Quick Recap — Gravity, Field & Its Variation

  • Newton's law: F=Gm1m2r2F=\dfrac{Gm_1m_2}{r^2}; surface gravity g=GMR2=43πGρRg=\dfrac{GM}{R^2}=\dfrac{4}{3}\pi G\rho R.
  • Variation: with height gh=g(RR+h)2≈g ⁣(1−2hR)g_h=g\left(\dfrac{R}{R+h}\right)^2\approx g\!\left(1-\dfrac{2h}{R}\right); with depth gd=g ⁣(1−dR)g_d=g\!\left(1-\dfrac{d}{R}\right) (zero at the centre). Rotation makes gg smallest at the equator.
  • Field & potential: field =GMr2=\dfrac{GM}{r^2}; potential V=−GMrV=-\dfrac{GM}{r}; potential energy U=−GMmrU=-\dfrac{GMm}{r} (both negative). Inside a uniform shell the field is zero.
  • Kepler: orbits are ellipses (1st); areal velocity is constant (2nd); T2∝r3T^2\propto r^3 (3rd).

Worked mini-example. At a height h=Rh=R, gh=g(R2R)2=g4g_h=g\left(\dfrac{R}{2R}\right)^2=\dfrac{g}{4} — gravity is one-quarter of its surface value.