Kepler's laws of planetary motion

Johannes Kepler's primary contributions to astronomy/astrophysics were his three laws of planetary motion. Kepler, a nearly blind though brilliant German mathematician, derived these laws, in part, by studying the observations of the keen-sighted Danish astronomer Tycho Brahe. The article on Johannes Kepler gives a less mathematical description of the laws, as well as a treatment of their historical and intellectual context.

Solution for the motion as a function of time

The Keplerian problem assumes an orbit with semimajor axis a, semiminor axis b, and eccentricity ε. To convert the laws into predictions, Kepler began by adding the orbit's auxiliary circle (that with the major axis as a diameter) and defined these points:

Related Topics:
Keplerian problem - Eccentricity

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• c center of auxiliary circle and ellipse
• s sun (at one focus of ellipse);
• p the planet
• z perihelion
• x is the projection of the planet to the auxiliary circle; then
• y is a point on the circle such that

and three angles measured from perihelion:

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• true anomaly , the planet as seen from the sun
• eccentric anomaly , x as seen from the centre
• mean anomaly , y as seen from the centre

Then

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:mbox{area }cxz=mbox{area }cxs+mbox{area }sxz=mbox{area }cxs+mbox{area }cyz

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:rac{a^2}2E=aarepsilonrac a2sin E+rac{a^2}2M

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giving Kepler's equation

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:M=E-arepsilonsin E.

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To connect E and T, assume r=mbox{length }sp then

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:aarepsilon+rcos T=acos E and rsin T=bsin E

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:r=rac{acos E-aarepsilon}{cos T}=rac{bsin E}{sin T}

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: an T=rac{sin T}{cos T}=rac barac{sin E}{cos E-arepsilon}=rac{sqrt{1-arepsilon^2}sin E}{cos E-arepsilon}

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which is ambiguous but useable. A better form follows by some trickery with trigonometric identities:

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: anrac T2=sqrtrac{1+arepsilon}{1-arepsilon} anrac E2

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(So far only laws of geometry have been used.)

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Note that mbox{area }spz is the area swept since perihelion; by the second law, that is proportional to time since perihelion. But we defined mbox{area }spz=rac abmbox{area }cyz=rac abrac{a^2}2M and so M is also proportional to time since perihelion—this is why it was introduced.

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We now have a connection between time and position in the orbit. The catch is that Kepler's equation cannot be rearranged to isolate E; going in the time-to-position direction requires an iteration (such as Newton's method) or an approximate expression, such as

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:Epprox M+left(arepsilon-rac18arepsilon^3 ight)sin M+rac12arepsilon^2sin 2M+rac38arepsilon^3sin 3M

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via the Lagrange reversion theorem. For the small ε typical of the planets (except Pluto) such series are quite accurate with only a few terms; one could even develop a series computing T directly from M.http://info.ifpan.edu.pl/firststep/aw-works/fsII/mul/mueller.html

Related Topics:
Lagrange reversion theorem - Pluto

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