## Deep Space Flight and Communications: Exploiting the Sun as a Gravitational LensThe majority of books dealing with prospects for interstellar flight tackle the problem of the propulsion systems that will be needed to send a craft on an interstellar trajectory. The proposed book looks at two other, equally important aspects of such space missions, and each forms half of this two part book. Part 1 looks at the ways in which it is possible to exploit the focusing effect of the Sun as a gravitational lens for scientific missions to distances of 550 AU and beyond into interstellar space. The author explains the mechanism of the Sun as a gravitational lens, the scientific investigations which may be carried out along the way to a distance of 550 AU (and at the 550 AU sphere itself), the requirements for exiting the Solar System at the highest speed and a range of project ideas for missions entering interstellar space. Part 2 of the book deals with the problems of communicating between an interstellar spaceship and the Earth, especially at very high speeds. Here the author assesses a range of mathematical tools relating to the Karhunen-Ločve Transform (KLT) for optimal telecommunications, technical topics that may one day enable humans flying around the Galaxy to keep in contact with the Earth. This part of the book opens with a summary of the author’s 2003 Pešek Lecture presented at the IAC in Bremen, which introduces the concept of KLT for engineers and ‘newcomers’ to the subject. It is planned to include a DVD containing the full mathematical derivations of the KLT for those interested in this important mathematical tool whilst the text itself will contain the various results without outlines of the mathematical proofs. Astronautical engineers will thus be able to see the application of the results without getting bogged down in the mathematics. |

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### Contents

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112 Uniform motion | 185 |

113 Decelerated motion | 188 |

114 Checking the KLT of decelerated motion by Matlab simulations | 194 |

115 Total energy of the noisy signal from relativistic spaceships in decelerated and uniform motion | 195 |

exploiting the KLT to detect an alien spaceship approaching the Earth in decelerated motion | 199 |

117 References | 200 |

KLT of radio signals from relativistic spaceships in hyperbolic motion | 203 |

123 Total energy of signals from relativistic spaceships in hyperbolic motion | 205 |

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233 Cosmology | 25 |

234 Solar system studies | 26 |

241 Dust | 27 |

242 Plasma and energetic particle distributions | 28 |

245 Plasma waves | 29 |

26 References | 30 |

Magnifying the nearby stellar systems | 33 |

33 Keplerian theory of simple hyperbolic flybys | 36 |

34 The flyby of the Sun performed by the FOCAL spacecraft | 43 |

35 References | 45 |

Astrodynamics to exit the solar system at the highest speed | 47 |

421 Elementary background planar problem | 48 |

422 Optimization of a single Jupiter flyby | 49 |

423 Two optimized Jupiter flybys plus one intermediate Sun flyby | 50 |

43 A chemically powered closeSun flyby? | 51 |

44 Theory of the Sun Flyby enhanced by a perihelion boost | 52 |

45 Determining the perihelion boost by knowing the target star the time to get 550 AU and the Sun approach | 53 |

46 References | 57 |

SETI and the FOCAL space mission | 58 |

52 The narrowband assumption SETI | 60 |

53 A short introduction to the KLT | 63 |

54 Mathematics of the KLT | 64 |

advantages of the KLT for the FOCAL space mission | 67 |

GLSETI gravitational lensing SETI Receiving far ETI signals focused by the gravity of other stars | 71 |

62 Only two types of SETI searches from the Earth up to 2001 | 72 |

623 Searches | 73 |

625 Allsky survey | 74 |

626 Common requirements | 75 |

632 Summary | 80 |

64 Maccones equation relating to 1 magnification of lensing star 2 distance of the ET transmitter and 3 power of the ET transmitter | 81 |

the Search for ExtraTerrestrial Visitation | 83 |

66 References | 84 |

The gravitational lenses of Alpha Centauri A B C and of Barnards Star | 85 |

72 The Suns gravity+plasma lens as a model for the nearby stars | 86 |

73 Assumed data about Alpha Centauri A B C and Barnards Star | 90 |

74 Gravitational lens of the naked Sun | 93 |

75 Gravitational lens of the naked Alpha Centauri A | 98 |

76 Gravitational lens of the naked Alpha Centauri B | 101 |

77 Gravitational Lens of the naked Alpha Centauri C Proxima | 104 |

78 Gravitational lens of the naked Barnards Star | 107 |

79 Conclusions | 110 |

The Coronal Plasma pushing the focus of the gravity+plasma lens far beyond 550AU | 113 |

82 The refraction of electromagnetic waves in the Sun Coronal Plasma | 115 |

83 Summary of the Sun pure gravity naked Sun lightbending theory | 116 |

focal axis intercept for any ray passing at distance b from the Sun | 118 |

85 Asymptotic z straight light path | 122 |

closeSun middistance and atinfinity LK and F Corona respectively | 123 |

87 Focal distance vs height and minimal focal distance for any assigned frequency | 127 |

88 The two causes of the gravity+plasma lens of the sun | 130 |

89 Observing frequencies for the closeSun middistance and atinfintity approxiamtions | 131 |

810 References | 134 |

NASAs Interstellar Probe ISP20102070 and the Cosmic Microwave Background CMB | 135 |

2010 to 2055 | 136 |

93 Looking at the 2728 K Cosmic Microwave Backround through the Suns gravity lens by virtue of NASAs Interstellar Probe ISP | 137 |

94 The effective minimal focal distance for the gravity+plasma lens looking at the 27K Cosmic Microwave Background is 763 AU which NASAs Int... | 142 |

95 Improving COBEs angular resolution by nine orders of magnitude by looking at the 27K Cosmic Microwave Background by virtue of NASAs Int... | 145 |

96 Conclusions | 146 |

98 References | 147 |

KLToptimized telecommunications | 148 |

A simple introduction to the KLT | 151 |

103 A heuristic derivation of the KL expansion | 152 |

104 The KLT finds the best basis eigenbasis in Hilbert space spanned by the eigenfunctions of the autocorrelation of Xt | 155 |

105 Continuous time vs discrete time in the KLT | 157 |

just a linear transformation in the Hilbert space | 158 |

The Final Variance theorem | 159 |

108 BAM Bordered Autocorrelation Method to find the KLT of stationary processes only | 162 |

109 Developments in 2007 and 2008 | 168 |

1010 KLT of stationary white noise | 169 |

1011 KLT of an ET sinusoidal carrier buried white cosmic noise | 170 |

1012 Analytic proof of the BAMKLT | 172 |

1013 KLT signaltonoise SNR as a function of the final T eigenvalue index n and alien frequencies v | 174 |

1014 How to eavesdrop on alien chat | 175 |

1015 Conclusions | 176 |

1016 Acknowledgments | 177 |

KLT of radio signals from relativistic spaceships in uniform and decelerated motion | 180 |

124 KLT for signals emitted in asymptotic motion by Matlab simulations | 206 |

125 Checking the KLT of asymptotic hyperbolic motion by motion by Matlab simulations | 210 |

126 Signal total energy as a stochastic process of T | 211 |

preparatory calculations | 214 |

128 KL expansion for the instantaneous energy of the noise emitted by a relativistic spaceship | 220 |

129 Conclusion | 221 |

KLT of radio signals from relativistic spaceships in arbitrary motion | 223 |

132 Arbitrary spaceship acceleration | 225 |

1322 KL expansion of the Gaussian noise emitted by a spaceship having an arbitrary acceleration profile | 227 |

1323 Total noise energy | 229 |

1324 KL expansion of noise instantaneous energy | 230 |

133 Asymptotic arbitrary spaceship acceleration | 232 |

1332 Asymptotic KL expansion for noise | 234 |

1333 Asymptotic total noise energy | 236 |

134 Powerlike asymptotic spaceship | 238 |

1342 Powerlike asymptotic KL expansion for noise | 239 |

1343 Approximated powerlike asymptotic KL expansion for noise | 241 |

1344 Powerlike asymptotic total noise energy | 242 |

1345 Powerlike asymptotic KL expansion for noise instantaneous energy | 243 |

1346 Approximated powerlike asymptotic KL expansion for noise instantaneous energy | 246 |

135 Conclusion | 247 |

136 References | 248 |

Genetics aboard relativistic spaceships | 249 |

142 Diffusion partial differential equation for Xt | 250 |

143 Firstpassage time for Xt | 252 |

144 Relativistic interstellar flight | 254 |

145 Timerescaled Brownian motion | 255 |

146 Genetics | 256 |

147 Relativistic genetics | 258 |

148 A glance ahead | 259 |

149 References | 260 |

Engineering tradeo s for the FOCAL spacecraft antenna | 262 |

Reference | 264 |

FOCAL Sun flyby characteristics | 269 |

Mission to the solar gravitational focus by solar sailing | 278 |

C2 Example sailcraft for SGF mission | 282 |

C3 Trajectory profile for SGF mission | 283 |

C4 Conclusions | 290 |

FOCAL radio interferometry by a tethered system | 293 |

D2 References | 296 |

Interstellar propulsion by Sunlensing | 298 |

E2 Highlights on research areas in interstellar propulsion by Sunlensing | 300 |

light from Sirius naked Sun gravity lens and relevent solar sail size | 301 |

E4 Conclusions | 305 |

Brownian motion and its time rescaling | 307 |

F2 Brownian motion essentials | 308 |

F3 KLT of Brownian motion | 310 |

F4 White noise as the derivative of Brownian motion with respect to time | 311 |

F5 Introduction to time rescaling | 313 |

F7 Time rescaling and Gaussian properties of Xt | 315 |

F8 Orthogonal increments for nonoverlapping time intervals | 317 |

F10 References | 324 |

Maccone First KLT Theorem KLT of all timerescaled Brownian motions | 325 |

G3 Solution of the integral equation for eigenfunctions | 328 |

G4 A simpler formula for Bessel Function order | 334 |

G5 Stability criterion for eigenfunctions | 335 |

G6 References | 337 |

KLT of the Bt2H timerescaled Brownian motion | 338 |

H3 KL expansion of BpHt | 341 |

H4 Total energy of BpHt | 346 |

H5 References | 349 |

Maccone Second KLT Theorem KLT of all timerescaled square Brownian motions | 351 |

13 KLT of any zeromean timerescaled square process | 352 |

14 KLT of square Brownian motion | 356 |

I5 Checking the KLT of the square Brownian motion by Matlab simualtions | 361 |

KLT of the B˛t˛ᵸ timerescaled square Brownian motion | 363 |

J2 Preparatory calculations about B˛t˛ˣ+š | 366 |

J3 KL expansion of the square process B˛t˛ᵸ | 371 |

J4 Checking the KLT of B˛t˛ᵸ | 373 |

J5 References | 374 |

A Matlab code for KLT simulations | 375 |

K3 The file input_data_togglem | 377 |

K4 The file Brownian_Autocorrelationm | 379 |

K5 The file process_pathm | 380 |

K7 The file analytic_KLTm | 382 |

K8 The file ANALYTIC_KLT_square_brow_motionm | 385 |

K9 The file ANALYTIC_KLT_uniform_relm | 386 |

K10 Conclusions | 389 |

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### Other editions - View all

Deep Space Flight and Communications: Exploiting the Sun as a Gravitational Lens Claudio Maccone No preview available - 2010 |

Deep Space Flight and Communications: Exploiting the Sun as a Gravitational Lens Claudio Maccone No preview available - 2009 |