Flare in the Galactic stellar outer disc detected in SDSS-SEGUE data
aa r X i v : . [ a s t r o - ph . GA ] M a y Astronomy&Astrophysicsmanuscript no. fit c (cid:13)
ESO 2018August 1, 2018
Flare in the Galactic stellar outer disc detected in SDSS-SEGUEdata
M. L ´opez-Corredoira , , J. Molg´o Instituto de Astrof´ısica de Canarias, E-38205 La Laguna, Tenerife, Spain Departamento de Astrof´ısica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain GRANTECAN S.A., E-38712, Brea Baja, La Palma, SpainReceived xxxx; accepted xxxx
ABSTRACT
Aims.
We explore the outer Galactic disc up to a Galactocentric distance of ≈
30 kpc to derive its parameters and measure the magnitudeof its flare.
Methods.
We obtained the 3D density of stars of type F8V-G5V with a colour selection from extinction-corrected photometric dataof the Sloan Digital Sky Survey – Sloan Extension for Galactic Understanding and Exploration (SDSS-SEGUE) over 1,400 deg ino ff -plane low Galactic latitude regions and fitted it to a model of flared thin + thick disc. Results.
The best-fit parameters are a thin-disc scale length of 2.0 kpc, a thin-disc scale height at solar Galactocentric distance of 0.24kpc, a thick-disc scale length of 2.5 kpc, and a thick-disc scale height at solar Galactocentric distance of 0.71 kpc. We derive a flaringin both discs that causes the scale height of the average disc to be multiplied with respect to the solar neighbourhood value by a factorof 3 . + . − . at R =
15 kpc and by a factor of 12 + − at R =
25 kpc.
Conclusions.
The flare is quite prominent at large R and its presence explains the apparent depletion of in-plane stars that are oftenconfused with a cut-o ff at R &
15 kpc. Indeed, our Galactic disc does not present a truncation or abrupt fall-o ff there, but the stars arespread in o ff -plane regions, even at z of several kpc for R &
20 kpc. Moreover, the smoothness of the observed stellar distribution alsosuggests that there is a continuous structure and not a combination of a Galactic disc plus some other substructure or extragalacticcomponent: the hypothesis to interpret the Monoceros ring in terms of a tidal stream of a putative accreted dwarf galaxy is not onlyunnecessary because the observed flare explains the overdensity in the Monoceros ring observed in SDSS fields, but it appears to beinappropriate.
Key words.
Galaxy: structure — Galaxy: disc — Galaxy: stellar content
1. Introduction
We know many things about the structure of our Galaxy, andwe have an approximate idea of the functional shape of its com-ponents: thin and thick disc, bulge, long bar, halo or spheroid,spiral arms, ring. In the past decades, di ff erent large-area sur-veys in visible or near-infrared wavelengths have allowed us toknow our Galaxy much better. However there are some parts ofthe Galaxy that are not well known; one of them is the outer partof the disc, with Galactocentric distances beyond R =
15 kpc.We know this component is warped and flared, but the details ofits shape are still a topic to develop further.We are interested in analyse this outer part of stellar disc inour Galaxy in this paper, using available visible data of the SloanDigital Sky Survey (SDSS) in low Galactic latitude regions.Several authors (Bilir et al. 2006, 2008; Juri´c et al. 2008; Jia et al.2014) have previously used the SDSS to measure the parametersof the disc, but they have explored the high Galactic latitudes,so they could not access the outer disc. Other authors have ana-lyzed the disc using the Two Micron All Sky Survey (2MASS)either at low latitudes (L´opez-Corredoira et al. 2002, 2004) orat high latitudes using red-clump giants (Cabrera-Lavers et al.2005, 2007; Chang et al. 2011) or within the whole sky usinga given luminosity function (Polido et al. 2013), but, even us-ing near-plane regions, the depth of 2MASS is not enough to
Send o ff print requests to : [email protected] analyze the outer disc; it is dominated by stars with smaller dis-tances than the outer disc we would like to analyze.Our purpose here is to use a survey like the SDSS ( § § ff erent compo-nents of the Galaxy, but the stellar flare at R &
15 kpc is stillpoorly known. The flare of HI was investigated for instance byNakanishi & Sofue (2003), Levine et al. (2006), or Kalberla etal. (2007). It was modelled by Kalberla et al. (2007) in terms ofa dark matter ring, by Saha et al. (2009) in terms of a lopsidedhalo, or by L´opez-Corredoira & Betancort-Rijo (2009) in termsof accretion of intergalactic matter onto the disc. Another exam-ple of modelling is given by Narayan & Jog (2002), who used athree-component (HI, H and stars) disc gravitationally coupledto calculate the scale height of each component and their flares,and they derived a good prediction for the gas flare; the result ofmild stellar flaring up to R =
12 kpc was obtained in the modelby Narayan & Jog (2002), and it agreed well with the observa-tional data on flaring provided by Kent et al. (1991). The stellarflare was also observed by Alard (2000), L´opez-Corredoira etal. (2002), Yusifov (2004), Momany et al. (2006), or Reyl´e etal. (2009) for the outer disc, but limited to R .
20 kpc and withlarge uncertainties over 15 kpc or for the di ff erences between theflare of the thin and thick disc. A flare in the thick disc is posited (Hammersley & L´opez-Corredoira 2011; Mateu et al. 2011; L´opez-Corredoira et al.2012), and this may constitute an explanation for the Monocerosring instead of the tidal stream hypothesis. This is an additionalreason to pursue this research: to know whether we are able tofit our star counts without needing some new extragalactic com-ponent. The fact that a flared thin + thick disc alone fits our data( § §
5) gives a positive answer, which is discussed and com-pared with other works of the literature in § §
7, together withthe considerations derived from the morphological informationobtained in this paper.
2. Data
The SDSS (Sloan Digital Sky Survey)-DR8 release (Aiharaet al. 2011) contains imaging of 14 555 deg of the sky infive filters (u,g,r,i,z), with a completeness higher than 95% for m g ≤ .
2. Within this survey, the subsample SEGUE (SloanExtension for Galactic Understanding and Exploration) includesmany Galactic plane regions. For this paper, in which we areinterested to explore the outer disc, we used the regions with | ℓ | > ◦ (to avoid the regions of the inner Galaxy), | b | ≤ ◦ (dominated by disc stars), taking all the point-like SDSSsources with the flags and constraints for the g and r filters:((flags r & 0x10000000) ! = = =
0) or (psfmagerr r¡ = =
0) or (flags r &0x1000) = r larger than 0.2, stars with poor detection, or cases withcosmic rays. All the magnitudes were corrected for extinction bythe SDSS team using the Galactic extinction model of Schlegelet al. (1998). In total, we cover an area of 1 745 deg . As we ex-plain in the next section, we reduced this area by avoiding theregions very close to the plane with high extinction to reduce theerrors due to the correction of extinction.
3. Method
The simplest method of determining the stellar density along aline of sight in the disc is by isolating a group of stars with thesame colour and absolute magnitude M within a colour magni-tude diagram. This allows the luminosity function to be replacedby a constant in the stellar statistics equation and the di ff eren-tial star counts for each line of sight, A ( m ), can be immediatelyconverted into density ρ ( r ): A ( m ) ≡ dN ( m ) dm = ln ωρ [ r ( m )] r ( m ) , (1) r ( m ) = [ m − M + / , where ω is the area of the solid angle in radians and r is thedistance in parsecs.In the near-infrared, red-clump giants have been success-fully used as standard candles, particularly for the innermost15 kpc from the Galactic centre (L´opez-Corredoira et al. 2002).However, red-clump stars in the outer disc at the distance of in-terest would appear at m k ≥
14, where the local dwarfs withthe same colour would completely dominate the counts (L´opez-Corredoira et al. 2002). Therefore, we have to use somethingdi ff erent here. g l=183 o , b=21 o Fig. 1.
Extinction-corrected colour-magnitude diagram for theregion ℓ = ◦ , b = ◦ . The region between the dashed linescontains F8V-G5V dwarfs.For our extinction-corrected areas, it is possible to use starcounts in visible. An examination of the HR diagram shows thatwhen the extinction is low, the late F and early G dwarfs canbe isolated using colour with only minimal contamination fromother sources with the same colour, but di ff erent absolute mag-nitudes. For this work we selected the sources between F8V andG5V. There will be su ffi cient stars detected in the outer Galaxyto give meaningful statistics, which would not be the case if asmaller range in absolute magnitudes were used. Earlier sourceswere not included as these sources would belong to a youngerpopulation with a far lower scale height; the absolute magni-tude also changes far more rapidly with colour. Later sourceswould have significant giant contamination, and again the abso-lute magnitude changes more rapidly with colour.The F8V-G5V dwarfs have a range of g − r of 0.36 to 0.49and a range of absolute magnitudes M g = m g . . M g = ff ect, and theabove approximation remains valid: see L´opez-Corredoira et al.2002, § ff erence between a nar-row Gaussian distribution and a Dirac delta: it leads to an errorin the scale length of the order of 2% assuming an r.m.s. in theGaussian distribution of 0.3 mag.; although the application ofL´opez-Corredoira et al. is for red-clump giants, it is valid for anykind of population. We did not take into account the possibilitythat some of these stars might indeed be binary systems. SeeSiegel et al. (2002) or Bilir et al. (2009, § ff ects of a high ratio of binary stars to derive the parametersof the disc: they may produce ∆ M g ∼ . § ff erence for the most extreme cases of lower metallicity for the highest values of R and z ([ Fe / H ] ∼ ∆ M R ≈ . R − I = .
38, which isthe corresponding transformation from SDSS to Johnson filters(Jordi et al. 2006) of the color of our population with average( r − i ) = .
13 (Bilir et al. 2009)]. The variation of M g is similar(Juri´c et al. 2008, Fig. 3). Therefore, the fact that we did not takethis metallicity gradient into account may produce an overesti-mate of the distance of the stars at the highest R or z of ∼
20% .Its e ff ects are explored in § of our SDSS data are 6 506 360 stars with g − r (corrected for extinction) between 0.36 and 0.49. Thesource densities in these regions are expected to be very low,therefore we require regions of more than 0.5 square degrees ofsky to be covered to provide su ffi cient counts to give reason-able statistics. We divided them into multiple space bins with ∆ ℓ × ∆ b = ◦ × ◦ , ∆ m g = . . < m g < .
85, and, sincewe know their distances and coordinates (hence, we know theirposition in 3D space), given the di ff erential star counts A ( m ), wecan derive through Eq. (1) the density ρ ( R , φ, z ) (in cylindricalGalactocentric coordinates). This average stellar density is plot-ted in Fig. 2, excluding the bins with | b | < ◦ .We excluded these in-plane regions (bins with | b | < ◦ ),which reduced our area to 1 396 deg , to reduce the errorsin the extinction correction: with extinctions of h A ( g ) i . . | b | ≥ ◦ (Schlegel et al. 1998), E ( g − r ) = . A ( g ) and a relative uncertainty of the redden-ing of ∼
10% (Schlegel et al. 1998; M¨ortsell 2013), we de-rive relative errors of ∆ ( g − r ) . .
04 for each region. With a dM g d ( g − r ) ≈ . − . . − . = .
2, we compute uncertainties in the distancedetermination of . §
7. At present, we do not carry out a deconvolve the line of sightdistribution with the spread of distances because of the r.m.s. ofthe reddening, which would provide a better determination of themorphology, because one would need a very accurate expressionof the r.m.s. of the extinction, which is not available (see also dis-cussion in § ff the plane weassumed that this extinction is local.
4. Fitting free parameters of a disc model
After deriving the stellar density ρ ( R , φ, z ), we can fit a discmodel to obtain its best free parameters. We used the followingmodel, which contains flared axysymmetric thin + thick discs: ρ disc ( R , z ) = ρ thin ( R , z ) + ρ thick ( R , z ) , (2) ρ thin ( R , z ) = ρ ⊙ h z , thin ( R ⊙ ) h z , thin ( R ) exp R ⊙ h r , thin + h r , hole R ⊙ ! × exp − Rh r , thin − h r , hole R ! exp − | z | h z , thin ( R ) ! ,ρ thick ( R , z ) = f thick ρ ⊙ h z , thick h z , thick ( R ) exp R ⊙ h r , thick + h r , hole R ⊙ ! × exp − Rh r , thick − h r , hole R ! exp − | z | h z , thick ( R ) ! , ρ ) −11 −10 −9 −8 −7 −6 −5 −4 −3 log (star pc −3 R (kpc) −10−50+5+10+15−15 z ( kp c ) Fig. 2.
Average stellar density of F8V-G5V stars from availableSDSS data as a function of the cylindrical coordinates R (kpc), φ (deg.), z (kpc) for the range 0 < R <
30 kpc, | z | <
15 kpc,constrained within Galactic latitudes 8 ◦ ≤ | b | ≤ ◦ . The circlestands for R = z = φ (for the Sun, it would be φ = h z , thin ( R ) = h z , thin ( R ⊙ ) + X i = k i , thin ( R − R ⊙ ) i , (3) h z , thick ( R ) = ( h z , thick ( R ft ) R < R ft h z , thick ( R ft ) (cid:16) + P i = k i , thick ( R − R ft ) i (cid:17) R ≥ R ft . This expresses an exponentially flared disc (L´opez-Corredoiraet al. 2002) with a hole or deficit of stars in the inner in-planeregion (L´opez-Corredoira et al. 2004). Since we did not exam-ine the inner disc, we kept h r , hole constant: h r , hole = .
74 kpc(L´opez-Corredoira et al. 2004). We also kept constant the ratio ofthick to thin disc stars in the solar neighbourhood, f thick = . length of the thin disc h r , thin , the scale height of the thin discat solar Galactocentric distance h z , thin ( R ⊙ ), the scale length ofthe thick disc h r , thick , the scale height of the thin disc at solarGalactocentric distance h z , thick ( R ⊙ ), the two parameters definingthe flare of the thin disc k , thin and k , thin , the two parametersdefining the flare of the thick disc k , thick , k , thick , and the scale atwhich the thick disc flare starts, R ft .Bovy et al. (2012) suggested that there is no thick disc thatsensibly can be characterized as a distinct component becausemono-abundance sub-populations, defined in the [alpha / Fe]-[Fe / H] space, are well described by single-exponential spatial-density profiles in both the radial and the vertical direction;therefore, any star of a given abundance would be associ-ated with a sub-population of a given scale height and therewould be a continuous and monotonic distribution of disc thick-nesses. This result is contradicted by other authors however (e.g.,Haywood et al. 2013), who showed a bimodal distribution in the[alpha / Fe]-[Fe / H] space, or that thin and thick disc can be distin-guished because of the rotation and vertical dispersion, age, orother indicators of metallicity. Here we assumed that there aremorphologically two discs and do not engage in the discussionabout their populations.We also included a halo, but did not try to fit any of its pa-rameters; we just took the fixed density distribution provided byBilir et al. (2008) using SDSS data. Its contribution to our countsis lower than the disc contribution (20% for the highest R and z of our range). ρ halo ( R , z ) = . × − ρ ⊙ exp (cid:20) . (cid:18) − (cid:16) R sp R ⊙ (cid:17) / (cid:19)(cid:21) ( R sp / R ⊙ ) / , (4) R sp = p R + . z . Our model does not take the stellar warp into account. Thewarp moves the position of the Galactic plane: the plane canbe shifted away from the expected position of b =
0. It can beclearly seen in the counts within a few kpc of the Sun (L´opez-Corredoira et al. 2002, Momany et al. 2006, Reyl´e et al. 2009)and its e ff ect is stronger towards the outer edge of the disc. Themodels of the warp are normally simple, based on tilted rings,and in general represent the star count well. Nonetheless, itsaverage e ff ect is not important for | b | ≥ ◦ (see Fig. 15 / topof L´opez-Corredoira et al. 2002 with ratio of positive / negativecounts close to unity): some regions are overdense in the posi-tive latitudes with respect to the negative latitude for the northernwarp region (the southern warp is not visible with SDSS), but onaverage for our fit both positive and negative latitude data canceleach other out and only a minor second-order e ff ect may remainin the fit, which we neglected. In Fig. 2, no significant signs ofthe northern warp are detected (the southern warp is not visiblein our SDSS data). A more detailed discussion on the influenceof the warp is given in the last paragraph of the next subsectionand in § We used the selected regions for the fit, excluding bins with adensity lower than 3 × − star pc − . For the fit of the scalelengths and scale heights ( h r , thin , h z , thin ( R ⊙ ), h r , thick , h z , thick ( R ft )),we used the regions with | z | ≤ R <
15 kpc, i.e., we neglectthe flare; after we obtained these four parameters we fitted therest of them for the flare in the regions with 1 . < | z (kpc) | ≤ . −40 −20 0 20 40 60 80 100 120φ (deg)−1.0−0.50.00.51.0 ( ρ o b s − ρ m o d ) / ρ o b s −40 −20 0 20 40 60 80 100 120φ (deg)−1.0−0.50.00.51.0 ( ρ o b s − ρ m o d ) / ρ o b s Fig. 4.
Residuals of ρ with respect to the best fit for the dataextracted from the star counts of F8-G5V stars in the SDSS. R > . | z | < . z >
0. Botton: z < . < R (kpc) <
30. Within these ranges, the Galaxy isdominated by the disc rather than the halo star counts.The parameters that produce the best weighted fit are givenin Table 1. Figs. 3 and 4 show how this model fits the data.The error bars (1 σ ) of these parameters were derived usingthe method of Avni (1976) for four free parameters (the firstfit of the scales) and five free parameters (the fit of the flare): ∆ χ = .
72 and 5.89 respectively; where we normalized χ suchthat χ = N (number of data points) to take into account thatthe Poissonian error is only a small part of the total error.It can be observed how the flare becomes strong for high R and z . In Fig. 5, we plot the functional shape of h z , thin ( R ) and h z , thick ( R ) in this best-fit model. The error bars for the flare ampli-tude are high, but they exclude the non-flared solution. Note thatthe errors for the thin disc include the free variation of the thickdisc to compensate for the respective variations within those er-rors and vice versa. If we fixed one of the discs, the errors bars ofthe other disc would be much lower than the case with its varia-tion. When we combine both discs, we also expect to reduce theerror of the average disc (the errors of the average are also de-rived through the χ analysis), from which we conclude that weneed a flare to fit our data, although distinguishing between thinand thick disc component is only possible with low significance.It is clear from Fig. 3 that an extrapolation of the disc fitted at R <
15 kpc without flare does not work at larger R : the shape ofthe density presents a more complex form than the dashed linein the log-linear plot of Fig. 3.We can see how the variation of the density with z for high R becomes quite weak: green extends to almost all regions with -9 -8 -7 -6 -5 -4 -3 ρ ( ⋆ / p c ) |z|=0.5 kpc 5 10 15 20 25 30R (kpc)10 -9 -8 -7 -6 -5 -4 ρ ( ⋆ / p c ) |z|=1.1 kpc5 10 15 20 25 30R (kpc)10 -9 -8 -7 -6 -5 -4 ρ ( ⋆ / p c ) |z|=1.7 kpc 10 15 20 25 30R (kpc)10 -9 -8 -7 -6 ρ ( ⋆ / p c ) |z|=2.3 kpc5 10 15 20 25 30R (kpc)10 -9 -8 -7 -6 -5 ρ ( ⋆ / p c ) |z|=2.9 kpc 5 10 15 20 25 30R (kpc)10 -9 -8 -7 -6 -5 ρ ( ⋆ / p c ) |z|=3.5 kpc Fig. 3.
Best fit (solid line) for the data of ρ ( R , z ) extracted from the star counts of F8-G5V stars in the SDSS. The dashed line standsfor the best fit of the disc within R <
15 kpc without any flare, and extrapolated for R ≥
15 kpc. Error bars stand only for Poissonianerrors in the star counts and do not include other possible factors such as errors in the extinction. Note that in the lowest | z | binsthere are no points at high R because of our constraint of | b | > ◦ (see Fig. 2). Table 1.
Disc parameters of the best fit (see text). Note that there are nine independent parameters in the fit; the last six parametersdepend on the previous ones. The constraint | φ | ≤ ◦ explores the region where the warp amplitude is very low. The metallicitygradient is modelled following Eq. (5). parameter All data, no grad. metal. | φ | ≤ ◦ , no grad. metal. All data, with grad. metal. h r , thin (kpc) 2 . + . − . h z , thin ( R ⊙ ) (kpc) 0 . + . − . h r , thick (kpc) 2 . + . − . h z , thick ( R ⊙ ) (kpc) 0 . + . − . k , thin (kpc − ) -0.037 0.090 -0.190 k , thin (kpc − ) 0.052 0.043 0.070 k , thick (kpc − ) 0.021 1.448 0.000 k , thick (kpc − ) 0.006 -0.055 0.010 R ft (kpc) 6 . + . − . h z , thin (15 kpc) / h z , thin ( R ⊙ ) 3.3 + . − . h z , thin (20 kpc) / h z , thin ( R ⊙ ) 8.1 + . − . h z , thin (25 kpc) / h z , thin ( R ⊙ ) 15.5 + . − . h z , thick (15 kpc) / h z , thick ( R ⊙ ) 1.5 + . − . h z , thick (20 kpc) / h z , thin ( R ⊙ ) 2.3 + . − . h z , thick (25 kpc) / h z , thick ( R ⊙ ) 3.4 + . − . high R in Fig. 2, and the few blue areas (very low densities) rep-resent the few fluctuations due possibly to some errors in the ex-tinction or warp presence (because the lowest latitudes are morea ff ected by these fluctuations). This is precisely what we obtainin our fit, which indicates the existence of a conspicuous flare: avery significant increase of the scale height of the disc both forthe thin and thick discs. In Fig. 3, we also observe the e ff ect ofthe flare: at R ≈
15 kpc and high z the density becomes almostconstant with little decrease with R because of a combination of the exponential decrease with radius and the increase of thedensity at high z due to the increase of the scaleheight.Fig. 4 shows some residuals in the observed stellar densitywith respect to the best fit. If the northern warp e ff ect were sig-nificant, it would produce an excess density at 60 ◦ . φ . ◦ at z > z < ff erent e ff ects such as metallicitygradients, a di ff erent contamination of the halo than expected (at h z ( R ) ( kp c ) Average discThin discThick disc
Fig. 5.
Scale height of the thin and thick discs according to ourbest fit. The average is defined as the − (cid:16) d ln ρ disc ( R , z = d | z | (cid:17) − and takesinto account the increasing ratio of thick disc stars outwards. h z ( R ) ( kp c ) Average disc, all data, no grad. metal.Average disc, | φ |<30 o , no grad. metal.Average disc, all data, with grad. metal. Fig. 6.
Scale height of the average defined as in Fig. 5 for thebest fits of Table 1.high | z | ), some irregularities in the distribution of the extinction,some degree of lopsidedness, or a variation of h z with φ (L´opez-Corredoira & Betancort-Rijo 2009).
5. Exploring the effects of the warp and metallicitygradients
We carried out two numerical experiments to determine the ef-fects of the warp and the metallicity gradients. Since there arelarge uncertainties for both, we do not expect to derive directconclusions from the results of the following best fits, but theyshould serve as an estimate of the typical variation in our param-eters under these changes.Warp: instead of the whole data set, we used only those with | φ | ≤ ◦ , where the amplitude of the warp is lowest. Wefitted ρ ( R , z ) in the same way as before and the parametersobtained are those given in Table 1 and Fig. 6. They are com-patible with the previous values, confirming our guess thatthe warp does not strongly change our results. Metallicity gradient: we attributed to each star a Galactocentricdistance R ′ , φ ′ and vertical position z ′ corresponding to theircoordinates and a distance r ′ ( m ) given by r ′ ( m ) = [ m − M ′ + / , (5) M ′ = . − . ∆ [ Fe / H ] , ∆ [ Fe / H ]( R ′ [kpc] , z ′ [kpc]) = ( − . R ′ − R ⊙ ) − . | z ′ | , R ′ ≤ f ( z ′ ) − , R ′ > f ( z ′ ) , f ( z ′ ) = R ⊙ + . − . | z ′ | . As said in §
3, this stems from the variation of the abso-lute magnitude of F8V-G5V stars estimated by Siegel et al.(2002) for a variation of matallicity with respect to the Sun[ ∆ M R ≈ . ∆ [ Fe / H ] = − R − I = .
38, whichis the corresponding transformation from SDSS to Johnsonfilters (Jordi et al. 2006) of the color of our population withaverage ( r − i ) = .
13 (Bilir et al. 2009); ∆ M g ≈ ∆ M R (Juri´cet al. 2008, Fig. 3)], and considering a variation of metal-licity from the combination of radial and vertical gradientsof metallicity for the disc given by Rong et al. (2001) andAk et al. (2007); assuming the same gradient for thin andthick discs and that it remains constant for the farthest discfor a metallicity lower than the solar one. This is probablyan overestimation of | ∆ [ Fe / H ] | , which should be lower thanunity at least for the thin disc (Andreuzzi et al. 2011), butit serves as a limit of the strongest e ff ect of this gradient.Given that ∆ [ Fe / H ]( R ′ , z ′ ) depends on the position and theposition depends on this variation of metallicity, we carriedout the calculation with an iterative process.Then, we fitted ρ ( R ′ , z ′ ) in the same way as before and theparameters obtained are those given in Table 1 and Fig. 6.The results for the flare parameters are totally compatiblewithin the error bars with those obtained without taking intoaccount any gradient of metallicity. As said in §
3, a lowermetallicity in the farthest parts of the disc may overestimatethe distance of the stars (unfortunately, we do not know byhow much since there are no accurate measurements of themetallicity of stars at those Galactocentric distances; we es-timate an error of no more than 20% [ §
6. Comparison with other works
The scale lengths and scale heights can be compared with thosein other publications, although mainly for low R . Table 2 givessome of the values of the literature. Jia et al. (2014, Table 1) re-ported many other values. In general, the relative trends betweenthin and thick disc are similar in all these papers and our results,but there is some variation in the absolute scales that might arisebecause of di ff erent observed populations, di ff erent techniquesof distance estimation, or di ff erent regions of application, apart,of course, from possible systematic errors. Jia et al. (2014) haveshown how the parameters, especially the scale height of the thindisc, depend on the absolute magnitude of the main-sequencestars used, indicating that di ff erent populations have di ff erent ve-locity dispersions. The numbers of Juri´c et al. (2008) are some-what higher than ours, possibly because they represent a range of Table 2.
Some values of the scale lengths and scale heights from the literature (units in kpc), derived either with SDSS (visible) or2MASS (near infrared red).
Reference source spatial range h r , thin h z , thin ( R ⊙ ) h r , thick h z , thick ( R ⊙ )L´opez-Corredoira et al. (2002) 2MASS low Gal. latitudes 3.3 0.28 – –Cabrera-Lavers et al. (2005) 2MASS high Gal. latitudes – 0.27 – 1.1Bilir et al. (2006) SDSS interm. Gal. latitudes 1.9 0.22 – –Cabrera-Lavers et al. (2007) 2MASS interm. Gal. latitudes – 0.19 – 0.96Bilir et al. (2008) SDSS high Gal. latitutes – 0.19 – 0.63Juri´c et al. (2008) SDSS r < . + SCUSS interm. Gal. latitude – 0.20 – 0.60 smaller R , or possibly because of their method of distance deter-mination. Bilir et al. (2006) showed that the scale length dependson Galactic longitude. Here, we derived its average value but didnot examine any dependence on Galactic coordinates.The flare was previously observed by Alard (2000), L´opez-Corredoira et al. (2002), Yusifov (2004), Momany et al. (2006),or Reyl´e et al. (2009). Polido et al. (2013) also introduced aflare in their model, but did no fit their parameters. The valuesof the scale height at high R from our fit (see Fig. 5) are high:with a thin disc scaleheight around 0.8 kpc at R =
15 kpc, lowerthan the extrapolation of L´opez-Corredoira et al. (2002) or thatof Yusifov (2004), and higher than the values of Alard (2000),Momany et al. (2006) and Reyl´e et al. (2009). Between 20-25kpc we derive a thin disc scale height of 2-4 kpc, which is alsohigher than the values by Momany et al. (2006) and Reyl´e et al.(2009) (for the rest of the authors, there are no values at suchhigh values of R ). For the thick disc, there are few or no studiesto compare our studies with: there is a hint with unconclusiveresults by Cabrera-Lavers et al. (2007), which was constrainedwithin R <
10 kpc and obtained the opposite sign in the in-crease of scale height for the solar neighbourhood; it is interest-ing that the values of the flare in the thick disc that we obtainedare those that are needed to explain the Monoceros ring in termsof Galactic structure (Hammersley & L´opez-Corredoira 2011).Note that in our results at R &
17 kpc the thin disc becomes“thicker” than the so-called thick disc. This should motivate usto change the nomenclature: maybe instead of thin disc + thickdisc we should speak of disc 1 and disc 2. In any case, our modelis just an exercise of fitting stellar densities and, within the errorbars we are unable to see which disc has a larger scale height atlarge Galactocentric radius. We do not aim here to distinguishamong di ff erent populations. We see from our results that weneed a flare to interpret the global density, but, with the presentanalysis, we are not able to distinguish the populations that areflared at high R . We expect that at high R there should be no sig-nificant di ff erence between both discs thicknesses and we haveonly one mixed component. An average disc as plotted in Fig. 5represents this average old population of type F8V-G5V stars. Inthis average disc, at high R the thick disc (or better: disc 2) has ahigher ratio of stars than at R ⊙ : while at R ⊙ it is 9% of the starsof the thin disc (or better: disc 1), at R =
25 kpc it is 50% of thestars of the thin disc, because the scale length of the thick disc islarger than that of the thin disc, and consequently the fall-o ff ofthe density is slower.The theoretical explanation of the observed features is be-yond the scope of this paper. The origin of the thick disc andits flare need to be predicted by a model that aims to under-stand these observations. One of the hypotheses for the forma-tion of thick discs is through minor mergers, which predicts a scale length of the thick disc larger than the scale length of thethin disc, a flare of increasing scale height in the thick disc, anda constant scale height for the stellar excess added by the merger(Qu et al. 2011). If we assumed that the mixture of the thin discof the primary galaxy plus the stellar excess due to the accretedminor galaxy produces what we observe as the thick disc, ourresults would be fitted by those predictions. Nonetheless, ouranalysis is very rough, and without a necessary study of the pop-ulations we are unable to confirm this scenario.
7. Discussion and conclusions
Our method of deriving the 3D stellar distribution is quitestraightforward, although it may contain some errors due mainlyto inappropriate extinction estimate (some systematic error in thescales may be produced, but we do not expect it to exceed 18%;see § ff ect of the warp. Even tak-ing into account these factors, we have not observed features thatsuggested that the derived morphology might be very di ff erent:the scales might change slightly, but the presence of the flare isunavoidable.Our results show that the stellar density distribution of theouter disc (up to R =
30 kpc) is well fitted by a component ofthin + thick disc with flares (increasing scale height outwards).From our diagrams, it is clear that there is no a cut-o ff of thestellar component at R = −
15 kpc as stated by Ruphy et al.(1996) or Minniti et al. (2011); we only examined o ff -plane re-gions ( | b | ≥ ◦ ) so we cannot judge what occurs in the in-planeregions, but from our results and by interpolating the results inthe z -direction one can clearly conclude the reason why Ruphyet al. or Minniti et al. appreciated a significant drop-o ff of starsat R = −
15 kpc: the flare becomes important at those galac-tocentric distances, and consequently, the stars are distributed ina much wider range of heights, producing this apparent deple-tion of in-plane stars. Indeed, our Galactic disc does not presenta cut-o ff there but the stars are spread in o ff -plane regions, evenat z of several kpc up to a Galactocentric distance of 15 scalelengths. Assuming that our fit is correct, for a constant lumi-nosity function along the disc, the flux of the Milky Way seenobserved face-on would follow a dependence (from Eq. (2), ne-glecting the hole of the inner disc, which is totally insignificantfor R > R ⊙ ) F ( R ) ∝ Z ∞−∞ dz ρ disc ( R , z ) (6) ≈ ρ ⊙ " exp − Rh r , thin ! + f thick exp − Rh r , thick ! . It is clear that in F ( R ) there is no radial truncation in the exploredrange, and if this F ( R ) did not represent our Galaxy, we wouldnot derive as good a fit as we did. This does not mean that radialtruncations are not possible in spiral galaxies: there are othergalaxies in which it is observed (van der Kruit & Searle 1981;Pohlen et al. 2000), but the Milky Way is not one of them.The smoothness of the observed stellar distribution also sug-gests that there is a continuous structure and not a combinationof a Galactic disc plus some other substructure or extragalac-tic component. The hypothesis of interpreting the Monocerosring in terms of a tidal stream of a putative accreted dwarfgalaxy (Sollima et al. 2011; Conn et al. 2012; Meisner et al.2012; Li et al. 2012) is not only unnecessary (as stated byMomany et al. 2006; Hammersley & L´opez-Corredoira 2011;L´opez-Corredoira et al. 2012), but appears to be quite inappro-priate: we see in Fig. 2 no structure overimposed on the Galacticdisc. Instead, the observed flare explains the overdensity in theMonoceros ring observed in the SDSS fields (Hammersley &L´opez-Corredoira 2011).Given these results, it would be interesting for future worksif the dynamicists explained the existence of the observed flaresin the Galactic disc, and further observational research on thespectroscopical features of the disc stars at very high R and high z is necessary to know more about the origin and evolution ofthis component. Acknowledgements.
Thanks are given to A. Cabrera-Lavers, S. Zaggia and theanonymous referee for useful comments that helped us to improve this paper.MLC was supported by the grant AYA2012-33211 of the Spanish Ministry ofEconomy and Competitiveness (MINECO). Thanks are given to Astrid Peter(language editor of A&A) for proof-reading of the text.Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation,the Participating Institutions, the National Science Foundation, and theU.S. Department of Energy O ffi ce of Science. The SDSS-III web site ishttp: // / . SDSS-III is managed by the Astrophysical ResearchConsortium for the Participating Institutions of the SDSS-III Collaborationincluding the University of Arizona, the Brazilian Participation Group,Brookhaven National Laboratory, Carnegie Mellon University, University ofFlorida, the French Participation Group, the German Participation Group,Harvard University, the Instituto de Astrofisica de Canarias, the MichiganState / Notre Dame / JINA Participation Group, Johns Hopkins University,Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics,Max Planck Institute for Extraterrestrial Physics, New Mexico State University,New York University, Ohio State University, Pennsylvania State University,University of Portsmouth, Princeton University, the Spanish Participation Group,University of Tokyo, University of Utah, Vanderbilt University, University ofVirginia, University of Washington, and Yale University.
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