Astron. Astrophys. 349, 1017–1026 (1999) ASTRONOMY...

Astron. Astrophys. 349, 1017–1026 (1999) ASTRONOMYAND

ASTROPHYSICS

On the physics of cold MHD winds from oblique rotators

S.V. Bogovalov

Moscow Engineering Physics Institute, Kashirskoje Shosse 31, 115409 Moscow, Russia ([email protected])

Received 29 March 1999 / Accepted 18 June 1999

Abstract. I show that the self-consistent solution of the prob-lem of MHD plasma flow in the magnetosphere of an obliquerotator with an initially split-monopole magnetic field is reducedto the solution of a similar problem for the axisymmetric rotator.All properties of the MHD cold plasma flows from the axisym-metric rotators with the initial split-monopole magnetic field arevalid for the oblique rotators as well. Rotational losses of theoblique rotator do not depend on the inclination angle and thereis no temporal evolution of this angle. Self - consistent analyticaland numerical solutions for the axisymmetric plasma flows ob-tained earlier show that the rotators can be divided on both fastrotators (σ0/U

20 > 1) and slow rotators (σ0/U

20 < 1), whereσ0

is the ratio of the Poynting flux over the matter energy flux inthe flow at the equator on the surface of the star,U0 = γ0v0/c,v0 andγ0 being the initial velocity and the Lorentz-factor of theplasma. The self-consistent approximate analytical solution forthe plasma flow from the oblique rotator is obtained under theconditionσ0/U

20 1. Implications of these results for radio

pulsars are discussed. In particular, I argue that all radio pulsarsare apparently the slow rotators ejecting the Poynting dominatedrelativistic wind.

Key words: Magnetohydrodynamics (MHD) – stars: pulsars:general – ISM: jets and outflows

1. Introduction

An analysis of the relativistic plasma flow is necessary to under-stand the processes taking place in radio pulsar magnetospheres,compact galactic objects and in AGNs (Arons 1996; Mirabel &Rodriguez 1996; Pelletier et al. 1996). In the present paper, asin a previous one (Bogovalov 1997), we concentrate fully on theproblem of the relativistic plasma flow in the conditions typicalto radio pulsars.

In spite of systematic research in the field of the physicsof radio pulsars in the last years, the structure of the magne-tosphere and the mechanisms for the acceleration of plasma inthese objects remain to a large extent vague (Lyubarsky 1995).One of the most important unsolved problems of the physicsof radio pulsars is the problem of the energetics of the ejectedwind of relativistic plasma. For example, the kinetic energy of

the plasma accelerated in the inner magnetosphere of the Crabpulsar is not sufficient to explain the energetics of the relativis-tic wind exiting the synchrotron nebula surrounding the pulsar(Arons 1996).

The basis of the theory of plasma production in radio pul-sar magnetospheres was initiated by Sturrock (1971). This the-ory was developed in more detail for different conditions onthe stellar surface by Ruderman & Sutherland (1975) and byArons (1981). Primary electrons are accelerated in the so-called“electrostatic gaps” and produce dense relativistic plasma ofsecondary particles with Lorentz-factorγ0 ∼ 102–103. Thisplasma screens the accelerating electric field and limits the po-tential drop by the valueγgapmc

2 with γgap ∼ 107. The totalflux of the kinetic energy of all particles appears to be far interiorto the total rotational losses of the fast rotating radio pulsars dueto this limitation. Almost all energy is carried out by the electro-magnetic field. A wind with similar characteristics is formed inthe outer gap model (Cheng et al. 1986). The wind of relativisticplasma from radio pulsars can be characterized by the ratio ofthe Poynting flux over the flux of the kinetic energy of the plasmaσ. The plasma is Poynting flux dominated, whenσ > 1, andis kinetic energy dominated, ifσ < 1. Electrostatic gaps giveσ > 1000 for the Crab pulsar. At the same time, interpretationof the observations of the Crab Nebula compels us to concludethat this ratio is very small at large distances from the pulsar.This conclusion is based on the assumption that the subsonicflow of the plasma terminated by the shock wave at the interac-tion of the wind with the interstellar medium can be consideredas the flow of ideal plasma with the only dissipative process ofsynchrotron cooling. Under this assumption, the observed ex-pansion of the outer edge of the Crab Nebula, the synchrotronand TeV gamma-ray emission of the nebula produced via In-verse Compton Scattering of the relativistic particles on 2.7-KMBR emission can be explained under the unique choice ofσ = 3 · 10−3 (Atoyan & Aharonian 1996).

Recently Begelman (1998) reviewed the key assumption ofthe theory by Kennel & Coroniti (1984). He argued that evena wind with σ ∼ 1 in the pre-shock region can give the ob-servable properties of the Crab Nebula if one takes into accountdissipative processes in the nebula. It follows from the theory ofrelativistic MHD shocks that a wind withσ ∼ 1 in the pre-shockregion creates the flow in the post-shock region withσ > 1. But

1018 S.V. Bogovalov: On the physics of cold MHD winds from oblique rotators

this flow must be strongly unstable. The instability provides aneffective transformation of the magnetic field into the kinetic en-ergy of the plasma accompanied by acceleration of particles inthe nebula. In both (Kennel & Coroniti or Begelman) scenariosit follows that some unspecified mechanism for the transforma-tion of the Poynting flux into the flux of the kinetic energy existsas the plasma travels from the star to infinity. This mechanism isapparently the basic mechanism for plasma acceleration sinceit ensures the transformation of at least 50% (in the Begelmanscenario) and up to 99.7% (in the Kennel & Coroniti model) ofthe rotational energy of the neutron star into the kinetic energyof the relativistic wind.

One of the possible mechanisms of the acceleration is themagnetic acceleration of the plasma by the rotating magneto-sphere. Unfortunately this mechanism appeared non effectivefor the axisymmetric rotators with a typical pulsar’s parameters.No effective acceleration was found neither in the nearest zone(Bogovalov 1997), nor in the far zone of the rotator (Bogovalov& Tsinganos 1999). At the same time, until now no solution de-scribing the MHD plasma flow from the oblique rotator exists.Usually it is believed that the inclination of the magnetic mo-ment of the star to the axis of rotation could be important for theplasma acceleration. In this paper we study this possibility inthe model of the oblique rotator with the initial split-monopolemagnetic field on the surface of the star.

2. Basic assumptions

The axis of rotation for real radio pulsars is not directed alongthe magnetic moment. The solution of the problem of the plasmaflow in the magnetosphere of this object is extremely compli-cated. To simplify the problem, several models of radio pulsarswere proposed. The sequence of these models is presented inFig. 1. Firstly Goldreich & Julian (1969) proposed an axisym-metrically rotating star with an initially dipole magnetic field(step 1 in Fig. 1). The rotational losses of the axisymmetricallyrotating star ejecting relativistic plasma are comparable to therotational losses of the oblique dipole in vacuum. The energyof rotation is carried out near the surface of the axisymmetricrotator by the Poynting flux. This is why this model can be usedfor study of the process of the Poynting flux transformation intothe kinetic energy of the plasma.

However, even with such simplification, the structure of theaxisymmetric flow of the plasma from the star with the initialdipole magnetic field appears too complicated. There were alot of attempts to solve this problem in massless approximation(Michel 1973b; Beskin et al., 1983; Lyubarskii 1990; Contopou-los et al. 1999). To simplify the problem, Michel (1969) wasthe first to use the rotator with the prescribed split-monopolepoloidal magnetic field for the investigation of the relativisticplasma flow in the magnetosphere of the axisymmetrical rotator(step 2). It follows from Fig. 1 that there are no closed field linesin the split-monopole model. All lines go to infinity. This allowsone to simplify the analysis of the flow considerably.

Michel (1969) considered the flow of the plasma in the pre-scribed split-monopole magnetic field. The affect of the moving

4

ΩΜ

ΩΩ Ω Μ

ΜΩΜΩ

1

2

3

Fig. 1. The sequence of models introduced to simplify the solution ofthe problem of the plasma outflows from radio pulsars. Step 1 is thetransition from the oblique rotator with a dipole magnetic field to theaxisymmetric rotator. Step 2 is the transition from the axisymmetricrotator with the dipole magnetic field to the axisymmetric rotator withthe split-monopole magnetic field. In this paper, the transition fromthe axisymmetric split-monopole to the oblique split-monopole modelis done (step 3) together with step 4, connecting this model with realpulsars.

plasma on the poloidal magnetic field was not taken into ac-count. In the self-consistent solution, the plasma and the elec-tromagnetic field affect each other. The problem of the self-consistent plasma flow from the rotator with the initial split-monopole magnetic field was investigated in nonrelativistic andrelativistic limits in the papers by Sakurai (1985), Bogovalov(1992) Bogovalov (1997), Bogovalov & Tsinganos (1999). Thephrase “initial split-monopole magnetic field” means that thenormal component of the magnetic field on the surface of thestar is constant but changes sign on the magnetic equator. Inthis paper we firstly consider the model of the oblique rotatorwith the split-monopole magnetic field on the surface of the star(step 3). This model allows one to investigate the plasma flowin conditions more typical to real radio pulsars than it occursin the axisymmetric model and to connect the model with thesplit-monopole magnetic field with the real pulsars (step 4).

Since the density of the relativistic plasma produced in thepulsar magnetosphere is high enough to screen the electric fieldparallel to the magnetic field, the magnetohydrodynamical ap-proximation is used in this paper to describe the flow of theplasma. The plasma is considered cold as a first approximation(Lyubarsky 1995).

3. Reduction of the oblique rotator problemto the axisymmetrical problem

System of time dependent equations defining the temporal evo-lution of the flow of the relativistic plasma (Akhiezer et al. 1975)

mn(∂γv∂t

+ (v∇)γv) = q · E +1c

j × H, (1)

S.V. Bogovalov: On the physics of cold MHD winds from oblique rotators 1019

∂Hc∂t

= curl E, (2)

curl H =4πc

j +∂Ec∂t

, (3)

div H = 0, (4)

div E = 4πq, (5)

∂n

∂t+ div(vn) = 0. (6)

together with frozen-in conditionE + 1cv × H = 0 gives, in

spherical coordinate system, the equations of motion

mn(∂γvr

∂t+ (v∇)γvr − γ(v2

θ + v2ϕ)

r) =

q · Er +14π

(

1r sin θ

∂Hr

∂ϕ− 1r

∂(rHϕ)∂r

)Hϕ −

−(∂(rHθ)r∂r

− ∂Hr

r∂θ)Hθ +

1c(Hθ

∂Eϕ

∂t−Hϕ

∂Eθ

∂t), (7)

mn(∂γvθ

∂t+ (v∇)γvθ +

γ(vrvθ − v2ϕ cot θ)

r) =

q · Eθ +14π

(∂(rHθ)r∂r

− ∂Hr

r∂θ)Hr −

− Hϕ

r sin θ(∂ sin θHϕ

∂θ− ∂Hθ

∂ϕ) +

+1c(Hϕ

∂Er

∂t−Hr

∂Eϕ

∂t), (8)

mn(∂γvϕ

∂t+ (v∇)γvϕ +

γ(vrvϕ + vθvϕ cot θ)r

) =

q · Eϕ +14π

1r sin θ

(∂ sin θHϕ

∂θ− ∂Hθ

∂ϕ)Hθ −

−(1

r sin θ∂Hr

∂ϕ− ∂rHϕ

r∂r)Hr +

+1c(Hr

∂Eθ

∂t−Hθ

∂Er

∂t), (9)

and the induction equations:

∂Hr

c∂t= − 1

r sin θ(∂(sin θEϕ)

r∂θ− ∂Eθ

∂ϕ), (10)

∂Hθ

c∂t= −(

1r sin θ

∂Er

r∂ϕ− ∂(rEϕ)

r∂r), (11)

∂Hϕ

c∂t= −(

∂(rEθ)r∂r

− ∂Er

r∂θ). (12)

The laws of conservation of the magnetic field and the matterflux, together with the Coulomb law take the following form

1r2∂(r2Hr)∂r

+1

r sin θ∂ sin θHθ

∂θ+

1r sin θ

∂Hϕ

∂ϕ= 0. (13)

∂n

∂t+

1r2∂(r2nvr)

∂r+

1r sin θ

∂ sin θnvθ

∂θ

+1

r sin θ∂nvϕ

∂ϕ= 0. (14)

1r2∂(r2Er)∂r

+1

r sin θ∂ sin θEθ

∂θ+

1r sin θ

∂Eϕ

∂ϕ= 4πq. (15)

The frozen-in conditions are

Er +1c(vθHϕ − vϕHθ) = 0, (16)

Eθ +1c(vϕHr − vrHϕ) = 0, (17)

Eϕ +1c(vrHθ − vθHr) = 0. (18)

In this systemq is the induced space electric charge density,θis the polar angle andϕ is the azimuthal angle.

Boundary conditions on the surface of the star with radiusR∗ for the system of equations above are:

1. The Lorentz-factorγ0 is specified constant;2. The normal component of the density of the matter flux is

specified constant and uniform;3. The normal component of the poloidal magnetic fieldH0

does not depend on coordinates or time for every point ofthe surface of the star and it changes sign on the magneticequator.

4. The tangential component of the electric field is continuouson the star’s surface;

It is assumed that the velocity of the plasma exceeds fast mag-netosonic velocity (the flow is super sonic) at the infinity. Thissystem of equations together with the boundary conditions de-scribes the stationary axisymmetric flow as well.

Let’s assume that the self-consistent solution for the problemof the plasma flow from the axisymmetric rotator with an initialmonopole-like poloidal magnetic field is specified. The phrase“initial monopole-like magnetic field” means that the normalcomponent of the poloidal magnetic field on the surface of thestar is constant and does not change sign on the magnetic equa-tor. Actually this field has no magnetic equator at all. Althoughsuch magnetic fields can not be created in reality, this rather ar-tificial mathematical model is convenient to construct solutionsfor more realistic plasma flows. The monopole-like magneticfield differs from the split-monopole magnetic field only bydirection of the field lines in one of the hemispheres and was in-troduced by Michel (1969). In the model with the monopole-likemagnetic field, all field lines are coming out from the surfaceof the star as it is shown in Fig. 2. There is no current sheet inthis flow in contrast to the model with the split-monopole mag-netic field which contains the current sheet in the equatorialplane. The current sheet is a tangential discontinuity (Landau& Lifshitz 1963) in ideal MHD approximation. The magneticfield changes direction during the passage through the currentsheet. It is obvious that the solution for the split-monopole canbe obtained from the solution for the monopole-like magnetic


Current sheet

Monopole-like magnetic field Split-monopole magnetic field

Fig. 2. The solution in the model with the split-monopole magneticfield can be obtained from the solution for the monopole-like magneticfield by a change of sign of the magnetic field in the lower hemisphere.The same is valid for the reverse operation. There is a current sheet atthe equatorial plane in the split-monopole model which is a tangentialdiscontinuity in an ideal MHD approximation

field by formally reversing the sign of the magnetic fields in thelower hemisphere. Similar operation can be done in the case ofthe oblique rotator as well.

The flow and the monopole-like poloidal magnetic field areperturbed by rotation in the self-consistent axisymmetric flow(Bogovalov & Tsinganos 1999). We assume that this perturbedself-consistent solution is known. Let’s introduce the velocityV(r), the poloidalBp(r) and toroidalBϕ(r) magnetic fieldsand the density of the plasmaN(r) in the self-consistent ax-isymmetric flow with the initial monopole-like magnetic field.The poloidal electric fieldE and the poloidal magnetic field areconnected in the axisymmetric flowE = (r sin θΩ/c)Bp × eϕ

(Weber & Davis 1967), whereeϕ is the unit vector directed inthe azimuthal direction. The toroidal electric field is equal to0. These variables depend only on coordinates. Dependence ontime is absent for the stationary axisymmetric flow. This solu-tion automatically satisfies all the equations of the system (7-18)and corresponding boundary conditions.

Now we show how the self-consistent solution for theplasma flow in the magnetosphere of the oblique rotator with theinitial split - monopole magnetic field can be obtained from theknown solution for the axisymmetric problem. Let’s considerthe following transformation of the axisymmetric solution. Thesame velocity and the density values as for the axisymmetricrotator are used:

v(r, t) = V (r), n(r, t) = N(r). (19)

The poloidal magnetic fieldHp(r, t) for the oblique rotator isobtained from the poloidal magnetic field of the axisymmetricrotator as follows

Hp(r, t) = η(r, t)Bp(r). (20)

The same procedure gives us the toroidal magnetic field

Hϕ(r, t) = η(r, t)Bϕ(r), (21)

where η(r, t) is an unknown function to be specified. Thepoloidal electric fieldE is defined as in the axisymmetric case

Eθ = −r sin θΩc

Hr, (22)

ϕ

X

Y

ΩZ

M

e

χθ

Fig. 3.The initial position of the star and the geometry used in Eq. (29).

Er =r sin θΩ

cHθ, (23)

and the toroidal electric fieldEϕ = 0. It is easy to show thatthis is indeed the solution of the problem for the oblique rotatorand to specify the functionη.

The frozen-in conditions (16-18) are fulfilled automaticallyfor the solution (19-23). At the stationary rotation, the de-pendence ofη on ϕ and t can be presented by one variableξ = ϕ− Ωt in the spherical system of coordinates. Thus,η de-pends on three variablesr, θ andξ. Then, the induction equation(10) is reduced to the equation

∂

∂ξ(Eθ

r sin θ+

ΩHr

c) = 0 (24)

which is fulfilled due to condition (22). The same is valid forthe induction equation (11) which is reduced to the equation

∂

∂ξ(Er

r sin θ− ΩHθ

c) = 0 (25)

which is satisfied due to condition (23). The third inductionequation (12) after the substitution of the conditions (22,23) isreduced to the flux conservation condition (13) which takes theform

(B · ∇η) = 0. (26)

since the total magnetic fieldB of the axisymmetric flow sat-isfies the equation∇ · B = 0. This implies that functionη isconstant on the field line of the magnetic filed of the axisym-metric flow.

Fig. 3 shows the initial position of the star with the magneticmomentM inclined to the axis of rotation at the angleχ att = 0. It follows from Eq. (20) and the boundary conditions thatfunctionη is equal to 1 for the points on the surface of the starwhere the field lines leave the surface of the star and is equal


to −1 for the points where the field lines enter the surface. It isconvenient to introduce the following function

D(x) =

1, ifx ≥ 1−1, ifx < 1. (27)

The sign of the functionη on the surface of the star varies withthe sign of the product(e · eM ), wheree is the unit vectordirected to the point on the surface of the star,eM is the unitvector directed along the magnetic moment. This product canbe presented as

(e · eM ) = sinχ sin θ sinϕ+ cosχ cos θ. (28)

Then, on the surface of the star, the functionη is

η(θ, ϕ− Ωt) = D(sinχ sin θ sin(ϕ− Ωt) + cosχ cos θ). (29)

Actually this is the boundary condition for Eq. (26). The equa-tion can be presented in the form

Br∂η

∂r+Bθ

∂η

r∂θ+Bϕ

∂η

r sin θ∂ϕ= 0. (30)

The equations for the characteristics of this equation are

dr

Br=r sin θdϕBϕ

(31)

and

dr

Br=rdθ

Bθ. (32)

Therefore the general solution is

η(r, θ, ϕ, t) = f(θ −∫ r Bθdr

rBr, ϕ−

∫ r Bϕdr

r sin θBr), (33)

where the integrals over r are taken along the field line of thepoloidal magnetic field of the axisymmetric solution,f is an ar-bitrary function. The solution satisfying the boundary condition(29) has the form:

η(r, θ, ϕ, t) = D(sin(χ) sin(θ −

∫ r

R∗

Bθdr

rBr)×

× sin(ϕ− ∫ r

R∗Bϕdr

r sin θBr− Ωt) +

+ cos(θ − ∫ r

R∗BθdrrBr

) cosχ). (34)

It follows from this solution thatη2 = 1 andη changes signwhen the magnetic field changes direction. It is easy to shownow that equations of motion (7-9) are also satisfied for thesolution (19-23) at the functionη defined by (34). Notice thaton the left hand side of these equations there is no functionη. Inthe right hand side of the equations of motion, functionη comesin the combinationAiη

∂ηBk

∂xl, whereAi andBk are arbitrary

components of fields of the axisymmetric solution, andxl is aspatial or time coordinate in 4-space. This relationship can bepresented as

Aiη∂ηBk

∂xl= η2Ai

∂Bk

∂xl+AiBk

12∂η2

∂xl= Ai

∂Bk

∂xl. (35)

Z Ω

Fig. 4.Top panel shows the structure of field lines and the current sheet(thick wave-like line) in the poloidal plane. The lower panel shows thesame in the equatorial plane. Arrows show the direction of the magneticfield lines. The direction of the field lines changes on the current sheet.

Therefore, the functionη disappears in the equations of mo-tion. Here we ignore the difference in the dynamics of the cur-rent sheet and the surrounding plasma assuming that the currentsheet is the mathematical discontinuity as usual in ideal MHD.This assumption can be violated for the oblique rotators at largedistances from the star. But, at large distances, the dynamicsof the current sheet can be considered particularly in WKB ap-proximation (Coroniti 1990; Michel 1994).

Thus, we obtain the self-consistent solution for the obliquerotator from the known self-consistent solution for the axisym-metric rotator. The sketch demonstrating the structure of the coldwind from the oblique rotator is presented in Fig. 4. The struc-ture of the plasma flow is symmetric in relation to the equator.The form of the poloidal field lines is the same as for the axisym-metric rotator. In general there is a collimation of the plasmaflow to the axis of rotation, although the effect of the collima-tion depends on the parameters of the problem (Bogovalov &Tsinganos 1999). In the axisymmetric flow the current sheetdividing the magnetic fluxes of opposite directions is locatedon the equator. In the wind from the oblique rotator, the currentsheet takes the form of a wave. In the poloidal plane the poloidalmagnetic field lines change direction on the current sheet. Atfirst glance it seems that this behavior contradicts magnetic fluxconservation. The bottom panel of Fig. 4 shows the structure ofthe field lines in the equatorial plane. It is seen that there is nocontradiction with the magnetic flux conservation since the totalmagnetic field depends on the azimuthal angleϕ. The velocity


and the density of the plasma do not depend onχ andϕ and arethe same as for the axisymmetric rotator. Only the magnetic andthe electric fields are modulated with the period of rotation. Thesign of these fields is modulated in the sector limited by the fieldlines which have roots on the magnetic equator. The squares ofthese fields are not modulated. The wave is simply the wave ofthe tangential discontinuity (Landau & Lifshitz 1963) which isconvected with the velocity of the plasma.

The rotational losses of the oblique rotator can be calculatedas the total Poynting flux through a sphere surrounding the star.The Poynting flux is defined by the formulacE×H

4π . This formulais biquadratic on the components of the electromagnetic field.Since for the oblique rotator these components differ from thecomponent for the axisymmetric rotator only by functionη, thePoynting flux appears independent on functionη and, therefore,the rotational losses are independent on the angleχ.

There is also no alignment of the magnetic moment along theaxis of rotation as it happens in vacuum approximation (Ostriker& Gunn 1969, Pachini 1968, Michel & Goldwire 1970, Davis& Goldstein 1970) or disalignment as is seen in the solution byBeskin et al. (1983). The evolution of the inclination angleχ isdefined by torques on the surface of the star (Michel & Goldwire1970)

Ly =∫

[Trθ cosϕ− Trϕ cos θ sinϕ]d2S, (36)

Lx = −∫

[Trθ cosϕ− Trϕ cos θ sinϕ]d2S, (37)

and

Lz =∫Trϕ sin θ d2S, , (38)

where Trθ and Trϕ are the components of the energy-momentum tensorTrθ. Integration in (37) is performed overthe surface of the star. The energy-momentum tensor is also bi-quadratic on the components of the electromagnetic field (Lan-dau & Lifshitz 1975). Therefore it does not depend on the func-tion η and the anglesχ andϕ. The torques for the oblique rota-tor are the same as for the axisymmetric rotator. Integration inEq. (37) over the surface of the star givesLx = Ly = 0. Thereare no torques which can change the inclination angle of theoblique rotator with the initial split-monopole magnetic field.This is not a totally unexpected conclusion. The torques on thestar rotating in a vacuum differ from the torque on the star eject-ing ideal plasma. The torque on the star rotating in a vacuumalways align the rotational and magnetic axes (Soper 1972). Thetemporal evolution of the inclination angleχ of the star ejectingplasma depends on the distribution of the magnetic field on thesurface of the star. It was shown by the perturbation theory infirst order approximation onΩ that magnetic and rotation axestend to align when the magnetic flux is more concentrated at themagnetic poles and evolve toχ = 90 when there is less fluxdensity at the poles than at the equator (Mestel & Selley 1970).Thus, the result obtained here for the arbitrary angular velocityof the rotator with the split-monopole magnetic field is due to

the combination of two conditions: by the ejection of the plasmasimultaneously with the uniform magnetic flux distribution onthe surface of the star.

4. The flow of the plasma atσ0U2

0 1

The solution of the problem of the cold plasma flow from therotator with the initial split-monopole magnetic field is definedby the following parameters and variables:H0, mn0, v0, Ω, c,r, θ, ϕ, R∗, χ and t. Lower index “0” denotes the values onthe surface of the star. Since the dependence onχ, ϕ andt isknown, we can consider only the dependence of the solution forthe axisymmetric flow on the other 8 parameters. There are 3parameters in this set with independent dimensions. Therefore,according to the theory of dimensions (Barenblatt 1979), the so-lution in dimensionless variables depends only on 5 parameters.Let us write this dependence as

A = f(Rstar/a, α1, α2, r/a, θ). (39)

HereA is the vector of the state of the plasma including density,velocity and magnetic field,f is the unknown vector-functionand a is some parameter with the dimension of length. It isreasonable to consider a limiting case when the dimensionlessradius of the star goes to zero. It allows one to decrease thenumber of variables in the solution by one. Now the solutioncrucially depends only on two parameters and two variables inthe dimensionless presentation.

Let us measure the velocity of the plasma in the units of theinitial velocity v0 and all geometrical variables measured withthe initial radius of the fast mode surface (FMS)rf . The FMSis the surface where the poloidal velocity of the plasma equalsthe local velocity of the fast mode MHD perturbations. In thecomoving coordinate system where the electric field is equal tozero, this condition takes the form

vp =H

′2

4πmn′c2 +H ′2 , (40)

wheren′

andH′

are the density of the plasma and the magneticfiled in the comoving system. Taking into account thatH2 −E2

is invariant (Landau & Lifshitz 1975),rf can be calculated withΩ = 0 as follows

r2f =H2

0R2∗

4πmcn0v0U0. (41)

In these units the radius-vector isX = r/rf . The electricand magnetic fields can be measured in the units of the initialpoloidal magnetic field on the FMS. The density of the plasmacan also be measured in the units of the initial density on theFMS.

The parametersα1 andα2 can be chosen rather arbitrarily.In particular, it is convenient to chooseα1 = α = (rfΩ/v0γ0)andα2 = ε = (rfΩ/c). Let’s consider the dependence of thesolution on these parameters.

A nonrelativistic limit corresponds toε → 0 andv0 c.The solution depends only on one parameter in this case. The


axisymmetrical nonrelativistic plasma flow was investigated nu-merically by Bogovalov & Tsinganos (1999) for the parameterα achieving the value up to 4.5. It was found that there is aclear division of the rotators between fast and slow ones witha dependence on the value ofα. The slow rotators correspondto α < 1 and the fast rotators correspond toα > 1. There isa strong physical difference between them. The plasma in out-flows from the slow rotators is not accelerated considerably andis not collimated in the subsonic region. In the case of the fastrotators the plasma is effectively accelerated and collimated tothe axis of rotation already in the subsonic region of the flow.

The flow of the relativistic plasma depends crucially on twoparameters. But in this case it is also possible to divide the rota-tors between slow and fast ones depending on the value of theparameterα. The solution can be presented as a vector functionA with componentsAl which are the density of plasma, thecomponents of velocity, the magnetic field, etc. Let’s considerthe expansion of this function by powers ofα in the pointα = 0at fixedε /= 0. Physically, this means that we consider the rela-tivistic limit of the problem atγ0 → ∞. The expansion can bepresented as

Al = f(0)l (ε,X)(1 + αf (1)(ε,X) + (α)2f (2)(ε,X) + ....).(42)

The first termf(0)l (ε,X) in this expansion has a very sim-

ple presentation when using the arbitrary parameterε /= 0. Itwas shown that, at the limit under consideration, the functionf

(0)l (ε,X) is as follows using ordinary (dimension) variables

(Bogovalov, 1997)

Ur = U0, Uϕ = 0, Uθ = 0, n(r) = n0(r0r

)2. (43)

Hϕ = −(rΩ sin θ

c)Hp, E = (

rΩ sin θc

)Hp, (44)

and

Hr =

H0( r0r )2, ifθ ≤ π/2

−H0( r0r )2, ifθ > π/2

, Hθ = 0. (45)

This is exactly the solution obtained by Michel (1973a) for themassless plasma. It is easy to show that the full system of theequations of motion is satisfied for this solution except in thecase of the frozen-in condition (18). The residual in this condi-tion on the FMS is

δvϕ = ε(α/ε)2

(1 + v0/c). (46)

The residual shows that the corrections to the solution are indeedproportional to powers ofα, and can be neglected in the regionlimited by the FMS, provided thatα 1. This residual alsoshows that expansion (42) contains the terms(α/ε)k, where kis integer number. Therefore the first term in expansion (42)gives an incorrect result for the limitα → 0, ε → 0 providedthatα/ε = const which corresponds to the slow rotation of thestar at arbitrary velocityv0. Mathematically, this means that thepoint (α = 0, ε = 0) is a particular point of the expansion. The

terms(ε/α)k can be eliminated from expansion (42). Let’s todefine the function

gl(α/ε,X) = limα → 0

α/ε = const

Al(α, ε,X)

f(0)l (ε,X)

. (47)

Then the expansion can be presented as

Al = f(0)l

(ε,X)(gl(α/ε,X) +

∑k=1

αkg(k)l (ε,X)

). (48)

The expansion∑

k=1 αkg

(k)l (ε,X) already does not contain

terms(α/ε)k and functionFl = f(0)l (ε,X)(gl(α/ε,X) gives

a correct first term forα → 0 with the arbitrary parameterεincluding pointε = 0. This function was calculated analyticallyby Bogovalov (1992). The corrected solution of the problemfor the limitα → 0 with arbitraryε coincides with the solution(43-45) with the exception of the toroidal magnetic field. It isreplaced by

Hϕ = −(rΩ sin θv0

)Hr. (49)

According to this solution the plasma moves radially in thesplit-monopole poloidal magnetic field with constant poloidalvelocity and without any toroidal motion. At large distancesthe collimation of the plasma to the axis of rotation and thecorresponding acceleration of the plasma are also very weak(Bogovalov & Tsinganos 1999). It is reasonable to call all rota-tors withα < 1 slow rotators, and all rotators withα > 1 fastrotators, independent of the value ofε.

Let us consider the physical sense of the parametersα andε. It is clear thatε is the ratio of the initial radius of the fast modesurface over the light cylinder radius. On the other hand

ε2 = σ0 =H2

0

4πn0mc2γ0(R∗Ωv0

)2. (50)

For the slow rotatorsσ0 is the ratio of the Poynting flux overthe matter energy flux (kinetic energy flux for the relativisticplasma) at the equator and coincides with the well known mag-netization parameter (Arons 1996) for the slow rotators. It iseasy to show that

α2 =σ0

U20. (51)

Therefore the slow rotators correspond to the conditionσ0U2

0 1.

The diagram in Fig. 5 shows different regimes of the flowfrom rotators with the initial split-monopole magnetic field incoordinatesσ0 andU0. The lineσ0 = 1 divides the rotatorswith the Poynting flux dominated flow near the surface of thestar (above this line) from the rotators with the matter energydominated flow (below this line). The slow rotators are locatedbelow the lineσ0 = U2

0 while the fast rotators are located abovethis line. In the wide range of the parameters, the regimes ofthe axisymmetric flows have been already investigated. Now itis evident that the obtained solutions are valid for the oblique


10-4

10-2

1

10 2

10 4

10 6

10-2

10-1

1 10 102

103

σ = U

F A S

T R

O T

A T

O R

S

S L O

W R

O T

A T

O R

So ,

σr ο

U

o2 6

23

1

4

7

5

σ

o

Fig. 5. Regimes of the cold plasma flows. The locations of 6 pulsarsemitting observable gamma-rays above 100 MeV: Crab -1, PSR 1508-59 - 2, Vela - 3, PSR 1706-44 - 4, PSR 1951+32 - 5, Geminga -6 andPSR 1055-52 - 7 are shown by dashed lines for all possible values ofU0. All the pulsars are located in the region of slow rotators ejectingthe Poynting flux dominated wind provided thatU0 > 200.

rotators also. For the slow rotators with the strong matter en-ergy dominated flow, (σ0 → 0) an approximate analytical solu-tion with accuracy on the orderα2 was obtained by Bogovalov(1992). With the increase ofσ0 at constantU0, rotators withthe nonrelativistic flow firstly cross the lineσ0 = U2

0 and be-come fast rotators. Numerical solutions for the nonrelativisticplasma flows (σ0 → 0) from fast rotators were obtained up toα = 4.5 (Bogovalov & Tsinganos 1999). It was found that dueto the strong collimation of the plasma to the axis of rotation,the flow becomes subsonic in some region along the axis of ro-tation and therefore should be absolutely unstable well beforethe rotator reaches the lineσ0 = 1 (see also Lyubarskii (1992)and Begelman (1998)).

The rotators with the initial relativistic plasma flow firstcross the lineσ0 = 1 with the increase ofσ0 at constantU0 andbecame the slow rotators ejecting the Poynting flux dominatedwind. With the increase ofσ they also cross the lineσ = U2

0and become the fast rotators. The relativistic stationary plasmaflow from the fast rotators was investigated recently by Beskinet al. (1998). Surprisingly, the authors found that the perturba-tion of the poloidal magnetic field go to zero asσ0/U

20 → 0,

and acceleration of the plasma remains very ineffective (likein the model by Michel 1969 with the prescribed monopole-like poloidal magnetic field). The Lorentz-factor of the plasmareaches the valueγ∞ ∼ (σ0γ0)1/3.

The plasma flow from the slow oblique rotator becomes es-pecially simple. In particular, the functionη for the slow rotators(α 1) is

η(r, θ, ϕ, t) = D(sinχ sin θ sin(ϕ− Ω(t− r/v0))+

+ cos θ cosχ). (52)

Rotational losses of the slow rotators with the initial split-monopole magnetic field are defined by the total Poynting fluxthrough the surface of the star

Erot =23H2

0R4∗Ω

2

v0, (53)

and they do not depend on the angleχ. It is useful also to expressthe rotational losses through the total magnetic fluxψ of the openfield lines of one direction (for example leaving the star).

Erot =1

6π2

ψ2Ω2

v0, (54)

whereψ = 2πH0R2∗ for the split-monopole magnetic field.

5. Implications for radio pulsars

How the model considered above is connected with the realradio pulsars? It is reasonable to assume that the rotator withthe split-monopole magnetic field is equivalent to a real radiopulsar if it provides the plasma flow similar to the plasma flowfrom the real pulsar at distances larger than the light cylinder.Therefore the equivalent split-monopole rotator should have thesame matter flux, the same poloidal magnetic field flux from oneof the poles, the sameγ0, v0, Ω, and the same inclination angleχ as the real pulsar.

The magnetic flux from one of the poles of the split-monopole rotator isψ = 2πR2

∗H0. The magnetic flux of theopen field lines from the radio pulsar with a dipole magnetic

field is defined by the radius of the polar capRp = R∗√

R∗Ωc

(Goldreich & Julian 1969). Therefore, the flux of the open fieldlines from one of the polar caps of the radio pulsar is estimatedas

ψ = πH0R2∗(R∗Ωc

). (55)

The ratio of the poloidal magnetic field over the matter flux den-sityHp/nvp is constant on the field line. Therefore this ratio forthe split-monopole rotator and for the radio pulsar should be thesame. These conditions allow one to define all parameters. Thesplit-monopole rotator, equivalent to the pulsar with the mag-netic field on the polar capH0, angular velocityΩ and ejectingplasma with initial densityn0 andγ0, has the magnetizationparameter

σr =H2

0

8πn0mc2γ0(R∗Ωc

)3. (56)

Below we consider the relativistic plasma withv0 = c. Therotational losses of this object are

Erot =16H2

0R6∗Ω

4

c3(57)

These rotational losses equal the losses of the dipole rotating invacuum at the angle of inclinationχ = 300.


The initial density of the plasman0 is defined by the pro-cesses of multiplication of plasma in the electromagnetic cas-cades initiated by primary particles with the Lorentz-factorγgap ∼ 2 · 107 in the magnetosphere of the pulsar. In the innergap theories, (Ruderman & Sutherland 1975; Arons 1981) theprimary beam has Goldreich-Julian density

nGJ ∼ ΩH0

2πec, (58)

We neglect here the dependence ofnGJ on the inclinationangle. The primary particles produceλ secondary electronsand positrons. Therefore the initial density of the plasma isn0 = λnGJ . Calculations of cascades performed by Daugh-erty & Harding (1982) and by Gurevich & Istomin (1985) showthatλ ∼ 104. The outer gap model (Cheng & Ruderman 1986;Romani 1996) gives similar values ofn0. After substitution of(58) in (56),σr takes a form

σr =eH0

4λmcΩγ0(R∗Ωc

)3. (59)

The estimates ofγ0 are less definite. The flow of the plasmaformed in the electromagnetic cascade consists of two compo-nents. One of them is the beam of the primary particles whichcan lose a remarkable amount of energy. Another component isthe plasma of the secondary particles. It is reasonable to take theaverage Lorentz-factor of the secondary particles asγ0 becauseit is this plasma which screens the electric field and ensuresthe frozen-in condition. The average Lorentz-factor of the sec-ondary particles strongly depends on the model of the gap. Thisparameter lies in the range20 < γ0 < γgap/λ. To be definite,we assume here that

γ0 = 0.1γgap

λ∼ 200. (60)

This value agrees closely with the results of the cascade simula-tions performed by Daugherty & Harding (1982). For otherγ0’sthe magnetization parameter is scaled asσr = σ200(200/γ0).The location of all radio pulsars observed by EGRET in gamma-rays above 100 MeV (Crab, Vela, Geminga, PSR 1706-44, PSR1509-58, PSR 1951+32 and PSR 1055-55; Thompson 1998)in the parameter spaceσr, U0 for all possibleU0 is shown inFig. 5 by dashed lines. For the estimates of the magnetic fieldH0, Eq. (57) was used. It was assumed also that the momentumof inertia of all pulsars is1045g · cm2. It is seen that all thepulsars appear as slow rotators provided thatU0 ≥ 200. Thisimplies that they do not accelerate plasma and do not collimateit considerably to the axis of rotation beyond the light cylinderif the flow of the plasma is dissipativeless (Bogovalov 1998;Bogovalov & Tsinganos 1999).

6. Conclusion

This paper clearly demonstrates that the oblique rotators do notdiffer strongly from the axisymmetric rotators. In the particu-lar case considered in this paper, the dynamics of the plasmaejected by the oblique rotator is exactly the same as the plasmadynamics of the wind from the axisymmetric rotator. It allows us

to apply all results obtained for the cold axisymmetric plasmaflows to the oblique rotators. But these results show that forparameters typical of radio pulsars the acceleration of the rel-ativistic plasma is not effective. Apparently radio pulsars arebasically slow rotators. The obliqueness of the rotators does notchange this conclusion.

Our results indicate that only violation of the ideal MHD canprovide effective acceleration of the winds from the radio pul-sars since neither the magneto-centrifugal mechanism nor theacceleration by the gradient of the toroidal magnetic field at thecollimation of the plasma to the axis of rotation are effective forthe pulsars. The role of the dissipative processes in the relativis-tic winds has been under discussion for many years (Coroniti1990, Michel 1994, Mestel & Shibata 1994). Recently Melatos& Melrose (1996) tried to argue that the violation of the frozen-in condition takes place in the wind from an oblique rotator andit can provide the acceleration of the wind. Our self-consistentsolution of the problem for the particular case of the obliquerotator with the initial split-monopole magnetic field shows noevidence of the violation of the ideal MHD in the wind.

Mestel & Shibata (1994) argued that the frozen-in condi-tion can be violated right after the light cylinder due to theformation of singularities in the stationary axisymmetric flow.This assumption is not confirmed by Bogovalov’s (1997) andby Contopoulos et al. (1999) latest results.

It follows from all data obtained recently in the physicsof relativistic plasma outflows that only dissipative processescan ensure plasma acceleration under pulsar conditions. One ofthese mechanisms was proposed by Coroniti (1990) and Michel(1994). They argued that, due to geometrical reasons, the dissi-pative processes can ensure the transformation of the Poyntingflux into the kinetic energy of the plasma, provided that the in-clination angle is high enough (∼ 80). The Poynting flux istransformed into the thermal energy of the plasma which is thanaccelerated by the thermal pressure gradient.

Another possibility is the Poynting flux transformation in thevolume of the flow. This can happen if the Poynting flux domi-nated plasma flow is unstable in relation to MHD perturbations.The anomalously low conductivity due to the MHD turbulencewould provide the violation of the ideality in the flow with thecorresponding transformation of the Poynting flux into the en-ergy of the plasma. The advantage of this approach is that itcan provide effective acceleration even with the axisymmetricrotation. But the question about the stability of the Poynting fluxdominated plasma flow is still open and should be investigatedin detail separately.

Acknowledgements.The author acknowledges discussions with col-leagues from the Astrophysics group of the MPI fur Kernphysik and Dr.Yu.E.Lyubarsky. The author also thanks Dr. F.A.Aharonian for readingthe manuscript and useful comments, as well as the MPI fur Kernphysikfor their warm hospitality and support given during the work on thepaper. This work is partially supported by the Ministry of Education ofRussia in framework of the program “Universities of Russia - basic re-search”, project N 897. The author is also grateful to unknown refereefor useful remarks on the paper.


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