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Chapter 20: Problem 42

A star in a circular orbit about the black hole at the center of the Milky Way (whose mass \(M_{\mathrm{BH}}=8 \times 10^{36} \mathrm{kg}\) ) has an orbital radius of 0.0131 light-year \(\left(1.24 \times 10^{14}\) meters). What is the \right. average speed of this star in its orbit? (Hint: You may want to refer back to Chapter \(4 .\)

Short Answer

Expert verified
The average speed of the star is approximately \(2.075 \times 10^{6} \; m/s\).

Step by step solution

01

- Identify the necessary formula

To find the average speed of a star in a circular orbit, use the formula for orbital velocity: \[ v = \sqrt{\frac{GM}{r}} \] where \(G\) is the gravitational constant, \(M\) is the mass of the black hole, and \(r\) is the orbital radius.
02

- Input given values

From the problem, we have: \(M_{\mathrm{BH}} = 8 \times 10^{36} \; \mathrm{kg}\)\(r = 1.24 \times 10^{14} \; \mathrm{m}\)Also, the gravitational constant \(G\) is \(6.674 \times 10^{-11} \; \mathrm{m^3 \; kg^{-1} \; s^{-1}}\).
03

- Calculate the orbital velocity

Substitute the values into the formula: \[ v = \sqrt{\frac{6.674 \times 10^{-11} \times 8 \times 10^{36}}{1.24 \times 10^{14}}} \]First, multiply \(G\) and \(M\):\[ 6.674 \times 10^{-11} \times 8 \times 10^{36} = 5.3392 \times 10^{26} \]Next, divide by \(r\):\[ \frac{5.3392 \times 10^{26}}{1.24 \times 10^{14}} = 4.305161 \times 10^{12} \]Finally, take the square root of the result:\[ v = \sqrt{4.305161 \times 10^{12}} \approx 2.075 \times 10^{6} \; \mathrm{m/s} \]

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Key Concepts

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

Black Hole Mass
The mass of a black hole is an essential factor when calculating the orbital velocity of an object in its gravitational field.
In the given exercise, the black hole at the center of the Milky Way has a mass of 8 x 1036 kg. This immense mass causes a substantial gravitational force, pulling objects like stars into orbit.

Black holes are regions in space where gravity is so strong that nothing, not even light, can escape. The mass of a black hole is typically determined by observing the motion of nearby objects.
  • Stars, gas clouds, and other phenomena provide clues about the black hole's mass.
  • The mass directly influences how strongly the black hole attracts objects.
It's important to note that more massive black holes exert a stronger gravitational pull, affecting orbital speeds.
Gravitational Constant
The gravitational constant, denoted as G, is a key parameter in the formula for calculating gravitational forces.
Its value is approximately 6.674 x 10-11 m3 kg-1 s-1. This small number indicates the relative weakness of the gravitational force compared to other fundamental forces in nature.

In calculations involving celestial bodies, the gravitational constant allows us to determine the strength of the gravitational attraction between two masses.
  • It is crucial for calculating the orbital velocity of objects around massive entities like black holes.
  • Using G helps find how fast an object needs to move to balance the gravitational pull of a massive body, remaining in a stable orbit.
Without the gravitational constant, we couldn't accurately describe the motion of planets, stars, or satellites in their orbits.
Orbital Radius
The orbital radius is the distance from the center of the massive object (like a black hole) to the object in orbit (like a star).
In this exercise, the star's orbital radius around the black hole is 1.24 x 1014 meters, equivalent to 0.0131 light-years.

The orbital radius is crucial because:
  • It determines the distance over which gravity acts between the two objects.
  • It influences the orbital velocity needed to maintain the orbit.
Using the orbital radius in our formula allows us to find out how fast the star must travel to stay in its circular path around the black hole without spiraling in or flying away.
Calculating the orbital velocity with the correct radius ensures an accurate representation of the star's motion.

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Most popular questions from this chapter

The best evidence for the presence of dark matter in the Milky Way comes from the observation that the rotation curve a. is flat at great distances from the center. b. rises swiftly in the interior. c. falls off and then rises again. d. has a peak at about 2,000 light-years from the center.

Detailed observations of the structure of the Milky Way are difficult because a. the Solar System is embedded in the dust and gas of the disk. b. the Milky Way is mostly dark matter. c. there are too many stars in the way. d. the galaxy is rotating too fast (about \(200 \mathrm{km} / \mathrm{s}\) ).

Old stars in the inner disk of the Milky Way have higher abundances of massive elements than those of young stars in the outer disk. Explain how this difference might have developed.

Parallax measurements of the variable star RR Lyrae indicate that it is located 230 pc from the Sun. A similar star observed in a globular cluster located far above the galactic plane appears 160,000 times fainter than RR Lyrae. a. How far from the Sun is this globular cluster? b. What does your answer to part (a) tell you about the size of the galaxy's halo compared to the size of its disk?

Halo stars are found in the vicinity of the Sun. What observational evidence does not distinguish them from disk stars? a. the direction of their motion b. their speed c. their composition d. their temperature
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