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Astron. Astrophys. 332, 55-70 (1998)

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5. Distribution of the thermal 3/4 keV XRB

In Sect. 4 we found evidence that the observed intensity variation of the [FORMULA] keV XRB radiation is a function of galactic longitude. Since the differential photoelectric absorption cannot cause the apparent intensity variation between the galactic center and anticenter we conclude that the distant galactic XRB plasma component is the source of the intensity variation.

It is important to establish the distribution of the distant X-ray plasma located beyond the bulk of the galactic neutral hydrogen layer. Nousek et al. (1982) confronted three different models with [FORMULA] keV observations which had been obtained on Aerobee rocket flights:

1) An isotropic background component [FORMULA] affected by absorption plus a local unabsorbed foreground component [FORMULA]:


2) A layer parallel to the galactic plane [FORMULA] plus an extragalactic isotropic component [FORMULA], both absorbed by neutral matter in the Galaxy:


3) A spherical galactic halo model as an uniformly emitting sphere [FORMULA] affected by absorption:


where [FORMULA] is the radius of the halo in units of the solar galactocentric distance and [FORMULA].

Nousek et al. (1982) found that their models 1 and 2 gave acceptable fits to the X-ray data available at that time. They concluded in favor of a disk-like galactic halo (model 2), mainly because such a model is easier to understand in a physical context.

In the following we perform an analysis similar to that of Nousek et al. (1982) using the RASS and the Leiden/Dwingeloo H data. With the new H and X-ray data it is possible to constrain the models more strongly.

5.1. Modeling the 3/4 keV XRB

We start the analysis with model 1. Based on the results given in Sect. 3 and Sect. 4, we assume that the 3/4 keV foreground component is [FORMULA]. The extragalactic XRB component is [FORMULA] and the distant galactic XRB plasma count rate is [FORMULA] in the direction of [FORMULA].

Using these values we calculate the expected count rate distribution using Eq. (5) and compare the count rates with the observations (thin lines in Fig. 4). On small angular scales of [FORMULA], the [FORMULA] keV intensity distribution can be reproduced. For example, the deep X-ray absorption minima near [FORMULA], [FORMULA], and [FORMULA] close to the Orion-Eridanus Bubble can be modelled, demonstrating that the fraction of the H column to the total absorbing column density is sufficient to model the absorption of the 3/4 keV radiation even in this molecular-gas-rich area. Even the humps in Fig. 5 between [FORMULA] and [FORMULA] at [FORMULA] and [FORMULA] are predicted by the model. This demonstrates that Eq. (4) is meaningful in describing the observed [FORMULA] keV count rate distribution on angular scales of [FORMULA] towards the high-latitude sky.

[FIGURE] Fig. 4. Comparison of the observed [FORMULA] keV count rates (points) with the halo model 1 (isotropic XRB intensity distribution, thin line) for selected latitude strips towards the northern (upper panel) and southern galactic hemispheres (lower panel). The isotropic count rate [FORMULA] is the superposition of the extragalactic component with an intensity of [FORMULA] and a distant X-ray plasma component with a temperature of k [FORMULA] and an X-ray intensity of [FORMULA]. Both components are absorbed by all neutral matter distributed along the line of sight. The shaded area marks the position of the Orion-Eridanus Bubble. The thick line represents an intensity-variable plasma component as a function of galactic longitude in addition to the constant extragalactic XRB. At [FORMULA] the thermal component decreases linearly from [FORMULA] ([FORMULA]) down to [FORMULA] ([FORMULA]). The amplitude of the [FORMULA] keV XRB intensity variation decreases with increasing latitude. The difference between the observed [FORMULA] keV intensity distribution and the models is plotted in the lower half of each diagram. The longitude-dependent plasma model fits the observations best.

[FIGURE] Fig. 5. Intensity variation of the [FORMULA] keV XRB predicted by the flattened-halo model (Sect. 5.2) with [FORMULA] and [FORMULA] (left) and by the spherical-halo model with a halo radius of [FORMULA] (right). The [FORMULA] keV intensity variation is plotted against longitude for latitudes between [FORMULA] and [FORMULA]. The intensity level of the extragalactic XRB is marked by the horizontal dashed line. Note that the intensity variation is not caused by photoelectric absorption of the interstellar matter along the line of sight, but represents the unabsorbed [FORMULA] keV XRB brightness.

Nevertheless, deviations of the observed count rates from the modelled [FORMULA] keV count rates exist on larger angular scales. All [FORMULA] keV latitude slices reveal a count rate minimum in the direction of the galactic anticenter (e.g. Fig. 4, [FORMULA]). The intensity contrast between the galactic anticenter region and the neighbouring longitudes decreases with increasing latitude.

Such a behaviour cannot be reproduced by the isotropic model (model 1), nor can it be reproduced by a plane-parallel galactic halo (model 2) (see e.g. Fig. 5, [FORMULA] and [FORMULA]). To overcome this problem we introduce a galactic-longitude dependence to the model. We demonstrate that this is necessary by modeling the observations first with a simple linear dependence of the [FORMULA] keV intensity on galactic longitude, assuming a linear decrease of the [FORMULA] keV distant galactic component between [FORMULA] and [FORMULA].

Furthermore, we have to take into account that the amplitude of the intensity modulation decreases as latitude increases from the galactic equator to the galactic poles. As is obvious in Fig. 5, such a linear dependence of the [FORMULA] keV intensities on galactic longitude (thick lines in Fig. 4) reproduces the observed [FORMULA] keV count-rate distribution better than the models discussed above. It is also relevant that the dependence of the [FORMULA] keV radiation on galactic longitude is the same for both galactic hemispheres. This indicates that the brightness of the distant galactic XRB plasma is the same on both sides of the Galaxy.

Hence we conclude that a thermal X-ray halo model must fulfill the following conditions:

  • The halo X-ray count rate modelled should decrease smoothly from the galactic center towards the galactic anticenter.
  • The halo X-ray count rate should decrease smoothly from the galactic plane to the galactic poles.
  • The halo model should be largely symmetric with respect to the galactic equator.

Recently Freyberg (1994, 1997) analyzed the [FORMULA] keV RASS data and came to the conclusion that a spherical-halo model (model 3) with [FORMULA] gives the best fit to the data. Indeed such a model (see Fig. 5 right) is consistent with the conditions mentioned above.

Another model which also accounts for the above conditions follows from analysis of the Leiden/Dwingeloo survey and describes the distribution of H gas at large z distances. We explain this model first before comparing it with the spherical galactic halo model.

5.2. A halo hydrostatic-equilibrium model

Recent investigations by Kalberla et al. (1997) tentatively suggest the existence of H at large z distances which can be described by hydrostatic equilibrium conditions between gravitation, gas pressure, and magnetic pressure. H gas with a velocity dispersion of [FORMULA] was found to be consistent with model assumptions of a turbulent gas layer with a scale height of 2 kpc or more. Such a high-dispersion H halo agrees with the predictions of Boulares & Cox (1990) but is apparently in disagreement with models proposed by Spitzer (1956), Bloemen (1987), and Wolfire et al. (1995).

In the following, we assume that the galactic X-ray halo may be caused by a hot gas at a temperature of k [FORMULA] which is in hydrostatic equilibrium with the gravitational potential of the Galaxy. The pressure [FORMULA] is assumed to balance the gravitational potential [FORMULA]:


The function [FORMULA] according to Taylor & Cordes (1993) defines a volume-density gradient as function of galactocentric radius, R:


where [FORMULA] is the radial scale length and [FORMULA]. Taylor & Cordes (1993) assumed a scale length of [FORMULA] for the Reynolds (1991) layer. We used the potential [FORMULA] derived by Kuijken & Gilmore (1989) which is in agreement with the potential derived by Bienaymé et al. (1987). Such a potential implies a low fraction of hidden mass in the Galactic disk (see Boulares & Cox 1990 and Crézé 1991). The normalization of the density [FORMULA] is determined by the emission measure, [FORMULA] at [FORMULA], taken from the spectral-fitting results described in Sect. 3.

Most of the H gas is confined at [FORMULA] pc (Dickey & Lockman 1990). Since in the spectral analysis the foreground plasma component is fit well by k [FORMULA] keV (Sect. 3), no [FORMULA] keV emission is expected from the foreground component. Taking this observational result into account, we assumed that the hot gas causing the XRB is located entirely beyond [FORMULA] pc. Since our analysis is restricted to [FORMULA] this assumption holds true for the local environment of the Sun and not for the entire galactic disk. Such an assumption affects the results of our modeling only to a minor degree, due to the necessary renormalization with respect to the observed emission measure towards the galactic poles. We found no evidence for any significant X-ray plasma temperature variation, hence we assume that the X-ray halo can be described by an isothermal distribution. In this case the X-ray halo temperature of k [FORMULA] (Sect. 3) corresponds to a scale height of [FORMULA].

In summary, the parameters needed for our model are: a temperature, k [FORMULA] ; a mid-plane density, [FORMULA], reproducing the observed emission measure, EM ; and a scale length, [FORMULA], which still has to be determined. The count rate dependence on galactic coordinates for a scale length of [FORMULA] is drawn in Fig. 5 (left). Comparison with the count rate distribution of the spherical-halo model (Fig. 5, right) shows that both models reproduce the basic conditions mentioned in Sect. 5. The differences are found in the detailed l variations, as discussed in the following section.

5.3. Flattened-halo model versus spherical-halo model

To decide which model fits the [FORMULA] keV X-ray data best, we calculated the rms deviations, [FORMULA], of the residual between observations and model for each smoothed galactic latitude strip (Fig. 6). For the flattened models we varied the scale-length parameter [FORMULA] between [FORMULA] and [FORMULA], while the spherical-halo parameter [FORMULA] was varied between 1.5 and 5. The data-points with the largest deviations correspond to the latitude strips [FORMULA], where generally the largest deviations from our simple radiation transfer model are expected to occur in any case, because of the additional X-ray radiation from the galactic disk.

[FIGURE] Fig. 6. Mean rms deviations ([FORMULA]) between the [FORMULA] keV RASS data and the different X-ray halo models evaluated for each latitude strip ([FORMULA]) from [FORMULA] to [FORMULA] (smoothed to [FORMULA]). The first four models are thick-disk models with [FORMULA] and [FORMULA] 10, 15, 20, and 25 kpc, respectively; models 5 to 8 represent spherical-halo models with [FORMULA] 1.5, 2.5, 3.5, and 5, respectively. The diamond symbols mark the mean rms deviation averaged over all latitude strips. Obviously the mean rms deviations of flattened-halo models are in general smaller than those of the spherical halo models. The flattened-halo model with the lowest rms deviation is number 2, corresponding to [FORMULA].

The spherical-halo models (Fig. 6) reveal in general a mean rms deviation (marked with diamonds in Fig. 6) larger than that of the flattened-halo models. Of the spherical-halo models, the one with a normalized radius of [FORMULA] appears to fit the observations best. This is roughly consistent with the result of Freyberg (1994, 1997) who determined [FORMULA] as best-fit value.

The flattened-halo models (Fig. 6, models 1-4) have, however, generally lower mean rms deviations; the scatter between the different latitude strips is also significantly lower, indicating a systematically better fit of the observations by the flattened-halo models than is realized by the spherical-halo ones. The best-fit scale length is [FORMULA], similar to the scale length found for the flattened H halo suggested by Kalberla et al. (1997).

For the preferred flattened X-ray halo model with [FORMULA] we obtain an emission measure EM at the galactic poles of


with a mid-plane density of [FORMULA] (Eq. 8). The resulting variations with longitude and latitude, expressed as [FORMULA] keV count rates, are shown in Fig. 5 (left).

5.4. Comparison of the best-fit flattened-halo model with the observations

We have to modify Eq. (4) according to the results derived above, by introducing a functional dependence of [FORMULA] on galactic coordinates:


In Fig. 7 we show data from all analyzed latitude strips, smoothed to an angular resolution of [FORMULA]. The thin line within the upper part of each latitude panel represents the modelled [FORMULA] keV count rate. The thick line in the lower half of the corresponding panel indicates the difference between model and observations. Especially the amplitude of the observed [FORMULA] keV intensity variation is well reproduced in each latitude strip.

[FIGURE] Fig. 7. Comparison of the [FORMULA] keV RASS data (points) with the count rates predicted by the flattened-halo model with [FORMULA] and [FORMULA] (thin line). In the lower part of each panel the differences between data and model are plotted. The latitude strips cover a range of [FORMULA] centered at the latitude noted within each panel. The data are smoothed in longitude to [FORMULA] angular resolution. In the southern galactic hemisphere the grey-shaded area indicates the Orion-Eridanus X-ray enhancements. The Loop I region, as well as the region of [FORMULA] (where no adequate H 21-cm line data are yet available) are excluded in this figure. In the ideal case that the observed and modelled [FORMULA] keV XRB intensity profiles would match each other perfectly, the thick line would show a white-noise response.

The largest deviations between observations and model, with [FORMULA], are located near the galactic plane ([FORMULA]) between [FORMULA] and [FORMULA]. Since [FORMULA] is larger than zero, corresponding to too low predicted count rates, excess emission is located here.

Figs. 8 and 9, maps in Hammer-Aitoff projections centered on the galactic center and the galactic anticenter, respectively, demonstrate our results. These maps show the entire sky observed by the Leiden/Dwingeloo H survey, but exclude the region of the galactic equator itself. The maps have been smoothed to an angular resolution of [FORMULA]. Patterns of excess emission at low latitudes are visible in the difference maps, panels (e), representing the observed-minus-modelled [FORMULA] keV distribution. The difference maps suggest that the excess emission X-ray features may be characteristically oriented perpendicular to the galactic plane, consistent with the existence of energetic events originating near the galactic disk and extending into the lower galactic halo. Certainly, one has to consider that the interstellar environment is more complex closer to the galactic plane. The radiation transport equation used, Eq. (11), may not represent this complex situation well.

[FIGURE] Fig. 8. Comparison of the modelled and observed [FORMULA] keV and [FORMULA] keV XRB maps centered on the galactic center in Hammer-Aitoff projection. The area within [FORMULA] of the galactic plane is blanked out. a Intensity distribution of the distant galactic X-ray plasma as predicted by the flattened X-ray halo model ([FORMULA], [FORMULA]) smoothed to an angular resolution of [FORMULA]. The count rates are normalized to the XRB intensities observed within the [FORMULA] keV energy range. To obtain the total unabsorbed diffuse XRB intensity the constant extragalactic XRB radiation has to be added to this map. b H column density distribution as measured in the Leiden/Dwingeloo H survey. The area close to the galactic equator is blanked, as is the area at [FORMULA] not covered by the Leiden/Dwingeloo survey. The H column density distribution quantitatively traces the X-ray absorbing interstellar matter which modulates the unabsorbed XRB intensity distribution shown in panel a. c Modelled [FORMULA] keV XRB following Eq. (11). This modelled [FORMULA] keV XRB distribution has to be compared with the observed XRB intensity distribution, shown in panel d. e Difference map showing observed minus the modelled XRB intensity distribution. The grey-scale coding is optimized to accentuate the differences between the modelled and observed XRB maps. Solid contour lines mark areas of too-low predicted X-ray intensities (10 and [FORMULA] levels); too-high predicted X-ray intensities ([FORMULA] level) are indicated by dashed contour lines. The X-ray features contributing to the strong residuals are discussed in the text. f Modelled XRB intensity distribution of the [FORMULA] keV energy range smoothed to an angular resolution of [FORMULA]. The X-ray foreground intensities are given in Table 2. To calculate this figure we scaled the unabsorbed XRB map shown in panel a by a constant factor to the intensity level within the [FORMULA] keV energy range. This unabsorbed XRB intensity distribution is attenuated by the total H column density distribution shown in panel b. Map f has to be compared quantitatively with the observed XRB map in panel g. h Difference map between the observed and modelled XRB intensity distribution. Solid contour lines enclose areas of too-low predicted X-ray intensities; dashed contour line encircle regions of too-high predicted X-ray intensities. The dominant features in the residual maps are Loop I and the Orion-Eridanus Bubble. The dynamic range of the difference maps is much smaller than those of the XRB maps, thus the modelled XRB intensity distributions fits the observational data well.

[FIGURE] Fig. 9. Comparison of the modelled and observed [FORMULA] keV and [FORMULA] keV XRB maps centered on the galactic anticenter in Hammer-Aitoff projection (cf. Fig. 8). a [FORMULA] keV intensity distribution of the distant galactic X-ray plasma as predicted by the flattened X-ray halo model ([FORMULA], [FORMULA]). b H column density distribution derived from the Leiden/Dwingeloo H survey. c Modelled [FORMULA] keV XRB intensity distribution, which has to be quantitatively compared with the observed XRB intensity distribution shown in panel d. The difference map representing observed minus modelled XRB intensities is shown in panel e. Solid contour lines mark areas of too-low predicted X-ray intensities ([FORMULA] and [FORMULA] levels); too-high predicted X-ray intensities ([FORMULA] level) are indicated by dashed contour lines. e Modelled XRB intensity distribution of the [FORMULA] keV energy range. g Observed [FORMULA] keV intensity distribution. The difference between the observed and modeled X-ray intensity distribution is shown in panel h. Prominent in the [FORMULA] keV as well in the [FORMULA] keV difference maps are the Orion-Eridanus Bubble, located near [FORMULA], and the northern part of the "Monogem ring", located near [FORMULA]. Other residual X-ray features are discussed in the text.

Other enhanced residual-emission areas may be associated with X-ray features described in Sect. 4.2. The Orion-Eridanus Bubble is centered near [FORMULA] ; the enhancements near the galactic center are associated with Loop I and the galactic bulge. Comparison of the [FORMULA] keV and 1.5 keV RASS maps published by Snowden et al. (1995) reveal a rough correlation between [FORMULA] keV and 1.5 keV enhancements, suggesting the existence of an additional hot plasma component (k [FORMULA]) towards both Loop I and the Orion-Eridanus Bubble, as well as generally towards the galactic plane.

Extended areas where the predicted count rates are too high are observed at a level of [FORMULA] in the smoothed latitude strips, e.g. in the strip [FORMULA] to [FORMULA] at [FORMULA] (Fig. 7). Comparing the residual maps in Figs. 8 and 9 with the survey exposure map shown by Snowden et al. (1995) indicates that these deviations may be attributed to the instrumental scanning direction of the RASS (see the [FORMULA] keV map by Snowden et al., 1995).

In view of the above, we conclude that our flattened-halo model with [FORMULA] and [FORMULA] can reproduce the observed [FORMULA] keV RASS data down to the present accuracy limit of the X-ray data. Additional emission close to the galactic plane may be caused by localized galactic features.

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998