The spectrum, directivity and state of polarization is presented of the bremsstrahlung radiation expected from a beam of high energy electrons spiraling along radial magnetic field lines toward the photosphere. The results are used for calculation of the characteristics of the reflected plus direct flux

Langer, S. H.

Petrosian, V.

English

NASA Technical Reports Server

General Disclaimer 
One or more of the Following Statements may affect this Document 
 
 This document has been reproduced from the best copy furnished by the 
organizational source. It is being released in the interest of making available as 
much information as possible. 
 
 This document may contain data, which exceeds the sheet parameters. It was 
furnished in this condition by the organizational source and is the best copy 
available. 
 
 This document may contain tone-on-tone or color graphs, charts and/or pictures, 
which have been reproduced in black and white. 
 
 This document is paginated as submitted by the original source. 
 
 Portions of this document are not fully legible due to the historical nature of some 
of the material. However, it is the best reproduction available from the original 
submission. 
 
 
 
 
 
 
 
Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19770010033 2020-03-22T10:51:17+00:00Z
Impulsive Solar X-Ray Bursts.
III. Polarization and Directivity
of Bremsstrahlung Radiation
from a Beam of ElectronsT
D	 7
r	 0
1	 ^ r.•
Directed Toward the Photosphere
by
'	 Steven H. Langer
L N --	 Vahe Petrosion
I	 'r i-	 y:24 `V
July 1976
L
i C.
^ U
4	 ^
n
J
SUIPR Report No. 668
National Aeronautics and
Space Administration
Grants NSF 7092 and
NG L 05-020-272
FEB 1977
N	 RECEIVED
NASA SP FACILITY
INPUT BRANCH
)UN/p^Q 
G^i
`	 N
• ^CAhato
INSTITUTE FOR PLASMA RESEARCH
STANFORD UNIVERSI TY, STANFORD, CALIFORNIA
AW
s1
1
IMPULSIVE SOLAR X-RAY BURSTS. III. POLARIZATION AND
DIRECTIVITY OF DREMSSTRAHLUNG RADIATION FROM A DEA51
OF ELECTRONS DIRECTED TOWARD THE PHOTOSPHERE
by
Steven 11. Langer and Vand Petrosian
July 1976
SUIPR Report No. 666
National Aeronautics and Space Administration
Grant NSG 7092
1	 and
Grant NGL 05-020-27P
Institute for Plasma Research
S.tanf.ord. University 	
-
Stanford, California
i
IMPULSIVE SOLAR X-RAY BURSTS. III. POLARIZATION AND DIRECTIVITY OF
BREMSSTRAHLUNG RADIATION FROM A BEAM OF ELECTRONS DIRECTED
TOWARD THE PHOTOSPHERE
Steven H. Langer* and Vahe Petrosian*
Institute for Plasma Research
Stanford University
ABSTRACT
We present the spectrum, directivity and state of polarization
of the bremsstrahlung radiation expected from a beam of high energy
electrons spiraling along radial magnetic field lines toward the
.
	
	
u
photosphere. The results from this paper are used in the accom-
panying paper for calculation o1 the characteristics of the reflected
plus direct flux.
1
1
a
1
1
a
a
a
Also Department of Applied Physics
y
r
{
Ire1	
__ — _^_^.e.^,
	
_	
..».:	
,n:=rrmc..'Y^,Z<^Ai.3¢x9rss?c.:+vw+c'-cwrwsr+'e
u I.	 INTRODUCTION
r
The observed properties of impulsive x-ray bursts have led us to a
^j model in which the bulk of the x-rays are emitted in the form of brems-
E strahlung	 from n beam of high energy electrons directed toward the
.,
photosphere (c.f. Petrosian 1973, Paper I, and 1975) •
	In Paper I we
f calculated the spectrum and angular distribution of the hard x-rays
(> 10 keV) produced by an electron beam directed radially into the
;j photosphere.
1
The purpose of this paper is to extend these earlier calculations.
	
a
to include the effer	 of the electrons spiraling around magnetic field
1
4 lines (non-zero pitch angles) and to calculate the polarization of the
x-rays.	 This	 'y.	 problem has previously been considered by Haug . (1972) and
' Brown (1972).
	 Haug's results are for a thin target model and Brown
	
{
,i
only published results for the polarization of x-rays directed away from
the sun.	 However, to calculate the spectrum, angular dependence and polari-
zation of the x-rays reflected by the photosphere we need the characteristics
of the bremsstrahlung x-rays emitted in all directions.
	 We briefly discuss
theprooedure used In this calculation (Sec. II)and present the results_
(See, UI)..	 These results are then used in the accompanying paper to
.; calculate the characteristics of the radiation reflected from the photo-
t
sphere and the total observable flux .(direct plus reflected).
'j
<<
II.
	
CALCULATIONS	
-
The bremsstrahlung cross sections. nee:dedin -our calculations have
j been derived. by May (1951) and by Gluckstern and Hull (2.953) [to which we
apply Elwert.'s (1939) Coulomb_ correction].. The cross sections
	 dal (E,	 k,'C^)
(^
I
1
i
L	 4
REPROnuc>BILrrY or
+^RIe[NAL PAGIr IS P00r
and dull (E, k) 0) are the cross sections for the emission of photons
polarized perpendicular and parallel to the plane of the electron-photon
momenta as a function of initial electron energy E, photon energy k and
•'I	 -1
the angle 0 between the electron and photon momenta p and k. However,
for non-zero pitch angles this plane ks constantly changing as tae electron
spirals around the magnetic field lines, which are assumed tobeuniform,
fixed and radial (the coordinate system is chosen so that the magnetic
field vector H is in the z-direction). It is convenient to fix the
direction of observation with respect to the field lines in which case
we need cross sections da.i and doll/ for emission of photons with
polarization perpendicular and parallel to the plane of the magnetic
-1
field and the photon momentun k = k(sine, o, cose). As shown by Haug
(1972)
da.L + doll' = dot + dall
dui - doll' = cos 2,U(dal - dull)
where	 the angle between the two planes, is given by:
	
p x k.	 H x k _ (sine cosh - nose sine) costp) 2- sin 2n sin 2mcos	
_ 
^B x kl	 IH x kl (sing cosh - cosB sinT^ costp) 2+ in'll sin2cp	 (2)
Here) p the initial momentum of an electron with pitch angle 71 ) is
p = p(sin7l cosrp, sinjj sinp, cosq where cp rims from 0 to 2u for one
revolution. With this geometry, which is the same as that described by
(1)
where
r	 ^
a IBrown (1972) ;
y -I
cos 0 =	 p k	 = cosTI cose + cos[p sinTj sin g
	(3)
^p^
In paper I ( equation 20) we showed that for a power law electron
spectrum ( dS = KE 6-1dE) the integral and differential spectra of
the photon energy flux ( all energies expressed in units of m ec and K has
unit of sec-1) are given by
J(k) = 6 (2rt1nA/,/
	
E1-S(3/2F(E',k)dE'
k
//'	
(4)
J(kre) 
= 6 (2rt nA )Jk m E, 
613 f(E^)k,B)dE'
I
27t
f E k B
	
1f
 dm k . da^t(E^k,^O) +.^n( E CIQ j( ° ° ) _ ^2 	 2rt	 dk dQ	 dk d(Z J
p 
it
	
F ( E,k) ° fi /
	
f(E,k,B)sinB de
To .calculate the degree of polarization we need
	
Q( k^ e ) = b (21rinAV 	 k, B) dE'	 (6)
3
t{
(5)
7
.2
f
E
J
1
g(E^k,B)	 1 2^ 21r2rt k 
dQl E k 0) _dull (	 0)1 cos 2*
^0 JJ0	
dk eC! J
In the above equations P is the electron velocity in units where the
speed of light c = 1, a is the fine structure constant, r0 is the
classical electron radius, and A is given by equations (2) and (3) of
paper I.
We define the directivity1 and degree of polarization as
D( kj q ) = JTkke)
(S)
1 1 B
The total flux, J(k), is independent of the pitch angle I
because the amount of radiation emitted by an electron is independent of
the direction of travel as long as the thick target assumption holds.
The quantity R 2 anA k6-1J(k) is shown in figure 3 of paper I. The
x-ray spectrum is essentially a power law (but with an index about one
smaller than the electron spectral index -b).
The directivity is shown in figures la, b, and c for a b = 4
electron spectrum and three values of the pitch angle 1. In all cases
the radiation is strongly beamed into the photosphere. The shape of the
beam changes with pitch angle with the maximum directivity dropping and
1D is the same as the function R, of paper I. the new notation is	 -
adopted to be consistent with the work of Brown (1972)) Henoux (1975)
and others.
4
s	 ^
r	 _	
I
(7)
the peak moving away from the magnetic field direction as r increases.
The fraction of the radiation beamed into the photosphere decreases from
about 85% at 11- 0 to 80% at T1 = 300 for a photon energy of 46.4 keV
and radial Yield lines.
The degree of polarization is shown in figures 2a ) b and c for a
1
g 4 power law and three pitch angles. .',t low energies the radiation
is polarized in the plane containing the observer, the flare site and the
i
I
center of the sun. As the energy increases the polarization decreases and i
finally changes sign at high enough energies ) so that the plane of polari-
zation 13 normal to the plane found at lower energies. The magnitude of
the polarization decreases as the pitch angle increases. Figures 3a and b
show the variation of the degree of polarization with pitch angle at two
different photon energies.. The variation is slow and smooth.
Figures 4a and b show the variation of the polarization with the power
law index at zero pitch angle. The degree of polarization is a monotonically
increasing function of the power law index.
In the accompanying paper we use these results to calculate the
characteristics of the radiation which is reflected from the photosphere
via Compton scattering. The reflected radiation is then added to that which
escapes directly from the production region to yield a prediction for the
characteristics of the observed x-ray flux.
ACKNOIVLEDGMENT
We would like to thank the National Center for Atmospheric Research,
.which is sponsored by the National Science Foundation . for computer time
used in this calculation. We would. also like to thank Mr. G. Langer of
5
Y
ae
L,_ ,
a
NCAR for assistance in the computations. This work was supported by the
National Aeronautics and Space Administration under grants NSG 7092
and NGL 05-020-272.
i
I
I/
f
REFERENCES
Brown, J. C. 1972, Solar Phys., 26, 441.
Elwert, G. 1939, Ann. Phys., 34, 17e,
Gluckstern, R. L., and Hull, M. H. 1953, Phys. Rev., 90, 1030.
Haug, E. 1972, Solar Phys., E5, 425.
Henoux, J. C. 1975, Solar Phys., 42, 219.
May, M. M. 1951, Phys. Rev., 84, 265.
Petrosian, V. 1973, Ap. J., 186, 291 (Paper I).
Petrosian, V. 1975) Ap. J. 197, 235•	 j
7
s
t
t FIGURE CAPTIONS
Figure 1 Directivity of the x-ray energy flux for nn electron spectrum
with index b = 4. Curves are labelled by the value of the
photon energy in keV. CosO = 1 for photons ;Irected radially
into the photosphere.	 (a) Pitch angle (1 = 0; (b) 7j = 300;
( e )	 = 450•
Figure 2 Same as figure 1 except for degree of linear polarization:
( a)	 = 0; ( b )	 300; ( c)	 = 450•
Figure 3 Degree of polarization for 6 = 4 and various values of the
Pitch angle in degrees as indicated. (a) Photon energy
k = 21.5 keV; (b) Photon energy k = 100 keV.
Figure 4 Same as figure 3 except	 = 0 and for d = 2 1 3y 4, 5 as
labelled. (a) k = 21.5 keV; (b) k = 100 keV.
.01
t
A6 A'
/A6A^	
lv
i
100
10
. 3
1
wi
a
.I
ii
i
1
Ja
1
-1.0 -.8 -.6 -.4 -.2 .0
	 .2	 .4	 .6	 .8	 1.0
COSO
'
]1+	
Figure la
9
i
.01 1	 f	 1	 1 _i	 1	 i	 1	 1	 f	 I	 i	 1	 I	 1	 I	 I _II
-1.0 -.8 -.6 -.4 -.2 .0
	 .2	 .4	 .6	 8	 1.0
COSO
Figure 16
t1
t
1 Y
100
a
c
r'
10
.01
-1.
	 U	 .2	
.4	
.6	 .8	 1.0
COSU
10C
10
N
D
Figure lc
ll
^EPROATJCIBII,1'1'Y C1T+' '!`?.^.
ORI81NAL PAGE IS POni'
.4
COSO
C .2O
vN
.0
v
0CL
`+- -.2 
O
aD
O
M -.4
C
-.6
-.8
964.
goo
F6,
1.0	 -.8	 -.6	 -.4	 -.2	 .0 .2	 .4	 .6	 8	 1.(
r.
.4
O
d
N_
.0
O
O
Q_
-.2
O
N
-.4O
-.6
r,
-.8
i
0
COSO
Figure 2h
-1000
^_-969
p	
Ipp
-1.0	 -.8	 .6	 -.4	 -.2	 .0	 .2	 .4	 .6	 8	 1
^=	 .2
O
d
N_
O
O
`^-	 -.2
O
N
N
-.4
N
-.6
.4
1000
969
10	 Z^
q6. q
-.8
-1.0 -.8 -.6 -.4 -.2 .0	 .2	 .4	 .6	 .8	 1.0
COSO
Figure ?c
j	
14
J
d
COSO
Figure 3a
"TT'—^T ^^Tr—rte
95
3p
a^
1.0	 -.8	 -.6	 -.4	 -.2	 .0 .2	 .4
,i
. 
4
tq
U
.2
`'	 D
N
C3
U
Q_
^^	
N
N
-.4
yiy
l
i
i
1
3
J
45
30
20
0
1.0
	 -.8
	 -.6
	 -.4
	 -.2
	 .0	 2	 .4	 .6	 .8	 1.0
.4
C	
.2O
N
"i .0
_a
O
`'- -.2 O
G	
^,
W
N
L
-.4
D
-.6
-.8
!f
COSO
Figure 36. .
16
2
ii
.4
.2C
c
'^- . 0
_G
O
n
`+- -.2 
O
N
N
G^ .4
N
-	 D
-.6
-.8
T^ ^_—r_TT
—^ T r—^— —r— r— r—
r
s ^'
-1.0 -.8 -.6 -.4 --.2 .0	 .2	 .4	 .6	 .8	 1.0
COSO
Figure 4a	 -
17
0
._•,Kn._.bw.•.nww	 '^---re..,'"%--.,-r .. --ra^..,	 tea...	 rw^
YCOSO
rim,xc 4n
1fl
i
2
a
S
1.0	 -.8	 -.6	 -.4	 -.2	 .0	 .2	 .4	 .6	 .8	 1.
4
0
s-- • 2O
O
.N
i .0
_U
U
`+- -.2 O
`.	 U
U
L
-.4UO
-.6
-.8
t


Impulsive solar X-ray bursts. 3: Polarization and directivity of bremsstrahlung radiation from a beam of electrons directed toward the photosphere

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server