Relativistic scaling laws for the light curve in supernovae

In order to explain light curve (LC) for Supernova (SN) we derive a classical formula for the conversion of the flux of kinetic energy into radiation. We then introduce a correction for the absorption adopting an optical depth as function of the time. The developed framework allows to fit the LC of type Ia SN 2005cf ( B and V ) and type IIp SN 2004A (B,V,I and R ). A relativistic formula for the flux of kinetic energy is also derived in terms of a Taylor expansion and the application is done to the LC of GRB 050814. The decay of the radioactive isotopes as a driver the LC for SNs is also reviewed and a new formulation is introduced. The Arnett's formula for bolometric luminosity is corrected for the optical depth and applied to SN 2001ay.


Introduction
The term light curve (LC) for Supernova (SN) usually denotes the behavior of the apparent/absolute visual magnitude as function of the time. The development of the multiwavelength astronomy fixes the wavelength passband, i.e BVRI, or the frequency ν, i.e. 15.2 GHz, or the energy, i.e. 1kev. Further on in gamma, X and radio astronomies the flux or the count rate are recorded rather than the magnitude, see as an example Fong et al. (2012). The first model to be considered is connected with the radioactive decay where L and L λ,0 are the luminosities at time t and at t = 0 respectively, λ is the considered wavelength and τ n is the typical lifetime, see Bowers and Deeming (1984). On introducing the apparent magnitude m λ , the previous formula becomes m λ = k ′ λ + 1.0857( where k ′ λ is a constant. The most important radioactive isotopes are 56 Ni with τ n = 8.757 d and 56 Co with τ n = 111.47 d. The analysis of many authors has shown that the decay of one of these two radioactive isotopes fit only few days of a typical LC, see Smith and McCray (2007). At the same time the spectral index in the radio of SN 1993J is constant after 700 days, see Figure 8 in Martí-Vidal et al. (2011) and this observational fact points toward the presence of synchrotron emission having flux , F (ν) ∝ ν −α with α ≈ 0.7. The hypothesis of the synchrotron emission in SNs is not new and we now report some applications among others: GRBs, see Preece et al. (2002); Beniamini and Piran (2013); Burgess et al. (2014) and Supernovae Remnants (SNRs), see Katsuda et al. (2010); Miceli et al. (2013). The presence of the synchrotron emission makes attractive the analysis of a turbulent cascade from the large scale to the small scales where presumably the relativistic electrons are accelerated. Insofar we have isolated two completely different physical mechanism for the source of radiation in the LC: (i) the number of radioactive isotopes as function of the time, (ii) the flux of mechanical kinetic energy which is the driver for the power injected in the turbulent cascade. The fact that the spectral index in the optical regime varies considerably with the time points toward a variable optical thickness as function of the time and of the considered passband. The basic idea is that the optical thickness is low at the beginning of the LC and it increases its value with time. A series of questions can now be posed • Can we build a formula for the flux/magnitude versus time relationship in the framework of conversion of the mechanical luminosity into radiation?
• Can we introduce the correction for variable optical thickness introducing a dependence of the optical thickness with time?
• Can the new developed framework be applied to the various LCs which arise from the various typologies of LCs such as type Ia, Ib, Ic or type IIb, II-L, II-p, IIn?
• Can the radioactive model in it's various versions model the most common LCs?

Preliminaries
Here we first introduce an elementary equation of motion and then we assume a linear relationship between the mechanical and the observed luminosity at a given frequency ν.

The simplest equation of motion
The equation of the expansion of a SN in the first ten years can be modeled by a power law of the type where R is the radius of the expansion, t is the time, R 0 is the radius at t = t 0 and α is an exponent which can be found from a numerical analysis. The velocity is As an example in the case of SN 1993J we have α = 0.828. The rate of transfer of mechanical energy, L m is We now assume that the density in front of the advancing expansion scale as where d is a parameter which allows to match the observations; this assumption is not new and as an example Nagy et al. (2014) quotes d=3. The mechanical luminosity for the power law dependence of the radius becomes where L 0 is the luminosity at t = t 0 .

The emitted radiation
The energy fraction of the mechanical luminosity, L ν , deposited in the frequency ν is assumed to be proportional to the mechanical luminosity through a constant ǫ ν L ν = ǫ ν L m .
The flux at frequency ν and distance D is The problem of the absorption can be parametrized introducing a slab of optical thickness τ nu . The emergent intensity I ν after the entire slab is where S ν is a uniform source function. The integration gives see formula 1.30 in Rybicki and Lightman (1991). A first model for the optical thickness assumes a power law dependence where a 1,ν and a 2,ν are two coefficients which can be found from the astronomical data. The flux corrected for absorption in the power law case, F ν , is An expression for the flux of the first model can be obtained inserting the simplest equation of motion for the R − t dependence in the mechanical luminosity as given by (7) F where F ν,0 is the flux a t = t 0 . This formula is useful when we have the flux versus the time, as an example Jansky versus JD. The absolute/apparent magnitude version of for the first model is where m k is a constant of calibration. This formula is useful when we have the absolute/apparent magnitude versus time as in the case of optical LC in SN. The asymptotic approximation is A second model for the optical thickness assumes an exponential law dependence where a 1,ν , a 2,ν and a 3,ν are three coefficients which can be found from the astronomical data. The absolute/apparent magnitude version for the second model is 2.5 ln (t) αd − 12.5α ln (t) + 7.5 ln (t) − 2.5 ln 1 − e −a1,ν 1−e −a 2,ν t a 3 ,ν ln (2) + ln (5) (18) +m k .
A third model for the optical thickness makes a comparison between the observed intensity I obs and the theoretical intensity I th through the optical depth The optical depth is The observed intensity as function of the time is an astronomical quantity and the theoretical intensity can be the mechanical luminosity or the momentary number of radioactive isotopes. Once the temporal behavior of τ ν is derived we search for the best fit as function of time. A fit already used is the power law fit as represented by eqn. (12). Another type of fit is the logarithmic polynomial approximation of degree M, The presence of the logarithm allows to cover the oscillatory behavior of τ ν over many decades in time.

Astrophysical results
The time is usually expressed in JD or seconds and a subtraction of the initial JD or seconds relative to the considered phenomena should be done in order to have zero at the beginning of the temporal scale. We start by analyzing SN 2005cf in NGC 5812 which is of type Ia, it's distance is 29.4 Mpc and the distance moduli µ = 32.51, see Pastorello et al. (2007). Figure 1 reports the temporal evolution of the V visual magnitude of SN 2005cf for the power law model as well the interpolating curve and Figure 2 the asymptotic approximation; data as in Table 1. The quality of the fits is measured by the merit function χ 2 where m th and m obs are the theoretical and observed magnitude, respectively. The (B-V) color evolution of SN 2005cf for the exponential law model (data as in Table 2) is reported in Figure 3.
The SNs of type IIp are characterized by a flat LC for a long period of time, i.e. 100 days. We therefore analyzed SN 2004A in NGC 6207 , which is Table 1: Numerical values of the adopted parameters and χ 2 for the optical LC in SN in the case of optical thickness with a power law dependence, α = 0.828 everywhere.   Hendry et al. (2006);Tsvetkov (2008). Figure 4 reports the temporal evolution of the V visual magnitude of SN 2004A for the exponential law model as well the interpolating curve, data as in Table 2. We now apply the developed theory to model the radio flux density of SN 1993J at 15.2 GHz, see Pooley and Green (1993); Ho et al. (1999), with data available at http://www.mrao.cam.ac.uk/∼dag/sn1993j.html . In this radio-case we plot the flux version of the first model as given by eqn. (14), see Figure 5 and Table  3. Table 3: Numerical values of the adopted parameters and χ 2 for the radio LC in SN 1993J and gamma LC in GRB 050814in the case of optical thickness with a power law dependence, α = 0.828 everywhere. The theory is now applied to GRB 050814at 0.3-10 kev in the time interval 10 −5 − 3 days, see Jakobsson et al. (2006) with data available at    http://www.swift.ac.uk/xrt curves/00150314/. Figure 6 reports the LC, in this case the flux, as function of the elapsed time since Burst Alert Telescope (BAT) trigger and Table 3 reports the involved parameters.

Relativistic model
The density, ρ, of the ISM at a distance r from the SN is here modeled by a Lane-Emden (n = 5) profile where b represents the scale. The relativistic conservation of momentum for the thin layer approximation in presence of a the Lane-Emden (n = 5) profile is given by the following differential equation where r 0 is the initial radius of the advancing sphere, v 0 is the initial velocity at r 0 , c is the light velocity and β 0 = v0 c . The relativistic transfer of energy through a surface, A, is where p is the pressure, for sake of simplicity we take p=0, and the Lorentz factor γ is  Table 4.
We now assume the following power law behavior for the density in the advancing thin layer and we obtain We can now derive L m,r in two ways: (i) from a numerical evaluation of r(t) and v(t), (ii) from a Taylor series of L m,r (t) of the type L m,r (t) = 3 n=0 a n (t − t 0 ) n .
The coefficients are . Figure 7 compares the numerical solution for the luminosity and the series expansion for the luminosity about the ordinary point t 0 . The flux at frequency ν and distance D is The flux corrected for absorption in the relativistic case is

The radioactive model
Here we consider the decay of a radioactive isotope, a radioactive chain and the Arnett's rule for the bolometric luminosity.

Decay of one element
The decay of a radioactive isotope is modeled by the following equation where τ n is a constant and the negative sign indicates that dN is a reduction in the number of nuclei , see Yang and Hamilton (2010). The integration of this differential equation of the first order in which the variables can be separated gives: where N 0 is the number of nuclei at t = 0. The half life is T 1/2 = ln(2) τ n . The absolute magnitude version of the previous formula is where M is the absolute luminosity, C and k are two constants. This means that we are waiting for a straight line for the absolute magnitude versus time relationship. At the same time the observational fact that the spectral index in the radio varies considerably but becomes constant, β ≈ −0.7, after ≈ 700 days, see Figure 8 in Martí-Vidal et al. (2011), asks the absorption.

Radioactive chains
The isotope 56 Ni is unstable and decays (τ 1 = 8.757 d, T 1/2 =6.07 d) into 56 Co emitting gamma photons. The isotope 56 Co is unstable and decays (τ 2 = 111.47 d, T 1/2 = 77.27 d) into 56 Fe through electron capture and β-decay. The decay rates of of the two species, species 1 is 56 Ni and species 2 is 56 Co, is modeled by the following equations The two solutions obtained inserting as initial conditions N 1 (0) = N 0,1 and N 2 (0) = 0 are The sum of the two species, N(t), is according to formula (8.5) in Rust et al. (2010) where C 1 and C 2 are two adjustable parameters. This linear sum is associated with the LC in SNs assuming that the γ-rays are thermalized in the ejecta and emerge in the various bands. The logarithmic form,M(t) , is associated with the magnitude evolution where where k is a constant and M d = ln(2) + ln(5) .
We plot the decay of the LC of SN 2001el , which is of type Ia, adopting a distance modulus of 31.65 mag, see Krisciunas et al. (2003), the nuclear decay which according to equation (35) is a straight line, and the theoretical curve of the two species as represented by equation 39, see Figure 9.

Bolometric luminosity
The bolometric LC after Arnett (1982); Arnett et al. (1985) has been associated with the combined radioactive decays of the isotopes 56 Ni and 56 Co. A formula of practical use is given by L(t R ) = α (6.45 × 10 43 )e −tR/8.8 + (1.45 × 10 43 )e −tR/111.3 erg s , where t R is the elapsed time from the explosion to the maximum of the LC and α is ≈ 1, see formula (2) in Krisciunas et al. (2011). The previous formula represents the optically thin case. According to the comparison method developed in Section 2.2 a corrected bolometric luminosity for the absorption, L c (t R ), is and Figure 10 reports the comparison between observed and theoretical bolometric luminosity.

Conclusions
Classical and relativistic flux of energy: The classical flux of kinetic energy can be easily parametrized in the case of a radius-time relationship represented by a power law, see eqn. (7). Conversely is more complex to derive the relativistic flux of kinetic energy which requires a relativistic law of motion. In the framework of Lane-Emden (n = 5) profile as given by eqn. (22) and momentum conservation in a thin layer we can deduce an analytical solution for the relativistic flux of energy in terms of a Taylor series, see the four coefficients in (30). Light curve: Assuming a linear relationship between the luminosity in the various astronomical bands and the classical or relativistic flux of mechanical kinetic energy we can easily deduce a theoretical time dependence for the LC, Nuclear Decay: The LC of a SN is often model by the decay of the radioactive isotope 56 Ni, but in order to follow the LC with time we should introduce a radioactive chain, see eqn. (38). Further on some classical approach to the bolometric luminosity must be corrected for the optical depth, see eqn.(43).
Comparison with astronomical data: The framework of conversion of the classical flux of mechanical kinetic energy into the various optical bands coupled with a time dependence for the optical depth allows to simulate the various morphologies of the LC: for a type Ia we chosen SN 2005cf, see Figure 1 for V band and Figure 3 for (B-V) color. The enigmatic behavior of type IIp SNs, here represented by SN 2004A, can also be modeled, see Figure 4 for the V band and Table 1 for B, I and R bands. The opposite sides of the electro-magnetic spectrum can also be simulated: for the radio band of SN 1993J see Figure 5 and for the gamma/X spectrum of GRB 050814 see Figure 6. More complex is the derivation of the relativistic flux of energy here parametrized by a series expansion. The coupling of the previous series with a logarithmic polynomial approximation allows to model fine details such as the oscillation in LC visible at ≈ 1000 s in GRB 050814, see Figure 8. All the fits here presented report the χ 2 , see Tables 1 and 2. This means that other types of functions for the optical depth versus time have a reference for comparison.