🔭 astrophysics

Lunar tidal response-induced clock frequency variations on the Moon

This paper investigates how lunar tidal response-induced fractional frequency shifts between clocks on the Moon, analyzed via Keplerian expansions for elliptical orbits, reveal that longitude differences affect both amplitude and phase while latitude differences affect only amplitude, ultimately quantifying a significant daily clock discrepancy between lunar and Earth-based clocks to inform future lunar timekeeping and synchronization systems.

Original authors: Hongbin Zhang, Baocheng Zhang

Published 2026-10-27

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CC BY 4.0

Original authors: Hongbin Zhang, Baocheng Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Time is not a universal constant that ticks the same way everywhere. In the realm of modern physics, time is a fluid thing, stretched and squeezed by gravity and speed. A clock placed deep in a strong gravitational field will tick slower than one floating in empty space, and a clock moving at high speed will fall behind a stationary one. As humanity prepares to return to the Moon with ambitious plans for bases, rovers, and satellites, these subtle differences in time are becoming a practical engineering problem. To navigate the lunar surface or coordinate a network of instruments, we need a unified lunar time. However, the Moon is not a static, rigid rock; it is an elastic body that flexes under the gravitational pull of the Earth and the Sun. This flexing, known as a tidal response, slightly alters the Moon's shape and its gravitational field, which in turn affects how fast a clock runs. With the newest atomic clocks becoming so precise that they can detect changes in time as small as one part in a hundred quintillion, these tiny wobbles in the Moon's shape are no longer just a theoretical curiosity; they are a source of error that must be accounted for.

Researchers at the China University of Geosciences have taken a close look at how this lunar flexing influences the time kept by clocks on the Moon. They focused on a specific scenario where the Moon travels in its actual elliptical orbit around the Earth, rather than a simplified perfect circle, to get a more realistic picture. Using a mathematical approach that breaks down the Moon's complex motion into manageable parts, they calculated how the Earth's gravity pulls on the Moon, causing it to bulge and then relax as the distance between the two bodies changes. This deformation creates a shifting gravitational potential that ripples across the lunar surface. The team found that this shifting potential causes a measurable difference in the ticking rate of two clocks placed at different locations on the Moon. The effect is incredibly small, occurring at a level of one part in 10 to the power of 18, but this is exactly the level of precision that today's best clocks can detect. If we ignore this effect, our future lunar timekeeping systems will drift out of sync.

The study revealed a fascinating pattern in how the Moon's shape affects time depending on where you stand. The researchers discovered that the difference in longitude between two clocks changes both the size and the timing of the frequency shift. As the Moon moves along its orbit, the signal of this time difference waxes and wanes, shifting its phase in a predictable rhythm. In contrast, the difference in latitude between two clocks only changes the size of the effect, not its timing. If two clocks are placed at the same latitude but different longitudes, the time difference between them will grow and shrink as the Moon orbits. However, if they are at the same longitude but different latitudes, the time difference simply scales up or down without changing its rhythm. The researchers also noted that if two clocks were placed at the exact same spot, or if they were both at the very top of the Moon's poles, the tidal response would cause no difference in their timekeeping at all.

Beyond the surface, the team extended their calculations to include clocks on satellites orbiting the Moon and even compared a clock on the lunar surface to one on Earth. They estimated that the tidal response of the Moon causes a clock on the lunar surface to lag behind a clock on Earth by approximately 5.68 × 10⁻⁷ microseconds per day. This is a significant amount when considering the goal of nanosecond-level synchronization for future missions. The study also showed that as a satellite moves farther away from the Moon, the influence of the Moon's tidal flexing on its clock diminishes, eventually stabilizing at a specific difference relative to the surface clock. These findings are not just theoretical exercises; they provide a necessary correction factor for the upcoming era of lunar exploration. As nations like the United States and China prepare to land astronauts and establish infrastructure, understanding and correcting for the Moon's elastic heartbeat will be essential to ensure that the clocks on the ground, in the sky, and back on Earth all agree on what time it is.

Technical Summary: Lunar Tidal Response-Induced Clock Frequency Variations on the Moon

Problem Statement
As atomic clock technology advances, the instability and uncertainty of fractional frequency shifts (FFS) have reached the $10^{-18}$ level. At this precision, the FFS induced by the lunar tidal response (LTR)—the deformation of the Moon's elastic body due to external gravitational fields—becomes comparable to the intrinsic uncertainties of state-of-the-art clocks. While previous studies have addressed relativistic frequency shifts and tidal effects on Earth or treated the Moon's orbit as circular, the specific impact of LTR on high-precision clock comparisons and lunar time construction within a realistic elliptical orbit configuration has not been systematically investigated. This gap is critical for the emerging need to establish a standard lunar time and synchronize devices for upcoming missions (e.g., Artemis, China's lunar landing plans).

Methodology
The authors model the LTR within the Lunar Celestial Reference System (LCRS), adhering to International Astronomical Union (IAU) resolutions. The study employs the following methodological framework:

Relativistic Framework: The spacetime metric is constructed in the LCRS using a two-potential form (scalar and vector potentials) up to the post-Newtonian order $c^{-2}$ . The vector potential is shown to be negligible ( $<10^{-18}$ ) for the analysis.
Keplerian Expansions: Unlike previous circular-orbit approximations, the Moon's motion is modeled as an elliptical orbit. The authors use Keplerian expansions to express the Earth-Moon distance ( $r_{mE}$ ) and the true anomaly ( $T$ ) in terms of the mean anomaly ( $M$ ) and orbital eccentricity ( $e$ ).
Tidal Potential and Response: The Earth's tidal potential ( $\Phi_{tE}$ ) is expanded to the second order. The LTR is introduced as a correction to the lunar gravitational potential ( $\Phi'_m = \Phi_m + \Phi_{tr}$ ), where $\Phi_{tr} = k^m_2 \Phi_t$ . The lunar Love number ( $k^m_2 = 0.02405$ ) is used to quantify the elastic response. Solar tidal contributions are calculated and found to be below the $10^{-18}$ threshold, thus neglected.
Approximations: Tidal dissipation and lag effects are excluded, as the study focuses on short-term clock comparisons rather than long-term orbital evolution. Calculations are maintained at a precision level of $10^{-18}$ .

Key Contributions

Elliptical Orbit Modeling: The paper provides a more realistic description of LTR by incorporating the Moon's elliptical orbit, analyzing how the mean anomaly ( $M$ ) affects the frequency shift.
Position-Dependent Analysis: The study derives analytical expressions for the FFS between two clocks at arbitrary locations on the lunar surface, explicitly separating the effects of longitude differences ( $\Delta\lambda$ ) and latitude differences ( $\Delta\phi$ ).
Quantification of LTR Magnitude: The authors quantify the LTR-induced FFS and time differences for various scenarios, including clocks on the lunar surface, clocks on lunar satellites, and comparisons between lunar and Earth surface clocks.

Results

Magnitude of FFS: The LTR-induced FFS between two clocks on the lunar surface is on the order of $10^{-18}$ , making it measurable with current technology.
Geometric Dependence:
- Longitude Difference: When two clocks share the same latitude, the FFS amplitude and phase vary periodically with the Moon's motion. The FFS is symmetric around the 90-degree longitude and decreases as the longitude difference moves away from this point.
- Latitude Difference: When two clocks share the same longitude, the FFS amplitude changes with latitude, but the phase remains constant relative to the Moon's motion. The FFS increases with latitude difference but decreases as the clocks' longitudes approach 90 degrees.
- Pole Effect: At the lunar poles (latitude 90°), the FFS vanishes regardless of longitude differences.
Time Differences:
- Surface to Surface: The time difference between a clock at the highest point on the lunar surface and one at the lowest point, caused by LTR, is calculated to be $6.43148 \times 10^{-7}$ $\mu$ s/day.
- Surface to Satellite: As a satellite moves away from the Moon, the time difference between its clock and a surface clock increases (in absolute value) due to the diminishing influence of LTR on the satellite.
- Surface to Earth: When a clock on a lunar satellite is moved to an infinite distance (effectively comparing the lunar surface to the Earth's surface), the LTR-induced time difference converges to $-5.68 \times 10^{-7}$ $\mu$ s/day.

Significance
The paper asserts that the LTR is a non-negligible factor for next-generation high-precision timing applications near the Moon. Because the magnitude of the LTR-induced FFS ( $10^{-18}$ ) is comparable to the instability of modern atomic clocks, it must be accounted for in:

The construction of a consistent and high-precision lunar time standard.
The synchronization of clocks between multiple devices on the lunar surface and in lunar orbit.
Accurate relativistic corrections for future lunar missions (e.g., Chang'e, Blue Ghost) and the establishment of lunar relay communication networks.

The study concludes that ignoring LTR could introduce systematic errors in lunar timekeeping that exceed the capabilities of future clock networks, necessitating its inclusion in the relativistic framework for lunar time.

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