Appendix B. Star Centroids Under Atmospheric Turbulence
Atmospheric turbulence causes starlight to scintillate (random fluctuations in intensity) and randomly changes direction ("seeing").
Over a long term, "seeing" causes the FWHM of a star's point spread function to spread symmetrically in the radial direction. The spread can be as low as 0.5 arcsec on high mountain tops, but at sea level, 2 arcsec is considered "average" seeing. Notice that the Dawes Limit for telescopes are smaller than 2 arcsec even for refractors that have relatively small apertures — even a 60 mm refractor will see a 40% star "bloat" on an average night. The variation between good and poor "seeing" is greater with telescopes of larger apertures.
Shorter exposures will show that the instantaneous "shape" of a star is non symmetrical. Under "average" seeing, a very short exposure shows a shift the centroid of a star randomly by over an arc second. To average out this instantaneous shift of the centroid, a guide exposure often needs to be 2 seconds or longer for most nights.
As shown in the "sawtooth" in Section 5.1, the amount the centroid of a guide star moves (the peak-to-peak amplitude of the "sawtooth") in direct proportion to the slope of its periodic error and also to the duration of the guide exposure. A 2-second exposure is likely to be unusable to pretty much all strain wave gear mounts.
Fortunately, there is an alternative to using long exposures to average out the "seeing" error.
Notice that CMOS camera integration (and also binning) is non-coherent. I.e., integrating for 2 seconds will only improve the SNR by 3 dB instead of 6 dB. Assuming that the process is statistically stationary, this means that we can expose for 1 second, then later come back and exposure for another 1 second, and combine the two exposures, the result is no different (statistically) from exposing for 2 second (this is why combining sub exposures work for images of the sky).
Furthermore, if the atmospheric turbulence causes different stars to behave statistically independently, and assuming the statistical process is ergodic (ensemble averages is equal to time averages), we can measure the movement of the centroid of one star for 1 second, and then measure the movement of the centroid of a different star for another second, the result would still be statistically the same as measuring one star for 2 seconds.
Additionally, again assuming stationarity (statistics is not dependent on when the exposure is taken) and statistical independence (the stars "twinkle" independently) we can measure the two stars from the same 1 second exposure!
Similarly, measuring the centroid of 3 stars in a single 1-second guide frame is statistically the same as measuring the centroid of 1 star in a 3 second exposure.
Thus, by using the centroid of 6 stars in a single 0.5 second exposure would give the same statistical error as exposing a single star for 6 seconds. The "however" is this — the stars must all be statistically identical, and they all have the same brightness, SNR, etc. In practice, even 12 stars could produce the equivalence of just averaging the centroids of 3 or 4 stars.
With many current implementations of multi-centroid guiding, the multiple centroids are accumulated as a weighted sum, whose weights are the Signal-to-Noise Ratio (SNR) of each star. The brightest star carries the most weight. As shall see, this may be a poor choice of weights.
Note that in astrometry, the signal itself (since it arrives as quanta of photons) contains shot noise, so that the signal itself also appears in the denominator of SNR equation as its square root, and accumulated with the intensity of the other noise sources.
The signal (numerator of the equation B.1) consists of the product of P, the photon flux; Qe, the quantum efficiency of the sensor; and T, the integration (exposure) time. The numerator is measured in ADU units. Since the ADU measures intensity, there is no such thing as negative ADUs.
As explained above (shot noise), the denominator includes the square root of the signal term. Additionally, B is the shot noise contribution from the sky background, which also has to include the quantum efficiency of the sensor. Nd is the sensor dark current noise, which is a function of integration time, but not of the quantum efficiency. The shot noise from the signal, the sky background and the sensor dark noise are all scaled by exposure time T. Finally, the denominator includes the read noise term Nr2 (not a function of T, since read noise occurs only once per exposure).
If we now make the assumption that all the stars in a guide frame have the same amount of noise from the sky background and the camera sensor, we can write the following for all the stars for the guide frame,
ADUk is the ADU of the k-th star in the guide frame, and SNRk is its SNR.
We now look at a small patch of sky that is centered at Right Ascension of 0h (RA of the Sun at the Spring equinox) and Declination of 0 (Celestial Equator). The patch has a width of 1.7º and height of 1.1º (a frame from a 250 mm focal length guide scope that has a sensor size of 7.5mm by 5 mm).
Fig. B.1 above shows the stars in the guide frame, sorted by 8-bit ADU units. The brightest star has a magnitude of 6.5. The distribution of star brightness is quite typical — one or two very bright star(s), with the rest more evenly distributed in brightness. The smallest ADU in the above chart is 11.1. The stars were taken from the HYG (Hipparcos) catalog.
The figure below shows the SNR of the 12 brightest stars from Fig B.1, by setting the constant noise term of Equation B.2 to produce a SNR of 10 for the brightest star.
The ordinate of Fig B.2 is scaled so that the SNR of all the 12 stars sum to 1.
Recall that the centroid of each stars is weighted relative to the SNR of the star. Notice that the star with the largest SNR has a relative weight of 0.28. That brightest star will produce 28% of the "seeing" error from a single star. Had all the stars share the same SNR, that number would have been 0.08.
The reciprocal of that weight, in this case 3.6, represents an approximate number of stars that are used to estimate the final centroid, had they all share the same SNR. A 0.5 second exposure with these 12 stars would at be equivalent to a 1.8 second exposure from a single star, instead of producing a 12 fold (6 second exposure) improvement.
The problem becomes worse when the SNR is poorer, e.g., when a smaller guide scope or a noisier camera is used.
Fig. B.3 shows the relative SNR for the same 12 stars, but this time the constant noise term for Equation B.2 has been set to reduce the SNR of the brightest star to 4 instead of 10 for the above case. Notice that the brightest star now hogs almost 0.31 of the total sum of SNR weights, thus producing the equivalent to averaging the centroids from only 3.3 stars.
There is probably a better set of weights for centroid averaging — it is a topic that deserves further study. The DONUTS paper (published 2013 in the Publications of the Astronomical Society of the Pacific) indicates for example that when the centroids of all stars are equally weighted, a precision of 0.18 pixels can be achieved.
Using fewer guide stars definitely does not help. The figure below shows the relative SNR for the 6 brightest stars from above, with the brightest star again having a SNR of 10.
In this case, the brightest star now contributes an even larger weight of 0.386, or the equivalent of averaging the centroids of only 2.6 stars with identical SNR. A smaller guide scope, or an OAG, will also produce this (fewer stars) condition.
Because of the distribution of stellar magnitudes, there is a way to not let a single star dominate the weighted centroid. As mentioned earlier when discussing Fig. B.1, the distribution of star ADU that is shown in Fig. B.1 is very common — where a single bright star in a guide frame is much brighter than the rest. The solution is to ignore that outlier.
Most centroid estimate algorithms will ignore saturated stars. Because of that, we can simply increase the gain of the camera slightly so that the brightest star saturates, and thus gets ignored. And even better, increasing the camera gain so that the next brightest star is close to saturation, but does not.
Using the same stars as in Fig. B.1, but ignoring the brightest star, we get the SNR distribution that is shown in Fig. B.5.
The brightest of the remaining stars now accounts for only 0.16 of all the weights, or equivalent to the centroids of about 6 stars with equal SNR. The maximum SNR (using the same noise term as the one used in Fig. B.2) is now reduced from 10 to 4.6, so this method does require a good guide scope and/or low noise guide camera. With this, we are starting to take full advantage of the multi-star centroid method as a countermeasure for poor "seeing."
For certain regions of the sky, a larger dynamic range (using 10- or 12-bit cameras instead of an 8-bit guide camera) will also help. Dynamic range has never been a problem with single star guiding. However, many programs have kept using a bit depth of 8 bits when guiding with multiple stars.