A n t i g r a One hundred years ago, Albert Einstein came up with a new theory of v i t gravitation. Later he added to his field equations the cosmological y constant which is accountable for the accelerated expansion of the universe. Thus a natural question arises concerning what is the energy source for this expansion. This monograph is about the novel concept of antigravity and antigravitational forces. We present various observational arguments showing the expansion on small cosmological scales. In particular, the Solar system and single galaxies expand at a rate comparable to the Hubble constant. The novelty of our ideas is that traditionally the expansion of the universe according to Edwin Hubble has been assumed to take place only between galaxies. The boldness of our approach is that it points to a weak violation of the law of conservation of energy. We claim that any system of free bodies that interact gravitationally with delays expands on average. We suggest that this is due to gravitational aberration effects resulting from a finite speed of gravity. Our book should be especially useful for scientists who look for the origin of dark matter and dark energy. Michal Křížek Filip Křížek Lawrence Somer Michal Křížek is a senior researcher at the Institute of Mathematics of the Czech Academy of Sciences and Professor at Charles University. Filip Křížek is a junior researcher at the Nuclear Physics Institute of the Czech Antigravity - Its Origin and Academy of Sciences. Lawrence Somer is Professor of Mathematics at the Catholic University of America in Washington, D.C. Manifestations K ř í ž e k , K ř í ž e k , S o m 978-3-659-79834-4 e r ! ! "#$%&"#$%& "'( "'( ! " # $ !% ! & $ ' ' ($ ' # % %) %* % ' $ ' + " % & ' !# $, ( $ - . ! "- ( % . % % % % $ $ $ - - - - // $$$ 0 1"1"#23." 4& )*5/ +) * !6 !& 7!8%779:9& % ) - 2 ; ! * & /- <:=9>4& )*5/ +) "3 " & :=9> Dedicated to those who look for the origin of dark matter and dark energy. LATEX typesetting prepared by Hana B´ılkov´a The effort to understand the universe is one of the very few things that lifts human life a little above the level of farce, and gives it some of the grace of tragedy. Steven Weinberg Contents Listofsymbolsandconstants ..............................x Foreword .................................................xii Part 1: Newton’s theory of gravity and the problem of dark matter 1. Astronomy and mathematics: cross-fertilization over the millennia . 3 1.1. Introduction ...........................................3 1.2. Kepler’slaws...........................................7 1.3. Someusefulrelations ..................................10 1.4. ConsequencesofKepler’ssecondlaw ..................11 2. The role of the protractor in understanding the universe . .14 2.1. Anglemeasurementdevices ...........................14 2.2. Measurement of relative distances in the Solar system. .16 2.3. Establishmentofabsolutedistances ...................18 2.4. Establishment of relative distances of inner planets . 20 2.5. Improvement of the accuracy of the Earth-Sun distance .............................................. 21 iii 2.6. Further improvement of the accuracy of the Earth-Sun distance................................................23 2.7. Slowing-down of Earth’s rotation . .24 2.8. Annualparallaxoftheneareststars ...................26 2.9. Measurementofthespeedoflight .....................27 2.10. Spherical trigonometry . 29 2.11. Deflection of light in a gravitational field . 32 3. OnKepler’sequation ......................................36 3.1. True and eccentric anomaly . 36 3.2. The relationship between the true and eccentric anomaly...............................................38 3.3. Kepler’s equation for the eccentric anomaly . 38 3.4. Keplerianorbitalelements ............................40 4. The law of gravity — discovery of the millennium .. 42 4.1. Newton’stheorems ....................................42 4.2. The most important discoveries and applications . 45 4.3. The size of a constant in Kepler’s third law . 47 4.4. MassoftheSun .......................................49 4.5. MassofMars .........................................49 4.6. Falling into the Sun . 50 4.7. Thesizeofescapevelocities ...........................51 4.8. Flight altitude of geostationary satellites . 53 4.9. TheflighttimeonMars ...............................53 4.10. MeanmassdensityoftheSun .........................55 4.11. SpeedofHalley’scomet ...............................56 4.12. The validity of the gravitational law outside the Solar system ................................................57 iv 4.13. Determination of the distances of exoplanets from their motherstars ..........................................58 4.14. Mass of the black hole at our Galaxy center . 58 4.15. Physical characteristics of the planets . 62 5. The N-bodyproblem ......................................65 5.1. Introduction ..........................................65 5.2. Thetwo-bodyproblem ................................66 5.3. Thethree-bodyproblem ..............................69 5.4. The N-bodyproblem .................................73 5.5. Totalapproximationerror .............................75 6. Eclipsesandtheaberrationoflight.........................81 6.1. The importance of eclipses in exploring the universe . .81 6.2. A briefhistoryofeclipses .............................83 6.3. Theoriginandperiodicityofeclipses ..................83 6.4. Why lunar eclipses are less frequent than solar eclipses . 86 6.5. Consequences of light aberration during total eclipses ..87 7. Zwicky’spostulationofdarkmatter ........................90 7.1. FritzZwicky ..........................................90 7.2. TheVirialtheorem ...................................93 7.3. Zwicky’s application of the Virial theorem to the Comacluster .........................................95 8. Theproblemofmissingmatter............................100 8.1. Analysis of Zwicky’s method . 100 v 8.2. Analysisofcurrentdata ............................. 108 8.3. Reduction of the virial mass of the Coma cluster . 112 8.4. What is the mass of dark matter inside the Coma cluster? ..............................................120 9. Vera Rubin and rotational curves of spiral galaxies . 122 9.1. VeraRubin ..........................................122 9.2. Spiral galaxies do not rotate according to Kepler’s laws .................................................123 9.3. Orbital velocity around a spherically symmetric body . 127 9.4. Orbitalvelocityarounda flatdisk ................... 130 9.5. Orbital velocity around galaxy bulges and halo . .134 9.6. Arguments against dark matter . 136 Part 2: Antigravity and dark energy 10. The accelerating expansion of the universe . 143 10.1. The 2011 Nobel Prize for Physics . 143 10.2. An expanding universe and the Hubble constant . 145 10.3. TypeIasupernovae— standardcandles .............150 10.4. Measurements of cosmological parameters . 153 10.5. Historicalnotes ......................................161 11. RecessionofMarsfromtheSun ...........................163 11.1. Antigravity and the law of conservation of energy . 163 11.2. The rate of expansion of the Solar system . 165 11.3. RiversonMars ......................................167 11.4. Mars from the perspective of the Stefan–Boltzmann law ..................................................173 vi 12. RecessionoftheMoonfromtheEarth ....................178 12.1. Measurement of the Earth-Moon distance . 178 12.2. The paradox of tidal forces of the Moon . .180 12.3. A remarkablecoincidence ............................181 12.4. Recession speed of the Moon from the Earth due to tides .................................................182 12.5. A time dependent momentum of inertia of the Earth . 188 12.6. The paradox of the large orbital angular momentum oftheMoon .........................................189 13. RecessionoftheEarthfromtheSun ......................191 13.1. ThefaintyoungSunparadox ........................191 13.2. Theexpansionoftheecosphere ......................192 13.3. Analysis of growth patterns on fossil corals from solar data .................................................195 13.4. Analysis of growth patterns on fossil corals from lunar data .................................................197 13.5. Consequences for the variable Earth-Moon distance . 200 13.6. Prolongation of the sidereal year of the Earth . 200 13.7. Elimination of other possibilities for the large recessionspeed ......................................202 13.8. Why other authors obtained much smaller values for recessionspeeds ......................................206 13.9. Generation of energy by the Earth-Sun system . 209 14. Antigravityandtheanthropicprinciple ...................211 14.1. Theanthropicprinciple ..............................211 14.2. Two-sidedestimates .................................213 14.3. Will antigravity protect the Earth from an expanding Sun? . 219 14.4. The probability of the appearance of life . 220 vii 15. ExpansionoftheSolarsystem ............................224 15.1. Fast satellites . 224 15.2. Where was Larissa billions of years ago? . 226 15.3. Satellites of Uranus . 228 15.4. Falling Phobos . .229 15.5. DelayingNeptune ....................................233