  The primary goal of any future hadron collider is to discover new physics
(NP) associated with a high mass scale, $M$, beyond the range of the LHC. In
order to maintain the same {\it relative} mass reach for NP, $M/{\sqrt s}$, as
$\sqrt s$ increases, Richter recently reminded us that the required integrated
luminosity obtainable at future hadron colliders (FHC) must grow rapidly, $\sim
s$, in the limit of naive scaling. This would imply, eg, a $\sim 50$-fold
increase in the required integrated luminosity when going from the 14 TeV LHC
to a FHC with $\sqrt s=100$ TeV, an increase that would prove quite challenging
on many different fronts. In this paper we point out, due to the scaling
violations associated with the evolution of the parton density functions (PDFs)
and the running of the strong coupling, $\alpha_s$, that the actual luminosity
necessary in order to maintain any fixed value of the relative mass reach is
somewhat greater than this scaling result indicates. However, the actual values
of the required luminosity scaling are found to be dependent upon the detailed
nature of the NP being considered. Here we elucidate this point explicitly by
employing several specific benchmark examples of possible NP scenarios and
briefly discuss the (relatively weak) search impact in each case if these
luminosity goals are not met.
