Variables

Symbol	Quantity	Unit
r	Orbital radius	m
p	Semi-latus rectum	m
e	Eccentricity
θ	True anomaly	deg

What it means

This is the standard conic section equation for orbital motion, describing the radius as a function of true anomaly. The semi‑latus rectum p = h²/μ. This equation is derived from the conservation of energy and angular momentum. It is used to determine positions along an orbit and to solve for orbital elements. Understanding this equation is fundamental to celestial mechanics and astrodynamics.

Worked example

Orbit Equation – Two Examples

Real‑World

Scenario: Circular orbit p = 6.678×10⁶, e = 0. Find radius at θ = 0°.

Parameter	Value
p	6.678×10⁶
e	0
θ	0°

1r = p/(1+e·cosθ) = 6.678e6/(1+0) = 6.678×10⁶ m

Result 6,678 km ✓ Circular

Scenario: p = 2.4419×10⁷, e = 0.72, θ = 0°. Find r (perigee).

Parameter	Value
p	2.4419×10⁷
e	0.72
θ	0°

1r = 2.4419e7/(1+0.72×1) = 2.4419e7/1.72 = 1.4197×10⁷ m

Result 14,197 km ✓ Perigee

Key insight: The orbit equation gives radius as a function of true anomaly – conic sections.

Common mistakes

Orbit equation (semi‑latus rectum form): r = p / (1 + e·cos(θ)).
p: Semi‑latus rectum = a(1−e²).
e: Eccentricity.
θ: True anomaly.
Gives radial distance as a function of angle.

Applications

The orbit equation in semi‑latus rectum form, r = p/(1 + e·cosθ), describes the shape of a conic orbit. It is fundamental to determining the trajectory of a spacecraft. Engineers use this to compute position as a function of true anomaly, to design interplanetary trajectories, and to predict occultation or illumination. By understanding the orbit equation, aerospace engineers can plan spacecraft operations, such as Earth‑to‑Mars transfers, and ensure accurate navigation.

Orbital trajectory propagation and prediction
Interplanetary mission design and targeting
Navigation and guidance for planetary flybys
Spacecraft attitude and payload pointing analysis
Computation of ground track and coverage

Orbit Equation (Semi-Latus Rectum Form)