2 Observations and data reduction

   
2.1 Gemini/Hokupa'a data

We analysed H and $K^\prime$ images of the Arches cluster center obtained in the course of the Gemini science demonstration at the Gemini North 8 m telescope located on Mauna Kea, Hawai'i, at an altitude of 4200 m above sea level.

Gemini is an alt-azimuth-mounted telescope with a monolithic primary mirror and small secondary mirror optimised for IR observations. The telescope is always used in Cassegrain configuration with instruments occupying either the upward looking Cassegrain port or one of three sideward facing ports. The University of Hawai'i adaptive optics (AO) system Hokupa'a is a 35 element curvature sensing AO system (Graves et al. 2000), which typically delivers Strehl ratios between 5% and 25% in the K-band.

Hokupa'a is operated with the near-infrared camera QUIRC (Hodapp et al. 1996), equipped with a $1024 \times 1024$ pixel HgCdTe array. The plate scale is 19.98 milliarcsec per pixel, yielding a field of view (FOV) of 20.2 $^{\prime \prime }$. The images are shown in Fig. 1, along with the HST/NICMOS images used for calibration.

Figure 1: Upper panels: Gemini/Hokupa'a H (left) and $K^\prime$ (right) images, 20 $^{\prime \prime }$$\times$ 20 $^{\prime \prime }$(0.8 pc $\times$ 0.8 pc) with a spatial resolution of less than 0.2 $^{\prime \prime }$. North is up and East is to the right. Lower panels: HST/NICMOS F160W and F205W images from Figer et al. (1999), geometrically transformed to the Gemini FOV.

The observations were carried out between July 3 and 30, 2000. 12 individual 60 s exposures, dithered in a 4 position pattern with an offset of 16 pixels (0.32 $^{\prime \prime }$) between subsequent frames, were coadded to an H-band image with a total integration time of 720 s. In $K^\prime$, the set of 34 dithered 30 s exposures obtained under the best observing conditions was coadded to yield a total exposure time of 1020 s. The full width at half maximum (FWHM) of the point spread function (PSF) was 9.5 pixels (0.19 $^{\prime \prime }$) in H and 6.8 pixels (0.135 $^{\prime \prime }$) in $K^\prime$. The observations were carried out at an airmass of 1.5, the lowest airmass at which the Galactic Center can be observed from Hawai'i.

The H-band data were oversampled, and $2\times 2$ binning was applied to improve the effective signal-to-noise ratio per resolution element, allowing a more precise PSF fit. A combination of long and short exposures has been used to increase the dynamic range. For the short exposures, 3 frames with 1 s exposure time have been coadded in H, and 16 such frames in $K^\prime$. See Table 1 for the observational details. In the long exposures, the limiting magnitudes were about 21 mag in H and 20 mag in $K^\prime$. Note that the completeness limit in the crowded regions was significantly lower. The procedure used for completeness correction will be described in detail in Sect. 2.1.6.

   
2.1.1 Data reduction

The data reduction was carried out by the Gemini data reduction team, F. Rigaut, T. Davidge, R. Blum, and A. Cotera. The procedure as outlined in the science demonstration report was as follows: sky images, obtained after the short observation period when the Galactic Center was in culmination, were averaged using median clipping for star rejection, and then subtracted from the individual images. The frames were then flatfielded and corrected for bad pixels and cosmic ray hits. After inspecting the individual frames with respect to signal-to-noise ratio and resolution, and background adjustment, the images with sufficient quality were combined using sigma clipped averaging. The final images were scaled to counts per second. For the analysis presented in this paper, this set of images reduced by the Gemini reduction team has been used.

   
2.1.2 Photometry

The photometry was performed using the IRAF (Tody 1993) DAOPHOT implementation (Stetson 1987). Due to the wavelength dependence of the adaptive optics correction and anisoplanatism over the field, the H and $K^\prime$ data have been treated differently for PSF fitting. While in H the PSF radius increases significantly with distance from the guide star, with a radially varying FWHM in the range of 0.18 $^{\prime \prime }$ to 0.23 $^{\prime \prime }$ (see the science demonstration data description), the $K^\prime$ PSF was nearly constant over the field (0.125 $^{\prime \prime }$ to 0.135 $^{\prime \prime }$). This behaviour is expected from an AO system, as the isoplanatic angle $\theta_0$ varies as $\lambda^{6/5}$, yielding a 1.4 times larger $\theta_0$ in $K^\prime$ than in H, resulting in a more uniform PSF in $K^\prime$ across the field of view.

As obscuration due to extinction decreases with increasing wavelength, many more stars are detected in $K^\prime$ than in H. For comparison, the number of objects found with $K^\prime < 20$ mag and uncertainty $\sigma_{K^\prime} < 0.2$ mag was 1017 (1020 s effective exposure time), while for H < 20 mag and $\sigma_{\rm H} < 0.2$ mag we detected only 391 objects (720 s effective exposure time), where in both cases visual inspection led to the conclusion that objects with photometric uncertainties below 0.2 mag were real detections. On the other hand, the increased stellar number density in $K^\prime$ leads to increased crowding effects, such that we decided to use a non-variable PSF for the $K^\prime$-band data after thorough investigation of the results of a quadratically, linearly or non-varying PSF. It turns out that, due to a lack of isolated stars for the determination of the PSF variation across the field, the mean uncertainty is lower and the number of outliers with unacceptably large uncertainties reduced when a non-variable PSF is used. Thus, the 5 most isolated stars on the $K^\prime$ image, which were well spread out over the field, were used to derive the median averaged PSF of the long exposure. In the case of the short exposure, where due to the very short integration time faint stars are indistinguishable from the background, leaving more "uncrowded'' stars to derive the shape of the PSF, 7 isolated stars could be used. In $K^\prime$, the best fitting function was an elliptical Moffat-function with $\beta = 2.5$.

 

 

Table 1: Gemini/Hokupa'a and HST/NICMOS observations.

Date	Filter	single exp.	${n_{\exp}}$	exp. total	det. limit	$\sigma_{\rm back}$	resolution	diffraction limit
Gemini
05/07/2000	H	1 s	3	3 s	18.5 mag	7.74	0.17 $^{\prime \prime }$	0.05 $^{\prime \prime }$
05/07/2000	H	60 s	12	720 s	21 mag	0.19	0.20 $^{\prime \prime }$	0.05 $^{\prime \prime }$
30/07/2000	$K^\prime$	1 s	16	16 s	17.5 mag	3.73	0.12 $^{\prime \prime }$	0.07 $^{\prime \prime }$
09/07/2000	$K^\prime$	30 s	34	1020 s	20 mag	0.22	0.13 $^{\prime \prime }$	0.07 $^{\prime \prime }$
HST
14/09/1997	F160W			256 s	21 mag	0.04	0.18 $^{\prime \prime }$	0.17 $^{\prime \prime }$
14/09/1997	F205W			256 s	20 mag	0.15	0.22 $^{\prime \prime }$	0.21 $^{\prime \prime }$

Due to the lower detection rate on the H-band image crowding is less severe, while the PSF exhibits more pronounced spatial variations than in $K^\prime$. We thus used the quadratically variable option of the DAOPHOT psf and allstar tasks for our H-band images, with 27 stars to determine a median averaged PSF function and residuals. The best fitting function was a Lorentz function on the binned H-band image. In both filters, the average FWHM of the data has been used as the PSF fitting radius, i.e., the kernel of the best-fitting PSF function, to derive PSF magnitudes of the stars.

The short exposures have been used to obtain photometry of the brightest stars, which are saturated in the long exposures. The photometry of the long and short exposures agreed well after atmospheric extinction correction in the form of a constant offset had been applied. The saturation limit was 13.0 mag in H and 13.3 mag in $K^\prime$. At fainter magnitudes, the photometry of both images was indiscernible within the uncertainties for the bright stars, and the better quality long exposure values were used. Furthermore, a comparison of the bright star photometries was used to estimate photometric uncertainties (see Sect. 2.1.6 and Table 2).

Figure 2: Formal DAOPHOT photometric uncertainties. Top panel: Gemini/Hokupa'a H and $K^\prime$ photometry transformed to HST/NICMOS F160W and F205W filter magnitudes, m160 and m205. Bottom panel: HST/NICMOS m160, m205 photometry of the same $20\times 20\hbox {$^{\prime \prime }$ }$ field (shown in Fig. 1).

   
2.1.3 Photometric calibration

To transform instrumental into apparent magnitudes, we used the HST/NICMOS photometry of Figer et al. (1999) as local standards. The advantage of this procedure lies in the possibility to correct for remaining PSF deviations over the field, e.g., due to a change in the Strehl ratio with distance from the guide star or due to the increased background from bright star halos in the cluster center. Indeed, as will be discussed below, the spatial distribution of photometric residuals shows a mixture of these effects.

We were able to use approximately 380 stars to derive colour equations. The residuals obtained for these stars after calibration allow a detailed analysis and correction of field variations. The colour equations to transform Gemini instrumental H and $K^\prime$magnitudes to magnitudes in the HST/NICMOS filter system were determined using the IRAF PHOTCAL package, yielding:

$$m160 = H_{\rm inst} + 0.001 (\pm 0.017) \cdot (H - K^\prime)_{\rm inst} - 0.028 (\pm 0.031)\ {\rm mag}$$

$$m205 = K^\prime_{\rm inst} + 0.023 (\pm 0.008) \cdot (H - K^\prime)_{\rm inst} - 1.481 (\pm 0.016)\ {\rm mag},$$

where $H_{\rm inst}$ and $K^\prime_{\rm inst}$ are the Gemini instrumental magnitudes, and m160 and m205 correspond to magnitudes obtained with the NICMOS broadband filters F160W and F205W, respectively. After the transformation had been applied, it turned out that the residual magnitudes in the two independent fitting parameters $H - K^\prime$and $K^\prime$ still varied systematically over the field. As can be seen in Fig. 3, the variation is not a simple radial variation increasing with distance from the guide star (GS), but a mixture of displacement from the GS position (shown in the contour map in Fig. 3 as a cross), and the position with respect to the cluster center or bright stars in the field. Fortunately, the variation was well behaved in the Y-direction, and could be fitted by a fourth order polynomial. Close inspection showed that two areas on the QUIRC array showed a remaining photometric offset compared to the HST photometry. In the region 400 < X < 700 pixels and Y < 250 pixels, the magnitude was underestimated by 0.1 mag. For X > 850 pixels and Y > 850 pixels, i.e., the upper right corner, the $K^\prime$ magnitude was overestimated by 0.25 mag (however, note that there are only 7 stars in this region). For a discussion of these effects, see Sect. 2.1.4. We corrected for these offsets before deriving the Y-correction, which then showed remarkable homogeneity over the entire field. This smooth correction function is probably due to discrepancies in the dome flat field illumination versus sky exposures. The correction was then applied to transformed $K^\prime$ and $H - K^\prime$ magnitudes, denoted $K^\prime_{\rm trans}$ and  $(H-K^\prime)_{\rm trans}$ from now on. The $H_{\rm trans}$ magnitude was calculated from the corrected $K^\prime_{\rm trans}$ and $(H-K^\prime)_{\rm trans}$ values.

An additional advantage of this procedure is the independence on uncertainties in colour transformations at large reddening and non-main sequence colours, as opposed to colour transformations derived from typical main sequence standard stars. Using the HST photometry as local standards, we are naturally in equal colour and temperature regimes, allowing the direct comparison of the Gemini and HST photometry. For most parts of the paper, we remain in the HST/NICMOS system. We use the colour equations obtained in Brandner et al. (2001, hereafter BGB) to transform typical main sequence colours and theoretical isochrone magnitudes into the HST/NICMOS system where indicated. This allows us to transform mainly unreddened main-sequence stars, for which the BGB colour transformations have been established. The only exceptions are the two-colour diagram (Sect. 3.4) and the derivation of the extinction variation from colour gradients (Sect. 3.1), where the extinction law is needed to determine the reddening path. We will use the notation "m160'' and "m205'' as in FKM for magnitudes in the HST/NICMOS filters, and " $H_{\rm trans}$'' and " $K^\prime_{\rm trans}$'' for the Gemini/Hokupa'a data calibrated to the NICMOS photometric system. HST magnitudes transformed to the ground-based 2MASS system will be denoted by $(JHK_{\rm s})_{\rm 2MASS}$ or simply $JHK_{\rm s}$.

Figure 3: Map of residuals of NICMOS vs. Gemini photometry (orientation as in Fig. 1). Left panels: $m205 - K^\prime _{\rm trans}$, right panels: $m160 - H_{\rm trans}$. Stars have been binned into areas of $50 \times 50$ pixels, and the value displayed shows the average of all stars in each bin. Statistical fluctuations are large due to the varying number of stars in each pixel intervall, but the overall trends are clearly visible. The position of the guide star is marked by a cross, the cluster center as determined from the HST/NICMOS F205W image as an asterisk, and stars denote stars resolved in the Gemini images with magnitudes brighter than $K^\prime _{\rm trans}=13$ mag. The strong correlation between the position of bright stars and positive residuals reveals the tendency of PSF fitting photometry applied to AO data to overestimate the flux of bright sources, and underestimate the flux of faint sources in their vicinity (see text for discussion).

   
2.1.4 Discussion of the residual maps

The behaviour of the residual of the HST/NICMOS vs. Gemini magnitudes, $m_{\rm NICMOS} - m_{\rm Gemini,trans}$, can be analysed in more detail when studying the residual map and the smoothed contour plot. In Fig. 3, positive (negative) residuals correspond to overestimated (underestimated) flux. From the map we denote a general tendency to overestimate the flux. The contour maps show that positive residuals are correlated with the position of bright stars on the $K^\prime$ image, both in the crowded cluster center as well as in the area to the lower left, where a band of bright stars is located (see Fig. 1). This is the area where the $K^\prime$ magnitudes were found to be underestimated in Sect. 2.1.3, and thus the flux overestimated. This suggests that the increased background due to the uncompensated seeing halos of bright stars (cf. Sect. 2.1.5), inherent to AO observations, causes an overestimation of the flux of bright ( $K^\prime\la 13$ mag) sources. On the other hand, points with negative residuals are mainly correlated with fainter stars ($K \ga 16$ mag), suggesting that this enhanced background leads to an oversubtraction of the individually calculated background of nearby fainter stars. The result is an underestimate of the flux of companion stars in the vicinity of bright stars. The correlation of positive residuals with the position of bright stars seems to be less pronounced in the H-band image (Fig. 3). In H, the distance from the guide star is supposed to be more important due to the smaller size of the isoplanatic angle at shorter wavelengths and consequently more pronounced anisoplanatism. Indeed, the smoothed residual contour plot shows a symmetry in the residuals around the guide star, with close-to-zero residuals in the immediate vicinity of the guide star, where the best adaptive optics correction can be achieved. With increasing distance from the guide star, the residuals increase not only towards the bright cluster, but also to the west (left in Fig. 1) of the field, indicating that remaining distortion effects from the AO correction are mixed with the problem of the proximity to bright stars as seen in $K^\prime$.

   
2.1.5 Strehl ratio

The Strehl ratio (SR) is defined to be the ratio of the observed peak-to-total flux ratio to the peak-to-total flux ratio of a perfect diffraction limited optical system. This definition allows to compare the quality and photometric resolution of different optical systems using a single characteristic quantity.

$$SR = (F_{\rm peak}/F_{\rm total})_{\rm obs} / (F_{\rm peak}/F_{\rm total})_{\rm theo}$$

where $F_{\rm peak}$ is the maximum flux value of the PSF, and $F_{\rm total}$ is the total flux, including the uncompensated halos induced by the natural seeing. The labels obs and theo refer to the observed and the theoretically expected flux, respectively. Diffraction limited theoretical PSFs have been calculated using the imgen task of the ESO data analysis package eclipse (Devillard 1997). The Strehl ratio in the Gemini H-band 720 s exposure is found to be 2.5% compared to 95% in the HST F160W image, and 7% in $K^\prime$ (1020 s) compared to $\sim$90% in F205W. The low SRs measured in the Gemini science demonstration data indicate that more than 90% of the light of a star is distributed into the resolution pattern of the AO PSF and the halo around each star induced by the natural seeing. It has turned out that these halos cause a significant limitation to the resolution and depth of the observations in a crowded field, as faint stars can be lost to the enhanced background in the vicinity of bright objects. When comparing the HST and Gemini luminosity functions (Sect. 4), this effect causes the main difference between both data sets.

In the case of very low Strehl ratios, the SR does not directly indicate the fraction of the flux concentrated in the FWHM area of the PSF. A much larger fraction of the source flux can be used for PSF fitting in this case, although the spatial resolution is limited by the large FWHM as compared to diffraction limited observations (cf. Table 1). The ratio of the flux in the FWHM kernel of the compensated stellar image to the total flux,

$$FR_{\rm obs} = F_{\rm FWHM}/F_{\rm total},$$

may be determined by creating curves of growth for individual stars (Stetson 1990). The larger number of nearly isolated stars found on the H-band image (used for PSF creation) allowed a reliable determination of $FR_{\rm obs}$ only in H, although many stars on the H-band image were still too influenced by neighbours to study the aperture curve of growth in detail. We were able to create well-behaved curves of growth for 7 stars. As in the PSF fitting routine, we have used the FWHM of the PSF as kernel radius and as the reference flux for the flux ratio determination. These ratios range from 0.47 to 0.58, with an average of $0.53 \pm 0.05$, i.e., $\sim$50% of the integrated point source flux are used for PSF fitting. In addition, the variation of the flux ratio $FR_{\rm obs}$ over the field serves as an indicator of the Strehl ratio variation. If the AO correction mechanism is the dominant factor determining the concentration of the flux into the FWHM kernel, the Strehl ratio and thus $FR_{\rm obs}$ should decrease with distance form the guide star, as the seeing correction worsens. No correlation of the flux ratio with distance from the guide star is found. Though these are small number statistics, this supports the suggestion that the sensitivity variations over the field are not predominantly due to increasing distance from the guide star (Sect. 2.1.4).

   
2.1.6 Photometric uncertainties and incompleteness calculation

For the incompleteness correction, artificial frames were created with randomly positioned artificial stars. Magnitudes were also assigned automatically in a random way. Due to the very crowded field, only 40 stars were added to each artificial frame in order to avoid significant changes in the stellar density. A total of 100 frames was created for both the H and $K^\prime$ deep exposures, leading to a total of 4000 artificial stars used in the statistics. In addition to the individual incompleteness in each band, the loss of sources due to scatter of the main sequence generated by the more uncertain photometry in the dense parts of the cluster was estimated. For this purpose the artificial $K^\prime$ stars were assigned a formal instrumental "colour'' of $(H - K^\prime) = 0.33$ mag (corresponding to 1.745 mag after photometric transformation), derived from the average observed instrumental colour of the main sequence, and via this transformed into instrumental H magnitudes. Artificial stars were inserted at the same positions in the H and $K^\prime$frames. In this way the procedure also accounts for stars lost due to the matching of H and $K^\prime$ data. The artificial stars were calibrated using the colour equations shown in Sect. 2.1.3 , thus allowing us to estimate the loss of stars in the mass function derivation due to the applied main sequence colour selection (see Sect. 5). This resulted in significantly larger corrections as compared to the individual filter recoverage without matching and colour selection. As an example, the results for the mass function calculation performed on the artificial stars are displayed in Fig. 4.

As the recovery rate depends strongly on the stellar density and thus radial distance from the cluster center, the incompleteness correction will be determined in dependence of the radial bin analysed when radial variations in the MF are studied (Sect. 5.3).

Figure 4: Incompleteness tests performed on the Gemini H and $K^\prime$ data. The plot demonstrates the necessity to include the matching between single filter observations to obtain a realistic estimate on the incompleteness, when matching of objects is needed to create colour-magnitude- and two-colour-diagrams.

In addition to luminosity and mass function corrections, the artificial star tests were used to estimate the real photometric uncertainties by comparing inserted to recovered magnitudes of the artificial stars. The median difference between the original and the recovered magnitude of the artificial stars, $\Delta m_{\rm arti} =~<\!\!m_{\rm added} - m_{\rm DAOPHOT}\!\!>$, has been used as an estimate of the real photometric uncertainty. To obtain the median uncertainty, the intervalls $0 < \Delta m_{\rm arti} < 1$ and $-1 < \Delta m_{\rm arti} < 0$ have been treated individually, and the mean of the absolute value of both median values, weighted with the number of objects in each intervall, is quoted in Table 2. The overall flux deviation, including positive and negative deviations, is close to zero for stars brighter than 20 mag (<$\pm 0.004$), and for fainter stars becomes -0.17 and -0.13 in H and $K^\prime$, respectively, showing a tendency to underestimate the flux of faint stars. This tendency is more serious in the cluster center, where comparably large uncertainties are already observed for magnitudes fainter than 18 mag in both pathbands.

In a second test, photometry of the bright stars in the short exposures was compared to the magnitudes of the deep exposures, yielding $\Delta m_{\rm sl} =\ <\!\!m_{\rm short} - m_{\rm long}\!\!>$using the same procedure as for artificial stars. Note, however, that the quality of the short exposures is much worse than the one of the long exposures due to the high background noise. Therefore, the artificial star experiments, for which only the deep exposures have been used, yield a more realistic estimate of the photometric uncertainties. The results of both tests are summarised in Table 2.

The resulting uncertainty is roughly a factor of 2 to 3 larger than the theoretical magnitude uncertainty $\sigma_{\rm DAO}$ determined from detector characteristics by DAOPHOT. The photometry in the cluster center shows a larger uncertainty than the photometry in the outskirts. As expected in a crowding-limited field, this also implies a reduced detection probability of faint sources in the center of the cluster.

 

 

Table 2: Photometric uncertainties derived from artificial star experiments ( $\Delta m_{\rm arti}$) and the comparison of short and long exposures ( $\Delta m_{\rm sl}$). The photometric uncertainties determined by DAOPHOT are given for comparison. Note that for the brightest bin, 10-12 mag, only the photometry of the short exposures was available. The higher starting bin in H is due to the calibration procedure of the inserted artificial stars. Magnitudes are given in the NICMOS system (F160W, F205W).

Band	mag	$\Delta m_{\rm arti}$	$\Delta m_{\rm sl}$	$\sigma_{\rm DAO}$	$\Delta m_{\rm arti}$	$\Delta m_{\rm sl}$	$\sigma_{\rm DAO}$	$\Delta m_{\rm arti}$	$\Delta m_{\rm sl}$	$\sigma_{\rm DAO}$
		all		R < 10 $^{\prime \prime }$		R > 10 $^{\prime \prime }$
H	12-14	0.007	0.059	0.004	0.012	0.052	0.005	0.004	-	0.002
	14-16	0.015	0.061	0.008	0.025	0.065	0.010	0.011	0.035	0.004
	16-18	0.044	0.128	0.015	0.077	0.120	0.017	0.038	0.134	0.009
	18-20	0.119	-	0.036	0.167	-	0.039	0.098	-	0.030
	>20	0.264	-	0.142	0.514	-	0.142	0.265	-	0.144
$K^\prime$	10-12	0.004	-	0.005	0.004	-	0.005	0.003	-	0.006
	12-14	0.008	0.042	0.005	0.011	0.048	0.005	0.007	0.028	0.005
	14-16	0.027	0.072	0.007	0.041	0.086	0.007	0.020	0.042	0.006
	16-18	0.071	0.121	0.016	0.125	0.166	0.017	0.057	0.088	0.016
	18-20	0.206	-	0.060	0.280	-	0.073	0.189	-	0.056
	>20	0.411	-	0.274	0.520	-	-	0.360	-	0.274

   
2.2 HST/NICMOS data

HST/NICMOS observations have been obtained in the three broad-band filters F110W, F160W and F205W, roughly equivalent to J, H and K. The basic parameters are included in Table 1. For a detailed description of the HST data and their reduction see Figer et al. (1999).