The exotic quantum process of photon splitting has great potential to explain the softness of emission in soft gamma repeaters (SGRs) if they originate in neutron stars with surface fields above the quantum critical field B_{\rm cr}=4.413\times 10^{13}Gauss. Splitting becomes prolific at such field strengths: its principal effect is to degrade photon energies, initiating a cascade that softens gamma-ray spectra. Uniform field cascade calculations have demonstrated that emission could be softened to the observed SGR energies for fields exceeding about 10^{14}Gauss. Recently, we have determined splitting attenuation lengths and maximum energies for photon escape in neutron star environments including the effects of magnetospheric dipole field geometry. Such escape energies \erg_{esc} suitably approximate the peak energy of the emergent spectrum, and in this paper we present results for \erg_{esc} as a function of photon emission angles for polar cap and equatorial emission regions. The escape energy is extremely insensitive to viewing perspective for equatorial emission, arguing in favour of such a site for the origin of SGR activity

Baring, M G

Harding, A K

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PHOTON SPLITTING IN SOFT GAMMA REPEATERS
MATTHEW G. BARING

and ALICE K. HARDING
NASA Goddard Space Flight Center, Code 665,
Greenbelt, MD 20771, U.S.A.
Abstract. The exotic quantum process of photon splitting  !  has great potential to
explain the softness of emission in soft gamma repeaters (SGRs) if they originate in neutron
stars with surface elds above the quantum critical eld B
cr
= 4:41310
13
Gauss. Splitting
becomes prolic at such eld strengths: its principal eect is to degrade photon energies,
initiating a cascade that softens gamma-ray spectra. Uniform eld cascade calculations
have demonstrated that emission could be softened to the observed SGR energies for
elds exceeding about 10
14
Gauss. Recently, we have determined splitting attenuation
lengths and maximum energies for photon escape in neutron star environments including
the eects of magnetospheric dipole eld geometry. Such escape energies "
esc
suitably
approximate the peak energy of the emergent spectrum, and in this paper we present
results for "
esc
as a function of photon emission angles for polar cap and equatorial
emission regions. The escape energy is extremely insensitive to viewing perspective for
equatorial emission, arguing in favour of such a site for the origin of SGR activity.
Key words: Soft Gamma Repeaters { neutron stars { magnetic elds { gamma-rays
1. Introduction
Soft gamma repeaters (SGRs), have recently become extremely topical in the
high energy astrophysics community, largely due to the identication of mul-
tiple counterparts to these gamma-ray transients in other wavebands. These
associations include radio supernova remnants (e.g. G10.0-0.3 for SGR1806-
20, see Kulkarni and Frail, 1993) and a compact X-ray region within the
X-ray/radio map of the LMC remnant N49 for the 5th March 1979 repeater
(Rothschild et al., 1994). The cumulative eect of these and other observa-
tions has been a strengthening of the idea that SGRs are of neutron star ori-
gin. If such an association is admitted and the eight second periodicity of the
soft gamma-ray tail to the Mar. 5, 1979 repeater is assumed to result from
magnetic dipole radiation spin-down throughout the life of its parent neu-
tron star, then a eld estimate (Duncan and Thomson, 1992) of B  610
14
Gauss is obtained using the estimated age of N49 of 5400 years. Such elds,
exceed the quantum critical eld strength B
cr
= m
2
c
3
=eh = 4:413  10
13
Gauss, and dwarf those found in most radio pulsars.
In such conditions, the exotic process of magnetic photon splitting  !
 acts eectively (Baring, 1991) below the threshold of single photon pair
production  ! e
+
e
 
around 1 MeV, attenuating gamma-rays and degrad-
ing them to soft gamma-ray energies while at the same time polarizing
the emission. Recently, Baring (1995) outlined the thermostatic action of
splitting as a possible reason for why SGR emission is so soft:  !  is

Compton Fellow, USRA; Email: Baring@lheavx.gsfc.nasa.gov
2 M. G. BARING & A. K. HARDING
unavoidable in supercritical elds, and the emergent spectra are then largely
insensitive to the eld strength above  5B
cr
. Baring (1995) observed that
a eld estimate of B  4B
cr
would be required to roughly t the ICE obser-
vations (Fenimore et al., 1994) of SGR1806-20. As a further exploration of
these uniform eld calculations, Baring and Harding (1995) calculated split-
ting attenuation lengths for photons as a function of their emission point
and propagation angle near the neutron star surface, including the eect of
dipole eld geometry. An aspect of this analysis is presented in this paper,
namely the computation of the maximum energy for photon escape "
esc
from
neutron star magnetospheres (below which the SGR emission will generally
appear), as a function of photon emission angles for polar cap and equa-
torial emission regions. Our analysis strongly suggests that SGR emission
emanates from equatorial regions if it is the result of splitting cascades.
2. Photon Splitting and Escape Energies
Photon splitting is forbidden in eld-free regions, but has a substantial non-
zero rate and is a powerful polarizing mechanism in a magnetized vacuum.
For "B
<

B
cr
mc
2
and B
<

B
cr
, where " is the incident photon energy, the
splitting attenuation coecients (i.e. rates divided by c ) for dierent pho-
ton polarization modes assume simple forms (using rates from Adler 1971,
Baring 1991): T
sp
/ "
5
(B=B
cr
)
6
sin
6

kB
, where 
kB
is the angle between
the photon momentum and the magnetic eld vectors. Reducing 
kB
or B
dramatically increases the photon energy required for splitting to operate;
such sensitivities suggest that considerations of magnetospheric geometry
are important for analyses of splitting, motivating the work presented here.
High eld (B
>

B
cr
) corrections (e.g. see Adler 1971, Baring 1995) to the
above formula for splitting diminish its dependence on B .
The simplest quantity that denes the eectiveness of photon splitting in
SGRs from neutron star magnetospheres is its attenuation length L , dened
to be the path length over which the optical depth of  !  is unity. We
consider photons that originate on the neutron star surface, and propagate
(generally non-radially) outward. Attenuation lengths are explored in detail
by Baring and Harding (1995), where L was generally found to be shorter
for polar (as opposed to equatorial) emission where the surface eld strength
is higher. At low energies the splitting rate is low, and photons can escape
the magnetosphere. For each set of initial conditions, i.e. photon location
and propagation direction, there is a critical energy "
esc
, called the escape
energy, below which the optical depth to innity is always less than unity
and photons escape the magnetosphere; this occurs because of the decay of
the dipole eld away from the stellar surface.
Figure 1 depicts escape energies for polar (  = 0

) and equatorial emis-
sion (  = 90

) as a function of the initial propagation angle 
k
to the dipole
axis, where 90

   is the latitude of emission. The neutron star radius was
eslab29bh.tex - Date: June 6, 1995 Time: 15:59
PHOTON SPLITTING IN SOFT GAMMA REPEATERS 3
Fig. 1. The energy, below which photons escape from the magnetosphere without splitting,
for photons of observable polarization (? and k are dened in the text) originating at
the pole (left) and the equator (right), as functions of the polar angle 
k
of the photon.
The curves tend to merge above 10B
cr
due to the saturation of the splitting rate at high
elds, and therefore are not displayed. The curves for polar emission diverge near 
k
= 0

because the photons are almost parallel to the eld lines throughout their path.
set at 10
6
cm. The plots discriminate between the two polarization states
of the photons, namely ? or k , where the photon's electric eld vector is
respectively parallel or orthogonal to kB . In the polar case, "
esc
clearly
declines with both 
k
and B . In the case of equatorial emission, the results
in Fig. 1 indicate a remarkable insensitivity to 
k
. This is largely due to the
fact that even if photons start out moving along the eld, they soon prop-
agate obliquely to the eld because of its curvature and so cannot escape
unless they are below the 
k
= 90

escape energy. The curves in Fig. 1
indicate that for low B , photons with ? polarization are more likely to
split than those in the k state, while for B  B
cr
this situation is reversed.
Also note that for 
k
 90

, the escape energies for polar and equatorial
emission are similar when B  B
cr
; this arises because then the splitting
of photons above "
esc
occurs in regions away from the stellar surface.
3. Discussion
Our calculations of the photon splitting escape energy in a neutron star
dipole eld give a realistic estimate of the degree of spectral softening achiev-
able through a photon splitting cascade. They go beyond the work of Baring
(1995), who assumed a homogeneous eld of nite extent 2  10
6
cm and
found that the escaping photon spectrum is quasi-thermal and peaks roughly
at the escape energy. Our escape energies for polar emission at 
k
= 90

are
 2:5 times those of Baring, primarily because a dipole eld decreases signif-
eslab29bh.tex - Date: June 6, 1995 Time: 15:59
4 M. G. BARING & A. K. HARDING
icantly over the region that he assumed to be homogeneous. The surface eld
strength required to t observed SGR spectra with photon splitting cascade
spectra will therefore be somewhat higher than the B  4B
cr
obtained by
Baring in tting spectra from SGR1806-20.
Fig. 1 exhibits a strong variation in escape energies for photons emitted
at dierent angles near the magnetic pole of the neutron star, but almost
no variation for photons emitted at the dipole equator. This result is purely
a function of the geometry of the dipole eld curvature: near the pole, the
eld lines are diverging rapidly and dierent emission directions sample very
dierent eld orientations, whereas at the equator, the eld looks nearly
the same in all directions. This behaviour has important implications for
splitting cascade models of SGR spectra. It predicts that the cascade spectra
from emission near the equator will be insensitive to observer angle, yielding
the same peak energy in all directions. Thus, the spectra would not vary
from burst to burst, even if the neutron star orientation changes (i.e. the
star rotates). This is consistent with what is observed in dierent SGR
bursts (e.g. Fenimore, Laros and Ulmer, 1994; Kouveliotou et al., 1994 for
SGR1806-20), and also with phase-resolved spectroscopy of the periodic soft
gamma-ray tail of the Mar. 5, 1979 outburst (Mazets et al., 1982). Hence the
principal conclusion of this paper is that emission near the dipole equator
would be strongly favoured for photon splitting cascade models of SGRs.
The initial angular distribution of the photons, which depends on the
emission mechanism operating, will play a crucial role in determining the
escape energy. For example, resonant Compton upscattering (e.g. Dermer,
1990) produces -ray photons from soft thermal photons above a neutron
star surface that are beamed at small angles to the eld. Cyclotron radiation
from relativistic electrons in low Landau levels has similar beaming. Escape
energies for these photons will be much higher at the pole than at the equator
(see Fig. 1), where even photons propagating initially parallel to the eld
have low "
esc
. Consideration of such angular distributions forms the next
stage of our program of research into splitting cascades and SGR emission.
References
Adler, S. L.: 1971, Ann. Phys. 67, 599.
Baring, M. G.: 1991, Astron. & Astrophys. 249, 581.
Baring, M. G.: 1995, Astrophys. J. (Lett.) 440, L69.
Baring, M. G. and Harding, A. K.: 1995, to appear in \High Velocity Neutron Stars and
Gamma-Ray Bursts" eds. Rothschild, R. et al., AIP, New York.
Dermer, C. D.: 1990, Astrophys. J. 360, 197.
Duncan, R. C. and Thompson, C.: 1992, Astrophys. J. (Lett.) 392, L9.
Fenimore, E. E., Laros, J. G. and Ulmer, A.: 1994, Astrophys. J. 432, 742.
Kouveliotou, C., et al.: 1994, Nature 368, 125.
Kulkarni, S. and Frail, D.: 1993, Nature 365, 33.
Mazets, E. P. et al.: 1982, Astrophys. Space Sci. 84, 173.
Rothschild, R. E., et al.: 1994, Nature 368, 432.
eslab29bh.tex - Date: June 6, 1995 Time: 15:59


Photon splitting in soft gamma repeaters

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