Orbit Equation (Semi-Latus Rectum Form) Calculator
Solve the fundamental conic-section orbit equation r = p / (1 + e · cos θ) for any of its four unknowns, with step-by-step math, a live orbit schematic, and companion orbital metrics including periapsis and apoapsis radii, semi-major axis, and the velocity at any point on the conic.
One equation, all conics
The single equation r = p/(1 + e·cos θ) describes every possible Keplerian orbit — circle, ellipse, parabola, and hyperbola. The eccentricity e is what distinguishes them, and p sets the overall size. This is why it is sometimes called the "universal orbit equation".
p is the size scale
At θ = 90°, cos θ = 0 and r = p exactly. This is why p is called the semi-latus rectum: it is literally the distance from the focus to the orbit when the radius vector is perpendicular to the major axis. For a circular orbit, p = r. For any other conic, p sets the entire scale of the curve.
e controls the shape
At e = 0, r is constant and the orbit is a circle. As e increases toward 1, the orbit becomes increasingly elongated, with periapsis rp = p/(1+e) shrinking and apoapsis ra = p/(1−e) growing. At e = 1 the apoapsis is at infinity (parabola). For e > 1, the orbit is a hyperbola and the spacecraft escapes.
θ sweeps the orbit
The true anomaly θ is the angle that sweeps out the trajectory. As θ increases from 0° (periapsis) to 180° (apoapsis) and back to 360° (periapsis again for an ellipse), the spacecraft traces the entire path. For a hyperbola, θ is limited to |θ| < arccos(−1/e) — at those limits the radius goes to infinity.
How to Use This Calculator
The orbit equation links four quantities: the radius from the focus, the semi-latus rectum, the eccentricity, and the true anomaly. Pick which one you want the tool to solve, then fill in the others.
1. Choose what to solve for
Pick r, p, e, or θ — the tool rearranges the relation automatically. Solving for θ uses the inverse cosine with a domain check on (p/r − 1)/e.
2. Use realistic presets
Radius presets span LEO to GEO. Semi-latus rectum presets cover LEO, GPS, and GEO. Eccentricity presets span 0 (circle) to 1.5 (hyperbola). True anomaly presets span 0° to 270°.
3. Read the result and steps
The result card shows the solved quantity with engineering notation. The step-by-step panel walks through the arithmetic plus the orbit classification (circle, ellipse, parabola, or hyperbola).
4. Check the conic schematic
The diagram shows the focus, the orbit path, the radius vector r, the semi-latus rectum p (drawn perpendicular to the major axis), and the true anomaly θ — all updating live with the current values.
5. Switch to Advanced for depth
Advanced mode adds a chart of r vs. eccentricity, a 1% sensitivity check, and companion metrics including periapsis radius, apoapsis radius, semi-major axis, orbital period, specific angular momentum, and the velocity at the current point.
6. Cross-check with the reference table
The reference section lists typical (p, e) pairs for LEO, GTO, Molniya, GPS, GEO, and escape trajectories, plus a lookup grid across the eccentricity–anomaly envelope.
What This Tool Offers
Built around the universal conic-section relation used throughout astrodynamics and mission design.
Four-way solving
Solve for radius, semi-latus rectum, eccentricity, or true anomaly — whichever the problem gives you.
Live conic schematic
A focus at the origin with the orbit path drawn from the equation, the radius vector, the semi-latus rectum, and the true anomaly — all updating live.
Dual comparison charts
r vs. true anomaly at three eccentricities, plus r vs. eccentricity at three semi-latus rectum values.
Companion orbital metrics
Advanced mode reports periapsis and apoapsis radii, semi-major axis, orbital period, specific angular momentum, specific energy, and the current velocity from vis-viva.
Built-in sensitivity check
See how much r moves for a 1% change in p, e, or θ. The sensitivity to θ depends on sin θ, so it's zero at periapsis and apoapsis.
Reference tables
(p, e) values for LEO, GTO, Molniya, GPS, GEO, and escape trajectories, plus a lookup grid of r across the eccentricity–anomaly envelope.
Reference Tables
Representative (p, e) pairs for common missions, plus a lookup grid of r across the eccentricity–anomaly envelope.
r quick lookup (in Mm, p = 7.0×10⁶ m)
r = p / (1 + e · cos θ) for various eccentricities and true anomalies.
| e | θ = 0° | θ = 45° | θ = 90° | θ = 135° | θ = 180° |
|---|
Typical Orbit Parameters
Earth-centred orbits. p in km, e dimensionless. Elliptical orbits have periapsis and apoapsis; hyperbolic ones have only periapsis and "escape".
| Orbit | p (km) | e | rp (km) | ra (km) | Type |
|---|---|---|---|---|---|
| LEO 400 km circular | 6,778 | 0.000 | 6,778 | 6,778 | Circle |
| Sun-synchronous 700 km | 7,071 | 0.001 | 7,064 | 7,078 | Near-circle |
| GTO (typical transfer) | 24,600 | 0.730 | 6,678 | 42,164 | Ellipse |
| Molniya (12-hr) | 26,600 | 0.740 | 6,900 | 46,300 | Ellipse |
| GPS constellation | 26,560 | 0.010 | 26,295 | 26,825 | Near-circle |
| GEO | 42,164 | 0.000 | 42,164 | 42,164 | Circle |
| Escape (parabolic) | 2·rp | 1.000 | rp | ∞ | Parabola |
| Hyperbolic escape | > 2·rp | > 1 | rp | ∞ | Hyperbola |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| r | Radius from focus | 6.4×10⁶ – 5×10⁸ | m |
| p | Semi-latus rectum | 6.4×10⁶ – 5×10⁸ | m |
| e | Eccentricity | 0 – 3 | — |
| θ | True anomaly | −180° – 360° | degrees |
| rp | Periapsis radius = p/(1+e) | — | m |
| ra | Apoapsis radius = p/(1−e) (if e<1) | — | m |
| a | Semi-major axis = p/(1−e²) | — | m |
| h | Specific angular momentum = √(μ·p) | — | m²/s |
What Does the Orbit Equation Measure?
The orbit equation is the fundamental relation between the distance of an orbiting body from its central body and the angle it makes with the periapsis direction. It applies to every Keplerian conic — circle, ellipse, parabola, and hyperbola — with the eccentricity alone determining which. The semi-latus rectum p sets the overall size scale, while θ is the true anomaly, the angle swept out along the orbit. Every position, velocity, and orbital element for a two-body problem can be derived from this equation plus the vis-viva relation.
Orbit Equation Formula
The defining relationship is:
Where the semi-latus rectum is related to the semi-major axis and specific angular momentum by:
Key special values are:
Apoapsis (θ = 180°, e < 1): ra = p / (1 − e)
Semi-latus (θ = 90° or 270°): r = p
Rearranged to solve for p, e, or θ:
Orbit Equation Variables
| Symbol | Meaning | Notes |
|---|---|---|
| r | Radius from focus | Distance from the central body to the spacecraft |
| p | Semi-latus rectum | Sets the overall size of the conic; r = p at θ = 90° |
| e | Eccentricity | Determines the shape: 0 circle, <1 ellipse, 1 parabola, >1 hyperbola |
| θ | True anomaly | Angle from the periapsis direction, measured at the focus |
How Each Parameter Moves the Radius
The radius is directly proportional to p and depends on e and θ through the term (1 + e·cos θ). A 1% increase in p raises r by 1% uniformly at every point on the orbit. The dependence on e is more subtle: increasing e always decreases r at periapsis and increases it at apoapsis, while at θ = 90° the value of r is unchanged. The dependence on θ is through cos θ, so r is maximum when cos θ is minimum (θ = 180° for e > 0) and minimum when cos θ is maximum (θ = 0°). Near θ = 90° and 270°, cos θ changes slowly, so r is relatively insensitive to small changes in θ. This is why circular orbits (e ≈ 0) stay at nearly constant radius no matter what θ is, while highly elliptical orbits sweep through enormous radius changes over a small range of true anomaly near periapsis.
Orbit Equation Calculation Examples
cos(90°) = 0
1 + e·cos θ = 1 + 0.1×0 = 1
r = p / 1 = 7,000,000 m
rp = p/(1+e) = 7,000,000/1.1 = 6,363,636 m
ra = p/(1−e) = 7,000,000/0.9 = 7,777,778 m
a = p/(1−e²) = 7,000,000/0.99 = 7,070,707 m
cos(45°) = 0.7071
1 + 0.15 × 0.7071 = 1 + 0.1061 = 1.1061
p = r × 1.1061 = 8,000,000 × 1.1061 = 8,848,800 m
Frequently Asked Questions
Because the parameter p in the equation has a very specific geometric meaning: it is the semi-latus rectum — the distance from the focus to the orbit along a line perpendicular to the major axis. Equivalently, it is the value of r at θ = 90°. Using p as the size parameter (rather than the semi-major axis a) has the advantage that the same equation works for every conic without modification: an ellipse, parabola, and hyperbola all satisfy r = p/(1+e·cos θ). This is why it's sometimes called the "universal conic equation".
The radius r goes to infinity. This occurs when cos θ = −1/e, which is only possible if e ≥ 1 (parabolic or hyperbolic orbits). The angle at which this happens, θ∞ = ±arccos(−1/e), is called the true anomaly at infinity. It marks the boundary of the physically accessible region of the trajectory. For a hyperbola, the spacecraft travels from θ = −θ∞ (far inbound) through θ = 0 (periapsis) to θ = +θ∞ (far outbound). For a parabola, θ∞ = 180°, and the trajectory is unbounded at that direction.
By the fundamental relation p = h²/μ, where h is the specific angular momentum and μ is the gravitational parameter of the central body. This is why p is often the most convenient way to express orbital size: it comes directly from the conserved quantity h, which does not change along the orbit. If you know h and the central body's μ, you immediately know p — and then e and θ determine the entire geometry via r = p/(1 + e·cos θ).
Because the true anomaly θ is measured from the periapsis direction, which is the direction of closest approach. At periapsis, θ = 0 so cos θ = 1, which makes the denominator (1 + e·cos θ) = 1 + e, its maximum value — and therefore r = p/(1+e), the minimum radius. At apoapsis, θ = 180° so cos θ = −1, making r = p/(1−e), the maximum radius (for e < 1). Using sin θ instead would put the periapsis at θ = 90°, which is a less convenient convention. The cosine convention anchors the geometry to the direction of closest approach, which is also the direction of the eccentricity vector.
The true anomaly θ is the actual angle between the radius vector and the periapsis direction, measured from the focus. The eccentric anomaly E is an auxiliary angle measured from the centre of the ellipse rather than the focus, and it is used in Kepler's equation M = E − e·sin E to relate mean anomaly to true anomaly. For a circle (e = 0) they coincide. For an ellipse they differ by an amount that depends on e; the relationship is tan(θ/2) = √((1+e)/(1−e))·tan(E/2). This calculator works with the true anomaly, which is the natural angle appearing in the orbit equation.
For elliptical orbits, the semi-major axis is a = p/(1 − e²). At e = 0, a = p (circular orbit). At e → 1, a → ∞ (parabolic). For hyperbolic orbits, the convention is that a is defined as negative: a = p/(1 − e²) still works, but the result is negative. The periapsis distance is rp = p/(1 + e) = a(1 − e), and for an ellipse the apoapsis is ra = p/(1 − e) = a(1 + e). The semi-latus rectum form is often more compact because p stays positive for all conics, whereas a changes sign for hyperbolic orbits.
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| Parameter | Value |
|---|---|
| p | 6.678×10⁶ |
| e | 0 |
| θ | 0° |
| Parameter | Value |
|---|---|
| p | 2.4419×10⁷ |
| e | 0.72 |
| θ | 0° |
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
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