Twin satellites (GRACE / GRACE-FO) fly nose-to-tail in the same near-polar orbit, ranging their separation with a microwave/laser link accurate to about a micron. A buried or surface mass anomaly of magnitude Δm perturbs the local gravitational acceleration; treating it as a point mass at perpendicular distance h below the spacecraft gives the vertical attraction:
g(x) = G·Δm·h / (x² + h²)^1.5
G = 6.674×10⁻¹¹ m³ kg⁻¹ s⁻²
x = along-track distance from the anomaly (m)
h = orbit altitude above the anomaly (m)
As the leading satellite (A) crosses the anomaly first, it is tugged forward/back a fraction of a second before the trailing satellite (B) feels the same pull. That timing difference stretches or compresses the ranging baseline — the raw observable GRACE actually measures:
Δg_LOS(t) = g(x_A) − g(x_B) (line-of-sight gravity gradient)
ΔL(t) ∝ ∫ Δg_LOS(t) dt (measured inter-satellite range change)
The top panel plots g(x) along the whole orbit track as the pair sweeps past, with markers showing exactly where GRACE-A and GRACE-B sit right now. The bottom panel is the same range-rate integral the 3D version shows as geoid bulge, drawn here as a running strip-chart of ΔL(t) — this is literally how satellite geodesy tracks Greenland and Antarctic ice-sheet loss, groundwater depletion in aquifers, and post-glacial crustal rebound.
- Δm slider — strength and sign of the buried/surface mass change (Gt = 10¹² kg).
- Altitude h — raising the orbit widens and weakens the footprint, exactly as the 1/(x²+h²)^1.5 falloff predicts.
- Baseline L — the along-track spacing between the two spacecraft; GRACE-FO flies ≈220 km.
- Readouts — g under each satellite, their difference, and the cumulative range-rate integral the ranging instrument actually reports.