THE ASTRONOMICAL JOURNAL, 124:1570-1584, 2002 September
© 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.

We have used the Two Micron All Sky Survey and Palomar Observatory Sky Survey photographic plates, digitized by the US Naval Observatory's Precision Measuring Machine program, to derive proper motions over a 100 square degree, spatially complete region centered on the Praesepe open cluster. Proper-motion measurements spanned the magnitude range R 1219, which covers most of the lower main sequence in Praesepe. The incidence of H emission from moderate-resolution (2600), red spectroscopy of 434 faint candidate members and 126 field control stars demonstrates that 60%80% of candidates within 2° of the cluster center are genuine Praesepe members. Spectral index ratios from TiO and CaH absorption features show a well-defined sequence with J magnitude for high-probability candidates, relative to that of field stars. The inferred membership data have allowed us to estimate Praesepe's mass function and general projected structure for stars with mass 0.11.0 M. The mass function, when fitted with a power law of the form dN/dm m^-, rises ( 1.6) from 1.0 to 0.4 M, where it distinctly becomes flat ( 0) down to 0.1 M. The mass function also exhibits a marginal (1 ) radial dependence, which, when compared with recent N-body simulations, suggests that the core is depleted of low-mass stars. The outer halo of Praesepe is diffuse and shows a slightly elliptical shape among the highest probability members. We place an upper limit on the surface density of escaped members at 2.2 stars deg^-2 outside the limiting radius.

Key words: astrometryopen clusters and associations: individual (Praesepe)stars: activitystars: low-mass, brown dwarfs

On-line material: machine-readable tables

     ¹ Visiting Astronomer, Kitt Peak National Observatory, National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.

     In addition to providing a basis for the study of star formation, stellar evolution, and astronomical distances, open star clusters are laboratories for the study of small (N 1001000) stellar dynamical systems. Open clusters can be modeled with direct N-body numerical simulations. Aarseth (1999) includes a review of such N-body code. Extensive computations of simulated open clusters have been presented by Terlevich (1987), Kroupa (1995), de la Fuente Marcos & de la Fuente Marcos (2000), Portegies Zwart et al. (2001), and Kroupa, Aarseth, & Hurley (2001). Typically, these simulations model the role of multiple systems, mass segregation and velocity distribution, halo structure, and ultimately the evolution of the mass function from loss of stars through escape processes. However, empirical measurements of the properties of open clusters required to constrain the models depend on the study of faint, low-mass stars in clusters and thus are an observational challenge.

     Praesepe, a nearby (180 pc), intermediate-age (600900 Myr) open cluster, provides an opportunity for the study of stellar and dynamical evolution. Praesepe has been the object of several proper-motion searches. Klein Wassink (1927) and Artyukhina (1966a, 1966b) presented pioneering studies of bright proper-motion members, while Jones & Cudworth (1983) and Jones & Stauffer (1991) carried proper-motion detection ultimately down to V 18, corresponding to 0.3 M. Hambly et al. (1995b, hereafter HSHJ) performed a large, sensitive search to 0.1 M over 19 deg². In general, these previous studies were limited in areal coverage of the cluster at low mass, where a large number of stars contribute substantially to the dynamical mass. A study of the full extent of the cluster is required in order to determine its fundamental properties, such as the mass function and shape of the halo. This paper presents the results of the largest proper-motion study in Praesepe to date, using the Two Micron All Sky Survey (2MASS) and digitized Palomar Observatory Sky Survey (POSS) plate material to reach 0.1 M over 100 deg². The purpose of this work is to reveal the structure of the cluster and determine the nature of the mass function. We also compare the results with open cluster simulations described in Portegies Zwart et al. (2001).

     This study used 2MASS working database point-source data in the Praesepe field, which provided JHK_s photometry and positions on a uniform reference frame and also indicated the presence of any optical counterpart in the USNO-A catalog (Monet et al. 1996). The 2MASS data exceeded a signal-to-noise ratio of 10 at J = 15.8, H = 15.1, and K_s = 14.3. Internal positional precision of 2MASS is 01, and positional accuracy is better than 02.

     The USNO's Precision Measuring Machine (PMM) program (Monet 1998) scanned and digitized POSS-I E plates and POSS-II F plates. The PMM detections provided first-epoch point-source positions (POSS-I) and instrumental R_E and R_F magnitudes. Errors in positions and magnitudes should be comparable to those in the USNO-A catalog, about 025 and 0.25 mag, respectively, in the magnitude range 1219 (Monet et al. 1996). The F detections were used only to confirm E detections. In order to eliminate spurious POSS-I sources, we correlated E and F detections in the Praesepe field using a 5 search radius over a 6° radius around the nominal cluster center of R.A. = 8^h40^m, decl. = 19°59 (Lyngå 1987).

     We aligned the E positions onto the 2MASS reference frame using a linear least-squares fit. We then used 2MASS positions (epoch 19972000) and POSS-I positions (epoch 19511955) to calculate relative proper motions over a 45 yr baseline. We chose this method of computing proper motions because the high internal precision, longer time baseline, and smaller systematic errors associated with the 2MASS data yielded slightly better results for the proper-motion dispersion of HSHJ stars than did proper motions computed from POSS-II detections (Adams et al. 2001) or an average of results from both these methods. The final database of 2MASS and POSS correlations contained 280,000 sources with 2MASS positions, R_ER_FJHK_s magnitudes down to K_s = 15, and relative proper motions.

     Broad color selection reduced the number of field stars in the database. The design of the color cuts was based on the location of known Praesepe members from HSHJ and brighter stars listed in the Open Cluster Database,² such as those from Jones & Cudworth (1983) and Jones & Stauffer (1991), in K_s versus R_E-K_s and K_s versus J-K_s color-magnitude diagrams. The color selection accommodated both photometric error and binarity. Color selection was similar to that in Adams et al. (2001), but the magnitude range spanned 8.5 K_s < 15. Figures 1 and 2 show the color loci of known Praesepe members and regions selected for this study. Figure 1 shows that there is a systematic error in R_E-K_s colors at the brighter K_s magnitudes. However, this plot was used only to empirically select the regions where new candidate members will lie, rather than to derive any physical properties of stars. Thus, an external calibration and systematic correction for R_E magnitudes was unnecessary.

FIG. 1.Color-magnitude diagram using instrumental RE and 2MASS Ks magnitudes for all correlated RERFJHKs detections in the Praesepe field. The open squares represent colors for a sample of previously published Praesepe candidates from the Open Cluster Database, while the filled squares represent colors for HSHJ stars. Two typical error bars are shown near Ks = 9 and Ks = 14 to represent error with magnitude and color. The bounded area indicates the region in which sources were selected for proper-motion analysis.

FIG. 2.The 2MASS (Ks, J-Ks) color-magnitude diagram for the sources described in Fig. 1. Symbols and bounded region are analogous to those in Fig. 1.

     In total, 25,000 sources survived color selection. Source counts against magnitude indicated that the completeness of the sample reached R_E 18, which corresponds to mass 0.2 M for Praesepe members. Figure 3 shows the vector point diagram for these color-selected sources. The cluster lies near (-29, -07) per century, but it is contaminated by the distribution of field stars. The dispersion of cluster stars in the vector point diagram is due to astrometric measurement errors; the internal velocity dispersion of the cluster stars is negligible (<1 km s^-1; Jones 1971). The Praesepe region appears relatively more diluted by field stars than the corresponding Pleiades region (Adams et al. 2001), primarily for two reasons. First, Praesepe has a smaller average proper motion than the Pleiades, in a region where the field star density is roughly exponential in form. Second, this work pushes the color selection to a fainter magnitude in order to further cover magnitudes where the Praesepe luminosity function is likely rising or peaking.

FIG. 3.Vector point diagrams for color-selected stars. Most field stars lie near (0, 0) per century, whereas the cluster stars are distributed around (-29, -07) per century. The internal velocity dispersion of the cluster is negligible compared with astrometric error. Left, vector points for stars within a 3° radius around the nominal center of Praesepe; right, including vector points over the full field (6° radius).

     ² The Open Cluster Database, as provided by C. F. Prosser (deceased) and J. R. Stauffer, currently may be accessed at http://cfa-www.harvard.edu/~stauffer/opencl, or by anonymous ftp to cfa-ftp.harvard.edu (131.142.10.30), cd /pub/stauffer/clusters/.

     Functional forms can often describe the distribution of cluster and field stars in the vector point diagram. When fitted to the data, the distribution functions yield a probability of cluster membership for any data on the vector point diagram. The usefulness of the membership probabilities is limited by the surface density ratio of cluster stars to field stars in the vicinity of the cluster's proper motion, which depends on the magnitude of proper motion, the number of cluster and field stars detected, and the precision of the proper-motion measurements.

     We adopted a modified form of the maximum likelihood technique developed by Sanders (1971), first applied to Praesepe by Jones & Stauffer (1991), to compute the distribution functions and proper-motion membership probabilities. We used the same procedure outlined in Adams et al. (2001) for computing membership probabilities of stars near the mean Praesepe motion. In contrast to Jones & Stauffer (1991), we refrained from weighting membership probabilities by radial distance, in order to determine the structure of the outer portion of the cluster. The distribution function fitting procedure utilized 5519 sources within a boxed region of size 5 × 5 per century around the mean cluster motion, in the magnitude range 8.5 K_s 15. While, ideally, the distribution function ought to be fitted as a function of magnitude (Hambly, Hawkins, & Jameson 1993), the large area covered by this work in the outskirts of the cluster results in a large number of field stars in the vicinity of the cluster in the vector point diagram, which prevents our fitting the distribution function independently at the faint end of the sample. The membership probabilities are therefore biased toward stars brighter than about K_s 14.

     Table 1 lists positions, 2MASS magnitudes and errors, instrumental R_E magnitudes, and proper-motion membership probabilities for all sources in Praesepe's region of the vector point diagram. In total, we have detected 1159 sources in the field with proper-motion membership probability p 0.2; of those, 833 sources lie within 4° of the cluster center. Assuming the cluster radius is 4°, this implies that 30% of the stars within 4° are field stars, and thus we have detected 573 Praesepe stars with p 0.2.

TABLE 1     MAGNITUDES AND PROPER-MOTION MEMBERSHIP PROBABILITIES FOR SOURCES IN PRAESEPE'S REGION OF FIGURE 3

     The fraction of HSHJ stars that are included in our cluster sample provides a basic measure of the completeness of this work. The cluster center in our Figure 3 was well within 1 of the cluster center measured by HSHJ. The region in the vector point diagram used to fit the distribution functions contained 92% of 515 Praesepe candidates identified originally by HSHJ. We consider these results to be in good agreement with HSHJ. However, because of the large area we covered at low cluster surface density, our individual probabilities are less robust. Figure 4 shows a histogram of membership probabilities, as well as the distribution of our probabilities for HSHJ stars. This probability histogram, when compared with analogous plots in the HSHJ study (Fig. 10 of HSHJ), demonstrates that our individual membership values are on average lower than those of HSHJ. This means that our sample of Praesepe candidates contains more field star contamination. We found 313 (61%) HSHJ stars with our p 0.2. In addition, we have identified 152 new candidates with p 0.2 in the region and magnitude range covered by HSHJ. In general, however, most new candidates lie outside the HSHJ region. Section 3.2.1 contains a spectral comparison between new candidates from this work and candidates identified by HSHJ.

FIG. 4.Histogram of proper-motion membership probabilities. The dotted line represents the histogram for our membership probabilities of HSHJ stars.

     The stellar models of Baraffe et al. (1998) provided a means of estimating masses of Praesepe candidates using mass-luminosity relations. Recent distance modulus estimates range from 6.16 (Pinsonneault et al. 1998) to 6.28 (Robichon et al. 1999). We chose to adopt the distance modulus of 6.28, but using a lower estimate of 6.16 will show little difference in our results in comparison; both distance estimates are accurate enough for our purposes. Color relations from Carpenter (2001) provided a transformation of 2MASS JHK_s photometry into the standard CIT system. Interpolation of the Baraffe et al. models converted the resulting J magnitudes into masses for the range 0.11.0 M. This explicitly ignores the presence of binaries in our sample.

     We have also cross-correlated probable Praesepe members in the Open Cluster Database. In total, 199 members in that database were not recovered by this search; most were brighter than V 12, corresponding to 1 M. The Baraffe et al. models provided a mass-to-luminosity conversion for the Open Cluster Database members fainter than V = 11.5, while mass-magnitude relations from Henry & McCarthy (1993) and Allen (1973) provided mass estimates for brighter stars.

     Proper-motion studies of faint stars in nearby open clusters suffer from field star contamination. Observations at low to moderate resolution can provide useful membership information for a large sample of stars, albeit without definitive membership determination from high-resolution measurements of radial velocity or chemical abundances. This section presents spectroscopic results from such moderate-resolution observations of many Praesepe candidates. The primary goals were to assess the fraction of obvious field stars in the Praesepe sample and consequently identify viable candidates for future high-resolution studies. In addition, the spectroscopic results will serve as a sample with which to more reliably calibrate isochrones of more distant clusters against Praesepe isochrones than with proper-motion selection alone. Necessary, but not sufficient, criteria for membership include the presence of H in emission for stars later than type M2 (Williams et al. 1994). Less discriminating criteria include appropriate spectral type with location on the Praesepe main sequence. Thus, absence of H emission, or inappropriate spectral type, in low-mass candidates indicates that they belong to the field star population.

     Our goal was to obtain low- to moderate-resolution spectra in order to observe a relatively large number of Praesepe candidates. To achieve this goal, we used the Hydra multiobject, fiber-fed spectrograph and T2KC CCD camera (Barden et al. 1994) on the WIYN telescope³ at Kitt Peak National Observatory. Observations focused on Praesepe targets in WIYN fields near the center of the cluster. A simple scheme determined WIYN field coordinates systematically around the cluster. The circular fields, each about 1° in diameter, partially overlapped one another so that every point in the inner part of the cluster fell within at least one WIYN field. Table 2 lists the coordinates of the centers of the WIYN fields and the corresponding exposure times. Nine configurations targeted faint (R = 1620) stars. We also used two configurations to observe bright (R = 1216) candidates and several spectroscopic standards, which received some priority, in order to obtain standards over a sufficient range of spectral type. The Hydra program also included several fields in the Pleiades to obtain observations of faint standards reported by Steele & Jameson (1995). The exposure times on the Pleiades fields were also typically 6090 minutes each.

TABLE 2     COORDINATES FOR WIYN HYDRA SPECTROGRAPH FIELDS

     Within a field, Hydra targets were prioritized according to membership probability. Typically, WIYN fields nearest to the center of the cluster contained 5060 proper-motion candidates, while fields farther from the center contained fewer than 50 candidates. The Hydra fields included a control sample of field stars with similar colors to the Praesepe candidates and large proper-motion vectors (>1 per century), but which were well outside (>3 per century) the Praesepe vector distribution. Each field contained more than 10 sources in the control sample, except the first field, which was configured for spectral standards and bright candidates. Table 2 also gives the number of candidates and control stars in each Praesepe field. The telescope tracking system used at least five guide stars to maintain fiber orientation. Remaining Hydra fibers pointed to blank-sky positions or were moved to idle positions. In total, the program obtained observations for 434 candidates and 126 control stars.

     Our Hydra configuration consisted of the red fiber bundle, a grating with 600 lines mm^-1 and blaze angle 139, and a blocking filter to eliminate photons with wavelength < 5000 Å. This setup provided a spectral resolution of 2600 and spectral dispersion of 1.4 Å pixel^-1 over a wavelength region 62009000 Å. The T2KC CCD format was configured to read out an area containing 1200 × 2000 pixels, which covered all of the array illuminated by the fibers. For each field, dome flats were taken, as well as CuAr wavelength calibration images.

     We used standard CCD reductions and analysis routines in IRAF⁴ to process the CCD images, trace and flat-field the spectra, sum each spectrum over its aperture, determine the sky background and subtract it from each object, and compute the wavelength calibrations. The IRAF routine FITPROFS extracted the H equivalent width W_H for each one-dimensional object spectrum over a spectral region of 65556569 Å.

     ³ The WIYN Observatory is a joint facility of the University of WisconsinMadison, Indiana University, Yale University, and the National Optical Astronomy Observatory.
     ⁴ IRAF is distributed by the National Optical Astronomy Observatory.

3.2.1. H Equivalent Widths

     Chromospheric activity is more prevalent among late-type star cluster members, relative to field stars of similar type. All Praesepe members later than M2 are expected to show H in emission due to chromospheric activity (Williams et al. 1994). Chromospheric activity is less frequent in field stars, cumulatively at 22% for types M2M6 (Hawley, Gizis, & Reid 1996).

     Table 3 shows the dependence of W_H on proper-motion membership probability p, J magnitude, and radial distance r from the cluster center. Many candidates with low proper-motion membership probability are also unlikely to be cluster members, based on no or weak H emission. The onset of H emission in Praesepe stars occurs in the interval 12 J 13; thus, we limited Table 3 to candidates with J 12.5. For candidates with J 12.5 and p 0.2, 16% show no or weak H emission. We applied a correction to this value based on the fraction of dMe stars in the field (22%; Hawley et al. 1996) in order to estimate the total number of field stars. If there are 32 stars with p 0.2 and absent or weak H emission, then the total number of field stars must be 41, which yields a contamination fraction of 20% based on the number of candidates with H in emission from Table 3. The contamination fraction increases to 40% if candidates with p < 0.2 are included. The sample brighter than J = 12.5 will be less contaminated, because of fewer field stars of comparable magnitude, better astrometric precision, and more extensive information available from previous studies.

TABLE 3     EQUIVALENT WIDTH DEPENDENCES

     There was little difference in incidence of H emission between recovered candidates from HSHJ and our new, high-probability (p 0.2) candidates. For 145 recovered HSHJ stars with J 13, 85% had H in emission. Similarly, 88% of 62 new candidates with J 13 and p 0.2 showed H in emission. Thus, the sample in this work seems to have comparable contamination and completeness relative to the HSHJ sample in the inner regions of Praesepe.

     Color selection of Praesepe candidates determines, in large part, their range of spectral classes. The 2MASS colors lie near J-K_s 0.9, for both nearby field stars and cluster members. Inappropriate spectral class for a Praesepe candidate with these colors, due to, for example, reddening, indicates that it probably belongs to the field population rather than the cluster.

     The relative strengths of spectral features due to atomic and molecular absorption are a standard means of classifying M dwarfs (see, e.g., Kirkpatrick, Henry, & McCarthy 1991; Prosser, Stauffer, & Kraft 1991; Hamilton & Stauffer 1993). The primary molecular species in early to mid M dwarfs are TiO, CaOH, and CaH; VO features arise in spectra of late M dwarfs. In this paper, we adopt the technique used by Prosser et al. (1991) to classify Pleiades M dwarfs, because it is most compatible with the observed spectral resolution. The relevant spectral regions are their R2 (65076598 Å), R3 (66356718 Å), R4 (67506844 Å), and R7 (70007068 Å). R5 and R6 lie shortward of 6200 Å, at or beyond the limit of our Praesepe spectra. The flux ratio indices in these regions, R3/R2 and R4/R7, measure relative strengths of TiO and CaH absorption, which vary with spectral type in early to mid M dwarfs. We chose not to use R4/R2 because of its large scatter with spectral type and frequent inconsistency with the other ratios.

     The flux ratio R3/R2 was computed with the H line clipped in R2. The computation rejected minimum and maximum data points in each region to eliminate outliers that would produce spurious ratios. Poisson statistics determined the errors in each ratio. Figure 5 shows that the fractional errors were typically on the order of 10%.

FIG. 5.Errors R3/R2 and R4/R7 for spectral indices R3/R2 and R4/R7, respectively.

     Figure 6 contains plots of spectral indices R3/R2 and R4/R7 versus J magnitude for all spectra observed in this work. High-probability members show a well-defined sequence of spectral index with magnitude, but lower probability candidates show more scatter below J = 13, many of which appear to follow the field star ratios.

FIG. 6.Relationship between spectral indices R3/R2 and R4/R7 and J magnitude. Data points are distinguished as low-probability Praesepe candidates (open circles), high-probability candidates (filled circles), and field stars (crosses). WH refers to sources with H in emission.

     Spectral standards from Allen & Strom (1995) and Steele & Jameson (1995) provided a basis to calibrate the indices into spectral types. Table 4 lists the standards observed in this study. The table includes several faint Pleiades candidates whose spectra were taken during observations described in Adams et al. (2001). Figure 7 shows the flux ratios measured for the set of standards, where negative M subtype implies a K subtype (-1 = K7, etc.). Figure 7 also displays the best third-order polynomial fits relating adopted spectral types to observed flux ratios. A ² test indicated strong agreement between the fits and the data (p-values greater than 0.999). These fits determined a spectral type corresponding to each value of R3/R2 and R4/R7 for all candidate and control stars in the spectroscopic sample. The average over these two values of spectral type then gave a final, single spectral type for each candidate and control star. The standard deviation of the difference between our derived spectral types and for the standard stars and their previously published types was 0.6 subtypes.

TABLE 4     STANDARDS IN PRAESEPE AND THE PLEIADES USED TO CALIBRATE SPECTRAL INDICES INTO SPECTRAL TYPES

FIG. 7.Spectral indices R3/R2 and R4/R7 for standard stars of given M types (Table 4). Negative M subtypes represent K subtypes (-1 = K7, etc.). Solid lines show the best third-order polynomial fits.

     Tables 5 and 6 present 2MASS positions, magnitudes, and errors, as well as instrumental R_E magnitudes, proper-motion membership probabilities, spectral indices R3/R2 and R4/R7, spectral types, and H equivalent widths for all Praesepe candidates and field control stars observed at WIYN. Table 7 reports the incidence of H emission in high-probability candidates with spectral type and compares it with the same for control stars and field stars from Hawley et al. (1996). The high incidence of H emission among stars of type M3M5 indicates a cumulative field star contamination rate of 10%20%. Note these results are confined to a 2° radial distance from the cluster; substantially worse contamination rates are expected at larger radial distances. The lower rate of emission of our control stars relative to the field dMe fraction from Hawley et al. (1996) could be due to our selection criteria in color and large proper motions, which perhaps favored older, less chromospherically active disk or halo stars with larger, more random velocities.

TABLE 5     DATA FOR ALL PRAESEPE CANDIDATES OBSERVED AT WIYN

TABLE 6     DATA FOR ALL FIELD CONTROL STARS OBSERVED AT WIYN

TABLE 7     NUMBER AND FRACTION OF CANDIDATES WITH p 0.2 AND H IN EMISSION (WH 1 Å)

     Because of the difficult nature of gathering membership data for nearby clusters and interpreting their dynamical state with low number statistics, we utilized model data from the published N-body study of Portegies Zwart et al. (2001). The models served as a guide for identifying those effects of dynamical evolution that can be observed in real clusters, within the limits of projection effects and field star contamination. In addition, the observations provide empirical tests of the models.

     The models include the processes of binary formation and stellar evolution, as well as a Galactic tidal field resulting from circular orbit around the Galactic center. Specifically, we used models W4 and W6 at ages of 600 Myr, which lie at 6 and 12 kpc, respectively, from the Galactic center. At this age, W4 and W6 contained 661 and 777 single stars, plus 383 and 703 binary systems, respectively, comprising total masses of 770 and 991 M. Table 8 summarizes these model characteristics.

TABLE 8     CHARACTERISTICS OF N-BODY MODELS USED IN THIS STUDY

     To directly compare the model results with observations, we "observed" the model clusters at the location and distance of Praesepe. All binary systems were unresolved, whereby their total anticipated J-band luminosity was converted into an effective, observed mass using the same mass-luminosity relations and distance modulus that converted observed magnitudes into masses (see § 2.4).

     A sum of candidates provided a crude estimate of the cluster mass from the combined list from this study and the Open Cluster Database. We iterated the mass sum to provide agreement between the mass interior to a given radius and the mass M_c expected from the classical limiting radius of a cluster in circular orbit around the Galactic center:



(King 1962), where A and B are the Oort constants, here taken to be A = 14.4 km s^-1 kpc^-1 and B = -12.0 km s^-1 kpc^-1 (Kerr & Lynden-Bell 1986). We used a projected limiting radius of 38 for the mass estimate, corresponding to 12 pc at the distance of Praesepe, where the enclosed mass for high-probability candidates (p 0.2) within r agreed with that expected from the above equation. The total mass within this radius was 600 ± 19 M, without correction for systematic errors such as incompleteness, field star contamination, and unresolved binarity, each of which is estimated to be 10%30%. This mass estimate is consistent with previous measurements of the radial extent (Mermilliod et al. 1990) and total mass from various membership catalogs (Holland et al. 2000).

     Figure 8 shows the spatial distribution of our high-probability proper-motion candidates (p 0.2), with primarily brighter, more massive members from the Open Cluster Database overlaid as larger dots. Figure 8 also contains boxes that represent the regions surveyed by HSHJ. Figure 9 gives the surface number density profile of high-probability Praesepe candidates and stars in the projected models. The radial profiles follow similar slopes, with the exception that the cluster contains a constant surface beyond 3° because of field star contamination. This implies that, as expected, Praesepe is dynamically evolved and contains a spatial density distribution similar to the ones in the models. The field surface density outside a 4° radius places an upper limit of 2.2 stars deg^-2 on the evaporated population in this field.

FIG. 8.Spatial distribution of Praesepe candidates with proper-motion membership probability p 0.2 (smaller dots). Larger dots represent primarily more massive (>1 M) Praesepe members from the Open Cluster Database that were too bright for detection by this work. Boxed regions show the area spanned by the HSHJ study.

FIG. 9.Radial profile of surface density fs in stars per square degree for Praesepe candidates (p 0.2), and projected models W4 and W6. The relatively flat surface density level beyond 3° shows the presence of a field star component in the Praesepe field. Error bars were omitted from the W6 points for clarity.

     We attempted to fit the surface density of Praesepe stars with mass 0.11.0 M using a standard three-parameter King model (King 1962). The best ² fit gave a projected core radius of r_c = 11 ± 01 and a projected tidal radius of r_t = 5.3 ± 0.3 pc, corresponding to r_c 3.5 pc and r_t 16 pc at the distance of Praesepe. This estimate of the tidal radius is somewhat larger than the 12 pc measured by Mermilliod et al. (1990), perhaps as a result of field stars near r = 4°, where uncertainties from field stars limit the accuracy of the King model fit. Table 9 summarizes the fundamental properties of the Praesepe cluster.

TABLE 9     CHARACTERISTICS OF THE PRAESEPE CLUSTER

     The technique of principal components analysis, which Raboud & Mermilliod (1998a) applied to higher mass stars in the Pleiades, provides a suitable way of quantifying the spatial distribution in terms of its ellipticity and the position angle of its major axis. We measured the ellipticity e of the Praesepe distribution and the position angle of its major axis cumulatively with radius, normalized to r_lim = 38. Major-axis position angles with increasing radius, in the range 0 < r/r_lim < 0.8, vary widely in both Praesepe and the models. This is a signature of an approximately spherical shape in those regions. Beyond 0.8r/r_lim, a slight amount of ellipticity (e 0.10.2) is visible for candidates with p 0.4, but not for those with p 0.2. Figure 10 also shows the ellipticity of projected models W4 and W6. In the models, ellipticity with consistent orientation is detectable only near and beyond the tidal radius, where field stars dominate the observations. Note that the ellipticity of W4 (e 0.20.5) is larger than that of W6 (e 0.10.3), because of the stronger Galactic tidal field imposed on W4. The weaker ellipticity in the observations, a factor of 12 less than the ellipticity in the models, is probably an artifact of field star contamination. Better membership information at large distances from the cluster center is required to improve the ellipticity determination.

FIG. 10.Ellipticity e and major-axis position angle of the distribution of Praesepe candidates projected in the sky, plotted cumulatively in radius normalized to a projected limiting radius rlim = 38. Also shown are those of N-body model snapshots, at ages of 600 Myr, from Portegies Zwart et al. (2001). The W4 and W6 models are projected onto the sky at the distance of Praesepe and normalized to their projected limiting radii of 41 and 45, respectively. Error bars are from Poisson statistics; only one set of error bars is shown, for clarity.

     The presence of mass segregation has been well established in Praesepe (see, e.g., Jones & Stauffer 1991; Raboud & Mermilliod 1998b). Here we attempt to quantitatively compare mass segregation in Praesepe with the results of models W4 and W6. Figure 11 summarizes the dependence of mean stellar mass with radial distance for these cases, with Praesepe stars limited to p 0.2. In all cases, the mean mass decreases with radius by 30% from the inner to the outer regions. Also note that the standard deviation for the mean masses also decreases, as a result of the presence of a larger number of higher mass (m > 1 M) stars near the projected cluster centers.

FIG. 11.Mean stellar mass m and standard deviation for high-probability cluster candidates (solid lines) and model stars (models W4 and W6; dotted and dashed lines, respectively), binned into concentric annuli.

     The mass function is defined as (m) = dN/dm, where (m) is the number of stars in the mass interval [m, m + dm]. We chose to construct the mass function using mass bins 0.1 M in width in the range 0.11.0 M. Candidates from this study were cut at p = 0.01, in order to accommodate greater dispersion among proper-motion vectors of faint sources. As Figure 8 shows, a large number of field stars are present in the Praesepe proper-motion sample. The surface density of stars in an annulus 4°6° from the nominal center provided a basis for estimating the number of field stars in each mass bin. Figure 12 shows the mass function for Praesepe candidates, within r_lim = 38 of the nominal center of the cluster, after subtraction of the estimated number of field stars in each bin. The mass function is affected by detection incompleteness in the 0.10.2 M bin, as well as by the presence of unresolved binary stars in all mass bins below 1 M. Figure 12 also shows the scaled Pleiades mass function (Adams et al. 2001) for comparison.

FIG. 12.Mass function of Praesepe candidates at low mass, with proper-motion membership probability p 0.01, within a radius of 38 from the nominal center. Within each bin, the estimated number of field stars has been subtracted. The dashed line represents a brokenpower-law fit. Also shown is the mass function for candidates sorted into two concentric annuli, each 1 projected half-mass radius (125) in size, around the cluster center. The dotted line shows the Pleiades mass function from Adams et al. (2001), scaled by a factor of 1.4, without field star subtraction.

     The adopted form of the mass function was a power law, broken at 0.4 M. Each segment of the fit took the form (m) m^-. The best fits for each segment were



Thus, the mass function contains a rising exponent ( = 1.6) from 1.0 to 0.4 M, changing to one that is flat ( 0) below 0.4 M. These indices are much shallower than the Salpeter index of = 2.35 (Salpeter 1955). If the mass function remains flat below 0.1 M, the cluster contains 100 brown dwarfs, which would make up only 1%2% of the total cluster mass. However, this work does not reach the hydrogen burning mass limit and cannot exclude a change in the value of in the brown dwarf regime.

     Figure 12 also shows a 1 increase in the slope of the mass function with radial distance for candidates with p 0.2. A similar radial dependence of the mass function is seen in the Pleiades (Adams et al. 2001).

     Both this work and analogous work on the Pleiades show a marginal flattening of the mass function in the cluster cores. Figure 13 shows the radial dependence of the mass functions for models W4 and W6. Both cases show a flattening of the mass function at the 1 level due to the relative depletion of low-mass stars in the cores. Projection of the models onto the plane of the sky causes many halo stars to be included in the core populations, which implies that more depletion in the core is present than the mass functions reveal in projection. These results indicate that Praesepe and the Pleiades have experienced some limited form of mass segregation over their lifetimes. Dynamical interactions cause the massive stars to lose velocity and sink toward the center of the cluster, or to remain near the center if they formed close to the center. Low-mass stars preferentially populate halo orbits because of their larger velocities, acquired through interactions with higher mass stars and binary systems in the core. Tidal truncation of the velocity distribution in the halo would then expedite cluster evaporation (Mathieu 1985). We can infer from the models that Praesepe was much more populated at its origin, perhaps by a factor of 2, according to the evaporation rates caused by the static tidal field. A consequence of mass segregation is that the escaped members must preferentially be of low mass.

FIG. 13.Mass functions of projected models W4 and W6. The plots also contain the mass functions for the inner two concentric annuli, each 1 half-mass radius in size. The half-mass radii in projection are 11 and 12 for W4 and W6, respectively.

     Praesepe and the Hyades share comparable size and mass (400600 M), as well as age and space velocity (Eggen 1992). Thus, a comparison of their present-day mass functions is important. The slope of the Praesepe mass function from this work appears to more closely resemble the slope of the Hyades mass function than do slopes from previous Praesepe studies. While previous, spatially limited work in Praesepe has suggested that the mass function continues to rise below 0.4 M (Williams, Rieke, & Stauffer 1995; Hambly et al. 1995a), this work has shown a kink in the mass function at 0.4 M. Leggett, Harris, & Dahn (1994) and Reid & Hawley (1999) have shown that the Hyades mass function is flat at low mass. Isolated brown dwarfs are scarce in the Hyades (Gizis, Reid, & Monet 1999; Dobbie et al. 2002). Better observations are required to look for a similar trend in Praesepe, since this work is incomplete below 0.2 M.

     Previous studies of Praesepe have led to speculation on its dynamical interaction with external objects. Raboud & Mermilliod (1998b) mentioned that the distribution of brighter stars traced a structure that was more round than the Pleiades, because of, for example, an incidental collision with a massive object (e.g., a giant molecular cloud) that has tidally stripped an already tenuous halo (Wielen 1974). Eggen (1992) noticed a slight deviation in Praesepe's space velocity from the Hyades' and suggested its original velocity was closer to the Hyades velocity in the past, while perhaps meantime something has altered the motion of Praesepe. Holland et al. (2000) suggest that the irregular shape of Praesepe seen in surface density contours of previously cataloged members is due to a collision between Praesepe and a small (30 M) cluster, causing a disruption of Praesepe. This study shows little deviation of the observed Praesepe structure from the numerical simulation results, which suggests a normal evolutionary history within the errors and limitations of the data.

     Although the data presented in this study cover an unprecedented area in Praesepe, they are limited in many ways. Subsequent studies of proper-motion candidates outside the cluster are required in order to discriminate escaped members. The observed mass segregation in Praesepe and the Pleiades is consistent with the results of the N-body models. This consistency is a necessary, but not sufficient, criterion for evaluating the tidal field in the models. For example, introducing a time-dependent tidal field (e.g., Murali & Weinberg 1997; Combes, Leon, & Meylan 1999) into the models may affect the structure and evaporation rates of the models. Sufficient criteria for evaluating the models require better observations, including resolution of the internal distance and velocity dispersions within nearby open clusters, as well as the identification of escaping stars. The upcoming generation of space-based astrometry missions, for example, SIM (Unwin 2002) and GAIA (Perryman et al. 2001), will achieve such observational capabilities.

     We have used 2MASS and POSS photographic plates, digitized by the US Naval Observatory's PMM program, to derive proper motions over a 100 deg², spatially complete region centered on the Praesepe open cluster. Proper-motion detections spanned the magnitude range R 1219, which covers most of the lower main sequence in Praesepe. Using magnitude, color, and proper-motion selection criteria, we have identified 830 candidate members that lie within 4° of the cluster center, many in areas previously unstudied at low mass. A determination of the field star surface density outside the cluster suggests that about 573 candidates with p 0.2 are genuine Praesepe members, and about 313 were previously identified by Hambly et al. (1995b) over a limited area.

     Moderate-resolution (R 2600) spectroscopy over the 62009000 Å wavelength region of a sample of 434 Praesepe candidates and 126 field stars provided further statistical assessment of the proper-motion sample. The incidence of H emission shows that 20% of candidates within 2° of the nominal Praesepe center with p 0.2 are field stars. Our proper-motion sample has completeness and contamination comparable to those of the Hambly et al. (1995b) study in the core of Praesepe. Flux ratios in spectral regions containing primarily TiO and CaH absorption bands show a tighter correlation with J magnitude for candidates selected by proper-motion and H properties than field stars. We used the flux ratios to classify candidates and field stars by spectral type. A list of the spectral and photometric properties of candidates provides a database to select individual targets for future study.

     We estimate Praesepe's mass at 600 M, excluding correction for systematic errors such as unresolved binarity, field star contamination, and incompleteness, each of which is expected to be of order 10%30%.

     In projection, N-body models W4 and W6, and the Praesepe cluster, have a spherical shape inside 0.6 limiting radii. The models have an elliptical shape in the outer halo, due to the effects of the imposed Galactic tidal field. Only the highest probability Praesepe candidates suggest elliptical structure at large radial distances. In general, Praesepe shows lower ellipticity than the models by a factor of 12 in its outer parts, probably as a result of field star contamination.

     The mean mass of Praesepe candidates decreases with radial distance from the inner to the outer regions by about 30%. This result agrees with the presence of mass segregation in the projected N-body models.

     The Praesepe mass function, when fitted with a power law of the form dN/dm m^-, rises ( 1.6) from 1.0 to 0.4 M, where it distinctly becomes flat ( 0) down to 0.1 M. A radial dependence of the mass function at the 1 level suggests a depletion of low-mass stars in the core, which is expected from theoretical and numerical expectations. This result implies that escaped members are preferentially of low mass.

     More discriminating observations of Praesepe are required to confirm membership of individual stars, and to identify tidal streams or escaped populations. Future N-body models should include a more realistic, time-dependent Galactic tidal field.

     This work was supported by Boston University and NSF grant AST 97-31656. S. P. Z. is a fellow of the Royal Dutch Academy of Science. S. P. Z. was supported by NASA through Hubble Fellowship grant 1740-6. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research has also made use of the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA. The National Geographic SocietyPalomar Observatory Sky Atlas (POSS I) was made by the California Institute of Technology with grants from the National Geographic Society. The Second Palomar Observatory Sky Survey (POSS II) was made by the California Institute of Technology with funds from the National Science Foundation, the National Geographic Society, the Sloan Foundation, the Samuel Oschin Foundation, and the Eastman Kodak Corporation. The Digitized Sky Survey was produced at the Space Telescope Science Institute under US government grant NAGW-2166. The images of these surveys are based on photographic data obtained using the Oschin Schmidt telescope on Palomar Mountain and the UK Schmidt Telescope. The plates were processed into the present compressed digital form with the permission of these institutions. Thanks to Jun Makino, Steven McMillan, and Piet Hut. Part of this work used the GRAPE-4 and GRAPE-6 facilities at the University of Tokyo, Drexel University, Indiana University (Bloomington), MIT, and the American Museum of Natural History. This research has made use of NASA's Astrophysics Data System Abstract Service.

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