# Constraints on the Stellar/Substellar Mass Function in the Inner Orion Nebula Cluster

Lynne A. Hillenbrand
(lah@astro.caltech.edu)

### and

John M. Carpenter
(jmc@astro.caltech.edu)

## Abstract

We present the results of a 0.5-0.9" FWHM imaging survey at K (2.2 micron) and H (1.6 micron) covering ~ 5.1' × 5.1' centered on Theta¹C Ori, the most massive star in the Orion Nebula Cluster (ONC). At the age and distance of this cluster, and in the absence of extinction, the hydrogen burning limit (0.08 Mo) occurs at K~13.5 mag while an object of mass 0.02 Mo has K~16.2 mag. Our photometry is complete for source detection at the 7 sigma level to K~17.5 mag and thus is sensitive to objects as low-mass as 0.02 Mo seen through visual extinction values as high as 10 magnitudes. We use the observed magnitudes, colors, and star counts to constrain the shape of the inner ONC stellar mass function across the hydrogen burning limit. After determining the stellar age and near-infrared excess properties of the optically visible stars in this same inner ONC region, we present a new technique that incorporates these distributions when extracting the mass function from the observed density of stars in the K-(H-K) diagram. We find that our data are inconsistent with a mass function that rises across the stellar/sub-stellar boundary. Instead, we find that the most likely form of the inner ONC mass function is one that rises to a peak around 0.15 Mo, and then declines across the hydrogen-burning limit with slope N(log M) ~ M0.57+/-0.05. We emphasize that our conclusions apply to the inner 0.71 pc × 0.71 pc of the ONC only; they may not apply to the ONC as a whole where some evidence for general mass segregation has been found.

[preprint] [Table 1]

Presence of super-planetary mass objects as suggested by Lucas & Roche (2000) in the area covered by our Keck/NIRC imaging survey is not readily supported by our analysis. While our photometry agrees with their J- and H-band data in that faint, moderately reddened point sources are present, interpretation of these sources as free-floating super-planetary mass objects is suspect for several reasons:

1. field star contamination is not insignificant, and in fact dominates the source counts at faint magnitudes (see, e.g. our Figures 5 & 7);
2. theoretical predictions of the effective temperatures and luminosities of sub-stellar/super-planetary mass objects are poorly understood; and
3. transformations from the theoretical to the observational plane at such low masses are highly uncertain.
The statistical analysis presented by us does not exclude the hypothesis that a small number of these faint sources could indeed be physically associated with the Orion Nebula Cluster and hence plausibly in the 10-20 M(Jupiter) range; however, spectroscopy combined with better models is needed to thoroughly explore such ideas.

Figures
(click on thumbnails to see enlarged versions)

 Figure 1: Images of our H and K-band mosaics from Keck/NIRC along with an extinction map derived from the molecular column density data of Goldsmith, Bergin, & Lis (1997). The pixel size of the infrared mosaics is 0.15" and the angular resolution of the extinction map is 50". Contours in the extinction map begin at AV = 5 mag and are spaced at 10 magnitude intervals. Figure 2: Spatial distribution of ONC stars within our NIRC mosaics. The x's indicate stars whose photometry we could not derive, x's surrounded by open circles indicate stars with photometry at K but not H, *'s indicate stars with photometry at H but not K, and filled circles indicate stars with photometry at both K and H. Large + signs indicate the optically brightest stars, for orientation. Figure 3: Internal (IRAF) errors in photometry at H- and K-band, and in H-K color. Figure 4: Open circles represent all positional matches < 1" between our NIRC sources and 2MASS sources while filled circles represent a set of relatively bright, isolated stars (those used to derive the aperture corrections). At K, the standard deviation per point about the mean is 0.19 mag for the full sample but 0.08 mag for the isolated stars. At H, the standard deviations are 0.22 mag and 0.09 mag for the full sample and for the isolated stars. In H-K, the values are 0.17 mag and 0.19 mag. Figure 5: Distribution of K magnitudes for stars photometered with NIRC. The open histogram represent all stars with measured K magnitudes while the hatched histogram represents a reduced sampled of stars used in the mass function analysis. See text for explanation of the second sample. Short-dashed line represents the Galactic model of Wainscoat et al. (1992); long-dashed line represents the same model but reddened for stars located behind the cloud by the extinction map shown in Figure 1. Figure 6: K vs H-K diagram for stars photometered with NIRC. Also shown is the 100 Myr isochrone (equivalent to the zero-age main sequence for masses M > 0.35 Mo) and the 1 Myr pre-main sequence isochrone from D'Antona & Mazzitelli (1997, 1998) translated into this color-magnitude plane (solid lines). Reddening vectors (dashed lines) originate from the 1 Myr isochrone at masses of 2.5 Mo, 0.08 Mo, and 0.02 Mo. We believe that the source detection is 90% complete at the 7 sigma theshold to K > 17.5 mag. Internal errors in the K magnitudes are indicated; errors in the H-K color are larger than those in K band alone. The limit for 10% photometry occurs at K~17.3 mag and H~17.4 mag. Figure 7: Hess format K-(H-K) diagram for our data (left panel) and an appropriately reddened field star model (right panel). To generate the contours for the observations, individual stars were smoothed by an elliptical gaussian corresponding to their photometric errors as described in the text. Similarly, the field star model was convolved with the typical photometric error as a function of magnitude. The white solid/dotted line is the 1 Myr pre-main sequence isochrone with the transition from a solid to dotted occuring at the hydrogen burning limit of 0.08 Mo. The lowest mass represented by the isochrone is 0.017 Mo. The reddening vector for AV < 50 mag is indicated by red dashed lines. The color stretch is identical for both panels, with the data plot containing 658 stars and the field star model containing 34 stars down K=17.5 and 43 stars down to K < 18 mag. These figures demonstrate that field stars make a negligible contribution to the ONC star counts except at K > 16 mag (see also Figure 5 by K > 17 mag the field stars dominate cluster members. Figure 8: Distribution of ages for optically visible ONC stars with M < 1.5 Mo located within the boundaries of our NIRC mosaics. This Figure was constructed using the data in Hillenbrand (1997) but the transformations between observational and theoretical quantities, and the pre-main sequence evolutionary calcuations adopted in this paper. For the current analysis we assume an age distribution which is uniform in log between 0.1 Myr and 1 Myr, shown as the solid line, and we also consider an age distribution which is uniform in log between 0.03 Myr and 3 Myr, shown as the dashed line. Figure 9: Distribution of K and H-K excesses. The top panel shows a histogram of H-K color excesses for ONC stars located within the field of view of our NIRC mosaics, calculated using data from Hillenbrand et al. (1997, 1998). The solid curve is a half-gaussian fit to the distribution and has a dispersion sigma=0.4 mag. The bottom panel shows the correlation between K band excess and H-K color excess for stars in Taurus, calculated using data from Strom et al. (1989) and Kenyon & Hartmann (1995). The solid line is the best fit to these data, Delta K = 1.785 × Delta (H-K) + 0.134 with the dashed lines indicating ± 0.25 mag scatter. In analyzing the ONC mass function we assume the distribution of H-K excess shown in the top panel, and the K band excess correlation with H-K excess shown in the bottom panel. Figure 10: ONC mass spectrum derived using the {\it optical} data of Hillenbrand (1997). The input photometry and spectroscopy are the same in all three panels, and represent stars over 30' × 34' of the ONC. In the top panel we show the mass function produced by the theoretical description of luminosity and effective temperature evolution with mass of D'Antona & Mazzitelli (1997,1998) and the transformations between observational and theoretical quantities adopted in this paper. In the middle panel we show the same tracks with the observational-theoretical calibrations adopted by Hillenbrand (1997). In the bottom panel we show the mass function produced by the D'Antona & Mazzitelli (1994) calculations and the calibrations adopted by Hillenbrand (1997). Note the dramatic difference in shape of the mass function below 0.2 Mo between these three panels. Figure 11: Model K-(H-K) diagrams for various assumptions about the age and near-infrared excess distributions. The mass function is log-uniform between 0.017 and 3.0 Mo. The left panel shows the K-(H-K) distribution of two single-aged populations at 0.1 Myr and 1 Myr with no near-infrared excess. The middle panel shows a population distributed log-uniform in age between 0.1 Myr and 1 Myr, as we adopt for the ONC (see Figure 8), and again with no near-infrared excess. The right panel shows the same log-uniform age distribution but now includes the near-infrared excess distribution adopted for the ONC (see Figure 9). Figure 12: Simulations of the K-(H-K) diagram using the age distribution assumed from Figure 8, the near-infrared excess distribution assumed from Figure 9, and an extinction distribution which is uniform in the interval AV=0-5 mag. The middle panel shows the log-normal form of the Miller-Scalo mass function while the right panel shows a shallow power law mass function (N(log M) ~ M^-0.35). Our data are shown in the left panel, which is the subtraction of the field star model in Figure 7b from the observations in Figure 7a. The models suggest that a falling mass function like that of Miller-Scalo better represents the peak in the observed ONC star counts than does an increasing mass function like the shallow power-law. Although there appear to be some more highly extincted stars in the data than in these models, broadening the AV distribution in the models dilutes the peak; this suggests that the bulk of the ONC stars are found at relatively low extinction, AV < 10 mag. Figure 13: Illustrative mass probability functions derived using our methodology. Left panels show stars with H-K=0.5 and right panels show stars with H-K=3.0, both columns of panels decreasing in brightness top to bottom from K=9 to K=18. The de-reddening model uses the same distributions in age and in near-infrared excess as employed elsewhere in this paper. Note the tails upward at the lower and upper mass extrema in the panels for K=16, H-K=0.5 and K=9, H-K=3.0, respectively. These are caused by our imposition of integrated probability equal to unity over the mass range 0.02-3.0 Mo. Figure 14: Tests of the ability of our method to recover an input mass function. Solid lines represent the input mass function while crosses represent the recovered mass function. Tests using the Miller-Scalo mass function appear in the left panels and those using a shallow power-law mass function N(log M/Mo) ~ (M/Mo)^-0.35 in the right panels; the age distribution in both the left and right panels is log-uniform between 0.1 and 1 Myr. From top to bottom the panels indicate a) no extinction and no near-infrared excess; b) extinction uniformly distributed AV=0-5 mag and no near-infrared excess; and c) extinction uniformly distributed between A$_V$=0-5 mag and near-infrared excess distributed using the half-Gaussian function described elsewhere. In every case we are able to distinguish between the slowly falling and the slowly rising mass functions. Figure 15: Tests of the ability of our method to recover an input mass when we intentionally assume an incorrect age or near-infrared distribution. Solid lines represent the input mass function while crosses represent the recovered mass function. The Miller-Scalo mass function is tested in the left panels while a shallow power-law mass function N(log M/Mo) ~ (M/Mo)^-0.35 is tested in the right panels. In all panels the input age distribution is log-uniform between 0.1 and 1 years, the input near-infrared excess distribution is the half-Gaussian function discussed elsewhere, and the input extinction distribution is uniform between AV=0-5 mag. From top to bottom we have varied the assumptions in recovering the mass functions to test incorrect ages (0.1 Myr, 1 Myr, and log-uniform between 0.03 and 3 Myr), and to test an incorrect near-infrared excess assumption (no infrared excess). For reference, we also show in the fourth set of panels from top, the results when the correct age and the correct near-infrared excess distributions are assumed. Figure 16: Derived ONC mass spectrum under three different extinction cuts. The nonlinearity/saturation limit of our observations means that we are fully sensitive to stars with M < 1.5 Mo only while the full sensitivity low-mass mass limit is M = 0.02 Mo, for AV < 10 mag. A Miller-Scalo function normalized to the total number of stars in the AV < 10 mag distribution is shown for comparison (dashed line). Our data indicate that the mass function in the inner ONC declines across the hydrogen burning limit into the brown dwarf regime, perhaps with a somewhat narrower log-normal distribution than Miller-Scalo. Figure 17: Comparison of the ONC mass spectrum derived from optical spectroscopic techniques with that derived here using infrared photometric techniques. Filled circles are the same spectroscopic data as in the top panel of Figure 10, now limited to AV < 2.5 mag leaving 758 stars. Open circles represent that portion of the spectroscopic data located within the same spatial area as our NIRC data, also limited to AV < 2.5 mag leaving 120 stars. Histogram is the NIRC mass function for extinction AV < 2.5 mag. No normalization has been applied to these curves. Note the general agreement between the optical spectroscopic results and the near-infrared photometric results in the mass completeness and the spatial area regimes where they overlap (open circles vs hatched histogram). Note also the disagreement between the shape of the mass spectrum derived for the inner ONC (r < 0.35 pc; open circles) vs the greater ONC (r < 2.5 pc; filled circles). Figure 18: This object is located approximately 15' northeast of our mosaic center and was observed as a local standard for the purpose of atmospheric extinction calibration. The observations plotted were taken 12-15 minutes apart and show variations at the 0.05-0.1 mag level. Similar variability on similar timescales may be a common feature of the young stellar objects the ONC.