2 The model

The axisymmetric, stationary and ideal MHD wind solution provides the density, velocity and temperature for each volume element along the field. Prescribing the jet mass flow rate $\dot{M}_{\rm j}$ together with the shape of the field line, these solutions give a unique set of parameters of the flow defined by the regularity condition across the magnetosonic points (see FG01 for details).

The calculated dynamical parameters are our starting point to obtain the X-ray spectra of the jet. Here, we refer to the solution S3 of FG01 obtained for a collimating field line z(x) = 0.1 (x-x₀)^6/5, x being the normalized cylindrical radius, x₀ the foot point of the field line at the equatorial plane, and z the height above the disk. Length scales are normalized to the gravitational radius $r_{\rm g} = 7.4 \times 10^5$cm $(M/5\,{M}_{\odot})$. For completeness, we show the radial profiles of poloidal velocity, density, temperature, and emitting volume along the field line in Fig. 1.

The jet geometry consists of nested collimating conical magnetic surfaces with sheets of matter accelerating along each surface. The sheet cross section becomes larger for larger distances from the origin. The distribution of the 5000 volume elements along the jet is such that velocity and density gradients are small within the volume. We have 63 volumes in $\phi$ direction defining an axisymmetric torus (i.e. 5000 tori along the magnetic surface).

We distinguish two parts of the inner jet flow. One is for a temperature range T = 10^6.6-10⁹K, where we calculate the optically thin continuum (bremsstrahlung) and the emission lines. The other is for T = 10⁹-10¹²K, where only bremsstrahlung is important. Any pair processes are neglected and no ( $\rm e^-e^-$)-bremsstrahlung will be considered, although that might be dominant at the highest temperatures. Such unphysically high temperatures are to a certain degree caused by the use of a non-relativistic equation of state. Employing a relativistically correct equation of state (Synge 1957) one would expect gas temperatures an order of magnitude lower (Brinkmann 1980). These temperatures belong to the intermediate region between disk and jet. The injection radius,which is, in fact, the boundary condition for the jet flow, is located at $R_{\rm i} = 8.3\,r_{\rm g}$and at a height above the disk (and the foot point of the field line) of $0.74\,r_{\rm g}$. For the chosen MHD solution the temperature at this point is $T_{\rm i} = 10^{10.2}$K. With $R_{\rm i} \simeq 6 \times 10^6$cm $(M/5\,{M}_{\odot})$we investigate a region of about $2.5 \times 10^{-5}$ $(M/5\,{M}_{\odot})$AU. Having determined the emissivities of single volume elements, these can be put together obtaining a rest-frame spectrum where any motion is neglected. However, the knowledge of the MHD wind velocities allows us to determine the Doppler shift of the spectral energies and the boosting of the luminosity for each volume. We finally obtain a total spectrum of the inner jet considering these effects in a differential way for each volume element. The final spectra of course depend also from the jet inclination.

We emphasize that our approach is not (yet) a fit to certain observed spectra. In contrary, for the first time, for a jet flow with characteristics defined by the solution of the MHD wind equation, we derive its X-ray spectrum. Our free parameters are the mass of the central object M defining the length scales, the jet mass flow rate $\dot{M}_{\rm j}$ and the shape of the poloidal field lines. In the end, from the comparison of the theoretical spectra with observations, we expect to get information about the internal magnetic structure of the jet close to the black hole and the jet mass flow rate.

Figure 2: X-ray luminosities of jet-tori of 63 volume elements with different temperatures: T = 106.64K, T = 107K, T = 108K, T = 109K (from left to right). The jet mass flow rate considered here is $\dot{M}_{\rm j} = 10^{-10}\,{M}_{\odot}\,{\rm yr}^{-1}$ for a $5\,{M}_{\odot }$ central object.