\varphi(\mathbf x)
=
\int_{\text{earth}}
- -G \frac{\rho(\mathbf x)}{\|\mathbf x-\mathbf y\|}
+ -G \frac{\rho(\mathbf y)}{\|\mathbf x-\mathbf y\|}
\ \text{d}y,
@f]
then $\mathbf g(\mathbf x) = -\nabla \varphi(\mathbf x)$. If we assume that
the density $\rho$ is constant throughout the earth, we can produce an
- analytical expression for the gravity vector:
-
-XXX
- Of course, within this problem, we are only interested in the branch that
- pertains to within the earth.
-
+ analytical expression for the gravity vector (don't try to integrate above
+ equation somehow -- it leads to elliptic integrals; the simpler way is to
+ notice that $-\Delta\varphi(\mathbf x) = 4\pi G \rho
+ \chi_{\text{earth}}(\mathbf x)$ and solving this
+ partial differential equation in all of ${\mathbb R}^3$ exploiting the
+ radial symmetry):
+ @f[
+ \mathbf g(\mathbf x) =
+ \left\{
+ \begin{array}{ll}
+ -\frac{4}{3}\pi G \rho \|\mathbf x\| \frac{\mathbf x}{\|\mathbf x\|}
+ & \text{for} \ \|\mathbf x\|<R_1, \\
+ -\frac{4}{3}\pi G \rho R^3 \frac{1}{\|\mathbf x\|^2}
+ \frac{\mathbf x}{\|\mathbf x\|}
+ & \text{for} \ \|\mathbf x\|\ge R_1
+ \end{array}
+ \right.
+ @f]
+ The factor $-\frac{\mathbf x}{\|\mathbf x\|}$ is the unit vector pointing
+ radially inward. Of course, within this problem, we are only interested in
+ the branch that pertains to within the earth, i.e. $\|\mathbf x\|<R_1$. In
+ the program, we therefore only consider the expression
+ @f[
+ \mathbf g(\mathbf x) =
+ -\frac{4}{3}\pi G \rho \|\mathbf x\| \frac{\mathbf x}{\|\mathbf x\|}
+ =
+ -\frac{4}{3}\pi G \rho \mathbf x
+ =
+ - 9.81 \frac{\mathbf x}{R_1} \frac{\text{m}}{\text{s}^2},
+ @f]
+ where we can infer the last expression because we know Earth's gravity at
+ the surface (where $\|x\|=R_1$).
There are two problems with this, however: (i) The Earth is not homogenous,
i.e. the density $\rho$ depends on $\mathbf x$; in fact it is not even a
- function that only depends on the radius $r$. (ii) The density, and by
+ function that only depends on the radius $r$. In reality, gravity therefore
+ does not always decrease as we get deeper: because the earth core is so much
+ denser than the mantle, gravity actually peaks at around $10.7
+ \frac{\text{m}}{\text{s}^2}$ at the core mantle boundary (see <a
+ target="_top" href="http://en.wikipedia.org/wiki/Earth's_gravity">this
+ article</a>). (ii) The density, and by
consequence the gravity vector, is not even constant in time: after all, the
problem we want to solve is the time dependent upwelling of hot, less dense
material and the downwelling of cold dense material. This leads to a gravity