3. Analysis of the models
Following the model suggested by Mills (1956) we have assumed that the trough is produced by the absorption of an HII region in front of, or immersed in, a broad and bright source which appears as two maxima. To analyze the double peaked structure we have studied temperature profiles taken along the line connecting the peaks, and also along the equator. Dish and filled-array observations show along the equator a broad and strong feature several degrees wide superposed on a smooth and intense background that tapers off with distance from the center (Fig. 1). To determine this background, that will be a part of our models, we filter out the high spatial frequency components from the temperature profile using the method of Sofue & Reich (1979), somewhat modified. The profile along the line connecting the peaks, which at 45 MHz and below runs slightly inclined to the equator, shows the two peaks and the trough in between them. In an attempt to retrieve the magnitude of the absorbed source we fit a gaussian curve to the wings of the profile, that is, to the part not absorbed. We have defined this gaussian as the expected profile, the nomenclature for which is illustrated schematically in Fig. 2. The corresponding gaussian fits are shown in Fig. 3. Table 3 presents the measured parameters as defined in Fig. 2 and the angular size represented by the FWHM corrected for beam smoothing assuming a gaussian beam. The absorption depth at 45 and 85.7 MHz is 4 and 10 , respectively, being the rms dispersion of the fits. We found that the temperature of the base of the gaussian fit, , and that of the diffuse background, which should be the same, differ by less than 10%, and this has been considered satisfactory. For the 408-MHz profile, at , a good fit was obtained with two gaussians rather than one, so we have associated the wide gaussian (FWHM = 3.1 ) with the source we will define as the broad source while we have associated the narrow gaussian (FWHM = 0.8 ) with an emission that replaces the absorption dip ( = -1100 K). In the case of the 408-MHz profile corresponds to the peak temperature.
Table 3. Expected profile parameters
3.1. Preliminary model
There is evidence for the existence of an HII region in the direction of the galactic center. Jones & Finlay (1974) studying discrete absorption regions at 29.9 MHz along the galactic plane, found only one such a region within and from the G.C. Its location at (-0.4, 0.1 ) coincides fairly well with the position of the trough, as shown in Table 2, therefore we assume that it is the object that causes the trough. The size of the region was estimated as 1.5 x 1.0 (Jones & Finlay 1974), which is the value we will adopt to compute later the subtended solid angle. Also, Downes & Maxwell (1966) observing the galactic center region at 3 GHz found an extended thermal source, or congregation of unresolved sources, covering , . To the best of our knowledge, the only work related to the physical properties of the central HII region is that by Matthews et al. (1973 a, b). From symmetry considerations and evidence that the HII region lies behind both the 3 kpc arm and the nuclear disk, those authors conclude that the thermal source is at the galactic center. They also determined an electronic temperature of 9800 K, first from 19.7 MHz free-free absorption and 5.0 GHz continuum emission (1973a), then from 166 hydrogen recombination line observations (1973b). Following Matthews et al. we will assume that the HII region is at the G.C. and we will adopt K for its electronic temperature.
As a first approximation we assume at 408 MHz a simple model consisting of a non-thermal source of amplitude and radius a, concentric with a smaller HII region. The antenna beam (0.8 ) is about the angular size of the HII region at 408 MHz. We have also assumed that the two sources are spherical and embedded in a diffuse non-thermal background, at temperature , that pervades the whole Galaxy. The expression for the absorption depth, , can be written:
In this equation L is the radius of the Galaxy, assumed 20 kpc, and R is the distance from the Sun to the G.C., taken as 8.5 kpc. Knowing , and at 408 MHz (Table 3), we compute the optical depth ; with this optical depth we determine from the relationship and from Eq. (1) we finally obtain the absorption depth at 85.7 MHz. The absorption depth so obtained is 11800 K, which is considerably larger than the 8100 K given by the expected profile. Since we consider this difference too large to be attributable to errors we conclude that the preliminary model does not work.
3.2. Proposed model
We have seen that the amplitude of the source defined by the expected profile at 85.7 MHz is insufficient to account for the depth of absorption calculated in the preliminary model. We have then postulated the existence of a non-thermal source with angular size equal or smaller than the trough, that is, than the HII region, and which we define as the narrow source. We have assumed it to be non-thermal since an increase in intensity is needed to account for the low frequency observations. It should be noted that the existence of this source would help in solving the problem found by Little (1974) observing at 408 MHz: modeling the G.C. as a combination of a thermal source, whose temperature he extrapolated from microwave frequencies, and a non-thermal source, whose temperature he extrapolated from 85.7 MHz, he found about 1000 K he could not account for.
Our model consists of four sources, for simplicity assumed spherical and concentric with the G.C., embedded in a diffuse non-thermal background at temperature (Fig. 4). The largest of the four is a non-thermal and broad source of radius a and temperature . Then we have an HII region of radius r and electronic temperature . Uniformly mixed with the thermal region is a narrow non-thermal source of temperature and size equal or smaller than that of the HII region. The last condition comes from the fact that at 408 MHz the width of the narrow gaussian (0.8 ) constitutes an upper bound for the size of the narrow non-thermal source. The narrow source should not be confused with the narrow gaussian since, at 408 MHz, the latter is formed by the narrow source plus the HII emission. The model is completed with Sgr A complex, that we will consider as a point source, located at the very center of the Galaxy. The geometry of the model at 85.7 and 408 MHz is shown in Fig. 4a, where is the antenna beam width. Fig. 4b shows the same model for 45 MHz. Here is the angular size of the thermal source. We recall that at 85.7 and at 408 MHz . The equation of transfer valid for any of these two frequencies, and corresponding to Fig. 4a, is:
Here is the temperature of Sgr A complex and is the temperature observed with the antenna pointing to the center of the absorption region, that is, practically to the G.C. The equation for 45 MHz corresponding to Fig. 4b is:
where and is solid angle.
In order to check the consistency of our model we need to determine the nature (spectrum) of the narrow and broad sources. We will investigate first the narrow source.
3.2.1. The narrow source
In using Eq. (2) to compute , the temperature of the narrow source, the radius a can be obtained from the angular size (FWHM) given in Table 3 so, since K, the unknowns are and . To determine the optical depth, and following Little(1974), we extrapolate down to 408 MHz the microwave temperatures between 14.5 and 1.41 GHz, quoted by Downes & Maxwell (1966), which fit quite accurately a -1.99 spectral index. The temperature we obtain is 740 K (somehow Little obtained 940 K). From the relation ) and the adopted we get . Finally, the contribution from Sgr A complex is obtained from the spectrum given by Pedlar et al. (1989). Even though this spectrum corresponds only to the halo of the complex, it is recognized that the halo makes the most significant contribution, compared to Sgr A East and Sgr A West. At 408 MHz the spectrum gives a flux density of 300 Jy therefore, since the sensitivity of the Haslam et al. survey is 1.42 Jy/K, the temperature contribution from the Sgr A complex is 211 K. Replacing numerical values in Eq. (2) we get T = 202 K.
The calculation is similar for 85.7 MHz, except that at this and lower frequencies the Sgr A term is negligible compared to the others. can be computed from . Inserting numerical values in Eq. (2) for 85.7 MHz we obtain K.
To compute at 45 MHz with Eq. (3) we need to know and k. From Jones & Finlay (1974) we take x 1.0, while the antenna beam x 2.4 so . As before, can be computed from . Since at this frequency the contribution from Sgr A is negligible we obtain K. This value should be considered as an upper bound only because of the uncertainties in the measurements.
Fig. 5 shows the spectrum of the narrow source that exhibits a spectral index -2.4, clearly indicating a non-thermal source. To check this result we determined from the surveys the spectrum of the well known supernova remnant W28, which is nearby at approximately (6.2, 0.1 ). This supernova is fairly strong, it is clearly seen at 29.9, 34.5 and 85.7 MHz, and it does not seem to be much affected by absorption at 45 MHz and above (Finlay & Jones 1973; Milne & Hill 1969). The spectrum of the peak amplitude observed with 0.8 resolution, shown in Fig. 5, has an index of -2.1, within the range expected for a supernova remnant. Therefore we confirm that the spectral index of the narrow source corresponds to a non-thermal object.
Next we investigate the nature of the broad source.
3.2.2. The broad source
In order to determine the spectrum of the broad source we plotted the amplitude of the expected profiles at 408, 85.7, 45 and 30 MHz. We have assumed that the wings of the profiles, where the gaussian were fitted, have no contributions from the other sources. Unfortunately, it was not possible to have data points with the same angular resolution at the different frequencies. To circumvent this problem we determined spectral indices from data between adjacent frequencies, taking care of equalizing the angular resolutions. Thus, the 408 and 85.7-MHz points have the original 0.8 resolution, and the spectral index they determine is -2.7. To obtain the spectrum between 85.7 and 45 MHz, the 85.7 datum was degraded to the 45-MHz resolution; the point thus degraded and the 45-MHz point gave a spectral index-1.8. A similar procedure was followed for the spectrum between 45 and 30 MHz, where the 45-MHz point was degraded to the 30-MHz resolution obtaining an index +0.3.
The spectrum is shown in Fig. 5 where the piece between 85.7 and 45 MHz was obtained by drawing a line with slope -1.8 through the 85.7 MHz point, and the piece between 45 and 30 MHz was obtained by drawing a line with a slope +0.3 through the upgraded 45-MHz point at 73000 K. It should be noticed that the temperature of the point at 85.7 MHz, with 0.8 resolution, does not include the effect of absorption since the source was restored from its non-absorbed flanks; however, the temperature of the lower frequency points, obtained with lower resolution, do include the absorption because the wide beam does not resolve the trough. This effect may partly explain the turn over of the spectrum at the lower frequencies, and lead us to believe that the index -2.7 is representative of the spectrum, therefore we conclude that the broad source is non-thermal. From Fig. 5 we see also that as the frequency decreases thermal absorption sets in at about 45 MHz.
By examining the area around the G.C. in 408, 150, 85, 45, and 30 MHz, we observe that the broad non-thermal source is elongated in the direction of the galactic equator. Fig. 6 shows the 45-MHz data smoothed out to 11, the resolution of the 30-MHz map. We notice that the contours are fairly elliptical out to 10 from the center, with an axial ratio of 1.7 and the major axis slightly inclined to the equator. The center of symmetry is seen to be offset with respect to the G.C. at (1.9, 0.0 ). A similar situation is present in the 30 MHz map where the center of the contours is located roughly at (1, -1 ). However at 408 and 85 MHz the source is seen centered at (0, 0 ).
Another characteristic of the broad source is that the width of the expected profile, corrected for beam smoothing, increases with decreasing frequency; thus, the FWHM at 408, 85.7, 45 and 30 MHz are 3.1, 3.2, 8 and 14, respectively.
It is interesting to know the integrated flux density at 45 MHz. To compute it we integrated numerically the map previously smoothed to 11 (Fig. 6) in order to filter out some sources in the range 6 , that do not belong to the broad source. The integration was done inside the contour 35000 K which seems a reasonable source boundary (see Fig. 6).The resulting flux density is 4.4 10-22 W m-2 Hz-1, which gives a spherically radiated power of 9.4 1019 W Hz-1. At 45 MHz this power is equivalent to 1740 Crab nebulae and 14 Cass A's. Ilovaiski & Lequeux (1972) have estimated that the luminosity of the Galaxy at 150 MHz is 6 1021 W Hz-1. From this value, and using a spectral index of -0.38 (Howell 1970), we derive for 45 MHz a total luminosity of 9.5 1021 W Hz-1, which means that at this frequency the broad source emits 1% of the power of our whole Galaxy.
3.2.3. The HII region
We have computed the emission measure (EM) from the relation , (where is in MHz and EM in pc cm-6), by: a) using = 0.077 derived from the work of Downes & Maxwell (1966); b) adopting K from the work of Matthews et al. (1973a, b); and c) assuming . The result gives pc cm-6, in very good agreement with Matthews et al. who obtained 2.6 - 3.2 104 pc cm-6.
© European Southern Observatory (ESO) 1997
Online publication: April 6, 1998