A CLASSICAL AND A RELATIVISTIC LAW OF MOTION FOR SPHERICAL SUPERNOVAE

The absorption features of supernovae (SN) allow the determination of their expansion velocity, v. We select, among others, some results. The spectropolarimetry (CA ii IR triplet) of SN 2001el gives a maximum velocity of ≈26, 000 km s⁻¹; see Wang et al. (2003). The same triplet, when searched in seven SNe of type Ia, gives $10{,}400 {{\,\rm km\,s}^{-1}}\le v \le 17{,}700 {{\,\rm km\,s}^{-1}}$; see Table 1 in Mazzali et al. (2005). A time series of eight spectra in SN  2009ig allows us to assert that the velocity at the CA ii line, for example, decreases in 12 days from 32,000 km s⁻¹ to 21,500 km s⁻¹; see Figure 9 in Marion et al. (2013). A recent analysis of 58 type Ian SNe gives $9660 {{\,\rm km\,s}^{-1}}\le v \le 14{,}820 {{\,\rm km\,s}^{-1}}$ in Si ii; see Table 2 in Childress et al. (2014). The previous analysis allows us to say that the maximum velocity observed so far for SNe is (v/c) ≈ 0.1, where c is the speed of light; this observational fact points to a relativistic equation of motion.

We now briefly review the shocks and the Kompaneyets approximation in special relativity (SR). A similar solution for strong relativistic shocks in a circumstellar medium (CSM) which varies with the radius was found by Blandford & McKee (1976). Relativistic shocks are commonly used for gamma-ray bursts (GRB) in order to explain the production of non-thermal electrons; see Baring (2011). The interactions between the shock and the ambient density fluctuations can produce turbulence with a significant component of magnetic energy; see Inoue et al. (2011). Relativistic radiation-mediated shocks can produce GRBs with typical parameters similar to those observed; see Nakar & Sari (2012). Trans-relativistic shocks have been used to produce high-energy neutrinos and gamma-rays in SNe; see Kashiyama et al. (2013). The Kompaneyets approximation is usually developed in a Newtonian framework (see Kompaneyets 1960; Olano 2009) and has been derived in SR by Shapiro 1979; Lyutikov 2012. The temporal observations of SNe such as SN 1993J establish a clear relation between the instantaneous radius of expansion r and the time t: r∝t^0.82 (see Marcaide et al. 2009) and therefore allows the variants of the thin layer approximation to be explored. The previous observational facts excludes SN propagation in a CSM with constant density: two solutions of this type are the Sedov solution, which scales as r∝t^0.4 (see Sedov 1959; McCray 1987), and the momentum conservation in a thin layer approximation, which scales as r∝t^0.25 (see Dyson & Williams 1997; Padmanabhan 2001). Previous efforts to model these observations in the framework of the thin layer approximation in a CSM governed by a power law (see Zaninetti 2011) or in the framework in which the CSM has a constant density but swept mass regulated by a parameter called porosity (see Zaninetti 2012) have been successfully explored. An important feature of the various models is based on the type of CSM that surrounds the expansion. As an example in the framework of classical shocks, Chevalier (1982a 1982b) analyzed self-similar solutions with a CSM of type r^-s, which means an inverse power-law dependence. In the framework of Kompaneyets' equation (see Kompaneyets 1960), for the motion of a shock wave in different plane-parallel stratified media, Olano (2009) considered four types of CSM. It is therefore interesting to take into account a self-gravitating CSM, which gives a physical basis to the considered model. The relativistic treatment has concentrated on the determination of the Lorentz factor, γ, for the ejecta in GRB; we report some research in this regard: Granot & Kumar (2006) found 30 < γ < 50 for a significant number of GRBs; Pe'er et al. (2007) found γ = 305 for GRB 970828 and γ = 384 for GRB 990510; Zou & Piran (2010) found high values for the sample of the GRBs considered 30.5γ < 900; Aoi et al. (2010), in the framework of a high-energy spectral cutoff originating from the creation of electron–positron pairs, found γ ≈ 600 for GRB 080916C; and Muccino et al. (2013) found γ ≈ 6.7 × 10² for GRB 090510. The last phase of the stellar evolution predicts the production of ⁵⁶Ni (see Truran et al. 1967; Bodansky et al. 1968; Matz & Share 1990; Truran et al. 2012) and therefore this type of decay has been used to model the light curve of SNe (see among others Mazzali et al. 1997; Elmhamdi et al. 2003; Stritzinger et al. 2006; Magkotsios et al. 2010; Krisciunas et al. 2011; Okita et al. 2012; Chen et al. 2013) as well the reddening measurements of the supernova remnant (SNR) Cassiopeia A (see Eriksen et al. 2009). These theoretical and observational efforts are interested in exploring the modification of ⁵⁶Ni decay due to time dilation.

In this paper, we review the standard two-phase model for the expansion of an SN (Section 2), and three density profiles (Section 3). In Section 4, we derive the differential equations that model the thin layer approximation for an SN in the presence of three types of media. Section 4 also contains a model in which the center of the explosion does not coincide with the center of the polytrope. A relativistic treatment is carried out in Section 5. The application of the developed theory to SN 1993J is split into the classical case (Section 6), and the relativistic case (Section 7).

An SN expands at constant velocity until the surrounding mass is of the order of the solar mass. This time, t_M, is

$$\begin{equation} t_M= 186.45\,{\frac{\sqrt[3]{{M_{\odot } }}}{\sqrt[3]{{n_0}}{v_{10000}}}} \; \rm yr, \end{equation} \tag{ 1 }$$

where M_☉ is the amount of matter in solar masses in the volume occupied by the SN, n₀ is the number density expressed in particles cm⁻³, and v₁₀₀₀₀ is the initial velocity expressed in units of 10,000 km s⁻¹; see McCray (1987). A first law of motion for the SN is the Sedov solution

$$\begin{equation} R(t)= \left({\frac{25}{4}}\,{\frac{{E}\,{t}^{2}}{\pi \,\rho }}\right)^{1/5}, \end{equation} \tag{ 2 }$$

where E is the energy injected into the process and t is the time; see Sedov (1959) and McCray (1987). Our astrophysical units are: time, (t₁), which is expressed in years; E₅₁, the energy in 10⁵¹ erg; n₀, the number density expressed in particles cm⁻³ (density ρ = n₀m, where m = 1.4 m_H). In these units, Equation (2) becomes

$$\begin{equation} R(t) \approx 0.313\,\sqrt[5] {{\frac{{E_{51}}\,{{t_1}}^{2}}{{n_0}}} }\, {\rm pc}. \end{equation} \tag{ 3 }$$

The Sedov solution scales as t^0.4. We are now ready to couple the Sedov phase with the free expansion phase

$$\begin{equation*} R(t) = \left\lbrace\begin{array}{@{}ll}0.0157t\,{\rm pc} & {{\rm if} t \le 2.5 {\rm yr} } \\ 0.0273\,\sqrt[5] {t}^2\, {\rm pc} & {{\rm if} t > 2.5 {\rm yr}.} \end{array}\right.\nonumber \end{equation*}$$

This two-phase solution is obtained with the following parameters M_☉ = 1, n₀ = 1.127 × 10⁵, E₅₁ = 0.567 and Figure 1 presents its temporal behavior as well as the data. A similar model is reported in Spitzer (1978) with the difference that the first phase ends at t = 60 yr against our t = 2.5 yr. A careful analysis of Figure 1 reveals that the standard two-phase model does not fit the observed radius–time relation for SN 1993J.

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This section introduces three density profiles for the CSM: a Plummer-like profile, a self-gravitating profile of Lane–Emden type, and a power-law profile.

3.1. The Plummer Profile

The Plummer-like density profile, after Plummer (1911), is

$$\begin{equation*} \rho (r;R_{\rm flat}) =\rho _c \left({\frac{R_{\rm flat}}{\left({R_{\rm flat}^2 +r^2}\right)^{1/2}}}\right)^{\eta },\nonumber \end{equation*}$$

where r is the distance from the center, ρ is the density, ρ_c is the density at the center, R_flat is the distance before which the density is nearly constant, and η is the power-law exponent at large values of r; see Whitworth & Ward-Thompson (2001) for more details. The following transformation, $R_{\rm flat}= \sqrt{3} b$, gives a Plummer-like profile, which can be compared with the Lane–Emden profile

$$\begin{equation} \rho (r;b) = \rho _c \left(\frac{1}{1+ \frac{1}{3}\,{\frac{{r}^{2}}{{b}^{2}}}} \right) ^{\eta /2}. \quad \end{equation} \tag{ 4 }$$

At low values of r, the Taylor expansion of the Plummer-like profile can be performed:

$$\begin{equation*} \rho (r;b) \approx \rho _{{c}}\left(1-1/6\,{\frac{\eta \,{r}^{2}}{{b}^{2}}}\right), \end{equation*}$$

and at high values of r, the behavior of the Plummer-like profile is

$$\begin{equation*} \rho (r;b) \sim \rho _c (\sqrt{3}\,b) ^{\eta }\left(\frac{1}{r}\right) ^{ \eta }. \end{equation*}$$

The total mass M(r; b) between 0 and r is

$$\begin{equation} M(r;b) = \int _0^r 4 \pi r^2 \rho (r;b) dr = \frac{{\rm PN}}{{\rm PD}}, \end{equation} \tag{ 5 }$$

where

$$\begin{eqnarray*} {\rm PN} \!&=&\!-\!\sqrt{3}b\rho _{{c}}\pi \left(\!4 {{_2{\rm F}_1}\left(\!\eta /2,-3/2\!+\!\eta /2; n\!-\!1/2\!+\!\eta /2; -3\,{\frac{{b}^{2}}{{r}^{2}}}\!\right)}\right. \nonumber \\ &&\left. \times \Gamma (-\eta /2+5/2) \Gamma (\eta /2) \cos (1/2\,\pi \,\eta) {r}^{3-\eta }{3}^{1/2+\eta /2}{b}^{\eta - 1}\right. \nonumber \\ &&\left. -9\,{\pi }^{3/2}{b}^{2}\eta +27\,{\pi }^{3/2}{b}^{2}\right) \nonumber \end{eqnarray*}$$

and

$$\begin{equation*} {\rm PD}=3\,\cos (1/2\,\pi \,\eta) \Gamma (-\eta /2+5/2)(\eta -3) \Gamma (\eta /2) \nonumber, \end{equation*}$$

where ₂F₁(a, b; c; z) is the regularized hypergeometric function (Abramowitz & Stegun 1965; Olver et al. 2010). When η = 6, M(r; b)₆, the above expression simplifies to

$$\begin{eqnarray*} M(r;b)_6 &=& {\frac{27\,\rho _{{c}}\pi \,{b}^{7}\sqrt{3}}{2\,(3\,{b}^{2}+{r }^{2}) ^{2}}\arctan \left(1/3\,{\frac{r\sqrt{3}}{b}}\right) }\nonumber \\ &&+9\,{\frac{\rho _{{c}}\pi \,{b}^{5}\sqrt{3}{r}^{2}}{ (3\,{b}^{2}+{r}^{2}) ^{2}}\arctan \left(1/3\,{\frac{r \sqrt{3}}{b}}\right) }\nonumber \\ &&+3/2\,{\frac{\rho _{{c}}\pi \,{b}^{3}\sqrt{3}{r}^{4}}{(3\,{b}^{2}+{r}^{2}) ^{2}}\arctan \left(1/3\,{ \frac{r\sqrt{3}}{b}}\right) }\nonumber \\ &&-{\frac{27\,\rho _{{c}}\pi \,{b}^{6}r }{2\,(3\,{b}^{2}+{r}^{2}) ^{2}}}+9/2\,{\frac{\rho _{{c}} \pi \,{b}^{4}{r}^{3}}{(3\,{b}^{2}+{r}^{2}) ^{2}}}. \nonumber \end{eqnarray*}$$

The astrophysical version of the total mass is

$$\begin{equation*} M(r_{\rm pc};b_{\rm pc}) = \frac{{\rm PNA}}{{\rm PDA}} \,M_{\odot }, \nonumber \end{equation*}$$

with

$$\begin{eqnarray*} {\rm PNA}&=& {-} 2.47\,10^{-10}\,{b_{{{\rm pc}}}}^{3}n_{{0}} \left[ 1.02\,10^{10}\,\arctan \left(1.73\,{\frac{b_{{{\rm pc}}}}{r _{{{\rm pc}}}}}\right) {b_{{{\rm pc}}}}^{4}\right. \nonumber \\ &&\left. + 6.8\,10^9\,\arctan \left(1.73\,{\frac{b_{{{\rm pc}}}}{r_{{{\rm pc}}}}}\right) {b_{{{\rm pc}}}}^{2}{r_{{{\rm pc}}}}^{2}\right.\nonumber \\ &&\left. + 1.13\,10^9\, \arctan \left(1.73\,{\frac{b_{{{\rm pc}}}}{r_{{{\rm pc}}}}}\right) {r_{{{\rm pc}}}}^{4}- 1.6\,10^{10}\,{b_{{{\rm pc}}}}^{4}\right. \nonumber \\ &&\left. + 5.89\,10^9\,{b_{{{\rm pc}}}}^{3}r_{{{\rm pc}}}- 1.06\,10^{10}\,{b_{ {{\rm pc}}}}^{2}{r_{{{\rm pc}}}}^{2}\right. \nonumber \\ &&\left. - 1.96\, 10^9\,b_{{{\rm pc}}}{r_{ {{\rm pc}}}}^{3}- 1.78\,10^9\,{r_{{{\rm pc}}}}^{4} \right] \nonumber \end{eqnarray*}$$

and

$$\begin{equation*} {\rm PDA}=\left(3.0\,{b_{{{\rm pc}}}}^{2}+{r_{{{\rm pc}}}}^{2} \right) ^{2}, \nonumber \end{equation*}$$

where b_pc is b expressed in pc, r_pc is r expressed in pc, and n₀ is the same as in Equation (3). The relationship between the FWHM and b_pc is

$$\begin{equation*} {\rm FWHM}=1.766\,b_{{{\rm pc}}}. \nonumber \end{equation*}$$

3.2. The Lane–Emden Profile

A self-gravitating sphere of polytropic gas is governed by the Lane–Emden differential equation of the second order

$$\begin{equation*} {\frac{d^{2}}{d{x}^{2}}}Y(x) +2\,{\frac{{\frac{d}{dx} }Y(x) }{x}}+(Y(x)) ^{n}=0, \nonumber \end{equation*}$$

where n is an integer; see Lane (1870), Emden (1907), Chandrasekhar (1967), Binney & Tremaine (2011), and Zwillinger (1989).

The solution Y(x)_n has the density profile

$$\begin{equation*} \rho = \rho _c Y(x)_n^n, \nonumber \end{equation*}$$

where ρ_c is the density at x = 0. The pressure P and temperature T scale as

$$\begin{equation} P = K \rho ^{1 +\frac{1}{n}}, \end{equation} \tag{ 6 }$$

$$\begin{equation} T = K^{\prime } Y(x), \end{equation} \tag{ 7 }$$

where K and K' are two constants; for more details, see Hansen & Kawaler (1994).

Analytical solutions exist for n  =  0, 1, and 5; that for n = 0 is

$$\begin{equation*} Y(x) = \frac{\sin (x)}{x}, \nonumber \end{equation*}$$

and therefore has an oscillatory behavior. The analytical solution for n = 5 is

$$\begin{equation*} Y(x) ={\frac{1}{{\left(1+ \frac{{x}^{2}}{3}\right)^{1/2}}} }, \nonumber \end{equation*}$$

and the density for n = 5 is

$$\begin{equation} \rho (x) =\rho _c {\frac{1}{{\left(1+ \frac{{x}^{2}}{3}\right)^{5/2}}} }. \end{equation} \tag{ 8 }$$

The variable x is non-dimensional and we now introduce the new variable x = r/b

$$\begin{equation*} \rho (r;b) =\rho _c {\frac{1}{{\left(1+ \frac{{r}^{2}}{3b^2}\right)^{5/2}}} } \nonumber. \end{equation*}$$

This profile is a particular case, η = 5, of the Plummer-like profile as given by Equation (4). At low values of r, the Taylor expansion of this profile is

$$\begin{equation*} \rho (r;b) \approx \rho _{{c}} \left(1-5/6\,{\frac{{r}^{2}}{{b}^{2}}}\right), \end{equation*}$$

and at high values of r, its behavior is

$$\begin{equation} \rho (r;b) \sim 9\,{\frac{\rho _{{c}}\sqrt{3}{b}^{5}}{{r}^{5}}}. \end{equation} \tag{ 9 }$$

The FWHM is

$$\begin{equation*} {\rm FWHM}=1.95\,b_{{{\rm pc}}}. \nonumber \end{equation*}$$

The gradient here is assumed to be local: it covers lengths smaller than 1 pc, and is not connected with the gradient which regulates the equilibrium of a galaxy that extends over a region of a couple of kpc. The interaction of the progenitor star with the CSM, through stellar winds, creates CSM. The dynamics of this medium can be far from equilibrium. The main astrophysical assumption adopted here is that the density of the CSM decreases smoothly at low values of distance, as ρ ≈ A − B r², due to previous stellar winds, and ρ ≈ C r⁻⁵ in the far regions is not contaminated by previous activity, with A, B, and C being constants. In view of the behavior of this self-gravitating profile at high r, in Section 3.3 we will analyze a power-law dependence for the CSM.

The total mass M(r; b) between 0 and r is

$$\begin{equation} M(r;b) = \int _0^r 4 \pi r^2 \rho (r;b) dr =\frac{ 4\,{b}^{3}{r}^{3}\rho _c\,\pi \,\sqrt{3} }{ (3\,{b}^{2}+{r}^{2}) ^{3/2} }, \end{equation} \tag{ 10 }$$

or in solar units

$$\begin{equation*} M(r_{\rm pc};b_{\rm pc}) = \frac{ { 2.2\times 10^{55}}\,{b_{{{\rm pc}}}}^{3}{r_{{{\rm pc}}}}^{3} n_{{0}} }{ \left({ 2.85\times 10^{37}}\,{b_{{{\rm pc}}}}^{2}+{ 9.52\times 10^{36}}\,{r_{{{\rm pc}}}}^{2} \right)^{3/2} } \,M_{\odot }. \nonumber \end{equation*}$$

The total mass of the profile can be found by calculating the limit r → ∞ in Equation (10)

$$\begin{equation} M(\infty ;b) = \lim _{r\rightarrow \infty }M(r;b) = 4\,{b}^{3}{\rho _c}\,\pi \,\sqrt{3}. \end{equation} \tag{ 11 }$$

Another interesting physical quantity deduced in the framework of the virial theorem is the mean square speed of the system, which according to formula (4.249a) in Binney & Tremaine (2011) is

$$\begin{equation} \langle v^2\rangle = \frac{GM}{r_g}, \end{equation} \tag{ 12 }$$

where M is the total mass, r_g is the gravitational radius as defined in Equation (2.42) in Binney & Tremaine (2011), and G is the Newtonian gravitational constant. In the case of a Lane–Emden profile as given by Equation (9), the gravitational radius is

$$\begin{equation} r_g = \frac{32\,b\sqrt{3}}{3\,\pi }. \end{equation} \tag{ 13 }$$

The mean square speed of the system according to formulae (13) and (11) is

$$\begin{equation} \langle v^2\rangle = \frac{ 3\,G{b}^{2}\rho _{{c}}{\pi }^{2} }{ 8 }. \end{equation} \tag{ 14 }$$

The relationship between gravitational radius, r_g, and half-mass radius, r_h, is

$$\begin{equation} \frac{r_h}{r_g}= 0.16647, \end{equation} \tag{ 15 }$$

and this allows a more simple definition for the mean square speed

$$\begin{equation} \langle v^2\rangle = 0.16647\,\frac{GM}{r_h}. \end{equation} \tag{ 16 }$$

The astrophysical version of the square root of the mean square speed as given by formula (14) is

$$\begin{equation} \sqrt{\langle v^2\rangle } = 233.81 \,\sqrt{{b_{{{\rm pc}}}}^{2}n_{{7}}} \,\, {\rm km \,s}^{-1}, \end{equation} \tag{ 17 }$$

where b_pc is the scale parameter expressed in pc, n₇ represents the number density expressed in 10⁷cm⁻³ units, and G = 6.67384  m³ kg⁻¹ s⁻²; see Mohr et al. (2012).

3.3. A Power Law for the CSM

We now assume that the CSM around the SN scales with the following piecewise dependence (which avoids a pole at r = 0)

$$\begin{equation} \rho (r;r_0,d) = \left\lbrace\begin{array}{@{}ll}\rho _c & {{\rm if} r \le r_0} \\ \rho _c (\frac{r_0}{r})^d & {{\rm if} r > r_0.} \end{array}\right. \end{equation} \tag{ 18 }$$

The mass swept, M₀, in the interval [0, r₀] is

$$\begin{equation*} M_0 = \frac{4}{3}\,\rho _{{0}}\pi \,{r_{{0}}}^{3}. \nonumber \end{equation*}$$

The total mass swept, M(r; r₀, d), in the interval [0, r] is

$$\begin{eqnarray*} &&M (r;r_0,d)= -4\,{r}^{3}\rho _{{c}}\pi \,\left({\frac{r_{{0}}}{r}}\right) ^{d} (d -3) ^{-1} \nonumber \\ &&+\,4\,{\frac{\rho _{{c}}\pi \,{r_{{0}}}^{3}}{d- 3}} + \frac{4}{3}\,\rho _{{c}}\pi \,{r_{{0}}}^{3}. \nonumber \end{eqnarray*}$$

or in solar units

$$\begin{eqnarray*} &&M(r_{\rm pc}{r_{{0,{\rm pc}}}},d) \\ &&\quad = \frac{ 3.14\,n_{{0}} \left(0.137\,{r_{{{\rm pc}}}}^{3} \left({\frac{r_{{0,{\rm pc}}}}{r_{{{\rm pc}}}}} \right) ^{d}- 0.0459\,{r_{{0,{\rm pc}}}}^{3}d \right) }{ 3-d } \,M_{\odot }, \nonumber \end{eqnarray*}$$

where r_{0, pc} is r₀ expressed in pc. The FWHM is

$$\begin{equation*} {\rm FWHM}= \frac{ 2\,r_{{0,{\rm pc}}} }{ {{\rm e}^{-{\frac{\ln \left(2 \right) }{d-2}}}} }. \nonumber \end{equation*}$$

This section reviews the standard equation of motion in the case of the thin layer approximation in the presence of a CSM with constant density and derives the equation of motion under the conditions of each of the three density profiles for the density of the CSM. A simple asymmetrical model is introduced.

4.1. Motion with Constant Density

In the case of a constant density, ρ_c, CSM, the differential equation which models momentum conservation is

$$\begin{equation*} \frac{4}{3}\,\pi \,(r(t)) ^{3}\rho _{{c}}{\frac{d }{dt}}r(t) -\frac{4}{3}\,\pi \,{{r_0}}^{3}\rho _{{c}}v_{{0}}=0, \nonumber \end{equation*}$$

where the initial conditions are r = r₀ and v = v₀ when t = t₀. The variables can be separated, and the radius as a function of the time is

$$\begin{equation*} r(t)= \sqrt[4] {4\,{r_{{0}}}^{3}v_{{0}}(t-t_{{0}}) +{r_{{0}}}^{4}}, \nonumber \end{equation*}$$

and its behavior as t → ∞ is

$$\begin{eqnarray*} &&r(t) =\sqrt{2}{r_{{0}}}^{3/4}\sqrt[4] {v_{{0}}}\sqrt[4] {t-t_{{0}}}+ \frac{1}{16}\,{ \frac{\sqrt{2}{r_{{0}}}^{7/4}}{{v_{{0}}}^{3/4}(t-t_{{0}}) ^{3/4}}}. \nonumber \end{eqnarray*}$$

The velocity as a function of time is

$$\begin{equation*} v(t)= \frac{ {r_{{0}}}^{3}v_{{0}} }{ \left(4\,{r_{{0}}}^{3}v_{{0}}(t-t_{{0}}) +{r_{{0}}}^{4 }\right) ^{3/4} }. \nonumber \end{equation*}$$

4.2. Motion with Plummer Profile

The case of CSM with a Plummer-like profile as given by (4) when η = 6 produces the differential equation

$$\begin{equation} \frac{d}{dt}r(t) = \frac{{\rm NDEP}}{{\rm DNEP}}, \end{equation} \tag{ 19 }$$

where

$$\begin{eqnarray*} {\rm NDEP} &=& \left(9\,\sqrt{3}\arctan \left(1/3\,{\frac{{r0}\,\sqrt{3}}{b }}\right) {b}^{4}\right. \nonumber \\ &&\left. +6\,\sqrt{3}\arctan \left(1/3\,{\frac{{r0}\, \sqrt{3}}{b}}\right) {b}^{2}{{r0}}^{2}\right. \nonumber \\ &&\left. +\sqrt{3}\arctan \left(1 /3\,{\frac{{r0}\,\sqrt{3}}{b}}\right) {{r0}}^{4}-9\,{b}^{3}{r0}+3\,b{{r0}}^{3}\right)\nonumber \\ &&\times {v0}\,(3\,{b}^{2}+ (r(t)) ^{2}) ^{2}, \nonumber \end{eqnarray*}$$

and

$$\begin{eqnarray*} {\rm DNEP}& =& (3\,{b}^{2}+{{r0}}^{2}) ^{2}\left(9\,\sqrt{3} \arctan \left(1/3\,{\frac{r(t) \sqrt{3}}{b}}\right) {b}^{4}\right. \nonumber \\ &&\left. +6\,\sqrt{3}\arctan \left(1/3\,{\frac{r(t) \sqrt{3}}{b}}\right) {b}^{2}(r(t)) ^{2}\right. \nonumber \\ &&\left. +\sqrt{3}\arctan \left(1/3\,{\frac{r(t) \sqrt{3}}{b} }\right)(r(t)) ^{4}\right. \nonumber \\ &&\left. -9\,{b}^{3}r (t) +3\,b(r(t)) ^{3}\right). \nonumber \end{eqnarray*}$$

There is no analytical solution to this differential equation, but the solution can be found numerically.

4.3. Motion with the Lane–Emden Profile

In the case of variable density for the CSM as given by the profile (8), the differential equation which models momentum conservation is

$$\begin{equation} 4\,{\frac{{b}^{3}(r(t)) ^{3}\rho _{{c}} \pi \,\sqrt{3}{\frac{d}{dt}}r(t) }{(3\,{b}^{2}+ (r(t)) ^{2}) ^{3/2}}}-4\,{\frac{{ b}^{3}{r_{{0}}}^{3}\rho _{{c}}\pi \,\sqrt{3}v_{{0}}}{\left(3\,{b}^{2 }+{r_{{0}}}^{2}\right) ^{3/2}}}=0. \end{equation} \tag{ 20 }$$

The variables can be separated and the solution is

$$\begin{equation} r(t;r_0,v_0,t_0,b) = \frac{N}{D}, \end{equation} \tag{ 21 }$$

where

$$\begin{eqnarray*} N &=& \sqrt{2}{r_{{0}}}^{3/4}\left ({r_{{0}}}^{13/2}+2\,{r_{{0}}}^{11/2} (t-t_{{0}}) v_{{0}}\right. \nonumber \\ &&\left. +{r_{{0}}}^{9/2}(t-t_{{0}}) ^{2}{v_{{0}}}^{2}+6\,{b}^{2}{r_{{0}}}^{9/2}\right. \nonumber \\ &&\left. +18\,{b}^{2}{r_{{0 }}}^{7/2}(t-t_{{0}}) v_{{0}}+\sqrt{A}{r_{{0}}}^{4}+ \sqrt{A}{r_{{0}}}^{3}(t-t_{{0}}) v_{{0}} \right.\nonumber \\ &&\left. +9\,{b}^{4}{r_{ {0}}}^{5/2}+36\,{b}^{4}{r_{{0}}}^{3/2}v_{{0}}(t-t_{{0}})\right. \nonumber \\ &&\left. +9\,\sqrt{A}{b}^{2}{r_{{0}}}^{2}+18\,\sqrt{A}{b}^{4} \right)^{1/2} \nonumber \end{eqnarray*}$$

and

$$\begin{eqnarray*} &&D=2\left(3\,{b}^{2}+{r_{{0}}}^{2}\right) ^{3/2} \nonumber, \end{eqnarray*}$$

with

$$\begin{eqnarray*} A(t-t_0)&=& {r_{{0}}}^{3}(t-t_{{0}}) ^{2}{v_{{0}}}^{2}+36\,{b}^{4} (t-t_{{0}}) v_{{0}} \nonumber \\ &&+18\,{b}^{2}{r_{{0}}}^{2}(t-t_{{0}}) v_{{0}} \nonumber \\ &&+2\,{r_{{0}}}^{4}(t-t_{{0}}) v_{{0} }+9\,{b}^{4}r_{{0}}+6\,{b}^{2}{r_{{0}}}^{3}+{r_{{0}}}^{5} \nonumber. \end{eqnarray*}$$

This is the first solution and has an analytical form. The analytical solution for the velocity can be found from the first derivative of the analytical solution as represented by Equation (21),

$$\begin{equation} v(t;r_0,v_0,t_0,b) = \frac{d}{dt}r(t;r_0,v_0,t_0,b). \end{equation} \tag{ 22 }$$

The previous differential Equation (20) can be organized as

$$\begin{equation} \frac{d}{dt} r(t) = f (r;r_0,v_0,t_0,b), \end{equation} \tag{ 23 }$$

and we seek a power series solution of the form

$$\begin{equation} r(t) = a_0 +a_1 (t-t_0) +a_2 (t-t_0)^2+a_3 (t-t_0)^3 + \cdots ; \protect \protect \protect \end{equation} \tag{ 24 }$$

see Tenenbaum & Pollard (1963) and Ince (2012). The Taylor expansion of Equation (23) gives

$$\begin{eqnarray*} && f (r;r_0,v_0,t_0,b)= \nonumber \\ && \quad b_0 +b_1 (t-t_0) +b_2 (t-t_0)^2+b_3 (t-t_0)^3 + \dots, \nonumber \end{eqnarray*}$$

where the values of b_n are

$$\begin{eqnarray} b_0&=& f (r_0;r_0,v_0,t_0,b) \nonumber \\ b_1&=& \frac{\partial }{\partial t} f (r_0;r_0,v_0,t_0,b) \nonumber \\ b_2&=& \frac{1}{2!}\frac{\partial ^2}{\partial t^2} f (r_0;r_0,v_0,t_0,b) \\ b_3&=& \frac{1}{3!}\frac{\partial ^3}{\partial t^3} f (r_0;r_0,v_0,t_0,b) \nonumber \\ \ldots &&\dots \dots \nonumber \end{eqnarray} \tag{ 25 }$$

The relation between the coefficients a_n and b_n is

$$\begin{eqnarray*} a_1&=& b_0 \nonumber \\ a_2&=& \frac{b_1}{2} \nonumber \\ a_3&=& \frac{b_2}{3} \nonumber \\ \ldots &&\dots \dots \nonumber \end{eqnarray*}$$

The higher-order derivatives plus the initial conditions give

$$\begin{eqnarray} a_0&=& r_0 \nonumber \\ a_1&=& v_0 \nonumber \\ a_2&=& -\frac{ 9\,{{v_0}}^{2}{b}^{2}}{ 2\left(3\,{b}^{2}+{{r_0}}^{2}\right) {r_0}}\nonumber \\ a_3&=&\frac{ 9\,{{v_0}}^{3}{b}^{2}\left(7\,{b}^{2}+{{r_0}}^{2}\right) }{2\,{{r_0}}^{2}\left(3\,{b}^{2}+{{r_0}}^{2}\right) ^{2} } \\ \ldots &&\dots \dots \nonumber \end{eqnarray} \tag{ 26 }$$

These are the coefficients of the second solution, which is a power series.

A third solution can be represented by a difference equation which has the following type of recurrence relation

$$\begin{eqnarray} r_{n+1} =& r_n + v_n \Delta t \nonumber \\ v_{n+1} =& \frac{{r_{{n}}}^{3}v_{{n}}\left(3\,{b}^{2}+{r_{{n+1}}}^{2}\right) ^{3/2}}{\left(3\,{b}^{2}+{r_{{n}}}^{2}\right) ^{3/2}{r_{{n+1}}}^{3}}, \end{eqnarray} \tag{ 27 }$$

where r_n, v_n, and Δt are the temporary radius, the velocity, and the interval of time.

The physical units have not yet been specified: pc for length and yr for time are the units most commonly used by astronomers. With these units, the initial velocity v₀ is expressed in pc yr⁻¹, 1 yr = 365.25 days, and should be converted into km s⁻¹; this means that v₀ = 1.02 × 10⁻⁶v₁ where v₁ is the initial velocity expressed in km s⁻¹. In these units, the speed of light is c = 0.306 pc yr⁻¹. The previous analysis covers the case of symmetrical expansion. In the framework of the spherical coordinates (r, θ, φ) where r is the radius (r = 0 refers to the center of the expansion), θ is the polar angle with range [0 − π], and φ is the azimuthal angle with range [0 – 2π]. The symmetrical expansion is characterized by the independence of the advancing radius from θ and φ. We now cover the case of asymmetrical expansion in which the center of the explosion is at r = 0 in spherical coordinates but does not coincide with the center of the polytrope, which is at the distance z_asymm. The density profile of the polytrope, which now has a shifted origin, now depends on the direction of the radius vector. So, the supernova's shell would evolve in non-spherical forms. We align the polar axis with the line 0 − z_asymm. The symmetry is now around the z-axis and the expansion will be independent of the azimuthal angle φ. The advancing radius will conversely depend on the polar angle θ. The case of an expansion that starts from a given distance, z_asymm, from the center of the polytrope cannot be modeled by the differential Equation (20), which is derived for a symmetrical expansion. It is not possible to find R analytically and a numerical method should be implemented. The advancing expansion is computed in a 3D Cartesian coordinate system (x, y, z) with the center of the explosion at (0, 0, 0). The degree of asymmetry is evaluated by introducing R_eq, R_up, and R_down which are the momentary radii in the equatorial plane, polar direction up, and polar direction down. The percentage of asymmetry is defined by the two ratios

$$\begin{equation} a_{\rm up} = \left|\frac{ R_{\rm up} - R_{\rm eq}}{ R_{\rm eq}} \right| *100, \end{equation} \tag{ 28 }$$

and

$$\begin{equation} a_{\rm down} = \left|\frac{ R_{\rm down} - R_{\rm eq}}{ R_{\rm eq}} \right| *100. \end{equation} \tag{ 29 }$$

As a reference the measured asymmetry of SN 1993J is under 2%; see Marcaide et al. (2009).

4.4. Motion with a Power-law Profile

The differential equation that models momentum conservation when the density has a power-law behavior, as given by (18), is

$$\begin{eqnarray} &&\left(-4\,{\frac{(r(t)) ^{3}\rho _{{c}} \pi }{d-3}\left({\frac{r_{{0}}}{r(t) }}\right) ^{d}} +4\,{\frac{\rho _{{c}}\pi \,{r_{{0}}}^{3}}{d-3}}+4/3\,\rho _{{c}}\pi \, {r_{{0}}}^{3}\right) {\frac{\rm d}{{\rm d}t}}r(t) \nonumber \\ &&\quad -4/3 \,\rho _{{c}}\pi \,{r_{{0}}}^{3}{v_0}=0. \end{eqnarray} \tag{ 30 }$$

A first solution can be found numerically; see Zaninetti (2011) for more details. A second solution is a truncated series about the ordinary point t = t₀, which to fourth order has coefficients

$$\begin{eqnarray} a_0&=& r_0 \nonumber \\ a_1&=& v_0 \nonumber \\ a_2&=& \frac{-3\,{{v0}}^{2}}{2\,{r_0}} \nonumber \\ a_3&=& \frac{(d+7) {{v_0}}^{3} }{2\,{{r_0}}^{2} }. \end{eqnarray} \tag{ 31 }$$

A third approximate solution can be found assuming that $3 r_0^d r^{4-d}$≫ $-(4 r_0^3 d-r_0^3 d^2)r$

$$\begin{eqnarray*} r(t) &=& \left({r_{{0}}}^{4-d}-\frac{1}{3}d{r_{{0}}}^{4-d} (4-d)\right. \nonumber \\ &&\left. + \frac{1}{3} (4-d) v_{{0}}{r_{{0}}}^{3-d}(3-d) (t-t_{{0}})\right) ^{\frac{1}{4-d}}. \nonumber \end{eqnarray*}$$

This is an important approximate result because, given the astronomical relation r(t)∝t^α, we have d = 4 − (1/α).

The thin layer approximation assumes that all mass swept during the travel from the initial time, t₀, to the time t resides in a thin shell of radius r(t) with velocity v(t). Assuming a Lane–Emden dependence (n = 5), the total mass M(r; b) between 0 and r is given by Equation (10). The relativistic conservation of momentum (see French 1968; Zhang 1997; Guéry-Odelin & Lahaye 2010) is formulated as

$$\begin{equation*} M(r_0;b) \gamma _0 \beta _0 = M(r;b) \gamma \beta, \nonumber \end{equation*}$$

where

$$\begin{equation*} \gamma _0 = \frac{1}{ \sqrt{1-\beta _0^2} } ; \qquad \gamma = \frac{1}{ \sqrt{1-\beta ^2} }, \nonumber \end{equation*}$$

and

$$\begin{equation*} \beta _0 =\frac{v_0}{c} ; \qquad \beta =\frac{v}{c}. \nonumber \end{equation*}$$

The relativistic conservation of momentum is easily solved for β as a function of the radius:

$$\begin{equation*} \beta = \frac{ \sqrt{A(3\,{b}^{2}+{r}^{2}) }(3\,{b}^{2}+{r}^{2}) {r_{{0}}}^{3}\beta _{{0}} }{A}, \nonumber \end{equation*}$$

with

$$\begin{eqnarray*} A &=& {-}27\,{b}^{6}{r}^{6}{\beta _{{0}}}^{2}+27\,{b}^{6}{\beta _{{0}}}^{2}{r_{{0 }}}^{6}-27\,{b}^{4}{r}^{6}{\beta _{{0}}}^{2}{r_{{0}}}^{2}\nonumber \\ && +27\,{b}^{4}{r }^{2}{\beta _{{0}}}^{2}{r_{{0}}}^{6} -9\,{b}^{2}{r}^{6}{\beta _{{0}}}^{2}{r_{{0}}}^{4}+9\,{b}^{2}{r}^{4}{\beta _{{0}}}^{2}{r_{{0}}}^{6}+27\,{b}^{6}{r}^{6}\nonumber \\ &&+27\,{b}^{4}{r}^{6}{r_{{0}}}^{2}+9\,{b}^{2}{r}^{6}{r_{{0}}}^{4}+{r}^{6}{r_{{0}}}^{6} \nonumber. \end{eqnarray*}$$

Inserting

$$\begin{equation*} \beta = \frac{1}{c}{\frac{d}{dt}}r(t), \nonumber \end{equation*}$$

the relativistic conservation of momentum can be written as the differential equation

$$\begin{equation} \frac{ 4\,{b}^{3}(r(t)) ^{3}\rho \pi \,\sqrt{3 }{\frac{d}{dt}}r(t) }{ (3\,{b}^{2}+(r(t)) ^{2}) ^{ 3/2}c\sqrt{-{\frac{({\frac{d}{dt}}r(t)) ^{2}}{{c}^{2}}}+1} } \,{=}\, \frac{ 4\,{b}^{3}{r_{{0}}}^{3}\rho \pi \,\sqrt{3}\beta _{{0}} }{ \left(3\,{b}^{2}+{r_{{0}}}^{2}\right) ^{3/2}\sqrt{-{\beta _{{0}}}^{ 2}+1} }. \end{equation} \tag{ 32 }$$

This first order differential equation can be solved by separating the variables:

$$\begin{equation} \int _{r_0}^r \frac{A}{ \sqrt{A(3\,{b}^{2}+{r}^{2}) }(3\,{b}^{2}+{r}^{2}) {r_{{0}}}^{3}\beta _{{0}} } \,dr = c (t-t_0). \end{equation} \tag{ 33 }$$

The previous integral does not have an analytical solution and we treat the previous result as a non-linear equation to be solved numerically. The differential equation has a truncated series solution about the ordinary point t = t₀ which to fifth order is

$$\begin{equation} r_s(t)= \sum _{n=0}^{4}a_{{n}}{(t-t_0)}^{n}. \end{equation} \tag{ 34 }$$

The coefficients are

$$\begin{eqnarray} a_0&=& r_0 \nonumber \\ a_1&=& c\beta _{{0}} \nonumber \\ a_2&=& \frac{9\,{b}^{2}{\beta _{{0}}}^{2}{c}^{2} \left({\beta _{{0}}}^{2}-1\right)}{ 2\,r_{{0}}\left(3\,{b}^{2}+{r_{{0}}}^{2}\right) } \\ a_3&=&\frac{9\,{c}^{3}(\beta _{{0}}-1)(\beta _{{0}}+1) \left(12\,{b}^{2}{\beta _{{0}}}^{2}-7\,{b}^{2}-{r_{{0}}}^{2}\right) {\beta _{{0}}}^{3}{b}^{2}}{2\left(3\,{b}^{2}+{r_{{0}}}^{2}\right) ^{2}{r_{{0}}}^{2}} \nonumber \\ a_4&=&\frac{9\,{b}^{2}(\beta _{{0}}-1)(\beta _{{0}}+1) B{\beta _{{0}}}^{4}{c}^{4}}{8\,{r_{{0}}}^{3}\left(3\,{b}^{2}+{r_{{0}}}^{2}\right) ^{3}}, \nonumber \\ {\rm where}\, B&=& 756\,{b}^{4}{\beta _{{0}}}^{4}-927\,{b}^{4}{\beta _{{0}}}^{2}-117\,{b}^{ 2}{\beta _{{0}}}^{2}{r_{{0}}}^{2}\nonumber\\ &&+231\,{b}^{4}+69\,{b}^{2}{r_{{0}}}^{2} +4\,{r_{{0}}}^{4}. \nonumber \end{eqnarray} \tag{ 35 }$$

The velocity approximated to the fifth order is

$$\begin{equation} v_s(t) = \sum _{n=1}^{4}{\frac{a_{{n}}{(t-t_0)}^{n}n}{(t-t_0)}}. \end{equation} \tag{ 36 }$$

The presence of an analytical expression for β as given by Equation (32) allows the recursive solution

$$\begin{eqnarray} r_{n+1} &= & r_n + c \beta _n \Delta t \nonumber \\ \beta _{n+1}& =& \frac{ \sqrt{A\left(3\,{b}^{2}+{r_{{n+1}}}^{2}\right) } \left(3\,{b}^{2} +{r_{{n+1}}}^{2}\right) {r_{{n}}}^{3}\beta _{{n}} }{ A } \\ {\rm with}\ A &=& 27\,{b}^{6} {\beta _{{n}}}^{2}{r_{{n}}}^{6}-27\,{b}^{6} {\beta _{{n}}}^{2}{r_{{n+1}}}^{6}+27\,{b}^{4} {\beta _{{n}}}^{2}{r_{{n+1}}}^{2}{r_{{n}}}^{ 6} \nonumber \\ &&-27\,{b}^{4}{\beta _{{n}}}^{2}{r_{{n+1}}}^{6}{r_{{n}}}^{2} +9\,{b}^{2}{\beta _{{n}}}^{2}{r_{{n+1}}}^{4}{r_{{n}}}^{6}\nonumber \\ &&-9\,{b}^{2}{\beta _{{n}}}^{2}{r_{{n+1}}}^{6}{r_{{n}}}^{4}+27\,{b}^{6}{r_{{n+1}}}^{6} +27\,{b}^{4}{r_{{n+1}}}^{6}{r_{{n}}}^{2} \nonumber \\ &&+9\,{b}^{2}{r_{{n+1}}}^{6}{r_{{n}}}^{4}+{r _{{n+1}}}^{6}{r_{{n}}}^{6} \nonumber, \end{eqnarray} \tag{ 37 }$$

where r_n, β_n, and Δt are the temporary radius, the relativistic β factor, and the interval of time, respectively. Up to now we have taken the time interval t − t₀ to be that seen by an observer on Earth. For an observer moving on the expanding shell, the proper time τ* is

$$\begin{equation*} \tau ^* = \int _{t_0}^{t} \frac{dt}{\gamma } = \int _{t_0}^{t} \sqrt{1-\beta ^2} dt ; \protect \protect \protect \nonumber \end{equation*}$$

see Larmor (1897), Lorentz (1904), Einstein (1905), and Macrossan (1986). In the series solution framework, β = (v_s/c), where v_s is given by Equation (36). A measure of the time dilation is given by

$$\begin{equation*} D = \frac{\tau ^*}{t-t_0}, \nonumber \end{equation*}$$

with 0 < D < 1. It is interesting to point out that the time dilation analyzed here is not connected with the "cosmological time dilation in GRB" which, conversely, is related to the cosmological redshift; see Kocevski & Petrosian (2013) and Zhang et al. (2013) for two diametrically opposed points of view. An application of the time dilation is in modeling the decay of a radioactive isotope usomg the following law for remnant particles in the laboratory framework

$$\begin{equation*} N(t) = N_0 e^{-\frac{(t-t_0)}{\tau }}, \end{equation*}$$

where τ is the proper lifetime, N₀ is the number of nuclei at t = t₀, and the half life is T_1/2 = ln(2) τ. In a frame that is moving with the shell, the decay law is

$$\begin{equation*} N(t) = N_0 e^{-\frac{\tau ^*}{\tau }}. \end{equation*}$$

This theory is used to explain the lifetime of the muons in cosmic rays: in that case, γ ≈ 8.4 and τ = 2.196 × 10⁻⁶ s; see Rossi & Hall (1941); Frisch & Smith (1963); Beringer et al. (2012). In SR, the total energy of a particle is

$$\begin{equation*} E = m c^2 = m_0 \gamma c^2, \nonumber \end{equation*}$$

where m₀ is the rest mass. The relativistic kinetic energy is

$$\begin{equation*} {\rm KE} = m_0 c^2 (\gamma -1), \nonumber \end{equation*}$$

where the rest energy has been subtracted from the total energy. In order to have a simple expression for the velocity as a function of time, we deduce a series expansion, limited to the third order, for the radius as a function of time. The relativistic kinetic energy is therefore

$$\begin{equation*} {\rm KE} = m_0 c^2 \left({\frac{1}{\sqrt{1 -\left(c\beta _{{0}}+9\,{\frac{{b}^{2}{\beta _{{0}} }^{2}{c}^{2}({\beta _{{0}}}^{2}-1)(t-t_{{0}}) }{r_{{0}}(3\,{b}^{2}+{r_{{0}}}^{2}) }}\right) ^{2}{c}^{-2}}}} -1\right). \nonumber \end{equation*}$$

This section introduces: the SN chosen for testing purposes, the astrophysical environment connected with the selected SN, two types of fit commonly used to model the radius–time relation in SNe, the application of the results obtained for the Lane–Emden density profile to the selected SN, the asymmetric explosion, and the case of CSM characterized by a power law.

6.1. The Data

The data of SN 1993J, radius in parsecs and elapsed time in years, can be found in Table 1 of Marcaide et al. (2009). The instantaneous velocity of expansion can be deduced from the formula

$$\begin{equation*} v_i = \frac{r_{i+1} - r_i}{t_{i+1} - t_i}, \end{equation*}$$

where r_i is the radius and t_i is the time at the position i. The uncertainty in the instantaneous velocity is found by implementing the error propagation equation; see Bevington & Robinson (2003). A discussion of the thickness of the radio shell in SN 1993J in the framework of a reverse shock Chevalier (1982a, 1982b) can be found in Bartel et al. (2007). The thickness of the radio shell can also be explained in the framework of image theory; see Section 6.3 in Zaninetti (2011).

6.2. Astrophysical Scenario

6.3. Two Types of Fit

The quality of the fits is measured by the merit function χ²

$$\begin{equation*} \chi ^2 = \sum _j \frac{(r_{\rm th} -r_{\rm obs})^2}{\sigma _{\rm obs}^2}, \end{equation*}$$

where r_th, r_obs and σ_obs are the theoretical radius, the observed radius, and the observed uncertainty, respectively. A first fit can be done by assuming a power-law dependence of the type

$$\begin{equation*} r(t) = r_p t^{\alpha _p}, \end{equation*}$$

where the two parameters r_p and α_p as well their uncertainties can be found using the recipes suggested in Zaninetti (2011). A second fit can be done by assuming a piecewise function as in Figure 4 of Marcaide et al. (2009)

$$\begin{equation*} r(t) = \left\lbrace \begin{array}{ll}r_{\rm br}(\frac{t}{t_{\rm br}})^{\alpha _1} & {{\rm if} t \le t_{\rm br} } \nonumber \\ r_{\rm br}(\frac{t}{t_{\rm br}})^{\alpha _2} & {{\rm if} t > t_{\rm br}. } \nonumber \end{array}\right.. \end{equation*}$$

This type of fit requires determining four parameters: t_br the break time, r_br the radius of expansion at t = t_br, and the exponents α₁ and α₂ of the two phases. The parameters of these two fits as well the χ² can be found in Table 1.

Table 1. Numerical Values of the Parameters of the Fits and χ²; N Represents the Number of Free Parameters

N	Values	χ2
	Power law as a fit
2	αp = 0.82 ± 0.0048	6364
	$r_p = (0.015 \pm 0.00011)\ {\mathrm{\rm pc}}$
	Piecewise fit
4	α1 = 0.83 ± 0.01	32
	α2 = 0.78 ± 0.0077;
	$r_{\rm br} = 0.05\, {\mathrm{\rm pc}}; t_{\rm br}=4.10 \,{\rm yr}$
	Plummer profile, η = 6
2	$b = 0.0045\, {\mathrm{\rm pc}} ; r_{0} = 0.008\, {\mathrm{\rm pc}};v_0={\rm 19{,}500}{\, \rm km s}^{-1}$	265
	Lane–Emden profile
2	$b\,=\,0.00367\, {\mathrm{\rm pc}}; r_{0} = 0.008\, {\mathrm{\rm pc}};v_0={\rm 19{,}500}{\, \rm km s}^{-1}$	471
	Power-law profile
2	$d=2.93; r_{0} = 0.0022\, {\mathrm{\rm pc}};$	276
	t0 = 0.249 yr; v0 = 100, 000 km s−1

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6.4. The Lane–Emden Case

The radius of SN 1993J, which represents the momentum conservation in a Lane–Emden density profile, is reported in Figure 2; r₀ and t₀ are fixed by the observations and the two free parameters are b and v₀.

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Figure 3 compares the theoretical solution and the series expansion about the ordinary point t₀. The range of time in which the series solution approximates the analytical solution is limited. Figure 4 compares the theoretical solution and the recursive solution as represented by Equation (27). The recursive solution approximates the analytical solution over all the ranges of time considered and the error at t = 10 yr is ≈0.6% when Δt = 0.05 yr and ≈0.1% when Δt = 0.0083 yr. The time evolution of the velocity is reported in Figure 5. The asymmetrical case in which there is some distance, z_asymm, from the center of the polytrope and the center of the expansion is clearly outlined in Figure 6. In this figure, we have two sections of the expansion in the plane connecting z_asymm with the center of the expansion, with z_asymm increasing from 0 (the circular section) to 0.00367 pc (the asymmetrical section).

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Figure 7 reports the complex structure of the 3D advancing surface. The point of view of the observer is parameterized by the Euler angles (Φ, Θ, Ψ).

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6.5. Plummer and Power-law Cases

The numerical solution of the differential equation connected with the Plummer-like profile, η = 6, is reported in Figure 8 when the data of Table 1 are adopted.

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A comparison with the power-law behavior for the CSM is reported in Figure 9 which is built from the data in Table 1. The series solution for the power-law dependence of the CSM with coefficients as given by Equation (31) is not reported because the range in time of reliability is limited to t − t₀ ≈ 0.0003 yr.

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We now apply the relativistic solutions derived so far to SN 1993J. The initial observed velocity, v₀, as deduced from radio observations (see Marcaide et al. 2009), is v₀ ≈ 20, 000  km s⁻¹ at t₀ ≈ 0.5 yr. We now reduce the initial time t₀ and we increase the velocity up to the relativistic regime, t₀ = 10⁻⁴ yr and v₀ = 100, 000  km s⁻¹. This choice of parameters allows the observed radius–time relation that should be reproduced to be fitted. The data used in the simulation are shown in Table 2. The relativistic numerical solution of Equation (33) is reported in Figure 10, the relativistic series solution as given by Equation (34) is reported in Figure 11, and the recursive solution as given by Equation (38) is contained in Figure 12.

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Table 2. Numerical Values of the Parameters Used in Three Relativistic Solutions

Parameters
$t_0=10^{-4}\, \mathrm{yr} ; r_0=0.0033\,\mathrm{\rm pc}; \beta _0=0.3333; b=0.004\, \mathrm{\rm pc}$

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Figure 13 reports a 2D map of the parameter D which parameterizes the time dilation.

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Figure 14 reports the temporal evolution in the number of ⁵⁶Ni (τ = 8.757 days) in the laboratory frame at rest and in the frame which is moving with the SN. Figure 15 reports the temporal evolution of the relativistic kinetic energy for a proton.

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Classic case. The thin layer approximation, which models the expansion in a self-gravitating medium of the Lane–Emden type (n = 5), can be modeled by a differential equation of the first order for the radius as a function of time. This differential equation has an analytical solution represented by Equation (21). A power-law series (see Equation (24)) can model the solution of Lane–Emden type for a limited range of time; see Figure 3. Conversely, a recursive solution for the first order differential equation, as represented by Equation (27), approximates quite well the analytical solution of the Lane–Emden type and at the time t = 10 yr a precision of four digits is reached when Δt = 10⁻³ yr. The goodness of the results as given by the solutions of the three differential equations is evaluated in Table 1. The smallest χ² is obtained by the Plummer-like (η = 6) profile, followed by the power-law profile and the Lane–Emden type (n = 5) profile. The two-piece fit has the smallest χ² but requires four parameters and does not have a physical basis. The disadvantages of the power-law dependence in the CSM are the following. (1) There is a two-piece dependence at r = r₀ which was introduced in order to avoid a pole, (2) there is no analytical solution, and (3) the series solution has a narrow range of reliability.

Self-gravitating medium. The previous analysis raises a question: is the CSM around an SN really self-gravitating? In order to answer this question, the CSM should be carefully analyzed by astronomers in order to detect the presence of gradients in density or pressure or temperature. In the case of a Lane–Emden (n = 5) CSM, the density of the medium around SN 1993J, see Equation (8), decreases by a factor ≈21 going from r₀ = 0.008 pc to r = 0.1 pc. Over the same distance, the pressure (see Equation (6)) decreases by a factor ≈40 and the temperature (see Equation (7)) by a factor ≈1.8. The presence or absence of a magnetic field should be also confirmed.

Nature of the CSM. The total mass obtained in the three models can be interpreted as the stellar mass ejected by the pre-SN right before the explosion or the stellar mass involved with the interaction of the binaries; see Table 3. The size of the pre-SN 1993J envelope, i.e., the FWHM, is an important data that could be related to the size of the Roche lobes of the hypothetical binary system; see Table 3. Here we have chosen, in the three models, a central density of n₀ = 10⁷ cm⁻³. As a comparison, Suzuki & Nomoto (1995), in order to model the X-ray observations, quotes a central density of 5.7 10⁻⁴g cm⁻³ at a distance of ${\approx } 10^{-5} \mathrm{\rm pc}$ which means n₀ = 2.45 10¹⁰cm⁻³. An astrophysical evaluation of n₇ can be done by imposing that the swept mass is a fraction f, 0 < f < 1, of the mass of the progenitor which is 5 M_☉ (see Woosley et al. 1994); in the case of the Emden profile, n₇ = 13.57 f. With this evaluation the velocity dispersions to maintain the dynamic equilibrium, as given by formula (17), are of the order of $3.161\,\sqrt{f} {\, \rm km\, s}^{-1}$. The previous table allows a fast evaluation of the final velocity, v, in the framework of the momentum conservation v = v₀((M(0.008; 0.00367)/M(0.1; 0.00367)) = 15437(0.1778/0.3686) km s⁻¹ = 7447 km s⁻¹.

Table 3. Numerical Values of the Swept Total Mass When n₀ = 10⁷cm⁻³ and the FWHM

Model	Total Mass	FWHM
Plummer profile, η = 6	M(0.1; 0.0045) = 0.402 n7 M☉	$0.0079\, {\mathrm{\rm pc}}$
Emden profile	M(0.008; 0.00367) = 0.178 n7 M☉	$0.0071\, {\mathrm{\rm pc}}$
Emden profile	M(0.1; 0.00367) = 0.368 n7 M☉	$0.0071\, {\mathrm{\rm pc}}$
Power-law profile	M(0.1; 0.0022, 2.93) = 0.217 n7 M☉	$0.0092\, {\mathrm{\rm pc}}$

Note. The parameter n₇ represents the number density expressed in 10⁷cm⁻³ units.

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Asymmetry. In the binary system scenario, both stars probably had a common envelope centered at the gravitational center of the system. Hence, the SN exploded at a distance ≈ a/2 (assuming two stars of similar masses) with respect to the center of the density distribution of the CSM, where a is the distance between the stars of the binary system. In the case of SN 1993J, the expansion shows a circular expansion over 10 yr and the asymmetry is under 2%. This means, in the binary system scenario, that the distance, d, between the initial point of the expansion and the center of the Lane–Emden (n = 5) CSM should be d ⩽ 0.037 b or d ⩽ 0.00013 pc. The distance a/2 is therefore equal to our parameter d which means a ⩽ 0.00026 pc.

Relativistic case. The temporal evolution of an SN in a self-gravitating medium of the Lane–Emden type can be found by applying the conservation of relativistic momentum in the thin layer approximation. This relativistic invariant is evaluated as a differential equation of the first order; see Equation (32). Three different relativistic solutions for the radius as a function of time are derived: (1) a numerical solution (see Equation (33)) which covers the range 10⁻⁴yr < t < 10 yr and fits the observed radius–time relation for SN 1993J ; (2) a series solution (see Equation (33)) which has a limited range of validity, 10⁻⁴yr < t < 1.2 × 10⁻² yr; and (3) a recursive solution (see Equation (38)) in which the desired accuracy is reached by decreasing the time step Δt. The relativistic results here presented model SN 1993J and are obtained with an initial velocity of v₀ = 100, 000 km s⁻¹, β₀ = 0.333, or γ = 1.06. The time dilation is evaluated and then applied to the decay of ⁵⁶Ni; see Figure 13. The relativistic kinetic energy of a proton is computed and the temporal evolution in Mev outlined; see Figure 15.