Timekeeping on the Moon

Artemis is about to return humans to the Moon. At some point we’ll establish a permanent base, most likely at the Moon’s south pole. In preparation, scientists are beginning to consider the question: How will we keep time on the Moon?

https://www.space.com/does-moon-need-its-own-time-zone

During the Apollo lunar missions, timekeeping was tied to ground control back on Earth at the Johnson Space Center in Houston, TX. The official time on the Apollo spacecraft, and on the lunar surface, was that of Houston. This allowed for precise timing of burns and maneuvers. It was a simple system that did the job.

But for long term missions around the Moon (like NASA’s planned Gateway), and for stations or colonies on the lunar surface, selenocentric (Moon-based rather than geocentric) timekeeping makes a lot more sense. Lunar colonies will need to establish a lunar GPS for navigation on and around the Moon, and as on Earth, it will need to be integrated with a precise timekeeping system based on atomic clocks.

All GPS satellites need to correct for the fact that their clocks run at a different rate than clocks on the ground — a consequence of relativity. This is called time dilation and Einstein deduced how it works. First, time runs slower when the clock is in a gravitational field. You get more dilation with stronger gravity. So the GPS satellites, located at an altitude of 22,200 km (12,550 miles) experience weaker gravity then clocks on the ground, and that speeds them up by an average of 45 μs (microseconds) per day.

The gravitational time dilation equation is shown in the graphic. “G” is the universal gravitational constant, “M” is the mass producing the gravity, “r” is the distance from the center of that mass, and “c” is the speed of light.

Simultaneously, the GPS satellites are moving at a speed of 3.9 km/s as they orbit. Einstein showed that this would slow down their clocks relative to a clock at rest by an amount calculated from the Lorentz time dilation formula, the second equation in the graphic. For GPS satellites this amounts to around 7 μs per day.

The two dilation effects combine to make the GPS satellite clocks run a total of 38 μs fast per day. The dilation must be accounted for in order to synchronize the satellite clock with clocks on the ground and get the kind of accuracy we enjoy with GPS devices.

Fortunately, atomic clocks are accurate to around 20–30 ns (nanoseconds) per day. Since 1 μs = 1000 ns, the clocks are accurate enough to notice and correct for time dilation effects.

When we set up selenocentric navigation and timekeeping systems it will likely be based around a small constellation of satellites orbiting the Moon. Whether we count lunar days, which are roughly 29.5 Earth-days, or use standard 24 hour Earth-days, or simply tie it directly to UTC, we need to keep that timekeeping system synchronized with Earth. And this will again require correction for time dilation, because gravity on the Moon is weaker than Earth gravity, and on Moon the clock is moving at 1.022 km/s relative to Earth.

I won’t bore you with the calculations, but all the variables in these two time dilation formulas are available online, and I encourage readers to try these calculations for themselves. For an atomic clock on Earth that measures 1.0 second of time (Δt in the formulas), an atomic clock on the Moon will measure:

Gravitational time dilation from Earth’s gravity: Δt’ = 1.00000000068664 seconds (clock runs faster)
Gravitational time dilation from the Moon’s gravity: Δt’ = 1.00000000003152 seconds (clock runs slower)
Lorentz time dilation from the Moon’s orbital speed: Δt’ = 0.99999999999419 seconds (clock runs slower)
Combined dilation effect: Δt’ = 1.0000000006493 seconds (clock runs faster) = 56 μs per Earth-day

So an atomic clock on the Moon will run faster than one on Earth, but it’s a minuscule effect. It would shorten an otherwise 100 year life span by only about 2 seconds. And on a video call between Earth and Moon, neither party would notice anything different — save for the 2.5 second back and forth signal delay. Nonetheless, to keep Earth and Moon timekeeping systems synchronized this effect would need to be accounted for.

When we colonize Mars similar adjustments in timekeeping will be necessary. On Mars, there are four seasons but the year lasts 687 Earth-days, and a Mars day (called a “sol”) lasts 24:39:35 Earth-hours. Mars will also need GPS, but the need to synchronize with the Earth-Moon system is less important. Time runs faster on Mars too because of its lower gravity, but the real issue is that radio signals from Earth can take anywhere from 5–20 minutes to cross the vast distance, depending on where Mars is in its orbit.

Next Week in Sky Lights ⇒ Why Astronauts Stumbled on the Moon

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