Orbit Determination

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In this post, I’ll dive into orbit determination — the science of estimating the position and velocity of a satellite in space. This is part of my ongoing research and contributions to an open-source project called Nyx, a Rust-based astrodynamics toolkit.

Why Is Orbit Determination Important?

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Orbit determination is essential for space missions. It allows us to track and predict the movement of satellites using measurements from ground stations. The “state” of a satellite includes its position and velocity — both of which are needed to determine its orbit.

Satellite Position

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Definition

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The satellite’s position is defined by a position vector — a 3D vector from the center of the Earth to the satellite. It’s typically expressed in the Earth-Centered Inertial (ECI) frame, which considers Earth’s rotation.

To determine this vector, we often rely on three observational measurements from ground stations:

• Azimuth: The angle between the satellite and the North, measured clockwise (in degrees).
• Elevation: The angle between the satellite and the horizon (in degrees).
• Range: The straight-line distance between the ground station and the satellite (in kilometers).

These values are obtained via radio signals. The round-trip time of the signal gives us the range, assuming the speed of light is constant.

Example Calculation

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Let’s say our satellite is located at point A(-2.8, 4.13, 4). Using measured azimuth, elevation, and range, we can derive this position with trigonometry:

Given:

• Distance = 6.39 km
• Elevation = 38.74°
• Azimuth = 235.86°

Altitude (z-axis)

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let distance = 6.39;
let elevation = 38.74;
let altitude = distance * elevation.to_radians().sin();

Result: altitude ≈ 3.99 km (close to actual z = 4)

x Coordinate

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let azimuth = 235.86;
let x = distance * elevation.to_radians().cos() * azimuth.to_radians().sin();

Result: x ≈ -2.8 km

y Coordinate

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let azimuth = 63.26;
let y = distance * azimuth.to_radians().cos() * elevation.to_radians().cos();

Result: y ≈ 4.13 km

This shows how trigonometric relations using elevation and azimuth angles help convert observations into position vectors.

Satellite Velocity

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Doppler Effect

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The Doppler effect is the change in frequency of a wave due to the relative motion between the transmitter (satellite) and the receiver (ground station). When a satellite moves toward a ground station, the received frequency increases; when it moves away, the frequency decreases.

This shift can be used to determine radial velocity — the component of velocity along the line of sight. By combining Doppler measurements with position data, we can compute the full velocity vector.

(WIP)