Bruns formula

Remarks

The proof is a one-term Taylor expansion of $U$ along the ellipsoidal normal. Normal gravity points inward along this normal, so $U(P) \approx U(Q) - \gamma(Q) N = W_0 - \gamma(Q) N$. Since $W(P) = W_0$, this gives $T(P) = W(P) - U(P) \approx \gamma(Q) N$. The neglected terms are of order $N^2$ times the second derivative of $U$, which is negligible because $N$ is at most about $100$ m. Example: a disturbing potential $T(P) = 98\ \mathrm{m^2/s^2}$ with $\gamma \approx 9.8\ \mathrm{m/s^2}$ gives a geoid height of $10$ m.

Going deeper

For a student

Bruns formula turns the geoid problem into a problem about the harmonic function $T$. Combined with the boundary condition $\partial T/\partial r + 2T/R = -\Delta g$ for the gravity anomaly, it leads to the geodetic boundary value problem and to Stokes' formula. It is also used in reverse, to turn a global spherical harmonic model of $T$ into geoid heights.