The Kennicutt-Schmidt Law and Gas Scale Height in Luminous and Ultra-Luminous Infrared Galaxies

TL;DR

This study analyzes high-resolution ALMA data of luminous and ultra-luminous infrared galaxies, revealing a super-linear Kennicutt-Schmidt relation, nearly constant gas scale height, and implications for star formation efficiency and turbulence models.

Contribution

It provides new empirical measurements of the KS relation slope, gas scale height, and star formation efficiency in extreme star-forming galaxies, challenging some feedback-driven turbulence models.

Findings

KS relation slope ~1.74 for high gas surface densities

Gas scale height remains nearly constant at 150-190 pc

Star formation efficiency per free-fall time is 5-7%

Abstract

A new analysis of high-resolution data from the Atacama Large Millimeter/submillimeter Array (ALMA) for 5 luminous or ultra-luminous infrared galaxies gives a slope for the Kennicutt-Schmidt (KS) relation equal to $1.74^{+0.09}_{\rm -0.07}$ for gas surface densities $\Sigma_{\rm mol}>10^3\;M_\odot$ pc$^{-2}$ and an assumed constant CO-to-H$_2$ conversion factor. The velocity dispersion of the CO line, $\sigma_v$, scales approximately as the inverse square root of $\Sigma_{\rm mol}$, making the empirical gas scale height determined from $H\sim0.5\sigma^2/(\pi G\Sigma_{\rm mol})$ nearly constant, 150-190 pc, over 1.5 orders of magnitude in $\Sigma_{\rm mol}$. This constancy of $H$ implies that the average midplane density, which is presumably dominated by CO-emitting gas for these extreme star-forming galaxies, scales linearly with the gas surface density, which, in turn, implies that the gas dynamical rate (the inverse of the free-fall time) varies with $\Sigma_{\rm mol}^{1/2}$, thereby explaining most of the super-linear slope in the KS relation. Consistent with these relations, we also find that the mean efficiency of star formation per free-fall time is roughly constant, 5%-7%, and the gas depletion time decreases at high $\Sigma_{\rm mol}$, reaching only $\sim 16$ Myr at $\Sigma_{\rm mol}\sim10^4\;M_\odot$ pc$^{-2}$. The variation of $\sigma_v$ with $\Sigma_{\rm mol}$ and the constancy of $H$ are in tension with some feedback-driven models, which predict $\sigma_v$ to be more constant and $H$ to be more variable. However, these results are consistent with simulations in which large-scale gravity drives turbulence through a feedback process that maintains an approximately constant Toomre $Q$ instability parameter.

Tables

Table 1. Table 1: (Ultra-)Luminous Infrared Galaxies observed with ALMA

Galaxy	DistanceaaFrom redshift (corrected to the 3K CMB reference frame) and assuming $H_o=70.5$ km s-1 Mpc-1. For NGC 7469, SN Type Ia distance from Ganeshalingam et al. 2013.	Map areabbThe area of high signal-to-noise emission used in this analysis; the ALMA maps detect emission over a larger area, especially in CO.	beam FWHM	${\sigma }_{\mathrm{cont}}$	${\sigma }_{\mathrm{CO}\left(1-0\right)}$	binned pixel
	(Mpc)	(kpc2)	(′′)	(mJy beam-1)	(mJy beam-1 km s-1)	size (pc)
IRAS 17208-0014	182	1.3	0.5	0.05	0.18	397
Arp 220	79	1.1	0.95$\times$0.60	0.10	0.18	345
IRAS 13120-5453	134	3.4	1.1	0.08	0.14	650
NGC 3256	44	5.2	2.2	0.05	0.12	512
NGC 7469	66	1.6	0.95	0.015	0.043	418

Equations

$$X_{\mathrm{CO}}=\min (4,6.75/W_{\mathrm{CO}}^{0.32})\times 10^{20}{\mathrm{cm}}^{-2}({\mathrm{K}\mathrm{km}\mathrm{s}}^{-1}{)}^{-1}$$

$$H=\frac{{\Sigma }_{\mathrm{gas}}}{2{\rho }_{\mathrm{mid}}}=\frac{{\sigma }_{\mathrm{v}}^2(1+\alpha +\beta )}{\pi G{\Sigma }_{\mathrm{total},\mathrm{GL}}}.$$

$$\frac{{\bar{g}}_{\mathrm{galaxy}}}{\bar{g}}=0.5\left(\frac{M_{\mathrm{galaxy}}}{M_{\mathrm{disk},\mathrm{GL}}}\right)\left(\frac{H}{R}\right),$$

$$\left(\frac{H}{R}\right)\left(1+\frac{H}{2\mu R}\right)=\frac{\xi }{\mu }$$

$$\frac{H}{R}={\left({\mu }^2+2\xi \right)}^{1/2}-\mu .$$

$$H=\frac{{\sigma }_{\mathrm{v}}^2}{\pi G{\Sigma }_{\mathrm{gas}}}\times \left(\frac{1+\alpha +\beta }{1+{\bar{g}}_{\mathrm{galaxy}}/\bar{g}}\right)\times \left(\frac{{\Sigma }_{\mathrm{gas}}}{{\Sigma }_{\mathrm{total},\mathrm{GL}}}\right).$$

$$H\simeq 0.5\frac{{\sigma }_v^2}{\pi G{\Sigma }_{\mathrm{mol}}}.$$

$$t_{\mathrm{ff}}=\sqrt{\frac{3\pi }{32G{\rho }_{\mathrm{mid}}}}=\frac{\sqrt{3}}{4G}\frac{{\sigma }_v}{{\Sigma }_{\mathrm{mol}}}.$$

$$t_{\mathrm{dep}}=\frac{{\Sigma }_{\mathrm{mol}}}{{\Sigma }_{\mathrm{SFR}}}.$$

$${ϵ}_{\mathrm{ff}}=t_{\mathrm{ff}}\frac{{\Sigma }_{\mathrm{SFR}}}{{\Sigma }_{\mathrm{mol}}}=\frac{\sqrt{3}}{4G}\frac{{\sigma }_v{\Sigma }_{\mathrm{SFR}}}{{\Sigma }_{\mathrm{mol}}^2}$$

$$\log (t_{\mathrm{dep}})=-1.04\log (\sigma )+1.71$$

$$t_{\mathrm{dep}}=t_{\mathrm{ff}}/{ϵ}_{\mathrm{ff}}={\left(\frac{3\pi }{32G\rho }\right)}^{1/2}/{ϵ}_{\mathrm{ff}}$$

$${\sigma }_v=7.7\times 10^{17}{\rho }^{1/2}(\mathrm{cgs}\mathrm{units}).$$

$$H=0.5\frac{{\sigma }_v^2}{\pi G{\Sigma }_{\mathrm{gas}}}=2.97\times 10^{35}\frac{\rho }{\pi G{\Sigma }_{\mathrm{gas}}}=2.97\times 10^{35}\left(\frac{{\Sigma }_{\mathrm{gas}}}{2H\pi G{\Sigma }_{\mathrm{gas}}}\right)=7.08\times 10^{41}/H$$

Full text

galaxies: ISM — galaxies: star formation — galaxies: starburst

††journal: ApJ††facilities: ALMA††software: astropy (Astropy Collaboration et al., 2013), CASA (McMullin et al., 2007)

1 Introduction

The Kennicutt-Schmidt (KS) relation describes the observed correlation between the star formation rate per unit area, $\Sigma_{\rm SFR}$, and the surface density of gas, $\Sigma_{\rm gas}$ and is a power law for the main disk regions of spiral galaxies. Because star formation is expected to follow the gas, a slope close to unity, as found for CO emission by Bigiel et al. (2008) and Leroy et al. (2008) or HCN emission by Gao & Solomon (2004), might not be surprising. However, star formation is a dynamical process involving the rate of conversion of gas into stars, so a mass dependence alone (as in the linear law) cannot be the full story. There has to be a time component, and for gravitating systems, that means a volume density is involved. The linear laws only depend on the gas surface density, rather than the volume density, so these laws presumably arise from selection effects in surveys that observe sub-regions of gas at a characteristic density, depending on the molecular transition used (Krumholz & Thompson, 2007; Narayanan et al., 2008). The timescale is then the collapse time at that selected density, i.e., a constant. In contrast, the total gas should have a continuum of densities that widely participates in a gravity-driven condensation into dense clouds (Elmegreen, 2015, 2018). If the average density increases with $\Sigma_{\rm SFR}$, then the KS slope will be steeper than linear, such as $1.4$ in the observations by Kennicutt (1998), de los Reyes & Kennicutt (2019), and others.

For a disk with gas surface density $\Sigma_{\rm gas}$ and scale height $H$, the average midplane gas density is $\rho_{\rm mid}=\Sigma_{\rm gas}/(2H)$, so the observed total-gas slope of $\sim 1.4$ can result from a gravity-driven model with a rate $(G\rho_{\rm mid})^{0.5}$, provided that the disk scale height is about constant (Madore, 1977; Larson, 1988; Elmegreen, 2018). In the Milky Way, the thickness of the molecular layer is indeed about constant inside the solar radius (Heyer & Dame, 2015), but there is no direct view yet of the disk thickness in other galaxies where the KS relation has been measured.

The KS relation for starbursts and (ultra)-luminous infrared galaxies (U/LIRGS) has about the same $\sim 1.5$ slope for CO as the total gas relation in galaxy disks (Kennicutt, 1998; Gao & Solomon, 2004; Krumholz et al., 2012; Gowardhan et al., 2017; Shi et al., 2018). This is presumably because most of the gas in starbursts is dense enough to emit CO and that molecule is no longer a sparse tracer subject to selection effects. The similar slope implies that even with extremely high star formation rate densities, the balance between feedback and self-gravity produces a vertical equilibrium with a relatively constant gas thickness, i.e., much more constant than the range of surface densities.

The purpose of this paper is to examine more closely the KS relation in the starburst regime and to estimate the disk thickness from the observed molecular gas velocity dispersion and surface density. From these we determine the average midplane density, free-fall time, gas consumption time, and efficiency per free-fall time. The results confirm the super-linear KS slope found previously for starbursts, and they also reveal a nearly constant disk thickness, confirming the most basic model in which three-dimensional density primarily determines the rate at which gas turns into stars (Madore 1977; Silk 1987; Katz 1992; Elmegreen 1994, 2002; Krumholz & McKee 2005; Bacchini et al. 2019; see review in Krumholz 2014).

In what follows, Section 2 describes the observations and data processing. Section 3.1 derives the KS law, Section 3.2 determines the disk scale heights, and Section 3.3 derives the gas depletion time, free-fall time, and efficiency per free-fall time. Section 4 considers our observations in the context of various theoretical predictions and Section 5 presents the conclusions.

2 Observations and data processing

To study the KS relation at high star formation rates, we searched the ALMA archive for U/LIRGs for which suitable observations of the CO J=1-0 line were available (Table 2). For each project, the raw uv-data were calibrated using the scripts retrieved from the archive and the CASA version used in the original calibration. All further processing was carried out in CASA versions 5.0 to 5.4. Continuum subtraction was performed on the uv-data using line-free channels. Cleaned image cubes were made using Briggs weighting with robust=0.5 and channel widths of 20 km s*-1* (26.4 km s*-1* for NGC 3256, 40 km s*-1* for Arp 220). Continuum images were made with the same weighting using the line-free channels. For three galaxies where the CO and the continuum images used different ALMA data sets, a common minimum uv-distance cutoff was used for both datasets and a taper was applied to roughly match the resulting beams. Finally, the continuum image and line cube were smoothed to have identical resolution. More details of the image processing are given in Wilson et al. (2019).