2019年IOAA理论第4题-改进普通的反射望远镜

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61 CYGNI讨论 | 贡献2020年7月31日 (五) 20:16的版本 (解答部分更新)

英文题目

4. Improving a common reflecting telescope (10 p)

A student has an average quality Cassegrain telescope, with primary and secondary mirrors having $$𝜀1 = 91%$$ reflectivity aluminium layers.

a) What will be the change in the limiting magnitude of this telescope by replacing the mirror coatings with "premium" quality $$𝜀2 = 98%$$ reflectivity ones? (5 p)

b) Assuming the student also uses a star diagonal mirror, also with reflectivity 𝜀1 with the original telescope - what will be the improvement if he/she also replaces this piece with an $$𝜀3 = 99%$$ reflectivity (“dielectric” mirror) model, combined with the new $$𝜀2$$ mirrors? (3 p)

(A star diagonal mirror is a flat mirror, inclined to the optical axis by 45°.)

c) Is this difference obviously detectable by the human eye? Mark "YES" or "NO" on the answer sheet. (2 p)

Consider the whole visual band and disregard any wavelength dependence and geometric effects.

中文题目

4. 改进普通反射望远镜(10p)

一个学生有一台普通的卡塞格林望远镜,其主镜和副镜均镀有反射率$$𝜀1 = 91%$$的铝层。

a)如果用反射率更好的$$𝜀2 = 98%$$的镀膜换掉原来的,这台望远镜的极限星等会发生什么变化?(5p)

b)假设学生在原来的望远镜上使用天顶镜,原始反射率为$$𝜀1$$——- 如果在用新的$$ε2$$的望远镜时,他/她也用$$ε3= 99%$$反射率的天顶镜替换原来这个的话,这台望远镜相比于原来将会有什么改进 ? (3 p)

c)这种差异能被人类肉眼明显地感受到吗?在答题纸上的“是”或“否”上作出标记。(2p)

考虑整个视觉波段,忽略任何与波长有关的修正和几何效应。

解答

考虑一颗亮度为E的恒星

改造前望远镜星等$$m1$$满足

$$m_{1} =-2.5\lg_{}{\varepsilon _{1 }^{2} E } $$

改造后星等$$m2$$满足

$$m_{2} =-2.5\lg_{}{\varepsilon _{2}^{2} E } $$

因此改造前后星等差有

$$m_1- m_{2} =-2.5\lg_{}{\frac{\varepsilon _{1}^{2}}{\varepsilon _{2}^{2}}} $$

代入数据,计算得

$$(待更新)