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Astron. Astrophys. 355, 863-872 (2000)

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3. Discussion

3.1. The spectrum of the Whole Source

McGee et al. (1955) found a strong positive curvature in the spectrum of Cen A below about 1000 MHz, as did Cooper et al. (1965) who also claimed that the spectrum steepens with decreasing frequency. The fact that any significant deviation from a synchrotron spectrum has physical implications for either the source or the intervening medium (or both), prompted us to undertake an exhaustive study of the spectrum of Cen A. Furthermore, the spectrum has been studied thus far over a rather narrow range of frequencies. The present work, incorporating data found in the literature, extends this range from 1420 up to 4750 MHz. Table 4 gives the spectral indices for the Whole Source determined by different authors. Fig. 2 shows the spectrum of the Whole Source between 4.7 and 4750 MHz. Above 4.7 MHz the fluxes fit well a power law with spectral index [FORMULA], and a correlation coefficient of 0.998. Below 10 MHz there are indications of absorption, as already noticed by Ellis & Hamilton (1966), so the straight line spectrum could be extrapolated down to 4.7 MHz (Hamilton & Haynes 1968). We see that Table 4 contains a spread of values for the spectral index of the Whole Source, ranging approximately between -0.6 and -1.0. This spread in results is probably attributable to the reduced frequency range or to the insufficient number of observed frequencies comprising each measurement.


[TABLE]

Table 4. Spectral index of the Whole Source


To obtain the fitted spectra presented in this work all of the data points were given equal weight.

A most interesting feature in Fig. 2 is the lack of any spectral break. Except for the measurement of Harris & Roberts (1960) at 960 MHz, the best fit line passes through or touches the ends of the error bars. A flattening of the spectrum in the high frequency range could imply new ejections replenishing the highest energy electrons, while a steepening could imply losses; however, neither of these two effects is observed. Since the spectrum is straight, we submit that the electrons are undergoing some kind of continuous reacceleration.

3.2. The spectrum of the Inner Lobes

Table 2 lists the integrated flux densities of the individual ILs. The widest frequency range in which the spectral indices have been given is 18.3 to 5000 MHz, and the highest frequency at which the flux has been measured is 43 GHz. By collecting published fluxes we have produced a spectrum extending from 80 MHz to 43 GHz. The spectra of the individual ILs are shown in Fig. 3. The best fit lines give an index of [FORMULA] for both ILs and correlation coefficients of 0.994 and 0.996 for the ILNE and ILSW, respectively. Thus, we confirm the results of Slee et al. (1983) who found -0.70 for both indices between 80 MHz and 10.7 GHz. We see that the fluxes of the ILNE are larger than those of the ILSW. The equality of the indices supports the hypothesis that the lobes were produced by electrons coming from a common population. Table 5 presents the spectral indices determined by different authors.

[FIGURE] Fig. 3. Integrated flux density spectrum of the Inner Lobes. The straight lines represent the least-squares fits to the data corresponding to the NE and SW lobes, as given by the equations [FORMULA] and [FORMULA], respectively


[TABLE]

Table 5. Spectral indices of the Inner Lobes.
Notes:
a) Presumably includes both ILs.
b) From collected data.
c) Cited by Johnson (1963).
d) For peak of centroids.


3.3. The spectrum of the Giant Outer Lobes

Even though a determination of the spectra of the individual Giant Outer Lobes is of much interest to understand the evolution of this radio galaxy, their flux densities have been measured at only four frequencies: 408 and 1410 MHz by Cooper et al. (1965), 960 MHz by Bolton & Clark (1960), and 4670 MHz by Junkes et al. (1993). Furthermore their spectra have not been determined. We have added 2650 MHz by undertaking graphical integration of the brightness temperature contour maps published by Cooper et al. (1965) (see Table 3).

We have also repeated the calculations of Cooper et al. at 406 and 1410 MHz to ensure the use of the same criterion to separate the GLN, the GLS and the CR. The Northern Loop is not shown in the 5000 MHz map by Haynes et al. (1983) and for this reason these data have not been considered. In this study we have included the Northern Middle Lobe and the Northern Loop as parts of the GLN.

As mentioned earlier, the major problem at frequencies of 1410 MHz and below is the separation of the GLN from the CR which, due to lack of sufficient angular resolution, merges with the Northern Middle Lobe. This complication does not exist in the GLS because there is no southern counterpart of the Northern Middle Lobe and so the contours can be better estimated. In spite of these difficulties we believe that the separation can be achieved with reasonable accuracy thus: 1) by following the trend of the isophotes of the different components, 2) by assuming that the contours of the CR are symmetrically elongated, 3) by assuming that the GLN and GLS contours lie only north and south of the source's center declination, and 4) confirming that the sum of the fluxes of the three parts is close to the integrated flux obtained for the Whole Source. The flux densities and the spectra are shown in Table 3 and Fig. 4, respectively.

[FIGURE] Fig. 4. Integrated flux density spectrum of the Giant Outer Lobes. The straight lines represent the least-squares fits to the data corresponding to the N and S lobes, given by the equations [FORMULA] and [FORMULA], respectively

Cooper et al. (1965) apparently calculated the flux of the Whole Source at 960 MHz since they state that, with the Caltech flux density scale, the flux obtained by Bolton & Clark (1960) at the same frequency would come "closely into line with our own flux density at this frequency". However, this flux density is not given in their paper. From their map we have calculated a value of [FORMULA] Jy, which is well below the values given by both Bolton & Clark (1960) and Harris & Roberts (1960); see Table 1. Also, our flux determinations at 960 MHz for the GLN and GLS from the map of Cooper et al. (431 and 613 Jy, respectively) are below the 508 and 731 Jy determined by Bolton & Clark (1960). Interestingly, however, the flux ratios GLN/GLS are the same. We conclude, however, that the 960 MHz map of Cooper et al. (1965) has a large temperature scale offset, which we have not attempted to estimate. We do not consider these data further.

Our flux determinations at 1410 MHz are in agreement with those of Cooper et al. (1965). Also their GLS value at 406 MHz is in good agreement with ours. However, there is disagreement between the GLN fluxes (see Table 3). We believe this is due to the problem of separating the large scale components.

In Fig. 4 it is seen that, except for the 406-MHz determinations of Cooper et al. (1965), the mean value of the GLS is higher than that of the GLN at all frequencies. The best fit lines through the mean values of the GLS and GLN give spectral indices of [FORMULA] and [FORMULA], respectively. Thus, the spectral indices are equal, within the precision of the fit. The correlation coefficients for these data are 0.987 and 0.972, respectively.

The average spectral indices of the Whole Source ([FORMULA]) and of the ILs ([FORMULA]) are considerably more reliable than those of the GLs because they were obtained in a much wider frequency range and with a significantly larger number of data. This is reflected in the high correlation coefficients: 0.998 (Whole Source), 0.994 (ILNE), and 0.996 (ILSW). However, the spectral indices of the GLs have been determined in a narrower range of frequencies, with smaller number of measurements and data with larger error bars due to the difficulty in separating the CR from the GLs. Since the GLs contribute 73% of the luminosity of the Whole Source (see Sect. 3.4) we would expect the spectral index of the GLs to be very close to that of the Whole Source. For both GLs the errors of fit make this expectation possible.

In the search for the spectral indices of the individual GLs, we have used the work of Combi & Romero (1997) who studied the spatial distribution of the spectral index over the source at both 408 and 1410 MHz. From a graphical integration of their Fig. 2 we have deduced averaged indices of -0.75 and -0.70 for the GLN and GLS, respectively. To separate the lobes in their figure we arbitrarily drew a constant declination line through [FORMULA], which is the declination of the center of the source. The mean error of [FORMULA] in the map of indices, quoted by the original authors, is sufficient to bring the values we computed into agreement with the indices discussed earlier. Table 6 shows the spectral indices of the Giant Outer Lobes.


[TABLE]

Table 6. Spectral indices of the Giant Outer Lobes.
Notes:
a) Averaged over spatial distribution.
b) Errors quoted by original authors.
c) Based on data from Combi & Romero (1997).


In their review, Ebneter & Balick (1983) state that "the spectral index of the northern lobe is noticeably different from that in the southern lobe"; however, they give no measurements or references.

Examining the spectral indices of the ILs and GLs we see that they are similar. This precludes the idea, sustained by some authors, that the outer lobes are a sort of halo since this would be expected to have a much steeper spectrum. For example, Shain (1958) noticed a difference in the spectrum of the extended and central sources, with indices of -1.25 and -0.6, respectively, and interpreted that "...the very high frequency observations refer only to the central source...". This is disclaimed by our findings. The similarity of the spectral indices of the ILs and GLs also supports the hypothesis that the pairs of lobes were formed by two energetic electron ejections. These ejections may have come at different times from the same electron parent population or, if the electrons were generated at the time of the ejection, the mechanism produced the same distribution of energies.

3.4. Luminosities

Table 7 presents the luminosities of the analyzed Cen A components. To compute these luminosities we have assumed that the power law spectra fitted to the mean values of the data, as shown in Figs. 2, 3 and 4, are valid between 4.7 MHz and 43 GHz. This may not be so for the GLs at high frequencies nor for the ILs at low frequencies; ranges where there are no data. If the assumption is not valid the consequences would be more noticeable for the GLs, since the luminosity for a power law spectrum with a negative index is determined by the high-frequency end (when the two ends are distant). We are also assuming that the power is radiated isotropically. The adopted distance of 3.5 Mpc is the most recent value we have found (Hui et al. 1993). We have not applied K corrections since the redshift is only 0.002.


[TABLE]

Table 7. Luminosity between 4.7 MHz and 43 GHz


Table 7 shows several interesting results. The GLs contribute with 73% of the total luminosity while the contribution of the ILs is 18%; that is, the GLs are four times more luminous than the ILs. Taking the luminosities in Table 7 at face value, the Nuclear Region contributes approximately 10% of the total luminosity. It is seen that, of the GLs, the southern one is the more luminous while the opposite is true of the ILs; the ratio of luminosities are ILSW/ILNE=0.68 and GLS/GLN=1.20. It is interesting that, in spite of the fact that these luminosity ratios are neither unity nor close to it, the spectral indices of each pair are the same. In the double ejection hypothesis this could imply that, in both epochs, the fast electrons came from the same parent population but that either the amounts or the densities of ejected particles, were different. Further, the numbers of ejected particles were different not only in the first and second ejections, but also between the two opposing jets. The fact that the much older GLs have maintained their original spectral indices, at least up to 4.75 GHz, indicates that they have not suffered significant losses.

In the literature we have found very few calculations related to the luminosity of Cen A. Burbidge & Burbidge (1957) assumed a core-halo structure with three different spectral indices between 18 and 3000 MHz, and a distance of 2.5 Mpc, obtaining [FORMULA] erg s-1. Moffet (1975) quotes 5.0 [FORMULA] erg s-1 between 10 MHz and 10 GHz, and gives no indication of the spectrum, while Rogstad & Ekers (1969) give 3.5 [FORMULA] erg s-1 between 10 MHz and 100 GHz, for a distance of 4 Mpc and an unspecified spectrum. Matthews et al. (1964) find 7.4 [FORMULA] between 10 MHz and 100 GHz, adopting a distance of 4.7 Mpc. We have discussed elsewhere (Alvarez et al. 1993) the dangers of computing luminosities from data over a short range of frequencies, at a single frequency or, even worse, assuming a spectral index for the whole spectrum. Considering this, and the different distances and frequency ranges adopted, the above values are consistent. The luminosities of Table 7 are reduced by one half if computed up to 4.75 GHz rather than 43 GHz.

3.5. Spectral aging

We have seen that the spectra of the GLs and ILs are straight over all the observed range of frequencies up to 4.75 and 43 GHz, respectively. We can estimate upper limits to the ages of the fast electrons responsible for the lobes if we assume, first, that their spectra break down precisely at 4.75 and 43 GHz, though there are no observations to see the steepening, and second, that only synchrotron and inverse Compton losses occur above those frequencies. Following Perola (1981), the age, in years, is given by:

[EQUATION]

where B is the magnetic field strength in the source (G), [FORMULA] is the equivalent magnetic field (G), [FORMULA] is the break frequency (Hz), and z is the cosmological redshift. For the cosmic microwave background [FORMULA]G; the multiplying constant is derived by taking a radiation energy density of [FORMULA] erg[FORMULA]. For Cen A we will assume [FORMULA].

The magnetic field strength has been inferred from x-ray observations. For the GLs, Cooke et al. (1978) and Harris & Grindlay (1979) give 0.7 and [FORMULA]G, respectively, while Marshall & Clark (1981) give [FORMULA]G. These authors assume that one and the same distribution of relativistic electrons produces the x-ray emission by inverse Compton mechanism and radio emission by the synchrotron process. They take special care to remove the contribution from the Central Region. Feigelson & Berg (1983) do not detect x-ray emission neither from the GLs nor from the ILs, and estimate that the field in the GLs and ILs must be larger than 1.6 and [FORMULA]G, respectively. From minimum pressure arguments, Burns et al. (1983) obtain 12 and [FORMULA]G for the ILNE and ILSW, respectively. The corresponding ages are [FORMULA] and [FORMULA] years. It should be noted that these calculations refer only to the centroids of the ILs.

Eq. (1) is doubled-valued for B, that is, for each [FORMULA] there are two Bs that satisfy it, except for [FORMULA]G, where a maximum of [FORMULA] years occurs for [FORMULA] (assuming [FORMULA]).

Adopting for the GLs [FORMULA]G and [FORMULA] GHz we obtain that their ages are less than [FORMULA] years. For the ILs with [FORMULA]G and [FORMULA] GHz the upper limit for the age yields [FORMULA] years. The estimated maximum ages are of the same order of magnitude than those found by Mack et al. (1998) for giant radio galaxies, and by Feretti et al. (1998) for tailed radio galaxies.

Because of the way the x-ray workers computed the magnetic field, it is reasonable to use Eq. (1) complete. Had we assumed only synchrotron losses [FORMULA] the upper limits for the GLs and ILs would have been [FORMULA] and [FORMULA] years, respectively. We see that the presence of inverse Compton losses can make a large difference in the estimation of ages. Future radio observations at frequencies higher than the actual break frequencies should reveal the shape of the spectrum and should give information about the radiating mechanism.

We can calculate the equipartition field, [FORMULA]. This is given by Perola (1981), in c.g.s. units, by:

[EQUATION]

where K is the ratio of relativistic protons to relativistic electrons, c is a constant that depends on the radio spectral index and on the frequencies between which the luminosity L is calculated (Pacholczyk 1970), [FORMULA] is a filling factor, and R is the radius of the source. We will assume [FORMULA] and [FORMULA]. For the GLs we have calculated c for [FORMULA] and for the end frequencies 4.7 and 4750 MHz, obtaining [FORMULA]. We have adopted an angular size of [FORMULA] which, assuming a spherical source, gives a radius of 320 kpc. Finally, from Table 7 we obtain the luminosities. In the case of the GLs the given luminosity should be divided approximately by 2 because the known spectrum extends only up to 4.75 GHz. For each of the ILs we have adopted an angular extent of [FORMULA] and end frequencies 4.7 MHz and 43 GHz, while [FORMULA] for [FORMULA]. With these numbers we obtain [FORMULA]G and [FORMULA]G for the GLs and ILs, respectively. We see that the values adopted for the magnetic field in our estimations of age are of the same order of magnitude as those derived from equipartition arguments.

3.6. P-[FORMULA] correlation

Extended extragalactic structures have been divided by Fanaroff & Riley (1974) into two luminosity classes. The weaker sources brightest at the center and fading toward the edge are defined as FRI class. The more luminous sources are limb-brightened and are defined as FRII class; they often show hot-spots at the farthest edges of the structure. For powerful, extended, double radio galaxies, Laing & Peacock (1980) found a correlation between radio luminosity (defined by them as P) at 1400 MHz and spectral index, and also between redshift (z) and spectral index. The correlation holds for FRI sources and for the extended structure of FRII sources, observing a smooth continuity between the two classes in the P-[FORMULA] diagram. Even though Cen A cannot be considered as a powerful radio galaxy, its luminosity is within the range of the sample selected by Laing & Peacock (1980), so we can use it to check the correlation. We have found that Cen A fits quite well the P-[FORMULA] correlation, as illustrated in Fig. 3 of Laing & Peacock (1980), which contains all identified sources in the 178 MHz sample (with the central source not removed). Originally, there was some uncertainty as to whether the real correlation was with the luminosity or with the redshift; however, Gopal-Krishna (1988) has given evidence that the correlation is primarily with radio luminosity. Since the lowest z in the Laing & Peacock's sample is 0.03, the much smaller z of Cen A (0.002) offers a good opportunity to test the z-[FORMULA] correlation. We have found that Cen A definitely does not fit this correlation, in support of Gopal-Krishna's (1988) result.

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Online publication: March 21, 2000
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