The bulk kinetic power of the jets of GRS 1915+105

We calculate the minimum value of the power in kinetic bulk motion of the galactic superluminal source GRS 1915+105. This value far exceeds the Eddington luminosity for accretion onto a black hole of 10 solar masses. This large value severely limits the possible carriers of the kinetic luminosity at the base of the jet, and favours a jet production and acceleration controlled by a magnetic field whose value, at the base of the jet, exceeds $10^8$ Gauss. The Blandford and Znajek process can be responsible of the extraction of the rotational energy of a Kerr black hole, if lasting long enough to provide the required kinetic energy. This time, of the order of a day, implies that the process must operate in a stationary, not impulsive, mode.


I N T R O D U C T I O N
GRS 1915105 was discovered in 1992 with the WATCH telescope on board the GRANAT satellite (Castro-Tirado, Brandt & Lund 1992). A radio counterpart was subsequently identi®ed and bipolar out¯ows with apparent superluminal motions were observed; the standard interpretation of this phenomenon in terms of relativistic jets (Rees 1966) places the source at a distance D 12:5 kpc at an angle i 708 to the line of sight (Mirabel & Rodriguez 1994). The source cannot be observed in the optical band, because of the heavy extinction in the Galactic plane, but the X-ray luminosity, often well above the Eddington luminosity for a neutron star, suggests the presence of a ,10-M ( black hole. Other circumstantial evidence comes with the similarities with the other galactic superluminal source GRO J1655À40, which has been shown unambiguously to harbour a compact object of .7 M ( (Orosz & Bailyn 1997). Since its discovery, GRS 1915105 has displayed an extraordinary richness in variability in the X-ray (Greiner, Morgan & Remillard 1996;Belloni et al. 1997;Chen, Swank & Taam 1997), infrared and radio bands . Recent (Eikenberry et al. 1997;Mirabel et al. 1998) simultaneous multiwavelength observations of this superluminal source show a strict link between activity in the inner accretion disc and plasma ejections.
In this paper we estimate the minimum kinetic power associated with the ejections and investigate the possible energy transport mechanism from the compact source to distances where the radio blobs are resolved, showing that the more suitable solution implies the presence of a strong magnetic ®eld at the footpoint of the jet.
In Section 2 we summarize the existing radio and infrared data for the ejection events. In Section 3 we directly calculate the minimum kinetic power during the ejection. In Section 4 we investigate the possible origin of the kinetic power. In Section 5 we summarize the main results of the paper and discuss their consequences.

O B S E RVAT I O N A L C O N S T R A I N T S D U R I N G T H E O U T F L OW S
Radio observations show that GRS 1915105 in its active state is characterized by the increase of the¯ux level from S n # 10 mJy to S n , 100 mJy, the so called plateau state, on top of which radiō ares are superimposed, with durations from days to a month and uxes up to a few Jy (Foster et al. 1996). The spectra during the plateau state are generally¯at or inverted, a # 0 (with S n~n Àa ), probably indicating synchrotron self-absorption. On the other hand, the spectra of radio¯ares reveal a transition from optically thick to thin (a > 0) emission during the rise stage, with spectra harder (a , 0:5) when the¯uxes reach the maximum and softer (a , 1) during the subsequent decay, generally interpreted as an indicator of synchrotron radiation from an expanding radio cloud.
Observations with the Very Large Array and multiwavelength campaigns (with VLA and UKIRT simultaneously) pointed out that some strong radio¯ares are related to the ejection of radio clouds. There are two different kind of ejections: (i) major ejections, such as the prominent radio outburst of 1994 March (Mirabel & Rodriguez 1994), where the ejecta actually moved for several weeks along a direction forming an angle v 708 to the line of sight, with bulk velocity 0:92c and expansion at ,0:2c, with a spectral index change from a 0:49 (when the blobs could not be resolved) to a 0:84 (when the two condensations had moved apart), and (ii) so-called`baby jets' (Eikenberry et al. 1997;Mirabel et al. 1998): quasi-periodic oscillations in the radio¯ux, coupled with similar oscillations in the X-ray and infrared bands and interpreted as scaled-down (in space and time) ejections of synchrotronemitting expanding blobs. The analysis of quasi-periodic oscillations (QPOs) in the infrared and radio bands clearly shows that there is a time shift of¯ux peaks, with short wavelengths peaking ®rst: the infrared (2-mm) peak precedes the 2-cm peak, which precedes the 3.6-and 6-cm peaks.
It must be emphasized that on one occasion (1995 July) a jet was also observed in the near-infrared K band (l 2:2 mm), with the same position angle as the radio jet and a near-infrared magnitude of K=13.9 (slightly brighter than the total source magnitude at its weakest measured value of K=14.3), and separated from the central source by 0.3 arcsec (Sams, Eckart & Sunyaev 1996).

M I N I M U M K I N E T I C P OW E R C O N D I T I O N
In order to deduce the minimum kinetic luminosity L k related to a major ejection we consider the 1994 March event, for which the observational data are the most exhaustive. As we are interested more in power than in energy estimates, we follow an alternative method rather than the standard one, which consists of determining ®rst the internal energy of the blob via the minimum energy criterion and then obtaining an estimate of the kinetic power, by making some (highly uncertain) assumptions about the energization time.
We instead directly calculate the kinetic power, equal to the energy¯ux through a cross-section of the jet (see also Ghisellini & Celotti 1999). This energy¯ux can be carried by particles and by the toroidal magnetic ®eld, and correspondingly we have where m is either m p in the case of a`normal' e±p plasma or hgim e for e 6 pairs, G is the Lorentz factor of the bulk motion, R is the cross-section radius of the jet, n H is the comoving particle density, U B is the magnetic energy density measured in the comoving frame and hgi is the mean Lorentz factor of the electrons, as measured in the comoving frame. A lower limit on n H can be estimated from the observed synchrotron emission L syn of the blob. Assuming a spherical emitting volume of radius R, the number density of leptons producing the observed radiation is where hg 2 i is averaged over the relativistic electron distribution, and d G1 À b cos v À1 is the Doppler factor. The B À2 dependence of the estimated particle density allows us to minimize the total power L k;p L k;B with respect to the magnetic ®eld, because both the bulk Lorentz factor G ( 2:55) and the viewing angle v ( 728) are known: yielding the value of the magnetic ®eld B min corresponding to the minimum power, where f is a parameter directly related to the shape of the electron energy distribution, and therefore to the shape of the synchrotron radiation they emit: As the radio emission is a power law with spectral index a , 0:5, the particle distribution Ng~g À2 between g 1 and g 2 , yielding f , lng 2 =g 1 =g 2 f1 m =m e g 1 lng 2 =g 1 g. With R , 7´10 15 cm (geometric average of the size of blob of GRS 1915105, assuming a distance of 12.5 kpc), L syn 10 33 erg s À1 and b 0:92 (corresponding to G 2:55), we obtain B min;ep 0:12 G and B min;e 6 0:036 G if f 1:8 (g 1 1 and g 2 10 3 , as required for production of the observed synchrotron photons at n , 300 GHz). A low-energy cut-off in the particle distribution would decrease B min somewhat (B min;ep 0:054 G and B min;e 6 0:031 G for g 1 30), while the high-energy cut-off is consistent with the production of the observed radiation at 300 GHz (with B B min ). However, the dependence of B min on the extremes of the electron distribution is rather weak (B min~f 1=4 ). Note that the value of B min;ep 0:12 is almost a factor of 3 greater than the one obtained by Mirabel & Rodriguez (1995) and Liang & Li (1995), while it more-or-less agrees with the value estimated from the requirement of optical transparency of the plasmoid with respect to synchrotron self-absorption (Atoyan & Aharonian 1999).

7
These internal energies can be compared with the corresponding kinetic energies in bulk motion E k given by where t is the time needed for the blob to cross the jet section.

O R I G I N O F T H E K I N E T I C L U M I N O S I T Y
The kinetic power calculated above refers to the radio-emitting blob, at a huge distance from the putative black hole and accretion disc. The most economic assumption is that this power is approximately conserved along the jet, with very small dissipation giving rise, e.g., to the random particle energy responsible for the emission. If not, we would have to assume that at the base of the jet an even larger value of L k is produced. In principle there are several possible energy carriers: e 6 pairs, p±e À , pure magnetic ®eld or a mixture of these components. Consider also that the particles can in principle be`hot or`cold: at large distances at least some of the electrons are`hot' (indeed, in our estimates we assumed that all the L38 M. Gliozzi, G. Bodo and G. Ghisellini electrons are`hot', because this is the most economic assumption), however these`hot' particles cannot come directly from the inner region (see below), but need to be accelerated or reaccelerated, therefore for the inner jet energy carriers one has to consider both cases. Furthermore, we note that in the`hot' case, the particle random energy, increasing the mass, can contribute to the kinetic power. Let us examine the possible cases.

4.1`Cold' e 6 pairs
If the kinetic power is carried by a pure e 6 cold pair plasma, we can calculate the corresponding pair density and scattering optical depth at some jet radius R close to the base of the jet: , 4:2´10 2 L k 2:9´10 39 erg s À1 10 7 cm R : 9 As the annihilation time-scale is of the order of R=ct 6 , cold pairs cannot survive annihilation.

4.2`Hot' e 6 pairs
If the pairs are relativistic, with average random Lorentz factor hgi, the above estimate of t 6 decreases by a factor hgi. In addition, the annihilation cross-section decreases. However these relativistic pairs are embedded in a dense radiation ®eld, produced by the accretion disc, and they cool on time-scales shorter than the dynamical time-scale. In a region of size R d , the radiation energy density resulting from the accretion disc luminosity L d produced within R d is of the order of U d L d =4pR 2 d c. The ratio between the inverse Compton cooling time and the dynamical time R d =c for a particle of Lorentz factor g is A large fraction of L k would be lost and converted into radiation, contrary to what is observed. We therefore conclude that a pure e 6 pair plasma cannot carry the derived jet kinetic power, irrespective of whether it is hot or cold.

4.3`Normal' e±p plasma
The case of electrons with an average random Lorenz factor hgi greater than m p =m e and cold protons is analogous to the above case: inverse Compton scattering of seed accretion disc photons cools the electrons in a time shorter than R d =c.
The case of`hot', relativistic protons is instead immune to radiative losses, and our main concern is with the required con®ning pressure. If the kinetic power is carried by protons with an average Lorentz factor < g p >, the corresponding (comoving) pressure is of the order of their energy density. If the con®nement is magnetic, the required value of the magnetic ®eld is which is equal to the value that the magnetic ®eld must have if it is the main carrier of the kinetic power. If both electrons and protons are cold, the electron-scattering optical depth is a factor m p =m e smaller than that of equation (9), i.e. of the order of a few. There are no severe limits in this case: the Compton drag is not suf®cient to slow down the plasma in the jet. It is, however, useful to estimate the predicted power emitted by the Compton drag process, because it can turn out to be important in sources where the jet is more aligned with the line of sight. For this simple estimate we will assume that the jet is Compton-thick (i.e. t T > 1), so that the effective cross-section of the process is the geometrical one, i.e. pR 2 . As before, we assume that the accretion disc produces the luminosity L d within the region R d > R, at the typical frequency n d . With these hypotheses, the observed bulk Compton luminosity L bC is Most of this luminosity is observed at the frequency n , dGn d . With R=R d , 1=10, L d , 10 39 erg s À1 and d 0:57 we obtain L bC , 1:7´10 36 erg s À1 and n , n d . This radiation is therefore unobservable in the case of the known Galactic superluminals, which are both observed with large viewing angles, but may be very important for sources with d , G q 1 (see also Sikora et al. 1997).

Poynting vector
The value of the magnetic ®eld needed to carry L k in the vicinity of the black hole is equal to that given in equation (11), if the bulk of the magnetic ®eld is moving with G, while the B value before acceleration is a factor of G greater, corresponding to B , 3´10 8 G at R 10 7 cm. It must be remarked that the standard theory of accretion discs (Shakura & Sunyaev 1973) predicts that the maximum possible magnetic ®eld at a given R=R S (where R S is the Schwarzshild radius) in a radiation-dominated disc is B~M=M ( À1=2 , so that microquasars will have magnetic ®elds ,10 4 times stronger than those in quasars. Therefore, the theoretical estimate B , 10 2 ±10 4 G obtained with the Blandford & Payne model (Blandford & Payne 1982) near the massive black holes in active galactic nuclei (AGN) clearly shows that in microquasars B , 10 8 G can easily be attained.
Note that this value of the magnetic ®eld would be of the same magnitude as the magnetic ®eld required by the Blandford±Znajek (Blandford & Znajek 1977) process to produce L k by extracting the rotational energy of a 10-M ( Kerr black hole: L rot 10 41 B 2 9 M 2 1 erg s À1 ; 13 where B 10 9 B 9 G and M 10M 1 M ( . From the above arguments, we conclude that e 6 or relativistic electrons with hgi greater than m p =m e cannot carry L k in the inner region of the jet, while protons and the Poynting vector can. An alternative solution could be that the energy that is carried in the inner region by a`normal' cold plasma or toroidal magnetic ®eld is converted into e 6 pairs at a large distance from the black hole and accretion disc. Pairs are produced most ef®ciently through photon±photon collisions, but this requires a powerful g-ray continuum. As the maximum ef®ciency in converting pairs in this way is of the order of 10 per cent (Svensson 1987), the high-energy luminosity has to exceed 10 40 erg s À1 , to produce a kinetic power of the order of 10 39 erg s À1 . Such a high luminosity is not observed. Therefore we conclude that e 6 pairs do not play any role as energy carriers along the jet, and the minimun kinetic luminosity involved in major ejection events is given by the value estimated for an e±p plasma (,3´10 40 erg s À1 ).
There are no strong observational arguments to decide between protons and the Poynting vector, unless we observe a new Galactic superluminal at a small viewing angle. Note however that if L k is initially carried by a large magnetic ®eld there is the possibility of tapping a great reservoir of energy (the rotational energy of the black hole) and accelerating the plasma to relativistic speeds.
We therefore conclude that the kinetic power carried initially by the magnetic ®eld is the most economic way to explain the observed energetics. Differential accretion disc rotation or rotating black holes can amplify magnetic ®elds to the required values, and the magnetic ®eld will then accelerate particles to relativistic speeds.
The amount of matter present at the base of the jet may not be negligible, especially if magnetic ®eld lines help to channel particles from the disc to the jet. We can compare the in¯ow rate Ç M in of the accretion process with the out¯ow rate Ç M out necessary to account for the kinetic power, in particles, of the radio blob. If we write the accretion luminosity as L acc h Ç M in c 2 , as usual, and the kinetic power as L k G À 1 Ç M out c 2 , we derive As h , 0:1 and L k , 10L acc , we have that the in¯owing and out¯owing mass rates are comparable. This in turn suggests that most of the matter in the jet may come from the accretion disc.

T H E E J E C T I O N T I M E
Another important question concerns the duration of the ejection events. When the blob becomes visible in the radio, it has a size of a few light-days. It seems unlikely that it corresponds to an ejection duration lasting longer than that, while a shorter ejection time may be possible. In the latter case, the requirement on the initial kinetic power correspondingly increases. Note that direct radio observations of¯are events do not directly constrain the ejection time, because the radio¯ux eventually produced in the ®rst parts of the jet is heavily self-absorbed. Again, the minimum power requirement criterion favours ejection events that last for t out , a few d. This corresponds to ,10 9 R S;10 =c, where R S;10 is the Schwarzchild radius for a 10-M ( black hole. The immediate consequence is that the ejection event must be considered as a continuous and stationary process. Scaling for a superluminal AGN, we would have, for a 10 9 -M ( black hole, an ejection phase lasting for 2´10 5 yr.

D I S C U S S I O N
We have obtained a reliable lower limit to the kinetic power corresponding to major ejection events in superluminal galactic sources, in particular for GRS 1915105. This limit is of the order of 3´10 40 erg s À1 , much greater than the observed radiative luminosity, which is probably Eddington-limited to values of the order of 10 39 erg s À1 , corresponding to a black hole of 10 M ( . This by itself suggests that the jet acceleration mechanism cannot be radiative.
We have investigated the role of e 6 pairs, normal plasma and the magnetic ®eld as energy carriers of the kinetic power in the inner jet regions, excluding an important role for the e 6 pairs, and favouring a scenario in which the inital acceleration phase is controlled by a magnetic ®eld of the order of 10 8 G or more, able to channel and accelerate accretion disc matter in the jet.
Conservation of kinetic power dictates that the injection timescale is of the order of a day, a very long time-scale if measured in units of the light-crossing time of a 10-M ( Schwarzchild radius, indicating a stationary, not impulsive, process. Shorter injection times, although possible, correspond to larger kinetic powers, exacerbating the problem of how to obtain them. The rough equality between the value of the magnetic ®eld needed to carry the kinetic power in the inner jet and the value necessary to tap the rotational energy of a Kerr black hole via the Blandford & Znajek process can be regarded as circumstantial evidence that this process is indeed the one responsible for jet formation and acceleration. This process is also a candidate to power the jets in radio-loud quasars, and we can compare the results obtained here with the corresponding estimates of the kinetic power of AGN derived by Celotti, Padovani & Ghisellini (1997) for a sample of radio-loud sources. For those AGN, the above authors ®nd a kinetic power between 10 45 and 10 48 erg s À1 , of the same order as the luminosity needed to ionize the broad-line region of the same objects, and in agreement with the power required by the existence of the outer radio lobes. In the AGN case the kinetic and accretion luminosities are roughly equal, while in Galactic superluminal objects the kinetic power dominates. Another obvious difference is the Lorentz factor of the bulk motion, which is much greater in the AGN case. This results in a ratio of the out¯owing to infalling mass rate Ç M out = Ç M in of order unity for galactic superluminals and two orders of magnitude smaller for AGN. These estimates make Galactic superluminal sources the most ef®cient engines to produce collimated relativistic bulk motion, with the possible exception of gamma-ray bursts.