2018年IOAA理论第2题-距离

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英文题目

(T2) Distance (10 points)

An observer measured trigonometric parallaxes of stars in a star cluster. Due to random errors, the measured parallax values are distributed symmetrically around the expected value with standard deviation equal to 0.05 mas (milliarcsec). Assume there are no systematic errors and assume all stars in the said cluster have the same luminosity. It is known that the distance of this cluster from us is R= 5 kpc.

He gave the data table to 4 of his students (A, B, C and D) and they estimated the distance to the cluster in the following ways:

A. Convert each parallax measurement into distance and then find the average distance (R_A)

B. Take the average of all parallaxes first and then convert the average parallax into distance. (R_B)

C. Convert each parallax measurement into distance and then take the median distance value. (R_C)

D. Find the median value of the parallaxes and then convert the median value into distance. (R_D)

State if the following statements are true or false. In case a given mathematical relation is false, give the correct relation.

(l) If the ith star gave the smallest value of parallax and the jth star gave the highest value of parallax, in all likelihood R_i-R > R-R_j

(m) R_A = R (i.e. there is a high chance that the distance estimated by A fairly matches the true distance)

(n) R_B = R (i.e. there is a high chance that the distance estimated by B fairly matches the true distance)

(o) R_C < R (i.e. there is a high chance that the distance estimated by C will be systematically lower than the true distance)

(p) R_D = R (i.e. there is a high chance that the distance estimated by D fairly matches the true distance)

中文翻译

(T2)距离 (10分)

一个观测员测量一个星团中恒星的三角视差.由于随机误差,测量到的视差值对称地分布在精确值的两边,分布的标准差是0.05毫角秒.假设没有系统误差,并且这个星团中的恒星光度相同.已知这个星团距离我们 R = 5 kpc.

他把数据表给了他的四个学生(A, B, C 和D),要求他们用以下四种方法估计到星团的距离.

A. 把每个测量视差转换成距离,然后计算平均值(R_A)

B. 先计算视差的平均值,然后把它转换成距离(R_B)

C. 把每个测量视差转换成距离,然后计算中位数(R_C)

D. 先计算测量视差的中位数,然后把它转换成距离(R_D)

判断以下叙述的对错.如果给出的数学关系是错误的,请给出正确的关系.

(l) 如果第i颗恒星的视差值最小,第j颗恒星的视差值最大,那么总有R_i-R > R-R_J

(m) R_A = R(即A估计的距离有很高的几率和实际距离相符)

(n) R_B = R(即B估计的距离有很高的几率和实际距离相符)

(o) R_C < R(即C估计的距离有很高的几率系统地低于实际距离)

(p) R_D = R(即D估计的距离有很高的几率和实际距离相符)