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astro-ph/
12 Nov 95
A&A manuscript no. (will b e inserted by hand later) Your thesaurus co des are: 10(03.13.6; 04.19.1; 10.08.1; 10.11.1; 10.19.2; 10.19.3)
ASTRONOMY
AND
ASTROPHYSICS
Structure and kinematical prop erties of the Galaxy at
intermediate galactic latitudes
D.K.Ojha^1 , O.Bienayme^2 , A.C.Robin^2 ;^3 , M.Creze^2 , and V.Mohan^4 (^1) Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India (^2) Observatoire de Strasb ourg, CNRS URA 1280, 11 rue de l'Universite, F-67000 Strasb ourg, France (^3) Observatoire de Besancon, 41 bis, Av. de l'Observatoire, BP 1615, F-25010 Besancon Cedex, France (^4) U. P. State Observatory, Manora Peak 263 129, Nainital, India
Received xxxx, 1995, accepted xxxx, 1995
Abstract. We have carried out a sample survey in UBVR photometry and prop er motions in di erent directions in the Galaxy, as part of an investigation of galactic struc- ture and evolution. Three elds in the direction of galactic
anticentre (l = 167 ^ , b = 47 ^ ), galactic centre (l = 3 ^ ,
b = 47 ^ ) and galactic antirotation (l = 278 ^ , b = 47 ^ )
have b een surveyed. Using our photographic photometry, we determine photometric distances for a sub-sample of stars in the color range 0.3(B-V)0.9. The stellar space velo cities (U, V and W) are derived directly from the mea- sured prop er motions and distances. Using our new data together with wide-area surveys in other elds available to date, we discuss the radial and ver- tical structure of the Galaxy. We have derived the density laws for stars as a function of distance from the galactic plane for each absolute magnitude interval. The density laws for stars with MV 3.5 follow a sum of two exp o- nentials with scale heights of 260 50 p c (thin disk) and 760 50 p c, resp ectively. This second exp onential corre-
sp onds to a thick disk with a lo cal density of 7.4+2 1 ::^55 %
relative to the thin disk. The scale lengths for these two p opulations are resp ectively 2.30.6 kp c and 3 1 kp c. The kinematical distribution of F and G{typ e stars have b een prob ed to distances up to 3.5 kp c ab ove the galactic plane. A new value for the solar motion has b een determined from mo derately distant stars (1<z< 2 kp c). It is consistent with lo cal determinations and implies that there is no large motion of the LSR relative to the mean motion of stars at 1-2 kp c ab ove the galactic plane. The rotational velo city curve is found at in the solar neigh- b orho o d. The radial gradient in velo city disp ersions has b een determined for the thin disk p opulation. The thick disk app ears as a kinematically distinct p opulation from the thin disk and shows no vertical gradient. A multivari- ate discriminant analysis is also used to distinguish the thick disk from the thin disk and to estimate its asym-
Send o print requests to : D.K.Ojha (dko@iucaa.ernet.in)
metric drift. It is found to b e 53 10 km/s, indep endent of the galactic radius. Of the many mo dels that have b een prop osed for the origin of the thick disk, the evidence at present seems to favour a mo del in which thick disk formed through the rapid dynamical heating of an early disk by satellite accretion into the disk.
Key words: Galaxy: Kinematics and dynamics { Galaxy: Stellar content { Galaxy: structure { Metho ds: statistical { Surveys
- Intro duction
The concept of stellar p opulations has played a critical role in the development of mo dern studies of stellar evo- lution, galactic structure and galactic evolution. To un- derstand fully the stellar p opulations and their di erences would take us a long way toward a complete theory of galactic structure and evolution. Yet, there are many dif- cult questions that have b een very unyielding to nal solutions, such as, how can we de ne the p opulations in a way that is physically meaningful, how many discrete p opulations really are there, and how can we unambigu- ously separate the stellar constituents of the p opulations for detailed analyses? It was only with the discovery of the asymmetric drift of high-velo city stars (Stromb erg 1924; Oort 1922, 1926), it b ecame clear that multiple overlapping comp onents would b e needed if an ellipsoidal mo del were to provide an adequate representation of the data. After the identi ca- tion of the galactic centre (Shapley 1918), and the discov- ery of the di erential rotation of the galactic disk (Oort 1922, 1927; Lindblad 1925, 1927, 1959; Stromb erg 1924), the multicomp onent concept was taken to the limit with Lindblad's 1927 (see also 1959) prop osed division of the
Galaxy \into an in nite numb er of `sub-systems' of vary- ing mean sp eed of rotation". Stromb erg's (1924) compre- hensive analysis of the available data on stellar distances, prop er motions, and radial velo cities in the mid-1920s was among the rst to show the di erences in the velo city el- lipsoids of various stellar groups. His kinematic separation into groups was the b eginning of the classi cation of the galactic comp onents that eventually led to the p opulation concept. The pioneering work on p opulations was carried out by Baade during the 1940's and 1950's using mostly ob- servations of galaxies in the Lo cal Group. In 1944, Baade intro duced the idea of two basic stellar p opulations (Baade 1944, 1958a). Unlike the kinematic comp onents discussed previously by Stromb erg (1924) and Oort (1922, 1926), Baade's two p opulations were de ned according to the morphology of the resp ective color-magnitude diagrams. Population I stars are stars with photometric character- istics similar to those found in op en clusters and stel- lar asso ciations (e.g. luminous early-typ e stars, classical Cepheids), and Population I I stars are similar to the stars found in globular clusters (e.g. RR Lyrae stars, sub dwarfs, Population I I Cepheids). At the conclusion of the 1957 Vatican conference on stellar p opulations, ve stellar p opulations were advo cated : extreme p opulation I, older p opulation I, disk p opu- lation, intermediate p opulation I I, and halo p opulation I I. Excellent reviews on the sub ject have b een written by King (1971), van den Bergh (1975), Mould (1982), Sandage (1986) and others. Throughout the last decade there has b een further de- bate over the structure and history of our Galaxy. One of the main controversies concerns the existence of a thick disk, i.e., a comp onent with characteristics intermediate to those of the thin disk and halo p opulations, and whether or not three comp onents (rather than two, ve or more comp onents) are needed to account for star counts, prop er motion distributions and other observations. There are now strong evidences that our Galaxy is not simply de- scrib ed by the traditional disk and a halo, but that there must exist a third comp onent, a thick disk, which is dom- inated over distances on the order of kiloparsecs ab ove the plane, and intermediate in kinematics and metallic- ity b etween the disk and halo p opulations. The idea of \third comp onent" has apparently b een around in one form or another for decades, at least since the time of the 1957 Vatican conference (see Blaauw 1965). The con- cept of a thick disk came into existence after Gilmore & Reid (1983) showed need for it through their inability to t two-comp onent mo dels to starcount data. Since Gilmore & Reid (1983), other starcount surveys have reached sim- ilar conclusions (Yoshii et al. 1987; Sandage 1987; Fenkart 1988). Prop onents of a three-comp onent mo del have raised questions ab out the nature of the thick disk, in particu- lar, whether it is kinematically discrete (Wyse & Gilmore 1986; Sandage & Fouts 1987; Gilmore, Wyse & Kuijken
1989; Soubiran 1993ab) and/or chemically discrete (Car- ney et al. 1989; Carney, Latham & Laird 1990). Ma jewski (1992) have shown that an intermediate comp onent must exist in order to account for a bimo dal distribution in b oth the V-velo city and UV-excess up to 5.5 kp c ab ove the galactic plane, far b eyond the region dominated by the traditional thin disk of scale height 260 p c or less. The p opulation of \disk" globular clusters and the appar- ent excess of intermediate - metallicity stars with vertical scale height hz 1 kp c, represent also the evidence for a thick disk. (Baade 1958b; Morgan 1959; Zinn 1985; Ar- mandro & Zinn 1988). It should b e stressed here that starcount analysis alone do not warrant the reality of thick disk stars as a third ma- jor comp onent of the Galaxy, b ecause adding new comp o- nent(s) in mo dels always gives a sup erior t to the data. In fact, current interpretations of thick disk stars have b een ranging b etween a separate comp onent (Gilmore & Wyse 1985; Bienayme et al. 1990, 1992) and a high-velo city tail of the old disk (Norris 1987ab; see also Norris & Green 1989). This third discrete p opulation could b e the signa- ture of a merger event early in the Galaxy's history o c- curing shortly after the disk had formed (Carney et al. 1989). In fact, the arguments regarding the existence of this third p opulation b ear on problems of galaxy forma- tion and evolution. For example, \the G dwarf problem", that the lo cal numb er of metal-p o or long-lived stars is to o small to b e consistent with simple mo dels of chemical evolution, may b e solved straightforwardly by use of an intermediate-metallicity \reservoir" disp ersed ab ove and b elow the plane (Gilmore & Wyse 1986). Studies of stellar p opulations often take as their start- ing p oint large-scale surveys, with selection criteria involv- ing one or more observable variables. The de nition of the survey has to b e de ned in order to ensure a go o d retrieval of the p opulation parameters we lo ok for. The physical lo- cation of the surveys is arguably the most imp ortant con- straint. Obviously several survey lo cations need to b e cho- sen in order to vary the mixing prop ortions of the various p opulations and to determine the scale heights, lengths and relative densities. On the other hand, the survey must prob e to great distances so that the regions dominated by each of the Galaxy's p opulations are within the limits of the survey. For example it is necessary to reach V magni- tude ab out 19 in order to reach domains dominated solely by the halo p opulations. Finally, in order to b e con dent that the data are free of systematic bias, it must b e com- plete, i.e. all stars within a region of the sky must b e surveyed without any selection criteria which b ear on the problems b eing studied. With these p oints in mind, we have carried out a sam- ple survey in UBVR photometry and prop er motions in 3 directions in the Galaxy. The 3 elds chosen are in the
direction of galactic anticentre (l = 167 ^ , b = 47 ^ ; Ojha et
al. 1994a, 1995; hereafter GAC1,2), galactic centre (l = 3 ^ ,
b = 47 ^ ; Ojha et al. 1994b; hereafter GC) and galactic an-
MV = (
2 : 31 1 :04(B V )
and MV = 0 0 : 6 < 0 The UV excess - metallicity relation was studied by Carney (1979), and can b e approximated as :
p
0 : 01191 0 :05353[F e=H ]
So we obtain a MV correction that dep ends on z. The absolute magnitudes calculated from the MV versus B-V relation was therefore corrected by these amounts. In practice, a new distance is calculated, a revised MV correction applied, and the pro cedure was rep eated to con- vergence.
2.2. Errors in distance measurements
A variety of systematic errors a ect the determination of stellar distances. The rst source of errors could b e from B and V magnitudes. This a ects the distances of individ- ual stars, but should not a ect mean distance of a sample of stars, since the photometric errors should b e randomly distributed. More imp ortant are the systematic errors that could b e pro duced by the contamination by subgiant, gi- ant and binary stars in our samples.
2.2.1. Subgiant contamination
We used the main-sequence color-absolute magnitude re- lation to derive the distances. For the distance determi- nation, it is rstly assumed that all stars are unevolved. This is certainly not applicable to all stars in our survey, but without sp ectroscopy or photometric gravity indica- tors there is no a priori way to separate main-sequence dwarfs from evolved subgiant stars at the main-sequence turno. In our selection of subsamples of F and G{typ e
stars (3.5MV 6 or 0.3B-V0.9), the contamination by
giant stars should b e small b ecause at the ab ove absolute magnitude and color intervals, distances and velo cities im- plied for red giant stars are improbable. However contam-
ination of main sequence (4MV 6) by subgiants with
2 MV 4 (with same color) could cause a serious under-
estimate of distance. We have made a selection by color, by raising the
blue limit of the color range (B-V0.9) b eyond the re-
gion where subgiant stars are exp ected to contribute sub- stantially and we can greatly reduce their contamination. Only drawback of this selection is that we loss the farthest distance and most of the halo stars. The Besancon mo del of p opulation synthesis (Robin & Creze 1986; Bienayme et al. 1987) has b een used to check the prop ortion of sub- giants in our subsample of stars. If we supp ose that there
exists an intermediate-metallicity p opulation of stars with
scale height hz 750 p c and lo cal density of 6% relative to
the thin disk, then we would exp ect 30% of the intermedi-
ate subgiant p opulation (MV 4) to have (14V16) and
the bulk to have V17. An old p opulation I distribution
(hz 250 p c) would have subgiants (MV 4) contributing
mostly at V14.5. The numb er of p ossible disk and thick
disk subgiants in our survey is minimal, since the total numb er of stars in the appropriate magnitude and color range is ab out a 10%. For the halo, the distribution is fairly broad. We would exp ect the largest contribution from halo subgiants at
15.5V19.5. The worst contamination of the main se-
quence by spheroid giants probably o ccurs at 0.6B-
V0.8 or 5.9MV 6.7 (Sandage 1982, Bahcall et al.
1983). Recently, a sp ectroscopic observations have b een p erformed on a sample of stars in a eld of 9.6 square
degrees in the galactic centre direction (l ' 3 ^ , b ' 45 ^ )
by Perrin et al. (1995). The most striking result of these observations is the very low prop ortion of Population I I giants in this direction, ab out 2 times less than the pre- dictions of the current galactic mo dels. Similar result has b een found in other intermediate latitude elds by di er- ent authors (Friel 1987, Morrison 1993). It is p ossible that some ranges of (V, MV ) are not seriously contaminated. For example, if in the range
4 MV 6, the luminosities of the subgiants are close to
their main sequence values then the mis-identi ed evolved stars do not come from a much larger volume (Tritton & Morton 1984). Also at distances where the disk density dominates over the spheroid, subgiant or giant contami- nation should not b e a problem.
2.2.2. Binary contamination
Another systematic error a ecting the distance determi- nation is the problem of unresolved binaries. Every star has b een treated as if it were a single ob ject, but this is certainly not the case for a signi cant p ortion of the sample \but unknown". Only binaries with comp onents of approximately equal mass are problematical. In this case, the distance error will b e 40%, while for slightly un- equal mass stars, there is a concern for color errors in the primary (a 10% error in color results in 0.6 mag er- ror in estimated absolute magnitude). The binary fraction of Population I stars is supp osed to b e relatively large- approximately 55%. The thick disk binary fraction is es- sentially unknown, but it is likely to b e somewhere b e- tween that of the thin disk and the halo. In summary, sys- tematic errors in the distance determination due to the presence of binaries are probably most severe for young disk stars. Clearly this problem would b e solved with a systematic sp ectroscopic survey for the sample of stars at mo derately high resolution to identify sp ectroscopic bina- ries. Recently, such a programme has b een initiated by Soubiran (1994) for subsamples of stars in our surveys.
2.2.3. Photometric errors
The ma jor factors contributing to distance errors are the
photometric errors in the color B V and magnitude V.
To roughly estimate the contribution from all of these, it is necessary to go back to the observable quantities, as
some of these errors are correlated, e.g. B V and [F e=H ]
with MV. From our in situ surveys, we exp ect the mean
errors in B V = 0.1 and V = 0.07 for V = 11 to 18.
From these values, our estimate of error to b e ab out 20 % in distance seems to b e realistic up to z = 3 kp c. However,
we have estimated that for farthest distances (z 3 kp c),
there may b e an error in our distances as large as 30 %.
- Tangential velo cities
The cardinal comp onents of the stellar space velo city (in km/s), U, V and W (where U is de ned as p ositive in the direction of the Galactic anticentre, V is p ositive in the direction of Galactic rotation, and W is p ositive in the direction of NGP) were derived from prop er motions, l
and b (in arcsec year ^1 ), and distance d (in p c) for the
three elds as follows (Murray 1983) :
U
V
W
A = RG
4 : 74 d l 4 : 74 d b Vr
A
The matrix RG for each eld is as follows :
GAC 1 ; 2
A
GC
A
GAR
A
The unknown radial velo city contributions (Vr ) to UVW are as follows and are neglected in the following study :
GAC1,2 : UVr = 0.661 Vr ; VVr = 0.147 Vr ; WVr = 0.738 Vr
GC : UVr = -0.681 Vr ; VVr = 0.032 Vr ; WVr = 0.731 Vr
GAR : UVr = -0.093 Vr ; VVr = -0.680 Vr ; WVr = 0.728 Vr
It is clear from the ab ove 3 matrices that we are mea- suring the velo cities (U-W,V) in the direction of GAC1,2,
(U+W,V) in the direction of GC and (U,V-W) in the di- rection of GAR eld. We should p oint out here that the
measured prop er motions are directly converted to U W
and V velo cities. For GC or GAC1,2 eld :
U W
p
' 4 : 74 d b and V ' 4 : 74 d l
and we de ne :
2 U; W =
2 U + W^2
- Space density
We are interested in deriving the structural parameters (e.g. scale height and scale length) of the thin and thick disk p opulations using our data sets at intermediate lati- tude. For this we have calculated the stellar space density as a function of b oth absolute magnitude and distance from the Galactic plane. The calculation is straightfor- ward : The space density of total stars (Population I + In- termediate Population + Population I I) of the luminosity group (M 1 < M < M 2 ) falling into the partial volume
4 V 1 ; 2 is given by :
( r ) = (r 1 ; r 2 ) = N 1 ; 2 = 4 V 1 ; 2
N 1 ; 2 b eing the total numb er of stars of the lumi-
nosity group in question, and partial volume 4 V 1 ; 2 =
( =180)^2 (ut=3)(r 32 r 31 ). r 1 and r 2 are the limiting dis-
tances; ut : eld size in square degrees; r = ((r 13 + r 32 )=2)^1 =^3
: centrod distance of 4 V 1 ; 2.
4.1. Density laws
The space density distribution can b e approximated by a double exp onential :
(R; z ) = (R 0 )exp
R R 0
hR
exp
jz j
hz
R 0 = 8.090.3 kp c (Pont et al. 1994) is the solar dis-
tance from the galactic centre. (R 0 ) is the solar \neigh- b orho o d" normalisation. R is the galacto centric distance pro jected up on the galactic plane and z is the height ab ove the galactic plane. hz and hR are the scale height and scale length, resp ectively. The density can b e expressed as a function of the dis- tance d along the line of sight for a eld of galactic co or- dinates (l ,b) with,
R = (R^20 + d^2 cos^2 b 2 R 0 d cosb cosl )^1 =^2
jz j = d sinb
Table 1. The b est tted structural parameters of the thin and thick disk stars with 3.5MV 5 derived from the two data sets (GAC1,2 and GC)
Field Thin Disk Thick Disk Thin Disk : Thick Disk 3.5MV 5 hz (p c) hz (p c) density ratio GAC1,2 336 807 100 : 7. GC 226 674 100 : 9.
Table 2. The b est tted structural parameters of the thin and thick disk stars with 5 MV 6 derived from the two data sets (GAC1,2 and GC)
Field Thin Disk Thick Disk Thin Disk : Thick Disk 5 MV 6 hz (p c) hz (p c) density ratio GAC1,2 267 849 100 : 8. GC 217 726 100 : 9.
certainty in the determination of density laws, we built a more consistent mo del. We t simultaneously data in each eld with the same values of parameters (exp ect for the thin disk scale length that has not b een tted). For each disk, we have used the same lo cal normalization of density indep endently of directions. In this case we may solve b oth parameters (hR & hz ) together. The observed density is tted by the sum of two exp onentials shown in gure 3. The same pro cedure (describ ed in the previous section) was applied to derive the b est tted structural parameters. The results are shown in table 3.
4.3. Discussion
The b est value for the scale height of thin disk is hz = 260 50 p c. In contrast with the generally adopted value around 325 p c, we nd a lower value for the scale height. This low value is in agreement with the scale height of 249 p c tted by Kuijken & Gilmore (1989) from a K dwarf photometric parallax study in the direction of the South Galactic Pole. Haywo o d (1994, 1995) showed that the overall vertical density pro le of the galactic disk is closed to an exp onential with scale height hz ' 250 p c. Ng et al. (1995) recent determination gives hz = 250 p c for the scale height of thin disk based on a sample of stars toward the galactic centre. The thick disk characteristics are, hz = 760 50
p c and lo cal density = 7.4+2 1 ::^55 % relative to the
thin disk. Robin et al. (1995) recent determination gives hz =760 50 p c, with a lo cal density of 5.61.0 % rela- tive to the thin disk. This last determination has b een done with available B, V photometric data taken quite extensively from literature. They use a di erent approach to analyse data using a synthetic mo del repro ducing ob- servable quantities (magnitude and color counts). This
Fig. 3. The density distribution for stars (3.5MV 5) in three elds (GAC1,2, GC and GAR) as a function of distance ab ove the galactic plane. The continuum line represents the sum of two exp onentials with scale height 260 p c and 760 p c and cor- resp onding to the thin disk and thick disk, resp ectively
metho d is exp ected to avoid systematic bias that can b e encountered in inversing the pro cess. Two p oints can ex- plain the small di erences b etween Robin et al. (1995) and present determinations. First, the density law used by Robin et al. (1995) is slightly di erent from our own. They use a mo di ed exp onential with a null derivative at z = 0, in order to conform to the p otential. Our density law is strictly exp onential thus the lo cal density deduced from the counts at z> 1 kp c is slightly overestimated in our case
Table 3. The b est tted structural parameters for the thin disk and thick disk stars with 3.5MV 5 derived from the three data sets (GAC1,2, GC and GAR). The scale length of the thin disk was not well determined due to p o or statistics of the data in the nearer distance bins (z< 800 p c)
Thin Disk Thick Disk Thin Disk : Thick Disk hz (p c) hz (p c) hR (p c) density ratio 260 50 760 50 3800 500 100 : 7.4+2 1 ::^55
compared to Robin et al. (1995) result. Second, there is a slight correlation b etween scale height and density when determined using star counts (Reid & Ma jewski, 1993). However, Robin et al. (1995) used a larger magnitude in- tervals (from V = 12 to 22), thus raising the indetermina- tion. Our scale height determinations are mainly based on the shap e of densities on gures 1-3, while the scale lengths dep end on ratio of counts b etween opp osite elds. Small relative photometric errors b etween elds will not intro- duce large change on the shap e of gures, but will af- fect relative counts b etween elds and mo dify the scale length. This e ect is larger when we select sample with small extent in B V (e.g. the values of the scale length of thick disk range b etween 2 and 3 kp c for the second sample (5MV 6), and 3.70.5 kp c from the rst sample (3.5MV 5)). A conservative estimate of the errors con- sists to consider the maximal uctuations of our various determinations for di erent samples and di erent tting pro cesses. We found that the scale length is not accurately measured but conversly the exact values have no large in- uence on the other measured quantities (scale height and lo cal densities). We deduce that the scale length of the thick disk is hR = 3 1 kp c. Robin et al. (1995) determination gives 2.80.8 kp c, which is based partly on the same data. These results can b e compared to recent determination given by Soubiran (1993ab), based on a sample of stars with prop er motions towards the NGP : hz ;thin disk = p c, hz ;thick disk =700 p c. Von Hipp el & Bothun (1993) gives hz ;thin disk 290 p c, hz ;thick disk =860 90 p c based on a faint Stromgren photometric survey.
- Kinematics of stellar p opulations : SEM (Sto chastic-Estimation-Maximizati on) al- gorithm
The kinematical prop erties of each stellar p opulation are related to their spatial distributions. Scale height, velo city disp ersions and asymmetric drift are linked by the Boltz- mann equation. The prop ortion of each p opulation varies with the distance ab ove the galactic plane and the selec- tion of a stellar sample at a given distance allows to opti- mize the prop ortion of one p opulation. Since kinematical data allow to improve this identi cation of p opulations, we
have minimized bias from mutual contamination of each p opulation at a given height by p erforming a kinemati- cal separation of p opulations. For that purp ose we have assumed that the kinematic of each p opulation is well ap- proximated by a maxwellian velo city distribution. In case of an isothermal p opulation and separation of vertical and radial motions, this is particularly suited, since no kine- matical gradient is exp ected, mean velo city disp ersion and asymmetric drift remain constant with height ab ove the plane. This allows to recognize one p opulation at various heights and to measure its density along the line of sight. To p erform the kinematical separation, we have used a maximum likeliho o d metho d (SEM algorithm : Celeux & Dieb olt 1986) in order to deconvolve the multivariate gaussian distributions and estimate the corresp onding pa- rameters. The aim of the SEM algorithm is to resolve the nite mixture density estimation problem under the max- imum likeliho o d approach using a probabilistic teacher step. Full details can b e found in a review pap er published by Celeux & Dieb olt (1986). Through SEM one can obtain the numb er of comp onents of a gaussian mixture (without any assumption on this numb er), its mean values, disp er- sions and the p ercentage of each comp onent with resp ect to the whole sample. This metho d has already b een used by Gomez et al. (1990) and Soubiran (1993ab) to charac- terize the (U,V,W) parameters of the stellar p opulations. The samples of stars in 2 elds (GAC1,2 and GC) have b een devided in 6 or 7 bins of distance, and in each bin of distance a t has b een p erformed with a SEM algorithm to separate the 2-D gaussian distributions to identify the three comp onents (thin disk, thick disk and halo) of the Galaxy. As can b e seen in Ojha et al. (1994a) (tables 11 & 12) and Ojha et al. (1994b) (tables 10 & 11), the p er- centage ratio of 2 p opulations (thin disk/thick disk) varies with the distance ab ove the galactic plane and therefore with the line of sight distance. The thin disk p opulation has b een identi ed as a mixture of comp onents (probably young and old disks) and thus we observe a gradient in velo city disp ersions as a function of z. The thick disk p op- ulation (table 12 in Ojha et al. (1994a) and table 11 in Ojha et al. (1994b)) has b een identi ed as a discrete and distinct comp onent. We do not nd any gradient either radial or vertical in the velo city ellipsoid of the thick disk p opulation. This proves that the thick disk is an isother- mal p opulation. The halo p opulation has b een identi ed
prop er motion surveys towards the NGP. Their results can b e compared to our results since selection criteria are very similar. Applying our distance determination (see x2) and SEM to their samples, we obtain nearly the same values for disp ersions and asymmetric drift. Then we obtain a mean U = 71 5 km/s from measurements towards the NGPs and from our eld (GAR). We mention that this value is consistent with the dynamical constraint of the asymmetric drift relation (see x6). V had b een obtained in four elds (three di erent directions : GC, GAC1,2 and NGP) and is apparently constant : mean V = 57 4 km/s. It is dicult to extract W from our determinations of U;W and U using the relation (^2) U;W = ( (^2) U + (^2) W )=2, since errors from b oth determinations are added and W is the (small) di erence of two large quantities. The rela- tive error on W is then to o large. We prefer to apply an indirect determination in order to estimate W. The scale height of an isothermal p opulation is related to its vertical velo city disp ersion. The exact relation can b e determined if the vertical p otential is known. In fact this metho d has b een frequently applied to determine the vertical p oten- tial from samples of disk stars with known vertical density distribution and velo city disp ersions. Most recent results (Bienayme et al. 1987; Kuijken & Gilmore 1989; Flynn & Fuchs 1994) have shown that the vertical p otential can b e explained by the visible mass (stellar and IMS) without advo cating for a dark matter thin disk. Such determina- tions are based on disk stars not very distant from the galactic plane, where the vertical force changes linearly, and where the scale height of these p opulations is nearly prop ortional to their velo city disp ersions. For thick disk stars considered in this pap er, their distances range from 1 to 2 kp c, where the vertical force is nearly constant, and so the scale height of the p opulation in this range of distance is nearly prop ortional to (^) W^2. So W is less de- p endent on the exact value of the p otential. Taking the p otential obtained by Bienayme et al. (1987), Kuijken & Gilmore (1989) and Flynn & Fuchs (1994), we determine a value of W = 40 km/s. Assuming the p otential of Oort (1965) with 30 p ercent of unseen matter in the disk, we would obtain W = 45 km/s. The accuracy on measured disp ersions dep ends on the numb er of stars in the sample. The samples towards the NGP are relatively small and result in larger errors (error bars given by Soubiran (1993ab) re ect directly the num- b er of stars in her sample). Errors on Kharchenko et al. (1994) data must b e similar (5 to 10 km/s). A second e ect comes from the accuracy of prop er motion that has b een exactly determined for all these surveys, and range from 1 to 2 mas/year or 10 to 40 km/s for stars at distance of 1 to 2 kp c. From the observed prop er motion distributions and with the help of the Besancon galactic mo del, we de- termine the ellipsoid of the velo city distribution corrected from errors on the prop er motions (table 5). Remark : All previous determinations have b een done assuming that the main axis of the velo city ellipsoid re-
mains parallel to the plane. The exact situation is not clear while it is sometimes claimed that the ellipsoid should p oint towards the plane in a direction b eyond the galac- tic centre. If the ellipsoid is oriented in such a way, the velo city disp ersion observed on the GC eld would imply a smaller U in this direction, and a slightly larger in the GAC eld. This implies a gradient of U , with increas- ing values of U outwards. This apparently contradicts the fact that we observe a small negative gradient for the V-disp ersions (that is not mo di ed by an ellipsoid incli- nation), and since U and V-disp ersions are kinematically linked.
Table 5. The mean kinematic parameters of thick disk (in km/s) derived from 4 elds (GAC1,2, GC, GAR & NGP). W is determined for the most probable vertical p otential (see text). VLag is with resp ect to the Sun
U V W VLag Thick disk 67 4 51 3 40 -53 10
It should b e noted that since two years the accuracy of the kinematical measurements of the thick disk p opulation has greatly improved and the results are well in agreement with each others (see table 6). It remains a controversy ab out a p ossible vertical gradient claimed by Ma jewski (1992) but not seen in data from Soubiran (1993ab), Ojha et al (1994ab) and others. The determination of the gradi- ent is of great consequence on the scenario of formation for this p opulation (see x7). The controversy seems to come from the way p opulations are distinguished from each oth- ers. The SEM algorithm (used by Soubiran 1993ab and Ojha et al. 1994ab) allows to well separate the thick disk and to evaluate its kinematics within a go o d accuracy in distance bins where it is prep onderant.
5.2.2. Multivariate discriminant analysis Because the kinematics and metallicity of thick disk p op- ulation is supp osed to di er from the thin disk and halo, it may b e p ossible to pick out thick disk stars on the basis of the kinematical di erences b etween the three p opulations. We therefore try here to nd new constraints on thick disk p opulation using samples at intermediate latitude which include photometry and prop er motions. We have used multivariate discriminant analysis (MDA) to qualify the thick disk using observations in multidimensional space (V, B-V, U-B, l & b ). A MDA is used to search for the b est discriminant axes to pro ject the data in multi-dimensional space to seek the optimal separation b etween di erent p opulations. The dis- tribution of stars along the discriminant axis (combination of the observed axis) shows the clearest separation b e- tween the thin disk, thick disk and halo. We have applied
Table 6. Determinations of the velo city ellipsoid of thick disk p opulation (in km/s)
Author(s) Thick disk
U V W VLag Wyse & Gilmore 1986 80 60 60 100 Carney & Latham 1986, 1989 { { { 30 Norris 1986,1987ab { { 35 20 Sandage & Fouts 1987 75 35 42 31
Ratnatunga & Freeman 1989 77 16 54 { {
Morrison et al. 1990 55 40 { 35 10
Ma jewski 1992 ^ 35-130 35-70 { 21-
Soubiran 1993ab 56 11 43 6 { 41 16
Kharchenko et al. 1994 67 1 54 1 { 63 1
Bartasiute 1994 64 5 49 3 42 3 39 5
Beers & Larsen 1994 63 7 42 4 38 4 20
Layden 1994 59 12 43 10 42 10 34 10
(^) The author advo cates a vertical kinematical gradient of the intermediate p opulation stars
this metho d to our data sets (GC and GAC1,2), using sim- ulations from the Besancon mo del of p opulation synthesis to separate the thick disk among other p opulations and to investigate the circular velo city of this p opulation. The advantage of this metho d is that we can extract the kine- matic parameters of stellar p opulations without estimat- ing the stellar distances i.e. by using a mo del of p opulation synthesis. Using mo del predictions, we select a subsample of stars (where the thick disk stars are in ma jority), chosen as B-
V0.8 and 14 V15.5. We used the mo del simulations
to nd the b est discriminant axes where to pro ject the data in order to separate the thick disk p opulation from the thin disk and halo. This was done using a multivari- ate discriminant analysis under the MIDAS^1 environment. Mo del simulations have b een made assuming di erent cir- cular velo cities of thick disk. The characteristics of each tested mo del are shown in Ojha et al. (1994b) (table 8). To avoid to o large Poisson noise in the Monto-Carlo simu- lations, we computed at least 10 simulations of 100 square degrees for each of the mo dels tested in our analysis. The rst discriminant axis for the circular velo city of 180 km/s of thick disk in the direction of GC is given by :
x = 0 :024(B V )+0:139(U B ) 0 : 079 V 0 : 310 l 0 : 069 b
It should b e noted that the discriminant axis varies when we change the circular velo city in mo del simulations. The resulting discriminant axis is dominated by the prop er motion along the rotation (l ) and by the U-B color index
(^1) MIDAS is an acronym for M unich Image Pro cessing
and Data Analysis System develop ed by Europ ean Southern Observatory.
due to metallicity di erences b etween the thin disk, thick disk and halo. To quantitatively estimate the adequacy of the mo dels with various circular velo cities, we applied a ^2 test to compare the distribution of the sample on the discriminant axis with a set of mo del predicted distributions assuming di erent circular velo cities of thick disk. The ^2 is given by the following formula :
^2 =
X^ n
i=
(bi ai )^2
ai
Where n is the numb er of bins and a & b are the num- b er of counts in each bin in the mo del and observed data sets, resp ectively. The resulting ^2 distribution may b e approximated by a normal distribution given by the ex- pression (Wilson & Hilferty 1931) :
f(
^2
n 1
)^1 =^3 +
9(n 1)
1 g(
9(n 1)
)^1 =^2
The ab ove expression is approximately normally dis- tributed around zero mean with unit variance (Kendall & Start 1969). Table 7 and gure 5 show the values of the probability (in sigmas) of each mo del to come from the same distri- bution as the observed sample. The most probable value for the lag or asymmetric drift of thick disk comes out
to b e -55 10 km/s (see gure 5), which shows that the
circular velo city of thick disk is of the order of 180 10
km/s (where VLS R = 229 km/s and V = 6.3 km/s in the mo del). We notice that the mo del predictions are at 3 sigmas of the data. This is due to the fact that the statis- tics of the errors in the data is not a Poisson statistics,
Using the density function of Equ. (1) the star count
ratio b etween the two elds GC and GAC1,2 (where jR
R 0 j ' jz j), assuming that the luminosity function is the
same, can b e written as :
AGC (m)=AGAC 1 ; 2 (m) = exp(+2 jR R 0 j=hR ) =
exp(
p
2 dlos =hR )
or
hR =
p
2 dlos l og (AGC (m)=AGAC 1 ; 2 (m)) By comparing the star count ratio b etween the two data sets in each distance bin, we obtain the scale length
of thin disk is hR = 2.60.6 kp c and for the thick disk
hR = 3.60.5 kp c. It app ears from the test with sim-
ulated data that SEM results were not very stable when two p opulations have the same prop ortions. This limit the accuracy of density determination in some distance bins of transitions. However the results obtained on scale length is compatible with the results obtained directly from star
counts in x 4 (table 3), comforting the separation of p opu-
lation made from the SEM metho d.
Fig. 7. The observed numb er of thin disk stars as a function of d distances obtained from SEM algorithm from the two data
sets (GAC1,2 and GC) in 0.3B-V0.9 color interval
- Solar motion and asymmetric drift relation
6.1. Solar motion
Velo city comp onents of the solar motion are usually de- duced directly from the mean motion of nearby stars rela- tive to the Sun. The tangential comp onent V of the solar
Fig. 8. The observed numb er of thick disk stars as a function of d distances obtained from SEM algorithm from the two data sets (GAC1,2 and GC) in 0.3<B-V<0.9 color interval
motion in a circular rotating co ordinate frame is based on the extrap olation of the asymmetric drift relation (, (^2) U ), where is the mean of the V velo city comp onents of a stellar sample, (^2) U is the variance. Here, we measure the solar motion from distant sample of stars (1 to 3 kp c) and compare it to the lo cal determinations. Since our samples of stars are non lo cal, we deduce the asymmetric drift relation from a general form (see Binney & Tremaine, 1987) and we obtain the expression :
Vlag = Vc (R) + Vr ot (R) =
R 2 U
2 Vc
@ l n @ R
@ l n (^2) U @ R
R
V^2
U^2
U^2
@ < U W >
@ z
or shortly
Vc (R) + Vr ot (R) =
U^2
D
under the following assumptions that the Galaxy is sta- tionary and axisymmetrical, and Vlag << Vc. Vc (R) is the circular velo city curve and Vr ot (R) is the mean rotational velo city of stars with radial velo city disp ersion U. The mean apparent velo city of a stellar sample relative to the Sun is given by :
< V >obs = Vr ot (R) Vc (R ) V (3)
combining the equations (2 & 3), we obtain :
< V >obs
@ Vc (R) @ R
(R R ) V +
2 U
D
In gure 9, we have plotted the (, (^) U^2 ; W ) p oints (with (^2) U; W = ( (^2) U + (^2) W )=2) obtained from the observa- tional data sets in two directions (GC and GAC1,2) for F and G-typ e stars (0.3<B-V<0.9). We nd that, for U; W < 19 km/s, the (, (^2) U; W ) p oints di er signi cantly from the straight line (see also Mayor 1974). In two pap ers, Mayor (1970, 1972) have shown that density waves or lo cal p erturbations disturb essentially the stellar p opulations with low velo city disp ersions. Then discarding these sub- p opulations having a low velo city disp ersions and within the limits of statistical uncertainties, the extrap olation of = 0 gives a mean V = 6.61.6 km/s. Joined with the mean values of the U and W comp onents, we obtain the solar motion : U = -1.06.0 km/s, V = 6.61.6 km/s and W = 6.46.0 km/s. The derived standard solar motion is typical for F and G-typ e disk dwarfs. These values can b e compared to those of Mayor (1974) : (-10.31.0, 6.30.9, 5.90.4) km/s and Oblak (1983) : (-8.21.8, 5.00.7, 5.50.4) km/s, determined from the lo cal sample of stars in a radius of 200 p c around the Sun. There is no signi cant di er- ences b etween the values of solar motion derived from lo- cal and distant samples of stars, implying no visible mean motion of lo cal stars relatively to distant one on the merid- ional plane.
6.2. Galactic rotation curve
The value of D dep ends on quantities that are equal in b oth opp osite elds (GC and GAC1,2). The contribution of the gradient of the circular velo city curve has an opp o- site sign in the two elds, and comparison of them gives a direct measure of rVc (R). On gure 9, the relation (< V >; 2 ) has nearly the same value for b oth elds, indi- cating a null gradient for the velo city curve. We determine rVc (R) = +0: 7 1 km/s based on a range of distances of 1 : 5 kp c and conclude that the rotation curve is at around the solar radius. Our analysis is in agreement with various determinations of the outer rotation curve (Fich & Tremaine, 1991) based on more distant tracers.
6.3. Asymmetric drift
The gradient of the velo city curve b eing null, the expres- sion of the asymmetric drift can b e written in a simpler form: < V >obs = V + ^
(^2) U D Simpli ed expressions for D are frequently found and are generally based on assumptions without any strong theoretical supp ort. A careful analysis can b e found in Fux & Martinet (1994). Current assumptions concern the last term in the bracket of Equ. (2) that is frequently replaced by its value for a spherical or a plane parallel p otential, but not by its more probable value which may b e intermediate. For z = 0, this last term can b e written
(Fux & Martinet 1994) as ( RR )(1 ^
(^2) W (^2) U^ )^ with^0 ^ ^ ^ 1.
Exact value of dep ends on unknown quantities like the attening of the dark halo. In the following discussion we will cho ose = 0 :7, according to Fux & Martinet (1994). The other assumption concerns the kinematical scale lengths that is currently put equal to the density scale length. This is partly based on Lewis and Freeman (1989) kinematical observations. Their kinematical gradi- ent seems to b e con rmed by few existing measurements including those presented in this pap er. However the value they adopt for the density scale length is now very ques- tionable and is much larger than the most recent direct (non-kinematical) determinations. Then we cannot assess equality of density and kinematical scale length for our own Galaxy. If z 6 = 0, the vertical derivative of density is not zero. Since we are analysing samples of stars far outside the mid-plane of the Galaxy, a new term app ears inside the bracket of Equ. (2). For an exp onential disk of the form (z ) = exp( (z =h)n^ ), we obtain :
2 Vc Vlag R (^) U^2
2(A B ) D
=
H
H 2 U
R
(1
2 V
U^2
(R)
R
(1 n(
z h
)n^ )(1
2 W
2 U
where A and B are Oort's constants and H , H (^2) U are density and kinematical scale lengths. This relation shows that for an isothermal p opulation, the asymmetric drift do es not change with the distance ab ove the galactic plane if and only if the p otential is cylindrical ( = 0). We can estimate for typical values ( = 0 :7, n=1) that the lag of a p opulation with U = 30 km/s will vary by only 1 km/s on one scale height, and for U = 70 km/s by 5 km/s.
6.3.1. Thick disk
We have informations for all dominant terms present in the asymmetric drift relation for the thick disk p opula- tion. We have measured the scale length directly from star counts, the asymmetric drift, V in all elds, U; W for GC and GAC1,2 elds, and we get U from indep en- dent prop er motion surveys. So we can determine one of these quantities from the others, and try to determine the self-consistency of observed quantities. Since the scale height of thick disk is 760 p c and the distances of observed stars range b etween 1 to 2 kp c ab ove the galactic plane where the Kz force is nearly constant and the density is exp onential, we get n = 1 and z =h 1 : 5 3 in the last term of the asymmetric drift relation. We apply the asymmetric drift relation in order to obtain an estimate of U by replacing all the other known quantities. Velo city gradients are null for the thick disk p opulation. Substituting VLag = -53 km/s (with resp ect to the Sun), Vc =220 km/s, hR =2.8 kp c, R=8.09 kp c and = 0 : 7 in Equ. (2), we obtain U = 72{82 km/s, in agreement with
disk o ccurs due to kinematically heating of thin disk. This heating may have b een violent, p erhaps from the accre- tion of a satellite galaxy (Carney et al. 1989, Quinn et al. 1993). So the merging of one or more smaller galaxies with our Galaxy is a p ossible explanation for the forma- tion of a thick disk. In such a scenario, the thick disk could b e either the direct relic of the merged galaxy, dissolved and smo othly distributed in our Galaxy, or an old thin disk formed b efore the merger event(s) and `pu ed up' by the gravitational p erturbations of the merging galaxy. In case of mergers b etween disks and satellites, b oth the radial and vertical structures of the disk are altered. The radial heating and re-arrangement of material by angu- lar momentum transp ort results in an asymmetric drift (Wyse 1994). The vertical heating increases the disk scale height by ab out a factor of two. Direct evidence for an on- going merger event has b een recently discovered towards the galactic bulge by Ibata et al. (1994). Recently Quinn et al. (1993) (hereafter QHF) have pre- sented a mo del of thick disk, which is based on a scenario of satellite accretion by a spiral galaxy which may pro- duce a thick disk if the event o ccurs at the b eginning of the life of the thin disk. According to this mo del the satel- lite accretions do gradually thicken disks by dep ositing ordered kinetic energy into random motions of disk stars. The abruptness of accretion origin implies a kinematical, chemical and age disjointedness from the thin disk. Of course the N-b o dy simulations of QHF is only illus- trative, but our measurements repro duce most of features of their numerical simulations. Asymmetric drift ( 50 km/s) and scale height (760 p c) of the thick disk are in go o d agreement with the QHF mo del. Our measurement of a vertical velo city disp ersion of 40 km/s for the thick disk is also well in agreement with the QHF accretion mo del. Also the velo city disp ersion gradient of the QHF's thick disk is much smaller than for the disk, and is nearly zero in the outer part. This corresp onds to the null gradi- ent observed. Scale lengths of thin disk and thick disk are also in agreement if we are in the outer part of the QHF mo del. This estimate is suggestive, however, that the thick disk mo del, which has a vertical velo city disp ersion of 45 km/s, could b e formed from a thin disk with vertical disp ersion of 20 km/s by accretion of a satellite of ab out 10% of the mass of the disk. Our measurement of the scale height of thick disk ( p c) is also in p erfect agreement with QHF mo del. How- ever, our data do not allow to observe the correlation of scale height with the galacto centric radius as QHF have predicted in their N-b o dy simulations. Finally the lo cal surface density of the thick disk is ab out 18% of thin disk, this gives a crude estimate of the total mass ratio of thick disk and thin disk stars. This im- plies that the merging o ccurs quite early in the Galaxy life when 5-10% of the disk was formed. If the star formation rate has b een nearly constant (Haywo o d et al. 1995) at the ep o ch of formation, this indicate that the merge o ccurs 2
to 4 billions years after the formation of disk comp onent (of course this primordial-disk has b een transformed in thick disk). The main feature we determine for the thick disk is its isothermality and its clear separation from thin disk and halo. This comp onent is clearly simple and not a mixture of sub-comp onents as for the thin disk and there is no gradient in the kinematic neither vertically nor radially. Similar feature has b een obtained by Gilmore et al. (1995) and Robin et al. (1995), who nd a null vertical gradient in metallicity : this last result can b e understo o d as a consequence of the isothermality of the thick disk. Concerning alternative explanations, we mention Burkert et al. (1992) who build a 1-D (vertical) mo del of collapse of our Galaxy and formation of galactic disk. Their self-regulated chemical and dynamical evolution of an initially hot, gaseous proto disk leads to the formation of a thick disk. This is the only \top-down" scenario of formation for the thick disk where some discontinuity b e- tween thin and thick disk is predicted with an acceptable agreement with our observations. However a more accu- rate chemico-dynamical mo dels (Samland, 1994), with a 2-D resolution show a more continuous age, metallicity and kinematic gradient b etween the various disk comp o- nents. To sum up, the results emerging from the present study of the correlations b etween photometry and kinematics give a mounting evidence in favour of merging pro cess of satellite galaxies with the disk of our Galaxy for the formation of the thick disk.
- Conclusion
In this pap er we have used star counts and kinematical data to constrain the galactic structure parameters. We have also shown that a prop er statistical analysis of the data in the 5-dimensional space (V, B-V, U-B, l , b ) in comparison with the mo del of p opulation synthesis allows one to constrain the physical mo del parameters. We obtain a value of -0.180.03 kp c ^1 for the galac- tic radial gradient of velo city disp ersion ( @ ln (^2) U; W @ R )^ for^ the thin disk p opulation. Determined from mo derately distant stars, a new measurement of the solar motion is obtained. We found no systematic motion of the LSR relative to dis- tant stars on the galactic meridional plane. The rotational velo city curve is found at in the solar neighb orho o d. Our results con rm that the thin disk has a relatively short scale length of 2.30.6 kp c and scale height of 260 50 p c. The thick disk p opulation is distinct from other p opula- tions based on their kinematical and spatial distributions. The data constrain the asymmetric drift of the interme- diate p opulation, which is found to b e 53 10 km/s with resp ect to the Sun. No radial or vertical gradient is found in the rotational velo city and velo city disp ersions of the thick disk p opulation. We therefore conclude that the for- mation of the thick disk did not o ccur as a smo othly con-
tinuous transitional phase b etween formation of the halo and formation of the thin disk during the collapse of the Galaxy (Eggen, Lynden-Bell & Sandage 1962). Rather, it supp ort the mo del of a scenario of satellite accretion by our Galaxy which may pro duce a thick disk if the event o ccurs at the b eginning of the life of the thin disk (Quinn et al. 1993). The most probable value of scale height for the thick disk comp onent is determined to b e hz ' 760 50
p c with a lo cal density of Athick = 7.4+2 1 ::^55 % relative to
the thin disk. The ratio of the numb er of thick disk stars in our galactic centre region to that in anticentre region yields hR 3 1 kp c for the scale length of thick disk. These values are in p erfect agreement with the recent determina- tion given by Robin et al. (1995) based on the analysis of a large set of available photometric catalogues with accurate photometry.
Acknow ledgements. This research work was partially sup- p orted by the Indo-French Centre for the Promotion of Ad- vanced Research (IFCPAR) / Centre Franco-Indien Pour la Promotion de la Recherche Avancee (CFIPRA), New Delhi (India). We esp ecially thank referee Dr. Gerry Gilmore for his comments. We also thank Caroline Soubiran and Elena Schilbach for letting us use their data.
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