Frame Transformations¶

The astrojax.frames module provides functions for transforming satellite state vectors between common coordinate frames using the full IAU 2006/2000A CIO-based model and fixed inertial rotations.

Reference Frames¶

Key reference frames in astrodynamics:

Frame	Full Name	Rotation
GCRF (ECI)	Geocentric Celestial Reference Frame	Fixed with respect to the stars
ITRF (ECEF)	International Terrestrial Reference Frame	Rotates with the Earth
Ecliptic	Ecliptic Coordinate Frame	Fixed; equator = ecliptic plane

GCRF and ITRF share the same origin (Earth's centre of mass), but their axes differ due to Earth's rotation, precession, nutation, and polar motion. The ecliptic frame shares the GCRF origin but its fundamental plane is the ecliptic (Earth's orbital plane), tilted by ~23.44° from the GCRF equator.

The IAU 2006/2000A Model¶

The full transformation between GCRF and ITRF involves three components:

\[ R_{\text{GCRF} \to \text{ITRF}} = W(t) \; R(t) \; Q(t) \]

Component	Effect	Model
\(Q(t)\)	Bias-precession-nutation (GCRF -> CIRS)	IAU 2006/2000A
\(R(t)\)	Earth rotation (CIRS -> TIRS)	IAU 2000 ERA
\(W(t)\)	Polar motion (TIRS -> ITRF)	IERS conventions

All functions require an eop: EOPData parameter providing Earth Orientation Parameters (polar motion, UT1-UTC, celestial pole offsets). Use zero_eop() for a quick approximation or static_eop(...) for known EOP values.

Rotation Matrices¶

rotation_gcrf_to_itrf and rotation_itrf_to_gcrf compute the 3x3 direction-cosine matrix at a given epoch:

from astrojax import Epoch
from astrojax.eop import zero_eop
from astrojax.frames import rotation_gcrf_to_itrf, rotation_itrf_to_gcrf

eop = zero_eop()
epc = Epoch(2024, 1, 1, 12, 0, 0)

R_gcrf2itrf = rotation_gcrf_to_itrf(eop, epc)  # 3x3
R_itrf2gcrf = rotation_itrf_to_gcrf(eop, epc)  # transpose of the above

ECI/ECEF aliases (rotation_eci_to_ecef, rotation_ecef_to_eci) are provided for backward compatibility.

State Vector Transformations¶

For satellite state vectors \([\mathbf{r}, \mathbf{v}]\), position and velocity must be transformed differently. The velocity includes a correction for the rotating frame:

\[ \mathbf{v}_{\text{ITRF}} = W \left( R \, Q \, \mathbf{v}_{\text{GCRF}} - \boldsymbol{\omega} \times (R \, Q \, \mathbf{r}_{\text{GCRF}}) \right) \]

where \(\boldsymbol{\omega} = [0, 0, \omega_\oplus]^T\) is Earth's angular velocity vector (7.292x10^-5 rad/s).

import jax.numpy as jnp
from astrojax import Epoch
from astrojax.eop import zero_eop
from astrojax.constants import R_EARTH, GM_EARTH
from astrojax.frames import state_gcrf_to_itrf, state_itrf_to_gcrf

eop = zero_eop()
epc = Epoch(2024, 1, 1, 12, 0, 0)
sma = R_EARTH + 500e3
v_circ = jnp.sqrt(GM_EARTH / sma)
x_gcrf = jnp.array([sma, 0.0, 0.0, 0.0, v_circ, 0.0])

# Transform to ITRF
x_itrf = state_gcrf_to_itrf(eop, epc, x_gcrf)

# Transform back to GCRF (roundtrip)
x_back = state_itrf_to_gcrf(eop, epc, x_itrf)

Ecliptic-ICRF Transformations¶

The ecliptic coordinate frame is related to ICRF by a fixed rotation about the x-axis by the J2000 mean obliquity (\(\varepsilon \approx 23.44°\)). Since both frames are inertial, no epoch, EOP data, or Coriolis velocity correction is needed:

\[ \mathbf{r}_{\text{ICRF}} = R_x(-\varepsilon) \; \mathbf{r}_{\text{ecl}} \]

import jax.numpy as jnp
from astrojax.frames import (
    rotation_ecliptic_to_icrf,
    state_ecliptic_to_icrf,
    state_icrf_to_ecliptic,
)

# Rotation matrix (fixed, no epoch needed)
R = rotation_ecliptic_to_icrf()  # 3x3

# State vector transformation
x_ecl = jnp.array([7000e3, 0.0, 0.0, 0.0, 7.5e3, 0.0])
x_icrf = state_ecliptic_to_icrf(x_ecl)

# Roundtrip
x_back = state_icrf_to_ecliptic(x_icrf)

JAX Compatibility¶

All functions use JAX primitives and work with jax.jit and jax.vmap:

import jax

# JIT compilation
jit_transform = jax.jit(state_gcrf_to_itrf)
x_itrf = jit_transform(eop, epc, x_gcrf)

# Batch transformation over multiple states
states = jnp.stack([x_gcrf, x_gcrf + 100.0])
batched = jax.vmap(state_gcrf_to_itrf, in_axes=(None, None, 0))(eop, epc, states)