Question

Be sure to show all calculations clearly and state your final answers in complete sentences. A Black Hole II? You've just discovered another new X-ray binary, which we will call Hyp-X2 ("Hyp" for hypothetical). The system Hyp-X2 contains a bright, G2 main-sequence star orbiting an unseen companion. The separation of the stars is estimated to be 12 million kilometers, and the orbital period of the visible star is 5 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 \(\left(M_{\mathrm{Sun}}=2.0 \times 10^{30} \mathrm{kg}\right)\) b. Determine the mass of the unseen companion. Is it a neutron star or a black hole? Explain. (Hint: A G2 mainsequence star has a mass of \(1 M\) sun.)

Short answer

The sum of masses is \(10.44 M_{\text{Sun}}\), with the unseen companion likely a black hole at \(9.44 M_{\text{Sun}}\).

Step-by-step solution

Understand the problem

We are given an X-ray binary system, Hyp-X2, with a visible G2 main-sequence star and an unseen companion. The distance between the stars is 12 million kilometers, and the visible star's orbital period is 5 days. We need to find the sum of the masses of the two stars using Newton's version of Kepler's Third Law, and then determine if the unseen companion is a black hole or a neutron star.

Using Newton's version of Kepler's Third Law

Newton's form of Kepler's Third Law is \( p^2 = \frac{4\pi^2}{G(M_1 + M_2)}a^3 \), where:- \( p \) is the orbital period.- \( a \) is the semi-major axis (average distance between the two stars).- \( G \) is the gravitational constant \( (6.674 \times 10^{-11} \text{Nm}^2/ ext{kg}^2) \).- \( M_1 + M_2 \) is the sum of the masses.Converting periods and distances:- \( p = 5 \text{ days} = 5 \times 24 \times 3600 = 432000 \text{ seconds} \).- \( a = 12 \times 10^6 \text{ km} = 12 \times 10^9 \text{ m} \).We solve for \( M_1 + M_2 \).

Calculating the mass sum

Rearranging \( p^2 = \frac{4\pi^2}{G(M_1 + M_2)}a^3 \) gives \( M_1 + M_2 = \frac{4\pi^2a^3}{Gp^2} \). Substituting values:\[M_1 + M_2 = \frac{4\pi^2(12 \times 10^9)^3}{6.674 \times 10^{-11} \times (432000)^2}\]Calculating, we find:\(M_1 + M_2 = 2.088 \times 10^{31} \text{ kg}\).

Converting to Solar Masses

To convert the mass sum to solar masses, use \( 1 M_{\text{Sun}} = 2.0 \times 10^{30} \text{ kg} \).\[\text{Mass sum in solar masses} = \frac{2.088 \times 10^{31}}{2.0 \times 10^{30}} = 10.44 M_{\text{Sun}}\]

Determining the mass of the unseen companion

The mass of the visible G2 main-sequence star is \( 1 M_{\text{Sun}} \). Thus, the mass of the unseen companion is:\[10.44 M_{\text{Sun}} - 1 M_{\text{Sun}} = 9.44 M_{\text{Sun}}\].

Classifying the unseen companion

Black holes typically have masses greater than about 3 solar masses, while neutron stars are usually less. Given the unseen companion's mass of \( 9.44 M_{\text{Sun}} \), it is too massive to be a neutron star and is therefore likely a black hole.

Key concepts

X-ray binary system

An X-ray binary system consists of two stars orbiting each other, where one star is a compact object like a black hole or a neutron star, and the other is often a regular star. These systems are particularly fascinating because they emit strong X-rays. This occurs when material from the normal star spirals onto the compact object due to its strong gravity.
X-ray binaries are crucial for astronomers to study. They help us understand the extreme physics acting close to black holes and neutron stars:

• The regular star in the system is often a massive star, whose gravitational bound is compromised by the nearby dense object, a condition creating fascinating X-ray emissions.
• The material forms an 'accretion disk' around the compact object, heating up to incredible temperatures as it spirals inward, resulting in X-ray radiation.

These systems make it possible to measure the mass of the unseen companion through observations of the visible star and any spectral lines shifted due to its motion.

Newton's version of Kepler's laws

Newton’s version of Kepler’s laws is an extension of Kepler's initial laws of planetary motion. It incorporates the gravitational force and allows predictions beyond just our solar system. Specifically, the Third Law is pivotal when considering the orbits of binary systems.
Kepler’s Third Law can be expressed as
\[p^2 = \frac{4\pi^2}{G(M_1 + M_2)}a^3\]
Where:

• \(p\) is the orbital period.
• \(a\) is the semi-major axis (average distance between the two bodies in orbit).
• \(G\) is the gravitational constant.
• \(M_1 + M_2\) is the combined mass of both objects.

Using this formula, one can find the total mass in a binary system. This approach was applied in the problem to ascertain the total mass of the Hyp-X2 system, providing a key insight into its mysterious unseen member.

Black holes

Black holes are regions in space where the gravitational pull is so intense that not even light can escape them. They are formed when massive stars collapse under their own gravity at the ends of their life cycles. In the context of X-ray binaries, black holes can have significant impacts.
Key features of black holes in such systems include:

• They usually possess masses more than three solar masses. This tremendous mass density warps space and time merging into what’s called a 'singularity'.
• Due to their nature, black holes in binary systems allow us to estimate their mass based on the orbit of the companion star, as seen in the Hyp-X2 system example where the unseen companion is a black hole due to its mass of \(9.44 M_{\text{Sun}}\).
• Scientifically, studying black holes in binary systems has helped confirm theories of general relativity and much about high-energy processes in the universe.

Neutron stars

Neutron stars are the remnants of supernova explosions of massive stars. They are incredibly dense, with masses greater than that of the Sun but limited to approximately 1.4 to 3 solar masses. Neutron stars are not as dense as black holes but significantly denser than ordinary matter.
In X-ray binaries, neutron stars can be distinguished from black holes due to their distinct characteristics:

• They are incredibly compact, often only about 20 kilometers in diameter, which means a sugar-cube-sized amount of neutron-star material would weigh about a billion tons.
• Neutron stars may exhibit intense magnetic fields and can rotate at very high speeds, resulting in pulsar phenomena.
• In the case of the Hyp-X2 system, the unseen companion was not a neutron star because its mass greatly exceeded the typical neutron star mass range, indicating it was instead a black hole.

This distinction is crucial when analyzing X-ray binaries and helps astronomers explore the diversity of stellar evolution.