HOT ELECTROMAGNETIC OUTFLOWS. I. ACCELERATION AND SPECTRA

Our first interest here is in how such a "hot electromagnetic outflow" is accelerated. We focus on stationary, axisymmetric flows in the ideal MHD limit, but add a radiation field with a prescribed source radius that can have arbitrary size with respect to the light cylinder. The inertia of the magnetic field dominates that of the matter to which it is tied, so that the outflow can achieve relativistic speeds. The magnetofluid is accelerated by two mechanisms: the Lorentz force (which operates in cold MHD winds); and scattering off the radiation field (which operates in thermal fireballs).

In the optically thin regime, the radiation field is self-collimating, so that a relativistically moving frame can be defined in which the radiation exerts a vanishing net force on the matter. Radiation pressure dominates matter pressure, and with the possible exception of a small region close to the engine, the radiation temperature lies far below the rest energy of the advected particles. The entrained matter can therefore be assumed cold outside the transparency surface, and scattering operates in the Thomson limit. In this first paper, we follow Goldreich & Julian (1970) in restricting the poloidal magnetic field to a monopolar geometry. This limits the efficiency of MHD acceleration (the fast critical point of the cold MHD flow sits at infinity). A much stronger Lorentz force can arise through differential decollimation of magnetic flux surfaces (e.g., Komissarov et al. 2007; Tchekhovskoy et al. 2009), but even then some parts of a cold MHD outflow may not have the requisite geometry. A companion paper considers jet geometries and allows for differential decollimation of the jet with respect to the radiation field (Russo & Thompson 2013, Paper II).

The efficiency of radiative acceleration outside the fast magnetosonic point was noted by Thompson (2006) in the context of GRBs. Previous work by Li et al. (1992) and Beskin et al. (2004) considered the interaction of a relativistic MHD outflow with a quasi-isotropic radiation field. In a first approximation, the outflowing matter feels a net drag force. Beskin et al. (2004) focused on a slightly decollimating outflow and analyzed the influence of radiation drag in integral form through the changes imparted to the energy and angular momentum. They considered changes in the shape of the magnetic surfaces, and therefore in the MHD acceleration rate, imparted by the radiation field, but the first-order effect addressed in this paper—the outward acceleration due to the strong radial anisotropy of the radiation field—was absent in their calculations.

We focus on time-independent outflows, because acceleration by radiation pressure becomes important before effects associated with the radial structure of the flow. The acceleration of a magnetized slab in planar geometry was studied numerically by Granot et al. (2011). The slab is initially static; its mean Lorentz factor quickly reaches $\sigma ^{1/3}_0$, where σ₀ is the initial magnetization, and then continues to grow. This effect may be relevant to GRBs after a jet has made a transition to a planar geometry at a distance r ≳ cΔt ∼ 3 × 10¹¹(Δt/10 s) cm, where Δt is the duration of the prompt phase. If the shell is already moving relativistically, only the outermost fraction ∼1/2Γ² of the shell will experience this type of acceleration. The interaction of this thin layer with an external plasma shell, which absorbs momentum from the magnetic field (e.g., Thompson 2006), must also be taken into account. Typically the bulk of the shell receives momentum from the radiation field before it comes into causal contact.

We do not address how the outflow might accelerate below its scattering photosphere. For example, periodic reversals of a toroidal field might induce magnetic reconnection, break the degeneracy between magnetic pressure gradient and curvature forces, and thereby induce a net outward acceleration (Drenkhahn & Spruit 2002). The high efficiency claimed for this mechanism depends on neglecting the input of enthalpy, and therefore inertia, into the magnetofluid as the magnetic field reconnects. When an enthalpy source term is included, one only obtains relativistic bulk motion if a high fraction of the unsigned magnetic flux is removed by reconnection. This requires, at a minimum, that the net (signed) magnetic flux carried by the outflow is small compared with the unsigned flux, and that the non-radial magnetic field maintains a strictly uniform (e.g., toroidal) direction across current sheets. Our focus here and in Paper II is therefore on ideal MHD effects.

The second focus of this paper is on the non-thermal spectrum of photons that are scattered by the outflowing matter. We ignore any effects of internal dissipation in the outflow, and focus purely on the spectral signature of bulk relativistic motion. We find a broad and flat extension of the seed spectrum when the outflow is rapidly accelerated by the Lorentz force, so that the seed photon beam is wider than the Lorentz cone of the MHD outflow. The smoothness of the scattered spectrum—in particular, the presence or absence of a residual bump at the seed thermal peak—is shown to depend on the net optical depth that is seen by the broader, unscattered photon beam.

Existing calculations of non-thermal "photospheric" emission from GRB outflows (Giannios 2006; Beloborodov 2011) generally assume that the Lorentz factor of the outflow has saturated at the transparency surface, as do calculations of the low-frequency spectral tail of thermal photospheres observed at oblique angles (e.g., Pe'er 2008; Lazzati et al. 2011). We broaden this approach by noting that locking between the advected thermal photons and the relativistic outflow will be broken in the presence of a somewhat slower component of the outflow. For example, material with a Lorentz factor ∼3–5 times smaller than the magnetofluid would be present as the result of the interaction of a jet with a star (Thompson 2006). In a neutron star binary merger, a relativistic magnetofluid could interact with a sub-relativistic, neutron-rich wind (Bucciantini et al. 2012). The slower material scatters advected X-ray photons into a broader beam, which continues to interact strongly with the faster magnetofluid even at low optical depth. It has long been realized that relativistic material moving into an isotropic bath of very low-frequency (optical-UV) photons will upscatter them as it loses energy to Compton drag (e.g., Sikora et al. 1994; Ghisellini et al. 2000). In that case a spectral slope F_ν ∼ ν^−1/2 is generated by the decaying peak energy of the upscattered photons—a different effect from that considered here, and a spectrum somewhat softer than that observed in the low-energy tails of GRBs.

The plan of the paper is as follows. Section 2 explains the efficiency of radiation-driven acceleration beyond the fast magnetosonic point, and explains further our critique of acceleration by magnetic reconnection. The relativistic wind equations including radiation pressure are described in Section 3, and in Section 4 they are specialized to a monopole magnetic field. The flow properties near the fast critical point are analyzed and the numerical method described. Numerical results are presented in Section 5. The spectrum of scattered photons is calculated in Section 6 by a Monte Carlo method, and the low- and high-frequency components of the spectrum discussed in the context of GRBs. Our results are summarized in Section 7, and the effect of rotation in the photon source is briefly discussed in the Appendix.

An essential feature of a steady magnetohydrodynamic (MHD) flow is that the equation for the fluid speed, obtained by combining the continuity and momentum equations, becomes singular where it matches a normal mode of the fluid. In the absence of significant matter pressure, the relevant normal modes are the Alfvén mode and the fast mode.

In a relativistic outflow, the inertia of the advected toroidal magnetic field rapidly becomes insignificant outside the fast critical point. Here we consider the case where the poloidal magnetic flux surfaces do not experience the differential bending needed for efficient MHD acceleration. Then, when the radiative force is strong enough to provide significant acceleration at the fast point, it also controls the terminal acceleration of the outflow, as if the magnetic field were not present. In spite of this, the Poynting flux carried by the magnetic field can continue to dominate the kinetic energy flux: the matter acts as a couple between the radiation field and the magnetic field (Thompson 2006).

To see this, focus on the zone far outside the Alfvén critical point. The solution to the induction equation is Ferraro (1937)

$$\begin{equation} \frac{B_{\phi }}{B_{r}} = \frac{v_{\phi } - \Omega _f r\sin \theta }{v_{r}}, \end{equation} \tag{ 8 }$$

where Ω_f is the rotational angular frequency of the magnetic footpoint at the boundary of the engine, and is constant along a magnetic flux surface. Equation (8) implies a uniform rate of transport of toroidal magnetic flux

$$\begin{equation} {\dot{\Phi }}_\phi = v_r r B_\phi \simeq -\Omega_f \sin \theta r^2 B_r = {\rm const}, \end{equation} \tag{ 9 }$$

since the toroidal velocity has decreased to v_ϕ ≪ v_r ≃ c. We now allow an external (e.g., radiation) force to be applied to the fluid while imposing this constraint, as well as the constancy of mass flux

$$\begin{equation} {d\dot{M}\over d\Omega } = \Gamma \rho v_r r^2 = {\rm const}. \end{equation} \tag{ 10 }$$

The change in the radial energy flux is, at a fixed radius,

$$\begin{eqnarray} \delta T_{tr} &=& \delta \left(\Gamma ^2 \rho c^2 v_r + {E_\theta B_\phi \over 4\pi } c\right) = {1\over r^2}\delta \left({d\dot{M}\over d\Omega }c^2 - {\dot{\Phi }}_\phi ^2{1\over v_r}\right)\simeq {\delta \Gamma \over r^2}\left({d\dot{M}\over d\Omega }c^2 - {{\dot{\Phi }}_\phi ^2\over \Gamma ^3 c}\right). \end{eqnarray} \tag{ 11 }$$

The right-hand side here vanishes at the fast point, where

$$\begin{equation} \Gamma ^3 = \Gamma _c^3 = {{\dot{\Phi }}_\phi ^2\over c^3 d\dot{M}/d\Omega } = \sigma. \end{equation} \tag{ 12 }$$

The (toroidal) magnetic field dominates the inertia of the outflow in between the Alfvén and fast points, but the small flux of ordinary matter dominates outside the fast point.

Our main focus is on flows that are sub-fast magnetosonic at the transparency surface, and are accelerated through the critical point by the radiation force as well as the Lorentz force. For our adopted field geometry, this occurs when Γ(r_τ) ≲ σ^1/3, and requires a strongly magnetized outflow.

A second type of flow is super-fast magnetosonic at the transparency surface, having already been accelerated through the critical point, Γ(r_τ) ≳ σ^1/3. The dominant acceleration mechanism outside the photosphere then depends on the poloidal field geometry. When the poloidal flux surfaces do not diverge from each other, the case studied in this paper, the Lorentz force freezes out and the remaining acceleration is by the radiation force. However, in a jet geometry there is a much more extended competition between MHD and radiation stresses outside the fast critical point, as we discuss in Paper II.

Since the Alfvén mode has the lower speed, its critical point sits closer to the rotating "engine." Inside the Alfvén critical point, the magnetic field lines are effectively rigid and guide the outflow of matter and radiation. Outside, the rotation of the engine causes the magnetic field lines to be bent back into a Parker spiral, and the magnetic field is mainly toroidal. After a choice is made for the poloidal field profile, we wish to test the insensitivity of the large-distance flow solution to details close to the Alfvén point. In fact, we obtain an essentially unique outer solution without explicitly requiring it to pass through the Alfvén point. If the solution were not unique in this way, it would be suspect: the magnetic field configuration near the Alfvén point maps directly onto the engine, and details of the inner flow solution depend, in turn, on details of plasma heating close to the engine.

The acceleration of a cold, spherical MHD outflow is inefficient due to a cancellation between the magnetic pressure gradient and curvature forces. Rather than considering deviations from radial flow, Drenkhahn & Spruit (2002) suggest that reconnection of a reversing toroidal magnetic field would convert Poynting flux to a large-scale relativistic expansion. Inside the scattering photosphere, they impose conservation of energy flux per steradian,

$$\begin{equation} {dL\over d\Omega } = r^2\Gamma ^2v_r \left[h + {(B_\phi /\Gamma)^2\over 4\pi }\right]. \end{equation} \tag{ 13 }$$

Here h ≃ ρc² is the bulk-frame enthalpy per unit volume, and B_ϕ/Γ the bulk-frame toroidal magnetic field. A sink term for the magnetic flux is imposed, c∂_r(rB_ϕv_r) = −rB_ϕv_r/τ, but no corresponding source of h. With these assumptions, the dissipation of Poynting flux must be compensated by a growth in kinetic energy. A small decrease in Poynting flux corresponds to a large increase in Γ in a cold, high-σ outflow.

More realistically, reconnection heats the particles (e.g., Giannios & Spruit 2005; Thompson 2006; McKinney & Uzdensky 2012). In a high-σ outflow there is a large increase in the particle inertia. Even allowing for rapid radiative cooling, the photons are trapped by the outflow, and

$$\begin{equation} \delta h \sim - \delta \left[{(B_\phi /\Gamma)^2\over 8\pi }\right]. \end{equation} \tag{ 14 }$$

Reconnection also creates a radial magnetic field, through the appearance of multiple magnetic X-points. The inertia of this small-scale magnetic field also dominates the particle rest mass. Taking both of these effects into account, one sees that dissipation of even half the magnetic energy can generate only mildly relativistic bulk motion.

The radiation emanates from a spherical static "emission surface" of radius r_s (x = 1). We adopt simplified spectral and angular distributions: the unscattered radiation is monochromatic, uniform at angles 0 < θ < π/2 at the emission surface, and streams freely outward. At x > 1, the cone of the radiation field contracts and

$$\begin{eqnarray} &&I_\nu = I_0\nu _0\delta (\nu -\nu _0) \quad {\rm for}\; \theta < \theta _s \equiv \sin ^{-1}\left({r_s\over r}\right) = \sin ^{-1}\left({1\over x}\right); \quad I_\nu = 0\quad {\rm for}\; \theta > \theta _s. \end{eqnarray} \tag{ 22 }$$

This accurately represents an isotropically emitting star that is surrounded by an optically thin wind. It still produces qualitatively correct results if the outflow is optically thick near the engine, and experiences nearly linear growth of Lorentz factor with radius, driven by radiation pressure (Section 3.2).

Because ω ≫ 1 typically, a non-rotating emission surface is a reasonable approximation. The effect of adding modest rotation to the photon source is addressed quantitatively in the Appendix, and the corrections to the flow solution are shown to be small.

The radiation interacts with electrons (and positrons) via Thomson scattering, and the radiative force is taken to be unperturbed by this interaction. The radiation force per scattering charge in a global inertial ("lab") frame is related to the force ${\bf F}^{\rm rad\,^{\prime }}$ in the flow rest frame by

$$\begin{equation} {\bf F}^{\rm rad} = \left[{\bf F}^{\rm rad\,\prime } - ({\bf F}^{\rm rad\,\prime }\cdot {\boldsymbol {\beta }})\frac{\boldsymbol {\beta }}{\beta ^{2}}\right]\Gamma ^{-1} + ({\bf F}^{\rm rad\,\prime }\cdot {\boldsymbol {\beta }})\frac{\boldsymbol {\beta }}{\beta ^{2}}, \end{equation} \tag{ 23 }$$

where primes denote quantities in the frame comoving with the fluid. Letting $\hat{k}$ be the unit wave vector of the incoming photon, we have the transformations

$$\begin{eqnarray} &&I_{\nu }^{\prime } = I_{\nu }\Gamma ^{3}(1-{\boldsymbol {\beta }}\cdot \hat{k})^{3};\quad d\nu ^{\prime } = d\nu \Gamma (1-{\boldsymbol {\beta }}\cdot \hat{k});\quad d\Omega ^{\prime } = \frac{d\Omega }{\Gamma ^{2}(1-{\boldsymbol {\beta }}\cdot \hat{k})^{2}};\quad \hat{k}^{\prime } = \frac{\hat{k} + (\Gamma -1)(\hat{\beta }\cdot \hat{k})\hat{\beta } - {\boldsymbol {\beta }}\Gamma }{\Gamma (1-{\boldsymbol {\beta }}\cdot \hat{k})}.\nonumber\\ \end{eqnarray} \tag{ 24 }$$

The radiation force in the fluid frame is given by

$$\begin{eqnarray} &&{\bf F}^{\rm rad\,\prime }\equiv \frac{d{\bf p}^{\prime }}{dt^{\prime }} = \frac{\sigma _{T}}{c}\int I_{\nu }^{\prime }\hat{k}^{\prime }d\Omega ^{\prime }d\nu ^\prime= \frac{\sigma _{T}I_{0}\nu _0}{c}\int \Gamma ^{2}(1-{\boldsymbol {\beta }}\cdot \hat{k}) \left[\hat{k}\Gamma ^{-1} + (1-\Gamma ^{-1})\left(\frac{{\boldsymbol {\beta }}\cdot \hat{k}}{\beta ^{2}}\right){\boldsymbol {\beta }}-{\boldsymbol {\beta }}\right]d\Omega, \end{eqnarray} \tag{ 25 }$$

and combining with Equation (23) gives

$$\begin{equation} {\bf F}^{\rm rad} = \frac{\sigma _{T}I_{0}\nu _0}{c}\int (1-{\boldsymbol {\beta }}\cdot \hat{k})[\hat{k} -{\boldsymbol {\beta }}\Gamma ^{2}(1 - {\boldsymbol {\beta }}\cdot \hat{k})]d\Omega. \end{equation} \tag{ 26 }$$

The radiation force (26) can be evaluated analytically by integrating over the emission surface. Defining dimensionless functions R, P, by

$$\begin{equation} F^{\rm rad}_r = \chi _s {\bar{m} c^2\over r_s} R(r,\Gamma);\; F^{\rm rad}_\phi = \chi _s {\bar{m} c^2\over r_s} P(r,\Gamma), \end{equation} \tag{ 27 }$$

one finds

$$\begin{eqnarray} &&\hspace{-1.5pc}R = -\frac{8}{3}u\Gamma + \frac{1+2u^{2}}{x^{2}} + \beta _{r}(2\Gamma ^{2}+v^{2})\sqrt{1-\frac{1}{x^{2}}} + \beta _{r}\left(\frac{2}{3}\Gamma ^{2} - v^{2}\right)\left(1-\frac{1}{x^{2}}\right)^{3/2}; \end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray} &&P= -\beta _{\phi }\left[\frac{8}{3}\Gamma ^{2}-\frac{2u\Gamma }{x^{2}} - (1+2\Gamma ^{2}+v^{2})\sqrt{1-\frac{1}{x^{2}}}+\frac{1}{3}(1-2u^{2}+v^{2})\left(1-\frac{1}{x^{2}}\right)^{3/2}\right], \end{eqnarray} \tag{ 29 }$$

where the radial and non-radial four-velocities are

$$\begin{equation} (u,v)\equiv (\Gamma \beta _r,\Gamma \beta _{\phi }). \end{equation} \tag{ 30 }$$

At large radius and Lorentz factor, expressions (28) and (29) simplify to

$$\begin{equation} R\simeq \frac{1}{4x^2\Gamma ^2} - \frac{\Gamma ^2}{12x^6}; \quad P\simeq -\frac{\beta _{\phi }}{2x^2}\left(\frac{1}{2\Gamma ^2} + \frac{1}{x^2} + \frac{\Gamma ^2}{6x^4}\right). \end{equation} \tag{ 31 }$$

The first term in the radial force is due to photons propagating nearly parallel to the fluid; the second is the drag caused by photons which are aberrated into the anti-radial direction by the relativistic particle motion. The photon field seen by the particle is nearly isotropic in a frame were R = 0. At large χ_s and x not too large, the matter therefore tends toward the equilibrium Lorentz factor

$$\begin{equation} \Gamma _{\rm eq}\simeq 3^{1/4}\left({r\over r_s}\right) = 3^{1/4}x. \end{equation} \tag{ 32 }$$

This frame only exists due to the extended nature of the source.

In many cases, the outflow moves relativistically at its scattering photosphere, so that the radiation field is already collimated at the base of the transparent zone. Then the emission surface becomes a virtual one. Setting the photospheric Lorentz factor to the equilibrium value (32), the emission surface is pushed inward to a radius

$$\begin{equation} r_{s,\rm eff} = 3^{1/4}\,{r_\tau \over \Gamma (r_\tau)}. \end{equation} \tag{ 33 }$$

A radiation field that emerges from a relativistically moving photosphere is not cut off sharply at an angle θ_s = r_{s, eff}/r_τ = 3^1/4/Γ(r_τ), but has a somewhat smoother cutoff. Consider, instead, an intensity that is isotropic in the matter rest frame, $I^{\prime }_{\nu ^{\prime }} = I_0^R\nu _0^R \delta (\nu ^{\prime }-\nu _0^R)$, and compare with the top-hat spectral distribution (22) by normalizing to a fixed flux in the frame of the engine, F = 2π∫dμ∫dνI_ν(μ). Then the frequency-integrated intensity in the lab frame is

$$\begin{eqnarray} &&I(\theta) = \int d\nu I_\nu (\theta) = {I_0^R\nu _0^R\over [\Gamma (1-\beta \mu)]^4} \simeq {I(\theta =0)\over (1 + \Gamma ^2\theta ^2)^4}= {3^{3/2}\over \left(1 + \sqrt{3}\theta ^2/\theta _s^2\right)^4} {F\over \pi \theta _s^2} \quad \left(\Gamma = \Gamma _{\rm eq} = {3^{1/4}\over \theta _s}\right),\qquad \end{eqnarray} \tag{ 34 }$$

as compared with $I = F/\pi \theta _s^2$ for θ < θ_s.

These two angular distributions yield the same photon force for a static charge at r ≫ r_{s, eff} and, by construction, the same Lorentz factor for which the radiation force vanishes. The power with which an electron scatters photons in the bulk frame is nearly identical as well: $P^{\prime } = 4\pi \int d\nu ^{\prime } \sigma _T I^{\prime }_{\nu ^{\prime }} = 3\sigma _TF/4\Gamma ^2$ for the isotropic bulk-frame intensity, and

$$\begin{eqnarray} &&P^{\prime } = \left(1 + \sqrt{3}{\Gamma ^2\over \Gamma _{\rm eq}^2} + {\Gamma ^4\over \Gamma _{\rm eq}^4}\right){\sigma _TF\over 4\Gamma ^2} = \left({1\over 2} + {\sqrt{3}\over 4}\right){\sigma _TF\over \Gamma _{\rm eq}^2} \quad(\Gamma = \Gamma _{\rm eq}) \end{eqnarray} \tag{ 35 }$$

for the top-hat distribution. The spectrum of photons scattered by a cold electron flow is proportional to P', and differs in normalization by ∼20% in the two cases.

More generally, if the outflow interacts with slower relativistic material—as discussed by Thompson (2006) in the context of GRBs—then the beam of photospheric photons is scattered into a cone of width >1/Γ. This motivates our considering outflows with (1) significant scattering optical depth at r = r_s; but (2) Γ(r_s) > 1. Given the potential importance of such additional effects for GRBs, we stick with the simplest model of a photon emission surface with uniform intensity, with the understanding that this is a virtual surface in some circumstances. See Section 6 for calculations of the scattered photon spectrum and the application to GRBs.

As a test of our formalism, we revisit the acceleration of cold matter by an intense radiation field, setting σ = 0. Then Equations (15), (27), and (28) combine to give

$$\begin{eqnarray} &&{d\Gamma \over dx} = \chi _s\left(R + \frac{v}{u}P\right);\quad {d\beta _{\phi }\over dx} = -\frac{\beta _{\phi }}{x} - \chi _s\left[\frac{\beta _{\phi }}{\Gamma }R - \frac{\left(1-\beta _{\phi }^{2}\right)}{u}P\right], \end{eqnarray} \tag{ 36 }$$

and simplify further to

$$\begin{eqnarray} &&{d\Gamma \over dx} \simeq \chi _s R\simeq \frac{\chi _s}{4x^2}\left(\frac{1}{\Gamma ^2} - \frac{\Gamma ^2}{3x^4}\right); \quad {d\beta _{\phi }\over dx} \simeq -\frac{\beta _{\phi }}{x} - \frac{\chi _s\beta _{\phi }}{2x^2\Gamma }\left(\frac{1}{\Gamma ^2}+\frac{1}{x^2}\right) \end{eqnarray} \tag{ 37 }$$

for large x, Γ. The matter Lorentz factor tracks Γ_eq until the local compactness χ ∼ χ_s/Γ³x drops below ∼1 (see Figure 1). The asymptotic Lorentz factor is

$$\begin{equation} \Gamma _{\infty }\simeq 1.086\chi _s^{1/4} \end{equation} \tag{ 38 }$$

as long as baryon loading is sufficiently low ($\eta \equiv L_\gamma /\dot{M}c^2 \gg \chi _s^{1/4}$); otherwise Γ_∞ ≃ η.

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The radiation force has a stronger effect when the flow remains optically thick out to a large distance from the engine, and develops at least a moderate Lorentz factor before the photons begin to stream freely. Returning to the case of a magnetized outflow, the constraint L_γ < L_P would correspond to a limit on the photon compactness as measured at the photosphere,

$$\begin{equation} \chi (r_\tau) \equiv {L_\gamma \sigma _T\over 4\pi r_\tau m_ec^3} = {r_s\over r_\tau } \chi _s, \end{equation} \tag{ 39 }$$

namely,

$$\begin{equation} \chi (r_\tau) \lesssim 6\Gamma ^2(r_\tau) \sigma \end{equation} \tag{ 40 }$$

(see Equation (20)). Now the parameter χ_s describes a "virtual" photon source that is buried at large scattering depth, and is already collimated at the photosphere. The terminal Lorentz factor increases to

$$\begin{equation} \Gamma _{\infty,\chi } \;\sim \; \chi _s^{1/4} = [3^{-1/4}\,\Gamma (r_\tau)\,\chi (r_\tau)]^{1/4}. \end{equation} \tag{ 41 }$$

If the fast point were to sit in the transparent zone, then Γ(r_τ) ≲ σ^1/3 and one has the bound

$$\begin{equation} \chi _\tau \lesssim 6\sigma ^{5/3};\quad \Gamma _{\infty,\chi } \lesssim 1.3\sigma ^{1/2}. \end{equation} \tag{ 42 }$$

The terminal Lorentz factor is higher if the outflow passes through the fast critical point before reaching its photosphere. The radiation field can drive significant supplemental acceleration outside the photosphere for any value of Γ(r_τ) below σ.

These MHD wind equations have critical points where the flow matches the speed of a cold MHD mode. The Alfvén critical point corresponds to a flow speed u = σ(1 − ω²x²)/ω²x², and therefore sits inside the light cylinder.

Since we are looking for robust flow solutions which do not depend on the magnetic field structure in this inner zone (Section 2.2), our focus is on the fast critical point, where

$$\begin{equation} u = \frac{\sigma }{x^2\omega ^2}\left(1-\omega ^2x^2+\frac{B_{\phi }^2}{B_r^2}\right). \end{equation} \tag{ 50 }$$

This is the singularity appearing in Equation (46). A regular flow solution passing through this point must satisfy two conditions:

$$\begin{equation} \Gamma ^{\prime }_{\sigma }(x_c) + \Gamma ^{\prime }_{\chi }(x_c) + \Gamma ^{\prime }_{\sigma \chi }(x_c) = 0; \quad \mu _{\rm eff}(x_c) = 0. \end{equation} \tag{ 51 }$$

Here x_c is the (so-far undetermined) radius of the fast point. Equations (51) generate a one parameter family of flow solutions.

The specific angular momentum evolves according to the simple equation $d\mathcal {L}/dx = (x/\beta _r)\chi _s P$. We have tested Equations (44) and (45) by combining them and re-deriving this equation. This means that the critical point of the equation for dβ_ϕ/dx does not impose independent constraints on the flow solution: a solution with a smooth Γ(x) profile automatically satisfies the $\beta ^{\prime }_{\phi,\sigma }(x_c) + \beta ^{\prime }_{\phi,\chi }(x_c) + \beta ^{\prime }_{\phi,\sigma \chi }(x_c) = 0$.

The presence of a cross term in the wind equations, involving both the magnetization σ and the compactness χ_s, deserves some comment. Inside the fast magnetosonic point, and outside the Alfvén point, the radial motion of the fluid has effectively a negative inertia: a positive external radial force extracts energy from the outflow. This negative inertia arises from the response of B_ϕ, which dominates the energy integral (21), to changes in the radial flow speed. The scaling $B_\phi \sim \beta _r^{-1}$ implies a decrease in toroidal field energy with increasing β_r. On the other hand, the angular momentum (21) is also dominated by the electromagnetic field. A change in B_ϕ creates unbalanced toroidal stresses, which are a source for β_ϕ, and are proportional to σχ_s. Changes in β_ϕ and β_r of opposing signs allow B_ϕ to remain nearly constant. The net change in Lorentz factor is positive, δΓ = Γ⁻³(β_rδβ_r + β_ϕδβ_ϕ) > 0, if 1 + β_ϕB_ϕ/β_rB_r > 0. Close to the light cylinder, where the term ∼ − σ/ux²ω² in μ_eff dominates the term −σ/u³, the effective inertia has the usual sign.

When radiation fields are absent and Γ ≫ 1 our equations reduce to the cold limit of the system studied by Goldreich & Julian (1970). The angular momentum ${\cal L}$ (Equation (21)) is conserved, and determines

$$\begin{equation} \beta _{\phi } = \frac{x\omega (\mathcal {L}\omega \beta _r - \sigma)}{x^{2}\omega ^{2}u - \sigma }. \end{equation} \tag{ 52 }$$

The fast point lies at infinite radius, the Lorentz factor being limited to its critical value

$$\begin{equation} \Gamma _{\infty,\sigma } = \sqrt{1 + \sigma ^{2/3}} \simeq \sigma ^{1/3}. \end{equation} \tag{ 53 }$$

The unique solution passing through the Alfvén and fast critical points is the minimum-energy solution found by Michel (1969). The slow acceleration of radial flows is an artifact of the near perfect cancellation of the outward magnetic pressure gradient force and the inward curvature force. Faster acceleration is possible through a faster-than-spherical divergence of the outflow (Begelman & Li 1994), or differential bending of the poloidal field lines (Komissarov et al. 2007; Tchekhovskoy et al. 2009), the effect of which we examine in Paper II.

For small but finite χ_s, the Lorentz factor at the fast point remains unchanged from the pure MHD solution, Γ_c ≃ σ^1/3, but the critical point is brought in to a finite radius. The pure magnetocentrifugal and radiation terms in the wind equations dominate at large x, Γ, and simplify to

$$\begin{eqnarray} &&{\Gamma^{\prime}_{\sigma}} \simeq -\frac{v\sigma }{ux^2\omega }\left(2 - \frac{\beta _{\phi }x\omega }{\beta _r^2}\right) \simeq -\frac{v\sigma }{ux^2\omega }; \quad \Gamma ^{\prime }_{\chi } \simeq \chi _s R\simeq \frac{\chi _s }{4x^2\Gamma ^2}. \end{eqnarray} \tag{ 54 }$$

Applying the regularity condition $\Gamma ^{\prime }_{\sigma }+\Gamma ^{\prime }_{\chi }\simeq 0$ at the critical point gives

$$\begin{eqnarray} x_{c}\simeq \frac{4\sigma ^{5/3}}{\chi _s \omega ^2}; \quad \beta _{\phi,c} \simeq \frac{1}{x_c\omega } \simeq \frac{\chi _s \omega }{4\sigma ^{5/3}}; \quad \mathcal {L}_{c} &\simeq& \Gamma _cx_c\beta _{\phi,c} + \frac{\sigma }{\omega \beta _{r,c}} \simeq {\sigma ^{1/3}\over \omega }(1+\sigma ^{2/3}) \quad ({\rm low}\, \chi _s). \end{eqnarray} \tag{ 55 }$$

For very large χ_s the fluid is locked to the radiation field while crossing the fast point at a finite radius,

$$\begin{equation} \Gamma _c \simeq \Gamma _{\rm eq} \,{\Rightarrow}\, x_c \simeq 3^{-1/4}\Gamma _c\quad \quad ({\rm high}\, \chi _s). \end{equation} \tag{ 56 }$$

Here B_r need not be small compared with B_ϕ. Approximating β_ϕ ≪ xω and making use of Equation (8), Equation (50) becomes

$$\begin{equation} u^{3}\simeq \sigma \left(1+\Gamma ^{2}\frac{B_{r}^{2}}{B_{\phi }^{2}}\right) \simeq \frac{B_{\phi }^{2}+\Gamma ^{2}B_{r}^{2}}{4\pi u\rho c^{2}}. \end{equation} \tag{ 57 }$$

Substituting u_c ≃ Γ_c ≃ 3^1/4x_c gives the critical point parameters

$$\begin{equation} \Gamma _{c}\simeq \sigma ^{1/3}\left(1+\frac{\sqrt{3}}{\omega ^2}\right)^{1/3};\quad \mathcal {L}_c \simeq \frac{\sigma }{\omega }\left(1 + \frac{1}{2}\Gamma _{c}^{-2}\right) \quad ({\rm high}\, \chi _s). \end{equation} \tag{ 58 }$$

The transition between the low- and high-χ_s scalings for the fast point occurs where $x_{c,\rm low}\simeq x_{c,\rm high}$, at a compactness χ_s ≃ 4σ^4/3ω⁻².

The singularity at the fast point is difficult to handle with standard integration methods, especially when the wind equations become stiff, as they do when χ_s ≫ 1. We employ two different integration schemes: a shooting method for low χ_s, and a relaxation method for high χ_s. The position x_c of the critical point is unknown a priori, since the regularity condition provides only one constraint on the two coupled ordinary differential equations (ODEs). Once a candidate value of x_c is chosen, the flow variables at the critical point are uniquely determined, as is the flow solution interior to it. However, this interior solution generally diverges at small radius. A strong divergence is avoided only for a narrow range of x_c, and even then the solution tends to develop sharp gradients in Γ and β_ϕ. Avoiding such gradients leads to an essentially unique choice of x_c and a robust flow solution. We have found that these smooth solutions have the property that the terms in dβ_ϕ/dx which depend explicitly on the radiation field should sum to zero at x = 1,

$$\begin{equation} (\beta ^{\prime }_{\phi,\chi }+\beta ^{\prime }_{\phi,\sigma \chi })_{x=1}=0. \end{equation} \tag{ 59 }$$

This recovers Michel's minimum-energy solution (Michel 1969) in the limit of vanishing radiation field, χ_s → 0. The resulting initial values of Γ and β_ϕ are plotted in Figure 2 for σ = 10³ and ω = 2.

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At low χ_s, the flow solution interior to x_c is obtained by first determining the locus of critical points, and the corresponding values of Γ(x_c), β_ϕ(x_c). We then shoot inward from x = x_c using a fifth-order Runge–Kutta algorithm with adaptive step size (see Sections 7.3, 7.5 of Kiusalaas 2010). The value of ${\cal L}_c$—which, unlike x_c, is single valued—is iterated until the required small-x behavior is obtained. This method fails when χ_s ≳ 100, since the equations become extremely stiff near x = 1, and machine precision becomes inadequate to distinguish values of {x_c, Γ(x_c), β_ϕ(x_c)} that lead to converging and diverging solutions.

At high χ_s, we use the relaxation method described in London & Flannery (1982) for transonic hydrodynamic flows. The wind equations are replaced by finite-difference equations on a grid. Starting with a simple trial solution, and an initial guess for x_c, the inner boundary condition (59) is applied along with the regularity condition (51) at the critical point. Since the location of the critical point is unknown, an independent variable q which labels mesh points is introduced, along with a mesh-spacing function Q(x) = Ψq (where Ψ is an unknown number). We choose Q(x)∝ln x, which tightly packs the grid points near x = 1 (where the equations are stiff), and spreads them out near the critical point (so as to avoid divergences induced by the singularity). This necessitates adding two more ODEs, for x(q) and Ψ. The error in the initial guess at grid point i is quantified in terms of E_i = y_i − y_{i − 1} − (x_i − x_{i − 1})dy/dx and a new solution is obtained by a multi-dimensional Newton–Raphson method (see Section 18.3 of Press et al. 2007). The matrix inversion is performed using a Gaussian pivoting algorithm detailed in Kiusalaas (2010). The critical point must be approached from below with this method and so the trial solution is typically cut off at ∼90% of the expected critical radius. This method fails in the low χ_s regime since the critical point is at large radius where dΓ/dx ≃ 0, and even a slight overshoot in Γ_c will be catastrophic.

In both the low and high χ_s regimes, the solution is completed by shooting outward from the critical point.

In a hot magnetized outflow, the fluid acts to couple the radiation and magnetic field, allowing energy to be exchanged between them as the flow accelerates. To study this exchange we write the total energy in terms of kinetic, Poynting and radiation luminosities as

$$\begin{equation} \frac{L_{K}+L_{P}+L_{\gamma }}{\dot{M}c^{2}} = \Gamma - \frac{1}{x\omega \sin \theta }\frac{B_{\phi }}{B_{r}}\sigma + \frac{x\chi (x)}{6\tau _{{\rm es,s}}\Gamma _{s}^{2}} \end{equation} \tag{ 60 }$$

where we have taken Γ∝r in calculating the optical depth. For simplicity, our solutions (being calculated in the optically thin regime) assume a constant radiation flux and thus do not strictly conserve energy. Nevertheless, we can impose conservation using the obtained profiles for the kinetic and Poynting fluxes to study the qualitative features of the energy exchange. The results are shown in Figure 6 for various values of photon compactness while taking the optical depth at r = r_s to be unity. The radiation luminosity generally increases sharply at small radius where photons are upscattered by the highly relativistic flow. This must be accompanied by a corresponding decrease in the magnetic energy since the Lorentz force is still accelerating the flow. To see why this occurs note that well outside the light cylinder the Poynting luminosity in a monopolar outflow is $L_P\sim \beta _r^{-1}\sigma \dot{M}c^2$ which can decrease significantly only if the flow is not yet extremely relativistic at the photosphere. In this case field lines are coiled tight and transfer their magnetic energy to the radiation field as they unwind. The matter acts mainly as a catalyst in this process, with relatively small changes in kinetic energy.

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Outside the fast point the Poynting flux is essentially constant and further changes in the kinetic energy come directly from the interaction with the photon field. (In Figure 6 the positive/negative slopes of the energy profiles are plotted as dotted/dashed the corresponding colors.) When χ_s is high, the fast point marks a sharp transition in the energy exchange: conversion of magnetic to radiation energy inside; conversion of radiation to kinetic energy outside. Accounting for the increase in photon luminosity at small radius would enhance the acceleration outside the fast point.

We briefly consider the application of our results to GRBs. When a jet emerges from a Wolf–Rayet star, or from a cloud of neutron-rich debris, much of the jet material may experience a strong outward Lorentz force (Tchekhovskoy et al. 2010). Even in that circumstance, the radiation pressure force can enhance or impede the acceleration of the jet material outside its photosphere (Paper II). Other parts of the jet may have unfavorably curved flux surfaces and feel a weaker Lorentz force, more typical of the poloidal field geometry examined in this paper.

Although our solutions are found in the zone exterior to a static, photon-emitting surface, the photon field that emerges from a displaced photosphere will have a similar effect on the outflow exterior to it, if we define an effective source radius r_{s, eff} as in Section 3.2. A high photon intensity also has the effect of driving the flow profile to a linear relation Γ(r)∝r inside the fast critical point. In that case, the flow profile does not depend on whether the outflow is optically thick or thin inside the critical point. The conditions for this to be the case are outlined at the end of Section 3.3.

To fix some numbers, consider a gamma-ray outflow with an (isotropic) luminosity 4πdL_γ/dΩ = 10⁵¹ L₅₁ erg s⁻¹ and a photosphere of radius r_τ = 10¹⁰r_{τ, 10} cm. The corresponding photospheric compactness (1) is $\chi (r_\tau) = 1\times 10^8 L_{51}r_{\tau,10}^{-1}(m_p/\bar{m})$. Transparency at r ≳ r_τ is guaranteed if Γ(r_τ) ≳ 10, and pairs have mostly annihilated. The effective source radius, obtained by equating the bulk Lorentz factor (32) of the radiation field with Γ(r_τ), is r_{s, eff} ∼ r_τ/Γ(r_τ) < 10⁹r_{τ, 10} cm. The compactness scaled to this radius, which determines the amplitude of the radiation force in the wind equations (46)–(49), is $\chi _s \gtrsim 10^9L_{51}r_{\tau,10}^{-1}[\Gamma (r_\tau)/10](m_p/\bar{m})$.

Since we are considering a strictly monopolar poloidal magnetic field, the asymptotic Lorentz factor is limited to Γ_{∞, σ} ≃ σ^1/3 at small radiation compactness. Higher Lorentz factors are possible at large χ_s, where acceleration is dominated by radiation pressure. First consider the case where the outflow is fully transparent at r ∼ r_s, and moves transrelativistically near the inner boundary. Then $\Gamma _{\infty,\chi } \simeq 1.086\chi _s^{1/4}$ (Section 3.3). In the example at the end of the preceding section, this corresponds to Γ_{∞, χ} ∼ 200.

We find a smooth transition between these two limits as shown in Figure 7 for σ = 10³ and ω = 2, 10. This is well described by the function

$$\begin{equation} \Gamma _\infty ^n = \Gamma _{\infty,\sigma }^n + \Gamma _{\infty,\chi }^n, \end{equation} \tag{ 61 }$$

with n ≃ 2.5. In this situation, strong radiative acceleration requires a photon source that is not sourced internally by the outflow. Otherwise, L_γ ≲ L_P and χ_s is smaller than (40).

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We are interested in the power radiated from a steady flow, with a fixed particle density at a given radius. In the case of an impulsive event such as a GRB, of duration Δt, the flow is effectively steady near the transparency surface if Γ²(r_τ) ≫ r_τ/cΔt. In this situation, there is a weak correction between the time coordinate of an observer sitting at a large (but non-cosmological) distance from the engine, and the time coordinate t of the engine rest frame.

We start with a monochromatic photon source I_ν = I₀ν₀δ(ν − ν₀), and the top-hat angular distribution (22) corresponding to a (virtual) emission radius r_s. More general source spectra are then considered by a convolution. The condition of elastic (Thomson) scattering is

$$\begin{equation} \tilde{\nu }_{\rm em}\equiv {\nu _{\rm em}\over \nu _0} \;=\; {1-\beta \mu \over 1-\beta \mu _{\rm em}} \;=\; {1+\beta \mu _{\rm em}^{\prime }\over 1+\beta \mu ^{\prime }}. \end{equation} \tag{ 62 }$$

The direction cosines of the incident and emitted photons are measured with respect to the radial direction (we neglect any small non-radial motion of the matter), $\mu = \hat{k}\cdot \hat{r}$, $\mu _{\rm em} = \hat{k}_{\rm em}\cdot \hat{r}$. They take the range

$$\begin{equation} \sqrt{1-{1\over x^2}} \le \mu \le 1; \quad -1 \le \mu _{\rm em} \le 1. \end{equation} \tag{ 63 }$$

Primed quantities are measured in the rest frame of a scattering charge.

A scattered photon reaches the maximum frequency

$$\begin{equation} \nu _{\rm em,max} = {1-\beta \mu _{\rm min}\over 1-\beta }\nu _0 \simeq \left(1+{\Gamma ^2\over x^2}\right)\nu _0 \quad (x,\Gamma \gg 1). \end{equation} \tag{ 64 }$$

Hence the scattered spectrum develops a significant tail at frequencies above ν₀ if the outflow has a Lorentz factor Γ ≳ Γ_eq (Equation (32)). This non-thermal tail is more pronounced at lower values of the compactness χ_s, where the Lorentz force dominates the acceleration of the flow. The scattered spectrum also extends to a low frequency ν_{em, min} = [(1 − β)/(1 + β)]ν₀ ∼ ν₀/4Γ². Sample profiles of ν_{em, max} and ν_{em, min} are given in Figure 8, and the asymmetry of the rest-frame photon flux in Figure 9.

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The frequency ν_em of the emitted photon is hardest in the flow direction. Softer photons are emitted off this axis, and also see a larger optical depth to scattering. In a spherically symmetric outflow, a photon emitted at radius r sees a scattering depth

$$\begin{eqnarray} &&\tau _{\rm es}(r,\mu _{\rm em}) = {\sigma _T\over c}{d\dot{N}\over d\Omega }\ \int _r^\infty \left[1-\beta (r_2)\mu (r_2)\right]{dr_2\over \beta (r_2) \mu (r_2) {r_2}^2}. \end{eqnarray} \tag{ 65 }$$

The direction cosine evolves from the emission radius r to r₂ > r according to

$$\begin{equation} 1 - \mu (r_2)^2 = \left({r\over r_2}\right)^2\left(1-\mu _{\rm em}^2\right) \quad (r_2 > r). \end{equation} \tag{ 66 }$$

It is straightforward to calculate the emergent photon spectrum by a Monte Carlo method. The direction cosines of the input photons are drawn randomly from the uniform distribution (63) at x = 1. The optical depth Δτ_es to the first (next) scattering is determined by randomly picking $1-e^{-\Delta \tau _{\rm es}}$, followed by a step-by-step integration of τ_es along the ray. Scattering is performed in the rest frame of the cold flow, by transforming μ' = (μ − β)/(1 − βμ), picking rest frame scattering angles $\theta _s^{\prime }$, $\phi _s^{\prime }$ with respect to this axis, and then determining the direction cosine of the outgoing photon via $\mu _{\rm em}^{\prime } = \mu ^{\prime }\cos \theta _s^{\prime } + (1-\mu ^{\prime 2})^{1/2}\sin \theta _s^{\prime }\cos \phi _s^{\prime }$. The photon escapes if Δτ_es exceeds the total optical depth along the ray. Working in spherical symmetry, we record the frequency but not the direction of the outgoing photon.

To illustrate some of the main effects, we show in Figure 10 the spectrum resulting from an outflow with a simple linear profile Γ(r) = 5 (r/r_s), and various values of the scattering depth experienced by the most obliquely propagating photons. In this case, the bulk frame of the seed photons moves at a somewhat lower Lorentz factor than the magnetofluid, Γ_slow ∼ 0.2 Γ. The scattered spectrum therefore peaks well above the seed frequency, ν_{em, max} ∼ 30 ν₀ (Equation (64)), driven by bulk Comptonziation.

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First consider the output from a monochromatic source spectrum (the black lines in Figure 10). The output spectrum is fairly flat, F_ν ∼ ν^0.5, over a decade in frequency below the peak, steepening to F_ν ∼ ν at ν ∼ ν₀. The scattered photons see a small optical depth (suppressed by a factor ∼(Γ_slow/Γ)²) down to fairly low frequencies, where the slope approaches a Rayleigh–Jeans value.

As the optical depth of the seed photons is increased, the spectrum steepens slightly at high frequencies. The spectra in Figure (10) are labeled by the optical depth seen by seed photons at the maximum angle θ_s ∼ (r/r_s)⁻¹ and minimum direction cosine $\mu _{\rm min} \simeq 1-\theta _s^2/2$. Requiring that τ_es(r_s, μ_min) ≳ 1 allows the radial optical depth to remain small if Γ · θ_s ≫ 1:

$$\begin{eqnarray} &&\tau _{\rm es}(r_s,\mu _{\rm min}) \sim [1 + \Gamma ^2(r_s)]\,\tau _{\rm es}(r_s,1) \,{\Rightarrow }\, \tau _{\rm es}(r_s,1)\gtrsim {1\over 1 + \Gamma ^2(r_s)}. \end{eqnarray} \tag{ 67 }$$

It is straightforward to calculate the output spectrum for a broadband source with spectrum F_ν0(ν₀) via

$$\begin{equation} F_{\nu _{\rm em}}(\tilde{\nu }_{\rm em}) \rightarrow \int {F_{\nu _0}(\nu _0^{\prime })\over F_0} F_{\nu _{\rm em}}\left({\nu _{\rm em}\over \nu _0^{\prime }}\right) {d\nu _0^{\prime }\over \nu _0^{\prime }}. \end{equation} \tag{ 68 }$$

Here $F_{\nu _{\rm em}}(\tilde{\nu }_{\rm em})$ is the Greens function response of the scattering outflow to a line photon source with total energy flux F₀.

The result for a blackbody seed is shown in the left panel of Figure 10, and for a thermal seed with a GRB-like spectrum at low frequencies, $F_\nu = {\rm constant}\times e^{-h\nu /kT_0}$, in the right panel.⁶ In both cases, the seed temperature is normalized so that F_ν peaks at ν = ν₀. In the second case, the upscattered portion of the spectrum at ν > ν₀ connects smoothly with the seed spectrum at ν < ν₀.

The total spectrum, including the attenuated source, is shown in Figure 11. The seed thermal peak is apparent if the seed photons see a maximum optical depth τ(r_s, μ_min) ∼ 1–3. It is subdominant for larger optical depths.

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In Figures 12–15 we show spectra calculated using the outflow profiles obtained in Section 5, with flow parameters σ = 1000, ω = 10. A distinct feature of the spectra is a prominent high-energy tail appearing in outflows with moderate χ_s ≲ 10³—compare the spectra for χ_s = 10² and 10⁶ in Figure 12. At low compactness, the outflow acceleration is dominated by MHD forces, and Γ ≫ Γ_eq at the base of the outflow. When the acceleration is dominated by radiation pressure, increasing the optical depth only causes small changes in the peak of the scattered spectrum (as for χ_s = 10⁶). A direct comparison of outflows with a range of χ_s is made in Figure 13.

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The spectral index of the χ_s = 10² outflow is shown in Figure 14, for both blackbody and GRB-like seed spectra. As the optical depth at the base of the outflow increases, the dip in the spectral index near the seed peak is reduced. In the blackbody case, the low-energy tail has a fairly constant, Rayleigh–Jeans slope, becoming slightly steeper at high compactness, as discussed in Section 6.5.

The decomposition of the output spectrum into the components emitted at different radii is shown in Figure 15. The contribution from large radius is in the optically thin regime, and has a harder spectrum than the (dominant) contribution near the photosphere. The overall normalization of the spectrum remains essentially constant, $F_{\nu _{\rm em}} \sim F_0$, at ν_em ∼ ν₀ when τ_es(r_s, μ_min) ≳ 1.

Here we examine in more detail the effect that radiation transfer near a relativistic photosphere will have on the low-frequency spectrum. A Rayleigh–Jeans spectral slope arises from side scattering monochromatic radiation in a locally spherical outflow, and is therefore maintained for a blackbody source (Section 6.3). The optical depth of low-frequency photons scales as $\tau _{\rm es}(r,\theta _{\rm em}) \propto \theta _{\rm em}^2 \propto \Gamma ^{-2}\nu _{\rm em}^{-1}$ and emission time $t \propto \nu _{\rm em}^{-1}$ for photons emitted off the axis to the observer. The increase in optical depth also pushes the off-axis photosphere out to a radius $r_\tau (\nu _{\rm em}) \propto \Gamma ^{-2}\nu _{\rm em}^{-1}$, where the rate of scatterings in a volume ${\sim }r_\tau ^3$ is proportional to (1 − β)(Γρ)n_γ∝Γ⁻²r_τ(ν_em)⁻¹∝ν_em. In the optically thin regime, one therefore finds a low-frequency spectrum $F_{\nu _{\rm em}} \propto t^{-1} \propto \nu _{\rm em}$, which hardens to $F_{\nu _{\rm em}} \propto \nu _{\rm em}^2$ if every frequency is emitted from its photosphere.

It is also worth examining briefly how this result for a steady outflow would be modified in the case where the outflow is impulsive. Consider a slightly simpler situation in which the seed photons flow radially (and are monochromatic). Then the frequency of the outgoing photon is purely a function of scattering angle, and we can consider the number of photons scattered the bulk frame of the magnetofluid in a time interval dt',

$$\begin{equation} d^2N_\gamma = n_{\gamma 0}^{\prime } c {d\sigma _T\over d\mu _{\rm em}^{\prime }} d\mu _{\rm em}^{\prime } dt^{\prime }. \end{equation} \tag{ 69 }$$

Here $n_{\gamma 0}^{\prime } \sim n_\gamma /\Gamma$ is the bulk frame photon density, and the differential scattering cross section varies mildly with the rest-frame scattering angle. Since $\nu _{\rm em}/\nu _0 = (1-\beta)/(1-\beta \mu _{\rm em}) = \Gamma ^2(1-\beta)(1+\beta \mu _{\rm em}^{\prime })$, we have

$$\begin{equation} {d^2N_\gamma \over d\tilde{\nu }_{\rm em}dt^{\prime }} = {n_{\gamma 0}^{\prime } c\over \Gamma ^2\beta (1-\beta)} {d\sigma _T\over d\mu _{\rm em}^{\prime }}. \end{equation} \tag{ 70 }$$

The observed arrival time of a photon depends on emission angle and therefore frequency, $dt_{\rm obs} = \Gamma (1-\beta \mu _{\rm em})dt^{\prime } = \tilde{\nu }_{\rm em}^{-1}\Gamma (1-\beta)dt^{\prime }$. But integrating over the entire history of a pulse, the total number of photons emitted is dN_γ/dln ν_em∝ν_em. At very low frequencies, one must compensate for the expanded photosphere and reduced scattering rate, as outlined above.

In the Monte Carlo evaluation of the scattered spectrum, we tested a simplified evaluation of the optical depth integral, using the small-angle approximation and assuming linear growth of the Lorentz factor, Γ(r₂) = (r₂/r)Γ(r) ≫ 1. Then Equation (65) becomes

$$\begin{equation} \tau _{\rm es}(r,\mu _{\rm em}) \simeq {\sigma _T\over 6cr}\,{d\dot{N}\over d\Omega }\, \left[{1\over \Gamma ^2(r)} + \theta _{\rm em}^2\right]. \end{equation} \tag{ 71 }$$

The output spectra differ only in detail from those displayed in Figures (10), (11).

It is also useful to work out the spectrum of singly scattered photons in the case where the seed photons see a small to modest optical depth. Since the matter is cold, it is simplest first to transform into its rest frame and consider the power scattered into solid angle $d\Omega _{\rm em}^{\prime }$,

$$\begin{equation} {d^3E^{\prime }\over d\nu _{\rm em}^{\prime }d\Omega _{\rm em}^{\prime }dt^{\prime }} = {3\sigma _T\over 16\pi }\int I_{\nu ^{\prime }}^{\prime }[1+(\hat{k}^{\prime }\cdot \hat{k}_{\rm em}^{\prime })^2]d\Omega ^{\prime }. \end{equation} \tag{ 72 }$$

The rest-frame frequency and spectral intensity are ν' = ν₀/Γ(1 + βμ') and $I_{\nu ^{\prime }}^{\prime } = I_\nu /[\Gamma (1+\beta \mu ^{\prime })]^3$. One sees in Figure 9 that the radiation field flows both forward and backward in the bulk frame close to the engine. At large distances the radiation field continues to collimate even as Γ saturates, and $I_{\nu ^{\prime }}^{\prime }$ is concentrated in the forward (anti-radial) direction.

The power of the scattered radiation from a single charge is

$$\begin{eqnarray} &&{d^2E\over d\nu _{\rm em}dt} = \int {1\over \Gamma } {d^3E^{\prime }\over d\nu _{\rm em}^{\prime }d\Omega _{\rm em}^{\prime }dt^{\prime }} d\Omega _{\rm em}^{\prime } = {3\sigma _T\over 16\pi }\int d\Omega ^{\prime } d\Omega _{\rm em}^{\prime } {I_0\nu _0\over \Gamma ^4(1+\beta \mu ^{\prime })^3} \delta \left({1+\beta \mu ^{\prime }\over 1+\beta \mu _{\rm em}^{\prime }}\nu _{\rm em} - \nu _0\right) [1+(\hat{k}^{\prime }\cdot \hat{k}^{\prime }_{\rm em})^2]. \end{eqnarray} \tag{ 73 }$$

Using

$$\begin{eqnarray} &&\int d\phi ^{\prime } d\phi ^{\prime }_{\rm em} [1+(\hat{k}^{\prime }\cdot \hat{k}_{\rm em}^{\prime })^2]= 2\pi ^2\left[3 + 3{\mu ^{\prime }}^2{\mu ^{\prime }_{\rm em}}^2 - {\mu ^{\prime }}^2 - {\mu ^{\prime }_{\rm em}}^2\right] \end{eqnarray} \tag{ 74 }$$

gives

$$\begin{equation} {d^2E\over d\nu _{\rm em}dt} = {3\pi \sigma _T\over 8}{I_0\nu _0\over \beta \Gamma ^4}\tilde{\nu }_{\rm em}\int d\mu ^{\prime } {3 + 3{\mu ^{\prime }}^2{\mu ^{\prime }_{\rm em}}^2 - {\mu ^{\prime }}^2 - {\mu ^{\prime }_{\rm em}}^2\over (1+\beta \mu ^{\prime })^2}, \end{equation} \tag{ 75 }$$

where $\mu _{em}^{\prime } = \mu _{\rm em}^\prime (\mu ^{\prime },\tilde{\nu }_{\rm em})$ from Equation (62). The range of integration over μ' is restricted if $\tilde{\nu }_{\rm em}> 1$. Since $1\,{+}\,\beta \mu ^{\prime } = (1\,{+}\,\beta \mu ^{\prime }_{\rm em})/\tilde{\nu }_{\rm em}$, we have $(1-\beta)/\tilde{\nu }_{\rm em}\le 1+\beta \mu ^{\prime } \le (1+\beta)/\tilde{\nu }_{\rm em}$, and more generally

$$\begin{eqnarray} &&{\rm max}\left[{1-\beta \over \tilde{\nu }_{\rm em}},\; {1\over \Gamma ^2(1-\beta \mu _{\rm min})}\right] \;\le \; 1+\beta \mu ^{\prime } \le (1+\beta)\,{\rm min}\left(1,\;{1\over \tilde{\nu }_{\rm em}}\right). \end{eqnarray} \tag{ 76 }$$

The flux of scattered radiation measured at a large distance is

$$\begin{equation} F_{\nu _{\rm em}}(r) = {1\over r^2}\int ^\infty {dr\over \beta (r)c} {d\dot{N}\over d\Omega } {d^2E\over d\nu _{\rm em}dt}. \end{equation} \tag{ 77 }$$

Normalizing to the incident radiation flux F₀ = πI₀ν₀/x² gives

$$\begin{eqnarray} &&{F_{\nu _{\rm em}}\over F_0} = {3\sigma _T\over 8cr_s}{d\dot{N}\over d\Omega }\tilde{\nu }_{\rm em}\int d\mu ^{\prime } dx {3 + 3{\mu ^{\prime }}^2{\mu ^{\prime }_{\rm em}}^2 - {\mu ^{\prime }}^2 - {\mu ^{\prime }_{\rm em}}^2 \over \Gamma ^4\beta ^2(1+\beta \mu ^{\prime })^2}. \end{eqnarray} \tag{ 78 }$$

Following Equation (71), the prefactor can be written as

$$\begin{equation} {3\sigma _T\over 8cr_s}{d\dot{N}\over d\Omega } \;=\; {9\over 4}\Gamma ^2(r_s)\;\tau _{\rm es}(r_s,1). \end{equation} \tag{ 79 }$$

At first sight, one might associate the high-energy tail of the scattered photon spectrum with the observed high-energy tails of GRBs, and the seed photon energy with the observed spectral peak energy E_pk. But the calculated spectrum is relatively hard compared with the high-energy tails of GRBs, and it is limited in spectral width.

Instead it appears more promising to identify the high-energy peak of the scattered spectrum with the measured E_pk—at least in some bursts or possibly some phases of the burst emission. Then an additional source of dissipation, which is left out of our calculations, is needed to generate the high-energy tail. The seed blackbody photons, generated deep in the outflow, provide a buffer that suppresses bursts with low E_pk, but upscattering allows a range of higher E_pk values. We leave open here the nature and origin of the low-frequency seed spectrum, except to say that a hard Rayleigh–Jeans slope is by no means guaranteed if thermalization occurs at an intermediate optical depth.

In this context, it is worth recalling some features of the Amati et al. (2002) relation between E_pk and the apparent isotropic energy E_iso of GRBs. This relation is suggestive of jet breakout from the core of a Wolf–Rayet star (e.g., Thompson 2006), but it appears to represent a boundary in the E_pk–E_iso plane. The observed bursts have E_pk lying on or above the Amati et al. line, with a strong deficit mainly below the line: for example, a significant proportion of BATSE burst spectra are too hard to be consistent with this relation, independent of the (unknown) redshift (Nakar & Piran 2005).

Evidence for a low frequency blackbody component has been found in some GRBs (e.g., Axelsson et al. 2012). It is natural to try to accommodate such components in the present calculation, where the thermal seed should be partially transmitted. The measured soft blackbody component sits a factor ∼1/30 below the burst peak, which requires a large broadening of the seed photons (by a factor ∼Γθ_seed  ∼  5 in angle, where θ_seed is the angular width of the seed component). If the decrease in the peak frequency during the burst evolution were due mainly to a drop in Γθ_seed, then one would expect to see a strong re-emergence of the seed thermal peak alongside the scattered peak, which is generally not observed. When considering the emission from relativistic outflows, one must always keep in mind the basic degeneracy between temporal and spectral degrees of freedom, and the angular degree of freedom. Therefore the presence of a soft thermal component in the spectrum may signal the presence of off-axis emission that does not, necessarily, interact with the emission zone of the harder component.

Finally, one should keep in mind that the scattered thermal radiation emitted by an electromagnetic outflow could represent a subcomponent of the burst emission, with some other non-thermal (e.g., synchrotron) process dominating (Pe'er et al. 2006; Zhang & Yan 2011). In that case, the photon field would have a smaller (but not necessarily negligible) influence on the outflow acceleration.

We find that a hot electromagnetic outflow typically experiences a strong radiation force before it expands far enough that radial inhomogeneities become important. The acceleration of a static, bounded magnetic slab, studied by Granot et al. (2011), could be relevant for the later stages of impulsive GRB outflows, outside a radius ∼cΔt. But if a magnetized shell already moves relativistically at this radius, only a thin outer layer, comprising a fraction ∼1/2Γ² of the shell, would experience a strong outward magnetic pressure gradient force. Its interaction with slower material, swept up from a Wolf–Rayet star or a preceding neutron-rich wind, must then be taken into account.

Other acceleration mechanisms have been suggested which depend on more complicated, non-ideal MHD effects, such as the creation of a (net) outward pressure gradient force by reconnection of a toroidal magnetic field (Drenkhahn & Spruit 2002). Zones of alternating B_ϕ, separated by current sheets, are indeed present in the force-free solution to the oblique rotator (Spitkovsky 2006), in a zone straddling the rotational equator. An active dynamo operating in a GRB engine could also lead to stochastic reversals in the wind magnetic field (Thompson 2006). But when the increase in flow inertia associated with particle heating and radiation is taken into account, we have argued that canceling even half the magnetic flux leads only to mildly relativistic radial motion. Magnetic reconnection plays a more natural role in the GRB phenomenon by modifying the gamma-ray spectrum, via particle heating and stochastic bulk motions.