Abstract
Celestial mechanics is the topic of this chapter. The 2-body solution is given, the restricted 3-and n-body solutions discussed, and the effects of perturbations on the orbital elements are treated in detail. Tidal friction and its effects in the Earth-Moon system, spin-orbit and orbit-orbit resonances are discussed.
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Notes
- 1.
Newton’s second law of motion: \( {\displaystyle \sum \overrightarrow{F}}=m\overrightarrow{a} \). The acceleration, \( \overrightarrow{a} \), of a body is proportional to the sum of the forces, \( {\displaystyle \sum \overrightarrow{F}} \), acting on it and inversely proportional to its mass, m.
- 2.
The latest values for physical constants, generally accepted worldwide, may be found on the National Institute for Standards and Technology (NIST) website, http://physics.nist.gov/cuu/Constants/. At current writing (May 2013), these are the 2010 CODATA recommended values, where CODATA is the Committee on Data for Science and Technology.
- 3.
More strictly, we should write Kepler’s first “law” as we do in Chap. 1, because Kepler’s “laws” are actually empirical approximations to the true motions of planets. Planetary orbits differ from ellipses because of the perturbing effects of the other planets and asteroids in the solar system, and the effects of General Relativity. However, we here follow common usage and forego the quotation marks.
- 4.
named after Joseph Louis Lagrange (1736–1813).
References
Applegate, J.H., Douglas, M.R., Gursel, Y., Sussman, G.J., Wisdom, J.: The outer solar system for 200 million years. Astron. J. 92, 176–194 (1986)
Bradstreet, D.H., Steelman, D.P.: Binary Maker 3.0, Contact Software, Norristown, PA (2004)
Brouwer, D., Clemence, G.M.: Methods of Celestial Mechanics. Academic Press, New York (1961)
Consolmagno, G.J., Schaefer, M.W.: Worlds Apart: A Textbook in the Planetary Sciences. Prentice Hall, Englewood Cliffs, NJ (1994)
Danby, J.M.A.: Fundamentals of Celestial Mechanics, 1st edn. McMillan, New York (1962) (Reprinted in 1964)
Danby, J.M.A.: Fundamentals of Celestial Mechanics, 2nd edn. Willmann-Bell, Richmond, VA (1988)
Laskar, J., Gastineau, M.: Existence of collisional trajectories of Mercury, Mars, and Venus with the Earth. Nature 459, 817–819 (2009)
Milone, E.F., Wilson, W.J.F.: Solar System Astrophysics: Planetary Atmospheres and the Outer Solar System, 2nd edn. Springer, New York (2014)
Montenbruck, O.: Practical Ephemeris Calculations. Springer, New york (1989) (Tr. by Armstrong A.H. of Montenbruck’s Grundlagen der Ephemeridenrechnung 3. Verlag Sterne und Weltraum Dr Vehrenberg GmbH, Munich (1987))
Moulton, F.R.: An Introduction to Celestial Mechanics, Second revised edn. MacMillan, New York (1914) 10th Printing (1958)
Murray, C.D., Dermott, S.F.: Solar System Dynamics. University Press, Cambridge (1999/2001).
Neutsch, W., Scherer, K.: Celestial Mechanics: An Introduction to Classical and Contemporary Methods. B.I. Wissenschaftsverlag, Mannheim (1992)
Newhall, X.X., Standish, E.M., Williams, J.G.: DE 102: a numerically integrated ephemeris of the moon and planets spanning forty-four centuries. Astron. Astrophys. 125, 150–167 (1983)
Nobili, A.M.: Long term dynamics of the outer solar system: review of LONGSTOP project. In: The Few Body Problem, Procs. of IAU Colloquium 96. pp. 147–163 (1988)
Peirce, B.O.: A Short Table of Integrals, 4th edn. Ginn and Co., Boston, MA (1957)
Schlosser, W., Schmidt-Kaler, Th., Milone, E.F.: Challenges of Astronomy: Hands-On Experiments for the Sky and Laboratory. Springer-Verlag, New York (1991/1994)
Smart, W.H.: Celestial Mechanics. Wiley, New York (1953)
Sussman, G.J., Wisdom, J.: Numerical evidence that the motion of Pluto is chaotic. Science 241, 433–437 (1988)
Szebehely, V.G.: Adventures in Celestial Mechanics. University of Texas Press, Austin, TX (1989)
Wisdom, J., Holman, M.: Symplectic maps for the N-body problem. Astron. J. 102, 1528–1538 (1991)
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Challenges
Challenges
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[3.1]
Derive Kepler’s third law from Newton’s gravitational and motion laws. [Hint: Consider areal speed and make use of (2.37).]
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[3.2]
Suppose a projectile is shot vertically from a site on the equator with initial velocity less than escape velocity by, say, 1 km/s. Discuss what happens to the projectile.
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[3.3]
Assume that the initial orbit for an Earth satellite is 100 km above the Earth’s (mean) equator and that its final orbital radius is that of a geo-synchronous satellite. (a) Compute the orbital elements and other parameters for the Earth satellite transfer orbit. Compute the velocities of (b) the circular orbits and (c) the transfer orbit at points of thrust and thus the velocity difference required to achieve the changed orbit. For these purposes, you can ignore other perturbations.
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[3.4]
(a) Compare the orbital properties of synchronous satellites on Mars and Earth and (b) compute the orbit of a synchronous satellite of the Moon. (c) Demonstrate the feasibility or non-feasibility of such a lunar satellite.
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[3.5]
There have been discussions about spacecraft paths that make full use of the gravitational attraction of solar system objects and thus minimize thrusts and use of chemical fuel. (a) Discuss the celestial mechanics involved in the design of such a ‘highway’ to the outer planets; (b) write down an expression for the net acceleration on the spacecraft at some instant; and (c) describe an iterative process which can be used to predict its future path without additional use of its rockets.
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[3.6]
If we now know the masses of all the planets to high precision, why is it difficult to predict the exact positions of Earth-crossing asteroids a few decades into the future?
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[3.7]
Derive an approximate expression for the maximum distance the Moon can recede from the Earth and retain its satellite status, and compute that distance. [Hint: Use (3.55) and set Δr equal to the distance between the Earth and the Moon. If all else fails, you can consult the more exact solution in Sect. 13.2 of Milone and Wilson (2014).]
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Milone, E.F., Wilson, W.J.F. (2014). Celestial Mechanics. In: Solar System Astrophysics. Astronomy and Astrophysics Library. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8848-4_3
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