Gravitation Inside Earth — Question 6

Question 6

A dense core can make interior gravity exceed surface gravity

A spherical planet has radius $R$, core radius $c=R/2$, core density $4\rho$ and mantle density $\rho$. Let $C=4\pi G\rho/3$.

1. Find $g(r)$ in the core and mantle, and compare $g(c)$ with $g(R)$.

2. Locate any stationary point in the mantle and determine the global maximum of $g$.

Question 6 - Solution

A dense core can make interior gravity exceed surface gravity

(a) Derivation and calculation

With $C=4\pi G\rho/3$, the core has $M(r)=4(4\pi\rho/3)r^3$. The mantle contains the core plus its own shells: $$\boxed{g(r)=\begin{cases}4Cr,&0\le r\le c,\\C(r+3c^3/r^2),&c\le r\le R.\end{cases}}$$ For $c=R/2$, $$g(c)=4Cc=2CR,\quad g(R)=C(R+3R/8)=\frac{11}{8}CR.$$

(b) Result and interpretation

In the mantle, $$g'(r)=C(1-6c^3/r^3)=0 \quad\Longrightarrow\quad r_*=6^{1/3}c=(3/4)^{1/3}R.$$ $$g''(r)=18Cc^3/r^4>0,$$ so $r_*$ is a minimum. The core rises monotonically, the exterior field decreases, and $g(c)>g(R)$; therefore $$\boxed{g_{\max}=2CR\quad\text{at }r=R/2}.$$

Physical check

The field is continuous at the core boundary even though its derivative changes sign there.