Calculations of orbits and related quantities.

Modules
	The Kepler Problem

Functions
template<typename units >
units::realT	mx::astro::orbitPeriod (typename units::realT m1, typename units::realT m2, typename units::realT a)
	Calculate the period of an orbit, given the masses and semi-major axis.

template<typename units >
units::realT	mx::astro::orbitSemiMaj (typename units::realT m1, typename units::realT m2, typename units::realT P)
	Calculate the semi-major axis of an orbit, given the masses and Period, using SI units.

template<typename realT >
realT	mx::astro::orbitMeanAnol (realT t, realT t0, realT P)
	Calculate the mean anomaly at time t, given the time of pericenter passage t0 and period P.

template<typename realT >
long	mx::astro::orbitElements (realT &r, realT &f, realT E, realT D, realT e, realT M, realT a, realT tol=KEPLER_TOL, long itmax=KEPLER_ITMAX)
	Calculate the separation and true anomaly for an orbit at the specified mean anomaly.

template<typename realT >
realT	mx::astro::orbitPhaseCosine (realT f, realT w, realT inc)
	Get the orbital phase at true anomaly f. Calculates the cos(alfa) where alfa is the orbital phase.

template<typename realT >
realT	mx::astro::orbitLambertPhase (realT cos_alf)
	Get the lambertian phase function at orbital phase specified as cos(alfa), where alfa is the phase angle.

template<typename units >
units::realT	mx::astro::orbitRV (typename units::realT m1, typename units::realT m2, typename units::realT inc, typename units::realT a, typename units::realT e, typename units::realT w, typename units::realT f)
	Calculate the radial velocity of an orbiting body inclined relative to an observer.

template<typename vectorT , typename realT >
int	mx::astro::orbitCartesianWork (vectorT x, vectorT y, vectorT z, vectorT r, vectorT rProj, vectorT f, vectorT cos_alf, vectorT phi, const vectorT &t, const size_t N, realT a, realT P, realT e, realT t0, realT i, realT w, realT W)
	Calculate various quantities of an orbit given the keplerian elements and a vector of times.

template<typename vectorT , typename arithT >
int	mx::astro::orbitCartesian2DPhi (vectorT &x, vectorT &y, vectorT &r, vectorT &rProj, vectorT &phi, const vectorT &t, const size_t N, arithT a, arithT P, arithT e, arithT t0, arithT i, arithT w, arithT W)

Function Documentation

◆ orbitCartesian2DPhi()

template<typename vectorT , typename arithT >

int mx::astro::orbitCartesian2DPhi	(	vectorT &	x,
		vectorT &	y,
		vectorT &	r,
		vectorT &	rProj,
		vectorT &	phi,
		const vectorT &	t,
		const size_t	N,
		arithT	a,
		arithT	P,
		arithT	e,
		arithT	t0,
		arithT	i,
		arithT	w,
		arithT	W
	)

Calculate the cartesian x-y position and the Lambert phase function of an orbit given Keplerian elements and a vector of times.

Template Parameters

vectorT	a type whose elements are accessed with the [] operator.
arithT	the type in which to perform arithmetic

Return values

-1	on error calculating the orbit (from orbitCartesianWork)
0	on success.

Parameters

[out]	x	the projected x positions of the orbit. Must be at least as long as t.
[out]	y	the projected y positions of the orbit. Must be at least as long as t.
[out]	r	the separation of the orbit. Must be at least as long as t.
[out]	rProj	the projected separation of the orbit. Must be at least as long as t.
[out]	phi	the Lambert phase function. Must be at least as long as t.
[in]	t	the times at which to calculate the projected positions
[in]	N	the number of points contained in N, and allocated in nx, ny, nz
[in]	a	the semi-major axis of the orbit
[in]	P	the orbital period
[in]	e	the eccentricity of the orbit
[in]	t0	the time of pericenter passage of the orbit
[in]	i	the inclination of the orbit
[in]	w	the argument of pericenter of the orbit
[in]	W	the longitude of the ascending node of the orbit.

Definition at line 310 of file orbitUtils.hpp.

References mx::astro::orbitCartesianWork().

◆ orbitCartesianWork()

template<typename vectorT , typename realT >

int mx::astro::orbitCartesianWork	(	vectorT *	x,
		vectorT *	y,
		vectorT *	z,
		vectorT *	r,
		vectorT *	rProj,
		vectorT *	f,
		vectorT *	cos_alf,
		vectorT *	phi,
		const vectorT &	t,
		const size_t	N,
		realT	a,
		realT	P,
		realT	e,
		realT	t0,
		realT	i,
		realT	w,
		realT	W
	)

Calculate various quantities of an orbit given the keplerian elements and a vector of times.

Only those quantities with non-null pointers are actually calculated.

The orbital parameters with dimensions, (a, P, and t0) must be in consistent units.

Template Parameters

vectorT	a type whose elements are accessed with the [] operator. No other requirements are placed on this type, so it could be a native pointer (i.e. double *).
realT	the type in which to perform calculations. Does not need to match the storage type of vectorT.

Return values

-1	on error calculating the orbit (from rf_elements)
0	on success.

Parameters

[out]	x	[optional] If not NULL, will be the projected x positions of the orbit. Must be at least as long as t.
[out]	y	[optional] If not NULL, will be the projected y positions of the orbit. Must be at least as long as t.
[out]	z	[optional] If not NULL, will be the projected z positions of the orbit. Must be at least as long as t.
[out]	r	[optional] If not NULL, will be the separation of the orbit. Must be at least as long as t.
[out]	rProj	[optional] If not NULL, will be the projected separation of the orbit. Must be at least as long as t.
[out]	f	[optional] If not NULL, will be the true anomaly of the orbit. Must be at least as long as t.
[out]	cos_alf	[optional] If not NULL, will be the phase angle of the orbit. Must be at least as long as t.
[out]	phi	[optional] If not NULL, will be the Lambertian phase function. Must be at least as long as t.
[in]	t	the times at which to calculate the projected positions
[in]	N	the number of points contained in t, and allocated in nx, ny, nz. Avoids requiring a size() member (i.e. if vectorT is native pointers).
[in]	a	the semi-major axis of the orbit.
[in]	P	the orbital period.
[in]	e	the eccentricity of the orbit.
[in]	t0	the time of pericenter passage of the orbit.
[in]	i	the inclination of the orbit [radians].
[in]	w	the argument of pericenter of the orbit [radians].
[in]	W	the longitude of the ascending node of the orbit [radians].

Definition at line 201 of file orbitUtils.hpp.

References mx::astro::orbitElements(), and mx::astro::orbitMeanAnol().

Referenced by mx::astro::orbitCartesian2DPhi().

◆ orbitElements()

template<typename realT >

long mx::astro::orbitElements	(	realT &	r,
		realT &	f,
		realT	E,
		realT	D,
		realT	e,
		realT	M,
		realT	a,
		realT	tol = `KEPLER_TOL`,
		long	itmax = `KEPLER_ITMAX`
	)

Calculate the separation and true anomaly for an orbit at the specified mean anomaly.

Template Parameters

realT is the type used for arithmetic

Returns: the numer of iterations in solving Kepler's equation, if successful.; -1 if itmax is exceeded.

Parameters

[out]	r	is the orbital separation
[out]	f	is the true anomaly
[out]	E	is the eccentric anomaly, provided since it is free
[out]	D	is the error after the last iteration of the Kepler solution.
[in]	e	is the eccentricity
[in]	M	is the mean anomaly
[in]	a	is the semi-major axis
[in]	tol	is the desired tolerance
[in]	itmax	is the maximum number of iterations

Definition at line 77 of file orbitUtils.hpp.

References mx::astro::solve_kepler().

Referenced by mx::astro::orbitCartesianWork().

◆ orbitLambertPhase()

template<typename realT >

realT mx::astro::orbitLambertPhase ( realT cos_alf )

Get the lambertian phase function at orbital phase specified as cos(alfa), where alfa is the phase angle.

Uses cos(alfa) to save an operation after calculating orbit phase.

Template Parameters

realT is the type used for arithmetic

Returns: the lambert phase function at alpha

Parameters

[in] cos_alf the cosine of the phase angle

Definition at line 154 of file orbitUtils.hpp.

References mx::math::six_fifths().

◆ orbitMeanAnol()

template<typename realT >

realT mx::astro::orbitMeanAnol	(	realT	t,
		realT	t0,
		realT	P
	)

Calculate the mean anomaly at time t, given the time of pericenter passage t0 and period P.

Definition at line 62 of file orbitUtils.hpp.

References mx::math::six_fifths().

Referenced by mx::astro::orbitCartesianWork().

◆ orbitPeriod()

template<typename units >

units::realT mx::astro::orbitPeriod	(	typename units::realT	m1,
		typename units::realT	m2,
		typename units::realT	a
	)

Calculate the period of an orbit, given the masses and semi-major axis.

Template Parameters

units specifies the unit system and precision, see Unit Conversions.

Returns: the period in the specified units

Parameters

[in]	m1	mass of one object.
[in]	m2	mass of the other object (can be 0).
[in]	a	the semi-major axis of the orbit.

Definition at line 28 of file orbitUtils.hpp.

References mx::math::six_fifths().

◆ orbitPhaseCosine()

template<typename realT >

realT mx::astro::orbitPhaseCosine	(	realT	f,
		realT	w,
		realT	inc
	)

Get the orbital phase at true anomaly f. Calculates the cos(alfa) where alfa is the orbital phase.

Only calculates cos(alfa) to save an operation when possible.

Template Parameters

realT is the type used for arithmetic

Returns: the cosine of the phase angle

Parameters

[in]	f	is the true anomoaly at which to calculate alfa
[in]	w	the argument of pericenter of the orbit
[in]	inc	is the inclinatin of the orbit

Definition at line 136 of file orbitUtils.hpp.

◆ orbitRV()

template<typename units >

units::realT mx::astro::orbitRV	(	typename units::realT	m1,
		typename units::realT	m2,
		typename units::realT	inc,
		typename units::realT	a,
		typename units::realT	e,
		typename units::realT	w,
		typename units::realT	f
	)

Calculate the radial velocity of an orbiting body inclined relative to an observer.

Returns: the radial velocity of the body with mass m1 as it orbits the barycenter of its orbit w.r.t. m2.

Parameters

[in]	m1	the mass of the body for which the RV is computed
[in]	m2	the mass of the other body in the orbiting pair
[in]	inc	the inclination of the orbit of m2.
[in]	a	the semi-major axis of the orbit
[in]	e	the eccentricity of the orbit
[in]	w	the argument of pericenter
[in]	f	the true anomaly at which the RV is to be calculated.

Definition at line 166 of file orbitUtils.hpp.

◆ orbitSemiMaj()

template<typename units >

units::realT mx::astro::orbitSemiMaj	(	typename units::realT	m1,
		typename units::realT	m2,
		typename units::realT	P
	)

Calculate the semi-major axis of an orbit, given the masses and Period, using SI units.

Template Parameters

units specifies the unit system and precision, see Unit Conversions.

Returns: the semi-major axis in the specified units

Parameters

[in]	m1	mass of one object.
[in]	m2	mass of the other object (can be 0).
[in]	P	the period of the orbit.

Definition at line 45 of file orbitUtils.hpp.

References mx::math::six_fifths().

Modules

Functions

Function Documentation

◆ orbitCartesian2DPhi()

◆ orbitCartesianWork()

◆ orbitElements()

◆ orbitLambertPhase()

◆ orbitMeanAnol()

◆ orbitPeriod()

◆ orbitPhaseCosine()

◆ orbitRV()

◆ orbitSemiMaj()