For Cyg \(\mathrm{X}-1,\) the most likely value of the mass function is $$\left(M_{\mathrm{x}} \sin i\right)^{3} /\left(M_{\mathrm{x}}+M_{\mathrm{opt}}\right)^{2}=0.25 \mathrm{M}_{\odot}$$ For an inclination angle of \(30^{\circ},\) and an optical star mass of \(33 \mathrm{M}_{\odot}\), find the mass of the compact object.

When studying binary star systems, determining the mass of the compact object (such as a black hole or neutron star) is crucial. This mass is often denoted as \( M_{\text{X}} \). In the context of Cyg X-1, this is the object we are trying to find. To determine the mass, we use observational data coupled with equations that describe the system's dynamics. One of these key equations is the mass function equation we will discuss later. By analyzing how the compact object impacts its companion star's motion, we can derive its mass.

The mass function equation is a vital tool in astronomy for calculating the mass of an unseen companion in a binary system. It is given by:

\[\frac{(M_{\text{X}} \sin i)^{3}}{(M_{\text{X}} + M_{\text{opt}})^{2}} = f \]

In this equation:

• <\( M_{\text{X}} \): Mass of the compact object
• <\( M_{\text{opt}} \): Mass of the optical star (companion star)
• <\( \sin i \): Sine of the inclination angle
• <\( f \): Mass function value, which is derived from observations

For Cyg X-1, the mass function value is given as 0.25 \( \text{M}_{\text{sun}} \). Inserting known values into the mass function equation allows us to rearrange and solve for \( M_{\text{X}} \), the mass of the compact object.

The inclination angle, denoted as \( i \), is the angle at which we view the orbit of the binary system. It plays a significant role in determining the exact mass of the compact object. The sine of the inclination angle \( \sin i \) acts as a multiplier in the mass function equation. For instance, in Cyg X-1, the inclination angle is given as 30 degrees. Hence, \( \sin 30^{\circ} = 0.5 \). This value is crucial as it directly affects our calculations. A smaller or larger inclination angle would change the calculated mass of the compact object.

The optical star mass, \( M_{\text{opt}} \), is the mass of the visible star in a binary system. In our problem, the mass of the optical star of Cyg X-1 is given as 33 \( \text{M}_{\text{sun}} \). This value is also a critical component in the mass function equation. It interacts with the mass of the compact object \( M_{\text{X}} \) to dictate the system’s dynamics. Knowing the optical star's mass allows us to use the mass function equation effectively to solve for the mass of the compact object once all other factors are considered. Together with the inclination angle and the observed mass function value, this gives a clear path to finding the compact object's mass. By inserting these values into our simplified equation, we can solve for \( M_{\text{X}} \) numerically.