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 nu, Keplerian const &kepl)

Real	nu_to_E (Real nu, Keplerian const &kepl)

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

Real	M_to_E (Real M, Keplerian const &kepl)

Real	M_to_H (Real M, Keplerian const &kepl)

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

Real	E_to_nu (Real E, Keplerian const &kepl)

Real	E_to_M (Real E, Keplerian const &kepl)

Real	H_to_nu (Real H, Keplerian const &kepl)

Real	H_to_M (Real H, Keplerian const &kepl)

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

Real	L_to_lambda (Real L, Real nu, Real M)

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

Real	lambda_to_L (Real lambda, Real nu, Real M)

void	cartesian_to_keplerian (Cartesian const &cart, Real mu, Keplerian &kepl)

void	keplerian_to_cartesian (Keplerian const &kepl, Anomaly const &anom, Real mu, Cartesian &cart)

void	equinoctial_to_cartesian (Equinoctial const &equi, Anomaly const &anom, Real mu, Cartesian &cart)

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

void	equinoctial_to_cartesian (Equinoctial const &equi, Anomaly const &anom, Factor I, Real mu, Cartesian &cart)

void	equinoctial_to_keplerian (Equinoctial const &equi, Keplerian &kepl)

Function Documentation

◆ cartesian_to_keplerian()

void Astro::OrbitalElements::cartesian_to_keplerian	(	Cartesian const &	cart,
		Real	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.

◆ E_to_M()

Real Astro::OrbitalElements::E_to_M	(	Real	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_nu()

Real Astro::OrbitalElements::E_to_nu	(	Real	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 \( \nu \).

◆ equinoctial_to_cartesian() [1/2]

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

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

Parameters

[in]	equi	The equinoctial orbital elements.
[in]	anom	The orbital anomalies.
[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,
		Anomaly const &	anom,
		Real	mu,
		Cartesian &	cart )

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

Parameters

[in]	equi	The equinoctial orbital elements.
[in]	anom	The orbital anomalies.
[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	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	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,
		Anomaly const &	anom,
		Real	mu,
		Cartesian &	cart )

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

Parameters

[in]	kepl	The keplerian orbital elements.
[in]	anom	The orbital anomalies.
[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	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	L,
		Real	nu,
		Real	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	L,
		Keplerian const &	kepl,
		Factor	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	lambda,
		Real	nu,
		Real	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	lambda,
		Keplerian const &	kepl,
		Factor	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()

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_H()

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

Compute the mean anomaly \( M \) from the mean anomaly \( M \) as \( M = e\sinh(H) - H \text{.} \). 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	M,
		Keplerian const &	kepl,
		Factor	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	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	nu,
		Keplerian const &	kepl,
		Factor	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()

Real Astro::OrbitalElements::nu_to_M	(	Real	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 \( M \).

Classes

Functions

Function Documentation

◆ cartesian_to_keplerian()

◆ E_to_M()

◆ E_to_nu()

◆ equinoctial_to_cartesian() [1/2]

◆ equinoctial_to_cartesian() [2/2]

◆ equinoctial_to_keplerian()

◆ H_to_M()

◆ H_to_nu()

◆ keplerian_to_cartesian()

◆ keplerian_to_equinoctial()

◆ L_to_lambda()

◆ L_to_nu()

◆ lambda_to_L()

◆ lambda_to_M()

◆ M_to_E()

◆ M_to_H()

◆ M_to_lambda()

◆ nu_to_E()

◆ nu_to_L()

◆ nu_to_M()