Exercise 3.2 Prove that the gradient of equation 3.4 yields equation 3.1
3.2 The Potential of Distributions of Mass
Gravitational potential obeys the principle of superposition: The gravi- tational potential of a collection of masses is the sum of the gravitational attractions of the individual masses. Hence, the net force on a test par- ticle is simply the vector sum of the forces due to all masses in space.
The superposition principle can be applied to find the gravitational at- traction in the limit of a continuous distribution of matter. A continuous distribution of mass m is simply a collection of a great many, very small masses dm = p(x, y, z) dv, where p(x, y, z) is the density distribution.
Applying the principle of superposition yields
(3.5) where integration is over V, the volume actually occupied by mass. As usual, P is the point of observation, Q is the point of integration, and r is distance between P and Q. Density p has units of kilogram-meter"3 in SI units and gram-centimeter"3 in the cgs system. The conversion between the two systems is 1 kg-m~3=10~3 g-cm"3.
First consider observation points located outside of a mass distribution (Figure 3.2). If density is well behaved, integral 3.5 converges for all P outside of the mass (Kellogg [146]), and differentiation with respect to x, y, and z can be moved inside the integral. For example, the partial derivative of U with respect to x is
dU(P) __
dx
3.2 The Potential of Distributions of Mass
P(x,y,z)
47
Fig. 3.2. Gravitational attraction at point P due to a density distribution p.
Repeating the differentiation of equation 3.5, once with respect to y and once with respect to z, and adding the three components will provide the attraction outside of any distribution of mass:
(3.6) Second-order derivatives can be derived in similar fashion; for example, the x component is
d2u _ f :
dx2 7 / rv 3
Repeating for the y and z components and adding the three results yields d2U d2U d2U
dx2
= 0,
dy2 dz2
(3.7) and the gravitational potential is harmonic at all points outside of the mass.
What about the potential inside distributions of mass? If P is inside the mass, the integrand in equation 3.5 is singular, and the integral is improper. Nevertheless, the integral can be shown to converge. In fact,
Kellogg [146] shows that the integral
is convergent for P inside V and is continuous throughout V if n < 3, V is bounded, and p is piecewise continuous. Hence, U(P) and g(-P) exist and are continuous everywhere, both inside and outside the mass;
so long as the density is well behaved. Kellogg [146] also shows that g(P) = VU(P) for P inside the mass. This last point is not obvious because derivatives cannot be moved inside improper integrals.
The Helmholtz theorem (Section 2.2.2) tells us that if g satisfies g = VU and vanishes strongly at infinity, then
£/= J- / ^—? dv . (3.8)
4?r J r
Comparing the integrand of equation 3.8 with the integrand of equa- tion 3.5 suggests that
V2U(P) = - 4 T T7P ( P ) . (3.9) Equation 3.9 is Poisson's equation, which describes the potential at all points, even inside the mass distribution. Laplace's equation is simply a special case of Poisson's equation, valid for mass-free regions of space.
Although the foregoing is not a rigorous proof of the relationship between equations 3.5 and 3.9, the example in Section 3.2.2 will demonstrate the validity of Poisson's equation.
The following theorems can be stated in summary:
1. The Newtonian potential U and the acceleration of gravity g exist and are continuous throughout space if caused by a bounded distribution of piecewise-continuous density.
2. The potential U is everywhere differentiate so equation g = VU is true throughout space.
3. Poisson's equation V2U = — Air^p describes the relationship between mass and potential throughout space. Laplace's equation V2C/ = 0 is a special case of Poisson's equation valid in regions of space not occupied by mass.
Surface and Line Distributions
It is sometimes useful, as will be seen in the next sections, to consider the gravitational attraction and potential of mass distributions that are
3.2 The,Potential of Distributions of Mass 49 spread over vanishingly thin surfaces and along vanishingly narrow lines.
The potential of a mass distribution spread over surface S and viewed at a point P not on the surface is given by
U(P)=-y f^-dS, (3.10) s
where a is the surface density with units of mass per unit area. The potential of a mass concentrated along a line / is given by
U{P)=1 ^dl, (3.11)
where A is the line density with units of mass per unit length. The gravitational attractions of these hypothetical distributions are easily derived from g = VU.
3.2.1 Example: A Spherical Shell
To investigate some of the points of the previous sections, consider the gravitational effects of a thin-walled, spherical shell of radius a and uni- form surface density a. We simplify the task (Figure 3.3) by arranging the coordinate system in order to take advantage of the symmetry of the problem: The origin is placed at the center of the sphere, and one axis is oriented so that it passes through P.
For P outside the shell, the potential is given by equation 3.10,
=w/7—<»*,.
s7/-:
0 0
The distance from P to any point on the sphere is r = [R2 + a2 - ^ so
dr aR sin 0
P
Fig. 3.3. Thin-walled, spherical shell with radius a observed at point P.
Substituting yields
R+a 27T^fO~a f
U(P) = / dr
R-a
= 7- R (3.12)
M
where M is the total mass of the shell. Therefore, the gravitational po- tential at any point outside a uniform shell is equivalent to the potential of a point source located at the center of the shell with mass equal to the total mass of the shell It follows, therefore, that the gravitational attraction at points outside the shell is equivalent to the attraction of a point mass,
M
and that
V2U(P) = 0 .
a-\-t
/
3.2 The Potential of Distributions of Mass 51 Now consider P inside the shell. The previous derivation can be re- peated but with slightly different limits of integration; that is,
a+R dr
a-R
•a (3.13) M
All quantities in equation 3.13 are constant, so the gravitational potential is constant everywhere inside a uniform shell. Consequently, no gravita- tional forces exist inside the hollow shell because
/ M
V
a= 0.
Obviously, V2U = 0 within the shell because U is uniform throughout its interior.
Exercise 3.3 Equation 3.13 is easy to understand when P is located at the center of the shell. Observed at the center, the attraction due to any patch of the shell is exactly canceled by the attraction of an identical patch on the opposite side, so it seems reasonable that g = 0 at the center. Less obvious is the fact that g = 0 at points away from the center. Explain in terms of geometry and solid angles why all forces cancel at any point inside the shell.
3.2.2 Example: Solid Sphere
Equations 3.12 and 3.13 provide an easy way to investigate the gravi- tational effects of a solid sphere. For P outside the sphere, the problem is simple. A solid sphere of radius a is just a collection of concentric, thin-walled shells with radii ranging from 0 to a. The superposition principle states that the gravitational potential of the entire set of con- centric shells is the sum of their individual potentials, which, according to the previous section, are each equivalent to a point mass at their cen- ters. Consequently, the potential of a solid sphere appears at all external points as a single point of mass located at the center of the sphere with magnitude equal to the total mass of the sphere; that is,
i&p-, (3.14)
Fig. 3.4. Observation point P inside a sphere. Point P lies within a narrow spherical cavity between radius r — | and r + | .
and V2U(P) = 0 everywhere outside the sphere. Computer subrou- tine B.I in Appendix B provides a Fortran subroutine that calculates the gravitational attraction at external points due to a sphere with ho- mogeneous density.
To investigate the potential at points inside the sphere, we place P in a narrow, spherical cavity of radius r and thickness e concentric about the center of the sphere (Figure 3.4). The potential at P is due to two sources: (1) That part of the sphere with radius less than r — | and (2) the concentric shell with radius greater than r + | . Equation 3.14 gives the potential of the inner sphere:
We know from equation 3.13 that the potential of the outer shell must be constant because each concentric, thin-walled shell is a constant. Equa- tion 3.13 can be integrated to provide the potential of the entire outer shell:
a da
r+i
3.2 The Potential of Distributions of Mass 53 Adding U\(P) and Uo(P) and letting e —>• 0 provide the potential inside a spherical mass:
= §777,9 [3a2 - r2] . (3.15)
The gravitational attraction is given by
and £fte attraction at internal points of a uniform sphere is proportional to the distance from the center. The Laplacian (in spherical coordinates) of equation 3.15 yields
which is Poisson's equation.
Exercise 3.4 Show that U(P) and g(P) are continuous across the surface