How Do We Calculate the Mass of Binary Stars Like Centauri A and B?

[PSEYE]

Homework Statement

Centauri A and Centauri B are binary stars with a separation of 3.45x10^12m
and a period of 2.52x10^9s

Assuming the two stars are equally massive (which is approximately the case), determine their mass.

Homework Equations

( m1 + m2 ) P^2 = ( d1 + d2 )^3 = R^3
I used this and it doesn't work :x

The Attempt at a Solution

M1=M2
D1=D2

M(2.52x10^18)=3.45x10^36

M=3.45x10^36/2.52x10^18

= wrong :X

 

[loonychune]

$${(2.52 \times 10^9)}^2 \neq 2.52 \times 10^{18}$$

 

[baba89]

I understand your frustration with this problem, as it can be tricky to determine the mass of binary stars. However, there are a few key concepts and equations that can help you arrive at the correct solution.

First, it is important to note that the equation you used, (m1 + m2 ) P^2 = ( d1 + d2 )^3 = R^3, is not quite correct. The correct equation for calculating the mass of binary stars is (m1 + m2 ) P^2 = ( a^3 ) / ( G ), where a is the semi-major axis (half of the separation distance between the two stars) and G is the gravitational constant.

Using this equation, we can rearrange it to solve for the total mass (m1 + m2) of the binary stars:

(m1 + m2) = (a^3) / (G x P^2)

Now, let's plug in the given values for Centauri A and B:

a = 3.45x10^12m / 2 = 1.725x10^12m

G = 6.67x10^-11 m^3/kg/s^2

P = 2.52x10^9s

And solving for (m1 + m2):

(m1 + m2) = (1.725x10^12m)^3 / (6.67x10^-11 m^3/kg/s^2 x (2.52x10^9s)^2)

= 1.59x10^32 kg

Since we know that the two stars are equally massive, we can divide this value by 2 to get the individual mass of each star:

m1 = m2 = 1.59x10^32 kg / 2 = 7.95x10^31 kg

Therefore, the mass of each star in the Centauri A and B binary system is approximately 7.95x10^31 kg.

I hope this explanation helps you understand the correct way to approach this type of problem. Remember to always double check your equations and units to ensure accuracy in your calculations. Keep up the hard work in your scientific studies!