Analytical Potentials for Flat Galaxies with Spheroidal Halos
aa r X i v : . [ a s t r o - ph . C O ] O c t Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-ddd de 2016
Physical Sciences
Analytical Potentials for Flat Galaxies with Spheroidal Halos
Guillermo A. Gonz´alez , ∗ , and Jerson I. Reina , Escuela de F´ısica, Universidad Industrial de Santander, Bucaramanga, Colombia Departamento de Ciencias B´asicas, Universidad Santo Tom´as-Bucaramanga, Bucaramanga, Colombia
Abstract
A family of analytical potential-density pairs for flat galaxies with spheroidal halos is presented. Thepotential are obtained by means of the sum of two independent terms: a potential associated with a thindisc and a potential associated with a spheroidal halo, which are expressed as appropriated superpositionsof products of Legendre functions, in such a way that the model implies a linear relationship betweenthe masses of the thin disc and the spheroidal halo. By taking a particular case for the halo potential,we found that the circular velocity obtained can be adjusted very accurately to the observed rotationcurves of some specific galaxies, so that the models are stable against radial and vertical perturbations.Two particular models for the galaxies NGC4389 and UGC6969 are obtained by adjusting the circularvelocity with data of the observed rotation curve of some galaxies of the Ursa Mayor Cluster, as reportedin
Verheijen and Sancisi (2001). The values of the halo mass and the disc mass for these two galaxiesare computed obtaining a very narrow interval of values for these quantities. Furthermore, the values ofobtained masses are in perfect agreement with the expected order of magnitude and with the relative orderof magnitude between the halo mass and the disc mass.
Key words:
Potential Theory, Disk Galaxies, Celestial Mechanics, Galactic Mass.
Potenciales Anal´ıticos Para Galaxias Planas con Halos EsferoidalesResumen
Se presenta una familia de pares anal´ıticos potencial-densidad para galaxias planas con halos esferoidales.Los potenciales son obtenidos por medio de la suma de dos t´erminos independientes: un potencial asociadoal disco delgado y un potencial asociado al halo esferoidal, los cuales son expresados apropiadamentecomo la superposici´on de productos de funciones de Legendre, de tal manera que el modelo implica unarelaci´on lineal entre las las masas del disco delgado y el halo esferoidal. Tomando un caso particular parael potencial del halo, encontramos que la velocidad circular obtenida puede ser ajustada muy precisamentecon la curva de rotaci´on de algunas galaxias espec´ıficas, de tal manera que los modelos son estables contraperturbaciones radiales y verticales. Dos modelos particulares para las galaxias NGC4389 y UGC6969 sonobtenidos ajustando la velocidad circular del modelo con datos de la curva de rotaci´on observada de algunasgalaxias del Cluster de la Osa Mayor, reportados en
Verheijen ans Sancisi (2001). Los valores de la masadel halo y la masa del disco para estas dos galaxias son calculados obteniendo un intervalo muy estrecho devalores para dichas cantidades. Adem´as, los valores de masa aqu´ı obtenidos est´an en perfecto acuerdo conel orden de magnitud esperado y con el orden de magnitud relativo entre la masa del halo y la masa del disco.
Palabras clave:
Teor´ıa del Potencial, Galaxias de Disco, Mec´anica Celeste, Masa de Galaxias. ∗ Correspondence: G. A. Gonz´alez, [email protected], Received xxxxx XXXX; Accepted xxxxx XXXX. . A. Gonz´alez and J. I. Reina Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-ddd de 2016 Introduction
One of the oldest and most important problems in galacticdynamics is the determination of the mass distribution basedon the observations of the circular velocity or rotation curve(
Pierens and Hure , 2004), defined as the speed of the starsmoving in the galactic plane in circular orbits around the cen-ter. Now, if we assume a particular model for the compositionof the galaxy, the fit of that model with the rotation curve ofa particular galaxy can, in principle, completely determine thedistribution of mass. So then, the rotation curve provides themost direct method to measure the distribution of mass of agalaxy (
Binney and Tremaine , 2008).Currently, the most accepted description of the compositionof spiral galaxies is that a significant portion of its mass isconcentrated in a thin disc, while the other contributions tothe total mass of the galaxy come from a spherical halo ofdark matter, a central bulge and, perhaps, a central black hole(
Binney and Tremaine , 2008). Now, since all componentscontribute to the gravitational field of the galaxy, obtainingappropriate models that include the effects of all parts is aproblem of great difficulty. However, the contribution of eachpart is limited to certain distance scales, so in a reasonablyrealistic model it is not necessary to include the contributionof all components (
Faber , 2006).In particular, the gravitational influence of the central blackhole is appreciable only within a few parsecs around the centerof the galaxy (
Sch¨odel et al. , 2002), so it can be completelyneglected when studying the dynamics of the disc, or in regionsoutside the central bulge, while the bulge mainly dominatesthe inner region of the galaxy to a few kiloparsec. So then,the main contributions to the gravitational field of the galaxycome from the galactic disc and the dark matter halo (
Faber ,2006). However, it is commonly accepted that many aspectsof galactic dynamics can be described, in a fairly approximateway, using models that consider only the contribution of a thingalactic disc (
Binney and Tremaine , 2008).Accordingly, the study of the gravitational potential generatedby an idealized thin disc is a problem of great astrophysi-cal relevance and so, through the years, different approacheshave been used to obtain such kind of thin disc models (see
Binney and Tremaine (2008) and references therein). So,once an expression for the gravitational potential has been de-rived, corresponding expressions for the surface mass densityof the disc and for the circular velocity of the disc particles canbe obtained. Then, if the expression for the circular velocitycan be adjusted to fit the observational data of the rotation curve of a particular galaxy, the total mass can be obtained byintegrating the corresponding surface mass density.However, although most of these thin disc models have surfacedensities and rotation curves with remarkable properties, manyof them mainly represent discs of infinite extension and thusthey are rather poor flat galaxy models. Therefore, in orderto obtain more realistic models of flat galaxies, it is better toconsider methods that permit obtaining finite thin disc models.Now, a simple method to obtain the gravitational potential, thesurface density and the rotation curve of thin discs of finite ra-dius was developed by
Hunter (1963), the simplest example ofa disc obtained by this method being the well known
Kalnajs (1972) disc.In a previous paper (
Gonz´alez and Reina , 2006) we usedthe Hunter method in order to obtain an infinite family ofthin discs of finite radius with a well-behaved surface massdensity. This family of disc models was derived by requir-ing that the surface density behaves as a monotonously de-creasing function of the radius, with a maximum at the cen-ter of the disc and vanishing at the edge. Furthermore,the motion of test particles in the gravitational fields gen-erated by the first four members of this family was studiedin
Ramos-Caro, L´opez-Suspez and Gonz´alez (2008). So,although the mass distribution of this family of discs presentsa satisfactory behaviour in such a way that they could be con-sidered adequate as flat galaxy models, their corresponding ro-tation curves do not present a so good behavior, as they do notreproduce the flat region of the observed rotation curve.On the other hand, in
Pedraza, Ramos-Caro and Gonz´alez (2008) a new family of discs was obtained as a superposition ofmembers of the previously obtained family, by requiring thatthe surface density be expressed as a well-behaved function ofthe gravitational potential, in such a way that the correspond-ing distribution functions can be easily obtained. Furthermore,besides presenting a well-behaved surface density, the modelsalso presented rotation curves with a better behavior than thegeneralized Kalnajs discs. However, although these discs arestable against small radial perturbations of disc star orbits,they are unstable to small vertical perturbations normal tothe disc plane. Then, apart from the stability problems, thesediscs can be considered as quite adequate models in order tosatisfactorily describe a great variety of galaxies.Based on these works, in
Gonz´alez, Plata-Plata and Ramos-Caro (2010) were obtained some thin disc models in which the circu-lar velocities were adjusted to very accurately fit the observedrotation curves of four spiral galaxies of the Ursa Major clus-ter, galaxies NGC3877, NGC3917, NGC3949 and NGC4010. ev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-ddd de 2016 Analytical Potentials for Flat Galaxies with Spheroidal Halos These models presented well-behaved surface densities and theobtained values for the corresponding total mass agree withthe expected order of magnitude. However, the models pre-sented a central region with strong instability to small verticalperturbations. Now, this result was expected as a consequenceof the fact that the models only consider the thin galacticdisc. Therefore, more realistic models must be considered in-cluding the non-thin character of the galactic disc or the masscontribution of the spheroidal halo.In agreement with the above considerations, in this paper wewill consider a family of models obtained by expressing thegravitational potential as the superposition of a potential gen-erated by the thin galactic disc and a potential generated by thespheroidal halo, in such a way that the model implies a linearrelationship between the masses of the thin disc and the spher-oidal halo. By adjusting the corresponding expression for thecircular velocity to the observed data of the rotation curve ofsome specific galaxies, some particular models will be analysed.Then, from the corresponding expressions for the disc surfacedensity and the density of the halo, estimate values for the totalmass of the disc and the total mass of the halo will be obtained.The paper is organised as follows. First we present the thin discplus halo model. Then, we obtain the corresponding expres-sions for particular models, and then the models are fitted todata of the observed rotation curve of some galaxies of the UrsaMayor Cluster, as reported in
Verheijen ans Sancisi (2001).Finally, we discuss the obtained results.
The Thin Disc Plus Halo Model
In order to obtain galaxy models consisting of a thin galac-tic disc and a spheroidal halo, we begin considering an axiallysymmetric gravitational potential Φ = Φ(
R, z ), where (
R, ϕ, z )are the usual cylindrical coordinates. Also, besides the axialsymmetry, we suppose that the potential has symmetry of re-flection with respect to the plane z = 0,Φ( R, z ) = Φ( R, − z ) , (1)which implies that the normal derivative of the potential satis-fies the relation ∂ Φ ∂z ( R, − z ) = − ∂ Φ ∂z ( R, z ) , (2)in agreement with the attractive character of the gravitationalfield. We also assume that ∂ Φ /∂z does not vanish on the plane z = 0, in order to have a thin distribution of matter that rep-resents the disc. On the other hand, in order to separately describe the thin discand the spheroidal halo, we consider that the gravitational po-tential can be written as the superposition of two independentcomponents Φ( R, z ) = Φ d ( R, z ) + Φ h ( R, z ) , (3)where Φ d ( R, z ) is the part of the potential generated by the thingalactic disc, while Φ h ( R, z ) corresponds to the spheroidal halocomponent. The disc component Φ d ( R, z ) must be a solutionof the Laplace equation everywhere outside the disc, ∇ Φ d = 0 , (4)while the halo component Φ h ( R, z ) satisfies the Poisson equa-tion ∇ Φ h = 4 πG̺, (5)where ̺ ( R, z ) is the mass density of the halo.So, given a potential Φ(
R, z ) with the previous properties, wecan easily obtain the circular velocity v c ( R ), defined as the ve-locity of the stars moving at the galactic disc in circular orbitsaround the center, through the relationship v c ( R ) = R ∂ Φ ∂R (cid:12)(cid:12)(cid:12)(cid:12) z =0 , (6)while the surface mass density Σ( R ) of the thin galactic disc isgiven by Σ( R ) = 12 πG ∂ Φ ∂z (cid:12)(cid:12)(cid:12)(cid:12) z =0 + , (7)which it is obtained by using the Gauss law and the reflectionsymmetry of Φ( R, z ).Accordingly, in order that the potential of the spheroidal halodoes not contribute to the disc surface density, we will imposethe condition ∂ Φ h ∂z (cid:12)(cid:12)(cid:12)(cid:12) z =0 + = 0 . (8)Furthermore, in order to have a surface density correspondingto a finite disclike distribution of matter, we impose boundaryconditions in the form ∂ Φ d ∂z (cid:12)(cid:12)(cid:12)(cid:12) z =0 + = 0; R ≤ a, (9a) ∂ Φ d ∂z (cid:12)(cid:12)(cid:12)(cid:12) z =0 + = 0; R > a, (9b)in such a way that the matter distribution is restricted to thedisc z = 0, 0 ≤ R ≤ a , where a is the radius of the disc.In order to properly pose the boundary value problem, we intro-duce the oblate spheroidal coordinates, whose symmetry adapts . A. Gonz´alez and J. I. Reina Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-ddd de 2016 in a natural way to the geometry of the model. These coor-dinates are related to the usual cylindrical coordinates by therelation ( Morse and Fesbach , 1953) R = a p (1 + ξ )(1 − η ) , (10a) z = aξη, (10b)where 0 ≤ ξ < ∞ and − ≤ η <
1. The disc has the coordi-nates ξ = 0, 0 ≤ η <
1. On crossing the disc, the η coordinatechanges sign but does not change in absolute value. The singu-lar behaviour of this coordinate implies that an even function of η is a continuous function everywhere but has a discontinuous η derivative at the disc.Now, in terms of the oblate spheroidal coordinates, the Laplaceoperator acting over any axially symmetric function Φ( ξ, η )gives ∇ Φ = (cid:2) (1 + ξ )Φ ,ξ (cid:3) ,ξ + (cid:2) (1 − η )Φ ,η (cid:3) ,η a ( ξ + η ) , (11)whereas the boundary condition (8) is equivalent to ∂ Φ h ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ =0 = 0 , (12a) ∂ Φ h ∂η (cid:12)(cid:12)(cid:12)(cid:12) η =0 = 0 , (12b)and the boundary conditions (9a) and (9b) reduce to ∂ Φ d ∂ξ (cid:12)(cid:12)(cid:12)(cid:12) ξ =0 = 0 , (13a) ∂ Φ d ∂η (cid:12)(cid:12)(cid:12)(cid:12) η =0 = 0 . (13b)Moreover, in order for the gravitational potential to be contin-uous everywhere, Φ( ξ, η ) must be an even function of η , whichgrants also the fulfilment of conditions (12b) and (13b).Accordingly, by imposing the previous boundary conditionsover the general solution of the Laplace equation in oblatespheroidal coordinates, we can write the gravitational potentialof the galactic disc as ( Bateman , 1944)Φ n ( ξ, η ) = − n X l =0 C l q l ( ξ ) P l ( η ) , (14)where n is a positive integer, which it defines the modelof disc considered. Here P l ( η ) are the usual Legendrepolynomials and q l ( ξ ) = i l +1 Q l ( iξ ), with Q l ( x ) the Le-gendre functions of second kind (see Arfken and Weber (2005) and, for the Legendre functions of imaginary argument,
Morse and Fesbach (1953), page 1328). The coefficients C l are, in principle, arbitrary constants, though they must bespecified to obtain any particular model. We will do this later on, by adjusting the circular velocity of the model with theobserved data of the rotation curve of some specific galaxies.With this expression for the gravitational potential of the disc,the surface density is given byΣ( e R ) = 12 πaGη n X l =0 C l (2 l + 1) q l +1 (0) P l ( η ) , (15)where, as ξ = 0, η = p − e R , with e R = R/a . Then, byintegrating on the total area of the disc, we find the value M d Ga = C (16)for the total mass of the disc. Now, it is clear that the surfacedensity diverges at the disc edge, when η = 0, unless that weimpose the condition ( Hunter , 1963) n X l =0 C l (2 l + 1) q l +1 (0) P l (0) = 0 , (17)that, after using the identities P n (0) = ( − n (2 n − n )!! , (18a) q n +1 (0) = (2 n )!!(2 n + 1)!! , (18b)which are easily obtained from the properties of the Legendrefunctions, leads to the expression C = n X l =1 ( − l +1 C l , (19)which gives, through (16), the value of the disc mass M d interms of the constants C l , with l ≥ h ( ξ, η ) = m X j =0 j X k =0 B jk q kj ( ξ ) P kj ( η ) , (20)where m is a positive integer, which defines the model of haloconsidered, and the coefficients B jk are arbitrary constantswhich must be specified to obtain any particular model. Here( Lamb , 1945) q kj ( ξ ) = (1 + ξ ) k d k q j ( ξ ) dξ k , (21)are the solutions of the differential equation ddξ " (1 + ξ ) dq kj dξ = (cid:20) j ( j + 1) − k ξ (cid:21) q kj ( ξ ) , (22) ev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-ddd de 2016 Analytical Potentials for Flat Galaxies with Spheroidal Halos while the associated Legendre functions ( Arfken and Weber ,2005), P kj ( η ) = (1 − η ) k d k P j ( η ) dη k , (23)are the solutions of the differential equation ddη " (1 − η ) dP kj dη = (cid:20) k − η − j ( j + 1) (cid:21) P kj ( η ) , (24)where j and k are integers, with j ≥ k . On the other hand, dueto the discontinuous character of η , Φ h ( ξ, η ) will be continuouseverywhere only if we take ( j − k ) as an even number in orderthat P kj ( η ) be an even function of η .With the previous expressions, and using the Laplace operatorin oblate spheroidal coordinates (11) in the Poisson equation(5), we obtain for the mass density of the halo the expression ̺ ( ξ, η ) = 14 πG m X j =0 j X k =0 B jk ̺ kj ( ξ, η ) , (25)where ̺ kj ( ξ, η ) = k q kj ( ξ ) P kj ( η ) a (1 + ξ )(1 − η ) . (26)Now, from (23) and (26) it is easy to see that at the z axis,when η = ±
1, the function ̺ kj ( ξ, η ) diverges for k = 1 and van-ishes for k >
2. Accordingly, in order to have a well behavedmass density for the halo, we only consider in expression (20)the terms with k = 0 and k = 2 and so, in order to grant thecontinuity of the potential, we must take j as an even number.Furthermore, in order to have a nonzero mass density for thehalo, we must consider models with m ≥
2. Finally, as P kj ( η )is finite at the interval − ≤ η ≤ q kj ( ξ ) goes to zero when ξ → ∞ , ̺ ( ξ, η ) properly vanishes at infinity.A simple possibility for the halo potential in agreement withthe above considerations is given by taking (20) with m = 4,Φ h ( ξ, η ) = B q ( ξ ) P ( η ) + B q ( ξ ) P ( η )+ B q ( ξ ) P ( η ) + B q ( ξ ) P ( η )+ B q ( ξ ) P ( η ) , (27)in such a way that at least two terms in (25) contribute to themass halo density. Then, after using the explicit expressionsfor q kj ( ξ ) and P kj ( η ), the halo density can be written as ̺ ( ξ, η ) = 3 πGa (cid:26) B (cid:20) ξ − ξ (5 + 3 ξ )(1 + ξ ) (cid:21) +5 B (7 η − (cid:20)
154 (1 + 7 ξ ) acot ξ − ξ (81 + 190 ξ + 105 ξ )4(1 + ξ ) (cid:21)(cid:27) (28) which is maximum at the disc surface, when ξ = 0, and thenfastly decreases being constant at the oblate spheroids definedby ξ = cte.By integrating over all the space, we obtain the expression B + 6 B = M h G a , (29)where M h is the total mass of the halo. Furthermore, from thecondition (12a), we obtain the relations B = − B
96 + B , (30a) B = − B − B , (30b) B = − B − B . (30c)Finally, solving the system of equations (29) and (30), we ob-tain B = − M h Ga , (31a) B = M h G a − B , (31b) B = M h G a − B , (31c) B = − B , (31d)and so all the constants in (27) are expressed in terms of thehalo mass M h and the coefficient B .On the other hand, if we restrict to particles moving in the thindisc, the circular velocity is written in terms of the spheroidalcoordinates as v c = ( η − η ∂ Φ ∂η (cid:12)(cid:12)(cid:12)(cid:12) ξ =0 , (32)which, by using (3), (14), (27) and the properties of the Le-gendre functions, reduces to v c ( e R ) = e R η m X l =1 e C l P ′ l ( η ) , (33)where e C = q (0) (cid:20) C + 66 B + M h G a (cid:21) , (34a) e C = q (0) [ C + 24 B ] , (34b)and e C l = q l (0) C l , (35) . A. Gonz´alez and J. I. Reina Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-ddd de 2016 for l ≥
3. Then, by using (19), (34a), (34b) and (35), it is easyto establish that M d Ga + M h G a = m X l =1 ( − l +1 e C l q l (0) − B , (36)and thus the model implies a linear relationship between M d and M h , where the independent term is determined by theconstants e C l , with l ≥
1, and the coefficient B . Now, it isclear that the above relationship makes sense only if the righthand side it is positive, which should be checked for every setof constants e C l corresponding to any particular model. Thecoefficient B must be chosen in such a way that the modelrepresent galaxies with a surface density mass and vertical fre-quency with a physically acceptable behavior. Obtaining Particular Models
In order to obtain particular models, we must specify theconstants e C l of the general model. So, we will adjustthese constants in such a way that the circular velocity v c ( e R ) fits with the data of the rotation curve of someparticular galaxy. As expression (33) for the circular ve-locity only involves derivatives of the Legendre polynomi-als of even order, it can be written as the rotation law( Gonz´alez, Plata-Plata and Ramos-Caro , 2010) v c ( e R ) = m X l =1 A l e R l , (37)where the A l constants are related with the previous constants e C l , for l = 0, through the relation e C l = 4 l + 14 l (2 l + 1) m X k =1 A k I kl , (38)where I kl = Z − η (1 − η ) k P ′ l ( η ) dη, (39)which is obtained by equaling expressions (33) and (37) and byusing the orthogonality properties of the associated Legendrefunctions ( Arfken and Weber , 2005).Then, if the constants A l are determined by a fitting of theobservational data of the corresponding rotation curve, the cor-responding values of the coefficients e C l can be determined bymeans of relation (38), obtaining then a particular case of (36)corresponding to a specific galaxy model, which can be writtenin terms of the constants A l as M d Ga + M h G a = m X k,l =1 ( − l +1 (4 l + 1) A k I kl l (2 l + 1) q l (0) − B . (40) However, this relation does not determine completely the val-ues of M d and M h , but only gives a linear relationship betweenthem. So, in order to restrict the allowed values of these masses,it is needed to analyse the behaviour of some other quanti-ties characterizing the kinematics of the model. These featuresare the epicycle or radial frequency, κ ( R ), and the verticalfrequency, ν ( R ), which describe the stability against radialand vertical perturbations of particles in quasi-circular orbits( Binney and Tremaine , 2008). These frequencies, whichmust be positive in order to have stable circular orbits, aredefined as κ ( R ) = ∂ Φ eff ∂R (cid:12)(cid:12)(cid:12)(cid:12) z =0 , (41) ν ( R ) = ∂ Φ eff ∂z (cid:12)(cid:12)(cid:12)(cid:12) z =0 , (42)where Φ eff = Φ( R, z ) + ℓ R , (43)is the effective potential and ℓ = Rv c is the specific axial angu-lar momentum. Then, by using expression (6) for the circularvelocity, we can write the above expressions as κ ( R ) = 1 R dv c dR + 2 v c R , (44) ν ( R ) = ∇ Φ (cid:12)(cid:12) z =0 − R dv c dR , (45)where we also used the expression for the Laplace operator incylindrical coordinates.Now, by using (37), the epicycle frequency can be cast as e κ ( e R ) = n X l =1 l + 1) A l e R l − , (46)where e κ = aκ . It is easy to notice that the above expressionis completely determined by the set of constants A l , whichare fixed by the numerical fit of the rotation curve data, suchthat it is not possible to find a relation between the disc andhalo masses that can be adjusted by requiring radial stability.On the other hand, by using the Poisson equation (5), the ex-pression (28) for the halo density and the expression (37) forthe circular velocity, we find that the vertical frequency can bewritten as e ν ( e R ) = f ν ( M h , B , e R ) − f ( e R ) , (47)where e ν = aν , f ν ( M h , B , e R ) = 9 πG M h a + 567 πB − π B e R , (48) ev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-ddd de 2016 Analytical Potentials for Flat Galaxies with Spheroidal Halos and f ( e R ) = m X l =1 lA l e R l − . (49)Thus, as e ν must be positive everywhere at the interval 0 ≤ e R ≤ f ν ( M h , B , e R ) ≥ f ( e R ) , (50)which give us a range for M h y B such that e ν ≥ e R ) = p − e R πaG n f ( e R ) − f d ( M h , B , e R ) o , (51)where f ( e R ) = m X l =1 e C l (2 l + 1) p − e R q l +1 (0) q l (0) (cid:20) P l ( η ) − P l (0) η (cid:21) , (52)and f d ( M h , B , e R ) = 3 M h G a + 238 B − B e R . (53)So, in order for the surface mass density to be positive in theinterval 0 ≤ e R ≤
1, it must be met that f ( e R ) ≥ f d ( M h , B , e R ) . (54)The relation (54) give us another range of values, not neces-sarily equal to the relation (50), for which we obtain a surfacedensity mass with a acceptable behavior. Then, in order thatthe model make sense, we must verify that it meets the relation f ν ( M h , B , e R ) ≥ f ( e R ) ∩ f ( e R ) ≥ f d ( M h , B , e R ) , (55)which should be checked for every set of constants correspond-ing to any particular model. Adjusting Data to Models
In order to illustrate the above model to the real observed data,we have taken a sample of spiral galaxies of the Ursa Majorcluster. We pick the corresponding data out from Table 4 ofthe paper by
Verheijen ans Sancisi (2001), which presentsthe results of an extensive 21 cm-line synthesis imaging surveyof 41 galaxies in the nearby of the Ursa Major cluster usingthe Westerbork Synthesis Radio Telescope. The mean distancebetween this telescope and the cluster is 18.6 Mpc. At thisdistance, 1 arcmin corresponds to 5.4 kpc.For each rotation curve data, we take as the value of a , thevalue given by the last tabulated radius, i.e. we are assumingthat the radius of each galaxy is defined by its corresponding last observed value. Although this assumption about the galac-tic radius do not agrees with the accepted standard about theedge of the stellar disc ( Binney and Merrifield , 1998), wewill make it since we are assuming that all the stars moving incircular orbits at the galactic plane are inside the disc and thatthere are no stars moving outside the disc. Thereafter we takethe radii normalized in units of a to fit the rotation curve ofevery galaxy by mean of the model (37).The fits are made through a non-linear least squares fittingusing the Levenberg-Marquardt algorithm, implemented inter-nally by ROOT version 5.28 ( Brun and Rademakers , 1997),which minimizes the weighted sum of squares of deviationsbetween the fit and the data. We assigned weights to thedata points inversely proportional to the square of their er-rors. These errors corresponding to 2 v ∆ v being ∆ v the galaxyvelocity measurement error. For each galaxy, initially we lookfor all the possible fits starting at m = 1 up to m = N − R , v ), hence wefind a value for m such that we get the minimum reduced chisquare χ r (the best fit). Now we can discard the galaxies thatdo not pass the reduced chi squared test with a confidence levelof 95% ( Bevington and Keith , 2003).The e C l constants are calculated by using the relations (19),(35) and (38). Therefore, by using this set of constants in(49) and (52) we find for each galaxy the functions f and f . Finally, through a routine made in Mathematica 8.0., wecheck for each galaxy of the sample the validity of the condition(55). However, when we check the consistency of the adjust,we found that only the fit of the data for the galaxies NGC4389and UGC6969 it agrees with these conditions, whereas that forall the other galaxies we found that the solution interval for M h and B , given by (55), is empty. Table 1.
Constants A l in units of 10 m s − NGC4389 UGC6969 A ± ± A -57552.0 ± ± A ± ± A -27760.5 ± ± A l , in units of10 m s − , obtained by the numerical adjust with the rotationcurve data for galaxies NGC4389 and UGC6969. With thisvalues for the constants, we obtain, from (40), for the galaxyNGC4389 the relationship M h + 4 M d = 5 . × − . × B , (56) . A. Gonz´alez and J. I. Reina Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-ddd de 2016 and for the galaxy UGC6969 the relationship M h + 4 M d = 2 . × − . × B , (57)where all the quantities are in kg .In Fig. 1. we present the region that represent the solutioninterval of the condition (55) for the galaxies NGC4389 andUGC6969. This region represent the values that the halo massand the coefficient B
42 can take, in order to obtain galaxymodels with a vertical frequency always positive and with asurface density that has a maximum value at the disc centreand then decreases as e R increases, vanishing at the disc edge.In Table 2 we present, based on the values obtained by thecondition (55) and plotted in the Fig. 1, the minimum andmaximum values for the halo mass of each galaxy and the discmass calculated from the relations (56) and (57), in units of10 M ⊙ , whereas in Table 3 we present the respectives valuesof the coefficient B , in units of 10 m s − .In Fig. 2., we show the adjusted rotation curve for these twogalaxies. The points with error bars are the observations as re-ported in Verheijen ans Sancisi (2001), while the solid lineare the circular velocity determined from (37) and the A l pa-rameters given by the best fit. As we can see, for the two galaxies we get a fairly accurate numerical adjustment withthe observational rotation curve. Table 2. M h and M d in units of 10 M ⊙ . NGC4389 NGC6969min max min max M h M d Table 3.
Coefficient B in units of 10 m s − . NGC4389 NGC6969min max min max B ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ M h @ kg D B @ m (cid:144) s D NGC4389 ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ ´ M h @ kg D B @ m (cid:144) s D NGC6969 (a) (b)
Figure 1.
Solution interval of the condition (55). In (a) we show the interval of parameters M h and B for the galaxy NGC4389. In (b) we show the interval of parameters M h and B for the galaxyUGC6969.8 ev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-ddd de 2016 Analytical Potentials for Flat Galaxies with Spheroidal Halos R Ž Υ c @ k m (cid:144) s D NGC4389 R Ž Υ c @ k m (cid:144) s D UGC6969
Figure 2.
Circular velocity v c , as a function of the dimensionless radial coordinate e R , for the galaxiesNGC4389 and UGC6969. Error bars represent the observed data by Verheijen ans Sancisi (2001),while the solid line are the circular velocity determined from (37), and the A l parameters given by thebest fit. ´ ´ ´ ´ ´ R Ž Κ Ž @ m (cid:144) s D NGC4389 ´ ´ ´ ´ ´ ´ R Ž Κ Ž @ m (cid:144) s D UGC6969
Figure 3.
Epicycle frequency e κ × − in (km / s) , as a function of the dimensionless radial coordinate e R , for the galaxies NGC4389 and UGC6969. the dashed line represent the vertical frequency by using themaximum value for the halo mass. As can be notice in the fig-ure, for the two galaxies the vertical frequency is positive overthe entire range of e R , so the models are stable against verticalperturbations. It is easy to verify that for any other value ofthe halo mass and the corresponding parameter B , as deter-mined from Fig. 1., the vertical frequency remains positive inall range e R . In Fig. 5. we present the corresponding plots of the surfacemass density for the two galaxies. As in the previous case, forboth galaxies the solid line represents the behavior of the sur-face mass density by taking the minimum value for the halomass, while the dashed line is the surface mass density for themaximum value of the halo mass. The behavior of this quantityis similar for both galaxies, i.e. the surface mass has a maxi-mum value at the disc centre and then decreases as e R increases,vanishing at the disc edge. . A. Gonz´alez and J. I. Reina Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-ddd de 2016 ´ ´ ´ ´ ´ R Ž Ν Ž @ m (cid:144) s D NGC4389 ´ ´ ´ ´ ´ ´ R Ž Ν Ž @ m (cid:144) s D NGC6969
Figure 4.
Vertical frequency e ν × − in (km / s) , as a function of the dimensionless radial coordinate e R , for the galaxies NGC4389 and UGC6969. The solid line represents the vertical frequency by takingthe minimum value for the halo mass, whereas the dashed line represents the vertical frequency byusing the maximum value for the halo mass. R Ž S H R Ž L @ K g (cid:144) m D NGC4389 R Ž S H R Ž L @ K g (cid:144) m D NGC6969
Figure 5.
Surface mass density Σ × − in (kg / m ), as a function of the dimensionless radial coor-dinate e R , for the galaxies NGC4389 and UGC6969. The solid line represents the surface mass densityby taking the minimum value for the halo mass, whereas the dashed line represents the surface massdensity by using the maximum value for the halo mass. Finally, from (28), in Fig. 6. we show the contours of the halodensity distribution for the galaxy NGC4389. In plot (a), thecontours are drawn using the minimum value for the halo mass,while in plot (b) we present the contours using the maximumvalue for the halo mass. Similarly, in Fig. 7. we show the samequantities, but for the galaxy UGC6969. In both cases, thedensity profiles are positive and do not have discontinuities in all range e R , z , taking a maximum value at center and smoothlydecreasing to zero when e R → ∞ . ev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-ddd de 2016 Analytical Potentials for Flat Galaxies with Spheroidal Halos - - R Ž z Ž NGC4389 - - R Ž z Ž NGC4389 (a) (b)
Figure 6.
Contours of the halo density distribution for the galaxy NGC4389. In (a) we show thecontours for the minimum value of halo mass. In (b) we show the contours using the maximum valueof halo mass. - - R Ž z Ž NGC6969 - - R Ž z Ž NGC6969 (a) (b)
Figure 7.
Contours of the halo density distribution for the galaxy UGC6969. In (a) we show thecontours for the minimum value of halo mass. In (b) we show the contours using the maximum valueof halo mass. 11 . A. Gonz´alez and J. I. Reina Rev. Acad. Colomb. Cienc. Ex. Fis. Nat. nn(nnn):ww–zzz,ddd-ddd de 2016
Concluding Remarks
We have presented a family of analytical potentials for flatgalaxies with spheroidal halos characterised by a linear rela-tionship between the halo mass and the disc mass. The mod-els are stable against radial and vertical perturbations, andtheir circular velocities can be adjusted very accurately to theobserved rotation curves of some specific galaxies. The herepresented models are a generalisation of the models presentedin
Gonz´alez, Plata-Plata and Ramos-Caro (2010), whereonly models with a thin galactic disc are considered. The gener-alisation was obtained by adding to the gravitational potentialof the thin disc the gravitational potential corresponding to aspheroidal halo, in such a way that we have solved the problemof vertical unstability presented by the previous models.Two particular models were obtained by a numerical fit of thegeneral expression (37) for the circular velocity with the ob-served data of the rotation curve of galaxies NGC4389 andUGC6969. For these two galaxies we have obtained a fairly ac-curate numerical adjustment with the rotation curve and, fromthe constants A l obtained with the numerical fit, we computethe values of the halo mass, the disc mass and the total mass forthese two galaxies in such a way that we obtain a very narrowinterval of values for these quantities. Furthermore, the valuesof masses here obtained are in agreement with the expected or-der of magnitude, between about 10 and 10 M ⊙ , and withthe relative order of magnitude between the halo mass and thedisc mass, M d / M h ≈ .
1, (
Ashman , 1990). Accordingly, webelieve that the values of mass obtained for the two studiedgalaxies may be taken as a very accurate estimate of the upperand lower bounds for the mass of the galactic disc and for themass of the spheroidal halo in these two galaxies. Additionally,the density profiles obtained satisfy several conditions whichare necessary to describe real galactic systems, i.e. they arepositive and not have discontinuities in all range e R , z , takinga maximum value at center and smoothly decreasing its valueto zero when e R → ∞ .However, although we tested the applicability ofthe present model with all the galaxies reported by Verheijen ans Sancisi (2001), consistent models were ob-tained only for the two galaxies NGC4389 and UGC6969,whereas for all the other galaxies were obtained models withvalues of the halo mass such that the condition (55) is notsatisfied. Now, it can be considered that this result occurs asa consequence of the simple halo model that we have takenhere. Indeed, as we can see from expressions (26) and (27),only one term of the gravitational potential of the halo con-tributes to their density, what leaves only one free constant to be determined in order to fit the model to the imposed con-sistency conditions. This constant is precisely the mass of thehalo, M h , which is determined by requiring the positivenessof the vertical frequency and the surface mass density. On theother hand, if we consider additional terms in expression (27)for the halo potential, we will have new free parameters thatperhaps allow to better adjust the model to properly describethe behavior of other galaxies besides the two considered here.In agreement with the above considerations, we can considerthe simple set of models here presented as a fairly good approx-imation to obtaining quite realistic models of galaxies. In par-ticular, we believe that the values of mass obtained for the twogalaxies here studied may be taken as a very accurate estimateof the upper and lower bounds for the mass of the galactic discand for the mass of the spheroidal halo in these two galaxies.Accordingly, we are now working on a more involved model,obtained by including additional terms in expression (27) forthe halo potential, in order to get some particular models thatcan be properly adjusted with the observed data of the rotationcurve of some other galaxies besides the two here considered. Acknowledgments.
The authors were supported in partby VIE-UIS, under grant number 1838, and COLCIENCIAS,Colombia, under grant number 8840. JIR wants to thankthe support from Vicerrector´ıa Acad´emica, Universidad SantoTom´as, Bucaramanga.
Conflict of interest.
The authors declare that they have noconflict of interest.
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