Exploration of the Interaction of Type Ia Supernovae with the Circumstellar Environment Paul Dragulin

Florida State University Libraries

2015 Exploration of the Interaction of Type Ia Supernovae with the Circumstellar Environment Paul Dragulin

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COLLEGE OF ARTS AND SCIENCES

EXPLORATION OF THE INTERACTION OF TYPE IA SUPERNOVAE WITH THE

CIRCUMSTELLAR ENVIRONMENT

PAUL DRAGULIN

A Dissertation submitted to the Department of Physics in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

2015

Paul Dragulin defended this dissertation on November 3, 2015. The members of the supervisory committee were:

Peter Hoeﬂich Professor Directing Dissertation

Wei Yang University Representative

Kevin Huﬀenberger Committee Member

Winston Roberts Committee Member

Joseph Owens Committee Member

The Graduate School has veriﬁed and approved the above-named committee members, and certiﬁes that the dissertation has been approved in accordance with university requirements.

ii ACKNOWLEDGMENTS

I would like to thank everyone who make this possible for me, and especially my parents Victor and Rodica Dragulin, whom I could always count on for their support and encouragement. This dissertation is in their honor. I would also like to thank my adviser Dr. Peter Hoeﬂich for his guidance and support, without which I would be no where. I would also like to thank Jesus Christ the sin-pardoning Saviour who did nothing less than carry me throughout this whole journey. My aim in life and particularly with this PhD is to bring glory to God and contribute to the real upliftment of humanity. The work presented in this dissertation has been supported by the NSF projects AST-1008962, “Interaction of Type Ia Supernovae with their Environment”, and AST-0708855, “Three-Dimensional Simulations of Type Ia Supernovae: Constraining Models with Observations”.

iii TABLE OF CONTENTS

ListofTables...... vi ListofFigures ...... viii List of Abbreviations ...... xiv Abstract...... xv

1 Introduction 1 1.1 SN Ia and Possible Progenitors ...... 1 1.2 SN Ia Explosion Scenarios ...... 9 1.3 Which Progenitor and Explosion Scenarios are responsible for SN Ia? ...... 10 1.4 CluesfromtheEnvironmentsofSNIa ...... 14 1.5 Theoretical Background: the Π Theorem and Self-Similar Solutions ...... 16 1.5.1 Buckingham’s Π Theorem ...... 16 1.5.2 Application of the Π Theorem: Self-Similar Solutions ...... 19 1.6 Application of Self-Similar Solutions to SNe Ia ...... 19

2 Theory and Assumptions 21 2.1 Self-Similar Solutions for Power-law ISM Density Proﬁle ...... 21 2.2 Self-Similar Solution for s=0 and Large Mach Numbers: the Zero Ambient Pressure (ZAP)Model ...... 24 2.3 Self-Similar Solutions for s=2 ...... 25

3 Boundary Conditions for s=0 29 3.1 Limitations of the ZAP Model ...... 29 3.2 TheCAPModel ...... 32

4 The Existence of Solutions 39 4.1 Case s =0...... 39 4.2 Case s =2...... 40

5 Time series of Solutions 45 5.1 s=0TimeSeries ...... 45 5.2 s=2TimeSeries ...... 47

6 Applications: Environments of Type Ia Supernova Progenitors 52 6.1 ParameterizedStudy...... 53

7 Results 55 7.1 CaseI:ConstantISMDensity...... 55 7.1.1 Case Ia: Fast Winds from an Accretion Disk ...... 55 7.1.2 Case Ib: MS Star Winds ...... 57 7.1.3 CaseIc:Slow,RG-likeWinds...... 57

iv 7.1.4 Case Id: Fast Wind from an Accretion Disk Combined with Mass Loss from RG Donor Star or Super-Eddington Accretion ...... 59 7.1.5 ExtremelyLowDensityEnvironments ...... 60 7.2 Case II: Environments Produced by Winds from Prior Mass Loss ...... 60 7.2.1 Case IIa: Fast Wind from an Accretion Disk and a Non-RG Donor Star . . . 60 7.2.2 Case IIb: Fast Wind from an Accretion Disk Combines with a “RG-like” Wind 61

8 Application to SN2014J and other SN Ia 69 8.1 Background of SN2014J ...... 69 8.2 Analysis of SN2014J ...... 71 8.3 Analysis of Other SNe Ia ...... 76 8.3.1 PTF 11kx ...... 77 8.3.2 SN2011fe ...... 79 8.3.3 SN2007le ...... 80 8.3.4 SN2006X ...... 81 8.3.5 SN without Time-Varying Narrow Lines ...... 82 8.3.6 A Collection of SNe Ia Spectra ...... 83 8.3.7 Sternberg et al. Analysis ...... 85

9 Conclusion 98

Appendices A SPICE 104 A.1 SPICEDescription ...... 104 A.2 README for SPICE ...... 105 A.2.1 File Glossary ...... 105 A.2.2 System Requirements and Compilation ...... 107 A.2.3 RunningSPICE ...... 107 A.2.4 Understanding SPICE ...... 111

B Additional Time Series for s=0 127

C Lookup Table (s=0) 129

D Lookup Table (s=2) 136

Bibliography ...... 144 BiographicalSketch ...... 166

v LIST OF TABLES

3.1 For constant interstellar medium (ISM) and t . tp, the relations for the radius of the contact discontinuity RC , the forward and reverse shock, R1 and R2, the ﬂuid velocity at the shell uc and its mass column density τm, and the density of the inner void, n2, all as a function of the density of the environment n0, the mass lossm ˙ , its wind velocity v , and duration t. For n , I assumed (R /R )3 1 (see Eq. 2.21). w 2 2 C ≪ Asymptotic (t ) values of the reverse shock R ∞, and the particle density n ∞ → ∞ 2, 2, are obtained from the CAP model and given below. p0 is the ambient pressure. . . . . 30

4.1 Same as Table 3.1 but for an environment produced by wind from prior mass loss (s=2). The index 1 corresponds to the prior wind. The relations are valid for high velocity winds running into environments produced by low velocity winds (see Fig. −4 4.1). For RC , R1, and u1, the relations are valid form/ ˙ m˙ 1 . 10 . For R2 and ρ2, it is valid form/ ˙ m˙ . 10−4 and v /v between 10 and 170. τ is calculated by 1 w w,1 ∼ m formal integration in SPICE...... 42

7.1 Interaction of winds of an accretion disk (AD) with a constant ISM (s=0). Numerical results are given for parameters typical for those winds for the CAP model (see text and Table 3.1). In addition, I give the velocity dispersion of the shell σu and the optical depth τ and equivalent width EW of the NaID line. The reference model for this category of models is marked by ...... 63 ∗ 7.2 Same as Table 7.1 but for an “RG-like” wind. The reference model for this category of models is marked by ...... 64 ∗ 7.3 Same as Table 7.1 but for the combination of an AD- and RG-like wind (see text). . 65

7.4 Same as Table 7.1 but for a “MS-like” wind for the solution including the ambient pressure (CAP models). My reference model is marked by +. Values are given for the radius of the contact discontinuity and the inner shock using parameters typical for the wind of a main sequence star, and using the solution including the ambient pressure (CAP models). For times (t) much larger than tp, the location of the reversed shock becomes stationary and close to the distance R2,PE expected from equilibrating the pressure at the reversed shock with the ambient medium. In addition, models without outer pressure are given and marked by ’Rem.’. This shows the importance of the pressure term for t/t 1 in the extreme case. However, for winds relevant in p ≫ SNe, the diﬀerence is of the order of several % (see text, and Fig. 3.1)...... 66

7.5 Same as tables 7.1 and 7.4 but for very low density ISM typical for the galactic halo and elliptical galaxies...... 67

7.6 Interaction of winds with an environment produced by RG wind originating from the progenitor WD (vw,1 = 30 km/s, s = 2). The index 1 corresponds to the prior mass loss. Models 1-7 and 8-9 show the results for an AD-wind and a combination of a

vi AD and RG-like wind, respectively (Tables 7.1 & 7.3). The reference model for this category of models is marked by ...... 68 ∗

C.1 Lookup table for the s = 0 case. The outer Mach number M, K01, K0C , and K02, are here given numerically as a function of ΠT .ΠT = 2.50 corresponds to t = tp (see Chapter 3). The entire table is available in electronic form as part of the SPICE package...... 129

C.2 Continuation of Table C.1, for ΠT = 30.3 through ΠT = 9.5...... 130

C.3 Continuation of Table C.1, for ΠT = 9.2 through ΠT = 2.9...... 131

C.4 Continuation of Table C.1, for ΠT = 2.8 through ΠT = 0.87...... 132

C.5 Continuation of Table C.1, for ΠT = 0.84 through ΠT = 0.26...... 133

C.6 Continuation of Table C.1, for ΠT = 0.25 through ΠT = 0.080...... 134

C.7 Continuation of Table C.1, for ΠT = 0.077 through ΠT = 0.036...... 135

D.1 Lookup table for the s = 2 case. K21, K2C , and K22 are here given in numerical

form, as a function of the Π groups Πm˙ and Πvw (see Chapter 2). The entire table is available in electronic form as part of the SPICE package...... 136

D.2 Continuation of Table D.1, for Π = 6.27 and Π = 5.40...... 137 m˙ − m˙ − D.3 Continuation of Table D.1, for Π = 4.53 and Π = 3.67...... 138 m˙ − m˙ − D.4 Continuation of Table D.1, for Π = 2.80 and Π = 1.93...... 139 m˙ − m˙ − D.5 Continuation of Table D.1 for Π = 1.07 and Π = 0.20...... 140 m˙ − m˙ − D.6 Continuation of Table D.1, for Π = 0.67 and Π = 1.53...... 141 m˙ − m˙

D.7 Continuation of Table D.1, for Πm˙ = 2.40 and Πm˙ = 3.27...... 142

D.8 Continuation of Table D.1 for Πm˙ = 4.13 and Πm˙ = 5.00...... 143

vii LIST OF FIGURES

1.1 Artist depictions of the single degenerate (SD) scenario (left) and the double degenerate (DD) scenario (right). (CREDIT: (left) NASA/CXC/M. Weiss, (right) NASA/GSFC/T. Strohmayer) ...... 3

1.2 Plot of H-accretion rate vs. WD mass taken from [146]. For accretion rates lower than the solid line, degenerate H shells on the surface are thermally unstable and nova eruptions occur (vertical dashed lines give the maximum shell mass in M⊙). For accretion rates above the dotted dashed line, the accumulated envelope expands and produces a red giant star and emanates a strong wind. The dotted line gives the Eddington accretion rate. The regime for stable accretion is marked out in the center. 5

1.3 Figure 1 from a review by Wang and Han 2012 [232] illustrating possible evolutionary paths for a double MS binary system to eventually transition into a MS+WD binary system and a SN Ia. FGB stands for “first giant branch”, RLOF stands for “Roche lobe overflow,” CE means “common envelope,” EAGB signifies “early asymptotic giant branch,” and TPAGB stands for the “thermal pulsing asymptotic giant branch” stage. The black disks are degenerate stellar cores...... 6

1.4 Figure 2 from Wang and Han 2012 [232] illustrating possible evolutionary paths for a double MS binary system to eventually transition into a RG+WD binary system and a SN Ia. RLOF stands for “Roche lobe overﬂow,” CE means “common envelope,” and TPAGB stands for the “thermal pulsing asymptotic giant branch” stage...... 7

1.5 Similar to Fig. 1.2 but for He-accretion. The figure is taken from [252] using the numerical figures given in [144]: The steady He-shell burning upper limit is −6 (dM/dt) = 7.2 10 (M /M⊙ 0.6) M⊙/yr while the lower limit is 0.4(dM/dt) . RHe × WD − RHe Lower than that is an unstable burning regime prone to degenerate He shell flashes. −8 For accretion rates below 5 10 M⊙/yr, a He detonation of the WD can occur ∼ × and produce a “double detonation” SN ([121], see also text)...... 8

1.6 (left) Chandra and (right) Hubble Space Telescope (HST) images of SN1572 or Tycho’s Supernova. Tycho G, the companion star for SN1572 argued by [193], is pictured on the right. (CREDIT: NASA) ...... 11

1.7 The 400-year-old SN Ia remnant 0509-67.5, located in the LMC. Within the 1.4 ∼ arc-sec radius circle in the center is the 3σ conﬁdence region where any companion star from the progenitor system is likely to be. No stars are detected in the region down to 0.04 L⊙ in the visual. [200] argue that this strongly eliminates the possibility of a SD progenitor for this SN. This image is a composite of HST in B, V, I, Hα, and Chandra inX-rays...... 12

2.1 Velocity, pressure and density structure of a typical model shows four distinct regions dominated by the wind (I), the reversed shock (II), a shell (III), and the environment

viii (IV) both for constant (left) and wind (right) environments. In the figures of this work, density is shown in particle density n = ρNA/µ, where NA is Avogadro’s number and µ is the mean molecular weight. In this work, µ is set equal to 1. For different µ, ρ µρ inallequations...... 22 → 2.2 The scale free variables U, Ω, and P , as defined by Eqs. 2.5 - 2.8 plotted as a function of r/RC . These quantities are functions of r/RC only–these curves are invariant for all times during which the self-similar solution holds...... 24

2.3 Optical HST image of the Cat’s Eye planetary nebula, or NGC6543, representing the ending evolutionary stage of a Red Giant star. A few phases of mass loss are clearly evident. At the center, a fresh, axial-symmetric outburst 1000 yr old is ≈ seen expanding into an ambient medium, most likely produced by prior mass loss. (X-ray images of the inside region reveal hot structures of 106 K, most likely ∼ resulting from the interaction of diﬀerent mass loss phases.) On the far outside is a ﬁlamentary structure–the remnants of a much older mass loss episode that had broken into fragments due to instabilities and mixing...... 26

3.1 Structure feature comparisons of ZAP and CAP models for given sets of parameters as a function of t/tp for the reference models of the AD-, RG-like and MS wind (right to left). Dependence of the radii of the outer and inner shock R1,2, of the contact discontinuity RC , and the ram pressure pram/p0 are shown as a function of the duration of the wind t normalized to the time tp at which the inner and outer pressure are equal. The functions for the zero ambient and constant pressure model are indicated by the small and large symbols, respectively...... 31

3.2 The scale-free quantities U, P , and Ω as deﬁned in Eqs. 2.5-2.8, plotted as a function of r/RC according to the CAP model. Here they are shown for various values of ΠT and the corresponding Mach number M, from ΠT = 10 (M = 4.87), the corresponding curves of which closely resembles those in Fig. 2.2, to ΠT = 0.5 and M = 1.385, which isnearthecriticalMachnumber...... 36

3.3 Proportionality factors for the characteristic distances as a function of ΠT for constant density environments and the CAP model. Note that all factors are constant for ZAP models with values corresponding to large ΠT (see Chapter 3)...... 37

3.4 Structure of model 3 for AD wind (see Table 7.1). I show the ZAP (left) with t = 3 105yr and at t = t (middle), and the CAP model at t = 3 105yr. The overall × p × structure is similar within the parameter range...... 37

3.5 Same as Fig. 3.4 but for RG-like winds (see Table 7.2)...... 38

3.6 Fractional difference in RC between models of zero ambient pressure (RC,s) and finite pressure models (RC,p) in the parameter space of the mass lossm ˙ , the wind velocity vw, and the environment density n0 as a function of log(t/tp). The plots visualize the (Π-theorem) as discussed in Chapter 3: The difference depends on (t/tp) only. The ratios between scale-free variables are constant throughout the parameter space. The

ix Π-theorem applies also to R1. In the lower right is shown the fractional diﬀerence of RC (red), R1 (blue) and R2 (magenta) between the ZAP and CAP models as a function of log(t/tp)...... 38

4.1 Value of K as a function of v and log K (Π , Π ) where Π =(v v )/v ) 2C w 2 m˙ vw vw w − w,1 w,1 and Πm˙ = (˙m/m˙ 1). K2C is close to constant in regime II, pertaining to low mass winds running into high mass loss wind, i.e. log(Πm˙ ) < 0 (see text for details). . . . . 41

4.2 Value of K as a function of v and log K (Π , Π ) where Π =(v v )/v ) 21 w 2 m˙ vw vw w − w,1 w,1 and Πm˙ =(˙m/m˙ 1). K21 is close to constant in the regime of low mass winds running into high mass loss wind, i.e. log(Πm˙ ) < 0...... 42

4.3 Value of K as a function of v and log K (Π , Π ) where Π =(v v )/v ) 22 w 2 m˙ vw vw w − w,1 w,1 and Πm˙ =(˙m/m˙ 1). K22 is close to constant in the regime of low mass winds running into high mass loss wind, i.e. log(Πm˙ ) < 0...... 43

4.4 K2(C,1,2) as a function of the ratio between inner and outer wind velocity described −6,−4,−2,−1,0,1 by Πv,w as obtained by cuts at log(Πm˙ ) of 10 indicated by progressively thicker lines. The exact solutions are given in red. I give the solutions for K in comparison to the ﬁts. The approximations are given (magenta and blue, dotted). . . 44

5.1 Time series for a typical AD wind in the s =0case(seetext)...... 50

5.2 Time series for a typical AD wind in the s =2case(seetext)...... 51

7.1 Hydrodynamic proﬁle for a wind typical for an accretion disk (left), vw = 3000 km/s −8 and mass-loss rate of 10 M⊙/yr, and a RG-like wind (right), 30 km/s and mass-loss −7 −3 rate of 10 M⊙/yr, running into an ISM of constant, 1 cm particle density after a time of 300, 000 yr. The contact discontinuity is at 21.7 and 5.45 ly, respectively. Fluid velocity u (magenta) is normalized to 100 km/s, pressure p (blue) is normalized to the pressure p1 just inside the outer shock, and particle density n is unnormalized. 57

7.2 Same as Fig. 7.1 but for a mixture of AD plus RG-like winds. Left and right are shown models no. 21 and 22 from Table 7.3, respectively...... 59

7.3 Hydrodynamic profile for a wind typical of an accretion disk running into an r−2 density profile of prior mass loss. In the left figure, a wind of velocity vw = 3000 km/s −8 and mass-loss rate ofm ˙ = 10 M⊙/yr runs into another wind of mass lossm ˙ 1 = −7 10 M⊙/yr and vw,1 = 30 km/s for a time of 300, 000 yr. The right figure is the −10 same except the ongoing mass loss ratem ˙ = 10 M⊙/yr. The contact discontinuity is at 240 and 67.7 ly, respectively. Fluid velocity u (magenta) is normalized to 100 km/s, pressure p (blue) is normalized to the pressure p1 just inside the outer shock, and particle density n is normalized to 10−6 cm−3. Note that the particle density normalization used is much smaller than in the constant density case...... 61

7.4 Same as Fig. 7.3 but for a mixture of AD plus RG-like winds. Left and right are shown models no. 8 and 9 from Table 7.6, respectively...... 62

x 8.1 Potassium K I λ7665 line evolution for SN2014J. Colored lines designate observations from various epochs, from purple (earliest) to red. Vertical line marks show corresponding velocities in km/s (negative velocities mean moving toward the observer). The absorption features at 144 km/s and 127 km/s clearly show a steady decline − − in strength throughout the course of measurement. The Na I D1 spectrum is shown with the dotted line, clearly showing an absorption feature also at 144 km/s. From − Graham et al. [64]...... 72

8.2 Allowed duration of the wind for SN2014J as a function of Rc and R˙c, for constant density environment, s=0 (left), and environments formed from prior mass loss, s = 2 (right)...... 73

2 8.3 Determination of µ =mv ˙ w /n0 by observations. µ is given as a function of the velocity of the shell and the shell radius in the range given by [64]. I assume particle densities of the ISM between 0.1 and 10 cm−3. I ﬁnd a range of possible solutions µ(SN2014J) 2 2 3 between 15.11 and 95.85, in units of (M⊙/yr)(km /s )(cm ). Here, the range of µ is given by the extrema in shell velocity and radius...... 76

8.4 Parameters for SN2014J in them ˙ -vw-space as constrained from the spectra and the X-ray and radio luminosities. The lines of constant luminosities Lν are given for δ = log10(˙m/vw) which can be obtained by eqs. 8.1 and 8.2. The red, striped area is the forbidden region based on radio and X-ray observations of SN2014J (δ & 11.15). − Note that the radio limit from 2011fe gives δ & 11.22 [23] (see Fig. 8.8). The bright − green area indicates the allowed range for SN2014J for constant density environments with particle densities between 0.1 and 10 cm−3. A RG wind prior to the formation of the WD may produce a r−2 power law (region I) and a pre-existing constant density surrounding (region II) as low as 0.009 cm−3 (see Chapter 8 and Fig. 2.1). The thick, curved lines correspond progenitor winds running into region I of a prior RG wind −7 −5 with (m/v ˙ w) in (M⊙/yr)/(km/s) of (10 /60) and (10 /20) which produce low and high CSM, respectively. These lines bracket the range of possible prior RG winds. Their widths indicate the range of solutions of possible AD or RG-like progenitor winds (see text). The boxes indicate the range observed of ongoing RG, MS and AD winds as discussed in Chapter 6. Note that the variations in wind properties are not wellknown...... 87

8.5 Photo-ionization calculation for a total density n = 2.4 107 cm−3 and 12 days after × maximum light. The ionization fraction is given in terms of an ionization parameter (see [212]) and the distance from the system which is given on the top axis. Hydrogen is completely ionized up to 0.003 pc (0.98 ly). More than 50% of sodium is ionized ≈ to Na I for distances less than 10 pc (33 ly). From Simon et al. [212]...... 88 ≈ 8.6 Schematic taken from [38] showing their interpretation of the observations of the spectral features of PTF 11kx. See text for explanation...... 88

8.7 Same as Fig. 8.4 but for PTF 11kx. The left and right plots correspond to the inner and outer shells, respectively, that were suggested by [38]. The velocity of the inner

xi (outer) shell, R˙ C , is set to 100 km/s (65 km/s), and the range radii are selected to be R = 0.01 ly (0.02 R 1 ly) For s = 2 models in the right ﬁgure, the wind C ≤ C ≤ parameter range from prior mass loss is chosen to be the same as in Fig. 8.4. For the inner shell (left), the range of prior mass loss rates is chosen to be the same values as in Fig. 8.4, but the prior wind velocity range is chosen within the range of inner and outer shell speeds as illustrated in Fig. 8.3.1: 75 km/s v 90 km/s. The red, ≤ w,1 ≤ striped area in the left plot is the forbidden region based on radio observation. . . . . 89

8.8 Same as Fig. 8.4 but for SN2011fe. The Radio limit for the immediate environment (. 1016 cm) was obtained by [23]. No time-varying narrow lines for this SN have been observed and stationary narrow lines are attributed to uncorrelated clouds or distant ISM [162]. Therefore, the existence of shells cannot be inferred from the spectra. . . 90

8.9 Same as Fig. 8.4, but for SN2007le. A shell is assumed to be at a distance of 0.3 ly, moving outward at 10 km/s, according to the analysis of [212]. No X-ray or Radio ≈ observations were conducted and hence no corresponding limits are known. Due to the slow velocity of the shell, wind on wind (s = 2) interaction with an RG-like wind (20

8.10 Time evolution of the spectra of 2006X, taken from [159]. The diﬀerent curves correspond to various epochs: -2 days (black), +14 days (red), and +61 days (blue), all with respect to maximum light. The Na I D2 features are shown top and Ca II K is shownbottom...... 92

8.11 Same as Fig. 8.4 but for 2006X. The region ruled out by the radio observation given by [219] is shown in red strips. A shell is assumed to be moving at 50 km/s at a distance range of 0.03 to 0.05 ly, as deduced from the multi-epoch, high resolution spectra given by [159]. The prior mass loss wind environment for the s = 2 case is −7 −5 deﬁned by a mass loss rate range of 10 < m˙ 1 < 10 M⊙/yr and wind velocities of 20 < vw,1 < 40 km/s. Prior mass loss velocities greater than 40 km/s are not considered since they will not produce interaction with a 50 km/s shell. High and low −5 −7 CSM densities correspond tom ˙ 1/vw,1 = 10 /20 andm ˙ 1/vw,1 = 10 /40 in units of −1 −1 (M⊙ yr /km s ),respectively...... 93

8.12 Distribution of various Na I D components, or “bumps” in the spectra, with various log of column density, (log(N), where N is in atoms per square centimeter along the line of sight), and velocity in km/s relative to the inferred location of the progenitor system in velocity space. Plotted are components belonging to SNe Ia, core collapse (CC) SNe, and objects used as a control sample for the Milky Way (MW). Averages are shown with solid horizontal lines. Uncertainties in the average is shown by dashed lines. Among the SNe Ia components, 67% are blueshifted (negative). From [218] . . . 94

8.13 Sample line proﬁles from the SNe analyzed by [170]. For these SNe both Na I D and K I absorption features were observed or detected, and are given here. Measurements are in black and models from a ﬁtting program in red. Components are marked with redlines.From[170]...... 95

xii 8.14 Sample line proﬁles from the SNe analyzed by [170]. For these SNe only Na I D features were observed or detected, and are shown here. Measurements are in black and models from a ﬁtting program in red. Components are marked with red lines. From [170] ...... 96

8.15 The observed SNe I spectral component data of[218] as shown in Fig. 8.12 (blue triangles) plus reference models from this work (open shapes). The horizontal axis is the relative velocity from the progenitor system, and the vertical axis is Log of the number column density for sodium (cm−2). The mean of the data is the blue solid line. The uncertainty of the mean is given by the dashed lines. The purple circle represents reference model no. 2 from Table 7.1, for the s=0, AD wind case, and the orange circle represents reference model no. 9 from Table 7.2 for the s=0, RG wind case. (The reference model for AD winds, model no. 4 in Table 7.6 would lie oﬀ the plot at a relative velocity of 239 km/s and log(N) = 3.8 107 cm−2). The reference − × models give results within the range of observed number column densities. For these models, higher velocities are possible for shorter duration winds, which imply either faster accretion rates of He or C/O or faster peculiar velocities of the system, or from faster winds, and/or higher mass loss rates, and/or lower density ISM. The fastest, most blueshifted components in this ﬁgure may be due to ISM as they are consistent with typical rotational velocities in spiral galaxies...... 97

A.1 Components of SPICE and their dependencies ...... 105

A.2 Flowchart for the SPICE code package. All input is entered at the command line. . . 122

A.3 Flowchart for the SPICE code package. Example path for calculating a single structure for s = 0, CAP, is highlighted in red ...... 123

A.4 Flowchart for the plot scripts in the SPICE package. Input is entered from the com- mandline...... 124

A.5 Flowchart for the SPICE code package. Example path for calculating a s = 2 time seriesishighlightedinred...... 125

A.6 Flowchart for the SPICE code package. Example path for ﬁnding the wind parameters from observables by inversion is highlighted in red...... 126

−3 −7 B.1 Time sequence for a RG: n0 = 1 cm ,m ˙ = 10 M⊙/yr, and vw = 30 km/s. t = 2.36 104 yr. The ﬁrst eight snapshots were in an unphysical regime where p × R2 >RC (See Chapter 3 and Eq. 3.1)...... 127

−3 −14 B.2 Time sequence for a MS wind: n0 = 1 cm ,m ˙ = 10 M⊙/yr, and vw = 500 km/s. 128

xiii LIST OF ABBREVIATIONS

AD accretion disk AGB asympotic giant branch CAP constant ambient pressure CSM circumstellar medium C/O carbon/oxygen DD double degenerate DDT delayed-detonation transition DM dynamical merger EW equivalentwidth HGB horizontal giant branch ISM interstellar medium IR infrared MS mainsequence RG red giant RGB red giant branch RT Rayleigh Taylor RLOF Roche lobe overﬂow SD single degenerate SN(e) supernova(e) SN(e) Ia type Ia supernova(e) SNR supernova remnant SPICE Supernova Progenitor Interaction Calculator for parametereized Environments UV ultraviolet WD(s) white dwarf(s) ZAP zero ambient pressure

xiv ABSTRACT

The identities of the progenitors of type Ia supernova (SN Ia) has long been under study and remains an unsolved problem of astrophysics. The answer to this question will impact cosmology and subfields such as galactic evolution. To help resolve this issue and determine what systems give rise to SN Ia, the relationships between progenitor systems, their winds, and their environments are here considered, and a theoretical tool is created to model the consequences. I present theoretical semi-analytic models for the interaction of stellar winds with the interstellar medium (ISM). To investigate a wide range of possible winds and environments, I developed and employ piecewise, semi-analytical descriptions implemented in the code SPICE1 (Supernovae Progenitor Interaction Calculator for parameterized Environments), assuming spherical symmetry and power-law ambient density profiles. It is shown that a wide class of solutions can be found using the Buckingham Π-theorem. Semi-analytic solutions allow us to test a wide variety of configurations, their dependencies on the wind and environment parameters, and find non-unique solutions within a set of observational constraints. SPICE may be used to model such interactions in different types of Supernovae (SNe), stellar winds, as well as modeling realistic feedback in star formation and large scale galactic evolution simulations. As one of the many potential applications for SPICE, here I study pre-conditioning of the environment of Type Ia Supernovae (SNe Ia), which may originate from two merging WDs, known as the double degenerate scenario (DD), or an accreting white dwarf star (WD) from a non-degenerate companion, known as the single degenerate scenario (SD). The wind of the progenitor systems may originate from the progenitor, a donor star, or an accretion disk (AD). The environment is determined by the ISM and/or the wind of the donor star or the wind of the progenitor star during a prior epoch. The free parameters are: the a) mass lossm ˙ , b) wind velocity vw, c) density distributions r−s of the ISM, and d) the duration of the wind prior to the supernova explosion. ∝ I discuss the observational signatures with respect to light curves and high resolution spectra as tools to probe the environment of SNe Ia. The specific properties and evolution of the progenitor systems are found to leave unique imprints. During the progenitor evolution and with typical parameters in the SD scenario, the winds

1available on request

xv create a low density bubble surrounding the progenitor system and a high-density shell. It is also found that accretion disk winds dominate the environment formation. Within a distance of several light-years (ly), the densities are smaller by factors of 102...4 compared to the environment. This explains the general lack of observed interaction in late time Supernova (SN) light curves for, at least, several years. The overdensities of the shells are between a factor of 4 to several hundred in case of constant density ISM and environments produced by stellar winds, respectively. The expansion velocity and width of the shell are typically 1-10 % of both vw and the contact discontinuity RC and may produce narrow spectral lines as observed in some SNe Ia. Typically, narrow circumstellar lines of equivalent width 100mA˚ are found for uniform ISM typical in Spiral galaxies and 1mA˚ ≈ ≈ for wind environments. The outer layers of a SNe Ia expands with velocities of 10 to 30 % of the speed of light and we may expect some interaction with the shells several years after the explosion. I apply the analysis to SN2014J and discuss several scenarios. For SN 2014J, the environment is likely formed by the AD wind running into a region produced by the Red Giant (RG) wind from the progenitor star prior to its WD stage. The delay times between the formation of the WD and the explosion is suggested to be short, 105 yr. Finally the same analysis is repeated with other ∼ well-observed SN, including SN2001fe, PTF 11kx, SN2006X, and SN2007le.

xvi CHAPTER 1

INTRODUCTION

Type Ia supernovae (SNe Ia) allow us to study the Universe at large and have proven invaluable in cosmological studies and the understanding of the origin of elements. They are laboratories to study physics such as: hydrodynamics, radiation transport, non-equilibrium systems, nuclear and high energy physics. They are thermonuclear explosions known for having luminosities comparable to an entire galaxy. Due to their brightness and their remarkably uniform characteristics, SNe Ia observations were used in the famous discovery of the accelerating expansion of the universe, driven by the so-called “dark energy” [187, 166] and also have been employed in determining cosmological parameters. Despite their fame and utility in astrophysics and cosmology, certain important questions about SNe Ia remain to be answered. Perhaps the most important of these questions is what the progenitors are. If we knew what systems give rise to this phenomena, we would be able to put tighter constraints on their observation and gain greater accuracy in the above fields were SN Ia are used, including the nature of dark energy. Low red-shift SN were used to calibrate the Phillips relation [168], which was the main agent in the dark energy discovery. Therefore, it would impact cosmology if the nature of SN Ia varied with redshift, e.g. if different progenitor channels were more favored at different times, or if different environmental conditions changed the observational signatures at high redshift. It is this question of environmental interaction–progenitor system im- pacts on their environments, and subsequently the environmental impact on the SNe themselves, that will be the over-arching theme of this work.

1.1 SN Ia and Possible Progenitors

SN Ia have been historically referred to as “thermonuclear explosions,” to distinguish them from the other main branch of stellar explosions, core-collapse supernovae. Core-collapse SNe originate from massive stars (& 8M⊙) and will leave behind either a neutron star or a black hole. Type II, Type Ib, and Type Ic SNe are all believed to be core-collapse explosions. In contrast, thermonuclear explosions originate from lower mass stars (. 8M⊙) which expel their outer envelopes during Helium

1 shell burning and become small, roughly Earth sized carbon/oxygen white dwarf (WD) stars. WDs are sufficiently dense that Fermi repulsion between electrons dominate the pressure that supports their structures. The resulting equation of state (p ρ5/3) means that stellar structure is insensitive ∝ to heat change and the material is an efficient heat conductor. If left to itself, the WD will gradually radiate all of its energy away and cool down, while maintaining its size. Chandrasekhar discovered, however, that as WD mass increases, there exists a finite point where electrons must transition to the relativistic regime, the pressure behavior transitions to p ρ4/3 where no stable stellar ∝ structure can exist [17]. This is referred to as the Chandrasekhar limit, given by M 1.4 M⊙. Ch ≈ No WD has ever been observed with a mass greater than MCh. The result is that when an accreting

WD reaches or exceeds MCh, it will collapse in on itself, ignite a thermonuclear runaway reaction that will consume the entire WD on the timescale of 1 s and explode the star [97]. SN Ia are ∼ defined by the absence of hydrogen and helium features in their spectra along with the presence of blueshifted, broad signatures of silicon, iron, and calcium around maximum light [49]. It is also generally agreed that they originate from close binary stellar systems [232], since this is the most likely way that mass can accumulate onto a WD to make it reach Mch. Potential progenitor systems may either consist of two WD, the so called double degenerate system (DD), or a single WD along with a Main Sequence (MS), Red Giant (RG) or Helium star, the so called single degenerate system (SD). For an illustration see Fig. 1.1. The Double Degenerate Scenario Within this picture [98], the detonation may be triggered by a direct impact between two WD under the influence of neighboring stars [5], or by the merging of two WD in a binary system after expelling angular momentum via gravitational radiation. The resultant merged WD may or may not exceed Mch and multiple explosion scenarios are possible within this class of systems (see 1.2). The merging process might tidally disrupt the less massive WD, causing an accretion disk to form around the companion star. The variation in potential WD mass combinations within the DD scenario is expected to give rise to a wider variation of explosion energies, with merged masses varying from 0.9 M⊙ to 2.2 M⊙. Also, the high angular momenta ∼ ∼ and off-center ignition involved in the merger scenario are expected to produce asymmetric explosions with detectable polarization ([238, 128]). There is little observable environmental interaction expected within this scenario, since the binary pair have long shed their outer envelopes, and any material left over from the merging process will be quickly overrun by the SN ejecta. Also, due to

2 Figure 1.1: Artist depictions of the single degenerate (SD) scenario (left) and the double degenerate (DD) scenario (right). (CREDIT: (left) NASA/CXC/M. Weiss, (right) NASA/GSFC/T. Strohmayer)

the mutual destruction of both WDs, there will obviously not be any companion star left after the explosion. DD systems and mergers have been modeled by [123, 139, 31]. See [31] for a parameter study. In the case of DD progenitors, we may expect long evolutionary time scales after the formation of the WDs, compared to the accretion phases in MCh and the double detonation explosion scenarios. The time scales depend on the unknown initial separation and mass of the binary WDs and the decay of the orbits due to gravitational waves (possibly modiﬁed during a common envelope phase). The time scale of angular momentum loss by gravitational waves scales with the fourth power of 8 the separation [119]. For example, the orbits of two 1 M⊙ WDs at 1 R⊙ will decay in 6 10 yr ≈ × representing a period with no or little mass loss. However, we may expect wind just prior to the explosion when the WDs ﬁll their Roche lobes. The size of the Roche lobe corresponds to a separation of 13, 000 km [44] which translates to mass loss at most some months prior to the ≈ merging [77], and material close to the system which will be quickly overrun by the SN material. In DD, therefore, we expect no ongoing wind with the exception of a brief period just prior to the dynamical merging. Thus, the environment of a DD system may be dominated by the ISM the system has moved into which depends on its peculiar velocity and the delay between WD formation and explosion. [125] argued that the observed evidence of SNe Ia rates favors a bimodal distribution

3 of the delay times between star formation and explosion with about 50 % of all explosions take place after 0.1 Gyr and 3 Gyr, respectively. Using the star formation rates and assuming that ≈ all SNeIa originate from DD systems, [175] concluded that 50% of all DD systems explode within 4 108 yr, with a long tail comparable to the age of the universe. Recent studies show that the × distribution of delay times is more continuous (see [128] and references therein). The Single Degenerate Scenario Within this class of systems [243], the larger companion star sheds mass via either the wind, the Lagrange points of the system, or Roche-Lobe overflow (RLOF) due to stellar expansion during the course of stellar evolution. Particularly for Roche- lobe mass transfer, an accretion disk forms around the WD. Material must lose kinetic energy and angular momentum in order to be accreted onto the WD surface. It does so via a magneto- hydrodynamic viscosity mechanism that is not entirely understood [207, 177, 111]. Some of this angular momentum and energy is expelled via a fast, low mass-loss wind. The remaining material is available to rain onto the WD. Nomoto et al. [146] discuss the different scenarios for varying accretion rates of hydrogen rich matter based on models. The results are summarized below and in Fig. 1.2. For low accretion rates, Hydrogen-rich material builds a layer on the stellar surface. Once a critical mass is reached this material may ignite and be unbound and ejected from the system in what is known as a nova eruption [216]. It is unclear whether, or under what conditions, WDs undergoing recurrent nova eruptions like this can gain mass over time, or even experience a net mass loss [68, 161, 199]. For higher accretion rates ([146]), the accreted material will undergo nuclear burning on the WD surface, preventing nova-like eruptions later on. If the rate is large enough, the contact burning is sufficiently luminous to halt further accretion via driving away in-falling material by radiation pressure [144]. In the case of He-rich accretion, the region of stable accretion is larger, due to the smaller energy yield of burning He into C. SN Ia explosions originating from

SD progenitor systems would exhibit greater consistency in that all ignitions take place near MCh, releasing about the same amount of energy, and the WD ignite at or near their centers, producing a more symmetric morphology. CSM interaction is expected with material left over from the accretion process, a likely denser ISM, and possibly with the companion star, which will be left over after the explosion. SD systems have been modeled by [79, 174]. They analyzed varied conditions including diﬀerent accretion rates and materials.

4 Figure 1.2: Plot of H-accretion rate vs. WD mass taken from [146]. For accretion rates lower than the solid line, degenerate H shells on the surface are thermally unstable and nova eruptions occur (vertical dashed lines give the maximum shell mass in M⊙). For accretion rates above the dotted dashed line, the accumulated envelope expands and produces a red giant star and emanates a strong wind. The dotted line gives the Eddington accretion rate. The regime for stable accretion is marked out in the center.

Various binary evolution scenarios have been considered for SD systems. For an informative review of these, see Wang and Han 2012 [232]. Variables that differentiate the different evolutionary paths include initial binary separation (and hence orbital period) and the stellar masses. Fig. 1.3 is taken from [232] and shows a few channels whereby a pair of main sequence stars in a close binary system ultimately triggers the explosion. Fig. 1.4 shows the evolutionary channel for the WD+RG channel. In [232] there is a similar figures for the WD+He star channels. Every channel features an instance where the more massive companion out-evolves the “secondary” main sequence star, expands and fills its Roche lobe, and material overflows the Lagrange point between the stars onto the secondary. This overflow is dynamically unstable–the stars will be drawn further together from the mass transfer, the Roche lobe will shrink, and more material will

5 Figure 1.3: Figure 1 from a review by Wang and Han 2012 [232] illustrating possible evolutionary paths for a double MS binary system to eventually transition into a MS+WD binary system and a SN Ia. FGB stands for “first giant branch”, RLOF stands for “Roche lobe overflow,” CE means “common envelope,” EAGB signifies “early asymptotic giant branch,” and TPAGB stands for the “thermal pulsing asymptotic giant branch” stage. The black disks are degenerate stellar cores.

spill over. In the model this then induces a common envelope (CE) phase, which is then ejected, leaving behind a WD and the secondary star, which later evolves, expands, ﬁlls its Roche lobe, and material overﬂows onto the WD until it reaches MCh and explodes. The left-most channel from Fig. 1.3 is considered the most common scenario for producing SNe Ia (e.g. [233]). The time delay from zero age main sequence to explosion for the SD scenario is approximately given by the main sequence (hydrogen burning) evolutionary time of the primary star, since the subsequent burning timescale is about an order of magnitude less. The mass of the primary is 10 between 1 8 M⊙ For a 1 M⊙ star the delay time tτ 10 yr, and for an 8 M⊙ star it is ∼ − H ≈ 7 tτ 2 10 yr [112]. For wide binaries (separation & 1000 R⊙) there will be negligible mass H ≈ ×

6 Figure 1.4: Figure 2 from Wang and Han 2012 [232] illustrating possible evolutionary paths for a double MS binary system to eventually transition into a RG+WD binary system and a SN Ia. RLOF stands for “Roche lobe overﬂow,” CE means “common envelope,” and TPAGB stands for the “thermal pulsing asymptotic giant branch” stage.

transfer up until the ﬁnal period of accretion, hence the stars evolved as single stars. The onset of accretion is given by the time it takes the companion star to ﬁll its Roche Lobe after hydrogen burning ceases in its core, and it becomes a Red Giant star. For close binaries (MS companions) the timescale is shortened by mass transfer from one star to the other, but will remain within the same order of magnitude [232].

The accretion timescale is given by t =∆M/m˙ , where ∆M 1 M⊙ is the mass growth of acc acc ∼ the WD star until it reaches MCh, and macc must be in the stable range for the accreting material. Using the ﬁgures in Fig. 1.2, for H accretion this timescale can range from t 105 yr 107 yr. acc ∼ − [68, 161, 199] argue, however, that there is no stable H accretion that can grow the WD to to

MCh. Thus, it is possible that the longer timescales might be unrealistic. However, strength of the

7 shell burning flashes might be overestimated in spherical models of WD accretion, and it has been suggested that the strength of the flashes might be reduced due to rotation. For the SD scenario with RG companion stars Hachisu et al 2008 [71] find that, following, a brief period of star formation, explosions on average take place between 0.4 Gyr and 15 Gyr later. Assuming an optically thick accretion wind for high accretion rates, and mass stripping of the companion star from this wind, they obtain a delay time distribution (DTD) of t−1 that matches observations (e.g. [225]). Similar DD scenario modeling also shows an observed t−1 dependency of the DTD.

Figure 1.5: Similar to Fig. 1.2 but for He-accretion. The figure is taken from [252] using the numerical figures given in [144]: The steady He-shell burning upper limit is −6 (dM/dt) = 7.2 10 (M /M⊙ 0.6) M⊙/yr while the lower limit is 0.4(dM/dt) . RHe × WD − RHe Lower than that is an unstable burning regime prone to degenerate He shell flashes. For −8 accretion rates below 5 10 M⊙/yr, a He detonation of the WD can occur and produce ∼ × a “double detonation” SN ([121], see also text).

8 1.2 SN Ia Explosion Scenarios

The observed explosion energies of SN Ia are 1051 erg, which is equivalent to the amount of ∼ energy released from the fusion of 1 M⊙ of carbon and oxygen into iron group elements. This ∼ lends support to the view that these are explosions of degenerate C/O WD stars [224]. The WD is completely unbounded and destroyed, leaving no remnant such as a neutron star or black hole. A large quantity of 56Ni is produced, the decay of which powers the optical/infrared light curves. How the explosion is triggered varies, depending on the speciﬁc conditions and the progenitor system involved. Here, the main scenarios are discussed.

MCh explosions. Sn Ia explosions may result from accretion onto a WD that is near the

Chandrasekhar limit (MCh), either from a RG or MS companion via Roche-lobe overflow within the SD scenario, or from tidal disruption of another WD within the DD scenario. As it nears MCh, the degenerate electron velocities transition into the relativistic regime, which changes the equation of state and increases the compressibility [114]. As the WD compresses, heat is released from gravitational collapse. Heat is usually conducted efficiently through the degenerate material to the WD surface and radiated away. As the WD compresses further, the high densities induce electron capture within the WD, and hence energy is carried away via neutrino losses [112]. Eventually the compressional heating rate will exceed both conduction and neutrino cooling rates, and the WD will ignite at or near the center. The details of the ignition and explosion depend on the accretion rate. Dynamical merger explosions. Another explosion possibility is the merging of two WD in a dynamical merger within the DD scenario [171, 227, 208]. The stars combine on a hydrodynamical timescale of 1 sec. The large accretion rate overwhelms any cooling mechanism and the pair ∼ ignites and explodes, even for combined masses below MCh. One major argument used against dynamical merger explosions has been that due to the large accretion rates and off-center ignition, the C/O mixture burns as a deflagration to oxygen and neon instead of iron-group elements. The O/Ne mixture subsequently undergoes electron capture which then saps the supporting degenerate pressure, leading to what is known as an “accretion induced collapse,” and a neutron star. The detectability of such an event would be significantly lower than a SN Ia [145, 196, 92, 208].

9 Throughout this work, the first class of explosions will be referred to as MCh explosions which may originate from either SD or DD systems, and dynamical mergers (DM) which may originate from DD systems. For overviews, see [10, 148, 36, 32, 232, 82]. Double Detonation Scenario Recently, a third explosion progenitor system/trigger mechanism has been revived which is known as double detonation scenario or He-detonation in a sub-MCh mass WD [143, 121, 248, 85, 117, 246]. In this picture, a C/O WD star accretes He-rich material at a low rate to prevent burning. Explosions are triggered from ignition of the surface He layer with masses of about a few hundredths to 0.1 M⊙. The resulting strong shock wave may trigger 56 a detonation of underlaying C/O. Previous calculations produced a few 0.01 M⊙ of Ni which is inconsistent with the early time spectra, because the photosphere is formed within the outer −3...−2 10 M⊙ [93]. Modern recalculations have utilized a smaller He shell mass and obtain better agreement with observations [117, 246], though, the problem with the outer layers still persists. Recent studies of helium detonations including curvature and expansion effects may be in better agreement with the observations [140, 192, 255]. For this class of explosions, we may expect accretion disk winds similar to those in MCh scenarios as discussed below. Using the figures in Fig. 1.5, for He accretion the accretion timescale can range from t acc ∼ 104 yr to 106 yr. The timescale for mass growth hence can be significantly smaller for accretion ∼ of He rather than H [144], see Fig. 1.5. C accretion from a tidally disrupted WD companion in the DD scenario does not involve burning, and hence the rate can be much much larger. This permits extremely short accretion timescales, up until the dynamical merger timescale of 1 s. ∼

1.3 Which Progenitor and Explosion Scenarios are responsible for SN Ia?

Candidate progenitor systems have been observed for both the SD and DD scenarios: supersoft X-ray sources [65, 230, 181, 103] showing accretion onto the WD from an evolved companion, and WD binary systems with the correct period to merge in a Hubble time and an appropriate total mass [137]. At least for SN1572, the Tycho SN, the donor star has been identiﬁed as a G0 MS star strongly supporting the single degenerate progenitors [193, 61, 109] (see Fig. 1.6). For two other supernova remnands (SNR), 0509-75 & 0519-69.0, no massive companion star candidates have been found. These results may favor double degenerate progenitors [200, 43] (see Fig. 1.7).

10 Figure 1.6: (left) Chandra and (right) Hubble Space Telescope (HST) images of SN1572 or Tycho’s Supernova. Tycho G, the companion star for SN1572 argued by [193], is pictured on the right. (CREDIT: NASA)

Studies have shown that accretion from a Helium star can produce MCh mass WDs and, based on binary population synthesis, that the Galactic SNe Ia rate is consistent from this channel [234, 231]. Other recent work on rates and the delay time distribution may favor the double degenerate scenario [126, 129, 127]. Nevertheless, theoretical work on the explosion, spectra and light curves continues to favor the single degenerate scenario, with some contribution of double degenerate scenario [85, 196, 247, 138, 195, 208]. For overviews see Refs. [11, 149], and Refs. [232, 35, 33]. From the observation, there are several candidate SD progenitor systems such as super-soft X-ray sources, and cataclysmic variables including novae systems. Within the SD evolution, the emerging consensus picture seems to be that these diﬀerent systems may actually be diﬀerent evolutionary phases of the same basic system [34]. However, as discussed above, even the evolution of a DD system may lead to MCh WD explosions if the material is accreted from tidally disrupted

WD because the explosion would still lead to a central ignition of a WD close to MCh . In contrast,

11 Figure 1.7: The 400-year-old SN Ia remnant 0509-67.5, located in the LMC. Within the ∼ 1.4 arc-sec radius circle in the center is the 3σ conﬁdence region where any companion star from the progenitor system is likely to be. No stars are detected in the region down to 0.04 L⊙ in the visual. [200] argue that this strongly eliminates the possibility of a SD progenitor for this SN. This image is a composite of HST in B, V, I, Hα, and Chandra in X-rays.

dynamical merging of two WDs will likely result in progenitors with masses ranging from well below

MCh to about 2 MCh.

The favorable view of MCh mass explosions as most likely explosion scenario for the majority of SNe Ia is based on the homogeneity in light curves and spectra, though there is strong evidence of contributions of both to the SNe Ia population, including Super-MCh mass explosions [85, 155, 4, 96, 178, 249, 190, 67]. For reviews, see Refs. [155, 9, 80]. In particular, the chemical structure in the supernovae remnant S Andromedae and IR line proﬁles obtained several hundred days after 9 3 maximum light strongly favor explosions at densities 10 g/cm , the hallmark of MCh explosions [84, 141, 48, 59, 124]. Delayed-detonation models [110, 250], those possessing a transition from

12 a deflagration to detonation front (DDT) have been found to reproduce the optical and infrared light curves and spectra of “typical” SNe Ia reasonably well, including the time evolution [85, 91, 87, 50, 152, 242, 120]. Here the burning starts as a well subsonic deflagration and then turns to a nearly sonic, detonative mode of burning. The amount of burning prior to the transition from deflagration to detonation is the main factor which determines the production of radioactive 56Ni which powers the light curves (see also Refs. [56, 57, 176]. When the detonation front propagates through the WD, the density of burning decreases with distance and we see a layered chemical structure consistent with observations in both the abundance pattern and velocity distribution of individual objects [85, 4, 91, 87, 50, 152, 242, 120, 2, 83]). The absolute brightness at maximum light and the rate of the post-maximum decline over 15 days are related through what is known as the brightnes-decline relation ∆m15 . The∆m15 relation plays a key role both for cosmology and understanding of the explosion physics [168, 169, 60]. From 56 theory, ∆m15 is well understood: light curves (LCs) are powered by radioactive decay of Ni [28]. More 56Ni increases the luminosity and causes the envelopes to be hotter. Higher temperature means higher opacity and, thus, longer diffusion time scales and slower decline rates after maximum light [86, 151, 229, 106]. The ∆m15 relation holds up for virtually all explosion scenarios as long as there is an excess amount of stored energy to be released [86, 3]. The tightness of the relation observed for Branch-normal SNe Ia is about 0.3m [169, 75, 188, 221, 167], consistent with explosions of models of similar mass, but hardly consistent with the entire range of masses for two WDs undergoing dynamical merging. However, MCh explosions have difficulties as well. Although the brightness of the relation can be understood within the framework of the single degenerate scenario and spherical delayed detonation models [217, 86, 88], it falls apart when taking into account burning instabilities and mixing during the deflagration phase [56, 86, 106, 189]. Some important piece of physics appears to be missing still which may restore a “close-to-1D” layered chemical structure.

For MCh explosions, some progress has been made in understanding variations among SNe Ia, with suggestions that some of the spectral diversity is due to diﬀerences in metallicity, central density, WD rotation and asymmetries [4, 124, 88, 90, 95, 253, 254, 136, 163]. It is widely accepted that the exact condition of the WD at the time of explosion is a key solving the problems of current

13 explosion models and understanding the diversity in SNe Ia [84, 56, 106, 3, 136, 89, 226, 228, 14, 122, 104, 105, 191, 211, 206, 256]. A set of extremely luminous SNe Ia may lend support for dynamical merging with progenitors well above the Chandrasekhar mass [96, 198, 222, 94]. I note, however, that the inferred brightness 56 depends on a unique relation between the Ni mass MNi, and the intrinsic color at maximum light.

At least, in a few cases, the apparent brightness can be understood within the framework of MCh mass WDs with an intrinsically red color rather than a boost in brightness by assuming a large interstellar reddening correction [178].

1.4 Clues from the Environments of SN Ia

One of the distinguishing characteristics of the progenitor systems are the imprints that they leave on their environments. As noted above, a competent understanding of the relationship between SN Ia and their environments might also critically aﬀect their application as cosmological distance indicators. For these reasons, a study of SN Ia environments is important and will be pursued further in this work. The stellar environment will shed light on the evolutionary history of the progenitor, supernovae light curves, and spectra, with X-rays and radio emission being the probes (e.g. [26, 42, 21, 16, 55]). As discussed below, the limits on density for the environment of a typical SNe Ia are well below those of the solar neighborhood and one of the goals is to probe whether SD and DD systems may create this environment. In case of DD progenitors, we may expect long evolutionary time scales ([125], see above) and, likely, the binary system has moved far away from its place of birth and the stellar evolution prior to the WD stage. Any circumstellar material at large distances should be uncorrelated to the DD system. In most cases, we may expect a low density environment consistent with observations, although in some there may be ISM nearby by chance. The environment of SD systems can be expected to consist of three main components: 1) Some matter bound in the progenitor system at the time of the explosion that may originate from the accretion disk or be shed from the donor star; 2) the wind from the WD, accretion disk or donor star; and 3) the interstellar medium (ISM).

Within the scenario of MCh WD explosions, hydrodynamic calculations have shown that the expanding supernova ejecta wraps around the companion star and may pull oﬀ several tenths

14 of a solar mass of material in case of a RG donor [133, 106]. Besides the donor star, another source of matter is the accretion disk material [58] lifted during a pulsational phase during the explosion, or debris from the merging of two WDs [85, 179]. There has been some observed evidence for interaction between the explosion and the immediate environment. Although H-lines like in SN 2003ic are rare, a common observed feature is a high-velocity CaII line which, ﬁrst, was prominently seen in events like SN1995D, SN2001el, SN2003du, and SN2000cx. It is now known to be a feature present in almost all SNe Ia [78, 51, 236, 52, 210]. This line may be attributed to the material which did not undergo nuclear burning and may be part of the progenitor system [59, 179].

At intermediate distances of up to several light-years, in the case of MCh explosions, the environment may be dominated by the wind from the donor star, the accretion disk or, for high accretion rates, the wind from the WD, or the interstellar material (ISM). A number of possible interaction signatures has been studied, including X-rays, Radio, and narrow H and He lines, but no evidence −5 has been found, with an upper limit of 10 M⊙ for the mass loss [25, 203, 202, 30, 24]. In late-time light curves, kinematic interaction should result in excess luminosity over the time scale of 1-10 yr but, in general, is not seen. No sign of an interaction has been found even in SN1991T, which has been observed up to day 1000. This implies particle densities 1 less than 10−3cm−3[204]. ≈ At large distances, from few tenths to several light-years, the environment is determined by the ISM. It is known that Type Ia SNe generally explode away from star forming regions [237]. This can be partly attributed to the long stellar evolutionary lifetimes of the low-mass stars in the progenitor systems, allowing them suﬃcient time to move away from their place of birth. It is also known that SNe Ia occur in elliptical and spiral galaxies, including galactic disks, the bulge and the halo. One may expect the explosion to occur in ISM particle densities of 10−3...1 cm−3 ≈ [47]. Light echoes from SNe Ia have been used to probe their environments, and showed that many SNeIa have circumstellar dust shells at distances ranging from a few up to several hundred parsecs [76, 186, 1, 158, 29, 239, 185]. Most evidence for a link between SNe Ia and their environment comes from the observations of narrow, blue-shifted, possibly time-dependent Na I D and K I absorption lines which, for a

1In this work, particle density is deﬁned as the number of atoms per cubic centimeter if the gas is composed of pure atomic hydrogen. The code SPICE assumes pure atomic hydrogen of 1 atomic mass unit (AMU, see Appendix A.2).

15 significant fraction of all SNe Ia, indicates strong outflows [39, 160, 7, 54, 218, 170]. In addition, extinction laws derived from SNe Ia seem to be different from the interstellar medium in our galaxy, suggesting a component linked to the environment of SNe Ia rather than the general host galaxy [15, 115, 45, 150, 157]. Possibly, the hydrodynamical impact of the SN ejecta will produce additional emission and may modify the outer structure of the envelope and, thus, the Doppler shift of spectra features. Light emitted from the photosphere of the supernovae may heat up matter in the environment, which, in turn may change the ionization balance or the dust properties [182, 118, 213, 164]. This effect for different dust formation in the host galaxy may lead to extinction laws different from the Milky Way as commonly observed in SNe Ia [62, 116, 53, 107, 13]. The dust properties may effect the light echoes which could in turn change the extinction laws [235]. The following picture of the environment emerges: SNe Ia are surrounded by a cocoon with a much lower density environment than the ISM. Sometimes, narrow ISM lines indicate clumps or surrounding shells. However, the diversity of supernova and progenitor channels leaves open a huge parameter space which cannot be covered by numerical simulations. To cover the parameter space, I use a semi-analytic approach similar to those developed by Parker [156], Weaver et al. [240] for stellar wind/ISM interaction, Chevalier & Imamura [20] for stellar wind/stellar wind interaction, and Chevalier [18] for supernova remnants. In this study, I make use of the Π theorem [12, 205] to study the classes of self-similar solutions for the environments of SNe Ia.

1.5 Theoretical Background: the Π Theorem and Self-Similar Solutions 1.5.1 Buckingham’s Π Theorem

The theoretical work for constructing my models will be framed by the Buckingham Π The- orem [12, 205]. This fundamental theorem of dimensional analysis states that 1) dimensionless parameters can only depend on other dimensionless parameters, and 2) the number of independent dimensionless parameters that can be formed from n parameters is n m, where m is the number − of independent dimensions represented in the n parameters. The dimensionless groups thus formed are called Π groups. I now give three examples of using the Π theorem. Example 1 A mass is dropped to the ground under inﬂuence of constant gravity only. We wish to determine the functional form of the downward velocity as it depends on time. The

16 relevant parameters of the motion are the time, t and the gravitational acceleration g. We wish to determine v so we add it to the parameter set. The units are: [t] = time, [g] = length/time2, and [v] = length/time. That gives us three parameters and two dimensions represented. This allows 1 Π group (3 parameters minus 2 dimensions): v/(gt). This Π group can only be expressed as a function of other Π groups, but here there is only one. Therefore it is equal to an arbitrary constant. This yields v/gt = const and v(t)= const gt. By the kinematic equations the constant × is found to be 1. Example 2 We now consider the object being thrown to the ground with an initial velocity v0. This will become one more relevant parameter, bringing the number to four, but since it has dimensions of length/time, it does not increase the number of represented dimensions in the problem. Now 2 Π groups (4 parameters minus 2 dimensions) are chosen: Π1 = v/gt, and Π2 = v0/gt. Since any member of the group can only depend on the other group members, we have: Π1 = K(Π ), where K is an arbitrary function. This yields v/gt = K(v /gt) and v(t)= gt K(v /gt). 2 0 × 0 Although the Π theorem could only take us this far, we may learn about the character of K by physical intuition. We infer that K must increase monotonically with Π2, since the dependence of the initial downward velocity v0 is entirely contained within it, and increasing it must always make v increase. However, since this group contains the time also, this means that as t increases, K must decrease. As t , however, it will have the same eﬀect as v 0, and we must reproduce → ∞ 0 → the case where the mass is dropped from rest. Therefore K must converge to the same arbitrary constant as in the previous example. Also, for t 0, we require v v , therefore K v /gt. In → → 0 → 0 summation, K(Π2) = Π2 for small time and steadily decreases to a constant value as t increases.

Comparing with the kinematic equations gives: K(v0/gt)=1+ v0/gt, or K(Π2) = 1 + Π2. Example 3 Finally, we re-do the previous example except that now we would like to take into account air resistance in the form of a drag force: F = bv where b has units of mass/time. First, D − let us assume that the relevant parameters are simply the same as before but with the addition of the coefficient of drag: t, g, v, v0, and b. Here are five parameters and three dimensions (mass, length, and time), giving two Π groups: Π1 = v/gt,Π2 = v0/gt. These are the same groups as in the previous example, and b cannot find a place among them since it is the only parameter that depends on mass. This is problematic since the drag cannot now enter the physics of the falling object. One more parameter is needed to make one more dimensionless group that incorporates

17 the dimension of mass. It is the mass of the object, m. There are now six parameters and three dimensions, yielding three Π groups: Π1 = v/gt,Π2 = v0/gt, and Π3 = bt/m. We are able to choose the groups such that v appears in only one of them, therefore we can derive an analytic expression for it. Π = K(Π , Π ), which means v(t) = gt K(v /gt, bt/m). This far the Π 1 2 3 × 0 theorem takes us. We can infer, however that K must increase with greater Π2 to an upper bound for the same reasons as in the previous case. K must decrease with greater Π3, since decreasing the drag coefficient b should cause v to increase. Both of these dependencies tell us that K will always decrease as t increases. However, for t , the dependency on Π should vanish as → ∞ 2 it did in the previous example, giving us v(t) = gt K(Π ). Also, K cannot decrease without × 3 bound, since it cannot become negative. In fact, for large t we expect v to converge to some finite positive value vT , where the magnitudes of the gravitational force and the drag force are equal. This convergence can only happen if K 1/t for large times, which, being a function of Π = bt/m, ∝ 3 means that K m/bt = 1/Π in this regime. This means v( ) = v mg/b. As t 0, we ∝ 3 ∞ T ∝ → require v v , therefore K v /gt = Π , just like in the previous example. Again, K(Π , Π ) → 0 → 0 2 2 3 starts at Π for small time and steadily decreases to const/Π as t . Comparing with the 2 3 → ∞ result from Newton’s Second Law gives K(v /gt, bt/m) = m/bt(1 e−bt/m)+ v /gt e−bt/m, or 0 − 0 × K(Π , Π ) = 1/Π (1 e−Π3 ) + Π e−Π3 . The constant again is equal to 1. 2 3 3 − 2 The Π theorem works by utilizing a fundamental symmetry inherent in all physical laws: the qualitative behavior of a system does not depend on the system of units being used to describe it. Taking the second example, we obtained that v(t)/gt = K(v0/gt). Let us write it instead as v(t)/gt = K(v0/gt, v0,g,t). The left-hand side (LHS) is dimensionless, meaning that if we changed the system of units, its numerical value would stay the same. The individual parameters v0, g, and t on the right-hand side (RHS) however would change numerical values. As a result, v/gt cannot depend on them individually. Only the dimensionless quantity v0/gt remains invariant under a change of units, therefore the dimensionless quantity v(t)/gt can depend on v0/gt. To implement the Π theorem successfully it is critical that one determines exactly what parameters are “relevant.” If, in the above examples, we decided that the gravitational acceleration g was not relevant, but the initial height h0 was, we would have obtained very different, and incor- rect, expressions for the velocity. The strength of the Π theorem lies in its ability to reduce the dependencies of a problem from n independent parameters to n m Π groups. As an illustration, −

18 if we ran a computer simulation of the final example, we could numerically solve for v(t) using various values of tg, m, v0, and b independently. If we took a range of r values for each parameter, r5 calculations would be necessary. If we instead use the Π theorem and reduce the dependencies to just two dimensionless parameters, only r2 calculations would be necessary, cutting down the overhead significantly. A useful application of this capability of the Π theorem is in solving partial differential equations (PDEs). If it so happens that one of the Π groups of a problem contains more than one of the independent variables, and those variables does not appear in any other Π group, then the number of independent variables becomes reduced by one. In this way it is possible to transform PDEs into ordinary differential equations (ODEs) under the right conditions. This idea will be explained in more detail in the following subsection.

1.5.2 Application of the Π Theorem: Self-Similar Solutions

The Buckingham Π Theorem can be used to ﬁnd self-similar solutions for a given problem. Say, a function F can be expressed in the form: F (txα). Then F has the property that as t scales up by a factor d > 1, it has the same eﬀect as scaling x with d1/α < 1. Thus the morphology of F remains constant in time, only that it expands outward in x. Therefore, functions of this type are called self-similar. How this can come about is apparent from the last section: if both independent variables can be contained entirely within one Π group, this group will contain the entire dependency on both variables, and the problem will then only have one independent variable given by this group. The application of this concept to the hydrodynamical equations for 1D will be explored in Chapter 2.

1.6 Application of Self-Similar Solutions to SNe Ia

The current state of the research leaves some important questions unresolved. How can we understand the ubiquitously low density environment, their general structure, and their link to the progenitor systems? Do SNe Ia all originate from merging WDs? Which of the wide variety of progenitor systems are compatible with the observations and the range of parameters? What other possible signatures might be seen due to the interaction of the explosion within the possible progenitor systems? The recent, nearby SN2014J has given us a premier opportunity to probe the progenitor system. Searches for X-ray and radio from SN2014J have yielded upper limits that seem

19 to favor dynamical merging scenarios [130, 165], while observed narrow ISM lines for the same event seem to favor MCh mass explosions [64]. For SN2014J, can we find a class of progenitor systems which is consistent with both sets of observations? To address the questions, I developed a parameterized model in Chapters 2-5 using fluid mechanics. In Chapters 6 and 7, I present the application of semi-analytic models and the code SPICE as an analysis tool in the framework of environment of SNe Ia. I evaluate the imprint of different environments and wind properties. In chapter 8, I apply the framework to SN2014J as an example and discuss the results. Other SN are then also analyzed in similar fashion, including PTF 11kx, 20011fe 2007le, 2006X. In chapter 9, final discussions and conclusions are presented.

20 CHAPTER 2

THEORY AND ASSUMPTIONS

2.1 Self-Similar Solutions for Power-law ISM Density Proﬁle

I develop here a model for wind-environment interaction from basic fluid mechanics using the Π Theorem introduced in the last chapter. This formalism was developed, for example, by Parker [156] and Weaver et al. [240]. Here, I discuss the details which lay the foundation for the newly developed closure relation needed for this study. In the case of spherical symmetry and adiabatic flows, the hydrodynamic equations take the form: ∂ρ 1 ∂ + (r2ρu) = 0 (2.1) ∂t r2 ∂r ∂u ∂u 1 ∂p + u + = 0 (2.2) ∂t ∂r ρ ∂r ∂p ∂p γp ∂ + u + (r2u) = 0 (2.3) ∂t ∂r r2 ∂r where u is the fluid velocity, ρ is the mass density, p is the pressure, and γ = 5/3 is the adiabatic index. For the systems we study, a constant, isotropic wind of velocity vw and mass-loss rate m˙ emanates from the center into an ambient medium where the density is given by a power- −s law: ρ(r) = ρcr . The hydrodynamical structures have four characteristic regions (Fig. 2.1): I) undisturbed wind emanating from the source between r = 0 and an inner shock front R2, II) the inner shocked region of accumulated wind matter between R2 and the contact discontinuity RC ,

III) an outer region of swept-up interstellar gas between RC and the outer shock R1, and IV) the outermost, undisturbed, ambient medium. The solutions for regions I and IV are trivial.

I now use the Π theorem to simplify the problem for region III. The contact discontinuity RC behaves as a solid piston that separates the outer and inner regions. If the wind velocity originating from the center is much faster than the expansion of the outer region, then there will be a constant 2 mechanical luminosity in the interior, Lw =mv ˙ w/2. As the wind material shock-heats and slows down, it will do work as it expands and pushes against RC and the outer region. Thus Lw will directly aﬀect the dynamics of region III. All of the relevant parameters in this region are therefore

21 3 2 Log [u/(100 km/s)] 2 Log [u/(100 km/s)] Log (p/p1) Log (p/p ) Log n RC R1 1 1 -6 -3 1 Log [n/(10 cm ) ] 0 -1 R 2 0 -2 I II III IV I II III IV -3 -1 R2 RC R1 -4 -5 -2

-6 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 r/Rc r/RC

Figure 2.1: Velocity, pressure and density structure of a typical model shows four distinct regions dominated by the wind (I), the reversed shock (II), a shell (III), and the environment (IV) both for constant (left) and wind (right) environments. In the ﬁgures of this work, density is shown in particle density n = ρNA/µ, where NA is Avogadro’s number and µ is the mean molecular weight. In this work, µ is set equal to 1. For diﬀerent µ, ρ µρ in all equations. →

the ﬂuid density ρ, u, p, r, t, Lw, γ,and ρc. These are eight parameters with three total dimensions, giving ﬁve Π groups. γ is a dimensionless number and will form its own group. It will not vary in this work, so its individual dependence will be ignored. Most importantly, r and t form one dimensionless group with other, constant parameters. This group I write as

(s−5)/3 1/3 Π0 = r t(Lw/ρc) (2.4)

Π t/r5/3 and t/r for s = 0 and s = 2, respectively. The remaining Π groups are typically 0 ∝ s 2 2−s chosen to be: Πu = ut/r,Πρ = ρr /ρc, and Πp = pt /(r ρc) [156]. Each Π group can be written as a function of the other groups, and by substituting them into each other, the arbitrary functions can be re-written to depend only on Π , or the combination η = tr−λ, where λ = (5 s)/3. These 0 − functions are called U, Ω, and P below:

r u = U(η) (2.5) t −s ρ = Ω(η)ρcr (2.6)

2−s −2 p = r t ρcP (η) (2.7)

22 η = tr−λ. (2.8)

Parker [156] implemented this similarity transformation and solved for the structure of region III. He applied his solution to the dynamics of the solar wind. The contact discontinuity will move with the structure and be represented by a constant value of Π0:

R (˙mv 2/ρ )1/(3λ)t1/λ (2.9) C ∝ w c

λ = 5/3 for s = 0 and λ = 1 for s = 2 (See Eqs. 2.17 and 2.23 for exact expressions of RC ). P (η) Substituting these into equations 2.2, 2.1 and 2.3, making the substitution χ = γ Ω(η) , and re- arranging gives us the following:

dχ γχ(1 λU)[2 + U(1 3λ 3γ + λγ) + 2λγU 2]+ λχ2(sγ 2γ + 2 s 2λ + 2λγU) = − − − 2 2 − − − d log(η/η1) γ(1 λU)[(1 λU) λ χ] − − − (2.10)

dU γU(1 U)(1 λU)+ χ(2λ + s 2 3λγU) = − − − − (2.11) d log(η/η ) γ(1 λU)2 γλ2χ 1 − −

d log(P/P ) 2+ U(s 2 2λ + λγ 3γ)+ λU 2(2 s + 2γ)+ λχ(s 2) 1 = − − − − − (2.12) d log(η/η ) (1 λU)2 λ2χ 1 − − The boundary conditions at the outer shock are obtained from the Rankine-Hugoniot shock jump conditions for large Mach numbers:

2γ(γ 1) χ = − (2.13) 1 λ2(γ + 1)2 2 P = (2.14) 1 λ2(γ + 1) 2 U = . (2.15) 1 λ(γ + 1)

Integration is carried out with respect to U from the outer shock R1 where U = U1 to the

RC contact discontinuity RC where the ﬂuid velocity u is equal to dRC /dt = λt (by deﬁnition), and hence Uc = 1/λ. The functions U, Ω, and P are plotted in Fig. 2.2.

23 1 log U log P log Ω 0.5

-0.5

-1 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 1.16

r/RC

Figure 2.2: The scale free variables U, Ω, and P , as deﬁned by Eqs. 2.5 - 2.8 plotted as a function of r/RC . These quantities are functions of r/RC only–these curves are invariant for all times during which the self-similar solution holds.

2.2 Self-Similar Solution for s=0 and Large Mach Numbers: the Zero Ambient Pressure (ZAP) Model

The solution for region III is self-similar because the only relevant parameters are the mechanical luminosity of the wind Lw emanating from the origin and the outer density constant ρ0. There is no way to obtain a parameter with dimension of length or time using those parameters, i.e. using 1 2 2 mv˙ w and ρ0, no Π group can be formed of r or t separately. This is not the case in region II, where the relevant parameters are ρc, as well asm ˙ and vw, individually. Therefore, in the case of s = 0 a self-similar solution is not possible in region II, as was shown by Weaver et al. [240]. They did however obtain useful analytic relations directly from the hydrodynamic equations by noting that region II is approximately isobaric. Their results for the locations of the inner shock, contact discontinuity, outer shock, the velocity, pressure, and density are the following:

24 3/10 1/10 2/5 R2(t) = 0.747 (m/ρ ˙ 0) vw t (2.16)

2 1/5 3/5 RC (t) = 0.660 mv˙ w/ρ0 t (2.17) mv˙ 2 1/5 R (t) = 0.769 w t3/5 (2.18) 1 ρ 0 11 R (t)3 4 r u(r, t)= C + (2.19) 25 r2t 25 t 2 4 3 1/5 −4/5 p(t) = 0.126 m˙ vwρ0 t (2.20) − m˙ 2ρ3 1/5 r3 8/33 ρ(r, t) = 0.628 0 t−4/5 1 (2.21) v6 − R (t)3 w C after correcting a small typographical error in their given expression for the velocity. This is the solution for the structure of the interaction region for the s = 0 (constant IS density) case as the Mach number goes to inﬁnity, i.e. the ambient pressure is neglected. Hereafter, this will be referred to as the zero-ambient pressure (ZAP) model. One notable feature about these structures is that the density goes to zero as r approaches

RC from above but diverges to infinity upon approach from below (see Fig. 2.1, left). Pressure and fluid velocity are finite and continuous across the boundary. Using the analytic expression for ρ(r) (equation 2.21) we may define a characteristic width of the density peak in the inner region by ∆r R r(2ρ ) = R [1 2−33/8(1 R 3/R 3)]1/3 , where r(ρ) is the inverse of equation ρ ≡ C − 2 C − − 2 C ρ(r) as given in 2.21. The width of region III is given by the integration of the equations 2.10 and

2.11; it is ∆rIII = 0.165RC . The temperature varies with the inverse of the density. In reality the extreme temperature discontinuity will smooth out due to ﬁnite heat conduction and Rayleigh Taylor (RT) instability resulting in mixing. Fig. 2.3 shows an example in nature of outﬂow from a stellar system mixing with the ambient environment by instability.

2.3 Self-Similar Solutions for s=2

Chevalier and Imamura [20] found self-similar solutions for the interaction regions of colliding winds emanating from a common point source. Their work is similar to what I do here. In the case

m˙ 1 of s = 2, the density in region IV is of the form ρ0(r)= 2 , assuming it is of a prior stellar wind 4πvw,1r with parametersm ˙ 1 and vw,1. In Eq. 2.9, an expression was found for RC given the assumption

25 Figure 2.3: Optical HST image of the Cat’s Eye planetary nebula, or NGC6543, representing the ending evolutionary stage of a Red Giant star. A few phases of mass loss are clearly evident. At the center, a fresh, axial-symmetric outburst 1000 yr old is seen expanding ≈ into an ambient medium, most likely produced by prior mass loss. (X-ray images of the inside region reveal hot structures of 106 K, most likely resulting from the interaction ∼ of diﬀerent mass loss phases.) On the far outside is a ﬁlamentary structure–the remnants of a much older mass loss episode that had broken into fragments due to instabilities and mixing.

that vw >> R˙ C . Although it gives the correct form of RC in that regime, a more general form is allowed by the Π theorem for s = 2. The parameter set (excluding γ) is vw,m ˙ , t, RC ,m ˙ 1, and vw,1.

The parametersm ˙ 1 and vw,1 must appear separately and not in the combinationm/v ˙ w,1. This condition must be imposed because the ambient medium is moving outward with velocity vw,1 and so a Galilean transformation depending on vw,1 will need to be implemented on both the outer and inner shock jump conditions. The wind velocities will therefore have a separate eﬀect. Only for vw >> vw,1 will the individual dependence of vw,1 vanish and Eq. 2.9 will be applicable for s = 2. The independent variables r and t for s = 2 can combine to form one Π group:

26 Π2 = r/(vwt). (2.22)

The characteristic scales RC , R1, and R2 will be at certain speciﬁc values of Π2, up to a dimensionless proportionality constant. In this case, however, two extra Π groups come out of the parameter set:m/ ˙ m˙ 1 and vw/vw,1. Therefore, the characteristic scales (and the structures) will depend on these ratios. I thus obtain the following expression for the contact discontinuity, the outer shock and the reverse shock:

RC = K2C (Πm˙ , Πvw )vwt (2.23)

R1 = K21(Πm˙ , Πvw )vwt (2.24)

R2 = K22(Πm˙ , Πvw )vwt (2.25)

with

Πm˙ =m/ ˙ m˙ 1 (2.26)

vw Πvw = 1. (2.27) vw,1 −

vw K2C,1,2 are functions to be determined numerically, and the combination 1 is chosen vw,1 − instead of vw/vw,1 to make up Πvw . The boundary conditions become

2γ(γ 1) v t 2 χ = − w,i 1 (2.28) i (γ + 1)2 R − i 2 v t 2 P = w,i 1 (2.29) i γ + 1 R − i 1 v t U = 2+(γ 1) w,i (2.30) i γ + 1 − R i where the subscript i is either 1 or 2, referring to the outer and inner shock front boundaries, respectively, and here v = v . Since the R s are t, the time dependence cancels out of the w,2 w i ∝ boundary conditions.

27 By requiring pressure continuity across RC , I acquire an analytic expression for the inner shock radius as a function of time:

−1 + tvw,1 P2 Pc ρc,1 R2 = vwt 1+ 1 . (2.31) − R P − P ρ " 1 s c 1 c,2 # The outer shock radius can be given by

+ − R1 =(ηc /η1)(η2/ηc )R2, (2.32) and likewise + RC =(η1/ηc )R1. (2.33)

− + + − The quantities (P2/Pc ), (Pc /P1), (ηc /η1) and (ηc /η2) are found from integrating equations 2.10, 2.11 and 2.12 in either region III (+) or II ( ). However, in order to calculate them, initial guesses − of R1, R2 and RC are required. A consistent solution is obtained by damped fixed-point iteration. As Fig. 2.1 shows, the structures are qualitatively different for s = 2 than for when s = 0. The density goes to infinity as one approaches RC from either side while the pressure is finite. Formally, the temperature therefore goes to zero.

28 CHAPTER 3

BOUNDARY CONDITIONS FOR S=0

3.1 Limitations of the ZAP Model

As discussed above, the self-similar ZAP solution depends on the ambient density, ρ0, and the kinetic energy flux, Lw at the inner boundary. For constant density ISM (s=0), the ZAP solutions are not valid for t 0 and t . This is because physical assumptions break down and the → → ∞ results are unphysical solutions. For small times, this can be seen as follows: As shown in Table 3.1, R t3/5 and R t2/5. C ∝ 2 ∝ This implies the velocity of the reverse shock would go against infinity. No interaction is possible if the wind cannot overrun R2, therefore the description becomes unphysical. Also, the reverse shock R2 is greater than the contact discontinuity RC for small enough t. Therefore, no self-similar ˙ m˙ solutions exist for R2 vw and R2 RC , i.e. times shorter than t 3 . This critical time ≈ ≈ ∼ ρ0vw can be defined as when R2 and RC , as defined by Eqs. 2.16 and 2.17, areq equal:

m˙ tcr = 1.86 3 . (3.1) sρ0vw For my application to SN environments at times greater than when interaction takes place within the progenitor system, this hardly poses a limitation. For the reference model in the accretion disk wind case, the MS wind, and the RG-like wind (see Tables 7.1, 7.2 & 7.3), tcr is 3.8 yr, 0.17 yr, and 1.2 104 yr, respectively. These times are short compared to the wind duration times of the × progenitor system 3 105 yr (see Chapter 6). ∼ × For large times, the solution depends on the outer boundary condition, namely the pressure of the ambient medium. In the following, I will consider the validity of solutions at large times, and develop approximations which allow us to study environmental properties. I will compare solutions with and without ambient pressure, referring to those as zero-ambient pressure (ZAP) and constant-ambient pressure models (CAP), respectively. For t , R goes out indeﬁnitely according to the self-similar solution as external pressure is → ∞ C neglected. In reality, the outer pressure will increasingly conﬁne the expansion of the structure and,

29 thus, R . In the self-similar solution without ambient pressure, p decreases with R 2/t2 t−4/5 C 1 1 ∝ (Eq. 2.20) and, eventually, it will drop below the ambient pressure of the physical medium. As reference, I deﬁne the pressure-equilibration time tp as the time at which the pressure just inside

R1 equals the ambient (constant) pressure, p0. It is given by

− mv˙ 2 p 5/4 t = 1.23 w λ2(γ + 1) 0 . (3.2) p ρ × ρ s 0

Table 3.1: For constant interstellar medium (ISM) and t . tp, the relations for the radius of the contact discontinuity RC , the forward and reverse shock, R1 and R2, the ﬂuid velocity at the shell uc and its mass column density τm, and the density of the inner void, n2, all as a function of the density of the environment n0, the mass lossm ˙ , its wind velocity v , and duration t. For n , I assumed (R /R )3 1 (see Eq. 2.21). Asymptotic w 2 2 C ≪ (t ) values of the reverse shock R ∞, and the particle density n ∞ are obtained from → ∞ 2, 2, the CAP model and given below. p0 is the ambient pressure.

− 1 2 1 3 5 5 R n vwm˙ 5 t 5 C ∝ 0 − 1 2 1 3 5 5 R n vwm˙ 5 t 5 1 ∝ 0 − 3 1 3 2 10 10 R n vw m˙ 10 t 5 2 ∝ 0 − 1 2 1 2 5 5 − u n vwm˙ 5 t 5 c ∝ 0 3 − 6 2 4 5 5 − n n vw m˙ 5 t 5 2 ∝ 0 4 2 1 3 5 5 τ n vwm˙ 5 t 5 m ∝ 0 3 5 − 1 − t n 4 v m˙ 2 p 4 p ∝ 0 w 0

mv˙ w R2,∞ 0.30 ≈ p0 2 n ∞ 3.9pq/v 2, ≈ 0 w

In Fig. 3.1, I give the evolution of the basic physical quantities as a function of time for models with parameters typically for AD-, RG-like and MS-star winds. For the reference models (see Tables 7.1, 7.2 and 7.4), assuming an ideal gas ambient pressure characterized by a temperature

30 T = 104 K, we obtain in the AD wind case, the RG-like wind, and the MS wind t = 7.45 105 0 p × yr, t = 2.36 105 yr, t = 124 yr, respectively. p × p

4 1 4 1 4 1

0.5 0.5 0.5

RCS RCS 3 R1S 0 3 0 3 R1S 0 R2S R2S (pram/p0)S (pram/p0)S RCP RCP R R 1P -0.5 -0.5 1P -0.5 R2P R2P ) ) (pram/p0)P ) (pram/p0)P 0 0 0 /p /p /p (ly) ram ram ram (ly) + 2 (ly) + 3 10 2 -1 2 -1 2 -1 (p (p (p 10 10 10 10 10 Log Log Log Log Log Log -1.5 -1.5 -1.5

RCS 1 -2 1 R1S -2 1 -2 R2S (pram/p0)S RCP R -2.5 1P -2.5 -2.5 R2P (pram/p0)P

0 -3 0 -3 0 -3 -0.4-0.2 0 0.2 0.4 0.6 0.8 1 -0.4-0.2 0 0.2 0.4 0.6 0.8 1 -1 -0.5 0 0.5 1 1.5 2 2.5 3

Log10(t/tp) Log10(t/tp) Log10(t/tp)

Figure 3.1: Structure feature comparisons of ZAP and CAP models for given sets of parameters as a function of t/tp for the reference models of the AD-, RG-like and MS wind (right to left). Dependence of the radii of the outer and inner shock R1,2, of the contact discontinuity RC , and the ram pressure pram/p0 are shown as a function of the duration of the wind t normalized to the time tp at which the inner and outer pressure are equal. The functions for the zero ambient and constant pressure model are indicated by the small and large symbols, respectively.

As an extreme case and benchmark for modiﬁcations, I use a MS wind with parametersm ˙ and vw similar to the Sun (see reference model in Table 7.4). tp is some 124 years only. In the ZAP model, the contact discontinuity RC and the location of the reversed shock are about 23 and 20 ly, respectively. The solar wind has similar properties but the termination shock is at about 75 to 95 AU (0.0012 to 0.0015 ly) based on Voyager 1 [209]. The discrepancies can be understood due to the ambient pressure not being taken into account. Moreover, for times much larger than tp, R˙ c where c is the ambient sound speed. We would therefore expect turbulent instabilities C ≪ s s

31 which results in mixing. Ignoring ambient pressure for the MS (solar) model results in the contact discontinuity overrunning the the heliopause in about 10 years. Therefore, it is imperative that we consider how to account for ﬁnite ambient density in order to get realistic solutions.

A ﬁrst order estimate for the solution may be obtained by stopping the time integration at tp for a model without ambient pressure, and neglecting the further evolution. For RC and R2 and with this crude approximation, I obtain the right order of magnitude with R2 = 190 AU compared to the solar value of 75 to 95 AU.

3.2 The CAP Model

In the following, I will construct physically motivated boundary conditions for moderate log(t/tp) < 1 ... 3, and discuss the uncertainties estimated by a comparison between the resultant models and the ZAP model.

Besides simply truncating the solution at tp, there is a way to approximately incorporate the ambient pressure in a way that retains the self similar solution at any time, although with a modified ambient pressure profile. In order to see how, we first notice the Rankine-Hugoniot jump conditions (in shock rest frame):

2ρ u 2 (γ 1)p p′ = 0 0 − − 0 (3.3) γ + 1 2γp +(γ 1)ρ u 2 u′ = 0 − 0 0 (3.4) (γ + 1)ρ0u0 (γ + 1)ρ 2u 2 ρ′ = 0 0 (3.5) 2γp +(γ 1)ρ u 2 0 − 0 0 where the 0 subscripts denote pre-shock and the primed variables are post-shock quantities. After applying the substitutions in 2.5, 2.6, and 2.7, we see that the boundary conditions can remain −4/3 constant with respect to space and time if p0 follows a spatial power law with r (or, equivalently, a time power-law of t−4/5. This is because R t3/5.). An effective pressure power-law environment 1 ∝ can be defined by requiring that, at a certain final time tf , the thermal energy contained within

R1(tf ) in our eﬀective environment is equal to the thermal energy in the physical, constant ambient

5p0 tf 4/5 density at the same radius. This is found by volume integration along R1(t): peﬀ (t)= 9 t .

Although tf can be thought of as a constant parameter used to deﬁne the environment, in practice

32 5p0 tf = t and peﬀ (t) = 9 . However, I note that, as the solution advances forward in time, this means that tf and the boundary condition will vary as well, meaning the solution will not be truly self-similar, i.e. the morphology changes with time (see Fig. 5.1). The solution obtained in this way is, in fact, a series of snapshots of self-similar solutions where tf is equal to the instantaneous time t. Using this parameterisation, the Buckingham Π theorem gives us the following:

mv˙ 2 t3 1/5 R = K (Π ) w (3.6) C 0C T ρ 0 mv˙ 2 t3 1/5 R = K (Π ) w (3.7) 1 01 T ρ 0 2 1/5 mv˙ w ΠT = (3.8) ρ (R T )5/2t2 0 s 0 where the Ks are to be determined numerically and the ideal gas relation was used: Rs is the gas constant divided by the mean molecular weight and T0 is the physical ambient, constant tempera- 4 ture. T0 will be taken to be 10 K typical for ISM gas [154]. Note that, using Eqs 3.8 and 3.2, we −2/5 have ΠT = 2.50(t/tp) . The outer boundary conditions of the ODEs are then given as:

(2γ (γ 1)/M 2)(2/M 2 + γ 1) χ = − − − (3.9) 1 λ2(γ + 1)2 (2γ (γ 1)/M 2) P = − − (3.10) 1 λ2γ(γ + 1) 1 1/M 2 U = 2 − (3.11) 1 λ(γ + 1) where the Mach number M is given by:

R (t) 9ρ M = 1 0 . (3.12) λt 5γp r 0 An initial guess of R1 is required in order to numerically solve the ODEs and obtain the structure, therefore iteration is necessary in order to obtain a consistent solution. Following the method of Weaver et al. [240] and using this modiﬁed boundary conditions, I obtain the functions

U, Ω, and P for ﬁnite Mach numbers (ﬁnite ΠT ). They are plotted for various values of ΠT in Figs.

3.2. According to Fig. 3.2, the morphologies of the curves for ΠT = 10 and 5 are similar to those in Fig. 2.2.

33 We obtain the following expressions for the radii and inner structure proﬁle:

m˙ 3/10 R (t) = 1.13(βα)3/2 v 1/10t2/5 (3.13) 2 ρ w 0 mv˙ 2 1/5 R (t)= α/21/5 w t3/5 (3.14) 1 ρ 0 RC (t)= βR1 (3.15) 11 R 3 r u(r, t)= C + (3.16) 25 r2t 25t 5 p(r, t)= (1/4˙m2v 4ρ 3)1/5t−4/5 (3.17) 22π(βα)3 w 0 − 0.274 m˙ 2ρ 3 1/5 r3 8/33 ρ(r, t)= 0 t−4/5 1 (3.18) (αβ)3 v 6 − R 3 w C 1/5 −1/λ 1 5 ηc + where α = + , β = , and Pc and ηc are evaluated at the contact discontinuity. β 22πPc η1 Comparison of Eqs. 3.14, 3.15 with Eqs. 3.7, 3.6 give K = α and K = βα . We can then 01 21/5 0C 21/5 deﬁne a proportionality for the reverse shock:

m˙ 3/10 R = K (Π ) v1/10t2/5 (3.19) 2 02 T ρ w 0 where 3/2 K02 = 1.39K0C . (3.20)

Hereafter, I refer to this model as the constant-ambient pressure (CAP) model. For p 0, 0 → M , α 0.88, β 0.86, and the ZAP model is reproduced (Eqs. 2.16-2.21) For the reference → ∞ → → models for an AD-, RG-like and MS-wind, a comparison of the basic properties between the ZAP and CAP models as a function of t/tp is shown in Fig. 3.1. Qualitatively, the main diﬀerences are as follows: For CAP models, R2 and RC are smaller and R1 is larger than for ZAPs, and R2 goes to a constant value for large t/tp. For the range shown and for the RG-like wind, R2 becomes larger than R at about t/t 0.5 marking the regime of “unphysical” solutions already discussed C p ≈ − above for the ZAP model. The functional relations appear to be similar and, in fact, they are identical as a consequence of the Π theorem. The relative shifts between the various quantities are given by proportionality factors which, in turn, depend only on the basic parameters, namely vw, m˙ and n0. For the CAP model, the proportionality constants have to be determined numerically (Fig. 3.3). A further consequence of the Π theorem is that the diﬀerences are only a function of

34 t/tp and do not depend individually onm ˙ , vw and n0 (Fig. 3.6). For a constant density medium, the characteristic parameters can be directly obtained using Fig. 3.3. The detailed solutions for the reference models are shown in Figs. 3.4 & 3.5. The morphology of the envelopes does not change for a wide range of parameters and time. As discussed above in case of the MS star wind, however, we must expect strong mixing for t/tp >> 1 in CAP models.

The solution becomes unphysical for r & RC in the regime of a weak shock. Both the ZAP and CAP models share one basic limitation discussed above, namely the divergent velocities and the crossover of R and R as t 0. 2 C →

35 1 1 log U Pi = 10. M = 4.87 log U Pi = 2. M = 1.45 log P T log P T log Ω log Ω 0.5 0.5

0 0

-0.5 -0.5

-1 -1 1 1.05 1.1 1.15 1.2 1 1.2 1.4 1.6 1.8 2 2.2

r/RC r/RC

1 1 log U Pi = 5. M = 2.55 log U Pi = 1. M = 1.387 log P T log P T log Ω log Ω 0.5 0.5

0 0

-0.5 -0.5

-1 -1 1 1.05 1.1 1.15 1.2 1.25 1.3 1 2 3 4 5 6

r/RC r/RC

1 2 log U Pi = 2.5 M = 1.56 log U Pi = 0.5 M = 1.385 log P T 1.5 log P T log Ω log Ω 0.5 1 0.5 0 0 -0.5 -0.5 -1 -1.5 -1 -2 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 2 4 6 8 10 12 14 16 18

r/RC r/RC

Figure 3.2: The scale-free quantities U, P , and Ω as deﬁned in Eqs. 2.5-2.8, plotted as a function of r/RC according to the CAP model. Here they are shown for various values of ΠT and the corresponding Mach number M, from ΠT = 10 (M = 4.87), the corresponding curves of which closely resembles those in Fig. 2.2, to ΠT = 0.5 and M = 1.385, which is near the critical Mach number.

36 1 0.8 Log(K0C) 0.6 Log(K01) 0.4 Log(K02) 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -0.5 0 0.5 1 1.5 Π Log( T)

Figure 3.3: Proportionality factors for the characteristic distances as a function of ΠT for constant density environments and the CAP model. Note that all factors are constant for ZAP models with values corresponding to large ΠT (see Chapter 3).

3 3 3 2 Log [u/(100 km/s)] 2 Log [u/(100 km/s)] 2 Log [u/(100 km/s)] Log (p/p1) Log (p/p1) Log (p/p1) 1 Log n 1 Log n 1 Log n 0 0 0 -1 -1 -1 -2 -2 -2 -3 -3 -3 -4 -4 -4 -5 -5 -5 n = 2.99x10-4 cm-3, R = 13.7 ly, R = 15.9 ly n = 3.63x10-4 cm-3, R = 11.8 ly, R = 13.8 ly n = 5.20x10-4 cm-3, R = 11.4 ly, R = 22.0 ly -6 2 c 1 -6 2 c 1 -6 2 c 1 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2

r/Rc r/Rc r/Rc

Figure 3.4: Structure of model 3 for AD wind (see Table 7.1). I show the ZAP (left) with t = 3 105yr and at t = t (middle), and the CAP model at t = 3 105yr. The overall × p × structure is similar within the parameter range.

37 1 1 1 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] 0.5 Log (p/p1) 0.5 Log (p/p1) 0.5 Log (p/p1) Log n Log n Log n 0 0 0

-0.5 -0.5 -0.5

-1 -1 -1

-1.5 -1.5 -1.5

-2 -2 -2 n = 5.02x10-2 cm-3, R = 5.45 ly, R = 6.35 ly n = 8.01x10-1 cm-3, R = 1.18 ly, R = 1.38 ly n = 3.75x10-1 cm-3, R = 2.75 ly, R = 20.3 ly -2.5 2 c 1 -2.5 2 c 1 -2.5 2 c 1 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

r/Rc r/Rc r/Rc

Figure 3.5: Same as Fig. 3.4 but for RG-like winds (see Table 7.2).

0.5 RC

) R1 s 0 R2 |/R p -0.5 - R s -1

-1.5 Log( |R -2

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Log(t/tp)

Figure 3.6: Fractional difference in RC between models of zero ambient pressure (RC,s) and finite pressure models (RC,p) in the parameter space of the mass lossm ˙ , the wind velocity vw, and the environment density n0 as a function of log(t/tp). The plots visualize the (Π-theorem) as discussed in Chapter 3: The difference depends on (t/tp) only. The ratios between scale-free variables are constant throughout the parameter space. The Π- theorem applies also to R1. In the lower right is shown the fractional difference of RC (red), R1 (blue) and R2 (magenta) between the ZAP and CAP models as a function of log(t/tp).

38 CHAPTER 4

THE EXISTENCE OF SOLUTIONS

Here, I will provide the range for which self-similar solutions exists using the Π theorem.

4.1 Case s =0

For CAP and s = 0, there is one Π group given by Eq. 3.8. In Fig. 3.3, the proportionality factors K0C , K01, and K02 pertaining to the characteristic radii RC , R1, and R2 are given as a function of Π . The character of these K functions changes at the point Π 3 or, equivalently, T T ≈ t t . We refer to the parameter space where Π is larger than 3 as regime I. The region of ≈ p T ≈ parameter space where Π is less than 3 is referred to as regime II. They are discussed below. T ≈ Regime I: As T 0, Π . In this limit the ZAP model is reproduced, and the K factors 0 → T → ∞ converge to their respective constants of that model (see Eqs. 2.16, 2.17, and 2.18). Regime II: As Π 0, we go to the limit where T and/or t . Now, as t , T → 0 → ∞ → ∞ → ∞ the dependence is expected to change. As the outer region sweeps up increasing amount of mass from the ISM, the contact discontinuity will slow down until the effect of the outer pressure p0 dominates over that of the outer density ρ0. Thus, p0 will become a relevant parameter while ρ0 ceases to be relevant. The dependency of R will then take on a different form: R (˙mv2 t/p )1/3. C C ∝ w 0 This implies that K Π2/3 in this regime. Likewise, as R˙ lowers to below the ambient sound 0C ∝ T C speed, cs, R˙ 1 will slow to a constant velocity of order of the sound speed cs. This implies that K 1/Π in this regime. R˙ is actually a little larger than c , according to the CAP model. There 01 ∝ T 1 s is a minimum Mach number, M = 1.3849, (see Eq. 3.12) below which no solution exists. Finally, the pressure in the inner region II will lower until it converges to p0. The shock jump condition at the inner boundary gives for the pressure p ρ v2 where ρ m/˙ (v2 R2). This means that for 2 ∼ 2 w 2 ∼ w 2 constant p = p , R (˙mv /p )1/2, and K Π . These power law dependencies can be seen 2 0 2 ∼ w 0 02 ∝ T as linear relationships in log space of Fig. 3.3 on the left hand side. The effective pressure profile defined in Chapter 3 therefore gives physically reasonable results for the location of the shock radii and the contact discontinuity. However, as stated in Chapter 3, ambient turbulence will play a real

39 role in altering the structures. For t>>tp (small ΠT ), mixing in the outer layers with the ambient environment by turbulent ISM will render the RC and R1 values unphysical.

4.2 Case s =2

Two Π groups exist for s = 2, thus possible solutions are a combination of Π groups with −2 K2 = K2(Πm˙ , Πvw ). given by equations 2.26 and 2.27. For r ambient density proﬁles, the shell velocity range is suﬃcient to determine the relation betweenm ˙ 1 and vw,1 for the prior mass loss.

In Figs. 4.1, 4.2 & 4.3, I show K2C ,K21 and K22 as a function of the wind parameters covering the entire range discussed in this paper. The character of the K functions can be described with a few dividing lines in the Π space. For v v , the interaction will be small, and it is expected that the contact discontinuity and the w ∼ w,1 shock radii will all propagate with velocity v . Thus, K 1 for small Π . This region is below ∼ w → vw Log(Π ) 1, or at the very bottom of the ﬁgures. This regime is not of interest to us here since vw ≈− the interaction will be small and not observable. The regions of interest are as follows:

Regime I: For R˙ C >> vw,1, the individual dependence on vw,1 should vanish from the set of relevant parameters since the contribution to the boundary conditions will be small (see Eqs. 2.28,

2.29, and 2.30). It should however remain in the combination ρc =m ˙ 1/vw,1 (see Eqs. 2.5, 2.6, and 2.7), which describes the density of the ambient medium. Thus, in this regime, the K functions should depend solely on one Π group mv˙ w,1 = Π /(Π +1) instead of the ratiosm/ ˙ m˙ and v /v m˙ 1vw m˙ vw 1 w w,1 separately. The K functions are then invariant for constant values of this group, i.e. Π Π +1. m˙ ∝ vw This symmetry is seen in the diagonal contours on the upper right of the ﬁgures. This region is to the right of the line log Π log Π . For log Π & 0 and Π large, K is greater than 1. This vw ∼− m˙ vw m˙ 21 is due to the sound speed in the shocked outer region, causing R1 to propagate even faster than vw. In Regime I, we have no power law relation between the wind and environmental parameters, and the values of K2C ,K21 and K22 need to be interpolated in the ﬁgures or can be calculated by SPICE.

Regime II: Form/ ˙ m˙ 1 small and vw/vw,1 not too large, the inertia of the outer wind region will be too great for the ongoing wind to accelerate, and R˙ C should be close to vw,1. This means K (Π +1)−1 in this regime, which is independent of Π . This is seen in the horizontal contours → vw m˙ on the left of Figs. 4.1, 4.2 & 4.3. This region is to the left of the line log Π log Π . vw ∼− m˙

40 In Fig. 4.4, the variation of the cuts for various Πm˙ . K2C and K21 can be well described by single functions. K22 needs two descriptions valid at low and high ratios of vw/vw,1 separated at Π 0.5. The resulting power law dependencies are given in Table 4.1. vw ≈

Figure 4.1: Value of K as a function of v and log K (Π , Π ) where Π = (v 2C w 2 m˙ vw vw w − vw,1)/vw,1) and Πm˙ =(˙m/m˙ 1). K2C is close to constant in regime II, pertaining to low mass winds running into high mass loss wind, i.e. log(Πm˙ ) < 0 (see text for details).

41 Figure 4.2: Value of K as a function of v and log K (Π , Π ) where Π = (v 21 w 2 m˙ vw vw w − vw,1)/vw,1) and Πm˙ =(˙m/m˙ 1). K21 is close to constant in the regime of low mass winds running into high mass loss wind, i.e. log(Πm˙ ) < 0.

Table 4.1: Same as Table 3.1 but for an environment produced by wind from prior mass loss (s=2). The index 1 corresponds to the prior wind. The relations are valid for high velocity winds running into environments produced by low velocity winds (see Fig. 4.1). −4 For RC , R1, and u1, the relations are valid form/ ˙ m˙ 1 . 10 . For R2 and ρ2, it is valid for m/˙ m˙ . 10−4 and v /v between 10 and 170. τ is calculated by formal integration 1 w w,1 ∼ m in SPICE.

R v t C ≈ w,1 R v t 1 ≈ w,1 1.42 vw,1 R2 1.21 vwt ≈ vw−vw,1 u γ(γ + 1)−1v c ≈ w,1 −2.84 −4 vw,1 −2 n2 0.385mv ˙ t ≈ w vw−vw,1 −1−1 τm =m ˙ 1vw,1t K2,τ (Πm˙ , Πvw )

42 Figure 4.3: Value of K as a function of v and log K (Π , Π ) where Π = (v 22 w 2 m˙ vw vw w − vw,1)/vw,1) and Πm˙ =(˙m/m˙ 1). K22 is close to constant in the regime of low mass winds running into high mass loss wind, i.e. log(Πm˙ ) < 0.

43 0 0

Π . Π . m = 10 m = 10 -0.5 Π . -0.5 Π . m = 1 m = 1 ) Π . ) Π . 21 2C m = 0.1 m = 0.1 -1 -1 Π . Π . m = 0.01 m = 0.01 Log(K Log(K Π . -4 Π . -4 m = 10 m = 10 -1.5 -1.5 Π . -6 Π . -6 m = 10 m = 10 Π -1 Π -1 K2C = ( v +1) K21 = ( v +1) -2 w -2 w -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Log(Π ) Log(Π ) vw vw

0 0

-0.5 -0.5 Π . m = 10 -1 -1 ) Π . ) m = 1 22 22 -1.5 Π . = 0.1 -1.5 m Π . -6 m = 10 Log(K Π . = 0.01 Log(K -2 m -2 K = (Π +1)-1 22 vw Π . = 10-4 m K = 1.21(Π )-1.42 -2.5 -2.5 22 vw Π . -6 m = 10 -3 -3 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Log(Π ) Log(Π ) vw vw

Figure 4.4: K2(C,1,2) as a function of the ratio between inner and outer wind velocity −6,−4,−2,−1,0,1 described by Πv,w as obtained by cuts at log(Πm˙ ) of 10 indicated by progressively thicker lines. The exact solutions are given in red. I give the solutions for K in comparison to the ﬁts. The approximations are given (magenta and blue, dotted).

44 CHAPTER 5

TIME SERIES OF SOLUTIONS

In this chapter is shown time series of hydrodynmaical structures for wind interaction with environments pertaining to the two cases, s = 0 and s = 2. Examining the characteristics of these structures will reveal the character of interaction with a SN, and could allow one to infer the age of the system. Other time series for s = 0 are available in the appendix and are not discussed in this chapter, although many of the same principles apply to those as well.

5.1 s=0 Time Series

Fig. 5.1 shows a time series of a typical AD structure using the CAP model (s = 0), running into −3 an ambient environment of particle density n0 = 1 cm . It has the same vw, andm ˙ parameters as the reference model for the AD wind case (Tab. 7.1). The time range is from t = 0.01 tp to t = 100 t , where t for this set of parameters is 7.45 105 yr (Eq. 3.2), and the time increment is p p × linear in log t. The time series begins at the upper left and then proceeds down the column, where the times are marked on the bottom right of the ﬁgures. The horizontal axis scales with the outer shock radius R1. The characteristic four-region structure described in Chapter 2 is preserved throughout the entire time sequence. The scales vary, however. The outside region expands rapidly as the quantity R /R goes from constant in time to t2/3 (see Chapter 4). This beings to happen at about 1 C ∝ t 0.2 Myr, or t 0.2 t . ∼ ∼ p Behavior of density For this model, there is a particle underdensity in the inside region (II) compared to the outside region (III). The density in the inner region decreases as a function of time. It begins at about 10−3 cm−3, and as the time progresses, the inner density structure (red dashed line) drops steadily and uniformly until t 1.68 Myr, or t 2 t , where it reaches a constant value ∼ ∼ p of 4 10−5 cm−3. The density in region III monotonically increases from r = R to R , with ∼ × C 1 the density at R1 slowly decreasing with time, starting at 4 times the outer density. The outer density structure “stretches” as R1/RC increases with time. As noted in Chapter 2, formally, there

45 is an extreme discontinuity of density (and temperature) at RC . Although the curves appear to converge from above and below RC , the inner density spikes at RC while the outer density vanishes. In reality, heat conduction and the RT instability will smooth this discontinuity out, so it should not be observable in nature. Molecular diﬀusion of particles from the outer layers into region II will also lower the density contrast between the two regions.

Behavior of velocity For t << tp, the velocity structure (magenta) monotonically decreases throughout the entire interaction region. However, starting at about t t At the inner shock, a ∼ p minimum begins to form at r R , which then moves inward to R as time develops. The fluid ≈ 1 C velocity continues to steadily increase from the minimum to the velocity at R1. At the reverse shock R2, the fluid velocity drops to 1/4 vw, independent of time. The fluid velocities at RC and

R1 steadily decrease with time. Behavior of pressure The pressure curve (blue) remains almost ﬂat throughout the entire interaction region for all times. The pressure in region II is exactly ﬂat by an a priori assumption that the region is isobaric (see Chapter 2). The pressure in region III increases slightly with r until

R1 is reached. However, the pressure curve is normalized to the pressure p1 just inside the outer shock, which varies initially with t−4/5 (Eq. 2.20). As time progresses, this pressure decreases ∝ to a constant value. Observables Radiative absorption eﬀects are much more likely to be seen in the outer layers, where the density is higher. The density in region III drops with time, however the physical width increases much more quickly. Therefore, the mass column density (τmass)in region III will increase with time, as will the optical depth and equivalent width of the lines (see Chapter 7, also Table 7.1). To a ﬁrst order, τ ρ ∆R, where ρ is the average density in the region and ∆R is the width mass ∼ avg avg of the region. For the earliest and latest time snapshots this is 10−6g/cm2 and 10−3 g/cm2, ∼ ∼ respectively. For the snapshot at t = 0.977 Myr (t tp) it is estimated at τ 10−5 g/cm2. ∼ mass ∼ SPICE can calculate τmass by direct integration.

The (mass weighted) velocity dispersion (σu) in region III decreases with time until it reaches a minimum at t 1.6 t after which it begins to increase again. From the ﬁgures, σ is given by ≈ p u the width of the velocity curve in the region of highest density inside of region III. For the earliest time snapshot it is estimated at σ 5 km/s. For the snapshot at t = 0.977 Myr (t tp) it is u ∼ ∼ estimated at σ 0.1 km/s. For the ﬁnal snapshot σ 0.2 km/s. u ∼ u ∼

46 In conclusion, for this model, the total absorbed power of the absorption lines due to region III will increase with the age of the system. The width of the lines due to velocity dispersion will correspondingly decrease. As the ﬂuid velocity of region III decreases with time, the narrow lines will be less and less blueshifted. However, as noted in Chapter 3, mixing in the outer layers will signiﬁcantly change the structure of region III for t>>tp. There may also be observable kinematic interaction of the SN ejecta with the structure of region II. The SN will produce streaming particles at the 10 MeV scale, capable of ionizing atoms by impact. If the thermal equilibrium timescale in region II is long compared to the recombination timescale, recombination can produce emission up into X-rays. This will depend on the density, which, as already stated, decreases with time. The interaction will persist until the ejecta has reached R , which will happen 100 yr after the explosion. At this point hydrodynamic interaction C ∼ with the dense region III will heat up the material and create thermal emission.

5.2 s=2 Time Series

Fig. 5.2 shows a time series of the same AD wind discussed above, with the exception that −7 the outer environment is defined by wind from prior mass loss withm ˙ 1 = 10 M⊙/yr and vw,1 = 30 km/s. The time range is from t = 3000 yr to 3 107 yr, and the time increment is linear in × log t. The horizontal axis co-moves with the structure and changes scale accordingly. The same characteristic four-region structure is seen in this case (Chapter 2) and, unlike in the s = 0 scenario, the entire interaction region is self similar and scales uniformly with time. As time progresses, the length scale grows much larger in this time series than for the s = 0 series. Behavior of density For this model, the particle density curves are normalized to 10−6 cm−3. The density of the inner region (II) is consistently lower than that of the outer region (III) by about an order of magnitude, and both densities are lower this time series than for any snapshot in the s = 0 time series of Fig. 5.1. The density structure features a spike at RC with width of order 0.03 R . At the earliest time sequence, the inner density is at 10−6 cm−3 while the outer density C ≈ is 10−5 cm−3. Both densities lower as time progresses. In the final time snapshot, the density ≈ drops to 10−14 cm3 and 10−13 cm3 for the inner and outer, densities, respectively. ≈ ≈ Behavior of velocity The velocity profile of the model does not change with time, besides scaling with the solution. In fact, this is true for any model for the s = 2 case. This is because of

47 the functional form of Eq. 2.5: the r/t factor is time independent since the size of the structure scales with time, therefore u(r, t) U(r/t). At the inner shock R , the velocity drops by a factor ∝ 2 of 4, and steadily decreases until r = R , where the velocity drops again by a factor of 8. The ∼ 1 ∼ velocities of R2, RC , and R1 are all constant with time. Behavior of pressure The pressure curve also does not change with time but this is due to the normalization to the outer shock pressure p . The pressure p t−2 for s = 2 (see Eq. 1 ∝ 2.7), therefore it will uniformly decrease with time. The curve is nearly ﬂat throughout the entire interaction region. Observables Radiative absorption is more likely to be seen in the outer region III, where the density is higher, especially in a thin spike just around RC . The mass column depth τmass is given roughly by the width of the shell times the density. The physical width increases t as the ∝ density decreases t−2 (see Eq. 2.6, with r t) therefore τ decreases t. For the earliest ∝ ∝ mass ∝ time snapshot it is estimated at τ 10−11 g/cm2. For the time snapshot at t = 0.39 Myr, mass ∼ τ 10−13g/cm2, and for the ﬁnal snapshot at t = 30 Myr, τ 10−14g/cm2. These values mass ∼ mass ∼ are much smaller than for the s = 0 model studied above. The optical depth and equivalent width will also decrease with time.

The velocity dispersion σu is constant in time (as is true for all s = 2 models). It is difficult to estimate its value from the figures because the profile is so flat in the region where the density is highest, from r = R to r R . It will be very small in region III. C ≈ 1 In conclusion, for this model, absorption lines will be very weak and very thin. If they can be observed at all, their intensity will diminish with time as their width in velocity space remains constant. Their blueshift will be constant, also. The kinematic interaction of the SN ejecta in region II, discussed in the previous section, will be much weaker in this case since the densities are much lower than for the model in Fig. 5.1. The hydrodynamic interaction with the structure will also be much weaker. The realization of this model is limited to situations where the pre-blown (s = 0) RG wind structure is larger than the outer shock front R1, i.e. the interaction is in region I of the interaction region of prior mass loss. Very few of the s = 0 RG models discussed in this work have R2 greater than a few light-years (see Table 7.2). Therefore any structures evolving beyond the first few time sequences for this model are unlikely to be realized in nature. What is more likely is, as discussed

48 in later chapters, is that the interaction region proceeds into the region II of prior mass loss, which is approximately of constant density up until RC . This will simulate the interaction with an s = 0 environment, but with lower densities compared to typical ISM. SN interaction could be observed to come from both the ongoing wind interaction region and the region III of the RG wind from prior mass loss. This would create a two sets of narrow lines, corresponding to the two regions.

49 2 2 2 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] 1 Log (p/p1) 1 Log (p/p1) 1 Log (p/p1) Log n Log n Log n 0 0 0

-1 -1 -1

-2 -2 -2

-3 -3 -3

-4 -4 -4 time = .00745 Myr time = .33058 Myr time = 8.53186 Myr -5 -5 -5 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 30 0 100 200 300 400 500 600 r (ly) r (ly) r (ly)

2 2 2 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] 1 Log (p/p1) 1 Log (p/p1) 1 Log (p/p1) Log n Log n Log n 0 0 0

-1 -1 -1

-2 -2 -2

-3 -3 -3

-4 -4 tp = .745 Myr -4 time = .01280 Myr time = .56829 Myr time = 14.66689 Myr -5 -5 -5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 10 20 30 40 50 0 200 400 600 800 1000 r (ly) r (ly) r (ly)

2 2 2 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] 1 Log (p/p1) 1 Log (p/p1) 1 Log (p/p1) Log n Log n Log n 0 0 0

-1 -1 -1

-2 -2 -2

-3 -3 -3

-4 -4 -4 time = .03785 Myr time = .97693 Myr time = 25.21345 Myr -5 -5 -5 0 1 2 3 4 5 6 7 8 0 10 20 30 40 50 60 70 0 200 400 600 800 1000 1200 1400 1600 1800 r (ly) r (ly) r (ly)

2 2 2 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] 1 Log (p/p1) 1 Log (p/p1) 1 Log (p/p1) Log n Log n Log n 0 0 0

-1 -1 -1

-2 -2 -2

-3 -3 -3

-4 -4 -4 time = .06507 Myr time = 1.67942 Myr time = 43.34374 Myr -5 -5 -5 0 2 4 6 8 10 0 20 40 60 80 100 120 0 500 1000 1500 2000 2500 3000 r (ly) r (ly) r (ly)

2 2 2 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] 1 Log (p/p1) 1 Log (p/p1) 1 Log (p/p1) Log n Log n Log n 0 0 0

-1 -1 -1

-2 -2 -2

-3 -3 -3

-4 -4 -4 time = .11186 Myr time = 2.88706 Myr time = 74.51102 Myr -5 -5 -5 0 2 4 6 8 10 12 14 16 0 50 100 150 200 0 1000 2000 3000 4000 5000 r (ly) r (ly) r (ly)

2 2 Log [u/(100 km/s)] Log [u/(100 km/s)] 1 Log (p/p1) 1 Log (p/p1) Log n Log n 0 0

-1 -1

-2 -2

-3 -3

-4 -4 time = .19230 Myr time = 4.96306 Myr -5 -5 0 5 10 15 20 0 50 100 150 200 250 300 350 r (ly) r (ly)

Figure 5.1: Time series for a typical AD wind in the s = 0 case (see text). 50 2 2 2

0 0 0

-2 -2 -2

-4 -4 -4 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] Log (p/p1) Log (p/p1) Log (p/p1) -6 Log [n/(10-6 cm-3)] -6 Log [n/(10-6 cm-3)] -6 Log [n/(10-6 cm-3)] -8 -8 -8

-10 time = .00300 Myr -10 time = .13308 Myr -10 time = 3.44446 Myr

1 1.5 2 2.5 3 60 80 100 120 140 1500 2000 2500 3000 3500 r (ly) r (ly) r (ly)

2 2 2

0 0 0

-2 -2 -2

-4 Log [u/(100 km/s)] -4 Log [u/(100 km/s)] -4 Log [u/(100 km/s)] Log (p/p1) Log (p/p1) Log (p/p1) -6 Log [n/(10-6 cm-3)] -6 Log [n/(10-6 cm-3)] -6 Log [n/(10-6 cm-3)] -8 -8 -8

-10 time = .00521 Myr -10 time = .22862 Myr -10 time = 5.84953 Myr

2 2.5 3 3.5 4 4.5 5 5.5 80 100 120 140 160 180 200 220 240 2000 2500 3000 3500 4000 4500 5000 5500 6000 r (ly) r (ly) r (ly)

2 2 2

0 0 0

-2 -2 -2

-4 -4 -4 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] Log (p/p1) Log (p/p1) Log (p/p1) -6 Log [n/(10-6 cm-3)] -6 Log [n/(10-6 cm-3)] -6 Log [n/(10-6 cm-3)] -8 -8 -8

-10 time = .01538 Myr -10 time = .39365 Myr -10 time = 10.16532 Myr

6 8 10 12 14 16 150 200 250 300 350 400 4000 5000 6000 7000 8000 9000 10000 11000 r (ly) r (ly) r (ly)

2 2 2

0 0 0

-2 -2 -2

-4 -4 -4 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] Log (p/p1) Log (p/p1) Log (p/p1) -6 Log [n/(10-6 cm-3)] -6 Log [n/(10-6 cm-3)] -6 Log [n/(10-6 cm-3)] -8 -8 -8

-10 time = .02612 Myr -10 time = .67627 Myr -10 time = 17.26319 Myr

10 15 20 25 300 400 500 600 700 6000 8000 10000 12000 14000 16000 18000 r (ly) r (ly) r (ly)

2 2 2

0 0 0

-2 -2 -2

-4 -4 -4 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] Log (p/p1) Log (p/p1) Log (p/p1) -6 Log [n/(10-6 cm-3)] -6 Log [n/(10-6 cm-3)] -6 Log [n/(10-6 cm-3)] -8 -8 -8

-10 time = .04499 Myr -10 time = 1.10948 Myr -10 time = 29.99999 Myr

15 20 25 30 35 40 45 400 500 600 700 800 900 1000 1100 1200 10000 15000 20000 25000 30000 r (ly) r (ly) r (ly)

2 2

0 0

-2 -2

-4 -4 Log [u/(100 km/s)] Log [u/(100 km/s)] Log (p/p1) Log (p/p1) -6 Log [n/(10-6 cm-3)] -6 Log [n/(10-6 cm-3)] -8 -8

-10 time = .08111 Myr -10 time = 2.00042 Myr

30 40 50 60 70 80 800 1000 1200 1400 1600 1800 2000 2200 r (ly) r (ly)

Figure 5.2: Time series for a typical AD wind in the s = 2 case (see text). 51 CHAPTER 6

APPLICATIONS: ENVIRONMENTS OF TYPE IA SUPERNOVA PROGENITORS

I will explore winds emanating from the progenitor system and interacting with the ISM of mass loss of the system prior to the supernova explosion. Winds will be considered from each source separately, and I address the question of which component is mostly responsible for the formation of the environment, and the typical structure to be expected. Subsequently, I discuss the link between observables and progenitor systems, and analyze SN2014J. I employ spherical, semi-analytical models constructed by piecewise, scale-free analytic solutions. Scales enter the system via the equation of state, the boundary, and the jump conditions.

The free parameters are: 1) the velocity vw, 2) mass loss ratem ˙ from the central object, the ambient medium density characterized either by 3) n0 = const or a mass loss ratem ˙ 1 with vw,1, i.e. n r−s, and 4) the duration of the wind interaction t. As a result, I obtain the density, velocity and ∝ pressure as a function of time, namely ρ(r, t),v(r, t) and p(r, t) which can be linked to observables. I use typical parameters to discuss the diﬀerent regimes which may occur in nature. For actual ﬁts of observations, appropriate solutions can be constructed by tuning these parameters with SPICE. The wind may originate from the AD, the donor star which may be a MS or a RG-, horizontal- and asymptotic-branch star, the WD during a phase of over-Eddington accretion, or a combination of AD with a RG-like wind. As shown below, the time scale for the accretion and, thus, the progenitor, is an important factor in formation of the environment. The time scales t vary widely depending on the scenario and chemical composition of the accreted material and the initial mass of the progenitor (e.g. [220, 172] and reviews cited in the introduction). For hydrogen accretion, the rates for stable hydrogen burning are between M˙ 2 10−8...−6 ≈ × M⊙/yr depending on the metallicity [144, 73]. The upper and lower limits for M˙ are set by the Eddington limit for the luminosity and the minimum amount of fuel needed for steady burning, respectively. However, it is under discussion whether and at which accretion rates steady H-burning can continue until the WD approaches MCh. It depends on the chemical composition, rotation of

52 the WD, and details of the approximations used [144, 215, 230, 70, 173, 251, 197, 74, 8]. For a recent review, see [128]. In this study, the wide range of accretion rates are used in order to avoid restricting possible solutions. Thus, I consider time scales t between 105 and 4 107 yr. × Larger rates of mass overﬂow result in over-Eddington luminosity and a strong wind from the progenitor WD with properties typical of RG winds [69, 72]. Subsequently, the high-density, low velocity winds will be referred to as “RG-like”. Accretion of He and C/O -rich matter allow much shorter timescales down to the dynamical times of merging WDs. For instance, helium accretion −7 −6 rates are stable between 4 10 and 8 10 M⊙/yr, giving corresponding timescales between ≈ × × t = 3.8 104 and 2.5 106 yr. × × −8 −10 For accretion disk winds the mass loss rate ranges from 10 to 10 M⊙/yr; and the wind velocities originating from the disks are believed to be from 2000 to 5000 km/s [102]. −6 −8 Mass loss in ’RG-like’ stars are typically between 10 and 10 M⊙/yr with wind velocities between 10 and 60 km/s [184, 101, 180]. Main sequence star winds are similar to the solar wind [245]. For solar wind the velocity is −14 between 400 km/s and 750 km/s and the mass loss is 2...3 10 M⊙/yr [142, 46, 135]. ×

6.1 Parameterized Study

In the following, I assume typical wind parameters as follows: In the ’RG-like’ case, mass loss −6 −7 −8 ratesm ˙ of 10 , 10 and 10 M⊙/yr, and a wind velocity of 30 km/s will be used. Similar winds can be expected for WDs with high accretion rates. For the case of a MS star donor, mass −13 −14 −15 loss rates of 10 , 10 , and 10 M⊙/yr, and vw = 500km/s will be used. For AD winds the mass loss rate ranges from 10−8, 10−9, and 10−10 solar masses per year; a typical rate has been −9 measured to be 10 M⊙/yr. A wind velocity of 3000 km/s is used. For the duration of the winds, I consider t between 1.5 105 and 6 108 yr with 3 105 yr for the references. × × × The outer environment of the system depends on its history including the delay time between the formation of the WD and the onset of the accretion phase. For long delays, I assume an ISM with constant density, i.e. s = 0. For short delay times of the system and small peculiar velocities, the outer environment may be created during the ﬁnal stage of the progenitor evolution, namely the red giant branch (RGB), horizontal giant branch (HGB) and the asymptotic giant branch (AGB) phase. For systems with large peculiar velocities, an extra eﬀect needs to be considered:

53 the system will create a bubble “tube” as it passes through undisturbed ISM. When such a system is observed from the front or the side, respective to the direction of motion, the distance from the shell radius will be less than that of a similar stationary system, and so it will eﬀectively have a shorter duration. The shell velocity towards the observe will also be greater, on average.

54 CHAPTER 7

RESULTS

7.1 Case I: Constant ISM Density

I ﬁrst consider scenarios where the wind blows out into a medium of constant density for a wide range of parameters (Tables 7.1-7.3). The structures are characterized by I) an undisturbed, inner layer dominated by the stellar wind, II) an inner, shocked region with almost constant, low density and a velocity declining with distance, III) a slowly expanding shell of high density swept up material, and IV) the ISM. The overall solution is representative for all cases as has been shown in Sect. 2. For the estimate of the equivalent width EW of the Na I doublet at 5890/5896 A,˚ solar abundances are used, X = 3.34 10−5 X . EW is estimated according to [214, 41]. For Na × H potassium lines, the corresponding equations apply.

7.1.1 Case Ia: Fast Winds from an Accretion Disk

Table 7.1 contains calculated results from several cases with diﬀerent parameters but the same time-scales t. The reference, model 2 of Table 7.1, is shown in Fig. 7.1. It has a mass lossm ˙ of 10−8

M⊙/yr and a wind velocity vw = 3, 000 km/s. For the duration of the wind, I choose a duration of 3 105 years which, within the SD scenario, corresponds to the evolutionary time for a low mass × −6 WD to grow to M at an accretion rate of 2 10 M⊙/yr. Up to about 2.3 ly, the environment Ch ≈ × is dominated by the on-going wind. In this region, the particle density drops below 1(100) cm−3 at a distance of 0.005(0.0005) ly 1000(100) AU 1016(15) cm which will be overrun by the SN ≈ ≈ ejecta within 5(0.5) days. Particle densities below 100 cm−3 in this region will hardly aﬀect the light curves or spectra because the swept up mass will be small. Using the same argument, the low densities within 20 ly are too low to aﬀect the hydrodynamics of the SN envelope. A high density, outer shell expands at a velocity of 11 km/s with a velocity dispersion of 13% 2 km/s. This ≈ ≈ shell would produce a narrow line Doppler shifted by about 11 km/s. For an interstellar medium, the equivalent width would be about 165 mA˚ for the Na I D line well comparable to values found by [170] who found between 27 and 441 mA˚ in a sample of some 30 SNe Ia.

55 The morphology of the structures are hardly aﬀected by variations in the wind parameters as expected from the Π theorem. However, the actual size of the regions and the densities vary (see Table 7.1) with dependencies as expected from Table 3.1. Within the framework of SNeIa, the ram pressure dominates the ambient pressure which, therefore, hardly aﬀects the solution. Typically, wind from an accretion disk will produce an inner cavity between 5 and 30 ly surrounded by an expanding shell with a velocity of 5...20 km/s. EW (Na I D) is 100 to 200 mA.˚ The ≈ ≈ combination of line shifts and strength allows us to derive the wind parameters. −6 In the table, I assumed an accretion rate of 2 10 M⊙/yr which is at the upper limit ≈ × allowed for stable accretion of hydrogen rich matter. Higher mass loss can be expected for over- Eddington accretion or He or C/O accreting WDs (see Sect. 1). Higher and lower mass loss rates will increase/decrease the size of the cavity but the dependency is relatively weak, m˙ 1/5. ∝ However, the actual size of the region does depend sensitively on the time t of the progenitor evolution as the size of the cavity goes like t3/5. In hydrogen accreters, the rate of accretion may ∝ be smaller by a factor of 100 and, thus, t may be larger by the same factor which, in turn, will increase the size of the cavity by 16 and decrease shell velocities by about a factor of 6. Shorter durations can be realized if we start with a WD of 1.2 M⊙, the upper end of mass range for a C/O WD (see Introduction). Indeed, there is some evidence and theoretical arguments that the progenitors originate close to the upper end of that mass range [146, 81, 194] which may reduce the amount of accreted material from 0.8 to 0.2 M⊙. Thus, the duration of the accretion t may ≈ be correspondingly shorter which, in turn, reduces the size of the cocoon and increases the shell velocities by 2 and 1.5, respectively. ≈ On the other hand, rates for He and C/O accretion can be larger than hydrogen accreters by, at least, the same factor of 100 [244, 172, 234, 231], reducing the size of the shell and increasing the velocity of the shell lines by the same factors. High shell velocities may indicate He or C/O accreters. Despite the line shifts, systems with high shell velocities can be expected to have smaller low density regions. The SN ejecta have velocities up to about 10 to 20% of the speed of light. For such SNe Ia, we may expect interaction on time scales from 1 to 10 years for the set of parameters in this work.

56 2 0.5 Log [u/(100 km/s)] 1 Log (p/p1) Log n 0 0

-1 -0.5 -2 Log [u/(100 km/s)] Log (p/p1) -3 Log n -1 -4

-5 -1.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

r/RC r/RC

Figure 7.1: Hydrodynamic proﬁle for a wind typical for an accretion disk (left), vw = −8 3000 km/s and mass-loss rate of 10 M⊙/yr, and a RG-like wind (right), 30 km/s and −7 −3 mass-loss rate of 10 M⊙/yr, running into an ISM of constant, 1 cm particle density after a time of 300, 000 yr. The contact discontinuity is at 21.7 and 5.45 ly, respectively. Fluid velocity u (magenta) is normalized to 100 km/s, pressure p (blue) is normalized to the pressure p1 just inside the outer shock, and particle density n is unnormalized.

7.1.2 Case Ib: MS Star Winds

MS star winds produced by a donor star are expected to show low velocities vw = 500 km/s and −14 a very low mass lossm ˙ = 10 M⊙/yr. Although the morphology of the shells remain the same as above, the corresponding regions will be overrun and dominated by wind from the accretion disk.

The radius of the reverse shock R2 show little change for time scales much longer than tp (see Table

7.4). As discussed above, RC becomes unphysical due to mixing in a regime of weak shocks. For SNe Ia, the MS wind component can be neglected.

7.1.3 Case Ic: Slow, RG-like Winds

These winds may be produced during the RG phase of the donor star or the progenitor prior to the formation of the progenitor WD, or as a result of over-Eddington mass overﬂow (Table 7.2). Those will be referred to as RG-like winds. The resulting structures are very similar to Case Ia but the densities are higher by an order of magnitude (Fig. 7.1). In particular, the density contrast n /n 0.1 (see also Table 3.1). The cocoons are smaller and less pronounced in this case. 2 0 ≈ For typical properties of the environment, we have to distinguish between winds for an RG donor, RG-like winds from over-Eddington accretion, and the prior RG-phase of the progenitor.

57 In Table 7.2, models are shown for RG-like winds for various n0,m ˙ and t. As expected, the size of the cavity decreases with n0 (model 8 vs. 9). t is chosen in the range for stable hydrogen accreters (models 8 to 14). As duration t increases, the location of R2 “stalls” and n2 is hardly aﬀected because the outer and inner pressure equilibrates (t/tp >> 1) as discussed in the Sect. 3).

See also Table 3.1. For the same reason, the size of the cavity increases withm ˙ but n2/n0 hardly changes. Typically, RG-like winds will produce an inner cavity between 1 and 10 ly surrounded by an expanding shell with a velocity of 5 km/s. EW (Na I D) is a few hundred mA.˚ Note that the ≈ scales of RG winds are smaller by an order of magnitude compared to those of AD winds unless those have much lower mass loss than considered in the example above. An RG wind is more likely to form a combined AD-RG wind as discussed below. RG winds from the progenitor prior to the WD phase (models 15-20) will produce a similar −6 structure as an RG donor in a system with high accretion rates ( 10 M⊙/yr) but are system- ∼ atically larger because the longer duration t. This wind may form an environment for subsequent winds. Progenitor systems may run into this environment if the delay time between the formation of the WD and the onset of the accretion is suﬃciently short, and if the peculiar velocity of the system is small. The peculiar velocities of stars show a wide range with a typical value of 25-50 km/s for the Galactic plane ([66], and references therein). Here, the duration of the wind is given by the evolutionary time scale of the RG phase rather than the time to reach MCh. For models

15-18, I use t corresponding to the post-main sequence life time of 5 and 8 M⊙ stars with a mass −7 loss rates of 10 M⊙ The resulting total mass loss is 1.3 and 0.3 M⊙ for the 5 and 8 M⊙ star, respectively [201, 22]. The environment formed by a wind consists of inner region with s 2 with a size R and a ≈ 2 low density void (s 0) of size R . The resulting size of the void, R , is typically 10-20 ly. The ≈ C C duration t can be increased by lower main sequence masses for the progenitor. However, then, the −7 amount of mass lost prior of forming a WD of M(WD) 0.6 M⊙. Using a mass loss of 10 M⊙ ≈ and v = 30km/s, the maxima in R and R is produced by a 3.6M⊙ star: 33 ly and 3.7 ly, w C 2 ≈ ≈ respectively (models 19,20, Table 7.2). Models with durations of 3, 13 and 50 Myr correspond to progenitor stars of 8, 5, and 3.6 M⊙. It is noted that long durations may also be produced during the evolution of the progenitor −8 system if the hydrogen accretion rate is close to the lower limit of 10 M⊙/yr, though this low ∼

58 a rate may not allow for stable accretion for WD close to MCh [172]. Models with RG-like winds may correspond to systems with over-Eddington accretion and a MS donor star (models 15-20). For those long duration RG-like winds, the resulting cavities can be 15 to 33 ly and large EW of ≈ 500mA.˚ ≈ 7.1.4 Case Id: Fast Wind from an Accretion Disk Combined with Mass Loss from RG Donor Star or Super-Eddington Accretion

If we have a system with both a dense RG donor wind and an accretion disk wind, the two will combine. The inner interaction region will be Rayleigh-Taylor (RT) unstable and mix fast (Fig. 7.2). They can be expected to form a uniform wind with an acceleration region. Assuming momentum conservation and the reference models for the AD wind and “RG-like” winds, I obtain −5,−6,−7 a total mass loss rate of 10 M⊙ and wind velocity of 33, 60 and 300 km/s (see models ≈ ≈ 21-23, Table 7.3). This case applies in system with high peculiar velocity. There, the system has moved out of the environment formed during the stellar evolution of the progenitor. The cavities are somewhat larger by about a factor of 2 compared to the RG-like wind cavities, surrounded by an expanding shell of a similar velocity and EW (Na I D). The larger cavity will result in a slower evolution of the narrow lines.

1 0.5 0.5 Log [u/(100 km/s)] Log [u/(100 km/s)] Log (p/p1) Log (p/p1) 0 Log n 0 Log n -0.5 -0.5 -1 -1.5 -1 -2

-2.5 -1.5 -3 -3.5 -2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 1 1.5 2 2.5

r/RC r/RC

Figure 7.2: Same as Fig. 7.1 but for a mixture of AD plus RG-like winds. Left and right are shown models no. 21 and 22 from Table 7.3, respectively.

59 7.1.5 Extremely Low Density Environments

In case of long delay times of SNe Ia, the progenitor system may have wandered away from the galactic disk into the halo, in a regime of very low densities [63], or in a hot-ISM [48], or in the constant density region II of the cavity from prior mass loss. In Table 7.5, some examples are given for constant density with particle densities between 10−2 to 10−3 cm−3. Most likely, the donor star will not be a RG star. The inner, low density region is expected to be of the order of or larger than 50 100 ly. EW(Na I D) are between 20 to 100 mA˚ and show increased Doppler shifts of 30 to − ≈ 50 km/s or larger for even lower density than found in the vicinity of SN1885 [48]. In conclusion, of all of wind components analyzed separately, the AD wind component will dominate the formation of cocoons.

7.2 Case II: Environments Produced by Winds from Prior Mass Loss

Now I consider the scenario where the wind of the progenitor system runs into a nearby environment produced by a prior mass loss (s = 2). “Nearby” means that the reverse shock R2 produced by the prior mass loss must be larger than the size of the region produced by the wind of the system. Otherwise the interaction region will move into a region of constant density, shocked RG wind, or the ISM. Moreover, the speed of the ongoing wind must exceed the outer wind. For our discussion, I disregard the eﬀect of prior mass loss by a MS wind because it produces a cocoon of < 0.005 ly (Table 7.4) and consider parameters only in which the prior wind can be produced by an RG star.

7.2.1 Case IIa: Fast Wind from an Accretion Disk and a Non-RG Donor Star

If the donor star is a compact object like a MS or He-star, the AD wind will dominate. Examples −6 are shown in Table 7.6. As reference, some typical parameters arem ˙ 1 = 10 M⊙/yr, vw,1 = −10 5 30 km/s,m ˙ = 10 M⊙/yr, v = 3000 km/s and a total run time of 1.5 10 yr. Again, we w × can identify the diﬀerent zones as above (Fig. 7.3). The density contrast between the inner bubble −2 within RC is smaller than in the constant density case, i.e. a factor of 10 , but the density at RC is signiﬁcantly lower than the ISM. This results in an “ultra-low” density bubble of n 4 10−8 cm−3. ≈ ×

60 Moreover, this solution shows a qualitative diﬀerent feature compared to the constant density ISM: We see a very thin and dense shell with particle densities 102 cm−3 and a thickness of light ≤ weeks to months (Fig. 7.3). The resulting shell produces a narrow, optically thick Na I D line with small equivalent width EW of 0.24 mA,˚ Doppler shifted by about uc = 240 km/s and a width of 23 km/s. I note that, to ﬁrst order, EW m˙ (model 1, Table 7.6). ≈ ∝

3 3 Log [u/(100 km/s)] Log [u/(100 km/s)] 2 2 Log (p/p1) Log (p/p1) -6 -3 Log [n/(10-6 cm-3) ] Log [n/(10 cm ) ] 1 1

0 0 -1 -1 -2 -2 -3 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2

r/Rc r/Rc

Figure 7.3: Hydrodynamic profile for a wind typical of an accretion disk running into −2 an r density profile of prior mass loss. In the left figure, a wind of velocity vw = −8 3000 km/s and mass-loss rate ofm ˙ = 10 M⊙/yr runs into another wind of mass loss −7 m˙ 1 = 10 M⊙/yr and vw,1 = 30 km/s for a time of 300, 000 yr. The right figure is the −10 same except the ongoing mass loss ratem ˙ = 10 M⊙/yr. The contact discontinuity is at 240 and 67.7 ly, respectively. Fluid velocity u (magenta) is normalized to 100 km/s, pressure p (blue) is normalized to the pressure p1 just inside the outer shock, and particle density n is normalized to 10−6 cm−3. Note that the particle density normalization used is much smaller than in the constant density case.

7.2.2 Case IIb: Fast Wind from an Accretion Disk Combines with a “RG-like” Wind

If we have a system with a dense RG donor and wind plus an accretion disk wind, they would form a uniform wind with an acceleration region (Fig. 7.4). I use the same parameters as in Sect. 7.1 but omit the mixed wind with the highest mass loss because its velocity will be comparable to the wind velocity of the surrounding medium. As expected, the cavities are smaller than the AD-wind case, and EW(Na I D) and its Doppler shift are larger.

61 Comparing interaction regions of environments created by winds with constant density ISM, the typical scales are larger in the former and the resulting narrow lines show a larger blue-shift with smaller EW.

3 Log [u/(100 km/s)] 4 Log [u/(100 km/s)] 2.5 Log (p/p1) Log (p/p1) 2 Log [n/(10-6 cm-3) ] 3 Log [n/(10-6 cm-3) ] 1.5 2 1

0.5 1 0 0 -0.5 -1 -1 0.7 0.8 0.9 1 1.1 1.2 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1

r/Rc r/Rc

Figure 7.4: Same as Fig. 7.3 but for a mixture of AD plus RG-like winds. Left and right are shown models no. 8 and 9 from Table 7.6, respectively.

62 Table 7.1: Interaction of winds of an accretion disk (AD) with a constant ISM (s=0). Numerical results are given for parameters typical for those winds for the CAP model (see text and Table 3.1). In addition, I give the velocity dispersion of the shell σu and the optical depth τ and equivalent width EW of the NaID line. The reference model for this category of models is marked by . ∗

No n0 vw m˙ t tp RC R1 R2 −3 (cm ) (km/s) (M⊙/yr) (Myr) (Myr) (ly) (ly) (ly)

1 0.1 3000 10−8 0.3 2.36 33.34 41.94 4.89 2 1.0 3000 10−8 0.3 0.745 20.11 28.53 2.29 ∗ 3 10.0 3000 10−8 0.3 0.236 11.39 22.02 0.98 4 1.0 3000 10−9 0.3 0.236 11.39 22.02 0.98 5 1.0 3000 10−10 0.3 0.0745 5.80 20.47 0.36 6 1.0 3000 10−8 0.15 0.745 13.70 17.84 1.82 7 1.0 3000 10−8 0.4 0.745 23.44 35.07 2.50

No u1 uc σu n2 τmass log(τ) EWNaI (km/s) (km/s) (km/s) (cm−3) (g/cm2) (mA)˚

1 16.56 19.99 0.85 2.07 10−5 2.60 10−6 1.34 157 × × 2 9.45 12.06 0.62 9.45 10−5 1.91 10−5 2.34 165 ∗ × × 3 5.53 6.83 0.32 5.20 10−4 1.71 10−4 3.59 192 × × 4 5.53 6.83 0.32 5.20 10−5 1.71 10−5 2.59 98.4 × × 5 4.50 3.48 0.62 3.94 10−5 1.84 10−5 2.33 163 × × 6 13.34 16.42 0.76 1.50 10−4 1.13 10−5 2.03 180 × × 7 8.16 10.54 0.56 7.95 10−5 2.42 10−5 2.49 157 × ×

63 Table 7.2: Same as Table 7.1 but for an “RG-like” wind. The reference model for this category of models is marked by . ∗

No n0 vw m˙ t tp RC R1 R2 −3 (cm ) (km/s) (M⊙/yr) (Myr) (Myr) (ly) (ly) (ly)

8 0.1 30 10−7 0.3 0.0745 5.80 20.47 3.55 9 1.0 30 10−7 0.3 0.0236 2.75 20.28 1.16 ∗ 10 1.0 30 10−5 0.3 0.236 11.39 22.02 9.77 11 1.0 30 10−6 0.3 0.0745 5.80 20.47 3.55 12 1.0 30 10−8 0.3 0.0075 1.28 20.27 0.37 13 1.0 30 10−7 0.15 0.0236 2.16 10.17 1.15 14 1.0 30 10−7 2.0 0.0236 5.20 135 1.16 15 0.1 30 10−7 3 0.0745 12.8 203 3.69 16 1.0 30 10−7 3 0.0236 5.96 203 1.17 17 0.1 30 10−7 13 0.0745 20.9 878 3.70 18 1.0 30 10−7 13 0.0236 9.71 878 1.17 19 0.1 30 10−7 50 0.0745 32.8 3377 3.70 20 1.0 30 10−7 50 0.0236 15.2 3382 1.17

No u1 uc σu n2 τmass log(τ) EWNaI (km/s) (km/s) (km/s) (cm−3) (g/cm2) (mA)˚

8 4.50 3.48 0.62 4.19 10−2 1.84 10−6 1.33 114 × × 9 4.37 1.65 0.94 3.75 10−1 1.94 10−5 2.17 231 ∗ × × 10 5.53 6.83 0.32 6.62 10−1 1.71 10−5 2.59 98.4 × × 11 4.50 3.48 0.62 4.19 10−1 1.84 10−5 2.33 163 × × 12 4.36 0.77 1.03 3.66 10−1 1.97 10−5 2.14 251 × × 13 4.41 2.60 0.79 3.92 10−1 9.46 10−6 1.94 181 × × 14 4.37 0.47 1.05 3.65 10−1 1.32 10−4 2.96 329 × × 15 4.36 0.77 1.03 3.66 10−2 1.97 10−5 2.14 251 × × 16 4.36 0.36 1.05 3.64 10−1 1.98 10−4 3.13 353 × × 17 4.36 0.29 1.05 3.64 10−2 8.58 10−5 2.77 309 × × 18 4.36 0.13 1.05 3.63 10−1 8.58 10−4 3.77 500 × × 19 4.36 0.12 1.05 3.63 10−2 3.30 10−4 3.35 391 × × 20 4.38 0.05 1.06 3.64 10−1 3.31 10−3 4.35 809 × × 64 Table 7.3: Same as Table 7.1 but for the combination of an AD- and RG-like wind (see text).

No n0 vw m˙ t tp RC R1 R2 −3 (cm ) (km/s) (M⊙/yr) (Myr) (Myr) (ly) (ly) (ly)

21 1.0 300 1.1 10−7 0.30 0.247 11.68 22.17 3.21 × 22 1.0 60 1.01 10−6 0.30 0.150 8.85 21.03 4.73 × 23 1.0 33 1.001 10−5 0.30 0.082 6.15 20.50 3.70 ×

No u1 uc σu n2 τmass log(τ) EWNaI (km/s) (km/s) (km/s) (cm−3) (g/cm2) (mA)˚

21 5.62 7.00 0.33 5.33 10−3 1.71 10−5 2.57 100 × × 22 4.88 5.30 0.33 1.17 10−1 1.75 10−5 2.59 100 × × 23 4.52 3.69 0.58 3.49 10−1 1.83 10−5 2.36 154 × ×

65 Table 7.4: Same as Table 7.1 but for a “MS-like” wind for the solution including the ambient pressure (CAP models). My reference model is marked by +. Values are given for the radius of the contact discontinuity and the inner shock using parameters typical for the wind of a main sequence star, and using the solution including the ambient pressure (CAP models). For times (t) much larger than tp, the location of the reversed shock becomes stationary and close to the distance R2,PE expected from equilibrating the pressure at the reversed shock with the ambient medium. In addition, models without outer pressure are given and marked by ’Rem.’. This shows the importance of the pressure term for t/t 1 p ≫ in the extreme case. However, for winds relevant in SNe, the diﬀerence is of the order of several % (see text, and Fig. 3.1).

No n0 vw m˙ t tp RC R2 R2,PE −3 (cm ) (km/s) (M⊙/yr) (Myr) (yr) (ly) (ly) (ly)

1 0.1 500 10−14 40 393 0.921 0.00477 0.00390 Rem. ” ” ” ” - 20.0 0.481 2+ 1 500 10−14 40 124 0.427 0.00151 0.00123 Rem. ” ” ” ” 12.6 0.241 0.0496 3 10 500 10−14 40 39.3 0.198 0.000477 0.000390 Rem. ” ” ” ” 7.94 0.121 0.0312 4 1 500 10−13 40 393 0.921 0.00477 0.00390 5 1 500 10−15 40 3.93 0.198 0.000477 0.00039 6 1 500 10−14 2 124 0.157 0.00151 0.00123 7 1 500 10−14 600 124 1.05 0.00151 0.00123 8 0.1 500 10−14 0.3 393 0.180 0.00477 0.00390 9 1 500 10−14 0.3 124 0.0836 0.00151 0.00123 10 10 500 10−14 0.3 3.93 0.0388 0.000477 0.000390 11 1 500 10−15 0.3 3.93 0.0388 0.000477 0.000390 12 1 500 10−14 0.4 124 0.0921 0.00151 0.00123

66 Table 7.5: Same as tables 7.1 and 7.4 but for very low density ISM typical for the galactic halo and elliptical galaxies.

No n0 vw m˙ t tp RC R1 R2 −3 (cm ) (km/s) (M⊙/yr) (Myr) (Myr) (ly) (ly) (ly)

1 0.01 3000 10−8 0.4 2.36 63.8 77.2 11.21 2 0.001 3000 10−8 0.4 23.6 102 121 22.7 3 0.01 30 10−7 50 0.236 70.8 3368 11.7 4 0.001 30 10−7 50 0.745 152 3377 36.9

No u1 uc σu n2 τmass log(τ) EWNaI (km/s) (km/s) (km/s) (cm−3) (g/cm2) (mA)˚

1 24.36 28.67 1.08 1.72 10−6 4.67 10−7 0.49 104 × × 2 39.6 45.87 1.58 4.21 10−7 7.18 10−8 -0.48 24.0 × × 3 4.32 0.25 1.05 3.62 10−3 3.29 10−5 2.35 272 × × 4 4.36 0.55 1.04 3.65 10−4 3.29 10−6 1.36 193 × ×

67 Table 7.6: Interaction of winds with an environment produced by RG wind originating from the progenitor WD (vw,1 = 30 km/s, s = 2). The index 1 corresponds to the prior mass loss. Models 1-7 and 8-9 show the results for an AD-wind and a combination of a AD and RG-like wind, respectively (Tables 7.1 & 7.3). The reference model for this category of models is marked by . ∗

No vw,1 m˙ 1 vw m˙ tRC R1 R2

(km/s) (M⊙/yr) (km/s) (M⊙/yr) (Myr) (ly) (ly) (ly)

AD 1 30 10−6 3000 10−9 0.3 67.7 74.2 16.4 2 30 10−7 3000 10−9 0.3 122 138 39.3 3 30 10−8 3000 10−9 0.3 240 278 106 4 30 10−7 3000 10−8 0.3 240 278 106 ∗ 5 30 10−7 3000 10−10 0.3 67.7 74.2 16.4 6 30 10−7 3000 10−9 0.15 60.8 69.1 19.6 7 30 10−7 3000 10−9 0.4 162 184 52.4 AD+RG-like 8 30 10−7 300 1.1 10−7 0.3 114 129 91.8 × 9 30 10−7 60 1.01 10−6 0.3 51.3 54.9 50.0 ×

No uc n2 τm log(τ) EWNaI (km/s) (cm−3) (g/cm2) (mA)˚

AD 1 67.7 1.7 10−7 1.59 10−8 -2.02 5.63 × × 2 122 2.9 10−8 1.18 10−9 -3.34 0.420 × × 3 239 4.0 10−9 6.96 10−11 -4.80 0.0248 × × 4 239 4.0 10−8 6.96 10−10 -3.80 0.248 ∗ × × 5 67.7 1.7 10−8 1.59 10−9 -3.02 0.566 × × 6 122 1.2 10−7 2.36 10−9 -3.02 0.837 × × 7 122 1.6 10−8 8.86 10−10 -3.45 0.315 × × AD+RG-like 8 114 5.9 10−6 1.59 10−9 -2.92 0.565 × × 9 51.3 9.1 10−4 5.05 10−9 -1.86 1.79 × ×

68 CHAPTER 8

APPLICATION TO SN2014J AND OTHER SN IA

I will now explore the application of my semi-analytic models to the cases of speciﬁc SNe including SN2014J and SN2011fe.

8.1 Background of SN2014J

This object was discovered in January of 2014 in a high density region of the nearby M82 ( 3.5 Mpc). The higher than average ISM densities and the small proximity of the host make ≈ SN2014J an excellent candidate for investigating the interaction of the SN with the environment. This provides a premier opportunity to implement these models and make predictions. Observational constraints for the environment and progenitor are obtained from searches for X-rays [130], time-dependence of high-resolution spectra of narrow potassium lines formed in the environment [64] (shown in Fig. 8.1), IR imaging [108], and radio [165]. X-rays and radio provided the most stringent constraints on the average density of ions in the environment. In the case of SN2014J, X-rays and radio luminosities at maximum light probe the wind from the progenitor system (region I in Fig. 2.1). When the SN material propagates into the circumstellar surrounding, a shock is formed and leads to the acceleration of partially or fully relativistic electrons with a power-law distribution −p ne(γ) = n0γ with p between 2 and 3 which produce X-ray and radio emission. From radio observations of SNe Ib/c, [19] ﬁnd p 3. Because the outer layers of a SNe Ia and SN Ibc have ≈ similar structure, abundances and velocities, we use this value in the following.

For X-rays [131], Lν is given by

0.64 −1.36 2 m/v˙ w t −1 Lbol erg Lν = 16500 ǫe − − ν − . (8.1) M⊙ yr 1 km 1 s days erg s 1 s Hz For the radio [165], Lν is given by

69 1.37 −1.55 −9 1.71 1.07 m/v˙ w t −1 erg Lν = 5.81 10 ǫe ǫB − − Tbright ν . (8.2) × M⊙ yr 1 km 1 s days s Hz

The parameters are the fraction of relativistic electrons ǫe, the energy fraction in the magnetic ﬁeld

ǫB, and the brightness temperature Tbright which, based on observations, can be expected to be 1011K [183]. [130] and [165] use a value of 0.1 for both ǫ and ǫ . ≈ e B If the supernova shell runs into a constant density environment, the X-ray luminosity is given by

− n 0.5 t 0.45 L erg L = 6.44 10−5 ǫ2 0 ν−1 bol . (8.3) ν × e cm−3 d erg s−1 s Hz For late times when the SN shock runs into region II (Fig. 2.1), the radio luminosity is given by

L T ǫ0.86 ǫ1.07 n1.27 t0.88. (8.4) ν ∝ bright e B ISM It is noted that region II has a constant density. Thus, we expect the radio luminosity to increase with time. Densities in my models of region II (see Fig. 2.1) are two to four orders of magnitude smaller than limits discussed above. However, radio must be expected when the SN shock front hits the shell 10 to 100 years after the explosion. In case of SN2014J running into a constant density medium and from radio observations, [165] 3 found an upper limit n0 . 1.3 cm . The same luminosity can be produced in a wind withm ˙ . −10 7.0 10 v /(100 km/s) M⊙/yr. The neutral lines and the IR emission indicate shells at × × w distances between 10 and 20 ly expanding at velocities between 120 140 km/s. Margutti et − al. [130] and Perez-Torres et al. [165] concluded from their observations that a DD system was the likely progenitor. However the ﬁndings of [64] led them to favor a SD progenitor, although excluding a Red Giant as donor. In light of the diﬀerent conclusions, I apply my analytic description to explore the wide range of parameters within the observational limits from the radio- and X-rays. For the outer environment, I consider both an interstellar medium with constant density, s=0, and one consisting of a wind, s=2, produced during the stellar evolution history.

70 The X-ray and radio observations provide limits on the far-inside region (region I) within the contact discontinuity (Figs. 7.1 and 7.3). In order to apply this constraint to our parameter space I will consider our three cases: RG-like wind, MS star wind, and wind originating from the accretion disk. Case I: If the environment is produced by an ongoing RG-like wind with a velocity −10 of 30 km/s, the mass loss would have to bem ˙ . 2.1 10 M⊙/yr which is much too low for × RG wind. Ongoing RG-like wind from the donor star is also very unlikely to be consistent with the IR imaging [108]. If the environment is formed by a RG-like wind, it must originate from the progenitor star prior to the formation of the WD. Case II: If the wind originates from a MS star −9 with 500 km/s, radio limits would allowm ˙ . 3.5 10 M⊙/yr. Case III: For an AD wind with × −8 3000 km/s, this meansm ˙ . 2.1 10 M⊙/yr, well within limits discussed in the introduction. × Narrow circumstellar lines have been used to probe the environment close to the contact discontinuity (e.g. Fig. 2.1). Using ionization models, [64] attributed the blueshifted K I and Na I D absorption features to circumstellar shells at 10 20 ly at velocities of 120 140 km/s. ∼ − ∼ − A further reduction of the parameter space comes from the limits on X-rays and radio. Using our models, we can infer what kind of progenitor wind (and system) could have produced such shells expanding at the proper velocity and distance with the duration of the wind as a free parameter.

8.2 Analysis of SN2014J

In the following, I want to apply my analysis by combining the tools of the previous chapters to the observations of SN2014J. For ﬁnding allowed parameters, I use the given distance and velocity of the shells in combination with wind parameters for the AD-, MS-, RG-like and “RG-like plus AD” winds. 1.) As a ﬁrst step, we want to determine the allowed duration of the wind t using the shell velocities and distance as given by [64]. Assuming the shells are at RC and the expansion velocity at R˙C , we can use equation 2.9 to estimate the allowed range for the duration of the wind (Fig. 8.2).

2.) In a second step, we make use of the Π-theorem to determine combinations of n0,m ˙ and vw. Because the solutions depend on one and two Π groups for constant ISM and wind-environments, respectively, we have to consider two cases.

71 Figure 8.1: Potassium K I λ7665 line evolution for SN2014J. Colored lines designate observations from various epochs, from purple (earliest) to red. Vertical line marks show corresponding velocities in km/s (negative velocities mean moving toward the observer). The absorption features at 144 km/s and 127 km/s clearly show a steady decline in − − strength throughout the course of measurement. The Na I D1 spectrum is shown with the dotted line, clearly showing an absorption feature also at 144 km/s. From Graham − et al. [64].

Case I: For the constant density environment (s=0), the solutions can be described in the Π 2 space by a unique parameter µ = mv˙ w . We have to ﬁnd µ from the observations and step one as n0 described above (Fig. 8.3). Assuming a particle density of the ISM between 0.1 and 10 cm−3, the allowed range for s = 0 in the M˙ -vw-space is shown in Fig. 8.4. Note that the allowed range has linear boundaries because the exponential dependence on the free parameter (see Table 2). The minimum and maximum of µ are found by the values in the lower left and upper right of Fig. 8.3, respectively. In Fig. 8.4, the allowed range of µ in them ˙ v space is shown in green. A constant − w density may also be produced in region II of prior winds (Fig. 2.1). Case II: For ISM environments produced by a wind (s=2, region I), the allowed solutions depend on two Π groups, Πm˙ and Πvw (Eq. 2.26 and 2.27) which relate the mass loss rates and velocities of the interacting winds. Possible solutions are linear in time (Eq. 2.9). Thus,

K2C = RC /(vwt) = R˙C /vw. Possible solutions depend only on the shell velocity which can be measured directly from the spectra [64]. Note that the velocity RC equals the matter velocity. Because it varies slowly in this region (see Fig. 7.3), the Doppler shift of lines is a good measure ˙ of RC . Knowing vw, the original wind velocity is given for any Πvw .Πm˙ gives the ratio between

72 Figure 8.2: Allowed duration of the wind for SN2014J as a function of Rc and R˙c, for constant density environment, s=0 (left), and environments formed from prior mass loss, s = 2 (right).

the ongoing and the outer wind. I assume that the progenitor wind has the parameters given by a RG star as discussed in Chapter 6. To solve for the implicit condition imposed given by the observations and possible model solutions, a standard rejection technique is applied. I use a Monte

Carlo Scheme to sample the parameter space of the ongoing wind for acceptable solutions for vw,1. For a given wind parameter of the outer wind, I test each position in Fig. 4.1 (or the lookup table) for whether the resulting wind velocity vw,1 is consistent with an RG wind. If it is, I accept the possible solution. The lines of solutions found in this manner are shown in Fig. 8.4. The line widths are given by the range of RG wind properties accepted to be consistent with a solution. 3.) In a third step, ﬁnally, I compare the allowed range in parameter space with the ranges consistent with our three cases. The ranges ofm ˙ and vw for each type of wind is presented as a box. A comparison between the predicted ranges for various winds and the allowed range for SN2014J constrains the possible progenitor properties. In addition, I marked the parameter range excluded from the lack of X-rays and radio detections. The possible wind parameters for SN 2014J and the location of typical RG, MS and AD winds are shown in Fig. 8.4. In addition, I show the allowed region for an environment produced by an RG wind of the progenitor prior to the WD formation. For constant ISM environments (Case I), there is obviously no overlap between regions allowed from the observations and the parameter space of solutions. Constant particle density environments

73 ranging from 0.1 to 10 cm−3 cm−3 can be ruled out. However, a very low density ISM of . 0.01 cm−3 is consistent. I ﬁnd two possible solutions–both of which require an environment formed by a prior RG-phase of the progenitor (Case II). In either case, we will need a short delay time between the formation of the WD and the explosion. As discussed above, a wind from the progenitor prior to the WD phase will produce an inner structure with a density close to r−2 (region I, Fig. 2.1), and an outer cocoon of constant, low ∝ density (s=0, region II) which is lower than the ISM by about an order of magnitude. The lower limit for the particle density in region II is 0.009 cm−3 which widens the regime of solutions for ≈ s=0. The AD wind may either interact in the r−2 or constant density region. In either case, we can exclude RG or RG-like winds from the progenitor system. A RG star as the donor star can be ruled out, too, by the X-rays and radio. This ﬁnding is also consistent with the lack of a RG remnant at the location of SN 2014J.

Within the scenario of MCh explosions, AD winds are consistent with both interaction in region I and II. AD winds are also supported by the high Doppler shifts of the narrow lines reported by [108]. For further constraints, I want to discuss the formation of the necessary wind environment formed from prior mass loss which created the RG cocoon to, at least, 10 to 20 ly. As discussed −5 −7 above, I use vw = 20 to 60 km/s, mass loss rates between 10 and 10 km/s, and particle −3 densities n0 between 0.1 and 10 cm for the initial ISM. Using the relation for RC of Table 3.1 and the reference model for normalization, I obtain for the duration needed for the wind

5 −2 −1 1/3 tRG = const [ RC n0 vw m˙ ]

Wind durations are obtained between 1.3 104 and 1.4 106 yr prior to the explosion. × × This provides an upper limit for the duration of the AD disk wind and hence the accretion. Within the picture of an MCh mass explosion starting from an 0.6 to 1.1M⊙ C/O WD, some 0.7...0.25M⊙ must be accreted. As discussed above, hydrogen is only stable between 2 × −8...−6 10 M⊙/yr. Thus, only the long time scale is consistent with a hydrogen accreter (assuming very small delay time). This makes it more likely that the donor was a He star or a C/O accretion

74 −3 disk. Assuming an ISM density of 0.1 to 1 cm , between 0.5 and 10 M⊙ would have been swept up by the ISM. The delay time between the formation of the WD and the onset of accretion is also constrained: tRG is the sum of the accretion time and the delay time, tD. At the very most, tD cannot exceed tRG, but it will likely be signiﬁcantly lower in order to allow a suﬃciently long phase for accretion. The largest t possible is given by the longer timescale and fastest accretion rate: t < 1.3 106yr D D × −6 6 for H accretion. The fastest He accretion rate of 8 10 M⊙/yr gives t = 1.4 10 yr. However, × D × for a more representative duration of t = 1.3 105 yr, H accretion is not possible, and using a RG × −6 4 fast He accretion rate of 8 10 M⊙/yr gives a more typical t = 9.2 10 yr. × D × I note that, with this constraint, the region within R2, with s = 2 is much too small in all models (see Table 7.2). The most likely solution is therefore an AD wind interaction within region II. For SN2014J, the following picture emerges as our most likely solution: An enviroment is produced during the RG phase of the WD progenitor, consisting of a low density cavity surrounded by a shell, or a very low density ISM which would obviously not be expected for the starburst galaxy M82. Wind from the accretion disk (AD-wind) runs into this environment and produces a shell structure responsible for the variable narrow lines observed. Consistent solutions can be obtained if the AD wind interacts either with region I (case A, s=2) or region II (case B, s=0) of the prior RG wind. This wind runs into an ISM of 0.03 and 0.1 cm−3, respectively. The RG winds have a mass loss rate, wind velocity and duration which are −7 6 −7 6 8 10 M⊙/yr, 30 km/s, 10 yr; and 10 M⊙/yr, 60 km/s and 2 10 yr, respectively. For × × case A, the location of the reversed shock is larger than 20 ly. For case B, the densities in region II are about 0.01 cm−3. Obviously, a lower ISM density by a factor of 10 is a possible solution without the need for an RG wind.

The most likely solutions for case A and B is an AD wind with (m ˙ [M⊙/yr],vw[km/s]) of (10−8, 3000) and ( 1.5 10−8, 5000) with a duration t of 35000 yr and 20000 yr, respectively. The × −4 −4 inner corresponding void contains 3.5 10 M⊙ and 2.9 10 M⊙ of material at a density ≈ × ≈ × of 4 10−5 and 3 10−5 cm−3. The corresponding density is 4 10−5 cm−3, the outer shell ≈ × × ∼ × width is 2 ly and its density is 0.02 cm−3. For case B, the shell has a thickness of about 2 ly ≈ ∼ and a density of 0.03 cm−3. For case A, a thin shell is formed (Fig. 2.1) with densities decreasing ≈

75 from 10 to 0.1 cm−3 over some 0.015 ly. The equivalent width for Na I D is EW = 27 mA˚ with a velocity dispersion of 14 km/s for case A, and 27 mA˚ and 4.2 km/s for case B.

2 Figure 8.3: Determination of µ =mv ˙ w /n0 by observations. µ is given as a function of the velocity of the shell and the shell radius in the range given by [64]. I assume particle densities of the ISM between 0.1 and 10 cm−3. I ﬁnd a range of possible solutions 2 2 3 µ(SN2014J) between 15.11 and 95.85, in units of (M⊙/yr)(km /s )(cm ). Here, the range of µ is given by the extrema in shell velocity and radius.

8.3 Analysis of Other SNe Ia

Besides SN2014J, there exist a handful of other SNe Ia that are known to have blueshifted, time-varying features, that were observed with high resolution spectrographs. These are PTF 11kx [38], 2007le [212], and SN2006X [159]. SN1999cl was also shown to have time-varying Na I but with low resolution measurement [6]. As a result, it cannot be determined whether the absorption

76 is blueshifted or not. Time varying, narrow, blueshifted absorption lines are a significant evidence that nearby, out-flowing material from the progenitor system is interacting with the supernova light. By interpreting these lines as originating from dense shells arising from wind/wind or wind/ISM interaction, we can apply SPICE to analyze these SNe as was done with SN2014J in the above section. PTF 11kx, SN2011fe, SN2007le, and SN2006X will be explored first, then some other SN for which high resolution spectra have been taken will be analyzed.

8.3.1 PTF 11kx

PTF 11kx was observed in January of 2011 at a luminosity distance of 207 Mpc [38]. Radio ∼ −6 1.65 observation of this SN gave an upper limit of 4.5 10 M⊙/yr (v /10 km/s) , assuming the ∼ × w SN ejecta ran into prior wind of mass loss m and velocity v . For v = 30 km/s this gives ⊙ w w −5 m⊙ 2.8 10 M⊙/yr which is weakly constraining in ourm ˙ v parameter space. However, ∼ × − w PTF 11kx exhibited time-varying features in its spectra, indicative of CSM interaction. Much stronger absorption was shown in Ca II H&K lines than in Na I D, which is anomalous for typical interstellar medium [241]. This observation by itself suggests that the progenitor system interacted with its environment. The spectra also revealed narrow Na I, Fe II, Ti II, and He I absorption lines, where the Na I, He I, and some of the Fe II lines were all time-varying, located at a blueshift of 65 km/s with respect to the progenitor system, and with velocity dispersion of 10 km/s. ∼ ∼ Other H and Ca features were observed at 100 km/s. At day 20, these features transitioned ∼ ∼ from narrow absorption to broad emission. Dilday et al [38] argued that the observations were best explained by two nova shells running into wind previously blown oﬀ from the RG companion star. They proposed that the inner shell contained H and Ca and was non-uniform, moving outward from the system at 100 km/s and, assuming that the SN ejecta ran into the inner shell at day 20, ∼ ∼ that the distance at the time of explosion was 1016 cm. The outer shell with the Na, H, Ca, Fe, ∼ and Ti moved outward from the system at a speed of 65 km/s. Their schematic showing this ∼ is reproduced in Fig. 8.3.1. The distance of this outer shell can be constrained by the upper limit of 10 ly, beyond which photoionization is assumed to be negligible (For a sample calculation ∼ of how photoionization from the SN varies with distance, see Fig. 8.5). The corresponding wind parameter space given by SPICE for these two shells is shown in Fig. 8.7. For the inner shell, Fig. 8.7 (left) shows that for wind interaction with prior mass loss (s = 2), AD winds are most likely. The wind duration will be 104 yr. Since the outside shell moves at ∼

77 65 km/s, the wind that runs into it must be faster, i.e. it is too fast to be RG-like. For constant ∼ density (s=0) interaction, AD winds are more likely than MS winds but there is small overlap with either region. For AD winds and MS winds, this would require durations of order 40 yr. The corresponding Na I equivalent widths however are 30 mA for AD winds and 0.03 mA for the MS ≈ ≈ winds. Since the latter is far below observational limits, MS progenitor systems can then be ruled out for constant density interaction in PTF 11kx, leaving AD winds as more likely. However, since we are exploring a two shell model, my interpretation of the results of SPICE must be reasonable for the two shell system. For AD winds in a constant density environment the required outer density would be large, on the order of 10 cm−3. This constant density would have to come from region II from the outer shell’s interaction region (see 2.1). This would require an outer interaction region contact discontinuity R moving at 65 km/s with inner density n 10 cm−3, and reverse shock C ∼ 2 ∼ R2 . 0.01 ly. I now consider the possibilities that the outer interaction region (outer shell) itself represents wind/ISM or wind/wind interaction: 1) If the outer interaction region is wind/ISM, this would require (see Table 3.1), for a wind velocity of 100 km/s, a set of parameters n 1012 cm−3, 0 ∼ 4 m˙ 0.1 M⊙/yr, and t 3 10 yr. It is not possible for the progenitor system to have blown off ∼ ∼ × 3 12 −3 10 M⊙ into a medium of 10 cm and be consistent with either the observations or reality. ∼ The scenario of the inner shell interacting with region II of another wind/ISM shell will therefore be discarded. 2) If the outer interaction region is wind on wind (s = 2), an extended inner region II of ∼ 10 cm−3, an R˙ of 65 km/s, and a reverse shock at . 0.01 ly is possible, granted the wind velocity C ∼ between the shells (called hereafter the “interim wind”) was between 65 km/s and 100 km/s. ∼ The mass loss rate of the interim wind can be approximated by the hard shock jump condition: 2 −10 ρ =m/ ˙ (πv R ) where ρ is the density. For these conditions it meansm ˙ 6 10 M⊙/yr. 2 w 2 2 ∼ × Putting these parameters into SPICE, I find that the far outside wind parametersm/v ˙ w must be −9 −11 M⊙/yr in the range of 10 /60 to 10 /20 in units of km/s . This wind should be RG-like and have been blown off from the WD star before it became a WD. These parameters are not in the appropriate range for RG-like winds discussed in Chapter 6, therefore the scenario of the inner shell interacting with region II of an outer s = 0 interaction region is also unlikely. Fig 8.7 reveals that RG-like winds, are largely allowed by the radio limits. However, they are very unlikely to produce the shells as they would be required to blow faster than the shell velocity

78 of 100 km/s. We can then conclude that for the inner shell, an AD wind interacting with RG-like wind from prior mass loss is the most likely scenario. Since there is at least one other shell further away from the progenitor system, we cannot conclude that the wind duration corresponding to the inner shell represents the entire accretion phase. The outer shell is constrained to be within the range where H can be ionized by the initial UV burst of the SN (see Fig. 8.5). It can be anywhere from a few hundredths of a light-year to about a light-year. distant from the progenitor system. For this reason, the allowed region is fairly large in Fig. 8.7 (right). However, due to the constraint from the two shell picture that the interim wind velocity between the shells must lie in the range of the outer and inner shell speeds, only that portion of the velocity range between 65 and 100 km/s is possible. Only a RG-like wind with ≈ ≈ very high velocity can be responsible. Both s = 0 and s = 2 interaction cases are possible for this shell. The corresponding duration is in the range t 102 yr to t 103 yr. ∼ ∼ The conclusion that seems most likely for the two shell model of [38] is that the outside shell was produced by a fast, 80 km/s RG-like wind (the interim wind) running into either a slower ∼ RG-like wind or constant ISM environment. The inside shell was produced by a much faster AD wind running into the 80 km/s interim velocity wind, i.e. region I of the outside shell structure. ∼ However, as is duly noted by [38], the inner shell was very likely not uniform. Since SPICE assumes spherical symmetry, this possibility casts doubt on the accuracy of this analysis. Ignoring the inner “shell” would then mean disregarding the left plot in Fig. 8.7, and for the right plot ignoring the constraint that the wind velocity must be 65 v 100 km/s. In that case, Fig. 8.7 (right) ≤ w ≤ reveals that AD winds are by far most likely, either interacting with a constant ISM or prior mass loss from RG-like wind. MS winds are marginally possible, but as noted above, the Na I EW would then be very low and likely not detectable. The duration required is very small, from 102 to ∼ 103 yr, pointing possibly either at He or C/O accretion, or large peculiar velocity (see Chapter ∼ 6).

8.3.2 SN2011fe

It may be interesting to apply this method to SN2011fe, one of the best observed SNeIa at a distance of ( 6.4 Mpc) [153]. This is another nearby, well-observed, very normal SN Ia. Similar ≈ to SN2014J, environmental density constraints have been set for SN2011fe using X-rays by [132] and radio by [23]. They ﬁnd upper limits, for a wind CSM environment, ofm ˙ . 6 10−10 × ×

79 v /(100 km/s) M⊙/yr. For an AD wind of 3000 km/s this limit corresponds tom ˙ . 1.8 w × −8 10 M⊙/yr. These parameters are consistent with the wind from an accretion disk with a SD progenitors as discussed above, as shown in Fig. 8.8. For constant CSM environment, [23] found −3 n0 . 6 cm which is above typical density found in our models within the bubble. Unlike for SN2014J, no time varying narrow lines have been observed for this event [162], which prevents us from specifying shell velocities or distances in conducting our analysis. The narrow lines may either attributed to the ISM or distances signiﬁcantly larger than in SN2014J which remains unaﬀected by the light front (see e.g. Fig. 5 in [64]). If there is a shell produced in a CSM, the bubble must be larger than & 40 ly. For typical AD wind parameters and a constant density environment of 1 cm−3, this requires durations of t & 106 yr or an environment produced by prior mass loss and a duration larger than in SN02014J by a factor of 2 to 4.

8.3.3 SN2007le

SN2007le was observed on October of 2007 [212] in NGC 7721 at a distance of 23 Mpc. No ≈ X-ray or Radio observations were conducted for this SN. It did however exhibit several Na I D features in the spectra, one of which was time-varying with increasing absorption. This line was at 10 km/s blueshifted with respect to the progenitor system, and based on photoionization ≈ calculations, at a distance of 0.3 ly away. The other, non-time varying Na I lines are shifted ≈ further in the blue. It is possible that they represent multiple shells from quasi-static mass loss episodes that show no variation due to the shells being at larger distances from the progenitor system. If there were other shells, however, we expect that they would be at lower velocities since they would be slowed down by the ISM. Therefore they should occupy a more redshifted portion of the spectrum. That is not the case here, so it is probable that they arise from uncorrelated clouds in the ISM. Assuming the prior discussed scenario that the innermost shell is indeed from a static mass-loss episode, Figure 8.9 shows the result of the SPICE analysis for this SN. According to Fig. 8.9, AD, RG, and MS winds are all about equally, but marginally, likely. The region indicating wind/ISM interaction just barely touches the parameter space regions for the diﬀerent considered progenitor wind cases. AD and RG winds are possible for large ambient −3 density n0 & 10 cm , and MS winds are possible for interaction within the low density region II of a prior mass loss (see Fig. 2.1). The Na I D EW for such an interaction, however, is expected to be . 1 mA˚ in such a case, and should not be detectable. MS wind interaction is therefore

80 again considered more unlikely than the AD and RG winds. The wind duration required for s = 0 interaction in SN2007le is t 5400 yr, which is far too short for a H accretion phase (see Chapter ∼ 6) This is possible for He or C/O accretion onto the WD, or for high peculiar velocities of the system (see Chapter 6). For this shell in SN2007le, only wind interaction with constant density (s=0) ISM is possible.

This is because for wind on wind interaction, the outside wind vw,1 must be slower than the shell speed of 10 km/s in order for there to be an interaction. RG-like winds are not slower than ≈ 10 km/s (see Chapter 6), therefore there is no interaction for the s = 2 case. For SN2007le, I conclude that a brief, t 103 yr outburst of either RG-like or AD winds, interacting with constant, ∼ high density ISM, could have created a shell which gave rise to the time varying narrow feature observed by [212].

8.3.4 SN2006X

SN2006X was the first SN Ia where time-varying narrow lines were discovered. This SN was observed February 2006 in M100, 20 Mpc away Early radio observations gave an upper limit of a −7 few 10 M⊙/yr for a wind velocity of 100 km/s [219]. This corresponds to δ & 8 (see Fig. 8.4). − SWIFT satellite measurements yielded no observed X-rays in the 0.2 10 keV band [99]. High − resolution spectra were observed during 5 epochs spanning a time period starting at 2 days before maximum light until 121 days after. These spectra revealed time-varying Na I D features starting from the closest observable blueshift of 15 km/s and spanning out to a blueshift of 120 km/s. ≈ ≈ This is shown in Fig. 8.10. [159] identify four distinct features, labeled A-D in Fig. 8.10, which exhibit complex evolution: first components B, C, and D strengthen with time while A does not change, then B stops evolving when C and D strengthen again, and A strengthens greatly, after which the evolution stops. As in the case of 2007le, we are prone to assume that the nearer components should be more time-varying and are more blueshifted than the further off ones. Given the complex spectral evolution it is difficult to determine exactly what might be going on, especially if we assume that the components represent shells that are from quasi-static mass loss and can be described by SPICE. The sudden increase of features C and D at date +61 may be attributable to SN ejecta running into the CSM material [159].

81 To pursue the analysis, a one shell model is chosen to approximate the entire range of varying components, with R˙ C = 50 km/s. [159] argued using photoionization calculations that the distance of the material responsible must be a few 1016 cm, or a few 0.01 ly away from the SN. A range of values 0.02

8.3.5 SN without Time-Varying Narrow Lines

We now shift focus to a category of SN Ia which show Na I D lines but do not vary noticeably during the interval of observation. Because the lines do not vary with time, there is less certainty that they are correlated with the system at all, and may arise from the ISM. However, I will explore some possibilities. Sternberg el at. [218] analyzed a sample of 35 SNe Ia for which high resolution spectra were available. They showed a statistical preference for blueshifted absorption of 54.6%, dominating over 22.7% for redshifted, and 22.7% for symmetric/neither, according to

82 their classification. This argues that outflows are responsible for at least some of the absorption features observed. Fig. 8.12, taken from their work, shows the distribution of spectral components in velocity-column density space. According to that figure, 67% of components from SN Ia (blue triangles) have negative or blushifted velocities, within that sample. [218] concluded from their analysis that the SD progenitor channel was at least a significant contributor to the total population of SNe Ia. Phillips et al. [170] examined a sample of 32 SNe Ia, all having high resolution spectra and well-observed light curves associated with them. [170] show in their work that SNe Ia with a particular measurable “diffuse interstellar band” (DIB)–absorption features arising from the ISM, show extinction derived from the SN colors consistent with dust extinction implied from the DIB. From this they argue that the extinction likely comes from dust ISM of the host galaxy and not CSM which is associated with the progenitor system. The Na I D lines observed for those cases seem to originate from the ISM and not from material blown off from the progenitor system. This might appear to challenge the conclusion of [218] discussed above, yet [170] and [218] give complimentary results, both shedding light on the issues at hand. Sample spectra from [170] is shown in Figs. 8.13 and 8.14. SN2006X from Fig. 8.13 and SN2007le from Fig. 8.14 were previously discussed above as having time-varying features. I now proceed to a general analysis of the SNe absorption components shown in Figs. 8.13, 8.14 and 8.12.

8.3.6 A Collection of SNe Ia Spectra

Figs. 8.13 and 8.14 show line proﬁles for several SNe Ia, with red lines to mark out distinct components. I will not go in depth describing the possibilities for every SN, but just touch on the general picture. Fig. 8.13 show a variety of spectral features for several SNe Ia, with varying complexity. For example, SN2007sr, shows but one peak per doublet line, whereas SN2012cg shows several components, one blueshifted and 6 redshifted, of varying strengths. Redshifted features are very likely due to uncorrelated ISM clouds along the line of sight and so will be ignored in this analysis. The blueshifted feature of SN2012cg lies at about 50 km/s with respect to the strongest line, taken to be the location of the progenitor system in velocity space. The top SN is 2006X, which is discussed above. Underneath is SN2006cm, showing a strong saturated Na I absorption component(s) with a small, blueshifted peak on the side. The corresponding K I λ 7665

83 profile reveals the underlying structure: two components blueshifted to about 20 and 40 km/s. SN2008fp reveals one, distinct, fairly strong absorption feature at 45 km/s (and what appears ≈ to be an extremely weak or blended feature at 30 km/s), 2009ig a significantly weaker one ≈ at 8 km/s, while 2009le shows several features overall, two of which are blueshifted at about ≈ 10 km/s and 18 km/s. SN2010ev shows no blueshifted or redshifted features. It may be significant that SN2006X, which we know showed time-varying Na I features, has its blueshifted feature much wider in velocity space than those features for the other SN. Fig. 8.14 shows 14 SNe, all of which have no K I lines detected or observed. SNe 2001cp, 2001da, 2002cr, and 2007af only show redshifted extra components. The absence of blueshifted lines may indicate the absence of shells altogether. This would leave a violent merger DD progenitor system as most likely (see Chapter 1). Another possibility is that shells do exist but are too weak to be detected. This is possible for s = 2 models with long wind durations (see Table 4.1). This however is unlikely for reasons discussed above: the physical size of the region where this interaction takes place is small and hence such a low EW from long duration is unlikely to be realized. The shell would move into a region of constant density and the EW would cease to follow the scaling relation. For s = 0 models the EW grows with time (see Table 3.1), so we would be inclined to suggest that short durations within the s = 0 picture would give a non-detection of a structure. Another possibility is MS winds. Due to the low mass loss rates they are unlikely to produce observable shells. For s = 2 models particularly, the velocity ratio vw/vw,1 between MS winds and the outer RG-like winds is also less than for AD winds and therefore the interaction will be weaker for this reason as well. Some of the SNe in Fig. 8.14 do show blueshifted features. SN2002bo shows a strong component at 70 km/s and a weaker one at 85 km/s, SN2002ha exhibits three blushifted, fairly weak ∼ ∼ components at 50, 70, and 90 km/s. SN2002jg shows two fairly strong components at 30 and 75 km/s, and SN2007fb has one weak component at 10 km/s. SN2007fs has one feature at ≈ 8 km/s, SN2007kk shows a few close components at 40 km/s, I skip SN2007le because it was ≈ ≈ discussed independently above, and SN2008C shows several blueshifted components in multiple groupings, at 8, 12, 50, 90, 105, 140 and 150 km/s. SN2008C has more components than any ≈ other thus far discussed, and spans the greatest range in velocity. SN2008ec shows a few weak

84 components, at 5 km/s, 10 km/s, and 50 km/s, while SN2009ds has two weak features, at 10 and 61 km/s. Of the 20 SNe considered in this discussion from [170], 5 have no blueshifted components, 5 SNe have 2 blueshifted components, 3 SNe have 3, and 1 SN has 8 blueshifted features. Typical component velocities are 10-90 km/s but can range from less than 10 km/s up to 140 km/s. Velocity dispersions are about 10 20 km/s, which is a little high compared the models in this − work (see e.g. Table 7.1). SNe with individual components can be given a fuller analysis, and be treated with a single shell model, like SN2014J. Sne with groups of blueshifted components can be treated as having a cluster of shells, and be analyzed as PTF 11kx was above. These analyses should also include the constraint that the shells are likely too far from the SN to show time-variation in the ionization balance. In the following section we will conduct a bief analysis of a collection of components and see how the reference models of this work compare.

8.3.7 Sternberg et al. Analysis

Fig. 8.12 (top) shows that 67% of the Na I D absorption components seen in the SNe Ia sample that are classiﬁed as either blueshift or redshift, are in the blueshift category. About 30% of blueshifted components have velocities less than 30 km/s. If these arise from shells produced by progentior system winds, they cannot represent the s = 2 case. A shell must be faster than the wind speed from prior mass loss, and for RG-like winds this velocity is 30 km/s. It is the s = 0 ∼ case that produces shells of 10 km/s so therefore those components can only arise from wind ∼ interaction with an ISM of constant density. The column density shown on the vertical column also provides limits on the age and origin of the shells. The mean for the log of column density shown is 11.6 cm−2 (solid horizontal blue line). Fig. 8.15 shows the same SNe Ia data points as in Fig. 8.12 (blue triangles), with the addition of the s=0 AD and RG reference models from this work: model no. 2 Table 7.1 and model no. 9, respectively (Table 7.2). The AD reference model for the s=2 case (model 4 in Table 7.6) has a number column density of 3.8 107 cm−2 and a velocity of × 239 km/s, therefore it does not show up within the plotted range of Fig. 8.15. Fig. 8.15 shows − that the sodium number column density for my models lie within the observed range. The velocities of the models, however, are both fairly low. Faster velocities may be attained by lower duration winds, either from fast He or C/O accretion or large peculiar velocities of the system, or it may be achieved by either faster winds, larger mass loss rates, or lower density ISM. Lower duration

85 3/5 wind will also lower the column density as τm scales with t (see Table 3.1). The s=2 models in Table 7.6 have column densities too small by a factor of about 103 to be in the observed range in Fig. 8.12, and some velocities are too high. To increase the Na column density, the duration must −1 decrease here also, as τm scales with t (see Table 4.1). The shell velocities can be lowered by having either a lower ratio of inner mass loss to outer mass lossm ˙ 2/m˙ 1, or a lower ratio of inner wind velocity to outer wind velocity vw/vw,1 (i.e. Πm˙ and Πvw ).

86 Figure 8.4: Parameters for SN2014J in them ˙ -vw-space as constrained from the spectra and the X-ray and radio luminosities. The lines of constant luminosities Lν are given for δ = log10(˙m/vw) which can be obtained by eqs. 8.1 and 8.2. The red, striped area is the forbidden region based on radio and X-ray observations of SN2014J (δ & 11.15). Note − that the radio limit from 2011fe gives δ & 11.22 [23] (see Fig. 8.8). The bright green area − indicates the allowed range for SN2014J for constant density environments with particle densities between 0.1 and 10 cm−3. A RG wind prior to the formation of the WD may produce a r−2 power law (region I) and a pre-existing constant density surrounding (region II) as low as 0.009 cm−3 (see Chapter 8 and Fig. 2.1). The thick, curved lines correspond progenitor winds running into region I of a prior RG wind with (m/v ˙ w)in(M⊙/yr)/(km/s) of (10−7/60) and (10−5/20) which produce low and high CSM, respectively. These lines bracket the range of possible prior RG winds. Their widths indicate the range of solutions of possible AD or RG-like progenitor winds (see text). The boxes indicate the range observed of ongoing RG, MS and AD winds as discussed in Chapter 6. Note that the variations in wind properties are not well known.

87 Figure 8.5: Photo-ionization calculation for a total density n = 2.4 107 cm−3 and 12 days × after maximum light. The ionization fraction is given in terms of an ionization parameter (see [212]) and the distance from the system which is given on the top axis. Hydrogen is completely ionized up to 0.003 pc (0.98 ly). More than 50% of sodium is ionized to Na ≈ I for distances less than 10 pc (33 ly). From Simon et al. [212]. ≈

Figure 8.6: Schematic taken from [38] showing their interpretation of the observations of the spectral features of PTF 11kx. See text for explanation.

88 Figure 8.7: Same as Fig. 8.4 but for PTF 11kx. The left and right plots correspond to the inner and outer shells, respectively, that were suggested by [38]. The velocity of the inner (outer) shell, R˙ C , is set to 100 km/s (65 km/s), and the range radii are selected to be R = 0.01 ly (0.02 R 1 ly) For s = 2 models in the right ﬁgure, the wind C ≤ C ≤ parameter range from prior mass loss is chosen to be the same as in Fig. 8.4. For the inner shell (left), the range of prior mass loss rates is chosen to be the same values as in Fig. 8.4, but the prior wind velocity range is chosen within the range of inner and outer shell speeds as illustrated in Fig. 8.3.1: 75 km/s v 90 km/s. The red, striped area ≤ w,1 ≤ in the left plot is the forbidden region based on radio observation.

89 Figure 8.8: Same as Fig. 8.4 but for SN2011fe. The Radio limit for the immediate environment (. 1016 cm) was obtained by [23]. No time-varying narrow lines for this SN have been observed and stationary narrow lines are attributed to uncorrelated clouds or distant ISM [162]. Therefore, the existence of shells cannot be inferred from the spectra.

90 Figure 8.9: Same as Fig. 8.4, but for SN2007le. A shell is assumed to be at a distance of 0.3 ly, moving outward at 10 km/s, according to the analysis of [212]. No X-ray or ≈ Radio observations were conducted and hence no corresponding limits are known. Due to the slow velocity of the shell, wind on wind (s = 2) interaction with an RG-like wind (20

91 Figure 8.10: Time evolution of the spectra of 2006X, taken from [159]. The diﬀerent curves correspond to various epochs: -2 days (black), +14 days (red), and +61 days (blue), all with respect to maximum light. The Na I D2 features are shown top and Ca II K is shown bottom.

92 Figure 8.11: Same as Fig. 8.4 but for 2006X. The region ruled out by the radio observation given by [219] is shown in red strips. A shell is assumed to be moving at 50 km/s at a distance range of 0.03 to 0.05 ly, as deduced from the multi-epoch, high resolution spectra given by [159]. The prior mass loss wind environment for the s = 2caseisdeﬁnedbyamass −7 −5 loss rate range of 10 < m˙ 1 < 10 M⊙/yr and wind velocities of 20

93 Figure 8.12: Distribution of various Na I D components, or “bumps” in the spectra, with various log of column density, (log(N), where N is in atoms per square centimeter along the line of sight), and velocity in km/s relative to the inferred location of the progenitor system in velocity space. Plotted are components belonging to SNe Ia, core collapse (CC) SNe, and objects used as a control sample for the Milky Way (MW). Averages are shown with solid horizontal lines. Uncertainties in the average is shown by dashed lines. Among the SNe Ia components, 67% are blueshifted (negative). From [218]

94 Figure 8.13: Sample line proﬁles from the SNe analyzed by [170]. For these SNe both Na I D and K I absorption features were observed or detected, and are given here. Measure- ments are in black and models from a ﬁtting program in red. Components are marked with red lines. From [170]

95 Figure 8.14: Sample line proﬁles from the SNe analyzed by [170]. For these SNe only Na I D features were observed or detected, and are shown here. Measurements are in black and models from a ﬁtting program in red. Components are marked with red lines. From [170]

96 Figure 8.15: The observed SNe I spectral component data of[218] as shown in Fig. 8.12 (blue triangles) plus reference models from this work (open shapes). The horizontal axis is the relative velocity from the progenitor system, and the vertical axis is Log of the number column density for sodium (cm−2). The mean of the data is the blue solid line. The uncertainty of the mean is given by the dashed lines. The purple circle represents reference model no. 2 from Table 7.1, for the s=0, AD wind case, and the orange circle represents reference model no. 9 from Table 7.2 for the s=0, RG wind case. (The reference model for AD winds, model no. 4 in Table 7.6 would lie oﬀ the plot at a relative velocity of 239 km/s and log(N) = 3.8 107 cm−2). The reference models give results within − × the range of observed number column densities. For these models, higher velocities are possible for shorter duration winds, which imply either faster accretion rates of He or C/O or faster peculiar velocities of the system, or from faster winds, and/or higher mass loss rates, and/or lower density ISM. The fastest, most blueshifted components in this ﬁgure may be due to ISM as they are consistent with typical rotational velocities in spiral galaxies.

97 CHAPTER 9

CONCLUSION

I presented theoretical, semi-analytic models for the interaction of stellar winds with the ISM and its implementation in my code SPICE 1. Spherical symmetry and power-law ambient density profiles are assumed. The free parameters are: the a) mass lossm ˙ , b) wind velocity vw, c) the particle density n0 of the in case of a constant density ISM or the mass loss and wind velocity for environments produced by a prior stellar wind, and d) the duration of the wind from the progenitor system. My approach provides an efficient means of studying a wide range of parameters well beyond what is feasible with complex numerical simulations. Using the Π-theorem allows us to test a wide variety of configurations, properties of the solutions along with their sensitivity and dependencies on the wind and environment parameters. Using these dependencies, I showed how to use observations to find possible solutions. The formalism presented and SPICE may be used for a wide variety of objects, including stellar winds. The speed of the semi-analytic approach produces solutions with low computational overhead. This allows us to evaluate a large parameter space for individual objects, and to include realistic feedback by many objects in star-formation and galactic large-scale simulations. Here, the formalism has been applied to study SNe Ia and to constrain the properties of progenitor systems. As discussed in the introduction, SNe Ia may originate from a wide range of scenarios and progenitor channels which lead to an ongoing discussion about the nature of these objects, and their diversity. Most of the channels are not well understood and, thus, my range may be wider than realized in nature.

I studied a variety of winds within the scenario of MCh explosions. The winds may originate from the accretion disk, MS and RG donor star, and over-Eddington accretion of H/He rich matter within this scenario. 1The code SPICE can be obtained by request

98 The environment considered may be produced by the ISM or may be produced by winds during the stages of the evolution of the progenitor prior to becoming a WD for both double or single degenerate systems. Within the MCh scenarios, we studied wind from the accretion disk, MS and RG donor stars, and over-Eddington accretion of H/He rich matter within parameters suggested from literature as discussed in Chapter 6.

Within the MCh explosions, I find that the wind from the accretion disk dominates the environment, or the combined wind from an RG donor and the AD. Such wind leads to a low-density “cocoon” of the order of light years in size. The actual size depends on the duration of the progenitor evolution. The calculations reveal that these cocoons are characterized by interior regions with particle densities often as low as 10−4 cm−3 and which are surrounded by a thin shell. This explains why most SNe Ia appear to explode in low density environment, although SN Ia are observed in the galactic halo, the disk and the bulge. The lack of ongoing interaction in SNe Ia may well be understood in the framework of MCh mass explosions whether they originate from accretion from a MS, RG, He-star or a tidally disrupted WD. If the wind of the progenitor system interacts with a constant, interstellar medium, we expect narrow lines produced by the shell. My calculations show shell velocities ranging from 10 to 100 km/s. For the s = 0 models, the narrow lines are expected to have an equivalent width of 100 mA˚ and are Doppler shifted by about 10 to 20 km/s (see ≈ Tables 7.1, 7.2 & 7.3). In contrast, an environment dominated by a prior stellar wind will result in weaker lines with EW lower by one to two orders of magnitude, i.e. 0.5 to 5 mA˚ (see Table 7.4), which is beyond current observational limits for most SNe Ia. For small cocoons, the narrow lines may show variations in strength and velocity on time scales of months due to the radiation from a SNe Ia. Radiation pressure of the SNe Ia light may accelerate nearby shells (as seen in, e.g. SN1993J). As a separate effect, the SN ejecta may interact with the shell as discussed in Chapter 1 The outer layers of a SNe Ia expands with velocities of 10 to 30 % of the speed of light and we may expect some interaction on time scales between years and several decades (see Tables 7.1, 7.3 & 7.4). For more details, see [82, 40]. Note the possible implications for SN-remnants and their evolution. I studied the effect of winds in ultra-low density environments. If there is a long delay time between the formation of the WD and the explosion, the progenitor system may have moved into

99 a low density environment. Within MCh mass explosions, the size of the cocoon will be larger by an order of magnitude and we would neither expect narrow lines nor interaction (Chapter 7). In general, dynamical mergers are expected to often explode in low density environments due to their long delay times but, also, are unlikely to produce a long-duration wind. Thus, some of these objects are expected to show ongoing interaction for suﬃciently large samples of SNe Ia. A step-by-step scheme has been developed to use the narrow lines for the analysis of SNe Ia progenitors. I applied my models to SN2014J using both the limits from the X-ray and radio, and the observation of narrow lines (Chapter 8), and discussed the allowed range of progenitor properties using the analytic relations (Fig. 8.4). I found the observations to be consistent with an environment produced by a stellar wind. I applied both of my analytic solutions to produce the environment and the wind from the progenitor system. We require an environment which has been created recently with times between 1.3 104 and 1.4 106 yr prior to the explosion. × × Within the picture of an MCh explosion starting from an initial 0.6 to 1.1 M⊙ C/O WD, some

0.7...0.25M⊙ must be accreted. As discussed in the reviews quoted in the Introduction, hydrogen −8...−6 is only stable between 2 10 M⊙/yr. Thus, only the upper limit for the evolutionary times × is consistent with a hydrogen accreter. This makes it more likely that the donor was a He star or a C/O accretion disk as a result of a tidally disrupted WD in a DD system. As discussed in the introduction, He-triggered or double degenerate explosions may show winds similar to AD and the

MCh models. However, to build up a suﬃciently large a He-layer seems to require low accretion rates inconsistent with the short delay times between the RG phase and the explosion. I note that a future generation of double degenerate models may void this argument. My analysis of SN2014J is consistent with other constraints: No RG donor star has been found

[108], optical to mid-infrared light curves and spectra are consistent with an Branch-normal MCh 56 mass explosion and 0.6 M⊙ of Ni [134, 223, 27, 37, 100], and the lack of polarization [164]. Within the class of single degenerate systems we found conﬁgurations for SN 2014J which are consistent with both the limits from X-rays and radio and the narrow lines. Our solutions invoke a wind from the accretion disk running into an unusually low density ISM environment, . 0.01 cm−3, or a low density cavity created by a RG wind some 20, 000 to 35, 000 yr prior to the explosion (see chapter 8).

100 I want to place these findings in context of the possible Hα emission discussed in the introduction. Its detectability depends on the amount of hydrogen, the mechanism of ionization, and the sensitivity and timing of the observations. Possible mechanisms include hard radiation from the shock breakout, 56Ni-decay, surface burning of the WD prior to the explosion, interaction in the forward shock with the CSM, and the reversed shock as a result of the interaction between SN and the CSM. The results are very model dependent with various regimes. Based on ionization calculations and the deflagration model W7 [147] , [30] studied in detail pre-ionization of the progenitor winds by supersoft X-ray sources [65, 181, 103] and the Hα emission in the outer supernovae ejecta ionized by the reversed shock interacting with the progenitor wind. They parameterized their −1 −1 study in terms of M/v˙ w [M⊙ yr km s] with vw being the progenitor wind velocity. [30] put their results into context for the observability of Hα emission for local SNe Ia at distances similar to SN 2014J. For M/v˙ . 1.9 10−8, supersoft X-ray sources dominate the reversed shock mech- w × anism. I note that the actual flux from supersoft X-ray sources may be significantly lower because of material in the progenitor system [59]. The Hα emission in the reversed shock region depends on the ionization fraction of hydrogen. [30] found the ionization fraction less than 0.01, at which the emission becomes inefficient for . M/v˙ = 1.5 10−8, and full ionization for M/v˙ = 1.5 10−6. w × w × For local supernovae with a reversed shock in a H-rich region, Hα should be observable in high −5 quality spectra for a RG wind with M˙ 10 M⊙/yr and v = 10 km/s. Note that the result of ∼ w [30] is consistent with the finding of [59]. Using the observed light curves, [59] put strong limits on the ongoing interaction. Without a strong interaction, there is reversed shock or too weak of one to ionize hydrogen. I note that the dominance of the ionization mechanism will depend on the class of explosion. For example, excitation by 56Ni-decay or the shock breakout can be expected to be small for deflagration models such as W7 or pulsating delayed-detonation models with a significant C/O shell reducing the γ emission, however the role of 56Ni may become important in double-detonation/HeD explosions which show some 56Ni in the outer layers and may produce He-lines instead [85]. For the progenitor wind and environment we found for SN2014J, we do not expect early, observational Hα emission. The mass loss from ADs is smaller by some 3 to 4 orders of magnitude compared to the limits by [30] for local SNe Ia, and the forward shock runs into low −3...−4 density material of some 10 M⊙ in the cavity or the ISM.

101 However, we may be able to see late-time, narrow Hα emission if the the supernova ejecta runs into the shell of the cavity. Due to the low masses of the cavity, the SN ejecta will remain largely unmodiﬁed as it travels through the void and produce Hα emission on impact by both the forward and the reversed shock if we have H in the outer layers of the SN. The velocities of the outer layers are about 1/3 c and, for SN2014J, we may expect emission in about 50 yr. The impact will be earlier for higher density, more compact shells or shells with inner clumps. While we wait to see hydrodynamic interaction for SN2014J, we could observe young SN remnants for this signature. One such example is the Branch-normal SNe Ia SN1972E in NGC 5253, 3.3 Mpc away [113]. Within an increasing number of well observed SNe Ia, it also becomes increasingly obvious that SNe Ia are not “all the same”. These conclusions on SN 2014J alone should not be generalized. PTF 11kx, SN2007le, SN2006X, SNe which were also observed to have time-varying lines, were similarly analyzed, and yielded varying results. For PTFkx, I concluded that both s=0 and s=2 cases were possible. For SN2007le, it was likely the s=0 case involved with AD or RG winds possible. For SN2006X, AD winds were most likely with s=2 more likely than s=0. The required durations for these three cases were in the 102..3 yr range, suggesting either fast accretion of He or C/O, or a system with high peculiar velocity, viewed head-on or from the side. The narrow line proﬁles shown in [170] show varying shapes and features, pointing to possibly complex shell structures, and the component analysis of [218] in conjunction with my models shows a plausible agreement, but suggesting, once again, that lower durations resulting from either He or C/O accretion or large peculiar velocities of the system is required to match the data better. Better observation of narrow line systems are also a key to the environment of SNe Ia and the diversity. Here, my method has been applied to SNe Ia but studies of other types of stellar explosions, namely core-collapse SNe, and SNe interacting with Wolf-Rayet star mass loss, are under way. SPICE may also be used in modeling other hydrodynamic phenomena and has the potential to include feedback from stellar winds on a subgrid scale in star formation and large scale galactic evolution simulations. This brings us also to the limit of this analysis. Although these analytic models provide a practical tool for individual shells or shells from well separated phases of mass loss, in reality nature may be more complicated. The mass loss can be expected to be time dependent. If an environment is formed by this wind, it will deviate from an r−2 law. The mass loss may come

102 in phases, namely brief periods of “superwinds” during the RG phase. Moreover, mass loss from the progenitor system may change over time. In fact, multiple narrow lines have been observed e.g. for SN 2014J [64]. However, the narrow lines may be produced both in the vicinity of the SN or at any distance by unrelated clouds in the ISM. Without time evolution of the narrow lines or measurements of dust components, the origin of the systems of narrow lines remain unclear. Obviously, detailed calculations for spectra and light curves are needed to quantify the intensity of the narrow lines. Moreover, multi-dimensional effects need to be considered for a full analysis, for example in instabilities and mixing, non-symmetric winds, and motion of the progenitor system through the ambient medium. A combination of semi-analytic solutions provide a good starting point for more complex, multi-D calculations with more detailed physics, which are under way, to be presented in forthcoming papers. What role and influence dust may have, especially in the outer layers, is another question deserving close attention. In light of the wide range of progenitor scenarios and properties of the resulting environment, this approach has been justified in order to find the right ballpark in the vast sea of parameters. In reality, however, multi-dimensional affects such as asymmetric winds will become important, including variable winds. Moreover, proper cooling functions and equation of states for the gas are to be taken into account for detailed analysis of high quality data such as SN2014J.

103 APPENDIX A

SPICE

A.1 SPICE Description

In this section is given a basic description of the code that was written to investigate the interactions of SN progenitor winds with their environments. Supernova Progenitor Interaction Calulator for parameterized Environments, or SPICE, is a piecewise, semi-analytic hydrodynamics code written in Fortran 90. It solves for the interaction regions of winds running into an ambient environment of power law density proﬁle ρ r−s. SPICE may be used for other applications than ∝ investigating SN progenitor systems. These applications include stellar wind modeling and for including realistic feedback by many objects in star-formation and galactic large-scale simulations. The assumptions in SPICE are that the interaction is spherically symmetric, self-similar (except in the region noted in Chapter 2, and with the caveat mentioned in Chapter 3), stable, vw >> cs, where cs is the ambient sound speed, the wind parametersm ˙ , vw, p0 (for s = 0) are constant in time, the outer environment is stationary in time for s = 0, and moving steadily away from the origin at velocity vw,1 and with mass lossm ˙ for s = 2, the ﬂow and shocks are adiabatic, and the viscosity is zero. For s = 0 and the Constant Ambient Pressure model (CAP, see Chapter 3), the time must be greater than tcr given by Eq. 3.1 in order to get a sensical solution (See Chapter 3).

Also for the CAP model, as t/tp becomes large (See Eq. 3.2), the solution becomes unphysical due to the mixing that would occur in reality, although the CAP model of SPICE gives a self-consistent solution at those times. For CAP, however, no solution exists for Mach numbers M lower than 1.38496, corresponding to t = . SPICE may need increasingly more CPU time to calculate the ∞ structure as M approaches the lower limit. The s = 0 program within the SPICE package allows the user to choose either the CAP or Zero Ambient Pressure (ZAP) model which corresponds to ambient pressure being neglected or, equivalently, very large outer Mach numbers. Integration of the transformed ODEs is performed with the fourth order Runga Kutta method.

104 A.2 README for SPICE

Figure A.1: Components of SPICE and their dependencies

A.2.1 File Glossary

README.pdf (SPICE/): User guide for SPICE •

Scripts

compile.sh (SPICE/): Bash script to compile the f90 ﬁles in SPICE: spice_s0.f90, •

105 spice_series_s0.f90, spice_s2.f90, spice_series_s2.f90, s0_inverter.f90, and s2_inverter.f90.

spice_s0.sh (SPICE/spice_s0/): Bash script that uses the lookup table and ﬁnd the inter- • action structure for the s=0 case.

spice_s2.sh (SPICE/spice_s2/): Bash script that uses the lookup table and ﬁnd the inter- • action structure for the s=2 case.

time_series_s0.sh (SPICE/spice_s0/exec/): Bash script used to create the time series of • models for the s = 0 case.

time_series_s2.sh (SPICE/spice_s2/exec/): Bash script used to create the time series of • models for the s = 2 case.

inverter.sh (SPICE/analysis_tools/): Bash script used to invert the lookup tables and • give wind/environment parameters compatible with inputted observations.

plot_s0.sh (SPICE/spice_s0/): Bash script used to plot s=0 structures using gnuplot. • plot_s2.sh (SPICE/spice_s2/): Bash script used to plot s=2 structures using gnuplot. • plot_inverted.sh (SPICE/analysis_tools/): Bash script used by inverter.sh to plot the • s0_inverted and s2_inverted ﬁles and create parameter_space.png

Source F90 Files

spice_s0.f90 (SPICE/spice_s0/exec/): Source f90 ﬁle for spice_s0, used by spice_s0.sh • to calculate the single structure for the s = 0 case.

spice_s2.f90 (SPICE/spice_s2/exec/): Source f90 ﬁle for spice_s2.f90, used by spice_s2.sh • to calculate the single structure for the s = 2 case.

s0_inverter.f90 (SPICE/analysis_tools/exec/): Source f90 ﬁle for s0_inverter, used • by inverter.sh to invert the lookup table for the s = 0 case.

s2_inverter.f90 (SPICE/analysis_tools/exec/): Source f90 ﬁle for s2_inverter, used • by inverter.sh to invert the lookup table for the s = 2 case.

spice_series_s0.f90 (SPICE/spice_s0/exec/): Source f90 ﬁle for spice_series_s0, used • by time_series_s0.sh to calculate the time series for the s = 0 case.

spice_series_s2.f90 (SPICE/spice_s2/exec/): Source f90 ﬁle for spice_series_s2, used • by time_series_s2.sh to calculate the time series for the s = 2 case.

106 Executables

spice_s0 (SPICE/spice_s0/exec/): Executable from spice_s0.f90 • spice_s2 (SPICE/spice_s2/exec/): Executable from spice_s2.f90 • s0_inverter (SPICE/analysis_tools/exec/): Executable from s0_inverter.f90 • s2_inverter (SPICE/analysis_tools/exec/): Executable from s2_inverter.f90 • spice_series_s0 (SPICE/spice_s0/exec/): Executable from spice_series_s0.f90 • spice_series_s2 (SPICE/spice_s2/exec/): Executable from spice_series_s2.f90 •

Lookup Tables

s0_lookup_table (SPICE/spice_s0/exec/): The lookup table for s = 0. • s2_lookup_table (SPICE/spice_s2/exec/): The lookup table for s = 2. •

A.2.2 System Requirements and Compilation

SPICE requires a Linux or UNIX-based operating system with bash shell, a fortran 90 compiler, and the plotting utility gnuplot. Compiling is performed by a short bash script compile.sh, located in the main directory. To select your compiler, open compile.sh with a text editor and change the value of compiler to whatever compiler you would like to use. Then run the script: ./compile.sh The following executables are generated: spice_s0 from spice_s0.f90, spice_series_s0 from spice_series_s0.f90, spice_s2 from spice_s2.f90, spice_series_s2 from spice_series_s2.f90, s0_inverter from s0_inverter.f90, and s2_inverter from s2_inverter.f90. They are located in the same directories as their parent source ﬁles.

A.2.3 Running SPICE

Fig. A.2 gives a concise logical ﬂow to the package. SPICE has two principle modes of operation: getting observables from wind/environment parameters or ﬁnding wind/environment parameters from observables. The following examples are given to guide the user through SPICE.

107 Example: s=0 single structure Fig. A.3 highlights the path one would take to calculate a CAP model for s=0 for given wind and environment parameters. To execute this from the command line, first navigate to the spice_s0 directory (see Fig. A.1). Then, execute the bash script spice_s0.sh. The wind/environment parameters must be inputted from the command line like in the following example: ./spice_s0.sh 1.0 1.E-8 3000. 3.E5 From left to right, the arguments are: the ambient density n_0 in particles per cubic centimeter (SPICE assumes that all particles are 1 AMU), the ongoing wind velocity v_w in km/s, the ongoing wind mass loss m_dot in M⊙/yr, and the duration t in years. All values are floats and must have decimal points in them. Exponentiation is indicated by e or E. spice_s0 will then output to the screen, and then prompt the user if they would like to calculate the interaction profile. Select y. A prompt will follow asking if the user desires the CAP or ZAP model. Select y for CAP. spice_s0 now calculates the structure, and then outputs again to the screen. There are two outputs files generated, located in a newly directory within the /spice_s0/outputs directory. The directory will be named like the following: 1._3000._1.E-8_3.E5_2015_Jul19_18:47:31 The first four numbers separated by the underscores are the input parameters. The next number is the year, then the date, followed by the time. This directory contains the following files: plotfile_s0, which contains the profile to be plotted, spice_s0.log, which contains a copy of the screen output, and plot_s0.sh is the plotting script, copied from the spice_s0 directory.

Example: plotting s=0 structure Fig. A.4 is a flowchart for plotting the profiles. To plot the structure we just created in the example, just run the plot_s0.sh script inside of the data directory within /spice_s0/outputs: ./plot_s0.sh 0 1.5 -5.5 2 file where the arguments are, from left to right: xmin and xmax, which are different values of r/Rc; ymin, ymax, and the filename. The output is file.eps. It should look identical to Fig. 7.1a. Example: s=2 single structure

108 To evaluate a single structure with an s=2 environment, navigate to the spice_s2 directory and evaluate the spice_s2.sh script like in the following example: ./spice_s2 30. 1.E-7 3000. 1.E-8 3.E5 where, from left to right, the arguments are: the ambient wind velocity v_w1 in km/s, the ambient mass loss m_dot1 in M⊙/yr, the ongoing wind velocity v_w in km/s, ongoing mass loss m_dot in M⊙/yr, and duration t in years. All values are floats and must have decimal points in them. Exponentiation is indicated by E or e. After some screen output, the user will be prompted whether they would like to calculate the structure. Select y to do so. spice_s2 will output some results of the calculation onto the screen. It will also create a directory inside of the /spice_s2/outputs directory with the output files inside of it. The directory will be named like the following: 30._1.E-7_3000._1.E-8_3.E5_2015_Jul20_14:38:40 The first five numbers in the directory name are the input parameters. They are followed by the year, date, and time. Inside of this directory are the following files: plotfile_s2, which contains the profile to be plotted, spice_s2.log, which contains a copy of the screen output, and plot_s2.sh is the plotting script, copied from the spice_s2 directory.

Example: plotting s=2 structure Following Fig. A.4, run the plot_s2.sh script contained in the output directory: ./plot_s0.sh 0.3 1.3 -2.5 3 file where the arguments are, from left to right: xmin and xmax, which are diﬀerent values of r/Rc; ymin, ymax, and the ﬁlename. The output is file.eps. It should look identical to Fig. 7.3a.

Example: s=0 time series To execute a time series for s=0 environments, first navigate to the spice_s0 directory. Execute the time_series_s0.sh bash script like in the following example: ./time_series_s0.sh 1. 3000. 1.E-7 3.E5 3.E6 4 -5.5 2 where the first three input values are n_0, v_w, and m_dot, just like in the single structure case (see example above); the next values are the initial time (yr), final time (yr), and step number, followed by the plot variables ymin and ymax. Upon execution, time_series_s0.sh starts outputting to the screen, and generates an output directory within /spice_s0/outputs. It will be named like the following:

109 SERIES:1._3000._1.E-7_3.E5_3.E6_4_2015_Jul20_15:32:35 The directory name in this case begins with SERIES:, which is followed by the six input parameters, the year, date, and time. Within this directory are four EPS files, one for each time step, created with gnuplot. Within this directory also are four other directories, one for each series step. Each directory contains a spice_s0.log file which contains a copy of the screen output, a plotfile_s0 file containing the numerical profile for each structure, and a copy of the plot_s0.sh bash script.

Example: s=2 time series Fig. A.5 gives the path for executing a time series for s = 2 environments. First, navigate to the spice_s2 directory. Execute the time_series_s2.sh bash script like in the following example: ./time_series_s2.sh 30. 1.E-7 3000. 1.E-7 3.E5 3.E6 4 -5.5 2 where the first three input values are n_0, v_w, and m_dot, just like in the single structure case (see example above); the next values are the initial time (yr), final time (yr), and step number, followed by the plot variables ymin and ymax. Upon execution, time_series_s0.sh starts outputting to the screen, and generates an output directory within /spice_s2/outputs. It will be named like the following: SERIES:1._3000._1.E-7_3.E6_3.E6_4_2015_Jul20_15:32:35 The directory name in this case begins with SERIES:, which is followed by the six input parameters, the year, date, and time. Within this directory are four EPS files, one for each time step, created with gnuplot. Within this directory also are four other directories, one for each time step. Each directory contains a spice_s0.log file which contains a copy of the screen output, a plotfile_s0 file containing the numerical profile for each structure, and a copy of the plot_s0.sh bash script.

Example: Finding wind parameters from observables Fig. A.6 gives the path one would follow in order to ﬁnd wind/environment parameters from measured shell velocities and shell radii. To execute this task, run the inverter.sh script located in the analysis_tools directory. The following is an example of its use: ./inverter.sh 120. 140. 20. 60. 1.E-7 1.E-5 10. 20. where the arguments are, from left to right: the minimum shell velocity Rc_min in km/s, maximum shell velocity Rc_max in km/s, minimum ambient wind velocity v_w1_min in km/s (s=2), maximum ambient wind velocity v_w1_max in km/s, minimum ambient mass loss in M⊙/yr, maximum

110 ambient mass loss in M⊙/yr, minimum shell radius Rc_min in ly, maximum shell radius Rc_maxin ly. inverter.sh outputs to the screen and creates an output directory within /analysis_tools/outputs named like the following: 120._140._20._60._1.E-7_1.E-5_10._20._2015_Jul20_19:05:44 where the numbers separated by the underscores are the input parameters in the same order, followed by the year, date, and time. This directory contains five output files: s0_inverted, a file containing the minimum and maximum mu values for s=0. s0_inverter.log, a copy of screen output from the s0_inverter.f90 program s2_inverted, a file containing parameter space points for s=2. s2_inverter.log, a copy of screen output from the s2_inverter.f90 program parameter_space.png, a plot of the corresponding parameter space.

A.2.4 Understanding SPICE

Here the process of SPICE will be described in greater detail, in context of the examples of the previous subsection.

s=0 single structure When ./spice_s0.sh 1.0 1.E-8 3000. 3.E5 is executed, SPICE searches s0_lookup_table for the correct output parameters for the CAP model. The lookup table consists of seven columns of numbers: the π group ΠT , the Mach number M, K01, K0C , K02, the proportionality factor for the velocity dispersion, K0σ; and the proportionality factor for τmass, K0τmass . SPICE calculates ΠT from the input variables and then ﬁnds the matching row in the lookup table. The following is output to the screen:

n_0 (cm^-3) = 1.0000 v_w (km/s) = 3000.0000 m_dot (M_sun/yr) = 1.0000E-08 t (Myr) = 0.3000

t_p (yr) = 745434.4033 t/t_p = 0.4024 t_cr (yr) = 6.9876 t/t_cr = 42933.2129

111 Mach number from lookup = 1.949265

From the lookup table (CAP) : Rc (ly) = 20.1086 R_1 (ly) = 28.5250 R_2 (ly) = 2.2916 u_1 (km/s) = 9.4514 u_c (km/s) = 12.0568 sigma_u (km/s) = 0.6247 n_2 (cm^-3) = 9.4495E-05 tau_mass (g/cm^2) = 1.9085E-05 Log optical_depth = 2.3436 EW (mA) = 164.7920

Would you like to generate the hydrodynamical profile?

The top four rows are the input parameters. The next four rows give the pressure equilibration time t_p = tp, the critical time t_cr = tcr, and the ratios of t over those values. The values following are all from the lookup table. They are: the contact discontinuity radius R_C, outer shock radius R_1, reverse shock radius R_2, ﬂuid velocity at the outer shock u_1, ﬂuid velocity at the contact discontinuity (the same as the speed of the contact discontinuity) u_c, the velocity dispersion of region III simga_u, particle density at the inner shock front n_2, mass column density for region III tau_mass, the base ten logarithm of the Na I optical depth for region III, and the equivalent width due to Na I EW.

If t is signiﬁcantly greater than tp, SPICE will output a warning message:

************************************************************* WARNING, t>>t_p. Mixing will modify outer structure. *************************************************************

If t is too close to the lower limit t_cr, SPICE will give the warning:

************************************************************ WARNING, time is close to t_cr. Nearing the lower limit of validity for solution. ************************************************************

If t is at or less than t_cr, SPICE outputs the following error and then terminates:

112 ************************************************************* ERROR. time is less than t_cr. Unphysical solution. Bye. *************************************************************

Moving further, SPICE then prompts the user if they would like to calculate the interaction proﬁle:

Would you like to generate the hydrodynamical profile?

If n is selected, the program terminates, creates an output directory within SPICE/spice_s0/outputs, and outputs there a log of the screen output, spice_s0.log. If y is selected, the user is then prompted if they would like the Finite Ambient Pressure model (CAP), or Zero Ambient Pressure (ZAP) model.

Press ’y’ for CAP, ’n’ for ZAP (ZAP sets M = 1000)

The default is CAP. ZAP is attained by using the CAP model, and setting the Mach number M=1000. For either option, SPICE proceeds to calculate the structure of region III, for CAP taking the Mach number from the lookup as input. When it is complete, it outputs the following to the screen (for CAP):

You have selected ’Yes.’ Calculating...

step count for region III = 999999

From SPICE calculation:

Rc (ly) = 20.1086 R_1 (ly) = 28.5250 R_2 (ly) = 2.2916 u_1 (km/s) = 9.4514 u_c (km/s) = 12.0568 u_mean (km/s) = 10.0188 sigma_u (km/s) = 0.6247 n_2 (cm^-3) = 9.4361E-05 tau_mass (g/cm^2) = 1.9085E-05 Log optical_depth = 2.3436 EW (mA) = 164.7918

113 Region III mass (M_sun) = 6.8744E+01 Expected region III mass (M_sun) = 6.8744E+01 Error in region III mass (%) = 0.0002

Region II mass (M_sun) = 2.9966E-03 Expected region II mass (M_sun) = 2.9977E-03 Error in region II mass (%) = 0.0358

Region II energy (log erg) = 47.0838 Region III energy (log erg) = 47.4988 Total energy (log erg) = 47.6401 Expected total energy (log erg) = 47.6400 Error in Total Energy (%) = 0.0001

The number of integration steps for region III is given. The target is 1000000. The target is given by the integer variable n in spice_s0.f90 and can be changed by the user (the package will need to be recompiled afterwards). The actual number of steps will vary, since SPICE “watches” for a certain condition to be fulfilled in order to stop the integration. The number of steps may be greater, especially for large t/tp. The same quantities are then also given as from the lookup, with the exception of one extra: the mean mass-weighted velocity u_m, which gives the center of the blue-shifted line in velocity space. If SPICE has found a consistent solution, the other values match close to those obtained from the lookup. As a further internal consistency check, the amount of mass of regions III and II are integrated numerically and compared to their expected values from the swept up ISM material and of the wind. The total energy of the structure is integrated and compared to the sum of the kinetic energy of the swept up wind and thermal energy of the swept up ISM. For this example the deviation is very small. In the output directory within outputs are two output files: spice_s0.log and plotfile_s0. The former is a copy of the screen output. The latter contains the structure profile for regions I-IV, in the format of four columns which are, from left to right, the radial coordinate r normalized to RC , fluid velocity u normalized to 100 km/s, fluid pressure p normalized to the outer shock pressure, and particle density n in particles per cm3. The portion of the file containing the data for region III contains three extra columns pertaining to the scale free variables (Π groups) U = ut/r, 2 2 P = pt /(r ρ0), and Ω = ρ/ρ0. The first line in plotfile_s0 gives the values of R1, RC , and R2, in ly.

114 s=2 single structure When ./spice_s2 30. 1.E-7 3000. 1.E-8 3.E5 is executed, SPICE searches s2_lookup_table for the model compatible with the input parameters. The lookup table consists of three columns of numbers in two divisions of lines. In the ﬁrst division the columns are, from left to right:

K21(Πvw + 1), K21/K2C , and the scale free pressure variable at the contact discontinuity in region + III, (PC ) . In the second division the columns are, from left to right: K22, K2C , and the scale − free pressure variable at the contact discontinuity in region II, (PC ) . There is one consistent solution which is given by the conditions: (P )+ =(P )− Π /(Π + 1), and K =[K (Π + C C × m˙ vw 2C 1 vw −1 −1 1)][K21/K2C ] [K21/K2C ] . SPICE ﬁnds the match and outputs the following to the screen:

v_w1 (km/s) = 30.0000 m_dot1 (M_sun/yr) = 1.0000E-07 v_w (km/s) = 3000.0000 m_dot (M_sun/yr) = 1.0000E-08 t (Myr) = 0.3000

Pi_m_dot = 1.0000E-01

Pi_vw = 9.9000E+01

From the lookup table: Rc (ly) = 239.6164 R_1 (ly) = 278.3916 R_2 (ly) = 106.2789 u_c (km/s) = 239.4506 sigma_u (km/s) = Need to calculate (press ’y’) n_2 (cm^(-3)) = 3.9828E-08 tau_mass (g/cm^2) = Need to calculate (press ’y’) Log optical_depth = Need to calculate (press ’y’) EW (mA) = Need to calculate (press ’y’)

Would you like to generate the hydrodynamical profile?

The input parameters are given ﬁrst, followed by the Π groups Π =m/ ˙ m˙ and Π = v /v 1. m˙ 1 vw w w,1 − The results are then given from the lookup table: the contact discontinuity radius, R_C; outer shock radius, R_1; inner shock radius, R_2; ﬂuid velocity at the contact discontinuity, u_c; and particle density at the inner shock, n_2. The velocity dispersion for both regions II and III, sigma_u, the mass column density tau_mass, the base ten logarithm of the optical depth for Na I, and the

115 equivalent width EW cannot be determined by the lookup table and need to be calculated by SPICE. If the structure and/or those values are desired, select y at the prompt. Otherwise the program terminates and outputs a copy of the screen output, spice_s2.log to the output directory within outputs. If y is selected, SPICE runs the integration and then outputs to the screen:

You have selected ’Yes.’ Calculating...

step count for region III = 999992 step count for region II = 999996

From SPICE calculation:

Rc (ly) = 239.6080 R_1 (ly) = 278.3803 R_2 (ly) = 106.2794 u_c (km/s) = 239.4423 u_mean (km/s) = 232.5758 sigma_u (km/s) = 26.7329 n_2 (cm^(-3))= 3.9827E-08 tau_mass (g/cm^2) = 6.9747E-10 Log optical_depth = -3.7249 EW (mA) = 0.2481

Region III mass (M_sun) = 2.4821E-01 Expected region III mass (M_sun) = 2.4819E-01 Error in region III mass (%) = 0.0081

Region II mass (M_sun) = 2.8939E-03 Expected region II mass (M_sun) = 2.8938E-03 Error in region II mass (%) = 0.0041

Region II energy (log erg) = 46.9141 Region III energy (log erg) = 47.2500 Total energy (log erg) = 47.4148 Expected total energy (log erg) = 47.4147 Error in Total Energy (%) = 0.0000

The number of integration steps for region III and region II are given. The target is 1000000. The target number of steps is given by the integer variable n in spice_s2.f90 and can be changed by

116 the user (the package will need to be re-compiled afterwards). The same quantities are again given as from the lookup, with the exception of one extra: the mean mass-weighted velocity of the entire structure, u_mean. The calculated mass weighted velocity dispersion sigma_u is over both regions II and III together. If SPICE has found a consistent solution, the values given from the lookup for the characteristic radii, the velocity of the contact discontinuity u_c and the inside particle density n_2 will be close to those obtained from the calculation. As a further internal consistency check, the amount of mass in regions III and II are integrated numerically and compared to their expected values given by the amount of wind swept up from within and without. The total energy of the structure is integrated and compared to the sum of the kinetic energies of the ongoing and ambient winds. The percent deviation for this example is close to zero. In the newly created output directory within outputs are two output files: spice_s2.log and plotfile_s2. The former is a copy of the screen output. The latter contains the structure profile for regions I-IV. The format is in four columns, from left to right: the radial coordinate r normalized to RC , fluid velocity u normalized to 100 km/s, fluid pressure p normalized to the outer shock pressure, and particle density n in units of 10−6 particles per cm−3. The output for regions II and III include three extra columns pertaining to the scale free variables (Π groups) U = ut/r, 2 2 P = pt /ρc, and Ω = ρr /ρc, where ρc =m/ ˙ (4πvw) for region II and ρc =m ˙ 1/(4πvw,1) for region

III. As in the s = 0 plot ﬁle, the ﬁrst line in plotfile_s2 gives the values of R1, RC , and R2 in ly.

Finding wind parameters from observables: s = 0 inversion When ./inverter.sh 120. 140. 20. 60. 1.E-7 1.E-5 10. 20. is executed, SPICE begins by ﬁnding the wind parametersm ˙ and vw that are consistent with typical constant density (s = 0) environments (Note: The ﬁnal two arguments of inverter.sh, the minimum and maximum shell radii, are optional. The user may omit them or set both equal to 0.). Using the expression

R˙ C = 3/5ΠT √RsT0, where Rs is the gas constant and T0 the ambient temperature, SPICE uses the lookup table s0_lookup_table to ﬁnd the consistent solution. The associated value of ΠT gives the input parameter combination for this solution. By constraining the range of variation for T and duration t, one can shorten the appropriate Π to µ mv˙ 2 /ρ . The ambient temperature 0 T ≡ w 0 4 T0 is set to 10 K (the user can change this value in s0_inverter.f90). If the minimum and maximum shell radii are speciﬁed in the input, the range of duration t is constrained by the

117 relation: t = 3/5(RC /R˙ C ), which will in turn constrain the µ parameter. For the above example, SPICE will output the following to the screen:

s=0 Inversion:

Input parameters: Rc_dot_min (km/s) = 120.0000 Rc_dot_max (km/s) = 140.0000 Rc_min (ly) = 10.0000 Rc_max (ly) = 20.0000

For constrained t (t = 3Rc/(5Rc_dot)) [units of (M_sun/yr)(km^2/s^2)/(cm^-3)]:

mu = m_dot * vw^2 / n_0

Inversion results:

mu max = 9.5774E+01 t(mu max) (yr) = 2.5696E+04

mu min = 1.5096E+01 t(mu min) (yr) = 1.4990E+04

global maximum t for s=0 (yr): 2.9979E+04

global minimum t for s=0 (yr): 1.2848E+04

mu min * n_0 for n_0 = 0.1 cm^-3:

1.5096E+00

mu max * n_0 for n_0 = 10 cm^-3:

9.5774E+02

118 The input parameters for the s = 0 process are given. The mu parameter is deﬁned, and then the maximum and minimum mu values are given from the lookup, along with the associated durations t, with each mu. The global maximum and minimum duration values for the entire range of the input parameters. To plot a relation betweenm ˙ and vw, values of the ambient density must be chosen. A typical range is 0.1 cm−3

s=0 Inversion:

Input parameters: Rc_dot_min (km/s) = 120.0000 Rc_dot_max (km/s) = 140.0000 Rc_min (ly) = 0.0000 Rc_max (ly) = 0.0000

For unconstrained t (Rc and t free parameters), [units of (M_sun/yr)(km^2/s^2)/(cm^-3 yr^2)]

mu/t^2 = m_dot * vw^2 /( n_0 * t^2)

Inversion results:

(mu/t^2) max = 1.4504E-07

(mu/t^2) min = 6.7187E-08

(mu/t^2) min * n_0 * t^2 for n_0 = 0.1 cm^-3, t = 1.5 x 10^5 yr:

1.5117E+02

119 (mu/t^2) max * n_0 * t^2 for n_0 = 10 cm^-3, t = 4 x 10^5 yr:

2.3207E+05

The output format is similar. The diﬀerences are as follows: The values for Rc_min and Rc_max are both zero, meaning they are unconstrained. As a result, the duration t is also unconstrained. The inversion process therefore yields the quantity mu/t^2. A numerical relation betweenm ˙ and vw requires setting a range for the duration as well as for the ambient density. Typical ranges are 0.1 cm−3 < n < 10 cm−3 and 1.5 105 yr

Finding wind parameters from observables: s = 2 inversion After SPICE inverts the solution for s=0, it proceeds to invert the s=2 process. It does this by using s2_lookup_table. The columns in the lookup table correspond to K21(Πvw + 1), K21/K2C , + − and (PC ) (ﬁrst division); and K22, K2C , and (PC ) (second division). First, a set of values for − K22, K2C , and (PC ) are chosen, A possible Πvw is determined by a Monte Carlo method from ˙ the input parameters using: vw = Rc/K2C , and Πvw +1 = vw/vw,1. Then, a set of values for + K21(Πvw + 1), K21/K2C , and (PC ) are chosen at random. Using the previously obtained value of Πvw + 1 and the chosen value of R2C , the value of K21 is checked from the two expressions: K = [K (Π + 1)]/(Π + 1), and K = [K /K ] K . The desired solution will have 21 21 vw vw 21 21 2C × 2C these two values close to each other within a pre-set limit. The correct set of values is found again using the Monte Carlo method. Finally, the corresponding value ofm ˙ is found by the expression: + − − m˙ =m ˙ 1(Πvw + 1)(PC ) /(PC ) . Then a new set of values of K22, K2C , and (PC ) are chosen, and the process is repeated. Thus, a set of vw andm ˙ are obtained that are consistent with the input parameters. These are output in a ﬁle called s2_inverted. SPICE also outputs to the screen (for the case of constrained Rc_min, Rc_max:

s=2 Inversion:

120 Input parameters: Rc_dot_min (km/s) = 120.0000 Rc_dot_max (km/s) = 140.0000 v_w1_min (km/s) = 20.0000 v_w1_max (km/s) = 60.0000 m_dot1_min (M_sun/yr) = 1.0000E-07 m_dot1_max (M_sun/yr) = 1.0000E-05 Rc_min (ly) = 10.0000 Rc_max (ly) = 20.0000

t_min for Rc_dot_min (yr): 24982.7057 t_max for Rc_dot_min (yr): 49965.4115 t_min for Rc_dot_max (yr): 21413.7478 t_max for Rc_dot_max (yr): 42827.4955

The entire list of input parameters are given, along with the entire range of possible durations, given by t = RC /R˙ C . This screen output is stored in the output file s2_inverter.log. SPICE then creates a graphic representing the entirem ˙ v parameter space, the sections represented − w by red giant, main sequence, and accretion disk winds, the curves represented by µmin, µmax for s=0, and the entire list of points generated by SPICE for s=2. An extra region in the space is filled, corresponding to an s=0 environment characterized by wind running into the low density region II of a prior mass loss cavity. The “ambient” density n0 is chosen in this case to be the 2 −3 asymptotic value for a high velocity red giant wind: n0,∞ = 3.9p0/vw = 0.009 cm , with p0 being the ambient density. This graphic file is called parameter_space.png (First an EPS file is generated but due to its size it is converted to PNG. If the user desires access to the original EPS file, the one corresponding to the last run is contained in the analysis_tools/exec folder.). This file, together with the outputs s0_inverted, s0_inverter.log, s2_inverted, and s2_inverter.log, are moved to a directory within the outputs directory named something like: 120._140._20._60._1.E-7_1.E-5_10._20._2015_Jul23_20:02:15 where the usual format applies: all of the input variables are in the name followed by the year, date, and time.

121 Figure A.2: Flowchart for the SPICE code package. All input is entered at the command line.

122 Figure A.3: Flowchart for the SPICE code package. Example path for calculating a single structure for s = 0, CAP, is highlighted in red

123 Figure A.4: Flowchart for the plot scripts in the SPICE package. Input is entered from the command line.

124 Figure A.5: Flowchart for the SPICE code package. Example path for calculating a s = 2 time series is highlighted in red.

125 Figure A.6: Flowchart for the SPICE code package. Example path for ﬁnding the wind parameters from observables by inversion is highlighted in red.

126 APPENDIX B

ADDITIONAL TIME SERIES FOR S=0

1 1 1 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] 0.5 0.5 0.5 Log (p/p1) Log (p/p1) Log (p/p1) Log n Log n Log n 0 0 0 -0.5 -0.5 -0.5 -1 -1 -1 -1.5 -1.5 -1.5 -2 -2 -2 t = .0236 Myr -2.5 p -2.5 -2.5 time = .01797 Myr time = .09129 Myr time = .46380 Myr -3 -3 -3 0.6 0.8 1 1.2 1.4 1 2 3 4 5 6 5 10 15 20 25 30 r (ly) r (ly) r (ly)

1 1 1 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] 0.5 0.5 0.5 Log (p/p1) Log (p/p1) Log (p/p1) Log n Log n Log n 0 0 0 -0.5 -0.5 -0.5 -1 -1 -1 -1.5 -1.5 -1.5 -2 -2 -2 -2.5 -2.5 -2.5 time = .03089 Myr time = .15694 Myr time = .79731 Myr -3 -3 -3 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2 4 6 8 10 10 20 30 40 50 r (ly) r (ly) r (ly)

1 1 1 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] 0.5 0.5 0.5 Log (p/p1) Log (p/p1) Log (p/p1) Log n Log n Log n 0 0 0 -0.5 -0.5 -0.5 -1 -1 -1 -1.5 -1.5 -1.5 -2 -2 -2 -2.5 -2.5 -2.5 time = .05310 Myr time = .26980 Myr time = 1.37064 Myr -3 -3 -3 0.5 1 1.5 2 2.5 3 3.5 4 2 4 6 8 10 12 14 16 18 20 20 40 60 80 100 r (ly) r (ly) r (ly)

1 Log [u/(100 km/s)] 0.5 Log (p/p1) Log n 0 -0.5 -1 -1.5 -2 -2.5 time = 2.35624 Myr -3 20 40 60 80 100 120 140 160 r (ly)

−3 −7 Figure B.1: Time sequence for a RG: n0 = 1 cm ,m ˙ = 10 M⊙/yr, and vw = 30 km/s. t = 2.36 104 yr. The ﬁrst eight snapshots were in an unphysical regime where R >R p × 2 C (See Chapter 3 and Eq. 3.1).

127 1 1 1 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] 0.5 0.5 0.5 Log (p/p1) Log (p/p1) Log (p/p1) Log n Log n Log n 0 0 0 -0.5 -0.5 -0.5 -1 -1 -1 -1.5 -1.5 -1.5 -2 -2 -2 -2.5 -2.5 -2.5 time = .00000124 Myr time = .00005509 Myr time = .00142197 Myr -3 -3 -3 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 6 7 8 9 r*100 (ly) r*100 (ly) r*100 (ly)

1 1 1 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] 0.5 0.5 0.5 Log (p/p1) Log (p/p1) Log (p/p1) Log n Log n Log n 0 0 0 -0.5 -0.5 -0.5 -1 -1 -1 -1.5 -1.5 -1.5

-2 -2 tp = .000124 Myr -2 -2.5 -2.5 -2.5 time = .00000213 Myr time = .00009471 Myr time = .00244448 Myr -3 -3 -3 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 2 4 6 8 10 12 14 16 r*100 (ly) r*100 (ly) r*100 (ly)

1 1 1 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] 0.5 0.5 0.5 Log (p/p1) Log (p/p1) Log (p/p1) Log n Log n Log n 0 0 0 -0.5 -0.5 -0.5 -1 -1 -1 -1.5 -1.5 -1.5 -2 -2 -2 -2.5 -2.5 -2.5 time = .00000630 Myr time = .00016282 Myr time = .00420224 Myr -3 -3 -3 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.2 0.4 0.6 0.8 1 1.2 0 5 10 15 20 25 r*100 (ly) r*100 (ly) r*100 (ly)

1 1 1 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] 0.5 0.5 0.5 Log (p/p1) Log (p/p1) Log (p/p1) Log n Log n Log n 0 0 0 -0.5 -0.5 -0.5 -1 -1 -1 -1.5 -1.5 -1.5 -2 -2 -2 -2.5 -2.5 -2.5 time = .00001084 Myr time = .00027990 Myr time = .00722395 Myr -3 -3 -3 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 0.5 1 1.5 2 0 10 20 30 40 50 r*100 (ly) r*100 (ly) r*100 (ly)

1 1 1 Log [u/(100 km/s)] Log [u/(100 km/s)] Log [u/(100 km/s)] 0.5 0.5 0.5 Log (p/p1) Log (p/p1) Log (p/p1) Log n Log n Log n 0 0 0 -0.5 -0.5 -0.5 -1 -1 -1 -1.5 -1.5 -1.5 -2 -2 -2 -2.5 -2.5 -2.5 time = .00001864 Myr time = .00048117 Myr time = .01241850 Myr -3 -3 -3 0 0.05 0.1 0.15 0.2 0.25 0 0.5 1 1.5 2 2.5 3 0 10 20 30 40 50 60 70 80 r*100 (ly) r*100 (ly) r*100 (ly)

1 1 Log [u/(100 km/s)] Log [u/(100 km/s)] 0.5 0.5 Log (p/p1) Log (p/p1) Log n Log n 0 0 -0.5 -0.5 -1 -1 -1.5 -1.5 -2 -2 -2.5 -2.5 time = .00003205 Myr time = .00082717 Myr -3 -3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 1 2 3 4 5 r*100 (ly) r*100 (ly)

−3 −14 Figure B.2: Time sequence for a MS wind: n0 = 1 cm ,m ˙ = 10 M⊙/yr, and vw = 500 km/s. 128 APPENDIX C

LOOKUP TABLE (S=0)

Table C.1: Lookup table for the s = 0 case. The outer Mach number M, K01, K0C , and K02, are here given numerically as a function of ΠT .ΠT = 2.50 corresponds to t = tp (see Chapter 3). The entire table is available in electronic form as part of the SPICE package.

ΠT M K01 K0C K02

99.9984844202 47.9736087686 0.7693887609 0.6602730058 0.7470756187 96.0952537492 46.1016193341 0.7693980994 0.6602675110 0.7470662932 92.3366502131 44.2990189287 0.7694082872 0.6602615175 0.7470561211 88.7323120524 42.5704165258 0.7694191095 0.6602551482 0.7470453110 85.2685506015 40.9092663225 0.7694309943 0.6602481561 0.7470334445 81.9331444151 39.3096899709 0.7694437261 0.6602406638 0.7470207289 78.7346006187 37.7757773429 0.7694575188 0.6602325479 0.7470069550 75.6607977412 36.3017198792 0.7694725861 0.6602236851 0.7469919135 72.7008722837 34.8823013010 0.7694889259 0.6602140739 0.7469756020 69.8623954856 33.5211508442 0.7695065381 0.6602037144 0.7469580207 67.1346311709 32.2131141586 0.7695254208 0.6601926053 0.7469391676 64.5132115973 30.9561186791 0.7695462163 0.6601803777 0.7469184162 61.9889126373 29.7457162488 0.7695684934 0.6601672756 0.7468961811 59.5681565764 28.5850016816 0.7695926802 0.6601530533 0.7468720451 57.2417767449 27.4695796296 0.7696189890 0.6601375869 0.7468457980 55.0061001171 26.3976827231 0.7696474175 0.6601208751 0.7468174380 52.8575816211 25.3676125569 0.7696781781 0.6601027944 0.7467867552 50.7928156844 24.3777370003 0.7697114828 0.6600832209 0.7467535397 48.8085306664 23.4264876098 0.7697475433 0.6600620310 0.7467175817 46.9015835446 22.5123571447 0.7697865711 0.6600391008 0.7466786711 45.0689422588 21.6338971789 0.7698289914 0.6600141837 0.7466363899 43.3077194064 20.7897158053 0.7698748013 0.6599872793 0.7465907370 41.6151163541 19.9784754310 0.7699244250 0.6599581407 0.7465412944 39.9917479143 19.2004761437 0.7699782883 0.6599265232 0.7464876468 38.4283210640 18.4512498629 0.7700363844 0.6598924241 0.7464297899 36.9288200902 17.7327236023 0.7700995680 0.6598553552 0.7463668955 35.4846566979 17.0407708375 0.7701680442 0.6598151906 0.7462987508 34.0995421324 16.3771712744 0.7702420244 0.6597718106 0.7462251533 32.7655278024 15.7381138347 0.7703221411 0.6597248456 0.7461454759 31.4860251306 15.1252421775 0.7704090327 0.6596739321 0.7460591032

129 Table C.2: Continuation of Table C.1, for ΠT = 30.3 through ΠT = 9.5.

ΠT M K01 K0C K02

30.2562053244 14.5362368916 0.7705031168 0.6596188249 0.7459656201 29.0741184234 13.9701685757 0.7706052409 0.6595590377 0.7458642019 27.9379164279 13.4261440213 0.7707156084 0.6594944505 0.7457546467 26.8480175788 12.9043703859 0.7708350662 0.6594245806 0.7456361369 25.7981608406 12.4018500110 0.7709648806 0.6593486971 0.7455074339 24.7910657029 11.9198830103 0.7711052559 0.6592666886 0.7453683512 23.8229205371 11.4566464561 0.7712574566 0.6591778299 0.7452176605 22.8922086613 11.0114124365 0.7714223201 0.6590816447 0.7450545566 21.9974678728 10.5834813277 0.7716008947 0.6589775371 0.7448780320 21.1372774089 10.1721806953 0.7717948668 0.6588645515 0.7446864695 20.3119922848 9.7776716319 0.7720044289 0.6587425898 0.7444797069 19.5184784980 9.3984628670 0.7722318967 0.6586103379 0.7442555210 18.7555060778 9.0339610070 0.7724783052 0.6584672207 0.7440129419 18.0233673783 8.6843127824 0.7727453289 0.6583123094 0.7437504023 17.3193113765 8.3481972574 0.7730346284 0.6581446776 0.7434663389 16.6422272324 8.0250906658 0.7733487125 0.6579629290 0.7431583944 15.9924030195 7.7151265922 0.7736885994 0.6577665301 0.7428256754 15.3673766998 7.4171346859 0.7740576280 0.6575536270 0.7424650524 14.7674180241 7.1312413941 0.7744572249 0.6573234762 0.7420752804 14.1902617384 6.8563678528 0.7748909176 0.6570741429 0.7416530990 13.6361535351 6.5926336952 0.7753609598 0.6568044486 0.7411965316 13.1041033728 6.3395677167 0.7758704320 0.6565127573 0.7407028309 12.5921082972 6.0962159727 0.7764236598 0.6561967478 0.7401680942 12.0993785469 5.8622055708 0.7770247388 0.6558542677 0.7395887105 11.6271146683 5.6381090321 0.7776754359 0.6554845266 0.7389633785 11.1724765204 5.4225791085 0.7783829467 0.6550836877 0.7382856509 10.7365909783 5.2161497363 0.7791489829 0.6546510741 0.7375544321 10.3168277872 5.0175788176 0.7799825538 0.6541819471 0.7367617700 9.9142383458 4.8273644027 0.7808859247 0.6536754370 0.7359062626 9.5263948869 4.6443609406 0.7818694617 0.6531262110 0.7349789815

130 Table C.3: Continuation of Table C.1, for ΠT = 9.2 through ΠT = 2.9.

ΠT M K01 K0C K02

9.1542753272 4.4690331004 0.7829363696 0.6525330413 0.7339779461 8.7971432317 4.3010342971 0.7840939616 0.6518925038 0.7328974836 8.4535593542 4.1396927004 0.7853538263 0.6511989505 0.7317281921 8.1229310881 3.9847324354 0.7867255646 0.6504480082 0.7304628480 7.8053635522 3.8362062922 0.7882167942 0.6496365664 0.7290963823 7.5008825900 3.6941313407 0.7898358304 0.6487613071 0.7276234047 7.2075633727 3.5576119451 0.7916021621 0.6478131419 0.7260288544 6.9261204024 3.4269866100 0.7935224850 0.6467901683 0.7243098053 6.6553462933 3.3017027197 0.7956172240 0.6456834707 0.7224515930 6.3958603382 3.1820498674 0.7978934066 0.6444916222 0.7204521875 6.1459331358 3.0672397395 0.8003809769 0.6432016177 0.7182902012 5.9056124944 2.9573045658 0.8030969449 0.6418078377 0.7159567279 5.6748951806 2.8522516449 0.8060590563 0.6403048496 0.7134432552 5.4532267367 2.7518393637 0.8092941295 0.6386832966 0.7107348110 5.2400753581 2.6558391722 0.8128325867 0.6369329429 0.7078150904 5.0353984300 2.5642465698 0.8167004780 0.6350467047 0.7046731923 4.8386560510 2.4768352038 0.8209358592 0.6330127690 0.7012905001 4.6497741045 2.3935892258 0.8255713888 0.6308232605 0.6976551399 4.4682064369 2.3142875120 0.8306555071 0.6284643443 0.6937455569 4.2934182891 2.2387220843 0.8362456048 0.6259200425 0.6895369401 4.1257254877 2.1670551458 0.8423770024 0.6231864347 0.6850247088 3.9645566243 2.0990686903 0.8491196521 0.6202461875 0.6801824233 3.8097520576 2.0347268214 0.8565372688 0.6170874561 0.6749930908 3.6611310967 1.9739866178 0.8647006602 0.6136981427 0.6694396991 3.5180954585 1.9166417027 0.8737157827 0.6100550290 0.6634875337 3.3804359898 1.8626538790 0.8836826151 0.6061415798 0.6571134624 3.2483173447 1.8121311027 0.8946806202 0.6019528902 0.6503138537 3.1214946418 1.7650180736 0.9068248857 0.5974738191 0.6430689923 2.9997192279 1.7212606953 0.9202438243 0.5926884822 0.6353587004 2.8823388729 1.6806686507 0.9351341712 0.5875619803 0.6271331801

131 Table C.4: Continuation of Table C.1, for ΠT = 2.8 through ΠT = 0.87.

ΠT M K01 K0C K02

2.7699359499 1.6434749536 0.9515469679 0.5821136011 0.6184304569 2.6614826933 1.6093620643 0.9697660592 0.5762884339 0.6091708383 2.5577032525 1.5785622040 0.9898021916 0.5701232058 0.5994215160 2.4576854470 1.5507830548 1.0119559674 0.5635658910 0.5891098717 2.3615055592 1.5260109984 1.0363478676 0.5566228431 0.5782568524 2.2694918135 1.5042409735 1.0629813877 0.5493312644 0.5669316793 2.1805811360 1.4851098243 1.0922528511 0.5416216525 0.5550387146 2.0955927627 1.4686455648 1.1239499566 0.5335869864 0.5427340803 2.0138495986 1.4545242564 1.1583259930 0.5251970824 0.5299839373 1.9354885923 1.4425624318 1.1953108690 0.5165036763 0.5168795805 1.8591110033 1.4323539713 1.2356113869 0.5073828908 0.5032490662 1.7870434600 1.4239805185 1.2779262762 0.4981644115 0.4895964925 1.7169664878 1.4169425351 1.3235101788 0.4886106683 0.4755800931 1.6492869253 1.4111041281 1.3721440696 0.4788179046 0.4613545654 1.5846658319 1.4063345900 1.4232717574 0.4689402293 0.4471523188 1.5240235760 1.4025074056 1.4758776105 0.4591977729 0.4332902429 1.4638494555 1.3992682665 1.5329974633 0.4490763799 0.4190439449 1.4057622233 1.3966131422 1.5933130847 0.4388774651 0.4048500270 1.3524371038 1.3945387336 1.6536758004 0.4291458376 0.3914592924 1.3004139622 1.3928124136 1.7177021995 0.4193139699 0.3780839549 1.2472905443 1.3913179984 1.7889394447 0.4089327199 0.3641304867 1.1990960511 1.3901695393 1.8593051070 0.3992187904 0.3512333222 1.1525739707 1.3892260000 1.9330405391 0.3895771145 0.3385863180 1.1077181538 1.3884520000 2.0101962751 0.3800360470 0.3262243979 1.0643771235 1.3878160000 2.0910924070 0.3705899883 0.3141374932 1.0230464703 1.3873000000 2.1747627800 0.3613744826 0.3024931012 0.9829477474 1.3868740000 2.2627857829 0.3522397211 0.2910963386 0.9444339125 1.3865260000 2.3544708139 0.3432853366 0.2800671224 0.9077767807 1.3862440000 2.4490490089 0.3345968244 0.2695019874 0.8720473646 1.3860100000 2.5489607991 0.3259706814 0.2591475220

132 Table C.5: Continuation of Table C.1, for ΠT = 0.84 through ΠT = 0.26.

ΠT M K01 K0C K02

0.8375647985 1.3858180000 2.6535339490 0.3174957403 0.2491070991 0.8059884910 1.3856680000 2.7571932967 0.3096034668 0.2398766487 0.7733732242 1.3855360000 2.8731979591 0.3013163048 0.2303102142 0.7437456560 1.3854340000 2.9874336328 0.2936660140 0.2215949111 0.7151960838 1.3853500000 3.1064992137 0.2861806471 0.2131766543 0.6864571720 1.3852780000 3.2363863216 0.2785294961 0.2046850108 0.6612207526 1.3852240000 3.3597765807 0.2717113483 0.1972154415 0.6350006460 1.3851760000 3.4983854478 0.2645251789 0.1894435290 0.6078493019 1.3851340000 3.6545401215 0.2569694444 0.1813850456 0.5849797632 1.3851040000 3.7973305897 0.2505110766 0.1745901146 0.5625463886 1.3850788000 3.9486897078 0.2440884651 0.1679191141 0.5406575075 1.3850578000 4.1084927617 0.2377345643 0.1614052819 0.5195767257 1.3850405500 4.2751333564 0.2315300572 0.1551280449 0.4991422854 1.3850263000 4.4501077201 0.2254321742 0.1490400885 0.4797091201 1.3850147500 4.6303443515 0.2195529833 0.1432479001 0.4610731446 1.3850053000 4.8174639639 0.2138379652 0.1376912903 0.4430380498 1.3849975000 5.0135439848 0.2082318104 0.1323121914 0.4255280934 1.3849910500 5.2198208303 0.2027142882 0.1270883645 0.4087860349 1.3849858000 5.4335808859 0.1973664172 0.1220925421 0.3930885044 1.3849816000 5.6505474599 0.1922846245 0.1174075665 0.3780081663 1.3849781500 5.8759569646 0.1873378393 0.1129061269 0.3625993757 1.3849751500 6.1256445944 0.1822142480 0.1083060582 0.3490407805 1.3849729000 6.3635867824 0.1776450785 0.1042579112 0.3353005625 1.3849709500 6.6243496685 0.1729535391 0.1001551835 0.3216075866 1.3849693000 6.9063841042 0.1682135868 0.0960662540 0.3099487732 1.3849681000 7.1661635464 0.1641242796 0.0925845456 0.2980746783 1.3849670500 7.4516289947 0.1599062041 0.0890383741 0.2861337632 1.3849661500 7.7625948349 0.1556073335 0.0854720935 0.2746482301 1.3849654150 8.0872150272 0.1514154664 0.0820416921 0.2638398448 1.3849648270 8.4185099198 0.1474169254 0.0788134382

133 Table C.6: Continuation of Table C.1, for ΠT = 0.25 through ΠT = 0.080.

ΠT M K01 K0C K02

0.2535504924 1.3849643500 8.7601391136 0.1435592606 0.0757401337 0.2436830973 1.3849639598 9.1148585373 0.1398103942 0.0727928024 0.2340902431 1.3849636363 9.4883768379 0.1361169455 0.0699274257 0.2248735924 1.3849633718 9.8772647913 0.1325204645 0.0671743814 0.2161662758 1.3849631586 10.2751257890 0.1290772820 0.0645734464 0.2077212386 1.3849629822 10.6928661054 0.1256933312 0.0620508315 0.1994555596 1.3849628352 11.1359901965 0.1223365172 0.0595817725 0.1916901778 1.3849627176 11.5871089148 0.1191403572 0.0572621427 0.1843537680 1.3849626235 12.0482203349 0.1160785583 0.0550690090 0.1770774099 1.3849625431 12.5432978062 0.1130037841 0.0528955002 0.1702857368 1.3849624792 13.0435738601 0.1100955843 0.0508667630 0.1636268498 1.3849624256 13.5743889307 0.1072064501 0.0488776840 0.1571082768 1.3849623810 14.1376028870 0.1043399683 0.0469305131 0.1509829861 1.3849623453 14.7111567165 0.1016100742 0.0451008159 0.1449900154 1.3849623155 15.3192226181 0.0989031761 0.0433106410 0.1393623361 1.3849622917 15.9378376306 0.0963271081 0.0416295806 0.1340096573 1.3849622724 16.5744342500 0.0938445445 0.0400306645 0.1286994112 1.3849622560 17.2583091652 0.0913487740 0.0384444219 0.1235913201 1.3849622426 17.9716035456 0.0889154107 0.0369185644 0.1189387443 1.3849622322 18.6746060067 0.0866697055 0.0355287747 0.1142746515 1.3849622233 19.4368054967 0.0843888465 0.0341355439 0.1097665150 1.3849622160 20.2350795452 0.0821545686 0.0327888988 0.1055479761 1.3849622101 21.0438346234 0.0800359489 0.0315287601 0.1014836745 1.3849622053 21.8866153274 0.0779679220 0.0303146939 0.0973927585 1.3849622011 22.8059475249 0.0758582738 0.0290926773 0.0938003431 1.3849621980 23.6793817559 0.0739812177 0.0280195699 0.0900674155 1.3849621953 24.6607956529 0.0720051682 0.0269044893 0.0864446635 1.3849621930 25.6942885450 0.0700611613 0.0258223195 0.0830787942 1.3849621912 26.7352715537 0.0682305171 0.0248168843 0.0798128863 1.3849621897 27.8292669905 0.0664304557 0.0238413089

134 Table C.7: Continuation of Table C.1, for ΠT = 0.077 through ΠT = 0.036.

ΠT M K01 K0C K02

0.0767266125 1.3849621885 28.9486795478 0.0647066993 0.0229193928 0.0737114939 1.3849621875 30.1328056101 0.0630002146 0.0220187318 0.0708000141 1.3849621867 31.3719445543 0.0613301626 0.0211490292 0.0680283764 1.3849621860 32.6501120760 0.0597189226 0.0203210995 0.0654321360 1.3849621855 33.9456152691 0.0581896789 0.0195455635 0.0629016709 1.3849621850 35.3112100423 0.0566795875 0.0187896756 0.0603757135 1.3849621847 36.7885360725 0.0551518502 0.0180351342 0.0581120984 1.3849621844 38.2215437781 0.0537645830 0.0173589583 0.0558759529 1.3849621841 39.7511630307 0.0523763471 0.0166909881 0.0535160000 1.3849621839 41.5041130243 0.0508910028 0.0159860347 0.0514978493 1.3849621838 43.1306188503 0.0496033818 0.0153831828 0.0495204568 1.3849621836 44.8528598724 0.0483253503 0.0147925060 0.0475910833 1.3849621835 46.6712240742 0.0470618482 0.0142161731 0.0457694893 1.3849621834 48.5287065067 0.0458531636 0.0136720357 0.0439767543 1.3849621833 50.5070040723 0.0446478633 0.0131365188 0.0420681925 1.3849621833 52.7984203555 0.0433465422 0.0125664027 0.0405282409 1.3849621832 54.8046019692 0.0422821505 0.0121063960 0.0389136820 1.3849621832 57.0784874890 0.0411516089 0.0116241030 0.0374987898 1.3849621832 59.2321545442 0.0401479512 0.0112014535 0.0359823874 1.3849621831 61.7283697043 0.0390581691 0.0107484813

135 APPENDIX D

LOOKUP TABLE (S=2)

Table D.1: Lookup table for the s = 2 case. K21, K2C , and K22 are here given in numerical form, as a function of the Π groups Πm˙ and Πvw (see Chapter 2). The entire table is available in electronic form as part of the SPICE package.

LogΠm˙ LogΠvw K21 K2C K22

-8.0000000000 -1.0000000000 0.9087756031 0.9087742423 0.8949447487 -8.0000000000 -0.7333333333 0.8443076437 0.8443053751 0.8210713382 -8.0000000000 -0.4666666667 0.7452670677 0.7452635020 0.7083353907 -8.0000000000 -0.2000000000 0.6135884036 0.6135833367 0.5603542356 -8.0000000000 0.0666666667 0.4621754016 0.4621688936 0.3943684829 -8.0000000000 0.3333333333 0.3176255055 0.3176179563 0.2432692400 -8.0000000000 0.6000000000 0.2005922651 0.2005841163 0.1310147530 -8.0000000000 0.8666666667 0.1205539008 0.1205455110 0.0641439387 -8.0000000000 1.1333333333 0.0695451368 0.0695366676 0.0289614286 -8.0000000000 1.4000000000 0.0379772058 0.0379687278 0.0118738684 -8.0000000000 1.6666666667 0.0213320423 0.0213235528 0.0050329164 -8.0000000000 1.9333333333 0.0116338679 0.0116254108 0.0020333120 -8.0000000000 2.2000000000 0.0062956252 0.0062871206 0.0008096550 -8.0000000000 2.4666666667 0.0035208634 0.0035123602 0.0003381609 -8.0000000000 2.7333333333 0.0018576286 0.0018491817 0.0001293251 -8.0000000000 3.0000000000 0.0010680875 0.0010596496 0.0000560622 -7.1333333333 -1.0000000000 0.9085046773 0.9085009715 0.8946308615 -7.1333333333 -0.7333333333 0.8432192429 0.8432130468 0.8198227046 -7.1333333333 -0.4666666667 0.7456932823 0.7456836567 0.7088107872 -7.1333333333 -0.2000000000 0.6124293950 0.6124156014 0.5590535579 -7.1333333333 0.0666666667 0.4624293141 0.4624116049 0.3946296070 -7.1333333333 0.3333333333 0.3191200388 0.3190995800 0.2447622644 -7.1333333333 0.6000000000 0.2005010092 0.2004789671 0.1309186343 -7.1333333333 0.8666666667 0.1207851219 0.1207624565 0.0643077198 -7.1333333333 1.1333333333 0.0696537888 0.0696307709 0.0290184956 -7.1333333333 1.4000000000 0.0379332843 0.0379102519 0.0118469530 -7.1333333333 1.6666666667 0.0213640036 0.0213409912 0.0050390458 -7.1333333333 1.9333333333 0.0118107909 0.0117876756 0.0020757124 -7.1333333333 2.2000000000 0.0063945580 0.0063715835 0.0008259368 -7.1333333333 2.4666666667 0.0036165659 0.0035936629 0.0003499736 -7.1333333333 2.7333333333 0.0021080222 0.0020851199 0.0001546471 -7.1333333333 3.0000000000 0.0011403067 0.0011177690 0.0000606785

136 Table D.2: Continuation of Table D.1, for Π = 6.27 and Π = 5.40. m˙ − m˙ −

LogΠm˙ LogΠvw K21 K2C K22

-6.2666666667 -1.0000000000 0.9091666263 0.9091566484 0.8953839048 -6.2666666667 -0.7333333333 0.8447217528 0.8447051621 0.8215286462 -6.2666666667 -0.4666666667 0.7462972799 0.7462712755 0.7094775450 -6.2666666667 -0.2000000000 0.6123712609 0.6123338587 0.5589626024 -6.2666666667 0.0666666667 0.4622145663 0.4621667883 0.3943654757 -6.2666666667 0.3333333333 0.3189190853 0.3188637279 0.2445250703 -6.2666666667 0.6000000000 0.2004030058 0.2003435194 0.1307975136 -6.2666666667 0.8666666667 0.1206918621 0.1206306610 0.0642080103 -6.2666666667 1.1333333333 0.0698578423 0.0697958427 0.0291180123 -6.2666666667 1.4000000000 0.0383827639 0.0383203349 0.0120411430 -6.2666666667 1.6666666667 0.0214074491 0.0213451607 0.0050405113 -6.2666666667 1.9333333333 0.0119377574 0.0118759048 0.0020987670 -6.2666666667 2.2000000000 0.0067503026 0.0066883031 0.0008883320 -6.2666666667 2.4666666667 0.0037249176 0.0036638018 0.0003601643 -6.2666666667 2.7333333333 0.0022069235 0.0021469007 0.0001615951 -6.2666666667 3.0000000000 0.0013738970 0.0013155955 0.0000774896 -5.4000000000 -1.0000000000 0.9095418732 0.9095149136 0.8957954257 -5.4000000000 -0.7333333333 0.8450128943 0.8449679492 0.8218287126 -5.4000000000 -0.4666666667 0.7453470579 0.7452759049 0.7083496097 -5.4000000000 -0.2000000000 0.6141045808 0.6140035572 0.5608222581 -5.4000000000 0.0666666667 0.4620244385 0.4618946338 0.3940738469 -5.4000000000 0.3333333333 0.3183894720 0.3182391316 0.2438953086 -5.4000000000 0.6000000000 0.2017143387 0.2015525438 0.1318808271 -5.4000000000 0.8666666667 0.1209622149 0.1207950274 0.0643322118 -5.4000000000 1.1333333333 0.0709481513 0.0707800099 0.0297139234 -5.4000000000 1.4000000000 0.0396149420 0.0394462862 0.0125704073 -5.4000000000 1.6666666667 0.0219385362 0.0217701514 0.0051934854 -5.4000000000 1.9333333333 0.0130947133 0.0129281535 0.0023844362 -5.4000000000 2.2000000000 0.0074836054 0.0073178514 0.0010174365 -5.4000000000 2.4666666667 0.0044117250 0.0042509722 0.0004501713 -5.4000000000 2.7333333333 0.0029305676 0.0027757211 0.0002375369 -5.4000000000 3.0000000000 0.0019384999 0.0017947053 0.0001234784

137 Table D.3: Continuation of Table D.1, for Π = 4.53 and Π = 3.67. m˙ − m˙ −

LogΠm˙ LogΠvw K21 K2C K22

-4.5333333333 -1.0000000000 0.9095048262 0.9094318317 0.8956998828 -4.5333333333 -0.7333333333 0.8436574093 0.8435339688 0.8201894043 -4.5333333333 -0.4666666667 0.7469056071 0.7467141690 0.7099778766 -4.5333333333 -0.2000000000 0.6152190138 0.6149455135 0.5618698807 -4.5333333333 0.0666666667 0.4634749382 0.4631222044 0.3953940207 -4.5333333333 0.3333333333 0.3199735048 0.3195657398 0.2452328800 -4.5333333333 0.6000000000 0.2033223950 0.2028849342 0.1330851493 -4.5333333333 0.8666666667 0.1227213680 0.1222709267 0.0654521343 -4.5333333333 1.1333333333 0.0719494585 0.0714942408 0.0301539519 -4.5333333333 1.4000000000 0.0416941816 0.0412407664 0.0134199985 -4.5333333333 1.6666666667 0.0243363071 0.0238857507 0.0059618126 -4.5333333333 1.9333333333 0.0151106406 0.0146670765 0.0028775793 -4.5333333333 2.2000000000 0.0092028477 0.0087770540 0.0013347777 -4.5333333333 2.4666666667 0.0065111367 0.0060979320 0.0007731854 -4.5333333333 2.7333333333 0.0045856923 0.0042026098 0.0004425045 -4.5333333333 3.0000000000 0.0031064380 0.0027777724 0.0002377997 -3.6666666667 -1.0000000000 0.9104775910 0.9102803968 0.8966744928 -3.6666666667 -0.7333333333 0.8470797325 0.8467523086 0.8238695256 -3.6666666667 -0.4666666667 0.7496369066 0.7491236916 0.7127095238 -3.6666666667 -0.2000000000 0.6196022710 0.6188673934 0.5662379126 -3.6666666667 0.0666666667 0.4676574782 0.4667089650 0.3992562811 -3.6666666667 0.3333333333 0.3242990775 0.3232040818 0.2489074126 -3.6666666667 0.6000000000 0.2104067847 0.2092312401 0.1388355795 -3.6666666667 0.8666666667 0.1291105391 0.1278971871 0.0697817685 -3.6666666667 1.1333333333 0.0777234130 0.0765028808 0.0332897089 -3.6666666667 1.4000000000 0.0474625464 0.0462610065 0.0159079291 -3.6666666667 1.6666666667 0.0291741905 0.0280028278 0.0075540466 -3.6666666667 1.9333333333 0.0192391641 0.0181176019 0.0039461854 -3.6666666667 2.2000000000 0.0130003753 0.0119694594 0.0021232131 -3.6666666667 2.4666666667 0.0097269905 0.0087735943 0.0013339789 -3.6666666667 2.7333333333 0.0074924155 0.0066386345 0.0008784652 -3.6666666667 3.0000000000 0.0058046613 0.0050655539 0.0005861608

138 Table D.4: Continuation of Table D.1, for Π = 2.80 and Π = 1.93. m˙ − m˙ −

LogΠm˙ LogΠvw K21 K2C K22

-2.8000000000 -1.0000000000 0.9127437608 0.9122233736 0.8989063006 -2.8000000000 -0.7333333333 0.8506918036 0.8498211522 0.8273792434 -2.8000000000 -0.4666666667 0.7553329734 0.7539716482 0.7182052013 -2.8000000000 -0.2000000000 0.6251938128 0.6232216834 0.5710913319 -2.8000000000 0.0666666667 0.4802266682 0.4776991971 0.4111125671 -2.8000000000 0.3333333333 0.3379362933 0.3350019228 0.2608783964 -2.8000000000 0.6000000000 0.2256482933 0.2224995812 0.1510096898 -2.8000000000 0.8666666667 0.1438614383 0.1406480786 0.0798404283 -2.8000000000 1.1333333333 0.0912813683 0.0880966164 0.0408556369 -2.8000000000 1.4000000000 0.0613026201 0.0581956208 0.0223043422 -2.8000000000 1.6666666667 0.0408786384 0.0379678382 0.0118734589 -2.8000000000 1.9333333333 0.0305573151 0.0278315689 0.0074853819 -2.8000000000 2.2000000000 0.0219203006 0.0195336750 0.0044152034 -2.8000000000 2.4666666667 0.0170151850 0.0149140286 0.0029511976 -2.8000000000 2.7333333333 0.0136577653 0.0118312878 0.0020871084 -2.8000000000 3.0000000000 0.0110858302 0.0095153577 0.0015059251 -1.9333333333 -1.0000000000 0.9190264528 0.9176990672 0.9051976770 -1.9333333333 -0.7333333333 0.8608645916 0.8586481559 0.8374777109 -1.9333333333 -0.4666666667 0.7710518025 0.7675280523 0.7335878585 -1.9333333333 -0.2000000000 0.6498523948 0.6447813866 0.5951720019 -1.9333333333 0.0666666667 0.5099140650 0.5033707601 0.4389497743 -1.9333333333 0.3333333333 0.3726731849 0.3650536114 0.2917405093 -1.9333333333 0.6000000000 0.2589799010 0.2508169236 0.1776259696 -1.9333333333 0.8666666667 0.1799875219 0.1717464407 0.1056833810 -1.9333333333 1.1333333333 0.1227264111 0.1147700270 0.0597938223 -1.9333333333 1.4000000000 0.0892358863 0.0817531855 0.0366652521 -1.9333333333 1.6666666667 0.0657575491 0.0590067874 0.0227673591 -1.9333333333 1.9333333333 0.0510925121 0.0450712486 0.0153074536 -1.9333333333 2.2000000000 0.0407611874 0.0354663247 0.0107320892 -1.9333333333 2.4666666667 0.0323350495 0.0278369074 0.0074875223 -1.9333333333 2.7333333333 0.0249923563 0.0213450764 0.0050404817 -1.9333333333 3.0000000000 0.0210598374 0.0178946208 0.0038737771

139 Table D.5: Continuation of Table D.1 for Π = 1.07 and Π = 0.20. m˙ − m˙ −

LogΠm˙ LogΠvw K21 K2C K22

-1.0666666667 -1.0000000000 0.9322490415 0.9291364411 0.9183452984 -1.0666666667 -0.7333333333 0.8829723864 0.8777284335 0.8593276858 -1.0666666667 -0.4666666667 0.8070011783 0.7987117968 0.7690489433 -1.0666666667 -0.2000000000 0.7014548638 0.6893178065 0.6451525028 -1.0666666667 0.0666666667 0.5750667465 0.5592731748 0.5001853063 -1.0666666667 0.3333333333 0.4518209033 0.4333006426 0.3634666798 -1.0666666667 0.6000000000 0.3394153243 0.3194943067 0.2451621506 -1.0666666667 0.8666666667 0.2555331369 0.2358354877 0.1634412701 -1.0666666667 1.1333333333 0.1937849984 0.1752051666 0.1086500020 -1.0666666667 1.4000000000 0.1498769850 0.1329898680 0.0737556195 -1.0666666667 1.6666666667 0.1181121930 0.1032082841 0.0513500591 -1.0666666667 1.9333333333 0.0916574263 0.0791425881 0.0349665690 -1.0666666667 2.2000000000 0.0746945032 0.0639466186 0.0256161206 -1.0666666667 2.4666666667 0.0601584643 0.0511938747 0.0184703493 -1.0666666667 2.7333333333 0.0486119193 0.0412002766 0.0134004971 -1.0666666667 3.0000000000 0.0389705806 0.0329348843 0.0096174778 -0.2000000000 -1.0000000000 0.9546028690 0.9484653095 0.9405846168 -0.2000000000 -0.7333333333 0.9210343854 0.9105770528 0.8970152253 -0.2000000000 -0.4666666667 0.8686879653 0.8520401212 0.8299167932 -0.2000000000 -0.2000000000 0.7925022947 0.7675743030 0.7336404511 -0.2000000000 0.0666666667 0.6983514454 0.6651585982 0.6180027368 -0.2000000000 0.3333333333 0.5930727889 0.5534488087 0.4937688247 -0.2000000000 0.6000000000 0.4906783723 0.4479982670 0.3791597150 -0.2000000000 0.8666666667 0.4021564817 0.3598392614 0.2863480966 -0.2000000000 1.1333333333 0.3264894586 0.2871225461 0.2128746136 -0.2000000000 1.4000000000 0.2637012751 0.2288343816 0.1568941624 -0.2000000000 1.6666666667 0.2148258886 0.1845227116 0.1167606967 -0.2000000000 1.9333333333 0.1748082956 0.1491094114 0.0867082794 -0.2000000000 2.2000000000 0.1422457468 0.1207352889 0.0642873102 -0.2000000000 2.4666666667 0.1146126172 0.0969496490 0.0469277849 -0.2000000000 2.7333333333 0.0937853569 0.0791574769 0.0349762411 -0.2000000000 3.0000000000 0.0761474469 0.0641739963 0.0257498570

140 Table D.6: Continuation of Table D.1, for Π = 0.67 and Π = 1.53. m˙ − m˙

LogΠm˙ LogΠvw K21 K2C K22

0.6666666667 -1.0000000000 0.9801307461 0.9705220710 0.9659936154 0.6666666667 -0.7333333333 0.9648753990 0.9483671036 0.9404716510 0.6666666667 -0.4666666667 0.9402213384 0.9132592897 0.9000963917 0.6666666667 -0.2000000000 0.9027601657 0.8620497067 0.8413700483 0.6666666667 0.0666666667 0.8495068928 0.7938964933 0.7635661202 0.6666666667 0.3333333333 0.7839080422 0.7154843722 0.6746535644 0.6666666667 0.6000000000 0.7061730709 0.6302026362 0.5788799271 0.6666666667 0.8666666667 0.6219034241 0.5448300951 0.4842885788 0.6666666667 1.1333333333 0.5360442923 0.4632866649 0.3955704091 0.6666666667 1.4000000000 0.4569537951 0.3912195897 0.3189937390 0.6666666667 1.6666666667 0.3869142648 0.3291998117 0.2549830156 0.6666666667 1.9333333333 0.3202893179 0.2713598430 0.1974387933 0.6666666667 2.2000000000 0.2646420639 0.2235729038 0.1520012606 0.6666666667 2.4666666667 0.2192044037 0.1848441549 0.1170423367 0.6666666667 2.7333333333 0.1793869264 0.1510718893 0.0883159724 0.6666666667 3.0000000000 0.1458833863 0.1227646816 0.0658322641 1.5333333333 -1.0000000000 0.9984192071 0.9863395338 0.9842343923 1.5333333333 -0.7333333333 0.9968889774 0.9760138727 0.9723249367 1.5333333333 -0.4666666667 0.9934241494 0.9588585692 0.9525535677 1.5333333333 -0.2000000000 0.9867494971 0.9337941224 0.9237015245 1.5333333333 0.0666666667 0.9739136121 0.8994602553 0.8842518942 1.5333333333 0.3333333333 0.9484338794 0.8539172000 0.8320638987 1.5333333333 0.6000000000 0.9100199253 0.8020507950 0.7728521344 1.5333333333 0.8666666667 0.8543683579 0.7402757885 0.7026871604 1.5333333333 1.1333333333 0.7874209671 0.6746772252 0.6286911934 1.5333333333 1.4000000000 0.7103515931 0.6042073677 0.5499253268 1.5333333333 1.6666666667 0.6288803438 0.5324829220 0.4707425307 1.5333333333 1.9333333333 0.5489617531 0.4634809373 0.3957804785 1.5333333333 2.2000000000 0.4691233663 0.3953676153 0.3233436575 1.5333333333 2.4666666667 0.3976108089 0.3347303882 0.2606082639 1.5333333333 2.7333333333 0.3339275792 0.2809316767 0.2067911898 1.5333333333 3.0000000000 0.2775173994 0.2333601585 0.1611299528

141 Table D.7: Continuation of Table D.1, for Πm˙ = 2.40 and Πm˙ = 3.27.

LogΠm˙ LogΠvw K21 K2C K22

2.4000000000 -1.0000000000 1.0078863676 0.9944251898 0.9935647021 2.4000000000 -0.7333333333 1.0134989533 0.9902256501 0.9887181543 2.4000000000 -0.4666666667 1.0218835118 0.9831932786 0.9806047966 2.4000000000 -0.2000000000 1.0323253873 0.9727616049 0.9685753493 2.4000000000 0.0666666667 1.0425042082 0.9574833532 0.9509694428 2.4000000000 0.3333333333 1.0474890216 0.9378960137 0.9284203782 2.4000000000 0.6000000000 1.0408966536 0.9118560490 0.8984843133 2.4000000000 0.8666666667 1.0201523726 0.8797540505 0.8616497249 2.4000000000 1.1333333333 0.9869175604 0.8423641901 0.8188527717 2.4000000000 1.4000000000 0.9381971226 0.7955873081 0.7654905383 2.4000000000 1.6666666667 0.8797349793 0.7432938484 0.7061049125 2.4000000000 1.9333333333 0.8080454254 0.6811371970 0.6359506357 2.4000000000 2.2000000000 0.7318224919 0.6161103109 0.5631692668 2.4000000000 2.4666666667 0.6469724686 0.5442315405 0.4836366612 2.4000000000 2.7333333333 0.5605126665 0.4712755371 0.4041815085 2.4000000000 3.0000000000 0.4877484309 0.4099926351 0.3387451325 3.2666666667 -1.0000000000 1.0118093220 0.9978615475 0.9975312658 3.2666666667 -0.7333333333 1.0203599206 0.9962354119 0.9956541388 3.2666666667 -0.4666666667 1.0338817581 0.9935588551 0.9925648066 3.2666666667 -0.2000000000 1.0519163368 0.9895004148 0.9878813336 3.2666666667 0.0666666667 1.0732614191 0.9834814654 0.9809372979 3.2666666667 0.3333333333 1.0921998857 0.9755252315 0.9717615163 3.2666666667 0.6000000000 1.1037991878 0.9648638403 0.9594725559 3.2666666667 0.8666666667 1.1053053197 0.9513956679 0.9439583914 3.2666666667 1.1333333333 1.0957551532 0.9339103723 0.9238352346 3.2666666667 1.4000000000 1.0761670656 0.9116555117 0.8982541547 3.2666666667 1.6666666667 1.0458984251 0.8828398353 0.8651873279 3.2666666667 1.9333333333 1.0066391257 0.8480048562 0.8253022205 3.2666666667 2.2000000000 0.9572176873 0.8054549642 0.7767307414 3.2666666667 2.4666666667 0.9011125423 0.7577872721 0.7225324335 3.2666666667 2.7333333333 0.8287337085 0.6966478907 0.6534105192 3.2666666667 3.0000000000 0.7510422398 0.6312096261 0.5800074157

142 Table D.8: Continuation of Table D.1 for Πm˙ = 4.13 and Πm˙ = 5.00.

LogΠm˙ LogΠvw K21 K2C K22

4.1333333333 -1.0000000000 1.0134116850 0.9991982595 0.9990743966 4.1333333333 -0.7333333333 1.0232409732 0.9985855259 0.9983670322 4.1333333333 -0.4666666667 1.0385238376 0.9975853130 0.9972123828 4.1333333333 -0.2000000000 1.0599867897 0.9960347440 0.9954225328 4.1333333333 0.0666666667 1.0852773375 0.9937857316 0.9928266640 4.1333333333 0.3333333333 1.1101108374 0.9907630587 0.9893383254 4.1333333333 0.6000000000 1.1294993954 0.9867447101 0.9847017557 4.1333333333 0.8666666667 1.1407785218 0.9814484379 0.9785923673 4.1333333333 1.1333333333 1.1439605364 0.9742893505 0.9703368222 4.1333333333 1.4000000000 1.1400538571 0.9654387560 0.9601349969 4.1333333333 1.6666666667 1.1295628593 0.9531700002 0.9460017108 4.1333333333 1.9333333333 1.1116144112 0.9361520692 0.9264142092 4.1333333333 2.2000000000 1.0891628962 0.9163075399 0.9035984814 4.1333333333 2.4666666667 1.0563383216 0.8881605086 0.8712873872 4.1333333333 2.7333333333 1.0194337703 0.8568657150 0.8354379844 4.1333333333 3.0000000000 0.9691295549 0.8144335979 0.7869646460 5.0000000000 -1.0000000000 1.0139380934 0.9997024065 0.9996564273 5.0000000000 -0.7333333333 1.0241682293 0.9994743459 0.9993931328 5.0000000000 -0.4666666667 1.0403269249 0.9991008378 0.9989619290 5.0000000000 -0.2000000000 1.0628878986 0.9985278410 0.9983004331 5.0000000000 0.0666666667 1.0899957309 0.9976819328 0.9973239253 5.0000000000 0.3333333333 1.1167982273 0.9965783920 0.9960500434 5.0000000000 0.6000000000 1.1394382220 0.9950614738 0.9942991169 5.0000000000 0.8666666667 1.1546624035 0.9930532703 0.9919812999 5.0000000000 1.1333333333 1.1631167073 0.9903131357 0.9888191059 5.0000000000 1.4000000000 1.1656255684 0.9867638745 0.9847239371 5.0000000000 1.6666666667 1.1638463721 0.9819126545 0.9791278178 5.0000000000 1.9333333333 1.1584791971 0.9755558715 0.9717968909 5.0000000000 2.2000000000 1.1501325581 0.9675738109 0.9625954512 5.0000000000 2.4666666667 1.1378849405 0.9567225324 0.9500930540 5.0000000000 2.7333333333 1.1187706442 0.9403164363 0.9312053196 5.0000000000 3.0000000000 1.0958674590 0.9209180992 0.9088969480

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165 BIOGRAPHICAL SKETCH

Paul Dragulin earned a Bachelor of Science degree in physics at Portland State University in Portland, OR, in 2007, and a Master of Science degree in physics from the same University in 2008. He came to Florida State University in the Fall of 2009, where he joined the astrophysics group working under the direction of Dr. Peter Hoeﬂich, studying the physics of Type Ia Supernovae.

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