Briefly, an image of a star is theoretically a point, but Earth's atmosphere and telescope's optics spread the point to a blob with Gaussian central part and an outer parts showing an "aureole". The problem of the aperture photometry is our ignorance of outer radii which may include an important fraction of a total flux. If we will ignore the aureole's flux, any magnitudes will depends on atmospheric conditions (seeing spreads stars with dependence on its air mass or a night) and on magnitude of the magnitude (A golden rule of differential photometry is to compare objects with similar magnitude). It means that precise photometry requires more sophisticated approach.
Described problems bothers CCD's photometrics for a long time. They are described in articles of classics at 80'. Howell(1989) points to properties of the aperture photometry for differently bright stars. He shows that total flux of bright stars is determined relative precise, but any small error in determination of background for any faint star strongly affects total fluxes. The solution of the problem did published Stetson(1990) in that great article. He developed a method on base of cumulative distribution of magnitudes of stars (in astronomical dialect the grow-curve method). Please read the paper, I have no time to transcribe it here. The most important ones (from my point of view) are: The grow-curve method is complement to PSF method (PSF <=> precise relative magnitude, grow <=> precise calibrated magnitude). The model of cumulative profile of the star as an integral of PSF can be described by a sum of analytic functions (Gaussian, exponential and Moffat - Moffat is generalisation of Lorentz).
The grow-curve method for a large radii can be described by Moffat's function ~1/(1+r^2)^A (A=2 for Lorentz). We suppose that all stars on the image has the same cumulative (grow-curve) profile (integral of PSF modeled by Moffat). To construct of the mean grow-curve for all stars with no matter to its brightness, Stetson uses ratio of fluxes in et sequentia of apertures, eg. the following differences of aperture's magnitudes.
I try it on image of the M 67 open stellar cluster exposed on 2008-02-26 at R filter via 0.4m telescope at HaP MK Brno. The standard Munipack run has been done. About 250 stars has been found. The results included in graph in fashion of the Stetson's article are included.
I take out of these basic fact from the graphs:
- The scatter of the points is relative great over five magnitudes (not all are included). It is due to random errors, disturbance of magnitudes of overlap stars, wrong determination of background for faint stars.
- A pretty description of magnitudes displays frequency of magnitudes in an interval of difference magnitude and apertures. There is a spine visualising of most frequent values of the variables. I'd pleasure from behaviour of the spine at large radii where the values converges to zero. It means that the robust mean algorithm used in Munipack to determine of sky level is really good.
- The graph show differences about 0.1 magnitudes at radii usually applied to photometry. The function is not a profile itself. It means that the difference between a real and determined magnitude will a few tenth of magnitude. In other word, the calibration of Munipack's aperture magnitudes will depends on actual observation conditions and the calibrations derived from the magnitudes will contain an systematic error which will be depend on magnitude of a star.