The annihilation spectrum of pairs with 1-D thermal distributions in the presence of a strong magnetic field is calculated. Numerical analysis of the spectrum are performed for mildly relativistic temperatures and for different angles of emission with respect to field lines. Teragauss magnetic fields are assumed so that conditions are typical of gamma ray burst and pulsar environments. The spectra at each viewing angle reveal asymmetric line profiles that are signatures of the magnetic broadening and red shifting of the line: these asymmetries are more prominent for small viewing angles. Thermal Doppler broadening tends to dominate in the right wing of the line and obscures the magnetic broadening more at high temperatures and smaller viewing angles. This angular dependence of the line asymmetry may prove a valuable diagnostic tool. For low temperatures and magnetic field strengths, useful analytic expressions are presented for the line width, and also for the annihilation spectrum at zero viewing angle. The results presented find application in gamma ray burst and pulsar models, and may prove very helpful in deducing field strengths and temperatures of the emission regions of these objects from line observations made by Compton GRO and future missions

Baring, Matthew G.

Harding, Alice K.

English

NASA Technical Reports Server

N92-21905
TWO-PHOTON ANNIHILATION OF THERMAL PAIRS
IN STRONG MAGNETIC FIELDS
MATTHEW G. BARING
Department of Physics
North Carolina State University
and
ALICE K. HARDING
Laboratory for High Energy Astrophysics
NASA/Goddard Space Flight Center
ABSTRACT
The annihilation spectrum of pairs with one-dimensional thermal distributions in the
presence of a strong magnetic field is calculated. Numerical computations of the spectrum
are performed for mildly relativistic temperatures and for different angles of emission with
respect to field lines. Teragauss magnetic fields are assumed so that conditions are typical
of gamma-ray burst and pulsar environments. The spectra at each viewing angle reveal
asymmetric line profiles that are signatures of the magnetic broadening and redshifting
of the line: these asymmetries are more prominent for small viewing angles. Thermal
Doppler broadening tends to dominate in the right wing of the line and obscures the
magnetic broadening more at high temperatures and smaller viewing angles. This angular
dependence of the line asymmetry may prove a valuable observational diagnostic tool. For
low temperatures and magnetic field strengths useful analytic expressions are presented
for the line width, and also for the annihilation spectrum at zero viewing angle. The
results presented find application in gamma-ray burst and pulsar models, and may prove
very helpful in deducing field strengths and temperatures of the emission regions of these
objects from line observations made by Compton/GRO and future missions.
INTRODUCTION
There is substantial observational motivation for detailed calculations of the amfihi-
lation spectrum of thermal pairs in the presence of a strong magnetic field. If the features
observed around 400 keV in gamma-ray burst (GRB) spectra (see Mazets ct al., 1981) are
interpreted as red-shifted 511 keV lines (for an extensive list of observations see Liang,
1986), then pair annihilation is taking place near a neutron star surface and will thus be
influenced by a strong magnetic field. Further, recent observations (see Agrinier ct al.,
1990) by the FIGARO II experimellt of an emission feature at about 400 keV in the spec-
trum of the Crab pulsar are suggestive of a red-shifted annihilation line in this neutron
star source. A study of the influence that a strong magnetic field has on the annihilation
spectrum is therefore of significant interest to astrophysicists.
There are two important effects of the magnetic field on two-I)hoton pair annihilation:
(i) because of the very high cyclotron emission rates, the v_t majority of pairs will annihi-
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late from the ground Landau state with a one-dimensional (1D) momentum distribution,
and (ii) since transverse momentum is not conserved in a strong magnetic field, there is an
intrinsic broadening of the two-photon annihilation line due to an increased availability of
phase space for this interaction. These effects become significant when the magnetic field
exceeds about a few percent of the quantum critical field Bc = 2 3rn_c /(eh) = 4.413 x 1013
Gauss, which corresponds to conditions in neutron star environments. To a first approxima-
tion, it is reasonable to assume that the annihilating pairs have a 1D thermal distribution.
The effects of the field on the shape of the thermal annihilation spectrum are potentially
the most useful observational diagnostic tool: the total rate of annihilation is not altered
much by the presence of the field when B G 4 x 1012 Gauss (Wunner, 1979).
Due to the intrinsic geometrical asymmetry imposed by the field, both the thermal
and magnetic contributions to the width of the annihilation line will be very angle depen-
dent. Intuitively, at viewing angles 01 "- 0 ° to the field, thermal Doppler broadening is
expected to be a maximum while magnetic broadening is a minimum. For viewing angles
01 "-, 90 o to the field this situation is reversed so that the magnetic broadening should be
more significant in comparison with the Doppler broadening. These expectations are borne
out by the computational results that are presented here. The prediction that the width of
observed lines at some angles could be due primarily to the magnetic field is observationally
significant. Annihilation spectra produced by pairs with a 1D thermal distribution have
been calculated by Kaminker et al. (1990), revealing significantly reduced Doppler broad-
ening at 01 "_ 90 o • However, they did not include the effects of magnetic broadening. In an
earlier paper, Kaminker et al. (1987) presented spectra for the annihilation of pairs at re,t
in the presence of a field. Here we present the first calculation of two-photon annihilation
spectra h'om a 1D thermal pair distribution, including full magnetic broadening effects
using the QED second-order annihilation cross section of Daugherty and Bussard (1980,
hereafter DB80). Of particular interest are the angle-dependent asymmetries of the line
profiles introduced by field broadening, signatures that may be detectable by the BATSE
instrument aboard the GRO mission and future high-resolution detectors (e.g. TGRS and
NAE).
MAGNETIC TWO-PHOTON ANNIHILATION
It is convenient to use a simple dimensionless notation for various quantities in rel-
ativistic magnetized environments, in order to express analytic results more compactly.
Hereafter, all magnetic fields will be expressed in units of the critical field Bc, and the
parallel pair temperature parameter T will represent the quantity kBT_/(m¢c2), where
kB is Boltzmann's constant. All photon and particle energies and momenta will be scaled
by m¢c 2 and m_c, respectively.
The conservation relations for energy and also for momentum parallel to the magnetic
field during pair annihilation can be simply written down. Let p_rnec and p+m¢c be the
momenta of the annihilating electron and positron, oppositely directed along the field, and
e_m¢c _ and e+rn¢c 2 be their energies. The produced photons have energies wlmec 2 and
w2m¢c 2 , and propagate at angles 01 and 02 to the field. Using the abbreviated notation
246
ci = cos0i, si = sin0i, i = 1,2, the conservation relations are
e_ +e+ =wl +o22
p_ -- p+ -- Og lcl -J_0.,' 2c 2
(1)
Alternatively these relations can be expressed in terms of the solutions for the pair
momenta p+ (see DB80). The kinematical requirement that these momenta be real gives
a restriction on the energy-angle phase space of the emitted photons:
w, cl +w2c2} _< <(w, +w2)2-4 (2)
For the annihilation of pairs with momenta p+m,c along the field, the distribution
function for the pairs that is needed for the computation of thermal annihilation spectra
can be obtained from equation (23) of Harding (1986):
1
rl+(p+) 2K_(1/T) exp{ -¢+.L ,- T. , (3)
where T is the dimensionless temperature of the pairs and I(1 is a modified Bessel func-
tion. The thermal annihilation spectrum can then be calculated by inserting this into the
expression in eq. (24) of DB80, who performed the QED calculation for the annihilation
cross-section from first principles. Their result is too small by a factor of two (c.f. Wun-
net, 1979), and also contains some minor typographical errors. The corrected annihilation
spectrum taken from DB80 is then
R27(Wl) = __ g+ d_2 Q27 (4a)
where
22 }n+n_c _wl _wls.1 +w2s 2 c_ +e+Q2-_- 4K_ti/T) 3 7r -ff exp 2B T E M2 (4b)
pol
Here at _ 1/137 is the fine structure constant, Ac = h/(rnec) and n+ are the particle
number densities. The matrix element .M can be obtained from the results of DB80. The
sum is over tile polarization states of the photons.
In obtaining eq. (4) a change of variables was performed: the energy 0, 2 and angle
02 of the unobserved photon were expressed in terms of the momenta P:t: of the pairs via
eq. (1). This transformation is convenient for both numerical and analytic computations,
yielding simple integration limits: with it is associated a Jacobian given by w2dw2 de2 =
[p_s+ +p+e_ [/(e_e+)dp_dp+. The azimuthal angle _2 of the second photon is the third
integration variable. In the limit of small 0a, the dominant contribution to the annihilation
spectrum occurs when the exponential factor in eq. (4b) is maximized.
247
NUMERICAL RESULTS
The results of numerical computations of the spectrum in eq. (4) are displayed in
Figures 1-3. The principle magnetic effectsare demonstrated in Fig. 1, where the pairs
are assumedto be at rest, i.e. zero temperature. This correspondsto the casestudied
numerically by DB80 and Kaminker et al. (1987), and indeed the results presented in
Fig. 1 are in close agreement with these earlier studies. With thermal broadening absent,
Fig. 1 clearly illustrates the magnetic broadening effect in the left wing of the two-photon
annihilation line. There are two prominent features to notice. First, the widths of the
annihilation lines are dependent on the strength of the field and the angle of emission
relative to field. These magnetic widths can be expressed approximately (for B << 1 ) as
(Ao,1)u { B/2, s, < x/-2B,B,/-ff 2, > (5)
remembering that sl = sin 01. These estimates can be obtained from eq. (4) or from the
results of Kaminker et al. (1987). Second, observe the appearance of angular-dependent
kinematic cutoffs, eliminating the right (blue) wing of the line. These occur at
2
o,, - , (6)
i -I- I_ll
as can be determined from eq. (2). Due to the presence of this kinematic cutoff at 0,1 > 1,
most of the broadening occurs on the left (red) side of the line at angles other than 900 .
Note that a summation of the spectra over emission angles removes the evidence of the
line asymmetry and the kinematic cutoffs, producing a symmetric line profile.
The inclusion of thermal broadening for the case of finite T is illustrated in Figs. 2-
3. The basic nature of Doppler broadening is similar to the 1D field-free case (Kaminker
et al., 1990), and the results show that the thermal contribution to the line broadening is
greatest for small viewing angles. The first order contribution to the Doppler width is
(/NO,l) D = ClV_ , (7)
when T << 1. The thermal broadening smooths out the line profile, and dominates the
blue wing of the line, but evidence of the magnetically-broadened left wing remains for
larger viewing angles. A comparison of eqs. (5) and (7) reveals that magnetic broadening
will dominate over first-order Doppler broadening when
61/C1 ' S1 > X'/_'
T _ 0.25B _2 2
confirming the immerical spectral results that the Doppler mechanism is more effective at
broadening the line for small viewing angles.
The calculations performed here are for conditions expected to be typical of gamma-
ray burst or pulsar environments. It is anticipated that pairs in the ground state are likely
to be in relatively cool quasi-thermal distributions. Cyclotron resonant scattering can
248
01 =0o 01 =45o
I0"12 I 10"13 I :\
B=0 _o_51 /_.J-/_t,o II
:: / ,I'
 0.F f / !Io-L_ _ / : ,
O0 0,5 l.O 0,0 0.5 1.0
COl Ol
10-13
01 = 900
B = 0.33
0.I /'
/
/
/
/
/
I
.033
'!
FIGURE 1 - Two-photon annihilation spectra for pairs at rest in the ground state, for
different magnetic fields B and viewing angles 01 from numerical integration of the cross
section over the energy w2 and angle 02 of the unobserved photon. The kinematic cutoffs
occur when Icos021= 1 (el. Eq. 6).
B=O.O1, 01 =0° B=O.1, 01 =0°
1043 , , lOq3 ......
"T' 1044
1045
©
"7
1 046
C¢3
O
_-. 10-17
cq
/
003t
1 0 -14
10-15
1046
10-17
.003
104s 10-Is ......
0.0 0.5 1.0 1,5 2.0 0.0 0.5 1.0 1.5
031 031
2.0
FIGURE 2 - Numerical annihilation spectra at an angle of 0 ° to the field direction for
pairs in a 1D thermal distribution with parallel temperature T (in units of m_c 2 ) at two
different field strengths.
249
"7,
"7,
;>
o
"7
o_
O'3
e'l
I0-13 ,
O1 = 0 O,
0.33
T = .01 T = .01
1.5
C01
2.0
1043
0t =00, T= 0.1
T
"7
10-14
"7,
;>
o
10 -15
"7
eel
1046
O
cq 10d7
"- .01
10-18 -- _ , , ,
0.0 0.5 1.0 1.5 2.0
C01
FIGURE 3 - Numerical annihilation spectra at different viewing angles 01 and parallel
pair temperatures T, showing the effect of combined magnetic and thermal broadening.
cool pairs in the ground state before they annihilate (Preece and Harding, 1991) to tem-
peratures T ,,, B/4 (Lamb et al., 1990). In this case magnetic broadening is dominant for
tan 2 01 > 1 or at viewing angles 01 > 45 o • This angular effect shows significant potential
as a spectral diagnostic for line observations in astrophysical sources.
250
ANALYTIC SPECTRAL FORMS
It is useful to develop analytic approximations to the annihilation spectrum. These
can be simply obtained in the low T, low B limit, and when the viewing angle is small:
01 _ 0 ° • The integrations in eq. (4) are then analytically tractable because of the effective
restriction to the w2- cos 02 (or equivalently the p_ -p+ ) phase space. For low T, we set
p_ = p+ = 0 in .M. Further, specializing to .51 _ 0 simplifies A/f dramatically (only one
term of a series remains -- see DB80 for details). In this case, M = N "(1) + N "(2) , with
N'O)
.if(z)
-- _.01 C 1 +
B +Od 1
2 {w2c2e_)e(2_)_
- B+w2
(9)
and the polarization vector components ez, 5:t: are given in DB80. The integration over
azimuthal angle turns out to be trivial, yielding
/?..-? { .. .}w2s 2 e_ + e+ - , (10a)R2-y(wl ) _ dp_.___+A exp 2B T
_- _+
oo oo
where
A -- n+n_c327r_-_-_001TB E { A/'(a) 2+ N,(2 ) 2} (10b)
pol
Since the pair temperature is low, the integrations over p_ and p+ can be performed by
the method of steepest descent. However it must be noted that there are two major regions
of contribution to these integrals: in the neighbourhood of p_ = -p+ = 0, corresponding
to c2 = -wlcl/w2 and a_2 = 2 - 001 , and also near c2 = -1 and a,'2 = 1/001 . The total
spectrmn is, in general, well approximated by the sum of these two contributions.
The contribution R_ from p+ _ 0 for 01 = 0 ° occurs only in the left wing of the
line. Evaluation of eq. (10) is then straightforward, resulting in
2 2 2Wl w_--NWl +2
R,(00,) B exp B '
This contribution is magnetically broadened, and is generally the dominant contribution
when T _< B 2 . It is consistent with the numerical plots, and reproduces the exact cold
plasma, zero viewing angle approximation derived in eq. (14) of Kaminker et al. (1987).
The second component R_,, for c2 _-1, p+ _ 0 and p_ _wl - 1/001, occurs in both
wings of the line and is a Doppler broadened contribution. When B £ v_, eq. (10) gives
COl' exp -
/"_li(_l) _ n+n_CO{_/_2e _/_ To) 1
which agrees with the B = 0, 1D thermal result in eq. (30) of Kaminker et al. (1990).
251
The results in equations (11) and (12), which can be extended to provide a more
complicated approximation to the spectrum when T ,-, B 2 , indicate that the annihilation
line is thermally broadened in the right wing (0.,1 > 1 ), but both thermally and magnet-
ically broadened in the left wing. This concurs with the numerical results plotted in the
figures. The widths obtainable from eqs. (11) and (12) agree with those in eqs. (5) and (7),
and from eq. (8) it can be deduced that when B << 1, small temperatures T _ B2/8 are
necessary for magnetic broadening to be important when 81 ,-_ 0 ° . Work is in progress to
extend these analytic results to the regime of larger angles 81.
CONCLUSIONS
The results presented in this paper show that two-photon annihilation of electrons
and positrons with 1D thermal distributions can be significantly influenced by a strong
magnetic field. Even for temperatures as high as T = 0.1, at large viewing angles to the
field, magnetic broadening dominates for B _ 1012 G. More specifically, in gamma-ray
burst models where the cooling of pairs by cyclotron resonant scattering is important,
the widths of annihilation features seen at viewing angles 8 > 45 ° will be determined
primarily by the magnetic field. At most viewing angles, the magnetic field preferentially
broadens the red side of the annihilation line. An integration over viewing angles removes
this asymmetry. Analytic approximations for zero viewing angle are obtained: the spec-
trum is well-described by a simple sum of magnetically-broadened and Doppler-broadened
contributions. For low T, the numerical computations are found to agree well with the
cold annihilation results of Kaminker et al. (1987). The results presented in this paper
have significant observational implications: the widths and spectral shapes of observed
annihilation features in gamma-ray burst and pulsar spectra may provide diagnostics of
both the magnetic field strength and viewing angle in these sources.
REFERENCES
Agrinier, B., et al.: 1990 A._trophys. J. 355, 645
Daugherty, J. K., and Bussard, R. W.: 1980 A_trophys. J. 238, 296 (DB80)
Harding, A. K.: 1986 Astrophys. J. 300, 167 (Corr. 462)
Kaminker, A. D., Pavlov, G. G. and Mamradze, P. G.: 1987 Astrophys. Sp. Sci. 138, 1
Kaminker, A. D., Pavlov, G. G. and Mamradze, P. G.: 1990 Astrophys. Sp. Sci. 174,241
Lamb, D. Q., Wang, J. C. L. and Wasserman, I.: 1990 Astrophys. J. 363, 670
Liang, E. P.: 1986 Astrophys. J. 304, 682
Mazets, E. P., et al.: 1981 Nature 290, 378
Preece, R. D. and Harding, A. K.: 1991 Astrophys. J., in press.
Wunner, G.: 1979 Phys Rev. Left. 42, 79
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Two-photon annihilation of thermal pairs in strong magnetic fields

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