# What Is the Lower Mass Limit of an Invisible Binary Companion?

• Kinetic
In summary, by using Newton's adaptation of Kepler's third law and the kinetic energy equation, you can calculate the lower mass limit of the unseen binary companion of a low-mass main sequence star by using the observed velocity variations and period.
Kinetic
Hey guys, got this question that's been giving me some trouble.

The spectral lines in a low-mass main sequence star are observed to show sinusoidal velocity variations with an amplitude of 500km/s and a period of ten hours. Calculate the lower mass limit of the unseen binary companion.

Here's how far I've managed to get:

I started with Newton's adaptation of kepler's third law

1. M1+M2=(4(PI)2a3)/(GP2)

where 'a' is the separation of the two masses

2. a=r1+r2

I don't know the radius of either orbit but i do know the velocity of one and the period.

3. V1=(2(PI)r1)/P
4. V2=(2(PI)r2)/P

I then rearranged 3 and 4 to get r in terms of P and V and substituted into 2 to get

5. a=(P/2(PI))*(V1+V2)

Then substituting 5 into 1 gives

6. M1+M2=(P/2(PI)G)*(V1+V2)3

I now want to get rid of the V2.

V2/V1=r2/r1=M1/M2

So i make V2= V1*(M1/M2)

Substitute this into 6 and with a bit of fiddling i get

7. M23/(M1+M2)2=(PV13)/(2(PI)G)

I've been told the visible star is a 'low-mass main sequence star' so i can make a rough estimate of M1. Now my only unknown is M2...
However! I've been playing with 7 for ages and simply cannot isolate M2

So I've either messed up somewhere along the way to 7 or my algebra is failing me.

Any suggestions or help would be great guys!

Kinetic

energy = P*V^2/(4*pi^2)Using the kinetic energy equation, you can solve for M2. You know the values for P and V, so you just need to solve for M2. You can then use this value to calculate the lower mass limit of the unseen binary companion.

## 1. What is a mass of an invisible binary companion?

The mass of an invisible binary companion refers to the amount of matter that makes up the unseen object in a binary star system. This object is typically a white dwarf, neutron star, or black hole, and its mass can range from a few times the mass of the sun to several times the mass of the sun.

## 2. How is the mass of an invisible binary companion determined?

The mass of an invisible binary companion is determined through various methods, such as measuring the orbital period and velocity of the visible star and using Kepler's laws of planetary motion, or through radial velocity measurements and the use of mass-luminosity relations.

## 3. Why is the mass of an invisible binary companion important to study?

The mass of an invisible binary companion is important to study because it provides valuable information about the formation and evolution of binary star systems. It also helps scientists understand the properties and behavior of compact objects, such as neutron stars and black holes.

## 4. Can the mass of an invisible binary companion change over time?

Yes, the mass of an invisible binary companion can change over time, particularly in systems where mass transfer occurs between the two stars. This can happen when the visible star expands and transfers material to the companion, altering its mass and potentially affecting the overall dynamics of the system.

## 5. How does the mass of an invisible binary companion affect the visible star?

The mass of an invisible binary companion can have a significant impact on the visible star in a binary system. Depending on the mass ratio between the two stars, the companion can influence the visible star's size, temperature, and evolution. In some cases, it can even lead to the destruction of the visible star through processes such as mass transfer or tidal interactions.

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