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An overview of kepler's laws of planetary motion, including the elliptical shape of planetary orbits around the sun, the equal area swept out in equal times, and the relationship between the orbital period and the semi-major axis. The document also discusses the mechanical energy and orbit families of two-body systems, and the conservation of angular momentum in planetary orbits.
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Consider an asteroid of mass m in (an arbitrary) orbit around a much larger planet of mass M. The mechanical energy of the two- body system is a conserved quantity that determines the nature of the orbit. Different families of orbits result from different signs of E mec
mec
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family E mec eccentricity e orbit bound < 0 < 1 (0) ellipse (circle) just unbound = 0 = 1 parabola really unbound > 0 > 1 hyperbola
If two bodies of masses m and M are in a gravitationally bound orbit, the mechanical energy determines the size of the orbit , defined as the semi-major axis a , of the two-body system while the angular momentum L determines the shape of the orbit, defined by the eccentricity e
mec
mec
2
2
2
For a set of bodies in circular orbits around a large mass M , the square of the orbital speed decreases inversely with distance r
2
http://cfa-www.harvard.edu/~bmcleod/castle.html Images of Lensing removed
