2017年IOAA理论第7题-外星生命

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IOAA赛场上通常是使用英文回答问题，所以建议各位用英文回答本题

英文原题

(T7) Life on Other Planets [20 marks]

One place to search for life is on planets orbiting main sequence stars. A good starting point is the planets that have an Earth-like temperature range and a small temperature fluctuation.Assume that for a main sequence star, the relation between the luminosity L and the mass M is given by

L∝M^3.5

You may assume that the total energy E released over the lifetime of the star is proportional to the mass M of the star. For the Sun, it will have a main sequence lifetime of about 10 billion years. The stellar spectral types are given in the table below. Assume that the spectral subclasses of stars (0-9) are assigned on a scale that is linear in log M .

Spectral Class	O5V	B0V	A0V	F0V	G0V	K0V	M0V
Mass（M⊙）	60	17.5	2.9	1.6	1.05	0.79	0.51

a) If it takes at least 4×10⁹ years for an intelligent life form to evolve, what is the spectral type (accurate to the subclass level) of the most massive star in the main sequence around which astronomers should look for intelligent life? [6]

b) Assume that the target planet has the same emissivity ε and albedo a as the Earth. In order to have the same temperature as the Earth, express the distance d, in au, of the planet to its parent main sequence star, of mass M. [6]

c) The existence of a planet around a star can be shown by the variation in the radial velocity of the star about the star-planet system centre of mass. If the smallest Doppler shift in the wavelength detectable by the observer is (Δλ / λ) =10^-10 , calculate the lowest mass of such a planet in b), in units of Earth masses, that can be detected by this method, around the main sequence star in a). [8]

中文翻译

外星生命

从围绕主序星公转的行星上寻找生命是探索地外文明的热点。其中一个很好的突破口就是那些和地球气温相似，且温差变化较小的行星。已知对于任意主序星，它的光度 L 和质量 M 满足L∝M^3.5。

假定恒星一生辐射出的总能量 E 与它的质量 M 成正比。以太阳为例，它在主序期的时长为100亿年。下表给出了不同光谱型的恒星质量。假设恒星同一光谱型中每级子类的分布与 logM 线性相关。

光谱型	O5V	B0V	A0V	F0V	G0V	K0V	M0V
质量（M⊙）	60	17.5	2.9	1.6	1.05	0.79	0.51

（a）如果行星上进化出智慧生命至少需要40亿年，那么天文学家重点搜寻的恒星对应的光谱型属于哪个范围？（光谱型应精确到具体的子类）

（b）假设目标行星的辐射率 ε 和反照率 a 与地球相同，温度也非常接近，其主序母星质量为M，请以天文单位 au 为单位，未出这种行星到其母星距离 d 的表达式。

（c）如果一颗恒星周围有一颗行星围绕它转动，恒星的视向速度就会发生微小的变化。如果观测者探测到的波长中的最小多普勒位移是(Δλ / λ) =10^-10，请计算围绕（a）问中这类光谱型的恒星运动的（b）问中这类行星的最小质量，以地球质量为单位。

（（a）问的题目有一些问题。实际上答案给出来的确实是一个范围，但是确是临界情况——也就是什么光谱型的恒星恰好满足这40亿年的条件）