# Stokes Problem

## Stokes’ Problem

The problem of determining the external gravitation field of a planet from the planet’s external equipoten-tial surface *S*, the mass within *S*, and the angular velocity of rotation about some axis is sometimes referred to as Stokes’ problem. G. G. Stokes showed that the problem can be solved and provided an approximate solution for an oblate spheroid with a relative error of the order of the square of the spheroid’s flattening. He treated the problem as a first boundary value problem. An exact solution of Stokes’ problem for an ellipsoid was put forth by the Italian mathematician P. Pizzetti and by M. S. Molodenskii.

To an arbitrary shape of *S* there corresponds the boundary condition

and the equation for φ

under the condition

∫ φ *ds* = 0

Here, ζ is the height of *S* above the reference ellipsoid *S*_{0} containing the given mass, *T* is the disturbing potential defined by the equation

φ is the density of the simple layer at *S*, *W*_{0} is the gravitational potential at the point where ζ = 0 on the intersection of *S* and *S*_{0}, U_{0} is the corresponding potential on *S*_{0}, γ is the gravity in the field of the ellipsoid, *r* is the distance between the element *ds* and the point on *S* with height ζ, and r_{0} is the distance between *ds* and the point where ζ = 0. The axes of rotation of *S* and *S*_{0} coincide. The equation for φ can be replaced by a system of linear algebraic equations. The determination of φ solves the problem known as Stokes’ problem.

The solution presented above is also applicable to the case where *S* is a nonequipotential surface and ζ is the height of the quasi-geoid (*see*GEOID).

### REFERENCES

Molodenskii, M. S., V. F. Eremeev, and M. I. Iurkina.*Metody izucheniia vneshnego gravitatsionnogo polia i figury Zemli. (Tr. Tsentr. n.-i. in-ta geodezii, aeros’emki i kartografii*, fasc. 131.) Moscow, 1960.

Stokes, G. G. “On Attractions and on Clairaut’s Theorem.”

*Cambridge and Dublin Mathematical Journal*, 1849, vol. 4.

M. I. IURKINA