Calculating masses and cosmic velocities

Newton refined Kepler’s third law: if the central body has mass M and the satellite mass m ≪ M, then T² = 4π² a³ / (G M), so M = 4π² a³ / (G T²), where G = 6.674 · 10⁻¹¹ N·m²/kg² is the gravitational constant. So by measuring a satellite’s orbital radius and period we can “weigh” the central body. For instance the Moon (a = 384,400 km, T = 27.32 days) gives the Earth’s mass as ≈ 6.0 · 10²⁴ kg; the Earth’s motion round the Sun gives the Sun’s mass as ≈ 2.0 · 10³⁰ kg. For comparing masses the ratio method is handy: M₁ / M₂ = (a₁ / a₂)³ · (T₂ / T₁)². Because two bodies attract each other equally, both circle their common centre of mass, so the exact formula uses M + m.