Vertical gradients of azimuthal velocity in a global thin disk model of spiral galaxies NGC 2403, NGC 4559, NGC 4302 and NGC 5775
Joanna Jałocha, Łukasz Bratek, Marek Kutschera, Piotr Skindzier
aa r X i v : . [ a s t r o - ph . GA ] A p r Printed 15 November 2018 MNL A TEXstyle
Vertical gradients of azimuthal velocity in a global thin disk model ofspiral galaxies NGC 2403, NGC 4559, NGC 4302 and NGC 5775.
Joanna Jałocha , Łukasz Bratek , Marek Kutschera , , Piotr Skindzier Institute of Nuclear Physics, Polish Academy of Sciences, Radzikowskego 152, PL-31342 Krak´ow, Poland Institute of Physics, Jagellonian University, Reymonta 4, PL-30059 Krak´ow, Poland
15 November 2018
ABSTRACT
We estimate the vertical gradient of rotational velocity for several spiral galaxies in the frameworkof a global thin-disc model, using the approximation of quasi-circular orbits. We obtain gradientshaving a broad range of values, in agreement with measurements, for galaxies with both low andhigh gradients. To model the gradient, it suffices to know the rotation curve only. We illustrate,using the example of galaxy NGC 4302 with particularly high gradients, that mass models ofgalactic rotation curves that assume a significant spheroidal mass component reduce the predictedgradient value, which may suggest that the mass distribution is dominated by a flattened disc-likecomponent. We conclude that the value and behaviour of the vertical gradient in rotationalvelocity can be used to study the mass distribution in spiral galaxies.
The definitive version is available at http://onlinelibrary.wiley.com/doi/10.1111/j.1365-2966.2010.17906.x/abstract
Key words: galaxies: kinematics and dynamics galaxies: spiral galaxies: structure.
Rotation curves carry important, though very limited, informationabout the kinematics of spiral galaxies. Until recently, rotation wasdetermined mainly for matter moving in the direct neighbourhoodof the galactic mid-plane. However, thanks to the constantly improv-ing quality of the measurements, for several galaxies it has becomepossible to determine the vertical structure of the rotation above thegalactic disc (Fraternali et al. 2002; Barbieri et al. 2005; Heald at al.2006a,b; Heald et al. 2007; Oosterloo, Fraternali & Sancisi 2007;Levine, Heiles & Blitz 2008). This enables researchers to examine themass distribution in spiral galaxies more accurately. The observationsreveal that the azimuthal component of the galactic rotation falls offwith the vertical distance from the galactic mid-plane. The directionof the fall-off distinguished may indicate that the galaxy dynamics isgoverned mainly by the gravity of masses concentrated close to thegalactic mid-plane, forming a flattened, disc-like object. As for thehigh gradient values, one can hypothesize that the flattened galaxycomponent should comprise the gross dynamical mass of the galaxy.Having this phenomenology in mind, we decided in our previouspaper (Jałocha et al. 2010) to predict the gradient value in the MilkyWay Galaxy. A flattened mass component was represented in ourmodel by a thin disc, while the spheroidal mass components were rep-resented by a spherically symmetric model. We analysed how the gra-dient estimates depended on the relative dynamical masses ascribed tothese components. We observed that both the value and the behaviourof the predicted fall-off were strongly dependent on the assumed massmodel. We obtained correct gradient values when the gross dynami-cal mass was attributed to the disc component, in agreement with ourhypothesis. We also found that similar agreement with gradient mea- surements was possible in the global disc model of galaxy NGC 891.We therefore decided in the current work to carry out similar esti-mates for other four galaxies (NGC 2403, 4559, 5775 and 4302) inwhich the rotation structure has been determined above the galacticmid-plane.In the current work we show that the thin-disc model, despiteits simplicity, predicts correct values for the vertical gradient of rota-tional velocity, in agreement with the gradient measurements in thegalaxies examined. In particular, this model predicts high gradientvalues that are difficult to explain by alternative gradient models (forexample, entirely ballistic models of disc-halo flow give much lowervalues than observed; see Heald et al. (2007)).Furthermore, the different gradient behaviour predicted by vari-ous mass models for the same rotation curve offers one the opportu-nity to test the qualitative properties of the mass distribution in spi-ral galaxies. High gradient values imply a flattened mass distributionrather than a spheroidal one. This is particularly evident when thegradient is measured close to the mid-plane and beyond the centre,further away from the galactic bulge. In this region the shape of thegravitational field is determined mainly by the properties of the massdistribution in the galactic disc and halo, while the structure of thebulge is not very important there, owing to the fact that the contribu-tion from the higher gravitational multipoles of a nearly sphericallysymmetric bulge falls off quickly with distance from the centre andthe bulge can effectively be treated there as a point mass located inthe centre. A qualitative argument for the suggestion that flatteningof the mass distribution increases the gradient is provided by the fol-lowing reasoning. In the case of a pure thin disc, one is more likelyto obtain high gradient values in the mid-plane vicinity than in theopposite case of a purely spherical mass distribution. The reason is that, for the same rotation curve, the predicted gradient under the as-sumption of spherical symmetry is zero at the mid-plane, whereas it isnon-zero in the disc model. Consequently, for a disc-like (i.e. mainlyflattened) mass distribution, it is easier to produce high gradient val-ues close to the disc plane, whereas a feature of a nearly sphericallysymmetric mass distribution is that close to the galactic plane its con-tribution to the overall gradient value is low. Thus, when the massmodel is changed so as to reduce the disc mass component at thecost of increasing the contribution from the spheroidal mass compo-nent (making the overall mass distribution more spheroidal rather thanmore flattened and keeping sufficient the amount of mass required forthe observed rotation velocity), the predicted gradient value will havebeen decreased, especially at larger radii (where the mass distributionbecomes in this case even more dominated by the spheroidal com-ponent) and closer to the mid-plane. Therefore, to ensure high gradi-ent values, the disc mass contribution to the mass function cannot betoo small. An example of this interplay between various mass com-ponents will be discussed later in the example of galaxy NGC 4302with its high measured gradient.
The gross dynamical mass distribution in a flattened galaxy may beapproximately described by a thin disc, assuming axial symmetry andcircular concentric orbits. In addition, it can be also assumed that theazimuthal component of the rotation above the galactic mid-planecan be determined by equating the radial component of the grav-itational force to the centrifugal force (in cylindrical coordinates).This is called the quasi-circular orbits approximation. Validity ofthis approximation was substantiated in the case of the Milky WayGalaxy by analysing the motions of test bodies in the gravitationalfield of a thin disc (Jałocha et al. 2010). In spite of its simplicity, themodel turned out to work strikingly well for the gradient descriptionin the Galaxy (this approximation was discussed in more detail inJałocha et al. 2010). We expect that this simplified description shouldbe applicable in general for similar galaxies.In the disc model, the column mass density associated with thegalactic mass distribution is represented by the equivalent surfacemass density of a substitute thin disc, in accord with the followingformula (Jałocha, Bratek & Kutschera 2008b) σ ( r ) = 1 π G P r Z v σ ( χ ) (cid:18) K (cid:0) χr (cid:1) r χ − rχ E (cid:0) χr (cid:1) r − χ (cid:19) d χ + . . .. . . + ∞ Z r v σ ( χ ) E (cid:0) rχ (cid:1) χ − r d χ . (1)Here, the integration is understood in the principal value sense (bothintegrands are divergent, however the result of integration is finite).This substitute surface mass density can be used for an approximatedescription of the gravitational field produced by a flattened galaxy.This in turn, assuming the circular orbits approximation, provides uswith a simple tool for predicting the azimuthal velocity componentabove the galactic mid-plane and, consequently, to determine the ve-locity fall-off rate as a function of the vertical distance from the mid-plane.Given a surface mass density, the azimuthal component of thevelocities of test particles in the gravitational field of a thin disc canbe estimated in the circular orbits approximation, using the followingexpression (Jałocha et al. 2010): v ϕ ( r, z ) = ∞ Z Gσ ( χ ) χ d χ p ( r − χ ) + z (cid:18) K [ X ] − χ − r + z ( r + χ ) + z E [ X ] (cid:19) , X = − r rχ ( r − χ ) + z < . (2)The model gradient value calculated as ∂ z v ϕ ( r, z ) , is a function of r and z , whereas the gradient measurements are presented by giving asingle number (and its error), being some average over an extendedregion. Therefore, to compare with, in a given region one could deter-mine a mean value of ∂ z v ϕ ( r, z ) over that region and the correspond-ing deviation from the mean.Other methods of vertical gradient estimation seem more rele-vant in the context of realistic gradient measurements, however. Tomimic such measurements, one can specify in a given region an arrayof azimuthal velocities predicted by (2), and calculated for variouspairs ( r, z ) . For each specified variable z , separately, one can calcu-late the mean velocity by carrying out a summation over all radii inthe region and then finding the slope of a linear fit to the mean-valuedata found at various z . We call this gradient estimation the method-I .Another estimation of the gradient value is obtained by finding theslope of a linear fit to the velocity values calculated for various z ,separately at each fixed r , and then finding an average of the slopestaken over all radii in this region. This we call the method-II gra-dient. When gradient ∂ z v ϕ ( r, z ) is almost constant in a given region,both the methods should give results consistent with each other. When ∂ z v ϕ ( r, z ) is not constant in that region, the mean values over that re-gion, determined based on both methods, could be still regarded asdescribing the observed fall-off rate well, provided the dispersion ofthe slopes discussed was comparably small.There are other, more or less complicated methods of gradientestimation possible, which we used in (Jałocha et al. 2010), but allthey led to similar results in the case of the Milky Way Galaxy. Weexpect that various method of gradient estimation should also giveresults consistent with each other for other spiral galaxies, thereforewe decided here to use only the two methods described above. We use two rotation curves for galaxy NGC 2403, published inSofue at al 1999 and Fraternali et al. 2002. We assume the distanceof .
24 Mpc , similarly as in Sofue at al 1999, and find the integratedmass in disk model to be . × M ⊙ . Unfortunately, owing tothe insufficiently high inclination, it was impossible to determine pre-cisely the vertical extent of the gas clouds, the velocity of which wasmeasured above the cold disc Fraternali et al. 2002. It was only pos-sible to estimate the gas-layer thickness to be of about and thecold-disc thickness of about . (Fraternali et al. 2002). There-fore, we assumed the region z ∈ (0 . ,
3) kpc for the gradient es-timation. In Fig. 1 are shown the rotation curves from (Sofue at al1999) and (Fraternali et al. 2002), measured at the mid-plane. Boththe curves are consistent with each other. Also shown is the rotationcurve of the anomalous gas observed above the disc, somewhere inthe interval mentioned. In between the curves are shown the rotationcurves predicted in our model at various heights above the mid-plane.Our results may suggest that (Fraternali et al. 2002)’s rotation curveof the anomalous gas, was measured even slightly above .We estimated the gradient value using the method-II in the region r ∈ (1 , .
5) kpc defined by the boundary points of the anomalousgas measurements. We assume the radial step size of . . Thevertical interval is z ∈ (0 . , .
4) kpc , with the step size of . .We did not use the method-I, which might have led to large errorsowing to very extended radial region (method-I is best for the regionsover the flat fragment of rotation curve, where the rotation above thedisk is roughly independent of the radial variable).The method-II gradient value is − ± − kpc − , consis-tent with Fraternali (2009)’s result of about −
12 km s − kpc − (the ertical gradients of azimuthal velocity in spiral galaxies v [ k m / s ] R[kpc]
Figure 1.
Galaxy NGC2403: disc rotation curve (solid circles) and anomalousgas rotation curve above the disk (open circles), both from Fraternali et al.(2002). The solid lines are rotation curves predicted by our model at variousheights above the mid-plane at z = 0 . , . , . , . and . . The top-most solid line is the Sofue at al (1999)’s rotation curve. v e r t i c a l g r ad i en t [ k m s - k p c - ] z[kpc] Figure 2.
Galaxy NGC2403. Gradient values predicted in the global discmodel at various heights above the mid-plane. error was not published). Fig. 2 illustrates the gradient behaviour atvarious heights above the mid-plane.
The vertical gradient in this galaxy was observed in the region z ∈ (0 . ,
4) kpc (Barbieri et al. 2005) and was determined in the sameway as for galaxy NGC 2403. At the assumed distance of . to this galaxy, the integrated mass corresponding in our model toBarbieri et al. (2005)’s rotation curve, is . × M ⊙ . In Fig.3 are shown the rotation curves of the cold disk of the anomalousgas and the rotational velocities predicted in our model at variousheights above the mid-plane, out to z = 4 kpc , a region in whichthe rotation of gas has been observed. As previously, the exact heightof measurements is unknown but is located somewhere in the inter-val z ∈ (0 . ,
4) kpc . Our predictions are consistent with the mea- v [ k m / s ] R[kpc]
Figure 3.
Galaxy NGC 4559: disc rotation curve (solid circles) and the anoma-lous gas rotation curve above the disk (open circles), both from Barbieri et al.(2005). The solid lines are rotation curves predicted by our model at variousheights above the mid-plane at z = 0 . , . , . , . , . and . . Thedashed line is the rotation curve predicted at z = 4 kpc . v e r t i c a l g r ad i en t [ k m s - k p c - ] z[kpc] Figure 4.
Galaxy NGC 4559: gradient values predicted in the global diskmodel at various heights above the mid-plane. surements; in particular, the predicted rotation at z = 4 . over-laps within the error bars with the rotation curve of the anomalousgas. This may also suggest that the gas was measured slightly above . . The method-II gradient was determined using the predictedrotation curves within the radial interval r ∈ (1 . , .
7) kpc usingthe same steps as for the rotation measurement points, and within thevertical interval z ∈ (0 . , .
7) kpc with the assumed step size of . .Method-I is not used for the same reasons as for galaxy NGC2403. The method-II gradient value is − . ± . − kpc − .This result is lower than that given by (Fraternali 2009) of about −
10 km s − kpc − (again, the error was not published). Since thisresult must also have an error, we are justified to say that our predic-tion is consistent with gradient observations. Fig. 4 shows the gradientbehaviour at various heights above the mid-plane. v [ k m / s ] z [kpc] Figure 5.
Galaxy NGC 4302: azimuthal velocity as a function of the verticaldistance from the mid-plane. The points represent the velocity values averagedover the interval r ∈ (2 . , kpc , the bars show the standard deviation in thatinterval. The rotation curve of galaxy NGC 4302 was taken from Heald et al.(2007). The integrated mass at the assumed distance of . in the disk model is . × M ⊙ . The galaxy has a large in-clination; therefore the gradient measurement of the azimuthal ve-locity was made analogously to that for NGC 891 (Heald et al.2007). Heald et al. (2007) measured the gradient in the region z ∈ (0 . , .
4) kpc , r ∈ (2 . ,
6) kpc and obtained the gradient value ofabout −
30 km s − kpc − ( − ± . − kpc − in the southside). For the gradient calculation in the disk model, we assumedthe same z − r region as in Heald et al. (2007), with assumed z -step size of . and r -step size of . . In figure 5 the az-imuthal velocity averaged over r is shown as a function of the verticaldistance. The method-I gradient is − . ± . − kpc − andthe method-II gradient is − . ± . − kpc − . These valuesoverlap within the errors with those for Heald et al. (2007)’s measure-ments. Fig. 6 shows the rotational velocity above the mid-plane pre-dicted in the global disc model and in a two-component model. Fig. 7shows the gradient behaviour at various heights above the mid-plane.NGC 4302 illustrates well the influence of the spherical componenton the gradient properties. To contrast this with the previous situationof purely a thin disc, consider a spherical halo with the density profile ρ ( r, z ) = a o b o b o + r + z comprising most of the galaxy mass, with the remaining mass at-tributed to an exponential thin disc. Now, the method-II gradient isstrongly reduced to − . ± . − kpc − . Fig. 8 shows theleast-squares fit of the two-component model. Surely a mass modelwith arbitrarily assumed halo and disc-mass profile, cannot accuratelyaccount for the rotation curve. With a maximal halo model one canaccount for the rotation perfectly, but the corresponding method-IIgradient value is even smaller, − . ± . − kpc − . Accord-ingly, for mixed mass models with various relative mass profiles, theexpected gradient value is between − . − kpc − for a morespherical-like mass distribution and − . − kpc − for a moredisc-like mass distribution. To see better the influence of a heavyhalo, let us estimate the gradient value in the mid-plane vicinity us-ing the rotation data at points r ∈ (3 . , in steps of . , V [ k m / s ] R[kpc]
Figure 6.
The measured rotation curve of galaxy NGC 4302 (Heald et al.2007, solid circles), the rotational velocity above the mid-plane predicted inthe global disk model, shown at various heights above the mid-plane in theinterval z ∈ (0 . , .
6) kpc in steps of . (solid lines), and the rota-tional velocity above the mid-plane predicted by the two-component modelpresented in Fig. 8, shown in the same interval at the same heights above themid-plane (dashed lines). v e r t i c a l g r ad i en t [ k m s - k p c - ] z[kpc] Figure 7.
Galaxy NGC 4302: gradient values predicted in the global discmodel at various heights above the mid-plane. and z ∈ (0 . , . in steps of . . The method-II gradientchanges only a little in the disk model − . ± . − kpc − ,but for a ”disk plus halo” model it reduces substantially, to − . ± . − kpc − . The greater the radial distance, the greater isthe halo influence and the less important the disc component. Atsmall z , the gradient strongly decreases for the spherical distribu-tion, whereas it is still high for the disc-like distribution. It is worthof noting the high gradient dispersion relative to the gradient value − . ± . − kpc − in this region. The gradient varies morewith r in the spherical model than in the disk model and therefore withincreasing halo and decreasing z the gradient will be more dependenton r . The observations, however, do not confirm such a behaviour.Therefore, to see the gradient features better it would be worth re- ertical gradients of azimuthal velocity in spiral galaxies V [ k m / s ] R[kpc]
Figure 8.
The least-squares fit of a two-component model of galaxy NGC 4302considered in the text, fitted to the rotation curve (Heald et al. 2007, solid line).The model rotation curve is decomposed to the exponential disc (dotted line)and to the dark halo component (dashed line). peating the rotation measurements in galaxy NGC 4302 at larger radii,closer to the mid-plane.
We use the rotation curve of galaxy NGC 5775 published in Irwin(1994). The integrated mass is . × M ⊙ at the assumed dis-tance of . . The gradient measurements of the azimuthal ve-locity was carried out for this galaxy in Heald at al. (2006a) in a veryextended region covering r ∈ (0 ,
12) kpc and z ∈ (1 . , .
6) kpc .This excludes the use of method I, since the azimuthal velocitiesat such a wide range of radii would differ too much, and the dis-persion of the gradient estimation might be larger than the gradientvalue. Therefore, we used only the method-II, which gives − . ± . − kpc − . The assumed r − z region is the same as thatused for the measurements, and the assumed steps are ∆ z = 0 . and ∆ r = 1 . . The measured gradient value given in Heald at al.(2006a) is ± − kpc − and is consistent within errors withour predicted value. Figure 9 shows the gradient behaviour at variousheights above the disk. Fig. 10 shows the rotational velocity above themid-plane predicted in the disc model. The vertical gradient in the rotational velocity of spiral galaxies can besatisfactorily explained in the framework of a global thin-disc model.The observed direction of the fall-off in rotational velocity (in thedirection vertical to the galaxy mid-plane) and the high observed gra-dient values suggest that the gross mass distribution in spiral galaxiesmight be flattened, i.e. disc-like rather than spheroidal. With this hind-sight, assuming that the disc comprises most of the dynamical massinferred from the rotation curve, we calculated the vertical gradient ofthe azimuthal velocity in the quasi-circular orbits approximation.We examined four spiral galaxies, obtaining gradient val-ues ranging from − . ± . − kpc − to − . ± . − kpc − . We observed a general tendency in the gradientbehaviour: for sufficiently large radii, the gradient values tend to beindependent of the height above the galactic mid-plane.Our predictions are consistent within errors with the measured v e r t i c a l g r ad i en t [ k m s - k p c - ] z[kpc] Figure 9.
Galaxy NGC 5775: gradient values predicted in the global diskmodel at various heights above the mid-plane. v [ k m / s ] R[kpc]
Figure 10.
Galaxy NGC 5775. From the top – the rotation curve from Irwin(1994), rotation curves predicted in disk model at various heights above themid-plane for z ∈ (0 . , .
6) kpc in steps of . kpc . gradient values. This agreement shows that the simple model ofa global thin disc is sufficient to account for the measured, of-ten high, gradient values. It is worth recalling that the global discmodel gives the high gradient values required for galaxies like NGC4302 or NGC 891 (see Jałocha et al. 2010), which are difficult toexplain by other models such as the ballistic model, the latter of-ten predicting gradient values several times lower than those ob-served. Depending on the parametrization, the ballistic model predictsthe gradient values of − − kpc − for NGC 4302, but morerealistic parameters give lower values of about − − kpc − (Heald et al. 2007), much smaller than the measured ones, evenreaching −
30 km s − kpc − , whereas the averaged rotation fall-off we obtain for this galaxy is − . ± . − kpc − .For galaxy NGC, 5775 the ballistic model gives from − . to − . − kpc − (Heald at al. 2006a), again less than in ourmodel (we obtain − . ± . − kpc − ), but, since the mea-sured gradient is comparably low ( − ± − kpc − ), the bal-listic model prediction also overlaps with the measurement. However, for galaxies with high gradient values our gradient modelling givesmuch better results than the ballistic model. We stress that here wesimply compare, using the example of particular galaxies, the resultsof our model with the results of the ballistic model (with assumed pa-rameters similar to those in the cited papers), and we do not make anyassessment of the ballistic model with respect to the concepts under-lying it.It is important to stress that, based only on the mass distri-bution precisely related to a given rotation curve, the disc model,apart from accounting for high gradient values, gives the correctlow gradient when the measured gradient is low. Of course, othermodels are also capable of giving correct results for galaxies withlow gradients. For example, in the case of galaxy NGC 5775, thedisc model predicts − . ± . − kpc − , the maximal halomodel − . ± . − kpc − , and the ballistic model gives from − . − kpc − to − . − kpc − . All of these valuesagree with the measurement − ± . It is therefore important to haveprecise rotation measurements in the mid-plane vicinity at large radii,since they could discriminate between flattened and halo-dominatedmass distributions. This is especially important for galaxy NGC 4302with its high gradient.The results of the current paper, and of our previous paper con-cerning the gradient study in the Milky Way Galaxy (Jałocha et al.2010), provide strong arguments that the gross mass distribution, atleast in some spiral galaxies, might be flattened disc-like rather thanspheroidal, unlike the suggestion of massive spheroidal dark halomodels. Another conclusion is that the gas motion above the galacticdiscs of spiral galaxies might be governed by the gravitational poten-tial of the galactic disc alone rather than by non-gravitational physicssuch as matter fountains. REFERENCES
Barbieri C. V., Fraternali F., Oosterloo T., Bertin G., Boomsma R.,Sancisi R., 2005, A&A, 439, 947Fraternali F., 2009, in Anderson J., Bland-Hawthorn J., Nordstr¨omB., eds. IAU Symp. 254, The Galaxy Disk in Cosmological Con-text. Kluwer, Dordrecht, p. 255Fraternali F., van Moorsel G., Sancisi R., Oosterloo T., 2002, AJ,123, 3124Heald G. H., Rand R. J., Benjamin R. A., Collins J. A., Bland-Hawthorn J., 2006a, ApJ, 636, 181Heald G. H., Rand R. J., Benjamin R. A., Bershady M. A., 2006b,ApJ, 647, 1018Heald G. H., Rand R. J., Benjamin R. A., Bershady M. A., 2007,ApJ, 663, 933Irwin J. A., 1994, ApJ, 429, 618Jałocha J., Bratek Ł., Kutschera M., 2008, ApJ, 679, 373Jałocha J., Bratek Ł., Kutschera, M., Skindzier P., 2010, MNRAS,407, 1689Levine E. S., Heiles C., Blitz L., 2008, ApJ, 679, 1288Oosterloo T., Fraternali F., Sancisi R., 2007, AJ, 134, 1019Sofue Y., Tutui Y., Honma M., Tomita A., Takamiya T., Koda J.,Takeda Y., 1999, ApJ, 523, 136 V [ k m / s ] R[kpc] .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.065707580859095100105110 v [ k m / s ]]