Supernova remnant model theory¶
For the calculation of SNR models we follow Kaastra & Jansen (1993). But also here we make some modifications in the implementation.
The X-ray spectrum of a SNR is characterised in lowest order by its average temperature \(T\), ionisation parameter \(U\) and emission measure \(Y\). If one calculates a specific model however, these average values are not known a priori. A relevant set of input parameters could be the interstellar density, the shock radius and the explosion energy. These are the parameters used by KJ. A disadvantage is that this choice is not suitable for spectral fitting: changing one of the parameters gives correlated changes in \(T\), \(U\) and \(Y\), which is an undesirable feature in spectral fitting. KJ solved this by inputting a guess of \(T\), \(U\), and \(Y\) and using these guess parameters as the fitting parameters. After fitting, the model had to be calculated another time in order to find the relation between guessed and true parameters. here we take another choice. We will select input parameters which scale with \(T\), \(U\) and \(Y\) respectively but are connected to the SNR parameters independent of the hydrodynamical model. The first parameter is the shock temperature
the second the shock ionisation parameter defined by
where \(n_{\mathrm e}\) is the electron density before the shock (for a completely ionised gas), and \(t\) the age of the remnant. Further we define the shock reduced emission measure by
where \(r_s\) is the main shock radius, and \(d\) the distance to the source. We have introduced here also the electron density \(n_e\) instead of the more natural Hydrogen density, because this allows an efficient use of the model for spectral fitting: if one fits a spectrum with variable chemical abundances, then using the current scaling we need to calculate (for a given \(T_s\), \(U_s\) and \(Y_s\)) the relative ion concentrations only once; for other abundances, the relative concentrations may be simply scaled without a need to redo completely the ionisation balance calculations. Finally we introduced the distance in the definition of \(Y_s\) in order to allow an easy comparison with spectra observed at earth. These 3 input parameters \(T_s\), \(U_s\) and \(Y_s\) will serve as input parameters for all hydrodynamical models. They are linked to other parameters as follows:
where \(\epsilon_\rho\) is a dimensionless factor representing the average mass in units of a proton mass per Hydrogen atom (\(\rho = \epsilon_\rho m_{\mathrm p}n_{\mathrm H}\) with \(m_{\mathrm p}\) the proton mass), and \(f_{\mathrm p}\) is another dimensionless factor, representing the ratio of electron plus ion density over the Hydrogen density. Further, \(f_{\mathrm T}\) is a model-dependent dimensionless scaling factor which is given by
with \(f_v\) the velocity scaling discussed below and \(\eta\) the density jump at the main shock front, which depends upon the Hydrodynamical model. Another scaling law links the gas velocity to the temperature:
with \(v_2\) the post-shock gas velocity (not to be confused with the shock velocity!). The velocity scaling factor is given by
with the understatement that for the Sedov and Solinger model one should take formally the limit \(n=5\).
The explosion energy \(E_0\) is linked to the other parameters by
where \(\alpha\) is the dimensionless energy integral introduced by Sedov. The value of \(\alpha\) depends both upon the hydrodynamical model and the value for \(s\) and \(n\), the density gradient scale heights of the interstellar medium and the stellar ejecta.
If we express \(T_s\) in keV, \(U_s\) in \(10^{20}\) \(\mathrm{m}^{-3}\)s, \(Y_s\) in \(10^{20}\) \(\mathrm{m}^{-5}\), \(r_s\) in units of \(10^8\) m, \(n_{\mathrm H}\) in \(10^{20}\) \(\mathrm{m}^{-3}\), \(d\) in units of \(10^{22}\) m and \(t\) in s, we obtain from (1) the value for the proton mass of \(m_{\mathrm p}\) = \(1.043969\times 10^5\) \(\mathrm{keV}^2\)(\(10^{16}\) \(\mathrm{m}^{2}\))\(^{-1}\). Using this value and defining \(f_m = f_T\epsilon_\rho m_{\mathrm p}/f_{\mathrm p}\), the scaling laws for SNRs may be written as
where \(M\) is the mass integral, and \(\beta\) is the dimensionless mass integral (defined by \(\beta = 4\pi\int \frac{\rho}{\rho_s}(\frac{r}{r_s})^2{\mathrm d}\frac{r}{r_s}\)).