Galactic Winds Research Articles

An origin for multiphase gas in galactic winds and haloes

The physical origin of high-velocity cool gas seen in galactic winds remains unknown. Following work by B. Wang, we argue that radiative cooling in initially hot thermally-driven outflows can produce fast neutral atomic and photoionized cool gas. The inevitability of adiabatic cooling from the flow's initial 107–108 K temperature and the shape of the cooling function for T ≲ 107 K imply that outflows with hot gas mass-loss rate relative to star formation rate of |$\beta =\dot{M}_{\rm hot}/\dot{M}_\star \gtrsim 0.5$| cool radiatively on scales ranging from the size of the energy injection region to tens of kpc. We highlight the β and star formation rate surface density dependence of the column density, emission measure, radiative efficiency, and velocity. At rcool, the gas produces X-ray and then UV/optical line emission with a total power bounded by ∼10−2 L⋆ if the flow is powered by steady-state star formation with luminosity L⋆. The wind is thermally unstable at rcool, potentially leading to a multiphase medium. Cooled winds decelerate significantly in the extended gravitational potential of galaxies. The cool gas precipitated from hot outflows may explain its prevalence in galactic haloes. We forward a picture of winds whereby cool clouds are initially accelerated by the ram pressure of the hot flow, but are rapidly shredded by hydrodynamical instabilities, thereby increasing β, seeding radiative and thermal instability, and cool gas rebirth. If the cooled wind shocks as it sweeps up the circumgalactic medium, its cooling time is short, thus depositing cool gas far out into the halo. Finally, conduction can dominate energy transport in low-β hot winds, leading to flatter temperature profiles than otherwise expected, potentially consistent with X-ray observations of some starbursts.

SUPERNOVA FEEDBACK AND THE HOT GAS FILLING FRACTION OF THE INTERSTELLAR MEDIUM

Supernovae (SN), the most energetic stellar feedback mechanism, are crucial for regulating the interstellar medium (ISM) and launching galactic winds. We explore how supernova remnants (SNRs) create a multiphase medium by performing 3D hydrodynamical simulations at various SN rates, $S$, and ISM average densities, $\bar{n}$. The evolution of a SNR in a self-consistently generated three-phase ISM is qualitatively different from that in a uniform or a two-phase warm/cold medium. By travelling faster and further in the low-density hot phase, the domain of a SNR increases by $>10^{2.5}$. Varying $\bar{n}$ and $S$, we find that a steady state can only be achieved when the hot gas volume fraction $f_{\rm{V,hot}}\lesssim 0.6 \pm 0.1 $. Above that level, overlapping SNRs render connecting topology of the hot gas, and the ISM is subjected to thermal runaway. Photoelectric heating (PEH) has a surprisingly strong impact on $f_{\rm{V,hot}}$. For $\bar{n}\gtrsim 3 \cm-3 $, a reasonable PEH rate is able to suppress the thermal runaway. Overall, we determine the critical SN rate for the onset of thermal runaway to be $S_{\rm{crit}} = 200 (\bar{n}/1\cm-3)^k (E_{\rm{SN}}/10^{51}\erg)^{-1} \kpc^{-3} \myr-1$, where $k = (1.2,2.7)$ for $\bar{n} \leq 1$ and $> 1\cm-3 $, respectively. We present a fitting formula of the ISM pressure $P(\bar{n}$, $S$), which can be used as an effective equation of state in cosmological simulations. Despite the 5 orders of magnitude span of $(\bar{n},S)$, the average Mach number varies little: $\mathcal{M} \approx \ 0.5\pm 0.2, \ 1.2\pm 0.3,\ 2.3\pm 0.9$ for the hot, warm and cold phases, respectively.

Gusty, gaseous flows of FIRE: galactic winds in cosmological simulations with explicit stellar feedback

We present an analysis of the galaxy-scale gaseous outflows from the FIRE (Feedback in Realistic Environments) simulations. This suite of hydrodynamic cosmological zoom simulations resolves formation of star-forming giant molecular clouds to $z=0$, and features an explicit stellar feedback model on small scales. Our simulations reveal that high redshift galaxies undergo bursts of star formation followed by powerful gusts of galactic outflows that eject much of the ISM and temporarily suppress star formation. At low redshift, however, sufficiently massive galaxies corresponding to L*-progenitors develop stable disks and switch into a continuous and quiescent mode of star formation that does not drive outflows far into the halo. Mass-loading factors for winds in L*-progenitors are $\eta \approx 10$ at high redshift, but decrease to $\eta \ll 1$ at low redshift. Although lower values of $\eta$ are expected as halos grow in mass over time, we show that the strong suppression of outflows with decreasing redshift cannot be explained by mass evolution alone. Circumgalactic outflow velocities are variable and broadly distributed, but typically range between one and three times the circular velocity of the halo. Much of the ejected material builds a reservoir of enriched gas within the circumgalactic medium, some of which could be later recycled to fuel further star formation. However, a fraction of the gas that leaves the virial radius through galactic winds is never regained, causing most halos with mass $M_h \le 10^{12} M_{\odot}$ to be deficient in baryons compared to the cosmic mean by $z=0$.

Galactic wind in NGC 4460: New observations

NGC4460 is an isolated lenticular galaxy, in which galactic wind has been earlier discovered as a gas outflow associated with circumnuclear regions of star formation. Using the results of observations in the Halpha line with the scanning Fabry-Perot interferometer on the SAO RAS 6-m telescope, we studied the kinematics of the ionized gas in this galaxy. The parameters of gas outflow from the plane of the galactic disk were refined within a simple geometric model. We show that it is impossible to characterize the wind by a fixed velocity value. Characteristic outflow velocities are within 30..80 km/s , and they are insufficient to make the swept-out matter ultimately leave the galaxy.

Distribution of streaming rates into high-redshift galaxies

We study the accretion along streams from the cosmic web into high-redshift massive galaxies using three sets of AMR hydrocosmological simulations. We find that the streams keep a roughly constant accretion rate as they penetrate into the halo centre. The mean accretion rate follows the mass and redshift dependence predicted for haloes by the EPS approximation, |$\dot{M} \propto M_{\rm vir}^{1.25} \ (1 + z)^{2.5}$|⁠. The distribution of the accretion rates can well be described by a sum of two Gaussians, the primary corresponding to ‘smooth inflow’ and the secondary to ‘mergers’. The same functional form was already found for the distributions of specific star formation rates in observations. The mass fraction in the smooth component is 60–90 per cent, insensitive to redshift or halo mass. The simulations with strong feedback show clear signs of reaccretion due to recycling of galactic winds. The mean accretion rate for the mergers is a factor 2–3 larger than that of the smooth component. The standard deviation of the merger accretion rate is 0.2–0.3 dex, showing no trend with mass or redshift. For the smooth component it is 0.12–0.24 dex.

Multiwavelength features of Fermi bubbles as signatures of a Galactic wind

Using hydrodynamical simulations, we show for the first time that an episode of star formation in the centre of the Milky Way, with a star formation rate (SFR) ∼0.5 M⊙ yr−1 for ∼30 Myr, can produce bubbles that resemble the Fermi bubbles (FBs), when viewed from the solar position. The morphology, extent and multiwavelength observations of FBs, especially X-rays, constrain various physical parameters such as SFR, age, and the circumgalactic medium (CGM) density. We show that the interaction of the CGM with the Galactic wind driven by star formation in the central region can explain the observed surface brightness and morphological features of X-rays associated with the FBs. Furthermore, assuming that cosmic ray electrons are accelerated in situ by shocks and/or turbulence, the brightness and morphology of γ-ray emission and the microwave haze can be explained. The kinematics of the cold and warm clumps in our model also matches with recent observations of absorption lines through the bubbles.

INDIRECT EVIDENCE FOR ESCAPING IONIZING PHOTONS IN LOCAL LYMAN BREAK GALAXY ANALOGS

A population of early star-forming galaxies is the leading candidate for the re-ionization of the universe. It is still unclear what conditions and physical processes would enable a significant fraction of the ionizing photons to escape from these gas-rich galaxies. In this paper we present the results of the analysis of HST COS far-UV spectroscopy plus ancillary multi-waveband data of a sample of 22 low-redshift galaxies that are good analogs to typical star-forming galaxies at high-redshift. We measure three parameters that provide indirect evidence of the escape of ionizing radiation: (1) the residual intensity in the cores of saturated interstellar low-ionization absorption-lines. (2) The relative amount of blue-shifted Lyman alpha line emission, and (3) the relative weakness of the [SII] optical emission lines. We use these diagnostics to rank-order our sample in terms of likely leakiness, noting that a direct measure of escaping Lyman continuum has recently been made for one of the leakiest members of our sample. We then examine the correlations between our ranking and other proposed diagnostics of leakiness and find a correlation with the equivalent width of the Lyman alpha emission-line. Turning to galaxy properties, we find the strongest correlations with leakiness are with the compactness of the star-forming region and the speed of the galactic outflow. This suggests that extreme feedback- a high intensity of ionizing radiation and strong pressure from both radiation and a hot galactic wind- combines to create significant holes in the neutral gas. These results not only shed new light on the physical mechanisms that can allow ionizing radiation to escape from intensely star-forming galaxies, they also provide indirect observational indicators that can be used at high-redshift where direct measurements of escaping Lyman continuum radiation are impossible.

KINETIC ENERGY FROM SUPERNOVA FEEDBACK IN HIGH-RESOLUTION GALAXY SIMULATIONS

We describe a new method for adding a prescribed amount of kinetic energy to simulated gas modeled on a cartesian grid by directly altering grid cells' mass and velocity in a distributed fashion. The method is explored in the context of supernova feedback in high-resolution ($\sim 10$ pc) hydrodynamic simulations of galaxy formation. Resolution-dependence is a primary consideration in our application of the method and simulations of isolated explosions (performed at different resolutions) motivate a resolution-dependent scaling for the injected fraction of kinetic energy that we apply in cosmological simulations of a $10^9$ Msun dwarf halo. We find that in high density media ($\gtrsim$ 50 cm$^{-3}$) with coarse resolution ($\gtrsim 4$ pc per cell), results are sensitive to the initial kinetic energy fraction due to early and rapid cooling. In our galaxy simulations, the deposition of small amounts of supernova energy in kinetic form (as little as 1%) has a dramatic impact on the evolution of the system, resulting in an order of magnitude suppression of stellar mass. The overall behavior of the galaxy in the two highest resolution simulations we perform appears to converge. We discuss the resulting distribution of stellar metallicities, an observable sensitive to galactic wind properties, and find that while the new method demonstrates increased agreement with observed systems, significant discrepancies remain, likely due to simplistic assumptions that neglect contributions from Type Ia supernovae and stellar winds.

Which processes shape stellar population gradients of massive galaxies at large radii?

AbstractWe investigate the differential impact of physical mechanisms, mergers (stellar accretion) and internal energetic phenomena, on the evolution of stellar population gradients in massive, present-day galaxies employing a set of high-resolution, cosmological zoom simulations. We demonstrate that negative metallicity and color gradients at large radii (>2Reff) originate from the accretion of metal-poor stellar systems. At larger radii, galaxies become typically more dominated by stars accreted from satellite galaxies in major and minor mergers. However, only strong galactic winds can sufficiently reduce the metallicity content of the accreted stars to realistically steepen the outer metallicity and colour gradients in agreement with present-day observations. In contrast, the gradients of the models without winds are inconsistent with observations (too flat). In the wind model, colour and metallicity gradients are significantly steeper for systems which have accreted stars in minor mergers, while galaxies with major mergers have relatively flat gradients, confirming previous results. This analysis greatly highlights the importance of both energetic processes and merger events for stellar population properties of massive galaxies at large radii. Our results are expected to significantly contribute to the interpretation of current and up-coming IFU surveys (like MaNGA and Califa), which in turn can help to better constrain still uncertain models for energetic processes in simulations.

Supernova-driven outflows in NGC 7552: a comparison of H α and UV tracers

We investigate the supernova-driven galactic wind of the barred spiral galaxy NGC 7552, using both ground-based optical nebular emission lines and far-ultraviolet absorption lines measured with the Hubble Space Telescope Cosmic Origins Spectrograph. We detect broad (~300 km/s) blueshifted (-40 km/s) optical emission lines associated with the galaxy's kpc-scale star-forming ring. The broad line kinematics and diagnostic line ratios suggest that the H-alpha emission comes from clouds of high density gas entrained in a turbulent outflow. We compare the H-alpha emission line profile to the UV absorption line profile measured along a coincident sight line and find significant differences. The maximum blueshift of the H-alpha-emitting gas is ~290 km/s, whereas the UV line profile extends to blueshifts upwards of 1000 km/s. The mass outflow rate estimated from the UV is roughly nine times greater than that estimated from H-alpha. We argue that the H-alpha emission traces a cluster-scale outflow of dense, low velocity gas at the base of the large-scale wind. We suggest that UV absorption line measurements are therefore more reliable tracers of warm gas in starburst-driven outflows.

DISSECTING THE GASEOUS HALOS OFz∼ 2 DAMPED LyαSYSTEMS WITH CLOSE QUASAR PAIRS

We use spectroscopy of close pairs of quasars to study diffuse gas in the circumgalactic medium (CGM) surrounding a sample of 40 Damped Lya systems (DLAs). The primary sightline in each quasar pair probes an intervening DLA in the redshift range 1.6 < z_DLA < 3.6, such that the second quasar sightline then probes Lya, CII, SiII, and CIV absorption in the CGM transverse to the DLA to projected distances $R_{\perp} < 300$ kpc. Analysis of the Lya profiles in these CGM sightlines constrains the covering fraction (f_C) of optically thick HI (having column density N_HI > 10^17.2 cm^-2) to be greater than ~30% within $R_{\perp} < 200$ kpc of DLAs. Strong SiII 1526 absorption with equivalent width W_1526 > 0.2 Ang occurs with an incidence f_C (W_1526 > 0.2 Ang) = 20(+12/-8)% within $R_{\perp}<100$ kpc, indicating that low-ionization metal absorption associated with DLAs probes material at a physical distance R_3D < 30 kpc. However, we find that strong CIV 1548 absorption is ubiquitous in these environments (f_C (W_1548 > 0.2 Ang) = 57(+12/-13)% within $R_{\perp} < 100$ kpc), and in addition exhibits a high degree of kinematic coherence on scales up to ~175 kpc. We infer that this high-ionization material arises predominantly in large, quiescent structures extending beyond the scale of the DLA host dark matter halos rather than in ongoing galactic winds. The Lya equivalent width in the DLA-CGM is anticorrelated with $R_{\perp}$ at >98% confidence, suggesting that DLAs arise close to the centers of their host halos rather than on their outskirts. Finally, the average Lya, CII and CIV equivalent widths are consistent with those measured around z~2 Lyman Break Galaxies. Assuming that DLAs trace a galaxy population with lower masses and luminosities, this finding implies that the absorption strength of cool circumgalactic material has a weak dependence on dark matter halo mass for M_h < 10^12 M_sun.

The neutral hydrogen cosmological mass density atz= 5

We present the largest homogeneous survey of $z>4.4$ damped Lyman-$\alpha$ systems (DLAs) using the spectra of 163 QSOs that comprise the Giant Gemini GMOS (GGG) survey. With this survey we make the most precise high-redshift measurement of the cosmological mass density of neutral hydrogen, $\Omega_{\rm HI}$. At such high redshift important systematic uncertainties in the identification of DLAs are produced by strong intergalactic medium absorption and QSO continuum placement. These can cause spurious DLA detections, result in real DLAs being missed, or bias the inferred DLA column density distribution. We correct for these effects using a combination of mock and higher-resolution spectra, and show that for the GGG DLA sample the uncertainties introduced are smaller than the statistical errors on $\Omega_{\rm HI}$. We find $\Omega_{\rm HI}=0.98^{+0.20}_{-0.18}\times10^{-3}$ at $\langle z\rangle=4.9$, assuming a 20% contribution from lower column density systems below the DLA threshold. By comparing to literature measurements at lower redshifts, we show that $\Omega_{\rm HI}$ can be described by the functional form $\Omega_{\rm HI}(z)\propto(1+z)^{0.4}$. This gradual decrease from $z=5$ to $0$ is consistent with the bulk of HI gas being a transitory phase fuelling star formation, which is continually replenished by more highly-ionized gas from the intergalactic medium, and from recycled galactic winds.

Interpretation of Voyager 1 data on low energy cosmic rays in galactic wind model

The local interstellar energy spectra of galactic cosmic rays down to a few MeV/nucleon were directly measured in the experiment on the board of the Voyager 1 spacecraft. We suggest interpretation of these data based on our models of cosmic ray acceleration in supernova remnants and the diffusion in galactic wind where diffusion coefficient is determined by the cosmic ray streaming instability. The dependence of wind velocity on distance above the Galactic disk is determined.

The creation and persistence of a misaligned gas disc in a simulated early-type galaxy

Massive early-type galaxies (ETGs) commonly have gas discs which are kinematically misaligned with the stellar component. These discs feel a torque from the stars and the angular momentum vectors are expected to align quickly. We present results on the evolution of a misaligned gas disc in a cosmological simulation of a massive ETG from the feedback in realistic environments project. This galaxy experiences a merger which, together with a strong galactic wind, removes most of the original gas disc. The galaxy subsequently reforms a gas disc through accretion of cold gas, but it is initially 120° misaligned with the stellar rotation axis. This misalignment persists for about 2 Gyr before the gas–star misalignment angle drops below 20°. The time it takes for the gaseous and stellar components to align is much longer than previously thought, because the gas disc is accreting a significant amount of mass for about 1.5 Gyr after the merger, during which the angular momentum change induced by accreted gas dominates over that induced by stellar torques. Once the gas accretion rate has decreased sufficiently, the gas disc decouples from the surrounding halo gas and realigns with the stellar component in about six dynamical times. During the late evolution of the misaligned gas disc, the centre aligns faster than the outskirts, resulting in a warped disc. We discuss the observational consequences of the long survival of our misaligned gas disc and how our results can be used to calibrate merger rate estimates from observed gas misalignments.

THREE-DIMENSIONAL HYDRODYNAMICAL SIMULATIONS OF THE SUPERNOVAE-DRIVEN GAS LOSS IN THE DWARF SPHEROIDAL GALAXY URSA MINOR

As is usual in dwarf spheroidal galaxies, today the Local Group galaxy Ursa Minor is depleted of its gas content. How this galaxy lost its gas is still a matter of debate. To study the history of gas loss in Ursa Minor, we conducted the first three-dimensional hydrodynamical simulations of this object, assuming that the gas loss was driven by galactic winds powered only by type II supernovae (SNe II). The initial gas setup and supernova (SN) rates used in our simulations are mainly constrained by the inferred star formation history and the observed velocity dispersion of Ursa Minor. After 3 Gyr of evolution, we found that the gas removal efficiency is higher when the SN rate is increased, and also when the initial mean gas density is lowered. The derived mass-loss rates are systematically higher in the central regions (<300 pc), even though such a relationship has not been strictly linear in time and in terms of the galactic radius. The filamentary structures induced by Rayleigh-Taylor instabilities and the concentric shells related to the acoustic waves driven by SNe can account for the inferred mass losses from the simulations. Our results suggest that SNe II are able to transfer most of the gas from the central region outward to the galactic halo. However, other physical mechanisms must be considered in order to completely remove the gas at larger radii.

STAR-FORMING GALAXIES AS THE ORIGIN OF THE ICECUBE PeV NEUTRINOS

Star-forming galaxies, due to their high star-formation rates and hence large number of supernova remnants therein, are huge reservoirs of cosmic rays (CRs). These CRs collide with gases in the galaxies and produce high-energy neutrinos through $pp$ collisions. In this paper, we calculate the neutrino production efficiency in star-forming galaxies by considering realistic galaxy properties, such as the gas density and galactic wind in star-forming galaxies. To calculate the accumulated neutrino flux, we use the infrared luminosity function of star-forming galaxies obtained by {\em Herschel} PEP/HerMES survey recently. The intensity of CRs producing PeV neutrinos in star-forming galaxies is normalized with the observed CR flux at EeV ({1\,EeV=$10^{18}\,$eV}), assuming that supernova remnants or hypernova remnants in star-forming galaxies can accelerate protons to EeV energies. Our calculations show that the accumulated neutrino emission produced by CRs in star-forming galaxies can account for the flux and spectrum of the sub-PeV/PeV neutrinos under reasonable assumptions on the CR confinement time in these galaxies.

GALACTIC ANGULAR MOMENTUM IN THE ILLUSTRIS SIMULATION: FEEDBACK AND THE HUBBLE SEQUENCE

We study the stellar angular momentum of thousands of galaxies in the Illustris cosmological simulation, which captures gravitational and gas dynamics within galaxies, as well as feedback from stars and black holes. We find that the angular momentum of the simulated galaxies matches observations well, and in particular two distinct relations are found for late-type versus early-type galaxies. The relation for late-type galaxies corresponds to the value expected from full conservation of the specific angular momentum generated by cosmological tidal torques. The relation for early-type galaxies corresponds to retention of only ~30% of that, but we find that those early-type galaxies with low angular momentum at z=0 nevertheless reside at high redshift on the late-type relation. Some of them abruptly lose angular momentum during major mergers. To gain further insight, we explore the scaling relations in simulations where the galaxy formation physics is modified with respect to the fiducial model. We find that galactic winds with high mass-loading factors are essential for obtaining the high angular momentum relation typical for late-type galaxies, while AGN feedback largely operates in the opposite direction. Hence, feedback controls the stellar angular momentum of galaxies, and appears to be instrumental for establishing the Hubble sequence.

THE VLT SINFONI Mg ii PROGRAM FOR LINE EMITTERS (SIMPLE). II. BACKGROUND QUASARS PROBING $Z\sim 1$ GALACTIC WINDS

The physical properties of galactic winds are of paramount importance for our understanding of galaxy formation. Fortunately, they can be constrained using background quasars passing near star-forming galaxies (SFGs). From the 14 quasar$-$galaxy pairs in our VLT/SINFONI Mgii Program for Line Emitters (SIMPLE) sample, we reobserved the 10 brightest galaxies in H$_{\alpha}$ with the VLT/SINFONI with 0.7" seeing and the corresponding quasar with the VLT/UVES spectrograph. Applying geometrical arguments to these ten pairs, we find that four are likely probing galactic outflows, three are likely probing extended gaseous disks, and the remaining three are not classifiable because they are viewed face-on. In this paper we present a detailed comparison between the line-of-sight kinematics and the host galaxy emission kinematics for the pairs suitable for wind studies. We find that the kinematic profile shapes (asymmetries) can be well reproduced by a purely geometrical wind model with a constant wind speed, except for one pair (towards J2357$-$2736) that has the smallest impact parameter b = 6 kpc and requires an accelerated wind flow. Globally, the outflow speeds are $\sim$ 100 km/s and the mass ejection rates (or $\dot M _{\rm out}$) in the gas traced by the low-ionization species are similar to the star formation rate (SFR), meaning that the mass loading factor, $\eta$ = $\dot M _{\rm out}$/SFR, is $\sim$1.0. The outflow speeds are also smaller than the local escape velocity, which implies that the outflows do not escape the galaxy halo and are likely to fall back into the interstellar medium.

Evolution of the gas mass fraction in galaxy clusters

The mass fraction of hot gas in clusters is a basic quantity whose level and dependence on the cluster mass and redshift are intimately linked to all cluster X-ray and SZ measures. Modeling the evolution of the gas fraction is clearly a necessary ingredient in the description of the hierarchical growth of clusters through mergers of subclumps and mass accretion on the one hand, and the dispersal of gas from the cluster galaxies by tidal interactions, galactic winds, and ram pressure stripping on the other hand. A reasonably complete description of this evolution can only be given by very detailed hydrodynamical simulations, which are, however, resource-intensive, and difficult to implement in the mapping of parameter space. A much more practical approach is the use of semi-analytic modeling that can be easily implemented to explore a wide range of parameters. We present first results from a simple model that describes the build up of the gas mass fraction in clusters by following the overall impact of the above processes during the merger and accretion history of each cluster in the ensemble. Acceptable ranges for model parameters are deduced through comparison with results of X-ray observations. Basic implications of our work for modeling cluster statistical properties, and the use of these properties in joint cosmological data analyses, are discussed.

Supernova feedback in an inhomogeneous interstellar medium

Supernova (SN) feedback is one of the key processes shaping the interstellar medium (ISM) of galaxies. SNe contribute to (and in some cases may dominate) driving turbulence in the ISM and accelerating galactic winds. Modern cosmological simulations have sufficient resolution to capture the main structures in the ISM of galaxies, but are typically still not capable of explicitly resolving all of the small-scale stellar feedback processes, including the expansion of supernova remnants (SNRs). We perform a series of controlled three-dimensional hydrodynamic (adaptive mesh refinement) simulations of single SNRs expanding in an inhomogeneous density field with statistics motivated by those of the turbulent ISM. We use these to quantify the momentum and thermal energy injection from SNe as a function of spatial scale and the density, metallicity, and structure of the ambient medium. We develop a series of analytic formulae that we fit to the simulations. These formulae can be used as a basis for a more predictive sub-resolution model for SN feedback for galaxy formation simulations. We then use simulations of multiple, stochastically driven SNe that resolve the key phases of SNRs to test the sub-resolution model, and show that it accurately captures the turbulent kinetic energy and thermal energy in the ISM. By contrast, proposed SN feedback models in the literature based on `delayed cooling' significantly overpredict the late-time thermal energy and momentum in SNRs.