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Survey for Transiting Extrasolar Planets in Stellar Systems: I. Fundamental Parameters of the Open Cluster NGC 1245

We derive fundamental parameters for the old, rich open cluster NGC 1245 using BVI photometry from the MDM 1.3m and 2.4m telescopes. Based on detailed isochrone fitting, we find NGC 1245 has a slightly sub-solar metallicity, [Fe/H]=-0.05 +- 0.03 (statistical) +- 0.08 (systematic), and an age of 1.04 +- 0.02 +- 0.09 Gyr. In contrast to previous studies, we find no evidence for significant differential reddening. We determine an extinction of A_V=0.68 +- 0.02 +- 0.09 mag and a distance modulus of (m-M)o=12.27 +- 0.02 +- 0.12 mag, which corresponds to a distance of 2.8 +- 0.2 Kpc. We derive a logarithmic mass-function slope for the cluster of a=-3.12 +- 0.27, where a Salpeter slope is a=-1.35. Fits to the radial surface-density profile yield a core radius of r_c=3.10 +- 0.52 arcmin (2.57 +- 0.47 pc). NGC 1245 is highly relaxed and contains a strongly mass segregated population. The mass function for the inner cluster has a very shallow slope, b=-0.56 +- 0.28. In contrast, the outer periphery of the cluster is enriched with low mass members and devoid of high mass members out to the tidal radius, r_t=20 arcmin (16.5 pc). Based on the observed surface-density profile and an extrapolated mass function, we derive a total cluster mass, M=1300 +- 90 +- 170 M_(solar).


INTRODUCTION
Open clusters are excellent laboratories for many different aspects of astrophysics. First, open clusters form a coeval set of stars with homogeneous properties. This homogeneity makes it possible to determine with relative ease the age and metallicity of the cluster (Yi et al. 2001). Second, owing to their relatively small relaxation times, open clusters provide an opportunity to study stellar systems in various stages of dynamical evolution (Binney & Tremaine 1987). Finally, theory predicts the star formation process within an open cluster is strongly affected by dynamical interactions, supernovae explosions, and UV radiation (Adams & Myers 2001). Characterizing the fraction of cluster members with companions from stellar to planetary masses for a cluster in comparison to the field provides valuable insight into how these processes affect stellar and planetary formation.
The large angular size (tens of arcminutes to several degrees) of open clusters has traditionally made it difficult and time consuming to observe a substantial fraction of an open cluster's members. However, recently available large-format CCD imagers now allow for complete studies of an open cluster with relatively small amounts of observing time. In this paper we present wide-field BV I photometry of the open cluster NGC 1245 using the MDM 8K Mosaic imager on the MDM 2.4m Hiltner telescope and supplemental BV I photometry obtained under photometric conditions on the MDM 1.3m McGraw-Hill telescope.
NGC 1245 is a rich, old open cluster with an approximately solar metallicity population (Janes, Tilley, & Lyngȧ 1988;Wee & Lee 1996). The cluster is located toward the galactic anticenter (ℓ = 147 • , b = −9 • ) at a distance, R ∼ 2.5 kpc (Janes, Tilley, & Lyngȧ 1988;Wee & Lee 1996). With its relatively large Galactocentric distance, NGC 1245 is particularly useful in constraining the Galactic metallicity gradient (Friel 1995;Wee & Lee 1996). Here, we improve on the observations for this cluster by covering a six-times greater area, obtaining the first CCD I-band photometric data, and acquiring the first CCD data under photometric conditions. With these significant improvements over earlier studies, we are able to determine more precisely the physical parameters of the cluster as a whole, rather than just the inner cluster core. In addition to the observational improvements, we improve the interpretation of the observations by calculating the physical parameters of NGC 1245 using a chi-square fit to the observations, and we make every effort to quantify the systematic errors in the derived physical parameters.
Because NGC 1245 is a rich open cluster and readily observable from our primary observing site, it is the first target of the Search for Transiting Extrasolar Planets in Stellar Systems (STEPSS) project (Burke et al. 2003). The goal of the STEPSS project is to assess the frequency of close-in extrasolar planets around main sequence stars in several open clusters. Part of the BV I data we are presenting in this paper were taken during a 19 night observing run to search for transiting extrasolar planets in NGC 1245. This transit data consist of I-band only observations obtained every seven minutes with the MDM 8K Mosaic imager on the MDM 2.4m telescope. Results of the search for extrasolar planets in NGC 1245 using this transit data will be presented in an upcoming paper (B. S. Gaudi et al., in preparation). An additional paper presenting the variable star content of our NGC 1245 field based on light curves from the transit data is also in progress (J. Pepper et al., in preparation). The main goal of this paper is to determine the fundamental physical parameters of NGC 1245. With the known properties for this cluster and subsequent clusters in the STEPSS project, we will gain insight into how metallicity, age, and stellar density affect planetary formation, migration, and survival.
We describe our observations and data reduction in §2. In §3, we derive the physical parameters of NGC 1245 from isochrone fits to the BV I photometry. We derive the radial profile of NGC 1245 in §4, the mass function in §5, and the total mass in §6. We summarize and conclude in §7.

Observations
We observed NGC 1245 on two occasions. The first set of observations was obtained in November 2001 using the MDM 8K mosaic imager on the MDM 2.4m Hiltner telescope. The MDM 8K imager consists of a 4x2 array of thinned 2048x4096 SITe ST002A CCDs (Crotts 2001). This instrumental setup yields a 26 ′ x26 ′ field of view and a 0.36 ′′ per pixel resolution employing the 2x2 pixel binning mode. Table 1 shows for each night of observations the number of exposures in each filter, exposure time, median full width at half maximum (FWHM), and a brief comment on the observing conditions. This observing run is optimized for a transiting extrasolar planet search, thus there are 960 I-band images obtained during this run that are not listed in Table 1. Since the emphasis of the present study is the color-magnitude diagram of NGC 1245, we chose only the I-band images on each night that were taken concurrent with the other two filters and the same number of images to have similar signal to noise characteristics for each filter. None of the nights were photometric. Therefore, we reobserved NGC 1245 in February 2002 using the MDM 1.3m McGraw-Hill telescope with the 2048x2048 "Echelle" imaging CCD. This CCD provides a 17 ′ x17 ′ field of view with 0.5 ′′ per pixel resolution. Several nights were photometric, and we use Landolt (Landolt 1992) standard star observations to calibrate our photometry. Table 2 details the images taken with the MDM 1.3m telescope. The relative placement between the 1.3m and 2.4m field of views is shown in Figure 1. Figure 1 shows a false-color image of NGC 1245 and shows the overlap between the MDM 2.4m and 1.3m datasets. The grid of eight numbered rectangles shows the field of view for the eight CCDs that make up the MDM 8K Mosaic imager. The large inscribed square is the field of view of the MDM 1.3m observations. The blue, green, and red color channels for the false-color image result from a median combined image of the BV I passbands, respectively. The circle denotes the cluster center derived in Section 4.

Data Reduction
We use the IRAF 1 CCDPROC task for all CCD processing. We describe the 2.4m data reduction first. The stability of the zero-second image over the course of the 19 nights allows us to median combine 95 images to determine a master zero-second calibration image. For master flat fields, we median combine 11, 19, and 66 twilight sky flats taken throughout the observing run in the BV I passbands, respectively. We quantify the errors in the master flat field by examining the night to night variability between individual flat fields. The small-scale, pixelto-pixel variations in the master flat fields are ∼ 1%, and the large-scale, illumination-pattern variations reach the 3% level. The large illumination-pattern error results from a strong sensitivity in the illumination pattern to telescope focus. However, such large-scale variations do not affect photometry where the point-spread-function (PSF) scale is the most relevant.
We use the DAOPHOT package (Stetson 1987) within IRAF for PSF fitting photometry, and we perform the photometry on each frame individually. To identify stars and to calculate stellar positions, we designate the highest signal-to-noise I-band image as input to the DAOFIND task. We transform the stellar positions determined on the high signal-to-noise I-band image to all the other images as initial guesses for photometry. To calculate the model PSF, we select 25 bright, isolated stars evenly distributed on the reference image. We use the same PSF stars for all images. Slight offsets between images result in at least twenty stars in the PSF model calculation for all the images. Modeling the spatially constant PSF model involves three iterations with faint neighbors to the PSF stars being subtracted from the image between iterations. Using the resulting PSF model, the IRAF ALLSTAR task calculates the instrumental magnitudes by fitting the PSF model to all stars simultaneously. The reference image photometry defines our instrumental magnitude system, and we apply a zeropoint offset to the individual image photometry to align them with the reference image photometry. A robust weighted mean using the errors output by ALLSTAR gives the final instrumental magnitude.
In order to place the MDM 2.4m instrumental magnitudes onto a standard system, we use the IRAF FIT-PARAMS task to determine the transformation to the photometric MDM 1.3m dataset. The rms scatter for an individual star around the best fit photometric transformation is ∼0.03 mag in all passbands. This error represents how well an individual star approximates the pho-tometric system defined by the 1.3m data described below. The data taken on CCDs 1 and 3 of the MDM 8K mosaic imager suffered from a nonlinearity that reaches a level of 0.14 mag near full well. With the aid of the 1.3m dataset, this nonlinearity is correctable such that the errors in the photometric calibration for these CCDs are similar to the other CCDs not affected by the nonlinearity. The bottom panels of Figure 2 show the resulting color magnitude diagrams (CMDs) for the MDM 2.4m dataset. The stars shown have sample standard deviations in the BV I photometry of less than 0.08 mag and at least 3 (B), 3 (V), and 5 (I) photometric measurements return from the photometry package without numerical convergence errors. The error bars in Figure 2 show the median of the sample standard deviations in the photometric color determination for stars in bins of 0.5 mag at V. These error bars represent the expected deviation from the mean of a color measurement from a single set of BV I exposures with similar signal to noise properties as our dataset. Our ability to determine the mean color for a star requires dividing these errors by the √ N of the number of measurements.
An alternative method to analyze the MDM 2.4m dataset is to combine the images and perform the photometry on this master combined image. The combined image enhances the signal to noise of the faintest stars. This enhanced signal to noise allows DAOFIND to detect and ALLSTAR to fit the model PSF for significantly fainter stars at the expense of having to trust the error output from the photometry package. We median combine 15, 30, and 31 BV I images, respectively, and perform the identical photometry procedure outlined above. Initial guesses for the stellar positions come from the combined I-band image. The instrumental magnitudes are fit to the calibrated MDM 1.3m dataset with a simple zeropoint and color term. The resulting CMDs are shown in Figure 3. As one can see these CMDs go significantly fainter, but the field star contamination becomes increasingly a problem, especially for the V, B − V CMD. The low contrast of the cluster's main sequence against the field star contamination limits the utility of these deeper CMDs in determining the physical parameters of NGC 1245. In this study we only use these deep CMDs for extending our mass function toward fainter magnitudes (see Section 5).
The analysis procedure for the MDM 1.3m dataset is similar to the MDM 2.4m dataset. The master zerosecond image consists of a median combination of 49 images, and a master sky flat consists of a median combination of 12, 15, and 4 images for BV I passbands, respectively. The PSF has strong spatial variations, so we choose the same set of ∼ 80 stars evenly distributed across a reference image to calculate a quadratically varying PSF model for each image individually. We employ the stand-alone ALLFRAME program (Stetson et al. 1998) to calculate the instrumental magnitudes for all images simultaneously. The aperture corrections also are spatially variable. We fit a third order 2-D polynomial to the aperture corrections calculated on the evenly distributed PSF stars. The IRAF XYZTOIM task provides the best fit solution for the 2-D polynomial aperture correction model. The standard deviation of the aperture correction is ∼ 0.02 mag.
Aperture photometry measurements of the Landolt field SA101 (Landolt 1992) taken at three different airmasses, X =1.2, 1.5, and 1.9, on two separate photometric nights (see Table 2) provide a determination of the airmass coefficient. The airmass coefficients for the two nights are statistically identical; thus, we average both airmass coefficient determinations for the final calibration. Fixing the airmass coefficients, we calculate color terms for each night using observations of the Landolt fields G44-27, G163-50/51, and PG1407 (Landolt 1992). The color coefficients from the individual nights are also statistically identical, and we use the mean of both color coefficient determinations for the final calibration. The standard deviations in the calibrations are 0.015, 0.015, and 0.02 mag for the BV I passbands, respectively. This represents how well an individual star approximates the photometric system outlined by Landolt (1992).
There are 2, 2, and 3 images in the BV I passbands, respectively, of NGC 1245 that are photometric. After applying the aperture corrections to the ALLFRAME instrumental magnitudes for the photometric data, we take a weighted mean of the measurements and then apply the photometric calibration to derive an initial standard-star catalog. The final standard-star catalog is obtained by applying the color term and aperture corrections to all nonphotometric MDM 1.3m data, fitting for the zeropoint offset to the initial standard-star catalog, and calculating the robust weighted mean of all measurements.
In the process of determining the zero-point fit for the nonphotometric MDM 1.3m data to the initial standardstar catalog, we uncovered a 1.5% amplitude residual that correlates with spatial position on the CCD. The residual is negligible over most of the CCD, but the residual abruptly appears for stars with a position x 500 in pixel coordinates. The residual linearly grows toward decreasing x position on the detector reaching the 1.5% level at the edge. Due to the abrupt onset of this residual, even a cubicly varying PSF model did not reduce the amplitude of the residual. To reduce the effect of this systematic spatial variability, we limit the zero-point fit for the nonphotometric MDM 1.3m data to stars in the middle of the detector defined by 500 < x < 1500 and 500 < y < 1500 in pixel coordinates.
The top panels in Figure 2 show the resulting CMDs for the MDM 1.3m dataset. Only stars that have at least 3 photometric measurements return from the photometry package without numerical convergence errors and a sample standard deviation < 0.08 mag in all three passbands are shown.

Comparison with Published Photometry
There is some confusion in the literature regarding the photometric calibration of NGC 1245. Carraro & Patat (1994) perform the first CCD detector study of NGC 1245. They calibrate their photometry by comparison with the photoelectric observations of Hoag et al. (1961). Unfortunately, the data of Carraro & Patat (1994) available on the widely used WEBDA open cluster database 2 is incorrect since it lacks the photometric zeropoint as used in publication (G. Carraro private communication). Thus, using the incorrect data from WEBDA, Wee & Lee (1996) concluded the cali-bration of Carraro & Patat (1994) was incorrect. Our independent photometric calibration agrees with the earlier photoelectric studies. For 5 and 3 stars in common with the photoelectric studies of Hoag et al. (1961) and Jennens & Helfer (1975), respectively, we find V -band differences of 0.03 ± 0.06 and −0.012 ± 0.007 mag, respectively, and (B − V ) color differences of −0.01 ± 0.01 and −0.001 ± 0.02 mag, respectively. The (B − V ) colors of the stars used in the comparison span the range from 0.6-1.2 mag. Thus, the photometric calibration for NGC 1245 is well determined.
The recent study by Subramaniam (2003) does not find any differences with the Carraro & Patat (1994) photometry in the V band but does find a linear trend in the (B −V ) color difference as a function of V magnitude with an amplitude of 0.35 mag. It is unclear if the comparison with the Carraro & Patat (1994) photometry in Subramaniam (2003) is with the incorrect data from WEBDA. However, we do not find any (B-V) color difference trends with the Carraro & Patat (1994) data. Additionally, inspection of the Subramaniam (2003) CMD finds the giant clump is 0.35 mag redder in (B − V ) than in our CMD. The above comparisons call into question the Subramaniam (2003) color calibration.

PHYSICAL PARAMETERS
NGC 1245 is located in the Galactic plane (b = −9 • ), and therefore, the observed field contains a large number of foreground and background disk stars complicating the process of fitting theoretical isochrones to the CMDs. We attempted to statistically subtract the field-star population with a procedure similar to Mighell, Sarajedini, & French (1998). Unfortunately, because the contrast of the main sequence above the background field stars was relatively low (in comparison to the work by Mighell, Sarajedini, & French (1998)), we were unable to achieve a satisfactory background field subtraction.
Rather than statistically subtracting field contamination, we select stars for the main sequence isochrone fit by the following procedure. Since binaries and differential reddening affect the (B − V ) color of a star less than its (V − I) color, the V, (B − V ) CMD has a tighter main sequence than the V, (V − I) CMD. The tighter main sequence in the V, (B − V ) CMD has a higher contrast above the contaminating field stars. Thus, to select main sequence cluster members for the isochrone fit, we trace by eye the blue main sequence edge in the V, (B − V ) CMD. Shifting this trace redward by 0.12 mag defines the red edge selection boundary. Finally, shifting the trace 0.02 mag blueward defines the blue edge selection boundary. These shifts were selected by eye to contain a high fraction of the single-star main sequence. The vertical light solid lines in Figure 4 show the resulting main sequence selection boundaries in the V, (B − V ) CMD.
Since the difference in color between the red-giant clump and the main sequence turnoff in the CMD places strong constraints on the cluster age, we designate a box in the V, (B − V ) CMD to select red giant members for the isochrone fit. The region with boundaries 13.7 < V < 14.4 and 1.09 < (B − V ) < 1.21 defines the red-giant clump selection. The light solid box around the red-giant clump in the V, (B −V ) CMD in Figure 4 shows the selection criteria for these stars. The red-giant clump is saturated in the MDM 2.4m dataset, and therefore, we fit isochrones to the MDM 1.3m photometry only. There are 1004 stars that meet our main sequence and red-giant clump selection criteria.
For the isochrone fitting, we use the Yale-Yonsei (Y 2 ) isochrones (Yi et al. 2001), which employ the Lejeune, Cuisinier, & Buser (1998) color calibration. The Y 2 isochrones provide an interpolation scheme to calculate isochrones for an arbitrary age and metallicity within their grid of calculations. For a given set of isochrone parameters (metallicity, age, distance modulus, A V , and R V ), we define the goodness of fit as where the sum is over all stars selected for isochrone fitting, and χ 2 i is the contribution from star i, (2) In the preceding equation, BV I pred are the magnitudes predicted by the isochrone, using the stellar mass, m i , as the independent variable, and BV I obs,i are the observed stellar magnitudes. We use the Brent minimization routine (Press et al. 2001) to determine the stellar mass that minimizes χ 2 i for each star. Theoretical isochrones and color calibrations contain systematic uncertainties that make it impossible to find a consistent fit to CMDs of the best studied open clusters over a wide range of colors and magnitudes (de Bruijne, Hoogerwerf, & de Zeeuw 2001;Grocholski & Sarajedini 2003).
In order to avoid overemphasizing the main sequence turnoff, where the photometric errors are the smallest, we adopt an equal weighting to all stars.
To find the best fit isochrone parameters, we perform a grid search over age and metallicity. At fixed age and metallicity, we use the Powell multidimensional minimization routine (Press et al. 2001) to determine the best fit distance modulus and reddening, A V , by minimizing χ 2 tot . To determine the reddening in the other passbands, A B and A I , we must assume a value for R V , the ratio of total to selective extinction (Cardelli, Clayton, & Mathis 1989). For our best fit solution, we fix R V = 3.2. We discuss the reasoning for this choice and its impact on the results in the Section 3.1. We calculate χ 2 tot over a 40x40 evenly-log-spaced grid of metallicity and age that ranges −0.26 ≤ [Fe/H] ≤ 0.13, and 0.7943 ≤ Age(Gyr) ≤ 1.4125.
We determine our confidence limits on the model parameters by scaling the resulting chi-square statistic, where χ 2 min is the minimum χ 2 tot and ν is the number of degrees of freedom. Figure 5 is a contour plot showing the confidence region for the joint variation in metallicity and age. The three solid contours represent the 1-, 2-, and 3-σ confidence limits corresponding to ∆χ 2 = 2.3, 6.14, and 14.0 for the two degrees of freedom, respectively. To further refine the best fit isochrone parameters, we ran another 40x40 metallicity and age grid at finer resolution: −0.17 ≤ [Fe/H] ≤ 0.03 and 0.9550 ≤ Age(Gyr) ≤ 1.2303. We quantify the 1-σ error bounds by fitting a paraboloid to the ∆χ 2 surface as a function of the four fit parameters [Fe/H], Age, (m−M ) 0 , and A V . The 1-σ errors are then the projection of the ∆χ 2 = 1.0 extent of this paraboloid on the parameter axes. We note that this procedure assumes that the ∆χ 2 surface near the minimum is parabolic, which we find to be a good approximation. Also, because we minimize on A V and (m − M ) 0 at fixed age and metallicity, we are ignoring the additional variance from these parameters. Therefore, our errors are slightly underestimated. The resulting best fit parameters are [Fe/H] = −0.05 ± 0.03, Age = 1.035 ± 0.022 Gyr, (m − M ) 0 = 12.27 ± 0.02 mag, and A V = 0.68 ± 0.02 mag. The best fit isochrone is overplotted on the V, (B − V ) and V, (V − I) CMDs in Figure 4, and the best fit metallicity and age are shown as the filled square in Figure 5.
The isochrone fits reasonably well in the V, (B − V ) CMD, deviating slightly too red toward the lower main sequence. We investigate whether an adjustment to the photometric calibration color term can reconcile the observed main sequence and the isochrone by first calculating the color residual between the stars and the isochrone at fixed V mag. A line fit to the color residuals as a function of (B − V ) color has a slope, b = −0.065 ± 0.007. This calculation assumes that the V-band color term is correct, and only the B-band color term needs adjusting. The error in determining the color term, 1 − σ ∼ 0.004 in all passbands, suggests the large, b = −0.065, color term adjustment required to reconcile the observed main sequence and theoretical isochrone does not result from an incorrect determination of the color term alone.
The fit is poorer in the V, (V − I) CMD. Visually, the best fit isochrone exhibits a tendency to lie redward of the main sequence in the V, (V − I) CMD. This tendency toward the red partly results from the broader and more asymmetric main sequence in the V, (V − I) CMD. Additionally, the overly red isochrone results from systematic errors in the shape of the isochrone. The isochrone fits best near (V −I) ∼ 0.9 and deviates redward of the main sequence toward fainter and brighter magnitudes. A line fit to the (V − I) color residual between the stars and isochrone at fixed V mag as a function of (V − I) color does not reveal any significant trend. Again, as with the V, (B −V ) CMD, the theoretical isochrones are unable to fit the detailed shape of the cluster main sequence from the turnoff down to the lowest observed magnitudes. The isochrone deviates from the observed main sequence by as much as 0.06 mag in (V − I) color at fixed V mag.

Systematic Errors
There are four main contributions to the systematic uncertainties in the derived cluster parameters: uncertainty in R V , photometric calibration errors, theoretical isochrone errors, and binary star contamination. We address the effect of each of these uncertainties on the derived physical parameters in turn.
We can, in principle, determine R V by additionally minimizing χ 2 tot with respect to R V . However, in doing so we find a best fit value of R V = 2.3 and a metallicity of [Fe/H] = −0.26. Values of R V this low are rare in our galaxy (however, see Gould, Stutz, & Frogel (2001)), and metallicity measurements based on medium resolution spectroscopic indices for NGC 1245 rule out a metallicity this low (J. Marshall et al., in preparation).
We also can use infrared photometry from the 2MASS All-Sky Data Release Point-Source Catalog (Cutri et al. 2003) 3 to place more robust constraints on R V . As a side note, 2MASS photometry does not reach nearly as faint as our BV I photometry. Thus, we do not include it directly into our best fit solution for the physical parameters of NGC 1245. To aid in determining R V , we constrain the sample of stars employed in the isochrone fit to stars with V < 16, which corresponds to K s ∼ 14.5 mag for turnoff stars. At this limiting magnitude, the typical photometric error in the 2MASS photometry is σ Ks 0.07. We transform the 2MASS K s passband magnitudes to the K-band system defined by Bessell & Brett (1988) (the system employed in the Lejeune, Cuisinier, & Buser (1998) color calibration used by the Y 2 isochrones) by adding 0.04 mag to the 2MASS photometry (Cutri et al. 2003). The photometric transformation of the 2MASS photometry from K s to K does not require a color-term (Cutri et al. 2003). Using the BV IK magnitudes of the V < 16 turnoff stars, we derive a best fit to the isochrones with R V ∼ 3.2. However, the resulting best fit age is ∼ 1.2 Gyr, somewhat older than the age of ∼ 1.0 Gyr derived from the deeper BV I data alone. Conversely, fixing the age at ∼ 1.0 Gyr in our isochrone fit to the turnoff stars, we find R V ∼ 3.4 is the best fit solution. Figure 4 shows the best fit isochrone to the BV I data systematically deviates from the observed main sequence at the turnoff. Since the 2MASS photometry limits our BV IK isochrone fit to within 1.5 mag of the turnoff, we suspect the difficulty in determining R V with an age consistent with our best fit solution arises from systematic errors in the isochrones. We therefore adopt a fiducial value of R V = 3.2 for our best isochrone fit but explore the impact a systematic uncertainty in R V of 0.2 has on the cluster parameters in the following paragraph.
We quantify the effect of the systematic uncertainty in R V on the cluster parameters by recalculating the isochrone fit assuming R V = 3.0, the 1-σ lower limit in our derived value of R V . The resulting best fit is [Fe/H] = −0.1, Age=1.036 Gyr, (m − M ) 0 = 12.29 mag, and A V = 0.68 mag. The open triangle in Figure 5 shows the resulting best fit metallicity and age for R V = 3.0. Since the (B − V ) color is more sensitive to metallicity variations than the (V − I) color, the color difference between the main sequence in the V, (B − V ) CMD and V, (V − I) CMD is a good indicator of metallicity. Unfortunately, for a fixed A V , variations in R V also alter the the relative (B − V ) and (V − I) color of the main sequence. This degeneracy between R V and metallicity in determining the main sequence color limits our ability to determine the metallicity.
To assess the systematic error associated with the photometric calibration uncertainty, we refit the isochrones assuming the I-band photometry is fainter by 0.02 mag. The resulting best fit is [Fe/H] = 0.0, Age=1.021 Gyr, (m − M ) 0 = 12.31, and A V = 0.62. The I-band offset solution is shown as the starred point in Figure 5.
We attempt to investigate the systematic uncertainties in the isochrones by fitting the Padova group isochrones (Girardi et al. 2000) to the CMDs. We find that these isochrones do not match the shape of the observed main sequence. Systematic differences between the isochrones and data can reach up to 0.08 mag in (B −V ) and (V −I) colors. The mismatch in shape to the observed main sequence prevents precise isochrone fits with the Padova isochrones, and we therefore cannot reliably quantify the systematic errors in the isochrones from these fits. However, we find that the solar-metallicity, 1 Gyr isochrone of the Padova group is broadly consistent with the observed data.
As an alternative method to quantify the systematic errors in the isochrones, we calculate the physical parameters for the well studied Hyades open cluster using an identical procedure to the NGC 1245 data. The Hyades CMD data are selected for membership based on proper motion and eliminated of binary star contaminants based on spectroscopic observations (see Pinsonneault et al. 2003 for a discussion of the data and membership selection). Using a metallicity and age grid with resolution ∆[Fe/H] = 0.01 dex and ∆ log(Age) = 0.006 dex, we find best fit isochrone parameters of [Fe/H] = 0.10, Age = 670 Myr, (m − M ) 0 = 3.23 mag, and A V = 0.03 mag. The recent study by Paulson, Sneden, & Cochran (2003) and many previous studies determine a value of [Fe/H] = 0.13 ± 0.01 for the metallicity of the Hyades using high resolution spectroscopy. Perryman et al. (1998) find an age of 625 ± 50 Myr using an independent set of isochrones. The Hipparcos distance to the Hyades is (m − M ) 0 = 3.33 ± 0.01 mag (Perryman et al. 1998), and the extinction for the Hyades is commonly cited as negligible. The difference in the physical parameters of the Hyades from our isochrone fit to the more accurate determinations of these parameters provides the relative systematic error in our isochrone fitting technique due to systematic errors in the isochrones.
The final systematic error source we address is the error resulting from contamination due to unresolved binaries. Unresolved binaries tend to lie redward of the single-star main sequence, and this binary "reddening" has a larger effect on the (V − I) color than the (B − V ) color. Thus, the main sequence in a V, (V −I) CMD is not as tight and shows greater intrinsic scatter than the V, (B − V ) CMD because the unresolved binaries are separated in (V − I) color from the single-star main sequence more than they are in (B − V ) color. An additional selection of the main sequence in the V, (V − I) CMD provides a sample of probable cluster members with reduced contamination by unresolved binaries. For determining the additional main sequence selection in the V, (V − I) CMD, we trace by eye the blue boundary of the main sequence. The red main sequence boundary is defined by offsetting the blue boundary by 0.08 mag in color. A 0.15 mag color offset is needed to select most of the stars that meet the original main sequence selection based on the V, (B − V ) CMD alone. This additional main sequence selection reduces the stellar sample to 584 from the original 1004. The best fit isochrone using stars that meet the main sequence selection criteria in both CMDs has parameters [Fe/H] = −0.05, Age = 1.052 Gyr, (m − M ) 0 = 12.33 mag, and A V = 0.62 mag. The above solution is shown as the empty square in Figure 5.
To define the 1-σ systematic error on the fitted parameters, we use the difference between the best fit parameters and the parameters determined during the above discussion of the three sources of systematic error: uncertainty in R V , photometric calibration and binary star contamination. For the 1-σ systematic error in the theoretical isochrones, we use the relative difference between our isochrone fit parameters for the Hyades and the quoted values from the literature. To derive an overall systematic error in the cluster parameters, these four sources of systematic error are added in quadrature. The resulting overall 1-σ systematic errors are σ [Fe/H] = 0.08, σ Age = 0.09 Gyr, σ (m−M)0 = 0.12 mag, and σ AV = 0.09 mag. The photometric calibration and R V uncertainty dominate the systematic uncertainty in metallicity. The systematic uncertainty in the isochrone dominates the systematic uncertainties in the age and distance. Unresolved binary contamination and photometric calibration dominate the uncertainty in the reddening.

Comparison with Other Determinations
The best fit metallicity we derive from isochrone fitting to NGC 1245 agrees with our independent metallicity determination using spectroscopy of individual red-giant members (J. Marshall et al., in preparation). Mediumresolution spectroscopic indices calibrated from highresolution spectroscopy indicate [Fe/H] = −0.06 ± 0.12, where the error is the 1-σ systematic error in the spectralindex-metallicity calibration. Wee & Lee (1996) measure the metallicity of NGC 1245 using Washington photometry and obtain [Fe/H] = −0.04 ± stat 0.05 ± syst 0.16. We note that NGC 1245 is commonly quoted in the literature as having a super-solar metallicity of [Fe/H] = +0.14 as given by the Lyngȧ (1987) open cluster database 4 . The origin of this high metallicity for NGC 1245 is unclear; the source for the high metallicity as given in Lyngȧ (1987) does not contain observations for or even discuss NGC 1245.
Our best fit age for NGC 1245 is in agreement with the Age = 1.1 ± 0.1 Gyr found by Wee & Lee (1996). Carraro & Patat (1994) and Subramaniam (2003) find a younger age of Age = 800 Myr for NGC 1245. In the case of Carraro & Patat (1994), the younger age most likely results from their assumption of [Fe/H] = +0.14 for the cluster. In the case of Subramaniam (2003), the difference in color between the main sequence turnoff and the red-giant clump requires the younger age. We believe the red-giant clump of Subramaniam (2003) is too red by 0.35 mag in (B − V ) color (as discussed in section 2). Our smaller color difference between the main sequence turnoff and the red-giant clump results in the older age for NGC 1245.
The main differences between this study and previous investigations of NGC 1245 are that we find a lower overall reddening and we find no evidence for significant differential reddening. We find E(B − V ) = 0.21 ± 0.03, where the error includes the systematic uncertainty in R V and A V . Previous studies find higher reddenings E(B − V ) = 0.28 ± 0.03 (Wee & Lee 1996), marginally consistent with our determination. The previous claims for differential reddening across NGC 1245 have not been highly significant and even conflicting. Carraro & Patat (1994) determine a north/south (B −V ) color gradient of 0.04 mag arcmin −1 with redder colors toward the south, and (Wee & Lee 1996) find a north/south (B − V ) color gradient of 0.03± 0.16 mag arcmin −1 , except the color gradient is in the opposite sense, with redder colors to the north.
We follow the same procedure as in Carraro & Patat (1994) and Wee & Lee (1996) to determine the differential reddening across NGC 1245 using the 1.3m dataset. To quantify the differential reddening across NGC 1245, we start with the same sample of stars that were used for the isochrone fit. We further restrict the sample to stars with 15.0 V 17.0 mag. The differential reddening is modeled as a linear trend between the (B − V ) color of stars as a function of north/south position. We formally find a north/south (B − V ) color gradient of 0.0042± 0.0006 mag arcmin −1 with redder colors to the north. The color gradient corresponds to a total change of ∆(B − V ) ∼ 0.03 over 9 arcmin. The color gradient in the east/west direction is half the north/south color gradient with a similar error. Therefore, we find a color gradient that is smaller by an order of magnitude than previous investigations. Furthermore, although formally significant, we are not convinced of the color gradient's reality because the two-dimensional aperture correction map and PSF variations both contain errors of order 0.02 mag.

RADIAL PROFILE
Stellar encounters drive an open cluster toward equipartition of kinetic energy resulting in mass segregation of the cluster members (Binney & Tremaine 1987). The survey of Nilakshi et al. (2002) demonstrates that the prominent cores of open clusters generally contain the most massive stars, but 75% of a cluster's mass lies in a surrounding corona containing only low mass members of the cluster. In this section, we search for the signatures of mass segregation in NGC 1245 by studying its radial surface-density profile as a function of the magnitude of the cluster stars.
The first step in deriving the radial profile is to locate the cluster center. Using the stellar positions for stars with high-quality photometry shown in the 1.3m CMD (Figure 2), we spatially smooth the number counts with a Gaussian of radius ∼ 75 ′′ . We do not use the 2.4m dataset to calculate the cluster center because gaps in the CCD array bias the derived center. The open circle in Figure 1 denotes the cluster center. Our derived cluster center is located 48 ′′ east and 75 ′′ south of the center determined by Carraro & Patat (1994) and 36 ′′ east and 36 ′′ south of the center determined by Subramaniam (2003). Since neither study gives exact details on how the cluster center was determined, we cannot determine the significance of these differences in the cluster center. However, we believe that our estimate for the cluster center is likely to be more robust, due to our larger field of view and fainter limiting magnitude.
We construct a radial surface-density profile by determining the surface density of stars in concentric annuli of width ∼ 25 ′′ . The top left panel in Figure 6 shows the surface-density profile derived from both the 1.3m and 2.4m datasets. We approximate the errors as N 1/2 i , where N i is the number of stars in each bin. Area incompleteness sets in due to the finite size of the detector for annuli 7.3 ′ in the 1.3m dataset and 11.6 ′ for the 2.4m dataset, thereby increasing the errors. Generally, the surface-density profiles derived from the two datasets agree well, except near the center of the cluster and a small overall scaling. The scaling difference results from the fainter limiting magnitude in the 2.4m dataset (see Figure 2). The discrepancy in the inner annuli arises from the fact that the cluster center falls very close to a gap in the CCD array for the 2.4m detector. Therefore, incompleteness affects the first few radial bins of the 2.4m surface-density profile. We disregard the first two radial bins for all surface-density profiles derived from the 2.4m dataset.
As expected, the radial profile of the cluster exhibits a gradual decline to a constant field-star surface density. We fit the surface-density profile to the model, where Σ f is the surface density of field stars, Σ 0 is the cluster surface density at r = 0, and r c is the core radius.
In fitting Equation 4, we require a positive Σ f . We fit this profile to both the 1.3m and 2.4m datasets. Although the 2.4m surface-density profile extends to larger radii, the incompleteness near the center generally results in less well-constrained parameters, in particular the core radius r c . Both fits are excellent: for the 1.3m dataset, χ 2 = 15.9 for 28 degrees of freedom (dof), whereas for the 2.4m dataset, χ 2 = 39.3 for 36 dof. Table 3 tabulates the best fit parameter values and 1-σ uncertainties for all datasets. A parabola fit to the envelope of χ 2 values as a function of the parameter of interest determines the parameter's error. The solid line in the top left panel of Figure 6 shows the best fit model to the 1.3m profile. The logarithmic slope of the surface density is β = 2.36±0.71, and the core radius is 3.10 ± 0.52 arcmin. The field and central surface densities are Σ f = 4.0 ± 0.8 arcmin −2 and Σ 0 = 14.7 ± 2.9 arcmin −2 , respectively. At the derived distance to the cluster, d = 2850 pc, the conversion from angular to physical distances is 1.207 arcmin/pc. Thus, NGC 1245 has a core radius of ∼ 2.6 pc. The core radius agrees with the previous determination of 2.7 ± 0.13 arcmin from Nilakshi et al. (2002).
To investigate mass segregation, we next divide the high-quality photometry sample of stars into two subsamples: bright stars with V < 17.8 and faint stars with V > 17.8. There are roughly an equal number of cluster stars in each subsample. The surface-density profiles for both the 1.3m and 2.4m data are shown in the bottom left and middle left panels of Figure 6. We find a significant difference between the best fit logarithmic slopes of the bright and faint subsamples. For the 1.3m dataset, we find β = 3.9 ± 1.3 for the bright sample, and we find β = 1.3±0.9 for the faint sample. This provides evidence, at the ∼ 2 − σ level, that the faint cluster stars in NGC 1245 are considerably more spatially extended than the brighter stars. The fainter stars being spatially extended is almost certainly due to the effects of mass segregation, which we expect to be important for a cluster of this mass and age. We provide additional evidence for mass segregation in Section 5 and discuss the implications of this mass segregation more thoroughly in Section 6.
The radial profile of NGC 1245 implies that the faint cluster members may actually extend significantly beyond our field of view. The radial profile for the 1.3m data predicts that, at 1-σ, 40 to 100% of the faint (V > 17.8) stars outside 10 ′ of the cluster center and within the field of view are cluster members. The strong cluster contribution in the outer periphery of our observed field hampers our ability to obtain an accurate statistical subtraction of the field star contamination necessary to determine the mass function for NGC 1245. We discuss this additional complication and our solution in Section 5.

MASS FUNCTION
Typically an assumed mass-luminosity relation allows the transformation of the observed luminosity function to derive the present day mass function (Chabrier 2003). In contrast, we assign a mass to each star individually using our best fit isochrone. Minimizing a star's individual χ 2 i (Equation 2) using the best fit isochrone parameters determines the star's mass. To determine the mass for each star, we use the high quality 2.4m photometry shown in the bottom panel CMDs of Figure 2 as it covers a much larger field of view. We reduce field star contamination of the cluster mass function by restricting the sample to stars with χ 2 i < 0.04. This χ 2 i cutoff selects a region in the V, (B − V ) CMD roughly similar to the main sequence selection used in the isochrone fit but naturally selects stars based on their proximity to the isochrone in all three passbands.
Despite the large field of view of the 2.4m data, the radial profile of the cluster suggests that there exists a significant number of low mass cluster members out to the limits of our field of view. Lacking observations of a field offset from the cluster, the Besançon theoretical galaxy model (Robin & Crézé 1986) provides an alternative method to correct the observed mass function for field star contamination. We obtain from the Besançon galaxy model electronic database the stellar properties and photometry for a one-square-degree field centered on the cluster's galactic coordinates without observational errors. A spline fit to the median photometric error in 0.5 mag bins in each passband allows us to simulate the observed errors in the theoretical galaxy field photometry. We add a zero-mean, unit-standard-deviation Gaussian random deviate scaled to the observed photometric error model to the theoretical galaxy field photometry. We apply the same error selection criteria (σ < 0.08) as was applied to the observed CMD and place a saturation cutoff at V=14.0 mag to the theoretical CMD.
To make a qualitative comparison between the theoretical field CMD and the observed CMD, we adjust the theoretical CMD for the smaller observed 2.4m field of view. In the comparison, a star is included in the theoretical CMD only if a uniform random deviate is less than the ratio between the two field of views. Figure 7 compares the CMDs of the theoretical galaxy model in the top panels and the observations for stars beyond 12.7 arcmin of the cluster center for the 2.4m data in the bottom panels. The dashed line shows the best fit isochrone for reference. As shown in Figure 7, the theoretical and observed field CMDs are qualitatively very similar.
To calculate the field star contamination in the mass function, we first obtain the mass for the stars in the theoretical field CMD and apply the same χ 2 i < 0.04 selection in the same manner as the observed field CMD. For both the theoretical field CMD and observed cluster CMD, we calculate mass functions using even log-spaced intervals of 0.071 log(M ⊙ ). Number counts from the entire one-square-degree field of view provide the basis for the theoretical galaxy mass function. The larger field of view of the theoretical galaxy model than the observed field requires scaling of the number counts and corresponding Poisson error of the theoretical galaxy mass function by the ratio of field of views. Subtracting the scaled theoretical galaxy field number counts from the observed number counts provides the cluster mass function. The cluster mass function is shown as the heavy solid line in the upper panel of Figure 8. The top axis shows the median apparent V magnitude for the corresponding mass bin. The vertical dashed line delineates mass bins that are complete to better than 90%. Comparing the cluster mass function to a mass function calculated by subtracting the theoretical galaxy number counts without applying the photometric error selection criteria determines the completeness of a mass bin. The long-dashed line labeled 'Salpeter' in Figure 8 illustrates the typical observed mass function slope, α = −1.35. A power law in linear-mass, linear-number-count space fit with weights to the mass function bins not affected by incompleteness yields a slope steeper than Salpeter, α = −3.12 ± 0.27.
To verify the validity of using the theoretical galaxy field in place of an observed off-cluster field, we calculate an alternative mass function using the observed field outside 12.7 arcmin of the cluster center for field contamination subtraction. The entire 2.4m field of view again provides the basis for the observed number counts. The smaller field of view of the outer periphery requires scaling the field contamination number counts and corresponding Poisson error by the relative area factor, f = 6.4. The light-solid line in the upper panel of Figure 8 shows the resulting cluster mass function using the outer periphery for determining the field contamination. The mass bin positions are offset slightly for clarity. The two mass functions agree within 1 − σ for all bins with log M > 0 but disagree significantly at the low mass end. The theoretical galaxy subtracted mass function contains greater number counts at the low mass end, consistent with our conclusion that the outer regions of our observed field of view contain a significant number of low mass cluster stars. Both mass functions contain a drop in the number counts at log(M ) = 0.071. We question the significance of this drop in the number counts. The drop occurs at a magnitude with some of the heaviest field star contamination, making the field star contamination subtraction less certain.
To further the evidence for mass segregation in NGC 1245, we calculate the mass function inside and outside a radius of 4.2 arcmin from the cluster center. According to the cluster radial profile, this radius roughly separates half of the cluster members. The inner and outer mass functions are shown as the solid and short-dashed lines, respectively, in the lower panel of Figure 8. The fit to the inner region mass function yields a shallower slope that is consistent with Salpeter, α = −1.37 ± 0.2. Ignoring the highest mass bin, the best fit slope for the inner region is even shallower, α = −0.56 ± 0.28. We find that the outer region is highly enriched in low mass stars. The outer region mass function has a very steep slope, α = −7.1± 1.2. The mass bins with the down pointing arrows have negative values, and the point represents the 2-σ upper confidence limit. Since the mass function is fit in linear space, the fit includes bins with negative values.
The radial profile and mass function demonstrate that a significant fraction of the low mass cluster members reside in the outer periphery of the cluster. Previous studies of NGC 1245 generally find Salpeter and shallower slopes for the mass function (Carraro & Patat 1994;Subramaniam 2003). As warned in Subramaniam (2003), the smaller field of view of these previous studies from which to determine the field contamination biases the mass function against low mass members that reside in the outer periphery of the cluster. The mass functions from the previous studies agree well with our mass function for the interior of NGC 1245, but we conclude that a steeper slope than Salpeter, α = −3.12 ± 0.27, is more appropriate for NGC 1245 as a whole down to the completeness limit of M = 0.8M ⊙ .
Our deeper CMD obtained from combining the MDM 2.4m data (Figure 3) allows us to extend the mass function to even lower masses. Determining the mass function to lower masses improves our mass estimate of the cluster, calculated in the following section. To calculate the mass function, we follow the same procedure as outlined above. At these fainter magnitudes, the theoretical galaxy field overpredicts the number of stars. Thus we must resort to using the outer periphery of the MDM 2.4m field of view for estimating the field contamination. The heavy solid line in Figure 9 shows the resulting mass function based on the deeper CMD. The light solid line reproduces the equivalent mass function from Figure 8 based on the shallower CMD data. The vertical dashed line represents the completeness limit at M = 0.56M ⊙ . The deep mass function has a slope, α = −0.5 ± 0.4, significantly steeper than the slope for the shallow mass function, α = −2.8 ± 0.8. The significantly differing slopes suggest there is a turnover in the mass function at log(M/M ⊙ ) = −0.05. Fitting to the last three complete mass bins in the deep mass function yields a slope, α = 1.8 ± 0.5. The low mass end of the deep mass function likely does not fall as steeply as measured, since we use the outer periphery of the MDM 2.4m field of view to quantify the field contamination, which likely contains many low mass cluster members. Thus, the measured slope at the low mass end represents an upper limit.

TOTAL MASS AND DYNAMICS
Numerous dynamical processes take place in open clusters. Interactions between the cluster members dramatically affect binary and planetary systems, mass segregation within the cluster, and the timescale for cluster dissolution (Hurley & Shara 2002a,b;Giersz & Heggie 1997;Portegies Zwart et al. 2001). The cluster's initial total mass sets the timescale for these dynamical processes and governs its dynamical evolution. Unfortunately, several difficulties prevent a completely empirical determination of the total mass of NGC 1245. First, as we show in Section 4, the surface-density profile suggests the cluster extends significantly beyond our field of view. Second, a substantial population of low mass cluster members exists below the completeness limit of our mass function. Third, we do not resolve binary systems. Therefore, the total mass in observed cluster members (∼ 820M ⊙ ) only places a lower limit to the current cluster mass. Finally, even overcoming the heretofore diffi-culties, stellar evolution and evaporation results in additional complications for determining the initial birth mass of the cluster.
In order to provide a rough estimate of the current total cluster mass, we must adopt a number of assumptions in order to extrapolate the observed mass to the total mass. We must account for stars outside our survey area and stars fainter than our magnitude limit. The total cluster mass determined by integrating the radial profile model we fit in Section 4 slowly converges and is unrealistically large for an integration to infinity. In reality, the tidal field of the Galaxy truncates the outer radius of the cluster. Therefore, we refit the observed dataset to the King profile (King 1962), where r t is the tidal radius of the cluster. Note that, as r t → ∞, this profile reduces to the radial profile given in Equation 4 for β = 2. For a given set of parameters; Σ 0 , r c , and r t ; the integral over Equation 5 determines the total number of stars, where x t ≡ (r t /r c ) 2 (King 1962). The total mass is then M = m obs N , where m obs = 1.00M ⊙ is the average mass of observed cluster stars.
Unfortunately, King profile fits to either dataset yield only lower limits for the value of r t . The lower limit only for r t is primarily due to the covariance between r t and Σ f . Therefore, we need an additional constraint on either the surface density of field stars or the tidal radius in order to determine the total cluster mass. Attempting to constrain the fit using values of Σ f determined from the theoretical galaxy field star counts (see Section 5) results in unrealistically large values of r t > 40 arcmin. Thus, we adopt a different approach.
The tidal radius of the cluster is set by both the local tidal field of the Galaxy and the total mass of the cluster. We can estimate the tidal radius as the radius of the Lagrange point (Binney & Tremaine 1987), where D is the distance of the cluster from the center of the Galaxy, M is the total mass of the cluster, and M MW is the mass of the Galaxy interior to D. For our derived distance of d = 2850 pc, assuming R 0 = 8 kpc, and Galactic coordinates (l = 147 • .6, b = −8 • .9), the Galactocentric distance of NGC 1245 is D = 10.50 kpc. For a flat rotation curve with v = 220 km s −1 , we find M MW = 1.18 × 10 11 M ⊙ , and thus r t ≃ 20 pc(M/2500M ⊙ ) 1/3 . We use equations (6) and (7) to solve for the total cluster mass M and tidal radius r t while simultaneously constraining r c , Σ f , and Σ 0 from fitting Equation 5 to the observed radial surface-density profile. However, we must first account for low mass stars that are below our detection limit. To do this, we use the best fit mass function (dN/d log M ∝ M α , with α = −3.12) to the turnover in the mass function at 0.85M ⊙ . For stars between the hydrogen-burning limit and 0.85M ⊙ , we assume α = 1.0. Thus, our assumed cluster mass function is The adopted mass function slope at the low mass end, α = 1.0 is a compromise between our measured lower limit to the slope, α = 1.8 (see Section 5), and other clusters that have an observed mass function slope as shallow as α ∼ 0.4 at the low mass end (Prisinzano et al. 2001;Bouvier et al. 2003). This yields a total cluster mass (in our field of view) of ∼ 1312M ⊙ and a correction factor of 1.61 to the observed mass in cluster stars. We use an iterative procedure to determine M and r t . We assume a value of r t and then fit the King profile (Equation 5) to the observed radial density profile. The best fit King profile yields trial values of r c , Σ f , Σ 0 , and the total number of cluster stars via Equation 6. We then apply the correction factor to account for stars below our completeness limit to determine the total cluster mass, M = 1.61 m obs N . We then use Equation 7 to predict a new value for r t . The procedure iterates until convergence with a tidal radius of r t ≃ 20 arcmin (16.5 pc) and a total mass of M = 1312 ± 90M ⊙ , where the error is the 1-σ statistical error from the fit. We determined the total cluster mass using the shallow 2.4m dataset as this is the dataset used to determine the mass correction factor and surface-density profile.
The mass correction factor, sensitive to the mass function slope at the low mass end, dominates the uncertainty in the cluster mass. For example, varying the logarithmic slope of the mass function below 0.85M ⊙ in the range 0.6 ≤ α ≤ 1.4 changes the derived total mass by +200 − 140M ⊙ .
Our derived mass does not account for the presence of unresolved binaries. Without knowledge of the properties of the binary population, it is difficult to assess the magnitude of the correction. However, for a large binary fraction, the correction could be as large as 50%. We have also not attempted to correct the derived presentday mass for evolutionary effects, such as evaporation or stellar evolution, to derive an initial cluster mass. For our adopted mass function (Equation 8), the correction to the total cluster mass due to stellar evolution is small, ∼ 12%. About 10% of the cluster stars are lost to the Galaxy per relaxation time (Spitzer 1987;Portegies Zwart et al. 2001). Since the current age of NGC 1245 is approximately 8 times larger than the current relaxation time (see below), it is possible that the cluster has lost an appreciable amount of mass over its lifetime. The numerical N-body simulations of Portegies Zwart et al. (2001) predict the initial cluster mass based on the current number of red-giant stars. Using their Equation 1, an age of 1 Gyr for NGC 1245 and a total of 40 observed red-giant stars yields an initial total cluster mass, M o = 3120M ⊙ .
The relevant quantity that determines the timescale for dynamical evolution of a stellar cluster is the half-mass relaxation timescale (Spitzer 1987), Here r hm is the radius enclosing half of the total mass of the cluster, m ≃ 0.63M ⊙ is the average mass of cluster members, and ln Λ ≃ ln 0.4N is the Coulomb logarithm. This is roughly the average time over which the velocity of a typical cluster member changes by order unity due to random encounters with other cluster members. For NGC 1245, r hm = 3.8 pc, and t rlx ≃ 130 Myr. Since the age of NGC 1245 is ∼ 1 Gyr, we expect this cluster to be dynamically quite evolved. Therefore, the evidence of mass segregation found in Sections 4 and 5 is not surprising.

CONCLUSION
In this paper, we report our results based on BV I photometry of the open cluster NGC 1245 using the MDM 1.3m and 2.4m telescopes. Here, we improve on the observations for this cluster by covering a sixtimes greater area, obtaining the first CCD I-band photometric data, and acquiring the first CCD data under photometric conditions. With these significant improvements over earlier studies, we are able to determine more precisely the physical parameters of the cluster as a whole, rather than just the inner cluster core. Our photometric data resolve some of the confusion in the literature regarding the correct photometric zeropoint for this cluster (Wee & Lee 1996;Subramaniam 2003). Based on isochrone fits employing the Y 2 calculations (Yi et al. 2001), we confirm the findings of Wee & Lee (1996) and (J. Marshall et al., in preparation) that this cluster has a slightly subsolar metallicity, [Fe/H] = −0.05 ± 0.03 (statistical) ± 0.08 (systematic). Our best fit age is 1.04 ± 0.02 ± 0.09 Gyr. In contrast to previous studies, we do not find evidence for significant differential reddening. We find a V -band extinction of A V = 0.68 ± 0.02 ± 0.09, and with the aid of 2MASS K s photometry, we constrain the ratio of absolute to selective extinction to be R V = 3.2 ± 0.2. The resulting absolute distance modulus is (m − M ) 0 = 12.27 ± 0.02 ± 0.12.
With the large field of view provided by the MDM 8K Mosaic imager, we find NGC 1245 is a highly relaxed cluster; a majority of the low mass cluster members reside in an extended halo. The mass function slope, α = −3.12 ± 0.27, down to M = 0.85M ⊙ is steeper than the Salpeter value of α = −1.35, found by earlier studies. The previous studies did not have a sufficient field of view to detect the low mass cluster members that preferentially reside in the outer periphery of the cluster. Based on the observed stellar surface-density profile and an extrapolated mass function, we derive a total cluster mass, M = 1300 ± 90 ± 170M ⊙ . This paper is the first in a series for the Search for Transiting Extrasolar Planets in Stellar Systems (STEPSS) project. The main goal of the STEPSS project is to determine the fraction of stars with closein, Jupiter-sized planets using the photometric transit technique. By searching for extrasolar planets in open clusters of known metallicity, stellar density, and age, we hope to constrain the impact of these parameters on the resulting planetary systems. The accurate physical parameters for NGC 1245 derived in this paper will be used in the analysis of the 19 night run targeting NGC 1245 to search for transiting extrasolar planets (B. S. Gaudi et al., in preparation). The NGC 1245 transit search, consisting of 5 minute time resolution I-band photometry Oct. 27 9 V 300 1.4 overcast · · · 9 I 300 1.1 · · · Nov. 7 7 V 300 1.8 partly cloudy · · · 7 I 300 1.2 · · · 8 5 B 240 1.6 cirrus · · · 5 V 240 1.5 · · · · · · 5 I 240 1.3 · · · 10 5 B 240 1.4 cirrus · · · 5 V 240 1.6 · · · · · · 5 I 240 1.3 · · · 11 5 B 120 1.1 clear · · · 5 V 120 1.1 · · · · · · 5 I 120 1.0 · · · · · · 1 I 240 0.9 · · · taken over 19 nights, will also provide valuable information on the variable star content of NGC 1245 (J. Pepper et al., in preparation). Mosaic imager (the 4x2 array of CCDs are numbered) on the MDM 2.4m telescope. The large inscribed square is the field of view for the MDM 1.3m telescope data obtained with a single CCD imager. The blue, green, and red color channels of the false color image consist of combined images in the BV I passbands, respectively. The circle shows the cluster center. 2.0 non-photometric · · · 3 V 60 1.9 · · · · · · 3 I 30 1.8 · · · · · · 3 I 20 1.8 · · · Feb. 5 6 B 120 1.7 non-photometric · · · 4 V 60 1.6 · · · · · · 6 I 20 2.0 · · · Feb. 7 2 B 90 1.6 photometric · · · 2 V 60 1.4 · · · · · · 3 I 20 1.7 · · ·    -Left panels show the radial surface-density profiles of stars in the NGC 1245 field for three different samples of stars. The top row shows the radial profile for all stars. The middle panel shows the profile for the subset of these stars with V < 17.8, while the lower panel is for V > 17.8. The solid circles are derived from the 1.3m data, while the open squares are from the 2.4m data. In each panel, the solid line shows the best fit model to the 1.3m data, which is composed of a cluster profile (dotted line) plus a constant field-star surface density (dashed line). The panels on the right show the 68%, 95%, and 99% confidence regions on the fitted logarithmic slope of the density profile β and core radius for the 1.3m data. The filled point shows the best fit model for the 1.3m data, whereas the open square is the best fit for the 2.4m dataset. The shaded region is the 68% confidence region for the fit to the 2.4m dataset. In the upper-right panel, we also show the 68% confidence regions for the bright and faint samples.  (Robin & Crézé 1986). Photometric errors and selection criteria are included in the theoretical galaxy to match the observed CMDs shown in the bottom panels. Bottom Panels: Observed V, (B − V ) and V, (V − I) CMDs for stars beyond 12.7 arcmin of the cluster center using the 2.4m data. In all panels, the dashed line shows the best fit isochrone.