latom.utils.keplerian_orbit¶

@authors: Alberto FOSSA’ Giuliana Elena MICELI

Functions

`coe2sv_vec`(a, e, i, raan, w, ta, gm)	Change of Coordinates from Classical Orbital Elements to State Vector in body-centred inertial reference frame.
`per2eq`(raan, i, w)	Rotation matrix from perifocal reference frame to inertial, body-centred equatorial reference frame.
`root`(fun, x0[, args, method, jac, tol, …])	Find a root of a vector function.

Classes

`KepOrb`(a, e, i, raan, w, ta, gm)	KepOrb defines a Keplerian Orbit.
`TwoDimOrb`(gm, **kwargs)	Defines a two-dimensional orbit from its keplerian parameters.

class latom.utils.keplerian_orbit.TwoDimOrb(gm, **kwargs)[source]¶

Bases: object

Defines a two-dimensional orbit from its keplerian parameters.

Parameters

gm (float) – Central body standard gravitational parameter [m^3/s^2]

Other Parameters

T (float) – Orbital period [s]
a (float) – Semi-major axis [m]
e (float) – Eccentricity [-]
ra (float) – Apoapsis radius [m]
rp (float) – Periapsis radius [m]

Variables

GM (float, optional) – Central body standard gravitational parameter [m^3/s^2]
T (float) – Orbit period [s]
a (float) – Semi-major axis [m]
e (float) – Eccentricity [-]
ra (float) – Apoapsis radius [m]
rp (float) – Periapsis radius [m]
va (float) – Velocity at the apoapsis [m/s]
vp (float) – Velocity at the periapsis [m/s]
h (float) – Specific angular momentum magnitude [m^2/s]
energy (float) – Specific energy of the orbit in 2BP [m^2/s^2]

static coe2polar(gm, ta, **kwargs)[source]¶

Transforms the classical orbital elements into polar coordinates.

Parameters

gm (float) – Central body standard gravitational parameter [m^3/s^2]
ta (float or ndarray) – True anomaly [rad]

Other Parameters

a (float) – Semi-major axis [m]
e (float) – Eccentricity [-]
h (float) – Specific angular momentum magnitude [m^2/s]

Returns

r (float or ndarray) – Position in polar coordinates [m]
u (float or ndarray) – radial velocity [m/s]
v (float or ndarray) – tangential velocity [m/s]

static polar2coe(gm, r, u, v)[source]¶

Transforms the polar coordinates into classical orbital elements.

Parameters

gm (float) – Central body standard gravitational parameter [m^3/s^2]
r (float or ndarray) – Position in polar coordinates [m]
u (float or ndarray) – radial velocity [m/s]
v (float or ndarray) – tangential velocity [m/s]

Returns

a (float) – Orbit semi-major axis [m]
e (float) – Orbit eccentricity [-]
h (float) – Orbit specific angular momentum magnitude [m^2/s]
ta (float or ndarray) – True anomaly [rad]

static polar2energy(gm, r, u, v)[source]¶

Returns the energy given the polar coordinates.

Parameters

gm (float) – Central body standard gravitational parameter [m^3/s^2]
r (float or ndarray) – Position in polar coordinates [m]
u (float or ndarray) – radial velocity [m/s]
v (float or ndarray) – tangential velocity [m/s]

Returns

(u**2 + v**2)*0.5 - gm/r – Specific energy of the orbit in 2BP [m^2/s^2]

Return type

float or ndarray

static propagate(gm, a, e, tai, taf, nb, tp=0.0)[source]¶

Propagates the orbit states in polar coordinates for the orbital period duration.

Parameters

gm (float) – Central body standard gravitational parameter [m^3/s^2]
a (float) – Orbit semi-major axis [m]
e (float) – Orbit eccentricity [-]
tai (float) – True anomaly initial angle [rad]
taf (float) – True anomaly final angle [rad]
nb (int) – number of point where the True Anomaly is calculated [-]
tp (float) – Time at periapsis passage [s]

Returns

t (ndarray) – Time vector [s]
states (ndarray) – List of the states values propagated in time [m, m/s, m/s]

class latom.utils.keplerian_orbit.KepOrb(a, e, i, raan, w, ta, gm)[source]¶

Bases: object

KepOrb defines a Keplerian Orbit.

Parameters

a (float) – Semi-major axis [m]
e (float) – Eccentricity [-]
i (float) – Inclination [rad]
raan (float) – Right Ascension of the Ascending Node [rad]
w (float) – Argument of Periapsis [rad]
ta (float) – True anomaly [rad]
gm (float) – Central body standard gravitational parameter [m^3/s^2]

Variables

eps (float) – Smallest number such that 1.0 + eps != 1.0
a (float) – Semi-major axis [m]
e (float) – Eccentricity [-]
i (float) – Inclination [rad]
W (float) – Right Ascension of the Ascending Node [rad]
w (float) – Argument of Periapsis [rad]
ta (float) – True anomaly [rad]
GM (float, optional) – Central body standard gravitational parameter [m^3/s^2]
n (float) – Mean motion [rad/s]
R (ndarray) – Position vector [m]
V (ndarray) – Velocity vector [m/s]
H (ndarray) – Specific angular momentum vector [m^2/s]
E (ndarray) – Eccentricity vector [-]
h (float) – Specific angular momentum magnitude [m^2/s]
r (float) – Position vector magnitude [m]
vr (float) – Radial velocity [m/s]

set_state_vector(r, v)[source]¶

Set the spacecraft state vector and compute the corresponding COE, H, E.

Parameters

r (ndarray) – Position vector [m]
v (ndarray) – Velocity vector [m/s]

set_classical_orbital_elements(a, e, i, raan, w, ta)[source]¶

Set the spacecraft COE and compute the corresponding state vector, H, E.

Parameters

a (float) – Semi-major axis [km]
e (float) – Eccentricity [-]
i (float) – Inclination [rad]
raan (float) – Right Ascension of the Ascending Node [rad]
w (float) – Argument of Periapsis [rad]
ta (float) – True anomaly [rad]

set_true_anomaly(ta)[source]¶

Set the spacecraft true anomaly and update the corresponding state vector.

Parameters: ta (float) – True anomaly [rad]

compute_mean_motion()[source]¶: Computes the spacecraft mean motion.

compute_angular_momentum()[source]¶: Computes the specific angular momentum vector.

compute_eccentricity()[source]¶: Compute the spacecraft eccentricity vector.

compute_classical_orbital_elements()[source]¶: Computes the spacecraft classical orbital elements, specific angular momentum vector and eccentricity vector from its state vector.

compute_state_vector()[source]¶: Computes the spacecraft state vector, specific angular momentum vector and eccentricity vector from its COE.

compute_eccentric_anomaly(ta)[source]¶

Compute the eccentric anomaly from a given true anomaly.

Parameters: ta (float) – True anomaly [rad]
Returns: ea – Eccentric anomaly [rad]
Return type: float

compute_true_anomaly(ea)[source]¶

Compute the true anomaly from a given eccentric anomaly.

Parameters: ea (ndarray or float) – Eccentric anomaly [-]
Returns: ta – True anomaly [rad]
Return type: ndarray or float

compute_periapsis_passage(ta, t)[source]¶

Compute the time at periapsis passage given the current time and true anomaly.

Parameters

t (float) – Time [s]
ta (float) – True anomaly [rad]

Returns

tp – Time at periapsis passage [s]

Return type

float

propagate(ta, t_vec, mode)[source]¶

Propagate the orbit forward or backward in time solving the Kepler’s time of flight equation.

Parameters

ta (float) – Initial true anomaly [rad]
t_vec (ndarray) – Time vector [s]
mode (str) – fwd for forward propagation or back for backward propagation

Returns

r_vec (ndarray) – Position vector time series [m]
v_vec (ndarray) – Velocity vector time series [m/s]

static kepler_eqn(ea, e, n, t, tp)[source]¶

Kepler’s Time of Flight equation.

Parameters

ea (float) – Eccentric anomaly [rad]
e (float) – Eccentricity [-]
n (float) – Mean motion [rad/s]
t (float) – Time [s]
tp (float) – Time at periapsis passage [s]

Returns

f – Current value of Kepler’s equation written as E - e*sin(E) - Me = 0

Return type

float

static kepler_eqn_prime(ea, e, n, t, tp)[source]¶

First derivative of Kepler’s time of flight equation wrt eccentric anomaly.

Parameters

ea (float) – Eccentric anomaly [rad]
e (float) – Eccentricity [-]
n (float) – Mean motion [rad/s]
t (float) – Time [s]
tp (float) – Time at periapsis passage [s]

Returns

fprime – First derivative of Kepler’s equation wrt eccentric anomaly

Return type

float

static kepler_eqn_second(ea, e, n, t, tp)[source]¶

Second derivative of Kepler’s time of flight equation wrt eccentric anomaly.

Parameters

ea (float) – Eccentric anomaly [rad]
e (float) – Eccentricity [-]
n (float) – Mean motion [rad/s]
t (float) – Time [s]
tp (float) – Time at periapsis passage [s]

Returns

fsecond – Second derivative of Kepler’s equation wrt eccentric anomaly

Return type

float