This short document calculates the degree of crowding in images which contain a given number of randomly distributed stars over an area of 2000 by 2000 pixels. Remember that real stars show more clumpiness than a random distribution, so the numbers shown herein are underestimates of the true degree of crowding.
I considered the case of a CCD chip with 2000 rows by 2000 columns (close to that of the Mark IV). Inside this area, I placed a number of stars at random. The number of stars ran from 1,000 to 9,000, in steps of a thousand. This probably covers the range of most Mark IV images taken from suburban sites, though images of the Milky Way taken at a dark site might show more.
After placing the stars at random, I found the neighbor nearest to each star, and calculated its distance in pixels. This is the center-to-center distance; in real images, stars are blobs, not points of light. I repeated each simulation 100 times, and averaged the results in bins one pixel wide.
One way to show the results is by plotting the distribution of this nearest-neighbor-distance for the different stellar densities. I have normalized the distributions so that the peak reaches a value of 1.0 in all cases. The figure below has three panels, to show clearly the results for low-, medium-, and high-density fields. Note the change in both scales from panel to panel.
Another way to describe the results is to calculate the fraction of stars with nearest neighbors less than X pixels away. In the table below, I show the cumulative fraction of all stars, and the number of stars, with nearest neighbors less than 3, 5, 8 and 12 pixels away.
Number of stars in frame: 1000 dist in pixels <= 3.0 cumul_fract 0.012 cumul_num 12 dist in pixels <= 5.0 cumul_fract 0.026 cumul_num 26 dist in pixels <= 8.0 cumul_fract 0.059 cumul_num 59 dist in pixels <= 12.0 cumul_fract 0.124 cumul_num 124 Number of stars in frame: 2000 dist in pixels <= 3.0 cumul_fract 0.024 cumul_num 48 dist in pixels <= 5.0 cumul_fract 0.055 cumul_num 111 dist in pixels <= 8.0 cumul_fract 0.122 cumul_num 244 dist in pixels <= 12.0 cumul_fract 0.234 cumul_num 469 Number of stars in frame: 3000 dist in pixels <= 3.0 cumul_fract 0.038 cumul_num 113 dist in pixels <= 5.0 cumul_fract 0.081 cumul_num 243 dist in pixels <= 8.0 cumul_fract 0.172 cumul_num 517 dist in pixels <= 12.0 cumul_fract 0.326 cumul_num 979 Number of stars in frame: 4000 dist in pixels <= 3.0 cumul_fract 0.048 cumul_num 193 dist in pixels <= 5.0 cumul_fract 0.106 cumul_num 426 dist in pixels <= 8.0 cumul_fract 0.224 cumul_num 896 dist in pixels <= 12.0 cumul_fract 0.409 cumul_num 1635 Number of stars in frame: 5000 dist in pixels <= 3.0 cumul_fract 0.062 cumul_num 312 dist in pixels <= 5.0 cumul_fract 0.134 cumul_num 668 dist in pixels <= 8.0 cumul_fract 0.273 cumul_num 1363 dist in pixels <= 12.0 cumul_fract 0.485 cumul_num 2424 Number of stars in frame: 6000 dist in pixels <= 3.0 cumul_fract 0.073 cumul_num 438 dist in pixels <= 5.0 cumul_fract 0.156 cumul_num 936 dist in pixels <= 8.0 cumul_fract 0.317 cumul_num 1905 dist in pixels <= 12.0 cumul_fract 0.547 cumul_num 3283 Number of stars in frame: 7000 dist in pixels <= 3.0 cumul_fract 0.084 cumul_num 585 dist in pixels <= 5.0 cumul_fract 0.179 cumul_num 1250 dist in pixels <= 8.0 cumul_fract 0.358 cumul_num 2509 dist in pixels <= 12.0 cumul_fract 0.604 cumul_num 4225 Number of stars in frame: 8000 dist in pixels <= 3.0 cumul_fract 0.096 cumul_num 765 dist in pixels <= 5.0 cumul_fract 0.201 cumul_num 1611 dist in pixels <= 8.0 cumul_fract 0.397 cumul_num 3178 dist in pixels <= 12.0 cumul_fract 0.652 cumul_num 5215 Number of stars in frame: 9000 dist in pixels <= 3.0 cumul_fract 0.107 cumul_num 962 dist in pixels <= 5.0 cumul_fract 0.225 cumul_num 2022 dist in pixels <= 8.0 cumul_fract 0.436 cumul_num 3921 dist in pixels <= 12.0 cumul_fract 0.696 cumul_num 6262
For the sake of argument, let's guess that a separation of less than 5 pixels might cause problems; the FWHM of stars in some recent Mark IV images is about 2.5 pixels, so this seems reasonable. The table above indicates that, if the total number of stars in the frame is 3000, about 8 percent might be "crowded;" if the total number is 6000 (as Tom mentioned in an E-mail earlier today), then about 15 percent might be "crowded".
Andrew Bennett wrote:
I just got out the backs of several old envelopes and came up with 15% standard deviation for confusion for 6000 stars on a 2kx2k image with optimal estimation and Neff = 12 pixels (the latest and best images.)
.... so it looks like his envelopes are pretty accurate!
Again, remember that real stars are more clumped than a random set of objects.