THE LUMINOSITY AND MASS FUNCTIONS OF LOW-MASS STARS IN THE GALACTIC DISK. II. THE FIELD

The SDSS (York et al. 2000; Stoughton et al. 2002) employed a 2.5 m telescope (Gunn et al. 2006) at Apache Point Observatory (APO) to conduct a photometric survey in the optical ugriz filters (Fukugita et al. 1996; Ivezić et al. 2007). The sky was imaged using a time-delayed integration technique. Great circles on the sky were scanned along six camera columns, each consisting of five 2048 × 2048 SITe/Tektronix CCDs with an exposure time of ∼54 s (Gunn et al. 1998). A custom photometric pipeline (Photo; Lupton et al. 2001) was constructed to analyze each image and perform photometry. Calibration onto a standard star network (Smith et al. 2002) was accomplished using observations from the "Photometric Telescope" (PT; Hogg et al. 2001; Tucker et al. 2006). Further discussion of PT calibrations for low-mass stars can be found in Davenport et al. (2007). Absolute astrometric accuracy is better than 01 (Pier et al. 2003). Centered on the northern Galactic cap, the imaging data span ∼10,000 deg², and is 95% complete to r ∼ 22.2 (Stoughton et al. 2002; Adelman-McCarthy et al. 2008). When the north Galactic pole was not visible from APO, ∼300 deg² were scanned along the δ = 0 region known as "Stripe 82" to empirically quantify completeness and photometric precision (Ivezić et al. 2007). Over 357 million unique photometric objects have been identified in the latest public data release (DR7, Abazajian et al. 2009). The photometric precision of SDSS is unrivaled for a survey of this size, with typical errors ≲0.02 mag (Ivezić et al. 2004, 2007).

We queried the SDSS catalog archive server (CAS) through the casjobs Web site (O'Mullane et al. 2005)⁶ for point sources with the following criteria.

• 1.  
  The observations fell within the Data Release 6-Legacy (DR6; Adelman-McCarthy et al. 2008) footprint. The equatorial and Galactic coordinate maps of the sample are shown in Figure 1.
• 2.  
  The photometric objects were flagged as PRIMARY. This flag serves two purposes. First, it implies that the GOOD flag has been set, where GOOD ≡ !BRIGHT AND (!BLENDED OR NODEBLEND OR N_CHILD = 0). BRIGHT refers to duplicate detections of bright objects and the other set of flags ensures that stars were not deblended and counted twice. The PRIMARY flag indicates that objects imaged multiple times are only counted once 2.⁷
• 3.  
  The object was classified morphologically as a star(TYPE = 6).
• 4.  
  The photometric objects fell within the following brightness and color limits:

  $$\begin{eqnarray*} &&i < 22.0, z < 21.2\\ &&r - i \ge 0.3, i -z \ge 0.2. \end{eqnarray*}$$

  The first two cuts extend past the 95% completeness limits of the survey (i < 21.3, z < 20.5; Stoughton et al. 2002), but more conservative completeness cuts are enforced below. The latter two cuts ensure that the stars have red colors typical of M dwarfs (Bochanski et al. 2007b; Covey et al. 2007; West et al. 2008).

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This query produced 32,816,619 matches. To ensure complete photometry, we required 16 < r < 22. These cuts conservatively account for the bright end of SDSS photometry, since the detectors saturate near the 15th magnitude (Stoughton et al. 2002). At the faint end, the r < 22 limit is slightly brighter than the formal 95% completeness limits. 23,323,453 stars remain after these brightness cuts.

SDSS provides many photometric flags that assess the quality of each measurement. These flags are described in detail by Stoughton et al. (2002) and in the SDSS Web documentation.⁸ With the following series of flag cuts, the ∼23 million photometric objects were cleaned to a complete, accurate sample. Since only the r, i, and z filters were used in this analysis, all of the following flags were only applied to those filters. The r-band distribution of sources is shown in Figure 2, along with the subset eliminated by each flag cut described below. The color–color diagrams for each of these subsets are shown in Figure 3.

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Saturated photometry was removed by selecting against objects with the SATURATED flag set. As seen in Figure 2, this cut removes mostly objects with r < 15. However, there were some fainter stars within the footprint of bright, saturated stars. These stars are also marked as SATURATED and not included in our sample. NOTCHECKED was used to further clean saturated stars from the photometry. This flag marks areas on the sky where Photo did not search for local maxima, such as the cores of saturated stars. Similarly, we eliminated sources with the PEAKCENTER set, where the center of a photometric object is identified by the peak pixel and not a more sophisticated centroiding algorithm. As seen in Figure 2, both of these flags composed a small fraction of the total number of observations and are more common near the bright and faint ends of the photometry. Saturated objects and very low signal-to-noise observations will fail many of these tests.

The last set of flags examines the structure of the point-spread function (PSF) after it has been measured. The PSF_FLUX_INTERP flag is set when over 20% of the star's PSF is interpolated. While Stoughton et al. (2002) claim that this procedure generally provides trustworthy photometry, they warn of cases where this may not be true. Visual inspection of the (r − i, i − z) color–color diagram in Figure 3 confirmed the latter, showing a wider locus than other flag cuts. The INTERP_CENTER flag is set when a pixel within three pixels of the center of a star is interpolated. The (r − i, i − z) color–color diagram of objects with INTERP_CENTER set is also wide, and the fit to the PSF could be significantly affected by an interpolated pixel near its center (Stoughton et al. 2002). Thus, stars with these flags set were removed. Finally, BAD_COUNTS_ERROR is set when a significant fraction of the star's PSF is interpolated over, and the photometric error estimates should not be trusted. Table 3 lists the number of stars in the sample with each flag set. For the final "clean" sample, we defined the following metaflag:

clean = (!SATURATED_r,i,z AND !PEAKCENTER_r,i,z AND !NOTCHECKED_r,i,z AND !PSF_FLUX_INTERP_r,i,z AND !INTERP_CENTER_r,i,z AND !BAD_COUNTS_ERROR_r,i,z AND (16< psfmag_r <22)).

After the flag cuts, the stellar sample was composed of 21,418,445 stars.

The final cut applied to the stellar sample was based on distance. As explained in Section 4.1, stellar densities were calculated within a 4 × 4 × 4 kpc³ cube centered on the Sun. Thus, only stars within this volume were retained, and the final number of stars in the sample is 15,340,771.⁹

In Figure 4, histograms of the r − i, i − z, and r − z colors are shown. These color histograms map directly to absolute magnitude, since color–magnitude relations (CMRs) are used to estimate absolute magnitude and distance. The structure seen in the color histograms at r − i ∼ 1.5 and r − z ∼ 2.2 results from the convolution of the peak of the LF with the Galactic stellar density profile over the volume probed by SDSS. Removing the density gradients and normalizing by the volume sampled constitutes the majority of the effort needed to convert these color histograms into an LF. The (g − r, r − i) and (r − i, i − z) color–color diagrams are shown in Figure 5, along with the model predictions of Baraffe et al. (1998) and Girardi et al. (2004). It is clearly evident that the models fail to reproduce the stellar locus, with discrepancies as large as ∼1 mag. These models should not be employed as color–absolute magnitude relations for low-mass stars.

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With any deep photometric survey, accurate star–galaxy separation is a requisite for many astronomical investigations. At faint magnitudes, galaxies far outnumber stars, especially at the Galactic latitudes covered by SDSS. Star–galaxy identification is done automatically in the SDSS pipeline, based on the brightness and morphology of a given source. Lupton et al. (2001) investigated the fidelity of this process, using overlap between HST observations and early SDSS photometry. They showed that star–galaxy separation is accurate for more than >95% of objects to a magnitude limit of r ∼ 21.5. Since the present sample extends to r = 22, we re-investigated the star–galaxy separation efficiency of the SDSS pipeline. We matched the SDSS pipeline photometry to the HST Advanced Camera for Surveys (ACS) images within the COSMOS (Scoville et al. 2007) footprint. The details of this analysis will be published in a later paper (J. J. Bochanski et al., 2010, in preparation). In Figure 6, we plot the colors and brightnesses of COSMOS galaxies identified as stars by the SDSS pipeline (red filled circles), along with a representative subsample of 0.02% of the stars in our sample. This figure demonstrates that for the majority of the stars in the present analysis, the SDSS morphological identifications are adequate, and contamination by galaxies is not a major systematic.

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The nearby star survey (D. A. Golimowski et al. 2010, in preparation) targeted stars with the colors of low-mass dwarfs and precise trigonometric parallaxes. The majority of targets were drawn from the Research Consortium on Nearby Stars (RECONS) catalog (e.g., Henry et al. 1994, 2004; Kirkpatrick et al. 1995). Most of the stars selected from the RECONS sample are within 10 pc, with good parallactic precision (σ_π/π ≲ 0.1). In addition to RECONS targets, the nearby sample included K dwarfs from the Luyten (1979) and Giclas et al. (1971) proper motion surveys. Parallax measurements for these additional stars were obtained from the Hipparcos (ESA 1997) or General Catalogue of Trigonometric Stellar Parallaxes (the "Yale" catalog; van Altena et al. 1995) surveys.

Near-infrared JHK_s photometry was obtained from the 2MASS Point Source Catalog (Cutri et al. 2003). Acquiring ugriz photometry proved more problematic. Since typical SDSS photometry saturates near r∼ 15, most of the nearby stars were too bright to be directly imaged with the 2.5 m telescope. Instead, the 0.5 m PT was used to obtain (ugriz)' photometry¹¹ of these stars. The PT was active every night; the 2.5 m telescope was used in imaging mode during the SDSS, observing patches of the nightly footprint to determine the photometric solution for the night, and to calibrate the zero point of the 2.5 m observations (Smith et al. 2002; Tucker et al. 2006). D. A. Golimowski et al. (2010, in preparation) obtained (ugriz)' photometry of the parallax sample over 20 nights for 268 low-mass stars. The transformations of Tucker et al. (2006) and the Davenport et al. (2007) corrections were applied to the nearby star photometry to transform the "primed" PT photometry to the native "unprimed" 2.5 m system (see Davenport et al. 2007 for more details).

To produce a reliable photometric parallax relation, the following criteria were imposed on the sample. First, stars with large photometric errors (σ>0.1 mag) in the griz bands were removed. Next, high signal-to-noise 2MASS photometry was selected, by choosing stars with their ph_qual flag equal to "AAA." This flag corresponds to a signal-to-noise ratio >10 and photometric uncertainties <0.1 mag in the JHK_s bands. Next, a limit on parallactic accuracy of σ_π/π < 0.10 was enforced. It ensured that the bias introduced by a parallax-limited sample, described by Lutz & Kelker (1973), is minimized. Since many of the stars in the nearby star sample have precise parallaxes (σ_π/π < 0.04), the Lutz–Kelker correction is essentially negligible (<−0.05; Hanson 1979). Finally, contaminants such as known subdwarfs, known binaries, suspected flares, or white dwarfs were culled from the nearby star sample.

To augment the original PT observations, we searched the literature for other low-mass stars with accurate parallaxes and ugriz and JHK_s photometry. The studies of Dahn et al. (2002) and Vrba et al. (2004) supplemented the original sample and provided accurate parallaxes (σ_π/π ≲ 0.1) of late M and L dwarfs. Several of those stars were observed with the SDSS 2.5 m telescope, obviating the need for transformations between the primed and unprimed ugriz systems. Six late M and L dwarfs were added from these catalogs, extending the parallax sample in color from r − i ∼ 2.5 to r − i ∼ 3.0 and in M_r from 16 to 20. Our final sample is given in Bochanski (2008).

Multiple color–absolute magnitude diagrams (CMDs) in the ugriz and JHK_s bandpasses were constructed using the photometry and parallaxes described above. The CMDs were individually inspected, fitting the main sequence with linear, second-, third-, and fourth-order polynomials. Piecewise functions were also tested, placing discontinuities by eye along the main sequence. There is an extensive discussion in the literature of a "break" in the main sequence near spectral type M4 (or V − I ∼ 2.8; see Hawley et al. 1996; Reid & Gizis 1997; Reid & Cruz 2002; Reid & Hawley 2005). Certain colors, such as V − I, show evidence of a break (Figure 10 of Reid & Cruz 2002), while other colors, such as V − K, do not (Figure 9 of Reid & Cruz 2002). We did not enforce a break in our fits. Finally, the rms scatter about the fit for each CMD was computed, and the relation that produced the smallest scatter for each color–absolute magnitude combination was retained. Note that the rms scatter was dominated by the intrinsic width of the main sequence, as both our photometry and distances have small (∼2%) uncertainties.

We present three different CMRs: (M_r, r − i), (M_r, r − z), and (M_r, i − z) in Table 4. M_r was used for absolute magnitude, as it contains significant flux in all late-type stars. The r − z color has the longest wavelength baseline and small residual rms scatter (σ ≲ 0.40 mag). Other long baseline colors (g − r, g − z) are metallicity sensitive (West et al. 2004; Lépine & Scholz 2008), but most of our sample does not have reliable g-band photometry. The adopted photometric parallax relations in these colors did not include any discontinuities, although we note a slight increase in the dispersion of the main sequence around M_r∼ 12. The final fits are shown in Figures 7, 9, and 10, along with other published photometric parallax relations in the ugriz system.

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Table 4. Color–Absolute Magnitude Relations in the ugriz System

Abs. Mag.	Color Range	Best Fit	$\sigma _{M_r}$
Mr	0.50 < r − z < 4.53	5.190 + 2.474 (r − z) + 0.4340 (r − z)2 − 0.08635 (r − z)3	0.394
Mr	0.62 < r − i < 2.82	5.025 − 4.548 (r − i) + 0.4175 (r − i)2 − 0.18315 (r − i)3	0.403
Mr	0.32 < i − z < 1.85	4.748 + 8.275 (i − z) + 2.2789 (i − z)2 − 1.5337 (i − z)3	0.481

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To assemble an (R, Z) density map, an accurate count of the number of stars in a given R, Z bin, as well as the volume spanned by each bin, was required. A cylindrical (R, Z, ϕ) coordinate system was taken as the natural coordinates of stellar density in the Milky Way. In this frame, the Sun's position was set at R_☉ = 8.5 kpc (Kerr & Lynden-Bell 1986) and Z_☉ = 15 pc above the plane (Cohen 1995; Ng et al. 1997; Binney et al. 1997). Azimuthal symmetry was assumed (and was recently verified by Jurić et al. 2008 and found to be appropriate for the local Galaxy). The following analysis was carried out in R and Z. We stress that we are not presenting any information on the ϕ = 0 plane. Rather, the density maps are summed over ϕ, collapsing the three-dimensional SDSS volume into a two-dimensional density map.

The coordinate transformation from a spherical coordinate system (ℓ, b, and d) to a cylindrical (R, Z) system was performed with the following equations:

$$\begin{equation} R = \sqrt{ (d \cos b)^2 + R_\odot (R_\odot - 2d \cos b \cos \ell)} \end{equation} \tag{ 2 }$$

$$\begin{equation} Z = Z_\odot + d \sin (b - \arctan (Z_\odot / R_\odot)), \end{equation} \tag{ 3 }$$

where d was the distance (as determined by Equation (1) and the (M_r, r − z) CMR), ℓ and b are the Galactic longitude and latitude, respectively, and R_☉ and Z_☉ are the positions of the Sun, as explained above.¹² The density maps were binned in R and Z. The bin width needed to be large enough to contain many stars (to minimize Poisson noise) but small enough to accurately resolve the structure of the thin and thick disks. The R, Z bin size was set at 25 pc. An example of the star counts as a function of R and Z is shown in Figure 11.

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The volume sampled by each R, Z bin was estimated with the following numerical method. A 4 × 4 × 4 kpc³ cube of "test" points was laid down, centered on the Sun, at uniform intervals 1/10th the R, Z bin size (every 2.5 pc). This grid discretizes the volume, with each point corresponding to a fraction of the total volume. Here, the volume associated with each grid point was k = 2.5³ pc³ point⁻¹ or 15.625 pc³ point⁻¹. The volume of an arbitrary shape is straightforward to calculate: simply count the points that fall within the shape and multiply by k. The α, δ, and distance of each point were calculated and compared to the SDSS volume. The number of test points in each R, Z bin was summed and multiplied by k to obtain the final volume corresponding to that R, Z bin. This process was repeated for each absolute magnitude slice. The maximum and minimum distances were calculated for each absolute magnitude slice (corresponding to the faint and bright apparent magnitude limits of the sample), and only test points within those bounds were counted. The same volume was used for all stars within the sample. The bluer stars in our sample were found at distances beyond 4 kpc, but computing volumes at these distances would be computationally prohibitive. Furthermore, this method minimizes galaxy contamination, which is largest for bluer, faint objects (see Section 2.3). Since the volumes are fully discretized, the error associated with N_points is Poisson-distributed. A fiducial example of the volume calculations is shown in Figure 12.

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After calculating the volume of each R, Z bin, the density (in units of stars pc⁻³) is simply,

$$\begin{equation} \rho (R,Z) = \frac{N(R,Z)}{V(R,Z)}, \end{equation} \tag{ 4 }$$

with the error given by,

$$\begin{equation} \sigma {_\rho } = \rho \sqrt{ \left(\frac{\sqrt{N(R,Z)}}{N(R,Z)} \right)^2 + \left(\frac{k\sqrt{N_{\rm points}(R,Z)}}{V(R,Z)} \right)^2}, \end{equation} \tag{ 5 }$$

where N(R, Z) is the star counts in each R, Z bin and V(R, Z) is the corresponding volume. Fiducial density and error maps are shown in Figures 13 and 14. Note that the error in Equation (5) is dominated by the first term on the right-hand side. While a smaller k could make the second term less significant, it would be computationally prohibitive to include more test points. We discuss systematic errors which may influence the measured density in Section 5.

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Using the method described above, (R, Z) stellar density maps were constructed for each 0.5 mag slice in M_r, from M_r = 7.25 to M_r = 15.75, roughly corresponding to spectral types M0–M8. The bin size in each map was constant, at 25 pc in the R and Z directions. For R, Z bins with density errors (Equation (5)) of <15%, the following disk density structure was fitted:

$$\begin{equation} \rho _{\rm thin}(R,Z) = \rho _{\circ } f e^{\left({ - \frac{ R - R_\odot }{R_{\rm \circ,thin}}}\right)} e^{ \left(- {\frac{\mid Z \mid - Z_\odot }{Z_{\rm \circ,thin}}}\right)} \end{equation} \tag{ 6 }$$

$$\begin{equation} \rho _{\rm thick}(R,Z) = \rho _{\circ } (1-f) e^{\left(-{ \frac{ R - R_\odot }{R_{\rm \circ,thick}}}\right)} e^{\left(- {\frac{\mid Z \mid - Z_\odot }{Z_{\rm \circ, thick}}}\right)} \end{equation} \tag{ 7 }$$

$$\begin{equation} \rho (R,Z) = \rho _{\rm thin}(R,Z) + \rho _{\rm thick}(R,Z), \end{equation} \tag{ 8 }$$

where ρ_° is the local density at the solar position (R_☉ = 8500 pc, Z_☉ = 15 pc), f is the fraction of the local density contributed by the thin disk, R_°,thin and R_°,thick are the thin and thick disk scale lengths, and Z_°,thin and Z_°,thick are the thin and thick disk scale heights, respectively. Since the density maps are dominated by nearby disk structure, the halo was neglected. Furthermore, Jurić et al. (2008) demonstrated that the halo structure is only important at |Z|>3 kpc, well outside the volumes probed here. Restricting the sample to bins with density errors <15% ensures that they are well populated by stars, have precise volume measurements, and should accurately trace the underlying Milky Way stellar distribution. Approximately 50% of the R, Z bins have errors ≳15%, while containing >90% of the stars in the sample.

The density maps were fitted using Equation (8) and a standard Levenberg–Marquardt algorithm (Press et al. 1992), using the following approach. First, the thin and thick disk scale heights and lengths, and their relative scaling, were measured using 10 absolute magnitude slices, from M_r = 7.25to11.75. These relatively more luminous stars yield the best estimates for GS parameters. Including lower-luminosity stars biases the fits, artificially shrinking the scale heights and lengths to compensate for density differences between a small number of adjacent R, Z bins. The scale lengths and heights and their relative normalization were fitted for the entire M_r = 7.25–11.75 range simultaneously. The resulting GS parameters (Z_o,thin, Z_o,thick, R_o,thin, Z_o,thick, f) are listed in Table 5 as raw values, not yet corrected for systematic effects (see Section 5 and Table 6). After the relative thin/thick normalization (f) and the scale heights and lengths of each component are fixed, the local densities were fitted for each absolute magnitude slice, using a progressive sigma clipping method similar to that of Jurić et al. (2008). This clipping technique excludes obvious density anomalies from biasing the final best fit. First, a density model was computed, and the standard deviation (σ) of the residuals was calculated. The R, Z density maps were refitted and bins with density residuals greater than 50σ are excluded. This process was repeated multiple times, with σ smoothly decreasing by the following series: σ = (40, 30, 20, 10, 5). An example LF, constructed from the local densities of each absolute magnitude slice and derived from the M_R, r − z CMR, is shown in Figure 15.

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A star with low metallicity will have a higher luminosity and temperature compared to its solar-metallicity counterpart of the same mass, as first described by Sandage & Eggen (1959). However, at a fixed color, stars with lower metallicities have fainter absolute magnitudes. Failing to account for this effect artificially brightens low-metallicity stars, increasing their estimated distance. This inflates densities at large distances, increasing the observed scale heights (e.g., King et al. 1990).

Quantifying the effects of metallicity on low-mass dwarfs is complicated by multiple factors. First, direct metallicity measurements of these cool stars are difficult (e.g., Woolf & Wallerstein 2006; Johnson & Apps 2009), as current models do not accurately reproduce their complex spectral features. Currently, measurements of metallicity-sensitive molecular band heads (CaH and TiO) are used to estimate the metallicity of M dwarfs at the ∼1 dex level (see Gizis 1997; Lépine et al. 2003; Burgasser & Kirkpatrick 2006; West et al. 2008), but detailed measurements are only available for a few stars. The effects of metallicity on the absolute magnitudes of low-mass stars are poorly constrained. Accurate parallaxes for nearby subdwarfs do exist (Monet et al. 1992; Reid 1997; Burgasser et al. 2008), but measurements of their precise metal abundances are difficult given the extreme complexity of calculating the opacity of the molecular absorption bands that dominate the spectra of M dwarfs. Observations of clusters with known metallicities could mitigate this problem (Clem et al. 2008; An et al. 2008), but there are no comprehensive observations in the ugriz system that probe the lower main sequence.

To test the systematic effects of metallicity on this study, the ([Fe/H],ΔM_r) relation from Ivezić et al. (2008) was adopted. We note that this relation is appropriate for more luminous F and G stars, near the main-sequence turnoff, but should give us a rough estimate for the magnitude offset. The adopted Galactic metallicity gradient is

$$\begin{equation} [{\rm Fe/H}] = -0.0958 - 2.77 \times 10^{-4} |Z|. \end{equation} \tag{ 9 }$$

At small Galactic heights (Z ≲ 100 pc), this linear gradient produces a metallicity of about [Fe/H] = −0.1, appropriate for nearby, local stars (Allende Prieto et al. 2004). At a height of ∼2 kpc (the maximum height probed by this study), the metallicity is [Fe/H] ∼−0.65, consistent with measured distributions (Ivezić et al. 2008). The actual metallicity distribution is probably more complex, but given the uncertainties associated with the effects of metallicity on M dwarfs, adopting a more complex description is not justified. The correction to the absolute magnitude, ΔM_r, measured from F and G stars in clusters of known metallicity and distance (Ivezić et al. 2008) is given by

$$\begin{equation} \Delta M_r = -0.10920 - 1.11[{\rm Fe/H}] - 0.18[{\rm Fe/H}]^2. \end{equation} \tag{ 10 }$$

Substituting Equation (9) into Equation (10) yields a quadratic equation for ΔM_r in Galactic height. After initially assigning absolute magnitudes and distances with the CMRs appropriate for nearby stars, each star's estimated height above the plane, Z_ini, was computed. This is related to the star's actual height, Z_true, through the following equation:

$$\begin{equation} Z_{\rm true} = Z_{\rm ini}10^{\frac{-\Delta M_r(Z_{\rm true})}{5}}. \end{equation} \tag{ 11 }$$

A star's true height above the plane was calculated by finding the root of this nonlinear equation. Since ΔM_r is a positive value, the actual distance from the Galactic plane, Z_true, is smaller than the initial estimate, Z_ini. As explained above, this effect becomes important at larger distances, moving stars inward and decreasing the density gradient. Thus, if metallicity effects are neglected, the scale heights and lengths are overestimated.

In Figure 16, the systematic effects of metallicity-dependent CMRs are shown. The first is the extreme limit, shown as the red histogram, where all stars in the sample have an [Fe/H] ∼ −0.65, corresponding to a ΔM_r of roughly 0.5 mag. All of the stars in the sample are shifted to smaller distances, greatly enhancing the local density. This limit is probably not realistic, as prior LF studies (e.g., Reid & Gizis 1997; Cruz et al. 2007) would have demonstrated similar behavior. The effect of the metallicity gradient given in Equation (9) is shown with the solid blue line. Note that local densities are increased, since more stars are shifted to smaller distances.

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The extinction and reddening corrections applied to SDSS photometry are derived from the Schlegel et al. (1998) dust maps and an assumed dust law of R_V = 3.1 (Cardelli et al. 1989). The median extinction in the sample is A_r = 0.09, while 95% of the sample has A_r < 0.41. Typical absolute magnitude differences due to reddening range up to ∼1 mag, producing distance corrections of ∼40 pc, enough to move stars between adjacent R, Z bins and absolute magnitude bins. This effect introduces strong covariances between adjacent luminosity bins, and implies that the final LF depends on the assumed extinction law. Most of the stars in our sample lie beyond the local dust column, and the full correction is probably appropriate (Marshall et al. 2006). To bracket the effects of extinction on our analysis, two LFs were computed. The first is the (M_r, r − z) LF, which employs the entire extinction correction. The second uses the same CMR, but without correcting for extinction. The two LFs are compared in Figure 17. When the extinction correction is neglected, stellar distances are underestimated, which increases the local density. This effect is most pronounced for larger luminosities. The dominant effect in this case is not the attenuation of light due to extinction, but rather the reddening of stars, which causes the stellar absolute magnitudes to be underestimated.

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The statistical error in a given LF bin is quite small, typically ≲0.1%, and does not represent a major source of uncertainty in this analysis. The assumed CMR dominates the systematic uncertainty, affecting the shape of the LF and resulting MF. To quantify the systematic uncertainty in the LF and GS, the following procedure was employed. The LF was computed five times using different CMRs: the (M_r, r − z) and (M_r, r − i) CMRs with and without metallicity corrections, and the (M_r, r − z) CMR without correcting for Galactic extinction. The LFs measured by each CMR are plotted in Figure 18, along with the unweighted mean of the five LF determinations. The uncertainty in a given LF mag bin was set by the maximum and minimum of the five test cases, often resulting in asymmetric error bars. This uncertainty was propagated through the entire analysis pipeline using three LFs: the mean, the "maximum" LF, corresponding to the maximum Φ in each magnitude bin, and the "minimum" LF, corresponding to the lowest Φ value. We adopted the mean LF as the observed system LF and proceeded to correct it for the effects of Malmquist bias and binarity, as described below.

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Malmquist bias (Malmquist 1936) arises in flux-limited surveys (such as SDSS), when distant stars with brighter absolute magnitudes (either intrinsically, from the width of the main sequence, or artificially, due to measurement error) scatter into the survey volume. These stars have their absolute magnitudes systematically overestimated (i.e., they are assigned fainter absolute magnitudes than they actually possess), which leads to underestimated intrinsic luminosities. Thus, their distances will be systematically underestimated. This effect artificially shrinks the observed scale heights and inflates the measured LF densities. Assuming a Gaussian distribution about a "true" mean absolute magnitude M_°, classical Malmquist bias is given by

$$\begin{equation} \bar{M}(m) = M_{\circ } - \frac{\sigma ^2}{{\rm log} e} \frac{dA(m)}{dm}, \end{equation} \tag{ 12 }$$

where σ is the spread in the main sequence (or CMR), dA(m)/dm is the slope of the star counts as a function of apparent magnitude m, and $\bar{M}(m)$ is the observed mean absolute magnitude. Qualitatively, $\bar{M}(m)$ is always less than M_° (assuming dA(m)/dm is positive), meaning that the observed absolute magnitude distribution is skewed toward more luminous objects.

Malmquist bias effects were quantified by including dispersions in absolute magnitude of $\sigma _{M_r} = 0.3$ and $\sigma _{M_r} = 0.5$ mag and a color dispersion of σ_{r − z,r − i} = 0.05 mag. These values were chosen to bracket the observed scatter in the color–magnitude diagrams (see Table 4). The LF measured with the Malmquist bias model is shown in Figure 19. The correction is important for most of the stars in the sample, especially the brightest stars (M_r < 10). Stars at this magnitude and color (r − i ∼ 0.5, r − z ∼ 1; see Figure 4) are very common in the SDSS sample because they span a larger volume than lower-luminosity stars. Thus, they are more susceptible to having over-luminous stars scattered into their absolute magnitude bins. However, the dominant factor that produced the differences between the raw and corrected LFs was the value of the thin disk scale height.

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For all but the widest pairs, binaries in our sample will masquerade as a single star. The unresolved duo will be over-luminous at a given color, leading to an underestimate of its distance. This compresses the density maps, leading to decreased scale heights and lengths, as binary systems are assigned smaller distances appropriate to single stars.

Currently, the parameter space that describes M dwarf binaries, binary fraction, mass ratio, and average separation, is not well constrained. However, there are general trends that are useful for modeling their gross properties. First, the binary fraction (f_b) seems to steadily decline from ∼50% at F and G stars (Duquennoy & Mayor 1991) to about 30% for M dwarfs (Fischer & Marcy 1992; Delfosse et al. 2004; Lada 2006; Burgasser et al. 2007). Next, the mass ratio distribution becomes increasingly peaked toward unity at lower masses. That is, F and G stars are more likely to have a companion from a wide range of masses, while M dwarfs are commonly found with a companion of nearly the same mass, when the M dwarf is the primary (warmer) star (Burgasser et al. 2007). The average separation distribution is not well known, but many companions are found with separations of ∼10–30 AU (Fischer & Marcy 1992), while very low mass stars have smaller average separations (Burgasser et al. 2007). At the typical distances probed by the SDSS sample (hundreds of pc) these binary systems would be unresolved by SDSS imaging with an average PSF width of 14 in r.

We introduced binaries into our simulations with four different binary fraction prescriptions. The first three (f_b = 30%, 40%, and 50%) are independent of primary star mass. The fourth binary fraction follows the methodology of Covey et al. (2008) and is a primary-star mass–dependent binary fraction, given by

$$\begin{equation} f_b({\cal M}_p) = 0.45 - \frac{0.7 - {\cal M}_p}{4}, \end{equation} \tag{ 13 }$$

where ${\cal M}_p$ is the mass of the primary star, estimated using the Delfosse et al. (2000) mass–luminosity relations. This linear equation reflects the crude observational properties described above for stars with ${\cal M}_p < 0.7\,{\cal M}_{\odot }$. Near 1 ${\cal M}_{\odot }$, the binary fraction is ∼50%, while at smaller masses, the binary fraction falls to ∼30%. Secondary stars are forced to be less massive than their primaries. This is the only constraint on the mass ratio distribution.

An iterative process, similar to that described in Covey et al. (2008), is employed to estimate the binary-star population. First, the mean observed LF from Figure 18 is input as a primary-star LF (PSLF). A mock stellar catalog is drawn from the PSLF, and binary stars are generated with the prescriptions described above. Next, the flux from each pair is merged, and new colors and brightnesses are calculated for each system. Scatter is introduced in color and absolute magnitude, as described in Section 5.4. The stellar catalog is analyzed with the same pipeline as the data, and the output model LF is compared to the observed LF. The input PSLF is then tweaked according to the differences between the observed system LF and the model system LF. This loop is repeated until the artificial system LF matches the observed system LF. Note that the GS parameters are also adjusted during this process, and the bias-corrected values are given in Table 6. The thin disk scale height, which has a strong effect on the derived LF, is in very good agreement with previous values. As the measured thin disk scale height increases, the density gradients decrease, and a smaller local density is needed to explain distant structures. This change is most pronounced at the bright end, where the majority of the stars are many thin disk scale heights away from the Sun (see Figure 19). The preferred model thin disk and thick disk scale lengths were found to be similar. This is most likely due to the limited radial extent of the survey compared to their typical scale lengths. Upcoming IR surveys of disk stars, such as APOGEE (Allende Prieto et al. 2008), should provide more accurate estimates of these parameters.

SDSS observations form a sensitive probe of the thin disk and thick disk scale heights, since the survey focused mainly on the northern Galactic cap. Our estimates suggest a larger thick disk scale height and smaller thick disk fraction than recent studies (e.g., Siegel et al. 2002; Jurić et al. 2008). However, these two parameters are highly degenerate (see Figure 1 of Siegel et al. 2002). In particular, the differences between our investigation and the Jurić et al. (2008) study highlight the sensitivity of these parameters to the assumed CMR and density profiles, as they included a halo in their study and we did not. The Jurić et al. (2008) study sampled larger distances than our work, which may affect the resulting Galactic parameters. However, the smaller normalization found in our study is in agreement with recent results from a kinematic analysis of nearby M dwarfs with SDSS spectroscopy (J. S. Pineda et al. 2010, in preparation). They find a relative normalization of ∼5%, similar to the present investigation. The discrepancy in scale height highlights the need for additional investigations into the thick disk and suggests that future investigations should be presented in terms of stellar mass contained in the thick disk, not scale height and normalization.

The iterative process described above accounts for binary stars in the sample and allows us to compare the system LF and single-star LF in Figure 20. Most observed LFs are system LFs, except for the local volume-limited surveys. However, most theoretical investigations into the IMF predict the form of the single-star MF. Note that for all binary prescriptions, the largest differences between the two LFs are seen at the faintest M_r, since the lowest-luminosity stars are most easily hidden in binary systems.

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Figure 21 demonstrates a clear difference between the single-star LF and the system LF. The single-star LF rises above the system LF near the peak at M_r ∼ 11 (or a spectral type ∼M4) and maintains a density about twice that of the system LF.¹³ This implies that lower-luminosity stars are easily hidden in binary systems, but isolated low-luminosity systems are intrinsically rare. The agreement between the system and single-star LFs at high luminosities is a byproduct of our binary prescription, which enforced that a secondary be less massive than its primary-star counterpart. Since our LF does not extend to higher masses (G and K stars), we may be missing some secondary companions to these stars, which would inflate the single-star LF at high luminosities. However, only ∼700,000 G dwarfs are present in the volume probed by this study. Even if all of these stars harbored an M-dwarf binary companion, the resulting differences in a given bin would only be a fraction of a percent.

Since many traditional LF studies have not employed the r band, our ability to compare our work to previous results is hampered. The most extensive study of the M_r LF was conducted by Jurić et al. (2008), using 48 million photometric SDSS observations, over different color ranges. Figure 19 compares the M_r system LF determined here to the "joint fit, bright parallax" results of Jurić et al. (2008, their Table 3), assuming 10% error bars. The two raw system LFs broadly agree statistically, although the Jurić et al. (2008) work only probes to M_r∼ 11, due to their red limit of r − i∼ 1.4. We compare system LFs, since the Jurić et al. (2008) study did not explicitly compute an SSLF and their reported LF was not corrected for Malmquist bias.

We next converted M_r to M_J using relations derived from the calibration sample described in Bochanski (2008). The J filter has traditionally been used as a tracer of mass (Delfosse et al. 2000) and bolometric luminosity (Golimowski et al. 2004) in low-mass stars, since it samples the spectral energy distribution (SED) near its peak. The largest field LF investigation to date, Covey et al. (2008), determined the J-band LF from M_J = 4 to M_J = 12. In Figure 22, our transformed system M_J LF (given in Table 9) is plotted with the M_J LF from Covey et al. (2008). The shape of these two LFs agrees quite well, both peaking near M_J = 8, although there appears to be a systematic offset, in that our M_J LF is consistently lower than the one from the Covey et al. (2008) study. This is most likely due to the different CMRs employed by the two studies. Covey et al. (2008) used an (M_i, i − J) CMR, as opposed to the various CMRs employed in the current study.

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Figure 22 also compares our single-star M_J LF (Table 10) to the LF of primaries and secondaries, first measured by Reid & Gizis (1997) and updated by Cruz et al. (2007). These stars are drawn from a volume-complete sample with d < 8 pc. A total of 146 stars in 103 systems are found within this limit. The distances for the stars in this volume-complete sample are primarily found from trigonometric parallaxes, with <5% of the stellar distances estimated by spectral type. There is reasonable agreement between our M_J LF and the LF from the volume-complete sample, within the estimated uncertainties, indicating that the photometric and volume-complete methods now give similar results. Furthermore, it indicates that our assumed CMRs are valid for local, low-mass stars. Finally, the agreement between the single-star LFs validates our assumed corrections for unresolved binarity.

A single analytic description of the IMF over a wide range in mass may not be appropriate. Figure 27 shows the derived MFs from this study, and those from the Reid & Gizis (1997) sample and the Pleiades (Moraux et al. 2004). The lognormal fit from this study is extended to higher masses, and it clearly fails to match the Pleiades MF. Therefore, it is very important to only use the analytic fits over the mass ranges where they are appropriate. Extending analytic fits beyond their quoted bounds can result in significant inaccuracies in the predicted number of stars.

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Any successful model of star formation must accurately predict the IMF. The measured field MF traces the IMF of low-mass stars averaged over the star formation history of the Milky Way. Thus, the field MF is not a useful tool for investigating changes in the IMF due to physical conditions in the star-forming regions, such as density or metallicity. However, it does lend insight into the dominant physical processes that shape the IMF. Recent theoretical investigations (see Elmegreen 2007, and references therein) have mainly focused on three major mechanisms that would shape the low-mass IMF: turbulent fragmentation, competitive accretion and ejection, and thermal cooling arguments.

Turbulent fragmentation occurs when supersonic shocks compress the molecular gas (Larson 1981; Padoan et al. 2001). Multiple shocks produce filaments within the gas, with properties tied to the shock properties. Clumps then form along these filaments and collapse ensues. In general, the shape of the IMF depends on the Mach number and power spectrum of shock velocities (Ballesteros-Paredes et al. 2006; Goodwin et al. 2006) and the molecular cloud density (Padoan & Nordlund 2002). Turbulence readily produces a clump distribution similar to the ubiquitous Salpeter IMF at high masses (>1 ${\cal M}_\odot$). However, the flattening at lower masses is reproduced if only a fraction of clumps are dense enough to form stars (Moraux et al. 2007).

An alternative model to turbulent fragmentation is accretion and ejection (Bate & Bonnell 2005). Briefly, small cores form near the opacity limit (∼0.003 ${\cal M}_\odot$), which is set by cloud composition, density, and temperature. These clumps proceed to accrete nearby gas. Massive stars form near the center of the cloud's gravitational potential, thus having access to a larger gas reservoir. Accretion ends when the nascent gas is consumed or the accreting object is ejected via dynamical interactions. The characteristic mass is set by the accretion rate and the typical timescale for ejection, with more dense star-forming environments producing more low-mass stars. This method has fallen out of favor recently, as brown dwarfs have been identified in weakly bound binaries (Luhman 2004b; Luhman et al. 2009), which should be destroyed if ejection is a dominant mechanism. Furthermore, if ejection is important, the spatial density of brown dwarfs should be higher near the outskirts of a cluster compared to stars, and this is not observed in Taurus (Luhman 2004a, 2006) or Chamaeleon (Joergens 2006).

Larson (2005) suggested that thermal cooling arguments are also important in star formation. This argument has gained some popularity, as it predicts a relative insensitivity of the IMF to initial conditions, which is supported by many observations (e.g., Kroupa 2002; Moraux et al. 2007; Bastian et al. 2010; and references therein). This insensitivity is due to changes in the cooling rate with density. At low densities, cooling is controlled by atomic and molecular transitions, while at higher densities, the gas is coupled with dust grains, and these dust grains dominate the cooling. The result is an equation of state with cooling at low densities and a slight heating term at high densities. This equation of state serves as a funneling mechanism and imprints a characteristic mass on the star formation process, with little sensitivity to the initial conditions.

The general shape of the IMF has been predicted by star formation theories that account for all of these effects (Chabrier 2003a, 2005). In particular, the high-mass IMF is regulated by the power spectrum of the turbulent flows (Padoan & Nordlund 2002; Hennebelle & Chabrier 2009) and is probably affected by the coagulation of less massive cores, while the flatter, low-mass distribution can be linked to the dispersions in gas density and temperatures (Moraux et al. 2007; Bonnell et al. 2006). As the IMF reported in this study is an average over the star formation history of the Milky Way, changes in the characteristic shape of the IMF cannot be recovered. However, our observational IMF can rule out star formation theories that do not show a flattening at low masses, with a characteristic mass ${\sim }0.2\,{\cal M}_\odot$. Recent numerical simulations have shown favorable agreement with our results (Bate 2009); however, most numerical simulations of star formation are restricted in sample size and suffer significant Poisson uncertainties. Analytical investigations of the IMF (Hennebelle & Chabrier 2008) are also showing promising results, reproducing characteristic masses ${\sim }0.3\,{\cal M}_\odot$ and lognormal distributions in the low-mass regime.