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84 lines
3.8 KiB
TeX
84 lines
3.8 KiB
TeX
\documentclass[../Main.tex]{subfiles}
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\graphicspath{{\subfix{Assets/img/}}}
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\begin{document}
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In his dissertation \cite{RaoDissertation} briefly examines the "survival rates" of a satellite constellation.
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I've applied this to my model and extended the results.
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This approach allows us to construct a elasticity of survival and satellite additions, i.e. an elasticity
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of risk.
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%I should probably look up how to analyze changes in risk level and quantitative representations etc.
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% Marginal survival.
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The survival rate for a constellation $i$ is defined as $R^i = 1-l^i(\cdot)$, i.e. the proportion of satellites-
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that were not lost (degraded nor destroyed) between period $t$ and $t+1$.
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Thus the marginal survival rate represents the additional loss of
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satellites due to a slightly larger constellation or fleet stock.
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To extend this definition to all fleets, we can measure the total number of
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satellites that survive.
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This can be calculated as the weighted sum of survival rates.
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\begin{align}
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R =& \frac{\sum_{i=1}^n s^i_t R^i}{\sum_{i=1}^n s^i_t}
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\end{align}
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\subsubsection{Marginal Survival Rates}
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We can find the marginal survival rate with respect to a given constellation $s^i_t$ as:
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\begin{align}
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\parder{R}{s^i_t}{} =& \parder{}{s^i_t}{}\frac{\sum_{j=1}^n s^j_t R^i}{\sum_{j=1}^n s^i_t} \\
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=& \left(\frac{1}{\sum_{j=1}^n s^j_t}\right)^2
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\left[
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\left(\sum^n_{j=1}s^j_t\right) \left(\parder{}{s^i_t}{}\sum^n_{j=1} s^j_t R^j\right)
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- \sum^n_{j=1} s^j_t R^j
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\right] \\
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=& \left(\frac{1}{\sum_{j=1}^n s^j_t}\right)
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\left[
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\left(\sum^n_{j \neq i} s^j_t \parder{R^j}{s^i_t}{}\right)
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+ \left( R^i + s^i_t \parder{R^i}{s^i_t}{}\right)
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- R
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\right]
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\\
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=&
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\left(\sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{}\right)
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+ \left(\frac{ R^i - R }{\sum_{j=1}^n s^j_t}\right)
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\\
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\parder{R}{s^i_t}{}
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=&
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\sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{}
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+ \left(\frac{ R^i - R }{\sum_{j=1}^n s^j_t}\right) \label{EQ:MarginalSurvivalRelation}
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\end{align}
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This can also be written in differential form as
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\begin{align}
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d{R}
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=&
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\sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) d{R^j}
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+ \left(\frac{ R^i - R }{\sum_{j=1}^n s^j_t}\right) d{s^i_t}\label{EQ:differentialSurvivalRelation}
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\end{align}
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From \cref{EQ:MarginalSurvivalRelation,EQ:differentialSurvivalRelation},
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we can see that the fleetwide marginal survival rate
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is made up of two components.
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\begin{itemize}
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\item $\sum^n_{j=1} \left(\frac{s^j_t}{\sum_{j=1}^n s^j_t}\right) \parder{R^j}{s^i_t}{}$
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represents the effect on each satellite constellation, and is always negative because
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$\parder{R^j}{s^i_t}{} < 0$ by assumption.
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Thus each constellations' survival rate will decrease as satellites are added to
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any constellation.
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\item $\frac{ R^i - R }{\sum_{j=1}^n s^j_t}$,
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represents the effect of averaging out marginal survival rates.
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Intuitively, when a constellation has a higher survival rate
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than the fleet's survival rate, adding a satellite to that fleet contributes
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less colision risk than if it were given to another
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Note that it is positive but only when $R^i > R$.
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Additionally, it disappears quickly as the total number of satellites increase.
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Thus when there are a large number of satellites in orbit, regardless of who
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owns them, it is almost certain that any increase in satellite stocks will
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lead to a reduction in the survival rate.
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\footnote{I believe Rao makes this an assumption, I show it is a result}
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\end{itemize}
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Consequently, we can see that in many cases, the marginal survival rate will be negative.
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\end{document}
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