Differences

This shows you the differences between two versions of the page.

--- courses:ast201:7 [2023/08/22 01:13] – asad
+++ courses:ast201:7 [2023/11/26 00:02] (current) – [3.3 Pros and cons] asad
@@ Line 96: / Line 96: @@
 The corrector plate then removes SA. The marginal rays ( $\rho$  large) converge more strongly than the axial rays ( $\rho$  small, closer to axis) as you saw in Section 5.3 of Chapter 6. A Schmidt corrector has higher power for the axial rays than for the marginals, this bringing all rays to a common focal plane.
 ===== - Space telescopes =====
 ==== - Resolution ====
+ $\theta = 2\alpha = \frac{2.44 \lambda}{D}$
 ==== - Sensitivity ====
-==== - Background ====
+The number of photons detected by a detector is the **Signal**
-==== - Visible sky area ====
+$$ S = \frac{\pi D^2}{4} \frac{\lambda}{hc} f_\lambda tQ $$
-==== - Atmosphere ====
+where the factor  $hc/\lambda$  converts from the energy units to number of photons,  $Q$  is a factor dependent on the bandwidth and efficiency of the detector,  $t$  is the exposure (integration) time, and  $f_\lambda$  is the actual flux from the object at a particular wavelength.
-==== - Environment ====
+A star is detectable if  $S$  is about the same size as its uncertainty. The measurement
-==== - JWST ====
+$$ M = S+B $$
+where  $B$  is the **background** containing everything other than the star in the image. The background can actually be both in front of the star or behind star or at the same distance as the star. The background
+ $B = \frac{\pi D^2}{4} \frac{\lambda}{hc} \frac{\pi\theta^2}{4} b_\lambda tQ$
+where  $\theta$  is the size of the star in the image and  $b_\lambda$  the actual background flux. Now according to Poisson statistics the uncertainty in counting photons, the noise,
+ $N = \sigma(S) = \sqrt{(S+B)+B} \approx \sqrt{2B} = \frac{\pi D\theta}{4} \left(\frac{2\lambda b_\lambda tQ}{hc}\right)^{1/2}$
+and, thereby, the signal-to-noise-ratio (SNR)
+ $\frac{S}{N} \approx \left(\frac{\lambda Qt}{hcb_\lambda}\right) \frac{D}{\theta} f_\lambda$
+for a star that is marginally detectable. Purring  $S/N=1$  results in the flux of such a star
+ $f_\lambda = \sqrt{\frac{hc}{\lambda Q}} \sqrt{\frac{b_\lambda}{t}} \frac{\theta}{D}$
+for a telescope whose image results in the diameter of a star to be  $\theta$ . In case of ground-based telescope, the size is that of the **seeing disk**. For space telescope putting  $\theta=2.44\lambda / D$  gives
+ $f_{d,space} = \sqrt{\frac{hc\lambda}{Q}} \sqrt{\frac{b_\lambda}{t}} \frac{2.44}{D^2}$
+which means, increasing the aperture size gives a huge advantage in case of space telescopes because  $f_d \propto D^{-2}$  in that case.
+==== - Pros and cons ====
+Advantages are as follows.
+  - Background  $b_\lambda$ : the darkest background for HST is  $23.3$  mag arcsec $^{-2}$ , and the background for the darkest ground-based telescopes is  $22.0$  mag arcsec $^{-2}$ . For JWST, the sky at 5  $\mu$ m is 12 mag darker in space than on the darkest ground.
+  - Atmospheric transmission
+  - Visible sky area
+  - Gravitational and frictional stress
+Disadvantages are as follows.
+  - Cost. One 8-m Gemini telescope cost 100 million USD, and one 2.4-m HST telescope cost 2 billion USD.
+  - Thermal stress
+  - Harmful radiation, high-energy particles
 ===== - Ground-based telescopes =====
+The [[wp>Hale telescope]] had 5-m mirror made of Pyrex glass and weighed 14 tons. The paraboloidal mirror needs a moving support structure that weighs 530 tons. Completed in 1949, it set the standard of the time. There are 3 stages of making a mirror.
+  - Casting: molten glass poured into a cylindrical mold
+  - Annealing
+  - Grinding and polishing
+{{:courses:ast201:mirror-types.png?nolink&250|}}
 ===== - Adaptive optics =====