# How to Determine Masses in a Binary Star System?

• pierce15
In summary, the problem is as follows:Sirius is a visual binary with a period of 49.94 years. Its measured parallax is .37931"## \pm ##.00158", and the angular extent of the semimajor axis of the reduced mass is 7.61". The ratio of the distances of Sirius A and B to the center of mass is ## a_A / a_B = .466 ##. Find the masses of the two stars, assuming that the motion is in the plane of the sky.
pierce15
Hello,

I wasn't sure whether I should post this is the homework section since it's technically a textbook problem, but I figured I'd get better responses here. The problem is as follows:

Sirius is a visual binary with a period of 49.94 years. Its measured parallax is .37931"## \pm ##.00158", and the angular extent of the semimajor axis of the reduced mass is 7.61". The ratio of the distances of Sirius A and B to the center of mass is ## a_A / a_B = .466 ##. Find the masses of the two stars, assuming that the motion is in the plane of the sky.

First, you can use the ratio to get ## m_A / m_B = 1/.466 = 2.146##. I'm pretty sure I next have to use the 7.61", but I don't know how. After that, I would have all the unknowns in Kepler's third except the masses, so I could solve the system. So how do I get the semimajor axis of the smaller star?

piercebeatz said:
I wasn't sure whether I should post this is the homework section since it's technically a textbook problem, but I figured I'd get better responses here.
Textbook questions belong to the homework section. I moved it with a redirect in the original forum.

First, you can use the ratio to get ## m_A / m_B = 1/.466 = 2.146##.
Okay.
I'm pretty sure I next have to use the 7.61", but I don't know how.
This is related to the true semi-major axis of the system, if you know the distance. There is another parameter given that allows to calculate the distance.

So how do I get the semimajor axis of the smaller star?
Find the semi-major axis of the reduced mass first.

mfb said:
This is related to the true semi-major axis of the system, if you know the distance. There is another parameter given that allows to calculate the distance.

Using the parallactic angle yields ## d [pc] = 1/p" = 1/.37921 = 2.6363 pc##. Now what?

By the way, the "reduced mass" just refers to the star with lower mass, right?

Last edited:
piercebeatz said:
Using the parallactic angle yields ## d [pc] = 1/p" = 1/.37921 = 2.6363 pc##. Now what?
You got an angle (as seen from earth) and a distance...

By the way, the "reduced mass" just refers to the star with lower mass, right?
No.

Yeah, my bad... 7.61" = a / 2.636 pc ---> a = 3.00 E12 after converting 7.61" to rad and 2.636 to m. So is this the same a that goes in kepler's third equation? Or do I have to go back and use the semimajor axis ratio that I was given

So is this the same a that goes in kepler's third equation?
Should be. Check the link to the reduced mass.
Or do I have to go back and use the semimajor axis ratio that I was given
There was no given semi-major axis, you had to calculate it.

mfb said:
Should be. Check the link to the reduced mass.
There was no given semi-major axis, you had to calculate it.

Got it. Thank you very much.

## 1. How do you calculate the mass of a binary star system?

The mass of a binary star system can be calculated using the formula M = (4π^2a^3) / (GT^2), where M is the total mass of the system, a is the distance between the two stars, G is the gravitational constant, and T is the orbital period of the stars.

## 2. What is a spectroscopic binary star system?

A spectroscopic binary star system is a pair of stars that cannot be seen as separate entities with a telescope, but their presence can be detected by analyzing the light spectrum of the system. This is because the stars are orbiting each other so closely that they appear as a single point of light.

## 3. How does the mass ratio of binary stars affect the calculation?

The mass ratio of binary stars, which is the ratio of the mass of the smaller star to the mass of the larger star, can affect the calculation of the total mass of the system. This is because the gravitational force between the stars is inversely proportional to the square of the distance between them, so the smaller star will have a greater impact on the orbital period and therefore the calculated mass.

## 4. Can binary star masses change over time?

Yes, binary star masses can change over time due to a variety of factors such as mass transfer between the stars, collisions, or interactions with other nearby stars. These changes can affect the orbital period and therefore the calculated mass of the system.

## 5. How accurate are calculations of binary star masses?

The accuracy of calculations of binary star masses depends on the quality and quantity of observational data available. With precise measurements of the orbital period and distance between the stars, the calculated mass can be accurate to within a few percentage points. However, there are many variables and uncertainties in binary star systems that can make the calculations less accurate.

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