The formula of MacCullagh is an important formula to describe the revolved field of of a nonspherical body at a distance sufficiently large but not very large compared to linear dimensions of the body. It is obtained on the basis of the basic definition of the potential gravific, without carrying out a general development of Laplace whose it represents a particular case limited to an approximation of order 2. It is very much used in theoretical geodesy and Géophysique.

General expression of the potential gravific

In Coordinated spherical, let us indicate the radial component of the vector-position R of a point potentié P by R, and that of the vector-position R ′ of a point potentiant P ′ by R ′. The origin O of the frame of reference is for the moment still arbitrary. The angle ranging between the vectors R and R ′ is ψ and the element of mass out of P ′ is DM ( R ′ ), the total mass of the body B being Mr. Under these conditions, the potential gravific V (P) generated at the point P by the body B can be written in any general information

V (P) = G ∫_M |  R ′ - R  |^-1 DM ( R ′ ),

with

|  R ′ - R  |^-1 = r^{-1 -1/2}. Constant G indicates the Constante gravitation of Newton.

Like (1 + X) ^-1/2 = 1 - ½ X + ⅜ X ² + O (X ³) when - 1 ≤ X ≤ +1, one finds, by posing X = - 2 (R ′ /r) cos ψ + (R ′ /r) ²:

^-1/2 = 1 + (R ′ /r) cos ψ - (1/2) (R ′ /r) ² + (3/2) (R ′ /r) ² cos ² ψ + O. 

Consequently, one has

R d^-1 (P, P ′) = 1 + (R ′ /r) cos ψ + (R ′ /r) ² - (3/2) (R ′ /r) ² sin ² ψ +….

The potential gravific out of P is thus

V (P) = G r^-1 ∫_M DM + G r^-2 ∫_M R ′ cos ψ DM + G r^-3 ∫_M R ′ ² DM - (3/2) G r^-3 ∫_M R ′ ² sin ² ψ DM +… .

Theorem of MacCullagh

One sees thus that with the approximation of order 2, the potential gravific V (P) can break up into four distinct terms V₀ (P), V₁ (P), V₂ (P), V₃ (P), that is to say

V (P) = V₀ (P) + V₁ (P) + V₂ (P) + V₃ (P).

The first term,

V₀ = G r^-1 ∫_M DM = GM r^-1,

corresponds to the potential of a spherical distribution, which is the same one as that of a specific mass equal to the mass included/understood in all the sphere. With long distance, it is obviously the term dominating, since its decrease in 1/r is slower than that of the other terms. It is the traditional Newtonian potential used to establish the Keplerian trajectories of planets. One says of him that it corresponds to a Monopôle , or with a distribution of monopolar mass .

The second term,

V₁ = G r^-2 ∫_M R ′ cos ψ DM,

to a distribution of dipolar mass corresponds, in other words with a Dipôle . By choosing the origin of the coordinates in the center of mass of the body, one cancels this term. Indeed, while referring to the figure opposite, one can write

V₁ = G r^-2 ∫_M R ′ cos ψ DM = G r^-2 ∫_M X DM = GM r^-2 x_o,

where x_o indicates the component along an axis OX of the position of the center of mass.

The third and fourth terms, V₂ and V₃, refer to a distribution of quadrupolar mass , in other words with a Quadrupôle . For V₂ one finds successively

V₂ = G r^-3 ∫_M R ′ ² DM = G r^-3 ∫_M (X ² +y ² +z ²) DM = ½ G r^-3 ∫_M DM = ½ G r^-3 (A+B+C),

where has, B, C indicate the three moments of inertia compared to axes OX, OY, OZ, respectively. By defining the average moment of inertia Ī by

Ī = ⅓ (has + B + C),

one thus has

V₂ = (3/2) G r^-3 Ī.

For V₃, one finds

V₃ = - (3/2) G r^-3 ∫_M R ′ ² sin ² ψ DM = - (3/2) G r^-3 ∫_M R ′ ² cos ² α DM = - (3/2) G r^-3 I,

where I is the moment of inertia around the direction OP. One leads thus to the formula (sometimes also called theorem ) of MacCullagh :

V (P) = GM r^-1 - (3/2) G r^-3 (I - Ī).

This one applies very usefully to almost spherical bodies like the Earth and planets, and it is a still valid approximation remotely sufficient for a body of unspecified symmetry. There exists obviously an implicit limitation with its validity by the fact that the series was truncated with order 2: harmonics higher than degree 2 are of course necessary to represent in a more precise way the external field of gravity in any general information.

Formulate of MacCullagh for the bodies with symmetry of revolution

The Moment of inertia I around an axis passing by the center of mass O of a body can be written in terms of the principal moments of inertia has, B, C obtained by preliminary Diagonalisation of the matrix 3x3 representing the Tenseur inertia by means of the relation

I = n_x ² has + n_y ² B + n_z ² C.

Here, n_x, n_y, n_z designate the cosine directors of the axis OP compared to the main axes of inertia OX, OY, OZ, respectively.

By making the assumption that the body is of revolution around the axis OZ, i.e. it has a axial symmetry of axis OZ, one has = B. Indicate by φ the angle which the direction OP with the Oxy plan. Then n_z = sin φ and, because of the relation binding n_x ² + n_y ² + n_z ² = 1 the cosine directors, one finds

sin ² φ = 1 - n_x ² - n_y ².

With has = B, the formula of MacCullagh becomes

V (P) = GM r^-1 - ½ G r^-3 (3 I - 2 has - C).

In the case of an axial symmetry, the moment of inertia I takes itself the form

I = has (n_x ² +n_y ²) + C sin ² φ = has (1 - sin ² φ) + C sin ² φ.

The formula of MacCullagh, for a body of revolution, thus takes finally the following form:

V (P) = GM r^-1 + ½ G r^-3 (C - A) (1 - 3 sin ² φ).

Geodynamic factor of form

In addition, always on the assumption of an axial symmetry, the general development of the potential gravific in multipôles limited under the terms of order 2 is written

V (P) = GM r^-1 - J₂ GM has ² r^-3 P₂ (cos θ),

where

P₂ (cos θ) = (3/2) sin ² φ - 1/2

is the polynomial of Legendre of degree 2. One thus has

V (P) = GM r^-1 - ½ J₂ GM has ² r^-3 (1 - 3 sin ² φ)

In the case of the Earth, the zonal geopotential coefficient of degree 2, namely J₂, is often called “factor of geodynamic form”. One obtains it by identification of this last formula with that of MacCullagh, that is to say

J₂ = (C - A) (My ²).

Certain geodesists call also this last relation “formulates of MacCullagh”, but this use does not seem very judicious.