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Astron. Astrophys. 356, 975-988 (2000)

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7. The end of the jet

The energy transported by the jets of microquasars is enormous. This energy will be continuously deposited at the end of the jets and may lead to significant radiation from this region depending on how it is dissipated. In the following we investigate the fate of the energy transported by the jets of microquasars as predicted by our model. The discussion is based on the work by Leahy (1991).

7.1. Momentum balance

In order for the jets to expand they have to accelerate the surrounding ISM and push it aside. The velocity of the contact surface between the front end of the jet and the ISM, [FORMULA], is given by balancing the momentum or `thrust' of the jet material with the ram pressure of the receding ISM

[EQUATION]

where [FORMULA] and [FORMULA] are the adiabatic indices of the jet material and the ISM respectively, [FORMULA] is the internal Mach number of the jet flow, [FORMULA] is the Mach number of the contact surface with respect to the sound speed in the ISM and [FORMULA]. Here [FORMULA] is the mass density of the jet material while [FORMULA] is the density of the ISM. This expression is strictly valid only for non-relativistic jet velocities. However, since the bulk velocity of the shocked jet material, [FORMULA], is only mildly relativistic in our fiducial model, [FORMULA], we take Eq. (22) to be a good approximation. Note that the velocity of the jet material in front of the shock is even lower than [FORMULA].

For [FORMULA] the jet material does not decelerate strongly at the end of the jet. This implies that little of the kinetic energy transported by the jet is dissipated. Even for large internal Mach numbers it is then unlikely that a strong shock will develop in the jet flow close to the contact surface. This occurs when the jet is overdense, i.e. [FORMULA] and so the jet flow is close to being ballistic. For underdense jets, [FORMULA], the ratio [FORMULA] can become considerably smaller than 1. In this case a strong deceleration of the jet ensues and much of its kinetic energy is dissipated. For [FORMULA] a strong shock will form and can act as a site of efficient acceleration of relativistic particles. Examples for this are the powerful extragalactic radio sources of type FRII (Fanaroff & Riley 1974) with their very bright radio hot spots at the end of their jets. The diffuse radio lobes enveloping their jets are the remains of the shocked jet material left behind by the advancing contact surface. In the transonic regime, [FORMULA], only weak shocks may form at the jet end and particle acceleration is less efficient. The less powerful jets of FRI objects fall in this class.

7.2. Application to our fiducial model

In the model developed in the previous sections we have assumed the jets of microquasars to be conical with a constant opening angle. This implies [FORMULA] and, because of the adiabatic expansion of the jet material, [FORMULA]. The bulk velocity of the jet material is high in our fiducial model and unless the jet material is very hot ([FORMULA] K in the case of a proton-electron jet) the internal Mach number of the jet flow will always greatly exceed 1. Since [FORMULA] is a strongly decreasing function of R, we expect from Eq. (22) that the ratio [FORMULA] will always fall significantly below unity for large values of R. This means that the jets of microquasars should end eventually in strong shocks which may be detectable in the radio. In the source XTE J1748-288 a region of bright radio emission was observed to slow down and brighten at the same time some distance from the centre of the source (Hjellming et al. 1999). In our model this is interpreted as an internal shock reaching the end of the jet where the termination shock further boosts the relativistic particle population which was pre-accelerated by the internal shock. After passing through the termination shock the jet material may inflate a radio lobe very similar to extragalactic FRII objects if [FORMULA] (see also Levinson & Blandford 1996a, b). It has been suggested that the diffuse radio emission region W50 around SS433 is the radio lobe inflated by the jets of this source (Begelman et al. 1980). Other radio lobes were detected around 1E 1740.7-2942 (Mirabel et al. 1992), GRS 1758-258 (Rodríguez et al. 1992) and possibly GRO J1655-40 (Hunstead et al. 1997), but not in the vicinity of GRS 1915+105 (Rodríguez & Mirabel 1998). The absence of a radio lobe in GRS 1915+105 may indicate that the jet in this source is relatively young and has not yet reached the point at which it becomes underdense with respect to the ISM. In the following we estimate the distance out to which the jets in this source may travel without the formation of a strong termination shock.

A lower limit for [FORMULA] can be derived from our fiducial model assuming that the jets consist only of the relativistic particles responsible for the synchrotron emission plus the particles needed for charge neutrality. Thus

[EQUATION]

where [FORMULA] is the mass of the average particle in the jet. An upper limit for [FORMULA] can be derived from the assumption that the rate at which mass is ejected along the jet can not exceed the mass accretion rate within the disk powering the jet. Fits to the X-ray spectrum of GRS 1915+105 suggest an accretion rate of order [FORMULA] kg s-1 (Belloni et al. 1997). We then find

[EQUATION]

Note that this upper limit does not depend on the nature of the jet material. Using Eqs. (23) and (24) and assuming [FORMULA] kg m-3, corresponding to a particle density of 1 cm-3, we find [FORMULA] for a proton-electron jet and [FORMULA] for a pair plasma jet. The lower limit for the pair plasma jet assumes that the pairs are cold. Because of pair annihilation it is unlikely that the material of a pair plasma jet is cold (e.g. Gliozzi et al. 1999) and so this lower limit is used here for illustrative purposes only. Relativistic thermal motion of the pairs would raise this lower limit.

For the reasonable assumptions [FORMULA] and [FORMULA] Fig. 3 shows the ratio [FORMULA] as calculated from Eq. (22). We see that even if the mass transport rate of the jet is equal to the mass accretion rate a termination shock should form about [FORMULA] m away from the core of the source. This distance is reached by the jet material traveling at [FORMULA] c in less than two years.

[FIGURE] Fig. 3. The advance speed of the termination shock at the end of a hypersonic, conical jet in units of the bulk velocity of the jet material. Solid line: The case of maximum mass transport rate. Dashed line: Minimum mass transport rate of a proton-electron jet. Dot-dashed line: Minimum mass transport rate for a pair plasma jet.

GRS 1915+105 was discovered as a bright X-ray source on the 15th of August 1992 (Castro-Tirado et al. 1992). Given the availability of X-ray monitoring satellites before 1992 it is not likely that this source was very active before this date. The subsequent radio monitoring with the Green Bank Interferometer (e.g. Foster et al. 1996) shows that after its discovery GRS 1915+105 produced radio outbursts every few months. In the frame of the internal shock model described here this implies that jet production must have been reasonably steady since 1992. Assuming that the bulk velocity of the jet material did not vary strongly, the end of the jet must have reached a distance of roughly [FORMULA] m from the core by the end of 1997. This is the time of the radio observation of the large scale surroundings of GRS 1915+105 by Rodríguez & Mirabel (1998) who did not find any evidence for a termination shock of the jet or radio lobes (but see Levinson & Blandford 1996a).

The estimation of the position of the termination shock depends crucially on the overdensity of the jet material with respect to the ISM. It is possible that the gas density in the vicinity of GRS 1915+105 is lower than assumed here. However, given its location in the galactic plane this is rather unlikely. A further possibility is that the jets of microquasars are not conical for their entire length. The jets of extragalactic FRII objects are believed to pass through a very oblique reconfinement shock which brings them into pressure equilibrium with their environment (e.g. Falle 1991). These shocks are not very efficient in accelerating relativistic particles and so are often undetectable. This scenario is also confirmed for FRII sources by numerical simulations of their jets (e.g. Komissarov & Falle 1998). The same process may recollimate the jets of microquasars as well. In this case they may stay overdense with respect to the ISM much longer and this would enable them to travel out to much larger distances before terminating in a strong shock. In this respect it is interesting to note that Rodríguez & Mirabel (1998) found a compact non-thermal emission region located 16.3' away from GRS 1915+105. The feature is elongated and its major axis is aligned with one of the jets. If this feature is caused by the jet pointing in its direction then it must have been ejected by the core roughly 280 years ago. This may be the time scale on which GRS 1915+105 becomes active and produces jets.

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Online publication: April 17, 2000
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