Track and stack images

Imaging faint objects requires a minimum exposure time to get an useful signal to noise ratio (SNR). But if a faint object moves fast the exposure time is limited by the moving speed and the 0.5-sigma radius of the object. This maximum exposure time is reached, when the object moves a distance of its own diameter or 1 sigma of its own gaussian shape. Unfortunately - imaging FMOs this maximum exposure time is often to short to detect the object.

For example:
The 1-sigma-diameter value may be 2 arcsec (which is a typical value at Calar Alto) and the moving speed is 0.8 arcsec/minute. The maximum useful exposure time is limited to 2/0.8 = 240 seconds. To take an useful image of a 22mag object at Calar Alto a exposure time of 720 seconds is required, but a single exposure must be less than 240 seconds. So we have to divide the required exposure time of 720 seconds into 4 single 180 seconds exposures. After taking this 4 frames the images will be stacked by software, resulting in a composite with greatly improved SNR of the target. The SNR increases with the Sqrt(N) where N is the numer of stacked frames.

The following comparision shows improving SNR using track and stack images:

single image - 40 s exposured

N=1 --- Peak SNR = 7.5 --- FWHM = 2.4"

composite of 8 images. Exposure 8*40 s = 320 s

N=8 --- Peak SNR = 21.2 --- FWHM = 2.6"

The SNR gain is 21.2/7.5 = 2.827 which is very close to the theoretical value of sqrt(8) = 2.828 ! Note the quite well fitting of the point spread function to the gaussian shape in images above. The fitting of the composite is nearly perfect (Fit RMS = 0.045).

The magnitude of the object is direct proportional to the integrated light inside the 3-sigma-radius area of the object. Using the FWHM we can calculate the magnitude gain for the example above. The gain is 2.5*log10 ((FWHM2^2*SNR2)/(FWHM1^2*SNR1)). The composite allows us to keep 1.3 mag fainter objects.

The next example shows the relationchips using a really faint and noisy object:

single image - 60 s exposured

N=1 --- Peak SNR = 3.1 --- FWHM = 0.8"

composite of 16 images. Exposure 16*60 s = 960 s

N=16 --- Peak SNR = 6.5 --- FWHM = 2.8"

The theoretical SNR gain must be sqrt(16) = 4, but we are measured only 6.5/3.1 = 2.1. Whats happen ? The PSF fitting of the single image failed, because the object is very noisy. The Fit RMS of the PSF is 0.28 only, and thats results in bad values for peak SNR and FWHM.

Assuming that the real FWHM is around 2 arcsec, the peak SNR is approximatly 1.6 for the single image. Using this value a SNR gain of 3.8 calculated ( less than sqrt(16)=4 ). This is a very rough approximation only.

Compare the measured magnitudes. The magnitude measurement (V = 20.4) of the single image is not plausible. The magnitude measurement of the composite (V = 21.5) is near the prediction.

The images above are scaled different to optimize the visibilty of the object. To compare the visible impression of the improved SNR look at the composite below, which is scaled nearly as the single image.