The normal gravity of the earth. More...

#include <GeographicLib/NormalGravity.hpp>

Public Member Functions
Setting up the normal gravity
	NormalGravity (real a, real GM, real omega, real f_J2, bool geometricp=true)

	NormalGravity ()

Compute the gravity
Math::real	SurfaceGravity (real lat) const

Math::real	Gravity (real lat, real h, real &gammay, real &gammaz) const

Math::real	U (real X, real Y, real Z, real &gammaX, real &gammaY, real &gammaZ) const

Math::real	V0 (real X, real Y, real Z, real &GammaX, real &GammaY, real &GammaZ) const

Math::real	Phi (real X, real Y, real &fX, real &fY) const

Inspector functions
bool	Init () const

Math::real	EquatorialRadius () const

Math::real	MassConstant () const

Math::real	DynamicalFormFactor (int n=2) const

Math::real	AngularVelocity () const

Math::real	Flattening () const

Math::real	EquatorialGravity () const

Math::real	PolarGravity () const

Math::real	GravityFlattening () const

Math::real	SurfacePotential () const

const Geocentric &	Earth () const

Static Public Member Functions
static const NormalGravity &	WGS84 ()

static const NormalGravity &	GRS80 ()

static Math::real	J2ToFlattening (real a, real GM, real omega, real J2)

static Math::real	FlatteningToJ2 (real a, real GM, real omega, real f)

Friends
class	GravityModel

Detailed Description

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, (a² − b²)^1/2, where a is the equatorial radius of the ellipsoid and b is its polar semi-axis. In the case of an oblate ellipsoid, the mass is concentrated on a "focal rod" of length 2(b² − a²)^1/2. 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 b/a ∈ [0.01, 100] or f ∈ [−99, 0.99].

Definitions:

V₀, the gravitational contribution to the normal potential;
Φ, the rotational contribution to the normal potential;
U = V₀ + Φ, the total potential;
Γ = ∇V₀, the acceleration due to mass of the earth;
f = ∇Φ, the centrifugal acceleration;
γ = ∇U = Γ + f, the normal acceleration;
X, Y, Z, geocentric coordinates;
x, y, 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, 4, 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).
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 Normal gravity.

Example of use:

// Example of using the GeographicLib::NormalGravity class
 
#include <iostream>
#include <exception>
#include <GeographicLib/NormalGravity.hpp>
#include <GeographicLib/Constants.hpp>
 
using namespace std;
using namespace GeographicLib;
 
int main() {
  try {
    NormalGravity grav(Constants::WGS84_a(), Constants::WGS84_GM(),
                       Constants::WGS84_omega(), Constants::WGS84_f());
    // Alternatively: const NormalGravity& grav = NormalGravity::WGS84();
    double lat = 27.99, h = 8820; // Mt Everest
    double gammay, gammaz;
    grav.Gravity(lat, h, gammay, gammaz);
    cout << gammay << " " << gammaz << "\n";
  }
  catch (const exception& e) {
    cerr << "Caught exception: " << e.what() << "\n";
    return 1;
  }
}

Definition at line 79 of file NormalGravity.hpp.

Constructor & Destructor Documentation

◆ NormalGravity() [1/2]

GeographicLib::NormalGravity::NormalGravity	(	real	a,
		real	GM,
		real	omega,
		real	f_J2,
		bool	geometricp = `true`
	)

Constructor for the normal gravity.

Parameters

[in]	a	equatorial radius (meters).
[in]	GM	mass constant of the ellipsoid (meters³/seconds²); this is the product of G the gravitational constant and M the mass of the earth (usually including the mass of the earth's atmosphere).
[in]	omega	the angular velocity (rad s⁻¹).
[in]	f_J2	either the flattening of the ellipsoid f or the the dynamical form factor J2.
[out]	geometricp	if true (the default), then f_J2 denotes the flattening, else it denotes the dynamical form factor J2.

Exceptions

if	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 (geomtricp = true), the ellipsoid is defined by the flattening f = (a − b) / a, where a and b are the equatorial radius and the polar semi-axis. The parameters should obey a > 0, f < 1. There are no restrictions on GM or omega, in particular, GM need not be positive.
physically (geometricp = false), the ellipsoid is defined by the dynamical form factor J₂ = (C − A) / Ma², where A and C are the equatorial and polar moments of inertia and M is the mass of the earth. The parameters should obey a > 0, GM > 0 and J2 < 1/3 − (omega²a³/GM) 8/(45π). There is no restriction on omega.

Definition at line 61 of file NormalGravity.cpp.

◆ NormalGravity() [2/2]

GeographicLib::NormalGravity::NormalGravity ( )

inline

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.

Definition at line 151 of file NormalGravity.hpp.

Member Function Documentation

◆ SurfaceGravity()

Math::real GeographicLib::NormalGravity::SurfaceGravity ( real lat ) const

Evaluate the gravity on the surface of the ellipsoid.

Parameters

[in] lat the geographic latitude (degrees).

Returns: γ the acceleration due to gravity, positive downwards (m s⁻²).

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.

Definition at line 159 of file NormalGravity.cpp.

References GeographicLib::Math::LatFix(), GeographicLib::Math::sind(), and GeographicLib::Math::sq().

Referenced by GeographicLib::GravityModel::Circle(), and GeographicLib::GravityModel::GeoidHeight().

◆ Gravity()

Math::real GeographicLib::NormalGravity::Gravity	(	real	lat,
		real	h,
		real &	gammay,
		real &	gammaz
	)		const

Evaluate the gravity at an arbitrary point above (or below) the ellipsoid.

Parameters

[in]	lat	the geographic latitude (degrees).
[in]	h	the height above the ellipsoid (meters).
[out]	gammay	the northerly component of the acceleration (m s⁻²).
[out]	gammaz	the upward component of the acceleration (m s⁻²); this is usually negative.

Returns: U the corresponding normal potential (m² s⁻²).

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, gammax = 0. The function includes the effects of the earth's rotation. When h = 0, this function gives gammay = 0 and the returned value matches that of NormalGravity::SurfaceGravity.

Definition at line 237 of file NormalGravity.cpp.

References U().

◆ U()

Math::real GeographicLib::NormalGravity::U	(	real	X,
		real	Y,
		real	Z,
		real &	gammaX,
		real &	gammaY,
		real &	gammaZ
	)		const

Evaluate the components of the acceleration due to gravity and the centrifugal acceleration in geocentric coordinates.

Parameters

[in]	X	geocentric coordinate of point (meters).
[in]	Y	geocentric coordinate of point (meters).
[in]	Z	geocentric coordinate of point (meters).
[out]	gammaX	the X component of the acceleration (m s⁻²).
[out]	gammaY	the Y component of the acceleration (m s⁻²).
[out]	gammaZ	the Z component of the acceleration (m s⁻²).

Returns: U = V₀ + Φ the sum of the gravitational and centrifugal potentials (m² s⁻²).

The acceleration given by γ = ∇U = ∇V₀ + ∇Φ = Γ + f.

Definition at line 228 of file NormalGravity.cpp.

References Phi(), and V0().

Referenced by GeographicLib::GravityModel::Circle(), Gravity(), GeographicLib::GravityModel::SphericalAnomaly(), and GeographicLib::GravityModel::U().

◆ V0()

Math::real GeographicLib::NormalGravity::V0	(	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.

Parameters

[in]	X	geocentric coordinate of point (meters).
[in]	Y	geocentric coordinate of point (meters).
[in]	Z	geocentric coordinate of point (meters).
[out]	GammaX	the X component of the acceleration due to the gravitational force (m s⁻²).
[out]	GammaY	the Y component of the acceleration due to the
[out]	GammaZ	the Z component of the acceleration due to the gravitational force (m s⁻²).

Returns: V₀ the gravitational potential (m² s⁻²).

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.

Definition at line 165 of file NormalGravity.cpp.

References GeographicLib::Math::pi(), GeographicLib::Math::sq(), and std::swap().

Referenced by U().

◆ Phi()

Math::real GeographicLib::NormalGravity::Phi	(	real	X,
		real	Y,
		real &	fX,
		real &	fY
	)		const

Evaluate the centrifugal acceleration in geocentric coordinates.

Parameters

[in]	X	geocentric coordinate of point (meters).
[in]	Y	geocentric coordinate of point (meters).
[out]	fX	the X component of the centrifugal acceleration (m s⁻²).
[out]	fY	the Y component of the centrifugal acceleration (m s⁻²).

Returns: Φ the centrifugal potential (m² s⁻²).

Φ is independent of Z, thus fZ = 0. This function NormalGravity::U sums the results of NormalGravity::V0 and NormalGravity::Phi.

Definition at line 221 of file NormalGravity.cpp.

References GeographicLib::Math::sq().

Referenced by GeographicLib::GravityModel::Circle(), GeographicLib::GravityModel::Phi(), U(), and GeographicLib::GravityModel::W().

◆ Init()

bool GeographicLib::NormalGravity::Init ( ) const

inline

Returns: true if the object has been initialized.

Definition at line 264 of file NormalGravity.hpp.

◆ EquatorialRadius()

Math::real GeographicLib::NormalGravity::EquatorialRadius ( ) const

inline

Returns: a the equatorial radius of the ellipsoid (meters). This is the value used in the constructor.

Definition at line 270 of file NormalGravity.hpp.

References GeographicLib::Math::NaN().

Referenced by GeographicLib::GravityModel::EquatorialRadius().

◆ MassConstant()

Math::real GeographicLib::NormalGravity::MassConstant ( ) const

inline

Returns: GM the mass constant of the ellipsoid (m³ s⁻²). This is the value used in the constructor.

Definition at line 278 of file NormalGravity.hpp.

References GeographicLib::Math::NaN().

Referenced by GeographicLib::GravityModel::GravityModel(), and GeographicLib::GravityModel::ReferenceMassConstant().

◆ DynamicalFormFactor()

Math::real GeographicLib::NormalGravity::DynamicalFormFactor ( int n = 2 ) const

inline

Returns: J_n the dynamical form factors of the ellipsoid.

If n = 2 (the default), this is the value of J₂ used in the constructor. Otherwise it is the zonal coefficient of the Legendre harmonic sum of the normal gravitational potential. Note that J_n = 0 if n is odd. In most gravity applications, fully normalized Legendre functions are used and the corresponding coefficient is C_n0 = −J_n / sqrt(2 n + 1).

Definition at line 293 of file NormalGravity.hpp.

References GeographicLib::Math::NaN().

◆ AngularVelocity()

Math::real GeographicLib::NormalGravity::AngularVelocity ( ) const

inline

Returns: ω the angular velocity of the ellipsoid (rad s⁻¹). This is the value used in the constructor.

Definition at line 300 of file NormalGravity.hpp.

References GeographicLib::Math::NaN().

Referenced by GeographicLib::GravityModel::AngularVelocity().

◆ Flattening()

Math::real GeographicLib::NormalGravity::Flattening ( ) const

inline

Returns: f the flattening of the ellipsoid (a − b)/a.

Definition at line 307 of file NormalGravity.hpp.

References GeographicLib::Math::NaN().

Referenced by GeographicLib::GravityModel::Flattening().

◆ EquatorialGravity()

Math::real GeographicLib::NormalGravity::EquatorialGravity ( ) const

inline

Returns: γ_e the normal gravity at equator (m s⁻²).

Definition at line 314 of file NormalGravity.hpp.

References GeographicLib::Math::NaN().

◆ PolarGravity()

Math::real GeographicLib::NormalGravity::PolarGravity ( ) const

inline

Returns: γ_p the normal gravity at poles (m s⁻²).

Definition at line 321 of file NormalGravity.hpp.

References GeographicLib::Math::NaN().

◆ GravityFlattening()

Math::real GeographicLib::NormalGravity::GravityFlattening ( ) const

inline

Returns: f* the gravity flattening (γ_p − γ_e) / γ_e.

Definition at line 328 of file NormalGravity.hpp.

References GeographicLib::Math::NaN().

◆ SurfacePotential()

Math::real GeographicLib::NormalGravity::SurfacePotential ( ) const

inline

Returns: U₀ the constant normal potential for the surface of the ellipsoid (m² s⁻²).

Definition at line 335 of file NormalGravity.hpp.

References GeographicLib::Math::NaN().

◆ Earth()

const Geocentric & GeographicLib::NormalGravity::Earth ( ) const

inline

Returns: the Geocentric object used by this instance.

Definition at line 341 of file NormalGravity.hpp.

Referenced by GeographicLib::GravityModel::Circle(), GeographicLib::GravityModel::Disturbance(), GeographicLib::GravityModel::GeoidHeight(), GeographicLib::GravityModel::Gravity(), and GeographicLib::GravityModel::SphericalAnomaly().

◆ WGS84()

const NormalGravity & GeographicLib::NormalGravity::WGS84 ( )

static

A global instantiation of NormalGravity for the WGS84 ellipsoid.

Definition at line 66 of file NormalGravity.cpp.

References GeographicLib::Constants::WGS84_a(), GeographicLib::Constants::WGS84_f(), GeographicLib::Constants::WGS84_GM(), and GeographicLib::Constants::WGS84_omega().

◆ GRS80()

const NormalGravity & GeographicLib::NormalGravity::GRS80 ( )

static

A global instantiation of NormalGravity for the GRS80 ellipsoid.

Definition at line 74 of file NormalGravity.cpp.

References GeographicLib::Constants::GRS80_a(), GeographicLib::Constants::GRS80_GM(), GeographicLib::Constants::GRS80_J2(), and GeographicLib::Constants::GRS80_omega().

◆ J2ToFlattening()

Math::real GeographicLib::NormalGravity::J2ToFlattening	(	real	a,
		real	GM,
		real	omega,
		real	J2
	)

static

Compute the flattening from the dynamical form factor.

Parameters

[in]	a	equatorial radius (meters).
[in]	GM	mass constant of the ellipsoid (meters³/seconds²); this is the product of G the gravitational constant and M the mass of the earth (usually including the mass of the earth's atmosphere).
[in]	omega	the angular velocity (rad s⁻¹).
[in]	J2	the dynamical form factor.

Returns: f the flattening of the ellipsoid.

This routine requires a > 0, GM > 0, J2 < 1/3 − omega²a³/GM 8/(45π). A NaN is returned if these conditions do not hold. The restriction to positive GM is made because for negative GM two solutions are possible.

Definition at line 250 of file NormalGravity.cpp.

References GEOGRAPHICLIB_PANIC, GeographicLib::Math::NaN(), GeographicLib::Math::pi(), and GeographicLib::Math::sq().

◆ FlatteningToJ2()

Math::real GeographicLib::NormalGravity::FlatteningToJ2	(	real	a,
		real	GM,
		real	omega,
		real	f
	)

static

Compute the dynamical form factor from the flattening.

Parameters

[in]	a	equatorial radius (meters).
[in]	GM	mass constant of the ellipsoid (meters³/seconds²); this is the product of G the gravitational constant and M the mass of the earth (usually including the mass of the earth's atmosphere).
[in]	omega	the angular velocity (rad s⁻¹).
[in]	f	the flattening of the ellipsoid.

Returns: J2 the dynamical form factor.

This routine requires a > 0, GM ≠ 0, f < 1. The values of these parameters are not checked.

Definition at line 289 of file NormalGravity.cpp.

References GeographicLib::Math::sq().

Friends And Related Function Documentation

◆ GravityModel

friend class GravityModel

friend

Definition at line 83 of file NormalGravity.hpp.

The documentation for this class was generated from the following files:

Public Member Functions

Static Public Member Functions

Friends

Detailed Description

Constructor & Destructor Documentation

◆ NormalGravity() [1/2]

◆ NormalGravity() [2/2]

Member Function Documentation

◆ SurfaceGravity()

◆ Gravity()

◆ U()

◆ V0()

◆ Phi()

◆ Init()

◆ EquatorialRadius()

◆ MassConstant()

◆ DynamicalFormFactor()

◆ AngularVelocity()

◆ Flattening()

◆ EquatorialGravity()

◆ PolarGravity()

◆ GravityFlattening()

◆ SurfacePotential()

◆ Earth()

◆ WGS84()

◆ GRS80()

◆ J2ToFlattening()

◆ FlatteningToJ2()

Friends And Related Function Documentation

◆ GravityModel