Ground-based astrometry with wide field imagers. V. Application to near-infrared detectors: HAWK-I@VLT/ESO
M. Libralato, A. Bellini, L. R. Bedin, G. Piotto, I. Platais, M. Kissler-Patig, A. P. Milone
aa r X i v : . [ a s t r o - ph . I M ] J a n Astronomy&Astrophysicsmanuscript no. ms c (cid:13)
ESO 2018October 11, 2018
Ground-based astrometry with wide field imagers
V. Application to near-infrared detectors: HAWK-I@VLT/ESO ⋆ M. Libralato , , ,⋆⋆ , A. Bellini , L. R. Bedin , G. Piotto , , I. Platais , M. Kissler-Patig , , A. P. Milone Dipartimento di Fisica e Astronomia, Universit`a di Padova, Vicolo dell’Osservatorio 3, Padova, I-35122, Italye-mail: [email protected];[email protected] Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD-21218, USAe-mail: [email protected] INAF-Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, Padova, I-35122, Italye-mail: [email protected] Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD-21218, USAe-mail: [email protected] Gemini Observatory, N. Aohoku Place 670, Hilo, Hawaii, 96720, USAe-mail: [email protected] European Southern Observatory, Karl-Schwarzschild-Str. 2, Garching b. M¨unchen, D-85748, Germany Research School of Astronomy and Astrophysics, The Australian National University, Cotter Road, Weston, ACT, 2611, Australiae-mail: [email protected]
Received 11 June 2013 / Accepted 18 December 2013
ABSTRACT
High-precision astrometry requires accurate point-spread function modeling and accurate geometric-distortion corrections. This paperdemonstrates that it is possible to achieve both requirements with data collected at the high acuity wide-field K -band imager (HAWK-I), a wide-field imager installed at the Nasmyth focus of UT4 / VLT ESO 8 m telescope. Our final astrometric precision reaches ∼ ffi cacy of our approach, we combined archival material takenwith the optical wide-field imager at the MPI / ESO 2.2 m with HAWK-I observations. We showed that we are able to achieve anexcellent separation between cluster members and field objects for NGC 6656 and NGC 6121 with a time base-line of about 8 years.Using both
HST and HAWK-I data, we also study the radial distribution of the SGB populations in NGC 6656 and conclude that theradial trend is flat within our uncertainty. We also provide membership probabilities for most of the stars in NGC 6656 and NGC 6121catalogs and estimate membership for the published variable stars in these two fields.
Key words.
Instrument: Infrared Detectors – Techniques: Geometric Distortion Correction – Astrometry – Photometry – GlobularCluster: NGC 104, NGC 6121, NGC 6388, NGC 6656 – Galaxy: Bulge – NGC 6822 – LMC
1. Introduction
Multiple fields within astronomy are driving the executionof larger and yet larger surveys of the sky. Over the last twodecades, this scientific need has stimulated the constructionof instruments equipped with mosaics of large-format digi-tal detectors for wide-field imaging at both the optical andnear-infrared (NIR) wavelengths. The most recent generationof these wide-field imagers now competes with the oldertechnology of Schmidt telescope and photographic plates interms of number of resolution elements on sky but does so withorder-of-magnitude greater sensitivity and e ffi ciency.A list of some widely-used wide-field imagers was given byAnderson et al. (2006, hereafter Paper I). Since then, however,many wide-field imagers have been upgraded or decommis-sioned, and additional new wide-field imagers have begun theiroperations. In the top-half of Table 1, we provide a brief list of ⋆ Based on observations with the 8 m VLT ESO telescope. ⋆⋆ Visiting Ph.D. Student at STScI under the 2013 DDRF program. the major operative wide-field imagers on 3 m + telescopes (wealso included the [email protected] m MPI / ESO as reference).In addition, two wide-field imagers mounted on 1 m tele-scopes should be mentioned. The LaSilla-QUEST Variabilitysurvey is a project that uses the ESO 1.0-m Schmidt Telescope atthe La Silla Observatory of the European Southern Observatoryin Chile with the new large area QUEST camera. It is a mosaicof 112 600 × ◦ . × ◦ .
6. The camera, commissioned in early 2009 hasbeen built at the Yale and Indiana University. La Silla-QUESTsurvey is expected to cover about 1000 square degrees per nightrepeated with a 2-day cadence (Hadjiyska et al. 2012).The Panoramic Survey Telescope and Rapid ResponseSystem (Pan-STARRS) also is of great interest for the astro-nomical community. The Pan-STARRS survey will cover thesky using wide-field facilities and provide astrometric and pho-tometric data for all observed objects. The first Pan-STARRStelescope, PS1, is located at the summit of Haleakala on Maui,Hawaii and began full time science observations on May 13,
Table 1.
List of the major operative wide-field imagers on 3 m + telescopes. The [email protected] m MPI / ESO has been included asreference.
Name Telescope Detectors Pixel Scale [ ′′ / pixel] FoVOPTICAL REGIME WFI 2.2 m MPI / ESO 8 × (2048 × ′ × ′ Prime Focus Camera William Herschel Telescope 2 × (2048 × ′ . × ′ . × (2048 × ′ × ′ Suprime-Cam Subaru Telescope 10 × (2048 × ′ × ′ MOSA KPNO Mayall 4 m 8 × (2048 × ′ × ′ LAICA Calar Alto 3.5 m Telescope 4 × (4096 × ′ . × ′ . × (2048 × ′ . × ′ . × (2048 × ′ × ′ DECam CTIO Blanco 4 m 62 × (2048 × + × (2048 × ′ × ′ NIR REGIME
GSAOI Gemini 4 × (2048 × ′ . × ′ . × (2048 × ′ . × ′ . × (2048 × ′ . × ′ . × (2048 × ′ . × ′ . × (2048 × ′ . × ′ . × (2048 × ′ . × ′ . × (2048 × ′ . × ′ . × (2048 × ′ . × ′ . × (2048 × ′ . × ′ . ∼ (Large Synoptic Survey Telescope) represents the most signifi-cant step forward for wide-field imagers in modern astrophysics.It will be a 8.4 m wide-field ground-based telescope with a FoVof about 9.6 square degrees. With its 189 4k ×
4k CCDs, it willobserve over 20 000 square degrees of the southern sky in sixoptical bands. Construction operations should begin in 2014;the survey will be taken in 2021.While a great number of papers have presented photometryobtained with these facilities over the last decade, their astro-metric potential has remained largely unexploited. Our team iscommitted in pushing the astrometric capabilities of wide-fieldimagers to their limits. Therefore, we have begun to publishin this Journal a series of papers on
Ground-based astrometrywith wide field imagers . In
Paper I , we developed and appliedour tools to data collected with the [email protected] m MPI / ESOtelescope. The techniques used in
Paper II (Yadav et al. 2008)and
Paper III (Bellini et al. 2009) produced astro-photometriccatalogs and proper motions of the open cluster M 67 and of theglobular cluster NGC 5139, respectively. In
Paper IV (Bellini &Bedin 2010), we applied the technique to the wide-field cameraon the blue focus of the LBC@LBT 2 × ×
2k arrays are now mounted at the foci of varioustelescopes (bottom-half of Table 1).These wide-field imagers enable wide surveys, such as theVISTA variables in the Via Lactea (VVV, Minniti et al 2010). . The VVV is monitoring the Bulge and the Disk of the Galaxy.The survey will map 562 square degrees over 5 years (2010-2014) and give NIR photometry in Z , Y , J , H , and K S bands.The first data set of the VVV project has already been releasedto the community (Saito et al. 2012). It contains 348 individualpointings of the Bulge and the Disk, taken in 2010 with ∼ stars observed in all filters. Typically, the declared astrometricprecisions vary from ∼
25 mas for a star with K S =
15, to ∼ K S =
18 mag.Ground-based telescopes are not alone in focusing theirattention on this kind of detector. The James Webb SpaceTelescope (JWST) will be a 6.5-meter space telescope opti-mized for the infrared regime. It will orbit around the Earth’ssecond Lagrange point (L2), and it will provide imaging andspectroscopic data. The wide-field imager, NIRCam, will bemade up by a short- (0.6 – 2.3 µ m) and a long-wavelength (2.4– 5.0 µ m) channel with a FoV of 2 ′ . × ′ . Fig. 1.
HAWK-I layout. The labels give the dimensions in arcsecand arcmin (and in pixels) of the four detectors and of the gaps.Numbers in square brackets label the chip denomination used inthis work (note that the choice is di ff erent from that of Fig. 9 ofKissler-Patig et al. 2008). In each chip, we indicate the coordi-nate of the chip center. This is also the reference position thatwe used while computing the polynomial correction describedin Sect. 5.1. The black cross in the middle shows the center ofthe field of view in a single exposure that we used in Fig. 2.respectively. Section 9 shows some possible applications of ourcalibrations. Finally, we describe the catalogs that we releasewith this paper in Sect. 10.
2. HAWK-I@VLT
An exhaustive description of HAWK-I is given in Kissler-Patiget al. (2008). Here, we only provide a brief summary.The HAWK-I focal plane is equipped with a mosaic of four2048 × − , resulting in a total FoV of about7 ′ . × ′ . ∼ ′′ between the detectors). A sketchedoutline of the HAWK-I FoV layout is shown in Fig. 1. The detec-tors and the filter wheel unit are connected to the second stage ofthe Closed Cycle Cooler and operated at a temperature close to75–80 K. The remaining parts of the instrument are cooled to atemperature below 140 K. The acquisition system is based on theIRACE system (Infrared Array Control Electronics) developedat ESO. HAWK-I also is designed to work with a ground-layeradaptive optics module (GRAAL) as part of the Adaptive OpticsFacility (Arsenault et al. 2006) for the VLT (scheduled to beinstalled in the second half of 2014). HAWK-I broad band filtersfollow the Mauna Kea Observatory specification. Fig. 2.
Outline of the relative positions of pointings in ouradopted dither-pattern strategy. The 25 images are organized ina 5 × ff er-ent locations of the detectors as possible. This enables us to self-calibrate the geometric distortion. The zoom-in in the blue panelshows an example of the adopted dither between two pointings.As described in Table 2, the shift step can change from field tofield.
3. Observations
In Table 2 we provide a detailed list of the observations.All of the HAWK-I images used here were collectedduring the instrument commissioning, when several fieldswere observed with the aim of determining an average opticalgeometric-distortion solution for HAWK-I and for monitoringits stability in the short- and mid-term.To this end, the fields were observed with an observing strat-egy that would enable a self-calibration of the distortion. Briefly,the strategy consists of observing a given patch of sky in asmany di ff erent parts of the detectors as possible. Each observingblock (OB) is organized in a run of 25 consecutive images. Theexposure time for each image was the integration time (DIT ins) times the number of individual integrations (NDIT). Figure 2shows the outline of the adopted dither-pattern strategy .Important by-products of this e ff ort are astrometric standardfields (i.e., catalogs of distortion-free positions of stars), whichin principle could be pointed by HAWK-I anytime in the futureto e ffi ciently assess whether the distortion has varied and byhow much. Furthermore, these astrometric standard fields mightserve to calibrate the geometric distortions of many othercameras on other telescopes (including those equipped with AO,MCAO, or those space-based). However, the utility of our fieldsdeteriorate over time since a proper motion estimate for stars in Note that this strategy had been modified for some fields. We spec-ify these changes, when necessary, in the following subsections. 3ibralato, M. et al.: Ground-based astrometry with wide field imagers. V.
Table 2.
List of the HAWK-I@VLT data set used for this work. N dither is the number of dithered images per observing block.“Step” is the dither spacing (shift in arcsec from one exposure to the next one). The integration time (DIT) times the number ofindividual integrations (NDIT) gives the exposure time. σ (Radial residual) gives an assessment of the astrometric accuracy reached(see Sect. 5.5 for the full description). Filter N dither
Step Exposure Time Image-quality Airmass σ (Radial residual) (arcsec) (NDIT × DIT) (arcsec) (sec z ) (mas) Commissioning 1 , August 3-6, 2007
Bulge — Baade’s Window ( J
25 95 (6 ×
10 s) 0 . .
07 1.038-1.085 4.5 H
25 95 (6 ×
10 s) 0 . .
76 1.081-1.149 4.3 K S
25 95 (6 ×
10 s) 0 . .
45 1.042-1.091 2.8
Bulge — Baade’s Window (Rotated by 135 ◦ ) ( K S
25 95 (6 ×
10 s) 0 . .
85 1.015-1.044 5.6
NGC 6121 (M 4) J ×
25 95 (6 ×
10 s) 0 . .
04 1.010-1.540 6.5 K S ×
10 s) 0 . .
51 1.050-1.056 3.8
NGC 6822 J ×
10 s) 0 . .
83 1.028-1.049 5.3 K S ×
10 s) 0 . .
75 1.050-1.082 4.8
Commissioning 2 , October 14-19, 2007
NGC 6656 (M 22) K S
25 47.5 (6 ×
10 s) 0 . .
41 1.252-1.420 3.1
NGC 6388 J
25 95 (6 ×
10 s) 0 . .
94 1.287-1.428 9.7 K S
25 95 (6 ×
10 s) 0 . .
75 1.436-1.637 12.2
JWST calibration field (LMC) J
25 95 (6 ×
10 s) 0 . .
65 1.408-1.412 5.3 K S
24 95 (6 ×
10 s) 0 . .
60 1.411-1.429 4.8
Commissioning 3 , November 28-30, 2007
NGC 104 (47 Tuc) J
25 47.5 (6 ×
10 s) 0 . .
82 1.475-1.479 7.1 K S
23 47.5 (6 ×
10 s) 0 . .
01 1.475-1.483 15.0 our catalogs is only provided for those stars that are in commonwith UCAC 4 catalog. In this paper, we make these astrometricstandard fields available to the community.
The first selected astrometric field is located in Baade’sWindow. Our field is centered on coordinates ( α, δ ) J2000 . ∼ (18 h m s . , − ◦ ′ ′′ . K S exposure are typically separated by afew arcseconds, so that there are many of them in each field.In general, however, they are su ffi ciently isolated to allow us tocompute accurate positions.The choice of a cumulative integration time of 60 s wasdriven by two considerations. First, we wanted to have theupper main sequence in a CMD of all chosen targets to beoptimally exposed with low-luminosity RGB stars still below the saturation threshold. Second, the large-scale semi-periodicand correlated atmospheric noise (with estimated scale length at ∼ ′ ) noted by Platais et al. (2002, 2006) essentially disappearsat exposures exceeding 30 s. Thus, 60 s was a good compromiseof integration time. According to the formula developed byLindegren (1980) and Han (1989), a standard deviation due toatmospheric noise on the order of 15 mas is expected over theangular extent of HAWK-I FoV. This certainly is an upper limitof the actual standard deviation because the seeing conditionsof our NIR observations were 2-3 times better than those con-sidered by the aforementioned authors for visual wavelengths.The images were taken close to the zenith in an e ff ort tominimize di ff erential refraction e ff ects, which plague ground-based images (and consequently a ff ect the estimate of thelow-order terms of the distortion solution).The Baade’s Window field is the main field we use to derivethe geometric-distortion solution that is tested for stability–or refined– with the other fields. In Sect. 5, we derive the Fig. 3. ( From Left to Right ): The
First Panel is a depth-of-coverage map of 25 K S HAWK-I’s images collected in Baade’s Windowduring the run of August 3-6, 2007. The gray-scale goes linearly from 1 to 25. The green box is the 7 ′ . × ′ . Second Panel shows the resulting stack of the 25 images. The dark spot on thetop-right is the signature left by the “shadow” of the probe, which pick-up the star used for the simultaneous Active Optic correctionof the VLT / UT4’s primary mirror. On the bottom-left, there is the globular cluster NGC 6522. Note that neither the dark spotnor NGC 6522 are inside the region enclosed by the green box. The
Third Panel focuses on the green region and shows that thedistribution of stars in this field is remarkably homogeneous. The
Fourth Panel is a zoom-in of a representative sub-set of the field(indicated by the 10 ′′ × ′′ red box in all panels), which is able to show a better resolved image.geometric-distortion solution in this field for each of the threeavailable filters, J , H and K S , using 25 images dithered with astep of about 95 ′′ . In addition to this, we also collected 25 K S images of the same field but with the de-rotator at a positionof 135 ◦ clockwise. We used this field to perform a check of thedistortion with di ff erent angles (see Sect. 5.8).In Fig. 3, we show a summary of one of these observingruns in filter K S from left to right: the overlap of the di ff erentpointings, the stacked image, a zoom-in of the region actuallyused to calibrate the geometric distortion (the region highlightedin green), and a further zoom-in at a resolution able to revealindividual pixels (region indicated in red in the other panels). The tangential internal motions of Bulge stars is on average100 km s − , and assuming an average distance of 8 kpc, thisyields a proper motion dispersion of ∼ − (see, forexample, Bedin et al. 2003). In just a few years, proper motionsthis large can mask out systematic distortion trends that haveamplitudes below the 3-mas-yr − level (such as those discussedin Sect. 5.2). It is therefore important in some applications tohave more stable astrometric fields.For this reason, we also observed four globular clusters.Stars gravitationally bound in globular clusters have an internalvelocity dispersion .
20 km s − in their cores and are evensmaller in their outskirts. Although the systemic motion of starclusters is usually di ff erent to (and larger than) the Galactic fielddispersion, their common rest-frame motions are generally morethan 10 times smaller than the internal motions of Bulge stars,so clusters members can be expected to serve as astrometricstandards with much smaller internal proper motions. The second field was centered on the globular clusterNGC 6656 (M 22). At a distance of about 3.2 kpc, M 22( α, δ ) J2000 . = (18 h m s . , − ◦ ′ ′′ .
1, Harris 1996, 2010edition) is one of the closest globular clusters to the Sun. These data were impacted by an internal reflection of theMoon in the optics, causing an abnormally-high sky value onthe rightmost 300 pixels of the detector. In spite of this, theexquisite image quality of these data makes them among thebest in our database. We used this field to test the solution of thegeometric distortion (see Sect. 5.7 for detail).
The third field is centered on globular cluster NGC 6121 (M 4),( α, δ ) J2000 . = (16 h m s . , − ◦ ′ ′′ .
7, Harris 1996, 2010edition). It is the closest globular cluster to the Sun, and its richstar field has a small angular distance from the Galactic Bulge.The observing strategy for the J -filter is similar to that de-scribed before. Each OB is organized in a run of 25 consecutiveexposures and the the same block was repeated four times infour di ff erent nights, shifting the grid by few arcsec each time.This field was also observed in the K S -filter but with a ditherpattern completely di ff erent from the others. There are only fiveexposures dithered with steps of 100 ′′ , which are taken with thepurpose of estimating stars’ color. NGC 6388 is a globular cluster located in the Galactic Bulge at( α, δ ) J2000 . = (17 h m s . , − o ′ ′′ .
8) (Harris 1996, 2010edition). Some exposures of this field show the same dark spotdue to the probe as in the Baade’s window (see Fig. 3).
The last globular cluster observed during the HAWK-Icommissioning is NGC 104 (47 Tuc), ( α, δ ) J2000 . = (00 h m s . , − ◦ ′ ′′ .
6) (Harris 1996, 2010 edition).Two of the 25 pointings of the K S -filter data were not usable. Extra-galactic fields are more stable than Galactic fields, sincetheir internal proper motions are negligible compared to fore- ground stars, even with a 10-yr time baseline. The downside ofsuch extra-galactic fields is the need to increase the integrationtime to compensate for the faintness of the targets.
The first extra-galactic field is centered on the Local Groupdwarf irregular galaxy NGC 6822 at a distance of ∼
500 kpc(Madore et al. 2009).For this galaxy, we took fewer pointings (9 in a 3 × =
12 andDIT =
10 s.
In 2005, a field near the center of the Large Magellanic Cloud(LMC) was selected as reference field to solve for the geometricdistortion and to eventually help calibrate the relative positionsof JWST’s instruments in the focal plane. This field is in theJWST continuous viewing zone and it can be observed when-ever necessary. In 2006, it was observed with the AdvancedCamera for Surveys (ACS) Wide Field Channel (WFC) to createa reference catalog in F606W.The JWST calibration field is centered at ( α, δ ) J2000 . = (5 h m s . , − o ′ ′′ . × K S -filter data set was not usable.
4. PSF-modeling, fluxes and positioning
In our reductions, we used the custom-made software tools. It isessentially the same software used in the previous papers of thisseries. We started from a raw multi-extension FITS image. Eachmulti-extension FITS image stores all four chips in a datacube.We kept this FITS format up to the sky-subtraction phase.First, we performed a standard flat-field correction on allthe images. In the master flat fields, we built a bad-pixel maskby flagging all the outliers respect to the average counts. Weused the bad-pixel-mask table to flag warm / cold / dead pixelsin each exposure. Cosmic rays were corrected by taking theaverage value of the surrounding pixels if they were not insidea star’s region ; bad columns were replaced by the averagebetween the previous and following columns.Digital saturation in our images starts at 32 768 counts.To be safe, we adopted a saturation limit of 30 000 countsto minimize deviations from linearity close to the saturationregime (accordingly to Kissler-Patig et al. 2008) and flat-fielde ff ects. Each pixel for which the counts exceed the saturationlimit was flagged and not used.Finally, we subtracted the sky from the images, computingthe median sky value in a 10 ×
10 grid and then subtracting thesky according to the table (bi-linear interpolation was used The correction was performed using a single master flat field foreach of the three filters. We did not use a flat field tailored to each epochbecause some of them were not collected. Cosmic rays close to the star’s center increase the apparent fluxand shift the center of the star, resulting in a large
QFIT value (seeAppendix A for detail). to compute the sky value in a given location). After the skysubtraction, we split each multi-extension FITS file in fourdi ff erent FITS files, one per chip. The next step was to computethe PSF models.HAWK-I’s PSF is always well sampled, even in the best-seeing condition. To compute PSF models, we developed thesoftware img2psf HAWKI in which our PSF models are com-pletely empirical. This is derived from the [email protected] reductionpackage (Paper I). They are represented by an array of 201 × P i , j in the vicin-ity of a star of total flux z ∗ that is located at position ( x ∗ , y ∗ ) is: P i , j = z ∗ · ψ ( i − x ∗ , i − y ∗ ) + s ∗ , where ψ ( ∆ x , ∆ y ) is the instrumental PSF, or specifically, thefraction of light (per unit pixel area) that falls on the detector ata point o ff set ( ∆ x , ∆ y ) = ( i − x ∗ , j − y ∗ ) from the star’s center,and s ∗ is the local sky background value. For each star, wehave an array of pixels that we can fit to solve for the tripletof parameters: x ∗ , y ∗ , and z ∗ . The local sky s ∗ is calculated asthe 2.5 σ -clipped median of the counts in the annulus between16 and 20 pixels from the location where the star’s center falls.The previous equation can be inverted (with an estimate of theposition and flux for a star) to solve for the PSF: ψ ( ∆ x , ∆ y ) = P i , j − s ∗ z ∗ . This equation uses each pixel in a star’s image to providean estimate of the 2-dimensional PSF at the location of thatpixel, ( ∆ x , ∆ y ). By combining the array of sampling from manystars, we can construct a reliable PSF model. As opposed to thepioneering work of Stetson and his DAOPHOT code (Stetson1987) that combines an empiric and semi-analytic PSF model,we created a fully-empirical PSF model, as described in Paper I.The software img2psf HAWKI iterates to improve boththe PSF model and stellar parameters. The starting point isgiven by simple centroid positions and aperture-based fluxes. Adescription of the software is given in detail in Paper I.To model the PSFs in both the core and the wings, we useonly stars with a high S / N (signal-to-noise ratio). This is doneby creating a list of stars that have a flux of at least 5000 countsabove the local sky (i.e., S / N > × magnitudes, and another quantity called quality-of-PSF-fit( QFIT , which represents the fractional error in the PSF-model fitto the star). For each pixel of a star within the fitting radius (2.5pixels), the
QFIT is defined as the sum of the absolute value ofthe di ff erence between the pixel values P i , j (sky subtracted) andwhat the local PSF model predicts at that location ψ ( i − x ∗ , j − y ∗ ),normalized with respect to the sky-subtracted P i , j : QFIT = X i , j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (P i , j − sky) − z ∗ · ψ ( i − x ∗ , j − y ∗ )P i , j − sky (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where ( x ∗ , y ∗ ) is the star’s center. The QFIT is close to zero forwell-measured stars and close to unity for ones that are badly-measured (or not star-like). Typically we found
QFIT . .
05 forwell-measured stars in our images. Saturated stars are also mea-sured in our pipeline. For these, stars we only fitted the PSF onthe wings of the stars using unsaturated pixels. In this way, weare able to measure a flux and a position for saturated stars, evenif they are less accurate (high
QFIT ) than for unsaturated stars.
5. Geometric distortion correction
In this section, we present a geometric-distortion solution forthe HAWK-I in three broad band filters ( J , H , and K S ) derivedusing exposures of the Baade’s Window field. No astrometricreference data is available for the Baade’s Window field, so weiteratively constructed our own.Adopting the observing strategy described in Sect. 3, thesystematic errors in the measure of stars’ positions from oneexposure to the other have a random amplitude and the stars’averaged positions provide a better approximation of their truepositions in the distortion-free master frame.To build the master frame, we cross-identified the starcatalogs from each individual HAWK-I chip. Conformaltransformations (four-parameter linear transformations, whichinclude rigid shifts in the two coordinates, one rotation, andone change of scale, so the shape is preserved) were used tobring the stars’ positions, as measured in each image, intothe reference system of the master frame. We considered onlywell-measured, unsaturated objects with a stellar profile andmeasured in at least three di ff erent images.Our geometric-distortion solution for HAWK-I is madeup of five parts: (1) a linear transformation to put the fourchips into a convenient master frame (Hereafter, we refer to thetransformation from chip k of the coordinate system of image j to the master system T j , k .), (2) two fifth-order polynomials todeal with the general optical distortion (hereafter, the “P” cor-rection), (3) an analytic correction for a periodic feature alongthe x-axis, as related to the detector read-out amplifiers (the “S”correction), (4) a fine-tuning to correct second-order e ff ects onthe x-residuals of the S correction (the “FS” correction), and (5)a table of residuals that accounts for both chip-related anomaliesand a fine-structure introduced by the filter (the “TP” correction).The final correction is better than ∼ ∼ ff erentforms: a FORTRAN subroutine and a set of FITS files for eachfilter / chip / coordinate. Since focus, flexures, and general condi-tions of the optics and telescope instrumentation change duringthe observations (within the same night and even between con-secutive exposures), we derive an average distortion correction.Here, we describe our correction procedure for filter K S .The procedure for filters J and H is identical, and the results arepresented in the Sect. 5.6. We followed the method given in Anderson & King (2003) forthe Wide-Field Planetary Camera 2 (WFPC2). This methodwas subsequently used to derive the distortion correction forthe ACS High Resolution Channel (Anderson & King 2004)and for the Wide Field Camera 3 (WFC3) Ultraviolet-Visual(UVIS) channel (Bellini & Bedin 2009; Bellini, Anderson &Bedin 2011). The same strategy was also used by two of us tocalibrate the blue prime-focus camera at the LBT (Paper IV).We treated each chip independently, and we solved the 5 th -orderpolynomial that provided the most correction. We chose pixel(1024,1024) in each chip as a reference position and solved forthe distortion with respect to it.The polynomial correction is performed as follows: – In each of the list of unsaturated stars found in each chip ofeach exposure (4 ×
25 lists), we first selected stars with aninstrumental magnitude brighter than K S ≃ −
11 and with a
QFIT lower than 0.05, to ensure that the master list wouldbe free from poorly measured stars, which would harm thedistortion solution. – We computed the linear transformation ( T j , k ) between starsin each chip of each exposure and the current master frame. – Each star in the master frame was conformally transformedin the raw-coordinate system of each chip / image ( T − j , k ) andcross-identified with the closest source. Each such cross-identification generates a pair of positional residuals ( δ x , δ y ),which correspond to the di ff erence between the observed po-sition and the transformed reference-frame position. – These positional residuals were distilled into a look-up ta-ble made up of 12 ×
12 elements of 170.7 × – We performed a linear least-square fit of the average posi-tional residual of each of the 144 cells to obtain the coe ffi -cients for the two fifth-order polynomials in each chip (seePaper IV for a detailed description). – We applied this P correction to all stars’ positions. – Finally, we iterated the entire process, deriving a new andimproved combination of a master frame and distortion so-lution. The residuals improved with each iteration.The iterative process was halted when the polynomial coe ffi -cients from one iteration to the next di ff ered by less than 0.01%.The final P correction reduced the average distortion resid-uals (from the center of the detector to the corner) from ∼ ∼ . th percentileof the σ (Radial residual), see Sect. 5.5 for detail) improvesfrom ∼ ∼ ∼ ∼ Fig. 4. ( Left ): Residual trends for the four chips when we use uncorrected stars’ positions. The size of the residual vectors ismagnified by a factor of 500. For each chip, we also plot the single residual trends along X and Y axes. Units are expressed inHAWK-I raw pixel. (
Right ): Residuals after our polynomial correction is applied. The size of the residual vectors is now magnifiedby a factor 5000.
Our P correction reveals a high-frequency, smaller-amplitudee ff ect, which is a periodic pattern in the x-positional residualsas a function of the x positions in all HAWK-I chips. This e ff ectis clearly shown in the distortion map after the P correction isapplied ( δ x vs. X panels in Fig. 4). For every 128 columns, stars’positions have positive residuals (about 0.075 HAWK-I pixel) inthe first 64 pixels and negative residuals (about 0.045 HAWK-Ipixel) in the second 64 pixels (see panels (a) of Fig. 5). At firstglance, this residual pattern has the appearance of being causedby irregularities in the pixel grid of the detectors. However, a de-tailed analysis (see Sect. 6) leads us to conclude that it is insteada pattern caused by a “periodic lag” in the readout process, whichis o ff set in opposite directions in alternating 64 pixel sections ofthe detector addressed by each of the 32 read-out amplifiers.We adopted an iterative procedure to empirically correct forthis periodic pattern that shows up only along the x axis. Westarted with the master frame made by using catalogs correctedwith our P correction. We then transformed the position of eachstar ( i ) from the master frame back into the raw coordinatesystem of each chip ( k ) of each image ( j ). We determined thequantity: δ x i = x raw i − x P − (T − j , k ) i , j , where x raw i are the raw x-coordinates, and x P − (T − j , k ) i , j are thex-coordinates on the master frame transformed to the raw coor-dinate system and corrected with the inverse P correction. We as-sumed that the periodic trend had a constant amplitude across thedetector. Panels (a) in Fig 5 show δ x vs x raw for each chip (from1 to 4) and δ x vs. x raw modulus 128, in which we collect togetherall the residuals (panel 5) before applying the S correction. To model the trend in the residuals, we used a square-wavefunction (panel (5a) of Fig. 5). The amplitude of this functionis defined as the 3 σ -clipped median value of the residualsbetween pixels 2.8–62.2 and 66.8–126.2. To model the averageperiodicity between 62.2 ≤ x raw ≤ − pixel.Combining the two corrections (S + P, applied in this order tothe raw coordinates), we are able to reduce the 68 . th percentileof the σ (Radial residual) down to ∼ . In panels (b) of Fig. 5, we show the residual trend after the S + Pcorrection is applied. Looking at panels (1–4b), it is obviousthat the amplitude of the δ x periodicity pattern is not constantfrom chip to chip. In addition to this, there is still a polynomialresidual that needs to be removed. For this reason, we applied afine-tuned residual correction as follows.We first computed a master frame by applying the S + Pcorrection to the raw positions of each chip / exposure. We thendetermined the residuals as the di ff erence between the rawx-coordinates corrected with the S correction and x P − (T − j , k ) i , j . Next,we divided each chip into 32 bins of 64 pixels each along thex axis, computed the 3 σ -clipped average value of the residuals,and subtracted the 75% of it from the δ x residuals in each bin.Then, we iterated the procedure until the di ff erence between the Fig. 5. ( Top ): δ x as a function of X in units of HAWK-I pixels before S correction. ( Middle ): As above but after S correction.(
Bottom ): Same as above but after S and FS corrections. In the left panels (from 1 to 4), we took 32 bins of 64 pixels each andcomputed the median of residuals in each bin (red squares). In the right panels (5), we show the periodogram with a period of 128columns containing all the points plotted in the left panels. The red dashed lines show + .
05, 0, and − .
05 HAWK-I pixel.3 σ -clipped average value of the residuals in all bins of all chipsfrom one iteration to the next one was smaller than 10 − pixel.In panels (c-1) to (c-4) of Fig. 5, we show the residual trends forall chips after our S + FS + P corrections are applied. This approach was able to provide accuracies (68 . th percentile of the σ (Radial residual)) down to 0.035 pixel ( ∼ Fig. 6. ( Left ): Example of cell and grid-point locations on thebottom-left area of chip[1]. Dotted lines mark the 170.7 × ∗ ), we used the four surrounding closest grid point to performthe bi-linear interpolation (sketched with the arrows) and eval-uate the residual geometric distortion in that location of the de-tector. ( Right ): Bi-linear interpolation outline. Each grid pointP , . . . , P is weighted by the corresponding area A , . . . , A toassociate the correction in ∗ . The final step of our distortion-solution model consists offour look-up tables (one for each chip) to minimize all theremaining detectable systematic residuals that were left. Weconstrained the look-up tables using the same procedure thatBellini, Anderson & Bedin (2011) used to derive the distortioncorrection for the WFC3 / UVIS camera.First, we corrected all stars’ positions by applying the S,FS, and P corrections (in this order). We then built a new masterframe and computed the residuals, as described in Sect. 5.1. Wesubdivided again each chip into 12 ×
12 square elements. We usedthe stars’ residuals within each cell to compute a 3 σ -clipped me-dian positional residuals and assigned these values to the corre-sponding grid points (open circles in Fig 6). When a cell adjoinsdetector edges, the grid point is displaced to the edge of the cell,as shown. For the grid point on the edges, the value of the medianonly at the first iteration is computed at the center and shifted tothe edge. Then, we iteratively found the value that the grid-pointelement on the edge should have to remove the systematicerrors. We built a look-up table correction for any given locationof the chip, using a bi-linear interpolation among the surround-ing four grid points. Figure 6 shows an example of the geometryadopted for the look-up table and of the bi-linear interpolation.We corrected stars’ positions using only 75% of the recom-mended grid-point values, computed an improved master frame,and calculated new (generally smaller) residuals. We calculatednew grid-point values and added them to the previous values.The procedure was iterated until the bi-linear interpolationo ff ered negligible improvement of the positional residuals r.m.s.from one iteration to the next. In Fig. 7, we show the final HAWK-I distortion map after weapplied our full distortion solution (S + FS + TP + P). To have areliable assessment of the errors in the distortion correction,we computed the r.m.s. of the position residuals of each star ( i )observed in each chip ( k ) of the image ( j ), which have been dis-tortion corrected and conformally transformed into the master-frame reference system ( x T j , k i , j , k , y T j , k i , j , k ). The di ff erence betweenthese positions and the distortion-free positions ( X master i , Y master i )directly quantifies how close we are to reach the ideal distortion-free system. We defined the σ (Radial residual) as: σ (Radial residual) i , j = s ( x T j , k i , j , k − X master i ) + ( y T j , k i , j , k − Y master i ) . In Fig. 8, we show the size of these σ (Radial residual) versusinstrumental K S magnitude after each step of our solution. Totest the accuracy of the geometric-distortion solution, we onlyused unsaturated stars with an instrumental magnitude K S ≤− . QFIT ≤ σ -clipped 68 . th -percentile value of these residuals isshown on the right of each panel. The 3 σ clipping rule excludesoutliers, which can bias the percentile value. These outliers canhave di ff erent explanations. For example, most of the outliersfor the Bulge field are close to the edge of the FoV, where thedistortion solution is less constrained. In the case of NGC 6656,most of these outliers are close to the center of the cluster(crowding e ff ects) or are located in the region a ff ected by theinternal reflection of the Moon in the optics. Hereafter, we referin the text with σ perc to the 3 σ -clipped 68 . th -percentile valueof the σ (Radial residual). The distributions of the r.m.s. is verynon-Gaussian and the 68 . th -percentile is an arbitrary choice torepresent the errors. Although it is not absolutely correct mathe-matically, it gives a good indications of where an outlier will lie.In the bottom panel, we plot the σ (Radial residual) ob-tained using more-general 6-parameter linear transformationsto compute the master-frame average positions. These trans-formations also include other two terms that represent thedeviation from the orthogonality between the two axes and thechange of relative scale between the two axes (the shape is notpreserved anymore). When general linear transformations areapplied, most of the residuals introduced by variations in thetelescope + optics system and di ff erential atmospheric refractionare removed, and σ perc further reduces to 0.027 pixel ( ∼ . J and H filters Each HAWK-I filter constitutes a di ff erent optical element,which could slightly change the optical path and introducechanges in the distortion. To test the filter-dependency of our K S -based distortion solution, we corrected the positions mea-sured on each J - and H -filter images of Baade’s Window fieldwith our K s -filter-derived distortion solution and studied theresiduals. We found σ (Radial residual) significantly larger thanthose obtained for the K s -band images. We also tried to applythe K s -filter distortion solution plus an ad-hoc table of residual(TP correction) for each filter without significant improvements.For these reasons, we decided to independently solve for thedistortion for the J and H images. Fig. 7.
As Fig. 4 but after we have applied all the distortion corrections. The size of the residual vectors are now magnified by afactor 10 000.We built the J -filter master frame using only stars with aninstrumental magnitude brighter than J ≃ − . QFIT ≤ H -filter, we built the master frame usingonly stars with H ≃ − . QFIT ≤ J -and H -filter, respectively). The distortion corrections wereperformed as described in the previous sections.As shown in Table 2, the image quality of the J -filterBulge images changed dramatically during the night of theobservation, reaching 1.07 arcsec. We initially used all J -filterimages to compute the distortion correction and obtained a σ perc (using general transformations) of ∼ = σ (Radial residual) before andafter applying the distortion correction for the J and H images.The σ perc for well-measured unsaturated stars is shown on theright of each panel. Using general linear transformations, weobtained σ (Radial residual) of ∼ ∼ J and H filter, respectively.On the left panel in Fig. 10, we compare the residual trendsobtained by applying the K S -filter correction to J -filter Bulgeimages (blue vectors) to the residual trends obtained by applyingthe J -filter correction instead (red vectors). In the right panel, we Fig. 8. σ (Radial residual) versus instrumental K S magnitude af-ter each step of our solution. The red solid horizontal line showsthe σ perc ; the red dashed vertical line indicates the magnitudecut-o ff K S = − . H -filter case. A clear residualtrend (up to 0.1 pixel) of is present when a correction made fora di ff erent filter is applied to a given data set of images. The op-tical system performances are di ff erent at di ff erent wavelengths,so it is not surprising that the K S solution is not completelysuitable for the J - and H -filter data. The filter also introduces anadditional optical element that leads to a di ff erent distortion onthe focal plane. Both σ (Radial residual) and distortion maps tellus that an auto-calibration for the distortion correction in eachfilter is required for high-precision astrometry. Di ff erent factors (e.g., the contribution given by light-pathdeviations caused by filters, alignment errors of the detector onthe focal plane) change the HAWK-I distortion over time. Toexplore the stability of our derived distortion solution over time,we observed the astrometric precision obtained by applying ourdistortion correction to images taken several months apart.We applied our distortion solution to images of NGC 6656(M 22) taken during the second commissioning. The σ perc (computed as described in Sect. 5.5) was found to be ∼ ∼ σ (Radial residual) after we applied the Bulge-baseddistortion correction (bottom) and the newly made NGC 6656-based correction (top). The di ff erence between these distortionsolutions is only 0.003 pixel. Therefore, our Bulge distortioncorrection should be stable at a 3-mas level on a 3-month Fig. 9.
In each half of the figure, we show the σ (Radial resid-ual) vs. instrumental magnitude before and after we applied ourdistortion correction for the J ( Top ) and H ( Bottom ). The redlines have the same meaning as in Fig. 8 but the red dashed ver-tical lines are set at J = − . Top ) and H = − . Bottom ),respectively.scale for general uses. In Fig. 12, we show the distortion-mapcomparison. There are systematic trends when the K S -filterBulge solution is applied to this data set. Nevertheless, for high-precision astrometry, we suggest auto-calibrating the distortioncorrection for each data set, as it is continuously evolving.We note that the positions of the nodes of the periodic trenddid not change over this trim baseline, adding support to ourconclusion that this periodic residual is linked to the detector’s properties and is not a function of the telescope, epoch, filter, orimage quality (see Sect. 6). As done in the previous section for the case of NGC 6656,we apply the distortion correction obtained by self-calibrationof Bulge images (hereafter, Bulge ff erent data set,which is collected for the same field, but with the de-rotator ata di ff erent position angle at ∼ ◦ (hereafter, Bulge σ perc of ∼ σ perc ∼ σ perc of ∼ + telescope conditions). The distortionmaps obtained applying the two solutions to the same data set(Fig. 13) highlight di ff erent trends (even if the residuals arelower than 0.05 pixel), in the upper-right corner of chip[4],which recommends again the auto-calibration for the distortionsolution of each data set for high-accuracy astrometry.Nevertheless, even if rotated Bulge σ perc ∼ σ perc tells us how accurately we Fig. 10. ( Left ): J -filter distortion map comparison. In blue, we plot the vectors when the K S -filter correction is applied; in red, weshow when the J -filter correction is used. We plot the single residual trends along X and Y axes with the same colors, but we donot plot the single stars to not create confusion in the plot. ( Right ): The same but for the H filter. The size of the residual vectors ismagnified by a factor 6000. Fig. 11.
Comparison of NGC 6656 σ (Radial residual) after ap-plication of the Bulge-based ( Bottom ) and the NGC 6656-based(
Top ) correction. The red solid horizontal line shows the 3 σ -clipped value of the σ (Radial residual); the red dashed verticalline indicates the magnitude limit K S = − . relative position of a star among dif-ferent dithered images. However, these are internal estimates ofthe error, and do not account for all of the sources of systematicerrors. For a better estimate of the uncertainty on the relative po-sition of stars, we compared the two calibrated master frames ofBulge ff erential atmospheric refractions, etc. For this reason,we transformed the two master frames of Bulge th percentile of the ∆ X distribution is about10.7 mas, while that of ∆ Y is about 9.4 mas. Thus, the non-linearterms of our distortion solution can be transferred between ob-serving runs at the 10 mas level.
Fig. 12.
As in Fig. 10 but for the NGC 6656 case. In blue, weplot the vectors when the Bulge correction is applied, and in red,we show when the NGC 6656-made solution is applied. The sizeof the residual vectors is magnified by a factor 6000.
We release FORTRAN routines to correct the geometric dis-tortion, using the solution computed for the Bulge (Sect. 5.1, 5.2, 5.3 and 5.4). There are three di ff erent routines http://vizier.u-strasbg.fr/viz-bin/VizieR . 13ibralato, M. et al.: Ground-based astrometry with wide field imagers. V. Fig. 13.
As in Fig. 10 but for the Bulge
Fig. 14. ( Top ): ∆ x vs. X between the two Bulge fields. We plottedonly bright unsaturated stars. ( Bottom ): as on
Top but with ∆ y vs.Y. The red solid line is set at 0 mas, while the dashed lines areset at ± ± J , H , K S ). They require x raw and y raw coordinates and the chip number. In output, the codesproduce x corr and x corr corrected coordinates. Both raw andcorrected coordinates are in the single-chip reference frame(1 ≤ x raw / corr , y raw / corr ≤ / chip / filter) to make the distortion solutions alsoavailable for other program languages. Bi-linear interpolationmust be used to compute the amount of the distortion correction Fig. 15.
Local flat ratio of K S chip[3] flat field. Dashed red linesmark the boundaries of the possible discontinuities. There arenot significant local flat ratio variations at the 1 st , 64 th , and 128 th columns (highlighted by red open circles).in inter-pixel locations. We refer to Appendix B for a briefdescription of these corrections. Furthermore, we release FITSimages (one per chip / filter) that could be used to correct thevariation in the pixel area across the field of view. These imagesare useful for improving the HAWK-I photometry.
6. A possible explanation of the periodicity
In Sect. 5.2, we have corrected for the periodic trend observedin the δ x positions. At first, this component might suggestthe presence of some irregularities in the pixel grid, due tomanufacturing defects, such as an imperfect alignment in theplacement of the lithographic stencils that established the pixelboundaries on the detector. Examples include, the well-known34 th -row error found by Anderson & King (1999) in the caseof the CCDs of the WFPC2 of the Hubble Space Telescope ( HST ), or, for a more recent example, the pattern observed onthe detectors of the
HST ’s WFC3 / UVIS channel (see Bellini,Anderson & Bedin 2011 for details). If the square wave that wesee here in the HAWK-I δ x residuals (from a-1 to a-4 panels ofFig. 5) is due to a geometric e ff ect, then the variation in pixelspacing would cause periodic features in the flat fields, sincewider pixels collect more light when the detector is illuminatedby a flat surface brightness. In this case, the observed δ x -residualtrend would imply that the 64 th and the 65 th pixels in each rowwould be physically smaller than the 128 th and the 129 th pixels.To verify this hypothesis, we computed the local flat ratioas described in Bellini, Anderson & Bedin (2011). We took theratio of the pixel values over the median of the 32-pixel valueson either side along X direction (independently for each of theamplifiers). We computed the median value of this ratio for allpixels within 400 < y raw < < x raw < K S -filter flat field is shown in Fig. 15 as example. The variation ofthe flat ratio of all chips in the vicinity of the columns 1, 64, and128 is lower than 0.25%, thus suggesting an uniform pixel grid.Another possible explanation of this e ff ect can be ascribedto the readout process of the Rockwell detectors. The RockwellHgCdTe detectors are designed to have three output modes. Itcan use 1, 4 or 32 amplifiers. The HAWK-I detector is set up touse all 32 amplifiers, and it takes 1.3 s to read out the entire chip. Fig. 16.
Example of the weeding process for the 47 Tuc catalog.(
Top ): Magnitude o ff set versus radial o ff set for sources foundaround bright stars. We plotted the spurious objects that havebeen rejected in red. ( Middle ): Y vs. X separation of the re-jected (left) and accepted (right) detections from the correspond-ing bright stars. (
Bottom ): On the left, we show all the objects(red circles) in a ∼ ×
100 pixels region in the 47 Tuc catalog,and on the right, we highlight the detections that have been ac-cepted (green circles). As described in the text, a faint star nearthe bright one pointed by the red arrow on the bottom has beenflagged even if it was not an artifact.In the 32-amp mode, the chip is divided into 32 64-pixel-widestrips, each fed into a di ff erent amplifier. The adopted operatingmode performs the read-out from left-to-right in even amplifiersand from right-to-left in odd amplifiers.An apparent shift of the stars’ position along the x axis mayhappen if there is a “periodic lag” during the read out, since theamplifier reads the pixels in sequence. This e ff ect is very similarto the “bias shift” observed in ACS / WFC of the
HST after partof the electronics, in particular the new amplifiers, has beenreplaced during service mission 4 (Golimowski et al. 2012). Asfor ACS, the readout electronics of HAWK-I’s detectors takea while to settle to a new value when the charge of anotherpixel is loaded. Without waiting an infinite amount of time tosettle down, there is some imprint left from the previous pixel.Although HAWK-I’s NIR-detectors are very di ff erent from theACS / WFC CCDs, a similar e ff ect, or, an inertia of dischargingthe capacitors to reset to a new value, could cause the observed periodicity. Furthermore, each amplifier in the 32-amp modereads 64 pixels, and 64 pixels is the observed periodicity of thee ff ect that we found. This suggests that the cause of the periodictrend we see in the distortion could be related to the amplifiers’setup.
7. Weeding out spurious objects
We applied our Bulge-based distortion correction, derived as inSect. 5, to the entire data set with the exceptions of the Bulge J ≥ − QFIT ≥ ff erence and the distance from the closest brightstar (e.g. J ≤ −
11 for the case of 47 Tuc) out to 15 pixels. Wethen plotted those magnitude di ff erences as a function of the ra-dial distance (top panel of Fig. 16). Di ff erent clumps show up onthe plot. We drew-by-hand a region around them that enclosesmost of these spurious objects (in red). In this way, we built amask (one for each filter / field) used to purge PSF artifacts.The selection we made is a compromise between missingfaint objects near bright stars and including artifacts in thecatalog. In the bottom panel of Fig. 16, the red circles show allthe detected objects in the 47 Tuc catalog. In the bottom rightpanel of Fig. 16, the green circles highlights the objects thathave been finally accepted as real stars. The bright star closeto the bottom of the figure has a faint neighbor (pointed by thearrow) that clearly is not an artifact but has been unfortunatelyflagged-out by our mask. These flagged stars represent onlya very small fraction with respect to the total number of PSFartifacts removed by our procedure.In each final catalog, we added a column for each filtercalled F weed . The flag F weed is equal to 0 for those objectsrejected by our mask. The only exception is the K S -filter catalogof Bulge
8. Photometric calibration
In this section, we provide two calibrations of the zero-points.The first calibration was performed using the 2MASS photo-metric system (Skrutskie et al. 2006) and the second calibrationusing the native system of the HAWK-I filters.
Table 3.
List of the HAWK-I filter zero-points, r.m.s., number of stars used, and zero-point formal uncertains ( σ/ √ N −
1) to which2MASS error of the stars used to calibrate should be added in quadrature. The values listed in columns from (3) to (6) are thoseobtained in the 2MASS system, and from (7) to (10) in the MKO system.
Field Filter Zero-point σ N σ/ √ N − Zero-point σ N σ/ √ N − Bulge — Baade’s Window ( J − .
31 0.07 995 0.01 − .
25 0.07 966 0.01 H − .
60 0.05 57 0.01 − .
57 0.05 37 0.01 K S − .
01 0.09 543 0.01 − .
98 0.09 543 0.01Bulge — Baade’s Window ( K S − .
77 0.09 543 0.01 − .
75 0.09 542 0.01NGC 6121 (M 4) J − .
76 0.06 298 0.01 − .
71 0.06 298 0.01 K S − .
69 0.09 86 0.01 − .
67 0.09 86 0.01NGC 6822 J − .
40 0.04 28 0.01 − .
36 0.04 28 0.01 K S − .
59 0.07 26 0.01 − .
57 0.07 26 0.01NGC 6656 (M 22) K S − .
95 0.15 83 0.02 − .
93 0.15 81 0.02NGC 6388 J − .
46 0.06 289 0.01 − .
41 0.06 286 0.01 K S − .
73 0.05 229 0.01 − .
72 0.05 229 0.01JWST calibration field (LMC) J − .
78 0.07 233 0.01 − .
74 0.07 226 0.01 K S − .
84 0.09 122 0.01 − .
82 0.09 123 0.01NGC 104 (47 Tuc) J − .
26 0.05 80 0.01 − .
22 0.05 80 0.01 K S − .
71 0.05 128 0.01 − .
69 0.05 128 0.01
Fig. 17.
Magnitude di ff erence between HAWK-I and 2MASS asfunction of the 2MASS magnitude. The black dots represent allthe stars matched between HAWK-I and 2MASS with good pho-tometry. Red crosses show stars from the saturation limit ( K S = − .
46) to two magnitudes fainter. The red solid line is the zero-point (median of the magnitude di ff erence of the red crosses); thedashed line are set at a zero-point ± σ (defined as the 68.27 th percentile of the distribution around the median). The label onthe top left corner gives the zero-point ± σ/ √ N −
1, where N isthe number of stars used to compute the zero-point. The first photometric calibration was performed using the2MASS catalog. Since 2MASS is a shallow survey, we only gota small overlap between unsaturated stars in HAWK-I imagesand 2MASS data covering a very narrow magnitude range nearthe faint limit of 2MASS. Therefore, we can apply a singlezero-point calibration only. We selected well-measured brightunsaturated stars within two magnitudes from saturation in ourcatalogs to calculate these photometric zero-points. In Fig. 17, we show the case of the Bulge K S -filter, we first registered the zero-point to thatof the Bulge σ ), the num-ber of stars used to compute the zero-points ( N ) and σ/ √ N − / field and not one for eachchip / filter / field. We registered all chips in the flat-fielding phaseto the common reference system of chip[1], and while buildingthe master frame, we iteratively registered the zero-point of allchips to that of the chip[1].As clearly visible in Fig. 17, this calibration is not perfect,but there is a more conceptual limitation of this calibration. Thefilters of HAWK-I are in the Mauna Kea Observatory (hereafter,MKO) photometric system, which have pass-bands slightlydi ff erent from the 2MASS pass-bands and are likely to containa color term. As suggested by the Referee, determining the zero-points in thenative MKO photometric system would be a more rigorous zero-point calibration.Therefore, we transformed the 2MASS magnitudes into theMKO system using the transformations described in the 2MASSSecond Incremental Release website for 2MASS stars in com-mon with our catalogs:( K S ) = ( K ) MKO + (0 . ± . + (0 . ± . J − K ) MKO , .16ibralato, M. et al.: Ground-based astrometry with wide field imagers. V. Fig. 18.
Full set of CMDs of the fields used in this paper. The dotted gray lines set the saturation threshold in K S filter.( J − H ) = (1 . ± . J − H ) MKO + ( − . ± . , ( J − K S ) = (1 . ± . J − K ) MKO + ( − . ± . , ( H − K S ) = (0 . ± . H − K ) MKO + (0 . ± . . We used only 2MASS stars that, once transformed in theMKO system, were in the color range − . < ( J − K ) MKO < . ff erence between the MKO and2MASS zero-points of 0.05, 0.03, and 0.02 mag in J -, H -, and K S -filter, respectively. For the K S filter in NGC 6121 catalog,we did not have enough stars to compute the zero-point in thecolor range in which the transformations are valid, so we addedthe average K S -filter o ff set (0.02 mag) between the two systems to the 2MASS-based zero-point. In Table 3, we also list theMKO zero-points (with their σ , N , and σ/ √ N −
9. Applications: NGC 6656 and NGC 6121
Figure 18 shows a full set of CMDs with one for each field.For the HAWK-I data used in this section, the photometriczero-points are those obtained in the 2MASS system (Sect. 8.1).We chose two possible targets among the observed fields toillustrate what can be done with HAWK-I. The two closeglobular clusters NGC 6656 and NGC 6121 have high propermotions relative to the Galactic field. In the ESO archive, wefound [email protected] m MPI / ESO exposures of the same fields taken
Fig. 19. ( Left ): Calibration fits and equations for B , V , and I filters for NGC 6656. ( Right ): Same as on the left but for B , V , and Rc filters for NGC 6121. ∼ We downloaded multi-epoch images of NGC 6656 from theESO archive (data set 163.O-0741(C), PI: Renzini), takenbetween May 13 and 15 1999 in B , V , and I filters with [email protected] m MPI / ESO. These images were not taken for astro-metric purposes and only have small dithers, thus preventingus from randomizing the distortion-error residuals. Photometryand astrometry were extracted with the procedures and codesdescribed in Paper I. Photometric measurements also werecorrected for sky concentration e ff ects (light contaminationcaused by internal reflections of light in the optics, causing a re-distribution of light in the focal plane) using recipes in Paper III.The WFI photometry was calibrated matching our catalogs withthe online secondary-standard stars catalog of Stetson (Stetson2000, 2005) using well-measured, bright stars, and least-squarefitting. We found that a linear relation between our instrumentalmagnitudes and Stetson standard magnitudes was adequate toregister our photometry. The calibration equations are shown inFig. 19 in the left panels.As for NGC 6656, we downloaded the NGC 6121 imagesfrom the ESO archive taken with the [email protected] m MPI / ESObetween August 16 and 18 1999, in B , V , and Rc filters. Weperformed the photometric calibration as described above. Thecorresponding calibration fit and equations are shown in Fig. 19in the right panels.In the following subsections, we explore some applicationsin which the photometry has been corrected for di ff erentialreddening. We performed a di ff erential reddening correctionfollowing the iterative procedure described in Milone et al.(2012). As described in detail by Milone et al., the correction toapply to a given star is measured from the di ff erential reddening Fig. 20.
Zoom-in of the K S vs. ( J − K S ) CMD of NGC 6121.We show the CMDs before and after the di ff erential reddeningcorrection is applied (left and right panels respectively). The redarrow indicates the reddening direction.of the selected reference stars that are spatially close to thetarget. The number of stars to use should be a compromisebetween the need to have an adequate number of reference starsto compute the correction and the need for spatial resolution.We chose the nearest 45 reference stars from the faint part ofthe red giant branch (RGB) to the brighter part of the mainsequence (MS) to compute the correction. In Fig. 20, we presentan example to demonstrate how the CMDs change by takingthe di ff erential reddening into account and correcting for it. Weshow a zoom-in of the NGC 6121 K S vs. ( J − K S ) CMD before(left panel) and after (right panel) the correction. Around the MSturn-o ff ( K S ∼ ∆ ( J − K S ) ∼ ∼ ff erential reddening. Fig. 21. ( Top panels ): Proper motion vector-point diagram with a ∼ Bottom panels ): V vs. ( V − K S ) color-magnitude diagram. ( Left ):The entire sample. (
Center ): Stars in VPD with proper motion within 4 mas yr − around the cluster mean. ( Right ): Probable back-ground / foreground field stars in the area of NGC 6656 studied in this paper. The ellipse that encloses most of the field stars iscentered at ( − . − with major and minor axes of 12.5 and 14.8 mas yr − , respectively. To compute proper motions, we followed the method describedin Paper I, to which we refer for the detailed description of theprocedure. For the WFI images, we only used those chips thatoverlap with the HAWK-I field and with an exposure time of ∼
239 s for a total of 19 first-epoch catalogs that include B , V , and I filters. With the 100 catalogs for the second epoch (HAWK-I)in K S band, we computed the displacements for each star. As described in Paper I, the local transformations used to transformthe star’s position in the 1 st epoch system into that of the 2 nd epoch minimize the e ff ects of the residual geometric distortion.In Fig. 21, we show our derived proper motions forNGC 6656. We show only stars with well-measured propermotions. In the left panels of Fig. 21, we show the entiresample of stars; the middle panels display likely cluster mem-bers. The right panels show predominantly field stars. In themiddle vector-point diagram (VPD), we drew a circle around Fig. 22. ( a ): SGB zoom-in on B vs. ( B − K S ) CMD of NGC 6656.The four points (and the two straight lines) used to perform thelinear transformation are plotted in red. ( b ): Rectified SGB. Thered horizontal lines are set at ‘Ordinate’ 0 and 1; the gray solidline is set at ∆ ‘Abscissa’ =
0. ( c c c
1) but in the range3.0-9.0 arcmin. ( c e , d ): Same as in panels ( a , b , c c
2) but forthe
HST data in the m F275W vs. ( m F275W − m F814W ) plane.the cluster’s motion centroid of radius 4 mas yr − to selectproper-motion-based cluster members. The chosen radius isa compromise between missing cluster members with largerproper motions and including field stars that have velocitiesequal to the cluster’s mean proper motion. This example demon-strates the ability of our astrometric techniques to separate fieldand cluster stars. To enclose most of the field stars, we drew anellipse centered at ( − . − in the right VPD withmajor and minor axes of 12.5 and 14.8 mas yr − , respectively.We analyzed the impact of the di ff erential chromatic e ff ectsin our astrometry for this cluster as done in Anderson et al.(2006). Using unsaturated stars and with a color baseline ofabout 3 mag, the e ff ects seem to be negligible (less than 1 mas / yrin each direction) within the airmass range of our data set. Thuswe assumed to be negligible and did not correct it. The sub-giant branch (SGB) based on HAWK-I data remainsbroadened even after the di ff erential reddening correction.This is not surprising since NGC 6656 is known to have asplit SGB (Piotto et al. 2012). The large FoV of our data setallowed us to study the behavior of the radial trend of the ratioˆp fSGB = N fSGB / (N fSGB + N bSGB ), where N bSGB and N fSGB arethe number of stars belonging to the bright (bSGB) and the faintSGB (fSGB), respectively. First of all, we computed this ratiofor SGB stars between 1.5 and 3.0 arcmin from the center ofthe cluster (we adopted the center given by Harris 1996, 2010edition), and between 3.0 and 9.0 arcmin (close to the edgeof the FoV). We chose these two radial bins to have about thesame number of SGB stars in both samples. Since the innermostregion (within 1 . ′ Fig. 23.
Radial trend of ˆp fSGB . The numbers 1, 2, and 3 cor-respond to panels (c) in Fig. 22. The points are placed at theaverage distance of the SGB stars used to compute the ratioin each radial bin. In blue, we plotted the ratios obtained withthe HAWK-I data set; in red, we show the ratio obtained withthe
HST data set. The vertical error bars are computed as de-scribed in the text. The horizontal error bars cover the radial in-tervals. The two vertical lines indicate the core radius and half-light radius (1 . ′
33 and 3 . ′
36 respectively; from Harris 1996, 2010edition). In the top-right panels, the cyan region highlights theHAWK-I field. The cluster center is set at (0,0). The three cir-cles have radius 1.5, 3.0, and 9.0 arcmin. The black parallelo-gram represents the field covered by the
HST data. The regionsused to compute the ratios are labeled with the numbers 1, 2, and3, respectively.To compute the N fSGB / (N fSGB + N bSGB ) ratio, we rectifiedthe SGBs using an approach similar to that described in Miloneet al. (2009). In this analysis, we used only cluster memberswith good photometry. The results are shown in Fig. 22. Westarted by using HAWK-I data only. As described by Milone etal., we need four points (P1b, P1f, P2b, and P2f in Fig. 22) torectify the SGBs. Once selected these points, we transformedthe CMD into a new reference system in which the pointsP1b, P1f, P2b, and P2f have coordinates (0,0), (1,0), (0,1), and(1,1), respectively. We drew by hand a line to separate the twosequences. We derived a fiducial line for each SGB by dividingit into bins of 0.12 ‘Ordinate’ value and fitting the 3.5 σ -clippedmedian ‘Abscissa’ and ‘Ordinate’ for each of them with a spline.We rectified the two sequences using the average of the twofiducials. The rectification was performed by subtracting, fromthe ‘Abscissa’ of each star, the ‘Abscissa’ of the fiducial line atthe same ‘Ordinate’ level (panel (b)). In panels (c1) and (c2),we show the resulting final ∆ ‘Abscissa’ histogram (between ∆ ‘Abscissa’ − .
19 and + .
19 and ‘Ordinate’ 0 and 1) for starsbetween 1.5 and 3.0 arcmin (c1) and between 3.0 ad 9.0 arcmin(c2) from the cluster center. The individual Gaussians for thebright and the faint SGB are shown in blue and red, where thesum of the two in black. After this complicated procedure, we
Fig. 24.
Same as Fig. 21 but for NGC 6121. The radius of the circle centered on the origin of the VPD is 4 mas yr − , while theellipse in the right VPD defining probable field stars is centered at (9.5,14.0) mas yr − with major and minor axes of 17.6 and 14.3mas yr − , respectively. The ellipse mainly encloses stars in the outer part of the Bulge.were finally able to estimate the fraction of stars belonging tothe fSGB and bSGB. We used binomial statistics to estimatethe error σ associated with the fraction of stars. We defined σ ˆ p fSGB = p ˆ p fSGB (1 − ˆ p fSGB ) / ( N fSGB + N bSGB ).As noted by Piotto et al. (2012), points P1b-P2b and P1f-P2fdefine a mass interval for stars in the two SGB segments. If wewant to calculate the absolute value of the ratio ˆp fSGB , we needto make sure that the same mass interval is selected in the twoSGBs and at all radial distances. Due to the lack of appropriateisochrones for the HAWK-I data, this was not feasible. Still,we can estimate the radial trend of ˆp fSGB by taking advantage of both HST (for the inner region) and HAWK-I (for the outerregion) data by making sure that we use the same mass intervalfor the bSGB (and the same mass interval for the fSGB) in bothdata sets.For this reason, we cross-correlated our HAWK-I catalogwith that of Piotto et al. (2012). First, we selected the sample ofSGBs stars in the B vs. ( B − K S ) CMD between the P1b, P1f,P2b, and P2f points of panel (a) in Fig. 22. In the m F275W vs.( m F275W − m F814W ) CMD, we selected the same stars. In thisCMD, we fixed four points that enclose these stars, used themto rectify the SGBs, and then calculated the ratio by following the same procedure as described for HAWK-I data (panels (c3),(d), and (e) in Fig. 22). We emphasize that these intervals arenot the same as the ones used in Piotto et al. (2012), but weused approximately (within the uncertainties due to di ffi culty toselect the limiting points) the same mass intervals in calculatingthe three SGB population ratios.The trend of the ˆp fSGB ratio is shown in Fig. 23. To give amore reliable estimate of the error bars, we also included thehistogram binning uncertainty (We computed the ratio varyingthe starting point / bin width in the histogram and estimate the σ of these values.) and quadratically added them to σ ˆ p fSGB . In anycase, these error bars still represent an underestimate of the totalerror because other sources of uncertainty should to be takenin account (e.g., the uncertainty in the location of the limitingpoints of the selected SGB segments). The error bars are largerfor the two HAWK-I points because of the smaller number ofobjects in the sample.Our conclusion is that the radial trend of the two SGBpopulations within the error bars is flat. As before, we chose all WFI images with an exposure timeof about 180 s and only used the chips overlapping HAWK-Idata. In this way, we had 36 catalogs for the first epoch. For thesecond HAWK-I epoch, we had 400 catalogs (100 images × J ∼ K S ∼ − of the clustermean motion, while we drew an ellipse centered at (9.5,14.0)mas yr − with major and minor axes of 17.6 and 14.3 mas yr − for field stars (right VPD), respectively.Unlike NGC 6656, we did not estimate the di ff erentialchromatic refraction e ff ects, since the color baseline is notlarge enough to study the e ff ect using only unsaturated stars.Saturated stars’ proper motions are less precise, and we couldconfuse di ff erential chromatic refraction with systematic trendsin saturated stars’ proper motions. For the two globular clusters with new proper motions,NGC 6656 and NGC 6121, we calculated cluster membershipprobability, P µ , for each star. Recently, these two clusters havebeen analyzed by Zloczewsky et al. (2012) but, instead of givingmembership probabilities, these authors simply divided all starswith measured proper motions into field stars, possible clustermembers, and likely cluster members. This approach can bejustified on the grounds of a clear separation between field andcluster in the VPD (Fig. 21, 24). However, a more rigorouscluster membership calculation technique would help to bettercharacterize each star’s membership probability. We selected awell-tested local sample method (e.g., van Altena 2013, Chapter25). In this method, a limited subset of stars is selected for eachtarget star with properties close to those of a target. Then, acluster membership probability, P µ of a star is calculated usingthe density functions defined by the local sample. This approach delivers more accurate membership probabilities over the entirerange of magnitudes. In the case of globular clusters, the poten-tial bias in P µ at various magnitudes is less significant becausethe cluster stars dominate a relatively small number of fieldstars. In the presence of a highly varying precision of calculatedproper motions (ranging from 0.2 to 5.5 mas yr − for NGC 6656and NGC 6121), however, using an aggregate density functionfor a cluster and field can produce unreliable membershipprobabilities for low-precision proper motions. This is dueto a significant widening of cluster’s density function at thelow-precision end of proper motions. Therefore, we adopted themean error σ µ of proper motions as a single parameter allowingus to find a local sample, which is similar to what was appliedto the catalog of proper motions in ω Cen (Paper III). Thereare a few di ff erences from the study of ω Cen. First, we used afixed window in the error distributions with the total width notexceeding 0.75 mas yr − so that a target star is located in themiddle of this window. The total number of stars in local samplenever exceeds 3000, hence the window size for well-measuredproper motions, which dominate the catalog, can be as smallas 0.1 mas yr − . At the extreme values of proper-motion errors,the window size is fixed and the placement of a target mayno longer be in the middle of this window. Second, we used amodified mean σ µ = q ( µ α cos δ ) + µ δ / √
2. Third, the Gaussianwidth of a cluster density function was interpolated by using anempirical relationship: σ c = (0 . × ( K s − + × σ µ , where K S is the measured near-infrared magnitude of a target star. Inaddition, σ c was never let to be lower than 0.7 mas yr − .While the cluster density function is always a 2-D Gaussian,it is often convenient to use a flat sloping density function forthe field-star distribution in the VPD. This is related to thebinning of VPD. The adopted size of a binning area, centeredon the cluster, is 5 σ c × σ c which formally should contain allcluster members. If a star’s σ µ < − , then it also meansthat the binning area never reaches the center of a field-starcentroid in the VPD for both globular clusters. In the regimeof high proper motion errors ( > − ), the distributionof field stars is so di ff use that a significant portion of its wingsfalls outside the VPD area covered by proper motions. The fewfree parameters of both cluster and field density distributions, Φ c and Φ f , are calculated according to Kozhurina-Platais et al.(1995) but the resulting cluster membership probability P µ isdefined by Eq. 25.8 from van Altena (2013).We note that likely clusters stars have P µ > P µ < ∼
24% for NGC 6656and ∼
9% for NGC 6121. This should be considered whenexamining the astrometric cluster membership of rare stars,such as variables, blue stragglers, and horizontal branch stars.
Kaluzny & Thompson (2001) published a catalog of 36 variablestars in the central field of NGC 6656. We cross-identified thesesources in our catalog and found 27 stars in our proper motioncatalog. In Table 4, we report the membership probability forthese stars (ID KT are Kaluzny & Thompson labels; ID L13 arethe identification labels in our catalog.). In Fig. 25, we show V vs. ( V − K S ) CMD (bottom-left panel), V vs. P µ (bottom-right Table 4.
Membership probability for the NGC 6656 variable starcatalog of Kaluzny & Thompson (2001). ID KT is the ID used inKaluzny & Thompson (2001), and ID L13 is in our catalog. ID KT ID L13 P µ ID KT ID L13 P µ Members
M22 V02 45566 99 M22 V29 161746 96M22 V04 56333 99 M22 V33 181768 86M22 V10 93320 89 M22 V34 181631 80M22 V16 104996 98 M22 V36 185551 84M22 V20 131248 99 M22 V45 155768 79M22 V23 144995 95 M22 V51 77720 95M22 V28 155692 87 M22 V55 148685 94
Probably Members
M22 V14 102946 2 M22 V37 193535 37
Non Members
M22 V03 47877 0 M22 V18 113038 0M22 V05 57746 0 M22 V42 79935 0M22 V07 68695 0 M22 V46 157407 0M22 V08 74698 0 M22 V48 183116 0M22 V12 101344 0 M22 V54 138212 0M22 V15 106319 0 panel) and the VPD (top panel) for all stars in our sample witha membership probability measure. We set two thresholds inP µ (P µ =
2% and P µ = µ < ≤ P µ <
75% and likely cluster members withP µ ≥ µ <
2% (green triangles); twostars have 2% ≤ P µ <
75% (yellow squares), and the remaining14 stars have P µ ≥
75% (azure circles). The two variable starswith 2% ≤ P µ < We can similarly use our proper motion data to assign mem-bership probabilities to candidate variable star members ofNGC 6121. Shokin & Samus (1996) cataloged 53 NGC 6121variable stars from the literature and provided equatorialcoordinates. We cross-checked our proper motion catalog withthat provided by the authors, and we found 42 sources incommon. Figure 26 shows these variable stars in J vs. ( B − J )CMD and VPD. As for NGC 6656, we set two thresholds atP µ =
2% and P µ = S andID L13 are the labels in Shokin & Samus and in this paper,respectively.). All cross-identified variable stars are saturated
Table 5.
Membership probability for the NGC 6121 variable starcatalog of Shokin & Samus (1996). ID S is the ID used in Shokin& Samus (1996); ID L13 is in our catalog. ID S ID L13 P µ ID S ID L13 P µ Members
V1 176589 99 V28 30831 99V2 173396 98 V30 16737 99V5 164050 97 V36 (NE) 167082 91V8 148455 87 V36 (SW) 167583 96V9 146061 92 V37 125921 94V10 136435 93 V38 119757 98V12 129429 98 V39 111350 99V14 127336 91 V41 87105 99V15 123997 99 A381 98514 95V16 121622 78 A382 96072 99V18 111157 95 A505 98523 85V20 107627 98 A519 112683 99V22 100193 93 L1610 142443 87V24 94766 91 L1717 115979 96V25 87432 97 L2630 155330 98V26 76423 86 L3602 60119 99V27 68948 97 L3732 97216 85
Probably Members
V6 149664 39 V19 109197 20V7 149106 6 V31 14721 71V11 133287 41
Non Members
V17 114412 0 A246 174902 0V23 98450 0 in our proper motion catalog. Five stars (V6, V7, V11, V19,and V31) have 2% ≤ P µ <
10. Catalogs
We constructed eight di ff erent catalogs for our seven fields.We split the Baade’s Window field into two di ff erent cata-logs (Bulge .We also converted pixel-based coordinates into equatorialcoordinates using the UCAC 4 catalog as reference. We onlyused bright, unsaturated stars to compute the coe ffi cientsof the 6-parameter linear transformation between HAWK-Iand UCAC 4 frames. The choice of using 6-parameter lin-ear transformations (see Sect. 5.5 for a description of thesetransformations) to transform star positions in our catalogsinto the UCAC 4 reference system allows us to solve not onlyfor shift, orientation, and scale, but it also minimizes mostof the telescope + optics-system residuals, as well as most ofthe atmospheric refraction e ff ects. Furthermore, linking ourcatalogs to the UCAC 4 catalog, we did not only provide the http://vizier.u-strasbg.fr/viz-bin/VizieR . 23ibralato, M. et al.: Ground-based astrometry with wide field imagers. V. Fig. 25. ( Bottom-left panel ): NGC 6656 V vs. ( V − K S ) CMDfor all stars in our catalog (black dots) that have a membershipprobability measure. We plotted variable stars from Kaluzny &Thompson (2001) with green triangles that are cross-identifiedin our catalog with P µ < ≤ P µ < µ ≥ Bottom-right panel ): V vs. P µ . ( Top ): VPD. We alsodrew proper motion error bars for matched variable stars.equatorial coordinates for all stars but we determined the linearterms of our distortion. The equatorial coordinates are truly ourbest calibrated coordinates.The first eight columns are the same in all catalogs. Column(1) contains the ID of the star; columns (2) and (3) give J2000.0equatorial coordinates in decimal degrees. Note that positionsare given at the epoch of HAWK-I observations because ofproper motion. Columns (4) and (5) contain the pixel coordinatesx and y of the distortion-corrected reference frame. Columns (6)and (7) contain the corresponding positional r.m.s.; column (8)gives the number of images where the star was found.Columns (9) and the following columns contain the pho-tometric data and proper motions when present. Values areflagged to − + ) of each catalog. Fig. 26. ( Bottom-left panel ): NGC 6121 J vs. ( B − J S ) CMD forall stars with a membership probability measure. As in Fig. 25,all stars are shown with black dots. We used azure circles, yellowsquares, and green triangles to highlight cross-identified stars inthe Shokin & Samus (1996) with P µ ≥ ≤ P µ < µ < Bottom-right panel ): J vs. P µ . ( Top ):VPD. We show proper motion error bars for Shokin & Samusvariable stars. Star V17 (P µ = Baade’s Window (Bulge
Columns (9) to (23) contain thephotometric data: i.e., J , H , K S (both with 2MASS- and MKO-based zero-points added) , their errors, the number of imagesused to compute the magnitude of the star in the master frame,and the QFIT in this order. Columns (24) and (25) contain a flagto weed out PSF artifacts from the J and H filter (see Sect. 7). Allstars in this catalog have a measure of the magnitude in K S filter. NGC 6822, NGC 6388, LMC, and 47 Tuc.
Columns (9) to(18) contain the J and K S photometric data; columns (19) and(20) contain the weed-out flags (see Table 7). In the NGC 6822,LMC, and 47 Tuc catalogs, all stars have a J -magnitude mea-surement, and NGC 6388 has a K S -magnitude measurement. Baade’s Windows rotated by 135 ◦ (Bulge Columns (9) to(14) contain K S magnitudes, errors, the number of stars used tocompute the average magnitudes, QFIT and the weed-out flagvalues, respectively (Table 8).
NGC 6656.
Columns (9) to (26) contain the photometric datain K S , B , V , and I band. Finally, Columns (27) to (31) containthe proper motion data: µ α cos δ (27), σ µ α cos δ (28), µ δ (29), σ µ δ (30), and the number of pairs of images in which a given star’sproper motion was measured (31). Finally, column (32) containsthe membership probability (Table 9). Stars measured in onlyone exposure in either B , V , or I filters have photometric r.m.s.values of 9.9. As for Bulge K S filter. NGC 6121.
Columns (9) to (32) contain the photometric datain J , K S , B , V , and Rc bands. Columns (33) to (37) contain theproper motion data and column (38) is the membership proba-bility (Table 10). As in the NGC 6656 catalog, a r.m.s. equal to9.9 is used for those stars measured in only 1 exposure in K S , B , V , or Rc filter. In this catalog, all stars have a J -magnitudemeasurement.
11. Conclusions
We derived an accurate distortion solution in three broadband filters for the HAWK-I detector and release the tools tocorrect the geometric distortion with our solution. We alsoproduced astro-photometric catalogs of seven stellar fields.We release catalogs with astrometric positions, photometry,proper motions, and membership probabilities of NGC 6121(M 4) and NGC 6656 (M22), while the remaining fields (theBaade’s Window, NGC 6822, NGC 6388, NGC 104, and theJames Webb Space Telescope calibration field) studied in thepresent paper only contains astrometry and photometry. Thesecatalogs are useful for selecting spectroscopic targets, and canserve as distortion-free frames with respect to which one cansolve for the geometric distortion of present / future imagers. Theastronomical community has started to focus its attention onwide-field cameras equipped with NIR detectors, and the quan-tity and quality of NIR devices have improved considerably.This is a first e ff ort to develop the expertise with these detectorsto fully exploit the data coming from large-field NIR surveys,such as the VVV survey taken with VIRCAM@VISTA. Finally,an additional goal of this work is to get ready for the upcomingJames Webb Space Telescope, whose imagers define the state-of-the-art in astrometry, in particular in crowded environmentsnot reachable by GAIA.We analyzed both photometric and astrometric performanceof the NIR mosaic HAWK-I@VLT using images of seven di ff er-ent fields observed during commissioning in 2007. We computeda geometric-distortion solution for each chip of HAWK-I in threedi ff erent broad band filters ( J , H , K S ). Our dithered-observationstrategy using the self-calibration technique allowed us to ran-domize the systematic errors and to compute the average stars’positions that provide an approximation of the true positionsin the distortion-free master frame. A fifth-order polynomialsolution highlighted a periodic pattern in the distortion residuals.We have demonstrated that this pattern is not a geometric e ff ect(as it is the case for the WFPC2 or the WFC3 / UVIS@
HST ) butit is a periodic lag introduced by alternating readout amplifiers.To remove it, we used a square-wave function and a 64-pixelstep table of residuals. Finally we used four additional look-uptables (one per chip) to perform a bi-linear interpolation to takeall uncorrected residuals into account and to further improveour solutions. Thanks to our 5-step distortion correction, we areable to reach a positional r.m.s. of ∼ ff erent images, the e ff ects due to telescope + instrument andatmosphere are absorbed, and the σ (Radial residual) furtherdecreases, reaching ∼ ff erentialposition of a star in multiple images of the same field.We have also shown that the non-linear terms of ourdistortion solution can be transferred between observing runsat the 10 mas level. The astrometric accuracy contained in thepixel-coordinate system degrades moving toward the edges ofthe FoV because the stars’ positions were obtained as the aver-age of fewer images than in the center of the field. Therefore,the average positions are more vulnerable to poorly-constrainedtransformations of the individual exposures into the masterframe. The accuracy can decrease from ∼
10 mas to ∼
100 mas( ∼ ff ects of residuals in the geometric distortioncorrections as described in Paper I.In the second part of the paper, we showed the potentialapplications of our astrometric techniques and computed the relative proper motion of stars in the field of the globularclusters NGC 6656 and NGC 6121. With a time baseline ofabout 8 years, we have clearly separated cluster members fromfield stars. Accuracy of proper-motion measurements is limitedby the depth and the precision of first-epoch data set. We notethat the stellar positions in our catalogs have been derived fromonly a single epoch of HAWK-I data. A second-epoch HAWK-I(or another wide-field infrared camera) data set is needed toprovide proper-motion solutions that allow these data to beextended with confidence to arbitrary future epochs. We exploitphotometry and proper motions of stars in NGC 6656 to studyits stellar populations. We find that the bimodal SGB, previouslydiscovered from visual and ultraviolet HST photometry (Piottoet al. 2012), is also visible in the K S versus ( B − K S ) CMD.We combined information from HAWK-I observation of theouter part of NGC 6656 and from HST images of the innermostcluster region (Piotto et al. 2012) to study the radial distributionof the two SGBs. To do this, we calculated the number ratio ofthe faint SGB ˆp fSGB for stars at di ff erent radial distances fromthe cluster center to 9 ′ ( ∼ Acknowledgements. ML and GP acknowledge partial support by the Universit`adegli Studi di Padova CPDA101477 grant. ML acknowledges support bythe STScI under the 2013 DDRF program. APM acknowledges the financialsupport from the Australian Research Council through Discovery Project grantDP120100475. We thank Dr. Jay Anderson for careful reading of the manuscriptand for thoughtful comments. We thank the anonymous referee for the usefulcomments and suggestions that considerably improved the quality of our paper.
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Table 6.
Bulge
Column α Right Ascension [ ◦ ](3) δ Declination [ ◦ ](4) x x-master frame position [pixel](5) y y-master frame position [pixel](6) σ x r.m.s. error in the x-position [pixel](7) σ y r.m.s. error in the y-position [pixel](8) n pos Number of images where the star was found in used to compute the master-frame position(9) J Calibrated J magnitude in 2MASS system(10) H Calibrated H magnitude in 2MASS system(11) K S Calibrated K S magnitude in 2MASS system(12) J Calibrated J magnitude in MKO system(13) H Calibrated H magnitude in MKO system(14) K S Calibrated K S magnitude in MKO system(15) σ J r.m.s. error in J photometry(16) σ H r.m.s. error in H photometry(17) σ K S r.m.s. error in K S photometry(18) n J Number of images where the star was found in used to compute the J magnitude(19) n H Number of images where the star was found in used to compute the H magnitude(20) n K S Number of images where the star was found in used to compute the K S magnitude(21) QFIT J Quality of J PSF-fit(22)
QFIT H Quality of H PSF-fit(23)
QFIT K S Quality of K S PSF-fit(24) weed J J weed-out flag (1 = star, 0 = PSF-artifact, − = star not found in J exposures)(25) weed H H weed-out flag Table 7.
NGC 6822, NGC 6388, LMC and 47 Tuc catalogs.
Column J Calibrated J magnitude in 2MASS system(10) K S Calibrated K S magnitude in 2MASS system(11) J Calibrated J magnitude in MKO system(12) K S Calibrated K S magnitude in MKO system(13) σ J r.m.s. error in J photometry(14) σ K S r.m.s. error in K S photometry(15) n J Number of images where the star was found in used to compute the J magnitude(16) n K S Number of images where the star was found in used to compute the K S magnitude(17) QFIT J Quality of J PSF-fit(18)
QFIT K S Quality of K S PSF-fit(19) weed J J weed-out flag (1 = star, 0 = PSF-artifact, − = star not found in J exposures)(20) weed K S K S weed-out flag Table 8.
Bulge
Column K S Calibrated K S magnitude in 2MASS system(10) K S Calibrated K S magnitude in MKO system(11) σ K S r.m.s. error in K S photometry(12) n K S Number of images where the star was found in used to compute the K S magnitude(13) QFIT K S Quality of K S PSF-fit(14) weed K S K S weed-out flag (1 = star, 0 = PSF-artifact, − = star not found in K S exposures) 27ibralato, M. et al.: Ground-based astrometry with wide field imagers. V. Table 9.
NGC 6656 catalog.
Column K S Calibrated K S magnitude in 2MASS system(10) K S Calibrated K S magnitude in MKO system(11) B Calibrated B magnitude(12) V Calibrated V magnitude(13) I Calibrated I magnitude(14) σ K S r.m.s. error in K S photometry(15) σ B r.m.s. error in B photometry(16) σ V r.m.s. error in V photometry(17) σ I r.m.s. error in I photometry(18) n K S Number of images where the star was found in used to compute the K S magnitude(19) n B Number of images where the star was found in used to compute the B magnitude(20) n V Number of images where the star was found in used to compute the V magnitude(21) n I Number of images where the star was found in used to compute the I magnitude(22) QFIT K S Quality of K S PSF-fit(23)
QFIT B Quality of B PSF-fit(24)
QFIT V Quality of V PSF-fit(25)
QFIT I Quality of I PSF-fit(26) weed K S K S weed-out flag (1 = star, 0 = PSF-artifact, − = star not found in K S exposures)(27) µ α cos δ Proper-motion value along µ α cos δ [mas yr − ](28) σ µ α cos δ r.m.s. of µ α cos δ [mas yr − ](29) µ δ Proper-motion value along µ α cos δ [mas yr − ](30) σ µ δ r.m.s. of µ δ [mas yr − ](31) n pairs Number of pairs of first-second epoch images used to compute the proper motion of the star(32) P µ Membership probability
Table 10.
NGC 6121 catalog.
Column J Calibrated J magnitude in 2MASS system(10) K S Calibrated K S magnitude in 2MASS system(11) J Calibrated J magnitude in MKO system(12) K S Calibrated K S magnitude in MKO system(13) B Calibrated B magnitude(14) V Calibrated V magnitude(15) Rc Calibrated Rc magnitude(16) σ J r.m.s. error in J photometry(17) σ K S r.m.s. error in K S photometry(18) σ B r.m.s. error in B photometry(19) σ V r.m.s. error in V photometry(20) σ Rc r.m.s. error in Rc photometry(21) n J Number of images where the star was found in used to compute the J magnitude(22) n K S Number of images where the star was found in used to compute the K S magnitude(23) n B Number of images where the star was found in used to compute the B magnitude(24) n V Number of images where the star was found in used to compute the V magnitude(25) n Rc Number of images where the star was found in used to compute the Rc magnitude(26) QFIT J Quality of J PSF-fit(27)
QFIT K S Quality of K S PSF-fit(28)
QFIT B Quality of B PSF-fit(29)
QFIT V Quality of V PSF-fit(30)
QFIT Rc Quality of Rc PSF-fit(31) weed J J weed-out flag (1 = star, 0 = PSF-artifact, − = star not found in J exposures)(32) weed K S K S weed-out flag(33) µ α cos δ Proper-motion value along µ α cos δ [mas yr − ](34) σ µ α cos δ r.m.s. of µ α cos δ [mas yr − ](35) µ δ Proper-motion value along µ α cos δ [mas yr − ](36) σ µ δ r.m.s. of µ δ [mas yr − ](37) n pairs Number of pairs of first-second epoch images used to compute the proper motion of the star(38) P µ Membership probability28ibralato, M. et al.: Ground-based astrometry with wide field imagers. V.
Fig. A.1.
Left : 10 ×
10 PSFs for the whole HAWK-I detector (5 × Right : Spatial variation of the PSFs. To each lo-cal PSF, we subtracted a single average PSF for the whole detec-tor. A darker color means less flux than the average PSF, and alighter color means more flux.
Appendix A: Study of the HAWK-I PSF
A.1. PSFspatialvariability
As in the case of the [email protected] m (Paper I), the PSF shape forthe HAWK-I@VLT detector is di ff erent from one chip to theother and from side to side within the same chip. To fully takethis spatial variation into account, we decided to solve for anarray of 25 PSFs per chip (5 across and 5 high). A bi-linearinterpolation is used to derive the proper PSF model in eachlocation of the detector (see Paper I). The left panel of Fig. A.1shows these 5 × × × = ff erentPSF-array solutions: (1) one PSF per chip; (2) four PSFs perchip (at the corners); (3) six PSFs per chip, (4) nine PSFs perchip; and (5) the full 5 × QFIT across the image: better PSFmodels provide smaller
QFIT values.The top panel of Fig. A.2 shows
QFIT values as a func-tion of the instrumental magnitude for chip[1] of exposureHAWKI.2007-08-03T01:41:29.785.fits. This is an image in thefield of M 4 taken through J filter. The instrumental magnitudeis defined as − . × log( P counts), where P counts is the sum of the total counts under the fitted PSF. The red line in the figureindicates when stars start to be saturated .For well-exposed stars (e.g., with instrumental magnitude J between −
14 and − QFIT values are typically below 0.05and increase for fainter or saturated stars. However, there are afew sources with anomalously high
QFIT in this interval. To findout what kind of outliers these sources are, we selected two ofthem (highlighted in yellow in Fig. A.2). Their location on theimage is shown in the bottom-left panels of Fig. A.2 in yellow.Blue circles mark all stars for which we were able to measurea position and a flux. White pixels are those flagged using thebad-pixel mask. We did not find stars too close to these badpixels. Bottom-right panels in the figure show the correspondingsubtracted images. We can see that our PSF-fitting procedureis able to leave very small residuals in the subtracted image,except for saturated stars. The first of the two high-
QFIT starswe selected (see top-right panel) is in close proximity of asaturated star; it has been poorly measured because of the lightcontamination from the neighboring star, and therefore has alarge
QFIT value. The second star has a cosmic ray event closeto its center, increasing its total apparent flux and shifting itscenter on the image. This star has been over-subtracted (seebottom-right panel), resulting again in a larger
QFIT value.To better quantify how our PSF models adequately representstar profiles across the detector, we perform the following test.A total of 100 exposures in the field of M 4 were taken betweenAugust 3 and 5, 2007 in four runs of 25 images each. We usedhere only the first 25 images taken consecutively on August3, 2007 during a time span of about 34 minutes. We derivedan array of 5 × k bright, unsaturated stars (instrumentalmagnitude J < − × ff er from real star profiles.We extracted 11 ×
11 pixel rasters around each star k (i.e., ± i , j pixel values per star. We subtracted the local sky value to all ofthem, which is computed as the 2 σ -clipped median value of thecounts in an annulus between 8 and 12 from the star’s center.This is the net star’s flux at any given location on the raster. Thefractional star’s flux is obtained by dividing these values by thetotal star’s flux z . Besides Poisson errors, these values shouldreflect what our PSF models predict for those pixels ( ψ i , j ), sothat we should always have in principle:P ki , j − sky k z k − ψ ki , j = . Deviations of these values from zero tell us how much our PSFsover- or underestimate the true star’s profile. Results of this testare reported in Fig. A.3 for chip[1]. We divided the 2048 × × σ -median valuesof the residuals for each pixel of the raster. Pixel values arecolor-coded as shown on top of Fig. A.3. From the Fig. A.3, The maximum central pixel value of the PSFs for this exposure is0.058 (i.e., 5.8% of the star’s flux falls within its central pixel). We setsaturation to take place at 30 000 counts, which means at instrumentalmagnitude − . × log(30000 / . ≃ − .
28. 29ibralato, M. et al.: Ground-based astrometry with wide field imagers. V.
Fig. A.2.
Top : QFIT parameter as a function of the instru-mental magnitude J for chip[1] of exposure HAWKI.2007-08-03T01:41:29.785.fits. In yellow, we highlighted two sourceswith anomalously high QFIT . The red line shows the saturationlimit.
Middle / Bottom-left : Location of the two sources (in yel-low) on the image. Blue circles mark all stars for which we wereable to measure a position and a flux. White pixels are thoseflagged according to the bad-pixel mask. We do not find starstoo close to these bad pixels.
Middle / Bottom-right : The corre-sponding subtracted images. The first of the two high-
QFIT stars(middle-right panel) we selected is in close proximity to a satu-rated star; it has been poorly measured and therefore has a large
QFIT value. The second star (bottom-right panel) has a cosmicray event close to its center, increasing its total apparent fluxand shifting its center on the image. This star has been over-subtracted, resulting again in a larger
QFIT value.we can easily see that PSF residuals are in general smaller than0.05% even in the central pixel (where Poisson noise is moste ff ective). This proves that our spatial-dependent PSF modelsare able to adequately represent a star profile at any givenlocation of the chip. A.2. PSFtimevariability
Ground-based telescopes su ff er from varying seeing andairmass conditions, telescope flexures, and changes in focus.These are all e ff ects that may severely alter the shape of thePSF. Figure A.4 illustrates how much the seeing can actuallya ff ect the PSFs. In the figure, we show the first 25 exposures Fig. A.3.
PSF spatial variability for chip[1]. Pixel values arecolor-coded as shown on top.of M 4 that were consecutively taken on August 3, 2007. Thetotal time baseline is 34 minutes. For each of our PSF models,we considered the value of its central pixel as a function ofthe exposure sequence, starting from the first exposure. Onaverage, chip[4] PSFs are sharper, while chip[2] stars have theleast amount of flux in their central pixels. As a reference, wehighlight a central PSF value of 0.05 (i.e., 5% of the total star’sflux in its center pixel) in blue. Within the same exposures,central PSF values can range from 0.03 to 0.07 (see also Fig. A.1for the PSF to PSF variation).In a time span as short as half an hour, we can already seesome interesting PSF time-variability e ff ects. First of all, centralPSF values vary in an inhomogeneous way across the detector.For instance, there are specific locations on the detector (e.g.,the top PSFs of chip[4]) where central PSF values can changeby up to 40%. On the other hand, central PSF values are morestable in di ff erent locations (e.g., the bottom PSFs of chip[2]).Moreover, while for some PSFs (e.g., the one labeled as4-(1,5) on chip[4]), we have a general decrease of the centralvalues. For other PSFs (e.g., 1-(2,1) on chip[1]), we have a de-crease of the central values during the first 15 minutes, followedby an increase afterwards. [Note that no focus adjustments havebeen made during this 34 minutes.]Figure A.4 clearly shows that there are large variations inthe PSF shape even from one exposure to the other, and thisvariation is not constant across the field. HST ’s PSFs are verystable over time with variations on the order of at most a fewpercent, mostly due to the so-called telescope breathing . For HST , one spatially-constant perturbation PSF for is generally HST focus is known to experience variations on the orbital timescale, which are attributed to thermal contraction / expansion of the HST optical telescope assembly as the the telescope warms up during its or-bital day and cools down during orbital night.30ibralato, M. et al.: Ground-based astrometry with wide field imagers. V.
Fig. A.4.
Central PSF values as function of the exposure sequence. We highlighted the central PSF value of 0.05 in blue.enough to take into account this e ff ect (e.g., ACS / WFC PSFs,Anderson & King 2006). To achieve high-precision astrometryand photometry with the HAWK-I camera, we have to derive aspecific set of PSFs for each individual exposure.To further infer the e ff ects of time variation on our PSF models,we performed the following additional analysis using the 100images in the field of M 4. As already mentioned, these imageswere taken in blocks of 25 consecutive exposures in fourdi ff erent runs, spanning three nights. Because each observingrun lasted about 30 minutes, we can safely assume that focusvariations have played a little role in changing the shape ofPSFs, if compared to airmass and seeing variations. Seeingshould actually be the most important factor in changing thePSF shape from one exposure to the next one. We focused on the centermost four PSFs, namely: 1-(5,5); 2-(1,5); 3-(5,1); and4-(1,1), following labels of Fig. A.4. In Fig. A.5, we plot thecentral value of these PSFs as a function of the image quality(i.e., the average stars’ FWHM as measured directly on theexposures). Di ff erent observing runs are marked with di ff erentcolors and symbols. On the bottom right panel of the figure,we plot the variation in the image quality during the four runs.Here, we want to emphasize that these variations occurredwithin 30 minutes within the same run. As we expected, there isa strong correlation between our PSF shapes and image quality.The correlation between PSF shapes and airmass is shown inFig. A.6. Airmass variations seem to play a secondary role inchanging PSF shape with respect to image quality. Fig. A.5.
The central PSF value of these PSFs as a function ofthe image quality.
Fig. A.6.
Same as Fig. A.5 but for airmass variations.
Appendix B: Geometric distortion: 2-D maps andsize of the corrections
In this appendix, we report the size of the distortion correctionsreleased in this paper. In Table B.1, B.2, and B.3, we showthe minimum and maximum values of each correction in bothcoordinates for all chips. The largest correction is applied withthe P corrections, which decrease from the corner to the centerof the detector. The S and FS corrections are only applied to thex-coordinates. Figure B.1 demonstrates there is no δ y periodicpattern, so an FS correction along this axis is not necessary. Theresidual distortion is corrected with the TP correction. While theS correction is the same for all chips and has only two values,the FS correction changes from chip to chip and varies acrossthe same chip.In Fig. B.2, we show a 2-D map of the correction foreach chip / filter. In the four left boxes of each row, we plotthe correction of each chip for the x-coordinates in the rightboxes for the y-coordinates. The polynomial correction createsthe radial pattern that changes from the corner to the centerin all chips. The S correction is visible only in the left boxes(x-coordinate corrections) and creates a striped pattern.Another important correction that we are going to release(as FITS images) is the correction for the pixel area variationacross the detector. This is a useful tool for improving HAWK-Iphotometry. On average the size of the pixel area varies up to0.7% across the detector. This value is reached at a point closeto the edge and to the center of the detector. We only applied thepolynomial correction, since it gives the maximum correction.Note that the corrections of the periodic-lag e ff ect should notbe included in the pixel area correction. The periodic lag is dueto charges left in the amplifiers, so the area of the pixel itself isnot modified on sky. This is di ff erent to what happens with theoptics + filters distortion. In Fig. B.3, we show three maps of thecorrection with one for each HAWK-I filter. Fig. B.1. δ y as function of Y in units of HAWK-I pixels for all chips. The red lines is set at 0 HAWK-I pixel. Table B.1.
Size of the J corrections. All the values are given in pixel. Chip X-axis Y-axis X-axis X-axis X-axis Y-axis
Min Max Min Max Min Max Min Max Min Max Min Max
P Correction S Correction FS Correction TP Correction − − − − − − − − − − − − − − − − − − − − − − − − Table B.2.
As in Table B.1 but for the H corrections (in pixel). Chip X-axis Y-axis X-axis X-axis X-axis Y-axis
Min Max Min Max Min Max Min Max Min Max Min Max
P Correction S Correction FS Correction TP Correction − − − − − − − − − − − − − − − − − − − − − − − − Table B.3.
As above but for the K S corrections (in pixel). Chip X-axis Y-axis X-axis X-axis X-axis Y-axis
Min Max Min Max Min Max Min Max Min Max Min Max
P Correction S Correction FS Correction TP Correction − − − − − − − − − − − − − − − − − − − − − − − − Fig. B.2.
Maps of the corrections. From top to bottom, J -, H -, and K S -filter corrections. For each filter, the four chips on theleft show the X-correction, while the four chips on the right show the Y-correction. A linear scale is used. Red means positivecorrections; purple are negative corrections. The values in the color bar are expressed in pixels. For the J filter, the x-correctionsvaries between − .
95 and 2.96 pixels across the whole detector, while the y-corrections between − .
88 and 3.95 pixels. For the H filter, the minimum and maximum corrections for the x-coordinate are − .
40 and 2.99 pixels, while the corrections for the y-coordinate are − .
00 and 3.59 pixels. The minimum and maximum x-corrections for the K S -filter solution are − .
67 and 2.85 pixels;for y-corrections, the minimum and maximum are − .
77 and 3.42 pixels.
Fig. B.3.
Maps of the pixel area corrections. From top to bottom, J -, H -, and K S -filter corrections. The values in the color barrepresent the corrected area of the pixels. Before the correction, all pixels have an area of 1 pixel . The scale in the images is linear.. The scale in the images is linear.