Orbital Mechanics Glossary

Local Reference Frame

r̂ Radial unit vector

Points outward from Earth's centre through the satellite. One of two orthogonal directions defining the local frame at any point on the orbit.

θ̂ Transverse unit vector

Perpendicular to r̂, in the direction of increasing θ (ahead in the orbit). The component of velocity along θ̂ is what generates angular momentum.

r₀ Injection radius

"Injection" is the moment the rocket engine cuts off. Up to that point the rocket is thrusting — not in a natural orbit. The instant the engine stops, the satellite becomes a free-falling body governed only by gravity. That moment of "injection into orbit" fixes r₀, v₀, and β₀ as the three starting conditions.

r₀ is measured from Earth's centre, not its surface, because gravity acts as if all of Earth's mass is concentrated at the centre (shell theorem). So r₀ = R + altitude, where R ≈ 6,371 km. In the textbook example r₀/R = 2, meaning the satellite is one Earth-radius above the surface.

Launch Velocity

v₀ Speed at burnout

Magnitude of the satellite's velocity at engine cutoff. Together with r₀ and β₀, it completely determines the orbit. Related to χ by v₀ = √(χK/r₀).

β₀ Heading angle

Angle from the local transverse direction θ̂ to the velocity vector v₀, measured toward r̂. β₀ = 0 means purely transverse launch — the most efficient injection. β₀ ≠ 0 adds a radial component that shifts where on the orbit injection occurs.

v₀ cos β₀ → angular momentum h. v₀ sin β₀ → radial velocity ṙ₀ (how fast r is changing at injection).

γ Flight-path angle

Angle from θ̂ to v at any point on the orbit (not just at injection). tan γ = e sin θ / (1 + e cos θ). γ = 0 at perigee and apogee; γ ≠ 0 everywhere else on an ellipse.

Energy Parameter

χ Dimensionless energy parameter (chi)

The simplest way to read χ: it is the square of your speed relative to circular orbit speed at r₀.

Because a circular orbit at r₀ requires exactly v² = K/r₀, you get χ = 1 for free if you launch at that speed. Faster than that → bigger orbit. Twice the kinetic energy (χ = 2) → escape.

The four orbits in the diagram are launched from the same point (orange) with increasing speed — only χ changes, β₀ = 0 throughout.

χ < 1 — slower than circular → subcircular ellipse (falls back)
χ = 1 — exactly circular speed → circle
1 < χ < 2 — faster than circular but not escape → ellipse
χ = 2 — escape velocity (v = √2 · v_circ) → parabola
χ > 2 — faster than escape → hyperbola

Angular Momentum

h Specific angular momentum

The angular momentum per unit mass. It is conserved throughout the orbit (Kepler's second law) because the gravitational force has no torque about the focus. h is fixed entirely by the transverse velocity at burnout: h = r₀v₀ cos β₀.

The shaded triangle swept in time δt has area ½ r · r δθ = ½ h δt — equal areas in equal times.

Orbit Shape

p Semi-latus rectum

The orbit radius at θ = 90° from perigee — the half-width of the orbit at the focus. It sets the overall scale of the orbit and depends only on angular momentum h and the gravitational parameter K.

e Eccentricity

Measures how much the orbit deviates from a circle. e = 0 is a circle; 0 < e < 1 is an ellipse; e = 1 is a parabola (just barely escapes); e > 1 is a hyperbola (escapes with speed to spare).

e² = (χ − 1)²cos²β₀ + sin²β₀ — both χ and β₀ affect eccentricity.

θ True anomaly

The angle measured at the focus from the perigee direction (apse line) to the satellite's current position. Increases in the direction of motion. θ₀ is the true anomaly at injection — it tells you where on the orbit the satellite is when the engine cuts out.

tan θ₀ = (e sin θ₀) / (e cos θ₀) = (χ sin β₀ cos β₀) / (χ cos²β₀ − 1)

Orbit Size

rₚ Perigee distance

Closest approach to Earth's centre. rₚ = p / (1 + e). The satellite moves fastest here (Kepler's second law).

rₐ Apogee distance

Farthest point from Earth (elliptic orbits only). rₐ = p / (1 − e). The satellite moves slowest here. For hyperbolas, there is no apogee.

a Semi-major axis

Half the longest diameter of the ellipse. a = (rₚ + rₐ)/2 = p/(1 − e²). Sets the orbital period via Kepler's third law: T² = 4π²a³/K.

Orbit Changes

Δv Velocity increment

The change in velocity produced by a short rocket burn (impulse approximation). |Δv| is the mission cost — proportional to propellant consumed via the rocket equation.

Δvₜ Tangential component

Component of Δv along θ̂. Changes the orbit energy and semi-major axis. Δvₜ = v₂ cos γ₂ − v₁ cos γ₁.

Δvₙ Normal (radial) component

Component of Δv along r̂. Rotates the velocity vector without directly changing speed. Δvₙ = v₂ sin γ₂ − v₁ sin γ₁. Dominant when the apse line must be preserved but γ differs between orbits.

Apse line

The line connecting perigee and apogee — i.e. the major axis of the ellipse, passing through the focus. An impulse applied off the apse line generally rotates it. The scenarios in Thomson §4.13 specifically seek Δv that leaves the apse line unchanged.

Also called the line of apsides. θ is measured from this line.