Synodic Period (and Synodic Frequency)

Author: Tianjiang Shuo

Website: https://cislunarspace.cn

Definition

The synodic period $S$ of two bodies is the time after which they return to the same relative geometry as seen from a third point (usually the primary). If their sidereal orbital periods are $T_1$ and $T_2$, then

The reciprocal $\omega_\text{syn} = 2\pi/S$ is the synodic frequency (angular). Synodic period and sidereal period are easily confused: sidereal period is measured against inertial space (the fixed stars); synodic period is measured against the moving partner.

The lunar synodic period

The lunar sidereal period (one revolution against the stars) is 27.3217 d. The Earth has itself moved $\sim 27°$ along its orbit in that time, so the Moon must travel another $\sim 27°$ to return to the same Sun–Earth–Moon angle. The result is the lunar synodic period, i.e. the cycle of lunar phases (one synodic month):

This is the natural clock for any phenomenon driven by the Sun–Earth–Moon angle: tides, lunar-phase-dependent lighting, solar-exclusion windows in sensor coverage, and the design of resonant distant retrograde orbits. For example, a 2:1 resonant DRO has a period of $S_\text{Moon}/2 \approx 14.77$ d (Welch et al. 2015). Cislunar situational-awareness simulations typically span an integer number of synodic periods so that solar-exclusion statistics are representative (Vendl & Holzinger 2021).

The solar synodic frequency in the Earth–Moon frame

When the Earth–Moon barycentric rotating frame is used (with angular rate $n_\text{EM} = 2\pi/S_\text{Moon}$), the Sun does not stand still: it completes one revolution in that frame every Earth sidereal year, which translated into the rotating frame becomes one cycle per synodic period of the Sun as seen from the Earth–Moon binary — about 29.5 d × correction. In the bicircular and quasi-bicircular models of the Earth–Moon–Sun system, this solar synodic frequency $\omega_\odot$ enters as the leading external forcing frequency, modulating otherwise-autonomous Earth–Moon dynamics and producing the quasi-periodic perturbation that governs long-term stability near $L_4/L_5$ and along ballistic-capture trajectories (Gómez et al. 2001, vol. II).

Inter-satellite phasing

For coplanar circular rendezvous, the same formula gives the synodic period of an interceptor and a target: $S_\text{ph} = 2\pi/(\omega_\text{int} - \omega_\text{tgt})$. The wait time until the next phasing opportunity is

where $\vartheta$ is the required phase-angle separation and $k$ counts the revolutions (Vallado 2022, §6.5). Two satellites in similar orbits have a very long synodic period; two in widely separated orbits phase quickly — the counter-intuitive rule that drives launch-window design.

Related entries

• Circular Restricted Three-Body Problem (CR3BP)

• Synodic Frame (Rotating Frame)

• Jacobi Constant

References

• Vallado, 2022, Fundamentals of Astrodynamics and Applications, §6.5 and §3.4 — synodic period in inter-satellite phasing; barycentric time scales.

• Szebehely, 1967, Theory of Orbits, §1.5 — dimensionless synodic system and the role of the angular rate $n$.

• Gómez, Jorba, Llibre, Masdemont, Simó, 2001, Dynamics and Mission Design near Libration Points, vol. II — solar synodic perturbation in the bicircular model.

• Welch, Barden, Howell, 2015, Mission Considerations for Transfers to a Distant Retrograde Orbit — 2:1 resonant DRO at half the lunar synodic period.

• Vendl & Holzinger, 2021, "Cislunar periodic orbit analysis for persistent space object detection capability" — synodic-period cadence for sensor coverage.

• Thornton et al., 2022 — simulation baseline spanning full synodic cycles.