Chapter 5.2, Problem 14E

Reminder Round all answers to two decimal places unless otherwise indicated.

Binary Stars Binary stars are pairs of stars that orbit each other. The period $p$ of such a pair is the time, in years, required for a single orbit. The separation $s$ between such a pair is measured in seconds of arc. The parallax angle $a$ (also in seconds of arc) for any stellar object is the angle of its apparent movement as the Earth moves through one half of its orbit around the sun. Astronomers can calculate the total mass $M$ of a binary system using

$M=s^3{a}^{-3}{p}^{-2}$

Here $M$ is the number of solar masses.

a. Alpha Centauri, the nearest star to the sun, is in fact a binary star. The separation of the pair is $s=17.6$ seconds of arc, its parallax angle is $a=0.76$ second of arc, and the period of the pair is $80.1\text{years}$. What is the mass of the Alpha Centauri pair?

b. How would the mass change if the separation angle were doubled, but the parallax and period remained the same as for the Alpha Centauri system?

c. How would the mass change if the parallax angle were doubled. but the separation and period remained the same?

d. How would the mass change if the period doubled, but the parallax angle and separation remained the same?