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ABSTRACT

We present an analysis of the kinematics of Galactic disc white dwarf (WD) stars in the solar neighbourhood using data from Gaia Data Release 2. Selection of WDs based on parallax provides the first large, kinematically unbiased sample of solar neighbourhood WDs to date. Various classical properties of the solar neighbourhood kinematics have been detected for the first time in the WD population. The disc WD population exhibits a correlation between absolute magnitude and mean age, which we exploit to obtain an independent estimate of the solar motion with respect to the local standard of rest. This is found to be (U, V, W)⊙ = (9.5 ± 1.2, 7.5 ± 1.2, 8.2 ± 1.2) km s−1. The UW components agree with studies based on main-sequence stars, however the V component differs and may be affected by systematics arising from metallicity gradients in the disc. The velocity ellipsoid is shown to vary strongly with magnitude, and exhibits a significant vertex deviation in the UV plane of around 15 deg, due to the non-axisymmetric Galactic potential. The results of this study provide an important input to proper motion surveys for WDs, which require knowledge of the velocity distribution in order to correct for missing low-velocity stars that are culled from the sample to reduce subdwarf contamination.

white dwarfs, Galaxy: disc, Galaxy: kinematics and dynamics, solar neighbourhood

1 INTRODUCTION

The stars in the vicinity of the Sun move under the combined gravitational influence of all the other stars, gas, and dark matter in the Galaxy, and by analysing their motions it is possible to infer aspects of the dynamical structure, evolution, and physical processes at work in the Milky Way as a whole. These include the kinematic heating of stars through both secular processes and merger events, the local circular velocity and slope of the rotation curve, and the shape of the gravitational potential. Knowledge of the velocity distribution of stars is also relevant from a survey astronomy point of view, as it is a vital input when quantifying the completeness of proper motion surveys (e.g. Rowell & Hambly 2011, hereafter RH11; Lam et al. 2019).

This field has a long history, but went through something of a renaissance with the release of data from the Hipparcos mission (e.g. Feast & Whitelock 1997; Dehnen & Binney 1998; Mignard 2000; van Leeuwen 2007; Schönrich, Binney & Dehnen 2010) which provided stringent constraints on various parameters of the Milky Way derived for the first time from absolute parallaxes. In particular, Dehnen & Binney (1998) (hereafter DB98) used a sample of 16 054 main-sequence (MS) stars to measure the solar motion with respect to the local standard of rest (LSR), an important quantity for transforming heliocentric velocities to a Galactic frame. This classic technique exploits the correlation between colour and mean age that exists for MS stars bluewards of the turn-off of the old disc at B − V ∼ 0.6; the break in the trend of velocity dispersion against colour that exists at this point is known as Parenago’s discontinuity. However, as explained by Schönrich et al. (2010) the DB98 methodology is undermined by metallicity gradients within the disc that lead to complex kinematic substructure in the colour–magnitude plane. This field is now set to be truly revolutionized by the data from the Gaia mission (Gaia Collaboration 2016), the successor to Hipparcos, which ultimately aims to measure absolute parallaxes and proper motions at the microarcsecond level for more than one billion stars in the Milky Way down to G ≈ 20. The second data release (DR2 – see Gaia Collaboration 2018a) was published on 2018 April 25, and already contains over 1.3 billion sources with parallax accuracies of the order 700 μas at the G = 20 faint end. Much progress is still to be made at reducing systematic errors by improving various instrument calibrations, but early studies demonstrate the enormous potential of the Gaia data. These include Bovy (2017), who measured the Oort Constants that parametrize the local stellar streaming velocity, obtaining the first significant estimates of the C and K terms. Gaia Collaboration (2018b) used a subsample of 6.4 million FGK stars with full 6D phase space coordinates to draw 3D maps of the stellar velocity distribution with unprecedented accuracy, revealing rich substructure and complex streaming motions in the Galactic disc. On a larger scale, Kawata et al. (2019) used a sample of 218 Cepheids with Gaia proper motions to estimate several parameters of the Galactic rotation, including the centrifugal velocity, the slope of the circular speed curve, the local velocity dispersion and the Sun’s peculiar velocity.

One avenue that has recently been opened by the Gaia DR2 is the possibility of conducting a kinematic study using the white dwarf (WD) stars in the solar neighbourhood. Both Hipparcos and Gaia DR1 (TGAS) contain only very small numbers of WDs that cannot be used for a systematic study. Instead, the largest quantifiably complete catalogues of WDs to date have been derived using the reduced proper motion (RPM) method applied to ground-based survey data. These include the Sloan Digital Sky Survey (Kilic et al. 2006; Munn et al. 2017), the SuperCOSMOS Sky Survey (RH11), and PanSTARRS 1 (Lam et al. 2019), each of which contain of order ∼10 000 WDs. The problem with these catalogues, from a kinematic point of view, is that the RPM method necessitates the use of a low tangential velocity threshold of around 30 km s−1, below which stars are rejected in order to limit contamination from subdwarfs. This results in a catalogue that is suitable for luminosity function studies but is inherently kinematically biased. Note that the SDSS spectroscopic WD catalogue (Kleinman et al. 2013; Kepler et al. 2015), which contains around 30 000 WDs in total, is biased by the complex selection function for fibre allocation.

The Gaia DR2 on the other hand contains several hundred thousand WDs (Gentile Fusillo et al. 2019) that can be reliably identified based on parallax. The need to construct a robust sample within well-characterized survey limits requires use of stricter selection criteria that reduces the useful sample size, but even so this is orders of magnitude larger than any previous comparable WD survey. Although in any magnitude-limited survey, such as that conducted by Gaia, WDs will exist in much smaller numbers than MS stars, analysis of their kinematics is still of importance for the following reasons:

  • In a volume-limited sample such as the solar neighbourhood the WDs make up a much more significant fraction of the population, and any study that omits them can only be an incomplete picture.

  • The location of a WD in colour–magnitude space is independent of the progenitor star metallicity, which may avoid certain systematics that arise in similar studies that use MS stars, as described in Schönrich et al. (2010); briefly, metal-rich stars generally formed at R < R0 and visit the solar neighbourhood close to the apocentre of their orbit where their azimuthal velocities lag behind the local circular speed. As metallicity displaces the MS, this introduces substructure in the colour–magnitude diagram that correlates with kinematics.

  • For solar neighbourhood WDs there exists a correlation between absolute magnitude and mean age, analogous to the colour/age correlation that exists for MS stars, which can be exploited to obtain an independent determination of the motion of the Sun with respect to the LSR.

The final point holds for any WD population that did not form in a single burst, and is due mainly to the monotonic cooling law (Mestel 1952). This magnitude/age correlation exists for essentially all realistic star formation histories, though the strength of the correlation varies and requires detailed modelling to quantify. In Fig. 1 [reproduced from Rowell (2013)] we depict the mean total stellar age as a function of bolometric magnitude for a range of simulated WD populations of different star formation history. The correlation is strongest in the Mbol  = 13–15 range and for the less burst-like star formation histories. The Gaia G magnitude is expected to show a similar trend due to the extremely broad wavelength range leading to small bolometric corrections of ΔMbol ≤ 0.09 for both H and He atmosphere WDs over the G  = 13–15 range (Bergeron, private communication).

The mean total stellar age as a function of bolometric magnitude for a range of simulated WD populations of different star formation rate ψ(t). The solid and dot–dashed lines are for a constant ψ with a total age T0 = 10 and 13 Gyr, respectively; the dashed line is for ψ(t) ∝ exp (−t/τ) with τ = 3 Gyr and T0 = 13 Gyr; the dotted line is for ψ(t) ∝ (1 + exp ((t − t′)/τ)) − 1 with t′ = 10 Gyr, τ = 3 Gyr, and T0 = 13 Gyr. This figure is reproduced from Rowell (2013).
Figure 1.

The mean total stellar age as a function of bolometric magnitude for a range of simulated WD populations of different star formation rate ψ(t). The solid and dot–dashed lines are for a constant ψ with a total age T0 = 10 and 13 Gyr, respectively; the dashed line is for ψ(t) ∝ exp (−t/τ) with τ = 3 Gyr and T0 = 13 Gyr; the dotted line is for ψ(t) ∝ (1 + exp ((t − t′)/τ)) − 1 with t′ = 10 Gyr, τ = 3 Gyr, and T0 = 13 Gyr. This figure is reproduced from Rowell (2013).

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In this study we use Gaia DR2 to assemble a catalogue of 78 511 WDs in the solar neighbourhood. This is the largest kinematically complete sample of WDs to date. We use this to investigate several classical features of the solar neighbourhood kinematics, including the asymmetric drift, LSR and age–velocity dispersion relation. Note that this study is independent of WD cooling models, being based on only direct observational quantities. In Section 2 we outline the construction of our WD sample. Section 3 explains the methods used to determine the kinematics, which are based on the proper motion deprojection method presented in DB98. We present our results in Section 4 with further discussion in Section 5, and draw conclusions in Section 6.

2 WHITE DWARF SELECTION

The sample of WDs used in this work was obtained from the Gaia data by selection in the Hertzsprung–Russell diagram (HRD). We applied initial filters on the signal-to-noise of the photometry and astrometry to reject objects with relative parallax errors greater than 20 per cent and errors on GBP/GRP flux greater than 10 per cent. This reduces the scatter in the HRD and allows the absolute magnitude to be determined based on simple inversion of the parallax. We also place cuts on the flux excess factor to eliminate objects whose photometry is compromised by blending issues, and we reject objects further than 250 pc from the Sun for reasons explained in Section 3. This resulted in an initial sample of 3061 480 objects drawn from the Gaia archive; the ADQL query that performs this selection is presented in Appendix  A.

We applied additional astrometric quality criteria to reduce contamination from objects with poor astrometric solutions, by rejecting objects with a Renormalized Unit Weight Error [RUWE; described in Lindegren (2018)] greater than 1.4. This has the effect of removing objects lying between the MS and WD locus, which includes many partially resolved close pairs that the Gaia astrometric solution is not yet capable of handling properly. We also reject objects with tangential velocity greater than 150 km s−1 as these are likely not disc members. This reduces the initial sample to 2644 657 well-measured solar neighbourhood disc stars. The HRD of these objects is presented in Fig. 2.

Hertzsprung–Russell diagram of 2644 657 well-measured disc stars within 250 pc. The red line marks the WD selection region in the Gaia bands defined by Gentile Fusillo et al. (2019), which contains 79 225 WDs. Sources have been filtered using the RUWE (see text).
Figure 2.

Hertzsprung–Russell diagram of 2644 657 well-measured disc stars within 250 pc. The red line marks the WD selection region in the Gaia bands defined by Gentile Fusillo et al. (2019), which contains 79 225 WDs. Sources have been filtered using the RUWE (see text).

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We select WDs from the lower left region of the MG/GBP − GRP plane using the same criteria as Gentile Fusillo et al. (2019). This region is marked by a red boundary in Fig. 2, and contains 79 225 WDs. There is also clearly a small amount of contamination from unresolved WD + dM systems lying in the region between the MS and WDs, which will add some degree of noise to our results in the MG range ≈13–15. Although the astrometry for these objects is presumably reliable, having passed our selection criteria, the photometry will be compromised. The fraction of contaminating sources as a function of MG can be estimated from the red tail of the colour distribution; we estimate that the contamination is limited to less than a few per cent at all magnitudes. We do not expect this level of contamination to have a significant effect on our results. We also restrict the analysis to WDs with absolute magnitudes in the range [10:15.5] for the following reasons. Brighter than MG = 10 there are very few WDs and the correlation with age is non-existent. Fainter than MG = 15.5, the WD population starts to become heavily contaminated by thick disc and spheroid objects that have different kinematics, as well as low- and high-mass WDs that have not followed the same evolutionary path as the rest of the population and so do not obey the same magnitude / mean age correlation. This results in a clean sample of 78 511 WDs. The sky distribution of these is presented in Fig. 3. Note that the lower density of stars towards the Galactic centre is not due to dust extinction, as our sample is relatively local. Instead this is most likely caused by various issues that Gaia faces in crowded regions, in particular blending of objects in both the photometric and astrometric instruments, but also challenging cross-match and loss of faint stars due to onboard resource limitations.

Sky distribution in Galactic coordinates of the 78 511 WDs that pass our selection criteria. The lower density of sources towards the Galactic centre is most likely due to problems that Gaia faces in crowded regions, as explained in the text.
Figure 3.

Sky distribution in Galactic coordinates of the 78 511 WDs that pass our selection criteria. The lower density of sources towards the Galactic centre is most likely due to problems that Gaia faces in crowded regions, as explained in the text.

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3 METHODS

The WDs in Gaia DR2 have parallax and proper motion estimates but no radial velocities. Even the final Gaia catalogue will have few radial velocities for WDs, due to their intrinsic faintness and the fact Gaia’s radial velocity spectrograph is tuned to the calcium triplet that does not appear in the spectra of the majority of WDs. Although the lack of radial velocities means the 3D space motion of individual stars cannot be recovered, we can still calculate the moments of the 3D velocity distribution by means of the proper motion deprojection technique introduced by DB98. In this method, we first correct the proper motions of stars for the effects of Galactic rotation using equation (1) from DB98, with values of the Oort constants A = 15.3 and B = −11.9 taken from Bovy (2017). The remaining tangential velocity of each star|$\underline{p}$|, which is estimated from the parallax and corrected proper motion, is related to the 3D space motion |$\underline{v}$| according to
(1)
where the projection matrix |${\boldsymbol {\sf A}}$| is defined by |$\mathbb {I} - \underline{\hat{\boldsymbol{ r}}} \otimes \underline{\hat{\boldsymbol{ r}}}$|⁠, with |$\underline{\hat{\boldsymbol{ r}}}$| the unit vector towards the star and ⊗ denotes the outer product. Obviously, this equation cannot be inverted to obtain |$\underline{v}$| and as such the matrix |${\boldsymbol {\sf A}}$| is singular. However, by averaging over a large number of stars distributed widely across the whole sky, the mean matrix 〈|${\boldsymbol {\sf A}}$|〉 is non-singular and the equation can be inverted to obtain the mean velocity |$\left\langle \underline{v}\right\rangle$| of the population relative to the Sun. The residual velocities of stars can be used to estimate the scalar velocity dispersion, and an extension of this method allows the full velocity dispersion tensor to be computed. We refer the reader to DB98 for further details. As explained in McMillan & Binney (2009), the method is strictly only applicable when the lines of sight to individual stars are uncorrelated with the stars’ velocities. This criterion holds for stars within a limited distance from the Sun, beyond which the velocity dispersion varies with both radius and distance from the plane. For this reason we restrict the analysis to stars within 250 pc. In any case, the Gaia apparent magnitude limit of |$G\mathbf {\approx }21$| means that fainter than around |$M_G\mathbf {\approx }13.7$| all our stars are within 250 pc anyway, and according to Rowell (2013) the correlation between magnitude and total age is very weak brighter than |$M_G\mathbf {\approx }13$|⁠.

Note that this method is superior to simply assuming zero radial velocity and computing the sample mean and covariance, which is done in some studies. The assumption of zero radial velocity causes the mean velocity to be underestimated by (on average) one third. The velocity dispersion is also underestimated by up to one third due to a combination of the loss of the peculiar velocity along the line of sight and the bias in the estimated mean.

We apply the kinematic analysis independently to bins of different G magnitude. We tried bins containing equal numbers of stars, but this conceals important structure around the faint end of the range where there are relatively few stars. After some tests we settled on bins of a fixed width of ΔMG = 0.125.

4 RESULTS

We applied the methods described in Section 3 to our sample of 78 511 WDs, independently on 44 bins of ΔMG = 0.125 between MG = 10 and MG = 15.5. Table 1 presents the mean velocity and velocity dispersion tensor for each magnitude bin. In this section we highlight some selected results from these data.

Table 1.
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For each magnitude bin, we present the mean stellar velocity relative to the Sun (columns 3–5) and the velocity dispersion tensor (columns 6–11). Units are km s−1 and km2 s−2 respectively, and quantities are referred to the Galactic coordinate frame with principal axes U (towards Galactic centre), V (in the direction of rotation), and W (towards the north Galactic pole).

Bin centreNMean velocity (km s−1)Velocity dispersion (km2 s−2)
MG (mag)(stars)VUVVVWΣUUΣVVΣWWΣUVΣUWΣVW
10.06382−15.88−20.84−9.491846.13627.04227.49562.22461.50−207.45
10.188119−2.05−18.56−9.031331.90685.98259.71225.83−341.21146.84
10.313166−13.20−19.16−7.691381.77550.74140.47263.70129.24321.92
10.438223−6.95−22.92−9.361407.75626.93250.58−1.8522.8070.62
10.563349−7.81−22.40−7.301171.79522.92417.5615.1627.7082.71
10.688500−9.28−19.68−8.951118.97471.78266.60350.7773.7066.07
10.813635−9.31−19.47−7.891248.14592.03295.02165.80−40.0718.47
10.938786−10.01−18.75−8.881257.76498.45341.81299.97−112.5312.59
11.0631048−8.19−20.91−9.151212.53550.68258.01162.8872.62105.13
11.1881186−8.00−18.66−8.281328.73508.88308.12232.6885.83116.67
11.3131390−9.55−20.28−8.001199.62542.13256.45193.32−53.75−21.61
11.4381574−9.34−20.77−7.731036.90516.40281.83118.3987.60−19.37
11.5631690−10.25−20.02−8.251254.47567.40314.16384.9745.03−11.15
11.6882034−10.35−20.11−6.941245.43553.62313.30195.89−0.5316.19
11.8132102−9.89−20.07−8.431124.08510.88301.71249.5910.0931.51
11.9382243−8.29−19.67−7.141196.07510.06275.11162.786.94−19.39
12.0632224−9.97−19.05−7.281158.61485.98249.41244.89−28.9417.43
12.1882371−8.99−18.32−6.201177.38544.20248.29266.9027.85−7.90
12.3132500−8.34−18.70−8.001197.14536.58281.03231.8441.8418.75
12.4382545−8.65−20.56−8.271147.77581.53339.41268.6225.2443.65
12.5632801−8.89−19.74−7.071112.51542.88302.89297.0013.54−14.77
12.6882891−10.58−20.55−7.381151.91553.23314.96257.9240.6512.79
12.8133133−10.27−20.95−8.481200.32564.73337.15178.5244.5315.18
12.9383200−7.81−20.41−7.241248.96533.83302.04198.8516.3775.86
13.0633287−8.95−21.30−6.391280.43525.50278.14287.9112.1337.62
13.1883319−7.85−19.86−6.911246.33536.79307.74289.4071.2936.38
13.3133221−8.86−21.29−8.071229.89527.36317.62291.31−8.1663.57
13.4383326−8.38−21.79−6.771315.21607.53336.40206.8168.2947.08
13.5633218−7.87−21.90−7.411289.66645.94362.45234.8470.5763.60
13.6883142−8.52−21.65−7.701449.37526.23344.46254.5439.1844.32
13.8132953−9.72−22.54−6.611411.39598.96357.67192.26−11.5084.65
13.9382865−7.95−23.69−7.201472.20616.75357.78255.87−34.5651.51
14.0632675−10.31−24.21−7.641520.92684.45355.90315.226.1359.11
14.1882388−7.97−24.07−7.291543.36653.46441.34278.35−66.8168.07
14.3131965−11.16−24.42−7.821679.18653.07489.92284.11196.4867.31
14.4381690−8.29−25.41−7.321621.61762.39404.23195.83−55.39−23.96
14.5631513−10.15−25.80−6.881697.80715.78477.75299.89−51.76−58.32
14.6881351−9.30−27.65−7.721849.70684.15544.75150.33−94.65−52.84
14.8131225−7.39−28.81−7.732002.02774.65569.74400.7165.81147.53
14.938931−8.34−28.43−5.511989.77795.08682.89418.77−183.8859.79
15.063687−8.77−32.44−5.422259.52971.15656.68344.8370.88157.15
15.188475−14.64−33.43−6.472255.08823.56823.38271.3461.09106.76
15.313300−14.50−41.76−0.982439.981153.29935.6843.4014.91−32.18
15.438188−10.10−46.06−1.032078.841463.10998.57−103.49545.14−228.99
Bin centreNMean velocity (km s−1)Velocity dispersion (km2 s−2)
MG (mag)(stars)VUVVVWΣUUΣVVΣWWΣUVΣUWΣVW
10.06382−15.88−20.84−9.491846.13627.04227.49562.22461.50−207.45
10.188119−2.05−18.56−9.031331.90685.98259.71225.83−341.21146.84
10.313166−13.20−19.16−7.691381.77550.74140.47263.70129.24321.92
10.438223−6.95−22.92−9.361407.75626.93250.58−1.8522.8070.62
10.563349−7.81−22.40−7.301171.79522.92417.5615.1627.7082.71
10.688500−9.28−19.68−8.951118.97471.78266.60350.7773.7066.07
10.813635−9.31−19.47−7.891248.14592.03295.02165.80−40.0718.47
10.938786−10.01−18.75−8.881257.76498.45341.81299.97−112.5312.59
11.0631048−8.19−20.91−9.151212.53550.68258.01162.8872.62105.13
11.1881186−8.00−18.66−8.281328.73508.88308.12232.6885.83116.67
11.3131390−9.55−20.28−8.001199.62542.13256.45193.32−53.75−21.61
11.4381574−9.34−20.77−7.731036.90516.40281.83118.3987.60−19.37
11.5631690−10.25−20.02−8.251254.47567.40314.16384.9745.03−11.15
11.6882034−10.35−20.11−6.941245.43553.62313.30195.89−0.5316.19
11.8132102−9.89−20.07−8.431124.08510.88301.71249.5910.0931.51
11.9382243−8.29−19.67−7.141196.07510.06275.11162.786.94−19.39
12.0632224−9.97−19.05−7.281158.61485.98249.41244.89−28.9417.43
12.1882371−8.99−18.32−6.201177.38544.20248.29266.9027.85−7.90
12.3132500−8.34−18.70−8.001197.14536.58281.03231.8441.8418.75
12.4382545−8.65−20.56−8.271147.77581.53339.41268.6225.2443.65
12.5632801−8.89−19.74−7.071112.51542.88302.89297.0013.54−14.77
12.6882891−10.58−20.55−7.381151.91553.23314.96257.9240.6512.79
12.8133133−10.27−20.95−8.481200.32564.73337.15178.5244.5315.18
12.9383200−7.81−20.41−7.241248.96533.83302.04198.8516.3775.86
13.0633287−8.95−21.30−6.391280.43525.50278.14287.9112.1337.62
13.1883319−7.85−19.86−6.911246.33536.79307.74289.4071.2936.38
13.3133221−8.86−21.29−8.071229.89527.36317.62291.31−8.1663.57
13.4383326−8.38−21.79−6.771315.21607.53336.40206.8168.2947.08
13.5633218−7.87−21.90−7.411289.66645.94362.45234.8470.5763.60
13.6883142−8.52−21.65−7.701449.37526.23344.46254.5439.1844.32
13.8132953−9.72−22.54−6.611411.39598.96357.67192.26−11.5084.65
13.9382865−7.95−23.69−7.201472.20616.75357.78255.87−34.5651.51
14.0632675−10.31−24.21−7.641520.92684.45355.90315.226.1359.11
14.1882388−7.97−24.07−7.291543.36653.46441.34278.35−66.8168.07
14.3131965−11.16−24.42−7.821679.18653.07489.92284.11196.4867.31
14.4381690−8.29−25.41−7.321621.61762.39404.23195.83−55.39−23.96
14.5631513−10.15−25.80−6.881697.80715.78477.75299.89−51.76−58.32
14.6881351−9.30−27.65−7.721849.70684.15544.75150.33−94.65−52.84
14.8131225−7.39−28.81−7.732002.02774.65569.74400.7165.81147.53
14.938931−8.34−28.43−5.511989.77795.08682.89418.77−183.8859.79
15.063687−8.77−32.44−5.422259.52971.15656.68344.8370.88157.15
15.188475−14.64−33.43−6.472255.08823.56823.38271.3461.09106.76
15.313300−14.50−41.76−0.982439.981153.29935.6843.4014.91−32.18
15.438188−10.10−46.06−1.032078.841463.10998.57−103.49545.14−228.99
Table 1.
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For each magnitude bin, we present the mean stellar velocity relative to the Sun (columns 3–5) and the velocity dispersion tensor (columns 6–11). Units are km s−1 and km2 s−2 respectively, and quantities are referred to the Galactic coordinate frame with principal axes U (towards Galactic centre), V (in the direction of rotation), and W (towards the north Galactic pole).

Bin centreNMean velocity (km s−1)Velocity dispersion (km2 s−2)
MG (mag)(stars)VUVVVWΣUUΣVVΣWWΣUVΣUWΣVW
10.06382−15.88−20.84−9.491846.13627.04227.49562.22461.50−207.45
10.188119−2.05−18.56−9.031331.90685.98259.71225.83−341.21146.84
10.313166−13.20−19.16−7.691381.77550.74140.47263.70129.24321.92
10.438223−6.95−22.92−9.361407.75626.93250.58−1.8522.8070.62
10.563349−7.81−22.40−7.301171.79522.92417.5615.1627.7082.71
10.688500−9.28−19.68−8.951118.97471.78266.60350.7773.7066.07
10.813635−9.31−19.47−7.891248.14592.03295.02165.80−40.0718.47
10.938786−10.01−18.75−8.881257.76498.45341.81299.97−112.5312.59
11.0631048−8.19−20.91−9.151212.53550.68258.01162.8872.62105.13
11.1881186−8.00−18.66−8.281328.73508.88308.12232.6885.83116.67
11.3131390−9.55−20.28−8.001199.62542.13256.45193.32−53.75−21.61
11.4381574−9.34−20.77−7.731036.90516.40281.83118.3987.60−19.37
11.5631690−10.25−20.02−8.251254.47567.40314.16384.9745.03−11.15
11.6882034−10.35−20.11−6.941245.43553.62313.30195.89−0.5316.19
11.8132102−9.89−20.07−8.431124.08510.88301.71249.5910.0931.51
11.9382243−8.29−19.67−7.141196.07510.06275.11162.786.94−19.39
12.0632224−9.97−19.05−7.281158.61485.98249.41244.89−28.9417.43
12.1882371−8.99−18.32−6.201177.38544.20248.29266.9027.85−7.90
12.3132500−8.34−18.70−8.001197.14536.58281.03231.8441.8418.75
12.4382545−8.65−20.56−8.271147.77581.53339.41268.6225.2443.65
12.5632801−8.89−19.74−7.071112.51542.88302.89297.0013.54−14.77
12.6882891−10.58−20.55−7.381151.91553.23314.96257.9240.6512.79
12.8133133−10.27−20.95−8.481200.32564.73337.15178.5244.5315.18
12.9383200−7.81−20.41−7.241248.96533.83302.04198.8516.3775.86
13.0633287−8.95−21.30−6.391280.43525.50278.14287.9112.1337.62
13.1883319−7.85−19.86−6.911246.33536.79307.74289.4071.2936.38
13.3133221−8.86−21.29−8.071229.89527.36317.62291.31−8.1663.57
13.4383326−8.38−21.79−6.771315.21607.53336.40206.8168.2947.08
13.5633218−7.87−21.90−7.411289.66645.94362.45234.8470.5763.60
13.6883142−8.52−21.65−7.701449.37526.23344.46254.5439.1844.32
13.8132953−9.72−22.54−6.611411.39598.96357.67192.26−11.5084.65
13.9382865−7.95−23.69−7.201472.20616.75357.78255.87−34.5651.51
14.0632675−10.31−24.21−7.641520.92684.45355.90315.226.1359.11
14.1882388−7.97−24.07−7.291543.36653.46441.34278.35−66.8168.07
14.3131965−11.16−24.42−7.821679.18653.07489.92284.11196.4867.31
14.4381690−8.29−25.41−7.321621.61762.39404.23195.83−55.39−23.96
14.5631513−10.15−25.80−6.881697.80715.78477.75299.89−51.76−58.32
14.6881351−9.30−27.65−7.721849.70684.15544.75150.33−94.65−52.84
14.8131225−7.39−28.81−7.732002.02774.65569.74400.7165.81147.53
14.938931−8.34−28.43−5.511989.77795.08682.89418.77−183.8859.79
15.063687−8.77−32.44−5.422259.52971.15656.68344.8370.88157.15
15.188475−14.64−33.43−6.472255.08823.56823.38271.3461.09106.76
15.313300−14.50−41.76−0.982439.981153.29935.6843.4014.91−32.18
15.438188−10.10−46.06−1.032078.841463.10998.57−103.49545.14−228.99
Bin centreNMean velocity (km s−1)Velocity dispersion (km2 s−2)
MG (mag)(stars)VUVVVWΣUUΣVVΣWWΣUVΣUWΣVW
10.06382−15.88−20.84−9.491846.13627.04227.49562.22461.50−207.45
10.188119−2.05−18.56−9.031331.90685.98259.71225.83−341.21146.84
10.313166−13.20−19.16−7.691381.77550.74140.47263.70129.24321.92
10.438223−6.95−22.92−9.361407.75626.93250.58−1.8522.8070.62
10.563349−7.81−22.40−7.301171.79522.92417.5615.1627.7082.71
10.688500−9.28−19.68−8.951118.97471.78266.60350.7773.7066.07
10.813635−9.31−19.47−7.891248.14592.03295.02165.80−40.0718.47
10.938786−10.01−18.75−8.881257.76498.45341.81299.97−112.5312.59
11.0631048−8.19−20.91−9.151212.53550.68258.01162.8872.62105.13
11.1881186−8.00−18.66−8.281328.73508.88308.12232.6885.83116.67
11.3131390−9.55−20.28−8.001199.62542.13256.45193.32−53.75−21.61
11.4381574−9.34−20.77−7.731036.90516.40281.83118.3987.60−19.37
11.5631690−10.25−20.02−8.251254.47567.40314.16384.9745.03−11.15
11.6882034−10.35−20.11−6.941245.43553.62313.30195.89−0.5316.19
11.8132102−9.89−20.07−8.431124.08510.88301.71249.5910.0931.51
11.9382243−8.29−19.67−7.141196.07510.06275.11162.786.94−19.39
12.0632224−9.97−19.05−7.281158.61485.98249.41244.89−28.9417.43
12.1882371−8.99−18.32−6.201177.38544.20248.29266.9027.85−7.90
12.3132500−8.34−18.70−8.001197.14536.58281.03231.8441.8418.75
12.4382545−8.65−20.56−8.271147.77581.53339.41268.6225.2443.65
12.5632801−8.89−19.74−7.071112.51542.88302.89297.0013.54−14.77
12.6882891−10.58−20.55−7.381151.91553.23314.96257.9240.6512.79
12.8133133−10.27−20.95−8.481200.32564.73337.15178.5244.5315.18
12.9383200−7.81−20.41−7.241248.96533.83302.04198.8516.3775.86
13.0633287−8.95−21.30−6.391280.43525.50278.14287.9112.1337.62
13.1883319−7.85−19.86−6.911246.33536.79307.74289.4071.2936.38
13.3133221−8.86−21.29−8.071229.89527.36317.62291.31−8.1663.57
13.4383326−8.38−21.79−6.771315.21607.53336.40206.8168.2947.08
13.5633218−7.87−21.90−7.411289.66645.94362.45234.8470.5763.60
13.6883142−8.52−21.65−7.701449.37526.23344.46254.5439.1844.32
13.8132953−9.72−22.54−6.611411.39598.96357.67192.26−11.5084.65
13.9382865−7.95−23.69−7.201472.20616.75357.78255.87−34.5651.51
14.0632675−10.31−24.21−7.641520.92684.45355.90315.226.1359.11
14.1882388−7.97−24.07−7.291543.36653.46441.34278.35−66.8168.07
14.3131965−11.16−24.42−7.821679.18653.07489.92284.11196.4867.31
14.4381690−8.29−25.41−7.321621.61762.39404.23195.83−55.39−23.96
14.5631513−10.15−25.80−6.881697.80715.78477.75299.89−51.76−58.32
14.6881351−9.30−27.65−7.721849.70684.15544.75150.33−94.65−52.84
14.8131225−7.39−28.81−7.732002.02774.65569.74400.7165.81147.53
14.938931−8.34−28.43−5.511989.77795.08682.89418.77−183.8859.79
15.063687−8.77−32.44−5.422259.52971.15656.68344.8370.88157.15
15.188475−14.64−33.43−6.472255.08823.56823.38271.3461.09106.76
15.313300−14.50−41.76−0.982439.981153.29935.6843.4014.91−32.18
15.438188−10.10−46.06−1.032078.841463.10998.57−103.49545.14−228.99

4.1 Mean velocity and the asymmetric drift

In Fig. 4 we plot the U, V, W components of the mean solar motion with respect to the WDs, along with the quantity S from DB98 that is a scalar measure of the velocity dispersion. The most obvious feature is the rise in the V component towards faint magnitudes, which is a manifestation of the asymmetric drift. This is accompanied by a corresponding rise in the velocity dispersion as some of the original circular motion is scattered randomly into the U, W components. At magnitudes brighter than MG ≈ 13 there is no trend in V or S; in this range the correlation between magnitude and mean age is non-existent. No break analogous to Paranego’s discontinuity is present; there is no equivalent of the MS turnoff colour in the WD population. The U, W components show no systematic variation with MG as expected. Less obvious features include the deviation in U and W beyond MG ≈ 15; this is likely due to contamination by spheroid and thick disc WDs that make up a larger fraction of the population at these magnitudes.

The U, V, W components of the mean solar motion with respect to the WDs, including a measure of the scalar velocity dispersion S. The trend in V and S is a manifestation of the asymmetric drift arising from the magnitude / age correlation.
Figure 4.

The U, V, W components of the mean solar motion with respect to the WDs, including a measure of the scalar velocity dispersion S. The trend in V and S is a manifestation of the asymmetric drift arising from the magnitude / age correlation.

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4.2 Velocity dispersion and the local standard of rest

Classically, the main utility of measurements of the mean motion versus velocity dispersion is in estimating the solar motion with respect to the LSR. This exploits the linear relationship between the velocity dispersion S2 and rotational lag first identified empirically by Strömberg (1946); the modern theoretical framework for the asymmetric drift relation is described in Binney & Tremaine (2008) (see equation 4.228). In practise this is done by extrapolating the mean velocity to S2 = 0, which provides an estimate of the solar motion relative to a (hypothetical) population of stars with zero velocity dispersion. Such a population represents newly born stars on closed, circular orbits, the motion of which defines the LSR in the solar neighbourhood.

Figs 5–7 plot the U, V, W components of the mean solar motion against S2 along with linear fits. In these plots we have omitted the four points fainter than MG = 15.0 for the reasons outlined in Section 4.1. For the U and W components there is no strong correlation and the fit is flat within 1σ, whereas for V there is a clear slope. Extrapolating to S2 = 0 yields (U, V, W)⊙ = (9.50 ± 1.18, 7.47 ± 1.21, 8.22 ± 1.19) km s−1 for the solar motion with respect to the LSR. Note that relative to similar studies that use MS stars we have few points at low S2 due to the WDs having a certain minimum age, which results in larger uncertainties due to extrapolating over a longer interval.

U component of the mean solar motion versus velocity dispersion; the black line is a linear fit and the grey lines indicate 1σ confidence intervals.
Figure 5.

U component of the mean solar motion versus velocity dispersion; the black line is a linear fit and the grey lines indicate 1σ confidence intervals.

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V component of the mean solar motion versus velocity dispersion.
Figure 6.

V component of the mean solar motion versus velocity dispersion.

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W component of the mean solar motion versus velocity dispersion.
Figure 7.

W component of the mean solar motion versus velocity dispersion.

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4.3 Velocity ellipsoid

In Fig. 8 we plot the diagonal components of the velocity dispersion tensor against magnitude.1 In all components the velocity dispersion increases towards faint magnitudes, although it seems the rate of increase in the V component is smaller. The velocity dispersion is in general larger than that of MS stars, as the mean age of WDs is larger due to the lack of young, newly formed stars: towards the faintest magnitudes nearly all the WDs are very old.

Components of the velocity dispersion tensor along the axes of the Galactic coordinate frame.
Figure 8.

Components of the velocity dispersion tensor along the axes of the Galactic coordinate frame.

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The velocity dispersion tensor has some significant off-diagonal elements, indicating that the velocity ellipsoid is not aligned with the U, V, W axes of the Galactic frame. This can be quantified by the vertex deviation lXY, which is the angle by which one has to rotate the velocity ellipsoid in the XY plane to bring it into alignment. The vertex deviation in each of the UV, UW, and VW planes is plotted in Fig. 9. The deviations in the UW and VW planes, which involve the velocity dispersion in the direction out of the plane, are consistent with zero, whereas a non-zero deviation of around 15° is observed in the UV plane. These results are similar to those of DB98, who find lUV ≈ 20° for early spectral types and 10° ± 4° for old disc stars, although we find that the vertex deviation in UV is roughly constant with no obvious trend with magnitude and, equivalently, mean age. Figs 10–12 present projections of the velocity dispersion tensor into each of the UV, UW, and VW planes, which helps to visualize the velocity ellipsoid orientation, scale and evolution with magnitude. Ellipses are drawn at the 1σ level.

Vertex deviation in each of the UV, UW, and VW planes. The deviations in the UW and VW planes, which involves the velocity dispersion in the direction out of the plane, are consistent with zero, whereas a non-zero deviation of around 15 deg is observed in the UV plane.
Figure 9.

Vertex deviation in each of the UV, UW, and VW planes. The deviations in the UW and VW planes, which involves the velocity dispersion in the direction out of the plane, are consistent with zero, whereas a non-zero deviation of around 15 deg is observed in the UV plane.

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Projection of the velocity ellipsoid on to the UV plane. Ellipses are drawn at the 1σ level.
Figure 10.

Projection of the velocity ellipsoid on to the UV plane. Ellipses are drawn at the 1σ level.

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Projection of the velocity ellipsoid on to the UW plane, as per Fig. 10.
Figure 11.

Projection of the velocity ellipsoid on to the UW plane, as per Fig. 10.

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Projection of the velocity ellipsoid on to the VW plane, as per Fig. 10.
Figure 12.

Projection of the velocity ellipsoid on to the VW plane, as per Fig. 10.

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5 DISCUSSION

Our values for U⊙ and W⊙ agree within 1σ with the values obtained by both Schönrich et al. (2010) and DB98. However, our value for V⊙ = 7.47 ± 1.21 differs from the Schönrich et al. (2010) value of V⊙ = 12.24 ± 0.47 by ≈3.7σ and from the Kawata et al. (2019) value of V⊙ = 12.4 ± 0.7 by a similar amount, being more in line with the value of V⊙ = 5.25 ± 0.62 obtained by DB98. As explained in Schönrich et al. (2010) the DB98 value for V⊙ is affected by systematics arising from the metallicity gradient in the disc; briefly, metal-rich stars generally formed at R < R0 and visit the solar neighbourhood close to the apocentre of their orbit where their azimuthal velocities lag behind the local circular speed. Increasing metallicity shifts the MS to the right in the colour–magnitude plane, introducing a velocity gradient across the MS and biasing the colour/age correlation. The fact that our value of V⊙ lies between the DB98 and Schönrich et al. (2010) value could indicate that our WD sample is affected to a lesser extent by similar systematics. Although the location of a WD in the colour–magnitude diagram is independent of the progenitor metallicity to first order, in a population of stars the metallicity-dependent progenitor lifetimes could play a role in introducing kinematic substructures in the WD colour–magnitude plane similar to those described by Schönrich et al. (2010). In addition, over the magnitude range of interest to this study the Gaia WD sequence splits into two branches, due mostly to atmospheric composition but also to the presence of massive WDs possibly formed through stellar mergers (El-Badry, Rix & Weisz 2018; Kilic et al. 2018). The question of to what extent this affects the kinematics as a function of MG is complex. The WD cooling rate depends on both atmosphere composition and mass, so both of these properties will affect the age distribution of stars as a function of MG. While the kinematics are likely to be independent of atmosphere type (DA / non-DA), massive WDs formed from merged binary systems could conceivably have different kinematics. Ultimately, both of these effects could lead to kinematic substructure in the WD colour–magnitude diagram and be responsible for systematic errors in our results. However, a thorough investigation of this is outwith the scope of our study. It is also possible that large scale streaming motions within the disc are affecting our results, due to the local nature of our sample. For example, Bovy et al. (2015) finds that the solar neighbourhood is likely affected by a streaming motion of the order of 10 km s−1 in the direction of rotation, which would bias our estimate of V⊙.

The non-zero vertex deviation in the UV plane is understood to be caused by the non-axisymmetric Galactic potential in the disc, likely due to the effect of the spiral arms. The fact that the vertex deviation is independent of magnitude indicates that this has been stable for a long period. No such asymmetry in the potential in the UW and VW planes is evident.

Previous WD proper motion surveys and luminosity function studies that used RPM selection generally assume a constant value for the reflex solar motion and WD velocity distribution when computing correction factors for stars that fall below the vtan ≈ 30 km s−1 selection threshold. This includes Harris et al. (2006; RH11) and Lam et al. (2019). As shown in this study, there is a strong correlation between WD absolute magnitude and both the mean and dispersion of the heliocentric velocity, which will cause the correction factors to be somewhat overestimated towards the faint end of the luminosity function, leading to an overestimation of the spatial density for faint WDs.

For example, the study of RH11 adopted a mean velocity of (U, V, W)⊙ = (8.62, 20.04, 7.1) km s−1 and velocity dispersion (σU, σV, σW) = (32.4, 23.0, 18.1) km s−1 with no off-diagonal terms. In conjunction with their lower tangential velocity threshold of 30 km s−1 this leads to a sky-averaged discovery fraction of ≈57 per cent, which is then used to correct the estimated spatial density to account for the missing low velocity stars. However for faint WDs both the mean velocity and velocity dispersion are significantly larger than these values; adopting instead the values that we measure for the MG bin centred on 14.563 we get a sky-averaged discovery fraction of ≈68 per cent, meaning that the RH11 study will to first order overestimate the spatial density of WDs of these magnitudes by around a factor ≈1.2, although the true figure depends on the complex effects of the magnitude-dependent proper motion limits when computing the generalized survey volume, as described in Lam, Rowell & Hambly (2015).

6 CONCLUSIONS

We have presented the first detailed analysis of the kinematics of Galactic disc WD stars in the solar neighbourhood, based on a large, kinematically unbiased sample of WDs with absolute astrometry provided by Gaia DR2. Various classical properties of the solar neighbourhood kinematics have been detected for the first time in the WD population. These are found to be largely consistent with expectations based on MS star studies, although form an independent measure. The results of this study provide an important input to proper motion surveys for WDs, which require knowledge of the velocity distribution in order to correct for missing low-velocity stars that are culled from the sample to reduce subdwarf contamination.

ACKNOWLEDGEMENTS

This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

Footnotes

1

In this figure and the ones that follow, the plotted quantities are non-linear functions of our estimated values: the (asymmetric) error bars depict the 15.87 and 84.13 percentiles of the error distribution, obtained from Monte Carlo integration.

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APPENDIX: ADQL QUERY

The ADQL query used to select our initial sample from the Gaia archive is shown below. This returns a table containing 3061 480 rows, which is subject to additional filtering as described in Section 2.

SELECT *

FROM gaiadr2.gaia_source

WHERE parallax_over_error > 5.0

 AND phot_bp_mean_flux_over_error > 10

 AND phot_rp_mean_flux_over_error > 10

 AND astrometric_n_good_obs_al > 5

 AND phot_bp_rp_excess_factor BETWEEN

   1.0 + (0.03*POWER(bp_rp, 2.0))

   AND 1.3 + (0.06*POWER(bp_rp, 2.0))

 AND 1000.0/parallax < 250

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