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eigenvectors the eigenvalues can easily be found from expression (17.59).

Problem b: Take twice the inner product of (17.59) with the eigenvector v(j) to show

^

that

= v(j) T v(j) :

^ ^ (17.61)

j

Problem c: Use this with expression (17.58) to show that the eigenvalues are given by:

Z

(r) 2x2 y2 z 2 dV

x= ; ;

Z

(r) 2y2 x2 z 2 dV

y= (17.62)

; ;

Z

(r) 2z 2 x2 y2 dV :

=

z ; ;

It will be useful to relate these eigenvalues to the Earth's moments of inertia. The moment

of inertia of the Earth around the z-axis is de ned as:

Z

(r) x2 + y2 dV

C (17.63)

whereas the moment of inertia around the x-axis is de ned as

Z

(r) y2 + z 2 dV :

A (17.64)

By symmetry the moment of inertia around the y-axis is given by the same moment

A. These moments describe the rotational inertia around the coordinate axes as shown

17.7. THE QUADRUPOLE FIELD OF THE EARTH 259

C

z-axis

y-axis

x-axis

A

A

Figure 17.8: De nition of the moments of inertia A and C for an Earth with cylinder

symmetry around the rotation axis.

in gure 17.8. The eigenvalues in (17.62) can be related to these moments of iner-

tia. Because of the assumed axi-symmetric density distribution, the integral of y2 in

(17.62) is equal to the integral of x2 . The eigenvalue x in (17.62) is therefore given by:

R; R;

x = (r) x2 z 2 dV = (r) x2 + y2 y2 z 2 dV = C A.

; ; ; ;

Problem d: Apply a similar treatment to the other eigenvalues to show that:

=C A A)

x= y = (17.65)

z

; ;2(C ;

Problem e: Use these eigenvalues in (17.59) and use (17.56) and expression (3.7) for the

unit vector ^ to show that the quadrupole term of the potential is given by:

r

G

Vqua (r) = 2r3 (C A) 3 cos2 1: (17.66)

; ;

1;

The Legendre polynomial of order 2 is given by: P20 (x) = 2 3x2 1 . The quadrupole

;

term can therefore also be written as:

G

Vqua(r) = r3 (C A) P20 (cos ) : (17.67)

;

The term C A denotes the di erence of the moments of inertia of the Earth around

;

the rotation axis and around an axis through the equator, see gure 17.8. If the Earth

would be a perfect sphere, these moments of inertia would be identical and the quadrupole

term would vanish. However, the rotation of the Earth causes the Earth to bulge at the

equator. This departure from spherical symmetry is responsible for the quadrupole term

in the Earth's potential.

If the Earth would be spherical, the motion of satellites orbiting the Earth would satisfy

Kepler's laws. The quadrupole term in the potential a ects a measurable deviation of the

trajectories of satellites from the orbits predicted by Kepler's laws. For example, if the

potential is spherically symmetric, a satellite will orbit in a xed plane. The quadrupole

term of the potential causes the plane in which the satellite orbits to precess slightly.

Observations of the orbits of satellites can therefore be used to deduce the departure of

CHAPTER 17. POTENTIAL THEORY

260

the Earth's shape from spherical symmetry 35]. Using these techniques it has been found

that the di erence (C A) in the moments of inertia has the numerical value 58]:

;

J = (C A) = 1:082626 10;3 : (17.68)

;

2 Ma2

In this expression a is the radius of the Earth, and the term Ma2 is a measure of the av-

erage moment of inertia of the Earth. Expression (17.68) states therefore that the relative

departure of the mass distribution of the Earth from spherical symmetry is of the order

10;3 . This e ect is small, but this number carries important information about the dy-

namics of our planet. In fact, the time-derivative J_2 of this quantity has been measured 68]

as well! This quantity is of importance because the rotation rate of the Earth slowly de-

creases because of the braking e ect of the tidal forces. The Earth adjusts its shape to this

deceleration.. The measurement of J_2 therefore provides important information about the

response of the Earth to a time-dependent loading.

17.8 Epilogue, the fth force

Gravity is the force in nature that was understood rst by mankind through the discovery

by Newton of the law of gravitational attraction. The reason the gravitational force was

understood rst is that this force manifests itself in the macroscopic world in the motion

of the sun, moon and planets. Later the electromagnetic force, and the strong and weak

interactions were discovered. This means that presently, four forces are operative in nature.

In the 1980's, geophysical measurements of gravity suggested that the gravitational

eld behaves in a di erent way over geophysical length scales (between meters and kilo-

meters) than over astronomical length scales (>104 km). This has led to the speculation

that this discrepancy was due to a fth force in nature. This speculation and the obser-

vations that fuelled this idea are clearly described by Fishbach and Talmadge 23]. The

central idea is that in Newton's theory of gravity the gravitational potential generated by

a point mass M is given by (17.40):

VN (r) = GM : (17.69)

r ;

The hypothesis of the fth force presumes that a new potential should be added to this

Newtonian potential that is given by

GM e;r= :

V5 (r) = (17.70)

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