Lastly we apply the fitting formula to data release 8 of Sloan Digital Sky Survey (SDSS, York et al., 2000) for stars that were observed spectroscopically with the SEGUE Survey (Yanny et al., 2009). Stellar parameters for the stars were estimated using the SEGUE Stellar Parameters Pipeline (SSPP, Lee et al., 2008a,b; Allende Prieto et al., 2008). In contrast to the Geneva- Copenhagen-Survey, which combined radial velocity measurements with the Hipparcos paral-
160 170 180 190 200 210 220 230 240 0 0.5 1 1.5 2
V / kms
-1z / kpc
using the adiabatic potential simple adiabic approximation
Figure 7.2: Comparison of the mean rotational velocities against altitude above the plane with and without the correction of the adiabatic potential. We used an adaption of the formalism of Binney (2010), using the potential from McMillan (2011).
altitude/pc 0−400 400−800 800−1200 1200−1700 1700−2300 z>2300 num. stars 1002 7243 7630 5601 3637 3217 mean z/pc 433 595 985 1423 1971 2808 ln(P) −80.10 −195.23 −93.64 −88.22 −88.35 −87.1 1 - p-level 0.999 1 0.734 0.194 0.606 0.672 σ0 32.6 31.6 34.1 37.75 40.59 39.42 h0 432.8 594.6 856.1 691.7 651.3 472.7 adoptedα 1.38 1.08 0.72 0.28 −0.27 −1 v2R 32.02 35.44 39.60 45.89 52.41 55.0 v2Rdata 42.63 45.79 49.12 55.25 60.39 68.3
Table 7.1: Fit parameters and observers at different altitudes in the SEGUE DR8 sample. Dis- persion values are from stars with|U|<200 km s−1and|W|<80 km s−1.
0 -100 -50 0 50 100 150 200 250 300 V/kms-1 0 100 -100 -50 0 50 100 150 200 250 300 V/kms-1 0 -100 -50 0 50 100 150 200 250 300 V/kms-1 1 10 100 -100 -50 0 50 100 150 200 250 300 counts V/kms-1 |z| < 0.4 kpc fit at |z| = 0.32 kpc z > 2.3 kpc fit at |z| = 2.81 kpc 1 10 100 1000 -100 -50 0 50 100 150 200 250 300 counts V/kms-1 0.4 kpc < |z| < 0.8 kpc fit at |z| = 0.62 kpc 0.8 kpc < |z| < 1.2 kpc fit at |z| = 0.98 kpc 1 10 100 -100 -50 0 50 100 150 200 250 300 counts V/kms-1 1.2 kpc < |z| < 1.7 kpc fit at |z| = 1.42 kpc 1.7 kpc < |z| < 2.3 kpc fit at |z| = 1.97 kpc
Figure 7.3: Velocity distributions from SEGUE DR8 versus the fits with parameters from table 7.1 at different altitude bins. Error bars give the Poisson noise, but do not account for systematic or velocity errors.
laxes and proper motions, SEGUE only contains line-of-sight velocities, proper motions and stellar parameters. The survey is far deeper and covers a larger volume reaching higher altitudes, but due to the larger distance proper motion errors translate into larger velocity errors. As there are no parallaxes, distances are highly uncertain and every study of kinematics is complicated by non-Gaussian errors arising from the distance uncertainties. As a first approach we adopt the main sequence calibration of Ivezic et al. (2008) (Eq. A7 in their Appendix) to obtain photo- metric parallaxes for stars with gravities log(g)>4.1 determined in the SSPP. An alternative that would reduce the loss of available objects would be to use the method of Burnett & Bin- ney (2010), however, we have sufficient numbers of stars so that we can afford to limit the risk of systematic mis-selections by restricting the sample to main sequence stars with a tight sur- face gravity cut. The selection of stars, distance determination and derivation of kinematics are done as in Sch¨onrich, Asplund & Casagrande (2011a) (see esp. their appendix). In contrast to Sch¨onrich, Asplund & Casagrande (2011a) we do not make use of the calibration sample, but use instead all objects that fulfil the target selection criteria of the categories: F turnoff, low metal- licity, K dwarf, F/G dwarfs, M sub-dwarfs, G dwarfs in SEGUE1 and of the categories: MS turnoff, Low metallicity for SEGUE2. This gives 191522 unique entries among which 28830 pass our quality criteria plus a cut for 0.2<(g−i)0<0.7 (to remove red and very blue stars), have [Fe/H]>1.0, are considered to be within 4 kpc distance and satisfy the surface gravity cut log(g)>4.1.
system, where we adopt again the proper motion of the Sun from Sch¨onrich, Binney & Dehnen (2010), a standard rotation velocity of 220 km s−1and a galactocentric radius of R=8 kpc. Ve- locities were transformed into the local coordinate systems at the stars’ supposed position. The necessity for the latter transformation is given by the extension of the sample and reduces the systematic drift in radial velocities.
For our tentative fits we cut the sample into six altitude regions and performed separate fits. At each altitude we fit the sum of a simple Gaussian halo component and a disc component with the presented analytical function including the adiabatic correction and re-correction. As there are very few halo stars in this sample, it contains insufficient information for a determination of the halo velocity distribution and hence we use the values from Sch¨onrich, Asplund & Casagrande (2011a) fixing the halo velocity distribution at Galactic rest with an azimuthal velocity dispersion of 70 km s−1. Its normalisation is the only free fitting parameter applied to this component. As errors are relatively large, information inherent in the velocity distribution is limited and so we fix the disc scale length at Rd=2.5 kpc, the adiabatic correction toα=1.7−0.01|z|/pc. As we
presented in Sch¨onrich, Asplund & Casagrande (2011a) very reasonable fits without the vertical energy re-correction, we this time show results making use of it.
Due to the large uncertainties of distances and proper motions an elaborate error analysis is es- sential to get at least a rough picture on the real distribution of velocities underlying the observed distribution. We identify three sources of errors in the distribution: The ”direct” errors from proper motions and the radial velocity determinations, the uncertainties in distance determina- tions directly stretching the V velocities and velocity ”cross-overs” (cf. Schoenrich et al. in prep.) from the other velocity components by distance errors.
The first category is easily accounted for: At each altitude we performed an error propagation of the line-of-sight velocity errors and proper motion errors onto the velocity components. This error propagation delivers the error distribution from proper motion errors as a sum of Gaussians (for different error amplitudes), which is then folded onto the theoretical distribution. We can write this error function caused by the proper motion errors as:
errpm(∆V) =Nerr,pm
∑
igi(∆V,σV,i,0) (7.5)
where the sum runs over all stars going into the distribution and gi((∆V,σV,0) is the unbiased
Gaussian error of withσV,i for star i. We thus estimate the width of the error for a single star
to be the square root of the sum of the squared proper motion errors and squared radial velocity errors on the velocity component in question, which are derived by inserting the errors instead of their observables into the terms connecting the velocity components to the observables. The normalisation Nerr,pm just ensures that errpm(v)has weight 1:
Z
errpm(∆V)d(∆V) =1 (7.6)
The second major source of errors is the uncertainty of distances, leading to a second error distri- bution errd(v). Apart from the systematic uncertainties in the choice of the locus of isochrones,
decreasing uncertainty by influence of metallicity on the isochrones with decreasing metallicity is partly balanced by an increased intrinsic scatter in the parameter determinations. As described in Sch¨onrich, Asplund & Casagrande (2011a) the distance error from a Gaussian distance mod- ulus error is highly non-Gaussian, but has a prolonged tail towards larger distances. We obtain the distance error distribution by transforming the Gaussian distance modulus error into relative distance space. For the azimuthal velocity V we calculate the average part of the squared V ve- locity component in the sample that is carried by proper motions, ζV. As the sample is rather
high up in the sky and additionally concentrates away from the the apex and antapex of a circu- lar orbit, the average proper motion partition on V approaches 1 (cf. also Sch¨onrich, Asplund & Casagrande, 2011a). The distance error distribution translates into a map of the observed V velocity distribution onto itself that we to zeroth order approximate as:
V →V+ (V−V⊙)ζV∆
s
s (7.7)
where∆s/s is the relative distance mis-estimate, and(V−V⊙)is the stellar velocity in a heliocen- tric frame. This is not the only term. Additionally there will be small crossovers from the U and V velocities, that give a minor error term to the V velocities, which should again be shaped as a sum of Gaussians. We approximate this by a single Gaussian error term errcv(V) =g(∆V,7 km s−1,0)
(7 km s−1 would correspond to a 70 km s−1 velocity dispersion at a conversion factor of 0.5 and an effective distance error of 20%. This has also the pleasant effect to waive a tiny numeri- cal instability that happens by the integration of errd at the solar velocity V⊙. The entire error
calculation is done by first folding errpm, then errd and finally errcvonto the data:
fobs=errcv◦errd◦errpm◦f (7.8)
where f is the theoretical velocity distribution and fobs is its estimated observational represen-
tation. The folding with the error functions is numerically described with matrices mapping the velocity space onto itself to keep the calculation budget at a reasonable value. Due to their dif- ferent positions in the sky we treat the halo and disc components separately and take all stars with V <50 km s−1for assessing the halo errors and all stars with V >80 km s−1as basis for an assessment of the disc velocity errors. We checked that this analytic approach is equivalent to the results of a Monte Carlo scattering of the stars in distances and the observables. Numerically (to speed up the calculations) the error propagation was solved by a transition matrix in V velocity at 1 km s−1resolution.
Table 7.1 shows fitting parameters and observables for the data sets at different altitudes binned in 10 km s−1intervals. Fits were performed in the azimuthal velocity range−250 km s−1≤V ≤
280 km s−1, the left limit fully covering the halo velocity distribution to get the right normalisa- tion, the right limit near the upper edge of the theoretical velocity distribution and thus placed inwards of the high velocity edge of the observed distribution to reduce the influence of mere measurement error on the fits.
For the fits we used four free parameters, namely the normalisation of disc and halo, the local horizontal dispersionσ0 and the local scale height h0. The p-levels given are taken in the disc
component region and hence they are a bit below what could be attained if the fit was performed on that range exclusively. Yet it can be seen, that in the two lowest bins the fit does not yield a proper result, mostly by the fitting function being 3−5 km s−1shifted against the observations. This can have three reasons: First it is an indication that a mixture of more than one disc com- ponent could be required in this region (indeed by introducing another disc component the result can be improved). Second we expect large and partly systematic distance errors in the formula we used. This can to some extent shift the peak towards solar velocity (distance underestimates) and lead to a wrong shape in the distribution. And third the adiabatic re-correction for vertical energy could exaggerate the effect, offsetting the distribution too much towards the low velocity side.
At the altitude bins above 800 pc the fits are of very good quality, indicated also by p-levels that exclude any significant deviation from the model prediction. The local scale height differs a bit from the expected values, yet it’s value is uncertain and depends on the assumption for the vertical energy correction.
In an upcoming study we will make use of our improved distances to get a better characterisation of both SEGUE and RAVE data and will publish the above in the revised framework.