Question

If the moon has a core with a radius of 400km and the uniform density of the overlying mantle is 3300kgm-3, what is the density of the core, hence calculate the mass of the core.

Answer 1

The radius of the Moon is $1738\ km$ and its mean density is $3347 \ kg/m^3$ . So,

mass of core $m_c= \frac{4}{3}\pi r_c^3ρ_c$

volume of mantle $V=\frac{4}{3}\pi R^3−\frac{4}{3}\pi r_c^3$

mass of mantle $m_m= \frac{4}{3}\pi (R^3-r_c^3)\rho_m$

mass of planet $m=m_m+m_c$

$\frac{4}{3}\pi R^3\rho= \frac{4}{3}\pi (R^3-r_c^3)\rho_m+ \frac{4}{3}\pi r_c^3ρ_c\to$

$\frac{\rho-\rho_m}{\rho_c-\rho_m}=(\frac{r_c}{R})^3\to\frac{3347-3300}{\rho_c-3300}=(\frac{400}{1738})^3\to$

$\rho_c\approx7155\ kg/m^3$ . Answer