orbix.equations.orbit#

Common equations for orbital mechanics.

Attributes#

`two_pi_G`
`_TOL_E`
`_TOL_N`
`_ARC_EPS`

Functions#

`period_a`(a, mu)	Orbital period from semi-major axis and standard gravitational parameter.
`period_n`(n)	Orbital period from mean motion.
`mean_motion`(a, mu)	Mean motion from semi-major axis and standard gravitational parameter.
`period_to_sma`(T, Ms)	Semi-major axis from orbital period via Kepler's third law.
`semi_amplitude`(T, Ms, Mp, e, i)	Semi-amplitude of the radial velocity curve from base quantities.
`semi_amplitude_reduced`(T, Ms, minimum_mass, ...)	Semi-amplitude of the radial velocity curve from pre-calculated quantities.
`mean_anomaly_t0`(t, n, M0, t0)	Mean anomaly at time t (can be vector) from epoch.
`mean_anomaly_tp`(t, n, tp)	Mean anomaly at time t (can be vector) from periapsis passage.
`AB_matrices`(a, e, i, W, w)	Compute the A and B matrices for a given set of orbital elements.
`AB_matrices_reduced`(a, sqrt_one_minus_e2, sini, cosi, ...)	Compute the A and B matrices from the trig values of the orbital elements.
`thiele_innes_constants`(W, i, w)	Compute the Thiele-Innes constants from the orbital angles.
`thiele_innes_constants_reduced`(sinW, cosW, sinw, cosw, ...)	Compute the Thiele-Innes constants from the orbital angles.
`_safe_norm`(x)	Vector norm with a finite (zero) gradient at `x == 0`.
`state_vector_to_keplerian`(r, v, mu)	Convert state vectors (r, v) to Keplerian elements using JAX.

Module Contents#

orbix.equations.orbit.two_pi_G#

orbix.equations.orbit.period_a(a, mu)[source]#

Orbital period from semi-major axis and standard gravitational parameter.

Parameters:

a – Array Semi-major axis
mu – Array Standard gravitational parameter

Returns:

Array: Orbital period

Return type:

orbix.equations.orbit.period_n(n)[source]#

Orbital period from mean motion.

Parameters:

n – Array Mean motion

Returns:

Array: Orbital period

Return type:

orbix.equations.orbit.mean_motion(a, mu)[source]#

Mean motion from semi-major axis and standard gravitational parameter.

Parameters:

a – Array Semi-major axis
mu – Array Standard gravitational parameter

Returns:

Array: Mean motion

Return type:

orbix.equations.orbit.period_to_sma(T, Ms)[source]#

Semi-major axis from orbital period via Kepler’s third law.

Parameters:

T – Orbital period (days). Scalar or array.
Ms – Stellar mass (kg). Scalar or array.

Returns:

Semi-major axis (AU). Scalar or array.

Return type:

orbix.equations.orbit.semi_amplitude(T, Ms, Mp, e, i)[source]#

Semi-amplitude of the radial velocity curve from base quantities.

Parameters:

T – Array Orbital period
Ms – Array Mass of the star
Mp – Array Mass of the planet
e – Array Eccentricity
i – Array Inclination

Returns:

Array: Semi-amplitude of the radial velocity curve

Return type:

orbix.equations.orbit.semi_amplitude_reduced(T, Ms, minimum_mass, sqrt_one_minus_e2)[source]#

Semi-amplitude of the radial velocity curve from pre-calculated quantities.

Parameters:

T – Array Orbital period
Ms – Array Mass of the star
minimum_mass – Array Mass of the planet multiplied by sin(i)
sqrt_one_minus_e2 – Array Square root of (1 - eccentricity^2)

Returns:

Array: Semi-amplitude of the radial velocity curve

Return type:

orbix.equations.orbit.mean_anomaly_t0(t, n, M0, t0)[source]#

Mean anomaly at time t (can be vector) from epoch.

Requires that all units are consistent and does NOT clip the mean anomaly to the range [0, 2pi).

Parameters:

t – Array Time
n – Array Mean motion
M0 – Array Mean anomaly at epoch
t0 – Array Epoch

Returns:

Array: Mean anomaly at time t

Return type:

orbix.equations.orbit.mean_anomaly_tp(t, n, tp)[source]#

Mean anomaly at time t (can be vector) from periapsis passage.

Parameters:

t – Array Time
n – Array Mean motion
tp – Array Time of periapsis passage

Returns:

Array: Mean anomaly at time t

Return type:

orbix.equations.orbit.AB_matrices(a, e, i, W, w)[source]#

Compute the A and B matrices for a given set of orbital elements.

In keplertools Dmitry defines these as: “inertial frame components of perifocal frame unit vectors scaled by orbit semi-major and semi-minor axes.” and I wouldn’t dare disagree with him on this.

Parameters:

a – Array Semi-major axis
e – Array Eccentricity
i – Array Inclination
W – Array Longitude of the ascending node
w – Array Argument of periapsis

Returns:

jnp.ndarray: A matrix
B: jnp.ndarray: B matrix

Return type:

orbix.equations.orbit.AB_matrices_reduced(a, sqrt_one_minus_e2, sini, cosi, sinW, cosW, sinw, cosw)[source]#

Compute the A and B matrices from the trig values of the orbital elements.

Parameters:

a – Semi-major axis
sqrt_one_minus_e2 – Square root of (1 - eccentricity^2)
sini – Sine of the inclination
cosi – Cosine of the inclination
sinW – Sine of the longitude of the ascending node
cosW – Cosine of the longitude of the ascending node
sinw – Sine of the argument of periapsis
cosw – Cosine of the argument of periapsis

Returns:

jnp.ndarray: A matrix
B: jnp.ndarray: B matrix

Return type:

orbix.equations.orbit.thiele_innes_constants(W, i, w)[source]#

Compute the Thiele-Innes constants from the orbital angles.

Parameters:

W – Longitude of the ascending node
i – Inclination
w – Argument of periapsis

Returns:

A constant B: B constant F: F constant G: G constant

Return type:

orbix.equations.orbit.thiele_innes_constants_reduced(sinW, cosW, sinw, cosw, sinwcosi, coswcosi)[source]#

Compute the Thiele-Innes constants from the orbital angles.

Parameters:

sinW – Sine of the longitude of the ascending node
cosW – Cosine of the longitude of the ascending node
sinw – Sine of the argument of periapsis
cosw – Cosine of the argument of periapsis
sinwcosi – Sine of the argument of periapsis times cosine of the inclination
coswcosi – Cosine of the argument of periapsis times cosine of the inclination

Returns:

A constant B: B constant F: F constant G: G constant

Return type:

orbix.equations.orbit._TOL_E = 1e-09#

orbix.equations.orbit._TOL_N = 1e-09#

orbix.equations.orbit._ARC_EPS = 1e-07#

orbix.equations.orbit._safe_norm(x)[source]#

Vector norm with a finite (zero) gradient at x == 0.

jnp.linalg.norm has a NaN gradient at exactly the zero vector (x / norm(x) is 0 / 0 there). This computes the same value but evaluates the norm on a nonzero stand-in whenever x is zero, so the local derivative is finite; the jnp.where then zeroes the corresponding cotangent, giving a well-defined (zero) gradient.

orbix.equations.orbit.state_vector_to_keplerian(r, v, mu)[source]#

Convert state vectors (r, v) to Keplerian elements using JAX.

Robust implementation handling edge cases (circular, equatorial, and non-bound orbits) using jnp.where for JIT compatibility.

Unit-agnostic: r, v, and mu must be expressed in one consistent unit system (e.g. meters / m/s / m^3 s^-2, or the AU / day units used elsewhere in this library); the function does not enforce or convert any particular convention, and a is returned in the same length unit as r.

Parameters:

r – Stellar-centric position vector (3,).
v – Stellar-centric velocity vector (3,).
mu – Gravitational parameter G * M_total.

Returns:

(a, e, i, W, w, M) – semi-major axis (same length: unit as r), eccentricity, inclination [rad], longitude of ascending node [rad], argument of periapsis [rad], mean anomaly [rad].

Return type:

tuple