Harvard Astronomy 201b

ARTICLE: Self-Regulated Star Formation in Galaxies via Momentum Input from Massive Stars

In Journal Club, Journal Club 2011 on April 7, 2011 at 5:32 am

Read the Paper by P.F. Hopkins, E. Quataert, and N. Murray (2010)

Summary by Dylan Nelson and Josh Suresh


Feedback from massive stars is believed to play a critical role in shaping the galaxy mass function, the structure of the interstellar medium (ISM) in galaxies, and the slow conversion of gas into stars over many dynamical times. This paper is the first in a series studying stellar feedback in galaxy formation. We present a new numerical method for implementing stellar feedback via the momentum imparted to the ISM by radiation pressure, supernovae, and stellar winds. In contrast to the majority of the results in the literature, we do not artificially suppress cooling or ‘turn off’ the hydrodynamics for a subset of the gas: the gas can cool to <100K and so the ISM inevitably becomes highly inhomogeneous. For reasonable feedback efficiencies galaxies reach an approximate steady state in which gas collapses due to gravity to form giant molecular clouds and feedback disperses these dense regions back into the more diffuse ISM. This is true for a wide range of galaxy models, from SMC-like dwarfs and Milky-way analogues to z~2 clumpy disks. The resulting star formation efficiencies are consistent with the observed global Kennicutt-Schmidt relation. Moreover, the star formation rates in our galaxy models are nearly independent of the numerically imposed high-density star formation efficiency and density threshold. This is a consequence of star formation regulated by stellar feedback; it enables our method to be more predictive than previous treatments. By contrast, without stellar feedback, the ISM experiences runaway collapse to very high densities and the global star formation rates exceed those observed by 1-2 orders of magnitude. This highlights the critical role that momentum in stellar feedback plays regulating star formation in galaxies.

In cosmological numerical simulations without strong stellar feedback, gas in galaxies generically becomes very cold and collapses, forming stars at a much higher rate than is observed in real galaxies.  Furthermore, simulations which can resolve the formation of giant molecular clouds (GMCs) often do not include the physics to allow these structures to be disrupted.  In past work, the most common approach to dealing with these issues is to include (only) thermal gas heating from supernovae.  However, since this gas cools relatively quickly, simulators are forced to artificially turn off cooling for some time, which is difficult to justify on physical grounds. Furthermore, most simulators neglect the importance of momentum supplied by stellar feedbackWith these limitations of past work in mind, the main motivation of this paper is to explore the effect of stellar momentum feedback on star formation and ISM turbulence.

Big Ideas:

  • Stellar feedback important for ISM turbulence and star formation
  • Some examples: protostellar jets, HII regions, stellar winds, radiation pressure, supernovae
  • Goal: Explore effects of momentum deposition from massive stars

This work carries out several simulations using the TreeSPH code GADGET (Springel 2005). The numerics include stars, dark matter, and gas, with a new implementation of stellar feedback, described below. There is no consideration of black holes, AGN growth or feedback.

Numerical Method (summarized from paper):

  • Isolated (non-cosmological) galaxy evolution using SPH code
  • Standard star formation prescription relates SFR to gas density above some critical threshold:
  • Momentum input model:
    1. Associate each gas particle with nearest star forming clump (Friends-of-friends algorithm)
    2. Calculate “clump radius” and enclosed “clump mass”
    3. Use Starburst99 simple stellar population (SSP) with Kroupa02 IMF for total clump luminosity
    4. Rate of momentum deposition in the gas calculated as:
    5. Use an empirical star cluster size-mass relation to calculate the escape velocity
    6. Assign a stochastic kick probability for each SPH particle based on escape speed as:
    7. Finally, apply momentum kicks at each simulation timestep

The authors generate four distinct, idealized models of disk galaxies, in order to test a wider range of galactic parameter space. Each of the four models includes an extended dark matter halo (NFW), a stellar bulge (Hernquist), and stellar and gaseous disks (exponential). See below for descriptions of each individual model. The figure blow shows each galaxy type during the quasi-steady feedback-regulated phase that sets in after a few dynamical times (with large scale and zoomed-in views).

Galaxy Models:

  • HiZ: z~2-3, strongly unstable and self gravitating. Represents a massive, high redshift, and strongly unstable disk forming stars at a very high rate. System is baryon dominated out to large radii. Also initialized with a stable Q = 1.
  • Sbc: intermediate mass, relatively gas rich star-forming disk in local universe (LIRG). Initialized with a Toomre Q = 1 at all radii. Vertical support is from thermal pressure. The total baryonic mass is \sim 10^{10} solar masses, and the halo is \sim 10^{11}. This gives a cosmologically motivated mass-to-light ratio.
  • MW: Milky Way like disk. Disk and halo parameters tuned to observationally motivated values from the Milky Way. Dark matter dominated.
  • SMC: Dwarf galaxy model, initialized with values similar to the SMC. Dark matter dominated at all radii outside the center.

From Fig. 1 of the paper. Each of the four galaxy types during the quasi-steady feedback-regulated phase that sets in after a few dynamical times (with large scale and zoomed-in views).

The Star Formation Rate (SFR) in the four models:

  • Without feedback (red lines), compare each simulation to the same galaxy run without stellar feedback (shown as red lines in the following figure of star formation rate with time). In models without feedback, note that the SFR increases to a peak value on a single dynamical time. The SFR then remains at this high value until the gas is depleted. Problematically, this peak value of the SFR is a factor of 10 greater than what is observed in real galaxies. These models without stellar feedback also lie off the observed Kennicutt-Schmidt relation (see last figure on this page).
  • With feedback (blue lines), each simulation rapidly reaches a maximum SFR and then remains at this quasi-steady state for several dynamical times. Of course the SFR must eventually decline due to gas depletion, but this decline is much more gradual when stellar feedback is included. This case is also reasonably consistent with the observed KS relation (see last figure on this page). The conclusion is that stellar feedback is critical to regulating star formation in galaxies.

The properties of the ISM are then analyzed in detail for the HiZ model as a function of time and radius.

ISM Structural Properties in the HiZ model:

The HiZ model, representing a galaxy at high redshift, has a significantly higher density than the other galaxy types.  Hence, it represents the strongest test for whether momentum feedback can stabilize star formation in galaxies.  Hence, the authors look at the ISM of their HiZ model in close detail, producing the plots shown below.

  • (Top left) Vertical velocity dispersion – This plot shows the vertical velocity dispersion (which can be interpreted as vertical support in the disk).  Note that initially, the disk is initially thermally supported, but this thermal energy rapidly radiates away. At late times, however, the feedback from massive stars revives the vertical support and reaches a quasi-equilibrium with Q~1 (see next point).
  • (Top center) Toomre’s Q – Note that turbulent support maintains quasi-constant Q~1 (for the gas)
  • (Top right) Gas density distribution – The dotted blue line shows the gas density without feedback.  Note that with no feedback, the gas piles up at the highest resolvable densities.  However, when momentum feedback is implemented, there is a natural peak density which arises that is significantly lower.  Also note that the simulations are well-converged, in the sense that they agree very well on the peak of the density distribution.
  • (Bottom left) Integrated momentum input – The dash-dotted line denotes the feedback generated by stars that are less than 1 Myr old.  Note that the dash-dotted line actually dominates the total feedback.  This shows that, in the simulations, very young stars immediately disrupt their host gas clouds, transferring the most feedback to nearby gas.  Older stars do not have much gas around them (since it was all disrupted while they were young), hence they do not contribute much to the total integrated momentum feedback.
  • (Bottom center) Mean kick velocities – Note that the mean kick velocity is about 150-200 km/s, as expected given the mass of the star-forming clumps.  This is significantly larger than the disk dispersion because the particles immediately share this momentum with other particles (i.e. the particles are not unphysically ejected from the disk).
  • (Bottom right) Resolved IR optical depths of gas clumps – Note that for most of the lifetime of the disk, the IR optical depth ranges from 30-50.  This is consistent with the observed surface densities of star clusters on roughly parsec scales.

Comparison with the Observed Global Kennicutt-Schmidt Law

We saw in the above six plots comparisons between different resolutions and momentum feedback parameter \eta_p (black, red, and green lines). The work discusses in detail numerical convergence testing with regards to resolution, the feedback model, and the star formation prescription. They conclude that the results summarized above do not depend sensitively on either of the main free parameters in their simulations. We do not include the numerous plots showing the details of this independence on model parameters. Finally, the work compares the global Kennicutt-Schmidt law predicted by the simulations with and without stellar feedback, to observations. This is shown below:

Primary Results:

  • Without feedback the simulations predict a SFR surface density well in excess of the observed KS law. However, simulations with feedback lie close to the observed relation at essentially all times. Furthermore, varying the simulation parameters shifts the systems along the relation, and does not offset them from the relation. This agreement holds over a large dynamic range of gas surface density from 10^7 - 10^{10} M_{\odot} \rm{kpc}^{-2}.
  • In this model, the single key parameter is the momentum supplied to high density gas around star clusters. The resulting effects on the ISM are a prediction of the model, as opposed to many previous works. However, the model remains “sub-grid” on the scale of individual molecular clouds.
  • With feedback included, GMCs dissociate quickly and the galaxy develops a turbulent, multiphase ISM. Thus galaxies self-regulate and approach a quasi steady state SFR, maintaining the ISM at Q ~ 1.
  • The SFR is nearly independent of numerical factors (SF efficiency, threshold, ηv, ηp). Specifically, less than a factor of two change in the SFR is found over an order of magnitude change of ηp, the normalization of the momentum injection rate, the key parameter which determines the efficiency of the stellar feedback.


The findings of this paper are fairly robust, since their model is physically motivated and they never “turn off” the hydrodynamics as do many past simulations of thermal feedback.  Furthermore, their method is generalizable to both Lagrangian and Eulerian codes.

Reading this paper, we noticed a couple caveats, however.  First of all, the feedback that is implemented is strictly radiation pressure from young stars; the momentum flux from supernovae and massive stellar winds, which can also be significant, are neglected.  Furthermore, there is no feedback from black holes or AGNs, and there is no thermal feedback.  Since all of these processes interact non-linearly, it is not always clear how to interpret the results of a simulation which only includes one type of feedback.  Another caveat, which the authors note, is that the disks in these simulations are isolated.  Hence the paper does not explore the effect of momentum feedback over cosmologically interesting timescales.

Nevertheless, the agreement with the observed global Kennicutt-Schmidt relation is impressive, so it seems that this is a realistic implementation of the important mechanism of momentum feedback from young stars.


  1. […] Galaxy evolution cannot be properly understood without accounting for strong feedback from massive stars. Specifically, in cosmological models that don’t include feedback processes, the star formation rates in simulated galaxies are much too high, as gas quickly cools and collapses. Additionally, these simulations find that the total amount of gas present in galactic disks is too high. Both of these problems can be solved by including local outflows and galactic superwinds that remove baryons from the disks, slowing star formation and bringing simulations in line with observations. Such winds are difficult to include in simulations, however, because they have their origins in stellar feedback processes, which occur on small scales. Most simulations are either too low resolution to properly model these processes, or they make simplifying assumptions about the physics that prevent accurate modeling of winds. Thus, although we have seen observational evidence of such winds in real galaxies, until recently simulations have not been able to generate galactic winds from first principles and have instead added them in manually. Hopkins, Quataert, and Murray for the first time present a series of numerical simulations that successfully reproduce galactic winds that are consistent with observations for a wide range of galaxy types. Unlike previous work, their simulations have both sufficiently high resolution to focus on small-scale processes in giant molecular clouds (GMCs) and star forming regions and the physics to account for multiple types of stellar feedback, not just thermal heating from supernovae. These simulations are also discussed in two companion papers (Hopkins et al 2011 and Hopkins et al 2012), which focus on the star formation histories and properties of the galactic ISM of simulated galaxies and outline rigorous numerical tests of the models. The 2011 paper was discussed in the 2011 Ay 201b journal club and is summarized nicely here. […]

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