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The curvature radiation of cosmic monopoles
Chin.Astron.Astrophys.2 (1985) Chin.J.Sp.Sci.
Pergamon Press. Printed in Great Britain 0275-1062/85$10.00+.00
339-342
-5 (1985) 161-167
THE CURVATURE
RADIATION
OF COSMIC
MONOPOLES
LI Xiao-qing
Purple Mountain
LI Zhong-yuan
Department of Earth & Space Sciences, Science & Technology of China
Observatory,
Academia
Sinica University
of
Received 1984 August 1
ABSTRACT It is shown that curvature radiation of magnetic monopole is suppressed in a plasma medium.
Since Dirac proposed the possible existence of the magnetic monopole in 1931, [l], serious attempts have been made by theoretical physicists and astrophysicists on its detection and the effects it produces. An important topic in astrophysics is this: as the monopole passes through space and objects, how much energy loss does it suffer and does it leave any signature in radiation? Some interesting results have already been obtained in this regard. Also, in a cosmic plasma having an external magnetic field, the monopole will in general move along a curved path, so it must produce curvature radiation, which means some loss of energy, and the mutual "approach" leading to annihilation of monopole pairs must also first be related with the curvature radiation. Hence, there is a need to consider in some detail the curvature radiation of cosmic monopoles. As a monopole moves along a curved path. its velocity vector changes continuously while its speed may remain constant. Let the unit vector along the curve be b. We expand its orbit r(t) to the third-order term, + .L,,pw 6 JII'
r(t)
(1)
r(r) a r. + ,crb + $- (ac)Wb
+ + (,!k)Wb . v’)b,
d+(r) __ du _ ~-------------_, dr' dr
(3)
where B=v/c. The magnetic current generated by moving monopoles is j,dr,
1) =
gu(tMr
-
rb)),
(4)
where 4 is the magnetic charge of the monopole and 6 is the Dirac function. The Fourier components of (4) are L((,,,h> -
J+Ed!J--” exptr~w -L . r(,)lji,,,j,(r, ,) -_(2n) (2a)' +m
=g
. u(r)erp{i[or - R * r(t)]}
&,.
1_w
c5)
Substituting (3) in (5) and neglecting thirdorder small quantities in the amplitude, we have
where a, = &b, 4, = (Bc)‘(b
and write
. v>b
. v)b,
w, = o - ,YCR- b, db _ at
< *a = - + (Br)‘k * (b . v)b,
db dl dl dt
(7)
,ms= - + (Bc)‘k * (b . v)‘b.
or
It is easy to transform (6) into T
= u'(b . v)b,
(2)
where & is the arclength along the curved down a similar orbit. We can write 3 Then (1) becomes expression for d'r(t)/dt'.
i.(r,L)-g.ex~{iA--~.r~}II_~,[(r,-_~,)
+
where
(111
I
* exp{i(co,r’
+
at)},
(8)
LI and LI
P(k) (91
cf,
B=l
-
hj(w,
&)I (8. J*(m)
&)I 3
or 2. Hence we have
U"(R)- 2(2~,[--&]'
*R *
U"(R)= 2(2,y f--&]'
. R .
Hence (8) can be expressed in terms of Macdonald's functions of orders l/3 end Z/3, [8;* - j(d,
L)l[e:
(16b)
- j*(of,~)l.
Again, if n is the unit vector along the orbital curvature p, then we have
where
(be v)b = =/PI
(17al
(be VI% = (b . v)E(b - v)bJ = (a * v)(n/p)
(11)
In virtue of the Maxwell equations, c&l?-4xpf,
188
rotE=-----
c
1
4%.
c Jmr
8:
BU
(121
4s
divB-4=pm, rotB=~8;-+
tit.
I
dt
J
drj.,(+,
r>B(r,
[(b
. v)*J
r>;
=
$ (-
$)
ri
-
$_
(17b)
Let us choose the orbital plane as the xyplane, and let K be in the xz-plane, making an angle 9 with the z-axis, and let the direction b of the instantant motion be along the x-axis, and let n point to the negative y direction. In these coordinates, K-n=0 and from (7) and (17) we see wz= 0, and es is along the y-direction, hence 8; . b = cosf?, l: an=-I,
the current generated by monopoles moving in a plasma will induce a field B, and work is done by the current on the field so that the energy of the field is increased, that is, radiation is produced. ~0 ==-
= $
K-b-
sin8,
Noting these, and substituting (10) into (16a) and (16b), we easily obtain, for the total emitted energy, V(R) = u"(R)+ P(k)
(13) The velocity of the super-heavy magnetic monopole is always non-relativistic, Bc
and we may obtain, [6]
(dJ)2 = b.? pe
+
k2c2
where R=
W4 hence R=1/2.
t&w, R) = E;,(% RMkM~), U” =
1&U*(b); .
Also,
(15b) (15cl
wo is the frequency of the emitted wave, and it is a function of R(=k/k), cii(w, k) is the dielectric tensor of the plasma medium and co(k) is the unit polarization vector of the electric field of the wave, with two independent polarizations, eel and e"2. Here, K, ey, ez form an orthogonal triad, and the polarization tensor is
we have replaced k by &/c in (ZO), since, for very small values, Macdonald's functions rapidly decrease, the emitted intensity tends rapidly to zero, making no contribution to the total emission. Hence from
341
Curvature Radiation
we obtain
Let P(8 , t) be the power per unit solid angle. We have
and, for the Macdonald function itself, we take the asympotic formula for large argument and obtain
From now on, we shall omit the super-fixa in i&P. Applying inverse Fourier transform to Maedonald's functions in U(W), e.g.,
*
case* ctg6.
(28)
We now integrate (27) and (28) over all angles and obtain substituting into (22), and interchanging the order of integrations,we get
Applying the inverse transformation,
where 4" (2m -i-%)/W! l
(29’1
Using the approximation gives
4 1.2* 1 - r g, (l--x) 2 and the following known integral, [6],
x WI-2*-uI k , (Rerc>O, Rt#?>O), --T-ST 0 Y Integrating both sides with respect to w from wpelwc to m, and interchange the order of integrations on the right side, we have
we obtain
Similarly, we obtain where
Similarly, we have
Therefore, the total power is P-P’+-PU,
For the last integral in (25), it is sufficient to use the asymptotic expression,
(321
LI and LI
W’CI that is, when fZu+ wPe) > wo', the radiation is completely suppressed, while if Y-2 >>wPe2/W2, then waves satisfying the inequality w
In general, we have wp, >
Generally speaking, it is possible for
d ,
pulsars to capture a quantity of cosmic monopoles. In this case, p%107cm,and
hence we have
for
w to be a radio frequency (%lO*), we require
(36) If we do not include the effect of plasma, that is, in the case wpebOo' the total power is, approximately,
Y%lOO.
On the other hand, wpe<
this corresponds to ne"3, very far from the region of polar caps. Therefore, it is impossible for a relativistic monopole to emit radio photons in the polar caps of pulsars, and, in regions far from the polar
Based on the above results, we present the following comments: 1) Formula (36) shows that the curvature radiation of monopoles is strongly suppressed by the plasma. We may say that it is almost impossible for the nonrelativistic monopoles to produce curvature radiation in a plasma medium. 2) The %nnihilation" of monopole pairs must be preceded by their mutual "approach", but the latter event is possible only after they have lost some energy through curvature radiation. In an environment of plasma, this pre-condition for annihilation is not satisfied and the positive and negative monopoles first pass each other by. 3) If the monopole has a relativistic velocity, then
cap, the rapid increase in p means a drastic decrease in the frequency w& and the curvature radiation of the monopole is, in general, even more suppressed. A similar situation should also apply to electrons.
REFERENCES
PI Dirac, P.A.,
proc.
Phys. Sot., Vol. A133,
p. 60, 1931.
PI Cabrera, B., Phys. Rev. Lett., Vol. 46, p. 1378, 1982. I31 Parker, E.N., Ap. J., Vol. 163, p. 225, 1971. t41 Parker, E.N., Ap. J_, Vol. 166, p. 295, 1971. PI Martel'yanov, V.P. and Khakimov S. Kh., 2h.E.T.F.
62 (1972) 35.
PI Gradshtein I.S. and Rizhik I.M., "Table of Integrals etc." (1971) p. 353.
where, in the last step, we have put K*b= sin0 =l, because, from the above estimate we know that the decay characteristicsof Macdonald's function at large argument mean that the largest contribution comes from the ease of K and b parallel. The argument z is then
In this case, we need not repeat the calculation following (18) in order to see that, similar to (30) and (31), the frequency here, uoc should be replaced by
Pergamon Press. Printed in Great Britain 0275-1062/85$10.00+.00
339-342
-5 (1985) 161-167
THE CURVATURE
RADIATION
OF COSMIC
MONOPOLES
LI Xiao-qing
Purple Mountain
LI Zhong-yuan
Department of Earth & Space Sciences, Science & Technology of China
Observatory,
Academia
Sinica University
of
Received 1984 August 1
ABSTRACT It is shown that curvature radiation of magnetic monopole is suppressed in a plasma medium.
Since Dirac proposed the possible existence of the magnetic monopole in 1931, [l], serious attempts have been made by theoretical physicists and astrophysicists on its detection and the effects it produces. An important topic in astrophysics is this: as the monopole passes through space and objects, how much energy loss does it suffer and does it leave any signature in radiation? Some interesting results have already been obtained in this regard. Also, in a cosmic plasma having an external magnetic field, the monopole will in general move along a curved path, so it must produce curvature radiation, which means some loss of energy, and the mutual "approach" leading to annihilation of monopole pairs must also first be related with the curvature radiation. Hence, there is a need to consider in some detail the curvature radiation of cosmic monopoles. As a monopole moves along a curved path. its velocity vector changes continuously while its speed may remain constant. Let the unit vector along the curve be b. We expand its orbit r(t) to the third-order term, + .L,,pw 6 JII'
r(t)
(1)
r(r) a r. + ,crb + $- (ac)Wb
+ + (,!k)Wb . v’)b,
d+(r) __ du _ ~-------------_, dr' dr
(3)
where B=v/c. The magnetic current generated by moving monopoles is j,dr,
1) =
gu(tMr
-
rb)),
(4)
where 4 is the magnetic charge of the monopole and 6 is the Dirac function. The Fourier components of (4) are L((,,,h> -
J+Ed!J--” exptr~w -L . r(,)lji,,,j,(r, ,) -_(2n) (2a)' +m
=g
. u(r)erp{i[or - R * r(t)]}
&,.
1_w
c5)
Substituting (3) in (5) and neglecting thirdorder small quantities in the amplitude, we have
where a, = &b, 4, = (Bc)‘(b
and write
. v>b
. v)b,
w, = o - ,YCR- b, db _ at
< *a = - + (Br)‘k * (b . v)b,
db dl dl dt
(7)
,ms= - + (Bc)‘k * (b . v)‘b.
or
It is easy to transform (6) into T
= u'(b . v)b,
(2)
where & is the arclength along the curved down a similar orbit. We can write 3 Then (1) becomes expression for d'r(t)/dt'.
i.(r,L)-g.ex~{iA--~.r~}II_~,[(r,-_~,)
+
where
(111
I
* exp{i(co,r’
+
at)},
(8)
LI and LI
P(k) (91
cf,
B=l
-
hj(w,
&)I (8. J*(m)
&)I 3
or 2. Hence we have
U"(R)- 2(2~,[--&]'
*R *
U"(R)= 2(2,y f--&]'
. R .
Hence (8) can be expressed in terms of Macdonald's functions of orders l/3 end Z/3, [8;* - j(d,
L)l[e:
(16b)
- j*(of,~)l.
Again, if n is the unit vector along the orbital curvature p, then we have
where
(be v)b = =/PI
(17al
(be VI% = (b . v)E(b - v)bJ = (a * v)(n/p)
(11)
In virtue of the Maxwell equations, c&l?-4xpf,
188
rotE=-----
c
1
4%.
c Jmr
8:
BU
(121
4s
divB-4=pm, rotB=~8;-+
tit.
I
dt
J
drj.,(+,
r>B(r,
[(b
. v)*J
r>;
=
$ (-
$)
ri
-
$_
(17b)
Let us choose the orbital plane as the xyplane, and let K be in the xz-plane, making an angle 9 with the z-axis, and let the direction b of the instantant motion be along the x-axis, and let n point to the negative y direction. In these coordinates, K-n=0 and from (7) and (17) we see wz= 0, and es is along the y-direction, hence 8; . b = cosf?, l: an=-I,
the current generated by monopoles moving in a plasma will induce a field B, and work is done by the current on the field so that the energy of the field is increased, that is, radiation is produced. ~0 ==-
= $
K-b-
sin8,
Noting these, and substituting (10) into (16a) and (16b), we easily obtain, for the total emitted energy, V(R) = u"(R)+ P(k)
(13) The velocity of the super-heavy magnetic monopole is always non-relativistic, Bc
and we may obtain, [6]
(dJ)2 = b.? pe
+
k2c2
where R=
W4 hence R=1/2.
t&w, R) = E;,(% RMkM~), U” =
1&U*(b); .
Also,
(15b) (15cl
wo is the frequency of the emitted wave, and it is a function of R(=k/k), cii(w, k) is the dielectric tensor of the plasma medium and co(k) is the unit polarization vector of the electric field of the wave, with two independent polarizations, eel and e"2. Here, K, ey, ez form an orthogonal triad, and the polarization tensor is
we have replaced k by &/c in (ZO), since, for very small values, Macdonald's functions rapidly decrease, the emitted intensity tends rapidly to zero, making no contribution to the total emission. Hence from
341
Curvature Radiation
we obtain
Let P(8 , t) be the power per unit solid angle. We have
and, for the Macdonald function itself, we take the asympotic formula for large argument and obtain
From now on, we shall omit the super-fixa in i&P. Applying inverse Fourier transform to Maedonald's functions in U(W), e.g.,
*
case* ctg6.
(28)
We now integrate (27) and (28) over all angles and obtain substituting into (22), and interchanging the order of integrations,we get
Applying the inverse transformation,
where 4" (2m -i-%)/W! l
(29’1
Using the approximation gives
4 1.2* 1 - r g, (l--x) 2 and the following known integral, [6],
x WI-2*-uI k , (Rerc>O, Rt#?>O), --T-ST 0 Y Integrating both sides with respect to w from wpelwc to m, and interchange the order of integrations on the right side, we have
we obtain
Similarly, we obtain where
Similarly, we have
Therefore, the total power is P-P’+-PU,
For the last integral in (25), it is sufficient to use the asymptotic expression,
(321
LI and LI
W’CI that is, when fZu+ wPe) > wo', the radiation is completely suppressed, while if Y-2 >>wPe2/W2, then waves satisfying the inequality w
In general, we have wp, >
Generally speaking, it is possible for
d ,
pulsars to capture a quantity of cosmic monopoles. In this case, p%107cm,and
hence we have
for
w to be a radio frequency (%lO*), we require
(36) If we do not include the effect of plasma, that is, in the case wpebOo' the total power is, approximately,
Y%lOO.
On the other hand, wpe<
this corresponds to ne"3, very far from the region of polar caps. Therefore, it is impossible for a relativistic monopole to emit radio photons in the polar caps of pulsars, and, in regions far from the polar
Based on the above results, we present the following comments: 1) Formula (36) shows that the curvature radiation of monopoles is strongly suppressed by the plasma. We may say that it is almost impossible for the nonrelativistic monopoles to produce curvature radiation in a plasma medium. 2) The %nnihilation" of monopole pairs must be preceded by their mutual "approach", but the latter event is possible only after they have lost some energy through curvature radiation. In an environment of plasma, this pre-condition for annihilation is not satisfied and the positive and negative monopoles first pass each other by. 3) If the monopole has a relativistic velocity, then
cap, the rapid increase in p means a drastic decrease in the frequency w& and the curvature radiation of the monopole is, in general, even more suppressed. A similar situation should also apply to electrons.
REFERENCES
PI Dirac, P.A.,
proc.
Phys. Sot., Vol. A133,
p. 60, 1931.
PI Cabrera, B., Phys. Rev. Lett., Vol. 46, p. 1378, 1982. I31 Parker, E.N., Ap. J., Vol. 163, p. 225, 1971. t41 Parker, E.N., Ap. J_, Vol. 166, p. 295, 1971. PI Martel'yanov, V.P. and Khakimov S. Kh., 2h.E.T.F.
62 (1972) 35.
PI Gradshtein I.S. and Rizhik I.M., "Table of Integrals etc." (1971) p. 353.
where, in the last step, we have put K*b= sin0 =l, because, from the above estimate we know that the decay characteristicsof Macdonald's function at large argument mean that the largest contribution comes from the ease of K and b parallel. The argument z is then
In this case, we need not repeat the calculation following (18) in order to see that, similar to (30) and (31), the frequency here, uoc should be replaced by