Orbital Mechanics: Kepler's Laws, Orbits & Velocities

How satellites stay up. Learn Newton's gravitation and the two-body problem, circular and elliptical orbits, Kepler's three laws, orbital elements (a, e, i, Ω, ω, ν), the vis-viva equation, orbital period, escape velocity, types of orbits (LEO, MEO, GEO, SSO, Molniya, HEO), ground tracks and perturbations (J2, drag) — with Python computing ISS, GPS and GEO orbits.

A satellite is falling around the Earth: it moves sideways so fast (7.7 km/s in low orbit) that the ground curves away beneath it as quickly as it falls. There is gravity in orbit — astronauts float because they and their spacecraft fall together.

Orbital mechanics (astrodynamics) was worked out by Kepler and Newton four centuries ago, yet it governs every modern space mission, from Starlink constellations to Mars landers.

An analogy

Newton's cannonball: fire a cannonball horizontally from a mountain top — faster and faster — and it lands further away each time, until at orbital speed the Earth curves away as fast as the ball falls and it never lands.

Kepler's laws

Three empirical laws, later derived by Newton:

Orbital elements

Six numbers define an orbit and the satellite's position: size (a) and shape (e); orientation (inclination i, right ascension of the ascending node Ω, argument of perigee ω); and position along the orbit (true anomaly ν, or time since perigee). Two-line element sets (TLEs) publish these for tracked objects.

Orbit types

Chosen by mission:

Perturbations

Real orbits drift: Earth's oblateness (J2) precesses the node (used deliberately for sun-synchronous orbits) and the perigee; atmospheric drag lowers LEO orbits (solar activity changes density 10×); lunar and solar gravity and solar radiation pressure perturb GEO satellites, requiring station-keeping. Precise orbit determination fuses GNSS, laser ranging and radar tracking; space-debris conjunction assessment is now routine.

Formulas

Worked example: ISS velocity

Given: h = 420 km

Find: v and period.

Answer: 7.66 km/s, 93 min

Worked example: Elliptical orbit speeds

Given: Perigee 300 km, apogee GEO (r_a = 42 164 km)

Find: Perigee speed.

Answer: ≈ 10.15 km/s

Worked example: Escape velocity

Given: Earth surface

Find: v_esc.

Answer: 11.18 km/s

Circular orbits, GEO and escape velocity

import math
mu, Re = 3.986004418e14, 6378.137e3
def circ(h):
    r = Re + h
    v = math.sqrt(mu / r)
    T = 2 * math.pi * math.sqrt(r**3 / mu)
    return v, T
for name, h in (("ISS", 420e3), ("GPS", 20200e3)):
    v, T = circ(h)
    print(f"{name}: h={h/1000:.0f} km  v={v/1000:.2f} km/s  period={T/60:.1f} min")
T_sid = 86164.1
r_geo = (mu * (T_sid / (2 * math.pi)) ** 2) ** (1 / 3)
print(f"GEO radius {r_geo/1000:.0f} km (altitude {(r_geo-Re)/1000:.0f} km), v={math.sqrt(mu/r_geo)/1000:.2f} km/s")
v_esc = math.sqrt(2 * mu / Re)
print(f"escape velocity at surface {v_esc/1000:.2f} km/s")
# ISS: h=420 km  v=7.66 km/s  period=93.0 min
# GPS: h=20200 km  v=3.87 km/s  period=718.7 min
# GEO radius 42164 km (altitude 35786 km), v=3.07 km/s
# escape velocity at surface 11.18 km/s

GPS satellites orbit twice per sidereal day (≈ 718 min); geostationary satellites match Earth's rotation (one sidereal day, 23 h 56 min), so they hover over one longitude at 35 786 km. Higher orbits are slower.

Common mistakes

Applications

Key takeaways

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