Orbital insertion: getting captured

On 10 March 2006, after seven months in space, NASA’s Mars Reconnaissance Orbiter fired its six main engines for 27 minutes. For part of that time it was behind Mars, out of radio contact: nobody on Earth could know whether it had worked. If the burn had failed, the spacecraft would have flown past Mars and into an endless orbit around the Sun.

Arriving is not enough. You have to be caught.

Look

A spacecraft does not “stop” at a planet. It falls towards it, speeding up as it gets closer, on an open curve called a hyperbola. Left alone, it would whip around the planet and climb away again, just as fast as it came in.

To stay, it has to lose speed at exactly the right moment, the closest approach:

1. Capture burn. Engines fire against the direction of motion. A modest burn turns the open curve into a long, closed ellipse.
2. Aerobraking. Instead of burning more fuel, controllers lower the orbit’s low point into the thin top of the atmosphere. Every pass, a little air drag slows the spacecraft and the high point sinks.
3. Science orbit. After hundreds of passes over about five months, MRO reached a low, nearly circular orbit about 250 by 316 km up, and could start mapping Mars.

Understand

Coming from Earth on a Hohmann-like path, the spacecraft reaches Mars about 2.65 km/s slower than Mars. From Mars’s point of view that is its “speed at infinity”, v∞. Falling to 426 km above the surface, it speeds up to

v = √(v∞² + 2μ/r) = √(2.65² + 2 × 42,828 / 3,822) ≈ 5.4 km/s

On an ellipse from 426 km to 44,500 km, the speed at the low point must be 4.56 km/s (vis-viva). The ideal, instantaneous capture burn is therefore about 0.87 km/s. A real 27-minute burn is less efficient because it is spread along the curve; MRO’s slowed the spacecraft by about 1 km/s.

Capturing straight into a low circular orbit would cost about 2 km/s (see the Try it). Aerobraking saves most of that difference, and the saved fuel becomes spacecraft and science.

Master

A hyperbola has semi-major axis a = −μ/v∞² and eccentricity e = 1 + r_p v∞²/μ. The flight time from far away to periapsis is set by the hyperbolic Kepler equation M = e sinh F − F.

The capture Δv at periapsis r_p into an ellipse with apoapsis r_a is

Δv = √(v∞² + 2μ/r_p) − √(2μ r_a / (r_p (r_p + r_a)))

which is smallest for a very long ellipse (r_a → large): the spacecraft is barely captured. That is why arrivals first enter a loose orbit, and why aerobraking or several smaller burns follow. For crewed Mars missions, aerocapture (a single deep pass through the atmosphere behind a heat shield) could replace the capture burn altogether.