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In the X-ray binary system A0620-00, the radial orbital velocities for the normal star and the compact object are \(v_{s, r}=457 \mathrm{km} \mathrm{s}^{-1}\) and \(v_{c, r}=43 \mathrm{km} \mathrm{s}^{-1},\) respectively. The orbital period is 0.3226 day. (a) Calculate the mass function (the right-hand side of this equation) $$\begin{aligned} &\frac{m_{c}^{3}}{\left(m_{s}+m_{c}\right)^{2}} \sin ^{3} i\\\ &\frac{m_{2}^{3}}{\left(m_{1}+m_{2}\right)^{2}} \sin ^{3} i=\frac{P}{2 \pi G} v_{1 r}^{3} \end{aligned}$$ where \(m_{s}\) is the mass of the normal star, \(m_{c}\) is the mass of its compact companion, and \(i\) is the angle of inclination of the orbit. What does this result say about the mass of the compact object? (Note that the value of \(v_{c, r}\) was not needed to obtain this result.) (b) Now use the value of the orbital radial velocity of the compact object to determine its mass, assuming \(i=90^{\circ} .\) What does this result say about the mass of the compact object? (c) The X-rays are not eclipsed in this system, so the angle of inclination must be less than approximately \(85^{\circ} .\) Suppose that the angle of inclination were \(45^{\circ} .\) What would the mass of the compact object be then?

Step by step solution

01

Calculate the Mass Function

The mass function is given by the formula: \[ \frac{m_c^3}{(m_s+m_c)^2} \sin^3 i = \frac{P}{2\pi G} v_{s,r}^3 \] where \(v_{s,r} = 457 \text{ km/s}\), \(P = 0.3226 \text{ day}= 0.3226 \times 24 \times 3600 \text{ s}\), and \(G = 6.674 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2}\). Convert \(v_{s,r}\) to \(\text{m/s}\) and calculate: \[\frac{0.3226 \times 24 \times 3600}{2\pi \times 6.674 \times 10^{-11}} (457000)^3\]

02

Use Assumptions of Inclination Angle

Assume \(i = 90^\circ\) initially, which means \(\sin^3 i = 1\). This simplifies the mass function to: \[ \frac{m_c^3}{(m_s+m_c)^2} \] Calculate the value using the output from Step 1.

03

Calculate Compact Object's Mass with Provided Velocity

Use the relationship: \[ v_{c,r} = \frac{m_s}{m_c} v_{s,r} \] given that \(v_{c,r} = 43 \text{ km/s} = 43000 \text{ m/s}\). From this, \(\frac{m_s}{m_c} = \frac{43}{457}\) and combine with the previous function to solve for \(m_c\) given \(i= 90^\circ \).

04

Consider Effect of Inclination Angle Being 45 Degrees

For \(i = 45^\circ\), \(\sin^3 i\) changes to \((\sin 45^\circ)^3 = \left(\frac{\sqrt{2}}{2}\right)^3\). Recalculate the mass function using this new value of \(\sin^3 i\) and solve for \(m_c\) again.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radial Velocities

In X-ray binary star systems, radial velocities are vital for understanding and determining the properties of the system. Radial velocity refers to the speed at which an object moves towards or away from an observer.
In the context of binary stars, we measure the radial velocities of both the visible star and its compact companion. This measurement is vital to interpreting the binary's orbital dynamics. For example:

• The radial velocity of the normal star in system A0620-00 is measured at 457 km/s.
• The compact object, meanwhile, has a radial velocity of 43 km/s.

These velocities help to develop equations that can ultimately solve for masses of the system members, as they are a function of how the stars move around their mutual center of mass.

Orbital Period

The orbital period in a binary star system is the time it takes for the stars to complete one full orbit around each other. This is crucial for understanding the dynamics of the system.
For A0620-00, the period is 0.3226 days. Converting this to seconds is essential when using it in calculations. The formula to convert days to seconds is:
\[ P = 0.3226 \times 24 \times 3600 \]
By calculating the period in seconds, we can plug the value into various equations related to the system’s dynamics, like the mass function, which helps in estimating the masses of the objects involved.

Mass Function

The mass function is a critical concept in understanding the mass distribution in binary systems. It relates the masses of the stars and the inclination angle of their orbit. The equation is given by:

\[\frac{m_c^3}{(m_s+m_c)^2} \sin^3 i = \frac{P}{2\pi G}v_{s,r}^3\]
This function allows us to calculate the mass of the compact object when the masses of both stars are unknown. Importantly, it depends on the inclination angle \(i\), making it sensitive to our knowledge of the orbit’s tilt relative to our line of sight.

• For \(i = 90^\circ\), the problem simplifies since \(\sin i = 1 \).
• This is the most straightforward scenario but also often unrealistic in real astronomical observations.

It essentially restricts the mass solution range for the compact star given all other known values.

Compact Object Mass Estimation

Estimating the mass of a compact object in a binary system helps identify its nature—whether it is a neutron star, black hole, or another type of body.
Using the mass function and radial velocities, we get the foundation to start these estimations. For our calculations, when assuming \(i=90^\circ\), we find the simplest estimate of the mass. Additionally, however, X-ray observations provide further constraints. Since the X-rays are not eclipsed, the inclination must be less than \(85^\circ\), implying:

• For \(i = 45^\circ\), recalculating \(\sin^3 i\) becomes essential.
• With a lower inclination, the mass of the compact object is effectively increased when deriving masses through the mass function.

This provides a more comprehensive mass range and illustrates the importance of exact inclination measurements to solid mass estimation in astrophysical studies.