Gravitation Inside Earth — Question 5

Question 5

An interior gravity profile can be inverted to obtain density

For a spherical body, the inward gravitational field magnitude is measured to be $g(r)=Ar+Br^3$ for $0<r\le R$, with $A>0$. Assume no point mass at the center.

1. Derive the local density profile.

2. For $B<0$, find the restriction on $B$ required for density to remain nonnegative.

Question 5 - Solution

An interior gravity profile can be inverted to obtain density

(a) Derivation and calculation

Invert the shell-theorem relation: $$M(r)=\frac{r^2g(r)}G=\frac{Ar^3+Br^5}G.$$ Differentiate and divide by shell volume per unit radius: $$4\pi r^2\rho(r)=\frac{dM}{dr}=\frac{3Ar^2+5Br^4}G,$$ $$\boxed{\rho(r)=\frac{3A+5Br^2}{4\pi G}}.$$

(b) Result and interpretation

If $B<0$, then $d\rho/dr=10Br/(4\pi G)<0$ for $r>0$. The minimum occurs at $R$: $$\rho(R)\ge0\quad\Longleftrightarrow\quad 3A+5BR^2\ge0.$$ $$\boxed{-\frac{3A}{5R^2}\le B<0}.$$ At the limiting value, the density vanishes only at the surface.

Physical check

The condition also ensures $g(r)>0$ for $0<r\le R$; there is no hidden central point mass.