3 Field geometries

Figure 1: Evolution of an initial dipole field by a radial outflow (schematic). Top left: initial field configuration. Top right: view on a larger scale, when the outflow, of finite duration $\tau$, now moves in a shell of width $c\tau$. It has stretched the field interior to the shell into a radial field. Lower left: reconnection in the low-density interior region restores the dipole field near the source. Lower right: as the shell moves out its thickness becomes small compared the distance travelled, and the further evolution depends on reconnection processes inside the shell

A burst of duration $\tau$ produces a shell of width $d=c\tau$, travelling outward at a speed $\beta c$. The observed radiation from this shell is produced when it has expanded to a radius $r\gg d$. This thin shell carries with it magnetic field lines from the source, which we call "trapped field lines''. For the rest of this section, we assume a relativistic outflow, $\beta\approx 1$.

3.1 Trapped fields

If reconnection processes inside the shell can be neglected, the field lines are frozen in the expanding shell. If $\vec B = (B_{\rm r}, B_\theta, B_\phi)$ (in spherical coordinates), and the width of the shell is constant, the components then vary with distance as $B_{\rm r} \sim r^{-2}$, $B_\theta \sim B_\phi\sim r^{-1}$. This is because the radial component is divided over the surface of the spherical shell, while the components parallel to the shell surface decrease with the circumference. More formally, the induction equation $\partial_{\rm t} \vec B = \nabla\times(\vec u\times\vec B)$ has the components

 

$$\partial_{\rm t} B_\theta (r\sin\theta)^{-1} \partial_\phi (u_\theta B_\phi - u_\phi B_\theta)$$ =

$$- r^{-1} \partial_{\rm r}(r (u_{\rm r} B_\theta - u_\theta B_{\rm r}))$$ (9)

$$\partial_{\rm t} B_\phi r^{-1} (\partial_{\rm r}(r (u_\phi B_{\rm r} - u_{\rm r} B_\phi))$$ =

$$- \partial_\theta (u_\theta B_\phi - u_\phi B_\theta)).$$ (10)

Assuming constant radial outflow $(u_{\rm r}={\rm const.}, u_\theta = u_\phi=0)$ and spherical symmetry $(\partial_\theta = \partial_\phi = 0)$ the time evolution in a comoving fluid element is

$$\frac{{\rm d} B_{\theta,\phi}}{{\rm d}t} =\partial_{\rm t}... ...eta,\phi} =-r^{-1} \frac{{\rm d}r}{{\rm d}t} B_{\theta,\phi}.$$ (11)

Hence the tangential field $\vec B_{\rm t} = (B_\theta, B_\phi) \sim r^{-1}$. Since the expansion factor is very large between the source and the radius from which the emitted radiation reaches us, the radial component (varying as $\sim$r^-2) can be neglected and the field is almost exactly parallel to the surface of the shell (Fig. 1). If the width of the shell is constant, the magnetic energy in the shell $e_{\rm m} = \int B_{\rm t}^2 {\rm d}S$is constant. The magnetic field thus carries a constant fraction of the kinetic energy of the outflow. Depending on how large this fraction is, the trapped field can be sufficient to produce the synchrotron emission proposed in internal shock models, without the need for "in situ'' field generation mechanisms (Medvedev & Loeb 1999).

How many field lines are trapped in the outflow, and hence which fraction of the outflow energy is magnetic, depends on conditions near the source. We consider here the representative possibilities.

3.2 A "passive'' magnetic field

First consider a fireball expanding from the surface of a central object (not specified further) of radius R and a dipolar magnetic field of moment $\mu$. We assume that the magnetic field plays no role in the driving of the outflow, but that it can confine plasma up to its own energy density $B^2/8\pi$.

Assume first that the expansion is kinematic, i.e. that the magnetic field is weak and its backreaction on the flow can be neglected. The number of field lines trapped in the outflow is equal to the flux of field lines outside crossing the dipole's equator outside the radius R:

$$\Phi=\frac{\pi\mu}{R}\cdot$$ (12)

This flux is a Lorentz invariant, so that the field strength of the shell (as measured in the frame of the central engine) is of the order $B=\Phi/(2\pi rc\tau)$, or

$$B\approx B_0\frac{R^2}{rc\tau}$$ (13)

where $B_0=\mu/R^3$. Whether this is a large field strength, compared with the kinetic energy density $\Gamma\rho c^2$, depends on the assumed dipole moment $\mu$. An upper limit to the dipole moment follows from the requirement that the energy of the burst should be able to open the field lines of the central object. Let the energy of the burst be $\Gamma Mc^2$, where $\Gamma$ the asymptotic Lorentz factor of the outflow and M the baryon load. If this energy was initially confined in a region of size R (the size of the central engine), the magnetic field strength B₀ in the confining dipole field of the source must satisfy

$$\frac{B_0^2}{8\pi} < \frac{\Gamma Mc^2}{\frac{4}{3}\pi R^3}$$ (14)

in order for the field to be opened up by the fireball. With (13), the magnetic energy density at distance r then satisfies

$$\frac{B^2}{8\pi} < e_{\rm k} \frac{R}{3c\tau},$$ (15)

where $e_{\rm k}$ is the kinetic energy density in the shell:

$$e_{\rm k}=\frac{\Gamma Mc^2}{4\pi r^2c\tau}\cdot$$ (16)

For most central engine models considered, the duration of the burst is long compared to the light travel time across the source, $R/c\tau\ll 1$. A "passive'' model, in which the magnetic field does not play a role in driving the outflow, therefore can only yield field strengths in the shell which are small compared with kinetic energy density. Even at such a low field strength, however, the magnetic field can become important for synchrotron emission in internal shocks, as discussed below in Sect. 4.

3.3 Active magnetic fields

In magnetic models for GRB engines, the magnetic field serves to extract rotation energy from a rapidly rotating relativistic object. The details of such magnetic extraction (especially three-dimensional ones) are still somewhat uncertain, but basic energetic considerations are simple. Rotation of the mass-loaded field lines induces an azimuthal field component $B_\phi$. Let the distance from the rotating object where this component is equal to the radial field be r₀. Except for cases with large baryon loading that are probably not relevant for GRB engines, r₀ is of the order of the Alfvén radius, which can lie anywhere between the surface of the object Rand the light cylinder $r_{\rm L}=c/\Omega$. The energy output Ltransmitted by the azimuthal magnetic stress ( $B_\phi B_{\rm r}/4\pi$) is then of the order $L\approx\Omega r_0^3B_{\rm r}^2(r_0)$. For a field dominated by its dipole component $\mu$, this yields a luminosity

$$L=\Omega\mu^2/r_0^3.$$ (17)

For $r_0=r_{\rm L}$, this yields (by order of magnitude) the pulsar spin-down formula for an inclined dipole rotating in vacuum, emitting an electromagnetic wave from its light cylinder.

Figure 2: Field configuration in quasi-spherical magnetic outflow driven by a perpendicular rotator ("pulsar-like'' case) (schematic). Left: view in the equatorial plane, with dots and pluses indicating field lines into and out of the plane of the drawing. Right: top view from the rotational pole. Bottom right: same view on larger scale, at a later time $t\gg \tau$

The energy estimate (17) does not tell what the field configuration in the outflow looks like. The possibilities for dissipation of magnetic energy inside the outflow depend strongly on this configuration, which in turn depends on the nature of the magnetic field of the central engine. An important distinction is whether the rotating magnetic field is axisymmetric or nonaxisymmetric. In the nonaxisymmetric case, the outflow contains magnetic fields varying on the rather small scale $\pi c/\Omega$, the wavelength of the outgoing wave. In such a field, internal dissipation turns out to be much more likely to be important than in an axisymmetric field, where the length scale of the field is of the order of the distance r. In the following, this is illustrated with a few specific cases.

3.3.1 Ratio of magnetic to kinetic energy

Estimate (17) only gives the total luminosity. Which fraction of it is in the form of kinetic energy and which in magnetic energy depends on more physics. At one extreme is Michel's (1969) model, in which the outflow consists of cold (pressureless) matter accelerated exclusively by the magnetic field. In this case, the ratio $\alpha = E_{\rm m}/E_{\rm k}$ of magnetic to kinetic luminosity is of the order $\xi\sim \Gamma^2\gg 1$. At large Lorentz factors, the energy is almost entirely in electromagnetic form, in this model. This is probably a property of the special geometry of the model, in which the flow is limited to the equatorial plane. If outflow in other directions is considered, much smaller values of $\xi$ result (Begelman & Li 1994). At the other extreme, consider a case in which most magnetic energy is released inside the light cylinder, in the form of a dense pair plasma (by some form of magnetic reconnection, for example). The result would then be just like the hydrodynamic expansion of a simple (Goodman 1986; Paczynski 1986) pair plasma fireball, and the resulting outflow would have $\xi\ll 1$. If some but not all energy is dissipated close to the source, intermediate values of $\xi$ would result. Since these important questions have not been resolved yet, we keep $\xi$ as a free parameter in the following. Where necessary we assume that it typically has values or order unity.

   
3.4 Nonaxisymmetric quasi-spherical outflow

Assume we have a perpendicular rotator, i.e. the rotating object has its dipole axis orthogonal to the rotation axis. At the source surface r₀, the rotating field is then strongest at the equator, and one expects the energy flux to be largest near the equatorial plane. With each rotation, a "stripe'' consisting of a band of eastward and one of westward azimuthal field moves outward (Fig. 2, see also Coroniti 1990; Usov 1994). Assuming the outflow to be relativistic, the width $\lambda$ of such a stripe in the rest frame of the central object is $\lambda = 2\pi c/\Omega$. The (absolute value of the) azimuthal flux $\Phi$ in each half-wavelength is of the order of the poloidal flux outside the source surface r₀, $\Phi\approx 2\pi\mu/r_0$. For spherical expansion of this amount of azimuthal flux, the field strength at distance r is then

$$B_\phi\approx\frac{\Phi}{r\lambda} =\frac{\mu}{r_0 r_{\rm L} r}$$ (18)

while the the total (magnetic plus kinetic) energy density $e_{\rm k}$ is

$$e_{\rm k}+e_{\rm m}=\frac{L}{4\pi r^2c}\cdot$$ (19)

Hence with (17) the ratio of magnetic energy density to total energy density (in the lab frame) is of the order

$$e_{\rm m}/(e_{\rm k}+e_{\rm m}) \approx\frac{r_0}{r_{\rm L}}\cdot$$ (20)

   
3.5 Jet

Consider a collimated outflow along the axis of rotation (Fig. 3). This might be achieved by magnetic models in which the azimuthal field collimates the flow towards the axis by hoop stress (Bisnovatyi-Kogan & Ruzmaikin 1976; Blandford & Payne 1982; Sakurai 1985). In such a model, one assumes an axisymmetric (about the rotation axis) poloidal field, which is wound up into an azimuthal field wrapped around the axis. Let the opening angle of the outflow be $\theta$ (assumed constant), and $\varpi$ the cylindrical radius. At the source surface $\varpi_0 = \theta z_0$ the poloidal and azimuthal field components are equal, and in the absence of magnetic reconnection processes

$$B_\phi B_{\rm p0}(\varpi_0/\varpi)= B_{\rm p0}(z_0/z),$$ = (21)

$$B_{\rm p} B_{\rm p0}(z_0/z)^2.$$ = (22)

If the collimation angle is small, the field as seen in a frame comoving with the jet is a slowly varying, nearly azimuthal field. Such a field is known to be highly unstable to kink instabilities (e.g. Bateman 1980). They operate on a time scale $\tau_{\rm k}$ of the order of the Alfvén crossing time, i.e. $\tau_{\rm k} = \varpi/v_{\rm A}$. Though the details of this process have not been worked out for jets (see however Lucek & Bell 1996,1997), it is likely that the release of magnetic energy operates in two steps. In the first step, kink instability transforms the axisymmetric configuration into a nonaxisymmetric, helical, configuration. For an application to jets, see Königl & Choudhuri (1985). This step is fast, operating on the Alvén crossing time. At this stage, the field has already lost much of its collimating ability, since the average azimuthal field strength has decreased in favour of a less ordered field component. During the instability, the so-called magnetic helicity is conserved, however, so that only a fraction of the magnetic energy is released. The further release of magnetic energy depends on reconnection processes. As discussed in Sect. 5, this is also likely to proceed on a time scale proportional to the Alfvén crossing time, though somewhat slower than the kink process itself. As in the quasi-spherical outflow case, we ignore this internal dissipation of the magnetic energy for the moment, and return to it in Sect. 5. For a jet expanding with fixed opening angle $\theta$ (see Fig. 3), the field strength then varies as r^-1 and the magnetic energy is constant with distance. Since only a fraction of the magnetic energy is released in the kinking process, the ratio of magnetic to kinetic energy density is still of the order found in axisymmetric magnetic jet calculations (e.g. Camenzind 1987), i.e. of order unity:

$$e_{\rm m}/e_{\rm k}\sim O(1).$$ (23)

Figure 3: Jet-like outflow of finite duration, magnetically driven by an axially symmetric rotator. Left: configuration near the source, showing how the field in the outflow is wound into a toroidal (azimuthal) field. Right: large scale view after a time long compared to the duration $\tau$. The field in the outflow is now a "pancake'' of toroidal flux. Sketch ignores nonaxisymmetric processes like kink instability and subsequent reconnection, which can release energy from this configuration