Heating from free-free absorption and the mass-loss rate of the progenitor stars to supernovae
aa r X i v : . [ a s t r o - ph . H E ] A p r Heating from free-free absorption and the mass-loss rate of theprogenitor stars to supernovae
C.-I. Bj¨ornsson and P. Lundqvist , [email protected]@astro.su.se ABSTRACT
An accurate determination of the mass-loss rate of the progenitor stars to core-collapse supernovae is often limited by uncertainties pertaining to various modelassumptions. It is shown that under conditions when the temperature of thecircumstellar medium is set by heating due to free-free absorption, observationsof the accompanying free-free optical depth allow a direct determination of themass-loss rate from observed quantities in a rather model independent way. Thetemperature is determined self-consistently, which results in a characteristic timedependence of the free-free optical depth. This can be used to distinguish free-free heating from other heating mechanisms. Since the importance of free-freeheating is quite model dependent, this also makes possible several consistencychecks of the deduced mass-loss rate. It is argued that the free-free absorptionobserved in SN 1993J is consistent with heating from free-free absorption. Thededuced mass-loss rate of the progenitor star is, approximately, 10 − M ⊙ yr − for a wind velocity of 10 km s − . Subject headings: radiation mechanisms: non-thermal — stars: mass-loss —supernovae: general
1. Introduction
The properties of the circumstellar medium into which the explosion of a core-collapsesupernova expands are determined by several different effects. The density structure is Department of Astronomy, AlbaNova University Center, Stockholm University, SE–106 91 Stockholm,Sweden. The Oskar Klein Centre, AlbaNova, SE-106 91 Stockholm, Sweden
2. Heating due to free-free absorption of a synchrotron spectrum
Free-free absorption leads to heating of the absorbing gas. Consider a fully ionized gaswith density n , temperature T , and a radiation field described by an intensity I ( ν, θ ). Theheating can then be written d E d t = Z d ν dΩ I ( ν, θ ) µ ( ν ) , (1)where E = 3 nkT / µ is the absorption coefficient. Mostof the heating of the circumstellar gas occurs within one scale height in front of the forwardshock, i.e., during a time t ≈ R/v sh , where R is the shock radius and v sh is the shock velocity.The energy is absorbed by the electrons so that T is the electron temperature. The effectsof the nuclei are accounted for by introducing a factor f e so that E = 3 n e kT f e /
2, where n e is the density of electrons. Here, 1 ≤ f e ≤ f e = 2for hydrogen and equipartition between electrons and protons, while f e ≈ π , while integrationover frequency (d ν ) gives, roughly, ν . Hence,3 f e n e k T d t ≈ πνI ( ν ) µ. (2)The synchrotron spectrum is characterized by a self-absorption frequency ν abs . Assume thatthe optical depth to free-free absorption at ν abs is less than unity, i.e., τ ff ( ν abs ) <
1. In thiscase most of the absorbed energy comes from around ν abs . With I ( ν abs ) ≈ kT b ( ν abs /c ) ,where T b is the brightness temperature, equation (2) leads tod T d t ≈ π ν T b µc n e f e . (3) 4 –In the absence of cooling, equation (3) can be integrated for a given mass element (i.e., n e = constant). With µ = 1 . × − g ff Z n i n e /T / ν , the temperature of the gas at theshock front varies with time as T sh ≈ . × (cid:18) g ff Z f e n i , ν abs , t T b , η (cid:19) / . (4)Here, g ff , Z , and n i are, respectively, the free-free Gaunt factor, the charge and density ofthe ions. Furthermore n i , ≡ n i / (10 cm − ), ν abs , ≡ ν abs / (10 Hz), t ≡ t/ (10 days), and T b , ≡ T b / (10 K). The factor η accounts for the transition to the regime where heating isnot important and is given by η = 1 + ( T o . × ( g ff Z f − n i , ν abs , t T b , ) / ) / , (5)where T o is the initial, constant temperature of the circumstellar medium. The radial vari-ation above the shock front of the gas temperature at a given time can be calculated in asimilar manner.When the mass-loss rate of the progenitor star is constant, the electron density in thecircumstellar medium is given by n e = 4 . × ˙ M − v w , v , t µ e , (6)where ˙ M and v w are, respectively, the mass-loss rate and wind velocity of the progenitor star, µ e is the mean molecular weight of the electrons, and R = v sh t has been used. Furthermore,˙ M − ≡ × − M ⊙ yr − , v w , ≡ v w / (10 km s − ), and v sh , ≡ v sh / (10 km s − ). This can beused in equation (4) to obtain T sh ≈ . × g ff Zf e µ e ˙ M − ν abs , T b , ηv w , v , t ! / . (7)
3. The free-free optical depth
In situations when the main heating mechanism is free-free absorption, the resultingfree-free optical depth can be determined self-consistently. Since both the density and thetemperature of the electrons decrease away from the shock front, τ ff ≈ µ ( R ) R . At thesynchrotron self-absorption frequency, this yields τ ff ( ν abs ) ≈ . × − g ff Z n ( R ) Rν T / ≈ . Z g ff ) / f / µ / T / , v / , t / ν / , η / ˙ M − v w , ! / , (8)where equations (6) and (7) have been used. For a constant brightness temperature, ν abs ∝ B ,where B is the magnetic field. The strength of the magnetic field is usually taken to scaleinversely with either t or R . Under such conditions equation (8) shows that τ ff ( ν abs ) variesonly slowly with time; in particular, it may even increase.With the use of equation (8), the temperature in equation (7) can also be expressed as T sh ≈ . × (cid:18) g ff Zf τ ff ( ν abs ) t ν , T , η v sh , (cid:19) / . (9)When an appreciable amount of free-free absorption (i.e., τ ff ( ν abs ) < ∼
1) is present, equation(9) shows the expected temperature of the gas to be around 10 K for typical supernovaparameters.It is seen from equation (8) that for a supernova shock expanding into a slow progenitorwind (i.e., v w , ∼ τ ff ( ν abs ) > τ ff ( ν abs ) >
1, the heating rate is proportional to the absorptioncoefficient µ , while the duration of the heating is roughly proportional to µ − ; hence, theenergy absorbed is that for τ ff ( ν abs ) ≈
1. This gives a maximum value of the temperature,which is obtained from equation (3) and using t ≈ /µv sh instead of t ≈ R/v sh . T sh , max ≈ π δ f e T b ν n e v sh c ≈ . × µ e δf e T b , v w , v sh , t ν , ˙ M − (10)The factor δ takes into account the extra heating that occurs for an optically thin synchrotronspectrum with spectral index α <
1, where I ( ν ) ∝ ν − α . In this case, ν I ( ν ) increases withfrequency so that most energy is absorbed at a frequency ν ff such that τ ff ( ν ff ) ≈
1. Hence, δ ≈ (cid:18) ν ff ν abs (cid:19) − α ≈ τ ff ( ν abs ) (1 − α ) / , (11)where τ ff ( ν ) ∝∼ ν − have been used.
4. Numerical results
The synchrotron spectrum used in the numerical calculations is approximated by thatcoming from a spherical shell of radius R . Furthermore, the value of ν abs is assumed to be 6 –independent of the angle θ . The intensity is then given by I ( ν ) = S ( ν )[1 − exp {− τ synch ( ν ) } ],where S ( ν ) ∝ ν / is the source function. Since µ ∝ ν − , the main heating in equation (2)occurs somewhat below the frequency where the intensity peaks. In order to simplify thenotation, ν abs is taken to be the frequency corresponding to τ synch ( ν abs ) = 1. The synchrotronoptical depth is then given by τ synch ( ν ) = ( ν/ν abs ) − ( p +4) / , where N ( γ ) ∝ γ − p is the distribu-tion of electron Lorentz factors γ . The peak intensity occurs at a synchrotron optical depthobtained from exp τ peaksynch − p + 45 τ peaksynch , (12)which corresponds to a frequency ν peak . The brightness temperature is defined by the inten-sity at ν peak ; i.e., I ( ν peak ) ≡ kT b ( ν peak /c ) . The synchrotron spectrum can then be writtenas I = 2 kT b f ( ν peak /ν abs ) c ν / ν / [1 − exp {− (cid:16) ν abs ν (cid:17) ( p +4) / } ] , (13)where f ( x ) = x / [1 − exp {− x − ( p +4) / } ]. In the calculations below p = 2, which implies x = 1 .
413 and, hence, f = 0 . ν abs islarger than that at ν peak by a factor (1 − e − ) /f = 1 .
78. In order to illustrate the accuracyof the analytical approximations as compared to the numerical results, in Figures (1) and(2) this correction factor has been applied to the brightness temperature of the former.With the use of this intensity, the heating of the circumstellar gas is integrated in timeas given by equation (1). In the numerical models, we take into account the angle-averagedoptical depth for the attenuation of the intensity, and not just the radial optical depth asin equation (8). The gas consists of hydrogen and helium, which are assumed to be fullyionized. In addition to the heating, we also include free-free and Case B free-bound coolingas given by Ferland et al. (1992). The initial temperature of the circumstellar gas, when thefree-free heating sets in, is a free parameter. The mass-loss rate of the progenitor star isassumed to be constant, which, in the standard self-similar model (Chevalier 1982a), impliesthat the radius of the forward shock varies as R ∝ t ( n − / ( n − , where n is the power lawindex of the density structure of the supernova ejecta.In Figures (1) and (2) the numerical results for n = 30 are compared to the analyticalapproximations in Sections 2 and 3. The heating is assumed to start instantaneously at 5 daysand the scaling of R is such that it coincides with the value used in Fransson & Bj¨ornsson(1998) at 10 days. Furthermore, T b , = 1, v w , = 1 and, for simplicity, the gas is assumed toconsist of hydrogen only. Since the equipartition timescale (e.g., Spitzer 1978) for the tem-peratures and densities implied by equations (4) and (6) is much shorter than the dynamicaltimescale, f e = 2 is used. For the chosen parameters cooling has a minor effect on the tem-perature and was not included. More detailed calculations are presented in Section 5.2.1 in 7 –which both cooling and helium are included. The radial optical depth shown in Figure (2)corresponds to the center of the source. The synchrotron self-absorption frequency is an im-portant parameter. Its value is taken to coincide with that derived in Fransson & Bj¨ornsson(1998) at 10 days and vary with radius as ν abs ∝ /R . Anticipating the discussion in Sec-tion 5.2.1 of SN 1993J, the parameter values have been chosen close to those deduced inFransson & Bj¨ornsson (1998). For other parameter values the results can be obtained fromFigures (1) and (2) by using equations (7) and (8) as scaling relations.It is seen in Figure (1) that the main heating occurs over one scale height ahead of theshock so that it takes, roughly, one dynamical time after the heating has started beforethe temperature immediately in front of the shock has reached the regime where equation(7) is valid. Except for the initial rise in temperature, which was not included in Sections2 and 3, the agreement between the analytical and numerical results for the temperatureand free-free optical depth is rather good. Although the analytical approximations give asomewhat too high a temperature, its variation with time is well described by the analyticalexpressions. It may be noticed that neglecting the correction factor for the analytical resultsdiscussed above give good agreement also for the amplitude of the temperature. The analyt-ical approximations use the temperature at the shock to calculate the free-free optical depth.This overestimates the average temperature above the shock and, hence, gives a value of thefree-free absorption which is too low. As the temperature at the shock approaches T o therelative importance of the radial variation of the density increases. Since the analytical result(equ. (8)) assumes constant density within one scale height above the shock, it overestimatesthe average density, which then leads to an overestimate of the free-free optical depth atlater times. For the case shown in Figure (2), these two effects together cause the decline ofthe free-free optical depth with time to be underestimated.Another effect not included in Sections 2 and 3 is the attenuation of the incident intensityby the free-free absorption itself; i.e., the circumstellar medium is artificially assumed to beoptically thin. This overestimates the heating and, hence, the temperature. In order toillustrate the effects of the free-free optical depth, the heating has also been calculated byartificially assuming the incident spectrum to be unaffected by free-free absorption. Theresult is shown as the optically thin models in Figures (1) and (2). Figure (2) shows thatalthough the optical depth in this case is lower, which is due to the higher temperature,its variation with time is similar to the physically realistic situation. The reason is that, inthis case, the value of τ ff ( ν abs ) varies only slowly with time so that the scaling between theoptically ”thin” and ”thick” cases is roughly constant. In general, when the optical depth islarge enough to affect the heating, variations of, for example, ν abs and v sh will influence thetime evolution of τ ff ( ν abs ) (cf. equ. (8)). When τ ff ( ν abs ) decreases (increases) with time, theaverage heating ahead of the shock decreases slower (faster) than for the optically ”thin” 8 –case. This leads to a slower (faster) decline of the average temperature and, hence, to asteeper (flatter) decline of the free-free absorption.
5. The initial temperature of the circumstellar medium
As discussed in Section 3, for typical supernova parameters the maximum temperatureof the circumstellar medium due to heating from free-free absorption is around 10 K andoccurs for optical depths of order unity (cf. equ. (9)). Hence, in order for free-free absorptionto be an important heating mechanism, the heating of the circumstellar medium prior to theradio emitting phase must not give temperatures higher than this.
The breakout of the radiation mediated supernova shock from below the stellar surfaceis expected to give rise to an initial flash of energetic radiation that is able to ionize and heatthe surrounding medium. Except for low shock velocities at breakout, v s < ∼ . c (Weaver1976; Sapir et al. 2013), the radiation is likely to deviate from black body. The calculationsby Sapir et al. (2013) assumed the photons to be produced by bremsstrahlung and that localCompton equilibrium was reached by all photons which were not absorbed. However, close tothe breakout, there is also the possibility for the bremsstrahlung photons to escape. Althoughenergetically these escaping photons are likely to be unimportant, their low frequencies couldmake them dominate the ionization and, hence, the temperature of the circumstellar mediumat distances large enough for recombination not to occur before reached by the viscous shock.This requires that they are not all absorbed as they diffuse out from behind the shock atbreakout.Let hν I ≪ kT Comp , where ν I is the ionization frequency of hydrogen and T Comp is thetemperature behind the shock. The number of bremsstrahlung photons with frequencies ≈ ν I produced per unit area per second within a distance ∆ r behind the shock is N ( ν I ) ≈ (cid:18) π (cid:19) / (cid:18) mc kT Comp (cid:19) / c α f σ T n s , e n s , i g ff Z ∆ r, (14)where α f is the fine structure constant, σ T is the Thomson cross section, n s , e and n s , i are,respectively, the densities of electrons and ions behind the shock. In a steady state situation,the minimum distance (∆ r I ) ahead of the shock that can be ionized is then obtained by 9 –assuming negligible contribution to the ionization from other sources, e.g., collisions, N ( ν I ) ≈ n e n p β ∆ r I , (15)where n p is the density of protons and β is the recombination coefficient. By introducing thecorresponding scattering optical depths, τ b ≡ σ T n s , e ∆ r and τ I ≡ σ T n e ∆ r I , equation (15)can be written τ I τ b ≈ . g ff T / , β − Z n s , i n p , (16)where T Comp , ≡ T Comp / (10 K) and β − ≡ β/ (10 − cm sec − ). In a radiation mediatedshock and a medium dominated by hydrogen and helium Z n s , i /n p = 7(1 + Y ), where Y isthe mass fraction of helium. Since the number of escaping photons is much higher than thenumber of electrons ahead of the shock, these photons are expected to heat the electronsto a temperature similar to T Comp without changing the emerging spectrum significantly(Sapir et al. 2013). As shown by Sapir et al. (2013), the value of T Comp is in the range10 − K for v s /c ≈ .
1. With β − ≈ T Comp , ≈ g ff ∼
1, it is seen from equation(16) that τ I > ∼ τ b . The weak temperature dependence of T / β implies that τ I > τ b shouldapply to most shock breakouts independent of supernova type.The value of τ b is determined by the condition that the Compton parameter is roughlyunity, i.e., 4 kT Comp mc τ ≈ . (17)The temperature immediately behind the shock is determined by the density of photons.In a steady state situation this is roughly determined by the photons produced within onediffusion length. The resulting value of T Comp increases very rapidly with v s . For an acceler-ating shock close to breakout, this corresponds to the number of photons produced duringone dynamical time. The velocity gradient behind the shock gives in this case also a con-tribution of photons diffusing up to the shock front from regions downstream with higherphoton density. The relative importance of this latter contribution depends on the deviationfrom LTE behind the shock, which, in turn, increases with shock velocity. As shown bySapir et al. (2013), photon diffusion causes the variation of T Comp with v s to be more mod-erate. Since the scattering optical depth at shock breakout is τ s ≈ c/v s , it is interesting tonote that T Comp /v increases only slowly with v s (see Fig. 3 in Sapir et al. 2013); hence, τ b /τ s decreases slowly with v s . Furthermore, τ b > ∼ τ s for typical supernova parameters. The valueof T Comp decreases with decreasing density, since the number of available photons increasesdue to a lower free-free absorption frequency. Only for the highest densities, correspondingto Wolf-Rayet progenitor stars, is τ b ≈ τ s .The bremsstrahlung photons, which escape before reaching Compton equilibrium, arelocated right behind the shock front and, hence, are part of the initial phase of the shock 10 –breakout flash. The fraction that is absorbed while diffusing out is approximately τ s /τ I .From equation (16) and the discussion above, for progenitor stars with a hydrogen domi-nated atmosphere this is at most 10%. The main point is that the initial phase of the flashhas a significant fraction of photons with a bremsstrahlung spectrum. Hence, at a distancewhere the recombination time is longer than the dynamical time of the viscous shock, thetemperature of the circumstellar medium is determined by ionization with a bremsstrahlungspectrum. For a power law distribution of photons with spectral index α and a photoioniza-tion cross-section varying as ν − , the average photoionization frequency is ( α + 3) / ( α + 2) ν I .The corresponding temperature for a bremsstrahlung spectrum ( α = 0) is 7 . × K. Amore accurate treatment of the frequency dependence of the photoionization cross-sectiongives a temperature somewhat lower; for example, Lundqvist (1992) finds 5 - 6 × K for aflat spectrum.
The viscous shock propagating into the circumstellar medium produces high energyradiation. This will Compton scatter and thereby heat the circumstellar medium ahead ofthe shock if α < ∼
2. The most likely emission mechanisms of this radiation is bremsstrahlungand Comptonization of the photospheric photons. Both of these are sensitive to densityand, hence, to ˙
M /v w . The resulting temperature is rather model dependent, since its valuedepends on cooling, which, in turn, is determined by the ionization. When this heating isimportant the rise in temperature ahead of the shock is likely to be most prominent at smallradii; hence, the temperature is expected to decline with radius and at some point thereshould be a transition to the constant temperature set by the radiation from the breakoutshock. This assumes a fully ionized circumstellar medium, which may not be achieved forlarge mass-loss rates (e.g., Lundqvist & Fransson 1988). The radio spectrum from SN 1993J showed clear evidence for free-free absorption duringthe first 100 days (Fransson & Bj¨ornsson 1998; P´erez-Torres et al. 2001). In Fransson & Bj¨ornsson(1998) this was modeled by parameterizing the temperature variation as an initial declinewith a later transition to a constant value. As compared to the temperature structure cal-culated in Fransson et al. (1996), the deduced parameters indicated a much slower initialdecline and a transition to the constant temperature part at a considerable later date. Therapid temperature decline in Fransson et al. (1996) results in a roughly constant free-free 11 –optical depth at a given frequency ( τ ff ( ν )) until day 15 after which the constant tempera-ture range caused it to decline approximately as t − . The free-free absorption derived inFransson & Bj¨ornsson (1998) for SN 1993J instead indicates a transition to a constant tem-perature at ≈
80 days. However, the deduced parameters are such that an approximatepower law dependence is obtained over the whole observed time interval with τ ff ( ν ) ∝∼ t − . .As mentioned above, the temperature structure is sensitive to model assumptions; hence,these discrepancies do not necessarily compromise the conclusions drawn in Fransson et al.(1996) and Fransson & Bj¨ornsson (1998). However, since the temperature of the circumstel-lar medium heated by free-free absorption declines considerably slower (see Fig. 1) than thatdeduced in Fransson et al. (1996), it is interesting to compare the time variation of τ ff ( ν )expected in the two cases. Since τ ff ( ν ) = τ ff ( ν abs )( ν abs /ν ) it is seen from equation (8) that τ ff ( ν ) ∝ t − / ν − / v − / η − / . From Fransson & Bj¨ornsson (1998) one finds ν abs ∝ t − . and v sh ≈ constant up to day 100, which leads to τ ff ( ν ) ∝ t − . η − / . Hence, neglecting forthe moment the effects of a finite initial temperature of the circumstellar medium, this isconsistent with that found in Fransson & Bj¨ornsson (1998). The implied mass-loss rate isthen significantly smaller than assumed in Fransson & Bj¨ornsson (1998); for example, usingthe same wind velocity gives a mass-loss rate roughly a factor of five lower (i.e., ˙ M − ≈ M − < ∼ τ ff ( ν ) ∝ t − ) depends on the initial temperature. Figure (1) shows that a transitionat ≈
80 days, as deduced for SN 1993J, corresponds roughly to an initial temperature of50 000 K. This is consistent with ionization and heating by a bremsstrahlung spectrum fromthe initial flash of radiation from the breakout of the supernova shock (cf. Sec. 5.1).Although cooling is only marginally important for ˙ M − ∼
1, it is seen from Figure (2)that optical depth effects are significant, which leads to somewhat lower temperatures. Aself-consistent inclusion of the free-free optical depth is therefore needed in order to derivea reliable estimate of the mass-loss rate. This is done in the calculations presented inFigure (3), where, also, a helium-to-hydrogen ratio of 0.3 (Shigeyama et al. 1994) is usedand, for completeness, cooling is included. The free-free optical depth in Figures (3b) and(3d) is now angle-averaged over the source. 12 –It is seen in Figure (3b) that the agreement between the average decline of the free-freeoptical depth in SN 1993J and the analytical approximations is fortuitous for n = 30. Thetransition to the constant temperature regime of the circumstellar medium steepens the de-cline to the extent that it no longer is consistent with that deduced by Fransson & Bj¨ornsson(1998). If this average decline reflects the true properties of the circumstellar medium, itwould imply a temperature gradient larger than expected for free-free heating alone. It couldbe caused by a small amount of additional heating corresponding to the beginning of theradio phase; for example, due to X-ray emission from the viscous shocks and/or from thebreakout shock.However, the usefulness of a detailed comparison between the deduced average timevariation of τ ff ( ν ) with that expected in a free-free heating scenario may be limited by thesensitivity to the shock velocity. Appreciable free-free absorption in SN 1993J was observedto be present mainly during the initial phase when the measured shock velocity was constant(roughly, the first 100 days). The transition to a constant temperature of the circumstellarmedium at ≈
80 days, which led to a steeper time dependence of τ ff ( ν ), was compensatedby the start of a decreasing shock velocity at 100 days. Hence, the value of the exponent inthe approximate relation τ ff ( ν ) ∝∼ t − . in Fransson & Bj¨ornsson (1998) is influenced by theshock velocity during the time when no free-free absorption could be measured.In the discussion above it has tacitly been assumed that the shock velocity also corre-sponds to the observed expansion velocity of the outer boundary of the synchrotron source.This is not necessarily the case. Relaxing this assumption gives an alternative way to recon-cile the observed time variation of τ ff ( ν ) with that expected from a free-free heating scenario.It is seen from equation (8) that a shock velocity decreasing with time results in a slowervariation of the optical depth. Observations by Brunthaler et al. (2010) and Bietenholz et al.(2011) show that the velocity of the outer boundary of the synchrotron source varies withtime roughly as t − . after 100 days, which corresponds to n = 7 in the standard self-similarmodel. Figures (3c) and (3d) show the temperature at the shock front and the optical depthfor a model with n = 7 also during the first 100 days. In this case the scaling is such thatthe radius at 100 days is the same as the value used in Fransson & Bj¨ornsson (1998). It isseen that the variation of the optical depth with time agrees well with that expected forfree-free heating; in particular, the agreement is better with an initial temperature of thecircumstellar medium corresponding to 50 000 K rather than 20 000 K. It may be noticed thatthe slopes of the various curves in Figures (3b) and (3d) are independent of the mass-lossrate. Instead they are determined by the combined time variations of the shock velocity, thesynchrotron self-absorption frequency and the transition to the initial, constant temperatureof the circumstellar medium ( T o ). Once the slope agrees with observations, the mass-lossrate is obtained from the amplitude of the free-free absorption (cf. equ. (8)). 13 –The value of the mass-loss rate deduced from Figure (3d) depends on the brightnesstemperature. The brightness temperature observed in SN 1993J was considerably lowerthan expected for a standard homogeneous source. In the model by Fransson & Bj¨ornsson(1998) this was accounted for by a combination of a magnetic field much stronger thanthe equipartition value and cooling of the relativistic electrons. Together they lowered thebrightness temperature by roughly a factor of three, with the contribution from the latterslowly decreasing with time due to the decreasing importance of cooling. If the temperatureof the circumstellar medium in SN1993J is set by free-free heating, its density is substan-tially smaller than assumed in Fransson & Bj¨ornsson (1998). As a result the low brightnesstemperature is unlikely due only to a strong magnetic field and/or cooling. A direct way oflowering the brightness temperature is by invoking inhomogeneities (e.g., Bj¨ornsson 2013).However, even for a homogeneous circumstellar medium, the heating could be patchy dueto a varying local brightness temperature. The use of an average value of the brightnesstemperature in such situations may limit the accuracy to which the free-free optical depthcan be modeled.The most accurate estimate of the mass-loss rate is, therefore, obtained by comparingthe free-free optical depths at 100 days, when the observed brightness temperature is only afactor 2 − n = 7 and n = 30) shown in Figure (3) give approximately the same result at this time. Witha standard brightness temperature somewhat lower than 10 K, the mass-loss rate shouldbe estimated using an effective brightness temperature T b , ≈ . − .
3. For a given curveof the average optical depth in Figures (3b) and (3d), the mass-loss rate and brightnesstemperature are related by ˙ M − ∝ T / , (cf. equ. (8)). Since the best fit corresponds to˙ M − ≈ . v w , = 1) for T b , = 1, correcting for the value observed in SN 1993J then yields˙ M − ≈ . − . ∼
50 000 K. The implications of this for the deducedproperties of the viscous shock will be discussed in a forthcoming paper.
6. Discussion and conclusions
When the temperature of the circumstellar medium is determined by free-free absorp-tion, observation of the free-free optical depth makes it possible to directly relate the mass-loss rate to observed quantities only. This gives a method to obtain a value of the mass-loss 14 –rate which is considerably less model dependent than those normally used. It should benoticed that the brightness temperature is often not directly observable. Although for astandard synchrotron model its value is quite insensitive even to rather large variationsof source parameters, the presence of inhomogeneities, large deviations from equipartitionbetween magnetic fields and relativistic particles and/or cooling of the latter can have non-negligible effects on the deduced mass-loss rate.The analytical expression for the mass-loss rate (equ. (8)) gives a rather good estimatefor moderate optical depths. Although the temperature immediately ahead of the shock isadequately described by the analytical expressions, increasing the optical depth causes theheating to be localized closer to the shock. This leads, on average, to a lower temperaturein the region where most of the free-free absorption occurs. Hence, using the analyticalexpression for the mass-loses rate would then give a value larger than the actual one.Since the time variation of the free-free optical depth is expected to be the best way todistinguish free-free absorption from other heating mechanisms, these analytical approxima-tions should allow to evaluate the importance of the former. They can also be used to makeconsistency checks of the deduced mass-loss rate. For large mass-loss rates several effectscan invalidate one or more of the assumptions leading up to the results in Sections 2 and 3;for example, at some density cooling will balance free-free heating and thereby reduce themaximum temperature of the circumstellar gas ahead of the shock. Likewise, recombinationmay become important at distances corresponding to the initial phases of radio emission.When this occurs, reionization by radiation with a spectrum hard enough would result in atemperature larger than that possible for free-free heating, making free-free heating negligi-ble. Also, the Comptonized radiation from behind the shock hardens with increasing densityand could provide an additional heating mechanism (Fransson et al. 1996). The mass-lossrates, where these various effects set in, are model dependent and, in particular, the shockvelocity is important.A prerequisite for free-free heating to leave a distinct mark on the free-free absorptionis an initial temperature not much larger than 10 K. As discussed above, this requires anionizing spectrum that is not too hard; for example, a black body or a Wien spectrum wouldgive too high a temperature. Hence, observations of free-free absorption can be used toconstrain the spectral distribution of the radiation in the initial flash associated with thebreakout of the supernova shock. It was argued in Section 5.2.1 that the free-free absorptionin SN 1993J could plausibly be explained by a phase where free-free heating was important.This would imply that a significant fraction of the bremsstrahlung seed photons behind thebreakout shock did not reach Compton equilibrium.The mass-loss rate of the progenitor stars to supernovae is an essential quantity not 15 –only for understanding the later evolutionary stages of massive stars but also as an inputparameter for the physics governing the conditions behind the viscous shock.The non-thermal properties of the shocked gas are an important aspect particularly ofthe forward shock. These are not well known. The reason is not only limited observations butalso that the relevant physics is only partly understood; for example, the injection problem(e.g., Blandford 1994) is still not solved and the amplification of the magnetic field behindshocks cannot yet be calculated from basic physics. The determination from observations ofthe fractions of the thermal energy input behind the shock which go into relativistic electronsand magnetic fields could give important constraints to ongoing attempts to understand bothof these processes (e.g., Caprioli & Spitkovsky 2013; Ellison et al. 2013). Although, in prin-ciple, the energy densities of relativistic electrons and magnetic fields can be obtain from ananalysis of the synchrotron emission, their fraction of the total energy density behind theshock depends on the density of the circumstellar medium, i.e., the mass-loss rate of the pro-genitor star. A related issue is the occurrence of non-linear shocks (eg., Berezhko & Ellison1999; Blasi et al 2005) in which the pressure behind the shock is dominated by relativisticparticles. Again, the efficiency of injecting particles into the acceleration process is an im-portant parameter determining the characteristics of such shocks. A reliable determinationof the mass-loss rate could then contribute to a better understanding of a wide range ofissues.In conclusion, the main points of the present paper can be summarized as follows:1) When the heating of the circumstellar medium is by free-free absorption, its temper-ature can be calculated self-consistently. As a result the density of the circumstellar mediumcan be obtained directly from the observed free-free optical depth in a rather model inde-pendent way. The maximum temperature is around 10 K for typical supernova parametersand occurs for optical depths of order unity.2) It was argued that for a hydrogen rich circumstellar medium, the temperature result-ing from ionization by the initial flash of radiation from the shock breakout is lower than10 K. Hence, for such supernovae the main heating mechanism of the circumstellar mediummay be free-free absorption.3) The observed time dependence of the free-free optical depth in SN 1993J can beaccounted for by heating due to free-free absorption. The resulting value of the mass-lossrate of the progenitor star is estimated to be (0 . − . × − M ⊙ yr − for a wind velocityof 10 km s − .P.L. acknowledges support from the Swedish Research Council. 16 – REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
18 – l og [ T e m p e r a t u r e ( K )] log [Time (days)]Temperature at shock front Analytical Red is for optically thin modelsBlue is for optically thick models
Fig. 1.— The temperature of the circumstellar medium at the shock front is shown for˙ M − /v w , = 1 and various initial temperatures as colored curves: 50 000 K (solid lines),20 000 K (long-dashed lines), and 10 000 K (short-dashed lines). The curves labelled as ”thin”correspond to the artificial case when the effects of the free-free absorption on the incidentspectrum (see text) is neglected in the heating, while for the ”thick” curves it is included(i.e., the physically relevant case). The black solid line is the analytical approximation inequation (7) for T o = 50 000 K and the black dashed line is for T o = 0 K. 19 – l og [ R a d i a l op ti ca l d e p t h ] log [Time (days)]Optical depth at n abs Analytical Red is for optically thin modelsBlue is for optically thick models
Fig. 2.— The radial free-free optical depth of the circumstellar medium is shown for˙ M − /v w , = 1 and various initial temperatures of the circumstellar medium as colored curves:50 000 K (solid lines), 20 000 K (long-dashed lines), and 10 000 K (short-dashed lines). Thecurves labelled as ”thin” correspond to the artificial case when the effects of the free-freeabsorption on the incident spectrum (see text) is neglected in the heating, while for the”thick” curves it is included (i.e., the physically relevant case). The black solid line is theanalytical approximation in equation (8) for T o = 50 000 K and the black dashed line is for T o = 0 K. 20 –Fig. 3.— The temperature of the circumstellar medium at the shock front and the averagefree-free optical depth are shown for various mass-loss rates in units of M ⊙ yr − (assuminga wind velocity of 10 km s − ) and initial temperatures of the circumstellar medium. Thetwo different models (labelled n = 7 and nn