Astronomy & Celestial Mechanics
the sky, computed
Concept

Specific energy — where it appears

The total energy per unit mass of an orbit, equal to −μ/2a, which fixes the semi-major axis and nothing else about the shape. Two orbits with the same energy and wildly different eccentricities have the same period, which is the fact a phasing manoeuvre and a multi-revolution transfer both exploit.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The energy budget of an orbit at e = 0.7. Kinetic, potential and total energy per unit mass against distance from the primary, in units where GM = 1. The total is a horizontal line — it depends only on the semi-major axis — and the vis-viva relation is that statement solved for the speed.

One equation for the speed anywhere, and the eccentricity is not in it

The vis-viva relation gives the speed at any point of any orbit from two numbers. What it leaves out is the surprise — the shape of the orbit does not appear at all.

orbits · Vis-viva Total Δv against the radius ratio. Total transfer Δv, in units of the starting circular speed, against the ratio of the two circular radii. The Hohmann transfer is cheapest at small ratios; the bi-elliptic transfers overtake it, and the limiting one — an intermediate apoapsis taken to infinity — crosses at a ratio of 11.94. Above about 15.6 every bi-elliptic transfer beats the Hohmann.

Going too far in order to arrive cheaply

The Hohmann transfer is the cheapest two-burn route between circular orbits. Past a radius ratio of 11.94 the cheapest route is three burns, and it goes far beyond the destination first.

spaceflight · Orbital transfer

Named alongside it

The objects these essays reach for when they reach for this one.

ΔvOberth effectOrbital energyVis-vivaAngular momentumCharacteristic energyEccentricityGravity lossHohmann transferLaunch windowPatched conicsPeriapsis

All concepts