Transverse Quasilinear Relaxation in Inhomogeneous Magnetic Field

Home , Peter Goldreich

arXiv:physics/9802026v1 [physics.plasm-ph] 13 Feb 1998 h eoatparticle, resonant the ra ait fteosre usrphenomena. pulsar observed the cyclotron-Cheren of that variety [6]), broad (e.g. a shown been has It light. of speed where atri h usrframe, pulsar the in factor antshr a ensgetdb 2 n 3 n eeoe lat Doppler anomalous developed the and at [3] develops and instability cyclotron-Cherenkov [2] by th pulsar suggested of of been instability lines cyclotron-Cherenkov has the magnetosphere field of open possibility The the on [1]). (e.g. conditions physical the th field describes magnetic This strong the along propagating beam relativistic highly rnvreqaiierrlxto nihmgnosmagne inhomogeneous in relaxation quasilinear Transverse nti ae ecnie uslna eaaini nooeeu mag inhomogeneous in relaxation quasilinear consider we paper this In hoeia srpyis aionaIsiueo Technolog of Institute California Astrophysics, Theoretical ω ( k n ue r ossetwt h n bevdfo pulsars. from observed one the with consistent are fluxes and mtdb h oe a ihteseta index spectral the with law power the by can imated range frequency broad a in but dependence, frequency gener law intensities wave resulting cons The the to invariant. due adiabatic field inhomogeneous the in arising force plas the pair by magnetized strongly through propagating beam a of cyclotron-Ch kinetic the which in states quasilinear magnetosphere find We pulsar of field magnetic inhomogeneous the in stefeunyo h omlmode, normal the of frequency the is ) rnvreqaiierrlxto ftecyclotron-Chere the of relaxation quasilinear Transverse ω B = q ω stecag ftersnn particle, resonant the of charge the is | ( q k | ) B/mc − .INTRODUCTION I. k k v 2 aur 1998) January (25 ai Lyutikov Maxim stenneaiitcgyrofrequency, nonrelativistic the is k − s ω γ 1 B for 0 = k sawv vector, wave a is < s − .Teeegn spectra emergent The 2. ,Psdn,Clfri 91125 California Pasadena, y, (1) 0 lyhv nonpower have ally rno instability erenkov o ntblt a explain can instability kov m kvinstability nkov ai saturated is ma sconsidered. is s ewl approx- well be raino the of ervation resonance rb 4,[] 6.The [6]. [5], [4], by er sisms and mass its is emi h pulsar the in beam e og arplasma. pair a rough v stevlct of velocity the is γ magnetospheres ei edo a of field netic steLorentz the is i field tic c sthe is Close to the stellar surface, where the initial beam is produced and accelerated, the particles quickly reach their ground gyrational state due to the sychrotron emission in a superstrong magnetic field, so that their distribution becomes virtually one dimensional [1].

In the outer parts of magnetosphere it becomes possible to satisfy the anomalous Doppler resonance - the cyclotron-Cherenkov instability develops bringing about the diffusion of particles in transverse moments. The relevant saturation mechanism then determines the final spectrum (which can be later modified be the absorption processes).

The nonlinear saturation of the cyclotron-Cherenkov instability due to the diffusion of the resonant particles has been previously considered by several authors. Kawamura & Suzuki [2] neglected the possible stabilizing effects of the radiation reaction force due to the cyclotron emission at the normal Doppler resonance and the force arizing in the inhomogeneous magnetic field due to the conservation of the adiabatic invariant. These forces result in a saturation of the quasilinear diffusion. Lominadze et al. [3] were the first to notice the importance of the radiation reaction force due to the emission at the normal Doppler resonance on the saturation of the quasilinear diffusion. Unfortunately, Lominadze et al. [3] used an expression for the cyclotron damping rates which is applicable only for the nonrelativistic transverse motions, when γψ (psi is the pitch angle) is much less than unity. In the pulsar magnetospheres the development of the cyclotron-Cherenkov instability results in a diffusion of particles in transverse moments, quickly increasing there transverse energy to relativistic values.

In a review paper Lominadze et al. [7] took a correct account of the radiation reaction force due to the emission at the normal Doppler resonance and pointed out the importance of the the force arizing in the inhomogeneous magnetic ﬁeld due to the conservation of the adiabatic invariant ( G force Eq. (10)). When considering the deceleration of the beam

Lominadze et al. [7] has incorrectly neglected the radiation reaction force due to the emission at the anomalous Doppler resonance in comparison with the radiation reaction force due to the emission at the normal Doppler resonance. In this work we reconsider and extend the treatment of the quasilinear stage of the

2 cyclotron-Cherenkov instability. We have found a state, in which the particles are constantly slowing down their parallel motion, mainly due to the component along magnetic ﬁeld of the radiation reaction force of emission at the anomalous Doppler resonance. At the same

time they keep the pitch angle almost constant due to the balance of the force G and the component perpendicular to the magnetic ﬁeld of the radiation reaction force of emission at the anomalous Doppler resonance. We calculate the distribution function and the wave intensities for such quasilinear state.

In the process of the quasilinear diffusion the initial beam looses a large fraction of its initial energy 10%, which is enough to explain the typical luminosities of pulsars. Though ≈ the quasisteady wave intensities are not strictly power laws (see Eq. (38)), they can well approximated by a power law with a spectral index F (ν) 2 (F (ν) is the spectral flux ∝ − density in Janskys) which is very close to the observed mean spectral index of 1.6 [8]. The − predicted spectra also show a turn off at the low frequencies ν 300MHz and a flattering ≤ of spectrum at large frequencies ν 1GHz which may be related to the possible turn-up in ≥ the flux densities at mm-wavelengths [9].

II. QUASILINEAR DIFFUSION

A particle moving in a dielectric medium in magnetic ﬁeld with the velocity larger than the velocity of light in a medium is emitting electromagnetic waves at the anomalous Doppler resonance ( s< 0 in Eq. (1)) and at the normal Doppler resonance ( s> 0 in Eq. (1)). The

radiation reaction due to the emission at the normal Doppler resonance slows the particle’s motion along magnetic field and decreases its transverse momentum. The radiation reaction due to the emission at the anomalous Doppler resonance increases its transverse momentum and also slows the particle’s motion along magnetic field [10]. As the particle propagates into the region of lower magnetic field, the force G decreases the its transverse momentum

and increases the parallel momentum. The stationary state in transverse moments may be reach when the actions of the G force and radiation reaction due to the emission at the

3 normal Doppler resonance is balanced by the radiation reaction due to the emission at the anomalous Doppler resonance. The quasisteady stage may also be considered in terms of a detailed balance of for the

particle transitions between the Landau levels. The quasisteady stage is reached when the number of induced transitions up in Landau levels due to the emission at the anomalous Doppler resonance is balanced by the number of the spontaneous transitions down in Landau levels due to the emission at the anomalous Doppler resonance.

The equations describing the quasilinear diﬀusion in the magnetic ﬁeld are

df(p) 1 ∂ ∂ ∂ = sin ψ Dψψ + Dψp f(p) dt sin ψ ∂ψ " ∂ψ ∂p! #

1 ∂ 2 ∂ ∂ p Dpψ + Dpp f(p) (2) p2 ∂p " ∂ψ ∂p! #

2 Dψψ (∆ψ)   dk   = w(s, p, k)n(k) (3)  Dψp = Dpψ  3  (∆ψ)(∆p)    s<0 Z (2π)     X      2   Dpp   (∆p)      dn(k)  ∂ cosψ (kv/ω) cos θ ∂ = dpw(s, p, k) n(k)¯h + − f(p) (4) dt 0 ∂p p sin ψ ∂ψ ! ! s

hω¯ h¯(ω cos ψ k v) ∆p = ∆ψ = − k (5) v pv sin ψ E2(k) n(k)= (6) hω¯ (k) 8π2q2R (k) w(s, p, k)= E e(k) V(s, p, k) 2 δ(ω(k) sω /γ k v ) (7) hω¯ (k) | · | − B − k k s ′ V(s, p, k)= v⊥ Js(z), iσsv⊥Js(z) , vkJs(z) (8) z −

where E2(k)dk/(2π)3 is the energy density of the waves in the unit element range of k-space. In the Table I we give the dimensions of the main used quantities. In Eq. (2) we neglected the spontaneous emission processes at the anomalous Doppler resonance and the induced emission processes at the normal Doppler resonance. The net

eﬀect of the spontaneous emission at the normal Doppler resonance is treated as a damping

4 force acting on each particle in the Boltzman-type left hand side of equation (9). To be exact, we should have treated the eﬀects of spontaneous emission at the normal Doppler resonance as stochastic terms in the Fokker-Plank-type terms on the right hand side of

equation (9). But the fact that the emission at the normal Doppler resonance occurs on very high frequencies at which the presence of a medium is unimportant in the dispersion relation of the waves and can be neglected allows one to integrate the corresponding terms over angles and sum over harmonics to obtain a classical synchrotron radiation damping

force, that can be treated using the Boltzman approach. Thus, the total time derivative of the distribution function is

df(p) ∂f(p) ∂f(p) ∂ q = + v + G + F + (v B0) f(p) (9) dt ∂t ∂r ∂p c ×

where G is the force due to the conservation of the adiabatic invariant

2 2 mc Gk = βγψ , G⊥ = βγψ, β = (10) − − RB

Here R 109 cm is the radius of curvature and F is the radiation damping force due to B ≈ the spontaneous synchrotron emission at the normal Doppler resonance:

2q2ω2 F = αγ2ψ2, F = αψ 1+ γ2ψ2 , α = B (11) k − ⊥ − 3c2 From (10) and (12) we ﬁnd that

F α F α k = γ, ⊥ = γψ2, for γψ 1 (12) Gk β G⊥ β ≫ where r = c/ω is a Larmor radius and r = q2/(mc2)=2.8 10−13cm is a classical radius L B e × of an electron. The dimensionless ration in (12) is

α 2RBre −4 −6 = 2 =5 10 RB,9R9 (13) β rL ×

9 9 9 RB,9 = RB/10 cm is the radius of curvature in units of 10 cm, R9 = R/10 cm is the distance from the neutron star surface in units of 109 cm.

5 Using (13) we ﬁnd that for the primary particles with γ 107 ≈ F k 1 Gk ≫ 2 F⊥ rL −2 1, for ψ 2RB reγ 10 (14) G⊥ ≪ ≪ r ≈

Then the total derivative (9) may be written as

df(p) ∂f(p) ∂f(p) 1 ∂ 1 ∂ = + v + (sin ψG f(p)) + p2F f(p) (15) dt ∂t ∂r p sin ψ ∂ψ ⊥ p2 ∂p k We are interested in the quasilinear diffusion of the particles due to the resonant interac- tion with the waves at the anomalous Doppler effect. We expand the transition currents (8) in small v keeping only s = 1 terms: V( 1, p, k) = v /2 (1, iσ, 0). Then for the waves ⊥ − − ⊥ propagating along magnetic field e(k)=(1, 0, 0) we find

π2q2v2 w( 1, p, k)= ⊥ δ(ω(k) sω /γ k v ) (16) ± hω¯ (k) − B − k k

where we took into account that R (k) 1/2. E ≈ We now can find the diffusion coefficients in the approximation of a one dimensional spectrum of the waves.

2πδ(θ) k2 k n(k)= 2 n(k), n(k)= dΩ 2 (17) k sin θ Z (2π) n(k) We ﬁrst note that we can simplify the change in the pitch angle (5) in the limit ψ2 δ ≪ and 1/γ2 δ ≪ hωδ¯ ∆ψ (18) ≈ −pv sin ψ

We then ﬁnd

δ 2 D 2 E  γ k |k=kres  Dψψ     2 2 2  ψmc  π q π re =  D E2  ,D = = (19)  Dψp = Dpψ   γ k k=kres  2 3    − |  m c mc          Dpp   ψ2m2c2     D E2     δ k k=kres   |      6 where

k2dΩ 2 k k Ek =¯hω(k)n(k)= 2 hω¯ ( )n( ) (20) Z (2π) is energy density per unit of one-dimensional wave vector and we assumed that ω(k) is an isotropic function of k. We next solve the partial diﬀerential equation describing the evolution of the distribution

function by sucsessive approximations. We first expand equation (2) in small ψ assuming that ∂/∂ψ 1/ψ. We also neglect the convection term assuming that the characteristic ≃ time for the development of the quasilinear diffusion is much smaller that the dynamic time of the plasma flow. Then we assume that it is possible to separate the distribution function

into the parts depending on the ψ and p:

f(p)= Y (ψ,p)f(p) (21)

with

f(p)=2π sin ψdψf(p), dpp2f(p)=1 (22) Z Z In the lowest order in ψ we obtain an equation:

1 ∂ 1 ∂ ∂Y (ψ) (sin ψG⊥Y (ψ)) = sin ψDψψ (23) − p sin ψ ∂ψ sin ψ ∂ψ " ∂ψ # which has a solution

2 2 2 2 2 1 ψ 2 DmcδEk DRBδEk π δRBreEk Y (ψ)= 2 exp 2 , ψ0 = 2 = 2 = 2 2 (24) πψ0 −ψ0 ! βγ cγ γ mc

The next order in ψ gives

∂f(p) ∂ 1 ∂ ∂f(p) 1 ∂ 2 ∂f(p) + Fkf(p) = sin ψDψp + p Dpψ (25) ∂t ∂p sin ψ ∂ψ " ∂p # p2 ∂p " ∂ψ # By integrating (25) over ψ with a weight ψ we ﬁnd the equation for the parallel distribution function:

∂f(p) ∂ 2 ∂ AE2γ2f(p) = pDm2c2E2f(p) (26) ∂t − ∂p k p2 ∂p k 7 where

2 2 2 2 2 2 4 2 2 αψ0 2q ωBψ0 2π ωBq RBδ 2π RBre δ A = 2 = 2 2 = 2 2 6 = 2 2 (27) Ek 3c Ek 3γ m c 3γ rL

The term containing A describes the slowing of the particles due to the radiation reaction force and the term containing D describes the slowing of the particles due to the quasilinear diﬀusion, or, equivalently, due to the radiation reaction force of the anomalous Doppler resonance. To estimate the relative importance of these terms we consider a ratio

3 Aγ αδγ 2δγ RBre = = 2 1 (28) Dmc β 3 rL ≪

Neglecting the second term on the left hand side of (28) we ﬁnd

∂f(p) 2 ∂ pDm2c2E2f(p) ==0 (29) ∂t − p2 ∂p k If the cyclotron quasilinear diﬀusion has time to fully develop and reach a steady state, then the distribution function of the resonant particles is

1 f(p) 2 (30) ∝ p Ek

Next we turn to the equation describing the temporal evolution of the wave intensity (4). Neglecting the spontaneous emission term and the wave convection we ﬁnd

∂E2 k = ΓE2f(γ) (31) ∂t − k res where

1 ∂ cos ψ (kv/ω) cos θ ∂ Γ= dpw(s, p, k) h¯ + − f(p) (32) f(γ)res s ∂p p sin ψ ∂ψ ! ! X Z and we introduced

f(γ)γ2dγ = f(p)p2dp (33)

We will estimate this growth rate for the emission along the external magnetic ﬁeld for distribution (22), (24). Neglecting ∂/∂p and assuming that ψ2 2δ (so that most of the ≪ particles are moving with the superluminal velocity) we ﬁnd for s = 1 − 8 πω2 Γ= p,res γ2 (34) 2ω res

(It is important to note that in the limit ψ2 2δ the growth rate does not depend on the ≪ scatter in pitch angles). Equations (29) and (34) may be combined to a quasilinear expression

∂ 2 ∂ pDm2c2E2 f(γ)+ k =0 (35) ∂t p2 ∂p Γ !!

Which after integration gives

2 ∂ γDE2 f(γ) k = f 0(γ) (36) − γ2 ∂γ Γ !

Neglecting the initial density of particles in the region of quasilinear relaxation and using Eqs. (30) and (36) we can ﬁnd a distribution function and the asymptotic spectral shape:

1 1 1/2 f(γ)= 3 (37) 2γ log(γmax/γ) log(γmax/γmin)! 2 2 1/2 4 3 1/2 2 mc δrLγ log(γmax/γ) mc δ log(γmaxωrL/(cδ)) Ek = 2 = 2 2 (38) 2πrerS log(γmax/γmin)! 2πω rerLrS log(γmax/γmin) !

It is noteworthy, that a simple power law distribution for the spectral intensity and distribution function cannot satisfy both Eqs. (30) and (36). The particle distributions function and the energy spectrum of the waves are displayed in Figs. 1 and 2.

9 10 f7.5 20 5 2.5 10 0 6 0 2. 10 gamma_perp 6 4. 10 6 -10 gamma6. 10 6 8. 10 -20

15 f 10 -6 5 2. 10 0 6 0 2. 10 psi 6 4. 10 6 gamma6. 10 -6 6 -2. 10 8. 10 FIG. 1. Asymptotic distribution functions in γ p and γ ψ spaces in arbitrary units − ⊥ − 7 for γmax = 10 . The spike at the γ = γmax is an artifact of the initial distribution function f(γ)0 = δ(γ γ ). The divergence at γ = γ is weak (logarithmic) and would be removed if − max max the more realistic nitial distribution function was used.

10 2 Ek 0.3

0.25

0.2

0.15

0.1

0.05

gamma 6 6 6 7 4. 10 6. 10 8. 10 1. 10 Jy 300

250

200

150

100

MHz 400 500 600 700 800 900 1000 FIG. 2. Asymptotic one dimensional energy density in the waves in the γ-space (arbitrary units), and the predicted observed ﬂux in Janskys.

We can now estimate the ﬂux per unit frequency:

4 3 1/2 2 mc δ log(γmaxωrL/(cδ)) F (ν)=2πEk = 2 2 , (39) ω rerLrS log(γmax/γmin) !

11 characteristic pitch angle

1/2 1/4 πRBrL log(γmax/γ) −6 ψ0 = δ 2 10 , (40) rS ! log(γmax/γmin)! ≈

(which remarkably stays almost constant for a broad range of particles’ energies and also

for diﬀerent values of γmin), and the total energy density in the waves

νmax mc2γ E = F (ν)dν max , (41) tot 2 1/2 Zνmin ≈ 4√πrerS log (γmax/γmin) This total energy can be compared with the kinetic energy density of the beam:

Etot π 2 (42) γbmc nGJ ≈ slog(γmax/γmin)

It means that some considerable fraction of the beam energy can be transformed into waves.

We can also estimate the energy ﬂux (39) at the Earth. Assuming that distance to the pulsar is d 1 kpc, we ﬁnd ≈ ν −2 F obs(ν) 300 Jy (43) ≈ 400MHz

With time, the value of γmin decreases as the particles are slowed down by the radiation reactions force. Since at the given radius, the particles with lower energies resonate with waves having larger frequencies, more energy will be transported to higher frequencies hard- ening the spectrum. The lower frequency cutoﬀ is determined by the initial energy of the

beam. No energy is transported to frequencies lower than

ωB ωmin = (44) γbδ

This simple picture, of course, will be modiﬁed due to the propagation of the ﬂow in the

inhomogeneous magnetic ﬁeld of pulsar magnetosphere.

III. CONCLUSION

In this work we investigated the new saturation mechanism for the cyclotron-Cherenkov instability of a beam in a inhomogeneous magnetic ﬁeld. We showed that for the typical

12 parameters of the pulsar magnetosphere it is possible to reach quasisteady state, in which the transverse motion of particles is determined by the balance of a radiation reaction force due to the emission at anomalous Doppler effect and the force arising in the inhomogeneous magnetic field due to the conservation of adiabatic invariant. The resulting wave intensities are sufficient to explain the observed fluxes from radio pulsars.

ACKNOWLEDGMENTS

I would like to thank Roger Blandford, Peter Goldreich and Gia Machabeli for their useful comments.

[1] Arons J., ApJ, 266, 215, (1983)

[2] Kawamura K. & Suzuki I., ApJ, 217, 832, (1977)

[3] Lominadze J.G., Machabeli G.Z. & Mikhailovsky A.B., Sov. J. Plasma Phys., 5, 748, (1979)

[4] Machabeli G.Z. & Usov V.V, Sov. Astron. Lett., 15 , 393, (1989)

[5] A.Z. Kazbegi, G.Z. Machabeli, G.I Melikidze, MNRAS, 253, 377 (1991)

[6] Lyutikov M., PhD thesis, Caltech, (1998)

[7] Lominadze J.G., Machabeli G.Z. & Usov V.V., Astroph.Space Sci., 90, 19, (1983)

[8] Lorimer D.R. et al., MNRAS, 273, 411, (1995)

[9] Kramer M. et al., A&A, 306, 867, (1996)

[10] Ginzburg V.L. & Eidman V.Ya., Sov. Phys. JETPh, 36, 1300, (1959)

TABLE I. Dimensions of the main used quantities

13 2 2 2 E (k) Ek n(k) n(k) α, βR Dψψ Dψp Dpp D A f(p)p dp, f(γ)dγ

erg 1 erg 1 erg erg2 sec cm2 erg cm2 1 cm2 cm sec cm cm2 erg sec cm 1