The local stellar sample

The best knowledge about the IMF we glean from the local volume-limited stellar sample. The basic technique is to construct the stellar luminosity function (LF),

V(Mp), where MP is the stellar absolute magnitude in some photometric pass band (we are stuck with magnitudes rather than working with the physically more intuitive luminosities due to our historical inheritance). The number of stars within the complete volume in the magnitude interval MP, MP + dMP is then dN = V (MP) dMP. These are the same stars as enter dN = % (m )dm above, and thus results our master equation

d MP

The observable is V, which we get from the sky. Our target is % and the hurdle is the derivative of the stellar mass-luminosity relation, dm /d MP. This is quite problematical, because we can get at dm /dMP only by either constructing observational mass-luminosity relations using extremely well-observed binary stars with known Kepler solutions that nevertheless have uncertainties that are magnified when considering the derivative, or resorting to theoretical stellar models, which give us well-defined derivatives but depend on theoretically difficult processes within stellar interiors (opacities, convection, rotation, magnetic fields, the equation of state, nuclear-energy-generation processes).

There are two basic local LFs.

(I) We can count all stars within a trigonometrically defined distance limit such that the stellar sample is complete, i.e. we can see all stars of magnitudes in the range MP, MP + dMP within a distance rt. The volume-limited sample for Solartype stars having excellent parallax measurements extends to rt ~ 20 — 30 pc, while for the faintest M dwarfs rt ~ 5 — 8 pc (Reid et al. 2002). Tests of completeness are made by comparing the number of stars with MP within rt with the number of such stars in a volume element further out (Henry et al. 1997), finding that faint stars remain to be discovered even within a distance of 5 pc. For this nearby LF, Vnearby, the stars are well scrutinised on an individual-object basis, and geometric distances are known to within about 10%. At the faint end Vnearby is badly constrained, resting on only a few stars.

(II) Prompted by the 'discovery' of large amounts of dark matter in the MW disc (Bahcall 1984),2 novel deep surveys were pioneered by Reid & Gilmore (1982). This second type of sampling can be obtained by performing deep, pencil-beam-photographic or CCD-imaging surveys through the Galactic disc. From the 105 images the typically 100 or so main-sequence stars need to be gleaned using automatic image-, colour- and brightness-recognition systems. The distances of the stars are determined using the method of photometric parallax, which relies on estimating the absolute luminosity of a star from its colour and then calculating its

2 The evidence for dark matter within the Solar vicinity disappeared on closer scrutiny (Kuijken & Gilmore 1991).

Figure 24.1. Stellar luminosity functions (LFs) for Solar-neighbourhood stars. The photometric LF corrected for Malmquist bias and at the mid-plane of the Galactic disc (^phot) is compared with the nearby LF (^near). The average, ground-based ^phot (dashed histogram, data pre-dating 1995, Kroupa (1995a)) is confirmed by Hubble Space Telescope star-count data, which pass through the entire Galactic disc and are thus less prone to Malmquist bias (solid dots, Zheng et al. (2001)). The ground-based volume-limited trigonometric-parallax sample (dotted histogram) systematically overestimates ^near due to the Lutz-Kelker bias, thus lying above the improved estimate provided by the Hipparcos-satellite data (solid histogram, JahreiB & Wielen (1997) and Kroupa (2001b)). The depression/plateau near MV = 7 is the Wielen dip. The maximum near MV « 12, Mj « 9 is the KTG peak. The thin dotted histogram at the faint end indicates the level of refinement provided by recent stellar additions (Kroupa 2001b), demonstrating that even our star-count for the immediate neighbourhood within 5.2 pc of the Sun probably remains incomplete at the faintest stellar luminosities.

Figure 24.1. Stellar luminosity functions (LFs) for Solar-neighbourhood stars. The photometric LF corrected for Malmquist bias and at the mid-plane of the Galactic disc (^phot) is compared with the nearby LF (^near). The average, ground-based ^phot (dashed histogram, data pre-dating 1995, Kroupa (1995a)) is confirmed by Hubble Space Telescope star-count data, which pass through the entire Galactic disc and are thus less prone to Malmquist bias (solid dots, Zheng et al. (2001)). The ground-based volume-limited trigonometric-parallax sample (dotted histogram) systematically overestimates ^near due to the Lutz-Kelker bias, thus lying above the improved estimate provided by the Hipparcos-satellite data (solid histogram, JahreiB & Wielen (1997) and Kroupa (2001b)). The depression/plateau near MV = 7 is the Wielen dip. The maximum near MV « 12, Mj « 9 is the KTG peak. The thin dotted histogram at the faint end indicates the level of refinement provided by recent stellar additions (Kroupa 2001b), demonstrating that even our star-count for the immediate neighbourhood within 5.2 pc of the Sun probably remains incomplete at the faintest stellar luminosities.

distance from the distance modulus. The resulting flux-limited sample of stars has photometric-distance limits within which the counts are complete. The distance limits are smaller for fainter stars.

Clearly, while only one nearby LF exists, many photometric LFs can be constructed for different fields of view. Each observation yields a few dozen to hundreds of stars, so the overall sample size becomes very significant. The various surveys have shown ^phot to be invariant with direction. This should, of course, be the case, since the Galactic-field stars with an average age of about 5 Gyr have a velocity dispersion of about 25-50 pcMyr-1 such that within 200 Myr a volume with a dimension of the survey volumes (a few hundred parsecs) is completely mixed.

It therefore came as a surprise that ^near and ^phot are significantly different at faint luminosities (Figure 24.1).

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