Stellar Photometry Techniques Giampaolo Piotto Dipartimento di Astronomia Universita’ di Padova

1. Stellar Photometry Techniques Giampaolo Piotto Dipartimento di Astronomia Universita’ di Padova These lectures have been inspired by a set of lectures given at the “V Escola Avancada de Astrofisica”in Aguas de Sao Pedro, Brazil, in 1989, by Peter B. Stetson, the maestro virtuoso of technique of stellar photometry on digital images, whose work has made the job of a generation of astronomers so much more pleasant and productive.

2. Why do we need accurate stellar photometry?

3. Color-magnitude diagrams: We need to measure fluxes (and colors) Comparison with the models. Age 17 mag range! Evolutionary sequences

4. The problem of the double MS and of the multiple SGBs and TO in Omega Centauri Sometimes accurate photometry is of fundamental importance… Bedin, Piotto et al. 2004, ApJL, 605, L125

5. Luminosity functions Evolutionary clock We need to count stars Mass functions

6. Astrometry We need to measure stellar positions …to determine membership. NGC6121=M4

7. M4: (U,V,W)LSR = ( 53+- 3, -202+-20, 0+- 4)Km/s (P, Q,Z)LSR = ( 54+- 3, 16+-20, 0+- 4)Km/s …to meausure Absolute proper motions

8. INTERNAL DYNAMICS (Bedin et al. 2003)

9. Astronomers do not like easy jobs! It is the ability to count stars, measure fluxes, and positions in crowded environments which makes stellar photometry an art!

10. Stellar Photometry Packages RICHFILD Tody 1981 KPNO ROMAFOT Buonanno et al. 1983 A&A, 126, 278 WOLF Lupton, Gunn 1986 AJ, 91, 317 STARMAN Penny, Dickens 1986 MNRAS, 220, 845 DAOPHOT Stetson 1987 PASP, 99, 191 ALLSTAR Stetson 1994 PASP, 106, 250 ALLFRAME Stetson 1994 PASP, 106, 250 LUND Linde 1989 1st ESO/ECF data analysis WS DoPHOT Schechter,Mateo 1993 PASP 105, 1342 Shara ePSF Anderson, King 2000 PASP, 112, 1360 Plus a number of generic photometry softwares (INVENTORY, SEXTRACTOR, etc.) DAOPHOT

11. Fundamental tasks for stellar photometry (*) FINDcrude estimate of star postion and brightness PSF determine stellar profile (point spread function) FIT fit the PSF to multiple, overlapping stellar images (and sky) SUBTRACT subtract stellar images from the frame ADD add artificial stellar images to the frame (*) accurate stellar photometry needs accurate astrometry.

12. Before starting….. • There are at least five things you should tell to your computer • before starting: • Read out noise; • Conversion factor (electrons to ADU); • Maximum linear signal (physical or electronic saturation level); • Map of bad pixels, rows, columns; • Size of typical stellar images (seeing, FWHM in pixels)

13. FIND(1) • First you must FIND the stellar-appearing objects in the frame. • Each program has its own method - sometimes several methods – • of performing this, but the basic idea is to produce an initial list • of approximate centroid positions for all stars that can be • distinguished in the two dimensional data array. • The star finder must have at least some ability to tell the difference • between a single star, a blended clump of stars, • a galaxy (or extended object), and a noise spike in the data. • Second, for the following steps one needs a crude estimate of each • star's apparent brightness at the same time, e.g. with • some simple aperture PHOTOMETRY algorithm. Note: From here on, I will indicate DAOPHOT commands using upper case MAGENTA color

14. FIND (2) • Basic idea: A star is brighter than its sorrounding; • Simple method: set a brightness threshold at some • level above the sky brightness level; • Complications: • The sky brightness might vary across the frame • Blended objects, extended objects, artifacts, • cosmic rays must be recognized.

15. A possible solution:Once given a numerical value for the FWHM, DAOPHOT's FIND routine then assumes that the stellar profile is a circular Gaussian function with that full-width at half-maximum, and just goes through the entire picture fitting Gaussian profiles to a small region around every single pixel (excluding a narrow border around the frame). For each pixel (i0, j0) it performs the following fit, with D(i,j) the counts on the pixel: where with It is a least square solution, so:

16. Example of a gaussian fit to the original data. • Note: • The broad galaxy is suppressed • by the convolution; • The background value of C is 0 • even if originally the background • is nonzero and sloping; • The blended pair is better • separated; C Note that the stellar images are critically sampled (FWHM=2.4), i..e. it is hard to distinguish a star from a cosmic ray. • A star image • Blended pair • Galaxy • Cosmic ray hit • Low value bad pixel

17. Two parameters to help eliminating non-stellar objects: SHARPNESS di0,j0 = Di0,j0 /<Di,j >, with (i,j) near (i0,j0), but different from (i0,j0) SHARP=d i0,j0/Ci0,j0 ROUNDNESS ROUND=2*(Cx-Cy)/(Cx+Cy) Cx from the monodimensional Gaussian fit along the x direction Cy from the monodimensional Gaussian fit along the y direction

18. Aperture Photometry • It is the most accurate flux measurement, for • non-crowded images; • Simply integrate counts within a given aperture • (possibly circular), and subtract the background • counts estimated in a nearby region; • The crucial (and most delicate problem) is the • estimate of the sky background.

19. Background evaluation The background measurement can be rather tricky (because of the crowding) A good estimate of the local sky brightness is the mode of the distribution of the pixel counts in an annular aperture around the stars. Poisson errors make the peak of the histogram rather messy. A good guess of the background level is: mode=3x(median)-2x(mean) (which is striclty true for a gaussian distribution) NOTE: this background estimate is rather important, as it is the only background measurement in DAOPHOT

20. The stellar profile model: the PSF Ideally, the model stellar profile should come from: 1. The BEST, most luminous, most isolated stars 2. NUMEROUS stars (to reduce noise), well spread within the frame (to measure PSF spatial variations). When the PSF is fitted to some arbitrary stars in the digital image,the uncertainty of the fit will be dominated by the quality of the data for the program star, not by the quality of the model profile. In order to construct a model profile from the average of several stars, the observed data for those stars must be registered to the same centroid and to a constant background level and peak intensity.

21. Well sampled stars: ideal case Badly undersampled.Star profile strongly depends on the position of the center within the central pixel. The problem is worsened by the intra-pixel sensibility variation.

22. In the case of badly undersampled stars the PSF determination becomes very difficult. Still, an appropriate PSF is crucial, expecially for astrometry. For a detailed description of the problem, and for a possible solution, see Anderson and King (2000, PASP, 112, 1360). A&K solution is optimized for very accurate astrometry.

23. The stellar profile model: the PSF The detailed shape of the average stellar profile in a digital frame must be encoded and stored in a format the computer can read and use for the subsequent fitting operations. There are two possible approaches: 1. The analytic PSF. E.g. a gaussian, or, better a Moffat function: 2. The empirical PSF. i.e. a matrix of numbers representing the stellar profile.

24. The analytic PSF • Advantages: • Once the parameters of the analytic function are known, • the profile fitting is quick and accurate; • 2. The PSF can be integrated over finite pixels in • undersampled (FWHM < 2.5 pixels) images; • 3. Multiple PSF stars automatically registered and scaled. • Problems: • 1. Not very flexible: has difficulty modelling complex • profiles caused by optical aberrations or tracking errors • 2. If one tries to include too many parameters in the • model, convergence of the model may prove difficult. • This is the approach of ROMAFOT and STARMAN

25. The Empirical PSF Advantages: 1. It is able of encoding arbitrarely complex stellar profiles. Problems: 1. Relies on numerical interpolation techniques  may loses accuracy (*) in undersampled images (but see ePSF). 2. Requires accurate registration and scaling before averaging multiple PSF stars. (*) Application of the empirical PSF requires 2 interpolations in matching the model stars to the program stars : i) The stars defining the PSF must be interpolated to a common pixel grid before they can be averaged; ii) the model PSF must be interpolated to the program star pixel grid for the fitting to take place; This is the approach of RICHFLD, WOLF, ePSF

26. The Hybrid PSF 1. First, fit the best possible analytic profile to the PSF stars; 2. Then subtract these best analytic profiles from the images of the PSF stars 3. Register and scale the array of the residuals remaining after these subtractions, and average them together to form an empirical look-up table of corrections from the analytic profile to the BEST PSF

27. The Hybrid PSF Advantages: 1. The experience teaches that the analytic profile contains ~97% of the information in the stellar profile (i.e. image anomalies due to aberrations and tracking errors represent only about 3% of the stellar flux); 2. The analytic profile can be integrated over finite pixels: it is not very sensitive to undersampling; 3. The empirical look-up table of corrections is still subject to interpolation errors, but these now amount to a small fraction of ~3% of the stellar profile. Problems: 1. It is a little more work (for the computer!), but it is worth it! This is the approach of DAOPHOT

28. Variable PSF Especially with large, modern CCDs, optical aberrations in the telescope can cause the shape of the stellar profile to change with position in the field. What to do? If there is a sufficent number of PSF stars well distributed in the field, we can use roboust least-squares techniques to replace each element of the look-up table of corrections with a function (most easily, a polynomial) of the star position within the field: ai,j = (ai,j + bi,jXk + ci,jYk +…) , where (Xk,Yk) is the position of the star k in the frame.

29. And now…the real stuff…: INPUT PARAMETERS: GENERAL INPUT PARAMETERS: APERTURE PHOTOMETRY

30. The PSF stars must be BRIGHT and CLEANED Contaminating stars must be removed This shows the relevance of the SUBTRACT routine

31. The PSF determination is an iterative process! The PSF model after three iterations

32. After the starting guesses of the centroids (FIND) and brightness (PHOTOMETRY) are measured, and the PSF model determined (PSF), the PSF is first shifted and scaled to the position and brightness of each star, and each profile is subtracted, out to the profile radius, from the original image. This results in an array of residuals containing the sky brightness, random noise, and systematic errors due to inaccuracies in the estimate of the stellar parameters. From the PSF we know the first derivatives of the model profile with respect to the (x,y)-centroid, and knowing the star brightness, first order corrections to the stellar parameters are computed by least square solutions of the system of equations:

33. Example of a crowded image in the outkirts of a globular cluster

34. The same image after the file fitting and subtraction routines to the original star list. Many secondary components and blended doubles are present

35. The same image after two passes through the find-fit-subtract loop

36. Improving our photometry DAOPHOT fits multiple stars with partially overlapping profiles (distances smaller than 1 PSF radius+1fitting radius). An improvement of the original program is in ALLSTAR, which performs the simultaneous determination of position and brightness estimate for every star in a digital image. In other words, ALLSTAR calculates the first order incremental correction to each star estimated position and brightness. The huge 3Nx3N matrix is inverted piecewise: The order in which stars are considered is sorted so that the matrix is block diagonal  only stars which actually have pixels in common within their fitting regions are treated in the same submatrix inversion. Having calculated the new incremental correction, ALLSTAR goes back to the original image and subtracts the stars with the improved values of position and brightness. ITERATE! When positional and brightness corrections become negligible, the star is permanently subtracted from the original image, and the parameters stored in a file.

37. Matching stars between different digital images Important astronomical information is often extracted from multiple images of the program object(s). These images could be taken with different pointings, orientations, filters, and even at different telescopes. Once a list of common stars is constructed, the determination of the geometrical transformation parameters is a simple least-square problem. The real problem is to find an efficent way to match many thousands of stars located in dozens of images.

38. The triangle method The basic idea is that any translation, rotation, scale change, or flip is not going to change the basic shape of a triangle, although of course it will change the size and orientation. The method, then, is to take the stars in each star list in groups of threes, and intercompare the shapes ofthe triangles that result. When you allow for the fact that there may be arbitrary translations, rotations, scale changes, or flips of the positional coordinate system, each triangle contains two independent, invariant shape parameters. There are a number of ways that these parameters could be defined. One possibility is to choose: parameter 1 as the ratio of the length of the triangle's intermediate side to its longest side,b/a, and parameter 2 as the ratio of the shortest side to the longest side, c/a.

39. By definition: Which implies also: Given some triangle defined by three arbitrary points in some (x, y)space, that triangle can be represented by a point in two-dimensional (b/a, c/a) space. Because of the obvious definitions just given, not all parts of (b/a, c/a)-space can be occupied, but the same three stars - no matter how you shift, rotate, expand, contract, or flip the coordinate system - will always be projected to the same point in (b/a, c/a) space. One starts by sorting in magnitude the star lists. He chooses the three brightest stars and check whether they form a similar triangle. Then he keeps adding stars, till a sufficent number of similar triangles (I.e. matches) are found. This method is used by DAOMATCH (P. Stetson).

40. The final list of stars Once a provisional list of common stars is identified, it can be used to obtain a provisional geometric transformation matrix. The program starts off by considering the first input list as a "master" list. Taking each star in turn from the second input list, it applies the provisional transformations derivedto determine the star's position in the coordinate system of the master list. It then goes through the master list, looking for that star which lies closest to the transformed position of the star from list 2. If it does find a star in the master list which is within some critical distance of the transformed position (initially several pixels) the star from list 2 is provisionally identified with that star in the master list. If that star in the master list had already been provisionally identified with some other star from list 2, whichever star has a transformed position closer to the "master" position will remain provisionally identified with it; the other gets "bumped" and must go off looking for some other star in the master list that it can be identified with. Then, ITERATE (reducing the critical distance parameter) to improve geometric transformation. An efficent program which does the job is DAOMASTER (P. Stetson).

41. Going deeper….! Once we have the final geometric transformation we…are ready to start it all over, if we are really interested to reach the limiting magnitude of our data set and measure the faintest objects in our images!! The idea is very simple. We use ALL OF THE IMAGES of the same field, independently from the pointing and filter; we use the geometric transformation obtained from the cross-correlation of the star lists from our fitting photometry software (ALLSTAR for the aficionados) in order to align all images to the same reference system, and sum them, in order to obtain the highest S/N image (use MONTAGE2, by P. Stetson!). We then run FIND, and impose the new starlist (properly tranformed to the appropriate reference frame) to our preferred fitting photometry software. ALLSTAR, if you like it. Or even better….

42. Further improvements I think ALLSTAR produces results which approach the best one can do using only the information available in a single digital image. However, we already realized that most of the important problems in stellar photometry require combining information from multiple images (colors, variability, etc.). Usually, every single image is reduced independently from the others. This might not be the best solution. A typical PSF core radius(fitting radius) of 2 pixels gives 12 pixels to estimate 3 parameters: the (x,y) position of the centroid and the brightness. If we impose to any given star to be in the same position in two frames (registered to the same reference system), we will have to estimate four parameters (instead of six) from 24 data points: the best average (x,y) centroid position and the magnitudes for the two epochs. ALLFRAMEallows simultaneous reduction of multiple images

43. Advantages of ALLFRAME • In itself, transforming the two frame example from one with 18 • independent degrees of freedom to one with 20 degrees of freedom is not • a major improvement. But there are many other advantages: • In crowded fields there will not be 12 independent pixels per stars: • star profiles partially overlap, and some pixels will be held in common. • 2. Simultaneous reduction allows to impose a self-consistent star list on • all the images since the beginning of the reduction process, and the • decision whether to retain or reject marginal detections need no longer to • be made independently for each image: a blended double can be reduced • as a blended double in all the images! • 3. It is no longer necessary that each detection be statistically significant • in every frame for every frame data to be used. • 4. Simultaneous reduction allows to lower the weight of pixels affected • by cosmic ray hits.

44. ALLFRAME 1. ALLFRAMEuses a separate appropriate PSF model, read out noise, gain, and fitting radius for each image. 2. ALLFRAMEperiodically determines (as ALLSTAR) a new estimate of the underlying diffuse sky background for each star, from the median distribution of the counts in a region at and around the star location after all the stars have been provisionally subtracted from the image. 3. ALLFRAMEincludes as an option the possibility of making modest corrections of the input geometric transformation equations from all the stellar centroids. 4. ALLFRAMEabandones the concept of stellar groups (not all the stars in a group are necessarely within all the frames). The least-square matrix is completely diagonalized and inverted. More computer time, but same accuracy, and it easier to assign a sky level to each star.

45. And now let’s see what we have done Once our measurements are done, we need to know how accurate our magnitudes, color, positions, and counts are. It is the time for. artificial star experiments. We can use the routine ADD to add to the original image a bounce of new stars, and see how good we are to recover them, with the appropriate magnitude and position. Input stars Output CMD Original CMD There is a systematic tendency to measure brighter magnitudes Photometric errors

46. We can also estimate the completeness C of our star counts: C=(found stars)/(added stars) This is very important for the determination of the luminosity functions. NOTE: do not apply compl. corrections greater than 2! NOTE: pay attention also to the magnitude migration,