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Next: Calculus of Variations Up: Ellipsoidal Potential Theory Previous: Analysis

Exercises

  1. Demonstrate that the volume of an ellipsoid whose bounding surface satisfies

    $\displaystyle \frac{x_1^{\,2}}{a_1^{\,2}}+ \frac{x_2^{\,2}}{a_2^{\,2}}+\frac{x_3^{\,2}}{a_3^{\,2}}=1,$

    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

    $\displaystyle I_1$ $\displaystyle =\frac{M}{5}\,(a_2^{\,2}+a_3^{\,2}),$    
    $\displaystyle I_2$ $\displaystyle = \frac{M}{5} \,(a_1^{\,2}+a_3^{\,2}),$    
    $\displaystyle I_3$ $\displaystyle = \frac{M}{5} \,(a_1^{\,2}+a_2^{\,2}).$    

  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

    $\displaystyle {\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

    $\displaystyle {\mit\Psi}(x_1,x_2,x_3)$ $\displaystyle \simeq -\frac{G\,M}{r} - \frac{G\,M}{5\,r^{\,5}}\,a_1^{\,2}\left[x_1^{\,2}-\frac{1}{2}\,(x_2^{\,2}+x_3^{\,2})\right]$ (D.37)
      $\displaystyle \phantom{=}- \frac{G\,M}{5\,r^{\,5}}\,a_2^{\,2}\left[x_2^{\,2}-\f...
...\,r^{\,5}}\,a_3^{\,2}\left[x_3^{\,2}-\frac{1}{2}\,(x_1^{\,2}+x_2^{\,2})\right].$    

  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

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

    where

    $\displaystyle \alpha_0$ $\displaystyle =\int_\lambda^\infty \frac{du}{\mit\Delta},$    
    $\displaystyle \alpha_i$ $\displaystyle =\int_\lambda^\infty \frac{du}{(a_i^{\,2}+u)\,{\mit\Delta}},$    

    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

    $\displaystyle \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).


next up previous
Next: Calculus of Variations Up: Ellipsoidal Potential Theory Previous: Analysis
Richard Fitzpatrick 2016-03-31