Question

Consider two planets with uniform mass distributions. The mass density and the radius of planet 1 are rho1 and R1, respectively, and those of planet 2 are rho2 and R2. What is the ratio of their masses?

Answer 1

Answer:

$\frac{R_1^3 * \rho _1}{R_2^3 * \rho _2}$

Explanation:

The mass density of an object with uniform mass distribution its defined as

$\rho = \frac{mass}{volume}$.

So, if we know the volume, and, we can obtain the mass of the object:

$mass = volume * \rho$.

Now, we can take the planets as spheres, of course, this is only an approximation, but good enough for us. The volume of a sphere of radius r its:

$Volume_{sphere} = \frac{4}{3} \pi r^3$

So, for our planets, the mass its given by:

$mass = volume * \rho\\mass = \frac{4}{3} \pi r ^3 \rho$,

so

$mass_{planet1} = \frac{4}{3} \pi R_1^3 \rho_1$

$mass_{planet2} = \frac{4}{3} \pi R_2^3 \rho_2$

Now, we can take the ratio:

$\frac{mass_{planet1}}{mass_{planet2}} = \frac{ \frac{4}{3} \pi R_1 ^3 \rho_1 }{ \frac{4}{3} \pi R_2 ^3 \rho_2 }$

Now, we can just cancel the $\frac{4}{3} \pi [\tex] that appear in both sides of our fractions, and finally obtain: [tex]\frac{R_1^3 * \rho _1}{R_2^3 * \rho _2}$.