Hunting a black hole. Observations of the light from a certain star indicate tha

Question

Hunting a black hole. Observations of the light from a certain star indicate that it is part of a binary (two-star) system. This visible star has orbital speed v = 260 km/s, orbital period T = 29.5 days, and approximate mass m1 = 6.5Ms, where Ms is the Sun's mass, 1.99 x 1030 kg. Assume that the visible star and its companion star, which is dark and unseen, are both in circular orbits (see the figure). Find the ratio of the approximate mass m2 of the dark star to Ms.

I have tried this problem but cant get it right.

Explanation / Answer

The force exerted on the visible star, due to the unseen object is:

F= Gm1m2/r2

The acceleration of the star is just the centripetal acceleration:

a=w2r1

Now let's use Newton's Second Law for the visible star:

F=ma

Gm1m2/r2 = m1 w2r1

If we let the origin lie at the CM, then we can write the CM as

0= (r1m1- r2m2)/ (m1+ m2)

Using the definition of r in this CM expression we can rewrite it as:

r= r1 (m1 + m2) /m2

Now, let's use this for r and 2p/ T for w in our N2 equation. After much algebra, we get:

Let r= r1 + r2

The force exerted on the visible star, due to the unseen object is:

F= Gm1m2/r2

The acceleration of the star is just the centripetal acceleration:

a=w2r1

Now let's use Newton's Second Law for the visible star:

F=ma

Gm1m2/r2 = m1 w2r1

If we let the origin lie at the CM, then we can write the CM as

0= (r1m1- r2m2)/ (m1+ m2)

Using the definition of r in this CM expression we can rewrite it as:

r= r1 (m1 + m2) /m2

Now, let's use this for r and 2p/ T for w in our N2 equation. After much algebra, we get:

Remember, m2 is the mass of the unseen object and r1 is the distance from the star to the center of mass,

and both of these are unknown, but we do know r1 from the speed of a particle moving in a circular orbit:

v= 2pr1/ T => r1= v T/ 2p

Once we substittue this in, along with the known value of m1, we find that:

Remember, m2 is the mass of the unseen object and r1 is the distance from the star to the center of mass,

and both of these are unknown, but we do know r1 from the speed of a particle moving in a

circular orbit: v= 2pr1/ T => r1= v T/ 2p

Once we substittue this in, along with the known value of m1, we find that:

m2^3/(m1+m2) = v^3T/2piG

m2^3/6.5Ms+m2)^2 = (2.6*10^5)^3*(2.95)*(86400) / 2pi(6.67*10^-11) = 1.06893*10^31KG = 5.371Ms

blech. This is a cubic equation. Eventually you would find that m2 12Ms.

The data here is approximately that of the binary system LMC X-3 in the Large Magellanic Cloud. From other data, the dark object is known to be especially compact; it may be a star that collapsed under its own gravitational pull to become a neutron star or a black hole. Since a neutron star cannot have a mass larger than about 2Ms, the result here strongly suggests that the dark object is a black hole.

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