A&A 381, L5-L8 (2002)
H.-Y. Chang - C.-H. Lee - I. Yi
Korea Institute for Advanced Study,
207-43 Cheongryangri-dong Dongdaemun-gu, Seoul 130-012, Korea
Received 18 September 2001 / Accepted 6 November 2001
We investigate consequences of a continuously energy-injecting central engine of gamma-ray burst (GRB) afterglow emission, assuming that a highly magnetized pulsar is left beaming in the core of a GRB progenitor. Beaming and continuous energy-injection are natural consequences of the pulsar origin of GRB afterglows. Whereas previous studies have considered continuous energy-injection from a new-born pulsar to interpret the deviation of afterglow light curves of GRBs from those with the simple power law behavior, a beaming effect, which is one of the most important aspects of pulsar emissions, is ignored in earlier investigations. We explicitly include the beaming effect and consider a change of the beaming with time due to a dynamical evolution of a new-born pulsar. We show that the magnitude of the afterglow from this fireball indeed first decreases with time, subsequently rises, and declines again. One of the most peculiar optical afterglows light curve of GRB 970508 can be accounted for by continuous energy injection with beaming due to a highly magnetized new-born pulsar. We discuss implications on such observational evidence for a pulsar.
Key words: gamma rays: bursts - pulsar: general - stars: magnetic fields
Gamma-ray bursts (GRBs) are widely accepted to be produced when fast-moving, relativistic shells ejected from a central source in a relatively short period collide with slowly moving, yet relativistic shells that were ejected at an earlier time (Rees & 1994; Paczy ski & Xu 1994; Kobayashi et al. 1997; Daigne & Mochkovitch 1998, 2000). In connection with the so-called internal shock model, the external shock model also prevails as a possible origin of the GRB afterglows. In the external shock model the relativistic material is assumed to be decelerated via interactions with the surrounding medium. A shock wave results in heating the ambient matter to relativistic temperatures, and emitting photons in longer wave lengths (Rees & 1992; & Rees 1993; Paczy ski & Rhoads 1993; Sari et al. 1996; Vietri 1997). GRBs and their afterglows seem to result from the dissipation of bulk energy in the relativistic outflows, which are in the form of a narrow beam rather than a spherical shell. Even though the origin of the observed GRBs are still unknown, from the observations of several GRB afterglows the evidence of beamed GRBs has been accumulated (Sari et al. 1999; Halpern et al. 1999; Rhoads 1999). There are several works on models for the geometry of GRBs (e.g., Chang & Yi 2001 and references therein) and their environments (e.g., Scalo & Wheeler 2001 and references therein). Much of the current research on GRBs is aimed at determining the nature and the origin of the central engine (Duncan & Thompson 1992; Narayan et al. 1992; Woosley 1993; Katz 1994, 1997; Usov 1992, 1994a; Shaviv & Dar 1995; & Rees 1997b; Yi & Blackman 1997; Blackman & Yi 1998; Paczy ski 1998; MacFadyen & Woosley 1999; Portegies Zwart et al. 1999; Li 2000; Wheeler et al. 2000; Zhang & Fryer 2001). Although the simple cosmological fireball afterglow model is in a good agreement with the observed light curves of a power law decay (e.g., Wijers et al. 1997; Waxman 1997a,b), the afterglow of GRB970228 observed by HST, for instance, shows a deviation from the simple power law behavior (Fruchter et al. 1997). Several works have been done to further investigate more subtle effects that can change the afterglow characteristics (Katz & Piran 1997; & Rees 1997a, 1999; Rhoads 1999; Berger et al. 2000; Dai & Lu 2000; Kumar & Panaitescu 2000a,b; Panaitescu & Kumar 2000; Dai & Lu 2001).
Most fireball models assume that the energy injection into the fireball occurs in a short period of time compared with the lifetime of the afterglows (Rees & 1998; Kumar & Piran 2000; Sari & 2000). However, in some types of central engines, such as a fast rotating new-born pulsar with the strong magnetic field (magnetar), a significant energy input into the fireball may in principle continue for a significantly longer timescale, and accordingly the temporal decay of the afterglow will be slower. Hence, it is worthwhile to investigate a continuously fed fireball in more details as a probe of the central engine of GRBs. We here consider the central engine that emits both an initial impulsive energy input and a continuous power. In fact, recently there were such attempts to provide an explanation for the deviation of the afterglow light curve from the simple power law (Dai & Lu 1998a,b; Zhang & 2001a,b). Even though previous studies have considered continuous injection from a highly magnetized millisecond pulsar to interpret the deviation of afterglow light curves of some GRBs from the power law behavior, one of the most important aspects of pulsar emissions is ignored, that is, beaming. Beaming and continuous powering are the clear consequences of the pulsar origin of the GRB afterglows. In this letter, this feature is included and a change of the beaming due to a dynamical evolution of the new born pulsar is explicitly taken into account. We find that the deviations suggested by previous studies indeed occur, but that the shape of the light curve is significantly modified due to a beaming effect. If such observational evidence for beaming is found, the corresponding pulsar origin for GRBs would imply that activities resulting in too massive objects to be a neutron star might be ruled out as a central engine of GRBs. Therefore, the question of whether or not a signature of beaming can be found in the afterglows thus extremely important.
The total luminosity emitted from a young millisecond pulsar (MSP)
has two important terms: an electromagnetic (EM) radiation term
and a gravitational wave (GW) radiation term.
Given that the spin-down is mainly due to electromagnetic dipolar
radiation and to gravitational wave radiation, the spin-down law is
The differential energy conservation relation
for the self-similar blast wave can be written
and t are the energy and time measured in the fixed frame
and q' and
are constants (Cohen & Piran 1999).
The first term denotes the continuous luminosity injection,
and the second term takes into account radiative energy losses
in the blast wave. For t>t0, the bulk Lorentz factor of
the fireball scales with time as
with m and
(Cohen et al. 1998).
If m=3, it corresponds to the adiabatic case (Blandford & McKee 1976).
In the observer frame, the time T is related to the fixed
frame t by
The differential energy conservation relation
in the observer frame is now given by
and can be integrated as
The total energy of the blast wave given by
Eq. (2) may be dominated either by the continuous injection
or by the initial impulsive term
), subject both to the relative values of
the two indices and to the values of L0 and
(Dai & Lu 1998a,b; Zhang &
One may classify three regimes according to
the relative values of the two indices as discussed by
We are interested in the case where
a pulsar signature is no longer observable even if there is
a pulsar in the central engine.
the first term in Eq. (2)
will eventually dominate over the second term after a critical .
The injection-dominated regime begins at a critical time
by equating the injection and energy-loss terms in Eq. (2),
To obtain the temporal decay index of the afterglow light curve
for which the MSP is responsible
we adopt the cylindrical geometry instead of the spherical geometry,
which may accommodate elongated beaming configurations.
The rotational axis of the MSP coincides with the z-axis of the geometry.
For a fireball blastwave decelerated by a homogeneous external medium with
particle number density n, the energy conservation equation
at time t=r/c is given by
If the continuous injection term becomes dominant over the impulsive
term after ,
the afterglow light curves rises after
again after some time, that is, about
Therefore, there may be two types of afterglow patterns
for continuous injections according to a appropriate combination.
conditions which allow to detect a signature for a pulsar
and concluded that physical parameters are consistent with those
of a magnetar in case of the afterglow features in case of GRB000301c.
In practice, however,
is very short, can be even shorter
unless ambient matter density is very high,
and is therefore unlikely to be observed.
q changes -1 to 0 and 0 to -2, respectively.
These scaling laws represent a change from the standard adiabatic
case to an EM-loss dominated regime as shown above (see also
For the forward shock, the temporal decay index
to 3 and returns to
For the reverse shock, the temporal decay index
to 3 and
A continuous energy injection signature in the GRB afterglow light curve may directly provide diagnostics about the nature of the injection as well as information on the GRB progenitor. Therefore, the question of whether or not a bump in the afterglow light curve is such an observational evidence for beaming from a pulsar is thus extremely important. In this sense, the correct geometry should be applied to the model calculation. We have discussed the case where a pulsar is continuously injecting energy cylindrically rather than spherically. We suggest that a possible explanation for the deviation of the afterglow light curve of GRB 970508 is due to a beaming from a central pulsar. The optical afterglow of GRB 970508 has been explained as evidence of gravitational lensing event (Loeb & Perna 1998; Dado et al. 2001). So far the gravitational lensing can only account for the fact that the GRB afterglow shows a rise and a fall. We show in this letter that with a correctly assumed geometry of energy injection the afterglow of GRB970508 can be explained by a strongly magnetized fast-rotating pulsar with the continuous energy injection in a beam. In this calculation we ignore effects of the lateral expansion of the jet which may occur in the later time of the jet evolution. The light curve should be modified and becomes that of the spherical geometry case if the sideways expansion occurs in a timescale comparable to those we discussed after a neutron star formed. For instance, if the jet is expanded much faster at earlier stage than the light cone evolution, the cylindrical geometry effect becomes less obvious.
If the bump is caused by such a pulsar indeed, it puts constraints on the GRB progenitor models. That is, the corresponding pulsar origin for GRBs would imply that activities resulting in too massive objects to be a neutron star, such as, neutron star mergers, black hole formation, should be ruled out as a central engine of GRBs. During the birth of the neutron star, an initial fireball may occur through electromagnetic processes (Usov 1992). This has led to models in which GRBs are powered by rapidly spinning compact objects with strong magnetic fields (Blackman & Yi 1998; Blackman et al. 1996; Usov 1994b; Yi & Blackman 1997, 1998).
We are grateful to the referee, Robert Mochkovitch, for useful comments and suggestions. We are grateful to Ethan Vishniac for hospitality while visiting Johns Hopkins University where this work began.