Outflows from Compact Objects in Supernovae and Novae

Outﬂows from compact objects in supernovae and novae by Andrey Dmitrievich Vlasov

Submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2017 c 2017 Andrey Dmitrievich Vlasov All Rights Reserved Abstract

Outﬂows from compact objects in supernovae and novae by Andrey Dmitrievich Vlasov

Originally thought of as a constant and unchanging place, the Universe is full of dramas of stars emerging, dying, eating each other, colliding, etc. One of the first transient phenomena noticed were called novae (the name means ”new” in Latin). Years later, supernovae were discovered. Despite their names, both novae and supernovae are events in relatively old stars, with supernovae marking the point of stellar death. Known for thousands of years, supernovae and novae remain among the most studied events in our Universe. Supernovae strongly influence the circumstellar medium, enriching it with heavy elements and shocking it, facilitating star formation. Cosmic rays are believed to be accelerated in shocks from supernovae, with small contribution possibly coming from novae. Even though the basic physics of novae is understood, many questions remain unanswered. These include the geometry of the ejecta, why some novae are luminous radio or gamma-ray sources and others are not, what is the ultimate fate of recurrent novae, etc. Supernova explosions are the primary sources of elements heavier than hydrogen and helium. The elements up to nuclear masses A 100 can form ≈ through successive nuclear fusion in the cores of stars starting with hydrogen. Beyond iron, the fusion becomes endothermic instead of exothermic. In addition, for A > 100 the temperatures required to overcome the Coulomb ∼ barriers are so high that the nuclei are dissociated into alpha particles and free nucleons. Hence all elements heavier than A 100 should have formed ≈ by some other means. All nuclear species heavier than A 100 are formed ≈ by neutron capture on seed nuclei close to or heavier than iron-group nuclei. Depending on the ratio between neutron-capture timescale and beta- decay timescale, neutron-capture processes are called rapid or slow (r- and s-processes, respectively). The s-process, which occurs near the valley of stable isotopes, terminates at Bi (Z = 83), because after Bi there is a gap of four elements with no stable isotopes (Po, At, Rn, Ac) until we come to stable Th. The significant abundance of Th and U in our Universe therefore implies the presence of a robust source of r-process. The astrophysical site of r-process is still under debate. Here we present a study of a candidate site for r-process, neutrino-heated winds from newly-formed strongly magnetized, rapidly rotating neutron stars (”proto-magnetars”). Even though we find such winds are incapable of synthesizing the heaviest r-process elements like U and Th, they produce substantial amounts of weak r-process (38 < Z < 47) elements. This may lead to a unique imprint of rotation and magnetic fields compared to such yields from otherwise analogous slowly rotating non-magnetized proto-neutron stars. Novae explosions are not as powerful as those of supernovae, but they occur much more frequently. The standard model of novae assumes a one-stage ejection of mass from the white dwarf following thermonuclear runaway. The discovery by the Fermi space telescope of gamma-rays from classical novae made the existence of shocks in novae outflows evident. The presence of shocks in novae was considered well before the discovery of gamma-ray emission; however, little previous theoretical work acknowledged the over- whelming effect of shocks on observed emission and ejecta geometry. Here we present the calculations of synchrotron radio emission from the shocks as they propagate down the density gradient and peak at the timescale of a few months. The model satisfactory fits observations and has several implications for the physics of novae. Contents

List of Figures iv

List of Tables vii

1 Introduction 1 1.1 Supernovae ...... 1 1.1.1 Neutrino-heated winds as the source of r-process . . . .5 1.1.2 Neutrino-heated winds from rotating proto-magnetars .9 1.2 Novae ...... 10 1.2.1 Shock acceleration of protons and electrons ...... 15

2 Supernovae 24 2.1 Introduction ...... 24 2.2 Methods ...... 28 2.2.1 Hydrodynamic model description ...... 28 2.2.2 Wind Equations ...... 31 2.2.3 Neutrino microphysics ...... 32 2.2.4 Magnetosphere model ...... 35 2.2.5 Conservation checks ...... 37 2.2.6 Shooting method ...... 37 2.2.7 Nucleosynthesis ...... 38 2.3 Results ...... 47 2.3.1 Validity of the Force-Free Approximation ...... 48

i 2.3.2 Spherical versus Magnetized Rotating Trajectories . . 53 2.3.3 Spherical vs Magnetized Rotating Nucleosynthesis yields 59 2.4 Discussion ...... 71 2.4.1 r-Process Nucleosynthesis ...... 71 2.4.2 Gamma-Ray Burst Outﬂows ...... 81 2.5 Conclusions ...... 89

3 Novae 94 3.1 Introduction ...... 94 3.2 Shocks in Novae ...... 99 3.2.1 Evidence for Non-Thermal Radio Emission ...... 103 3.2.2 Condition for Radio Maximum ...... 107 3.2.3 Is the Forward Shock Radiative or Adiabatic? . . . . . 109 3.2.4 Thermal X-ray Emission ...... 111 3.3 Synchrotron Radio Emission ...... 113 3.3.1 Leptonic Emission ...... 115 3.3.2 Hadronic Emission ...... 117 3.4 Radio Emission Model ...... 118 3.4.1 Pressure Evolution of the Post-Shock Gas ...... 118 3.4.2 Electron and Positron Spectra ...... 119 3.4.3 Radiative Transfer Equation ...... 119 3.5 Results ...... 124 3.5.1 Analytical estimates ...... 124 3.5.2 Radio lightcurves and spectra ...... 130 3.5.3 Fits to Individual Novae ...... 134 3.6 Discussion ...... 136 3.6.1 Eﬃciency of Relativistic Particle Acceleration . . . . . 136 3.6.2 The slow matter in not main-sequence progenitor wind 138 3.6.3 Light Curve of V1723 Aquila ...... 140 3.6.4 Radio Emission from Polar Adiabatic Shocks ...... 142

ii 3.7 Conclusions ...... 145

Bibliography 150

Appendix: Non-Thermal Electron Cooling 181

iii List of Figures

1-1 Schematic view of a shock ...... 17

2-1 Geometry of neutrino-heated winds from proto-magnetar . . . 29 2-2 Magnetic field profiles for polar and last open field lines . . . . 34 2-3 Density trajectory with ρ t−3 extrapolation ...... 43 ∝ 2-4 Ye,eq(t) and Lν(t) evolution from [22] used for nucleosynthesis calculation ...... 46 2-5 Energy density profiles of hydrodynamical trajectories . . . . . 51 2-6 Minimum surface B required for validity of the force-free approximation ...... 52 2-7 Pressure profiles in closed and open zones ...... 54 2-8 Radial profiles of v, ρ, T for the fiducial solution ...... 60

2-9 Radial proﬁles of Ye,q ˙ and s for the ﬁducial solution . . . . . 61

2-10 texp versus r for polar, last open ﬁeld line and spherical outﬂows 62

2-11 texp versus S for diﬀerent models with threshold ...... 63 2-12 M˙ versus polar angle θ for P = 2 ms ...... 64 2-13 r-process ﬁgure of merit η versus polar angle θ for P = 2 ms . 65

52 −1 2-14 X(Z) for diﬀerent polar angles θ for Lν = 10 ergs s , P =1 and 3 ms ...... 67

2-15 Mej in individual elements and its ratio to spherical case . . . 69 ¯ 2-16 A versus P and Ye for ﬁxed Ye models ...... 70

2-17 r-process figure of merit η versus Lν for different periods . . . 72 ˙ 2-18 Mtot versus Lν for different periods ...... 75 iv 2-19 Rates of magnetar birth versus P ...... 79 2-20 Nucleosynthesis yields compared to abundances of metal-poor stars ...... 82

2-21 Magnetosphere-integrated Lorentz-factor versus Lν for diﬀer- ent P ...... 83 2-22 Angular distribution of maximal Lorentz factor for P = 2 ms . 85

2-23 Time-integrated Xα, and Xn,Xp at 100 s versus period . . . . 87

3-1 Scheme of X-ray, UV and radio production regions in novae outﬂow ...... 96 3-2 Scheme of forward shock, reverse shock, cold shell and unshocked regions ...... 101 3-3 Radio light curves of V1324 Sco, V1723 Aql and V5589 Sgr . . 104 3-4 Brightness temperature of the early component of the three radio novae ...... 105 3-5 X-ray luminosities and temperatures kT of V5589 Sgr . . . . . 113 3-6 Peak thermal X-ray luminosity and peak 10 GHz radio lumi-

nosity νLν ...... 114

3-7 Free-free optical depth τν versus gas temperature T behind the shock ...... 121 3-8 Cooling function Λ(T ) from [247] for our ﬁducial ejecta composition ...... 122 3-9 Energy spectrum and synchrotron emissivity of relativistic leptons ...... 124 3-10 10 GHz brightness temperature in the post-shock region versus gas temperature for diﬀerent models ...... 127 3-11 Peak brightness temperature of non-thermal emission as func-

tion of the fraction of shock energy used for ionization fEUV,

density scaleheight H, shock velocity v8, and fraction of the

shock energy placed into relativistic electrons, e ...... 131

v 3-12 Peak brightness temperature of thermal emission as function

of density scaleheight H and shock velocity v8 ...... 132 3-13 Example model radio lightcurves and corresponding spectral indices ...... 133 3-14 Synchrotron shock best-ﬁt models for V1324 Sco, V1723 Aql and V5589 Sgr ...... 134 3-15 Example light curves for a steep density gradient H = 5 1012 × cm with smearings of diﬀerent widths ...... 141 3-16 Ratio of nonthermal peak radio luminosity to peak thermal

X-ray luminosity versus shock velocity vsh ...... 144

vi List of Tables

2.1 Summary of Wind Models for RNS = 12 km, M = 1.4M ... 49

2.2 Summary of magnetosphere-integrated quantities for RNS =

12 km, M = 1.4M ...... 50 2.3 Summary of Time-integrated Models ...... 67

3.1 Commonly used variables and their definitions...... 100 3.2 Best-fit parameters for the thermal radio emission models . . . 106 3.3 Multiwavelength properties of novae ...... 106 3.4 Best-fit parameters of synchrotron model...... 135 Acknowledgement

I would like to thank our Lord and God and Saviour Jesus Christ for His continued support throughout my life and throughout the PhD program. I am greatly indebted to my advisor, Brian David Metzger, for his continuous help and guidance ever since we ﬁrst met. I learned from him not only great deal of science, but also patience. I am grateful to my parents for everything they have done for me. I am grateful to my wife Boguslava Bojidarova Vlasova for her support. I am grateful to my kids for providing me the opportunity to get distracted from science and then come back with fresh mind. I am grateful to Andrei Beloborodov, Indrek Vurm, Yuri Levin for fruitful discussions.

viii 1

Chapter 1

Introduction

Originally thought of as a constant unchanging place, space is full of evolution which can be seen in human life, sometimes in hours, minutes and seconds. One of the ﬁrst variable phenomena noticed in European civilization was supernova (SN) 1572, which was coined ’nova’ - new in Latin. Supernovae and novae were not distinguished from each other for a long time. Only much later it was realized that the events are clearly distinct in all parameters - total energy, luminosity, rate, ejected mass, etc. Novae constitute frequent transients (around 50 per year in our Galaxy) with moderate total energies of 1045 ergs, while supernovae are much rarer (1-5 per 100 years ≈ in our Galaxy) and much more energetic (around 1049 ergs in radiation). Supernovae and, to less extent, novae, are major factors contributing to evolution of interstellar medium: they inject radiation, kinetic energy, products of their nucleosynthesis, seeding the next generation of stars from the ashes of the previous. The shock waves of supernovae are thought to trigger star formation in giant molecular clouds [1].

1.1 Supernovae

The term ”supernovae” unites phenomena which are observationally similar, but not necessarily similar in their mechanism of explosion. Supernovae reach 2 peak optical luminosities of 1043 ergs s−1 and typically last for 50 100 ≈ − days. Historically, supernovae were divided into subclasses by their spectra [2]. Type I supernovae do not exhibit hydrogen in their spectrum, but Type II do. Type Ia displays intensive silicon lines, while Types Ib and Ic do not. The supernovae of Type Ic, unlike Ib, lack also helium lines in their spectra. Type II supernovae are divided on II-L (linear lightcurve decay) and II-P (plateau in the lightcurve), IIn (narrow emission lines), IIb (after peak, their spectrum transitions to Ib-like spectrum). So far, this classiﬁcation has been based mainly on observational properties and the association between the above-mentioned classes and physics of the event has not been unambiguously established for all classes [2]. Supernovae Ia are associated with thermonuclear runaway of a white dwarf [2] which, by one mechanism or another, reached central density and temperature enough for runaway carbon burning. Type II supernovae originate from collapse of the core of massive (M > 10M ) star [2]. An ambi- ∼ guity still exists with physics of some further subclasses of SN Ia,Ib and Ic [2]. Sometimes, the supernova ejecta runs into the circumstellar matter [3] or into the material ejected by the same progenitor star prior to its explosion [4, 5]. These interactions result in additional emission lines in the spectra and increase bolometric luminosity by dissipating kinetic energy. The result of the explosion is also notably diﬀerent for core-collapse and white dwarf runaway cases. Core-collapse supernova leaves behind either a neutron star or a black hole, while in the case of thermonuclear runaway of a white dwarf the star is fully destroyed. Hereafter, we focus on core-collapse supernovae.

Core-collapse supernovae occur when the core of massive (M > 10M ) ∼ star runs out of nuclear fuel hence loses its energy source and pressure support. Typical progenitor star has a mass (10 15)M . The abundance ≈ − structure of the progenitor is onion-like: the layers outwards are iron, sul- 3 fur/silicon, neon/magnesium/oxygen, carbon/oxygen, helium. Type II supernovae progenitors also have a hydrogen envelope on the outside, but Type I supernova progenitors have lost the hydrogen envelope. For a long time, the main paradigm was that the hydrogen envelope is stripped by line-driven winds [6]. However, now the theories have shifted to the model where binary interactions strip the envelope (see [7] for review). Regardless of the presence of a hydrogen envelope, fusion reactions terminate at iron-group nucleus because iron-group nuclei are the most stable nuclei of all (they have the most binding energy per nucleon). Hence, after iron group, nucleosynthesis cannot produce energy by either fusion or ﬁssion anymore and gravitational contraction of the core is inevitable. The subsequent fate of the core depends on its mass. If its gravitation can be balanced by degenerate electron pressure alone, then the star sheds outer envelope and the core peacefully transitions to a cooling white dwarf. If the mass of the core is too large to be supported by electron degeneracy pressure

(or, in other words, its mass is close to the Chandrasekhar limit 1.4M ), ≈ then the collapse will follow. Although by the time of collapse the star has an age of few times 107 years, the collapse itself happens in fractions of a second. As a result of collapse, iron-group nuclei are dissociated into free nucleons. Electrons are ”pushed into” the protons (e+p n+νe). The core → becomes optically thick to neutrinos at this stage and they cannot carry out energy efficiently. Dissociation of nuclei into nucleons greatly (by factor of 60) multiplies the number of particles and hence the number of degrees of ≈ freedom. Consequently, the equation of state stiffens and collapse is stopped at a density 2-3 times the nuclear density. This ”bounce” creates a sound wave which travels down the density gradient and steepens into a shock at radius of 10 20 km. This shock produces a lot of neutrinos, but they do ≈ − not escape to infinity because the shock is optically thick initially. Neutrino 4

53 −1 luminosity Lν experiences a spark of (2 3) 10 ergs s for 10 20 ms when − · − the shock becomes optically thin to neutrinos as it propagates outward and then Lν declines monotonically. The naive and beautiful idea that it is this shock which produces a supernova has proven wrong. According to [8, 9], the outward moving shock stalls into an accretion shock after having traveled out to a distance of around 100 km. The shock stalls because of (1) neutrino losses - even though the neutrino do not escape to inﬁnity yet, they still carry the energy away from immediate post-shock region and (2) the energy loss to photodissociation of nuclei. How to revive this shock and make a supernova explode is still an open question despite decades of intensive investigation [10]. This problem is especially complicated because it involves almost exact cancellation: only 1051 ergs come out in the form of kinetic energy of the ejecta of the total 3 1053 ergs liberated by gravitational collapse and leaving · in form of neutrinos. Several scenarios have been proposed to facilitate the supernova explosion. The scenarios which rely on neutrino-heating of pre- shock region are called neutrino-driven mechanisms [11, 12, 13, 14]. Another wide group of explosion scenarios are magneto-rotational scenarios [15, 16, 17, 18]. In those, the explosion is facilitated by centrifugal force and magnetic pressure. By one mechanism or another (or by some combination of them, or by some completely diﬀerent mechanism), in nature the core-collapse supernovae do explode. The shock wave heats up the outer layers of the star and sets them expanding. It is these outer layers that are observed as optical supernova. Both observations of optical spectra line widths [19, 20] and simulations [21], show that velocity of the shock at the breakout from the star is v 20, 000 km s−1 and ejection velocities 10, 000 km s−1. ≈ ≈ After the core bounce, the hot proto-neutron star deleptonizes and cools down on the timescale of 100 s [22]. The protoneutron star in the center ≈ 5 is left to cool and contract from the radius 50 km to normal neutron star ≈ radius 12 km. Most of gravitational energy will be radiated as neutrinos ≈ at this stage. Outer layers of proto-neutron star are heated up by neutrinos and evaporate giving rise to a neutrino-driven wind. Neutrino-driven (or, alternatively, neutrino-heated) winds are one of promising astrophysical sites for rapid neutron capture, or r-process [23, 24]. Chapter 2 considers the feasibility of neutrino-driven winds from rotating magnetized proto-neutron stars as the sites for r-process.

1.1.1 Neutrino-heated winds as the source of r-process

The primary form of nuclear reactions in the Universe is thermonuclear fusion occurring in the central regions of ordinary stars. It powers their radiated luminosity. However, nuclear fusion, even in explosive environments, is unable to produce nuclei heavier than A 100. The reason is that with increasing ≈ nucleus charge, the Coulomb barrier also increases. For nuclei with A > 100, ∼ the temperatures required to overcome Coulomb barrier are so high that the nuclei are dissociated into alpha-particles or free nucleons instead of fusing. However, we measure non-zero abundance of A > 100 nuclei, up to 238U, on Earth. That means that these heavy nuclei are produced by some other process. Seminal works [25, 26] establish the origin of the elements heavier than A 100 (and some lighter isotopes as well) as neutron-capture process. ≈ Neutrons do not have to overcome Coulomb barrier and can easily be captured by nuclei. After neutron capture, nuclei become neutron-rich and they β-decay back to stability. So, in any environment where neutron capture happens, β-decay also has to happen. Depending on speed ratio between these two processes, rapid (r) and slow (s) neutron capture processes are distinguished. For s-process, the neutron capture rate is much slower than beta 6 decay rate. S-process is known to happen in convective shells of Asymp- totic Giant Branch (AGB) stars [27]. Comprehensive models of s-process predicting isotopic yields are available [28]. S-process track lies very close to stability and s-process cannot step over the gap of A values with no stable isotopes. For example, between Bi (Z=83) and Th (Z=90) there is gap with no stable elements. So, U and Th can only be produced by r-process. In addition, because r-process and s-process tracks are different, they produce different isotopic patterns in lighter elements. Analysis of chemical composition of Earth [29], meteorites [30], other stars [31, 32] confirms necessity of r-process for explaining measured abundances. However, the astrophysical site where r-process occurs remains a matter of debate [33, 34, 35, 36]. The two most promising sites are neutrino-driven winds of core-collapse supernovae [37] or neutron star mergers (NSM) [38]. NSMs naturally eject neutron-rich matter which then undergoes r-process nucleosynthesis up to A > 200 and beyond [39, 40, 41, 42, 43, 44]. Here ∼ we concentrate on another candidate site, the neutrino-heated winds from rotating magnetized proto-neutron stars [37]. Neutrino-heated winds were proposed as most probable r-process site within core-collapse paradigm [45, 37, 46, 47]. However, the current con- sensus is that the neutrino-heated winds are unable to provide conditions necessary for formation of third r-process peak at A 200 in non-rotating ≈ non-magnetized spherically-symmetric models [37, 48, 49, 50, 51, 52, 53, 54, 22, 55]. Also, the studies of 244P u in the deep sea sediments exclude the possibility of all core-collapse supernovae producing heavy r-process elements [29], (however, see [30]). These two factors suggest that r-process happens not in neutrino-heated winds of all core-collapse supernovae, but in some subset of them. Several variations of standard scenario were proposed to obtain successful r-process production by neutrino-heated winds like shock 7 waves leading to additional heating [56, 57], extreme mass neutron stars

(M > 2M - see [58]) or nonconventional physics, like sterile neutrinos with ∼ eV mass [59, 60]. ≈ In chapter 2, we study another modiﬁcation of standard neutrino-heated wind, the neutrino-heated wind from rapidly rotating (P < 10 ms) strongly

15 16 ∼ magnetized (Bsurface > 10 10 G) protoneutron star, or ”proto-magnetar” ∼ − [23, 61]. Magnetar births account for at least 10 % of all neutron star births in the Galaxy [62]. So, if the successful r-process happens in proto-magnetar winds, this would not contradict [29] which find that production of heavy r-process elements by all supernovae is ruled out. We calculate hydrodynamic trajectories of outflows with different parameters (polar angle θ, luminosity Lν, period P ). In order to quickly estimate feasibility of a particular hydrodynamic trajectory for r-process and to compare our trajectories with trajectories from other works, we use a simple analytic criterion developed by [63]: r-process will reach A 200 ≈ 3 3 if r-process ”figure of merit” η s /(τ0.5MeVY ) exceeds a threshold value ≡ e 3 3 −1 −3 of ηthr = 2 10 k s nucleon , where s is entropy in kB per nucleon, · B Ye = nprotons/(nprotons + nneutrons) is the electron fraction and τ0.5MeV is the expansion time at the temperature of α-particle formation, Tα 0.5 MeV. ≈ This criterion can be understood as following. At the conditions of interest, the outflow is radiation-dominated and hence the entropy per nucleon is proportional to T 3/ρ. To high precision, α-formation occurs at approximately

12 fixed temperature Tα 0.5 MeV. Formation of C from three α-particles ≈ proceeds by the reaction sequence 4He(αn, γ)9Be(α, n)12C. Because 9Be is much less bound that 12C, the second α-particle capture happens much faster. Therefore this process is effectively four-body reaction (three α’s and one neutron). Hence the reaction rate falls very fast with density. That’s why we can assume that the reaction happens only during first τ0.5MeV after 8

α-formation. The amount of 12C produced can be thus estimated by:

3 3 X12 Xn(ρα) τ0.5MeV Xn(Xαρ) τ0.5MeV = C ∝ ∝ 3 3 3 3 = Xn(2YeT /S) τ0.5MeV XnY τ0.5MeV/S (1.1) α ∝ e taking into account that Xα = 2Ye for neutron-rich winds (Ye < 0.5). For r-process to proceed up to the nuclei with mass A and charge Z, the neutron- to-seed ratio has to exceed mean number of captured neutrons in the resulting nucleus,

X n > A 2Z. (1.2) X12C − r-process terminates at A 200 due to the lack of more massive stable ≈ nuclei. For A 200, combination of (1.2) and (1.1) yields: ∝

Xn Xn 3 3 3 3 −1 −3 3 3 = S /(Ye τ0.5MeV) > ηthr = 2 10 kBs nucleon (1.3) X12C ∝ XnYe τ0.5MeV/S ·

For Ye < 0.38, the final composition has non-zero neutron abundance and the criterion (1.2) is modified. The modified version of the criterion is given in the formula (2.24). Neutrino-heated winds from rapidly-rotating magnetized proto-neutron stars have been studied before [57, 64], but the geometry of the magnetic field was approximated to be that of a simple monopole with B 1/r2. [57] ∝ found that rotation and magnetic fields enhance η, but not to extent to imply successful third-peak nucleosynthesis. In [57], part of the energy to escape the star was supplied to the wind material by the rotational energy instead of neutrino heating, hence reducing the amount of heating and asymptotic entropy. 9

1.1.2 Neutrino-heated winds from rotating proto-magnetars

In Chapter 2 we use more realistic geometry motivated by force-free mag- netospheric solution from [65]. Simulations of [65] study magnetosphere of aligned rotator, that is, the pulsar with rotation and magnetic axes aligned. Under this assumption, the magnetosphere is static. [65] also assume that the matter inertia is negligible as compared to magnetic field density (this assumption is called force-free condition). The characteristic size of the magnetosphere is the light cylinder radius, RLC = c/(2πΩ), where Ω is the angular frequency of rotation. The light cylinder radius is the radius where corotation speed equals to the speed of light. Hence, beyond light cylinder no corotation of matter is possible and the field lines have to be open, leaving to infinity. The region filled with field lines reaching infinity is called open zone. Inside the light cylinder, there will be a region of corotating field lines which do not leave to infinity. This region is called a closed zone. Very close to the proto-neutron star (PNS) surface, the magnetic field is similar to that of a standard magnetic dipole. The geometry of an aligned rotator magnetosphere is shown at the Figure 2-1. In aligned rotator geometry, the matter first moves along open lines which all originate in the polar cap. Near the surface, all field lines point almost radially outwards. Hence, the impact of centrifugal force in the heating region close to the surface is minimal. The resulting entropies range from spherically-symmetric values for polar outflows almost to equatorial values from [57] for last open field lines. τ0.5MeV is determined close to the light cylinder. In that region, the last open field line direction has a substantial horizontal component and the outflow is highly affected by centrifugal force (see Fig. 2-1). The force-free geometry of an aligned dipole yields the centrifugal reduction in τ0.5MeV and Ye which helps r-process nucleosynthesis. The result is that the force-free outflows are more favorable for r-process 10 than equatorial ones, but still not enough for a successful third peak A 200 ≈ nucleosynthesis. Nucleosynthesis yields are also calculated using nucleosynthesis network SkyNet [66], allowing us to quantitatively access the influence of magnetic fields and rotations on yields of individual elements. Chapter 2 also addresses implications of proto-magnetars as central engines of GRB on baryon loading of GRB jets and UHECR composition.

1.2 Novae

Novae explosions are the result of thermonuclear runaways (TNR) from the surface of white dwarfs [67, 68, 69, 70, 71]. Novae occur in binary systems where one of the stars is a white dwarf, and the other one is main- sequence star or a giant. The hydrogen-rich material from outer layers of non-degenerate companion star accretes on the white dwarf, either through the inner Lagrange point after Roche lobe overflow or through stellar wind of giant star captured by white dwarf. After enough material has accumulated, the temperature and pressure at the base of the accreted layer are sufficient to ignite p-p hydrogen burning. At this time, the electrons in most of hydrogen envelope are degenerate. Hence, the release of energy and increase of temperature do not imply increased pressure and no rapid expansion occurs. In addition, accreted matter is not governed by its self-gravitation, but rather the gravitation of the central white dwarf. Hence, to reduce gravitational force the shell has to expand substantial fraction of the WD radius, not the substantial part of its own size (so-called thin-shell instability). Eventually, the degeneracy is lifted and pressure raises, but at this point the temperature is growing so fast that thermonuclear runaway occurs. At T > 2 107 K, CNO cycle overtakes p-p reaction and at T > 108 · K hot CNO cycle dominates. Hot CNO cycle dominates the total energy 11 production of TNR. During hot CNO cycle, large fraction of nuclei capable of capturing a proton (CNONeMg...) do capture a proton and become β+ unstable nuclei (13N, 14O, 15O, 17F). The rate of energy generation is limited then by temperature and pressure independent β+ decay rates [72]. During hot CNO operation, convectional zone spans from the bottom of burning zone almost to the surface, bringing β radioactive nuclei to cold outer layers. Later β decay produces isotopic ratios which are very different from equilibrium operation of CNO cycle and from the solar abundances. Observational evidence exists that the WD material gets mixed in the burning zone and part of it is ejected together with nova ejecta [73, 74, 75]. Because of this dredging, WD in classical novae systems decrease their mass in subsequent novae explosions and accretion in between the explosions. Hence, they cannot serve as a progenitor of Ia supernovae [76, 73].

−4 3 −1 As a result of TNR, 10 M is ejected with v 10 km s . For a ≈ ≈ long time, thermonuclear runaway was thought to produce single ejection of matter which then expanded freely to large radii, interacting only with ISM [72]. However, growing evidence from across the electromagnetic spectrum suggests that the shocks are common, if not ubiquitous, in novae. Shocks may originate from collision of two outflows: slower outflow launched earlier in the outburst and faster outflow which then catches up with the slow outflow [77, 24]. Alternatively, shocks may originate from the collision of main nova outflow with pre-existing material around binary system, possibly accelerated away by super-Eddington luminosity of white dwarf during the outburst [78]. In either case, we call the outer ejecta Dense External Shell (DES) while the faster outflow is called fast wind, following [79] and [24]. Optical spectroscopy reveals few system of lines corresponding to different negative velocities [80],[81],[78]. The most prominent is wide (v (1 2) 103 ≈ − · km s−1) P Cygni profile which is visible on the timescale of few weeks after 12 the outburst. The P Cygni profile implies spherical outflow for fast wind. Another system (sometimes multiple systems) shows narrow absorption lines with negative radial velocities of order 200 600 km s−1. These correspond − to DES. DES absorption systems are observed in the majority of novae [78] implicating that DES possesses high covering fraction. In some novae, absorption line systems are seen to accelerate outwards with time [81]. Fast spherical P Cygni wind collides with DES and that is indeed what is observed. After 3-4 weeks, both P Cygni and absorption lines disappear and instead, one wide system of emission lines appears [81, 78]. This phase is called nebular phase of nova and it is accompanied by forbidden emission lines, which require ionization source harder than the central white dwarf [82]. v 1000 ≈ km s−1 shocks naturally provide such ionization source. Velocity of nebular system between P Cygni velocity and absorption lines velocities also naturally arises from the shock model. IR observations of novae reveal dust formation in some cases (see [83] for a review). The cold shell behind the radiative shocks provides natural site for cold neutral gas shielded from ionizing radiation where dust can form [84]. Hard thermal X-ray emission with kT > few keV observed for some novae ∼ [85] provide other evidence for shocks. This emission is too hard to originate from a WD, which emits as a supersoft X-ray source with energies < 0.1

33 35 −1 keV. The fact that hard X-ray luminosity LX 10 10 ergs s is ≈ − 37 38 much less than bolometric luminosity of the nova Lbol 10 10 ergs ≈ − s−1 seemed to imply that shocks play subdominant, almost negligible role in novae energetics. For that reason, shocks in novae outﬂows have been studied as an aside phenomenon for a long time with many comprehensive models of novae explosions not mentioning shocks at all [86, 87, 72]. Shocks were used to explain some features in the lightcurve [88] or spectrum [82],[89], but their paramount importance for producing lightcurve was not acknowledged. 13

Dominant role of shocks in novae was fully realized following the discovery of high-energy (> 100 MeV) γ-rays emission from novae by Fermi gamma-ray ∼ space telescope [90]. While lower energy, 1 MeV, γ radiation resulting from ≈ γ-decays of TNR products and Compton down-scattering of positron-electron annihilation had been predicted beforehand [91], the only natural explanation of > 100 MeV γ-rays are high-energy particles from shocks [92, 93, 90, 79, 94]. ∼ First γ-ray detection was from nova V407 Cyg 2010 [95, 96, 93]. That nova belongs to a rare class of symbiotic novae which are embedded in dense wind of an M giant. Hence, the γ-ray emission from novae was thought to be rare [95] and the material in which the nova outﬂow runs into (DES) was thought to be the wind of M giant. However, since that time, Fermi- LAT detected in > 100 MeV γ-rays ﬁve classical novae (V959 Mon 2012, ∼ V1324 Sco 2012, V339 Del 2013 - see [90], V1369 Cen 2013, V5668 Sgr - see [97]). These discoveries show that classical novae also possess DES with high covering fraction. Estimates based on both leptonic and hadronic scenarios for γ-ray emission [79, 98] show that γ-luminosity cannot exceed 2 % of ≈ total shock power. So, the γ-rays observations tell us lower limit on total

38 36 −1 shock power Lshocks > Lγ/0.02 10 10 ergs s . Because of high ∼ ≈ − density environment, the shocks are radiative, i. e. they radiate all of the shock power right away: Lopt,shock = Lshock. Then one ﬁnds that the shock power and hence optical luminosity of shocks are a big fraction, up to 100 %

38 −1 of optical luminosity of novae, Lopt,tot 10 ergs s . Now, that we know ≈ true power of shocks, [77] propose that hard X-ray luminosity of shocks is small not because their overall power is small, but because of column of neutral gas ahead of them which eﬀectively absorbs X-ray emission. Indeed, hard X-ray spectra of novae reveal absorbing columns 1021 1023 cm−2 (for − V5589 Sgr see [99], for V1723 Aql see [100], for review see [85]). Evidence of shocks is also observed in radio waves. The most prominent 14 feature in radio lightcurve is late (t 200 500 days) peak produced by ≈ − thermal bremsstrahlung of electrons in expanding ionized spherical gas shell with T 104 K [101]. Some novae show early-time < 100 days non-thermal ≈ ∼ 5 6 radio peak [88, 100, 102, 103, 104] with brightness temperature Tb 10 10 ≈ − K. This peak has been interpreted as shock-powered synchrotron emission [88, 100, 102, 103, 104]. Its emergence at early times implies small size. Together with ﬂux comparable or greater than the late thermal peak, this

5 6 implies much higher brightness temperatures Tb 10 10 K. Note that ≈ − photoionization heat balance keeps the temperature of nova ejecta almost constant at the level 104 K [105]. Hence, high brightness temperatures of ≈ the early peak cannot be explained by gradually expanding spherical shell of 104 K. Instead, they imply either nonthermal emission mechanisms or gas temperatures 105 106 K, in either case being signature of shocks. In − addition, at early times the spectral index of optically thick expanding hot sphere should be α = 2 and the flux should be raising as the observed surface area t2. Therefore, the presence of early peak in numerous novae, for ∝ example, Nova Vul 1984, V1324 Sco, V1723 Aql, V5589 Sgr and evolution of spectral index: α > 2.4 in Nova Vul 1984 [88], α from 2.5 to 0.7 for V1324 Sco [106], from 1.5 to -0.2 for V1723 Aql [103] or from 0.77 to -0.08 for V5589 Sgr [99] cannot be reconciled with expanding hot sphere model. Radio imaging using EVLA and other interferometers reveal the geometry of nova outflow and the location of non-thermal emission. The geometry is consistent with DES being the dense torus concentrated in equatorial plane and fast wind being more spherically-symmetric outflow (see [107, 108] for V959 Mon, [103] for V1723 Aql). Optical line profile modeling supports this picture for V959 Mon [109]. For V959 Mon, the nonthermal emitting knots are observed just where the DES and fast wind collide [107]. In the picture where DES is ejected during the outburst, DES of dense torus shape 15 can originate from gravitational interaction of companion star with originally spherical outflow [24]. Alternatively, the TNR itself can be asymmetric ejecting more on the equator (Levin 2016, private communication). In the picture [78] where DES is present even before the outburst, it naturally has toroidal shape because DES is the gas which flew out of outer Lagrange point of the companion. In all these scenarios, the orientation of dense torus is coincident with the orbital plane. In some novae this feature was confirmed observationally (for example, V959 Mon - see [109]). Early radio peak and nonthermal emission seen by EVLA can be explained as emission of shock-accelerated high-energy electrons ([24], see also Chapter 3). Shock accelerated ultrarelativistic electrons produce synchrotron radiation which is observed as early radio peak. Ultrarelativistic electrons emit synchrotron radiation in the magnetic field amplified by the shock - this is leptonic scenario. Alternatively, ultrarelativistic protons collide with thermal protons and produce ultrarelativistic pions. Charged pions then decay to electrons/positrons which, in turn, produce synchrotron radiation - this is a hadronic scenario. In Chapter 3 we present the model of synchrotron radiation of electrons in novae shocks. The model can explain observed in- tensities of synchrotron radiation and, in some cases, the correct shape of the lightcurve. For radio synchrotron radiation, quite high electron acceleration efficiencies e 0.01 and magnetic field energy fractions B 0.01 are ≈ ≈ implied, ruling out the hadronic scenario. The model also rules out the main sequence (MS) stellar wind as DES, because MS stellar wind is too dilute to produce observed radio emission.

1.2.1 Shock acceleration of protons and electrons

Most of the astrophysical studies of particle acceleration in shocks use supernovae remnants (SNR). Numerous models of increased complexity were 16 proposed to explain cosmic rays (CR) by shock acceleration in SNR [110, 111, 112, 113, 114, 115, 116]. We will comment on SNR shock model explaining CR origin because it is the main source of information about particle acceleration in astrophysical shocks, and then we will make connections of these developments to the shocks in novae.

Energy budget of SNR

First, we consider if SNR can indeed supply required CR energy. CR energy is injected in the Galaxy at a rate 4.3 1034 ergs s−1 [117]. The CR energy ≈ × budget is explained if the following fraction of SNR kinetic energy goes into CR energy:

−1 −1 f = 0.14R100E51 , (1.4)

where R100 is supernovae rate per 100 years and E51 is ejected kinetic energy 51 in foe (10 ergs). Taking into account that R100 is usually estimated to be a few (1 3) and given the acceleration eﬃciencies up to 10 % of shock power − seen in modern simulations (e. g. [113]), this number is not unreasonable.

Acceleration mechanism

The general outline of particle acceleration by successive bounces was ﬁrst proposed by Fermi [118, 119]. He explained the emergence of cosmic rays by random wandering of energetic particles between moving magnetized interstellar clouds [118, 119]. The particle energy gain for every bounce is proportional to cloud velocity β = v/c, and the prevalence of head-on over head-tail collisions is also proportional to β. Hence the acceleration eﬃciency is proportional to β2 and the process is called second-order Fermi acceleration. In shock environment however the particle gains energy every time it 17

Figure 1-1: Schematic view of a shock. v1 is the upstream and v2 is the downstream region. The downstream region is more dense and the velocity in it is slower. For strong shock, downstream region is 4 times denser and velocity is 4 times less than in the upstream. bounces back and forth between upstream and downstream of the shock. This is because the upstream region of the shock moves towards the downstream region in downstream rest frame and vice versa, downstream moves towards upstream in the upstream rest frame (see Figure 1-1). Because energy gain of the particle every time is proportional to β, the Fermi acceleration at the shock is called first-order Fermi acceleration. The particles are believed to experience small deflections rather than reflections as a mechanism of bouncing back and forth. Small deflections can be represented as diffusion in the phase space. Hence another name for the mechanism is Diffusive Shock Acceleration (DSA). 18

Upon relativistic particle collision with background particle of the same type, substantial fraction of particle’s energy is lost. Hence, if the collisions of particles are effective, then they will damp energy of CR-to-be much faster than it is acquired. The necessary condition for CR acceleration is then that the shock is collisionless, i. e. mediated not by collisions, but by some other means. In astrophysics, collisionless shocks are usually mediated by magnetic fields. Natural question then arises what are the values of magnetic fields in SNR shocks. Tightly related question is what is the maximal energy of CR accelerated by SNR shock.

Maximum energy of a CR in a shock

The maximum energy of CRs is limited by the ability of the accelerating region to eﬀectively scatter energetic particle, and the scattering is mediated by magnetic ﬁeld turbulence. [120] have argued that the maximum energy attainable in shock acceleration SNR is:

Emax = sBRSNR(v1 v2)Ze, (1.5) − where RSNR and B are the characteristic size and magnetic ﬁeld in SNR, s is the factor showing the eﬃciency of scattering s = rL/(mean free path) and rL is the Larmor radius. In what follows, we adopt s = 1. After the time T0 when the mass swept by the ejecta equals the ejecta mass (”sweep time”), the ejecta begins slowing down and shock power falls accordingly. Hence, it is reasonable to calculate (1.5) at the sweep time T0:

14 B 1/2 −1/6 −1/3 Emax = 1.8 10 E M n Z eV (1.6) · 3µG 51 ej, H

Here Mej, is the ejected mass in solar masses and nH is interstellar density in cm−3. The value of B is normalized to typical interstellar magnetic field: 19 the particles have to be confined not only in downstream region, but also in the upstream region. Modest variations from the expectation value of the equation (1.6) were obtained in [121, 122]. Observed CR spectrum has almost uniform power law dE E−2.7 up to ∝ 15 the ”knee” region, Eknee 4 10 eV [117]. We see that simple estimate ≈ × (1.6) fails to reach this observed energy. Moreover, for E > Eknee, CR spectrum falls steeper than E−2.7, but it is still measurable up to 5 1019 ≈ · eV. A satisfactory theory of Galactic CRs acceleration has to provide CRs at least up to the knee and possibly to 5 1019 eV. Based on (1.6), [120] ≈ · argued that shocks in SNR are unfavorable for producing CRs. The way out of this difficulty is that the magnetic field to be substituted in the formula (1.5) is much bigger than typical interstellar magnetic field.

CR-driven instabilities

The reason for the confining magnetic field to be bigger than interstellar magnetic field is that cosmic rays themselves create instabilities leading to magnetic turbulence and magnetic field amplification in the precursor layer of the shock [123, 124, 125, 126, 127]. Precursor layer is the layer of upstream media which is not disturbed by the shock itself yet, but it is disturbed by propagating CRs and instabilities driven by CRs. At first, the resonant CRs driven instability was thought to be the main mode for creating magnetic turbulence [123, 124, 125, 126]. Later however the non-resonant Bell instability [127] was shown to grow faster in linear regime than resonant CRs instability. The magnetic fields in the shock precursor are observed to be amplified by factors 30 60 as compared to far upstream both in SNR [128] and in − simulations [114]. This higher magnetic field is sufficient to push maximal energy (1.6) little bit above the knee region. Above the knee, the CRs are dominated by ions heavier than protons [117] with significant contribution 20 up to Fe. For heavy nuclei, additional factor Z increases the maximal energy (1.6).

Predicted particle spectrum

DSA also makes prediction about the shape of accelerated spectrum. The spectral shape can be understood from simple argument as follows [117]. If we denote the compression ratio σ = ρ2/ρ1, then mass conservation implies 0 v1 = v2σ (see Figure 1-1). At every bounce, new energy E of ultrarelativistic particle is

0 E = γ(E + p cos(θ)(v1 v2)), (1.7) − where E, p are the energy and momentum of the particle before scattering, θ is the angle between its momentum direction and the shock normal, and γ is Lorentz factor for v1 v2. Since it’s ﬁrst order Fermi acceleration, we can − set γ = 1. After proper averaging over angles, we obtain

δp 4 v1 v2 4 v2 = − = (σ 1) (1.8) p 3 c 3 − c for two reflections corresponding to one bounce back and forth. It is important to note that the scattering has to be isotropic in the frame of the fluid in order for this analysis to be true. On the other side, particles are lost at every bounce by being advected downstream. The velocities of individual particles are c: that is much bigger than the velocity of shock front in the upstream frame, v1. However, the net flow of particles downstream through fixed fluid element is zero because downstream is also filled with CRs and they come from far downstream to shock surface, too. Hence, the CR particles are advected downstream with rate ∆n = nCRv1 per unit area of the shock with nCR being the number density of CRs. Number of particles crossing the shock 21

surface per unit time per unit area is just n = nCRc cos(θ) /2 = nCRc/4. h i Hence,

∆n v = 4 2 (1.9) n − c

Suppose the population of n0 particles starts with isotropically distributed momenta of absolute value p0. Then, after N shock crossings, it will on average evolve to:

N v2 n = n0 1 4 (1.10) − c N 4 v2 p = p0 1 + (σ 1) (1.11) 3 − c

Excluding N from the above equations, we can incorporate populations of particles with diﬀerent N’s into single formula:

n 3 p ln = ln (1.12) n0 −σ 1 p0 − − 3 n p σ−1 (1.13) ∝ − σ+2 dn p σ−1 dp (1.14) ∝

For strong shocks, σ = 4 and the last line becomes simply dn = p−2dp. However, overall power going into CR acceleration can be as big as 10 % of total shock power, altering the parameters of the shock itself and compression ratio [113]. CRs make compression ratio smaller than 4, hence the resulting spectrum of CRs is softer with slope 2.1. CRs observed on Earth have ≈ − spectrum dn p−2.7dp in the region 1011 4 1015 eV [117]. In fact, spallation ∝ − · analysis of CRs yields travel times from the source E−γ with γ between ∝ 0.4 and 0.6 [117]. Injection spectrum E−2.1 times traveling (”trapping”) time naturally yields the observed E−2.7 - this is one of attractive features of SNR 22 shock acceleration models.

Novae shocks as particle accelerators

In this work, we study the consequences of particle acceleration in novae shocks. They are diﬀerent from SNR shocks in a few aspects:

1. The density is 6-8 order of magnitude higher than in supernovae shocks (but novae shocks are still collisionless, see [79])

2. The undisturbed magnetic field is not an ISM magnetic field, but the frozen-in WD magnetic field drawn by DES from the surface of WD

High density of novae shocks precludes direct determination of magnetic field by detecting self-absorption synchrotron frequency of relativistic electrons. Free-free absorption of the medium dominates synchrotron self-absorption. Hence, in radio synchrotron observations magnetic field amplification is de- 3/4 generate with particle acceleration efficiency (we can measure only eB , see Sect.3.6.1). Another consequence of high density is that far upstream of the shock is neutral. Naively, this would preclude the possibility of particle acceleration by novae because non-resonant Bell instability is damped efficiently by ion-neutral damping [129]. However, exactly because of high density, the power of the shock ionizing radiation is enough to ionize quite wide layer ahead of the shock [79, 94, 24]. Bell instabilities, magnetic turbulence, energetic particle acceleration upstream are all confined to this ionized layer and hence its thickness should be used instead of ejecta radius in the formula (1.5) for maximal achievable energy (see also [94]). The magnetic field in DES is another factor different from SNR. In SNR context, the upstream magnetic field is interstellar field which is often ho- mogeneous on the scale of the remnants. So, the shocks probe all possible inclination angles. Novae ejecta has traveled 103 104 WD radii by the time − 23 we observe shocks in radio, therefore the ”frozen in” field is stretched to be in tangential direction irregardless of its original direction at the surface of WD. The shocks propagate radially outwards, so the naive expectation is that the shocks propagate perpendicularly to the magnetic field (quasi-perpendicular shocks). Quasiperpendicular shocks do not accelerate protons or electrons, according to simulation [130]. The matter of inclination angle is discussed in more detail in Sect.3.3.1. The initial value of magnetic field in novae also can be different than for supernovae shocks. However, the fact that we observe high energy gamma-ray and radio synchrotron emission from novae tells us unambiguously that particle acceleration does happen in at least some novae. 24

Chapter 2

Supernovae

2.1 Introduction

All neutron stars are born as hot ‘proto-neutron stars’ (PNSs), which radiate their gravitational binding energy and lepton number in neutrinos over the first 10 100 seconds of their life [131]. A small fraction of ∼ − this luminosity is absorbed by matter in the atmosphere just above the PNS surface, powering an outflow of mass known as the ‘neutrino-driven wind’ [45, 37]. Initial interest in neutrino winds arose due to their potential as a primary source of rapid neutron capture (r-process) nucleosynthesis [132, 46, 133, 47]. Neutrino-heated outflows are, however, also of great importance in other contexts, such as their role in determining the maximum Lorentz factor (baryon loading) and nuclear composition of gamma-ray burst outflows [134, 135, 57, 136, 137, 138, 139]. Past studies of neutrino-driven winds have been primarily focused on spherically symmetric, non-rotating winds accelerated purely by thermal pressure [48, 49, 50, 51, 140, 53, 141, 54, 55]. This body of work has led to the conclusion that normal PNS winds fail to achieve the conditions necessary for nucleosynthesis to reach the third r-process peak at mass number 25

A 195. The latter requires an outflow with a combination of high en- ∼ tropy S, short expansion timescale texp, and low electron fraction Ye [37, 63] as it passes through the radii where seed nuclei form. Proposed mechanisms to move beyond the standard scenario in order to achieve r-process success include postulating additional sources of heating (e.g. damping of convectively-excited waves; [56, 57]) or by resorting to extreme parameters, such as massive > 2.2M neutron stars [58]. ∼ The class of neutron stars known as ‘magnetars’ possess very strong magnetic fields > 1014 1015 G and account for at least 10 per cent of neutron ∼ − ∼ star births [62]. Basic estimates show that magnetar-strength fields are dynamically important in PNS winds [142], even in the first few seconds after core bounce when the neutrino heating and wind kinetic energy are highest. Strong magnetic fields confine the outflow to only a fraction of the PNS surface threaded by open field lines [142]. Rotation, when coupled with a strong magnetic field, also provides an additional source of acceleration in the wind via magneto-centrifugal forces [134]. Strong magnetic fields and rapid rotation do not represent separate assumptions if rapid rotation at birth is necessary to produce the strong field via dynamo action [143, 144]. [57] calculated the steady-state structure of rotating magnetized PNS winds in ideal MHD under the assumption of an equatorial (one-dimensional)

2 outflow with an assumed monopolar radial field Br 1/r . [57] found that ∝ strong magnetic fields > 1014 1015 G and rapid rotation (spin period P < ∼ − ∼ few ms) act to increase both the power and the mass loss rate of the wind as compared to the spherical case (see also [134]). The conditions for a successful

3 r-process, as quantified by the critical ratio S /texp we found to be moderately improved from the spherical case. However, in some cases of extreme rotation the conditions were in fact worse because although magneto-centrifugal acceleration increases the outflow velocity (reducing texp), the residence time 26 of matter in the heating region and hence its final entropy was likewise − − reduced in a compensatory way. A successful r-process can take place even without high entropy if the asymptotic electron fraction of the outflow Ye is substantially lower than the equilibrium value Ye,eq > 0.4 set by neutrino irradiation (eq. [2.10]). This is ∼ conceivable if the outflow expands from the surface, where Ye starts much lower, sufficiently rapidly as the result of centrifugal acceleration to avoid entering such an equilibrium. In practice, however, both simple analytic estimates ( 2.3.2) and numerical simulations show that Ye Ye,eq is achieved § only for extremely rapid rotation (P < 1 ms) near the centrifugal break-up ∼ limit [136, 145]. Extreme rotation necessarily increases the total mass ejected per event by a factor > 10 100 [136], placing such sources in tension with ∼ − studies of Galactic chemical evolution that suggest more frequent, lower- mass pollution events are more consistent with the abundance distributions of metal poor stars ([146, 147, 148]; however, see [149]). Most Galactic magnetars cannot be born with extremely short spin periods P 1 ms, ∼ as their high birth rate would overproduce the Galactic r-process [150] and the energy of the explosion accompanying their birth would exceed those measured for supernova remnants hosting magnetars [151]. Here we present calculations of neutrino-heated outflows from magnetized, rotating PNSs in the more realistic geometry of an aligned (axisymmetric) dipole. Our solutions are calculated under the force-free assumption, namely that mass flows along the magnetic field lines, with the latter prescribed by a self-consistent solution of the cross-field force balance [152, 153, 65]. Once the hydrodynamical trajectories are obtained, they are processed through nuclear reaction network SkyNet [66]. A series of one- dimensional solutions calculated along flux tubes corresponding to different polar field lines are then pieced together to determine the global properties 27 of the flow and total nucleosynthesis yields at a given neutrino luminosity and rotation period. Although the rotation axes of pulsars are not in general aligned with their magnetic fields, alignment is expected if the field is generated as the result of a convective dynamo [143] and the results of our calculations are expected to be insensitive to the inclination angle as long as it is small. The case of aligned rotator is also the first step in exploring generic inclination angles. The wind calculations presented here are more accurate than previous works, which were one dimensional and assumed a poloidal field the scaled with radius as 1/r2 or 1/r3 to mimick those ∝ ∝ characterizing a monopolar or dipolar field structure [134, 57]. Our formal- ism allow us to assess, for the first time, possible latitudinal dependencies of the wind properties and nucleosynthesis yields. A primary conclusion of this work is that proto-magnetar outflows are more favorable for the r-process than slowly rotating, unmagnetized PNSs

3 by a factor up to 4 in S /texp. This improvement results largely due to ∼ the faster expansion rate of the outflow from centrifugal slinging, which is accompanied by a more modest decrease in entropy than in the equatorial monopole case. These improved r-process conditions are found to extend to spin periods P > 5 ms, indicating that core collapse events with extremely ∼ high angular momentum are not necessary for a marked improvement. Nu- cleosynthesis yields generally confirm this conclusion. Beyond the r-process, our results have implications for the viability of millisecond proto-magnetars as potential central engines of GRBs [154, 155, 134, 137] and the composition of UHECR, if UHECR are indeed produced by GRB jest from proto-magnetars [156]. A long-standing theoretical mystery is why GRB jets accelerate to bulk Lorentz factors Γ in the relatively narrow range of 100 1000 [157], as this requires a finely tuned baryon loading ∼ − [158]. Here we confirm previous results that proto-magnetar outflows with 28 spin periods P 1 2 ms in the range necessary to explain the energetics ∼ − of GRBs come naturally ‘loaded’ with the correct baryon flux to achieve the observed values of Γ. Our results also confirm the prediction that the nuclear composition of GRB jets may be dominated by heavy nuclei instead of protons [156]. There are some indications that ultra-high energy cosmic rays (UHECRs; [159]) may be dominated by heavy nuclei ([160]; however, see e.g. [161]). This would be naturally explained if millisecond magnetar birth is the origin of UHECRs: in all our models nuclei with A¯ 100 have ≈ mass fraction of order of unity (see Fig. 2-23 and Fig.2-16).

2.2 Methods

2.2.1 Hydrodynamic model description

We calculate steady-state outflows along open field lines corresponding to the solutions to the Grad Shafranov equations for an aligned dipole field [65] and then we calculate nucleosynthesis yields on these trajectories using nuclear reaction network SkyNet [66]. The steady-state approximation is valid for hydrodynamical trajectories because the properties of the PNS (rotation period P , neutrino luminosity Lν, radius Rns) change slowly compared to the time for matter to flow to the radii of interest. In force-free outflows the matter flows along the prescribed field geometry. This approximation is justified if the pressure P and kinetic energy density ρv2/2 are much less than magnetic energy density B2/8π (where ρ and v are the density and velocity of the outflow, respectively), a condition we check a posteriori from our derived solutions. The simplification that matter moves along the field lines reduces the 2D magneto-hydrodynamics problem to a series of 1D calculations along flux tubes centered around each field line. Calculations are performed for different 29

Figure 2-1: Geometry of neutrino-heated winds from magnetized, rotating proto-neutron stars. Outflows occur from the ‘open zone’, along field lines −1 with polar angles θ < θmax, where θmax = sin [Rns/RY] is the angle of the last closed field line, Rns is the PNS radius, RY < RL is the radius at which the last closed field line crosses the equator (the∼ ‘Y point’), RL = c/Ω is the light cylinder radius, and Ω is the angular rotation rate of the PNS. In the illustration we have assumed RY = RL. The variable l defines the distance measured along the field line from the surface, with the local field line direction denoted as ˆl (the toroidal component of which is not shown). Note that l differs from the spherical radius r measured to the PNS center. The wind is composed of free nucleons at small radii, which recombine into α-particles and heavier seed nuclei once the temperature decreases to < 7 GK several neutron star radii above the surface. ∼ 30 initial polar angles θ, with results then integrated over the open magneto- −1 p sphere. The latter includes angles θ < θmax, where θmax = sin Rns/RY is the angle of the last closed field line, RY is the radius at which the last closed field line crosses the equator (the ‘Y point’). Even if the force-free approximation is valid in the wind itself, the location of the Y point, and hence the size of the open zone, can be affected by the pressure of the closed zone ([162]; 2.2.4, 2.3.1). The standard spherically-symmetric case is ac- § § commodated within our framework by assuming a monopole magnetic field

2 Br 1/r and zero rotation. A schematic diagram of the wind configuration ∝ is shown in Figure 2-1 for the case RY = RL, where RL = c/Ω = cP/2π is the light cylinder radius. The wind dynamics is calculated under the (Newtonian) approximation of non-relativistic velocities, even though this assumption fails near the light cylinder. This inaccuracy is tolerable because we aim to calculate hydrodynamic trajectories interior to where T = 0.5 MeV, as typically occurs within < 10 PNS radii above the surface where special relativistic effects play no ∼ appreciable role. For the very same reason, we use nuclear statistical equilibrium (NSE) equation of state for hydrodynamic trajectories calculations, because NSE is good approximation before alpha particles formation. Nucle- osynthesis network, however, uses density trajectories beyond our calculated hydrodynamical trajectories and equation of state which takes into account all nuclear species present. ρ(t) prescription and nuclear equation of state used in nucleosynthesis network are described in detail in sect.2.2.7. For simplicity we also neglect general relativistic (GR) effects, except as included in an approximate way in a few test cases. Past studies have shown that GR tends to increase the entropy of the flow [163, 50], but this effect is expected to be similar for rotating and non-rotating stars, and our main concern here is a comparison between these cases. 31

2.2.2 Wind Equations

The steady-state axisymmetric wind equations in cylindrical coordinates (x, z) are given by

∂ ρv = 0; (2.1) ∂l B

∂v 1 ∂p GM v + xΩ2(ˆl xˆ) (ˆl rˆ) = 0; (2.2) − ∂l · − ρ ∂l − x2 + z2 ·

dY v e = λ(ρ, T, Y ); (2.3) dl e

dS vT =q ˙(ρ, T, Y ), (2.4) dl e where l is the distance measured along the ﬁeld line. Equation (2.1) expresses mass continuity, where ρ is the gas density, v is the velocity along

q 2 2 the magnetic field, B = Bp + Bφ is the magnetic field strength (where Bφ and Bp are the toroidal and poloidal components, respectively). In 2.2.4 we § describe how the field strength and outflow geometry are calculated. Equa- tion (2.2) expresses momentum conservation, where the second term is the centrifugal force1, Ω = 2π/P is the rotation rate of the PNS, ˆl is the unit vector along the field line, andr ˆ is the unit vector from the center of the PNS (see Fig. 2-1). The pressure p, entropy S and other thermodynamic quantities are calculated from the Helmholtz equation of state [164].

Equation (2.3) evolves the electron fraction Ye due to weak interactions,

1The Coriolis force does not affect the motion of gas because it is perpendicular to the velocity and hence to the magnetic field, the latter of which is parallel to the flow velocity in the force-free approximation. 32

which occur at the rate λ(ρ, T, Ye). Equation (2.4) evolves the entropy of the outﬂowing gas due to neutrino heating and cooling, whereq ˙(ρ, T, Ye) is the net heating rate. Tabulated values of λ andq ˙ are calculated as a function of density, temperature, neutrino luminosity Lν, and neutrinosphere temperature Tν ( 2.2.3). § Equation (2.1) can be manipulated to read

∂ρ ρ ∂v d ln(B) = + ρ , (2.5) ∂l −v ∂r dl which upon substitution into equation (2.2) gives

2 2 cs v dv 2 d ln(B) 1 ∂p GM ˆ 2 ˆ − = cs + + 2 2 (l rˆ) rΩ (l xˆ) (2.6) v dl dl ρ ∂l ρ r + x · − ·

2 ∂p where cs ∂ρ and the second term on the right hand side can be ≡ S,Ye written in terms of the other thermodynamic variables as follows

∂p = ∂l ρ ( ) ∂p dS ∂p dY q˙ ∂p ∂p + e = v + λ , ∂S dl ∂Y dl T ∂S ∂Y ρ,Ye e ρ,S ρ,Ye e ρ,S (2.7) where we have used equations (2.3) and (2.4).

2.2.3 Neutrino microphysics

Neutrino microphysics enters the wind evolution through the functions λ and q˙ (eqs. [2.3],[2.4]). Included are the dominant charged current reactions:

+ ν¯e + p n + e ; νe + n p + e (2.8) ↔ ↔ 33

Cooling also includes neutral-current interactions [165]. We neglect the effects of strong magnetic ﬁelds on the rate of neutrino heating and cooling in the PNS atmosphere [166, 167], as the corrections (primarily due to quanti- zation of the lepton phase space into Landau levels) are found to be relatively minor for surface ﬁeld strengths < 1016 G. The temperature of the neutri- ∼ nosphere is calculated from the total electron neutrino luminosity according to

7σ L + L = 4πR2 T 4. (2.9) νe ν¯e ns 8 ν

Competition between the absorption of neutrinos and antineutrinos by nucleons in the wind (as dominate reactions well above the PNS surface) drive the electron fraction towards the equilibrium value

2 −1 Lν¯e ν¯e 2∆ + 1.2∆ / ν¯e Ye,eq = 1 + h i − 2 h i , (2.10) Lν ν + 2∆ + 1.2∆ / ν e h e i h e i set by the luminosities Lν /Lν¯ and mean energies ν¯ / ν of the elec- e e h e i h e i tron neutrinos and antineutrinos, respectively, where ∆ = 1.293 MeV is the proton-neutron mass diﬀerence [37].

Nucleosynthesis is extremely sensitive to the asymptotic value of Ye, which in most of our wind solutions is found to approach Ye,eq to high accuracy.

The value and temporal evolution of Ye,eq, however depends on theoretical uncertainties in the neutrino spectrum of the cooling PNS [54, 141]. Our calculations of hydrodynamic trajectories take the mean neutrino and antineutrino energies to be equal, ν¯ = ν , at the common value deter- h e i h e i mined from the neutrinosphere temperature (eq. [2.9]). The ratio of Lνe and eq Lν¯ is then chosen such that Y 0.45 (eq. [2.10]), as motivated by recent e e ' calculations which ﬁnd a value of this order at early times in the PNS cooling evolution (t < 5 10 s after core bounce; e.g. [141], their Fig. 5). Please ∼ − 34

2.00

1.80

3 1.60 ns R 0 1.40 Light cylinder 1.20

T=0.5 Mev Polar field line Maximum heating Sonic point 1.00 , normalized to B

3 0.80 Last open field line Br 0.60 r-1 0.40

106 107 Radius, cm

Figure 2-2: Magnetic field strength B normalized to its surface value B0, along two field lines, corresponding to a polar outflow (θ = 0) and the last ◦ open field line (θ = θmax 21 ), calculated for a PNS with spin period P = 2 ms. Near the surface both≈ lines obey B 1/r3 (dipole), while at larger radii the polar and last open field line follow∝B 1/r2 and 1/r, respectively. Shown for comparison are the locations of the∝ sonic point∝ (cross), maximum net heating (plus), alpha particle formation radius R0.5MeV (asterisk), and 52 −1 light cylinder radius RL for our solution with Lν = 10 erg s . note that neutrino microphysics described above is used for calculation of hydrodynamic trajectories only. Nucleosynthesis network runs after formation of α-particles and hence above-mentioned processes are not relevant. Nu- cleosynthesis network takes into account only neutrino emission reactions.

Neutrino microphysics and initial Ye values used in nucleosynthesis calculations are described in detail in sect.2.2.7. 35

2.2.4 Magnetosphere model

The magnetic field strength B and the field line trajectories (x,z) are calculated as a function of the distance along the field line l according to (e.g. [65])

B(l) = ξ/x (2.11) dx dz ˆ ˆ −1 , = l x,ˆ l zˆ = ξ ( ∂zΨ, ∂xΨ) (2.12) dl dl · · − z∂xΨ x∂zΨ ˆl rˆ = − , (2.13) · ξ√x2 + z2 where

s 4π 2 ξ ( Ψ)2 + I2, (2.14) ≡ ∇ c and Ψ(x, z), I(x, z) are functions proportional to the magnetic flux and current, respectively, flowing through a ring of constant cylindrical coordinates (x, z). Coordinates (x, z), quantities l, ˆl, r, rˆ are illustrated at Figure 2-1. We employ the solutions2 Ψ(x, z) and I(Ψ) of [65], focusing on the standard case in which the last closed field line (θ = θmax) crosses the equatorial plane at the light cylinder (RY = RL). As will be discussed further in 2.3.1, the § open zone may be larger than this (RY < RL) due to the high pressure of the closed zone. The geometry of the force-free magnetosphere is invariant to an overall rescaling of Ψ and I. This implies that our solutions do not depend on the strength of the magnetic field, as long as it is sufficiently high to justify the force-free condition ( 2.3.1). In contrast, our results do depend on the § PNS rotation period, as the latter determines the physical radius of the light cylinder.

2Under force-free conditions the current can be written as a function of the ﬂux, such that I(Ψ). 36

Figure 2-2 shows the magnetic field strength B as a function of spherical radius r, calculated for a PNS spin period P = 2 ms along two different field lines, corresponding to a polar outflow (θ = 0) and the last open field line

◦ (θ = θmax 21 ). Close to the PNS surface, the poloidal component of the ≈ 3 ﬁeld is dipolar (Bp 1/r ), with the toroidal component Bφ being small in ∝ comparison. At larger radii near the light cylinder and beyond, the poloidal

2 field becomes monopolar (Bp 1/r ), while the toroidal field approaches the ∝ dependence Bφ 1/r expected given the magneto-statics analogy between ∝ the polar current and an infinite wire. These expectations are borne out in Figure 2-2 by the magnetic field strength along the last open field line, which varies as 1/r3 close to the PNS surface (poloidal dominated) before ∝ transitioning to 1/r (toroidal dominated) near the light cylinder. The polar ∝ field line θ = 0, by contrast, is dominated by the poloidal component at all radii, which slowly transitions from dipolar 1/r3 to monopolar 1/r2. ∝ ∝ The faster the magnetic field strength decreases, the faster the matter expands. The magnetic field near the surface decreases faster in the proto- magnetar case B 1/r3 than for a spherical outflow B 1/r2. Non-polar ∝ ∝ outflows also experience additional centrifugal acceleration, which causes the matter to expand even more rapidly along field lines with larger θ ( 2.3). In § 2.3 we show that centrifugal effects dominate the purely geometric effect of § the diverging field geometry in terms of its influence on the conditions for a successful r-process. The necessity of employing a realistic description for the field geometry is made clear by the locations of various radii in the flow which are critical to the nucleosynthesis. As shown by symbols in Figure 2-2, the sonic point, location of maximum heating, and alpha particle formation radius often occur at radii where the field geometry is not well-described by any limiting case. For nucleosynthesis network calculations, we need to know ρ(t) beyond 37 calculated hydrodynamic trajectories, to several light cylinder radii. After ρ(t) from the hydrodynamical trajectory ends, we extrapolate the density dependence as ρ t−3 (see Fig. 2-3). The motivation for this extrapolation ∝ is described in sect. 2.2.7.

2.2.5 Conservation checks

Equation (2.1) shows that the ratio of mass to magnetic ﬂux,

ρv ˙ , (2.15) M ≡ B is a conserved quantity. It is automatically conserved at the level of our numerical scheme because in our solution we express density from velocity using (2.15). Conservation of momentum, energy, and lepton fraction, on the other hand, serve as independent checks of our solutions. Momentum and energy conservation (eqs. [2.2],[2.4]) can be combined into the integral constraint (‘Bernoulli integral’) that

ρv v2 p GM 1 Z l ρ(l0)q ˙(l0) + + r2Ω2 dl0, (2.16) 2 2 0 B 2 ρ − √r + z − 2 − l0 B(l ) is constant at each distance l across the radial grid, where is the internal energy and l0 is an arbitrary starting point.

2.2.6 Shooting method

Equations (2.3), (2.4), and (2.6) are solved using a ‘shooting’ method by which the eigenvalue is iteratively guessed and corrected in order to converge upon an accurate solution. Equation (2.6) has a singularity at the sonic point (v = cs), which can only be avoided for a transonic solution if the right hand side of the equation simultaneously goes to zero. The value of ˙ that M 38

zeroes the right hand side at v = cs is not known a priori but must instead be determined simultaneous with the full solution. Several considerations are used to determine whether a given guess for ˙ is bigger or smaller than the critical value. First, if dv/dl is negative at M any point along the integration, then this implies that ˙ is smaller than the M transonic value. On the other hand, if dv/dl exceeds an order-of-magnitude estimate of its value by a large factor (taken to be 50) then dv/dl is assumed ∼ to be approaching inﬁnity, indicating that ˙ is bigger than the transonic M value. Such a ‘guess and check’ technique is used iteratively to determine ˙ and the structure of the subsonic solution up to the point where v is a M few percent less than cs. Above this velocity dv/dl is taken to be constant in integrating the other equations (2.3,2.4) across the sonic point. Finally, once v is a few percent above cs, then the full set of equations (2.3,2.4,2.6) are again used to determine the supersonic structure. Our solutions are found to conserve energy/momentum (eq. [2.16]) to an accuracy of < 3%. ∼ In addition to the transonic condition, boundary conditions on the temperature, electron fraction, and density at the PNS surface are calculated according to the following criteria [57]:

1. Zero net heating (q ˙ = 0.)

2. Weak equilibrium (dYe/dt λ = 0). ' 3. Density ρ = 1012 g cm−3, as found to produce an integrated neutrino

optical depth from inﬁnity τν 1 (thus deﬁning the surface as the ∼ ‘neutrinosphere’).

2.2.7 Nucleosynthesis

We start our reaction network calculations just after free nucleons recombine into α-particles at a temperature of T 7 109 K. At small radii, where ' × 39 the magnetic field is dynamically strong compared to the thermal or kinetic energy densities, the force-free approximation is valid and we employ calculated density trajectories. At sufficiently large radii, matter inertia comes to dominate the energy density of the magnetic field, and the outflows should approach a spherical outflow. The density profile of a steady-state wind which has reached a constant asymptotic velocity, should approach 1/r2 1/t2, ∝ ∝ while at larger radii where internal velocity gradients become important the profile will approach ρ 1/r3 1/t3 appropriate for a freely expanding ∝ ∝ homologous outflow. As shown in Figure 2-3, we interpolate directly between our trajectories at small radii and the asymptotic 1/t3 dependence at large radii. This simplification is motivated by the fact that the qualitative features of the abundance patterns are robust to the detailed density trajectory outside of the radii where charged particle processes cease and the neutron-to-seed ratio has been determined. In order to test the importance of an alternate density extrapolation, we have run a handful of trajectories with ρ 1/r2 extrapolation. The difference in the asymptotic abundances ∝ between the 1/r3 and 1/r2 density proles is less than 1% for most cases, and none of the conclusions of this chapter are changed. Another simplification is that we neglect heating or deceleration of the wind by the reverse shock, which is produced as the wind interacts with the surrounding supernova ejecta. This is justified in part3 because the radius

1/2 of the pulsar wind termination shock Rrs (vej/c) Rej (e.g. [168]), where ' vej is the mean velocity of the SN ejecta, is generally much larger than the radius where the formation of seed nuclei occurs. The latter generally occurs close to the light cylinder radius, which determines the radial scale of outﬂow

3In addition, highly magnetized winds experience much weaker compressional heating in the shock than unmagnetized winds due to the additional support from magnetic pressure. 40

divergence. At time t since explosion we have Rej vejt and thus '

3/2 −1 Rrs vej t P 37 4 −1 . (2.17) RL ≈ 10 km s 1s 1 ms

We thus have Rrs RL for times t > 1 s and rotation periods P < 10 ms ∼ ∼ of interest, implying that seed formation occurs well inside the termination shock. We start the nucleosynthesis network after the formation of α-particles. By this time, the key parameters of the outﬂow (entropy s, expansion timescale ˙ τdyn, electron fraction Ye and mass loss rate M) have already been determined in the free nucleon zone at small radii, where nuclear statistical equilibrium (NSE) provides a good approximation for equation of state. Furthermore, after α-particle formation, neutrino heating and cooling have become negligible and hence are not included in the network reactions. Only neutrino emissions are included in network calculations. Figure 2-8 shows that our calculations begin at radii 5 7 times larger than the location where the − neutrino heating rate reaches its maximum just above the NS surface. The locations of maximum neutrino heating are marked by stars in this ﬁgure, while the locations where T = 0.5 MeV, which approximately corresponds to the locations of α-formation, are marked by circles. Hydrodynamic trajectories do not account for entropy gain from α-particle formation, which we therefore increase by hand from its initial value s0 according to

s = s0 + Xα(s, ρ, Ye) ∆sα, (2.18) · where Xα(s, ρ) is mass fraction of α under NSE with given entropy, ∆sα =

Qα/Tα is entropy gain from α formation, Qα is the heat of α-formation, and

Tα is the temperature of α formation. Since the latter depends weakly on 41

−1 other wind parameters, we adopt a fixed value of ∆sα = 10kB nucleon , corresponding to Tα = 8 GK. The starting temperature for the reaction network is determined from the density and the entropy according to equation (2.18), with subsequent evolution tracked self-consistently from the density trajectory and radioactive heating. Finally, we must specify the initial outflow electron fraction used in our network calculations, which equals the “final” electron fraction Ye,final set by processes near the PNS surface. In standard thermally-driven PNS winds, this value is determined by the competition between electron neutrino and electron anti-neutrino capture reactions on free nucleons (2.8). These reactions have frozen-out (become slow compared to the expansion rate) at the large radii where our calculations would begin, resulting in an electron fraction close to the equilibrium value set by neutrino absorption reactions (2.10). For relatively slowly rotating proto-magnetars, P > 2 ms, the final electron fraction is very similar to the normal thermally-driven case, i.e.

Ye,final Ye,eq. However, in the most rapidly spinning cases, P < 2 ms, we see ' ∼ from our hydrodynamical trajectories that Ye,final < Ye,eq due to rapid magne- tocentrifugal acceleration (see also [134, 57, 169]), which causes a premature freeze-out of the neutrino absorption processes before Ye is raised completely from its low value near the PNS surface. More quantitatively, introducing the definition (∆Ye)cent Ye,final Ye,eq, we find that (∆Ye)cent 0.14 is ≡ − ' − maximum in our trajectories. The values of (∆Ye)cent are reported in Table

2.1. We keep the dependence of (∆Ye)cent on Lν = 0.5(Lνe + Lν¯e ) and θ and we neglect the dependence on the other parameters (Ye,eq,Lνe , νe , ν¯e ). In sect.2.2.3, we mentioned that for hydrodynamic trajectories we always set Lνe /Lν¯e so that Ye,eq = 0.45. However, in nucleosynthesis calculations we use Ye,eq(t) using the time evolution of Lν ,Lν¯ , ν¯ , ν from e e h e i h e i 42

PNS cooling calculations of [22]. This is justiﬁed because hydrodynamical trajectories depend mainly on the amount of heating which is weakly dependent on Ye,eq. Indeed, heating results from charged-particle reactions

(2.8). Lνe /Lν¯e in our calculations is never different from 1 by factor more than 5, so if we replace some protons by neutrons, they will just capture neutrino instead of antineutrino and heating effect would be roughly the same. As the initial conditions for our network calculations we therefore take Ye,final = Ye,eq(t) + (∆Ye)cent[Lν, θ] for P = 1, 2 ms and Ye = Ye,eq(t) for all other periods. Given the uncertainties in neutrino radiation transport [55, 22, 54], especially in the essentially unexplored case of very rapid rotation (however, see [170, 171]) and given the effects of ultra-strong magnetic fields on the microphysics [172, 166], we concentrate on comparing the properties rotating proto-magnetar winds to the conventional unmagnetized spherical wind case. In this way, we isolate the effects arising exclusively due to the novel outflow geometry and magneto-centrifugal acceleration.

Reaction Network Calculations

We start our nucleosynthesis calculations at the point where hydrodynamical trajectory reaches T = 7 109 K, corresponding to the approximate temper- × ature of α particle formation. Actual starting temperature is slightly larger − because we take into account the entropy enhancement from α-formation (eq. 2.18). We use the nuclear reaction network SkyNet [66] for the nucleosynthesis calculation. The composition starts out in nuclear statistical equilibrium (NSE), a good approximation for the initial temperature T > 7 109 K. ∼ × Given the extrapolated density as a function of time (see 2.2.1), SkyNet then § evolves the abundances of 7843 nuclear species under the inﬂuence of over 140,000 nuclear reactions. The evolved species range from free neutrons and protons to 337Cn (Z = 112). SkyNet also evolves the entropy and tempera- 43

107 Spherical

P=3 ms, θ=θmax 0 106 P=3 ms, θ=1 -3 105 ρ ~ t-3

104 Density, g cm ρ ~ t-3

103

102 0.0001 0.001 0.01 0.1 Time from α-particle formation, s

52 −1 Figure 2-3: Density profiles of wind calculations for Lν = 10 ergs s , for both rotating proto-magnetars with P = 3 ms (blue and green lines) and spherical wind (blue line). At large radii we extrapolate the density profile ρ t−3 (black lines), as expected at very late times for homologous expansion.∝ 44 ture, which change due to the expansion of the material and energy released by the nuclear reactions. SkyNet uses a modified version of the Helmholtz equation of state [173], which treats every nuclear species as a separate Boltz- mann gas, including also electron-positron gas with arbitrary degree of rel- ativism and photon gas. In some trajectories the initial temperature after taking into account the entropy gain from α-formation is above 1010 K. In these cases, SkyNet uses nuclear statistical equilibrium for evolution of the network down to 1010 K, below which it switches to full network evolution with all reactions. The rates of the strong reactions come from the JINA REACLIB database [174], but only the forward rates are used and the inverse rates are computed from detailed balance. This is to ensure consistency with NSE, which depends on the nuclear masses. Spontaneous and neutron-induced fission rates are taken from [175], [176], [177], and [178]. Most of the weak rates come from [179], [180], and [181] whenever they are available, and otherwise the REACLIB weak rates are used. We used the nuclear masses and partition functions from the WebNucleo XML file distributed with REACLIB, which contains experimental data where available and finite-range droplet macro- scopic model [182] data otherwise.

For Ye > 0.5, charged particle process nuclei (Z 38 40) can also be ' − formed through νp process (e.g. [183, 140]); however, even though SkyNet supports including neutrino interactions, we do not include these in the present calculations. We leave an exploration of charged particle process nuclei formation in proton-rich proto-magnetar winds to future work.

Classes of Abundance Models

The nucleosynthesis products of proto-magnetar winds vary as a function of the latitude of the open flux tube at a fixed time. They also vary in a 45 global sense, integrated over the entire magnetosphere at a fixed time, or averaged over all times in the PNS cooling evolution. We cover the range of possible diagnostics of the wind nucleosynthesis by calculating the abundance patterns in three general cases:

1. Individual flux tubes along different latitudes θ at a fixed time (or equivalently, neutrino luminosity), producing abundance yields as a

function of (θ, Lν,Ye,P ).

2. The entire wind at a ﬁxed time, by integrating individual ﬂux tubes over the solid angle of the open magnetosphere, producing abundance

yields as a function of (Lν,Ye,P ).

3. The entire wind (integrated over the open magnetosphere) averaged

over the Kelvin-Helmholtz cooling epoch Lν(t) (Fig. 2-4), producing abundance yields as a function of just the rotation period P .

In the angle-integrated cases, we weight the abundances by the mass loss rate along each given open flux tube. In the time-integrated case, they are weighted also by the total ejected mass from the entire magnetosphere at each epoch t. In the latter case, by fixing the rotation period, we have assumed that the magnetic spin-down time of the pulsar is longer than Kelvin- Helmholtz cooling timescale of seconds; this approximately is justified for

16 surface magnetic dipole ﬁeld strengths of Bd < 10 G. ∼ In the time-integrated case, we also consider two methods for treating the time evolution of the electron fraction Ye. In one case, we ﬁx Ye at a constant value throughout the cooling epoch. In the second case we derive its value as described in 2.2.1 using the equilibrium electron fraction (eq. 2.10) from the § 46

0.55 1053 Ye, eq 0.54 _ (Lν +Lν )/2 e e 0.53 0.52 1052 0.51 )/2 e ν 0.5 _ e Y +L e

0.49 ν (L 0.48 1051 0.47 0.46 0.45 0.44 1050 0.1 1 10 Time, s

Figure 2-4: The blue line (right axis) shows the time evolution of the electron neutrino luminosity Lν(t) = (Lνe + Lν¯e )/2 used in our calculations, based on the electron neutrino luminosity Lνe (t) and antineutrino luminosity Lν¯e (t) from the Kelvin-Helmholtz cooling calculations of [22]. A red line (left axis) shows the corresponding equilibrium value of the electron fraction Ye,eq (eq. 2.10) used as input to our nucleosynthesis calculations (we also down-correct Ye for centrifugal eﬀects for P 2 ms, but this is not shown here). ≤

time evolution of Lν,Lν¯, ν¯ , ν from the PNS cooling calculations of [22], h e i h e i as shown on Figure 2-4. Since we are focused on the r-process, we integrate only over epochs when Ye,eq < 0.5, i.e. at times 0.1 < t < 5 s, and all reported ∼ ∼ wind properties (e.g., ejecta mass, mass fractions, etc. - all the data in Table 2.1) refer to just this time period. This is a reasonable approximation to the total yield of the wind, since the bulk of the total mass loss is occurs at early times t 1 2 s. ∼ − 47

2.3 Results

Solutions are calculated for a range of total neutrino luminosities Lν (Lν + ≡ e 52 51 51 51 −1 Lν¯ )/2 = 10 , 6 10 , 3 10 , 10 erg s , as are physically achieved at e × × times t 1, 3, 6 and 15 seconds after core bounce according to the PNS ∼ cooling calculations of [184]. For each luminosity, solutions are calculated for different values of the PNS spin period P = 1, 2, 3, 5, 10 ms and for different angles θ of the open field lines as a fraction of the angle θmax of the last open line (Fig. 2-1). For every hydrodynamic trajectory, corresponding nucleosynthetic yield is also calculated using SkyNet network. All results discussed in this section, as summarized in Figures 2-6 2-18 − and Table 2.1, are calculated assuming a PNS of ‘standard’ mass M = 1.4M and radius Rns = 12 km (except where otherwise noted). Because only a fraction of the PNS surface is open to outflows in the magnetized rotating case, a spherically equivalent mass loss rate can be defined according to

˙ 2 M 4πR ρ0v0, (2.19) ≡ ns where ρ0 and v0 are the density and velocity, respectively, at the PNS surface. The value of M˙ for each ﬁeld line solution is reported in Table 2.1 and Figure 2-12. The spherically equivalent mass loss rate M˙ is not to be confused with the true total mass loss rate

Z ˙ Mtot = ρ0v0dS, (2.20)

where ρ0, v0 are surface density and velocity, and the integration extends over the PNS surface threaded by open magnetic ﬂux, θ [0, θmax]. The ∈ ˙ magnetosphere-integrated values of Mtot are reported reported in Table 2.2 48 and Figure 2-18.

2.3.1 Validity of the Force-Free Approximation

Figure 2-5 compares the pressures associated with matter and radiation Pgas 2 to that of the magnetic field PB = B /8π as a function of radius, calculated 52 −1 for our fiducial solution (Lν = 10 erg s ; P = 2 ms) and an assumed surface 15 magnetic field strength B0 = 10 G. Thermal pressure exceeds magnetic pressure at the PNS surface, as expected since otherwise the field could not be anchored to the star. Gas pressure decreases rapidly with radius, such that magnetic pressure comes to dominate just 2 3 km above the ∼ − surface. Magnetic pressure continues to dominate until the kinetic energy grows to a comparable size near the light cylinder, at which point the force- free approximation breaks down (our assumption of non-relativistic velocity is violated at a similar location). As already discussed, such inaccuracies are tolerated at large radii because we are interested is the outflow structure at temperatures T > 0.5 MeV (marked as a circle in Fig. 2-5), at which ∼ point the value of Pgas/Pmag is still < 10 per cent. The ratio Pgas/Pmag at ∼ T = 0.5 MeV for each of our solutions is provided in Table 2.1, for the

15 same assumed surface ﬁeld strength B0 = 10 G. For the nucleosynthesis network calculations we use ρ(t) the hydrodynamic trajectory up to the light cylinder. The part of ρ(t) near the light cylinder can be viewed as the interpolation between the inner region where force-free approximation is valid and ρ(t) t−3 approximation which we use after the end of hydrodynamic ∝ trajectory.

Figure 2-6 shows the minimum surface field strength Bmin for which the force-free criterion (defined as Pgas/Pmag < 0.1 at T = 0.5 MeV) is satisfied for all outflow angles θ < θmax, as a function of the neutrino luminosity and PNS spin period. At the highest neutrino luminosities 1052 ergs s−1, ∼ 49

Table 2.1: Summary of Wind Models for RNS = 12 km, M = 1.4M

(a) θ ˙ (b) (c) (d) (e) 3 3 (f) Pgas (g) PLν M S texp (∆Ye) η = S /texpY R | θmax cent e 0.5MeV Pmag R0.5MeV 51 −1 −1 −1 (h) (ms) (10 erg s )-(M s ) (kb n ) (ms) - % of (ηthr ) (km) -

∗N/A 1 ∗N/A 3.8 · 10−6 100 0.1837 0.0039 0.7 47 ∗N/A ∗N/A 3 ∗N/A 7.1 · 10−5 77 0.0424 0.0028 1.5 68 ∗N/A ∗N/A 6 ∗N/A 4.5 · 10−4 67 0.0199 0.002 2.1 89 ∗N/A ∗N/A 10 ∗N/A 1.7 · 10−3 61 0.013 0.0013 2.4 114 ∗N/A 2 1 0.0 5.8 · 10−6 94 0.1301 0.0038 0.9 36 2.0 · 10−2 2 3 0.0 1.1 · 10−4 74 0.0262 0.0025 2.2 48 1.0 · 10−1 2 6 0.0 7.4 · 10−4 65 0.0109 0.0017 3.5 58 3.3 · 10−1 2 10 0.0 2.9 · 10−3 60 0.0066 0.0008 4.6 70 9.5 · 10−1 2 1 0.3 7.4 · 10−6 68 0.0347 0.0034 1.2 28 5.4 · 10−3 2 6 0.3 9.3 · 10−4 48 0.0023 -0.0003 6.8 41 6.6 · 10−2 2 10 0.3 3.6 · 10−3 45 0.0017 -0.0017 7.9 49 2.4 · 10−1 2† 10 0.7 7.7 · 10−4 67 1.0 0.0019 40.6 35 3.3 · 10−2 2 3 1.0 2.5 · 10−4 37 0.0016 -0.0041 4.8 27 9.7 · 10−3 2 10 1.0 6.2 · 10−3 33 0.0006 -0.0102 9.0 38 2.7 · 10−1 2 1 1.0 1.3 · 10−5 45 0.0092 0.0006 1.4 23 2.4 · 10−3 2 3 1.0 2.6 · 10−4 37 0.0015 -0.0046 4.8 27 9.7 · 10−3 2 6 1.0 1.7 · 10−3 33 0.0008 -0.0085 7.4 32 4.7 · 10−2 2 10 1.0 6.5 · 10−3 32 0.0006 -0.011 9.1 38 2.8 · 10−1 3 1 0.1 5.8 · 10−6 95 0.132 0.0037 0.9 36 9.3 · 10−3 3 3 0.1 1.1 · 10−4 75 0.0267 0.0025 2.2 48 4.7 · 10−2 3 6 0.1 7.4 · 10−4 66 0.011 0.0017 3.6 58 1.5 · 10−1 3 10 0.1 2.9 · 10−3 61 0.0065 0.0009 4.8 69 4.4 · 10−1 3 1 0.3 6.7 · 10−6 77 0.0562 0.0036 1.1 31 3.9 · 10−3 3 3 0.3 1.3 · 10−4 61 0.0093 0.0019 3.3 39 1.5 · 10−2 3 6 0.3 8.4 · 10−4 54 0.0038 0.0007 5.8 46 4.7 · 10−2 3 10 0.3 3.2 · 10−3 51 0.0027 -0.0001 7.2 56 1.7 · 10−1 3 1 1.0 9.2 · 10−6 58 0.0198 0.0028 1.4 26 1.7 · 10−3 3 6 1.0 1.2 · 10−3 42 0.0014 -0.0025 7.6 36 2.4 · 10−2 3 10 1.0 4.5 · 10−3 39 0.0009 -0.0043 9.5 43 1.1 · 10−1 4 1 0.1 5.8 · 10−6 95 0.1332 0.0037 0.9 36 5.3 · 10−3 4 3 0.1 1.1 · 10−4 75 0.0269 0.0024 2.2 48 2.7 · 10−2 4 6 0.1 7.3 · 10−4 66 0.0111 0.0017 3.6 58 8.8 · 10−2 4 10 0.1 2.9 · 10−3 61 0.0065 0.0009 4.8 69 2.5 · 10−1 4 1 1.0 7.9 · 10−6 66 0.0311 0.0033 1.3 28 1.4 · 10−3 4 3 1.0 1.5 · 10−4 52 0.0049 0.001 4.1 34 5.0 · 10−3 4 6 1.0 10.0 · 10−4 47 0.0021 -0.0007 7.1 40 1.7 · 10−2 4 10 1.0 3.9 · 10−3 44 0.0014 -0.0022 9.0 47 6.7 · 10−2 5 1 0.1 5.8 · 10−6 95 0.1339 0.0037 0.9 36 3.4 · 10−3 5 6 0.1 7.3 · 10−4 66 0.0112 0.0017 3.6 58 5.7 · 10−2 5 10 0.1 2.9 · 10−3 61 0.0065 0.0009 4.9 69 1.6 · 10−1 5 1 1.0 7.4 · 10−6 71 0.0411 0.0035 1.2 29 1.1 · 10−3 5 3 1.0 1.4 · 10−4 56 0.0066 0.0015 3.8 36 4.3 · 10−3 5 6 1.0 9.3 · 10−4 50 0.0027 0.0001 6.5 42 1.4 · 10−2 5 10 1.0 3.6 · 10−3 47 0.0018 -0.0013 8.3 50 5.0 · 10−2 10 1 0.1 5.7 · 10−6 95 0.1311 0.0037 0.9 36 8.8 · 10−4 10 3 0.1 1.1 · 10−4 75 0.0264 0.0024 2.2 48 4.5 · 10−3 10 10 0.1 2.8 · 10−3 61 0.0065 0.0008 4.8 69 4.1 · 10−2 10 1 1.0 6.1 · 10−6 86 0.0876 0.0037 1.0 34 6.0 · 10−4 10 3 1.0 1.2 · 10−4 68 0.0159 0.0023 2.7 43 2.6 · 10−3 10 6 1.0 7.7 · 10−4 60 0.0064 0.0014 4.7 51 8.3 · 10−3 10 10 1.0 3.0 · 10−3 56 0.0039 0.0005 6.3 60 2.5 · 10−2 1 3 0.0 1.1 · 10−4 72 0.0233 0.0024 2.2 48 3.2 · 10−1 1 6 0.0 7.4 · 10−4 63 0.0099 0.0015 3.5 59 9.9 · 10−1 1 10 0.0 2.9 · 10−3 57 0.0062 0.0008 4.3 72 2.8 · 100 1 6 0.4 1.7 · 10−3 32 0.0007 -0.0094 6.9 32 1.6 · 10−1 1 3 1.0 5.4 · 10−3 14 0.0003 -0.1309 7.2 20 2.6 · 10−1 1 6 1.0 3.2 · 10−2 14 0.0002 -0.1367 9.8 25 2.8 · 100 1 10 1.0 1.1 · 10−1 14 0.0002 -0.1281 10.4 32 2.7 · 101

∗ † (a) Spherical solutions; Calculated using Pacynzsky-Wiita gravitational potential to approximate eﬀects of GR. Lν ≡

(b) (c) (d) (Lνe +Lν¯e )/2; Mass outﬂow rate (equivalent spherical; eq. [2.19]); Final entropy Expansion timescale at T = 0.5 (e) (f) MeV (eq. [2.23]); Centripetal Ye lowering from equilibrium Ye = 0.45; Radius at which T = 0.5 MeV, approximate

(g) 2 location of alpha particle formation; Ratio of ‘matter pressure’ Pgas to magnetic pressure Pmag = B /8π (for an

15 assumed surface ﬁeld B0 = 10 G) at T = 0.5 MeV, where Pgas includes gas pressure P , thermal energy density e, kinetic energy density (rest energy subtracted); (h)Threshold ratio for outﬂow to attain third-peak r-process (eq. [2.24]) 50

Table 2.2: Summary of magnetosphere-integrated quantities for RNS = 12 km, M = 1.4M (a) ˙ (b) (c) (d) 3 3 (e) PLν Mtot Mej Γmax hηi = hS /texpYe i 51 −1 −1 (ms) (10 erg s )(M s )(M ) - (% of ηthr)

∗N/A 1 3.8 · 10−6 3.5 · 10−5 ∗N/A 0.7 ∗N/A 3 7.1 · 10−5 2.6 · 10−4 ∗N/A 1.5 ∗N/A 6 4.5 · 10−4 9.4 · 10−4 ∗N/A 2.1 ∗N/A 10 1.7 · 10−3 3.3 · 10−3 ∗N/A 2.4 1 1 5.5 · 10−6 5.1 · 10−5 438.5 1.2 1 10 5.5 · 10−3 1.0 · 10−2 1.4 8.9 1 3 2.7 · 10−4 9.8 · 10−4 10.1 6.2 1 6 2.1 · 10−3 4.4 · 10−3 2.2 9.6 2 1 6.3 · 10−7 5.8 · 10−6 239.7 1.4 2 10 3.1 · 10−4 5.9 · 10−4 1.5 8.9 2 3 1.3 · 10−5 4.6 · 10−5 13.1 4.4 2 6 8.1 · 10−5 1.7 · 10−4 2.9 7.0 3 1 3.8 · 10−7 3.5 · 10−6 79.1 1.3 3 10 1.9 · 10−4 3.5 · 10−4 1.2 8.5 3 3 7.0 · 10−6 2.6 · 10−5 5.2 3.9 3 6 4.8 · 10−5 10.0 · 10−5 1.6 6.6 4 1 2.8 · 10−7 2.6 · 10−6 34.9 1.1 4 10 1.4 · 10−4 2.6 · 10−4 1.1 7.9 4 3 5.5 · 10−6 2.0 · 10−5 2.7 3.6 4 6 3.5 · 10−5 7.3 · 10−5 1.3 6.0 5 1 2.4 · 10−7 2.2 · 10−6 17.3 1.1 5 10 1.2 · 10−4 2.2 · 10−4 1.0 7.2 5 3 4.7 · 10−6 1.8 · 10−5 1.8 3.4 5 6 3.0 · 10−5 6.3 · 10−5 1.1 5.7 10 1 9.0 · 10−8 8.2 · 10−7 3.7 0.9 10 10 4.4 · 10−5 8.3 · 10−5 1.0 5.7 10 3 1.7 · 10−6 6.5 · 10−6 1.1 2.5 10 6 1.1 · 10−5 2.4 · 10−5 1.0 4.2

∗ (a) (b) Spherical Solutions. Lν ≡ (Lνe +Lν¯e )/2; Total mass loss rate, integrated across the open zone from (c) θ = 0 − θmax (eq. [2.20]). Total ejected mass at luminosities ∼ Lν , estimated using the PNS cooling evolution from [184]. (d)Maximum Lorentz factor achieved by the proto-magnetar outﬂow, calculated 16 (e) assuming a surface ﬁeld B0 = 10 G. Threshold for r-process (eq. [2.24]), in a mass-weighted average over the open zone from θ = 0 − θmax. 51

51 -1 Polar outflow of Lν=3 10 ergs s , P=2 ms 1032

1030 -3 28 10 P 2 B /8 π

1026 Rotational KE 1024

1022

Energy density, ergs cm Radial KE 1020

1018 106 107 Radius, cm

Figure 2-5: Energy densities as a function of radius along the polar direction 51 −1 θ = 0 for our ﬁducial solution with Lν = 3 10 erg s and P = 2 ms, 15 · assuming a surface ﬁeld B0 = 10 G. and hence earliest times, our solutions provide an accurate description of the

15 outflow at radii of interest only for extremely strong fields B0 > 3 10 G × 51 −1∼ (for P > 2 ms). However, at later times when Lν 10 erg s , even weaker ∼ ∼ 14 fields B0 3 10 G are sufficient to satisfy the force-free condition. ∼ × Even if the force-free approximation is valid in the outflow for magnetar- strength fields, the same may not be true in the closed zone due to the large hydrostatic pressure of the co-rotating atmosphere. If the closed zone pressure Pclosed exceeds that which can be confined by the magnetosphere, the field will ‘tear’ open, enlarging the extent of the open zone (RY < RL) ∼ compared to the standard case RY = RL [134, 185]. 51 Figure 2-7 compares the radial profile of Pclosed and Popen for Lν = 3 10 × −1 erg s and P = 2 ms along the last closed field line θ = θmax, to the pressure 52

1017 P=1 ms P=2 ms P=3 ms P=4 ms P=5 ms 1016 , G min B

1015

1014 1 2 3 4 5 6 7 8 9 10 51 -1 Lν, 10 ergs s

Figure 2-6: Minimum surface magnetic field strength Bmin required for validity of the force-free approximation at all angles and radii interior to the location of α-particle formation (T = 0.5 MeV), calculated for different values of the neutrino luminosity Lν and spin period P . The force-free condition 2 is defined as Pgas/Pmag 0.1, where Pmag = B /8π is the magnetic pressure ≤ and Pgas is the ‘matter pressure’, the latter of which includes gas pressure P , thermal energy density e, and kinetic energy density (rest energy subtracted). 53

2 15 of the poloidal magnetic ﬁeld PB = Bp/8π, calculated for B0 = 10 G. The closed zone pressure is determined by solving the equations of hydrostatic, thermal and weak interactions equilibrium (v = 0;q ˙ = 0; dYe/dt = λ = 0).

The analytic model of [162] predicts that the closed zone RY extends to the location where PB = Pclosed, which according to Fig. 2-7 occurs at a radius 15 r 30 km for B0 = 10 G which is a factor of 3 times smaller than in ∼ ∼ standard case (RY = RL = 100 km). Although most models in this chapter assume RY = RL, the true extent of the open zone θmax could be larger than 1/2 our assumed value by a factor (RY/RL) < 2. ∼

2.3.2 Spherical versus Magnetized Rotating Trajecto- ries

3 3 Based on S /(τYe ) criterion of [63], the most promising period for nucleosynthesis is P = 1 ms. However, the rotational energy of a P = 1 ms magnetar exceeds the kinetic energy of a normal SN of order 1051 ergs - hence their birth should be observable as a hyper-energetic supernova or a hypernova. However, these events are rare [186, 187] and hence P = 1 ms magnetars are rare. We then concentrate at P = 2 ms magnetars in this section. This section provides a comparison between the properties of our rotating PNS

52 −1 wind solutions (for fiducial parameters Lν = 10 erg s , P = 2 ms) and the spherical non-rotating solutions corresponding to the same neutrino luminosity. In particular, Figures 2-8 and 2-9 show the radial profiles of wind properties along two field lines, corresponding to a polar outflow (θ = 0) and

◦ one along the last open ﬁeld line (θ = θmax 21 ). This section also com- ≈ pares the properties of our rotating solutions, such as the entropy/expansion timescale (Fig. 2-11) and mass loss rate (Fig. 2-12), across a range of rotation periods and neutrino luminosities. 54

1032

1030

-3 Pclosed 1028

B2/8π 1026

1024

Energy density, ergs cm Popen 22 10 RL

1020 106 107 Radius r, cm

Figure 2-7: Comparison between the hydrostatic thermal pressure of the 2 closed zone Pclosed (red solid) and the poloidal magnetic field pressure Bp/8π (green dashed) along the last open field line (θ = θmax), calculated for a PNS 51 −1 with neutrino luminosity Lν = 3 10 erg s , spin period P = 2 ms, and × 15 surface magnetic field strength B0 = 10 G. Also shown for comparison is the thermal pressure in the outflow itself Popen (blue dotted). The closed zone pressure exceeds that of the confining magnetosphere at radii > 30 km smaller than the light cylinder radius (RL 100 km), which according∼ to the criterion of [162] indicates that the true open≈ portion of the magnetosphere may be larger than the force free assumption, i.e. the Y-point radius should obey RY < RL (see Fig. 2-1). 55

Centripetal lowering of Ye below Ye,eq

The electron fraction Ye quickly rises and saturates just a few kilometers above the PNS surface (Fig. 2-9, top panel). The values of (∆Ye)cent are reported in Table 2.1. For most of trajectories, (∆Ye)cent < 0.01. This is | | because in each case the outﬂow composition enters equilibrium with neutrino absorptions, which drives Ye to Ye,eq (eq. [2.10]). Electron and positron captures play no signiﬁcant role in the freeze-out process because their rate T 6 decreases rapidly with temperature above the PNS surface, making ∝ them quickly negligible compared to neutrino absorptions. Low value of

(∆Ye)cent justifies neglecting it for P > 3 ms (see Sect.2.2.7). It is inevitable that the outflow composition enter equilibrium with the neutrinos if the latter dominate at freeze-out. Each nucleon must absorb an energy > GMmp/Rns 160 MeV (for M = 1.4M and Rns = 12 km) ∼ ≈ to escape the gravitational potential well of the PNS. Because this energy greatly exceeds that of an average neutrino ν 4kTν 15 MeV, each h i ' ≈ outflowing nucleon must absorb 10 neutrinos. This process renders its ∼ initial identity as a proton or neutron irrelevant (e.g. [136]). Centrifugal acceleration alters the above argument by providing an additional source of wind energy independent of neutrino heating. In practice,

eq however, the asymptotic value of Ye is appreciably reduced below Ye only for extremely rapid rotation near the centrifugal break-up limit [136]. In our most rapidly rotating model with P = 1 ms, (∆Ye)cent reaches values as low as 0.14 for the largest outﬂow angles θ = θmax (Table 2.1). However, for − slower rotation P > 3 ms, Ye varies by less than a percent from its spherical, ∼ non-rotating value across the entire open zone. 56

Mass loss rate

The spherically-equivalent mass loss rates of our fiducial polar (θ = 0) and ˙ −3 −1 −3 inclined (θ = θmax) solutions are M = 1 10 M s and 2 10 M × × −1 ˙ −4 −1 s , respectively, as compared to M = 7 10 M s for the spherical, × non-rotating case. These nearly equal values again result because the mass loss per unit surface area is set largely by the neutrino energy absorbed near the surface [37]. The structure of the outflow, being subsonic in the heating region, is determined mainly by hydrostatic equilibrium and hence is relatively insensitive to the precise outflow geometry. This expectation is borne out in the radial profiles of density (Fig.2- 8; middle panel) and heating (Fig.2-9; bottom panel) just above the PNS surface, which are similar between the spherical and rotating magnetized ˙ outflows. The slightly larger value of M in the θ = θmax case is the result of centrifugal acceleration, which increases the density scale-height in the net heating region, to which M˙ is exponentially sensitive. This increase is not as great as for a equatorial monopolar outflow with the same PNS rotational period [57], because of the lower surface rotational velocity vφ,0 < ∼ RnsΩ sin θmax as compared to the equatorial value vφ,0 = RnsΩ. Figure 2-12 compares the values of M˙ for different rotation rates and neutrino luminosities. As expected, the mass loss rate depends sensitively ˙ 5/3 10/3 5/2 on the neutrino luminosity, approximately as M Lν ν Lν , with ∝ h i ∝ a weaker dependence on other parameters [37]. The largest enhancement in M˙ due to rotation is for the shortest rotation period (P = 1 ms) and largest ˙ angle (θ = θmax), for which M is a factor 10 times larger than the polar ≈ or spherical cases. Although the mass loss rate per surface area is enhanced by magnetic fields and rotation, this is offset by the smaller fraction of the 57

PNS surface fopen which is open to outﬂows in the magnetized case:

2 −1 2πθmax 1 Rns RL P fopen 0.13 , (2.21) ≈ 4π ≈ 2 RY ≈ RY ms

−1 p where in the ﬁnal two equalities we have approximated θmax = sin Rns/RY ≈ p Rns/RY and have assumed Rns = 12 km. ˙ A numerical estimate of the total mass loss rate Mtot integrated across the open zone (eq. [2.20]) for each neutrino luminosity and spin period are given ˙ in Table 2.2. The value of Mtot varies from a factor of 10 times larger to ∼ 30 times smaller than the spherical case as the spin period increases from ∼ P = 1 ms to 10 ms. This transition occurs as the centrifugal enhancement

−1 at small P is oﬀset by the shrinking open fraction fopen P for large P . ∝

Final entropy

The asymptotic values of the entropy for our ﬁducial rotating solutions are

−1 −1 S = 87 kb nucleon and 51 kb nucleon for the θ = 0 and θ = θmax, −1 respectively, as compared to S = 87 kb nucleon for the equivalent spherical case. These similarities and diﬀerences can be understood from the fact that the entropy increase of the outﬂowing matter is approximately given by

Z dQ ∆Q ∆S , (2.22) ∝ T ≈ T¯ where dQ is the net heating per nucleon and T¯ is the average temperature in the heating region. To first order, the entropy of the polar rotating outflow is similar to that in the spherical case for the same reason that M˙ is similar: heating is concentrated in the hydrostatic region and the value of ∆Q is essentially fixed by that required to escape the gravitational well of the PNS.

In the case of inclined ﬁeld lines θ = θmax the entropy is reduced because less neutrino heating ∆Q is necessary to escape the potential well given the 58 additional source of centrifugal acceleration. The value of S in this case approaches its value for an equatorial outﬂow in the monopolar case with the same neutrino luminosity and spin period [57]. Figure 2-11 shows the asymptotic entropy across a range of PNS properties. The strongest trend is that entropy decreases with increasing neutrino

−1/6 −1/3 −1/4 luminosity, approximately as S Lν ν Lν [37]. This depen- ∝ h i ∝ dence can also be understood from equation (2.22): the mean temperature in the region of net heating is smaller for lower neutrino luminosities because ¯ T is to ﬁrst order set by the balance between neutrino heating Lν and ∝ neutrino cooling T 6. ∝

Expansion timescale

Matter ﬂows across the range of radii at which alpha particles form on the expansion timescale [63]

1 d ln T texp = 1.28 (2.23) v dl T=0.5MeV

The expansion timescale for our ﬁducial rotating solutions are texp = 11 ms and 0.6 ms for the θ = 0 and θ = θmax, respectively, as compared to texp = 13 ms for the equivalent spherical case (Fig. 2-9; bottom panel). Faster expansion in the magnetized case results from more rapid acceleration (Fig. 2-8; top panel) due to the faster divergence of the dipolar areal function (Fig. 2-2) and the related fact that for similar entropy a faster decrease in density results in T = 0.5 MeV being achieved closer to the PNS surface, where the radius (and hence expansion time) is likewise smaller (Fig. 2-8; bottom panel). The expansion time is further reduced by a large factor for θ > 0 due to centrifugal acceleration, which increases the outﬂow velocity as compared to 59

the purely thermally-driven case. The value of texp in Table 2.1 shows that at large angles θ θmax centripetal acceleration is more important than the ≈ purely geometric effect of areal divergence for spin periods P 4 ms. For our ≤ fiducial solution, for instance, areal divergence reduces texp from 13 ms for spherical solution to 11 ms for polar outflows (θ = 0). In contrast, centrifugal acceleration decreases the dynamical timescale from 11 ms for polar outflow to 0.6 ms for outflows along the last open field line.

Figure 2-11 compares texp for solutions with diﬀerent rotation rates and neutrino luminosities. The strongest trends are that (1) texp decreases with increasing neutrino luminosity (as also occurs in spherical, non-rotating winds;

[51]); (2) at ﬁxed neutrino luminosity, texp is smaller for shorter spin periods P and larger outﬂow angles θ, again due to centrifugal acceleration.

2.3.3 Spherical vs Magnetized Rotating Nucleosynthe- sis yields

Variation in wind properties across the magnetosphere

P = 1 ms is found (see Sect.2.3.2) to be the most promising period for

3 heavy r-process nucleosynthesis based on s /τexp criterion of [63], however, even in P = 1 ms the third abundance peak (A 200) is not reached. ≈ Nucleosynthesis calculations conﬁrm this conclusion. We focus our analysis in this section on models with P = 1 and 3 ms. The latter case plays important role because the rotational energy of the magnetar with P = 3 ms,

2 51 Erot = IΩ /2 10 ergs, is comparable to the kinetic energy of a normal ≈ SN. Hence magnetars born with P > 3 ms could in principle be ‘hidden’ ∼ among the normal population of normal SNe without violating constraints on the observed kinetic energies of their ejecta from SN spectra and the total energies of their remnants (as already discussed, hypernovae are known to 60

L=1052 ergs s-1 P=2 ms 1011 1012

11 θ=θ 10 1010 max θ=0 1010 -1 109 Spherical case -3 109

108 8 , g cm 10 ρ 107 7 RL 6 Spherical case

10 Density 10 Velocity v, cm s T=0.5 MeV θ=θmax Sonic point 105 106 Maximum heating 104 θ=0

105 103 106 107 106 107 Radius, cm Radius, cm

10 -

Spherical case θ θ = max

-1 θ 10 =0 Temperature T, MeV

10-2 106 107 Radius, cm

Figure 2-8: Radial profiles of velocity v (top left), density ρ (top right), and 52 −1 temperature T (bottom) for our fiducial solution with Lν = 10 erg s and P = 2 ms, shown for different field lines corresponding to a polar outflow (θ = 0), maximally inclined outflow (θ = θmax). Shown for comparison is a spherical, non-rotating outflow for the same neutrino luminosity. Critical radii are marked with symbols, including the light cylinder RL (square), radius of α particle formation R0.5Mev (circle), sonic point (diamond), and point of maximum− net heating (asterisk). 61

52 -1 L=10 ergs s P=2 ms 1023 0.50 0.45 1022 -1

0.40 s -1 0.35 1021 0.30

e 0.25 Y θ θ 20 0.20 Magnetar, =1.0 max 10 RL 0.15 T=0.5 MeV

Sonic point ergs g Net heating q˙, 19 0.10 10 Maximum heating θ θ 0.05 Magnetar, =0.0 max Spherical case 0.00 18 6 7 10 10 10 106 107 Radius, cm Radius, cm

1010 -1 n B

109 Entropy s, k

108 106 107 Radius, cm

Figure 2-9: Radial proﬁles of the electron fraction Ye (top left), net heating rateq ˙ (top right), and entropy s (bottom) for our ﬁducial solution with Lν = 1052 erg s−1 and P = 2 ms (same as Fig. 2-8). Radii marked with symbols are the same as in Figure 2-8. 62

L=1052 ergs s-1 P=2 ms 101

RL T=0.5 MeV 100 Sonic point Maximum heating

10-1

, s Spherical case exp t 10-2 θ=0

-3 10 θ θ = max

10-4 106 107 Radius, cm

Figure 2-10: Radial profiles of the local expansion timescale texp (eq. [2.23]) 52 −1 for our fiducial solution with Lν = 10 erg s and P = 2 ms, shown for different field lines corresponding to a polar outflow (θ = 0), maximally inclined outflow (θ = θmax). Shown for comparison is a spherical, non-rotating outflow for the same neutrino luminosity. 63

P=2 ms 100 1e+51 -1 10 3e+51 θ increase 6e+51 1e+52 10-2 , s 0

exp GR, θ=14 t 10-3

rd 3 peak threshold for Ye=0.45 10-4

10-5 30 40 50 60 70 80 90 100 110

S, kB/nucleon

Figure 2-11: Expansion timescale texp (eq. [2.23]) versus asymptotic entropy S, calculated for a PNS with spin period P = 2 ms and shown for differ- 52 −1 51 −1 ent neutrino luminosities: Lν = 10 erg s (red square), 6 10 erg s (blue asterisk), 3 1051 erg s−1 (green cross), and 1051 erg s−1×(purple plus). Different field line× angles θ are shown for each luminosity, with θ increasing to the lower left hand corner of the plot. Circles of the same color show the corresponding equivalent spherical, non-rotating solutions with the same neutrino luminosity. The solid black line shows the threshold for a successful third peak r-process η = ηthr (for an assumed electron fraction Ye = 0.45). 64

P=2 ms 10-2

1052 10-3 6 1051 -1

s 51 ⊙ 10-4 3 10 M˙, M M˙,

10-5 1051

10-6 0 0.2 0.4 0.6 0.8 1 θ θ / max

˙ 2 Figure 2-12: Spherically equivalent mass loss M = 4πr0ρ0v0 (eq. [2.19]) as a function of the ﬁeld polar angle, calculated for a PNS with spin period P = 2 ms for diﬀerent neutrino luminosities. Shown for comparison with horizontal lines are the mass loss rates of the spherical solutions at the same luminosities. Luminosities in ergs s−1 are written near each line. 65

P=2 ms 0.1 0.09 0.08 0.07 0.06 thr

η 0.05 / η 0.04 0.03 0.02 0.01 0 0 0.2 0.4 0.6 0.8 1

θ/θmax

3 3 Figure 2-13: Critical ratio η S /Y texp of wind properties for a successful ≡ e r-process in units of the required threshold value ηthr (eq. [2.24]) as a function of the field polar angle, calculated for a PNS with spin period P = 2 ms for 52 −1 different neutrino luminosities: (from top to bottom) Lν = 10 erg s (red square), 6 1051 erg s−1 (blue asterisk), 3 1051 erg s−1 (green cross), and 51 −×1 × 10 erg s (purple plus). Note that GR will act to enhance η/ηthr by a factor 3 4 over the values shown. Shown for comparison with horizontal lines are∼ the− values of η of the spherical solutions of the same luminosities. 66 be rare; e.g. [186, 187]). Figure 2-14 shows the mass fraction as a function of nuclear charge Z for different outflow latitudes θ, calculated for P = 1 ms (left panel) and P = 3 ms (right panel). For all values of θ, the abundance distribution extends to higher masses Z than the otherwise equivalent spherical wind with the same Lν and Ye, which is shown for comparison with a purple line. Part of this effect is due to the shorter expansion time through the seed formation region caused by the faster diverging outflow areal function 1/r3 in the ∝ dipole magnetic field, as compared to the spherical wind case (areal function 1/r2), as well as centrifugal force from rotation. ∝ Figure 2-14 also shows that the abundance distribution proceeds to heavier elements with increasing θ, due to the additional acceleration caused by magneto-centrifugal acceleration along field lines inclined with respect to the rotation axis. The heaviest nuclei are synthesized in those outflows which graze the closed zone and pass near the equatorial plane outside the light cylinder. Shown for comparison with a dashed blue line are the abundances integrated over the open zone of the entire magnetosphere, from which it is apparent that the flow properties near the last open field line (θmax) also dominate the total abundance of the wind. This is expected because the mass loss rate per unit surface area is enhanced by magneto-centrifugal acceleration for larger θ [134, 57] and, to a lesser extent, because outflows with larger θ contribute a greater fraction of the total open solid angle of the magnetosphere.

Time-integrated models

Figure 2-15 shows the total yield of proto-magnetar winds, integrated across the entire Kelvin-Helmholtz cooling evolution using the Ye(t) evolution from 67

52 -1 52 -1 P=1 ms, Lν=10 ergs s , Ye, eq=0.45 P=3 ms, Lν=10 ergs s , Ye=0.45 100 Spherical Spherical θ=0.90, Y =0.45 -1 θ 0 0 e 10 =0.9 -1 θ=20.7 , Y =0.40 θ 0 10 0 e =5.4 θ=30.6 , Y =0.33 θ=11.70 e 0 Integrated -2 θ=18.0 10 Integrated 10-2

-3 -3 10 X 10 X

-4 10-4 10

-5 10-5 10

10-6 10-6 15 20 25 30 35 40 45 50 55 15 20 25 30 35 40 45 50 55 Z Z

Figure 2-14: Mass fraction of nuclei X(Z) synthesized in the proto-magnetar 52 −1 wind of neutrino luminosity Lν = 10 ergs s , and neutrino equilibrium electron fraction Ye = 0.45. Results are shown for two rotation periods: P = 1 ms (left panel) and P = 3 ms (right panel); in the former case, the value of Ye is lower than its equilibrium value set by neutrino absorptions due to centrifugal acceleration effects, as marked in the key. Different lines correspond to the abundances synthesized in the wind along field lines with different polar angles θ at the NS surface, ranging from near the pole (θ 0) ' to the last open field line θmax (depending on the rotation period). Shown for comparison with a blue dashed line are the mass-weighted abundances integrated over the solid angle of the open magnetosphere. A purple line shows the abundances from an otherwise identical calculation of a spherical non-rotating non-magnetized wind.

Table 2.3: Summary of Time-integrated Models (a) (a) † † P, ms A¯ Z¯ Xα Xn Xp Mej,M -∗ 87.34 37.71 0.561 9.24 · 10−7 3.65 · 10−7 9.98 · 10−4 1 98.94 42.61 0.429 4.83 · 10−4 5.58 · 10−5 3.45 · 10−3 2 98.73 42.53 0.685 3.42 · 10−4 4.07 · 10−5 1.79 · 10−4 3 97.81 42.13 0.719 2.18 · 10−4 2.55 · 10−5 1.06 · 10−4 4 97.36 41.94 0.714 1.90 · 10−4 2.22 · 10−5 7.80 · 10−5 5 95.07 40.96 0.688 1.62 · 10−4 1.90 · 10−5 6.64 · 10−5 10 94.07 40.53 0.675 9.34 · 10−5 1.08 · 10−5 2.52 · 10−5

(a) Mean mass number for all elements except H and He; ∗Spherical solutions; † Mass fraction of free neutrons and free protons at t = 100 s, i.e. prior to the decay of free neutrons. 68

[22] (Fig. 2-4) and our calculated (∆Ye)cent for P = 1, 2 ms. We show both the total mass (left panel) as well as in ratio of abundances to those calculated in the otherwise equivalent case of a spherical wind (right panel). One clear trend is that the abundance distribution extends to heavier elements in the rotating case, with a larger number of heavy elements synthesized with decreasing rotation period as compared to the otherwise equivalent spherical case. However, strong magnetic fields also reduce the mass of the ejecta in light elements. In part, this reduction is due to the fact that only a small fraction of the PNS surface is open to outflows, such that the total ejecta mass from each event is typically smaller than the spherical case by purely geometric factor (2.21), which we show as a dashed line in Fig. 2-15. Another factor decreasing the mass ejected in light elements is the nucleosynthesis pattern: when the peak shifts to heavier elements, the abundances of light elements are reduced. The only period that exhibits a significant enhancement compared to geometric factor across most elements is P = 1 ms because of the large centrifugal enhancement in the mass loss rate per unit surface area, which overcomes the (comparatively modest) reduction due to fopen 0.1 in this ≈ fastest spinning case. Figure 2-16 shows the abundance-weighted mass A¯ of synthesized nuclei as a function of the magnetar rotation period (red crosses, bottom axis), from which it is clear that A¯ decreases monotonically with increasing spin period P . For comparison a red line shows the result for a spherical wind using the same neutrino-cooling evolution from [22] as in the magnetized case. The mean value of A¯ for P = 1 10 ms is 97, but the A¯ in spherical − ¯ ¯ case is Asph = 87: hence, the rotation with P = 1 10 ms lowers A by 10 − units. On the top axis we also show the value of A¯ calculated in the spherical non-rotating case assuming a temporally fixed value of Ye. 69

104 Spherical P=1 ms

⊙ -4 P=1 ms 3 P=2 ms 10 P=2 ms 10 P=3 ms P=3 ms P=4 ms P=4 ms 102 P=5 ms 10-5 P=5 ms P=10 ms P=10 ms 101

10-6 100

-1 10-7 10

10-2 10-8 10-3 Ejected mass in individual element, M Ejected mass/Ejected mass in spherical case 10-9 10-4 15 20 25 30 35 40 45 50 55 15 20 25 30 35 40 45 50 55 Z Z

Figure 2-15: Left: Total wind ejecta mass in individual elements, time- integrated across the Kelvin-Helmholtz cooling epoch using the Ye(t) evolution from [22] (Fig. 2-4) and our calculated (∆Ye)cent for P = 1, 2 ms. Right: Ratio of the wind ejecta mass of each element in proto-magnetar models relative to those in the otherwise equivalent spherical non-rotating wind models. The dashed lines of same color shows the purely geometric factor fopen (eq. 2.21) arising from fraction of the PNS magnetosphere open to outﬂows.

The fact that the ejecta mass in individual elements exceeds those produced in the spherical case by a factor up to 102 (or 103 in case of P = 1 ms) has the striking implication that millisecond magnetars possess unique nucleosynthetic signatures, which would be measurable even if only one in 100(1000) neutron stars were born strongly magnetized with P < 4 ms ∼ (P 1 ms). ≈ Table 2.3 summarizes important quantities from our time-integrated models. We see that centrifugal eﬀects enhance the values of A¯ and Z¯ moving to shorter periods. Enhanced centrifugal mass loss overpowers the geometric

−3 factor for P = 1 ms, such that the total ejecta mass of Mej 3 10 M ≈ × for P = 1 ms is about three times higher than in the spherical case. 70

Ye 0.445 0.45 0.455 0.46 0.465 0.47 0.475 0.48 0.485 0.49 0.495 100

90 Spherical case _ A 85

70 2 4 6 8 10 P, ms

Figure 2-16: Abundance-weighted mass A¯ of synthesized nuclei, excluding H and He. Bottom Axis: Red crosses show the results for time-integrated models with Ye(t) evolution from [22] (Fig.2-4) for diﬀerent rotation periods, while the solid line shows the result for the non-rotating spherical wind case for the same Ye evolution. Top Axis: Blue asterisks show time-integrated models of a spherical wind which instead assume a temporally constant value of Ye as marked on the top axis. 71

2.4 Discussion

2.4.1 r-Process Nucleosynthesis

Free nucleons recombine into α-particles once the temperature decreases to T 5 109 K, as our solutions show occurs several tens of kilometers ∼ × above the PNS surface. Heavier elements start to form once the temperature decreases further, via the reaction 4He(αn,γ)9Be(α,n)12C. After 12C forms, additional α captures produce heavy ‘seed’ nuclei with characteristic mass A¯ 90 120 and charge Z¯ [132]. The r-process occurs as remaining free ' − neutrons are captured onto these seed nuclei. The maximum mass Amax to which the r-process proceeds depends on the ratio of free neutrons to seed nuclei following completion of the α-process. Because 12C production is the rate-limiting step to forming seeds, the neutron to seed ratio in turn depends on the electron fraction Ye, entropy S, and expansion timescale texp (eq. [2.23]) of the outﬂow [188].

The condition for r-process to reach the third mass peak (Amax > 190) ∼ can be expressed as [63]:

 9 −1 3 −1 S3  8 10 (kBn ) s ,Ye 0.38 η > η · ≥ (2.24) 3 thr 9 (1−2Ye) −1 3 −1 ≡ Y texp ≈ 10 2 , (k n ) s ,Y < 0.38 e ∼  3 0.33 1 B e  Ye − ( Ye−0.17 ) (2Ye)2 where the ratio η/ηthr thus serves as a ‘ﬁgure of merit’ for the potential success of a given r-process site. Previous studies of the r-process in spherically symmetric, non-rotating PNS winds typically ﬁnd that η ηthr (e.g. [51]), thus disfavoring such events as sources of heavy r-process nuclei.

Figure 2-13 shows η/ηthr calculated for our P = 2 ms solutions corresponding to different field lines and neutrino luminosities. Figure 2-17 shows the (mass flux weighted) average value of η/ηthr over the entire outflow for 72

10-1 Increasing time

10-2 , threshold value -3 e Y -1 P=1 ms exp

t P=2 ms 3

S P=3 ms P=4 ms P=5 ms Spherical case 10-3 1 2 3 4 5 6 7 8 9 10 51 -1 Lν, 10 ergs s

3 3 Figure 2-17: Critical ratio η S /Y texp of wind properties required to ≡ e achieve third-peak r-process in units of the threshold value ηthr (eq. [2.24]), mass-averaged over the open field lines as a function of the PNS neutrino luminosity for different rotation periods P = 1 ms (plus), 2 ms (cross), 3 ms (asterisk), 4 ms (open square), and 5 ms (filled square). Shown for comparison with dot-centered circles are the equivalent spherical wind solution for the same neutrino luminosity. Note that GR will act to enhance η by a factor 3 4 over the values shown. ∼ − different periods and neutrino luminosities. Shown for comparison in each case are the values of η/ηthr for our spherical solutions of the same luminosities.

Our most promising solutions for a PNS of mass 1.4 M are for Lν = 52 −1 10 ergs s , θ θmax and near break-up spin period P = 1 ms, for which ≈ η is 10 per cent of the threshold value ηthr. Although these solutions are ≈ still well below the threshold, they are a factor > 4 times higher than the ∼ spherical, non-rotating wind solution of the same neutrino luminosity. This 73

enhancement results from the combination of a much lower values of Ye and texp due to centrifugal flinging and the dipolar geometry ( 2.3.2). In most § promising cases, entropy was considerably smaller than the spherical case, similar to that in the case of equatorial outflows of [57]. In light of our conclusion that magnetic fields and rotation enhance the prospects for the r-process in PNS winds, we note several caveats. First, our calculations do not include general relativity (GR). Past studies of neutrino- heated winds have found that GR effects increase the final entropy4 and decrease the dynamical timescale. If we adopt the 50% increase in entropy ∼ found by [51] (see also [163, 50]), then the resulting increase in η by a factor of 3 is nearly sufficient to push our most promising models for M = 1.4M to within a factor of 2 3 of the threshold. In order to explore the effects ∼ − 52 −1 of GR further, one of our most promising solutions Lν = 10 ergs s ,P =

2 ms, θ = 0.7θmax was recalculated with a [189] gravitational potential. The value of η of the corresponding solution was found to be 4 times higher ∼ than in the Newtonian case (see Table 2.1). Another caveat is that we assume a small PNS radius of 12 km, even at the highest neutrino luminosities corresponding to the earliest times (t 1 ∼ s) in the PNS cooling evolution. At such early stages after the explosion, the PNS is inflated by thermal pressure support relative to its final size [184]. A larger PNS radius results in a lower value of η than would be calculated assuming a fully contracted PNS (the rotation rate prior to full contraction is also likely to be lower due to angular momentum conservation) due to the shallower gravitational potential well (eq. [2.22]). We note, however, that [184] do not include the effects of convective cooling, which may act to enhance the rate of PNS cooling and contraction [190, 191].

4Higher entropy results primarily from the deeper eﬀective gravitational potential well of the PNS in GR, which requires a larger amount of neutrino heating ∆Q to escape, thus increasing S according to equation (2.22). 74

Proto-magnetars as Galactic r-process site

In this subsubsection, we shall proceed under the assumption that the r- process is indeed successful in proto-magnetar winds with moderately rapid rotation P few ms, due to the inclusion of GR and other potential eﬀects ∼ not taken into account, and consider the resulting implications. Our most promising models have luminosities L = 1052 ergs s−1 and P < 5 ms. ∼ Producing all of the heavy r-process elements in the Galaxy over its ˙ −7 −1 lifetime requires a mean production rate Mr 5 10 M yr [192], thus ∼ × requiring an ejecta mass per event of

˙ −1 −1 ¯ Mr −3 fr Mej = 1.5 10 M R (2.25) fr ≈ × 0.1 0.1 CC R R for an event rate , which we have normalized to the estimated rate of core R collapse supernovae of 3 per century [193], where fr 1 2Ye 0.1 0.2 ∼ ≈ − ∼ − is the fraction of the ejecta mass placed in r-process nuclei (as opposed to 4He).

−4 Our models with P = 2 5 ms have total ejecta masses Mej 10 M − ∼ (Table 2.2), which could be enhanced by a factor > fopen RL/RY 3 ∼ ∝ ∼ (eq. [2.21]) due to the ﬁnite pressure of the closed zone (eq. [2-7]) or due to coupling between the proto-magnetar outﬂow and the surrounding envelope of the star [135]. Combining these estimates for the ejecta with equation (2.25), we conclude that > 30% of the Galactic r-process could in principle originate from ∼ proto-magnetars with birth periods P < 5 ms at a rate that is > 10% of the ∼ core collapse rate [62]. Their contribution could be greater if the r-process receives contributions from extremely rapidly spinning neutron stars with

P = 1 ms (which contribute higher Mej but are probably much rarer; see also [145]). 75

10-2 Increasing time

10-3

-1 10-4 s ⊙ , M

tot -5

M˙ 10

P=1 ms P=2 ms 10-6 P=3 ms P=4 ms P=5 ms Spherical case 10-7 1 2 3 4 5 6 7 8 9 10 51 -1 Lν, 10 ergs s

˙ Figure 2-18: Total mass loss rate Mtot integrated over open field lines (eq. [2.20]) as a function of the PNS neutrino luminosity for different rotation periods P = 1 ms (plus), 2 ms (cross), 3 ms (asterisk), 4 ms (open square), and 5 ms (filled square). Shown for comparison with dot-centered circles are the equivalent spherical wind solution for the same neutrino luminosity. 76

A population of moderately rapidly-rotating magnetar r-process sources is compatible with a variety of other observations. A rotation period P > 3

51 ∼ ms corresponds to a total rotational energy Erot < 2 10 ergs, which is too ∼ × low to be excluded as accompanying the birth of Galactic magnetars based on the energetics of their SN remnants [151]. Magnetar birth, though likely not as common as normal core collapse supernovae (however, see [194]), is more common than binary neutron star mergers, the other commonly discussed r-process site. An r-process source from magnetar birth would thus be in better agreement with studies of Galactic chemical evolution that claim to disfavor infrequent, high-Mej events [146, 195]. Magnetar birth in core collapse supernovae could also provide prompt enrichment, as would accompany even the earliest stages of Galactic chemical evolution, another observation in tension with the NS merger scenario. The discovery of magnetars in massive clusters suggest that they result from the core collapse of very massive stars > 40M [196, 197, 198], which are ∼ expected to produce massive neutron stars more favorable to the r-process [58]. On the other hand, massive progenitors may be disfavored by the lack of correlation between Fe production and third peak r-process elements based on the observed abundances in metal-poor stars (e.g. [199]). Finally, the birth of a millisecond magnetar is one of the leading theoretical models for explaining recently discovered class of hydrogen-poor ‘super-luminous’ supernovae [200, 201, 202], events which are indeed observed to occur in very metal poor galaxies (e.g. [203]) with conditions similar to those encountered in the early history of the Galaxy.

Rate constraints on the birth of magnetars

If millisecond proto-magnetars are found to produce large quantities of rare isotopes, one could in principle place constraints on their birth periods based 77 on their nucleosynthetic yields compared to those in our Galaxy as inferred from the solar abundances. An upper limit on birth rate of magnetars of a given rotation period is given by

X M X M R min i, gas = i, gas , (2.26) ≤ Mej,it Xi,windMejtgal where Xi,wind is mass fraction of an element (or an isotope) in the PNS wind,

Mej is the total wind ejecta mass, Xi, is the mass fraction of the same 10 element (or isotope) on the Sun, Mgas 3.3 10 M is the total mass of ≈ × 10 gas in the Galaxy when the Sun formed, and tgal 10 years is the age of ≈ the Galaxy when the Sun formed. In other words, the total ejecta mass of a given element or isotope under consideration cannot exceed the total mass of that element contained in the gas from which the Sun formed (assuming the Sun formed from “ordinary” gas with abundances representative of the Galactic mean). Figure 2-19 shows the result of such a calculation of the maximum allowed birth rate in events per century, which is approximately also the fraction of neutron star births. A purple line shows the rate derived from our time- integrated calculation of proto-magnetar winds, as a function of the magnetar rotation period. Shown for comparison with a blue line is the rate limit for the standard non-rotating spherical wind case. It may be surprising that, for standard spherical winds, the allowed event rate is very low, < 0.05 per century. This is due to the well-known fact that ∼ standard spherical winds overproduce the charged particle process nuclei with

Z = 38 40 (e.g. [140]) for the neutron-rich wind conditions Ye < 0.5 found − ∼ using contemporary PNS cooling calculations [22, 54, 55]. It may also be surprising that the rate constraints are weaker on proto- magnetar winds than on spherical winds, despite the fact that the nucleosyn- 78 thesis of proto-magnetars extends to higher Z elements, which are rarer in the solar system. This is again explained by the fact that only a small fraction of the PNS surface is open to outflows, such that the total ejecta mass from each event is typically smaller than the spherical case by the geometric factor fopen (eq. 2.21). The green line in Fig. 2-19 shows the rate constraints one would derive if the composition of a spherical wind was attenuated by the purely geometric factor fopen. The resulting rate constraint is weaker than the proto-magnetar case including the full effects of the magnetized wind dynamics on the composition itself (purple line); this shows that the rare higher Z elements which are synthesized proto-magnetar winds do tighten the rate constraint significantly, but not enough to overcome the purely geometric suppression factor fopen. The net result of all of this is that the effects of magnetic fields and rotation are sufficiently modest that - given also current uncertainties in the electron fraction of the wind - one cannot at present place meaningful constraints even on the birth rate of non-rotating PNS, much less on their birth periods and magnetic field strengths. Still, our results show that at least under our assumption for the electron fraction of the wind, the birth rate of magnetars with millisecond periods cannot exceed 1 10% of the core ∼ − collapse SNe rate, depending on rotation period. Reassuringly, this number exceeds the total birth rate of Galactic magnetars [62] and is consistent with the lower millisecond magnetar birth rate inferred if they power hydrogen- poor superluminous supernovae [204]. It is possible that neutron stars are generically born with magnetar- strength fields. This could occur, for instance, if strong fields are generated by a convective dynamo in the PNS, which later decay away in the majority of cases by the time they are observed as radio pulsars. We note that, in such a case the reduction in the open fraction of the PNS surface due to con- 79

1.4 Magnetosphere rates 1.2 Spherical composition Spherical 1

0.8

0.6

0.4 Event rate, event per century 0.2

0 2 4 6 8 10 P, ms

Figure 2-19: Upper limit on rates of magnetar birth as a function of the birth rotation period P so as not to overproduce solar system abundances of r-process nuclei (eq. 2.26). Purple crosses show the results based on our abundance calculations of proto-magnetar winds (Fig. 2-15). Green crosses show the rate constraint that would result if we used the composition from the spherical model, but down-correcting the ejecta mass Mej to account for the purely geometric correction fopen (eq. 2.21) resulting from the small solid angle of the open magnetosphere.

ﬁnement by a dynamically-important magnetic ﬁelds [64] would be generic to all winds and thus could help alleviate the current overproduction of the charged particle process nuclei Ye < 0.5 wind models (even if rapid rotation itself is comparatively rare). Also note that including GR will increase the depth of potential well for matter to escape and lower neutrino energy because of gravitational redshift. Both these factors will lower the ejecta mass, also alleviating constraints on the event rate. 80

As sources of light r-process nuclei in metal-poor stars

Figure 2-20 shows the abundances of our time-integrated models (Fig. 2-15) compared to the abundances of two Galactic metal-poor stars, HD 122563 [31] and HD 88609 [32] which show a relative dearth of “heavy” r-process nuclei (Z > 56) compared to the solar system; all abundances have been ∼ normalized to the Ru (Z = 44) abundance in our spherical model. Although the spherical wind models produce large quantities of charged particle process nuclei (Z = 38 40), they underproduce the solar abundances of the weak − r-process nuclei (Z = 41 55) in HD 122563 and HD 88609. By contrast, − our rotating magnetar models produce larger abundances of weak r-process nuclei with Z > 40, while comparatively underproducing the lighter charged ∼ particle process nuclei. Our results provide a potential explanation, within the PNS wind paradigm, for the fact that light r-process nuclei with Z < 56 in metal-poor stars show ∼ greater star-to-star variation than the heavier r-process nuclei [205]. We propose that such star-to-star variation could be understood if the charged particle process and weak r-process nuclei are produced by a combination of normal (slowly-rotating and/or weakly magnetized) NSs and a subclass of magnetars with a range of birth rotation periods. Magnetars with very short birth rotation periods of P < 1 ms have also ∼ been discussed as a source of heavy r-process nuclei, mainly by the ejection of low-Ye matter during the early dynamical phases of the bipolar explosion [169, 145, 206]. Although such extreme magnetars are probably rare, potentially disfavoring this channel as the dominant Galactic source of heavy r-process nuclei, magnetars with less extreme birth periods of 2 5 mil- ∼ − liseconds could produce light r-process nuclei in a larger fraction of events. The rapid core rotation rates of the massive progenitor stars giving rise to millisecond proto-magnetars are also likely to be more common at the low 81 metallicities which characterized early epochs in the chemical evolution of our Galaxy [207]. This work considers the winds from aligned proto-magnetars, for which the rotation axis coincides with magnetic axis. Studies of the more general case of inclined rotators are less developed because the problem is inherently non-stationary (for review, see e.g. [208]). Numerical studies have shown that the equatorial region of the magnetar outflow in this case is characterized by ‘stripes’ of alternating magnetic fields, which are separated by current sheets and are prone to magnetic reconnection [153]. Dissipation of magnetic energy, for instance as these stripes reconnect outside the light cylinder radius [209], provides an additional possible source of heating in the wind, which would act to both increase the entropy of the flow and contribute to its acceleration. If this heating occurs near the seed formation radius, typically close to the light cylinder in our models, this would substantially enhance the prospects for synthesizing third-peak r-process nuclei in misaligned rotators compared to the aligned case studied here. The magnetic inclination angle therefore provides an another parameter, in addition to the rotation period, which could impart diversity to the r-process yields of proto-magnetar winds. More work is clearly needed to distinguish the magnetar hypothesis from other proposed sites of the light r-process elements, such as binary neutron star mergers [210, 43, 211, 212, 44, 213, 214].

2.4.2 Gamma-Ray Burst Outﬂows

Lorentz Factors (Baryon Loading)

Millisecond proto-magnetars are considered promising central engines for powering gamma-ray bursts (GRBs) [154, 155, 134, 137]. One striking feature of GRB outﬂows are their large Lorentz factors 100 1000 [157], which ∼ − 82

100 Spherical P=1 ms P=2 ms 10-1 P=3 ms P=4 ms P=5 ms P=10 ms -2 HD 88609 10 HD 122563 X

10-3

10-4

10-5 25 30 35 40 45 50 Z

Figure 2-20: Abundances in time-integrated models compared to abundances of Galactic metal-poor stars HD 122563 [31] and HD 88609 [32]. Abun- dances are normalized to the Ru (Z = 44) abundance of our spherical time- integrated model. After Z = 55, our models produce very small abundances and they are not able to match the abundances observed in HD 122563 and HD 88609. 83

103 Increasing time

102

P=1 ms

max P=2 ms

Γ P=3 ms

101

100 1 2 3 4 5 6 7 8 9 10 51 -1 Lν, 10 ergs s

Figure 2-21: Maximum angle-averaged Lorentz Γmax achieved by proto- magnetar outflows as a function of neutrino luminosity calculated according to equation (2.28) for an assumed surface field strength B = 1016 G and based on the baryon loading determined from our wind solutions, shown for spin periods P = 1 ms (plus), 2 ms (cross), and 3 ms (asterisk). Lower neutrino luminosities correspond to later times following core bounce, indicating that proto-magnetars with P < 2 ms achieve Γmax 100 1000 on timescales of several seconds. ∼ ∼ − 84 require a specific amount of entrained baryonic mass (‘baryon loading’). In the proto-magnetar model the GRB jet is powered by the electromagnetic extraction of the rotational energy of the PNS, as occurs at the rate

µ2Ω4 B 2 P −4 E˙ = 4 1049 0 erg s−1 (2.27) c3 ≈ × 1015G ms

3 where µ = B0Rns/2 is the magnetic dipole moment. The maximum Lorentz factor (angle-averaged) that the outflow can achieve at a given time is set by the ratio of the outflow power to its rest mass flux

˙ ˙ 2 E + Mtotc σ = Γmax = . (2.28) ˙ 2 Mtotc

We have assumed RY = RL in equation (2.27), but Γmax will be moderately ˙ higher if the open zone is larger (RY < RL) because to first order M is pro- ∼ portional to the open magnetic flux (neglecting centrifugal enhancements), while E˙ is proportional to the open flux squared. ˙ Figure 2-21 shows the angle-averaged Γmax calculated using Mtot from our solutions with P = 1 3 ms (Fig. 2-18; Table 2.2) as a function of neutrino − 16 luminosity for an assumed magnetic field B0 = 10 G (this is the field strength required to produce a jet with power according to eq. [2.27] similar to those of the most luminous observed GRBs). Our results show that at high luminosities, corresponding to early times after core bounce, Γmax is relatively small < 10. However, at lower luminosities corresponding to later times of ∼ several seconds after core bounce, Γmax increases to 100. Our results thus ∼ confirm previous findings [134, 57, 137] that proto-magnetars with short spin periods P 1 2 ms are in principle capable of producing outflows with ∼ − the correct range of Lorentz factors over the appropriate timescale to power long GRBs. By contrast, slower rotating magnetars with P 3 ms produce ∼ outflows with lower luminosities and Γmax 10, which could be responsible ∼ 85

P=2 ms 103

102 max Γ

101

100 0 10 20 30 40 50 60 70 80 90 Asymptotic θ

Figure 2-22: Maximum Lorentz factor Γmax of the proto-magnetar outflow along individual field lines, shown as a function of the asymptotic polar angle θ of the outflow, calculated for a PNS with spin period P = 2 ms and surface field strength B = 1016 G and shown for different neutrino luminosities: 52 −1 51 −1 (from top to bottom) Lν = 10 erg s (red square), 6 10 erg s (blue asterisk), 3 1051 erg s−1 (green cross), and 1051 erg s−1×(purple plus). The × Γmax(θ) distribution depicted only represents the true final angular distribution of the outflow only in the case of ‘naked’ magnetar birth, with little or no surrounding matter at larger radii (as in the case of accretion-induced collapse, but unlike the core collapse case). 86 for less energetic or less relativistic phenomena such as low luminosity GRBs or X-ray flashes [137]. Figure 2-22 shows the maximum Lorentz factor along each field line as a function of asymptotic polar angle θ, calculated for a proto-magnetar with

16 B0 = 10 G and P = 2 ms and shown for diﬀerent neutrino luminosities.

In general Γmax peaks in the rotational equator (θ = π/2), with its value falling to half of the peak value by θ < 50◦. Keep in mind that for an aligned ∼ rotator, the power per solid angle obeys dP/dΩ sin2 θ. Magnetar winds ∝ are thus more powerful, and more highly relativistic, along the rotational equator than in the polar direction. However, translating these predictions for the bare magnetar wind into the angular structure of the resulting relativistic jet in a GRB is non-trivial, because the wide-angle magnetar outflow must interact with and become collimated into a much narrower jet by the surrounding ejecta from the exploding star. This is accomplished by magnetic stresses resulting from the toroidal magnetic field that accumulates behind the outgoing supernova ejecta, which redirect the equatorial wind into a collimated bipolar flow that burrows its way out of the star along the polar rotation axis [135]. The angular distribution Γmax(θ) shown in Fig. 2-22 should however, apply directly to the case of ‘naked’ magnetar birth, such as might occur following the accretion-induced collapse of a white dwarf.

Ultra-High Energy Cosmic Rays

A surprising discovery by the Pierre Auger Observatory (PAO) is that the composition of ultra high energy cosmic rays (UHECRs) appears to be dominated by heavy nuclei (with mass similar to iron) at the highest cosmic ray energies > 5 1019 eV (e.g. [160]; [215]; however, see e.g. [161]). Such a ∼ × heavy-dominated composition is unexpected for most astrophysical sources 87

100

10-1 Xα -2 10 Xn at 100 s Xp at 100 s 10-3 X 10-4

10-5

10-6

10-7 2 4 6 8 10 P, ms

Figure 2-23: Time-integrated mass fractions of α particles, free neutrons and free protons in proto-magnetar winds. We report− the mass fractions of neutrons and protons at t = 100 s, since at that time free neutrons have not yet decayed to protons. The horizontal lines of corresponding colors show these quantities in the spherical model. In spherical wind, the expansion is slower and hence neutrons have more time to capture onto seed nuclei during the r-process. This results in much lower value of Xn for spherical winds. such as AGN, but it could arise naturally if UHECRs are accelerated in GRB jets as a result of nucleosynthesis in proto-magnetar outflows [156, 216]. Figure 2-23 shows the composition of proto-magnetar jets is predicted to be roughly XHe 0.6 by mass in helium, with the remainder Xh 1 XHe ≈ ' − ≈ 0.4 in heavy nuclei of average mass A¯ 95 (Fig. 2-16, Table 2.1). The ≈ latter result is non-trivial, because if the expansion time of the outflow were sufficiently short, then neutron capture reactions could in principle freeze-out with an order unity mass fraction of free neutrons. However, we find that the

−3 free neutron mass fraction at 100 s is always small Xn < 10 (Fig. 2-23). ∼ 88

Such a low free neutron fraction would disfavor models in which the GRB prompt emission is powered by the relative kinetic energy in the jet between its neutral (neutron) and charged (protons, nuclei) constituents [217], unless the nuclei are destroyed in the jet by photo-disintegration before the radius at which neutrons collisionally decouple (see below). The prediction of UHECRs dominated by nuclei with A 100 has im- ∼ portant implications for the energy spectrum and pathlength of UHECRs through the intergalactic medium. The normal Grezin-Zatsepin-Kuzmin (GZK) cut-off in the cosmic ray spectrum occurs due to inelastic pion production by high energy protons which interact with the cosmic microwave background or extragalactic background light [218, 219]. At a fixed (measured) cosmic ray energy E, a nucleus of mass A has a bulk Lorentz-factor which is a factor of A times smaller than a proton of the same energy. Naively, one would therefore expect the effective GZK cut-off energy of a nucleus (interacting with background radiation of a fixed temperature like the CMB) to be A time larger than that for a proton. However, nuclei suffer from other loss processes, the most important one being Giant Dipole Resonances (GDR), which once excited cause the nucleus to shed free nucleons or α particles, − reducing its energy.

20 [156] show that a nucleus of initial mass A = 56A56, energy E = 10 E20 eV travels a distance of

−1.5 1.3 −1 χ75 = 170E20 A56 n¯ej Mpc (2.29) before losing 25 % of its initial energy, wheren ¯−1 1 is the mean number ej ≈ of nucleons ejected per GDR excitation. For an iron nucleus, the mean free path with respect to EBL interaction is coincidentally the same as mean free path for protons with respect to CMB pair production, resulting in a 89

1.3 similar GZK cut-off to the proton case. However, χ75 A implies that for ∝ A = 95 nuclei from our models (Fig. 2-16), the mean free path is 2 times larger than for protons. Thus we expect the GZK-like cut-off in the cosmic ray energy spectrum to extend farther for proto-magnetar wind composition than the usual cut-off predicted for protons or iron nuclei. By contrast, the mean free path for helium is much shorter than iron, such the composition arriving at Earth is expected to be dominated by heavy nuclei at the highest energies. As already mentioned, the highest energy UHECRs do appear to be heavier than protons [160], although the precise composition is model- and calibration-dependent and hence remains uncertain.

2.5 Conclusions

We have calculated nucleosynthesis yields of steady-state neutrino-heated outflows from magnetized, rotating proto-neutron stars (‘proto-magnetars’) under the assumption of an axisymmetric force-free dipolar geometry (Figs. 2- 1, 2-2). The wind and abundances calculations presented here are more accurate than previous one-dimensional models, that adopted a poloidal field that scaled as either 1/r2 or 1/r3 at all radii [134, 57]. This allows us to ∝ ∝ assess, for the first time, the latitudinal dependencies of the wind properties. Our conclusions are summarized as follows:

The force-free approximation is justiﬁed in magnetized proto-neutron • stars outﬂows at temperatures T > 0.5 MeV relevant to heavy element

∼ 14 15 nucleosynthesis for surface magnetic fields B0 > 10 10 G similar to ∼ − those of Galactic magnetars (Fig. 2-6) at neutrino luminosities achieved over the first 10 seconds following core bounce. At later times (lower ∼ neutrino luminosities), even lower magnetic field strengths < 1013 G ∼ similar to those of normal radio pulsars are sufficient for force-free con- 90

ditions.

The force-free assumption breaks down in the closed zone due to the • high pressure of the co-rotating, hydrostatic atmosphere, as ﬁrst pointed out by [134]. This is estimated to increase the fraction of the surface

15 open to outﬂows (Y point radius RY RL/3 for B0 = 10 G; Fig. 2-7) ∼ as compared to the standard case RY = RL.

Nucleosynthesis yields of magnetized rotating PNS outflows differ from • the standard spherical case as a result of the more rapid divergence of the dipolar field geometry and the effects of magneto-centrifugal acceleration. The latter dominates the influence on the expansion timescale

of the outflow texp for spin periods P < 4 ms. ∼ The asymptotic value of the electron fraction is substantially reduced • below its neutrino equilibrium value (eq. [2.10]) only for very short rotation periods, P 1 ms, near centrifugal break-up. This confirms the ∼ results of previous works employing a pure monopolar field geometry [169].

The mass loss rate per surface area is enhanced by centrifugal effects • along low latitude field lines (Fig. 2-12), although this enhancement is not as great as in the equatorial monopole case [57]. The smaller fraction of the PNS surface open to outflows as compared to the spherical case (eq. [2.21]) more than compensates for this increase, resulting in the total mass loss rate being smaller by a factor > 10 than the equiv- ∼ alent luminosity spherical case for P > 2 ms (Fig. 2-18, Table 2.2). ∼ Such suppression in the outflow fraction could be a generic feature of PNS winds if strong, dynamically-important magnetic fields are transient but ubiquitous, e.g. as would be the case if they are present only 91

during the convective phases of the PNS cooling evolution. Such uncer-

tainties, in addition to the uncertain Ye evolution make it challenging to place constraints on the birth rate of magnetars based on their nucleosynthesis abundances (Fig. 2-19). For the most rapidly rotating case P = 1 ms, the total mass loss is instead enhanced by a factor of 3 over the spherical case due to magneto-centrifugal acceleration (Table 2.3).

Neutrino-heated winds from millisecond magnetars with rotation pe- • riods P 1 10 ms produce heavy element abundance distributions ∼ − that extend to higher atomic number than that from otherwise equiv-

alent spherical winds with the same Ye(t) (Fig. 2-15). This increase in the neutron-to-seed ratio is driven mainly by the faster expansion rate in proto-magnetar winds caused by the faster divergence of the area function and due to additional magneto-centrifugal acceleration. For the fastest rotation periods, P = 1 2 ms, it is also driven by − the lower value of Ye resulting from centrifugal acceleration. Unfor- tunately, a direct detailed comparison of our predicted abundances to data (e.g. in the solar system or on metal-poor stars) is hindered by

the larger uncertainties in the time evolution of Ye in the wind during the PNS cooling phase.

The heaviest elements are synthesized by outflows emerging along flux • tubes with latitude θ θmax, i.e. those in outflows which graze the ≈ closed zone and pass near the equatorial plane outside the light cylinder (Fig. 2-14). These fields lines also dominate the total mass budget of the wind due to their larger fraction of the total solid angle and, to a lesser degree, enhancements in the mass loss rate per unit area due to centrifugal acceleration. 92

Due to dependence of the charged particle process and weak r-process • pattern on the magnetic ﬁeld strength and rotation rate of PNSs, natural variations in these quantities between diﬀerent core collapse events could contribute to the diversity of abundances observed on metal-poor stars (Fig. 2-20). This r-process site in the PNS phase (post-explosion) is notably distinct from that discussed in the context of MHD super-

novae [145]. The latter, which invoke low-Ye ejecta, have the potential to produce a greater total r-process yield; however, such extreme magnetars are likely rarer than those discussed here due to the larger angular momentum of the stellar progenitor core required to produce a maximally-spinning PNS versus the slower 2 5 ms periods described ∼ − here.

Additional diversity in the r-process abundances of proto-magnetar winds, not considered in detail here, could result from variations in the magnetic inclination angle. Magnetic dissipation within the resulting striped wind could result in additional wind heating and concomitant entropy gain prior to seed formation, facilitating a heavier r-process than the aligned case focused on here.

Magnetars with birth periods P > 4 ms represent an appealing site • ∼ for Galactic r-process source, consistent with a variety of observational constraints, including (1) the requirement to not overproduce the energies of magnetar hosting supernova remnants; (2) within uncertainties, the mass of r-process nuclei per event needed to explain observed galactic abundances (eq. [2.25]); (3) low mass per event, consistent with previous studies of Galactic chemical evolution [195]; (4) recent evidence for the birth of millisecond magnetars in low metallicity environments based on the discovery of superluminous supernovae. 93

If proto-magnetars are the central engines of GRBs, their relativis- • tic jets should contain an order unity mass fraction of heavy nuclei with A¯ 100. Subsequent particle acceleration in such a jet could ≈ produce UHECRs with a heavy composition [160] and an energy spectrum that extends roughly a factor of 2 above the nominal GZK cut-off for protons or iron nuclei [220]. Better statistics and modeling of the UHECR energy spectrum, as well as a more firm measurement of the composition-dependent UHECR spectrum, is needed to test this prediction and its subtle differences with the normal proton-dominated model predictions. 94

Chapter 3

Novae

3.1 Introduction

Classical and symbiotic novae are luminous transients, powered by runaway thermonuclear burning of a hydrogen-rich layer accreted from a binary companion (e.g., [67, 68, 69, 70, 71]). The resulting energy release causes the

−5 −4 white dwarf atmosphere to inflate, ejecting 10 10 M of CNO- ∼ − enriched matter at hundreds to thousands of kilometers per second (e.g., [221, 222]). Radio and optical imaging [102, 223], and optical spectroscopy [109];[224], suggests that a nova outburst proceeds in at least two stages (Figure 3-1). The runaway is first accompanied by a low velocity outflow concentrated in the equatorial plane of the binary, perhaps influenced by the gravity of the companion star, as occurs in common envelope phases of stellar evolution (e.g., [225, 226]). Given the uncertain nature of the slow ejecta, we use the agnostic term ‘dense external shell’ (DES; [77]). The outflowing DES is then followed by a more continuous wind (e.g., [227]) with a higher velocity and a more spherical geometry. A collision between this fast outflow and the slower DES results in strong internal shocks, 95 which are most powerful near the equatorial plane where the density contrast is largest. If the fast component expands relatively unimpeded along the polar direction, this creates a bipolar morphology [102, 79]. Such a scenario does not exclude fast shocks within the polar region, characterized by lower densities and hard X-ray emission (Fig. 3-1). Several lines of evidence support shocks being common features of nova outbursts. Nova optical spectra exhibit complex absorption lines with multiple velocity components (e.g., [80, 81, 78]). In addition to broad P Cygni lines indicating high velocity > 1000 km s−1 matter, narrower absorption ∼ features ( 500 900 km s−1) are observed near optical maximum and may ≈ − be created within the DES. A large fraction of novae show such narrow lines. This indicates that the absorbing material has a high covering fraction [78] and hence is likely to impact the faster P Cygni outflow. Many novae are accompanied by thermal X-ray emission of luminosity

33 35 −1 LX 10 10 erg s and temperatures > keV (e.g., [228, 229, 230, 231, ∼ − ∼ 232, 233, 234, 100, 102]; see [85] for a summary of Swift observations). This emission is too hard to be thermal radiation from the white dwarf surface [235], but is readily explained as free-free emission from > 1000 km s−1 shocks ∼ (e.g. [236]). The X-ray emission is often delayed by weeks or longer after the optical maximum, perhaps due to absorption by the DES [77]. Although the presence of shocks in novae have been realized for some time, their energetic importance was only recently revealed by the Fermi LAT discovery of > 100 MeV gamma-rays, coincident within days of the ∼ optical peak and last for weeks [90]. Gamma-rays were ﬁrst detected in the symbiotic nova V407 Cyg 2010, in which the target material for the shocks could be understood as the dense wind of the companion red giant [95, 96, 93]. However, six additional novae have now been detected by Fermi-LAT, at least four of which show no evidence for a giant companion and hence were likely 96

Figure 3-1: Proposed scenario for the locations of X-ray and radio emitting shocks. A slow outflow is ejected first, its geometry shaped into an equatorially-concentrated torus (blue). This is followed by a faster outflow or continuous wind with a higher velocity and more spherical geometry (red). The fast and slow components collide in the equatorial plane, producing powerful radiative shocks (Fig. 3-2) which are responsible for the gamma-ray emission on timescales of weeks and the non-thermal radio emission on a timescale of months. Adiabatic internal shocks within the fast, low density polar outflow power hard keV thermal X-rays and possibly radio emission at very early times. Slower equatorial shocks produce the non-thermal radio peak on a timescale of months and softer X-rays (kT < keV), which are challenging to detect due to their lower luminosity and∼ confusion with supersoft X-rays from the white dwarf. The direction of the binary orbital angular velocity Ω is indicated by an arrow. 97 ordinary classical novae with main sequence companions. The DES is clearly present even in binary systems that are not embedded in the wind of an M giant or associated with recurrent novae, supporting the internal shock scenario.

35 36 −1 The high gamma-ray luminosities Lγ 10 10 erg s require shocks ∼ − with kinetic powers which are at least two orders of magnitude larger, i.e.

37 38 −1 Lsh 10 10 erg s , approaching the bolometric output of the nova [79]. ∼ − Gamma-rays are produced by the decay of neutral pions created by collisions between relativistic protons and ambient protons in the ejecta (hadronic scenario), or by inverse Compton or bremsstrahlung emission from relativistic electrons (leptonic scenario). Hadronic versus lepton emission scenarios cannot be distinguished based on the gamma-ray spectra alone [90], although [79] cite evidence in favor of a hadronic scenario. Additional evidence for shocks comes at radio wavelengths. Novae produce thermal radio emission from the freely expanding photoionized ejecta of temperature 104 K, which peaks as the ejecta becomes optically thin to ∼ free-free absorption roughly a year after the optical outburst [101]. However, a growing sample of novae show an additional peak in the radio emission at earlier times (< 100 days; [88, 100, 102, 103]; Fig. 3-4) with brightness ∼ temperatures 105 106 K higher than that of photo-ionized gas. This addi- − tional early radio peak requires sudden heating of the ejecta (e.g., [237, 77]) or non-thermal emission [88, 103], in either case implicating shocks. [77] (Paper I) developed a one-dimensional model for the forward-reverse shock structure in novae and its resulting thermal X-ray, optical, radio emission. This initial work provided an acceptable fit to the radio light curves of the gamma-ray nova V1324 Sco, under the assumption that the dominant cooling behind the shock was provided by free-free emission. However, for low velocity shocks < 103 km s−1 line cooling of the CNO-enriched gas can ∼ 98 greatly exceed free-free cooling. For cooling rates determined by collisional ionization equilibrium (CIE), this additional cooling reduces the peak brightness temperature of thermal models to values < 104 K, which are too low to ∼ explain the early radio peak, unless line cooling is suppressed by non-LTE effects. In this work, we extend the [77] model to include non-thermal synchrotron radio emission, which we demonstrate can explain the observed emission, even for low shock velocities. In addition to providing information on the structure of the ejecta and the nova outburst mechanism, synchrotron emission provides an alternative probe of relativistic particle acceleration in these events, complementary to that obtained from the gamma-ray band. In leptonic scenario, relativistic electrons accelerated directly at the shock power the radio emission. Radio-emitting e± pairs are also produced in hadronic scenarios by the decay of the charged pions. Radio observations can in principle help disentangle leptonic from hadronic models. This chapter is organized as follows. We begin with an overview of shocks in novae ( 3.2), including the collision dynamics, observational evidence for § shocks, the analytic condition for radio maximum, the radiative versus adiabatic nature of the shock, and thermal X-ray emission. In 3.3 we describe § key features of synchrotron emission, including leptonic and hadronic sources of non-thermal particles and their cooling. In 3.4 we provide a detailed de- § scription of our model for radio emission from the forward shock. In 3.5 we § describe our results, including analytic estimates for brightness temperature, and fits to the radio lightcurves of three novae: V1324 Sco, V1723 Aql, and V5589 Sgr. In the discussion ( 3.6) we use the radio observations to con- § strain the acceleration efficiency of relativistic particles and magnetic field amplification in the shocks and their connection to gamma-ray emission. We also discuss outstanding issues, including the unexpectedly monochromatic 99 light curve of V1723 Aql and the role of adiabatic X-ray producing shocks. In 3.7 we summarize our conclusions. §

3.2 Shocks in Novae

Following [77], we consider the collision between a fast outflow from the white dwarf of velocity v1 and the DES of velocity v4 < v1 and unshocked density n4, as shown in Figure 3-2. We assume that the DES of velocity v4 is ejected at t = 0, corresponding to the time of the first optical detection. The fast outflow of velocity v1 is ejected after a delay of time ∆t. The fast outflow and DES collide at a radius and time given, respectively, by

v1 v1v4 t0 = ∆t; R0 = ∆t (3.1) v1 v4 v1 v4 − − The mean density of the DES at the time of the collision is

−3 −3 MDES 8 MDES t0 v4 −3 n¯0 3 10 −4 3 −1 cm , (3.2) ≈ 4πR0fΩmp ∼ 10 M 60 d 10 km s where fΩ 0.5 is the fraction of the total solid-angle subtended by the ∼ outﬂow and for purposes of an estimate we have assumed the thickness of the DES is R0 t0v4. We deﬁne the ejecta number density as n ρ/mp, ∼ ∼ ≡ where ρ is the mass density. This interaction drives a forward shock through the DES and a reverse shock back through the fast ejecta (see Fig. 3-2). We assume spherical symmetry and, for the time being, that the shocks are radiative ( 3.2.3). For § radiative shocks the post shock material is compressed and piles up in a central cold shell sandwiched by the ram pressure of the two shocks. Neglecting non-thermal pressure, the shocked gas cools by a factor of 103, its vol- ∼ ume becoming negligible. Hence the shocks propagate outwards at the same 100

Table 3.1: Commonly used variables and their deﬁnitions.

Variable Definition v1 Velocity of fast outflow (Fig. 3-2) v4 Velocity of DES n1 Density of (unshocked) fast outflow n4 Density of (unshocked) DES T4 Temperature of DES in the photo-ionized layer just ahead of the shock T3 Temperature immediately behind the forward shock n3 Density immediately behind forward shock vshock Velocity of forward shock in the white dwarf frame vshell Velocity of central shell in the white dwarf frame 8 −1 v˜f 10 v8 cm s Velocity of the shock in the frame of the DES ≡ 14 H = 10 H14 cm Density scale height of DES Tb Observed brightness temperature, corrected for pre-shock screening by the photo-ionized layer τff,4 Free-free optical depth of unshocked DES αff,4 Free-free absorption coefficient of unshocked DES tcool Cooling time of gas in the post shock region ∆ion Thickness of ionized layer ahead of the shock (eq. [3.5]) η3 tcool/tfall Ratio of post-shock cooling timescale to shock expansion time (eq. [3.18]) ≡ npk,∆ Density of unshocked DES at time of radio peak (τff,4 = 1) for case when ∆ion < H (eq. [3.12], upper line) npk,H Density of unshocked DES at time of radio peak (τff,4 = 1) for case when ∆ion > H (eq. [3.12], lower line) Mej Total ejecta mass Rsh Radius of cool central shell radius of shock ∼ fEUV = 0.1fEUV,−1 Fraction of shock power placed into hydrogen-ionizing radiation p = 0.1p,−1 Fraction of shock power placed into relativistic protons e Fraction of shock power placed into relativistic electrons and positrons γpk Lorentz factor of the electrons which determine peak synchrotron emissivity Tν Brightness temperature of emission at generic location behind the shock Tν,sync Brightness temperature of synchrotron emission (eq. [3.38]) τν Optical depth at arbitrary location behind the shock th Tν,pk Peak observed radio brightness temperature due to thermal emission nth Tν,pk Peak observed radio brightness temperature of synchrotron emission th Tν,H,pk Thermal contribution to the peak brightness temperature for adiabatic shocks (eq [3.46]) nth Tν,H,pk Synchrotron contribution at the peak brightness temperature of adiabatic shocks (eq. [3.48]) 101

��

Figure 3-2: Shock interaction between the fast nova outﬂow (Region 1) and the slower dense external shell [DES] (Region 4). A forward shock is driven into the DES, while the reverse shock is driven back into the fast ejecta. The shocked ejecta (Region 2) and shocked DES (Region 3) are separated by a cold central shell containing the swept up mass. Observed radio emission originates from the forward shock, since emission from the reverse shock is absorbed by the cold central shell. The ionized layer of thickness ∆ion (eq. 3.5) in front of the forward shock is also shown.

velocity as the central shell, vshell = vshock vsh. The velocity of the cold ≡ central shell is determined by equating the rate of momentum deposition from ahead and from behind, as described in [77]. In what follows, we deﬁne the shock velocity in the upstream frame,

8 −1 v˜f = vsh v4 = 10 v8 cm s , (3.3) −

normalized to a characteristic value of 1000 km s−1. Radio emission is assumed to originate from the forward shock because the reverse shock emission is highly attenuated by free-free absorption within 102 the cold central shell. The forward shock heats the gas to a temperature

2 3µmpv˜f 7 2 T3 1.7 10 v K (3.4) ' 16k ≈ × 8 and compresses it to a density n3 = 4n4, where µ is the mean molecular 8 −1 weight and v8 v˜f /10 cm s . We assume solar chemical composition ≡ with enhanced abundances as follows: [He/H]=0.08, [N/H]=1.7, [O/H]=1.3, [Ne/H]=1.9, [Mg/H]=0.7, [Fe/H]=0.7, typical of nova ejecta (e.g., [238]), which corresponds to µ = 0.76. However, our qualitative results are not sensitive to the precise abundances we have assumed. Absent sources of external photo-ionization, gas well ahead of the forward shock is neutral due to the short timescale for radiative recombination. The upstream is, however, exposed to ionizing UV and X-ray radiation from the shock, which penetrates gas ahead of the shock to a depth, ∆ion. The latter is set by the balance between photo-ionization and recombination, similar to an HII region ([77]; Fig. 3-2), and is approximately given by

−1 14 n4 3 ∆ion 2 10 fEUV,−1 v cm, (3.5) ≈ × 107 cm−3 8

3 where fEUV,−1 = fEUV/0.1 is the fraction of the total shock power Lsh n4v ∝ sh placed into ionizing radiation and absorbed by the neutral layer. Although the neutral gas upstream of the shock eﬀectively absorbs soft UV and X-ray photons, harder X-rays can escape from this region. A minimum ionizing fraction of

2 Ryd −2 fEUV = 0.02 v8 (3.6) ≈ kT3 is obtained in the limit that free-free emission is the sole source of ionization, where Ryd = 13.6 eV. The value of fEUV can in principle greatly exceed this 103 minimum due to line emission and from the reprocessing of higher-energy photons to lower frequencies by the neutral gas ahead of the shock or in the central shell.1

3.2.1 Evidence for Non-Thermal Radio Emission

Figure 3-3 shows the radio light curves of three novae with early-time shock signatures: V1324 Sco [239], V1723 Aql [100, 103], and V5589 Sgr [99]. Ther- mal free-free emission from the photo-ionized nova ejecta peaks roughly a year after the outburst [101]. In V1324 Sco and V1723 Aql, the light curve also shows a second, earlier peak on a characteristic timescale of a few months. This early peak is also present in V5589 Sgr, although the late thermal peak in this event is only apparent at the highest radio frequencies [99]. Thermal emission from an expanding sphere is initially characterized by

2 an optically-thick, Rayleigh-Jeans spectrum Fν ν , with the total ﬂux in- ∝ creasing with the surface area R2 t2. Then, starting at high frequencies, ∝ ∝ the ejecta begins to become optically thin and the radio lightcurve decays as the radio photosphere recedes back through the ejecta. The shape of the radio lightcurve near its maximum depends on the density proﬁle of the ejecta, which we take to be that of homologous expansion, n t−1r−2. Finally, ∝ after the entire ejecta becomes optically thin, the spectrum approaches that

−0.1 −3 of optically thin free-free emission, Fν ν t . ∝ In order to isolate the non-thermal, shock-powered contribution to the radio emission, we ﬁrst remove the thermal emission from the photo-ionized ejecta, using a model for the latter as outlined in [240]. We assume that shocks make no contribution to the emission at late times, t > 100 120 ∼ − 1Ionizing X-rays from the white dwarf are likely blocked by the neutral central shell in the equatorial plane, although they may escape along the low density polar region (Fig. 3-1). Many novae are not detected as supersoft X-ray sources until after the early shock-powered radio emission has peaked [234]. 104

10 36.5 GHz 10 27.5 GHz 17.5 GHz 13.3 GHz 7.4 GHz 4.6 GHz 1 1 , mJy , mJy F F

0.1 36.5 GHz 28.2 GHz 6.8 GHz 5.2 GHz 0.1 0.01 10 100 10 100 1000 Time from optical outburst, days Time from optical outburst, days

10 32.1 GHz 14.0 GHz 7.8 GHz 4.8 GHz 1.9 GHz 1.3 GHz 1 , mJy F

0.1 40 50 60 70 80 90 100 110 120 130 140 Time from optical outburst, days

Figure 3-3: Radio light curves of V1324 Sco (top left panel; [239, 106]), V1723 Aql (right panel; [100, 103]), and V5589 Sgr (bottom left panel; [103]). In V1723 Aql, only frequencies with full time coverage are shown. In V1324 Sco and V1723 Aql, we show for comparison our best-ﬁt thermal models (Table 3.2). In V1324 Sco and V1723 Aql, the maximum uncertainties in the data are 0.3 and 0.1 mJy, respectively, and hence the error bars are not discernible for most of the data points. In V5589 Sgr, we do not attempt to ﬁt a thermal model, even though some of the emission after day 100 could in principle be thermal. 105

106 36.5 GHz 27.5 GHz 17.5 GHz 13.3 GHz 105 7.4 GHz 4.6 GHz , K b T

104

103 20 40 60 80 100 120 140 Time from optical outburst, days

36.5 GHz 106 28.2 GHz 6.8 GHz 5.2 GHz , K b T 105

104 20 40 60 80 100 120 140 Time from optical outburst, days

Figure 3-4: Brightness temperature of the early component of the radio emission for V1324 Sco (top panel) and V1723 Aquila (bottom panel), calculated by subtracting the thermal emission component off the raw fluxes (Fig. 3-3) and assuming an emitting radius for the non-thermal emission corresponding to the fastest velocity of the ejecta inferred from the thermal fits. In V1723 Aql, only frequencies with full time coverage are shown. In V1324 Sco and V1723 Aql, the maximum uncertainties in the data are 0.3 and 0.1 mJy, respectively, and hence the error bars are not discernible for most of the data points. 106

Table 3.2: Best-ﬁt parameters for the thermal radio emission models. The ﬁts are based on observations at t > 100 120 days in order to exclude contributions from the early radio peak.∼ −

(a) (b) (c) (d) (e) Nova D vmin vmax T Mej ∆t −1 −1 kpc km s km s K M days V1324 Sco 7.8 550 1300 1.2 104 2.7 10−4 15 V1723 Aq 6.1 270 1600 1.0 · 104 2.2 · 10−4 -2.2 · · (a)Minimum velocity. (b)Maximum velocity. (c)Ejecta temperature. (d)Ejecta mass. (e)Time delay between outﬂow ejection and optical outburst.

Table 3.3: Multiwavelength properties of novae † nth th L10GHz,pk ‡ q Name D L L LX, pk kT Lγ, pk Orientation 10GHz, pk 10GHz, pk LX,pk pk (kpc) (erg s−1) (erg s−1) (erg s−1) - (keV) (erg s−1) - V1324 Sco 6.5(l) − 8(7.8) 3.4 × 1030(d) 1.4 × 1030(d) < 1034(h) > 3 × 10−4 - 2.7 × 1036(m) ? V1723 Aq 5.3 − 6.1(6.1)(e) 2.5 × 1030(d) 6.3 × 1029(d) 1.2 × 1034(f) 2.1 × 10−4 1.8 − 3(f) - I V5589 Sgr 3.2 − 4.6(4.0)(b) 9 × 1029(b) < 3.6 × 1028(b) 1.2 × 1034(b) 7.8 × 10−5 0.14 − 32.7(b) - ? V959 Mon 1.0 − 1.8(1.4)(a) < 1.1 × 1029(c) 1.8 × 1029(c) 2.4 × 1033(g) < 5 × 10−5 3.2(g) 5.3 · 1034(m) E V339 Del 3.9 − 5.1(4.5)(i) < 2.3 × 1028(k) 9.2 × 1029(k) 1.9 × 1033(j) < 1.2 × 10−4 > 0.8(j) 2.9 × 1035(m) F †Estimated uncertainty range of distance, followed in paranthesis by the ﬁducial value adopted in calculating luminosities; ‡Temperature of X-ray emission; qApproximate inclination of binary (E = edge on; F = face on; I = intermediate; ? = unknown) (a) [108] (b) [99] (c) [107] (d) [241] (e) [103] (f) [100], correct to unabsorbed value (g) [242] (h) for kT ∼ 2 − 25 keV, [243], (i) [244] (j) [245] (k) Justin Linford, private communication (l) [239] (m) [90]

−4 days. Our best-fit parameters for the ejecta mass of Mej 2 3 10 M ≈ − × −1 and maximum ejecta velocities of vmax 1300 1600 km s are compiled in ≈ − Table 3.2. Our values are consistent with those found by [103] for V1723 Aql. After subtracting the thermal component from the raw fluxes (Fig. 3-3), the remainder should in principle contain only the shock-powered emission. Our model for the late thermal emission is admittedly simplified, as it assumes a homologous spherically symmetric and isothermal outflow. However, we are primarily interested in modeling the shape of late thermal emission near the time of early radio peak, at around 80-100 days. At this time the ejecta is still optically-thick and thus our only essential assumption is that the outer ejecta is isothermal and expanding ballistically.

Figure 3-4 shows the brightness temperature Tb of the early shock-powered radio emission, which we have calculated assuming that the shock-powered 107 emission covers the entire solid angle of the outﬂow and originates from a radius equal to that of the fastest ejecta inferred from the thermal ﬁt,

5 6 R = vmaxt. The brightness temperature peaks at values 10 10 K which ∼ − greatly exceed the temperature 104 K of the photo-ionized ejecta inferred ∼ at late times [103]. The outer DES through which the forward shock is propagating at time t, can be modeled as an exponential density proﬁle,

n4 = n0 exp[ v˜f (t t0)/H], (3.7) − − where n0 is a ﬁducial density at the time t0 of the collision (eq. [3.1]) and H deﬁnes the density scaleheight. The light curves of V1324 Sco and V5589 Sgr exhibit a rapid post-maximum decline on a timescale of tfall weeks much ∼ shorter than the timescale of the collision t0 (Fig. 3-3). This suggests that ∼ the forward shock is propagating down a steep radial gradient,

13 tfall H v˜f tfall 6 10 v8 cm, (3.8) ∼ ≈ × week

15 which is smaller than the collision radius R0 10 cm (eq. [3.1]). ∼

3.2.2 Condition for Radio Maximum

The detection of gamma-rays within days of the optical maximum [90] demon- strates that shocks are present even early in the nova eruption. However, at such early times, the density of the shocked matter is highest, and radio emission from the shocks is absorbed by the photo-ionized gas ahead of the shock.

The observed brightness temperature, Tb, is related to the unscreened 108

value just ahead of the shock, Tν shock, according to [77] |

−τff,4 −τff,4 Tb = Tν shocke + (1 e )T4, (3.9) | − where τff,4 = αff,4∆ion is the free-free optical depth of the ionized layer ahead of the shock of thickness ∆ion (eq. [3.5]) and

−2 −3/2 2 αﬀ,4 α0 ν T n (3.10) ≈ 4 4

4 is the free-free absorption coeﬃcient [246], where T4 2 10 K is the ≈ × temperature in the photo-ionized layer. For conditions in photo-ionized layer,

5 3/2 −2 α0 = 0.1 cm K s .

Although the unscreened ﬂux Tν is initially large, the emission reaching the observer is suppressed by a factor of e−τﬀ,4 1. Shock-powered radio emission peaks at frequency ν once the shock propagates down the density gradient until [77]

τﬀ,4(ν) = αﬀ,4(ν) ∆ion 1. (3.11) ≈

Combining equations (3.5) and (3.11), radio emission peaks once the upstream density decreases to a value of

 6 2 −1 −3 −3  npk,∆ 1 10 ν10 fEUV,−1 v8 cm npk = max ≡ × (3.12) 6 −1/2 −3  npk,H 5 10 ν10 H cm ≡ × 14

14 where ν = 10 ν10 GHz, H = H1410 cm, and the second line accounts for cases when the entire scaleheight is ionized (∆ion > H). q The dependency npk ν , where q = 1 2, illustrates that the time ∝ − delay (∆tpk) between the maximum ﬂux at diﬀerent frequencies ν2 and ν1 is 109 typically comparable to the rise time,

H ν2 ∆tpk = q log , (3.13) v˜f ν1 as was obtained by combining equations (3.7) and (3.12).

In summary, for characteristic parameters v8 0.5 2, ν 3 30 ∼ − ∼ − GHz, the radio light curve peaks when the shock reaches external densities

6 8 −3 of npk 10 10 cm . These are generally lower than the mean density ∼ − of the shell at the time of the collision,n ¯0 (eq. [3.2]), and those required to produce the observed γ ray emission [79]. We now consider properties of − the forward shock at the radio maximum, deﬁned by n4 = npk.

3.2.3 Is the Forward Shock Radiative or Adiabatic?

Whether the shock is radiative or adiabatic near the radio peak depends on the cooling timescale of the post-shock gas,

3kT3 tcool = , (3.14) 2µΛn3,pk where n3,pk = 4npk (eq. [3.12]) and

2 2 Λn = (Λlines + Λﬀ )n (3.15) is the cooling rate per unit volume (in ergs s−1 cm−3). For two-body cooling processes, Λ depends on temperature only and it is called the cooling function. Λ receives contributions from emission lines, Λlines, and from free-free emission

−27 1/2 3 −1 Λﬀ 3 10 (T/K) erg cm s . (3.16) ≈ × 110

Figure 3-8 shows tabulated cooling function from [247] for our assumed ejecta abundances and power-law approximation to the line cooling rate,

δ −22 T 3 −1 Λlines 10 Λc,−22 erg cm s , (3.17) ≈ 1.7 107K × with Λc,−22 = 1.5, δ = 0.7. The cooling rate is larger than for solar metal- − licity gas due to the enhancements of CNO elements. The ratio of the cooling timescale (eq. [3.14]) at the time of radio maximum to the characteristic light curve decay timescale, tfall = H/v˜f (eq. [3.8]) is given by

 7.4 −1 −2 tcool 3kT3v˜f  4.8 v8 H14 fEUV,−1ν10 η3 = min (3.18) ≡ tfall 8µΛnpkH ≈ 4.4 −1/2 −1 n4=npk  1.3 v8 H14 ν10

where the ﬁnal equality uses equations (3.4), (3.12). Radiative shocks (tcool tfall) and adiabatic shocks (tcool tfall) are divided sharply atv ˜f 1000 km ≈ −1 2 s , due to the sensitive dependence of tcool/tfall on the shock velocity. As we will discuss, in a radiative shock, only the immediate post-shock material contributes to the thermal or non-thermal emission, allowing for faster light curve evolution. The rapid post-maximum decline of V1324 Sco and V5589 Sgr (tfall t) is consistent with radiative shocks in these systems [77].

2Note that η also equals the ratio of the post-shock cooling length

Lcool =v ˜f tcool = η3H (3.19) to the scale-height, such that the condition for radiative shocks (η3 1) can also be written as Lcool H. 111

3.2.4 Thermal X-ray Emission

Thermal X-rays are diagnostic of the shock velocity and hence of their radiative nature. Figure 3-5 shows that the X-ray light curve of V5589 Sgr peaks

34 −1 at a luminosity of LX 10 erg s a few weeks after the optical outburst ∼ [99]. The temperature is very high kT > 30 keV initially, decreasing to 1 ∼ ∼ keV by the radio peak around day 50. Large velocities v8 1 are required to produce the high X-ray temperatures at the earliest times, clearly implicating adiabatic shocks. However, the X-ray and radio emission may not originate from the same shocks, while even in a given band diﬀerent shocks may dominate the emission at diﬀerent times. The X-ray luminosity of the forward shock is given by ([77]; their eq. [38])

2 4πfΩRshv˜f 3 n4 Λff LX kT3 (3.20) ≈ 1 + 5η3/2 2 µ Λ where fΩ is the fraction of the solid angle of the outflow covered by shocks. −1 The factor (1 + 5η3/2) accounts for the radiative efficiency of the shock and the factor Λff /Λ accounts for the fraction of the shock power emitted as free-free emission in the X-ray band (line emission occurs primarily in the UV). X-ray emission can be attenuated by bound-free absorption. For our adopted composition we find that the bound-free opacity is reasonably ap-

−2 2 −1 proximated as κ 2000(EX /keV) cm g across the range of X-ray en- ≈ ergies EX of interest [248]. The X-ray optical depth at the energies corre- 2 sponding to the peak of the free-free emission (EX = kT3 1.4v keV) is ≈ 8 thus given by

−2 EX n4 −4 n4 τX 3.3 H14 1.8v H14, (3.21) ≈ keV 107 cm−3 ≈ 8 107 cm−3 112

Analogous to the radio light curve, the X-ray luminosity from the shock propagating down a density gradient peaks at τX 1. This occurs once the ∼ density of the pre-shock gas decreases to a value of

6 −1 4 −3 nX 6 10 H v cm . (3.22) ≈ × 14 8

Figure 3-6 shows the maximum X-ray luminosity as a function of the shock velocity, obtained by combining nX with equation (3.20) for fΩ = 1 and 15 assuming a shock radius of Rsh 10 cm. Although this maximum value is ∼ calculated assuming that the density of the shocked matter equals or exceeds nX , this assumption is unlikely to be valid for the highest shock velocities because the density required to achieve τX > 1 becomes unphysically high. ∼ For high velocity shocks (v8 > 1), the predicted X-ray luminosity is several ∼ orders of magnitude greater than the observed range in classical novae, LX ∼ 1033 1035 erg s−1 ([236, 233, 85]; see Table 3.3). This indicates that the − observed X-ray producing shocks either (a) cover a small fraction of the outﬂow solid angle fΩ 0.01 0.1, or (b) are produced in regions of the ∼ − outﬂow with much lower densities, n nX , than the mean valuesn ¯ ∼ 107 109 cm−3 (eq. [3.2]). − The second condition is challenging to satisfy at early times when the densities are probably highest, while both conditions are at odds with the high covering fractions fΩ 1 and high shock densities needed to power the ∼ gamma-ray luminosities [79]. We therefore postulate that the hard keV X-ray emission may not originate from the same shocks responsible for the gamma-ray and non-thermal radio peak, but instead from the low density polar region (Fig. 3-1). In such a scenario, the early radio peak could be powered either by the same fast polar shocks, or by lower velocity (v8 < 1) radiative shocks in ∼ 113

1x1035 100

10 -1 Time of radio max

1x1034 , ergs s kT, keV X L 1

1x1033 0.1 15 20 25 30 35 40 45 50 55 60 65 70 Time from outburst, days

Figure 3-5: X-ray luminosities (black circles; left axis) and temperatures kT (blue circles; right axis) of V5589 Sgr as measured by Swift XRT. Blue triangles indicate temperature lower limits. the higher density equatorial region (Fig. 3-1). In the latter case the radio- producing shocks still produce thermal X-rays; however, being comparable in brightness, yet much softer in energy, they may not be readily observable. This would also be true if the X-ray luminosity of radiative shocks is suppressed due to the role of thin-shell instabilities [249].

3.3 Synchrotron Radio Emission

Ampliﬁcation of the magnetic ﬁeld is required to produce the observed synchrotron emission and is expected to result from instabilities driven by cosmic ray currents penetrating the upstream region (e.g. [127, 114]). The strength 114

Figure 3-6: Thermal X-ray luminosity at the time of X-ray peak (black line - left axis; n = nx) and 10 GHz radio luminosity νLν at the time of non-thermal radio peak (red line - right axis; n = npk) as a function of shock velocity. Both luminosities are maximum allowed values, because they are calculated assuming τX = 1 and τν=10 GHz = 1, respectively. We have assumed ﬁducial parameters for the density scale-height H = 1014 cm and shock microphysical 15 parameters e = B = 0.01, and a common radius Rsh = 10 cm and covering fraction fΩ = 1 of the shock. Also shown is the division between radiative and adiabatic shocks (vertical dashed blue line) and the range of observed radio luminosities (dashed red line) and X-ray luminosities and shock velocities corresponding to the observed X-ray temperature (grey region; Table 3.3, Fig. 3-5). 115 of the post shock magnetic ﬁeld can be estimated as

1/2 1/2 1/2 Bsh = (24πP3B) 0.22n v8 G, (3.23) ≈ 4,7 B,−2

7 −3 −2 where n4,7 n4/10 cm , B = 10 B,−2 is the fraction of the post- ≡ shock energy density in the magnetic ﬁeld. Substituting n4 = npk npk,∆ ≈ (eq. [3.12]) into equation (3.23) gives the ﬁeld strength when the radio emission reaches its peak,

−2 1/2 −1/2 −1/2 Bpk = 8.2 10 f v ν10 G. (3.24) × B,−2 EUV,−1 8

An electron or positron of Lorentz factor γ produces synchrotron emission mostly near a characteristic frequency

2 2 1 eBγ 1/2 −1/2 −1/2 γ νsyn = 92 GHz B,−2v8 fEUV,−1ν10 . (3.25) 2π mec ≈ 200

There thus exists a special electron Lorentz factor,

−1/4 1/4 1/4 γpk 210 v f , (3.26) ≈ B,−2 8 EUV,−1 which determines the peak emission (νsyn = ν) and, remarkably, is independent of the observing frequency.3

3.3.1 Leptonic Emission

The relativistic leptons responsible for synchrotron radiation originate either from the direct acceleration of electrons via diﬀusive shock acceleration (e.g.,

3 The ‘Razin’ effect suppresses synchrotron radiation below a critical frequency νR = νpγ 2 1/2 1/4 5/4 1/4 [246], where νp = (npke /πme) . For γ = γpk we thus have ν/νR = 4.5 B,−2v8 fEUV,−1. Since for parameters of interest we have ν > νR, modifications from the Razin effect should be weak and are hereafter neglected. ∼ 116

[112]), or from electron-positron pairs produced by the decay of charged pions from inelastic proton-proton collisions. Both theory and simulations predict the spectrum of accelerated electrons (dN/dE)dE E−pdE, with ∝ 2 p > 2 (when E mec ) in the case of strong shocks (Figure 3-9, left panel). ∼ Two key parameters are the fractions of the shock kinetic power placed into non-thermal electrons and ions, e and p, respectively. Particle-in-cell (PIC) and hybrid kinetic simulations (e.g., [250], [113],

[114], [115], [251], [252]) indicate that the values of e and p depend on the Mach number of the shock, and the strength and the inclination angle θ of the upstream magnetic field with respect to the shock normal. Shocks which propagate nearly perpendicular to the direction of the magnetic field (θ 90◦) do not efficiently accelerate protons or electrons [130], while those ≈ propagating nearly parallel to the magnetic field (θ 0◦) accelerate both ≈ [251, 252]. What is the geometry of the magnetic field in the ejecta? On timescales of months, the ejecta has expanded several orders of magnitude from its initial size at the base of the outflow [79]. This expansion both dilutes the magnetic field strength via flux freezing and stretches the field geometry to be perpendicular to the radial direction in which the shocks are likely propagating. Based on the above discussion, such a geometry would appear to disfavor hadronic scenarios. However, leptonic scenarios are also strained

−2 because the values of e > 10 required to explain the γ-ray emission in ∼ V1324 Sco [79] greatly exceed the electron acceleration eﬃciencies seen in current numerical simulations [251, 252]. Due to global asymmetries, or inhomogeneities in the shocked gas caused by radiative instabilities [79], the shocks may not propagate perpendicular to the magnetic ﬁeld everywhere, allowing hadronic acceleration to operate across a fraction of the shock surface. This possibility motivates considering 117 alternative, hadronic sources for the radio-emitting leptons.

3.3.2 Hadronic Emission

Relativistic protons accelerated near the shock collide with thermal protons in the upstream or downstream regions, producing neutral (π0) and charged (π−, π+) pions. The former decay directly into γ rays, while π± decay − into neutrinos and muons, which in turn decay into relativistic e± pairs and neutrinos. In addition to electrons accelerated directly at the shock, these secondary pairs may contribute to the observed radio emission. Figure 3-9 shows the e± spectrum from pion decay, calculated for a ﬂat input proton energy spectrum (p = 2). Radio emission near the time of

2 peak flux is produced by electrons of energy γpkmec 100 MeV, where ≈ γpk 200 (eq. [3.26]). By coincidence, this is close to the pion rest energy ≈ 2 ± mπc 140 MeV and hence to the peak of the e distribution produce by ≈ pion decay. Thus, secondary pairs can contribute significantly to the radio emission if they are deposited in regions where their radiation is observable. Although pion-producing collisions occur both ahead of the shock and in the post-shock cooling layer, pairs produced in the cold central shell cannot contribute to the radio emission due to the high free-free optical depth in this region (see Fig. 3-7). Only e± pairs produced in the first cooling length behind the shock contribute to the observable radio emission, but these contain only a small fraction of the shock power,

tcool −4 3.4 e p fπ = 4 10 (p/0.1)(fπ/0.1)v8 , (3.27) ∼ tπ × where fπ is the fraction of proton energy per inelastic collision placed into −1 −26 pions, tπ = (n3σπc) is the timescale for pion production, and σπ 4 10 ≈ × cm2 is the characteristic inelastic p-p cross section [253]. 118

Pairs are also produced by p-p collisions upstream of the shock, which then radiate after being advected into the magnetized downstream. However, proton acceleration via Diﬀusive Shock acceleration (DSA, see Introduction) is conﬁned to the narrow photo-ionized layer ahead of the shock [94], the narrow radial thickness of which, ∆ion (eq. [3.5]), limits the radial extent of the pion production region ahead of the shock.4

3.4 Radio Emission Model

3.4.1 Pressure Evolution of the Post-Shock Gas

Gas compresses behind the shock due to radiative cooling at approximately constant pressure (neglecting the eﬀect of thermal instabilities; [254]). The total pressure includes both the thermal gas and relativistic ions (‘cosmic rays’),

n T n 4/3 Ptot = Pth + PCR = Pth,0 + PCR,0 , (3.28) n3 T3 n3

The energy fraction which goes into relativistic ions is twice the pressure ratio between relativistic ions and total pressure p = 2PCR,0/(Pth,0 + PCR,0).

Our numerical calculations use p = 0.2, although our results are not sensitive to this assumption because the observed emission is dominated by that originating within the ﬁrst cooling length behind the shock, i.e. before compression becomes important.

4 Very high energy protons of energy Emax > 10 GeV leak out of the DSA cycle, es- caping into the neutral upstream with a nearly∼ mono-energetic∼ distribution with a comparable energy ﬂux to the power-law spectrum ultimately advected downstream (e.g. [113]). 2 However, the protons leaving upstream inject pairs at multi-GeV energies γpkmec (eq. [3.26]) which radiate most of their synchrotron emission at higher frequencies than that responsible for the bulk of the radio emission. 119

3.4.2 Electron and Positron Spectra

The electron and e± energy spectra, N(E), in both leptonic and hadronic scenarios are normalized such that a fraction e of shock power is placed into relativistic electrons. In the leptonic scenario, we assume that

−p N(E)= CEE ,Emin < E < Emax (3.29)

2 2 6 12 where E = γmec , Emin = 2mec 10 eV and Emax = 10 eV [94]. The ≈ normalization constant is given by

 p−2 p+2  (p 2)Emin 2 3  − , p = 2 9mpn4v˜f e n  −p+2 C = 1 (Emax/Emin) 6 (3.30) E 8 n − 3 ×  −1 [ln(Emax/Emin)] , p = 2,

(p+2)/3 where the term (n/n3) accounts for adiabatic heating as gas com- ∝ presses downstream. As shown in Appendix , most sources of cooling of relativistic electrons in the post shock thermal cooling layer are negligible, with the possible exception of Coulomb losses, which we include in a few select cases as described below. In the hadronic scenario, we model the spectrum of e± pairs from pion decay following [253], assuming an input power-law spectrum of relativistic

−p protons Np(E) E with the same energy distribution in the leptonic case. ∝

3.4.3 Radiative Transfer Equation

The radio emission downstream of the shock is governed by the radiative transfer equation, which can be conveniently written in terms of the brightness temperature Tν [246],

dTν = T Tν + Tν,syn, (3.31) dτν − 120 where T is the gas temperature,

2 jν,sync Tν,syn = 2 . (3.32) 2αﬀ kν

R is the synchrotron brightness temperature, and τν = αff dz is the free-free optical depth. Here z is the depth behind the shock and αff is the free-free absorption coefficient (eq. [3.10]), where the relevant pre-factor in is now

5 3/2 −2 α0 = 0.28 cm K s for the higher temperatures appropriate to the post- shock gas. We neglect synchrotron self-absorption, which is only relevant for

11 relativistic brightness temperatures of Tν > 10 K, much higher than those ∼ observed in novae.

It is useful to change variables from free-free optical depth τν to temperature, according to [77]

dT 5kα n v˜ dτ = α = 0 4 f dT (3.33) ν ﬀ dT/dz 2ν2T 3/2Λ(T )

2 using the temperature gradient dT/dz = 2n Λ(T )/(5kn4v˜f ) behind the shock set by cooling. As both the free-free absorption coeﬃcient and the cooling rate scale as n2, this allows for a one-to-one (up to a coeﬃcient) mapping ∝ τν(T ) given Λ(T ). This mapping is notably independent of the details of how the gas compresses behind the shock.

Figure 3-7 shows the evolution of the optical depth τν/τν,0 as a function 2 of temperature T behind the shock, where τν,0 = n4,7v8/ν10 and τν = 0 is deﬁned at the forward shock surface (T = T3; eq. [3.4]). If only free-free cooling is included, then the radio photosphere (τν = 1) can occur at higher 5 temperatures of > 10 K, depending on τ0,ν. However, if full line cooling is ∼ included [247], then the photosphere temperature is much lower, 104 K. ≈ The observed brightness temperatures > 105 K of nova shocks (Fig. 3-4) can ∼ therefore only be thermal in origin if line cooling is highly suppressed from 121

0.1 Free-free cooling ,0 / 0.01

Full line cooling 0.001

0.0001 4 4.5 5 5.5 6 6.5 7 7.5

Log10(T/K)

Figure 3-7: Free-free optical depth τν as a function of gas temperature T behind the shock. The optical depth is normalized to a ﬁducial value 7 −3 2 τ0,ν (n4/10 cm )v8/ν10. Diﬀerent lines correspond to cases calculated for a cooling≡ function including full line cooling (purple) and one including just free-free cooling (green).

its standard CIE value [77]. When line cooling is present, additional non- thermal synchrotron emission above the photosphere is needed to reproduce the observations. The synchrotron emissivity is given by an integral over the electron distribution function,

1 Z Emax jν,syn = P (ν, E) N(E)dE, (3.34) 4π Emin where P (ν, E) is the emissivity for a single electron. For analytical estimates 122

10-19

10-20 -1 s 3 10-21 , erg cm

10-22 ~ T-0.7

10-23 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

Log10(T/K)

Figure 3-8: Cooling function Λ(T ) from [247], adopted for our ﬁducial ejecta composition enriched in CNO elements. A green line shows the power-law Λ T −0.7 (eq. [3.17]). ∝ 123 it is convenient to use the so-called delta-function approximation,

4 2 2 P (ν, E) cσT UBγ δ ν γ νB , (3.35) ≈ 3 − where νB = eB/(2πmec). This yields reasonably accurate results for broad and smooth electron distributions. Nevertheless, our numerical calculations use the exact expression for synchrotron emissivity. For a power-law distribution of electrons,

p−1 − 2 1 ˜ ν jν,syn = CEhαf νB , (3.36) 9 νB

˜ 2 −p+1 where αf is the ﬁne structure constant and CE = CE(mec ) is a rescaled normalization constant of the electron distribution. We assume that the magnetic ﬁeld does not decay as the plasma cools and compresses downstream of the shock, its strength increasing as B =

Γ Bsh(n/n3) , where Γ = 2/3 for the flux freezing of a tangled (statistically isotropic) field geometry. Our results are not sensitive to this choice because, for any physical value of Γ, the integrated emission is dominated by the first

(p+2)/3 cooling length behind the shock ( 3.5.1). Using CE n appropriate § ∝ (2p+3)/3 for adiabatic compression, the emissivity scales as jν,syn n . Figure ∝ 3-9 compares the frequency dependence of the emissivity for electrons from direct shock acceleration and pion decay. 124

1 10-23 p=2.0 electron power-law p=2.0 power-law pion decay from p=2.0 protons pion decay from p=2.0 protons Thermal

-24

-1 10 Hz -1 -3 str

-1 -25

0.1 s 10 -3 E dN/dE, cm , erg cm

-26 j 10

0.01 10-27 101 102 103 10-1 100 101 102 103 E, MeV , GHz

Figure 3-9: Left Panel: Energy spectrum of relativistic leptons, calculated for direct shock acceleration of electrons (electron index p = 2) and for secondary pairs from pion decay (proton index p = 2). Right Panel: Synchrotron emissivity as a function of frequency for the particle spectra in the left panel. The thermal free-free emissivity is shown for comparison. Calculations were performed adopting the best-ﬁt shock parameters for V1324 Sco (Table 3.4) of v8 = 0.63, e = 0.08, fEUV = 0.05 at the 10 GHz maximum (t = 71.5 days, 6 −3 n4 = 4.5 10 g cm ). · 3.5 Results

3.5.1 Analytical estimates

The formal solution of equation (3.31) for the brightness temperature at the forward shock surface (T = T3) is given by

Z 0 −τν Tν = (Tν,syn + T )e dτν, (3.37) ∞ where, using equations (3.10), , (3.30), (3.32), and (3.36),

2p−3 3−p 1−p 11+p p+1 3/2 3 9 − 4 2 2 4 T n Tν,syn = 2 10 n4,7 ν10 v8 B,−2e,−2 K, (3.38) × T3 n3 and the prefactor varies between 1.4 2 109 for p = 2 2.5 due to the − × − complex dependence of CE (eq. [3.30]). 125

Radiative Shocks

For low velocity, radiative shocks (v8 < 1) line cooling dominates free-free ∼ cooling behind the shock, such that the radio photosphere τν = 1 is reached at gas temperatures T < 104 K (Fig. 3-7). Because the latter is significantly ∼ lower than the observed brightness temperatures (Fig. 3-4), non-thermal radiation with Tν,syn T must dominate the emission. This justifies neglecting T compared to Tν,syn in equation (3.37). As justified below, a further simpli- R fication is allowed because Tν,syndτν is dominated by the first cooling length 6 behind the shock, where T > 10 K and τν 1, allowing us to approximate ∼ e−τν 1 in equation (3.37). Setting dT = T in equation (3.33), the optical ≈ depth accumulated over a cooling length centered around temperature T is given by

1 −δ− 2 5α0kn4v˜f −3 −2δ −2 −1 T ∆τν = 2 1/2 = 2.3 10 n4,7v8 ν10 Λc,−22 , (3.39) 2ν T Λ(T ) × T3 where we have used the power-law approximation Λ T δ for the line cool- ∝ ing function in equation (3.17) assuming v8 < 1. From equations (3.38) ∼ and (3.39), the brightness temperature accumulated over a cooling length centered around temperature T is given by

nth Tν Tν,syn∆τν ≈ 2p−3 ≈ 1+p p+3 11+p−4δ p+1 1−δ 3 6 4 − 2 2 4 −1 T n 5 10 n4,7 ν10 v8 B,−2e,−2Λc,−22 K ≈ × T3 n3 6−2p−3δ T 3 . (3.40) −1 n/n3=(∝T/T3) T3

Because T nth is an increasing function of temperature for p < (6 3δ)/2 4, ν − ≈ this illustrates that most of the ﬂux is accumulated over the ﬁrst cooling length behind the shock. 126

The thermal contribution to the brightness temperature at post-shock temperature T is likewise approximately

1 2 −δ th 4 2−2δ −2 −1 T Tν ∆τνT = 3.5 10 n4,7v8 ν10 Λc,−22 K. (3.41) ≈ × T3

If free-free emission dominates the cooling (δ = 0.5), then the emission receives equal contributions from each decade in temperature behind the shock [77]. For line cooling (δ = 0.7) the first cooling length again provides the − dominant contribution. Figure 3-10 shows how the 10 GHz brightness temperature grows as a function of temperature behind the shock for different assumptions about the cooling and emission of the post-shock gas. At high optical depths, the radiation and gas possess the same temperature (Tν = T ) because synchrotron emission is not present in the cold central shell. However, differences in Tν become pronounced as the shock front is approached at T = T3. When line cooling is included, only the first cooling length behind the shock significantly contributes to the final brightness temperature at the shock surface. Coulomb losses of relativistic electrons only noticeably impact the non- thermal emission well downstream of the shock, in regions where the gas temperature is < 105 K (eq. [A.2]). Coulomb losses therefore significantly im- ∼ pact the observed radiation temperature only if free-free cooling dominates,

1/6 for which contributions to the emission from the post-shock gas Tν T ∝ (eq. [3.40]) vary only weakly with gas temperature. Even in this case, however, the observed temperature reduced only by a factor of 2 compared to ∼ an otherwise identical case neglecting Coulomb losses. Coulomb corrections are completely negligible in our ﬁducial models when line cooling is included because the bulk of the emission comes from immediately behind the shock, where the temperature is high and the density is low. 127

108

Free-free cooling

107 shell

Free-free cooling, Coulomb 3

, K 6

=T 10 T b T

Cold central shell, T Free-free cooling, thermal Shock front, T 105 Line cooling

Line cooling, thermal 104 104 105 106 107 T, K

Figure 3-10: 10 GHz brightness temperature in the post-shock region as a function of gas temperature T < T3 behind the shock (T = T3). Different lines correspond to different assumptions about (1) whether the emission accounts only for thermal free-free emission [dashed lines] or also includes non-thermal synchrotron emission [dash-dotted lines], and (2) whether the assumed cooling function is just free-free emission [blue, purple] or also includes emission lines [red]. A purple line shows the non-thermal case with free-free cooling, including the effect of Coulomb losses on the emitting relativistic lep- 6 tons. All calculations were performed for a pre-shock density of n4 = 4.5 10 g cm−3 corresponding to the 10 GHz peak time, adopting the best-fit shock· parameters for V1324 Sco (Table 3.4) of v8 = 0.63, e = 0.08, fEUV = 0.05. 128

Thus far, we have focused on the emission temperature at the forward shock (the ‘unscreened’ temperature, Tν shock in eq. [3.9]), and we have not | yet accounted for free-free attenuation by the ionized layer ahead of the shock. The observed brightness temperature Tb only equals that of the shock,

Tν shock, after the upstream density has become suﬃciently low (n4 < npk; | ∼ eq. [3.9]). The measured peak brightness temperature, Tν,pk, is thus obtained by substituting n4 = npk (eq. [3.12]) into equations (3.40) and (3.41) for

T = T3, n = n3. Multiplying the resulting expression by a factor of 1/e

(since at the radio peak τν = 1) yields

 −2δ−1 −1 −1 th  1800 v8 fEUV,−1Λc,−22 K Tν,pk max (3.42) ≈ 2−2δ −1 −1/2 −1  6900 v8 ν10 H14 Λc,−22 K in the thermal case and

 p+1 19−p−8δ p+1 5 − 4 −1 4 4 −1  3.9 10 f ν v e,−2Λ K T nth max × EUV,−1 10 8 B,−2 c,−22 (3.43) ν,pk − p+5 11+p−4δ p+1 − p+1 ≈ 6 4 2 4 −1 8  1.0 10 ν v e,−2Λ H K × 10 8 B,−2 c,−22 14 in the non-thermal case. For p = 2, and for our ﬁducial power-law ﬁt to the line cooling function

(Λc,−22 = 1.5, δ = 0.7), equation (3.43) becomes −  5 −3/4 −1 5.7 3/4 nth  2.6 10 fEUV,−1ν10 v8 B,−2e,−2 K, Tν,pk max × (3.44) ≈ 5 −7/4 7.9 3/4 −3/8  6.9 10 ν v e,−2H K. × 10 8 B,−2 14

Figures 3-11 and 3-12 compare our analytic expressions for the peak observed brightness temperature (eqs. [3.42, 3.43]) and the peak value resulting from a direct integration of equation (3.37).

nth The top panel of Figure 3-11 shows how Tν,pk depends on the density scaleheight H and the ionized fraction of the preshock layer, fEUV. When the value 129

of H is small, or fEUV is suﬃciently high, then the entire DES scaleheight is ionized (H = ∆ion) and the peak brightness temperature is independent of fEUV. On the other hand, when ∆ion < H, then the brightness temperature is independent of H but decreases with increasing ionizing radiation as T nth ν,pk ∝ f −(p+1)/4 = f −0.75 (eq. [3.43]). The bottom panel of Figure 3-11 shows how EUV p=2 EUV the peak temperature depends on the shock velocity,v ˜f , and the electron acceleration eﬃciency, e. Figure 3-12 compares the brightness temperature calculated in a purely

th thermal model (e = 0) to our analytic estimate of Tν,pk (eq. [3.42]). For a broad range of parameters, the peak temperatures fails to exceed 105 K, making thermal models of radiative shocks challenging to reconcile with observations (see Fig.3-4).

Adiabatic shocks

8 −1 For higher velocity v8 > 10 cm s , adiabatic shocks, the brightness temper- ∼ ature is again estimated by assuming that the first scaleheight H behind the shock dominates the emission, i.e. neglecting ongoing emission from matter shocked many dynamical times earlier. However, unlike with radiative shocks, this assumption cannot be rigorously justified without a radiation hydrodynamical simulation, an undertaking beyond the scope of this work. The peak brightness temperature in the adiabatic case is estimated using the expressions for radiative shocks from 3.5.1, but replacing the first cooling § length behind the shock tcoolv˜f with H. For purely thermal emission, the brightness temperature is

th 5 2 −2 −1 T = 1.1 10 n H14ν v K (3.45) ν,H × 4,7 10 8

Substituting n4 = npk into equation (3.45) gives a peak thermal temperature 130 of

T th = 1.2 104v−1K, (3.46) ν,H,pk × 8 where we have assumed that the scaleheight is fully ionized (npk = nH,pk), which is generally well satisﬁed for high velocity shocks. For adiabatic shocks, the peak thermal brightness temperature is a de-

−3/2 creasing function of the shock velocity due to the αﬀ T dependence ∝ of free-free absorption. Comparing equations (3.44) and (3.46), a maximum thermal brightness temperature of 104 K is thus obtained for the inter- ≈ mediate shock velocity v8 1 separating radiative from adiabatic shocks. ∼ The fact that this falls well short of observed peak brightness temperatures (Fig. 3-4) again disfavors thermal models for the early radio peak.. For non-thermal emission from adiabatic shocks, the brightness temperature and its peak value are given, respectively, by

5+p p+3 p+5 p+1 nth 7 4 − 2 2 4 T = 1.2 10 n ν v e,−2H14 K (3.47) ν,H × 4,7 10 8 B,−2 p+1 5+p p+1 3−p nth 6 − 4 2 4 8 T = 1.4 10 ν v e,−2H . (3.48) ν,H,pk × 10 8 B,−2 14

3.5.2 Radio lightcurves and spectra

The shape of the radio light curve and spectrum are driven by the eﬀects of free-free opacity, which are independent of the emission mechanism and hence apply both to thermal [77] and non-thermal models. Figure 3-13 shows an example model light curve (left panel) and corresponding evolution of the spectral index across two representative frequency ranges (right panel). At high frequencies the light curve peaks earlier and reaches a larger maximum

−2 ﬂux, due to the frequency dependence of free-free absorption, αﬀ ν , ∝ 131

106

H14=0.1

H14=0.5 , K , pk T

H14=2.2

H14=10 105 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

fEUV

1010 -4 E=10 -3 E=10 -2 E=10 109 E=0.1 , K 108 , pk T

107

106 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Shock velocity, 108 cm s-1

Figure 3-11: Peak brightness temperature of non-thermal emission as func- nth tion of shock parameters. The top panel shows Tν,pk as a function of the fraction of shock energy used for ionization fEUV and density scaleheight H. nth The bottom panel shows Tν,pk as a function of the shock velocity, v8, and fraction of the shock energy placed into relativistic electrons, e. Results of the full calculation are shown as symbols, while the analytic estimates from equation (3.40) are shown as lines. The values of parameters not varied in these ﬁgures are taken as best-ﬁt values for V1324 Sco (Table 3.4). 132

4.5 H14=0.1 H =0.3 4 14 H14=0.8 H14=2.2 3.5 K 4

, 10 3 , pk T 2.5

1.5 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Shock velocity, 108 cm s-1

Figure 3-12: Same as Figure 3-11 but showing the observed thermal brightness temperature as a function of the shock velocity v8 and scale-height H14. Results of our full calculation are shown as symbols, while the analytic estimates from equation (3.41) are shown as lines. 133

4.5 10 Max of 3 GHz 4 Max of 10 GHz 3.5 Max of 30 GHz 3 1 2.5

2 , mJy F 1.5 Spectral index 0.1 1

0.5

0.01 -0.5 30 40 50 60 70 80 90 100 110 120 30 40 50 60 70 80 90 100 110 120 Time from optical outburst, days Time from optical outburst, days

Figure 3-13: Left: Example model radio lightcurves at radio frequencies of 3 GHz (black), 10 GHz (blue), and 30 GHz (red), calculated for our best- fit parameters for V1324 Sco (Table 3.4). The lightcurve does not include thermal emission from the cold central shell or other sources of photo-ionized ejecta. Right: Measured spectral index β between the 3 and 10 GHz bands (red) and the 10 and 30 GHz bands (blue). The time of light curve maxima are marked with symbols for 3 GHz (circle), 10 GHz (square), and 30 GHz (triangle). The plateau at β = 2 at early times corresponds to when the photo-ionized layer ahead of the shock is still optically thick. and the monotonically declining shock power. When such behaviour is not observed, this indicates that our one zone model is inadequate or that free- free opacity does not determine the light curve maximum ( 3.6.3). § The spectral index β initially rises to exceed that of optically-thick isothermal gas (β > 2) because the higher frequency emission peaks first. Then, once the lower frequency emission peaks, the shock becomes optically thin and hence the spectral index approaches the spectral index of optically thin synchrotron emission. Importantly, a flat spectral index does not itself provide conclusive evidence for non-thermal synchrotron emission, even though the optically thin spectral indices are different for synchrotron and thermal bremsstrahlung emission. The emission mechanism is instead more accu- rately distinguished from thermal emission based on the higher peak brightness temperatures which can be achieved by non-thermal models. Also, at late times the radio emission should eventually come to be dominated by 134

10 10 36.5 GHz 27.5 GHz 17.5 GHz 13.3 GHz 1 7.4 GHz 4.6 GHz 1 0.1 , mJy , mJy 36.5 GHz F F 28.2 GHz 25.6 GHz 0.01 20.1 GHz 8.7 GHz 6.8 GHz 5.2 GHz 0.1 0.001 20 40 60 80 100 120 140 20 40 60 80 100 120 140 Time from optical outburst, days Time from optical outburst, days

10 32.1 GHz 14.0 GHz 7.8 GHz 4.8 GHz 1.9 GHz 1.3 GHz 1 , mJy F

0.1 20 40 60 80 100 120 140 160 Time from optical outburst, days

Figure 3-14: Synchrotron shock models fit to V1324 Sco (top left panel), V1723 Aql (right panel) and V5589 Sgr (bottom left panel). Parameters of the fits are provided in Table 3.4. In all cases the late-time thermal emission has been subtracted prior to the fit. Uncertainties are cited in the caption of Fig. 3-3. thermal emission of the photo-ionized ejecta and hence the spectral index will again rise to β = 2. This feature is not captured by Figure 3-13 because we have not included thermal emission from the cool central shell or other sources of photo-ionized ejecta.

3.5.3 Fits to Individual Novae

We ﬁt our model to the radio light curves of three novae with early-time coverage, V1324 Sco, V1723 Aql and V5589 Sgr (Figs. 3-3, 3-4). We employ a χ- squared minimization technique across 9 free parameters: v1, v4, n4/n1, ∆t, H, n0, p, e, fEUV .

We assume a shock covering fraction of fΩ = 1 and take B = 0.01, as the 135

Table 3.4: Best-ﬁt parameters of synchrotron model.

(a) (b) (c) (d) (e) (f) (g) (h) (i) Nova v1 v4 n4/n1 v˜f ∆t H n0 e fEUV (km s−1) (km s−1) - (km s−1) (days) (cm) (cm−3)-- V1324 Sco 1700 670 0.43 630 12 3.7 × 1013 7.8 × 109 0.081 0.051 V1723 Aql 2300 770 0.66 830 3.2 9.5 × 1013 6.7 × 107 0.015 0.31 V5589 Sgr 2200 190 2.3 810 5 6.8 × 1013 4.1 × 108 0.014 0.07

(a)velocity of fast outflow, (b)velocity of slow outflow (DES), (c)ratio of densities of DES and fast outflow, (d)velocity of (e) the shock in the frame of the upstream gas, calculated from v1, v4, n4/n1 using equation (3.3), time delay between launching fast and slow outflows, (f)scale-height of slow outflow, (g)normalization of density profile of slow outflow (eq. [3.7]), (h)fraction of shock power placed into power-law relativistic electrons/positrons, (i)fraction of shock power placed into hydrogen-ionizing radiation (eq. [3.5])

latter is degenerate with e. Table 3.4 provides the best-fit parameters of each novae. Although the electron power-law index is a free parameter, in practice it always converges to a best-fit value of p 2. ' Our best fit to V1324 Sco is shown in the upper panel of Figure 3-14. The best-fit shock velocity of 640 km s−1 is within the range of radiative shocks, consistent with the rapid post-maximum decline. For the parameters of our best-fit model, npk,∆/npk,H = 0.96ν10, implying that for frequencies above(below) 10 GHz the thickness of ionization layer is less than(greater than) the scale height at the time of peak flux. The transition between these two regimes can be seen as a small break in the 7.4 GHz light curve around day 68. Substituting our best-fit parameters into equation (3.1), we find that the collision occurred 20 days after the start of the outflow, corresponding to 7 days after the onset of the gamma-ray emission [90]. This discrepancy is not necessarily worrisome, as the colliding ‘shells’ may be ejected over days or longer. V1723 Aql was more challenging to fit (Fig.3-14, middle panel), mainly because different frequencies peak at nearly the same time. This is contrary to the the expectation that high frequencies will peak first if free-free absorption indeed controls the light curve rise, as our model assumes (see 3.6.3 for § alternative interpretations). Although we cannot fit V1723 Aql in detail, we can nevertheless reproduce the magnitudes of the peak fluxes for reasonable 136 parameters. Our best-fit model for V5589 Sgr is shown in the bottom panel of Figure 3-14, although the data is more sparse than for the other two events.

3.6 Discussion

3.6.1 Eﬃciency of Relativistic Particle Acceleration

The non-thermal radio emission from novae probes relativistic particle acceleration and magnetic field amplification at non-relativistic shocks, as pa- rameterized through e and B. Considering all three novae, we infer electron acceleration efficiencies in the range e 0.01 0.08 for B = 0.01 (Table ∼ − 3.4). However, the peak radio luminosity is largely degenerate in e, B and v˜f . By combining the peak 10 GHz luminosity of V1324 Sco, V1723 Aql, and V5589 Sgr with analytic estimates of the peak brightness temperature (eq. [3.43]) and assuming a shock radius equal to that of our best fit model

(Rsh = v4tpeak; Table 3.4), we obtain the following limits,

7.9 3/4 −5 −4 fΩv e 7 10 1.2 10 , (3.49) 8 B ≈ × − × where we have again assumed radiative shocks (v8 < 1). ∼ Now consider some implications of these constraints. Making the very 3/4 conservative assumption that e + B < 1, i.e. eB < 0.5, we ﬁnd ∼ ∼

−0.13 −0.25 v8 > 0.4fΩ kT > 0.2fΩ keV. (3.50) ∼ ⇒ ∼

This lower limit on the shock temperature ensures, for example, that thermal free-free emission from the shocks will fall within the spectral window of Swift and Chandra. The radio emitting electrons should produce a measurable X- ray signature, even if its too weak to detect or is overpowered by faster 137 adiabatic shocks.

Next, using our best ﬁt parameters for the shock velocities v8 of each nova (Table 3.4) and assuming an error on this quantity of 10 per cent, each event separately provides a constraint

−3 3/4 −3 1.2 10 fΩe 5.4 10 V1324 Sco (3.51) × ≤ B ≤ × −4 3/4 −3 2.3 10 fΩe 10 V1723 Aql (3.52) × ≤ B ≤ −4 3/4 −4 2 10 fΩe 9 10 V5589 Sgr (3.53) × ≤ B ≤ ×

−3 −2 For physical values of B < 0.1 [114], we thus require e > 10 10 . ∼ − Such high acceleration eﬃciencies disfavor hadronic scenarios for the

−4 radio-emitting leptons, which are estimated to produce e < 10 (eq. [3.27]). ∼ They are also in tension with PIC simulations of particle acceleration at

−4 non-relativistic shocks [251, 252] which find e 10 when extrapolated to ∼ 3 −1 shock velocities v8 < 10 km s , modeling of supernova remnants ([116]; −4 ∼ −3 e 10 ), and galactic cosmic ray emission ([255]; e 10 ). On the ∼ ∼ other hand, the inferred cosmic ray efficiencies are dependent on the shock fraction of the accelerated electrons which escape the supernova remnant. Modeling of unresolved younger radio supernovae typically find higher values of e [256, 257], consistent with our results. Given these observational and theoretical uncertainties, we tentatively favor a leptonic source for the radio-emitting electrons. In the above, we have assumed that the shocks cover a large fraction of the outflow surface (fΩ 1). However, if instead we have fΩ 1, then the ∼ required values of e and B would be even higher than their already strained values. If the radio-emitting shocks are radiative, they must therefore possess a large covering fraction fΩ 1. ∼ Lower values of e and B are allowed if the radio emission is instead 138

dominated by high-velocity v8 > 1, adiabatic shocks covering a large solid ∼ fraction fΩ 1. The sensitive dependence of the brightness temperature on ∼ the shock velocity implies that even a moderate increase inv ˜f can increase the radio flux by orders of magnitude for fixed e, B. On the other hand, the adiabatic shocks responsible for the hard X-rays appear to require a small covering fraction fΩ 1 so as not to overproduce the observed X-ray luminosities ( 3.2.4). § −2 In V1324 Sco, [79] place a lower limit of nth = p + e > 10 on the total fraction of the shock energy placed into non-thermal particles. Assuming that the microphysical parameters of the radio and gamma-ray emitting shocks are identical, and that the gamma-rays are leptonic in origin (nth = e ∼ 0.01 0.1), then by combining this constraint with our constraints on V1324 − Sco from equation (3.51), we find

−3 −2 3 10 B 2 10 V1324 Sco, (3.54) × ≤ ≤ × providing evidence for magnetic ﬁeld ampliﬁcation.

3.6.2 The DES is not a MS progenitor wind

Gamma-rays are observed not only in novae with red giant companions (symbiotic novae), but also in systems with main-sequence companions (classical novae). Could the DES required for shock-powered radio and gamma-ray emission be the companion stellar wind? In this section we estimate the radio emission from the fast nova ejecta interacting with the (assumed stationary) stellar wind of the binary companion. The density proﬁle of a spherically symmetric, steady-state wind is given by

˙ ˙ M M−10 −3 nw 2 = 300 2 cm , (3.55) ' 4πr vwmp r14vw,8 139

14 8 −1 where r = r1410 cm is radius, vw = vw,810 cm s is the wind speed, and ˙ ˙ −10 −1 M = M−1010 M yr is the mass-loss rate normalized to one speciﬁc for main-sequence stars. The density and radius of the radio photosphere are given by equating

(3.55) with npk,H (eq. [3.12]) for a density scaleheight H r, ∼

8 4/3 1/3 ˙ −1/3 nw,pk = 1.4 10 ν v M (3.56) × 10 w,8 −10 11 − −2/3 ˙ 2/3 rw,pk = 1.5 10 ν 2/3v M (3.57) × 10 w,8 −10

Although nw,pk is several orders larger than in our ﬁducial DES models, the radius of the photosphere is 3 orders of magnitude smaller. Under these conditions the shock is adiabatic and hence the peak brightness temperature is found by substituting nw,pk and rw,pk into equation (3.47), which gives (for p = 2)

6 ˙ 1/12 3.5 3/4 −5/6 −1/12 Tw,pk = 1.7 10 M v e,−2ν v K (3.58) × −10 8 B,−2 10 w,8

6 Although the value of Tw,pk 10 K is comparable to those of observed radio ∼ emission, the much smaller radius of the photosphere rw,pk compared to that in our ﬁducial DES models of 1014 1015 cm would result in a peak radio ∼ − 2 ﬂux r Tw,pk which is approximately 8 orders of magnitude smaller than ∝ w,pk the observed values. Radio emission from the interaction of the nova ejecta with the main sequence progenitor wind is thus undetectable for physical values of M˙ , unless the mass loss rate is comparable to that of the nova eruption itself. 140

3.6.3 Light Curve of V1723 Aquila

Our model does not provide a satisfactory ﬁt to the light curve of V1723 Aql because all radio frequencies peak nearly simultaneously (Fig. 3-14), contrary to the expectation if the light curve peaks due to the decreasing free- free optical depth ( 3.5.3). Here we describe modiﬁcations to the standard § picture that could potentially account for this behavior.

Steep density gradient

The V1723 Aql light curves are consistent with all frequencies peaking within a time interval ∆tpk < few days. From equation (3.13) this requires a scale ∼ height of thickness

13 ∆tpk H < ∆tpkv˜f = 2.6 10 v8 cm, (3.59) ∼ × 3 day

14 15 which is much smaller than the shock radius of Rsh 10 10 cm. Such ∼ − a steep gradient should also result in a short rise and fall time, which is inconsistent with the relatively broad peak observed in V1723 Aql. This contradiction could be alleviated if the shock reaches the critical density of peak emission n = npk at different times across different parts of the ejecta surface. In this case, the total light curve, comprised of the emission from all locations, would be “smeared out” in time while still maintaining the nearly frequency-independent peak time set by the steep gradient. Figure 3-15 shows the light curves of our single-zone model (for an assumed scale height of H = 5 1012 cm) convolved with a Gaussian profile of × different widths σ, in order to crudely mimic the effects of non-simultaneous shock emergence. The top panel shows the unaltered radio lightcurve, i.e. assuming simultaneous shock emergence, which again makes clear that the high frequencies peak first ( 3.2.2). The middle and bottom panels show the § 141

3 GHz 3.0 GHz 10 GHz 1 10.0 GHz 30 GHz 30.0 GHz 1

0.1 , mJy , mJy F F 0.1

0.01

0.01 15 20 25 30 35 40 15 20 25 30 35 40 Time from optical outburst, days Time from optical outburst, days

0.1 , mJy F

3 GHz 10 GHz 30 GHz 0.01 15 20 25 30 35 40 Time from optical outburst, days

Figure 3-15: Example light curves (with late-time thermal emission subtracted) for a steep density gradient H = 5 1012 cm, which has been convolved with a Gaussian of different widths ×σ = 0 (top left panel), σ = 2 days (right panel), and σ = 5 days (bottom left panel). This illustrates how a shock propagating down a steep density profile, but reaching the DES surface at different times at different locations across the ejecta surface, can produce the appearance of an achromatic light curve. same light curve, but smeared over time intervals of σ = 2 days and σ = 5 days, respectively. In this way, the difference between the times of maximum at different frequencies can be substantially smaller than the rise or fall time, as observed in V1723 Aql.

Collision With Optically Thin Shell

Another possibility is that the radio-producing shock occurs in a dilute thin shell, which has a lower density than the mean density of the DES, i.e.

7 −1 n0 npk 10 cm . Absorption then plays no part in the lightcurve rise, ∼ 142 which is instead controlled by the time required for the shock to cross the radial width of the shell.

14 The free-free optical depth of a shell of radial thickness D = 10 D14 cm 4 and temperature Tsh = 2 10 K is given by ·

2 −2 τsh = 3.5n4,7D14ν10 , (3.60)

The density and radiative eﬃciency of the shell can be written as

6 1/2 −1/2 −3 n4 = 1.7 10 τ D ν10 cm (3.61) × sh,−1 14 tcool 4.4 −1/2 −1/2 −1 η = 3.1v8 τsh,−1D14 ν10 , (3.62) ≡ tsh where τsh = 0.1τsh,−1 is normalized to a sufficiently low value so as not to influence the radio lightcurve significantly.

Although the parameter space for an optically thin shell (τsh < 0.1) which is radiative (η < 1) is not large, this represents a viable explanation for the ∼ behavior of V1723 Aquila.

3.6.4 Radio Emission from Polar Adiabatic Shocks

Figure 3-6 shows the maximum X-ray luminosity and the maximum 10 GHz radio luminosity as a function of the shock velocity, spanning the range from radiative (v8 < 1) to adiabatic (v8 > 1) shocks. Both radio and X-ray lumi- ∼ ∼ nosities increase by many orders of magnitude across this range. However, their ratio LR/LX coincidentally varies only weakly with v8, as shown in Fig- ure 3-16. Absolute luminosities depend on the covering fraction and radius of the shock, but this ratio is obviously independent of these uncertain quantities. Reasonable variations in the density scaleheight H, ionization fraction fEUV, and the CNO abundances result in a factor < 3 variation in this ratio ∼ 143

(Fig. 3-16, top panel), which is instead most sensitive to the microphysical 3/4 parameters, e for p = 2 (Fig. 3-16, bottom panel). Any shock produc- ∝ B ing X-rays will therefore also produce radio emission of intensity comparable to the observed values. Fig. 3-16 shows for comparison the measured value of

LR/LX, or limits on its value, using the observed peak X-ray and non-thermal radio luminosities for ﬁve novae.

The LR/LX ratio can deviate from its value shown in Fig. 3-16 if the density of the shocked shell is suﬃciently low that the optical depth of the

DES to radio or X-rays is < 1, i.e. if its central density obeysn ¯ nX, npk. ∼ The ratio of the density of maximum X-ray emission nX (eq. [3.22]) to that of maximum radio emission npk (eq. [3.12]) is given by

nX −1 −1/2 4 1.2ν10 H14 v8 (3.63) npk ≈

For high velocity v8 1 adiabatic shocks we can therefore have npk < n¯ < ∼ ∼ nX , in which case the X-ray emission is optically thin at peak and hence the ratio LR/LX will be higher than its value in Figure 3-16. Likewise, when n¯ < npk, nX (both X-rays and radio are optically thin at peak), we have that ∼ −1/4 LR/LX n¯ also increases with decreasingn ¯. Thus, the LR/LX ratio in ∝ Fig. 3-16 is a conservative minimum, achieved in the limit of a high density DES. The early-time X-rays in some novae are too hard to originate from radiative shocks (Table 3.3, Figure 3-5) and thus could instead originate from fast adiabatic shocks in the low density polar regions (Fig. 3-1; 3.2.4). § If all the observed X-rays originate from fast polar shocks, and all radio emission from slower equatorial radiative shocks, then the measured ratio

LR/LX = LR,rad/LX + LR,ad/LX, comprised of contributions from both radiative and adiabatic shocks, should exceed the ratio shown in Figure 3-16. 144

Figure 3-16: Ratio of nonthermal peak 10 GHz radio luminosity (eq. [3.44]) to peak thermal X-ray luminosity (eq. [3.20]) as a function of shock velocity vsh. The X-ray luminosity is calculated for the preshock density n4 = nX (eq. [3.22]), the latter calculated using the bound-free cross sections from [248] for our assumed ejecta composition. The top panel shows that the ratio of luminosities varies by only a factor of a few for realistic variations in the density scale-height H, the ionized fraction fEUV, or if the CNO mass fraction is doubled. The bottom panel shows the more sensitive dependence of the luminosity ratio on the shock microphysical parameters e and B, calculated for H14 = 1, fEUV = 0.1, and standard CNO abundances. Shown for comparison are measurements (squares), or limits (triangles) on the luminosity ratio of the novae compiled in Table 3.3. 145

3/4 −3 For V1324 Sco, V1723 Aql, V5589 Sgr this requires that e 10 , B ≈ consistent with our best-ﬁt values found earlier. The non-detections of non- thermal radio emission in V339 Del and V959 Mon also do not contradict

3/4 −3 e 10 . We conclude that, within the uncertainties, the thermal X-ray B ≈ and non-thermal radio emission in novae is consistent with originating from distinct shocks.

3.7 Conclusions

We have explored non-thermal synchrotron radio emission from radiative shocks as a model for the early radio peaks in novae by means of a one- dimensional model. Broadly speaking, we find that the measured brightness temperatures can be explained for physically reasonable parameters of the shock velocity and microphysical parameters e and B (Table 3.4). The presence of a detectable early radio peak requires the presence of DES of density > 106 107 cm−3 on a radial scale of 1014 1015 cm from the ∼ − ≈ − white dwarf. The emission from the nova ejecta colliding with a progenitor ˙ −9 −1 wind of mass loss rate M < 10 M yr is not sufficient to explain the ∼ observed radio peaks ( 3.6.2). § The thin photo-ionized layer ahead of the forward shock plays a key role in our model, as its free-free optical depth determines the time, intensity and spectral indices near the peak of the emission (see 3.2.2). One-dimensional § models robustly predict that higher frequency emission peaks at earlier times, imprinting a distinct evolution on the spectral index (Figure 3-13) which is independent of whether the emission mechanism is thermal or non-thermal. V1324 Sco and V5589 Sgr do exhibit this behavior, but in V1723 Aql the light curves at all frequencies peak at nearly the same time. Possible explanations include the nova outflow colliding with a shell of much lower density than the 146 mean density of the nova ejecta ( 3.6.3). Alternatively, the evolution may § appear achromatic if the shock breaks out of the DES at different times across different parts of its surface, as would occur if the DES is not spherically symmetric (Fig. 3-15). Because the light curve evolution is controlled by free-free opacity effects, one cannot readily distinguish thermal from non-thermal emission based on the spectral index evolution alone. Thermal emission models exhibit spectral index behaviour [77] which is very similar to the non-thermal model described here (Figure 3-13). Non-thermal emission is best distinguished by the much higher brightness temperatures which can be achieved than for thermal emission. The relativistic electrons (or positrons) responsible for the non-thermal synchrotron emission originate either from direct diffusive acceleration at the shock (leptonic scenario), or as secondary products from the decays of pions produced in proton-proton collisions (hadronic scenario). In either case, the radio-emitting leptons are identical, or tightly related to, the particles which power the observed γ-ray emission. For instance, if the observed > 100 ∼ MeV gamma-rays are produced by relativistic bremsstrahlung emission (as favored in leptonic scenario by [79]), then the energy of the radiating electrons (also > 100 MeV) are very close to those which determine the peak of the ∼ radio synchrotron emission at later times (eq. [3.26]). Likewise, gamma-rays from π0 decay in hadronic scenarios are accompanied by electron/positron pairs with energies > 100 MeV in the same range, as determined by the ∼ pion rest mass. Once produced, relativistic leptons evolve approximately adiabatically downstream of the shock for conditions which characterize the radio maximum. The only possible exception is Coulomb cooling in cases when free-free emission dominates lines in cooling the post-shock thermal gas (Fig. 3-10). 147

In hadronic scenarios, only a small fraction of relativistic protons produce pions in the downstream before being advected into the central cold shell (from which radio emission is heavily attenuated by free-free absorption). Relativistic protons therefore provide considerable pressure support in the post-shock cooling layer, preventing the gas from compressing by more than an order of magnitude ( 3.4.1). The possible role of clumping due to § thermal instability of the radiative shock on this non-thermal pressure support deserves further attention, as the central shell may provide a shielded environment which is conducive to dust and molecule formation in novae [84]. The small number of pions produced in the rapidly-cooling post-shock layer implies that of the < 10% of the total shock power placed into rel-

∼ −4 ± ativistic protons, only a tiny fraction e 10 (eq. [3.27]) goes into e ∼ pairs capable of producing detectable radio emission. For physical values of

3/4 −5 B < 0.1, the critical product eB < 10 which controls the brightness ∼ ∼ temperature is a few orders of magnitude below that required by data of 10−4 10−3 (eq. [3.51-3.53]). We therefore disfavor the hadronic scenario ∼ − for radio-producing leptons. This does not rule out a hadronic origin for the γ-ray emission, but it does suggest that the direct acceleration of electrons is occurring at the gamma-ray producing shocks. 3/4 Within leptonic scenarios, the values of eB required by the radio data are typically higher than those measured from PIC simulations [113] and by

3/4 −4 modeling Galactic SN remnants [116]. However, values of e 10 B ∼ − 10−3 do appear similar to those inferred by modeling young radio supernovae (e.g. [257]). We confirm the finding of [77] that when CIE line cooling is included, thermal emission from radiative shocks cannot explain the early high brightness temperature radio peak. Perhaps the best evidence for non-thermal radio emission comes from cases like V1324 Sco, where no X-ray emission 148 is detected. An X-ray non-detection implies a low velocity, radiative shock, which would be especially challenged to produce a significant radio flux in the early peak without a non-thermal contribution. We can also rule out thermal emission from an adiabatic shock, at least under the assumption that only the first scaleheight behind the shock dominates that contributing to the observed emission ( 3.6.4). Future radiation hydrodynamical simu- § lations of adiabatic shocks and their radio emission in the case of a steep density gradient are needed to determine whether thermal adiabatic shocks can be completely ruled out. We have derived analytic expressions for the peak brightness temperature of non-thermal synchrotron emission (eq. [3.44]). Our model can reproduce the observed fluxes of V5589 Sgr, V1723 Aql, V1324 Sco and, in the case of V1324 Sco, details of the radio lightcurve. Non-detections of early radio peak from V959 Mon and V339 Del place upper limits on e, B. The upper limits for V339 Del and V959 Mon are broadly consistent with the inferred values of e, B values for novae with detected early radio peak. In summary, radio observations of novae provide a important tool for studying particle acceleration and magnetic field amplification in shocks which is complementary to γ-rays observations. They also inform our under- standing of the structure of the nova ejecta and internal shocks, which may vary considerably with time and as a function of polar angle relative to the binary axis.

Acknowledgements

We thank Laura Chomiuk, Tom Finzell, Yuri Levin, Koji Mukai, Michael Ru- pen, Jeno Sokoloski, and Jennifer Weston for helpful conversations. We thank Jeno Sokoloski for providing us with the numerical tools to ﬁt the thermal 149 radio emission. ADV and BDM gratefully acknowledge support from NASA grants NNX15AU77G (Fermi), NNX15AR47G (Swift), and NNX16AB30G (ATP), and NSF grant AST-1410950, and the Alfred P. Sloan Foundation. 150

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Appendix: Non-Thermal Electron Cooling in the Post-Shock Cooling Layer

In calculating the radio synchrotron emission from nova shocks, we assume that relativistic electrons and e± pairs evolve adiabatically in the post-shock cooling layer. This assumption is justiﬁed here by comparing various sources of cooling (Coulomb, bremsstrahlung, inverse Compton) to that of the background thermal plasma, tcool (eq. [3.14]). We again focus on electrons or positrons with Lorentz factors of γpk 200 (eq. [3.26]), as these determine ∼ the synchrotron ﬂux near the light curve peak, where n3 = 4n4 = 4npk. First note that thermal electrons and protons are well coupled behind the shock by Coulomb collisions and thus share a common temperature. At radio maximum (n4 = npk,∆) the Coulomb equilibration timescale for the thermal 3/2 plasma, te−p = 3.5T3 /npk s ([258]), is short compared to the thermal cooling timescale tcool (eq. [3.14]),

−0.2 te−p T −0.4 0.03 v8 . (A.1) tcool ≈ T3

This ratio evolves only weakly as gas cools behind the shock. Relativistic electrons experience Coulomb losses on thermal background 182 electrons on a timescale which is given by

−1 te−e 2γpk −3.4 γpk ln Λ = = 10 v8 . (A.2) tcool 3cσTn3 ln Λ tcool 200 20

Coulomb losses can thus be potentially important behind the shock where n npk. The cooling rate due to relativistic bremsstrahlung emission from interaction with thermal protons and electrons is well approximated byγ ˙ rb ≈ 1.2 (5/3)cσTαfsn3γ , an expression which is accurate to 10 20% between ∼ − 3 γ 10 10 , where αfs 1/137 is the ﬁne structure constant. This yields ∼ − '

−0.2 trb γpk 3 −3.4 γpk = 0.2 45 v8 . (A.3) tcool γ˙ rb tcool ' 5cσTαfsn3γpk tcool ≈ 200

2 The ratio of the synchrotron cooling timescale tsyn = 6πmec/(σT Bshγ) for electrons with γ = γpk to the thermal cooling time is given by

−1 tsyn −1 −5.4 γpk 100 B,−2v8 . (A.4) tcool ≈ 200

Finally, the characteristic timescale for inverse Compton cooling on the shock-produced thermal X-rays is tIC = 3mec/(4σT Uradγpk), such that

−1 tIC 3mec 3 −6.4 γpk = 1.7 10 v8 , (A.5) tcool 4σT Uradγpktcool ≈ × 200

5 where Urad = (3/8)(˜vf /c)n3kT3/µ is the X-ray energy density. In summary, relativistic radio-emitting leptons cool in the post-shock gas on a timescale which is generally much longer than that of the thermal background plasma for the range of velocities v8 < 1 of radiative shocks. ∼ This justiﬁes evolving their energies adiabatically behind the shock, with the

5Soft X-rays are in the Thomson regime for interacting with γ 100 electrons. ∼ 183 possible exception of Coulomb losses, which are unimportant immediately behind the shock but may become so as gas compresses to higher densities n n3.