A&A 375, 863-889 (2001)
DOI: 10.1051/0004-6361:20010918
R. D. Jeffries
1,
-
M. R. Thurston2 -
N. C. Hambly3
1 - Department of Physics, Keele University, Keele, Staffordshire
ST5 5BG, UK
2 -
School of Physics and Astronomy,
University of Birmingham, Edgbaston, Birmingham B15 2TT,
UK
3 -
Institute for Astronomy, University of Edinburgh, Blackford
Hill, Edinburgh EH9 3HJ, UK
Received 15 May 2001 / Accepted 26 June 2001
Abstract
We present the results of a 0.86 square degree CCD photometric survey
of the open cluster NGC 2516, which has an age of about 150 Myr and
may have a much lower metallicity than the similarly-aged Pleiades. Our
survey of cluster members is complete to
and is
used to select a preliminary catalogue of 1254 low mass
(
)
cluster candidates, of which about 70-80 percent
are expected to be genuine. After applying corrections for
contamination by non-members and adding data for higher mass stars from
the literature, we investigate the cluster binarity, luminosity and
mass function, mass segregation and total mass. We find a binary
fraction of
percent, for A to M-type systems with mass ratios
between 0.6 and 1, which is very similar to the Pleiades. The mass
function is metallicity and evolutionary-model dependent, but
consistent with a Salpeter-like law (
,
or
for the solar and
half-solar metallicity models of Siess et al. 2000, and
for the solar metallicity models of
D'Antona & Mazzitelli 1997), for
.
At lower masses
(
)
there is a sharp fall in the mass function, with
or
(for the solar and
half-solar metallicity models of Siess et al.), and
(for the solar metallicity models of D'Antona &
Mazzitelli). The true stellar mass function might have
values up to 0.4
larger if account were taken of low mass stars in
unresolved binary systems with mass ratios less than 0.6. The falling mass
function of NGC 2516 at lower masses seems inconsistent with the much
flatter mass functions derived from comparable data in the Pleiades and
field populations. This deficit of lower mass, fainter stars is also
seen in the observed luminosity function. We rule out incompleteness
as the cause of this discrepancy, but demonstrate that mass segregation
is clearly present in NGC 2516, with more than half the low-mass
(<
)
stars likely to lie outside our survey area, but the
vast majority of high-mass (>
)
stars included. Taking
this into account, it is probable that the whole-cluster mass functions
for NGC 2516 and the Pleiades are similar down to 0.3
.
The
mass of NGC 2516 stars with
inside our survey is
,
depending on metallicity and what corrections are
applied for unresolved binarity. Correcting for mass segregation
increases this to
,
about twice the total mass
of the Pleiades. If NGC 2516 and the Pleiades do have similar mass
functions, then less massive stars and brown dwarfs contribute about a
further 15 percent to the mass of NGC 2516 and we predict a cluster
population of about 360-440 brown dwarfs with
.
Key words: open clusters and associations: individual: NGC 2516 - stars: luminosity function, mass function - stars: binaries: general
Observations of co-eval stars in open clusters play a vital rôle in investigating the low-mass stellar initial mass function and defining the physical processes that drive the evolution of rotation, magnetic activity and photospheric light element abundances in cool stars with convective envelopes. The ability to study samples with common, well determined, ages, distances and compositions is the key to their usefulness. A careful census and identification of cluster members is usually a prerequisite of all such studies: (a) in order to prevent contamination of sample properties by interloping non-members that have different properties to the cluster stars and (b) to select samples of cluster stars that are unbiased with respect to the properties which are under investigation (e.g. magnetic activity).
Prior to 1990, most work in this area focussed on the nearby, well studied clusters such as the Hyades and Pleiades, for which there was ample pre-existing photometry and proper motion information. However, in the last decade it has been realised that a wider range of clusters need to be studied in detail. The reasons for this are:
It is this possible low metallicity, together with its richness that makes NGC 2516 so interesting. There are plausible reasons why dynamo generated activity, rotational spindown and light element depletion could be profoundly affected by a low photospheric metallicity and thus differing convection zone properties. The compact size and numerous cluster members make NGC 2516 an ideal target for fibre spectroscopy and for X-ray satellites such as XMM and Chandra, which have limited fields of view. Recently, new X-ray studies by the ROSAT high resolution imager (Micela et al. 2000), Chandra (Harnden et al. 2001) and XMM-Newton (Sciortino et al. 2001) have devoted considerable amounts of time to observing NGC 2516. These more recent studies have used the photometric catalogue described in this paper to identify cluster X-ray sources.
So far, systematic membership studies in NGC 2516 have been limited to fairly bright stars or small areas. An unbiased, but severely incomplete list of cluster members down to about V=15 is presented by Jeffries et al. (1997), based upon position in the V vs. B-Vcolour-magnitude diagram (CMD). Hawley et al. (1999) present photometry and low resolution spectroscopy in the cluster down to much fainter limits, though over a very small area. Jeffries et al. (1998) present high resolution spectroscopy of photometrically selected candidates between 11<V<14.5, establishing definite membership through radial velocity measurements as well as lithium abundances. This work on brighter objects has been continued recently by Terndrup et al. (2001).
In this paper we present a catalogue of CCD photometry and astrometry down to faint magnitudes (V=20) and over a wide area (0.86 square degrees) in NGC 2516. This catalogue will be an invaluable tool for selecting unbiased samples of F, G, K and M stars for further study, interpreting X-ray observations and studying the dynamical state and mass function of NGC 2516. A preliminary version of this work appeared in Thurston (1999). We will restrict ourselves in this paper to a description of the data, the construction of a photometric catalogue and a membership classification on the basis of this photometry. Section 2 outlines the collection and reduction of the photometric data, and deals with the astrometric calibration, catalogue completeness and comparison with the previous literature. Section 3 describes a method for selecting candidate cluster members, which will need to be refined as more observational data become available. Section 4 uses this catalogue to look at the luminosity and mass function of NGC 2516 and search for evidence of mass segregation. The catalogue itself can be obtained in electronic format from the Centre de Données astronomiques de Strasbourg.
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Figure 1: An optical picture from the Space Telescope Science Institute digitized sky survey showing the region of sky included in our CCD photometry. |
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The data used in this paper were collected at the 0.9-m telescope of
the Cerro Tololo Interamerican Observatory (CTIO) on the nights of 8, 9 and 12 January 1995 (lunar phase 0.52-0.88). A CCD photometric survey was
completed over an approximate 0.86 square degrees, centred upon RA
,
Dec
(J2000.0), which
comprised of 25 overlapping frames of
square arcminutes,
taken with a
pixel Tektronix CCD at the Cassegrain focus. The
centres of the fields were separated by 10.5 arcmin, leading to a
square survey area,
on a side (see Fig. 1).
A series of short (30 s, 15 s, 15 s) and long (200 s, 100 s, 100 s) exposures
were taken through a set of Harris
filters respectively. In
addition to these target exposures, on each night a set of high quality
twilight flat fields were obtained through each filter and many Landolt
(1992) standard fields were observed over the entire range of airmass
for which the cluster data were taken.
All of the frames were bias subtracted and flat-fielded with standard methods using the Starlink FIGARO package (Shortridge et al. 1999). The only complications were that the CCD was read out using two amplifiers with slightly different bias, gain and readout noise characteristics. This was dealt with by reducing the two halves separately and then correcting for the gain ratio by requiring that the flat fields were continuous across the amplifier boundary. The flat-fielding was tested by flat-fielding median stacked night sky exposures and found to successfully remove most of the structure in the sky at a level better than a few tenths of one percent, apart from several CCD cosmetic defects (including a number of bad columns) and a narrow (10 arcsec) strip around the outside of the field. A mask was constructed so that these defective pixels were not considered in the subsequent reduction.
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Figure 2:
Top: the V vs. B-V CMD for our survey. The solid line shows
the Siess et al. (2000) solar metallicity
isochrone we use to select members of the cluster (see
Sect. 3.2). Bottom: objects selected as cluster members
(see Sect. 3.3)
are shown as triangles (where both B-V and
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Figure 3:
Top: the V vs.
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Figure 4:
Top: the
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The Starlink PHOTOM package (Eaton et al. 2000)
was used to determine aperture
photometry of the standards in a 6 arcsec radius (the stellar FWHM for all our
data was around 1.3-1.7 arcsec), which contained about 95-98 percent of
the stellar flux. Nightly transformation coefficients, extinction and
zero points were determined using iterative weighted least squares fits
to photometry of about 100 Landolt (1992) standards, rejecting stars
greater than
from the fit at each step until the solution
converged.
Equations of the form
V eqn | ![]() |
KV | ZV | N | rms |
8 Jan. | 0.018 | 0.158 | 23.337 | 88 | 0.015 |
9 Jan. | 0.011 | 0.152 | 23.327 | 101 | 0.016 |
12 Jan. | 0.022 | 0.227 | 23.277 | 111 | 0.024 |
B-V eqn | ![]() |
KBV | ZBV | N | rms |
8 Jan. | 0.928 | 0.106 | -0.205 | 87 | 0.017 |
9 Jan. | 0.933 | 0.120 | -0.206 | 102 | 0.019 |
12 Jan. | 0.934 | 0.146 | -0.205 | 110 | 0.017 |
V-I eqn | ![]() |
KVI | ZVI | N | rms |
8 Jan. | 1.003 | 0.130 | 0.965 | 100 | 0.024 |
12 Jan. | 1.001 | 0.091 | 0.968 | 86 | 0.018 |
9 Jan. | 1.003 | 0.130 | 0.965 | 100 | 0.024 |
12 Jan. | 1.016 | 0.117 | 0.968 | 99 | 0.018 |
Each target frame was then analysed. The first step was to mask out regions that were saturated due to the presence of very bright stars. The routine DAOFIND from the DAOPHOT II package (Stetson 1987; Eaton & Privett 1996) was then executed using a 5 sigma threshold and with shape rejection parameters set to exclude extended objects and cosmic ray spikes. The number of sources found per field ranged from 100-200 in the short Bexposures to about 3000 in the long I exposures.
Aperture photometry was performed using the PHOTOM routines. Both 3 and 6 arcsec radius apertures were used. For the brighter stars, where the statistical errors were smaller than 0.01 mag (about V=17), the larger aperture results were used and these also calibrate the aperture correction from the small to the larger aperture for the fainter stars. The sky estimation was taken from the mode in a surrounding annulus. At this stage stars were rejected if masked pixels fell within the object aperture and if they were close to the frame edges.
The next stage combined the separate filter measurements in each field,
treating the short and long exposure sequences separately. The
instrumental b,v,i measurements for a particular field were combined
by matching the pixel coordinates and allowing for small translations
between the fields. The v object list was used as the reference frame
(i.e. an object must be detected in v and at least one other filter
to be incorporated in the final
catalogue). Mean airmasses were calculated for the B,V and V,Ipairs and the catalogue of measurements for that field transformed onto
the standard magnitude system. If no B-V measurement was available (faint, red
objects), then we used a mean relationship between B-Vand
in order to estimate B-V for the purposes of
transformation to a standard V magnitude (see Sect. 3.3).
This latter procedure
should add no more than
mag to the V error because
the colour term,
,
is small.
A preliminary astrometric solution was found for the centroided CCD positions in the long V exposure of each field. This was obtained by identifying many stars from the Guide Star Catalogue Version 1.1 (Lasker et al. 1990). A 6-coefficient fit to these reference stars, resulted in solutions good at the level of 1 arcsec. The x,y pixel coordinates of objects identified in the long exposures were then transformed to RA and Dec using this solution. Of course some bright stars (V<14) were saturated in the long exposures and their positions must be taken from the short V exposure. This was achieved by fitting a linear translation between the long and short exposures, using stars detected in both frames.
Objects were then matched between the catalogues from the short and long exposures. Where objects appear in both, a weighted mean of the photometry was taken. There were generally about 200 well measured stars common to both the short and long catalogues in each field. The magnitudes of these stars were in very close agreement. The biggest differences we ever found were about 0.02 mag, which confirmed our preliminary assessment at the telescope that these were very good photometric nights.
We followed a similar procedure to deal with stars in the overlapping regions between fields. Again, where two (or even more) measurements existed, a weighted mean of the photometry was taken. Analysis of the discrepancies between magnitudes of stars in these overlap regions is our primary estimate of the internal accuracy of our photometry. We found no evidence for systematic variation in the photometric calibrations between nights or between fields at a level greater than 0.02 mag. Table 2 shows the internal error estimates (the rms values for the overlap discrepancies) as a function of magnitude. Beyond V=20 the statistical errors in the photometry rise rapidly.
Possible causes of error are (of course) the statistical errors, but also variations in the point spread function, and hence the aperture correction, over the CCD field of view (especially in frames not precisely in focus). This would contribute a term (in quadrature) which was the same at all magnitudes - which is approximately what we see. Although the short and long exposures in each field were taken in one observing sequence, some of the overlapping fields are separated by hours or even nights. It is therefore quite plausible that some genuine variability also contributes to these errors. That the errors in the V magnitudes are larger than the colour indices, suggests that errors in correcting measured magnitudes to the standard star aperture values near the edges of the CCD fields are the more likely culprit. If this is the case, the errors in Table 2 are likely to be overestimates for the majority of stars with V<18. For V>18, the statistical errors dominate.
V data | B-V data | V-I data | ||||
V | rms | N | rms | N | rms | N |
11-12 | 0.078 | 15 | 0.025 | 12 | 0.036 | 13 |
12-13 | 0.021 | 25 | 0.012 | 25 | 0.036 | 25 |
13-14 | 0.042 | 62 | 0.013 | 58 | 0.027 | 60 |
14-15 | 0.029 | 80 | 0.015 | 78 | 0.016 | 80 |
15-16 | 0.032 | 163 | 0.021 | 156 | 0.033 | 156 |
16-17 | 0.040 | 253 | 0.026 | 241 | 0.027 | 249 |
17-18 | 0.050 | 395 | 0.041 | 318 | 0.042 | 360 |
18-19 | 0.075 | 598 | 0.046 | 257 | 0.050 | 401 |
19-20 | 0.099 | 534 | 0.052 | 65 | 0.055 | 255 |
The preliminary positions determined from 6-coefficient fits to
objects in the Guide Star Catalogue (rms typically 1 arcsec) were
improved upon using pre-release data from the SuperCOSMOS Sky Survey
(Hambly et al. 2001). The UK Schmidt
plate from field 124
was used (plate number J2978, epoch 1977.223). These data consisted of
a catalogue of
1.4 million objects to
.
At
the time these data were used, the global astrometric plate reductions
were based on standards from the Tycho-AC catalogue (Urban et al. 1998)
and had typical residuals per standard of
0.3 arcsec in either
co-ordinate (for more details of the astrometric reductions for
SuperCOSMOS Sky Survey data, see Hambly et al. 2001). For each CCD
frame, objects in the photometric catalogue were matched with objects
found on the photographic plate. A matching radius of 5 arcsec was
used. A 6-coefficient linear transformation was then applied to the
CCD coordinates with an additional cubic radial distortion coefficient
as a further free parameter. The optical axis was assumed to be the
centre of the CCD. Parameters were adjusted to get the smallest rms
when compared with tangent-plane positions on the photographic plate
(the radial distortion term was almost negligible). The typical
zero-point shifts applied to the preliminary positions were about
0.7 arcsec, with final rms values of around 0.3 arcsecs, which we
expect are largely dominated by uncertainties in the photographic
positions.
The quality of the astrometry has recently been tested with a fibre-spectroscopy run on the low-mass cluster candidates (see Sect. 3.3), using HYDRA on the CTIO 4-m telescope. Excellent results were achieved over a 40 arcmin diameter field, using brighter cluster members as the fiducial acquisition stars and 2 arcsec diameter fibres (Jeffries et al. in preparation).
The photometric/astrometric catalogue is given in Table 5
(available from the Centre de Données astronomiques de Strasbourg) and consists of
15495 stars with V magnitudes, B-V and
colours when
available, along with their J2000.0 positions and flags indicating
their membership and binarity status (see Sect. 3.3).
The complete V vs. B-V and V vs.
colour-magnitude diagrams (the BV and VI CMDs) are shown in Figs. 2 and
3 and the colour-colour diagram is shown in Fig. 4.
We refer hereafter to this as the CTIO catalogue.
Table 3:
The photometric catalogue for NGC 2516, including J2000.0
positions, V, B-V and
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In Fig. 6, the apparent V magnitude luminosity function of
the whole catalogue is shown in 0.25 mag bins.
There is a clear turnover in the apparent luminosity
function beyond V=19.25. Catalogue completeness has been investigated
by adding simulated stars to our observed frames and then searching for
these stars using the same DAOFIND parameters. We find that there
are three possible causes for the apparent turnover in the luminosity
function.
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Figure 5:
The apparent V luminosity function for all the stars
detected in our survey (solid squares) compared with the luminosity
function for those stars with
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Figure 6: A comparison of photometry between this paper and photoelectric photometry from Dachs & Kabus (1989) [solid symbols] and Eggen (1972) [open symbols]. The straight line simply indicates equality and is not a fit to the data. |
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Figure 7: A comparison of photometry between this paper and photographic photometry from Dachs & Kabus (1989). The straight line simply indicates equality and is not a fit to the data. |
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Figure 8: A comparison of photometry between this paper and CCD photometry from Hawley et al. (1999). The straight line simply indicates equality and is not a fit to the data. |
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A number of other photometric studies have been published on NGC 2516,
though none that go both as deep and cover as large an area as this
one. We have compared our work with photoelectric (BV) photometry published
by Eggen (1972), photographic and photoelectric (BV) photometry published by
Dachs & Kabus (1989) and CCD (
)
photometry published by
Hawley et al. (1999).
The Eggen (1972) and Dachs & Kabus (1989) BV photometry is discussed extensively by Jeffries et al. (1997). In that paper an optical catalogue is compiled from these sources, in which small corrections (of order 0.02 mag) are made to the Eggen photoelectric and the Dachs & Kabus photographic values, to put them on a consistent system defined by the Dachs & Kabus photoelectric values. Dachs & Kabus estimate 0.02 mag errors in their photoelectric values and Jeffries et al. deduced errors of about 0.06 mag for their photographic values. The final catalogue contains 568 objects and was estimated to be complete to about V=13.5. We find that there are 500 matches with our new CTIO catalogue. The remaining objects either lie outside the area covered by the CTIO survey or were bright objects and hence saturated. Hawley et al. (1999) present a list of 155 candidate members of NGC 2516 that are within our survey area. We were able to find 130 matches to these within 1.2 arcsec. The remaining unmatched stars are very faint and were unlikely to be detected in our survey.
Graphical comparisons between the catalogues are shown in Figs. 7-9 where we have divided the comparison between: the photoelectric data from Eggen (1972) and Dachs & Kabus (1989) (which are given different symbols); the photographic photometry from Dachs & Kabus; and the CCD photometry from Hawley et al. (1999). The lower panels in these figures show the discrepancies between the CTIO and comparison photometry. In all plots, the solid line simply represents equality between the CTIO and comparison data, rather than a fit.
Figure 7 shows that there is an increasing discrepancy towards fainter V magnitudes among the the Dachs & Kabus (1989) and Eggen (1972) photometry in comparison with the CTIO data. This appears to be the case to a much severer degree in the photographic Dachs & Kabus data (Fig. 8). We are confident that our photometry is excellently calibrated in this V magnitude range, with no trends at all to V=16. We note that the two labelled outliers, DK427 and DK865, are quite close (<6 arcsecs) to companion stars which might easily have made them appear brighter in the poorer quality photographic photometry. In contrast the B-V data show very good agreement with our data, although there is some indication that the photographic colours are too blue by about 0.05. We have inspected our images and can see no problems with our photometry of the star called E3 by Eggen (labelled in Fig. 7). This may well have been misidentified in Eggen (1972), which is reinforced by the agreement of our value with that obtained by Hawley et al. (1999) for the same star. The scatter in the residuals for Figs. 7 and 8 are in line with our estimates of the errors in these datasets, although the Eggen photoelectric photometry appears to show more scatter than that of Dachs & Kabus.
Turning to Fig. 9 we again have reasonable
agreement. There is a systematic offset of about 0.04 mag in V(in the sense that the Hawley et al. photometry is brighter) and a
suggestion that this reverses for the faintest stars in the sample. We
note that as Hawley et al. only published data for cluster candidates,
all these very faint stars are also very red. It therefore suggests
that there may be minor problems with either their or our
colour-dependent terms in the transformation equations. We re-iterate
that we did observe Landolt (1992) stars as red as
and we are certainly confident in our calibration to this point
(and the colour term is very small in any case for our CCD and filter
combination). This pattern is repeated for the
comparison, where Hawley et al.'s colours are blue by about 0.04 mag compared with ours, with a definite reversal in the
brightest stars. The scatter in the residuals of both comparisons is
almost precisely in accord with the error estimates given by Hawley et
al. and in this paper. We have no explanation for the very discrepant
point HTR103 (labelled on the diagram), other than perhaps a major
stellar flare occurring during Hawley et al.'s observation.
We have used the photometric catalogue to attempt a preliminary selection of cluster members based only on photometric criteria. This selection procedure is especially useful because it does not rely on any characteristic of the cluster members which one might choose to investigate - for instance coronal X-ray emission. Such a catalogue of cluster candidates will be unbiased with respect to magnetic activity, rotation rate or lithium depletion and therefore provides an ideal starting point for such investigations see (Jeffries et al. 1998; Micela et al. 2000; Harnden et al. 2001; Sciortino et al. 2001).
We must provide a caveat here; our selection procedure is arbitrary to some degree, and will exclude some genuine members and will include some non-members. We will fashion our photometric selection criteria so as to avoid excluding the vast majority of cluster members. The interested reader should easily be able to generate membership catalogues using their own (possibly more restrictive) criteria. Ideally these catalogues should then be refined using other unbiased indicators of membership such as proper-motions or radial velocities.
Our CCD data combined with the SuperCOSMOS scan of the 1977 epoch Schmidt plate (see Sect. 2.2) did allow a preliminary attempt at proper-motion selection for the fraction of the catalogue which had good photographic positions. We found that the best accuracies achievable were about 6 milli-arcsecyear-1 in each coordinate. We calibrated the cluster mean proper motion using known members from Jeffries et al. (1998), and found it to be essentially zero within the errors. It was soon discovered that even for the most accurate data, we were unable to exclude more than about 10 percent of the general field background contamination whilst including more than 90 percent of the cluster members. We will not present these preliminary proper motion results in this paper, but will await a second epoch CCD survey which should be capable of producing more precise and useful results.
Our selection philosophy is therefore restricted to using the two CMDs
(V vs. B-V and V vs.
)
to select stars close to a
cluster isochrone and then to check the colour-colour diagram for
consistency.
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Figure 9: The Pleiades BV and VI CMDs showing how we defined the isochrone fits using Siess et al. (2000) models and a distance modulus of 5.6 (see text). |
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Figure 10: A comparison of isochrones generated from the three models discussed in Sect. 3.2. The solid lines are from the solar metallicity model of Siess et al. (2000), the dashed lines are the half-solar metallicity models of Siess et al. (2000) and the dotted lines (virtually indistinguishable from the solid line) are from the solar metallicity models of D'Antona & Mazzitelli (1997). The isochrones have been shifted by different distance moduli to match the observational data (not shown for clarity - see Figs. 2 and 3). |
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We used 150 Myr approximately solar metallicity (z=0.02)
isochrones generated from the models of
Siess et al. (2000) over the mass range
.
These isochrones were provided in the
form of bolometric luminosity versus effective temperature and
had to be transformed into the observational plane. We did not use
the photometric colours provided with the models.
The critical step here is the relationship between colour and effective
temperature, which is highly uncertain, especially among cooler
stars.
The procedure we adopted is as follows: We obtained a catalogue of BVI photometry of the Pleiades (courtesy of the Open Cluster Database
operated by J. Stauffer and C. Prosser) and supplemented this with
photometry of cooler Pleiades members presented by Stauffer et al. (1998a).
V-I indices on the Kron system were converted to the
Cousins values using transformations in Bessell & Weis (1987). We
posit that there is a single colour-
relationship
which applies to all stars of approximately the same age as the
Pleiades, for which we assume an age of 120 Myr, a distance modulus of 5.6, a reddening of
E(B-V)=0.04,
and extinction
of
AV=0.13 (see Stauffer et al. 1998a; Stauffer et al. 1998b).
Note that the
exact values of these quantities have very little influence on the
membership selection for NGC 2516 but could affect the deduced distance
modulus or metallicity for the cluster from isochrone fitting.
An empirical isochrone is fitted (by eye) to the Pleiades
data (in both BV and VI CMDs), ensuring that the isochrone is not
biased upward by the presence of binary cluster members (see
Fig. 10).
At about 25 fiducial points along these empirical isochrones
we determine absolute magnitudes and colours, which are then converted
to bolometric luminosities using bolometric correction-colour
relationships from polynomial fits to the empirical data in Flower (1996)
for B-V, and a combination of empirical data from Leggett et al. (1996) for
and the atmospheric models of Bessell et al. (1998)
for
.
Judging by the scatter around these
relationships and by comparisons with other bolometric correction-colour
relationships (e.g. Monet et al. 1992) we estimate systematic errors
of no more than a few hundredths of a magnitude (at
least over the colour range we are interested in). This contrasts
markedly with the uncertainties if one tries to use a bolometric
correction-
relation, which is often what has been done in
the literature to convert models to observables.
Having obtained luminosity as a function of intrinsic colour, we then
use the assumed age of the Pleiades to obtain the
value
appropriate for a particular luminosity, by interpolating along the
120 Myr model isochrone. This in turn defines a
set of colour-
points which can then be used to transform
any other model isochrone.
The net result is that we have well calibrated isochrones over the
colour range defined by the fiducial Pleiades data (or the mass range
of the models - whichever is more restrictive). In practice this
means
-0.15<(B-V)0<1.7 and
.
The hotter
limit to the
range is defined by a lack of
photometry for hotter Pleiades stars. For the sake of defining an
isochrone for membership selection, we have extended this data back to
by transforming the hot star B-V data into
using relationships defined by Johnson (1966)
(between B-V and Johnson V-I) and then Bessell (1979) (between
Johnson V-I and
.
The 150 Myr isochrones were transformed into the BV and VICMDs, and reddening and extinction applied for NGC 2516 (
AV=0.38,
E(B-V)=0.12 - see Dachs & Kabus (1989) and Jeffries et al. (1997, 1998)
- and an assumed
E(V-I)=0.15). We then adjusted the intrinsic distance
modulus of the isochrones to match the data, particularly for B-V<0.9in the BV CMD and for
and
in the VI CMD, where contamination by field stars appears to be
small (see Sect. 3.4). The results are shown in Figs. 2 and 3, where a
distance modulus of 8.10 has been applied to both the BV and VI CMDs, with estimated uncertainties of
.
This procedure is quite robust to the assumed ages of the Pleiades and NGC 2516, because for all but the very hottest and coolest stars, changing the age by as much as 50 Myr makes little difference to the isochrones. It is also robust to the choice of evolution model, because we require the models to fit the Pleiades at a similar age. We obtain essentially an identical fit to the NGC 2516 data using 150 Myr isochrones from D'Antona & Mazzitelli (1997) and with the same distance moduli as above (see Fig. 11).
Of more consequence is the possibility that the metallicity of NGC 2516 is
sub-solar by a factor of two (Cameron 1985;
Jeffries et al. 1998; Pinsonneault et al. 2000). If this is the case then the
distance moduli we have obtained will be overestimated because lower
metallicity stars are fainter at the same colour. The effect should be larger
in the BV CMD than the VI CMD because the opacity caused by
metal-lines causes blanketing in the B band (Alonso et al. 1996). Pinsonneault
et al. (1998)
have calibrated this effect for F and G stars and indeed
used it to calculate the metallicities (and metallicity-corrected
distance moduli) for a number of open clusters. Jeffries et al. (1998)
and Pinsonneault et al. (2000) used the same approach to calculate a
metallicity for NGC 2516 of
and -0.26 respectively
on the basis of preliminary
photometry of known members (from
this dataset).
We have checked the effects of a low metallicity by
using a 150 Myr isochrone from Siess et al. (2000)
with z=0.01 (approximately half-solar metallicity,
).
This is again calibrated using the
Pleiades photometry. A question arises as to whether the
colour-
relation derived empirically from the Pleiades is suitable for
a lower metallicity cluster. The B-V index is sensitive to
metallicity for warm stars with partially ionized metal lines and is
partly the reason that the BV CMD may be changed more by metallicity than the VICMD. However for hot stars (B-V<0.2) and cool stars
(B-V>1.3) this is not likely to be the cases
(see Castelli 1999). Leggett et al. (1996) also show that the
colour index is a good temperature indicator with
relatively little metallicity sensitivity for cool stars
(
)
and line blanketing in the V band is not
expected to be very important for hotter stars. Thus to first order,
this approach to generating a low metallicity observational isochrone
should be valid.
We find that the solar and half-solar metallicity isochrones yield
comparable fits to the data.
As expected, the distance moduli required to fit the
data are smaller for the lower metallicity models. We find distance
moduli of
and
for the BV and VI CMDs
respectively. These distance moduli are in excellent
agreement with the
in Jeffries et al. (1997) and
found by Jeffries et al. (1998), but a little larger than
the
deduced by Pinsonneault et al. (2000) and the Hipparcos
distance of
found by Robichon et al. (1999). We emphasize
that our errors are underestimated because they do not take into
account uncertainties in the metallicity or reddening.
The isochrones are compared in Fig. 11, where we also include the solar-metallicity D'Antona & Mazzitelli (1997) model. The isochrones have been shifted to the distance moduli required to give a reasonable fit to the observational data (which is not shown for clarity of comparison between the models). The most important point to make is that the shapes of the models are extremely similar. The main discrepancy occurs in the cool part of the VI CMD, where low metallicity stars lie just less than 0.1 mag below solar metallicity stars of the same colour. This is fortunate, because it means irrespective of which model/metallicity we choose, the selection of cluster members by photometric means is almost unaffected - although cluster properties such as the mass function are (see Sect. 4.2). A more detailed investigation of the metallicity, distance and reddening is left to another paper that uses a more complete sample of spectroscopically confirmed NGC 2516 F and G stars (Terndrup et al. 2001).
To select the members we apply the following criteria, based on the Siess et al. (2000) solar metallicity isochrones:
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Figure 11:
An example of our fitting procedure to determine the level of
contamination among our candidate members (see text). The y-axis shows
the number of star per unit magnitude found in strips a distance
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Whilst we are reasonably confident that our membership criteria should have included the vast majority of true cluster members (within the bounds of our photometry completeness), it is clear looking at Figs. 2 and 3 that we have also included contaminating foreground and background sources. As we have taken no offset fields of the cluster, we attempt to estimate membership probabilities and quantify this contamination by interpolating the distribution of cluster non-members in the VI CMD.
We proceed by removing the cluster candidates from the catalogue and then select stars in several strips in the VI CMD, both above and below (but parallel to) the fiducial cluster isochrone (from 1.2 mag below to 2.0 mag above), but beyond the limits defining membership. We fit a smooth function to the density (number of stars per V magnitude interval) of contaminating stars along vertical strips (colour ranges) in the CMD. We achieved the best looking fits using the sum of a constant density and an exponential decay with an e-folding length of 0.5-0.9 mag. An example is shown in Fig. 12. The fitted function is then integrated over the range in which cluster members were selected, in order to estimate how many contaminating stars we expect to fall within the membership selection region. Because the contamination tends to decrease quite sharply as we move above the cluster isochrone we subdivide this range further in an attempt to estimate how much contamination there would be among single and binary star candidates separately. This was achieved by integrating the the interpolating function above and below a line 0.3 mag above the cluster isochrone (see Sect. 3.3). These figures are then reduced by the number of objects which passed test 1, but failed tests 2 or 3, because we argue that these must be part of the contaminating sample. Finally we subtract the remaining contaminants from the number of cluster candidates in the same magnitude range and divide by the number of cluster candidates to estimate the probability that a cluster candidate is a genuine cluster member. At the same time we can see what advantage has accrued from using more than one colour in assessing cluster membership, by finding what fraction of the contamination in the VI CMD has been rejected using B-V.
The results are given in Table 3, where we have split the sample into colour ranges which roughly correspond to 1 mag V intervals for single stars in the cluster. Such crude binning is necessary in order to have enough stars (especially in the CMD strips above the cluster isochrone) to yield reliable fits for the interpolating function. We have also assumed that all hotter candidates (roughly brighter than V=11.5) are members.
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0.53-0.70 | 0.70-0.87 | 0.87-1.05 | 1.05-1.33 | 1.33-1.68 | 1.68-2.04 | 2.04-2.42 | 2.42-2.73 | 2.73-3.02 |
Observed | |||||||||
Single Candidates | 31 | 61 | 51 | 115 | 67 | 66 | 119 | 146 | 106 |
Binary Candidates | 17 | 41 | 37 | 62 | 27 | 23 | 40 | 40 | 47 |
Predicted | |||||||||
Single contaminants | 6 | 38 | 40 | 63 | 14 | 19 | 25 | 63 | 31 |
Binary contaminants | 5 | 18 | 25 | 55 | 15 | 11 | 22 | 39 | 26 |
Rejected | |||||||||
Single contaminants | 6 | 22 | 16 | 32 | 13 | 13 | 17 | 26 | 1 |
Binary contaminants | 3 | 8 | 4 | 27 | 11 | 11 | 5 | 19 | 11 |
Membership Prob. | |||||||||
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Binary candidates | ![]() |
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The results in Table 3 can now be used to correct statistical ensembles for contamination (e.g. in investigating the luminosity function - see Sect. 4.1). We caution however that these membership probabilities are averages over our field of view. Because the cluster is centrally concentrated (see Fig. 1 and Sect. 4.5) and we expect that the background contamination has a uniform spatial distribution, membership probability will be higher for a cluster candidate close to the cluster centre and lower for a candidate near the edge of our survey area.
The contamination is worst for
(
12.5<V<15.5) and a glance at Fig. 3 confirms that this is
where the two "fingers'' of contamination caused by main sequence and
background giant stars respectively, cross the cluster
isochrone. Candidates from
(
15.5<V<18.0)
suffer very little contamination, but note that this is largely because
the addition of B-V data removes the majority of the contamination in
the VI CMD. For
(V>18.0) there is growing
incompleteness in the B band as well as a growing number of
contaminants from the VI CMD alone, resulting in a drop in the
discrimination in our membership selection. We expect that the level of
contamination is underestimated for the coolest bin
(approximately V>19.5) in Table 3, because the data below
the cluster isochrone in the VI CMD is incomplete. The
errors in the membership probabilities are estimated assuming Poisson
errors in the numbers of candidate members, the numbers of candidate
members rejected on the basis of their B-V and the numbers of
predicted contaminants in the VI CMD. This latter error is an
overestimate, because the contaminant numbers arise from modelling
populations several times larger. In any case, these errors are similar
(as a fraction) to the simple Poissonian errors in the numbers of
cluster candidates in each colour bin. In what follows we will simply
assume that the contamination fraction is known accurately, but will
compare our results with what would have been obtained without any
correction for contamination.
It is of interest to compare the estimates above with information
appearing in the literature. Jeffries et al. (1998) used a similar
photometric selection technique to choose a cluster candidate
list. These were subsequently followed up with spectroscopy. In that
paper, 22 out of 31 objects with
12.5<V<15.0 were confirmed as
members based on their radial velocities. This is perfectly consistent
with the estimates in Table 3.
Hawley et al. (1999) did do an offset field (about 1 degree from
the cluster centre) covering 225 square arcminutes in V and Ionly. It is not clear (see Sect. 4.5) that this is far
enough from the cluster centre to guarantee no cluster members and
indeed Hawley et al. spectroscopically identified a few low mass NGC
2516 candidates in this field. However, we can use similar membership
criteria to those in our survey to find how many bogus NGC 2516 members
we might expect in a 225 square arcminute area. This is then multiplied
by 13.7 to match our survey area. The total number of contaminants
expected in the VI CMD (to be compared with the sum of
rows 3 and 4 of Table 3)
are
for
(compared with 212 in
Table 3), and
for
(compared with 179
in Table 3). The former estimate is in good agreement with our own,
but the latter is a little higher, perhaps indicating that there are
indeed some low mass NGC 2516 members even 1 degree (
7 pc)
from the cluster centre.
The CTIO catalogue of candidate members, together with the estimates of contamination in Table 3, allow us to make a preliminary investigation of the luminosity and mass functions (LF and MF) of NGC 2516 (that is the number of stars per V magnitudes interval and the number of stars per logarithmic mass interval).
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Figure 12:
The V band luminosity function (LF) of NGC 2516. The solid histogram
shows the LF for candidate members identified from either B-V or
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Figure 13: The corrected absolute V magnitude luminosity function (LF - see text) of NGC 2516 (shown with error bars) compared with LFs for the field and the Hyades. The field and Hyades LFs are shown as histograms and are normalised to match the NGC 2516 LF at 5<Mv<7. The error bars on the field LF are about the same as in NGC 2516 at the same absolute magnitude, whereas the Hyades LF error bars are about a factor of two bigger (but with twice as many bins). The field and Hyades LFs were taken from Reid & Hawley (2000) and references therein. |
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The CTIO candidate membership catalogue is incomplete for V>20, but
is also incomplete for brighter stars because of saturation in the CCD
frames. As discussed in Sect. 2.4, there were 68 objects from the
catalogue of bright stars in the NGC 2516 field compiled by Jeffries et al.
(1997),
which are not present in the CTIO catalogue. 55 of these objects (which
have V and B-V photometry) reside
within our CCD frames and 48 have photometry consistent with main
sequence cluster membership. These should be included when calculating the
LF and MF, and, because Jeffries et al. (1997)
estimate that this bright catalogue is complete to V=13.5, we then
have an almost complete catalogue of cluster members from 5.8<V<20,
covering a mass range of approximately
(the upper
limit being rather age, model and metallicity dependent).
Figure 13 shows the LF of the candidate
members without a correction for contamination. The plot shows the LF
due to candidates selected using both B-V and V-I and also
the complete candidate list (including the brighter stars discussed
above and those stars classed as members for which either B-V or
are unavailable). We also mark on the plot the approximate
faint magnitude completeness limits for these two samples (discussed in
Sect. 2.3).
The rapid fall off at faint magnitudes coincides with our estimates of
where the survey begins to be significantly incomplete. There
appears to be a minimum in the LF for
15.5<V<17.0, which was also
commented upon by Hawley et al. (1999). However, we might suspect that
this feature merely results from the inclusion of many background
contaminants at slightly brighter magnitudes (see Table 3
and Fig. 3), resulting
in the peak at
13.5<V<15.5. An NGC 2516 LF
corrected for contamination by non-members and plotted with Mv on
the x-axis (assuming an apparent distance modulus of 8.3) appears in
Fig. 14. We compare this with LFs from the field
(Reid & Hawley 2000)
and the Hyades (Reid 1992; Reid & Hawley 2000), which are
normalised to match NGC 2516 for 5<Mv<7. The correction for
contamination in NGC 2516 is applied by making each star worth in the luminosity histogram according to it's
colour and
whether it is a photometric binary candidate, using the fraction of
such stars which are likely to be members according to Table 3.
For instance, a candidate member with
and which is flagged as
a probable binary system will only contribute 0.43 to the LF bin
corresponding to its V magnitude. All the very bright candidate
members without
are assumed to be genuine members.
The NGC 2516 LF in Fig. 14 exhibits both similarities and
differences when compared with the LFs of the field, the Pleiades (which is
almost identical to the field LF - Hambly et al. 1993;
Meusinger et al. 1996)
and the Hyades. The LF is consistent with a monotonically rising
curve at least as far as Mv=7 (
in NGC 2516). There are relatively more bright stars in NGC 2516 than the
field because it is a very young cluster.
There is then, despite the correction for contamination, a
significant minimum at
7.7<Mv<8.7 (16<V<17), that coincides
with the "Wielen dip'' (see Wielen 1974; Upgren & Armandroff 1981; Bahcall 1986) -
that is also seen (at the level of 30-40 percent) in
the LFs of the field, the Pleiades and the young
Persei
cluster at
(Prosser 1992; Meusinger et al. 1996; Belikov et al. 1998; Reid & Hawley 2000)
and in the Hyades at Mv=8.5 (Reid 1992). This dip is
defined with reasonable clarity in NGC 2516, due to the large numbers
of cluster members. There is some evidence that the overall rise in
the LF towards fainter magnitudes then levels off for Mv>9 and
certainly there is no sign of the very steep rise in the LF of a factor
2-3 between
9<Mv<10, that is seen in the Pleiades and nearby field
stars. In this respect, NGC 2516 is much more similar to the Hyades.
We do not believe that incompleteness can be responsible for a lack of faint stars in NGC 2516. We have calculated that the LF is complete to at least Mv=11.7, in the sense that stars are not missed because they were not detected. Perhaps then the LF of NGC 2516 is different to that of the Pleiades and the field, but there are other possibilities: first, it may be that that the contamination fraction is increasingly overestimated for V>17 or underestimated for 14<V<17. We regard this as unlikely and in fact for the last column in Table 3, we believe the contamination fraction is probably underestimated (see Sect. 3.4). Second, we may have set our membership criteria too tightly and missed a significant fraction of fainter cluster members that lie either above or below our membership bounds in the VI CMD. We already regard our membership criteria as quite generous and thus the factor of approximately two increase in the LF that would be required between 10<Mv<12, seems unlikely to be explained in this way. Third, it might be that mass segregation has been successful in removing lower mass stars from the central cluster regions and hence our survey. This has been put forward as an explanation of the differences between the Hyades and field LFs (Reid & Hawley 2000) and although NGC 2516 is much younger than the Hyades, we have only surveyed the central regions. Mass segregation is discussed in more detail in Sect. 4.5. Lastly, it is possible that the mass function in NGC 2516 is similar to that in the Pleiades, even though the LF is not. This might be the case if the mass-luminosity relationship were markedly different at low masses because NGC 2516 has a low metallicity compared with the Pleiades. This possibility is discussed in Sect. 4.2.
The cluster MF can be explored in a preliminary way by calculating the mass of each candidate member using a mass-colour relationship derived from the isochrones we used to fit the cluster VI CMD. Unresolved binarity is of course a problem here. The net result of assuming that unresolved binaries are single stars, will be that the true MF will be slightly steeper than that derived (in the sense that an assumed single star of a given mass is actually two stars, one of which could have a much lower mass - see below). In principle, the numbers of low mass stars could be doubled by hiding them in binary systems with more luminous companions.
The binarity in NGC 2516 and its effect on the derived MFs is discussed
further in Sect. 4.3, but we note that
in Sect. 3.3 we identified candidate members which are
approximately equal mass binary systems based on their position above
the cluster isochrone. Using the BV CMD we can also identify
candidate binaries from among the bright stars added to the CTIO sample
from Jeffries et al. (1997).
Each identified binary system is counted as if it consisted of two
identical components with the same colour. This is an approximation,
because for objects with mass ratios less than unity, one component
will be cooler. However, in the absence of better binarity information,
a more complex approach is not warranted. Corrections for
contamination by non-members are applied by using the fractional
membership probabilities in Table 3. Each candidate will
contribute a fraction to the MF bin corresponding to its mass,
according to the value of
(or B-Vif
is unavailable) and whether it is a
candidate binary.
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Figure 14: The corrected mass functions for NGC 2516 (see text), derived from the solar metallicity model of Siess et al. (2000) (top), the solar metallicity model of D'Antona & Mazzitelli (1997) (middle) and the half-solar metallicity model of Siess et al. (2000) (bottom). The dashed lines indicate the power law fits discussed in Sect. 4.2 and the dash-dot lines indicate the approximate mass at which our survey becomes incomplete. |
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The mass-colour relationships
are derived from the same evolutionary models and empirical
isochrones used to select the cluster candidates in the first place.
The uncertain metallicity of NGC 2516 causes some difficulties
here. Stars of a given colour have smaller masses at lower
metallicities. We perform our calculations using the solar
and half-solar metallicity models of Siess et al. (2000) and the solar
metallicity model of D'Antona & Mazzitelli (1997), which were
discussed in Sect. 3.2. Note that we
choose to use
,
rather than B-V, as the primary
indicator of
and hence mass, because it is relatively
insensitive to metallicity (Leggett et al. 1996) and covers a wider range
of masses. Only the hottest stars have their masses calculated from B-V, where B-V is an almost metallicity-independent
proxy (Castelli 1999), so the fact that we have
ignored the metallicity dependence of the B-V-
relationship in deriving the low-metallicity isochrone should be
unimportant (see Sect. 3.2).
The corrected MFs are shown in Fig. 15, expressed as
the number of stars per unit logarithmic mass interval, as a function
of logarithmic mass (in solar units). The approximate lower mass limit
at which our survey is complete (corresponding to V=20) is about
for the solar metallicity models and
for the half-solar metallicity model.
In these diagrams, the canonical stellar initial MF of Salpeter (1955)
would be a straight line of
the form
with
.
Our MF for NGC 2516 is well defined between 3
(where any
uncertainty in the age of NGC 2516 would start to be a factor and also where small number
statistics become important) and 0.25-0.3
(where
incompleteness sets in). Note that below the completeness limit we
cannot simply say that the points are lower limits because there is
also the uncertain level of contamination by cluster non-members to consider.
For each of the two metallicities, there is a clear Salpeter-like
rise in the MF as the mass decreases, followed by a peak at
and
a turnover towards lower masses. We have checked that the
corrections for contamination by
non-members and for binaries have very little effect on this overall shape.
The exact form of the MF might of course be age, model and
metallicity-dependent. Barrado y Navascués et al. (2001) investigated the
MF of M 35, a rich northern hemisphere cluster of similar age to the
Pleiades and NGC 2516, using a variety of ages and (solar metallicity) evolutionary
models. Their results show that above
,
derived MFs are
virtually identical in all cases.
We have parameterised our derived MFs in terms of two power law fits in
the range 0.3-0.7
(or 0.25-0.7
for the half-solar
metallicity model) and 0.7-3.0
.
For the higher mass range
we find
and
for the solar
and half-solar metallicity models of Siess et al. (2000), and
for the solar metallicity model of
D'Antona & Mazzitelli (1997). For the lower mass
range we find
and
for the
solar and half-solar metallicity models of Siess et al., and
for the solar metallicity model of D'Antona &
Mazzitelli. The
systematic difference between the models is caused by the change in
the colour-mass relationship. At lower metallicities, stars of the same
colour have lower masses, increasing the value of
,
but there
is also some modest model-dependence.
For simplicity of discussion in what follows we shall only use the two Siess et al.
models. We have also done
all our calculations for the solar metallicity D'Antona & Mazzitelli (1997) model
and the results it yields in later sections are in reasonable agreement
with the Siess et al. solar metallicity model.
The power law fits to the higher mass range are in good agreement with
the Salpeter value, agree well with the average value of
found for intermediate mass stars in many open
clusters by Phelps & Janes (1993) and are close to the "universal'' field
initial MF of
for
found by
Kroupa (2001). We also note that these quoted results neglect the
effects of binarity and used a single relationship between V and mass
to calculate the MF from the LF. This has the effect of making the MF
slightly less steep at high masses because binary systems are then
treated as one star with a slightly higher mass. We have
investigated what difference this makes by simply treating our data in
the same way. If we were to adopt a single relationship between V and
mass, irrespective of binary status, then the slopes of our derived
MFs would be smaller by about 0.2 and hence in even better agreement
with the previously published values.
Determinations of the MF in the Pleiades (Meusinger et al. 1996) and M 35
(Barrado y Navascués et al. 2001) clusters find MF slopes of
and
for stars with
.
Meusinger et
al. then find that
in the Pleiades for
0.3-1.0
,
whilst Hambly et al. (1999) estimate
for
in the Pleiades and Barrado y Navascués et al. finds
for M 35 between 0.8 and 0.2
.
The
latter two slopes take no account of binarity and should probably be
increased by
0.2 (see above)
before comparison with our results. At very low
masses it is likely that the MF falls again. Bouvier et al. (1998) and
Moraux et al. (2001) find
and -0.5 across the brown dwarf
boundary at
0.05-0.2
in the Pleiades and Barrado y
Navascués et al. estimate a more extreme slope of -1.8 below
0.2
in M 35. This behaviour is mirrored in the field MF, where
a Salpeter-like slope is found above
,
a relatively flat MF
with an
of -0.1 to +0.3 down to
,
and then
a decline into the brown dwarf regime with
between -0.5and -1.0 (Gould et al. 1997; Reid et al. 1999; Chabrier & Baraffe 2000;
Kroupa 2001).
If NGC 2516 has a solar metallicity, then its MF drops much
more sharply towards lower masses than occurs in either the Pleiades or
field for
.
This steep slope is
of course directly related to the deficit of low luminosity stars in
the NGC 2516 LF with respect to the Pleiades and the field which we
remarked upon in Sect. 4.1.
Any of the explanations offered there (such as mass-segregation) might make
the drop in the MF below 0.7
less extreme. A Pleiades-like
MF could still be recovered for the solar metallicity scenario
if the number of NGC 2516 members between 0.3 and 0.5
were roughly doubled with respect to the more massive stars.
If NGC 2516 has a half-solar metallicity then the discrepancy with
the Pleiades and field MFs is reduced but still present.
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Figure 15:
Binary fraction in NGC 2516 for systems with approximately
0.6<q<1 (see Sect. 4.3) as a function of
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The frequency of binary systems and the distribution of their mass
ratios are important constraints on star formation theories.
So far in this work we have made a partial correction for binarity in
determining the cluster MF, by identifying cluster candidates with near
equal mass ratios (
). Unresolved binarity will also influence the
deduced total mass of the cluster (see Sect. 4.4). We have also not
considered the possibility of triple systems in this paper, which could
place an object up to 1.2 mag above the cluster isochrone (for equal
mass components). However, the numbers of missed triple systems should be
quite rare - less than 2 percent of systems in the Pleiades
(Mermilliod et al. 1992).
Figure 16 shows the fraction of photometrically
identified binaries as a function of
colour in NGC 2516, where membership was
determined from the solar metallicity isochrone of Siess et al. (2000).
The binary fraction is defined as the number of probable binary
systems divided by the total number of systems. The fractions are
calculated using our membership catalogue, the binarity flags and the
membership probabilities in Table 3. The weighted mean binary fraction
is
2 percent with marginal evidence (at the 90 percent
confidence level) for a decrease in the binary fraction towards lower
masses. This may be more significant, as we expect the lowest mass bin
in Fig. 16 to have an overestimated binary fraction,
because incompleteness sets in and will bias against single star
detection. It is also the case that low mass M-dwarfs are more prone
to flaring, which might result in some fraction of the lowest mass
single stars being misclassified as binaries (see Stauffer et al. 1984). The
confidence level for a significant downward trend increases to 98 percent if this last point is excluded. On the other hand, the error in the binary
fraction should be increased to take into account the somewhat
arbitrary nature of the isochrones we have used to define the cluster
single star sequence. If we allow these isochrones to move up or down
by
0.1 mag (see Sect. 3.2), the mean binary
fraction changes by
percent. As we discussed in
Sect. 3.2, the shape of the half-solar metallicity
isochrone is slightly different for
,
which would
result in a differential increase in the binary fraction of cool stars
with respect to the hot stars of about 5 percent. This would be enough
to remove the possible trend just discussed, so it is premature to
claim to have seen any dependence of binary fraction on mass.
The binary fraction estimated in this way is a lower limit.
The mass ratios to which our photometric identification technique is
sensitive, can be estimated using the results and equations found in
Kähler (1999). We note that the q sensitivity limit
is not quite independent of colour for stars 0.3 mag above the single
star locus, because the gradient of the V vs.
cluster isochrone is not constant. Using Eq. (5) in
Kähler (1999) and the gradients determined from the cluster
isochrone we find that we are sensitive to
0.59<q<1.0 for
,
0.55<q<1.0 for
,
0.62<q<1.0 for
and
0.60<q<1.0 for
.
Given an
approximately flat distribution of q or one that rises towards lower
q values, any small variations in/apparent binary fraction seen in
Fig. 16 could also be partially explained by this varying
sensitivity.
In field stars, Duquennoy & Mayor (1991) found that the mass ratio
distribution increased towards lower mass ratios, perhaps peaking at
q=0.3, and that the binary fraction of solar-type stars was about 70
percent for all mass ratios, and 21 percent for 0.6<q<1.0. Work on
the Pleiades solar-type stars is in broad agreement with these results
(Mermilliod et al. 1992; Bouvier et al. 1997). There is some
evidence that the binary fraction among lower mass field objects is
smaller (30-40 percent), but that they are more inclined to be found in
systems (Fisher & Marcy 1992; Reid & Hawley 2000). Using
a similar photometric selection technique to the one used here,
Stauffer et al. (1984) found that 26 percent of Pleiades low mass stars
(
0.7<V-I<2.1) were more than 0.3 mag above a single star isochrone
and that this was similar to the 22 percent photometrically determined
binary fraction in hotter stars found by Bettis (1975).
The main result of this subsection is that the binary fraction in NGC 2516 for stars with q in the approximate range 0.6-1 is similar to that of the Pleiades or the field, when the binaries are identified in the same way. If the distribution of q were that proposed for the field by Duquennoy & Mayor (1991), down to q=0, then the total binary fraction in NGC 2516 would be about 85 percent. If the distribution of q were flat, then the total binary fraction would be 65 percent.
A subsidiary consideration is estimating what effect unresolved
binarity might have on the derived mass functions. Our identification
of q>0.6 binaries, where the companion star can significantly change
the colours and magnitudes of the system as a whole, results in MFs
which have slopes about 0.2 larger than those where this is ignored
(see Sect. 4.2). These MFs have the merit of being obtained from
observables which are in principle comparable with other estimates
derived from photometry data in other clusters, and do not rely on
knowledge of the mass ratio distribution.
However, even for those systems with q<0.6 there can be a
significant number of "hidden'' lower mass stars, resulting in a MF
for all stars which is even steeper still. Sagar & Richtler (1991) investigated
this effect in 2-14
stars in LMC clusters. They found, for
true MFs with
,
a binary fraction between 0.5 and 1,
and binary companions drawn randomly from the same MF, that the
observed MF (estimated from a mass-magnitude relation) had a slope that
was 0.3-0.4 smaller. The effect was even larger for smaller values of
.
A true
of +0.5 might appear as
for a
binary fraction of 1.0. Similar simulations by Kroupa (2001) yield
differences between the system and single star MFs largely in agreement
with these results.
We point out to the reader that by using a colour-mass relation
and identifying and dealing with q>0.6 binaries, we have partially
alleviated this problem. To try and gauge by how much the true
stellar MF slopes might be further increased over the quoted MF
slopes in this paper, we randomly added binary companions to a fraction
of the "single'' stars. Using a total binary fraction of 65 percent
and a flat q distribution we find that the true single-star MF
is increased by a further 0.05 for
and 0.3 for
(using the Siess et al. 2000 solar
metallicity models). Using the q distribution
proposed for field binaries by Duquennoy & Mayor (1991) and assuming a
total binary fraction of 85 percent, results in
increasing by 0.1 and 0.4 in these two mass ranges.
We can integrate our corrected MF (from the solar metallicity
isochrone) to yield an estimate of the total
mass of the cluster (down to about 0.3 )
of 1105
.
More than half of this mass is contained in stars with 0.6-2
with decreasing contributions at lower and higher masses.
The equivalent calculation for the MF derived from the half-solar metallicity
isochrone is 945
(complete to 0.25
).
Unresolved binarity will increase the derived cluster mass. We have
made a partial correction for this in our work so far, by identifying
near equal mass-ratio binary systems. If we had not done so, our
deduced cluster mass would have been only 880
(solar
metallicity). To estimate the maximum likely contribution that could
arise from binaries with q<0.6, we can use the q distribution
proposed for field binaries by Duquennoy & Mayor (1991) and assume that the
total binary fraction is 85 percent and independent of primary
mass. Integrating this distribution, we find that the cluster would be
35 percent more massive than if all the stars were single, and thus the
total cluster mass would be 1190
(solar metallicity), for primaries with
.
A correction could also be applied for stars less
massive than this. Integrating an extrapolated MF derived from the
Siess et al. (2000) solar metallicity isochrone (see Sect. 4.2)
from say 0.0-0.3
,
yields only another 73
.
The
corresponding additional mass for the half-solar metallicity model is
(0.0-0.25
). Thus irrespective of the
cluster metallicity, the contribution of low-mass stars and brown
dwarfs to the total cluster mass inside our surveyed area is
likely to be less than 10 percent unless there were a sharp upturn in
the MF below 0.3
.
The cluster mass is similar, but a little higher, than that found for
the Pleiades in the same mass range. Meusinger et al. (1996) quote a
figure of 800
for
and Pinfield et al. (1998) derive
735
for all masses down to the substellar limit. However, we
have to be careful to compare like with like. The quoted Pleiades
results are extrapolations for all stars out to the cluster tidal radius.
Pinfield et al. (1998) give an approximate expression for the tidal radius of
a cluster in a circular Galactic orbit, close to the Sun as
Using a distance modulus of 7.9, our survey of the cluster covers a
square area, 6.2 pc on a side. Thus we expect significant numbers of
cluster members outside the area covered by our
survey. Pinfield et al. (1998)
find that the Pleiades has
pc and that
although the high mass stars are highly concentrated within a few pc of
the cluster centre, there are significant numbers of low mass stars at
much greater distances, such that only half the cluster mass was
contained within 3.66 pc. If NGC 2516 is analogous to the Pleiades in
terms of the spatial distribution of its members and the amount of mass
segregation present (see Sect. 4.5), then the total cluster mass
might be significantly greater than just what we have observed in our limited survey.
Dynamical evolution and mass segregation can have a significant effect
on the shape of the present day LF and MF of NGC 2516. Equipartition
leads to a growing core radius with decreasing mass and the
preferential evaporation of low-mass cluster members (see for example
de la Fuente 1995; Kroupa 1995; de la Fuente Marcos & de la Fuente Marcos 2000;
Kroupa et al. 2001). The evidence for
dynamical effects on the MFs of open clusters older than a few hundred Myr
is strong (e.g. Reid 1992; Montgomery et al. 1993; Sarajedini et al. 1999).
The segregation
effects are less obvious, but still seen, in younger clusters like the
Pleiades and M 35 (Pinfield et al. 1998; Barrado y Navascués et al. 2001),
with a hint that equipartition has not been achieved in the lowest mass
stars, perhaps because these clusters have ages similar to their
dynamical relaxation timescales.
![]() |
Figure 16: The radial distribution of stellar surface density in NGC 2516, split into 4 approximate mass bins (see text). The x-axis values assumes a cluster distance modulus of 7.9. The horizontal dashed lines indicate the level of (assumed uniform) contamination expected in each of these samples. The solid lines are King profile fits for assumed tidal radii of 14.6 pc (see Table 4). |
Open with DEXTER |
As a step towards understanding how much mass segregation might have
taken place in NGC 2516, we examine the radial surface density profiles
of candidate cluster members
split up into bins according to their
colours
and hence masses. We exclude binary candidates at this stage, because
their colours are not reliable mass indicators - they could be up to
twice as massive. We have to acknowledge
though that we must have included some binary stars with q<0.6, which
will have the effect of blurring any distinction between different mass
bins, perhaps lessening the effects of mass segregation. In
order to have sufficient stars to draw statistically significant
conclusions we divide the stars into just 4 mass bins (see Table 4).
The division is made preserving the colour
boundaries in Table 3, so that estimates of field star contamination
can still be made. We do not include the lowest mass interval from
Table 3 because of possible uncertainties in the exact level of
contamination (see Sect. 3.4). Note that the values for
the boundaries of the mass bins are derived from the solar metallicity
Siess et al. (2000) isochrone. The corresponding mass boundaries in
Table 4 for the half-solar metallicity would be:
1.38
,
0.79
,
0.58
and 0.27
.
The centre of mass for each bin is determined by minimising the projected moment of the stars about a point. The centres drift south-west by about 2.5 arcmin between the highest and lowest mass bins. This small drift has no effect on our results, which would be almost identical if we had fixed the cluster centre at a single position. The surface density of stars as a function of radius (assuming a cluster distance modulus of 7.9) is then calculated for each mass bin, taking account of the geometry of the surveyed area.
The results are shown in Fig. 17. The horizontal dashed
line in each plot indicates the (assumed uniform) density of
contaminating objects. It now becomes clear that the probability that
a candidate member is actually a contaminating object shrinks
drastically if we only consider objects within 15 arcmin
(
1.7 pc) of the cluster centre. Conversely, if we consider
objects in the outer part of our survey, the probability of membership
is lower than the average values in Table 3.
To get some parameterisation of the mass segregation we have
fitted empirical profiles of the form (see King 1962)
The fits are all reasonable, with
values of between 7.7 and 13.4, with 10 degrees of freedom in each case. The fit results,
including 68 percent confidence intervals in
,
and the assumed
surface density of contaminants, are reported in Table 4.
The King profile modelling suggests that the core radius undergoes a
dramatic enlargement as the mass decreases from
to
and that this is independent of the assumed value of
.
It could be argued that this might be explained in terms of an
underestimation of the contamination level in the two lower mass bins. Such
an underestimation would lead to a "flatter'' distribution of surface
density that would then be fitted with a large value of
.
To
test this, we re-fitted the two lowest mass bins with a King profile
plus a variable uniform surface density, which simulates an extra
unaccounted for level of contamination. The core radius was fixed at
0.9 pc, corresponding to the value of
for stars with mass
.
We found that the
values were
similar with this model (
values of 13.6 and 11.5
respectively for the same numbers of degrees of freedom), but the
levels of contamination implied by the fits (i.e. the extra constant
surface density terms) were factors of 3-5 higher than we estimated in
Sect. 3.4. This would mean that our estimates of the
contamination levels were wildly in error and that only about 35
percent of the cluster candidates with
were
members in contrast to our present estimate of 84 percent. We do
not believe that errors of this extent are possible.
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Mass | Centre | ![]() |
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subset | (![]() |
(J2000.0) | (pc) | (stars/arcmin2) | |||||
<0.53 | >1.48 | RA = 7 58 03.2 | 14.6 | 0.84+0.17-0.14 | 0.223 | 0.0 | 7.7 | 92 | 113 |
Dec = -60 45 29 | 18.4 | 0.81+0.16-0.14 | 0.218 | 0.0 | 7.8 | 92 | 121 | ||
0.53-1.05 | 0.87-1.48 | RA = 7 58 05.6 | 14.6 | 0.90+0.23-0.17 | 0.228 | 0.013 | 10.9 | 103 | 126 |
Dec = -60 46 48 | 18.4 | 0.86+0.21-0.16 | 0.225 | 0.013 | 11.0 | 103 | 135 | ||
1.05-1.68 | 0.65-0.87 | RA = 7 57 56.0 | 14.6 | 1.91+0.56-0.40 | 0.141 | 0.010 | 11.9 | 150 | 196 |
Dec = -60 47 00 | 18.4 | 1.80+0.53-0.37 | 0.132 | 0.010 | 11.9 | 150 | 216 | ||
1.68-2.73 | 0.35-0.65 | RA = 7 57 48.0 | 14.6 | 3.52+0.99-0.64 | 0.279 | 0.017 | 13.4 | 280 | 637 |
Dec = -60 47 00 | 18.4 | 3.26+0.87-0.56 | 0.238 | 0.017 | 13.1 | 280 | 725 |
We conclude that the evidence for mass segregation is strong when
comparing stars above and below about 0.8
.
This coincides
with the break in slope of the MF and lead us to suspect that
the downturns in the LF and MF of NGC 2516 at low masses may be
explained by mass segregation (see Sects. 4.1 and
4.2). We can test this by
extrapolating and integrating the surface density profiles beyond the
extent of the current survey. King (1962) gives an expression for the
integral of equation 5 out to the tidal radius, which we use to
calculate the total number of cluster stars in each mass bin,
,
in Table 4. This number can be compared with the actual number
of cluster members (corrected for contamination),
,
to estimate what
fraction of cluster stars lie outside our survey. Naturally, such an
extrapolation is reliant on a knowledge of the surface density
profile outside the area we have surveyed!
The fraction of cluster stars inside our survey decreases from
81 percent for
to 44 percent for
,
if
pc.
These fractions decrease to 76 percent and 39 percent for the larger
value of the cluster mass and
.
There are two major implications of this result. (1) The total cluster
mass is probably greater than we first estimated. Most of the cluster
mass is concentrated in stars between 0.6-2
so it is likely
that the cluster is more massive by about a factor of 1.3. (2) The
decrease in the MF towards lower masses will be flattened
off. Earlier we stated that an increase of roughly a factor of two in
the numbers of stars below 0.5
(with respect to the numbers
of more massive stars) could bring the LF of NGC 2516 roughly
into agreement with that of the Pleiades. The results in Table 4 show
that within our survey area the ratio of stars with
to those with
is 3.0, but that
this ratio for all cluster stars is probably as large as 5.6-6.0;
approximately the increase required. Taking the results from the lowest
two mass bins, and assuming that the correction to the MF will vary
linearly (with log mass) from about 1.3 at 0.7
to 2.3 at 0.3
,
the fitted values of
(see Sect. 4.2)
would increase from
to
and from
to
for the solar and half-solar metallicity models of
Siess et al. (2000) respectively.
It is difficult to provide precise corrections to the MF and total
cluster mass because our analysis is dependent to a great extent on the
assumed form of the surface density distribution outside of the region
surveyed and to a lesser extent on the assumed non-member contamination fraction in each
bin. For example, if the level of contamination for low-mass stars is
higher than we have assumed then: the numbers of low mass stars inside
our survey will be smaller; the fitted core radius will decrease and
the fraction of cluster members inside the survey will increase; the
cluster mass inside the survey will be smaller; the MF will drop more
steeply and the total cluster mass will be smaller. To give an idea of
the possible size of these effects we ran through our analysis again,
but systematically increased the number of contaminants predicted in
rows 3 and 4 of Table 3, for the columns with
(
), by twice their Poisson errors. The results
from this were: that the MF slope below 0.6
changed from
to
;
the cluster mass inside the survey
decreased slightly from
to 1050
;
the core
radius in the lowest mass bin decreased from 3.5 pc to 2.8 pc; and
the fraction of stars with
inside our survey
increased from 44 to 52 percent. Changes of this order do not affect
any of our major conclusions.
NGC 2516 has been referred to as the "southern Pleiades'' on the basis of their similar ages and apparent richness of high mass stars (Eggen 1983). The central questions which we can attempt to answer on the basis of the results presented here are: (1) how similar are NGC 2516 and the Pleiades, in terms of their LFs, MFs, total mass, mass segregation and binarity, especially among low-mass stars? (2) What are the prospects for finding even lower mass objects and brown dwarfs in NGC 2516?
Of key importance in any comparison with the Pleiades is to make sure
we are comparing similar surveys, in terms of their area and
completeness. We have seen that our NGC 2516 survey is complete to V=20,
(
)
and covers the central
pc. The Pleiades is approximately 3 times closer
and thus surveys of the Pleiades must cover 9 times the area of our NGC 2516 survey to be equivalent.
Dachs & Kabus (1989) presented the LF for bright stars (Mv<5) in NGC 2516, from a 1 degree diameter central region. They showed that this LF was similar in shape to that of the Pleiades with approximately a factor of two more early-type stars in NGC 2516. There was marginal evidence for mass segregation in their sample, but few stars at this luminosity were estimated to lie outside their survey. On this basis they claimed that NGC 2516 was about twice as massive as the Pleiades, implicitly assuming that the two clusters have similarly shaped MFs at all masses.
The Pleiades survey of Meusinger et al. (1996) covered the central 16.5 square degrees
(about
pc) and was complete to Mv=12. The LF shows a
relatively sharp increase of a factor
2 at Mv=10 and the MF
is quite flat, with
(see Sect. 4.2),
between
.
The LF and MF were simply corrected by a
linear factor for stars of all masses and luminosities to account for
Pleiades stars which lay outside the survey area. Thus no account of
mass segregation was taken other than a crude correction to the cluster
mass, which resulted in
down to stars
of
.
Hambly et al. (1999) derived the MF from a wider 36 square degree survey (
pc) and obtained
below 0.5
.
Using survey material complete to radii of about 6 pc, Pinfield et al. (1998)
show that core radii in the Pleiades increased from about
1 pc for stars of 3-5
solar masses to 3 pc at
0.3
.
As a result we can say that Meusinger et al.'s work
must have been affected by mass segregation, with roughly one third of
the low-mass members lying outside their survey, but essentially all
the Pleiades stars with
included. Hambly et al.'s
work fares better, with perhaps only
20 percent of stars
with
missing. In both cases then, the true value of
is likely to be a little higher than derived from these spatially
limited surveys, although the effect should not be nearly as large as
calculated for NGC 2516 in Sect. 4.5.
Pinfield et al. estimate that the total
mass of the Pleiades, after accounting for this segregation is
735
.
The core radius values in the Pleiades are comparable with the NGC 2516
values that we determined in Table 4, but it does seem that the growth
of
with decreasing mass is more rapid in NGC 2516. A simple
power-law fit indicates that
with
,
rather than the 0.5 found for the Pleiades by
Pinfield et al. (1998). In a virialised
system, we might expect
(see
Pinfield et al. 1998 for an analytic argument and Spitzer & Shull 1975
for numerical simulations). This is perhaps a hint
that at least some of the mass segregation in NGC 2516 is
primordial, as has been supposed for some younger clusters
(Sagar et al. 1988; Bonnell & Davies 1998), with the high-mass stars being initially more centrally
concentrated than the low-mass stars.
This would be a mildly surprising result, because
dynamical mass segregation should have removed the signature of initial
conditions on the cluster relaxation timescale - which is about 100 Myr
for clusters of the size of NGC 2516 and the Pleiades.
Both the total mass and the MF of NGC 2516 are dependent on what is
assumed for the metallicity of the cluster and also to a lesser extent
on which stellar evolution models are used. If the cluster has a solar
metallicity then the mass inside our survey area (for stars with
)
is
1100-1200
,
depending on how unresolved binarity is
treated. The equivalent figure for a half-solar metallicity model is
950-1050
(for stars with
).
If the cluster does have a solar metallicity then
the MF drops sharply below 0.7
,
with
.
A
half-solar metallicity yields a shallower slope of
.
We have established that most high mass (>
)
stars are
included in our survey, but that more than half of the low mass
(<
)
stars of the cluster may lie outside this region -
depending on the exact form of the density distribution beyond our
surveyed area. Correcting the total cluster mass for this segregation
leads to an estimate of about 1240-1560
- twice the mass of the
Pleiades, in agreement with the earlier prediction made by
Dachs & Kabus (1989). This implies that the shapes of the MFs of NGC 2516
must be reasonably similar (at least over the mass range which
contributes most to the total cluster mass). As we discussed in
Sects. 4.2 and 4.5, a factor of two increase in the
numbers of low mass stars relative to high mass stars could increase
at low masses and bring the NGC 2516 MF into into
agreement with the flat (
)
Pleiades MF between
.
Mass segregation appears to provide just
this correction, increasing
by about 0.7.
Recent simulations of an evolving star cluster
by Kroupa et al. (2001) show both the total MF and the MF for stars within
2 pc of the centre of a cluster similar in size to the Pleiades, and
at an age of 100 Myr. Their Figs. 14 and 15, demonstrate that mass
segregation can indeed produce a downturn in the MF of the central regions below
0.7 ,
even if the whole-cluster MF is flat. A thorough survey
for low-mass stars outside the area discussed in this paper will be
vital to constrain the surface density of members, the whole-cluster
mass function and the total cluster mass.
On the basis of a comparison of high mass ratio (q>0.6), unresolved
binary systems, the Pleiades and NGC 2516 have a similar binary
fraction of
percent. It remains to be seen whether this similarity
persists for lower mass ratio binary systems.
In summary, subject to a wider survey confirming the presence of an extended population of low-mass objects in NGC 2516 (see the offset field of Hawley et al. 1999 for some preliminary evidence), we believe that NGC 2516 and the Pleiades are similar in most respects, but that NGC 2516 has twice the mass and numbers of stars.
Any apparent difference between the Pleiades and NGC 2516 MFs at
low masses could be very significant for those searching for even lower
mass stars and BDs in NGC 2516. If we take our observed NGC 2516 solar
metallicity MF below
,
with
(see
Sect. 4.2), and simply extrapolate to lower masses, we find
that there would only be about 100 brown dwarfs (with
)
in the area we have surveyed. The half-solar
metallicity MF, with
,
predicts around 220.
The total number of brown dwarfs in the cluster could be much larger if
mass segregation continues to lower masses. At the very least these
numbers should be doubled to account for the segregation that is
already apparent in
stars.
If instead we assume a Pleiades-like MF that is flat between 0.7
and 0.2
and then decreases with
below this, we
calculate that there would be 360-440 brown dwarfs in the solar and
half-solar metallicity cases respectively. Even with this MF, the total mass in
the form of brown dwarfs would be less than 3 percent of the cluster
mass and the extra contribution in the form of stars with
would be about 15 percent. A search for brown dwarfs and low mass
stars in the outer regions of NGC 2516 will tell us much about the
whole-cluster mass function and the cluster dynamics.
In this paper we have presented a large, accurate and uniform
survey of stars covering 0.86 square degrees (
pc) of NGC 2516, and which is almost complete to V=20,
.
Thanks to the relatively low contamination of the
cluster main sequence by foreground and background objects we have been
able to select a sample of candidate cluster members. NGC 2516 probably
has a lower metallicity than the similarly aged Pleiades and our list
of candidate members list will be an important source of
optically selected low-mass targets for further investigations of
magnetic activity, elemental abundances, lithium depletion and rotation
rates in convective stars.
We have made a preliminary investigation of the luminosity function, mass function, binarity, mass segregation and total mass of NGC 2516, and compared our results with the Pleiades, other young clusters and the field. Because the metallicity of NGC 2516 may be as low as half the solar value, we have performed two sets of calculations, one for a mass fraction of heavy elements, z=0.02 and another with z=0.01. Some of our results are somewhat dependent on the assumed metallicity and others depend on extrapolating to lower masses or larger spatial areas. We can summarize the results of our investigation as follows:
Acknowledgements
We would like to thank the director and staff of the Cerro Tololo Interamerican Observatory, operated by the Association of Universities for Research in Astronomy, Inc., under contract to the US National Science Foundation. RDJ and MRT acknowledge the financial support of the UK Particle Physics and Astronomy Research Council (PPARC) throughout a large proportion of this work. Computational work was performed on the Keele, Birmingham and Edinburgh nodes of the PPARC funded Starlink network.