Chapter 12.8, Problem 9P

To determine

To calculate: The total mass of a double star is directly proportional to the cube of maximum distance between the stars and inversely proportional to the square of the period.

Answer to Problem 9P

The total mass of the Capella is: $5.30$ cm.

Explanation of Solution

Given information:

The total mass of a double star is directly proportional to the cube of maximum distance between the stars and inversely proportional to the square of the period.

The double star series has a period of $50.0$ years ,a maximum distance of $41.0$ AU, and a total mass of $3.45$ . What is the total mass of the Capella if its period is $0.285$ years and the maximum distance is $1.51$ AU.

Formula used:

For varies directly and inversely:

  $M=K\frac{d^3}{t^2}$

Calculation:

Consider that, The double star series has a period of $50.0$ years ,a maximum distance of $41.0$ AU, and a total mass of $3.45$ . What is the total mass of the Capella if its period is $0.285$ years and the maximum distance is $1.51$ AU.

Recall that varies directly and inversely:

  $M=K\frac{d^3}{t^2}$

The values are:

  $M=3.45,d=41.0,t=50.0$

Therefore, the value of $k$ is,

  $\begin{array}{c}M=K\frac{d^3}{t^2}\\ 3.45=K\frac{{\left(41.0\right)}^2}{{\left(50.0\right)}^2}\\ K=\frac{3.45\times {50.0}^2}{{41.0}^2}\\ K=\frac{3.45\times 2500}{68921}\end{array}$

So simplified further the value of $k$ is,

  $\begin{array}{l}K=\frac{3.45\times 2500}{68921}\\ K=\frac{8625}{68921}\\ K=0.1251\end{array}$

Now to find the value of $r$ ,

  $M_2=K\frac{d_2{}^3}{t_2{}^2}$

The values are:

  $K=0.1251,d=\left(1.51\right),t=0.285$

Hence, when:

  $\begin{array}{c}{M}_2=K\frac{d_2{}^3}{t_2{}^2}\\ M=0.1251\times \frac{{\left(1.51\right)}^3}{{\left(0.285\right)}^2}\\ M=0.1251\times \frac{3.442951}{0.081225}\\ M=0.1251\times 41.38782395\end{array}$

So,

  $\begin{array}{c}M=0.1251\times 41.38782395\\ M=5.177616776\\ M=5.30\end{array}$

Thus, the total mass of the Capella is: $5.30$ cm.