Astro::OrbitalElements Namespace Reference

The namespace for the orbital elements definition and conversion.

Classes
struct	Anomaly
	Structure container for the orbital anomalies. More...
struct	Cartesian
	Structure container for the cartesian orbital elements. More...
struct	Equinoctial
	Struct container for the (modified) equinoctial orbital elements. More...
struct	Keplerian
	Structure container for the (modified) Keplerian orbital elements. More...
struct	Quaternionic
	Structure container for the quaternionic orbital elements. More...

Functions
Real	nu_to_M (Real const nu, Real const e)
Real	nu_to_M (Real const nu, Keplerian const &kepl)
Real	nu_to_E (Real const nu, Keplerian const &kepl)
Real	nu_to_L (Real const nu, Keplerian const &kepl, Factor const I)
Real	M_to_E (Real M, Real const e)
Real	M_to_E (Real M, Keplerian const &kepl)
Real	M_to_H (Real const M, Keplerian const &kepl)
Real	M_to_lambda (Real const M, Keplerian const &kepl, Factor const I)
Real	E_to_nu (Real const E, Real const e)
Real	E_to_nu (Real const E, Keplerian const &kepl)
Real	E_to_M (Real const E, Real const e)
Real	E_to_M (Real const E, Keplerian const &kepl)
Real	H_to_nu (Real const H, Keplerian const &kepl)
Real	H_to_M (Real const H, Keplerian const &kepl)
Real	L_to_nu (Real const L, Keplerian const &kepl, Factor const I)
Real	L_to_lambda (Real const L, Real const nu, Real const M)
Real	lambda_to_M (Real const lambda, Keplerian const &kepl, Factor const I)
Real	lambda_to_L (Real const lambda, Real const nu, Real const M)
Real	cartesian_to_keplerian (Cartesian const &cart, Real const mu, Keplerian &kepl)
void	keplerian_to_cartesian (Keplerian const &kepl, Real const nu, Real const mu, Cartesian &cart)
void	equinoctial_to_cartesian (Equinoctial const &equi, Real const L, Real const mu, Cartesian &cart)
void	keplerian_to_equinoctial (Keplerian const &kepl, Factor const I, Equinoctial &equi)
Real	cartesian_to_equinoctial (Cartesian const &cart, Real const mu, Equinoctial &equi)
void	equinoctial_to_cartesian (Equinoctial const &equi, Real const L, Factor const I, Real const mu, Cartesian &cart)
void	equinoctial_to_keplerian (Equinoctial const &equi, Keplerian &kepl)

Function Documentation

cartesian_to_equinoctial()

Real Astro::OrbitalElements::cartesian_to_equinoctial	(	Cartesian const &	cart,
		Real const	mu,
		Equinoctial &	equi )

Convert the cartesian state (position and velocity) vectors to the equinoctial orbital elements.

Parameters

[in]	cart	The cartesian state (position and velocity) vectors.
[in]	mu	The gravitational parameter.
[in]	I	The posigrade (+1)/retrograde (-1) factor.
[out]	equi	The equinoctial orbital elements.

Returns

The true longitude \( L \).

Note

The transformation is only valid for posigrade orbits (I = +1).

cartesian_to_keplerian()

Real Astro::OrbitalElements::cartesian_to_keplerian	(	Cartesian const &	cart,
		Real const	mu,
		Keplerian &	kepl )

Convert the cartesian state (position and velocity) vectors to the keplerian orbital elements.

Parameters

[in]	cart	The cartesian state (position and velocity) vectors.
[in]	mu	The gravitational parameter.
[out]	kepl	The keplerian orbital elements.

Returns

The true anomaly \( \nu \) (rad).

E_to_M() [1/2]

Real Astro::OrbitalElements::E_to_M	(	Real const	E,
		Keplerian const &	kepl )

Compute the mean anomaly \( M \) from the eccentric anomaly \( E \) as

\[ M = E - e\sin(E) \text{.} \]

Parameters

[in]	E	The eccentric anomaly \( E \).
[in]	kepl	The keplerian orbital elements.

Returns

The true anomaly \( \nu \).

E_to_M() [2/2]

Real Astro::OrbitalElements::E_to_M	(	Real const	E,
		Real const	e )

Compute the mean anomaly \( M \) from the eccentric anomaly \( E \) as

\[ M = E - e\sin(E) \text{.} \]

Parameters

[in]	E	The eccentric anomaly \( E \).
[in]	e	The eccentricity \( e \).

Returns

The true anomaly \( \nu \).

E_to_nu() [1/2]

Real Astro::OrbitalElements::E_to_nu	(	Real const	E,
		Keplerian const &	kepl )

Compute the true anomaly \( \nu \) from the eccentric anomaly \( E \) as

\[ \tan\left(\frac{\nu - E}{2}\right) = \displaystyle\frac{\sin(E)}{\beta - \cos(E)} \text{,} \]

with \( \beta = \displaystyle\frac{1 + \sqrt{1 - e^2}}{e} \). That is equivalent to

\[ \nu = E + 2\arctan\left(\frac{\sin(E)}{\beta - \cos(E)}\right) \text{.} \]

Note

Refer to R. Broucke and P. Cefola, A note on the relations between true and eccentric anomalies in the two-body problem, Celestial Mechanics, Vol. 7, pp. 300-389, 1973.

Parameters

[in]	E	The eccentric anomaly \( E \).
[in]	kepl	The keplerian orbital elements.

Returns

The true anomaly \( \nu \).

E_to_nu() [2/2]

Real Astro::OrbitalElements::E_to_nu	(	Real const	E,
		Real const	e )

Compute the true anomaly \( \nu \) from the eccentric anomaly \( E \) as

\[ \tan\left(\frac{\nu - E}{2}\right) = \displaystyle\frac{\sin(E)}{\beta - \cos(E)} \text{,} \]

with \( \beta = \displaystyle\frac{1 + \sqrt{1 - e^2}}{e} \). That is equivalent to

\[ \nu = E + 2\arctan\left(\frac{\sin(E)}{\beta - \cos(E)}\right) \text{.} \]

Note

Refer to R. Broucke and P. Cefola, A note on the relations between true and eccentric anomalies in the two-body problem, Celestial Mechanics, Vol. 7, pp. 300-389, 1973.

Parameters

[in]	E	The eccentric anomaly \( E \).
[in]	e	The eccentricity \( e \).

Returns

The true anomaly \( \nu \).

equinoctial_to_cartesian() [1/2]

void Astro::OrbitalElements::equinoctial_to_cartesian	(	Equinoctial const &	equi,
		Real const	L,
		Factor const	I,
		Real const	mu,
		Cartesian &	cart )

Convert the equinoctial orbital elements to cartesian state (position and velocity) vectors.

Parameters

[in]	equi	The equinoctial orbital elements.
[in]	L	The true longitude \( L \).
[in]	I	The posigrade (+1)/retrograde (-1) factor.
[in]	mu	The gravitational parameter.
[out]	cart	The cartesian state (position and velocity) vectors.

equinoctial_to_cartesian() [2/2]

void Astro::OrbitalElements::equinoctial_to_cartesian	(	Equinoctial const &	equi,
		Real const	L,
		Real const	mu,
		Cartesian &	cart )

Convert the equinoctial orbital elements to the cartesian state (position and velocity) vectors.

Parameters

[in]	equi	The equinoctial orbital elements.
[in]	L	The true longitude \( L \).
[in]	mu	The gravitational parameter.
[out]	cart	The cartesian state (position and velocity) vectors.

equinoctial_to_keplerian()

void Astro::OrbitalElements::equinoctial_to_keplerian	(	Equinoctial const &	equi,
		Keplerian &	kepl )

Convert the equinoctial orbital elements to the keplerian orbital elements.

Parameters

[in]	equi	The equinoctial orbital elements.
[in]	I	The posigrade (+1)/retrograde (-1) factor.
[out]	kepl	The keplerian orbital elements.

H_to_M()

Real Astro::OrbitalElements::H_to_M	(	Real const	H,
		Keplerian const &	kepl )

Compute the mean anomaly \( M \) from the hyperbolic anomaly \( H \) as

\[ M = e\sinh(H) - H \text{.} \]

Parameters

[in]	H	The hyperbolic anomaly \( H \).
[in]	kepl	The keplerian orbital elements.

Returns

The mean anomaly \( M \).

H_to_nu()

Real Astro::OrbitalElements::H_to_nu	(	Real const	H,
		Keplerian const &	kepl )

Compute the true anomaly \( \nu \) from the hyperbolic anomaly \( H \) as

\[ \nu = 2\arctan\left(\sqrt{\frac{e + 1}{e - 1}}\tanh\left(\frac{H}{2}\right)\right) \text{.} \]

Note

Refer to (p. 167) "An Introduction to the Mathematics and Methods of Astrodynamics" by R. H. Battin, AIAA Education Series, 1999.

Parameters

[in]	H	The hyperbolic anomaly \( H \).
[in]	kepl	The keplerian orbital elements.

Returns

The true anomaly \( \nu \).

keplerian_to_cartesian()

void Astro::OrbitalElements::keplerian_to_cartesian	(	Keplerian const &	kepl,
		Real const	nu,
		Real const	mu,
		Cartesian &	cart )

Convert the keplerian orbital elements to the cartesian state (position and velocity) vectors.

Parameters

[in]	kepl	The keplerian orbital elements.
[in]	nu	The true anomaly \( \nu \).
[in]	mu	The gravitational parameter.
[out]	cart	The cartesian state (position and velocity) vectors.

keplerian_to_equinoctial()

void Astro::OrbitalElements::keplerian_to_equinoctial	(	Keplerian const &	kepl,
		Factor const	I,
		Equinoctial &	equi )

Convert the keplerian orbital elements to the equinoctial orbital elements.

Parameters

[in]	kepl	The keplerian orbital elements.
[in]	I	The posigrade (+1)/retrograde (-1) factor.
[out]	equi	The equinoctial orbital elements.

L_to_lambda()

Real Astro::OrbitalElements::L_to_lambda	(	Real const	L,
		Real const	nu,
		Real const	M )

Compute the mean longitude \( \lambda \) from the true longitude \( L \) as

\[ \lambda = L - \nu + M \text{.} \]

Parameters

[in]	L	The true longitude \( L \).
[in]	nu	The true anomaly \( \nu \).
[in]	M	The mean anomaly \( M \).

Returns

The mean longitude \( \lambda \).

L_to_nu()

Real Astro::OrbitalElements::L_to_nu	(	Real const	L,
		Keplerian const &	kepl,
		Factor const	I )

Compute the true anomaly \( \nu \) from the true longitude \( L \) as

\[ \nu = L - \omega - I\Omega \text{.} \]

Parameters

[in]	L	The true longitude \( L \).
[in]	kepl	The keplerian orbital elements.
[in]	I	The posigrade (+1)/retrograde (-1) factor.

Returns

The true anomaly \( \nu \).

lambda_to_L()

Real Astro::OrbitalElements::lambda_to_L	(	Real const	lambda,
		Real const	nu,
		Real const	M )

Compute the true longitude \( L \) from the mean longitude \( \lambda \) as

\[ L = \lambda + \nu - M \text{.} \]

Parameters

[in]	lambda	The mean longitude \( \lambda \).
[in]	nu	The true anomaly \( \nu \).
[in]	M	The mean anomaly \( M \).

Returns

The true longitude \( L \).

lambda_to_M()

Real Astro::OrbitalElements::lambda_to_M	(	Real const	lambda,
		Keplerian const &	kepl,
		Factor const	I )

Compute the mean anomaly \( M \) from the mean longitude \( \lambda \) as

\[ M = \lambda - \omega - I\Omega \text{.} \]

Parameters

[in]	lambda	The mean longitude \( \lambda \).
[in]	kepl	The keplerian orbital elements.
[in]	I	The posigrade (+1)/retrograde (-1) factor.

Returns

The mean anomaly \( M \).

M_to_E() [1/2]

Real Astro::OrbitalElements::M_to_E	(	Real	M,
		Keplerian const &	kepl )

Compute the eccentric anomaly \( E \) from the mean anomaly \( M \) through a the solution of the nonlinear equation \( E = M + e\sin(E) \) for \( E \). The solution is found through a basic Newton method.

Parameters

[in]	M	The mean anomaly \( M \).
[in]	kepl	The keplerian orbital elements.

Returns

The eccentric anomaly \( E \).

M_to_E() [2/2]

Real Astro::OrbitalElements::M_to_E	(	Real	M,
		Real const	e )

Compute the eccentric anomaly \( E \) from the mean anomaly \( M \) through a the solution of the nonlinear equation \( E = M + e\sin(E) \) for \( E \). The solution is found through a basic Newton method.

Parameters

[in]	M	The mean anomaly \( M \).
[in]	e	The eccentricity \( e \).

Returns

The eccentric anomaly \( E \).

M_to_H()

Real Astro::OrbitalElements::M_to_H	(	Real const	M,
		Keplerian const &	kepl )

Compute the mean anomaly \( M \) from the hyperbolic anomaly \( H \). \( M = e\sinh(H) - H \text{.} \). The solution is found through a basic Newton method.

Note

The mean anomaly \( M \) must not be clamped in the range \( [0, 2\pi] \)!

Parameters

[in]	M	The mean anomaly \( M \).
[in]	kepl	The keplerian orbital elements.

Returns

The mean anomaly \( M \).

M_to_lambda()

Real Astro::OrbitalElements::M_to_lambda	(	Real const	M,
		Keplerian const &	kepl,
		Factor const	I )

Compute the mean longitude \( \lambda \) from the mean anomaly \( M \) as

\[ \lambda = M + \omega + I\Omega \text{.} \]

Parameters

[in]	M	The mean anomaly \( M \).
[in]	kepl	The keplerian orbital elements.
[in]	I	The posigrade (+1)/retrograde (-1) factor.

Returns

The mean longitude \( \lambda \).

nu_to_E()

Real Astro::OrbitalElements::nu_to_E	(	Real const	nu,
		Keplerian const &	kepl )

Compute the true anomaly \( \nu \) from the eccentric anomaly \( E \) as

\[ \tan\left(\frac{\nu - E}{2}\right) = \displaystyle\frac{\sin(\nu)}{\beta + \cos(\nu)} \text{,} \]

with \( \beta = \displaystyle\frac{1 + \sqrt{1 - e^2}}{e} \). That is equivalent to

\[ E = \nu - 2\arctan\left(\frac{\sin(\nu)}{\beta + \cos(\nu)}\right) \text{.} \]

Note

Refer to R. Broucke and P. Cefola, A note on the relations between true and eccentric anomalies in the two-body problem, Celestial Mechanics, Vol. 7, pp. 300-389, 1973.

Parameters

[in]	nu	The true anomaly \( \nu \).
[in]	kepl	The keplerian orbital elements.

Returns

The eccentric anomaly \( E \).

nu_to_L()

Real Astro::OrbitalElements::nu_to_L	(	Real const	nu,
		Keplerian const &	kepl,
		Factor const	I )

Compute the true longitude \( L \) from the true anomaly \( \nu \) as

\[ L = \nu + \omega + I\Omega \text{.} \]

Parameters

[in]	nu	The true anomaly \( \nu \).
[in]	kepl	The keplerian orbital elements.
[in]	I	The posigrade (+1)/retrograde (-1) factor.

Returns

The true longitude \( L \).

nu_to_M() [1/2]

Real Astro::OrbitalElements::nu_to_M	(	Real const	nu,
		Keplerian const &	kepl )

Compute the mean anomaly \( M \) from the eccentric anomaly \( E \) as

\[ M = \nu - 2\arctan\left(\frac{\sin(\nu)}{\beta + \cos(\nu)}\right) - \displaystyle\frac{e\sqrt{1 - e^2}\sin(\nu)}{1 + e\cos(\nu)} \text{,} \]

with \( \beta = \displaystyle\frac{1 + \sqrt{1 - e^2}}{e} \). That is equivalent to

Note

Refer to Broucke and P. Cefola, Celestial, A note on the relations between true and eccentric anomalies in problethe two-body m" b Mechnics, Vol. 7, pp. 300-389, 1973.

Parameters

[in]	nu	The true anomaly \( \nu \).
[in]	kepl	The keplerian orbital elements.

Returns

The mean anomaly \( M \).

nu_to_M() [2/2]

Real Astro::OrbitalElements::nu_to_M	(	Real const	nu,
		Real const	e )

Compute the mean anomaly \( M \) from the eccentric anomaly \( E \) as

\[ M = \nu - 2\arctan\left(\frac{\sin(\nu)}{\beta + \cos(\nu)}\right) - \displaystyle\frac{e\sqrt{1 - e^2}\sin(\nu)}{1 + e\cos(\nu)} \text{,} \]

with \( \beta = \displaystyle\frac{1 + \sqrt{1 - e^2}}{e} \). That is equivalent to

Note

Refer to Broucke and P. Cefola, Celestial, A note on the relations between true and eccentric anomalies in problethe two-body m" b Mechnics, Vol. 7, pp. 300-389, 1973.

Parameters

[in]	nu	The true anomaly \( \nu \).
[in]	kepl	The keplerian orbital elements.

Returns

The mean anomaly \( M \).