Time Aligned Orbit

• Time aligned orbit is a specially designed orbital configuration where the onboard clock’s proper time matches the lunar geoid or coordinate time, leveraging general relativistic corrections.
• The approach requires a nearly circular, low-eccentricity orbit to minimize periodic time offsets, achieving sub-nanosecond synchronization over extended intervals.
• This configuration enables simultaneous realization of multiple timescales for enhanced lunar positioning, navigation, and timing, offering an efficient alternative to traditional space-time reference systems.

A time-aligned orbit is a specifically engineered orbital configuration (notably around the Moon, but generalizable to other bodies) in which the proper time accumulated by an onboard clock matches—over long intervals and to high precision—that of a designated reference clock, commonly located on the geoid (“selenoid” for the Moon), or aligns with a body-fixed coordinate time (such as Lunar Coordinate Time, TCL). The concept enables the simultaneous realization and transfer of both coordinate timescales and geoid-based proper times from orbit, with implications for lunar positioning, navigation, and timing (PNT) infrastructure, clock networks, and time transfer science (Seyffert, 10 Sep 2025, Yang et al., 28 Dec 2025).

1. Theoretical Foundations of Time-Aligned Orbits

Time-aligned orbits solve for orbital elements such that a satellite’s proper time matches (to within a desired tolerance) a natural timescale, such as the proper time on a lunar geoid or TCL. The problem is rooted in general relativistic time transformations:

• For the Moon, the Lunar Celestial Reference System (LCRS) defines coordinate time TCL, with the transformation between proper time $\tau$ (for a clock with lunar-centric position $\mathbf{Y}$ and velocity $\dot{\mathbf{Y}}$) and TCL (Yang et al., 28 Dec 2025):

$$\mathrm{TCL} - \mathrm{TCL}_0 = \tau - \tau_0 + \frac{1}{c^2} \int_{\tau_0}^{\tau} \left[ U_M(\mathbf{Y}) + \frac{1}{2} \dot{\mathbf{Y}}^2 \right] d\tau + \mathcal{O}(c^{-4})$$

• For an ideal selenoid (lunar geoid), the proper time $\tau_s$ is linearly related to TCL with

$$\mathrm{TCL} = (1 + L)(\tau_s - \tau_{s0}) + \mathrm{TCL}_{s0} + \mathcal{O}(c^{-4}),$$

where $L = W_{M0}/c^2$ with $W_{M0}$ the Newtonian potential at the selenoid (Yang et al., 28 Dec 2025).

To achieve time alignment, one determines an orbital configuration such that the secular drift between the onboard clock’s proper time and the geoid proper time is nulled (or made arbitrarily small), accounting for both gravitational redshift and kinematic time dilation (Seyffert, 10 Sep 2025, Yang et al., 28 Dec 2025).

2. Analytical Derivation and Orbital Criteria

For a clock in a lunar-circular orbit (semi-major axis $\bar{a}_p$, inclination $\bar{i}_p$), the proper time is given by

$$\mathrm{TCL} = (1 + L_P) (\tau_p - \tau_{p0}) + \mathrm{TCL}_{p0} + \mathcal{O}(c^{-4}),$$

where

$$L_P = \frac{3}{2} \frac{GM_M}{c^2 \bar{a}_p} \left[ 1 + \frac{7}{3} J_2^M \left( \frac{R_M}{\bar{a}_p} \right)^2 (1 - \frac{3}{2} \sin^2 \bar{i}_p) \right].$$

Setting $L_P = L$ ensures the same ticking rate as the selenoid. This leads to the time-aligned semi-major axis:

$$\bar{a}_p = \frac{3GM_M}{2c^2 L} \left[ 1 + \frac{28}{27} J_2^M L^2 \left( \frac{GM_M}{c^2 R_M} \right)^{-2} (1 - \frac{3}{2} \sin^2 \bar{i}_p) \right]$$

which numerically for the Moon yields $\bar{a}_p \approx 1.5 R_M \approx 2606$ km, weakly dependent on inclination (Yang et al., 28 Dec 2025).

A key criterion is that circularity ($e=0$) nulls first-order periodic terms in the time offset, so eccentricity must be minimized to achieve sub-nanosecond alignment over operational intervals (Seyffert, 10 Sep 2025).

3. Relativistic Corrections and Perturbation Modeling

High-fidelity realization of time-aligned orbits requires modeling higher-order perturbations:

• Lunar gravity field harmonics (up to degree/order $N$), Sun/planet third-body effects, solar radiation pressure.
• Secular and periodic clock offsets are computed via post-Newtonian expansions, with the central term for proper time drift in lunar orbit:

$$\dot{\Delta\tau}_{sec} = -\frac{GM}{c^2} \left( \frac{1}{r_0} - \frac{3}{2a} \right)$$

where $r_0$ is selenoid radius (Seyffert, 10 Sep 2025).

Numerical simulations (including perturbations) for orbits at $a \sim 1.5\,R_M$, $e=0$, and various inclinations yield maximal proper time desynchronization from the selenoid within 190 ns after a year, with residual frequency offsets $\leq 6 \times 10^{-15}$. After correcting for mean element deviations, desynchronization is further reduced ($\leq 13$ ns, $4 \times 10^{-16}$ fractional) (Yang et al., 28 Dec 2025). For comparison, the intrinsic frequency spread from lunar topography is $1.6 \times 10^{-13}$.

4. Simultaneous Realization of Multiple Timescales

A salient feature of the time-aligned orbit is the ability to realize both Lunar Coordinate Time (O1) and lunar geoid proper time (O2) with a single orbital clock. The transformation is an exact linear relation:

$$t_{O1} = \alpha t_{O2} + \beta, \quad \alpha = 1 + L$$

so the clock in a time-aligned orbit provides a reference for both coordinate and geoid-based temporal frameworks (Yang et al., 28 Dec 2025). This approach generalizes to other terrestrial planets by replacing $M_M$, $R_M$, and $J_2^M$ with the pertinent planetary parameters.

Practical deployment enables one orbital asset to serve both lunar navigation and science (TCL realization), as well as synchronization for users on the selenoid with only a scaling adjustment.

5. Comparison to Other Temporal Reference Concepts

Time-aligned orbits differ from classical space-time reference (STR) satellites, which link onboard clocks to ground-based atomic references via two-way lasercom, steering their clocks to global standards (UTC, TAI) but not necessarily realizing geoid or coordinate timescales intrinsically via orbital design (Berceau et al., 2014, Berceau et al., 2015). STR systems achieve picosecond-level time alignment via high-stability atomic clocks, femtosecond optical combs, precision ground-space laser time transfer, and mm-level orbit determination.

In contrast, TAO design leverages orbital dynamics—gravity and velocity profiles—so that the onboard clock naturally traces the chosen timescale, minimizing the need for active frequency steering relative to the local environment (Seyffert, 10 Sep 2025, Yang et al., 28 Dec 2025).

6. Applications, Limitations, and Generalization

Time-aligned orbits are foundational for next-generation lunar (and planetary) PNT architectures. Their ability to maintain sub-nanosecond synchronization between orbital and surface clocks supports interoperable timescales, accurate ranging, and scientific observation (e.g., tests of fundamental physics, geodesy) (Seyffert, 10 Sep 2025, Yang et al., 28 Dec 2025).

Limitations include sensitivity to orbit determination error, noncentral gravity field, and residual perturbation-induced drift. Eccentricity and mean element maintenance ($e \lesssim 10^{-3}$, $\Delta a \lesssim 1$ km) are required to sustain sub-ns alignment. Extensions to Venus, Mars, and Mercury are possible, with the time-aligned semi-major axis scaling as $a \approx 1.5 R$ for each (Yang et al., 28 Dec 2025).

Body	Radius $R$ (km)	TAO Semi-Major Axis $a$ (km)
Mercury	2440	3660
Venus	6052	9080
Earth	6371	9560
Mars	3390	5085

7. Context Within Broader Astrophysical Usage

The term “aligned orbit” also arises in exoplanetary studies, where it refers to the coplanarity (small obliquity) between a planet’s orbital plane and the host star’s rotation—measured, for example, via transit light curves, gravity-darkening, and spectroscopic techniques (Singh et al., 2023). This meaning is distinct from the time-aligned orbit described above, but both contexts exploit systematic alignment—either spatial or temporal—for precise characterization and synchronization within astrophysical systems. In exoplanet parlance, alignment is typically quantified via sky-projected and true obliquity parameters.

• (Berceau et al., 2014) Laser time-transfer and space-time reference in orbit
• (Berceau et al., 2015) Space-Time Reference with an Optical Link
• (Singh et al., 2023) CHEOPS observations of KELT-20 b/MASCARA-2 b: An aligned orbit
• (Seyffert, 10 Sep 2025) Relativistic Time Modeling for Lunar Positioning Navigation and Timing
• (Yang et al., 28 Dec 2025) Two birds with one stone: simultaneous realization of both Lunar Coordinate Time and lunar geoid time by a single orbital clock