Tumbling Dice: Radio Constraints on the Presence of Circumstellar Shells around Type Ia Supernovae with Impact Near Maximum Light
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Tumbling Dice: Radio Constraints on the Presence of Circumstellar Shells around Type IaSupernovae with Impact Near Maximum Light
Chelsea E. Harris , Laura Chomiuk , and Peter. E. Nugent
2, 3 Center for Data Intensive and Time Domain Astronomy, Department of Physics and Astronomy, Michigan State University, EastLansing, MI 48824, USA Lawrence Berkeley National Laboratory, 1 Cyclotron Road, MS 50B-4206, Berkeley, CA 94720, USA Department of Astronomy, University of California, Berkeley, CA 94720-3411, USA (Received January 29, 2021; Accepted February 25, 2021)
Submitted to ApJABSTRACTThe progenitors of Type Ia supernovae (SNe Ia) are debated, particularly the evolutionary state ofthe binary companion that donates mass to the exploding carbon-oxygen white dwarf. In previouswork, we presented hydrodynamic models and optically thin radio synchrotron light-curves of SNe Iainteracting with detached, confined shells of CSM, representing CSM shaped by novae. In this work,we extend these light-curves to the optically thick regime, considering both synchrotron self-absorptionand free-free absorption. We obtain simple formulae to describe the evolution of optical depth seen inthe simulations, allowing optically thick light-curves to be approximated for arbitrary shell properties.We then demonstrate the use of this tool by interpreting published radio data. First, we consider thenon-detection of PTF11kx – an SN Ia known to have a detached, confined shell – and find that the non-detection is consistent with current models for its CSM, and that observations at a later time wouldhave been useful for this event. Secondly, we statistically analyze an ensemble of radio non-detectionsfor SNe Ia with no signatures of interaction, and find that shells with masses (10 − − . M (cid:12) located(10 − ) cm from the progenitor are currently not well constrained by radio datasets, due to theirdim, rapidly-evolving light-curves. Keywords:
Type Ia supernovae(1728) — Circumstellar gas(238) — Shocks(2086) INTRODUCTIONThermonuclear Type Ia supernovae (SNe Ia) are oneof the most mature and precise cosmological tools inmodern astronomy, and have revealed the acceleratingexpansion of the universe (Riess et al. 1998; Perlmutteret al. 1999). SNe Ia are the explosion of a carbon-oxygenwhite dwarf that has merged with or accreted mass froma companion star. However, wecurrently remain ignorant of the identity of the com-panion star, which affects the timescale of explosion,explosion trigger, properties of the white dwarf at timeof explosion, and local environment.
Corresponding author: [email protected]
It has long been recognized that characterizing thecircumstellar material (CSM) around SNe Ia constrainsthe nature of their companions (Branch et al. 1995).For example, main sequence and red giant compan-ions of the “single-degenerate” channel will have winds.Growth of the white dwarf happens through accre-tion, either Roche-lobe overflow or directly from thecompanion wind (i.e., a symbiotic system). Instabili-ties in this mass-transfer (e.g., novae) can create denseshells of hydrogen-rich CSM. In contrast, the “double-degenerate” channel, where explosion is triggered by themerger of two white dwarfs, is expected to have a cleanenvironment.Radio observations are sensitive probes of the CSMaround SNe, as synchrotron emission is produced whenthe SN blast wave shocks surrounding gas, accelerateselectrons to relativistic speeds, and amplifies the mag-netic field in the shocked region (Chevalier 1982). This a r X i v : . [ a s t r o - ph . H E ] F e b Harris et al. emission will be subject to absorption, and of particu-lar relevance to this work is the absorption caused bythe CSM itself: within the shock region, radio emis-sion is affected by synchrotron self-absorption, and ra-dio emission emerging from the shock region is furthersubject to free-free absorption from outlying, unshockedCSM. Despite extensive observations of SNe Ia at radiowavelengths, there are no published radio detections ofSNe Ia to date, even for those known to be interact-ing. Upper limits on radio luminosity imply that theCSM around typical SNe Ia is substantially lower den-sity than observed around most core-collapse SNe, as-suming the CSM is a continuous medium like a wind(Weiler et al. 2002; P´erez-Torres et al. 2014; Chomiuket al. 2016; Lundqvist et al. 2020).In recent decades, the picture of SN Ia environmentsand the single-degenerate channel has become muddiedby the discovery of what was long-sought: SNe Ia withsignatures of hydrogen (from CSM interaction) in theirspectra, dubbed SNe Ia-CSM by Silverman et al. (2013).SNe Ia-CSM can be broken into two groups. The firstand most common are events like SN 2005gj, which werehistorically grouped with the canonical CSM interactionclass of SNe IIn but have distinct underlying SN Ia fea-tures. Radio non-detections are expected for such eventssince light at radio frequencies will be totally absorbedby the outlying CSM that has not yet been shocked. Theother case is more rare, where an SN Ia transforms froma normal event into an interacting event (which we forshorthand call SNe Ia;n, Harris et al. 2018). The proto-type is PTF11kx (Dilday et al. 2012), though SN 2002icmay also have been an SN Ia;n (Wood-Vasey et al. 2004).A search for more instances of SNe Ia;n discovered in-teraction in SN 2015cp (Graham et al. 2019). The CSMof SNe Ia;n may be shaped by nova outbursts or otherinstabilities in the mass-transfer process that sweep anyexisting material into a distant shell.SNe Ia;n are of particular interest because they areso disruptive to our current theoretical understandingand abilities, and to SN Ia observational traditions —yet they are a clear path forward to understanding thesingle-degenerate channel. They disrupt our theoreticalunderstanding because novae should interrupt the massgrowth of the carbon-oxygen white dwarf (see, e.g., thediscussion and references in Branch et al. 1995). Yet theCSM mass observed for PTF11kx (Graham et al. 2017)was too low to have come from an expelled common en-velope of the double-degenerate scenario (Livio & Riess2003). They disrupt theoretical ability because the well-established tools for interpreting interaction with a windor other continuous medium cannot be applied (Cheva-lier 1982). SNe Ia;n are furthermore extremely diffi- cult to detect via traditional SN Ia observation methods,which only cover the phase near maximum light duringwhich the CSM will not be visible (even in spectra, aswas the case for SN 2015cp). Furthermore, the interac-tion may be very short-lived, eluding even observationsat late times — and the fact that the time between massejection and supernova is unknown means the locationof the shell is unknown and potentially random. Finally,as is the case with the SN IIn-like events, SNe Ia;n comefrom the rare “shallow silicon” or “SN 1991T-like” sub-group of SNe Ia, making the chance of discovery evensmaller since SN Ia surveys usually attempt to recreatethe underlying distribution of SN Ia properties. Despiteall of these difficulties, SNe Ia;n are the clearest pathforward to understanding the single-degenerate chan-nel, because we can constrain the ejecta properties frompre-interaction data to alleviate degeneracies in the in-teraction modelling, and because the CSM mass is toolow to be explained by a double-degenerate origin (asaforementioned).The potential for SNe Ia;n to illuminate SN Ia progeni-tors motivates the alleviation of the theoretical obstaclesfacing their study. Harris et al. (2016, Paper I) mod-eled SNe Ia interacting with low-mass, confined shells ofCSM in the months following maximum light. The opti-cally thin radio light-curves from these models were thenstudied, and a parameterization was created to allow forlight-curves to be created for an arbitrary CSM shellconfiguration. These light-curves can be used to limitCSM shell properties from radio non-detections partic-ularly for the very thin, low-mass shells expected fromsingle nova eruptions, distant shells that will be very lowdensity, or radio observations taken after interaction hasended (Harris et al. 2018; Cendes et al. 2020; Pellegrinoet al. 2020).However, there is a sizeable sample of SNe Ia with ra-dio observations near maximum light, probing shells ata distance r ∼ (10 − ) cm (Chomiuk et al. 2016),and the use of this dataset is currently limited by theoptically thin assumption, which is only applicable toshells of density (cid:46) − g cm − or after the shock hascrossed the shell. In order to study interaction within r ∼ cm and to incorporate lower-frequency observa-tions, the optically thin light-curves of Paper I must beextended into the regime of synchrotron self-absorptionand external free-free absorption, which is our aim forthis work. With absorption accounted for, we can usethe radio sample to constrain the presence of nova-likeshells around SNe Ia for higher shell masses than waspreviously possible.This work is organized as follows. In §
2, we summa-rize the main results of Paper I for the reader’s conve-
N Ia Shell CSM §
3. We ac-count for synchrotron self-absorption ( τ ssa ) and free-free( τ ff ) absorption. In § τ ssa ( t ) and τ ff ( t ) that we give in §
5. In § SUMMARY OF PAPER IPaper I presented a suite of one-dimensional hydro-dynamic models of a typical SN Ia interacting with alow-mass, confined shell of CSM. This section providesa brief summary of the Paper I results and reiterates itslimitations for the reader’s convenience.The SN Ia ejecta have mass M ej = 1 . M (cid:12) and en-ergy E ej = 10 erg. Before impact with the CSM shell,they are in free expansion. This is equivalent to as-suming that any CSM within the detached shell is toolow density to affect the dynamics of the ejecta. Theejecta mass-density profile is assumed to have a brokenpower-law structure with ρ ej ∝ r − in the inner regions( v (cid:46) ,
000 km s − ) and ρ ej ∝ r − in the outer re-gions.The CSM is assumed to be confined to a constant-density shell with density ρ csm between radii R in and(1 + f R ) R in . The parameter f R ∈ [0 . ,
1] is called the“fractional width” of the shell, since f R = ∆ R/R in , (1)where ∆ R is the width of the shell.The ejecta impact the CSM at time t imp after explo-sion. The time of impact is related to R in through Pa-per I Equation 5, R in = (1 . × cm) (cid:18) t imp
100 days (cid:19) . × (cid:18) ρ csm − g cm − (cid:19) − . . (2)This scaling ensures that the density ratio between theCSM and ejecta at the point and time of first contact isfixed to 0.33, which defines the “fiducial model set.” The models are invalid when (1) R in /t imp > ,
000 km s − , i.e., they imply an unphysically largeejecta speed, (2) R in /t imp < ,
000 km s − , i.e., interac-tion is with the inner ejecta, or (3) ρ csm > − g cm − where cooling and photon trapping are likely to be im-portant, i.e., the adiabatic assumption does not hold.At a given R in , the first constraint places a lower limiton the allowed CSM densities, whereas the second twoplace upper limits on the CSM density. These limita-tions are summarized in Figure 1 of Paper I.The hydrodynamics are evolved assuming adiabaticevolution using the one-dimensional Lagrangian solverof SEDONA (Roth & Kasen 2015). The hydrodynamicbehavior of this system is as follows. Initially, the shock“ramps” up in the CSM—energy density grows, as doesthe width of the shock region. Before it can reach theself-similar limit, the forward shock reaches the edge ofthe CSM shell—the “end” of interaction. The hot, accel-erated CSM is uncontained by any external material andtherefore rapidly expands—a rarefaction wave crossesback toward the ejecta. The energy density plummets.Paper I assumes the relativistic electron population inthe shocked gas is distributed as n e ( E ) dE = C E E − p dE (3)where E is the electron energy, and p = 3 is assumed.The normalization factor is determined by assumingthat the energy density in relativistic electrons is 10%of the total shocked gas energy density, i.e. (cid:15) e = 0 . t p . Paper I Equation 7gives t p /t imp = 0 . f R ) . , (4)which can be used to produce an expression for theevolution of the forward shock radius ( R f ), since R out /R in = (1 + f R ), R f /R in = 1 . t/t imp ) . . (5)The peak luminosity scales like L ν, thin ,p ∝ (cid:15) e (cid:15) B ν − ρ / R / × [1 − (1 + f R ) − . ] , (6)as can be seen in Paper I Equations 11 and 37, where (cid:15) B is the ratio of the magnetic field energy density densityto the gas energy density and is typically assumed tobe (cid:15) B = 0 . Harris et al.
The shape of the light-curves is described by anasymptotic rise (Paper I Equation 10) followed by acomplex decline. The decline is described by the f R -dependent time it takes the light-curve to reach charac-teristic fractions of the peak luminosity (Paper I Table 1and Equation 12). ESCAPE FRACTION OF PHOTONS FROM ATHIN SPHERICAL SHELLIn this work we consider the absorption of radio emis-sion from synchrotron self-absorption in the emitting re-gion itself as well as external absorption by the free-free(Bremsstrahlung) process in the external, unshockedCSM. The strategy for obtaining optically thick radiolight-curves from the optically thin light-curves param-eterized in Paper I is to simply find the escape fractionof radio photons, the ratio of the optically thick to op-tically thin luminosity.The expression for the escape fraction depends on thegeometry of the emitting and absorbing gasses. In ourcase, the emitting (and self-absorbing) region is a thin,spherical shell. The external, absorbing medium is alsoa thin, spherical shell and only exists before the forwardshock overtakes the edge of the shell.Weiler et al. (1990) provide an expression for correct-ing optically thin luminosity for internal absorption inthe emitting medium, L = L thin (cid:18) − e − τ τ (cid:19) , (7)which is the calculation for a planar slab geometry and τ is the optical depth of the slab along the line of sight.In Appendix A, we show that the full solution for a thinshell geometry has the same asymptotic behavior as theslab approximation so long as one uses an appropriateexpression for τ . Given the synchrotron self-absorptionextinction coefficienct ( α ssa ) and the volume-to-surface-area ratio of the emitting sphere ( (cid:102) ∆ r ), the appropriate τ to capture the effect of synchrotron self-absorption is τ ssa = 4 α ssa (cid:102) ∆ r (8)such that the escape fraction in the absence of an exter-nal absorbing medium can be approximated by L ν / L ν, thin = 1 − e − τ ssa τ ssa , (9)as derived in Appendix A.1. In the presence of an ab-sorbing medium with extinction coefficient α ff and radialwidth ∆ r ext , we take the free-free optical depth to be τ ff = α ff ∆ r ext (10) -4 -3 -2 shell mass [ M fl ] -20 -19 -18 -17 -16 -15 C S M s h e ll d e n s it y [ g c m − ] ( ρ c s m ) ρ csm R in s h e ll i nn e r r a d i u s [ c m ] ( R i n ) f R Figure 1.
Summary of the CSM shell properties for thesimulations presented in this work.
Circles show the CSMdensity (left axis) and squares represent the inner radius(right axis), and color illustrates the shell width (color bar).Axis limits are set to separate the density and radius points. and approximate the escape fraction as L ν / L ν, thin = 1 − e − τ ssa τ ssa e − τ ff , (11)as discussed in Appendix A.2. As noted in the appen-dices, the error incurred by using these approximationsin lieu of the exact integral depends on the extent of themedia and their optical depth but is typically small.Thus, the goal of this work is to determine, from thesimulations, the time evolution of α ssa (cid:102) ∆ r and α ff ∆ r ext . CALCULATION OF OPTICAL DEPTH FROMSIMULATIONSIn this section we describe how we calculate the opticaldepths τ ssa and τ ff from hydrodynamic models. Figure 1shows the CSM shell properties of the models, whichcover a range of shell masses through variations in theshell location, extent, and density.First we must calculate the extinction coeffi-cient of synchrotron self-absorption, α ssa , in eachshocked resolution element for each simulationsnapshot. We perform these calculations with rad tools.SynchrotronCalculator of csmpy . The synchrotron extinction coefficient ( α ssa ) in eachresolution element of the simulation is calculated accord- https://github.com/chelseaharris/csmpy N Ia Shell CSM α ssa ( ν ) = ν − ( p +4) / √ q e πm e (cid:18) q e πm e c (cid:19) p/ × C E (2 B/π ) ( p +2) / (12) × Γ (cid:18) p + 212 (cid:19) Γ (cid:18) p + 2212 (cid:19) , where p = 3 is the electron distribution power-law in-dex, q e is the electron charge, m e the electron mass, c the speed of light, C E is the normalization of theelectron distribution (Equation 3, B = (cid:112) π(cid:15) B u gas themagnetic field strength (the factor of 2 /π multiplying B in α ssa accounts for the pitch angle term as in Pa-per I), and Γ is the gamma function (calculated using scipy.special.gamma ).Equation 11 assumes a constant extinction coefficientin the self-absorbing shell. In reality, especially afterthe shock crosses the shell and it begins to expand, theextinction coefficient may be different across the shockedgas. The representative α we use in our optical depthcalculations is the radial average value, (cid:104) α ssa (cid:105) = (cid:80) k α ssa ,k dr k (cid:80) k dr k , (13)where k indicates the index of a resolution element in theshock. We exclude the five resolution elements closestto the contact discontinuity in our calculation of (cid:104) α ssa (cid:105) because mass-density is a factor in the α ssa calculationsand is incorrect near the contact discontinuity due to theunaddressed Rayleigh-Taylor instability, as noted, e.g.,in Chevalier (1982).The representative shell thickness (cid:102) ∆ r (Equation A6)can be computed directly from the contact discontinuityradius ( r ) and the forward shock radius ( r ).Thus for each time snapshot of the simulation we candetermine τ ssa = 4 (cid:104) α ssa (cid:105) (cid:102) ∆ r . (14)The free-free extinction coefficient ( α ff ) must describethe preshock CSM, which we assume is hydrogen rich,isothermal, constant density, and fully ionized by the ra-diation field of the shock. Therefore, although there aremany resolution elements of preshock CSM in the hydro-dynamic simulation, for the purposes of radiation trans-port it is one-zone model. The extinction coefficientis calculated using the formulae in Rybicki & Lightman(1979) via the rad tools.BremCalculator.calc al BB function of csmpy which assumes the electrons are ther-mally distributed and uses the gaunt factors calculatedby van Hoof et al. (2014). Extinction by the CSM is inthe Rayleigh-Jeans limit and the formula used is α ff = 0 . Z g ff T − / n e n I ν − (15) where Z = 1 is the ion charge, g ff is the gaunt factorat the target frequency, and n e and n I are the electronand ion number densities, which we estimate simply as n e = n I = ρ csm /m p in this work unless stated otherwise,where m p is the proton mass. The width of the pre-shockCSM is ∆ r ext = R out − R f , (16)and thus τ ff is known from Equation 10.The evolution of τ ssa and τ ff calculated from the mod-els are shown in Figure 2. In these calculations, we haveassumed (cid:15) B = 0 . , T csm = 10 K , ν = 4 . , µ e = µ I = 1 , and Z = 1. The sharp elbow on the decline ofeach curve marks t p , the time when the forward shockcrosses the outer edge of the CSM. In the next section,we will discuss what drives the normalization. Here wewill point out that for most of the time that the shockis in the shell, τ ssa and τ ff are nearly constant. Inde-pendent of f R , the models with significant absorptionby either process have ρ csm (cid:38) − g cm − . Since τ ssa ( t ) and L ν, thin ( t ) are driven by the synchrotron pro-cess, they have similar shapes, with a long tail after theshock has crossed the shell and shells following the samerise independent of f R . Contrary to this, the τ ff is onlyimportant while the shock is in the shell and higher- f R have higher optical depths at all times. In some of thelater impact time models, we see that the τ ff curves jumpto low values at certain time steps. This is a numeri-cal artifact of the shock front identification process anddoes not affect our later results; for the sake of trans-parency in our methods and because it does not have alarge illustrative impact, we have chosen to not to editthe τ ff ( t ) curves of these models. PARAMETERIZATION OF OPTICAL DEPTHEVOLUTIONIn this section, we present a parameterization to al-low synchrotron self-absorption and free-free (external)absorption optical depths to be reconstructed for an ar-bitrary CSM shell using Equation 11. An example ofthe light-curves created with this method can be seenin Figure 3, which shows the light-curve with no ab-sorption (dotted curves), τ ssa only (dashed curves), andboth sources of absorption (solid curves) for models oftwo different densities.The variable for time we will use is x ≡ t/t imp , (17)i.e., time is normalized to the time of impact. The timethat the forward shock crosses the edge of the CSM shellis the time of peak luminosity in the optically-thin radiolight-curves and is denoted by t p in Paper I; therefore, Harris et al.
10 10010 -6 -3 τ ff
10 100 days since explosion -6 -3 τ ss a parameterization0.1 1 f R Figure 2.
Calculation of τ ff (top) and τ ssa (bottom) fromthe models shown in Figure 1, using (cid:15) B = 0 . § f R affectsthe shape of the curve. (Note that the jaggedness of some τ ff ( t ) curves is numerical, not physical) The black dashedcurves show an example of applying the parameterizationgiven in § ρ csm = 8 . × − g cm − , f R = 1 , R in =9 . × cm). here we will use x p = t p /t imp . As in Paper I, we willprovide a functional form for the evolution of τ ssa and τ ff while x ≤ x p , and evaluate times that τ ssa reachescharacteristic values for times x > x p (at which pointthere is no external absorption because the CSM hasbeen swept over).5.1. Synchrotron Self-Absorption
First we determine the normalization of τ ssa . UsingEquation 52 of Paper I, the normalization of the extinc-tion coefficient is α ssa ∝ ρ − u / ν − / (cid:15) / B . (18) From Equation 7 of Paper I, the time evolution of theforward shock radius, R f , and shock speed, v s , while theshock is in the shell is x = 0 . R f /R c, ) . (19) ⇒ R f = R in x . (20)and (21) v s = R in t imp x − . . (22)Equation 2 gives R in in terms of t imp and ρ csm . Assum-ing u gas ∝ ρ csm v s , we now have (dropping factors of x because we are interested in the normalization) α ssa ( t ) ∝ ρ / t − . ν − / (cid:15) / B . (23)The radial term in τ ssa ( (cid:102) ∆ r ) is the volume-to-area ratio(Equation A6) and should roughly evolve like R f if theshell is thin and has width ∆ R ∝ R f . Then the expectednormalization of τ ssa should scale like τ ssa ∝ α ssa ( t ) R f ( t ) ∝ ρ / t − / ν − / (cid:15) / B . (24)and we find that, indeed, the evolution of τ ssa ( x ) is thesame for all models when normalized by this factor.Therefore, τ ssa ( x ) at times after impact but before theshock crosses the outer edge of the CSM (i.e., 1 ≤ x ≤ x p ) can be described by the asymptotic function τ ssa ( x ) = 13 . (cid:18) ρ csm − g cm − (cid:19) / (cid:18) t imp
100 days (cid:19) − / × (cid:16) ν . (cid:17) − / (cid:16) (cid:15) B . (cid:17) / (25) × x − . (1 − x − . ) . (26)The normalization factor and the exponents in theasymptotic function x − . (1 − x − . ) were determinedusing scipy.optimize.curve fit .As with the optically thin luminosity, we fit the evolu-tion of τ ssa after the shock has crossed the outer edge ofthe CSM (i.e., x > x p ) by determining the time at which τ ssa reaches characteristic fractions of τ ssa ,p ≡ τ ssa ( x p )and connecting the points with power-laws (i.e., linearinterpolation in logarithmic space). The characteristicpoints are x ( τ ssa = 0 . τ ssa ,p ) = 1 . f R ) . (27) x ( τ ssa = 0 . τ ssa ,p ) = 1 . f R ) . (28) x ( τ ssa = 10 − τ ssa ,p ) = 1 . f R ) . (29) x ( τ ssa = 10 − τ ssa ,p ) = 1 . f R ) . . (30) N Ia Shell CSM α ssa ∝ t − ( t − ) / ∝ t − . (31) (cid:102) ∆ r ∝ t (32) ⇒ τ ssa ∝ t − . . (33)where we have assumed the shell inner and outer radiievolve like r ∝ t , density evolves like ρ ∝ t − , and energydensity evolves like u ∝ V − / ∝ t − , with V being theshell volume. 5.2. Free-Free Absorption
For the external, free-free absorption, the evolution ofoptical depth reflects the radial evolution of the shock,i.e., τ ff ( x ) = α ff ∆ r ext ( x ) . (34)Using Equation 5, this can be estimated as∆ r ext ( x ) = R in [(1 + f R ) − x . ] , (35)where we have used 0 . ≈ x = 1rather than using the fit value, so, τ ff ( x ) = α ff R in [(1 + f R ) − x . ] , (36)with α ff as in Equation 15 and R in given by Equation 2.Note that τ ff ( x > x p ) = 0 because x p represents thetime at which the forward shock crosses the edge of theCSM shell, thus, all of the CSM has been shocked andthere is no “external” medium.5.3. Error of the Parameterization
The error incurred by using the fitting functions givenabove — i.e., comparing τ ssa ( x ) and τ ff ( x ) calculatedwith the given formulae versus from the simulationsthemselves — is small, (cid:46)
30% on each, near the peakof the optically thin light-curve. Very near the time ofimpact, when the system is changing rapidly, the errorcan be much larger. We also find that the adiabatic ap-proximation does not match the very late time behaviorwell, possibly due to deceleration from the “interstellarmedium” gas (of density 10 − g cm − ) that lies outsidethe shells. It is unlikely that either of these phases willbe of practical use to the interpretation of observations,since the (optically thin) luminosity of the shocked gasis so low at these times — (cid:46) .
1% of the optically thinpeak luminosity. Nevertheless, we caution that one takecare if interpretation of observed data hinges on the veryearly or late phases of the interaction. APPLICATION TO RADIO DATASETSIn this section we show how the parameterized light-curves can be applied to radio datasets. For these analy-ses, we assume T csm = 10 K when calculating the free-free absorption, i.e., that the preshock CSM is heatedsimilar to an HII region by the ionizing radiation of theshock. 6.1.
Testing Models of PTF11kx
Dilday et al. (2012) report a non-detection ofPTF11kx with the Karl G. Jansky Very Large Array(VLA) obtained on March 30, 2011 with a 1 σ root-mean-square image noise of 23 µ Jy. This is +61 dayssince B -band maximum (January 29, 2011). From theNRAO archive, we find that the central frequency of theobservation was 8.4 GHz.Consistent with Graham et al. (2017), in this anal-ysis, we assume a distance of 204.4 Mpc and that B -band maximum occurs 13 days after explosion, interac-tion began at t imp = 50 days, and interaction ended at t p = 500 days. Variations in these timings of ∼ f R from t p and t imp (Equation 4).The only other necessary model input is the density ofthe shell ρ csm , which can be combined with t imp to find R in .For PTF11kx, ρ csm can be estimated from its opticalspectra. The Ca II H&K absorption lines were satu-rated at early times, allowing an inference of the CSMcolumn density, N csm , assuming solar composition. Themass density, ρ csm , can be found from N csm if we as-sume the density is constant within the shell and if theextent of the CSM (∆ R = f R R in ) and mean particleweight ( ¯ m ) are known, as ρ csm = ¯ mN csm / ∆ R . Wetake ¯ m is 1.33 times the proton mass, as appropriatefor neutral material of solar abundance. Graham et al.(2017) derive N csm ≈ × cm − , significantly lowerthan the original estimate by Dilday et al. (2012) of N csm ≈ cm − . The lower estimate is probablycorrect, for two reasons. First, because two differentmethods for analyzing the line indicate a lower N csm (Graham et al. 2017). Second, the higher value of N csm creates an inconsistency—to create a saturated line re-quires that the CSM cover the SN photosphere, but fullcoverage implies a high CSM mass that is inconsistentwith the weak levels of interaction seen (Dilday et al.2012). Although we favor the lower density estimate,we will investigate both hypotheses.In Figure 3 we compare the radio limit for PTF11kxto the radio light-curves based on the current best de-scriptions of its CSM as described above. In calculat-ing τ ff , we have assumed µ I = 1 . , µ e = 1 .
18 to be
Harris et al.
SN age [days] L ν [ e r g s − H z − ] EVLA σ PTF11kx at 8.4 GHz un a b s o r b e d S S A o n l y log N csm = log N csm = Figure 3.
The VLA 3 σ radio limit for PTF11kx (blacktriangle) is consistent with models for either N csm ≈ cm − (blue) or N csm ≈ × cm − (orange) . Dotted curves show the optically thin light-curves, dashed curves include synchrotron self-absorption ( (cid:15) B = 0 . solid curves furthermore include free-free absorption (with Z = 1 , µ e = 1 . , µ I = 1 . , T csm = 10 K). A radio obser-vation at ∼
500 days may have distinguished between the N csm measurements. consistent with the compositional assumptions used fordetermining N csm . We find that with either estimateof N csm , our models are consistent with the radio non-detection of PTF11kx. Our optically thick light-curvesare needed to interpret the high- N csm scenario, whereasthe low- N csm case is subject to very little absorption. Ifa second observation had been taken around one year af-ter explosion, it would have been able to distinguish be-tween the N csm values. That is, under the assumption ofspherically distributed CSM. When one allows the CSMto be in a torus, the light-curves must be modified; thisshould roughly be a diminution of the luminosity by thecovering fraction of the CSM, if we saw PTF11kx edge-on as suggested by the saturated pre-impact absorptionlines, which would make the signal too dim to be seenby the VLA observation.6.2. The Allowed Fraction of SNe Ia with CSM Shells
Using the optically thick light-curve parameterization( §
3) we can explore the detection power of radio upper-limits for CSM shells. While a similar analysis has beencarried out for individual objects (Harris et al. 2018;Cendes et al. 2020; Pellegrino et al. 2020), this is the firstsuch analysis of a population of SNe Ia. In this analysiswe assume µ e = µ I = 1 , Z = 1 , and T csm = 10 K.Chomiuk et al. (2016) present VLA observations ofthermonuclear supernovae (SNe Ia) across all sub-groups (1-2) GHz8 SNe (4.5-6) GHz32 SNe1 10 10010 (6.8-9) GHz22 SNe 11kx 1 10 100(15-43) GHz7 SNe SN age [days] σ L ν li m it [ e r g s − H z − ] Figure 4.
Data (3 σ upper limits) used to determine thefraction of SNe Ia that may host shells. of the class. Of these, we use the “cool,” “shallow-silicon,” and “core-normal” groups. We also incorporatedata compiled in Lundqvist et al. (2020) and those pre-sented in Mooley et al. (2016) and Ryder et al. (2019).We group the three sub-types together since the cooland shallow-silicon groups are too sparsely sampled tobe analyzed independently. The sample of data, shownin Figure 4, covers a range of frequencies from 1–43 GHz.Observations span 1 −
365 days after explosion for a to-tal of 50 SNe among all observations independent of fre-quency (observations are grouped by frequency here forcomparison with Figure 5). We show PTF11kx in thisfigure for reference; this data point is not included inour following statistical analysis because it is not a use-ful limit, i.e., it would not be able to detect any modelin our set of interest.In this analysis, we characterize CSM shells by theirmass ( M csm ), inner radius ( R in ), and fractional width( f R ≡ ∆ R/R in ). These three parameters fully deter-mine a shell light-curve in our model framework. Weare interested in constraining the fraction of SNe Ia withCSM of a given M csm and f R , which we will call theCSM’s “configuration.”We must choose a distribution of R in to make this con-straint, which we do as follows. Note that we will use R in , = R in / (10 cm). Moore & Bildsten (2012) usedanalytic calculations to explore the CSM established byrecurrent nova eruptions in a binary system with a redgiant companion and significant associated winds. Theyfound that the nova ejecta sweep up the giant wind andquickly (within 20 years) and decelerate to a drasticallyreduced coasting speed of (cid:46)
100 km s − . The exact val-ues depend on the recurrence time and companion windmass-loss rate. Due to the low speed, the shells build upinto a thicker, more massive shell than would be formed N Ia Shell CSM R in , ∼ . −
10, depending onthe binary parameters. Traveling at (cid:46)
100 km s − , thethick shell will remain in the system for 10 − yr be-fore mixing into the interstellar medium — giving plentyof time for a massive shell to build up if the recurrentnovae continue.We consider the delay time between shell formationand SN explosion to be entirely unknown (i.e., that theSN event is equally likely to occur at any time afterthe start of the recurrent nova period begins). There-fore, the probability distribution for R in is determinedby the shell kinematics. Since the shells spend only 20years within r ∼ cm compared to the > yearsthey spend beyond this distance, we treat the probabil-ity of R in , < . R in , ≥ . R in , ∈ [0 . , t imp ∝ R in and f R (= ∆ R/R ∝ t/t ) is con-stant as it moves away from the binary.The radio light-curves for the shell configurations weconsider are summarized in Figure 5.In the top panel, we show all light-curves generated forjust one shell configuration in different frequency bins.In the actual analysis, a light-curve is generated at thethe frequency of each individual observation. To deter-mine if an observation has constraining power in a sit-uation where the location of the CSM is unknown, onemust not compare a luminosity limit to a single light-curve but instead look at the light-curve “roof” that iscreated by the set of possibilities (black dashed line).Observations under the roof have constraining power,whereas anything above the roof has no possibility ofdetecting any shell and therefore no statistical power.In the bottom panel, we show the roofs for other shellconfigurations, spanning M csm = 10 − − . M (cid:12) and f R = 0 . −
1. The dark green curve (farthest right)is the same as the black dotted line from the top panelat 5 GHz, the frequency with the most SNe observed.Observations in the shaded regions (i.e., under the roof)have constraining power on the configuration. The blackdashed line shows the typical luminosity limit of thedata, which is very close to the top of the “roof” formost models, and has best coverage for the higher-mass f R = 1 shells – i.e., these are the shells most suited forstudy by the radio observations. Again we note that thePTF11kx observation shown in Figures 3 and 4 is above SN age [days] L ν [ e r g s − H z − ] . M fl , f R = 1 R in ,
10 100
SN age [days] L ν [ e r g s − H z − ] f R − M fl , 0.1 − M fl , 1 − M fl , 0.1 − M fl , 1 − M fl , 0.1 − M fl , 10.1 M fl , 1typical L ν limit Figure 5. Top:
A grid of light-curves for a single shellconfiguration ( M csm = 0 . M (cid:12) , f R = 1), where R in , variesbetween 0.1–1 (denoted by color scale ranging from magentato yellow). The light-curve grid forms a “roof” at a given fre-quency (black dotted line) ; observations looking to samplethis shell configuration must be under the roof. Bottom: R in , ∈ [0 . , σ luminosity limit of the SN Ia sample (Figure 4),and the x-axis range has been chosen to span the observationtimes. Observations are primarily under the roofs of f R = 1models, so radio observations will be most constraining forthese shells. the roofs, so it is not useful to our analysis. Note thatCSM parameters that would violate model assumptions(as described in §
2) are not shown, which is one reasonthat the low-mass shell roofs look different from thoseof higher masses.The probability of detecting interaction with a CSMshell of mass M csm and fractional width f R in an SN Iaevent is the product of (a) the fraction of SNe Ia thathost such shells ( ξ ) and (b) the probability that ob-0 Harris et al. servations of the SN can detect the interaction signal( P i (det | occ)). The former term is the one of interest toour study, and it can range from ξ ∈ [0 , R in , ∈ [0 . , σ upper-limitsfor the observed SN sample (i.e., Figure 4). The num-ber of injected light-curves that would be detectablefor SN i ( N det ,i ) compared to the number in the grid( N ) is a good estimation of the detection probability forthe shell, provided that the number of models is highenough: P i (det | occ) = N det ,i N (37)We use N = 100 models. Thus for each SN the proba-bility of detecting interaction is P i (det) = ξ N det ,i N . (38)The probability that of S events, none discovered in-teraction with a shell is P (no dets) = S (cid:89) i =0 (cid:18) − ξ N det ,i N (cid:19) . (39)In a Bayesian framework, the probability density p ofa given value of ξ being true is p ( ξ | no dets , I ) ∝ p (no dets | ξ, I ) × p ( ξ | I ) , (40)where I represents our model assumptions. Since ourmodel assumptions do not depend on the fraction ofSNe Ia with CSM shells, p ( ξ | I ) = 1. The probabilitydensity p (no dets | ξ, I ) is proportional to P (no dets).Thus the observed non-detections can be transformedinto an upper limit on ξ via P ( ξ < ξ up ) = (cid:82) ξ up (cid:81) Si =0 (cid:16) − ξ (cid:48) N det ,i N (cid:17) dξ (cid:48) (cid:82) (cid:81) Si =0 (cid:16) − ξ (cid:48) N det ,i N (cid:17) dξ (cid:48) (41)and we can obtain the maximum allowed value of ξ at99.7%-confidence (3 σ ) by finding the ξ up at which Equa-tion 41 evaluates to 0.997.Figure 6 shows the results of the analysis, providing99.7% confidence limits on ξ , assuming (cid:15) B = 0 . (cid:15) B = 0 .
01 (small markers, dot-ted lines). We see that in all cases, the large sample ofradio non-detections is still consistent with a high frac-tion of SNe Ia having confined CSM shells — especiallyif (cid:15) B = 0 .
01 is the appropriate value for these shocks. -4 -3 shell mass [ M fl ] m a x i m u m a ll o w e d ξ R in , = f R † B Figure 6.
Maximum fraction of SNe Ia ( ξ ) that can haveCSM shells of a given mass and width, within R in = 10 cm.The value of ξ is calculated at 3 σ (99.7%) confidence. Largemarkers are calculations with (cid:15) B = 0 .
1, while small mark-ers represent (cid:15) B = 0 .
01. The shell with M csm = 0 . M (cid:12) and f R = 4 represents a PTF11kx-like configuration. Single nova outbursts. — The thickness of a single novaoutburst is predicted to be f R ∼ . ∼ − M (cid:12) Chomiuk et al. 2014). We see that nova-like shells (thinand low-mass) are currently largely unconstrained byradio observations. Essentially all SNe Ia could havea 10 − M (cid:12) , f R = 0 . f R = 1) are only constrained to (cid:46) (cid:15) B = 0 . Multiple novae. — Multiple nova eruptions could producea thicker, more massive shell. We see that (for (cid:15) B =0 . f R = 1 shells are constrained to be (cid:46)
50% acrossthe mass range explored. This is because, as can beseen in Figure 5, these models have similar requirementsfor their observability — higher mass shells are moreluminous (in the optically thin limit) but are also subjectto more absorption.
Very thick (PTF11kx-like) shells. — In our analysis, weinclude a 0 . M (cid:12) shell with f R = 4, representing aPTF11kx-like CSM (Graham et al. 2017). From the ra-dio limits alone, we find that up to 90% of SNe Ia couldhave CSM with a PTF11kx-like configuration. Lowermass, thick shells are more constrained because theyhave a lower free-free optical depth, yet we see that theradio non-detections are still consistent with a relativelyhigh fraction of SNe Ia having thick shells. N Ia Shell CSM
Comparison with Nebular H α Statistics
Recently, Tucker et al. (2020) used a sample of 111low-redshift SNe Ia to constrain the dominance of thesingle-degenerate channel in creating SNe Ia using thetheoretical framework of Boty´anszki et al. (2018). Thesemodels focus on the H α signature from hydrogen thathas been stripped off the companion envelope, and theyallow one to convert flux limits into limits on the massof stripped material (subject, of course, to a variety ofunderlying model assumptions), which can then be com-pared to theoretical expectations.One striking decision made in the Tucker et al. (2020)analysis was to exclude all known cases of SNe Ia withlate-time H α emission, even if those events looked nor-mal near maximum light and had observations in thesame time frame as the rest of the sample — the 91T-like(“shallow silicon”) PTF 11kx and the 91bg-like (“cool”)events SN 2018fhw and SN 2018cqj (Dilday et al. 2012;Kollmeier et al. 2019; Prieto et al. 2020). Another ex-ample of a 91T-like SN Ia with late-time H α emission,but that unfortunately does not have observations in thetime window considered, is SN 2015cp (Graham et al.2019). Tucker et al. (2020) essentially argue that theseevents ought to be excluded because they are abnormal;but, by definition, hydrogen emission is abnormal in anySN I. We note that there is a strong distinction betweenthe 91T-like delayed-interaction events and 91bg-likecases; pertinent to this discussion, 91bg-like cases havelow-luminosity line emission and may represent strippedcompanion material, whereas 91T-like cases have higherline luminosity and a distinct CSM origin. Therefore,it was sensible for Tucker et al. (2020) not to ana-lyze PTF11kx or SN 2015cp in the stripped-companionmodel framework, since the line signal was not of thisorigin. However, we note that the 91bg-like events hadestimated stripped masses of ∼ − − − M (cid:12) , andwe estimate that the Tucker et al. (2020) have 65 eventsthat probe a similar mass (their Figure 6), thereforehad these events been included in the nebular samplethe statistics would have been two detections among 67events, which under a simple binomial distribution anal-ysis results in a 3 σ limit of (0 . − . ∼ − − − M (cid:12) of (stripped) hydrogen.Our study, however, is not concerned with the signa-ture of stripped material, but rather with circumstellarmaterial. We do not yet have H α emission models forthe CSM shell scenario investigated in this work, so wecannot perform an analysis we have done for the radiousing the Tucker et al. (2020) data. However, what wecan say is that any of the Tucker et al. (2020) observa-tions would have been able to detect H α emission froma PTF11kx twin. Including the other three events into the statistics is complicated by their lower luminosity(SN 2018cqj), observations being earlier than the restof the sample (SN 2018fhw), or observations being laterthan the rest of the sample (SN 2015cp). If Tucker et al.(2020) had chosen to include PTF11kx in their sample,then they would have 104 normal, 91T-like, or 91bg-likeevents in their sample (the categories we analyze in thisstudy) and one detection of H α emission in the 3–15months after maximum light time window, resulting inan allowed fraction of of SNe Ia with CSM like PTF11kxof (0 . − . ∼
90% of SNe Iacould have a PTF11kx-like shell (because, for the ma-jority of the interaction, we predict the radio emissionis absorbed by the preshock CSM).One point of interest to both the radio and opticalstudies of delayed interaction is that hydrogen emissionin SNe Ia — regardless of its time of appearance —is so far associated with 91bg-like or 91T-like SNe Ia(see the above references for individual events with late-time hydrogen emission as well as Leloudas et al. 2015),which are relatively rare. Therefore, even large sam-ples, like those discussed in this work, will not provide astatistically significant number of events from these sub-groups. For example, if PTF11kx had been included inTucker et al. (2020), the sample size of 91T-like eventswould be six, with one detection, and the prevalenceof PTF 11kx-like objects constrained to (0.8–77.1)% of91T-like SNe Ia — or (0.6-79.9)%, if SN 2015cp is in-cluded also. For 91bg-like events, the sample would be-come ten events with two detections, and the fractionallimit constrained to (0.9-75.6)% of 91bg-like events sim-ilar to SN 2018fhw. If these subgroups represent thesingle-degenerate channel, as has been suggested on the-oretical grounds by Fisher & Jumper (2015), then thesestatistics highlight how little we know about SNe Ia thatdo come from the single-degenerate channel compared tothe constraints that have been made on the prevalence ofthe single-degenerate channel overall. Furthermore, noSN Ia with hydrogen emission fits neatly into the single-degenerate progenitor picture (having either too muchor too little hydrogen mass inferred, and nothing thatlooks like a normal stellar wind) which challenges ourpicture of this pathway to explosion — and therefore,challenges some of the very models used to constrain itsprevalence among SNe Ia. SUMMARYSNe Ia with detached, confined shells of CSM (whichproduce SNe Ia;n) provide a window into the single-degenerate channel and may represent SNe Ia impact-ing a CSM shaped by novae. However, the uncertain2
Harris et al. mass, extent, and location of these shells makes it chal-lenging to observe them in an interacting phase, creat-ing large uncertainty in the intrinsic prevalence of theseshells. Adding to this uncertainty, and what this workaims to alleviate, is the need for theoretical tools thatcan interpret SN Ia observations in the context of in-teraction with these shells — because observations aretaken during periods of hydrodynamic transition, pop-ular equations based on asymptotic solutions cannot beaccurately applied and new ones must be found. With-out appropriate modeling, the properties of these shellscannot be precisely determined (limiting studies of theirorigin), nor can their occurrence rate be assessed froman SN Ia survey.In Harris et al. (2016, Paper I, summarized in § § τ ssa ) and free-freeabsorption ( τ ff ) to obtain the escape fraction of the ra-dio photons. We find that around a shell density of ρ csm (cid:38) − g cm − , both sources of absorption beginto come into play. We then derive 4.9 GHz τ ssa and τ ff values from the hydrodynamic model suite, whichrequires finding the shock width and mean extinctioncoefficient as a function of time ( § § HNK16 tools.py ) availableonline. In § ∼ . M (cid:12) shell extending from ∼ cm to ∼ × cm. Wefind that the radio non-detection is consistent with thismodel, and the non-detection limit was well above the https://github.com/chelseaharris/csmpy maximum radio luminosity reached by the interactionat any phase. Dilday et al. (2012) originally proposeda higher density of CSM, and we show that (if this hadbeen spherically distributed) it would have reached adetectable level, but only at late times ( ∼ ξ , the fraction of SNe Ia thathost a shell of mass M csm and fractional width f R ata distance of 10 − cm. We consider M csm be-tween 10 − M (cid:12) and 0 . M (cid:12) with f R = 0 . , , and 4.Overall, we find that, at 99.97% statistical confidence,thick shells ( f R = 1 ,
4) of any mass < . M (cid:12) can bepresent in up to ξ ∼
60% of SNe Ia and still be consistentwith the radio non-detections. Thin shells are essentiallycompletely unconstrained. Surprisingly, PTF11kx-likeshells, which should be relatively easy to see in opti-cal spectra, are only constrained by radio data to bein (cid:46)
90% of SN Ia systems, because these relativelymassive and thick shells are more subject to free-freeabsorption. We further calculate the constraints underthe assumption of weaker magnetic field amplification (cid:15) B = 0 .
01, in which case the radio limits allow a largemajority of SNe Ia to host shells.ACKNOWLEDGMENTSC.E.H. and L.C. are grateful for support fromNSF through AST-1751874 and AST-1907790. C.E.H.also acknowledges support from the Packard Founda-tion.P.E.N. acknowledges support from the DOE un-der grant DE-AC02-05CH11231, Analytical Modelingfor Extreme-Scale Computing Environments.We thank the anonymous reviewer for their commentson this manuscript.This research used resources of the National EnergyResearch Scientific Computing Center, a DOE Office ofScience User Facility supported by the Office of Scienceof the U.S. Department of Energy under Contract No.DE-AC02-05CH11231.The National Radio Astronomy Observatory is a facil-ity of the National Science Foundation operated undercooperative agreement by Associated Universities, Inc.Michigan State University occupies the ancestral,traditional, and contemporary Lands of the Anishi-naabeg–Three Fires Confederacy of Ojibwe, Odawa, and
N Ia Shell CSM UNSHOCKED
EJECTA no e ff ect (assumed) SHOCKED
GAS emits & absorbs, α ssa UNSHOCKED
CSM absorbs, α ff no e ff ect r r r z z = r cos θ = r cos θ ʹ θ θ ʹ r r θ φ UNSHOCKED
EJECTA no e ff ect (assumed) SHOCKED
GAS emits & absorbs, α ssa No External Absorbing Medium External Absorbing Medium
Figure 7.
Schematic of the geometries considered for solving the radiation transport equation: the case with no external ab-sorption ( left ) and with external absorption ( right ). The ejecta absorption is assumed to have no effect because the contributionof the obscured emission region to the overall luminosity is small.
Potawatomi peoples. The University resides on Landceded in the 1819 Treaty of Saginaw.
Facilities:
Karl G. Jansky Very Large Array
Software:
Sedona (Kasen et al. 2006), SciPy (Joneset al. 2001), NumpPy (Oliphant 2006), Astropy (AstropyCollaboration et al. 2013), Matplotlib (Hunter 2007)APPENDIX A. RAY TRACING IN A SPHERICAL SHELL GEOMETRYIn the optically thin limit of isotropic emission, the spectral (or “specific”) luminosity of an emitting shell can besimply calculated as L ν, thin = 4 πj ν V, (A1)where j ν is the emissivity (units like erg s − Hz − cm − str − , value is assumed constant throughout the shell), V isthe volume (units like cm ), and the factor of 4 π accounts for the angle covered by the emission (thus has units of str[sterradians]). In the case that the shell both emits and absorbs light, however, one must solve the radiation transportequation. A.1. Internal Absorption Only
Consider a thin, spherical shell extending from radius r to r , with constant extinction coefficient ( α , units likecm − ) and j ν within the shell and no emission or absorption in the cavity r < r . This is the scenario if one assumes(1) absorption by the unshocked ejecta is negligible and (2) absorption by the unshocked CSM is negligible, either dueto the CSM column density or because the shock has already crossed the outer edge of the CSM shell. (Note thateven if the ejecta full absorb the radio emission, it will have a negligible effect on the overall radio luminosity becausemost of the emission comes from the projected inner edge of the radiating shell, as is familiar from spatially-resolvedexamples of interaction, such as the H α emission of SN remnants or radio interferometry of interacting SNe.)Then the solution to the radiation transport equation along a straight path through the sphere that makes an angle θ with the outward surface normal at r is I ν ( θ ) = j ν α { − exp[ − τ ( θ )] } = L ν, thin πV α { − exp[ − τ ( θ )] } , (A2)4 Harris et al. where I ν is the specific intensity (units like erg s − Hz − cm − str − ), τ is the optical depth ( α multipied by thepath length), and in the second expression we have substituted in the equation for optically thin luminosity. Defining τ ≡ αr and θ by sin θ ≡ r /r , and using the convention µ ≡ cos θ (thus µ = cos θ ), the optical depth is givenby τ ( θ ) = (cid:40) τ µ if µ ≤ µ τ ( µ − (cid:112) µ − µ ) µ > µ . (A3)The emerging luminosity from the surface of the sphere ( r = r ) is L ν = 4 πr F ν = 4 πr (cid:73) I ν ( µ ) µ dµ dφ, (A4)where φ is the angle in the plane perpendicular to the line of sight. Since we are considering isotropic, sphericalemission, I ν is independent of φ , and since there is only vacuum contributing to rays coming from π/ ≤ θ ≤ π , L ν = 2 π (4 πr ) (cid:90) I ν ( µ ) µ dµ = L ν, thin πr V α (cid:90) { − exp[ − τ ( µ )] } µ dµ . (A5)We here observe that the volume-to-surface-area can be used to define a characteristic width of the shell, (cid:102) ∆ r ≡ V πr = r (cid:34) − (cid:18) r r (cid:19) (cid:35) , (A6)so the escape fraction is L ν L ν, thin = 12 α (cid:102) ∆ r (cid:90) (1 − e − τ ) µ dµ . (A7)Here, L ν, thin is the luminosity the gas would have if it were optically thin. In general, given the form of τ ( θ )(Equation A3), this integral must be computed numerically. In the limit τ (cid:28)
1, this integral recovers L ν = L ν, thin (we note for the reader’s convenience in checking this result themselves that when evaluating the optically thin limit,it is helpful to define a factor f V = (cid:102) ∆ r/r and use αV = τ f V ). In the limit of high optical depth, the escape fractionis L ν / L ν, thin = 1 / (4 α (cid:102) ∆ r ).The shell solution has the same asymptotic behaviors as the slab approximation, L ν / L ν, thin = [1 − exp( − τ )] /τ , ifone uses τ = 4 α (cid:102) ∆ r . We find that the error on the escape fraction incurred by using the slab approximation versusnumerical integration depends on the thickness of the emitting region ( r /r ) and α (cid:102) ∆ r but in any case is < r /r (cid:38) .
8) and near the transition between optically thick and thin regimes( α (cid:102) ∆ r ∼ . L ν L ν, thin = 1 − exp( − α (cid:102) ∆ r )4 α (cid:102) ∆ r . (A8)A.2. Including Absorption by an External Medium
In this scenario we have the same emitting (and self-absorbing) shell as in the last case, but additionally there isabsorption from an external shell that extends from r (the edge of the emission region) to r (the edge of the CSMshell), representing the as-yet-unshocked CSM.The specific intensity along any path is I ν ( z ) = L ν, thin παV (1 − exp[ − τ ssa ( z )]) exp[ − τ ff ( z )] , (A9)where z is the height above the equator, τ ssa is the optical depth to synchrotron self-absorption (internal absorption;simply called “ τ ” in the previous calculation), τ ff is the optical depth to free-free absorption (external absorption),and all other variables are as before. We use z rather than θ here because in terms of z the integral to calculate flux N Ia Shell CSM θ as the angle relative tothe surface normal at r , we define θ (cid:48) to be the angle relative to the surface normal at r ; then z = r sin θ = r sin θ (cid:48) ,and µdµ = − r − zdz = − r − zdz . For this calculation, we are evaluating the flux at r rather than r . Then L ν = 4 πr L ν, thin αV (cid:90) r [1 − e − τ ssa ( z ) ] e − τ ff ( z ) zdzr (A10) L ν L ν, thin = 4 π αV (cid:90) r [1 − e − τ ssa ( z ) ] e − τ ff ( z ) zdz (A11)= 12 r α (cid:102) ∆ r (cid:90) r [1 − e − τ ssa ( z ) ] e − τ ff ( z ) zdz . (A12)(A13)Defining ζ = z/r , L ν L ν, thin = 12 α (cid:102) ∆ r (cid:90) [1 − e − τ ssa ( ζ ) ] e − τ ff ( ζ ) ζdζ . (A14)This is the exact solution for the escape fraction.In this work we have approximated this result by simply accounting for external absorption with an exponentialfactor such that L ν L ν, thin = 1 − exp( − α (cid:102) ∆ r )4 α (cid:102) ∆ r exp( − α ff ∆ r ext ) . (A15)where ∆ r ext = r − r is the radial width of the CSM and α ff is the free-free (Bremsstrahlung) extinction coefficient,which is assumed to be constant in the preshock CSM.In a case where the SSA optical depth is low, we computed the difference between the result of the numerical integraland this approximation for various values of α ff r (optical depth) and r /r (absorbing medium thickness). We findthat the error of the approximation increases as this α ff r increases, and that the error due to geometric effects islargest at r /r ∼ .
5. However, even for this worst case thickness, the error is ∼
10% at τ ∼ τ ∼