Energy Conservation in the thin layer approximation: IV. The light curve for supernovae
EEnergy Conservation in the thin layer approximation: IV. Thelight curve for supernovae
Lorenzo Zaninetti
Physics Department, via P. Giuria 1, I-10125 Turin, ItalyEmail: [email protected]
Abstract
The light curves (LC) for Supernova (SN) can be modeled adopting the conversion ofthe flux of kinetic energy into radiation. This conversion requires an analytical or a numerical lawof motion for the expanding radius of the SN. In the framework of conservation of energy for thethin layer approximation we present a classical trajectory based on a power law profile for thedensity, a relativistic trajectory based on the Navarro–Frenk–White profile for the density, and arelativistic trajectory based on a power law behaviour for the swept mass. A detailed simulationof the LC requires the evaluation of the optical depth as a function of time. We modeled the LCof SN 1993J in different astronomical bands, the LC of GRB 050814 and the LC GRB 060729 inthe keV region. The time dependence of the magnetic field of equipartition is derived from thetheoretical formula for the luminosity.
Keywords: supernovae: general, supernovae: (individual: SN1993j), gamma-ray burst: (individual: GRB050814), gamma-ray burst: (individual: GRB 060729)
The number of observational and theoretical analyses of the light curves (LCs) for supernovae (SN)has increased in recent years. We list some of the recent treatments. The LC of the type Ia supernova2018oh has an unusual two-component shape [1], the radio LC of SN 1998bw shows a double-peak profile,possibly associated with density variations in the circumstellar medium [2], the R-band LCs of 265 SNsfrom the Palomar Transient Factory were followed and a model-independent LC template was builtfrom this data-set [3], SN 2007D (which is a luminous type Ic supernova) has a narrow LC and highpeak luminosity that were explored with a multi-band model [4], evolutionary models for the LC wereintroduced using the STELLA software application [5], the conversion of the kinetic energy of ejecta toradiation at the reverse and forward shocks was introduced in [6], the LC was modeled in the frameworkof the radioactive decay of Co, Co and Fe [7], the cosmological importance of the LC was analysedby [8], and PS15dpn is a luminous rapidly rising Type Ibn SN which was modeled in the frameworkof the circumstellar interaction (CSI) model plus Ni decay [9]. The previous papers leave a series ofquestions unanswered. – Given the observational fact that the radius–time relation in young SNRs follows a power law, is itpossible to find a theoretical law of motion in the framework of the classical energy conservation? – Can we express the flux of kinetic energy in an analytical way in a medium which is characterizedby a decreasing density? – Can we parametrize the conversion of the analytical or numerical flux of kinetic energy into theobserved luminosity? – Can we model the double-peak profile for the LC in the framework of the temporal variations of theoptical thickness? – Can we apply the classical and relativistic approaches to the LC of SNs and Gamma Ray Bursts(GRBs)? – Can we model the evolution of the magnetic field?This paper is structured as follows. In Section 2 we explore the power law fit model. Section 3 reviewsthe classical and relativistic conversion of the flux of kinetic energy into luminosity. Section 4 presents a r X i v : . [ a s t r o - ph . H E ] F e b ome analytical results for a classical law of motion, Section 5 introduces two new relativistic equationsof motion, Section 6 presents the simulation of the LC for one SN and two GRBs and Section 7 presentsthe temporal evolution of the magnetic field as well some evaluations for the accelerating clouds due tothe Fermi II acceleration mechanism. This section presents the analysed SN and GRB, introduces the adopted statistics, and reviews the powerlaw model as a useful fit for the radius–time relation in SNs.
The first SN to be analysed is SN 1993J , for which the temporary radius of expansion has been measuredfor ≈
10 yr in the radio band [10,11]. Here we processed for the case of SN 1993J the LC for the R band as reported in Figure 5 in [12], the V band for a short number of days, ≈
63 days, which shows anoscillating behaviour, see Figure 4 in [13], the luminosity of the H − α plotted with the 2.0–8.0 keV LCas reported in Figure 5 in [14] and the radio flux density at 15.2 GHz as observed by the Ryle Telescope[15] with data available at .The second object to be analysed is GRB 050814 at 0.3–10 keV, which covers the time interval[10 − −
3] days, see [16] with data available at .The third object to be analysed is GRB 060729 observed by the Ultraviolet and Optical Telescope(UVOT) in the time interval [10 − −
26] days, see Figure 1 in [17].
The adopted statistical parameters are the percent error, δ , between the theoretical value and approxi-mate value, and the merit function χ evaluated as χ = N (cid:88) i =1 (cid:104) y i,theo − y i,obs σ i (cid:105) (1)where y i,obs and σ i represent the observed value and its error at position i , y i,theo is the theoretical valueat position i and N is the number of elements of the sample. The equation for the expansion of a SN may be modeled by a power law r ( t ) = Ct α fit , (2)where r is the radius of the expansion, t is the time, and α fit is an exponent which can be foundnumerically. The velocity is v ( t ) = Ct α fit − α fit . (3)As a practical example, the radius (pc) time (yr) relation in SN 1993J is r ( t ) = 0 . × t . pc , (4)when 0 . yr < t < . yr , see also Table 1. In these subsections we analyse the classical and relativistic conversion of the flux of kinetic energy intoluminosity. The absorption of the produced radiation is parametrized by the optical thickness. .1 Conversion of energy
In the classical case, the rate of transfer of mechanical energy, L m , is L m ( t ) = 12 ρ ( t )4 πr ( t ) v ( t ) , (5)where ρ ( t ), r ( t ) and v ( t ) are the temporary density, radius and velocity of the SN. We assume that thedensity in front of the advancing expansion scales as ρ ( t ) = ρ ( r r ( t ) ) d , (6)where r is the radius at t and d is a parameter which allows matching the observations; as an example,a value of d = 3 is reported in [18]. With the above assumption, the mechanical luminosity is L m ( t ) = 12 ρ ( r r ( t ) ) d πr ( t ) v ( t ) . (7)The mechanical luminosity in the case of a power law dependence for the radius is L m ( t ) = 2 ρ r d C fit − d +5 t − − d +5) α fit π α fit . (8)The energy fraction of the mechanical luminosity deposited in the frequency ν , L ν , is assumed to beproportional to the mechanical luminosity through a constant (cid:15) ν L ν = (cid:15) ν L m . (9)The flux at frequency ν and distance D is F ν = (cid:15) ν L m πD . (10)For practical purposes, we impose a match between the observed luminosity, L obs , and the theoreticalluminosity, L m , L obs = C obs L m , (11)where C obs is a constant which equalizes the observed and the theoretical luminosity and varies on thebase of the selected astronomical band. In a analogous way, the observed absolute magnitude is M obs = − log ( L m ) + k obs , (12)where k obs is a constant. In the relativistic case the rate of transfer of mechanical energy, L m,r , assumingthe same scaling for the density in the advancing layer, is L m,r ( t ) = 4 π r ( t ) ρ c β ( t )1 − β ( t ) (cid:16) r r (cid:17) d , (13)where β ( t ) = v ( t ) c , for more details, see [19].A useful formula is that for the minimum magnetic field density, B min , B min = 1 . η L ν V ) / ν / T , (14)where ν is the considered frequency of synchrotron emission, L ν is the luminosity of the radio sourceat ν , V is the volume involved, and η = (cid:15) total (cid:15) e is a constant which connects the relativistic energy ofthe electrons, (cid:15) e , with the total energy in non-thermal phenomena, (cid:15) total , see formula (16.50) in [20] orformula (7.14) in [21]. .2 Absorption The presence of the absorption can be parametrized introducing a slab of optical thickness τ ν . Theemergent intensity I ν after the entire slab is I ν = (cid:90) τ ν S ν e − t dt , (15)where S ν is a uniform source function. Integration gives I ν = S ν (1 − e − τ ν ) , (16)see formula 1.30 in [22]. In the case of an optically thin medium, τ ν = ∞ , the observed luminosity canbe derived with Equation (11), but otherwise, the following equation should be used: L obs = C obs L m (1 − e − τ ν ) , (17)where τ ν is a function of time. For the case of the apparent magnitude, we have m obs = − log ( L m ) − log (1 − e − τ ν ) + k obs . (18)The value of τ ν can be derived with the following equation: τ ν = − ln (cid:16) − e − ( m obs − m theo ) ln(10) (cid:17) (19)where m theo and m obs represent the theoretical and the observed apparent magnitude. Due to the com-plexity of the time dependence of τ ν , a polynomial approximation of degree M is used: τ ν ( t ) = a + a t + a t + · · · + a M t M , (20)with more details in [23]. In some cases we apply the logarithms to the pair of data, i.e. log ( x i ) andlog y i ); we call this the logarithmic polynomial approximation .The absorption in the relativistic case is assumed to be the same once the classical luminosity, L m ,is replaced by the relativistic luminosity L m,r L obs = C obs L m,r (1 − e − τ ν ) , (21)and m obs = − log ( L m,r ) − log (1 − e − τ ν ) + k obs . (22) Let us analyse the case of conservation of energy in the thin layer approximation in the presence of apower law profile of density of the type ρ ( r ; r ) = { ρ c if r ≤ r ρ c ( r r ) α if r > r . , (23)where ρ c is the density at r = 0, r is the radius after which the density starts to decrease and α > r ( t ) = 12 ( α − − r α − α − × (cid:16) − r v ( α −
5) ( t − t ) √ − α − ( α −
3) ( α − ( t − t ) v + 12 r (cid:17) − ( α − − , (24)and the asymptotic velocity v ( t ) = 2 (cid:16) − r v ( α −
5) ( t − t ) √ − α − ( α −
3) ( α − ( t − t ) v + 12 r (cid:17) − αα − × (cid:16) r α − α − √ − α + v r α − α − ( α −
3) ( α −
5) ( t − t ) (cid:17) ( α − − v . (25) able 1. Numerical values of the parameters for the fit and the theoretical models applied to SN 1993J . model values χ F it by a power law α fit = 0 . C = 0 . Classic power law profile α = 2 . r = 1 . − pc; 176 . t = 5 10 − yr ; v = 20000 kms Relativistic NF W b = 0 . r = 1 10 − pc; 823 t = 3 . − yr ; v = 269813 kms Relativistic NCD δ = 1 . r = 5 10 − pc; 9589 t = 1 . − yr ; v = 269813 kms Figure 1.
Theoretical radius as given by Eq. (24), v = 4000 kms , t = 10 yr and t = 5 10 yr . The model is theconservation of the classical energy in the presence of an inverse power law profile for the density. n example of trajectory is reported in Figure 1 with data as in Table 1.As a consequence, we may derive an expression for the theoretical luminosity in presence of an inversepower law profile, L theo , based on Equations (7) and (11) L theo = ρ r − d +5 α − α − v − dα − × (cid:16) − r v ( α −
5) ( t − t ) √ − α − ( α −
3) ( α − ( t − t ) v + 12 r (cid:17) d +10 − αα − π × (cid:18) r √ − α + v ( α −
3) ( α −
5) ( t − t )2 (cid:19) . (26)The above luminosity is based on theoretical arguments and no fitting procedure was used. The observedluminosity, L obs , can be obtained introducing L obs = C obs × L theo , (27)where C obs is a constant. Similarly, M obs = − log ( L theo ) + k obs . (28) The relativistic conservation of kinetic energy in the thin layer approximation as derived in [25] is M ( r ) c ( γ −
1) = M ( r ) c ( γ − , (29)where M ( r ) and M ( r ) are the swept masses at the two radii r and r respectively, γ = √ − β and β = v c . We assume that the medium around the SN scales as the Navarro–Frenk–White (NFW) profile: ρ ( r ; r , b ) = (cid:40) ρ c if r ≤ r ρ c r ( b + r ) r ( b + r ) if r > r , (30)where ρ c is the density at r = 0 and r is the radius after which the density starts to decrease, see [26].The total mass swept, M ( r ; r , b, ρ c ), in the interval [0,r] is M ( r ; r , ρ c , b ) = 4 ρ c π r ρ c ( b + r ) (( b + r ) ln ( b + r ) + b ) r πb + r − ρ c ( b + r ) (( b + r ) ln ( b + r ) + b ) r π . (31)Inserting the above mass in equation (29) makes it possible to derive the velocity of the trajectory as afunction of the radius as well as the differential equation of the first order which regulates the motion.The differential equation has a complicated behaviour which is not presented and Figure 2 displaysthe numerical solution. Conversely, we present an approximate solution as a third-order Taylor seriesexpansion about r = r r ( t ; r , v , t , b ) = 12 r c (cid:32) t − t ) ( − v + c ) ( v + c ) (cid:112) ( c − v ) − − c (cid:16) ( t − t ) c − t v + (2 tv + 2 / r v ) t − t v − / tr v − / r (cid:17)(cid:33) . (32)Figure 3 presents the Taylor approximation of the trajectory in the restricted range of time [4 10 − yr − − yr ]. igure 2. Numerical radius for the NFW profile (full line), with data as in Table 1. The model is the conservationof the relativistic energy in the presence of an NFW profile for the density.
Figure 3.
Numerical solution (full red line) and Taylor approximation (blue dashed line) for the NFW profilewith parameters as in Table 1. The model is the conservation of the relativistic energy in the presence of an NFWprofile for the density. .2 NCD case
We assume that the swept mass scales as M ( r ; r , δ ) = (cid:40) M if r ≤ r M ( rr ) δ if r > r , (33)where M is the swept mass at r = 0, r is the radius after which the swept mass starts to increase and δ is a regulating parameter less than 3. The differential equation of the first order which regulates themotion is obtained inserting the above M ( r ) in equation (29) dr ( t ; r , v , c, δ ) dt = ANAD , (34)where AN = (cid:32)
16 ( c − v ) c ( r − δ ( − / c + 5 / v )( r ( t )) δ + r − δ (1 / c − / v )( r ( t )) δ +( r ( t )) δ ( c − / v ) r − δ − / c + 1 / v )( c + v ) (cid:112) ( c − v ) − +(10 c − c v + 5 v ) r − δ ( r ( t )) δ − r − δ ( c − v ) ( c + v ) ( r ( t )) δ +( − c + 20 c v − v )( r ( t )) δ r − δ + 8 c − c v + v (cid:33) / c , (35)and AD = 2 c ( c − v )( c + v )( r − δ ( r ( t )) δ − (cid:112) ( c − v ) − + r − δ ( c − v )( r ( t )) δ + ( − c + 2 v )( r ( t )) δ r − δ + 2 c − v . (36)The above differential does not have an analytical solution and therefore the solution should be derivednumerically except about r = r where a third-order Taylor series expansion gives r ( t ; r , v , t , δ ) = r + v ( t − t )+ δ ( c − v ) ( c + v ) ( t − t ) cr (cid:16) c − c (cid:112) c − v − v (cid:17) √ c − v . (37)Figure 4 presents the numerical solution and Figure 5 the Taylor approximation of the trajectory. We introduce one SN and two GRBs which were processed.
In this subsection we adopt a classical equation of motion with a power law profile of density, see Section4. Figure 6 presents the decay of the R magnitude of SN 1993J , which is type IIb, as well our theoreticalcurve.We present the H − α with soft and hard band X-ray luminosities as well the theoretical luminosityin Figure 7. Figure 8 presents the radio flux density of SN 1993J at 15.2 GHz observed by the RyleTelescope as well the theoretical flux, which requires a time dependent evaluation of the optical depth τ ν , see Figure 9.Figure 10 presents the V-magnitude of SN 1993J for few days as well the theoretical magnitude andthe time evolution of the optical depth τ ν , see Figure 11. igure 4. Numerical solution of the differential equation (34) for the NCD case (full line), with data as in Table 1.The astronomical data of SN 1993J are represented with vertical error bars. The model is the conservation ofthe relativistic energy in the NCD case.
Figure 5.
Numerical solution (full red line) and Taylor approximation (blue dashed line) for the NCD case withparameters as in Table 1. The model is the conservation of the relativistic energy in the NCD case. igure 6.
The R LC of SN 1993J over 10 yr (empty stars) and theoretical curve in the classical framework ofa power law profile for the density as given by eq. (27) (full line). Parameters of the trajectory as in Table 1, d = 6, k obs = -11.5 and ρ = 1. The data were extracted by the author from Figure 5 in Zhang et al. (2004). Figure 7.
The H − α and the 2.0–8.0 keV luminosities of SN 1993J over 10 yr (empty stars) and theoreticalcurve in the classical framework of a power law profile for the density as given by eq. (28) (full line). Parametersof the trajectory as in Table 1, d = 2.5, C obs = 1 . and ρ = 1. The data were extracted by the author fromFigure 5 in [14]. igure 8. The radio flux density of SN 1993J over 443 days (empty stars) and theoretical behaviour in theclassical framework of a power law profile for the density evaluated with formula (17) (full line). Parameters ofthe trajectory as in Table 1 , d = 2.5, C obs = 8 .
45 10 and ρ = 1. Figure 9.
The time dependence of τ ν (empty stars) and a polynomial approximation of degree 6 (full line).Parameters as in Figure 8. igure 10. The V LC of SN 1993J over 63 days (empty stars) and theoretical curve in the classical frameworkof a power law profile for the density as given by eq. (18) (full line). Parameters of the trajectory as in Table 1, d = 6, k obs = -12.5 and ρ = 1. The data were extracted by the author from Figure 4 in [13]. Figure 11.
The time dependence of τ ν (empty stars) and a polynomial approximation of degree 10 (full line).Parameters as in Figure 10. igure 12. The XRT flux of GRB 050814 at 0.2–10 keV (empty stars) and theoretical curve with velocityand radius as given by the NFW relativistic numerical model. The theoretical luminosity, which is corrected forabsorption, is given by eq. (13) (full line).
Figure 13.
The time dependence of τ ν (empty stars) for GRB 050814 and a logarithmic polynomial approxima-tion of degree 7 (full line). Parameters as in Figure 12. .2 The case of GRB 050814 In this subsection we adopt a relativistic equation of motion with an NFW profile for the density, seeSection 5.1. Figure 12 presents the XRT flux of GRB 050814 and Figure 13 presents the temporalbehaviour of the optical depth.
In this subsection we adopt a relativistic equation of motion for the NCD case, see Section 5.2. Figure 14presents the LC of UVOT (U) apparent magnitude for GRB 060729 and Figure 15 presents the temporalbehaviour of the optical depth.
Figure 14.
The LC of UVOT (U) + HST (F330W) for GRB 060729 (empty stars) and theoretical curve withradius as given by the NCD relativistic numerical model with data as in Table 1. The theoretical luminosity isgiven by eq. (13) (full line).
Figure 15.
The time dependence of τ ν (empty stars) for GRB 060729 and a logarithmic polynomial approxi-mation of degree 10 (full line). Parameters as in Figure 12. Acceleration and magnetic field
The flux at frequency, S ν , in the radio band for SNs is parametrized by S ν = C ν ν − α r , (38)where α r is the observed spectral index and C ν is a constant. As a consequence, the luminosity, L ν , is L ν = 4 πD S ν , (39)where D is the distance. We now explain how it is possible to derive the magnetic field from the luminosity.The magnetic field for which the total energy of a radio source has a minimum is H min = 1 . c / L / (1 . k ) / Φ / R / gauss , (40)where c = 2 √ c ( − α r ) (cid:0) √ ν ν α r − ν α r √ ν (cid:1) c ( ν ν α r − ν ν α r ) ( − α r ) , (41)where α r is the spectral index, c and c are two constants, ν and ν are the lower and upper frequencyof synchrotron emission, L is the luminosity in erg s − , k is the ratio between energy in heavy particleand electron energy, Φ is the fraction of source’s volume occupied by the relativistic electrons and themagnetic field, and R is the radius of the source; for more details see formula (7.14) in [21] or formula(5.109) in [27]. The constant c is numerically evaluated in Table 8 of [21] and an example is presentedin Figure 16. The scaling of the magnetic of equipartition as given by equation (40) is Figure 16.
The constant c as a function of the spectral index α r when ν = 10 Hz , ν = 10 Hz (red fullline) and ν = 10 Hz (blue dashed line). H min ∝ L / R / . (42)The first example presents the temporal behaviour of H min for GRB 050814 in which we inserted thetheoretical luminosity corrected for absorption, see Figure 17.The second example presents the temporal behaviour of H min for SN 1993J in which we inserted thetheoretical luminosity as given by the power law fit, see Figure 18. igure 17. The time dependence (seconds) of the minimum magnetic field for GRB 050814 with theoreticalluminosity as in Figure 13 when H min = 1 gauss at t = t . The model is a fit to a power law. Figure 18.
The time dependence (years) of the minimum magnetic field for SN 1993J with theoretical luminosityas given by formula (7) when H min = 1 gauss at t = t . The model is a fit to a power law. n electron which loses its energy due to the synchrotron radiation has a lifetime of τ r ≈ EP r ≈ E − H − sec , (43)where E is the energy in ergs, H the magnetic field in Gauss, and P r is the total radiated power, see Eq.1.157 in [28]. The energy is connected to the critical frequency, see Eq. 1.154 in [28], by ν c = 6 . × HE Hz . (44)The lifetime for synchrotron losses is τ syn = 39660 1 H √ Hν yr . (45)Following [29,30], the gain in energy in a continuous form is proportional to its energy, E , dEdt = Eτ II , (46)where τ II is the typical time scale, 1 τ II = 43 ( u c )( cL II ) , (47)where u is the velocity of the accelerating cloud belonging to the advancing shell of the GRB, c is thespeed of light and L II is the mean free path between clouds, see Eq. 4.439 in [28]. The mean free pathbetween the accelerating clouds in the Fermi II mechanism can be found from the following inequalityin time: τ II < τ sync , (48)which corresponds to the following inequality for the mean free path between scatterers L < . u H / √ νc pc . (49)The mean free path length for a GRB which emits synchrotron emission around 1 keV ( 2 .
417 10 Hz )is
L < . − β H / (cid:112) E ( kev ) pc (50)where β is the velocity of the cloud divided by the speed of light. When this inequality is fulfilled, thedirect conversion of the rate of kinetic energy into radiation can be adopted. Figure 19 presents the aboveline in the framework of the fitted model. The mean free path length varies from 1 . − to 1 . − with respect to the numerical value of the advancing radius. Classical and relativistic flux of energy:
The classical flux of kinetic energy has an analytical expression in the case of energy conservation in thepresence of a power law profile for the density, see equation (26). The relativistic flux of energy in thetwo cases here analysed can only be found numerically.
Momentum versus energy:
The comparison of the trajectories for SN 1993J for the four possibilities, classic or relativistic, conser-vation of energy or momentum, is presented in Table 2.The best results are obtained for the energy conservation in the presence of a power law profile inthe present paper, see equation (24).
Light curve:
The luminosity in the various astronomical bands is here assumed to be proportional to the classical or igure 19.
The time dependence (years) of the mean free path length for SN 1993J when E = 1 keV. Themodel is a fit to a power law. Table 2.
Type of regime, conservation, model, χ and reference for SN 1993Jregime conservation model χ Referenceclassical momentum inverse power law 276 Figure 6 in [31]classical momentum Plummer profile 265 Figure 8 in [32]relativistic momentum Lane–Emden profile 471 Figure 10 in [32]relativistic energy power law profile 6387 Figure 4 in [25]relativistic energy exponential profile 13145 Figure 6 in [25]relativistic energy Emden profile 8888 Figure 8 in [25]classical energy power law profile 176.6 Figure 1 in this paperrelativistic energy NFW profile 823 Figure 2 in this paperrelativistic energy NCD 9589 Figure 4 in this paper elativistic flux of mechanical kinetic energy. This theoretical dependence is not enough and the conceptof optical depth should be introduced. Due to the complexity of the time dependence of the opticaldepth, a polynomial approximation of degree M with time as independent variable has been suggested,see equation (20) which is used in a linear or logarithmic form. Comparison with astronomical data:
The framework of conversion of the classical flux of mechanical kinetic energy into the various astro-nomical bands coupled with a time dependence for the optical depth allowed simulating the variousmorphologies of the LC of supernovae. In particular, in the case of SN 1993J we modeled: (i) the R LCof SN 1993J over 10 yr, see Figure 6, (ii) the H − α and the 2.0–8.0 keV luminosities over 10 yr, seeFigure 7, (iii) the radio flux density over 443 days, see Figure 8 and (iv) V LC over 63 days, see thedouble peak visible in Figure 10. The LC of of GRB 050814 at 0.2–10 keV was modeled in Figure 12 andthat of UVOT (U) + HST (F330W) for GRB 060729 was modeled in Figure 14.
Magnetic field
The minimum magnetic field depends on the luminosity and this allows to derive its theoretical depen-dence on time, see Figure 18. The above dependence allows deriving the distance for the mean free pathbetween accelerating clouds for the Fermi II mechanism when the relativistic electron emits synchrotronradiation in the keV region, see Figure 19.
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