Ground Track Velocity⚠ unverified

Understanding the Parameters

• The ground track — the curve traced on the Earth's surface by the line from the satellite to the planet's centre (the sub-satellite or nadir point). It is what a mapping or imaging satellite "paints" across the ground, and its speed sets how fast a push-broom sensor scans terrain.

• Why the ground point is slower — the satellite and its nadir point share the same angular rate about Earth's centre (they are on the same radius line). But linear speed is angular rate times radius, and the ground point sits at the smaller radius $R_\oplus$ while the satellite is at $R_\oplus + h$. So the ground speed is the orbital speed scaled down by $R_\oplus/(R_\oplus + h)$.

• How much slower — modest for LEO: at 400 km, $R_\oplus/(R_\oplus + h) = 6371/6771 = 0.941$, so the ground track moves at 94 % of orbital speed — about $7.2\ \text{km/s}$ for a $7.67\ \text{km/s}$ orbit. Higher orbits project down more: the ratio falls as altitude grows.

• The rotation caveat — the big omission. As the satellite orbits, the Earth turns beneath it ($\sim465\ \text{m/s}$ eastward at the equator). This makes each ground track shift westward on successive passes and changes the apparent speed and heading of the sub-satellite point relative to the ground. For precise coverage, revisit-time, and swath planning, the Earth's rotation must be added vectorially — which this formula does not do.

• Where it matters — ground-track speed drives the design of imaging missions: the scan rate of a sensor, the overlap between successive swaths, and the revisit time over a target. It also sets the Doppler shift and contact duration for a ground station watching an overhead pass.

Derivation (Approaching a Proof)

The satellite and its sub-satellite (nadir) point lie on the same line from the Earth's centre, so they share the same angular velocity $\omega$ about that centre. Linear speed is angular velocity times radius.

Satellite at orbital radius $r = R_\oplus + h$:

$$v_{orbit} = \omega\,(R_\oplus + h) \quad\Longrightarrow\quad \omega = \frac{v_{orbit}}{R_\oplus + h}.$$

Ground point at the surface radius $R_\oplus$, with the same $\omega$:

$$v_{gt} = \omega\,R_\oplus.$$

Substitute $\omega$:

$$v_{gt} = \frac{v_{orbit}}{R_\oplus + h}\,R_\oplus = v_{orbit}\,\frac{R_\oplus}{R_\oplus + h}. \qquad\blacksquare$$

The result is a pure similar-triangles (radial projection) scaling: a point closer to the centre of rotation moves slower in proportion to its radius. Adding Earth's rotation would replace the shared $\omega$ with the satellite's inertial $\omega$ minus Earth's rotation rate projected along the track — a vector correction the simple form omits.

Dimensional check. $R_\oplus/(R_\oplus + h)$ is $\text{m}/\text{m} = $ dimensionless, so $v_{gt}$ carries the m/s of $v_{orbit}$ ✓.

History and Development

• Ground tracks and reconnaissance. The concept became operationally central with the first imaging satellites — Corona (from 1960) and the weather satellites of the 1960s — where the sub-satellite point's motion determined film advance rates and coverage. Sun-synchronous orbits (see Sun Synchronous Inclination) were adopted precisely to control the ground-track geometry and lighting.

• Repeat ground tracks. Many Earth-observation satellites are placed in repeat-ground-track orbits, where the track pattern exactly repeats after a whole number of days — achieved by tuning the orbit so that the ground-track spacing (set by this speed and Earth's rotation together) closes up. Landsat's 16-day repeat cycle is a classic example, and it depends on combining ground-track speed with Earth rotation, the very term this formula drops.

• Doppler and station passes. For communications and tracking, the ground-track speed sets how quickly a satellite crosses the sky and the Doppler shift a ground station sees — parameters that shaped everything from early Transit navigation satellites (which used the Doppler shift to fix position) to modern LEO constellations.

Related Concepts: Circular Orbit Velocity, Orbital Velocity, Orbital Period, Sun Synchronous Inclination, Nodal Precession

Notes: Geometric radial projection — the nadir point shares the satellite's angular rate but sits at the smaller radius $R_\oplus$, so $v_{gt}=v_{orbit}R_\oplus/(R_\oplus+h)<v_{orbit}$ (94 % at 400 km). Ignores Earth's rotation (which shifts tracks westward and alters apparent speed/heading) and inclination/latitude; approximate. Drives sensor scan rate, swath overlap, revisit time, pass Doppler.