# 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 $$R\_{inner}=3840~\si{\km}$$ and outer radius $$R\_{outer}=6371~\si{\km}$$. The density is constant in the domain and set to $$\rho_0=3300~\si{\kg\per\cubic\metre}$$.

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 ($$r<R\_{inner}$$) and outside ($$r>R\_{outer}$$) the spherical shell (i.e. where $$\rho=0$$) the Poisson equation simplifies to the Laplace equation $$\Delta U=0$$: $$$\frac{1}{r^2} \frac{\partial }{\partial r} \left(r^2 \frac{\partial U}{\partial r} \right) = 0.$$$$The solution to this expression is:$$$$g=\frac{\partial U}{\partial r} = \frac{C}{r^2} \label{eq:app1},$$$$where$$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_{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$$, yielding$$$$g=\frac{\partial U}{\partial r} = \frac{4 \pi}{3} G \rho_0 r + \frac{A}{r^2},$$$$where$$A$$is another integration constant. At the inner boundary,$$r=R_{inner}$$and$$g=0$$, allowing$$A$$to be computed. Substituting in the value of$$A$$,$$$$g=\frac{\partial U}{\partial r} = \frac{4 \pi}{3} G \rho_0 \left(r - \frac{R\_{inner}^3}{r^2} \right). \label{eqgin}$$$$When$$r\geq R_{outer}$$, the gravitational potential is given by Eq. ([$eq:app1$][1]). Requiring the gravity field to be continuous at$$r=R_{outer}$$:$$$$g(r) = \frac{G M}{r^2} \label{eq:gout},$$$$where$$M=\frac{4 \pi}{3} \rho_0(R_{outer}^3-R_{inner}^3)$$is the mass contained in the shell. For$$r\ge R_{outer}$$, the potential is obtained by integrating Eq.([$eq:gout$][2]):$$$$U(r)=-\frac{GM}{r} +D,$$$$where$$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_{inner}\leq r \leq R_{outer}$$, is found by integrating Eq.&nbsp;[$eqgin$][3]:$$$$U(r)= \frac{4 \pi}{3} G \rho_0 \left(\frac{r^2}{2} + \frac{R\_{inner}^3}{r} \right) + E,$$$$where$$E$$is a constant. Continuity of the potential at$$r=R_{outer}$$requires that$$E=-2\pi\rho_0 G R_{outer}^2$$. Gravitational acceleration is zero for$$r\leq R_{inner}$$, so the potential there is constant and a continuity requirement yields$$$$U(r)=2\pi G \rho_0 (R\_{inner}^2-R\_{outer}^2).$$$

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 $$r=10,000~\si{\km}$$ (Figure 2). Arbitrarily, the latitude and longitude are both set to $$13\si{\degree}$$ so as to avoid potential measurement artifacts due to symmetry. The list of radii is defined as follows:




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 Fig. (2) and are in good agreement with the analytical profiles.