page_title: Geopotential model
section_title: The deviations of Earth's gravitational field from that of a homogeneous sphere
hierarchical_section_title: Geopotential model / The deviations of Earth's gravitational field from that of a homogeneous sphere
context_page_description: In geophysics, a geopotential model is the theoretical analysis of measuring and calculating the effects of Earth's gravitational field.
context_section_description: In reality, Earth is not exactly spherical, mainly because of its rotation around the polar axis that makes its shape slightly oblate. If this shape were perfectly known together with the exact mass density ρ = ρ(x, y, z), the integrals (1) and (2) could be evaluated with numerical methods to find a more accurate model for Earth's gravitational field. However, the situation is in fact the opposite. By observing the orbits of spacecraft and the Moon, Earth's gravitational field can be determined quite accurately and the best estimate of Earth's mass is obtained by dividing the product GM as determined from the analysis of spacecraft orbit with a value for G determined to a lower relative accuracy using other physical methods.
  From the defining equations (1) and (2) it is clear (taking the partial derivatives of the integrand) that outside the body in empty space the following differential equations are valid for the field caused by the body:
  Functions of the form 
  where (r, θ, φ) are the spherical coordinates which satisfy the partial differential equation (6) (the Laplace equation) are called spherical harmonic functions.
  They take the forms:
  where spherical coordinates (r, θ, φ) are used, given here in terms of cartesian (x, y, z) for reference:
  also P⁰ₙ are the Legendre polynomials and Pᵐₙ for 1 ≤ m ≤ n are the associated Legendre functions.
  The first spherical harmonics with n = 0,1,2,3 are presented in the table below.
  The model for Earth's gravitational potential is a sum
  where  and the coordinates (8) are relative the standard geodetic reference system extended into space with origin in the center of the reference ellipsoid and with z-axis in the direction of the polar axis.
  The zonal terms refer to terms of the form:
  and the tesseral terms terms refer to terms of the form:
  The zonal and tesseral terms for n = 1 are left out in (9).  The coefficients for the n=1 with both m=0 and m=1 term correspond to an arbitrarily oriented dipole term in the multi-pole expansion. Gravity does not physically exhibit any dipole character and so the integral characterizing n = 1 must be zero.
  The different coefficients Jₙ, Cₙᵐ, Sₙᵐ, are then given the values for which the best possible agreement between the computed and the observed spacecraft orbits is obtained.
  As P⁰ₙ(x) = −P⁰ₙ(−x) non-zero coefficients Jₙ for odd n correspond to a lack of symmetry "north–south" relative the equatorial plane for the mass distribution of Earth. Non-zero coefficients Cₙᵐ, Sₙᵐ correspond to a lack of rotational symmetry around the polar axis for the mass distribution of Earth, i.e. to a "tri-axiality" of Earth.
  For large values of n the coefficients above (that are divided by r in (9)) take very large values when for example kilometers and seconds are used as units. In the literature it is common to introduce some arbitrary "reference radius" R close to Earth's radius and to work with the dimensionless coefficients
  and to write the potential as
  The dominating term (after the term −μ/r) in (9) is the "J₂ term":
  Relative the coordinate system
  illustrated in figure 1 the components of the force caused by the "J₂ term" are
  In the rectangular coordinate system (x, y, z) with unit vectors (x̂ ŷ ẑ) the force components are:
  The components of the force corresponding to the "J₃ term"
  are
  and
  The exact numerical values for the coefficients deviate (somewhat) between different Earth models but for the lowest coefficients they all agree almost exactly.
  For JGM-3 the values are:
  μ = 398600.440 km³⋅s⁻²
  J₂ = 1.75553 × 10¹⁰ km⁵⋅s⁻²
  J₃ = −2.61913 × 10¹¹ km⁶⋅s⁻²
  
  For example, at a radius of 6600 km (about 200 km above Earth's surface) J₃/(J₂r) is about 0.002, i.e.
  the correction to the "J₂ force" from the "J₃ term" is in the order of 2 permille. The negative value of J₃ implies that for a point mass in Earth's equatorial plane the gravitational force is tilted slightly towards the south due to the lack of symmetry for the mass distribution of Earth's "north–south".
caption: Figure 1: The unit vectors. This is wrong. There should be a theta, not lambda 
    
      
        
          
            
              
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      {\displaystyle {\hat {\varphi }}\ ,\ {\hat {\theta }}\ ,\ {\hat {r}}}