Question

Be sure to show all calculations clearly and state your final answers in complete sentences. A Black Hole I? You've just discovered a new X-ray binary, which we will call Hyp-X1 ("Hyp" for hypothetical). The system Hyp-X1 contains a bright, B2 main- sequence star orbiting an unseen companion. The separation of the stars is estimated to be 20 million kilometers, and the orbital period of the visible star is 4 days. a. Use Newton's version of Kepler's third law to calculate the sum of the masses of the two stars in the system. (Hint: See Mathematical Insight $15.4.$ ) Give your answer in both kilograms and solar masses $(M_{\text{Sun }}=2.0\times {10}^{30}\mathrm{kg})$ b. Determine the mass of the unseen companion. Is it a neutron star or a black hole? Explain. (Hint: A B2 mainsequence star has a mass of about $10M_{\text{Sun. }}.)$

Short answer

The mass of the unseen companion is greater than 3 solar masses, indicating it is likely a black hole.

Step-by-step solution

Identify Given Values

In the problem, we are given the orbital separation $a=20\times {10}^6$ km, and the orbital period $T=4$ days. We will convert these to meters and seconds.\1 kilometer = ${10}^3$ meters, so $a=20\times {10}^6\times {10}^3$ meters.\1 day = $86400$ seconds, so $T=4\times 86400$ seconds.

Use Kepler's Third Law

Newton's version of Kepler's third law states that $$T^2=\frac{4{\pi }^2}{G(M_1+M_2)}{a}^3$$\Where $T$ is the period, $a$ is the semi-major axis (or average separation), and $G$ is the gravitational constant $6.674\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$. Rearrange to solve for $M_1+M_2$:\[ M_1 + M_2 = \frac{4\pi^2 a^3}{GT^2} \]

Calculate the Sum of the Masses

Substitute the converted values of $a$ and $T$ into the equation: $$M_1+M_2=\frac{4\times (3.1415{)}^2\times (20\times {10}^9{)}^3}{6.674\times {10}^{-11}\times (4\times 86400{)}^2}$$ \Perform the calculations to find $M_1+M_2$ in kilograms.

Convert to Solar Masses

Once we have calculated $M_1+M_2$ in kilograms, convert this into solar masses using $1M_{\text{Sun}}=2\times {10}^{30}$ kg. \[ M_1 + M_2 \text{ in solar masses} = \frac{M_1+M_2 \text{ (in kg)}}{2 \times 10^{30}} \] \Calculate to find the sum in solar masses.

Determine Mass of Unseen Companion

Given that the mass of a B2 star is approximately $10M_{\text{Sun}}$, subtract this from the total mass $M_1+M_2$ to find the mass of the unseen companion.

Classify the Unseen Companion

If the mass of the unseen companion exceeds approximately $3M_{\text{Sun}}$, it is likely a black hole, otherwise, it could be a neutron star.

Key concepts

Newton's version of Kepler's Third Law

Newton's version of Kepler's Third Law serves as a vital tool in understanding the dynamics of celestial bodies, especially in binary star systems. According to this law, the square of the orbital period of two objects orbiting each other is proportional to the cube of the semi-major axis of their orbit. We express this relationship as:$$T^2=\frac{4{\pi }^2}{G(M_1+M_2)}{a}^3$$Where:

• $T$ is the orbital period,
• $a$ is the semi-major axis (the average distance between the two objects),
• $G$ is the gravitational constant $6.674\times {10}^{-11}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\mathrm{s}}^{-2}$, and
• $M_1+M_2$ is the sum of the masses of the two objects.

By measuring the orbital period and the distance between the objects, we can use this equation to calculate the sum of their masses. In our exercise, this fundamental relationship allows us to determine the total mass of stars in the X-ray binary system Hyp-X1.

X-ray binary systems

X-ray binary systems are intriguing astronomical phenomena where a normal star orbits around an unseen companion, which is usually a neutron star or a black hole. These systems emit X-rays when material from the normal star gets drawn towards and accretes onto the compact companion. In Hyp-X1, the presence of X-ray emissions suggests that the unseen companion is either a neutron star or a black hole. The gravitational pull of such dense objects is strong enough to pull material from the visible star, generating intense X-rays in the process.
Understanding such systems aids astronomers in studying the evolution of binary stars and the properties of extreme states of matter found in neutron stars and black holes.

Mass of a black hole

The mass of a black hole is a crucial parameter in determining its influence in an X-ray binary system. Black holes have such strong gravitational fields that not even light can escape, hence the 'black' in their name.
When calculating the mass of an unseen companion like a black hole, it's important to ascertain whether the mass exceeds the limiting threshold commonly assigned to neutron stars.If calculated in the exercise and found to be over approximately $3M_{\text{Sun}}$, the unseen companion would be classified as a black hole. This significant mass typically suggests that the gravitational potential is strong enough to account for the observed X-ray emissions through the process of accretion from a companion star.

Mass of a neutron star

Neutron stars are the remnants of supernova explosions, composed mostly of neutrons. Despite being incredibly dense, the mass of a typical neutron star ranges from about 1.4 to 3 times that of our Sun ($M_{\text{Sun}}$).
In the context of the Hyp-X1 exercise, if the unseen companion's mass is determined through the use of Newton's version of Kepler's Third Law and found to be less than $3M_{\text{Sun}}$, it suggests the companion is likely a neutron star.

Neutron stars exhibit intense magnetic fields and rapid rotation, both contributing substantially to the X-ray emissions usually observed in binary systems. By calculating the masses involved and comparing them to established astrophysical thresholds, the nature of the unseen companion can be better understood.