Force Models

Overview

The force models in SatMAD are used generally within the numerical propagation algorithms (see Numerical Propagation). Currently, the only force model is the two-body acceleration.

Two-Body Force Model

The two-body acceleration is given by:

\[ \ddot{\vec{r}} = - \dfrac{\mu}{r^3} \vec{r} \]

where \(\ddot{\vec{r}}\) is the inertial acceleration, \(\vec{r}\) is the position vector (with \(r\) its norm). \(\mu\) is equal to the constant \(GM\) (Gravitational Constant times the Mass of the main attracting body).

The equation holds regardless of whether the orbit is elliptic, parabolic or hyperbolic.

For an object orbiting the Earth, this equation should be solved with the coordinates in the GCRS frame. For an object orbiting the Sun, the coordinates should be in the Heliocentric inertial frame, with the \(\mu\) selected as that of the Sun.

Reference/API

Force models generally for use with the numerical propagators.

satmad.propagation.force_models.two_body_accel(r, mu)

Two-body acceleration.

The input vectors as well as the output acceleration is in in the inertial frame of the central body. For the Earth this is GCRS.

The acceleration value is given by:

\[\ddot{\vec{r}} = - \dfrac{\mu}{r^3} \vec{r}\]

where \(\ddot{\vec{r}}\) is the inertial acceleration, \(\vec{r}\) is the position vector (with \(r\) its norm). \(\mu\) is equal to the constant \(GM\) (Gravitational Constant times the Mass of the main attracting body.)

Note

This method is used in the Scipy ODE solver, therefore it does not accept Quantity inputs. Care must be taken with the units.

Parameters

  • r (ndarray) – inertial position vector with 3 elements (km)

  • mu (float) – GM value (\(km^3 / s^2\))

Returns

inertial acceleration (\(km / s^2\))

Return type

float

satmad.propagation.force_models.two_body_energy(r, v, mu)

Two-body specific energy.

The input vectors are in in the inertial frame of the central body. For the Earth this is GCRS.

The specific energy value is given by:

\[\varepsilon = \dfrac{v^2}{2} - \dfrac{\mu}{r}\]

with \(v\) is the norm of the velocity vector, \(r\) is the norm of the position vector and \(\mu\) is equal to the constant \(GM\) (Gravitational Constant times the Mass of the main attracting body).

This is the formula for the specific energy, or the sum of potential and kinetic energy per unit mass.

Parameters

  • r (ndarray or Quantity) – inertial position vector with 3 elements (km)

  • v (ndarray or Quantity) – inertial position vector with 3 elements (km/s)

  • mu (float or Quantity) – GM value (\(km^3 / s^2\))

Returns

sp_energy – specific energy (\(km^2 / s^2\))

Return type

float or Quantity