I'm in desperate need of Physics help! Calculating the density of Jupiter?

Question

Question: Galileo discovered the four major moons of the planet Jupiter in 1610. The nearest one, Io, has a period of 1.7699 days and is 5.578 Jovian radii (Radius of Jupiter) from the center of the planet. Use this data (not from your book's table of planetary information) to calculate the density of Jupiter.

Okay, I've discovered that the Period in seconds is 152919.36 seconds.

Here are some relevant equations:
Density= Mass/volume
Volume= (4/3)*pi*r^3
Period^2=((4*pi^2)/(G(Mass of Jupiter)))*r^m

I know I need to figure out what the Mass of Jupiter is and the radius of Jupiter too. In addition, I need to then find the mass and volume in order to get the density.

Please help me if possible, any information is greatly appreciated!

What should my units be for the period? Do I keep it as days? Also, in the end, what should my units end up being?

Also, is there a way I can solve this without knowing the mass of Jupiter and the radius of it? Because I don't believe I'm able to use them.

Answer 1

I don't understand your equation for period. The formula for orbital period is

T^2 = 4*π^2*a^3 / G*M

where a = radius of orbit and M = mass of planet.

You are given that a = k*r, where r is planet radius and k = 5.578. Also M = (4/3)*π*r^3*rho, where rho is the planet density. Making the substitutions, you get

T^2 = (3/π)*k^3 / rho*G (Edited)

You can now get rho from the information you have.

rho = 3*k^3 / π*G*T^2

EDIT: If rho is kg/m^3, and G = 6.67*10^-11 m^3/(kg*sec^2), then the period is sec.

You get rho = 106.26 kg/m^3

Answer 2

μ = 5.578

>Density= Mass/volume
>Volume= 4π/3 R³
>Period²= (2π)² / (GM) (μR)³

Combine first and second equations:
Density= Mass/[4π/3 R³]
R³ = Mass/[4π/3 Density]

Substitude R³ into the third equation:
Period²= (2π)² / (GM) μ³ (Mass/[4π/3 Density])
Density = (2π)² / (GM) μ³ M /[4π/3 Period²]