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Chapter 13: Problem 44

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 \(30,000 \mathrm{~km} / \mathrm{s}\). (a) How far are these clumps from the center of the black hole? (b) 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. (c) What is the radius of its event horizon?

Short Answer

Expert verified
The orbit radius is approximately 7.4脳10^11 m,the black hole's mass is about 1.0 x 10^35 kg (or 5 million times the mass of our sun), and the radius of the event horizon on the black hole is roughly 2.2 x 10^8 m.

Step by step solution

01

Calculate the Orbital Radius

The velocity \(v\) of an object in circular motion can be found using the formula \(v = \sqrt{GM/r}\), where \(G\) is the gravitational constant, \(M\) is the mass of the central object, and \(r\) is the orbital radius. Given the rotational period \(T\), the velocity can also be computed by \(v = 2\pi r / T\). When both expressions for the velocity \(v\) are combined, an expression for the orbital radius \(r\) is obtained, \(r = \(\left( \frac{GMT^{2}}{4\pi^{2}} \right)^{\frac{1}{3}}\). Since \(T\) = 27 hours = 97200 seconds, \(G\) is the gravitational constant = \(6.67430 脳 10^{-11} m^{3}kg^{-1}s^{-2}\) and \(v\) = \(30,000 \mathrm{~km / s}\), we substitute these values into the equation to obtain the orbital radius.
02

Calculate the Black Hole's Mass

Rearrange the above equation to compute the mass of the black hole: \(M = \frac{v^2 r}{G}\). The values computed for \(v\) and \(r\) in step 1 are substituted into the equation for obtaining the mass. This value is then converted to solar masses by dividing the result by the sun's mass \(1.989 脳 10^{30} kg\).
03

Calculate the Event Horizon's Radius

The radius of the event horizon of a black hole, \(r_s\), is calculated by the formula \(r_s = 2GM/c^2\), with \(G\) the gravitational constant, \(M\) the mass of the black hole, and \(c\) the speed of light. Substitute the calculated mass of the black hole into the equation to calculate the radius.

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Key Concepts

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

Gravitational Constant
The gravitational constant, often symbolized as \( G \), is a fundamental constant that plays a crucial role in the field of physics, particularly in gravitation-related calculations. Its value is \( 6.67430 \, \times \, 10^{-11} \, \text{m}^3\text{kg}^{-1}\text{s}^{-2} \). This constant is utilized in Isaac Newton's law of universal gravitation, helping to determine the gravitational force between two masses.

Gravitation is a key concept when discussing black holes. The gravitational constant allows us to compute the gravitational force exerted by a black hole on nearby objects, like stars or clumps of matter. Since black holes have extremely strong gravitational fields, \( G \) is indispensable in quantifying these interactions.
  • In the context of the universe, it's one of the reasons why stars orbit around galaxies instead of flying away into space.
  • Without this constant, we wouldn't be able to calculate the orbits around massive celestial bodies, like black holes.
Orbital Radius
The orbital radius is the distance from the center of a gravitational body, such as a black hole, to an orbiting object. Calculating the orbital radius helps astronomers understand how far objects, like clumps of matter or stars, are orbiting a black hole.

In exercises involving black holes, the orbital radius can be derived using orbital velocity and the period of orbit. The formula \( r = \left( \frac{GMT^{2}}{4\pi^{2}} \right)^{\frac{1}{3}} \) is used to find the orbital radius, where:
  • \( G \) is the gravitational constant.
  • \( M \) is the mass of the central object (black hole).
  • \( T \) is the orbital period.
Discovering the exact distance helps astronomers confirm properties about black holes and test predictions made by theories such as general relativity.
Event Horizon
The event horizon of a black hole is a critical boundary. It marks the point of no return, beyond which nothing can escape the black hole's gravitational pull, not even light. This makes it invisible to observation, hence the term "black hole."

The radius of the event horizon, often referred to as the Schwarzschild radius, is calculated using the formula \( r_s = \frac{2GM}{c^2} \). In this equation:
  • \( G \) is the gravitational constant.
  • \( M \) is the mass of the black hole.
  • \( c \) is the speed of light, a constant \( 299,792,458 \, \text{m/s} \).
By determining this radius, scientists can understand the scale of the black hole. It allows them to measure how much space the black hole occupies and further explore the impacts of such a massive object on its surroundings.

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Most popular questions from this chapter

The star Rho \({ }^{1}\) Cancri is 57 light-years from the earth and has a mass 0.85 times that of our sun. A planet has been detected in a circular orbit around Rho \({ }^{1}\) Cancri with an orbital radius equal to 0.11 times the radius of the earth's orbit around the sun. What are (a) the orbital speed and (b) the orbital period of the planet of Rho " Cancri?

At the Galaxy's Core. Astronomers have observed a small, massive object at the center of our Milky Way galaxy (see Section 13.8). A ring of material orbits this massive object; the ring has a diameter of about 15 light-years and an orbital speed of about \(200 \mathrm{~km} / \mathrm{s}\) (a) Determine the mass of the object at the center of the Milky Way galaxy. Give your answer both in kilograms and in solar masses (one solar mass is the mass of the sun). (b) 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? (c) Many astronomers believe that the massive object at the center of the Milky Way galaxy is a black hole. If so, what must the Schwarzschild radius of this black hole be? Would a black hole of this size fit inside the earth's orbit around the sun?

In March \(2006,\) two small satellites were discovered orbiting Pluto, one at a distance of \(48,000 \mathrm{~km}\) and the other at \(64,000 \mathrm{~km}\). Pluto already was known to have a large satellite Charon, orbiting at \(19,600 \mathrm{~km}\) with an orbital period of 6.39 days. Assuming that the satellites do not affect each other, find the orbital periods of the two small satellites without using the mass of Pluto.

A particle of mass \(3 m\) is located \(1.00 \mathrm{~m}\) from a particle of mass \(m\). (a) Where should you put a third mass \(M\) so that the net gravitational force on \(M\) due to the two masses is exactly zero? (b) Is the equilibrium of \(M\) at this point stable or unstable (i) for points along the line connecting \(m\) and \(3 m,\) and (ii) for points along the line passing through \(M\) and perpendicular to the line connecting \(m\) and \(3 m ?\)

A narrow uniform rod has length \(2 a\). The linear mass density of the rod is \(\rho,\) so the mass \(m\) of a length \(l\) of the rod is \(\rho l\). (a) A point mass is located a perpendicular distance \(r\) from the center of the rod. Calculate the magnitude and direction of the force that the rod exerts on the point mass. (Hint: Let the rod be along the \(y\) -axis with the center of the rod at the origin, and divide the rod into infinitesimal segments that have length \(d y\) and that are located at coordinate \(y\). The mass of the segment is \(d m=\rho d y\). Write expressions for the \(x\) - and \(y\) -components of the force on the point mass, and integrate from \(-a\) to \(+a\) to find the components of the total force. Use the integrals in Appendix B.) (b) What does your result become for \(a \gg r ?\) (Hint: Use the power series for \((1+x)^{n}\) given in Appendix B.) (c) For \(a \gg r,\) what is the gravitational field \(g=\boldsymbol{F}_{g} / m\) at a distance \(r\) from the rod? (d) Consider a cylinder of radius \(r\) and length \(L\) whose axis is along the rod. As in part (c), let the length of the rod be much greater than both the radius and length of the cylinder. Then the gravitational ficld is constant on the curved side of the cylinder and perpendicular to it, so the gravitational flux \(\Phi_{g}\) through this surface is cqual to \(g A\), where \(A=2 \pi r L\) is the area of the curved side of the cylinder (see Problem 13.59 ). Calculate this flux. Write your result in terms of the mass \(M\) of the portion of the rod that is inside the cylindrical surface. How does your result depend on the radius of the cylindrical surface?
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