Black hole in X-Ray Binary.
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).
(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.
http://ocw.mit.edu/courses/physics/8-01-physics-i-classical-mechanics-fall-1999/assignments/sol8.pdf
a zillion thx
(a) To find the orbital period of the binary, we can use Kepler's third law of planetary motion, which states that the square of the orbital period is proportional to the cube of the semi-major axis of the orbit.
Let's start by finding the semi-major axis of the orbit. Since the orbits are circular, the semi-major axis (a) is equal to the radius (r) of the orbit.
The total mass of the binary system is the sum of the masses of the stars, m_1 and m_2. So, the mass of the binary system is given by:
M = m_1 + m_2
According to Newton's law of gravitation, the gravitational force between the two stars is given by:
F = (G * m_1 * m_2) / r^2
Since the stars are in circular orbits, the gravitational force provides the necessary centripetal force:
F = (m_2 * v^2) / r
where v is the orbital speed of the donor star.
Setting these two equations equal to each other, we get:
(G * m_1 * m_2) / r^2 = (m_2 * v^2) / r
Simplifying the equation, we can cancel out m_2 and rearrange the terms:
(G * m_1) / r = v^2
Now, we can solve for the orbital period, T. The orbital period is the time it takes for one complete revolution around the orbit. It can be expressed in terms of the circumference of the orbit and the orbital speed:
T = (2 * pi * r) / v
Plugging in the value for v from the question, we have:
T = (2 * pi * r) / 148 km/sec
To express the answer in terms of m_1, m_2, r_1, and r_2, we can substitute the relationship between the masses and the radii of the stars:
m_1 / r_1 = m_2 / r_2
Rearranging the equation, we get:
m_2 = (m_1 * r_2) / r_1
Substituting this into the equation for T, we have:
T = (2 * pi * r) / [148 km/sec * (m_1 * r_2) / r_1]
Simplifying further, we get the final expression for the orbital period in terms of m_1, m_2, r_1, and r_2:
T = (2 * pi * r_1 * r) / (148 km/sec * m_1 * r_2)
(b) To calculate the mass of the accreting compact object (m_1), given the orbital period (T) and the orbital speed (v), we can rearrange the equation for T:
T = (2 * pi * r) / v
Solving for r, we have:
r = (T * v) / (2 * pi)
Now, we can substitute this value for r in the equation for T:
T = (2 * pi * r_1 * [(T * v) / (2 * pi)]) / (148 km/sec * m_1 * r_2)
Simplifying further, we get:
T = r_1 * [(T * v) / (74 km/sec * m_1 * r_2)]
To calculate m_1, we can isolate it on one side of the equation:
m_1 = [(T * v) / (74 km/sec * T * r_2)] * r_1
Plugging in the given values, we can calculate m_1.