HAT-P-5 is a 12th magnitude star in the constellation Lyra, approximately 1000 light years away. Orbiting the star is a Jupiter-sized planet, designated HAT-P-5 b, discovered October 9, 2007. The planet revolves around the star every 2 days 18 hours 55 minutes. Due to the inclination of the planet's orbit Earth-bound observers can witness the planet transiting across the face of the star. The time to complete a transit is approximately 175 minutes.
Although the planet is the size of Jupiter the amount of starlight it blocks is very small, only 0.0142 magnitudes. That is only 1.3% of the star's light. Amateur astronomers typically use telescopes having a minimum aperture of 8 inches. My scope is a mere 71mm (a little under 3 inches). I wanted to see if it could be done. I was pleasantly surprised.
I am pondering the question, what can I do to decrease noise?
If I had deep pockets I would start with the camera. I'd choose a high QE sensor having a Full Well Depth of at least 50,000 electrons but preferably 100,000 electrons. Of course there is just a limited number of cameras that meet those criteria but probably they would have a pixel size of 8-10 microns which is important when selecting my new telescope's focal length by using the "CCD Suitability Calculator" from First Light Optics. Given the recommended focal length I would then search for a telescope. I'd choose the largest aperture I could afford having quality optics (no two-element refractors!) Finally, I'd look at the total weight of the OTA plus all of the accessories I plan to add. If my current mount can't swing it then I'd purchase a new one. As you can see I could easily spend $10,000 without breaking a sweat. Of course you need to weigh this against the quality of your site's dark skies and frequency of clear nights. For my location I can not justify spending $10,000.
So what can I do with my current kit? Well, there isn't much I can do except for increasing the exposure or stacking or both. Increasing the exposure isn't a viable alternative due to the fact that this 12th magnitude star would soon saturate. (That's the price I pay for purchasing a camera with a relatively low Full Well Depth of only 13,400 electrons.) The only other choice is to stack. According to my calculations this 12th magnitude star has a Signal-to-Noise Ratio of 71 with only a single 90-second sub. That is darn good if I was imaging a galaxy or nebula but I need a much higher SNR in order to detect very small changes in star magnitude. If I were to stack two adjacent frames then SNR increases to 101. If I stack three frames then it goes up to 124. I can't stack too many frames because then I lose the ability to detect the ingress and egress of the exoplanet transit. So it's a balancing act.
Once again a fabulous observation and the fascinating piece of writing about it. The immediate question that comes to my mind when reading about your problems with full well depth is : why not just defocus slightly to spread the light of the star over more pixels?
It's funny that you should suggest de-focusing. Last week I discovered the artifacts of an exoplanet imager who shared a frame that was severely de-focused since the host star was very bright. I composed an email asking if de-focusing helped reduce noise. I'm still awaiting a reply which at this point I don't think I'll get. (I've found lots of people are tight-lipped about their techniques.)
Just now I ran two instances of my SNR calculator side-by-side. I ensured that all inputs were identical except for the surface brightness of the star. Under poor seeing conditions like mine a magnitude 12.0 star has a surface brightness of 15.0 mags/arcsec^2. If I double the severity of the seeing conditions (i.e. simulating de-focusing) the surface brightness dims to 16.5 mags/arcsec^2. Not surprisingly I had to quadruple the exposure time from 90s to 360s in order to obtain the same peak pixel value. I was hoping that the SNR readout would be dramatically higher but it was the same. It took me a few minutes to ponder this but then it dawned on me that photometry is fundamentally different than astrophotography. Astrophotographers want to retain the sensor's pixel scale but photometrists do not. They are happy to integrate all of the pixels within the aperture into a single measure. As a result, signal grows linearly but noise grows in quadrature, so the SNR of a de-focused star must be higher! I like that solution better than stacking adjacent frames.
I'd like your expert opinion on software binning vs hardware binning. Is it true that software binning is effective but just not as effective as hardware binning? I understand that with hardware binning you are assured of paying the cost of Read Noise only once but that's not true of software binning. So that is an obvious difference. What benefits do they have in common? It seems to me that both give you a reduction in Shot Noise. Is that true?
I ask this because I now realize that integrating pixels within an aperture in Photometry is similar to software binning. Like I said, I will pay the cost of Read Noise per pixel but for a star the Read Noise isn't the largest noise component, it is Shot Noise.
True hardware binning – only possible on CCD cameras – does mean that you only pay the read noise price once per binned pixel. That's important if you're looking at the faint background stuff and doubly important if you have a CCD camera that has a high read noise of maybe 8e. If you do this sort of hardware binning at 2 x 2 then you are quadrupling the signal and leaving the noise unchanged (assuming that the read noise is the dominant source of noise).
Software binning, or any claimed 'hardware binning' on a CMOS camera pays the cost of the read noise for every physical pixel on the sensor. Since the read noise like any other noise adds in quadrature, that means that for software binning using a 2 x 2 bin well quadruple the signal but only double the noise – you come out ahead, but not as far ahead as on a CCD camera.
However, for your type of imaging I don't think that any of that matters. You are largely imaging close to saturation where you have maybe 10,000 electrons captured in each pixel. 10,000e gives a shot noise of 100e, which massively overwhelms the read noise to the extent that you can pretty much forget about it – particularly if you're using a low noise CMOS camera.
If you collect light over multiple pixels then the shot noise of the total collected should be just the square root of the total number of electrons summed over all pixels – that means that if you spread the light out over four times as many pixels and capture for four times as long to bring the brightness levels backup then you should make a gain of a factor of two in the signal-to-noise ratio.
