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Astron. Astrophys. 344, 295-309 (1999)

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4. Integrated characteristics of the X-ray radiation from non-spherical SNRs

SNRs are powerful sources of X-ray radiation, therefore X-ray observations of SNR give unique information about physical conditions inside the remnants. We calculate here the X-ray luminosities in the energy ranges [FORMULA] ([FORMULA]) and [FORMULA] ([FORMULA]) as well as the spectral index [FORMULA] at [FORMULA]

4.1. Plasma X-ray emissivity under ionization equilibrium condition

We assume the cosmic abundance (Allen 1973). The X-ray continuum energy emissivity per unit energy interval is (in [FORMULA])

[EQUATION]

where [FORMULA] is the photon energy in keV , [FORMULA] is the plasma temperature in [FORMULA], [FORMULA] is the electron number density. The approximation for total Gaunt factor [FORMULA] as the sum of Gaunt factors for free-free [FORMULA] free-bound [FORMULA] and two-photon [FORMULA] processes was taken from Mewe et al. (1986):

[EQUATION]

This approximation represents the continuum losses to an accuracy of [FORMULA] for [FORMULA] and of [FORMULA] for [FORMULA].

For calculation of continuum and line emission of plasma in the different energy ranges, we have approximated the Raymond & Smith (1977) data for plasma emissivity [FORMULA] (in [FORMULA]) for [FORMULA] as follows:

[EQUATION]

(accurate to [FORMULA] for [FORMULA]); for [FORMULA] as follows:

[EQUATION]

(accurate to [FORMULA] for [FORMULA]); for [FORMULA] as follows:

[EQUATION]

(accurate to [FORMULA] for [FORMULA]).

The total continuum flux [FORMULA] at photon energy [FORMULA] and luminosity [FORMULA] of the entire SNR can be calculated by integrating over the remnant volume V

[EQUATION]

[EQUATION]

The spectral index [FORMULA] is

[EQUATION]

4.2. Evolution of the total X-ray emission from aspherical SNRs

It is well-known that the Sedov (1959) solution for SNR characteristics in a uniform medium is determined by three parameters, e.g., the energy of the explosion [FORMULA] the initial number density [FORMULA] and time t. The shape of the integrated spectrum (in particular the spectral index [FORMULA]) depends only on one parameter (the shock temperature [FORMULA]) in case of collision ionization equilibrium (CIE) and on two ([FORMULA] and [FORMULA]) in case of NEI (for fixed abundance) (Hamilton et al. 1983). In order to estimate the total luminosity of a Sedov SNR in the NEI case, we need a third parameter, i.e. the explosion energy [FORMULA] (Hamilton et al. 1983). In the Sedov CIE case the luminosity depends only on two parameters [FORMULA] and [FORMULA]

If the SN explodes in a non-uniform medium, we have an additional fourth parameter which characterises the non-uniform density distribution. In our cases, it is the scale height H for exponential density distribution or [FORMULA] for power-law density distribution.

It is naturally to expect, that in cases of NSNR considered, the total X-ray luminosity [FORMULA] as well as spectral index [FORMULA] will strongly depend on non-uniformity characteristics, such as scale height H etc. Therefore, we calculate here extensive grid of models for evolution of X-ray radiation from NSNRs which evolve in non-uniform ISM and compare it with the Sedov case. Our grid covers the range of paramers which correspond to the adiabatic stage of SNR evolution, i.e. for SNR radii between [FORMULA] when the swept up mass of ISM becomes equal to the mass of ejecta [FORMULA] and [FORMULA] when the effective temperature [FORMULA] corresponds to the maximum value of plasma emissivity function [FORMULA] (Lozinskaya 1992). In a non-uniform medium, the value of [FORMULA] depends on the concrete density distribution in a selected direction (sector). So, if the density distribution in a sector is exponential [FORMULA] we may estimate [FORMULA] from the relation [FORMULA] For a uniform medium [FORMULA] we obtain [FORMULA] which is close to results of other authors (Lozinskaya 1992). In our calculation, the maximal time was estimated from adiabaticity violation in the [FORMULA] sector. For parameters of ISM considered here, maximal radii of SNRs do not exceed a few scale heights. Therefore, obtained results are not affected by shock acceleration, which is important only at considerably greater distances.

We show in Fig. 7 the evolution of X-ray luminosity for range [FORMULA] and spectral index [FORMULA] at [FORMULA] for SNR in an exponential medium. It is easy to see that both X-ray characteristics of NSNR evolve analogously to the Sedov case and similarity increases with decreasing surrounding medium density gradient (with increasing H) and decreasing explosion energy. The results presented in Fig. 7 of calculations of the Sedov case for [FORMULA] (Hamilton et al. 1983) reveal some differences between X-ray luminosities at low temperatures. They may be caused, in part, by atomic data and chemical composition differences (our model uses the Allen abundance, whereas in the paper of Hamilton et al. (1983) have been used the Meyer abundance). Our results for Sedov case coinside with of Leahy & Aschenbach (1996) and Kassim et al. (1993) results for model with Allen abundance within 25%.

[FIGURE] Fig. 7a and b. Evolution of luminosity [FORMULA] in range [FORMULA] and of spectral index [FORMULA] of SNRs in a uniform medium (line 1), in an exponential medium Eq. (11) with [FORMULA] (line 2) and [FORMULA] (line 3). In all cases [FORMULA]. Results of numerical calculations of Hamilton et al. (1983) for a uniform medium and [FORMULA] with Meyer abundance are also shown by dots. Hereafter [FORMULA].

We have calculated a grid of NSNR models (Tables 1-4) which confirms that this analogy in evolution is an intrinsic property of NSNR X-ray radiation. In fact, it may be seen from Tables 1-4 that for a wide range of number density at the point of explosion [FORMULA] the X-ray luminosity of adiabatic NSNR evolving in a medium with strong enough density gradient ([FORMULA]) in different energy ranges are not far from the X-ray luminosity of SNR in a uniform density medium with the same initial model parameters. The differences increase with age of the SNR and with decreasing [FORMULA]. But even for old NSNR (e.g., [FORMULA]) in a low density medium ([FORMULA]), the maximal difference is about [FORMULA] (for [FORMULA]). The differences in spectral index of adiabatic NSNR do not exceed [FORMULA]. Table 1 shows that luminosity in range [FORMULA] is close to the Sedov case also. These last two facts reveal the similarity of total spectra of Sedov SNR and non-spherical ones.


[TABLE]

Table 1. Luminosity in ranges [FORMULA] and [FORMULA] (in parenthesis) of SNR expanding in an exponential medium Eq. (11) with [FORMULA] (upper lines) in comparison with Sedov SNR (lower lines). The SN explosion energy is [FORMULA].



[TABLE]

Table 2. Spectral index [FORMULA] of NSNR in exponential medium Eq. (11) with [FORMULA] (upper lines) in comparison with Sedov SNR (lower lines). The SN explosion energy is [FORMULA].



[TABLE]

Table 3. The same as in Table 1 for [FORMULA] and range [FORMULA].



[TABLE]

Table 4. The same as in Table 2 for [FORMULA].


We suppose that the behavior of integral X-ray characteristics of NSNR in other cases of smooth continuous density distribution of surrounding medium will be similar considered above, since X-ray emission mainly depends on the emission measure [FORMULA] but both mass of swepted up gas M and volume of NSNR V remain close to those of Sedov SNR (Table 5). Moreover, the last fact allow us to introduce a characteristic shock temperature for NSNR [FORMULA], comparable to [FORMULA] in the Sedov case:

[EQUATION]

where [FORMULA] [FORMULA] with [FORMULA] and [FORMULA] The temperature [FORMULA] determined from the observed spectrum will be connected with [FORMULA] as in the Sedov case [FORMULA] (Itoh 1977). This characteristic temperature may be used to estimate the parameters of the whole remnant, such as mass, volume, etc.


[TABLE]

Table 5. The ratios of SNR characteristics (the volume V, swept-up mass M and characteristic temperatures [FORMULA]) in non-uniform medium (with exponential (E) Eq. (11) and power law (PL) Eq. (13) density distribution) to uniform ones.


The results of calculations show that integral X-ray characteristics weakly depend on the surrounding medium density gradient (e.g., H). Therefore, as in the Sedov case, a spectral index of equilibrium emission from NSNR depends approximately only on one parameter [FORMULA] and luminosity depends approximately only on two parameters: [FORMULA] and [FORMULA]. As confirmation of this, we show that maximal deviation from this rule for NSNR in exponential density distribution reaches few percent for spectral index and a few tens of percent for luminosity (Table 6).


[TABLE]

Table 6. Evolution of SNR luminosity [FORMULA] and spectral index [FORMULA] in uniform (S) and exponential (E) Eq. (11) media.


4.3. Discussion

From Fig. 7 and Tables 1-6 follows a remarkable fact of proximity of total fluxes and spectrum shapes of SNRs in uniform and non-uniform media even in case of considerable anisotropy of adiabatic NSNR. The reason of this phenomenon is mutual compensation of emission deficit from low density regions of NSNR and enhanced emission from high density regions. This explains the strange circumstance that integral X-ray characteristics of the majority of SNRs with evident asymetry in shape and/or surface brightness distribution are described with sufficient accuracy by the Sedov model of SN explosion in a uniform medium. For more correct modelling of physical conditions inside and outside the SNR, it is necessary to analyze the characteristics of X-ray emission distributed over the SNR surface.

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© European Southern Observatory (ESO) 1999

Online publication: March 10, 1999
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