2. Really fitting model profiles to star images
In subsection 1 above I cheated. I said that "I estimated the position of the star's centroid by eye, and obtained the average intensity within the star image as a function of the radial distance from the center by averaging the observed pixel values in narrow, concentric annuli." Now what's the point of using a computer if you're going to go around estimating things by eye? The real way to fit a model profile to a star image is to include the x, y coordinates of the star's centroid as parameters in the fit:
where
All you have to do is augment the four derivatives given above with
Since
so
and
then
Similarly,
Now we can solve for the six unknown parameters, but notice that this is done with the original two-dimensional data, not with the annular averages which I used for the radial profile fits of the previous section (for the annular averages x - x0 and y - y0 don't mean anything). A reduction of this sort is shown in Figs. 2-8 and 2-9. Fig. 2-8, again, is a photocopy of my computer printout documenting the convergence of the solution. The m.e.1 is larger here than it was before, because it now represents the standard error of a single pixel, rather than the error of an annular average determined from many pixels. As you can see, the final coordinates of the star's centroid have been determined with a formal uncertainty of only 0.01 pixel, which is about the best that modern astrometry can do. Fig. 2-9 shows the raw data (points), the initial guess at the profile (dashed curve), and the final solution (solid curve) for one slice through the star image at y = 456 - many more data went into the actual reductions than are illustrated here.
Enter number of terms:
6
Enter starting guesses for C, B, Xo, Yo, R, beta
6000 0 245 456 8 3
C B Xo Yo R beta m.e.1
6000.00 0.00 245.00 456.00 8.00 3.00 543.
-443.22 66.67 -1.48 0.08 2.33 0.89
5556.78 66.67 243.52 456.08 10.33 3.89 161.
671.18 -11.10 0.05 0.03 -2.16 -0.95
6227.96 55.57 243.57 456.12 8.17 2.94 96.
-19.14 15.16 -0.02 -0.02 0.79 0.45
6208.82 70.73 243.55 456.10 8.96 3.39 93.
6.00 -0.98 0.00 0.00 0.04 0.05
6214.82 69.75 243.56 456.10 9.00 3.43 93.
-0.47 -0.01 0.00 0.00 0.00 0.00
6214.35 69.73 243.56 456.10 9.01 3.44 93.
0.09 0.00 0.00 0.00 0.00 0.00
6214.44 69.73 243.56 456.10 9.01 3.43 93.
0.00 0.00 0.00 0.00 0.00 0.00
Converged
6214.44 69.73 243.56 456.10 9.01 3.43
+/-46.83 26.04 0.01 0.01 0.46 0.31
When one is performing automatic profile-fitting photometry for many stars in an image, it is not advantageous to solve for all six parameters for every star. Instead, if the image has been obtained with a linear detector, such as a CCD, and a telescope with good imaging properties and in good focus, it is usually possible to assume that all star images in the frame will have the same shape even though they will have different apparent magnitudes and will be located in different places, perhaps with different local sky brightnesses. A faint star image does not contain enough raw information to provide good constraints on six (or more) different fitting parameters: cross-correlations with uninteresting but nevertheless necessary shape parameters will increase the uncertainties of those parameters that we really want, namely the star's brightness and x, y position. (Remember how cross-talk reduced the accuracy of our abundance determinations for the isotopes with 4- and 7-minute half-lives in last lecture's radioactivity experiment?) Therefore, real-world practitioners of profile-fitting stellar photometry almost always begin by performing complete model fits to one or a few bright, well-isolated stars in each frame to solve for all fitting parameters. Then the boring but presumably constant shape parameters are fixed at their best values, and new profile fits are performed for all the stars allowing only those parameters which really differ from one star image to another to be determined.
|
| Figure 2-9. |
In the standard Moffat function, the shape of the star profile is
defined by the two
parameters R and
. Having fixed
R and at
the values derived from the
fit in Figs. 2-8
and 2-9, I then performed four-parameter fits
for the same star and two
other, fainter stars
in the same CCD image. The reductions are shown in
Figs. 2-10 -
2-12. Finally, the raw
data, the initial guess, and the final profile fit for a slice through
the faintest of the three
star images are illustrated in Fig. 2-13.
Enter number of terms:
4
Enter starting guesses for C, B, X, Y, R, beta
6000 0 245 456 9.01 3.43
C B Xo Yo m.e.1
6000.00 0.00 245.0O 456.00 517.
-151.33 138.60 -1.46 0.09
5848.67 138.60 243.54 456.09 125.
363.66 -71.43 0.02 0.01
6212.33 67.16 243.56 456.10 93.
0.05 -0.01 0.00 0.00
6212.38 67.16 243.56 456.10 93.
0.00 0.00 0.00 0.00
Converged
6212.38 67.16 243.56 456.10
+/-19.41 6.09 0.01 0.01
Do you think that you could now determine the abundances of three radioactive elements with unknown half-lives? Sure, you'll need a darn sight more than ten data points to do a good job of it, but can you at least set up the equations?
Enter number of terms:
4
Enter starting guesses for C, B, X, Y, R, beta
500 0 322 38 9.01 3.43
C B Xo Yo m.e.1
500.00 0.00 322.00 38.00 118.
122.15 45.33 2.72 1.25
622.15 45.33 324.72 39.25 42.
85.18 -16.54 -0.83 -0.33
707.33 28.79 323.89 38.92 23.
11.67 -2.29 0.13 0.03
719.00 26.50 324.02 38.96 22.
0.34 -0.07 -0.01 0.00
719.34 26.44 324.01 38.96 22.
0.00 0.00 0.00 0.00
Converged
719.34 26.44 324.01 38.96
+/-4.57 1.43 0.03 0.03
Enter number of terms:
4
Enter starting guesses for C, B, X, Y, R, beta
100 0 133 144 9.01 3.43
C B Xo Yo m.e.1
100.00 0.00 133.00 144.00 33.
-71.03 28.82 0.90 -0.72
28.97 28.82 133.90 143.28 20.
5.82 -0.82 2.44 -1.80
34.79 28.00 136.34 141.48 19.
5.45 -0.81 -0.40 0.51
40.24 27.19 135.94 141.99 19.
0.32 -0.05 0.04 -0.08
40.56 27.14 135.98 141.91 19.
0.01 0.00 0.00 0.00
40.57 27.14 135.98 141.91 19.
0.00 0.00 0.00 0.00
Converged
40.57 27.14 135.98 141.91
+/-3.86 1.20 0.42 0.46
|
| Figure 2-13. |