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 ∝ M3.5

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

寻找地外生命的一个基本思路就是从围绕主序星公转的系外行星下手。最好是从那些和地球气温相似,且温差变化较小的行星开始找起。对于任意主序星,它的光度 L 和质量 M 满足:

L ∝ M3.5

假定恒星在主序阶段辐射出的总能量 E 与它的质量 M 成正比。以太阳为例,它在主序阶段时长为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的每一亚级(数字)用 logM 的线性关系来分割:

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)如果一颗行星进化出智慧生命至少需要40亿年的话,那么天文学家可以搜寻的最重的恒星位于什么光谱型范围?(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。如果该行星的温度和地球相近,请以天文单位 au 为单位,列出该行星到其质量为 M 的母恒星的距离 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。如果观测者能探测到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绕转的(b)问中这类行星的最小质量,以地球质量为单位。

((a)问的翻译有修改。天爱上的翻译寻找的是着重搜寻什么光谱型的恒星,但很显然,看了英文原题就会发现翻错了。还是那句话,英语答题,看英文题目,可以有效避免领队翻译题目错误导致的分数损失。)

思路

(a)问要求我们寻找主序时长长于40亿年的恒星,准确说是质量小于某个阈值的恒星。这一题需要我们找到这个阈值是什么。题目已经给出了质光关系,那么根据这个推导出恒星的主序时长应当不难。

(b)问其实是斯特番-博尔兹曼定律的应用。

(c)问是多普勒效应和双星系统相关的知识的应用。