Meeting the neighbours: NStars and 2MASS

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Proper motion selection of candidate nearby dwarfs

1. Introduction

Figure 2.1a: The POSS I image (epoch 1952) of a 5x5 arcminute field centred near the nearby M8 dwarf, LHS 2924

Figure 2.1b: The same field from the POSS II IIIaF plate (epoch 1994). LHS 2924 is obvious from its large angular motion.

Proper motion is an extremely effective method of identifying stars within the immediate vicinity of the Sun. The standard relation between tangential velocity and proper motion is

mu = Vtan / (k . d) ..... (1)
where mu is the proper motion, in arcseconds/year; Vtan is the velocity, in km/sec, tangential to our line of sight to the star; d, the distance in parsecs; and k (actually kappa) is a constant, 4.74 for these units. Thus, the observed proper motion is directly dependent on the tangential velocity, and inversely dependent on distance.

Figure 2.2: The predicted tangential velocity distribution for a stellar population with kinematics matching the single-population disk Schwarzschild ellipsoid outlined below.

Proper motion would not be an effective indicator of proximity if stars could acquire any velocity with respect to the sun. Fortunately, that is not the case. The overwhleming majority of stars in the Solar Neighbourhood (around 99.8%) are members of the Galactic disk, sharing its rotational velocity around the Galactic centre. The kinematics of these stars are well characterised (or at least adequately characterised) as a Schwarzschild velocity ellipsoid in the Galactic (U, V, W) co-ordinate system, with a mean heliocentric velocity of [ -9, -22, -7 km/sec] and Gaussian dispersions of [43, 31, 25 km/sec]. As discussed in our PMSU analysis, subdivision into at least two kinematic sub-populations is a distinct possibility. In any case, the tangential velocity distribution is a two-dimensional projection of the three-dimensional Schwarzscjild ellipsoid, integrated over the celestial sphere. As Figure 2.2 shows, the resulting distribution peaks at a characteristic velocity, dependent on both the Solar Motion (systematic offset with respect to the Sun) and the velocity dispersions (random motion). In the case of the disk-like kinematics of the local M dwarf population, the characteristic velocity is ~60 km/sec.

Figure 2.3: The correlation between reduced proper motion and absolute magnitude for stars within 8 parsecs of the Sun. The dotted line plots the best-fit linear relation; the solid line, the linear relation derived if we set the slope at unity and adopt the constant given by the average tangential velocity.

The existence of coherent, restricted velocity distributions within stellar populations has several interesting consequences, which can be exploited to investigate various aspects of Galactic structure. In particular, the reduced proper motion diagram offers a powerful tool for segregating members of kinematically-distinct stellar populations. This parameter was used extensively by Luyten in analysing data from his proper motion surveys, but was coined (according to Luyten) by Hertzsprung. Hence, the reduced proper motion is denoted by H. The definition is

H = m + 5 + 5 log(mu) ...... (2)
analogous to the definition of absolute magnitude. It can be shown empirically that H is correlated absolute magnitude, and Luyten used this correlation to estimate absolute magnitudes for proper motion stars, and hence volume densities and stellar luminosity functions.

Luyten's final calibration of M(H) was

MH = -3.5 + 0.86 H ........ (3)
That correlation agrees well with modern observations: Figure 2.3 plots data for stars within 8 parsecs of the Sun; the best-fit linear relation is
MV = -2.56 + 0.89 HV ........, rms = 1.24 mag. (4)
However, consider the origin of this correlation: substitute for mu (annual proper motion) in equation (2) in terms of velocity and distance (equation (1)). Collecting terms, we have
H = m + 5 - 5 log (d) + 5 log ( Vtan ) + const. = M + 5 log(Vtan) - 3.38 ...... (5)
Thus, the slope of 0.89 in the best-fit relation implies the mean tangential velocity increases from ~18 km/sec at MV=0 to ~35 km/sec at MV=12. A variation on that scale is not entirely implausible, given the short lifetimes of higher-mass stars, but it doesn't seem reasonable to extend that variation through the long-libed M dwarfs, where one expects the same age distribution at all luminosities. Figure 2.2 suggests an alternative means of approaching the calibration: since there is a characteristic velocity for stars in a given population, H should be linearly correlated with absolute magnitude, M, for stars in that population. The average tangential velocity for the 8-parsec sample stars is 37 km/sec, giving
MV = -4.46 + HV ..... (6)
As figure 2.3 shows, this relation is well-matched to the observations.

Figure 2.4: The (HV, (V-I)) reduced proper motion diagram for stars with accurate V,I photometry from Luyten's Half-Second catalogue. The solid line plots the expected ridge-line for main-sequence stars with disk motions; the dotted line is the the ridge-line for the high velocity "thick disk" population; and the dashed line, the expected relation for halo stars. White dwarf stars are evident blueward of the halo sequence.

Equation (5) points to other uses for reduced proper motion: there is the potential to segregate both obejcts with similar kinematics, but different absolute magnitude calibrations; or objects with similar absolute magnitude distributions, but different kinematics. That segregation can be achieved in the reduced proper motion diagram (RPMD), where we plot H versus a colour. This is effectively the kinematic equivalent of the HR diagram, with H serving as a surrogate for luminosity. Luyten applied this technique extensively to identify degenerate stars; hot white dwarfs are up to 10 magnitudes fainter than main-sequence stars of the same temperature (colour), and therefore have correspondingly larger values of H. the white dwarf and red dwarf sequences merge in the (Hr, (mpg-mr) RPMD but, as with photometric parallax work, a suitable choice of colours ( (V-I), (R-J) etc) can preserve the offset between the degenerate and main-sequence disk sequences.

More recently, the RPMD has received more attention as a means of segragating main-sequence stars belonging to the different Galactic populations and sub-populations, notably members of the Galactic halo. This is particularly important in analyses of proper motion data, where the halo population can make a significant contribution. Consider a proper motion catalogue including stars with motions greater than a given limit, mulim; for a given kinematic population, those stars are drawn from an effective sampling volume which is defined by the value of the characteristic tangential velocity, VtanI. Consider a second population, with different kinematics; again, the sampling volume will depend on VtanII; the relative number of stars from Pop I and Pop II will be given by

NII / NI = [ VtanII / VtanI]3 x rhoII / rhoI
where rho is the local volume density of the population. In the case of the Galactic disk and halo, the local density ratio is ~500:1; but the average tangential velocity of members of the high velocity-dispersion, low-rotation halo is 210 km/sec, approximately 6 times that of the disk. Thus, if we consider a sample of stars which are proper-motion limited (that is, inclusion depends only on the proper motion, without regard to any other properties), the disk/halo ratio could be reach as low a value as ~2.5:1.

Figure 2.5: The predicted differential and cumulative distributions with distance of stars selected from three kinematic populations; the proper motion limit is set at 0.5 arcsec/year.

In practice, the contribution of halo subdwarfs to proper motion samples is seldom amplified by the full possible factor, usually because other factors, such as the apparent magnitude limit, truncate the halo sample. The latter consideration affects halo stars (or, more generally, high-velocity populations) to a greater extent than disk stars since those stars are sampled preferentially from larger distances. Figure 2.5 illustrates this for a particular 3-component model, with a low-velocity disk,

[U, V. W; sigmaU, sigmaV, sigmaW ] = [ -9, -22, -7; 43, 31, 25]
a thick disk, or Intermediate Population II, with
[ 0, -53, 0; 67, 51, 40]
and a halo, Population II, having
[ 0, -210, 0; 168, 102, 97]
We have imposed a proper motion cut-off of 0.5 arcseconds/year, which biases the sample towards higher velocities. The resultant average tangential velocities are Vtan = 81 km/sec (disk), 138 km/sec (IP II) and 362 km/sec (halo). (These values are used to plot the population ridge lines in figure 2.4.) As a result, the average distance of halo subdwarfs is ~4.5 times that of disk dwarfs. Note that this invalidates the (HV, MV) calibration for halo stars; their magnitudes are underestimated (too faint) by ~4 magnitudes. This accounts for the extended tail towards low luminosities (Mpg > 18) in Luyten's luminosity function analyses. Nonetheless, proper motion surveys are an extremely effective ways of finding halo subdwarfs, as emphasised by Figure 2.4.

2. Large-scale Proper Motion Surveys

The most extensive proper motion surveys currently available are the Lowell Observatory Proper Motion survey (Giclas et al, 1968), based on plates taken with the 13-inch refractor at Lowell Observatory, Flagstaff, the same telescope used by Tombaugh to discover Pluto, and the extensive series of surveys completed by Luyten using photographic material taken with the Palomar 48-inch Schmidt. The first-epoch plates for the Lowell survey, covering a field of view of y x y degrees at a plate sale of z arcsec/mm, were taken in parallel with the Pluto search, with the second-epoch material accumulated in the 1950s and 60s, giving a baseline of 30+ years (Giclas, 1959). The survey covers stars with photographic magnitudes (mpg between 8th magnitude and between 16th and 17th magnitude, the limiting magnitude of the plates. The proper motion catalogue, constructed based on visual comparison of first and second epoch plates in a blink comparator extends to annual motions exceeding 0.27 arcsec/yr, with an estimated uncertainty (probable error) of 0.021 arcsec/year for well-exposed stars. There are also, however, systematics; as Luyten (1974) originally pointed out, comparison with external sources (mainly Luyten's own work) tends to show that the Lowell motions are systematically larger, particularly for motions close to 0.8 arcsec/year, 0.5 arcsec/year and at the limit of the survey (see also Majewski, 1993). This can be attributed to unconscious subjective biases to identify objects with "large" motions (0.8 and 0.5), and/or meet the limiting requirement for inclusion in the catalogue. Nonetheless, the Lowell survey has served as the basis for many follow-up studies (Fouts & Sandage, 1986; Carney et al, 1990 and refs within).

2.1 The Luyten Surveys

Willem Luyten's surveys use plates taken for the Palomar Sky Survey (POSS I) as first-epoch material, matched against second-epoch plates taken with the same telescope in the early 1960s. As a result, the time baseline is only 10-15 years, but the plate-scale (67.1 arcsec/mm) is better than the Lowell plates, permitting proper motion measurements that are at least as accurate. The survey also extends to significantly fainter magnitudes - at least in the celestial regions accessible from Palomar, where the 10-minute 103aE second epoch plates give an effective limiting magnitude between 18th and 19th magnitude in mr. (Luyten's r-band magnitudes derived from 103aE emulsion plates, with an effective wavelength of ~6300 Angstroms and FWHM ~350 Angstroms.) At more southerly declinations (below -40o), Luyten's all-sky catalogues rest on data from the Bruce survey, plates taken in the 1920s, using the Harvard-run 24-inch Bruce refractor, sited at Bloemfontein in Namibia These plates have a limiting magnitude of mpg~ 15). Large-scale, deep southern proper motion surveys are only now being undertaken (eg Ruiz et al, 2001).

Luyten's proper motion data are collected together in two main publications: the Luyten Half-Second catalogue (LHS), including data for 3601 stars with annual motions of 0.5 arcsec/year or more; and the New Luyten Two-Tenths Catalogue (NLTT), with data for 58845 stars with mu > 0.18 arcsec/year. Each lists both astrometric data - positions (generally accurate to a few arcseconds, but sometime much more wayward) and proper motions (mu, theta) - photometry (mr, mpg) and crude spectral types (usually a, f, g, g-k, k, k-m, m and m+, based on the colours). Many of the plates were scanned in Luyten's Automated Plate Scanner, the measuring machine developed by YYY, providing automated detection of proper motion stars and measurement of red magnitudes; a subset, however, were blinked by eye, and all of the photographic magnitudes are visual estimates.

Figure 2.6: The number-magnitude distribution of stars in Luyten's LHS and NLTT catalogues; the sample is limited to stars north of declination -30o. The uppermost panel shows the proportion of stars with accurate (B)VI photometry.

Most of the names attached to stars in both the LHS and the NLTT are from Luyten's Palomar survey (LP nnn-nnn), although Luyten made a point of retaining the original nomenclature for prior discoveries (eg BD+58: 02605, Ross 248, VB 12, Wolf 359), with one conspicuous exception: none of the Giclas stars, from the Lowell survey, are identified explicitly. In general, the Luyten surveys are more complete and have more accurate astrometry than the Lowell survey (particularly since Luyten's postdate the Giclas publications).

2.2 The LHS and the NLTT

Not surprisingly, the higher-motion stars in the LHS catalogue have received most observational attention. That catalogue includes 73 stars (LHS 1-73) with mu > 2 arcsec/year, 455 stars (LHS 101-552) with 2 > mu > 1 arcsec/year and 3073 (LHS 1001-4058) with motions between 0.5 and 1 arcsec/year (the numbering system is not completely consecutive - there are interstitial stars, such as LHS 2397a). Data are given for a further 441 stars (LHS 5001-5413) with 0.5 > mu > 0.48 arcsec/year, and for 428 stars with lower motions which previously were listed as having motions exceeding 0.5 arcsec/year. Figure 2.6 plots the magnitude distribution for the "northern" (dec > -30o) subset of both LHS and NLTT. Broadband photometry exists for most of the brighter stars in the LHS, mainly due to the heroic efforts of Ed Weis, who has obtained VRI photometry for over 1190 LHS stars and a further 1930 stars from the NLTT (Weis, 1996 and refs within; see figure 2.7). Coverage is much sparser at faint magnitudes, although spectra are in hand for all LHS stars with mr > 16.5 (Reid & Gizis, in prep.). The faintest LHS stars are predominantly late-type M dwarfs (M5-M7, not ultracool dwarfs) and cool subdwarfs (sdM and esdM), with a few 4000-5000K white dwarfs.

In contrast to the LHS, the members of the NLTT catalogue have received little attention. This partly reflects their greater numbers, but also is tied to difficulties in locating the targets: Luyten provided (eventually) finding charts for the LHS stars; NLTT stars can be located only using the listed positions. For the most part, those positions are relatively accurate (a few arcseconds), allowing judicious selection based on the expected colour. However, a moderate-sized subset of sources have listed positions which are in error by 1 or more arcminutes - perhaps a typographical error, perhaps a mismeasure. Despite those problems, Weis has accumulated VRI data for 1930 stars, including almost all stars north of the celestial equator classed as type m, and with mr < 14.5.

Figure 2.7: Photometric data for LHS stars with VRI observations by Weis. The upper panels show a comparison between Luyten's mr magnitudes and accurate V, R photometry; the lowest panel plots the colour-magnitude distribution of the sample.

The magnitude distribution of the LHS stars peaks at mr = 15, while the NLTT sample continues rising to mr ~ 17. This behaviour does not reflect incompleteness at these relatively bright magnitudes - Dawson (1985) estimates that the LHS survey is 90% complete (or better) for mr < 18 and |b| > 10o. Rather, the magnitude distribution reflects the convolution of the volume element (defined from Figure 2.5) and the underlying luminosity functions of the different Galactic populations. This sampling is clearly evident in Figure 2.7, which shows the colour-magnitude distribution for the LHS stars observed by Weis; in general, proper motion stars at fainter apparent magnitudes have fainter absolute magnitudes ( modulo population kinematics, vide the halo subdwarf sequence at V > 12.) As a corollary, the linear relation in the uppermost panel in this figure, which compares Luyten's mr against V-band data, is actually measuring a colour term, rather than a scale error in magnitude.

Figure 2.8: The distribution of the NLTT stars on the celestial sphere for three magnitude ranges. The increasing incompleteness at southern declonations is obvious.

The distribution of stars in the NLTT catalogue is plotted in Figure 2.8. The increasing incompleteness at fainter magnitudes and southern declinations is obvious. However, it is also clear that the NLTT is significantly less complete close to the Galactic Plane, even at northern declinations covered by Palomar. This is not too surprising given the high star density, but clearly suggests the imposition of a cut in galactic latitude for any statistical survey.

Figure 2.9: Completeness at high latitudes in the NLTT catalogue. The uppermost figure plots number counts from the NLTT, the middle figure LHS number counts, both for an area centred on the NGP. The ratio between those counts is plotted in the lowermost panel.

Dawson's estimate of completeness in the LHS allows us to use the relative numbercounts between the LHS and the NLTT to estimate the effective completeness limit for the latter catalogue. Given invariant kinematics, the number of stars in a proper motion sample varies with the inverse cube of the proper motion limit, since the effective distance limit is inversely proportional to the proper motion limit (see Flynn et al, 2001). The characteristic distance of a proper motion star also scales inversely with the proper motion limit, so the typical distance modulus for the catalogue scales inversely with the proper motion limit squared.
Consider two proper motion samples, with proper motion limits p1 and [2. Then the relative volume sampled is given by

fV = Vol1 / Vol2 = (p2 / p1)3
The two surveys have effective distance moduli of (m-M)1 and (m-M)2. Defining
dM = (m-M)2 - (m-M)1
then we expect the relative number counts to scale as
N2 (m) = fV . N1(m - dM)
Plugging in the relevant numbers for the NLTT and LHS, we expect
NNLTT (m) = 21 . NLHS(m - 2.2)
Figure 2.9 plots the relevant data for LHS and NLTT stars with 10 < RA < 16 hours, -20 < dec < 50, a high-latitude sample centred on the NGP. The observed ratio lies within 1-sigma of the predicted value for mr(NLTT) < 16, falling below expectations (fewer NLTT stars) at fainter magnitudes. This suggests that the NLTT is effectively 90% complete to mr = 16th magnitude.

Given this relatively bright limiting magnitude, the NLTT is not likely to provide a complete census of ultracool, low-luminosity M dwarfs in the Solar neighbourhood. However, with 90% completeness at 16th magnitude (equivalent to Mr = 14.5 at 20 parsecs), this catalopgue provides an effective means of at least increasing completeness by searching for previously overlooked early- and mid-type M dwarfs (spectral types late K to ~M5) in the immediate Solar neighbourhood.

3. Proper Motion Surveys, 2MASS and NStars

The release of the 2MASS catalogue provides the first opportunity to fully exploit the NLTT catalogue. It is clear from Figure 2.7 that Luyten's mr magnitudes, even though machine-derived for the most part, are not particularly accurate - fainter than 12th magnitude, the dispersion in mr-RC is over +/-0.5 magnitudes, with a systematic offset of ~0.5 magnitudes. Nonetheless, (R-K) (or (mr-K)) provides a long baseline colour index, sufficient that we can tolerate random errors on this scale, and still use (mr-K) colours to identify late-type dwarfs liable to be within 20 parsecs of the Sun. Figures 2.10 and 2.11 illustrate our basic strategy.

Figure 2.10: The (Mr, (mr-K)) colour-magnitude diagram defined by nearby stars with well determined trigonometric parallaxes.

There is no simple, elegant relation between Luyten's mr and either V or RC photometry. However, most of the nearby stars with well-determined trigonometric parallaxes are ioncluded in the NLTT, and therefore have mr estimates. Figure 2.10 plots the resulting colour-magnitude diagram. As expected, the scatter is substantial, even amongst the lowest-luminosity stars on the degenerate portion of the main sequence. This needs to be taken into account in identifying candidate nearby stars.

Figure 2.11: The (mr, (mr-K)) main sequence for stars at a distance of 20 parsecs. The lines indicate the criteria we have adopted to pick out candidate nearby stars.

Figure 2.11 plots the (mr, (mr-K)) colour-magnitude diagram for the nearby stars shifted to a distance of 20 parsecs. The solid line is defined by the relations

mrlim = 2.174 (mr - K) + 3.65, for 3.5 < (mr-K) < 4.3
mrlim = 5.25 (mr - K) - 9.58, for 4.3 < (mr-K) < 4.7
mrlim = 1.48 (mr - K) + 8.15, for 4.7 < (mr-K) < 7
Our r < 20 parsec candidates are selected by requiring mr < mrlim and (mr-K) > 3.5 (the vertical yellow line); all stars with (mr-K) > 7 are included as potential ultracool nearby dwarfs.

These criteria are used to define target lists of candidate nearby stars. The application of these techniques to the NLTT catalgue, and the subsequent analysis is described on the following pages:


Index NStars proper motion star index

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page by Neill Reid, last updated 23/02/2002