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Kepler's laws

 Definition/Summary I. Each planet moves in an ellipse which has the Sun at one of its foci II. The radius vector of each planet passes over equal areas in equal intervals of time. III. The cubes of the mean distances of any two planets form the Sun are to each other as the squares of their periodic times.

 Equations $$\frac{a_1^3}{a_2^3} ~::~\frac{P_1^2}{P_2^2}$$

 Scientists Johannes Kepler (1571-1630)

 Recent forum threads on Kepler's laws

 Breakdown Physics > Astro Cosmo >> Celestial Mechanics

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 Extended explanation The Second law is also known as the "Law of areas" The Third law is also known as the "Harmonic law" The Third law is only approximate and only closely holds if the Sun is vastly more massive than the planets. If is not, then the relative masses of the Sun and planets must be taken into account and the relationship becomes: $$\frac{a_1^3}{a_2^3} ~::~\frac{P_1^2(M+m_1)}{P_2^2(M+m_2)}$$ where M is the mass of the Sun and m1 & m2 are the masses of the respective planets. Newton's law of gravitation: Kepler's laws combined with centripetal acceleration ($-\omega^2r$) enabled Newton (and others) to obtain the inverse-square law of gravitation: Kepler: $$\omega^2 :: 1/T^2 :: 1/r^3$$ Newton: $$F :: \omega^2r :: r/r^3 = 1/r^2$$

Commentary

 tiny-tim @ 04:58 AM Jan15-11 Added derivation of Newton's law of gravitation. Also fixed missing LaTeX (again! ).

 Redbelly98 @ 02:49 PM Jan1-09 In the Equations section, it just says:Click to see the LaTeX code for this imageClicking there shows some LaTex code, but there must be a problem with it as the actual equation is not displayed. EDIT (tiny-tim): I restored it with your edit-and-save trick: it doesn't just get rid of the white background!

 matematikawan @ 05:43 PM Sep5-08 To be more precise the third law should be state as: The squares of the periods of the planets are proportional to the cubes of the semi-major axes of their orbits.

 tiny-tim @ 03:36 AM Aug10-08 Fixed LaTeX