Orbital Mechanics: Key Equations

Basics and Constants

All the orbit equations below come from Newton's law of gravitation:

F = G · m₁ · m₂ / r²

In astrodynamics you usually work with the standard gravitational parameter μ = G·M. It is known much more precisely than G or M on their own.

Symbol	Meaning	Value
G	Gravitational constant	6.674 × 10⁻¹¹ m³/(kg·s²)
μ Earth	G · M of Earth	3.986 × 10¹⁴ m³/s²
μ Sun	G · M of the Sun	1.327 × 10²⁰ m³/s²
R Earth	Mean radius of Earth	6,371 km

Important: In all the formulas, r is the distance from the centre of the body, not the altitude. For an orbit 400 km above Earth, use r = 6,371 + 400 = 6,771 km.

Kepler's Three Laws

Orbits are ellipses, with the central body at one focus.
Equal areas in equal times: a planet moves faster when it is close to the Sun and slower when it is far away.
T² ∝ a³: the square of the orbital period is proportional to the cube of the semi-major axis.

Describing an Orbit (Orbital Elements)

Six numbers describe any orbit completely:

Element	Symbol	What it describes
Semi-major axis	a	Size of the orbit
Eccentricity	e	Shape: 0 = circle, 0–1 = ellipse, 1 = parabola, above 1 = hyperbola
Inclination	i	Tilt against the reference plane
Longitude of ascending node	Ω	Where the orbit crosses the reference plane going upward
Argument of periapsis	ω	Where the closest point lies within the orbit
True anomaly	ν	Where the object is on the orbit right now

Closest and farthest points of an ellipse:

Periapsis  r_p = a · (1 − e)
Apoapsis   r_a = a · (1 + e)
a = (r_p + r_a) / 2

The Most Important Equations

Circular Orbit Velocity

v = √(μ / r)

Escape Velocity

The minimum speed needed to leave a body's gravity for good:

v_esc = √(2μ / r) = √2 · v_circular

Vis-Viva Equation

This is the most useful equation in orbital mechanics. It gives your speed at any point of any orbit:

v² = μ · (2/r − 1/a)

For a circle (r = a), it becomes v = √(μ/r).
For escape (a → ∞), it becomes v = √(2μ/r).

Orbital Period

T = 2π · √(a³ / μ)

Specific Orbital Energy

ε = v²/2 − μ/r = −μ / (2a)

When the energy is negative, the object is in a closed (bound) orbit. At zero it is exactly on an escape path. When it is positive, the object follows a hyperbola and leaves.

Worked Examples (Earth)

Orbit	Radius r	Speed	Period
ISS (about 400 km altitude)	6,771 km	7.67 km/s	about 92 min
Geostationary (GEO)	42,164 km	3.07 km/s	23 h 56 min (one sidereal day)
Escape from the surface	6,371 km	11.19 km/s	–

Example calculation for the ISS:

v = √(3.986 × 10¹⁴ / 6.771 × 10⁶) ≈ 7,673 m/s
T = 2π · √((6.771 × 10⁶)³ / 3.986 × 10¹⁴) ≈ 5,545 s ≈ 92.4 min

Changing Orbits: The Hohmann Transfer

The most fuel-efficient way to move between two circular orbits in the same plane takes two engine burns:

At the inner orbit r₁, burn forward. This puts you on an ellipse whose farthest point touches r₂.
At r₂, burn forward again to make the orbit circular.

Δv₁ = √(μ/r₁) · ( √(2r₂ / (r₁ + r₂)) − 1 )
Δv₂ = √(μ/r₂) · ( 1 − √(2r₁ / (r₁ + r₂)) )
Transfer time = π · √( ((r₁ + r₂)/2)³ / μ )

Example: From a 300 km orbit (r₁ = 6,678 km) to GEO (r₂ = 42,164 km):

Step	Value
Δv₁	about 2.43 km/s
Δv₂	about 1.47 km/s
Total Δv	about 3.89 km/s
Transfer time	about 5.3 hours

This example ignores the plane change. A real launch from a site away from the equator also needs extra Δv to tilt the orbit.

How much fuel a Δv costs is covered in The Rocket Equation and Delta-v.

Sphere of Influence

The sphere of influence is the region where a planet's gravity matters more than the Sun's:

r_SOI ≈ a · (m / M)^(2/5)

Here a is the planet's distance from the Sun, m is the planet's mass and M is the Sun's mass. For Earth this gives about 925,000 km. Inside this region you can calculate as if only Earth existed; outside it, you switch to the Sun. This approach is called the "patched conics" method.