Assume that the Earth consists of a core of uniform density \(\rho_{\mathrm{c}}\), surrounded by a mantle of uniform density \(\rho_{\mathrm{m}}\), and that the boundary between the two is of similar shape to the outer surface, but with a radius only three-fifths as large. Find what ratio of densities \(\rho_{c} / \rho_{\mathrm{m}}\) is required to explain the observed quadrupole moment. (Hint: Treat the Earth as a superposition of two ellipsoids of densities \(\rho_{\mathrm{m}}\) and \(\rho_{\mathrm{c}}-\rho_{\mathrm{m}}\). Note that in reality neither core nor mantle is of uniform density.)

To find the quadrupole moment of an ellipsoid, we need to evaluate the integral $$ Q_{ij} = \int_V (3x_i x_j - r^2 \delta_{ij})\rho(\textbf{r}) d^3 \textbf{r} $$ where \(i, j = 1, 2, 3\) correspond to x, y, and z, and \(\delta_{ij}\) is the Kronecker delta. Since the Earth is axisymmetric and has a uniform density, the quadrupole moment terms except \(Q_{zz}\) are zero. Therefore, we focus on the \(Q_{zz}\) term: $$ Q_{zz} = \int_V (3z^2 - r^2)\rho(\textbf{r}) d^3 \textbf{r} $$ For an ellipsoid, we can rewrite the integral in terms of elliptical coordinates: $$ Q_{zz} = \int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{a} (3z^2 - r^2)\rho(\textbf{r}) r^2\sin\theta dr d\theta d\phi $$ where \(a\) is the semi-major axis of the ellipsoid.

Now we treat the Earth as a superposition of two ellipsoids. One has density \(\rho_m\) and the other has density \(\rho_c - \rho_m\). Using the result of Step 1, we can calculate the \(Q_{zz}\) term for both ellipsoids: $$ Q_{zz1} = \int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{a1} (3z^2 - r^2)\rho_m r^2\sin\theta dr d\theta d\phi $$ $$ Q_{zz2} = \int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{a2} (3z^2 - r^2)(\rho_c - \rho_m) r^2\sin\theta dr d\theta d\phi $$ Where \(a1\) is the semi-major axis of the larger ellipsoid representing the mantle and \(a2\) is the semi-major axis of the smaller ellipsoid representing the core. Since the boundary between the core and the mantle is three-fifths the radius of the Earth, we have: $$ a2 = \frac{3}{5}a1 $$ The total quadrupole moment is the sum of these two terms: $$ Q_{zzT} = Q_{zz1} + Q_{zz2} $$

To find the required ratio of densities \(\rho_c / \rho_m\), we first need to express \(Q_{zzT}\) in terms of \(\rho_c\) and \(\rho_m\). Since we need the ratio to explain the observed quadrupole moment, we will then solve for the ratio: $$ \rho_c / \rho_m = \frac{Q_{zzT}}{Q_{obs}} $$ Where \(Q_{obs}\) is the observed quadrupole moment. By substituting the integral expressions obtained in step 2 and the relationship between \(a1\) and \(a2\), we can find the ratio of densities that explains the observed quadrupole moment and complete the exercise.