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Generation of magnetic fields by nonuniform neutrino beams
25 August 1997 PHYSICS LETTERS A
Physics Letters A 233 (1997) 181-183
Generation of magnetic fields by nonuniform neutrino beams P.K. Shuklaa*‘, L. Stenflob, R. BinghamC, H.A. Bethe d, J.M. Dawson e, J.T. Mendoqa f a Fakultlltfiir Physik und Astronomie, Ruhr-Universitiit Bochum, D-44780 Bochum, Germany b Department of Plasma Physics, Umecf University, S-90187 Umed, Sweden ’ Rutheflord Appleton Laboratory, Chilton, Didcot, Oxon, OX11 OQX, UK e Department of Physics, Cornell University, Ithaca, NY 14853, USA e Department of Physics, University of Cabfomia, Los Angeles, CA 90024-1547, USA ’ Department of Physics, University of Lisbon, 1096 Lisbon, Portugal
Received 26 May 1997; acceptedfor publication 2 June 1997 Communicatedby V.M. Agranovich
Abstract
It is shown that the ponderomotive force of a nonuniform intense neutrino beam can generate large scale quasi-stationary magnetic fields in a dense electron plasma. This mechanismcan be responsiblefor the origin of magnetic fields in the early universe. @ 1997 Elsevier Science B.V. PACS:97.3O.Qd;95.3O.Cq;97.60.B~; 97.60.Gb
It is well known [ l-31 that there exist cosmic magnetic fields in the early universe whose origin is not understood. In the past, it has been suggested that the magnetic field in the Universe may have been created either by the Biermann-type battery effect [ 1 ] or by a galactic dynamo mechanism [ 21. In the Biermann effect a magnetic field is created as long as the electron temperature and density gradients are nonparallel, whereas the dynamo mechanism assumes the presence of a seed field ( ~10-20 G) which is subsequently amplified up to the observed level of microGauss in a spiral galaxy. Several alternative mechanisms for primordial magnetic fields are discussed in Ref. [3]. However, in the early universe high energy particles of different flavors are produced by stars and exploding supernovae. Thus, the background of most of the
high energy astrophysical plasmas, such as those in supernovae, star interiors, and neutron stars (pulsars), contains intense fluxes of neutrinos in addition to electrons, positrons, and quarks. The collective interactions between the electron neutrinos with the background plasma are of great importance with regard to the generation of electron plasma waves by parametric processes [ 41. In this Letter, it is shown that nonuniform neutrino beams propagating through a dense electron plasma can be responsible for the generation of quasi-static magnetic fields. Let us consider the propagation of an ensemble of random-phased electron neutrino wave packets in an electron plasma with a fixed ion background. The dispersion relation for the neutrino oscillations is [ 51
’ E-mail: ps@tp&uhr-uni-bochumde. 0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved.
PI1 SO375-9601(97)00450-7
(1)
182
l?K. Shukla et al./Physics
where E is the energy, p is the momentum, c is the speed of light, m is the neutrino mass, and V is equivalent to a potential energy defined by V = &Gn,; here G is the Fermi constant of the weak interaction, and n, is the electron number density. Assuming that the neutrinos have a particle- as well as a wave-like behavior, we identify E by &u,, and p by tik,, where w, is the frequency, k, is the wave vector, and li is the Planck constant. Thus, Eq. ( 1) yields w, = (k$* + m2c4/h2)‘/* + A( G/A)n,.
JZG fik,,c e’
(3)
where 0 = mc*/fi < k,c. Introducing the neutrino amplitude wave function @k, (not the wave function for a single neutrino) we can write the neutrino energy density as wk, = ck,,(]r,& ]*)/4r. On the other hand, expressing the neutrino power density Pk, in terms of W/cm*, we have 4, N wk,c. In a nonlinear dispersive medium, an ensemble of random-phased nonuniform neutrino beams exert a ponderomotive force [4] F = ( 1/8~) ck,( N 1) v(($k, I*) on the electrons as well as on the ions. In steady state, the sum of the electron and ion ponderomotive forces is balanced by the -j x B/c force. Here, ( ) denotes the ensemble average, j the plasma current density, and B the magnetic field. Accordingly, we have (4) where OG = Gm/fi and Q = (n,). By means of Ampere’s law VxB=Fj,
(5)
we rewrite Eq. (4) as (VxB)xB=
+
F Y
+(I#k.I*), Y
charge, Be is the azimuthal component of the magnetic field and me is the electron mass, we then obtain from Eq. (6)
Let us suppose that the intensity distribution of an ensemble of random-phased neutrino beams is of the form
(2)
The index of refraction N for the neutrinos is thus l-$2--n Y
Letters A 233 (1997) 181-183
(6)
Defining the electron gyrofrequency by w, = eBe/mc, where e is the magnitude of the electron
c(l+k.
I*) = WO exp(-r:/$?
(8)
kv
where ro is the effective radius of the neutrino beam and WO is the maximum neutrino energy density on the beam axis. Substituting Eq. (8) into Eq. (7)) we obtain W2 c = 4&J x [I-
WGWO rg -pe
ovEp r*
(l+$)q(
-$)I,
(9)
where wpe = (4rwe*/G)‘/* is the electron plasma frequency, and Ep = nomec2 is the plasma energy density. The gyrofr,ry at r = ro is thus roughly wpe (wG~o/w&) . To summarize, we have shown that the ponderomotive force of randomly distributed nonuniform neutrino beams can generate a quasi-stationary magnetic field. Physically [ 61, the radial gradient of the ponderomotive potential generates an electron current in the axial direction, which, in turn, becomes a source for the azimuthal magnetic field. By using a Gaussian distribution for random phased neutrino beams, we have obtained an explicit profile for the spontaneously generated stationary magnetic field. The latter is proportional to the square root of the neutrino power density. In conclusion, we suggest that the present model of neutrino driven magnetic fields could be successfully employed to understand the origin of primordial magnetic fields in the early universe. Acknowledgements
This work was supported by NSF PHY9 l-21052. One of the authors (R. Bingham) would like to thank
t?K. Shukla et al/Physics Letters A 233 (1997) 181-183
the Institute for Theoretical Physics, Santa Barbara, where some of this work was carried out. Referemxs [ 11 L,.Biermann, Z. Naturforsch. A 5 (1950) 65. [2] E.N. Parker, Cosmic magnetic fields (Clarendon, Oxford,
1979); Y.B. Zeldovich,
A.A. Ruzmaikiu and D.D. Sokoloff,
183
Magnetic fields in astrophysics (Gordon and Breach, New York, 1983). [31 R. Dpher and U.F. Wichoski, Phys. Rev. Len. 78 (1997) 787; K. Dimopoulos and A.C. Davis, Fhys. Lett. B 390 ( 1997) 87. 141 R. Bingham, H.A. Bethe, J.M. Dawson, PK. Shukla and 1.1. Su, Phys. Lett. A 220 (1996) 107. [51 H.A. Bethe, Phys. Rev. Lea. 56 (1986) 1305. [61 O.M. Gradov and L. Stenflo, Phys. L.&t. A 95 (1983) 233; PK. Shukla and MY. Yu, Plasma F’hys. Control. Fusion 26 (1984) 841.
Physics Letters A 233 (1997) 181-183
Generation of magnetic fields by nonuniform neutrino beams P.K. Shuklaa*‘, L. Stenflob, R. BinghamC, H.A. Bethe d, J.M. Dawson e, J.T. Mendoqa f a Fakultlltfiir Physik und Astronomie, Ruhr-Universitiit Bochum, D-44780 Bochum, Germany b Department of Plasma Physics, Umecf University, S-90187 Umed, Sweden ’ Rutheflord Appleton Laboratory, Chilton, Didcot, Oxon, OX11 OQX, UK e Department of Physics, Cornell University, Ithaca, NY 14853, USA e Department of Physics, University of Cabfomia, Los Angeles, CA 90024-1547, USA ’ Department of Physics, University of Lisbon, 1096 Lisbon, Portugal
Received 26 May 1997; acceptedfor publication 2 June 1997 Communicatedby V.M. Agranovich
Abstract
It is shown that the ponderomotive force of a nonuniform intense neutrino beam can generate large scale quasi-stationary magnetic fields in a dense electron plasma. This mechanismcan be responsiblefor the origin of magnetic fields in the early universe. @ 1997 Elsevier Science B.V. PACS:97.3O.Qd;95.3O.Cq;97.60.B~; 97.60.Gb
It is well known [ l-31 that there exist cosmic magnetic fields in the early universe whose origin is not understood. In the past, it has been suggested that the magnetic field in the Universe may have been created either by the Biermann-type battery effect [ 1 ] or by a galactic dynamo mechanism [ 21. In the Biermann effect a magnetic field is created as long as the electron temperature and density gradients are nonparallel, whereas the dynamo mechanism assumes the presence of a seed field ( ~10-20 G) which is subsequently amplified up to the observed level of microGauss in a spiral galaxy. Several alternative mechanisms for primordial magnetic fields are discussed in Ref. [3]. However, in the early universe high energy particles of different flavors are produced by stars and exploding supernovae. Thus, the background of most of the
high energy astrophysical plasmas, such as those in supernovae, star interiors, and neutron stars (pulsars), contains intense fluxes of neutrinos in addition to electrons, positrons, and quarks. The collective interactions between the electron neutrinos with the background plasma are of great importance with regard to the generation of electron plasma waves by parametric processes [ 41. In this Letter, it is shown that nonuniform neutrino beams propagating through a dense electron plasma can be responsible for the generation of quasi-static magnetic fields. Let us consider the propagation of an ensemble of random-phased electron neutrino wave packets in an electron plasma with a fixed ion background. The dispersion relation for the neutrino oscillations is [ 51
’ E-mail: ps@tp&uhr-uni-bochumde. 0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved.
PI1 SO375-9601(97)00450-7
(1)
182
l?K. Shukla et al./Physics
where E is the energy, p is the momentum, c is the speed of light, m is the neutrino mass, and V is equivalent to a potential energy defined by V = &Gn,; here G is the Fermi constant of the weak interaction, and n, is the electron number density. Assuming that the neutrinos have a particle- as well as a wave-like behavior, we identify E by &u,, and p by tik,, where w, is the frequency, k, is the wave vector, and li is the Planck constant. Thus, Eq. ( 1) yields w, = (k$* + m2c4/h2)‘/* + A( G/A)n,.
JZG fik,,c e’
(3)
where 0 = mc*/fi < k,c. Introducing the neutrino amplitude wave function @k, (not the wave function for a single neutrino) we can write the neutrino energy density as wk, = ck,,(]r,& ]*)/4r. On the other hand, expressing the neutrino power density Pk, in terms of W/cm*, we have 4, N wk,c. In a nonlinear dispersive medium, an ensemble of random-phased nonuniform neutrino beams exert a ponderomotive force [4] F = ( 1/8~) ck,( N 1) v(($k, I*) on the electrons as well as on the ions. In steady state, the sum of the electron and ion ponderomotive forces is balanced by the -j x B/c force. Here, ( ) denotes the ensemble average, j the plasma current density, and B the magnetic field. Accordingly, we have (4) where OG = Gm/fi and Q = (n,). By means of Ampere’s law VxB=Fj,
(5)
we rewrite Eq. (4) as (VxB)xB=
+
F Y
+(I#k.I*), Y
charge, Be is the azimuthal component of the magnetic field and me is the electron mass, we then obtain from Eq. (6)
Let us suppose that the intensity distribution of an ensemble of random-phased neutrino beams is of the form
(2)
The index of refraction N for the neutrinos is thus l-$2--n Y
Letters A 233 (1997) 181-183
(6)
Defining the electron gyrofrequency by w, = eBe/mc, where e is the magnitude of the electron
c(l+k.
I*) = WO exp(-r:/$?
(8)
kv
where ro is the effective radius of the neutrino beam and WO is the maximum neutrino energy density on the beam axis. Substituting Eq. (8) into Eq. (7)) we obtain W2 c = 4&J x [I-
WGWO rg -pe
ovEp r*
(l+$)q(
-$)I,
(9)
where wpe = (4rwe*/G)‘/* is the electron plasma frequency, and Ep = nomec2 is the plasma energy density. The gyrofr,ry at r = ro is thus roughly wpe (wG~o/w&) . To summarize, we have shown that the ponderomotive force of randomly distributed nonuniform neutrino beams can generate a quasi-stationary magnetic field. Physically [ 61, the radial gradient of the ponderomotive potential generates an electron current in the axial direction, which, in turn, becomes a source for the azimuthal magnetic field. By using a Gaussian distribution for random phased neutrino beams, we have obtained an explicit profile for the spontaneously generated stationary magnetic field. The latter is proportional to the square root of the neutrino power density. In conclusion, we suggest that the present model of neutrino driven magnetic fields could be successfully employed to understand the origin of primordial magnetic fields in the early universe. Acknowledgements
This work was supported by NSF PHY9 l-21052. One of the authors (R. Bingham) would like to thank
t?K. Shukla et al/Physics Letters A 233 (1997) 181-183
the Institute for Theoretical Physics, Santa Barbara, where some of this work was carried out. Referemxs [ 11 L,.Biermann, Z. Naturforsch. A 5 (1950) 65. [2] E.N. Parker, Cosmic magnetic fields (Clarendon, Oxford,
1979); Y.B. Zeldovich,
A.A. Ruzmaikiu and D.D. Sokoloff,
183
Magnetic fields in astrophysics (Gordon and Breach, New York, 1983). [31 R. Dpher and U.F. Wichoski, Phys. Rev. Len. 78 (1997) 787; K. Dimopoulos and A.C. Davis, Fhys. Lett. B 390 ( 1997) 87. 141 R. Bingham, H.A. Bethe, J.M. Dawson, PK. Shukla and 1.1. Su, Phys. Lett. A 220 (1996) 107. [51 H.A. Bethe, Phys. Rev. Lea. 56 (1986) 1305. [61 O.M. Gradov and L. Stenflo, Phys. L.&t. A 95 (1983) 233; PK. Shukla and MY. Yu, Plasma F’hys. Control. Fusion 26 (1984) 841.