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Antigravity and Antigravitational Forces 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 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 List of symbols and constants . 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's laws . 7 1.3. Some useful relations . .10 1.4. Consequences of Kepler's second law . 11 2. The role of the protractor in understanding the universe . .14 2.1. Angle measurement devices . 14 2.2. Measurement of relative distances in the Solar system. .16 2.3. Establishment of absolute distances . 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. Annual parallax of the nearest stars . 26 2.9. Measurement of the speed of light . .27 2.10. Spherical trigonometry . 29 2.11. Deflection of light in a gravitational field . 32 3. On Kepler's equation . 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. Keplerian orbital elements . 40 4. The law of gravity | discovery of the millennium .. 42 4.1. Newton's theorems . .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. Mass of the Sun . .49 4.5. Mass of Mars . 49 4.6. Falling into the Sun . 50 4.7. The size of escape velocities . 51 4.8. Flight altitude of geostationary satellites . 53 4.9. The flight time on Mars . .53 4.10. Mean mass density of the Sun . .55 4.11. Speed of Halley's comet . 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 mother stars . 58 4.14. Mass of the black hole at our Galaxy center . 58 4.15. Physical characteristics of the planets . 62 5. The N-body problem . 65 5.1. Introduction . 65 5.2. The two-body problem . 66 5.3. The three-body problem . 69 5.4. The N-body problem . 73 5.5. Total approximation error . .75 6. Eclipses and the aberration of light . 81 6.1. The importance of eclipses in exploring the universe . .81 6.2. A brief history of eclipses . 83 6.3. The origin and periodicity of eclipses . 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's postulation of dark matter . 90 7.1. Fritz Zwicky . 90 7.2. The Virial theorem . 93 7.3. Zwicky's application of the Virial theorem to the Coma cluster . 95 8. The problem of missing matter . 100 8.1. Analysis of Zwicky's method . 100 v 8.2. Analysis of current data . 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. Vera Rubin . 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. Orbital velocity around a flat disk . 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. Type Ia supernovae | standard candles . 150 10.4. Measurements of cosmological parameters . 153 10.5. Historical notes . 161 11. Recession of Mars from the Sun . 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. Rivers on Mars . 167 11.4. Mars from the perspective of the Stefan{Boltzmann law . 173 vi 12. Recession of the Moon from the Earth . 178 12.1. Measurement of the Earth-Moon distance . 178 12.2. The paradox of tidal forces of the Moon . .180 12.3. A remarkable coincidence . 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 of the Moon . 189 13. Recession of the Earth from the Sun . 191 13.1. The faint young Sun paradox . 191 13.2. The expansion of the ecosphere . 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 recession speed . 202 13.8. Why other authors obtained much smaller values for recession speeds . .206 13.9. Generation of energy by the Earth-Sun system . 209 14. Antigravity and the anthropic principle . 211 14.1. The anthropic principle . 211 14.2. Two-sided.
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