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Synodic Reference Frames

Brahe provides three synodic (two-body rotating) reference frames from NASA TP-202200148141: EMR (Earth-Moon Rotating), SER (Sun-Earth Rotating), and GSE (Geocentric Solar Ecliptic). Synodic frames rotate with the line between two primary bodies and are the natural frames for cislunar trajectory analysis, libration point missions, and comparing trajectories to circular restricted three-body problem (CR3BP) solutions.

All three frames share the same axis construction from the relative position \(\boldsymbol{r}_{12}\) and velocity \(\boldsymbol{v}_{12}\) of the secondary with respect to the primary:

\[ \hat{\boldsymbol{x}} = \frac{\boldsymbol{r}_{12}}{\|\boldsymbol{r}_{12}\|}, \qquad \hat{\boldsymbol{z}} = \frac{\boldsymbol{r}_{12} \times \boldsymbol{v}_{12}}{\|\boldsymbol{r}_{12} \times \boldsymbol{v}_{12}\|}, \qquad \hat{\boldsymbol{y}} = \hat{\boldsymbol{z}} \times \hat{\boldsymbol{x}} \]

The velocity transformation uses the exact time derivative of the rotation matrix (the GTDS/STK convention in TP-20220014814 §4.6.1), including the \(d\hat{\boldsymbol{z}}/dt\) term evaluated from the relative acceleration \(\boldsymbol{a}_{12}\), which Brahe computes by analytically differentiating the SPK ephemeris Chebyshev polynomials (see spk_acceleration).

EMR (Earth-Moon Rotating)

Primaries: Earth → Moon. Origin: the Earth-Moon barycenter (NAIF ID 3). The Moon lies permanently on the \(+\hat{\boldsymbol{x}}\) axis, the Earth on \(-\hat{\boldsymbol{x}}\). This is the standard frame for cislunar trajectory visualization and Earth-Moon libration point analysis.

SER (Sun-Earth Rotating)

Primaries: Sun → Earth. Origin: the Sun-Earth barycenter. The SEB has no NAIF ID or SPK ephemeris entry; Brahe computes it as the \(GM\)-weighted combination of the Sun and Earth SPK states and identifies it internally by the synthetic center ID SUN_EARTH_BARYCENTER_ID. The Earth lies on \(+\hat{\boldsymbol{x}}\), the Sun ~450 km from the origin on \(-\hat{\boldsymbol{x}}\).

GSE (Geocentric Solar Ecliptic)

Origin: Earth center. \(\hat{\boldsymbol{x}}\) points from the Earth to the Sun — the reversed sense relative to SER — and \(\hat{\boldsymbol{z}}\) is normal to the instantaneous ecliptic plane (~23.44° from the GCRF \(z\)-axis). GSE is common in space-weather and magnetospheric work. Because GSE is Earth-centered, converting between GCRF and GSE involves no translation.

Function Reference

Conversion Function
GCRF → EMR rotation_gcrf_to_emr, position_gcrf_to_emr, state_gcrf_to_emr
EMR → GCRF rotation_emr_to_gcrf, position_emr_to_gcrf, state_emr_to_gcrf
GCRF → SER rotation_gcrf_to_ser, position_gcrf_to_ser, state_gcrf_to_ser
SER → GCRF rotation_ser_to_gcrf, position_ser_to_gcrf, state_ser_to_gcrf
GCRF → GSE rotation_gcrf_to_gse, position_gcrf_to_gse, state_gcrf_to_gse
GSE → GCRF rotation_gse_to_gcrf, position_gse_to_gcrf, state_gse_to_gcrf

All three frames are also available through the frame router as ReferenceFrame.EMR, ReferenceFrame.SER, and ReferenceFrame.GSE, usable in rotation_frame_to_frame, position_frame_to_frame, state_frame_to_frame, and every provider's state_in_frame/states_in_frame. The de440s SPK kernel is auto-loaded on first use.

See Also

  1. Folta, D., Bosanac, N., Elliott, I., Mann, L., Mesarch, R., & Rosales, J. (2022). Astrodynamics Convention and Modeling Reference for Lunar, Cislunar, and Libration Point Orbits, NASA/TP-20220014814, §2.5 and §4.6. ↩