2. The jet/disk-symbiosis model
The model by Falcke & Biermann (1995) was derived from a simple Blandford & Königl (1979) model which calculates the synchrotron emission of a relativistic, conical jet as a function of jet power (here: of two cones) parametrized by the accretion disk luminosity , such that . Here we will make use of this parametrization only to recapitulate our earlier results and later express the simplified equations in terms of the jet power alone. In the end we will therefore be able derive the parameter without making any ab-initio assumptions about its value.
The maximally efficient model assumes equipartition between magnetic field and particles and between internal and kinetic energy (). In Falcke (1996b) we added a self-consistent description of the velocity field: the jet was considered to be a fully relativistic electron/proton plasma with a turbulent magnetic field treated as a photon gas leaving a nozzle and freely expanding into the vacuum (this is in contrast to large scale jet models which assume pressure equilibrium and confining cocoons). The longitudinal pressure gradient then leads to a modest, yet significant acceleration along the z-axis of the jet in the asymptotic (i.e. observable) regime given by the equation for the jet proper velocity
with and an adiabatic index (Falcke 1996b).
This model assumes that there is no additional acceleration beyond the nozzle and therefore just gives a lower limit to the terminal jet speed of order which seems to be just enough for some low-power nuclei. This will of course fail for powerful quasar jets since they exhibit significantly faster motion. We also point out that this does not make any assumption on the process of jet formation itself which is here treated as a "black box" of linear dimension so that , where Z is the distance from the black hole.
The radio spectrum of such a pressure driven jet (which is no longer perfectly conical) can then be calculated assuming energy conservation along the jet when assuming a certain energy distribution of the relativistic electrons as described in more detail in Falcke (1996b). For simplicity we assume that a fraction of the electrons gets accelerated up to an initial characteristic energy of (or possibly produced near this energy by -decay, e.g. Biermann et al. 1995). While the existence of quasi-monoenergetic electron distributions has been claimed for LLAGN (Duschl & Lesch 1994; Reuter et al. 1996), this choice here was made mainly to account for the possibility of a low-energy cut-off/break in the electron distribution (Celotti & Fabian 1993; Falcke & Biermann 1995). Since in almost all cases will have to be such that the characteristic frequency is close to the self-absorption frequency , all the results obtained here will also be roughly valid for a more realistic power-law distribution with low-energy break at . On the other hand it can also not be excluded that indeed electrons in radio cores start out with a very narrow energy distribution when they are injected and only along the way get accelerated into a power-law distribution.
2.2. Simplified equations
Using all the basic premises above, one arrives at a set of equations for the expected radio spectrum and typical size scale of a radio core for a given accretion disk luminosity (or jet power), inclination angle and electron energy, given as Eqs. (8-12) in Falcke (1996b) for the case of M 81*, which we reproduce here (using the parameters exactly as specified in that paper). The characteristic scale of the radio core is
where the parameters are AU, GHz, and (500, 1200,1100, 900, 700) AU for , while the total spectrum of the jet is given by
for Mpc, and ) respectively.
Unfortunately, these equations are not easy to handle since the power-law indices are a function of the inclination angle. We therefore derive here an approximate formula for the spectra and sizes predicted by such a model.
We note here once more that Eqs. 3-5 have two inherent constraints. First of all, by definition cannot be smaller than unity since it is defined as the ratio between the energy densities in protons and electrons plus one. For the sake of simplicity only, we will now ignore the relativistic proton content and set , so that we can substitute the relativistic electron fraction with
Secondly, one cannot increase indefinitely since at some point the flux would become negative, i.e. the jet would become completely self-absorbed and the simplifications would break down. Moreover, the equations are difficult to handle because of this sum, since for large changes in the sum also would become negative. Hence,, we formally introduce an arbitrary scaling relation
which allows us to simplify the equations further. The physical meaning is that electrons are pushed to somewhat higher energy with increasing to keep them in the optically thin part. A mechanism which indeed could lead to such an effect is the `synchrotron boiler' (Ghisellini et al. 1988) that describes the evolution of low-energy electrons in a self-absorbed system, but it is not clear whether this formally introduced relation here has any significance in the real world and therefore we will ignore it in the discussion of our results.
To further simplify the equations we have rounded the exponents typically to the 2nd digit after the decimal and factorized the equations. Even for the most strongly varying parameters, like which can vary over 6 orders of magnitude, the resulting error will be only some ten percent. Moreover, the exponents in the equations, which are a function of the inclination angle i of the jet, were fitted by 2nd and 3rd order polynomials in i to an accuracy of much better than a few percent over a large range of angles.
All these simplifications lead to the following expressions for the observed flux density and angular size of a radio core observed at a frequency as a function of jet power. For a source at a distance D, with black hole mass , size of nozzle region (in ), jet power , inclination angle i, and characteristic electron Lorentz factor (see Eq. 7) the observed flux density spectrum is given as
with the correction factors depending on the inclination angle i (in radians):
Likewise, the characteristic angular size scale of the emission region is given by
with the correction factors
where again the inclination angle i is in radians. We point out that in this model the characteristic size scale of the core region is actually equivalent to the offset of the radio core center from the dynamical center. This does not exclude the existence of emission in components further down the jet, which might be caused by shocks or other processes.
The equations are scaled to the typical values for Galactic jet sources like GRS1915+105 and the correction factors are normalized to an inclination angle of 1 rad (). We note that the approximations fail at small inclination angles where the accretion disk is seen face on and the jet points towards the observer (i.e. ). The benefits of these equations now are that they can be used to quickly compare observed radio core properties with the model predictions, especially since we have reduced the number of free parameters to the absolute minimum.
© European Southern Observatory (ESO) 1999
Online publication: December 22, 1998