Heyer et al. (2009)

The big idea¶

Larson (1981) proposed three scaling relations for molecular clouds: a size–linewidth law $\sigma_v\propto L^{0.38}$, virial equilibrium $2\sigma_v^2 L/GM\sim1$, and an inverse density–size law $n\propto L^{-1.1}$. Together these imply that all clouds share the same surface density — a cornerstone assumption of many cloud/star-formation models. Heyer+2009 re-test this with higher-quality, optically-thinner $^{13}$CO data (the BU-FCRAO Galactic Ring Survey) on the Solomon+1987 (SRBY) cloud sample, and find it fails: the size–linewidth coefficient is not universal but increases with surface density. Clouds are still close to self-gravitating equilibrium — but across a range of surface densities, not a single one.

Core result — the surface-density dependence¶

Define the structure-function coefficient

Larson’s relations require $v_0=$ const. Heyer+2009 instead find it scales with the cloud mass surface density $\Sigma$:

exactly as expected for clouds in self-gravitational (virial) equilibrium. To see why, write the virial parameter for a uniform sphere of mass $M=\pi R^2\Sigma$:

If $v_0^2\propto\Sigma$ (the Heyer relation), then $\alpha_\mathrm{vir}\approx$ constant ($\sim1$–2) across clouds — they self-regulate to virial equilibrium at every surface density. Only if one (wrongly) holds $v_0$ fixed at Larson’s constant does one get $\alpha_\mathrm{vir}\propto\Sigma^{-1}$.

Numbers worth pinning¶

• LTE masses (from $^{13}$CO) are typically ~5× smaller than SRBY virial masses.

• Median mass surface density of the Heyer sample: $\Sigma\approx 42\,M_\odot\,\mathrm{pc}^{-2}$.

• SRBY had assumed a constant $\Sigma(\mathrm{H_2})\approx 200\,M_\odot\,\mathrm{pc}^{-2}$ (corrected from the 170 quoted by SRBY for an updated galactocentric radius).

• Larson (1981): $\sigma_v\propto L^{0.38}$; Solomon+1987 (SRBY): $\sigma_v\propto R^{0.5}$ (steeper) — the convention progenax’s larson_sigma_v default ($\alpha=0.5$) follows.

Use in progenax¶

• The correct grounding for any $\sigma_v(R,\Sigma)$ or $\alpha_\mathrm{vir}(\Sigma)$ relation in the PN11 alternative path (collapse_threshold.py), which currently takes $\alpha_\mathrm{vir}$ as an explicit input.

• Confirms the Larson/Solomon size–linewidth exponents used by cluster/turbulence.py larson_sigma_v.

Notes¶

• Citation correction. The legacy cluster/gravoturbulent.py attributed an $\alpha_\mathrm{vir}\propto\Sigma^{-1}$ relation (with an unsourced $\Sigma_0=85$) to “Heyer & Dame (2015)”. That is wrong on three counts: (i) the relevant paper is Heyer+2009 (Heyer & Dame 2015 is a separate ARA&A review); (ii) Heyer+2009 refutes the constant-$v_0$ assumption that gives $\alpha_\mathrm{vir}\propto\Sigma^{-1}$, finding $v_0\propto\Sigma^{1/2}\Rightarrow\alpha_\mathrm{vir}\approx$ const; (iii) $\Sigma_0=85$ matches none of the paper’s values (42 sample median; 200 SRBY constant). That legacy module is being removed in the clean-room pass.

• $\alpha_\mathrm{vir}\approx1$ is marginally bound/virial; $\alpha_\mathrm{vir}\gg1$ is unbound (turbulence-dominated). This is the $\alpha_\mathrm{vir}$ that enters the PN11 critical density (Padoan & Nordlund (2011)).

• Caveats: $^{13}$CO LTE masses assume constant abundance and can be uncertain in cloud envelopes; cloud boundaries/areas carry definitional scatter.