Transcript Photometric techniques I

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.
Why do we need accurate
stellar photometry?
Color-magnitude diagrams:
We need to
measure fluxes
(and colors)
17 mag
range!
Evolutionary sequences
Comparison with the
models. Age
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
Luminosity functions
Evolutionary
clock
Mass functions
We need to
count stars
Astrometry
We need to
measure
stellar
positions
…to determine
membership.
NGC6121=M4
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
INTERNAL DYNAMICS
(Bedin et al. 2003)
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!
Stellar Photometry Packages
RICHFILD Tody
1981
ROMAFOT Buonanno et al. 1983
WOLF
Lupton, Gunn 1986
STARMAN Penny, Dickens 1986
DAOPHOT Stetson
1987
ALLSTAR Stetson
1994
ALLFRAME Stetson
1994
LUND
Linde
1989
DoPHOT
Schechter,Mateo 1993
Shara
ePSF
Anderson, King 2000
KPNO
A&A, 126, 278
AJ, 91, 317
MNRAS, 220, 845
PASP, 99, 191
DAOPHOT
PASP, 106, 250
PASP, 106, 250
1st ESO/ECF data analysis WS
PASP 105, 1342
PASP, 112, 1360
Plus a number of generic photometry softwares (INVENTORY,
SEXTRACTOR, etc.)
Fundamental tasks for stellar photometry (*)
FIND
crude 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.
Before starting…..
There are at least five things you should tell to your computer
before starting:
1. Read out noise;
2. Conversion factor (electrons to ADU);
3. Maximum linear signal (physical or electronic saturation level);
4. Map of bad pixels, rows, columns;
5. Size of typical stellar images (seeing, FWHM in pixels)
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
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:
1. The sky brightness might vary across the frame
2. Blended objects, extended objects, artifacts,
cosmic rays must be recognized.
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:
Example of a gaussian fit to the
original data.
Note:
1) The broad galaxy is suppressed
by the convolution;
2) The background value of C is 0
even if originally the background
is nonzero and sloping;
3) The blended pair is better
separated;
C
(a)
(b)
(c)
(d)
(e)
A star image
Blended pair
Galaxy
Cosmic ray hit
Low value bad pixel
Note that the stellar images are
critically sampled (FWHM=2.4),
i..e. it is hard to distinguish a star
from a cosmic ray.
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
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.
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
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.
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.
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.
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.
The analytic PSF
Advantages:
1. 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
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
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
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
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.
And now…the real stuff…:
INPUT PARAMETERS: GENERAL
INPUT PARAMETERS: APERTURE PHOTOMETRY
The PSF stars must be BRIGHT and CLEANED
Contaminating
stars must
be removed
This shows the
relevance of the
SUBTRACT
routine
The PSF determination is an iterative process!
The PSF model after
three iterations
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:
Example of a crowded image in the outkirts of a globular cluster
The same image after the file fitting and subtraction routines to the original
star list. Many secondary components and blended doubles are present
The same image after two passes through the find-fit-subtract loop
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.
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.
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 of
the 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.
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).
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 derived to 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).
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….
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.
ALLFRAME allows simultaneous reduction of multiple images
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:
1. 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.
ALLFRAME
1. ALLFRAME uses a separate appropriate PSF model, read out noise,
gain, and fitting radius for each image.
2. ALLFRAME periodically 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. ALLFRAME includes as an option the possibility of making modest
corrections of the input geometric transformation equations from all the
stellar centroids.
4. ALLFRAME abandones 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.
And now let’s see what we have done
Input stars
Output CMD
Photometric errors
Original CMD
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.
There is a systematic
tendency to measure
brighter magnitudes
We can also estimate the
completeness C of our star
counts:
C=(found stars)/(added stars)
NOTE: do not
apply compl.
corrections
greater than 2!
This is very important for
the determination of the
luminosity functions.
NOTE: pay
attention also
to the
magnitude
migration,