Analytical models of finite thin disks in a magnetic field
aa r X i v : . [ a s t r o - ph . GA ] S e p Analytical models of finite thin disksin a magnetic field
E Cardona-Rueda and G Garc´ıa-Reyes ∗ Universidad Tecnol´ogica de Pereira, Departamento de F´ısica, Pereira, Colombia
Abstract
Analytical models of axially symmetric thin disks of finite extensionin presence of magnetic field are presented based on the well-knownMorgan-Morgan solutions. The source of the magnetic field is cons-tructed separating the equation corresponding to the Ampere’s law ofelectrodynamics in spheroidal oblate coordinates. This produces twoassociated Legendre equations of first order for the magnetic potentialand hence that can be expressed as a series of associated Legendre func-tions of the same order. The discontinuity of its normal derivate acrossthe disk allows us to interpret the source of the magnetic field as a ring-like current distribution extended on all the plane of the disk. We alsostudy the circular speed curves or rotation curve for equatorial circularorbits of charged test particles both inside and outside the disk. Thestability of the orbits is analyzed for radial perturbation using a exten-sion of the Rayleigh criterion.
Keywords:
Classical gravitation and electrodynamics; Thin disks; Mag-netic fields
PACS Nos.: ∗ Corresponding Author, E-mail: [email protected] . Introduction Axially symmetric thin disks are important in astrophysics as models of galax-ies, accretion disks and certain stars. In the case of galaxies it is motivated bythe fact that the main part of the mass of the galaxies is concentrated in thegalactic disk [1]. Even though a realistic disk has thickness, in first approxima-tion these astrophysical objects can be considered to be very thin, e.g., in ourGalaxy the radius of the disk is 10 kpc and its thickness is 1 kpc. Thin disks inpresence of magnetic field are also of astrophysical importance. In fact, thereis strong observational evidence that magnetic fields are present in all galaxiesand galaxy clusters [2–4]. These fields are characterized by a strength of theorder of µ G. For example, the strength of the total magnetic field in our MilkyWay from radio synchrotron measurements is about 6 µ G [5]. Observationaldata also provide a strong support of the existence of extragalactic magneticfields of at least B ≃ − G on Mpc scales [6, 7]. Magnetic fields are alsopresent in accretion disks such as protoplanetary disks and play an importantrole in disks evolution. They induce disk accretion via magneto-rotationalinstability [8] and also produce disk winds and outflows via the magneto-centrifugal force [9] or the magnetic pressure [10]. A key parameter in theunderstanding of the evolution of accretion disks is the large-scale poloidalmagnetic field threading the disks [11]. Moreover, although for the generalscenario the magnetic fields present in the galaxies are toroidal, poloidal mag-netic fields have been also observed in the central regions of galaxies whichhave gaseous rings [12]. Finite size current disks are also used as models ofthe magnetic dipole field presents in stars [13]. Such fields are small enoughnot to influence the spacetime structure of these objects. Indeed, the generalrelativistic effects are important for very compact objects and strong magneticfields as in the case of highly magnetized neutron stars, or magnetars, wherethe magnetic field is characterized by a strength of the order of 10 G [14–16].Thus, the Newtonian approximation is valid for modeling above structures.A simple model of thin disk is the Kuzmin-Toomre disk [17, 18] whichrepresents a disk-like matter distribution with a concentration of mass in itscenter and density that decays as 1 /r on the plane of the disc. These modelsare constructed using the image method that is usually used to solve problemsin electrostatics. This structure has no boundary of the mass but as the surfacemass density decreases rapidly one can define a cut off radius, of the order ofthe galactic disk radius, and, in principle, to consider these disks as finite.Another simple set of models are the Morgan-Morgan disks [19]. Thesedisks have a mass concentration on their centers and finite radius. The modelsare constructed using a method developed by Hunter [20] based on obtainingsolutions of Laplace equation in terms of oblate spheroidal coordinates, which2re ideally suited to the study of flat disks of finite extension. Several classes ofanalytical models corresponding to thin disks have been obtained by differentauthors [21–28]. Thin disks with electromagnetic fields, specially in curvedspacetimes, also have been studied extensively [29–37].In this work we consider analytical models of finite thin disk in presence ofa magnetic field. The mass density is determined by Morgan-Morgan solutionsand the source of magnetic field is constructed using the Hunter method whichinvolves the use of oblate spheroidal coordinates. It produces as source anazimuthal (toroidal) electric current distribution which extends on all the planeof the disk, and hence a poloidal magnetic field. These models differ fromthose presented in foregoing references in that we assume that magnetic fieldstrength is known far away from the disk and then we determine the magneticfield structure near to the disk. The radial component of the magnetic fieldon the disk surface or the profile of the azimuthal electric current density arealso assumed.The paper is organized as follows. In Section 2 we present the generalformalism to construct models of thin disks with electric current. We alsoanalyse the motion of charged test particles around of the disks and we derivatethe stability condition of the system against radial perturbations using anextension of the Rayleigh criterion.In Section 3, we first present a summary of the pair potential-density asso-ciated to the Morgan- Morgan disks. Then we construct a family of finite thindisk with electric current in presence of a pure magnetic field. As a simpleexample, we consider a particular profile of azimuthal electric current densitywith a physically reasonable behavior and we present explicit expression forthe main physical quantities associated to the disks for the two first termsof the series of the solutions. Finally, in Section 4 we summarize the resultsobtained.
2. Disks and Maxwell equations
Exact solutions of Poisson’s equation representing the field of a thin disk at z = 0 can be constructed assuming the gravitational potential Φ continuousacross the disk, and its first derivative discontinuous in the direction normalto the disk. This can be written as[Φ ,z ] = 2 Φ ,z | z =0 + . (1)Since the disk is thin the Poisson’s equation can be written as ∇ Φ = 4 πG Σ( R ) δ ( z ) , (2)3here δ ( z ) is is the usual Dirac function with support on the disk and Σ( R )is the surface mass density. The mass density can be obtained, for example,by using the approach presented in Ref. [38]. Thus, written by the Laplacianoperator in cylindrical coordinates and integrating from z = 0 − to z = 0 + , weobtain Σ( R ) = 12 πG ∂ Φ ∂z (cid:12)(cid:12)(cid:12)(cid:12) z =0 + . (3)Similarly, thin disks with electric current in presence of a pure magneticfield can be obtained assuming the magnetic potential A continuous and itsfirst derivate discontinuous. That is[ A ,z ] = 2 A ,z | z =0 + . (4)The magnetic field is governed by the Maxwell’s equations ∇ · B = 0 , (5a) ∇ × B = µ J , (5b)where J is the electric current density vector. For axially symmetric fields themagnetic potential is A = A ˆ ϕ where A is function of R and z only and ˆ ϕ aunit vector in the azimuthal direction. Since B = ∇ × A , (6)Eq. (5b) can be written as ∇ A − R A = − µ j ϕ δ ( z ) . (7)where j ϕ is the surface azimuthal electric current density. It implies thatthe magnetic field is poloidal. Again written by the Laplacian operator incylindrical coordinates and integrating from z = 0 − to z = 0 + , we obtainj ϕ = − µ ∂A∂z (cid:12)(cid:12)(cid:12)(cid:12) z =0 + . (8)Now we analyse the motion of charged test particles around of the disks.For equatorial circular orbits the equations of motion of test particles read ∂ Φ ∂R − ˜ ev R ∂ ( RA ) ∂R = v R , (9)where ˜ e is the specific electric charge of the particles and v is the speed of theparticles which is given by v c = − ˜ e ∂∂R ( RA ) ± s(cid:18) ˜ e ∂∂R ( RA ) (cid:19) + R ∂ Φ ∂R . (10)4he positive sign corresponds to the direct orbits or co-rotating and the neg-ative sign to the retrograde orbits or counter-rotating.To analyse the stability of the particles of the disks in the case of circularorbits in the equatorial plane we use an extension of Rayleigh criteria of sta-bility of a fluid at rest in a gravitational field [39–41]. The method works asfollows. Any small element of the matter distribution analyzed (in our case atest particle in the disk) is displaced slightly from its path. As a result of thisdisplacement, forces appear which act on the displaced matter element. If thematter distribution is stable, these forces must tend to return the element toits original position.The first term on the left-hand side of the motion Eq. (9) is the gravi-tational force F g (per unit of mass), the second term the magnetic force F m ,and the term on the right-hand side the centrifugal force F c acting on the testparticle. So we have a balance between the total force F = F g + F L and thecentrifugal force. We consider the particle to be initially in a circular orbitwith radius R = R . In terms of specific angular momentum L = Rv + ˜ eAR , F c ( R ) = ( L − ˜ eAR ) /R . We slightly displace the particles to a higher orbit R > R . Since the angular momentum of particle remains equal to its initialvalue, the centrifugal force in its new position is ˆ F c ( R ) = ( L − ˜ eAR ) /R .In order that the particle returns to its initial position must be met that F ( R ) > ˆ F c ( R ), but according to the balance Eq. (9) F ( R ) = F c ( R ) so that F c ( R ) > ˆ F c ( R ), and hence ( L − ˜ eAR ) > ( L − ˜ eAR ) . It follows that thecondition of stability is L > L . By doing a Taylor expansion of L around R = R , we can write this condition in the form L dLdR > , (11)or, in other words, dL /dR > ∂∂R ( RA ) dR + ∂∂z ( RA ) dz = 0 . (12)Thus the Eq. RA = C , with C constant, gives the lines of force of themagnetic field.
3. Finite disks with magnetic field
Solutions representing the field of a finite thin disk can be obtained solvingthe Laplace equation in oblate spheroidal coordinates ( u , v ), which are defined5n terms of the cylindrical coordinates ( R , z ) by R = a (1 + u )(1 − v ) , (13a) z = auv, (13b)and explicitly √ u = q [( ˜ R + ˜ z − + 4˜ z ] / + ˜ R + ˜ z − , (14a) √ v = q [( ˜ R + ˜ z − + 4˜ z ] / − ( ˜ R + ˜ z − , (14b)where u ≥ − < v <
1, ˜ R = R/a , ˜ z = z/a , and a the radius of the disk. Thedisk is located in u = 0, − < v <
1, and when it is crossed the coordinate v changes of sign but not its absolute value, whereas u is continuous. Thisimplies that an even function of v is a continuous function everywhere but hasa discontinuous v derivative at the disk.In this coordinate system the Laplacian operator has the form ∇ = 1 a ( u + v ) (cid:20) ∂∂u (1 + u ) ∂∂u + ∂∂v (1 − v ) ∂∂v (cid:21) . (15)The general solution of Laplaces equation can be written asΦ = − ∞ X n =0 c n q n ( u ) P n ( v ) , (16)where c n are constants, P n are the Legendre polynomials of order 2 n and q n ( u ) = i n +1 Q n ( iu ) , (17)being Q n ( iu ) the Legendre functions of the second kind.The mass surface density Eq. (3) takes the formΣ( R ) = 12 πaGv ∞ X n =0 c n (2 n + 1) q n +1 (0) P l ( v ) (18)with v = p − R /a .For n = 0 we have the zeroth order Morgan-Morgan disk and, for the twofirst terms of the series, the first Morgan-Morgan disk [41]. In this case, thegravitational potential isΦ = − M Ga (cid:26) cot − ( u ) + 14 (cid:2) (3 u + 1) cot − ( u ) − u (cid:3) (cid:0) v − (cid:1)(cid:27) , (19)being M is the mass of the disks, and the surface mass densityΣ( R ) = 3 M πa r − R a . (20)6 .2. Current Disks We now consider a thin disk with electric current in presence of a pure magneticfield. Using the Laplace operator in the form of Eq. (15), making the changeof variable u = ix , using the algebraic relation x − v (1 − x )(1 − v ) = 11 − x − − v , (21)and letting A = X ( x ) V ( v ), Eq. (7) for the magnetic potential yields twoassociated Legendre equations of first order(1 − x ) d Xdx − x dXdx + l ( l + 1) X − − x X = 0 , (22a)(1 − v ) d Vdv − v dVdv + l ( l + 1) V − − v V = 0 . (22b)The solution which vanishes at infinity is A n = b n Q n ( − iu ) P n ( v ), where b n is a constant, P n ( v ) are the associated Legendre functions of the the firstkind of order 1 and Q n ( − iu ) the associated Legendre functions of the sec-ond kind of order 1 [42, 43]. But Q mn ( − z ) = ( − n + m +1 Q mn ( z ), hence A n = b n ( − n +2 Q n ( iu ) P n ( v ). Moreover, since A must be invariant to reflection inthe equatorial plane by symmetry of the problem, A ( u, v ) = A ( u, − v ), theparity property P mn ( − v ) = ( − n + m P mn ( v ) [44] shows that n is an odd integer.Thus the most general solutions for the magnetic potential can be written as A = ∞ X n =0 b n +1 ( − n +3 Q n +1 ( iu ) P n +1 ( v ) . (23)As A is a even function of v , it is continuous across the disk. Consequently,its normal derivate is an odd function of v and hence discontinuous across thedisks. This implies that the source of the magnetic field also is planar. Usingthe relation ddz Q mn ( z ) | z =0 = − ( n − m + 1) Q mn +1 (0) , (24)one finds that the surface current density Eq. (8) isj ϕ = 2 µ av ∞ X n =0 b n +1 (2 n + 1)( − n +3 iQ n +2 (0) P n +1 ( v ) , (25)where v = p − R /a .In order to determine the coefficients b n +1 , we write the current density asj ϕ = 2 µ a F ( v ) v (26)7ith F ( v ) = ∞ X n =0 a n +1 P n +1 ( v ) , (27)where a n +1 = b n +1 (2 n + 1)( − n +3 iQ n +2 (0). By using the orthogonalityproperty of the the associated Legendre functions of the the first kind, weobtain a n +1 = (4 n + 3)(2 n )!2(2 n + 2)! Z − F ( v ) P n +1 dv. (28)On the other hand, from Eq. (6) we have B R = − ∂A∂z , (29)so that Eq. (8) for the current density can be written asj ϕ = − µ B R | z =0 + . (30)Thus, using the identity iQ n +2 (0) = ( − n +2 (2) n +1 ( n + 1)!(2 n + 1)!! , (31)we get b n +1 = a (4 n + 3)[(2 n )!] ( − n (2) n +2 n !( n + 1)!(2 n + 2)! Z − vB R ( v ) P n +1 ( v ) dv. (32)This expression permits us to determinate all the coefficients b n +1 by givingthe radial component of the magnetic field on the disk or the profile of theazimuthal electric current density. As a simple example, we consider the azimuthal current density of the disk tobe j ϕ = j (cid:18) Ra (cid:19) (cid:18) − R a (cid:19) j , (33)where j and j are constants. In oblate coordinates we havej ϕ = j v j (1 − v ) . (34)Since the surface current density diverges at the disc edge, when v = 0, thisexpression satisfies the requirement that the current density must vanish there.8t also vanishes at the disk center, which can indicate the presence of a centralstar.By taking j as a odd positive integer the integral in Eq. (32) is even andhence b n +1 = µ aj (4 n + 3)[(2 n )!] ( − n +1 (2) n +2 n !( n + 1)!(2 n + 2)! Z v j +1 (1 − v ) P n +1 dv. (35)Using the expression [45]( − m m +1 Γ(1 − m + n ) Z x σ (1 − x ) m P mn ( x ) dx =Γ( + σ )Γ(1 + σ )Γ(1 + m + n )Γ(1 + σ + m − n )Γ( + σ + m + n ) , (36)which is valid for Re σ > − m a positive integer, we obtain b n +1 = µ aj (4 n + 3)[(2 n )!] ( − n n +4 n !( n + 1)!(2 n + 2)! × Γ(2 n + 3)Γ(1 + j )Γ( + j )Γ(2 n + 1)Γ( + j − n )Γ(3 + j + n ) , (37)where n ≥ j is an odd positive integer. The constant j can be deter-mined from a reference magnetic field B , and accordingly is the parameterthat controls the magnetic field. Since the field is poloidal, the exterior fieldto the disk asymptotically approaches at infinity a uniform vertical field whichis also measurable [46, 47], and hence this value of magnetic field can be takenas reference. Thus, from Eq. (6) we have that j = R B ∂ ( ˜ AR ) ∂R (cid:12)(cid:12)(cid:12) R , (38)where R is the radial distance at which the imposed magnetic field is known,and ˜ A = A/j . For protoplanetary disks an accepted value for this field is B = 10 µ G for R = 100 AU [11].We consider the two first terms of the series in Eq. (23) taking j = 1. For n = 0 have b = µ aj and for n = 1 b = − µ aj . Thus b = − b andthe parameter j takes the value j = 2 B b [ − u + (3 ˜ R −
2) cot − ( u )] , (39)where ˜ b = b /j , u = q ˜ R − R = R /a .9n this case the magnetic potential is given by A = b √ − v (cid:26) u [8 + (13 + 15 u )(5 v − √ u − [8 + 3(1 + 5 u )(5 v − √ u cot − ( u ) o , (40)and surface electric charge density isj ϕ = − b µ o a ( R/a ) r − R a . (41)In order to analyse the motion of charged test particles around of the disks,we consider the first Morgan-Morgan disk. In oblate coordinates, Eq. (10) forthe circular speed takes the form v c = − ˜ e (cid:18) A + ( v − v ∂A∂v (cid:19) ± s ˜ e (cid:18) A + ( v − v ∂A∂v (cid:19) + ( v − v ∂ Φ ∂v , (42)so that inside the disk the circular speed of particles is given by v c = −
58 ˜ ebπ ˜ R (3 ˜ R − ± r e b π ˜ R (3 ˜ R − + 3 π ˜ R , (43)whereas outside the disk is v c = −
54 ˜ eb ˜ R [ − u + (3 ˜ R −
2) cot − ( u )] ± (cid:18) e b ˜ R [ − u +(3 ˜ R −
2) cot − ( u )] + 34 [ − u + π ˜ R − R cot − ( u )] (cid:19) / , (44)with u = p ˜ R − ϕ for ˜ e = 1 anddifferent values of the parameter of magnetic fields b = 0 , 0 . .
3, and 0 . R . We see that j ϕ is zero in the center of the disk where thesurface mass density is greater (Eq. (20)), reaches a maximum and then fallsto zero at rim of the disk where there is no matter.In Fig. 2 we have plotted the curves of the circular speed v + and v − forthe motion of charged test particles inside the the disk (Figs. 2( a ) and 2( b ))and for particles outside the disk (Figs. 2( c ) and 2( d )), for ˜ e = 1 and thesame value of the other parameters. Inside the disk, we find that for directorbits the magnetic field increases the speed of particles to a certain values of˜ R , and then decreases it, whereas for retrograde orbits the contrary occurs.10utside the disk, we observe that for direct orbits the magnetic field decreasesthe speed of particles everywhere of the disk whereas for retrograde orbits thecontrary occurs.In Fig. 3 we have plotted the specific angular momentum L and L − forthe motion of test particles inside the the disk (Figs. 3( a ) and 3( b )) and forparticles outside the disk (Figs. 3( c ) and 3( d )), with ˜ e = 1 and the samevalue of the other parameters. Inside the disk, we find that for direct orbitsthe magnetic field can make less stable the motion of the particles againstradial perturbations, whereas for retrograde orbits the particles are alwaysstable. Outside the disk, we observe that for direct orbits the magnetic fieldcan stabilize the particles, whereas for retrograde orbits the magnetic fieldenhances the zone of instability.In Fig. 4 we have plotted the surface and level curves of the function RA that represents the magnetic field lines as a function of R and z for theparameter b = 0 .
5. We find that the lines are closed curves surrounding thedisk which suggests that the source of the magnetic field is a ringlike currentdistribution and that the field is poloidal.
4. Conclusions
In the present work we constructed a family of finite thin disks with electriccurrent in presence of a poloidal magnetic field by using oblate spheroidalcoordinates. These models permit to determine the magnetic field structurenear to disk knowing the experimental magnetic field strength far away fromthe disk. The profile of the azimuthal electric current density on the disksurface or the radial component of the magnetic field also need to be assumed.The models are illustrated for a particular profile of anular azimuthal elec-tric current density. For these structures we analyse the rotation curves forequatorial circular orbits of charged test particles both inside and outside thedisk and also its stability against radial perturbation using a extension of theRayleigh criterion. We find that the presence of the magnetic field can affectin different ways the movement of particles.
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