The M4 Core Project with HST -- V. Characterizing the PSFs of WFC3/UVIS by Focus
aa r X i v : . [ a s t r o - ph . I M ] J un Mon. Not. R. Astron. Soc. , 1–18 (2017) Printed 2 June 2017 (MN L A TEX style file v2.2)
The M 4 Core Pro ject with
HST – V. Characterizing thePSFs of WFC3/UVIS by Focus ⋆ J. Anderson and L. R. Bedin Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA Istituto Nazionale di Astrofisica - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, Padova, IT-35122
Accepted 2017 May 19. Received 2017 May 19; in original form 2017 April 19
ABSTRACT
As part of the astrometric
Hubble Space Telescope (HST) large program GO-12911,we conduct an in-depth study to characterize the point spread function (PSF) of the
Uv-VISual channel (UVIS) of the
Wide Field Camera 3 (WFC3) , as a necessary stepto achieve the astrometric goals of the program. We extracted a PSF from each of the589 deep exposures taken through the F467M filter over the course of a year and findthat the vast majority of the PSFs lie along a one-dimensional locus that stretchescontinuously from one side of focus, through optimal focus, to the other side of focus.We constructed a focus-diverse set of PSFs and find that with only five medium-brightstars in an exposure it is possible to pin down the focus level of that exposure. Weshow that the focus-optimized PSF does a considerably better job fitting stars thanthe average “library” PSF, especially when the PSF is out of focus. The fluxes andpositions are significantly improved over the “library” PSF treatment. These resultsare beneficial for a much broader range of scientific applications than simply theprogram at hand, but the immediate use of these PSFs will enable us to search forastrometric wobble in the bright stars in the core of the globular cluster M 4, whichwould indicate a dark, high-mass companion, such as a white dwarf, neutron star, orblack hole.
Key words: globular clusters: individual (M 4, NGC 6121)
One of the Hubble Space Telescope’s (
HST ’s) great advan-tages is that it has an extremely stable point-spread func-tion (PSF) relative to ground-based telescopes. The fact that
HST is in orbit high above the atmosphere means that thePSF that is delivered to the detector is free of turbulence-related variations: the PSF varies more spatially across thedetector than it does temporally. The spatial variations aredue to a combination of geometric optics and detector-related features (such as variable charge diffusion due tochanges in chip thickness, see Krist 2003).For the case of the
Uv-VISual channel (UVIS) of the
Wide Field Camera 3 (WFC3) —on which the present in-vestigation is focused—a set of 7 × ⋆ Based on observations with the NASA/ESA Hubble Space Tele-scope, obtained at the Space Telescope Science Institute, whichis operated by AURA, Inc., under NASA contract NAS 5-26555,under Large Program GO-12911. other words, the PSF can change significantly over about500 detector pixels. (Sabbi & Bellini 2013).Even though the
HST
PSF is not beset by atmosphericvariations, the fraction of light in the
HST
PSF core can varywith time by ±
3% due to “breathing”, the focus variationsthat result as
HST changes orientation relative to the Sunor goes into and out of the Earth’s shadow during its orbit.This has been explored in numerous documents (see Dressel2014 for a summary).A proper understanding of the
HST
PSF is critical formany high-precision
HST studies. With an accurate PSF,systematically accurate positions can be measured for well-exposed stars to better than 0.01 pixel ( ∼ HST
PSF has not beencharacterized very well thus far stems from the fact that itis undersampled. When a detector is not at least Nyquist-sampled, that means its pixels are too wide to capture allthe spatial information in the scene that the telescope isdelivering to the detector (see Lauer 1999). Each exposure
J. Anderson & L. R. Bedin gives us a limited amount of information about the scene,and we must combine multiple dithered exposures in order tofully represent all the information delivered by the telescopeto the detector. This is equally true for the scene and for thePSF: we need to combine multiple dithered images to fullyunderstand either of them.Unfortunately, it is particularly complicated to combinemultiple dithered exposures with
HST on account of its largeoptical distortion. Even dithers of 10 pixels cannot be com-bined without careful attention to the forward and reversedistortion solutions, since a shift of 10.0 pixels at the centerof the detector can correspond to 10.2 pixels at the edge.Anderson & King (2000, AK00) developed a detailedprocedure whereby a set of dithered exposures can be usedto extract a properly sampled PSF model from a series ofundersampled images. AK00 further demonstrate that thisPSF can be used to extract accurate and unbiased posi-tions for stars in a single exposure. This initial procedurewas constructed for
Wide Field Camera 2 (WFPC2) , butit has since been generalized to two out of three channelsof the
Advanced Camera for Surveys (ACS)’s, namely the
High Resolution Channel (HRC) and the
Wide Field Chan-nel (WFC, see Anderson & King 2004, Anderson & King2006) and to both the WFC3 channels (UVIS and the
NearInfra-Red, NIR , Anderson 2016). The AK00 procedure shows how to construct a PSFfor a particular data set, and the PSF should in principlebe valid only for the particular average state of the tele-scope during those exposures. Fortunately, in practice, itturns out that breathing affects the
HST
PSF in specificways that often do not affect our ability to do astrometryor photometry (if we allow for a possible shift in zeropointof ± general linear astrometric trans-formations). For this reason, Anderson has constructed aset of “library” PSFs for various filters and detectors fromdata-sets that are well suited for PSF reconstruction (a largenumber of high S/N unsaturated stars and several ditheredexposures). These PSFs can be used to extract differentialpositions good to better than 0.01 pixel and differential pho-tometry good to better than 0.01 magnitude (see Bedin etal. 2013).As such, many projects do not require tailor-madePSFs, but can be done with a static archive of library PSFs.Other projects, however, do require more than just sim-ple differential astrometry or photometry on well-separatedstars. Such more complicated projects rely on the PSF ei-ther to do simultaneous fitting to multiple overlapping stars(such as in crowded fields or resolved-binary studies) or todo a deconvolution-type fitting of resolved objects (such asin weak-lensing studies of field galaxies). For these purposes,better PSF models are necessary.Unfortunately, many of the projects for which staticlibrary PSFs are not adequate do not have enough bright,well-exposed stars in the field to allow construction of a fullspatially variable model of the PSF. (It typically requiresover 560 high-signal-to-noise stars in an image, i.e., enough ∼ jayander/ to get 10 stars in each of the 7 × HST
PSF varies in an irregular way from exposureto exposure, then there will be no way to do these star-pooryet PSF-dependent projects. However, if it can be shownthat the
HST
PSFs fall largely into a one-parameter familyregulated by the telescope focus and breathing, then it maybe possible to construct a set of library PSFs, parametrizedby focus. In that case, rather than needing enough stars inan image to extract a full spatially variable PSF, it may bepossible to specify only one parameter: the telescope focus.This would allow high-precision PSF-based analyses for agreat many exposures that heretofore have been amenableonly to less rigorous analyses.In this article, we will determine whether such a familyexists for F467M (the main filter of our
HST large programGO-12911, see Sect. 3) and, if so, how easy it will be topinpoint the focus, and thus specify the full PSF, for a givenexposure. If this procedure works for our data set in onefilter, then perhaps it could be performed for other filters inother data sets as well.
An initial study of how the PSF responds in detail to changesof focus is documented in Anderson et al. (2015, A15). Thisstudy made use of the fact that the upper-left corner of thedetector is known to be out of focus relative to the rest ofthe detector, thus making that corner more sensitive thanthe rest of the detector to variations of focus.When the telescope goes from one side of focus to theother, the PSF changes are symmetric to first order: someflux is transferred from the diffraction rings to the core andthen back again. To break this degeneracy, we need to ex-plore higher-order PSF changes. Models show that when thetelescope is on one side of focus, the PSF is slightly astig-matic in one sense, then on the other side of focus its astyg-matism is the opposite.It is worth noting that the undersampled nature of the
HST
PSF makes it particularly difficult to explore this as-symmetry, since all the changes happen within the star’sinner 3 × × Figure 1.
The 4 × ied along a simple “banana”-like path in this two-parameterspace (see Figure 7 of A15).The fact that there is such a tight empirical relation-ship between astigmatism and sharpness means that thePSFs apparently come from a simple one-parameter fam-ily. It was further found that all of the star images from agiven exposure clustered in a single location along this curveand stars from different exposures clustered in different lo-cations. This led us to conclude that the different locationsalong the curve correspond to different focus levels of thetelescope.The next step was to group together exposures that hadthe same focus and extract images of unsaturated but well-exposed stars (i.e., those with between 5 × and 3 × total counts) that were in the upper-left corner. A15 thenused these extracted images of actual stars to construct anaverage PSF for the upper-left corner, one for each of eightdifferent focus levels, thus arriving at an empirical picture ofhow the PSF changes. Figure 13 of A15 shows this variationvisually. It is clear that the PSF goes from being asymmetricin the +135 ◦ direction at their first focus level, to beingasymmetric in the +45 ◦ direction at the other side of focus,at their focus-level 8.This limited study of the F606W PSF in the upper-left-corner of the WFC3/UVIS detector was an encouragingindication that it should be possible to characterize the PSFacross the entire detector in terms of focus. This is clearlythe next step to take. Because of the undersampled nature of the PSF, we need tohave multiple observations of point sources at multiple sub-pixel locations if we hope to construct an accurate super-sampled model of the PSF. The fact that the PSF changesspatially across the detector means that we need many starsin each spatially-coherent zone. Anderson & King (2000)showed that, in addition to this, we also need a way to de-termine an accurate position for each PSF-contributing star.All of this means that we require multiple dithered images,each of which has a large number of stars. Globular clustershave served as the ideal targets for this purpose.As such, large program GO-12911 (PI: Bedin) is perfectfor such a study. Its main focus is to do high-precision as-trometry of stars in globular cluster M4 over the course of12 months with an aim to measure the astrometric wob-ble of main-sequence stars that may have heavy unseen binary companions (such as black holes, neutron stars, orwhite dwarfs). Bedin et al. (2013, Paper I) provides for anoverview.The observations are divided into 12 epochs, spacedroughly a month apart. Each epoch consists of ∼
49 deepobservations with WFC3/UVIS through the medium-bandF467M filter, wherein the turnoff is just below the level ofsaturation, thus providing the maximum number of highsignal-to-noise stars. The F467M filter was chosen so thatthe ideal exposure time would be just over 339s, which is thethreshold for efficient buffer-dumping with WFC3/UVIS.The program also took short exposures in filter F775W toprovide colors, as well as short exposures in F467M to pro-vide some handle on the evolved population. But we focushere on the 589 deep exposures through F467M taken in 120single-orbit visits between 9 October 2012 and 16 September2013.
Even though the upper-left corner is more sensitive tochanges in focus than the rest of the detector, in realitythe PSF across the entire detector changes when the focuschanges, and there is surely more information in the entiredetector than in just the upper left sixteenth of the detector.It is hard, though, to use the moment-based analysis thatworked in the upper-left corner across the entire chip, sincedifferent locations across the detector are at different placesalong the local focus curve. For this reason, we sought a wayto extract an estimate of the full spatially variable PSF foreach exposure.We noted above that we would need at least 560 well-exposed-but-unsaturated stars in an image if we hope toconstrain the array of 7 × × J. Anderson & L. R. Bedin with the fine-scale detector-related variations (traceable toissues such as static distortion, or chip thickness and chargediffusion or vignetting) while at the same time accountingfor the low-spatial-frequency variations due to focus.The “basis” PSF model for WFC3/UVIS is similar tothat constructed in 2006 for ACS/WFC in Anderson & King(2006), except that instead of using an array of 9 × × × ×
8) fiducial locations is represented by101 ×
101 super-sampled grid, as in AK00. The super sam-pling is × × × × × We compared the 7 × × i and j as: d ij = 156 × X IJ (cid:0)X XY | ψ IJXY ; i − ψ IJXY ; j | (cid:1) (1) , where X and Y go from 1 to 101 covering the ∆ x and∆ y domain of the PSF, and I ( J ) goes from 1 to 7 (8),corresponding to the spatial variation of the PSF. Thesesimple difference estimates should tell us which exposureshave PSFs that are more similar to each other (small d ij )and which have PSFs that are are more different from eachother (large d ij ). Figure 2 shows the distribution of the589 ×
589 ( ∼
350 000) d ij values. Most PSFs differ by about0.05 (equivalent a 5% shift of the flux from one place in thePSF to another), but there is a significant tail that differsby more than 10%, and some that differ by more than 20%.In the field of biology, it is common to characterizethe difference between two species in terms of the total“difference” of the DNA code that represents them. Mul-tiple species can be inter-compared on a “phylogram”, atwo-dimensional plot that represents graphically the multi-dimensional differences among the various species in terms ofdifferences between points in a 2-dimensional space. Such di-agrams show, for instance, how similar humans are to chim-panzees, dolphins, and paramecia.In the present study, we also have a large number ofdata sets that have been characterized by their differences insome quantitative domain. It is worth investigating whethersuch a two-dimensional graph could be useful in this case aswell.We explored several approaches to constructing suchan optimized phylogram-type plot. Of course no two-dimensional plot can do a perfect job representing the myr-iad of differences among the PSFs, but our aim is to con-struct the best possible two-dimensional plot that represetsthese differences. In phylogram-type plots, the distance be-tween species is representative of the log of the number ofdifferences in their DNA sequences. Here, we seek a diagramwhere the two-dimensional “distance” between two PSFs inour plot is representative of d ij , the difference computedabove.We devised several stragies for coming up with such adiagram and show one such strategy in Figure 3. The goal isto place each point such that its distance (in ∆PSF terms)represents how different the PSFs are. The x and y axes inthis space are arbitrary, but the distance between two pointsis representative of how similar the PSFs are.We start with the representative of how different thePSFs are. We start with the first exposure at ( x , y ) =(0 . ,0 . ). For the second exposure, we explore all possibletrial locations ( x , y ) between ( − . . , − . . d (which happens to be 0.025) and r (which we define tobe p ( x − x ) + ( y − y ) )). We identify the minimum of E ≡ | d − r | to be the best location to place the secondexposure relative to the first; the quantity E represents thedifference between the distance as measured in the plot andthe distance measured between the PSFs. The upper left plotin Figure 3 shows the contours of E . There is a circle (shownin green) of best-placement locations for exposure d equals r along the circumference of this circle, and E has its minimum value of 0.0. We adopt an arbitrary pointin this circle as the best placement location for exposure Figure 2.
The distribution of values for d ij , which represents the difference between the PSFs extracted from two different exposures.A value of 0.1 means that 10% of the flux from one PSF would have to be re-distributed to arrive at the other PSF. Once exposure E (defined to be P − n =1 | d n − r n | ). The previous pointsare shown in red and the lowest contour is shown in green.The third exposure (shown in blue) is placed at the loca-tion with the lowest value of E . After each new placement,we explore the local neighborhood around each point to seewhether it might have a lower value of E by shifting by 0.001in any direction.The other panels in Figure 3 show the optimal loca-tion to place exposures J. Anderson & L. R. Bedin
Figure 3.
The panels show the construction of the phylogram by adding one expsure at a time. Each panel shows the previously addedexposures in red. The contours identify the best location to place the current exposure with respect to the previous exposures such thatthe computed d ij (the distance between the PSFs) is as close as possible to r ij (distance between the points on the graph). The axesare arbitrary, but their scale represents that of d ij , namely how much flux would have to be moved to go from one PSF to another. different from the PSFs in the middle than they are fromeach other. This is in agreement with what we saw in thebanana plots from Figure 7 of A15: when the PSF is in fo-cus, it has a maximal fraction of its flux in its central pixel,but this fraction goes down in a similar way on both sidesof focus. The PSF is not identical, however, on the differentsides of focus, as indicated by distance between the two sidesof the wishbone.Since there are a few outliers from the trend, we in-vestigate several possible causes. We extracted the jitter filefor each exposure and determined an RMS with respect to the average pointing. In Fig. 5 observations with large jitterare shown in red. We also determined which of the obser-vations were able to be placed in a location on the plotthat was a good represtation of the difference in their PSFs.The two-dimensional plot allowed most observations to beplaced such that E ≡ P | d − r | is small, but some observa-tions (see left inset) were not. Perhaps this is an indicationthat a third dimension might exhibit even more order, butsince these observations are few, we chose to ignore themand focus on the those that followed the majority trend.Figure 6 shows the same phylogram-type plot, but in Figure 4.
The panels show how the placement of the exposures in the phylogram plot change as more point are added to the diagram.The open end of each black curve show the initial placement of each exposure (based on the preceding exposures), and the red dot at theother end shows the final, optimal resting place for the exposure, after its position has been allowed to creep slowly as more and morepoints are added. each of the small panels we highlight in red the observa-tions that correspond to a particular epoch. The epochs arespaced about a month apart and the visits within each epochspan between 21 and 45 hours. It is clear that the PSF is notconstant over an epoch, though the PSFs do typically varyover a relatively narrow part of the entire focus range dur-ing an epoch. There is no clear progression over time frommonth to month.Several visits in Epoch 2 do not follow the focus curve.Inspecting the images, we see that several of them sufferedfrom guide-star failures. Indeed, this is borne out by an in- spection of the jitter files. All the other epochs follow thewell-worn focus path to within 0.02 (corresponding to a 2%average difference between the PSFs, meaning that to getfrom one PSF to another one would have to rearrange about2% of the flux).Now that we are able to characterize individual expo-sures in terms of where they lie along the focus curve, wecan group together exposures at similar focus levels. Fig-ure 7 shows the same points as in Figure 5, but this timewe have color-coded those observations at each focus level.We have drawn in a fiducial line for the focus curves and
J. Anderson & L. R. Bedin
Figure 5.
This the final reference phylogram-type plot where each exposure is represented by a dot, and the dots are separated by“distances” that correspond to the difference between their PSFs. The axes are arbitrary, but units for this are in terms of the averageabsolute difference between the PSFs ( d ij , such that 0.1 means that to get from PSF to another you have to rearrange 10% of the flux.The inset at the lower left shows E min for each exposure. Stars with low E min have consistent placement with respect to most of theother exposures. Points are marked blue if they have a high value of E min . The inset in the lower right shows the jitter RMS for eachexposure. Exposures with more pointing jitter than typical are flagged in red. consider all observations within about 0.02 (98% PSF agree-ment) of the focus curve to be representative. We arbitrarilydivide the curve into 11 distinct focus zones and color-codeblack the odd zones and in different colors the even ones.The open circles denote the few observations that did notfollow the general focus trend as well as the others (mostlyduring second epoch) that suffered major guiding failures.The first focus group had 4 exposures and the last group5, but the other groups had between 7 and 150 representa-tive exposures. Focus groups at the extremes naturally have fewer exposures, reflecting the fact that most of the timesthe telescope is on focus. The next step was to take all the exposures associated witha given focus group and determine an average PSF for eachgroup. AK00 shows that in order to derive an accurate PSFfrom stellar profiles, we must have accurate positions andfluxes for each star in each exposure. In undersampled im-
Figure 6.
This plot shows in black the same points as in Figure 5, but for each of the 12 epochs it highlights the 49 exposures takenduring that epoch. ages, it is hard to determine accurate unbiased positions,so AK00 developed a procedure to take a dithered set ofdata and iterate between the solution for PSF and for thepositions and fluxes of the stars in a virtuous cycle.Here, we simply determined a position for each star us-ing the average “library” PSF and determined the flux bymeans of the total amount of light within a 5-pixel radius.This is considerably larger than the 5 × flc im- ages , we can extract a spatially variable PSF from the setof exposures associated with each focus zone. As mentionedearlier, there were between 4 and 150 exposures in each zone.Since we are now extracting detailed PSF models, it isbeneficial to illustrate specifically where each fiducial PSFis located on the detector. As we mentioned before, eachchip is covered by an array of 7 × The flc images are produced by the STScI archive pipeline.They are flt images corrected from imperfect charge-transferefficiency (CTE) with an algorithm essentially based on the onepresented in Anderson & Bedin (2010). J. Anderson & L. R. Bedin
Figure 7.
This plot shows the specified zones along thephylogram-type focus curve. edges and corners of each chip in order to avoid the needfor extrapolation. The PSF is linearly interpolated in be-tween fiducial locations but is not assumed to be continuousacross the intra-chip gap. Figure 8 shows the locations forthe fiducial PSFs for our UVIS PSF model.The next figure (Figure 9) provides a snapshot of thedata that went into our PSF models. From AK00, the PSFmodel tells us the fraction of a star’s light that should fallin pixel [ i, j ] relative to its center at ( x ∗ , y ∗ ). We can thusmodel the star with the following equation: P ij = z ∗ × ψ ( i − x ∗ , j − y ∗ ) + s ∗ , where z ∗ and s ∗ are the star’s total flux and background skyvalue, respectively. This is the equation for a line, where theslope is the flux and the intercept is the sky. We can invertthis to solve for the PSF at a single point in its domain(∆ x ,∆ y ) from the value of pixel [ i, j ]: ψ (∆ x, ∆ y ) = ( P ij − s ∗ ) /z ∗ , where ∆ x = i − x ∗ and ∆ y = j − y ∗ . Each pixel in each star’simage thus provides an estimate of the PSF at one particularlocation in its domain.When we allow for spatial and focus variations,we see that the PSF is now a 5-dimensional function: ψ (∆ x, ∆ y ; i, j ; f ). To visualize it, we will consider two di-mensions at a time. In Figure 9, we plot the samples fromthe center of the PSF ( | ∆ x | < .
25 and | ∆ y | < .
25) for themiddle focus level ( f =6) as a function of detector i coordi-nate for six horizontal slices across the detector (shown inFig 8).We see that even for the “optimal” focus level, the frac-tion of light in the central pixel can vary from 0.175 to 0.225,more than ± ∼ Figure 8.
This plot shows the locations of the fiducial PSFsacross the two WFC3/UVIS detectors. The useful part of eachchip is 4096 × × × The solid blue dots and connecting lines show the actualPSF model across each strip. The dark-blue line does notrepresent the data perfectly, but it is good to better than0.5%.On the right, we show the same connected points forthe central-focus sample ( f =6 in dark blue, as on the left)and also show the extracted-model points for the two mostextreme focus levels, f =1 and f =11 in green and cyan, re-spectively. These green and cyan curves are almost every-where lower than the dark-blue curve, which is consistentwith them being much more out of focus. The central valueof the PSF in the extreme focus curves varies from 0.13 to0.18.The previous figure showed how the central pixel of thePSF varies with position and focus. Figure 10 shows theentire central region of the PSF for the middle focus level( f = 6). In each of the 7 × × ×
21 PSF grid-points) in terms of their residual with respect to the aver-age PSF across the detector for f = 6. Black corresponds tomore flux than average and white corresponds to less flux. Itis clear that there is a large sweet spot in the middle of thedetector, and the PSF becomes less tight towards the edgesof the field, particularly in the upper-left corner, which weknow to be very sensitive to changes in focus. The sharp“happy bunny” feature from SB13 at the bottom is alsoclear.The next figure, Figure 11, shows how the PSF varies Figure 9.
The left plot shows the estimates of the central value of the PSF as a function of x within 6 horizontal slices across the chips(as shown in the previous figure). The individual points correspond to estimates of the central value of the PSF from individual starsin individual exposures that happen to be centered on a pixel to within ± x and y . The units on the left therefore refer tothe fraction of a star’s light that would land in its central pixel if the star is centered on that pixel. The individual exposures here areall taken from the group with focus level 6 (the middle of the focus range). The solid blue dots show the value of the function for thefiducial locations of the PSF. The line simply shows the linearly interpolated value between the fiducial locations. The right plot showsthese blue (focus=6) points along with the points for the most extreme focus location on one side of focus (green, focus=1) and the otherside of focus (cyan, focus=11). with focus for 7 different locations on the detector. It is clearthat in the center of the detectors (the top two panels) themiddle focus position has the sharpest PSF, and the PSFgets worse on either side of focus. This is true for the “happybunny” location as well (see the bottom row). The upper leftand right corners appear to be in their best focus closer tothe f = 1 extreme focus position.It is clear from all of this that the PSF behavior is complicated, both spatially and as a function of focus. Thereis no obvious way to reduce this from a 3-parameter family(x,y,f) to anything simpler. Even so, the behavior with focusand position do appear to be adequately characterized byour empirical modeling.We have saved our focus-diverse PSF model in a sim-ple four-dimensional fits image that is 101 × × ×
11, where the first two dimensions correspond to the PSF J. Anderson & L. R. Bedin
Figure 10.
The array of PSFs for the middle focus position( f =6), shown with respect to the average across the detector forthis focus level. Black corresponds to more flux. Figure 11.
Each row shows the PSF at a different location onthe detector, as indicated in the schematic at the left. The panelsfrom left to right then show the central region of the PSF as afunction of focus level, with focus level 1 in the leftmost columnafter the schematic and focus level 11 in the rightmost column.The PSFs are shown relative to the average over all focus levelsat that location. Black corresponds to more flux. model itself. The third dimension (56) corresponds to the7 × Now that we have a focus-diverse PSF model, we can fitit to the stars in a given exposure in order to empiricallydetermine the best focus for that exposure and consequentlythe best PSF for that exposure. The fact that we have a fullspatially variable model for each focus level means that wecan use stars from all over the detector to help us determinethe focus.
To determine the focus for each exposure, we identified aset of about 1000 bright isolated stars in the image. Wethen fit each star with the model PSF that is appropriatefor its location on the detector at a range of focus levelsbetween 1 and 11, stepping by 0.2 focus level (we used linearinterpolation to get the PSF between focus levels).For each fitted star at a given focus level, we determinedthe optimal (x,y) position and flux and then got an estimateof the quality of fit, which is simply the absolute value of theresiduals between observed pixels and PSF model, scaled bythe flux of the star. Well-fitted stars tend to have residualsof less than 3%. We determined an average quality of fit forthe stars for each focus level and identified the best focus asthe one that produced the smallest average residual for allthe stars in the image.We thus determined an empirical focus level for eachexposure with a precision of about a fifth of a focus level.Figure 12 shows the focus level determined for the time-sorted exposures within each epoch in the twelve labeledpanels. We connect with a solid line those exposures takenwithin 500 seconds of each other (indicating that they weretaken one after another). We reiterate that the focus level foreach exposure was determined entirely independently, thusthe trends we see reflect real and regular PSF variations.Some of the epochs exhibit extremely regular variationsthat repeat every orbit. Others show less regularity. Clearlythe focus variations depend both on the telescope’s insola-tion history and how the Sun heats it during its on-sourcepointing. It is beyond the scope of this study to try to under-stand these variations, rather our aim is simply to measurethe focus and arrive at the best possible PSF. Nevertheless,it is clear that this represents a powerful way to examine fo-cus changes due to short-term breathing or other, possiblylonger-term, phenomena such as focus drift.
Given the success of the fitting-focus by exposure above, itis worth asking how many stars are needed to pin-down thefocus. To examine this, we identified seven images that hadfocus levels of 1, 3, 5, 6, 7, 9, and 11. For each exposure, weselected 500 or so stars from across the detector that hadS/N of 300 or higher. For each individual star, we deter-mined an optimal focus level based on the focus parameterthat provided the best quality of fit to the star’s central 5 × Figure 12.
In each of the twelve panels, we show the extractedfocus determination for the ∼
50 exposures taken during the cor-responding epoch. Exposures are connected with a solid line ifthey are taken within 500 s of each other. in the image are excellent predictors of the focus: typicallyeach star can predict the focus to within half a focus level.To show this even more visually, in Fig.14 we show ex-plicitly the scaled residuals for six stars, probing three differ-ent pixel phases at each of two different focus levels. Whereasin A15 we had to restrict our analysis to stars that were cen-tered on pixels in one corner of the detector, here it is clearthat with a comprehensive model, the variatiation with fo-cus can be sensed and calibrated for all stars at all locationson the detector.We clearly do not need thousands of stars to pin-downthe focus level. Furthermore, it is not clear from Figure 13how a star’s signal-to-noise will affect our ability to deter-mine focus from it.To explore this, we took the image in the fifth panel( ic0532ecq ), found to have focus level ∼
7, and determinedthe focus level for individual isolated stars with S/N from400 ( m = −
13) to S/N 30 ( m = − . ± σ range for the distri-bution within each magnitude bin. It is clear that a handfulof stars of moderate flux can pin down the focus to wellwithin a focus level. This is good news, since even sparsefields (such as the UDF) have about 10 stars, several withreasonable S/N. Clearly with focus-diverse PSF models formany more filters it will be possible to characterize the focuslevel for a large fraction of HST images.
The
HST platescale is not constant. As
HST orbits theEarth and the Earth orbits the Sun, its space velocity iscontinuously changing. Relativistic velocity aberration (VA)
Figure 13.
In each panel, we show the focus level as determinedindependently by several hundred extremely bright but unsatu-rated isolated stars. The seven panels showcase the results fromseven images that span the observed focus range. can lead to changes in the platescale of up to one part in 10 (Cox & Gilliland 2003). The VAFACTOR keyword in the imageheaders provides a calculation of the VA based on Hubble’svelocity and pointing vectors averaged over the exposuretime.In addition to velocity aberration, breathing can alsoaffect an observation’s platescale, and the 589 exposures ofour field give us a unique opportunity to examine the ef-fect of breathing on platescale. For each exposure, we candetermine an average platescale by relating the distortion-corrected positions for stars in that frame ( x, y ) to the po-sitions of stars in the master frame (
X, Y ). The platescaleis simply √ AD − BC of that global linear transformationdefined by ( X = A x + B y + X ◦ Y = C x + D y + Y ◦ . (1)The left panel of Figure 16 plots the observed platescale (rel-ative to the average) as a function of the VAFACTOR keywordin the header of each image. Clearly, much of the platescalevariation can be explained by VA. The middle plot shows theresidual between the observed and VA-predicted platescale.Finally, the right-most plot shows this residual as a func-tion of our determined focus level. It is clear that the same“breathing” that causes changes in focus also causes changesin the amplitude of the platescale up to 0.04-pixels.
Now that we have identified the best-focus PSF for eachexposure, it is worth considering how well that PSF actu-ally fits the stars. To do this, we investigated four differentPSFs: (1) the “library” PSF we constructed for F467M that J. Anderson & L. R. Bedin ! " $ % & ’ " ( ) * $ ’ + ,- . / ’ % ’ , ’ : ’ ; ’ < ’ = ’ > ’ ? ’ @ ’ ,- ’ A ’ ,, ’ Figure 14.
The scaled fitting residuals for six stars, probing three different pixel phases and two images at different focus levels. The leftpanel shows the 5 × flc image centered on the star. These were the pixels that the focus-diverse PSF was fitted to.The next panel shows the quality-of-fit metric as a function of assumed focus level. Finally, the eleven rasters show the scaled residualsfor the eleven fiducial focus levels. For each fit, we minimized the residuals by allowing the center of the star and its flux to vary freely.Dark corresponds to more flux in the image than in the model. The direct star image on the left is shown scaled down by a factor of 10relative to the residuals. The top three rows of panels were taken from exposure ic0532ecq and the bottom three from ic050kc4q . Figure 15.
The focus level as determined by isolated stars at arange of brightness in image ic0532ecq (found to have focus level7). Clearly the bright stars are better predictors of focus, but evena few faint stars can determine the focus quite accurately.
Figure 16. (Left) Observed relative platescale plotted againstthe velocity-aberration factor keyword (
VAFACTOR ) calculated onthe base of the (predicted) telemetry, and provided in the header.(Middle) Difference beteen observed and predicted scale. (Right)Difference plotted against the determined focus-level for each ex-posure. The dotted lines correspond to ± was designed to represent average focus, (2) the fitted-focusPSF, (3) the 4 × × × x, y ) location and flux that minimize the resid-uals in a least-squares sense, taking into consideration the Figure 17. (Left) The average fractional star-fit residual foreach exposure, plotted as a function of extracted focus for fourdifferent PSF treatments, as labeled on right. (Right) Histogramsof star-fit residuals for the four treatments.
Poisson error in each pixel. Since it is not a time-consumingprocess, we simply do a grid search for ( x, y ) and at eachtrial position, determine the flux by simple aperture pho-tometry (knowing from the PSF what fraction of the star’slight should fall within the aperture based on its trial po-sition). Once we have a best-fit position, we determine thetotal absolute residual between the observations and modeland divide by the total flux to get a scaled residual.Figure 17 shows the average fitting residual for thebright stars within 1.5 magnitudes of saturation for eachof the 589 exposures as a function of fitted focus for thatexposure. The black points show the fitting error for theunperturbed temporally constant “library” PSF. The cyanpoints show the residuals for the fitted-focus PSF. The greenpoints show the perturbed library PSF, and the blue pointsfor the perturbed focus-fitted PSF. It is clear that the bluepoints have fitting residuals smaller than 2% for all focuslevels. The perturbed library PSF is almost as good, but itloses quality when the focus is considerably off nominal. Thefocus-fit PSF has residuals of about 3% everywhere, and thelibrary PSF has about 4% errors when the focus is good,but has a trail to well over 20% errors when the focus is off.The histograms on the right show the same data summedover all focus levels.One of the most obvious effects of breathing is that thefraction of light in the core of the PSF changes. When fluxesfor stars are determined by fitting a “library” PSF to thecore, breathing results in photometric zero-point shifts fromexposure to exposure, typically ± J. Anderson & L. R. Bedin
Figure 18.
The empirical photometric zeropoint shifts for the four different PSF treatments plotted against exposure number.
Figure 19.
Distribution of RMS photometric residuals (solid)and 2-D astrometric residuals (dotted) for the four different PSFtreatments. shifts of about ± × × Program GO-12911 also included a short 20s F775W expo-sure in nine out of ten of the orbits available within each ofthe twelve epochs, for a total of 108 short F775W exposures.This is enough for a preliminary examination of the varia-tion in that PSF. As above with F467M, we used a “library”F775W PSF to fit the stars in each exposure and developeda 4 × Figure 20.
The “phylogram” plot for F775W. It is clear that the PSF does not vary nearly as much for this redder filter.
PSF to another focus level, we would have to rearrange atmost 10% of the F775W PSF.An additional consideration that arises when modelingthe PSFs for a wide-band filter is the fact that blue star andred stars can have somewhat different effective wavelengthsdue to their different spectral-energy distributions (SEDs).If we are able to come up with a focus-diverse set of PSFsfor wide-band filters, it might be possible to introduce anadditional parameter to deal with the color of the stars, butthat is beyond the scope of this effort.
We have made use of the uniform dataset of GO-12911 toexplore and model the variation of the F467M PSF in greatdetail. We have shown that, by and large, the PSF varies inpredictable ways along a single-parameter curve, where thesingle parameter is related to the telescope focus.We have grouped together images that were taken atsimilar focus levels and have constructed for each of elevenfocus levels a full spatially variable representation of thePSF (i.e., with 7 × × J. Anderson & L. R. Bedin the chips, the PSF is actually sharpest at off-nominal focuspositions.Although in a previous work (Anderson et al 2015), wemade use the fact that the upper-left corner is particularlysensitive to focus changes in order to study PSF changeswith focus, we find here that the PSF across the entire de-tector is actually quite sensitive to focus. If we fit a typical,bright (S/N >
50) star with PSFs at a range of trial focuslevels, we find that we can identify the image focus to bet-ter than 1 focus level (out of eleven). With only a handfulof stars, it should be possible to pin-down the focus levelof any exposure. This will make it possible to construct atailor-made PSF for every deep exposure, since a typicaldeep field has at least five stars with reasonable signal.The PSF model we constructed here is for F467M, ad-mittedly an uncommon filter. However the procedure we de-veloped can be used for the entire archive: it does not de-pend on the fact that we are observing the same field in allthe exposures. Therefore it should be possible to construct afocus-diverse PSF model for all of the common WFC3/UVISfilters. The WFC3/UVIS instrument team is currently pur-suing this, as such a detailed understanding of focus changeswill allow the engineers to make better determinations ofwhen to re-adjust focus. Having access to a robust empiri-cal focus measurement will also help engineers at the
SpaceTelescope Institute to develop a more accurate model of howtelemetry and environmental data may be better able to pre-dict focus variations.Overall, we find that the focus-diverse PSFs representa significant improvement in the quality of fit for sourcesand in the absolute photometric zeropoint for exposures,the improvement is particularly significant when the tele-scope is out of focus. The new PSFs also represent a modest—although measurable— improvement in the photometricand astrometric precision. As such, they will clearly makethe biggest difference for point-source/galaxy discrimina-tion, detailed PSF-fitting to resolved or slightly resolved ob-jects, and for absolute-catalog work.The particular M 4 project at hand (GO-12911), whichhas enabled this detailed PSF study, will benefit of this im-proved PSFs, not only directly trough the marginally im-proved astrometric and photometric precision, but also be-cause of the significant improvement in the PSFs shape,which will help to better disentangle blends from isolatedstars. Therefore, we will use the perturbed focus-diverse inour high-precision astrometric wobble analysis, which is thenext step for the project.
ACKNOWLEDGMENTS
JA acknowledges support from STScI grant GO-12911. LRB ac-knowledges PRIN-INAF 2012 funding under the project enti-tled: ‘The M 4 Core Project with Hubble Space Telescope’ . Wewould like to thank Elena Sabbi, Linda Dressel, Kailash Sahu,and Matthew Bourque for many useful discussions in the courseof this work.