Mathematics of Kepler's Modified Third Law

Recall Kepler's third law relating a planet's period of revolution to its mean distance from the Sun:

Newton worked out a more precise version of Kepler's law, which is expressed mathematically as

Thus K = 4(pi)²/[G(M + M_p)], where M is the mass of the Sun, M_p is the mass of the planet, P is the orbital period, and a is its mean distance from the Sun. Notice that K is not quite a constant for all planets because the mass of each planet, although a small fraction of the solar mass, is not truly zero, as was Kepler in effect supposing it to be.

Example: Let us determine the mass of the Sun. Applying Newton's modification of Kepler's third law to the Earth and the Sun, we have

or

M + Me = 39.5(1.50 x 10¹³ cm³)/(6.67 x 10^-8 cm³/g*s²)(3.16 x 10⁷ s²) = 1.99 x 10³³ g,

where M is the mass of the Sun, M_e is the mass of the Earth, a is the Earth's mean distance from the Sun (1.50 x 10¹³ cm), G is the gravitational constant (6.67 x 10^-8 cm³/g*s²), and Pe is Earth's period of revolution around the Sun (3.16 x 10⁷ s). Thus the mass of the Sun is 1.99 x 10³³ g, since the Earth's mass is negligible.