The luminosity function of individually undetectable stars

A method is presented to derive the faint end of the stellar luminosity function in crowded fields. The method assumes that two images that look similar, have similar luminosity functions. A simulated image with known luminosity function is compared with an observed image, and the luminosity function is adjusted until the images look sufficiently similar. A number of tests were performed on simulated and real images, showing the effects of various error sources. The histogram of the image was used to determine similarity. It was possible to derive the luminosity function to a limit of approximately one source per beam, with the beam defined as times the square of the half width at half maximum of the point spread function. Defining the limit of detection for classical photometry as the magnitude where the completeness of detection dropped below 0.5, and for the luminosity function derived with this method at the magnitude where the uncertainty in the luminosity function exceeded the derived number of stars at this magnitude, it was possible to derive the luminosity function up to two magnitudes fainter than possible with classical photometry. The accuracy was limited by knowledge of the analog-digital converter effects, read-out noise, and point spread function.