8.1.7. 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

T_s,

the second the shock ionisation parameter defined by

U_s\equiv n_{\mathrm e}t

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

Y_s\equiv n_{\mathrm e}n_{\mathrm H}r_s^3/d^2

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:

(1)f_{\mathrm p} kT_s = f_{\mathrm T} \epsilon_\rho m_{\mathrm p}r_s^2/t^2

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

f_{\mathrm T} = \frac{f_v^2}{\eta - 1}

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:

v_2 = f_v r_s/t

with v_2 the post-shock gas velocity (not to be confused with the shock velocity!). The velocity scaling factor is given by

f_v \equiv \frac{(n-3)(\eta - 1)}{(n-s) \eta}

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

E_0 = \frac{\alpha r_s^5 \rho}{t^2},

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

n_{\mathrm e} = \frac{T_s^{1.5}U_s^3}{f_m^{1.5}Y_s d^2}

t = \frac{f_m^{1.5}Y_s d^2 n_{\mathrm e}}{ T_s^{1.5}U_s^2 n_{\mathrm H}}

r = \frac{f_mY_s d^2 n_{\mathrm e}}{U_s^2T_s n_{\mathrm H}}

v = \frac{f_vT_s^{0.5}}{f_m^{0.5}}

E_0 = [1.6726231\times 10^{33}\ {\mathrm J}] \frac{\alpha \epsilon_\rho f_m^{0.5}Y_s^2d^4 n_{\mathrm e}}{U_s^3T_s^{0.5}n_{\mathrm H}}

M = [8.409\times 10^{-14}\ \mathrm{M_{\odot}} ] \frac{\epsilon_\rho \beta \eta f_m^{1.5}Y_s^2 d^4 n_{\mathrm e}}{ U_s^3T_s^{1.5} n_{\mathrm H}}

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}).