# What is a general formula for Earth gravity as a function of radius, given a spherical shell model?

I'm trying to derive the formula for gravitational acceleration as a function of Earth radius $$g(r)$$, given a spherical shell model where each shell has a constant density $$\rho(r)$$. If I set $$r=0$$ to be the core-mantle boundary, I think it's something like:

$$g(r) = \frac{4\pi G}{3r^{2}}[ \int_{0}^{r}{\rho}(r)r^{2}.dr +F ]$$

Where $$F$$ accounts for core mass. But I would like confirmation/references if possible! Thanks!

Edit: Here's my thinking thus far: at a radius $$r$$,

$$g(r) = \frac{G}{r^{2}}\int_{V}\rho.dV$$

where $$V$$ is the volume below $$r$$. Since $$V = \frac{4\pi}{3} r^{3}$$, $$dV = 4\pi r^{2}.dr$$. So

$$g(r) = \frac{4\pi G}{r^{2}} \int_{0}^{r}\rho(r)r^{2}.dr$$, plus a constant for the mass of the core.

• Perhaps you should further discuss your reasoning (sounds like a hw question, which people may help with, but usually when the poster is showing good faith in talking through what they've done) Aug 10 at 9:39
• It's for a paper I'm writing... I genuinely can't find the answer after a literature trawl, only simplified solutions e.g. $\rho$ varies linearly with depth. Aug 10 at 11:02
• Your equation is dimensionally incorrect. This is a start over situation. I also suggest you do a better job of researching literature. A good place to start is the 40+ year old paper on the Preliminary Reference Earth Model, the many papers cited by the PREM paper, and the 11000+ papers that cite the PREM paper as a reference. Aug 10 at 11:19
• Thanks for your super-helpful comment; nowhere in the PREM paper do they explain how they calculated their gravity values. I am currently searching through the 11000+ papers, I just thought someone out there might know where to find the answer off the top of their head and be able to help me out. Aug 10 at 12:15
• Apologies, didn't realize it was complex, thought integrating over a shell was usually fairly straightforward and a type of problem I thought I remembered from schools days, but sounds like it's more complex than I realized Aug 11 at 3:49

$$g(r) = \frac{4\pi G}{(r+r_c)^2}[ \int_{0}^{r}{\rho}(r)r^{2}.dr +F ]$$
where your radial distance (which is the argument in the inverse square law) is measured from the center of the shells and thus includes the core radius $$r_c$$. Also you do not use the $$3$$. The mass of each shell is the surface area times the density times the shell thickness, and the surface area does not have the factor of $$1/3$$. Of course when you add the core mass, that is computed over a volume and thus does include the $$1/3$$ factor if you are calling the core a homogemeous ball of radius $$r_c$$ and density $$\rho_c$$. Thus $$F=\rho_cr_c^3/3$$.
• Edit: I will leave the denominator of the prefactor as $r^{2}$, and change the bottom limit of the integral to $R_{CMB}$, which is core-mantle boundary radius, and set $r=0$ to be the centre of the Earth - this makes more sense intuitively. Aug 11 at 9:31