54.5: Earth Density Model

Suppose we have a uniform, spherical body (such as a planet) of radius $R$ and mass $M$. What is the acceleration $g$ due to gravity as a function of $r$ for $r$ both inside and outside the body $(0 \leq r<\infty)$ ?

First, consider the case where we're inside the body $(r \leq R)$. In this case, the acceleration due to gravity at $r$ is $g(r)=G m / r^{2}$, where $m$ is the total mass inside a sphere of radius $r$ :

$$m=\frac{4}{3} \pi r^{3} \rho$$

where the (uniform) density $\rho=M /\left(\frac{4}{3} \pi R^{3}\right)$. Thus

$$g(r)=\frac{G M}{R^{3}} r \quad(0 \leq r<R)$$

so inside the body, $g \propto r$.

Second, consider the case where we're outside the body $(r>R)$. In this case, the total mass inside a sphere of radius $r$ is $M$, and so

$$g(r)=\frac{G M}{r^{2}} \quad(r \geq R)$$

so that outside the body, $g \propto 1 / r^{2}$. The maximum value of $g$ is at the surface, $g=G M / R^{2}$ at $r=R$ (Figure $\PageIndex{1}$).

Figure $\PageIndex{1}$: Acceleration due to gravity for a uniform sphere.

However, planetary bodies are generally not uniform. For example, the Earth has a higher density closer to its core, and its density decreases closer to the surface. One density model of the Earth given by Dziewonski and Anderson ${ }^{1}$ is shown in Figure $\PageIndex{2}$. We can use this density model to compute a more realistic model of $g(r)$ inside the Earth:

Figure $\PageIndex{1}$: Earth density model (Dziewonski and Anderson, 1981.)

$$g(r)=\int_{0}^{r} \frac{G \rho(r)}{r^{2}} d V=\int_{0}^{r} \frac{G \rho(r)}{r^{2}}\left(4 \pi r^{2}\right) d r=4 \pi G \int_{0}^{r} \rho(r) d r$$

The result is plotted in Figure $\PageIndex{3}$. We see that in a more realistic model of the Earth's interior, the maximum value of the acceleration to to gravity $g$ occurs just outside the outer core, where $g=10.7 \mathrm{~m} / \mathrm{s}^{2}$.

Figure $\PageIndex{1}$: Modeled acceleration due to gravity for Earth.

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${ }^{1}$ Dziewonski, A.M., and Anderson, D.L., Preliminary Earth reference model. Physics of the Earth and Planetary Interiors, 25 (1981) $297-356$.