_{1}

^{*}

Two new equations of motion for a supernova remnant (SNR) are derived in the framework of energy conservation for the thin-layer approximation. The first one is based on an inverse square law for the surrounding density and the second one on a non-cubic dependence of the swept mass. Under the assumption that the observed radio-flux scales as the flux of kinetic energy, two scaling laws are derived for the temporal evolution of the surface brightness of SNRs. The astrophysical applications cover two galactic samples of surface brightness and an extragalactic one.

The surface brightness versus diameter, (Σ-D), for supernova remnants (SNRs) was initially analysed from a theoretical point of view in the framework of the initial conditions for the time evolution of SNRs [

The astrophysical approach to the Σ-D has always mixed the observations with the theory and the statistics. We select some items among others: a catalog of 25 SNRs has been compiled by [

The conservation of kinetic energy in spherical coordinates in the framework of the thin-layer approximation, here taken to be an assumption, states that

1 2 M 0 ( r 0 ) v 0 2 = 1 2 M ( r ) v 2 , (1)

where M 0 ( r 0 ) and M ( r ) are the swept masses at r_{0} and r, while v_{0} and v are the velocities of the thin layer at r_{0} and r. We now present two equations of motion for SNRs and the back-reaction for one of the two.

The medium around the SN is assumed to scale as an inverse square law

ρ ( r ; r 0 ) = { ρ c if r ≤ r 0 ρ c ( r 0 r ) 2 if r > r 0 (2)

where ρ c is the density at r = 0 and r 0 is the radius after which the density starts to decrease. When the conservation of energy is applied, the velocity as a function of the radius is

v ( r ; r 0 , v 0 ) = − − ( 2 r 0 − 3 r ) r 0 v 0 2 r 0 − 3 r . (3)

The trajectory, i.e. the radius as a function of time, is

r ( t ; t 0 , r 0 , v 0 ) = 1 6 2 3 r 0 3 ( ( 9 t − 9 t 0 ) v 0 + 2 r 0 ) 2 / 3 + 2 3 r 0 , (4)

and the velocity as a function of time is

v ( t ; t 0 , r 0 , v 0 ) = 2 3 r 0 3 v 0 ( 9 t − 9 t 0 ) v 0 + 2 r 0 3 . (5)

More details can be found in [

L m ( t ) = 1 2 ρ ( t ) 4 π r ( t ) 2 v ( t ) 3 , (6)

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

ρ ( t ) = ρ 0 ( r 0 r ( t ) ) d , (7)

where r_{0} is the radius at t_{0} and d is a parameter which allows matching the observations. The mechanical luminosity is now

L m ( t ) = 1 2 ρ 0 ( r 0 r ( t ) ) d 4 π r ( t ) 2 v ( t ) 3 . (8)

In the case here analysed of the inverse square profile for density we have

L m ( t ; v 0 , t 0 , r 0 ) = D L M 81 v 0 t − 81 v 0 t 0 + 18 r 0 , (9)

where

D L M = ρ 0 6 d ( r 0 ( 2 3 r 0 3 ( ( 9 t − 9 t 0 ) v 0 + 2 r 0 ) 2 3 + 4 r 0 ) − 1 ) d × π ( 2 3 r 0 3 ( ( 9 t − 9 t 0 ) v 0 + 2 r 0 ) 2 3 + 4 r 0 ) 2 r 0 v 0 3 . (10)

We now assume that the observed luminosity, L ν , in a given band denoted by the frequency ν is proportional to the mechanical luminosity

L ν ( t ) = c o s t ∗ L m ( t ) , (11)

where L ν is the observed radio luminosity in a given band and c o s t a constant which resolves the mismatch between theory and observations. The surface brightness is the luminosity divided by the interested area

Σ = L ν ( t ) π r ( t ) 2 , (12)

which is

Σ ( t ; v 0 , t 0 , r 0 ) = c o s t 4 ρ 0 6 d ( r 0 2 3 r 0 3 ( ( 9 t − 9 t 0 ) v 0 + 2 r 0 ) 2 / 3 + 4 r 0 ) d r 0 v 0 3 9 v 0 t − 9 v 0 t 0 + 2 r 0 . (13)

The swept mass is assumed to scale as

M ( r ; r 0 , δ ) = { M 0 if r ≤ r 0 M 0 ( r r 0 ) δ if r > r 0 (14)

where M_{0} is the swept mass at r = r 0 , and δ is a regulating parameter less than 3. The differential equation of the first order which regulates the motion is obtained by inserting the above M ( r ) in Equation (1)

d r ( t ; r 0 , v 0 , δ ) d t = v 0 r δ r 0 − δ , (15)

which has as the solution

r ( t ; r 0 , v 0 , δ ) = exp ( E R δ + 2 ) (16)

where

E R = ln ( r 0 δ + 1 δ t v 0 − r 0 δ + 1 δ t 0 v 0 + 2 r 0 δ + 1 t v 0 − 2 r 0 δ + 1 t 0 v 0 + r 0 δ + 2 + r 0 δ δ 2 t 2 v 0 2 4 − r 0 δ δ 2 t t 0 v 0 2 2 + r 0 δ δ 2 t 0 2 v 0 2 4 + r 0 δ δ t 2 v 0 2 − 2 r 0 δ δ t t 0 v 0 2 + r 0 δ δ t 0 2 v 0 2 + r 0 δ t 2 v 0 2 − 2 r 0 δ t t 0 v 0 2 + r 0 δ t 0 2 v 0 2 ) . (17)

The velocity is

v ( t ; r 0 , v 0 , δ ) = N V ( δ + 2 ) ( δ t v 0 − δ t 0 v 0 + 2 t v 0 − 2 t 0 v 0 + 2 r 0 ) 2 (18)

where

N V = v 0 ( r 0 δ ( v 0 ( δ + 2 ) ( t − t 0 ) + 2 r 0 ) 2 ) ( δ + 2 ) − 1 × ( 24 − ( δ + 2 ) − 1 δ 2 t v 0 − 24 − ( δ + 2 ) − 1 δ 2 t 0 v 0 + 84 − ( δ + 2 ) − 1 δ t v 0 − 84 − ( δ + 2 ) − 1 δ t 0 v 0 + 4 δ + 1 δ + 2 δ r 0 + 84 − ( δ + 2 ) − 1 t v 0 − 84 − ( δ + 2 ) − 1 t 0 v 0 + 84 − ( δ + 2 ) − 1 r 0 ) . (19)

The mechanical luminosity is assumed to scale as in Equation (8) and therefore in the NCD case is

L m ( t ; r 0 , v 0 , t 0 , δ ) = 16 e E A v 0 3 ( r 0 e E B ) d π ρ 0 ( δ + 2 ) 3 ( v 0 ( δ + 2 ) ( t − t 0 ) + 2 r 0 ) 6 S 2 3 ( δ + 2 ) − 1 × ( S 1 4 δ + 1 δ + 2 + 4 − ( δ + 2 ) − 1 v 0 δ 2 ( t − t 0 ) ) 3 , (20)

where

E A = 1 δ + 2 ( − 4 ln ( 2 ) + 2 ln ( 4 v 0 ( δ + 2 ) ( t − t 0 ) r 0 δ + 1 + 4 r 0 δ + 2 + ( ( t 2 − 2 t t 0 + t 0 2 ) δ 2 + ( 4 t 2 − 8 t t 0 + 4 t 0 2 ) δ + 4 t 2 − 8 t t 0 + 4 t 0 2 ) v 0 2 r 0 δ ) ) , (21)

and

E B = 2 ln ( 2 ) − ln ( ( v 0 ( δ + 2 ) ( t − t 0 ) + 2 r 0 ) 2 r 0 δ ) δ + 2 , (22)

S 1 = ( δ + 1 ) ( t − t 0 ) v 0 + r 0 ( δ + 2 ) 2 , (23)

and

S 2 = ( v 0 ( δ + 2 ) ( t − t 0 ) + 2 r 0 ) 2 r 0 δ . (24)

The surface brightness is derived according to Equation (12) and in the NCD case is

Σ ( t ; r 0 , v 0 , t 0 , δ ) = c o s t 16 ρ 0 ( 4 − ( δ + 2 ) − 1 v 0 δ 2 t − 4 − ( δ + 2 ) − 1 v 0 δ 2 t 0 + S C 4 δ + 1 δ + 2 ) 3 e S A v 0 3 r 0 ( d + 3 ) δ + 2 d δ + 2 S B − 6 − 6 δ δ + 2 ( δ + 2 ) 3 , (25)

where

S A = − d ( − 2 ln ( 2 ) + 2 ln ( v 0 ( δ + 2 ) ( t − t 0 ) + 2 r 0 ) + δ ln ( r 0 ) ) δ + 2 , (26)

S B = v 0 ( δ + 2 ) ( t − t 0 ) + 2 r 0 (27)

and

S C = ( δ + 1 ) ( t − t 0 ) v 0 + r 0 ( δ + 2 ) 2 . (28)

A comparison of the two models here implemented for the trajectory is reported in

The radiative losses per unit length are assumed to be proportional to the flux of momentum as an assumption

− ε ρ s v 2 4 π r 2 , (29)

where ε is a constant and ρ s is the density in the thin advancing layer.

The volume, V, of the advancing layer is

V = 4 π r 2 Δ r (30)

with Δ r = r / 12 , therefore the above density for the advancing layer is

ρ s = M ( r ; r 0 , δ ) V . (31)

Inserting in the above equation the velocity to the first order as given by Equation (18) the radiative losses, Q ( r ; r 0 , v 0 , δ , ε ) , are

Q ( r ; r 0 , v 0 , δ , ε ) = − 12 ε M 0 v 0 2 r . (32)

The sum of the radiative losses between r_{0} and r is given by the following integral, L,

L ( r ; r 0 , v 0 , δ , ε ) = ∫ r 0 r Q ( r ; r 0 , v 0 , δ , ε ) d r = − 12 ε M 0 v 0 2 ln ( r ) + 12 ε M 0 v 0 2 ln ( r 0 ) . (33)

The conservation of energy in the presence of the back reaction due to the radiative losses is

2 π r 0 3 v 2 3 ( r r 0 ) δ + 16 ε π r 0 3 v 0 2 ln ( r ) − 16 ε π r 0 3 v 0 2 ln ( r 0 ) = 2 π r 0 3 v 0 2 3 . (34)

An analytical solution for the velocity to second order, v c ( r ; r 0 , c 0 , δ , ε ) , is

v c ( r ; r 0 , v 0 , δ , ε ) = r − δ 2 r 0 δ 2 24 ln ( r 0 ) ε − 24 ln ( r ) ε + 1 v 0 . (35)

The inclusion of the back reaction allows the evaluation of the SRS’s maximum length, r b a c k ( r 0 , δ , ε ) , which can be derived by setting the above velocity equal to zero.

r b a c k ( r 0 , δ , ε ) = e 24 ln ( r 0 ) ε + 1 24 ε . (36)

In the following we will process a sample of data x i , y i with i varying between 1 and N by a power law fit of the type

y ( x ) = C x α , (37)

where C and α are two constants to be found from the sample.

The first source for the observed Σ-D relationship for the SNRs of our galaxy can be found in

_{min}, the average diameter, D ¯ , the maximum diameter, D_{max}, the minimum Σ-D, Σ-D_{min}, the average Σ-D, Σ - D ¯ and the maximum Σ-D, Σ-D_{max} as well as the two parameters of the power law fit.

A second source for the Σ-D relationship is the Green’s catalog [_{1}, and the mayor and minor angular size in arcmin, θ , are reported for 295 SNRs. The surface-brightness in SI is

parameter | observed | theoretical |
---|---|---|

D min ( pc ) | 1.18 | 3.44 |

D ¯ ( pc ) | 37.17 | 37.4 |

D max ( pc ) | 227.17 | 86.708 |

Σ min ( pc ) | 5.96 × 10^{−}^{23} | 3 × 10^{−}^{20} |

Σ ¯ ( pc ) | 2.16 × 10^{−}^{19} | 2.24 × 10^{−}^{19} |

Σ max ( pc ) | 1.66 × 10^{−}^{17} | 2.54 × 10^{−}^{19} |

C | 3.83 × 10^{−}^{19} | 1.32 × 10^{−}^{17} |

α | −1.32 | −1.33 |

Σ = 1.181 × 10 − 19 S 1 θ W m 2 ⋅ Hz ⋅ sr SI , (38)

but after [

Σ = 1.505 × 10 − 19 S 1 θ W m 2 ⋅ Hz ⋅ sr astronomy , (39)

which will also be adopted here.

The probability density function (PDF), p ( z ) , to have an SNR as a function of the galactic height z is characterized by an exponential PDF

p ( z ) = 1 b exp − z b , (40)

with b = 83 pc [_{0} is chosen to have the following dependence with the galactic height

r 0 = 0.01 + 0.5 ( z z max ) , (41)

where z max = 687 is the maximum galactic height which belongs to an SNR. The above assumption coupled with an initial velocity for the inverse square law model for density of v 0 = 4000 km / s for all the SNRs will ensure a longer radius for SNRs with higher galactic heights, see

We now simulate the galactic Σ-D relationship for a number of theoretical SNRs, N, equal to that observed according to the following rules

1) We randomly generate N parameters z according to the exponential PDF (40).

2) At each random value of z we associate an initial parameter r_{0} according to the empirical Equation (41).

3) We randomly generate N times, t, according to the uniform distribution between a minimum value of time, t_{min} and a maximum value of time, t_{max}.

4) Given the parameters r_{0}, v_{0}, t_{0} and t we evaluate the radius according to Equation (4) for an inverse power law profile for density. The diameter is obtained doubling the above result.

5) Σ is now generated according to Equation (13) once the regulating parameter d is provided.

The theoretical display of the results is reported in ^{−}^{19}, −1.32) and for our simulation are (37.4, 2.24 × 10^{−}^{19}, −1.33).

The surface brightness of the Green’s catalog is reported in

A sample of SNRs in nearby galaxies with diameter in pc and flux at 1.4 GHz in mJy has been collected [

The theoretical Σ-D is now evaluated in the framework of the NCD model, see Equation (25), with a numerical procedure which is similar to that of the galactic case but with the difference that there is no dependence of r_{0} on z. The numerical value of r_{0} in the extragalactic case is now randomly generated according to the uniform distribution between a minimum value, r 0 , min , and a maximum value, r 0 , max , see

parameter | observed | theoretical |
---|---|---|

Σ min ( pc ) | 1.42 × 10^{−}^{25} | 1.27 × 10^{−}^{21} |

Σ ¯ ( pc ) | 1.13 × 10^{−}^{18} | 1.13 × 10^{−}^{18} |

Σ max ( pc ) | 4.34 × 10^{−}^{17} | 6.17 × 10^{−}^{17} |

parameter | observed | theoretical |
---|---|---|

D min ( pc ) | 0.51 | 4.05 |

D ¯ ( pc ) | 34.6 | 34.23 |

D max ( pc ) | 450 | 54.39 |

Σ min ( pc ) | 2.4 × 10^{−}^{22} | 1.87 × 10^{−}^{17} |

Σ ¯ ( pc ) | 6.13 × 10^{−}^{16} | 8.82 × 10^{−}^{16} |

Σ max ( pc ) | 8.6 × 10^{−}^{14} | 4.14 × 10^{−}^{14} |

C | 8.85 × 10^{−}^{16} | 2.64 × 10^{−}^{12} |

α | −3.1 | −2.96 |

When an analytical law of motion is available, a theoretical Σ-D relationship can be derived as a function of time. Here, in the framework of the energy conservation in the thin-layer approximation, we have derived one formula for Σ-D when the density of the surrounding ISM scales as an inverse square law, see Equation (13), and another formula for the NCD case, see Equation (25). The two formulae allow simulating:

1) The galactic Σ-D relationship as given by the data of [

2) The galactic Σ-D relationship as given by a catalog of SNRs [

3) An extragalactic Σ-D catalog [

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

Zaninetti, L. (2021) Energy Conservation in the Thin-Layer Approximation: V. The Surface Brightness in Supernova Remnants. International Journal of Astronomy and Astrophysics, 11, 252-264. https://doi.org/10.4236/ijaa.2021.112013