WASP-3 is a 10.7 magnitude star 3 degrees south of Vega in Lyra. WASP stands for Wide Angle Search for Planets. It consists of two robotic observatories in Spain and South Africa. https://en.wikipedia.org/wiki/Wide_Angl ... or_Planets
Exoplanet "b" was discovered in 2007. It is a Jupiter-like planet 1.3 times the size of Jupiter. It revolves around WASP-3 every 1.846835 days. The duration of the transit is 137 minutes and its depth is 0.0123 magnitudes.
This is my second exoplanet, the first being HAT-P-5 b. Here is a write-up on it: https://u235-varstar.now.sh/gallery/hat-p-5-b
You will notice that there is a lot of uncertainty in the data, nevertheless AstroImageJ was able to perform a transit model fit to my data.
Following that first experience I wanted to see what I could do to increase accuracy. Robin suggested de-focusing as a technique to effectively turn my $400 camera into a $4000 camera.
Tonight I was not able to capture the entire transit from beginning to end but I did capture some high quality data for the second half. I de-focused as much as I could but there was a nearby 13th magnitude star only 18 arc-seconds away. Thankfully I set the exposure right, just below 55,000 ADU which is where the CCD begins to go non-linear.
Here are the results. I am very pleased with it:
Brian
WASP-3 b Exoplanet Transit 2020-06-17 B Morgan.jpg (155.45 KiB) Viewed 1605 times
The next step would be to work out the relationship between three variables: Star magnitude, Magnitude error, and Aperture.
Aperture is equal to the amount of de-focus needed to achieve a certain star disc radius.
Magnitude error (the 3rd column in the spreadsheet) should be no greater than half the depth of transit, preferably smaller.
It should be possible to work this out mathematically but I think I will take the empirical approach to see how the data points cluster and then fit a curve to it.
The goal is to achieve consistent results instead of guessing. It's OK to rely on experience but it would be nice to have a heads-up in the planning stage to determine if a target is within the abilities of my equipment. For example, I can tell you that there is no way my kit could handle a 16th magnitude star with a transit depth of 0.005 magnitudes. It's not going to happen but there are other scenarios where I can pull it off. Let's say that the star is 8th magnitude then I know I can detect it just using the experience I've accumulated thus far. That's fine but I want a more rigorous approach.
The distance between the Earth and Sun is 93,000,000 miles.
The distance between Mercury and the Sun is 30,000,000 miles.
The distance between WASP-3b and WASP-3 is 3,000,000 miles.
WASP-3b is 1.3 times the size of Jupiter!
One has to imagine the great tidal forces between these two massive bodies.
Wikipedia says that "the planet's mass and radius indicate that it is a gas giant with a similar bulk composition to Jupiter".
Like I said, I am not an astrophysicist but how can a gas giant consisting mainly of hydrogen and helium (their words, not mine) exist so closely to a star without the light elements being blown away by the intense "solar" wind? Maybe in fact that is what has happened, and all that is left are the heavier elements like methane.
One thing that bothers me about my data is the seemingly abrupt step at the exit point of the transit. According to my math it should take about 12 minutes. This is equal to about 4 or 5 of my data points.
To show what I mean I annotated my chart to include what I expected the transition to look like, indicated in blue:
Notice that one annoying data point? If it were higher, within the margin of error, then I would be pleased. Unfortunately this is the price of looking at a single sample in the real world. There will always be uncertainty. The only remedy is to sample many more transits and average the results together.
WASP-3 b Exoplanet Transit 2020-06-17 (3rd and 4th contact) B Morgan.jpg (147.02 KiB) Viewed 1584 times
In terms of working out how accurate your measurements might be, I'm going to hazard a guess that the shot noise in the measured counter photons is going to be the largest source of error in each frame measurement. If you have done a sensor analysis on your camera (or you know the e/ADU anyway), then you should be able to work out the total number of photons in your defocused star image from summing the ADU values of each pixel. Take the square root of that and you have the shot noise and hence the SNR of the frame. You can then compare that to the actual interframe variation that you see in brightness measurement (remembering to take account of the shot noise in the comparison stars too). If the figures from the calculation line up with the observed variations then you should be able to model the effects of changing aperture or exposure time by working out how they will affect the number of photons you would collect in a frame.
I understand what you are getting at. I decided to start with my SNR calculator which has proven itself in astrophotography.
First, I measured the FWHM in a frame of WASP-3 and each individual surrounding star: 9 arc-seconds.
But before I go any farther, straight away this suggests that for any given target, regardless of its magnitude, if I de-focus to 9 arc-sec and then adjust the exposure until the maximum pixel value is 50,000 ADU then I can expect an error of approximately 0.0033 magnitudes (see spreadsheet previously posted.)
So now the question is: what if I need greater accuracy? The only lever I can pull to achieve it is to de-focus more than 9 arc-sec. What will the exposure be? I can use my SNR calculator to get a good estimate. But first let's plug in the numbers for WASP-3 to see if it is a viable tool:
One of the inputs to the calculator is the surface brightness of the target. Normally it would be a galaxy but I have used it successfully for stars too.
First, I converted 9 arc-seconds FWHM to arc-minutes: 0.15 arc-minutes.
Next, I used C2A to find the wideband (i.e. luminance) magnitude of WASP-3: 10.543.
I used a calculator to convert it to surface brightness: 15.05 mags/arc-sec squared.
I entered 15.05 into the SNR calculator and then all of the other properties of my telescope, camera, and bortle class, in addition to a simple model of the QE of my sensor.
Next, I input the exposure of 150 seconds.
The calculator spat out 8,308 signal electrons per pixel.
I divided that by my gain of 0.21 e-/ADU to arrive at: 39,562 ADU. Let's round it to 40,000 ADU.
Now compare that to 50,000 ADU for the maximum pixel value that I directly sampled from my frames.
Like my old friend who served in the Army would say "that's good enough for government work!"
So now I believe that the calculator is a viable tool.
Finally, let's put it to use. Let's say that I need to double the accuracy from 0.0033 magnitudes to 0.0016. To achieve that I believe that I need to double the FWHM from 9 arc-sec to 18 arc-sec. Let's see what happens:
Surface brightness goes from 15.05 to 16.56. Plug that into the calculator.
Now increase exposure until signal electrons per pixel reaches 8,308 or thereabouts.
The answer is 600 seconds which makes sense since it is four times 150 seconds.
Of course that is true if the new target is the same magnitude as WASP-3 but what if it is brighter? Let's say that it is magnitude 8.0.
Now the surface brightness is 14.01. Plug that in and adjust the exposure. The exposure is now 58 seconds.
So I've doubled the accuracy from 0.0033 to 0.0016 and reduced the exposure from 600 seconds to 58 seconds because the magnitude of the target went from 10.543 to 8.0. Let's see if that makes sense:
10^((8.0-10.543)/2.5)*600 = 57.67 seconds
Right away you can see I'm ignoring something vital: noise. Thankfully the calculator computes noise and SNR. To do this properly I should be keeping an eye on both the maximum pixel value and the SNR. I need to maintain the same SNR on a per-pixel basis. Due to the additional noise it may cause saturation so I would have to de-focus a bit more to compensate. In the example I gave above I would have to increase the exposure to 630 seconds in order to compensate for the additional noise (saturation not a problem in this case.)
I think I was imagining looking at the numbers from the other end...
Don't worry about the brightness of the star, the optics or the QE of the sensor, all that matters is
* The ADU level that you hit and
* The number of pixels the light is spread over
Let's go with 40000 ADU, which is ~8400e for your sensor. Imagine that the starlight is spread over 1000 pixels (a disc with diameter about 36 pixels). Assuming that the distribution is even over the disk then the total number of electrons captured is ~8,400,000. If the distribution isn't even then you just add the ADU values for all star pixels to get the correct value and multiply by the e/ADU.
Ok, so you've collected a total of 8,400,000 electrons for some combination of exposure, defocus, aperature and star brightness. Shot noise is just SQRT(N), so the shot noise is about 2900 electrons and the SNR is about 2900:1 for the brightness of the whole star (individual pixel SNR will be much lower, but I'm not sure we need to care much about those).
So, the theory is that the standard deviation of the brightness measurement will be about 1 part in 2900 of the actual brightness - this can be transferred back into magnitudes via the usual formulae.
If you want to increase the SNR of the whole star brightness measurement you have to collect more photons - quadruple the exposure to collect 4x more photons and also double the defocus radius (4x area) to keep away from saturation and you will now be collecting about 33 million photons and your SNR will be 5800:1. An alternative to quadrupling the exposure would be to double the aperture (collect photons over 4x the area).
Yes I see. Following your formulation I can get close to the magnitude error reported by AstroImageJ given the total ADU counts over the aperture.
The reason why I went with my solution was twofold:
1. The PSF of a star does not uniformly distribute ADU's across the width of an aperture.
2. I can easily measure the FWHM for a given amount of de-focus. Given that I can then calculate the surface brightness for input into my calculator.
In the end both of our approaches are valid, I think. There is only one way to tell is to try it. Fortunately it can be tested on a standard star field.
