point masses and Gauss’ Law of Gravity

My final year project involves searching for stable orbits for the gravitational n-body problem where we use Newton’s point mass approximation to describe the force field around masses assuming that these are approximately spherical. Now given that this model uses point masses we can’t model rotations. In fact, it’s generally assumed that they are negligible. 

However, I wondered whether this assumption can break down at some point. Consider the following:

  1. On the scale of millenia, the surface of our planet actually behaves like a fluid.
  2. If the distance between two planetary bodies is small and we assume zero rotation, this would imply a growing deviation from sphericity over time. It follows that the point-mass approximation would no longer make sense.
  3. In fact, our bodies would look something like blobs of gooey paint slowly dripping towards each other. 

While it’s true that the probability of zero rotation is close to zero the challenge of dealing with non-spherical masses is something that Newton wouldn’t have handled very well with his point-mass approximation. However, this is not quite the end of Newton’s model. 

Given that the gravitational potential in his model holds for point masses a natural solution would be to integrate over the volume enclosed by a compact body. This is exactly what Gauss did when he discovered the divergence theorem from which the Gauss Law of Gravity emerges as an immediate corollary. As this handles the case where our masses aren’t spherical the rest of my blog post will focus on the statement and proof of this theorem. 

Gauss Law of Gravitation:

Given U  as the gravitational potential and \vec{g} = -\nabla U  as the gravitational field, Gauss’ Law of Gravitation states:

\begin{aligned} \int_{V} (\nabla \cdot \vec{g}) dV =\int_{\partial V} \vec{g} \cdot d \vec{A}  \end{aligned}

In order to arrive at this result, I’ll first need to introduce two useful definitions:

flux:The flux \phi of a vector field through a surface is the integral of the scalar product of the field in a given point times the infinitesimal oriented surface at that point:

\begin{aligned} \phi =  \int_{\partial V} \vec{g} \cdot d \vec{A} \end{aligned}

where \vec{g} denotes the gravitational field, \partial V the boundary of our mass and d \vec{A} is our infinitesimal oriented surface.

divergence:The divergence represents volume density of the outward flux of a vector field from an infinitesimal volume around a given point:

\begin{aligned} \nabla \cdot \vec{g} \end{aligned}

The flux and divergence of a field are actually related in a local manner. Consider a box with dimensions \delta x_1,\delta x_2,\delta x_3 . Using Cartesian coordinates, let’s suppose that (x_1,x_2,x_3) denotes a corner on this box. Then the flux of \vec{g} projected onto \hat{x_3} is given by:

\begin{aligned} \phi_3 =(g_3(x_1,x_2,x_3+\delta x_3)-g_3(x_1,x_2,x_3)) \delta x_1 \delta x_2 \end{aligned}

For very small \delta x_3 ,

\begin{aligned} g_3(x_1,x_2,x_3+\delta x_3)-g_3(x_1,x_2,x_3) \approx \frac{\partial g_3}{\partial x_3} \delta x_3 \end{aligned}

where we get equality in the limit as \delta x_3 tends to zero.

As a result we have \forall i \in \{1,2,3 \}  :

\begin{aligned} \phi_i \approx \frac{\partial g_i}{\partial x_i} \delta x_1 \delta x_2 \delta x_3 =\frac{\partial g_i}{\partial x_i} \delta V  \end{aligned}

\begin{aligned} \phi =\sum_{i=1}^{3} \phi_i \end{aligned}

Now, we may deduce that the flux and divergence are related as follows:

\begin{aligned} \frac{\phi}{\delta V}\approx  \sum_{i=1}^{3} \frac{\partial g_i}{\partial x_i} \end{aligned}

\begin{aligned}\lim_{\delta V \to 0} \frac{\phi}{\delta V} = \nabla \cdot \vec{g} =  \sum_{i=1}^{3} \frac{\partial g_i}{\partial x_i} \end{aligned}

If we take an arbitrary solid and slice it into compact volumes V_i with associated surfaces \partial V_i we have:

\begin{aligned} \phi =\sum_{i=1}^{N} \phi_i =\sum_{i=1}^{N} \int_{\partial V_i} \vec{g_i} \cdot d \vec{A} =\int_{\partial V} \vec{g} \cdot d \vec{A} \end{aligned}

In addition, using the relationship between flux and divergence we have:

\begin{aligned} \phi =\lim_{N \to \infty} \lim_{V_i \to 0} \sum_{i=1}^{N} \nabla \cdot \vec{g_i} V_i = \int_{V} \nabla \cdot \vec{g} dV \end{aligned}

By equating the last two expressions we conclude the proof of this marvellous result by Gauss which is succinctly summarised on Wolfram as follows:

in the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or away from the region through its boundary

At this point it’s natural to ask how can we describe the gravitational field which is defined in terms of the gravitational potential. In general, for non-spherical masses we can no longer use the usual inverse radius gravitational potential due to Newton.

In some sense we’ll need an infinite sum of perturbations in the gravitational potential which can be done using spherical harmonics, another wonderful mathematical invention that has many applications. This shall be the subject of my next blog post. 



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