Exercises

1. Demonstrate that the volume of an ellipsoid whose bounding surface satisfies
   $$\frac{x_1^{\,2}}{a_1^{\,2}}+ \frac{x_2^{\,2}}{a_2^{\,2}}+\frac{x_3^{\,2}}{a_3^{\,2}}=1,$$
   is $V=(4/3)\,\pi\,a_1\,a_2\,a_3$ .

2. Demonstrate that the moments of inertia about the three Cartesian axes of a homogeneous ellipsoidal body of mass $M$ , whose bounding surface satisfies $(x_1/a_1)^2+(x_2/a_2)^2+ (x_3/a_3)^2=1$ , are

   $$I_1 =\frac{M}{5}\,(a_2^{\,2}+a_3^{\,2}),$$

   $$I_2 = \frac{M}{5} \,(a_1^{\,2}+a_3^{\,2}),$$

   $$I_3 = \frac{M}{5} \,(a_1^{\,2}+a_2^{\,2}).$$

   
   

3. According to MacCullagh's formula (Fitzpatrick 2012), the gravitational potential a relatively long way from a body of mass $M$ whose center of mass coincides with the origin, and whose principal moments of inertial are $I_1$ , $I_2$ , and $I_3$ (assuming that the principal axes coincide with the Cartesian axes), takes the form
   $${\mit\Psi}(x_1,x_2,x_3) \simeq -\frac{G\,M}{r} - \frac{G\,(I_1+I_... ...}} + \frac{3\,G\,(I_1\,x_1^{\,2}+ I_2\,x_2^{\,2}+I_3\,x_3^{\,2})}{2\,r^{\,5}},$$
   where $r=(x_1^{\,2}+x_2^{\,2}+x_3^{\,2})^{1/2}$ . Demonstrate that if the body in question is a homogeneous ellipsoid whose bounding surface satisfies $(x_1/a_1)^2+(x_2/a_2)^2+ (x_3/a_3)^2=1$ then

   $${\mit\Psi}(x_1,x_2,x_3) \simeq -\frac{G\,M}{r} - \frac{G\,M}{5\,r^{\,5}}\,a_1^{\,2}\left[x_1^{\,2}-\frac{1}{2}\,(x_2^{\,2}+x_3^{\,2})\right]$$ (D.37)

   $$\phantom{=}- \frac{G\,M}{5\,r^{\,5}}\,a_2^{\,2}\left[x_2^{\,2}-\f... ...\,r^{\,5}}\,a_3^{\,2}\left[x_3^{\,2}-\frac{1}{2}\,(x_1^{\,2}+x_2^{\,2})\right].$$

   
   

4. Show that the gravitational potential external to a homogeneous ellipsoidal body of mass $M$ , whose outer boundary satisfies $(x_1/a_1)^2+(x_2/a_2)^2+ (x_3/a_3)^2=1$ , takes the form

   $${\mit\Psi}(x_1,x_2,x_3) = - \frac{3}{4}\,G\,M\left(\alpha_0-\sum_{i=1,3}\alpha_i\,x_i^{\,2}\right),$$ (D.38)

   
   
   where

   $$\alpha_0 =\int_\lambda^\infty \frac{du}{\mit\Delta},$$

   $$\alpha_i =\int_\lambda^\infty \frac{du}{(a_i^{\,2}+u)\,{\mit\Delta}},$$

   
   
   and ${\mit\Delta} = (a_1^{\,2}+u)^{1/2}\,(a_2^{\,2}+u)^{1/2}\,(a_3^{\,2}+u)^{1/2}$ . Here, $\lambda$ is the positive root of
   $$\sum_{i=1,3} \frac{x_i^{\,2}}{a_i^{\,2}+\lambda} = 1.$$
   Demonstrate that, at large $\lambda$ , Equation (D.38) reduces to Equation (D.37).