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The light curves (LC) for Supernova (SN) can be modeled adopting the conversion of the flux of kinetic energy into radiation. This conversion requires an analytical or a numerical law of motion for the expanding radius of the SN. In the framework of conservation of energy for the thin layer approximation, we present a classical trajectory based on a power law profile for the density, a relativistic trajectory based on the Navarro-Frenk-White profile for the density, and a relativistic trajectory based on a power law behaviour for the swept mass. A detailed simulation of the LC requires the evaluation of the optical depth as a function of time. We modeled the LC of SN 1993J in different astronomical bands, the LC of GRB 050814 and the LC GRB 060729 in the keV region. The time dependence of the magnetic field of equipartition is derived from the theoretical formula for the luminosity.

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 supernova 2018oh has an unusual two-component shape [^{56}Co, ^{57}Co and ^{55}Fe [^{56}Ni decay [

· Given the observational fact that the radius-time relation in young SNRs follows a power law, is it possible 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 characterized by a decreasing density?

· Can we parametrize the conversion of the analytical or numerical flux of kinetic energy into the observed luminosity?

· Can we model the double-peak profile for the LC in the framework of the temporal variations of the optical 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 reviews the classical and relativistic conversion of the flux of kinetic energy into luminosity. Section 4 presents some analytical results for a classical law of motion, Section 5 introduces two new relativistic equations of motion, Section 6 presents the simulation of the LC for one SN and two GRBs and Section 7 presents the temporal evolution of the magnetic field as well as some evaluations for the accelerating clouds due to the Fermi II acceleration mechanism.

This section presents the analysed SN and GRB, introduces the adopted statistics, and reviews the power law 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 measured for ≈10 yr in the radio band [

The second object to be analysed is GRB 050814 at 0.3 - 10 keV, which covers the time interval [10^{−}^{5}-3] days, see [

The third object to be analysed is GRB 060729 observed by the Ultraviolet and Optical Telescope (UVOT) in the time interval [10^{−}^{2}-26] days, see

The adopted statistical parameters are the percent error, δ , between the theoretical value and approximate value, and the merit function χ 2 evaluated as

χ 2 = ∑ i = 1 N [ y i , t h e o − y i , o b s σ i ] 2 (1)

where y i , o b s and σ i represent the observed value and its error at position i, y i , t h e o is the theoretical value at 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 ) = C t α f i t , (2)

wherer is the radius of the expansion, t is the time, and α f i t is an exponent which can be found numerically. The velocity is

v ( t ) = C t α f i t − 1 α f i t . (3)

As a practical example, the radius (pc) time (yr) relation in SN 1993J is

r ( t ) = 0.0155 × t 0.828 pc , (4)

when 0.49 yr < t < 10.58 yr , see also

In these subsections we analyse the classical and relativistic conversion of the flux of kinetic energy into luminosity. The absorption of the produced radiation is parametrized by the optical thickness.

In the classical case, the rate of transfer of mechanical energy, L m , is

L m ( t ) = 1 2 ρ ( t ) 4 π r ( t ) 2 v ( t ) 3 , (5)

where ρ ( t ) , r ( t ) and v ( t ) are the temporary density, radius and velocity of the SN. We assume that the density in front of the advancing expansion scales as

ρ ( t ) = ρ 0 ( r 0 r ( t ) ) d , (6)

where r 0 is the radius at t 0 and d is a parameter which allows matching the observations; as an example, a value of d = 3 is reported in [

L m ( t ) = 1 2 ρ 0 ( r 0 r ( t ) ) d 4 π r ( t ) 2 v ( t ) 3 . (7)

The mechanical luminosity in the case of a power law dependence for the radius is

L m ( t ) = 2 ρ 0 r 0 d C f i t − d + 5 t − 3 + ( − d + 5 ) α f i t π α f i t 3 . (8)

The energy fraction of the mechanical luminosity deposited in the frequency ν , L ν , is assumed to be proportional to the mechanical luminosity through a constant ϵ ν

L ν = ϵ ν L m . (9)

The flux at frequency ν and distance D is

F ν = ϵ ν L m 4 π D 2 . (10)

For practical purposes, we impose a match between the observed luminosity, L o b s , and the theoretical luminosity, L m ,

L o b s = C o b s L m , (11)

where C o b s is a constant which equalizes the observed and the theoretical luminosity and varies on the base of the selected astronomical band. In a analogous way, the observed absolute magnitude is

M o b s = − log 10 ( L m ) + k o b s , (12)

where k o b s is a constant. In the relativistic case the rate of transfer of mechanical energy, L m , r , assuming the same scaling for the density in the advancing layer, is

L m , r ( t ) = 4 π r ( t ) 2 ρ 0 c 3 β ( t ) 1 − β ( t ) 2 ( r 0 r ) d , (13)

where β ( t ) = v ( t ) c , for more details, see [

A useful formula is that for the minimum magnetic field density, B min ,

B min = 1.8 ( η L ν V ) 2 / 7 ν 1 / 7 T , (14)

where ν is the considered frequency of synchrotron emission, L ν is the luminosity of the radio source at ν , V is the volume involved, and η = ϵ t o t a l ϵ e is a

constant which connects the relativistic energy of the electrons, ϵ e , with the total energy in non-thermal phenomena, ϵ t o t a l , see formula (16.50) in [

The presence of the absorption can be parametrized introducing a slab of optical thickness τ ν . The emergent intensity I ν after the entire slab is

I ν = ∫ 0 τ ν S ν e − t d t , (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 [

L o b s = C o b s 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 o b s = − log 10 ( L m ) − log 10 ( 1 − e − τ ν ) + k o b s . (18)

The value of τ ν can be derived with the following equation:

τ ν = − ln ( 1 − e − ( m o b s − m t h e o ) ln ( 10 ) ) (19)

where m t h e o and m o b s represent the theoretical and the observed apparent magnitude. Due to the complexity of the time dependence of τ ν , a polynomial approximation of degree M is used:

τ ν ( t ) = a 0 + a 1 t + a 2 t 2 + ⋯ + a M t M , (20)

with more details in [

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 o b s = C o b s L m , r ( 1 − e − τ ν ) , (21)

and

m o b s = − log 10 ( L m , r ) − log 10 ( 1 − e − τ ν ) + k o b s . (22)

Let us analyse the case of conservation of energy in the thin layer approximation in the presence of a power law profile of density of the type

ρ ( r ; r 0 ) = { ρ c if r ≤ r 0 ρ c ( r 0 r ) α if r > r 0 (23)

where ρ c is the density at r = 0 , r 0 is the radius after which the density starts to decrease and α > 0 , see Section 3.5 of [

r ( t ) = 12 ( α − 5 ) − 1 r 0 α − 3 α − 5 ( − 4 r 0 v 0 ( α − 5 ) ( t − t 0 ) 9 − 3 α − ( α − 3 ) ( α − 5 ) 2 ( t − t 0 ) 2 v 0 2 + 12 r 0 2 ) − ( α − 5 ) − 1 , (24)

and the asymptotic velocity

v ( t ) = 2 ( − 4 r 0 v 0 ( α − 5 ) ( t − t 0 ) 9 − 3 α − ( α − 3 ) ( α − 5 ) 2 ( t − t 0 ) 2 v 0 2 + 12 r 0 2 ) 4 − α α − 5 × ( 2 r 0 2 α − 8 α − 5 9 − 3 α + v 0 r 0 α − 3 α − 5 ( α − 3 ) ( α − 5 ) ( t − t 0 ) ) 12 ( α − 5 ) − 1 v 0 . (25)

An example of trajectory is reported in

model | values | χ 2 |
---|---|---|

Fit by a power law | α f i t = 0.828 ; C = 0.015 ; | 43 |

Classic power law profile | α = 2.5 ; r 0 = 1.0 × 10 − 5 pc ; t 0 = 5 × 10 − 4 yr ; v 0 = 20000 km / s | 176.6 |

Relativistic NFW | b = 0.00185 pc ; r 0 = 1 × 10 − 4 pc ; t 0 = 3.6 × 10 − 4 yr ; v 0 = 269813 km / s | 823 |

Relativistic NCD | δ = 1.16 ; r 0 = 5 × 10 − 5 pc ; t 0 = 1.8 × 10 − 4 yr ; v 0 = 269813 km / s | 9589 |

As a consequence, we may derive an expression for the theoretical luminosity in presence of an inverse power law profile, L t h e o , based on Equations (7) and (11)

L t h e o = ρ 0 128 r 0 − 2 d + 5 α − 15 α − 5 v 0 3 12 5 − d α − 5 ( − 4 r 0 v 0 ( α − 5 ) ( t − t 0 ) 9 − 3 α − ( α − 3 ) ( α − 5 ) 2 ( t − t 0 ) 2 v 0 2 + 12 r 0 2 ) d + 10 − 3 α α − 5 π × ( r 0 9 − 3 α + v 0 ( α − 3 ) ( α − 5 ) ( t − t 0 ) 2 ) 3 . (26)

The above luminosity is based on theoretical arguments and no fitting procedure was used. The observed luminosity, L o b s , can be obtained introducing

L o b s = C o b s × L t h e o , (27)

where C o b s is a constant. Similarly,

M o b s = − log 10 ( L t h e o ) + k o b s . (28)

The relativistic conservation of kinetic energy in the thin layer approximation as derived in [

M 0 ( r 0 ) c 2 ( γ 0 − 1 ) = M ( r ) c 2 ( γ − 1 ) , (29)

where M 0 ( r 0 ) and M ( r ) are the swept masses at the two radii r 0 and r respectively, γ 0 = 1 1 − β 0 2 and β 0 = v 0 c .

We assume that the medium around the SN scales as the Navarro-Frenk-White (NFW) profile:

ρ ( r ; r 0 , b ) = { ρ c if r ≤ r 0 ρ c r 0 ( b + r 0 ) 2 r ( b + r ) 2 if r > r 0 (30)

where ρ c is the density at r = 0 and r 0 is the radius after which the density starts to decrease, see [

M ( r ; r 0 , ρ c , b ) = 4 ρ c π r 0 3 3 + 4 ρ c ( b + r 0 ) 2 ( ( b + r ) ln ( b + r ) + b ) r 0 π b + r − 4 ρ c ( b + r 0 ) ( ( b + r 0 ) ln ( b + r 0 ) + b ) r 0 π . (31)

Inserting the above mass in Equation (29) makes it possible to derive the velocity of the trajectory as a function 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

solution.

Conversely, we present an approximate solution as a third-order Taylor series expansion about r = r 0

r ( t ; r 0 , v 0 , t 0 , b ) = 1 2 r 0 c ( 3 ( t − t 0 ) 2 ( − v 0 + c ) 2 ( v 0 + c ) 2 ( c 2 − v 0 2 ) − 1 − 3 c ( ( t − t 0 ) 2 c 2 − t 0 2 v 0 2 + ( 2 t v 0 2 + 2 / 3 r 0 v 0 ) t 0 − t 2 v 0 2 − 2 / 3 t r 0 v 0 − 2 / 3 r 0 2 ) ) . (32)

^{−}^{4} yr - 10^{−}^{3} yr].

We assume that the swept mass scales as

M ( r ; r 0 , δ ) = { M 0 if r ≤ r 0 M 0 ( r r 0 ) δ if r > r 0 (33)

where M 0 is the swept mass at r = 0 , r 0 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 the motion is obtained inserting the above M ( r ) in Equation (29)

d r ( t ; r 0 , v 0 , c , δ ) d t = A N A D , (34)

where

A N = ( 16 ( c − v 0 ) c ( r 0 − 2 δ ( − 5 / 8 c 2 + 5 / 8 v 0 2 ) ( r ( t ) ) 2 δ + r 0 − 3 δ ( 1 / 8 c 2 − 1 / 8 v 0 2 ) ( r ( t ) ) 3 δ + ( r ( t ) ) δ ( c 2 − 3 / 4 v 0 2 ) r 0 − 2 δ − 1 / 2 c 2 + 1 / 4 v 0 2 ) ( c + v 0 ) ( c 2 − v 0 2 ) − 1 + ( 10 c 4 − 15 c 2 v 0 2 + 5 v 0 4 ) r 0 − 2 δ ( r ( t ) ) 2 δ − 2 r 0 − 3 δ ( c − v 0 ) 2 ( c + v 0 ) 2 ( r ( t ) ) 3 δ + ( − 16 c 4 + 20 c 2 v 0 2 − 4 v 0 4 ) ( r ( t ) ) δ r 0 − δ + 8 c 4 − 8 c 2 v 0 2 + v 0 4 ) 1 / 2 c , (35)

and

A D = 2 c ( c − v 0 ) ( c + v 0 ) ( r 0 − δ ( r ( t ) ) δ − 1 ) ( c 2 − v 0 2 ) − 1 + r 0 − 2 δ ( c 2 − v 0 2 ) ( r ( t ) ) 2 δ + ( − 2 c 2 + 2 v 0 2 ) ( r ( t ) ) δ r 0 − δ + 2 c 2 − v 0 2 . (36)

The above differential does not have an analytical solution and therefore the solution should be derived numerically except about r = r 0 where a third-order Taylor series expansion gives

r ( t ; r 0 , v 0 , t 0 , δ ) = r 0 + v 0 ( t − t 0 ) + δ ( c − v 0 ) ( c + v 0 ) ( t − t 0 ) 2 2 c r 0 × ( c 2 − c c 2 − v 0 2 − v 0 2 ) 1 c 2 − v 0 2 . (37)

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 Section 4.

We present the H − α with soft and hard band X-ray luminosities as well the theoretical luminosity in

1993J at 15.2 GHz observed by the Ryle Telescope as well the theoretical flux, which requires a time dependent evaluation of the optical depth τ ν , see

In this subsection we adopt a relativistic equation of motion with an NFW profile for the density, see Section 5.1.

In this subsection we adopt a relativistic equation of motion for the NCD case, see Section 5.2.

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 π D 2 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 = 1.5368 c 12 2 / 7 L 2 / 7 ( 1.0 + k ) 2 / 7 Φ 2 / 7 R 6 / 7 gauss , (40)

where

c 12 = 2 c 1 ( − 1 + α r ) ( ν 1 ν 2 α r − ν 1 α r ν 2 ) c 2 ( ν 1 ν 2 α r − ν 2 ν 1 α r ) ( − 1 + 2 α r ) , (41)

where α r is the spectral index, c 1 and c 2 are two constants, ν 1 and ν 2 are the lower and upper frequency of synchrotron emission, L is the luminosity in ergs^{−}^{1}, k is the ratio between energy in heavy particle and electron energy, Φ is the fraction of source’s volume occupied by the relativistic electrons and the magnetic field, and R is the radius of the source; for more details see formula (7.14) in [

H min ∝ L 2 / 7 R 6 / 7 . (42)

The first example presents the temporal behaviour of H min for GRB 050814 in which we inserted the theoretical luminosity corrected for absorption, see

The second example presents the temporal behaviour of H min for SN 1993J in which we inserted the theoretical luminosity as given by the power law fit, see

An electron which loses its energy due to the synchrotron radiation has a lifetime of

τ r ≈ E P r ≈ 500 E − 1 H − 2 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 [

ν c = 6.266 × 10 18 H E 2 Hz . (44)

The lifetime for synchrotron losses is

τ s y n = 39660 1 H H ν yr . (45)

Following [

d E d t = E τ I I , (46)

where τ I I is the typical time scale,

1 τ I I = 4 3 ( u 2 c 2 ) ( c L I I ) , (47)

where u is the velocity of the accelerating cloud belonging to the advancing shell of the GRB, c is the speed of light and L I I is the mean free path between clouds, see Eq. 4.439 in [

τ I I < τ s y n c , (48)

which corresponds to the following inequality for the mean free path between scatterers

L < 16182.11 u 2 H 3 / 2 ν c 2 pc . (49)

The mean free path length for a GRB which emits synchrotron emission around 1 keV (2.417 × 10^{17} Hz) is

L < 3.2908 × 10 − 5 β 2 H 3 / 2 E ( keV ) pc (50)

where β is the velocity of the cloud divided by the speed of light. When this inequality is fulfilled, the direct conversion of the rate of kinetic energy into radiation can be adopted. ^{−}^{5} to 1.5 × 10^{−}^{4} 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 the presence of a power law profile for the density, see Equation (26). The relativistic flux of energy in the two 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, conservation of energy or momentum, is presented in

regime | conservation | model | χ^{2} | Reference |
---|---|---|---|---|

classical | momentum | inverse power law | 276 | |

classical | momentum | Plummer profile | 265 | |

relativistic | momentum | Lane-Emden profile | 471 | |

relativistic | energy | power law profile | 6387 | |

relativistic | energy | exponential profile | 13,145 | |

relativistic | energy | Emden profile | 8888 | |

classical | energy | power law profile | 176.6 | |

relativistic | energy | NFW profile | 823 | |

relativistic | energy | NCD | 9589 |

The best results are obtained for the energy conservation in the presence of a power law profile in the present paper, see Equation (24).

Light curve:

The luminosity in the various astronomical bands is here assumed to be proportional to the classical or relativistic flux of mechanical kinetic energy. This theoretical dependence is not enough and the concept of optical depth should be introduced. Due to the complexity of the time dependence of the optical depth, 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 astronomical bands coupled with a time dependence for the optical depth allowed simulating the various morphologies of the LC of supernovae. In particular, in the case of SN 1993J we modeled: (i) the R LC of SN 1993J over 10 yr, see

Magnetic field

The minimum magnetic field depends on the luminosity and this allows to derive its theoretical dependence on time, see

The author declares no conflicts of interest regarding the publication of this paper.

Zaninetti, L. (2021) Energy Conservation in the Thin Layer Approximation: IV. The Light Curve for Supernovae. International Journal of Astronomy and Astrophysics, 11, 37-58. https://doi.org/10.4236/ijaa.2021.111003