Orbital Mechanics

Draft

Space Fundamentals established what an orbit is. This section is about working with one: describing it precisely, moving between orbits, holding one against the forces trying to change it, and predicting where a spacecraft will be.

Reading order

The first three pages are prerequisites for everything else. Transfers and plane changes are the core manoeuvre theory. The last four are applications that draw on all of it.

The results this section turns on

Hohmann transfer

\Delta v_1 = \sqrt{\frac{\mu}{r_1}}\left(\sqrt{\frac{2r_2}{r_1+r_2}} - 1\right), \qquad \Delta v_2 = \sqrt{\frac{\mu}{r_2}}\left(1 - \sqrt{\frac{2r_1}{r_1+r_2}}\right)

The minimum-energy two-impulse transfer, and the reference against which every other manoeuvre is measured.

Combined plane and speed change

\Delta v = \sqrt{v_1^2 + v_2^2 - 2v_1v_2\cos\Delta i}

Rotating the plane and changing the speed in one burn. For a standard GTO-to-GEO insertion from 28.5°, this saves 1.15 km/s against doing them separately, which is roughly a third of a commercial satellite’s wet mass.

J2 nodal regression

\dot{\Omega} = -\frac{3}{2}\,n\,J_2\left(\frac{R_\oplus}{p}\right)^2\cos i

Earth’s oblateness precessing the orbital plane. A nuisance to be corrected in most orbits, and the entire basis of the sun-synchronous orbit in one.

Clohessy-Wiltshire equations

\ddot{x} - 3n^2x - 2n\dot{y} = 0, \qquad \ddot{y} + 2n\dot{x} = 0, \qquad \ddot{z} + n^2z = 0

Relative motion near a target, and the reason thrusting toward another spacecraft moves you away from it.

Three things that surprise people

Plane changes are ruinously expensive. A 90° change in low Earth orbit costs more than launching in the first place. Cost scales with orbital speed, which is why every plane change that can be moved to apoapsis is.

Drag makes satellites faster. It removes energy, which lowers the orbit, and a lower orbit is faster. The satellite loses energy and gains speed.

To catch up, slow down. Thrusting toward a target ahead of you raises your orbit, lengthens your period, and drops you further behind.

Pages

01Orbital elements

The six classical elements, conversion to and from state vectors, the singularities at zero eccentricity and zero inclination, equinoctial alternatives, and what a TLE really contains.

02Orbit types and regimes

LEO through cislunar compared, the J2 conditions behind sun-synchronous and Molniya orbits, and the 63.435 degree critical inclination.

03Delta-v budgets

Why delta-v is the currency, a launch budget including gravity and steering losses, a delta-v map for the inner solar system, and how deterministic, statistical and margin terms differ.

04Hohmann and bi-elliptic transfers

The minimum-energy transfer derived and worked, the 11.94 and 15.58 radius ratios where three burns beat two, and how a low-thrust spiral compares on propellant rather than delta-v.

05Plane changes and inclination

The most expensive routine manoeuvre in spaceflight, how combining it with an energy change recovers most of the cost, and three ways to get it for free.

06Rendezvous and docking

The Clohessy-Wiltshire equations, the 2:1 free-drift ellipse, the phases of an operational approach, and why passive abort safety shapes the trajectory.

07Station keeping

North-south and east-west control in GEO, drag makeup in LEO, frozen orbits that need no maintenance at all, and the disposal reserve.

08Orbital perturbations

Magnitudes for every term the two-body model omits, the J2 secular rates, why atmospheric density is the hard part of drag, and the split between special and general perturbation methods.

09Trajectory optimization

Lambert’s problem and porkchop plots, gravity assist geometry, direct versus indirect optimal control, low-thrust formulations, and the toolchains in operational use.

10Ground tracks and coverage

Ground track drift, the repeat condition, access geometry and why elevation mask drives constellation size, pass duration, and coverage figures of merit.