(sec:benchmarks:thick-shell-gravity)= # Thick shell gravity benchmark *This section was contributed by Cedric Thieulot.* This benchmark tests the accuracy of the gravity field and gravitational potential computed by the gravity postprocessor inside and outside an Earth-sized planet (without its core) of constant density. The domain is a spherical shell with inner radius $R_\text{inner}=3840~\text{ km}$ and outer radius $R_\text{outer}=6371~\text{ km}$. The density is constant in the domain and set to $\rho_0=3300~\text{ kg}/\text{m}^3$. First, let us calculate the exact profile which we expect the benchmark to reproduce. The gravitational potential $U$ of a spherically symmetric object satisfies the Poisson equation $\Delta U = 4\pi G \rho(\mathbf r)$. For a constant density shell, this equation can be solved analytically for the gravitational acceleration and potential inside and outside the planet. Inside ($rR_\text{outer}$) the spherical shell (i.e. where $\rho=0$) the Poisson equation simplifies to the Laplace equation $\Delta U=0$: {math} \frac{1}{r^2} \frac{\partial }{\partial r} \left(r^2 \frac{\partial U}{\partial r} \right) = 0.  The solution to this expression is: {math} :label: eq:app1 g=\frac{\partial U}{\partial r} = \frac{C}{r^2},  where $C$ is a constant of integration. In order to avoid an infinite gravity field at $r=0$ (where the density is also zero in this particular setup of a shell), we need to impose $C=0$, i.e. the gravity is zero for $r\leq R_\text{inner}$. Another way of arriving at the same conclusion is to realize that $g$ is zero at the center of the body because the material around it exerts an equal force in every direction. Inside the shell, $\rho=\rho_0$, yielding {math} g=\frac{\partial U}{\partial r} = \frac{4 \pi}{3} G \rho_0 r + \frac{A}{r^2},  where $A$ is another integration constant. At the inner boundary, $r=R_\text{inner}$ and $g=0$, allowing $A$ to be computed. Substituting in the value of $A$, {math} :label: eqgin g=\frac{\partial U}{\partial r} = \frac{4 \pi}{3} G \rho_0 \left(r - \frac{R_\text{inner}^3}{r^2} \right).  When $r\geq R_\text{outer}$, the gravitational potential is given by {math:numref}eq:app1. Requiring the gravity field to be continuous at $r=R_\text{outer}$: {math} :label: eq:gout g(r) = \frac{G M}{r^2},  where $M=\frac{4 \pi}{3} \rho_0(R_\text{outer}^3-R_\text{inner}^3)$ is the mass contained in the shell. For $r\ge R_\text{outer}$, the potential is obtained by integrating {math:numref}eq:gout): {math} U(r)=-\frac{GM}{r} +D,  where $D$ is an integration constant which has to be zero since we require the potential to vanish for $r\rightarrow \infty$. The potential within the shell, $R_\text{inner}\leq r \leq R_\text{outer}$, is found by integrating {math:numref}eqgin: {math} U(r)= \frac{4 \pi}{3} G \rho_0 \left(\frac{r^2}{2} + \frac{R_\text{inner}^3}{r} \right) + E,  where $E$ is a constant. Continuity of the potential at $r=R_\text{outer}$ requires that $E=-2\pi\rho_0 G R_\text{outer}^2$. Gravitational acceleration is zero for $r\leq R_\text{inner}$, so the potential there is constant and a continuity requirement yields {math} U(r)=2\pi G \rho_0 (R_\text{inner}^2-R_\text{outer}^2).  The gravity postprocessor in can be used to calculate the radial components of gravity ($g_r$ and $U$) at arbitrary points using the sampling scheme '*list of points*.' For this benchmark we calculate points along a line from the center of the planet to a distant point, $r=0$ to $r=10,000~\text{ km}$ ({numref}fig:grav-thick-shell1). Arbitrarily, the latitude and longitude are both set to $13\text{ \degree}$ so as to avoid potential measurement artifacts due to symmetry. The list of radii is defined as follows: {literalinclude} thick_shell.prm  The resulting measurements obtained for a mesh composed of 12 caps of $32^3$ cells (i.e., 393,216 total mesh cells) are shown in {numref}fig:grav-thick-shell2 and are in good agreement with the analytical profiles. {figure-md} fig:grav-thick-shell1 Gravity benchmark for a thick shell. Gravitational potential computed on a line from the center of a constant density shell to a radius of 10,000 km. The gray area indicates the region $R_{inner} \leq r \leq R_{outer}$ inside the shell, where the density is not zero.  {figure-md} fig:grav-thick-shell2 Gravity benchmark for a thick shell. Gravitational acceleration computed on a line from the center of a constant density shell to a radius of 10,000 km. The gray area indicates the region $R_{inner} \leq r \leq R_{outer}$ inside the shell, where the density is not zero.