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Posted on • Originally published at iwtlp.com

Orbital mechanics with Python, from circular orbits to Hohmann transfers

#aerospace #orbitalmechanics #python

Orbital mechanics sounds like it needs a graduate degree, but the core results are surprisingly reachable with high-school physics and a few lines of Python. If you can compute an orbital velocity and a transfer, you understand the foundation that mission planning is built on.

A circular orbit

For a circular orbit, gravity provides exactly the centripetal force, which gives a clean formula for orbital speed:

import math
MU = 3.986e14   # Earth's gravitational parameter, m^3/s^2

def circular_velocity(r):
    return math.sqrt(MU / r)   # m/s at orbital radius r (meters)
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A satellite in low Earth orbit (about 200 km up, so r is roughly 6,571 km) moves at close to 7.8 km/s. That single formula already explains why orbital speeds are so high: lower orbits need to move faster to stay up.

Orbital period

How long one lap takes follows from Kepler's third law:

def period(r):
    return 2 * math.pi * math.sqrt(r**3 / MU)   # seconds
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Plug in geostationary radius (about 42,164 km) and you get roughly 24 hours, which is exactly why those satellites appear to hang still over one spot on Earth. The formula predicts the orbit that makes GPS and communications satellites work.

The Hohmann transfer

The classic, fuel-efficient way to move between two circular orbits is the Hohmann transfer: one burn to enter an elliptical path that just touches both orbits, and a second burn to circularize at the new altitude. The two velocity changes come from the vis-viva equation:

def hohmann(r1, r2):
    v1 = math.sqrt(MU / r1)
    v2 = math.sqrt(MU / r2)
    a = (r1 + r2) / 2   # transfer ellipse semi-major axis
    vp = math.sqrt(MU * (2 / r1 - 1 / a))   # speed at perigee of transfer
    va = math.sqrt(MU * (2 / r2 - 1 / a))   # speed at apogee of transfer
    return abs(vp - v1) + abs(v2 - va)      # total delta-v
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That total delta-v is the fuel budget for the maneuver, the number a mission planner actually cares about. With these few functions you can already reason about real transfers, like raising a satellite from a parking orbit to its operational altitude.

Build the whole thing

This is the kind of problem where coding it is understanding it: change a radius and watch the velocity and delta-v respond. The aerospace with Python track builds orbital mechanics and much more from scratch, graded in your browser. The first project is free.

The math of getting to orbit is closer than you think. Start computing it.

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