arXiv:astro-ph/0001216v1 12 Jan 2000 e oe o h prlSrcueo h Galaxy. the of Structure Spiral the for Model New A n ehd epeetanwaayi ftelongitude-veloc I the of field. analysis new gravitational a galactic present we perturbed method, the ond in clouds, ISM optdteeouino nesml of ensemble these an to of corresponding evolution structure the the visible computed differen turn under sp To kinematics, the Cepheid of tions. parameters from free obtained p the were the waves and method, first curve the rotation In kinematics, galactic arms. stellar the spiral of w of study 4-armed positions of a modes). and tracing used: pure 2- are superposition besides methods of description, plementary model simplest a (the of harmonics terms in Way Milky ocuini htteSnhpest epatclya h co the at practically be to A happens structure. Sun disk the the that of is description conclusion good a is respectively) h -re n -re oehv ieetpthage (6 angles pitch different have mode mode 4-armed “self-sustained” and the 2-armed that the indicate methods Both gram. prlam nteglci ln nolcio hs rsi t in arms these po of converting locii into regions, plane HII galactic the galactic in of arms sample spiral the of diagram pc eerhDprmn,Rso tt nvriy Zorg 5 University, State Rostov Department, Research Space eivsiaetepsiiiyo eciigtesia pat spiral the describing of possibility the investigate We 40-0 ˜oPuo P rzl -al [email protected] E-mail: Brazil; SP, S˜ao Paulo, 04301-904 nttt srnoioeGosc aUP vMge Stefan Av.Miguel USP, da Geofisico Astronˆomico e Instituto uepsto f24amdpatterns 2+4-armed of Superposition 400 usa -al [email protected] E-mail: Russia; 344090, uNMsuo,S.Yu.Dedikov Yu.N.Mishurov, ABSTRACT J.R.D.L´epine and 1 N prils iuaigthe simulating -particles, ,Rostov-on-Don, e, agusp.usp.br 4200, o rmtr of arameters important n he rldensity iral oes we models, eno the of tern ,i which in l, h sec- the n w com- Two n direct and ◦ t ( ity assump- t iin of sitions l n 12 and rotation − v l − dia- ave v ◦ ) , circle. As an additional result of our study, we propose an independent test for localization of the corotation circle in a spiral galaxy: a gap in the radial distribution of interstellar gas has to be observed in the corotation region.
Subject headings: Galaxy: kinematics and dynamics – Galaxy: structure
2 1. Introduction the 2-armed component, so that the Galaxy looks 4-armed. However, there are theoret- A good understanding of the large-scale ical arguments discussed in the present pa- spiral structure of the Galaxy has not been per showing that the pitch angle for the 4- reached up to the present. Georgelin & Georgelin armed pattern should be different from that (1976), herafter GG, derived a 4-armed pat- of the 2-armed component. A Fourier analysis tern, based on an analysis of the distribution of external galaxies in terms of spiral modes, of giant H II regions. According to Vall´ee performed by Puerari & Dottori (1992) in- (1995) most researches support the 4-armed dicates as well that in the cases in which the pattern, although there are discordant opin- 2-armed and 4-armed patterns are prominent, ions; for instance Bash (1981) finds the pat- they indeed have different pitch angles. These tern to be 2-armed, similar to the first spi- arguments suggest that at least in a range of ral structure model proposed by Lin and Shu radius, the Galaxy might present a structure (1964). But even if we accept that there are like 2-arms plus 4-arms with different pitch strong observational evidences in favor of a 4- angles, which could look like 6 arms. An at- arms structure, theoretical and observational tractive aspect of such a model with arms of difficulties remain, and the observation of ex- different pitch angles is that it naturally ac- ternal galaxies strongly recommends that we counts for bifurcations of arms. In external look for more complex solutions, which should galaxies branching of arms is a widespread include arms with different pitch angles. phenomenon, e.g. the galaxy M 101, where It is now accepted that the distance to the there are both branching arms and bridges center is about 7.5 kpc (eg. Olling & Mer- between them. The structure of the Milky refield, 1998), which implies a smaller rota- Way is perhaps similarly complicated. An- tion velocity of the Local Standard of Rest other fact that stimulates us to look for more than previously estimated, and as a conse- refined solutions to the spiral structure is that quence, the rotation curve decreases beyond there are observed arms in the Galaxy that the solar radius (Amaral et al., 1996, Honma do not fit in a simple 4-arms structure. For & Kan-Ya, 1998). From the rotation curve, instance, outside the solar circle, around lon- the epicyclic frequency and the radius of the gitudes 210◦ - 260◦, three arms are clearly vis- inner and outer Lindblad resonances can be ible, in HI and in IRAS sources (Kerr, 1969, derived. Amaral and L´epine (1997, here- Wouterloot et al., 1990), but they are not after AL) concluded that the 2-armed pattern expected from a 4-arms model that fits the could exist between 2.5 and 12 kpc, which is tangential directions of the inner parts of the about the range of the observed spiral pat- Galaxy (e.g. Ortiz & L´epine, 1993). tern, but a 4-armed pattern could only exist The main goal of this paper is to inves- between about 6 to 11 kpc. AL proposed a tigate if a superposition of 2+4-armed wave representation of the galactic spiral structure harmonics is a good representation of the spi- as a superposition of 2- and 4-armed wave ral structure of the Galaxy. Two different ap- patterns, to account for the number of arms proaches are used. In one approach, we anal- and for the existence of arms over a wider yse the Cepheid kinematics, using a model range of radius. In AL’s model, two arms of that takes into consideration the perturba- the 4-armed component are coincident with
3 tion of the stellar velocity field by the spi- 2. Method of estimation of structural ral arms, to derive the structural parameters parameters for the gravitational field of the galactic disk. And, since the structure of the perturbation The derivation of the spiral wave param- potential does not necessarily correspond to eters is based on the statistical analysis of the visible structure, we performed a many- stellar motion in the Galaxy (see Cr´ez´e & particle simulation of gas dynamics in this po- Mennessier, 1973, Mishurov & Zenina, 1999, tential, to predict the visible structure. In hereafter MZ, Mishurov et al., 1997, hereafter the second approach, we directly analyse the MZDMR and papers cited therein). Note visible structure by means of the sample of that we look for the parameters of the struc- HII regions, since these objects are recognized ture of the galactic gravitational field, which as the best large-scale tracers of the galactic is not directly visible. However, the gravita- structure. We present a new analysis of the tional field determines the stellar motion, and observed longitude-velocity (l −v) diagram of in particular, the spiral perturbations of the the best up-to-date sample of H II regions, de- field deviate the stellar motion from rotation riving from it the position of the spiral arms, symmetry. Hence, analysing the stellar veloc- and we compare the observed l − v diagram ity field in the framework of a perturbation with the theoretical ones, computed from our model, we can derive both the parameters of gas dynamics simulations. the density waves and those of the galactic rotation. The Cepheid kinematics analysis is per- formed for two different models of 2-arms + 4- Unlike the above cited papers, let us rep- arms structures, and the results are compared resent the gravitational potential ϕG of the with previous calculations of pure 2-arms and Galaxy as the sum: pure 4-arms. An important result is that for ϕG = ϕ0 + ϕS2 + ϕS4 (1) the more reallistic model, the Sun lies very close to the corotation region. Furthermore, where ϕ0 is, as usual, the unperturbed ax- our simulation revealed the interesting effect isymmetric part of the potential that de- termines the Galaxy equilibrium as a whole of gas pumping out from the corotation. This 2 suggests that in spiral galaxies, there must (dϕ0/dR = Ω R, Ω(R) is the angular rotation be a gas deficiency in a region near the coro- velocity of galactic disk, R is the galactocen- tation circle. This phenomenon explains the tric distance) and ϕSm (m =2 or m = 4) are lack of atomic hydrogen in a ring-like region the m−th harmonics of the perturbations due near the solar circle derived by Kerr (1969) to spiral density waves. According to Lin et and by Burton (1976), which remained unun- al (1969): derstood up to now. We propose this effect ϕSm = Amcos(χm), (2) as an independent test for localization of a corotation circle in external spiral galaxies. where Am is the amplitude of the m−th har- monic, χm is the wave phase:
χm = m [cot(im) ln(R/R⊙) − ϑ]+ χ⊙m , (3) m is the azimuthal wave number, i.e. the number af arms for a given harmonic, im is
4 the corresponding pitch angle of the arms, R, the value Ωpm does not depend on m, and the ϑ are the polar coordinates with the origin at corotation radius does not depend on m ei- the Galactic center, R⊙ is the solar galacto- ther. centric distance, χ⊙m is the initial phase or Further, the nature of the perturbed grav- the wave phase at the Sun position. From itational field could be two-fold. First, in Lin Eq.(3) it is seen that this last value fixes the et al.(1969) theory, the spiral waves are self- m − th harmonic position relative to the Sun. sustained and the disturbances of the gravi- In accordance with Eq (1) we can write rational field are mainly due to the den- the perturbed stellar velocity for the radial sity waves which propagate in the galactic (directed along galactocentric radius)v ˜R and disk. The corresponding dispersion relation the azimuthalv ˜ϑ components as follows: imposes a connection between i2 and i4. In- deed, the solution of the dispersion relation v˜R = fR2 cos(χ2)+ fR4 cos(χ4) , (4) for spiral waves relative to the radial wave number km in the vicinity of corotation does v˜ϑ = fϑ2 sin(χ2)+ fϑ4 sin(χ4) , (5) not depend on m (Shu, 1970, Mark, 1976). where fRm and fϑm are the amplitudes of Since cot(im)= kmR/m, we have m − th harmonic. These quantities are re- lated to the parameters of the density waves cot(i2) = 2cot(i4) . (6) by some formulas (see Lin et al, 1969). Therefore, the two-armed pattern is tighter Substituting Eqs (4-5) into Eqs.(1,2) of MZ wound (about twice for small pitch angles) and using the statistical method described than the four-armed one. by MZ and MZDMR, we derive the parame- ′′ The above argument is strictly held in the ters of the rotation curve (Ω⊙, A, R Ω ) and ⊙ ⊙ vicinity of the corotation circle (according to the parameters of the spiral waves (f , f , Rm ϑm Cr´ez´e& Mennessier, 1973; MZ and MZDMR; i , χ⊙ ) over the observed stellar velocity m m AL, etc. the Sun is situated just near the field. Then by means of the density wave the- corotation). However, since the radial wave ory (Lin et al, 1969) we compute the differ- number and consequently the pitch angle are ence ∆Ω between the angular rotation ve- m slowly varying functions of R (Lin et al, 1969) locity of m − th pattern Ω and the rota- pm the relation (6) can be used for a sufficiently tion velocity of the Galaxy at the Sun po- wide region around the corotation radius. sition (∆Ωm = Ωpm − Ω⊙). By equating In the second approach the perturbations Ω(Rcm) = Ωpm we find the displacement ∆Rm of the Sun relative to the corotation radius of the gravitational potential are mainly due to a bar in the galactic center. According to Rcm (∆Rm = Rcm − R⊙, for computational details see MZ and MZDMR). the models in which 2 of the 4-arms compo- nents coincide with the 2-arms components Notice here some features of our task. It (eg. Englmaier & Gerhard 1999, Al): is widely believed that the spiral structure of our Galaxy is generated by a bar in the galac- χ4 =2χ2; i4 = i2. (7) tic center (e.g., Marochnik & Suchkov, 1984). So, the angular rotation velocity of the pat- In what follows we shall analyze both these tern is determined by the bar rotation. Hence, approaches. We call them the self-sustained
5 model (approach 1) and the bar-dominated both the line-of-sight velocities (Pont et al, model (approach 2). 1994, Gorynya et al, 1996, Caldwell & Coul- Another important peculiarity of our task son, 1987) and proper motions (HIPPAR- is that some of the wave parameters: fR2, fϑ2, COS, ESA, 1997) are available. Let us discuss fR4, fϑ4, i2 (i4 is fixed automatically in the the results derived for the above approaches both models), χ⊙2 and χ⊙4, obey nonlinear separately. statistics. In the self-sustained model χ⊙2 and Self-sustained model. For illustration in χ⊙4 are considered to be independent quan- Fig.1 the surfaces ∆(χ⊙2, χ⊙4) are given for tities; in the bar-dominated approach, these 3 values of i2.One can easily see the global ◦ values are connected by Eq (7). But if we fix minimum in the vicinity of i2 ≈ −6 . The fi- the pitch angles and the initial wave phases, nal values of the parameters with their errors the task becomes a linear one over other the are given in Table 1. parameters. So, the strategy to localize the First of all, we notice the significant de- 2 global minimum for the residual (δ ) in the crease of the residual min(δ2) in this case in second approach is just the same presented comparison with the one for the single har- by MZ and MZDMR. However, in the self- monic m = 2 or m = 4 (see Table 1 of MZ, sustained model it slightly changes. In this runs no 3 and 8). As it was shown in the case we have to fix 3 independent quantities: cited paper, the inclusion of the spiral per- i2, χ⊙2 and χ⊙4, and look for the minimum of turbation in stellar motion happens to be sig- residual over other the parameters by means nificant. However the authors could not make of the least square method (we denote this a choice between the two alternatives: pure 2- 2 minimum by ∆ = min i2,χ⊙2,χ⊙4 (δ )). Then or 4-armed patterns. we change values of i2, χ⊙2 and χ⊙4 and again Now by means of F-test we can show that derive ∆, and so on. After that, we construct the previous hypotheses that the pattern can the net ∆ as a function of i2, χ⊙2 and χ⊙4. Of be represented by only one harmonic (m =2 course, we cannot imagine this function vi- or m = 4), should be rejected in favor of the sually, but we can construct the surface, say, alternative hypothesis that the pattern is well ∆(χ⊙2, χ⊙4) for a set of values i2 and check by represented by superposition of 2+4-armed eye localization of the global minimum of the pattern. In other words, the representation residual. After the minimum is localized, by of the galactic structure by a superposition of means of a linearization procedure (Draper 2+4-harmonics is clearly preferable to the one & Smith, 1981) we define more exactly the that uses a single harmonic. searched parameters and the covariation ma- The quantities ∆Ω and ∆R can both trix of errors (see details in MZ). m m be calculated from the parameters obtained for m = 2 and for m = 4. It is not hith- 3. Results of Cepheid kinematics anal- erto obvious that for different m the quan- ysis tities occur to be the same as we have sup- The above described procedure was ap- posed above. But our calculations lead to plied to a sample of Cepheids which represent very close values for the corresponding quan- −1 −1 the best observational material for solving the tities: ∆Ω2 = 0.15 kms kpc and ∆Ω4 = −1 −1 above formulated problem. For the Cepheids 0.18kms kpc ; accordingly ∆R2=-0.03 kpc
6 and ∆R4=-0.04 kpc, the standard (i.e. 68%) results are quite different. The pattern ro- confidence intervals being for ∆Ω2 : −0.61 to tation velocities are: for m = 2 Ωp2 = −1 −1 −1 −1 1.02 kms kpc ; for ∆Ω4 : 0.13 to 0.24 km 35.0 kms kpc and for m = 4 Ωp4 = −1 −1 −1 −1 s kpc ; for ∆R2 : −0.21 to 0.13 kpc; for 29.2 kms kpc . So that the main require- ∆R4 : −0.05 to 0.03 kpc (the confidence in- ment of the model (independence of pattern tervals were estimated by means of numerical rotation velocity and the corotation radius experiments described by MZ). Hence, in the from m) is not held. Indeed, for parame- model under consideration the Sun is practi- ters of rotation curve of Tables 1 and the cally situated at the corotation circle, slightly above value for Ωp2 the corotation radius Rc2 beyond it. does not exist as a real number at all (this is ′′ We cannot estimate with any reasonable mainly because the second derivative R⊙Ω⊙ accuracy the values of the amplitudes of the is negative). On the other hand for m = 4 spiral gravitational field A2 and A4 (this is Rc4 =5.4 kpc. Further, in this case the ratio expected from linear perturbation theory). A2/A4 ≈ 8.21 is very large. So, the visible However, their ratio is derived very precisely: pattern happens to be 2-armed! 2 A2/A4 =0.79, the standard confidence inter- At last, the residual δ ≈ 220 is signifi- val being 0.77 to 0.80. cantly greater than in the previous approach
The locus of minima for ϕSm are shown in and occurs to be very close to the values for Fig. 2; they are the lines of constant phase pure m = 2 and pure m = 4 solutions (see χm on the galactic plane corresponding to Table 1 of MZ 1, runs no 3 and 8). Hence, it min ϕSm. From this figure the pattern may is impossible to make a choice between pure be thought to be 6-armed one. However, this m = 2, pure m = 4 or superposition of 2+4- is not the case in our model, in which the po- modes in the bar-dominated approach. tential perturbation is represented by a cosine The above statistical analysis of the large- functions. Indeed, simple computation shows scale stellar kinematics of Cepheid stars, leads that for the above derived parameters the sum us to the conclusion that the preferable solu- ϕS2 + ϕS4 has at most 4 minima over ϑ for tion for the spiral structure of the Galaxy is a a fixed R. It would only be possible to ob- superposition of self-sustained 2+4-harmonics tain 6 minima (in the frame of a 2+4 armed of density waves, and that the Sun is situated model) if the potential were represented by very close to the corotation circle. some function presenting sharp minima, con- trary to the cosine function. The visible struc- 4. Visible large-scale structure of the ture derived by means of particle-cloud sim- Galaxy ulations and given in Sec. 4 supports this point of view. Of course, the actual structure In the previous Section we derived the of the Galaxy may occur to be more compli- structural parameters for the gravitational cated e.g., due to higher wave harmonics or to field of the Galaxy. However, as it was men- special effects at the corotation (Mark, 1976). tioned above, it is not directly seen. To make However, these possibilities are beyond the visible the above derived structure, the evo- present investigation. lution of a gas cloud ensemble in the galac- tic gravitational field pertubed by spiral arms Bar-dominated model. In this case the will be next considered. Following Roberts &
7 Hausman (1984), we simulate the interstellar particle motion in the galactic plane. All com- clouds by ballistic particles moving in a given putations are performed in a frame of refer- gravitational field with a potential ϕG defined ence corotating with spiral arms. by Eqs (1-3). Let us briefly describe the for- The result of our simulation of particle- mulation of the task. cloud dynamics in the spiral gravitational At the initial moment of time (t = 0) the field for the superposion of 2+4 self-sustained spiral perturbations are assumed to be absent density wave harmonics is shown in Fig. 3. 4 (ϕSm = 0). N particles (N = 2 · 10 ) are There is good agreement in major features uniformly distributed over a disk within R< with Fig. 3 of Efremov (1998). In a signif- 13 kpc. Each particle is given the local ro- icant part of galactocentric distances the pat- tation velocity, disturbed by a chaotic veloc- tern looks like a 4-armed one. But we do not ity with one-dimensional dispersion 8 km s−1. face with the problem of too short arms, as it For t> 0 the spiral perturbation is ”switched would be in the case of pure m = 4 harmonic on”. Our task is to compute the reaction of (see AL). Our pattern reflects the complicated the system on this perturbation (the N-body picture often observed in external galaxies, problem for particles moving in an external e.g. arm bifurcation or their overlaping. field). The parameters both for unperturbed and 5. Gap in the galactic gaseous disk as for perturbed potential were taken from Table an indicator for the corotation cir- 1. Since the derived rotation curve is valid cle in a restricted region, for R > 9.4 kpc we One of the most important conclusion of continue it by a flat part. Sec. 3 is that the Sun lies very close to the The self-sustained galactic waves are well corotation radius. In this Section we present known to exist between the inner and outer a new test which turns possible to localize di- Lindblad resonances. Since we do not take rectly the position of the corotation circle in into consideration a bar in the galactic cen- a spiral galaxy. ter, the spiral perturbation was cut off for Many years ago Kerr (1969) payed atten- R < 2 kpc. For the spiral gravitational am- sion to a ring-like region which is markedly plitude A2 we assumed the ”standard” value: 2 2 deficient in neutral hydrogen, with radius 2A2cot(i2)/Ω ⊙R ⊙ = 0.05 (Lin et al, 1969). slightly greater than the solar distance from In the simulation, collisions between the in- the galactic center (see also Simonson, 1970). terstellar clouds occur, the collisions being This result was later supported in more de- energy-dissipating. For the computation of tail by Burton (1976). He showed that there this process, we used the method described in is a very clear gap in radial distribution of detail by Roberts & Hausman (1984). For the atomic hydrogen in our Galaxy at R ≈ 11 kpc, cloud cross-section we adopted a typical value whereas in the old scale, used in that paper, for the H I clouds. Mutual clouds gravitation, R⊙ = 10 kpc (see Fig. 6 of Burton, 1976). In or other effects like interaction of the clouds general, the gap reminds the Cassini gap in with expanding envelopes of supernovae, etc., Saturnian ring. were not taken into account. We also restrict ourselves to consideration of two-dimensional It is natural to connect this gap in the ra-
8 dial ISM distribution with the process occur- from the rising effect of a dark matter com- ing at corotation: the gas is pumped out from ponent at distances larger than the solar ra- the corotation under the influence of the grav- dius. Amaral et al. exclude these hypotheses, itational field of spiral arms (see also Suchkov, in their discussion of the nature of this min- 1978). The qualitative explanations is as fol- imum. Now, if there is a ring-like region de- lows. It is well known, that when the gas void of gas, in principle the rotation velocity flows through the galactic density waves, a could not be measured in that region, using a shock arises in the medium (Roberts, 1969; gas tracer. Similarly, if one selects short-lived Roberts & Hausman, 1984). Since the galac- stars as tracers, these stars are not expected tic disk rotates differentially and for R < Rc to form and to exist inside that region, be- Ω > Ωp, the gas in this region overtakes the cause of the gas deficiency and of the very spiral wave, entering it from the inner side. low velocity of the gas with respect to the In the shock the clouds are decelerated, and spiral pattern, which turns the gas compres- fall towards the galactic center. For R > Rc sion (the star-formation process) inefficient. Ω < Ωp, and the process is inverse. Here It is therefore probable that the gas clouds the wave overtakes the gas and pushes it. and the stars that we observe inside the gap So, the clouds pass to an orbit more remote are objects with non-circular orbits that in- from the galactic center (see also Goldreich vade the gap; they are observed close to their & Tremaine, 1978 and Gor’kavyi & Fridman, maximum elongation and therefore, present 1994). smaller velocity than the circular one, in the Our simulation of gas cloud dynamics in direction of rotation. spiral gravitational field directly demonstrates So, a ring-like gap in the galactic gaseous this phenomenon. We show in Fig.4 the radial disk, and possibly also a sharp minimum in gas distribution (
9 ones. We next discuss some of the features of ◦ The procedure adopted is to trace spiral the proposed structure. In the region l= 340 ◦ arms in the galactic plane, and to transform to 270 , which is the only region where clear X − Y positions along the arms into locus of arms were observed by GG, the structure re- the arms in the l − v diagram, by means of sulting from our study closely resembles that the rotation curve. By varying the parame- of these authors, presenting about the same ters of the arms, we looked for the best fit to tangential directions. Remark that the longi- the l − v diagram. It is well known that the tudes of tangential directions are not affected arms that are situated inside the solar circle, by a change of distance scale (GG used R0 = transform into narrow loops in the l − v dia- 10 kpc). Therefore, in this range of longitude, gram; the extremity of a loop corresponding it is almost impossible to distinguish between to a tangential direction in the galactic plane. the 2+ 4 arms model that results from our The observed pattern is represented by the study, and the empirical model of GG.This sum of the m = 2 system (two identical long is specially true if we adopt a smooth func- arms with phase difference of 180◦) and of the tion to represent the potential, like we did in m = 4 system (four identical short arms each previous sections, since close potential min- separated by 90◦ in phase) shifted by some ima tend to merge. A difference between GG phase angle from the first system. The ad- and our work is that GG did not indicate the justed parameters are the pitch angles, the existence of a tangential direction at about ◦ angle for the phase shift, the inner and the l=338 (our inner loop e), but obviously there outer radii of each of the two systems. are observed HII regions in that direction, We used the catalog of H II regions of at large negative velocities, that justify our Kuchar & Clark (1997). The l − v dia- model. In some other directions, the obser- gram with the fitted loops, is shown in Fig.5. vations favor our model as well. Remark for The rotation curve, that we adopted for the instance that there are concentrations of H II position-velocity transformation was derived regions near labels a and f in Fig. 5. These from the interstellar gas data of Clemens are well explained by a spiral arm that pass very close to the Sun, seen almost at l ≈ +90◦ (1985), reinterpreted in terms of R0 =7.5 kpc. ◦ In this transformation we also took into ac- and then at l ≈ −90 , since the velocities count a non-circular motion, due to the fact are almost zero in these directions, for dis- that when they form, H II regions have a sys- tances that are not too large, according to tematic velocity of about 10 kms−1 towards the well known expression v ∝ sin(2l). The the center of the Galaxy, that keep them in or- longitudes of the tangential directions indi- cate that it is an arm with small pitch angle bits which are close to the spiral arms (Bash, ◦ 1981). Note that the choice of this parame- (about 6 ). On the contrary, the wide loop label b, can only be well reproduced with a ter considerably helps in obtaining good fits ◦ of the observed locus of the inner HII regions. larger pitch angle (12 ).This emphazises the The loops are labeled a to i in the l − v dia- nedd for arms with different pitch angles. The gram; the corresponding positions of the tan- best fit of the l − v diagram of the observed gential points are indicated on the pattern in HII regions that we obtained with a simple 2 the galactic plane, in Fig. 6. + 4 armed model, although not perfect, re- produces the main features of the diagram,
10 and in particular, the main tangential direc- parison of theoretical diagrams with observed tions. This was obtained with pitch angles HII regions, but they are not perfect fits to the 6.6◦ for the 2-armed component and 12◦ for HII regions. In the HII regions diagram (Fig. the 4-armed component. 5) we can see many objects between loops c Let us now discuss the theoretical l − v and d, that are not well fitted by the loops. diagrams, that were computed by means of We can see many objects as well in this re- our particle simulations in Sec. 4 and are gion in Figure 8, the theoretical self-sustained shown in Figs. 7 to 9. Figs.7 and 8 are model. In other words, the theoretical model very similar to the observed diagram for H is closer to reality than the “empirical” fit II regions, in many aspects. In particular, represented by the lines. Around longitude ◦ ◦ the observed loops like b, c, d, e which are 240 (or -120 ), the theoretical l − v diagram interpreted in Fig. 5 as empirical locii of spi- of the self-sustained shows many objects with −1 ral arms, with corresponding tangential direc- velocities of the order of 30 kms . These ob- tions to the arms, can be clearly seen in Fig. jects seem to delineate a spiral arm that is 7 and 8 as well. A difference that appears not seen in the HII regions diagram. How- between the theoretical l − v diagrams and ever, a arm indeed exists at this position, as the observed ones for H II regions is a strong can be seen in the longitude-velocity diagram concentration of points along the line from of IRAS sources given by Wouterlout et al. l ≈ −60◦, v ≈ +100 kms−1 to l ≈ +60◦, v ≈ (1990). This arm is deficient in HII regions, −100 kms−1. This is an expected result since probably due to the proximity of corotation. the points corresponding to H I (recall that On the contrary, the 4 arms model (Fig. 9) the particles simulate the H I clouds, see Sec show stronger differences with observations. 4) are situated at large galactic radii, where The particles do not show loops a and b, but H I is known to exist, whereas there is lack of show a clear loop between loops c and d, that H II regions at large radii. is not present in the HII regions diagram. It is not surprising that the l − v dia- In summary, the pure 2-arms model and grams in Figs. 7 and 8 are so similar, since the self-sustained 2+4 arms model produce both models contain similar 2-arms compo- theoretical l − v diagrams that are similar be- nent, and this component is prominent over tween them and similar to that of observed a wider galactocentric region than the 4-arms HII regions. The only striking difference be- component of the 2+4 arms model. However, tween the theoretical and observed diagrams we can point out a number of differences. For is an expected one, due to the fact that we instance, the observed loop b, which is due to are comparing objects that behave differently a 4-arms component, is correctly reproduced (HI gas and HII regions). If we look into the by the particles of the self-sustained model details, the 2+ 4 arms self-sustained model is (Fig. 7) and does not appear in Fig.8. On favored. The choice in favor of the 2+4 arms the other hand, the arm that passes very close model, compared to the pure 2-arms, is more to the Sun, as discussed above (loops a and strongly dictated by the study of Cepheid f) appear more clearly in the 2-arms model kinematics. (Fig.8). We emphazise that the loops pre- sented in Figs. 7-9 are eye-guides for com-
11 7. Conclusion the theoretical l − v diagrams from the parti- cle simulations with the locii of arms derived In the present research a new approach to from HII regions.Although the differences be- the problem of the galacic spiral structure was tween l − v diagrams of the theoretical mod- proposed in order to construct a more realis- els (pure 2-arms, pure 4-arms, and 2 different tic picture like often seen in external galax- models of 2+4 arms) are not striking, the 2- ies: co-existing of different spiral systems in a arms model and the self-sustained 2+4 arms galaxy, arm bifurcation and their overlaping. models are the ones that produce theoretical Our theoretical considerations shows that su- l − v diagrams most similar to that of the HII perposition of self-sustained spiral wave har- regions. monics could explain some of the above fea- Of all the arguments that we examined, the tures, since different azimuthal wave harmon- significantly better fit of the kinematics (the ics have different pitch angles. 2 δ analysis) of Cepheids with the 2+4 armed, In the framework of the simplest model of self-sustained model, is the most convincing superposition of 2+4-armed spiral wave har- one, but clearly the analysis of the HII regions monics, we analysed the best up-to-date data sample gives support to our interpretation. on stellar kinematics, which is the sample of Although the Galaxy probably shows some Cepheid stars, with proper motions and par- deviations from any simplified model, the 2+4 allaxes determined by HIPPARCOS. We ex- armed model with different pitch angles is the amined two models, the self-sustained and the one that constitutes the best approach, being bar-dominated waves. This study comple- consistent with observations and with spiral ments the previous studies of pure 2-arms and waves theory. This is the simplest model that pure 4-arms presented by MZ ad MZDMR. can be proposed, apart from pure harmonic Of the four models, clearly the one that gives modes. Our model does not exclude the exis- the best fit to the Cepheid kinematics is the tence of higher modes, that are allowed to ex- self-sustained model, which is a superposition ist near corotation, but these modes are prob- two arms with pitch angle about 6◦ and four ◦ ably less significant than the first harmonics. arms with pitch angle about 12 . We per- It is interesting to remark that our model is formed N-particle simulations to make visible able to reconcile the first model of Lin & Shu the structure of the potential derived from the (1964), which has 2 arms with pitch angle 6◦ Cepheid kinetics for the four models, and to similar to our 2-arms harmonic, with the need construct the corresponding l − v diagrams. to satisfy Kennicut’s (1982) correlation be- As an independent test of a 2+4 arms tween pitch angle and maximum rotation ve- model,we performed a new analysis of the l−v locity, which predicts a pitch angle of about diagram of the galactic HII regions, which are 14◦ for our Galaxy, not very different from the best tracers of the large-scale spiral struc- that of our 4-arms harmonic. ture. We fitted the observed l − v diagram As a by-product of the study, our particle empirically with the locii of spiral arms, us- simulation shows that a deficiency of intertel- ing a 2+4 arms model. Coincidentally, the lar gas must occur near corotation. This ex- best empirical fit was again found using pitch ◦ ◦ plains the gap in the HI ditribution observed angles about 6 and 12 . We also compared by Kerr(1969) and by Burton (1976). This ef-
12 fect is also possibly related to the sharp min- Draper N.R., Smith H., 1981, Applied Re- imum in the rotation curve near the solar ra- gression Analysis, John Wiley and Sons. dius discussed by Amaral et al. (1996), also Inc., New York – Chichester – Brisbane – seen in the curve by Honma & Kan-ya (1998). Toronto – Singapore In external galaxies the ring-like gap in H I Efremov Yu.N., 1998, A&A Transact, 15, 3 distribution may directly show the localiza- tion of the corotation circle. Englmaier P., Gerhard O., 1999, MNRAS 304, 512. Our investigation of Cepheid kinematics points out that the Sun lies very close to ESA, 1997, The Hipparcos Catalogue. ESA, the corotation circle. This conclusion is sup- SP-1200 ported by the solar closeness to the ring-like Georgelin Y.M., Georgelin Y.P., 1976, A&A region of H I deficiency in the Galaxy derived 49, 57 (GG) by Kerr (1969) and by Burton (1976), as well Goldreich P., Tremaine S., 1978, Icarus 34, as by the position of Lindblad resonances (as 227 discussed in Section 1) and by direct measure- ment of the pattern speed using open clusters Gor’kavyi N.N., Fridman A.M., 1994, Physics (AL). of planetary rings: Celestial mechanics of continious medium, Moscow, Nauka Gorynya N.A., Samus N.N., Rastorguev A.S., Acknowledgements. Authors are grateful to Sachkov M.E., 1996, Astron. Letters (So- the Hipparcos astronomical team, who pre- viet) 22, 198 sented us the Hipparcos catalogue. Honma M., Kan-ya Y., 1998, ApJ 503, L139 REFERENCES Kennicut R.C., 1982, AJ 87, 286 Kerr F.J., 1969, Ann. Rev. Astron. Astro- Amaral L.H., L´epine J.R.D., 1997, MNRAS, phys. 7, 39 286, 885 Kuchar T.A., Clark F.O., 1997, ApJ 488, 224 Amaral L.H., Ortiz R., L´epine J.R.D., Maciel W.J., 1996, MNRAS 281, 339 Lin C.C., Shu F.H., 1964, ApJ 140, 646 Avedisova V.S., 1996, Astron. Letters (So- Lin C.C., Yuan C., Shu F.H., 1969, ApJ 155, viet) 22, 497 721 Bash F.N., 1981, ApJ 250, 551 Mark J.W.-K., 1976, ApJ 203, 81 Burton W.B., 1976, Ann. Rev. Astron. Ap Marochnik L.S., Suchkov A.A., 1984, The 14, 275 Galaxy. Moscow, Nauka (in Russian). Caldwell J.A.R., Coulson I.M., 1987, AJ 93, Mishurov Yu.N., Zenina I.A., Dambis A.K., 1090 Mel’nik A.M., Rastorguev A.S., 1997, A&A 323, 775 (MZDMR) Clemens D.P., 1985, A&A 295, 422 Mishurov Yu.N., Zenina I. A., 1999, A&A Cr´ez´eM., Mennessier M.O., 1973, A&A 27, 341, 81 (MZ) 281
13 Olling,R.P., Merrefield, M.R., 1998, MNRAS Figure captions 297, 943 Fig.1.- Surfaces of the ∆ as a function of χ⊙2 Ortiz R., L´epine J.R.D., 1993, A&A 279, 90 and χ⊙4 for 3 values of pitch angle i2.
Pont F., Mayor M., Burki G., 1994, A&A 285, Fig.2.- The locus of min ϕS2 and min ϕS4 on 415 the galactic plane. The scale is indicated in kpc. As discussed in the text, this does not Puerari I., Dottori H.A., 1992, A&A 93, 469 correspond necessarily to the visible struc- Roberts W.W., 1969, ApJ 158, 123 ture. Roberts W.W., Hausman M.A., 1984, ApJ Fig.3.- The visible structure of the Galaxy de- 277, 744 rived for the best model (superposition of 2+4 Shu F.H., 1970, ApJ 160, 99 self-sustained wave harmonics) by means of particle-cloud simulation. Simonson S.C., 1970, A & A 9, 163. Fig.4.- The radial distribution of cloud con- Suchkov A.A., 1978, Astron. Zhurn. (Soviet) centration
14 camparison, and not a fit to the m =4model.
15 Table 1: Model parameters and their errors derived by means of statistical analysis. ′′ 2 appro− Ω⊙ A R⊙Ω⊙ u⊙ v⊙ |i2| χ⊙2 fR2 fϑ2 |i4| χ⊙4 fR4 fϑ4 min δ km km km km km ◦ ◦ km km ◦ ◦ km km ach ( s kpc ) ( s kpc ) ( s kpc2 ) ( s ) ( s ) ( ) ( ) ( s ) ( s ) ( ) ( ) ( s ) ( s ) self− 26.3 17.5 9.9 −8.8 11.9 6.1 311. 0.4 −14.0 12.0 122. 0.8 −10.9 187. sust. ±1.3 ±0.8 ±1.9 ±1.0 ±1.1 ±0.4 ±11. ±3.0 ±3.0 ±0.8 ±15. ±3.3 ±2.9 bar− 25.7 9.8 −6.8 −13.6 9.3 12.6 184. 12.5 −19.4 12.6 8. 6.6 −10.1 220. domin. ±1.2 ±2.3 ±5.0 ±2.4 ±1.2 ±0.5 ±4. ±3.8 ±4.4. ±1.0 ±7. ±2.4 ±2.1
16 This figure "Fig1.jpg" is available in "jpg" format from:
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