Axially Symmetric Post-Newtonian Stellar Systems
aa r X i v : . [ g r- q c ] O c t Axially Symmetric Post-NewtonianStellar Systems
Camilo Akimushkin ∗ Javier Ramos-Caro † Guillermo A. Gonz´alez ‡ Escuela de F´ısica, Universidad Industrial de SantanderA. A. 678, Bucaramanga, Colombia
Abstract
We introduce a method to obtain self-consistent, axially symmetric,thin disklike stellar models in the first post-Newtonian (1PN) approx-imation. The models obtained are fully analytical and corresponds tothe post-Newtonian generalizations of classical ones. By introducingin the field equations provided by the 1PN approximation a knowndistribution function (DF) corresponding to a Newtonian model, twofundamental equations determining the 1PN corrections are obtained,which are solved using the Hunter method. The rotation curves ofthe 1PN-corrected models differs from the classical ones and, for thegeneralized Kalnajs discs, the 1PN corrections are clearly appreciablewith values of the mass and radius of a typical galaxy. On the otherhand, the relativistic mass correction can be ignored for all models.PACS numbers: 04.25.Nx, 98.10.+z
The stars observed in the universe tend to cluster in huge self-gravitatingsystems, being the galaxies among the most noticeable and studied of them. ∗ e-mail: [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] v/c ). For Minkowski’s background metric, the firstpost-Newtonian corrections (1PN) are included taking the terms g ≈ − g + g , (1) g i ≈ g i , (2) g ij ≈ δ ij + g ij , (3)where the upper index denotes the order of the power of ( v/c ) of the term.Using harmonic coordinates, the 1PN order potentials are defined as g ≡ − φ/c , (4) g i ≡ ζ i /c , (5) g ≡ − φ + ψ ) /c . (6)Henceforth we consider only the stationary case, so the explicit time-dependentterms along with the potential vector ζ disappear in all equations.The stationary field equations in the 1PN approximation are ∇ φ = 4 πGc T , (7) ∇ ψ = 4 πG ( T + T ii ) , (8)where, in the classical limit, potential ψ vanishes and φ tends to the New-tonian potential φ N . Whereas thatd v d t = −∇ φ − c (cid:2) ∇ (cid:0) φ + ψ (cid:1) + 4 v ( v · ∇ φ ) − v ∇ φ (cid:3) (9)2s the stationary equation of motion.A complete statistical description is achieved by knowing the distributionfunction (DF) of the system. The DF satisfies a continuity equation in thephase space called the Boltzmann equation. In the 1PN approximation,the collisionless Boltzmann equation (CBE) for a many identical particlessystem is given by [20] v i ∂F∂x i − ∂φ∂x i ∂F∂v i − c (cid:18) ∂φ∂x i (4 φ + v ) − ∂φ∂x j v i v j + ∂ψ∂x i (cid:19) ∂F∂v i = 0 . (10)According to Jeans theorem [21], the solution of colissionless Boltzmannequation is any function of the integrals of motion. Now, it is easy to verifythat two isolated integrals a 1PN system with axial symmetry are given by E = 12 v + Φ , (11)where Φ = φ + 2 φ + ψc , (12)and L z = Rv ϕ e − φ/c ≈ Rv ϕ (1 − φ/c ) , (13)which can be interpreted as the 1PN generalizations of energy and the z component of angular momentum, respectively.In addition, the DF must satisfy the condition of self-consistently gener-ating the macroscopic mean values. In the post-Newtonian approximation,the following components of the energy-momentum tensor are need, T ( x , t ) = c Z F ( x , v , t ) d v, (14) T + T ii = 2 Z ( v ( x , t ) + φ ( x , t )) F ( x , v , t ) d v. (15)Therefore, in the post-Newtonian approximation, self-consistent equilibriummodels are defined by two scalar potentials, φ and ψ , together with a DF thatsatisfies 1PN CBE and relations (14) - (15) generating self-consistenly the1PN components of the energy-momentum tensor. In this paper we presenta method to obtain post-Newtonian axially symmetric equilibrium models.The method uses thin disk models, allowing to solve the two differentialequations with the Hunter’s method [22], as shown in the next section. Thesolution of the field equations is obtained considering equations in vacuum,3n which case, the energy-momentum tensor vanishes and the content ofmatter is expressed as a boundary condition on the fields in the disk, asshown below. Finally, in section 3 the first axially symmetric models in the1PN approximation are presented. For thin disks of finite radius the components of the energy-momentumtensor can be written as T = c Σ( R ) δ ( z ) , (16) T + T ii = σ ( R ) δ ( z ) , (17)for 0 ≤ R ≤ a , where δ is the Dirac delta function, and being zero for R > a .Therefore, the field equations reduce to two Laplace equations for the fields φ and ψ . It is demanded that the fields are even functions of zφ ( R, z ) = φ ( R, − z ) , ψ ( R, z ) = ψ ( R, − z ) , (18)and therefore, that the first derivatives with respect to z are odd functionsof z .Using Gauss’ theorem with (16) and (17) gives,Σ( R ) = 12 πG (cid:18) ∂φ∂z (cid:19) z =0 + , (19) σ ( R ) = 12 πG (cid:18) ∂ψ∂z (cid:19) z =0 + . (20)The problem is defined with the following boundary conditions: fields vanishat infinity, and at z = 0 its derivatives depend on the energy-momentumtensor components according to (19) and (20) for 0 ≤ R ≤ a and vanishingfor R > a . Applying Hunter’s method to each of the 1PN field equations,one can obtain exact analytical expressions for the potentials. The Hunter’smethod consists on obtaining solutions of the Laplace equation in oblatespheroidal coordinates. The oblate coordinates are related to the cylindricalby R = a p (1 + ξ )(1 − η ) , (21) z = aξη, (22)4here 0 ≤ ξ < ∞ and − ≤ η ≤
1. The disk is placed at ξ = 0, where, η = 1 − R /a .Following Hunter [22], the general solution of each Laplace equationsatisfying previous boundary conditions can be written as φ ( ξ, η ) = − ∞ X n =0 A n q n ( ξ ) P n ( η ) , (23)for the φ potential, and ψ ( ξ, η ) = − ∞ X n =0 B n q n ( ξ ) P n ( η ) , (24)for the ψ potential, where A n and B n are the constants required for eachmodel, P n ( η ) are the Legendre polynomials and q n ( ξ ) = i n +1 Q n ( iξ ) areLegendre functions of second kind. Note that when using classical models,the expression for φ can be written taking constants of the form A n = C n + D n /c , (25)where the C n constants define the Newtonian potential φ N and the con-stants D n define the correction φ P N . So that taking the limit c → ∞ , φ = φ N + φ P N is reduced to the Newtonian part only. The correspondingexpressions for Σ and σ in oblate coordinates areΣ = 12 πaGη ∞ X n =0 A n (2 n + 1) q n +1 (0) P n ( η ) , (26) σ = 12 πaGη ∞ X n =0 B n (2 n + 1) q n +1 (0) P n ( η ) . (27)Accordingly, Σ can also be writen as the sum of a Newtonian part and apost-Newtonian correction: Σ = Σ N + Σ P N .In order to obtain self-consistent models of axially symmetric thin disksin equilibrium, it is used a DF of the form F = f ( R, v R , v ϕ ) δ ( z ) δ ( v z ), whichis zero for R > a . The DF reproduces Σ and σ through the equationsΣ( R ) = Z Z f ( R, v R , v ϕ ) dv R dv ϕ , (28) σ ( R ) = 4 Z Z Ef ( R, v R , v ϕ ) dv R dv ϕ − φ N Σ N , (29)5here E is the integral of motion defined by (11). As in the Newtonian case,one could define a relative energy and a relative potential as ε = − E + Φ , (30)Ψ = − Φ + Φ , (31)where Φ is a constant that is chosen so that ε and Ψ are always positive,i.e., such that f > < ε ≤ Ψ.The method to obtain self-consistent post-Newtonian models is to take(28) with (26) as the first fundamental equation and (29) with (27) as thesecond fundamental equation. In accordance with the order of magnitude ofthe approach, the second fundamental equation is solved using the Newto-nian terms and then the first fundamental equation is solved using the termsup to 1PN order. Finally, note that for the second fundamental equation,the constants can be obtained directly by B i = 4 i + 14 i + 2 2 πaGq i +1 (0) Z − P i ( η ) ησ d η, (32)where the integral depends on the particular model. On the other hand, inthe first fundamental equation, the constants also appear on the right sideof the equation (hence, it is necessary to obtain the constants differently foreach model) and, in general, we can not use an explicit expression as doneabove. However, it is found that for all models treated it is finally possibleto obtain expressions analog to (32) for the first fundamental equation, see(57) - (59) and (74) - (75). The DF in the 1PN approximation presents thesame functional dependence on the integrals of motion that in the Newto-nian models, that is, one uses the same DF but now with the 1PN energyand angular momentum. Of course, this is for the sake of correspondencewith the Newtonian limit. Note that the correspondence principle must besatisfied by both the DF and its integrals.It is possible to obtain alternative expressions for Σ and σ considering inparticular the case where the DF depends on a linear combination of energyand angular momentum called Jacobi’s integral, J = Ω L z − E . Jacobi’sintegral is interpreted classically as the energy measured from a frame ofreference rotating with angular speed Ω [21]. In terms of the relative energy(30), we can write Jacobi’s integral as J = ε + Ω L z − Ψ e (0) , (33)where Ψ e (0) is the 1PN relative-effective potential evaluated in η = 0, de-fined by Ψ e = Ψ + Ω R (cid:0) − φ/c (cid:1) . (34)6he Jacobi’s integral takes values between zero and J max , with J max = Ψ −
12 Ω a η (cid:18) − φc (cid:19) − Ω a φc . (35)Now, given that 2 πdJ = dv R dv ϕ , the relation (28) can be written asΣ = 2 π Z J max f ( J ) dJ. (36)Then, by using the expression Z Z v ϕ f dv R dv ϕ = 2 π h v ϕ i Z J max f ( J ) dJ, (37)the second fundamental equation can be written as σ = 2(2Φ − Ω a − φ N + 2 a Ω p − η h v ϕ i )Σ N − π Z J max J f ( J ) dJ, (38)with Σ N given by (26) with A n = C n .Finally, the circular speed necessary for the rotation curve can be ob-tained considering (9) for circular orbits. In that case, the circular speed isequal to v φ and perpendicular to the gradient of the field φ . Therefore, the1PN equation of motion (9) reduces to v ϕ R (cid:20) Rc ∂φ∂R (cid:21) z =0 = ∂∂R (cid:20) φ + 2 φ + ψc (cid:21) z =0 . (39)Then, in accordance with the 1PN order of approximation, the expressionfor the circular speed is v ϕ = s R ∂φ∂R (cid:18) φc − Rc ∂φ∂R (cid:19) + Rc ∂ψ∂R % z =0 . (40)Note that in the limit c → ∞ , the previous expression reduces to that ofNewtonian theory v ϕ = p R∂φ N /∂R . We will now apply the previous formalism to the family of GeneralizedKalnajs Disks (GKD) introduced by Gonz´alez and Reina in [9]. This familyis characterized by mass densities of the formΣ ( m ) N = 3 M πa η m − , (41)7here the index m of the model is any positive integer. The potential φ ( m ) N is given by (23) by taking the Newtonian limit in (25) and using C ( m )2 n = M Gπ / (4 n + 1)(2 m + 1)! a m +1 (2 n + 1)( m − n )!Γ( m + n + ) q n +1 (0) (42)for n ≤ m , and C ( m )2 n = 0 for n > m . m = 1 GKD
The first disk of the family, the disk with m = 1, is the well known Kalnajsdisk [6], with the mass distributionΣ (1) N = 3 M η πa . (43)Then, as was shown by Kalnajs [23], the DF depends on the Jacobi’s integralas f ( J ) = 3 M π a (cid:2) − Ω ) J (cid:3) − / (44)where Ω = 3 πGM/ a . Now, for the Kalnajs disk it holds that h v ϕ i = Ω R ,so that the disk spins like a rigid body.Inserting expressions (43) and (44) into the second fundamental equation(38), and integrating the DF, one easily obtains the following system ofequations B − B + B = 0 , (45)3 B − B = 6 M Ga Ω − πG M a , (46) B = 9 πG M a − GM Ω a. (47)Likewise, for the first fundamental equation we have D − D + D = 9 πG M Ω a (Ω o − Ω ) , (48) γD + ϑD = B − B , (49) χD = 3516 B − πG M a − GM Ω a, (50)8here γ = 24 a (Ω o − Ω ) − πM G πM G , (51) ϑ = 135 πM G/ − a (Ω o − Ω )9 πM G , (52) χ = 70[128 a (Ω o − Ω ) − πM G ]864 πM G . (53)Therefore, we have a system of linear equations with an upper triangularmatrix for the constants of each potential, ψ and φ P N .Solving for the constants explicitly, we obtain B = 2 aGM ( π − − G M π (15 π − a , (54) B = aGM (7 π − − G M π (21 π − a , (55) B = GM a (cid:16) GMπ a − a Ω (cid:17) , (56) D = − G π M a ( a Ω − GMπ ) − G π M a (128 a Ω − GMπ ) − G π (1+80 π ) M a − ( G M π − G M π ) a (8 a Ω − GMπ ) , (57) D = − G π (7 π − M a − ( G M π − G M π ) a (8 a Ω − GMπ ) , (58) D = G M π ( a +39 GMπ ) a (128 a Ω − GMπ ) , (59)which defines the potential ψ , and the correction φ P N , respectively. Hencewe can get all the parameters of interest, in particular, we observed that themass correction, Σ
P N , is negligible compared with the classical mass Σ N .In contrast, the 1PN rotation curve obtained with (40), is visibly separatedfrom the Newtonian one from a certain radius, the difference being maximumat the edge of the disk (see Fig. 1), where the difference between both reaches10 .
3% approximately. Those have been obtained with the typical values ofa galaxy such as the Milky Way. m = 2 GKD
The procedure for the second disk is pretty much the same that for the firstone. Again, the DF has a simple form when written as a function of Jacobi’s9igure 1: First we plot the Newtonian and 1PN mass density for the m =1 GKD. The two curves seem to be superposed meaning that the masscorrection is negligible. Then we plot the Newtonian (dashed line) and 1PN(full line) rotation curves for the same disk. The 1PN corrections are clearlynotorious reaching a maximum at the border of the disc. Parameter valuesare: M = 4 × kg , a = 3 × m , Ω = 2 × − Hz .integral, f ( J ) = 2 M √ a (cid:18) a G M π J (cid:19) / , (60)for a mean circular speed h v ϕ i = Ω R = p πGM/ a R . This DF could beeasily integrated to self-consistently obtain the mass density of the model,which also can be obtained from (41) with m = 2,Σ (2) N = 5 M πa (cid:18) − R a (cid:19) / = 5 M η πa . (61)the associated gravitational potential is given by the Newtonian limit φ N of(23) with (42) and m = 2.Again, we replace the Newtonian terms into the second fundamentalequation (38) to obtain: X n =0 B n (2 n + 1) q n +1 (0) P n ( η ) = X n =0 ˜ C i η i , (62)10here ˜ C = ˜ C = 0 and˜ C = (25 π G M ) / (160 a ) , (63)˜ C = − (75 π G M ) / (160 a ) (64)˜ C = (75 π G M ) / (2240 a ) . (65)Multiplying (62) with a Legendre polynomial, integrating with respect to η ,and using the orthogonality properties of the Legendre polynomials we get B n = X i =0 √ π ˜ C i − i − (4 n + 1)Γ(2 n + 1) q n +1 (0)(2 n + 1)Γ( i − n + 1)Γ( i + n + 3 / . (66)We also could have used the expression (32).After rearranging the terms, the first fundamental equation can be writ-ten as X n =0 { D n [ ϑ n P n ( η ) + q n (0) P n (0)] − B n q n (0) P n ( η ) − ˆ C n η n } = 0 , (67)where, ϑ n = π (2 j + 1) / (32 a q n +1 (0) − q n (0)) , (68)ˆ C = (675 π G M ) / (4096 a ) + ψ (0 , , (69)ˆ C = − (1575 π G M ) / (4096 a ) , (70)ˆ C = − (1125 π G M ) / (2048 a ) , (71)ˆ C = − (2025 π G M ) / (4096 a ) , (72)ˆ C = (2025 π G M ) / (8192 a ) . (73)Integrating and using orthogonality relations, the constants D n are givenby D n = X i =0 √ π ˆ C i − i − (4 n + 1)Γ(2 j + 1) ϑ n (2 n + 1)Γ( i − n + 1)Γ( i + n + 3 /
2) + B n q n (0) ϑ n , (74)11igure 2: We plot the Newtonian and 1PN mass density, as well as theNewtonian (dashed line) and 1PN (full line) rotation curves for the m = 2GKD, with parameter values: M = 4 × kg , a = 3 × m , Ω = 2 × − Hz . As we can see, the two graphics of mass density appear to besuperposed, while that the 1PN rotation curve behaves completely differentto the Newtonian one.for n >
0, and D = 1 ϑ + π/ " X i =0 √ π ˆ C i − i − Γ(2 j + 1)Γ( i + 1)Γ( i + 3 / − X i =1 D i q i (0) P i (0) , (75)which defines the correction φ P N to the Newtonian potential. The massdensities and rotation curves are plotted at Figure 2, note that the rota-tion curve is not only cuantitative, but cualitatively different, presenting itsmaximum value more closely to the center of the disk and going upwards atthe border. This is due to the fact that the 1PN curve includes more termsthan the Newtonian one.
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