Black hole in X-Ray Binary.

(2.75/11.0 points)

An X-ray binary consists of 2 stars with masses (the accreting compact object) and (the donor). The orbits are circular with radii and centered on the center of mass.

(a) Find the orbital period of the binary following the guidelines given in lectures. Express your answer in terms of , and (enter m_1 for , m_2 for , r_1 for , r_2 for , pi for and G for ).



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(b) In the case of Cyg X-1 (as discussed in lectures), the orbital period is 5.6 days. The donor star is a “supergiant" with a mass 30 times that of the sun. Doppler shift measurements indicate that the donor star has an orbital speed of about 148 km/sec. Calculate (in meters).



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(c) Calculate (in meters).

hint: Your calculations will be greatly simplified if instead of you set up your equations in terms of , and using some relation between the distances and the masses. Once you express your equation in terms of , you will find a third order equation in . Only one solution is real; the other two are imaginary. There are various ways to find an approximation for . You can find the solution by trial and error using your calculator, or you can plot the function.



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(d) Now calculate the mass of the accreting compact object (express that as ratio to the mass of the sun).



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As discussed in lectures, since this turns out to be substantially larger than 3 times the mass of the sun, it is strongly believed to be a black hole.

To find the orbital period of the binary, we can use Kepler's third law of planetary motion. According to this law, the square of the orbital period (T) is proportional to the cube of the semi-major axis of the orbit (a^3).

(a) Given that the orbits are circular, the semi-major axis of the orbit is equal to the radius of the orbit. So, we can write:

a = r1 + r2

Now, using the formula for the orbital period, we have:

T^2 = k * a^3

where k is a constant. Substituting the value of a, we get:

T^2 = k * (r1 + r2)^3

To express the answer in terms of m1, m2, r1, r2, pi, and G, we need to substitute the values of the masses and radii.

(b) Given that the orbital period is 5.6 days and the donor star has a mass 30 times that of the sun, we need to find the value of r1 + r2. Unfortunately, without the value of r2, we cannot calculate it directly.

(c) To find the value of r1 + r2, we can use the relation between the distances and the masses. Let's call the distance between the two stars as d. According to Newton's law of gravitation, we have:

G * m1 * m2 / d^2 = G * m1 * m2 / r1^2 + G * m1 * m2 / r2^2

Simplifying, we get:

1 / d^2 = 1 / r1^2 + 1 / r2^2

As the hint suggests, using this equation, we can rewrite the equation in terms of d and r1:

1 / r1^2 = 1 / d^2 - 1 / r2^2

Substituting this back into the equation for the orbital period, we have:

T^2 = k * (r1 + r2)^3
= k * (d^2 - 1 / r1^2)^3

Now, with the given value of the orbital period T, we can solve this equation to find the value of r1 + r2. However, we need more information to proceed.

(d) Finally, to calculate the mass of the accreting compact object, we can use the mass-radius relation for stars. This relation depends on various factors such as the equation of state and the composition of the star. Without these additional details, we cannot calculate the mass of the accreting compact object.

However, based on the information provided, since the calculated mass is substantially larger than 3 times the mass of the sun, it is strongly believed to be a black hole.