# “2019年IOAA理论第4题-改进普通的反射望远镜”的版本间的差异

## 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)

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)

## 解答

a)考虑一颗亮度为E的恒星

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

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

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

＄＄m_1-m_2=0.16^{m}

b）思路和a)问类似

＄＄m_1-m_3=-2.5\lg_{ }{\frac{\varepsilon_1^3}{\varepsilon_2^2\varepsilon_3^1} } ＄＄

＄＄m_1-m_3=0.25^m＄＄

c)对人眼来说，0.25等的亮度差距足以觉察