3 Energetics and spectra of microquasar CRs

   
3.1 Microquasar energetics

The best studied cases of microquasar activity show that large amounts of kinetic energy can be liberated: conservative equipartition estimates of the energy released in the major outbursts of GRS 1915+105 give $E_{\rm kin} > 2 \times 10^{44}~{\rm ergs}$ (Mirabel & Rodríguez 1999; Fender et al. 1999a), released over a period of a few days at most.

Existing radio monitoring data (Pooley & Fender 1997; Foster et al. 1996) show that GRS 1915+105 exhibits several giant flares per year, not all of which were observed with detailed campaigns (e.g., Fender et al. 1999a). This yields an estimated average kinetic power, and, since almost all of the energy will initially be deposited in the form of CRs, an estimated cosmic ray power of $L_{\rm kin} > 10^{37}~{\rm ergs\ s^{-1}}$ for GRS 1915+105 alone. In fact, GRS 1915+105 seems to release an even higher power in the form of microflares between major outbursts, estimated to exceed ${\rm few} \times 10^{37}~{\rm ergs\ s^{-1}}$ (Mirabel & Rodríguez 1999) and even $3\times 10^{38}\ {\rm ergs\ s^{-1}}$ (Fender & Pooley 2000).

Using the publicly available GBI monitoring data (http://www.gb.nrao.edu/fgdocs/gbi/gbint.html), we estimate that GRS 1915+105 spends in excess of 60% of its time at flux levels significantly enhanced over the baseline flux (GBI monitoring data of Cyg X-3 show a similar rate), with about 2 major outbursts per year (see Fig. 6 and also Fender et al. 1999a; Foster et al. 1996). Assuming that the observed radio flux in flares is proportional to the amount of kinetic energy released in the flare, and using the observed 1997 flare (Fender et al. 1999a) with a minimum kinetic energy estimate of $2\times 10^{44}~{\rm ergs}$, we estimate that over the period covered by GBI monitoring, the mean kinetic luminosity of GRS 1915+105 in flares is of order $L_{\rm kin} \sim 10^{38}~{\rm ergs~s^{-1}}$. If the baseline radio emission from GRS 1915+105 is also due to low level jet emission, this estimate increases by a factor $\sim$1.4.

Figure 6: Plot of the 2.25 GHz GBI monitoring data ( http://www.gb.nrao.edu/fgdocs/gbi/gbint.html) for GRS 1915+105 over the time span from June 1994 to August 2000. Shown as a hatched area is the flux considered above the baseline flux, i.e., the flux considered to originate from flares, which we integrated to arrive at the estimate for the average kinetic power carried by the jet. The flare analyzed by Fender et al. (1999a) is indicated by the mark "F99''. The large gaps are due to gaps in the monitoring campaign and were not included in the procedure.

The jets in SS433 are even more impressive: Reasonable estimates put the total, continuous kinetic power in excess of $L_{\rm kin} \sim few \times 10^{38}$- $few \times 10^{39}\ {\rm ergs\ s^{-1}}$(Marshall et al. 2002; Margon 1984; Spencer 1984), which is already of order 1-10% of the total Galactic CR luminosity. While the jets in SS433 are only mildly relativistic, and the production of observable, relativistic CRs thus falls under similar restrictions with regard to particle acceleration efficiency as supernovae, the example of SS433 does show that Galactic jets are capable of releasing impressive amounts of kinetic energy. Thus, jets from objects like SS433 (with mildly relativistic bulk speeds) might be an important source of sub-CRs in the Galaxy, influencing heating and ionization of the ISM.

Since the subject of Galactic microquasars is still relatively young, and many of the known sources have only been discovered in recent years, estimating the true Galactic rate of radio outburst events and thus the total Galactic power in relativistic jets is difficult. Taking the interval from 1994 through 2000, there were at least 7 well observed giant radio outbursts comparable in strength with GRS 1915+105 (corrected for Galactic distance) in the sources Cyg X-3 (Mioduszewski et al. 2001), GRO J1655-40 (Hjellming & Rupen 1995), GRS 1915+105 (Mirabel & Rodríguez 1994; Fender et al. 1999a), V4641 Sgr (Orosz et al. 2001), and XTE J1748-288 (Fender & Kuulkers 2001), giving a very conservative lower limit on the Galactic rate of $1 \ {\rm yr}^{-1}$. As with GRS 1915+105, we expect many giant flares to have gone unnoticed, and a more reasonable estimate of the event rate would be of the order of 10-100 Galactic events per year.

Cyg X-3, which is believed to be relativistically beamed, shows radio peak luminosities up to 200 times stronger than GRS 1915+105 (Fender & Kuulkers 2001), and often exhibits flaring activity (Ogley et al. 2001) at or above the peak level of GRS 1915+105 on timescales of $\sim$10 days. The other sources mentioned above are very similar to each other in peak radio luminosity (Fender & Kuulkers 2001), which we take as an indicator of kinetic power (most of these sources are not resolved and an estimate of the equipartition energy of the jet is thus not possible).

If indeed these sources operate on the same level as GRS 1915+105, the minimum kinetic luminosity of these seven sources together would be $L_{\rm kin} \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$>$ }}}10^{39}~{\rm ergs~s^{-1}}$. Since this estimate is based on the minimum energy estimates of $L_{\rm kin}$ in GRS 1915+105, the true kinetic luminosity of these sources might well be much larger.

Furthermore, there are many sources that are known to have been active at earlier epochs [e.g., V404 Cyg, Han & Hjellming 1992 and Cir X-1, Haynes et al. 1978], which exhibited flux levels comparable to the above mentioned sources. Many sources currently active might simply not have been detected yet. Similarly, many more X-ray sources are observed to be consistently active at lower radio fluxes (e.g., Cyg X-1, or GX 339-4, Fender 2001) than the brightest sources mentioned above.

During the past few years, it has become clear that radio emission from Galactic X-ray sources is a very common phenomenon. Radio emission is usually detected during state changes of the X-ray source (into or out of the low/hard state), including soft X-ray transients. While the powerful radio flares discussed above are associated with such transients, there are many more X-ray sources which are active at lower radio flux levels (e.g., Cyg X-1, or GX 339-4, Fender 2001).

These sources are observed to produce stationary, optically thick jet emission (as opposed to the already optically thin emission detected in typical radio flares of transient jets). It is not clear whether these jets are in fact relativistic and how much energy they carry. One might hope to estimate the kinetic power from the observed flux, scaling it to the peak flux observed in GRS 1915 as was done above for transient sources, but detailed kinematic modeling of the jet would be required to justify such a simple argument. In any event, because no complete sample of such sources exists, it is impossible to estimate the total fraction of mechanical jet power contained in low power sources. Furthermore, these sources might have shown transient activity in the past as well, given that GRS 1915+105 also shows steady, quiescent radio emission at comparable flux levels.

Other sources like 1E 1740-294 (Mirabel et al. 1992), GRS 1758-258 (Martí et al. 1998), and Cir-X1 (Stewart et al. 1993) show permanent extended structure resembling radio lobes in extragalactic radio sources, which are witness to past radio activity and must harbor a significant amount of CRs as well. We note here that estimating the total kinetic power from the presence or absence of radio lobes in microquasar sources (indicating past activity) is severely hampered by the fact that Galactic radio lobes are expected to have very low surface brightness (Heinz 2002).

The estimate for the kinetic energy output from GRS 1915+105 (Fender et al. 1999a), which we used as a template case to estimate the total Galactic energy in jets, is based on the assumption that the jet plasma is composed of a powerlaw of relativistic electrons and cold protons (for charge conservation). The electron spectrum was assumed to extend only over the range in frequency observed in the radio. While an IR detection of the jet indicates a high energy tail of the spectrum (Sams et al. 1996; Mirabel et al. 1996; Eikenberry et al. 1998), a low energy component (down to $\gamma \sim 1 {-} 10$) has never been observed in any jet due to lack of viable emission mechanisms to reveal such a component. The possibility of detecting this component via inverse Compton scattering has been discussed, for example, by Ensslin & Sunyaev (2001).

Finally, the physical structure of microquasar jets is still not known-they could be made of either discrete ejections or a continuous stream of matter. If the jet is not composed of discrete ejections, but instead is a continuous outflow with knots corresponding to internal shocks, Kaiser et al. (2000) have shown that, in the case of GRS 1915+105, the estimate of the total kinetic energy carried in the jet (and thus the total CR energy released in the working surface) is a factor of 10 higher than the above estimate (though the instantaneous kinetic power is reduced), corrections for the low energy end of the particle distribution notwithstanding.

All of this indicates that the lower limit of $L_{\rm kin} \sim few \times 10^{38}\ {\rm ergs\ s^{-1}}$ is conservative, and it might be that the kinetic luminosity from microquasars is of the order of 10% of the total Galactic cosmic ray power $L_{\rm CR}$. Clearly, the uncertainty in frequency and power of radio flares in microquasars warrants continued monitoring of these sources to answer the question of how important energy input by these sources really is.

   
3.2 The termination shock

In the following, we will assume that a standard working surface as depicted in Fig. 2 exists. The conversion of kinetic to internal energy will then take place either in the forward or the reverse shock.

3.2.1 Discrete ejections

If the jets are composed of discrete ejections, then each ejection will eventually convert its kinetic energy into some form of internal energy via interaction with the ISM, not unlike the external shock encountered in GRB afterglows. During the phase of relativistic propagation of the ejection, either the forward or the reverse shock (or both) must be relativistic. It is inside of the relativistic portion of the shock that most of the energy is dissipated.

For a cold ballistically expanding ejection, most of the energy is dissipated in the forward shock. This is implied by the small opening angle $\theta$ of the ejection: as long as $\theta \sim c_{\rm s}/ \Gamma_{\rm jet} c \ll 1/\Gamma_{\rm jet}$ (with $\Gamma _{\rm jet}$ being the bulk Lorentz factor of the ejection), the characteristic transverse size of the ejection R is always much smaller than the distance d over which it slows down, as seen in the frame of the blob: $R \sim \theta d \ll d/\Gamma_{\rm jet}$ (here, the factor of $1/\Gamma_{\rm jet}$ accounts for the Lorentz contraction in going to the frame of the ejection). Thus, the deceleration time $d/\Gamma_{\rm jet} c$ is much longer than the light crossing time of the ejection R/c, and the deceleration must occur in a sub-relativistic shock. This implies that the ejection is not heated to relativistic temperatures, while the forward shock must be relativistic, with shock velocity corresponding to $\Gamma _{\rm jet}$.

The particles passing through this shock will have energies of order $\Gamma_{\rm jet}~m_{\rm p}~c^2\ \leq\ \Gamma_{\rm jet,0}~m_{\rm p}~c^2$, where $\Gamma_{\rm jet,0}$ is the initial Lorentz factor of the ejection (before interaction with the ISM). Because the ejection is slowed down, particles of a spread in energies are created in the forward shock, though the spectrum will show a sharp turnover or cutoff at energies $e \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$>$ }}}\Gamma_{\rm jet,0}~m_{\rm p}~c^2$ (below this cutoff, it is plausible that the spectrum rises with ${\rm d}N/{\rm d}\gamma \propto \gamma^{-3}$, see Appendix A).

3.2.2 Continuous jets

However, there are reasons to believe that at least in the well studied cases of GRS 1915+105 (Kaiser et al. 2000, though this question is up for debate) and in SS 433, as well as in the persistent radio outflows of low-hard state sources like GX 339-4 and Cyg X-1 the jets are continuous streams of matter. In which part of the shock the dissipation occurs in a continuous jet depends on the jet thrust and the external density: the advance speed of the jet is roughly given by ram pressure balance. The transition from forward to reverse shock dissipation then occurs when

$$\frac{L}{A_{\rm shock}~ \rho_{\rm x}~ c^3} = \frac{2 ~ L_{38.5}}{l_{16}^2~ \theta_{0.1}^2~ n_{\rm ISM}} \sim 1,$$ (2)

where $L_{38.5}\equiv L_{\rm kin}/10^{38.5}\ {\rm ergs\ s^{-1}} \sim L_{\rm kin}/3\times 10^{38}\ {\rm ergs\ s^{-1}}$ is the kinetic power, $l_{16}=l_{\rm jet}/10^{16}\ {\rm cm}$ is the jet length, $A_{\rm shock}$the surface area of the shock, $\theta_{0.1}\equiv \theta/0.1$ the jet opening angle in units of 0.1 radian or 5.7$^\circ$, and $n_{\rm ISM} \equiv \rho_{\rm x}/m_{\rm p}$ the ISM particle density in units of $1\ {\rm cm^{-3}}$. Note that the working surface propagates, so l₁₆ is a measured quantity, given by the age of the source, its power, and the ISM density. Strictly speaking, the jet thrust will have to be measured in the shock frame, and the relativistic shock jump conditions applied, but in order to estimate which of the two shocks is going to be relativistic, the above approximation is sufficient.

The reverse shock is relativistic if the ratio on the left hand side of Eq. (2) is smaller than unity, the forward shock is relativistic if the ratio on the left hand side is larger than unity. For Galactic sources both cases can occur for appropriate external densities, depending on the length of the jet.

3.3 Particle acceleration in relativistic shocks

   
3.3.1 Nucleon acceleration

For non-relativistic shocks, it has long been known that multiple shock crossings can accelerate particles to ultra-relativistic energies. The most commonly accepted scheme is first order Fermi acceleration (e.g., Krymskii 1977; Blandford & Ostriker 1978; Bell 1978; Blandford & Eichler 1987), which typically produces a powerlaw distribution

$$f(\gamma) = f_0 \gamma^{-s}$$ (3)

with canonical index $s\sim2$.

For relativistic shocks the situation is not as clear cut. Several attempts have been made to solve the problem of particle acceleration at relativistic shocks, mostly in the limit of test particle acceleration. In general, it is found that powerlaw distributions with somewhat steeper spectra than in non-relativistic shocks can be produced (Ellison & Double 2002; Achterberg et al. 2001; Kirk & Schneider 1987).

It is certain, however, that acceleration of particles in the shock must take place: Particles crossing the shock are by nature already relativistic in the downstream rest frame, with a typical Lorentz factor of $\Gamma_{\rm rel}$, the relative Lorentz factor between upstream and downstream frames. The particle distribution leaving the shock is thus strongly anisotropic, and essentially mono-energetic. The randomization of this energy is then a question of the efficiency of the typical plasma processes often assumed to be present in populations of relativistic particles.

The simplest assumption is that the particle momenta are simply isotropized behind the shock. The shock acceleration kernel is then a delta function and a cold upstream plasma will be transformed into a narrow but relativistic energy distribution, the width $\sigma_{\Gamma}$ of which should roughly be given by the Lorentz transformed width $\sigma$ of the upstream energy distribution, $\sigma_{\Gamma} \sim \Gamma_{\rm rel} \beta~ \sigma$, where $\Gamma_{\rm rel}$ is the relative Lorentz factor between the upstream and downstream frames. The mean particle energy will be $\langle \gamma~m~c^2 \rangle \sim \Gamma_{\rm rel}~m~c^2 \sim \Gamma_{\rm jet}~m~c^2$.

If scattering by downstream turbulence or particle interactions is stronger, the particles can be thermalized, in which case a Maxwell-Boltzmann distribution according to the relativistic Rankine-Hugoniot jump conditions will be established. The main observational difference between these two cases will be the width of the energy distribution (see Fig. 5).

If a significant fraction of the particles can perform multiple shock crossings (which again hinges on effective scattering to return downstream particles to the shock), we expect a powerlaw-type distribution to be established. It is reasonable to assume that the first time escape fraction from the downstream region (i.e., the probability that a particle which crossed the shock only once) is of order $\mathcal P_{\rm esc} \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$<$ }}}90\%$ (e.g., Achterberg et al. 2001), which implies that most of the particles will only cross the shock once (note that the escape probability in non-relativistic shocks is generally very small for fast particles, though $\mathcal P_{\rm esc}$ is of order unity for thermal particles, Bell 1978). The escape fraction in the upstream region is much smaller and generally neglected in calculations.

The particles re-crossing the shock will pick up another factor of order $\Gamma_{\rm rel}^2$ in energy gain (Vietri 1995), which implies that a significant fraction of particles will be boosted to higher energies ( $\gamma \geq \Gamma_{\rm rel}^3$). This fraction of the particles will contribute a significant amount of pressure to the post shock gas, which will modify the shock structure accordingly. Thus, the amount of energy accessible to the bulk of the particles which cross the shock only once is of the order $[1 + \Gamma_{\rm rel}^2 (1/\mathcal P_{\rm esc} - 1)]^{-1} \sim 50\% {-} 90\%$.

The remaining fraction $1 - \mathcal P_{\rm esc}$ of the particles will perform true Fermi acceleration. The low energy turnover of this distribution should be located roughly at $\gamma \sim \Gamma_{\rm rel}^3$(Vietri 1995) and subsequent shock crossings will produce features at energies of $\Gamma_{\rm rel}^{2i + 1}$ (where i enumerates the number of shock crossings). The similarity of this process to Compton upscattering was already mentioned in Sect. 2.3.

The superposition of features from multiple shock crossing cycles will lead to the formation of a powerlaw at high energies, very similar to the powerlaw produced by optically thin inverse Compton scattering (where the Lorentz transformations of the photon distribution to and from the particle rest frame are replaced by Lorentz transformations to and from the upstream fluid rest frame, assuming that scattering is strong enough to isotropize the particle distribution. The optical depth $\tau_{\rm IC}$ is replaced by the return probability $1 - \mathcal P_{\rm esc}$.) The shock powerlaw slope $s_{\rm shock}$ is determined by $\Gamma_{\rm rel}$ and $\mathcal P_{\rm esc}$(e.g., Bell 1978):

$$s_{\rm shock} \sim 1 - \frac{\ln{\left(1-\mathcal P_{\rm esc}\right)}}{\ln{\left(\Gamma^2\right)}}\cdot$$ (4)

Thus, observationally, diffusive acceleration will lead to the presence of a second feature in the CR spectrum around $\sim$ $\Gamma_{\rm jet}^3$. This situation is shown in Fig. 5, where the powerlaw spectral component was plotted for $\Gamma _{\rm jet}=5$. The energy of this broad peak coincides nicely with the peak in the excess CR proton component required in the HEMN model (Strong et al. 2000) to explain the $\gamma$-ray EGRET excess above 1 GeV.

We note that Achterberg et al. (2001) argue that in the absence of an efficient scattering mechanism, the average energy gain per particle will be restricted to a factor of order unity (rather than $\Gamma_{\rm rel}^2$) for higher order shock crossings ($i \geq 2$), and that the low energy turnover of the powerlaw component might thus be located at $\sim$ $\Gamma_{\rm rel}^2$ (rather than $\Gamma_{\rm rel}^3$). The search for additional features in the CR spectrum at energies $\Gamma^2$ and $\Gamma^3$ might offer a potential way to test these predictions.

It is also possible that a population of relativistic protons already exists upstream of the shock. If the proton number density is equal to the electron number density, the bulk of the particles and of the energy will presumably be at the low energy end ( $\gamma \sim 1$), otherwise the estimates for kinetic energy flux in the jet would have to be increased accordingly, increasing the impact on the Galactic CR spectrum as well (by a factor of $\gamma_0$, the low energy cutoff of the distribution).

This population will be shifted to higher energies in the shock, and possibly experience further Fermi type acceleration. Thus, if a powerlaw of relativistic protons exists prior to the terminal shock with lower cutoff $\gamma_0$ and spectral index $s_{\rm jet}$, the terminal shock should shift the lower cutoff roughly to $\Gamma_{\rm jet} \gamma_{0}$, while the slope of the new powerlaw will be ${\rm Min}(s_{\rm jet}, s_{\rm shock})$, where $s_{\rm shock}$ is the powerlaw slope produced in the relativistic terminal shock.

While the particles produced in the shock must be relativistic at injection, the dynamical evolution of the shocked gas can reduce their energies significantly if adiabatic cooling is important before the particles can escape the shock region (radiative cooling of the nucleon distribution will be negligible). The diffusion of particles out of radio lobes and hot spots is a highly uncertain process and has not been studied in the necessary detail to answer this important question. Rather than discussing it at length, which would by far exceed the scope of this paper, we decided to include a short discussion in Appendix B, where we show that adiabatic losses do indeed pose a significant obstacle to particle escape (see also Fig. 5).

3.3.2 Electron acceleration

CR protons do not emit any measurable amount of synchrotron radiation. Inverse Compton scattering on protons is also very inefficient. However, if diffusive acceleration is operating efficiently in the working surface, then the powerlaw electrons produced there should be detectable through their radiative signature. Tentative evidence of this powerlaw electron component are the radio lobes in the microquasars 1E 1740-294 and GRS 1758-258.

The spectral index of the non-thermal emission from the lobes of 1E 1740-294 is $\alpha \sim 0.7 {-} 0.9$ (Mirabel et al. 1992), corresponding to an electron spectrum of slope of $s \sim 2.4 {-} 2.8$. If the protons are injected at the working surface with the same slope (this would require that the electron spectrum is unaffected by radiative cooling and that the powerlaw electrons are not just advected through the shock from upstream), then the observed CR slope near the earth should be steepened by $\Delta s = 1/2$ due to the energy dependence in the Galactic diffusion coefficient. This would yield a proton powerlaw slope of $s\sim 2.9 {-} 3.3$, somewhat steeper than the canonical CR slope of $s \sim 2.7$, though not significantly.

Part of the observed powerlaw electron distribution inside the tentative radio lobes of 1E 1740-294 and GRS 1758-258 could also have been advected from the jet. In fact, is is unclear whether electrons can be accelerated efficiently in a shock if protons are present, since the shock thickness will be set by the proton Larmor radius, which will be much larger than the electron Larmor radius (e.g., Achterberg et al. 2001). In this case, the electrons will not experience a shock at all, more likely, they will be accelerated adiabatically to a narrow component with energies of order $\Gamma_{\rm jet}~m_{\rm e}~c^2$. Such a component will not be detectable through synchrotron radiation.

A detailed model of the energetic history of the particles in the lobes of 1E 1749-294 would be required to answer these questions.