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The local standard of rest and the well in the velocity distribution

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Abstract

It is now recognised that the traditional method of calculating the LSR fails. We find an improved estimate of the LSR by making use of the larger and more accurate database provided by XHIP and repeating our preferred analysis from Francis and Anderson (New Astron 14:615–629, 2009a). We confirm an unexpected high value of \(U_0\) by calculating the mean for stars with orbits sufficiently inclined to the galactic plane that they do not participate in bulk streaming motions. Our best estimate of the solar motion with respect to the LSR \((U_0, V_0, W_0) = (14.1\, \pm \, 1.1, 14.6\, \pm \, 0.4, 6.9\, \pm \, 0.1)\) km s\(^{-1}\).

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Authors and Affiliations

  1. Jesus College, Cambridge, England

    Charles Francis

  2. 360, Iowa St., Ashland, OR, 97520, USA

    Erik Anderson

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  1. Charles Francis
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Correspondence to Charles Francis.

Appendices

Appendix A: Estimate of dispersion and eccentricity

Kepler’s second law is an expression of conservation of angular momentum, and holds for planar orbits in any axisymmetric potential. Assuming a well-mixed distribution in which orbital eccentricities are low, the greater density of stars towards the centre of the Galaxy will not greatly affect calculation. If the number of stars on the outer part of the orbit (lagging the LSR) outnumbers the number on the inner part (leading the LSR) by 70:30, it is because the time on the outer part of the orbit is greater by roughly this ratio, so the velocity is less and the radial distance is greater by the square root of this ratio, i.e. by a factor greater than 1.5 (contradicting the assumption that orbital eccentricities are low). Then, for galactic rotation \(\theta _0 \approx 200 \) km s\(^{-1}\), azimuthal velocities will range from \(\sim \!160 \) to \(\sim \!240 \) km s\(^{-1}\) and dispersion in \( V \) will be about 40 km s\(^{-1}\), around twice the observed value of 21 km s\(^{-1}\) for disc stars.

If eccentricity, \( e \), is defined such that the ratio of distance to apocentre to distance to pericentre is given, as for a Keplerian orbit, by

$$\begin{aligned} \frac{1+e}{1-e} = \frac{R_{\mathrm {max}}}{R_{\mathrm {min}}}>\sim 1.5 \end{aligned}$$
(7.1)

from which

$$\begin{aligned} e> \sim 0.2 \end{aligned}$$
(7.2)

But modal value of eccentricity in the solar neighbourhood is \( \sim 0.11 \) (Fig. 6). The modal value of eccentricity is insensitive to the figure used for the LSR.

Appendix B: Significance of the well

A simple estimate of the probability that the well at \( (U, V) = (-12.5, -14) \) km s\(^{-1}\) seen in Fig. 2 could arise by chance can be found by defining one hundred \(2\times 2\) km\(^2\) s\(^{-2}\) bins centred at even integer velocities in the ranges \( -24\le U \le -6\), \(-24\le V \le -6 \) km s\(^{-1}\). The bins contain 3 824 stars. The bin centred at (\(-\)12, \(-\)14) km s\(^{-1}\) contains only 21 stars. For a uniform distribution (the null hypothesis) the number of stars in each bin is given by a binomial distribution \( \mathsf {B} (3 824, 0.01) \), for which the probability of finding 21 or fewer stars in a bin is 0.0017, giving a 99.8 % significance level. The use of a uniform probability distribution underestimates the statistical significance of the well, because there is generally a greater probability of finding data towards the centre of a distribution.

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Francis, C., Anderson, E. The local standard of rest and the well in the velocity distribution. Celest Mech Dyn Astr 118, 399–413 (2014). https://doi.org/10.1007/s10569-014-9541-z

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