Emission Model

The expanding shock wave continuously encounters neutral ISM. Thermodynamical quantities just before and after the shock are linked by the Rankine-Hugoniot conditions (Landau & Lifshitz, 1974). Typical post shock temperatures of middle-aged SNR's range between K and the plasma can be considered optically thin. Detailed calculations of X-ray spectra are difficult because of the uncertainties of ionization and recombination rates of relevant elements. One of the most widely used emission codes is described in Raymond & Smith (1977). Differences in the input atomic physics rates lead to differences in computed spectra; for a discussion of the impact of the differences among spectral codes on spectral analysis of PSPC data, see the Appendix of Paper I.

The plasma CIE emission models assume the local equality of the electron kinetic temperature, , and the ionization temperature, , defined in terms of the ions population fractions. Because of the collisional nature of the ionization process, if a perturbation makes and different, the equilibrium is restored on a time scale depending on electron kinetic temperature through the ionization and recombination rates. In the case of middle-aged SNRs, we have K and K, since before the shock the ISM is in CIE condition at K. Hence, the post-shock plasma is under-ionized, and, for any given element, we have

where is the electron density, the ionization rates and M is the most abundant ionization stage in condition of CIE at the temperature T (we assume that ions 1,...,M-1 are completely ionized). For instance, with K, cm we have yr for Carbon, yr for oxygen, and yr for iron. A more detailed calculation of is given by Masai (1994). These estimates imply that we expect the plasma to be in CIE at a distance behind the shock, where is the plasma velocity relative to the shock. In the case of the Vela SNR, where km sec, pc for iron, and pc for oxygen. Note that this is a strict lower limit because this simple reasoning does not take into account the shock deceleration. In conclusion, a NEI treatment is required to properly model the Vela SNR X-ray emission.

To properly model the X-ray spectra from the observed shock region of Vela SNR, we have built a grid of NEI spectra with two free parameters: the electron temperature, , and the ionization time ,

where t=0 is the time when a small volume of plasma, of which we want to find the emission, is struck by the blast wave. Keeping fixed the plasma element abundances, , and a normalization factor, depending on the plasma density, are the only parameters which a NEI spectrum depends on. To solve the problem of computing NEI X-ray emission we have made two main assumptions:

1. We assume that once a small volume of plasma is struck by the blast wave, it moves rigidly within a region of constant density and temperature. In other words, we neglect the convective terms in the continuity equation of the population fraction. This is certainly a good assumption in the Sedov expansion phase, since matter moves rigidly with a speed equal to , but it could result not valid in case of ISM inhomogeneities with high density contrast. A more general approach should take into account the change in ionization structure due to displacement of the plasma moving across zones with different thermodynamical conditions. Such an approach, that requires a more general code, is under study and will be reported in a subsequent paper.
2. The electron temperature, , and density, , stay constant during the ionization process. This is equivalent to assume that the emerging spectrum depends only on the instantaneous electron temperature and density. This assumption has been discussed by Hamilton & Sarazin (1984) and is valid when the ionization rates vary slowly with temperature, i.e. in high temperature conditions, or alternatively when the analysis is limited to the shock region, since the Sedov model predicts slow variations in the region few parsecs behind the shock front of a Vela-like SNR (Itoh 1979, Cui & Cox 1992). This approximation is no longer valid when one considers the central regions of SNR's. Again, problems may arise if a ISM cloudlet with high density gradient is shocked by the blast wave, because strong temperature and density gradients will occur in the subsequent cloud evolution (Bedogni & Woodward, 1990).

An approach similar to our was used by HH85 in developing their NEI code based on matrix calculations. Our calculation scheme is also applicable to physical conditions with time-dependent temperature and density profiles. Although we have assumed constancy in this work, we plan to generalize our model to deal with such time-dependent physical conditions.

Throughout all our calculations, we used the Exchange Classical Impact Parameter (ECIP) rates (Summers, 1974, Raymond & Smith, 1977), which are more accurate, especially in NEI conditions, than the Lotz formula (Lotz, 1967) and the related ionization rates (Mewe & Groneschild, 1981, Shull & van Steenberg, 1981), as discussed by Hamilton, Sarazin and Chevalier (1983).

We have solved the population equations for ions of the most abundant cosmic elements (He, C, N, O, Ne, Mg, Si, S, Ar, Ca, Fe, Ni) using a fifth order Runge-Kutta method (Press et al. 1986) modified to take into account specific physical events and numerical problems like ions turning on and the ``stiffness" of the involved equations. The initial condition () is the impact between the blast-wave and the small volume of plasma. We have assumed that the ISM elements are all neutral at this time. We have also checked that the results of our spectral analysis are not affected by this choice: in particular, the spectral fitting results obtained with grids of spectra computed assuming ISM initial temperatures up to K do not differ appreciably from results obtained with our choice. Once the ionization stages have been found for a given couple of values (), we have inserted the ion relative abundances into the Raymond-Smith code (Raymond & Smith, 1977, Raymond, 1989) to compute the final spectrum which was saved as a table readable by the XSPEC spectral fitting code. The energy range of the synthetic spectra is 0.05-3 keV with a step of 0.007 keV, and the bin sizes of the STNEI emissivity table have been chosen in the following way: from 0.02 to 2.0 keV in steps of 0.02 keV for the temperature, and from to yr cm in logarithmic steps of 0.21 for the ionization time.

In order to test our capability to recover the original input plasma parameters when fitting ROSAT PSPC data with our STNEI model, we have performed several simulations of spectra in NEI conditions folded through the PSPC response, using various values of and of the number of total counts (Bocchino, Maggio & Sciortino 1996). On the basis of the fits of these simulated spectra, we concluded that optimal restoration of the free parameters T and is obtained using spectra with at least counts in the PSPC band (0.1-2.4 keV).

By comparing the results produced by the Raymond-Smith code in CIE conditions and the results of our numerical calculations, we have also verified that the code we developed correctly describes the asymptotic behavior of the ionization process. In fact, the CIE population fractions are reproduced by our code when (in yr cm).

We expect that, in each of our bins, the ionization time of the emitting plasma ranges from at the shock front to , corresponding to the most internal regions along the line of sight. It is not obvious that the PSPC spectra, including contributions from regions with different ionization times, can be described in terms of our single-tau single-temperature (STNEI) emission model, but we show in the following why such a description is likely feasible.

In order to estimate to which extent a STNEI model is appropriate in fitting spatially resolved spectra, we should take into account the variation of , and of the emission measure profile along our line of sight, since the plasma is optically thin.

Expected value. The expected value of depends on the spatial bin considered. Assuming at the outer edge of the most external annulus (annulus 7, see section 4.1 and figure 2), a shock speed km/sec, a post-shock density cm (see §5.1 and 5.3) we have yr cm for a 1 pc wide spherical shell whose external edge matches the shock position. More internal shells do not scale linearly because the shock decelerates, but in any case should not be less than yr cm in the outermost zones, and it is not greater than a few anywhere within the Vela SNR.

The NEI plasma emissivity. The X-ray emissivity in the ROSAT PSPC band is a strong function of . This is shown in Figure 1, which indicates, in the case of kT=0.2-0.3 keV, variations of a factor of 7 between the maximum emissivity (at ) and the emissivity in equilibrium condition (), and a factor of 30 between the maximum value and the value at .

Emerging spectrum of a Sedov SNR. The knowledge of the STNEI emissivity is not sufficient to characterize the emerging spectrum: the emission measure and ionization time distributions inside the remnant need to be estimated. Profiles of , density and temperature have been extensively derived in the case of Sedov solution and volume integrated emission. Hamilton & Sarazin (1984) worked on the characterization of NEI integrated spectra. They clearly show that, given a snapshot of a SNR, profiles and thermal history can be safely ignored when interpreting volume integrated spectra, since the only relevant ``observational" parameters are the emission measure-weighted , and the total emissivity, . As they puts it, `` ... Sedov type SNR will have similar spectra if they have the same , and ". This is essentially due to the fact that most of the emission originates from a thin post-shock layer corresponding to a density bump (see also Cui & Cox 1992). We expect that these considerations can be applied also to our spatially-resolved analysis, because the contribution to the observed emission in each spatial bin, due to different layers along the line of sight, is similar but certainly less complex than in the case of the volume-integrated analysis.

The usage of a single component STNEI model as the one we have developed is therefore justified to some extent, and should provide us with a reasonable estimate of and . We intend to further investigate this point using simulations in which Sedov STNEI spectra, integrated along the line of sight across a spherical emitting Sedov type SNR, are fitted with our STNEI model (Bocchino et al. 1996, in preparation). This approach will help us to understand how the best-fit values compare with the true local values of T and .

Alternative approaches. The justification of a STNEI model, even though not discussed in detail, is implicitly assumed in the literature. Single T - single fits were carried out with models very similar to our by, for instance, Hwang et al. (1993) and Yamauchi et al. (1993), and give typical values (in yr cm) in the range 2.5-4.0 together with an estimate of the metal abundances in N132D and RX04591+1547 respectively. Fitting to linear combinations of STNEI models are not feasible as an alternative approach for fitting purposes, because of the large number of free parameters. In fact, for each component of the combination we should let vary at least the temperature and the normalization, but PSPC data are not suited for fitting with more than 4 or 5 free parameters. For the same reason (limited data quality), a different approach based on the computation of the distribution of T and by deconvolution of the data is unfeasible as well.

In summary, in spite of the mentioned simplifications reflecting our limited knowledge of the SNR plasma thermal history, our STNEI model is a more realistic description than the ionization equilibrium model we have considered in Paper I. A similar model was also adopted by Vedder et al. (1986) to analyze Einstein FPCS observations of the Cygnus Loop, and by Hughes & Singh (1994) to determine the abundances in the G292.0+1.8 SNR.