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Jets are highly collimated flows of matter. They are present in a large variety of astrophysical sources: young stars, stellar mass black holes (microquasars), galaxies with an active nucleus (AGN) and presumably also intense flashes of gamma-rays. In particular, the jets of microquasars, powered by accretion disks, are probably small-scale versions of the outflows from AGN. Beside observations of astrophysical jet sources, also theoretical considerations have shown that magnetic fields play an important role in jet formation, acceleration and collimation. Collimated jets seem to be systematically associated with the presence of an accretion disk around a star or a collapsed object. If the central object is a black hole, the surrounding accretion disk is the only possible location for a magnetic field generation. We are interested in the formation process of highly relativistic jets as observed from microquasars and AGN. We theoretically investigate the jet collimation region, whose physical dimensions are extremely tiny even compared to radio telescopes spatial resolution. Thus, for most of the jet sources, global theoretical models are, at the moment, the only possibility to gain information about the physical processes in the innermost jet region. For the first time, we determine the global two-dimensional field structure of stationary, axisymmetric, relativistic, strongly magnetized (force-free) jets collimating into an asymptotically cylindrical jet (taken as boundary condition) and anchored into a differentially rotating accretion disk. This approach allows for a direct connection between the accretion disk and the asymptotic collimated jet. Therefore, assuming that the foot points of the field lines are rotating with Keplerian speed, we are able to achieve a direct scaling of the jet magnetosphere in terms of the size of the central object. We find a close compatibility between the results of our model and radio observations of the M87 galaxy innermost jet. We also calculate the X-ray emission in the energy range 0.2--10.1\,keV from a microquasar relativistic jet close to its source of 5 solar masses. In order to do it, we apply the jet flow parameters (densities, velocities, temperatures of each volume element along the collimating jet) derived in the literature from the relativistic magnetohydrodynamic equations. We obtain theoretical thermal X-ray spectra of the innermost jet as composition of the spectral contributions of the single volume elements along the jet. Since relativistic effects as Doppler shift and Doppler boosting due to the motion of jets toward us might be important, we investigate how the spectra are affected by them considering different inclinations of the line of sight to the jet axis. Emission lines of highly ionized iron are clearly visible in our spectra, probably also observed in the Galactic microquasars GRS 1915+105 and XTE J1748-288. The Doppler shift of the emission lines is always evident. Due to the chosen geometry of the magnetohydrodynamic jet, the inner X-ray emitting part is not yet collimated. Ergo, depending on the viewing angle, the Doppler boosting does not play a major role in the total spectra. This is the first time that X-ray spectra have been calculated from the numerical solution of a magnetohydrodynamic jet.