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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 \(H y p-X 2(" \mathrm{Hyp}"\) 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 \(\left(M_{\text {Sun }}=2.0 \times 10^{30} \mathrm{kg}\right) .\) b. Determine the mass of the unseen companion. Is it a neutron star or a black hole? Explain. (Hint: A G2 main-sequence star has a mass of \(1 M_{\text {Sun. }}\) )

Step by step solution

01

Use Kepler's Third Law

Kepler's Third Law formula is \[ T^2 = \frac{4\pi^2a^3}{G(M_1 + M_2)} \] where \(T\) is the period, \(a\) is the semi-major axis, \(G\) is the gravitational constant \(6.67 \times 10^{-11} \text{ Nm}^2/\text{kg}^2\), and \(M_1 + M_2\) is the sum of the masses of the two stars.First, convert the orbital period \(T\) from days to seconds:\[ T = 5 \times 24 \times 3600 = 432000 \text{ seconds.} \]Convert the separation (semi-major axis) \(a\) from kilometers to meters:\[ a = 12 \times 10^6 \times 10^3 = 1.2 \times 10^{10} \text{ meters.} \]

02

Solve for Sum of Masses

Rearrange the formula to solve for \(M_1 + M_2\):\[ \left(M_1 + M_2\right) = \frac{4\pi^2a^3}{GT^2} \]Substitute the values you have:\[ \left(M_1 + M_2\right) = \frac{4\pi^2 (1.2 \times 10^{10})^3}{6.67\times10^{-11} \times (432000)^2} \]Calculate to find \(M_1 + M_2\):\[ \left(M_1 + M_2\right) \approx 6.32 \times 10^{30} \text{ kg.} \]

03

Convert to Solar Masses

Convert the sum of the masses into solar masses by dividing it by the mass of the Sun:\[ \frac{6.32 \times 10^{30}}{2.0 \times 10^{30}} \approx 3.16 M_{\text{Sun}}. \]

04

Determine Mass of Unseen Companion

Given that the G2 main-sequence star has a mass of \(1 M_{\text{Sun}}\), calculate the unseen companion's mass:\[ M_2 = 3.16 M_{\text{Sun}} - 1 M_{\text{Sun}} = 2.16 M_{\text{Sun}}. \]

05

Classify the Unseen Companion

A minimum mass for a black hole can be approximately around \(3 M_{\text{Sun}}\). Since the unseen companion has a mass of \(2.16 M_{\text{Sun}}\), it is not enough to be a black hole.It's more consistent with a neutron star, which typically has a maximum mass of around \(2-3 M_{\text{Sun}}\).

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Key Concepts

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

X-ray binary

An X-ray binary system consists of a pair of stars, where one of these stars is typically a compact object. This compact object is either a black hole or a neutron star, that pulls material from its companion star. This material forms an accretion disk around the compact object. As the material spirals inward, it heats up and emits X-rays.

• The presence of strong X-ray emissions helps astronomers to infer the existence of a companion that is not easily visible, such as a black hole or a neutron star.
• In Hyp-X2, the visible star is a G2 main-sequence star, while the unseen companion is causing X-ray emissions.
• By analyzing these emissions and the dynamics of the star movements, astronomers can determine the characteristics of the unseen object.

X-ray binary systems are essential for understanding the extreme matter and strong gravitational forces present around black holes and neutron stars.

Black holes

Black holes are one of the most fascinating outcomes of stellar evolution. A black hole forms when a massive star's core collapses under the influence of its gravity, typically after the star has exhausted its nuclear fuel.

• The gravity of a black hole is so strong that it prevents anything, even light, from escaping its grasp.
• Black holes are characterized by their mass, charge, and spin, and they dramatically influence their surroundings.
• The typical threshold for stellar-mass black holes is greater than around 3 times the mass of the Sun (\(M_{\text{Sun}}\)), hence also acting as a tool for identification.

In the Hyp-X2 system, the unseen companion has a calculated mass of \(2.16 M_{\text{Sun}}\), which is below the minimum mass typically associated with a black hole, suggesting it may not be a black hole.

Neutron stars

Neutron stars are remnants of massive stars that have gone supernova. These stars are composed almost entirely of neutrons and are incredibly dense. They form from the remnants of stellar cores that are not massive enough to form black holes.

• Neutron stars have masses between 1.4 to about 3 times the mass of the Sun (\(M_{\text{Sun}}\)), and the radius is typically about 10 kilometers.
• They often exhibit strong magnetic fields and can rotate very rapidly, sometimes several hundred times per second.
• Observing an X-ray binary like Hyp-X2 with an unseen mass of \(2.16 M_{\text{Sun}}\) indicates it is likely a neutron star, given its mass falls within this range.

Neutron stars provide valuable insights into the properties of matter under extreme conditions, which cannot be replicated on Earth.

Newton's laws of motion

Newton's laws of motion are fundamental principles that describe the relationship between the motion of an object and the forces acting upon it. These laws also serve as the foundation for understanding celestial mechanics and operations within binary systems like Hyp-X2.

• First Law: An object remains in a state of rest or uniform motion unless acted upon by an external force.
• Second Law: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (\(F = ma\)).
• Third Law: For every action, there is an equal and opposite reaction.

In the context of Hyp-X2, Newton's laws help us understand the gravitational forces at play between the main-sequence star and its compact companion. These forces can be observed and calculated to infer properties such as the masses of the stars, as exemplified by using Newton's version of Kepler's Third Law in the original problem.