astrodynx.generalized_anomaly

astrodynx.generalized_anomaly(alpha, sigma, sigma0, deltat=0, mu=1)

Returns the generalized anomaly.

Parameters:

• alpha (ArrayLike) – The reciprocal of the semimajor axis.

• sigma (ArrayLike) – The sigma function at the current time.

• sigma0 (ArrayLike) – The sigma function at the initial time.

• deltat (ArrayLike) – (optional) The time since the initial time.

• mu (ArrayLike) – (optional) The gravitational parameter.

Return type:

Array

Returns:

The generalized anomaly.

Notes

The generalized anomaly is defined as:

\[ \chi = \alpha \sqrt{\mu} \Delta t + \sigma - \sigma_0 \]

where \(\chi\) is the generalized anomaly, \(\alpha = \frac{1}{a}\) is the reciprocal of semimajor axis, \(\sigma\) is the sigma function at the current time, \(\sigma_0\) is the sigma function at the initial time, \(\mu\) is the gravitational parameter, and \(\Delta t\) is the time since the initial time.

References

Battin, 1999, pp.179.

Examples

A simple example:

>>> import jax.numpy as jnp
>>> import astrodynx as adx
>>> alpha = 1.0
>>> sigma = 1.0
>>> sigma0 = 0.0
>>> mu = 1.0
>>> deltat = 1.0
>>> adx.generalized_anomaly(alpha, sigma, sigma0, deltat, mu)
Array(2., dtype=float32, weak_type=True)

With broadcasting:

>>> alpha = jnp.array([1.0, 1.0])
>>> sigma = jnp.array([1.0, 2.0])
>>> sigma0 = jnp.array([0.0, 0.0])
>>> mu = jnp.array([1.0, 1.0])
>>> deltat = jnp.array([1.0, 1.0])
>>> adx.generalized_anomaly(alpha, sigma, sigma0, deltat, mu)
Array([2., 3.], dtype=float32)