3. Application to individual sources
Application of these simplified equations to real radio cores is straight forward, the basic input parameters being the jet power, the characteristic electron energy, the inclination to the line of sight, the observed frequency, the distance, the black hole mass, and the relative size of the nozzle region. The latter two enter only weakly and hence need to be known only to an order of magnitude.
A few systems are so well studied that most of these parameters (especially i, D, & ) can be fixed with some confidence and where size and flux of their cores at a certain frequency are well known through VLBI observations. Even though this may be a subjective criterion, we believe that the radio cores in M 81, NGC 4258, and GRS 1915+105 are, for various reasons, perhaps the best studied and best constrained examples of low-power radio cores. Following the convention in Falcke (1996b) and Melia (1992) and in analogy to Sgr A*, we will identify the radio cores in these sources by adding an asterisks to their host galaxy or source name to clearly distinguish them from their hosts.
We have listed the sources and their parameters in Table 1. The observed quantities we have used as input parameters for the model are given in columns 2-7. Since in all cases, except Sgr A*, we have only two unknowns left (jet power and characteristic electron energy) to describe the two observed quantities of the radio cores (flux and size) we were able to solve the model equations for each source completely and determine and directly from the observations. These results and the predicted spectral indices for the radio spectrum are given in the three right-most columns of Table 1. For comparison with the jet power, we also listed the accretion disk luminosity of each system in column 8. In the following we will briefly discuss the data and the modelling of each source.
Table 1. Parameters for compact radio core in various sources. Columns 2-7 are observationally determined input parameters: distance D, inclination angle i of disk axis and jet to the line of sight, observing frequency , flux density of radio core , size of radio core, and black hole mass . The inferred disk luminosity is not an input parameter here and given in column 8 for comparison only. Uncertain values are given in brackets, but since the black hole masses do not enter strongly the uncertainties in the black hole mass for M 81 and GRS 1915 are actually irrelevant. Columns 9-11 are output parameters of the radio core model: jet power , characteristic electron Lorentz factor , and average spectral index () in the radio.
3.1. NGC 4258
The VLBI observations of megamaser emission has led to the detection of a molecular disk in NGC 4258 (Miyoshi et al. 1995) which can be used to determine the inclination angle of the system, the black hole mass , and the distance Mpc (Herrnstein et al. 1997a) almost directly from the observations. The variable central VLA radio core (Turner & Ho 1994), here called NGC 4258*, has a flux of roughly 3 mJy and was interpreted by Lasota et al. (1996) as emission from an ADAF while Falcke (1997) suggested a scaled down AGN jet origin. The latter picture was confirmed by Herrnstein et al. (1997b&1998) who discovered a nuclear jet in NGC 4258 offset by 0.35 to 0.46 mas from the dynamical center. Herrnstein et al. (1996) suggested that this offset could be interpreted within the framework of the Blandford & Königl (1979) model as being due to self-absorption in the inner jet cone. The search for radio emission directly at the dynamical center remained unsuccessful (Herrnstein et al. 1998) and required a revision of the Lasota et al. (1996) ADAF model (Gammie et al. 1998).
For our purposes NGC 4258* is an ideal system because all crucial parameters, especially the inclination angle, seem to be fixed. Using an average radio flux of 3 mJy at 22 GHz and the offset of the core from the dynamical center as the characteristic size scale of the system we find a jet power of for the nuclear jet, a characteristic electron Lorentz factor of , and predict an average spectral index (). The jet-power of the nuclear jet is consistent with the large scale emission-line jet in NGC 4258, since its kinetic power is also of the order erg s-1 - as derived from the mass () and velocity (km s-1) of the emission-line gas (Cecil et al. 1995). Moreover, this is also in line with the estimated nuclear accretion disk luminosity of erg s-1 (Stüwe et al. 1992; Wilkes et al. 1995; see also discussions in Herrnstein et al. 1997 and Gammie et al. 1998). Hence, all the activity in NGC 4258 can be described in a consistent way by a low-luminosity jet/disk-system and an accretion rate of the order yr. One caveat exists, however, because the interpretation of the offset of the core from the dynamical center as the characteristic scale of the model (and not the self-absorption size which is smaller in this model) actually implies that also the core size is of similar order. If it were smaller, e.g. 0.1 mas, this would reduce the, compared to other sources, relatively high value for to around 200 without significantly reducing the required jet power. A difference between offset and actual core size would occur if the jet were collimated in the inner region more than assumed in our model (i.e. were narrower than the Mach cone).
3.2. GRS 1915+105
Mirabel & Rodriguez (1994) discovered a compact radio jet in GRS 1915+105 with apparent superluminal motions, for which they were able to determine the jet speed () and the inclination angle () of the system. Moreover, in recent papers Fender et al. (1997), Pooley & Fender (1997), Mirabel et al. (1998) and Eikenberry et al. (1998) found an intriguing correlation between radio outbursts and X-ray flares and hence a symbiotic jet/disk-system as proposed in Falcke & Biermann (1995&1996) seems to be a good description for GRS 1915+105. The parameters erg s-1 and are discussed in the literature (e.g. Mirabel et al. 1997; Morgan et al. 1997) for this source, but we point out that the mass determination is extremely uncertain, yet is also not really critical for the modelling.
Dhawan et al. (1998) observed the central radio core in a relatively quiescent phase finding an intrinsic source size of mas (major axis) at 15 GHz and fluxes around 40 mJy (flat spectrum). For these parameters the jet model gives a jet power of erg s-1 and a of . In addition, the predicted scaling of the core size () is consistent with the observed one (roughly , taking out the scatter broadening). The observed time delay of min of outburst peaks between 3.5 and 2cm can be explained as the delay in time it takes for each outburst to reach the optically thin regime () at the angular distance where the outburst first becomes visible. For the parameters given here and in Table 1 we get 2 mas and the time delay between 2 and 3.5 cm is predicted by the model to be of order 3 mins. The velocity of the jet in the model grows asymptotically as determined by Eq. 1 (see also Falcke 1996b), yielding 0.92 at and at the scale of a few mas, where the radio emission is coming from. Considering that Eq. 1 is a no-fit asymptotical description of the velocity field in the jet this is a reasonably good prediction. Clearly, the pressure gradient effect must be at work at least to some degree here. Mirabel & Rodriguez (1994) found tentatively that-in addition to their advance speed-the blobs may also expand with at larger scales, thus perhaps finding direct evidence for a relativistic "sound speed" which is needed for the pressure gradient effect to be important. All in all the Falcke (1996b) model for M 81* seems to give a remarkable good description of GRS 1915+105 as well and it confirms the basic Hjellming & Johnston (1988) picture for compact radio cores in stellar mass black hole candidates.
3.3. M 81* and Sgr A*
Finally, for a consistency check, we will apply the model presented here also to M 81* and Sgr A* for which we have discussed very similar jet-models and their parameters already in earlier papers (Falcke, Mannheim, Biermann 1993 and Falcke 1996b) while adding a few recent results that have appeared in the literature.
For M 81* Ebbers et al. (1998) and Bietenholz et al. (1998) presented some new measurements confirming that indeed it most likely has a core-jet structure. The average flux and size of M 81* at 8.5 GHz was mJy and mas. Falcke (1996b) concluded that a range of inclinations between fitted the radio observations best and this was confirmed by the detection of a nuclear emission line disk in M 81 with similar inclination (Devereux et al. 1997). The jet power derived from our model of M 81* with these parameters is erg s-1 and . For comparison, Ho et al. (1996) give a bolometric nuclear luminosity for M 81 of erg s-1.
For the Galactic Center radio core Sgr A* observational results now convincingly demonstrate the presence of a black hole of mass (Ghez et al. 1998; Eckart & Genzel 1996) while the exact nature of this source remains ambiguous. For the intrinsic size of Sgr A* at 43 GHz Bower & Backer (1998) give a 2 value of mas and in a later paper Lo et al. (1998) indeed claim this to be the intrinsic size together with an elongated source structure they find. Assuming an arbitrary inclination angle of one can fit Sgr A* with a jet power of erg s-1 and . The predicted spectral index of is also consistent with observed values (Falcke et al. 1988) and with the electron energies we found one could in principle explain the (sub)mm-bump in the spectrum as emission arising from the inner nozzle region of the jet (see Falcke 1996a). On the other hand there is currently no evidence for any emission of Sgr A* at other wavelengths than the radio, suggesting that any accretion "disk" emission is well below erg s-1 and therefore-unlike in the other sources-is well below the required jet-power.
© European Southern Observatory (ESO) 1999
Online publication: December 22, 1998