Prominent spiral arms in the gaseous outer galaxy disks
aa r X i v : . [ a s t r o - ph . C O ] D ec Astronomy&Astrophysicsmanuscript no. BertinAmorisco c (cid:13)
ESO 2018August 14, 2018
Prominent spiral arms in the gaseous outer galaxy disks
G. Bertin and N.C. Amorisco Dipartimento di Fisica, Universit`a degli Studi di Milano, via Celoria 16, I-20133 Milano, Italye-mail: [email protected] Dipartimento di Fisica, Universit`a di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italye-mail: [email protected] ⋆ Received November, 2009;
ABSTRACT
Context.
Several spiral galaxies, as beautifully exhibited by the case of NGC 6946, display a prominent large-scale spiral structurein their gaseous outer disk. Such structure is often thought to pose a dynamical puzzle, because grand-design spiral structure istraditionally interpreted as the result of density waves carried mostly in the stellar disk.
Aims.
Here we argue that the outer spiral arms in the cold gas outside the bright optical disk actually have a natural interpretationas the manifestation of the mechanism that excites grand-design spiral structure in the main, star-dominated body of the disk: theexcitation is driven by angular momentum transport to the outer regions, through trailing density waves outside the corotation circlethat can penetrate beyond the Outer Lindblad Resonance in the gaseous component of the disk.
Methods.
Because of conservation of the density wave action, these outgoing waves are likely to become more prominent in the outerdisk and eventually reach non-linear amplitudes. To calculate the desired amplitude profiles, we make use of the theory of dispersivewaves.
Results.
If the conditions beyond the optical radius allow for an approximate treatment in terms of a linear theory, we show that fittingthe observed amplitude profiles leads to a quantitative test on the density of the disk material and thus on the dark matter distributionin the outer parts of the galaxy.
Conclusions.
This study is thus of interest to the general problem of the disk-halo decomposition of rotation curves.
Key words. galaxies: spiral – galaxies: structure – galaxies: halos – galaxies: kinematics and dynamics
1. Introduction
Deep HI observations of nearby galaxies have led to the dis-covery of a number of important phenomena that are changingour views on the structure and dynamics of galaxies. These in-clude the existence in early-type galaxies of regular and radi-ally extended HI disks (Oosterloo et al. 2007a), the presence inspiral galaxies of extraplanar gas characterized by slow rotation(for NGC 891 see Oosterloo et al. 2007b; for NGC 2403, seeFraternali et al. 2002), and the properties of small-scale struc-tures in the HI distribution (for NGC 6946, see Boomsma et al.2008). One interesting related discovery is the existence of reg-ular and prominent spiral arms in the gaseous outer disk, welloutside the bright optical disk (Shostak & van der Kruit 1984;Dickey et al. 1990; Kamphuis 1993; for NGC 2915, see Meureret al. 1996; for NGC 3741, see Begum et al. 2005). In this re-spect, a particularly impressive example is given by the case ofNGC 6946 (Boomsma, 2007; Boomsma et al. 2008; see Fig. 1),where a spectacular set of gaseous arms can be traced all theway out, with a significant degree of regularity and symmetryeven if the outer disk is clearly lopsided and characterized by afragmentary structure; the outer arms also appear to contain stars(see Ferguson et al. 1998, who also analyze the interesting casesof NGC 628 and NGC 1058, and Sancisi et al. 2008).The study of spiral structure in galaxies has received great at-tention in the past. It is now generally thought that grand-design ⋆ also at Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56125Pisa, Italy; as of October 2009, at Institute of Astronomy, MadingleyRoad, Cambridge CB3 0HA, UK structure is the manifestation of density waves, mostly carriedby the stellar component of galaxy disks. A general frameworkfor the interpretation of the observed morphologies, based on adensity wave theory, has received significant support from theobservations of spiral galaxies in the near infrared (see Bertin &Lin 1996 and references therein). It might thus appear as a puz-zle to find well-organized spiral patterns in the outer disk, in aregion where stars are practically absent.The above-mentioned deep HI studies also serve as interest-ing probes in view of defining an appropriate visible matter -dark halo decomposition of the gravitational field in galaxies. Inparticular, the studies of prominent spiral arms in the gaseousouter disks have raised two issues that separately point to thequestion of whether the outer disk is light or heavy. On the onehand, concerns have been raised about the applicability, in suchregion, of the criterion for the onset of star formation proposedby Kennicutt (1989), which relies on a threshold on the ax-isymmetric stability parameter Q = c κ/π G σ . Accordingly, lightdisks, with low σ , should be unable to make new stars; but inthe outer parts the disk might be flared and three-dimensionale ff ects may change the picture significantly (in this regard, seealso the comments made by Ferguson et al. 1998). On the otherhand, there is widespread belief (see Sancisi et al. 2008, p. 212)that light disks should be unable to support spiral structure (seeToomre 1981; Athanassoula et al. 1987; criticism against thisbelief can be found, e.g., in the article by Bertin et al. 1989a).In general, the arguments often put forward in favor of amaximum-disk decomposition of the rotation curves of spiralgalaxies (starting with van Albada & Sancisi 1986) still await a G. Bertin and N.C. Amorisco: Prominent spiral arms in the gaseous outer galaxy disks decisive measurement to remove the remaining degeneracy thatcharacterizes such a decomposition. Some projects, such as the“Disk Mass Project” (Verheijen et al. 2004, 2007), aim at makingfull use of three-dimensional gas and stellar dynamical data soas to decompose the field, much like in the classical problem ofthe disk thickness in the solar neighborhood (Oort 1932, Bahcall1984, Kuijken & Gilmore 1989, Cr´ez´e et al. 1998, Holmberg &Flynn 2000, 2004). Dark halos are generally thought to be madeof collisionless dark matter and to have spheroidal shape, butit would be important to have direct proof that the outer diskof spiral galaxies is indeed light, in contrast with the possibilitythat the disk be heavy because of large amounts of molecularmaterial (Pfenniger et al. 1994; see also Revaz et al. 2009).One interesting aspect of galactic dynamics is that modelsfor the interpretation of observed structures generally o ff er anindependent diagnostics of the overall mass distribution. In thispaper we will present one more example of this general aspect ofdynamics. Indeed, we will propose a model for the observed spi-ral structure in the outer disk and then will show how the modelcan be tested and applied to probe the structure of the outer diskin relation with the problem of the amount and distribution ofdark matter.In passing, we note that, in principle, one might test the prop-erties of the final model (i.e., disk-dark matter decomposition)identified by our technique in specific cases against the expecta-tions of non-Newtonian theories of gravity such as MOND; butof course, a discussion within MOND of the full problem, in-cluding the behavior of density waves, is currently not available.The picture presented in this paper is the following. Globalspiral modes are driven by the transfer of angular momentumto the outer regions (see Bertin et al. 1989a,b and referencestherein; see also Lynden-Bell & Kalnajs 1972, Bertin 1983).Outside the corotation circle, the transfer is performed by shorttrailing waves. At the Outer Lindblad Resonance, such outgoingwaves are fully absorbed in the stellar disk (Mark 1971, 1974),but only partially absorbed in the gaseous component (Pannatoni1983; Haass 1983), so that the signal can penetrate beyond suchresonance and propagate in the HI disk, if present. The outer spi-ral arms are thus interpreted as the natural extension in the outerdisk of the short trailing waves that are responsible for excitingthe global spiral structure in the star-dominated main body ofthe galaxy disk. The amplitude profile of such outer arms shouldsimply conform to the requirements dictated by the conservationof wave action (Shu 1970; Dewar 1972). Because of this conser-vation, the amplitude of the outer arms is expected to increasewith radius, in the regions where the inertia of the disk becomessmaller and smaller, much like ocean waves can reach high am-plitudes when moving close to the shore. In these outer regions,the density wave is thus carried by the gas, but the stars presentwould collectively respond and some new stars may be born be-cause of gas compression, following the gaseous arms.Of course, galaxies such as NGC 6946 or NGC 628 and theBlue Compact Dwarf NGC 2915 are very di ff erent objects; eachcase should thus be studied separately in detail and each individ-ual object may have its own special character. Here we wish too ff er one quantitative reference frame for a common mechanismthat in general should operate in the outer parts of galaxy disks.A fully non-linear, three-dimensional analysis of the proper-ties of density waves in the HI outer disk is not available, butwe can hope that under suitable conditions an approximate de-scription based on the laws of conservation of wave action forlow-amplitude density waves is viable. In any case, it should betested against the observations. Since fitting the data requires as-sumptions on the density associated with the spiral arms and the Fig. 1.
Total HI distribution of NGC 6946 superimposed to itsoptical image; courtesy of Filippo Fraternali (see Boomsma etal. 2008).density associated with the fluid basic state, a test of this sce-nario would be able to tell how much mass in the outer disk ispresent in the form of molecular gas.
2. Physical model and calculation of the amplitudeprofiles
We refer to a grand-design spiral galaxy dominated by a globalmode with m arms and pattern frequency Ω p and, for simplicity,consider a barotropic fluid model of an infinitesimally thin diskas an idealized representation of the gaseous outer regions of thegalaxy. In terms of standard polar cylindrical coordinates ( r , θ ),let σ = σ + σ be the disk mass density, ( r Ω + v ) −→ e θ + u −→ e r thefluid velocity field, and c the e ff ective sound speed of the fluid.The quantity Ω = Ω ( r ) is the di ff erential rotation. As a measureof the distance from the corotation radius, we then refer to thedimensionless quantity ν = m ( Ω p − Ω ) /κ , where κ is the epicyclicfrequency. In the following, subscript 0 refers to the axisymmet-ric basic state of the disk and subscript 1 to the perturbation,on such equilibrium, associated with the large-scale spiral struc-ture. In the notation just introduced, the relevant axisymmetricstability parameter is defined as Q = c κ/ ( π G σ ).According to the general picture of the modal theory of spiralstructure (see Bertin & Lin 1996, and references therein), outsidethe corotation circle the global mode is associated with an out-going (short) trailing wave. In general, for a normal spiral modethe corotation circle is expected to occur at the edge of the opti-cal disk (i.e., at r co ≈ − h , with h the exponential scale of thestellar disk), while for a barred spiral mode the corotation circleis expected to occur just outside the tip of the bar, in the middleof the optical disk, at r co ≈ h . In the gaseous outer disk, theoutgoing trailing wave can penetrate beyond the Outer LindbladResonance (which occurs at the radius r OLR where ν = r > r OLR . In this region the calcu-lation of the amplitude profile is particularly simple; instead, theproperties of spiral structure out to r ≈ r OLR are determined bythe processes that govern the main body of the disk (see Bertin& Lin 1996). . Bertin and N.C. Amorisco: Prominent spiral arms in the gaseous outer galaxy disks 3
In the linear theory of density waves, the calculation can beperformed in a straightforward manner by imposing the conser-vation of the density of wave action (see Shu 1970; Dewar 1972;Bertin 1983). Mathematically, this is equivalent to carrying outthe analysis that leads to the standard algebraic dispersion rela-tion for density waves D ( ν, | ˆ k | ) =
0, with D ( ν, | ˆ k | ) = ν − − Q ˆ k / + | ˆ k | (where the radial wavenumber k = ˆ k κ / (2 π G σ )is associated with the pitch angle of the spiral arms, tan i = m / ( rk )), to the next order in the WKB expansion (e.g., see Bertin2000, Chapter 17.3; see Eqs. (17.2), (17.22), (17.28), (17.30),and (17.42)).In particular, the linear density wave analysis of a zero-thickness barotropic fluid disk is best carried out in terms of theenthalpy perturbation h = ( c /σ ) σ . The WKB asymptoticanalysis then leads to a Schroedinger-like equation of the form u ′′ + g ( r , ω ) u = , (1)where, for tightly wound density waves, the function g is givenapproximately by g ( r , ω ) = κ c Q − + ν ! . (2)The function u in Eq. (1) is defined in terms of the function h insuch a way that the amplitudes of the two functions are relatedas | h | = κ | − ν | r σ | u | . (3)The WKB analysis of the turning-point equation for u showsthat, away from the turning points, the amplitude of u must scaleas | u | ∼ | g | − / , which is interpreted as the conservation of thedensity of wave action. A little algebra then shows that such re-lation is equivalent to the condition (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∝ G ( ν, Q ) r − κ σ − , (4)where G ( ν, Q ) ≡ ν − Q p + ( ν − Q . (5)The proportionality constant is independent of r . Thus Eq. (4)provides the desired expression for the amplitude profile of thedensity wave in the outer regions.In order to complete the description of the amplitude profilesof the spiral arms, we may then consider the linearized continu-ity equation: ∂σ ∂ t + r ∂∂ r ( r σ u ) + r ∂∂θ ( σ v + σ r Ω ) = . (6)Since the vorticity equation shows that v = ( i /ν )[ κ/ (2 Ω )] u , sothat v and u have the same order of magnitude, in the continuityequation we can neglect the v contribution consistent with thepresent WKB approximation, thus obtaining: σ ku ∼ νκσ . (7)By inserting here the solution for k associated with the (short)trailing wave branch, k = − κ π G σ ! Q (cid:16) + p + ( ν − Q (cid:17) , (8) we find: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u r Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ν (cid:18) κ Ω (cid:19) | rk | ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∝ (cid:18) κ Ω (cid:19) H ( ν, Q ) r − / σ − , (9)with H ( ν, Q ) ≡ ν Q √ G ( ν, Q )1 + p + ( ν − Q . (10)Again, the proportionality factor is independent of r . Note thatthe first part of Eq. (9) is an equality, not a proportionality re-lation. In other words, in the context of the linear WKB theory,the scale in velocity amplitude u is uniquely determined by thescale in the amplitude of the density wave σ . Note also thatthe perturbation u , together with v , will generate the “wiggles”in the velocity field that are characteristic of density waves (forM81, see Visser 1977); in the observed cases where the ampli-tudes are large, a non-linear theory is required for a quantitativecomparison with the observations. The analysis described previously can be generalized to includethe e ff ects of finite thickness of the disk. In the discussion ofthe dynamics of self-gravitating disks, these e ff ects are often ig-nored, but may actually play an important role; in our case, sig-nificant e ff ects would be naturally expected if in the outer partsthe disk is flared, as often observed. Qualitatively, these e ff ectsshould become significant for short waves, when the wavelengthof the density wave becomes comparable to the thickness of thedisk; in terms of local stability, finite-thickness e ff ects are knownto be stabilizing, because they e ff ectively “dilute” the gravityfield. Quantitatively, they are approximately incorporated by thefollowing dispersion relation (see Vandervoort 1970, Yue 1982): D ft ( ν, | ˆ k | , ˆ z ) = ν − − Q ˆ k + | ˆ k | + | ˆ k | ˆ z = , (11)where z represents the disk thickness (defined in such a waythat the disk surface density σ is related to the volume density ρ on the equatorial plane by the expression σ = ρ z ) andˆ z ≡ κ z π G σ . (12)The modification of the dispersion relation with respect tothe standard one, used earlier in Sect. 2, changes the expres-sion of the trailing wave-branch that carries angular momentumoutwards k = k S T = ˆ k S T κ / (2 π G σ ), with respect to Eq. (8).But the calculation of this wavebranch is straightforward, be-cause the new dispersion relation (11) is just a cubic in | k | . Hereˆ k S T = ˆ k S T ( ν, Q , ˆ z ).From the general theory of dispersive waves, we know thatthe flux of density wave action is F = c g A , with the groupvelocity c g ≡ − ( ∂ω/∂ k ) and the wave action density A ∝ σ ( ∂ D /∂ω ). Thus we have F ∝ ( ∂ D /∂ k ). In the derivation re-ported in the first part of the Section the factor ( ∂ D /∂ k ) enters inthe conservation equation (4), as ( ∂ D /∂ ˆ k ) = p + ( ν − Q ,with the latter derivative evaluated from the zero-thickness dis-persion relation on the short trailing wave-branch (8).Therefore, the desired conservation equation in the finitethickness case becomes: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ σ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∝ G ft ( ν, Q , ˆ z ) r − κ σ − , (13) G. Bertin and N.C. Amorisco: Prominent spiral arms in the gaseous outer galaxy disks where now we have G ft ( ν, Q , ˆ z ) ≡ ν − Q ( ∂ D ft /∂ ˆ k ) ; (14)here the partial derivative is evaluated at ˆ k = ˆ k S T .The corresponding expression for the amplitude profile in thevelocity field is obtained from Eq. (9) by replacing H ( ν, Q ) with H ft ( ν, Q , ˆ z ), with H ft ( ν, Q , ˆ z ) ≡ ν p G ft ( ν, Q , ˆ z ) | ˆ k S T ( ν, Q , ˆ z ) | . (15)In the following subsection it will be shown that flaring ef-fects are under control and do not change significantly the gen-eral predictions of the zero-thickness theory. To test the overall picture we may consider the following simplereference model. We refer to a two-armed spiral structure ( m =
2) in an outer disk characterized by a flat rotation curve, so that κ = √ Ω , Ω / Ω p = r co / r , and ν = √ r / r co − r co = h , in terms of the exponential scale h of the (inner) stellar disk. Thus the Outer Lindblad Resonancewill occur at r OLR ≈ . h . Therefore, at r in = h we are outsidethe circle associated with the Outer Lindblad Resonance, in aregion where we expect the disk to be gas dominated, so thatbeyond such radius we may proceed to apply our fluid model.As a further simplification, we take the conservative case inwhich σ = σ ( r in )(6 h / r ). This corresponds to a very gentledecline of the gas density profile. (For faster declining profiles,we expect that the relative strength of the spiral amplitude shouldbe more pronounced as we move outwards (cf. Eq. (4)), and thusgive rise to a stronger e ff ect.) In our simple reference model, thegas density is then proportional to the di ff erential rotation, sothat we find | σ /σ | ∝ G ( ν, Q ) / r and | u / ( r Ω ) | ∝ H ( ν, Q ) / r / .In a zero-thickness disk, within the adopted reference model,the pitch angle of the spiral arms depends only on ν and Q , be-cause (cf. Eq. (8)) | rk | ∝ Q (cid:16) + p + ( ν − Q (cid:17) , (16)where the proportionality constant is independent of radius. Inthis case we are thus left to discuss the dependence Q = Q ( r ).There are observational indications (cf. Boomsma et al. 2008,Fig. 6) that the gas velocity dispersion is steadily declining in theouter disk, while in the present simple model the quantity κ/σ is r -independent. So we might be led to argue that a decliningQ -profile would be realistic. However, the observed decline inthe gas velocity dispersion is just likely to reflect the fact that,because of the stabilizing role of thickness (see previous subsec-tion), the disk can get colder and colder (in terms of the standard Q ) and still remain on the margin of local axisymmetric insta-bility. Then, for the present zero-thickness reference model wethink it appropriate to consider Q ≈ constant, and, for simplic-ity, we take Q =
1. Note that this condition of marginal stabilityis not strictly necessary, because in the outer disk short trailingwaves can propagate even at higher values of Q .In conclusion, the present simple zero-thickness model ischaracterized by G ( ν, Q ) = ( ν − /ν , H ( ν, Q ) = p ν ( ν − / (1 + ν ), and rk ∝ (1 + ν ). By applying Eqs. (8), (4), and (9) we can pro-ceed to calculate the desired profiles i ( r ), | σ /σ | , and | u / ( r Ω ) | . Fig. 2.
For our simple reference fluid model, with σ ∝ / r and Ω ∝ / r , the thick solid line represents the marginal stabilitycondition Q = Q max (ˆ z ) and the dashed curve the correspondingcondition for the case in which the dilution of the gravity field(associated with the finite thickness of the disk) in the relevantdispersion relation is described by an exponential factor (insteadof the rational factor used in Eq. (11)). The thin rising curvesrepresent the function Q = Q fluid (ˆ z ) for the two cases of a fullyself-gravitating layer (upper curve) and for a non-self-gravitatinglayer (lower curve).In the finite thickness case, the wavenumber for short trail-ing waves k S T ( ν, Q , ˆ z ) is obtained from the dispersion relationEq. (11), which can also be used to calculate the two relevantquantities G ft ( ν, Q , ˆ z ) and H ft ( ν, Q , ˆ z ). The remaining pointthat requires discussion is the radial dependence of the two func-tions ˆ z = ˆ z ( r ) and Q = Q ( r ).As to the vertical thickness of the outer disk, for our refer-ence model, characterized by Ω ∼ / r and σ ∼ / r , it can beshown (see Appendix A in Bertin & Lodato 1999) that, in eachof the two opposite limits of a totally self-gravitating disk andof a non-self-gravitating layer, the thickness behaves as z ∼ r ,so that taking ˆ z = constant is a reasonable choice. In addition,since we are referring to a fluid, the relevant velocity dispersiontensor is isotropic ( c r = c z = c ), so that there is a one-to-onerelation between the value of Q and the value of ˆ z that we aregoing to take; let us denote this relation by Q = Q fluid (ˆ z ). Onthe other hand, for a proper comparison with the zero-thicknessreference model described earlier in this Subsection, we shouldassume that the disk is at marginal stability, i.e. Q = Q max (ˆ z )(see Eq. (15.22) in Bertin 2000). By combining the above re-quirements into Q fluid (ˆ z ) = Q = Q max (ˆ z ) we get a unique valuefor the pair (ˆ z , Q ), as demonstrated in Fig. 2. With this determi-nation, from Eqs. (11), (14), (15), and (13) we can calculate thedesired profiles i ( r ), | σ /σ | , and | u / ( r Ω ) | for the finite thick-ness model.The results of this analysis are illustrated in Fig. 3 in theradial interval 6 h < r < h . In this figure, we have assumedthat at r = h the relative amplitude of the density wave is σ /σ = .
15 and that the pitch angle of spiral structure at suchinner location is 15 degrees. The figure shows that even for thepresently assumed very gentle decline of the gas density distribu-tion the amplitude of the spiral wave steadily increases with ra-dius. Note that in the two limits, of a totally self-gravitating diskand of a non-self-gravitating disk, the finite-thickness e ff ects donot change the general picture; similar results and a similar con-clusion have been checked to hold using a dispersion relationalternative to Eq. (11), in which the dilution of the gravitationalterm (cid:12)(cid:12)(cid:12) ˆ k (cid:12)(cid:12)(cid:12) is exponential rather than rational. . Bertin and N.C. Amorisco: Prominent spiral arms in the gaseous outer galaxy disks 5 Fig. 3.
Relative amplitude profiles of the spiral arms in termsof density σ /σ and of radial velocity u / ( r Ω ) (dimensionless,left axis) and pitch angle i of spiral structure (degrees, rightaxis), for three cases of the simple reference model describedin Sect. 2.2: zero-thickness disk (thick lines), finite-thicknesstotally self-gravitating disk (dashed lines), finite-thickness non-self-gravitating disk (dash-dotted lines).
3. Discussion and conclusions
HI observations of the gaseous outer regions provide measure-ments of the HI gas density σ HI , of the rotation curve Ω , andof the turbulent velocity c HI , which we may identify with c . Wemay argue that the actual disk density be traced by the atomichydrogen, so that σ = f σ HI . In the simplest model we may take f ≈ .
4, i.e. take that the outer disk just contains the relevant pro-portion of helium. For the following discussion, we may assumethat the thickness z of the gaseous outer disk is not well con-strained by the observations, since we wish to consider objectswith prominent observed spiral arms, and therefore galaxies thatare not edge-on. On the other hand, instead of considering thethickness profile z = z ( r ) as an additional free function of theproblem, we may refer to a self-consistent estimate of such pro-file, which is readily available (e.g., see Appendix A in Bertin &Lodato 1999 and Sect. 2.2 in the present paper).Clearly, if the linear theory happens to be viable, the basicrelations are over-constrained by the data, since we only havesome leverage on the precise value of Ω p (which then sets theform of the function ν ( r )) and basically no other free parameters.Note that the morphology, or a suitable Fourier decomposition ofthe observed structure, would identify the dominant value of m and the pitch angle i ( r ), i.e., the function k ( r ). The linear the-ory makes specific predictions, such as Eqs. (8), (4), (9) (or thecorresponding equations recorded in Sect. 2.1 for the study thatincludes the e ff ects of finite thickness). Much as for the classicaltests of the density wave theory (for M81, see Visser 1977), wemay hope that the various observed quantities all fall within areasonable agreement with the theoretical expectations.Suppose that we start from such a straightforward compar-ison with the observations on the basis of the linear theory de-scribed in the first part of Sect. 2. If this attempt turned out to beunsatisfactory, we would have three levels of action in interpret-ing the data.(i) One possibility would be to make use of an HI frac-tion f constant, but significantly larger than unity. Physically,a choice of this kind would correspond to imagining a heavierouter gaseous disk, but still in proportion to the observed HIdisk, and thus with little relevance to the overall problem of darkmatter. For given values of σ HI and c HI , this would allow us to reduce the value of the wavenumber scale κ / (2 π G σ ) and ofthe axisymmetric stability parameter Q , while leaving the rela-tive density amplitude σ /σ unchanged on the left-hand-side ofEq. (4). Except for a small adjustment through the functions G and H , such a constant f would have little or no e ff ect on the fitsto the observed amplitude profiles dictated by Eqs. (4) and (9).(ii) A second possibility would be to make use of a free function f = f ( r ), with the general requirement that f > . ff erent from that ofthe cold atomic hydrogen. This may thus go in the direction ofan alternative picture (with respect to the standard picture of aspheroidal halo) for the general problem of dark matter (e.g., seePfenniger et al. 1994). Clearly a non-constant HI fraction f ( r )would change the character of the observed amplitude profilespredicted by Eqs. (4) and (9).(iii) Finally, it may well be that the model developed above,which relies on the predictions of a linear theory, turns out to beinadequate. In other words, we should develop a model in whichthe role of the non-linearities associated with the finite ampli-tude observed in the prominent spiral arms is properly taken intoaccount.While a more advanced model, of the kind outlined in item(iii) above, is being developed (in this respect, we are encour-aged by the fact that a model of the outer disk as a fluid disk islikely to be appropriate and that the physical picture considered,of an outgoing wavetrain, is relatively simple), in this paper weargue that a first test of the simple picture presented here is worthtrying. If a satisfactory agreement with the linear theory could beobtained with f ≈ .
4, we would have one additional convincingargument that the picture of a spheroidal dark halo indeed holdsalso for the outermost regions of spiral galaxies. In any case, theconsiderable e ff ort required by setting up such a test on a specificcase, e.g. for NGC 6946, would prepare the ground for a test ofthe more realistic non-linear theory that we plan to investigatesoon.Finally, given the picture provided by the simple referencemodel described in Sect. 2.2, we may argue that for those galax-ies in which the outer gaseous disk density declines too sharply,the non-linear e ff ects that would rapidly take place because ofthe sharp increase in relative amplitude of the density waves mayactually tend to break the grand design outer structure and resultin turbulent dissipation, as often argued in the physical discus-sion of the outer boundary condition for the establishment ofglobal spiral modes (e.g., see Bertin & Lin 1996, p. 222). Acknowledgements.
We wish to thank Renzo Sancisi, Rosemary Wyse, and JayGallagher for pointing out the interest in this problem and for several stimulatingdiscussions and Filippo Fraternali and Giuseppe Lodato for a number of usefulcomments. Special thanks to Filippo Fraternali for providing us with Fig. 1.
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