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Astron. Astrophys. 342, 213-232 (1999)

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3. Radiation transfer of the soft X-rays

Earlier investigations of the ROSAT data (Snowden et al. 1994b; Herbstmeier et al. 1995; Kerp et al. 1996) indicated that the brightnesses from both, the distant X-ray sources and from the Local Hot Bubble (LHB) vary across the sky. Because of the variations of both source terms, we first address some general properties of the [FORMULA] keV radiation transfer through the interstellar medium before attempting to identify imprints of HVCs on the SXRB radiation. We focus on the following questions:

  1. Are there significant variations of the SXRB source distribution on scales of [FORMULA] to tens of degrees?

  2. Is H I alone the tracer of the soft X-ray absorption by neutral matter? Do the H I /X-ray results require diffusely distributed [FORMULA] and/or [FORMULA] at high [FORMULA] as additional tracers of soft X-ray absorption?

  3. Can the velocity range of galactic H I accounting for the SXRB absorption be constrained: does the gas at conventional and intermediate galactic velocities suffice, or are HVCs also implicated?

To answer these questions, we sought an expression for the soft X-ray radiation-transfer equation which reveals simultaneously the intensity and positional distribution of the individual source terms.

3.1. The soft X-ray radiation-transfer equation

The soft X-ray intensity distribution is modulated by photoelectric absorption due to interstellar matter lying between the observer and the source of the X-rays. The effective photoelectric absorption cross section [FORMULA] depends on the chemical composition of the absorbing matter, normalized to a mean absorption cross section per neutral hydrogen atom (Morrison & McCammon 1983). Moreover, the absolute value of this cross section depends on the source spectrum and on the sensitivity function (bandpass) of the X-ray detector system. These dependences stem from the energy dependence of the absorption cross section [FORMULA]. This leads to stronger attenuation for the lower-energy X-ray photons than for the more energetic ones. The more absorbing matter is located along the line of sight the stronger the softer-energy end is attenuated relative to the harder-energy end. This situation leads to an apparent hardening of the source X-ray spectrum due to photoelectric absorption.

The dependence of photoelectric absorption cross section on [FORMULA] is shown in Fig. 1 for the LHB, for a galactic halo plasma, and for a power-law extragalactic X-ray spectrum. We discuss below the X-ray source spectra of these three components. At high [FORMULA], the ROSAT PSPC data suggest that, in addition to emission from the LHB, diffusely distributed X-ray emission originates beyond the bulk of the galactic H I gas layer (Herbstmeier et al. 1995; Kerp et al. 1996; Pietz et al. 1998a, 1998b; Wang 1998). This situation requires at least two source terms in the radiation-transfer equation.

[FIGURE] Fig. 1. ROSAT [FORMULA] keV photoelectric absorption cross section (Morrison & McCammon 1983) versus [FORMULA]. The effective cross section [FORMULA] depends on the X-ray spectrum. The solid line shows [FORMULA] for the LHB plasma with [FORMULA] K; the dashed line shows [FORMULA] for the distant galactic plasma with [FORMULA] K, based on the Raymond & Smith (1977) plasma code. The dotted line shows [FORMULA] for the extragalactic power-law spectrum of [FORMULA] (Gendreau et al. 1995).

The X-ray emission from the LHB evidently originates from a thermal plasma ([FORMULA]; McCammon & Sanders 1990), embedded in the local void of neutral matter. The local interstellar cavity is evidently an irregularly-shaped, low-volume-density region enclosing the solar neighborhood, where the X-ray intensity varies (roughly proportionally to the pathlength, in the range of 50 to 150 pc, through the local cavity) on scales of several tens of degrees (Cox & Reynolds 1987; Egger et al. 1996).

The distant soft X-ray emission is most likely the superposition of thermal plasma radiation ([FORMULA]; Kerp 1994; Sidher et al. 1996) from the galactic halo (Pietz et al. 1998a, 1998b) and emission from unresolved extragalactic point sources building up the extragalactic soft X-ray background ([FORMULA]; Hasinger et al. 1993). Accordingly, the soft X-ray transfer equation has the form:

[EQUATION]

3.2. The spectral properties of the distant source terms

The LHB term represents the thermal plasma radiation of the local gas. The intensity of the LHB varies across the entire sky: [FORMULA] = (2.5 - 8.2)[FORMULA] (Snowden et al. 1998). The distant soft X-ray term represents the superposition of the isotropically distributed intensity of the extragalactic background radiation and the distant galactic plasma radiation. The unabsorbed X-ray intensity contributed by the extragalactic radiation is about [FORMULA] = (2.3 - 4.4)[FORMULA] (Barber et al. 1996; Cui et al. 1996) while the unabsorbed distant X-ray intensity (assuming a patchy galactic X-ray halo) is about [FORMULA] = (4.0 - 30)[FORMULA] (Snowden et al. 1998). Thus, the distant galactic X-ray plasma will be of prime importance in studying [FORMULA] because it is the source term with the largest intensity range and with an unknown distribution across the fields of interest.

It is plausible to assume that all of the galactic ISM is available to absorb radiation from the extragalactic SXRB component. Pietz et al. (1998b) derive an exponential scale height of [FORMULA] kpc for the X-ray emitting halo, while Lockman & Gehman (1991) showed most of the conventional-velocity galactic H I gas is located at [FORMULA] kpc.

3.2.1. The source spectrum of [FORMULA]

The galactic halo X-ray emission is evidently due to thermal-plasma processes. Rocchia et al. (1984) found plasma emission from O+6 and O+7 ions. Hasinger (1991) found indications in deep PSPC observations for an emission bump in the X-ray spectrum near 0.6 keV, also indicating the presence of these ions. Kerp (1994) and Sidher et al. (1996) showed that a thermal-plasma spectrum fits PSPC data well. The PSPC data suggest that the distant X-ray plasma, approximated by the Raymond & Smith (1977) model, has a temperature [FORMULA] K. In view of these results and of those of Pietz et al. (1998b), we assume that the galactic halo plasma is in collisionally ionized equilibrium. (This assumption is a simplification of the plasma processes occurring at high [FORMULA], but is reasonable despite lacking detailed information about the X-ray spectrum.) Note that near [FORMULA] K, the absorption cross-section in the [FORMULA] keV band does not depend strongly on plasma temperature (Snowden et al. 1997).

3.2.2. The source spectrum of [FORMULA]

[FORMULA] is caused by the superposition of X-rays from extragalactic point sources (Hasinger et al. 1998). The spectrum of the extragalactic background is a matter of discussion (Gendreau et al. 1995; Georgantopoulos et al. 1996). The averaged spectrum of bright, discrete soft X-ray sources, together providing the extragalactic background in the ROSAT energy window, can be approximated by a power law [FORMULA] with an averaged spectral index of 2.1 - 2.2 (Hasinger et al. 1993, 1998). At lower fluxes, the contribution of faint emission-line galaxies dominates the spectral properties of the extragalactic background, leading to a flatter power-law slope (Almaini et al. 1996). Our investigation of the SXRB deals with lower source fluxes than those investigated by Almaini et al.; accordingly, a plausible value of the extragalactic spectral index is [FORMULA] (Gendreau et al. 1995).

3.3. The simplified soft X-ray radiation-transfer equation

We show that we may simplify Eq. (1) into an expression involving only two X-ray source terms for the [FORMULA] keV band, namely the LHB source term [FORMULA] and the distant source term [FORMULA], representing the superposition of the thermal plasma emission beyond the bulk of the galactic H I and the extragalactic background radiation: [FORMULA].

3.3.1. The [FORMULA] source term

The LHB source term varies approximately in proportion to the extent of the local cavity (Snowden et al. 1998). The ROSAT X-ray data considered here are limited in sensitivity (at the 3-[FORMULA] level) to [FORMULA] variations of about [FORMULA]. The moderate angular resolution of the data we chose limits the angular extent of the small-scale intensity variations, anti-correlated to small-scale [FORMULA] variations, to about [FORMULA]. Thus, a narrow H I filament with [FORMULA] will not be detectable. Taking these limitations into account, interstellar absorption-line measurements (Welsh et al. 1998) show that properties of the local cavity vary smoothly on angular scales of several tens of degrees. Therefore, [FORMULA] reveals a distribution of soft X-rays approximately smooth over tens of degrees. We start our analysis using the assumption that across each individual field [FORMULA] = const., and then show below that this conforms to the observed situation.

Because the effective photoelectric absorption cross section of the LHB plasma is larger than that of the galactic-halo plasma and of the extragalactic power-law spectrum (Fig. 1), deviations from the assumption of [FORMULA] = const. will be easily detected. A local cloud attenuating soft X-rays will be disclosed by a deeper soft X-ray shadow than would be the case if the same cloud were located outside the LHB (see Kerp & Pietz 1998).

3.3.2. The [FORMULA] source term

The [FORMULA] source term represents the sum of [FORMULA] and [FORMULA]. Fig. 1 suggests that the photoelectric absorption cross sections of the halo plasma and the extragalactic power-law spectrum have comparable values. The largest difference between the cross sections, amounting to some 20%, occurs in the range [FORMULA]. Evaluating [FORMULA], we see that such a difference corresponds to a 7% effect on [FORMULA], which is negligible to our purposes in view of the statistical limitations of the X-ray data. Moreover, as we show below (see Fig. 7), [FORMULA] in most regions of the high-[FORMULA] sky, so that the influence of the difference between the cross sections is reduced in proportion to the intensity contrast of both source terms. Hence we assume, for our purposes, that [FORMULA] towards high [FORMULA].

We assume thus that [FORMULA] = const. within each field examined. Deviations from this assumption will be revealed by failures of our model to account for the observed soft X-ray emission. [FORMULA] is dominated by the [FORMULA] term because [FORMULA] towards high-[FORMULA] directions. To separate [FORMULA] and [FORMULA], we would need supplementary ROSAT PSPC pointed data (see Barber et al. 1996; Cui et al. 1996; and our discussion in Sect. 5.1). We thus arrive at the simplified radiation-transfer equation

[EQUATION]

This equation is of the form earlier studied by Marshall & Clark (1984). In the following we show that Eq. (2) represents the observed situation well.

3.4. Evaluation of the radiation-transfer equation

3.4.1. The general approach

We evaluated the SXRB radiation-transfer equation (Eq. 2) using several different methods. Table 1 lists the [FORMULA] range for each field. Traced by [FORMULA], we evaluate [FORMULA] and the corresponding attenuation of [FORMULA]. A standard method (see e.g. Herbstmeier et al. 1995) involves fitting Eq. (2) to the data, plotted in the form of a scatter diagram of observed SXRB count rate versus total [FORMULA]. The disadvantage of such a method is that it neglects the positional information of the data.

An alternative method was introduced by Kerp et al. (1996), who questioned the assumption that all of the H I is located between the observer and the distant X-ray sources; the SXRB/[FORMULA] relation might depend on the kinematic range of integration entering [FORMULA]. They evaluated the modelled SXRB intensity distribution according to Eq. (2) for each image pixel. Hence they determined the deviation between the observed and the modelled SXRB intensity distribution, giving a measure of the degree of correlation or anti-correlation of observed and modelled SXRB images. By averaging the individual deviation values of the image pixels across the entire field, they calculated the brightness of the source terms in Eq. (2). The intensities [FORMULA] and [FORMULA] were tuned to minimize the difference between both images. This method accounts for the location of the X-ray absorbing clouds within the field and directly reveals the areas where the X-ray data significantly deviate from the modelled mean intensity values.

Here, we optimize the method of Kerp et al. (1996) with respect to evaluation of the derived count rate of the [FORMULA] component. We calculated the optimal [FORMULA] value using Eq. (2) individually for each image pixel. For instance, if the distant X-ray source is patchy or if the distribution of [FORMULA] does not correctly trace the amount of X-ray absorbing matter (perhaps due to neglecting the existence of [FORMULA] and [FORMULA]), then a very patchy modelled SXRB intensity pattern would have followed, whereas, in fact, it was determined as quite constant.

3.4.2. First results

Fig. 2 illustrates our results, comparing, for one of our fields, the SXRB distribution observed by the ROSAT PSPC with the modelled situation. In order to calculate this modelled map, we determined a constant [FORMULA] intensity level across the entire field. In our procedure we let a constant [FORMULA] X-ray background intensity penetrate through the absorbing neutral interstellar medium - Fig. 2c shows the [FORMULA] distribution as tracing absorption at [FORMULA] - and add the [FORMULA] emission, also assumed to be constant, to this attenuated SXRB map. We tuned both constant X-ray source intensity levels of Eq. (2) in order to obtain the best fit to the observations.

[FIGURE] Fig. 2a-f. Maps of the part of HVC complex C at both lower l and lower b (see Sect. 4.1.1). a  Observed [FORMULA] keV ROSATPSPC map. Dark colours denote low brightnesses [FORMULA]; bright colours, strong emission [FORMULA]. Solid lines indicate areas where more soft X-ray emission is observed than expected by the model of Eq. (2); dashed lines enclose areas where the X-ray emission is weaker than expected. The lowest contour represents the 4-[FORMULA] level; the contour step is 2 [FORMULA] [FORMULA]. b  Modelled situation which results if a constant SXRB, attenuated only by the [FORMULA] values (at [FORMULA]) shown in panel c , with [FORMULA], is combined with the unabsorbed local radiation of [FORMULA], also assumed constant across the field. The modelled map has the same maximum and minimum intensity values as the ROSAT image shown in panel a . All images have the same angular resolution of [FORMULA]. c  [FORMULA] distribution contributed from [FORMULA]: [FORMULA], [FORMULA]. d  Greyscale: [FORMULA] distribution contributed by HVC velocities, [FORMULA]: [FORMULA], [FORMULA]. The contours are as described in a . e  Greyscale: [FORMULA] distribution of the IVC velocity regime, [FORMULA]: [FORMULA], [FORMULA]. The contours are as described in a . f  Positional dependence (with galactic latitude [top] and longitude [bottom]) of the averaged modelled and observed SXRB intensity profiles for the lower-longitude end of HVC complex C. The solid lines represent the simulated soft X-ray intensity distribution, modelled as described in the text; the points mark the observed distribution and its corresponding 1-[FORMULA] uncertainties.The agreement of the modelled values with those observed indicates that the soft X-ray background radiation is smoothly distributed across the field, and that H I traces predominately the large-scale photoelectric absorption by the galactic interstellar medium.

To quantify this result, we tested the hypothesis that the differences between the observed and modelled intensity distributions are statistical deviations and not uncertainties introduced by the modelling of the X-ray data. The observed minus modelled X-ray intensity distribution was binned into a histogram (100 bins) showing the frequency of the deviation versus the deviation value. The histogram was quantitatively compared with a Gaussian distribution using a [FORMULA] test. We found [FORMULA], well below the acceptable value of [FORMULA] for 96 degrees of freedom, and a rejection threshold of 0.05. The hypothesis that both distributions are significantly different has to be rejected. This confirms that our approach of assuming constant [FORMULA] and [FORMULA] matches the observed situation well. Additionally, this finding confirms that H I is the best tracer of the photoelectric absorption and that H+ as well as H2 influence the soft X-ray radiation transfer on a much lower level compared to H I . Thus, we conclude that [FORMULA] can be approximated well by an intensity which is constant across the entire field: the distant soft X-ray background radiation is not patchy on angular scales of some tens of degrees. This finding was verified for all analyzed fields, distributed across the sky. The absolute value of [FORMULA] varies significantly, however, between the individual fields. Because [FORMULA] is plausibly constant across the entire sky, the large-scale variation of [FORMULA] is entirely attributed to [FORMULA]. This will be discussed in detail in Sect. 5.1.

3.4.3. Interpretation of the results

Following the procedures described below, we scaled the intensity of a constant-intensity X-ray background source beyond the entire [FORMULA] contribution shown in Fig. 2c. This yielded the image of the modelled SXRB intensity distribution shown in Fig. 2b. In Fig. 2a, we superposed, as contour lines, the deviations between the observed and the modelled SXRB intensity distribution, starting with the 4-[FORMULA] level and increasing in steps of 2[FORMULA]. Dashed lines indicate areas where the modelled SXRB intensity is too bright, or where we missed additional X-ray absorbers not traced by the H I radiation; solid lines mark areas where ROSAT detected more radiation than expected by the H I data. At these positions, we have either overestimated the amount of absorbing matter or we are observing true excess X-ray emission. This excess corresponds to some 25% of the total SXRB intensity. In general, an underestimate of the amount of matter attenuating the X-rays is more likely than an overestimate, because neither H2 nor H+ is represented by the 21-cm tracer. Thus, it seems likely that the dashed contours indicate the presence of additional absorbing matter, but that the solid contours indicate X-rays in excess of the average.

3.4.4. Evaluation of [FORMULA]

We evaluated the level of the modelled constant distant X-ray source intensity using three additional methods. First, we averaged the X-ray halo intensities across the entire map over areas of equal [FORMULA] in bins of [FORMULA]. This yielded the dependence of the X-ray halo intensities on the amount of absorbing H I shown in Fig. 3. The slope of the dependence is a function of the assumed [FORMULA] count rate. If the [FORMULA] count rate is underestimated, we obtain a correlation of X-ray halo intensity with [FORMULA]; in case of an overestimate of [FORMULA], we obtain an anti-correlation. We tuned the [FORMULA] value such that the dependence is minimized. This alignment corresponds to the assumption that the [FORMULA] keV radiation is independent of the amount of H I along the line of sight. Such is certainly not the case for specific areas of the galactic sky. For example, towards the North Polar Spur (Egger & Aschenbach 1995) the X-ray intensity is not distributed independently from the [FORMULA] structure.

[FIGURE] Fig. 3. To constrain the velocity-integration range for [FORMULA], [FORMULA] is plotted as a function of [FORMULA] for the HVC complex C ROSAT data presented in Fig. 2 (see Sect. 4.1.2). The histograms represent [FORMULA] versus [FORMULA] integrated over three different velocity ranges. The horizontal solid line marks the best-fit [FORMULA] intensity level, while the horizontal dashed lines indicate the uncertainty range of the modelled value. Within the uncertainties (the errorbar in the lower-right part of the figure), [FORMULA] can be considered as constant across the field of interest. This finding has two major implications; first, [FORMULA] is within the uncertainties also constant across the field of interest. Second, the WNM, already enclosed in the velocity brackets [FORMULA], determines the [FORMULA] intensity level. The more extreme velocity ranges introduce only a minor intensity variation, while the functional dependence of [FORMULA] on [FORMULA] is unaffected.

Second, we averaged both the [FORMULA] keV and the [FORMULA] data over l and b, respectively, and compared these mean observed intensity values with the model. This method allows searching for systematic uncertainties introduced by the modelling of the X-ray data. We tested the hypothesis that areas of the sky with the same [FORMULA] values correspond to unique [FORMULA] and [FORMULA] values, within the uncertainties of the X-ray data. We evaluated the dependence of the source terms in Eq. (2) on the galactic l and b profiles. The derived values for [FORMULA] and [FORMULA] agree with those calculated by the first method. Fig. 2f shows the dependence on l and b of the soft X-ray radiation-transfer equation solved with the same intensity values for the LHB and the galactic halo plasma as used to derive panel (b) of Fig. 2. There are no significant large-scale differences between the l and b distributions of the observed SXRB radiation and the modelled X-ray intensity derived from the [FORMULA] distribution. This indicates that the distant soft X-ray emission is constant, within the statistical limitations of the X-ray data, across each field.

Third, we averaged observed SXRB count rates with a given [FORMULA] in steps of [FORMULA] and plotted a simple scatter diagram of [FORMULA] versus [FORMULA]. This method is sensitive to the choice of the source term parameters in Eq. (2), and would reveal erroneous model parameters.

Thus, we confirmed the validity of the soft X-ray radiation-transfer solution using three independent methods. The second and, even more so, the third, method suffers from neglect of the positional information in the ROSAT maps. But they show that the [FORMULA] values returned are consistent with those of the first method, which does account for the positional information. This indicates that the [FORMULA] source term is, within the statistical limitations of the X-ray data, constant on angular scales of several tens of degrees.

We considered the uncertainties of the individual soft X-ray source terms by varying [FORMULA] or [FORMULA] independently in a way that the modelled and observed intensities fit within the statistical uncertainties of the data. Because the quantities are field-averaged, the corresponding uncertainties are low. For the local X-ray emission, [FORMULA]; [FORMULA], depending on the averaged [FORMULA] value across the field, and thus typically an order of magnitude higher than the LHB plasma.

3.5. X-ray absorption traced by H I

3.5.1. Velocity information in the H I data

Above, we described our investigation of the [FORMULA] and [FORMULA] source terms of Eq. (2). Now, we show how we determined the amount of H I absorbing the soft X-rays. The velocity information contained in the H I data gives an additional free parameter in Eq. (2). We can integrate the H I brightness temperatures over different velocity intervals, introducing a kinematic unravelling which may indicate also a spatial separation. Three separate velocity regimes are commonly, albeit somewhat arbitrarily, distinguished in the literature, namely as low-velocity (LV: [FORMULA]), intermediate-velocity (IV: [FORMULA]), and high-velocity (HV: [FORMULA]). The low-velocity regime not only samples all of the higher-[FORMULA] H I which belongs to the conventional galactic disk, it includes most of the H I which corresponds to the warm diffuse H I layer (e.g. Dickey & Lockman 1990, also denoted as warm neutral medium , WNM) as well. If we integrate the H I spectra over the low-velocity regime, we neglect some 10% of the total amount of H I distributed across the field, although this percentage varies from region to region. In some regions, there is as much emission from H I gas at extreme velocities as from LV matter; towards these lines of sight it is not feasible to evaluate the soft X-ray radiation transfer only with the low-velocity [FORMULA]. The Draco cloud (Herbstmeier et al. 1996) is an example of an IVC dominating [FORMULA]; in addition, it contains significant amounts of molecular matter. Finally, HVCs may also absorb the distant SXRB source radiation (see Herbstmeier et al. 1995).

3.5.2. Dependence of [FORMULA] and [FORMULA] on the [FORMULA] velocity interval

To test whether the choice of [FORMULA] velocity interval reveals a kinematic unravelling of the source of the SXRB, we integrated the H I emission separately over the LV range [FORMULA], and over two wider velocity ranges [FORMULA] and [FORMULA] Towards high galactic latitudes the latter range encompases all interstellar gas except the HVCs. The histograms in Fig. 3 represent [FORMULA] as a function of [FORMULA]. Within the uncertainties of the histogram data points (the error bar in Fig. 3), [FORMULA] can be considered a constant across the [FORMULA] range of [FORMULA] to [FORMULA]. The horizontal solid line represents our field-averaged best-fit value of [FORMULA], while the dashed lines mark the uncertainties of this best-fit value. Taking into account the uncertainties of both, the data and the modelling, the assumption of a constant [FORMULA] is justified. With Fig. 3 we can also constrain the expected intensity variation of the [FORMULA] source term, because to evaluate [FORMULA] as a function of [FORMULA] we a priori assumed [FORMULA] = const. Consequently, our finding [FORMULA] = const. implies [FORMULA]= const. within the uncertainties of the analysis.

The three histograms in Fig. 3 show that the functional dependence of [FORMULA] on [FORMULA] is independent of the extent of the velocity range used to evalute [FORMULA].

However, the mean level of [FORMULA] increases proportionally to the extent of the integration range of [FORMULA]. Nevertheless, all data points of the three histograms are within the uncertainty range of the modelled [FORMULA] intensity level. We conclude that the WNM in the Galaxy determines the mean intensity level of the distant soft X-ray background radiation. Towards high galactic latitudes, the WNM best represents the physical state of the major fraction of the interstellar matter. Accordingly, the H I belonging to the WNM traces the amount of soft X-ray absorbing matter, and determines the mean intensity level of the distant diffuse X-ray radiation. The bulk of the WNM is already enclosed in the velocity bracket [FORMULA] (Dickey & Lockman 1990). Accordingly, the additional [FORMULA] at more extreme velocities increases the mean [FORMULA] level, but does not change significantly the functional dependence of [FORMULA] on [FORMULA].

The discussion above implies that our modelling of the ROSAT X-ray data can be well approximated by constant [FORMULA] and [FORMULA] source terms across the extent of the fields of interest (question 1 of Sect. 3). The mean [FORMULA] X-ray intensity level is determined by the distribution of the WNM gas. The more extreme velocity ranges represent, on the average, only a minor fraction of the total interstellar gas. Accordingly, the [FORMULA] is sufficient to determine the [FORMULA] responsible for the attenuation of the distant diffuse X-ray sources (question 3 of Sect 3). However, our aim is to search for soft X-ray enhancements of HVCs, with respect to the finding shown in Fig. 3 the [FORMULA] yields the highest [FORMULA] intensity level. This attributes a maximum of the diffuse X-ray emission to [FORMULA] and introduces a systematic bias in our analysis for the non-detection of excess X-ray emission associated with HVCs.

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Online publication: December 22, 1998
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