Photospheric emission of Poynting dominated outflows

Poynting flux dominated jets challenged by their photospheric emission

Abstract

One of the key open question in the study of jets in general, and jets in gamma-ray bursts (GRBs) in particular, is the magnetization of the outflow. Here we consider the photospheric emission of Poynting flux dominated outflows, when the dynamics is mediated by magnetic reconnection. We show that thermal three-particle processes, responsible for the thermalization of the plasma, become inefficient at a radius  cm, far below the photosphere, at  cm. Conservation of the total photon number above combined with Compton scattering below the photosphere enforces kinetic equilibrium between electrons and photons. This, in turn, leads to an increase in the observed photon temperature, which reaches  MeV (observed energy) when decoupling the plasma at the photosphere. This result is weakly dependent on the free model parameters. We show that in this case, the expected thermal luminosity is a few % of the total luminosity, and could therefore be detected. The predicted peak energy is more than an order of magnitude higher than the observed peak energy of most GRBs, which puts strong constraints on the magnetization of these outflows.

1 Introduction

One of the key open questions in the study of relativistic outflows in general, and of gamma-ray bursts (GRBs) in particular, is the mechanism responsible for accelerating the plasma to the ultra-relativistic speeds observed, (for a review, see, e.g., Mészáros, 2006; Gehrels & Mészáros, 2012; Zhang, 2014). In the classical GRB “fireball” model (Paczynski, 1986, 1990; Rees & Meszaros, 1992; Piran et al., 1993; Rees & Meszaros, 1994), the outflow is accelerated by the radiative pressure of the photons produced during the initial phase of collapse and explosion. In this model, conservation of energy and entropy implies a linear increase of the jet Lorentz factor with radius until the jet reaches the saturation radius , above which the jet internal energy is comparable to its kinetic energy, and no further acceleration is possible. In this model, magnetic fields are sub-dominant (the energy density stored in the magnetic field is much smaller than the energy density in the thermal photon field, ).

On the other hand, it was proposed that GRB outflows be magnetically dominated, (Spruit et al., 2001; Drenkhahn, 2002; Drenkhahn & Spruit, 2002; Lyutikov & Blandford, 2003; Giannios, 2006). In this scenario, if the central engine is able to produce a highly variable magnetic field, magnetic reconnection may be the mechanism responsible for the jet acceleration. Under the assumption of steady energy transfer rate, the most efficient configuration of magnetic lines orientation leads to a slower increase in the bulk Lorentz factor with radius, below the saturation radius (Drenkhahn & Spruit, 2002; Drenkhahn, 2002; Mészáros & Rees, 2011).

Since the magnetic field is not directly observed, one has to deduce its significance indirectly. For example, in their analysis of GRB080916C, Zhang & Pe’er (2009) argued that at least part of the outflow energy has to be in magnetic form. Their argument is based on the absence of a thermal component in the spectrum, which originates from the photosphere, and must accompany any photon-dominated outflow.

A different result was recently claimed by Bromberg et al. (2014). They argued that highly magnetized jets are disfavored by many GRB observations, since they do not allow to reproduce the plateau in the distribution of the GRB duration. This is because Poynting flux dominated jets are stable and break the envelope of their progenitor star on a time that is significantly shorter than observed. On contrast, the break time of baryonic jets are in agreement with the duration of the observed plateau, favoring this last model.

Solving this controversy is indeed of high importance, as the magnetization of the outflow puts strong constraints not only on possible acceleration mechanisms, but also on the nature of GRB progenitors, as well as the central engines that power GRBs. In this paper, we propose a novel way of constraining the magnetization of GRB outflows, based on their observed spectra. The key is the study of photon production processes. As we show here, models in which GRB jets are strongly magnetized lead to suppression of photon production. The produced photons, in turn, are Compton up-scattered; due to their small number, the predicted spectral peak is at  MeV, more than an order of magnitude above the typical observed peak.

2 Dynamics of Poynting-flux dominated jets

The evolution of the hydrodynamic quantities in a Poynting flux dominated outflow was first derived by Drenkhahn (2002); Drenkhahn & Spruit (2002), and was further discussed by Giannios (2005, 2006); Giannios & Spruit (2005); Mészáros & Rees (2011). In this model, an important physical quantity is the magnetization parameter, , which is the ratio of Poynting flux to kinetic energy flux at the Alfvén point, 5. This quantity plays a similar role to that of the baryon loading, in the classical “fireball” model.

The magnetic field in the flow changes polarity on a small scale, , which is of the order of the light cylinder in the central engine frame (, where is the angular frequency of the central engine (presumably a spinning black hole; see Coroniti, 1990)). This polarity change leads to magnetic energy dissipation via reconnection process, that is modeled by a fraction of the Alfvén speed 6.

The dissipated magnetic energy is converted to kinetic energy of the outflow, leading to an acceleration of the plasma. The spatial evolution of the Lorentz factor, and of the comoving number density, , below the saturation radius are given by

(1)
(2)

where the terminal Lorentz factor . The saturation radius is given by , where is the characteristic frequency of the reconnection process (Drenkhahn, 2002). The outflow luminosity (both magnetic and kinetic per unit of solid angle ) is , where is the outflow mass flux.

The spatial evolution of the comoving magnetic field, can be calculated using Equation 1, the definition of the magnetic luminosity, , the definition of the saturation radius, and energy conservation . Using below the saturation radius (Giannios & Spruit, 2005), one obtains

(3)

where is taken in the last equality, and in cgs units is used here and below.

Deep enough in the flow, radiation and matter are in thermodynamic equilibrium, sharing the same temperature, . The thermal energy increases by magnetic energy dissipation, and simultaneously decreases due to adiabatic losses. As a consequence, only a fraction of the injected thermal energy appears as black body radiation at the photospheric radius where matter and radiation decouple.

The spatial evolution of the comoving temperature was calculated by Giannios & Spruit (2005), under the assumption of full thermalization. For completeness, we briefly repeat their arguments. For constant magnetic energy dissipation rate, the energy released at radii () is

(4)

where we used the formula for given above Equation 3, Equation 1 and the definition of . About half of this dissipated energy is used to accelerate the flow, and the other half increases its thermal energy (Spruit & Drenkhahn, 2004). Adiabatic losses in radiative dominated flow imply , using Equation 2. Using again the scaling of the Lorentz factor in Equation 1, one obtains . Therefore, by the time the plasma reaches some radius , only a fraction of the energy dissipated at radius is still in thermal form. Integrating over all radii, the thermal luminosity at radius is given by

(5)

The comoving temperature of the flow is calculated using , where is Stefan-Boltzmann constant, and is given by

(6)

where we normalized the temperature to natural units of .

Photons decouple the plasma once they reach the photosphere, at which the optical depth becomes smaller than the unity. Along the radial direction, , where is Thomson’s cross section. Integrating from to infinity and requiring , using Equations 1 and 2, the photospheric radius is given by (Abramowicz et al., 1991; Giannios & Spruit, 2005; Pe’er, 2008)

(7)

For the fiducial values of the free model parameters assumed, and , this radius is below the saturation radius,  cm. This implies that the photons decouple the plasma while it is still in the acceleration phase.

The results of Equation 6 imply that as long as the photons maintain thermal equilibrium, their comoving number density scales with radius as . Here, is the comoving thermal energy density, and is the average photon energy. If, however, photon production is suppressed above some radius (namely, the remaining photons are still coupled to the particles in the plasma), the scaling low derived above implies that the photon density changes with radius as . The photon number density in this case thus drops faster than in thermal equilibrium. These photons eventually decouple the plasma at the photosphere. As we show below, this different scaling law modifies the emerging spectra at the photosphere, and in particular the observed peak energy 7.

3 Photon production mechanisms

In the following, we consider photon production below the photosphere. The leading radiative processes are double Compton, bremsstrahlung, and cyclo-synchrotron. Other radiative mechanisms, such as radiative pair production and three-photon annihilation are discarded because the plasma is not relativistic (, see Equation 6).

The key question is whether the photon sources are capable of producing enough photons to enable full thermalization below the photosphere. The rate of the interactions considered below were discussed by Beloborodov (2013) and Vurm et al. (2013), and references therein. For each of these processes, the radius at which a given interaction freezes out is given by equating the photon production rate to the expansion rate,

(8)

where and is the photon number density obtained if the photons are in thermal equilibrium ( is the Riemann zeta function and is Planck’s constant).

Double Compton. The rate of photon production in double Compton process is given by (Lightman, 1981)

(9)

where is the fine-structure constant, is a fitted formula to the exact numerical result (Svensson, 1984) and is the threshold energy8. Using Equations 8 and 9, the radius at which double Compton freezes out is

(10)

Bremsstrahlung. The temperature at which Bremsstrahlung freezes out is not relativistic, hence the pair density is expected to be much smaller than the proton density. As a consequence, the dominant bremsstrahlung process is scattering between electrons and protons. The rate of photon production via bremsstrahlung can be derived, e.g., using formula (5.14) in (Rybicki & Lightman, 1979). Dividing by and using the normalized photon energy , the photon emission rate per unit volume per unit energy is , where the Gaunt factor can be approximated by (Novikov & Thorne, 1973; Illarionov & Siuniaev, 1975; Pozdnyakov et al., 1983). The total photon emission rate is calculated by integrating over all energies, from to ,

(11)

The lower boundary on the energy of emitted photons, is found by comparing the absorption time, to the typical time a photon gains sufficient energy (by inverse Compton scattering) to avoid re-absorption, (Vurm et al., 2013). Using standard formula for free free absorption in the Rayleigh-Jeans limit, is calculated by solving

(12)

where is the Compton wavelength. While an analytic solution to this Equation does not exist, it is easily checked numerically that for a wide range of relevant parameter space, , leading to .

Using these results in Equation 8 enables to calculate the radius at which bremsstrahlung freezes out. For , this radius is approximated by

(13)

Cyclo-synchrotron. The rate of photon emission via cyclo-synchrotron process from a thermal population of electrons is given by (Vurm et al., 2013, and references therein)

(14)

where is the energy at which up-scattering and re-absorption rates are equal. For , can be approximated by (Vurm et al., 2013)

(15)

where is the cyclotron energy in the comoving frame. Assuming , and using the equations above, one finds that the freeze-out radius for cyclo-synchrotron emission is

(16)

While in the derivation of equation 16 we assumed a thermal population of electrons, we do not expect this result to change if electrons are accelerated to high energies during the dissipation process. This is due to the fact that the typical energy of a synchrotron emitted photon is proportional to , where is the Lorentz factor associated with the random motion of the electrons, and the total radiated power is, similarly, proportional to . Thus, the rate of photon emission is independent on .

All radiative process freeze out at . For the fiducial values of the luminosity, magnetization and angular frequency, Equations 7, 9, 13 and 16 imply . As a result, thermal equilibrium can exist only at radii . Above this radius, photons are not emitted at a high enough rate to ensure full thermalization. However, below the photosphere, Compton scattering enforces kinetic equilibrium between electrons and photons, such that both components can be described by a single temperature. The photon distribution at therefore obeys a Wien statistics.

4 Consequences of photon starvation

The results of the previous section imply that to a good approximation, one can assume that at the total number of photons is conserved. The photons thus follow a Wien distribution, with average (co-moving) photon energy . Due to the strong coupling between photons and electrons below the photosphere, the comoving thermal energy is shared by the protons, electrons and photons. As the plasma is non-relativistic, the energy density at is , where , and the electron and photon densities are evaluated at .

At larger radii, , full thermalization cannot be achieved. Nonetheless, due to the strong coupling between electrons and photons, for radii not much above (see below) the photon distribution is close to thermal, with comoving temperature given by . The energy density is , 9 with given in Equation 5. Conservation of photon number at implies that the comoving number density evolves according to For , one therefore obtains , namely .

The electrons are continuously heated by the magnetic reconnection process above (see Equation 4). They simultaneously radiate their energy by synchrotron emission and inverse-Compton scattering the quasi-thermal photons. As long as the cooling rate is sufficiently high, efficient energy transfer between electrons and photons exit, and both populations can be characterized by (quasi-) thermal distributions with similar temperatures, . However, as the jet expands, the cooling rate decreases, and as a result, above some radius, (, see below) the cooling can not balance the heating. At this stage,a ’two temperature plasma’ is formed, with (see detailed discussion in Pe’er et al., 2005, 2006).

The radius at which radiative cooling balances heating is calculated as follows. The rate of energy transfer via magnetic reconnection to the plasma as it expands from radius to was calculated in Equation 4. We assume that about half of this energy is used to heat the particles (the other half is converted to kinetic energy), implying that the comoving energy gain rate per unit volume is , where .10 Assuming next that a fraction of this energy is used to heat the electrons (rather than protons), using Equations 1 and 4 the electrons heating rate is given by (Giannios, 2006)

(17)

The main radiative loss term of the electrons is Compton scattering the thermal photons.11 As the plasma is non-relativistic , and the power loss (per unit volume) at is thus (where we assume that the thermal energy is dominated by the photons at ). Equating the energy loss and the energy gain rates gives

(18)

when considering double Compton, Bremsstrahlung and cyclo-synchrotron, respectively as the main photon production processes. We can thus conclude that for the fiducial values of the free model parameters chosen, in all scenarios considered.

At radii , the electrons can no longer efficiently convert their gained energy to the photons12. The photon temperature thus freezes 13. The peak of the observed spectrum () can therefore be estimated as follow. First, as the photons conserve their Wien distribution, the (comoving) peak energy is slightly above the average photon temperature, . Second, due to the Lorentz boost, the observed energy of the photons that decouple the plasma at the photosphere is (for on-axis observer). The peak of the observed spectrum is therefore expected at , namely

(19)

Note the weak dependence on the luminosity and on , and the moderate dependence on the magnetization, .

The electron and photon temperatures are presented in Figures 1 and 2. In Figure 1 we present the spatial evolution of the comoving electron temperature. At , the electrons temperature decays, in accordance to Equation 6. At larger radii, , photon starvation leads to constant temperature, and at even larger radii it increases. For comparison, we show the results obtained when photon starvation is omitted, which is shown by the segments (1) and (2) [dash-dotted lines] in Figure 1. In this case, the temperature continues to decay at radii , in accordance with Equation 6 (segment (1)). At larger radii the temperature increases again (segment (2)), once the electrons cannot convert efficiently their gained energy to the photons.

The dependence of the observer (photon) temperature on the unknown magnetization parameter is displayed in Figure 2. As is shown, for any value of , the observed peak energy is greater than a few MeV, comparable only with the highest GRB peak energies observed. For comparison, we provide two examples: GRB 050717 having  MeV (Krimm et al., 2006) requires to be at most in the order of 20. On the other hand the extreme peak energy of the first seconds of GRB 110721A around  MeV (Axelsson et al., 2012) can be explained in a highly magnetized jet, having for . We stress though, that these results show that the vast majority of GRBs, having peak energy at  MeV, are inconsistent with having high magnetization parameter,  a few, at least below the photosphere.

Figure 1: Radial evolution of the co-moving electrons temperature for the fiducial parameters , and . Four segments are shown. (1) At small radii, the temperature is directly associated to the internal energy, and (see Equation 6). (2) At , coupling between the electrons and photons is weak resulting in inefficient electron cooling which, in turn, leads to increase of the electrons temperature. (3) When photon starvation is considered, the electron’s temperature is fixed at , . (4) Same as in segment (2), but when photon starvation is taken into account. Thus, we expect the temperature to evolve along segments (1) - (3) - (4) (solid lines), with dash-dotted lines (including segment (2)) for comparison only. With the parameters at hand, is given by cyclo-synchrotron process.

Within the framework of our model, a lower limit on the luminosity of the photosphere is derived by considering the thermal luminosity at , as Compton scattering of photons above only increases the photospheric luminosity. One obtains

(20)

Thus, we conclude that the thermal part of the spectrum should be at least a few % of the total burst luminosity. In fact, since the non-thermal part is spectrally broad, it is possible that if observing over a limited band, that the thermal component will carry a larger fraction of the observed luminosity than presented here. Such a component, although weak, may be detected by careful analysis. Finally, note that for GRB 110721A, the expected fraction thermal luminosity is expected to be very small, in the order of 0.5 percent of the total luminosity.

The relation between and as a function of at constant is shown in Figure 3. The higher the photospheric peak (corresponding to large ), the smaller the radiative efficiency of the photosphere.

Figure 2: Observed peak energy as a function of the magnetization for and different reconnection frequencies . Fiducial value is shown by the thick solid line, and other solid lines are for different values of , as indicated. The double dashed thin (black) line shows the separation between photon production domination by bremsstrahlung process at low and cyclo-synchrotron domination at higher values of . The expected peak energy for via the different processes: bremsstrahlung, double Compton and cyclo-synchrotron are represented by the thick dash-dotted lines (green, blue and red, respectively). For comparison, we add the observed peak energies of the two highest record GRBs, GRB 050717 and GRB 110721A, by the two horizontal dashed (purple) lines.
Figure 3: The peak energy is plotted against the ratio of thermal to non-thermal luminosity, () for different values of and of . The solid lines represent different values of for fixed values of , ranging from 0.3 to 5. The dashed lines represent different values of for constant , ranging from 0.3 to 1. Here we consider , however as both and are weakly dependent on , the result is not substantially different for other values of .

5 Discussion

In this work, we analyzed the expected photospheric signal from Poynting flux dominated outflows. As we show here, in these conditions full thermalization can only be achieved at small radii, . As a result of this photon starvation, the observed peak of the photospheric emission is expected above (see Equation 19). Moreover, we find that this value has only weak dependence on the unknown values of the outflow parameters. This value is inconsistent with the observed peak energy of the vast majority of GRBs, which is of the order of  keV in average (Ghirlanda et al., 2009; Goldstein et al., 2012) for time integrated spectra and rises up to 3 MeV in some exceptional cases. Thus, if this peak is due to emission from the photosphere, our results indicate that only the bursts with the highest peak might be marginally consistent with the photospheric emission of magnetically dominated outflows.

Understanding of GRB prompt emission has been revolutionized in the past few years, with evidence for thermal emission being widely accepted in many bursts (Ryde & Pe’er, 2009; Ryde et al., 2010; Axelsson et al., 2012; Guiriec et al., 2011, 2013; Iyyani et al., 2013). Still, a full understanding of the origin of the spectra and the outflow conditions (in particular, the magnetization) are far from being understood. One hint may be the correlation (the “Amati” correlation; Amati et al. (2002); Amati (2006)) that shows that high spectral peak energy correlate with high total energy release. Moreover, there are some evidence for high efficiency in the prompt emission in very energetic bursts (Lloyd-Ronning & Zhang, 2004; Pe’er et al., 2012). It was proposed that these results may indicate a photospheric origin of a substantial part of the observed spectra, including the peak itself (Thompson et al., 2007; Lazzati et al., 2013; Deng & Zhang, 2014). The results obtained here show, however, that if the flows are highly magnetized, the expected peak energy is too high to be consistent with the observed one, and the efficiency of photospheric emission is only a few percent (Equation 20).

Our results indicate a high energy peak, at  MeV, significantly higher than considered by Giannios (2006, 2012). This difference originates from the larger comoving temperature at obtained here, resulting from photon starvation. They are aligned with the results obtained by Beloborodov (2013), which were considerably less detailed, and were obtained under the assumption of initially similar thermal and kinetic luminosity. It thus implies that the ratio of the photon number density to the electron number density is over-estimated. 14

Alternatively, the photospheric emission may be sub-dominant, the dominant part of the prompt spectrum being non-thermal. However, in this case, the expected peak, at  MeV should contain a few percent of the burst luminosity, and should therefore be observed. Moreover, Beniamini & Piran (2014) studied jets in which the ratio of the Poynting luminosity to the total luminosity is large at the dissipation zone. By identifying the MeV peak with synchrotron emission, they found strong constraints on the dissipation radius and the Lorentz factor at the emission region. We thus conclude that magnetized jet models in which the photospheric component is sub-dominant have additional difficulties with explaining the observed spectra.

High magnetization implies a lower ratio of (see Equation 20). Thus, the non-detection of a thermal component may be a signal of highly magnetized outflow. Our results are therefore consistent with the previous analysis carried by Zhang & Pe’er (2009), and stress the need for a careful spectral analysis that could enable to constrain the magnetization of GRB outflows.

The conditions for thermalization of the plasma in the classical “fireball” (when Poynting flux is sub-dominated) were studied by Vurm et al. (2013). In this work, it was shown that in order to obtain full thermalization, the energy dissipation radius is limited to a relatively narrow range ( cm), and the Lorentz factor during the dissipation must be mild, . Interestingly, these results are aligned with the inferred values of the outflow parameters (Pe’er et al., 2007).

In the magnetized outflow scenario considered here, the dissipation results from magnetic reconnection, and is assumed to be continuous along the jet. Thus, one cannot constrain a particular dissipation radius. The approach taken here is therefore different: by prescribing the dynamics, we study its observational consequences, in particular the expected peak energy and efficiency of the photospheric emission.

DB is supported by the Erasmus Mundus Joint Doctorate Program by Grant Number 2011-1640 from the EACEA of the European Commission. AP acknowledges support by Marie Curie grant FP7-PEOPLE-2013-CIG #618499

Footnotes

  1. affiliation: University of Roma “Sapienza”, 00185, p.le A. Moro 5, Rome, Italy
  2. affiliation: ICRANet, 65122, p.le della Repubblica, 10, Pescara Italy
  3. affiliation: Erasmus Mundus Joint Doctorate IRAP PhD student
  4. affiliation: Physics Department, University College Cork, Cork, Ireland
  5. In Poynting flux dominated models, at the flow velocity is equal to the Alfvén speed. Acceleration takes place at , where is the saturation radius.
  6. Note that this prescription assumes constant rate of energy transfer along the jet. As the details of the reconnection process are uncertain, the value of is highly uncertain. Often a constant value is assumed in the literature. Further, note that the Alfvén speed is essentially equal to the speed of light in magnetically dominated outflows.
  7. We note that in the classical “fireball” model dynamics, where magnetic fields are sub-dominant, this does not hold: even if photon production is suppressed above a certain radius, the scaling laws of the photon number density below the photosphere is not affected.
  8. A photon of energy will be up-scattered to higher energy by single Compton scattering and avoid re-absorption by the inverse process. is found by equating the Compton parameter to the photon opacity. For the double Compton process, is such that .
  9. We omit the factor in the denominator, as is the luminosity per steradian.
  10. The factor is omitted since is already expressed in .
  11. At all radii, the photon energy density, .
  12. It can easily be checked by integrating from to infinity that the Compton parameter in this regime is in the order of the unity (e.g., Bégué, Siutsou & Vereshchagin, 2013).
  13. The photon temperature slightly increases below the photosphere, due to Compton scattering with the electrons. However, this effect is discarded in our computation, since it only increases the observed temperature.
  14. Beloborodov (2013) estimated  MeV, somewhat less than the results derived here. The origin of this discrepancy is his assumption of coasting Lorentz factor below the photosphere.

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  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description