**H:\EXCERC\GEOD98.EX3e.wpd**

**Department of Geophysics, University of Copenhagen.**

Juliane Maries Vej 30, 2100 København Ø.

**Geodesy Course Fall 1999. Exercise 3.**

**1. ** The centrifugal potential is given by

Beregn værdien på Ækvator udtrykt i m^{2}/s^{2}. Benyt værdien af på side 17 i Torge, og
ellipsoideparametre som på side 57, formel (3.68).

**2.** A point has the coordinates = 56^{o}, = 10^{o}, h = 0 m, as in exercise 1.1, where (X,Y,Z), r
etc. Has been calculated.

What is the value of (see 3.1) ?

**3.** Calculate the gradient of in the point in 3.2 expressed in local spherical coordinates,

What is the contribution to the gravity vector in the point ?

**4.** The Moon is considered a homogeneous sphere with mass M = 735 * 10^{20} kg, and radius
1717.7 km. What is the gravity on the Moon ?

**5.** What is the attraction of the Moon on 3 points situated on the line connecting the moon
and the Earth's center, where one point is the Earth's center and the two other on the surface
of the Earth (considered spherical, with radius 6371 km) ? The distance to the Moon is fixed
to 60 times the Earth radius. What is the potential of the attraction of the Moon in the 3 points
?

What is the magnitude of the tides in the two surface points ?

**6. ** The Earth is regarded as spherical as in 3.5, and homogeneous with GM = 3.98 * 10^{14}
m^{3}/s^{2}. Calculate the **gradient** in a point with spherical coordinates as used in 3.2.

**7.** Calculate the value of the Laplace operator applied on in 3.1

**8.** Which rotation velocity should the Earth have at Equator in order that the centrifugal
potential has an attraction equal to the attraction of the Earth (again regarded, spherical,
homogeneous) ?

**9. **Find the expression for P_{3}(t) using equation (2.46) in Torge.

**10.** Show that P_{00} og P_{10} are orthogonal on the unit-sphere.

**11.** Show that the function

is harmonic (fulfill the Laplace equation).

**12.** Express the function in 3.11 in Cartesian coordinates (X,Y,Z).

**13.** A potential is given by

with constants (GM, a, J_{2}, ) from GRS80.

Calculate the potential at the following points, all having the height above the ellipsoid equal to zero:

North pole, Equator with = 0 and the point in 3.2 .

Note the order of magnitude in the variations of W between the 3 points. What is the
corresponding heights of the geoid in GRS80.

**14. **Use the same potential as in 3.13.

Calculate the gravity vector in the 3 points and in the altitude of 100 m above the point on
Equator.