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Physics LibreTexts

5.12: Gravitational Potential of any Massive Body

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You might just want to look at Chapter 2 of Classical Mechanics (Moments of Inertia) before proceeding further with this chapter.

In figure VIII.26 I draw a massive body whose centre of mass is C, and an external point P at a distance R from C. I draw a set of Cxyz axes, such that P is on the z-axis, the coordinates of P being (0,0,z). I indicate an element δm of mass, distant r from C and l from P. I’ll suppose that the density at δm is ρ and the volume of the mass element is δτ, so that δm=ρδτ.

Figure 5.26.png
FIGURE V.26

The potential at P is

ψ=Gdml=Gρdτl.

But l2=R2+r22Rrcos2θ,

so ψ=G[1Rρdτ+1R2ρrcosθdτ+1R3ρr2P2(cosθ)dτ+1R4ρr3P3(cosθ)dτ...].

The integral is to be taken over the entire body, so that ρdτ=M, where M is the mass of the body. Also ρrcosθdτ=zdm, which is zero, since C is the centre of mass. The third term is

12R3ρr2(3cos2θ1)dτ=12R3ρr2(23sin2θ)dτ.

Now

2ρr2dτ=2r2dm=[(y2+z2)+(z2+x2)+(x2+y2)]dm=A+B+C

where A, B and C are the second moments of inertia with respect to the axes Cx, Cy, Cz respectively. But A+B+C is invariant with respect to rotation of axes, so it is also equal to A0+B0+C0, where A0, B0, C0 are the principal moments of inertia.

Lastly, ρr2sin2θdτ is equal to C, the moment of inertia with respect to the axis Cz.

Thus, if R is sufficiently larger than r so that we can neglect terms of order (r/R)3 and higher, we obtain

ψ=GM(2MR2+A0+B0+C03C)2R3.

In the special case of an oblate symmetric top, in which A0=B0<C0, and the line CP makes an angle γ with the principal axis, we have

C=A0+(C0A0)cos2γ=A0+(C0A0)Z2/R2,

so that ψ=GR[M+C0A02R2(13Z2R2)].

Now consider a uniform oblate spheroid of polar and equatorial diameters 2c and 2a respectively. It is easy to show that

C0=25Ma2.

Exercise 5.12.1

Confirm Equation 5.12.7.

It is slightly less easy to show (Exercise: Show it.) that

A0=15M(a2+c2).

For a symmetric top, the integrals of the odd polynomials of Equation 5.12.2 are zero, and the potential is generally written in the form

ψ=GMR[1+(aR)2J2P2(cosγ)+(aR)J4P4(cosγ)...]

Here γ is the angle between CP and the principal axis. For a uniform oblate spheroid, J2=C0A0Mc2. This result will be useful in a later chapter when we discuss precession.


This page titled 5.12: Gravitational Potential of any Massive Body is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Jeremy Tatum via source content that was edited to the style and standards of the LibreTexts platform.

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