Ch 13: Gravitation

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 13: Gravitation

Problem 40c

Chapter 13, Problem 40c

In 2005 astronomers announced the discovery of a large black hole in the galaxy Markarian 766 having clumps of matter orbiting around once every 27 hours and moving at 30,000 km/s. What is the radius of its event horizon?

Verified step by step guidance

Understand that the problem involves a black hole, and we need to find the radius of its event horizon, also known as the Schwarzschild radius.

Recall the formula for the Schwarzschild radius:

r

_s=2GMc2, where

G

is the gravitational constant,

M

is the mass of the black hole, and

c

is the speed of light.

Use the given orbital speed and period to find the mass of the black hole. The centripetal force required for the orbit is provided by the gravitational force:

\frac{G M m}{r^{2}} = \frac{m v^{2}}{r}

, where

v

is the orbital speed, and

r

is the orbital radius.

Solve for the mass

M

using the relationship

v = \frac{2 π r}{T}

, where

T

is the orbital period. Rearrange to find

M = \frac{v^{2} r}{G}

Substitute the calculated mass

M

into the Schwarzschild radius formula to find the radius of the event horizon.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Black Hole Event Horizon

The event horizon of a black hole is the boundary beyond which nothing can escape its gravitational pull, not even light. It is defined by the Schwarzschild radius, which depends on the mass of the black hole. Understanding the event horizon is crucial for determining the size of a black hole and the region from which no information can escape.

Orbital Mechanics

Orbital mechanics involves the study of the motion of objects in space under the influence of gravitational forces. In this context, the clumps of matter orbiting the black hole provide information about the gravitational field and mass of the black hole, which are essential for calculating the event horizon radius using Kepler's laws and gravitational equations.

Relativistic Speeds

Relativistic speeds refer to velocities that are a significant fraction of the speed of light, requiring the use of Einstein's theory of relativity to accurately describe motion and gravitational effects. The clumps of matter moving at 30,000 km/s around the black hole suggest relativistic effects, which must be considered when calculating the dynamics and properties of the black hole, including its event horizon.

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Textbook Question

Astronomers have observed a small, massive object at the center of our Milky Way galaxy. A ring of material orbits this massive object; the ring has a diameter of about 15 light-years and an orbital speed of about 200 km/s. Observations of stars, as well as theories of the structure of stars, suggest that it is impossible for a single star to have a mass of more than about 50 solar masses. Can this massive object be a single, ordinary star?

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Textbook Question

In 2005 astronomers announced the discovery of a large black hole in the galaxy Markarian 766 having clumps of matter orbiting around once every 27 hours and moving at 30,000 km/s. What is the mass of this black hole, assuming circular orbits? Express your answer in kilograms and as a multiple of our sun's mass.

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Textbook Question

In 2005 astronomers announced the discovery of a large black hole in the galaxy Markarian 766 having clumps of matter orbiting around once every 27 hours and moving at 30,000 km/s. How far are these clumps from the center of the black hole?

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