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--- train:lectures:telescopeoptics [2024/03/18 16:05] – Roy Prouty
+++ train:lectures:telescopeoptics [2024/03/24 12:54] (current) – Roy Prouty
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 {{url>https://docs.google.com/presentation/d/e/2PACX-1vRmP6pIwNDxbX5YSV0szjqsvHnA0E9LFaD1gDYr5s1Kh5wz8bS2L82NwJN0TXlfarabYbtMSrjnvyBK/embed?start=false&loop=false&delayms=3000}}
-=====Telescope Optics=====
+=====Telescope Optics I =====
 The Primary Optical device is the first lens or mirror that light touches. The diameter of the Primary Optical Device is also (generally) the Aperture Diameter. The light-gathering power of a telescope depends on the area of the aperture, and so this "light-gathering power" goes with the square of the diameter or radius.
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 ===Seeing Limited===
-Many ground-based telescopes \\dream\\ of being diffraction limited. Our functional angular resolution is much higher than this diffraction limit since we must also deal with a turbulent atmosphere. This turbulent atmosphere creates many interfaces on which incident star light can refract (even if only very slightly) and therefore change direction. Over the course of the exposure time of the image (the integration time), these slight deviations in direction have the effect of smearing the incident light over a region of the detector called the "seeing disk". We can think of each beam of light hitting the detector and creating an Airy Disk, but every fraction of a second the atmosphere changes the direction of the beam of light ever-so-slightly so that it hits another part of the detector and creates another Airy Disk, and so on through the entire integration time. Adding these all together gives us our seeing disk and a stellar profile that is much broader than any single Airy Disk.
+Many ground-based telescopes //dream// of being diffraction limited. Our functional angular resolution is much higher than this diffraction limit since we must also deal with a turbulent atmosphere. This turbulent atmosphere creates many interfaces on which incident star light can refract (even if only very slightly) and therefore change direction. Over the course of the exposure time of the image (the integration time), these slight deviations in direction have the effect of smearing the incident light over a region of the detector called the "seeing disk". We can think of each beam of light hitting the detector and creating an Airy Disk, but every fraction of a second the atmosphere changes the direction of the beam of light ever-so-slightly so that it hits another part of the detector and creates another Airy Disk, and so on through the entire integration time. Adding these all together gives us our seeing disk and a stellar profile that is much broader than any single Airy Disk.
+----
+Written by Roy Prouty 20240318\\
+Reviewed by
+----
+{{url>https://docs.google.com/presentation/d/e/2PACX-1vTEHCMWUA4nQNtqn0_hPP0udEtwWJI9Fpkzd_q3lscCkpoRWfn2jnZCpPfKapvcRhd-VUYA3TCzKpv6/embed?start=false&loop=false&delayms=3000}}
+----
+=====Telescope Optics II=====
+====Point-Spread Function====
+The point spread function (PSF) describes the response of a focused optical imaging system to a point source. The PSF is the mathematical model that is thought to be convolved with any point source to generate the final image. For simplicity, let's choose a Gaussian PSF for our purposes.
+====Effective Angular Resolution====
+By choosing a simple PSF, we can more easily probe key measures, such as the width of the PSF at a certain level. We chose a Gaussian, but we may use more complicated (and better fitting) profiles in the future. So let's avoid using the standard of deviation ($\sigma$) as this width parameter -- let's instead choose the full-width, half-max (FWHM) as the width parameter.
+For a Gaussian of the form $f(x;\mu,\sigma) = A\exp^{-(x-\mu)^2/\sigma^2/2}$, the half-width, half-max will satisfy the following (setting $\mu=0$ for simplicity and noting the maximum of the function to be at $e^0$)
+$$A\exp^{-x_0^2/\sigma^2/2} = \frac{1}{2}A\exp^{0}$$
+$$\exp^{-x_0^2/\sigma^2/2} = \frac{1}{2}$$
+$$\frac{-x_0^2}{2\sigma^2} = \ln{\frac{1}{2}}$$
+$$x_0^2 = \sigma^2 2\ln{2}$$
+$$x_0 = \pm\sqrt{\sigma^2 2\ln{2}}$$
+$$x_0 = \pm\sigma\sqrt{2\ln{2}}$$
+Leaving us with the FWHM of: $2 \sigma\sqrt{2\ln{2}}$
+This FWHM is what we use to report our effective angular resolution. The FWHM (similar to the standard of deviation, $\sigma$) is measured in pixels in image space, but is most useful to discuss in terms of seconds of arc ('').
+Determine the effective angular resolution of the "Original Airy" profile shown below. Assume the ASI 432 is placed on the main scope.\\
+[{{:train:lectures:20240321_airyseeing.png?800|Single Airy Profile with fit. Sum of many Airy Profiles with fit.}}]
+Consider a telescope unencumbered by pesky atmosphere and is therefore observing an Airy Disk when observing a point source. Take the $x$ axis to be the number of pixels relative to the center of the profile and estimate the Angular Resolution.
+====Seeing Limited====
+[{{https://www.astronomynotes.com/telescop/twmountn.gif|Better Seeing Geometry}}]\\
+[{{https://www.astronomynotes.com/telescop/twinkle.gif|Poor Seeing Geometry}}]\\
+Due to the turbulent and inhomogeneous atmosphere, rays of light have many opportunities to refract upon impinging a volume of atmosphere with differing optical properties (think: index of refraction from Snell's Law). These refraction events have the effect of displacing the center of the Airy Profile. By the Central Limit Theorem, the sum$^1$ of many displaced Airy Profiles approaches a Gaussian Profile. This gives us confidence in our choice of a Gaussian PSF (though there are better-matching PSF models; e.g., Moffat Profile).
+Determine the effective angular resolution of the "Sum of 3000 Disp. Airys" profile shown above. Assume the ASI 432 is placed on the main scope.
+An optical system whose angular resolution is limited by these repeated refraction events is said to be **Seeing Limited**.
+==== Determining the Effective Angular Resolution of Our Telescope ====
+  - The effective angular resolution ('seeing') stands to vary day-by-day
+  - We expect that the Planetary Boundary Layer is unstable near sunset and sunrise, giving rise to increased turbulence/larger (worse) seeing values
+  - We expect that the seeing depends on the temperature, humidity, ambient pressure, as well as any differences in these between the dome and the broader local atmosphere
+In order to measure our instantaneous angular resolution, we need to find the FWHM of a source profile. To generate a source profile, we need to observe a source:
+  - that is point-like; meaning:
+      - far away
+      - not a visual binary(+) star system
+  - long enough that the full range of displacements is sampled
+  - short enough that the pixels don't 'max-out' or 'saturate'
+With this observation, i.e., an image, we can load the image and treat it as a table of pixel values.
+The source in the image can be thought of as tracing out a 3-D surface. Two of the dimensions are the pixel location, and the final dimension is the pixel value.
+We can extract a profile from this surface by just taking a "slice" of the image table (python array).
+====CODE====
+[[https://colab.research.google.com/drive/1SocmlQtnegS2GoyTZN7gaM840_eHzxcI?usp=sharing|Google Colab for Telescope Optics II]]
+----
+$^1$:Visible telescopes are inherently non-coherent imagers. This allows us to assume linearity in the observing system
+Written by Roy Prouty
+Reviewed by