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GeographicLib::NormalGravity Class Reference

The normal gravity of the earth. More...

#include <NormalGravity.hpp>

Public Member Functions

Setting up the normal gravity
 NormalGravity (real a, real GM, real omega, real f_J2, bool geometricp=true)
 
 NormalGravity (real a, real GM, real omega, real f, real J2)
 
 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 MajorRadius () 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 Math::real FlatteningToJ2 (real a, real GM, real omega, real f)
 
static const NormalGravity & GRS80 ()
 
static Math::real J2ToFlattening (real a, real GM, real omega, real J2)
 
static const NormalGravity & WGS84 ()
 

Private Types

typedef Math::real real
 

Private Member Functions

void Initialize (real a, real GM, real omega, real f_J2, bool geometricp)
 
real Jn (int n) const
 

Static Private Member Functions

static real atan5series (real x)
 
static real atan7series (real x)
 
static real atanzz (real x, bool alt)
 
static real Hf (real x, bool alt)
 
static real Qf (real x, bool alt)
 
static real QH3f (real x, bool alt)
 

Private Attributes

real _a
 
real _aomega2
 
real _b
 
real _E
 
real _e2
 
Geocentric _earth
 
real _ep2
 
real _f
 
real _fstar
 
real _gammae
 
real _gammap
 
real _GM
 
real _J2
 
real _k
 
real _omega
 
real _omega2
 
real _Q0
 
real _U0
 

Static Private Attributes

static const int maxit_ = 20
 

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:

References:

For more information on normal gravity see normalgravity.

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.

Member Typedef Documentation

◆ real

typedef Math::real GeographicLib::NormalGravity::real
private

Definition at line 82 of file NormalGravity.hpp.

Constructor & Destructor Documentation

◆ NormalGravity() [1/3]

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

Constructor for the normal gravity.

Parameters
[in]aequatorial radius (meters).
[in]GMmass 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]omegathe angular velocity (rad s−1).
[in]f_J2either the flattening of the ellipsoid f or the the dynamical form factor J2.
[out]geometricpif true (the default), then f_J2 denotes the flattening, else it denotes the dynamical form factor J2.
Exceptions
ifa 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 src/NormalGravity.cpp.

◆ NormalGravity() [2/3]

GeographicLib::NormalGravity::NormalGravity ( real  a,
real  GM,
real  omega,
real  f,
real  J2 
)
Deprecated:
Old constructor for the normal gravity.
Parameters
[in]aequatorial radius (meters).
[in]GMmass 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]omegathe angular velocity (rad s−1).
[in]fthe flattening of the ellipsoid.
[in]J2the dynamical form factor.
Exceptions
ifa is not positive or the other constants are inconsistent (see below).

If omega is non-zero, then exactly one of f and J2 should be positive and this will be used to define the ellipsoid. The shape of the ellipsoid can be given in one of two ways:

  • geometrically, the ellipsoid is defined by the flattening f = (a − b) / a, where a and b are the equatorial radius and the polar semi-axis.
  • physically, 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.

If omega, f, and J2 are all zero, then the ellipsoid becomes a sphere.

Definition at line 66 of file src/NormalGravity.cpp.

◆ NormalGravity() [3/3]

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 180 of file NormalGravity.hpp.

Member Function Documentation

◆ 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 329 of file NormalGravity.hpp.

◆ atan5series()

Math::real GeographicLib::NormalGravity::atan5series ( real  x)
staticprivate

Definition at line 122 of file src/NormalGravity.cpp.

◆ atan7series()

Math::real GeographicLib::NormalGravity::atan7series ( real  x)
staticprivate

Definition at line 99 of file src/NormalGravity.cpp.

◆ atanzz()

static real GeographicLib::NormalGravity::atanzz ( real  x,
bool  alt 
)
inlinestaticprivate

Definition at line 87 of file NormalGravity.hpp.

◆ DynamicalFormFactor()

Math::real GeographicLib::NormalGravity::DynamicalFormFactor ( int  n = 2) const
inline
Returns
Jn 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 Jn = 0 if n is odd. In most gravity applications, fully normalized Legendre functions are used and the corresponding coefficient is Cn0 = −Jn / sqrt(2 n + 1).

Definition at line 322 of file NormalGravity.hpp.

◆ Earth()

const Geocentric& GeographicLib::NormalGravity::Earth ( ) const
inline
Returns
the Geocentric object used by this instance.

Definition at line 370 of file NormalGravity.hpp.

◆ EquatorialGravity()

Math::real GeographicLib::NormalGravity::EquatorialGravity ( ) const
inline
Returns
γe the normal gravity at equator (m s−2).

Definition at line 343 of file NormalGravity.hpp.

◆ Flattening()

Math::real GeographicLib::NormalGravity::Flattening ( ) const
inline
Returns
f the flattening of the ellipsoid (a − b)/a.

Definition at line 336 of file NormalGravity.hpp.

◆ 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]aequatorial radius (meters).
[in]GMmass 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]omegathe angular velocity (rad s−1).
[in]fthe 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 307 of file src/NormalGravity.cpp.

◆ 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]latthe geographic latitude (degrees).
[in]hthe height above the ellipsoid (meters).
[out]gammaythe northerly component of the acceleration (m s−2).
[out]gammazthe 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 255 of file src/NormalGravity.cpp.

◆ GravityFlattening()

Math::real GeographicLib::NormalGravity::GravityFlattening ( ) const
inline
Returns
f* the gravity flattening (γp − γe) / γe.

Definition at line 357 of file NormalGravity.hpp.

◆ GRS80()

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

A global instantiation of NormalGravity for the GRS80 ellipsoid.

Definition at line 91 of file src/NormalGravity.cpp.

◆ Hf()

Math::real GeographicLib::NormalGravity::Hf ( real  x,
bool  alt 
)
staticprivate

Definition at line 141 of file src/NormalGravity.cpp.

◆ Init()

bool GeographicLib::NormalGravity::Init ( ) const
inline
Returns
true if the object has been initialized.

Definition at line 293 of file NormalGravity.hpp.

◆ Initialize()

void GeographicLib::NormalGravity::Initialize ( real  a,
real  GM,
real  omega,
real  f_J2,
bool  geometricp 
)
private

Definition at line 21 of file src/NormalGravity.cpp.

◆ 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]aequatorial radius (meters).
[in]GMmass 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]omegathe angular velocity (rad s−1).
[in]J2the 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 268 of file src/NormalGravity.cpp.

◆ Jn()

Math::real GeographicLib::NormalGravity::Jn ( int  n) const
private

Definition at line 165 of file src/NormalGravity.cpp.

◆ MajorRadius()

Math::real GeographicLib::NormalGravity::MajorRadius ( ) const
inline
Returns
a the equatorial radius of the ellipsoid (meters). This is the value used in the constructor.

Definition at line 299 of file NormalGravity.hpp.

◆ 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 307 of file NormalGravity.hpp.

◆ Phi()

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

Evaluate the centrifugal acceleration in geocentric coordinates.

Parameters
[in]Xgeocentric coordinate of point (meters).
[in]Ygeocentric coordinate of point (meters).
[out]fXthe X component of the centrifugal acceleration (m s−2).
[out]fYthe 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 239 of file src/NormalGravity.cpp.

◆ PolarGravity()

Math::real GeographicLib::NormalGravity::PolarGravity ( ) const
inline
Returns
γp the normal gravity at poles (m s−2).

Definition at line 350 of file NormalGravity.hpp.

◆ Qf()

Math::real GeographicLib::NormalGravity::Qf ( real  x,
bool  alt 
)
staticprivate

Definition at line 130 of file src/NormalGravity.cpp.

◆ QH3f()

Math::real GeographicLib::NormalGravity::QH3f ( real  x,
bool  alt 
)
staticprivate

Definition at line 153 of file src/NormalGravity.cpp.

◆ SurfaceGravity()

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

Evaluate the gravity on the surface of the ellipsoid.

Parameters
[in]latthe 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 177 of file src/NormalGravity.cpp.

◆ 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 364 of file NormalGravity.hpp.

◆ 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]Xgeocentric coordinate of point (meters).
[in]Ygeocentric coordinate of point (meters).
[in]Zgeocentric coordinate of point (meters).
[out]gammaXthe X component of the acceleration (m s−2).
[out]gammaYthe Y component of the acceleration (m s−2).
[out]gammaZthe 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 246 of file src/NormalGravity.cpp.

◆ 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]Xgeocentric coordinate of point (meters).
[in]Ygeocentric coordinate of point (meters).
[in]Zgeocentric coordinate of point (meters).
[out]GammaXthe X component of the acceleration due to the gravitational force (m s−2).
[out]GammaYthe Y component of the acceleration due to the
[out]GammaZthe 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 183 of file src/NormalGravity.cpp.

◆ WGS84()

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

A global instantiation of NormalGravity for the WGS84 ellipsoid.

Definition at line 83 of file src/NormalGravity.cpp.

Friends And Related Function Documentation

◆ GravityModel

friend class GravityModel
friend

Definition at line 83 of file NormalGravity.hpp.

Member Data Documentation

◆ _a

real GeographicLib::NormalGravity::_a
private

Definition at line 84 of file NormalGravity.hpp.

◆ _aomega2

real GeographicLib::NormalGravity::_aomega2
private

Definition at line 84 of file NormalGravity.hpp.

◆ _b

real GeographicLib::NormalGravity::_b
private

Definition at line 85 of file NormalGravity.hpp.

◆ _E

real GeographicLib::NormalGravity::_E
private

Definition at line 85 of file NormalGravity.hpp.

◆ _e2

real GeographicLib::NormalGravity::_e2
private

Definition at line 85 of file NormalGravity.hpp.

◆ _earth

Geocentric GeographicLib::NormalGravity::_earth
private

Definition at line 86 of file NormalGravity.hpp.

◆ _ep2

real GeographicLib::NormalGravity::_ep2
private

Definition at line 85 of file NormalGravity.hpp.

◆ _f

real GeographicLib::NormalGravity::_f
private

Definition at line 84 of file NormalGravity.hpp.

◆ _fstar

real GeographicLib::NormalGravity::_fstar
private

Definition at line 85 of file NormalGravity.hpp.

◆ _gammae

real GeographicLib::NormalGravity::_gammae
private

Definition at line 85 of file NormalGravity.hpp.

◆ _gammap

real GeographicLib::NormalGravity::_gammap
private

Definition at line 85 of file NormalGravity.hpp.

◆ _GM

real GeographicLib::NormalGravity::_GM
private

Definition at line 84 of file NormalGravity.hpp.

◆ _J2

real GeographicLib::NormalGravity::_J2
private

Definition at line 84 of file NormalGravity.hpp.

◆ _k

real GeographicLib::NormalGravity::_k
private

Definition at line 85 of file NormalGravity.hpp.

◆ _omega

real GeographicLib::NormalGravity::_omega
private

Definition at line 84 of file NormalGravity.hpp.

◆ _omega2

real GeographicLib::NormalGravity::_omega2
private

Definition at line 84 of file NormalGravity.hpp.

◆ _Q0

real GeographicLib::NormalGravity::_Q0
private

Definition at line 85 of file NormalGravity.hpp.

◆ _U0

real GeographicLib::NormalGravity::_U0
private

Definition at line 85 of file NormalGravity.hpp.

◆ maxit_

const int GeographicLib::NormalGravity::maxit_ = 20
staticprivate

Definition at line 81 of file NormalGravity.hpp.

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

gtsam
Author(s):
autogenerated on Tue Jul 4 2023 02:46:12