that of a one nanometer thick monolithic sail.

Consider that such a sail which is gridded with the above  316.2  nanometer thick sail fiber construction made of STP H₂O density carbonaceous materials would have a mass of  100,000 metric tons for a capture area of 10⁸ square kilometers. Assuming that the space craft plus her sail had a mass of 208,440 metric tons, the craft would start out with an acceleration of F/M = a = 2.0844 N/208,440,000 kg = 10^-8 meter/s². After traveling 6 x 10¹⁵ seconds or about 200,000,000 years, the velocity of the space craft will be 0.20 C and the relativistic Lorentz transformation factor will be 1.0206 thus permitting the above Newtonian formula to apply.

For such a space craft that deployed a linear series of 1,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000 metric tons could be identically accelerated. The mass of the crew quarters would be 208, 439,000 metric tons.

For such a space craft that deployed a linear series of 1,000,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000,000  metric tons could be identically accelerated. The mass of the crew quarters would be 208, 439,000,000 metric tons. For cosmic journeys, this is not bad.

A series of 100 million tethered sails might conceivably pull a sail craft combination having a mass of 20,844,000,000,000 metric tons and a crew quarters having a mass of 20,843,900,000,000 metric tons.

The background interstellar and intergalactic matter might not erode even many of highly relativistic sail of sub-micron thickness.

The diametrical cross-sectional area of our observable universe is close to 10 ⁴⁷  square kilometers and the mass of the total mass energy of the observable universe is only about 10 ⁵⁰ metric tons of which only 4 percent is baryonic.  Thus,  an average column spanning the diameter of the entire visible universe would have an H2O STP matter thickness of only 25 micrometers for reactive matter.

However,  this is not a concern for the following reasons.

First, the sails could be replaceable grid sails and driven by optical, IR, microwave or rf radiation. The mass of such sails can be reduced by many orders of magnitude relative to monolithic sails that are only micrometer scales in thickness.

Second, sails having a very thick cable or thread like construction are conceivable where the cables or wires would be many times if not several orders of magnitude thicker than 25 microns. The sails could be mostly empty space to almost entirely empty space to reflect long wave rF phased array beams.

As for concerns about over burdening the conductive or super-conductive wires or cables used for such sails by extremely intense rF beams, note that such reflective members could be very conductive to superconductive to thereby yield near perfect reflection. The EM energy that was not reflected would largely pass through the openings in the sail grid.

Second, a magnetic and/or electric field based scoop or anti-scoop could divert the chargons away from the sail just as an extended electrodynamic scoop for an interstellar ramjet would. Electro-dynamic-hydro-dynamic-plasma-drive features could utilize the diverted plasma in a reactive and gainful manner.

The sail might be deployed in a manner that is orthogonal to the ship’s velocity vector.  The sail might be parallel to the space craft velocity vector and driven obliquely from behind. This way, the effective thickness of the sail could be thousands of miles and the sail could include electro-dynamic-hydro-dynamic-plasma-drive features.

Fourth, the above parallel sail could conceivably be made of negative refraction index materials that would be pulled forward by incident star light and highly blue-shifted CMBR, far infrared, and non-CMBR radio sources.

Fifth, the sail can simply be a deployed mag-sail or M2P2 type of sail or any other magnetic or plasma bottle sail. It is possible that a plasma affixed to the space craft to be driven by rf radation, and even source based laser light upon attainment of extreme space craft gamma factors could be easily reflected by such sails. Plasma makes an excellent rF reflector even at very small densities.

I have done a lot of writing on parallel sails such as negative refraction index monolithic and grid sails capable of extreme gamma factors.

Sixth, some sail materials such as any future forms of super-strong very conductive to super-conductive metallic hydrogen can be used as nuclear fusion fuel for fusion rockets upon degradation to useless levels.

Seventh, it has been proposed that very thin,  metallic,  very low gas density containing balloons might be used for nuclear warhead decoys and which could survive 100 meter proximity detonation to a one kiloton neutron bomb in the vacuum of space. The rate of radiative cooling would be tens of billions of Kelvins per second due to the extreme thinness of the balloon membranes and most of the neutrons would pass right through the balloon without interacting or by only depositing a very small portion of the particles kinetic energy into the balloon and enclosed gas. Interstellar chargons are more reactive to electronic shell structures but not by that much.

The general idea for obliquely oriented beams involves the beamed energy incident on both sides of the sail. The sail could include a surface of hair like cilia or any other surface contour that would work so as to much more effectively grab ahold of the light.

In addition, the sail could be fabricated from photovoltaic materials in order to provide power for electro-dynamic-hydrodynamic-plasma-drives or chargon rockets, or perhaps even photon rockets.

For extreme gamma factors, the CMBR and starlight will be highly blue-shifted and will be relativistically abberated to what would approach a point source in front of the space craft at gamma = infinity. A sail parallel to the space craft velocity vector made of a suitable negative electromagnetic refraction index material will be pulled forward even by light incident on the sail at a very shallow angle from in front of the space craft.

To enhance the negative refraction index sails capture of EM energy, the sails may have negative index hairs or cilia distributed along its length.

Negative refraction index materials have actually been measured to be pulled on by incident light. Duke University and other academic and government labs are researching the various aspects of negative refraction index materials.

I have no problem with space craft being pulled forward by forward incident light. After all, the paradigm of light speed velocity limits may or may not have been shattered with any future validation or not of the CERN superluminal neutrino results. The big bang may have been the most recent free lunch. There is no reason why the big bang could not have started with miniscule quantities of mass-energy.

A good abstract for a great paper on negative super-pressure of light acting on a negative refractive index material is

Henri Lezec
(Center for Nanoscale Science and Technology, NIST)

Forty years ago, V. Veselago derived the electromagnetic properties of a hypothetical material having simultaneously-negative values of electric permittivity and magnetic permeability [1]. Such a material, denominated “left-handed”, was predicted to exhibit a negative index of refraction, as well as a number of other counter-intuitive optical properties. For example, it was hypothesized that a perfect mirror illuminated with a plane wave would experience a negative radiation pressure (pull) when immersed in a left-handed medium, as opposed to the usual positive radiation pressure experienced when facing a dielectric medium such as air or glass. Since left-handed materials are not available in nature, considerable efforts are currently under way to implement them under the form of artificial “metamaterials” — composite media with tailored bulk optical characteristics resulting from constituent structures which are smaller in both size and density than the effective wavelength in the medium. Here we show how surface-plasmon modes propagating in a stacked array of metal-insulator-metal (MIM) waveguides can be harnessed to yield a volumetric left-handed metamaterial characterized by an in-plane-isotropic negative index of refraction over a broad frequency range spanning the blue and green. By sculpting this material with a focused-ion beam we realize prisms and micro-cantilevers which we use to demonstrate, for the first time, (a) in-plane isotropic negative-refraction at optical frequencies, and (b) negative radiation pressure. We predict and experimentally verify a negative “superpressure”, the magnitude of which exceeds the photon pressure experienced by a perfect mirror by more than a factor of two. 1) V. Veselago, \textit{ Sov. Phys. Usp. }10, p.509 (1968).

Available at:

http://meetings.aps.org/Meeting/MAR09/Event/93172

The sail might not need to  be held by guy lines. A strong magnetic field based coupling or electrical charged based connection might work.

Another option is to fabricate the sail guy lines out of graphene, carbon nanotubes, boron nitride nanotubes, graphene oxide paper, and the like. A cable constructed from such materials could stretch for about 20 to 50 kilometers yet still handle tens to hundreds of Earth G’s. The tensile strength of graphene is close to 18 million PSI for perfect forms.

Materials such as solid quarkoniums and somehow stabilized neutroniums, and perhaps even Higgsiniums would be better yet, but such materials may only exist in nature in extreme mass quantity states as of the present cosmic era.

The collection area of the sail can be very, very, large. A large electro-dynamic scoop could extent very far out from the sail.

Regarding nanotech self-assembly mechanisms, just simply greatly increase the capture area of a electrodynamic scoop to collect enough interstellar materials and use most of the collected interstellar material as an EHPD, an MHPD, or a combination of the two and use the rest of the materials for sail repair.

Regarding holding M2P2 plasma affixed to the ship under high gamma factor condition, simply increase the strength of the fastening fields.

Now regarding interstellar matter density near our solar system of one particle for every 10 cm³, the density would  work out to be a layer of hydrogen or helium atoms about one atom thick for a column that is one light-year long. Not a show stopper for light sails or sails that are electro-dynamically shielded or protected.

If extreme materials are used with excellent reflectance, we could simply use a sail that has a thickness of one millimeter or more and which is monolithic, or better yet,  use a sail with grid lines that are one millimeter or perhaps much greater in thickness. This way, a sail that has an area of only one square kilometer can intercept a beam having an equivalent black body temperature of several thousand Kelvins provided it is constructed of suitably refractive materials.

We could simply use electrodynamic methods of grabbing ahold of the interstellar gas and diverting around the space craft and sail. The power to operate the electrodynamic mechanisms can be supplied by beams. The electrodynamic methods can include lasers for ionization, or rf radiation where the gamma factors are suitably large, magnetic fields, electric fields, plasma fields affixed to the space craft, and the like.

Then there is always the possibilities for sails comprised of truly exotic materials such as somehow stabilized neutroniums, quarkoniums, higgsiniums, monopoliums, and perhaps even raw space-time-mass-energy forms such as the “Yelm” of mid-20th Century big bang theory.

Since one cubic meter of neutronium would have a mass of about 10¹⁵ tons. A 1,000 kilometer long thread of the stuff that has a cross-sectional area of 1,000,000 neutrons would have a mass of only one kilogram. A 1 kilometer long thread having a cross-sectional area of 1 billion neutrons would have a mass of only 1 kilogram. Lines made of quarkoniums could have the same length and cross-section but would be 10 to 1,000 times more massive. Higgsiniums would be all the more massive.

Provided such extreme materials could be developed, they could also serve as electric current carrying magnetic sail components. Anyhow magnetic sails can be made of any ordinary conducting or superconducting period table materials.

It is also conceivable that a hybrid sail can be used where a current carrying magsail would deflect plasma away from a monolithic and grid like light sail or rf sail.

Now, regarding the subject of sail erosion by exposure to interstellar or intergalactic gas, we must realize that the kinetic energy of a gas atom traveling at a velocity of 86.7 percent of the speed of light with respect to the sail would be equal to the binding energy of roughly 10 billion atoms within a sail of micron thickness. Thus, the fact that 10 billion atoms could be dislodged should all of the energy of the gas atom be deposited within the sail. Incident gas atoms having even higher associated gamma factors with respect to the star ship sail could potentially knock loose even more atoms. Perhaps, there is no reason to worry about sail erosion in spite of this for the following reasons.

First, extremely relativistic particles would likely deposit only a small portion of its energy within the sail thereby greatly lessening the number of atoms that would be knocked loose. This fact would apply to chargons as well as neutral incident particles.

Second, for sails of near micron thickness, atoms that were knocked loose would likely simply be re-assimilated by the bulk sail materials. Perhaps the only chance for an atom to be knocked loose would include atoms located on the backward side of the sail.  Atoms for which bonds where broken within the bulk sail material would tend to simply re-bond with adjacent atoms.

Third, since the incident gas or plasma particle would deposit only a small portion of its energy within the sail, the kinetic energy per particle for particles that are knocked loose may be only slightly in excess of the binding energy of the dislodged atoms. Basically, the kinetic energy of the dislodged atoms could likely be re-absorbed and/or radiated away thereby promoting rebinding of the dislodged atoms.

Fourth, for cases where the sail would completely absorb the kinetic energy of the incident gas or plasma particles such as an alpha particle, for the case of a one micron thick sail, the sail would obviously be able to complete stop the chargon without losing it. Thus, any atoms disbonded by the incident chargon would also likely be captured and prevented from leaving the sail material.

Fifth, for grid like sails, the grid lines might be positively chargeable so that incident interstellar or intergalactic ions are pushed away from the grid lines and through the openings within the grid like sails. The effect would be similar to the Vander walls force that keeps neutral atoms from being squeezed together to tightly.

We now perform a reality check on the above formulations.

Consider the space craft at a stationary state. The CMBR appears equally bright from all directions within about 1 part in 30,000.

Now, the apparent angle, θ_s,  of CMBR pre-incident on the space craft at an angle of 90 degrees or with respect to the length of the space craft relative to the source reference frame at v = zero C will appear to be incoming at an angle, θ_o,  of 90 degrees with respect to the space craft,  ship’s reference frame.

If we consider the effects of relativistic aberration, the general formula for apparent shift in angle of incidence of the CMBR from the ship’s perspective is

Cos θ_o = {[cos θ_s] – (v/C)}/{1 – [(v/C) cos θ_s]}

Now say we desire to find the range of CMBR angles incident on the space craft with respect to the space craft reference frame for space craft velocities of 0.20 C.

Now since we are considering an angle of θ_o = π/2, cos θ_o = zero. Using the above formula, we achieve Cos θ_o = {[cos θ_s] – (v/C)}/{1 – [(v/C) cos θ_s]} = Cos π/2 = {[cos θ_s] – (0.20)}/{1 – [(0.20) cos θ_s]} = zero = {[cos θ_s] – (0.20)}/{1 – [(0.20) cos θ_s]}.

Thus, (zero) {1 – [(0.20) cos θ_s]} = {[cos θ_s] – (0.20)} = zero.

Therefore, cos θ_s = 0.20 — > θ_s  = 78.463 degrees. We will make a first order assumption that the incident CMBR from behind has a frequency of f’ = f / {γ [1 + (β cosine θ)]} = f / {1.02062 [1 + [0.2 cosine ( 0)]]} = (0.816497161) f. Thus, we will assume that θ = 0 degrees for the following 5 scenarios where we assume that the CMBR is directly incident from behind.

The radiated power received by the sail will be [(78.463)²/(90²)] f’ = [(78.463)²/(90²)] f / {γ [1 + (β cosine θ)]} = [(78.463)²/(90²)] f / {1.02062 [1 + [0.2 cosine ( 0)]]} = [(78.463)²/(90²)] (0.816497161) f =

Once again, the temperature of a black body is T = {P/[(A)(σ)(e)]}^1/4 = {P/{(A)[5.670400(40) x 10^-8 W m^-2 K^-4]} ^1/4}. The light pressure incident from directly behind will be approximately equal to [(θ_s)²/(90²)] {2 {σ {{(T_cmbr)/{γ [1 + (β cosine θ)]}}}⁴/C}} =  [(78.463)²/(90²)] {2 {[5.670400(40) x 10^-8 W m^-2 K^-4] {[(T_cmbr) (0.816497161) ]⁴}/C}} = 4.632092076  x 10^-15 Newtons/m². In actuality, not all of the light is directly incident from the back and so there will be angular affects that result in loss of driving power. However, we will assume that all of the radiation is absorbed and then re-cycled and released as a perfect backwardly directed laser beam.

Now  E_gain = ʃF^odx = ʃ(0,10²⁵) F^odx = ʃ(0,10²⁵)(10¹⁴) [4.632092076  x 10^-15 N] ^odx = 4.632092076 x 10²⁴ Joules.

Now, a 208,440 metric invariant mass space craft traveling at a starting velocity of 0.2 C has a kinetic energy of {1.02062[M C²]} – [M C²] = {1.02062[208,440,000  C²]} – [208,440,000  C²] =  1.9146 x 10²⁵ Joules – 1.87596 x 10²⁵ Joules = 3.864 x 10²³ Joules. When  4.632092076 x 10²⁴ Joules is added, the total gamma factor becomes [5.01849 x 10²⁴ Joules + 1.87596 x 10²⁵ Joules]/ [1.87596 x 10²⁵ Joules] = 1.2675. The associated space craft velocity will be equal to 0.6142 C.

Likewise doing iterated numerical approximations with v = 0.6142 C to obtain another higher velocity and then repeating the steps over and over again will give a first order approximation for space craft terminal velocity.

So we have reasonably demonstrated that CMBR sails can drive very large space arks to velocities considered fast by interstellar propulsion physicists. Typically, fast interstellar travel occurs at a better part of the speed of light.

However, a much finer scale is needed to produce results for many such steps where the computed velocity would not significantly diverge from the actual velocity obtained.

Note that here, I neglect the effects of mass based astrodynamic drag. I have come up with several mechanisms by which massive astrodynamic drag can be almost entirely eliminated and will post on this subject later this month.

Now consider again that radiation pressure is equal to σT⁴/C and {2 [σT⁴/C]} for reflected radiation. However, for a space craft traveling through a black body radiation field, the apparent temperature of the radiation increases in proportion to γ and so the black body power impinging on the space craft from directly in front grows in proportion to T⁴ and thus  to γ⁴.

The temperature of a black body is T = {P/[(A)(σ)(e)]}^1/4 = {P/{(A)[5.670400(40) x 10^-8 W m^-2 K^-4]} ^1/4}. Therefore, the cosmic microwave background radiation pressure on a perfectly reflective flat bow  relativistic space craft is approximately equal to  {2 [σT_cmbr ⁴/C]} γ⁴ = {2 [σT_cmbr ⁴/C]} { {1/{1 – [(v/C)²]}^1/2}  ⁴} =   {2 [σT_cmbr ⁴/C]} {1/{1 – [(v/C)²]}²} where γ is constant and T_cmbr is constant. However, the light pressure incident from directly behind will be approximately equal to {2 [σT_cmbr ⁴/C]}/{1/{1 – [(v/C)²]}²}. In actuality, not all of the light is directly incident from the back and so there will be angular affects that result in loss of driving power. However, we will assume that all of the radiation is absorbed and then re-cycled and released as a perfect backwardly directed laser beam. We will assume that CMBR light which is forwardly incident completely passes through the sail.

Assuming that the velocity of the sail starts out at (Zero) C, the optical pressure will be equal to {{2 [σT_cmbr⁴/C]}/γ⁴} = {{2 [σT_cmbr⁴/C]}/ { {1/{1 – [(v/C)²]}^1/2} ⁴} } = {{2{[5.670400(40) x 10^-8 W m^-2 K^-4][(2.725 K)⁴]/(300,000,000 m/s)}}/(1²)} = [2.0844355 x 10^-14] Newtons/m². For a 10,000 km by 10,000 km sail, the drive force will be 2.0844 Newtons. A 100,000 km by 100,000 km sail will produce a driving force of 208.44 Newtons.

Now assume that the sail is monolithic, made of one nanometer thick carbonaceous, STP water density materials. The sail would have a mass of 10,000,000 thousand metric tons. Assuming that the space craft plus her sail had a mass of 20,844,000 metric tons, the craft would start out with an acceleration of F/M = a = 208.44 N/20,844,000,000 kg = 10^-8 meter/s². After traveling 6 x 10¹⁵ seconds or about 200,000,000 years, the velocity of the space craft will be 0.20 C and the relativistic Lorentz transformation factor will be 1.0206 thus permitting the above Newtonian formula to apply as a reasonable approximation.

Assume that the background gas and dust that contacts the sail over a path length of 1 light year has an invariant  mass of (0.06667 kg)( 10¹⁴){[ 3 x 10¹⁰]^-1} = 222.2 kg. The momentum of the gas and dust with respect to the sail will be (m)(v) = (222.2 kg)(50,000,000 m/s) =1.111 x 10¹⁰  kg m s. Now Force equal dP/dt. Therefore, the force on the sail is on average equal to F_0.05C  = dP_0.2C/dt = 71.677 Newtons.. For a velocity of 0.02 C, the net propulsive force is 208.44 N – 71.67 N = 136.76 N which will still obviously permit 0.2 C velocities.

For such a space craft that deployed a linear series of 1,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000 metric tons could be identically accelerated. The mass of the crew quarters would be 108,440,000 metric tons. For such a space craft that deployed a linear series of 1,000,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000,000  metric tons could be identically accelerated. The mass of the crew quarters would be 108,440,000,000  metric tons. For cosmic journeys, this is not bad. A series of 100 million tethered sails might conceivably pull a sail craft combination having a mass of 20,844,000,000,000 metric tons and a crew quarters having a mass of 10,844,000,000,000 metric tons. For a velocity of 0.2 C, the each of the latter massed space craft would have the same ratio of backward driving force and massive drag force. The caveat is simply the deployment of commensurate numbers of sails simultaneously in a spatial series along the space craft velocity vector.

Now assume that the sail is a gridded fabric or net made of STP water density conducting 0.4 nanometer wide, one nanometer thick,  carbonaceous fibers that are separated by 0.0005 meters such as in a judicious cross weave spacing. A one square meter portion of the net will have a mass of 0.8 x 10^-12 kilograms. A 0.08 kilogram sail will have a plan-form area of 10⁵ square kilometers. A 8,000 kilogram sail will have a plan-form area of 10¹⁰ square kilometers and will have an acceleration of F/M = A = 208.44 Newtons/8,000 kg = 0.026055 m/s².  A space craft having a total mass of  8,000 metric tons will have an initial acceleration of 0.000026055 m/s².   In 80,000 years, the velocity of the 8,000 metric ton  system will be about 0.215388 C assuming Newtonian approximations.

Now, the 10¹⁰ square kilometer plan-form area gridded sail will have a massive species contact area of [10¹⁰km²]/1,250,000 = 8,000 km². So the background gas and dust that contacts the sail over a path length of 1 light year would have an invariant  mass of (0.06667 kg)( 10¹⁴){[ 3 x 10¹⁰]^-1}/(1,250,000) = 0.00017776 kg. The momentum of the gas and dust with respect to the sail will be (m)(v) = (0.00017776 kg)(50,000,000 m/s) =8,888 kg m/s. Now Force equal dP/dt. Therefore, the force on the sail is on average equal to F_0.2C  = dP_0.2C/dt = 0.000057341Newtons.. For a velocity of 0.2 C, the ratio of the driving force to massive drag force is [208.44 N/ 0.000057341 N] = 3,635,095.

The last result in the above paragraph is intreguing because it suggest that an ordinary CMBR sail-material configuration  could gainfully accelerate even in an environment such as local interstellar space where the baryonic mass density may be as much as one million times that of the observable universe on average.

Now, the apparent angle, θ_s,  of CMBR pre-incident on the space craft at an angle of 90 degrees or with respect to the length of the space craft relative to the source reference frame at v = zero C will appear to be incoming at an angle, θ_o,  of 90 degrees with respect to the space craft,  ship’s reference frame.

If we consider the effects of relativistic aberration, the general formula for apparent shift in angle of incidence of the CMBR from the ship’s perspective is

Cos θ_o = {[cos θ_s] – (v/C)}/{1 – [(v/C) cos θ_s]}. Now assume θ_s = 14.48 degrees, and a gamma factor of 3, Cos θ_o = {[cos (19.47 degrees)] – (0.942809)}/{1 – [(0.942809) cos (19.47 degrees)]}~ 0 — > θ_o = 90 degrees. Now (14.48²)/(90²) => (0.0468)/(4.5⁴) = 0.000114128. Now the grid would need to expand in area by a factor of 0.000114128^-1 = 8,762.

The grid line spacing can be increased by a factor of 3 thus yielding an increase in grid area by a factor of essentially 2. I obtained the areal expansion factor of 2 by inspection of hand-drawn grids although I am certain topologists and geometers have long since figured out the general relationships for various factors of line distance expansion for square gridded figures. However, we still need to increase the grid area by another factor of 4,381. Simply deploying  4,381 + 1 expanded sails = 4,382 expanded sails each having a mass of 8,000 kilograms will produce a complete sail rigging having a mass of 35,056 metric tons. Include a tether sub-rigging to link the sails in a serial distribution along with the rest of the mass of the space craft to yield a total craft mass of 200,000 metric tons and we obtain a forward oriented driving force still equal to 208.44 N. So the background gas and dust that contacts the fully deployed rigging at a gamma factor of 3 over a path length of 1 light year has an invariant  mass of {(0.06667 kg)( 10¹⁴){[ 3 x 10¹⁰]^-1}/(1,250,000)} (4,382)   = [0.00017776  kg](4,382) =  0.778944 kg.

Now,  the formula for relativistic momentum of a massive particle  is M₀ v γ = M₀ v /{{1 – [(v/C)²]}^1/2}. However, the number of interstellar or intergalactic massive particles impinging on a relativistic space craft per unit of time, ship’s frame, t,  is proportional  to (γ)v = {1/{{1 – [(v/C)²]}^1/2}v. Now, dP/dt = F which is is expressed in Newtons. Therefore, the force acting on a space craft, ship’s frame,  from the interstellar massive background is equal to d[(M₀ v γ) (γ)(v/C)ǀ]/dt = d{{M₀ v /{{1 – [(v/C)²]}^1/2}} {1/{{1 – [(v/C)²]}^1/2}}v}/dt where M₀ is the incident mass over the  the constant distance interval of (Delta x) background reference frame. The drag energy is thus equal to {d[(M₀ v γ) (γ)(v/C)]/dt} (Delta x) = {d{{M₀ v /{{1 – [(v/C)²]}^1/2}} {1/{{1 – [(v/C)²]}^1/2}}v/C}/dt} (Delta x) where t is the ship time.  The quantity γv/C = {1/{{1 – [(v/C)²]}^1/2}}v/C is considered dimensionless and is of a constant scalar form.

The momentum of the 0.778944 kg invariant mass with respect to the space craft will be M₀ v γ = M₀ v /{{1 – [(v/C)²]}^1/2} = (0.778944 kg)(282,842,700 m/s)(3) = 6.60956  x 10⁸  kg m/s. The force acting on the space craft will be d[(M₀ v γ) (γ)(v/C)ǀ]/dt = d{{M₀ v /{{1 – [(v/C)²]}^1/2}} {1/{{1 – [(v/C)²]}^1/2}}v}/dt = {[(0.778944 kg  (282,842,700 m/s)(3) (3)[( 282,842,700 m/s) /(300,000,000 m/s)]}/{(31,000,000 s)/[(3)/(0.942816)]} = 191.889 Newtons. The net driving force would be 208.44 N – 191.889 N = 16.551 N.

Thus, it is safe to say that our space craft could accelerate to a velocity of 0.942816 C or to a gamma factor of 3  and still maintain a gainfull driving force. Now KE = ʃ F∙ dx = F(cos α)(Δx) = F(cos 0)(Δx) = (F)(Δx) for parallel force and velocity vectors. Assuming a driving force of 16.551 Newtons during the entire trip to a gamma factor of 3, we will further assume that (F)(Δx) = [3 MC² – MC²] = 2MC² = (2)(200,000,000 kg)[(300,000,000 m/s)²]  =  3.6 x 10²⁵  joules. Dividing the energy by F yields x = distance travelled = [3.6 x 10²⁵J]/(16.551 N) = 2.175 x 10²⁴ meters = 217,510,000 light years.

Now,  a high end carbonaceous super-material having a cross sectional area of 1.3708 x 10^-9 square meters can support 208.44 Newtons assuming a yield strength of 10⁷ kilograms per square inch. A cable having such a cross-section and having a mass of 10,000 metric tons would plausibly have a length of (10,000)[1.3708 x 10^-9] meters or 1.3708 x 10¹³  meters or 1.3708 x 10¹⁰ kilometers. However, a cable with a cross-sectional area 1,000 times as great having the same length would have a mass of 10,000,000 metric tons and could conceivably tether 1,000 of  the previously described 20,000 kilogram gridded sails each having a plan form area of 10¹⁰ square kilometers and each producing a driving force of  208.44 Newtons.  Such a tethered sail system could propel a space craft having a total mass of 20 million metric tons, of which 10,020,000 metric tons would exist as the sail rigging. Each sail would be linearly separated by 1.3708 x 10⁷ kilometers. A cable having a cross-sectional area that is ten time greater yet could power a space craft having a total mass of 200 million metric tons of which 100,200,000 metric tons would be incorporated into the sail rigging.For the latter example, the sails would be separated by 1.3708 x 10⁶ kilomters thus preventing all but trivial shadowing of the driving CMBR at velocities of 0.2 C and only moderate shadowing at a gamma factor of 3.

The value of gamma = 3 is close to the maximum value conceivable with purely backward impinging, CMBR driven planar or plane-like sails, that are oriented orthogonally to the space craft velocity vector where the sails are made of ordinary atomic elements based materials and are of gridded forms for  the next billion years or so. This is because the baryonic mass density of the observable universe will very only slightly over this time period thus at best promoting a slight decrease in drag for the above systems at a gamma factor of 3.  Significantly higher gamma factors with self repairing grids are possible for universal ages that several or more times that of the present universe due to intergalactic massive rareification.

In order to compute a gamma factor of 3, an interpolated value for back-ward red-shift of about 4.5 was assumed as an approximately average value. Since the mass specific capture area and the associated massive drag values used are not absolute requirements, the adjustment of the latter values by a few percent can compensate for inaccuracies in the former estimated average and permit the actual maximum possible gamma factor per given intergalactic massive density to very by perhaps as much as plus or minus a few percent. The point is that because of the subject degrees of freedom in the engineering and applied physics for the above conjectural specific examples, the maximum gamma factor of 3 is a very good ball park to aim for in any future real world systems we will design.

Now, in addition of electromagnetic negative refractive index materials which have been demonstrated within research facilities, it may be possible that massive particle and perhaps even gravitational wave negative refraction index materials could be fashioned into sails that are pulled forward by the incident mass-energies. No one at present really knows if the later two types of materials are possible to construct, however, such materials are tantalizing to consider because of the implications of perpetual and increasing pull sail accelerations,  as long as the mass-energy influxes would not thermally or mechanically over burdern the negative refraction index materials.

However, we do not necessarily need even the still controversial pull sail negative electromagnetic refraction index drives. Ordinary positive index materials that are one way transmissive and which are suitably contoured can provide much higher gamma factors in the present cosmic era even in consideration of massive astrodynamic drag. I will describe such  increasing more extreme scenarios in an ongoing series of posts on the subject of plausible one way transmissive, positive refraction index, material based sails involving intelligible speculations regarding such sails made of ordinary atomic elements.

I will post further on this subject later today.

Regards;

Jim

Copyright James M. Essig  January 26, 2011  All Rights Reserved.

That space and time are intimately tied to electromagnetic radiation is obvious when one considered the ubiquitous inclusion of the speed of light in vacuu as a constant in virtually all special and general relativistic formulations. Even in classical electromagnetic theory, the velocity of light is intimately related to the properties of space time including the magnetic permeability and electric permittivity of free space by the formula C = {1/[μ₀ ɛ₀]}^1/2.

I am sure that most of the concepts expressed within this post have been contemplated by others before.

By now the reader is aware of the concept of light sail(s) driven space craft that can reach relativistic velocities. A space craft traveling at extreme gamma factors using an ordinary beam sail will experience extreme astro-dynamic drag, and the sail would likely be ionized by the drag induced friction. This is largely due to the fact that most beam sail space craft contemplate beam sails that are orthogonally spread  with respect to the craft velocity vector and thus which have a very large surface area to experience forward drag.

Suppose a relativistic rocket was powered by energy captured by an attached square or rectangular CMBR  sail that is  oriented in a perpendicular to the velocity vector of the space craft. The equation for Doppler shifting of  CMBR acting on the sail would then be:  

1 + z = {1 + [ ν (cos θ)/C]}/{[1 - [(v/C)²]]^1/2}

or,

z  = {1 + [ ν (cos θ)/C]}/{[1 - [(v/C)²]]^1/2} – 1

f’ = f / {γ [1 + (β cosine θ)]}

which reduces to F’ = f/γ for a radiation source and space craft moving in a direction perpendicular to the line connecting these reference frames with respect to a space craft observer since cos (π/2) = zero where f represents frequency. Here, θ is the angle of view with respect to the space craft velocity vector or the perceived angle of  radiation incidence on the sail with respect to the direction of space craft travel,  with respect to the space craft.

Now,  the energy of a photon is as follows:

E = [h/(2 π)] ω = hf = hC/λ 

where h is the Planck Constant and λ  is the photon wave-length.

Therefore, the energies of the individual CMBR photons impinging on the light sail oriented in a direction perpendicular to it from the space craft’s perspective from directly behind are equal to:

E + =  hf/{γ [1 + (β cos  θ)]} = hf /{γ [1 + (β cos  (0)]}

which reduces to;

hf /{γ [1 + β ]}.

Now,  the CMBR power impinging on the space craft sail per differential unit of time element (space craft reference frame), per differential unit of angle of pre-incidence (space craft reference frame), per differential element of sail area (space craft reference frame) for black body radiation is a function of γ ⁴. This is because the black body radiation frequency curve peak is proportional to black body source temperature and an incident source photon’s frequency is proportional gamma. Since black body total power emission per unit of surface area is proportional to the  fourth power of the temperature of the black body, the above differential area element of the sail will receive a total power that scales with γ ⁴. Black body emitter frequency distribution scales as a function of gamma relative to a moving observer traveling at a factor of γ with respect to the source for directly approaching observers and 1/ γ for directly receding observers.

Planck’s Law states that

I(ѵ,T)dѵ = {[2hѵ³]/C²}{1/{[e^[(hѵ)/(kT)]] -1}}dѵ

λ_max = b/T

where λ max,  is a function only of the temperature.

P_net = P_emit – P_absorbed

Applying the Stefan–Boltzmann law,

P_net = A σ e (T⁴ – T₀⁴)

where sigma =  σ = (2π⁵k_B⁴)/(15 h³ C²) = (π²k_B⁴)/(60 ђ³ C²)

or  where sigma =  σ = 5.670400(40) x 10^-8 W m^-2 K^-4

Therefore, the apparent spectral temperature of the CMBR radiation incident on the sail per unit angle of CMBR incidence for a stationary sail is:

{[P_cmbr/(A σ e)] ^1/4}

The apparent spectral temperature of the CMBR radiation incident on the sail per unit of apparent angle of incidence of the CMBR with respect to the space craft reference frame-based observer(s) for a sail traveling at a given velocity for backwardly impinging radiation is:

T_app = {∫(0, π/2){{{[P_cmbr/(e σ)]^1/4}/{γ [1 +  [(v/C) cos θ]]}}⁴}{{∫(y₁,y₂){ ∫(x₁,x₂) dx} dy}^-1} dθ}^1/4

= {∫(0, π/2){{{[P_cmbr/(e σ)]^1/4} /{{1/{[1 – [(v/C)² ]] ^1/2}} [1 +  [(v/C) cos θ]]}}⁴}{{∫(y₁,y₂){ ∫(x₁,x₂) dx} dy}^-1} dθ}^1/4

= {∫(0, π/2){{{[Pcmbr/(e σ)]^1/4}/{{1/{[1 – [(v/C)² ]] ^1/2}} [1 + [(v/C) cos θ]]}}⁴}[(∫d A)^-1] dθ}^1/4

where P_cmbr is the background CMBR power incident on the sail, dA is the differential element of sail area with respect to the space craft reference frame, v is the velocity of the space craft with respect to the background, and θ is the angle of radiation incidence on the sail with respect to a sail based observer. Theta ranges from π/2 radians for radiation traveling in an orthogonal direction with respect to the ship velocity vector to zero radians for radiation traveling in a parallel direction with respect to the ship velocity vector.

The total power backwardly incident upon the sail with respect to the sail’s reference frame for a given gamma factor  is  therefore:

P = ∫(y₁,y₂) {∫(x₁,x₂) {∫ (0, π/2) {{{(T_cmbr) /{γ [1 + [(v/C) cos θ]]}}⁴} σ e} dθ}dx}dy

= ∫(y₁,y₂) {∫(x₁,x₂) {∫ (0, π/2) {{{(T_cmbr) /{{1/{[1 + [(v/C)² ]] ^1/2}}  [1 + [(v/C) cos θ]]}}⁴} σ e} dθ}dx}dy

= ∫ {∫ (0, π/2) {{{(T_cmbr) /{{1/{[1 – [(v/C)² ]] ^1/2}}  [1 + [(v/C) cos θ]]}}⁴} σ e} dθ} dA

Here, T_cmbr is the background CMBR temperature.

Note that in the above calculations and the ones that follow, all of the relevant backwardly incident background energies are assumed to be initially absorbed by the sail even if the sail acquires a temperature significantly above absolute zero and thereby produces thermal electromagnetic black body emissions. I describe potential methods of the absorption of nearly all incident radiations even in cases where relativistic aberration would otherwise cause the bulk of the impinging radiation to easily reflect off the sail because of increasingly shallow angles of incidence. The forwardly incident radiation is assumed to completely pass through the sail without exchange of momentum.

We can numerically integrate the relativistic  energy growth of the ship in small time steps as follows:

∫P₁dt₁ + ∫P₂dt₂ + ∫P₃dt₃ +, …, + ∫P_ndt_n

Thus, the following expression can be used to compute relativistic energy gain by the ship in terms of t.

Egain  = Σ (0,n)    { ∫ (t_ai, t_bi) { ∫  (y₁, y₂){ ∫  (x₁,x₂){ ∫ (0,  π/2) {{{T_cmbr /{{{1/{[1 – [(v_i/C) ²]] ^1/2}}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ}dx}dy} dt}      

Here,  t_ai, t_bi, and dt are the times in the background reference frame.

Note, the reason why I assume the latter three times are background reference frame times is such that for a space craft traveling at a velocity of just under 1 C, where gamma is held constant, the energy gain for the space craft will be proportional to the length of the path traveled by the space craft according to the background reference frame. The distance of space craft travel  is proportional to the time of space craft travel with respect to the background reference frame. The same is true for a space craft traveling at any velocity held constant, thus the reason for the performance of the numerical integration for each time step where the velocity is incrementally increased but held constant for each time step.

Alternatively, we can use the following series:

Egain =  Σ (1,n) { ∫ (t_ai, t_bi) {∫ {∫ (0, π/2)  {{{T_cmbr /{{1/{[1 – [(v_i/C) ²]] ^1/2}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ} dA}dt}    

where t_ai and t_bi and dt are the times in the background reference frame.

Now for constant acceleration ship time, T₀ = (c/g) ln {{[[ (C²) + (V₀ ²)] ^1/2]   –   [V₀/[[1 – [(V₀/C) ²]] ^1/2 ]]} { [(C ²) + [[(g)(t)  + [V₀ /[1 – [(V₀/C) ²]] ^1/2 ] ²]] ^1/2] + [(g)(t)] +  [V₀/[[1 – [(V₀/C) ²]] ^1/2]]} / (C ²)}. We can incorporate the expression for T₀ prefaced by the notation Delta to indicate the time steps,  ship time,  of uniform duration ship frame.

For computation in terms of T₀, we obtain:

Egain = Σ (0,n)    {{ ∫  (y₁, y₂){ ∫  (x₁,x₂){ ∫ (0,  π/2) {{{T_cmbr /{{{1/{[1 – [(v_i/C) ²]] ^1/2}}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ}dx}dy} {(Delta) {(c/g) ln {{[[ (C²) + (V_0i ²)] ^1/2]   –   [V_0i/[[1 – [(V_0i/C) ²]] ^1/2 ]]} { [(C ²) + [[(g_i)(t_i)  + [V_0i /[1 – [(V_0i/C) ²]] ^1/2 ] ²]] ^1/2] + [(g_i)(t_i)] +  [V₀/[[1 – [(V_0i/C) ²]] ^1/2]]} / (C ²)}}} {1/{[1 – [(v_i/C) ²]] ^1/2}}}  .     

where t_i is the time in the background reference frame and g_i is the ship acceleration in the ship’s reference frame.

Note that the above formulas provide precise calculations for many numerical iterations involving small increments for velocity increase and small time steps in the ship’s frame.

Alternatively, we can use the following series:

Egain =  Σ (1,n) { {∫ {∫ (0, π/2)  {{{T_cmbr /{{1/{[1 – [(v_i/C) ²]] ^1/2}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ} dA} {(Delta) {(c/g) ln {{[[ (C²) + (V_0i ²)] ^1/2]   –   [V_0i/[[1 – [(V_0i/C) ²]] ^1/2 ]]} { [(C ²) + [[(g_i)(t_i)  + [V_0i /[1 – [(V_0i/C) ²]] ^1/2 ] ²]] ^1/2] + [(g_i)(t_i)] +  [V_0i/[[1 – [(V_0i/C) ²]] ^1/2]]} / (C ²)}}} {1/{[1 – [(v_i/C) ²]] ^1/2}}}             

Another method entails integration with respect to space craft velocity with respect to the background and integration with respect to time as follows:

E_gain = ∫ (v₁, v₂)  {{∫  (t₁, t₂) { ∫ (y₁, y₂){ ∫ (x₁,x₂){ ∫ (0,  π/2) {{{T_cmbr {1/{[1 – [(v/C) ²]] ^1/2}}[1 + [(v/C) cos θ]]} ⁴} σ e}dθ}dx}dy} dt}/v}dv

Or alternatively,

E_gain =  ∫(v₁,v₂) {∫ (t₁,t₂){∫  {∫ (0,  π/2) {{{_Tcmbr/{{{1/{[1 – [(v/C) ²]] ^1/2}}[1 + [(v/C) cos θ]]}}⁴} σ e} dθ} dA}dt}/v}dv

where t₁ and t₂ and dt are the times in the background reference frame.

Now, E_gain in practice needs to take into account the radiative temperature of the sail.

Now, given that

P_net = A σ e (T⁴ – T₀⁴)

where T is the body temperature and T₀ is the surrounding temperature, we can re-interpret T as the impinging radiation’s black body temperature and T₀ as the emitted thermal radiation black body temperature. So in other words, if the impinging temperature is 10 times higher in Kelvins then the thermal radiative temperature, the net power input into the sail is 10⁴ or 10,000 times greater than the power loss through radiative emissions.

The net power delivered to the  sail will be equal to the power intake minus the power thermally radiated as

P = {∫(y₁,y₂) {∫(x₁,x₂) {∫ (0, π/2) {{{(T_cmbr) /{γ [1 + [(v/C) cos θ]]}}⁴} σ e} dθ}dx}dy} – {∫(y₁,y₂) {∫(x₁,x₂) (T₀⁴ σ e) dx}dy}

= {∫(y₁,y₂) {∫(x₁,x₂) {∫ (0, π/2) {{{(T_cmbr) /{{1/{[1 + [(v/C)² ]] ^1/2}}  [1 + [(v/C) cos θ]]}}⁴} σ e} dθ}dx}dy}  – {∫(y₁,y₂) {∫(x₁,x₂) (T₀⁴ σ e) dx}dy}

= { ∫ {∫ (0, π/2) {{{(T_cmbr) /{{1/{[1 – [(v/C)² ]] ^1/2}}  [1 + [(v/C) cos θ]]}}⁴} σ e} dθ} dA} -

- {∫(T₀⁴ σ e) dA}

The following expression can be used to compute relativistic energy gain by the ship in consideration of the black body emissions from the sail heated by CMBR.

From computation in terms of t, we obtain:

Egain  = {Σ (0,n)   { ∫ (t_ai, t_bi) { ∫  (y₁, y₂){ ∫  (x₁,x₂){ ∫ (0,  π/2) {{{T_cmbr /{{{1/{[1 – [(v_i/C) ²]] ^1/2}}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ}dx}dy} dt}}   – {Σ (0,n) {∫ (t_ai, t_bi) {∫(y₁,y₂) {∫(x₁,x₂) (T⁴_0i σ e) dx}dy}dt}}     

where t_ai and t_bi and dt are the times in the background reference frame.

Now ship time = T₀ = {(c/g_n) ln {[[ (C²) + (V₀ ²)] ^1/2]   –   [V₀/[[1 – [(V₀/C) ²]] ^1/2 ]]} { [(C ²) + [[(g_n)(t_n)  + [V₀ /[1 – [(V₀/C) ²]] ^1/2 ] ²]] ^1/2] + [(g_n)(t_n)] +  [V₀/[[1 – [(V₀/C) ²]] ^1/2]]} / (C ²)}}

Computation in terms of T₀, we obtain:

Egain  = {Σ (0,n)   {{ ∫  (y₁, y₂){ ∫  (x₁,x₂){ ∫ (0,  π/2) {{{T_cmbr /{{{1/{[1 – [(v_i/C) ²]] ^1/2}}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ}dx}dy} {(Delta) {(c/g_i) ln {{[[ (C²) + (V_0i ²)] ^1/2]   –   [V_0i/[[1 – [(V_0i/C) ²]] ^1/2 ]]} { [(C ²) + [[(g_i)(t)  + [V_0i /[1 – [(V_0i/C) ²]] ^1/2 ] ²]] ^1/2] + [(g_i)(t)] +  [V_0i/[[1 – [(V_0i/C) ²]] ^1/2]]} / (C ²)}}} {1/{[1 – [(v_i/C)² ]] ^1/2}}}}    – {Σ (0,n)  {{∫(y₁,y₂) {∫(x₁,x₂) (T⁴_0i σ e) dx}dy}    {(Delta) {(c/g_i) ln {{[[ (C²) + (V_0i ²)] ^1/2]   –   [V_0i/[[1 – [(V_0i/C) ²]] ^1/2 ]]} { [(C ²) + [[(g_i)(t_i)  + [V_0i /[1 – [(V_0i/C) ²]] ^1/2 ] ²]] ^1/2] + [(g_i)(t_i)] +  [V_0i/[[1 – [(V_0i/C) ²]] ^1/2]]} / (C ²)}}} {1/{[1 – [(v_i/C)² ]] ^1/2}} }} 

where T₀ is the ship time.

Alternatively, we can use the following series calculated with  t:

Egain  =  {Σ (1,n) { ∫ (t_ai, t_bi) {∫ {∫ (0, π/2)  {{{T_cmbr /{{1/{[1 – [(v_i/C) ²]] ^1/2}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ} dA}dt}}   -  {Σ (1,n) ∫ (t_ai, t_bi) {∫ (T⁴ _0i σ e) dA}dt}   

where t_ai and t_bi and dt are the times in the background reference frame.

Calculating with respect to T₀, we obtain:

Egain =  {Σ (1,n)  {∫ {∫ (0, π/2)  {{{T_cmbr /{{1/{[1 – [(v_i/C) ²]] ^1/2}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ} dA}}   -  {Σ (1,n) {∫ (T⁴ _0i σ e) dA}   {(Delta) {(c/g_i) ln {{[[ (C²) + (V_0i ²)] ^1/2]   –   [V_0i/[[1 – [(V_0i/C) ²]] ^1/2 ]]} { [(C ²) + [[(g_i)(t_i)  + [V_0i /[1 – [(V_0i/C) ²]] ^1/2 ] ²]] ^1/2] + [(g_i)(t_i)] +  [V_0i/[[1 – [(V_0i/C) ²]] ^1/2]]} / (C ²)}}} {1/{[1 – [(v_i/C)² ]] ^1/2}}}.

Integrating with respect to time and velocity;

the formulas for total kinetic energy  gain are:

E_gain = {∫ (v₁, v₂)  {{∫  (t₁, t₂) { ∫ (y₁, y₂){ ∫ (x₁,x₂){ ∫ (0,  π/2) {{{T_cmbr {1/{[1 – [(v/C) ²]] ^1/2}}[1 + [(v/C) cos θ]]} ⁴} σ e}dθ}dx}dy} dt}/v}dv} -  E_{rad lost}

or alternatively,

E_gain =  {∫(v₁,v₂) {{∫ (t₁,t₂){∫  {∫ (0,  π/2) {{{_Tcmbr/{{1/{[1 – [(v/C) ²]] ^1/2}}[1 + [(v/C) cos θ]]}}⁴} σ e} dθ} dA}dt}/v}dv} -  E_{rad lost}

where t_ai and t_bi , t_i, and dt are the times in the background reference frame, g_i is the ship acceleration in the ship’s reference frame, and V_0i is the starting velocity at the beginning of each time of Delta T₀,  or ship time.

Now, the CMBR incident on the light sail from behind will generally require either a monolithic light sail of near nanometer thickness or perhaps a grid like sail with a cross-weave for which the lines or fibers are separated by less than 0.25 millimeters in order to reflect the vast majority of the incident CMBR for space craft traveling at mildly relativistic velocities. For grid like sails, the advantage of sail porosity enables much higher mass specific capture areas. Since the Doppler blue shifted light incident from directly in front of the sail or nearly so will be much shorter in wavelength than the backwardly incident light for high gamma factor sails, the forwardly incident light can largely pass through the sail openings providing a means for the backwardly incident light to push the sail efficiently forward for cases where the sail is transmissive from front to back to a suitable degree.

Now radiation pressure is equal to σT⁴/C and {2 [σT⁴/C]} for reflected radiation. However, for a space craft traveling through a black body radiation field, the apparent temperature of the radiation increases in proportion to γ and so the black body power impinging on the space craft from directly in front grows in proportion to T⁴ and thus  to γ⁴.

The temperature of a black body is T = {P/[(A)(σ)(e)]}^1/4 = {P/{(A)[5.670400(40) x 10^-8 W m^-2 K^-4]} ^1/4}. Therefore, the cosmic microwave background radiation pressure on a perfectly reflective flat bow  relativistic space craft is roughly equal to  {2 [σT_cmbr ⁴/C]} γ⁴ = {2 [σT_cmbr ⁴/C]} { {1/{1 – [(v/C)²]}^1/2}  ⁴} =   {2 [σT_cmbr ⁴/C]} {1/{1 – [(v/C)²]}²} where γ is constant and T_cmbr is constant. However, the light pressure incident from directly behind will be approximately equal to {2 [σT_cmbr ⁴/C]}/{1/{1 – [(v/C)²]}²}. In actuality, not all of the light is directly incident from the back and so there will be angular affects that result in loss of driving power. However, we will assume that all of the radiation is absorbed and then re-cycled and released as a perfect backwardly directed laser beam.

Assuming that the velocity of the sail starts out at (Zero) C, the optical pressure will be equal to {2 [σT_cmbr ⁴/C]}/γ⁴ = {2 [σT_cmbr ⁴/C]}/{ {1/{1 – [(v/C)²]}^1/2} ⁴} = {2{[5.670400(40) x 10^-8 W m^-2 K^-4][(2.725 K)⁴]/(300,000,000 m/s)}}/(1²) = 2.0844  x 10^-14 Newtons/m². For a 10,000 km by 10,000 km sail, the drive force will be 2.0844 Newtons.

Now, how are we going to deploy such a sail in a meaningful manner? The solution is obvious my dear Watson! Use a grid.

Consider that a monolithic one nanometer thick sail made of STP H₂O density carbonaceous materials would have a mass of  100,000 thousand metric tons, assuming that the space craft plus her sail had a mass of 208,440 metric tons, the craft would start out with an acceleration of F/M = a = 2.0844 N/208,440,000 kg = 10^-8 meter/s². After traveling 6 x 10¹⁵ seconds or about 200,000,000 years, the velocity of the space craft will be 0.20 C and the relativistic Lorentz transformation factor will be 1.0206 thus permitting the above Newtonian formula to apply.

For such a space craft that deployed a linear series of 1,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000 metric tons could be identically accelerated. The mass of the crew quarters would be 108,440,000 metric tons.

For such a space craft that deployed a linear series of 1,000,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000,000  metric tons could be identically accelerated. The mass of the crew quarters would be 108,440,000,000  metric tons. For cosmic journeys, this is not bad.

Some supermaterials already in laboratory existence such as carbon nanotubes can in theory be used to construct large space elevators that would extend from the surface of the Earth near the Equator to locations significantly father than geosynchronous orbit. Such tethers would perhaps have the equivalent of 0.1 G or 1m/s² acceleration based force pulling on it which would be commensurate with a cable roughly 100,000 km long accelerated at 1 m/s². Thus,  a cable that is 10¹³ km long or one light-year long could in theory withstand 10^-8 m/s² levels of acceleration. A linear series of tethered leading sails numbering 1,000,000 where each sail would have a width of 10,000 kilometers and be serially spaced a distance of 100,000 kilometers would have a length of 10¹¹ kilometers.  Thus, a series of 100 million tethered sails might conceivably pull a sail craft combination having a mass of 20,844,000,000,000 metric tons and a crew quarters having a mass of 10,844,000,000,000 metric tons.

Some high-end carbonaceous super-materials include:

1) carbon nano-tubes;
2) boron-nitride nanotubes;
3) buckyball-sheets;
4) layered sheet arrangements of graphene;
5) graphene-oxide paper;
6) fabrics composed of a weave or knit on carbon atom chains;
7) diamond fiber-based fabric;
8) carbon nitride fiber-based fabric;
9) combinations of two or more of the above, and the like material

Metalization would help in these regards.

The sails could have nanotech self-repair mechanisms. An ideal mechanism would entail sails constructed of metallic hydrogen where the hydrogen would be captured from interstellar space and incorporated into the sail membrane(s) in order to re-supply sail atoms knocked loose by interstellar atom and molecular species.

However, much higher sail velocities are anticipatable with much greater accelerations as will be covered in the next post in this series.

However, we can also deploy gridded sails. For example, consider a sail that is comprised on one nanometer wide fibers in a cross-weave where adjacent parallel  fibers are separated by 200 microns. Also consider situations where the fibers are one side reflective and one side transmissive. A sail comprised of such a material will have a mass specific capture area that is 1/100,000 that of a one nanometer thick monolithic sail.

Considering that such a sail that is gridded with the above  one nanometer thick sail fiber construction made of STP H₂O density carbonaceous materials would have a mass of  one metric ton for a sail area of 10⁸ square kilometers, assuming that the space craft plus her sail had a mass of 208,440 metric tons, the craft would start out with an acceleration of F/M = a = 2.0844 N/208,440,000 kg = 10^-8 meter/s². After traveling 6 x 10¹⁵ seconds or about 200,000,000 years, the velocity of the space craft will be 0.20 C and the relativistic Lorentz transformation factor will be 1.0206 thus permitting the above Newtonian formula to apply.

For such a space craft that deployed a linear series of 1,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000 metric tons could be identically accelerated. The mass of the crew quarters would be 208, 439,000 metric tons.

For such a space craft that deployed a linear series of 1,000,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000,000  metric tons could be identically accelerated. The mass of the crew quarters would be 208, 439,000,000 metric tons. For cosmic journeys, this is not bad.

A series of 100 million tethered sails might conceivably pull a sail craft combination having a mass of 20,844,000,000,000 metric tons and a crew quarters having a mass of 20,843,900,000,000 metric tons.

Consider again that such sails which  are gridded with the above  one nanometer thick sail fiber construction made of STP H₂O density carbonaceous materials would have a mass of  one metric ton each, assuming that the space craft plus her sail had a mass of 208,440 metric tons, the craft would start out with an acceleration of F/M = a = 2084.4 N/208,440,000 kg = 10^-5 meter/s² for cases where the craft would utilized 1,000 tethered driving sails. After traveling 6 x 10¹²  seconds or about 200,000 years, the velocity of the space craft will be 0.20 C.

Now consider a space craft having a mass of 208,440 metric tons driven by 10,000 such one metric tons sails. For such a space craft that deployed a linear series of 10,000 tethered sails where each sail was separated by an efficient 10 sail widths, the space craft having a total mass of 208, 440 metric tons would start out with an acceleration of F/M = a = 2084.4 N/208,440,000 kg = 10^-4  meter/s². After traveling 6 x 10¹¹ seconds or about 20,000 years, the velocity of the space craft will be 0.20 C.

We can consider more robust gridded sails such as those made from 10 nanometer diameter fibers spaced 200 microns apart. Each such sail would have a mass of 100 metric  tons. Thus, a space craft having a total mass of 208,440 metric tons that is driven by 1,000 such sails would too start out with an acceleration of F/M = a = 2084.4 N/208,440,000 kg = 10^-5 meter/s² and achieve a velocity of 0.20 C after 200,000 years.

Cnsider a sail that is comprised on 316.2  nanometer wide fibers in a cross-weave where adjacent parallel  fibers are separated by 200 microns. Also consider situations where the fibers are one side reflective and one side transmissive. A sail comprised of such a material will have a mass specific capture area that is equal to that of a one nanometer thick monolithic sail.

Consider that such a sail which is gridded with the above  316.2  nanometer thick sail fiber construction made of STP H₂O density carbonaceous materials would have a mass of  100,000 metric tons for a capture area of 10⁸ square kilometers. Assuming that the space craft plus her sail had a mass of 208,440 metric tons, the craft would start out with an acceleration of F/M = a = 2.0844 N/208,440,000 kg = 10^-8 meter/s². After traveling 6 x 10¹⁵ seconds or about 200,000,000 years, the velocity of the space craft will be 0.20 C and the relativistic Lorentz transformation factor will be 1.0206 thus permitting the above Newtonian formula to apply.

For such a space craft that deployed a linear series of 1,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000 metric tons could be identically accelerated. The mass of the crew quarters would be 208, 439,000 metric tons.

For such a space craft that deployed a linear series of 1,000,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000,000  metric tons could be identically accelerated. The mass of the crew quarters would be 208, 439,000,000 metric tons. For cosmic journeys, this is not bad.

A series of 100 million tethered sails might conceivably pull a sail craft combination having a mass of 20,844,000,000,000 metric tons and a crew quarters having a mass of 20,843,900,000,000 metric tons.

The background interstellar and intergalactic matter might not erode even many of highly relativistic sail of sub-micron thickness.

The diametrical cross-sectional area of our observable universe is close to 10 ⁴⁷  square kilometers and the mass of the total mass energy of the observable universe is only about 10 ⁵⁰ metric tons of which only 4 percent is baryonic.  Thus,  an average column spanning the diameter of the entire visible universe would have an H2O STP matter thickness of only 25 micrometers for reactive matter.

However,  this is not a concern for the following reasons.

First, the sails could be replaceable grid sails and driven by optical, IR, microwave or rf radiation. The mass of such sails can be reduced by many orders of magnitude relative to monolithic sails that are only micrometer scales in thickness.

Second, sails having a very thick cable or thread like construction are conceivable where the cables or wires would be many times if not several orders of magnitude thicker than 25 microns. The sails could be mostly empty space to almost entirely empty space to reflect long wave rF phased array beams.

As for concerns about over burdening the conductive or super-conductive wires or cables used for such sails by extremely intense rF beams, note that such reflective members could be very conductive to superconductive to thereby yield near perfect reflection. The EM energy that was not reflected would largely pass through the openings in the sail grid.

Second, a magnetic and/or electric field based scoop or anti-scoop could divert the chargons away from the sail just as an extended electrodynamic scoop for an interstellar ramjet would. Electro-dynamic-hydro-dynamic-plasma-drive features could utilize the diverted plasma in a reactive and gainful manner.

The sail might be deployed in a manner that is orthogonal to the ship’s velocity vector.  The sail might be parallel to the space craft velocity vector and driven obliquely from behind. This way, the effective thickness of the sail could be thousands of miles and the sail could include electro-dynamic-hydro-dynamic-plasma-drive features.

Fourth, the above parallel sail could conceivably be made of negative refraction index materials that would be pulled forward by incident star light and highly blue-shifted CMBR, far infrared, and non-CMBR radio sources.

Fifth, the sail can simply be a deployed mag-sail or M2P2 type of sail or any other magnetic or plasma bottle sail. It is possible that a plasma affixed to the space craft to be driven by rf radation, and even source based laser light upon attainment of extreme space craft gamma factors could be easily reflected by such sails. Plasma makes an excellent rF reflector even at very small densities.

I have done a lot of writing on parallel sails such as negative refraction index monolithic and grid sails capable of extreme gamma factors.

Sixth, some sail materials such as any future forms of super-strong very conductive to super-conductive metallic hydrogen can be used as nuclear fusion fuel for fusion rockets upon degradation to useless levels.

Seventh, it has been proposed that very thin,  metallic,  very low gas density containing balloons might be used for nuclear warhead decoys and which could survive 100 meter proximity detonation to a one kiloton neutron bomb in the vacuum of space. The rate of radiative cooling would be tens of billions of Kelvins per second due to the extreme thinness of the balloon membranes and most of the neutrons would pass right through the balloon without interacting or by only depositing a very small portion of the particles kinetic energy into the balloon and enclosed gas. Interstellar chargons are more reactive to electronic shell structures but not by that much.

The general idea for obliquely oriented beams involves the beamed energy incident on both sides of the sail. The sail could include a surface of hair like cilia or any other surface contour that would work so as to much more effectively grab ahold of the light.

In addition, the sail could be fabricated from photovoltaic materials in order to provide power for electro-dynamic-hydrodynamic-plasma-drives or chargon rockets, or perhaps even photon rockets.

For extreme gamma factors, the CMBR and starlight will be highly blue-shifted and will be relativistically abberated to what would approach a point source in front of the space craft at gamma = infinity. A sail parallel to the space craft velocity vector made of a suitable negative electromagnetic refraction index material will be pulled forward even by light incident on the sail at a very shallow angle from in front of the space craft.

To enhance the negative refraction index sails capture of EM energy, the sails may have negative index hairs or cilia distributed along its length.

Negative refraction index materials have actually been measured to be pulled on by incident light. Duke University and other academic and government labs are researching the various aspects of negative refraction index materials.

I have no problem with space craft being pulled forward by forward incident light. After all, the paradigm of light speed velocity limits may or may not have been shattered with any future validation or not of the CERN superluminal neutrino results. The big bang may have been the most recent free lunch. There is no reason why the big bang could not have started with miniscule quantities of mass-energy.

A good abstract for a great paper on negative super-pressure of light acting on a negative refractive index material is

Henri Lezec
(Center for Nanoscale Science and Technology, NIST)

Forty years ago, V. Veselago derived the electromagnetic properties of a hypothetical material having simultaneously-negative values of electric permittivity and magnetic permeability [1]. Such a material, denominated “left-handed”, was predicted to exhibit a negative index of refraction, as well as a number of other counter-intuitive optical properties. For example, it was hypothesized that a perfect mirror illuminated with a plane wave would experience a negative radiation pressure (pull) when immersed in a left-handed medium, as opposed to the usual positive radiation pressure experienced when facing a dielectric medium such as air or glass. Since left-handed materials are not available in nature, considerable efforts are currently under way to implement them under the form of artificial “metamaterials” — composite media with tailored bulk optical characteristics resulting from constituent structures which are smaller in both size and density than the effective wavelength in the medium. Here we show how surface-plasmon modes propagating in a stacked array of metal-insulator-metal (MIM) waveguides can be harnessed to yield a volumetric left-handed metamaterial characterized by an in-plane-isotropic negative index of refraction over a broad frequency range spanning the blue and green. By sculpting this material with a focused-ion beam we realize prisms and micro-cantilevers which we use to demonstrate, for the first time, (a) in-plane isotropic negative-refraction at optical frequencies, and (b) negative radiation pressure. We predict and experimentally verify a negative “superpressure”, the magnitude of which exceeds the photon pressure experienced by a perfect mirror by more than a factor of two. 1) V. Veselago, \textit{ Sov. Phys. Usp. }10, p.509 (1968).

Available at:

http://meetings.aps.org/Meeting/MAR09/Event/93172

The sail might not need to  be held by guy lines. A strong magnetic field based coupling or electrical charged based connection might work.

Another option is to fabricate the sail guy lines out of graphene, carbon nanotubes, boron nitride nanotubes, graphene oxide paper, and the like. A cable constructed from such materials could stretch for about 20 to 50 kilometers yet still handle tens to hundreds of Earth G’s. The tensile strength of graphene is close to 18 million PSI for perfect forms.

Materials such as solid quarkoniums and somehow stabilized neutroniums, and perhaps even Higgsiniums would be better yet, but such materials may only exist in nature in extreme mass quantity states as of the present cosmic era.

The collection area of the sail can be very, very, large. A large electro-dynamic scoop could extent very far out from the sail.

Regarding nanotech self-assembly mechanisms, just simply greatly increase the capture area of a electrodynamic scoop to collect enough interstellar materials and use most of the collected interstellar material as an EHPD, an MHPD, or a combination of the two and use the rest of the materials for sail repair.

Regarding holding M2P2 plasma affixed to the ship under high gamma factor condition, simply increase the strength of the fastening fields.

Now regarding interstellar matter density near our solar system of one particle for every 10 cm³, the density would  work out to be a layer of hydrogen or helium atoms about one atom thick for a column that is one light-year long. Not a show stopper for light sails or sails that are electro-dynamically shielded or protected.

If extreme materials are used with excellent reflectance, we could simply use a sail that has a thickness of one millimeter or more and which is monolithic, or better yet,  use a sail with grid lines that are one millimeter or perhaps much greater in thickness. This way, a sail that has an area of only one square kilometer can intercept a beam having an equivalent black body temperature of several thousand Kelvins provided it is constructed of suitably refractive materials.

We could simply use electrodynamic methods of grabbing ahold of the interstellar gas and diverting around the space craft and sail. The power to operate the electrodynamic mechanisms can be supplied by beams. The electrodynamic methods can include lasers for ionization, or rf radiation where the gamma factors are suitably large, magnetic fields, electric fields, plasma fields affixed to the space craft, and the like.

Then there is always the possibilities for sails comprised of truly exotic materials such as somehow stabilized neutroniums, quarkoniums, higgsiniums, monopoliums, and perhaps even raw space-time-mass-energy forms such as the “Yelm” of mid-20th Century big bang theory.

Since one cubic meter of neutronium would have a mass of about 10¹⁵ tons. A 1,000 kilometer long thread of the stuff that has a cross-sectional area of 1,000,000 neutrons would have a mass of only one kilogram. A 1 kilometer long thread having a cross-sectional area of 1 billion neutrons would have a mass of only 1 kilogram. Lines made of quarkoniums could have the same length and cross-section but would be 10 to 1,000 times more massive. Higgsiniums would be all the more massive.

Provided such extreme materials could be developed, they could also serve as electric current carrying magnetic sail components. Anyhow magnetic sails can be made of any ordinary conducting or superconducting period table materials.

It is also conceivable that a hybrid sail can be used where a current carrying magsail would deflect plasma away from a monolithic and grid like light sail or rf sail.

Now, regarding the subject of sail erosion by exposure to interstellar or intergalactic gas, we must realize that the kinetic energy of a gas atom traveling at a velocity of 86.7 percent of the speed of light with respect to the sail would be equal to the binding energy of roughly 10 billion atoms within a sail of micron thickness. Thus, the fact that 10 billion atoms could be dislodged should all of the energy of the gas atom be deposited within the sail. Incident gas atoms having even higher associated gamma factors with respect to the star ship sail could potentially knock loose even more atoms. Perhaps, there is no reason to worry about sail erosion in spite of this for the following reasons.

First, extremely relativistic particles would likely deposit only a small portion of its energy within the sail thereby greatly lessening the number of atoms that would be knocked loose. This fact would apply to chargons as well as neutral incident particles.

Second, for sails of near micron thickness, atoms that were knocked loose would likely simply be re-assimilated by the bulk sail materials. Perhaps the only chance for an atom to be knocked loose would include atoms located on the backward side of the sail.  Atoms for which bonds where broken within the bulk sail material would tend to simply re-bond with adjacent atoms.

Third, since the incident gas or plasma particle would deposit only a small portion of its energy within the sail, the kinetic energy per particle for particles that are knocked loose may be only slightly in excess of the binding energy of the dislodged atoms. Basically, the kinetic energy of the dislodged atoms could likely be re-absorbed and/or radiated away thereby promoting rebinding of the dislodged atoms.

Fourth, for cases where the sail would completely absorb the kinetic energy of the incident gas or plasma particles such as an alpha particle, for the case of a one micron thick sail, the sail would obviously be able to complete stop the chargon without losing it. Thus, any atoms disbonded by the incident chargon would also likely be captured and prevented from leaving the sail material.

Fifth, for grid like sails, the grid lines might be positively chargeable so that incident interstellar or intergalactic ions are pushed away from the grid lines and through the openings within the grid like sails. The effect would be similar to the Vander walls force that keeps neutral atoms from being squeezed together to tightly.

We now perform a reality check on the above formulations.

Consider the space craft at a stationary state. The CMBR appears equally bright from all directions within about 1 part in 30,000.

Now, the apparent angle, θ_s,  of CMBR pre-incident on the space craft at an angle of 90 degrees or with respect to the length of the space craft relative to the source reference frame at v = zero C will appear to be incoming at an angle, θ_o,  of 90 degrees with respect to the space craft,  ship’s reference frame.

If we consider the effects of relativistic aberration, the general formula for apparent shift in angle of incidence of the CMBR from the ship’s perspective is

Cos θ_o = {[cos θ_s] – (v/C)}/{1 – [(v/C) cos θ_s]}

Now say we desire to find the range of CMBR angles incident on the space craft with respect to the space craft reference frame for space craft velocities of 0.20 C.

Now since we are considering an angle of θ_o = π/2, cos θ_o = zero. Using the above formula, we achieve Cos θ_o = {[cos θ_s] – (v/C)}/{1 – [(v/C) cos θ_s]} = Cos π/2 = {[cos θ_s] – (0.20)}/{1 – [(0.20) cos θ_s]} = zero = {[cos θ_s] – (0.20)}/{1 – [(0.20) cos θ_s]}.

Thus, (zero) {1 – [(0.20) cos θ_s]} = {[cos θ_s] – (0.20)} = zero.

Therefore, cos θ_s = 0.20 — > θ_s  = 78.463 degrees. We will make a first order assumption that the incident CMBR from behind has a frequency of f’ = f / {γ [1 + (β cosine θ)]} = f / {1.02062 [1 + [0.2 cosine ( 0)]]} = (0.816497161) f. Thus, we will assume that θ = 0 degrees for the following 5 scenarios where we assume that the CMBR is directly incident from behind.

The radiated power received by the sail will be [(78.463)²/(90²)] f’ = [(78.463)²/(90²)] f / {γ [1 + (β cosine θ)]} = [(78.463)²/(90²)] f / {1.02062 [1 + [0.2 cosine ( 0)]]} = [(78.463)²/(90²)] (0.816497161) f =

Once again, the temperature of a black body is T = {P/[(A)(σ)(e)]}^1/4 = {P/{(A)[5.670400(40) x 10^-8 W m^-2 K^-4]} ^1/4}. The light pressure incident from directly behind will be approximately equal to [(θ_s)²/(90²)] {2 {σ {{(T_cmbr)/{γ [1 + (β cosine θ)]}}}⁴/C}} =  [(78.463)²/(90²)] {2 {[5.670400(40) x 10^-8 W m^-2 K^-4] {[(T_cmbr) (0.816497161) ]⁴}/C}} = 4.632092076  x 10^-15 Newtons/m². In actuality, not all of the light is directly incident from the back and so there will be angular affects that result in loss of driving power. However, we will assume that all of the radiation is absorbed and then re-cycled and released as a perfect backwardly directed laser beam.

Now  E_gain = ʃF^odx = ʃ(0,10²⁵) F^odx = ʃ(0,10²⁵)(10¹⁴) [4.632092076  x 10^-15 N] ^odx = 4.632092076 x 10²⁴ Joules.

Now, a 208,440 metric invariant mass space craft traveling at a starting velocity of 0.2 C has a kinetic energy of {1.02062[M C²]} – [M C²] = {1.02062[208,440,000  C²]} – [208,440,000  C²] =  1.9146 x 10²⁵ Joules – 1.87596 x 10²⁵ Joules = 3.864 x 10²³ Joules. When  4.632092076 x 10²⁴ Joules is added, the total gamma factor becomes [5.01849 x 10²⁴ Joules + 1.87596 x 10²⁵ Joules]/ [1.87596 x 10²⁵ Joules] = 1.2675. The associated space craft velocity will be equal to 0.6142 C.

Likewise doing iterated numerical approximations with v = 0.6142 C to obtain another higher velocity and then repeating the steps over and over again will give a first order approximation for space craft terminal velocity.

So we have reasonably demonstrated that CMBR sails can drive very large space arks to velocities considered fast by interstellar propulsion physicists. Typically, fast interstellar travel occurs at a better part of the speed of light.

However, a much finer scale is needed to produce results for many such steps where the computed velocity would not significantly diverge from the actual velocity obtained.

Note that here, I neglect the effects of mass based astrodynamic drag. I have come up with several mechanisms by which massive astrodynamic drag can be almost entirely eliminated and will post on this subject later this month.

The total kinetic energy gain for the craft will be

E _totalgain = ʃF ₁^odx₁  +  ʃF ₂^odx₂ + ʃF ₃^odx₃ + … +ʃF _n^odx_n

= {{ʃ{[(θ_s1)²/ ((90 degrees) ²)] {2 {σ {{(T_cmbr)/{γ₁ [1 + (β₁ cosine θ)]}}}⁴/C}}}^odx₁}(A)}+  {{ʃ{[(θ_s2)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ₂ [1 + (β₂ cosine θ)]}}}⁴/C}}} ^odx₂}(A)} + {{ʃ{[(θ_s3)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ₃ [1 + (β₃ cosine θ)]}}}⁴/C}}} ^odx₃}(A)} + … +{{ʃ{[(θ_sn)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ_n [1 + (β_n cosine θ)]}}}⁴/C}}} ^odx_n}(A)}

= ʃ{[(θ_s1)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{{1/{1 – [(v₁/C)²]}^1/2} [1 + (β₁ cosine θ)]}}}⁴/C}}}^odx₁  +  ʃ{[(θ_s2)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v₂/C)²]}^1/2}  [1 + (β₂ cosine θ)]}}}⁴/C}}} ^odx₂ + ʃ{[(θ_s3)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v₃/C)²]}^1/2}  [1 + (β₃ cosine θ)]}}}⁴/C}}} ^odx₃ + … +ʃ{[(θ_sn)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v_n/C)²]}^1/2}  [1 + (β_n cosine θ)]}}}⁴/C}}} ^odx_n

= Σ (i = 1, i = n) ʃF _i^odx_i   = Σ (i = 1, i = n) ʃ{[(θ_si)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ_i [1 + (β_i cosine θ)]}}}⁴/C}}}^odx_i

=   Σ (i = 1, i = n) ʃ{[(θ_si)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v_i/C)²]}^1/2}  [1 + (β_i cosine θ)]}}}⁴/C}}} ^odx_i

The following numerical formula offers a first order approximation of space craft gamma factor  gain;

{[E_KEtotalgain] + [M_restC²]}/[M_restC²]= {[ʃF ₁^odx₁]  +  [ʃF ₂^odx₂]  + [ʃF ₃^odx₃ ] + … + [ʃF _n^odx_n] + [M_restC²]}/[M_restC²] 

= {{{ʃ{[(θ_s1)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ₁ [1 + (β₁ cosine θ)]}}}⁴/C}}}^odx₁}(A)}+  {{ʃ{[(θ_s2)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ₂ [1 + (β₂ cosine θ)]}}}⁴/C}}} ^odx₂ }(A)}  + {{ʃ{[(θ_s3)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ₃ [1 + (β₃ cosine θ)]}}}⁴/C}}} ^odx₃ }(A)} + … + {{ʃ{[(θ_sn)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ_n [1 + (β_n cosine θ)]}}}⁴/C}}} ^odx_n}(A)}+ [M_restC²]}/[M_restC²] 

= {{{ʃ{[(θ_s1)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{{1/{1 – [(v₁/C)²]}^1/2} [1 + (β₁ cosine θ)]}}}⁴/C}}}^odx₁  }(A)}  +  {{ʃ{[(θ_s2)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v₂/C)²]}^1/2}  [1 + (β₂ cosine θ)]}}}⁴/C}}} ^odx₂ }(A)} +   {{ʃ{[(θ_s3)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v₃/C)²]}^1/2}  [1 + (β₃ cosine θ)]}}}⁴/C}}} ^odx₃ }(A)} + … +  {{ʃ{[(θ_sn)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v_n/C)²]}^1/2}  [1 + (β_n cosine θ)]}}}⁴/C}}} ^odx_n}(A)}+ [M_restC²]}/[M_restC²] 

= {{Σ (i = 1, i = n) ʃF _i^odx_i }+ [M_restC²]}/[M_restC²]   = {{Σ (i = 1, i = n) {{ʃ{[(θ_si)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ_i [1 + (β_i cosine θ)]}}}⁴/C}}}^odx_i}(A)}}+ [M_restC²]}/[M_restC²] 

=   {{Σ (i = 1, i = n) ʃ{[(θ_si)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v_i/C)²]}^1/2}  [1 + (β_i cosine θ)]}}}⁴/C}}} ^odx_i}+ [M_restC²]}/[M_restC²] 

Now, v = C{[-[1/γ²] + 1]^1/2} according to Special Relativity. Consequently, the following formulas can be used to compute v by numerical trial and error.

v = C{{-{1/{{[E_KEtotalgain] + [M_restC²]}/[M_restC²]}²} – 1}^1/2}  

=  C{{-{1/{{[ʃF ₁^odx₁]  + [ʃF ₂^odx₂]  + [ʃF ₃^odx₃ ] + ʃF _n^odx_n]} + [M_restC²]}/[M_restC²]}  ²}} + 1}^1/2}

=  C{{-{1/{{{{ʃ{[(θ_s1)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ₁ [1 + (β₁ cosine θ)]}}}⁴/C}}}^odx₁}(A)}+   {{ʃ{[(θ_s2)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ₂ [1 + (β₂ cosine θ)]}}}⁴/C}}} ^odx₂ }(A)} +  {{ʃ{[(θ_s3)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ₃ [1 + (β₃ cosine θ)]}}}⁴/C}}} ^odx₃ }(A)}   + … + {{ʃ{[(θ_sn)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ_n [1 + (β_n cosine θ)]}}}⁴/C}}} ^odx_n}(A)}+ [M_restC²]}/[M_restC²]}  ²}} + 1}^1/2}

=  C{{-{1/{{{{ʃ{[(θ_s1)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{{1/{1 – [(v₁/C)²]}^1/2} [1 + (β₁ cosine θ)]}}}⁴/C}}}^odx₁  }(A)} +  {{ʃ{[(θ_s2)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v₂/C)²]}^1/2}  [1 + (β₂ cosine θ)]}}}⁴/C}}} ^odx₂ }(A)} +  {{ʃ{[(θ_s3)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v₃/C)²]}^1/2}  [1 + (β₃ cosine θ)]}}}⁴/C}}} ^odx₃ }(A)} + … +  {{ʃ{[(θ_sn)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v_n/C)²]}^1/2}  [1 + (β_n cosine θ)]}}}⁴/C}}} ^odx_n}(A)}+ [M_restC²]}/[M_restC²]}²}} + 1}^1/2}.   .

=  C{{-{1/{{{{Σ (i = 1, i = n) ʃF _i^odx_i }+ [M_restC²]}/[M_restC²]}   ²}} + 1}^1/2} 

=    C{{-{1/{{{{Σ (i = 1, i = n) {{ʃ{[(θ_si)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ_i [1 + (β_i cosine θ)]}}}⁴/C}}}^odx_i}(A)}}+ [M_restC²]}/[M_restC²]}  ²}} + 1}^1/2} 

=   C{{-{1/{{{{Σ (i = 1, i = n) {{ʃ{[(θ_si)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v_i/C)²]}^1/2}  [1 + (β_i cosine θ)]}}}⁴/C}}} ^odx_i}(A)}}+ [M_restC²]}/[M_restC²]}  ²}} + 1}^1/2}

When the to two sides of the above equations are equal, we will have thus computed relativistic velocity, v.   

As you can see, for cases where there is much natural variation in acceleration with respect to the space craft frame, and for travel over very long distances, many iterations or steps need to be used in numerical algorithms to get mil spec and super-mil-spec results. Such precision is needed when traveling near light speed otherwise mission disaster could happen. In actuality, the above formulations would not be fit for mil spec computations because of the mere approximation to the actual vehicular performance.

We now consider scenarios where the photon angle of incidence from behind is considered and where drag effects are neglected for total space craft energy gains, accrued gamma factors, and accrued velocities. Here, we consider only angular values of radiation incident on the sail for which the radiation exerts forward pressure. In otherwords, we only consider values of θ₀ less than or equal to 90 degrees or π/2 radians. We also assume perfect backward sail reflectivity or trivially imperfect backward reflectivity and trivial massive astrodynamic drag.

The total kinetic energy gain for the craft will be

E _totalgain = ʃF ₁^odx₁  +  ʃF ₂^odx₂ + ʃF ₃^odx₃ + … +ʃF _n^odx_n

= {Σ(i = 1, i = n) {{{[[(θ_01i,1²) - (θ_02i,1²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,1 , θ_s2i,1)}^-1}{{[dr_bi,1/dr_ai,1]}│(r_b1i,1 , r_b2i,1)}[cos [(θ_01i,1 +  θ_02i,1)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₁ [1 + [β₁cosine [(θ_01i,1 + θ_02i,1)/2]]]}}}⁴/C}}}^odx₁}/n}(A)}}}  

+  {Σ(i = 1, i = n) {{{[[(θ_01i,2²) - (θ_02i,2²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,2 , θ_s2i,2)}^-1}{{[dr_bi,2/dr_ai,2]}│(r_b1i,2 , r_b2i,2)}[cos [(θ_01i,2 +  θ_02i,2)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₂ [1 + [β₂cosine [(θ_01i,2 + θ_02i,2)/2]]]}}}⁴/C}}}^odx₂}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{[[(θ_01i,3²) - (θ_02i,3²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,3 , θ_s2i,3)}^-1}{{[dr_bi,3/dr_ai,3]}│(r_b1i,3 , r_b2i,3)}[cos [(θ_01i,3 +  θ_02i,3)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₃ [1 + [β₃cosine [(θ_01i,3 + θ_02i,3)/2]]]}}}⁴/C}}}^odx₃}/n}(A)}}} 

+ … +    {Σ(i = 1, i = n) {{{[[(θ_01i,m²) - (θ_02i,m²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,m , θ_s2i,m)}^-1}{{[dr_bi,m/dr_ai,m]}│(r_b1i,m , r_b2i,m)}[cos [(θ_01i,m +  θ_02i,m)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_m[1 + [β₂cosine [(θ_01i,m + θ_02i,m)/2]]]}}}⁴/C}}}^odx_m}/n}(A)}}} 

= {Σ(i = 1, i = n) {{{[[(θ_01i,1²) - (θ_02i,1²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,1 , θ_s2i,1)}^-1}{{[dr_bi,1/dr_ai,1]}│(r_b1i,1 , r_b2i,1)}[cos [(θ_01i,1 +  θ_02i,1)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{{1/{1 – [(v₁/C)²]}^1/2}  [1 + [(v₁/C)cosine [(θ_01i,1 + θ_02i,1)/2]]]}}}⁴/C}}}^odx₁}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{[[(θ_01i,2²) - (θ_02i,2²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,2 , θ_s2i,2)}^-1}{{[dr_bi,2/dr_ai,2]}│(r_b1i,2 , r_b2i,2)}[cos [(θ_01i,2 +  θ_02i,2)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₂/C)²]}^1/2}  [1 + [(v₂/C)cosine [(θ_01i,2 + θ_02i,2)/2]]]}}}⁴/C}}}^odx₂}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{[[(θ_01i,3²) - (θ_02i,3²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,3 , θ_s2i,3)}^-1}{{[dr_bi,3/dr_ai,3]}│(r_b1i,3 , r_b2i,3)}[cos [(θ_01i,3 +  θ_02i,3)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₃/C)²]}^1/2}  [1 + [(v₃/C)cosine [(θ_01i,3 + θ_02i,3)/2]]]}}}⁴/C}}}^odx₃}/n}(A)}}} 

+ … +    {Σ(i = 1, i = n) {{{[[(θ_01i,m²) - (θ_02i,m²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,m , θ_s2i,m)}^-1}{{[dr_bi,m/dr_ai,m]}│(r_b1i,m , r_b2i,m)}[cos [(θ_01i,m +  θ_02i,m)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_m/C)²]}^1/2} [1 + [(v_m/C)cosine [(θ_01i,m + θ_02i,m)/2]]]}}}⁴/C}}}^odx_m}/n}(A)}}} 

=  Σ(i = 1, i = m)   {Σ(i = 1, i = n) {{{[[(θ_01i,j²) - (θ_02i,j²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,j , θ_s2i,j)}^-1}{{[dr_bi,j/dr_ai,j]}│(r_b1i,j , r_b2i,j)}[cos [(θ_01i,j +  θ_02i,j)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_j [1 + [β_jcosine [(θ_01i,j + θ_02i,j)/2]]]}}}⁴/C}}}^odx_j}/n}(A)}}} 

= Σ(i = 1, i = m)   {Σ(i = 1, i = n) {{{[[(θ_01i,j²) - (θ_02i,j²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,j , θ_s2i,j)}^-1}{{[dr_bi,j/dr_ai,j]}│(r_b1i,j , r_b2i,j)}[cos [(θ_01i,j +  θ_02i,j)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_j/C)²]}^1/2}  [1 + [(v_j/C)cosine [(θ_01i,j + θ_02i,j)/2]]]}}}⁴/C}}}^odx_j}/n}(A)}}} 

= {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}

²} – {{cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₁/C)}/{1 – [(v₁/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,1 , θ_s2i,1)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₁ /C)}/{1 – [(v₁/C) cos θ_s]}}}}│[cos  θ_s1i,1 ,  cos θ_s2i,1]}{cos {{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}+  {cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₁ {1 + {β₁cosine {{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}+ {cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}}/2}}}}}}⁴/C}}}^odx₁}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}

²} – {{cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₂/C)}/{1 – [(v₂/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,2 , θ_s2i,2)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₂ /C)}/{1 – [(v₂/C) cos θ_s]}}}}│[cos  θ_s1i,2 ,  cos θ_s2i,2]}{cos {{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}+  {cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₂ {1 + {β₂cosine {{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}+ {cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}}/2}}}}}}⁴/C}}}^odx₂}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}

²} – {{cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₃/C)}/{1 – [(v₃/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,3 , θ_s2i,3)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₃ /C)}/{1 – [(v₃/C) cos θ_s]}}}}│[cos  θ_s1i,3 ,  cos θ_s2i,3]}{cos {{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}+  {cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₃ {1 + {β₃cosine {{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}+ {cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}}/2}}}}}}⁴/C}}}^odx₃}/n}(A)}}} 

+ … +   {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}

²} – {{cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_m/C)}/{1 – [(v_m/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,m , θ_s2i,m)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_m /C)}/{1 – [(v_m/C) cos θ_s]}}}}│[cos  θ_s1i,m ,  cos θ_s2i,m]}{cos {{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}+  {cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_m {1 + {β_mcosine {{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}+ {cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}}/2}}}}}}⁴/C}}}^odx_m}/n}(A)}}} 

= {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}

²} – {{cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₁/C)}/{1 – [(v₁/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,1 , θ_s2i,1)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₁ /C)}/{1 – [(v₁/C) cos θ_s]}}}}│[cos  θ_s1i,1 ,  cos θ_s2i,1]}{cos {{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}+  {cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₁/C)²]}^1/2}   {1 + {(v₁/C)cosine {{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}+ {cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}}/2}}}}}}⁴/C}}}^odx₁}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}

²} – {{cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₂/C)}/{1 – [(v₂/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,2 , θ_s2i,2)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₂ /C)}/{1 – [(v₂/C) cos θ_s]}}}}│[cos  θ_s1i,2 ,  cos θ_s2i,2]}{cos {{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}+  {cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₂/C)²]}^1/2}   {1 + {(v₂/C)cosine {{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}+ {cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}}/2}}}}}}⁴/C}}}^odx₂}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}

²} – {{cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₃/C)}/{1 – [(v₃/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,3 , θ_s2i,3)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₃ /C)}/{1 – [(v₃/C) cos θ_s]}}}}│[cos  θ_s1i,3 ,  cos θ_s2i,3]}{cos {{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}+  {cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₃/C)²]}^1/2}   {1 + {(v₃/C)cosine {{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}+ {cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}}/2}}}}}}⁴/C}}}^odx₃}/n}(A)}}} 

+ … +   {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}

²} – {{cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_m/C)}/{1 – [(v_m/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,m , θ_s2i,m)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_m /C)}/{1 – [(v_m/C) cos θ_s]}}}}│[cos  θ_s1i,m ,  cos θ_s2i,m]}{cos {{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}+  {cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_m/C)²]}^1/2}   {1 + {(v_m/C)cosine {{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}+ {cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}}/2}}}}}}⁴/C}}}^odx_m}/n}(A)}}} 

Σ ( j= 1, j = m) {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}

²} – {{cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_j/C)}/{1 – [(v_j/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,j , θ_s2i,j)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_j /C)}/{1 – [(v_j/C) cos θ_s]}}}}│[cos  θ_s1i,j ,  cos θ_s2i,j]}{cos {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+  {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_j {1 + {β_jcosine {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+ {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}}}}⁴/C}}}^odx_j}/n}(A)}}} 

= Σ ( j= 1, j = m) {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}

²} – {{cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_j/C)}/{1 – [(v_j/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,j , θ_s2i,j)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_j /C)}/{1 – [(v_j/C) cos θ_s]}}}}│[cos  θ_s1i,j ,  cos θ_s2i,j]}{cos {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+  {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_j/C)²]}^1/2}   {1 + {(v_j/C)cosine {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+ {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}}}}⁴/C}}}^odx_j}/n}(A)}}} 

The following numerical formula offers a first order approximation of space craft gamma factor  gain;

{[E_KEtotalgain] + [M_restC²]}/[M_restC²]= {[ʃF ₁^odx₁]  +  [ʃF ₂^odx₂]  + [ʃF ₃^odx₃ ] + … + [ʃF _n^odx_n] + [M_restC²]}/[M_restC²] 

= {{{Σ(i = 1, i = n) {{{[[(θ_01i,1²) - (θ_02i,1²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,1 , θ_s2i,1)}^-1}{{[dr_bi,1/dr_ai,1]}│(r_b1i,1 , r_b2i,1)}[cos [(θ_01i,1 +  θ_02i,1)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₁ [1 + [β₁cosine [(θ_01i,1 + θ_02i,1)/2]]]}}}⁴/C}}}^odx₁}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{[[(θ_01i,2²) - (θ_02i,2²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,2 , θ_s2i,2)}^-1}{{[dr_bi,2/dr_ai,2]}│(r_b1i,2 , r_b2i,2)}[cos [(θ_01i,2 +  θ_02i,2)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₂ [1 + [β₂cosine [(θ_01i,2 + θ_02i,2)/2]]]}}}⁴/C}}}^odx₂}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{[[(θ_01i,3²) - (θ_02i,3²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,3 , θ_s2i,3)}^-1}{{[dr_bi,3/dr_ai,3]}│(r_b1i,3 , r_b2i,3)}[cos [(θ_01i,3 +  θ_02i,3)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₃ [1 + [β₃cosine [(θ_01i,3 + θ_02i,3)/2]]]}}}⁴/C}}}^odx₃}/n}(A)}}} 

+ … +    {Σ(i = 1, i = n) {{{[[(θ_01i,m²) - (θ_02i,m²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,m , θ_s2i,m)}^-1}{{[dr_bi,m/dr_ai,m]}│(r_b1i,m , r_b2i,m)}[cos [(θ_01i,m +  θ_02i,m)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_m[1 + [β₂cosine [(θ_01i,m + θ_02i,m)/2]]]}}}⁴/C}}}^odx_m}/n}(A)}}}}  + [M_restC²]}/[M_restC²] 

= {{{Σ(i = 1, i = n) {{{[[(θ_01i,1²) - (θ_02i,1²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,1 , θ_s2i,1)}^-1}{{[dr_bi,1/dr_ai,1]}│(r_b1i,1 , r_b2i,1)}[cos [(θ_01i,1 +  θ_02i,1)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{{1/{1 – [(v₁/C)²]}^1/2}  [1 + [(v₁/C)cosine [(θ_01i,1 + θ_02i,1)/2]]]}}}⁴/C}}}^odx₁}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{[[(θ_01i,2²) - (θ_02i,2²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,2 , θ_s2i,2)}^-1}{{[dr_bi,2/dr_ai,2]}│(r_b1i,2 , r_b2i,2)}[cos [(θ_01i,2 +  θ_02i,2)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₂/C)²]}^1/2}  [1 + [(v₂/C)cosine [(θ_01i,2 + θ_02i,2)/2]]]}}}⁴/C}}}^odx₂}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{[[(θ_01i,3²) - (θ_02i,3²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,3 , θ_s2i,3)}^-1}{{[dr_bi,3/dr_ai,3]}│(r_b1i,3 , r_b2i,3)}[cos [(θ_01i,3 +  θ_02i,3)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₃/C)²]}^1/2}  [1 + [(v₃/C)cosine [(θ_01i,3 + θ_02i,3)/2]]]}}}⁴/C}}}^odx₃}/n}(A)}}} 

+ … +    {Σ(i = 1, i = n) {{{[[(θ_01i,m²) - (θ_02i,m²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,m , θ_s2i,m)}^-1}{{[dr_bi,m/dr_ai,m]}│(r_b1i,m , r_b2i,m)}[cos [(θ_01i,m +  θ_02i,m)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_m/C)²]}^1/2} [1 + [(v_m/C)cosine [(θ_01i,m + θ_02i,m)/2]]]}}}⁴/C}}}^odx_m}/n}(A)}}}}  + [M_restC²]}/[M_restC²] 

=  {{Σ(i = 1, i = m)   {Σ(i = 1, i = n) {{{[[(θ_01i,j²) - (θ_02i,j²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,j , θ_s2i,j)}^-1}{{[dr_bi,j/dr_ai,j]}│(r_b1i,j , r_b2i,j)}[cos [(θ_01i,j +  θ_02i,j)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_j [1 + [β_jcosine [(θ_01i,j + θ_02i,j)/2]]]}}}⁴/C}}}^odx_j}/n}(A)}}}}  + [M_restC²]}/[M_restC²] 

= {{Σ(i = 1, i = m)   {Σ(i = 1, i = n) {{{[[(θ_01i,j²) - (θ_02i,j²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,j , θ_s2i,j)}^-1}{{[dr_bi,j/dr_ai,j]}│(r_b1i,j , r_b2i,j)}[cos [(θ_01i,j +  θ_02i,j)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_j/C)²]}^1/2}  [1 + [(v_j/C)cosine [(θ_01i,j + θ_02i,j)/2]]]}}}⁴/C}}}^odx_j}/n}(A)}}} + [M_restC²]}/[M_restC²]   

= {{{Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}

²} – {{cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₁/C)}/{1 – [(v₁/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,1 , θ_s2i,1)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₁ /C)}/{1 – [(v₁/C) cos θ_s]}}}}│[cos  θ_s1i,1 ,  cos θ_s2i,1]}{cos {{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}+  {cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₁ {1 + {β₁cosine {{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}+ {cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}}/2}}}}}}⁴/C}}}^odx₁}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}

²} – {{cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₂/C)}/{1 – [(v₂/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,2 , θ_s2i,2)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₂ /C)}/{1 – [(v₂/C) cos θ_s]}}}}│[cos  θ_s1i,2 ,  cos θ_s2i,2]}{cos {{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}+  {cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₂ {1 + {β₂cosine {{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}+ {cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}}/2}}}}}}⁴/C}}}^odx₂}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}

²} – {{cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₃/C)}/{1 – [(v₃/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,3 , θ_s2i,3)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₃ /C)}/{1 – [(v₃/C) cos θ_s]}}}}│[cos  θ_s1i,3 ,  cos θ_s2i,3]}{cos {{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}+  {cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₃ {1 + {β₃cosine {{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}+ {cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}}/2}}}}}}⁴/C}}}^odx₃}/n}(A)}}} 

+ … +   {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}

²} – {{cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_m/C)}/{1 – [(v_m/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,m , θ_s2i,m)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_m /C)}/{1 – [(v_m/C) cos θ_s]}}}}│[cos  θ_s1i,m ,  cos θ_s2i,m]}{cos {{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}+  {cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_m {1 + {β_mcosine {{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}+ {cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}}/2}}}}}}⁴/C}}}^odx_m}/n}(A)}}}} + [M_restC²]}/[M_restC²]   

= {{{Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}

²} – {{cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₁/C)}/{1 – [(v₁/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,1 , θ_s2i,1)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₁ /C)}/{1 – [(v₁/C) cos θ_s]}}}}│[cos  θ_s1i,1 ,  cos θ_s2i,1]}{cos {{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}+  {cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₁/C)²]}^1/2}   {1 + {(v₁/C)cosine {{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}+ {cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}}/2}}}}}}⁴/C}}}^odx₁}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}

²} – {{cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₂/C)}/{1 – [(v₂/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,2 , θ_s2i,2)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₂ /C)}/{1 – [(v₂/C) cos θ_s]}}}}│[cos  θ_s1i,2 ,  cos θ_s2i,2]}{cos {{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}+  {cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₂/C)²]}^1/2}   {1 + {(v₂/C)cosine {{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}+ {cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}}/2}}}}}}⁴/C}}}^odx₂}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}

²} – {{cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₃/C)}/{1 – [(v₃/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,3 , θ_s2i,3)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₃ /C)}/{1 – [(v₃/C) cos θ_s]}}}}│[cos  θ_s1i,3 ,  cos θ_s2i,3]}{cos {{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}+  {cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₃/C)²]}^1/2}   {1 + {(v₃/C)cosine {{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}+ {cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}}/2}}}}}}⁴/C}}}^odx₃}/n}(A)}}} 

+ … +   {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}

²} – {{cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_m/C)}/{1 – [(v_m/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,m , θ_s2i,m)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_m /C)}/{1 – [(v_m/C) cos θ_s]}}}}│[cos  θ_s1i,m ,  cos θ_s2i,m]}{cos {{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}+  {cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_m/C)²]}^1/2}   {1 + {(v_m/C)cosine {{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}+ {cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}}/2}}}}}}⁴/C}}}^odx_m}/n}(A)}}}}  [M_restC²]}/[M_restC²]   

{{Σ ( j= 1, j = m) {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}

²} – {{cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_j/C)}/{1 – [(v_j/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,j , θ_s2i,j)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_j /C)}/{1 – [(v_j/C) cos θ_s]}}}}│[cos  θ_s1i,j ,  cos θ_s2i,j]}{cos {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+  {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_j {1 + {β_jcosine {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+ {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}}}}⁴/C}}}^odx_j}/n}(A)}}}} + [M_restC²]}/[M_restC²]     

= {{Σ ( j= 1, j = m) {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}

²} – {{cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_j/C)}/{1 – [(v_j/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,j , θ_s2i,j)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_j /C)}/{1 – [(v_j/C) cos θ_s]}}}}│[cos  θ_s1i,j ,  cos θ_s2i,j]}{cos {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+  {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_j/C)²]}^1/2}   {1 + {(v_j/C)cosine {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+ {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}}}}⁴/C}}}^odx_j}/n}(A)}}}} + [M_restC²]}/[M_restC²]    

For total accrued velocity, v, we have

v = C{{-{1/{{{Σ(i = 1, i = m)   {Σ(i = 1, i = n) {{{[[(θ_01i,j²) - (θ_02i,j²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,j , θ_s2i,j)}^-1}{{[dr_bi,j/dr_ai,j]}│(r_b1i,j , r_b2i,j)}[cos [(θ_01i,j +  θ_02i,j)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_j [1 + [β_jcosine [(θ_01i,j + θ_02i,j)/2]]]}}}⁴/C}}}^odx_j}/n}(A)}}}}  + [M_restC²]}/[M_restC²]}  ²}} + 1}^1/2}

= C{{-{1/{{{Σ(i = 1, i = m)   {Σ(i = 1, i = n) {{{[[(θ_01i,j²) - (θ_02i,j²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,j , θ_s2i,j)}^-1}{{[dr_bi,j/dr_ai,j]}│(r_b1i,j , r_b2i,j)}[cos [(θ_01i,j +  θ_02i,j)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_j/C)²]}^1/2}  [1 + [(v_j/C)cosine [(θ_01i,j + θ_02i,j)/2]]]}}}⁴/C}}}^odx_j}/n}(A)}}} + [M_restC²]}/[M_restC²]} ²}} + 1}^1/2}

= C{{-{1/{{{Σ ( j= 1, j = m) {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}

²} – {{cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_j/C)}/{1 – [(v_j/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,j , θ_s2i,j)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_j /C)}/{1 – [(v_j/C) cos θ_s]}}}}│[cos  θ_s1i,j ,  cos θ_s2i,j]}{cos {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+  {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_j {1 + {β_jcosine {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+ {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}}}}⁴/C}}}^odx_j}/n}(A)}}}} + [M_restC²]}/[M_restC²]}     

²}} + 1}^1/2}

= C{{-{1/{{{Σ ( j= 1, j = m) {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}

²} – {{cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_j/C)}/{1 – [(v_j/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,j , θ_s2i,j)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_j /C)}/{1 – [(v_j/C) cos θ_s]}}}}│[cos  θ_s1i,j ,  cos θ_s2i,j]}{cos {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+  {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_j/C)²]}^1/2}   {1 + {(v_j/C)cosine {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+ {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}}}}⁴/C}}}^odx_j}/n}(A)}}}} + [M_restC²]}/[M_restC²]  ²}} + 1}^1/2}

As you can see, attempts at analytic solutions and even non-computational numerical solutions would pose a proverbial night-mare.

Now consider again that radiation pressure is equal to σT⁴/C and {2 [σT⁴/C]} for reflected radiation. However, for a space craft traveling through a black body radiation field, the apparent temperature of the radiation increases in proportion to γ and so the black body power impinging on the space craft from directly in front grows in proportion to T⁴ and thus  to γ⁴.

The temperature of a black body is T = {P/[(A)(σ)(e)]}^1/4 = {P/{(A)[5.670400(40) x 10^-8 W m^-2 K^-4]} ^1/4}. Therefore, the cosmic microwave background radiation pressure on a perfectly reflective flat bow  relativistic space craft is approximately equal to  {2 [σT_cmbr ⁴/C]} γ⁴ = {2 [σT_cmbr ⁴/C]} { {1/{1 – [(v/C)²]}^1/2}  ⁴} =   {2 [σT_cmbr ⁴/C]} {1/{1 – [(v/C)²]}²} where γ is constant and T_cmbr is constant. However, the light pressure incident from directly behind will be approximately equal to {2 [σT_cmbr ⁴/C]}/{1/{1 – [(v/C)²]}²}. In actuality, not all of the light is directly incident from the back and so there will be angular affects that result in loss of driving power. However, we will assume that all of the radiation is absorbed and then re-cycled and released as a perfect backwardly directed laser beam. We will assume that CMBR light which is forwardly incident completely passes through the sail.

Assuming that the velocity of the sail starts out at (Zero) C, the optical pressure will be equal to {{2 [σT_cmbr⁴/C]}/γ⁴} = {{2 [σT_cmbr⁴/C]}/ { {1/{1 – [(v/C)²]}^1/2} ⁴} } = {{2{[5.670400(40) x 10^-8 W m^-2 K^-4][(2.725 K)⁴]/(300,000,000 m/s)}}/(1²)} = [2.0844355 x 10^-14] Newtons/m². For a 10,000 km by 10,000 km sail, the drive force will be 2.0844 Newtons. A 100,000 km by 100,000 km sail will produce a driving force of 208.44 Newtons.

Now assume that the sail is monolithic, made of one nanometer thick carbonaceous, STP water density materials. The sail would have a mass of 10,000,000 thousand metric tons. Assuming that the space craft plus her sail had a mass of 20,844,000 metric tons, the craft would start out with an acceleration of F/M = a = 208.44 N/20,844,000,000 kg = 10^-8 meter/s². After traveling 6 x 10¹⁵ seconds or about 200,000,000 years, the velocity of the space craft will be 0.20 C and the relativistic Lorentz transformation factor will be 1.0206 thus permitting the above Newtonian formula to apply as a reasonable approximation.

Assume that the background gas and dust that contacts the sail over a path length of 1 light year has an invariant  mass of (0.06667 kg)( 10¹⁴){[ 3 x 10¹⁰]^-1} = 222.2 kg. The momentum of the gas and dust with respect to the sail will be (m)(v) = (222.2 kg)(50,000,000 m/s) =1.111 x 10¹⁰  kg m s. Now Force equal dP/dt. Therefore, the force on the sail is on average equal to F_0.05C  = dP_0.2C/dt = 71.677 Newtons.. For a velocity of 0.02 C, the net propulsive force is 208.44 N – 71.67 N = 136.76 N which will still obviously permit 0.2 C velocities.

For such a space craft that deployed a linear series of 1,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000 metric tons could be identically accelerated. The mass of the crew quarters would be 108,440,000 metric tons. For such a space craft that deployed a linear series of 1,000,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000,000  metric tons could be identically accelerated. The mass of the crew quarters would be 108,440,000,000  metric tons. For cosmic journeys, this is not bad. A series of 100 million tethered sails might conceivably pull a sail craft combination having a mass of 20,844,000,000,000 metric tons and a crew quarters having a mass of 10,844,000,000,000 metric tons. For a velocity of 0.2 C, the each of the latter massed space craft would have the same ratio of backward driving force and massive drag force. The caveat is simply the deployment of commensurate numbers of sails simultaneously in a spatial series along the space craft velocity vector.

Now assume that the sail is a gridded fabric or net made of STP water density conducting 0.4 nanometer wide, one nanometer thick,  carbonaceous fibers that are separated by 0.0005 meters such as in a judicious cross weave spacing. A one square meter portion of the net will have a mass of 0.8 x 10^-12 kilograms. A 0.08 kilogram sail will have a plan-form area of 10⁵ square kilometers. A 8,000 kilogram sail will have a plan-form area of 10¹⁰ square kilometers and will have an acceleration of F/M = A = 208.44 Newtons/8,000 kg = 0.026055 m/s².  A space craft having a total mass of  8,000 metric tons will have an initial acceleration of 0.000026055 m/s².   In 80,000 years, the velocity of the 8,000 metric ton  system will be about 0.215388 C assuming Newtonian approximations.

Now, the 10¹⁰ square kilometer plan-form area gridded sail will have a massive species contact area of [10¹⁰km²]/1,250,000 = 8,000 km². So the background gas and dust that contacts the sail over a path length of 1 light year would have an invariant  mass of (0.06667 kg)( 10¹⁴){[ 3 x 10¹⁰]^-1}/(1,250,000) = 0.00017776 kg. The momentum of the gas and dust with respect to the sail will be (m)(v) = (0.00017776 kg)(50,000,000 m/s) =8,888 kg m/s. Now Force equal dP/dt. Therefore, the force on the sail is on average equal to F_0.2C  = dP_0.2C/dt = 0.000057341Newtons.. For a velocity of 0.2 C, the ratio of the driving force to massive drag force is [208.44 N/ 0.000057341 N] = 3,635,095.

The last result in the above paragraph is intreguing because it suggest that an ordinary CMBR sail-material configuration  could gainfully accelerate even in an environment such as local interstellar space where the baryonic mass density may be as much as one million times that of the observable universe on average.

Now, the apparent angle, θ_s,  of CMBR pre-incident on the space craft at an angle of 90 degrees or with respect to the length of the space craft relative to the source reference frame at v = zero C will appear to be incoming at an angle, θ_o,  of 90 degrees with respect to the space craft,  ship’s reference frame.

If we consider the effects of relativistic aberration, the general formula for apparent shift in angle of incidence of the CMBR from the ship’s perspective is

Cos θ_o = {[cos θ_s] – (v/C)}/{1 – [(v/C) cos θ_s]}. Now assume θ_s = 14.48 degrees, and a gamma factor of 3, Cos θ_o = {[cos (19.47 degrees)] – (0.942809)}/{1 – [(0.942809) cos (19.47 degrees)]}~ 0 — > θ_o = 90 degrees. Now (14.48²)/(90²) => (0.0468)/(4.5⁴) = 0.000114128. Now the grid would need to expand in area by a factor of 0.000114128^-1 = 8,762.

The grid line spacing can be increased by a factor of 3 thus yielding an increase in grid area by a factor of essentially 2. I obtained the areal expansion factor of 2 by inspection of hand-drawn grids although I am certain topologists and geometers have long since figured out the general relationships for various factors of line distance expansion for square gridded figures. However, we still need to increase the grid area by another factor of 4,381. Simply deploying  4,381 + 1 expanded sails = 4,382 expanded sails each having a mass of 8,000 kilograms will produce a complete sail rigging having a mass of 35,056 metric tons. Include a tether sub-rigging to link the sails in a serial distribution along with the rest of the mass of the space craft to yield a total craft mass of 200,000 metric tons and we obtain a forward oriented driving force still equal to 208.44 N. So the background gas and dust that contacts the fully deployed rigging at a gamma factor of 3 over a path length of 1 light year has an invariant  mass of {(0.06667 kg)( 10¹⁴){[ 3 x 10¹⁰]^-1}/(1,250,000)} (4,382)   = [0.00017776  kg](4,382) =  0.778944 kg.

Now,  the formula for relativistic momentum of a massive particle  is M₀ v γ = M₀ v /{{1 – [(v/C)²]}^1/2}. However, the number of interstellar or intergalactic massive particles impinging on a relativistic space craft per unit of time, ship’s frame, t,  is proportional  to (γ)v = {1/{{1 – [(v/C)²]}^1/2}v. Now, dP/dt = F which is is expressed in Newtons. Therefore, the force acting on a space craft, ship’s frame,  from the interstellar massive background is equal to d[(M₀ v γ) (γ)(v/C)ǀ]/dt = d{{M₀ v /{{1 – [(v/C)²]}^1/2}} {1/{{1 – [(v/C)²]}^1/2}}v}/dt where M₀ is the incident mass over the  the constant distance interval of (Delta x) background reference frame. The drag energy is thus equal to {d[(M₀ v γ) (γ)(v/C)]/dt} (Delta x) = {d{{M₀ v /{{1 – [(v/C)²]}^1/2}} {1/{{1 – [(v/C)²]}^1/2}}v/C}/dt} (Delta x) where t is the ship time.  The quantity γv/C = {1/{{1 – [(v/C)²]}^1/2}}v/C is considered dimensionless and is of a constant scalar form.

The momentum of the 0.778944 kg invariant mass with respect to the space craft will be M₀ v γ = M₀ v /{{1 – [(v/C)²]}^1/2} = (0.778944 kg)(282,842,700 m/s)(3) = 6.60956  x 10⁸  kg m/s. The force acting on the space craft will be d[(M₀ v γ) (γ)(v/C)ǀ]/dt = d{{M₀ v /{{1 – [(v/C)²]}^1/2}} {1/{{1 – [(v/C)²]}^1/2}}v}/dt = {[(0.778944 kg  (282,842,700 m/s)(3) (3)[( 282,842,700 m/s) /(300,000,000 m/s)]}/{(31,000,000 s)/[(3)/(0.942816)]} = 191.889 Newtons. The net driving force would be 208.44 N – 191.889 N = 16.551 N.

Thus, it is safe to say that our space craft could accelerate to a velocity of 0.942816 C or to a gamma factor of 3  and still maintain a gainfull driving force. Now KE = ʃ F∙ dx = F(cos α)(Δx) = F(cos 0)(Δx) = (F)(Δx) for parallel force and velocity vectors. Assuming a driving force of 16.551 Newtons during the entire trip to a gamma factor of 3, we will further assume that (F)(Δx) = [3 MC² – MC²] = 2MC² = (2)(200,000,000 kg)[(300,000,000 m/s)²]  =  3.6 x 10²⁵  joules. Dividing the energy by F yields x = distance travelled = [3.6 x 10²⁵J]/(16.551 N) = 2.175 x 10²⁴ meters = 217,510,000 light years.

Now,  a high end carbonaceous super-material having a cross sectional area of 1.3708 x 10^-9 square meters can support 208.44 Newtons assuming a yield strength of 10⁷ kilograms per square inch. A cable having such a cross-section and having a mass of 10,000 metric tons would plausibly have a length of (10,000)[1.3708 x 10^-9] meters or 1.3708 x 10¹³  meters or 1.3708 x 10¹⁰ kilometers. However, a cable with a cross-sectional area 1,000 times as great having the same length would have a mass of 10,000,000 metric tons and could conceivably tether 1,000 of  the previously described 20,000 kilogram gridded sails each having a plan form area of 10¹⁰ square kilometers and each producing a driving force of  208.44 Newtons.  Such a tethered sail system could propel a space craft having a total mass of 20 million metric tons, of which 10,020,000 metric tons would exist as the sail rigging. Each sail would be linearly separated by 1.3708 x 10⁷ kilometers. A cable having a cross-sectional area that is ten time greater yet could power a space craft having a total mass of 200 million metric tons of which 100,200,000 metric tons would be incorporated into the sail rigging.For the latter example, the sails would be separated by 1.3708 x 10⁶ kilomters thus preventing all but trivial shadowing of the driving CMBR at velocities of 0.2 C and only moderate shadowing at a gamma factor of 3.

The value of gamma = 3 is close to the maximum value conceivable with purely backward impinging, CMBR driven planar or plane-like sails, that are oriented orthogonally to the space craft velocity vector where the sails are made of ordinary atomic elements based materials and are of gridded forms for  the next billion years or so. This is because the baryonic mass density of the observable universe will very only slightly over this time period thus at best promoting a slight decrease in drag for the above systems at a gamma factor of 3.  Significantly higher gamma factors with self repairing grids are possible for universal ages that several or more times that of the present universe due to intergalactic massive rareification.

In order to compute a gamma factor of 3, an interpolated value for back-ward red-shift of about 4.5 was assumed as an approximately average value. Since the mass specific capture area and the associated massive drag values used are not absolute requirements, the adjustment of the latter values by a few percent can compensate for inaccuracies in the former estimated average and permit the actual maximum possible gamma factor per given intergalactic massive density to very by perhaps as much as plus or minus a few percent. The point is that because of the subject degrees of freedom in the engineering and applied physics for the above conjectural specific examples, the maximum gamma factor of 3 is a very good ball park to aim for in any future real world systems we will design.

Now, in addition of electromagnetic negative refractive index materials which have been demonstrated within research facilities, it may be possible that massive particle and perhaps even gravitational wave negative refraction index materials could be fashioned into sails that are pulled forward by the incident mass-energies. No one at present really knows if the later two types of materials are possible to construct, however, such materials are tantalizing to consider because of the implications of perpetual and increasing pull sail accelerations,  as long as the mass-energy influxes would not thermally or mechanically over burdern the negative refraction index materials.

However, we do not necessarily need even the still controversial pull sail negative electromagnetic refraction index drives. Ordinary positive index materials that are one way transmissive and which are suitably contoured can provide much higher gamma factors in the present cosmic era even in consideration of massive astrodynamic drag. I will describe such  increasing more extreme scenarios in an ongoing series of posts on the subject of plausible one way transmissive, positive refraction index, material based sails involving intelligible speculations regarding such sails made of ordinary atomic elements.

I will post further on this subject later today. It is time for me to log off, turn off my computer and catch some sleep.

Regards;

Jim

Copyright James M. Essig  January 27, 2011  All Rights Reserved.

That space and time are intimately tied to electromagnetic radiation is obvious when one considered the ubiquitous inclusion of the speed of light in vacuu as a constant in virtually all special and general relativistic formulations. Even in classical electromagnetic theory, the velocity of light is intimately related to the properties of space time including the magnetic permeability and electric permittivity of free space by the formula C = {1/[μ₀ ɛ₀]}^1/2.

I am sure that most of the concepts expressed within this post have been contemplated by others before.

By now the reader is aware of the concept of light sail(s) driven space craft that can reach relativistic velocities. A space craft traveling at extreme gamma factors using an ordinary beam sail will experience extreme astro-dynamic drag, and the sail would likely be ionized by the drag induced friction. This is largely due to the fact that most beam sail space craft contemplate beam sails that are orthogonally spread  with respect to the craft velocity vector and thus which have a very large surface area to experience forward drag.

Suppose a relativistic rocket was powered by energy captured by an attached square or rectangular CMBR  sail that is  oriented in a perpendicular to the velocity vector of the space craft. The equation for Doppler shifting of  CMBR acting on the sail would then be:  

1 + z = {1 + [ ν (cos θ)/C]}/{[1 - [(v/C)²]]^1/2}

or,

z  = {1 + [ ν (cos θ)/C]}/{[1 - [(v/C)²]]^1/2} – 1

f’ = f / {γ [1 + (β cosine θ)]}

which reduces to F’ = f/γ for a radiation source and space craft moving in a direction perpendicular to the line connecting these reference frames with respect to a space craft observer since cos (π/2) = zero where f represents frequency. Here, θ is the angle of view with respect to the space craft velocity vector or the perceived angle of  radiation incidence on the sail with respect to the direction of space craft travel,  with respect to the space craft.

Now,  the energy of a photon is as follows:

E = [h/(2 π)] ω = hf = hC/λ 

where h is the Planck Constant and λ  is the photon wave-length.

Therefore, the energies of the individual CMBR photons impinging on the light sail oriented in a direction perpendicular to it from the space craft’s perspective from directly behind are equal to:

E + =  hf/{γ [1 + (β cos  θ)]} = hf /{γ [1 + (β cos  (0)]}

which reduces to;

hf /{γ [1 + β ]}.

Now,  the CMBR power impinging on the space craft sail per differential unit of time element (space craft reference frame), per differential unit of angle of pre-incidence (space craft reference frame), per differential element of sail area (space craft reference frame) for black body radiation is a function of γ ⁴. This is because the black body radiation frequency curve peak is proportional to black body source temperature and an incident source photon’s frequency is proportional gamma. Since black body total power emission per unit of surface area is proportional to the  fourth power of the temperature of the black body, the above differential area element of the sail will receive a total power that scales with γ ⁴. Black body emitter frequency distribution scales as a function of gamma relative to a moving observer traveling at a factor of γ with respect to the source for directly approaching observers and 1/ γ for directly receding observers.

Planck’s Law states that

I(ѵ,T)dѵ = {[2hѵ³]/C²}{1/{[e^[(hѵ)/(kT)]] -1}}dѵ

λ_max = b/T

where λ max,  is a function only of the temperature.

P_net = P_emit – P_absorbed

Applying the Stefan–Boltzmann law,

P_net = A σ e (T⁴ – T₀⁴)

where sigma =  σ = (2π⁵k_B⁴)/(15 h³ C²) = (π²k_B⁴)/(60 ђ³ C²)

or  where sigma =  σ = 5.670400(40) x 10^-8 W m^-2 K^-4

Therefore, the apparent spectral temperature of the CMBR radiation incident on the sail per unit angle of CMBR incidence for a stationary sail is:

{[P_cmbr/(A σ e)] ^1/4}

The apparent spectral temperature of the CMBR radiation incident on the sail per unit of apparent angle of incidence of the CMBR with respect to the space craft reference frame-based observer(s) for a sail traveling at a given velocity for backwardly impinging radiation is:

T_app = {∫(0, π/2){{{[P_cmbr/(e σ)]^1/4}/{γ [1 +  [(v/C) cos θ]]}}⁴}{{∫(y₁,y₂){ ∫(x₁,x₂) dx} dy}^-1} dθ}^1/4

= {∫(0, π/2){{{[P_cmbr/(e σ)]^1/4} /{{1/{[1 – [(v/C)² ]] ^1/2}} [1 +  [(v/C) cos θ]]}}⁴}{{∫(y₁,y₂){ ∫(x₁,x₂) dx} dy}^-1} dθ}^1/4

= {∫(0, π/2){{{[Pcmbr/(e σ)]^1/4}/{{1/{[1 – [(v/C)² ]] ^1/2}} [1 + [(v/C) cos θ]]}}⁴}[(∫d A)^-1] dθ}^1/4

where P_cmbr is the background CMBR power incident on the sail, dA is the differential element of sail area with respect to the space craft reference frame, v is the velocity of the space craft with respect to the background, and θ is the angle of radiation incidence on the sail with respect to a sail based observer. Theta ranges from π/2 radians for radiation traveling in an orthogonal direction with respect to the ship velocity vector to zero radians for radiation traveling in a parallel direction with respect to the ship velocity vector.

The total power backwardly incident upon the sail with respect to the sail’s reference frame for a given gamma factor  is  therefore:

P = ∫(y₁,y₂) {∫(x₁,x₂) {∫ (0, π/2) {{{(T_cmbr) /{γ [1 + [(v/C) cos θ]]}}⁴} σ e} dθ}dx}dy

= ∫(y₁,y₂) {∫(x₁,x₂) {∫ (0, π/2) {{{(T_cmbr) /{{1/{[1 + [(v/C)² ]] ^1/2}}  [1 + [(v/C) cos θ]]}}⁴} σ e} dθ}dx}dy

= ∫ {∫ (0, π/2) {{{(T_cmbr) /{{1/{[1 – [(v/C)² ]] ^1/2}}  [1 + [(v/C) cos θ]]}}⁴} σ e} dθ} dA

Here, T_cmbr is the background CMBR temperature.

Note that in the above calculations and the ones that follow, all of the relevant backwardly incident background energies are assumed to be initially absorbed by the sail even if the sail acquires a temperature significantly above absolute zero and thereby produces thermal electromagnetic black body emissions. I describe potential methods of the absorption of nearly all incident radiations even in cases where relativistic aberration would otherwise cause the bulk of the impinging radiation to easily reflect off the sail because of increasingly shallow angles of incidence. The forwardly incident radiation is assumed to completely pass through the sail without exchange of momentum.

We can numerically integrate the relativistic  energy growth of the ship in small time steps as follows:

∫P₁dt₁ + ∫P₂dt₂ + ∫P₃dt₃ +, …, + ∫P_ndt_n

Thus, the following expression can be used to compute relativistic energy gain by the ship in terms of t.

Egain  = Σ (0,n)    { ∫ (t_ai, t_bi) { ∫  (y₁, y₂){ ∫  (x₁,x₂){ ∫ (0,  π/2) {{{T_cmbr /{{{1/{[1 – [(v_i/C) ²]] ^1/2}}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ}dx}dy} dt}      

Here,  t_ai, t_bi, and dt are the times in the background reference frame.

Note, the reason why I assume the latter three times are background reference frame times is such that for a space craft traveling at a velocity of just under 1 C, where gamma is held constant, the energy gain for the space craft will be proportional to the length of the path traveled by the space craft according to the background reference frame. The distance of space craft travel  is proportional to the time of space craft travel with respect to the background reference frame. The same is true for a space craft traveling at any velocity held constant, thus the reason for the performance of the numerical integration for each time step where the velocity is incrementally increased but held constant for each time step.

Alternatively, we can use the following series:

Egain =  Σ (1,n) { ∫ (t_ai, t_bi) {∫ {∫ (0, π/2)  {{{T_cmbr /{{1/{[1 – [(v_i/C) ²]] ^1/2}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ} dA}dt}    

where t_ai and t_bi and dt are the times in the background reference frame.

Now for constant acceleration ship time, T₀ = (c/g) ln {{[[ (C²) + (V₀ ²)] ^1/2]   –   [V₀/[[1 – [(V₀/C) ²]] ^1/2 ]]} { [(C ²) + [[(g)(t)  + [V₀ /[1 – [(V₀/C) ²]] ^1/2 ] ²]] ^1/2] + [(g)(t)] +  [V₀/[[1 – [(V₀/C) ²]] ^1/2]]} / (C ²)}. We can incorporate the expression for T₀ prefaced by the notation Delta to indicate the time steps,  ship time,  of uniform duration ship frame.

For computation in terms of T₀, we obtain:

Egain = Σ (0,n)    {{ ∫  (y₁, y₂){ ∫  (x₁,x₂){ ∫ (0,  π/2) {{{T_cmbr /{{{1/{[1 – [(v_i/C) ²]] ^1/2}}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ}dx}dy} {(Delta) {(c/g) ln {{[[ (C²) + (V_0i ²)] ^1/2]   –   [V_0i/[[1 – [(V_0i/C) ²]] ^1/2 ]]} { [(C ²) + [[(g_i)(t_i)  + [V_0i /[1 – [(V_0i/C) ²]] ^1/2 ] ²]] ^1/2] + [(g_i)(t_i)] +  [V₀/[[1 – [(V_0i/C) ²]] ^1/2]]} / (C ²)}}} {1/{[1 – [(v_i/C) ²]] ^1/2}}}  .     

where t_i is the time in the background reference frame and g_i is the ship acceleration in the ship’s reference frame.

Note that the above formulas provide precise calculations for many numerical iterations involving small increments for velocity increase and small time steps in the ship’s frame.

Alternatively, we can use the following series:

Egain =  Σ (1,n) { {∫ {∫ (0, π/2)  {{{T_cmbr /{{1/{[1 – [(v_i/C) ²]] ^1/2}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ} dA} {(Delta) {(c/g) ln {{[[ (C²) + (V_0i ²)] ^1/2]   –   [V_0i/[[1 – [(V_0i/C) ²]] ^1/2 ]]} { [(C ²) + [[(g_i)(t_i)  + [V_0i /[1 – [(V_0i/C) ²]] ^1/2 ] ²]] ^1/2] + [(g_i)(t_i)] +  [V_0i/[[1 – [(V_0i/C) ²]] ^1/2]]} / (C ²)}}} {1/{[1 – [(v_i/C) ²]] ^1/2}}}             

Another method entails integration with respect to space craft velocity with respect to the background and integration with respect to time as follows:

E_gain = ∫ (v₁, v₂)  {{∫  (t₁, t₂) { ∫ (y₁, y₂){ ∫ (x₁,x₂){ ∫ (0,  π/2) {{{T_cmbr {1/{[1 – [(v/C) ²]] ^1/2}}[1 + [(v/C) cos θ]]} ⁴} σ e}dθ}dx}dy} dt}/v}dv

Or alternatively,

E_gain =  ∫(v₁,v₂) {∫ (t₁,t₂){∫  {∫ (0,  π/2) {{{_Tcmbr/{{{1/{[1 – [(v/C) ²]] ^1/2}}[1 + [(v/C) cos θ]]}}⁴} σ e} dθ} dA}dt}/v}dv

where t₁ and t₂ and dt are the times in the background reference frame.

Now, E_gain in practice needs to take into account the radiative temperature of the sail.

Now, given that

P_net = A σ e (T⁴ – T₀⁴)

where T is the body temperature and T₀ is the surrounding temperature, we can re-interpret T as the impinging radiation’s black body temperature and T₀ as the emitted thermal radiation black body temperature. So in other words, if the impinging temperature is 10 times higher in Kelvins then the thermal radiative temperature, the net power input into the sail is 10⁴ or 10,000 times greater than the power loss through radiative emissions.

The net power delivered to the  sail will be equal to the power intake minus the power thermally radiated as

P = {∫(y₁,y₂) {∫(x₁,x₂) {∫ (0, π/2) {{{(T_cmbr) /{γ [1 + [(v/C) cos θ]]}}⁴} σ e} dθ}dx}dy} – {∫(y₁,y₂) {∫(x₁,x₂) (T₀⁴ σ e) dx}dy}

= {∫(y₁,y₂) {∫(x₁,x₂) {∫ (0, π/2) {{{(T_cmbr) /{{1/{[1 + [(v/C)² ]] ^1/2}}  [1 + [(v/C) cos θ]]}}⁴} σ e} dθ}dx}dy}  – {∫(y₁,y₂) {∫(x₁,x₂) (T₀⁴ σ e) dx}dy}

= { ∫ {∫ (0, π/2) {{{(T_cmbr) /{{1/{[1 – [(v/C)² ]] ^1/2}}  [1 + [(v/C) cos θ]]}}⁴} σ e} dθ} dA} -

- {∫(T₀⁴ σ e) dA}

The following expression can be used to compute relativistic energy gain by the ship in consideration of the black body emissions from the sail heated by CMBR.

From computation in terms of t, we obtain:

Egain  = {Σ (0,n)   { ∫ (t_ai, t_bi) { ∫  (y₁, y₂){ ∫  (x₁,x₂){ ∫ (0,  π/2) {{{T_cmbr /{{{1/{[1 – [(v_i/C) ²]] ^1/2}}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ}dx}dy} dt}}   – {Σ (0,n) {∫ (t_ai, t_bi) {∫(y₁,y₂) {∫(x₁,x₂) (T⁴_0i σ e) dx}dy}dt}}     

where t_ai and t_bi and dt are the times in the background reference frame.

Now ship time = T₀ = {(c/g_n) ln {[[ (C²) + (V₀ ²)] ^1/2]   –   [V₀/[[1 – [(V₀/C) ²]] ^1/2 ]]} { [(C ²) + [[(g_n)(t_n)  + [V₀ /[1 – [(V₀/C) ²]] ^1/2 ] ²]] ^1/2] + [(g_n)(t_n)] +  [V₀/[[1 – [(V₀/C) ²]] ^1/2]]} / (C ²)}}

Computation in terms of T₀, we obtain:

Egain  = {Σ (0,n)   {{ ∫  (y₁, y₂){ ∫  (x₁,x₂){ ∫ (0,  π/2) {{{T_cmbr /{{{1/{[1 – [(v_i/C) ²]] ^1/2}}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ}dx}dy} {(Delta) {(c/g_i) ln {{[[ (C²) + (V_0i ²)] ^1/2]   –   [V_0i/[[1 – [(V_0i/C) ²]] ^1/2 ]]} { [(C ²) + [[(g_i)(t)  + [V_0i /[1 – [(V_0i/C) ²]] ^1/2 ] ²]] ^1/2] + [(g_i)(t)] +  [V_0i/[[1 – [(V_0i/C) ²]] ^1/2]]} / (C ²)}}} {1/{[1 – [(v_i/C)² ]] ^1/2}}}}    – {Σ (0,n)  {{∫(y₁,y₂) {∫(x₁,x₂) (T⁴_0i σ e) dx}dy}    {(Delta) {(c/g_i) ln {{[[ (C²) + (V_0i ²)] ^1/2]   –   [V_0i/[[1 – [(V_0i/C) ²]] ^1/2 ]]} { [(C ²) + [[(g_i)(t_i)  + [V_0i /[1 – [(V_0i/C) ²]] ^1/2 ] ²]] ^1/2] + [(g_i)(t_i)] +  [V_0i/[[1 – [(V_0i/C) ²]] ^1/2]]} / (C ²)}}} {1/{[1 – [(v_i/C)² ]] ^1/2}} }} 

where T₀ is the ship time.

Alternatively, we can use the following series calculated with  t:

Egain  =  {Σ (1,n) { ∫ (t_ai, t_bi) {∫ {∫ (0, π/2)  {{{T_cmbr /{{1/{[1 – [(v_i/C) ²]] ^1/2}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ} dA}dt}}   -  {Σ (1,n) ∫ (t_ai, t_bi) {∫ (T⁴ _0i σ e) dA}dt}   

where t_ai and t_bi and dt are the times in the background reference frame.

Calculating with respect to T₀, we obtain:

Egain =  {Σ (1,n)  {∫ {∫ (0, π/2)  {{{T_cmbr /{{1/{[1 – [(v_i/C) ²]] ^1/2}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ} dA}}   -  {Σ (1,n) {∫ (T⁴ _0i σ e) dA}   {(Delta) {(c/g_i) ln {{[[ (C²) + (V_0i ²)] ^1/2]   –   [V_0i/[[1 – [(V_0i/C) ²]] ^1/2 ]]} { [(C ²) + [[(g_i)(t_i)  + [V_0i /[1 – [(V_0i/C) ²]] ^1/2 ] ²]] ^1/2] + [(g_i)(t_i)] +  [V_0i/[[1 – [(V_0i/C) ²]] ^1/2]]} / (C ²)}}} {1/{[1 – [(v_i/C)² ]] ^1/2}}}.

Integrating with respect to time and velocity;

the formulas for total kinetic energy  gain are:

E_gain = {∫ (v₁, v₂)  {{∫  (t₁, t₂) { ∫ (y₁, y₂){ ∫ (x₁,x₂){ ∫ (0,  π/2) {{{T_cmbr {1/{[1 – [(v/C) ²]] ^1/2}}[1 + [(v/C) cos θ]]} ⁴} σ e}dθ}dx}dy} dt}/v}dv} -  E_{rad lost}

or alternatively,

E_gain =  {∫(v₁,v₂) {{∫ (t₁,t₂){∫  {∫ (0,  π/2) {{{_Tcmbr/{{1/{[1 – [(v/C) ²]] ^1/2}}[1 + [(v/C) cos θ]]}}⁴} σ e} dθ} dA}dt}/v}dv} -  E_{rad lost}

where t_ai and t_bi , t_i, and dt are the times in the background reference frame, g_i is the ship acceleration in the ship’s reference frame, and V_0i is the starting velocity at the beginning of each time of Delta T₀,  or ship time.

Now, the CMBR incident on the light sail from behind will generally require either a monolithic light sail of near nanometer thickness or perhaps a grid like sail with a cross-weave for which the lines or fibers are separated by less than 0.25 millimeters in order to reflect the vast majority of the incident CMBR for space craft traveling at mildly relativistic velocities. For grid like sails, the advantage of sail porosity enables much higher mass specific capture areas. Since the Doppler blue shifted light incident from directly in front of the sail or nearly so will be much shorter in wavelength than the backwardly incident light for high gamma factor sails, the forwardly incident light can largely pass through the sail openings providing a means for the backwardly incident light to push the sail efficiently forward for cases where the sail is transmissive from front to back to a suitable degree.

Now radiation pressure is equal to σT⁴/C and {2 [σT⁴/C]} for reflected radiation. However, for a space craft traveling through a black body radiation field, the apparent temperature of the radiation increases in proportion to γ and so the black body power impinging on the space craft from directly in front grows in proportion to T⁴ and thus  to γ⁴.

The temperature of a black body is T = {P/[(A)(σ)(e)]}^1/4 = {P/{(A)[5.670400(40) x 10^-8 W m^-2 K^-4]} ^1/4}. Therefore, the cosmic microwave background radiation pressure on a perfectly reflective flat bow  relativistic space craft is roughly equal to  {2 [σT_cmbr ⁴/C]} γ⁴ = {2 [σT_cmbr ⁴/C]} { {1/{1 – [(v/C)²]}^1/2}  ⁴} =   {2 [σT_cmbr ⁴/C]} {1/{1 – [(v/C)²]}²} where γ is constant and T_cmbr is constant. However, the light pressure incident from directly behind will be approximately equal to {2 [σT_cmbr ⁴/C]}/{1/{1 – [(v/C)²]}²}. In actuality, not all of the light is directly incident from the back and so there will be angular affects that result in loss of driving power. However, we will assume that all of the radiation is absorbed and then re-cycled and released as a perfect backwardly directed laser beam.

Assuming that the velocity of the sail starts out at (Zero) C, the optical pressure will be equal to {2 [σT_cmbr ⁴/C]}/γ⁴ = {2 [σT_cmbr ⁴/C]}/{ {1/{1 – [(v/C)²]}^1/2} ⁴} = {2{[5.670400(40) x 10^-8 W m^-2 K^-4][(2.725 K)⁴]/(300,000,000 m/s)}}/(1²) = 2.0844  x 10^-14 Newtons/m². For a 10,000 km by 10,000 km sail, the drive force will be 2.0844 Newtons.

Now, how are we going to deploy such a sail in a meaningful manner? The solution is obvious my dear Watson! Use a grid.

Consider that a monolithic one nanometer thick sail made of STP H₂O density carbonaceous materials would have a mass of  100,000 thousand metric tons, assuming that the space craft plus her sail had a mass of 208,440 metric tons, the craft would start out with an acceleration of F/M = a = 2.0844 N/208,440,000 kg = 10^-8 meter/s². After traveling 6 x 10¹⁵ seconds or about 200,000,000 years, the velocity of the space craft will be 0.20 C and the relativistic Lorentz transformation factor will be 1.0206 thus permitting the above Newtonian formula to apply.

For such a space craft that deployed a linear series of 1,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000 metric tons could be identically accelerated. The mass of the crew quarters would be 108,440,000 metric tons.

For such a space craft that deployed a linear series of 1,000,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000,000  metric tons could be identically accelerated. The mass of the crew quarters would be 108,440,000,000  metric tons. For cosmic journeys, this is not bad.

Some supermaterials already in laboratory existence such as carbon nanotubes can in theory be used to construct large space elevators that would extend from the surface of the Earth near the Equator to locations significantly father than geosynchronous orbit. Such tethers would perhaps have the equivalent of 0.1 G or 1m/s² acceleration based force pulling on it which would be commensurate with a cable roughly 100,000 km long accelerated at 1 m/s². Thus,  a cable that is 10¹³ km long or one light-year long could in theory withstand 10^-8 m/s² levels of acceleration. A linear series of tethered leading sails numbering 1,000,000 where each sail would have a width of 10,000 kilometers and be serially spaced a distance of 100,000 kilometers would have a length of 10¹¹ kilometers.  Thus, a series of 100 million tethered sails might conceivably pull a sail craft combination having a mass of 20,844,000,000,000 metric tons and a crew quarters having a mass of 10,844,000,000,000 metric tons.

Some high-end carbonaceous super-materials include:

1) carbon nano-tubes;
2) boron-nitride nanotubes;
3) buckyball-sheets;
4) layered sheet arrangements of graphene;
5) graphene-oxide paper;
6) fabrics composed of a weave or knit on carbon atom chains;
7) diamond fiber-based fabric;
8) carbon nitride fiber-based fabric;
9) combinations of two or more of the above, and the like material

Metalization would help in these regards.

The sails could have nanotech self-repair mechanisms. An ideal mechanism would entail sails constructed of metallic hydrogen where the hydrogen would be captured from interstellar space and incorporated into the sail membrane(s) in order to re-supply sail atoms knocked loose by interstellar atom and molecular species.

However, much higher sail velocities are anticipatable with much greater accelerations as will be covered in the next post in this series.

However, we can also deploy gridded sails. For example, consider a sail that is comprised on one nanometer wide fibers in a cross-weave where adjacent parallel  fibers are separated by 200 microns. Also consider situations where the fibers are one side reflective and one side transmissive. A sail comprised of such a material will have a mass specific capture area that is 1/100,000 that of a one nanometer thick monolithic sail.

Considering that such a sail that is gridded with the above  one nanometer thick sail fiber construction made of STP H₂O density carbonaceous materials would have a mass of  one metric ton for a sail area of 10⁸ square kilometers, assuming that the space craft plus her sail had a mass of 208,440 metric tons, the craft would start out with an acceleration of F/M = a = 2.0844 N/208,440,000 kg = 10^-8 meter/s². After traveling 6 x 10¹⁵ seconds or about 200,000,000 years, the velocity of the space craft will be 0.20 C and the relativistic Lorentz transformation factor will be 1.0206 thus permitting the above Newtonian formula to apply.

For such a space craft that deployed a linear series of 1,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000 metric tons could be identically accelerated. The mass of the crew quarters would be 208, 439,000 metric tons.

For such a space craft that deployed a linear series of 1,000,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000,000  metric tons could be identically accelerated. The mass of the crew quarters would be 208, 439,000,000 metric tons. For cosmic journeys, this is not bad.

A series of 100 million tethered sails might conceivably pull a sail craft combination having a mass of 20,844,000,000,000 metric tons and a crew quarters having a mass of 20,843,900,000,000 metric tons.

Consider again that such sails which  are gridded with the above  one nanometer thick sail fiber construction made of STP H₂O density carbonaceous materials would have a mass of  one metric ton each, assuming that the space craft plus her sail had a mass of 208,440 metric tons, the craft would start out with an acceleration of F/M = a = 2084.4 N/208,440,000 kg = 10^-5 meter/s² for cases where the craft would utilized 1,000 tethered driving sails. After traveling 6 x 10¹²  seconds or about 200,000 years, the velocity of the space craft will be 0.20 C.

Now consider a space craft having a mass of 208,440 metric tons driven by 10,000 such one metric tons sails. For such a space craft that deployed a linear series of 10,000 tethered sails where each sail was separated by an efficient 10 sail widths, the space craft having a total mass of 208, 440 metric tons would start out with an acceleration of F/M = a = 2084.4 N/208,440,000 kg = 10^-4  meter/s². After traveling 6 x 10¹¹ seconds or about 20,000 years, the velocity of the space craft will be 0.20 C.

We can consider more robust gridded sails such as those made from 10 nanometer diameter fibers spaced 200 microns apart. Each such sail would have a mass of 100 metric  tons. Thus, a space craft having a total mass of 208,440 metric tons that is driven by 1,000 such sails would too start out with an acceleration of F/M = a = 2084.4 N/208,440,000 kg = 10^-5 meter/s² and achieve a velocity of 0.20 C after 200,000 years.

Cnsider a sail that is comprised on 316.2  nanometer wide fibers in a cross-weave where adjacent parallel  fibers are separated by 200 microns. Also consider situations where the fibers are one side reflective and one side transmissive. A sail comprised of such a material will have a mass specific capture area that is equal to that of a one nanometer thick monolithic sail.

Consider that such a sail which is gridded with the above  316.2  nanometer thick sail fiber construction made of STP H₂O density carbonaceous materials would have a mass of  100,000 metric tons for a capture area of 10⁸ square kilometers. Assuming that the space craft plus her sail had a mass of 208,440 metric tons, the craft would start out with an acceleration of F/M = a = 2.0844 N/208,440,000 kg = 10^-8 meter/s². After traveling 6 x 10¹⁵ seconds or about 200,000,000 years, the velocity of the space craft will be 0.20 C and the relativistic Lorentz transformation factor will be 1.0206 thus permitting the above Newtonian formula to apply.

For such a space craft that deployed a linear series of 1,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000 metric tons could be identically accelerated. The mass of the crew quarters would be 208, 439,000 metric tons.

For such a space craft that deployed a linear series of 1,000,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000,000  metric tons could be identically accelerated. The mass of the crew quarters would be 208, 439,000,000 metric tons. For cosmic journeys, this is not bad.

A series of 100 million tethered sails might conceivably pull a sail craft combination having a mass of 20,844,000,000,000 metric tons and a crew quarters having a mass of 20,843,900,000,000 metric tons.

The background interstellar and intergalactic matter might not erode even many of highly relativistic sail of sub-micron thickness.

The diametrical cross-sectional area of our observable universe is close to 10 ⁴⁷  square kilometers and the mass of the total mass energy of the observable universe is only about 10 ⁵⁰ metric tons of which only 4 percent is baryonic.  Thus,  an average column spanning the diameter of the entire visible universe would have an H2O STP matter thickness of only 25 micrometers for reactive matter.

However,  this is not a concern for the following reasons.

First, the sails could be replaceable grid sails and driven by optical, IR, microwave or rf radiation. The mass of such sails can be reduced by many orders of magnitude relative to monolithic sails that are only micrometer scales in thickness.

Second, sails having a very thick cable or thread like construction are conceivable where the cables or wires would be many times if not several orders of magnitude thicker than 25 microns. The sails could be mostly empty space to almost entirely empty space to reflect long wave rF phased array beams.

As for concerns about over burdening the conductive or super-conductive wires or cables used for such sails by extremely intense rF beams, note that such reflective members could be very conductive to superconductive to thereby yield near perfect reflection. The EM energy that was not reflected would largely pass through the openings in the sail grid.

Second, a magnetic and/or electric field based scoop or anti-scoop could divert the chargons away from the sail just as an extended electrodynamic scoop for an interstellar ramjet would. Electro-dynamic-hydro-dynamic-plasma-drive features could utilize the diverted plasma in a reactive and gainful manner.

The sail might be deployed in a manner that is orthogonal to the ship’s velocity vector.  The sail might be parallel to the space craft velocity vector and driven obliquely from behind. This way, the effective thickness of the sail could be thousands of miles and the sail could include electro-dynamic-hydro-dynamic-plasma-drive features.

Fourth, the above parallel sail could conceivably be made of negative refraction index materials that would be pulled forward by incident star light and highly blue-shifted CMBR, far infrared, and non-CMBR radio sources.

Fifth, the sail can simply be a deployed mag-sail or M2P2 type of sail or any other magnetic or plasma bottle sail. It is possible that a plasma affixed to the space craft to be driven by rf radation, and even source based laser light upon attainment of extreme space craft gamma factors could be easily reflected by such sails. Plasma makes an excellent rF reflector even at very small densities.

I have done a lot of writing on parallel sails such as negative refraction index monolithic and grid sails capable of extreme gamma factors.

Sixth, some sail materials such as any future forms of super-strong very conductive to super-conductive metallic hydrogen can be used as nuclear fusion fuel for fusion rockets upon degradation to useless levels.

Seventh, it has been proposed that very thin,  metallic,  very low gas density containing balloons might be used for nuclear warhead decoys and which could survive 100 meter proximity detonation to a one kiloton neutron bomb in the vacuum of space. The rate of radiative cooling would be tens of billions of Kelvins per second due to the extreme thinness of the balloon membranes and most of the neutrons would pass right through the balloon without interacting or by only depositing a very small portion of the particles kinetic energy into the balloon and enclosed gas. Interstellar chargons are more reactive to electronic shell structures but not by that much.

The general idea for obliquely oriented beams involves the beamed energy incident on both sides of the sail. The sail could include a surface of hair like cilia or any other surface contour that would work so as to much more effectively grab ahold of the light.

In addition, the sail could be fabricated from photovoltaic materials in order to provide power for electro-dynamic-hydrodynamic-plasma-drives or chargon rockets, or perhaps even photon rockets.

For extreme gamma factors, the CMBR and starlight will be highly blue-shifted and will be relativistically abberated to what would approach a point source in front of the space craft at gamma = infinity. A sail parallel to the space craft velocity vector made of a suitable negative electromagnetic refraction index material will be pulled forward even by light incident on the sail at a very shallow angle from in front of the space craft.

To enhance the negative refraction index sails capture of EM energy, the sails may have negative index hairs or cilia distributed along its length.

Negative refraction index materials have actually been measured to be pulled on by incident light. Duke University and other academic and government labs are researching the various aspects of negative refraction index materials.

I have no problem with space craft being pulled forward by forward incident light. After all, the paradigm of light speed velocity limits may or may not have been shattered with any future validation or not of the CERN superluminal neutrino results. The big bang may have been the most recent free lunch. There is no reason why the big bang could not have started with miniscule quantities of mass-energy.

A good abstract for a great paper on negative super-pressure of light acting on a negative refractive index material is

Henri Lezec
(Center for Nanoscale Science and Technology, NIST)

Forty years ago, V. Veselago derived the electromagnetic properties of a hypothetical material having simultaneously-negative values of electric permittivity and magnetic permeability [1]. Such a material, denominated “left-handed”, was predicted to exhibit a negative index of refraction, as well as a number of other counter-intuitive optical properties. For example, it was hypothesized that a perfect mirror illuminated with a plane wave would experience a negative radiation pressure (pull) when immersed in a left-handed medium, as opposed to the usual positive radiation pressure experienced when facing a dielectric medium such as air or glass. Since left-handed materials are not available in nature, considerable efforts are currently under way to implement them under the form of artificial “metamaterials” — composite media with tailored bulk optical characteristics resulting from constituent structures which are smaller in both size and density than the effective wavelength in the medium. Here we show how surface-plasmon modes propagating in a stacked array of metal-insulator-metal (MIM) waveguides can be harnessed to yield a volumetric left-handed metamaterial characterized by an in-plane-isotropic negative index of refraction over a broad frequency range spanning the blue and green. By sculpting this material with a focused-ion beam we realize prisms and micro-cantilevers which we use to demonstrate, for the first time, (a) in-plane isotropic negative-refraction at optical frequencies, and (b) negative radiation pressure. We predict and experimentally verify a negative “superpressure”, the magnitude of which exceeds the photon pressure experienced by a perfect mirror by more than a factor of two. 1) V. Veselago, \textit{ Sov. Phys. Usp. }10, p.509 (1968).

Available at:

http://meetings.aps.org/Meeting/MAR09/Event/93172

The sail might not need to  be held by guy lines. A strong magnetic field based coupling or electrical charged based connection might work.

Another option is to fabricate the sail guy lines out of graphene, carbon nanotubes, boron nitride nanotubes, graphene oxide paper, and the like. A cable constructed from such materials could stretch for about 20 to 50 kilometers yet still handle tens to hundreds of Earth G’s. The tensile strength of graphene is close to 18 million PSI for perfect forms.

Materials such as solid quarkoniums and somehow stabilized neutroniums, and perhaps even Higgsiniums would be better yet, but such materials may only exist in nature in extreme mass quantity states as of the present cosmic era.

The collection area of the sail can be very, very, large. A large electro-dynamic scoop could extent very far out from the sail.

Regarding nanotech self-assembly mechanisms, just simply greatly increase the capture area of a electrodynamic scoop to collect enough interstellar materials and use most of the collected interstellar material as an EHPD, an MHPD, or a combination of the two and use the rest of the materials for sail repair.

Regarding holding M2P2 plasma affixed to the ship under high gamma factor condition, simply increase the strength of the fastening fields.

Now regarding interstellar matter density near our solar system of one particle for every 10 cm³, the density would  work out to be a layer of hydrogen or helium atoms about one atom thick for a column that is one light-year long. Not a show stopper for light sails or sails that are electro-dynamically shielded or protected.

If extreme materials are used with excellent reflectance, we could simply use a sail that has a thickness of one millimeter or more and which is monolithic, or better yet,  use a sail with grid lines that are one millimeter or perhaps much greater in thickness. This way, a sail that has an area of only one square kilometer can intercept a beam having an equivalent black body temperature of several thousand Kelvins provided it is constructed of suitably refractive materials.

We could simply use electrodynamic methods of grabbing ahold of the interstellar gas and diverting around the space craft and sail. The power to operate the electrodynamic mechanisms can be supplied by beams. The electrodynamic methods can include lasers for ionization, or rf radiation where the gamma factors are suitably large, magnetic fields, electric fields, plasma fields affixed to the space craft, and the like.

Then there is always the possibilities for sails comprised of truly exotic materials such as somehow stabilized neutroniums, quarkoniums, higgsiniums, monopoliums, and perhaps even raw space-time-mass-energy forms such as the “Yelm” of mid-20th Century big bang theory.

Since one cubic meter of neutronium would have a mass of about 10¹⁵ tons. A 1,000 kilometer long thread of the stuff that has a cross-sectional area of 1,000,000 neutrons would have a mass of only one kilogram. A 1 kilometer long thread having a cross-sectional area of 1 billion neutrons would have a mass of only 1 kilogram. Lines made of quarkoniums could have the same length and cross-section but would be 10 to 1,000 times more massive. Higgsiniums would be all the more massive.

Provided such extreme materials could be developed, they could also serve as electric current carrying magnetic sail components. Anyhow magnetic sails can be made of any ordinary conducting or superconducting period table materials.

It is also conceivable that a hybrid sail can be used where a current carrying magsail would deflect plasma away from a monolithic and grid like light sail or rf sail.

Now, regarding the subject of sail erosion by exposure to interstellar or intergalactic gas, we must realize that the kinetic energy of a gas atom traveling at a velocity of 86.7 percent of the speed of light with respect to the sail would be equal to the binding energy of roughly 10 billion atoms within a sail of micron thickness. Thus, the fact that 10 billion atoms could be dislodged should all of the energy of the gas atom be deposited within the sail. Incident gas atoms having even higher associated gamma factors with respect to the star ship sail could potentially knock loose even more atoms. Perhaps, there is no reason to worry about sail erosion in spite of this for the following reasons.

First, extremely relativistic particles would likely deposit only a small portion of its energy within the sail thereby greatly lessening the number of atoms that would be knocked loose. This fact would apply to chargons as well as neutral incident particles.

Second, for sails of near micron thickness, atoms that were knocked loose would likely simply be re-assimilated by the bulk sail materials. Perhaps the only chance for an atom to be knocked loose would include atoms located on the backward side of the sail.  Atoms for which bonds where broken within the bulk sail material would tend to simply re-bond with adjacent atoms.

Third, since the incident gas or plasma particle would deposit only a small portion of its energy within the sail, the kinetic energy per particle for particles that are knocked loose may be only slightly in excess of the binding energy of the dislodged atoms. Basically, the kinetic energy of the dislodged atoms could likely be re-absorbed and/or radiated away thereby promoting rebinding of the dislodged atoms.

Fourth, for cases where the sail would completely absorb the kinetic energy of the incident gas or plasma particles such as an alpha particle, for the case of a one micron thick sail, the sail would obviously be able to complete stop the chargon without losing it. Thus, any atoms disbonded by the incident chargon would also likely be captured and prevented from leaving the sail material.

Fifth, for grid like sails, the grid lines might be positively chargeable so that incident interstellar or intergalactic ions are pushed away from the grid lines and through the openings within the grid like sails. The effect would be similar to the Vander walls force that keeps neutral atoms from being squeezed together to tightly.

We now perform a reality check on the above formulations.

Consider the space craft at a stationary state. The CMBR appears equally bright from all directions within about 1 part in 30,000.

Now, the apparent angle, θ_s,  of CMBR pre-incident on the space craft at an angle of 90 degrees or with respect to the length of the space craft relative to the source reference frame at v = zero C will appear to be incoming at an angle, θ_o,  of 90 degrees with respect to the space craft,  ship’s reference frame.

If we consider the effects of relativistic aberration, the general formula for apparent shift in angle of incidence of the CMBR from the ship’s perspective is

Cos θ_o = {[cos θ_s] – (v/C)}/{1 – [(v/C) cos θ_s]}

Now say we desire to find the range of CMBR angles incident on the space craft with respect to the space craft reference frame for space craft velocities of 0.20 C.

Now since we are considering an angle of θ_o = π/2, cos θ_o = zero. Using the above formula, we achieve Cos θ_o = {[cos θ_s] – (v/C)}/{1 – [(v/C) cos θ_s]} = Cos π/2 = {[cos θ_s] – (0.20)}/{1 – [(0.20) cos θ_s]} = zero = {[cos θ_s] – (0.20)}/{1 – [(0.20) cos θ_s]}.

Thus, (zero) {1 – [(0.20) cos θ_s]} = {[cos θ_s] – (0.20)} = zero.

Therefore, cos θ_s = 0.20 — > θ_s  = 78.463 degrees. We will make a first order assumption that the incident CMBR from behind has a frequency of f’ = f / {γ [1 + (β cosine θ)]} = f / {1.02062 [1 + [0.2 cosine ( 0)]]} = (0.816497161) f. Thus, we will assume that θ = 0 degrees for the following 5 scenarios where we assume that the CMBR is directly incident from behind.

The radiated power received by the sail will be [(78.463)²/(90²)] f’ = [(78.463)²/(90²)] f / {γ [1 + (β cosine θ)]} = [(78.463)²/(90²)] f / {1.02062 [1 + [0.2 cosine ( 0)]]} = [(78.463)²/(90²)] (0.816497161) f =

Once again, the temperature of a black body is T = {P/[(A)(σ)(e)]}^1/4 = {P/{(A)[5.670400(40) x 10^-8 W m^-2 K^-4]} ^1/4}. The light pressure incident from directly behind will be approximately equal to [(θ_s)²/(90²)] {2 {σ {{(T_cmbr)/{γ [1 + (β cosine θ)]}}}⁴/C}} =  [(78.463)²/(90²)] {2 {[5.670400(40) x 10^-8 W m^-2 K^-4] {[(T_cmbr) (0.816497161) ]⁴}/C}} = 4.632092076  x 10^-15 Newtons/m². In actuality, not all of the light is directly incident from the back and so there will be angular affects that result in loss of driving power. However, we will assume that all of the radiation is absorbed and then re-cycled and released as a perfect backwardly directed laser beam.

Now  E_gain = ʃF^odx = ʃ(0,10²⁵) F^odx = ʃ(0,10²⁵)(10¹⁴) [4.632092076  x 10^-15 N] ^odx = 4.632092076 x 10²⁴ Joules.

Now, a 208,440 metric invariant mass space craft traveling at a starting velocity of 0.2 C has a kinetic energy of {1.02062[M C²]} – [M C²] = {1.02062[208,440,000  C²]} – [208,440,000  C²] =  1.9146 x 10²⁵ Joules – 1.87596 x 10²⁵ Joules = 3.864 x 10²³ Joules. When  4.632092076 x 10²⁴ Joules is added, the total gamma factor becomes [5.01849 x 10²⁴ Joules + 1.87596 x 10²⁵ Joules]/ [1.87596 x 10²⁵ Joules] = 1.2675. The associated space craft velocity will be equal to 0.6142 C.

Likewise doing iterated numerical approximations with v = 0.6142 C to obtain another higher velocity and then repeating the steps over and over again will give a first order approximation for space craft terminal velocity.

So we have reasonably demonstrated that CMBR sails can drive very large space arks to velocities considered fast by interstellar propulsion physicists. Typically, fast interstellar travel occurs at a better part of the speed of light.

However, a much finer scale is needed to produce results for many such steps where the computed velocity would not significantly diverge from the actual velocity obtained.

Note that here, I neglect the effects of mass based astrodynamic drag. I have come up with several mechanisms by which massive astrodynamic drag can be almost entirely eliminated and will post on this subject later this month.

The total kinetic energy gain for the craft will be

E _totalgain = ʃF ₁^odx₁  +  ʃF ₂^odx₂ + ʃF ₃^odx₃ + … +ʃF _n^odx_n

= {{ʃ{[(θ_s1)²/ ((90 degrees) ²)] {2 {σ {{(T_cmbr)/{γ₁ [1 + (β₁ cosine θ)]}}}⁴/C}}}^odx₁}(A)}+  {{ʃ{[(θ_s2)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ₂ [1 + (β₂ cosine θ)]}}}⁴/C}}} ^odx₂}(A)} + {{ʃ{[(θ_s3)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ₃ [1 + (β₃ cosine θ)]}}}⁴/C}}} ^odx₃}(A)} + … +{{ʃ{[(θ_sn)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ_n [1 + (β_n cosine θ)]}}}⁴/C}}} ^odx_n}(A)}

= ʃ{[(θ_s1)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{{1/{1 – [(v₁/C)²]}^1/2} [1 + (β₁ cosine θ)]}}}⁴/C}}}^odx₁  +  ʃ{[(θ_s2)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v₂/C)²]}^1/2}  [1 + (β₂ cosine θ)]}}}⁴/C}}} ^odx₂ + ʃ{[(θ_s3)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v₃/C)²]}^1/2}  [1 + (β₃ cosine θ)]}}}⁴/C}}} ^odx₃ + … +ʃ{[(θ_sn)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v_n/C)²]}^1/2}  [1 + (β_n cosine θ)]}}}⁴/C}}} ^odx_n

= Σ (i = 1, i = n) ʃF _i^odx_i   = Σ (i = 1, i = n) ʃ{[(θ_si)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ_i [1 + (β_i cosine θ)]}}}⁴/C}}}^odx_i

=   Σ (i = 1, i = n) ʃ{[(θ_si)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v_i/C)²]}^1/2}  [1 + (β_i cosine θ)]}}}⁴/C}}} ^odx_i

The following numerical formula offers a first order approximation of space craft gamma factor  gain;

{[E_KEtotalgain] + [M_restC²]}/[M_restC²]= {[ʃF ₁^odx₁]  +  [ʃF ₂^odx₂]  + [ʃF ₃^odx₃ ] + … + [ʃF _n^odx_n] + [M_restC²]}/[M_restC²] 

= {{{ʃ{[(θ_s1)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ₁ [1 + (β₁ cosine θ)]}}}⁴/C}}}^odx₁}(A)}+  {{ʃ{[(θ_s2)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ₂ [1 + (β₂ cosine θ)]}}}⁴/C}}} ^odx₂ }(A)}  + {{ʃ{[(θ_s3)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ₃ [1 + (β₃ cosine θ)]}}}⁴/C}}} ^odx₃ }(A)} + … + {{ʃ{[(θ_sn)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ_n [1 + (β_n cosine θ)]}}}⁴/C}}} ^odx_n}(A)}+ [M_restC²]}/[M_restC²] 

= {{{ʃ{[(θ_s1)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{{1/{1 – [(v₁/C)²]}^1/2} [1 + (β₁ cosine θ)]}}}⁴/C}}}^odx₁  }(A)}  +  {{ʃ{[(θ_s2)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v₂/C)²]}^1/2}  [1 + (β₂ cosine θ)]}}}⁴/C}}} ^odx₂ }(A)} +   {{ʃ{[(θ_s3)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v₃/C)²]}^1/2}  [1 + (β₃ cosine θ)]}}}⁴/C}}} ^odx₃ }(A)} + … +  {{ʃ{[(θ_sn)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v_n/C)²]}^1/2}  [1 + (β_n cosine θ)]}}}⁴/C}}} ^odx_n}(A)}+ [M_restC²]}/[M_restC²] 

= {{Σ (i = 1, i = n) ʃF _i^odx_i }+ [M_restC²]}/[M_restC²]   = {{Σ (i = 1, i = n) {{ʃ{[(θ_si)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ_i [1 + (β_i cosine θ)]}}}⁴/C}}}^odx_i}(A)}}+ [M_restC²]}/[M_restC²] 

=   {{Σ (i = 1, i = n) ʃ{[(θ_si)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v_i/C)²]}^1/2}  [1 + (β_i cosine θ)]}}}⁴/C}}} ^odx_i}+ [M_restC²]}/[M_restC²] 

Now, v = C{[-[1/γ²] + 1]^1/2} according to Special Relativity. Consequently, the following formulas can be used to compute v by numerical trial and error.

v = C{{-{1/{{[E_KEtotalgain] + [M_restC²]}/[M_restC²]}²} – 1}^1/2}  

=  C{{-{1/{{[ʃF ₁^odx₁]  + [ʃF ₂^odx₂]  + [ʃF ₃^odx₃ ] + ʃF _n^odx_n]} + [M_restC²]}/[M_restC²]}  ²}} + 1}^1/2}

=  C{{-{1/{{{{ʃ{[(θ_s1)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ₁ [1 + (β₁ cosine θ)]}}}⁴/C}}}^odx₁}(A)}+   {{ʃ{[(θ_s2)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ₂ [1 + (β₂ cosine θ)]}}}⁴/C}}} ^odx₂ }(A)} +  {{ʃ{[(θ_s3)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ₃ [1 + (β₃ cosine θ)]}}}⁴/C}}} ^odx₃ }(A)}   + … + {{ʃ{[(θ_sn)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ_n [1 + (β_n cosine θ)]}}}⁴/C}}} ^odx_n}(A)}+ [M_restC²]}/[M_restC²]}  ²}} + 1}^1/2}

=  C{{-{1/{{{{ʃ{[(θ_s1)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{{1/{1 – [(v₁/C)²]}^1/2} [1 + (β₁ cosine θ)]}}}⁴/C}}}^odx₁  }(A)} +  {{ʃ{[(θ_s2)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v₂/C)²]}^1/2}  [1 + (β₂ cosine θ)]}}}⁴/C}}} ^odx₂ }(A)} +  {{ʃ{[(θ_s3)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v₃/C)²]}^1/2}  [1 + (β₃ cosine θ)]}}}⁴/C}}} ^odx₃ }(A)} + … +  {{ʃ{[(θ_sn)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v_n/C)²]}^1/2}  [1 + (β_n cosine θ)]}}}⁴/C}}} ^odx_n}(A)}+ [M_restC²]}/[M_restC²]}²}} + 1}^1/2}.   .

=  C{{-{1/{{{{Σ (i = 1, i = n) ʃF _i^odx_i }+ [M_restC²]}/[M_restC²]}   ²}} + 1}^1/2} 

=    C{{-{1/{{{{Σ (i = 1, i = n) {{ʃ{[(θ_si)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{γ_i [1 + (β_i cosine θ)]}}}⁴/C}}}^odx_i}(A)}}+ [M_restC²]}/[M_restC²]}  ²}} + 1}^1/2} 

=   C{{-{1/{{{{Σ (i = 1, i = n) {{ʃ{[(θ_si)²/(((90 degrees)²)] {2 {σ {{(T_cmbr)/{ {1/{1 – [(v_i/C)²]}^1/2}  [1 + (β_i cosine θ)]}}}⁴/C}}} ^odx_i}(A)}}+ [M_restC²]}/[M_restC²]}  ²}} + 1}^1/2}

When the to two sides of the above equations are equal, we will have thus computed relativistic velocity, v.   

As you can see, for cases where there is much natural variation in acceleration with respect to the space craft frame, and for travel over very long distances, many iterations or steps need to be used in numerical algorithms to get mil spec and super-mil-spec results. Such precision is needed when traveling near light speed otherwise mission disaster could happen. In actuality, the above formulations would not be fit for mil spec computations because of the mere approximation to the actual vehicular performance.

We now consider scenarios where the photon angle of incidence from behind is considered and where drag effects are neglected for total space craft energy gains, accrued gamma factors, and accrued velocities. Here, we consider only angular values of radiation incident on the sail for which the radiation exerts forward pressure. In otherwords, we only consider values of θ₀ less than or equal to 90 degrees or π/2 radians. We also assume perfect backward sail reflectivity or trivially imperfect backward reflectivity and trivial massive astrodynamic drag.

The total kinetic energy gain for the craft will be

E _totalgain = ʃF ₁^odx₁  +  ʃF ₂^odx₂ + ʃF ₃^odx₃ + … +ʃF _n^odx_n

= {Σ(i = 1, i = n) {{{[[(θ_01i,1²) - (θ_02i,1²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,1 , θ_s2i,1)}^-1}{{[dr_bi,1/dr_ai,1]}│(r_b1i,1 , r_b2i,1)}[cos [(θ_01i,1 +  θ_02i,1)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₁ [1 + [β₁cosine [(θ_01i,1 + θ_02i,1)/2]]]}}}⁴/C}}}^odx₁}/n}(A)}}}  

+  {Σ(i = 1, i = n) {{{[[(θ_01i,2²) - (θ_02i,2²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,2 , θ_s2i,2)}^-1}{{[dr_bi,2/dr_ai,2]}│(r_b1i,2 , r_b2i,2)}[cos [(θ_01i,2 +  θ_02i,2)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₂ [1 + [β₂cosine [(θ_01i,2 + θ_02i,2)/2]]]}}}⁴/C}}}^odx₂}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{[[(θ_01i,3²) - (θ_02i,3²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,3 , θ_s2i,3)}^-1}{{[dr_bi,3/dr_ai,3]}│(r_b1i,3 , r_b2i,3)}[cos [(θ_01i,3 +  θ_02i,3)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₃ [1 + [β₃cosine [(θ_01i,3 + θ_02i,3)/2]]]}}}⁴/C}}}^odx₃}/n}(A)}}} 

+ … +    {Σ(i = 1, i = n) {{{[[(θ_01i,m²) - (θ_02i,m²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,m , θ_s2i,m)}^-1}{{[dr_bi,m/dr_ai,m]}│(r_b1i,m , r_b2i,m)}[cos [(θ_01i,m +  θ_02i,m)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_m[1 + [β₂cosine [(θ_01i,m + θ_02i,m)/2]]]}}}⁴/C}}}^odx_m}/n}(A)}}} 

= {Σ(i = 1, i = n) {{{[[(θ_01i,1²) - (θ_02i,1²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,1 , θ_s2i,1)}^-1}{{[dr_bi,1/dr_ai,1]}│(r_b1i,1 , r_b2i,1)}[cos [(θ_01i,1 +  θ_02i,1)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{{1/{1 – [(v₁/C)²]}^1/2}  [1 + [(v₁/C)cosine [(θ_01i,1 + θ_02i,1)/2]]]}}}⁴/C}}}^odx₁}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{[[(θ_01i,2²) - (θ_02i,2²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,2 , θ_s2i,2)}^-1}{{[dr_bi,2/dr_ai,2]}│(r_b1i,2 , r_b2i,2)}[cos [(θ_01i,2 +  θ_02i,2)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₂/C)²]}^1/2}  [1 + [(v₂/C)cosine [(θ_01i,2 + θ_02i,2)/2]]]}}}⁴/C}}}^odx₂}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{[[(θ_01i,3²) - (θ_02i,3²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,3 , θ_s2i,3)}^-1}{{[dr_bi,3/dr_ai,3]}│(r_b1i,3 , r_b2i,3)}[cos [(θ_01i,3 +  θ_02i,3)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₃/C)²]}^1/2}  [1 + [(v₃/C)cosine [(θ_01i,3 + θ_02i,3)/2]]]}}}⁴/C}}}^odx₃}/n}(A)}}} 

+ … +    {Σ(i = 1, i = n) {{{[[(θ_01i,m²) - (θ_02i,m²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,m , θ_s2i,m)}^-1}{{[dr_bi,m/dr_ai,m]}│(r_b1i,m , r_b2i,m)}[cos [(θ_01i,m +  θ_02i,m)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_m/C)²]}^1/2} [1 + [(v_m/C)cosine [(θ_01i,m + θ_02i,m)/2]]]}}}⁴/C}}}^odx_m}/n}(A)}}} 

=  Σ(i = 1, i = m)   {Σ(i = 1, i = n) {{{[[(θ_01i,j²) - (θ_02i,j²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,j , θ_s2i,j)}^-1}{{[dr_bi,j/dr_ai,j]}│(r_b1i,j , r_b2i,j)}[cos [(θ_01i,j +  θ_02i,j)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_j [1 + [β_jcosine [(θ_01i,j + θ_02i,j)/2]]]}}}⁴/C}}}^odx_j}/n}(A)}}} 

= Σ(i = 1, i = m)   {Σ(i = 1, i = n) {{{[[(θ_01i,j²) - (θ_02i,j²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,j , θ_s2i,j)}^-1}{{[dr_bi,j/dr_ai,j]}│(r_b1i,j , r_b2i,j)}[cos [(θ_01i,j +  θ_02i,j)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_j/C)²]}^1/2}  [1 + [(v_j/C)cosine [(θ_01i,j + θ_02i,j)/2]]]}}}⁴/C}}}^odx_j}/n}(A)}}} 

= {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}

²} – {{cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₁/C)}/{1 – [(v₁/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,1 , θ_s2i,1)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₁ /C)}/{1 – [(v₁/C) cos θ_s]}}}}│[cos  θ_s1i,1 ,  cos θ_s2i,1]}{cos {{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}+  {cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₁ {1 + {β₁cosine {{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}+ {cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}}/2}}}}}}⁴/C}}}^odx₁}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}

²} – {{cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₂/C)}/{1 – [(v₂/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,2 , θ_s2i,2)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₂ /C)}/{1 – [(v₂/C) cos θ_s]}}}}│[cos  θ_s1i,2 ,  cos θ_s2i,2]}{cos {{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}+  {cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₂ {1 + {β₂cosine {{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}+ {cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}}/2}}}}}}⁴/C}}}^odx₂}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}

²} – {{cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₃/C)}/{1 – [(v₃/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,3 , θ_s2i,3)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₃ /C)}/{1 – [(v₃/C) cos θ_s]}}}}│[cos  θ_s1i,3 ,  cos θ_s2i,3]}{cos {{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}+  {cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₃ {1 + {β₃cosine {{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}+ {cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}}/2}}}}}}⁴/C}}}^odx₃}/n}(A)}}} 

+ … +   {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}

²} – {{cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_m/C)}/{1 – [(v_m/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,m , θ_s2i,m)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_m /C)}/{1 – [(v_m/C) cos θ_s]}}}}│[cos  θ_s1i,m ,  cos θ_s2i,m]}{cos {{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}+  {cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_m {1 + {β_mcosine {{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}+ {cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}}/2}}}}}}⁴/C}}}^odx_m}/n}(A)}}} 

= {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}

²} – {{cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₁/C)}/{1 – [(v₁/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,1 , θ_s2i,1)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₁ /C)}/{1 – [(v₁/C) cos θ_s]}}}}│[cos  θ_s1i,1 ,  cos θ_s2i,1]}{cos {{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}+  {cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₁/C)²]}^1/2}   {1 + {(v₁/C)cosine {{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}+ {cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}}/2}}}}}}⁴/C}}}^odx₁}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}

²} – {{cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₂/C)}/{1 – [(v₂/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,2 , θ_s2i,2)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₂ /C)}/{1 – [(v₂/C) cos θ_s]}}}}│[cos  θ_s1i,2 ,  cos θ_s2i,2]}{cos {{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}+  {cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₂/C)²]}^1/2}   {1 + {(v₂/C)cosine {{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}+ {cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}}/2}}}}}}⁴/C}}}^odx₂}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}

²} – {{cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₃/C)}/{1 – [(v₃/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,3 , θ_s2i,3)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₃ /C)}/{1 – [(v₃/C) cos θ_s]}}}}│[cos  θ_s1i,3 ,  cos θ_s2i,3]}{cos {{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}+  {cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₃/C)²]}^1/2}   {1 + {(v₃/C)cosine {{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}+ {cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}}/2}}}}}}⁴/C}}}^odx₃}/n}(A)}}} 

+ … +   {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}

²} – {{cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_m/C)}/{1 – [(v_m/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,m , θ_s2i,m)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_m /C)}/{1 – [(v_m/C) cos θ_s]}}}}│[cos  θ_s1i,m ,  cos θ_s2i,m]}{cos {{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}+  {cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_m/C)²]}^1/2}   {1 + {(v_m/C)cosine {{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}+ {cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}}/2}}}}}}⁴/C}}}^odx_m}/n}(A)}}} 

Σ ( j= 1, j = m) {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}

²} – {{cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_j/C)}/{1 – [(v_j/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,j , θ_s2i,j)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_j /C)}/{1 – [(v_j/C) cos θ_s]}}}}│[cos  θ_s1i,j ,  cos θ_s2i,j]}{cos {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+  {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_j {1 + {β_jcosine {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+ {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}}}}⁴/C}}}^odx_j}/n}(A)}}} 

= Σ ( j= 1, j = m) {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}

²} – {{cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_j/C)}/{1 – [(v_j/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,j , θ_s2i,j)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_j /C)}/{1 – [(v_j/C) cos θ_s]}}}}│[cos  θ_s1i,j ,  cos θ_s2i,j]}{cos {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+  {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_j/C)²]}^1/2}   {1 + {(v_j/C)cosine {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+ {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}}}}⁴/C}}}^odx_j}/n}(A)}}} 

The following numerical formula offers a first order approximation of space craft gamma factor  gain;

{[E_KEtotalgain] + [M_restC²]}/[M_restC²]= {[ʃF ₁^odx₁]  +  [ʃF ₂^odx₂]  + [ʃF ₃^odx₃ ] + … + [ʃF _n^odx_n] + [M_restC²]}/[M_restC²] 

= {{{Σ(i = 1, i = n) {{{[[(θ_01i,1²) - (θ_02i,1²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,1 , θ_s2i,1)}^-1}{{[dr_bi,1/dr_ai,1]}│(r_b1i,1 , r_b2i,1)}[cos [(θ_01i,1 +  θ_02i,1)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₁ [1 + [β₁cosine [(θ_01i,1 + θ_02i,1)/2]]]}}}⁴/C}}}^odx₁}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{[[(θ_01i,2²) - (θ_02i,2²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,2 , θ_s2i,2)}^-1}{{[dr_bi,2/dr_ai,2]}│(r_b1i,2 , r_b2i,2)}[cos [(θ_01i,2 +  θ_02i,2)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₂ [1 + [β₂cosine [(θ_01i,2 + θ_02i,2)/2]]]}}}⁴/C}}}^odx₂}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{[[(θ_01i,3²) - (θ_02i,3²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,3 , θ_s2i,3)}^-1}{{[dr_bi,3/dr_ai,3]}│(r_b1i,3 , r_b2i,3)}[cos [(θ_01i,3 +  θ_02i,3)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₃ [1 + [β₃cosine [(θ_01i,3 + θ_02i,3)/2]]]}}}⁴/C}}}^odx₃}/n}(A)}}} 

+ … +    {Σ(i = 1, i = n) {{{[[(θ_01i,m²) - (θ_02i,m²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,m , θ_s2i,m)}^-1}{{[dr_bi,m/dr_ai,m]}│(r_b1i,m , r_b2i,m)}[cos [(θ_01i,m +  θ_02i,m)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_m[1 + [β₂cosine [(θ_01i,m + θ_02i,m)/2]]]}}}⁴/C}}}^odx_m}/n}(A)}}}}  + [M_restC²]}/[M_restC²] 

= {{{Σ(i = 1, i = n) {{{[[(θ_01i,1²) - (θ_02i,1²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,1 , θ_s2i,1)}^-1}{{[dr_bi,1/dr_ai,1]}│(r_b1i,1 , r_b2i,1)}[cos [(θ_01i,1 +  θ_02i,1)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{{1/{1 – [(v₁/C)²]}^1/2}  [1 + [(v₁/C)cosine [(θ_01i,1 + θ_02i,1)/2]]]}}}⁴/C}}}^odx₁}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{[[(θ_01i,2²) - (θ_02i,2²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,2 , θ_s2i,2)}^-1}{{[dr_bi,2/dr_ai,2]}│(r_b1i,2 , r_b2i,2)}[cos [(θ_01i,2 +  θ_02i,2)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₂/C)²]}^1/2}  [1 + [(v₂/C)cosine [(θ_01i,2 + θ_02i,2)/2]]]}}}⁴/C}}}^odx₂}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{[[(θ_01i,3²) - (θ_02i,3²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,3 , θ_s2i,3)}^-1}{{[dr_bi,3/dr_ai,3]}│(r_b1i,3 , r_b2i,3)}[cos [(θ_01i,3 +  θ_02i,3)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₃/C)²]}^1/2}  [1 + [(v₃/C)cosine [(θ_01i,3 + θ_02i,3)/2]]]}}}⁴/C}}}^odx₃}/n}(A)}}} 

+ … +    {Σ(i = 1, i = n) {{{[[(θ_01i,m²) - (θ_02i,m²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,m , θ_s2i,m)}^-1}{{[dr_bi,m/dr_ai,m]}│(r_b1i,m , r_b2i,m)}[cos [(θ_01i,m +  θ_02i,m)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_m/C)²]}^1/2} [1 + [(v_m/C)cosine [(θ_01i,m + θ_02i,m)/2]]]}}}⁴/C}}}^odx_m}/n}(A)}}}}  + [M_restC²]}/[M_restC²] 

=  {{Σ(i = 1, i = m)   {Σ(i = 1, i = n) {{{[[(θ_01i,j²) - (θ_02i,j²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,j , θ_s2i,j)}^-1}{{[dr_bi,j/dr_ai,j]}│(r_b1i,j , r_b2i,j)}[cos [(θ_01i,j +  θ_02i,j)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_j [1 + [β_jcosine [(θ_01i,j + θ_02i,j)/2]]]}}}⁴/C}}}^odx_j}/n}(A)}}}}  + [M_restC²]}/[M_restC²] 

= {{Σ(i = 1, i = m)   {Σ(i = 1, i = n) {{{[[(θ_01i,j²) - (θ_02i,j²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,j , θ_s2i,j)}^-1}{{[dr_bi,j/dr_ai,j]}│(r_b1i,j , r_b2i,j)}[cos [(θ_01i,j +  θ_02i,j)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_j/C)²]}^1/2}  [1 + [(v_j/C)cosine [(θ_01i,j + θ_02i,j)/2]]]}}}⁴/C}}}^odx_j}/n}(A)}}} + [M_restC²]}/[M_restC²]   

= {{{Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}

²} – {{cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₁/C)}/{1 – [(v₁/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,1 , θ_s2i,1)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₁ /C)}/{1 – [(v₁/C) cos θ_s]}}}}│[cos  θ_s1i,1 ,  cos θ_s2i,1]}{cos {{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}+  {cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₁ {1 + {β₁cosine {{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}+ {cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}}/2}}}}}}⁴/C}}}^odx₁}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}

²} – {{cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₂/C)}/{1 – [(v₂/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,2 , θ_s2i,2)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₂ /C)}/{1 – [(v₂/C) cos θ_s]}}}}│[cos  θ_s1i,2 ,  cos θ_s2i,2]}{cos {{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}+  {cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₂ {1 + {β₂cosine {{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}+ {cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}}/2}}}}}}⁴/C}}}^odx₂}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}

²} – {{cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₃/C)}/{1 – [(v₃/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,3 , θ_s2i,3)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₃ /C)}/{1 – [(v₃/C) cos θ_s]}}}}│[cos  θ_s1i,3 ,  cos θ_s2i,3]}{cos {{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}+  {cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ₃ {1 + {β₃cosine {{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}+ {cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}}/2}}}}}}⁴/C}}}^odx₃}/n}(A)}}} 

+ … +   {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}

²} – {{cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_m/C)}/{1 – [(v_m/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,m , θ_s2i,m)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_m /C)}/{1 – [(v_m/C) cos θ_s]}}}}│[cos  θ_s1i,m ,  cos θ_s2i,m]}{cos {{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}+  {cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_m {1 + {β_mcosine {{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}+ {cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}}/2}}}}}}⁴/C}}}^odx_m}/n}(A)}}}} + [M_restC²]}/[M_restC²]   

= {{{Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}

²} – {{cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₁/C)}/{1 – [(v₁/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,1 , θ_s2i,1)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₁ /C)}/{1 – [(v₁/C) cos θ_s]}}}}│[cos  θ_s1i,1 ,  cos θ_s2i,1]}{cos {{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}+  {cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₁/C)²]}^1/2}   {1 + {(v₁/C)cosine {{{cos^-1 {{[cos θ_sli,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_sli,1]}}}+ {cos^-1 {{[cos θ_s2i,1] – (v₁/C)}/{1 – [(v₁/C) cos θ_s2i,1]}}}}/2}}}}}}⁴/C}}}^odx₁}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}

²} – {{cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₂/C)}/{1 – [(v₂/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,2 , θ_s2i,2)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₂ /C)}/{1 – [(v₂/C) cos θ_s]}}}}│[cos  θ_s1i,2 ,  cos θ_s2i,2]}{cos {{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}+  {cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₂/C)²]}^1/2}   {1 + {(v₂/C)cosine {{{cos^-1 {{[cos θ_sli,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_sli,2]}}}+ {cos^-1 {{[cos θ_s2i,2] – (v₂/C)}/{1 – [(v₂/C) cos θ_s2i,2]}}}}/2}}}}}}⁴/C}}}^odx₂}/n}(A)}}} 

+  {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}

²} – {{cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v₃/C)}/{1 – [(v₃/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,3 , θ_s2i,3)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v₃ /C)}/{1 – [(v₃/C) cos θ_s]}}}}│[cos  θ_s1i,3 ,  cos θ_s2i,3]}{cos {{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}+  {cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v₃/C)²]}^1/2}   {1 + {(v₃/C)cosine {{{cos^-1 {{[cos θ_sli,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_sli,3]}}}+ {cos^-1 {{[cos θ_s2i,3] – (v₃/C)}/{1 – [(v₃/C) cos θ_s2i,3]}}}}/2}}}}}}⁴/C}}}^odx₃}/n}(A)}}} 

+ … +   {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}

²} – {{cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_m/C)}/{1 – [(v_m/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,m , θ_s2i,m)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_m /C)}/{1 – [(v_m/C) cos θ_s]}}}}│[cos  θ_s1i,m ,  cos θ_s2i,m]}{cos {{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}+  {cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_m/C)²]}^1/2}   {1 + {(v_m/C)cosine {{{cos^-1 {{[cos θ_sli,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_sli,m]}}}+ {cos^-1 {{[cos θ_s2i,m] – (v_m/C)}/{1 – [(v_m/C) cos θ_s2i,m]}}}}/2}}}}}}⁴/C}}}^odx_m}/n}(A)}}}}  [M_restC²]}/[M_restC²]   

{{Σ ( j= 1, j = m) {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}

²} – {{cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_j/C)}/{1 – [(v_j/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,j , θ_s2i,j)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_j /C)}/{1 – [(v_j/C) cos θ_s]}}}}│[cos  θ_s1i,j ,  cos θ_s2i,j]}{cos {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+  {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_j {1 + {β_jcosine {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+ {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}}}}⁴/C}}}^odx_j}/n}(A)}}}} + [M_restC²]}/[M_restC²]     

= {{Σ ( j= 1, j = m) {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}

²} – {{cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_j/C)}/{1 – [(v_j/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,j , θ_s2i,j)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_j /C)}/{1 – [(v_j/C) cos θ_s]}}}}│[cos  θ_s1i,j ,  cos θ_s2i,j]}{cos {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+  {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_j/C)²]}^1/2}   {1 + {(v_j/C)cosine {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+ {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}}}}⁴/C}}}^odx_j}/n}(A)}}}} + [M_restC²]}/[M_restC²]    

For total accrued velocity, v, we have

v = C{{-{1/{{{Σ(i = 1, i = m)   {Σ(i = 1, i = n) {{{[[(θ_01i,j²) - (θ_02i,j²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,j , θ_s2i,j)}^-1}{{[dr_bi,j/dr_ai,j]}│(r_b1i,j , r_b2i,j)}[cos [(θ_01i,j +  θ_02i,j)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_j [1 + [β_jcosine [(θ_01i,j + θ_02i,j)/2]]]}}}⁴/C}}}^odx_j}/n}(A)}}}}  + [M_restC²]}/[M_restC²]}  ²}} + 1}^1/2}

= C{{-{1/{{{Σ(i = 1, i = m)   {Σ(i = 1, i = n) {{{[[(θ_01i,j²) - (θ_02i,j²)] /((90 degrees)²)]  {{[dθ₀/dθ_s]│(θ_s1i,j , θ_s2i,j)}^-1}{{[dr_bi,j/dr_ai,j]}│(r_b1i,j , r_b2i,j)}[cos [(θ_01i,j +  θ_02i,j)/2]]}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_j/C)²]}^1/2}  [1 + [(v_j/C)cosine [(θ_01i,j + θ_02i,j)/2]]]}}}⁴/C}}}^odx_j}/n}(A)}}} + [M_restC²]}/[M_restC²]} ²}} + 1}^1/2}

= C{{-{1/{{{Σ ( j= 1, j = m) {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}

²} – {{cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_j/C)}/{1 – [(v_j/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,j , θ_s2i,j)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_j /C)}/{1 – [(v_j/C) cos θ_s]}}}}│[cos  θ_s1i,j ,  cos θ_s2i,j]}{cos {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+  {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{γ_j {1 + {β_jcosine {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+ {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}}}}⁴/C}}}^odx_j}/n}(A)}}}} + [M_restC²]}/[M_restC²]}     

²}} + 1}^1/2}

= C{{-{1/{{{Σ ( j= 1, j = m) {Σ(i = 1, i = n) {{{{{{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}

²} – {{cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}²}} /((90 degrees)²)} {{{d{cos^-1 {{[cos θ_s] – (v_j/C)}/{1 – [(v_j/C) cos θ_s]}}}

/dθ_s}│(θ_s1i,j , θ_s2i,j)}^-1}{{[Δ (cos θ_s )]/{Δ {{[cos θ_s] – (v_j /C)}/{1 – [(v_j/C) cos θ_s]}}}}│[cos  θ_s1i,j ,  cos θ_s2i,j]}{cos {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+  {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}} {{{ʃ{{2 {σ {{(T_cmbr)/{ {1/{1 – [(v_j/C)²]}^1/2}   {1 + {(v_j/C)cosine {{{cos^-1 {{[cos θ_sli,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_sli,j]}}}+ {cos^-1 {{[cos θ_s2i,j] – (v_j/C)}/{1 – [(v_j/C) cos θ_s2i,j]}}}}/2}}}}}}⁴/C}}}^odx_j}/n}(A)}}}} + [M_restC²]}/[M_restC²]  ²}} + 1}^1/2}

As you can see, attempts at analytic solutions and even non-computational numerical solutions would pose a proverbial night-mare.

Now consider again that radiation pressure is equal to σT⁴/C and {2 [σT⁴/C]} for reflected radiation. However, for a space craft traveling through a black body radiation field, the apparent temperature of the radiation increases in proportion to γ and so the black body power impinging on the space craft from directly in front grows in proportion to T⁴ and thus  to γ⁴.

The temperature of a black body is T = {P/[(A)(σ)(e)]}^1/4 = {P/{(A)[5.670400(40) x 10^-8 W m^-2 K^-4]} ^1/4}. Therefore, the cosmic microwave background radiation pressure on a perfectly reflective flat bow  relativistic space craft is approximately equal to  {2 [σT_cmbr ⁴/C]} γ⁴ = {2 [σT_cmbr ⁴/C]} { {1/{1 – [(v/C)²]}^1/2}  ⁴} =   {2 [σT_cmbr ⁴/C]} {1/{1 – [(v/C)²]}²} where γ is constant and T_cmbr is constant. However, the light pressure incident from directly behind will be approximately equal to {2 [σT_cmbr ⁴/C]}/{1/{1 – [(v/C)²]}²}. In actuality, not all of the light is directly incident from the back and so there will be angular affects that result in loss of driving power. However, we will assume that all of the radiation is absorbed and then re-cycled and released as a perfect backwardly directed laser beam. We will assume that CMBR light which is forwardly incident completely passes through the sail.

Assuming that the velocity of the sail starts out at (Zero) C, the optical pressure will be equal to {{2 [σT_cmbr⁴/C]}/γ⁴} = {{2 [σT_cmbr⁴/C]}/ { {1/{1 – [(v/C)²]}^1/2} ⁴} } = {{2{[5.670400(40) x 10^-8 W m^-2 K^-4][(2.725 K)⁴]/(300,000,000 m/s)}}/(1²)} = [2.0844355 x 10^-14] Newtons/m². For a 10,000 km by 10,000 km sail, the drive force will be 2.0844 Newtons. A 100,000 km by 100,000 km sail will produce a driving force of 208.44 Newtons.

Now assume that the sail is monolithic, made of one nanometer thick carbonaceous, STP water density materials. The sail would have a mass of 10,000,000 thousand metric tons. Assuming that the space craft plus her sail had a mass of 20,844,000 metric tons, the craft would start out with an acceleration of F/M = a = 208.44 N/20,844,000,000 kg = 10^-8 meter/s². After traveling 6 x 10¹⁵ seconds or about 200,000,000 years, the velocity of the space craft will be 0.20 C and the relativistic Lorentz transformation factor will be 1.0206 thus permitting the above Newtonian formula to apply as a reasonable approximation.

Assume that the background gas and dust that contacts the sail over a path length of 1 light year has an invariant  mass of (0.06667 kg)( 10¹⁴){[ 3 x 10¹⁰]^-1} = 222.2 kg. The momentum of the gas and dust with respect to the sail will be (m)(v) = (222.2 kg)(50,000,000 m/s) =1.111 x 10¹⁰  kg m s. Now Force equal dP/dt. Therefore, the force on the sail is on average equal to F_0.05C  = dP_0.2C/dt = 71.677 Newtons.. For a velocity of 0.02 C, the net propulsive force is 208.44 N – 71.67 N = 136.76 N which will still obviously permit 0.2 C velocities.

For such a space craft that deployed a linear series of 1,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000 metric tons could be identically accelerated. The mass of the crew quarters would be 108,440,000 metric tons. For such a space craft that deployed a linear series of 1,000,000 tethered sails where each sail was separated by an efficient 10 sail widths, a space craft having a total mass of 208, 440,000,000  metric tons could be identically accelerated. The mass of the crew quarters would be 108,440,000,000  metric tons. For cosmic journeys, this is not bad. A series of 100 million tethered sails might conceivably pull a sail craft combination having a mass of 20,844,000,000,000 metric tons and a crew quarters having a mass of 10,844,000,000,000 metric tons. For a velocity of 0.2 C, each of the latter massed space craft would have the same ratio of backward driving force and massive drag force. The caveat is simply the deployment of commensurate numbers of sails simultaneously in a spatial series along the space craft velocity vector.

Now assume that the sail is a gridded fabric or net made of STP water density conducting nanometer wide carbonaceous fibers that are separated by 0.0005 meters such as in a judicious cross weave spacing. A one square meter portion of the net will have a mass of 2 x 10^-12 kilograms. A 0.2 kilogram sail will have an plan form area of 10⁵ square kilometers. A 20,000 kilogram sail will have an plan form area of 10¹⁰ square kilometers and will have an acceleration of F/M = A = 208.44 Newtons/20,000 kg = 0.010422 m/s².  A space craft having a total mass of  20,000 metric tons will have an initial acceleration of 0.000010422 m/s².   In 200,000 years, the velocity of the 20,000 metric ton  system will be about 0.215388 C assuming Newtonian approximations.

Now, the 10¹⁰ square kilometer plan form area gridded sail will have a massive species contact area of [10¹⁰km²]/500,000 = 20,000 km². So the background gas and dust that contacts the sail over a path length of 1 light year has an invariant  mass of (0.06667 kg)( 10¹⁴){[ 3 x 10¹⁰]^-1}/(500,000) = 0.0004444 kg. The momentum of the gas and dust with respect to the sail will be (m)(v) = (0.0004444 kg)(50,000,000 m/s) =22,220 kg m s. Now Force equal dP/dt. Therefore, the force on the sail is on average equal to F_0.05C  = dP_0.2C/dt = 0.000143354 Newtons.. For a velocity of 0.02 C, the ratio of the driving force to massive drag force is [208.44 N/ 0.000143354 N] = 1,454,000.

Regards;

Jim

Copyright James M. Essig January 26, 2012 All Rights Reserved.

That space and time are intimately tied to electromagnetic radiation is obvious when one considered the ubiquitous inclusion of the speed of light in vacuu as a constant in virtually all special and general relativistic formulations. Even in classical electromagnetic theory, the velocity of light is intimately related to the properties of space time including the magnetic permeability and electric permittivity of free space by the formula C = {1/[μ₀ ɛ₀]}^1/2.

I am sure that most of the concepts expressed within this post have been contemplated by others before.

By now the reader is aware of the concept of light sail(s) driven space craft that can reach relativistic velocities. A space craft traveling at extreme gamma factors using an ordinary beam sail will experience extreme astro-dynamic drag, and the sail would likely be ionized by the drag induced friction. This is largely due to the fact that most beam sail space craft contemplate beam sails that are orthogonally spread  with respect to the craft velocity vector and thus which have a very large surface area to experience forward drag.

Suppose a relativistic rocket was powered by energy captured by an attached square or rectangular CMBR  sail that is  oriented in a perpendicular to the velocity vector of the space craft. The equation for Doppler shifting of  CMBR acting on the sail would then be:  

1 + z = {1 + [ ν (cos θ)/C]}/{[1 - [(v/C)²]]^1/2}

or,

z  = {1 + [ ν (cos θ)/C]}/{[1 - [(v/C)²]]^1/2} – 1

f’ = f / {γ [1 + (β cosine θ)]}

which reduces to F’ = f/γ for a radiation source and space craft moving in a direction perpendicular to the line connecting these reference frames with respect to a space craft observer since cos (π/2) = zero where f represents frequency. Here, θ is the angle of view with respect to the space craft velocity vector or the perceived angle of  radiation incidence on the sail with respect to the direction of space craft travel,  with respect to the space craft.

Now,  the energy of a photon is as follows:

E = [h/(2 π)] ω = hf = hC/λ 

where h is the Planck Constant and λ  is the photon wave-length.

Therefore, the energies of the individual CMBR photons impinging on the light sail oriented in a direction perpendicular to it from the space craft’s perspective from directly behind are equal to:

E + =  hf/{γ [1 + (β cos  θ)]} = hf /{γ [1 + (β cos  (0)]}

which reduces to;

hf /{γ [1 + β ]}.

Now,  the CMBR power impinging on the space craft sail per differential unit of time element (space craft reference frame), per differential unit of angle of pre-incidence (space craft reference frame), per differential element of sail area (space craft reference frame) for black body radiation is a function of γ ⁴. This is because the black body radiation frequency curve peak is proportional to black body source temperature and an incident source photon’s frequency is proportional gamma. Since black body total power emission per unit of surface area is proportional to the  fourth power of the temperature of the black body, the above differential area element of the sail will receive a total power that scales with γ ⁴. Black body emitter frequency distribution scales as a function of gamma relative to a moving observer traveling at a factor of γ with respect to the source for directly approaching observers and 1/ γ for directly receding observers.

Planck’s Law states that

I(ѵ,T)dѵ = {[2hѵ³]/C²}{1/{[e^[(hѵ)/(kT)]] -1}}dѵ

λ_max = b/T

where λ max,  is a function only of the temperature.

P_net = P_emit – P_absorbed

Applying the Stefan–Boltzmann law,

P_net = A σ e (T⁴ – T₀⁴)

where sigma =  σ = (2π⁵k_B⁴)/(15 h³ C²) = (π²k_B⁴)/(60 ђ³ C²)

or  where sigma =  σ = 5.670400(40) x 10^-8 W m^-2 K^-4

Therefore, the apparent spectral temperature of the CMBR radiation incident on the sail per unit angle of CMBR incidence for a stationary sail is:

{[P_cmbr/(A σ e)] ^1/4}

The apparent spectral temperature of the CMBR radiation incident on the sail per unit of apparent angle of incidence of the CMBR with respect to the space craft reference frame-based observer(s) for a sail traveling at a given velocity for backwardly impinging radiation is:

T_app = {∫(0, π/2){{{[P_cmbr/(e σ)]^1/4}/{γ [1 +  [(v/C) cos θ]]}}⁴}{{∫(y₁,y₂){ ∫(x₁,x₂) dx} dy}^-1} dθ}^1/4

= {∫(0, π/2){{{[P_cmbr/(e σ)]^1/4} /{{1/{[1 – [(v/C)² ]] ^1/2}} [1 +  [(v/C) cos θ]]}}⁴}{{∫(y₁,y₂){ ∫(x₁,x₂) dx} dy}^-1} dθ}^1/4

= {∫(0, π/2){{{[Pcmbr/(e σ)]^1/4}/{{1/{[1 – [(v/C)² ]] ^1/2}} [1 + [(v/C) cos θ]]}}⁴}[(∫d A)^-1] dθ}^1/4

where P_cmbr is the background CMBR power incident on the sail, dA is the differential element of sail area with respect to the space craft reference frame, v is the velocity of the space craft with respect to the background, and θ is the angle of radiation incidence on the sail with respect to a sail based observer. Theta ranges from π/2 radians for radiation traveling in an orthogonal direction with respect to the ship velocity vector to zero radians for radiation traveling in a parallel direction with respect to the ship velocity vector.

The total power backwardly incident upon the sail with respect to the sail’s reference frame for a given gamma factor  is  therefore:

P = ∫(y₁,y₂) {∫(x₁,x₂) {∫ (0, π/2) {{{(T_cmbr) /{γ [1 + [(v/C) cos θ]]}}⁴} σ e} dθ}dx}dy

= ∫(y₁,y₂) {∫(x₁,x₂) {∫ (0, π/2) {{{(T_cmbr) /{{1/{[1 + [(v/C)² ]] ^1/2}}  [1 + [(v/C) cos θ]]}}⁴} σ e} dθ}dx}dy

= ∫ {∫ (0, π/2) {{{(T_cmbr) /{{1/{[1 – [(v/C)² ]] ^1/2}}  [1 + [(v/C) cos θ]]}}⁴} σ e} dθ} dA

Here, T_cmbr is the background CMBR temperature.

Note that in the above calculations and the ones that follow, all of the relevant backwardly incident background energies are assumed to be initially absorbed by the sail even if the sail acquires a temperature significantly above absolute zero and thereby produces thermal electromagnetic black body emissions. I describe potential methods of the absorption of nearly all incident radiations even in cases where relativistic aberration would otherwise cause the bulk of the impinging radiation to easily reflect off the sail because of increasingly shallow angles of incidence. The forwardly incident radiation is assumed to completely pass through the sail without exchange of momentum.

We can numerically integrate the relativistic  energy growth of the ship in small time steps as follows:

∫P₁dt₁ + ∫P₂dt₂ + ∫P₃dt₃ +, …, + ∫P_ndt_n

Thus, the following expression can be used to compute relativistic energy gain by the ship in terms of t.

Egain  = Σ (0,n)    { ∫ (t_ai, t_bi) { ∫  (y₁, y₂){ ∫  (x₁,x₂){ ∫ (0,  π/2) {{{T_cmbr /{{{1/{[1 – [(v_i/C) ²]] ^1/2}}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ}dx}dy} dt}      

Here,  t_ai, t_bi, and dt are the times in the background reference frame.

Note, the reason why I assume the latter three times are background reference frame times is such that for a space craft traveling at a velocity of just under 1 C, where gamma is held constant, the energy gain for the space craft will be proportional to the length of the path traveled by the space craft according to the background reference frame. The distance of space craft travel  is proportional to the time of space craft travel with respect to the background reference frame. The same is true for a space craft traveling at any velocity held constant, thus the reason for the performance of the numerical integration for each time step where the velocity is incrementally increased but held constant for each time step.

Alternatively, we can use the following series:

Egain =  Σ (1,n) { ∫ (t_ai, t_bi) {∫ {∫ (0, π/2)  {{{T_cmbr /{{1/{[1 – [(v_i/C) ²]] ^1/2}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ} dA}dt}    

where t_ai and t_bi and dt are the times in the background reference frame.

Now for constant acceleration ship time, T₀ = (c/g) ln {{[[ (C²) + (V₀ ²)] ^1/2]   –   [V₀/[[1 – [(V₀/C) ²]] ^1/2 ]]} { [(C ²) + [[(g)(t)  + [V₀ /[1 – [(V₀/C) ²]] ^1/2 ] ²]] ^1/2] + [(g)(t)] +  [V₀/[[1 – [(V₀/C) ²]] ^1/2]]} / (C ²)}. We can incorporate the expression for T₀ prefaced by the notation Delta to indicate the time steps,  ship time,  of uniform duration ship frame.

For computation in terms of T₀, we obtain:

Egain = Σ (0,n)    {{ ∫  (y₁, y₂){ ∫  (x₁,x₂){ ∫ (0,  π/2) {{{T_cmbr /{{{1/{[1 – [(v_i/C) ²]] ^1/2}}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ}dx}dy} {(Delta) {(c/g) ln {{[[ (C²) + (V_0i ²)] ^1/2]   –   [V_0i/[[1 – [(V_0i/C) ²]] ^1/2 ]]} { [(C ²) + [[(g_i)(t_i)  + [V_0i /[1 – [(V_0i/C) ²]] ^1/2 ] ²]] ^1/2] + [(g_i)(t_i)] +  [V₀/[[1 – [(V_0i/C) ²]] ^1/2]]} / (C ²)}}} {1/{[1 – [(v_i/C) ²]] ^1/2}}}  .     

where t_i is the time in the background reference frame and g_i is the ship acceleration in the ship’s reference frame.

Note that the above formulas provide precise calculations for many numerical iterations involving small increments for velocity increase and small time steps in the ship’s frame.

Alternatively, we can use the following series:

Egain =  Σ (1,n) { {∫ {∫ (0, π/2)  {{{T_cmbr /{{1/{[1 – [(v_i/C) ²]] ^1/2}}[1 + [(v_i/C) cos θ]]}} ⁴} σ e}dθ} dA} {(Delta) {(c/g) ln {{[[ (C²) + (V_0i ²)] ^1/2]   –   [V_0i/[[1 – [(V_0i/C) ²]] ^1/2 ]]} { [(C ²) + [[(g_i)(t_i)  + [V_0i /[1 – [(V_0i/C) ²]] ^1/2 ] ²]] ^1/2] + [(g_i)(t_i)] +  [V_0i/[[1 – [(V_0i/C) ²]] ^1/2]]} / (C ²)}}} {1/{[1 – [(v_i/C) ²]] ^1/2}}}