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the \dipole term."

Problem e: Compared to the monopole term, the dipole term decays as 1=r2 rather than

1=r. The monopole term does not depend on ^, the direction of observation. Show

r

that the dipole term varies with the direction of observation as cos and show how

the angle must be de ned.

CHAPTER 17. POTENTIAL THEORY

256

Problem f: You may be puzzled by the fact that the gravitational potential contains a

dipole term, despite the fact that there is no negative mass. Draw a gure similar

to gure 17.4 to show that a displaced mass can be written as an undisplaced mass

plus a mass-dipole.

Of course, one is completely free in the choice of the origin of the coordinate system. If

one chooses the origin in the center of mass of the body, then rg = 0 and the dipole term

vanishes.

We will now analyze the term Vqua (r) in (17.51). It can be seen from this expression

that this term decays with distance as 1=r3 . For this reason, this term will be called

the quadrupole term. The dependence of the quadrupole term on the direction is more

complex than for the monopole term and the dipole term. In doing so it will be useful to

use the double contraction between two tensors. This operation is de ned as:

X

(A : B) Aij Bij : (17.55)

ij

concept of the inner product of two vectors (a b) =

P adouble contraction generalizes thetwo. A double contraction occurs for example in the

The

i i bi to matrices or tensors of rank

following identity: 1 = (^ ^) =(^ I^) = (^^ : I), where I is the identity operator. Note

r r r r rr

that the term ^^ is a dyad. If you are unfamiliar with the concept of a dyad you may rst

rr

want to look at section 10.1.

Problem g: Use these results to show that Vqua(r) can be written as:

G rr

Vqua (r) = 2r3 (^^ : T) (17.56)

;

where T is the quadrupole moment tensor de ned as

Z

T= (r) 3rr Ir2 dV : (17.57)

;

Note that we renamed the integration variable r0 in the quadrupole moment tensor

as r.

Problem h: Show that T is in explicit matrix notation given by:

0222 1

Z B 2x 3xy z 2y2 3xy z2

y 3xz

C dV :

; ;

T= (r) @ x2 A

3yz (17.58)

; ;

2z 2 ; x2 ; y2

3xz 3yz

Note the resemblance between (17.56) and (17.54). For the dipole term the directional

dependence is described by the single contraction (^ rg ) whereas for the quadrupole term

r

directional dependence is now given by the double contraction ^^ : T. This double con-

rr

traction leads to a more rapid angular dependence of the quadrupole term than for the

monopole term and the dipole term.

To nd the angular dependence, we will use that the inertia tensor T is a real sym-

metric 3 3 matrix. This matrix has therefore three orthogonal eigenvectors v(i) with

^

17.7. THE QUADRUPOLE FIELD OF THE EARTH 257

corresponding eigenvalues i . Using expression (10.61) of section (10.5) this implies that

the quadrupole moment tensor can be written as:

X

3

T= iv

^ (i) v(i) :

^ (17.59)

i=1

Problem i: Use this result to show that the quadrupole term can be written as

GX 3

2

Vqua(r) = 2r3 i cos (17.60)

i

;

i=1

where the i denote the angle between the eigenvector v(i) and the observation

^

direction ^, see gure 17.7 for the de nition of these angles.

r

^ ^

v3 v1 r

Î¨3 Î¨1

Î¨2

^

v2

Figure 17.7: De nition of the angles i .

Since the dependence of these angles goes as cos2 i = (cos2 i + 1) =2 this implies that

the quadrupole varies through two periods when i goes from 0 to 2 . This contrast with

the monopole term, which does not depend on the direction, as well as with the dipole

term that varies according to problem e as cos . There is actually a close connection

between the di erent terms in the multipole expansion and spherical harmonics. This can

be seen by comparing the multipole terms (17.49)-(17.51) with expression (17.38) for the

gravitational potential. In the latter expression, the di erent terms decay with distance as

r;(l+1) and have an angular dependence Ylm ( '). Similarly, the multipole terms decay

as r;1 , r;2 and r;3 respectively and depend on the direction as cos 0, cos and cos 2

respectively.

17.7 The quadrupole eld of the Earth

Let us now investigate what the multipole expansion implies for the gravity eld of the

Earth. The monopole term is by far the dominant term. It explains why an apple falls

from a tree, why the moon orbits the Earth and most other manifestations of gravity

that we observe in daily life. The dipole term has in this context no physical meaning

whatsoever. This can be seen from equation (17.54) which states that the dipole term only

CHAPTER 17. POTENTIAL THEORY

258

depends on the distance from the Earth's center of gravity to the origin of the coordinate

system. Since we are completely free in choosing the origin, the dipole term can be made

to vanish by choosing the origin of the coordinate system in the Earth's center of gravity.

It is through the quadrupole term that some of the subtleties of the Earth's gravity eld

becomes manifest.

Problem a: The quadrupole term vanishes when the mass distribution in the Earth is

spherically symmetric Earth. Show this by computing the inertia tensor T when

= (r).

The dominant departure of the Earth's gure from aspherical shape is the attening

of the Earth due to the rotation of the Earth. If that is the case, then by symmetry

one eigenvector of T must be aligned with the Earth's axis of rotation, and the two

other eigenvectors are perpendicular to the axis of rotation. By symmetry these other

eigenvectors must correspond to equal eigenvalues. When we choose a coordinate system

with the z-axis along the Earth's axis of rotation the eigenvectors are therefore given by the

unit-vectors ^, x and y with eigenvalues z , x and y respectively. These last eigenvalues

z^ ^

identical because of the rotational symmetry around the Earth's axis of rotation, hence

y = x.

Let us rst determine the eigenvalues. Once the eigenvalues are known, the quadrupole

moment tensor follows from (17.60). The eigenvalues could be found in standard way by

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