- Email: [email protected]
Bremsstrahlung model of polarized radiation from magnetic white dwarfs
Volume
46A, number
4
PHYSICS
BREMSSTRAHLUNG
MODEL
FROM MAGNETIC
LETTERS
31 December
OF POLARIZED
1973
RADIATION
WHITE DWARFS
R.F. O’CONNELL Department of Physics and Astronomy,
Louisiana State University, Baton Rouge, Louisiana 70803, USA Received
Motivated by the desire to provide a model culate the intensities of right and left circularlv spheres of magnetic white dwarfs.
12 November
for the polarized radiation polarized bremsstrahltmg .
A new and very fruitful method of detecting stellar magnetic fields was initiated by Kemp [l] . He predicted that the continuous spectrum of light from magnetic white dwarfs should exhibit a fractional circular polarization 4, given by 4 = P+(w) -
P_(o)1 / [P+W + P_(w)1 = -(W(J). (1)
Hence P,(w) are the intensities of right and left circularly polarized light of angular frequency w, and s2 = (eB/2pc) is the Larmor frequency, where B is the magnetic field. This has led to the discovery of magnetic fields = ( lo6 - 107)G in at least four white dwarfs [2] . Nevertheless, Kemp’s prediction [l] that 4 - h (wavelength) is at variance with observations and it soon became apparent [2] that a more physical model than Kemp’s oscillator model [l] is required. Bremsstrahlung in a magnetic field has been considered by many authors [3-71 but, unfortunately, with no obvious agreement in the final results. Thus, we have carried out the calculation ab initio. For whitedwarf atmospheres, the electrons are nonrelativistic and the effects of the medium are negligible. As with [7] , we disagree with [S ] , but we also differ from [7] due to the fact that they used a free particle Green function whereas we used an exact function. Our results are consistent with those of [6] . In units fi = c = p = 1, the initial electron energy is E = (p2/2) + nwc,
n = 0,1,2,...
1973
(2)
where n denotes the Landau level, p the momentum along the magnetic field, and o, = 2R is the cyclotron frequency. Corresponding quantities in the final state will be denoted by a prime. With /3E (n,n’,p) we obtain,
observed from some white dwarfs, we calradiation emitted by electrons in the atmo-
for photon emission along B, CX3Z2n.
P,(d)
Q,(d)
z512,; (E-o-n’wc)V2
=-
’
(3)
where (Yis the fine-structure constant, ni is the number of ions per unit volume, Z is the atomic number,
Q+(M)
=
7 (tq2 dt ()
Q_(W3
=
rLJ ()
(ttX)2
J
(4)
nh’ nn”
h= (2wc)-‘(&2+2{(nn’)wc-w}]1’2-w}2,
(6)
J n’n = {(ntp
(7)
T(n’ntl,t)t(n’)1’27(n’-l,n,t)}
and, as given in [6] , Z(n, n’, t) = (n!r~‘!)-~‘~ eptj2 X t(n+n’)‘2 2Fo(-n’,--n;--t-1)
= S(n’,n, t) (8)
Except for a trivial factor of (2/n), our results agree with those of [6] , but are much different than the nonrelativistic limits of the results of [7] . We now use these results as the basis of a model of the polarized radiation from magnetic white dwarfs. It follows that the fractional polarization is given by 4(~,0)=
{~Q+-c~/(Q++Q_)~.
(9)
We see that 4 = 0 when n = n’. If wc x kT(T=104 K and B el O* G), then only a small number of the lowest 249
Volume 46A, number
PHYSICS
4
Landau levels are occupied. For X >> 1, we get q = 0 for n(n’) = 0 and n’(n) = 1. For h<< 1, we get q 22 --l for n(n’) = 0 and n’(n) = 1 i.e. only one type -of circularly polarized radiation is emitted. If kT >> w,, we may use Boltzmann statistics, then integrate over p and sum over n,n’ to get a Q,(w, 7) and hence a q(o, 7). These operations will have to be performed numerically. Details will appear elsewhere.
References [ll
J.C. Kemp,
2.50
Astrophys.,
J. 162 (1970)
169.
LETTERS
121 R.1:. O’Connell,
31 December
1973
Polarized radiation from magnetic white dwarfs, in Proc. Intern. Astronomical Union Symposium no. 53, Physics of dense matter, Boulder. Colorado, August 1972, eds. W. Brittin and C. Hansen (Reidel Publishers, 1973) gives a summary of observations and theoretical speculations. I31 R. Goldman and L. Oster, Phys. Rev. I29 ( 1963) 1469; R. Goldman, Phys. Rev. 133 (1964) A647. [41 R. Goldman and L. Oster. Phys. Rev. 136 ( 1964) A602. A606. Phys. Rev. 151 V. Canuto, I1.Y. Chiu and L. Fassio-Canuto, 185 (1969) 1607. [f31V. Canuto and H.Y. Chiu, Phys. Rev. A2 (1970) 5 18. and J. Arponen, Nuovo Cim. 11B (1972) [71 A. El-Gowhari 201.
46A, number
4
PHYSICS
BREMSSTRAHLUNG
MODEL
FROM MAGNETIC
LETTERS
31 December
OF POLARIZED
1973
RADIATION
WHITE DWARFS
R.F. O’CONNELL Department of Physics and Astronomy,
Louisiana State University, Baton Rouge, Louisiana 70803, USA Received
Motivated by the desire to provide a model culate the intensities of right and left circularlv spheres of magnetic white dwarfs.
12 November
for the polarized radiation polarized bremsstrahltmg .
A new and very fruitful method of detecting stellar magnetic fields was initiated by Kemp [l] . He predicted that the continuous spectrum of light from magnetic white dwarfs should exhibit a fractional circular polarization 4, given by 4 = P+(w) -
P_(o)1 / [P+W + P_(w)1 = -(W(J). (1)
Hence P,(w) are the intensities of right and left circularly polarized light of angular frequency w, and s2 = (eB/2pc) is the Larmor frequency, where B is the magnetic field. This has led to the discovery of magnetic fields = ( lo6 - 107)G in at least four white dwarfs [2] . Nevertheless, Kemp’s prediction [l] that 4 - h (wavelength) is at variance with observations and it soon became apparent [2] that a more physical model than Kemp’s oscillator model [l] is required. Bremsstrahlung in a magnetic field has been considered by many authors [3-71 but, unfortunately, with no obvious agreement in the final results. Thus, we have carried out the calculation ab initio. For whitedwarf atmospheres, the electrons are nonrelativistic and the effects of the medium are negligible. As with [7] , we disagree with [S ] , but we also differ from [7] due to the fact that they used a free particle Green function whereas we used an exact function. Our results are consistent with those of [6] . In units fi = c = p = 1, the initial electron energy is E = (p2/2) + nwc,
n = 0,1,2,...
1973
(2)
where n denotes the Landau level, p the momentum along the magnetic field, and o, = 2R is the cyclotron frequency. Corresponding quantities in the final state will be denoted by a prime. With /3E (n,n’,p) we obtain,
observed from some white dwarfs, we calradiation emitted by electrons in the atmo-
for photon emission along B, CX3Z2n.
P,(d)
Q,(d)
z512,; (E-o-n’wc)V2
=-
’
(3)
where (Yis the fine-structure constant, ni is the number of ions per unit volume, Z is the atomic number,
Q+(M)
=
7 (tq2 dt ()
Q_(W3
=
rLJ ()
(ttX)2
J
(4)
nh’ nn”
h= (2wc)-‘(&2+2{(nn’)wc-w}]1’2-w}2,
(6)
J n’n = {(ntp
(7)
T(n’ntl,t)t(n’)1’27(n’-l,n,t)}
and, as given in [6] , Z(n, n’, t) = (n!r~‘!)-~‘~ eptj2 X t(n+n’)‘2 2Fo(-n’,--n;--t-1)
= S(n’,n, t) (8)
Except for a trivial factor of (2/n), our results agree with those of [6] , but are much different than the nonrelativistic limits of the results of [7] . We now use these results as the basis of a model of the polarized radiation from magnetic white dwarfs. It follows that the fractional polarization is given by 4(~,0)=
{~Q+-c~/(Q++Q_)~.
(9)
We see that 4 = 0 when n = n’. If wc x kT(T=104 K and B el O* G), then only a small number of the lowest 249
Volume 46A, number
PHYSICS
4
Landau levels are occupied. For X >> 1, we get q = 0 for n(n’) = 0 and n’(n) = 1. For h<< 1, we get q 22 --l for n(n’) = 0 and n’(n) = 1 i.e. only one type -of circularly polarized radiation is emitted. If kT >> w,, we may use Boltzmann statistics, then integrate over p and sum over n,n’ to get a Q,(w, 7) and hence a q(o, 7). These operations will have to be performed numerically. Details will appear elsewhere.
References [ll
J.C. Kemp,
2.50
Astrophys.,
J. 162 (1970)
169.
LETTERS
121 R.1:. O’Connell,
31 December
1973
Polarized radiation from magnetic white dwarfs, in Proc. Intern. Astronomical Union Symposium no. 53, Physics of dense matter, Boulder. Colorado, August 1972, eds. W. Brittin and C. Hansen (Reidel Publishers, 1973) gives a summary of observations and theoretical speculations. I31 R. Goldman and L. Oster, Phys. Rev. I29 ( 1963) 1469; R. Goldman, Phys. Rev. 133 (1964) A647. [41 R. Goldman and L. Oster. Phys. Rev. 136 ( 1964) A602. A606. Phys. Rev. 151 V. Canuto, I1.Y. Chiu and L. Fassio-Canuto, 185 (1969) 1607. [f31V. Canuto and H.Y. Chiu, Phys. Rev. A2 (1970) 5 18. and J. Arponen, Nuovo Cim. 11B (1972) [71 A. El-Gowhari 201.