Supernovae Feedback in Galaxy Formation Miao Li

Supernovae Feedback in Galaxy Formation

Miao Li

Submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2017 c 2017

Supernovae Feedback in Galaxy Formation

Miao Li

Feedback – from stars and supermassive black holes – is the bottleneck of our understanding on galaxy formation: it is likely to be critical, but neither the working mechanism nor the impact is clear. Supernovae (SNe), the dominant feedback force associated with stars, is the subject of this thesis. We use high-resolution, 3D hydrodynamic simulations to study: (i) how a SN blast wave imparts energy to a multiphase ISM; (ii) how multiple SNe regulate a multiphase ISM; (iii) how SNe drive galactic outﬂows. We focus on better understanding the physics, quantifying the impacts, and testing the simulations against observations. Table of Contents

List of Figures v

List of Tables xii

1 Introduction 1 1.1 Importance of Feedback in Galaxy Formation ...... 1 1.2 Observations of Galactic Outflows Associated with Star Formation . . . . . 4 1.2.1 Warm/Cool outflows ...... 4 1.2.2 Hot outflows ...... 6 1.3 Current theoretical understanding of SNe feedback ...... 7 1.4 Open questions regarding supernova feedback ...... 9 1.5 Organization of the Thesis ...... 9

I Interaction of Supernova Feedback and Multiphase ISM 10

2 Single SN evolution and feedback 11 2.1 Background ...... 12 2.2 Numerical Methods ...... 13 2.3 SN feedback in uniform media ...... 14 2.3.1 Fiducial run: Setup and Results ...... 14 2.3.2 Dependence of results on input parameters ...... 19 2.4 SN feedback in hot-dominated multiphase media ...... 22 2.4.1 Model Description ...... 23

i 2.4.2 SN feedback in a medium with fV,hot = 0.6 ...... 25

2.4.3 SN feedback as a function of fV,hot ...... 30 2.5 Conclusions ...... 37

3 Multiple Supernova Remnants Regulating Multiphase ISM 39 3.1 Background ...... 40 3.2 Model Description ...... 41 3.3 Choice of Parameters ...... 43 3.4 Results ...... 46 3.4.1 Slices of some output ISM ...... 46 3.4.2 Evolution of the diﬀerent phases ...... 46

3.4.3 fV,hot as a function of (¯n, S)...... 53 3.4.4 Pressure as a function of (¯n, S)...... 58 3.4.5 Other output parameters ...... 61 3.5 Discussion ...... 63 3.5.1 Randomness of SN position on simulation box scale ...... 63 3.5.2 Connecting simulation data to real galaxy environments ...... 67 3.5.3 Periodic box ...... 68 3.5.4 Thermal conduction and magnetic ﬁelds ...... 69 3.6 Conclusions ...... 71

II Supernovae-Driven Multiphase Galactic Outﬂows 73

4 Quantifying Supernovae-Driven Multiphase Galactic Outflows 74 4.1 Background ...... 75 4.2 Numerical Methods ...... 76 4.2.1 Simulation Set-ups ...... 76 4.2.2 Gravitational fields ...... 77 4.2.3 SN Feedback models ...... 81 4.3 Results of fiducial runs ...... 82 4.3.1 Multiphase ISM and outflows ...... 82

ii 4.3.2 Velocity structure ...... 87 4.3.3 Loading factors ...... 93 4.4 Effects of several physical processes ...... 96 4.4.1 SNe scale height ...... 96 4.4.2 Photoelectric heating ...... 98 4.4.3 External gravitational field ...... 102 4.4.4 Enhanced SNe rates ...... 105 4.4.5 Comparison with other works ...... 107 4.4.6 A brief summary and some missing physics ...... 108 4.5 Discussion ...... 111 4.5.1 Loading factors: simulations vs observations ...... 111 4.5.2 Hot outflows and hot CGM ...... 112 4.5.3 Cool outflows ...... 113 4.5.4 Implications for cosmological simulations ...... 114 4.6 Conclusions ...... 115

5 OVI Emission from the Supernovae-Regulated ISM: Simulation vs Ob- servation 117 5.1 Background ...... 118 5.2 Methods ...... 120 5.3 Results ...... 121 5.3.1 OVI-emitting gas in simulations ...... 121 5.3.2 OVI in the solar neighborhood ...... 123 5.3.3 Scaling of OVI with SFR ...... 126 5.4 Conclusions ...... 131

III Conclusions 132

6 Conclusions 133 6.1 Summary of the Thesis ...... 133 6.2 Future work ...... 135

iii IV Bibliography 137

Bibliography 138

iv List of Figures

2.1 Evolution of a SNR of initial energy 1051 erg in a uniform medium with n = 1 cm−3. Snapshots are taken at t = 1.5 104, 4.6 104, 9 104, 2.9 105, × × × × 1.1 106, 2.6 106, 4.2 106 years, respectively. Numerical resolution is 0.75 × × × pc. Vertical axes show spherically-averaged quantities, and horizontal axes are radii. Dashed lines in the upper-left panel show the positions of the shock as predicted by the ST solution at the above timesteps. Pressure of the inner bubble drops precipitously once the thin shell forms, to even below that of the ISM. At later stage, the shell becomes thicker and keeps moving outward to R > 80 pc, while the hot bubble (T > 2 105 K) barely reaches 42 pc in × radius. Velocity is shown in absolute value...... 15

2.2 Radial position of the blast wave Rp vs time (left), energy (Etot,Eth,Ek) vs 51 Rp (middle), and momentum vs Rp (right) for a 10 erg SN in a uniform medium with n = 1 cm−3. Power-law fits are shown with lines. The power- law indices, as defined in Eq. 2.4, 2.5, 2.7-2.10 are given in the legend. . . 18 2.3 Slices of density, temperature, pressure and velocity of a 1051 erg SNR in ISM model “fvh0.60a” withn ¯ = 1 cm−3 (see Table 2.2 for full parameters). Snapshots are taken at 1.9 104 year (upper panels), 1.1 105 year (middle) × × and 2.6 105 year (lower), respectively. The velocity plot has a floor of 0.1 × km/s for log-scale display. Note that the first two rows have zoomed-in views. 26

v 2.4 Evolution of a 1051 erg SNR in the “fvh0.60a” multiphase ISM model with −3 n¯ = 1 cm . (a) radius of the blast wave Rp vs time; (b) energy (Etot,Eth,Ek)

vs time; (c) Etot vs Rp; (d) Eth vs Rp; (e) Ek vs Rp; (f) momentum vs Rp. Solid lines: “fvh0.60a” model; dashed lines: the uniform model with the same

mean density. The dotted lines in (a) show Rp t in a uniform medium with − n = 0.4 cm−3 and 0.004 cm−3, respectively, as predicted by the ST solution

up to their separate tcool (Eq. 2.2). In (f), the dotted and solid lines are for the total momentum and its radial component, respectively...... 28 2.5 Snapshots for a 1051 erg SNR evolving in ISM models “fvh0.66ss” (upper row), “fvh0.90a” (middle), “fvh0.82ss”(lower), respectively, at 3 104 year. × See Section 2.4.1 for model details. The velocity plot has a ﬂoor of 0.1 km/s for log-scale display...... 31 2.6 SN feedback in the 4 multiphase ISM models described in Section 2.4 vs a uniform medium with the same mean density n = 1 cm−3. Line styles and

the corresponding models are given in the legend in panel (c). (a) Rp vs time;

(b) momentum vs Rp; (c) Eth vs time; (d) Eth vs Rp; (e) Ek vs time; (f) Ek

vs Rp. In (b), only results of “fvh0.60a” and “fvh0.82a” are shown, and the thin and thick lines denote the total momentum and the radial component, respectively. The black dotted lines in each panel indicate the ﬁtting formula presented in Section 2.4.3...... 33

3.1 Slices of density, temperature, pressure and velocity of the ISM from simulations with varying mean densityn ¯ and SN rate S. Upper row:n ¯1 S200 (solar neighbourhood) at t = 104 Myr; middle:n ¯0.3 S100 at t = 46 Myr; lower:n ¯10 S3000 at t = 135 Myr. Note that the box sizes vary. See Section 3.2 for model details. Most volume is shared by the three typical phases with T 106, 104, 102 K, respectively. Depending onn ¯ and S, a certain phase can ∼ be missing. The clumping of gas varies withn ¯ and S...... 47

vi 3.2 Evolution of the three ISM phases forn ¯1 S200. Up to down, left to right:

volume-weighted pressure PV /kB, Mach number, volume fraction, mass fraction. Red triangles, yellow diamonds and blue circles denote the cold (T < 103 K), warm (103 K < T < 2 105 K ) and hot (T > 2 105 K) phases, × × respectively. The medium reaches a steady state and the three phases are in rough pressure equilibrium...... 48

3.3 Volume-weighted pressure PV of the three phases versus time for runsn ¯0.1 S50 (left) andn ¯10 S3000 (right). Forn ¯0.1 S50, the hot gas is over-pressured and the ISM is in a “thermal runaway” state. Forn ¯10 S3000, the hot phase is only present brieﬂy after SN explosions and is signiﬁcantly under-pressured. 51 3.4 Transition from a metastable state to “thermal runaway” forn ¯1 S200. Ini- tially, the ISM reaches the apparent steady state, when roughly half the volume is hot. The transition happens at t 55 Myr, when the hot gas ∼ quickly dominates pressure and > 90% of the volume, and pressure of all phases rises progressively...... 52 5 3.5 Volume fraction of the hot (T > 2 10 K) gas fV,hot as a function of the × mean gas densityn ¯ and SN rate S. Each circle represents one simulation.

Colors show the fV,hot . The blue solid line indicates S as a function ofn ¯ from MW mid-plane to halo (see Section 3.3 for a description). The green dashed line indicates S n¯1.5. (¯n, S) = (1, 200) is roughly for the MW-average. The ∝ shaded area shows the transitional region, i.e. fV,hot 0.6 0.1 – above it ≈ ± the hot gas dominates the volume and the ISM undergoes thermal runaway, and below it the SN play a subordinate role and most volume is warm/cold; within the transitional region the ISM is metastable. The gray solid line

indicates the nominal line to roughly represent Scrit above which thermal

runaway happens (Eq. 3.7). The black dotted line shows fV,hot = 0.6 from the simple analytical expectation Eq. 3.9...... 54 3.6 Comparison of the ISM from the runn ¯10 S1000 with diﬀerent photoelectric heating rates. Colors represent the three ISM phases...... 55

3.7 Same as Fig. 3.5, but colors show the volume-weighted pressure PV/kB. . . 57

vii 3.8 Equilibrium pressure (in unit of cm−3K) as a function of density n for T < 104 ∼ K gas for a variety of photoelectric heating rates. Red to green: 0.1, 0.5, 1, 2, 4, 10, 30, 100, 200 1.4E-26 erg/s per H atom. The calculation is based × on the cooling curve adopted in our simulation...... 58 3.9 Left: Cumulative distribution of 3D displacement of core collapse SN, relative to the birth places of their progenitors, due to the velocities of OB stars. Roughly 68% of core collapse SN have migrated more than 100 pc. Right: the projected 1D displacement. Nearly 10% of OB stars can migrate > 1 kpc. ∼ See Section 3.3 for the model description...... 64 3.10 Eﬀects of SN location on the resultant ISM forn ¯1 S200 (box size 144 pc). Left to right: mass fraction, volume fraction, and volume-weighted pressure of the three ISM phases. “Random”: SN explode at random positions. “HD”: SN explode only in high-density clumps. “HD + runaway”: SN have a displacement from the high-density clumps, due to the velocities of their progenitors. The ISM from the “Random” and “HD+runaway” are similar,

whereas that from “HD” has a much smaller fV,hot and more warm gas. . 66

4.1 Combinations of Σgas and Σ˙ SFR adopted in our simulations (blue triangles), plotted on top of Fig. 15 of Bigiel et al. 2008, which shows the observed correlations for nearby galaxies at sub-kpc scale...... 78 4.2 Comparison of g-ﬁelds adopted in literature and this work for the solar neigh-

bourhood. The solid lines end at zmax of each simulation box, which are 2.5kpc, 5kpc and 10kpc for this work, Walch et al. 2015 and Joung et al. 2009, respectively. ∆φ for each curve shows the gravitational potential

zmax ∆φ(zmax) = z=0 gtotdz. See Section 4.2.2 for details...... 80 4.3 Density andR temperature slices of the x-z plane for the four ﬁducial runs. Note that the physical scales of the slices are diﬀerent. The dimensions are shown at the bottom of the temperature slices, in format of “horizontal scale vertical”, in units of kpc...... 84 × 4.4 Phase diagram of the gas in Σ10-KS (solar neighborhood model). The color coding shows the fractional mass in each (density, pressure) bin...... 85

viii 4.5 Gas density (volume-weighted) for different phases as a function of z for Σ10-KS at t= 160 Myr...... 86 4.6 Fractional mass for gas with different z-velocity (absolute value), for the model Σ10-KS at t = 160 Myr. Different curves correspond to gas in different temperature ranges. Each curve is normalized to unity...... 88

4.7 Radius R that a parcel of gas with velocity v0 can reach from the center of the MW. See Section 4.3.2 for model details...... 89 4.8 Mass-weighted v and v of the outflows in different temperatures ranges. z Be Y-ticks on the right show R corresponding to the velocities on the left, as in Fig 4.7. See Section 4.3.2 for details...... 91 5 4.9 Mass-weighted vz and v for hot outflows (T > 3 10 K), as a function of Be × Σ˙ SFR, for the four fiducial runs. The power-law fits of the data are indicated in the box...... 92

4.10 Loading factors and volume fraction of each gas phase as a function of Σgas. The quantities are calculated for outﬂowing gas at z >1 kpc. “Hot” and | | “warm” here denote T > 3 105 K, and 104 < T < 3 105 K, respectively. × × See Section 4.3.3 for details...... 94

4.11 Loading factors for different SN scale heights hSN,cc for the model Σ55-KS. See Section 4.4.1 for details...... 97 2 4.12 Temperature and density slices of two runs with Σgas =55 M / pc : left - 4 fiducial (Σ55-KS); right - cooling curve has a cut-off at Tmin = 10 K (Σ55- KS-1e4K). The snapshots are taken at t = 41 Myr. The slices only include region at z < 325 pc. The white dashed lines indicate the scale height | | 4 of core collapse SNe hSN,cc =150 pc. Cut-off of the cooling curve at 10 K results in a much larger gas scale height. As a result, most SNe energy is lost through radiative cooling in the dense gas layer...... 100 2 4.13 Loading factors for different PEH rates, for the model with Σgas = 55 M /pc . See Section 4.4.2 for details...... 101

ix 4.14 External gravitational fields adopted for the solar neighbourhood (Σ10-KS) and the MW-average (Σ10-KS-4g). Different components are shown separately. See Section 4.4.3 for relevant discussions...... 103 4.15 Ratio of properties between two runs with different gravitational fields: Σ10- KS-4g (MW-average) and Σ10-KS (solar neighbourhood). See Section 4.4.3 for the discussion...... 104

4.16 Relative ﬂuxes of mass, energy and metal as a function of SN rate for Σgas = 2 1 M / pc . The ﬁducial run follows the KS relation, while the “enhanced- rate” runs have SN rates increased by 3 and 10 times, respectively. The dashed black line indicates a linear relation.See Section 4.4.4 for discussions. 106

5.1 Slices of temperature and OVI 1032A˚ photon emissivity from the midplane of the solar neighborhood model Σ10-KS. OVI emission mainly comes from the boundary between the hot and cool medium. The OVI-emitting gas is usually not well-resolved, with some emitting regions resolved by one computational cell. The dimensions are 350 pc 350 pc, and the resolution is 2 pc. The × color coding for the lower panel has a ﬂoor at 10−20s−1sr−1cm−3...... 122 5.2 Phase diagram for the run Σ10-KS, where the color coding shows the 1032A˚ emission rate from gas in that [temperature, density] bin. The calculation includes all gas in the simulation domain. The color coding has a ﬂoor at 1030 erg s−1 ...... 124 5.3 Upper left: projected gas density in the x-direction; upper right: projected OVI 1032A˚ emission in the x-direction (only –0.75 z 0.75 kpc is shown). Lower: projected OVI 1032A˚ emission in the z-direction...... 125 5.4 Purple stars: OVI 1032A˚ emission integrated over the z-direction in our

simulations, as a function of Σ˙ SFR. The correlation is roughly linear. Filled triangle: observation for the solar neighborhood (Otte et al. 2006), which detected OVI emission for about 40% of the sight lines. Hollow triangle: observation of SDSS J1156+5008 (Hayes et al. 2016), scaled to the solar abundance (the two triangles indicate O/H determined from two methods). 127

x 5.5 Purple stars: mean column density of OVI along the z-direction for the whole

simulation domain, as a function of Σ˙ SFR. Orange box: observed value for

the solar neighborhood (Bowen et al. 2008). Filled circles: observed NOVI for

starburst galaxies in Grimes et al. 2009. Hollow circle: observed NOVI for

J1156 (Hayes et al. 2016). For external galaxies, NOVI have been scaled to the solar abundance; the down arrows indicate these values should be an

upper limit for NOVI near the disk, since they also include OVI in the CGM. 129

xi List of Tables

2.1 Best-ﬁt parameters for the evolution of a SNR of 1051 erg in a uniform medium (deﬁnition of the symbols can be found in Section 2.3.1) ...... 20 2.2 Parameters of the multiphase ISM models in Section 2.4 ...... 24

3.1 Input/Output parameters of simulations in Chapter 3 ...... 45

4.1 Model description ...... 83

xii Acknowledgments

It has been a long journey, challenging but rewarding. Thank you all who have been with me along the way. My deep gratitude goes to my parents. From early on, you gave me all the freedom and support to feed my budding curiosity for the universe. You always listened to my questions with great interests, never discouraged my childish imagination, and provided me with whatever books I wanted to read. You also teach me, in everyday life, to be kind and responsible, no matter what I do. Your unconditional love is my ultimate strength to move forward. I thank my undergraduate thesis adviser Feng Yuan, who led me into the astrophysical research. The way he does research – focusing on the physics – has deeply impacted me. I am forever grateful for his passionate encouragement, which forges my initial confidence as a scientist. I am lucky enough to have two brilliant coaches, Greg Bryan and Jeremiah Ostriker, as my dissertation advisers. Each has a distinct way of advising. Greg accompanies me as I plod through the land of the unknown, and as I grow, he gradually loosens his hands, but always keeps me in sight. Jerry always goes a step ahead of me, raising the torch and showing me there is always a new way. Each perspective has helped me tremendously, and together they keep me balanced. Their broad knowledge in physics and astronomy, rigorous research style, great scientific insight, and independent minds have nourished me as a young scientist. I know what I have learned from them will continue to benefit me for many years to come. I also want to thank the rest of my thesis committee: David Schiminovich and Zoltan Haiman, who asked interesting questions and gave helpful feedback for our committee meet- ings in the past few years.

xiii A special thank you to the Astronomy department of Columbia University, which provides a rigorous training for the young researchers, while also maintaining a friendly scientiﬁc atmosphere. I am particularly grateful to Jules Halpern, for the many intriguing discussions about astronomy, theoretical or observational, and about other things in life, big or small. Thank you Millie Garcia and Ayoune Payne for helping me with the logistics, David Secrest for the computer support, and Nelson for your forever sunny way toward life and everyone you encounter. I thank numerous friends and colleagues, for having the journey together, be it long or short. Particularly among them are Maria Charisi and Liang Zhao, with whom I have shared so many special moments. I would like to say a heartfelt thank you to Rongxin Luo, for your love and support over the years. Finally, a thank you goes to the universe that we live in, which never ceases to surprise, perplex, and motivate me.

xiv To my mom and dad, and the universe we live in.

xv CHAPTER 1. INTRODUCTION 1

Chapter 1

Introduction

1.1 Importance of Feedback in Galaxy Formation

On large scales, the ΛCDM cosmology is highly successful in modeling the accelerated expansion of the universe, the cosmic microwave background and the inhomogeneous distribution of baryons in space [Dodelson, 2008]. On small scales where galaxy formation takes place, however, early ab initio cosmological simulations (which do not include effective feedback) reveal serious inconsistencies. Since baryons have only 1/6 the mass of dark matter (DM), one would think that the visible matter largely traces the DM. But it does not: The ΛCDM model yields far more satellite halos than the observed population of satellite galaxies [Klypin et al., 1999]. For the DM halos that do host galaxies, the mass of galaxies is a small fraction ( < 20%) of the available baryons in the halo [Guo et al., 2010; ∼ Moster et al., 2010], whereas in cosmological simulations, almost all baryons cool and settle down to the center of the halo and form “galaxies”. For the distribution of baryons within the galaxy, most mass is in extended thin disks for the majority of galaxies in the ﬁeld (i.e. outside galaxy groups/clusters), whereas in cosmological simulations, mass is mainly concentrated in dense spheroids [Scannapieco et al., 2008]. Star formation (SF) proceeds on time scales much longer than the free-fall time of self-gravitating gas [Kennicutt, 1998; Mac Low and Klessen, 2004; McKee and Ostriker, 2007]. All these discrepancies – that baryons are much more concentrated in simulations than in reality – suggest that the gravity dominates too much in the models, at galactic and sub-galactic scale. This leads to two CHAPTER 1. INTRODUCTION 2 generic possibilities for reconciliation: a) the cold DM model is incorrect on small scales; and/or (b) baryonic feedback, as a counter-gravity force, is not properly included. In my thesis, I focus on the feedback, and in particular, that from supernovae explosions. Supernovae are the most energetic feedback associated with stars. Most stars with mass

M > 8 M will end up in cataclysmic explosions as supernovae (SNe) [Heger et al., 2003], ∼ 51 releasing on average 10 erg energy in the remnant ejecta. Roughly 100-200 M of stars form for one star to become a supernova. Massive stars also have strong stellar winds, but the total energy the winds carry is an order of magnitude lower than SNe. Radiation from stars contains much more energy, but the energy/momentum is poorly coupled to the gas unless the infrared photons are trapped, which needs an extremely high column density of gas/dust and is only possible in the rare star-bursting regions [Murray et al., 2005]. Supermassive black holes at the center of galaxies provide another important feedback, but they likely dominate in massive elliptical galaxies, rather than Milky Way(MW)-like disk galaxies and dwarf systems where the black holes are much less massive. Therefore, for most disk galaxies, supernovae should be the dominant feedback energetically. Additionally, SNe are also the major producer of cosmic “metals” (meaning elements heavier than Helium). Metals drain energy from baryons via radiative cooling, and trace gas that has once been processed by stellar feedback, thus their role cannot be overestimated. Observational evidence of stellar feedback is abundant. Hot gas created by SNe per- meates in the ISM, as seen from OVI emission [Jenkins and Meloy, 1974; York, 1974; Hayes et al., 2016]. Metal-enriched X-ray corona are observed around galaxy disks [Martin et al., 2002; Li and Wang, 2013a]. Multiphase galactic-wide outflows beautifully display themselves in the nearby starbursting galaxy M82 [Strickland et al., 2004a; Griffiths et al., 2000; Walter et al., 2002; Leroy et al., 2015]. Blueshifted interstellar absorption lines – indicating galactic outflows – are ubiquitous in active star-forming galaxies [Steidel et al., 1996; Shapley et al., 2003; Martin, 2005; Weiner et al., 2009; Chen et al., 2010; Genzel et al., 2011; Heckman et al., 2015]. Galaxies retain less metals than they have produced [Tremonti et al., 2004; Erb et al., 2006], while the circum-galactic medium (CGM) and the intergalactic medium (IGM) are metal-enriched [Mitchell et al., 1976; Songaila and Cowie, 1996; Schaye et al., 2003] (and even dust-enriched [Ménard et al., 2010]), showing the broad CHAPTER 1. INTRODUCTION 3 impact of galactic outflows. Incorporating “feedback” makes the simulated galaxies more realistic. Outflows remove gas from galaxies and may suppress gas infall, therefore limiting a galaxy’s mass (e.g. [Springel and Hernquist, 2003]). Some ejected mass eventually falls back at the edge of the galaxy, making galaxies more extended and disk-like [Governato et al., 2007; Genel et al., 2015]. Outflows also funnel metals from galaxies to their surroundings [Mac Low and Ferrara, 1999; Fujita et al., 2004; Oppenheimer and Davé,2006], in line with the observed metal distribution. SNe drive turbulence and disperse dense gas clumps, thus suppressing star formation (see reviews by [Mac Low and Klessen, 2004; McKee and Ostriker, 2007]). However, the feedback models used in cosmological simulations are more phenomenological than physical. They usually adopt one of two tricks: turning off cooling, or invoking wind particles that decouple from gas dynamics. Each has free parameters that are fine- tuned to match the observations. As a result, it is very hard to make quantitative predictions from these simulations, other than suggesting that including feedback is likely necessary. Feedback is thus the bottleneck of cosmological simulation. The main reasons for the wide usage of such phenomenological models are: (i) insufficient resolution of cosmological simulations, which results in catastrophic radiative cooling that damps the impact of feedback, and (ii) lack of first-principle understanding of feedback physics, To overcome the above two deficiencies, this thesis uses high-resolution simulations, aim- ing at a better understanding of how supernova feedback works. In the following sections, I summarize the relevant observations of galactic outflows fueled by SF process (i.e. not due to super-massive black holes), the current theoretical knowledge of supernova feedback, major open questions, and the efforts I have made investigating these questions in this thesis. CHAPTER 1. INTRODUCTION 4

1.2 Observations of Galactic Outﬂows Associated with Star Formation

1.2.1 Warm/Cool outﬂows

1 In nearby galaxies with active star formation, warm/cool outflows are observed directly as extraplanar gas, or as blueshifted emission/absorption lines [Heckman et al., 1990; Bregman and Pildis, 1994; Martin, 1998; Rupke et al., 2002; Martin, 2005; Leroy et al., 2015; Heckman et al., 2015]. Outflows are seen in ionized, atomic, and even molecular phases. The velocities are generally in the range of 100-600 km/s. The cooler phases have lower velocities. The Mach numbers of the warm/cool outflows are around 10-50. For high-redshift star-forming galaxies, low-ionization blueshifted interstellar absorption lines are widely observed, with velocities of the same order as the local galactic outflows [Steidel et al., 1996; Shapley et al., 2003; Adelberger et al., 2003; Weiner et al., 2009; Chen et al., 2010]. Lyman α emission lines are also commonly seen in the same systems with similar redshifted velocities. Together they are interpreted as outflows from both sides of the star-forming regions. The detection rate of outflows has a clear trend with the strength of the star forming activity. More specifically, [Heckman et al., 2015] found that the maximum velocities of outflows correlates well with the SF intensity, i.e., the SF rate divided by the area of the SF region. The mass loading factor, defined by the mass outflow rate divided by the SF rate, is in the range of 0.01-10 [Veilleux et al., 2005]. But one should caution that this number is of great uncertainty, due to the lack of knowledge about the ionization fraction, geometry, metallicity, etc., which can easily lead to an uncertainty as large as a factor of 10 [Chisholm et al., 2016]. The origin of the warm/cool outflows is not clear and is an active research subject. Possible explanations include: (a) Warm/cool phase forms out of the cooling instabilities within hot outflows [Field, 1965; Efstathiou, 2000; Silich et al., 2003; Thompson et al., 2016]. This naturally explains the high velocity of the warm/cool phase. But since the hot outflows usually have a low mass loading factor, especially for high SFR cases (on the order of 0.1), it is hard to load ∼

1 5 4 In the thesis, we define “hot” as T∼ > 3 × 10 K, “warm” as around 10 K, and cool as around ∼< 1000 K. CHAPTER 1. INTRODUCTION 5 much mass this way (if the mass loading factor of the warm/cool outflows is indeed > 1). ∼ (b) Warm/cool outflows are propelled by the ram pressure of the hot outflows. There are two general scenarios: one is that warm/cool outflows are a (semi-)spherical shell that envolopes a high-pressured hot bubble powered by overlapping SNRs [Tomisaka and Ikeuchi, 1986; Mac Low et al., 1989; Koo and McKee, 1992; Yadav et al., 2017]. This is supported by the observed “superbubbles” and “supershells” in the Milky Way (MW) and nearby galaxies [Heiles, 1979; Heiles, 1984; Ochsendorf et al., 2015]; the other is that there are some low- density channels where the hot gas preferentially goes through, and the warm/cool outflows are the fragments of clouds that are bathed in and pushed outward by the hot outflows [Mac Low and Ferrara, 1999; Cooper et al., 2008]. This is supported by the filamentary structure as clouds are stripped by the hot outflows [Strickland et al., 2004a]. Either scenario has potential problems. For the former, > 90% of the energy of SNe still gets lost at the dense ∼ shell, and therefore it is very hard to accelerate shells to above 100 km/s [Yadav et al., 2017; Kim et al., 2017]. For the latter, warm/cool clouds in fast hot winds are subject to the Kelvin-Helmholtz instability, and can be dissolved before they are accelerated to high velocities [Klein et al., 1994; Zhang et al., 2015]. (c) Radiation pressure from SF region can provide extra acceleration force [Murray et al., 2005], but this generally requires infrared photons to be trapped, which is plausible for extreme starbursting regions with very high gas/dust column density (UV photons have much larger optical depths, but only supply roughly 1/10 of the momentum of what SNe can provide). In addition, Rayleigh-Taylor instability will develop even if the ISM is optically thick to IR photons and radiation largely vents through the low-density channels and is not coupled to the dense gas[Krumholz and Thompson, 2012], although more accurate radiation-hydrodynamic simulations suggest that acceleration may still be possible [Davis et al., 2014]. (d) Warm/cool outflows are driven by cosmic rays (CR). This is the least-explored scenario. Recent simulations including CRs as a second fluid have found that warm outflows can be launched as CRs diffuse relative to the thermal gas [Booth et al., 2013; Salem and Bryan, 2014; Girichidis et al., 2016a; Simpson et al., 2016], and a mass loading factor of unity can be achieved given an optimal diffusion coefficient. Much more work needs to be CHAPTER 1. INTRODUCTION 6 done to evaluate the role of cosmic rays in driving outflows.

1.2.2 Hot outﬂows

Hot outflows are unambigously detected in some nearby starbursting systems [Dahlem et al., 1998; Pietsch et al., 2000; Martin et al., 2002; McDowell et al., 2003; Huo et al., 2004; Grimes et al., 2005; Strickland and Heckman, 2009]. In M82, the hard X-ray emission is observed from the SF region, which is at least partly due to the supernovae-heated ISM [Griffiths et al., 2000]. Soft X-ray emission is more exteneded into the halo, and correlates very well spatially with the cooler outflows seen in Hα [Strickland et al., 2004a]. The soft X-ray emission is likely due to the shock-heating/mixing of a hotter “super wind” and warm clouds [Strickland et al., 2002]. Hot, metal-enriched outflows have also been inferred from the enhanced OVI in the CGM of SF galaxies at redshift 2-3 [Turner et al., 2015]. For nearby disk galaxies with “normal” star forming activities, soft x-ray emission is seen in the halo surrounding the disk (“galaxy corona”) [Read et al., 1997; Li and Wang,

2013a]. The X-ray luminosity Lx tightly correlates with the SF intensity in the disk over a wide range of SFRs [Strickland et al., 2004b; Grimes et al., 2005; Mineo et al., 2012; Li and Wang, 2013b]. The origin of the soft X-ray emission is still under debate, and the tight correlation of Lx to the SF activity can originate from (a) the cosmic hot-mode accretion feeding the star formation [Anderson and Bregman, 2011; Dai et al., 2012; Bogdán et al., 2013]; and/or (b) SNe feedback from the SF events driving hot outflows [Li and Wang, 2013b]. The former is likely more important for very massive and quiescent galaxies (massive 11 meaning the gravitational pull is large, usually M∗ > 3 10 M∗), while the latter dominates ∼ × in less massive but more active galaxies. Evidence supporting the latter is that the metal abundances in the corona is consistent with contamination by SNe [Martin et al., 2002; Yamasaki et al., 2009; Konami et al., 2011; Li and Wang, 2013a]. [Li et al., 2016] recently compiled about 50 highly-inclined disk galaxies, with a broad range of SFRs and galaxy masses. The nearly edge-on view ensures that the X-ray contami- −0.44 nation from the disk is minimal. They have found that LX /SF R Σ˙ , which, together ∝ SFR with the global wind model developed by [Chevalier and Clegg, 1985], implies the mass loading factor of the hot outflows declines with increasing Σ˙ SFR. (Here they assume the CHAPTER 1. INTRODUCTION 7 soft X-ray emission is from the volume-filling hot outflows, which can be true for galaxies less active in SF than the starburst galaxies like M82, in which the hot outflows have much higher temperatures). The soft X-ray temperature is about 0.5 keV, and increases very mildly with Σ˙ SFR. These give very important constraints on the SNe-driven hot outflows. The origin of the hot outflows associated with SF and the hot ISM is widely accepted to be supernova shock-heating [Cox and Smith, 1974; Larson, 1974; McKee and Ostriker, 1977; Cox, 2005].

1.3 Current theoretical understanding of SNe feedback

SNe feedback plays three dynamical roles in galaxy formation: (i) Shaping the ISM. [McKee and Ostriker, 1977] proposed the underlying theory of the three-phase ISM regulated by SN. Their theory is based on three basic assumptions: First, if the SN rate is sufficient to render the porosity of the ISM 1, then the ISM is ∼ inevitably multi-phase with the majority of volume being hot; Second, all three phases are in approximate pressure equilibrium [Spitzer, 1956]; Third, different phases are in dynamical equilibrium in terms of mass and energy: SN keep creating hot bubbles and compressing diffuse gas into dense clouds, while clouds are evaporated by the hot medium. Energy input from SNe balances dissipation in all phases. These principles yield a complete set of equations, and the solutions determine their pressure and mass/volume quota for each phase, given the mean gas density and SN rate. Their results were in rough agreement with observations of the ISM in the solar neighbourhood. Despite some imperfectness of the theory (for example, underestimating of the mass of warm gas, not considering the clustering of SN, etc.), their main ideas are still widely accepted as the standard picture of the local ISM [Cox, 2005]. Numerical simulations clearly confirm that three phases coexist in rough pressure equilibrium, with temperatures around 102, 104, and 106 K. The hot gas ocupies a significant fraction of volume but contains little mass [de Avillez and Breitschwerdt, 2004; Joung and Mac Low, 2006; Creasey et al., 2013; Gent et al., 2013; Walch et al., 2015; Kim et al., 2017]. The ISM is turbulent, with Mach number of about unity [van der Kruit and Shostak, 1982; Heiles and Troland, 2003]. SNe CHAPTER 1. INTRODUCTION 8 are one of the major drivers of the turbulences [Tamburro et al., 2009]. (ii) Driving galactic outflows. Massive stars tend to form in clusters, and SNe naturally correlate in space and time. Overlapping SNe create hot bubbles, and these bubbles can break out from gaseous disk and form outflows. This picture is seen in ISM simulations with stratified medium [McCray and Kafatos, 1987; Mac Low et al., 1989; Joung et al., 2009; Gent et al., 2013; Walch et al., 2015; Kim et al., 2017]. [Creasey et al., 2013] have studied the loading efficiencies of SNe-driven outflows from a multiphase ISM, and found that the mass loading factor is lower for higher SFR cases. Global evolution of the outflows has been studied analytically by [Chevalier and Clegg, 1985], a mass loading factor and an energy loading factor would determine the properties of the outflows. Their model is simple and neglected gravity and radiative cooling, which is suitable only for the “super winds” powered by extreme starbursting systems like M82. The 1D wind solution has been extended, including more physics, and a variaty of solutions have been found [Efstathiou, 2000; Silich et al., 2003; Bustard et al., 2016]. Global evolution of the outflows has also been studied using numerical simulations in 2D or 3D, where a multiphase ISM structure is adopted in the galaxy disk, in which outflows tend to vent through the easiest paths [Strickland et al., 2002; Cooper et al., 2008]. Clouds in the disk can be stripped and accelerated by the hot outflows. The cloud experiments have also been studied at much higher resolutions, to study the details of the shredding vs. acceleration of the clouds [Klein et al., 1994; Scannapieco and Brüggen,2015; Schneider and Robertson, 2015; McCourt et al., 2015]. (iii) Regulating star formation. Without feedback, self-gravitating gas will collapse into stars on a free-fall timescale, but the observed Kennicutt-Schmidt SF relation shows that at kpc scales, it takes about a hundred times longer [Kennicutt, 1998]. SNe drive turbulence into the ISM, thus preventing gas from collapsing (see reviews by [Mac Low and Klessen, 2004; McKee and Ostriker, 2007]). The SF activities are likely to be self- regulating: if the SF is too efficient, then the intense feedback can prevent further star formation; whereas if too little star formation proceeds, then feedback is too weak and therefore more gas will collapse into stars (e.g. [Shetty and Ostriker, 2012; Kim et al., 2013]). At the molecular cloud scale, other stellar feedback, such as ionizing photons, stellar CHAPTER 1. INTRODUCTION 9 winds, radiation pressure, also plays roles in regulating the gravitational collapse[Krumholz and Thompson, 2012; Gatto et al., 2016; Peters et al., 2017]. The inefficiency of SF is still not fully understood, but feedback should play a key role in the process. In this thesis, we focus on the first two aspects of SNe feedback, i.e., on how SNe regulate a diffuse (not self-gravitating) ISM, and drive outflows from it.

1.4 Open questions regarding supernova feedback

On the ISM scale, open questions are: – How does a SNR propagate in and impart energy/momentum to a multiphase ISM? – How do SNe collectively regulate the ISM? – What is the condition for outflows to occur? – How much mass/energy/metals are driven out from a multiphase ISM by SNe? On galactic scales: – What is the fate of the (different phases of) SNe-driven outflows? – How do SNe-driven outflows impact the CGM/IGM? – How do SNe-driven outflows interact with cosmic inflow? In this thesis, I focus on the questions on the ISM scale. The questions on larger scales will be addressed in future work (Chapter 6).

1.5 Organization of the Thesis

The thesis includes two parts. Part I (Chapters 2 & 3) investigates the interaction between SNe feedback and a multiphase ISM. Part II (Chapters 4 & 5) studies the multiphase outﬂows driven by SNe. Each chapter starts with the questions that we will focus on, followed by a brief background, then the design of our experiment, results, discussions, and a summary. The Conclusion part (Chapter 6) summarizes the thesis, and points to possible future directions. 10

Part I

Interaction of Supernova Feedback and Multiphase ISM CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 11

Chapter 2

Single SN evolution and feedback

This chapter is adapted from part of the publication in the Astrophysics Journal, 814:4, 2015. CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 12

Abstract

The primary question we address is: Given a multiphase ISM with a significant hot volume fraction fV,hot , how does a SNR evolve and impart feedback? In particular, how does the feedback differ from that in a uniform medium? We use high-resolution, 3D hydrodynamic simulations to study this question. We find that for a uniform medium, the energy of a SNR in the radiative phase shows a power- law decline with time, and the power-law indices show little variation over 4 orders of magnitude in ambient gas density. In a hot-dominated multiphase ISM, a SNR propagates in a fundamentally different way: the blast wave travels much faster and further in the tenuous hot phase with little radiative loss, and the impact domain of a SNR is enhanced by a factor of 300 in volume. We compare the results in both an idealized static multiphase ISM, and the ones that are self-consistently generated by multiple SNe. We find that the energy dissipation rate of SN is higher in the realistic ISM, possibly due to its turbulent nature. But to zeroth order of approximation, fV,hot is the major factor that determines the energetic feedback of SNR.

2.1 Background

The evolution of a SNR in a uniform medium is well-studied by many authors. Early 1D simulations (e.g. [Chevalier, 1974; Cioffi et al., 1988] showed that the evolution is characterized by four stages: free expansion, Sedov-Taylor, “pressure-driven snowplow” and momentum-conserving. [Thornton et al., 1998] revealed the feedback at later stages is dominated by kinetic energy while thermal feedback is negligible due to rapid cooling. But how fast SNR loses its energy (both thermal and kinetic), and how does this depend on the ambient gas density, which are important to quantify the feedback effect, have not been quantified by numerical studies. We will present this result, as well as the numerical convergence studies, in this chapter. SNRs evolve in a qualitatively different way when exploding in a clumpy medium. Pio- neer work by [Cowie et al., 1981] studied the remnant evolution and energy budget in a three phase medium with most volume hot, and found that the blast wave travels much faster CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 13 and further in the low-density, hot medium, and the loss of explosion energy happens later compared to the case when the medium is uniform. Recent 3D simulations by [Martizzi et al., 2015] and [Kim and Ostriker, 2015] confirmed that the evolution and energy feedback of a SNR in a clumpy medium is quite different from the uniform case, but the terminal radial momentum is affected little by the inhomogeneity of the ISM. [Walch and Naab, 2014] found similar results for molecular clouds, and the pre-SN ionization can slightly enhance the final momentum per SN. Recently works on SN feedback have mainly considered a relatively dense medium and neglect the third phase of the ISM – the tenuous hot gas. However, a medium with a significant fraction of volume hot exists widely, e.g. at the disk-halo interface and the galactic winds launching sites. Moreover, the hot phase being the most tenuous transports energy/momentum much faster (speed of sound/blast wave ρ−0.5) and cools very ineffi- ∝ ciently (cooling time ρ−1). It is therefore very interesting to investigate the SN feedback ∝ in a three-phase ISM, which we will do in this chapter. The multiphase ISM adopted in the studies of SN evolution has been largely set up by hand. However, we should note that while SNRs are affected by the lumpiness of the ISM, they are also the major player in shaping the overall properties of the gas. Thus it is very important to study SN feedback self-consistently. For the multiphase ISM, we compare the SN evolution in both idealized, static ISM and the ones self-consistently created by multiple SN (see Chapter 3).

2.2 Numerical Methods

The numerical experiments are performed on a uniform-mesh code which uses the shock- capturing total variation diminishing (TVD) scheme [Ryu et al., 1993] to solve the hydrodynamics equations. The ideal gas law is applied with the adiabatic index γ = 5/3. Cooling and heating are calculated using the Grackle library 1 as described in [Smith et al., 2008; Smith et al., 2011]. Metallicity is assumed to be solar, that is, 2% of gas mass is in metals. Cooling and heating rates due to H, He and metals (including ﬁne-structure lines) at a

1http://grackle.readthedocs.org/en/latest/ CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 14 given density and temperature are pre-calculated using the Cloudy code [Ferland et al., 1998], which determines the ionization equilibrium solution for atom/ion species computed under an incident spectrum. In reality, the radiation ﬁeld in a star formation region is very complicated and time-variable, and to be self-consistent one needs to appeal to radiative transfer. For simplicity, we do not take into account the ionizing radiation for this study. Photoelectric heating (PEH) through dust is included. We do not consider the heating of neutral gas due to cosmic rays or X-rays, since it is an order of magnitude smaller than the PEH [Draine, 2011]. At each time step (determined by the Courant condition), Grackle updates the internal energy of each cell by integrating the tabulated cooling/heating rates over that time step. Self-gravity, thermal conduction, molecule formation and cooling, and magnetic ﬁelds are not included in the current simulations.

2.3 SN feedback in uniform media

In this section, we present the simulations of a SNR in a uniform medium, and analyze how a SN imparts its energy and momentum to the gas. We also compare our results with other recent numerical works.

2.3.1 Fiducial run: Setup and Results

We set up our ﬁducial run as follows: initially the gas is uniform with n = 1 cm−3 and 3 51 T = 8 10 K. We inject ESN = 10 erg as purely thermal energy and 10 M mass into × a sphere which encloses about 40 M of the ISM. The initial sphere has a radius of 7.5 pc. Both energy and mass are evenly distributed. The spatial resolution is 0.75 pc, and the box is 225 pc on one side. Periodic boundary conditions are applied, although the SNR does not reach the boundary. Background PEH is set to be 1.4 10−26 erg/s per hydrogen atom, × which is similar to that in the solar neighbourhood [Draine, 2011]. Fig. 2.1 shows the physical quantities (spherically-averaged) as a function of radius. Dashed lines in the upper-left panel indicate the shock radius as predicted by the Sedov- Taylor (ST) solution for the 7 timesteps we show. The position of the shocks in our simulation coincide with theory very well in the energy conserving phase. The peak density at this CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 15

101 108

107 100 3 − K 3 6 − cm 10 1 10− cm / B k 105

2 10− 104 pressure/ number density/ 10 3 − 103

4 2 10− 0 20 40 60 80 100 120 10 0 20 40 60 80 100 120 radius/parsec radius/parsec

108

107

2 6 10 10 s / km K 105 T/

velocity/ 1 104 10

103

2 0 10 0 20 40 60 80 100 120 10 0 20 40 60 80 100 120 radius/parsec radius/parsec

Figure 2.1: Evolution of a SNR of initial energy 1051 erg in a uniform medium with n = 1 cm−3. Snapshots are taken at t = 1.5 104, 4.6 104, 9 104, 2.9 105, 1.1 106, × × × × × 2.6 106, 4.2 106 years, respectively. Numerical resolution is 0.75 pc. Vertical axes show × × spherically-averaged quantities, and horizontal axes are radii. Dashed lines in the upper- left panel show the positions of the shock as predicted by the ST solution at the above timesteps. Pressure of the inner bubble drops precipitously once the thin shell forms, to even below that of the ISM. At later stage, the shell becomes thicker and keeps moving outward to R > 80 pc, while the hot bubble (T > 2 105 K) barely reaches 42 pc in radius. × Velocity is shown in absolute value. CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 16 stage is 2.5n, somewhat smaller than that in the ST solution, i.e. 4n. This is likely due ∼ to the finite resolution and the application of spherical-averaging. After cooling becomes important, the blast wave slows down and lags behind the ST solution. A thin, dense and cool (T 104 K) shell forms, and the SNR enters the “pressure-driven snowplow” phase. ∼ We define R0.8Eth as the radial position of the shock when 20% of the thermal energy in ST 4 stage has been lost due to cooling. We find R0.8Eth = 22.1 pc at t = 4.1 10 year, in good × agreement with the analytical calculation by [Draine, 2011]:

n E R = 23.7 pc( )−0.42( SN )0.29, (2.1) cool 1 cm−3 1051 erg

4 n −0.55 ESN 0.22 tcool = 4.9 10 year( ) ( ) . (2.2) × 1 cm−3 1051 erg The equations are obtained by assuming the cooling rate Λ T −0.7 for 105−7.3K gas. Note ∝ that in [Draine, 2011], Rcool and tcool are defined somewhat differently, as the radius and time when 30% of the SN energy has been radiated away. While the shell has cooled and slowed down, the inner bubble is still filled with gas that is hot (T > 106 K) and fast-moving (v > 100 km/s). During this stage, the density of the hot bubble drops rapidly. This is because the fast-moving gas keeps running into the cool shell where it radiates energy very efficiently, which leads to substantial mass loss from the hot bubble. Accompanying the mass loss is a sharp pressure decrease, to even below that of the ambient gas. The precipitous decline of density and pressure makes the central engine quickly run out of steam. Thus, after a short-lived “pressure-driven snowplow” phase, a SNR enters the “momentum-conserving” stage. Without a propelling force, the cool shell moves forward due to its own inertia. The pressure of the shell, which is now beyond both the ISM and the inner bubble, drives itself to become thicker. Now the thick “shell” dominates the evolution of the SNR: while the outer edge of the shell still expands with a forward shock, the hot bubble it encompasses shrinks in size simultaneously. As shown in the temperature panel (lower-left), the radius of the hot bubble (T > 2 105 K) barely × reaches 42 pc, while most swept-up mass moves to R > 80 pc. We define φ as the ratio between the maximum size of the hot bubble and Rcool, i.e.

φ Rhot,max/Rcool . (2.3) ≡ CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 17

We find φ = 1.77 for the fiducial run. The parameter φ is important for the later calculation of the hot gas filling factor fV,hot (see Chapter 3). Eventually, the velocity of the front shock drops to the sound speed of the ISM, and the temperature of the shell decreases until the PEH balances the cooling. The bubble created by SN will be filled with the ambient gas again, as seen from the increase of the density of the central bubble at the end of the simulation. Regarding the feedback effects, we are interested in how fast the blast wave propagates in the ISM, and how much energy and momentum it injects into the ISM as the shock reaches different radii. We present the results in Fig.2.2. The radius of the blast wave Rp, defined as the radius where the density peaks, follows a piecewise power-law function with time. The turnover happens at Rp R0.8Eth. We define the power-law indices for the ST ∼ and the post-ST stage as ηST and ηp−ST, respectively, i.e.

ηST Rp t , for Rp < R0.8Eth, (2.4) ∝

ηp−ST Rp t , for RpR0.8Eth. (2.5) ∝

A simple linear ﬁt on log-log scale using the ordinary least squares method yields: ηST =

0.445 0.020 and ηp−ST = 0.292 0.004. The ST solution predicts ηST = 0.40. [Chevalier, ± ± 1974] found ηp−ST 0.31; [Blondin et al., 1998] had a somewhat higher value 0.33. ≈ The plot of energy vs Rp (middle panel of Fig. 2.2) indicates that the decline of kinetic energy Ek happens after that of thermal energy Eth. We use the parameter β to characterize this delay:

β R0.8Ek/R0.8Eth, (2.6) ≡ where R0.8Ek is deﬁned similarly as R0.8Eth. Our simulation indicates R0.8Eth = 22.1 pc,

R0.8Ek = 30.5 pc and thus β = 1.38. The numerical value of β is understandable, since this process can be seen as an inelastic collision between the cool shell and the ISM, during which Ek only undergoes signiﬁcant loss when the sweep-up mass is similar to that of the shell (38% increase in radius means roughly a 162% increase in volume and thus, mass).

The decline of Etot, Eth and Ek seem to occur in a power-law fashion, and we deﬁne the corresponding indices α as follows:

αEtot Etot R , for Rp > R0.8Eth, (2.7) ∝ p CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 18 p R (middle), and momentum vs p R ) vs k ,E th ,E tot E . Power-law ﬁts are shown with lines. The power-law indices, as 3 − = 1 cm n vs time (left), energy ( p R erg SN in a uniform medium with 51 Figure 2.2: Radial position(right) of for the a blast 10 wave deﬁned in Eq. 2.4, 2.5, 2.7-2.10 are given in the legend. CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 19

αEth Eth R , for Rp > R0.8Eth, (2.8) ∝ p

αEk Ek R , for Rp > R0.8Ek. (2.9) ∝ p

We ﬁnd αEtot = 3.30 0.04, αEth = 7.52 0.21, αEk = 3.16 0.03. The loss of thermal − ± − ± − ± energy is much more rapid than kinetic, indicating that at later stages of SN evolution, kinetic feedback dominates over thermal feedback [Thornton et al., 1998; Walch and Naab,

2014; Simpson et al., 2014; Kim and Ostriker, 2015]. We note that αEk 3 implies ∼ − 0.5 3 0.5 momentum = (2mEk) (R Ek) constant, that is, momentum conservation starts P ∝ p ∼ at Rp > R0.8Ek 30 pc. This is conﬁrmed by the vs Rp relation shown in the right panel ∼ P of Fig. 2.2.

The momentum shows a power-law rising with Rp, followed by a brief turn-over, and P then reaches a plateau when Rp > R0.8Ek. Again, we deﬁne the power-law index γP for the rising phase of the momentum as:

γP R , for Rp > R0.8Ek. (2.10) P ∝ p

3 We ﬁnd γP = 1.21 0.03. For the ST phase, when Ek is constant and m R , a simple ± ∝ p 0.5 1.5 43 scaling relation shows = (2mEk) R . The ﬁnal momentum end is 5.4 10 P ∝ p P × erg cm/s, consistent with Kim & Ostriker (2015, hereafter KO15). ·

2.3.2 Dependence of results on input parameters

The above scaling relations, especially the indices α, β, η and numerical values of end, P are very important for SN feedback models. We want to test how the results change as we vary the numerical resolution, as well as with other model inputs, since in real galaxies, SN explode in various environments with different ISM density, PEH rate, metallicity, clumpiness, etc. In this subsection we study the effect of gas density, PEH rate and numerical resolution, and leave the task of exploring ISM clumpiness to later sections. In Table 1 we listed the models and the outputs. For n = 1 cm−3, we run three additional simulations with resolutions of 0.25 pc, 0.46 pc and 1.5 pc, respectively. The output parameters show variance within 10%, and there seems no systematic change of the outputs as we improve the resolution by a factor of 6, with the exception that αEth seems somewhat CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 20 cm/s) · end g P 43 (10 1.601.32 10.4 11.0 1.651.69 5.3 5.3 > > > > β φ 8Ek . 0 R 8Eth . 0 R 0.005 7.9 11.4 1.44 1.67 3.8 0.006 21.6 29.4 1.36 1.58 5.3 0.003 201.00.006 210.0 280.00.006 1.39 290.00.012 58.0 1.38 0.003 57.8 80.00.004 21.6 1.38 84.50.004 21.4 1.64 1.46 29.8 21.8 1.5 1.38 29.9 8.3 0.005 1.40 30.1 8.5 7.9 1.38 1.77 11.4 1.44 5.3 1.67 3.8 0.005 8.3 11.4 1.37 1.67 3.7 ST ± − ± ± ± ± ± ± ± ± ± ± p 0.29 - - - - 5.9 η Output erg in a uniform medium (definition of the symbols can be found 0.08 0.293 0.03 0.321 0.14 0.334 0.09 0.328 0.12 0.335 0.07 0.298 0.03 0.290 0.03 0.292 0.09 0.284 0.07 0.294 1 0.08 0.277 51 ± Ek ± ± ± ± ± ± ± ± ± ± α -3.16 1.03 -3.20 0.13 -2.96 1.66 -2.78 0.23 -2.40 0.44 -2.33 0.14 -2.89 0.15 -3.14 0.21 -3.16 0.43 -3.00 1 0.44 -3.18 0.45 -3.10 ± ± ± ± ± ± ± ± ± ± ± Eth α 0.06 -6.40 0.13 -5.79 0.10 -6.14 0.22 -6.20 0.08 -6.46 0.04 -7.52 0.07 -7.64 0.10 -7.43 0.10 -10.01 0.06 -6.49 0.09 -7.33 ± ± - -5.69 ± ± ± ± ± ± ± ± ± Etot α -3.16 -2.95 -3.41 -3.89 -3.06 -3.09 -3.20 -3.30 -3.03 -3.10 -3.35 ) 1 − PEH (erg Res 0.25 1E-26 (pc) Model Input ) 3 measured from their Fig 1 − 1 1 0.25 .. 1 .. .. 1.4E-23 .. 10.0 ...... 0.46 0.75 ...... 3.96 .. 1.5...... 1.4E-25 n 10 0.56 .. 0.1 1.98 .. 0.005 5.0 1.4E-26 ( cm in Section 2.3.1) (KO15) Table 2.1: Best-fit parameters for the evolution of a SNR of 10 CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 21 higher in lower resolution runs. For the two highest-resolution runs, we have only lower limits on φ due to the limitation on the box size. Comparing four runs with different gas densities, i.e. n = 0.005, 0.1, 1.0 and 10.0 cm−3, we find that a denser ISM shows somewhat more rapid decrease of Eth with Rp, due to the more efficient cooling. But overall the difference is pretty small: decreasing n from 10 −3 to 0.1 cm makes αEth drop by only about 20%. KO15 also mentioned that the evolution is basically identical for different values of n when the radius and time are scaled relative −3 to their values at shell formation. For the n = 0.1 and 0.005 cm models, ηp−ST (0.33) is larger than in higher density runs, which usually have 0.28 0.30. φ, which quantifies the − volume of hot gas created by one SN, is in the range of 1.5-1.8.

end is very insensitive to n. As presented in KO15, this can be understood using a P simple analytical argument. Since

0.5 3 0.5 end = (2mEk) = [8π/3 nmH (βRcool) Ek] , (2.11) P where Ek is simply a constant before momentum-conserving phase. Combining Eq. 2.1 and 2.11 yields 43 n −0.13 ESN 0.94 β 1.5 end 5 10 g cm/s ( ) ( ) ( ) . (2.12) P ∼ × · 1 cm−3 1051 erg 1.4 In other words, in a uniform medium, SN (assuming a fixed total energy) inject an almost constant amount of momentum, independent of the density of the ISM. Our results agree −3 with the analytical calculation fairly well. αEk is somewhat larger than -3 for n = 0.1 cm , which implies that momentum is not quite conserved at this stage, but is still slowly increasing. In this case the hot inner bubble has some pressure left to push the shell. end P is measured when the post-shock velocity drops to the sound speed of the ambient gas, 10 km/s. We test how the PEH rate affects the result for n = 10 cm−3. From the data, we find there is no significant difference in the output parameters. One may notice that when we adopt a PEH rate 1000 times the MW value, αEth is somewhat smaller, which may seem unintuitive. This is because when we calculate Eth of the blast wave, we deduct the ISM energy (calculated by taking a sample outside the shocked region) from the total thermal energy enclosed in the blast wave domain. The high PEH rate keeps the ISM at 104 K, ∼ CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 22 while for the MW value, heating cannot balance cooling and the temperature of the ISM decreases with time. This makes the remaining Eth inside the blast wave smaller for higher PEH rates. Compared with KO15, where they use a resolution of 0.25 pc for n = 1 cm−3, we find very good agreement for αEk, ηp−ST and end. They have a somewhat smaller αEth 5.7, P ∼ − which can be due to the code and/or cooling curve we each adopt (they approximate the cooling curve by piece-wise power-law functions). In short, the evolution and feedback of a SNR in a uniform medium, as a function of time or Rp, seem to be well characterized by (piece-wise) power-law functions. Overall, the dimensionless numbers α, β, ηp−St and φ vary little over a large dynamic range of n.

2.4 SN feedback in hot-dominated multiphase media

In this section, we study how a hot-dominated multiphase medium (HDMM) affects SN feedback. For a medium that is predominantly two-phase (warm/cold), one can refer to the recent works by KO15. In principle, one needs many parameters to fully describe a multi-phase medium, for example, the size, density, shape of a single cloud; the location, number density and topology of clouds; the pressure of the inter-cloud medium; turbulence/velocity fields, etc, and to different extent, they all impact the evolution of a SNR. To exhaust the parameter space is not feasible. On the other hand, the above properties are not independent of each other, and more importantly, they are tightly correlated with SN feedback itself. (Other conditions, such as gravity, stellar outflows and radiation feedback, contribute to the ISM conditions as well, but arguably not as significantly as SN, and to consider these effects is beyond the scope of this paper.) For self-consistency, the ISM models adopted in this section are based on the results from Chapter 3, where we explore how multiple SN collectively shapes the ISM. Other recent works use different methods to construct multiphase media. For example, KO15 establish the warm/cold media through thermal instability; [Martizzi et al., 2015] impose a log-normal density structure; [Walch and Naab, 2014] use fractal molecular clouds, with some runs having pre-ionization. CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 23

2.4.1 Model Description

We evolve a SNR in four inhomogeneous ISM set-ups, which have the same mean density −3 1 cm but diﬀerent volume fractions of hot gas fV,hot, including two idealized medium and two self-consistently generated ISM by multiple SN:

i. “fvh0.60a” (“fvh0.60” denotes fV,hot = 0.6, and “a” artiﬁcial) ; ii. “fvh0.90a”; iii. “fvh0.66ss” (“ss” denotes self-consistent); iv. “fvh0.82ss”. In the artiﬁcial ISM models “fvh0.60a” and “fvh0.90a”, clouds are identical to each other, and each cloud is spherical with a core of cold medium and a layer of warm phase. All three phases are in pressure equilibrium. The warm and cold phases are thermally stable. The size and mass of the clouds in “fvh0.60a” are chosen based on the results of the runn ¯1 S200 from Chapter 3; whereas “fvh0.90a” represents a more extreme model where 90% of the volume is hot and 90% of the mass cold. The locations of clouds are random.

Detailed parameters of the three phases, such as the cloud radius Rcl, density n, pressure

P/kB and volume/mass fraction fV /fm are listed in Table 2.2 (a). The box size is 300 pc, and the resolution is 0.5 pc. The medium is static initially. For the self-consistently generated ISM, “fvh0.66ss” is the output from the runn ¯1 S200 of Chapter 3 at t = 1.1 108 year, and “fvh0.82ss” fromn ¯1 S3000 at t = 7.3 106 year. × × Statistics of the ISM such as the volume-weighted density nV , pressure PV /kB and fV /fm of each phase are shown in Table 2.2 (b). To avoid the new SNR propagating outside of the simulation domain, we duplicate eight times the periodic box, assembling the eight identical boxes to make a larger one, which is 286 pc on each side, twice the size of the original one. The resolution 1.12 pc are the same as the original runs. 51 We start by injecting 10 M mass and 10 erg energy into a sphere with Minit 200 M ≈ (Rinit 12.5 pc) at the center of the box. For the artiﬁcial medium, the initial energy ≈ partition is 30% kinetic plus 70% thermal. The injected thermal energy and mass are 2 distributed evenly within the sphere. The initial Ek of each grid is proportional to r , where r is the distance of grid from the explosion center. We also tried cases where we inject energy as pure thermal or pure kinetic, and the results converge after the blast wave CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 24 m f K) B 3 − Hot P/k m f ( cm ) 3 − K) B n 3 − /k 4.0E-33.2E-3 4.0E3 1.0E4 0.002 0.002 Hot V ( cm P ( cm m ) f 3 − V n 1.1E-36.7E-3 1.9E3 6.2E4 0.001 0.006 ( cm V f m f K) B 3 − P/k Warm V f ( cm ) 3 − K) B n 3 − /k V ( cm Warm P ( cm cl 5 1.0 1.0E4 0.089 0.075 12 0.4 4.0E3 0.394 0.165 ) R (pc) 3 − V n 0.54.3 3.0E3 4.5E4 0.322 0.152 0.177 0.756 m ( cm f m f V f V f K) B 3 − Cold P/k K) B ( cm 3 Table 2.2: Parameters of the multiphase ISM models in Section 2.4 − /k V ) Cold 3 P − ( cm n ) ( cm 3 − V 58 2.9E3 0.015 0.847 n 108 3.9E4 0.002 0.238 cl 3 133 4.0E3 0.006 0.833 ( cm 2.5 100 1.0E4 0.011 0.923 R (pc) Run Run “fvh0.90a” “fvh0.60a” “fvh0.66ss” “fvh0.82ss” (a) Artiﬁcial models (b) Self-consistent models CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 25 reaches about 25 pc. For a self-consistently generated ISM, the energy added is purely thermal, and the injected mass and energy are uniformly distributed in the initial sphere.

2.4.2 SN feedback in a medium with fV,hot = 0.6

We first take a detailed look at the SNR in the “fvh0.60a” model, to understand the physics of the evolution in a multiphase medium, and then compare results of all models in this section. Fig. 2.3 shows 2D snapshots centered on the SNR in the “fvh0.60a” model at t = 1.9 104, 1.1 105 and 2.6 105 years, respectively. The first two rows use zoomed-in × × × views. Visually the evolution is very different from that in the uniform ISM model. At t = 1.9 104 year, part of the blast wave has run into warm clouds where the shock velocity × slows down. The geometry of the remnant thus deviates from spherical symmetry. The blast wave continues to travel fast in the hot inter-cloud medium – the least obstructed channel. At the same time, the shock heats and strips the clouds. The warm clouds are relatively easy to disrupt and accelerate, and are mixed with the hot component. The dense cold clouds remain largely intact and static due to their larger inertia and smaller cross sections. There is a visible density enhancement at the shock front in the hot gas, but a dense and cool ( 104 K) shell, which features the “pressure-driven snowplow” phase ∼ of a SNR in a uniform medium, never forms. The post-shock region is also dissimilar to the Sedov-Taylor solution: the physical quantities inside the blast wave domain (excluding the clouds) are nearly uniform – it does not show the typical radial velocity distribution as v R, nor the progressively lower density and higher temperature toward the center of the ∝ blast wave. The shocked regions are turbulent, as a result of the violent interaction between the blast wave and the embedded clouds. At the end of the simulation, the warm clouds within R 50 pc are largely disrupted, while the cold cores have survived. Clouds in the ∼ outer part of the SNR are less distorted, as the power of the blast wave subsides when the energy is spread into a larger volume.

Fig. 2.4 shows the radial position of the shock Rp vs time and the energy/momentum feedback as a function of time and Rp. For comparison, the dashed lines indicate the SNR evolution in a uniform model with the same mean density n = 1 cm−3, same as what is CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 26 (see year 3 5 − 10 × 6 . = 1 cm n year (middle) and 2 5 10 × 1 . erg SNR in ISM model “fvh0.60a” with ¯ 51 year (upper panels), 1 4 10 × 9 . Figure 2.3: Slices of density, temperature, pressure and velocity of a 10 Table 2.2 for full parameters). Snapshots are taken at 1 (lower), respectively. The velocity plot hasviews. a floor of 0.1 km/s for log-scale display. Note that the first two rows have zoomed-in CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 27 shown in Fig. 2.2. It is obvious that the evolution in a HDMM is very different from that in a uniform case.

Panel (a) of Fig. 2.4 shows the evolution of Rp vs t. We determine Rp in the following way: since initially the medium is static, we divide the simulation domain into concentric spherical shells centered on the SN explosion site with the thickness of the cell size, and deﬁne Rp as the radius beyond which no shells has more than 10% of cells with v > 10 km/s. From the plot, the blast wave travels much faster and further in the multiphase medium.

For the uniform case, the power-law index of Rp-t turns from 0.4 to 0.29 after the SNR ∼ enters the “pressure-driven snowplow” phase; in contrast, in the multiphase model is more close to a single power-law function, with a slope 0.55. As a result, by t = 1.8 105 year, ≈ × the difference in shock radius is more than a factor of 3, and the volume enclosed differs by a factor of 30. The hot-dominated medium transfers the blast wave out much more ∼ efficiently. To see the importance of the clouds in determining the blast wave velocity, we plotted −3 −3 −3 Rp t for a uniform medium with n = nhot = 4 10 cm and n = nwarm = 0.4 cm , − × respectively (dotted lines in panel(a) of Fig. 2.4), as predicted by the ST solution up to their 5 separate tcool (Eq. 2.2). At t < 10 year, the curve for “fvh0.60a” lies in between the above ∼ cases; after that, the evolution almost coincides with that in a n = 4 10−3 cm−3 medium × without energy loss. This is because during the early stages, the blast wave is powerful and interacts more with clouds, thus the velocity is reduced, whereas at later stages, the shock largely bypass the clouds and travels more freely in the hot medium. This agrees with what we have seen in the slice snapshots. We estimate what fraction of a cloud interacts with the blast wave. For a cloud of the size 2Rcl, density ρcl and pressure Pcl that is located at a distance dcl from the center of a spherical blast wave, at a time t, the length scale of the cloud passed by the shock is t 0 0 Rsh(t) = vsh(t )dt , where the shock velocity in the cloud vsh(t) Psh(t)/ρcl t0(Rp=dcl) ∼ ∼ cs,cl Psh(Rt)/Pcl, Psh is downstream pressure of the shock, and cs,cl the soundp speed of the −3 pre-shockp cloud [Klein et al., 1994]. Assuming Psh is uniform within the SNR and R , ∝ p η and Rp t , then at t = tfade, when the blast wave decays to a sound wave, i.e. when ∝ CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 28 in a uniform . Solid lines: (Eq. 2.2). In t p R − cool p t (e) 2 2 R 10 10 uniform ”fvh0.60a” radial ”fvh0.60a” total /parsec /parsec . (a) radius of the blast wave p p 3 R R − ; (f) momentum vs p 1 1 = 1 cm R (f) 10 10 n vs 51 50 49 44 43 k

10 10 10 10 10 oetm g*cm/s momentum/ iei energy/erg kinetic E ; (e) p (c) (d) R 2 2 10 10 vs th E /parsec /parsec p p R R ; (d) p R uniform ”fvh0.60a” vs 1 1 10 10 tot E 51 50 49 51 50 49

10 10 10 10 10 10 oa energy/erg total energy/erg thermal , respectively, as predicted by the ST solution up to their separate 6 6 3 10 10 − (b) ) vs time; (c) k uniform ”fvh0.60a” 5 5 004 cm . ,E 10 10 erg SNR in the “fvh0.60a” multiphase ISM model with ¯ th 51 ,E and 0 time/year time/year tot 3 4 4 E − 10 10 tot th k E E E 4 cm . (a) 3 3 = 0 10 10 2 1 51 50 49

10 10

p R /parsec 10 10 10 energy/erg vs time; (b) energy ( p Figure 2.4: Evolution of a 10 “fvh0.60a” model; dashed lines: the uniformmedium model with with the same mean density.(f), The dotted the lines dotted in and (a) solid show lines are for the total momentum and its radial component, respectively. R CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 29

Psh Pcl, we have ≈ R 2 t d sh = ( fade ) [1 ( cl )(2−3η)/2η], (2.13) 2Rcl 2 3η ts,cl − Rfade − where Rsh/2Rcl is the fraction of a cloud being shocked, ts,cl is the sound crossing time of the cloud ( 2Rcl/cs,cl), and Rfade is the radius of the blast wave at tfade. Note that the closer ≡ a cloud is to the center of the SNR, the larger Rsh/2Rcl it has. Given the approximate value η 0.55, Rfade 200 pc, we have Rsh/2Rcl 0.5 and 1.2 for the cold and warm ∼ ∼ ≈ cloud, respectively, at dcl = 0. This means that only the warm clouds near the center can fully interact with the blast wave, and cold clouds are mostly bypassed. In other words, for most cold clouds encompassed by a SNR, the interior of the clump does not even know the existence of the blast wave before it fades away. Panel (b) of Fig. 2.4 shows the SN energy as a function of time. The SN energy is obtained by summing up all energy within Rp with the energy from the enclosed ISM deducted. The ISM energy density is acquired by averaging a sample outside of the SNR. The evolution of energy has two stages: an energy conserving phase followed by a radiative phase, divided by when roughly 20% of total energy is lost. For “fvh0.60a”, this transition 4 happens at t 5 10 year and Rp 65 pc. The time is similar to that in the uniform ∼ × ∼ case, while the corresponding Rp is about 2.7 times larger. The energy feedback shows interesting contrasts to the uniform case for both stages. Initially, after the deposition of energy as 30% kinetic and 70% thermal, Ek rises while Eth declines, deviating from the ST solution. Ek reaches its maximum of about 40% of SN energy at approximately the end of the energy conserving phase. The results do not show qualitative variation when we change the initial energy deposition to 100% thermal or 100% kinetic – Ek always exceeds that in the ST solution, though Ek/Etot when Ek peaks can vary by about 20%. This initial conversion of thermal to kinetic energy is reminiscent of free expansion, and is most likely due to expansion of the shock-heated clouds into the tenuous medium.

Despite the fact that the decline of Eth happens earlier than in the uniform case, the rate at which it decreases is much smaller in a HDMM. While in the uniform medium, Eth drops by a factor of 50 within 2 105 years, the remnant in “fvh0.60a” loses only about 50%. × The main reason for the signiﬁcantly reduced cooling rate is that the blast tends to choose a more tenuous path, which is cooling-ineﬃcient, and largely bypassing the dense clouds. CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 30

Another reason for the mild loss rate of Eth is that at later stages Ek can be converted to

Eth through turbulence. Ek, on the other hand, does not show very diﬀerent time evolution from that in the uniform phase.

The plots of energy vs Rp (panel (b)-(d)) contain no new information but simply represent the data from Rp t and E t. It is obvious that the energy is transported to a − − much larger volume by the fast-moving blast wave, and drops much more mildly with Rp in a HDMM, especially for Eth. In a uniform medium, SN energy is negligible after the catastrophic cooling at Rp = 65 pc, whereas in a multiphase ISM, when the shock fades to a sound wave at Rp > 130 pc, 25% of the SN energy is still maintained in the “fvh0.60a” ∼ model.

Momentum as a function of Rp is shown in panel (f) of Fig. 2.4. The momentum P does not increase as fast with Rp in a multiphase ISM. Given = √2mEk, the smaller P P and larger Ek implies much less mass is involved in sharing the kinetic energy in a HDMM. Note that no thin shell forms at the shock front which carries most of the radial momentum, as happens in a uniform medium. At the end of the simulation, the radial momentum in “fvh0.60a” is about 4.3 1043 g cm/s, and is still rising. But it is not likely to keep increasing × · for long, since the post-shock pressure is about to drop to that of the ISM and will no longer be converted into the outgoing momentum. This suggests that the total momentum injected in a clumpy medium is close to that in the uniform case 5.3 1043 g cm/s. We also plot × · the total momentum (dotted line), which initially coincides with the radial momentum, but later exceeds it since a non-radial component develops as the blast wave interacts with the clouds. At the end of the simulation, the total momentum is roughly 30% larger than the radial one in “fvh0.60a”.

2.4.3 SN feedback as a function of fV,hot

In this subsection, we compare SNR evolution in all 4 multiphase models, and present the fitting formulae to quantify SN feedback in a HDMM. The fitting formulae in this section 51 −3 are for ESN = 10 erg, and gas average densityn ¯ = 1 cm and fV,hot > 0.5. ∼ Fig. 2.5 shows the snapshots of the three multiphase ISM models “fvh0.66ss”, “fvh0.90a” and “fvh0.82ss” at t = 3 104 year. It is clear that the blast wave travels mostly in the × CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 31 erg SNR evolving in ISM models “fvh0.66ss” (upper row), “fvh0.90a” (middle), “fvh0.82ss”(lower), 51 year. See Section 2.4.1 for model details. The velocity plot has a floor of 0.1 km/s for log-scale display. 4 10 × Figure 2.5: Snapshots for a 10 respectively, at 3 CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 32 hot phase. For “fvh0.66ss” model, where the clouds form walls and hot bubbles do not fully connect to each other, the shock goes further in the low-density bubble and deviates from spherical symmetry. Within the SNR, the pressure is higher toward the clouds. For “fvh0.90a” and “fvh0.82ss”, where the clouds are smaller with nearly random locations, the blast waves are close to spherical. Fig. 2.6 compares SN feedback in all multiphase ISM models. For reference, the uniform case is shown with the blue dashed lines. For “fvh0.90a”, Rp, Eth , Ek are determined the same way as for “fvh0.60a”. For the self-consistently generated ISM models, which are turbulent and the SNR may have an irregular shape, we briefly describe how we determine the size and energy of the SNR. For “fvh0.66ss”, we adopt the following criterion to determine if a cell is within the SNR. A post-shock cell should satisfy at least one of the two conditions: a. v > 150 km/s; b. nT > 104.5 cm−3 K. The adopted velocity and pressure floors are beyond (but very close to) the maximum values in the pre-SN ISM. We then define the 1/3 “effective radius” Rp (3Vshock/4π) , where Vshock is the total volume of the post-shock ≡ region. For the “fvh0.82ss” model, since the ISM has higher pressure and velocities, and the spatial variations are larger than “fvh0.66ss”, it is difficult to obtain Rp in the same way. Taking advantage of the relatively regular shape of the SNR, we take snapshots and identify the Rp by eye. The SN energy is determined by running a reference simulation for each self-consistent ISM model where no SN is added. At each time step, the difference of

Ek and Eth between the two simulations are deemed those associated with the SN. In Fig. 2.6, one may ﬁrst notice that the four multiphase models show roughly similar

SNR evolution, but they are quite different from the uniform model (except panel (e) Ek- t). The comparable evolution is interesting because the four multiphase models have fairly different configurations: cloud shape, fV,hot, turbulence, and even resolution. This implies that those differences are secondary to the fact that they share the same feature of a HDMM, distinct from the uniform case.

Let us then look at the feedback in some details. Panel (a) of Fig. 2.6 shows Rp vs t. SNRs in all multiphase ISM models go much faster and further than the uniform medium.

For both the artiﬁcial and self-consistent ISM, Rp is larger for higher fV,hot at a given time, but overall the diﬀerence is small. For “fvh0.66ss”, at later stages, Rp reaches its maximum CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 33

1044

(a) (b) 102 /pc p R

43 momentum/(g cm/s) 10

total radial 101 104 105 101 102 time/year Rp/pc

1051 1051 (c) (d)

1050 uniform 1050 ”fvh0.60a” thermal energy/erg thermal energy/erg ”fvh0.90a” ”fvh0.66ss” ”fvh0.82ss” 1049 1049 104 105 101 102 time/year Rp/pc

1051 1051

(e) (f) kinetic energy/erg kinetic energy/erg

1050 1050

104 105 101 102 time/year Rp/pc

Figure 2.6: SN feedback in the 4 multiphase ISM models described in Section 2.4 vs a uniform medium with the same mean density n = 1 cm−3. Line styles and the corresponding models are given in the legend in panel (c). (a) Rp vs time; (b) momentum vs Rp; (c) Eth vs time; (d) Eth vs Rp; (e) Ek vs time; (f) Ek vs Rp. In (b), only results of “fvh0.60a” and “fvh0.82a” are shown, and the thin and thick lines denote the total momentum and the radial component, respectively. The black dotted lines in each panel indicate the fitting formula presented in Section 2.4.3. CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 34 of 95 pc, when the shock subsides and merges with the cloud boundaries of the bubbles. The SN feedback is confined by the walls, thus it does not influence as large volume as when hot gas has a connecting topology. Overall, Rp vs t is roughly represented by the following power-law function t 0.55 Rp = 22 pc( ) . (2.14) 7 103 year × The power-law index, 0.55, is larger than the ST solution. The corresponding curve is shown with the black dotted line (same linestyles are adopted for other fitting formulae presented in this subsection in Fig. 2.6).

Panel (c) of Fig. 2.6 shows the time evolution of Eth, which is characterized by an early onset of decline and a small cooling rate for all multiphase models. Self-consistent ISM models lose Eth at a somewhat larger rate, which may be due to their turbulent nature that enhances the dissipation rate. If we compare the two artiﬁcial ISM models or the self-consistent ones, a larger fV,hot yields a smaller cooling rate. Eth(t) has a power-law-like form, with power-law indices (absolute value) much smaller than the uniform case. We adopt a very simple formula to ﬁt the power-law indices for the two self-consistent models, which are inversely proportional to fV,hot:

0.65ESN for t 0.09tcool,¯n, ≤ Eth(t) =  (2.15) t −0.36×(0.8/f ) 0.65ESN( ) V,hot for t > 0.09tcool,¯n ,  0.09tcool,¯n

 4 −3 −0.55 51 0.22 where tcool,¯n = 4.9 10 year(¯n/1 cm ) (ESN/10 erg) , following Eq. 2.2. Note × that the functional form of the power-law index is chosen somewhat arbitrarily, with no direct physcial motivation. Only two values of fV,hot are fit, meaning that it is not strongly constrained; however, this is probably only of secondary importance for sub-grid modeling, as fV,hot does not vary strongly across HDMM conditions. We set a ceiling for it at 65% of the SN energy, to avoid the divergence at t = 0. The onset of cooling, which is about 10% of tcool,¯n, is much earlier than the uniform case, while the canonical value of the power-law index of Eth t for a HDMM, -0.36, is significantly smaller than the uniform case, 2. − ≈ − Panel (d) of Fig. 2.6 shows Eth vs Rp. The fitting formula can be obtained from CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 35 combining Eq. 2.14 and 2.15:

0.65ESN for Rp 0.73Rcool,¯n, ≤ Eth(Rp) =  (2.16) Rp −0.65×(0.8/f ) 0.65ESN( ) V,hot for Rp > 0.73Rcool,¯n,  0.73Rcool,¯n

 −3 −0.42 51 0.29 where Rcool,¯n = 23.7 pc(¯n/1 cm ) (ESN/10 erg) , following Eq. 2.1. Again, the

SN energy starts to cool at a somewhat smaller Rp, while the dependence of Eth on Rp is much less sensitive in a HDMM than in a uniform medium, which has a power-law index of about -7. Note that the evolution of Eth in our three-phase ISM simulations is different from that in two-phase models. [Martizzi et al., 2015] find that the the power-law slope of the declining phase of Eth vs Rp is similar to the uniform case (see their Fig 4). KO15 show that the energy evolution in a two-phase ISM is relatively close to the uniform case with the warm phase density (see their Fig 10), since the cold clumps only have a small volume fraction. In a three-phase medium, we find that evolution is more complicated. Although the late evolution of Rp is close to that in the hot medium only, the energy evolution cannot be rescaled to the uniform case with any density. It thus suggests that some new model is needed which takes into account both hot gas and warm clouds.

Panel (e) of Fig. 2.6 shows the time evolution of Ek , which in multiphase media does not show large deviations from that in the uniform medium, although in the energy conserving phase, Ek has a higher fraction of SN energy than 30%. A simple ﬁtting formula is

0.35ESN for t tcool,¯n, ≤ Ek(t) =  (2.17) t −0.87 0.35ESN( ) for t > tcool,¯n.  tcool,¯n  Ek vs Rp is shown in panel (f) of Fig 2.6. Similar to Eth , Ek is conveyed to a much larger volume in a HDMM. For the ﬁtting formula, combining Eq. 2.14 and 2.17, we get

0.35ESN for Rp 2.7Rcool,¯n, ≤ Ek(Rp) =  (2.18) Rp −1.58 0.35ESN( ) for Rp > 2.7Rcool,¯n.  2.7Rcool,¯n  Note that in the declining phase, the power-law index of Rp, -1.58, is less steep than the uniform case, -3, consistent with that the momentum is not conserved, but still rising when

Ek declines. CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 36

The momentum evolution is shown in panel (b) of Fig 2.6. Both the total momentum and the radial component are presented. We only plot the artiﬁcial ISM cases, since in a turbulent medium it is hard to isolate the momentum directly related to the new SNR. The SNR in the “fvh0.90a” model has a consistently smaller momentum than that in “fvh0.60a”, at a given an Rp, since less mass is involved in the blast wave in a more hot-dominated medium. As the shock goes to further distances, momentum is developed in non-radial directions and the post-shock region becomes more turbulent. For the most part of Rp, the momentum in a multiphase medium is much smaller than the uniform case, but the terminal momenta are likely to be similar. For the radial momentum, we adopt the following ﬁtting formula: a power-law increase with Rp followed by a plateau as in the uniform case:

43 −1 Rp 0.57 fV,hot −0.5 fV,hot 0.88 5 10 g cm s ( ) ( ) for Rp 7.5Rcool,¯n( ) , × · 7.5Rcool,¯n 0.6 ≤ 0.6 rad =  P 43 −1 fV,hot 0.88 5 10 g cm s for Rp > 7.5Rcool,¯n( ) .  × · 0.6 (2.19)  The power-law indices of ESN andn ¯ follow those of the uniform case (Eq. 2.11). Note that while a single power-law describes the evolution in “fvh0.60a” almost perfectly, it does not agree well with “fvh0.90a”, which may be better fitted by a broken power-law. Here we simply make fV,hot a factor that only influences the zero point of curve, to reflect that less momentum is developed at a given Rp in a medium with a larger fV,hot . Note that we have shown the uniform case with the same mean density of the HDMM models. Since the cold phase that contains most mass actually fills little of the volume, from a physical perspective, one could also compare to uniform models with the same median density. We do not show them here to avoid overcrowding the figures. However, as we have shown in Section 2.3, the SN feedback in a uniform medium is well described by power-law functions, and the power-law indices vary little for different gas densities. So, as a rule of thumb, the feedback curves simply shift horizontally for different densities.

In summary, the evolution of a SNR in a HDMM is characterized by a much larger impact volume, a reduced cooling rate, and a slowly developed momentum. When the medium is uniform in density, there is no preferential path and the shock encounters mass at the same rate in all directions; when the medium is clumpy, however, the shock tends to CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 37 choose the more tenuous regions where it travels much faster and further. The connected rarefied channels give vent to the blast wave energy, which largely goes around the clouds at later stages. Consequently, not all mass within Rp interacts significantly with the blast wave. A HDMM with connecting tenuous tunnels therefore greatly facilitates SN feedback. In cosmological simulations which cannot resolve the multiphase, turbulent medium, it has been commonly assumed that SN explode in a uniform medium with the mean cell density. The cooling rate can thus be severely overestimated. A more realistic and self-consistent model should consider a SN-modified multiphase ISM. We can define the domain of influence of a SN in 4D spacetime:

4π 3 V4D = R t0.2. (2.20) 3 0.2

If we define R0.2 and t0.2 as the Rp and time when 20% of SN is energy is left, then for a 51 5 −3 10 erg SN, R0.2 = 33.8 pc and t0.2 = 1.7 10 year in a uniform medium with 1 cm , and × 5 R0.2 = 190.0 pc and t0.2 = 3.5 10 year in a multiphase medium with same mean density × and fV,hot = 80%, according to Eq. 2.15 - 2.18. The impact volume is therefore 178 times larger and time 2.1 times longer, making a 374 times bigger V4D in a HDMM than in a uniform medium. In real galaxies, it is usually the collective effect of many SN that shape the ISM and launch galactic winds. A larger V4D for a SNR means that a new SN is more likely to explode within the V4D of an old one. Thus the chance for SNRs to overlap significantly increases, and energy from multiple SN are “grouped”, which may result in a big runaway explosion. It is therefore very interesting to investigate the question “Under what circumstances will a HDMM form?”, which we will do in the next chapter.

2.5 Conclusions

In this Chapter, we study the propagation of a SNR in a uniform medium and a hot- dominated multiphase ISM, respectively. The main results are summarized as follows: 1. For a uniform medium, the evolution of a SNR has four well-deﬁned stages after the initial free expansion: Sedov-Taylor, (transient) “pressure-driven snowplow”, momentum- conserving and close-up. Thermal energy drops precipitously once the shock becomes ra- CHAPTER 2. SINGLE SN EVOLUTION AND FEEDBACK 38

−6∼−8 −3 diative as Eth R , while kinetic energy drops later and more slowly as Ek R . ∝ p ∝ p The Rp t relation and the energy/momentum feedback vs Rp can be well described by − piece-wise power-law functions. The power-law indices vary little over a wide range of gas densities and photoelectric heating rates, and generally converge for different resolutions we have adopted (Table 2.3.2). 2. In a hot-dominated multiphase medium, a SNR evolves very differently from the uniform case or a two-phase medium. There are no distinct stages of evolution. The blast wave initially interacts with the clouds, but later travels much faster and mainly in between the clouds (Eq. 2.13). The SNR thus sweeps up much less mass, therefore developing less momentum at a given Rp (Fig. 2.6). A radiation-efficient thin shell never forms at the blast wave front, and the overall cooling rate is suppressed. The hot phase helps to retain the SN energy and facilitates energy transfer to much larger scales. The spatial-temporal domain a SN is enlarged by a factor of > 102.5 in a HDMM. 3. We compare the results in both idealized static multiphase ISM, and the ones that are self-consistently generated by multiple SNe. We find that the energy dissipation rate of SN is higher in the realistic ISM, possibly due to its turbulent nature. But to zeroth order of approximation, fV,hot is the major factor that determines the energetic feedback of SNR. CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 39

Chapter 3

Multiple Supernova Remnants Regulating Multiphase ISM

This chapter is adapted from part of the publication in the Astrophysics Journal, 814:4, 2015. CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 40

Abstract

In previous chapter, we demonstrated that the propagation of SN and its feedback is fundamentally different in a hot-dominated-multiphase ISM vs a uniform medium. It is thus very important to know under what conditions the ISM is hot-dominated. Since the major regulator of the ISM is SNe, in this chapter, we allow multiple SN to shape the ISM. We experiment with two parameters, the mean densityn ¯ and the SN rate S, and systematically analyze the resultant ISM. The numerical code and method are the same as in Chapter 2. Major conclusions in this chapter are the following: (i) There are two thermal conditions of the ISM: a steady state, where SNRs do not overlap and most of the SN energy is radiated away, and the ISM is a two-phase warm/cool medium; and a thermal-runaway state, where SNRs overlap, most of the volume is hot, and energy accumulates. A steady state can only be achieved when the hot gas volume fraction fV,hot < 0.6 0.1. Above ∼ ± that level, overlapping SNRs render connecting topology of the hot gas, and the ISM is subjected to thermal runaway. (ii) Photoelectric heating (PEH) has a surprisingly strong impact on fV,hot. (iii) The critical SN rate for the onset of thermal runaway to be Scrit = −3 k 51 −1 −3 −1 −3 200(¯n/1 cm ) (ESN/10 erg) kpc Myr , where k = (1.2, 2.7) forn ¯ 1 and > 1 cm , ≤ respectively. (iv) We present a fitting formula of the ISM pressure P (¯n, S), which can be used as an effective equation of state in cosmological simulations. (v) Despite the 5 orders of magnitude span of (¯n, S), the average Mach number varies little: 0.5 0.2, 1.2 M ≈ ± ± 0.3, 2.3 0.9 for the hot, warm and cold phases, respectively. ±

3.1 Background

As mentioned in the introduction of the thesis, [McKee and Ostriker, 1977] laid the foun- dation of the three-phase ISM regulated by SNe. They applied their theory primarily to the ISM around the solar neighborhood. In this work we want to extend the studies to very different environments, where the gas density and SN rate differ significantly from that is immediately surrounding us, such as in a tenous Galactic halo, a starburst region, etc, where the multiphase structure of the ISM can vary vastly. In particular, when the SN rate is high enough, an equilibrium state no longer exists and wind generation is inevitable CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 41

(e.g.[Joung et al., 2009; Gatto et al., 2015]). To our knowledge, there is no systematic exploration of the parameter space to determine on when the SNe could maintain an ISM with a significant volume in the hot phase. We will address this point in this chapter. An aspect of SN feedback that has long been neglected is that a significant fraction of SN progenitors are the so-called “OB runaways” [Gies and Bolton, 1986; Stone, 1991; Hoogerwerf et al., 2001; de Wit et al., 2005; Tetzlaff et al., 2011], thus SN can explode far from star formation sites. OB stars acquire this velocity when they reside in close binary systems and their companions explode as SN [Zwicky, 1957; Blaauw, 1961], and/or when they are in a crowded region of star clusters and get ejected due to dynamical interactions [Poveda et al., 1967]. [Stone, 1991] observed that 46% of O stars and 10% of B stars are in the “runaway” category with σ 30 km/s. This means that OB stars can travel 50 500 pc ∼ − before exploding as SN. SN can therefore deposit energy outside dense molecular clouds, and may go into the low density inter-spiral arm or halo. The spatial distribution of the pulsars independently confirms this idea: the scale height of pulsars at birth is much larger than that of the Galactic HI layer. [Narayan and Ostriker, 1990]. Consequently, the SN energy is much less prone to radiative losses due to low gas density. Type Ia SN, which are associated with the older stellar population with h 300 pc, have similar effects. The “runaway” ∼ OB stars together with Type Ia SN are thus very likely to contribute to the launching of galactic outflow out of proportion to their intrinsic frequency [Ceverino and Klypin, 2009; Kimm and Cen, 2014; Walch et al., 2015], but this effect has been explored very little. The velocities of OB stars also determine the spatial distribution of the core collapse SN, which is important for the resultant ISM properties. For example, [Gatto et al., 2015] found that random positioning of SN leaves most volume hot and thermal runaway, while SN exploding at density peaks result in little hot gas. We will address this point in the discussion section.

3.2 Model Description

The code and physical modules are the same as in Chapter 2. For this experiment, we have a series of boxes, each with two basic input parameters: gas mean density n, and SN rate S. The gas is initially uniform and static, and the sequential SNe would create and maintain CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 42 a multiphase ISM. More speciﬁcs of the setups are the following: The gas has an initial temperature of 3 103 K. Periodic boundary conditions are × applied. SN are assumed to have random locations (we test how good this assumption is later in Section 3.5.1, taking into account runaway OB stars). We explore the parameter space of (¯n, S) to study the resultant ISM systematically. Each simulation box contains 3 128 cells. The box size of each simulation varies withn ¯: the length is chosen to be 6Rcool (Eq. 2.1). The adaptive box size allows us to achieve higher resolution for a denser ISM, thus better capturing the evolution of SNRs and avoiding overcooling problems, which are frequently encountered in coarse resolution simulations. 51 Each supernova is injected as 10 erg thermal energy and 10 M mass, which are distributed evenly within a sphere. We choose the initial radius such that the enclosed mass −3 is close to but less than Minit, and we adopt Minit = 200 M forn ¯ 1 cm and 100 M ≤ forn ¯ > 1 cm−3. As for the duration of the simulation, we note that there is an intrinsic timescale after which ISM has been fully bombarded by random-located SNR, so that the initial uniform distribution of gas is destroyed entirely. This is roughly

1 S ¯n t = = 9.2 Myr ( )−1( )1.26. (3.1) b 3 −3 −1 −3 S (2Rcool) 1000 kpc Myr 1 cm

We therefore choose the simulation time to be tsim = 2.5 tb, so at the end of each simulation 66 SN have exploded in the box. The time span corresponds to about 10% of the gas ∼ −3 −3 depletion time ( nm¯ H/(S 100 M )) forn ¯ = 1 cm runs, and 40% forn ¯ = 30 cm . We ≡ · ﬁnd that the ISM reaches steady state (if there is one) long before the end of the simulation. We assume PEH rate scale linearly with the S, and adopt the following formula for a uniform PEH rate per H atom [Draine, 2011]:

−26 −1 S Γpe = 1.4 10 erg s ( ). (3.2) × · 100 kpc−3 Myr−1

We implicitly assume a constant dust/gas ratio, in line with a constant metallicity. Given a nominal dust photoabsorption cross section of FUV photons per H atom, 10−21cm2 [Draine, 2011], we ﬁnd that most of the simulation boxes are in the optically thin regime, except for then ¯ = 30 cm−3 runs where the total optical depth across the box is about 3.3. Therefore a linear scaling relation of PEH rate with S is, in general, not a bad approximation. CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 43

3.3 Choice of Parameters

The (¯n, S) combinations encountered in diﬀerent galaxy environments are not well-constrained, and are likely to be time-variable. Although on scales of kpc and above, we have the Ken- nicutt relation on the star formation (SF) rate and mean gas surface density [Kennicutt, 1998] (which has intrinsic scatters itself), on a smaller scale, like that of our simulation boxes, recent observations have shown large deviations from it. For example, by doing stellar counts for the star forming region in the MW, [Heiderman et al., 2010] found that the SF rate can be above the Kennicutt relation by at least an order of magnitude for −2 Σgas 100 M pc (although this is for small individual star-forming regions, not for large ∼ scale ISM). Besides, the Kennicutt relation uses the observables – the column densities of gas and SF rate, instead of the volume densities, and the scale height of the gas for external galaxies are usually not well-known. Furthermore, Type Ia SN, which are associated with the old stellar populations, do not correlate with gas properties. Lastly, Type II SN, because of the non-negligible velocities of their progenitor OB stars, can migrate out of the SF sites. All these contribute to the scatter ofn ¯ S relation in real galaxies on diﬀerent − length scales. Given our incomplete knowledge and the potentially big scatter in the relation ofn ¯ and S, we treat the two as free parameters, the relation between them being determined by physical modeling that depends on self-gravity, which has been ignored in this work (see, for example, [Kim et al., 2013] for self-consistent determination of ΣSFR as a function of

Σgas and Σstar for disk galaxies; for elliptical galaxies, SN are mostly Type Ia from old stellar populations, therefore such self-regulations do not exist.). In terms of choosing what parameter space to explore, we try to combine both observationally- and theoretically- motivated regimes. From the observational side, we cover the following regions and their neighbourhood – (1) MW disk-average (¯n 1 cm−3 and S 200 kpc−3 Myr−1); (2) MW ≈ ≈ halo: based on vertical distribution of SN and halo gas; (3) Scaling the MW disk-average 1.5 to “starburst” regime using the Schmidt relation S ρ/tdyn n¯ . Beyond the above ∝ ∝ domains, we also expand the parameter space after we analyze the results of the existing runs, to include those that are theoretically interesting. For example, given the density of the ISM, there is a critical SN rate Scrit above which no steady state exists. We list in CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 44

Table 3.1 the basic parameters for all simulations in this section. The symbolic name of a simulation, for example,n ¯0.1 S10, meansn ¯ = 0.1 cm−3 and S = 10 kpc−3 Myr−1. “Res” is short for “Resolution”. For the rest of this subsection, we explain how we estimate then ¯ - S correlation in the vertical direction of the MW disk. Observation of the 21 cm line indicates that the neutral medium lies within a ﬂat layer approximated by:

−3 2 2 nHI = 0.57 cm 0.7 exp[ (z/127 pc) ] + 0.19 exp[ (z/318 pc) ] + 0.11 exp( z /413 pc) , { − − −| | } (3.3) where z is the vertical distance from mid-plane [Dickey and Lockman, 1990]. [Gaensler et al., 2008] deduced from pulsar dispersion measure and diﬀuse Hα emission that the ionized warm medium of the MW follows an exponential function:

−3 ne = 0.014 cm exp( z /1830 pc). (3.4) −| |

The hot ionized gas is less constrained, and the density is very low, so we leave it out for this study. The sum of the above phases then yield the vertical distribution of the MW gas. The SN frequency for the MW is about 1/60 year−1 for core collapse SN and 1/250 year−1 for Type Ia [Cappellaro et al., 1997]. Type Ia SN come from the old stellar disk which has a scale height of 325 pc, therefore we adopt the following vertical distribution ∼ of Type Ia SN, as averaged over the MW disk (assuming 10 kpc radii in size):

−3 −1 SIa = 19.6 kpc Myr exp( z /325 pc). (3.5) −| |

For core collapse SN, the vertical distribution is inferred from that of pulsars at birth, which is a sum of two Gaussian functions [Narayan and Ostriker, 1990]:

−3 −1 2 2 SCC = 167 kpc Myr 0.75 exp(z/120 pc) + 0.25 exp(z/360 pc) . (3.6) { }

Putting Eq. 3.3-3.6 together, we get ann ¯ S relation along the vertical direction of the − MW. This is shown by the blue solid line in Fig. 3.5 and 3.7 that we will describe later. CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 45 1 TR heat t (Myr) Ek f V pred P P K) B 3 /k − V P 2.6E+053.2E+05 1.23.4E+05 1.04.8E+05 0.28 1.01.8E+06 0.36 0.9 0.133.7E+04 0.37 0.6 0.091.1E+05 0.39 N 2.8 0.071.2E+05 0.36 N 1.2 0.042.5E+05 0.46 N 1.1 0.033.6E+05 0.37 N 0.8 0.143.2E+03 0.31 m 0.6 0.127.7E+03 0.34 N 3.7 0.081.0E+04 0.46 N 3.6 0.063.3E+04 0.60 N 3.9 0.054.8E+04 0.41 m 1.6 0.209.0E+04 0.32 m 1.0 0.219.6E+02 0.40 N 0.8 0.191.6E+03 0.39 N 2.1 0.144.1E+03 0.34 N 2.5 0.121.6E+04 0.56 N 1.9 0.102.6E+04 0.49 N 1.3 0.396.5E+04 0.14 m 1.6 0.334.6E+02 0.13 N 0.8 0.272.3E+03 0.26 N 0.9 0.288.3E+03 0.23 m 0.7 0.232.1E+04 0.25 Y 0.5 0.179.1E+02 0.16 Y 0.9 0.764.6E+03 0.09 Y 0.4 0.546.7E+03 0.18 N 0.4 0.551.6E+04 0.18 m 0.6 0.363.5E+02 0.15 Y 0.4 0.641.4E+03 0.20 Y 0.2 0.616.1E+03 0.21 m 0.3 0.50 0.23 Y 0.3 0.44 0.18 Y 1.51 0.31 Y 0.89 m 0.60 Y Y ( cm K) 5 V 10 M / × V 2 f / > m f 0.01 /0.80 /0.51 0.02 /0.85 /0.52 0.01 /0.73 /0.47 0.04 /0.80 /0.69 0.004 /0.69 /0.85 0.002 /0.52 /0.70 0.003 /0.57 /0.90 0.001 /0.42 /0.68 0.001 /0.41 /0.66 0.003 /0.58 /0.72 0.001 /0.58 /0.42 0.002 /0.85 /0.32 0.004 /0.82 /0.48 0.006 /0.82 /0.51 0.003 /0.70 /0.47 0.004 /0.92 /0.31 0.008 /0.84 /0.47 0.003 /0.61 /0.46 0.009 /0.82 /0.45 0.004 /0.50 /0.55 2E-06 /0.01 /0.10 1E-05 /0.03 /0.22 1E-04 /0.11 /0.60 7E-04 /0.30 /0.83 1E-05 /0.03 /0.11 7E-04 /0.23 /0.67 5E-04 /0.26 /0.66 2E-05 /0.05 /0.12 2E-04 /0.16 /0.29 4E-04 /0.30 /0.39 4E-05 /0.07 /0.15 3E-04 /0.21 /0.32 3E-04 /0.20 /0.27 Hot ( V M / V f Warm / m f 1.0 /0.89 /1.00 1.0 /0.70 /1.35 1.0 /0.31 /1.73 1.0 /0.48 /1.28 1.0 /0.43 /1.80 0.19 /0.4 /1.25 0.86 /0.99 /0.55 0.98 /0.97 /0.73 0.43 /0.93 /0.72 0.89 /0.77 /0.76 0.98 /0.74 /1.01 0.16 /0.88 /1.28 0.33 /0.79 /1.20 0.37 /0.67 /1.18 0.78 /0.58 /1.14 0.96 /0.59 /1.16 0.99 /0.42 /1.11 0.16 /0.86 /1.40 0.20 /0.74 /1.44 0.19 /0.15 /1.03 0.30 /0.17 /1.24 0.77 /0.18 /1.33 0.20 /0.76 /1.30 0.18 /0.29 /1.25 0.14 /0.08 /0.89 0.74 /0.16 /1.21 0.30 /0.38 /0.92 0.52 /0.17 /1.21 0.86 /0.20 /1.33 0.96 /0.15 /1.56 0.44 /0.49 /0.93 0.81 /0.27 /1.03 0.96 /0.20 /1.95 K) V 3 M 10 / V < f / – /– /– – /– /– – /– /– – /– /– – / – /– –/–/– m f Cold ( 0.57 /0.04 /1.58 0.84 /0.07 /1.92 0.67 /0.04 /2.09 0.63 /0.03 /1.97 0.84 /0.08 /2.44 0.80 /0.05 /2.46 0.81 /0.02 /2.27 0.80 /0.04 /2.82 0.69 /0.008 /2.7 0.14 /0.006 /2.16 0.11 /0.003 /1.92 0.22 /0.006 /1.94 0.81 /0.008 /2.42 0.69 /0.007 /2.11 0.23 /0.002 /2.64 0.82 /0.008 /2.79 0.86 /0.005 /3.56 0.25 /0.003 /3.13 0.47 /0.006 /3.34 0.13 /0.002 /3.71 0.56 /0.007 /2.82 0.18 /0.002 /2.84 0.02 /5E-04 /2.35 0.02 /3E-04 /2.59 0.04 /9E-04 /2.58 0.01 /1E-04 /4.18 7E-03 /6E-05 /2.33 sim t (Myr) Table 3.1: Input/Output parameters of simulations in Chapter 3 Res (pc) ...... 35.9 .... 17.2 .... 7.2 .. 1.8 ...... 41.6 .... 20.6 .... 8.3 .. 4.1 ...... 129.4 .. 90.6 .... 30.2 .... 18.1 .. 9.1 ...... 226.5 .. 107.5 .... 45.3 .... 22.9 .. 7.5 ...... 115.2 .... 46.1 .. 9.2 ...... 25.8 .... 12.9 4.3 .. .. 32.9 3.0 35.7 0.28 72.5 54.3 0.42 134.8 90.0 0.70 301.9 (pc) 142.8 1.11 453.0 242.8 1.90 461.0 371.4 2.90 128.9 585.6 4.58 164.3 Box size 2 4 4 4 5 5 4 4 4 5 10 4 100 S 10 40 10 50 E E E E E E E E E S 100 500 100 300 E 50 S 2 4 8 2 8 1 2 5 1 3000 300 700 100 200 500 S S S S 1 1000 3000 5000 1000 3000 S S S S S S S S S S S S S S S S S S S S S 3 3 1 1 S S S S S 3 3 1 1 035 . . . . Run 1 035 . . . . . 3 1 1 1 3 035 3 . 0 0 0 0 3 3 3 1 1 ¯ n 30 30 30 30 10 10 10 10 30 . 0 0 0 0 0 10 ¯ ¯ ¯ ¯ ¯ n n n n n 0 ¯ ¯ ¯ ¯ ¯ n n n n n ¯ ¯ ¯ ¯ ¯ n n n n n 0 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ n n n n n n n n n n n n n n ¯ n ¯ n ¯ n TR = Thermal Runaway, Y = Yes, N = No, m = metastable state 1 CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 46

3.4 Results

3.4.1 Slices of some output ISM

Fig. 3.1 shows slices of density, temperature, pressure and velocity of three simulation examples. The upper panels are for the runn ¯1 S200, corresponding approximately to the solar neighbourhood. Gas settles into three distinct phases with contrasting density and temperature: Hot T 106.6 K, warm T 104 K, and cold T a few hundred K, which ∼ ∼ ∼ have been observed by a variety of techniques and summarized in [McKee and Ostriker, 1977]. These phases are in rough pressure equilibrium, with variations of a factor of 30, ∼ significantly smaller than the dynamical range of density and temperature. Velocity fields indicate that the ISM is turbulent, with the lighter component moving faster. Hot gas fills about 60% of the volume after reaching steady state. These results agree with the previous analytical and and numerical simulations [McKee and Ostriker, 1977; Joung and Mac Low, 2006; Gent et al., 2013; Gatto et al., 2015]. Middle panels are forn ¯0.3 S100, an example of a hot-dominated multiphase ISM (HDMM). SNRs overlap with each each other, rendering a connecting topology: 90% of the volume ∼ is filled with hot gas, and dense clouds are bathed in it. The clouds mainly consist of the warm phase. The deficiency of the cold medium is attributed to the less efficient cooling of less dense ISM, and the PEH is able to keep the temperature of most clouds to 104 K. ∼ Lower panels show the results ofn ¯10 S3000, corresponding to a star forming region. The SN rate is insufficient to retain a hot phase. Hot bubbles close up before they overlap with each other. SN do not play a significant role in shaping the thermal state of the ISM. The two-phase medium is maintained by the balance between cooling and PEH. On the other hand, the momentum input from SN drives turbulent motions in the gas, and is therefore important for shaping the dynamical state of the ISM, which may have strong implications for star formation [Mac Low and Klessen, 2004; McKee and Ostriker, 2007].

3.4.2 Evolution of the diﬀerent phases

(1)n ¯1 S200 (MW disk-average): example of reaching steady state and pressure equilibrium Fig. 3.2 presents the time evolution of the three phases for the runn ¯1 S200. We show CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 47 and n 3000 at S 10 n = 46 Myr; lower: ¯ t 100 at S 3 . 0 n , a certain phase can be missing. The clumping of gas varies with = 104 Myr; middle: ¯ S t and n 200 (solar neighbourhood) at S 1 n K, respectively. Depending on ¯ 2 10 , 4 10 , 6 . Upper row: ¯ 10 S . ∼ S T and = 135 Myr. Note that the box sizes vary. See Section 3.2 for model details. Most volume is shared by the three typical phases ¯ SN rate with Figure 3.1: Slices of density, temperature, pressure and velocity oft the ISM from simulations with varying mean density ¯ n CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 48

101 104

100 K)

3 3

− 10 1 (cm 10− B k / V

P 3 102 T < 10 K volume fraction 10 2 103 < T < 2 105 K − ∗ T > 2 105 K ∗ 101 10 3 20 0 20 40 60 80 100 120 − 20 0 20 40 60 80 100 120 − Time/Myr − Time/Myr 101 100

V 10−

100 2 10− Mass fraction Mach number

3 10− 1 10−

10 4 20 0 20 40 60 80 100 120 − 20 0 20 40 60 80 100 120 − Time/Myr − Time/Myr

Figure 3.2: Evolution of the three ISM phases forn ¯1 S200. Up to down, left to right: volume-weighted pressure PV /kB, Mach number, volume fraction, mass fraction. Red triangles, yellow diamonds and blue circles denote the cold (T < 103 K), warm (103 K < T < 2 105 K ) and hot (T > 2 105 K) phases, respectively. The medium reaches a × × steady state and the three phases are in rough pressure equilibrium. CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 49 in the four panels the volume-weighted pressure and Mach number, and the volume and mass fraction of each phase. Each data point represents one snapshot. The properties of the hot gas exhibit larger fluctuation, reflecting periodic explosions of SN. The evolution of the warm and cold phases, on the other hand, are smoother. Initially, the gas is uniformly 3000 K; within a few Myrs, cold clouds quickly build up and occupy > 80% of the mass, and the cold and warm media both settle into their separate thermally-stable temperatures. Thereafter, the mass fraction and temperature for these two phases keep almost constant. The separation of the two phases happens as a result of the thermal instability, facilitated by perturbations from SN. The Mach number is higher for colder phases: the cold and warm media are mildly supersonic with Mach number 2.3 and 1.3, respectively, while the hot ∼ gas is subsonic (the slightly supersonic cold/warm phase and subsonic hot phase turn out to be the case for most of the runs, see Table 3.1). All three phases are in approximate pressure equilibrium, with nT 3 4 103 cm−3 K. These results agree very well with the ∼ − × observation [Jenkins and Tripp, 2011]. Hot, warm and cold gas occupy 60%, 40%, 1.5% of the volume, respectively, consistent with the observations in the solar neighbourhood [Heiles and Troland, 2003; Murray et al., 2015]. Note that observations of 21 cm absorption-line yield about 60% of the mass of neutral hydrogen is warm [Heiles and Troland, 2003], which seems inconsistent with our result. However, we use T < 103 K as the criterion for being “cold”, different from the the observation, which is roughly T < 200 K. Also we do not ∼ distinguish molecular gas from the cold medium (nor do we track the ionization state) in our simulations. The mass ratio between cold HI and molecular is about 1.38 for the MW [Draine, 2011]. Taking this into consideration, the revised mass ratio of cold to warm in our simulations becomes 3.3 (even assuming all warm gas is neutral), still larger than the observed 0.67. Some effects that we do not include may account for the over-condensation of the cold phase, for example, UV from massive stars can ionize and heat the cold medium; thermal conduction by hot gas may evaporate some of the cold gas and form a warm component. Another possibility is that if a few SN happen to have exploded in the dense clumps, they can effectively turn the cold medium into warm and hot [Gatto et al., 2015]. The stellar winds from their progenitors can have a similar effect. We will discuss more on the position of SN in Section 5.1. CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 50

(2)n ¯0.1 S50 (MW disk-halo interface): example of a HDMM and “thermal runaway” The (¯n, S) of this run corresponds to the disk-halo interface of the MW (see Fig. 3.5 below). In the left panel of Fig 9, we show the volume-weighted pressure evolution for the three phases. Same as in Fig. 8, each data point is calculated for one snapshot. Hot gas is over-pressured compared to the other two phases, which reflects what we have found in the previous experiment: the blast wave travels preferentially in the hot medium. It would take roughly the sound-crossing time for the clouds to re-establish the pressure equilibrium. The overlapping SN determine the overall properties of the ISM. The medium does not reach a steady state at the end of the simulation, and it is not obvious that it will ever reach one. This is because, despite the rising density of the clouds which enhances cooling, the volume is increasingly occupied by low-density hot gas where cooling is negligible; since blast waves lose the majority of their energy in the clouds, cooling is inevitably delayed until the shock sweeps enough volume. Similar phenomena have also been observed in some of the simulations in [Gatto et al., 2015]. [Scannapieco et al., 2012] discovered a transitional 1D velocity dispersion 35 km/s for the turbulent ISM, above which the rarefied gas cannot ∼ cool efficiently and the ISM undergoes a thermal runaway. We can understand the thermal runaway state in this way: if the SN rate is high enough for multiple SN to overlap, the ISM will be multiphse where the hot bubbles connect to each other. As we have found in the experiment in which a single SN propagates in a HDMM, the volume encompassed by the blast wave is much larger, and the cooling is suppressed, thus the spatial-temporal domain of a single SN (V4D) is greatly increased. This makes the overlapping of the SN reinforces itself, which leads the ISM to a “runaway” state. In real galaxies, the persistent increase in pressure implies that the gas will expand, and hot gas can blow out and form winds (which are not captured in the periodic box simulations). (3)n ¯10 S3000 (SF region): example of a subordinate hot phase As shown in the right panel of Fig. 3.3, the ISM in this case has only warm and cold phases most of the time. Hot gas is present only briefly after SN explosions. SN are not determining the general thermal state of the ISM, since the pressure of the hot gas is significantly smaller than the other phases, and the time-averaged volume fraction is no more than a few percent (see Fig. 3.5 below). The reason for the under-pressure of the hot CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 51

104 105

104 K) K) 3 3 − −

(cm (cm 3 B B 10 k k / / V V P P T < 103 K 102 103 < T < 2 105 K ∗ T > 2 105 K ∗ 103 101 10 12 14 16 18 20 22 24 26 50 60 70 80 90 100 110 120 130 140 Time/Myr Time/Myr

Figure 3.3: Volume-weighted pressure PV of the three phases versus time for runsn ¯0.1 S50 (left) andn ¯10 S3000 (right). Forn ¯0.1 S50, the hot gas is over-pressured and the ISM is in a “thermal runaway” state. Forn ¯10 S3000, the hot phase is only present briefly after SN explosions and is significantly under-pressured. phase is this: once the SNR enters the “pressure-driven snowplow” phase, hot gas moves much faster than the shell, and quickly runs into it and loses energy, which leaves the hot bubbles at an extremely low density, and therefore low pressure (Fig. 2.1). Note that right after SN explosions, the hot gas is over-pressured, but it only lasts till the shock becomes radiative, and the time interval is very short compared to our sampling. In fact, according −3 to Eq. 2.2, tcool = 0.014 Myr for n = 10 cm , and yet the time interval between two consecutive SN is 2.08 Myr, and our sampling will not catch the over-pressured stage. In other words, if we could sample the data at a frequency larger than every 104 year, we would see the very spiky evolution of the hot gas pressure; if we could enlarge our box size (but keep S unchanged) so that at any given time, there are many SN exemplifying different stages of their evolution, we would see the mean pressure of hot gas go beyond that of warm/cold, because the spatially-averaging is biased towards regions with pressure that is orders-of-magnitude higher than the median. In other words, when we say hot gas pressure is lower than that of the other phases, we mean that at any given time, if we randomly choose a location that is above 2 105 K, it is most likely to be under-pressured. × (4) fV,hot 0.6 0.1: metastable state of the ISM and transition to “thermal runaway” ∼ ± CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 52

105 101

104 100 K) 3 − 3 1 (cm 10 10− B k / V P 3 Volume fraction 2 T < 10 K 2 10 10− 103 < T < 2 105 K ∗ T > 2 105 K ∗ 101 10 3 0 20 40 60 80 100 120 − 0 20 40 60 80 100 120 Time/Myr Time/Myr

Figure 3.4: Transition from a metastable state to “thermal runaway” forn ¯1 S200. Initially, the ISM reaches the apparent steady state, when roughly half the volume is hot. The transition happens at t 55 Myr, when the hot gas quickly dominates pressure and > 90% ∼ of the volume, and pressure of all phases rises progressively.

For the simulations which reach a steady state and have fV,hot 0.5 0.7, we sometimes ∼ − observe an interesting transition from “steady state” to “thermal runaway”. Fig. 3.4 exemplifies such a transition. This is for (¯n, S) = (1, 200), with the same set-ups as the fiducial run. The locations of the SN are random, but different from the fiducial one. Initially the ISM reaches a steady state (same as Fig. 3.2): Each of the three phases occupies a stable portion of the volume, and they are roughly in pressure equilibrium. Starting from t 55 Myr, however, hot gas quickly comes to dominate the space, while the ∼ volume fraction of the warm phase drops from 50% to a few percent. Meanwhile, the ∼ gas pressure rises progressively, and the equilibrium is disrupted – the hotter phase has a higher pressure. These phenomena are the same as those in a “thermal runaway” state. The transitions are somewhat stochastic, and can happen at any time after the steady state is established (the example shown in Fig. 3.2 is stable to the end of the run t = 110 Myr). The stochastic transition is not so surprising: the configuration of the ISM with half the volume hot is on the verge of changing from “isolated bubbles” to a “connecting” topology; any new SN may break the balance and lead to the topology transition. Once the hot bubbles CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 53

connect, any new SN will have a much larger V4D, and the overlapping will inevitably drive the ISM toward “thermal runaway”. Therefore, a steady state with fV,hot 0.6 0.1 is ∼ ± better termed as “metastable” due to its fragility.

3.4.3 fV,hot as a function of (¯n, S)

Fig. 3.5 shows how fV,hot of the ISM changes when we systematically varyn ¯ and S. Each simulation is represented by one circle on then ¯ S diagram, and the colors indicate fV,hot − . Each data point is obtained by averaging over 20 snapshots evenly sampled during the last 20% of the simulation time. From the ﬁgure, a lowern ¯ and higher S yield a higher fV,hot. As mentioned above, those with fV,hot 0.6 0.1 lie in the “transitional region” ≈ ± (grey-shaded), where the ISM is metastable and can change into “thermal runaway” with any new SN. Below the “transitional stripe”, SN do not play a major role in shaping the ISM, and can be signiﬁcantly under-pressured; above that, the hot gas has a “connecting” topology and is in “thermal runaway” state, with hot gas dominating the volume and an increasing pressure.

A simplest estimate of the fV,hot is

4π 3 fV,hot (φRcool) tcloseS. (3.7) ∼ 3

φRcool is the size of the hot bubble, where φ is about 1.6-1.8 (Section 2.3). tclose is the time for a SNR to close up:

tclose φRcool/cs. (3.8) ∼ Therefore,

φ 4 10km/s S n¯ −1.68 ESN 1.16 fV,hot 0.11( ) ( )( −3 )( −3 ) ( 51 ) . (3.9) ∼ 1.7 cs 100 kpc Myr−1 1 cm 10 erg

Eq. 3.7-3.9 only hold true when fV,hot < 0.6, i.e. SNRs do not overlap and thus thermal ∼ runaway does not come into operation. According to Eq. 3.9, given ann ¯, fV,hot should be proportional to S. This seems to be (very roughly) the case forn ¯1 S50,n ¯1 S100,n ¯1 S200.

However, the simulations show that for a higher density medium, the dependence of fV,hot on S is sublinear. For example, whenn ¯ = 3 cm−3, a factor of 10 increase of S – from 103 to 4 −3 −1 10 kpc Myr – only leads to a factor of 2 increase in fV,hot. Another way to look at ∼ CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 54

6 Color-coding: volume fraction of hot gas 10 0.9

0.8 105 0.7 1 − 104 hot-dominated 0.6 Myr 3 − 0.5 103 /kpc S 0.4 warm/cold-dominated 102 0.3 SN rate

0.2 101 Milky Way 0.1 S ∝ n¯ 1.5 0 10 2 1 0 1 2 10− 10− 10 10 10 3 ISM mean density n¯/cm−

5 Figure 3.5: Volume fraction of the hot (T > 2 10 K) gas fV,hot as a function of the mean × gas densityn ¯ and SN rate S. Each circle represents one simulation. Colors show the fV,hot . The blue solid line indicates S as a function ofn ¯ from MW mid-plane to halo (see Section 3.3 for a description). The green dashed line indicates S n¯1.5. (¯n, S) = (1, 200) is roughly ∝ for the MW-average. The shaded area shows the transitional region, i.e. fV,hot 0.6 0.1 ≈ ± – above it the hot gas dominates the volume and the ISM undergoes thermal runaway, and below it the SN play a subordinate role and most volume is warm/cold; within the transitional region the ISM is metastable. The gray solid line indicates the nominal line to roughly represent Scrit above which thermal runaway happens (Eq. 3.7). The black dotted line shows fV,hot = 0.6 from the simple analytical expectation Eq. 3.9. CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 55

3 Mass fraction Volume fraction Pressure/kB(cm− K) 1.4 1.4 hot warm 1.2 1.2 cold 105 1.0 1.0

0.8 0.8

0.6 0.6

0.4 0.4

104 0.2 0.2

0.0 0.0 1.4E-24 1.4E-25 1.4E-26 1.4E-24 1.4E-25 1.4E-26 1.4E-24 1.4E-25 1.4E-26 PEH rate (erg/s) PEH rate (erg/s) PEH rate (erg/s)

Figure 3.6: Comparison of the ISM from the runn ¯10 S1000 with diﬀerent photoelectric heating rates. Colors represent the three ISM phases.

it is to plot the corresponding curve of fV,hot = 0.6 using Eq. 3.9 (black dotted line on Fig. −3 3.5). Forn ¯ > 1 cm , the simulations withfV,hot 0.6 0.1 (grey-shaded) have an S above ≈ ± the simple theoretical expectation at a givenn ¯; whereas forn ¯ < 1 cm−3, the critical S to ∼ achieve a HDMM has a weaker dependence on n than expected. Overall, thermal runaway appears harder to achieve in dense gas. We argue that one key factor to explain the discrepancy is the photoelectric heating.

Recall that we have assumed Γpe S. If we consider a medium without SN (but with ∝ normal cooling and PEH), then the warm and cold phases would co-exist with the same pressure, each in its thermally stable state. If the mean density of the ISM is ﬁxed, a higher

Γpe means more mass in the warm phase. Note that the cold medium usually occupies a small volume whereas the warm phase permeate the space, so more mass in the warm phase suggests that a SNR will travel eﬀectively in a “denser” medium, in which it would have a smaller Rcool. This would have a substantial negative eﬀect on fV,hot when we consider 4 −1.68 multiple SNRs, since fV,hot depends on Rcool sensitively as fV,hot R n (Eq. 3.7 ∝ cool ∝ and 3.8).

To test this idea, we take the runn ¯10 S10000, and reduce its original PEH rate Γpe by a factor of 10 and 100, respectively. The latter case corresponds to a PEH rate similar to CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 56 the solar neighbourhood. The comparison is shown in Fig. 3.6.

The eﬀect of Γpe is obvious. Hot gas occupies more volume and has higher pressure for lower PEH rates. For the lowest PEH run, the ISM is clearly in a thermal runaway state, since the hot gas occupies most volume and determines the pressure. By ”determining the pressure” we mean that, if no SN explodes, the pressure of a thermally stable two- phase medium would be about 3 103 cm−3K for a PEH rate of 1.4 10−26 erg/s, but × × now the ISM pressure is 1-2 orders of magnitude higher than the equilibrium pressure. In this case the overlapping SNRs dominates the thermal state of the ISM. The mass fraction of the cold gas is also much higher in the low PEH run, which is likely due to two eﬀects: a smaller background heating rate that allows more gas in the cold phase, and compression from the overlapping SNRs. Interestingly, the volume-weighted pressure does not have a monotonic dependence on Γpe: the intermediate Γpe case has the lowest pressure among the three. This exhibits a subtle balance between the hot and warm phases: SN marginally overlap but are able to get rid of most of the energy; the pressure is slightly enhanced from that of a two-phase medium. PEH plays the leading role in determining the

ISM properties in the highest Γpe case, whereas SN wins in the lowest one. Note that in Section 2.3, we have shown that PEH rate does not impact the SN evolution in a uniform medium. It affects the ISM through regulating the warm/cold ratio in the gas reservoir. We emphasize that the PEH is a critical process in the ISM – it plays an important role in determining the portion of various phases and the overall pressure [Wolfire et al., 1995; Wolfire et al., 2003]. This may also have strong implications for the global structure of the disk and star formation [Tasker, 2011].

We give a simple ﬁtting formula for the critical SN rate Scrit as a function of n, above which thermal runaway occurs, i.e. fV,hot (¯n, Scrit) 0.6: ≈

n¯ k ESN −1 −3 −1 Scrit = 200( ) ( ) kpc Myr , (3.10) 1 cm−3 1051 erg where k = (1.2, 2.7) forn ¯ < 1, > 1 cm−3, respectively. We have shown this with a grey solid line in Fig. 3.5. CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 57

3 Color-coding: log(PV/kB/(cm− K)) 106 6.0

105 5.6

1 5.2 − 104

Myr 4.8 3 − 103 4.4 /kpc S 4.0 102

SN rate 3.6

101 Milky Way 3.2 S ∝ n¯ 1.5 2.8 0 10 2 1 0 1 2 10− 10− 10 10 10 3 ISM mean density n¯/cm−

Figure 3.7: Same as Fig. 3.5, but colors show the volume-weighted pressure PV/kB. CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 58

Figure 3.8: Equilibrium pressure (in unit of cm−3K) as a function of density n for T < 104 ∼ K gas for a variety of photoelectric heating rates. Red to green: 0.1, 0.5, 1, 2, 4, 10, 30, 100, 200 1.4E-26 erg/s per H atom. The calculation is based on the cooling curve adopted in × our simulation.

3.4.4 Pressure as a function of (¯n, S)

Fig. 3.7 is similar to Fig. 3.5, but the color-coding indicates the volume-weighted pressure

PV /kB for each simulation. The pattern of P (¯n, S) is clear. For a fixedn ¯, pressure increases with S, which is expected since higher S means higher heating rate. The scaling is nearly linear. The dependence of PV onn ¯ is less sensitive, and the correlation is mostly negative, at least for S < 104 kpc−3 Myr−1. ∼ Can we understand PV as a function of (¯n, S) more quantitatively? We have seen that it is not easy to obtain a very accurate fV,hot from a simple analytic argument, especially for the thermal runaway case when fV,hot keeps rising. On the other hand, if fV,hot is known, CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 59 can we predict pressure? In fact, if we neglect SN and only consider a two-phase medium, one can predict quantities such as temperature, pressure, density of each phase for a thermal equilibrium condition, given a cooling curve and a heating term [Draine, 1978] (note that under some circumstances, there may only be one phase). If the two phases coexist, they will reach pressure equilibrium. The equilibrium solution can be shown by a phase diagram, which is the correlation between any two of the three thermodynamic variables, density, temperature and pressure. We show in Fig. 3.8 examples of the equilibrium Peq n relation, − given our cooling curve and a variety of PEH rates. The equilibrium solution is thermally stable for the part of the curve with positive slope. The impact of different PEH rates on the equilibrium pressure of the neutral phase has been explored by [Wolfire et al., 1995;

Wolﬁre et al., 2003]. Our simple prescription predicts a very similar Peq n relation to their − more detailed model. Take the curve with the MW PEH rate for an instance (the third curve from the bottom): the unstable solution corresponds to roughly in between 0.4-15 cm−3, and most part outside is stable (except a second dip between 15 30 cm−3). When − −3 4 the two phases coexist, the warm phase has nw,eq 0.4 cm and T 10 K, and the ≈ ≈ −3 cold has nc,eq 15 cm and T 250 K. Another way to look at this relation is that, if a ≈ ≈ mean gas densityn ¯ is in the range [nw,eq, nc,eq], the medium will settle into two phases with 3 −3 pressure Peq/kB 3 10 cm K; ifn ¯ < nw,eq, the medium will be all warm and P n¯; ≈ × ∝ 0.5 ifn ¯ > nc,eq, the gas is all cold with roughly P n¯ . ∝ When we consider SN and the hot phase, the hot component can occupy a signiﬁcant volume but usually contains negligible mass, so the mean density for the warm/cold phase is simplyn/ ¯ (1 fV,hot). Therefore, given an n, PEH rate and fV,hot, the pressure can be − predicted in the following way. The equilibrium densities of warm and cold phase scale linearly with PEH:

−3 Γpe −3 Γpe nw,eq 0.4 cm ( ); nc,eq 15 cm ( ). (3.11) ≈ 1.4 10−26erg/s ≈ 1.4 10−26erg/s × × Then the pressure is (assuming the warm phase has a constant temperature 104 K):

n¯ 4 n¯ 10 K if nw,eq , 1−fV,hot 1−fV,hot  × ≤ 4 n¯ PV,pred/kB = nw,eq 10 K if nw,eq < < nc,eq, (3.12)  × 1−fV,hot  4 ¯n 0.5 n¯ nw,eq 10 K( /nc,eq) if nc,eq. × 1−fV,hot 1−fV,hot ≥    CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 60

In Table 3.1 we show the ratio between our analytic calculation and the simulation results

PV,pred/PV . They agree within a factor of 3 generally. Note that in the thermal runaway case, the hot gas pressure is higher than that in the warm and cold, but the ratio is on the order of unity. In cosmological simulations where the resolution in space and time is low, the multiphase structure of the ISM and individual SN are often not resolved, and the collective SN feedback may be modelled as a sub-grid eﬀective equation of state, if the the cell size is much larger than the spatial domain of a single SN, and the timestep is longer than tb (Eq.

3.1). Although Eq. 3.12 can predict the pressure, it has an extra parameter fV,hot which cannot be easily obtained. We are then tempted to ﬁt PV directly as a function ofn ¯ and

S. An advantage is that from Fig. 3.7, the pattern of PV (¯n, S) seems clear and simple and power-law-like. We thus ﬁt the output data using the least squares method, which yields:

3.47±0.05 −3 S 0.87±0.05 ¯n −0.33±0.07 PV /kB = 10 cm K( ) ( ) . (3.13) 100 kpc−3 Myr−1 1 cm−3

The indices show that PV scales almost linearly with S, while it is somewhat less dependant onn ¯. The fitting formula gives a reasonable prediction of pressure forn ¯ < 30 cm−3. We caution that the above fitting formula is time-independent. It is fine for an ISM that reaches equilibrium, but for those thermal runaway ones, the pressure is not constant but is still rising at the end of the simulation. To generalize the result to the thermal runaway cases, we extend Eq. 3.13 by adding a factor of t/tsim to allow for time evolution. Just to remind readers, tsim = 2.5tb, and tb is given in Eq. 3.1 as a function of (¯n, S). Thus,

3 −3 S 0.87 ¯n −0.33 2.9 10 cm K( −3 ) ( −3 ) if S < Scrit, × 100 kpc Myr−1 1 cm PV /kB =  3 −3 S 1.87 ¯n −1.59 t 9.6 10 cm K( −3 ) ( −3 ) ( ) if SScrit.  × 1000 kpc Myr−1 1 cm 10Myr (3.14)  Scrit is given in Eq. 3.10, which only depends onn ¯ and ESN. We emphasize that the above model holds only under the following conditions: (1) the spatial domain is at least a few times the size of a single SNR, and (2) the elapsed time t is on the order of a few tb. If t tb, the ISM is not fully impacted by SN, and if t tb and the ISM is in the thermal runaway state, the gas will probably expand due to the increasing pressure. In cosmological simulations, for a “star-forming” cell, given its gas density, S, and a time-interval, we can CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 61 use Eq. 3.14 to model the SN-modiﬁed pressure, provided that the above conditions are satisﬁed. Note that we do not take into account the “turbulence pressure” here. As we will show in the next subsection, the Mach number is close to unity, which means that the pressure from the random motions is similar to that of the thermal pressure [Joung et al., 2009]. In practice, a factor of 2 can be added to Eq. 3.14 to make it the total pressure. Finally, an interesting coincidence is that the box size and duration of the simulations are roughly on the same orders of the cell size ( 100 pc) and timestep ( 10 Myr) of ∼ ∼ cosmological simulations, respectively.

3.4.5 Other output parameters

Table 3.1 lists the input/output parameters for all simulations in this section. For each

ISM phase, we show the mass fraction fm, volume fraction fV and volume-weighted Mach number V . Again, the numbers are obtained by averaging over 20 snapshots evenly M sampled during the last 20% of the simulation time. The Mach number is deﬁned as

v/cs, (3.15) M ≡ where v and cs are the local velocity (in the simulation box frame) and the local sound speed, respectively, and

cs γkBT/µ, (3.16) ≡ p where the adiabatic index γ = 5/3, and the molecular weight µ = 1.22, 0.588mH for T < 104, > 104 K gas, respectively. The order of unity value of may not be surprising, since the blast waves induced by M SN explosions would dissipate until they decay to sound waves. Intriguingly, however, the deviation of is fairly small for the three phases, despite the large parameter space (¯n, S) M we have explored. We find V = 0.5 0.2, 1.2 0.3, 2.3 0.9 for the hot, warm and M ± ± ± cold phase, respectively. The corresponding mass-weighted m (not listed in the table) M are 0.5 0.3, 1.0 0.2, 2.6 1.2, respectively. The warmer phases have consistently lower ± ± ± . The Mach numbers of the cold phase lie within the observational range 1 4, M M ∼ − as derived from the thermal pressure distribution in the solar neighbourhood [Jenkins and Tripp, 2011]. However, we should caution that the cold clouds are usually resolved by < 10 ∼ CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 62 grids in radius, and the velocity dispersion of the gas might be suppressed due to the limited numerical resolution. Finally, for a given density, V for the warm/cold phase seems to M correlate (very) modestly with the S for eithern ¯10 cm−3 orn ¯0.1 cm−3. Note that for the formern ¯ range, the ISM is in a steady state (or metastable), and for the latter, thermal runaway (or metastable). Observationally, the velocity dispersion of Hα may have a mild correlation with star formation rate [Green et al., 2014; Arribas et al., 2014]. Whether the velocity dispersion of HI has such a correlation is not clear; evidence for either side has appeared in the literature [Tamburro et al., 2009; Rogers and Pittard, 2013; Stilp et al., 2013], but the correlation is modest at most, if any. Our results are broadly consistent with the observations. However, here we are considering only SN as the turbulence driver; other processes, such as gravitational and/or magneto-rotational instability, can also play roles in driving the turbulence. Also note that for runs with largern ¯, we use higher resolutions and smaller boxes. Thus the finer structures are increasingly resolved, especially for the dense phase. For the lowern ¯ runs, the resolution of the inner structure of the cold clumps may be limited by the finite cell size. In Table 3.1 we have also shown the fraction of the kinetic energy out of the total energy in the simulation domain, fEk. This is obtained by averaging over 30 snapshots for the last

3% of the simulation time. fEk is in the range of 15% - 50%, and the value is smaller for lowern ¯. This is consistent with that the denser media cool more eﬃciently, thus leaving a larger fraction of the energy in the kinetic form. We deﬁne a “heating time scale” as

etot,V etot,V theat = , (3.17) ≡ e˙SN S ESN · where etot is the volume-weighted energy density of the ISM at the end of the simulation, 51 e˙SN is the time-averaged SN heating rate, and ESN = 10 erg. If the ISM reaches the thermal equilibrium,e ˙SN is equal to the time-averaged cooling ratee ˙cool (the other heating source, PEH, is an order of magnitude weaker than the SN overall), so theat is approximately the cooling timescale. If the ISM is in the thermal runaway state, theat/tsim 1 e˙cool/e˙SN, ∼ − where tsim is the duration of the simulation. From the table, theat is generally smaller than 1 Myr, much smaller than tsim, indicating that cooling is eﬃcient in the ISM. Even in a thermal runaway state,e ˙cool/e˙SN is very close to unity. For extreme cases, such as CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 63

n¯0035 S100, theat/tsim 0.2, which implies that on average 20% of the energy from each ≈ SN is retained in the ISM, instead of being radiated away. For the majority of the thermal runaway media, the retained fraction of energy per SN is about 1-5%.

3.5 Discussion

3.5.1 Randomness of SN position on simulation box scale

In the simulations described in Section 4, we have kept the positions of SN random. In this section we discuss how good this assumption is. Type Ia SN, which are from old stellar populations, are not correlated with the gas properties, and are thus close to random. Core collapse SN are more complicated. They descend from OB stars, and are mostly associated with star-forming clusters, which have abundant dense molecular clouds. Recent work have shown that the SN position relative to the dense clumps has a significant impact on the feedback. [Iffrig and Hennebelle, 2015] found that a SN is most disruptive to a molecular cloud when it explodes inside; a SN outside has only moderate effect on the cloud, but the feedback influence is felt much further in the low-density inter-cloud medium. Considering multiple SN, [Gatto et al., 2015] discovered in a periodic box study that a random positioning leads to thermal runaway more easily than when the SN mostly explode within the clumps. [Hennebelle and Iffrig, 2014] used a set-up with stratified medium and found that random SN lead to a higher gas density and pressure at high galactic latitude – closer to what has been observed. The random positioning leaves most SN to explode in a rarefied medium, which enables an effective feedback on larger scales. To assess the SN position relative to the dense clumps, we emphasize that the non- negligible velocities of the OB stars should be taken into account. We note that some other effects can also lead core collapse SN to explode in a low-density environment. For example, molecular clouds disperse on a timescale of about 20 Myr, shorter than the life time of some SN progenitors, so even those in situ may explode in a rarefied gas; recent works [Bressert et al., 2012; Oey et al., 2013] suggested that some OB stars are indeed born in the field, rather than in crowded clusters. But here we simply focus on the effect of the OB velocities. In particular, it would be very interesting to know how far OB stars can CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 64

1.0 100

0.9

1 0.8 10− ) ) x x > > 0.7 d d ( ( P P 2 0.6 10−

0.5

0.4 10 3 0.00 0.05 0.10 0.15 0.20 −0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 x/kpc x/kpc

Figure 3.9: Left: Cumulative distribution of 3D displacement of core collapse SN, relative to the birth places of their progenitors, due to the velocities of OB stars. Roughly 68% of core collapse SN have migrated more than 100 pc. Right: the projected 1D displacement. Nearly 10% of OB stars can migrate > 1 kpc. See Section 3.3 for the model description. ∼ migrate before they explode as SN. To know the answer helps other important questions: (1) What fraction of core collapse SN explode inside molecular clouds versus outside? (2) Is the random positioning of SN assumed in our simulations a good approximation? To calculate the displacement distribution of core collapse SN progenitors, we do the following simple Monte Carlo simulation: We adopt the distribution of space velocities of OB stars compiled by [Stone, 1991], which is a sum of two Maxwellians with σ = 7.7 km/s and 28.2 km/s, respectively, and 46% of O stars and 10% of B stars belong to the high- velocity group. More recent work by [de Wit et al., 2005] conﬁrmed that roughly half of the O stars are “runaways”. Each core collapse SN progenitor is randomly assigned a mass −2.3 in the range 8 M < M∗ < 50 M according to the initial mass function dN/dM∗ M ∝ ∗ [Kroupa, 2001; Chabrier, 2003], a speed according to the distribution mentioned above

(if M∗ > 16 M , the progenitor is classiﬁed as an O star, otherwise a B star), and a random velocity direction. The life time of the progenitor is related to its mass by tlife = 10 −2.5 10 year(M∗/ M ) . The displacement distribution is shown in Fig. 3.9. In 3D space (left panel), about 96%, 82%, 70% of SN progenitors migrate away from their birth sites by distances greater than CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 65

10, 50, 100 pc, respectively. The giant molecular clouds appear filamentary [Molinari et al., 2010], and the size is on the order of 100 pc in the long dimension and an order of magnitude smaller on the shorter ones. Our result suggests that roughly < 5% of SN would explode ∼ within these clouds. Those SN may partly disperse the clouds. On larger scales, the external gravitational field from the dark matter halo and stellar disk may preferentially confine the motion of the progenitor stars to a certain direction, so it makes more sense to calculate the projected 1D displacement distribution (right panel of Fig. 3.9). Approximately 17%, 8.4%, 2.5% of the SN progenitors have displacements greater than 0.5, 1, 2 kpc, respectively. A significant fraction of runaway OBs are therefore likely to migrate out of the star formation regions, and explode in the low-density part of a galaxy, such as halo and inter-spiral arm regions, and may drive a wind there. The sizes of our simulation domain range from a few tens to a few hundred parsec, and we want to know how the runaway stars affect the randomness of the SN position on this scale. We conduct the following comparison experiments: we run two additional simulations ton ¯1 S200 (“Random”) – (1) the probability of where a SN progenitor is born is proportional to local gas density ρ1.5, and each SN is randomly assigned a displacement according to the displacement distribution in the left panel of Fig. 3.9 (“HD+runaway”, where HD stands for high-density); (2) same as (1) but no displacement, that is, SN explode where their progenitors are born (“HD”). All other input parameters, such asn ¯, S, the box size and resolution, remain the same. The results are shown in Fig. 3.10: the properties of the ISM produced by “HD+runaway” fall in-between those in the other two cases. In particular, displaced SN yield very similar volume/mass of each gas phase to random SN. The “HD” run produce more warm gas in mass and a much smaller volume fraction of the hot medium. This is understandable since the density peaks in the ISM consist mostly of the cold medium, and the energy released by a SN converts the cold medium to warm. The strong cooling in those dense clouds restrains −0.42 the power of SN (Rcool n ), so they cannot impact the ISM on larger scales. This ∝ explains the smaller hot gas fraction and lower pressure. Those results are qualitatively consistent with [Gatto et al., 2015]. We conclude that based on the observed OB star velocities, the positions of core collapse SN are nearly random on scales < 150 pc. ∼ CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 66

3 Mass fraction Volume fraction Pressure/kB(cm− K) 4500 1.4 1.4 hot 4000 warm 1.2 1.2 cold 3500

1.0 1.0 3000

2500 0.8 0.8

2000 0.6 0.6 1500 0.4 0.4 1000

0.2 0.2 500

0.0 0.0 0 Random HD+runaway HD Random HD+runaway HD Random HD+runaway HD

Figure 3.10: Eﬀects of SN location on the resultant ISM forn ¯1 S200 (box size 144 pc). Left to right: mass fraction, volume fraction, and volume-weighted pressure of the three ISM phases. “Random”: SN explode at random positions. “HD”: SN explode only in high- density clumps. “HD + runaway”: SN have a displacement from the high-density clumps, due to the velocities of their progenitors. The ISM from the “Random” and “HD+runaway” are similar, whereas that from “HD” has a much smaller fV,hot and more warm gas. CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 67

3.5.2 Connecting simulation data to real galaxy environments

We have plotted (¯n, S) encountered in the MW disk to halo on Fig. 3.5 (blue solid line), as described in Section 3.3. For the disk averagen ¯1 S200, the ISM is in the “transitional stripe”. Of course, the disk average does not take into account the clustering effect of the SN. OB stars are born mainly in OB associations, and eventually 60% of core collapse SN appear “grouped”. Since the disk average S makes the ISM in the transitional region, it is not surprising that any grouping in space and/or time can lead to thermal runaway, assuming the same gas mean density. Indeed, if SN explode within a sphere of R 100 pc ∼ every 0.3 Myr as for a super-bubble [Mac Low and McCray, 1988], which corresponds to S 800 kpc−3 Myr−1, it is well within the thermal runaway regime forn ¯ = 1 cm−3 (Fig. ∼ 3.5). As one goes up to the halo, the mean density is smaller, and the SN rate is such that fV,hot is at first larger and the ISM goes into the thermal runaway region. This suggests that at least part of the halo is not in hydrostatic equilibrium. Above a certain height, fV,hot drops again, which may suggest that the gaseous halo is convective, but a conclusion cannot be made without a simulation with stratified medium and an external gravitational field [Joung and Mac Low, 2006; Gent et al., 2013; Creasey et al., 2013]. We have also plotted the scaling relation of S andn ¯ as suggested by the Schmidt star formation relation S n¯1.5 (green dashed line). The relation is normalized such that ∝ the line passes through the MW averagen ¯1 S200. For data points that lie around this −3 relation, fV,hot is smaller for higher densities. Forn ¯ = 30 cm , the ISM around the

Schmidt relation has a fV,hot of only a few percent. Surprisingly, for ISM in star-burst regimes, little hot gas exists and the medium is thermally stable. Observationally, however, powerful winds are ubiquitously associated with such high star formation rates, and the mass loading ( M˙ outflow/M˙ SF) is on the order of unity or even higher [Heckman et al., 2000; ≡ Steidel et al., 2004]. Note that we have found the PEH rate to be a very important factor in determining the thermal state of the ISM. If the PEH rates we have adopted are reasonable, then the data seem to suggest that one needs other mechanism(s) to drive a wind for the high density regions. For example, “clustering” of SN may work, although a factor of 10 or more increase in S is needed forn ¯ > 3 cm−3 above the average value to have a ∼ CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 68 thermally-driven wind. Another factor is the pre-SN feedback, such as photo-ionization, radiation pressure and winds from massive stars, which can create low density tunnels, facilitating SN energy to leak out [Rogers and Pittard, 2013]. Alternatively, runaway OB stars, which can easily migrate more than a hundred parsecs, may lead to a significant fraction of core collapse SN exploding outside the immediate high density SF clouds. For a low density mediumn ¯ < 1 cm−3, the critical SN rate for the ISM to have a thermal ∼ runaway is much less stringent. Another possible mechanism for wind launching is by the cosmic rays. As the cosmic rays diffuse out, the pressure gradient exerts accelerating force on the baryonic gas. Recent simulations have shown that this mechanism is promising to drive winds with a reasonable mass loading [Uhlig et al., 2012; Hanasz et al., 2013; Salem and Bryan, 2014].

3.5.3 Periodic box

In the experiments where multiple SN shape the ISM, we have applied periodic boundary condition and initially uniform medium. We briefly discuss their applicability here. In a periodic box, the total mass is fixed, and the gas is not allowed to expand beyond the simulation domain. This means that such a box is good at capturing processes that have scales smaller than its size, which do not result in net ingoing/outgoing flux on the outer boundary. For our experiments, this is the case for those reaching steady states. For the thermal runaway cases, the pressure keeps rising and the box is accumulating energy. In real galactic environment, the hot gas is subject to expansion if the ISM pressure around is not sufficient to stop it. The expansion can lower the pressure and fV,hot [de Avillez and Breitschwerdt, 2004]. Such an evolution thus cannot be faithfully captured by a periodic box. On the other hand, we want to emphasize that the onset criterion for the thermal runaway, i.e. when the SNRs overlap, as given in Scirt(¯n), should not depend on whether the box is open or periodic – it only depends on the local density of the gas. Furthermore, even the outgoing flow is in an explosive state initially, it may adjust so that each patch of the flow comes to an equilibrium state. The length scale of the galactic winds is 1-10 ∼ kpc, much larger than Rcool of a single SN, and the dynamical time scale is Gyr, much ∼ CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 69 longer than what takes to reach a steady state. So when the gas on large scale is in a steady state, the effect of SN may be regarded as “microphysics” and incorporated in the effective equation of state in a cosmological simulation. A related issue is the density distribution in the box. In our simulations we keep gas uniform as the initial condition. We note that in real galaxies, especially disk galaxies, the ISM is stratified in the vertical direction. Therefore, the uniform assumption only holds when the processes we study have scales smaller than the scale height of the gas. The size of our simulation domain, which is 6Rcool, is generally smaller than the gas scale height in the MW, given a mean ISM density. For example, the thickness of the neutral gas layer is about 180 pc with a mean density 1 cm−3, the warm ionized medium has a scale height ∼ of about 1830 pc with the density 0.014 cm−3 [Gaensler et al., 2008]. For comparison, the box size is 142.2 pc forn ¯ = 1 cm−3, and 585.6 pc forn ¯ = 0.035 cm−3. So the non-stratified medium is probably not a bad assumption. That said, the gas scale height in extragalactic systems are usually not well-constrained, and SN themselves may play an important role in determining it [Shetty and Ostriker, 2012]. In disk galaxies, clustered SNR in a thermal runaway state can form “super-bubbles” that may break out of the disk and transfer materials into the halo [Tomisaka and Ikeuchi, 1986; Mac Low and McCray, 1988]. The outflowing gas forms galactic winds and/or fountains that regulate the circum-galactic environment [Heckman et al., 2000; Steidel et al., 2004]. These can only be captured in a larger box with more realistic set-up. For example, the medium is stratified, which has outflow outer boundary conditions in the vertical direction and external gravitational field (e.g. [Joung and Mac Low, 2006; Creasey et al., 2013]). We will postpone the study of SN feedback in such cases to future work.

3.5.4 Thermal conduction and magnetic ﬁelds

Thermal conduction can be an important process in a multiphase medium to drive mass and energy transfer among different phases. In the model of [McKee and Ostriker, 1977], evaporation through thermal conduction plays a major role in producing the warm phase from the dense, cold clumps in the post-shock region. Very small clouds bathed in a hot CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 70 medium can be quickly evaporated. As mentioned earlier, thermal conduction is not included in our simulation. The warm phase is maintained mainly through the balance of the shock heating/PEH and cooling. We did some calculations to estimate the role of conduction for our simulations. For the multiphase ISM models we adopt in Section 2.4, conduction actually causes hot gas to condense on the clouds [Cowie and McKee, 1977], and the evaporation is only important when very close to the explosion center, where the temperature is high and clouds are shock-stripped to small cold cores. The difference between our simulation and [McKee and Ostriker, 2007] arises from that our clouds, which are created and destroyed by SN interactions, are larger by a factor of a few than what was adopted in the [McKee and Ostriker, 2007] model, which was inspired by the HI observation at that time. Smaller clouds would result in a more effective mixing between the different phases in a number of ways, e.g. Kelvin-Hemholtz instability, and photo-evaporation by massive stars [McKee et al., 1984]. To fully evaluate the role of conduction, one also has to consider the magnetic field, which can cause significant anisotropy. Conduction is practically inhibited perpendicular to the field lines. When a blast wave travels in a HDMM, it tends to wrap the field lines around the clouds [Semenov and Bernikov, 1980; Dursi and Pfrommer, 2008], which suggests that the conduction between clouds and the hot medium may not be very important. The actual impact of conduction considering magnetic field is not clear and is definitely worth further study. The magnetic field itself can also be important dynamically. Local observations show that the magnetic pressure is a few times larger than gas thermal pressure [Heiles and Crutcher, 2005]. [Slavin and Cox, 1992] has found that in a uniform medium, the magnetic pressure at late stages of the SNR can thicken the shell and thus make the hot bubble smaller. This may change the critical SN rate for thermal runway at a given density. A MHD simulation is needed to quantify the role of the magnetic field in shaping the ISM [Shin et al., 2008; Hill et al., 2012]. CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 71

3.6 Conclusions

In this chapter we let SN self-consistently shape the ISM under various environments, and systematically studied the resultant ISM. We explored a large parameter space of the ambient gas density n and SN rate S. The main results are summarized as follows: 1. We study the ISM self-consistently shaped by multiple SN, covering a wide range of SN frequency S and gas average densityn ¯. We find that the classic view of the three-phase ISM shaped by SN, as envisioned by [McKee and Ostriker, 1977], needs to be extended, depending onn ¯ and S. The main differences we have found are the following: First, a certain phase can be absent, or exist only very briefly. Second, equilibrium solutions may not exist when SN explode at a sufficiently high rate, and the ISM undergoes thermal runaway. Third, the pressure of different phases may not be in equilibrium; the hot phase can be over- or under-pressured.

2. We find that fV,hot 0.6 0.1 marks the transition of ISM from a steady state ∼ ± to thermal runaway, along with the change of hot gas topology. When fV,hot < 0.6 0.1, ∼ ± SNRs evolve as individual bubbles; if fV,hot can reach 0.6 0.1, the hot bubbles connect to ± each other. The connecting topology makes any new SNR have a larger spatial-temporal domain of influence, which reinforces the overlapping of the blast waves. As a result, the overall heating dominates over cooling and the ISM undergoes thermal runaway. An ISM with fV,hot 0.5 0.7 is in a metastable state, and may transition into thermal runaway. ∼ − 3. The PEH has a surprisingly strong impact on fV,hot. For a fixed (¯n, S), fV,hot decreases as the PEH rate increases. In particular, forn ¯ > 3 cm−3, assuming that the PEH ∼ is proportional to S, fV,hot is confined to < 0.6 for a wide range of S, so that the medium ∼ is stable and does not form a wind (Fig. 3.5). The critical SN rates for the onset of the −3 k 51 −1 −3 −1 thermal runaway is roughly Scrit = 200(¯n/1 cm ) (ESN/10 erg) kpc Myr , where k = (1.2, 2.7) forn ¯ 1 and > 1 cm−3, respectively. ≤ 4. The pressure of the ISM relates to fV,hot, mean gas densityn ¯ and photoelectric heating rate Γpe through Eq. 3.11-3.12. The results do not depend on whether the ISM is in a steady state or thermal runaway. A simple fitting formula PV (¯n, S, t) can be used as CHAPTER 3. MULTIPLE SUPERNOVA REMNANTS REGULATING MULTIPHASE ISM 72 an effective equation of state considering SN feedback:

3 −3 S 0.87 ¯n −0.33 2.9 10 cm K( −3 ) ( −3 ) if S < Scrit, × 100 kpc Myr−1 1 cm PV /kB =  3 −3 S 1.87 ¯n −1.59 t 9.6 10 cm K( −3 ) ( −3 ) ( ) if S > Scrit.  × 1000 kpc Myr−1 1 cm 10Myr 5. The local Mach numbers of the three ISM phases show surprisingly small varia- M tions, despite the 5 orders of magnitude span of (¯n, S). We ﬁnd that V 0.5 0.2, 1.2 M ≈ ± ± 0.3, 2.3 0.9 for the hot, warm and cold phase, respectively. ± 6. We calculate the displacement distribution of the core collapse SN from the observed velocities of OB stars, which shows that on scales < 150 pc, SN explode almost at random ∼ positions, and nearly 10% of OB stars can migrate > 1 kpc (Fig. 3.9). The latter runaway ∼ OBs have a chance to migrate into low-density regions where it is easier to drive a wind. 73

Part II

Supernovae-Driven Multiphase Galactic Outﬂows CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 74

Chapter 4

Quantifying Supernovae-Driven Multiphase Galactic Outﬂows

This chapter is adapted from the publication in Astrophysics Journal, 841:101, 2017. CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 75

Abstract

In this paper, we study SNe-driven outflows from a stratified disk, with various gas surface densities and SN rates. We focus on the following questions: (1) How much energy, mass and metals can SNe launch out of the disk? How do these change along the Kennicutt- Schmidt relation? (2) What physical processes affect the energy, mass and metal loading? We explore the effects of runaway OB stars, photoelectric heating (PEH), gravitational field, enhanced SN rates, etc. While our simulations focus on regions around the disk ( ± 2.5 kpc), we discuss how our results connect to outflows on galactic scale. We perform 3D, high-resolution hydrodynamic simulations to study SNe-driven outflows from stratified media. The major conclusions are: (i) Assuming SN rate scales with gas surface density Σgas as in the Kennicutt-Schmidt (KS) relation, we find the mass loading factor, ηm , defined as the mass outflow flux divided by the star formation surface density, −0.61 2 decreases with increasing Σgas as ηm Σgas . Approximately Σgas < 50M / pc marks ∝ ∼ when ηm > 1. (ii) The energy and metal loading factors, defined as the fraction of en- ∼ ergy/metals produced by SN that goes into outflows does not show strong trend with Σgas. About 10-50% of the energy and 40-80% of the metals produced by SNe end up in the outflows. (iii) The tenuous hot phase (T > 3 105 K), which fills 60-80% of the volume × at mid-plane, carries the majority of the energy and metals in outflows. Hot outflows have super-solar metallicity. (iv)The relative scale height of gas and SNe is a very important factor in determining the loading efficiencies. (v) If gas is already hot dominated, enhancing SNe rate does not change the energy/metal loading factors, but can decrease the mass loading factor.

4.1 Background

Various papers have explored how SNe drive outflows from a stratified medium [de Avillez, 2000; Joung and Mac Low, 2006; Joung et al., 2009; Hill et al., 2012; Gent et al., 2013; Walch et al., 2015; Gatto et al., 2016; Kim and Ostriker, 2016; Peters et al., 2017]. Most works have focused on the solar neighbourhood, in which the mass loading factor of 0.3-3 is found. [Creasey et al., 2013] explored a wide parameter space of gas surface density and CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 76 external gravitational field. Assuming the SN rate correlates with the gas density via the empirical star formation law – the Kenniccut-Schmidt (KS) relation – they found that the mass loading decreases with increasing gas surface density. In a companion paper, [Creasey et al., 2015] also explored the metal entrainment in the outflows. The major differences in our models are the following: (i) Creasey et al.’s cooling model has a lower temperature cut at 104 K, but as we will show in Section 4.4.2, this would underpredict the loading efficiencies by orders of magnitude. (ii) Creasey et al.’s analysis does not distinguish warm outflows from the hot outflows, but treat all the gas that leaves the upper boundary (1 kpc above the plane) as “outflows” when calculating the loading factors. But as we show in Section 4.3.2, the warm outflows has a significantly smaller velocity, so they will largely be fountain flows and have much less impact than the hot outflows. In the literature, the external gravitational fields included in the models vary a lot, even the same “solar neighborhood” is claimed. As we will demonstrate later in this Chapter, the gravitational field would play an important role in determining the loading efficiencies. We will review the models in literature and compare to our own model, and discuss its implications in Section 4.2.2 and 4.4.3.

4.2 Numerical Methods

4.2.1 Simulation Set-ups

The simulations are performed using the Eulerian hydrodynamical code Enzo [Bryan et al., 2014]. We set up a rectangular box with z-dimension of 5 kpc ( 2.5 < z < 2.5 kpc). − The midplane of the disk is located at z = 0. The horizontal cross section, i.e., x-y plane, is a square. The length of the horizontal dimension, lx, varies with Σgas, as listed in Table 1.

The idea is that we adopt higher resolutions for larger Σgas , while keeping the corresponding lx smaller to gain computational efficiency (but sufficiently large to include many SNRs). The grid is refined near the midplane, with two refinement levels. Each refinement increases the resolution by a factor of two. The first level is within z < 1 kpc, and the second is | | z < 0.5 kpc. The finest resolution for each run is so chosen that the cooling radius of a | | SNR Rcool is resolved by approximately 12 cells for the initial midplane density ρmid. (For CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 77

the definition of Rcool, see Eq. 1 of Li et al. 2015.) [Kim and Ostriker, 2015] have shown that resolving Rcool by 10 cells is necessary to well capture the evolution of a SNR in the Sedov-Taylor phase. Once the ISM becomes multiphase, SNe exploding in the dense region could be under-resolved. But as we discuss in Section 4.4.6, this is likely a minor issue. The boundary conditions are periodic for the x- and y-directions, and outflowing for z. We use the finite-volume piece-wise parabolic method [Colella and Woodward, 1984] as the hydro-solver. We use the cooling curve as [Rosen and Bregman, 1995], for the temperature range of 300 109 K. Photoelectric heating (PEH) is time-independent and − uniform across the box. The rate of PEH scales linearly with the star formation surface density Σ˙ SFR; for the solar neighbourhood model Σ10-KS (see below), we adopt a PEH rate of 1.4 10−26erg s−1 per H atom [Draine, 2011]. We explore the variations of the PEH that × deviate from the fiducial settings in Section 4.4.2. We have four fiducial runs: Σ1-KS, Σ10-KS, Σ55-KS, Σ150-KS. The number after Σ 2 indicates the gas surface density in units of M /pc . The SFRs associated with those runs are along the KS relation. Fig 4.1 shows the (Σgas , Σ˙ SFR) adopted in our simulations, indicated by blue triangles. They are plotted on top of Fig 15 of [Bigiel et al., 2008], which shows the observed correlations of Σgas and Σ˙ SFR for nearby galaxies on sub-kpc scales. Table 4.1 summarizes the setups of the simulations. Σ10-KS is the model for the solar neighbourhood. Variations of fiducial runs are described in Section 4.4. The gravitational field, initial gas distribution and the model of SN feedback are detailed in the next two sub-sections. For the fiducial run Σ10-KS, we have carried out a resolution convergence check, where we lower the spatial resolution by a factor of 2. The results agree very well, including the ISM properties, volume fraction of difference gas phases, and outflow fluxes.

4.2.2 Gravitational ﬁelds

The gravitational field (“g-field” hereafter) has two components: a baryonic disk and a dark matter (DM) halo. The disk is modeled as self-gravitating with an iso-thermal velocity dispersion, so its g-field has the form g = 2πGΣ∗ tanh(z/z∗), where z∗ is the scale height 2 of the stellar disk: z∗ σ /(πGΣ∗), in which σ∗ and Σ∗ are the velocity dispersion and ≡ ∗ the surface density of stars, respectively. The height z∗ = 300 pc is observed for the solar CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 78

Figure 4.1: Combinations of Σgas and Σ˙ SFR adopted in our simulations (blue triangles), plotted on top of Fig. 15 of Bigiel et al. 2008, which shows the observed correlations for nearby galaxies at sub-kpc scale. CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 79 neighbourhood [Gilmore and Reid, 1983; Binney and Tremaine, 2008]; we keep it unchanged for all our runs. Since we do not include self-gravity of the gas, we multiply the stellar gravitational field by a factor of 1/f∗, where f∗ = Σ∗/(Σ∗ + Σgas). The g-field from the baryonic disk is 1 z gdisk(z) = 2πGΣ∗ tanh( ), (4.1) f∗ z∗ The g-field from the DM halo is modeled as an NFW profile projected to the z-direction,

GMDM(r) z gDM(z) = , (4.2) r3

3 where MDM(r) is the enclosed mass of a NFW halo within radius r, so MDM(r) = 4πρDMR ln(1+ s { 2 r/Rs) r/(r + Rs) , Rs = Rvir/c, ρDM = 200ρ ¯DM c(1 + c) ;ρ ¯DM is the mean cosmic DM − } density at redshift 0, and c is the concentration parameter. For the MW case, we take

Rvir = 200 kpc, and c = 12 [Navarro et al., 1997]. Note that r and z are related through 2 2 2 r = z + Rd, in which Rd is the distance from the location of the ISM patch we simulate, to the galactic center. The total gravitational ﬁeld is therefore

gtot = gdisk + gDM. (4.3)

Table 1 lists Σ∗ for all of our runs. We keep Rd = 8 kpc for all simulations, except for

Σ10-KS-4g which has Rd = 3 kpc (see Section 4.4.3 for details). In Table 1 we also include v∆φ (see table footnote for its definition) as an indicator for the total potential well for each simulation box. Note that in literature the adopted g-field can vary by a factor of a few even when the same “solar neighbourhood” is claimed. In Fig 4.2 we compare our value to a few others. [Walch et al., 2015] do not include the DM halo potential, so they have a smaller g; at z = 5 kpc, their g-field is about 1/3 of our value. [Joung et al., 2009] uses the observed g-field in the solar neighbourhood from [Kuijken and Gilmore, 1989], and extrapolates it into the halo. This works for z < 1-2 kpc, but above that a simple extrapolation is likely too ∼ large. [Hill et al., 2012] adopted a g-field that is similar to ours. ∆φ for each curve shows

zmax the gravitational potential ∆φ(zmax) = z=0 gtotdz. The numerical values of ∆φ are not negligible compared to the kinetic energiesR of the outﬂows, which are typically 100-500 km/s (see Section 4.3.2 below). Consequently, the gravitational ﬁeld is dynamically important. CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 80

Figure 4.2: Comparison of g-ﬁelds adopted in literature and this work for the solar neighbourhood. The solid lines end at zmax of each simulation box, which are 2.5kpc, 5kpc and 10kpc for this work, Walch et al. 2015 and Joung et al. 2009, respectively. ∆φ for each

zmax curve shows the gravitational potential ∆φ(zmax) = z=0 gtotdz. See Section 4.2.2 for details. R CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 81

Indeed, the value of g-field turns out to be important for the loading efficiencies of the outflows (Section 4.3.3). Any meaningful comparisons between different works should take into account the difference in g-fields. 4 Initially the gas has a uniform temperature T0 = 10 K. We set up the gas initial density to be in hydro-static equilibrium in the g-field gdisk(z), i.e.,

z 2α ρ = ρmid sech( ) , (4.4) { z∗ }

2 2 where α = γσ∗/(f∗cs,0), γ = 5/3 is the adiabatic index of the gas and cs,0 is the sound speed for T0. The power-law decay at large z can result in very low density, so we set up a density floor of 3 10−28 g cm−3. Due to the density floor and an extra gravitational field × from the DM, the gas is not in perfect hydrostatic equilibrium, but in practice this has little consequence, since the outflowing gas will soon dominate the space above the mid-plane.

4.2.3 SN Feedback models

The SN surface density is related to Σ˙ SFR by assuming one SN explosion per m0 =150

M star formation. There are some uncertainties associated with m0; diﬀerent works have adopted m0 = 100-200 M . The distribution of SNe over time is uniform. SNe are randomly located horizontally; in the z-direction, the distribution is stratiﬁed. We distinguish two components of SNe: Type Ia and core collapse SNe. Type Ia constitutes 10% of SNe occurrence and core collapse the rest. Type Ia SNe have an exponential distribution in z-direction, with a scale height of 325 pc, similar to the old stellar disk [Freeman, 1987].

Core collapse SNe have a Gaussian distribution vertically with a scale height hSN,cc = 150 pc. We note that due to runaway OB stars, core collapse SNe may explode outside of the dense gas layer. We test the sensitivity of the results on hSN,cc, as described in Section 4.4.1. 51 Each SN is implemented as injecting ESN = 10 erg energy, mSN = 10 M mass, and m0,met metals (metals are modeled as “color tracers” that passively follow the mass ﬂux, in arbitrary units), evenly distributed in a sphere. The energy added is 100% thermal. The injection radius Rinj varies for Σgas , and chosen to be 0.45-0.50 of the cooling radius for the initial midplane density ρmid. [Kim and Ostriker, 2015] argued that Rinj/Rcool < 1/3 CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 82 is the robust criterion to capture the evolution of a SNR in the Sedov-Taylor phase. Our choice is slightly larger than that.

4.3 Results of ﬁducial runs

Impacted by SN explosions, the stratified medium quickly becomes multiphase, in which the hot gas occupies a significant fraction of the volume while most mass is in cooler clouds. Cool gas settles down to near the midplane, while outflows are launched. In this section, we first examine the multiphase structure of the ISM, with the emphasis of the comparison between our solar neighbourhood run with the observations. Then we discuss the velocities of the outflows, and show that the hot phase has the strongest potential to travel to large radii in the DM halo and impact the CGM. Finally we show the mass, energy and metal loading factors as a function of Σgas .

4.3.1 Multiphase ISM and outﬂows

Fig 4.3 shows the slices of the four fiducial runs in the x-z plane. Note that the physical scales of the slices are different from each other. The horizontal lengths are same as the simulation boxes; the vertical dimensions are shown partially. The actual dimensions each slice represents are indicated at the bottom of the temperature slices. Most of the dense gas stays near the midplane. The medium has multiple phases – a cold phase at a few hundred K, a warm phase at around 104 K and a hot phase at T > 106 K. At ∼ the boundary between the hot and warm/cold phase, gas with intermediate temperature, 105−6 K, is also seen. For all four runs, the hot gas volume fraction is about 60-80% for the midplane. Hot gas occupies more volume in the halo for higher Σgas . Multiphase outflows are being launched from the midplane for all four runs. Cool clouds in the halo are clearly being stripped by the hot, faster gas. The hot phase appears hotter for higher Σgas . These qualitative results agree with previous works [McKee and Ostriker, 1977; Joung et al., 2009; Creasey et al., 2013]. Fig 4.4 shows the phase diagram for the run Σ10-KS at t = 100 Myr. The color coding indicates the fractional mass in each (density, pressure) bin. The three phases, CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 83 e sim t (Myr) inj R (pc) x l (pc) d ...... 240 .... 130 ...... 64 ...... 2.0 350 12.0 160 12.5 1200 80.0 300 0.75 150 4.5 40 0.60 122 3.0 15 (pc) Res c 4 min (K) T b φ ∆ v (km/s) cc , SN (pc) h ) ∗

Σ (M PEH (erg/s) yr) / Table 4.1: Model description a 2 kpc SFR / ˙ Σ

) 3 − mid ρ (cm dz = 150 M 0 tot ) g 2 m pc max / =0 1 0.011 1.26E-4 2.8E-28 0.5 150 52 300 ...... 3.78E-4.. 1.78 1.26E-3...... 180 ...... 178 0 .. 1.75E-24 150 ...... 300 0 .. .. 450 .. .. 150 ...... 10 z gas 10 0.82255 6.31E-3 8.2 1.4E-26 0.158 35 .. 3.5E-25 35 89 75 .. 117 .. z 150 26.3 0.708 1.6E-24 .. .. 160 300

R Σ , where (M 0 m ) = 2 500 pc / z < max SFR z ( ˙ Σ φ 2∆ Run ≡ Σ1-KS Σ1-3KS Σ10-KS Σ150-KS Σ1-10KS φ Σ10-KS-4g Σ55-KS-h75 2 ∆ Σ55-KS-h300 Σ55-KS-h450 Σ55-KS-1e4K simulation time temperature cut-oﬀ of the cooling curve resolution at v SN rate is Σ55-KS-5PEH Σ55-KS(-h150) Σ55-KS-noPEH e c b d a CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 84

Figure 4.3: Density and temperature slices of the x-z plane for the four ﬁducial runs. Note that the physical scales of the slices are diﬀerent. The dimensions are shown at the bottom of the temperature slices, in format of “horizontal scale vertical”, in units of kpc. × CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 85

Figure 4.4: Phase diagram of the gas in Σ10-KS (solar neighborhood model). The color coding shows the fractional mass in each (density, pressure) bin. CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 86

Figure 4.5: Gas density (volume-weighted) for diﬀerent phases as a function of z for Σ10-KS at t= 160 Myr.

hot, warm and cold, are clearly seen. Each of the three phases has some spread in the density distribution. But the majority of them are in rough pressure equilibrium, with 3 −3 P/kB 5 10 cm K. This is in good agreement with the observations near the solar ∼ × neighbourhood [Cox, 2005]. Some mass, which lies in between the two diagonal lines that indicate the standard “warm” and “cold” phases, is out of thermal equilibrium [Heiles and Troland, 2003]. Fig 4.5 shows the density, weighted by volume, of diﬀerent phases. Hotter phases have progressively smaller densities. Gas density for each phase near the plane is higher than that in the outﬂows. The warm-hot phase with 104 < T < 3 105 K has a slightly lower × density than the warm phase. As shown in Fig 4.3, the warm-hot phase is mostly at the CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 87 interface between warm clouds and hot gas. Even the coldest phase is seen at large z, even though the volume fraction can be very small. The densities for each phase agree with the observations of the local ISM [Draine, 2011]

4.3.2 Velocity structure

The velocity structure of the gas determines how far the gas can travel in a gravitational potential. Moreover, the velocity distribution can be observed from the profiles of emission/absorption lines. Since different gas phases have drastically different velocities, and observationally, they are detected through different line tracers, we hereby show the velocity distribution for each gas phase, separately. Fig 4.6 shows the z-velocity distribution for the run Σ10-KS at t = 160 Myr. The y-axis indicates the fractional mass in each velocity bin. Each curve is normalized to unity. Hotter phases have larger velocities, agreeing with the general observational trend [Heckman et al., 2001; Rupke et al., 2002], and other simulation works [Creasey et al., 2013; Girichidis et al., 2016b]. The hottest phase has the broadest range of velocities, up to > ∼ 600 km/s. A fraction of the warm phase can reach > 100 km/s. The velocities of cold phase remain small at < 50 km/s. ∼ Our simulations only capture the gas evolution that is relatively close to the midplane, i.e. z < 2.5 kpc. One way to relate the “local” outflows to their large-scale evolution | | is to estimate how far the gas can travel in a given potential. First, let us consider a ballistic evolution. For a parcel of gas with a velocity v0 at the bottom of a potential well, 2 the furthest distance it can reach, R, is simply determined by 1/2 v0 = ∆φ(R). We use function R(v0) to describe such a relation. Fig 4.7 shows R(v0) for the MW, for a single stream line that is perpendicular to the disk and goes through the center of the disk. The potential of the DM halo is the same as described in Section 5.2, and the disk is modeled 5 as a 2D razor-thin disk with a mass MD = 5 10 M and a radius of RD = 9.5 kpc. The × mass distribution within the disk is uniform. Thus, along the stream line mentioned above,

2 2 2 the g-ﬁeld from the disk has a simple analytic form: gD = 2GMD(1 z/ z + R )/R . − D D From Fig 4.7, gas with v0 > 620 km/s can escape from the DM halo; gasq with v0 = 300 ∼ km/s can travel to R 50 kpc, and so on. ∼ CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 88

Figure 4.6: Fractional mass for gas with different z-velocity (absolute value), for the model Σ10-KS at t = 160 Myr. Different curves correspond to gas in different temperature ranges. Each curve is normalized to unity. CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 89

200

150

100 R [kpc]

0 0 100 200 300 400 500 600 700 v0 [km/s]

Figure 4.7: Radius R that a parcel of gas with velocity v0 can reach from the center of the MW. See Section 4.3.2 for model details. CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 90

We now discuss what should be used as v0. The naive answer, the bulk velocity projected to the direction of g, may only give a lower bound. For a compressible fluid, it is likely that the gas motion is not ballistic, but thermal energy can later convert to bulk motions. According to the Bernoulli principle, the Bernoulli constant v2/2 + γ/(γ 1)P/ρ + φ, B ≡ z − remains unchanged along a stream line in a steady-state flow (for a constant γ). We thus define a modified “Bernoulli velocity” v √2 1/2, where φ. So for a parcel of Be ≡ B B ≡ B − gas with a bulk velocity v and a “Bernoulli velocity” v , the approximate range of radii it z e Be e can reach is roughly R(v0 = vz) R(v0 = v ). Note we only aim at a very rough estimate, ∼ Be ignoring cooling, interaction among different gas phases, etc, and assuming γ = 5/3. Fig 4.8 shows the mass-wighted v and v for different phases for the fiducial run Σ10- z Be KS. The y-axis on the right shows R corresponding to the velocities on the left axis. Only gas at z > 1kpc is included. The data are averaged over the last 20% of the simulation | | time. The error bars indicate time variations. The hot gas is affected more by each SN explosion, thus its properties vary stronger with time. Both v and v increase with gas z Be temperature. The hottest phase, given its large vz ( 150 km/s) and v ( 370 km/s), ∼ Be ∼ would travel much further into the halo, to about 30-70 kpc. Since the majority of the hot gas would not escape from the DM halo, large-scale fountain flows would form. The small velocities of the cool phase imply that they would fall back at below 10 kpc, unless being accelerated significantly. The velocity of the warm-hot phase, with T= 104 3 105K, is − × much closer to the warm phase than the hot. The ratio v /v is largest for the hot phase, Be z meaning that a significant fraction of the energy is thermal, which may convert to the bulk motion at large radii. For cooler phases, in contrast, most energy is kinetic. Note that the fiducial run is for the solar neighbourhood, and is not representative of the MW disk in general. We discuss the model for the MW-average, Σ10-KS-4g, in Section 4.4.3. Hot flows are much faster there than the solar neighborhood, and can thus have a much broader impact on the CGM (see Section 4.4.3 for details). Fig 4.9 shows the mass-weighted v and v for the hot outflows, as a function of Σ˙ for z Be SFR the four fiducial runs. Again, the error bars show time variation. Both v and v increase z Be with SFR. The velocities for the Σ1-KS run are 60-200 km/s, and rise to 600-900 km/s for Σ150-KS. The large velocities imply that the hot outflows can travel far, and even escape CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 91

Figure 4.8: Mass-weighted v and v of the outﬂows in diﬀerent temperatures ranges. Y- z Be ticks on the right show R corresponding to the velocities on the left, as in Fig 4.7. See Section 4.3.2 for details. CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 92

For hot phase: T > 3 105 K × ˙ 0.24 v ∝ ΣSFR v ∝ Σ˙ 0.16 103 SFR v B v ze Velocity [km/s]

102

4 3 2 1 0 10− 10− 10− 10− 10 2 Σ˙ SFR[M /kpc /yr]

5 Figure 4.9: Mass-weighted vz and v for hot outflows (T > 3 10 K), as a function of Be × Σ˙ SFR, for the four fiducial runs. The power-law fits of the data are indicated in the box. CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 93 from the halo potential. This suggests that hot outflows play a critical role on regulating the CGM and even the IGM. 0.24 0.16 We find that vz Σ˙ , and v Σ˙ . Observationally, while there is little direct ∝ SFR Be ∝ SFR constraint on velocities for hot gas, for the warm/cool phases, [Martin, 2005] and [Weiner et al., 2009] found v SFR0.3−0.35 for galactic-scale outflows. Our findings seem to indicate ∝ that the dependence of SFR for the hot gas velocity is weaker than for the cooler phases.

4.3.3 Loading factors

In this section, we discuss the loading of the outflows. We define outflows to be at z > 1 | | kpc and with outgoing z-velocity. We find in our simulations, the outflow fluxes show little variation with z at z > 1 kpc. | | The mass loading factor ηm is defined as the ratio between the outflowing mass flux and

Σ˙ SFR, that is, < ρvz > ηm . (4.5) ≡ Σ˙ SFR The outﬂow ﬂux includes both sides of the plane, and “<...>” denotes averaging over space

(1 < z < 2.5 kpc) and time (last 40% of tsim). | | The energy loading factor ηE is the ratio between the z-component of the energy ﬂux and the energy production rate by SNe, that is,

< (ek + eth)vz > ηE , (4.6) ≡ Σ˙ SFRESN/m0 where ek and eth are the kinetic and thermal energy per unit volume.

The metal loading factor ηmet is the ratio of the z-component of the metal ﬂux to the metal production rate by SNe.

< ρmetvz > ηmet , (4.7) ≡ Σ˙ SFR m0,met/m0 where ρmet is the density of metals, and m0,met is the mass of metals each SN produces (in arbitrary units, see Section 4.2.3 ). Note that we assume the metals are solely produced by SNe, and the ISM is otherwise pristine. Fig 4.10 summarizes the loading factors and the volume fraction of each gas phase in the outﬂows as a function of Σgas, for the four ﬁducial runs. For the loading factors we CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 94

101 Mass 0.5 Energy Total Hot 0.4

0.3

100

0.2 Loading Factor Loading Factor

0.1

10 1 0.0 − 1 10 55 150 1 10 55 150 2 2 Σgas[M /pc ] Σgas[M /pc ]

1.2 Metal 1.2 Warm 1.0 1.0 hot

0.8 0.8

0.6 0.6 Loading Factor 0.4 Volume fraction 0.4

0.2 0.2

0.0 0.0 1 10 55 150 1 10 55 150 2 2 Σgas[M /pc ] Σgas[M /pc ]

Figure 4.10: Loading factors and volume fraction of each gas phase as a function of Σgas. The quantities are calculated for outﬂowing gas at z >1 kpc. “Hot” and “warm” here | | denote T > 3 105 K, and 104 < T < 3 105 K, respectively. See Section 4.3.3 for details. × × CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 95 show the total loading, which includes all gas phases, as well as that of the hot outﬂows only. The error bars indicate the standard deviation of the time variation.

We ﬁnd that ηm decreases monotonically with increasing Σgas. The largest mass loading is about 6 for Σ1-KS. For the solar neighborhood case, i.e. Σ10-KS, our ηm is around 2-3.

For our highest density case Σ150-KS, ηm is only 0.2. The fraction of the mass loading contributed from the hot gas is about 1/3, except for Σ150-KS, where most of the mass flux is hot. The warm phase dominates the outflowing mass flux except when Σ˙ SFRis very high.

We ﬁt our mass loading factor by a simple power-law function of Σgas :

Σgas αml ηm = 7.4( 2 ) , αml = 0.61 0.03. (4.8) 1 M / pc − ±

For the hot gas, the mass loading factor

Σgas αml,h ηm,h = 2.1( 2 ) , αml,h = 0.61 0.03. (4.9) 1 M / pc − ±

[Creasey et al., 2013] have found a sharper decline, with α -1.1. Our results agree ≈ 2 with theirs for Σgas < 10 M / pc , but there are relatively large discrepancies at higher ∼ densities. See Section 4.4.5 for a discussion. The X-ray emission from the halo of edge-on galaxies suggests a decreasing mass loading of hot gas for higher SFR [Zhang et al., 2014; Bustard et al., 2016], consistent with our results.

The energy loading factor ηE shows surprisingly little dependence of Σgas . Despite a factor of 150 span of Σgas , ηE stays at about 10-30%. This means a signiﬁcant fraction of

SNe energy goes into the outﬂows. There is no obvious trend of ηE as a function of Σgas . The hot gas contains the majority, >90%, of the outﬂow energy.

The metal loading factor ηmet shows somewhat larger variation than ηE , although again we do not find an apparent dependence on Σgas . Overall, a quite large fraction of metals go into the outflows, about 40-90%. Hot outflows carry 35-60% of the metals produced by SNe. While the warm/cool phase may fall back to the disk later, the hot gas has the potential to travel much further, even escape the halo (see Section 4.3.2), and metals will be carried along. The mass-metallicity relation of galaxies implies that a significant fraction of metals ever produced are no longer in galaxies [Tremonti et al., 2004; Erb et al., 2006]. Our numbers agree with this general picture. CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 96

The volume in outﬂows is progressively occupied by the hot gas as Σgas becomes larger. The cold phase with T < 103 K has a negligible volume fraction, thus we omit it in the plot. For Σ1-KS, the volume is equally shared by warm and hot phase; for Σ55-KS, more than 90% of volume is hot; for Σ150-KS, the outﬂows are completely dominated by the hot phase.

4.4 Eﬀects of several physical processes

4.4.1 SNe scale height

Where SNe explode is critical for feedback efficiency. A SN exploding in a dense medium quickly radiates away its energy, and has little impact on the large-scale ISM, let alone contributing to driving winds [Girichidis et al., 2016b]. On the other hand, if a SN explodes in an environment dominated by tenuous gas, then the cooling is much less efficient, and a significant fraction of energy can be preserved [Li et al., 2015; Gatto et al., 2015; Simpson et al., 2014; Walch et al., 2015; Hennebelle and Iffrig, 2014]. One key factor to determine where SNe explode is the fact that a significant fraction of OB stars are “runaways”, that is, having high velocities. A simple calculation shows that OB runaways can migrate a few tens to a few hundred pc before exploding as SNe [Li et al., 2015]. This greatly facilitates SN feedback by allowing some of them to release their energy outside the dense SF regions. In principle, the locations of core collapse SNe depend on the velocities of OB stars, their lifetimes, the external gravitational field, close encounters with other stars, etc. One can also infer the SNe explosion sites from the spatial distribution and the velocities of pulsars [Narayan and Ostriker, 1990]. In this paper, we do not aim to model the location of SNe from first principles, but simply explore how sensitively the outflow properties depend on the vertical distribution of SNe. For the fiducial runs we have the hSN,cc =150 pc. Now we experiment with hSN,cc. We take the run of Σ55-KS and change hSN,ccto 75, 300, 450 pc, respectively. In Table 1, they are identified by names of Σ55-KS-h75, Σ55-KS-h300, and so on.

Fig 4.11 shows the mass, energy and metal loading factors for diﬀerent hSN,cc. Inter- estingly, ηm depends on hSN,cc in a diﬀerent way from ηE and ηmet . As hSN,cc increases, CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 97

2 Σgas = 55M /pc

Mass Energy 0 10 Metal

1 10− Loading Factor

2 10−

75 150 300 450 hSN [pc]

Figure 4.11: Loading factors for diﬀerent SN scale heights hSN,cc for the model Σ55-KS. See Section 4.4.1 for details. CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 98

ηm first increases, and reaches the peak at hSN,cc= 150 pc, and then declines. Loading factors ηE and ηmet increase monotonically with hSN,cc, and reaches plateau at hSN,cc > 150- ∼ 300 pc. For hSN,cc=75 pc, most SNe are buried in the mid-plane, and radiate their energy there, so the feedback is least efficient. As hSN,cc becomes larger, more SNe explode in the low density disk-halo interface, resulting in a more effective energy and metal loading. The non-monotonic dependence on hSN,cc of ηm can be understood in this way: when hSN,cc is too small, most energy radiates away in the disk, so the energy insufficiency is the limiting factor for the mass loading; when hSN,cc is too large, the outflowing mass is simply SNe ejecta, with little ISM involved. Consequently, the maximum ηm occurs in between those two extremes. We find that ηm achieves unity when hSN,cc=150 pc, while ηm < 0.2 for ∼ other cases. For hSN,cc > 300 pc, 40-50% of energy and 80% of metals produced by SNe ∼ end up in the outflows.

4.4.2 Photoelectric heating

In the absence of SN explosions, PEH maintains a two-phase warm/cold ISM [Draine, 1978;

Wolfire et al., 1995]. The value of the PEH rate ΓPEH determines the relative amount of mass in the two phases and the pressure of the ISM [Wolfire et al., 2003]. With SNe, ΓPEH is an important factor in determining whether the ISM is in a thermal runaway state or not [Li et al., 2015]. A higher ΓPEH keeps more gas in the warm phase and increases the ISM pressure, thus limiting the size of the hot bubbles of SNRs. As a result, SNRs may not effectively overlap, and each SNR loses the majority of its energy at the cooling stage.

Therefore little energy is left to drive an outﬂow. We thus expect that ΓPEH is important in determining the outﬂow properties.

In this section we study the eﬀect of diﬀerent values of ΓPEH. We note that ΓPEH depends on many factors: the far-UV background, dust abundance, work function of the dust grains, ionization fraction of the gas, etc [Draine, 2011]. A star forming region has a very complex structure with strong and time-varying radiation background with both ionizing and non- ionizing photons. Radiation background also varies in space, and is much more intense around OB stars. The exact condition is thus hard to determine. For simplicity, we keep

ΓPEH constant in time and uniform in space for each simulation, but just change ΓPEH to CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 99 explore its effect. Note that some previous works adopt a cooling curve with a cut-off at 104 K, which prohibits the formation of the cold phase. This is similar to the effect of a very high ΓPEH. We include a discussion of the cooling curve cut-off as well. 2 We compare four runs that have Σgas = 55 M / pc . The set-ups are identical (including the SN rate) except ΓPEH:

(a) ΓPEH = 0 (“Σ55-KS-noPEH”); −25 (b) ΓPEH = 3.5 10 erg/s (ﬁducial, “Σ55-KS”); × −24 (c) ΓPEH = 1.75 10 erg/s (“Σ55-KS-5PEH”); × 4 (d) cooling curve has a cut-oﬀ at Tmin = 10 K (“Σ55-KS-1e4K”).

4 In Fig 4.12 we show the slices for the fiducial run and Σ55-KS-1e4K. Adopting Tmin = 10 K results in a much larger scale height of gas (defined as enclosing 80% of the mass in the box), hgas 200 pc, in contrast to hgas 10 pc for the fiducial case. Assuming hydrostatic ∼ ∼ equilibrium, i.e., gravity is balanced by the thermal and turbulence pressure, 2 −9 2 1 + Tgas 5 10 cm/s hgas 140pc ( M )( )( × ), (4.10) ∼ 2 104K g where is the local Mach number of the gas, which is on the order of unity. Note that in M our simulations, SNe have a Gaussian distribution with hSN,cc=150 pc. So for the fiducial case, once the multiphase medium is formed, most SNe explode outside of the gas layer, whereas for Σ55-KS-1e4K, most SNe explode within the gas layer. For the latter, since the ISM is not in a thermal runaway state, most energy released from SNe is radiated 4 away. Therefore, a cooling curve with Tmin = 10 K gives much smaller energy, mass and metal loading. It is true that we are adopting a temperature cut of 300 K, and the actual temperature of the cold phase can be even lower. But our temperature cut is low enough to allow the ISM at midplane to undergo a thermal runaway – as mentioned in Section 4.3.1, all fiducial runs have a volume fraction of hot gas of 60-80% at midplane. We thus believe that our results do not suffer from a qualitatively erroneous cooling loss, while a temperature cut at 104 K may do so. Fig 4.13 compares the loading factors of all four runs in this section. The simulation Σ55-

KS-1e4K gives an energy loading two orders of magnitude smaller than the ﬁducial run; ηm is smaller by a factor of 10, and ηmet by a factor of 30. This indicates that if the formation of the CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 100

2 Figure 4.12: Temperature and density slices of two runs with Σgas =55 M / pc : left - 4 fiducial (Σ55-KS); right - cooling curve has a cut-off at Tmin = 10 K (Σ55-KS-1e4K). The snapshots are taken at t = 41 Myr. The slices only include region at z < 325 pc. The | | white dashed lines indicate the scale height of core collapse SNe hSN,cc =150 pc. Cut-off of the cooling curve at 104 K results in a much larger gas scale height. As a result, most SNe energy is lost through radiative cooling in the dense gas layer. CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 101

2 Σgas = 55M /pc

100

1 10− Loading Factor

2 10−

Mass Energy Metal 10 3 − 0 ﬁducial 5 cooling cut-off ﬁducial× at T=1e4K PEH rate

2 Figure 4.13: Loading factors for different PEH rates, for the model with Σgas = 55 M /pc . See Section 4.4.2 for details. CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 102 cold phase is prohibited, the power of SN feedback is severely underestimated. Comparing the three runs with different ΓPEH, we find that when ΓPEH is higher, the energy loading is smaller, as expected, since hgas is increasingly larger for stronger PEH. The mass and metal loadings do not show significant variation. This is likely due to the following two effects counteracting each other: a smaller ηE means that less energy is available to drive the mass out, while a larger hgas is favourable to loading more gas, as discussed in Section 4.4.1.

4.4.3 External gravitational ﬁeld

To explore the effect of external gravitational field on the outflows, we take the MW for an example. The fiducial run Σ10-KS uses the gravitational field in the solar neighbourhood, 2 2 which has Σgas = 10 M / pc ,Σ∗ = 35 M / pc , and a displacement Rd = 8 kpc from the center of DM halo. This g-field is smaller than the inner part of the MW disk. We set up a higher gravity run Σ10-KS-4g, which uses a g-field more typical for the inner MW disk, 2 2 with Σgas = 10 M / pc ,Σ∗ = 180 M / pc and Rd = 3 kpc. The g-field is approximately 4 times that of the solar neighbourhood. Fig 4.14 shows the different components of the two g-fields. Naively, one would think a larger gravity would make the feedback less effective, as gravity drags the outflows toward the disk. While this is generally true for regions away from the launching region, the situation near the disk is more complex. We plot the ratios of the loading factors between Σ10-KS-4g and Σ10-KS in Fig 4.15. The comparison is done for the time interval t = 40 64 Myr, and the error bar shows the standard deviation of − time variation. While ηm is indeed smaller by a factor of 3-4 in the higher gravity case, the energy loading ηE is, nevertheless, a factor of 3-5 larger. The metal loading ηmet is also larger by a factor of 1.5. How to understand this? It turns out that the dominant impact of larger g-field here is to reduce hgas. As we also show in Fig 4.15, a factor of 4 increase in gravity results in roughly the same factor of decrease in hgas. This is expected for gas in hydrostatic equilibrium (Eq. 4.10). Since we keep hSN,cc the same for the two runs, a smaller hgas exposes more SNe in the low-density halo. As discussed in Section 4.4.1, more SNe above the gas layer can lead to a smaller ηm while larger ηE and ηmet . Since less mass is heated CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 103

10 8 2.5 × −

2.0

1. disk,45M /pc2

] 2 2 1.5 2. disk,180M /pc

3. DM,Rd = 3 kpc cm/s [ 4. DM,Rd =8 kpc

g-ﬁeld 1.0 Near Sun: 1+4 MW-average: 2+3

0.5

0.0 0.5 1.0 1.5 2.0 2.5 z [kpc]

Figure 4.14: External gravitational ﬁelds adopted for the solar neighbourhood (Σ10-KS) and the MW-average (Σ10-KS-4g). Diﬀerent components are shown separately. See Section 4.4.3 for relevant discussions. CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 104

101 g

A 0 / 10 g 4 A

1 10− v hgas vz ηm ηE ηmet B A e

Figure 4.15: Ratio of properties between two runs with different gravitational fields: Σ10- KS-4g (MW-average) and Σ10-KS (solar neighbourhood). See Section 4.4.3 for the discussion. CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 105 by more energy, v of the outflowing gas is much larger in Σ10-KS-4g, by a factor of 3 Be than the solar neighbourhood. The values of v and v are about 175 km/s and 980 km/s, z Be respectively, the latter is even larger than the escape velocity of the MW halo 620 km/s. ∼ Thus the outflows for the MW-average is much more vigorous, which can broadly impact the CGM and even the IGM (see more discussion in Section 4.5.2).

4.4.4 Enhanced SNe rates

For our fiducial runs, we assume that Σ˙ SFR scales with Σgas as in the KS relation. Although the KS relation is well-established on scales > kpc, variation appears on smaller scales ∼ [Heiderman et al., 2010]. In particular, star formation tends to occur in groups, and the OB stars are clustered in space and time. The size of our simulation boxes are in the sub-kpc regime, so it will be interesting and relevant to discuss the variation on the SN rate. In this section, we discuss the effect of enhanced SN rates on the outflows. We are interested in how the energy, mass, and metal loading efficiencies depend on the SN rates. Since the interaction of blast waves is highly non-linear, it is non-trivial to predict whether the impact of multiple SNRs would be a simple add-up, or to reinforce, or to cancel out each other. We take the run Σ1-KS, and increase the SN rate by 3 and 10 , respectively. These × × runs are listed in Table 1 as “Σ1-3KS” and “Σ1-10KS”. Fig 4.16 shows the mass, energy and metal fluxes of the outflows relative to the fiducial run. The mass flux scales with the SN rate in a sub-linear manner. A factor of 3 and 10 increase in the SN rate only results in, on average, a factor of 1.5 and 3 enhancement in the mass flux, respectively. The energy and metal fluxes, on the other hand, show a roughly linear correlation with the SN rate. This means that the mass loading is less efficient when we increase the SN rate, while the energy and metal loading factors remain roughly constant for different SN rates. We caution that, even for the fiducial run, which has the lowest SN rate, most of midplane is in a hot-dominated multiphase state. The sub-linear dependence of the mass flux, and roughly linear dependence of the energy and metal flux are likely to be the feature in this regime. If, for example, one starts with a SN rate sufficiently small so that the SNRs in the disk would not overlap, then the enhancement of the SN rate would lead to a transition from a steady-state ISM to forming outflows. As a result, the dependence of CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 106

Σ1 linear mass 101 energy metal Flux relative to ﬁducial run

100 ﬁducial 3 10 Relative× SN rate ×

Figure 4.16: Relative fluxes of mass, energy and metal as a function of SN rate for Σgas = 2 1 M / pc . The fiducial run follows the KS relation, while the “enhanced-rate” runs have SN rates increased by 3 and 10 times, respectively. The dashed black line indicates a linear relation.See Section 4.4.4 for discussions. CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 107 all fluxes on the SN rate would be super-linear. But since we are, in this paper, interested in the regime where outflows are generated, we do not explicitly explore the parameter space that leads to a steady-state ISM. Indeed, for all the four fiducial runs, which are along the KS relation, the ISM at mid-plane is in a thermal runaway state and outflows are being launched. Therefore, the scaling relations as shown in Fig 4.16 should hold for other

Σgas cases as well. [Girichidis et al., 2016b] find that clustering of some SNe does not affect the mass outflow rate. We find a very mild increase in mass flux, although with large fluctuations. Within error bars our results are consistent with each other. Overall, once the ISM is hot-dominated, clustering of SNe does not help with the loading factors, and may even be negative for loading mass.

4.4.5 Comparison with other works

[Girichidis et al., 2016b] have found that for the solar neighbourhood, SNe can blow away most of the gas in the midplane, and drive outflows with a mass loading up to 10. This is higher than our value, which is around 2-3. Compared to our model, their SN scale height is smaller, 50 pc, which causes an “explosive” thermal-runaway at the midplane. Initially there is no leak of those hot gas, whose high-pressure propels the neutral gas layer up, like the formation of a super-bubble. Later on, the Rayleigh-Taylor instability will develop, the shell will fragment, and hot gas leaks to form winds [Mac Low and Ferrara, 1998]. For our case, hSN,cc is larger, meaning that SNe are more spread-out in z-direction. We do see warm shells of gas being driven out initially, but that does not involve too much mass, and would later either go beyond the box, or fragment and fall back. Our loading factors are calculated after those initial transient stage. Additionally, a weaker g-field (see Section 4.2.2) may also partially account for their relatively large mass loading. [Creasey et al., 2013] study SNe-driven outflows covering a broad parameter space of

Σgas and g-field. The g-field can be expressed using the gas fraction, fg Σgas/(Σgas + Σ∗). ≡ We here conduct on a one-on-one comparison between our fiducial runs and their models. We note that their boxes are smaller in the vertical direction, z < 0.5 kpc, and their outflow | | fluxes are measured at the outer boundaries; whereas ours are averaged over 1 < z < 2.5 | | CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 108 kpc. Our fiducial runs that overlap with their models are Σ10-KS, Σ55-KS, and Σ150-

KS, which corresponds to (Σgas , fg) = (10, 0.15), (55, 0.5), (150, 0.7), respectively. We 2 convert our g-field from the DM halo to an equivalent surface density of 25 M / pc . We ≈ interpolate their data if there is no direct comparison. For the solar neighbourhood, their mass loading factor is about unity, and energy loading (“thermalization factor” in their terminology) is around 0.1. These are slightly smaller, by a factor of 1.5-2, compared to our simulation. For the higher Σgas cases, however, the discrepancies are larger. For Σ55-KS and Σ150-KS, our ηm are 0.9 0.3 and 0.2 0.05, ± ± whereas theirs are 0.2 and 0.03, smaller by a factor of 4-7 than our values; for ηE , our values are around 0.2 for both cases, whereas theirs are around 0.05, smaller by a factor of 4. In a follow-up paper, [Creasey et al., 2015] measure the metal loading efficiency of the outflows for some runs in [Creasey et al., 2013]. Our model parameters only overlap with theirs for the solar neighbourhood, in which we have ηmet =0.65 0.2 and they have ± a smaller 0.2 . Note also that we both assume a KS relation to relate Σgas and Σ˙ SFR, ≈ but we convert the SFR to SN rate by assuming m0 = 150 M (definition of m0 in Section

4.2.3), whereas they have m0 = 100 M . This means the diﬀerence is even larger by a factor of 1.5. We attribute the discrepancies mainly to their adoption of a cooling cut-oﬀ at 104 K. As a result, the neutral gas layer in their runs is thicker, and most SNe lose their energy there, thus the loading factors are smaller (see Section 4.4.2 for detailed discussions).

4.4.6 A brief summary and some missing physics

Under the impact of many SN blast waves, the ISM becomes multiphase. The cooler, denser phase settle down near the midplane, whereas the hotter phases escape and form outflows. There are two regimes of the media: (i) a warm/cool-dominated ISM, where, if a SN explodes within, it would lose most energy by radiative cooling, and (ii) a hot-dominated ISM, where SN shock waves would propagate much faster and further, while the cooling is inefficient [Cowie et al., 1981; Li et al., 2015]. The fraction of SNe that explode in a hot-dominated ISM, fSN,h , is key in determining the efficiency of the loading efficiency of energy and metals. A stronger external g-field or a weaker PEH leads to a smaller hgas, leaving more SNe exploding in low-density medium, thus a more powerful loading of energy CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 109

and metals; a larger hSN,cc has a similar eﬀect. Simply enhancing the SN rate, without changing hSN,cc or hgas, yields unchanged energy and metal loading. The mass loading factor, on the other hand, has a more complex dependence on fSN,h : a fSN,h that is either too small or too large would result in a small mass loading factor. We brieﬂy discuss the possible impact of the physics that we do not include in this work.

Under-resolved SNe: The resolution and Rinj for each run are fixed, and are chosen based on Rcool for the initial ρmid. Later, when the ISM becomes multiphase, SNe exploding in the tenuous/hot phase continue to be well resolved. For all fiducial runs, the hot gas volume fraction at mid-plane is 60-80%. Since SNe are randomly located, a similar fraction of SNe would explode in the hot phase. The rest SNe which explode in the denser phase are likely under-resolved, but we argue that these SNe are unimportant to drive large-scale outflows. Take Σ10-KS for an example, the cold phase has a density of about 10 cm−3, corresponding to a Rcool 7 pc. So even when the evolution of the SNR is resolved spatially ∼ and temporally, it will lose the majority of its energy at Rcool. This means that these SNe only have a very localized impact, and contribute little to large-scale outflows. Magnetic fields: In the solar neighbourhood, the magnetic pressure is overall similar to the thermal pressure of gas, and even larger for dense phases [Heiles and Crutcher, 2005].

This extra pressure, if included, would be likely to enhance hgas, and also make SN bubbles smaller [Slavin and Cox, 1993]. The net effect on the loading factors would be similar to having a stronger PEH – the energy and metal loadings would be smaller, while the change of the mass loading is not certain. The magnetic field on the dynamics of diffuse ISM is rather mild for the solar neighbourhood [de Avillez and Breitschwerdt, 2005; Hill et al., 2012; Walch et al., 2015], except possibly for providing support for the vertical distribution of the gas [Cox, 2005]. Magnetic fields alone are unlikely to play an active role in driving the outflows. Self-gravity: We did not have self-gravity in our simulations. For the solar neighbourhood, only about 1% of the volume is in a self-gravitating state [Draine, 2011]. For larger gas surface density cases, however, the self-gravity is probably more important. Including self-gravity would make the cold phase smaller in volume, thus may facilitate feedback for those SNe that explode in the inter-cloud space [Girichidis et al., 2016b; CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 110

Kim and Ostriker, 2016]. But we note that (i) the external gravitational field in our simulations does include that from the gas, so the effect of “self-gravity” is not fully missed, at least in the vertical direction. (ii) As mentioned in Section 4.3.1, essentially in all runs, the volume fraction of hot gas at the midplane is 60-80%, thus we believe shrinking the volume fraction of cold gas by some percentage would not lead to a qualitative change in the loading factors of outflows. (iii) The self-gravity should be included with caution, that is, one should also resolve the counterbalancing force – feedback acting below the Jeans scale of the dense clumps, which is challenging given the resolution of the current ISM simulations; otherwise, most gas would collapse into a few small clumps, which is also not realistic. Cosmic rays (CRs): SNe are considered to be the main acceleration sites for CRs. Around 5%-15% of SNe energy may go into CRs [Hillas, 2005]. Recent simulations show CRs are promising candidates to drive galactic scale outflows, with a mass loading around 0.5 [Uhlig et al., 2012; Hanasz et al., 2013; Booth et al., 2013; Salem and Bryan, 2014]. Re- cent high-resolution, local simulations indicate that for the solar neighbourhood, both CRs and SN thermal feedback can drive an outflow with mass loading around unity [Girichidis et al., 2016a; Simpson et al., 2016]. CR-driven outflows are cooler, slower, and smoother than the thermally-driven ones. [Peters et al., 2015] point out that compared to the thermal feedback, CR-driven outflows give too little hot gas to match the soft X-ray background. Many questions remain open regarding how CRs propagate in, and interact with, a multiphase ISM. Further work is needed to determine whether thermal feedback or CRs are the dominant force launching outflows. Radiation pressure: Radiation pressure can propel gas out if gas is optically thick, especially for the infrared light. The photon energy from the stars can then be effectively utilized to drive outflows [Murray et al., 2005; Hopkins et al., 2012]. This can be the case for the extremely dense and dusty SF regions, such as in the ultra-luminous infrared galaxies

(ULIRGs). We do not actually explore those extreme cases (Fig 4.1). For the Σgas adopted in our models, the optical depth for the infrared is much less than unity, therefore the radiation pressure is not important. We also do not include the preprocessing of the ISM by other stellar feedbacks, such as stellar winds, ionizing photons, etc. In general these feedbacks alone do not contribute di- CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 111 rectly to launching outﬂows (unless the radiation is highly trapped), but they may make the ISM inhomogeneous. Therefore it is possible for SN to explode in a less dense environment, thus facilitating SN feedback to some extent [Gatto et al., 2016; Peters et al., 2017].

4.5 Discussion

4.5.1 Loading factors: simulations vs observations

−0.61 For the mass loading factor ηm, we have found in Section 4.3 that ηm Σ , for both total ∝ gas mass flux and the hot mass flux. Here we will compare the hot gas mass loading factor with observations, and discuss the cooler phases (which is a problem) in Section 4.5.3. Adopting ˙ 1.4 the Kennicutt relation that we assumed for our fiducial simulations ΣSFRΣgas, we then have −0.44 ηm Σ˙ . Recently, [Wang et al., 2016] compiled the X-ray data on the galactic coronae ∝ SFR −0.44 around roughly 50 highly inclined disks, and found that LX /SF R Σ˙ , which, together ∝ SFR with the global wind model developed by [Chevalier and Clegg, 1985], implies that the mass −0.48 loading factor of the hot outflows declines with increasing SF surface density as ηm Σ˙ . ∝ SFR If the soft X-ray emission in the halo is indeed from the volume-filling hot outflows, then the simulation results agree very well with the observations. The observations also show that the X-ray temperature increases very mildly with the SFR. In our simulation, the hot outflow right above the disk is clearly hotter for higher Σ˙ SFR. But without a global model we cannot compare directly with the observations. Future work that includes global evolution of the SNe-driven hot outflows shall give a more complete comparison with the observation. For the metal loading, we find the hot outflows take away roughly 50% of the metals ejected, and can potentially carry them to a few hundred of kpc. Observations indicate that roughly 50% of the metals produced are outside of galaxies [Tremonti et al., 2004; Ménard et al., 2010]. This again shows intriguing agreement. It is also worth discussing the metallicity of the outflows Zoutflows. In general, we have

ηmet 0.1 mmet 150M Zoutﬂow = ZISM + 0.04( )( )( )( ), (4.11) 0.5 ηm 1.2M m0 where ZISM is the metallicity of the ISM prior to SNe enrichment, and mmet is the average mass of metals ejected associated with each SN explosion. Here it is obvious that Zoutﬂow CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 112

is higher for smaller ηm. For the hot outflows with small mass loading (high SFR cases), the metallicity of outflows can be a few times solar metallicity. The super-solar metallicity of hot outflows is observed [Martin et al., 2002; Strickland and Heckman, 2009]. For the energy loading factor, it is much harder to directly compare to the observations. One possible way to indirectly constrain the loading efficiency is to compare the emission lines, especially the major coolants of the ISM (such as OVI), to the observations, since the feedback energy eventually either goes into outflows or gets lost via radiatve cooling. We will discuss the OVI emission in the simulations and its comparison with the observations in Chapter 5. Overall, the simulations again show consistency with available observations.

4.5.2 Hot outﬂows and hot CGM

Hot, X-ray emitting coronae have been observed around the MW [Snowden et al., 1998] and other star-forming disk galaxies, up to a few tens of kpc away from the galaxy [Strickland et al., 2004a; Tüllmann et al., 2006; Anderson and Bregman, 2011; Dai et al., 2012]. Hot gas around the MW is also detected through OVII, OVIII absorption lines [Fang et al., 2006; Bregman and Lloyd-Davies, 2007; Gupta et al., 2012].The COS-Halos survey finds the warm-hot metal-enriched gas out to > 100 kpc for MW-like galaxies, as traced by the OVI ∼ absorption lines [Tumlinson et al., 2011]. Our simulations indicate that the hot phase in the outflows has the largest v and v , and can potentially travel distances comparable to the z Be size of DM halos. Hot outflows carry 10%-50% of energy and 30-50% of metals produced by SNe (Section 4.3.3). Thus, the hot, metal-enriched CGM may at least be partly due to this SNe-driven outflows. Those outflows should have an important dynamic impact on the CGM [Hopkins et al., 2012]. Recently, [Faerman et al., 2016] create a two-phase phenomenological model of the halo gas for MW-like galaxies, which simultaneously fits the absorption features of OVI, OVII and OVIII. The two phases in their model have volume-weighted medium temperature of 3 105 K and 1.8 106 K, respectively. Nearly 90% of the mass is in the hotter phase. × × The total mass of the two phases are comparable to the baryons in the DM halo but not in the galaxy. Our hot phase in Σ10-KS and Σ10-KS-4g has a temperature comparable to that in their model. We note, though, that the cooling luminosity of the CGM in their CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 113

model is about twice the SNe heating rate from the MW, and given that ηE 0.45 for the ≈ MW-average, the total discrepancy is about a factor of 4.

4.5.3 Cool outﬂows

Warm/cool outflows are observed through interstellar absorption lines, which have velocities of about several hundred km/s [Steidel et al., 1996; Shapley et al., 2003; Martin, 2005; Weiner et al., 2009; Heckman et al., 2015]. Mass loading factors around a few are frequently reported. In our simulations, the cool phase in general does not have such high velocities, and the mass loading is usually smaller than unity, especially for high Σgas . We here discuss the possible reasons for this apparent discrepancy. We first note that observations usually focus on the star-bursting regime, such as (U)LIRGs and Lyman break galaxies, which we do not cover in our simulations (Fig 4.1). As mentioned before, radiation pressure may account for the potential high mass loading in those extreme environments, provided that the infrared light can be trapped by dust. Note that those bursting SF systems are very rare; even at higher redshifts when they are more common, they only account for < 10% of cosmic SF density [Rodighiero et al., 2011]. ∼ Second, the observed mass loading factors have large uncertainties, due to the uncon- strained metallicity, ionization fraction, geometry, etc. This can lead to an error bar as large as the mass loading factor itself. Recent work by [Chisholm et al., 2016], which claims a better constraints on the above quantities, suggests a small ηm 0.1. This is, in fact, not ≈ inconsistent with our value for the highest Σgas . On the simulation side, we note that even though we adopt high enough resolution to make sure the Sedov-Taylor phase of SNRs is well-resolved, the interaction between different gas phases may still not be sufficiently resolved, once the ISM becomes multiphase. To what extent this may affect the mass loading is not clear. Another apparent discrepancy is that we find that the cooler phases dominate less of the outflow mass flux for higher Σ˙ SFR. It is possible, however, that some cool clouds may form on the way due to thermal instability [Field, 1965; Thompson et al., 2016], which we do not capture because of the relatively small box size. But again, how much cool mass would form under this mechanism is not clear. CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 114

Overall, there remains much uncertainty about the observational interpretation and theoretical modelling of cool outﬂows.

4.5.4 Implications for cosmological simulations

Stellar feedback is one of the key ingredients for galaxy formation in cosmological simulations and semi-analytic models [Somerville and Davé,2015]. The general consensus is that strong feedback/outflows are needed to reproduce various observed scaling relations of galaxies [Springel and Hernquist, 2003; Stinson et al., 2006; Hopkins et al., 2012; Hummels and Bryan, 2012]. Due to the limited resolutions in cosmological simulations, however, multiphase ISM and individual SNR cannot be resolved. Ad hoc models are widely used in the field, with free parameters fine-tuned to match observations. Different groups adopt very different recipes, usually invoking shutting off cooling or hydrodynamics. To evaluate the real impact of feedback, however, a physically-grounded model is necessary. While formulating such a model is beyond the scope of this paper, we compare the loading factors found in our work to current cosmological simulations, and briefly discuss the implications. In cosmological simulations, a significant fraction of SNe energy is used to generate outflows, roughly 30-50% (e.g. [Oppenheimer and Davé,2008]). Our ηE varies from 15- 50%, broadly consistent with that fraction. To match the observed mass-matallicity relation of galaxies, a significant fraction of metals have to be driven out of the galaxies.

Our ηmet is about 40-80%, in general agreement with what observations require and cosmological simulation adopt. The main difference is the mass loading of the outflows, or in other words, the amount of mass that those energy and metals couple to. Our ηm is in general smaller than what is adopted in cosmological simulations, by roughly an order of magnitude. A smaller mass loading is not surprising for the simulations where the multiphase ISM is resolved: SN blast waves tend to vent through the least-obstructed channel, thus preferentially heating and accelerating the tenuous phase [Cowie et al., 1981; Li et al., 2015]. Cosmological simulations cannot resolve the multiphase ISM/outflows generally. Mixing the fast/tenuous and the slow/dense phase due to insufficient resolution would result in a slower, cooler and mass-loaded outflows. Therefore current cosmological CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 115 simulations may not accurately predict the impacts of galactic outflows on the CGM and galaxy formation. From our simulations, hot outflows, though not carrying a great amount of mass, may be able to suppress the inflows given their vigor (Section 4.3.2), therefore restricting galaxy masses. The implications of tenuous yet vigorous hot outflows to galaxy formation and the CGM are not clear (although see recent attempts by [Davé et al., 2016], [Fielding et al., 2017]).

4.6 Conclusions

In this paper, we use high-resolution simulations to study the multiphase outflows driven 2 by supernovae from stratified media. We cover a wide range of Σgas : 1-150 M /pc . We quantify the multiphase outflows by measuring the loading factors of energy, mass and metals. The fiducial runs assume Σgas scales with Σ˙ SFR as in the Kennicutt-Schmidt relation (Fig 4.1). We study the effects of various physics on the loading factors: SN scale height hSN,cc, photoelectric heating, external gravitational field, and enhanced SN rate. Here are our main conclusions: 1. The ISM quickly becomes multiphase under the impact of multiple SNe. The cold phase settles down near the midplane, whereas hotter phases preferentially escape and form outflows. 2. For the solar neighbourhood case, the gas pressure, volume fraction of hot gas, and mean densities of different gas phases agree well with the observations.

3. The mass loading factor ηm decreases monotonically with increasing Σgas as ηm ∝ −0.6 ˙ Σgas (Eq. 4.8). The outflowing flux is about 0.1-10 of ΣSFR. The energy and metal loading factors do not show significant variance with Σgas . Roughly 10-50% of the energy and 40-80% of the metals produced by SNe are carried away by the outflows (Fig 4.10). 5 4. More of the outflow volume is occupied by hot gas (T > 3 10 K) for larger Σgas . × The hot phase contributes to > 1/3 of the mass loading, >0.8 of the energy loading, and ∼ 0.5-0.9 of the metal loading (Fig 4.10 ). It has significantly larger v and v (see Section z Be 4.3.2 for definition) than the cooler phases. Hot outflows are very likely to have a broad impact on the CGM. Hot outflows also have super-solar metallicity (Eq. 4.11). CHAPTER 4. QUANTIFYING SUPERNOVAE-DRIVEN MULTIPHASE GALACTIC OUTFLOWS 116

5. Increasing hSN enhances the energy and metal loading, since more energy/metals are directly deposited into the low-density halo. The mass loading factor, on the other hand, does not show a monotonic dependence on hSN. The relative scale height of SNe and gas is a very important factor determining the loading efficiencies. Various physical processes affect the loading factors by changing hSN and/or hgas. 6. A stronger PEH makes SN feedback less effective, since it keeps more gas in the warm phase, thus a larger scale height of the neutral gas layer. In the extreme case where the cooling curve has a lower cut-off at 104 K, the feedback is severely suppressed, with the loading factors smaller by a factor of > 10. (Fig 4.12, 4.13). ∼ 7. A larger gravitational field, by lowering hSN,cc, may result in much stronger energy and metal loading (Fig 4.15). 8. If the SN rate is enhanced above the Kennicutt relation, the energy and metal fluxes roughly scale linearly with the SN rate, but the mass flux has a sub-linear dependence on the SN rate. Overall, once the ISM is hot-dominated, clustering SNe does not enhance the time-integrated properties of outflows. CHAPTER 5. OVI EMISSION FROM THE SUPERNOVAE-REGULATED ISM: SIMULATION VS OBSERVATION 117

Chapter 5

OVI Emission from the Supernovae-Regulated ISM: Simulation vs Observation

This chapter is adapted from the publication in Astrophysics Journal Letter, 835:1, 2017. CHAPTER 5. OVI EMISSION FROM THE SUPERNOVAE-REGULATED ISM: SIMULATION VS OBSERVATION 118

Abstract

The energy loading efficiency is a very important quantifier of SN feedback, but it is hard to constrain. The OVI λλ1032, 1038A˚ doublet emission traces collisionally ionized gas with T 105.5 K, where the cooling curve peaks for metal-enriched plasma. This ≈ warm-hot phase is usually not well-resolved in numerical simulations of the multiphase interstellar medium (ISM), but can be responsible for a significant fraction of the emitted energy. Comparing simulated OVI emission to observations is therefore a valuable test of whether simulations predict reasonable cooling rates from this phase. We calculate OVI λ1032A˚ emission, assuming collisional ionization equilibrium, for our small-box simulations of the stratified ISM regulated by supernovae (as described in Chapter 4). We find that the agreement is very good for our solar neighborhood model, both in terms of emission flux and mean OVI density seen in absorption. We explore runs with higher surface densities and find that, in our simulations, the OVI emission from the disk scales roughly linearly with the star formation rate. Observations of OVI emission are rare for external galaxies, but our results do not show obvious inconsistency with the existing data. Assuming the solar metallicity, OVI emission from the galaxy disk in our simulations accounts for roughly 0.5% of supernovae heating. We also find that the OVI emission tend to increase when we improve the numerical resolutions.

5.1 Background

Supernova (SN) explosions represent a key form of stellar feedback in galaxies, and produce copious hot gas in the ISM. The SN energy can regulate the ISM dynamics and launch galactic outflows [Cox and Smith, 1974; McKee and Ostriker, 1977; Cox, 2005]. One key factor to quantify the efficiency of SN feedback is the energy loading factor (or “thermalization factor”), which is the fraction of SN energy that is loaded into the outflows. The efficiency of radiative cooling in the multiphase ISM directly affects this loading efficiency. A temperature around 105.5 K is where the cooling curve peaks for metal-enriched gas (e.g.[Sutherland and Dopita, 1993]). The OVI λλ1032, 1038A˚ doublet is one major coolant around this temperature. These OVI resonant lines have been observed widely, from the CHAPTER 5. OVI EMISSION FROM THE SUPERNOVAE-REGULATED ISM: SIMULATION VS OBSERVATION 119 local ISM, which provides the first strong evidence of the prevalence of the hot phase in the ISM [Jenkins and Meloy, 1974; York, 1974], to the circumgalactic medium (CGM) and even the intergalactic medium (IGM) [Bregman et al., 2006; Tumlinson et al., 2011; Simcoe et al., 2002]. Most of these observations, though, focus on absorption instead of emission, except for the solar neighborhood (e.g.[Otte et al., 2003]). The origin of gas around this temperature is not totally clear. Thermal conduction/turbulent mixing at the interface between the hot plasma (T > 106 K) and warm clouds (T 104 K) [McKee and Ostriker, ∼ ∼ 1977; Weaver et al., 1977; Cowie et al., 1979; Begelman and Fabian, 1990], and cooling from hotter gas [Edgar and Chevalier, 1986; Slavin and Cox, 1993; Dopita and Sutherland, 1996; Heckman et al., 2002] have been suggested as possible mechanisms. In hydrodynamic simulations of astrophysical systems, the quantity of gas around 105.5 K can be affected by numerical mixing and diffusion. Depending on the resolution and technique used, hydrodynamic simulations can artificially promote or suppress the mixing and the amount of gas in the warm-hot phase. For example, the particle-based SPH scheme usually exhibits too little mixing, while the mesh-based techniques can have too much mixing [Agertz et al., 2007; Springel, 2010]. The magnitude of this numerical artifact is hard to gauge. This is a particular issue when multiple phases – with contrasting densities and temperatures – coexist at similar pressures, as is the case for the ISM. Even without explicitly including thermal conduction, gas with intermediate temperature 105 106 K can − show up around the interface between the hot and warm (or cold) phases, which is usually not well-resolved. Although higher numerical resolution helps, turbulence may drive the scale of the cold/hot interface down to small length scales and, until the Field length is resolved, improved resolution may simply result in more clouds without reducing the amount of material in the interface. Because of the high cooling efficiency in this warm-hot phase, it is important to test the simulation cooling rates. One way to do this is to compare against observed emission lines, which trace the cooling directly. We make such an attempt in this Letter. The OVI λλ1032,1038A˚ doublet falls in the far-UV band, which cannot be observed from the ground. Satellites with FUV spectrographs, such as Corpernicus, FUSE, SPEAR and HST, contribute to most of our knowledge about gas traced by OVI. For the local ISM, CHAPTER 5. OVI EMISSION FROM THE SUPERNOVAE-REGULATED ISM: SIMULATION VS OBSERVATION 120 the column density of OVI is obtained from absorption-line studies of bright stars [Jenkins, 1978a; Jenkins, 1978b; Oegerle et al., 2005; Savage and Lehner, 2006]. OVI line emission has also been detected in various sight lines [Dixon et al., 2006; Otte and Dixon, 2006; Welsh et al., 2007]. For external galaxies, the OVI interstellar absorption line is seen in nearly every starburst system observed (e.g.[Grimes et al., 2007]). The CGM around star forming galaxies shows absorption of OVI out to > 100 kpc [Tumlinson et al., 2011]. OVI emission from the disk ∼ or the CGM, on the other hand, has only been reported in a few cases [Otte et al., 2003; Grimes et al., 2007; Hayes et al., 2016]. In this Chapter, we calculate the emission of OVI 1032A˚ as well as the OVI column densities, for our ISM simulations with SN feedback. [de Avillez and Breitschwerdt, 2005; de Avillez and Breitschwerdt, 2012] simulated the SNe-regulated ISM for the solar neighborhood, and found OVI absorption agreeing well with the observations. We extend that study by calculating both the emission and the column density, for conditions with various star formation rates. We describe the simulations and calculation of OVI emission in Section 2, show the results and compare with the observations in 3, and summarize in 4.

5.2 Methods

The simulations presented in this paper are the four fiducial runs in [Li et al., 2016], hereafter referred to as Paper I. We briefly introduce the setup here – for details, see Paper I. Our simulations are performed using the Eulerian hydrodynamic code Enzo [Bryan et al., 2014]. We use the higher-order piece-wise parabolic method (PPM, [Colella and Woodward, 1984]) as the hydro-solver, along with the two-shock Riemann solver. The 3D, rectangular simulation boxes cover a fraction of the galaxy disk, with a square cross-section and the long z-axis perpendicular to the disk plane. Similar configurations have been widely adopted (e.g.[Dib et al., 2006; Joung and Mac Low, 2006; Creasey et al., 2013; Walch et al., 2015; Kim and Ostriker, 2016]). The four runs discussed in this paper have initial gas surface −2 densities Σgas = 1, 10, 55, 150 M pc , respectively, referred to as Σ1-KS, Σ10-KS, Σ55- KS, Σ150-KS. The simulation Σ10-KS is the model for the solar neighborhood. A static CHAPTER 5. OVI EMISSION FROM THE SUPERNOVAE-REGULATED ISM: SIMULATION VS OBSERVATION 121 gravitational field from gas, stars and the dark matter halo is included. The midplane is located at z = 0. The box size in the z-direction is 5 kpc, i.e., –2.5 z 2.5 kpc. The spatial resolution at z 0.5 kpc is 5.0, 2.0, 0.75, and 0.60 pc, respectively, | | so simulations with larger Σgas have higher resolutions. The resolution is coarsened by a factor of 2 for 0.5 z 1 kpc, and by another factor of 2 for z 1 kpc. An outflowing boundary | | | | condition is applied for the z-direction, and periodic boundaries for the x- and y-directions. −4 We have chosen the star formation surface density Σ˙ SFR to be 1.26 10 , 6.31 × × −3 −2 −1 10 , 0.158, 0.708 M kpc yr , respectively, for the four fiducial runs, such that the combinations of (Σgas , Σ˙ SFR) fall along the observed correlation for nearby galaxies at sub-kpc scale (see Figure 1 of Paper I, [Kennicutt, 1998; Bigiel et al., 2008]). The SN rate is related to Σ˙ SFR by assuming one SN explodes per 150 M of star formation. SNe are placed randomly in the disk, with a scale height of 150 pc for core collapse SNe (90%), and 325 pc for Type Ia SNe (10%). SNe are injected with 1051 erg thermal energy within a sphere. We use a cooling curve for the solar metallicity gas with a low-temperature cut-off at T = 300 K, which assumes collisional ionization equilibrium (CIE) above 104 K [Rosen and Bregman, 1995]. We apply photoelectric heating that scales linearly with the Σ˙ SFR. For the detailed model parameters, see Table 1 of Paper I. We do not explicitly include thermal conduction. To calculate the OVI density and emission, we assume CIE. The relative abundance of O/H is 5 10−4 by number (the “solar abundance” in this paper). The fraction of O in the × ionization stage of +5, as a function of temperature, is taken from [Sutherland and Dopita, 1993]. The collision strength for OVI 2s 2S 2p 2P, the transition of which results in the − 1032, 1038A˚ emission, is 5.0 [Osterbrock, 1989]. The emission flux of the 1032A˚ line is twice that of 1038A.˚

5.3 Results

5.3.1 OVI-emitting gas in simulations

We illustrate the distribution of OVI-emitting gas in simulations in both real and phase space. Figure 5.1 shows slices of temperature and OVI emissivity from the midplane of the CHAPTER 5. OVI EMISSION FROM THE SUPERNOVAE-REGULATED ISM: SIMULATION VS OBSERVATION 122

Figure 5.1: Slices of temperature and OVI 1032A˚ photon emissivity from the midplane of the solar neighborhood model Σ10-KS. OVI emission mainly comes from the boundary between the hot and cool medium. The OVI-emitting gas is usually not well-resolved, with some emitting regions resolved by one computational cell. The dimensions are 350 pc × 350 pc, and the resolution is 2 pc. The color coding for the lower panel has a ﬂoor at 10−20s−1sr−1cm−3. CHAPTER 5. OVI EMISSION FROM THE SUPERNOVAE-REGULATED ISM: SIMULATION VS OBSERVATION 123 simulation Σ10-KS. The x-y dimension is 350 pc 350 pc, with a resolution of 2 pc. OVI × emission mainly originates from the interface between the hot and cool medium. The gas that emits OVI is usually not well-resolved, and some is as thin as a single computational cell. This is generally seen in the simulations, even for the highest-resolution run Σ150-KS. Figure 5.2 shows the phase diagram for the same run Σ10-KS, in which the color coding indicates the total 1032A˚ emission from each [density, temperature] bin. The majority of the emission is from close to 3 105 K, where the ionization fraction of OVI is largest. × Gas with intermediate densities, 0.01 0.1 cm−3, contributes most of the emission. Below − 1.5 105 K and above 106 K, the emission is negligible. ×

5.3.2 OVI in the solar neighborhood

Figure 5.3 shows the integrated ﬂux along the x- and the z-directions for the OVI 1032A˚ emission. The snapshot is taken at t = 140 Myr for the run Σ10-KS. Most of the emission is from near the midplane, where gas is dense and where most SNe explode. Above the plane, gas is mostly hot (T > 106 K), and the oxygen atoms are ionized even further. It is generally the case that most of the emission is from near the disk in our simulations. Indeed, [Hayes et al., 2016] found emission from the SF disk is much stronger than from the halo. The 1032 A˚ emission outside of the disk is seen in and around the clouds. The hot gas can shock-compress and ablate the warm clouds, as well as mix with them, all of which may result in gas around 105.5 K. The x-projection of the emission shows some spatial variation. The regions where the emission is weaker have smaller gas column densities, as a result of hot, low-density bubbles that connect to the halo. The FUSE satellite has surveyed OVI 1032A˚ emission from the local ISM at various Galactic latitudes [Dixon et al., 2006; Otte and Dixon, 2006]. The emission is detected for about 40% of sight lines, with intensities of (1.9 8.6) 103 photon s−1cm−2sr−1. [Welsh et − × al., 2007] used the SPEAR satellite to look at the north Galactic pole region, and detected OVI 1032A˚ emission from 8 out of 16 survey regions, with intensities of 5 103 2 × − × 104 photon s−1cm−2sr−1. Note that the observed emission may be partly from the CGM, so these values may be an upper limit for the emission from near the disk. Our projection plot along the z-direction shows 1032A˚ emission of (0.5 6) 103 photon s−1cm−2sr−1, − × CHAPTER 5. OVI EMISSION FROM THE SUPERNOVAE-REGULATED ISM: SIMULATION VS OBSERVATION 124

Figure 5.2: Phase diagram for the run Σ10-KS, where the color coding shows the 1032A˚ emission rate from gas in that [temperature, density] bin. The calculation includes all gas in the simulation domain. The color coding has a ﬂoor at 1030 erg s−1 . CHAPTER 5. OVI EMISSION FROM THE SUPERNOVAE-REGULATED ISM: SIMULATION VS OBSERVATION 125

Figure 5.3: Upper left: projected gas density in the x-direction; upper right: projected OVI 1032A˚ emission in the x-direction (only –0.75 z 0.75 kpc is shown). Lower: projected OVI 1032A˚ emission in the z-direction. CHAPTER 5. OVI EMISSION FROM THE SUPERNOVAE-REGULATED ISM: SIMULATION VS OBSERVATION 126 broadly consistent with the observations. (The emission in the simulations includes the two sides of the midplane, whereas the observation is from within the plane, so there is a factor of 2 difference.) ∼ In addition to emission, we also briefly touch on OVI absorption. Our results show very good agreement with the observations. Absorption-line studies from stars with known distances reveal that the local OVI density nOVI in the disk to be about (2.2 1.0) ± × 10−8cm−3 [Jenkins, 1978b; Oegerle et al., 2005; Savage and Lehner, 2006; Bowen et al., 2008; −8 Barstow et al., 2010]. The nOVI in our solar neighborhood run is (2.4 0.3) 10 and ± × (1.8 0.3) 10−8 cm−3, averaged over z 100 and 200 pc, respectively. ± × | | The non-equilibrium effects will affect these results, generally by boosting the amount of OVI at lower temperatures. Recent simulations by [de Avillez and Breitschwerdt, 2012], which track the ionization stage of oxygen, i.e., relaxing the assumption of CIE, find that up 5 to 70% (by mass) of the OVI may be at T < 10 K, although their mean nOVI is consistent ∼ with ours. We expect that the non-equilibrium effects will affect the OVI absorption more strongly than emission, since low temperatures would lead to less efficient collision and excitation.

5.3.3 Scaling of OVI with SFR

Figure 5.4 shows the OVI 1032A˚ emission, integrated over the z-direction, as a function of

Σ˙ SFR. The quantity is averaged over the x-y plane. The error bar shows the time variation.

We ﬁnd the OVI emission scales roughly linearly with Σ˙ SFR. As mentioned before, the solar neighborhood model agrees well with the observations. Observations of OVI emission from external galaxies are limited to a few cases. We show the emission from SDSS J1156+5008 [Hayes et al., 2016]. The surface brightness reported in that paper is for the 1038A˚ line. To get that of the 1032A˚ , we have applied (i) a factor of 2 increase due to the relative intensity of 1032A˚ and 1038A˚ ; (ii) a factor of 2 increase to account for gas from both sides of the disk, since only the redshifted part is observed as emission. We have also scaled the emission to the solar abundance. [Hayes et al., 2016] calculated O/H using two methods, which diﬀer by a factor of 3. The two hollow triangles ∼ in the plot correspond to these two O/H values. Still, the observed surface brightness is CHAPTER 5. OVI EMISSION FROM THE SUPERNOVAE-REGULATED ISM: SIMULATION VS OBSERVATION 127

107 ]

1 6

− 10 sr 1 − s

2 5

− 10 cm [ 104 ˙ 0.90 ∝ ΣSFR 103 this paper 102 Otte+06 OVI 1032 emission Hayes+16 1 10 4 3 2 1 0 1 2 10− 10− 10− 10− 10 10 10 2 Σ˙ SFR[M /kpc /yr]

Figure 5.4: Purple stars: OVI 1032A˚ emission integrated over the z-direction in our simulations, as a function of Σ˙ SFR. The correlation is roughly linear. Filled triangle: observation for the solar neighborhood (Otte et al. 2006), which detected OVI emission for about 40% of the sight lines. Hollow triangle: observation of SDSS J1156+5008 (Hayes et al. 2016), scaled to the solar abundance (the two triangles indicate O/H determined from two methods). CHAPTER 5. OVI EMISSION FROM THE SUPERNOVAE-REGULATED ISM: SIMULATION VS OBSERVATION 128 likely a lower limit of the real emission for several reasons. First, the aperture of the COS spectrograph is 2.5 arcsec in diameter, corresponding to a physical scale of r 9.4 kpc at ∼ the galaxy’s redshift (0.235). The surface brightness of the 1038A˚ line is assumed to be uniform inside the aperture, but the star forming region is very compact – only about 1 kpc across (as seen from Hα). Thus the actual surface brightness in this starbursting region is probably higher. Second, the 1038A˚ line can be partially blended with the nearby CII absorption. Additionally, attenuation by dust in the host galaxy, especially in the disk, can also lower the emission. Based on these arguments, the reported emission intensity is probably a lower limit. Also, the CGM probably does not contribute much to the surface brightness in this measurement, since for J1156, the OVI emission in the halo is observed, with an intensity much smaller than the center. Apart from J1156, another galaxy, Haro 11, turns out to be very similar [Grimes et al., 2007; Hayes et al., 2016]. [Hayes et al., 2016] also compare J1156 with the stacked spectra of other starbursting galaxies from archival data, and ﬁnd an OVI emission comparable to J1156.

For high Σ˙ SFR, we cannot yet draw a ﬁrm conclusion about whether our results are consistent with the (very limited) observations. There can be agreement if, for the very high Σ˙ SFR cases, the actual emission is above the observed value for the reasons discussed above, and/or the OVI emission from the simulations increases more slowly for higher Σ˙ SFR, which we do not cover in our simulations.

Finally, the roughly linear scaling of the OVI surface brightness with Σ˙ SFR implies that the cooling rate from OVI is an approximately constant fraction of the SNe heating. We ﬁnd this fraction to be around 0.5%, including both 1032A˚ and 1038A˚ . Therefore cooling from OVI is not an important source of energy loss. Most (> 80%) cooling of the ISM comes from a broad warm-hot regime: 104 105.7 K. But OVI mainly exists within a narrow fraction − of it: 105.3 105.6 K; gas in this temperature range only contributes to a few percent of the − total cooling rate.

The predicted OVI column density NOVI (across the whole z-direction), as a function of Σ˙ SFR , is shown in Figure 5.5 (purple stars). The column density is averaged over the x-y plane. The error bar indicates the time variation. The correlation of NOVI with Σ˙ SFR CHAPTER 5. OVI EMISSION FROM THE SUPERNOVAE-REGULATED ISM: SIMULATION VS OBSERVATION 129

16 ˙ 0.19 10 NOVI ∝ ΣSFR this paper Hayes+16

] 15 2 10 Grimes+09 −

cm Bowen+08 [ OVI N 1014

13 10 4 3 2 1 0 1 2 10− 10− 10− 10− 10 10 10 2 Σ˙ SFR[M /kpc /yr]

Figure 5.5: Purple stars: mean column density of OVI along the z-direction for the whole simulation domain, as a function of Σ˙ SFR. Orange box: observed value for the solar neighborhood (Bowen et al. 2008). Filled circles: observed NOVI for starburst galaxies in Grimes et al. 2009. Hollow circle: observed NOVI for J1156 (Hayes et al. 2016). For external galaxies, NOVI have been scaled to the solar abundance; the down arrows indicate these values should be an upper limit for NOVI near the disk, since they also include OVI in the CGM. CHAPTER 5. OVI EMISSION FROM THE SUPERNOVAE-REGULATED ISM: SIMULATION VS OBSERVATION 130 is positive but the dependence is weak, with a power-law index of 0.19. For four orders of magnitude span in Σ˙ SFR, NOVI only diﬀers by a factor of 6. For the solar neighborhood, we show the observed value as in [Bowen et al., 2008] (orange box), with a factor of 2 increase applied to include OVI from both sides of the galaxy disk; the vertical range of the box indicates the statistical variation of NOVI , and the horizontal range denotes the uncertainty of Σ˙ SFR. Our result generally agrees with the observation.

[Grimes et al., 2009] have reported NOVI for 12 starburst galaxies (from absorption against galaxy disks). We plot these NOVI together with the corresponding Σ˙ SFR (Table 1 of [Heckman et al., 2015]). We applied a factor of 2 increase to the data to account for OVI from both sides of the disk, and scale NOVI to the solar abundance. Additionally, we plot

NOVI for J1156 [Hayes et al., 2016]. Those starburst systems have NOVI that are usually an order of magnitude larger than (the extrapolation of) our results. But note that a signiﬁcant fraction of NOVI can come from the CGM. For nearby Milky Way-like galaxies, NOVI of the CGM (from absorption against background quasars) ranges from 1014 1015cm−2 [Werk et − al., 2016]. The starburst systems may have even higher NOVI in the CGM for their much stronger SF/outﬂows activities. If OVI in the CGM indeed dominates the column density, then the extrapolation of our results may be consistent with observations.

We tested the effect of resolution for the two low-Σgas cases. The test runs include a midplane resolution of 4 pc for Σ10-KS (fiducial 2 pc), and 2.5 pc and 10 pc for Σ1-KS (fiducial 5 pc). The 1032A˚ emission agrees within the time variation (which can be as large as 80%). The mean values of OVI emission increase with improved resolution. Such a trend can be partly attributed to that resolving more cloud fractals increases the total area of boundary layers. The convergence rate is faster for Σ1-KS than Σ10-KS: improving resolution by a factor of 2 leads to 20% and 60% increase of the mean OVI emission, respectively. The total cooling rate shows less time variation ( 20%), and is consistent for ∼ different resolutions as well. CHAPTER 5. OVI EMISSION FROM THE SUPERNOVAE-REGULATED ISM: SIMULATION VS OBSERVATION 131

5.4 Conclusions

In this Letter we calculate the OVI 1032A˚ emission from our ISM simulations with SN feedback. This is done to test the radiative cooling from gas with T 105.5 K (which cools ∼ very efficiently but is usually not very well-resolved in simulations) against observations. We find that, for the solar neighborhood, our results agree well with the observations, both in terms of emission flux and mean OVI density in absorption. Most emission comes from the gas disk. The relatively dense (0.01-0.1 cm−3) gas around the interface between hot gas and warm clouds contributes the majority of the emission.

Changing Σgas and Σ˙ SFR along the Kennicutt relation, we ﬁnd that both NOVI and the

OVI surface brightness increase with Σ˙ SFR. For NOVI , the dependence is quite weak, while the surface brightness of OVI emission scales roughly linearly with Σ˙ SFR. OVI emission is approximately 0.5% of the SNe heating rate. It cannot yet be determined deﬁnitely whether our results give a reasonable emission for high Σ˙ SFR, because of the limited observational and numerical samples, although agreement is not unlikely. More observations of the OVI emission, for both low and intermediate-high Σ˙ SFR, are needed for a more complete comparison. 132

Part III

Conclusions CHAPTER 6. CONCLUSIONS 133

Chapter 6

Conclusions

6.1 Summary of the Thesis

In this thesis, we use high-resolution, 3D simulations to study the physics of supernovae feedback, about how they impart feedback to a multiphase ISM, how they collectively shape a patch of the ISM and drive the outflows. For a single SNR evolution, we found that a SN exploding in a multiphase ISM evolves fundamentally differently from that in a uniform medium. We studied in detail the evolution of a SNR in a multiphase ISM. SN blast waves travel much faster in a hot-dominated medium while cooling inefficiently, thus their impact domain is enhanced by 2 orders of magnitude. We presented fitting formulae that describe the energy and momentum evolution of the

SNR. The volume fraction of hot gas fV,hot is the major factor that determines the energetic feedback of SNR. These results have been applied in cosmological simulations as a sub-grid model for SN feedback [Núñez et al., 2017]. For multiple SNe shaping a multiphase ISM, we set up a series of simulations, each having two input parameters: gas mean density, n, and SN rate, S. We systematically investigated the properties of the resultant ISM. In general, there are two thermal conditions of the ISM: (i) a steady state, where SNRs do not overlap and most of the SN energy is radiated away; (ii) a thermal-runaway state, where SNRs overlap, most of the volume is hot, and energy accumulates. The latter state is when outflows can be launched. We found, for a given n, the critical S to drive the ISM into a thermal runaway (Eq. 3.10). We find that CHAPTER 6. CONCLUSIONS 134

fV,hot 0.6 0.1 marks the transition of ISM from a steady state to thermal runaway, ∼ ± along with the change of hot gas topology. We examined how gas pressure, volume/mass fraction of different gas phases, and the Mach numbers change with (n, S). Gas pressure scales roughly linearly with S (Eq. 3.13). The Mach numbers for all phases are around unity, and show very little dependence on n or S. For SNe-driven outflows, we quantified the energy, mass and metal loading factors of the outflows for different SF rates, which span 4 orders of magnitude — from a relatively tenuous dwarf galaxy environment to a star-burst regime. We found the mass loading ˙ −0.48 factors range from 0.2-10, and scale with ΣSFR , agreeing with recent X-ray observations of galaxy coronae [Wang et al., 2016]. The metal and energy loading factors do not show an obvious trend with Σ˙ SFR, and 10-35% of the energy and 40-80% of the metals produced by SNe end up in outflows. The volume fraction of the hot gas in an ISM is usually a significant fraction: 60-80%. We investigated how various physical processes, including photoelectric heating, OB runaway stars, external gravitational field, affect the loading factors. The relative scale height of SNe and gas is a very important factor. The outflows are highly inhomogeneous; the hot phase carries the majority of the energy and metals. Having velocities comparable to the escape velocities of galaxies, the hot phase would have a broad impact on the circumgalactic medium. The cooler phases, on the other hand, have considerably smaller velocities, and are less dominant for high Σ˙ SFR cases. We calculated the OVI emission from the SNe-regulated ISM/outflows in our stratified- medium simulations. The OVI doublet 1032/1038A˚ traces the warm-hot gas around 3 × 105K, where the cooling is most efficient for the metal-enriched gas. However, in ISM simulations, this warm-hot phase, usually occurring around the interface between the hot gas and cool clouds, is generally not well-resolved. Numerical diffusion could affect the amount of the warm-hot phase, thus may result in over- or under-cooling problem. To test the OVI emission against the observation is therefore critical to constrain this unresolved cooling. We generated emission maps for the OVI lines for the simulated ISM/outflows. Our model for the solar neighborhood agrees very well with the observations, in terms of both emission and absorption. For other SF environments, observations are limited to very few cases with extreme SF activity, but the extrapolation of my result is in general agreement CHAPTER 6. CONCLUSIONS 135 with the observations. We found the OVI emission scales roughly linear with the SFR, and accounts for 0.5% of the SN heating. More observations are necessary for a more complete comparison.

6.2 Future work

Galaxy formation is a multi-scale process. We have quantified the multiphase outflows driven from a multiphase ISM, it is natural then, to ask their evolution and impact on galactic scale and even beyond. Our simulations indicate that hot outflows are very powerful and have the potential to travel as far as a few hundred kpc (Chapter 4). As mentioned in the Introduction, current cosmological simulations, due to their use of ad hoc feedback models, cannot evaluate the real role of feedback in galaxy formation. Thus, it will be very useful to use a feedback model that is physically-based and calibrated by high-resolution simulations. This will help understand how SN feedback regulates the surroundings of galaxies, which can be tested against recent and future observations of the CGM, such as the COSHalos in optical and UV[Tumlinson et al., 2011], and the X-ray emission from galactic corona [Wang et al., 2016]. Furthermore, in a cosmological context, we can evaluate whether outflows can suppress the inflow, which is very important in determining the galaxy mass. Another potentially important aspect of SN feedback is cosmic rays (CRs). CRs are mainly produced at shock fronts by diffusive shock acceleration [Bell, 1978; Blandford and Ostriker, 1978]. SNe are considered to produce the majority of galactic CRs [Drury et al., 1994]. The CR pressure is comparable to that of the thermal gas and the magnetic field in the solar neighborhood, suggesting their essential role in the ISM. Thanks to inefficient cooling and their diffusion/streaming relative to thermal gas, CRs are promising candidates for driving galactic outflows [Booth et al., 2013; Salem and Bryan, 2014; Wiener et al., 2016]. It is poorly understood, however, how CRs launch outflows from a multiphase ISM. Moreover, the relative importance of SN thermal feedback vs. CRs is not clear. It is thus very interesting to incorporate CRs into the high-resolution ISM simulations. Of particular interests is whether CRs can help to drive cool outflows, which is known to be a theoretical challenge. Recent works by [Girichidis et al., 2016a; Simpson et al., 2016] suggest that CRs CHAPTER 6. CONCLUSIONS 136 may be able to do so, but much more work needs to be done to understand the role of CRs as a feedback force in galaxy formation. 137

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