\label{sec:expectations}
Pressure scaling: \[P \propto \rho \sigma_v^2\]
Density and velocity scale as power laws: \[\begin{aligned} \rho &\propto& L^{-a} \\ \sigma_v &\propto& L^b\end{aligned}\] Here \(\sigma_v\) is the 3d velocity dispersion, related to the 1d dispersion for isotropic fields by \(\sigma_{v,{\rm 1d}} = \sqrt{3} \sigma_{v,{\rm 3d}}\).
What values do we expect for the exponents \(a\) and \(b\)? Table \ref{tab:scaling} gives some possible examples. Combining these equations, we see that pressure scales as \[P \propto L^c; c=2b-a\] where \(-2 \lapprox c \lapprox 1\), with the lowest value for the exponent for thermal gas in a SIS envelope, and the highest value for turbulent, constant density gas. (Is the latter realistic? Probably not, limiting exponent to \(c < 0\)). Note Larson’s relations (updated for modern observations, and focusing on scales larger than core scales) give \(b \sim 0.5\) (Larson 1981, Solomon 198?, Goodman 199?, others?).
Upshot: we expect pressure to decrease toward small scales in clouds. We will test this with our data and simulations.
to do: think about: wouldn’t scaling expectations change as a function of size sampled? a cloud as a whole does NOT have the same density structure as a core – swiss cheese vs. bbs