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\begin{document}
\subsubsection{Defining the Proto-Kessler Region}
With the definitions of kessler syndrome and the law of debris given above, we can now 
explicitly describe the proto-kessler region.
\begin{align}
    \epsilon < (g - \delta) D_t  
            + \gamma \sum^n_{j=1} \left( 1-R^i(S_t,D_t) \right) s^i_t
            + \Gamma \sum^n_{j=1} x^j_t\}
\end{align}
As being in the proto-kessler region is a prerequesit to being in the kessler region, we see that
the kessler region depends on the collision rates of the constellation operators.

Although this is a straightforward result, I have not found it in any of the models I've examined so far.
I suspect it will impact optimal pigouvian taxation, but of course, I need to verify this in 
a computational example.
\end{document}
