On the Launching of Optically Thick Outflows from Massive Stars By

On The Launching of Optically Thick Outflows from Massive Stars

Stephen Ro

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Astronomy & Astrophysics University of Toronto

Abstract

On The Launching of Optically Thick Outﬂows from Massive Stars

Stephen Ro Doctor of Philosophy Graduate Department of Astronomy & Astrophysics University of Toronto 2017

For decades, massive stars have been seen to lose mass at every stage of evolution. In recent years, detections of outbursts and circumstellar relics uncover transient modes emerging moments before the supernova. The goal of this thesis is to bring theoretical attention into how massive stars launch outflows. In this thesis, we study two modes of mass loss seen during rare, but critical, phases of stellar evolution. The first considers dense, steady outflows from Wolf-Rayet stars. The second considers the eruptions and explosions seen from luminous blue variables (LBVs) and Type IIn supernova (SN) progenitors. Outflows from both systems are optically thick but either radiative or adiabatic in nature. Wolf-Rayet winds are driven by strong radiative pressure on metal lines. Suppressing the outflow is shown to drastically alter the stellar structure with peculiar features including an extended radiative cavity encased in a massive shell. We construct outflow models and conclude the galactic WR population does not harbour such structures. We derive a minimum mass loss rate to launch a transonic, optically thick outflow and find the iron opacity bump to be responsible for launching WR winds. LBVs and Type IIn SN progenitors are seen to abruptly expel mass. Observational inferences suggest the rates are both exceptional and unsustainable with durations exceeding the outer envelope dynamical time. Yet, these outflows are occasionally preceded by fast motions indicative of shock acceleration. In this thesis, we explain how waves can be responsible for these dynamics. We derive an analytic solution wave

ii steepening and shock formation for an inhomogeneous medium. The results suggest that any super-Eddington phenomenon driven by waves must involve shocks. Shocks form deep in the star where they are initially weak. We show how weak shocks are capable of accelerating to become strong, breakout, and produce fast ejecta using a revived semi-analytical approximation. We discuss how a train of shocks can heat a large volume of the envelope and eject signiﬁcant mass.

iii “You know, you blow up one sun and suddenly everyone expects you to walk on water.”

- Colonel Samantha Carter, Stargate SG-1

iv Acknowledgements

Graduate school can often require a spontaneous degree of diligence, discipline, and direction. Where I was unready, my friends, family, and colleagues were the reliable beacons that guided me through. Each one has given me a gift; a handful to which are described below. This thesis is dedicated to them and our fond memories.

When asked for compassion and perseverance, I see my mother. She has endowed me with the moral compass and ﬁbre to face the world. I love you, mom, and I am proud to be your son.

Chris is the mentor that people wish for. He has always made me feel welcome and equal. The freedom to ask any question and pursue whatever is most interesting (within reason) is an attitude I’ll keep with me. Chris, you are a phenomenal role model and friend. Thank you for everything.

Virginia is my relentless force for growth. She has witnessed my life and forged each lesson into a brick in my foundation. And, over several years, we’ve built quite the house. Wherever life takes us next, Virginia, you will always be with me.

To Charles, Ilana, Jamil, Mubdi, and Rob: I am a ridiculous man, and you are the best witnesses for this truth. It is rare to ﬁnd people that resonate in the ways that we do. You are my favourite people and I look forward to our next adventure together. Jamil, let’s pick one with fewer border guards and dubious knee surgeries. Also, Jimmy’s sounds good.

Where would I be without Chris, Joel, Krista, and Naoya? A lot less frustrated and a lot less funny. Heidi, Lauren, Liam, and Max, thank you for living it with me. I’ll miss our arguments and laughs in the library. Also, thank you, Lee, for letting us live there (with enthusiasm). Big praises to Chun, Brent, Steph, Nick, Frank, and Shakira for making me stronger. Thank you, Marten, Chris (Thompson), and Shelley for their lessons and guidance. And, thank you Mike (Reid), Yanqin, and all of the students I’ve TA’ed for helping me become a better teacher.

This is slightly unconventional, but I’d also like to acknowledge Samantha Carter, an astrophysicist from Stargate SG-1. You misled me to believe this profession uses guns and alien tech to solve problems. I am both disappointed and grateful.

I love my cat, Mo, and dog, Lucky.

v Contents

1 Introduction 1 1.1 Treatment of Mass Loss in Stellar Evolution ...... 2 1.2 Physics of Mass Loss ...... 3 1.2.1 Radiation and Metallicity ...... 3 1.2.2 Rotation ...... 10 1.2.3 Pulsations, Eruptions, and Explosions ...... 12 1.3 Summary ...... 15

2 Launching of Optically-Thick Radiatively Driven Winds 16 2.1 Chapter Overview ...... 16 2.2 Background ...... 17 2.3 Stellar Wind Models ...... 20 2.3.1 Structure equations ...... 21 2.3.2 Sonic point criteria ...... 22 2.3.3 Inner boundary: matching a hydrostatic star ...... 25 2.3.4 Regime of validity ...... 25 2.3.5 Numerical Method ...... 26 2.4 Results ...... 32

2.4.1 23 M Helium Star ...... 32 2.4.2 Other helium stars ...... 36 2.5 Inﬂation, Inversion, and Stability ...... 43 2.5.1 Convective instability ...... 43 2.5.2 Onset of Envelope Inﬂation and Extended Winds ...... 45

vi 2.5.3 The nature of weak WR winds ...... 49 2.5.4 Radiation-driven acoustic instabilities ...... 49 2.6 Discussion ...... 50 2.7 Addendum 1: The Phase Space of WR Wind Structures ...... 51 2.8 Addendum 2: Force Multiplier Approximation ...... 52

3 Shock Dynamics in Stellar Outbursts: I. Shock formation 55 3.1 Chapter Overview ...... 55 3.2 Background ...... 56 3.3 Propagation of a sound pulse ...... 58 3.3.1 Thermal diﬀusion ...... 60 3.4 Shock Formation ...... 61 3.4.1 Shock formation radius: heuristic derivation ...... 61 3.4.2 Detailed derivation of shock formation radius ...... 63 3.4.3 Maturation of the shockwave ...... 67 3.5 Hydrodynamic Simulations ...... 68 3.5.1 Planar Earth Atmosphere ...... 69 3.5.2 Spherical Polytropes ...... 72 3.6 Shock dissipation or radiative damping? ...... 72 3.7 Conclusion ...... 76

4 Shock Dynamics in Stellar Outbursts: II. Shock propagation 78 4.1 Chapter Overview ...... 78 4.2 Brinkley & Kirkwood Theory ...... 79 4.2.1 The Derivation ...... 80 4.2.2 Equations for Shock Propagation ...... 87 4.3 Numerical Simulation ...... 88 4.4 A Modiﬁed Brinkley & Kirkwood Theory ...... 90 4.5 Shock Dissipation and ‘Duds’ ...... 95 4.6 An Approximate Shock Formula ...... 98 4.7 Discussion ...... 98

vii 5 Shock Emergence in Supernovae 101 5.1 Chapter Overview ...... 101 5.2 Background ...... 101 5.3 Problem, method, and solutions ...... 103 5.3.1 Shock acceleration parameter and its limits ...... 104 5.4 Post-Shock Flow and Asymptotic Free Expansion ...... 106 5.4.1 The acceleration factor and its limits ...... 107 5.5 Discussion ...... 109

6 Conclusions & Future Work 114 6.1 Optically Thick Radiatively Driven Outﬂows ...... 114 6.2 Non-Terminal Eruptions and Explosions ...... 115 6.2.1 Optically Thick Adiabatic Outﬂows ...... 117 6.3 Future Work ...... 118

Appendices 120

A Acoustic Wave-Steepening 121

B Shock Maturation 124

Bibliography 127

viii List of Tables

3.1 Wave characteristics, planar simulations...... 69

5.1 Shock acceleration index β ...... 111 5.1 – Continued ...... 112

5.2 Shock acceleration factor vf /vs ...... 113

ix List of Figures

2.1 Stellar density proﬁles. Light blue is by Gräfener et al. (2012), and the remaining are wind solutions across a range of mass loss rates. Arrows indicate the sonic point location for each wind model. Dashed regions indicate where the Rosseland approximation is no longer valid...... 33

2.2 Temperature proﬁle for the stellar wind solutions. Arrows indicate the sonic point location. Dashed regions indicate where the diﬀusion approximation is no longer valid. Note the contrast in temperature scale height between weak and strong winds. Compact wind models are truncated to temperatures above the partial ionization zone of helium (104.8 K)...... 34

2.3 Radius and local sound speed at the sonic point location across a range of mass loss rates. The bifurcation in behaviour does not appear in the sonic point location or velocity...... 35

2.4 Wind velocity showing the bifurcation between strong, compact winds and weak, extended ones. The green line indicates the sonic point location and velocity, and dashed regions are where the Rosseland approximation

1/2 −1 becomes invalid. Note that the escape speed is 2100(2R /r) km s for

the 23 M ...... 37

2.5 Profiles of stellar wind models for M = 15, 20, 25, 30M helium stars. For figures (a) and (b), arrows indicate the respective sonic point location. For all figures, dashed-lines indicate where the Rosseland approximation is invalid...... 41

x 2.6 Shown is a diagram illustrating the onset of envelope inﬂation and extended

4 winds for a 23 M (or L∗/M∗ = 2.7 10 (L /M )) star. The density and × temperature structure of the MESA-generated model (thick red line) and stellar wind (colours) extends from the stellar interior (bottom-right) to the

surface (top-left). The stellar wind models trace the L∗κ(ρ, T )/(4πcGM∗) = 1 (greyscale) contour and become strongly radiation-dominated (decreasing

φ = Pg/(4Pr)) until reaching the sonic point (stars). Wind models that cross the line of inﬂation (thick-black) are extended in radius, as the temperature scale height and stellar radius become comparable (see eq. 2.35). Dashed regions of the wind models indicate where the Rosseland

approximation is no longer valid. A 14.25 M star with marginally extended winds is shown in Fig. 2.8...... 42

2.7 The numerical (points) and analytic (dashed line) estimates for the critical mass loss rate for bifurcation across stellar mass. For mass loss rates below this boundary, we ﬁnd slow, extended solutions that are incompatible with

successful WR winds. With our assumptions, the analytic estimate for M˙ b

converges to zero at M∗ = 14.04 M . The analytic model underestimates the critical mass loss rate by a small value towards more massive stars as the location of the iron opacity peak is further than the base of the wind. See eq. (2.36). Blue and red lines are estimates from Petrovic et al. (2006) and Gräfener et al. (2012), respectively...... 47

2.8 A structural diagram of the stellar winds for a 14.25M (or L∗/M∗ = 4 2.4 10 L /M ) star. The Γr = 1 contour recedes to higher densities × and away from the line of inﬂation for stars with decreasing L∗/M∗ or

stellar mass (see eq. (2.29)). At 14.04M , the Γr = 1 contour and line of inﬂation intersect at one point, and any non-zero mass loss rate is suﬃcient to prevent the formation of an extended wind (see eq. (2.36) and (2.37) for the exact criterion). Therefore, the critical mass loss rate for bifurcation increases with stellar mass...... 48

xi 2.9 Shown are the clumping-corrected galactic WR stars with Y 0.85 and ≥ Z = 0.0172 from Nugis & Lamers (2000). The red (blue) lines are for pure helium stars with solar (half-solar) metallicity. Stars below the dashed line (Eq. 2.36) are expected to generate slow extended winds; those above are compact (i.e., no envelope inﬂation or inversion) with strong winds. Solid line indicates the minimum mass loss rate to drive a WR wind due to the iron opacity bump...... 53

3.1 Deformation of a single sinusoidal impulse in a planar, isothermal atmosphere. The wavelength remains constant until a shock forms in the third frame. This indicates the end of stage one and beginning of stage two. The shockwave becomes fully developed and stage two completes once the wave peak (point) coincides with the shock location...... 63

3.2 Evolution of the wave front gradient ∂ru of four waves with diﬀerent grid resolutions. The wave may manifest across 3200 (AMR) grid cells for ∼ the ﬁnest resolution min(∆x) λ/(200 [1, 2, 4, 16]). All initial wave & × properties are A2 from Table 3.1. The dashed line is the analytic wave steepening prediction (Eq. 3.24)...... 70 3.3 Stages and evolution of the wave front gradient (solid black) and peak wave luminosity (purple). The thick coloured band indicates our predictions for Phase 2, which begins with shock formation and ends when the wave peak reaches the shock front. Once a shock fully develops, the peak acoustic wave

2 2 luminosity Lw = 2πr (pw p0) /(ρ0cs0) declines due to shock dissipation. 70 − 3.4 Numerical shock formation from vertically-propagating waves in a planar isothermal atmosphere (scale height 8.8 km). Waves within sets A and B have the same initial wave front gradient and diﬀerent wavelengths and amplitudes (see Table 3.1). See Fig. 3.2 and 3.3 for Figure descriptions. . 71 3.5 Numerical results from the launching of four waves of various strengths and frequencies in a n = 3 stellar polytrope. See Fig. 3.2 and 3.3 for ﬁgure descriptions. A shock becomes strong where the post-shock wave

luminosity Lw exceeds the maximum acoustic luminosity Lmax (black line). 73

xii 3.6 Wave propagation diagram of a core oxygen-burning blue supergiant model evolved in MESA. Filled contours show the exact (eq. 3.25) shock formation

radii for acoustic waves with frequency ω and peak luminosity L¯w launched −2.2 from the convective boundary ri = 10 R . The exact choice ri is indistinguishable from the stellar envelope. Also plotted are the Brunt-

Väisälä (N), acoustic cutoﬀ (ωac), and radiative damping (ωrad) frequencies.

Dashed lines are the critical shock frequency (ωsh), which approximates eq. (3.25) best in the stellar envelope...... 75

4.1 Strength of the acoustic waves and shocks versus stellar depth. Coloured points are from FLASH. Black solid lines are the modiﬁed BK theory. Black

points indicate where the shock has fully developed R = Rs. Coloured lines are the strengths if a shock does not form. Dashed lines represent the −β approximate shock formula amended with the Sakurai solution vs ρ ∝ 0 where the shock is strong (open points)...... 91

4.2 A comparison between the modiﬁed BK solutions and the FLASH simulations. Three iterations for corrections lead to the third panel. Eq. (4.47) is the ﬁrst correction. The second and third corrections involve (4.48); see the discussion for details. The last panel shows the approximate shock formula discussed in section 4.6...... 93

4.3 An example for the third correction for B is the black line. The green line

is the derivation for B without p0 terms (see eq. 4.51)...... 94

4.4 A set of solutions for the BK equations with various initial energies for a sample model. The percentage is the fraction of acoustic energy used as

the initial shock energy. Shocks with suﬃcient energy Emin & 0.75Ew have nearly identical evolutions. Shocks with insuﬃcient energy vanish before reaching the stellar surface...... 96

xiii 4.5 Shown is the shock heat deposited with respect to shock strength. The coloured lines in both panels are the solutions to equation (4.38) given the shock strengths from the FLASH simulations. The black line in the top panel is the prediction from BK theory with the best corrections for B. The black line in the bottom panel is also the solution for (4.38), but for

constant dEs/dt. Dashed lines are solutions to Eq. (4.54)...... 97

xiv Chapter 1

Introduction

Since the primordial universe contained no metal coolants, the ﬁrst stars were likely born massive (Umeda & Nomoto 2003). Hard UV radiation from these ‘Population III’ stars reionized neutral gas, carving HI bubbles to eventually engulf universe. The radiation is generated internally from the fusion of lighter elements until the core collapses or undergoes pair-instability. Either path leads to an explosion called a supernova (SN). The supernova ejecta enriches the galactic chemistry with heavier metals to both ﬁll the periodic table and enable the formation of smaller stars.

Present day massive stars (M∗ > 8 M ) compose less than 1% of newly born stars, yet they continue to act as shepherds in galaxy evolution. Massive stars typically exit the main sequence before low-mass stars enter the main-sequence. During this time, they heat and carve bubbles in the interstellar medium (ISM), shutting down local star formation. Strong radiation pressure on metal lines unbinds the outer stellar envelope to drive stellar winds, and the cumulative kinetic energy is comparable to SN ejecta (Lamers & Cassinelli 1999). Turbulence and shocks generated from these outﬂows are argued to both shut down (Hopkins et al. 2012; Bolatto et al. 2013) and trigger (Bodenheimer 2011) star formation, conditions for which are actively pursued. Absorption from dense outﬂows, dust formation, and molecular dissociation complicates radiative feedback calculations. Nevertheless, these ingredients are vital to study the epoch of reionization (Haiman & Loeb 1997), spectra from early galaxy formation, and star forming galaxies (Smith 2014; Mac Low et al. 2005).

1 Chapter 1. Introduction 2

Between the birth and death of massive stars is a dramatic life. Massive stars can expand in size and luminosity by a hundred-fold and, in most cases (Sana et al. 2012), interfere with their binary companion. The evolution is complex and eventually determined by every element of physics involved, the most fundamental and uncertain of which is stellar mass loss. This thesis explores physics of mass loss in a theoretical fashion. In the next section, I begin discussing how mass loss is incorporated stellar evolution codes and the problems related. Then, I discuss theoretical challenges in stellar mass loss before introducing two problems where advances are made here: (1) Wolf-Rayet stellar winds, and (2) stellar eruptions. The signiﬁcance of these problems to stellar evolution is motivated below.

1.1 Treatment of Mass Loss in Stellar Evolution

α Luminosity varies strongly with mass L∗ M , where α 3 4, for the majority of ∝ ∗ ∼ − massive stars on the main-sequence. Since mass loss is predominantly driven by radiation,

this implies the three parameters (L∗, M∗, M˙ ) are evolutionarily coupled. A variety of other physics also determine the evolution of massive stars: metallicity, opacity, equation of state, ionization and abundances, rotation, binaries, magnetic ﬁelds, energy transport mechanism (e.g., convection, radiation, pulsations), and more. The physics of mass loss is the most uncertain since it often requires an understanding of many, if not all, of these components. To circumvent this problem, stellar evolution codes have historically adopted empirical mass loss rates. Empirical relations are functional ﬁts to the apparent properties of a set of carefully observed stars that span the H-R diagram. For thirty years, the relations from de Jager et al. (1988) and Nieuwenhuijzen & de Jager (1990) for O-type stars and red super giants (RSGs), and Nugis & Lamers (2000) for Wolf-Rayet stars (Smith 2014) have been used. Once a stellar model is constructed, the model’s photospheric predictions are used to prescribe the mass to be removed from the stellar surface in the next time-step. Over time, a number of observations have raised concerns regarding the measurement

2 of the mass loss rate M˙ = 4πr ρ(r)v∞. Free-Free radio emission and recombination Chapter 1. Introduction 3

lines are often probes for the mass density ρ(r). Since the emissivity follows ρ2 , the h i inferred density appears larger due to inhomogeneities or ‘clumps’ in the wind (Lamers & Cassinelli 1999; Smith 2014). Polarization variations provide measures for ρ (St.-Louis h i 2 2 et al. 1993) and suggest clumping factors fcl = ρ / ρ of up to 10. This is consistent h i h i ∼ with more recent observations of O-stars in the Milky and both Magellenic clouds, which

suggest a reduction in empirical rates by √fcl 2 3 (Crowther et al. 2002; Hillier et al. ' − 2003; Massa et al. 2003; Evans et al. 2004; Bouret et al. 2005; Puls et al. 2006). The

clumping factor fcl is found to be the dominant uncertainty for mass loss in massive stellar evolution (Renzo et al. 2017). Empirical mass loss relations continue to serve stellar evolution codes because there is no alternative relation that is more accurate. These relations are often scaled by some constant clumping average. Vink et al. (2001, here on VKL) compute theoretical rates by following the fate of photons, and their line interactions, in a Monte-Carlo fashion (see next section for details). While these wind solutions are implicitly homogeneous, there is overall agreement with de Jager et al. (1988) for main sequence O-stars. But, what about the post-main sequence? Indeed, massive stars spend most of their lives in the hydrogen-burning phase. However, post-main sequence mass loss rates can increase by orders of magnitude. There is neither suﬃcient theory or computational resources to do such calculations. The predominant reason is that the wind-acceleration mechanism changes with stellar phase and each mechanism can be rich in complexity. In the next section, we discuss the basic physics beneath most mass loss mechanisms and introduce problems advanced in this thesis.

1.2 Physics of Mass Loss

1.2.1 Radiation and Metallicity

While hydrogen and helium are the most abundant elements in the universe, they do not contribute significantly to the radiative acceleration (Abbott 1982). Metal-poor stellar winds from the Magellanic clouds are seen to have slower terminal speeds than those from the Milky Way (Garmany & Conti 1985; Prinja 1987). If metal-rich stars have strong Chapter 1. Introduction 4 winds then the evolutionary consequences are accentuated after core hydrogen-burning when stars become more luminous and, therefore, drive more mass loss. NGC 6822 is an example where its metallicity (Z = 0.005) is lower than M31’s (Z = 0.036), but the maximum bolometric luminosity of RSGs is higher (Massey 1998; Maeder 2008). Since mass loss exposes deeper (or hotter) layers, the number ratio of blue supergiants to RSGs in clusters is seen to increase with metallicity, with hardly any RSGs found in the galactic center (Eggenberger et al. 2002). Massive stars that lose sufficient mass enter the Wolf-Rayet (WR) phase where their helium interiors are exposed (Crowther 2007). Nitrogen from the CNO-cycle (WN stars) and carbon from helium-burning (WC stars) are often present. For similar reasons to RSGs and BSGs, the fraction of WR/O, WC/WN, and WR/RSG stellar types are all found to increase with metallicity (Maeder & Conti 1994; Massey 2003); although, close binary interactions are argued to strongly influence the production of WRs as well (Crowther 2007). A handful of severely stripped WR stars

(WO stars) are found with little surface helium abundance Ys < 0.5 and no hydrogen

Xs = 0 (Tramper et al. 2015).

A goal of this thesis is to understand the driving mechanisms behind mass loss. The best understood systems for how metals partake in an outflow are O, B, and A-type stars. These stars emit UV photons that are either absorbed or scattered by metals lines near the stellar surface (Lamers & Cassinelli 1999). The radiative pressure on metal ions Coloumb drags the surrounding protons, helium ions, and electrons and drives gas away from the star. As the metals accelerate, the lines Doppler-shift and access untouched parts of the radiative spectrum. This causes a runaway acceleration until the radiative flux is negligible and the outflow reaches a terminal speed.

It has been thought that the maximum momentum (per second) of the wind Mv˙ ∞ was bound by the radiative momentum L∗/c (Lamers & Cassinelli 1999). In other words, the momentum transfer eﬃciency or eﬀective number of photon-scatterings

Mv˙ ∞ η = . 1. (1.1) L∗/c

The single-scattering limit is not a physical law, however. An outﬂow with an abundant Chapter 1. Introduction 5

and diverse set of metals will contain a dense population of lines. If the average gap

between these lines is less than 2(v∞/c)ν then multiple-scattering is likely. Castor et al. (1975) assume a single-scattering limit by considering an ensemble of lines that are well separated in frequency-space (i.e., no overlap). The total acceleration is then a multiple of single-line acceleration. CAK ﬁnd an analytic solution to the wind structure when solving the hydrodynamic equations with radiative forces, and is often referred to as the β-law:

β R∗ v(r R∗) = v∞ 1 , (1.2) ≥ − r

for constant β 0.5 and stellar radius R∗. As an aside, CAK argue the acceleration ' grows proportionally to (dv/dr)α with α 0.4 0.6. If we consider adding a sinusoidal ' − perturbation to a one-dimensional monotonically accelerating outflow, a differential velocity field leads to a deformation, clumps and, possibly, shock formation (Lamers & Cassinelli 1999).

The benefit of the CAK wind model is the analytic solution to the velocity structure for a steady homogeneous wind. It provides simple relations that connect the metallicity to the mass loss rate and terminal wind velocity (Puls et al. 1998). However, there are details besides wind instabilities that question the underlying assumptions (Krtička 2006). For example, the distribution of lines is not uniformly spaced, and so there is no guarantee that multiple line scattering is not important. Because of this, VKL follows photons in a Monte Carlo fashion from the inner stellar boundary (or photosphere) through the outflow where they can multiply-scatter. This is not a hydrodynamical calculation since a CAK velocity structure is prescribed, but global consistency is ensured (Vink et al. 2000; 2001). A semi-analytical treatment for the radiative acceleration permits a wide range of stellar parameters to be explored to derive a theoretical mass loss relation. As mentioned previously, their relation is in good agreement with the empirical rates for most O-stars (ignoring clumping effects) and is a testament to line-driven wind theory.

The VKL relation is, however, found in disagreement with smaller massive stars.

O-dwarfs (M∗ . 25 M ) and early-B stars are observed to have winds 100-fold weaker than predicted, which is referred to as the weak-wind problem (Martins et al. 2005; 2012). Chapter 1. Introduction 6

The discrepancy is found to be partly from the diagnostic quality of UV lines from cooler temperature stars. Analyses of the bow shock around ζ Oph (Gvaramadze et al. 2012) and X-ray lines (Huenemoerder et al. 2012) reduce the discrepancy by 10-fold, and clumping reduces the theoretical rates by 2 4; however, a discrepancy of 3-fold remains. −

Optically Thick Radiation-Driven Winds

5.2 Stars above 50 M or 10 L were not explored by VKL. Gräfener et al. (2011) and Vink et al. (2011) show that the mass loss rate becomes enhanced for very massive stars (Vink 2014) with luminosities beyond 50% their Eddington luminosity (the critical limit ∼ where radiation balances against gravity). This is empirically confirmed from Of (N III and He II emission) and WNh (hydrogen-rich) stars from 30 Doradus (Bestenlehner et al. 2014). And, this is confirmed by Monte Carlo calculations like VKL but for very massive stars by Vink & de Koter (2002) and Smith et al. (2004). A clue for the mass loss enhancement comes from the momentum transfer efficiency η. In the theoretical models, the photons surpass the single-scattering limit and can reach values of η 2.5; thus, the winds are becoming optically thick. This transition can be → explained in two ways. First, the stronger radiative pressure pushes the wind acceleration region deeper into the star where optical depths are higher. Second, the increasing stellar temperatures change the population of metal ions available for acceleration. The best example that demonstrates this is a WR star. Before explaining how, we first introduce an important parameter for optically thick outflow called the Eddington ratio

Γrad Lrad/LEdd. ≡ Consider the radiative acceleration on a spherical shell of gas with associated monochro-

matic opacity κν, transmitted radiative ﬂux Fν, for given photon frequency ν:

Z ∞ κνFν grad = dν. (1.3) 0 c

The Eddington luminosity mentioned previously is for electron or Thomson scattering

opacity κν = κes, which is constant. Thus, (1.3) is simply

L g = rad κ . (1.4) rad 4πr2c es Chapter 1. Introduction 7

Given Newton’s constant G, enclosed mass M, and Eddington luminosity Les = 4πcGM/κes =

Lrad/Γes (for electron scattering), the eﬀective gravity is then

GM κesLrad GM(1 Γes) geﬀ = + = − . (1.5) − r2 4πcr2 − r2

Since electron scattering occurs where there is ionized gas, it applies below and just above

the stellar surface. Therefore, massive stars with Γes & 0.5 have half the gravity due to electron scattering. This increases the effectiveness of line-driven acceleration and overall mass loss rate. Of course, metals also contribute to the opacity via free-free, bound-free, and bound- bound processes, for example. An effective mean opacity can be computed for the ensemble of frequency-dependent processes but must be done carefully. Since stars release heat, their interiors contain a temperature gradient and the radiation field is not isotropic. Despite this, a good approximation is to assume the gas is in local thermal equilibrium (LTE) and quasi-isotropic. The net radiative flux through a gas is the difference in

hotter and cooler black body spectrums or dB/dT , masked by the opacity κν. This is encapsulated by the following harmonic mean

R ∞ dBν (ρ,T ) 0 dT dν κRoss(ρ, T ) κ = , (1.6) ≡ h i R ∞ 1 dBν (ρ,T ) dν 0 κν dT

and is called the Rosseland mean opacity, which can be applied in the stellar interior

by substituting κes κRoss(ρ, T ). Extensive tables are provided by the OPAL (Iglesias → & Rogers 1996) and OP teams for κRoss (Badnell et al. 2005). Since electron scattering

is included in κRoss, the ‘total’ Eddington ratio for the very massive stars considered is higher. The Eddington ratio is connected to the mass loss rate by a unique constraint called the sonic point criterion. At the sonic point, the momentum equation becomes critical such that there is only one non-divergent solution that can lead to a wind. This criterion exists for all transonic outﬂows and was used to derive the CAK wind velocity structure (1.2). In Chapter 2, we discuss the details of how the sonic point manifests for a radiatively-driven optically thick wind. But to summarize, the criterion can be thought of as an outer Chapter 1. Introduction 8

boundary for the subsonic gas and inner boundary for the supersonic wind. Enforcing a self-consistent ‘boundary’ condition with the stellar interior can be numerically difficult, which is why most theoretical mass loss calculations have been limited to the supersonic region. The subsonic structure is often treated semi-analytically, but not necessarily self-consistently. The mass loss calculations for very massive stars predict multiple-line scatterings in the supersonic domain only, so the sonic point must reside at a higher optical depth and possibly below the atmosphere. An example of where this is likely true is WR stars. Their spectra are richly filled with strong broad emission lines with P-Cygni profiles indicative of 1,500-3,000 km/s outflow. The outflow is optically thick to the continuum for an extended radius, which suggests complex acceleration processes. The obscuration of the ‘hydrostatic’ surface prohibit measurements of basic stellar parameters such as the rotation, temperature, and radius.

Wolf-Rayet Winds

For twenty years, WR stars have driven the development of non-LTE atmosphere codes. Simplifications and approximations, like those by VKL, have been adopted since modeling an expanding atmosphere is a complex radiative-hydrodynamic problem. These homogeneous wind models decouple the radiative transfer and hydrodynamic calculations by imposing a CAK velocity structure, allowing radiative processes to be computed in a co-moving frame. Although this is a significant simplification, the task remains formidable considering the millions of (Doppler-shifted) lines and varying ion populations that can interfere with photon propagation. These one-dimensional codes have been critical in studying WR spectra. A summary of these codes is provided by Sander (2017). An iterative comparison between observed and synthetic spectra help constrain the parameters of otherwise invisible stars hidden below the winds. These codes infer surface

4.5 5.2 temperatures between T∗ 10 10 K, where heavy metals are known to contribute a ' − signiﬁcant density of lines. In particular, millions of bound-bound iron lines exist around ( 0.1 dex) 104.7 and 105.2 K, increasing the opacity up to a factor of 3. These regions ∼ of enhanced opacities are referred to as ‘iron opacity bumps’. Beneath the hot iron bump, the stellar structure is relatively simple as WR stars are eﬀectively helium-burning Chapter 1. Introduction 9

main-sequence stars (Crowther 2007). Hydrostatic models predict radii between 1-3 R

for the observed mass range 10-25 M (Schaerer & Maeder 1992). And yet, atmospheric codes predict radii up to 10 times larger.

The significant radius discrepancy highlights the disconnect between stellar models and atmosphere codes. And, it raises the question as to what a locally super-Eddington layer does to a star. Petrovic et al. (2006) and Gräfener et al. (2012) find that suppressing the outflow (i.e.,v(r) = 0) around the hot iron opacity bump inflates the stellar envelope by an order of magnitude in radius; a model they propose to resolve the radius discrepancy problem.

The proposed hydrostatic model contains very strange structural features. Surrounding the star is an energetically unbound radiation-dominated halo that is confined by a relatively thin and massive shell. To remain hydrostatic, a positive pressure gradient induces a density inversion to counteract the net outward force from a super-Eddington layer. While the structure is a valid solution (Joss et al. 1973), it is prone to one- dimensional (Paxton et al. 2013) and multi-dimensional instabilities (Jiang et al. 2015). Non-adiabatic stability analyses (Glatzel et al. 1993; Saio et al. 1998; Glatzel & Kaltschmidt 2002) find extended envelopes to be excited by violent ‘strange mode’ pulsations and transonic motions. Maeder (1992) discuss a number of hydrodynamic scenarios including strong turbulent motion, mechanical wave luminosity, eruptive geysers, and outflows as plausible scenarios. The stability of very massive or luminous stars rests on the stellar response to an opacity bump (Sanyal et al. 2015; Goodman & White 2016).

At some depth beneath the Eddington layer, the star becomes (relatively) stable. A near-Eddington layer will trigger convection (Joss et al. 1973), which may generate density inhomogeneities and wind clumps (Cantiello et al. 2009). However, the low envelope densities prevent convection from reducing the radiative luminosity to sub-Eddington values for more massive WR stars. And, the inertial acceleration from a background velocity field acts to stabilize against convection. Nugis & Lamers (2002) argue the sonic point sufficiently deep below the atmosphere where the gas is in LTE, but close to the surface where the envelope is effectively radiative. They conclude the iron opacity bump is responsible for launching WR winds. At its face, this could be an argument against Chapter 1. Introduction 10

the assumption of hydrostatic equilibrium around the iron opacity bump. However, the

critical sound speed at the iron opacity bump is the gas sound speed cs 30 km/s, which ' is quite small, and not the total (i.e., adiabatic) sound speed 300 km/s. Therefore, it is ∼ unclear whether an outflow will necessarily erase the extended structure. This question is answered in Chapter 2. We solve for the transonic outflow structure for a WR star that implicitly solves the sonic point criterion. We find the extended structure can persist for a slow steady wind up to a critical mass loss rate. When compared to the population of WR stars, we find all their winds to be too strong to retain the structure. In this chapter, we also add a discussion that is not in the published article (Ro & Matzner 2016). We derive a minimum mass loss rate for an opacity bump to harbour a sonic point or, in other words, launch an optically-thick outflow. WR stars are found to exceed this threshold and are indeed driven by iron lines. This threshold is highly sensitive to the iron abundance (or metallicity) and argued to be important to compute mass loss rates for metal-poor stars from the Magellanic clouds or early universe. A tangential application for this threshold is during the end of massive pulsating AGB stars. Lau et al. (2012) finds the last thermal pulse to be sufficiently strong to push gas above the helium-burning shell. The expanding gas cools to the temperature of the hot iron opacity bump, and strong radiative pressures are seen to drive a superwind until the code crashes.

1.2.2 Rotation

The initial rotation of massive stars comes from the accretion of star forming gas. While the initial spin of the gas is small, the specific angular momentum must be reduced by a factor of 105 106 for a star to form (Maeder 2008). The angular momentum is shed by − a combination of viscosity, bipolar outflows, and magnetic winds. Where these effects are inefficient or there is gravitational instability, the system may fragment into a binary and convert specific angular momentum into the orbit (Kratter et al. 2010). This is likely a common phenomenon considering how most massive stars are found in binaries that are close enough to interact and/or merge (see Smith (2014) for a list of sources). Once a star is formed, angular momentum is primarily gained from a companion (by accretion or Chapter 1. Introduction 11

merging; Sana et al. (2012)) and lost by mass loss (Maeder 2008).

Since mass loss is strongly dependent on metallicity, metal-poor stars are expected to be faster rotators. Fast rotators are seen at low metallicities (Maeder et al. 1999; Martayan et al. 2007), however, these spectroscopic observations may be partly skewed because the stars are in binaries. The early universe contains little to no metals, so most stars are expected to be fast rotators.

Massive stars are seen to rotate with a range of velocities (Huang & Gies 2006; Maeder 2008; Penny & Gies 2009; Huang et al. 2010), many of which are near critical. At the equator, the centrifugal acceleration acts against gravity and reduces the effective mass. Combined with radiative acceleration, the critical Eddington luminosity for a rotating star is reduced 2 4πcGM∗ Ω LEdd,Ω = 1 , (1.7) κ − 2πGρ¯M where ρ¯M is the mass-average density and Ω is the rotation rate (Maeder 2008). A rotating star is oblate since the centrifugal force is co-latitude dependent; however, mass loss is not necessarily strongest at the equator. The von Zeipel effect describes how the decreased effective gravity around the equator reduces the relative temperature and flux there (von Zeipel 1924). While this is confirmed from spectral modeling of fast-rotating stars (Domiciano de Souza et al. 2003; 2005), the effect is not consistent with direct imaging (Monnier et al. 2007). As a result, outflows are expected to be generally stronger at the poles; although, Maeder & Desjacques (2001) discuss the possibility of disk or cone ejection due to opacity bumps.

The consequences of mass and angular momentum transfer for stellar and binary evolution is truly vast. The computational resources are not available for a complete numerical study, and simplifications are necessary. But, there still remains basic uncertainties in the physics of mass loss for non-rotating stars. As the first step, this thesis only considers massive non-rotating stars where the companion is not actively influential. I suggest the reader see detailed reviews by Maeder & Meynet (2000) and Langer (2012) to find great discussions on rotation, binaries, and stellar evolution. Chapter 1. Introduction 12

1.2.3 Pulsations, Eruptions, and Explosions

Evidence for eruptive mass loss is substantial in massive stars. Progenitors of type IIn supernovae (SNe) are commonly (Ofek et al. 2014) found deeply enshrouded in dense circumstellar material (CSM) (Smith et al. 2011; Kiewe et al. 2012; Smith 2014; Moriya

−3 −1 et al. 2014; Margutti et al. 2016). The inferred mass loss rates (10 1 M yr ) are − unsustainably high which suggests a short-lived ejection event (Khazov et al. 2016; Yaron et al. 2017). Super-Eddington pre-supernova outbursts have been directly seen from SN 2006jc, SN 2009ip, SN 2010mc, SN 2015bh, and LSQ13zm within years to weeks before core-collapse (Pastorello et al. 2007; Foley et al. 2007; Margutti et al. 2014; Smith et al. 2014; Thöne et al. 2016; Tartaglia et al. 2016). Resolved circumstellar relics of giant eruptions are found around the luminous blue variables (LBVs) η Carinae and P Cyg (Smith & Hartigan 2006; Smith 2014). These are a rare class of stars that experience dramatic instabilities in the post-main sequence phase. They undergo irregular variability, or S Doradus outbursts, in both temperature (from peak UV to visual) and visual brightness (by 1-2 mag) (Smith 2014). And, their quiescent

5.4 mass loss rates are 10-fold larger than equally luminous O-stars (LLBV /L & 10 ). LBV eruptions are inferred to last months to decades in duration, ejecting up to tens of solar masses in gas (Smith et al. 2003; Smith 2006) and dust (Kochanek 2011), which would have strong consequences to the star’s evolution. Substantial increases in bolometric luminosities and extreme mass loss (Humphreys et al. 1999; Smith 2014) easily disguise these eruptions as SN explosions (e.g., SN 1961v Van Dyk & Matheson 2012a). SN 2009ip is the most appealing link between LBVs and pre-SN outbursts because of its multiple outbursts and S Dor-like variability before terminal explosion in 2012. The timing of pre-SN outbursts motivates a causal connection with the advanced nuclear burning stages, where a number of instabilities are expected. Low mass progenitors

( 9 11 M ) of electron-capture SNe have been predicted to contain excite pressure ∼ − pulses and eject the stellar envelope (Woosley & Heger 2015). Likewise, pulsational pair-

instability in very massive stars ( 95 135 M ) generate suﬃcient energy for multiple ∼ − shell ejections (Woosley et al. 2007). For intermediate masses, vigorous convection (Quataert & Shiode 2012; Shiode & Quataert 2014; Smith & Arnett 2014) and turbulent Chapter 1. Introduction 13

entrainment near a convective burning zone (Meakin & Arnett 2007) can excite waves with suﬃcient energy to unbind the stellar envelope. A nuclear energy source seems diﬃcult to adopt for LBV eruptions since the stars continue to survive after 102 to 104 years (Smith 2014). However, injection of fresh fuel into deep burning layers could generate pulsational instabilities (Podsiadlowski et al. 2010). Wave production from a binary merger or collision can also disrupt a stellar envelope (Smith 2011); although, another ejection mechanism may be necessary to produce multiple eruptions like η Car.

Pulsations are necessary to drive winds from cool stars (e.g., RSGs, AGBs, (thermal- pulse) TP-AGBs, OH/IR stars). These pulsations may come from convection (Kiss et al. 2006; Freytag & Höfner 2008) or instabilities from partial hydrogen ionization (Li & Gong 1994). A pulse drives over-densities above the photosphere where the temperature is suﬃciently low for dust condensation. Theoretical sonic point calculations for these winds are similar to those for WR stars since the grains are optically-thick. However, drag forces are necessary to pull hydrogen and helium into the wind. Without pulsations, the density and, thus, mass loss rate at the dust condensation radius would be too small (Lamers & Cassinelli 1999). Cool stars generate some of the strongest outﬂows (Smith 2014) with some RSGs experiencing ‘superwind’ phases (Yoon & Cantiello 2010). Three-dimensional simulations of AGB stars are found to generate strong plumes, pulsations, and shocks such that the dynamics dominate the envelope pressure support (Freytag et al. 2017).

The common transport mechanism amongst the variety of energy sources discussed is waves. Waves deﬁnitively propagate from the core to the surface on a sound-crossing time, which may span between hours to weeks for small and large stars. They can carry super- Eddington luminosities since waves are mechanical and not thermal. Quataert & Shiode (2012) and Shiode & Quataert (2014) argued that waves radiatively dampen to heat and unbind the outer stellar envelope. However, in Chapter 3, we derive how acoustic waves steepen in a medium with arbitrary structure (e.g., gradients in temperature, density, and adiabatic index) and the conditions for shock formation. We conclude super-Eddington waves must always steepen into weak shocks before radiative damping is eﬃcient. As a result, we argue wave-driven modes of super-Eddington mass loss must involve weak shocks. Chapter 1. Introduction 14

How do weak shocks drive mass loss? By deﬁnition, a weak shock carries relatively little energy in comparison to the thermal energy of the medium. A good approximation of a weak shock is an acoustic wave that converts wave energy into heat as it propagates. Detection of fast motions surrounding both 2009ip (2,000-5,000 km/s) and η Car (5,000 km/s) suggest a strong explosive mechanism (Smith 2008; Foley et al. 2011). This could be achieved by weak acoustic shocks if they accelerate as strong shocks do in supernova (i.e., shock breakouts, Matzner & McKee 1999). In Chapter 4, we revive a shock propagation theory by Brinkley & Kirkwood (1947) and suit it to treat shocks in an arbitrary medium. This theory is then used to predict the evolution and acceleration of a weak shock through a stellar polytrope until it becomes strong and emerge out of the star. While this is a pure hydrodynamic consideration, the breakout conditions and dynamical consequences of a weak eruption or explosion are discussed. We also discuss the minimum energy for a shock breakout to occur at all by considering shocks that disappear within the envelope. In Chapter 5, we derive the self-similar strong shock acceleration parameters by (Sakurai 1960) for arbitrary density gradients and adiabatic ﬂuids as well. This is necessary to guide the Brinkley & Kirkwood theory from the weak to the strong limits.

While weak shock acceleration may explain the fast motions seen around 2009ip and η Car, the bulk of the outflow expands at hundreds of km/s – speeds more typical of stellar winds. η Car’s giant eruption lasts for years, which is much longer than its dynamical time. Furthermore, Quataert et al. (2016) finds the steady deposition of super-Eddington rates of energy (of agnostic origin) generate an adiabatic outflow with bulk properties consistent with LBVs and type IIn outflows. These features are nevertheless consistent with the effects of multiple weak shocks. In Chapter 4, we explain how the shock dissipation rate is strongest when the shocks are weak. And, we discuss how the cumulative deposition of heat from a train of weak shocks may drive envelope expansion and an adiabatic outflow. While the adiabatic wind models by Quataert et al. (2016) are for local depositions of heat, an extension could be made to include non-local depositions of (shock) heat. Chapter 1. Introduction 15

1.3 Summary

Optically thick outflows can be radiative, adiabatic, or a mixture of both. This thesis explores two acceleration mechanisms that trigger one or more of these types of outflows from massive stars. The first regards a strong radiation field diffusing through gas with high opacity. In particular, we study the launching of WR winds due to the iron opacity bump and the stability of the outflow structure. The second considers the steepening of intense acoustic waves and the formation of weak shocks. Deep deposition of shock heat leads to adiabatic expansion, and possibly and outflow, so long as the photon diffusion speed is slow. Shock acceleration can lead to a shock breakout and the production of fast SN-like ejecta. In this thesis, I argue how the results presented here will lead to new projects regarding stellar wind theory. A summary can be found in the Conclusion. Chapter 2

Launching of Optically-Thick Radiatively Driven Winds

2.1 Chapter Overview

Hydrostatic models of Wolf-Rayet stars typically contain low-density outer envelopes that inflate the stellar radii by a factor of several and are capped by a denser shell of gas. Inflated envelopes and density inversions are hallmarks of envelopes that become super-Eddington as they cross the iron-group opacity peak, but these features disappear when mass loss is sufficiently rapid. We re-examine the structures of steady, spherically symmetric wind solutions that cross a sonic point at high optical depth, identifying the physical mechanism by which outflow affects the stellar structure, and provide an improved analytical estimate for the critical mass loss rate above which extended structures are erased. Weak-flow solutions below this limit resemble hydrostatic stars even in supersonic zones; however, we infer that these fail to successfully launch optically thick winds. Wolf- Rayet envelopes will therefore likely correspond to the strong, compact solutions. We also find that wind solutions with negligible gas pressure are stably stratified at and below the sonic point. This implies that convection is not the source of variability in Wolf-Rayet stars, as has been suggested; but, acoustic instabilities provide an alternative explanation. Our solutions are limited to high optical depths by our neglect of Doppler enhancements to the opacity, and do not account for acoustic instabilities at high Eddington factors; yet

16 Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 17

they provide useful insights into Wolf-Rayet stellar structures. See Ro & Matzner (2016) for the published article.

2.2 Background

The importance of mass loss in massive stellar evolution is most evident in the Wolf-Rayet (WR) stars, whose defining feature is an optically thick stellar wind. WR winds are an order of magnitude more dense than winds of O-type stars, which is sufficient to extend the line- and continuum-forming regions into the wind (Crowther 2007). A WR star’s wind enshrouds its hydrostatic interior, and hides fundamental stellar parameters such as mass, radius, and rotation from direct observation. This is problematic for the study of phenomena that hinge on these parameters, on the detailed stellar structure, or on the star’s evolution. Examples include binary evolution, tidal interactions, and WR populations within starburst and Wolf-Rayet galaxies (Schaerer et al. 1999). Should a WR star undergo core collapse and explode, the radius and structure of its outer envelope control the production of a shock breakout flash and the pattern of its fast ejecta (Matzner & McKee 1999; Ro & Matzner 2013) as well as the properties of its early light curve (Chevalier 1992a; Nakar & Sari 2010; Rabinak & Waxman 2011). The uncertain regions of WR structure are not small. Hamann et al. (2006) and

Crowther et al. (2006) estimate hydrostatic radii (R∗) by extrapolating the wind structure to a Rosseland optical depth of 20, assuming a β-law velocity structure (Castor et al. ∼ 1975):

β v = v∞(1 R∗/r) . −

Taking v∞ from observation and ﬁxing β = 1, these authors infer hydrostatic radii

( 3 10R ) up to an order of magnitude larger than those of reference models ( 1R ). ∼ − ∼ Although the β-law profile is uncertain, this raises the first question: what inflates WR structures? The strongest clue in this puzzle has been the discovery (by the OPAL opacity project: Rogers & Iglesias 1992) of an opacity peak at temperatures around 105.2 K due to bound- bound and bound-free transitions of iron nuclei. The peak, which joins another due to Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 18

He II at T 104.6−4.8 K, gained considerable support by resolving the ‘bump and beat’ ∼ mass discrepancies in Cephied variable models (Moskalik et al. 1992). In the WR context the Fe and He opacity peaks are especially important, as these stars are not far below the electron-scattering Eddington limit. The Eddington ratio Γ(r) = κ(r)L(r)/[4πGM(r)c] increases by a factor of several as temperatures cross through the Fe opacity peak, so that Γ approaches or even exceeds unity. Nugis & Lamers (2002) suggest this to be the root cause of these stars’ thick winds, which has been supported by wind models from Gräfener & Hamann (2005). Hydrostatic models of WR stars do indeed show inﬂated envelopes. Ishii et al. (1999), Petrovic et al. (2006), and Gräfener et al. (2012) construct such models using updated OPAL opacity tables, and discover a signiﬁcant redistribution of stellar material due to the Fe opacity bump. In these regions Γ approaches unity, and the pressure becomes

strongly dominated by radiation because dPgas(r)/dPrad(r) = 1/Γ(r) 1 (in hydrostatic, − radiative zones). The density scale height can also become very large, as it scales inversely

2 with the local effective gravity geff = (1 Γ)GM(r)/r . Gas density therefore declines only − slowly with radius, a feature which is not erased by the onset of convection. (Envelope inflation has also been observed in non-WR massive stellar evolution models: Köhler et al. 2015; Sanyal et al. 2015.)

A curious structure arises within one-dimensional hydrostatic models where Γ > 1. To balance the net outward force of radiation and gravity, gas pressure must rise towards the surface. For this reason, inflated envelope models experience a density inversion and are capped with a denser shell of gas. While the validity of such structures has been defended (Joss et al. 1973), strong instability is observed in one-dimensional evolutionary models (Paxton et al. 2013). Non-adiabatic stability analyses (Glatzel et al. 1993; Glatzel & Kaltschmidt 2002; Saio et al. 1998) find extended envelopes to be excited by violent ‘strange mode’ pulsations. It is very likely that non-hydrostatic, non-steady, or three- dimensional effects arise; Maeder (1992) considers strong turbulent motion, mechanical wave luminosity, eruptive geysers, and outflows as plausible scenarios. It is important to note that extended outer envelopes in hydrostatic models are often inconsistent with the mass outflow rates of WR stars. The assumptions of a hydrostatic Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 19

model are valid only where outﬂow motions are subsonic and carry a negligible fraction of the luminosity. The extended envelope of Petrovic et al. (2006)’s models reach densities of

−10 −3 −1 ρ(r) 10 g cm and isothermal sound speeds ci(r) 35 km s at radii 4R . The ' ' ∼ ˙ 2 ˙ −5.4 outflow Mach number v/ci = M/(4πr ρci) therefore exceeds unity for M & 10 M /yr. −5.5 −4.0 This is a low value for WR mass loss (M˙ 10 to 10 M /yr; Nugis & Lamers 2000). ∼ We therefore have a second question: how much mass loss alter the structure of WR envelopes? The question is not new. Kato & Iben (1992) suggested that an enhanced opacity bump can generate an optically-thick wind and extend the effective photospheric radius. Kato & Iben constructed an artificial opacity peak to test this hypothesis, and discovered a core-halo configuration in which the original compact stellar core is surrounded by an optically-thick outflow. Petrovic et al. (2006) propose an answer based on models in which mass loss is implemented within a stellar hydrodynamic code. Above a critical mass loss rate, which they identify with an outflow speed equal to the escape speed (as estimated from the hydrostatic density profile), extended model envelopes disappear. However, we are left with several questions. The escape speed is of order 30ci; so, why is the structure not altered by mass loss that is thirty times weaker? Second, what special conditions arise at the wind sonic point? Lastly, we are confused by the statement by Petrovic et al. (2006) that they remove ”a proportionate amount of mass from each shell” in the outer 40% of the stellar mass, as it is not clear how this corresponds to a steady outflow. A self-consistent model must include the dynamics of the transition between the envelope and the wind. Our goal, therefore, is to evaluate the impact of the opacity bump on dynamically self- consistent Wolf-Rayet stellar structures and winds, and to re-examine the consequences of mass loss for the survival of extended envelopes. Using OPAL opacities, Nugis & Lamers (2002) analyzed the opacity bump for its capacity to launch a transonic wind. They show that, within a radiation-dominated wind with diminishing radiative luminosity dLr/dr < 0, the sonic point (where v = ci) must reside where the opacity increases outward. Their examination of the sonic point Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 20

conditions showed the iron opacity bump and the smaller He II bump both have the capacity to launch optically-thick winds and to explain the observed mass loss rates of WR stars. But if opacity bumps are responsible both for envelope inﬂation and for wind launching, can envelope inﬂation ever coexist with a wind?

We aim to address this question by solving for the dynamical transition between envelope and wind implied by the iron opacity bump. We will capitalize on the high optical depths of WR winds by using tabulated Rosseland opacities to integrate through the wind sonic point; this is both a useful simplification, and a limitation of our results. In § 2.5.4, we discuss how acoustic instablities can generate density fluctuations, which can modify the effective opacity and alter the stellar structure. For simplicity, we do not include these effects. In § 2.3, we describe the assumptions and numerical methods used to construct our WR wind models, and re-examine the sonic point conditions. In § 2.4 we investigate a range of wind models for the same WR progenitors used by Gräfener et al. (2012), in order to understand the influence of dynamics upon an inflated envelope and to explicitly determine the maximal mass loss rate to retain such a structure. In § 2.4.2 we present wind models for a range of progenitor masses.

2.3 Stellar Wind Models

We will explore steady, one-dimensional, spherically symmetric models of Wolf-Rayet winds. We begin by enumerating the equations to be solved (§ 2.3.1) and then re-examine the sonic point conditions. Therefore, our models become inaccurate where the spherical flow is unstable and where the Rosseland approximation is not appropriate. Many WR winds are sufficiently optically thick that this approximation is valid through the sonic point. However, because we do not account for the increase in opacity due to Doppler shifting of the lines, we do not integrate through the wind photosphere. We cannot, therefore, solve self-consistently for the mass outflow rate. Instead we adopt a range of values for M˙ and study the structure of the outer envelope and deep wind for each value. A list of conditions is discussed in Section 2.3.4. Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 21

2.3.1 Structure equations

2 The total pressure P = Pg + Pr is composed of an ideal gas pressure Pg = ρkBT/µ = ρci 4 2 and radiation pressure Pr = arT /3 = ρcr, where T is temperature, µ the mean molecular weight (units of mass), and ar the Stefan-Boltzmann radiation constant. Note that ci is the isothermal sound speed.

We solve a simple set of equations for steady spherical ﬂow, equivalent to those adopted by Nugis & Lamers (2002): mass conservation

d r2ρv = 0, (2.1) dr

corresponding to a constant mass loss rate M˙ = 4πr2ρv (where ρ is the density, v is the velocity at radius r from the stellar centre); momentum conservation, in the form of the Euler equation dv 1 dP GM v + = ; (2.2) dr ρ dr − r2 and energy conservation,

Lr + M˙ = E˙ = constant, (2.3) B where Lr is the radiative luminosity in the ﬂuid frame at radius r,

1 = w + v2 v2 (2.4) B 2 − k

2 is the Bernoulli factor, or ratio of energy flux to mass flux, vk = GM/r is the negative 2 2 gravitational potential (square of the Kepler speed), w = 5ci /2 + 4cr is the specific enthalpy, and E˙ is the energy loss rate (not including rest energy).

We make several approximations in addition to the assumption of steady spherical ﬂow. First, we consider only an outer region of negligible mass, so we approximate the

enclosed mass with the total stellar mass, M(r) M∗. Second, as we concentrate on ' optically-thick regions without appreciable convective luminosity, we employ the radiation diﬀusion approximation dP L r = κρ r (2.5) dr − 4πr2c Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 22 where κ is the eﬀective opacity. Rewriting this in a convenient form,

1 dPr 2 = Γrv , (2.6) ρ d ln r − k where κL Γ = r (2.7) r 4πcGM is the local Eddington ratio. In our wind structure calculations we shall employ the

Rosseland approximation κ = κR and use tabulated values of κR from the OPAL project.

2.3.2 Sonic point criteria

The momentum equation contains a critical point, which supplies several constraints on the behaviour of the wind. These have already been discussed by Nugis & Lamers (2002), but we re-examine them to make a couple additional points. We substitute the

pressure gradient dP/dr = dPg/dr + dPr/dr in equation (2.2) and evaluate this using the temperature gradient implied by equation (2.6) and the density gradient from equation (2.1). We ﬁnd 2 2 2c v [1 Γr (1 + φ)] v0 = i − k − , (2.8) v2 c2 − i where φ Pg/(4Pr). (In terms of the more familiar quantity β Pg/P , φ = β/[4(1 β)].) ≡ ≡ − Here and elsewhere, a prime indicates a logarithmic derivative with respect to radius, e.g. v0 = d ln v/d ln r.

The critical point is the isothermal sonic point Rsp, where v(Rsp) = ci(Rsp), so that the denominator vanishes in equation (2.8). (We denote sonic-point values with the subscript

0 sp.) For vsp to be deﬁned, the numerator must also vanish; this shows that the sonic point can only exist where the radiative luminosity is sub-Eddington relative to the matter:

1 qi Γr,sp = − < 1 (2.9) 1 + φ

at r = Rsp, and this condition applies to accretion as well as outﬂow. Here, we deﬁne 2 2 2 2 qi 2c /v , and likewise qr 2c /v for upcoming derivations. In WR winds Γr,sp is only ≡ i k ≡ r k slightly below unity (cf. Nugis & Lamers 2002 eq. 40), because qi 1 and φ 1. Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 23

In fact, the value of qi is restricted by the fact that vk reﬂects the stellar central

temperature, which is moderated by the burning stage, and the fact that ci is determined

by the temperature of the opacity peak. Evaluating vk using the mass-radius relation 0.21 −1 of Schaerer & Maeder (1992), which implies vk 1900[M/(30 M )] km s , gives ' −3.25 5.2 0.42 qi 10 (T/10 K)(30 M /M) . However, in real WR stars the sonic point forms ' at a somewhat larger radius, so that qi can be a couple times larger than this estimate. The velocity gradient at the sonic point must be determined by l’Hôpital’s rule, as the ratio of derivatives of the numerator and denominator of equation (2.8). Following Nugis & Lamers (2002), we note that the denominator increases through the sonic point, and therefore the numerator must as well in order for the wind to accelerate outward (v0 > 0). Using equation (2.9), the radial derivative of the numerator is

2 2 dci dvk 2 dΓr 2 dφ 2 qi + v (1 + φ) + v Γr . dr − dr k dr k dr

2 2 The ﬁrst term can be evaluated with dc /dr = Γrφv /r (from eq. 2.6), and combined i − k with the second term, using dv2/dr = v2/r. Using equation (2.9) a second time, k − k d(numerator)/dr becomes

2 qi (2 3qi)φ dΓr dφ v − − + (1 + φ) + Γr . k (1 + φ)r dr dr

2 2 The ﬁrst term is small in magnitude, and negative if qi/(2φ) = 4cr/vk < 1. We note that the Bernoulli parameter is approximately 4c2 v2 at the sonic point. Therefore, B r − k for the ﬁrst term to be negative, the wind must be formally bound in the sense of having a negative Bernoulli parameter.

The second term tends to be negative if κ is constant, because Lr tends to decline outward as energy is converted to kinetic form. On the other hand, this term can be large and positive if κ increases sharply outward. The last term is negative if dφ/dr < 0, i.e. when ρT −3 decreases outward. Note, however, that when a radiation-dominated gas is stable against convection, ρT −3 must decrease outward. Therefore, this term is negative in a stably stratiﬁed wind. Our analysis therefore corroborates Nugis & Lamers’s conclusion that the wind sonic Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 24

point is almost certainly located where dκ/dr > 0 so that Γr is increasing outward.

Combined with the fact that Γr is only slightly below unity at the sonic point, it is highly likely that the ﬂow will be super-Eddington for some range of radii immediately outside the sonic radius.

While neither of these statements is absolute, we see that the sonic-point condition in a wind model is essentially identical to the condition for density inversion in a hydrostatic

model: namely, that Γr increase through unity. We hypothesize that density inversions are always erased by dynamical winds, and test this later with numerical models.

Our numerical solutions require a quantitative description of the sonic point, which we gain by using the partial derivatives of κ(ρ, T ), assuming κ does not depend on the 0 velocity gradient. Defining kρ ∂ ln κ/∂ ln ρ and kT ∂ ln κ/∂ ln T , we find that v ≡ ≡ sp satisfies a quadratic equation:

0 2 0 0 = (vsp) + Bvsp + C, (2.10) where

B = 2Ψ + kρ i + 4qrξ i(1 + φ) (2.11) W W

and

2 C = 6Ψ + 4Ψ 1 + 2 i (kρ + kT Ψ) − − W

+ξ i(3qiΨ + 8qi + 8qr 6), (2.12) W − with the following deﬁnitions:

˙ 2 Mvk 1 qi φ ξ = , i = − , Ψ = i, 2Lr W qi 1 + φW

all evaluated at the sonic point. We see that the solutions depend only on the local properties of the ﬂow and opacity gradients.

The roots of equation (2.10) are of the form v0 = ( √B2 4C B)/2, and equation sp ± − − Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 25

(2.11) shows that B > 0. Real solutions require 4C B2; if C > 0 then both solutions ≤ are negative, whereas if C < 0 then there exists one positive and one negative solution. We are primarily interested in winds that accelerate outward, i.e., those for which v0 0; sp ≥ this requires that C 0 and that ≤

p v0 = (B/2)2 C B/2. (2.13) sp − −

2.3.3 Inner boundary: matching a hydrostatic star

Rather than solving for the structures of the wind and star simultaneously, we identify the base of our wind model with conditions at a matching radius within a hydrostatic model. The exact boundary location is selected to satisfy the following conditions:

1. The total wind mass is negligible in comparison to the stellar mass;

2. The stellar model is locally chemically homogeneous, µ = 0; and ∇

3. The ﬂow speed in the wind model is much less than the gas sound speed, v ci. We have had no diﬃculty identifying radii at which all these conditions are met. We investigate hydrogen-free, chemically homogeneous winds composed of pure helium with solar metallicity Z = Z = 0.02, and consider a range of stellar masses M∗ =

(15, 20, 23, 25, 30)M are considered to study the phenomena of envelope inﬂation and

mass loss. We focus in particular on the 23M case presented by Gräfener et al. (2012).

2.3.4 Regime of validity

In order for our solutions to be valid, several requirements must be met. First, the ﬂow must be optically thick so that the diﬀusion approximation is valid; but this is essentially guaranteed in WR winds, so we ignore this constraint. Second, force enhancement due to the Doppler shifting of spectral lines (e.g. Castor et al. 1975, hereafter CAK) must not invalidate our use of the Rosseland opacities from the OPAL project. Nugis & Lamers (2002) have previously argued that the enhancement is negligible at the wind sonic point, but we revisit the issue throughout our solutions. Third, the subsonic portion of the Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 26

flow must be stable against convection; or, if convection sets in, it (and any waves it launches) must be too weak to alter the radiative flux. Fourth, any other instabilities of radiation-dominated fluids (e.g. Blaes & Socrates 2003) must also not invalidate the assumption of smooth spherical flow. These instabilities provide additional line broadening and may enhance wind acceleration. Our approach will be to obtain solutions assuming these conditions are met, and then check their validity after the fact. The Rosseland approximation degrades once absorption lines in the accelerating wind are Doppler-shifted beyond a thermal line-width across a photon mean-free-path. This occurs (Nugis & Lamers 2002) where the CAK optical depth parameter

σref v ρ t = e th (2.14) CAK dv/dr

ref 2 −1 falls below unity, where σe = 0.325 cm g is a reference electron scattering opacity and

vth = 0.8ci is the thermal velocity of protons. We use this criterion to highlight where the force enhancement due to line shifting is likely to be a signiﬁcant correction. Being non-rotating and homogeneous in composition, our ﬂows are unstable to convection where low-entropy matter lies above high-entropy matter according to the sense of the total acceleration (including gravity), i.e. when

rad ad (2.15) ∇ ≥ ∇ where rad = d ln T/d ln P is the radiative temperature gradient and ‘ad’ means the ∇ adiabatic gradient. The outwardly accelerating flow dv/dr > 0 enhances the total acceleration and stabilizes the flow against convection. Further discussion is found in Section 2.5. Finally, we evaluate the growth rates of modes identified by Blaes & Socrates (2003). These modes are radiation hydrodynamic instabilities, distinct from convection.

2.3.5 Numerical Method

The subsonic region of our ﬂow satisﬁes a two-point boundary value problem, between an inner matching location and the sonic point. Once the radius of the sonic point is Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 27

found, the supersonic region is solved separately as an initial value problem. Our notation and numerical methods are in close accordance to the models of hot Jupiter outﬂows by Murray-Clay et al. (2009).

Subsonic Region: Relaxation Method

In our work, we use the relaxation solver solvede from Numerical Recipes (Press et al. 1992). This routine interprets the system of differential equations as a multivariate root-finding problem, and requires equations to be in finite-difference (FD) form

dyi 0 = Eij ∆jyi ∆jx, ≡ − dx where yi are the i-th ﬂuid variables and ∆jx xj xj−1 at the j-th grid point. ≡ − The corresponding FD forms of equation (2.1), (2.8), and (2.5) are

dρ E1j ∆jρ ∆jr ≡ − dr ρ = ∆ ρ + (2 + v0) ∆ r, (2.16) j r j

dT E2j ∆jT ∆jr ≡ − dr 3κ(ρ, T )ρL = ∆ T + r ∆ r, (2.17) j 16πacr2T 3 j

and

dv E3j ∆jv ∆jr ≡ − dr 2 2 2ci vk [1 Γr (1 + φ)] v∆jr = ∆jv − − . (2.18) − v2 c2 r − i The Rosseland opacity κ is supplied by the OPAL opacity tables of Iglesias & Rogers (1996). The current system of equations cannot be solved without the location of the outer

boundary or sonic point radius r = Rsp. The power of the relaxation method is its ability Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 28

to treat the outer boundary location as a dependent variable, using the deﬁnition

z Rsp R∗, (2.19) ≡ − where R∗ is the inner boundary radius, which can be solved for simultaneously. Since there is only one outer boundary, z is a constant. We add the trivial FD equation

E4j ∆jz = 0. (2.20) ≡

We deﬁne the new independent spatial variable q [0, 1] and substitute all instances of ∈ radius,

r = R∗ + zq. (2.21)

The four dependent variables ρ, T, v, z are normalized (and non-dimensionalized) to −7 −3 6 the following ﬁducial set to maximize numerical precision: ρ0 = 10 g cm , T0 = 10 K, 7 −1 v0 = 10 cm s , z0 = 1R . Within solvede, the convergence parameter conv is set to 10−7 with the following weighting parameters or scalv used in the error measure: ρ: 10; T : 5; v: 1; z: 1.

The relaxation method is a multidimensional extension of Newton’s method, which estimates a set of ﬁrst-order corrections to the FD equations. This requires partial

derivatives of FD equations with respect to dependent variables ∂Eij/∂yi, i, j. We ∀ compute this with the diﬀerentiation package from GNU Scientific Library (Gough 2009). We explicitly use the opacity gradients ∂κ/∂ρ, ∂κ/∂T supplied by the OPAL opacity tables in all calculations.

Stellar Parameters and Boundary Conditions

The stellar wind requires four local boundary conditions as there are four dependent variables (ρ, T, v, z)(q). The boundary conditions are written in FD form and can con- veniently be deﬁned implicitly: we denote them B1 through B4, all of which equal zero when the boundary conditions are satisﬁed. The sonic point criteria supply two outer Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 29

boundary conditions at q = 1: 2 kBT B1 v , (2.22) ≡ − µ and

2 2ci B2 1 2 Γr (1 + φ) ≡ − vk − 2k T (R + z) = 1 B ∗ − GM∗µ κ(ρ, T )Lr 3kBρ 1 + 3 (2.23) − 4πcGM∗ 4aµT

The remaining two inner boundary conditions connect the stellar wind to the hydrostatic interior. The solution across the boundary cannot be deﬁnitively smooth nor continuous, since one domain is hydrostatic. However, the approximation becomes very good where the velocity at the base of the wind is small.

We choose the temperature to be continuous across the boundary and deﬁne

B3 T T0. (2.24) ≡ −

The temperature gradient cannot be continuous across the boundary unless the density and diffusive luminosity are as well. We adjust the diffusive luminosity such that it remains smooth across the boundary. In hydrostatic equilibrium (M˙ = 0), the diffusive luminosity is effectively unchanged beyond the regions of nuclear fusion and convection

Lcore = Lrad = E˙ . In a wind the diﬀusive luminosity from the core must be reduced to accelerate and lift the gas out of the potential well. For a WR star, lifting the gas out of the potential is the dominant source of energy lost; the remaining terms are negligible in comparison. Thus, from equation (2.3) we have

1 2 2 Lr = E˙ M˙ w + v v − 2 − k GM E˙ + M˙ ∗ (2.25) ' R∗ Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 30

The density is constrained implicitly with the mass continuity equation (eq. 2.1),

2 B4 M˙ 4πR ρv. (2.26) ≡ − ∗

In summary, the stellar parameters are the luminosity L∗, mass M∗, mass loss rate

M˙ , temperature T0, and molecular weight µ at the base of the wind R = R∗. Details of the chemical abundances and metallicity are only necessary in selecting the appropriate OPAL opacity tables.

Numerical Sonic Point Treatment

Any numerical method that does not respect a critical point is fortunate to converge at all, let alone be accurate. Simply increasing the resolution is self-defeating because

the numerator N(x1) or denominator D(x2) need not be zero at the same grid point

(ie. x1 = x2) or vanish at all! This is disastrous for iterative schemes as the diﬀerential 6 equations may either explode towards both positive and negative inﬁnities, become zero (ie. a breeze solution), oscillate in sign, or any combination of these.

We evaluate v0 in the manner of Murray-Clay et al. (2009):

0 dv dv vv+ = Fexact + (1 Fexact) , (2.27) dr dr exact − r

dv 0 where dr is given by equation (2.8), v+ is the positive root from equation (2.10), and exact

2 ci Fexact erf p 1 , (2.28) ≡ − − v2 where erf is the error function, and p = 100 is the transition width for the sonic point.

Generating the First Wind Model

A ‘good’ initial guess for the entire subsonic wind structure is required for the relaxation method to begin. The resulting solution can then be used as an initial guess for the next problem bearing a diﬀerent set of conditions. Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 31

We found this step to be very challenging, but eventually developed a viable strategy in which we ﬁrst solve a trivial problem and successively add additional physical terms. We

GM∗ begin with an isothermal wind with critical point Rsp = 2 , and impose the analytical 2ci solution (Cranmer 2004). We then allow the temperature to vary, and adjust the constant

diﬀusive luminosity L∗ and opacity κ0 until they are similar to the conditions within a

WR star. Third, we allow κ to vary linearly with radius as κ(q) = κ0 + (κ1 κ0)q, and − we vary the opacity limits as we include terms from the Bernoulli factor . Finally, we B transform from the artiﬁcial opacity κ(q) to the OPAL tables for κ(ρ, T ). Once the ﬁrst wind model is available, generation of subsequent wind models become trivially accessible.

Supersonic Region: Initial Value Problem

Since all of the wind variables are defined at the sonic point, we compute the supersonic component as an initial value problem. We use the Bulirsch-Stoer routine contained in the integration module odeint from Numerical Recipes (Press et al. 1992). The convergence parameter is set to EPS = 10−13. Integration is continued until either the flow becomes subsonic or approaches the next partial ionization zone of helium. We find that wind solutions that become subsonic do so for temperatures well above 7 104 K. The remaining × solutions are fast and certainly break the Rosseland approximation by this point.

Because we cannot solve for the region of low optical depth in which Doppler-enhanced line forces are signiﬁcant, we cannot integrate to inﬁnite radius, and we cannot choose a self-consistent value for M˙ . Nevertheless, we can explore wind and envelope structures across a range of mass loss rates. Often our solutions cross through the opacity peak but fail to accelerate to speeds above the escape velocity, and then decelerate and stall at some radius. If this occurs within the regime of validity of the Rosseland approximation, it indicates a physically inconsistent solution. However, if the Rosseland approximation fails at the radii for which the wind solution stalls, it is possible that line forces would have permitted the wind to escape. Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 32

2.4 Results

2.4.1 23 M Helium Star

We use Modules for Experiments in Stellar Astrophysics (MESA, Paxton et al. 2011; 2013)

to construct a 23 M pure helium star with solar metallicity. The remaining parameters 5.80 to deﬁne the stellar wind problem are the luminosity L∗ = 10 L and temperature

T0 at the base of the wind R0. The location of the inner boundary is chosen where log

T0/K = 6.0, well beneath the sonic point and the iron opacity bump. The radius of this

boundary is R∗ 1.44 R . ∼ Since our system is not hydrostatic, we artiﬁcially adjust the stellar luminosity L∗

at the base of the wind such that the diﬀusive luminosity Lr of the wind matches that

from MESA. The correction here is about 1 L . A discrepancy between the hydrostatic −5 and wind density is found of order (ρ0 ρMESA)/ρ0 10 at the inner boundary. The − ' discrepancy diminishes exponentially towards the interior, as expected by Lamers & Cassinelli (1999). Hydrostatic density proﬁles from MESA and Gräfener et al. (2012) are presented in Figure 2.1. The wind models converge towards the MESA solution in density and temperature (see Fig. 2.2) for ρ > 10−9 g cm−3 and T > 105.2 K. We note that wind solutions cannot converge exactly to the hydrostatic solution as the mass loss rate is not zero. However, the diﬀerences between all wind and hydrostatic solutions are proportional to the velocity, which diminishes rapidly towards the interior.

−5 −1 A bifurcation of wind models exists across M˙ b = 2 10 M yr . Winds with × M˙ > M˙ b rapidly decline in temperature and do not extend far beyond one stellar radius before dropping to temperatures and optical depths outside our range of validity. We refer to these as ‘compact winds’. In Figure 2.3, we see that the sonic point occurs deeper within the star for increasing mass loss rates; however Rsp varies smoothly with M˙ , so this is not the direct cause of the bifurcation.

Weaker winds (M˙ < M˙ b) are shallow in density and nearly isothermal throughout. We refer to these as ‘extended winds’. Their structure strongly resembles the hydrostatic models with envelope inﬂation. While they are not hydrostatic envelopes, as this zone Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 33

7 4.0 − −

4.2 )

− 1 ) − 3 8 4.4 − −

− yr 4.6

−

g cm 9 4.8 − − ρ/

5.0 ˙ (

− M/M ( 10 5.2

10 − 10 − log 5.4

− log 5.6 11 − 1 2 3 4 5 6 7 − r / R Figure 2.1: Stellar density proﬁles. Light blue is by Gräfener et al. (2012), and the remaining are wind solutions across a range of mass loss rates. Arrows indicate the sonic point location for each wind model. Dashed regions indicate where the Rosseland approximation is no longer valid. Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 34

6.0 4.0 −

4.2 )

5.8 − 1

4.4 − − 5.6 yr

K) 4.6

−

T/ 4.8 5.4 ( − 10 5.0 ˙

− M/M

5.2 (

log 5.2

− 10 5.0 5.4

− log 5.6 4.8 − 1 2 3 4 5 6 r / R Figure 2.2: Temperature proﬁle for the stellar wind solutions. Arrows indicate the sonic point location. Dashed regions indicate where the diﬀusion approximation is no longer valid. Note the contrast in temperature scale height between weak and strong winds. Compact wind models are truncated to temperatures above the partial ionization zone of helium (104.8 K). Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 35

1.9 40

1.8 38 ) 1 − 1.7

/R 36 (km s sp 1.6 R sp v 34 1.5

1.4 32 5.6 5.2 4.8 4.4 4.0 − − ˙ − 1− − log10(M/M yr− )

Figure 2.3: Radius and local sound speed at the sonic point location across a range of mass loss rates. The bifurcation in behaviour does not appear in the sonic point location or velocity. Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 36

is outside the sonic point, it is plausible that they connect smoothly to the hydrostatic solution in the case M˙ 0. The peak speeds of these weak, extended winds are lower → than those of the strong, compact winds (Figure 2.4) .

Figure 2.4 presents the structure of v and ci. All winds reach peak velocity with a maximum of v 400km s−1, a factor of ﬁve slower than the local escape velocity. Thus, ' no wind solutions are found to escape from radiation pressure alone for this star.

The Rosseland approximation is valid throughout our calculation of the structure of

−5.2 −1 weak, extended winds with M˙ < 10 M yr . These winds fail to reach escape velocity as they cross the Fe opacity bump, and rapidly decelerate at lower temperatures and larger radii. We conclude that these cannot be the interior to a successful wind solution. Winds with higher mass-loss rates, especially the strong, compact branch of solutions, exit the regime of validity of our Rosseland approximation. Because Doppler enhancement of the line opacities becomes strong, these are candidates for successful wind solutions.

We postpone our explanation of the bifurcation in wind models until § 2.5, where we shall analyze wind and envelope structures in the space of density and temperature.

2.4.2 Other helium stars

To extend our modelling to helium stars of other masses, we rely on the empirical relations of Schaerer & Maeder (1992) to supply the inner boundary conditions.

The stellar luminosity relation is

2 L∗ M∗ M∗ log10 = 3.03 + 2.70 log10 0.46 log10 , (2.29) L M − M which is accurate up to 0.1 dex for stellar masses between 3 M∗/M 65. The ± . . luminosity for a 23 M star from equation (2.29) is log (L∗/L ) = 5.85, and log (L∗/L ) = 5.80 from MESA. The luminosity at the base of the wind is also artiﬁcially adjusted such that the diﬀusive luminosity matches the luminosity from Schaerer & Maeder (1992). This correction is at most 10% across all stellar masses and increases proportionally with mass loss rate. Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 37

400 4.0 − 350 Strong, Compact Winds 4.2 − )

4.4 1 300 − − ) 4.6 yr 1 250 Weak, Extended Winds − −

4.8 200 − ˙

(km s 5.0 − M/M

150 ( v

5.2 10 100 − log 5.4 50 − 5.6 − 0 1 2 3 4 5 6 r / R

Figure 2.4: Wind velocity showing the bifurcation between strong, compact winds and weak, extended ones. The green line indicates the sonic point location and velocity, and dashed regions are where the Rosseland approximation becomes invalid. Note that the 1/2 −1 escape speed is 2100(2R /r) km s for the 23 M . Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 38

It is important to ensure the temperature and radius are consistent at the inner boundary. The hydrostatic radius relation found for a WR star is

R∗ M∗ log10 = 0.66 + 0.58 log10 , (2.30) R − M which is accurate to 0.05 dex. However, only the corresponding ‘surface’ temperature ± relation from the Stefan-Boltzmann law is available at this location. This temperature is not suitable as the surface is not freely emitting.

In the MESA-generated 23 M model, the temperature decreases by over an order of magnitude (105.4 K from 106.8 K) across 10% of the outer stellar envelope. We ﬁx the

temperature log (T0/K) = 5.8 at the inner boundary and construct three sets of stellar wind models with varying base radii of R0 = (1, 1.05, 1.10) R∗. This serves to ensure × the base of the wind is in close proximity to the true stellar conditions and determine how signiﬁcantly the wind structure is aﬀected by the depth of the potential well.

We present density, temperature and velocity proﬁles (ﬁg. 2.5a, 2.5b, and 2.5c) for

R0 = 1.05R∗. We find the difference in base radii to only affect the critical mass loss rate for extended winds and the onset of line-force amplification. In general, the results are

qualitatively similar to the 23 M example.

With increasing stellar mass, the extended wind models grow to larger radii and reach higher peak velocities. This structure is also more resilient to mass loss with increasing stellar mass. We ﬁnd the critical mass loss rate for wind bifurcation scales approximately

2 as M˙ b M . We found no extended wind solutions in stars less massive than 14 M . We ∝ ∗ discuss the physical origins of this limit further in Section 2.5.2.

Particular wind models for M∗ 30 M can approach the escape speed from radiation ≥ pressure, within the regime of validity of the Rosseland approximation and without line-force ampliﬁcation. Line-force ampliﬁcation, however, becomes important within a narrow range of wind velocities (150-200 km/s) for all stellar masses. Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 39

-71.0 4.0 15M 20M −

-8 4.2 −

-90.8 4.4 −

-10 ) 1 )

4.6 − 3 − − 0.6 -11 yr 4.8

g cm 1.0 1.5 2.0 0 2 4 6 8 − -7 ρ/ ˙

( 25M 30M M/M (

10 5.0 0.4 − -8 10 log

5.2 log -9 −

0.2 5.4 -10 −

5.6 -11 − 0.0 00.0 2 4 0.2 6 8 100.4 12 0 0.6 5 10 0.8 15 201.0 r / R ∗ (a) Density proﬁles of winds with strong, weak, and critical mass loss rates. The intermediate model shown bifurcates the weak, extended and strong, compact winds. All stars shown are capable of forming an extended wind. The radius extension grows with L∗/M∗, or stellar mass. Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 40

5.81.0 4.0 15M 20M −

5.6 4.2 −

5.40.8 4.4 − 5.2 ) 1

4.6 − 5.0 − 0.6 yr K) 4.8

T/ 1.0 1.5 2.0 0 2 4 6 8 − ( 5.8 10 25M 30M ˙ M/M 0.4 5.0 ( log −

5.6 10

5.2 log 5.4 −

0.2 5.2 5.4 − 5.0 5.6 − 0.0 00.0 2 4 0.2 6 8 100.4 12 0 0.6 5 10 0.8 15 201.0 r / R ∗ (b) Temperature proﬁles of winds with strong, weak, and critical mass loss rates. Strong winds are arbitrarily truncated at T = 104.8 K near the partial ionization zone of helium. Extended winds do not cool eﬀectively and become practically isothermal. Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 41

1.0 4.0 − 3 4.2 −

02.8 4.4 − ) 1

) 1

4.6 − 1 − − 0.6 yr

15M 20M 4.8 km s 1.0 1.5 2.0 0 2 4 6 8 − v/ ˙ ( M/M ( 10 03.4 5.0 − 10 log

5.2 log 2 −

0.2 5.4 − 1

5.6 25M 30M − 0.0 00.0 2 4 0.2 6 8 100.4 12 0 0.6 5 10 0.8 15 201.0 r / R ∗ (c) Velocity proﬁles for all mass loss rates. A green line indicates the sonic point location and p velocity, and the black line is the local Kepler speed vk = GM∗/r. Extended winds are found to always become supersonic, if the driving is by radiation pressure alone. Note that for higher stellar masses, radiation pressure alone is capable of accelerating the wind to near escape speeds. Simulations of higher mass stars M∗ > 30M not presented here support this.

Figure 2.5: Profiles of stellar wind models for M = 15, 20, 25, 30M helium stars. For figures (a) and (b), arrows indicate the respective sonic point location. For all figures, dashed-lines indicate where the Rosseland approximation is invalid. Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 42

11 2.00 4.0 − − 4.2 10 −

− 1.75 )

4.4 1

) − − 3

− 9 10 −4 4.6 yr − 10 − −3

1.50

r 4.8 g cm 8 − − Γ ρ/ ˙

( 5.0 1.25 − M/M ( 10 10 −1 7 10 − −2 5.2 10 log − log φ = 10 1.00 6 0 5.4 − − 5.6 − 4.8 5.0 5.2 5.4 5.6 5.8 log10(T/K) Figure 2.6: Shown is a diagram illustrating the onset of envelope inﬂation and extended 4 winds for a 23 M (or L∗/M∗ = 2.7 10 (L /M )) star. The density and temperature structure of the MESA-generated model× (thick red line) and stellar wind (colours) extends from the stellar interior (bottom-right) to the surface (top-left). The stellar wind models trace the L∗κ(ρ, T )/(4πcGM∗) = 1 (greyscale) contour and become strongly radiation- dominated (decreasing φ = Pg/(4Pr)) until reaching the sonic point (stars). Wind models that cross the line of inﬂation (thick-black) are extended in radius, as the temperature scale height and stellar radius become comparable (see eq. 2.35). Dashed regions of the wind models indicate where the Rosseland approximation is no longer valid. A 14.25 M star with marginally extended winds is shown in Fig. 2.8. Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 43

2.5 Inﬂation, Inversion, and Stability

We begin by dividing dP/dr (obtained from the momentum equation, eq. 2.2) by dPr/dr (from the diﬀusion equation, eq. 2.5) to obtain

2 0 dP v v −1 = 1 + 2 Γr . (2.31) dPr vk Equation (2.31) provides important information about structure and stability, especially in the context of a radiation-dominated outer WR envelope and wind, where M(r)/L(r) ' M∗/L∗ so that the Eddington factor Γr κL∗/(4πGM∗c) is almost exactly proportional ' to κ. (Indeed, M(r) is almost constant in our solutions and L(r) varies by at most 10%.) Furthermore κ is a function of density and temperature, at least where the Rosseland 2 0 2 approximation is valid. Finally, the inertial term v v /vk is negligible in subsonic regions, where the ﬂow is nearly hydrostatic, but becomes important in supersonic regions. However, we saw in Equation (2.9) that Γr takes a speciﬁc value, very close to unity, at the sonic point. Therefore, the structure of the outer stellar envelope, and the transition to a wind, can be related directly to the opacity law in the plane of density and temperature.

2.5.1 Convective instability

The criterion for convective instability, equation (2.15), can also be assessed within this

−1 plane. From equation (2.31) and the relation = 4(1 β)dP/dPr, along with the ∇ − expression for ad(β) in a monatomic gas (Kippenhahn & Weigert (1990); Eq. (13.21)), ∇ we ﬁnd that the ﬂow is unstable ( rad ad) where ∇ ≥ ∇

v2v0 Γr 1 + 2 Γc, (2.32) ≥ vk with 8(4 3β)(1 β) Γc − − < 1. (2.33) ≡ 8(4 3β) 3β2 − − Outﬂows inhibit convection

In the absence of any wind (v = 0), this criterion sets a very speciﬁc value of Γr = Γc(β) above which an envelope is unstable – eﬀectively dividing the phase space into stable Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 44

and unstable regions of ρ and T (for hydrostatic models). This instability condition is necessary but not always suﬃcient: an accelerating wind, with v = 0 and v0 > 0, is more 6 2 0 2 stable on account of the inertial term v v /vk in equation (2.32). It is therefore necessary to examine in some detail the convective instability at the sonic point. Evaluating equation (2.32) provides a condition for convective stability at the sonic

point, valid where φ, qi 1: 7φ + kρ + kT /3 < 0. (2.34)

Here φ and kρ are positive, but the sonic point forms at temperatures somewhat above 5.2 T 10 K where kT is sufficiently negative. As a result, we find that all the stellar wind ' models in this paper are stable against convection throughout their subsonic regions. We note that Cantiello et al. (2009) attribute WR star variability to convection driven by the iron opacity peak, on the basis that convection should set in near the wind sonic point. However our finding that the subsonic region and sonic point are stably stratified indicates that radiation-driven acoustic instabilities (Blaes & Socrates 2003) are a more likely cause. This does not imply, of course, that hydrostatic envelopes, or envelopes with very low

mass-loss rates, cannot contain convective regions. These envelopes pass through Γr,sp at subsonic speeds, rather than crossing a sonic point.

Hydrostatic models convect

Indeed, convection appears to be inevitable for radiation-dominated hydrostatic envelopes interacting with the Fe or He peak, as a consequence of equation (2.31) with v = 0,

rewritten as dPg/dP = 1 Γr; in words, gas pressure declines outward when Γr < 1. − It is impossible for a hydrostatic, low-β envelope to exist without convection in the presence of an opacity peak. Consider Figure 2.6 or Gräfener et al. (2012)’s Figure 5, which plot Γr in the space of ρ and T or P and Pr. Following a solution outward to

decreasing P and T , the density drops dramatically to skirt the hot side of the Γr > 1

zone. In the process, Γr self-consistently becomes very close to unity, because dPg/dPr

is very small along the Γr = 1 contour. On the cold side of the bump, however, ρ and

Pg increase rapidly along this contour (the density inversion). To remain close to this Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 45

contour requires Pg to rise, which requires Γr > 1; but this implies that Γr passed through

the convective threshold (Γr = Γc < 1) along the way.

2.5.2 Onset of Envelope Inﬂation and Extended Winds

The weak, ‘extended’ winds closely follow the Γr = 1 contour, but do so by becoming supersonic as they cross the opacity peak, and subsonic once again as they exit it. Strong, ‘compact’ winds, on the other hand, traverse the opacity peak more directly, plunging deep within the super-Eddington region. It is the inertial term which allows a wind to enter the opacity peak without developing a gas pressure or density inversion

2 0 2 Γr = (1 + v v /v )(1 dPg/dP ). k − What is the underlying cause of the bifurcation in wind behaviour? A major clue is that the bifurcation coincides with the dashed line on Figure 2.6, which denotes the

condition qr = 1/2, or 2cr = vk. (In the plot, vk is evaluated at the base of the wind

R∗ 1.44R .) The importance of this condition arises from fact that the temperature ∼ scale height HT = dr/d ln T can be evaluated, in any region governed by the radiation | | diﬀusion equation (eq. 2.6), as

2 2cr −1 HT = Γr r. (2.35) vk

Envelopes and winds that follow the Γr = 1 contour will contain an extended temperature

plateau in which HT > r. The result is an inflated envelope or a specimen of our weak, ‘extended’ wind class. On the other hand, if the density is sufficiently high the line of inflation is avoided. This causes the temperature to plummet through the opacity peak, with HT /r decreasing further as Γr exceeds unity; the result is a strong, ‘compact’ wind that accelerates rapidly. Importantly, this criterion depends only on the validity of the radiation diffusion equation, so it is equally valid within hydrostatic models as in winds; this explains why the weak, extended winds track the profile of the hydrostatic model. The wind model that traces the line of inflation through the opacity peak separates the weak ‘extended’ and strong ‘compact’ winds. We select the conditions where the line of Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 46

5.2 inﬂation (2cr = vk) and opacity peak (Tp = 10 K) intersect to estimate the bifurcating mass loss rate, ˙ 2 2 3GM∗ Mb = 4πrpρpvp = 4πR∗ 4 vp. (2.36) 4aR∗Tp

The location of the opacity peak rp is approximately at the base of the wind R∗. Since the

pressure gradient follows the line of inﬂation (dP/dPr = 1 + dPg/dPr 1 + 5φ), equation ' (2.8) and (2.31) supply a local estimate for the velocity

2 vp (1 + 5φ)Γr 1 2 2 = − . (2.37) v c qi + Γr (1 + φ) 1 p − i −

Finally, the Eddington factor Γr = κ(ρp,Tp)L∗/(4πcGM∗) is calculated with the OPAL opacity table and stellar luminosity relation equation (2.29).

Although our estimate of M˙ b is approximate, we ﬁnd excellent agreement with the wind models generated from the sequence of helium stars (see Figure 2.7). For the

−5 −1 23 M case generated with MESA, we predict M˙ b = 1.6 10 M yr , which is in × −5 −1 excellent agreement with our numerical results (2 10 M yr ) as well. M˙ b is marginally × underestimated towards more massive stars since the location of the opacity peak is further

from the base of the wind (ie. rp > R∗).

M˙ b is found to rapidly decline towards lower stellar mass. This is conﬁrmed with an

additional set of stellar wind models constructed for a M∗ = 14.25M helium star. The reason is seen from Figure 2.8 which displays the star on a ρ and T plane. In comparison

to Figure 2.6, the Γr = 1 contour recedes to higher densities for stars with lower mass

or L∗/M∗. This reduces the M˙ b necessary to avoid crossing the line of inﬂation. Higher mass stars can generate winds that extend tens of stellar radii and, likewise, increases the

M˙ b necessory to erase the structure.

−1 At exactly M∗ = 14.04M , the line of inflation and Γr = (1 + 5φ) 1) contour ' intersect at one point, and any non-zero mass loss rate will form a compact wind. Therefore, we find a minimum stellar mass or L∗/M∗ for envelope inflation from the iron opacity bump.

Petrovic et al. (2006) present a diﬀerent argument for approximating M˙ b. They state the inﬂated envelope is preserved, if the inertial term is smaller than gravitational Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 47

-4.0 -4.2

) -4.4 1

− -4.6 yr

-4.8

/M -5.0 b ˙

M -5.2 (

10 -5.4

log -5.6 -5.8

10 15 20 25 30 35 M ∗ Figure 2.7: The numerical (points) and analytic (dashed line) estimates for the critical mass loss rate for bifurcation across stellar mass. For mass loss rates below this boundary, we ﬁnd slow, extended solutions that are incompatible with successful WR winds. With our assumptions, the analytic estimate for M˙ b converges to zero at M∗ = 14.04 M . The analytic model underestimates the critical mass loss rate by a small value towards more massive stars as the location of the iron opacity peak is further than the base of the wind. See eq. (2.36). Blue and red lines are estimates from Petrovic et al. (2006) and Gräfener et al. (2012), respectively. Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 48

11 2.00 4.0 − − 4.2 10 −

− 1.75 )

4.4 1

) − − 3

− 9 10 4 4.6 yr − − −

1.50

r 4.8 g cm 8 − − Γ ρ/ 10 −3 ˙ ( 5.0 1.25 − M/M ( 10 10 −1 7 10 − −2 5.2 10 log − log φ = 10 1.00 6 0 5.4 − − 5.6 − 4.8 5.0 5.2 5.4 5.6 5.8 log10(T/K)

Figure 2.8: A structural diagram of the stellar winds for a 14.25M (or L∗/M∗ = 4 2.4 10 L /M ) star. The Γr = 1 contour recedes to higher densities and away from × the line of inflation for stars with decreasing L∗/M∗ or stellar mass (see eq. (2.29)). At 14.04M , the Γr = 1 contour and line of inflation intersect at one point, and any non-zero mass loss rate is sufficient to prevent the formation of an extended wind (see eq. (2.36) and (2.37) for the exact criterion). Therefore, the critical mass loss rate for bifurcation increases with stellar mass.

p acceleration (ie. v < GM∗/R∗). Evaluating M˙ b at the hydrostatic radius (Eq. 2.30)

and the minimum envelope density ρ = ρmin, as prescribed by Petrovic et al. (2006),

generates the blue line in Fig. 2.7. Since inﬂated envelopes trace the Γr = 1 contour, we

estimate ρmin at the opacity peak (see Fig. 2.6). We use Table 1 from Gräfener et al.

(2012) to generate the red line in Fig. 2.7. This is a similar approximation except M˙ b is reduced by 0.4 dex and evaluated at the envelope density minima location (see Eq. (33) from Gräfener et al. (2012)). These approximations suggest an inﬂated envelope becomes more robust for lower stellar mass, which is not in agreement with our models. Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 49

2.5.3 The nature of weak WR winds

We have found that stars with mass loss rates below M˙ b do not, within steady, spherically symmetric models, maintain the strong compact winds that we have identiﬁed as good candidates for WR winds. For mass loss rates below this limit, we ﬁnd weak, extended wind solutions that fail to launch winds by the iron opacity bump, because the Rosseland approximation remains valid as they decelerate. We infer these ‘winds’ fall back onto themselves, unless they are able to reach the helium opacity bump.

We note that extended envelopes in which 2cr > vk, Γr 1, and β 1 are formally ' unbound, in the sense that they have a positive Bernoulli parameter ( > 0 in equation B 2.3). However this is not relevant to the bifurcation in wind models, because diffusion is rapid enough that radiation is not trapped in WR winds. ˙ ˙ In the event where M . Mb, we suggest the stars will appear highly variable. The fallback of transonic material is indicative of inefficient acceleration and, as a result, non-radial, turbulent motions. Variations in density and temperature permits photons to preferentially diffuse through regions of lower optical depth, leading to further reduction in radiative acceleration. A porous envelope will have a radius smaller than the models with envelope inflation, but larger than the models by Schaerer & Maeder (1992). Large scale oscillations, like those seen from red supergiants (Freytag et al. 2017), have the capacity to launch shocks, adding interesting complications to radiative transport and motions beyond the photosphere.

2.5.4 Radiation-driven acoustic instabilities

The acoustic instability identiﬁed by Blaes & Socrates (2003) is a source of eﬀects not accommodated within our models. For a pure helium WR star with solar metallicity,

the instability occurs for log T/K . 5.7, which is achieved at radii beneath the sonic point. At log T/K = 5.7, Blaes & Socrates identify a wavelength of fastest asymptotic −1 growth approximately λmax/r = 2πqi/(Γrkρ) 10 , which exceeds the local pressure ∼ −2.3 scale height Hp/r 10 . We hypothesize that growth is suppressed for wavelengths ∼ larger than the pressure scale height, and evaluate the growth rate for λ = HP . The amplitude of this mode increases by 10 e-foldings from the point of instability to the sonic Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 50

point. Given initial perturbations greater than 10−4, the instability will become nonlinear in the subsonic domain and alter the conditions for wind launching. Jiang et al. (2015) perform three dimensional local radiation hydrodynamic simulations of an envelope patch at the iron opacity peak. They find the density inversions found in one dimensional simulations correspond to large-scale density fluctuations and supersonic turbulent velocity fields in three dimensions. Although the local simulations cannot determine whether a large-scale wind is initiated, the structural characteristics may be realized in Wolf-Rayet stars with weak extended winds. Density fluctuations may give rise to a clumped (or porous) atmosphere in which the effective opacity, and Eddington ratio, is modified and enhanced (or reduced). We anticipate that our analysis of outer WR envelopes and inner WR winds applies just as well to the modified opacity law as to the unmodified one. We direct the reader to Gräfener et al. (2012) and Gräfener & Vink (2013) for the effects of clumping on the structure of an inflated envelope and opacity enhancement.

2.6 Discussion

We draw several conclusions from our investigation of the transition from envelope to wind within WR stars. First, we ﬁnd that the inﬂation of stellar envelopes, caused by the iron opacity peak and observed within hydrostatic models of WR winds, extends into a class of weak,

‘extended’ winds. However, above the critical mass loss rate M˙ b, these are replaced by a strong, ‘compact’ class of solutions. Physically, this change in behavior arises from a

change in the ratio of the temperature scale height HT to the local radius. However our weak, extended winds fail to accelerate within the regime of validity of our Rosseland approximation. In contrast the strong, compact branch is compatible with acceleration to escape speeds (outside the regime of the Rosseland approximation). It is also compatible with the observed mass loss rates of WR stars. Second, we ﬁnd that continuum-driven WR winds are always convectively stable at the sonic point. Within a hydrostatic envelope, convection sets in at a critical Eddington Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 51

factor Γr that is slightly higher than the value of Γr at the wind sonic point; in a moving envelope, an inertial term raises this critical value further. Since the Eddington factor is increasing through the sonic point, the sonic point is always reached prior to the onset of convection (if it is reached at all).

Third, our adoption of the Rosseland approximation limits the applicability of our results in two ways. At large radii (usually outside the sonic point), our approach becomes invalid; the effective opacity is higher than the Rosseland mean, due to Doppler effects. We nevertheless probe the envelope-wind transition for a variety of mass loss rates in order to identify solutions compatible with the formation of a wind in the Doppler-enhanced regions. However, we also neglect acoustic instabilities that set in below the sonic point and may grow sufficiently to suppress the effective opacity relative to the Rosseland mean. While we do not predict the magnitude of this effect, we hypothesize that our analysis remains valid so long as the Rosseland opacity is replaced with the effective opacity.

Finally, we note that our results are not restricted to WR star winds, but apply to any object with a suﬃciently optically-thick, continuum-driven wind stimulated by an increase in the opacity.

2.7 Addendum 1: The Phase Space of WR Wind Struc- tures

After publishing this work, we realized the sonic point (SP) conditions applied onto an opacity bump sets a minimum mass loss rate for transonic ﬂow. A SP is satisﬁed where

v = ci and the Eddington ratio is near unity. If convection is not an eﬃcient transport

mechanism (i.e., Lr L∗) then the latter condition becomes a condition for the opacity ' κ(ρ, T ) κ∗ = 4πcGM∗/L∗. Nugis & Lamers (2002) argue via l’Hôpital’s rule that the ' SP must reside where the opacity gradient is positive or, equivalently, dκ/dT < 0.

2 The density ρ∗ and temperature T∗ (or sound speed v∗ = kBT∗/µ) reach a minimum

at the tip of the κ(ρ, T ) κ∗ contour. Assuming the SP radius is not far above the ' Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 52

hydrostatic radius Rsp R∗, we can estimate the minimum mass loss rate to be '

2 min(M˙ ) 4πR ρ∗v∗. (2.38) ' ∗

All stellar models containing an opacity bump bear a minimum mass loss rate for transonic flow. Therefore, Eq. (2.38) serves as a measure for the type of stellar outflow without requiring an explicit wind solution. This may be helpful to understand the outflows and growth of theoretical stellar populations such as Pop III and supermassive stars (Goodman & White 2016). Figure 2.9 compares the galactic population of WR stars, with surface helium abundances Y 0.85, to the minimum (Eq. 2.38) and critical mass loss rates (Eq. 2.36). We ≥ see that the observed population appears consistent with a compact interior and optically thick outflows launched by an iron opacity bump. The strong sensitivity of Eq. (2.38) to metallicity is shown for context. Porosity (clumping) reduces (increases) the effective opacity and would increase (decrease) the threshold.

2.8 Addendum 2: Force Multiplier Approximation

This analysis employed the proton thermal speed to estimate the optical depth tCAK of

a line (Eq. 2.14) instead of the thermal speed for iron. The thermal speed vth should be reduced by √56 7.5 due to the ratio of atomic weights. While this appears large, ' −α the force multiplier follows M(t) = k tcak with α 0.5. Thus, M(t) is enhanced by ∗ ' 561/4 2.7. ' Under these conditions, our wind solutions have M(t) 1 3 near the SP, with larger ' − values in the supersonic domain. The accuracy of our solutions is sensitive to the choices in the multiplier parameters k and α, which were adopted from Nugis & Lamers (2002). Around the sonic point, Nugis & Lamers (2002) ﬁnd “very small” values for the force

parameter 0.3 < M(tsp) < 0.6 relative to O-star winds. Since their values were for the proton thermal speed, they are in agreement with our results. We argue that our wind solutions underestimate the strength of the wind since we neglect enhancements in acceleration from Doppler shifted lines. As a result, the critical Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 53

3.5

4.0 ) 1

− 4.5 r y ¯

M 5.0 / ˙ M ( 0

1 5.5 g o l

6.0 Y = 1 Z, Z = Z − ¯ Y = 1 Z, Z = 0.5Z − ¯ 6.5 4.5 5.0 5.5 6.0 6.5 log10(L / L ) ∗ ¯

Figure 2.9: Shown are the clumping-corrected galactic WR stars with Y 0.85 and Z = 0.0172 from Nugis & Lamers (2000). The red (blue) lines are for pure helium≥ stars with solar (half-solar) metallicity. Stars below the dashed line (Eq. 2.36) are expected to generate slow extended winds; those above are compact (i.e., no envelope inﬂation or inversion) with strong winds. Solid line indicates the minimum mass loss rate to drive a WR wind due to the iron opacity bump. Chapter 2. Launching of Optically-Thick Radiatively Driven Winds 54

mass loss rate to erase envelope inflation becomes an overestimate. We expect envelope inflation to be more delicate than the presented models suggest. On the other hand, the dynamical role of clumps is not clear in driving the wind. If they manifest near the SP then their turbulent nature may increase the effective optical depth of a line and region where the Rosseland approximation is decent. Chapter 3

Shock Dynamics in Stellar Outbursts: I. Shock formation

3.1 Chapter Overview

Wave-driven outflows and non-disruptive explosions have been implicated in pre-supernova outbursts, supernova impostors, LBV eruptions, and some narrow-line and superluminous supernovae. To model these events, we investigate the dynamics of stars set in motion by strong acoustic pulses and wave trains, focusing here on nonlinear wave propagation, shock formation, and an early phase of the development of a weak shock. We identify the shock formation radius, showing that a heuristic estimate based on crossing characteristics matches an exact expansion around the wave front and verifying both with numerical experiments. Our general analytical condition for shock formation applies to one-dimensional motions within any static environment, including both eruptions and implosions. We also consider the early phase of shock energy dissipation. We find that waves of super-Eddington acoustic luminosity always create shocks, rather than damping by radiative diffusion. Therefore, shock formation is integral to super-Eddington outbursts. See Ro & Matzner (2017) for the published article.

55 Chapter 3. Shock Dynamics in Stellar Outbursts: I. 56

3.2 Background

Observational examples of progidious pre-supernova (pre-SN) mass loss now abound. Supernova impostors (Van Dyk & Matheson 2012b) are re-classiﬁed as intense luminous blue variable (LBV) outbursts once their progenitors are seen to survive. Some, like SN 2006jc, SN 2009ip, SN 2015bh, and LSQ13zm, undergo one or more eruptions before terminal explosion (Pastorello et al. 2007; Foley et al. 2007; Margutti et al. 2014; Smith et al. 2014; Thöne et al. 2016; Tartaglia et al. 2016). A few percent of core-collapse SNe exhibit narrow lines from dense circumstellar interaction, indicating intense phases of pre-SN mass loss (Smith et al. 2011; Kiewe et al. 2012; Smith 2014; Moriya et al. 2014). Ofek et al. (2014) ﬁnd that precursors are common in hydrogen-rich, narrow-line (type

IIn) SNe, and Margutti et al. (2016) ﬁnd evidence for outbursts ejecting 1 M in SN ∼ 2014C and similar events prior to 10% of type Ibc SNe. ∼ It is not always clear whether each mass-loss episode results from a single shock-driven outburst, an extended wind, or both. However, intense mass-loss is expected in a pre-SN

stellar evolution for both low and high progenitor masses. In the low-mass ( 9 11M ) ∼ − progenitors of electron-capture SNe, oxygen and silicon shell burning releases a sequence of pulses, building from 1049 to 1050 ergs over the last year of the star’s life, allowing for the possible ejection of the stellar envelope (Woosley & Heger 2015). At an order of

magnitude higher initial mass ( 95 130M ), pulsational pair instability is expected ∼ − to eject a series of massive shells (Woosley et al. 2007). Outside these mass windows, Quataert & Shiode 2012, Shiode & Quataert 2014, and Smith & Arnett (2014) argue that the enhanced pre-SN mass loss is driven by waves excited in zones of vigorous convection. Waves deposit their energy by either radiative damping or shock dissipation. Super- Eddington rates of acoustic dissipation can stimulate intense winds like those seen from LBVs and Type IIn SN progenitors. Quataert et al. (2016) envision radiative diffusion as the dominant form of wave dissipation in such events, but as we shall see, shock dissipation is more relevant. Moreover, the existence of 5000 km/s motions around η Carinae (Smith ∼ 2008) implies shock driving, as do 2000-7000 km/s speeds in the 2009ip precursor (Foley et al. 2011). These considerations motivate a detailed investigation of shocks within stars, which Chapter 3. Shock Dynamics in Stellar Outbursts: I. 57 we begin here by analyzing the birth and early phase of a radially-propagating shock front. Dessart et al. (2010) has noted that shocks may be responsible for many types of outbursts, and that shocks occur naturally when energy is released over a period shorter than the dynamical time. However, energy is usually released deep within a star where sound speeds are relatively large, so part of the deposited energy must first travel outward as a sound pulse or continuous wave. If the sound is sufficiently intense, it will convert into a shock at some point within the star. Indeed, shocks are a natural outcome of sound propagation. Barring reflection and dissipation by other means, all acoustic waves steepen into shocks in finite time (Landau & Lifshitz 1959).

Shocks launched by waves from the convective zone have long been considered as a heat source for the solar corona (Biermann 1946; 1948). Yet, with few exceptions, existing solutions for shock formation and evolution from acoustic waves are restricted to relatively simple cases, such as planar and homogeneous or isothermal atmospheres. We therefore seek more general solutions that can be applied to the stellar problems of interest, although we do consider only one-dimensional ﬂow, such as spherical symmetry.

Spherical symmetry may appear to be a drastic simpliﬁcation, as the Homunculus nebula, which surrounds the prototypical LBV η Car, is strongly bipolar. Moreover, aspherical strong explosions are known to develop strongly non-radial ﬂows near the stellar surface (Matzner et al. 2013; Salbi et al. 2014).

Nevertheless, a thorough understanding of the spherical problem is required for any detailed study of the non-spherical case, so this is where we begin. The spherical idealization was also adopted in numerical investigations by Wyman et al. (2004) and Dessart et al. (2010). It allows us to describe the problem in simple terms: we start with a

spherical hydrostatic stellar envelope of enclosed mass m(r), density ρ0(r), pressure p0(r),

and adiabatic sound speed cs0(r), where subscript ‘0’ denotes undisturbed quantities, and consider the evolution of an outgoing sound pulse or wave train. x Our ultimate goal is to predict (analytically, if possible) the entire sequence of events set in motion by a strong sound pulse from the stellar interior: its propagation as a sound wave; its steepening into a shock front; its strengthening into a strong shock, and arrival at the stellar surface; and the ensuing ejection and fall-back of matter, and release of light. Chapter 3. Shock Dynamics in Stellar Outbursts: I. 58

The strong-shock phase is pivotal to this sequence, because a normal strong shock must approach the stellar surface like the self-similar solutions identiﬁed by Sakurai

−β1 (1960). In these, the shock velocity follows vs(r) ρ0(r) , for an eigenvalue β1 0.2 ∝ ' that depends weakly on the density profile and the post-shock equation of state (Ro & Matzner 2013). However, to calculate the coefficient to this strong-shock law, and to determine the pattern of shock-deposited heat and momentum, we must first analyze shock formation and strengthening. We focus here on the precise radius of shock formation at the end of purely acoustic propagation (Phase 1), and provide a simple estimate of the initial phase of shock strengthening in which the wave peak catches up with the shock front (Phase 2). We postpone a detailed examination of shock evolution to a subsequent paper. We begin, therefore, by reviewing the nature of acoustic pulse propagation, before delving into a detailed analysis of weak shock formation and propagation. We then derive a general expression for the condition of shock formation in two ways. First, we use a wave action principle to generalize a classical derivation based on the crossing of sound front. Then, extending an analysis from the field of sonoluminescence, we use an expansion around the wave front to derive the same result. Numerical simulations validate our result and provide insight into the subsequent phase of weak shock evolution.

3.3 Propagation of a sound pulse

Let us begin by considering the propagation of a sound pulse, launched outward from the stellar core into the stellar envelope. Any such pulse can be decomposed into normal modes of the stellar envelope, and a pressure mode of angular momentum quantum

number ` and frequency ω can only propagate through a zone with sound speed cs0, if 2 2 2 2 they exceed the Lamb frequency ω > Sl = `(` + 1)cs0/r (e.g., Hansen & Kawaler 1994). Non-radial modes (` > 0) thus meet an angular momentum barrier and become evanescent inward of this radius. This suppresses their generation by subsonic motions in the stellar core; we consider only radial motions, for which there is no such barrier. There is, nevertheless, an outer turning radius for radial sound waves. Close to the Chapter 3. Shock Dynamics in Stellar Outbursts: I. 59

stellar surface, where c dH 1/2 H s0 1 2 , (3.1) . 2ω − dr the density scale height H is traversed by sound in a time less than ω−1 and the atmosphere responds quasi-statically, causing reflection (e.g., Aerts et al. 2010). (For later reference we designate ωac(r) as the local reflection or acoustic cutoff frequency.)

Away from its points of reﬂection, and in the absence of dissipation, a linear, outwardly-

propagating pressure wave carries a constant luminosity Lw. It is worthwhile to understand why and when Lw should be conserved, however; for this we rely on Dewar (1970).

Dewar averaged the Lagrangian of an adiabatic ﬂuid over a wave cycle, arriving at a

conservation law ∂nw/∂t + nwvg = 0 for the wave action density ∇ ·

Uw nw = . (3.2) ω k u0 − ·

Here vg is the group velocity (vg = u0 + cskˆ for sound waves), Uw is the wave energy

density, k is the wavevector, and u0 is the mean ﬂow velocity. In an otherwise motionless

stellar envelope (u0 = 0), Uw = ωnw is itself conserved, and if k is oriented radially outward, then the total wave luminosity

Lw = Uwcs0A(r)

across the area A(r) = 4πr2 will be constant (along the wave trajectory) as the wave

travels. Diﬀerent outgoing waves may nevertheless carry diﬀerent values of Lw.

Conservation of wave energy is a familiar feature of WKB theory. Note, however,

that the outward wave luminosity Lw is not conserved if: (1) the wave is reﬂected; (2)

the stellar envelope is in motion, so that k u0 varies; (3) non-adiabatic eﬀects lead to · dissipation that saps the wave energy; or (4) a shock forms, as shocks involve localized dissipation.

¯ 2 The mean wave energy density in a wave with peak velocity uw is Uw = ρ0uw/2, so Chapter 3. Shock Dynamics in Stellar Outbursts: I. 60

the mean wave luminosity in a spherical star is

2 2 ¯ 2 uw uw Lw = 4πr ρ0cs0 = 2 Lmax (3.3) 2 cs0 where

1 3 Lmax(r) A(r)ρ0c . (3.4) ≡ 2 s0

2 2 We see immediately that uw(r) /cs0(r) = L¯w/Lmax(r). Since supersonic wave motion

(uw > cs0) produces a shock very rapidly, sound cannot propagate in zones where

L¯w > Lmax – typically, the outer stellar envelope or atmosphere. However, wave dissipation by diﬀusion or shock formation sets in far before this condition is satisﬁed. A shock-driven

outburst is only possible if diﬀusion does not sap Lw prior to shock formation, so it is important to examine both processes in detail.

3.3.1 Thermal diﬀusion

Our estimate of losses due to thermal diﬀusion will be approximate, and similar to the analysis by Quataert & Shiode (2012). Below the stellar photosphere, this process is

described by the diffusion equation Frad = χ Urad, where Frad is the diffusive flux, χ is − ∇ the thermal diffusivity, and Urad is the portion of the total energy density Uth that can diffuse (i.e., the radiation energy density, if diffusion is due to photons). This equation applies to the outward diffusion of luminosity, so

Lrad LradHp,rad χ = 2 = 2 . 4πr Urad 4πr Urad |∇ |

Here Hp,rad = Urad/ Urad is the radiation pressure scale height, and Lrad is the diﬀusive |∇ | portion of the stellar luminosity L, so Lrad(r) L(r), where the equality holds in regions ≤ that are not convective.

If we consider the change of the wave luminosity Lw(ϕ) along a wave front (phase ϕ =const.) due to thermal diffusion, then considering that the thermal diffusion time is 2 2 cs0/(ω χ), we find 2 ˙ dLw(ϕ) Lw,rad(ϕ)χω Lw(ϕ) = cs0 2 . dr ' − cs0 Chapter 3. Shock Dynamics in Stellar Outbursts: I. 61

Here Lw,rad LwUrad/Uth is the part of the wave luminosity subject to diﬀusion. Using ' our expression for χ, the net loss across a distance Hp,rad is

2 2 dLw(ϕ)/dr ω Hp,radLrad Hp,rad | | 3 2 (3.5) Lw(ϕ) ' 4πcs0r Uth 1 2ωH 2 L = p,rad rad . 8γ(γ 1) cs0 Lmax −

2 The last step relies on the relation Uth = γ(γ 1)ρ0c for an ideal ﬂuid of adiabatic index − s γ, and yields a numerical prefactor in the range 0.1 to 0.3.

The form of (3.5) is convenient for determining whether linear acoustic waves will damp or reﬂect as they approach the stellar surface. However, our purpose is to quantify the non-linear process of shock formation, which we consider next. We return to radiative damping in section 3.6, where we derive a critical wave luminosity for shock formation.

3.4 Shock Formation

The evolution of an acoustic wave into a shock has two distinct stages: the creation of the shock discontinuity, and the driving of this shock by the acoustic pulse behind it. If this driving is successful, the shock will become strong and approach the stellar surface in the manner described by Sakurai (1960). We seek to predict the wave evolution through

the first and second stages. We first identify the radius Rsf at which the shock first forms; for this, we provide both a heuristic and a detailed calculation.

3.4.1 Shock formation radius: heuristic derivation

For a heuristic derivation of the location of shock formation, consider the fact that

acoustical information (such as the value of Lw) travels along outward-moving sound fronts (or characteristics), and that a shock forms when characteristics arrive at the same

location carrying conﬂicting information. Some variation of the propagation speed u + cs is inevitable if u is non-uniform, as the scalings of adiabatic linear perturbations (Landau Chapter 3. Shock Dynamics in Stellar Outbursts: I. 62

& Lifshitz 1959; Whitham 1974) imply

γ + 1 δ(u + c ) = δu. (3.6) s 2

Here δ represents the perturbation from the background state for a given ﬂuid element. In this section we generalize a classic result (Landau & Lifshitz 1959, §101) to non-planar and non-uniform environments.

Consider two nearby characteristics launched from an initial radius ri but separated

in space by a small amount ∆ri at an initial time ti – or equivalently, separated in time

(at a fixed ri) by ∆ti = ∆ri/cs0(ri). Each propagates outward at dr = (u + cs)dt, but − they move at different speeds because Lw differs slightly between them. At some larger radius, the difference in arrival times is

Z r dr0 Z r dr0 γ + 1 δu ∆t = ∆ti + ∆ ∆ti ∆ . ri u + cs ' − ri cs0 2 cs0

−1 −1 The second step uses equation (3.6), as well as (u + cs) = [cs0 + δ(u + cs)] ' −1 c [1 δ(u + cs)/cs0], which is correct to ﬁrst order in δ(u + cs)/cs0. s0 − We can now employ the conservation of wave luminosity, in the form

s u c L (r ) = s0 max i . u(ri) cs0(ri) Lmax

If we also write ∆u(ri) = (∂u/∂r)i∆ri, then after a little algebra we arrive at the shock formation condition

s ∂u Z Rsf γ(r) + 1 L (r ) dr max i = 1 (3.7) − ∂r i ri 2 Lmax(r) cs0(r) which corresponds to the crossing of characteristics: ∆t = 0 at r = Rsf . This is only a heuristic derivation, as we relied on Dewar’s phase-averaged conservation

law to infer the constancy of Lw, and from that, the propagation of characteristics within a single wave pulse. This procedure is hardly rigorous. However, we now show that equation (3.7) coincides perfectly with a more detailed calculation. Chapter 3. Shock Dynamics in Stellar Outbursts: I. 63

0.10

0.05

c 0.00 / u

0.05

0.10

1.00 0.75 0.50 0.25 0.00 ξ/λ

Figure 3.1: Deformation of a single sinusoidal impulse in a planar, isothermal atmosphere. The wavelength remains constant until a shock forms in the third frame. This indicates the end of stage one and beginning of stage two. The shockwave becomes fully developed and stage two completes once the wave peak (point) coincides with the shock location.

3.4.2 Detailed derivation of shock formation radius

The condition for shock formation from a sound pulse has been worked out in the context of sonoluminescence by Lin & Szeri (2001, hereafter LS01), and the solution is applicable to inertially-confined fusion and related topics. We generalize LS01’s analysis to account for a body force (due to the stellar gravity g) as well as the variations of fluid properties that define the stellar structure. We begin with the Euler equations,

αρu ∂ ρ + u∂ ρ + ρ∂ u + = 0, (3.8) t r r r 1 ∂tu + u∂ru + ∂rp = g, (3.9) ρ − αρuc2 ∂ p + u∂ p + ρc2∂ u + s = 0, (3.10) t r s r r where the density ρ, pressure p, sound speed cs, and ﬂuid velocity u reference the wave

properties, and ∂r and ∂t are partial derivatives with respect to space r and time t. In addition to the spherical case (α = 2) we allow for cylindrical and planar cases (α = 1 Chapter 3. Shock Dynamics in Stellar Outbursts: I. 64

and 0, respectively); in general A(r) = 2π(α+1)/2rα/Γ[(α + 1)/2].

We assume the structure of the quiescent gas is known (i.e., ρ0(r), p0(r), cs0(r)) and permit the quiescent adiabatic index,

2 dln(ρ0cs0) γ0 , (3.11) ≡ dln(ρ0)

to vary across the star: γ0 = γ0(r). This allows for a non-uniform composition in gas and radiation. We account for the fact that the instantaneous adiabatic index

dln(p) ρc2 γ = s (3.12) ≡ dln(ρ) p

differs from γ0 as a fluid element is perturbed. We neglect effects from ionization, which

may absorb heat, except insofar as they are described by γ0(r) and cs0(r). Taking equations (3.11) and (3.12) with the Euler equations (3.8)-(3.10), we choose

to substitute ρ with cs to work only with variables p, u, cs, and γ. This generates the following diﬀerential equations:

αqc u c dγ ∂ c + u∂ c + qc ∂ u + s = s , (3.13) t s r s s r r 2γ dt 2 cs ∂tu + u∂ru + ∂rp = g, (3.14) γp − αγpu ∂tp + u∂rp + γp∂ru = , (3.15) − r where dγ ∂tγ + u∂rγ = (γ γ0)∂ru, (3.16) dt ≡ − and q (γ 1)/2. We assume the body force is independent of perturbations (Cowling’s ≡ − approximation). The wave travels only in the radial direction, so refraction is ignored. Whitham (1974) found that a Taylor expansion of ﬂuid properties about the wave front generates a closed system of equations. From these equations, Lin & Szeri found velocity gradient ∂ru(r) evolution can be described in an explicit and analytic form until

shock formation ∂ru . While much of the following derivation is similar to LS01, → −∞ we include a body force (eg. gravity), cylindrical wave solutions, and a variable adiabatic Chapter 3. Shock Dynamics in Stellar Outbursts: I. 65

index.

The wave front r = F (t) propagates outward (or left to right) at the local quiescent sound speed,

0 F (t) = cs0(F (t)). (3.17)

The primes are derivatives with respect to independent variables. We deﬁne a new coordinate variable ξ = r F (t) around the wave front (ξ = 0) and expand the ﬂuid − variables for ξ < 0:

1 c (ξ, t) = c (F (t)) + ξc (t) + ξ2c (t) + ... , (3.18) s s0 s1 2 s2 1 u(ξ, t) = ξu (t) + ξ2u (t) + ... , (3.19) 1 2 2 1 p(ξ, t) = p (F (t)) + ξp (t) + ξ2p (t) + ... , (3.20) 0 1 2 2 1 γ(ξ, t) = γ (F (t)) + ξγ (t) + ξ2γ (t) + ... (3.21) 0 1 2 2

2 Integer subscripts represent the number of spatial derivatives taken (e.g., cs2 = (∂r) cs). Since the wave front is also a node, we use the quiescent gas values for variables with subscript 0. Variables with non-zero subscript are spatial gradients evaluated at the wave

0 front and are functions in only time. Therefore, our notation states u1(t) = du1/dt and 0 p0 = dp0/dr. Note that we assume the quiescent gas is initially static u0 = 0.

Next, we substitute the expanded variables into the set of diﬀerential equations (3.13)- (3.16). Since the derivatives are with respect to r and not ξ, we change the variables to generate a new derivative:

0 [∂t] = [∂t] + [∂t(ξ)] ∂ξ = ∂t F (t)∂ξ r ξ r −

= ∂t cs0(F (t))∂ξ. (3.22) −

We collect the ξ0 and ξ1 terms from each diﬀerential equation to obtain eight equations, which are listed in the Appendix. This requires a meticulous account of all variables.

Combining these equations presents a single diﬀerential equation for u1, which measures Chapter 3. Shock Dynamics in Stellar Outbursts: I. 66

the wave steepness or gradient, as a function of only the quiescent gas quantities:

0 0 2 0 γ0 αcs0 g 0 = 2u1 + (γ0 + 1) u1 + cs0 + cs0 + γ0 u1. (3.23) γ0 r − cs0

This is an example of a Bernoulli equation (Ince 1956), which has an analytic solution of the form

s s " Z r 0 0 # −1 Lmax(r) −1 γ0(r ) + 1 Lmax(ri) dr u1 (r) = u1 (ri) + 0 0 . (3.24) Lmax(ri) ri 2 Lmax(r ) cs0(r )

Although shock formation is a nonlinear process, our Taylor expansion is justiﬁed

by the fact that the shock forms at the wave node; only the ﬁrst term u1 appears in −1 −1/2 this solution. Insofar as the combination cs0 Lmax r tends to be much larger where a wave shocks than where it was launched (at least in the stellar context), it is reasonably

accurate to evaluate equation (3.24) with ri 0. →

Comparison to equation (3.7) shows that shock formation (u1 ) occurs precisely → ∞ where our heuristic analysis predicts (i.e., r = Rsf ). The wave front evolution is deﬁned

entirely by the structure of the quiescent gas, initial wave front gradient u1(ri) and location

ri = F (0), and not on the wave’s other properties (wavelength, amplitude, etc.). And, our result holds equally well for planar, cylindrical, or spherical symmetry, and for inward as well as outward propagation. Shock formation is used in ﬁelds as diverse as the heating of the Solar corona (Osterbrock 1961), the deﬂagration-to-detonation transition in type Ia supernovae (Charignon & Chièze 2013), and sonoluminescence (LS01), among others, so this general result should be widely applicable.

Although shock formation is a purely local process on the most rapidly compressive characteristic, we can relate it to the properties of a larger wave or pulse. Suppose a wave

is initially monochromatic, reaching its peak velocity amplitude uw a quarter wavelength

behind the wave node. The peak compression rate, max( u1) = max( ∂u/∂r) = − − ωuw/cs0, is achieved at a node. In our shock formation criterion, the combination p p p max( ∂ui/∂ri) Lmax(ri) is equivalent to ω L¯w(ri); but this equals ω L¯w elsewhere, − so long as the conditions discussed at the start of §3.3 hold. Condition (3.7) therefore Chapter 3. Shock Dynamics in Stellar Outbursts: I. 67

becomes s Z tsf γ + 1 L¯ 0 w ω dt = 1; (3.25) ti 2 Lmax ω dt is the change of phase angle. In other words, the wave propagates for

* s +−1 γ0 + 1 L¯w

2 Lmax

radians before producing a shock, where the bracket means a time average along the

−1/2 −1 wave front. The total propagation time is proportional to L¯w ω , so stronger and higher-frequency waves shock earlier.

3.4.3 Maturation of the shockwave

To estimate the point of intersection between the wave peak and shock front, we derive

their respective trajectories. First, suppose a monochromatic wave with luminosity L¯w

and frequency ω is led by a compressive edge (i.e., u1(ri) < 0). The wave peak initially

lags behind the wave front by a distance ri rw = λ/4 = πcs0/(2ω). −

The peak propagates with a speed vw = uw + cs,w, where cs,w is the compressed local sound speed. Assuming properties of the gas do not vary signiﬁcantly under

γ0 compression (i.e., constant p/ρ and γ = γ0), we can write the compressed sound speed

2 γ0 1/γ0 (γ0−1)/γ0 cs,w = γ0(p0/ρ0 ) pw in terms of the peak pressure pw. The thermodynamic 2 2 2 expression of the mean wave energy density U¯w = (pw p0) /2ρ0c = L¯w/4πr cs0 allows − s0 cs,w to be expressed in terms of conserved wave properties,

γ0−1  s  γ ¯ 0 2 2 Lw cs,w = cs0 1 + γ0  . Lmax

With the kinematic expression (3.3), the speed of the wave peak becomes

γ0−1 s  s  2γ0 vw L¯w L¯w = + 1 + γ0  . (3.26) cs0 Lmax Lmax Chapter 3. Shock Dynamics in Stellar Outbursts: I. 68

Before shock formation (r < Rsf ), the wave front simply travels at the local sound

speed cs0. For r > Rsf , the exact shock velocity requires numerical calculations to describe the arrival of the remaining wave pulse. We circumvent this calculation by approximating

the shock front strength z = ps/p0 with the wave peak properties (z pw/p0). With the ' following shock jump condition,

2 2γ0M (γ0 1) z = s − − , (3.27) γ0 + 1 we approximate the shock Mach number Ms = vs/cs0 to be

s ¯ 2 γ + 1 Lw Ms 1 + . (3.28) ' 2 Lmax

Thus, the wave peak and shock front converge at the same location r = Rs, once

Z Rs dr Z Rsf dr Z Rs dr = + (3.29) ri−λ/4 vw ri cs0 Rsf vs

is satisﬁed.

3.5 Hydrodynamic Simulations

To test our analytical predictions of shock formation and provide examples of shock evolution and strengthening, we turn to numerical simulations. We construct one-dimensional planar and spherical simulations in FLASH (Fryxell et al. 2000), a hydrodynamic adaptive mesh refinement (AMR) code. All quiescent structures begin in hydrostatic equilibrium with a gravitational acceleration g(r) that is independent of fluid perturbations (Cowling’s approximation). All simulations have inner reflecting and outer diode (outflow) boundary conditions. Two structures are considered: a planar isothermal atmosphere and a n = 3

stellar polytrope. The adiabatic index γ0 is held ﬁxed in all of our simulations. Chapter 3. Shock Dynamics in Stellar Outbursts: I. 69

Table 3.1: Wave characteristics, planar simulations.

Label λ (m) uw/a0 λa0/uw A1 50 0.02 2500 A2 250 0.10 2500 B1 50 0.04 1250 B2 250 0.20 1250

Note. — The atmospheric scale height is 8777 m and the base resolution corresponds to ∆x = 1.25 m.

3.5.1 Planar Earth Atmosphere

For planar shock formation we consider a vertically stratiﬁed, initially isothermal atmo-

sphere of gas (FLASH’s default Earth atmosphere) with γ0 = 1.4, in constant gravity. Lengths in this section should be compared to the 8.8 km scale height of the model, although the results can be scaled to any similar atmopshere. Upward-travelling waves are initialized by setting the isothermal atmosphere out of equilibrium; an example of the initial waveform and its evolution is shown in Figure 3.1. The initially sinusoidal wave steepens (Phase 1) and forms a shock at the wave node, after which the wave peak approaches (Phase 2) and merges with the shock front. In a grid-based simulation, a shock must span multiple grid cells of length ∆x; for a given velocity jump ∆u this sets an upper limit to the compression rate ∂u/∂r of order ∆u/∆x. Given this limitation, we expect the numerical solution to converge toward the analytical prediction (eq. 3.25 as ∆x 0; this veriﬁed in Figure 3.2 for one set of initial → conditions. Figure 3.3 shows that the peak wave luminosity is conserved during the acoustical phase of propagation (Phase 1), just as predicted in Dewar’s theory. Moreover it is diminished by less than 0.5% in the time between shock formation and the arrival of the wave peak at the shock front (Phase 2), and numerical dissipation is responsible for part of this loss. We launch four waves as described in Table 3.1. Within groups A and B, waves have

identical uw/λ and maximum compression rate. Equation (3.25) therefore states that Chapter 3. Shock Dynamics in Stellar Outbursts: I. 70

0.0

0.2 Maximum 0.4 Refinement Level

[s] 16 1 0.6 ×

− 1 4

u × 0.8 2 × 1 1.0 × 0.3 0.4 0.5 0.6 0.7 0.8 Height [km]

Figure 3.2: Evolution of the wave front gradient ∂ru of four waves with diﬀerent grid resolutions. The wave may manifest across 3200 (AMR) grid cells for the ﬁnest ∼ resolution min(∆x) & λ/(200 [1, 2, 4, 16]). All initial wave properties are A2 from Table 3.1. The dashed line is the analytic× wave steepening prediction (Eq. 3.24).

0.0 1.00

S 0.98 0.2 g h in o n c k e 0.96 0.4 p d e

] i e s s t s [

i S p 0.94

0.6 e a

1 v t i

− a 1 o 0.92

u n 0.8 W (normalized) w ¯ 0.90 L 1.0 0.88 1.2 Shock forms Rsf Rs Mature shock 0.86 0.4 0.6 0.8 1.0 1.2 Height [km]

Figure 3.3: Stages and evolution of the wave front gradient (solid black) and peak wave luminosity (purple). The thick coloured band indicates our predictions for Phase 2, which begins with shock formation and ends when the wave peak reaches the shock front. Once 2 2 a shock fully develops, the peak acoustic wave luminosity Lw = 2πr (pw p0) /(ρ0cs0) declines due to shock dissipation. − Chapter 3. Shock Dynamics in Stellar Outbursts: I. 71

0.0

0.2

0.4

[s] 0.6 1

− 1 A1 u 0.8 A2

1.0 B1 B2 1.2 0.3 0.4 0.5 0.6 0.7 0.8 Height [km]

(a) Wave front gradient

1.0 1.00

0.98 0.8 0.96

0.94 0.6 A1 A2 0.92 1.0 0.4 (normalized) 0.9 w ¯ L 0.8

0.70.2

0.6 B1 B2 0.5 0.0 0.00.2 0.40.2 0.60.4 0.8 0.6 1.0 0.8 1.2 1.0 Height [km]

(b) Peak wave luminosity

Figure 3.4: Numerical shock formation from vertically-propagating waves in a planar isothermal atmosphere (scale height 8.8 km). Waves within sets A and B have the same initial wave front gradient and diﬀerent wavelengths and amplitudes (see Table 3.1). See Fig. 3.2 and 3.3 for Figure descriptions. Chapter 3. Shock Dynamics in Stellar Outbursts: I. 72 waves in each group will steepen identically and form shocks (∂u/∂r)−1 = 0 at the same location; this is conﬁrmed in Figure 3.4a. A1 and B1 have shorter wavelengths and are more poorly resolved, so they obey the analytical prediction more poorly than A2 and B2.

3.5.2 Spherical Polytropes

We interpolate a n = 3 polytropic stellar model with a constant adiabatic index of γ0 = 4/3 onto a uniform grid of 130,000 cells. We disable AMR, as mesh refinement appeared to stimulate spurious oscillations in regions with short scale heights. We nevertheless observe small oscillations and a weak outflow of matter and corresponding inward-moving rarefaction wave due to imperfect force balance and the outer boundary conditions. (While density and pressure are formally zero at the stellar surface, simulation fluid variables cannot be defined zero. As a result, the grid boundary lies inside the stellar radius.) However, these have negligible impact on our results.

Our initial conditions generated both inward and outward travelling waves, so we measure the outgoing wave properties after it has separated from the ingoing wave. Waves with the shortest wavelength initially span 1% of the domain, contracting to 0.5% of the domain as they traverse regions of lower sound speed. However this is still highly resolved (650 cells). The wave luminosity is conserved to within 1%.

From Figure 3.5a, we observe equation (3.25) to successfully predict the location of shock formation for a n = 3 stellar polytrope in all of our simulations. Our estimate

for where the shock fully develops Rs is accurate to within Rs Rs,sim = 1 2 local | − | − −3 wavelengths or 6 10 R∗. . ×

3.6 Shock dissipation or radiative damping?

The results of the previous sections allow us to quantify which waves dissipate due to radiative damping, and which successfully convert into shocks. Returning to equation (3.30), we can deﬁne at each radius the critical frequency for waves that damp by radiative Chapter 3. Shock Dynamics in Stellar Outbursts: I. 73

s] 3 3

[10 4 1

− 1 5 u

7 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 r/R ∗

(a) Wave front gradient

9 Lmax

8 ]

7 L [ w ¯ L 6 log

3 0.0 0.2 0.4 0.6 0.8 1.0 r/R ∗

(b) Peak wave luminosity

Figure 3.5: Numerical results from the launching of four waves of various strengths and frequencies in a n = 3 stellar polytrope. See Fig. 3.2 and 3.3 for ﬁgure descriptions. A shock becomes strong where the post-shock wave luminosity Lw exceeds the maximum acoustic luminosity Lmax (black line). Chapter 3. Shock Dynamics in Stellar Outbursts: I. 74

diﬀusion in a single radiation pressure scale height:

2 2 cs0 Lmax ωrad 2γ0(γ0 1) 2 . (3.30) ' − Hp,rad Lrad

Above this frequency, waves dampen and are not capable of steepening into shocks. The

2 2 instantaneous damping time due to diﬀusion is tdamp Lw/ L˙ w = (Hp,rad/cs0)ω /ω . ≡ | | rad R t 0 We can also write the shock formation criterion dt /tshock = 1 where tshock = [2/(γ0 + ti 1/2 −1 1)](Lmax/Lw) ω . Evaluating the ratio of damping time to shock time at frequency

ω = ωrad, we ﬁnd 2 2 tdamp γ0(γ0 1)(γ0 + 1) Lw 2 = − . (3.31) t 2 Lrad shock ω=ωrad This analysis indicates a shock forms before radiative damping has had time to act, for any waves more luminous than the stellar radiative luminosity. Therefore, wave-driven outbursts that exceed the envelope Eddington luminosity must involve shock formation. This is especially true for super-Eddington outbursts, as the quiescent luminosity is usually well below the envelope’s Eddington limit.

In Figure (3.6) we examine acoustic propagation, radiative damping, and shock formation within a model star generated by the MESA code (r8118) from an initial

solar-metallicity object of 50 M . At the time of the ﬁgure, stellar winds have removed

all but 22 M and core oxygen burning has just begun. We plot the Brunt-Väisälä

(N), acoustic cutoﬀ (ωac), and radiative damping frequencies (ωrad), as well as the shock formation radius (eq. 3.25). As is clear from the ﬁgure, shock formation outpaces radiative ¯ 5.3 5.9 damping for Lw & 10 L , somewhat below the stellar luminosity of 10 L regardless of the wave frequency.

At each radius, we estimate the critical frequency for a wave to shock in a pressure

scale height tshock Hp/cs0: ' c2 L ω2 s0 max . (3.32) sh 2 ¯ ' 4Hp Lw Eq. (3.32), represented as dashed lines in Figure (3.6), is found to match with the shock formation condition (eq. 3.25) with high accuracy, especially in the stellar envelope. Chapter 3. Shock Dynamics in Stellar Outbursts: I. 75

8 ωrad 4

] ∗ s

[ 2 L

4 / ω w ¯ L g 0 o g l o

2 l

2 N ωac 0 4

2 2.0 1.5 1.0 0.5 0.0 0.5 1.0 log r [R ] ¯

Figure 3.6: Wave propagation diagram of a core oxygen-burning blue supergiant model evolved in MESA. Filled contours show the exact (eq. 3.25) shock formation radii for acoustic waves with frequency ω and peak luminosity L¯w launched from the convective −2.2 boundary ri = 10 R . The exact choice ri is indistinguishable from the stellar envelope. Also plotted are the Brunt-Väisälä (N), acoustic cutoﬀ (ωac), and radiative damping (ωrad) frequencies. Dashed lines are the critical shock frequency (ωsh), which approximates eq. (3.25) best in the stellar envelope. Chapter 3. Shock Dynamics in Stellar Outbursts: I. 76

3.7 Conclusion

Stars have previously been argued to respond to strong acoustic luminosity either by developing strong stellar winds powered by radiative wave dissipation (Quataert & Shiode 2012; Quataert et al. 2016), or alternatively by expanding in response to the increased wave pressure (Mcley & Soker 2014). A weak point of these analyses is the assumption

that shocks are only relevant where wave motions become supersonic (L¯w(r0) = Lmax(r)). If this were the case, wave pressure and radiative dissipation would indeed be important in the regime of pure acoustics. However, as we have seen, shocks form and then mature where the wave is strongly subsonic and, for appreciable Lw, before radiative diﬀusion becomes important. From this point onward, the eﬀect on the stellar envelope is a matter of shock propagation, which deposits both energy and momentum within the envelope.

At some point where L¯w(r) = Lmax(r) the shock does become strong, but this radius is

pushed outward by the decline of L¯w during weak shock propagation. The major result of this work, from which the above conclusion is drawn, is that the condition for shock formation within stars can be predicted with a single expression (equation 3.25). This is remarkably general, as it applies to any one-dimensional motion within a non-isentropic fluid; yet it is almost as simple as the classic result for planar, isentropic flows. Our exact derivation, which we obtained by generalizing an analysis from the sonoluminescence literature (Lin & Szeri 2001), matches a heuristic calculation of wave crossings that using wave luminosity conservation (which can be considered a consequence of the wave action principle due to Dewar 1970, in the absence of reflections). Because of its generality, our result applies equally well to a single wave pulse and to a continuous wave. Furthermore, the formation of a shock from one wave tends to be unaffected by the passage of earlier waves, because these deposit energy and momentum only after they have steepened into shocks. Therefore our criterion for shock formation remains reasonably valid even within stars that have been set into subsonic motion by a strong pulse or a steady acoustical flux. Dewar’s theory also predicts wave amplitudes also within a moving envelope, as in equation (3.2), so an extension of our results to steady wave-generated winds ought to be straightforward. We find that waves with super-Eddington luminosities necessarily involve shock Chapter 3. Shock Dynamics in Stellar Outbursts: I. 77 formation, rather than radiative dissipation. This distinction is important since shocks deposit wave energy and momentum in a distributed fashion. As a result, a train of shocks can conceivably drive envelope expansion, stellar winds, and mass ejection at the surface (i.e., shock breakout), all at the same time. The exact stellar response is decided along a range of radii that includes the shock formation radius, shock maturation radius, and where the shock becomes strong. Our next task is to predict in detail the propagation of weak shocks outside the shock formation radius, in order to understand the transition from weak to strong and to determine the patterns of heat and momentum deposition. We shall address this in subsequent papers. Chapter 4

Shock Dynamics in Stellar Outbursts: II. Shock propagation

4.1 Chapter Overview

In Chapter 3, we argue shock-driven mass loss is inevitable for sufficiently intense acoustic waves. And, a variety of stellar responses are possible. Previous investigations of acoustic shocks have been limited to heating of the solar chromosphere, which had the benefit of empirical support. Observations of eruptive mass loss, however, cannot directly reveal super-Eddington acoustic shocks because they would form below the photosphere. As a result, outflow models are required test whether eruptions are powered by super-Eddington acoustic sources. A one-dimensional hydrodynamic calculation can, in principle, compute a star’s response to intense acoustic waves. However, observations of eruptive mass loss are sparse and provide weak constraints on the stellar progenitor. Eruptions suggest a super- Eddington power source, yet, this has no bearing on the wave spectrum. Thus, the space of initial conditions renders numerical investigations to be explorative, rather than guided, which is difficult to justify unless it is thorough. In a sense, this has been a historic issue for studies of shock physics. For a variety of motivations, especially when computational resources were scarce (or non-existent), physicists have explored semi-analytic approximations to the fluid equations to study

78 Chapter 4. Shock Dynamics in Stellar Outbursts: II. 79

shocks. In this chapter, we study the treatment by Brinkley & Kirkwood (1947) because they adopt an acoustic interpretation of shock waves. We find this naturally extends the previous section up to the moment of shock breakout. While our attempt is not the first (Odgers & Kushwaha 1960; Kogure & Osaki 1962; Nadezhin & Frank-Kamenetskii 1965; Sachdev 1972), our calculations are not simplified. In fact, we find a missing term reflecting the density gradient, which is crucial to connect the weak and strong shock limits. Although this term is constrained empirically (due to the lack of time), there are clear indications for future theoretical work. The predictions are compared to the numerical simulations introduced in Chapter 3 and found to be quite accurate. We also find the treatment by Brinkley & Kirkwood (1947) suggests there exists is a minimum energy for an acoustic shock to reach the stellar surface and break out. In other words, shocks with insufficient energy become ‘duds’ and can vanish in the envelope. We do not confirm this with a hydrodynamic calculation and postpone this to future investigation.

4.2 Brinkley & Kirkwood Theory

Empirical data for aerial and underwater explosions were plentiful in the 1940s. Properties of the flow were observed to decay in a smooth, power-law fashion after the passing of a shock. Weak shocks are commonly referred to as sawtooth or N-waves since the excess post-shocked pressure and velocity appeared to decay linearly in time. And, strongly shocked gas cool at an exponential rate. For shocks of arbitrary strength, Brinkley & Kirkwood (1947) (hereon, BK) argue the decay evolution is effectively self-similar and normalizable by the immediate state behind the shock. As a result, they realize the evolution of the post-shocked gas does not require an explicit solution. This observation leads to a simpler set of equations describing shock propagation, which was attractive for many applications. Indeed, shocks in stars are an example (Odgers & Kushwaha 1960; Kogure & Osaki 1962; Nadezhin & Frank-Kamenetskii 1965; Sachdev 1968; Ulmschneider 1970). In the next section, we provide the derivation of the generalized BK equations for a stratified medium and summarize the physical motivations involved. A detailed derivation Chapter 4. Shock Dynamics in Stellar Outbursts: II. 80

is provided since the literature is often diﬃcult to decipher. Nevertheless, our work relies heavily on the contributions from numerous authors that have taken interest in this theory (Brinkley & Kirkwood 1947; Odgers & Kushwaha 1960; Kogure & Osaki 1962; Nadezhin & Frank-Kamenetskii 1965; Sachdev 1972; Lee 2016).

An important addition made here to this derivation is the allowance for a medium with variable density. This is critical for strong shocks since their propagation near the edge of a star is well understood (Sakurai 1960; Ro & Matzner 2013). This addition is not found to agree with Sakurai (1960), which we discuss in detail and attempt to correct for in section 4.4.

4.2.1 The Derivation

Consider a stratiﬁed medium with pressure p0(r), density ρ0(r), and sound speed cs0(r)

that is initially at rest u0(r) = 0 and distributed in either planar, cylindrical, or spherical

symmetry α = 0, 1, 2. A shock at radius R(t) moving with speed vs = dR/dt sweeps through an adiabatic p ργ, leaving it in a post-shock state deﬁned by the Rankine- ∝ Hugoniot (or shock-jump) conditions:

p ρ u v z s = 1 + 0 s s , (4.1) ≡ p0 p0 ρ v s = s , (4.2) ρ0 vs u − ps p0 1 1 hs h0 = − + , (4.3) − 2 ρs ρ0 with enthalpy γ p h = . (4.4) γ 1 ρ − Converse to the previous statement, the evolution of a shock can be determined from the post-shocked gas. Thus, our goal is to solve for the post-shocked state at every instance in time or shock radius. For example, a solution to the following:

dps ∂ps −1 ∂ps = + vs . (4.5) dR ∂Rs ∂t Chapter 4. Shock Dynamics in Stellar Outbursts: II. 81

With inspection, the shock-jump conditions show that the post-shocked states are deﬁned

by p0, ρ0, and any pair of ps, rhos, and us; we choose to solve us,

dus ∂us −1 ∂us = + vs . (4.6) dR ∂Rs ∂t

Therefore, four equations are required to solve for the four unknowns: ∂R(ps), ∂t(ps),

∂R(us), ∂t(us).

Across the shock, the ﬂuid must obey the continuity equation,

ρ ∂u 1 ∂p αu + 2 + = 0, (4.7) ρ0 ∂R ρc ∂t R

and momentum equation,

∂u 1 ∂(p p0) 1 dp0 + − + = g ∂t ρ0 ∂R ρ0 dR − ∂u 1 ∂p 1 dp + = 0 . (4.8) ∂t ρ0 ∂R ρ0 dR

The external force g is negated by assuming hydrostatic equilibrium.

The third equation follows from the chain rule,

du ∂u dp ∂u dp ∂u dρ s = s s + s 0 + s 0 , (4.9) dR ∂ps · dR ∂p0 · dR ∂ρ0 · dR

since u = u(p, p0, ρ0). Partial derivatives of the first jump condition (4.1) define the coefficients to be:

∂us ∂vs k ρ0vs = 1 ρ0us , (4.10) ≡ ∂ps − ∂ps ∂us ∂vs b ρ0vs = 1 + ρ0us , (4.11) ≡ ∂p0 − ∂p0 ∂us ∂vs w ρ0vs = usvs + ρ0us . (4.12) ≡ ∂ρ0 − ∂ρ0 Chapter 4. Shock Dynamics in Stellar Outbursts: II. 82

The three shock-jump equations can be combined to generate the following expression,

ρ v2 γ + 1 1 0 s = p + p , (4.13) γ 1 2(γ 1) s 2 0 − − which generates solutions to the partial derivatives above in terms of the shock strength:

∂ln(vs) z/2 = γ−1 , (4.14) ∂ln(ps) z + γ+1

and, ∂ln(vs) 1 = , (4.15) ∂ln(p0) γ+1 2 γ−1 z + 1 and, ∂ln(v ) 1 s = . (4.16) ∂ln(ρ0) −2

Conservation of Energy

As discussed in the beginning of this section, BK argues the post-shocked gas decays self-similarly. The nature of this decay introduces the fourth equation necessary to solve for shock propagation. In this section, we discuss how BK use the ‘energy’ of a shock to constrain the fourth equation. Connections to acoustic waves ﬁt naturally with our discussions on wave-steepening and shock formation.

Lee (2016) provides signiﬁcant clarity in understanding the motivations behind the BK theory. We adopt many of their steps here and suggest the reader to their work.

Suppose each layer of gas is a piston that acts only onto the gas above. A piston

at radius R0 = r(t < t0) is initially at rest u0 = 0 until a shock Rs(t0) = R0 sweeps

through at time t0. (In Paper 1 and in later parts of this chapter, we refer to Rs as the

location where an acoustic impulse has become a fully developed shock. Here, Rs(t) is momentarily deﬁned as the shock’s location at time t after it has fully developed.) The

piston’s position varies in time r(R0, t) and is referenced, in a Lagrangian sense, by its

initial coordinate R0. The immediate post-shocked gas properties p, ρ, u are labeled with

subscript s, and are unlabeled for t > t0. Chapter 4. Shock Dynamics in Stellar Outbursts: II. 83

Over time, the piston does work

Z t α W (R0, t) = kαr(R0, t) p(R0, t)u(R0, t)dt, (4.17) t0

to the shocked gas between the piston r(R0, t) and shock Rs(t),

Z Rs(t) 2 α u W (R0, t) = kαr ρ ∆e + dr. (4.18) r(R0,t) 2

Here, kα is the geometric coeﬃcient. The energy density is a combination of both the 2 internal ∆e = p/ρ(γ 1) p0/ρ0(γ 1) and kinetic u /2 components. By conservation of − − − α α mass, the shocked gas can be written in terms of it’s unshocked state kαr ρdr = kαR ρ0dR. Thus, the second integral (4.18) can be rewritten as

Z Rs(t) 2 α u W (R0, t) = kαR ρ0 ∆e + dR. (4.19) R0 2

Note that the above integral is evaluated at the time t. It is convenient to reference the

states at t when the kinetic energy vanishes and shock reaches inﬁnity Rs , → ∞ → ∞ Z ∞ α W (R0) W (R0, ) = kαR ρ0∆e∞dR. (4.20) ≡ ∞ R0

This is an important limitation when considering the BK theory to study ejecta production.

Eq. (4.17) can be written in terms of the initial pressure and over-pressure p = p0 +∆p:

Z ∞ Z ∞ α α W (R0) = kαr ∆pudt + kαr p0udt, t0 t0 Z r(R0,∞) α = Et(R0) + kαr p0dr. (4.21) R0

BK writes the last integral in terms of the piston location, since udt = dr, and splits the deﬁnite integral into parts:

Z r(R,∞) Z ∞ Z ∞ α α α W (R0) Et(R0) = kαr p0dr = kαr p0dr kαr p0dr. (4.22) − R0 R0 − r(R0,∞) Chapter 4. Shock Dynamics in Stellar Outbursts: II. 84

Since the ﬁrst integral on the right side is eﬀectively the thermal content of the unshocked gas, r is a dummy variable and can be substituted with R. Conservation of mass α α ρ∞r dr = ρ0R dR enables a substitution in the second integral, which is evaluated at t . Therefore, we have → ∞ Z ∞ α 1 1 W (R0) Et(R0) = kαR p0ρ0 dR. (4.23) − R0 ρ0 − ρ∞

With (4.20) and (4.23), we can solve for the over-pressure term:

Z ∞ Z ∞ α α 1 1 Et(R0) = kαR ρ0∆e∞dR kαR p0ρ0 dR R0 − R0 ρ0 − ρ∞ Z ∞ α p0 p0 = kαR ρ0 e∞ e0 + dR R0 − − ρ0 ρ∞ Z ∞ α = ER(R0) kαR ρ0 (h∞ h0) dR. (4.24) ⇒ ≡ R0 −

To solve for the last equation, BK suggests the post-shock pressure eventually returns to

its unshocked value p∞ p0. We note that this approximation is invalid for unbound gas. ' γ γ γ 1/γ By isentropy ps/ρ = p∞/ρ p0/ρ , the ﬁnal density becomes ρ∞ ρs(p0/ps) = s ∞ ' ∞ ' −1/γ ρsz , where z ps/p0. ≡

In conclusion, the total residual enthalpy of the shocked gas (4.25) above R0,

Z ∞ α γp0 ρ0 1/γ ER(R0) kαR zs 1 dR, (4.25) ' γ 1 ρs − R0 −

is the over-pressure work done by the piston at R0,

Z ∞ α Et(R0) = kαr ∆pudt. (4.26) t0

What are Et(R0) and Er(R0)?

In the weak shock limit z 1 1, the spatial derivative of Er(R0) becomes − dER 2 3 γ + 1 = kαRsp0(zs 1) 2 . (4.27) dRs − − 12γ Chapter 4. Shock Dynamics in Stellar Outbursts: II. 85

This expression is identically the weak shock dissipation rate derived by Mihalas & Weibel-Mihalas (1999). Therefore, the wave energy dissipated by a weak shock resides in the gas initially in the form of heat. Remarkably, the BK solution suggests the dissipation rate for all shock strengths is the spatial derivative of (4.25), assuming the gas returns to rest at its initial pressure.

We observe that the expression

Z ∞ α Et(R0) = kαr ∆pudt (4.28) t0

is simply the deﬁnition of acoustic wave energy, but for a shock. We refer to Et(R0) as

the ‘shock energy’ measured at R0. From the previous section, we conclude Et(R0) is also

the total shock heat deposited upstream R > R0.

The Fourth Equation

BK proposed the post-shocked gas properties, particularly the wave ‘luminosity’,

r α ∆p u f(Rs, τ) , (4.29) ≡ Rs ∆ps us

decays with some well-deﬁned structure, where

t t0 τ − 0, (4.30) ≡ µ(R0) ≥ Chapter 4. Shock Dynamics in Stellar Outbursts: II. 86

and µ(R0) is a decay constant for the gas R0. The total shock energy is calculated from

the initial shocked state, since it is constant in time for the gas at R0 = Rs(t0):

Z ∞ α Et(R0) = kαr ∆pudt t0 Z ∞ α α r ∆p u = kαRs ∆psus dt t0 Rs ∆ps us Z ∞ α = kαRs ∆psus f(R0, τ)dt t0 Z ∞ α = kαRs ∆psusµ(R0) f(R0, τ)dτ 0 α = kαRs ∆psusµ(R0)ν(R0). (4.31)

R ∞ The factor ν(R0) = 0 f(R0, τ)dτ is a constraint the energy-time curve, which is

self-similar when independent of R0. For strong shocks, the post-shocked gas is observed to decay exponentially; so, f(τ) = e−τ . This leads to ν = 1. Weakly shocked gas are seen to decay linearly in time. An approximation characterized by sawtooth or N-waves for the positive phase of the shock is f(τ 2) = (1 τ/2)2, and zero otherwise; this ≤ − generates ν = 2/3. In the work here, the value of ν is found to not inﬂuence the solution signiﬁcantly (see section 4.5).

The exponential decay constant µ(R0) is also a diﬀerential equation, since

1 1 ∂f = −µ(R0) f ∂t R0 αu 1 ∂∆p 1 ∂u = + + , r ∆p ∂t u ∂t

along some piston path r(R0, t). Evaluated along the shock path gives

1 αu 1 ∂∆p 1 ∂u = s + s + s . (4.32) − µ(Rs) Rs ∆ps ∂t us ∂t

The combination of (4.31) and (4.32) along the shock generates the fourth and ﬁnal equation: k Rα∆p u ν(R ) αu 1 ∂∆p 1 ∂u α s s s s = s + s + s . (4.33) − E(Rs) Rs ∆ps ∂t us ∂t Chapter 4. Shock Dynamics in Stellar Outbursts: II. 87

In the next section, we eliminate the subscript s since the variables are all deﬁned with respect to the initial shocked state.

4.2.2 Equations for Shock Propagation

With four equations (4.7), (4.8), (4.9), (4.33), we solve for the four unknowns partial derivatives:

2 α ∂p dp (p p0) uνR α(p p0)u dp0 2 = vs − − vs , (4.34) ∂t dR − E(R) − R − dR 2 α ∂p dp (p p0) uνR α(p p0)u dp0 2vs = vs + − + − + vs , (4.35) ∂R dR E(R) R dR 3 α ∂u 1 dp (p p0) νR α(p p0) ρ0 1 dp0 2 = − 2 − 1 + , (4.36) ∂t −ρ0 dR − (ρ0vs) E(R) − ρ0R − ρ ρ0 dR 3 α ∂u α(p p0) ρ0 (p p0) νR 1 K dp dp0 2vs = − (3 K) + K 1 + − 2 − , ∂R − ρ0R ρ − − (ρ0cs0) E(R) − ρ0 dR − dR where γ + 1 z 1 K − ≡ 2γ z

or Eq. (4.41). Note, the above solutions use the fact that d/dt = ∂t + vs∂R.

Solving for the four unknowns generates the coupled set of diﬀerential equations for

α+1 the shock strength and (normalized) energy y E(R)/(p0R ): ≡

2 dln(z 1) 2(z 1) νK ρ0 (F K) − = − α (2 K) + K − dln(R) −(γ + 1)z + γ 1 · y − ρs − − 2γ dln(p ) + + K 0 (γ + 1)z + z 1 dln(R) − dln(ρ ) +2 (1 + B) 0 , (4.37) dln(R) dy α+1 γ ρ0 1/γ dln(p0) = kαRs z 1 y α + 1 + ,(4.38) dln(R) γ 1 ρs − − dln(R) − Chapter 4. Shock Dynamics in Stellar Outbursts: II. 88 with adiabatic index γ, self-similarity parameter ν, density ρ, and

∂ln(v ) 1 B s = , (4.39) ≡ ∂ln(ρ0) −2 ps p0 ∂vs (z 1) F 2 2 − = 4 −γ−1 , (4.40) ≡ − vs ∂ps − z + γ+1 2 (ρ0vs) γ + 1 z 1 K 1 2 = − , (4.41) ≡ − (ρcs) 2γ z

ρ0 2(z 1) = 1 − . (4.42) ρs − (γ + 1)z + γ 1 − The parameters B, F , and K follow directly from the shock jump conditions. Please see Section 4.4 for a discussion on the parameter B.

4.3 Numerical Simulation

The FLASH simulations from Paper 1 (or Chapter 3.5.2) of steepening acoustic waves are continued until the shocks emerge out of the star. The star is a n = 3 stellar polytrope with adiabatic index γ0 = 4/3. A sinusoidal acoustic wave is added as a perturbation to the stellar structure. This addition is not perfect considering the background medium is not uniform and, as a result, the wave splits into inward and outward propagating components. We measure the outward propagating wave properties once the waves have separated to inform our wave-steepening calculations. Values of the wave properties are

not important except that the waves are locally weak u cs0 and compact λ Hp.A high grid resolution of 130,000 cells ensures the acoustic wave and shock are resolved. In Paper 1, we calculate the evolution of an acoustic wave front through an arbitrary

medium until a shock forms at a radius Rsf . Afterwards, the shock front leading the

acoustic pulse dissipates wave energy. Once the wave peak and shock front coincide (Rs), a self-similar shock structure forms as envisioned by Brinkley & Kirkwood (1947). At

this moment, we use the state of the shock (z, E) at R = Rs as initial conditions for the set of BK equations. p The initial shock strength z0 = z(Rs) = 1 + γ L¯w/Lmax(Rs) is approximated by

the wave peak amplitude at Rs, where Lmax is the maximum acoustic luminosity. The Chapter 4. Shock Dynamics in Stellar Outbursts: II. 89

estimate for Rs in Paper 1 is found to be accurate to within one to two local wavelengths.

An acoustic pulse conserves energy Ew = L¯w/(2ω) until a shock front forms at r = Rsf

(L¯w and ω are the average wave luminosity and frequency). An upper bound for the energy lost during the maturation phase of a weak shock can be estimated by assuming p the shock front strength z 1 + γ L¯w/Lmax is smaller than the wave amplitude for ≤ Rsf r Rs. Integration of the weak shock dissipation rate, ≤ ≤ 3/2 dE Lw γ + 1 1 p , (4.43) dRs ≤ − 3 2 cs0 Lmax(r) yields an equation that resembles the shock formation condition (see eq. 25 of Paper 1)

3/2 Rs Lw Z dr γ + 1 1 ∆E p ≤ − 3 Rsf cs0 2 Lmax(r) 3/2 Lw = (Z(Rs) Z(Rsf )) , (4.44) − 3 −

R r dr γ+1 −1/2 where Z(r) := Lmax and ri is the initial acoustic wave front radius. For an ri cs0 2 acoustic impulse, this equation is approximately

π 2 3/2 ∆E − L Z(Rsf ) (4.45) ' − 3 w

−1 p (see Appendix B). Using the results from Paper 1, Z(Rsf ) = (∂u/∂r) / Lmax(ri) − i ' 1/ω√Lw, we ﬁnd π 2 Lw π 2 ∆Esh − − Ew. (4.46) ' − 3 ω ' − 6

Thus, the energy remaining after the shock matures is between 0.81 . E(Rs)/Ew < 1. Integration of the BK equations with a range of possible initial shock energies generates two classes of solutions. Shocks with sufficient energy have nearly identical evolution. These shocks are capable of accelerating to exceptional speeds, ignoring the effects of a breakout. Shocks with insufficient energy vanish (i.e., z 1) before reaching the stellar → surface. The second class of shocks are called ‘duds’and are discussed further in section 4.5.

Fig. 4.1 shows the strength of the acoustic peak z = max(p)/p0 and shock z = ps/p0 Chapter 4. Shock Dynamics in Stellar Outbursts: II. 90

since inception. The black line is a prediction that uses only the initial acoustic wave properties and stellar structure. It describes the acoustic phase, which is well understood from the conservation of wave luminosity (i.e., Dewar 1970), and shock propagation phase, predicted by the BK equations. If wave-steepening and shock formation are ignored, the wave would continue to strengthen (see coloured lines) until z 2 3. Indeed, this was ∼ − the expected condition for shock formation before wave-steepening was considered in detail in Chapter 3. The prediction does not exactly represent equations (4.37) and (4.38). In fact, the parameter B = 1/2 (Eq. 4.39) is found to be inaccurate. The solution to equation − (4.37) with a modiﬁed deﬁnition for B is the black line shown Fig. 4.1. In the following section, we discuss the physical motivations for the discrepancy and suggest an empirically constrained relation as a temporary solution.

4.4 A Modiﬁed Brinkley & Kirkwood Theory

A comparison between the FLASH simulations (see Chapter 4.3) with the numerical solutions for equations (4.37-4.38) were found in agreement during the early phases

(z z0 1) of (weak) shock propagation using B = ∂ln(vs)/∂ln(ρ0) = 1/2. Afterwards, − − the shock strength is found to amplify faster than an acoustic pulse. Solving the BK equations with B = 0 (i.e., ignoring a density gradient) generates a

non-diverging solution that crudely traces the intermediate shock phase (z 1)/(z0 1) 3, − − ' but fails to reproduce the early and strong phases. The similarity of these solutions to those by Klimishin & Gnatyk (1981) and Nadezhin & Frank-Kamenetskii (1965), for example, suggests to us that B may have been mistreated or neglected in past studies. Sachdev (1972) claim to include B, but its deﬁnition is not stated. Sakurai (1960) argued that a strong planar shock sweeping through an inhomogeneous

−n gas with a power-law structure ρ0 x evolves self-similarly with depth x = (R∗ r) as ∝ − −β vs ρ . The acceleration parameter β is unique for every combination of n and γ (see ∝ 0 Chapter 5 for discussions and a table). This suggests the parameter B = ∂ln(vs)/∂ln(ρ0)

is not constant. The term B is deﬁned for ﬁxed pressures p0 and ps, so it is not necessarily Chapter 4. Shock Dynamics in Stellar Outbursts: II. 91

1 1) 0 − z

1 log (

No shock Acoustic and BK phases

2 dEs Constant dt w/ Sakurai phase FLASH results Fully developed shock 3 Sakurai transition Ms = 3

0.0 0.5 1.0 1.5 2.0 2.5 log (1 r/R ) − ∗

Figure 4.1: Strength of the acoustic waves and shocks versus stellar depth. Coloured points are from FLASH. Black solid lines are the modiﬁed BK theory. Black points indicate where the shock has fully developed R = Rs. Coloured lines are the strengths if a shock does not form. Dashed lines represent the approximate shock formula amended −β with the Sakurai solution vs ρ where the shock is strong (open points). ∝ 0 Chapter 4. Shock Dynamics in Stellar Outbursts: II. 92

β = dln(vs)/dln(ρ0) either. Yet after experimentation, B = 2β is found to follow the − − strong asymptotic z & 5 with reasonable accuracy.

The following correction

2γ B = 2β 1 (4.47) − − (γ + 1)z

is the first attempt to capture the intermediate and strong shock evolution from B ' 0 2β. And, it is found accurate to within O(z) 10% for all z. The error measure → − ∼ O(z 1), however, is more representative for the weak shock phase (see top panel of Fig. − 4.2). Using (4.47), we find deviations ranging 10 30% in some cases, which is significant − considering the weak shock dissipation rate follows the error measure to the third power

3 dEs/dR (z 1) . ∝ − Deviations in the weak limit are found to arise from neglecting B = 1/2. To include − this, we construct a transitional term between the weak and strong limits,

1 z z0 γ B = exp − 2β 1 . (4.48) −2 · −z0 1 − − z − A necessary adjustment of the second term from this addition leads to a signiﬁcant

improvement in two of the four models (middle panel of Fig. 4.2). Adopting the z0 from the two successful predictions for the unsuccessful predictions generates the last panel of Fig. 4.2 and the black lines of Fig. 4.1. The reason for why the last step is successful is not yet known. The corrections present here may produce suﬃciently accurate solutions for shock propagation in a n = 3 polytrope. Further investigation into B would prove beneﬁcial for shock propagation in other stellar structures or beyond stars in general.

As this document was written, an asymptotic expression motivated by the Sakurai solution was derived for B. Although this derivation is incomplete, it suggests a theoretically motivated connection between the weak and strong phases of shock evolution. For a strong planar shock (z 1, α = 0), Eq. (4.37) becomes

dln(z 1) dln(p0) dln(ρ0) (F K) − K + 2(1 + B) . (4.49) − dRs ' dRs dRs Chapter 4. Shock Dynamics in Stellar Outbursts: II. 93

1.4 1.3 Correction 1 1.2 1.1 1.0 0.9 0.8 1.4 1.3 Correction 2 1.2 1.1 1.0 0.9 0.8 1 1.4 − 1 1.3 H Correction 3 S − 1.2 A z

L 1.1 F

z 1.0 0.9 0.8 1.4 1.3 Sakurai phase 1.2 1.1 1.0 0.9 Constant dEs 0.8 dt 0.7 3 2 1 0 1 2 3 log(z 1) FLASH −

Figure 4.2: A comparison between the modiﬁed BK solutions and the FLASH simulations. Three iterations for corrections lead to the third panel. Eq. (4.47) is the ﬁrst correction. The second and third corrections involve (4.48); see the discussion for details. The last panel shows the approximate shock formula discussed in section 4.6. Chapter 4. Shock Dynamics in Stellar Outbursts: II. 94

0.3 Correction 3 0.2 (F K)(1 2β)/2 1 − − − 0.1 ) ) s 0 0.0 ρ V ln( ln( 0.1 ∂ ∂ 0.2

0.3

0.4 2 1 0 1 2 3 4 log(z 1) − Figure 4.3: An example for the third correction for B is the black line. The green line is the derivation for B without p0 terms (see eq. 4.51).

From the Rankine-Hugoniot condition

2 2 ρ0vs (γ 1) z = p0 − − , (4.50) γ + 1 we can estimate the left hand side of (4.49) to be

dln(z 1) dln(ρ0) dln(vs) dln(p0) dln(ρ0) dln(p0) − + 2 = (1 2β) , dRs ' dRs dRs − dRs − dRs − dRs

2 −β for ρ0v p0 and vs ρ for strong self-similar shock acceleration. Solving for B, we s ∝ have: −1 F K F dln(p0) dln(ρ0) B + 1 = (1 2β) − . (4.51) − 2 − 2 dRs dRs For a stellar polytrope, the gradients in the last term can be replaced with (1 + 1/n). The solution above does not exactly reproduce the empirically corrected value for B (i.e., eq.

4.48). But, it is close if the gas is assumed cold p0 = 0 (see Fig. 4.3). Setting γ = 4/3, B is close to, but not exactly, 2β in the strong limit. − Chapter 4. Shock Dynamics in Stellar Outbursts: II. 95

4.5 Shock Dissipation and ‘Duds’

Brinkley & Kirkwood adopt an acoustic interpretation of shocks. These perturbations have a strength z and energy E, which is equivalently the amplitude and period of an acoustic wave. Weak shocks dissipate wave energy in the form of heat, and the rate to which is set by the shock strength. This suggests there is a minimum energy for a shock to propagate and reach the surface.

We found the first term in (4.37) to be negligible for all of our models. This suggests the BK equations can be effectively decoupled. In other words, two shocks with the same strength (at some radius R) but different energy will propagate identically unless one shock dissipates all of its energy. When reducing the initial wave energy, the term eventually dominates and acts to terminate the shock. Figure 4.3 illustrates this for a range of initial shock energies. We call this artifact, from the BK theory, a shock ‘dud’. While this has not been confirmed with hydrodynamic simulation, we will certainly investigate this in the near future.

What are the acoustic conditions that lead to a shock dud? Given the initial shock location and strength, the shock strength can be determined from Eq. (4.37) by setting y−1 = 0, or the ﬁrst term, to be zero. Integrating equation (4.38) with the solution for z provides an estimate for the minimum energy for a shock to reach the stellar surface. Before a shock has fully developed, the acoustic wave led by a shock front also loses energy. In section 4.3, we argue the wave energy loss is less than 20%.

The ﬁrst panel of Figure 4.5 shows the relative amount of shock heat generated after

the shock has fully developed R Rs. We see that most of the shock energy is lost during ≥ the weak phase and not the strong phase. We also see that the correction for B is critical for the BK equations to accurately predict the initial rate of shock dissipation. Chapter 4. Shock Dynamics in Stellar Outbursts: II. 96

2 100% 90% 80%

0 1) − z 1 log (

10% 60% 70% 3

0.0 0.5 1.0 1.5 2.0 2.5 log (1 r/R ) − ∗

Figure 4.4: A set of solutions for the BK equations with various initial energies for a sample model. The percentage is the fraction of acoustic energy used as the initial shock energy. Shocks with suﬃcient energy Emin & 0.75Ew have nearly identical evolutions. Shocks with insuﬃcient energy vanish before reaching the stellar surface. Chapter 4. Shock Dynamics in Stellar Outbursts: II. 97

1.0

0.8

0.6

0.4 Correction 3 0.2 t a w e 1.0 h E E 0.8

0.6

0.4 Constant dEs 0.2 dt

0.0 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 log(z 1) − Figure 4.5: Shown is the shock heat deposited with respect to shock strength. The coloured lines in both panels are the solutions to equation (4.38) given the shock strengths from the FLASH simulations. The black line in the top panel is the prediction from BK theory with the best corrections for B. The black line in the bottom panel is also the solution for (4.38), but for constant dEs/dt. Dashed lines are solutions to Eq. (4.54). Chapter 4. Shock Dynamics in Stellar Outbursts: II. 98

4.6 An Approximate Shock Formula

A surprising result from this work is an approximate formula for weak shock propagation. We observe the shock dissipation rate

dE 2 3 γ + 1 = 4πR p0vs(zs 1) , (4.52) dt − s − 12γ2

to be roughly constant for in a n = 3 stellar polytrope. Since the initial shock properties p (z0 1 = γ Lw/Lmax) and location (r = Rs) are known from Paper 1, we can compute − dE/dt: dE γ + 1 3/2 −1/2 Lw Lmax (Rs), (4.53) dt Rs ' − 6 where we approximate vs cs0, although it is not necessary. Therefore, the shock strength ' is derived by inverting Eq. (4.52). When this calculation predicts the shock to be strong −β (e.g., vs/cs0 = 3), we transition to the Sakurai solution vs ρ . A comparison to the ∝ 0 FLASH results and BK equations is shown in Figures 4.1 and 4.2 (fourth panel). While the deviations are larger than the BK predictions, the calculation is trivial. And, the accuracy of the weak shock phase is better than our ﬁrst correction to the BK equations. The second panel of Figure 4.5 shows the estimated energy lost using Eq. (4.38). This result trivializes the calculation for the shock heat generated. Since dE/dt is constant, the heat lost is simply proportional to the propagation time:

Z t Z R dE γ + 1 3/2 −1/2 dr Eheat dt Lw Lmax (Rs) . (4.54) ' − dt Rs 0 ' 6 Rs cs0

The solution to the above are shown as dashed lines in Figure 4.5.

4.7 Discussion

Canonical examples of shocks in stars come from supernovae, unstable nuclear burning, and strong binary interaction. Indeed, shock formation is triggered from supersonic motions, but velocity is not the decisive condition. Ro & Matzner (2017) argue acoustic waves with very subsonic motions can steepen into shocks as well. Acoustic shocks are Chapter 4. Shock Dynamics in Stellar Outbursts: II. 99

distinct since the associated shock strengths and energies are generally weak. Weak shocks can, however, accelerate towards the surface and become strong. A simple theory for shock propagation has long been sought for to study the dynamics of terrestrial and astrophysical explosions. Matzner & McKee (1999) describe the early Sedov- Taylor and late Sakurai phases of strong shock evolution in stars with an approximate analytic expression. We desire a similar theoretical result for studies on weak eruptions and explosions. Brinkley & Kirkwood (1947) adopt an acoustic interpretation for shocks and present an attractive propagation theory to the acoustic shock problem. We generalize their theory to solve for inhomogeneous mediums with particular focus on a previously neglected term. With comparisons to hydrodynamical simulations, a number of conclusions are drawn. First, we suggest the BK theory is capable of predicting weak shock propagation with the correct asymptotic behaviour in the Sakurai phase. At the moment, the new term

B = ∂ln(vs)/∂ρ0 requires empirical constraints for the theory to succeed. However, we believe these corrections lead to the most successful application of the BK theory to date. And, we argue the definition of B is not arbitrary, but rather a connection between weak and strong explosions. Second, we introduce a hydrodynamic phenomenon called a shock dud: a shock wave that self-terminates from dissipating its reservoir of wave energy. The minimum energy to avoid a shock dud is trivial to calculate from the BK theory. Shocks with the same initial strength and location propagate towards the surface in a nearly identical fashion, so long as they both carry sufficient energy. These shocks are also capable of producing a shock breakout (i.e., an electromagnetic flash and fast ejecta) as seen from supernovae, if they do not radiatively dampen during the weak shock phase. Third, acoustic waves were previously argued to dissipate the majority of their energy within a scale height due to radiative damping. Ro & Matzner (2017) demonstrate this is

not the case for appreciable wave luminosities Lw & L∗/5. Furthermore, we find waves to steepen into weak shocks and dissipate most of their energy during the weak phase at much deeper envelope depths. Weak shocks are definitively small perturbations, so the shocked stellar envelope remains in quasi-static equilibrium. However, a continuous train Chapter 4. Shock Dynamics in Stellar Outbursts: II. 100 of weak shocks will distribute energy across an extended volume and trigger adiabatic expansion. Therefore, we do not expect acoustic waves with super-Eddington luminosities to initially satisfy the dynamical setup described by Quataert et al. (2016) (i.e., local, near-surface energy deposition), unless the acoustic source persists for several envelope dynamical timescales. In summary, acoustic shocks appear to have the capacity to explain many features of eruptions and non-terminal explosions. A pulsation can steepen into a weak shock and accelerate towards the surface to produce fast ejecta. A train of pulsations can heat the envelope to drive adiabatic expansion and initiate an outflow. Waves and shocks can fail to reach the surface if they have insufficient energy, which is partly dependent on details of the progenitor structure. This may, in part, explain the rarity of these events. Chapter 5

Shock Emergence in Supernovae: Limiting Cases and Accurate Approximations

5.1 Chapter Overview

We examine the dynamics of accelerating normal shocks in stratiﬁed planar atmospheres, providing accurate ﬁtting formulae for the scaling index relating shock velocity to the initial density and for the post-shock acceleration factor as functions of the polytropic and adiabatic indices which parameterize the problem. In the limit of a uniform initial atmosphere there are analytical formulae for these quantities. In the opposite limit of a very steep density gradient the solutions match the outcome of shock acceleration in exponential atmospheres. See Ro & Matzner (2013) for the published article.

5.2 Background

Shock emergence at the surface of an exploding star is an important moment in the life of a supernova. Shock and post-shock acceleration in the outer stellar envelope, and the breakout of post-shock radiation from a thin layer beneath the photosphere, can have a number of signiﬁcant consequences. The escaping ﬂash of radiation gives an energetic

101 Chapter 5. Shock Emergence in Supernovae 102

precursor which can signal the supernova’s existence (Klein & Chevalier 1978) and carries physical information about the explosion (Matzner & McKee 1999; Calzavara & Matzner 2004; Nakar & Sari 2012; 2010; Sapir et al. 2011; Katz et al. 2010; Suzuki & Shigeyama 2010; Piro et al. 2010); traveling outward, it can ionize a circumstellar nebula like the one surrounding SN 1987A (Lundqvist & Fransson 1996) and produce an infrared light echo as it encounters dust (Dwek & Arendt 2008). Shock emergence launches the fastest ejecta, the ﬁrst to host the supernova photosphere (Chevalier 1992b) and the ﬁrst to interact with circumstellar and interstellar matter, producing a synchrotron-emitting shell (Fransson et al. 1996). If they meet a companion star or dense circumstellar disk, an additional x-ray signal can be produced (Metzger 2010; Kasen 2010).

In particularly compact and energetic explosions the shock can become relativistic before emerging, and relativistic ejecta can create x-ray and γ-ray transients in their circumstellar collisions (Matzner & McKee 1999; Tan et al. 2001), and may produce light elements through spallation (Fields et al. 2002; Nakamura & Shigeyama 2004) and, potentially, ultra-high-energy cosmic rays (Wang et al. 2007; Budnik et al. 2008).

Potentially observable shock breakouts accompany several other types of astrophysical events, including the type Ia explosions (Piro et al. 2010) and accretion-induced collapses (Fryer et al. 1999; Tan et al. 2001) of white dwarfs, tidal disruptions of stars, jet and cocoon emergence in long-duration gamma-ray bursts, and (albeit in a less energy-conserving manner) superbubbles in galactic disks.

Underlying all these phenomena are the hydrodynamics of shock acceleration in the outer layers of a star, and anchoring these dynamics is the asymptotic problem of flow behind a normal, adiabatic shock accelerating through a planar medium which varies as a power law with depth. As Matzner & McKee (1999) first demonstrated, this asymptotic planar solution can be combined with the dynamics of a spherical, self-similar blastwave into an accurate approximate model for shock propagation and post-shock flow in a spherical explosion. This, in turn, can be used to predict the amount and upper speed limit of the fastest ejecta and properties of the breakout flash (Matzner & McKee 1999; Calzavara & Matzner 2004), transition to relativistic flow and aspects of the circumstellar interaction (Tan et al. 2001), and many other breakout-related phenomena. With advances Chapter 5. Shock Emergence in Supernovae 103

in the theory of photon-mediated shocks and emission around the time of breakout (e.g., Katz et al. 2010; Nakar & Sari 2010; Sapir et al. 2011; Katz et al. 2012), of the ultra- relativistic self-similar problem (Perna & Vietri 2002; Nakayama & Shigeyama 2005; Pan & Sari 2006; Kikuchi & Shigeyama 2007), and of the interaction of relativistic ejecta with a stellar wind (Nakamura & Shigeyama 2006), among others, there are ample opportunities for these approximate global models to be improved and extended. To advance this larger project we focus here on the planar, adiabatic, non-relativistic problem of an accelerating normal shock. Our goal is to provide ﬂexible yet highly accurate approximations for the most important ﬂow quantities, the shock acceleration index and the post-shock acceleration factor, as functions of the adiabatic and polytropic indices (γ

and γp, respectively) which parameterize the problem. A secondary goal is to demonstrate that although the flow quantities must typically be found as eigenvalues of the dynamical problem, they adhere to well-understood limiting forms in several asymptotic cases. The self-similar problem with a power-law atmosphere below vacuum was posed by Gandel’Man & Frank-Kamenetskii (1956) and solved in its Eulerian form by Sakurai (1960). We shall use Sakurai’s eigenvalue method to identify the shock acceleration index, but for the post-shock flow we employ the Lagrangian approach by Matzner & McKee (1999). This has the dual advantage that it continuously describes both the pre-breakout and post-breakout flow in a single function, and that it naturally connects each fluid element’s state at the shock front with those in the final state.

5.3 Problem, method, and solutions

Our problem involves one dimensional ﬂow with an altitude x relative to the stellar surface. n The initial density distribution of cold matter is ρ0(x) ( x) for x < 0 and ρ0(0) = 0 for ∝ − x > 0. Here n is the polytrope parameter, which is related to γp by the hydrostatic relation

γp with constant gravity g∗: if P = 0 at x = 0, then P (x) = g∗( x)ρ(x)/(n + 1) ρ(x) − ∝ with γp = 1 + 1/n. A strong adiabatic shock wave accelerates down this density gradient, reaching x = 0 at t = 0 with infinite velocity; neglecting radiative effects, this is the point of breakout. For t > 0 the shock disappears and matter expands into the region of positive Chapter 5. Shock Emergence in Supernovae 104 x. In the limit that all additional physical effects – curvature, gravity, and temperature of the star, finite depth, non-simultaneity of breakout, relativity, shock thickness, etc. – are negligible, the flow is self-similar. The shock velocity accelerates according to

−λ −β vs = x˙ s(t) ( x) ρ , where β = λ/n. The ﬂuid motion is a universal function of ∝ − ∝ self-similar variables like x/xs( t ) or x(m, t)/x0(m), in which each ﬂuid element (labeled | | by its mass coordinate m) accelerates from a post-shock velocity toward its terminal velocity, which is a unique multiple vf (m)/vs(m) of the shock velocity which crossed that

element. Our task is to ﬁnd β and vf /vs as functions of γ and γp.

5.3.1 Shock acceleration parameter and its limits

To find the shock acceleration index λ, or equivalently the velocity-density index β, we follow Sakurai (1960). Sakurai writes the conservation equations for mass, energy, and entropy in Eulerian form, introduces the self-similar ansatz, and arrives at a single, first-order differential equation for the spatial structure of the post-shock flow prior to breakout. This equation must pass smoothly from the conditions immediately behind the shock front, through a critical point at the sonic point of the flow; this is only possible for a unique value of λ or β, which we identify by a shooting method. We present the solution space β(n, γ) spanning log n = 6, ..., 6 and log (γ 1) = 5, ..., 6 in Table 10 − 10 − − 5.1. For the entire parameter space, the simple functional fit

" #−1 Bγ C β = A + with (5.1) γ 1 − 0.22 A = 2 , − 0.59n−9/8 + 1 2.31 B = 2 + , and 1.72n−1 + 1 1 0.0312 C = 2 − 1.1n−1 + 1

to be quite accurate: the root-mean-square error relative to the values in table 5.1 is 1.4%, and the error, which is concentrated at high n and low γ, is at most 4.0%. High accuracy is necessary, because β is the exponent of a number which becomes large around

2 γp/(2β) breakout. For instance, the energy in relativistic ejecta scale as Erel [Ein/(Mejc )] , ∝ Chapter 5. Shock Emergence in Supernovae 105

where Ein is the explosion energy and Mej is the total ejected mass; in the model for SN

1998bw discussed by Tan et al. (2001), an error of β leads to an error in Erel which is nine times greater. Higher accuracy can be obtained by interpolating our table, and the differential equations yield solutions to numerical accuracy. Several limiting forms of our fit to β(n, γ) are readily apparent. In the limit n 0 of an effectively uniform stellar envelope, β and its fit in equa- → tion (5.1) reproduce the approximate expression derived by Whitham (1958 and 1974, simplifying and improving upon results by Chisnell 1955, Chisnell 1957, and Chester 1960): " #−1 2γ 1/2 β 2 + . (n 0) → γ 1 → − Whitham arrived at this form by reasoning that quantities just behind the shock front should evolve similarly to those found along a forward-traveling sound wave, for which there is an exact equation. In general this is only a good approximation, because the shock moves more slowly than these forward characteristics and because conditions vary from one characteristic to another. In the limit n 0, however, there is no variation → among characteristics, and Whitham’s approximation becomes exact.

The isothermal limit γ 1 is characterized by →

β 0. (γ 1) → →

In this limit a strong shock is inﬁnitely compressive and governed by the conservation of momentum. The shock velocity is constant because the atmosphere above the shock front has negligible mass relative to the shell of post-shock material. We note that Whitham’s approximation is exact in this limit as well, because the shock no longer outruns forward characteristics.

In the limit n the stellar structure is isothermal (γp 1) and transitions from → ∞ → power-law to exponential in its depth dependence. It is reassuring, therefore, that equation (5.1) gives " #−1 4.321γ 0.469 β 1.78 + (n ) → γ 1 → ∞ − Chapter 5. Shock Emergence in Supernovae 106

in this limit. As Hayes (1968) notes, this limit coincides with the case of an exponential

atmosphere (γp = 1); we reproduce his solutions and those of Raizer (1964). Equation (5.1) demonstrates that the shock acceleration index is indistinguishable from the exponential

2 case for all n & 10 – or in practical terms, any time that the distance to the surface is very far when measured relative to the density scale height.

In the γ limit of incompressible ﬂow, β takes the deﬁnite form [A(n)+B(n)C(n)]−1. → ∞ We know of no physical explanation for this result.

We end this section by noting that for the specific case γ = 3/2 and n = 5, we find β = 1/5 and λ = 1 (at least to within a part in 108, while the fit of equation (5.1) gives λ = 1.01). This is unlikely to be a coincidence, although we have not identified any simplification in the dynamical equations for this case.

5.4 Post-Shock Flow and Asymptotic Free Expansion

After each parcel of gas has been swept into motion by the shock, it continues to accelerate

until its internal energy is spent and it has reached the terminal velocity vf (m). To describe this we employ the Lagrangian method of Matzner & McKee (1999). This

naturally provides quantities like vf (m)/vs(m), and continues smoothly through the point of breakout; however it does not yield eigenvalues like β as readily as Sakurai’s method. In order to correct a couple typos in Matzner & McKee’s Appendix (which do not aﬀect their results), we write out the equation. The Lagrangian self-similar time and space

coordinates are η = t/t0(m) and S = x/x0(m), where t0(m) is the time at which the

shock crosses x0(m). For a given fluid element both η and S decline from unity to −∞ as a fluid element accelerates outward; shock breakout (neglecting radiative effects) is at η = 0, and the element exits the boundaries of the progenitor somewhat later, when S = 0. As Matzner & McKee discuss, the pressure and density distributions and the resulting acceleration can be computed from these variables, and the equating resulting Chapter 5. Shock Emergence in Supernovae 107

00 2 ﬂuid acceleration to x¨(m) = S (η)x0(m)/t0(m) yields

2 γ 1γ (1+ λ )2S00(η) = − −γ + 1 γ + 1 × n 2λ λS0(η) + (λ + 1)ηS00(η) − γ(λ + 1)η (5.2) Σ(η)γ − Σ(η)γ+1 where Σ(η) S(η) (λ + 1)ηS0(η). Note that relative to Matzner & McKee’s equation ≡ − (A2) and its preceding discussion, all instances of λ 1 have been corrected to λ + 1. We − integrate equation (5.2) from the post-shock conditions S(1) = 1, S0(1) = 2/[(λ+1)(γ +1)] to a very large negative value of η (typically η > 1020). −

5.4.1 The acceleration factor and its limits

0 −(γ−1) 0 We use the asymptotic form [Sf S (η)] ( η) , valid for γ > 1 (where S = − ∝ − f 0 0 0 S (η )), to deduce vf (m)/vs(m) = (λ + 1)S /S (1). We present this in Table 5.2; → ∞ f the approximation

" # v (m) 2 2γ D f = + E where (5.3) vs(m) γ + 1 γ 1 − 1 1.832 1.655 D = + , 2 − (1.083/n)0.828 + 1 (1.5/n)0.883 + 1 0.803 E = 1 − (2.39/n)1.18 + 1

is reasonably accurate: over the values in Table 5.2, the rms error is 0.6% and the maximum error is 1.8%. We do not have solutions for very small values of γ 1, however, − so we cannot check accuracy in that regime. The form of equation (5.3) gives some insight into the nature of post-shock acceleration, because the initial factor 2/(γ + 1) expresses the immediate post-shock velocity in units D of vs, so the factor [2γ/(γ 1)] + E captures the subsequent acceleration from the − post-shock state to the ﬁnal state of free expansion. A clear limit of equation (5.3) is that in which the initial density distribution becomes uniform: " # v (m) 2 2γ 1/2 f + 1 . (n 0) vs(m) → γ + 1 γ 1 → − Chapter 5. Shock Emergence in Supernovae 108

This limit is a consequence of the isentropic nature of the post-shock ﬂow, which ensures

that the Riemann invariant v + 2cs/(γ 1) is conserved along outward-traveling sound − fronts from the post-shock state to the freely expanding state; here cs is the adiabatic sound speed.

In the opposite limit, corresponding to an exponential atmosphere,

" # v (m) 2 2γ 1/3 1 f + . (n ) vs(m) → γ + 1 γ 1 5 → ∞ −

to within 3%. Hayes (1968) and Grover & Hardy (1966) apply a Eulerian approach and do not provide this ratio. Raizer (1964) does computes post-shock acceleration in his Lagrangian treatment: Zel’dovich & Raizer (1967) report that he finds an increase of velocity, from the post-shock state to the time the shock blows out (vs ) by the factor → ∞ (1.54,1.85) for γ = 1.2, 5/3, respectively. Furthermore, Zel’dovich & Raizer report that Raizer finds practically no subsequent acceleration. According to equation (5.3), the ratio of final to post-shock acceleration is (2.43, 1.88) for the same values of γ: post-blowout acceleration is a strong function of γ.

In the isothermal limit γ 1, approximation (5.3) and the data of Table 2 indicate → −q that vf /vs diverges proportionally to (γ 1) for 1/5 q 1/2, as one might expect − . ≤ from the divergence of the internal energy. In the incompressible limit γ , the ﬁnal → ∞ velocity limits to 2D(n) + E(n) times the post-shock velocity, but the post-shock velocity

becomes negligible compared to vs; hence vf /vs 0. → One could describe the parameter dependence of other aspects of our planar shock emergence problem, such as the values of S and S0 at η = 0, which describe both the instant of breakout and the deep ejecta. Another example would be the coeﬃcient in the

−(γ−1) relation [vf (m) v(m, η)] S , which holds for ( η) 1 and from which one can − ∝ | | − express the influence of spherical expansion on the final velocity of fluid elements with

γ−1 ﬁnite values of ( x0 /R∗) (Kazhdan & Murzina 1992; Matzner & McKee 1999), where | | R∗ is the radius of curvature. But since these quantities are readily available from the

integration of equation (5.2) and are not as fundamental as β and vf /vs, we omit them. Chapter 5. Shock Emergence in Supernovae 109

5.5 Discussion

We intend our study of the parameter dependence and limiting cases of the shock emergence problem to be useful for future investigations, and we envision several possibilities.

First, real stellar structures are not perfectly characterized by ideal polytropes, and radial variations in the polytropic index near the stellar surface can inﬂuence the dynamics of shock emergence. Matzner & McKee (1999) propose the shock velocity approximation

1/2 β Ein mej vs = Λ 3 (5.4) mej ρ0r

for a spherical explosion of energy Ein traveling through a mass distribution described by

radius r, density ρ0, and enclosed ejecta mass mej. This form matches the properties of 2 an interior blastwave (an energy-conserving ﬂow in which vs(r) mej(r) Ein) with those ∝ of planar shock acceleration (a second-type similarity solution with eigenvalue indices). Matzner & McKee show that equation (5.4) achieves an accuracy of a few percent in spherical, adiabatic supernova models when Λ and β are taken from a representative Sedov blastwave and an n = 3 polytropic atmosphere, respectively. To improve the accuracy of the energy scale for relativistic ejection, Tan et al. (2001) adjust Λ according to the moments of the density distribution interior to r, while holding β ﬁxed. The information provided in Table 5.1 and equation (5.1) should allow for further improvements in which

1 vs better responds to the local conditions.

Second, for any hydrostatic envelope in which radiation pressure is initially negligible, there exists a range of shock strengths which are both strong and yet not dominated by radiation pressure in the post-shock state. These might develop in the steepening of ﬁnite-amplitude sound pulses (Wyman et al. 2004) or in the explosion launched by the impact of a comet or asteroid on the atmosphere (Chevalier & Sarazin 1994). Insofar as the post-shock gas is described by a characteristic value of γ diﬀerent from 4/3, our results indicate how shock emergence is changed.

1It may also be possible to incorporate Waxman & Shvarts (1993)’s second-type similarity solutions for spherical blastwaves in steep density distributions (d ln ρ/d ln r < 3), although the rapid variation of d ln ρ/d ln r in subsurface regions of stars poses a diﬃculty. − Chapter 5. Shock Emergence in Supernovae 110

Third, any investigation of the phenomena surrounding shock acceleration and shock emergence (e.g., shock instability; Luo & Chevalier 1994) must use the ideal planar solution as its reference state. Understanding this solution’s parameter dependence and limits therefore adds insight into the phenomenon under study. Chapter 5. Shock Emergence in Supernovae 111 3 / = 5 γ = 2 γ = 3 γ = 5 γ β = 10 γ 2 = 10 γ 3 = 10 γ Table 5.1. Shock acceleration index 4 = 10 γ 5 = 10 γ 6 = 10 0.292893150.29289304 0.292892600.29289202 0.29289250 0.292887140.29288175 0.29289147 0.29288704 0.292832510.29277948 0.29288120 0.29288601 0.29283241 0.292283290.29179950 0.29277893 0.29287574 0.29283138 0.29228319 0.28647450 0.29179895 0.29277346 0.29282110 0.29228215 0.28647438 0.27924077 0.29179337 0.29271871 0.29227175 0.28647322 0.27924064 0.26794918 0.29173755 0.29216823 0.28646167 0.27923935 0.26794903 0.24999998 0.29117637 0.28634660 0.27922645 0.26794755 0.24999981 0.23606796 0.28524584 0.27909810 0.26793281 0.24999810 0.23606777 0.27787244 0.26778608 0.24998100 0.23606593 0.26638903 0.24981098 0.23604750 0.24819942 0.23586431 0.23413390 0.26947121 0.26947050 0.26946332 0.26939150 0.26867090 0.26120469 0.25226650 0.23899134 0.21925258 0.20484213 0.26906781 0.26906709 0.26905989 0.26898795 0.26826612 0.26078961 0.25184438 0.23856746 0.21884069 0.20444712 0.26902677 0.26902605 0.26901886 0.26894690 0.26822495 0.26074742 0.25180152 0.23852446 0.21879895 0.20440713 0.26902266 0.26902194 0.26901475 0.26894279 0.26822083 0.26074320 0.25179722 0.23852015 0.21879477 0.20440312 0.26902225 0.26902153 0.26901434 0.26894238 0.26822041 0.26074278 0.25179679 0.23851972 0.21879436 0.20440272 6 5 4 3 2 1 2 3 4 5 6 − − − − − − 123 0.285022404 0.28096608 0.285021785 0.27852654 0.28096542 0.285015546 0.27690616 0.27852586 0.28095885 0.284953087 0.27575433 0.27690548 0.27851912 0.280893118 0.28432550 0.27489450 0.27575364 0.27689863 0.278451699 0.28023286 0.27422857 0.27773019 0.27489380 0.27574673 0.27683020 0.27777456 0.27369780 0.27331989 0.27422787 0.26962150 0.27488685 0.27567764 0.27614306 0.27326496 0.27070121 0.27369710 0.26488204 0.27422088 0.27481730 0.25718336 0.27498397 0.26897608 0.27326425 0.26210529 0.27369009 0.27415099 0.25205903 0.27411908 0.23791739 0.26775669 0.26029132 0.27325722 0.24911283 0.27361994 0.27344944 0.23244760 0.22335005 0.26685018 0.25901642 0.24721016 0.27318687 0.22938205 0.27291585 0.21779161 0.26615027 0.25807251 0.24588314 0.22743179 0.27248079 0.21472787 0.26559379 0.25734597 0.24490599 0.22608481 0.21279709 0.26514085 0.25676970 0.24415692 0.22509974 0.21147160 0.25630156 0.24356465 0.22434841 0.21050626 0.24308471 0.22375664 0.20977222 0.22327860 0.20919542 0.20873031 n γ 10 0.27290528 0.27290457 0.27289753 0.27282702 0.27211933 0.26476509 0.25591380 0.24268799 0.22288441 0.20834736 10 10 10 10 10 10 10 10 10 10 10 Chapter 5. Shock Emergence in Supernovae 112 5 − = 1 + 10 γ 4 − = 1 + 10 γ 001 . = 1 γ 01 . = 1 γ 06 . = 1 γ 10 . = 1 γ 15 . Table 5.1. – Continued = 1 γ 7 / = 9 γ 3 / = 4 γ 5 / = 7 γ 2 / = 3 0.22474485 0.21525042 0.20710676 0.19999998 0.16903939 0.14946752 0.12587822 0.06168015 0.02139325 0.00697212 0.00222610 0.19360194 0.18445447 0.17678575 0.17021368 0.14266166 0.12595038 0.10631033 0.05394810 0.01988142 0.00674689 0.00219684 0.22474466 0.21525022 0.20710656 0.19999978 0.16903920 0.14946734 0.12587807 0.06168008 0.02139324 0.00697212 0.00222610 0.19322367 0.18409167 0.17643698 0.16987761 0.14238391 0.12571071 0.10611667 0.05387344 0.01986630 0.00674458 0.00219654 0.22474274 0.21524825 0.20710457 0.19999778 0.16903728 0.14946556 0.12587652 0.06167943 0.02139311 0.00697210 0.00222610 0.19318538 0.18405496 0.17640170 0.16984362 0.14235584 0.12568649 0.10609710 0.05386590 0.01986477 0.00674435 0.00219651 0.22472354 0.21522862 0.20708471 0.19997783 0.16901808 0.14944769 0.12586101 0.06167292 0.02139185 0.00697191 0.00222607 0.19318155 0.18405129 0.17639817 0.16984022 0.14235303 0.12568406 0.10609514 0.05386514 0.01986461 0.00674433 0.00219650 0.22453274 0.21503352 0.20688741 0.19977969 0.16882748 0.14927042 0.12570727 0.06160839 0.02137936 0.00697008 0.00222584 0.19318116 0.18405092 0.17639782 0.16983988 0.14235275 0.12568382 0.10609495 0.05386507 0.01986460 0.00674432 0.00219650 0.22273535 0.21319978 0.20503642 0.19792385 0.16705409 0.14762734 0.12428800 0.06101711 0.02126501 0.00695327 0.00222365 6 5 4 3 2 1 2 3 4 5 6 − − − − − − 12 0.211735033 0.20213822 0.20619672 0.194002954 0.19667012 0.18697155 0.20318002 0.18862853 0.156992415 0.19371804 0.18170115 0.13850153 0.20129129 0.18574712 0.15236756 0.116575866 0.19187868 0.17889128 0.13440382 0.20000000 0.05791760 0.18395846 0.14995044 0.113190857 0.19062491 0.02066532 0.17715216 0.13228214 0.19906222 0.05659923 0.18274204 0.00686464 0.14846962 0.111452838 0.18971624 0.02040824 0.17597158 0.13098826 0.19835060 0.05592780 0.18186180 0.00682633 0.14747055 0.110397109 0.18902773 0.02027633 0.17511833 0.13011769 0.19779228 0.05552103 0.18119558 0.00221214 0.00680657 0.14675145 0.10968834 0.18848814 0.02019600 0.17447309 0.12949217 0.19734262 0.05524823 0.18067391 0.00220716 0.00679450 0.14620925 0.10917982 0.18805395 0.02014191 0.17396818 0.12902113 0.05505258 0.18025441 0.00220459 0.00678635 0.14578590 0.10879727 0.02010302 0.17356237 0.12865367 0.05490541 0.00220302 0.00678049 0.14544622 0.10849906 0.02007369 0.12835905 0.05479069 0.00220197 0.00677606 0.10826008 0.02005080 0.05469876 0.00220120 0.00677260 0.02003242 0.00220063 0.00676981 0.00220018 0.00219982 n γ 10 0.19697276 0.18769706 0.17990977 0.17322912 0.14516765 0.12811757 0.10806430 0.05462344 0.02001735 0.00676753 0.00219952 10 10 10 10 10 10 10 10 10 10 10 Chapter 5. Shock Emergence in Supernovae 113 7 / = 9 γ 3 / = 4 γ 5 / = 7 γ s /v f 2 v / = 3 γ 3 / = 5 γ = 2 γ = 3 Table 5.2. Shock acceleration factor γ = 5 γ = 10 0.452855820.45284794 0.860377530.45278311 0.86035969 1.366021180.45230353 0.86021382 1.36599636 1.999991360.44914536 0.85913741 1.36570632 1.99991857 2.427026490.43299067 0.85209731 1.36354654 1.99933390 2.42693295 2.75941695 0.81682384 1.34966085 1.99506907 2.42608170 2.75930702 3.03724468 1.28214995 1.96792794 2.41967881 2.75822288 3.03713155 3.27999086 1.84166283 2.37969643 2.74980213 3.03601982 3.27847912 3.49275349 2.19974497 2.69731006 3.02586515 3.27734354 3.49263769 2.46728053 2.96117950 3.26622378 3.49151902 2.68379133 3.19010855 3.47995904 2.86763036 3.39413426 3.02862713 0.271741840.26961772 0.519136390.26940604 0.51544773 0.826079180.26938403 0.51486690 0.81984467 1.200292940.26898345 0.51503932 0.81768673 1.19413475 1.43475741 0.51486171 0.81763521 1.19303976 1.42348349 1.60437973 0.81763006 1.19120321 1.42366157 1.59846719 1.73661580 1.18946787 1.42376162 1.59710106 1.72707902 1.84484855 1.42168664 1.59444347 1.72484391 1.84023222 1.93645313 1.59284112 1.72476060 1.83940673 1.93002834 1.72163227 1.83927471 1.92736575 1.83940534 1.92730526 1.92594338 6 5 4 3 2 1 2 3 4 5 6 − − − − − − 123 0.375065764 0.34455376 0.702030215 0.32711534 0.64686880 1.088209336 0.31607498 0.61597751 1.00657057 1.531167977 0.30851925 0.59655629 0.96228259 1.422306898 1.79844429 0.30304299 0.58330933 0.93475556 1.366269089 1.67401293 0.29889916 1.98896476 0.57372626 0.91608575 1.33209762 1.61234177 0.29565625 1.85312095 0.56648342 2.13717743 0.90262297 1.30913793 1.57533069 0.29305274 1.78782363 0.56082202 1.99201909 0.89246676 1.29263311 2.25915336 1.55065918 1.74914052 0.55627762 1.92386840 0.88454049 1.28029414 2.10581684 1.53304634 2.36311029 1.72353305 1.88391867 0.87818307 2.03523614 1.27061632 1.51985203 2.20243857 1.70533006 1.85762232 1.99423027 1.26289956 2.12966295 1.50960199 1.69172734 1.83899322 1.96737210 2.08770216 1.50140888 1.68117857 1.82510463 1.94840504 2.06033613 1.67263063 1.81435616 1.93429573 2.04106258 1.80579054 1.92338863 2.02675344 1.91470828 2.01570684 2.00691959 n γ 10 0.29091665 0.55255066 0.87297252 1.25662285 1.49471099 1.66575381 1.79880288 1.90763458 1.99976839 10 10 10 10 10 10 10 10 10 10 10 Chapter 6

Conclusions & Future Work

Large time-domain surveys such as LSST and PAN-STARRS are coming online and will gather vast catalogues of odd transients. Recent observations of pre-SN outbursts and circumstellar environments suggest the presence of new modes of mass loss. Therefore, it is important to invest effort to keep theory as sophisticated as the observations. Empirical rates for stellar mass loss remain superior to theoretical estimates. This is not indicative of a lack of theoretical effort, but rather the rich complexity of physics involved. We investigate optically thick outflows from stellar winds. These outflows can be either radiatively driven due to opacity enhancements from metals or adiabatically driven from acoustic shocks. This thesis has advances these problems by isolating and studying the underlying physics involved. Below I summarize the results from this thesis and discuss the tasks ahead.

6.1 Optically Thick Radiatively Driven Outﬂows

˙ −6 −1 Winds with mass loss rates above M & 10 M yr are suﬃciently dense to become optically thick (Gräfener et al. 2011; Vink et al. 2011). Non-LTE atmospheric codes use information from the supersonic or observable parts of the wind to constrain the invisible properties of the star. However, this has been problematic for WR stars when measuring basic properties such as the stellar radius. Petrovic et al. (2006) and Gräfener et al. (2012) found the hydrostatic solution for WR stars with an embedded iron opacity bump to contain a radiation dominated halo

114 Chapter 6. Conclusions & Future Work 115 capped by a density inversion. Lamers & Cassinelli (1999) argue that hydrostatic models are good approximations for the subsonic structure beneath a sonic point. However, the sonic point for WR winds is not the total sound speed, but the gas sound speed. This is because the photons diffuse faster than the outflow speed and, so, is decoupled from the gas. Therefore, we question whether these hydrostatic solutions persist in a (technically) supersonic flow.

We find the structures to persist up to a critical mass loss rate; a value that is • derived numerically and theoretically. Beyond this limit, the inflated structure becomes completely erased. Although the supersonic solutions we compute are not completely accurate since the flow is no longer in LTE, we argue that they underestimate the radiative acceleration by ignoring Doppler shifts; therefore, the critical rates are overestimates. Comparing this rate to the galactic population of WR stars, we find that all of their winds have rates that exceed our overestimated limit; so, they are not likely to contain these strange structures.

The condition for envelope inflation is found to be (approximately) equivalent to • having energetically positive gas. An alternative definition is where the temperature scale height is comparable to the stellar radius. These conditions translate into a boundary in ρ T space where envelope inflation occurs. Models that cross this − limit are expected to generate radiative-hydrodynamic instabilities.

A byproduct of this work is a theoretical estimate for the minimum mass loss rate • for an opacity bump to drive a transonic wind. This estimate is strongly sensitive to the metallicity and, so, is relevant for computing winds or build up of massive stars. This may be especially important when considering the ﬁrst stars or stability of supermassive stars.

6.2 Non-Terminal Eruptions and Explosions

Luminous blue variables and a handful of Type IIn progenitors are seen to erupt and explode non-terminally (either directly or indirectly). Here, I distinguish an explosion Chapter 6. Conclusions & Future Work 116

from an eruption when there are indications of shock acceleration. Eruptions could be driven by shocks, however, their outflow speeds and duration do not clearly indicate a single short-term event. Rather, they are more representative of winds that spontaneously turn on (Smith 2014). Exceptionally fast outflow is both a prediction and observation from core-collapse supernovae. There, a supernova shock accelerates due to the strong envelope density gradient and reaches a maximum speed upon breakout. This phenomenon motivated us to consider hydrodynamical energy transport mechanisms capable of forming shocks to explain the observations mentioned above. Quataert & Shiode (2012) and Shiode & Quataert (2014) argue waves from the advanced nuclear stages are capable of carrying sufficient energy to unbind the stellar envelope before the core collapses. Since acoustic waves are naturally unstable due to steepening, this inspired us to characterize this behaviour more carefully.

We derive an analytic solution that describes the steepening of an acoustic wave • front given an arbitrarily structured medium. So long as the wave remains planar, cylindrical, or spherical, the shock formation radius can be computed analytically. Waves are susceptible to dampening by radiative diffusion, which can prohibit the steepening process and shock formation. We find waves carrying super-Eddington luminosities must always form shocks before radiative dampening is efficient. There- fore, super-Eddington outbursts must involve shocks if they are powered by acoustic waves. This analytic solution can be a new test for hydrodynamical simulations.

We find the condition for a wave to steepen into a shock, rather than damp by • radiative diffusion, is when the wave luminosity exceeds a fraction 10 20% of − the local radiative luminosity. (The exact value to which depends on the stellar composition.) This suggests variable stars with flux variations above this fraction (e.g., RSGs, AGBs, Cepheids, β Cephs) may harbour shocks.

The majority of acoustic shocks are expected to be weak upon formation. Therefore, • the shocks must accelerate signiﬁcantly to generate the fast outﬂow seen from non- terminal stellar explosions. We re-derive the shock propagation theory by Brinkley Chapter 6. Conclusions & Future Work 117

& Kirkwood and suit to study inhomogeneous media. The theory does not approach the strong shock limit found by Sakurai (1960). So, we modify the equations to correctly obtain this behaviour. The sum of these modiﬁcations reproduce the

numerical results with . 10% error.

We derive the minimum energy for a shock to reach the surface of the star. Shocks • with insufficient energy become ‘duds’ and self-dissipate in the envelope. This calculation is purely hydrodynamic and ignores the conditions for breakout (i.e., radiative dampening). We also find shocks with the same strength (and location) but different energy propagate nearly identically.

For n = 3 stellar polytropes, we ﬁnd a formula that provides numerically trivial • calculations for weak shock propagation. The approximation gives an accurate estimate for where the shock becomes strong and supply the coeﬃcients for the Sakurai solution.

We solve for the strong shock self-similar acceleration parameter β for any density • power-law and adiabatic index in Chapter 5. This is necessary when modifying the Brinkley & Kirkwood equations to obtain the correct asymptotic limit, as well as considering strong shocks in general.

6.2.1 Optically Thick Adiabatic Outﬂows

Acoustic shocks are found to deposit most of their wave energy during the earliest phases of propagation. These shocks are initially weak (even if they transport super-Eddington luminosities) and inject relatively little heat into local envelope. However, the envelope may accumulate significant energy from a train of weak shocks. If the energy deposition timescale is longer than the thermal time then the envelope will convect the heat towards the surface. This continues until convection becomes inefficient due to low thermal capacities or porosity effects (Owocki et al. 2004), resulting in a optically thick radiatively-driven wind. Otherwise, the shocked gas will do p dV work, − expand adiabatically, and generate an optically thick outflow. Quataert et al. (2016) show Chapter 6. Conclusions & Future Work 118

how adiabatic outﬂows have properties that generally match the slow LBV outﬂows. One caveat is that the shock deposition occurs deep in the envelope over a distributed volume, rather than near the surface, which is not the setup considered. The sudden adiabatic expansion of gas could trigger a secondary acoustic wave that steepens into a shock. This is partly seen from MESA-hydrodynamical simulations when energy is injected into the envelope due to wave dissipation (Fuller 2017).

6.3 Future Work

The theoretical attention brought in this thesis raises a number of questions with exciting prospects for future work. A next step is to construct the stellar model for the entire subsonic domain up to the sonic point. One important limitation of the WR wind work is the assumption that the outflow is in LTE. We find that this is true for a marginal velocity range above the sonic point. As a result, we cannot determine whether a wind solution will escape to infinity or fall back onto the star without correctly solving for the supersonic structure. This is important since fallback may disrupt the assumption of steady flow. The addition of expanding opacities, or some treatment of Doppler expansion, may give sufficient insight into the invisible wind structure to anchor non-LTE atmospheric codes. The critical mass loss rate to erase envelope inflation is limited for the same reason. However, this is a criticism to all models of envelope inflation (static or not), if they are to be consistent with the observed mass loss rates from WR stars. Nevertheless, a ’steady’ sonic point boundary condition would provide a theoretical mass loss rate and serve as a useful reference. The minimum mass-loss rate for iron-driven winds is strongly sensitive to metallicity. A theoretical estimate is important to study the growth and appearance of novae winds and massive stars in the early universe (or low metallicity environments). As motivated in the Introduction, mass-loss regulates the angular momentum, so massive stars in the early universe are likely to rotate faster. Including rotation is the next step in deriving a sonic point criterion and determining the buildup and stability of rotating stars. All of the analyses on waves assume radial symmetry. A non-spherical wave or star will lead to partial wave transmittance and geometric wave deformation. Tidal interactions Chapter 6. Conclusions & Future Work 119

from a binary companion would also drive non-radial waves. Shock formation in either scenario introduces a means to deposit angular momentum into the gas, which may affect the stellar evolution or binary evolution. Non-radial shocks that reach the surface are known to have radical ejecta distributions (Matzner et al. 2013; Salbi et al. 2014). Extensions towards waves and shocks in disks may prove beneficial in studies of stellar mass accretion and planet formation. Calculations of weak shock propagation here assume a quasi-static progenitor. For slowly expanding gas, the calculations may be a good approximation to describe the energy deposition from a train of weak shocks. Deriving (and confirming) the conditions for when this is accurate would be very useful to construct a stellar wind model. The calculation, however, must degrade once a transonic outflow is triggered. Understanding energy deposition from acoustic shocks in an expanding medium is a rich numerical and theoretical problem. The benefits could lead to predictions for circumstellar structures around LBVs and Type IIn progenitors. Flash spectroscopy is the observation of spectra from circumstellar gas ionized by shock breakout radiation. Over time, observations are made closer and closer to the moment of breakout (Khazov et al. 2016; Yaron et al. 2017). Although rare, it is possible that one of these observations will be from a supernova imposter or pre-SN outburst. Using existing shock breakout emission analyses, constraints on the star and explosion (or wave) properties can be made with the shock analyses from Chapters 3 and 4. Such a calculation would provide the first constraints on non-terminal explosion mechanisms. The combination of a strong non-radial acoustic pulse, shock acceleration, and oblique shock breakout could produce a distribution of ejecta that resembles the circumstellar relic of η Car’s giant eruption. Appendices

120 Appendix A

Acoustic Wave-Steepening

Substituting the Taylor expanded ﬂuid variables into the ﬂuid equations generates the following collection of ξ0 and ξ1 order terms. Note that we simplify the notation by

substituting a cs. ≡

2 a0 p1 a0u1 = + g (A.1) γ0 p0 2 γ1 p1 0 2 2a0a1 p1 a0 p2 0 = ( a0u1 + g) + + u1 a0u2 + u1 + + (A.2) − γ0 p0 − γ0 p0 γ0 p0 0 p0 p1 a0 = a0 γ0u1 (A.3) p0 p0 − 0 p1 p2 αu1 p1 0 = a0 + γ1u1 + γ0 u2 + + (γ0 + 1)u1 (A.4) p0 − p0 r p0 0 γ0u1 = a0(γ γ1) (A.5) 0 − 0 γ1 = γ0u2 + γ2a0 (A.6) γ a0 = a + 0 u (A.7) 0 1 2 1

We leave out the ξ1 result from the continuity equation since it is egregious in length and 1 not used in the next derivation. The variables that emerged from the ξ result include γ0, 0 u1, u2, a0, a1, a2, and a1.

121 Appendix A. Acoustic Wave-Steepening 122

Derivation

0 The goal is to find a final differential equation with u1, u1 and any quiescent variables (subscript 0). Through a process of elimination using equations (A3), (A4), (A6), and the derivative of (A3), one can generate the following differential equation:

0 0 2 0 γ0 αa0 γ0g 0 = 2u1 + (γ0 + 1)u1 + a0 + a0 + u1. (A.8) γ0 r − a0

0 The quiescent gas is initially in hydrostatic equilibrium, which satisﬁes p = gρ0. Since 0 − 2 the local quiescent sound speed is a0 = γ0p0/ρ0, we can eliminate the body force (i.e., 0 gravity) by substituting γ0g/a0 = a0p /p0. − 0

With respect to u1(t) = y(t), this equation is known as the Bernoulli equation y0 + p(t)y + q(t)yn = 0 for n = 2. The solution can be written explicitly in the form

Z t −1 −φ(t) −1 γ0 + 1 φ(τ) u1 (t) = e u1 (0) + e dτ , (A.9) 0 2 where, Z t 0 0 1 0 γ0 αa0 a0p0 φ(t) a0 + a0 + dτ. ≡ −2 0 γ0 r − p0

We can rewrite the integrand in terms of the radius, since dτ = dr/a0, and integrate the expression to obtain

Z 0 0 0 r r a0 γ0 α p0 α α 3 2φ(r) + + dr = ln (γ0r a0p0) = ln r ρ0a0 − ≡ a0 γ0 r − p0 r0 − r0 s L (r ) = eφ(r) = max 0 . (A.10) ⇒ Lmax(r)

Thus, the wave front evolution can be solved analytically with the following expression

s s Z r ! −1 Lmax(r) −1 γ0 + 1 Lmax(r0) dr˜ u1 (r) = u1 (r0) + . Lmax(r0) r0 2 Lmax(˜r) a0

Notice that the body force g is never deﬁned explicitly. Since g is arbitrary, the analytic result must be valid for an arbitrary distribution of ﬂuid. This solution is valid for planar, Appendix A. Acoustic Wave-Steepening 123 cylindrical, and spherical waves (α = 0, 1, 2). Appendix B

Shock Maturation

The goal here is to show Z(Rs) (π 1)Z(Rsf ) for a weak impulse and Z(Rs) ' − ' R r γ+1 −1/2 dr (π/2)Z(Rsf ) for a weak wave train, where Z(r) := Lmax . ri 2 cs0

The shock reaches maturity at r = Rs once the peak (left-hand side) and wave/shock front (right-hand side) coincide. That occurs when

Z Rs dr Z Rsf dr Z Rs dr = + , (B.1) ri−λ/4 vw ri cs0 Rsf vs where λ0/4 is the initial separation of the wave peak and front, vs is the shock front velocity, and γ0−1 r r ! 2γ0 vw Lw Lw = + 1 + γ0 , (B.2) cs0 Lmax Lmax

is the wave peak’s phase velocity. We can use the above condition to solve for Z(Rs) by linearization and manipulating the bounds.

−1 The Taylor-expansion of vw is

c γ + 1 r L s0 1 w (B.3) vw ' − 2 Lmax

124 Appendix B. Shock Maturation 125

The left hand side of eq. (B.1) can be broken up:

Z Rs dr Z ri dr Z Rs dr = + ri−λ0/4 vw ri−λ0/4 vw ri vw " # r Z Rs γ + 1 Lw λ0 dr 1/2 1 + Z(Rs)Lw ' − 2 Lmax 4cs0 ri cs0 − Z Rs π dr 1/2 + Z(Rs)Lw . (B.4) ' 2ω ri cs0 −

p In the last step, we assume Lw/Lmax 1.

−1 1/2 From Paper 1, ω = Z(Rsf )Lw is related to the shock formation condition. Therefore, we have Z Rs Z Rs dr π 1/2 1/2 dr Z(Rsf )Lw Z(Rs)Lw + . (B.5) ri−λ0/4 vw ' 2 − ri cs0

Substituting this back into eq. (B.1), we can solve for Z(Rs):

Z Rs Z Rs 1/2 π dr dr Z(Rs)Lw + . (B.6) ' 2ω Rsf cs0 − Rsf vs

Therefore, Z(Rs) (π/2)Z(Rsf ) as long as the following two conditions hold: '

r L w 1 (B.7) Lmax Z Rs dr Z Rs dr , (B.8) Rsf vs ' Rsf cs0 which is the case for a train of waves.

2 p For a single impulse, we know that (vs/cs0) 1 + (γ + 1)/2 Lw/Lmax. Thus, '   Z Rs dr Z Rs dr Z Rs dr 1 1 q  c − v ' c − γ+1 Lw Rsf s0 Rsf s Rsf s0 1 + 4 Lmax Z Rs 1/2 γ + 1 −1/2 dr Lw Lmax ' Rsf 4 cs0 1/2 Lw = (Z(Rs) Z(Rsf )), (B.9) 2 − Appendix B. Shock Maturation 126 and

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