November 11, 2009 | 8

John Matson is an associate editor at Scientific American focusing on space, physics and mathematics. Follow on Twitter

It’s only fitting that on what would have been Carl Sagan’s 75th birthday, an organization started by the astronomer famous for his wondrous and elegant descriptions of the universe announced plans to test a wondrous and elegant way to explore the solar system and well beyond.

The nonprofit Planetary Society, which Sagan co-founded in 1980, unveiled plans at an event Monday on Capitol Hill for a 2010 launch of LightSail 1, a spacecraft pushed through space by sunlight. If successful, LightSail 1 would be followed by two higher-flying successors. The organization announced that it had received an anonymous $1 million donation that allowed the project to proceed.

LightSail 1 would comprise 32 square meters of Mylar sail, with a small control module and two more modules to contain the sail before deployment. The radiation pressure of sunlight photons striking LightSail 1 would exert a force on the sail surface, gently accelerating the spacecraft to potentially great speed over time. The Planetary Society estimates that a solar sail could speed a spacecraft to more than 200,000 kilometers per hour in three years’ time. Electric plasma engines such as that used on NASA’s Dawn probe operate on a similar principle, harnessing low-thrust rockets that, in the vacuum of space, eventually allow a spacecraft to attain great velocity.

The LightSail mission would constitute a mulligan of sorts for the Planetary Society, which attempted to launch a similar craft, Cosmos 1, in 2005. Cosmos 1 failed to reach orbit on a malfunctioning submarine-launched Russian rocket. The Planetary Society says it has not yet identified a vehicle to boost the new solar sail to orbit but that it could piggyback on several American and Russian launches in the works for 2010.

*Artist’s depiction of unfurled LightSail with satellite payload at center: The Planetary Society*

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I hope someone finally does it. But, one thing to add, Holy holes batman. Can it stay together long enough with enough surface survival to keep accelerating and also keep a package attached and aimed back here so we get the information back?

Link to thisWhat about the small particles that hurtle through space that we are all told are a threat to astronauts space suit on EVA’s? Wouldn’t these possibly damage the sails?

Link to thisThe solar sail has been considered by numerous researchers and has been a topic of science fiction stories. My favorite is "Sail 25" by the great science fiction and fantasy author Jack Vance, published in 1962. Check it out for a wonderful read.

Link to thisIt still is amazing to me that sails can "tack into the wind" and that even the solar sail can do so. At least I saw a newspaper illustration of the concept showing a solar sail heading into Venus or maybe even Mercury. Granted, a solar sail is going to work a lot better as it gets closer to the sun and the more intense radiation, but this does seem a lot like a salmon swimming upstream.

I can visualize how our atmosphere flows around a cloth sail to create the aerodynamic effect which allows sailing into the wind, but a sail boat depends on a deep keel which uses the water to leverage against the wind force and thus allow sailing into the wind. I don’t see photons as having quite the "flow around" properties of gas particles, nor can I imagine what in space could be leveraged against the pressure of outbound solar photons.

If the solar sailcraft is just going to be used to go outbound, to Mars, Jupiter, Saturn, etc., why is not one side of it a very dark color and the other side a very light color? As the spaceship passes Mars the light from the sun is growing steadily weaker, but the reflected light from Jupiter and its moons will exert a resistance to the approach of the sailcraft and so will starlight. Starlight seems weak, but cumulatively it does begin to match the intensity of our sun at the distance of Jupiter.

Link to thisKarl Schroeder summarizes his excellent concepts for Stellar Cycler manned starships at his website http://www.kschroeder.com/my-books/permanence/interstellar-cyclers

The Stellar Cycler would essentially reach a significant percentage of the speed of light at which point a charged member or other electrodynamically active member would be deployed wherein the Lorentz force imposed on the space craft by the background interstellar magnetic field would cause the craft to undergo a cyclical orbiting like path within the interstellar medium.

Imagine a Stellar Cycler that could utilize the Sun’s magnetic field or the solar magnetic field in such a manner that a Stellar Cycler craft would repeatedly dive in toward the Sun and approach the Sun within about 0.015 to 0.02 AU.

Upon reaching such a close encounter, a light sail would be deployed thus boosting the craft speed dramatically along with a powered gravity assist. For the inbound leg of each pass, the craft might use deployed photovoltaic elements to power ion, electron, or photon rockets, electro-hydrodynamic-plasma drive, magneto-hydrodynamic-plasma drive, electromagneto-hydrodynamic-plasma drive, magnetic field effect propulsion and the like. As the craft reached is closest point to the Sun, the light sail would be deployed thus giving the craft a good boost for each outward bound cycle.

As the craft velocity increased, the amount of electrical charge on the charged member could be increased to maintain the same radius of curvilinear tracking or the radius of curvature could be allowed to increase thus reducing the charging requirements for the craft.

It is interesting to consider the degree of relativistic velocity for a first solar sail pass that can be obtained using the ambient pressure of star light alone, or sunlight, in consideration of the case where the drag from the solar or stellar wind and interplanetary dust is neglected. It is also assumed that the pressure is exerted on a solar or stellar sail that is sufficiently reflective to the sun light or star light, and which is also sufficiently refractive with respect to the star light within the environment that is located within an order of magnitude or less greater distance from the star than the radius of the star; a tall order when once considers the use of Blue Super Giant or Blue Hyper Giant stars for propulsion since these stars can have a luminosity over 6 orders of magnitude greater than the Sun and are effectively black body emitters with surface temperatures of several tens of thousands of Kelvins. Any known materials in existence today would be immediately vaporized by such close proximity to these highly luminous hot stars.

Assuming fraction f of the starlight is reflected straight back and the sail moves radially outward,

the equation of motion is

B[(1 + (B EXP 2)]dB/[(1 B)EXP 2] {[1 (B EXP2)] EXP 3/2} = p [(R0/x) EXP 2](dx/Ro)

where B = v/c, v is the speed of the sail, x is the distance from the star, R0 is the initial distance from the star,

P = 2fA(u0)R0/[Mo(C EXP 2)]

Where;

A is the area of the sail, m0 is its rest mass, and u0 is the energy density of starlight at x = R0; thus, u(x) = (u0)[(R0/x) EXP 2]

Adopting f = 1, a value of M0/A = (10 EXP kg/(meter EXP 2) = the effective mass specific reflecting area of the sail craft, and u0 ~ L/[4(pi)(Ro EXP 2)C] with L the Suns luminosity and R0 = 0.03AU, I find p ~ 5 (10 EXP 2).

Note also that the equation of motion can be integrated analytically to find the terminal speed.

Just integrate from zero to its terminal value and x from R0 to infinity.

This yields for the terminal velocity:

{[(1 (B EXP 2)] EXP (1/2)} [7 14B + 11 (B EXP 2) + 2(B EXP 3)]/[(1 B ) EXP3](1 + B) = 7 + 15p

With p = 5 (10 EXP 2), the terminal velocity = 0.251 C.

We can also calculate the velocity gain of a Stellar Cycler Craft upon a first beam sail mode pass of a star with 100 solar luminosities.

I then calculated P using A = 1 square meter, f = 1, L = 4 x 10 EXP 28 watts ~ 100 solar luminosities, Ro = 1.5 x 10 EXP 10 meters, C = 3 x 10 EXP 8 meters/second, and Mo = 10 EXP – 8 kilograms to obtain p = 1.5719. Now, u0 ~ L/[4(pi)(Ro EXP 2)C] = 0.047157

Solving the above equation for B, I obtained B = v/C = .6389.

I then solved for P using L = 4 x 10 EXP 29 watts, all else remaining the same and obtained a value of B = 0.827131

We can also calculate first pass velocity gains that might be obtained from blue super giant or blue hyper giant stars, or perhaps red giant stars as well.

I then proceeded to solve for P using L = 4 x 10 EXP 32 Watts, Ro = 1.5 x 10 EXP 11 meters, and solving for B, I obtained B = 0.969. Since, u0 ~ L/[4(pi)(Ro EXP 2)C] = 4.7157 and P = 1,571.9

Note that solving the above analytical equation for B results in 2 real routes, one positive and one negative, and two complex roots or solutions for B. The real positive root is the proper solution.

It is interesting to consider the degree of relativistic velocity that can be obtained using the ambient pressure of QUAZAR light alone for a first pass in consideration of the case where the drag from the QUAZAR wind or interstellar dust and gas is neglected. It is also assumed that the pressure is exerted on a QUAZAR light sail that is sufficiently reflective to the QUAZAR light, and which is also sufficiently refractive with respect to the QUAZAR light within the environment that is located within an order of magnitude or less greater distance from the QUAZAR than the radius of the QUAZAR; a tall order when once considers the use of QUAZARS for propulsion since these bodies can have a luminosity over 14 orders of magnitude greater than the Sun and are effectively black body emitters with very high surface temperatures. Any known materials in existence today would probably be immediately vaporized by such close proximity to these highly luminous hot bodies.

Assuming fraction f of the QUAZAR light is reflected straight back and the sail moves radially outward,

the equation of motion is

B[(1 + (B EXP 2)]dB/[(1 B)EXP 2] {[1 (B EXP2)] EXP 3/2} = p [(R0/x) EXP 2](dx/Ro)

where B = v/c, v is the speed of the sail, x is the distance from the QUAZAR, R0 is the initial distance from the QUAZAR,

P = 2fA(u0)R0/[Mo(C EXP 2)]

Where;

A is the area of the sail, m0 is its rest mass, and u0 is the energy density of QUAZAR light at x = R0; thus, u(x) = (u0)[(R0/x) EXP 2]

Adopting f = 1, a value of M0/A = (10 EXP kg/(meter EXP 2) = the effective mass specific reflecting area of the sail craft, and u0 ~ L/[4(pi)(Ro EXP 2)C] with L the QUAZAR’s luminosity = 4 x 10 EXP 38 Watts = 10 EXP 12 solar luminosities and R0 = 15 x 10 EXP 15 meters, I find P ~ 15,719

Note also that the equation of motion can be integrated analytically to find the terminal speed.

Just integrate from zero to its terminal value and x from R0 to infinity.

This yields for the terminal velocity:

{[(1 (B EXP 2)] EXP (1/2)} [7 14B + 11 (B EXP 2) + 2(B EXP 3)]/[(1 B ) EXP3](1 + B) = 7 + 15p

With P = 15,719 the terminal velocity = 0.9875 C.

I then calculated P using A = 1 square meter, f = 1, L = 4 x 10 EXP 40 watts ~ 10 EXP 14 solar luminosities, Ro = 15 x 10 EXP 15 meters, C = 3 x 10 EXP 8 meters/second, and Mo = 10 EXP – 8 kilograms to obtain P = 1,571,900.

Solving the above equation for B, I obtained B = v/C = 0.998

Note once again that solving the above analytical equation for B results in 2 real routes, one positive and one negative, and two complex roots or solutions for B. The real positive root is the proper solution.

Note that other propulsion modes such as photoelectric powered electrical propulsion systems can greatly increase first pass velocities relative to first pass light sail mode alone and that powered gravity assists can be utilized to enhance these alternate propulsion modes effectiveness.

Now since we are talking about Karl’s brilliant idea of Stellar Cyclers, an idea I wish I had come up with, we are not limited to single pass cycle terminal velocities. To the extent that we can charge the space craft so that the Lorentz turning force keeps the space craft coming back for more, we can rely on the extremely Doppler blue shifted light for each inbound cycle to power photo-electric systems that perhaps operate somehow at ultra-violet or x-ray frequencies (requiring some exotic yet to be developed or invented materials) to power electrical propulsion systems such as the forms discussed herein, while on the outward bound legs of each cycle, the light sail would be deployed in full form thus given the craft repeated incremental kinetic energy boosts wherein each boost would instill a relativistic scale quantity of kinetic energy to the craft.

Assuming that some way is found to protect the ship from ionizing radiation, debris, and interstellar dust, and to mitigate astro-dynamic drag, there is no reason why the ship could not be commensurately electrically charged so that it can continue to repeat the accelerative cycling essentially for periods of cosmic time duration thus reaching outstanding relativistic gamma factors.

Now, within a given galaxy such as the Milky Way, a Stellar Cycler could in theory orbit the galaxy with ever greater relativistic velocities while at the same time receiving a boost from the electromagnetic emission of stars, mid range blackholes, and even the centrally located super massive black-holes which in the Milky Way have a mass on the order of 1 million to 10 million solar masses.

As long as the craft could maintain the requisite electrical charge and synchrotron radiation emissions by the craft could be kept at a level where the kinetic energy input to the craft is higher than synchrotron radiation and astrodynamic drag based losses, the craft should continue to accelerate essentially forever or at least over truly cosmic time intervals and achieve incredibly high gamma factors.

The use of photon, electron, proton, ion, neutrino and/or perhaps even tachyon rocket propulsion (if tachyons exist) could be made in space craft that would make powered gravity assists aground stars, mid-range, and super massive backhoes all the while that craft was using the Stellar Cycler type Lorentz turning force to keep it glued to its energy source, the electromagnetic and gravitational fields of the Milky Way Galaxy or other host galaxies.

Note that the relativistic Lorentz turning force is definable as follows:

For the ? = 1 component (x-component), the force is

The calculation of the ? = 2 or ? = 3 is similar yielding, where

or, in terms of the vector and scalar potentials A and

dP/Tau expresses a relativistic form of the Lorentz force when no other forces are present. One can see that as gamma = 1/{[1 [(v/C) EXP 2]] EXP (1/2)}, the Lorentz force grows. In the limit that C is reached, gamma goes to infinity and so does the Lorentz turning force even with a small but finite electrical charge of Q.

It is possible that the Lorentz force type of Stellar Cycler could take advantage of the Cosmic Microwave Background Radiation of CMBR where only the back side of a CMBR sail is reflective to CMBR and where additionally, the forward side of the sail is photo-electric to CMBR and which thus could absorb relativistic blue shifted CMBR and use the electrical energy produced to power photon, ion, proton, electron, neutrino, or perhaps even tachyon rockets, as well as electrodynamic hydrodynamic-plasma drive craft that utilized magnetic, electric, and/or electromagnetic fields as the field effect reaction mechanism to react against the interstellar and intergalactic plasma.

The same CMBR capture mechanisms could be utilized for CMBR blue-shifted to visible light, and ultraviolet light frequencies via the deployment of monolithic PV sails or perhaps mass saving PV grid type sails that are composed of PV electrically conducting materials in the form of a thread like weave or net comprised on PV nano-fibers separated by 0.05 to 0.1 microns. Such a sail could permit a mass savings of about 25 to 50 times that of a monolithic weave for hard ultraviolet photons. Such systems could also be used to collect solar or stellar energy to power electrical propulsion systems is Stellar Cycler dive fry runs as the relativistic gamma factor mounted.

Some forms of nanotechnology self assembly or an armada of small robotic devices could keep the sails in good condition at relativistic velocities.

The following limits suggest the theoretical mathematically extreme forms of stellar Cycler, QUAZAR Cycler, dive and fry iterations as well as CMBR Cycler iterations provided that either synchrotron radiation losses could be appropriately suppressed and the period of time for iterative behavior is unlimited, or the period of time for iterations is unlimited and the radius of orbit approaches infinity.

Given that gamma = [1 [(v/C) EXP 2]] EXP (-1/2), Nc = the number of cycles, t equals mission time background reference frame, I d(Es)/dt I is the modulus of the rate of synchrotron radiation energy loss for the ship, I d(Ed)/dt I is the modulus of the rate of astrodynamic drag energy induced kinetic energy losses for the ship., d(KE)/dt is the rate of kinetic energy gain for the ship due to the Cycler mechanisms active propulsion aspects

Lim gamma = infinity

Nc — > infinity, I d(Es)/dt I + I d(Ed)/dt I < d(KE)/dt

Lim Nc = infinity,

t — > infinity, cycle path length is finite, I d(Es)/dt I + I d(Ed)/dt I < d(KE)/dt

Lim v = C

Nc — > infinity, I d(Es)/dt I + I d(Ed)/dt I < d(KE)/dt

Lim v = C

t — > infinity, cycle path length is finite, I d(Es)/dt I + I d(Ed)/dt I < d(KE)/dt

As long as , I d(Es)/dt I + I d(Ed)/dt I < d(KE)/dt, the limits should

Some really far out limits are as follows.

Lim gamma = infinity

t — > Aleph 1 Planck Time Units: R = Aleph 0 Planck Length Units, I d(Es)/dt I + I d(Ed)/dt I < d(KE)/dt

Lim Nc = infinity,

t — > Aleph 1 Planck Time Units, R = Aleph 0 Planck Length Units, I d(Es)/dt I + I d(Ed)/dt I < d(KE)/dt

Lim v = C exactly

t — > Aleph 1 Planck Time Units, R = Aleph 0 Planck Length Units, I d(Es)/dt I + I d(Ed)/dt I < d(KE)/dt

The first and third of these later three limits assume that as t approaches Aleph 1 Planck Time units, the gamma factor will not grow beyond trans-classical-infinite values:- a situation which could lead to values of v greater than C.

Note that for gamma = infinity, the craft traveling at C exactly would travel an infinite number of light-years in one Planck Unit ship time and an infinite time into the future in one Planck Unit ship time. Note that the Planck Time Unit is tiny at tp = {[h/(2 pi)] G/[C EXP 5]} EXP (1/2).

With the negative refraction index metamaterials that are in the ongoing process of research and development within various laboratory settings including at Duke University,, in theory, stellar sails and even CMBR sails can be manufactured that are ironically pulled forward by the incoming incident light rather than being pushed by the light. Note that the theoretical ramifications and theoretical applications of EM negative refraction index materials are being evaluated, but as of yet, no commercial applications have arisen out of such metamaterials research.

Note that a negative refraction index stellar or CMBR sail would in theory pull the craft forward with ever greater force as the craft approached C due to Doppler blueshifting of the incident photons thus resulting in increases in impinging electromagnetic power and as a result, increases in the sail driving power. In theory, the following limits hold:

Lim Ep = Infinity

V—-> C;

Lim Fd = Infinity

V—-> C;

Lim a = Infinity

V—-> C

Lim [d(gamma)/dt] = Infinity

V—–> C;

Lim [d(Sp/dt)] = Infinity

V—–> C;

where V is the velocity of the space craft, C is the speed of light, Ep is the incident energy of the impinging photons, Fd is the drive force acting on the sail, a is the acceleration of the sail and space craft, gamma is the Relativistic Lorentz Transformation factor, and Sp is the forward oriented pressure of the incident EM radiation on the sail.

Note that limits imposed on the Doppler blueshifting of incoming light might exist as a result of the potential photon energy limit to the Planck Energy. The Planck Energy, Ep, is the energy of the photon at which its energy density would be commensurate with the photon becoming a blackhole. Such a black hole is theoretically the smallest possible due to Hawking Radiation based black hole decay and corresponds to a black-hole with a Schwarchilds radius roughly equal to the Planck Distance which is equal in length to Lp = [(h/(2 pi)) G/(C EXP 3)] EXP 1/2. Put another way, the Planck energy is the energy of a photon that has an energy density equal to the Planck Density which is equal to Dp = Ep/(Lp EXP 3)

Note that the above limits assume that the sail material can act or be put in condition to act on impinging photons of arbitrarily short wavelengths. In theory, meta materials can only act on photons that have a wavelength at least a half an order of magnitude or greater than the size of the active cells in the meta-materials, which in materials constructed out of ordinary molecular materials, e.g., materials out of atoms with normal valence structures, is limited to the scale of about one nanometer and no less.

Perhaps if metamaterials can be formed out of materials with somehow stably compressed electronic shell or cloud configurations, then the active cell size can be made to be smaller than one Angstrom. Note however that photons even with a wavelength as large as a few tens of nanometers can be said to be ionizing and so such photons would likely destroy the meta-materials they would interact with, especially over long periods of operation involving high power fluxes densities . Perhaps in the distant future, meta-materials made from some form of stabilized neutronium or other atomic nucleus density or greater density bulk material such as quarkonium, strange matter, or higgsinium would permit the required cell size, heat capacity, and radiation resistance to handle x-ray to soft gamma ray incident photons at high power flux densities in a negatively refractive manner.

Perhaps upon attaining such a high gamma factor, the space craft would act as a black hole ramjet wherein, due to relativistic Doppler wavelength decrease of the matter particles and photons, and neutrinos incident on the front of the space craft, the relative energy of these particles and thus the force by which they are pulled inward by the ship/blackhole would be many orders of magnitude greater than the force acting on the craft for particles pulling on the craft from the back due to the much lower relative energy of particles, most probably only photons and gravity waves, incident on the craft from behind. The faster the craft traveled, the greater the force for which the craft would be pulled forward, while the area specific force of the particles incident on the craft from behind and pulling the craft from behind would decrease.

In fact, in the limit that the craft’s velocity approached C, or the gamma factor of the craft climbed to infinity, the force per unit area pulling the craft forward would grow to infinity while the force per unit area pulling the craft from behind would decrease to zero. The net effect might be a runaway acceleration to velocities equal to C, and if the associated infinite energies can somehow be cloaked, perhaps to velocities much, much greater than C, perhaps even to values of Aleph 0 light years per second where Aleph 0 is the infinite number of positive integers. We can speculate even further and consider whether run away velocities equal to Aleph 1 light years per second where Aleph 1 is the number of real numbers which is an infinitely larger set then the number of counting numbers, or perhaps further to velocities Aleph i, where i = 2, 3, 4, … .

Note that the above extreme mathematical limits imply that the laws of special relativity apply for values that grow to infinity, and, also to infinite values of at least arbitrary but low non-zero ordinal values of cardinalities. The caveat here is that physical quantities can grow to higher order cardinal values. Note that we do not yet know that the cosmos is infinite in extent and in the event that it is finite, it is hard to see how a space craft as such could obtain states defined by infinite values, let alone higher cardinal infinite values.

Thus, the speculations in the previous few paragraphs imply that the universe has at the very least, an extent or size of arbitrarily non-zero but low ordinal values of cardinalities. In reality, it is possible that nature does not or would not work this way for us humans, and/or perhaps other and even more bazaar kinematical space time and mass-energy effects would be realized for such extreme physical values that we have not yet dreamed up, formulated, or otherwise contemplated. Then again, such non-zero ordinal cardinal infinite physical values might be achievable with the possibility that yet even more extreme, bazaar, and exotic kinematical, space time and mass- energy effects might be realized.

We may never know the true limits of our potential technological and scientific prowess and powers since technology and scientific development might be an eternal and never ending process simply because the entire cosmos might be of indefinably ordinal value of cardinality in extent, vastness and richness. From the purely sociological standpoint of a conservative Catholic who is not afraid to refer to GOD, since it is my opinion that God is a megalomaniac when it comes to HIS creative might, GOD perhaps imposes no limits on the size of HIS pride and joy, the entirely of all creation, a set of elements that I believe that GOD continues and will continue to create in ever greater numbers, by leaps and bounds.

I know this writing may seem a little far out and bazaar, but I am a firm believer that the ambient resources in interplanetary and interstellar space can enable us to ultimately travel about our universe in a literal sense of the word.

My suggestion for NASA is a very bold initiative or focus to eventually reach for the stars, and get there literally. Marc Millis, former head of the former NASA Breakthrough Propulsion Program has founded the 501 c(3) not for profit institute known as the Tau Zero Foundation with the bold mission of advocating the cause for manned missions to the stars. Paul Gilster is the head of the Tau Zero Centauri Dream Foundation, the news forum for the Tau Zero Foundation who post new threads daily on the subject of manned interstellar space travel and related issues and draws an active readership as is shown by the many comments he receives on the articles he posts. So I am not alone in this bold vision for humanity. On my website, I too advocate for manned interstellar travel and have posted hundreds of articles on various conjectural interstellar travel technologies, many of which have been proposed by other but which I extrapolate to extreme limits and others not so extreme that might be realizable this very century with a bold national or international funding program.

Check out Karl Schroeders website. He is much more familiar with his Stellar Cycler concepts then I am. After all, he is the one who came of with and developed the concepts of the Stellar Cycler.

Truly, with ideas like the Stellar Cycler. the Glory Days of Epic Sailing Voyages might begin anew, except in the modern case, we would all but realize Einstein’s youthful dream of riding along a beam of light.

Happy Sun Sailing;

Jim

Link to thisAssuming fraction f of the starlight is reflected straight back and the sail moves radially outward,

the equation of motion for a single pass Dive and Fry Solar Sail is

B[(1 + (B EXP 2)]dB/[(1 B)EXP 2] {[1 (B EXP2)] EXP 3/2} = p [(R0/x) EXP 2](dx/Ro)

where B = v/c, v is the speed of the sail, x is the distance from the star, R0 is the initial distance from the star,

P = 2fA(u0)R0/[Mo(C EXP 2)]

Where;

A is the area of the sail, m0 is its rest mass, and u0 is the energy density of starlight at x = R0; thus, u(x) = (u0)[(R0/x) EXP 2]

Adopting f = 1, a value of M0/A = (10 EXP kg/(meter EXP 2) = the effective mass specific reflecting area of the sail craft, and u0 ~ L/[4(pi)(Ro EXP 2)C] with L the Sun’s luminosity and R0 = 0.03AU, I find p ~ 5 × (10 EXP 2).

Note also that the equation of motion can be integrated analytically to find the terminal speed.

Just integrate from zero to its terminal value and x from R0 to infinity.

This yields for the terminal velocity:

{[(1 (B EXP 2)] EXP (1/2)} [7 14B + 11 (B EXP 2) + 2(B EXP 3)]/[(1 B ) EXP3](1 + B) = 7 + 15p

With p = 5 × (10 EXP 2), the terminal velocity = 0.251 C.

We can also calculate the velocity gain of a Stellar Cycler Craft upon a first beam sail mode pass of a star with 100 solar luminosities.

I then calculated P using A = 1 square meter, f = 1, L = 4 x 10 EXP 28 watts ~ 100 solar luminosities, Ro = 1.5 x 10 EXP 10 meters, C = 3 x 10 EXP 8 meters/second, and Mo = 10 EXP – 8 kilograms to obtain p = 1.5719. Now, u0 ~ L/[4(pi)(Ro EXP 2)C] = 0.047157

Solving the above equation for B, I obtained B = v/C = .6389.

I then solved for P using L = 4 x 10 EXP 29 watts, all else remaining the same and obtained a value of B = 0.827131

We can also calculate first pass velocity gains that might be obtained from blue super giant or blue hyper giant stars, or perhaps red giant stars as well.

I then proceeded to solve for P using L = 4 x 10 EXP 32 Watts, Ro = 1.5 x 10 EXP 11 meters, and solving for B, I obtained B = 0.969. Since, u0 ~ L/[4(pi)(Ro EXP 2)C] = 4.7157 and P = 1,571.9

Carl Schroeder in his Stellar Cycler concept anticipates a space craft that has an electrically charged member experiencing a cyclical orbital motion while traveling at relativistic velocities through the interstellar medium.

One can adopt this concept to repeated Dive and Fry iterations wherein the velocity of the craft should upon numerous dives approach C very closely. Brighter stars can also be used.

Link to thisAssuming fraction f of the starlight is reflected straight back and the sail moves radially outward,

the equation of motion is

B[(1 + (B EXP 2)]dB/[(1 B)EXP 2] {[1 (B EXP2)] EXP 3/2} = p [(R0/x) EXP 2](dx/Ro)

where B = v/c, v is the speed of the sail, x is the distance from the star, R0 is the initial distance from the star,

P = 2fA(u0)R0/[Mo(C EXP 2)]

Where;

A is the area of the sail, m0 is its rest mass, and u0 is the energy density of starlight at x = R0; thus, u(x) = (u0)[(R0/x) EXP 2]

Adopting f = 1, a value of M0/A = (10 EXP kg/(meter EXP 2) = the effective mass specific reflecting area of the sail craft, and u0 ~ L/[4(pi)(Ro EXP 2)C] with L the Sun’s luminosity and R0 = 0.03AU, I find p ~ 5 × (10 EXP 2).

Note also that the equation of motion can be integrated analytically to find the terminal speed.

Just integrate from zero to its terminal value and x from R0 to infinity.

This yields for the terminal velocity:

{[(1 (B EXP 2)] EXP (1/2)} [7 14B + 11 (B EXP 2) + 2(B EXP 3)]/[(1 B ) EXP3](1 + B) = 7 + 15p

With p = 5 × (10 EXP 2), the terminal velocity = 0.251 C.

We can also calculate the velocity gain of a Stellar Cycler Craft upon a first beam sail mode pass of a star with 100 solar luminosities.

I then calculated P using A = 1 square meter, f = 1, L = 4 x 10 EXP 28 watts ~ 100 solar luminosities, Ro = 1.5 x 10 EXP 10 meters, C = 3 x 10 EXP 8 meters/second, and Mo = 10 EXP – 8 kilograms to obtain p = 1.5719. Now, u0 ~ L/[4(pi)(Ro EXP 2)C] = 0.047157

Solving the above equation for B, I obtained B = v/C = .6389.

I then solved for P using L = 4 x 10 EXP 29 watts, all else remaining the same and obtained a value of B = 0.827131

We can also calculate first pass velocity gains that might be obtained from blue super giant or blue hyper giant stars, or perhaps red giant stars as well.

I then proceeded to solve for P using L = 4 x 10 EXP 32 Watts, Ro = 1.5 x 10 EXP 11 meters, and solving for B, I obtained B = 0.969. Since, u0 ~ L/[4(pi)(Ro EXP 2)C] = 4.7157 and P = 1,571.9

Using the Stellar Cycler concept of Carl Schroeder, repeated dive and fry passes by the Sun should enable velocities very close to C to be obtained and thus very high gamma factors.

The Stellar Cycler concept makes use of the relativistic Lorentz force to cause a relativistic craft to make a closed orbital motion path by deploying a charged member that reacts with the interstellar magnetic field.

Stars brighter than the Sun can be used for higher v.

Link to thisThe solar sail experiment is as backwards looking as NASAs continuing experiments with firecracker propulsion technologies. Leonardo DaVinci figured out how to take the sail concept to the next logical step 500 years ago by combining its physical principle with the inclined plane and rotating it around a central axis. What amazes me is how this dithering continues to be sold to the public as meaningful scientific research. My son figured it out when he was six. My only conclusion regarding the Planetary Society’s ‘great experiment’ is that it’s just a promotional hustle.

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