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 ).
correct
(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).
incorrect
(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.
To find the orbital period (T) of an X-ray binary, we can use the equation:
T = 2π * (r₁ + r₂)^(3/2) / (√(G * (m₁ + m₂)))
where r₁ and r₂ are the radii of the circular orbits, m₁ and m₂ are the masses of the two stars, G is the gravitational constant, and π is pi.
(a) In this case, we need to express the orbital period in terms of m₁, m₂, r₁, r₂, π, and G.
The orbital period is given by:
T = 2π * (r₁ + r₂)^(3/2) / (√(G * (m₁ + m₂)))
(b) In the case of Cyg X-1, we are given the orbital period T = 5.6 days, and the mass of the donor star m₂ = 30 times the mass of the sun. We need to calculate r₂.
To calculate r₂, we can rearrange the equation from part (a) as follows:
r₂ = (T * √(G * (m₁ + m₂)) / (2π))^(2/3) - r₁
Given the orbital period T = 5.6 days, we need to convert it to seconds:
T_sec = T * 24 * 60 * 60 (seconds in a day)
Substituting the known values, we can calculate r₂ in meters.
(c) To calculate m₁, we can use the Doppler shift measurement that gives the orbital speed of the donor star v₂ = 148 km/s. We need to calculate m₁ in kilograms.
The orbital speed is given by:
v₂ = 2π * r₂ / T
Rearranging the equation, we can solve for m₁:
m₁ = (v₂ * T) / (2π * G * r₁)
Substituting the known values, we can calculate m₁ in kilograms.