In Paper I, the results of the RS 2T model fitting were taken as a suggestion of NEI conditions of the emitting plasma. However, the attempt to fit the Vela SNR X-ray emission with a 1T STNEI model, carried out in Paper II yielded very low ionization times ( yr cm-3), difficult to support on physical grounds, and lead us to reconsider the 2T results in terms of ISM inhomogeneities. Similar results were obtained by studying the Vela SNR X-ray emission in the RP500015 image (Bocchino 1997), which is located in the same rim region of the RP200133 image, but slightly offset, thus allowing a better shell coverage.
In fact, the environment of the Vela SNR is expected to be inhomogeneous, because the wide field X-ray observations carried out using Einstein (Seward 1990) and ROSAT (Aschenbach 1993) show a patchy morphology, even though it is difficult to estimate quantitatively the size and density of the clumps embedded in it.
The observed spectrum originating from such environment comprises in general several components, whose contributions sum up along the line of sight because the plasma is optically thin. In particular, we expect contributions from: 1) plasma compressed in a thin shell just behind the shock front and likely to be out of ionization equilibrium conditions; 2) plasma in local ISM condensations (including material stripped from clouds by thermal evaporation); 3) lower density plasma in interior regions. Given the lower density of interior SNR regions predicted by the adiabatic expansion model, the latter contribution is expected to be negligible especially in the outer rings of our observations (see Sect. 4.1).
In this perspective, it is possible to interpret the two-component nature of the Vela SNR X-ray spectrum as a tracer of the presence of two phases in the ISM swept up by the shock. This interpretation, which has not been often investigated in the literature, could hold even for other middle-aged SNR, since they are expected to interact with inhomogeneous environments (for instance, for the Cygnus Loop see Decourchelle et al. 1997); successful fitting results with 2T models have been reported in the cases of IC443 (Asaoka & Aschenbach 1994), SN1006 (Willingale et al. 1996), G296.1-0.7 (Hwang & Markert 1994) and RX04591+5147 (Yamauchi et al. 1993) but the physical interpretation in terms of ISM inhomogeneities, as the one we propose, has not been investigated.
In this section, we shall follow the hypothesis of an association of the two X-ray emission components (X1 corresponding to the cooler component, and X2 to the hotter component) with two ISM phases having different densities. This association will allow us to estimate the SNR explosion energy and distance by applying the Sedov analysis to the X-ray component likely associated with the inter-cloud ISM phase. The results we obtain in this way are more accurate than those previously derived by spectral analysis based on single-temperature models.
4.1. Interpretation of the temperature components
In Table 4, we list the expected ISM phases together with their respective values of temperature, T, density, n and filling factor, f. The table is based on the work of McKee & Ostriker (1977) and subsequent revisions (Slavin & Cox 1993, Cox 1993, Shelton & Cox 1994). The presence of the low density phase IV is in doubt (Korpela et al. 1998), while the presence of molecular clouds is well established.
Table 4. Scheme of ISM expected phases according to the theoretical model of McKee & Ostriker (1977) and following updates.
The ISM model reported in Table 4 is our starting point for the interpretation of the two components detected in the X-ray emission of the Vela SNR rim. Let us consider a middle-aged SNR shock wave, which propagates in a ISM as described in Table 4, and let us assume that the shock velocity is km s-1 when the shock front is in phase III. The post-shock temperature will be in the range 1-5 K, and hence the plasma will emit in the X-ray band. As the shock encounters an inhomogeneity, it enters phase II which is times denser than phase III. The secondary shock transmitted in phase II is times slower ( according to McKee & Cowie 1975), thus heating the shocked material at a temperature 40 times lower, and driving the bulk of the emission outside the X-ray band. For the same reason, the shocks propagating in the phase I medium will achieve even lower temperatures.
Although, to be more realistic, continuous distributions of density should replace the "step variations" of Table 4, this simple scheme suggests that the temperature of the shocked plasma is somewhat determined by the density of the layers which the shock encounters during its propagation. From this point of view, we expect that X-ray observations probe the hotter and less dense plasma in the external regions of ISM inhomogeneities and the inter-cloud medium, while UV and optical observations trace cooler and more dense regions, associated with clouds internal layers. What it is still to be assessed now is a possible correspondence between the two X-ray emission components and some of the ISM phases described above. Our working hypothesis is that the hotter X-ray component (X2) is mainly associated with the "inter-cloud" plasma in which adiabatic expansion of the main shock occurs, while the cooler component (X1) originates from local ISM "inhomogeneities".
Our interpretation of the hotter X-ray component is not in contrast with the observed flat temperature profiles shown in Fig. 4, significantly different from the steep profile predicted by the Sedov model. This is because the X-ray emission we observe at any location of the spatial grid mainly originates from a thin layer behind the shock front, where the plasma density and hence the emission measure are the highest. The more internal (and hotter) spherical shell layers have much lower densities and do not contribute appreciably to the total X-ray emission. In fact, assuming a Sedov profile for the temperature and density of the plasma, we have estimated that more than 80% of the total emission is generated within a shell thickness % of the remnant radius, i.e. within pc in the case of the Vela SNR. Hence, in X-rays we see the external layer whichever direction we point at, measuring the same average temperature.
Finally, the fact that cannot be considered uniform while can be, is in agreement with our hypothesis, since the putative ISM clumps which give rise to the X1 component could have in principle very different densities, yielding different X-ray temperatures.
In the following, we will use the terminology "inter-cloud component" and "inhomogeneities component" with reference to the hotter and cooler X-ray component, respectively.
4.2. Densities and filling factors of the post-shock plasma components
We now use the fitting results to derive the plasma density and filling factor of each component. These quantities are useful to understand the relation between the X-ray morphology (in particular, the X-ray clumps as defined in Sect. 3.1) and the ISM phases. While the X-ray clumps are generally interpreted as tracers of ISM cloud, it is not clear how they are formed. In particular, there are at least 3 possible scenarios: clouds evaporation, bow shock, or secondary shock compression inside the cloud. In order to test these alternatives, an estimate of the density and filling factor is required.
Let be the volume of emitting plasma from which the emission we observe in a given spatial bin of our image originates. Here, A is the area subtended by the spatial bin, and l is the line of sight segment enclosed in the spherical SNR shell. The validity of the RS 2T model in a large fraction of the bins suggests that the two components are present in the volume V at the same time. We can therefore define a volume filling factor for each component in each spatial bin ( and for the component X1 and X2, respectively), analogous to the filling factor used in Table 4. The brightness of a given component is proportional to f and to the square of the density through the normalization factor (derived from the spectral fit):
where and are the density and volume of the plasma associated to the i-th emission component, and V is the total volume. Hence, the ratio of the normalization factors of the two components can be written as:
It is clear that the filling factor estimate has a twofold importance: first, it allows us a direct comparison with the theoretical expectation summarized in Table 4 ; second, it can be used in Eq. 1 to derive the density.
The calculation of for each spatial bin requires an independent estimate of the density ratio, and this can be obtained with the following reasoning, in the framework of our working hypothesis. If the X1 component is due to a strong bow or secondary shock propagating inside an ISM cloud, then we can use the results of McKee & Cowie (1975), which have shown that , where the subscripts refer to the two components. Since the square of the shock speed is proportional of the temperature behind the shock, we have
If instead the X1 component is due to material evaporated from ISM clouds, then we have , where the equal sign holds only if the evaporation is not violent, and hence quasi-isobaric. As we shall see in the following section, the physical conditions behind the Vela SNR shock are such that the evaporation parameter is well below unity, indicating that the evaporation, if present, is far from being saturated, and therefore is unlikely that we have strong pressure gradients. Under these circumstances, the relation may be valid in both the evaporation and strong shock scenario.
In this equation, is a function only of the best-fit quantities. We note that, even in the unlikely case of violent evaporation as discussed above, Eq. 4 provides us with an upper limit on the filling factor. In Fig. 7, we show a map of the parameter in each of the two images we are considering. The filling factor is always in the range 0.2-0.8: high values () are often associated with X-ray bright "clumps", as defined in Sect. 3.1, especially in RP200133; instead, in RP500015, sectors l, m, and n inside ring 4 have high brightness and , so that high surface brightness does not necessarily imply high values.
We stress that the filling factors reported in Fig. 7 were derived in the hypothesis that there are only two thermal components emitting in the volume V (i.e. ). In the case that an additional component is present, should be corrected by a factor , where is the volume filling factor of the third component. Detailed photometry and spectrophotometry of this region of the Vela SNR shell, which will be fully presented in a forthcoming paper, shows that a third component is actually present and it is responsible of the optical emission of filaments. Hester (1987) has shown that optical filaments arise from thin sheets of compressed material; if the width of the sheet is cm (Hester 1987 took cm), then the volume filling factor of the filaments in one of our spatial bin is . Therefore the filaments occupy a small fraction of the volume and is a good approximation.
Now, from the value of , we can derive the density in each spatial bin. Since and , where is the solid angle subtended by each spatial bin in Fig. 2, and d is the distance of the Vela SNR, we can rewrite Eq. 1 in the following way:
with a typical error of on and on , derived from the statistical uncertainties of . The dependence of the density on the the line of sight is not strong, and we have estimated l assuming that the emitting plasma is confined in a spherical shell of thickness 2 pc.
The possible presence of NEI effects, already addressed in Sect. 3.2.4, may introduce a further bias in the evaluation of the filling factor or density, as we have seen in Sect. 3.2.4 for T and EM. Following the same reasoning, we have evaluated the bias which would be introduced by our CIE spectral analysis if the plasma was effectively in NEI conditions. We have computed and n according to Eqs. 4 and 7 using the input and simulated values for F and T, deriving an "input" and n and their corresponding values derived from CIE fittings. We derived that the correction factor for () is 0.84, for () is 0.96, and for () is 0.79, which are near to unity, especially for .
In Fig. 8, we show the density map in each field of view, along with the histograms of and . The cooler X1 component density is on average a factor of 3 higher than the hotter X2 component density. The figure also shows that higher regions in RP200133 correspond in most cases to the brightness-defined "clumps" (Sect. 3.1, zone A and B in Paper I). Therefore, in these regions, the high brightness is due to high filling factors and also to high densities ( cm-3 in , , and , whereas cm-3 in surrounding regions, such as , and ). In RP500015, the luminosity "clumps" in sectors m and n also correspond to high density values ( cm-3), but many of the "clumps" have low densities ( cm-3). Therefore, in the case of RP500015, some of the brightness enhancements seem to be generated only by an increase of the filling factor of the inhomogeneities.
Since in all the spatial bins of both sequences, there is always a sizable fraction of volume occupied by the cooler gas associated to the inhomogeneities. This means that the ISM inhomogeneities affect not only the regions classified as "clumps", but also most of "diffuse emission" of the Vela shell, at least in the 7-8 pc covered by the two adjacent images. In particular, the determination of f and n allows us to better characterize the nature of inhomogeneities in some case. For instance, in the spatial bin of RP500015 which contains FilD we have and , while in the "Front 1" region of the same image (bins and ) the density is comparable with FilD, but the filling factor is sensibly higher. This suggests that FilD is probably an isolated cloud with a small extension along the line of sight and that Front 1 is a set of several clouds or a cavity wall with a greater extension along l. This is in agreement with the fact that the spectral analysis of the very bright region including FilD of Sect. 3.2.4, which is sensibly smaller than the spatial bin , gave a density of 0.88 cm-3 (Table 3) greater than the average density.
4.3. Clouds evaporation
The evaporative cloud scenario has been proposed by McKee & Cowie (1975) and discussed in the framework of SNRs by Charles et al. (1985), Ku et al. (1984), White & Long (1991), Dalton & Balbus (1993), and Bandiera & Chen (1994). Comparisons with observed SNRs, carried on, for instance, by Harrus et al. (1997) on W44, Craig et al. (1997) on CTB1, Fuerst et al. (1997) on G18.95-1.1, have shown that thermal evaporation is generally consistent with the radial profiles of plerionic and composite SNRs, but not with shell SNRs.
In this section, we will check if cloud evaporation is consistent with our data. To do that, we estimate the plasma density of the X1 component and we check if the evaporation efficiency is high enough to create the observed quantity of cool plasma.
For a given spatial bin, the total emitting mass of the X1 component is given by
which is derived by inverting Eq. 1. For instance, we typically have cm-5, and cm-2, yielding g for a distance between 200 and 350 pc. The evaporated mass in a time interval t from a cloud is given by , where is the proton mass. An analytical expression for has been derived by Cowie & McKee (1977) under classical and saturated conduction. In our case, the dimensionless saturation parameter is very low ( is 0.1 using our best-fit as inter-cloud medium temperature, as medium density and pc, in Eq. 32 of Cowie & McKee 1977), and the classical conduction formula applies:
where is the cloud radius in pc, () is the inter-cloud medium temperature, and assuming negligible magnetic fields. Using pc, estimated from the size of the clumps in the X-ray image, and K (as derived from the X2 component), we obtain g s-1, and therefore one cloud would take 600-2000 yr to accumulate the observed mass. Since we expect that the shock travels 1 pc (the effective spatial resolution of our grid) in yr, we conclude that our results are consistent with the evaporative model.
Inhomogeneities with radii smaller than 2 pc have lower evaporative loss rates but can be present in larger number, and therefore cannot be excluded. Larger clouds in principle yield enough mass, but their presence would have dramatic effects on the SNR dynamics, yielding clearly visible variations of the X-ray brightness over spatial scales larger than . This may be the case of the feature identified as Front 1 in RP500015, which extends over 4 pc or more.
4.4. The distance and explosion energy of the Vela SNR
We now apply the Sedov analysis to the X2 component, which we have associated to the main blast wave expanding in the inter-cloud medium, in order to derive the remnant characteristic parameters (e.g. the distance and the explosion energy).
The cloud filling factor is in about 60% of the bins. The main shock front is certainly distorted by the impact with the clouds, but several hydrodynamical simulations show that the distortions tend to disappear after the shock has overrun the cloud (Bedogni & Woodward 1990). Moreover, Cowie et al. (1981) have verified that clouds with 5 pc radius and inter-cloud density 0.3 cm-3 do not affect the whole expansion dynamics of the SNR and hence the average temperature profile. This conclusion certainly applies to our case, since the inhomogeneities we observe have sizes pc. It is therefore possible to apply the Sedov analysis to the inter-cloud component we have isolated in the X-ray emission. In practice, we have estimated according to Eq. 7, the pressure according to the relation , and the velocity from the relation . These quantities, derived from the X2 component, represent post-shock values. We adopted as our best estimate for a given observation, the average of each quantity across the whole set of bins in which the 2T model provides a statistically acceptable description of the spectral data. Moreover, since the values derived from RP200133 and RP500015 are compatible within the uncertainties, and the two sets of measurements are independent from each other, we adopted as our best estimates for both regions the mean of the average values. Summarizing the results and related uncertainties:
To derive the explosion energy and the SNR radius, we apply the Sedov analysis using the above shock speed and density, and the SNR age derived by the PSR0833-45 characteristic age. The connection between the pulsar and the SNR is rather established (Weiler & Panagia 1980, Weiler & Sramek 1988) and its age has been accurately assessed ( yr, Taylor et al. 1993). The Sedov analysis yields an explosion energy of erg, a value lower than the canonical one quoted for supernova explosions ( erg), but not unreasonably low. In fact, Nomoto et al. (1976) showed that the expected explosion energy could be as low as erg, and Aschenbach et al. (1991) derived erg for G18.95-1.1 on the basis of a similar Sedov analysis applied to a ROSAT All-Sky Survey observation. Another low value ( erg) was derived by Hughes & Singh (1994) for the SNR G292.0+1.8. Our derived energy is lower than previous estimates for the Vela SNR: for instance, Gorenstein et al. (1974) derived erg using a 1T model and low spatial resolution data. Our results should be more reliable since they are based on a more accurate model of the post-shock regions and on data with better spatial resolution. More recently, Jenkins & Wallerstein (1995) derived erg assuming a SNR distance of 250 pc, and a ram pressure of dyn cm-2; their estimate is based on absorption measurements along the line of sight in a region on the West part of the shell, very far from our pointings, and their value could be affected by the local physical conditions. Our estimate, instead, is an average on a large sky area of the X2 component and it is less (in principle not at all) affected by the clumpy environment of the Vela SNR.
Using the Sedov relations, as given for instance, by McKee & Hollenbach (1980), we derive the real SNR radius, which turns out to be pc. The apparent radius in the sky of the Vela shell has been recently measured on the basis of ROSAT All-Sky Survey observations by Aschenbach (1993), and it is of . We therefore derive a SNR best-estimate distance of 280 pc, and in any case in the range between 110 and 680 pc. Our derived best-estimate value is lower than the often quoted value of 500 pc based on the old estimate of Milne (1968), even if it is not incompatible considering the allowed range. A critical review of the Milne results made by Oberlack et al. (1994) has shown that Milne made a wrong assumption of the real shell radius, and that a new computation of the distance based on the Milne data is 230 pc, which is in good agreement with our best-estimate value. Oberlack et al. (1994) also stressed that independent preliminary results based on ROSAT data indicate a distance pc.
We have evaluated the influence of the NEI effects on the Vela SNR characteristic parameters, as outlined for T and EM in Sect. 3.2.4. The corresponding correction factors are: , and . The parameters values corrected for the NEI effects are: erg, pc and pc. The differences between these values and the ones derived assuming CIE conditions do not change our conclusions.
4.5. Energy equipartition between electrons and ions
Many of the considerations we have done are based on the hypothesis of instantaneous electron-ion energy equipartition. In fact, it is well known that the temperature derived by the X-ray spectral analysis is the electron temperature , and to derive the shock velocity applying the Sedov analysis we need the ion temperature . The equipartition therefore plays a crucial role in the study of SNRs and it is still the subject of many debates (Masai 1994, Laming et al. 1996). In particular, it is classically expected that just behind a strong shock and that the equipartition time is longer than the age of the oldest SNRs (Spitzer 1962). Since the SNR shell strongly emits thermal X-ray radiation even in the vicinity of the putative position of the blast wave, the classical theory should be reviewed and mechanisms based on plasma instabilities have been invoked to explain the apparent faster-than-classical equilibration (McKee 1974, Cargill & Papadopoulos 1988, Lesch 1990). On the other hand, strong observational constraints on the relation between and still lack. On the basis of our results, we can derive some hints on this relation in the Vela SNR shell.
with is the ion temperature in units of K, is in units of yr, in cm-3 and in erg. By inserting appropriate t and values, and using an upper limit for the explosion energy in units of erg (), we obtain a corresponding limit on . In the hypothesis of no instantaneous electron-ion energy equipartition, , where . For instance, if , which should be reasonable for typical supernova explosions (Lattimer et al. 1985), cm-3 and yr, we get K . This means that our data are not incompatible with deviation from energy equipartition, but we exclude large deviations, such as the one expected in case of slow Coulomb collision-driven equipartition of Spitzer (1962). Moreover, we stress that if and , the Sedov derived SNR distance would be 420 pc, which is larger than the recent constraints of the Vela SNR distance reported by Oberlack et al. (1994). If the plasma is in NEI, we have derived, following Sect. 3.2.4, that computed according to Eq. 10 should be increased by a factor 1.17. Therefore, we conclude that k should be in the range 1-2.5 with low values favored by consistency checks. Fast equipartition is also compatible with observation of some other SNRs, as reported by Laming et al. (1996) and Willingale et al. (1996).
4.6. Comparison of parameters derived with different emission models
Our determination of the remnant characteristic parameters provides us with the possibility to make a consistency check of our hypothesis about the association between the X2 component and the inter-cloud medium by comparison between the values derived from the 2T model fitting with the corresponding values derived assuming 1T models, and by the comparison with values published in literature. In Table 5 we summarize the Vela SNR parameters derived from the X2 component of our 2T model and those derived from the fitting results with the STNEI model of Paper II and with the 1T RS model of Paper I. We stress that, to produce Table 5, we have used only the fitting results in those spatial bins which are acceptable according to the test (at the 95% confidence level). We note that, while the normalization factors (F) derived by different models are similar, the explosion energy values () is unreasonably low for the RS 1T model and still very low for the 1T STNEI model, confirming that the X2-intercloud association represents the most realistic description. In fact, the energy derived with the 1T RS model, is always erg, i.e. below the lower limit quoted by Nomoto, Sugimoto & Neo (1976) for typical supernovae explosions, due to the low values of the plasma temperature ( in the Sedov model), obtained because the emission measure is dominated by the inhomogeneities. Table 5 also shows that the 1T spectral analysis, which has been performed in other middle-aged SNR (for instance, on G18.95-1.1 by Aschenbach et al. 1991, and on RX04591+5147 by Pfeffermann et al. 1991) could have yielded biased results, since the 2T nature of the X-ray emission may be very common in these objects. The derived pressure, dyn cm-2 also represents a problem with the 1T RS model, since, according to the results of Cui & Cox (1992), such a value is more appropriate for an age of yr rather than for the estimated Vela SNR age of yr, while the 2T model provides us with a more reasonable value.
Table 5. Vela SNR characteristic parameters derived from Sedov analysis of X2 component of the RS 2T model and from the RS 1T model.
4.7. Comparison with ISM models
Here we review our results about the nature of the sources of the Vela SNR X-ray emission and we compare them to the multi-phase ISM model introduced by McKee & Ostriker (1977, hereafter MO77) and subsequently updated. We stress, however, that both the X-ray emission model and the ISM model are based on a small number of discrete components and thus is just an approximation of the more realistic smooth variations of the physical parameters between different phases.
In Table 6, we report the physical parameters of the two X-ray components and their association with ISM phases. Our best-estimate for the inter-cloud medium density is 0.03 cm-3, significantly higher than the density of the phase IV reported by MO77 (0.004 cm-3).
Table 6. Comparison between X-ray derived results and ISM phases.
Therefore, we argue that the phase IV is incompatible with our results, in agreement with recent results which have posed strong constraints on the existence of a very low density and very hot ISM phase (Korpela et al. 1998) and the SNR ability to produce a detectable fraction of hot gas (Slavin & Cox 1993; Shelton & Cox 1994). We therefore associate the X2 component to a shock propagating in what we have called phase III.5 in Table 6, which have characteristics intermediate between the MO77 phases III and IV. In particular, the density of phase III.5 is always higher than 0.01 cm-3, and its pre-shock temperature is lower than that reported by MO77 for the phase IV, in agreement with the low abundance of O VI reported by Korpela et al. (1998).
The inhomogeneities associated to the X1 component, represents the boundary between the phase III.5 and the ISM clouds' corona (phase III of MO77). We note that, according to MO77, a typical ISM "cloud" has a density of 10 cm-3 or more, while the post-shock density derived by the X1 component is only 0.5 cm-3. This may somewhat favour the interpretation of the X1 component in terms of gas evaporated by dense clouds overrun by the blast wave, but on the other hand there is in principle no reason why the ISM clouds should have only densities of 10 cm-3 or higher, and hence the non-evaporative isolated cloud model is equally feasible.
The fact that we do not detect ISM inhomogeneities with density higher than 0.5-1 cm-3 is caused by the fact that the propagation of the Vela shock inside them would not produce detectable X-ray emission, but only UV and optical emission. It is therefore of great importance to analyze the optical emission of these rim regions, and to study the relative position of X-ray and optical filaments. This will be done in a subsequent paper, in which we shall discuss the physical characteristics of the most dense regions of the ISM inhomogeneities, as obtained from observations with IUE and from optical spectrophotometry of selected filaments with the ESO 2.2m telescope.
© European Southern Observatory (ESO) 1999
Online publication: February 23, 1999