Gravitational Time Dilation Calculator (GPS Clocks)
How many microseconds per day a clock gains at altitude from weaker gravity, how many it loses to orbital speed, and the net drift of a clock in that orbit.
Results
What this tool does
General relativity says clocks run faster where gravity is weaker. For GPS satellites, about 20 200 km up, that gain is roughly 45.7 microseconds a day, while their speed of nearly 3.9 km/s slows them by about 7.2, for a net gain near 38.5 µs per day. Left uncorrected, that would shift positions by more than 10 km a day, so the satellite clocks are deliberately set to tick slightly slow before launch. The ground clock here is taken as standing still at the surface, so Earth's rotation, which adds about 0.1 µs a day at the equator, is ignored, and the orbit is assumed circular.
Formula
τ/t = √(1 − 2GM ÷ rc²) · orbit: √(1 − 2GM ÷ rc² − v² ÷ c²), v² = GM ÷ r
Variables
| Symbol | Meaning | Unit |
|---|---|---|
mm | Mass of the planet | kg |
rr | Radius of the planet | km |
h | Altitude | km |
GD | Clock gain from weaker gravity | µs/day |
VD | Clock loss from orbital speed (circular orbit) | µs/day |
NT | Net drift of a clock in that orbit | µs/day |
OV | Speed of a circular orbit there | km/s |
Worked example
- Mass of the planet5.972e+24 kg
- Radius of the planet6371 km
- Altitude20,200 km
- Clock gain from weaker gravity45.72 µs/day
- Clock loss from orbital speed (circular orbit)-7.21 µs/day
- Net drift of a clock in that orbit38.51 µs/day
- Speed of a circular orbit there3.873 km/s
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- The default values are typical reference figures, not measurements of your situation. Replace them with your own data whenever you have it.