# Two-Body State Vector Propagator

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

> Canonical page: https://elysiatools.com/en/tools/two-body-state-vector-propagator

- **Category:** Science

- **Keywords:** two-body, state vector, orbit propagation, f-g functions, astrodynamics, orbital elements, universal variable, μ

## Overview

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.

## Inputs

- **Central Body (μ)** (select)
- **Custom μ (m³/s²)** (number): e.g. 3.986e14
- **Distance unit** (select)
- **Initial position r₀ (x,y,z; m)** (text): e.g. 7000000,0,0
- **Initial velocity v₀ (vx,vy,vz; m/s)** (text): e.g. 0,7.5,1
- **Time of flight Δt (s)** (number): e.g. 3600

## 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

- Enter the initial position vector r₀ and velocity vector v₀ as comma-separated x, y, z components.
- Select a central body and distance unit, or provide a custom gravitational parameter μ.
- Enter the time of flight Δt in seconds.
- The 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.

## Frequently asked questions

### 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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