# Astrophysics: Finding the mass of a hidden star in a binary system

• knowlewj01
In summary, the conversation discusses the calculation of a lower limit for the mass of an unseen binary companion of a low mass main sequence star, based on its spectral lines showing sinusoidal velocity variations. The equations used include M1 + M2 = (4π²a³)/GP², M1r1 = M2r2, and a = r1 + r2. The attempt at a solution involves calculating the distance traveled in one orbit, estimating the radius of the orbit, and assuming that the hidden object is much more massive. However, it is mentioned that this assumption may be incorrect and the problem may be solved without it.
knowlewj01

## Homework Statement

The spectral lines in a low mass main sequence star show sinusodal velocity variations with an amplitude of 500 km/s and a time period of 10 hours

calculate a lower limit to the mass of the unseen binary companion

## Homework Equations

M1 + M2 = $$\frac{4\pi^2a^3}{GP^2}$$

M1r1 = M2r2

a = r1 + r2

## The Attempt at a Solution

The redshifted and blueshifted spectral lines show that the star is traveling at 500km/s

in a time period of 10 hoiurs (= 36 000 seconds)

Distance traveled in 1 orbit = 18 000 000 km

radius of orbit = $$\frac{18 000 000}{2\pi}$$ = 28274334 km

assume that the hidden object is much more massive.

so a = radius of this orbit = 28274334000 m

M1 >> M2 therefore r1 << r2

M1 = $$\frac{4\pi^2r^3}{GP^2}$$

M1 = 1.30x10^13 kg

this is wrong,

think i may have made a mistake when i said that the hidden object is much more massive, can this be solved without making an assumption?

I think you can calculate ratio of both masses, then use information about mass of the observed star ("low mass main sequence star") to estimate mass of the other one. But that's just intuition, I can be easily wrong.

Yes, this can be solved without making the assumption that the hidden object is much more massive. The key is to use the second equation listed, which relates the masses and radii of the two objects in a binary system. Since we know the distance traveled by the visible star in one orbit (18,000,000 km), we can use this to calculate the radius of the visible star's orbit (r1) using the formula a = r1 + r2. Then, we can use this value for r1 in the equation M1r1 = M2r2 to solve for the mass of the hidden object (M2). This will give us a lower limit for the mass of the hidden object, since we are assuming that M1 >> M2.

## 1. How do scientists determine the mass of a hidden star in a binary system?

Scientists use a variety of techniques, such as radial velocity measurements, eclipsing light curves, and gravitational lensing, to calculate the mass of a hidden star in a binary system. These techniques involve analyzing the movement, brightness, and gravitational effects of the visible star in the system.

## 2. Why is it important to find the mass of a hidden star in a binary system?

Finding the mass of a hidden star in a binary system is crucial for understanding the overall structure and dynamics of the system. It also helps scientists determine the properties and evolution of the individual stars, as well as the nature of their interaction and potential for producing phenomena such as supernovae.

## 3. Are there any limitations to determining the mass of a hidden star in a binary system?

Yes, there are some limitations to the methods used for calculating the mass of a hidden star in a binary system. For example, the accuracy of the calculations can be affected by factors such as the orientation of the system, the presence of other nearby stars, and the level of precision of the instruments used.

## 4. Can the mass of a hidden star in a binary system change over time?

Yes, the mass of a hidden star in a binary system can change over time due to various factors such as mass transfer between the two stars, interactions with other objects, and nuclear reactions within the star itself. Scientists often need to continuously monitor and update their calculations to account for any changes in the mass of a hidden star.

## 5. How do scientists use the mass of a hidden star in a binary system to study other astrophysical phenomena?

The mass of a hidden star in a binary system can provide valuable insights into various astrophysical phenomena, such as the formation and evolution of galaxies, the production of gravitational waves, and the behavior of matter in extreme conditions. By studying the properties and interactions of these hidden stars, scientists can better understand the underlying principles and processes that govern the universe.

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