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/**
* \file NormalGravity.hpp
* \brief Header for GeographicLib::NormalGravity class
*
* Copyright (c) Charles Karney (2011-2022) <karney@alum.mit.edu> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
#if !defined(GEOGRAPHICLIB_NORMALGRAVITY_HPP)
#define GEOGRAPHICLIB_NORMALGRAVITY_HPP 1
#include <GeographicLib/Constants.hpp>
#include <GeographicLib/Geocentric.hpp>
namespace GeographicLib {
/**
* \brief The normal gravity of the earth
*
* "Normal" gravity refers to an idealization of the earth which is modeled
* as an rotating ellipsoid. The eccentricity of the ellipsoid, the rotation
* speed, and the distribution of mass within the ellipsoid are such that the
* ellipsoid is a "level ellipoid", a surface of constant potential
* (gravitational plus centrifugal). The acceleration due to gravity is
* therefore perpendicular to the surface of the ellipsoid.
*
* Because the distribution of mass within the ellipsoid is unspecified, only
* the potential exterior to the ellipsoid is well defined. In this class,
* the mass is assumed to be to concentrated on a "focal disc" of radius,
* (<i>a</i><sup>2</sup> − <i>b</i><sup>2</sup>)<sup>1/2</sup>, where
* \e a is the equatorial radius of the ellipsoid and \e b is its polar
* semi-axis. In the case of an oblate ellipsoid, the mass is concentrated
* on a "focal rod" of length 2(<i>b</i><sup>2</sup> −
* <i>a</i><sup>2</sup>)<sup>1/2</sup>. As a result the potential is well
* defined everywhere.
*
* There is a closed solution to this problem which is implemented here.
* Series "approximations" are only used to evaluate certain combinations of
* elementary functions where use of the closed expression results in a loss
* of accuracy for small arguments due to cancellation of the leading terms.
* However these series include sufficient terms to give full machine
* precision.
*
* Although the formulation used in this class applies to ellipsoids with
* arbitrary flattening, in practice, its use should be limited to about
* <i>b</i>/\e a ∈ [0.01, 100] or \e f ∈ [−99, 0.99].
*
* Definitions:
* - <i>V</i><sub>0</sub>, the gravitational contribution to the normal
* potential;
* - Φ, the rotational contribution to the normal potential;
* - \e U = <i>V</i><sub>0</sub> + Φ, the total potential;
* - <b>Γ</b> = ∇<i>V</i><sub>0</sub>, the acceleration due to
* mass of the earth;
* - <b>f</b> = ∇Φ, the centrifugal acceleration;
* - <b>γ</b> = ∇\e U = <b>Γ</b> + <b>f</b>, the normal
* acceleration;
* - \e X, \e Y, \e Z, geocentric coordinates;
* - \e x, \e y, \e z, local cartesian coordinates used to denote the east,
* north and up directions.
*
* References:
* - C. Somigliana, Teoria generale del campo gravitazionale dell'ellissoide
* di rotazione, Mem. Soc. Astron. Ital, <b>4</b>, 541--599 (1929).
* - W. A. Heiskanen and H. Moritz, Physical Geodesy (Freeman, San
* Francisco, 1967), Secs. 1-19, 2-7, 2-8 (2-9, 2-10), 6-2 (6-3).
* https://archive.org/details/HeiskanenMoritz1967PhysicalGeodesy
* - B. Hofmann-Wellenhof, H. Moritz, Physical Geodesy (Second edition,
* Springer, 2006). https://doi.org/10.1007/978-3-211-33545-1
* - H. Moritz, Geodetic Reference System 1980, J. Geodesy 54(3), 395-405
* (1980). https://doi.org/10.1007/BF02521480
*
* For more information on normal gravity see \ref normalgravity.
*
* Example of use:
* \include example-NormalGravity.cpp
**********************************************************************/
class GEOGRAPHICLIB_EXPORT NormalGravity {
private:
static const int maxit_ = 20;
typedef Math::real real;
friend class GravityModel;
real _a, _gGM, _omega, _f, _jJ2, _omega2, _aomega2;
real _e2, _ep2, _b, _eE, _uU0, _gammae, _gammap, _qQ0, _k, _fstar;
Geocentric _earth;
static real atanzz(real x, bool alt) {
// This routine obeys the identity
// atanzz(x, alt) = atanzz(-x/(1+x), !alt)
//
// Require x >= -1. Best to call with alt, s.t. x >= 0; this results in
// a call to atan, instead of asin, or to asinh, instead of atanh.
using std::sqrt; using std::fabs; using std::atan; using std::asin;
using std::asinh; using std::atanh;
real z = sqrt(fabs(x));
return x == 0 ? 1 :
(alt ?
(!(x < 0) ? asinh(z) : asin(z)) / sqrt(fabs(x) / (1 + x)) :
(!(x < 0) ? atan(z) : atanh(z)) / z);
}
static real atan7series(real x);
static real atan5series(real x);
static real Qf(real x, bool alt);
static real Hf(real x, bool alt);
static real QH3f(real x, bool alt);
real Jn(int n) const;
void Initialize(real a, real GM, real omega, real f_J2, bool geometricp);
public:
/** \name Setting up the normal gravity
**********************************************************************/
///@{
/**
* Constructor for the normal gravity.
*
* @param[in] a equatorial radius (meters).
* @param[in] GM mass constant of the ellipsoid
* (meters<sup>3</sup>/seconds<sup>2</sup>); this is the product of \e G
* the gravitational constant and \e M the mass of the earth (usually
* including the mass of the earth's atmosphere).
* @param[in] omega the angular velocity (rad s<sup>−1</sup>).
* @param[in] f_J2 either the flattening of the ellipsoid \e f or the
* the dynamical form factor \e J2.
* @param[out] geometricp if true (the default), then \e f_J2 denotes the
* flattening, else it denotes the dynamical form factor \e J2.
* @exception if \e a is not positive or if the other parameters do not
* obey the restrictions given below.
*
* The shape of the ellipsoid can be given in one of two ways:
* - geometrically (\e geomtricp = true), the ellipsoid is defined by the
* flattening \e f = (\e a − \e b) / \e a, where \e a and \e b are
* the equatorial radius and the polar semi-axis. The parameters should
* obey \e a > 0, \e f < 1. There are no restrictions on \e GM or
* \e omega, in particular, \e GM need not be positive.
* - physically (\e geometricp = false), the ellipsoid is defined by the
* dynamical form factor <i>J</i><sub>2</sub> = (\e C − \e A) /
* <i>Ma</i><sup>2</sup>, where \e A and \e C are the equatorial and
* polar moments of inertia and \e M is the mass of the earth. The
* parameters should obey \e a > 0, \e GM > 0 and \e J2 < 1/3
* − (<i>omega</i><sup>2</sup><i>a</i><sup>3</sup>/<i>GM</i>)
* 8/(45π). There is no restriction on \e omega.
**********************************************************************/
NormalGravity(real a, real GM, real omega, real f_J2,
bool geometricp = true);
/**
* A default constructor for the normal gravity. This sets up an
* uninitialized object and is used by GravityModel which constructs this
* object before it has read in the parameters for the reference ellipsoid.
**********************************************************************/
NormalGravity() : _a(-1) {}
///@}
/** \name Compute the gravity
**********************************************************************/
///@{
/**
* Evaluate the gravity on the surface of the ellipsoid.
*
* @param[in] lat the geographic latitude (degrees).
* @return γ the acceleration due to gravity, positive downwards
* (m s<sup>−2</sup>).
*
* Due to the axial symmetry of the ellipsoid, the result is independent of
* the value of the longitude. This acceleration is perpendicular to the
* surface of the ellipsoid. It includes the effects of the earth's
* rotation.
**********************************************************************/
Math::real SurfaceGravity(real lat) const;
/**
* Evaluate the gravity at an arbitrary point above (or below) the
* ellipsoid.
*
* @param[in] lat the geographic latitude (degrees).
* @param[in] h the height above the ellipsoid (meters).
* @param[out] gammay the northerly component of the acceleration
* (m s<sup>−2</sup>).
* @param[out] gammaz the upward component of the acceleration
* (m s<sup>−2</sup>); this is usually negative.
* @return \e U the corresponding normal potential
* (m<sup>2</sup> s<sup>−2</sup>).
*
* Due to the axial symmetry of the ellipsoid, the result is independent of
* the value of the longitude and the easterly component of the
* acceleration vanishes, \e gammax = 0. The function includes the effects
* of the earth's rotation. When \e h = 0, this function gives \e gammay =
* 0 and the returned value matches that of NormalGravity::SurfaceGravity.
**********************************************************************/
Math::real Gravity(real lat, real h, real& gammay, real& gammaz)
const;
/**
* Evaluate the components of the acceleration due to gravity and the
* centrifugal acceleration in geocentric coordinates.
*
* @param[in] X geocentric coordinate of point (meters).
* @param[in] Y geocentric coordinate of point (meters).
* @param[in] Z geocentric coordinate of point (meters).
* @param[out] gammaX the \e X component of the acceleration
* (m s<sup>−2</sup>).
* @param[out] gammaY the \e Y component of the acceleration
* (m s<sup>−2</sup>).
* @param[out] gammaZ the \e Z component of the acceleration
* (m s<sup>−2</sup>).
* @return \e U = <i>V</i><sub>0</sub> + Φ the sum of the
* gravitational and centrifugal potentials
* (m<sup>2</sup> s<sup>−2</sup>).
*
* The acceleration given by <b>γ</b> = ∇\e U =
* ∇<i>V</i><sub>0</sub> + ∇Φ = <b>Γ</b> + <b>f</b>.
**********************************************************************/
Math::real U(real X, real Y, real Z,
real& gammaX, real& gammaY, real& gammaZ) const;
/**
* Evaluate the components of the acceleration due to the gravitational
* force in geocentric coordinates.
*
* @param[in] X geocentric coordinate of point (meters).
* @param[in] Y geocentric coordinate of point (meters).
* @param[in] Z geocentric coordinate of point (meters).
* @param[out] GammaX the \e X component of the acceleration due to the
* gravitational force (m s<sup>−2</sup>).
* @param[out] GammaY the \e Y component of the acceleration due to the
* @param[out] GammaZ the \e Z component of the acceleration due to the
* gravitational force (m s<sup>−2</sup>).
* @return <i>V</i><sub>0</sub> the gravitational potential
* (m<sup>2</sup> s<sup>−2</sup>).
*
* This function excludes the centrifugal acceleration and is appropriate
* to use for space applications. In terrestrial applications, the
* function NormalGravity::U (which includes this effect) should usually be
* used.
**********************************************************************/
Math::real V0(real X, real Y, real Z,
real& GammaX, real& GammaY, real& GammaZ) const;
/**
* Evaluate the centrifugal acceleration in geocentric coordinates.
*
* @param[in] X geocentric coordinate of point (meters).
* @param[in] Y geocentric coordinate of point (meters).
* @param[out] fX the \e X component of the centrifugal acceleration
* (m s<sup>−2</sup>).
* @param[out] fY the \e Y component of the centrifugal acceleration
* (m s<sup>−2</sup>).
* @return Φ the centrifugal potential (m<sup>2</sup>
* s<sup>−2</sup>).
*
* Φ is independent of \e Z, thus \e fZ = 0. This function
* NormalGravity::U sums the results of NormalGravity::V0 and
* NormalGravity::Phi.
**********************************************************************/
Math::real Phi(real X, real Y, real& fX, real& fY) const;
///@}
/** \name Inspector functions
**********************************************************************/
///@{
/**
* @return true if the object has been initialized.
**********************************************************************/
bool Init() const { return _a > 0; }
/**
* @return \e a the equatorial radius of the ellipsoid (meters). This is
* the value used in the constructor.
**********************************************************************/
Math::real EquatorialRadius() const
{ return Init() ? _a : Math::NaN(); }
/**
* @return \e GM the mass constant of the ellipsoid
* (m<sup>3</sup> s<sup>−2</sup>). This is the value used in the
* constructor.
**********************************************************************/
Math::real MassConstant() const
{ return Init() ? _gGM : Math::NaN(); }
/**
* @return <i>J</i><sub><i>n</i></sub> the dynamical form factors of the
* ellipsoid.
*
* If \e n = 2 (the default), this is the value of <i>J</i><sub>2</sub>
* used in the constructor. Otherwise it is the zonal coefficient of the
* Legendre harmonic sum of the normal gravitational potential. Note that
* <i>J</i><sub><i>n</i></sub> = 0 if \e n is odd. In most gravity
* applications, fully normalized Legendre functions are used and the
* corresponding coefficient is <i>C</i><sub><i>n</i>0</sub> =
* −<i>J</i><sub><i>n</i></sub> / sqrt(2 \e n + 1).
**********************************************************************/
Math::real DynamicalFormFactor(int n = 2) const
{ return Init() ? ( n == 2 ? _jJ2 : Jn(n)) : Math::NaN(); }
/**
* @return ω the angular velocity of the ellipsoid (rad
* s<sup>−1</sup>). This is the value used in the constructor.
**********************************************************************/
Math::real AngularVelocity() const
{ return Init() ? _omega : Math::NaN(); }
/**
* @return <i>f</i> the flattening of the ellipsoid (\e a − \e b)/\e
* a.
**********************************************************************/
Math::real Flattening() const
{ return Init() ? _f : Math::NaN(); }
/**
* @return γ<sub>e</sub> the normal gravity at equator (m
* s<sup>−2</sup>).
**********************************************************************/
Math::real EquatorialGravity() const
{ return Init() ? _gammae : Math::NaN(); }
/**
* @return γ<sub>p</sub> the normal gravity at poles (m
* s<sup>−2</sup>).
**********************************************************************/
Math::real PolarGravity() const
{ return Init() ? _gammap : Math::NaN(); }
/**
* @return <i>f*</i> the gravity flattening (γ<sub>p</sub> −
* γ<sub>e</sub>) / γ<sub>e</sub>.
**********************************************************************/
Math::real GravityFlattening() const
{ return Init() ? _fstar : Math::NaN(); }
/**
* @return <i>U</i><sub>0</sub> the constant normal potential for the
* surface of the ellipsoid (m<sup>2</sup> s<sup>−2</sup>).
**********************************************************************/
Math::real SurfacePotential() const
{ return Init() ? _uU0 : Math::NaN(); }
/**
* @return the Geocentric object used by this instance.
**********************************************************************/
const Geocentric& Earth() const { return _earth; }
///@}
/**
* A global instantiation of NormalGravity for the WGS84 ellipsoid.
**********************************************************************/
static const NormalGravity& WGS84();
/**
* A global instantiation of NormalGravity for the GRS80 ellipsoid.
**********************************************************************/
static const NormalGravity& GRS80();
/**
* Compute the flattening from the dynamical form factor.
*
* @param[in] a equatorial radius (meters).
* @param[in] GM mass constant of the ellipsoid
* (meters<sup>3</sup>/seconds<sup>2</sup>); this is the product of \e G
* the gravitational constant and \e M the mass of the earth (usually
* including the mass of the earth's atmosphere).
* @param[in] omega the angular velocity (rad s<sup>−1</sup>).
* @param[in] J2 the dynamical form factor.
* @return \e f the flattening of the ellipsoid.
*
* This routine requires \e a > 0, \e GM > 0, \e J2 < 1/3 −
* <i>omega</i><sup>2</sup><i>a</i><sup>3</sup>/<i>GM</i> 8/(45π). A
* NaN is returned if these conditions do not hold. The restriction to
* positive \e GM is made because for negative \e GM two solutions are
* possible.
**********************************************************************/
static Math::real J2ToFlattening(real a, real GM, real omega, real J2);
/**
* Compute the dynamical form factor from the flattening.
*
* @param[in] a equatorial radius (meters).
* @param[in] GM mass constant of the ellipsoid
* (meters<sup>3</sup>/seconds<sup>2</sup>); this is the product of \e G
* the gravitational constant and \e M the mass of the earth (usually
* including the mass of the earth's atmosphere).
* @param[in] omega the angular velocity (rad s<sup>−1</sup>).
* @param[in] f the flattening of the ellipsoid.
* @return \e J2 the dynamical form factor.
*
* This routine requires \e a > 0, \e GM ≠ 0, \e f < 1. The
* values of these parameters are not checked.
**********************************************************************/
static Math::real FlatteningToJ2(real a, real GM, real omega, real f);
};
} // namespace GeographicLib
#endif // GEOGRAPHICLIB_NORMALGRAVITY_HPP