@ -4,8 +4,9 @@
\begin { document} \begin { document}
In his dissertation \cite { RaoDissertation} briefly examines the "survival rates" of a satellite constellation.
In his dissertation \cite { RaoDissertation} briefly examines the "survival rates" of a satellite constellation.
I've applied this to my model and extended the results.
I've applied this to my model and extended the results.
This approach allows us to construct a elasticity of survival and satellite additions, i.e. an elasticity
% This approach allows us to construct a elasticity of survival and satellite additions,
of risk.
% i.e. an elasticity of risk.
% I should probably look up how to analyze changes in risk level and quantitative representations etc.% I should probably look up how to analyze changes in risk level and quantitative representations etc.
% Marginal survival.% Marginal survival.
@ -18,7 +19,8 @@ To extend this definition to all fleets, we can measure the total number of
satellites that survive.
satellites that survive.
This can be calculated as the weighted sum of survival rates.
This can be calculated as the weighted sum of survival rates.
\begin { align} \begin { align}
R =& \frac { \sum _ { i=1} ^ n s^ i_ t R^ i} { \sum _ { i=1} ^ n s^ i_ t}
R =& \frac { \sum _ { i=1} ^ n s^ i_ t R^ i} { \sum _ { i=1} ^ n s^ i_ t} \\
% =& \frac { \text { Total Surviving Satellites} } \frac { \text { Total Starting Satellites} }
\end { align} \end { align}
\subsubsection { Marginal Survival Rates} \subsubsection { Marginal Survival Rates}
@ -59,25 +61,32 @@ This can also be written in differential form as
From \cref { EQ:MarginalSurvivalRelation,EQ:differentialSurvivalRelation} ,
From \cref { EQ:MarginalSurvivalRelation,EQ:differentialSurvivalRelation} ,
we can see that the fleetwide marginal survival rate
we can see that the fleetwide marginal survival rate
is made up of two components.
is made up of two components.
We'll call these the direct and relative survival effects,
corresponding to the $ dR ^ j $ and $ ds ^ i _ t $ terms respectively.
\begin { itemize} \begin { itemize}
\item $ \sum ^ n _ { j = 1 } \left ( \frac { s ^ j _ t } { \sum _ { j = 1 } ^ n s ^ j _ t } \right ) \parder { R ^ j } { s ^ i _ t } { } $
\item The direct survival effect,
represents the effect on each satellite constellation, and is always negative because
$ \sum ^ n _ { j = 1 } \left ( \frac { s ^ j _ t } { \sum _ { j = 1 } ^ n s ^ j _ t } \right ) \parder { R ^ j } { s ^ i _ t } { } $ ,
$ \parder { R ^ j } { s ^ i _ t } { } < 0 $ by assumption.
represents the effect of a new satellite on each constellation.
Thus each constellations' survival rate will decrease as satellites are added to
It is always negative because
any constella tion.
$ \parder { R ^ j } { s ^ i _ t } { } < 0 $ by assump tion.
\item $ \frac { R ^ i - R } { \sum _ { j = 1 } ^ n s ^ j _ t } $ ,
Thus each constellations' survival rate will decrease as satellites are added to
represents the effect of averaging out marginal survival rates .
any constellation .
Intuitively, when a constellation has a higher survival rate
\item The relative survival effect, found in
than the fleet's survival rate, adding a satellite to that fleet contributes
$ \frac { R ^ i - R } { \sum _ { j = 1 } ^ n s ^ j _ t } $ ,
less colision risk than if it were given to another
represents the effect of averaging out marginal survival rates.
Note that it is positive but only when $ R ^ i > R $ .
Intuitively, when a constellation has a higher survival rate
Additionally, it disappears quickly as the total number of satellites increase.
than the general fleet's survival rate, adding a satellite to
Thus when there are a large number of satellites in orbit, regardless of who
that constellation contributes less colision risk than if it were given
owns them, it is almost certain that any increase in satellite stocks will
to another constellation.
lead to a reduction in the survival rate.
Thus when there are a large number of satellites in orbit, regardless of who
\footnote { I believe Rao makes this an assumption, I show it is a result}
owns them, this effect is removed.
\end { itemize} \end { itemize}
Consequently, we can see that in many cases, the marginal survival rate will be negative.
Consequently, we can see that in most cases, the marginal survival rate will be negative.
In most models, this is either not examined or is assumed, but now we have the opportunity
to examine incentives in the case that it is not true.
One particular case where this may be important is when there is low utilization,
low internal risk, and near-monopolistic use of an orbital shell.
\end { document} \end { document}
youainti
5 years ago