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    . Some generic information on Σ-D can be found in reviews 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  with the conclusion that SNRs in the galactic halo have diameters greater than those in the disk, the evolutionary properties of SNRs have been deduced from observations with the Molonglo and Parkes radio telescopes , the Σ-D relationship was used to fix the scale for distances of SNRs , the diameters, luminosities, surface brightness, galactic heights for 231 SNRs were processed in the framework of the Sedov solution , a new high-latitude SNR  was analysed, an updated radio Σ-D relationship was derived  and the phases of some supernova remnant have been determined from the observations . In order to derive the Σ-D relationship, some hypotheses on the single equation of motion should be made, e.g., the conservation of the momentum . Here conversely we will apply the energy conservation in the framework of the thin-layer approximation and we will derive an analytical expression for the Σ-D relationship. This paper derives two new equations of motion in the framework of energy conservation for the thin-layer approximation, see Section 2, applies the two analytical expressions for the surface brightness to two galactic samples, see Section 3, and to an extragalactic catalog, see Section 4.
2. The Equations of Motion
The conservation of kinetic energy in spherical coordinates in the framework of the thin-layer approximation, here taken to be an assumption, states that
where and are the swept masses at r0 and r, while v0 and v are the velocities of the thin layer at r0 and r. We now present two equations of motion for SNRs and the back-reaction for one of the two.
2.1. The Inverse Square Law
The medium around the SN is assumed to scale as an inverse square law
where is the density at and 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
The trajectory, i.e. the radius as a function of time, is
and the velocity as a function of time is
More details can be found in . The rate of transfer of mechanical energy, , is
where , and are the instantaneous density, radius and velocity of the SN. We now assume that the density in front of the advancing expansion scales as
where r0 is the radius at t0 and d is a parameter which allows matching the observations. The mechanical luminosity is now
In the case here analysed of the inverse square profile for density we have
We now assume that the observed luminosity, , in a given band denoted by the frequency is proportional to the mechanical luminosity
where is the observed radio luminosity in a given band and a constant which resolves the mismatch between theory and observations. The surface brightness is the luminosity divided by the interested area
2.2. Non-Cubic Dependence
The swept mass is assumed to scale as
where M0 is the swept mass at , and is a regulating parameter less than 3. The differential equation of the first order which regulates the motion is obtained by inserting the above in Equation (1)
which has as the solution
The velocity is
Figure 1 reports the analytical solution for the NCD case as given by Equation (16) for SN 1993J.
Figure 1. Analytical solution for the NCD case with parameters , , and , which gives . The astronomical data of SN 1993J are represented with vertical error bars.
The mechanical luminosity is assumed to scale as in Equation (8) and therefore in the NCD case is
The surface brightness is derived according to Equation (12) and in the NCD case is
A comparison of the two models here implemented for the trajectory is reported in Figure 2.
2.3. Non-Cubic Dependence and Back Reaction
The radiative losses per unit length are assumed to be proportional to the flux of momentum as an assumption
Figure 2. Analytical solution for the inverse square model (dashed line) when , , , and NCD model (full line) with the same parameters and .
where is a constant and is the density in the thin advancing layer.
The volume, V, of the advancing layer is
with , therefore the above density for the advancing layer is
Inserting in the above equation the velocity to the first order as given by Equation (18) the radiative losses, , are
The sum of the radiative losses between r0 and r is given by the following integral, L,
The conservation of energy in the presence of the back reaction due to the radiative losses is
An analytical solution for the velocity to second order, , is
The inclusion of the back reaction allows the evaluation of the SRS’s maximum length, , which can be derived by setting the above velocity equal to zero.
Figure 3 reports the finite radius of the advancing SNR as a function of .
3. Galactic Application
In the following we will process a sample of data with i varying between 1 and N by a power law fit of the type
where C and are two constants to be found from the sample.
Table 1 reports the minimum diameter, Dmin, the average diameter, , the maximum diameter, Dmax, the minimum Σ-D, Σ-Dmin, the average Σ-D, and the maximum Σ-D, Σ-Dmax as well as the two parameters of the power law fit.
A second source for the Σ-D relationship is the Green’s catalog  where the flux density in Jy at 1 GHz, S1, and the mayor and minor angular size in arcmin, , are reported for 295 SNRs. The surface-brightness in SI is
Figure 3. Length of the SNR, , when as function of .
Figure 4. Observed Σ-D relationship (empty stars) and fitted power law (full line) with parameters as in Table 1.
Table 1. Statistics of the observed Σ-D galactic relationship and the two parameters of the power law fit. The theoretical parameters of the inverse square solution are , , , , and .
but after  the radio astronomers use the following conversion
which will also be adopted here.
The probability density function (PDF), , to have an SNR as a function of the galactic height z is characterized by an exponential PDF
with . The linear diameter (D) of an SNR increases with the galactic height see Figure 2 in  and in the framework of the model with an inverse square law model for density the parameter r0 is chosen to have the following dependence with the galactic height
where 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 for all the SNRs will ensure a longer radius for SNRs with higher galactic heights, see Figure 5.
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 r0 according to the empirical Equation (41).
3) We randomly generate N times, t, according to the uniform distribution between a minimum value of time, tmin and a maximum value of time, tmax.
4) Given the parameters r0, v0, t0 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 Figure 6, were we matched the three main parameters ( ) which for the observations are (37.17, 2.16 × 10−19, −1.32) and for our simulation are (37.4, 2.24 × 10−19, −1.33).
4. Extragalactic Application
A sample of SNRs in nearby galaxies with diameter in pc and flux at 1.4 GHz in mJy has been collected . The connected catalog is available at http://cdsweb.u-strasbg.fr/. The statistics of the Σ-D relationship for this extragalactic sample is reported in Table 3 and displayed in Figure 9.
Figure 5. Theoretical radius as given by the inverse square model, see Equation (2), as function of the galactic height; , and .
Figure 6. Theoretical Σ-D relationship (empty stars) and fitted power law (full line) with parameters as in Table 1.
Figure 7. Histogram of the Σ-D for 294 SNR as derived from the Green’s catalog .
Figure 8. Histogram for the Σ-D of our simulation in the framework of the inverse square model.
Figure 9. Observed Σ-D relationship for the extragalactic case (empty stars) and fitting power law (full line) with parameters as in Table 3.
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 r0 on z. The numerical value of r0 in the extragalactic case is now randomly generated according to the uniform distribution between a minimum value, , and a maximum value, , see Table 3 and Figure 10.
Figure 10. Theoretical Σ-D relationship for the extragalactic case in the framework of the NCD model (empty stars) and fitting power law (full line) with parameters as in Table 3.
Table 2. Statistics of the observed  and simulated Σ in the framework of the inverse square model.
Table 3. Statistics of the observed Σ-D extragalactic relationship and the two parameters of the power law fit. The theoretical parameters for the NCD case are , , , , , , , and .
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:
 Gull, S.F. (1973) A Numerical Model of the Structure and Evolution of Young Supernova Remnants. Monthly Notices of the Royal Astronomical Society, 161, 47-69.
 Caswell, J.L. and Lerche, I. (1979) Supernova Remnants—A Generalized Theoretical Approach to the Radio Evolution. Proceedings of the Astronomical Society of Australia, 3, 343-347.
 Long, K.S. (2017) Galactic and Extragalactic Samples of Supernova Remnants: How They Are Identified and What They Tell Us. In: Alsabti, A.W. and Murdin, P., Eds., Handbook of Supernovae, Springer, Berlin, 1-36.
 Clark, D.H. and Caswell, J.L. (1976) A Study of Galactic Supernova Remnants, Based on Molonglo-Parkes Observational Data. Monthly Notices of the Royal Astronomical Society, 174, 267-305.
 Kothes, R., Reich, P., Foster, T.J. and Reich, W. (2017) G181.1+9.5, a New High-Latitude Low-Surface Brightness Supernova Remnant. Astronomy & Astrophysics, 597, Article No. A116. (Preprint)
 Vukotic, B., Ciprijanovic, A., Vucetic, M., Onic, D. and Urosevic, D. (2019) Updated Radio Sigma-D Relation for Galactic Supernova Remnants—II. Serbian Astronomical Journal, 199, 23-37.
 Hu, Y.P., Zeng, H.A., Fang, J., Hou, J.P. and Xu, J.W. (2019) Theoretical Σ-D Relations for Shell-Type Galactic Supernova Remnants. Journal of Astrophysics and Astronomy, 40, Article No. 7.
 Zaninetti, L. (2012) Analytical and Monte Carlo Results for the Surface-Brightness-Diameter Relationship in Supernova Remnants. Astrophysical Journal, 746, Article No. 56. (Preprint)
 Zaninetti, L. (2020) Energy Conservation in the Thin Layer Approximation: I. The Spherical Classic Case for Supernovae Remnants International Journal of Astronomy and Astrophysics, 10, 71-88. (Preprint)
 Urosević, D., Pannuti, T.G., Duric, N. and Theodorou, A. (2005) The Σ-D Relation for Supernova Remnants in Nearby Galaxies. Astronomy & Astrophysics, 435, 437-447.