C. One last fling

Well, I'm starting to get near the end of my leash. Are you at the end of your rope yet?

Artificial-star tests are not as easy as they look. I just want to use the two programs that I discussed in the last subsection (for matching stars) to ask one of the nagging questions of artificial-star tests: When you use your model point-spread function to add artificial "stars" at various locations in a CCD frame, and you happen to put an artificial star right on top of a real star which was already there, and then when you reduce the new frame and find a star at that location, which one did you find, the real one or the artificial one? A stupid question, right? Well, computers are stupid.

In those artificial-star tests that Bill Harris and I did with our deep frames of M92, we added six sets of 250 randomly-placed stars to our seven original CCD images, producing an additional 42 synthetic frames (each containing over 3,000 stars). Let me call the original exposures of the M92 field frames 1, 2, ..., 7. We generated six sets of 250 stars, randomly distributed within the field, and with a color-magnitude diagram and luminosity function mimicking the lower part of M92's main sequence; let me call these six lists A, B, ..., F. Since at this point we knew both the positional transformations and the magnitude zero-point differences among the seven original frames, it was possible to add each of these lists of artificial stars to the original frames, with the correct positions and the correct magnitudes to simulate real stars that would reappear in exposure after exposure. That way, later on when we said, "We'll keep only those stars that are found in at least two blue and two visual frames," or whatever, we were able to apply precisely the same selection criteria to both real and artificial stars. Let me call the 42 synthetic frames produced by adding each of the six artificial-star lists to each of the seven original frames synthetic frames 1A, 1B, ... ,7E, 7F.

For artificial-star tests to be valid, you must reduce each synthetic frame exactly as though it were a real frame straight from the telescope, with no a priori knowledge of which detections are "real" and which are artificial. Only after you have gone through the whole reduction process from beginning to end (maybe I'll let you skip redetermining the point-spread function for each frame, provided all you've added are faint stars, and not many of them), are you allowed to pull out the secret lists of which stars were found in the original frames, and which were added at what locations and with what magnitudes. Bill and I had information for over 3,000 detections for each of 49 frames (7 real, 42 artificial), and 42 lists of input data for 250 stars per list. Who's going to sort all that out? I'm not. The computer is. How?

Well, let me tell you how I make the computer do it. What I do is I take the output data for, say, real frame 1 and append to it the input data for artificial star list A (these have been tagged by distinctive ID numbers), producing a single, augmented star list. I then use the star-matching program described in the previous section to intercompare this augmented star list with the output data from frame 1A. Each entry in list 1A then has every opportunity to choose whether it feels more like some detection that was in original frame 1, or a star which was added to frame 1 as part of synthetic-star list A. This is repeated for the of the 41 remaining combinations. Thus, the computer is enabled to place each detection in exactly one of five categories:

1. A star found in real frame 1 finds itself cross-identified with a star found in synthetic frame 1A, This is a rediscovery of a real star. In most cases, the photometric results for the star from the real frame and the synthetic one will be completely identical. However, if an artificial star has been added in the neighborhood, it may cause the photometric results to come out a little different.

2. A star in artificial-star list A which was added to real frame 1 finds itself cross-identified with a star found in synthetic frame 1A, This is a rediscovery of an artificial star. The difference between the input and output magnitudes of the star is a datum which will contribute to the determination of the systematic and random errors of the photometry as a function of apparent magnitude. (Ditto for the positions, if you are interested in astrometry.)

3. A star which was added to frame 1 as part of artificial-star list A is cross-identified with no star in output star list 1A, This is an artificial star which has been lost - a datum which will contribute to the determination of the discovery completeness as a function of magnitude.

So far so good. There is one worry, maybe. As I said above, unless an artificial star has been added in the immediate neighborhood, the reductions of a real star in frames 1 and 1A should proceed identically, and the two positions should match perfectly. I think this special ability of a real star to rediscover itself can be neglected, but it's something I want to think about some more. Certainly, as long as the artificial stars are comparatively faint, the readout noise, sky noise, and artificial Poisson noise should mask any bias due to the fact that artificial stars are put in and then fit with the same (that is, the model) point-spread function, whereas the real stars were put in with the unknown true PSF and fit with the model. But I said there were five categories, and so far I've mentioned only three.

4. A star which is discovered in synthetic frame 1A is cross-matched with no star in either real frame 1 or synthetic star list A.

5. A star which was discovered in real frame 1 is cross-matched with no star in synthetic frame 1A.

What do we do with these stars which pop up from nowhere or disappear into nowhere? I'm not sure. Sometimes your random-number generator puts an artificial star squarely on a real one, and the discovered object which results must be matched up with only one, not the other; sometimes this results in the artificial star getting lost, but sometimes it results in the real star getting lost. We include the one in our analysis, but as far as I know nobody has yet worried about the other. Is this a legitimate rediscovery of the artificial star or not? And when you find a new star in the synthetic frame which you know you didn't add? Should this be added to the list of "real" stars in the field or not? Details, details, ... Ah, me! Muito obrigado, y muchas gracias, a todos! Chao.

I am extremely grateful to the organizers of the V Escola Avancada de Astrofisica: Beatriz Barbuy, Eduardo Janot Pacheco, Sueli Viegas-Aldrovandi, and Antonio Mario Magalhaes. Thank you for this wonderful opportunity to visit Brazil and to meet so many South American astronomers! Sao Paulo and Aguas de Sao Pedro were beautiful and fascinating places. Thanks also to the other invited guest lecturers, Phil Crane, Endrik Krugel, and John Mathis, and to all the participants in the school for their cheerful companionship and stimulating conversation - not to mention a few cervejas and caipirinhas.