∑ The maths › Mission clocks · open the lab →

Mission clocks: the maths

The weak-field approximation of general relativity for clocks in circular orbit (the Moon base adds the Moon's own gravity). Computed live.

Equations

Δf/f ≈ GM/c² · (1/R − 1/r) − GM/(2rc²)

Rate against a ground clock: gravity (higher is faster) minus orbital speed (faster is slower); × 86,400 × 10⁶ for µs per day.

Δt = Δf/f × days

What the clock gains or loses over the mission.

Constants

GM	3.986004418 × 10¹⁴ m³/s²	Earth (IERS)
R	6,371 km	Earth's mean radius
c	299,792,458 m/s	exact

Worked examples ✓ checked on every change

ISS (408 km): rate per day	-24.6 µs/day	the formula above: speed wins
GPS (20,200 km): rate per day	+38.6 µs/day	Ashby (2003)
Six months on the ISS: total	-4.50 ms	about 4.5 thousandths of a second younger

The app's test suite puts the lab in each setup, reads the lab's own result, and fails if it strays from these values.

What's simplified

• Circular orbits; Earth doesn't rotate in the model, so the ground clock's own small motion is ignored.
• The Moon base uses the Moon's surface gravity plus its orbit around Earth.

Where it breaks

Near very strong gravity (neutron stars, black holes), where the weak-field formula stops being accurate.

Sources: N. Ashby, Living Rev. Relativ. 6, 1 (2003); IERS Conventions (2010).