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Two-Body State Vector Propagator

· Last updated：2026-07-31

Propagate a spacecraft/satellite state vector (r₀, v₀) to time t₀+Δt under two-body gravity via the universal-variable (f-and-g) method, and derive the classical orbital elements

Details

What this tool helps you do

Given an initial position vector r₀, velocity vector v₀, a central body's gravitational parameter μ, and a time-of-flight Δt, this tool propagates the body to time t₀+Δt using the analytical universal-variable method (Stumpff c/s functions + Newton iteration on the universal Kepler equation), which works for ALL conic-section orbit types (circle / ellipse / parabola / hyperbola) without branching.

It then derives the full set of classical orbital elements from the state vector:

  • Specific mechanical energy ε = v²/2 − μ/r and angular momentum h = |r×v|
  • Semi-major axis a (for bound orbits), eccentricity e (via the eccentricity vector), orbital period T
  • Inclination i, RAAN Ω, argument of periapsis ω, true anomaly θ
  • Periapsis / apoapsis radii

Different from kepler-orbit-solver, which takes a mean anomaly M + eccentricity e and solves for a single eccentric anomaly. This tool takes a STATE VECTOR and propagates it through time — the core of orbit determination. Inputs are in SI units (m, m/s, s); μ values are NASA/IAU standard. Formulas follow Vallado and Curtis.

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Written by the Elysia Tools team

Tool usage guide

Learn when to use this tool, what it supports, and how real users apply it.

Key facts

Category
Science
Input types
select, number, text
Output type
html
Sample coverage
4
API ready
Yes

Overview

The Two-Body State Vector Propagator advances a spacecraft or satellite from an initial position and velocity to a later time under two-body gravity, then derives classical orbital elements for the selected central body.

When to use

  • Propagate a satellite state vector forward or backward by a specified time of flight.
  • Estimate orbital elements from position and velocity vectors.
  • Analyze circular, elliptical, parabolic, or hyperbolic trajectories with one universal-variable method.

How it works

  1. 1Enter the initial position vector r₀ and velocity vector v₀ as comma-separated x, y, z components.
  2. 2Select a central body and distance unit, or provide a custom gravitational parameter μ.
  3. 3Enter the time of flight Δt in seconds.
  4. 4The tool applies the universal-variable f-and-g method and returns the propagated state and derived orbital elements.

Use cases

  • Satellite orbit determination from position and velocity measurements.
  • Preliminary analysis of spacecraft trajectories around planets or the Sun.
  • Comparison of bound orbits and escape or flyby trajectories using their orbital elements.

Examples

1. LEO satellite propagated one hour

Orbital analyst

Background

An analyst wants to estimate the future state of a satellite in a near-circular low Earth orbit.

Problem

Propagate a 7000 km orbit for 3600 seconds and inspect its orbital elements.

How to use

Select Earth and kilometers, enter r₀ as `7000,0,0`, v₀ as `0,7546,0`, and set Δt to `3600` seconds.

centralBody: earth; distUnit: km; r0: 7000,0,0; v0: 0,7546,0; dt: 3600

Outcome

The tool returns the propagated state and orbital elements for the near-circular bound orbit, which should remain nearly constant.

2. Hyperbolic Earth escape trajectory

Mission designer

Background

A mission designer is evaluating a high-speed trajectory departing from Earth.

Problem

Propagate a state vector at 7000 km radius with a 12000 m/s transverse velocity for 10000 seconds.

How to use

Select Earth and kilometers, enter r₀ as `7000,0,0`, v₀ as `0,12000,0`, and set Δt to `10000` seconds.

FAQ

What inputs does the propagator require?

It requires an initial position vector, an initial velocity vector, a time of flight, and a central body or custom gravitational parameter.

Which central bodies are available?

You can select the Sun, Earth, Moon, Mars, Jupiter, Venus, Mercury, or Saturn, or enter a custom μ.

What units should I use?

Time of flight is entered in seconds. Position supports meters or kilometers, and the underlying calculations use SI units.

What orbital elements are calculated?

The output can include energy, angular momentum, semi-major axis, eccentricity, period, inclination, RAAN, argument of periapsis, true anomaly, and periapsis or apoapsis radii.

Can it propagate hyperbolic trajectories?

Yes. The universal-variable method supports circular, elliptical, parabolic, and hyperbolic conic-section trajectories.

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centralBody: earth; distUnit: km; r0: 7000,0,0; v0: 0,12000,0; dt: 10000

Outcome

The propagator handles the unbound hyperbolic trajectory and reports its propagated state and corresponding orbital elements, including eccentricity and positive specific energy.

3. Custom central-body propagation

Astrodynamics student

Background

A student wants to study two-body motion using a gravitational parameter that is not one of the listed standard bodies.

Problem

Propagate a state vector under a user-supplied value of μ.

How to use

Choose `Custom μ`, enter the gravitational parameter in m³/s², provide the position and velocity vectors, and enter the time of flight in seconds.

Outcome

The tool propagates the state using the custom gravitational parameter and derives the resulting classical orbital elements.