Astronomy Reports, Vol. 45, No. 10, 2001, pp. 751–754. Translated from Astronomicheski˘ı Zhurnal, Vol. 78, No. 10, 2001, pp. 867–870. Original Russian Text Copyright c 2001 by Kondratyev. The Angular-Momentum Distribution in Axisymmetrical Galaxies B. P. Kondratyev Udmurtia State University, Izhevsk, Russia Received May 15, 2000 Abstract—We investigate the angular-momentum distribution in a classical homogeneous Maclaurin spheroid. An exact formula for the distribution as a function of the mass of a cylindrical column in the spheroid is obtained in both integral and differential forms. This expression is used to calculate the angular-velocity distributions in galaxies evolving from initially spheroidal configurations to steady-state configurations with observed density distributions. c 2001 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION Maclaurin spheroid (Section 2); in Section 3, the It is convenient to construct numerical and analyt- proposed technique is used to calculate the angular- ical models for the axisymmetrical figures ofequilib- velocity distributions in galaxies with real density dis- rium for gaseous systems and galaxies by specifying tributions. The method ofcomparing angular mo- the angular momentum L[M(R)] inside the system menta is efficient and simple, but has remained little as a function of the mass enclosed in a cylindrical used until now. surface with a specified value of L. This approach is physically more transparent than the frequently 2. THE FUNCTION L[M(R)] used in practice ofchoosing a priori some partic- FOR A MACLAURIN SPHEROID ular distribution for the angular velocity inside the configuration. Specifying the function L(R) is also Let us consider in cylindrical coordinates R = justified from the standpoint of evolution. Since L(R) x2 + y2 and z a homogeneous spheroid with den- is invariant during the evolution ofan isolated, ax- sity ρ, semi-axes a1 ≥ a3,andthesurfaceequation isymmetrical gravitating system free of turbulence R2 z2 and viscosity, the angular-momentum distributions 2 + 2 =1. (2) observed in galaxies can also shed light on the dy- a1 a3 namical states ofprotogalaxies. We note in this Let a cylindrical surface with radius R separate the connection the study ofCrampin and Houle [1], who volume V (R) in the spheroid (Fig. 1). This volume concluded that the distributions ofthe speci ficangu- corresponds to the shaded area in Fig. 1, and is equal lar momentum for eight flat galaxies derived from their to rotation curves coincided with the specificmomen- 2 3/2 tum l[M(R)] for a classical, homogeneous Maclaurin 4 2 − − R − 3 V (R)= πa1a3 1 1 2 = Vt(1 x ), spheroid to within 1% (see also [2]). These results 3 a1 were subsequently confirmed [3]. (3) In his detailed review [4], Antonov notes that many where authors adopt as the initial state a Maclaurin ellip- soid, for which H R2 4 x = = 1 − ,V= πa2a . 2 2 t 1 3 (4) /3 a3 a1 3 5 Lt − − M l[M(R)] = 1 1 . (1) ≤ ≤ 2 Mt Mt Obviously, 0 x 1. It follows from (3) that the mass ofthe shaded fractionofthe spheroid Here, l[M(R)] and M(R) have the same meaning 3 M(R)=Mt(1 − x ). (5) as above, and Lt and Mt are the total momentum and mass ofthe con figuration. Formula (1) is widely Further, the rotational momentum ofa con figura- known; in addition to the review [4], it is referenced, tion with a solid-body rotation and angular velocity Ω for instance, in the well-known study by Tassoul [5]. is Here, we first derive the formula for the mean L =ΩIz; (6) distribution ofthe angular momentum in a classical 1063-7729/01/4510-0751$21.00 c 2001 MAIK “Nauka/Interperiodica” 752 KONDRATYEV z It is apparent that 1 I = M R2 = πρR4H = πρR4a x. (9) cyl 2 cyl 3 A h R B The moment ofinertia fora spheroidal segment is H a3 1 1 π I = πρ R4dz = ρa4a (1 − x2)2dx. 0 R seg 2 2 1 3 H x C D (10) We can easily calculate the integral in (10): π 8 2 1 I = ρa4a − x + x3 − x5 , (11) Fig. 1. Oblate spheroid. The shaded cylindrical area seg 2 1 3 15 3 5 consists ofthe cylinder ABCD with radius R and two spheroidal segments with height h. where x is determined from (4).1 Thus, the moment ofinertia forthe entire delin- LR() eated part ofthe homogeneous spheroid is ------------ L t 4 4 − 3 5 1 I(R)= πρa1a3 2 5x +3x (12) 15 5 3 = I 1 − x3 + x5 , t 2 2 where It is the total moment ofinertia ofthe spheroid. Eliminating x from (12) using (5) and taking into account (6), we obtain the desired formula for 1 the angular-momentum distribution in a Maclaurin spheroid − 5 − M(R) 2 L(R)=Lt 1 1 (13) 2 Mt 3 M(R) 5/3 + 1 − , 2 Mt shown graphically in Fig. 2. 0 1 R/a1 3. APPLICATION TO A STRATIFIED INHOMOGENEOUS GALAXY Fig. 2. Angular momentum distribution in a Maclaurin spheroid as a function of R/a1:curve1 shows the func- We will compare two models ofgalaxies: an initial tion L(R)/Lt according to (13), and curve 2 shows the model A, which is less oblate and close to a Maclaurin distribution ofthe speci fic angular momentum according spheroid (this model is also considered in [1]), and a 3 2 final model B, which is essentially inhomogeneous to (19), 2 a1 l(R)= R 1 − R . 15 Lt a1 a1 and consistent with observations. The latter model displays barotropic differential rotation. As the galaxy evolves, each element conserves its angular where momentum, and there is no mixing ofelements with I = ρR2dV (7) z 1 V In the particular case x =0 (halfa spheroid), we obtain I = 2 ma2; besides, (11) yields for the case of a spherical is the moment ofinertia about the axis ofrotation Oz. z 5 1 In our case, ρ =const,andtheconfiguration under segment a1 = a1 = r consideration consists ofa cylinder with height 2H mh 20r2 − 15hr +3h2 I = , and two spheroidal segments (Fig. 1). The total mo- seg 10 3r − h ment ofinertia ofthe delineated part ofthe spheroid Iz where h is the height ofthe segment and m = πρh2 R − h is essentially the sum ofthe moments ofinertia ofthe 3 is its mass. The moment ofinertia fora spherical segment cylinder and two segments: given on p. 69 ofthe handbook is underestimated by a factor Iz = Icyl +2Iseg. (8) oftwo. ASTRONOMY REPORTS Vol. 45 No. 10 2001 ANGULAR-MOMENTUM DISTRIBUTION IN AXISYMMETRICAL GALAXIES 753 different momenta. The essence ofthe problem is z that, in this case, the two models will have virtually AB the same dependence L[M(R)]. Let us consider the specific case when, in the R course ofevolution, a protogalaxy is transformed R0 into a stratified, inhomogeneous spheroid with a 0 R barotropic rotation law Ω=Ω(R) and with the density distribution C D ρ = ρ(m). (14) For E galaxies, the density can be well approxi- Fig. 3. Astratified inhomogeneous spheroid. ABCD is mated by the law the delineated cylindrical volume ofradius R0.Thetorus cross section is shaded. ρ0 ⁄ ρ = , (15) Ω 1 – q23 2 3/2 ---- = ------------------- (1 + βm ) Ω˜ 2 m0 where the constant β is determined by comparison with photometric data. In the general case, the surfaces of constant density will form a family of spheroidal surfaces with various degree of oblateness: 20 z2 R2 + = m2a2, 0 ≤ m ≤ 1. (16) 1 − e2(m) 1 As noted above, it is important that the angular- momentum distribution (13), which for the sake of brevity can be written in the form 5 3 L(R)=L 1 − q + q5/3 ,q= x3, (17) t 2 2 10 will be invariant in the course ofevolution. Under these completely reasonable assumptions, we can de- rive the function Ω(R) specifying the rotation in the final model ofthe galaxy. To this end, we will delineate an elementary cylindrical envelope with radius R0 = a1m0 and unit thickness inside the spheroid (Fig. 3). Using the mathematical technique developed in [7], we can derive the angular momentum for the envelope: 0 dL(R) dL(m0) 1 m l(R )= = . (18) 0 0 dR a dm R=R0 1 0 Substituting (17) into this expression, we obtain after Fig. 4. Distribution ofrelative angular velocity inside an transformations inhomogeneous spheroidal galaxy, calculated according to (23), β = 150. 2 Lt − 2/3 l(m0)=10πa1m0 1 q Φ(m0), (19) Mt 1 where 3/2 dmρ(m) d 1 − e2(m) m2 − m2 1 dm 0 d m0 − 2 2 − 2 = . Φ(m0)= dmρ(m) 1 e (m) m m0 1 dm dmρ(m) d m3 1 − e2(m) m dm 0 0 (20) and On the other hand, for the same elementary cylin- M(R) drical envelope, assuming barotropic rotation in the q =1− (21) Mt galaxy, direct calculation yields the specificangular ASTRONOMY REPORTS Vol. 45 No. 10 2001 754 KONDRATYEV momentum velocity in a Maclaurin spheroid. This formula is 4 3 l(m0)=4πa m Ω(R0)Φ(m0). (22) new and convenient to use. The calculations are 1 0 in good agreement with observations ofrotational Comparing (22) and (19) yields the desired expres- motions in galaxies. Note also that, according to sion for the angular velocity: our calculations, the square ofthe angular velocity 2/3 1 − q in the center ofthe model increases by a factorof518, Ω(m0)=Ω 2 , (23) m0 whereas the density increases by a factor of 1855. Therefore, Maclaurin modeling of the momentum where the coefficient distributions in galaxies satisfies Poincare’s´ criterion 5 Lt Ω= (24) Ω2 ≤ 2πGρ.