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ASTR 5110, Majewski [FALL 2017]. Lecture Notes

ASTR 5110 (Majewski) Lecture Notes


PHOTOMETRY METHODS

(FROM PIXELS TO INSTRUMENTAL MAGNITUDES)

SOME REFERENCES:

  • Howell, Handbook of CCD Astronomy, Chapter 5.

  • Kitchin, Astrophysical Techniques, Section 3.2.

  • Buil, CCD Astronomy, Section 5.2.

  • Parts of Chapter 7 in Walker, Astronomical Observations: An Optical Perspective.

  • Chapter 9 of Birney. See especially pp. 131-147.


Methods for Measuring Photometric Flux:

The Single Channel Photometer

Photometry used to be done with a single channel, photoelectric photometer (generally a photomultiplier tube) with semiconductor substrate (basically one very large pixel):

The basic set-up of the device is as follows:

  • Mirror - directs light to eyepiece or photometer.
  • Aperture - plate with a hole in it to limit the area of the sky seen by photometer and isolate the object of interest. Can ruin photosensitive detector if too much light.
  • Filter - restricts wavelengths of light which strike the photomultiplier tube (PMT).
  • Aperture and filter are usually mounted on wheels so they can be changed easily.
  • Fabry Lens - a small lens used to make an image of the telescope pupil (i.e., objective) on the photocathode (first suggested to be used by French physicist Charles Fabry, 1867-1945). The lens must be placed behind the aperture mask, which isolates the target of interest, then that light is put through the Fabry lens. By imaging the pupil rather than the object image on the photocathode, image wander due to seeing variations do not cause the image to move across the photocathode, which may vary in sensitivity across its surface.
  • Dark slide - prevents light from hitting PMT.
  • Photomultiplier (the detector) - located in a cooled box (use dry ice or refrigeration).
  • Quantum Efficiency = 10-20%.
  • Cold box - place for refrigerant (usually dry ice) to lower dark current. In conjunction with small heating element ensures constant temperatures.

Simple version of single-detector photometry:

  1. Point to star and receive flux from both star and background sky for set integration time
  2. Point to blank sky nearby and measure sky flux in same time
  3. Subtract 1.-2. for star flux
  4. Normalize to 1 second
  5. Repeat procedure for standard star
  6. Compare measured counts/sec from standard star and target star -- get magnitude


Methods for Measuring Photometric Flux:

The Digital, Two-Dimensional Array Image

Can be from a digitized photograph or an array detector.

Some basic needs for analyzing such data:

  1. Find the locations of the image centers of objects to be photometered.

  2. Find sky level that provides the "pedestal" of background counts to entire image (including determining appropriate pixels to use for measuring sky flux).

    This is usually found as a level, B in ADUs, per pixel.

  3. Calculate the integrated source intensity, I, from each object of interest.

    This is accomplished by somehow summing the counts in a certain number of pixels, N, and subtracting the total estimated background from the total summed counts S in all of these pixels: I = S - NB.

    Many options for obtaining S exist (e.g., for stars, aperture versus PSF-fitting methods).
  4. The final magnitude of the source is given by:

    m = -2.5 log(I) + C

    where C sets the magnitude zero-point and is the basis of the full reduction to be described in the next web page.



Image Centroiding

However one decides how to measure the flux in a source, one generally must begin by finding the center of the source in the array image.

A simple way is by making use of marginal distributions of the images.

  • Imagine the array image of the source with intensity Ii,j (in ADUs) in pixel (i,j).

    Examples of marginal distributions. (Left) The image of a star on an array image. (Right) The marginal distributions in the X dimension. From Mike Bolte's lecture notes: http://www.ucolick.org/~bolte/AY257/ay257_2.pdf.
  • We can make marginal distributions of the image by extracting a 2L+1 pixel by 2L+1 subarray containing the source.

    The marginal distributions are taken by summing the Ii,j values in the subarray along rows and columns.

  • The mean intensities in each marginal distribution are given by

    which are then used to define an intensity weighted centroid of the source as:

  • This is usually reasonable for finding the center of a point source (e.g., a star), to about 1/5 or 1/10 of a pixel.

More elegant centroiding methods are also used, including fitting two-dimensional Gaussians to simultaneously derive the peak in each dimension at one time (if the size of the Gaussian is not the same in both dimensions, this is called fitting an "elliptical Gaussian").



Background Estimation

If one were to add up the ADU counts in the pixels containing the source of interest, we would not only collect source counts but counts from the background (which includes emissions from the Earth atmosphere, unresolved astronomical sources, dark current, etc.

Thus, we must estimate the typical level of the background, B, contained in each pixel, so that when we add the ADU counts in a set of N pixels around our source, we can subtract the background (taken as NB).

  • It is preferable to do a local estimation of the background.

    This accounts for possible local variations in the background (or in your image calibration).

  • In order to estimate the background, it is common to work in an annulus around the source (with the now well determined centroid), as shown by the dashed lined circles below.

    From Howell, Handbook of CCD Astronomy.

  • One could simply average the values of the pixels with centers in the annulus.

    But a median is more robust to bad pixels or cosmic rays.

    Or one could make a histogram of "sky" pixel ADU levels and fit a Gaussian to it.

    From Howell, Handbook of CCD Astronomy.



Aperture or Cookie-Cutter Photometry

The aperture photometry method makes no assumptions about the actual shape of the source, but simply counts flux falling within the selected aperture shape.

Note that in principle any aperture shape could be chosen, we will focus on the most common technique of using circular apertures centered on the centroid of each source.

Here we will assume that we are observing stars; a vast set of possible methods opens up for extended source photometry, which we cannot get into here.

  1. Use an aperture centered on the star; count sky + star flux
  2. Use an annulus to measure the sky level per pixel -- measure median, mode, or mean level of pixels in annulus. Median and mode preferable in case annulus has cosmic ray or neighboring star, bad pixel.
  3. Star flux = (total aperture flux) - (Npix)*(Median or mean sky/pixel), where Npix is the number of pixels in the aperture.

Problems and Challenges of Aperture Photometry:

  1. PROBLEM A: How big an aperture to use?
    • In principle, would be nice to measure all of the flux in the source (star).

    • However, stellar profiles extend to large radii (the extent depends on the quality of the optics, of course).
    • The profile of a star with logarithmic spatial and flux axes. From paper by Ivan King, 1971, PASP, 83, 199.

    • A Gaussian PSF has 99% flux contained within ~ 10σ.
    • Since for a Gaussian, the FWHM = 2.354σ, then one can see that most of the flux is contained within a radius equivalent to about 4 FWHM.

      From Howell, Handbook of CCD Astronomy.

    • However, measuring the flux entirely within an all inclusive aperture is not necessarily optimal in a S/N sense.

      • As your aperture grows, the total source counts, S, go up....

      • ... but so too does the noise, since, because, the area grows as r2, at large radius you are adding many pixels with little signal but relatively more sky and read noise.

      From Howell, Handbook of CCD Astronomy.

      Thus, there is a radius (smaller than the "100% light radius") that delivers an optimal S/N, and therefore optimal photometric precision, in the measurement of the flux within this radius.

    • But then, we are no longer measuring the full flux of the source!

      What do we do about the "missing light"? Two methods are followed --- both require us to measure a priori what the mean seeing PSF in the CCD frame is:

      1. OK to not measure all flux as long as you measure same percentage of flux in standard star calibrators (compare apples to apples!).
      2. Since all stars in a frame should, in principle, have identical PSFs, the fraction of total light within a given aperture size is same for all stars, all brightnesses

        Measure the mean seeing/PSF in each frame, and then scale the aperture as some multiple of the seeing FWHM, and use this same aperture size in both the target star and standard star frames (to ensure that the same fraction of light is always being used as a proxy for the total light).

        This is probably the easiest way to proceed.

        The IRAF task PHOT is well-matched to this type of photometry.

      3. Take advantage of the identicalness of the PSFs to determine what fraction of light is missing in each star in the same frame.

        This requires what is called a curve of growth (COG) analysis.

        One can measure the COG from a very well exposed, brighter star.

        Use this established COG (which should be the same for all stars in the frame) to correct the poorly measured, fainter stars for an aperture correction.

        From Howell, Handbook of CCD Astronomy.

  2. PROBLEM B: Cosmic rays (or uncorrectable bad pixels) landing in your aperture are detrimental -- typically fatal -- to a proper measurement of the stellar flux.

    Usually just throw out that measurement of that star on that frame. However:

    • Cosmic rays or bad pixels landing in aperture - if it lands in a high flux area of the stelal image, is often hard to even detect.
    • This is a real waste of data for a potentially recoverable situation using other methods (like PSF photometry -- see below).
    Note that bad pixels or cosmic rays landing in a sky annulis are usually less of a problem if you use a median or mode to gaue the sky level.

  3. PROBLEM C: Nearby stars contaminating aperture (or sky annulus).

    This is a similar issue to PROBLEM B, except that it is repeated on all frames you may take of the same piece of sky, and so the affected sta, may be impossible to measure accurately with aperture photometry.

    For both problems B and C, with aperture photometry there is not much one can do but throw out that measurement of the star. But you can try to identify the problem stars by using different size apertures and measuring ratios of flux in apertures.

    Uncontaminated stars should follow the characteristic curve of growth:
  4. (Left) A normal COG, (middle) the PSF for a star with a cosmic ray, and (right) the schematic COG for this CR-contaminated COG.

    From Mike Bolte's lecture notes: http://www.ucolick.org/~bolte/AY257/ay257_2.pdf.


Profile-Fitting Photometry

  • Based on the idea that all stars should have exact same shape.

    For example, assume stars on one image have a Gaussian shape with same standard deviation:

    • In general, the "size" of a stellar image on pictures takes from the ground on large telescopes is limited by the atmospheric seeing.
    • In general we are imaging a small enough area on the sky (less than a half degree or so) for a long enough period of time (longer than a second) that the entire "scene" we are imaging experiences the same foreground atmospheric perturbations on average. (Note that this is NOT true for very short integrations over large angular scales.)
    • Under the above circumstances every star will show the same Point Spread Function (PSF).
    • With good guiding of the telescope, the PSF will be round, and will show a clean radial variation only.
    • A Gaussian is a reasonable approximation to the shape of the typical PSF. The radial form of a two-dimensional Gaussian shape looks like:

      where

        I0 is the intensity peak of the profile,

        Ixy is the intensity at any other pixel xy

        is the width of the Gaussian

  • The basis and beauty of PSF-fitting photometry is that one may assume that all stars in one CCD image (observed under the above conditions) look like Gaussians with identical σ, and only varying I0 .

    • Thus, we can reduce the nature of the fits to a one-parameter family specific to each image.

      The brightnesses of stars are directly proportional to the integrated flux under the Gaussians (in this case, Gaussians of fixed σ but varying height I0 ).

      One can then work out that in the case of fixed σ, the brightnesses of stars are directly proportional to the height of the fitted Gaussian PSF.
    • We may then write that the magnitude difference between any two stars in the frame is given by:

    • Instead of counting photons, find the best-fitting Gaussian to Ixy. Gives you Io (know σ already):

      m = -2.5 log( Io ) + const.

  • Advantages of PSF-fitting photometry:
    1. More accurate:

      • Aperture photometry sums all pixels within the aperture to measure one number - total counts.

        Any single bad pixel in the sum (due, e.g., to a cosmic ray or bad pixel) conveys that "badness" to the final value of the total counts.
      • PSF photometry fits every pixel in profile to get best I0 (in effect, each pixel is one estimator of the I0, and the fit gives the most likely value for I0).

        The "many measures of Io" provided by the fit is more robust to bad pixels.
    2. So, PSF-fitting photometry is not as influenced by CRs, bad pixels, neighboring stars.
    3. Can measure crowded fields:

      • Because we know what the shape of each star should be, we can figure out how to apportion flux in situations where stars overlap.
      • We can iteratively remove the contributions of different stars and measure the magnitudes of overlapping images (see example below).
    (Left) PSF-fitting photometry fits to a star, a double star, and a galaxy. (Right) Residuals after subtracting the PSF from the brightest source in each subarray shown on left. As may be seen, the star is well-subtracted, the double-star residual leaves the fainter star behind (which may now be fit itself with the PSF), and the galaxy is poorly photometered because it is not a point source. Image from Mike Bolte's lecture notes: http://www.ucolick.org/~bolte/AY257/ay257_2.pdf.
  • Sometimes other mathematical functions are used in the fitting, other than a Gaussian, including the

  • Generally, any analytical function provides only a fair approximation to the true PSF.

    For example, in a poorly tracked image (!), the PSF will not be radially symmetric.

    A solution is to make a two-dimensional PSF template, empirically determined from some well exposed (but not saturated) stars in the frame.

    Then scale all detected sources to this template shape for the PSF.

    Especially useful for non-round images.

  • Commonly used PSF-fitting photometry packages:
    • DOPHOT - assumes profile shape
    • DAOPHOT - fit any profile - empirical
    • ROMAPHOT
  • Example of how PSF-fitting photometry works (e.g., with DAOPHOT):

      CCD image of the globular cluster M53:

      Plot showing the two-dimensional shape (i.e., "template") of the Point-Spread Function determined by averaging scaled versions of the images of a number of isolated stars in the M53 image above:

      The image below shows what happens when we:

      1. Fit each star in the original M53 image with the above PSF template to find the best scale factor for I0.
      2. Scale the template by the scale factor appropriate to each star. These scale factors give directly the magnitude of the star by the above equations.
      3. Subtract those scaled versions of each star from the original CCD image of the globular cluster M53.
      4. The degree to which many of the stars in the original image have disappeared shows how well the adopted template models the true PSF of the image.
      5. Note that the large, residual black spot below is from unresolved starlight in the core.

        Other, very bright stars with saturated cores were not well fit by the model PSF, so were not properly fit (leaving behind residuals).

      6. Note how the software can work in relatively crowded areas to get magnitudes of stars in the core. Ultimately, though, even this clever algorithm breaks down and cannot deal with the extremely crowded central region of the cluster.

  • Disadvantages of PSF-fitting photometry:

    • More laborious process, since have to first create a good template, and then you have to do statistical fits of template to EACH star of interest.
    • To put onto correct zero-point scale, still have to do aperture photometry on at least some of the stars to determine an "aperture correction" that accounts for the fraction (if any) of light that the scaled templates miss (a function of how good the template is).

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Thanks to Sangmo (Tony) Sohn for the M53 reduction images. All other material copyright © 2002, 2005, 2007, 2009, 2011, 2017 Steven R. Majewski. All rights reserved. These notes are intended for the private, noncommercial use of students enrolled in Astronomy 5110 at the University of Virginia.