Kepler's Third Law: $T^2 \propto R^3$
$$\frac{T_{Mars}^2}{T_{Mercury}^2} = \frac{R_{Mars}^3}{R_{Mercury}^3}$$Three laws discovered by Johannes Kepler (1609-1619): First law (Law of Ellipses): Every planet orbits the Sun in an ellipse with the Sun at one focus. Second law (Law of Equal Areas): A line joining the planet to the Sun sweeps equal areas in equal times. Consequence: planet moves faster when closer to Sun (perihelion), slower when farther (aphelion). Third law (Harmonic Law): $T^2 \propto R^3$ (for circular orbits $R$ = radius; for elliptical, $R$ = semi-major axis). Newton later derived all three laws from his law of gravitation.
For a circular orbit of radius $R$: gravitational force = centripetal force: $$ rac{GMm}{R^2} = rac{mv^2}{R}$$ Orbital speed: $v = \sqrt{GM/R}$. Period: $T = rac{2\pi R}{v} = 2\pi R \sqrt{ rac{R}{GM}} = rac{2\pi R^{3/2}}{\sqrt{GM}}$. Therefore: $$T^2 = rac{4\pi^2}{GM} R^3$$ This gives $T^2 \propto R^3$ with proportionality constant $ rac{4\pi^2}{GM}$ (same for all planets orbiting the same central body).
Orbital velocity at radius $R$: $v_o = \sqrt{GM/R}$. Decreases as $R$ increases (outer planets move slower). For Earth: $v_o pprox 7.9$ km/s at surface level. Escape velocity: $v_e = \sqrt{2GM/R} = \sqrt{2} \, v_o$. Escape velocity from Earth surface: 11.2 km/s. For Moon (small mass, small $g$): 2.38 km/s (no atmosphere because gas molecules can escape). For Jupiter: 59.5 km/s (keeps all gases, has thick atmosphere).
Geostationary orbit: satellite appears stationary relative to Earth. Must be in equatorial plane with $T = 24$ hours. Using Kepler third law: $R_{geo}^3 = rac{GM_E T^2}{4\pi^2}$. $R_{geo} pprox 42,240$ km from Earth's centre $pprox 36,000$ km above surface. Used for: TV broadcasting (INSAT), communication satellites, weather satellites (METEOSAT). GPS satellites: in medium Earth orbit (20,200 km), not geostationary.
For circular orbit of radius $R$: Kinetic energy: $KE = rac{1}{2}mv^2 = rac{GMm}{2R}$. Potential energy: $PE = - rac{GMm}{R}$. Total energy: $E = KE + PE = - rac{GMm}{2R}$. Total energy is negative (bound system). Binding energy = $|E| = rac{GMm}{2R}$. To increase orbit radius: energy must be added (paradox: adding energy decreases orbital speed — that's how Hohmann transfer works). To de-orbit: reduce energy (fire retrorocket).
In a satellite or freely falling body: apparent weight = 0. Both satellite and occupants are in free fall towards Earth (centripetal acceleration = $g$ at that orbit). No normal reaction force → weightlessness. Not because gravity is absent (gravity provides centripetal force). Astronauts in ISS: $g pprox 8.7$ m/s$^2$ (only slightly less than surface). But they're in continuous free fall. Weightlessness causes: bone density loss, muscle atrophy, fluid redistribution. Artificial gravity possible via rotation (centrifugal force simulates gravity).
$U = - rac{GMm}{r}$. Zero at infinity, negative everywhere else (bound). Change in PE when moving from $r_1$ to $r_2$: $\Delta U = GMm\left( rac{1}{r_1} - rac{1}{r_2} ight)$. Work done against gravity: $W = \Delta U$ (positive when moving away from Earth). For satellite in orbit: $E = - rac{GMm}{2R}$. For object on Earth surface: $E = - rac{GMm}{R_E}$. Energy needed to launch satellite to orbit: $\Delta E = rac{GMm}{2R} - rac{GMm}{R_E} \cdot ( ext{usually small compared to launch cost})$.
Standard $g = 9.8$ m/s$^2$ at sea level. Variation with altitude $h$: $g_h = g\left( rac{R}{R+h} ight)^2 pprox g\left(1 - rac{2h}{R} ight)$ for $h \ll R$. Variation with depth $d$: $g_d = g\left(1 - rac{d}{R} ight)$. At centre of Earth: $g = 0$. Variation with latitude $\phi$: $g_{eff} = g - \omega^2 R\cos^2\phi$. At equator: $g$ is minimum (both altitude and rotation reduce $g$). At poles: $g$ is maximum. Effect of Earth rotation: $g$ at equator $pprox 9.78$ m/s$^2$, at poles $pprox 9.83$ m/s$^2$.