54.5: Earth Density Model
- Page ID
- 92344
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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 \(\PageIndex{1}\)).
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:
\[g(r)=\int_{0}^{r} \frac{G \rho(r)}{r^{2}} d V=\int_{0}^{r} \frac{G \rho(r)}{r^{2}}\left(4 \pi r^{2}\right) d r=4 \pi G \int_{0}^{r} \rho(r) d r\]
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 \(g=10.7 \mathrm{~m} / \mathrm{s}^{2}\).
\({ }^{1}\) Dziewonski, A.M., and Anderson, D.L., Preliminary Earth reference model. Physics of the Earth and Planetary Interiors, 25 (1981) \(297-356\).