Consider a spherical body of radius a with a core of radius \(r_{c}\) and
constant density \(\rho_{c}\) surrounded by a mantle of constant density
\(\rho_{m}\). Show that the moment of inertia \(C\) and mass \(M\) are given $$
\begin{array}{l}
\text { TABLE 5-1 Values of the Dimensionless Polar Moment of Inertia, }
\mathrm{J}_{2} \text { , and the Polar Flattening for the Earth, } \\
\text { Moon, Mars, and Venus } \\
\begin{array}{lllll}
\text { Earth } & \text { Moon } & \text { Mars } & \text { Venus } \\
\hline C / M a^{2} & 0.3307007 & 0.3935 & 0.366 & 0.33 \\
J_{2} \equiv \frac{1}{M a^{2}}\left(C-\frac{A+B}{2}\right) & 1.0826265 \times
10^{-3} & 2.037 \times 10^{-4} & 1.96045 \times 10^{-3} & 4.458 \times 10^{-6}
\\\
f \equiv \frac{2}{(a+b)}\left(\frac{a+b}{2}-c\right) & 3.35281068 \times
10^{-3} & 1.247 \times 10^{-3} & 6.4763 \times 10^{-3} & \- \\
\hline
\end{array}
\end{array}
$$
by
$$
\begin{aligned}
C &=\frac{8 \pi}{15}\left[\rho_{c}
r_{c}^{5}+\rho_{m}\left(a^{5}-r_{c}^{5}\right)\right] \\
M &=\frac{4 \pi}{3}\left[\rho_{c}
r_{c}^{3}+\rho_{m}\left(a^{3}-r_{c}^{3}\right)\right]
\end{aligned}
$$
Determine mean values for the densities of the Earth's mantle and core given
\(C=8.04 \times 10^{37} \mathrm{~kg} \mathrm{~m}^{2}, M=\)
\(5.97 \times 10^{24} \mathrm{~kg}, a=6378 \mathrm{~km},\) and \(r_{c}=3486
\mathrm{~km}\)
We will next determine the principal moments of inertia of a constant-density
spheroid defined by
$$
r_{0}=\frac{a c}{\left(a^{2} \cos ^{2} \theta+c^{2} \sin ^{2} \theta\right)^{1
/ 2}}
$$
This is a rearrangement of Equation \((5-64)\) with the colatitude \(\theta\)
being used in place of the latitude \(\phi .\) By substituting Equations \((5-6)\)
and \((5-7)\) into Equations \((5-26)\) and \((5-29),\) we can write the polar and
equatorial moments of inertia as
$$
\begin{aligned}
C=& \rho \int_{0}^{2 \pi} \int_{0}^{r_{0}} \int_{0}^{\pi} r^{\prime 4} \sin
^{3} \theta^{\prime} d \theta^{\prime} d r^{\prime} d \psi^{\prime} \\
A=& \rho \int_{0}^{2 \pi} \int_{0}^{r_{0}} \int_{0}^{\pi} r^{\prime 4} \sin
\theta^{\prime} \\
& \times\left(\sin ^{2} \theta^{\prime} \sin ^{2} \psi^{\prime}+\cos ^{2}
\theta^{\prime}\right) d \theta^{\prime} d r^{\prime} d \psi^{\prime},
\quad(5-84)
\end{aligned}
$$
where the upper limit on the integral over \(r^{\prime}\) is given by Equation
\((5-82)\) and \(B=A\) for this axisymmetric body. The integrations over
\(\psi^{\prime}\) and \(r^{\prime}\) are straightforward and yield
$$
\begin{aligned}
C &=\frac{2}{5} \pi \rho a^{5} c^{5} \int_{0}^{\pi} \frac{\sin ^{3}
\theta^{\prime} d \theta^{\prime}}{\left(a^{2} \cos ^{2} \theta^{\prime}+c^{2}
\sin ^{2} \theta^{\prime}\right)^{5 / 2}} \quad(5-85) \\
A &=\frac{1}{2} C+\frac{2}{5} \pi \rho a^{5} c^{5} \int_{0}^{\pi} \frac{\cos
^{2} \theta^{\prime} \sin \theta^{\prime} d \theta^{\prime}}{\left(a^{2} \cos
^{2} \theta^{\prime}+c^{2} \sin ^{2} \theta^{\prime}\right)^{5 / 2}} \\
(5-86)
\end{aligned}
$$
The integrals over \(\theta^{\prime}\) can be simplified by introducing the
variable \(x=\cos \theta^{\prime}\left(d x=-\sin \theta^{\prime} d
\theta^{\prime}, \sin \theta^{\prime}=\right.\)
\(\left.\left(1-x^{2}\right)^{1 / 2}\right)\) with the result
$$
\begin{array}{l}
C=\frac{2}{5} \pi \rho a^{5} c^{5} \int_{-1}^{1} \frac{\left(1-x^{2}\right) d
x}{\left[c^{2}+\left(a^{2}-c^{2}\right) x^{2}\right]^{5 / 2}} \quad(5-87) \\
A=\frac{1}{2} C+\frac{2}{5} \pi \rho a^{5} c^{5} \int_{-1}^{1} \frac{x^{2} d
x}{\left[c^{2}+\left(a^{2}-c^{2}\right) x^{2}\right]^{5 / 2}}
\end{array}
$$
From a comprehensive tabulation of integrals we find
$$
\int_{-1}^{1} \frac{d x}{\left\\{c^{2}+\left(a^{2}-c^{2}\right)
x^{2}\right\\}^{5 / 2}}=\frac{2}{3} \frac{\left(2 a^{2}+c^{2}\right)}{c^{4}
a^{3}}
$$
$$
\int_{-1}^{1} \frac{x^{2} d x}{\left\\{c^{2}+\left(a^{2}-c^{2}\right)
x^{2}\right\\}^{5 / 2}}=\frac{2}{3} \frac{1}{c^{2} a^{3}}
$$
By substituting Equations \((5-89)\) and \((5-90)\) into Equations \((5-87)\) and
\((5-88)\), we obtain
$$
\begin{array}{l}
C=\frac{8}{15} \pi \rho a^{4} c \\
A=\frac{4}{15} \pi \rho a^{2} c\left(a^{2}+c^{2}\right)
\end{array}
$$
These expressions for the moments of inertia can be used to determine \(J_{2}\)
for the spheroid. The substitution of Equations \((5-91)\) and \((5-92)\) into the
definition of \(J_{2}\) given in Equation \((5-43)\), together with the equation
for the mass of a constant-density spheroid
$$
M=\frac{4 \pi}{3} \rho a^{2} c
$$
yields
$$
J_{2}=\frac{1}{5}\left(1-\frac{c^{2}}{a^{2}}\right)
$$
