Anomalies Conversions

In pykep we adopt the following naming for the various anomalies: \(M\) is the Mean Anomaly, \(E\) is the Eccentric Anomaly, \(L\) is the True Longitude, \(\lambda\) is the Mean Longitude, \(H\) is the Hyperbolic Anomaly, \(N\) is the Mean Hyperbolic Anomaly, \(\zeta\) is the Gudermannian, in functions or variable names these symbols can be spelled out (like zeta for \(\zeta\)) or made lowercase like m for \(M\).

Below we list various conversion functions that allow to convert from one anomaly to another, and their vectorized versions.

Normal

pykep.m2e(M, ecc)

Converts from Mean to Eccentric anomaly. Requires ecc < 1.

Args:

M (float): the Mean anomaly (rad.)

ecc (float): the eccentricity

Returns:

float: the Eccentric anomaly in [-pi, pi] (rad.)

Examples:

>>> import pykep as pk
>>> M = 1.2
>>> ecc = 0.1
>>> pk.m2e(M, ecc)
1.296254963787226

pykep.e2m(E, ecc)

Converts from Eccentric to Mean anomaly. Requires ecc < 1.

Args:

E (float): the Eccentric anomaly (rad.)

ecc (float): the eccentricity

Returns:

float: the Mean anomaly (rad.)

Examples:

>>> import pykep as pk
>>> E = 0.5
>>> ecc = 0.1
>>> pk.e2m(E, ecc)
0.4520574461395797

pykep.m2f(M, ecc)

Converts from Mean to True anomaly. Requires ecc < 1.

Args:

M (float): the Mean anomaly (rad.)

ecc (float): the eccentricity

Returns:

float: the True anomaly in [-pi, pi] (rad.)

Examples:

>>> import pykep as pk
>>> M = 0.32
>>> ecc = 0.65
>>> pk.m2f(M, ecc)
1.4497431281728277

pykep.f2m(f, ecc)

Converts from True to Mean anomaly. Requires ecc < 1.

Args:

f (float): the True anomaly (rad.)

ecc (float): the eccentricity

Returns:

float: the Mean anomaly in [-pi, pi] (rad.)

Examples:

>>> import pykep as pk
>>> f = -0.34
>>> ecc = 0.67
>>> pk.f2m(f, ecc)
-0.05065883735669101

pykep.e2f(E, ecc)

Converts from eccentric to true anomaly. Requires ecc < 1.

Args:

E (float): the Eccentric anomaly (rad.)

ecc (float): the eccentricity

Returns:

float: the True anomaly in [-pi, pi] (rad.)

Examples:

>>> import pykep as pk
>>> E = 0.5
>>> ecc = 0.1
>>> pk.e2f(E, ecc)
0.5502639747136633

pykep.f2e(f, ecc)

Converts from True to Eccentric anomaly. Requires ecc < 1.

Args:

f (float): the True anomaly (rad.)

ecc (float): the eccentricity

Returns:

float: the Eccentric anomaly in [-pi, pi] (rad.)

Examples:

>>> import pykep as pk
>>> f = 1.2
>>> ecc = 0.1
>>> pk.f2e(f, ecc)
1.1082931139529482

pykep.n2h(N, ecc)

Converts from Hyperbolic Mean to Hyperbolic anomaly. Requires ecc > 1.

Args:

N (float): the Hyperbolic Mean anomaly (rad.)

ecc (float): the eccentricity

Returns:

float: the Hyperbolic anomaly (rad.)

Examples:

>>> import pykep as pk
>>> N = 1.2
>>> ecc = 10.32
>>> pk.n2h(N, ecc)
0.12836469743916526

pykep.h2n(H, ecc)

Converts from Hyperbolic to Hyperbolic Mean anomaly. Requires ecc > 1.

Args:

H (float): the Hyperbolic anomaly (rad.)

ecc (float): the eccentricity

Returns:

float: the Hyperbolic Mean anomaly (rad.)

Examples:

>>> import pykep as pk
>>> H = 1.2
>>> ecc = 10.32
>>> pk.h2n(H, ecc)
14.377641187853621

pykep.n2f(N, ecc)

Converts from Hyperbolic Mean to True anomaly. Requires ecc > 1.

Args:

N (float): the Hyperbolic Mean anomaly (rad.)

ecc (float): the eccentricity

Returns:

float: the True anomaly

Examples:

>>> import pykep as pk
>>> N = 10.32
>>> ecc = 13.45
>>> pk.n2f(N, ecc)
0.7373697968359353

pykep.f2n(f, ecc)

Converts from True to Hyperbolic Mean anomaly. Requires ecc > 1.

Args:

f (float): the True anomaly (rad.)

ecc (float): the eccentricity

Returns:

float: the Hyperbolic Mean anomaly

Examples:

>>> import pykep as pk
>>> f = 1.2
>>> ecc = 5.7
>>> pk.f2n(f, ecc)
8.421335633880908

pykep.h2f(H, ecc)

Converts from Hyperbolic to True anomaly. Requires ecc > 1.

Args:

H (float): the Hyperbolic anomaly (rad.)

ecc (float): the eccentricity

Returns:

float: the True anomaly in [-pi, pi] (rad.)

Examples:

>>> import pykep as pk
>>> H = 10.32
>>> ecc = 4.5
>>> pk.h2f(H, ecc)
1.7948251330114304

pykep.f2h(f, ecc)

Converts from True to Hyperbolic anomaly. Requires ecc > 1.

Args:

f (float): the True anomaly (rad.)

ecc (float): the eccentricity

Returns:

float: the Hyperbolic anomaly

Examples:

>>> import pykep as pk
>>> f = 1.2
>>> ecc = 1.1
>>> pk.f2h(f, ecc)
0.30083016696826936

pykep.zeta2f(zeta, ecc)

Converts from Gudermannian to True anomaly. Requires ecc > 1.

See Battin: “An Introduction to the Mathematics and Methods of Astrodynamics” for a definition of zeta and the treatment of the resulting equations.

Args:

zeta (float): the Gudermannian (rad.)

ecc (float): the eccentricity

Returns:

float: the True anomaly

Examples:

>>> import pykep as pk
>>> zeta = 8.2
>>> ecc = 2.2
>>> pk.zeta2f(zeta, ecc)
2.3290929552114266

pykep.f2zeta(f, ecc)

Converts from True anomaly to Gudermannian. Requires ecc > 1.

Args:

f (float): the True anomaly (rad.)

ecc (float): the eccentricity

Returns:

float: the Gudermannian

Examples:

>>> import pykep as pk
>>> f = 0.5
>>> ecc = 3.3
>>> pk.f2zeta(f, ecc)
0.36923933496389816

Vectorized

pykep.m2e_v(Ms, eccs)

Converts from Mean to Eccentric anomaly (vectorized version). Requires ecc < 1.

Args:

Ms (numpy.ndarray or float): the Mean anomaly (rad.)

eccs (numpy.ndarray or float): the eccentricity

Returns:

numpy.ndarray or float: the Eccentric anomaly in [-pi, pi] (rad.)

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> Ms = np.linspace(-np.pi/2, np.pi/2, 100)
>>> ecc = 0.375
>>> Es = pk.m2e_v(Ms, ecc)
>>> np.shape(Es)
(100,)

pykep.e2m_v(Es, eccs)

Converts from Eccentric to Mean anomaly (vectorized version). Requires ecc < 1.

Args:

Es (numpy.ndarray or float): the Eccentric anomaly (rad.)

eccs (numpy.ndarray or float): the eccentricity

Returns:

numpy.ndarray or float: the Mean anomaly (rad.)

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> Es = np.linspace(-np.pi/2, np.pi/2, 100)
>>> ecc = 0.86345
>>> Ms = pk.e2m_v(Es, ecc)
>>> np.shape(Ms)
(100,)

pykep.m2f_v(Ms, eccs)

Converts from Mean to True anomaly (vectorized version). Requires ecc < 1.

Args:

Ms (numpy.ndarray or float): the Mean anomaly (rad.)

eccs (numpy.ndarray or float): the eccentricity

Returns:

numpy.ndarray or float: the True anomaly in [-pi, pi] (rad.)

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> Ms = np.linspace(-np.pi/2, np.pi/2, 100)
>>> ecc = 0.4
>>> fs = pk.m2f_v(Ms, ecc)
>>> np.shape(fs)
(100,)

pykep.f2m_v(fs, eccs)

Converts from True to Mean anomaly (vectorized version). Requires ecc < 1.

Args:

fs (numpy.ndarray or float): the True anomaly (rad.)

eccs (numpy.ndarray or float): the eccentricity

Returns:

numpy.ndarray or float: the Mean anomaly in [-pi, pi] (rad.)

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> fs = np.linspace(-np.pi/2, np.pi/2, 100)
>>> ecc = 0.4
>>> Ms = pk.f2m_v(fs, ecc)
>>> np.shape(Ms)
(100,)

pykep.e2f_v(Es, eccs)

Converts from eccentric to true anomaly (vectorized version). Requires ecc < 1.

Args:

Es (numpy.ndarray or float): the Eccentric anomaly (rad.)

eccs (numpy.ndarray or float): the eccentricity

Returns:

numpy.ndarray or float: the True anomaly in [-pi, pi] (rad.)

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> Es = np.linspace(-np.pi/2, np.pi/2, 100)
>>> ecc = 0.0256
>>> fs = pk.e2f_v(Es, ecc)
>>> np.shape(fs)
(100,)

pykep.f2e_v(fs, eccs)

Converts from True to Eccentric anomaly (vectorized version). Requires ecc < 1.

Args:

fs (numpy.ndarray or float): the True anomaly (rad.)

eccs (numpy.ndarray or float): the eccentricity

Returns:

numpy.ndarray or float: the Eccentric anomaly in [-pi, pi] (rad.)

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> fs = np.linspace(-np.pi/2, np.pi/2, 100)
>>> ecc = 0.23
>>> Es = pk.f2e_v(fs, ecc)
>>> np.shape(Es)
(100,)

pykep.n2h_v(Ns, eccs)

Converts from Hyperbolic Mean to Hyperbolic anomaly (vectorized version). Requires ecc > 1.

Args:

Ns (numpy.ndarray or float): the Hyperbolic Mean anomaly (rad.)

eccs (numpy.ndarray or float): the eccentricity

Returns:

numpy.ndarray or float: the Hyperbolic anomaly (rad.)

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> Ns = np.linspace(-np.pi/2, np.pi/2, 100)
>>> ecc = 4.5
>>> Hs = pk.n2h_v(Ns, ecc)
>>> np.shape(Hs)
(100,)

pykep.h2n_v(Hs, eccs)

Converts from Hyperbolic to Hyperbolic Mean anomaly (vectorized version). Requires ecc > 1.

Args:

Hs (numpy.ndarray or float): the Hyperbolic anomaly (rad.)

eccs (numpy.ndarray or float): the eccentricity

Returns:

numpy.ndarray or float: the Hyperbolic Mean anomaly (rad.)

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> Hs = np.linspace(-np.pi/2, np.pi/2, 100)
>>> ecc = 4.5
>>> Ns = pk.h2n_v(Hs, ecc)
>>> np.shape(Ns)
(100,)

pykep.n2f_v(Ns, eccs)

Converts from Hyperbolic Mean to True anomaly (vectorized version). Requires ecc > 1.

Args:

Ns (numpy.ndarray or float): the Hyperbolic Mean anomaly (rad.)

eccs (numpy.ndarray or float): the eccentricity

Returns:

numpy.ndarray or float: the True anomaly

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> Ns = np.linspace(-np.pi/2, np.pi/2, 100)
>>> ecc = 13.45
>>> fs = pk.n2f_v(Ns, ecc)
>>> np.shape(fs)
(100,)

pykep.f2n_v(fs, eccs)

Converts from True to Hyperbolic Mean anomaly (vectorized version). Requires ecc > 1.

Args:

fs (numpy.ndarray or float): the True anomaly (rad.)

eccs (numpy.ndarray or float): the eccentricity

Returns:

numpy.ndarray or float: the Hyperbolic Mean anomaly

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> fs = np.linspace(-np.pi/2, np.pi/2, 100)
>>> ecc = 5.7
>>> Ns = pk.n2f_v(fs, ecc)
>>> np.shape(Ns)
(100,)

pykep.h2f_v(Hs, eccs)

Converts from Hyperbolic to True anomaly (vectorized version). Requires ecc > 1.

Args:

Hs (numpy.ndarray or float): the Hyperbolic anomaly (rad.)

eccs (numpy.ndarray or float): the eccentricity

Returns:

numpy.ndarray or float: the True anomaly in [-pi, pi] (rad.)

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> Hs = np.linspace(-np.pi/2, np.pi/2, 100)
>>> ecc = 4.5
>>> fs = pk.h2f_v(Hs, ecc)
>>> np.shape(fs)
(100,)

pykep.f2h_v(fs, eccs)

Converts from True to Hyperbolic anomaly (vectorized version). Requires ecc > 1.

Args:

fs (numpy.ndarray or float): the True anomaly (rad.)

eccs (numpy.ndarray or float): the eccentricity

Returns:

numpy.ndarray or float: the Hyperbolic anomaly

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> fs = np.linspace(-np.pi/2, np.pi/2, 100)
>>> ecc = 5.7
>>> Hs = pk.n2h_v(fs, ecc)
>>> np.shape(Hs)
(100,)

pykep.zeta2f_v(zetas, eccs)

Converts from Gudermannian to True anomaly (vectorized version). Requires ecc > 1.

See Battin: “An Introduction to the Mathematics and Methods of Astrodynamics” for a definition of zeta and the treatment of the resulting equations.

Args:

zetas (numpy.ndarray or float): the Gudermannian (rad.)

eccs (numpy.ndarray or float): the eccentricity

Returns:

numpy.ndarray or float: the True anomaly

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> zetas = np.linspace(-np.pi/2, np.pi/2, 100)
>>> ecc = 2.2
>>> fs = pk.zeta2f_v(zetas, ecc)
>>> np.shape(fs)
(100,)

pykep.f2zeta_v(fs, eccs)

Converts from True anomaly to Gudermannian (vectorized version). Requires ecc > 1.

Args:

fs (numpy.ndarray or float): the True anomaly (rad.)

eccs (numpy.ndarray or float): the eccentricity

Returns:

numpy.ndarray or float: the Gudermannian

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> fs = np.linspace(-np.pi/2, np.pi/2, 100)
>>> ecc = 10.2
>>> zetas = pk.f2zeta_v(fs, ecc)
>>> np.shape(zetas)
(100,)