AN OBSERVED CORRELATION BETWEEN THERMAL AND NON-THERMAL EMISSION IN GAMMA-RAY BURSTS

Gamma-ray bursts (GRBs) are believed to arise from the deaths of massive stars or the coalescence of two compact stellar objects such as neutron stars or black holes. The resulting explosion gives rise to an expanding fireball with a jet pointed at the observer but hidden from the observer until the density of radiation and particles in this highly relativistic outflow is low enough for radiation to escape, a region called the photosphere (for a review, see Mészáros 2006). While the emission from this fireball is expected to be thermal (Goodman 1986; Paczynski 1986), observations over the past three decades suggest the prompt emission to be highly non-thermal (Mazets et al. 1981; Fenimore et al. 1982; Matz et al. 1985; Kaneko et al. 2006; Goldstein et al. 2012), with only a few exceptions (Ryde et al. 2010; Ghirlanda et al. 2013). The conversion of the fireball energy into non-thermal γ-ray radiation involves the acceleration of electrons in the outflow and their subsequent cooling via an emission process such as synchrotron radiation (Sari et al. 1998; Tavani 1996). Insight into these energy radiation emission processes in GRBs is obtained by comparing the observed γ-ray photon spectra directly to different radiation models. The Fermi Gamma-ray Space Telescope offers a broad energy range for these comparisons. Recent observations (Guiriec et al. 2010; Zhang et al. 2011; Axelsson et al. 2012; Guriec et al. 2013; Iyyani et al. 2013; Preece et al. 2014; Burgess et al. 2014) show that at least two mechanisms can be present: a non-thermal component that is consistent with synchrotron emission from accelerated electrons in the jet and a typically smaller blackbody contribution from the photosphere. This photospheric emission is released when the fireball becomes optically thin so that an observer may see a mixture of thermal and non-thermal emission with different temporal characteristics that, when viewed together, can probe the development and structure of the fireball jet. This simple photospheric model has been used to quantitatively interpret several observed correlations such as the Amati correlation (e.g., Thompson et al. 2007; Lazzati et al. 2011; Fan et al. 2012).

We are thus motivated to investigate correlations among spectral parameters derived by fitting the non-thermal component with a synchrotron photon model and the thermal component with a blackbody, an approach developed in previous investigations (Burgess et al. 2012, 2014). The synchrotron model consists of an accelerated electron distribution, containing a relativistic Maxwellian and a high-energy power-law tail that is convolved with the standard synchrotron kernel (Burgess et al. 2012, 2014; Rybicki & Lightman 1979). We find that the characteristic energies (E_p for synchrotron and kT for the blackbody) of the synchrotron and blackbody components are highly correlated across all the GRBs in our sample. We show that this correlation can be used to address the key question of how the energy of the outflow is distributed, i.e., whether the energy is in a magnetic field or is imparted as kinetic energy to baryons in the jet, and how this energy distribution evolves with time.

The Fermi Gamma-ray Burst Monitor (GBM; Meegan et al. 2009) has detected more than 1200 GRBs since the start of operations on 2008 July 14. A smaller number have been seen by the Fermi Large Area Telescope (LAT; Atwood et al. 2009) at energies greater than 100 MeV, but these are particularly interesting because they are among the brightest GRBs and offer the greatest opportunity for spectral analysis across a broad energy range. GRBs can last from a few milliseconds to hundreds of seconds or longer and have a variety of temporal profiles, from single spikes to multi-episodic overlapping pulses. Single-pulse GRBs exhibit the simplest spectral evolution, providing the "cleanest" signal for fitting physical models to the data (Burgess et al. 2012, 2014; Ryde & Pe'er 2009).

In this work, we analyze six bright, single-pulse GRBs detected by Fermi (see Table 1 and Figure 1) and find correlations between the E_p and kT values within each of these GRBs. The GRBs in our sample are GRB 081224A (Wilson-Hodge et al. 2008), GRB 090719A (van der Horst 2009), GRB 100707A (Wilson-Hodge & Foley 2010), GRB 110721A (Tierney & von Kienlin 2011), GRB 110920A, and GRB 130427A (von Kienlin 2013). The time histories of these GRBs are shown in Figure 1, with vertical dotted lines indicating the time binning used for the analysis of the spectral evolution of each spectral component. In a previous analysis (Burgess et al. 2014), the viability of fitting physical models to the Fermi GRB data was demonstrated for several GRBs and the spectral evolution of these models over the burst durations was investigated. The synchrotron model of Burgess et al. (2014) was constructed by convolving a shock-accelerated electron distribution of the form

$$\begin{equation} n_{\rm e}(\gamma) = n_{0} \left [\left({\gamma}\over {\gamma _{\rm{th}} } \right)^2\, e^{-\gamma /\gamma _{\rm{th}}} + \epsilon \, \left({\gamma}\over {\gamma _{\rm{th}}} \right)^{-\delta }\, \Theta \left({\gamma }\over {\gamma _{\rm{min}}} \right) \right]\, \end{equation} \tag{ 1 }$$

with the standard synchrotron kernel (Rybicki & Lightman 1979). Here, n₀ normalizes the distribution to the total number or energy, γ is the electron Lorentz factor in the fluid frame, γ_th is the thermal electron Lorentz factor, γ_min is the minimum electron Lorentz factor of the power-law tail, is the normalization of the power law, and δ is the electron spectral index. The function Θ(x) is a step function where Θ(x) = 0 for x < 1 and Θ(x) = 1 for x > 1. After convolution with the synchrotron kernel, the final fit parameters are the overall normalization of the spectrum, the νF_ν peak of the spectrum (E_p), and the electron spectral index, δ. These fits were found to be as good as those made with the empirical Band function (Band et al. 1993) that is the common choice for GRB spectroscopy. However, the Band function, being empirical, makes it difficult to deduce a more physical understanding. The fits with the synchrotron model provide a direct association of the observed spectrum with a physical emission mechanism and therefore the fit parameters can be used to study properties of the GRB jet without ambiguity.

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All of these GRBs were shown to be consistent with a physical model containing both a synchrotron and a blackbody component. For five of those GRBs we investigate herein correlations between the previously derived E_p and kT values, and we add to our sample the first pulse of the ultra-bright burst, GRB 130427A, for which a similar analysis has been performed (Preece et al. 2014). GRB 130427A is the brightest GRB detected by Fermi to date. Although its temporal structure is complex (Ackermann et al. 2014), it begins with a bright single pulse that is ideal for our physical modeling, which was used to show that internal shocks cannot explain the observed emission (Preece et al. 2014). GRB 081224A, GRB 110721A, and GRB 130427A were analyzed with GBM and LAT data; the rest of the sample were analyzed with GBM data alone. While this sample is limited by the number of bright, single-pulsed GRBs in the Fermi data set, this requirement allows reliable interpretation of the fits without confusion from overlapping pulses with different underlying spectra, which is essential to measure the evolution of the thermal and non-thermal components throughout the duration of the GRB.

Figure 2 shows an example of the spectral evolution of the two separate components. A strong correlation is found between E_p and kT, as illustrated in Figures 3 and 4. A power law of the form E_p∝T^α was fit to the E_p, kT pairs of the individual GRBs yielding values of α ranging from ∼1 to 2 (see Table 1). The general temporal trend of both E_p and kT is an evolution from higher to lower energies. As can be seen from Figure 2, the evolution of the flux of each component is not necessarily tied to the change in the characteristic energies. This is very evident during the rise phase of a pulse during which the flux rises while E_p and kT fall with time. However, during the decay phase of the pulse, the flux decreases along with the characteristic energies. Table 1 lists the ratio of the blackbody flux to the total flux for each burst.

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To interpret these observations, we assume an emission process in which the thermal and non-thermal emission occur in close proximity to one another with the non-thermal synchrotron emission arising in an optically thin region above the photosphere of the jet. The range of the indices observed in the correlation suggests that the relation between the thermal and non-thermal emission varies from burst to burst. One way to achieve this is to assume that the composition of GRB outflows vary in their ratio of magnetic content from being magnetically to baryonically dominated. In this scenario, the jet dynamics are parameterized by the dependence of the bulk Lorentz factor on the radius as Γ∝R^μ, from its initial launching radius of r₀ until the jet reaches its coasting Lorentz factor $\eta =L/\dot{M}c^2$ at the so-called saturation radius r_s, where L is the luminosity and $\dot{M}$ is the mass outflow rate. This will be approximately the jet's Lorentz factor until it is decelerated upon collision with the surrounding medium. For magnetically dominated jets μ ≈ 1/3 (Drenkhahn 2002; Drenkhahn & Spruit 2002; Kirk & Skjæraasen 2003), and in the baryonic case μ ≈ 1 (Mészáros et al. 1993). Intermediate values correspond to a mix of these components (Veres et al. 2013), and can be further modified by factors such as the topology of the magnetic field.

Under these assumptions, there are two regions of interest for which we can define the radial evolution of the bulk Lorentz factor:

$$\begin{equation} {\Gamma (r)}= \left\lbrace \begin{array}{@{}ll@{}}(r/r_0)^{\mu }&{\rm if} \ r<r_{\rm sat}\\ {\eta } &{\rm if} \ r_{\rm sat}<r. \end{array} \right. \end{equation} \tag{ 2 }$$

Here, $r_{\rm sat}= r_0 \eta ^{\frac{1}{\mu }}$ and is clearly larger when the jet is magnetically dominated. The emission of the blackbody is assumed to originate at the photospheric radius (r_ph), where the optical depth of the jet drops to unity. Following Mészáros et al. (1993), the photospheric radius is

$$\begin{equation} \frac{r_{\rm ph}}{r_0}=\left(\frac{ L\sigma _{\rm T}}{8\pi m_{\rm p} c^3r_0}\right)\frac{1}{\eta \Gamma _{\rm ph}^2}, \end{equation} \tag{ 3 }$$

where Γ_ph is the Lorentz factor of the outflow at r_ph. The value of Γ_ph depends on the magnetic content of the outflow; therefore, r_ph can take on two values,

$$\begin{equation} \frac{r_{\rm ph}}{r_0}=\eta _{\rm T}^{1/\mu }\left\lbrace \begin{array}{@{}ll@{}}(\eta _{\rm T}/\eta)^{1/(1+2\mu)}& {\rm if }\ \eta >\eta _{\rm T} \\ (\eta _{\rm T}/\eta)^3& {\rm if }\ \eta <\eta _{\rm T} \end{array} \right.. \end{equation} \tag{ 4 }$$

The introduction of the critical Lorentz factor,

$$\begin{equation} \eta _{\rm T}=\left(\frac{ L\sigma _{\rm T}}{8\pi m_{\rm p} c^3 r_0}\right)^{{\mu }/{(1+3\mu)}}, \end{equation} \tag{ 5 }$$

provides an important discriminator for the location of the r_ph relative to r_s. Outflows with η = η_T have their photospheres at the saturation radius. Typical observed Lorentz factors of GRBs derived via different methods indicate values of a few hundred (Lithwick & Sari 2001; Pe'er et al. 2007). In a magnetically dominated (μ = 1/3) case, we have $\eta _{\rm T}\simeq 150 L_{53}^{1/6} r_{0,7}^{-1/6}$. For physically relevant values of L = 10⁵³L₅₃ erg s⁻¹ and r₀ = 10⁷r_{0, 7} cm, η_T is low compared to observed values for η. Therefore, the photosphere is in the acceleration phase for a large segment of the parameter space. On the other hand, in baryonic cases (μ = 1), $\eta _{\rm T}\simeq 1900 L_{53}^{1/4} r_{0,7}^{-1/4}$, which is several orders of magnitude higher than observed Lorentz factors. Therefore, we assume that magnetically dominated jets have their photospheres in the acceleration phase and baryonically dominated jets have their photospheres in the coasting phase. With this critical assumption, we derive two cases for the behaviors of both E_p and kT.

Close above the photosphere, instabilities in the flow or magnetic field line reconnection can lead to mildly relativistic shocks and accelerate leptons, which in turn emit synchrotron radiation (Mészáros & Rees 2011; McKinney & Uzdensky 2012). The synchrotron peak energy is dependent on the baryon number density $n^{\prime }_b(r)=L/(4\pi r^2 m_p c^3 \Gamma (r) \eta)$ and the magnetic field $B^{\prime }\propto {n^{\prime }}_b^{1/2}$. The peak synchrotron energy is $E_{\rm p}=({3q_e B^{\prime }_{\rm ph}}/ {4 \pi m_e c}) \gamma _{\rm e,ph}^2 {\Gamma _{\rm ph}}$. From this expression we derive the following dependence on the input parameters:

$$\begin{eqnarray} E_{\rm p} \propto \left\lbrace \begin{array}{@{}ll@{}}L^{\frac{3\mu -1}{4\mu +2}} \eta ^{-\frac{3\mu -1}{4\mu +2}} r_0^{\frac{-5\mu }{4\mu +2}}&{\rm if }\ \eta >\eta _{\rm T}\\ L^{-1/2} \eta ^{3} &{\rm if }\ \eta <\eta _{\rm T}. \end{array} \right. \end{eqnarray} \tag{ 6 }$$

The acceleration of the jet is assumed to be adiabatic for r < r_sat, leading to a relation between the comoving temperature (T') and the comoving volume (V') of T'∝V'^−1/3. Since the expansion is along the radial direction of the jet, we have V' = d³x' = Γd³x ≡ 4πΓr²dr. Using Equation (2) for r < r_sat, we can write Γ∝r^μ. Therefore, the comoving temperature of the sub-dominant thermal component depends on radius as T'∝r^{−((μ + 2)/3)}. Above the saturation radius, the standard evolution of the temperature is T'∝r^−2/3. At the launching radius (r₀) the temperature is $T_0=(L/4\pi r_0^2 a c)^{1/4}$. Therefore, the observed temperature for the two scenarios is

$$\begin{eqnarray} kT_{\rm obs}(r_{\rm ph}) \propto \left\lbrace \begin{array}{@{}ll@{}}L^{\frac{14\mu -5}{12(2\mu +1)}} \eta ^{\frac{2-2\mu }{6\mu +3}} r_0^{-\frac{10\mu -1}{6(2\mu +1)}} & {\rm if }\ \eta >\eta _{\rm T} \\ L^{-5/12} \eta ^{8/3} r_0^{1/6} &{\rm if }\ \eta <\eta _{\rm T}. \end{array} \right. \end{eqnarray} \tag{ 7 }$$

It is unclear whether the evolution of the photosphere's luminosity or Lorentz factor, or some combination of both, drives the evolution of E_p and kT. One natural assumption is that the evolution of the photosphere's luminosity results in the observed variations in E_p and T as a burst proceeds. Therefore, considering the two types of jets; magnetically dominated (r_ph < r_s) and kinetic dominated (r_ph > r_s) we have two possibilities for the values of α.

• 1.  
  In the magnetic case, considering the appropriate powers of L, we have E_p∝T^{6(3μ − 1)/(14μ − 5)}. The exponent is singular at μ ≈ 0.36, but for values up to μ < 0.6 (these are the values of μ for which the photosphere will occur in the acceleration phase) we are able to explain values of α from 2 down to 1.4
• 2.  
  In the kinetic (baryonic) case, we have E_p∝T^1.2. This is observed in some GRBs.

The analysis of GRBs in the framework of this model can indicate whether the photosphere is in the acceleration or coasting phase, which in turn can be translated to the composition of the jet. We find that for exponents close to 2 the jet dynamics are dominated by the magnetic field while exponents close to 1 indicate baryonic jets. In our sample of six GRBs observed with Fermi, the exponents α of the relation between the characteristic energies of non-thermal and thermal components (Table 1) span the range of possible values, showing that energy content of GRB jets ranges from being dominated by the magnetic field to being contained mostly in the kinetic energy of baryons in the jet. A possible validation of this interpretation would be the future measurement of polarization in GRBs which will allow for the direct determination of the magnetization of GRB jets (see, for example, Lundman et al. 2013).

We note that the lack of a correlation between the ratio of the thermal flux to the total flux with the inferred magnetic content of the jet is puzzling (see Table 1). Naively, it is expected that a photosphere occurring deep in the acceleration phase of the outflow will have its thermal emission be much brighter than the non-thermal emission. A possible explanation for the weakness of the observed thermal component has been addressed by several authors (Zhang & Yan 2011; Hascöet et al. 2013). These works consider the effect of the magnetization parameter (σ = (B²/4πΓρc²)) on the intensity of the thermal component where ρ is the matter density of the outflow. For σ ≫ 1, most of the jet internal energy remains in the advected magnetic field, reducing the intensity of the observed thermal component from the photosphere. Another possibility for explaining the lack of correlation of the thermal flux ratios to the different jet modes is to consider that if the non-thermal flux is due to synchrotron following reconnection events above the photosphere, the amount of reconnection may not be simply given by the amount of magnetic energy and by the radius, but may depend also on the degree of tangledness of the field at that radius. For reconnection one needs field lines of opposite polarity near each other, and if the degree of randomness is stochastic (as it probably is), this could introduce a randomness in the amount of non-thermal electrons accelerated as well as the synchrotron flux produced. However, time-dependent simulations of magnetically dominated outflows in GRBs are not advanced enough to accurately test these assumptions and therefore the reduced intensity of the thermal component is still open to interpretation.

The Fermi GBM collaboration acknowledges support for GBM development, operations, and data analysis from NASA in the U.S.A and BMWi/DLR in Germany. We also thank the anonymous referee for very useful comments that aided in refining this work.