2020年GeCAA理论第3题-镜子

英文题目

3 Mirror

A bored cosmologist comes up with a thought experiment to determine the Hubble constant ($$H_0$$) for his model of a Steady-State-Universe. In this experiment, a large, fully reflecting flat mirror – carrying several gyroscopes that would maintain its spatial orientation in the same plane – would be placed at a distance $$D$$ from the Solar System in a region without gravitational influences. From the Earth, a laser beam would be directed towards that region for a long period of time. After a time T, the radiation would return and be detected, allowing the determination of the fixed constant $$H_0$$.

(a) (7 points) Find an expression for $$H_0$$ as a function of $$D$$, $$c$$ (speed of light) and $$T$$. Consider that the separation $$S$$ between the Solar System and the mirror increases only due to the expansion of the universe according to the law $$S = se^{H_0t}$$, where $$s$$ is the initial separation. You may use $$e^x ≈ 1 + x$$ for $$x≪ 1$$, if necessary.

(b) (3 points) Imagine that such a mirror is located in the vicinity of the star Vega (which also features on the logo of the 1st GeCAA). Vega was the first star outside the Solar System to be photographed and one of the first stars whose parallax ($$p = 0.125′′$$) was accurately measured in 1840 by G. W. von Struve. Estimate the total duration of this $$H_0$$ measurement experiment.