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Be sure to show all calculations clearly and state your final answers in complete sentences.Calculate the Schwarzschild radius (in kilometers for each of the following. a. A \(10^{8} M_{\text {Sun }}\) black hole in the center of a quasar b. A \(5 M_{\text {Sun }}\) black hole that formed in the supernova of a massive star c. A mini-black hole with the mass of the Moon d. A mini-black hole formed when a superadvanced civilization decides to punish you (unfairly) by squeezing you until you become so small that you disappear inside your own event horizon

Short Answer

Expert verified
1. A quasar's black hole: 295,000 km. 2. Supernova black hole: 14.8 km. 3. Moon-mass black hole: 0.111 mm. 4. Human-mass black hole: practically zero.

Step by step solution

01

Understanding the Schwarzschild Radius Formula

The Schwarzschild radius \( R_s \) can be calculated using the formula \( R_s = \frac{2GM}{c^2} \), where \( G \) is the gravitational constant \( (6.674 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2}) \), \( M \) is the mass of the object, and \( c \) is the speed of light \( (3 \times 10^8 \, \text{m/s}) \). To convert meters to kilometers, divide by 1000.
02

Calculate for a Quasar Black Hole

Given \( M = 10^8 M_{\text{Sun}} \), where \( M_{\text{Sun}} = 1.989 \times 10^{30} \, \text{kg} \). Substitute into the formula: \[ R_s = \frac{2 \, (6.674 \times 10^{-11}) \, (10^8 \, \times \ 1.989 \times 10^{30})}{(3 \times 10^8)^2} \] Calculating yields a Schwarzschild radius of approximately \( 2.95 \times 10^{11} \, \text{m} \), or \( 295,000 \, \text{km} \).
03

Calculate for a Supernova Black Hole

Given \( M = 5 M_{\text{Sun}} \). Substitute into the formula: \[ R_s = \frac{2 \, (6.674 \times 10^{-11}) \, (5 \, \times \ 1.989 \times 10^{30})}{(3 \times 10^8)^2} \] Calculating yields a Schwarzschild radius of approximately \( 1.48 \times 10^4 \, \text{m} \), or \( 14.8 \, \text{km} \).
04

Calculate for a Mini-Black Hole with Moon's Mass

Given \( M = 7.34 \times 10^{22} \, \text{kg} \) (Moon's mass). Substitute into the formula: \[ R_s = \frac{2 \, (6.674 \times 10^{-11}) \, (7.34 \times 10^{22})}{(3 \times 10^8)^2} \] Calculating yields a Schwarzschild radius of approximately \( 1.11 \times 10^{-4} \, \text{m} \), or \( 0.111 \, \text{mm} \).
05

Calculate for a Mini-Black Hole with Human Mass

Assume \( M = 70 \, \text{kg} \) (average human mass). Substitute into the formula: \[ R_s = \frac{2 \, (6.674 \times 10^{-11}) \, (70)}{(3 \times 10^8)^2} \] Calculating yields a Schwarzschild radius of approximately \( 1.04 \times 10^{-25} \, \text{m} \), which is an extremely small number, practically zero.

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Key Concepts

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

Black Holes
Black holes are one of the most fascinating objects in the universe, defined by an intense gravitational pull that prevents anything, even light, from escaping. They form when massive stars collapse under their own gravity at the end of their lifecycle. This collapse compresses the matter within the star to a point of infinite density known as a singularity. Surrounding this singularity is the event horizon, the boundary beyond which nothing can return. Black holes can vary in size from a few kilometers to several million kilometers, depending on their mass.
Different types of black holes include stellar black holes, formed from collapsing stars, and supermassive black holes, which reside in the centers of galaxies including quasars. Understanding black holes is crucial because they challenge and expand our understanding of general relativity and gravitational physics, pushing the limits of modern scientific knowledge.
Gravitational Physics
Gravitational physics is the study of the force of gravity, one of the four fundamental forces in the universe. Gravity is the force of attraction between masses, explained originally by Newton's laws of motion and gravitation. However, it was Einstein's theory of general relativity that revolutionized our understanding by describing gravity not as a force but as a curvature of spacetime caused by mass.
In the context of black holes, gravitational physics becomes intriguing as the forces involved are incredibly intense. For instance, the gravitational pull of a black hole can warp spacetime to such an extent that it creates a singularity. Concepts like the Schwarzschild radius arise from these theories, defining how compact an object needs to be to become a black hole. This application of gravitational physics helps scientists calculate the event horizon of these massive and mysterious objects.
Event Horizon
The event horizon is an essential concept in understanding black holes. It represents the point of no return—where the escape velocity is equal to the speed of light. Beyond this threshold, not even light can escape the gravitational pull of the black hole, rendering it invisible to the outside observer. This boundary is determined by the Schwarzschild radius, which can be calculated using the mass of the black hole.
For example, a black hole with a mass several times that of the Sun will have an event horizon with a radius of just a few kilometers. The event horizon is crucial when examining black holes since it signifies the limit at which an object can be observed before disappearing from view forever. Theoretical discussions often explore what happens inside the event horizon, but it's an area shrouded in mystery, as our current understanding of physics can't fully explain the phenomena beyond this limit.
General Relativity
Proposed by Albert Einstein in 1915, general relativity is a groundbreaking theory that illustrates gravity as the result of the warping of spacetime by mass. One of its most famous predictions is the existence of black holes—regions in space where spacetime is so severely curved that nothing can escape its grasp. This theory has passed numerous tests, predicting phenomena such as gravitational waves and the bending of light around massive objects.
In the framework of general relativity, black holes act as formidable proving grounds. They provide scenarios where the curvature of spacetime is extreme and testing predictions of the theory to its limits. For instance, the Schwarzschild solution to Einstein's equations offers insights into the structure of black holes, determining features like the Schwarzschild radius and helping us study event horizons. Thus, general relativity doesn't just explain black holes—it opens up a realm of possibilities and questions about the universe.

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Most popular questions from this chapter

Choose the best answer to each of the following. Explain your reasoning with one or more complete sentences. Which of these objects has the smallest radius? (a) a \(1.2 \mathrm{M}_{\mathrm{Sun}}\) white dwarf (b) a \(0.6 M_{\text {Sun }}\) white dwarf (c) Jupiter

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 \(1 M_{\text {Sun. }}\) )

The X-ray bursts that happen on the surface of an accreting neutron star are not powerful enough to accelerate the exploding material to escape velocity. Predict what will happen in an X-ray binary system in which the companion star eventually feeds more than 3 solar masses of matter into the neutron star's accretion disk.

Expressions such as "it disappeared into a black hole" are now common in popular culture. Give a few other examples in which the term black hole is used in popular culture but is not meant to be taken literally. In what ways are these uses correct in their analogies to real black holes? In what ways are they incorrect? Why do you think such an esoteric scientific idea as that of a black hole has captured the public imagination?

What are accretion disks, and why do we find them only in close binary systems? Explain how the accretion disk provides a white dwarf with a new source of energy that we can detect from Earth.

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