|Annu. Rev. Astron. Astrophys. 1999. 37:
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The high brigthness temperature, rapid variability, and linear polarization observed in the radio emission from X-ray binaries indicates a synchrotron origin. The time evolution of the radio emission has been modeled in terms of conical jets or expanding clouds of magnetized plasma (Hjellming & Johnston 1988, Martí et al 1992, Seaquist 1993).
In the simplest case of an adiabatically expanding spherical cloud in the optically thin regime, the van der Laan (1966) model is used, where the flux density is given by S (1-p)/2 r-2p, and the relativistic electrons have an energy distribution given by N(E) = KE-p, with K being a constant that is related to the density of the relativistic electrons. In this equation r is the radius of the cloud. Assuming that the cloud expands linearly with time, the flux density is given by S (1-p)/2 t-2p. Assuming a typical value of p = 2.4, one obtains S -0.7 t-4.8. This simple model fits the flux decrease reasonably well for several of the radio-emitting X-ray binaries (Ball 1996). However, in some of the best studied jet sources (SS 443, Hjellming & Johnston 1988, Vermeulen et al 1993, GRS 1915+105, Rodríguez & Mirabel 1999a), much less steep decreases are observed. This situation can be accounted for by making modifications to the simple expanding model. One possibility is to attribute this shallower drop of flux density with time to constrained expansion (the source cannot expand in 3 dimensions but only in 1 or 2 dimensions). In fact, the GRS 1915+105 maps with milliarcsec resolution by Dhawan et al (1999) show that the expansion of the clouds at hundreds of AU from the compact source is mostly in one direction. The flux density can be then approximately described as S -0.7 t-(2/3) pn, where n is the number of dimensions where expansion is allowed. Both in SS 433 (Hjellming & Johnson 1988) and in GRS 1915+105 (Rodríguez & Mirabel 1999a), a break in the power law that describes the decrease in flux as a function of time is observed. Remarkably, in both sources the decrease close to the source can be described with S t-1.3, while after a distance of ~ 2 × 1017 cm, S t-2.6 is observed. Hjellming & Johnston (1988) have proposed that these power laws can be explained as a result of an initial slowed expansion followed by free expansion in two dimensions. This steepening of the decrease in flux density with angular separation could be related to the similar tendency observed in the jets of some radio galaxies, where the intensity I declines with angular distance as I -x, with x = 1.2-1.6 in the inner regions and x ~ 4 in the outer regions of the jet (Bridle & Perley 1984).
It is also possible that continued injection of relativistic particles and/or magnetic field into the emitting plasma can produce shallower decreases with time of the flux density (Mirabel et al 1998). The particle injection could result from in situ acceleration as the moving gas shocks and entrains ambient gas or could result from beams or winds from the central energy source. The optically thick rise occurs very rapidly and has yet to be observed in detail for a proper comparison with the theoretical expectations.
It is possible to estimate the parameters of the ejected condensations using the formulation of Pacholczyk (1970) for minimum energy, correcting for relativistic effects and integrating the radio luminosity over the observed range of frequencies. Rodríguez & Mirabel (1999a) estimate for the bright 1994 March 19 event in GRS 1915+105 a magnetic field of about 50 mGauss and an energy of about 4 × 1043 ergs in the relativistic electrons. Assuming that there is one (non-relativistic) proton per (relativistic) electron, one gets a proton mass estimate in the order of 1023 g. To estimate the peak mechanical power during the ejection we need a value for the time over which the acceleration and ejection took place. Mirabel & Rodríguez (1994) conservatively estimate that the ejection event must have lasted 3 days, requiring a minimum power of ~ 5 × 1038 erg s-1, a value comparable with the maximum observed steady photon luminosity of GRS 1915+105, which is ~ 3 × 1038 erg s-1 (Harmon et al 1994).
The ejection events that preceded and followed the 1994 March 19 outburst are estimated to have masses in the order of 1021-22 g (Rodríguez & Mirabel 1999a, Gliozzi et al 1999). Finally, if the repetitive events observed with periods of tens of minutes in GRS 1915+105 (Rodríguez & Mirabel 1997, Pooley & Fender 1997, Mirabel et al 1998, Eikenberry et al 1998a) are interpreted as mini-ejection episodes, the mass associated with them is of order 1019 g. We crudely estimate that, on the average, GRS 1915+105 injects energy in the order of 1023 g year-1 in the form of relativistic (0.92c-0.98c), collimated outflows. This corresponds to an average mechanical energy of Lmech ~ 103 L. In contrast, SS 433 as a result of its more continuous jet flow, has Lmech ~ 105 L (Margon 1984) despite having a lower flow velocity than GRS 1915+105. The GRS 1915+105 bursts are thus very energetic but more sporadic.
Recently, there has been evidence that during some events the synchrotron emission in GRS 1915+105 extends from the radio into at least the near-infrared (Mirabel et al 1998, Fender & Pooley 1997). Then the synchrotron luminosity becomes significant, reaching values of 1036 erg s-1.
As emphasized by Hjellming & Han (1995), relativistic plasmas are difficult to confine and synchrotron radiation sources in stellar environments will tend to be variable in time. Then, one of the behaviors most difficult to account for is the relative constancy of the radio flux in some sources, of which Cyg X-1 is the extreme example. The presence of a steady outflow that is too faint to be followed up in time as synchrotron-emitting ejecta could be consistent with the lack of large variability in this type of source.