1. Why 8 parsecs?
The simplest method of deriving the stellar luminosity (and hence mass) function is to analyse a volume-limited sample; that is, a sample of stars which is complete to a particular distance limit. Choosing the appropriate distance limits is a balancing act: the nearer the distance limit, the greater than chance of being complete at even the faintest absolute magnitudes; but the nearer the distance limit, the smaller the sampling volume, the smaller the sample size and the larger the statistical uncertainties associated with the analysis. Traditionally (at least since van de Kamp), complete nearby star surveys (that is, surveys which set a common distance limit for all stars, rather than a magnitude-dependent limit) have chosen 5 parsecs as the appropriate limit.
5 parsecs is a fine limit for ensuring completeness, but not so fine for statistical analysis. There are only a total of 44 stars in 32 systems with distances of less than 5 parsecs. Those numbers are scarcely adequate for statistical analysis of parameters such as the multiplicity fraction, the mass-ratio and separation distributions of stars in binary systems, the luminosity function and the mass function.
So why expand to 8 parsecs? The change in distance limit corresponds to an increase in volume of a factor of 4 for an all-sky survey; the distance modulus at the limit (-0.48 mags) is such that the latest-type M dwarfs are still brighter than the effective limits of Luyten's LHS survey (mr ~ 18.5), at least the portion undertaken using plates taken on the Palomar 48-inch Schmidt; and, most important, Henry's speckle imaging survey (and, later, Oppenheimer's coronagraphic survey) for low-mass companions includes almost all M dwarfs with distances of less than 8 parsecs and declinations north of -25 degrees. The latter two considerations both suggest a qualifier - the 8-parsec sample should be limited to declinations accessible from northern hemisphere observatories.
The 8 parsec sample was compiled originally by Reid & Gizis (1997), with the main goal of understanding how binary stars affected the derivation of the luminosity function in photometric parallax surveys. As discussed elsewhere , deep photometric surveys provide one of the main tools for identifying late-type dwarfs and studying the stellar luminosity function; the derived luminosity functions, however, appeared to differ significantly from analyses of nearby stars (specifically, the Wielen et al, 1983, analysis of the Gliese CNS2 and supplement). One of the main complications in photometric parallax analysis is that the stars are at typical distances of 50 to 100 parsecs, and even moderately-separated binaries are likely to be unresolved, particularly on the photographic Schmidt plates used by most surveys. There are two main effects:
Figure E.1 plots the mass-ratio distribution for binaries in the initial sample. Reid & Gizis arrived at the conclusion that binarity isn't really a significant issue for those studies; rather, the problem lies with using a colour-magnitude calibration which actually matches the shape of the lower main-sequence (see Figure 1.8 here).
The initial 8-parsec sample consisted of 143 main-sequence stars (including the Sun), eight white dwarf degenerate stars and one brown dwarf (Gl 229B) in 106 systems. All lie north of a declination limit of -30o (B1950). Figure E.1 shows that those stars range from a handful of luminous A stars (Sirius, Vega) to the archetypical VLM dwarfs, VB 8 (M7) and VB 10 (M8). The 106 systems comprise 70 single stars (including 4 white dwarfs), 29 binaries (three with white dwarf components), six triple systems (where Gl 166B, or 40 Eri B, is a degenerate) and one quintuple system (Gl 643 \& Gl 644, the Wolf 629/630 system). not all systems had been surveyed to the same degree, but
As regarding the sample as a whole, the conclusion was that the sample might well be incomplete for stellar systems at MV > 14, but that that incompleteness was unlikely to be more than 10 %. In summary, it was regarded as unlikely that more than 10-12 stars in total, either undetected companions or newly-identified systems, remain to be added to the 8-parsec catalogue.
Figure E.2: The distribution of mass-ratios for the binary stars in the 8-parsec sample. The lower panel compares the observed distribution (solid points) against theoretical predictions for models assigning components by random selection from the same mass function. Those models predict a predominance high mass-ratio systems, rather than the bias toward near equal-mass ratio systems evident in the data.
Regarding binaries and biases for the luminosity function, Figure E.2 plots the mass-ratio (q) distribution for binaries in the initial sample. Those results generally support the binary model adopted by Reid (1991), with evidence of some bias toward equal-mass systems, rather than the predominance of high-q systems expected in a random association model. The observed binary fraction, and distribution of separations, amongst the field M dwarfs is generally consistent with the local sample. As a consequence, binarity isn't really a significant issue for luminosity function studies; rather, the problem in matching the photometric and nearby-star functions lies with the colour-magnitude calibration used on the lower main-sequence (see Figure 1.8 here).
One of the drawbacks of the initial 8-parsec sample was that spectroscopic parallaxes (mainly form the PMSU survey) provided the only distance estimates to a number of stars, while even some bright stars had trigonometric parallax measurements with substantial uncertainties. The ESA Hipparcos satellite transformed these circumstances. While observations were limited to relatively bright stars, and are complete (all-sky) to pnly 8th magnitude, known (or suspected) nearby stars were targetted specifically in the Input Catalogue. As a result, almost all of the 8-parsec stars with MV < 12 are included in the final catalogue. In addition, improved ground-based parallax measurements became available for the lower luminosity stars, while many of the hitherto-ignored lower luminosity stars have been subjected to more intense scrutiny for companions (Delfosse et al, 1999; Beuzit et al, 2001).
The net results are some additions and some subtractions from the 8-parsec sample, as summarised by Reid et al (1999):
|LP 476-207AabB||13||0.11+/-0.03||new component (1,2)|
|G 89-32AB||13||0.10+/-0.03||new component (1, 2)|
|LHS 2090||15||0.160+/-0.020||new system (4)|
|Gl 829B||11.9||106106||0.1483+/-0.0019||new component (2)|
|Gl 831C||15.4||0.126+/-0.023||new component (1)|
|Gl 896Aa||~13||116132||0.1601+/-0.0028||new component (2,3)|
|LTT 1445C||~14.5||0.127+/-0.025||new component (3)|
Comments: The first group of stars listed are rejected because the Hipparcos parallax
is less than 0.125 arcseconds.
LP 476-207 and G89-32 are eliminated because the discovery that they are binary/multiple systems implies that the apparent magnitude is enhanced, leading to a smaller inferred spectroscopic parallax.
Additions to the 8-parsec sample stem either from improved parallax data from the Hipparcos satellite, from new discoveries in surveys for multiple stars or, for LHS 2090, the addition of near-infrared data from 2MASS.
1 - Henry et al, 1997; 2 - Delfosse et al, 1999; 3 - Oppenheimer et al, 1999 LTT 1445 = LP 771-95/96 (RG97)
4 - Scholz et al (2002) - LHS 2090
The new data lead to the elimination of 7 systems (including 12 stars, although only 9 were listed in Reid & Gizis) from the sample, and the addition of five new systems and 5 new components of known systems (9 stars, one brown dwarf, Gl 570D). The Hipparcos catalogue includes five other stars which are both north of declination -30o and have nominal distances of less than 8 parsecs, although with substantial uncertainties. Those stars are BD-13:637B, Gl563.2A and B, BD+24:3192B and BD-15:6346B. Spectroscopy with the Double Spectrograph on the Hale 200-inch identifies all five as K-type stars, clearly incompatible with absolute magnitudes MV > 12, as implied by the Hipparcos parallax data.
Following these revisions, the 8-parsec sample includes a total of 153 stars and brown dwarfs in 104 systems: 143 main-sequence stars, eight degenerates and two brown dwarfs. Compared to the original sample, there has been a net loss of two systems and no change in the total number of main-sequence stars. Setting aside the four isolated white dwarfs, the multiplicity fraction for main-sequence systems is unchanged at 35+/-7 %.
Our primary motivation in extending the complete survey limits to 8 parsecs is obtaining better statistics. But are those gains spurious? Are we being overconfident, and sacrificing completeness as we push the survey limits beyond the traditional boundary? There are those who think so (e.g. Kroupa, 2001), pointing to low accuracy distance estimates for hypothetical sample members, and additions by the bucketful of previously-undetected, low-luminosity secondaries. However, those qualitative claims of low accuracy and gross incompleteness scarcely hold water when subjected to quantitative scrutiny.
Low accuracy distance measurements: rumour has it that a substantial fraction of the systems in the 8-parsec sample have distance estimates based on spectroscopic or photometric parallax estimates, rather than high-accuracy trigonomteric parallaxes. It is certainly the case that a fair number of systems in the original sample lacked trigonometric parallax data (although even then, the fraction was scarcely 30 %). Post-Hipparcos, and with the addition of new parallaxes from the USNO program, 100 of the 104 systems in the 8-parsec sample have distance determinations based on trigonometric data, usually with accuracies of much better than 5%. The four systems currently lacking trigonometric parallaxes are LHS 1723 (M4), G41-14AB (M3.5), LP 229-17 (M3.5) and the recent addition, LHS 2090 (M5.5). The 8-parsec sample is scarcely deficient with respect to the 5-parsec sample in this area.
New low-luminosity companions: given both advances in technology (higher-precision radial velocities, adaptive optics) and knowledge of which stars to look at, a substantial number of new companions have been discovered associated with nearby M dwarfs. However, the overwhelming majority of those discoveries are new companions to stars beyond the 8-parsec limit. As an example, Delfosse et al (1999) list 13 `new' companions, based on radial velocity monitoring of nearby M dwarfs. However, three of those companions were actually discovered by Reid & Gizis from their radial velocity observations; one is the planetary-mass companion to Gl 876; six are beyond the 8-parsec limit; leaving only three additions (all of which are included in the Reid et al (1999) revision). Similarly, while Beuzit et al (2001) have used adaptive optics to identify a further 13 new companions, none of the primaries are in the 8-parsec sample.
New low-luminosity systems: the availability of the DENIS and 2MASS near-infrared sky surveys provides an oportunity of searching for previously-unrecognised low-luminosity late-type systemsin the Solar neighbourhood (hence our NStars project). However, so far those observations have only turned up one system which is both closer than 8 parsecs and north of declination -30o - LHS 2090. There are additions to the southern sample - DENIS-P J1048-2956 (Delfosse et al, 2001) and LHS 1565 (Henry et al., 1997) - but we expect the southern sample to be less complete, particularly at those faint magnitudes.
Figure E.3: The cumulative distribution of systems as a function of distance from the Sun. Fitting an n3 relation to the observed density within 5 parsecs indicates a deficit of ~ 35 systems at d < 8 pc and 130 systems to d < 10 pc. Figure acquired from Todd Henry's site (see the RECONs pages ). Note that the predictions are for an all-sky sample.
So what evidence is there for any incompleteness? Glad you asked. The only datum which suggest significant incompleteness in the 8-parsec sample is the predicted number of stellar systems based on the extrapolation of the number density of stars within 5 parsecs of the Sun. Figure E.3, from Todd Henry's RECONS site, presents the evidence - the cumulative distribution as a function of parallax of stellar systems within 10 parsecs of the Sun. Galactic density laws have negligible variation within that limited volume, so the expectation is that those systems are effectively drawn at random for a uniform-density distribution. In that case, one expects a cumulative distribution that depends only on the volume surveyed, i.e.
So isn't that fairly conclusive? Well, yes and no - mainly no. The crucial point is which stellar systems are predicted as missing from the current 8-parsec sample. Consider the northern 8-parsec sample (i.e. Declination > -30o: Figure E.4 compares the luminosity function of single main-sequence stars and primaries in multiple systems for distance limits of 5 parsecs and 8 parsecs. The ratio between the two sampling volumes is 1:4; the observed numbers are 32 systems within 5 parsecs, and 101 systems within 8 parsecs. At face value, this suggests a deficit of 27 systems in the 8-parsec sample. On the other hand, the sampling statistics are not overwhelming: the formal Poisson uncertainties in the predicted numbers are 128+/-22.6; the uncertainties in the observed numbers are 101+/-10. Expressed in terms of the number density of stellar systems, there is barely a 1-sigma difference between the 5-parsec and 8-parsec results.
Figure E.4: Comparison between the luminosity function of main-sequence stellar systems for distancelimits of 8 parsecs (dotted line) and 5 parsecs (solid line).
Figure E.4 compares the luminosity function of the 8-parsec and 5-parsec samples, again limiting the datasets to main-sequence stars which are either single or primaries in multiple systems. Both functions are scaled to the same sampling volume. If the difference between the two datasets stems from incompleteness, one would expect the deficit to lie at the faintest absolute magnitudes, and it is clear that the 5-parsec sample does include proprotionately more stars fainter than MV > 13 than does the 8-parsec sample. On the other hand, the 8-parsec sample has twice as many G dwarfs than the 5-parsec sample, while the latter includes no stars with either 8 < MV < 9 or 12 < MV 13.Moreover, the bulk of the deficit at fainter magnitudes lies in stars with MV < 16; these stars have (V-R) colours exceeding 1.4 magnitudes, and at distances of less than 8 parsecs are expected to have R magnitudes brighter than ~14.
Figure E.5 plots the northern 5-parsec and 8-parsec luminosity functions with the associated formal sampling uncertainties. The discrepancies rarely exceed 1-sigma. The bulk of the deficit at fainter magnitudes lies in stars with MV < 16; these stars have (V-R) colours exceeding 1.4 magnitudes, and at distances of less than 8 parsecs are expected to have R magnitudes brighter than ~14. These are relatively bright stars, and one would expect most to have been identified by Luyten in his proper motion surveys, since even with all the obvious selection effects (see our NLTT survey ), Luyten's Two-Tenths catalogue shows little evidence for differential incompleteness at these magnitudes (Figure E.6). Weis has obtained VRI photometry of every class-m dwarf in the NLTT catalogue with mr < 13.5 and a declination north of the equator; no additions to the 8-parsec sample emerged from those observations.
Figure E.6: The distribution of NLTT stars on the celestial sphere as a function of magnitude. The lower star density in the Galactic Plane and at southerly declinations (covered by the Bruce survey) become significant only at mpg > 15.
Could the stars have lower proper motions? Not with radically changing the disk velocity distribution, since the proper motion limit corresponds to a tangential velocity of only 7.5 km/sec at 8 parsecs.
The bottom line is that while a straight comparison between the 5-parsec and 8-parsec samples points to a deficit in the latter regime, there is a problem in explaining where so many bright stars are hiding. It seems more likely (at least to this author) that, while there undoubtedly is some incompleteness at the faintest magnitudes, most of the deficit is covered by counting statistics - and there just happen to be a few more stars in the immediate vicinity of the Sun than within the 8-parsec sphere. Meanwhile, the onus rests with those who claim that there is a deficit to prove it by finding the missing stars.
We can use the 8-parsec sample to derive an estimate of the stellar mass function. This requires using a mass-luminosity relation to transform the observed absolute magnitudes (or luminosities) to mass. That relation is usually calibrated through observations of binary stars, either eclipsing systems or astrometric binaries. Those data and the resulting mass-luminosity calibrations are discussed as part of our PMSU survey, see the PMSU4 pages.
Figure E.7: The mass function derived from the 8-parsec sample. See this page for further discussion of the M/L calibration used.
Figure E.7 shows the results of applying one set of calibrations to the 8-parsec dataset. The distribution is relatively flat at low amsses, corresponding to a power-law index close to 1 (where 2.35 is the Salpeter value). This implies equal numbers of objects per decade in mass. Under those circumstances, the contribution towards the local mass density decreases with decreasing mass, and, if extrapolated into the substellar regime, indicates that brown dwarfs are remarkably unlikely to make any contribution to dark matter, whether in the Galactic disk, in galactic halos, or in galaxy clusters.
Data for the current 8-parsec sample are listed in the ascii data file linked below.
The format for the northern stars
is as follows (matching the appendices in Reid & Hawley, New Light on
Name, MV, (B-V), (V-R), (V-I), K, (J-H), (H-K), multiplicity, spectral type, distance, uncertainty in distance, Mbol
where VRI are on the Cousins system, and JHK on the CIT system. The multiplicity parameter is "1" for single stars, "2" for primaries, "3" for secondaries, tertiaries etc in multiple systems, and "6" for white dwarfs.
The data format for the southern stars is
NN, MU, MB, MV, MR, MI, MJ, MH, MK, TiO5, CaH2, CaH3, d, uncertainty, +/- mag, name