Chapter 4: Plasma Accretion in Gravitoelectromagnetic Field

5/12/2018 Chapter 4: Plasma Accretion in Gravitoelectromagnetic Field

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Chapter 4Plasma Accretion inGravitoelectromagnetic Field1

4.1 IntroductionNormal stars produce their energy by nuclear fusion. Another possible source of energyreleased by a star is via the conversion of gravitational potential energy into other formssuch as the electromagnetic radiation. For normal stars it was initially prosed that theprocesses of energy production consisted of slow shrinking in the size of the star andradiating away all the excess gravitational potential energy, This, however., could notaccount for the lifetime of a normal stellar source. Later proposal was for nuclear fusionas a source of energy for these stars. For a typical neutron star (mass ~. INfo! radiuslOkg) we find that (Treves et.al. 1989) the gravitational loss is about 0.15 units (restenergy being unity), that is by an appreciable amount of the total energy. Thus the in fallof matter into a compact star via accretion is a very efficient process for energy release,Accretion can be spherical in cases where matter is distributed isotropically around thestar. More often, matter distribution around a compact star is Dot isotropic, the deviationgives rise to the formation of an accretion disk having some angular momentum, There

IThis chapter is based on Mirza 2005.

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is a strong astrophysical evidence (Prendergast & Burbidge. 1968) that for most compactsources, gns from a large binary companion flows into the compact star in the formof an accretion disk, where gravitational energy lost is radiated away in the form ofx-rays, In this region the charged particles are more free to move while not accretedto relativistic velocities, and th e dynamics of these charged particles is governed notonly by the gravitational but also by the electromagnetic field of the star. Howeverthe electromagnetic field is itself effected by the geometry of the background spacetime.Therefore it is of importance and of astrophysical relevance that the two-fold effects;namely, the coupling of the gravitational and the electromagnetic field of the star, as wellas the direct. effects of the background spacetime on the motion of a charged particle, mustbe addressed together. Clearly to treat the problem of plasma accretion in compact starvicinities consistently, the effects of gravitational as well as electromagnetic field must beaddressed together,

In Schwarzschild spacetime the problem have been studied (Sengupta 1997, Prasana& Sengupta 1994, Perti 2004) in a combined toroidal, peloidal magnetic field. Theparticle trajectories are obtained by solving the equation of motion numerically, usingthe conservation of energy and the Conservation of angular momentum laws. Thesemodels are based on a solution of the Maxwell equations using a dipole expansion of theradiation field (Petterson 1974, Petterson 197.5,Chitre & Visheshwara 1975). It is foundthat particle jets observed for various compact sources in AGNs are due to the effect oftoroidal component of the magnetic field il l the polar regions which displaces the particleaway in the form of jet-like trajectories.

In this chapter we firt present the gravitoelectroma.gnetic approximation to the grav-itational field equations and to the geodesic equation. These are interpreted in terms ofthe Leuse-Thirring effect. Then an investigation is given for the gravitoelectric as well asgravitomagnetic effects 011 the dynamics of an accreted charged particle lying in the star'splasma atmosphere. The direct effects of the spacetime dragging on the particle dynam-ics are discussed in the gravitoelectrornagnetic (GEM) approximation to the geodesic

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equation (Ruggiero & Tartaglia 2002. Riudler 1997, Mirza 2(04). The GElVIapproxima-tion, which assumes the charged particle motion to be slow, is generally valid in the caseof stable compact stars (i.e. non-collapsing objects). Also for the case of a magnetizedcompact star with a dipole field the GEM approximation is applicable except at or verydose to the poles where particle velocities can be relativistic due to intense magneticfield. Generally the accretion occurs well below the relativistic limit and so the GEMapproximation can be used. Throughout we take the velocity four vector as defined byaZAMO. Further the electromagnetic field is assumed to be given by equations (3.110) to(3.113) and the star is assumed to be a sphere of homogeneous mass density. VVefirst setup the equations of motion for the charged particle in Cartesian coordinates in the nextsection, taking into account the effects of gravitational as well as electromagnetic field.Then in section 4.3 we study numerically the dynamics of the charged particle USingtheGEM approximation. Lastly, in section 4.4, we give a discussion and a summary of themain conclusions of the chapter.

4.2 The Gravitoelectromagnetic ApproximationThe effects gravitational field due to a rotating compact star must be described in termsof the general theory of relativity, As w.ehave described in chapter two, the very ex-istence of these star is due to general relativistic effects which become more importanttowards the terminal stages of a compact star ultimately forming a black hole. Oncea black hole is formed the nonlinear effects of the gravitational field equations play thekey role in every process occurring in the vicinity of the collapsed star. For a corn-pact star, however, the gravitational field is relatively weak. Moreover the star hasrotational speed well below the relativistic limits. Under these conditions an approxi-mation can be adequately used to linearize the field equations under the Lorentz gauge,The linearized equations; called the gravitoelectromagnetic fiekl equations, posses strik-ing formal similarities with the field equations of classical electromagnetic theory. For

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instance analogous to the electromagnetic waves the gravitoelectromagnetic equationspredict the existence of gravitational waves, For a rotating, gravitational source the grav~itoelectromagnetic theory leads to the existence of Lense-Thirring effect in terms of thegravitomagnetic force generated by the rotation of the mass (Thirring 1918, Lense &Thirring 1918). In fact any theory combining Newtonian gravity and Lorentz invarianceconsistently must include a gravitornagnetic force generated by mass current (Lense &Thirring 1918), The effect however is very small under normal terrestrial conditions.Recently attempt have been made to directly measure the Lense- Thirring frame dragdue to Earth using laser ranging observations of the artificial satellites which gives anaccuracy of 20% to 30%(Cillfolini) 1989, Lanunerzahl e t. at. 2001). However the grav-itomagneticallyinduced orbital precession of the artificial satellite is smaller than theeffects caused by other perturbations such as tides and the quadrupole moment of theEarth. It has therefore been proposed (Lanunerzahl d. a t. 2001, Will 2001) that dueto the intense gravitational field of compact stars means must be sought to observe theeffect of gravitouiagnetic force in astrophysical observations as well,

4.2.1 Gravitoelectromagnetic Field EquationsGenerally equation (2.26) can be integrated to give

, f / ' / TaB(0/3 = -4.. 7dV (4.1)For a rotating mass, neglecting the stresses and the product of source velocities we havefor the energy-momentum tensor

(4.2)where Uo: =1, u). In component form we have

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1 Ul 'Uz I~31/ ,! 0 0 0 (4.3)a !3 ;:::.PHz a 0 0' 1 1 . 3 0 0 0

It follows froni the equation (4.1) and the above energy-momentum tensor that

1 . f J ' J P=--(00=-_ -dV,4 T (4.4)and

(4,.5)where < D is the Newtonian "gravitoelectric'' potential and A is the "gravitomagnetic"potential.

Here it is evident that TOO = is the mass density, which for a homogeneous systemof total mass I V ! is given by

1 f l pdV =u, (4.6)thus r iO = Ji(= J) represents the "mass-current" density: related to the total angularrnomenturu of the system by

(4 ..7)With the above definitions of the potentials we already notice a certain similarity with

the classical electromagnetic theory. For instance the Lorentz gauge condition (2.21) canbe written in terms of the potentials q : . and A:

3 < } )o t + V,A =0, (4,8)

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the identification with the Lorentz gauge of the classical electromagnetic theory is strik-ing. Moreover if we define, analogous to the classical electromagnetic theory the "gravi-toelectrio" and "gravitornagnetic" fields as

G = -V', H = V'x4A. (4.9)respectively. It then follows from the gravitational field equations (4.1) and the Lorentzgauge condition that for a slowly rotating, weakly gravitating mass:

v .G = -4rrp, V x G=0, (4.10)

V H= 0 V x H =-4rrJ. (4.11)V ve notice that in contrast to the classical electromagnetism, these equations involve afactor of 4. The origin of this factor lies in the fact that the above Iinear approximationinvolves a spin-2 field whereas in classical electrodynamics we deal with a spin-l field.

4.2.2 Geodesic Equation In Gravitoelectromagnetic Approxi-mation

Let us make the plausible assumption that the test particles in a compact gravitationalsource are not accreted to relativistic velocities, that is. the velocity of light. Then theline element ds is identical to cit . Then for the geodesic equation we have

d2x i dx" d r i3. _ ri .... ,"v'-2~- - 0[3--dt . dt dt (4.12)For the Christoffel symbol we substitute from (2,11) and (2.14) into (4.12), then thegeodesic equation becomes

(4.13)

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where (= 1/4J ( r x / 3 zz; -h. Further more (= (00' Denote

we have ' / 1 .0 . : ' - -- (1,1/). Then the geodesic equation (4.1.3) reduces to

(4.15)As before we denote

(4.16)and

1A = --C'.< 1 0 (L l .17)Then the geodesic equation becomes

d2r~ =grad


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we have seen that it represents the usual Newtonian force of gravitation obtained fromthe gravitoelectric potential 1). However the second term in expression (4.19), called thegravitomagnetic force, is o f particular interest. Unlike the Newtonian force it dependson the rotation of the gravitational source also. It therefore has a purely general rcla-tivistic origin and represents the general relativistic effects where the gravitational fieldis sufficiently strong and the source is rotating, such as the compact stars. Moreover ithas been demonstrated (Ciufolini &Wheeler 1995; Ciufolini 1994) that the gravitornag-netic is independence from the choice of a particular coordinate system and has a generalphysical validity for rotating gravitational objects such as compact stars,

4.2.3 The Lense-Thirring EffectWe consider the gravitational field inside a rotating spherical shell of mass J11,radiusR, andangular velocity n, A s in classical electromaguetic theory, the gravitoelectricforce (analogous to the electric field) inside such a shell of mass M vanishes; whereas thegravitomag:netic force (analogous to the magnetic force) is given by

H =~_Mn3 R (4.20)

correspondingly a . test particle of unit mass moving with a velocity v experiences a force

2MFq=-Rvxn. 3 (4,21)which is theCoriolis force of classical mechanics.

For practical purposes the graviomagnetio field acting OIl a test mass in vicinity ofthe Earth can be approximated by a dipolar field, This corresponds to the case of acharged particle orbiting about a magnetic dipole, The gravitornagnetic approximationthan gives for the angular velocity precession (Ruggiero & Tartaglia 2002);

, . G .3r(r.S} - Sr2precession rate= (~)------c~ r5 (4,22)

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For the Earth this precession has value about 0.05 arcsec/year. As pointed out earlierthe effects, such 0.'3 due to the quadrupole moment of the Earth, are of greater magni-tude than gravitornagnetic ones. Recently however, data obtained from laser rangingsatellites LAGEOS and LAGEOS II had disclosed the gravitomagnetically induced pre-cession within 20 to 30 per cent accuracy. In astrophysics, observational evidence for theLeuse- Thirring frame dragging has been detected for the periastron precession rate ofthe binary pulsar PSR 1913+16 (Will 2001).

For the case of a slowly rotating gravitational mass such as a compact star the grav-itoelectroinagnetic analogy to the magnetic dipole is not generally applicable. Here forthe star, not rotating very fast, the effects of gravitoelectric force are dominant and henceto a high degree of approximation the star can be regarded as a rotating spherical massM with uniform mass density. Then analogous to the case of slowly rotating chargedsphere in classical electromagnetism. We thus have for the gra:vitoelectric force (Rindler2001):

(4.23)and for the gravitomagnetic

12 2 r 1nH=--111R (n.r---=----)5 r" 3 1'3 (4.24)This result was first given by Thirring and Lense (Lense & Thirring 1918).

4.3 Charge Accretion in Gravitoelectromagnetic Ap-proximation

The dynamics of a particle of charge q ill vicinity of a massive object endowed with anelectromagnetic field is determined by the equation of motion (Landau & Lifshitz 1969):

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(4.25)

where F a : i J is the electromagnetic field tensor and v a : = x " Id s is the particle velocityfour-vector. Here 1)0: ; (1 denotes the covariant derivative of the velocity four-vector withrespect to the coordinates Xe, it. is defined as

o,et. _ 'va' - + r Ct ",IL ,(3- 1 /3 .8,'" (4.26)With this definition equation (4.25) can be expressedas

(4.27)In the above expression We have for the the first term.

d2Xi : Y .ds2 . (4.28)We therefore obtain for the equations of motion for the charged particle

d? a d ;3 d "( d~x aX.X . oJ3X{3-.-=-r8~-+qF ~.ds2 . 'ds ds ds (4.29)For the first termon the right hand side of the last equation, it has been shown in section4.2 that the geodesic equation can be linearized by assuming that the metric tensor to beof the form ( i / Q { 3 - + h Q ( 3 ) ; where r ] a . ( 3 = d i a g ( -1,1, 1.;1) is the timelike Minkowski metrictensor and ha p is a small perturbation to the spacetime metric such that haj3


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implies that ds c: dt, thus for the stationary observer we have tto: = (-1,0,0,0). Withthis choice the electromagnetic field tensor has the following components

0 _1 _E2 _3pcx{3 = E1 0 B3 -B2 (4.31)E2 -B3 0 Bl

E3 B2 -e, 0This gives for the zeroth component of the electromagnetic force term E ' v =O. Cal-culating for the remaining three spatial components we obtain for the electromagneticforce the usual Lorentz force law

.d1:jFem =qF'J_' - ' =qeE + v x B)dsThus in thefield of E \. slowly rotating compact star, the net force acting on a particle ofmass In and charge q is the sum of the gravitational and the electromagnetic forces:

d2r .'/7I'-l2 = m(G + v x H)+q(E + v x B).tit (4~32)Further in the slow rotation approximation the deformation to the star due to the effectsof rotation are small compared to the radially attractive gravitoelectric force; hence wetake the star to bea slowly rotating sphere of homogeneous mass density. Then usinggrsvitoelectromagnetic analogy, we have

(4.33)and

12 . 2 r 1 nH= --l11R (O.r- - ~-),5 .,.531'3 (4.34)where J\,[ is the mass, R is the radi us and n is the angular velocity of the star.

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4.4 Numerical SolutionsTo specify the physical situation we study the motion of the charged particle in a Carte-sian coordinate system ( : t ; , y, z ) and take H to be along the positive 2'-


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(4.41)

(4.42)Then for the case of non-vanishing peloidal magnetic field we transform expressions(4.37)-( 4.39) and (4.40)-(4.42) into the Cartesian coordinates, andobtain the equationsof motion in the equatorial plane O . e. for e =7f /2):

d 2 ;; ; x. ( B,,)d y-:-;:)= -9- yE +H - . ~. ,d t - d td2 y _ "E)/ (H E Z ) dxdt2 _- 9 +.t.' - . - dt 1

(4.43)(4.44)

whereM9 : = : 1 G 1 = 2 2';1:' +y (4.45)

(4.46)

E X = E ' o = Y ,/ e x2 + y2)2 - 2JII(:c2 + y 2 )C l / 2 .EoBZ =r=,~~~~~~y x 2 + y2 _ 2.M j : t2 + y2

Here Eo = qAIA.i/m , Eo = qCdm and fb = 41V!R 2D/5. Notice that here the case of

(4.47)

(4.48)

vanishing peloidal magnetic field is obtained for Eo = 0 whereas requiring C 1 = 0 doesnot alter the equations of motion (4.4.3) and (4.44).

We numerically solve theequations of motion (4.43) and (4.44) for the initial. condi-tions :,,(0) = 10 = y(O) and dx/dt I t = o = 0 = dy/dt I t = o , and plot (in Figure (4.1) to (4.4))the solutions for various values of the parameters 1\;[, u; EOl and Bo.

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-J.Q/ d) Bo ~ ,0/

\\\\ .

y

\. e) Bn =0\ -"

o ,5 1..Sx

Figure 4-1: Trajectory of the charged particle for magnetic force of varying strength withBo =0,10,30.50,70, anclilI = 1, fJ , =0,1, Eo =1 ill gravitational units,

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-8.00 -lO~ oxFigure 4~2: Trajectory of the charged particle for electric force of varying strength withEo =0.1,1,5,10,20, and M = 1, P =0.1, Eo =10 iugravitational units.

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10

y

a) M = 10

9.5 11 11~S0.50 xFigure 4-.3: Trajectory of the charged particle for gravitoelectric force of varying strengthwith 111= 1,2,3; 5: 101 and p = 0.1, Eo = 1, Eo = 10 in gravitational units.

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y

a) p=fJ.8 \p.~O.3d) Ii.-:0.2d) li:= o .t

Figure 4-4: Trajectory of the charged particle for gravitomagnetic force of varyingstrength with J - l = 0.1,0.2.0.3,0.5: 0..8. and A{= 1, Eo = 1, Eo =20 in gravitationalunits.

] .OJx

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4.5 Discussion and ConclusionsIn this chapter we have considered the dynamics of a charged particle in a highly, C011-ducting plasma surrounding a slowly rotatingcompact gravitational source. The effectsof background spacetime have been discussed directly on the particle's motion il l anaxially symmetric electromagnetic field around the star.

Our numerical studies of the effects of gravitationaland electromagnetic field show(Fig. (4.3) to (4.6)) that whereas the electromagnetic effects dominate the overalldynamics of the charged particle at sufficiently large distances, the gravitational fieldalso becomes important closer to the surface of the star especially if the star is rotatingfast enough. Firstly we observe that the effects of the electric field is to lift a chargeparticle from close to the surface of the star (Fig. (4.:3)) and make it move along themagnetic field lines.

Here are now two possibilities: one, if the magnitude of the magnetic field is largerthan the magnitude of the electric: field the particle following a helical trajectory fallsinto the star (case (a): (b), and {c) in Fig. (4.4)); two, if the magnetic field strengthis weaker or is even comparable to the electric field strength, then the electrical effectsbecome dominant and the particle escapes from falling into the star (case (d), and (e) inFig.( 4.4)).

On the other had the effects of gravitational field can be regarded as opposing thatof the electromagnetic: field. In Figure (4.. 5 ) we notice that the GE force attracts theparticle towards the star's surface against the electric field.

Furthermore it is clear from Figure (4..6) that the gyro-radius ofthe helical trajectoryof the particle is increased due to the gravitomagnctic force; hence the gravitomagnet.icforce weakens the effects of the magnetic field.

Summing up these observations we note that the accretion of a charged particle invicinity of a compact star is mainly due to the gravitoelectric and magnetic effects. Theseeffects are especially dominant at a sufficiently large distances from the star. Howeverclose to the star the particle trajectory is also effected by the gravitomagnetic force as

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well as electric: held which cause accreted charged particles to follow open lines of force(Mirza 20005). For a sufficiently fast rotating star and an electric field comparable tothemagnetic field the above mechanism (gravitomagnetic + electric) may contribute toopposing the in-fall .ofthe plasma surrounding the compact star. The plasma equilibriumnear a rotating gravitational. source has been studied (Elsasser 2000) with the ideal fluidassumption based all a variational principle.

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