High Galactic latitude runaway stars as tracers of the spiral arms
aa r X i v : . [ a s t r o - ph . GA ] F e b Mon. Not. R. Astron. Soc. , ?? – ?? (2012) Printed 1 November 2018 (MN L A TEX style file v2.2)
High Galactic latitude runaway stars as tracers of thespiral arms
M. D. V. Silva ⋆ and R. Napiwotzki Centre for Astrophysics Research, STRI, University of Hertfordshire, College Lane, Hatfield AL10 9AB
ABSTRACT
A direct observation of the spiral structure of the Galaxy is hindered by our positionin the middle of the Galactic plane. We propose a method based on the analysis of thebirthplaces of high Galactic latitude runaway stars to map the spiral arms and deter-mine their dynamics. As a proof of concept, the method is applied to a local sampleof early-type stars and a sample of runaways stars to obtain estimates of the patternspeed (Ω p , local = 20 . ± . − kpc − and Ω p , runaway = 21 . ± . − kpc − )and the spiral arm’s phase angle ( φ local = − . ◦ ± . ◦ and φ runaway = − . ◦ ± . ◦ ).We also estimate the performance of this method once the data gathered by Gaia , inparticular for runaway stars observed on the other side of the Galaxy, is available.
Key words: stars: kinematics – stars: early-type – Galaxy: structure
The spiral arms are readily visible in photographic imagesof face-on spiral galaxies as exemplified by the galaxy M51.They are usually traced, in visible light, by young hot starsand respective H II regions or, in other words, are the pre-ferred sites of star formation. Our position within the Galac-tic disc does not permit a direct observation of the spiralarms as they appear superimposed along our line of sightand are heavily obscured by dust. Nevertheless, many at-tempts have been made to determine their precise geometryand dynamics. The results obtained in different studies ofthe spiral structure of our Galaxy, using a variety of tracers(gas, dust and stars), were summarised and synthesised in ametastudy by Vall´ee (2008). The picture that emerges is thatof a 4-arm spiral structure with pitch angle p = 12 . ◦ . How-ever, some controversy remains regarding the spiral struc-ture properties, in particular the number of arms and theirshape (pitch angle): L´epine et al. (2011), for example, claimthe existence of “square”-shaped spiral arms, correspondingto the 4:1 resonance, instead of the usual logarithmic spiralarms. In any instance, only nearby segments of the spiralarms appear to be well defined in most studies (e.g. Fig. 3of Russeil 2003 and Fig. 4 of L´epine et al. 2011), whereas thespiral structure behind the Galactic centre is virtually un-known ( Zona Galactica Incognita according to Vall´ee 2005).The determination of the spiral pattern speed Ω p is aclosely related problem that has been the subject of manydifferent studies. The different methods used to estimate thevalue of Ω p are reviewed by Gerhard (2011). The most direct ⋆ E-mail: [email protected] method consists in finding the birthplaces of open clustersor individual young stars by computing their orbits (e.g.Amaral & Lepine 1997; Fern´andez, Figueras & Torra 2001;Dias & L´epine 2005; Naoz & Shaviv 2007).In this paper we propose the use of a different tracer ofthe spiral arms structure: the Galactic population of highGalactic latitude runaway stars. It has already been shownthat high quality astrometry, in particular that obtainedwith
Hipparcos (see ? ), permits the recovery of the birth-places of runaway stars thus, the higher quality promisedby the Gaia mission, will allow the analysis of a muchlarger volume of the Galaxy. Although it is more commonto use larger structures as tracers, as the aforementionedH II regions (Russeil 2003) or open clusters (Dias & L´epine2005), single stars were also previously used (e.g. Chepeidstars by Majaess, Turner & Lane 2009 and O-B stars byFern´andez et al. 2001). However, “normal” early-type starsare confined to the disc and so are heavily affected by inter-stellar reddening, whereas high Galactic latitude runawaystars are not. Moreover, since runaway stars can travel largedistances (several kpc) away from their birthplaces, it is pos-sible to map portions of the spiral arms that are further awaythan the runaway stars.Nevertheless, the number of high Galactic latitude run-away stars and the spatial distribution of their birthplacesdoes not permit yet the application of this method withhigh accuracy, however the number of known stars of thistype will increase by at least one order of magnitude withthe
Gaia satellite mission, which will deliver high accuracyastrometric data. Thus, in this paper our objective is to con-duct a proof of concept study to show how the use of thebirthplaces of runaway stars may be used to trace the spiral c (cid:13) M. D. V. Silva and R. Napiwotzki arms so it can be applied to the larger population of runawaystars observed by
Gaia .In order to accomplish this objective we apply a vari-ation of the method used in other studies (usually withopen clusters as tracers, e.g. Amaral & Lepine 1997 andDias & L´epine 2005). This method consists in using kine-matical information about an object to compute its orbitin the Galaxy’s potential and thus its birthplace (given anage estimate). Since objects with different ages indicate theposition of the spiral arms in different time instants, thismethod also provides an estimate of the pattern speed Ω p .Because of the shortcomings of the runaway stars sample weapply this method to a sample of local early-type stars as itprovides a more accurate test (since it is larger and we mayuse the more accurate Hipparcos astrometry). After show-ing the results obtained with these two samples (local andrunaways), we discuss the expected improvement in perfor-mance when this method is applied to
Gaia data.
The sample of runaway stars under consideration in this pa-per is the sample presented in Silva & Napiwotzki (2011).This is a sample of 96 stars covering the brightness range6 . < V < .
5, and an altitude above the Galactic plane of0 . − . − , according to Fern´andez et al.(2001), which corresponds to a maximum systematic effectin the position of the birthplaces of ∼
600 pc for the olderstars (but less for the younger ones); this method is meantto be used with high Galactic latitude runaway stars and itwas already shown in Silva & Napiwotzki (2011) that thesestars are not very sensitive to changes in the disc Galac-tic potential, given that they leave the Galactic plane withlarge ejection velocities. The 51 birthplaces which were de-termined with a precision < . x and y direc-tions are shown in Fig. 1.In principle it would be possible to fit directly these 51points to determine the feasibility of the proposed methodto determine the spiral arms shape, position and patternspeed, however this sample is small and the distances andproper motions have relatively large measurements errors(no parallaxes). For this reason a second sample, consistingof local early-type main sequence stars, was selected allowingfor a better determination of birthplaces. This sample andthe runaway stars sample will be referred to as sample Aand sample B, respectively.Sample A was selected from a cross-match of the Hipparcos (new reduction by van Leeuwen 2007) andthe Bright Star (Hoffleit & Jaschek 1991) catalogues. Theselection criteria was: B − V colour < .
05 (corre-sponding to an effective temperature < (cid:0) (cid:1) (cid:2) X (kpc) (cid:3) (cid:4) (cid:5) Y ( k p c ) Figure 1.
Birthplaces of the runaway stars in the Galactic plane(frame centered in the centre of the Galaxy) including 1 σ errorbars. The concentric circles indicate distances of 5, 8 and 10 kpcto the centre. (cid:6) (cid:7) (cid:8) (cid:9) (cid:10) (cid:11) (B (cid:12) V) (cid:13) (cid:14) (cid:15) (cid:16) (cid:17) (cid:18) M V (cid:19) (cid:20) (cid:21) (cid:22) (cid:23) (cid:24)
12 M (cid:25)
15 M (cid:26)
20 M (cid:27)
Figure 2.
Deredenned colour-magnitude diagram of local early-type stars (sample A) with respective error bars. The evolution-ary tracks obtained through the bolometric corrections of Flower(1996) from the theoretical tracks by ? are also plotted. Napiwotzki, Sch¨onberner & Wenske 1993), and parallax > ∼
200 pc for stars with M V = 0). After the removal of starswith spectral type later than B (according to the Bright Starcatalogue classification) and of spectroscopic binaries, thesample was reduced to 516 stars. Note that the colours werederedenned using a procedure based on Str¨omgren uvbyβ photometry (Napiwotzki et al. 1993). The uvbyβ photome-try was obtained from the Hauck & Mermilliod (1998) cata-logue. In Fig. 2 the colour-magnitude of sample A is shown. The age of each individual star in sample A was esti-mated using a method analogous to the one used to com-pute the evolutionary ages of the runaway stars (sample B)(Silva & Napiwotzki 2011, also see eg. Irrgang et al. 2010; c (cid:13) , ?? – ?? unaway Stars Ramspeck et al. 2001). However, here the most recent evolu-tionary tracks by ? for non-rotating stars were used insteadof the older models by the Geneva group. These tracks werecomputed using updated input physics and the revised valueof the solar metallicity ( ? ). The tracks were first convertedfrom the L − T eff to the M V − ( B − V ) space and thenlinearly interpolated in order to obtain an evolutionary ageestimate.The conversion of the theoretical tracks was accom-plished in two steps: the absolute magnitudes in V bandcorresponding to the given luminosities were computed usingthe bolometric corrections by Flower (1996); then the ( B − V ) colours corresponding to the given effective tempera-tures were computed using the formulae in Napiwotzki et al.(1993). These converted tracks are shown in Fig. 2.The full space velocities were computed for all starsin both sample A and sample B were computed from thedistances, measured proper motions and radial velocities. Inthe case of sample A, the distances were derived from therespective Hipparcos parallaxes.
The method adopted to compute the pattern speed consistedin two steps: 1) tracing back the orbit of each star in order todetermine its birthplace (assumed to lay on a spiral arm);2) rotation of the position of the birthplace by the anglecorresponding to the pattern speed Ω p and the age of thestar. The best estimate of the pattern speed is the one thatminimises the distance of the birthplaces to a given modelof the spiral arms structure, for the sample. Note that otherparameters of the model may be determined if a suitablesample is provided. This is essentially the method used byAmaral & Lepine (1997) and Dias & L´epine (2005) on theirsample of open clusters. In our case we have adopted thecartographic model by Vall´ee (2008) as a representation ofthe present spiral arms structure as it based on previousstudies using a variety of tracers and it has a simple math-ematical representation (logarithmic spiral). Thus, in polarcoordinates: r ( θ ) = r e bθ + nπ/ φ , (1)where b = tan p , φ is the initial phase or phase angle,and n = 0 , , ,
3. The parameters of the spiral structure,as derived by Vall´ee (2008), are: the pitch angle p = 12 . ◦ ,the initial radius r = 2 . φ = − ◦ . However, these parameters were obtained assuming adistance of the Sun to the Galactic Centre, R ⊙ = 7 . R ⊙ = 8 kpc. Taking thisdifference into account, a fit of the spiral structure to thetangent directions indicated in Vall´ee (2008) gives an initialphase φ = − ◦ · · · − ◦ , for R ⊙ = 8 kpc keeping the otherparameters fixed. Given the nature of our samples (sample Aonly includes local stars and sample B is small), from theseparameters only the initial phase φ was fitted together withthe pattern speed Ω p .Thus, we used the following target function in our min-imisation procedure: F (Ω p , φ ) = X i σ x ( i ) + σ y ( i ) d n ( i ) , (2)where d n ( i ) = q ( x i (Ω p , t i ) − x s ( θ n , φ )) + ( y i (Ω p , t i ) − y s ( θ n , φ )) (3)is the minimum distance between the birthplace ro-tated by Ω p (with coordinates x i , y i ) and the closest spiralarm (with the closest point having coordinates x s , y s corre-sponding to an angle θ n ), t i is the estimated age of the star,and σ x ( i ) and σ y ( i ) are the uncertainties in the coordinatesof the birthplaces as derived through a Monte Carlo errorpropagation procedure from the uncertainties in the inputobservable parameters. Since the fitting procedure assumesthe spiral arms have no thickness (they are treated as lines)the uncertainty in the position of the stars was assumedto be a minimum of 200 pc (corresponding to a 3 σ thick-ness of 1 . F (Ω p , φ ), does not strictlyfollow a χ distribution, and hence does not have knownstatistical properties, the error in the derived parameterswas determined through a Monte Carlo procedure. Thus,2000 replicas of the original sample were created, varyingthe input parameters (colour, parallax, proper motion andbrightness) according to the respective errors distributions(assumed Gaussian). As new age estimates are computed ineach realisation of the Monte Carlo procedure, this techniqueis also a good way of propagating the age uncertainties. Although in principle it would be possible to constrain allthe parameters of the simple logarithmic spiral model givenby Equation (1), in the concrete case of our samples this isnot possible as the tracers used either cover only a very smallpart of the Galaxy (sample A), or are too few with largemeasurement errors and small spread in ages (sample B).Thus, only a fit to the phase angle, together with the patternspeed, was attempted.In order to look for the minimum of the target func-tion (Equation 2) a grid spanning the area of interest of theparameter space was prepared in order to find the locationof possible solutions which were later more precisely deter-mined using an implementation of the Nelder-Mead sim-plex algorithm in python . In Fig. 3 the contour plot corre-sponding to sample-A is shown. Only one solution appearsto exist (within the equivalent to 6 σ ) which is repeated forphases separated by 90 ◦ , i.e. when one arm switches to thenext one. However, care should be taken with contaminationby stars formed in regions unrelated to the spiral arms, e.g.the Gould Belt. According to Torra et al. (2000), the GouldBelt has an age of ∼
60 Myr, thus in order to remove thisfeature we computed a new grid considering only stars older http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.fmin.htmlc (cid:13) , ?? – ?? M. D. V. Silva and R. Napiwotzki (cid:28)
50 0 50 (cid:29) ( (cid:30) ) (cid:31) p ( k m s k p c ! ) F ( " p , ) Figure 3.
Contour plots of the χ values for sample A stars. $
50 0 50 % ( & ) ’ p ( k m s ( k p c ) ) F ( * p , + ) Figure 4.
Contour plots of the χ values for sample A stars olderthan 60 Myr (to partially remove the Gould Belt). than ∼
60 Myr. The resulting contour plot is shown in Fig. 4and it is visible that although there is still only one solutionit appears to become more uncertain.The reason for this increased uncertainty is understoodwhen the same procedure is applied to a sample of evenolder stars, as shown in Fig. 5. In this case the solution be-comes degenerate, showing that a sample of older stars doesnot provide a good constraint on the phase angle. On theother hand, the sample of stars younger than 60 Myr, shownin Fig. 6, provides a solution that behaves in the oppositeway, providing a better constraint on the phase angle butperforming worse in constraining the pattern speed. This issimply a reflection of the obvious fact that younger objectsare good at describing the present state of the spiral armsbut have not travelled enough to be able to carry informa-tion about the pattern speed, whereas the opposite is truefor older objects.In fact, if the same procedure is applied to the youngclusters ( < − ◦ than ,
50 0 50 - ( . ) / p ( k m s k p c ) F ( p , ) Figure 5.
Contour plots of the χ values for sample A stars olderthan 90 Myr.
50 0 50 ( ) p ( k m s k p c ) F ( : p , ; ) Figure 6.
Contour plots of the χ values for sample A starsyounger than 60 Myr. to the − ◦ derived by Vall´ee (2008). However, as we haveseen, most of this difference is attributable to a differencein the chosen distance of the Sun to the Galactic Centre( R ⊙ = 8 kpc in our case), and, in particular, the constrainsused by Vall´ee (2008) imply φ ≃ − ◦ assuming R ⊙ = 8 kpc.In Fig. 7 we have plotted the grid of the parameter spacefor this sample of clusters, and in Fig. 8 a comparison be-tween the two different angles and how they compare withthe sample of young clusters. Although a phase angle of − ◦ appears to be a better fit in some regions, a largerangle fits better other regions (Perseus arm, for example).Even though the agreement between our solution and theone obtained by Vall´ee (2008) (and hence with constraintsprovided by the tangential directions and the distance tothe Perseus arm obtained by Xu et al. 2006) is not perfectit would be possible to find a solution consistent with theconstraint, however that would require further analysis be-yond the scope of this paper. Note that in the plot shown inFig. 7 a phase angle of − ◦ is actually in the region of the c (cid:13) , ?? – ?? unaway Stars <
50 0 50 = ( > ) ? p ( k m s @ k p c A ) F ( B p , C ) Figure 7.
Phase angle estimate from sample of clusters youngerthan 7 Myr (from Dias et al. 2002). D E F X (kpc) G H I Y ( k p c ) Figure 8.
The spiral arms model by Vall´ee (2008) with φ = − ◦ (solid line) and φ = − ◦ (dashed line), derived for R ⊙ = 7 . R ⊙ = 8 kpc respectively, and φ = − . ◦ (dotted line). Thedifferent models are compared with the sample of open clustersyounger than < worst solutions, but a phase angle of − ◦ (the correctedvalue for R ⊙ = 8 kpc) is in the region of best solutions.The minimisation of the target function (Equation 2)around the solution found in the parameter space, (Ω p =20 , φ = − ◦ ), yielded the values seen in Table 1 for threedifferent samples: all of sample A, only stars in sample Athat are older than 60 Myr, and the sample B. Note thatthe solution in the case of sample B is quite degenerate,however there is a solution compatible with the one foundin the other two cases.The spiral arm structure with the phase estimated fromsample A is shown in Fig. 9, together with the birthplaces ofstars from samples A and B rotated from their original po- Table 1.
Pattern speed (Ω p ) and phase angle ( φ ) obtained forthree different samples: entire sample A (local Hipparcos stars),only stars older than 60 Myr in sample A, sample B (runawaystars).Sample Ω p (km s − kpc − ) φ ( ◦ )A (all) 20 . ± . − . ± .
1A ( >
60 Myr) 21 . ± . − . ± .
4B 21 . ± . − . ± . J K L X (kpc) M N O Y ( k p c ) A g e ( M y r ) Figure 9.
Distribution of the birthplaces in their present posi-tions, resulting from the derived pattern speed. Sample A andsample B (runaway stars) are plotted as coloured circles (withsizes proportional to the uncertainty in the position) and squareswith error bars, respectively. Only runaway stars with standarddeviations < . sitions by the derived (from sample A) pattern speed timesthe ages of the stars. It is interesting to note how the run-aways seem to have been born in many different arms, andin particular how well they appear to follow the “yellow”arm (to be identified with the Perseus arm). Estimates of the pattern speed in the literature cover awide range of values: using different variations of the “birth-place” determination method with samples of open clusters,Amaral & Lepine (1997) found Ω p ≈
20 km s − kpc − andDias & L´epine (2005) Ω p ≈ ± − kpc − ; by fit-ting a kinematical model of the Milky Way Fern´andez et al.(2001) found Ω p ≈
30 km s − kpc − (with an uncertaintybetween 2 km s − kpc − and about 7 km s − kpc − depend-ing on the specific tracer); Martos et al. (2004) obtainedΩ p ≈
20 km s − kpc − by fitting dust observations with adynamical model of the Galaxy. Moreover, Naoz & Shaviv(2007) estimate different pattern speeds, using a variationof the “birthplace” method with a sample of open clus-ters, for different structures (with an uncertainty close to ∼ − kpc − for all estimates): the Sagittarius-Carinaarm is actually a superposition of two spiral structures with c (cid:13) , ?? – ?? M. D. V. Silva and R. Napiwotzki Ω p , ≈ . − kpc − and Ω p , ≈ . − kpc − ,the Perseus arm with Ω p ≈
20 km s − kpc − , and theOrion “armlet” with Ω p ≈ . − kpc − . By fittingmodels of gas flow in the Milky Way to CO observations,Bissantz et al. (2003) obtained Ω p ≈
20 km s − kpc − .The different estimates of the pattern speed seem tosuggest (even though it is not clear how uncertain someof these estimates are) a value in the range Ω p ∼ −
25 km s − kpc − (see also Gerhard 2011), which would becompatible with our own estimates. Furthermore, there isindependent confirmation, from the observation of the pre-dicted ring-shaped gap in the radial H I density distribu-tion (Amˆores et al. 2009), of a corotation radius close tothe value of the distance of the Sun to the Galactic cen-tre, implying a pattern speed Ω p ∼ −
29 km s − kpc − .As we have seen, our best estimate gives a value of Ω p =20 . ± . − kpc − for the pattern speed, which, al-though consistent with other determinations, appears to beat odds with estimates of the corotation radius, particularlygiven the small uncertainty. However, we must bear in mindcertain caveats to this result: the uncertainty includes onlythe formal statistical contributions, ignoring systematic ef-fects; it is possible that the inclusion of regions of star for-mation unrelated with the spiral arms may introduce biasesin the estimates. Indeed, it can be seen in Table 1 that re-moving most of the Gould Belt from the sample producesan estimate (Ω p = 21 . ± . − kpc − ) closer to theupper limit of the suggested range for the pattern speed(Ω p ∼ −
25 km s − kpc − ), albeit with a larger uncer-tainty caused by the loss of the young stars constraining thecurrent position of the arms.For the same reason, it is possible that the uncertaintyon the phase angle determination was underestimated, be-cause the large number of young stars in the solar neigh-bourhood may introduce a bias in the position of the spiralarms, as they might have been formed in other regions (likethe Gould Belt, or interarm features). However, the valueswe obtained for the phase angle are close to the estimatesby Fern´andez et al. (2001) (varying between φ = − ◦ ± ◦ and φ = 8 ◦ ± ◦ , when assuming models with four spiralarms), and also with the estimate obtained from the sampleof open clusters (see Fig. 8). GAIA
The
Gaia mission will deliver trigonometric parallaxes, im-proved proper motions and spectral energy distributions formany early-type runaway stars. This will: 1) improve theachievable accuracy for the existing sample, and 2) allowthe selection of a much increased sample of runaway stars.In order to illustrate the expected increase in perfor-mance of our method once
Gaia data becomes available, wehave determined the birthplaces of two representative starsof our sample of runaways, one far away and one nearby,assuming the knowledge of their parallaxes and proper mo-tions with the predicted
Gaia accuracy. The results obtainedare summarised in Table 2 and contrasted with the previousdetermination. Our results for the nearby star HIP 59955 arealready fairly accurate.
Gaia measurements will reduce theoverall uncertainty of the determination of the birthplace toa level comparable with the typical width of a spiral arm. On
Table 2.
Precision of the determination of the birthplaces ofthe stars EC 04420-1908 and HIP 59955 using present data(Silva & Napiwotzki 2011) and the precision predicted using
Gaia data. Estimates of the distances and the coordinates of the birth-places obtained using present data are indicated with a p sub-script whereas the estimates obtained using
Gaia precision dataare indicated with a g subscript.EC 04420-1908 HIP 59955 V
13 9 . d p (kpc) 10 . +4 . − . . +0 . − . d g (kpc) 10 . ± . . ± . X p (kpc) − . +4 . − . − . +1 . − . X g (kpc) − . ± . − . ± . Y p (kpc) − . +2 . − . − . +0 . − . Y g (kpc) − . ± . − . ± . the other hand, the improvement for the star EC 04420-1908is dramatic, with the X coordinate uncertainty decreasingby ∼
500 per cent and the Y coordinate uncertainty decreas-ing by ∼
300 per cent. Although the uncertainty obtained inthis case is larger than the typical width of a spiral arm, itis now smaller than the expected interarm distance (Vall´ee2008).Noting that the distance uncertainty that would be ob-tained from the
Gaia parallax in the case of EC 04420-1908is ∼ V <
13 and
V <
14 (according tothe formula in the
Gaia
Science Performance website ). Itis possible to conclude from this plot that it will be possibleto determine directly – with high precision – the birthplacesof stars with V <
13 and
V <
14 as far as 10 kpc and8 kpc, respectively. For stars further away other methodswill be needed in order to attain a high precision, howeverit is important to note that distances of 14 kpc and 12 kpc,respectively for stars brighter than V = 13 and V = 14, willprovide parallax errors of ≈
15 per cent.
An issue always present in studies of runaway stars is thecontamination by low mass stars in advanced stages of evo-lution, in particular the post-AGB and core Helium burningHorizontal Branch, as they may mimic the atmospheric pa-rameters of main sequence stars (Tobin 1987; Martin 2004).However, as a direct measurement of the distance will beavailable with
Gaia it will be possible to select runawaystars based on their absolute magnitudes M V and effectivetemperatures T eff .From Fig. 10, we have already seen that it will possible (cid:13) , ?? – ?? unaway Stars d (kpc) P Q ( R a s ) Figure 10.
Parallax error as a function of distance correspondingto an error of 1 kpc (solid line), 2 kpc (dashed line) and 3 kpc(dotted line) in distance. The shaded areas correspond to theerrors for stars with
G <
15 (dark grey),
G <
14 (medium grey)and
G <
13 (light grey), that are predicted for
Gaia . to determine the distance with high precision for stars with G <
13, and, more generally, with an error <
15 per centfor stars with
G <
14, at least for distances up to 12 kpc.However, it can also be seen that an error <
30 per cent willbe possible for stars with
G <
15. Thus, considering thatthis error in the distance translates to an error of 0.65 inthe absolute brightness M V , it will be possible to use Gaia parallaxes to select a sample of early type runaway stars,even in those cases where the precision is not good enough tobe used directly, as can be seen in Fig. 12. This is because theabsolute magnitudes of stars with spectral type B7 or earlier(effective temperature ∼ M V >
1, for the same temperature. Note that this allowsthe selection of B7 stars as far as 12 kpc.Once the sample is selected, better estimates of the dis-tance could be obtained after a careful spectroscopic analysisin the case of the distant stars. Since the limiting factor ofspectroscopic distance estimates is usually the uncertainty inthe log g estimate, it is instructive to see how the error thatis expected from Gaia parallaxes compares with good spec-troscopic estimates, with small uncertainties in log g . Forexample, Nieva & Przybilla (2012) obtain a distance accu-racy of ∼
10 per cent in their study of nearby B stars, froman analysis where the uncertainty in the log g determina-tion is ∼ .
05 dex. The comparison between these estimates,and also the expected error if only the log g uncertainty (of ∼ .
05 dex) is considered, is shown in Fig. 11. Note how the
Gaia estimates have acceptable errors only up to distances of10 to 12 kpc. Moreover, since the radial velocities measuredwith
Gaia will have acceptable errors ( <
10 km s − ) onlyin the case of the brightest stars ( V = 13 . . . d (kpc) r e l a t i v ee rr o r Figure 11.
Comparison between the expected relative error, asa function of distance, for the distance estimates: from
Gaia par-allaxes for a
G <
13 star (solid line), from spectroscopy whenthe uncertainty in log g is 0 .
05 dex (dashed line), as obtained byNieva & Przybilla (2012), taking all error sources into account(dotted line).
These could be obtained from
Gaia spectrophotometry, ifwe assume it will be at least as accurate for effective tem-perature determination as medium band photometry, e.g.Str¨omgren-Crawford uvbyβ photometry. The expected er-ror would be then, according to Napiwotzki et al. (1993),about 3 per cent in the range 11000 K T eff Galex
UV photometric data (Martin et al., 2005).As was already suggested, the expected contaminationby Horizontal Branch stars, for stars with spectral type B7or earlier will be extremely low, if the appropriate selec-tion criteria is adopted. Adopting a criteria similar to theone used by Silva & Napiwotzki (2011) (dependent on es-timates of log g obtained from spectroscopy) should limitthe contamination to less than 10 per cent of the total. Infact, most contamination by Horizontal Branch stars can beeliminated by applying a cut in the T eff − M V space (e.g. T eff > M V < g information, since the distances of 0 . ⊙ and 2 . ⊙ stars of the same observed brightness differ by √
5. The contamination by post-AGB stars is more difficultto ascertain, however these stars are very rare and usuallyaccount for a very small amount of contamination in studiesof runaway stars (e.g. ∼ ∼ .
10 per cent.Furthermore, we will be able to use
Gaia astrometry to se-lect runaway stars with spectral types earlier than B7 upto a distance ∼
14 kpc and spectral types earlier than B3up to a distance ∼
22 kpc, assuming a brightness V = 15.However, follow-up spectroscopic analysis will be needed to c (cid:13) , ?? – ?? M. D. V. Silva and R. Napiwotzki T eff (K) S T U V W X M V Y Z [ \ ] ^
12 M _ ZAHBTAHBPAGB - 0.546 M ‘ Figure 12.
The Zero Age Horizontal Branch (ZAHB) and theTerminal Age Horizontal Branch (TAHB), from the models byDorman et al. (1993) for stars with an Helium mass fraction Y = 0 .
247 and a metallicity [Fe / H] = − .
48, and the theoreticalevolutionary track of a low mass Post-AGB star, from the modelsby Sch¨onberner (1979), in a M V − T eff diagram. The absolutemagnitudes for the Horizontal Branch models were obtained byapplying the bolometric corrections of Flower (1996). The reddots correspond to stars identified as Horizontal Branch in thesurvey by Brown et al. (2008). obtain good estimates of the distances in the case of thefainter stars. On the other hand, for stars brighter than V = 13 (corresponding to a distance ∼ ∼ Gaia parallaxes.
As we have seen, it will be possible to select a sample of run-away stars reaching distances up to ∼
14 kpc, creating theopportunity to study the spiral arms behind the centre ofour Galaxy. It is thus important to have an estimate of howmany runaway stars will be observed behind the Galacticcentre. The estimated number density of runaway stars highabove the Galactic plane is ∼ − (Silva & Napiwotzki2011). This corresponds to a total of ∼ > l = 0 ◦ ). The extinction for stars 3 kpc away from theGalactic plane depends on the distance under consideration,as the Galactic latitude b decreases for larger distances. InTable 3 we show the values of extinction A V obtained fromthe Schlegel maps for distances in the range 10 −
16 kpc. Adistance of 10 kpc should be enough to probe the Sagittarius-Carina arm, 12 kpc should probe the Perseus arm, 14 kpcshould probe the Cygnus arm, 16 kpc should probe the Crux-
Table 3.
Extinction predicted by the Schlegel et al. (1998) red-dening maps for different Galactic latitudes. The Galactic lati-tudes were computed for points 3 kpc above the Galactic planeand at the given distances ( d ). Also indicated are the apparentmagnitudes for stars with spectral types B3 and B7 when red-dened by the corresponding amounts. The values for the Northernand Southern Galactic hemispheres are distinguished by the useof parentheses, with the values inside parentheses correspondingto the Southern hemisphere. d (kpc) | b | ( ◦ ) A V V (B3) V (B7)10 17 1 . .
4) 15 . .
3) 16 . . . .
4) 15 . .
7) 16 . . . .
5) 16 . .
1) 17 . . . .
4) 16 . .
3) 18 . . Scutum arm, and should be close to the corotation radiuson the other side of the Galaxy. We have included the ex-pected extinction in the Northern and Southern Galactichemispheres separately because they differ by a large mar-gin due to the presence of the Rho Ophiuchi cloud complexin the Northern hemisphere. Also included in Table 3 arethe apparent magnitudes of B3 and B7 (reddened) stars atthe indicated distances. It should be noted, however, thatthese reddening estimates are actually overestimates, since
Gaia G is redder than Johnson- V making it less sensitive tointerstellar absorption.For a B star with V <
15, we can infer from Table 3 thatwe will be able to observe stars with spectral type earlierthan B3 as far as 12 kpc in both hemispheres and as far as14 kpc in the Southern hemisphere. On the other hand, starswith later spectral types (but earlier than B7) will be observ-able only in the Southern hemisphere but as far as 12 kpc.However, it will be possible to use B3 stars (and earlier) toprobe distance as far as 18 kpc in the Southern hemisphereand 14 kpc in the Northern hemisphere, although with aloss of accuracy. In the same manner, it will be possible touse B7 stars (and earlier) as far as 14 kpc in the Southernhemisphere and as far as 12 kpc in the Northern hemisphere,although with less accuracy.However, we should remember that we are limited todistances of ∼ ∼ Gaia parallaxes, as was previously explained. Nevertheless,as was also pointed out, it should be possible to obtain accu-rate distances from follow-up spectroscopic analysis but thisis within easy reach of modern medium-large telescopes. Itis also important to note that this does not invalidate that,at least for a fraction of the sample (i.e. stars with spec-tral types earlier than B3), it will be possible to use directdistance determinations from
Gaia .Thus, from the expected sample of ∼ ◦ of the direction of the Galactic centre, if they areuniformly distributed. Hence, given the previous considera-tions, we expect to have a sample of ∼
250 in the Southernhemisphere covering a distance up to 18 kpc and ∼
175 inthe Northern hemisphere covering a distance up to 14 kpc.The total number of stars should then be close to 400. c (cid:13) , ?? – ?? unaway Stars The determination of the birthplaces of early-type stars isa feasible method to trace the position of the spiral arms indifferent time instants. We were able to estimate the patternspeed and the (present) phase angle of the Galaxy’s spiralstructure from a sample of local stars and obtained values(Ω p = 20 . ± . − kpc − and φ = − . ◦ ± . ◦ ) thatare consistent with previous estimates. Moreover, althoughthese estimates are sensitive to systematic effects, in partic-ular biases introduced by stars born outside the spiral arms,these effects are not crippling and introduce an extra errorof ∼
10 per cent.From the sample of high Galactic latitude runawaystars (sample B) we also derived a consistent estimate ofthe parameters (Ω p = 21 . ± . − kpc − and φ = − . ◦ ± . ◦ ). However, given the nature of this sample(larger observational errors, small numbers and small spreadin ages) the solution is not as well defined as in the case ofthe one obtained from sample A and has thus a larger un-certainty.More importantly, we have shown that this method hasthe potential to be used with data obtained by Gaia to re-trieve the birthplaces of distant runaway stars and henceit will be possible to trace not only the closest spiral armsbut also the more distant ones, in particular the portionsbehind the Galactic centre, which are usually obscured byinterstellar reddening. We have estimated that the numberof runaway stars that will be available to trace these dis-tant portions of the spiral arms, in particular the far sidesof the Sagittarius-Carina and Perseus arms, will be ∼ ∼ > Gaia in the near future, itwill be possible to apply the method to a large number ofrunaway stars with high accuracy.
ACKNOWLEDGMENTS
M.S. gratefully acknowledges financial support by the Uni-versity of Hertfordshire. The authors wish to thank the ref-eree for helpful comments that helped to improve the paper.This research has made use of NASA’s Astrophysics DataSystem and of the SIMBAD database, operated at CDS,Strasbourg, France.
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