Originally posted by sonhouse
It occurred to me by just doing the cube root of the difference and comparing the real sizes, for instance if you did that for the sun Vs Jupiter, you could get a rough idea of the density difference. So the sun, 1.4 E6 Km diameter Vs 1.28E5 Km works out to be about 11 times bigger than Jupiter but if you take the cube root of that, the sun would be only ab ...[text shortened]... eal numbers are, but I bet it's not far off what I said, about 5 times more dense than the sun.
I don't think that works, the formula doesn't compare densities, because it was derived assuming the densities of the two compared planets were equal...If you want to compare densities based on size its going to be a whole new formula. I'll work it out for you.
p_A = density of planet A
p_B = density of planet B
the ratio, density of planet B: density of planet A = n'
n' = p_B/p_A (n' will be the number of times more dense or less dense "planet B" is than "planet A" )
now each of the respective densities of the planet are as follows
p_A = m_A/V_A ( that is the mass of planet A divided by the volume of planet A, and the same goes for planet B)
p_B = m_B/V_B
now we are still trying to find the ratio n', so we back substitute as follows.
n' = (m_B/V_B)/(m_A/V_A)
once again assuming a sphere
V_A = 4/3*PI*r_A^3
V_B = 4/3*PI*r_B^3
so we back substitute these equations into the most recent equation for n' and after simplification
n' = (m_B/m_A)*(r_A/r_B)^3
now we assume that planet B is n times more massive than planet A, which means
m_B = n*m_B
back substitute and simplify and we arrive at the final equation
n' = n*(r_A/r_B)^3
!AND A FURTHER NOTE!:
all the variables on the right you should know...that is to say, don't solve for r_A from the first equation we derived earlier when we assumed the planets A and B had equal densites. For If you do, n' will just be a bogus result.