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Chapter 22: Problem 45

Find the Schwarzschild radius for an object having a mass equal to that of the planet Saturn.

Short Answer

Expert verified
\(r_s\) is approximately 842715 meters or roughly 843 kilometers.

Step by step solution

01

Identifying the Known Variables

The mass of Saturn (M) is known, approximately 5.683 脳 10^26 kg. The gravitational constant (G) is 6.674脳10^-11 m^3 kg^-1 s^-2, and the speed of light (c), a fundamental physical constant, is about 3 x 10^8 m/s. These will be the values plugged into the formula.
02

Apply the Schwarzschild Radius Formula

We use the formula for the Schwarzschild radius \(r_s = 2GM/c^2\). Plugging in our known values, we get \(r_s = 2 * 6.674脳10^-11 m^3 kg^-1 s^-2 * 5.683 脳 10^26kg / (3 x 10^8 m/s)^2\).
03

Solving the equation

Performing the indicated operations, you obtain a numerical result for the Schwarzschild radius \(r_s\).

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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, denoted by the symbol \( G \), is a key part of Newton's law of universal gravitation. It quantifies the strength of the gravitational force between two masses. Essentially, it helps us calculate how strongly massive objects attract each other through gravity.

The value of the gravitational constant \( G \) is approximately \( 6.674 \times 10^{-11} \, \text{m}^3\, \text{kg}^{-1}\, \text{s}^{-2} \). This constant is crucial when exploring gravitational phenomena, especially when calculating forces in large systems such as planetary motion or even black holes.

Here's how \( G \) plays a role:
  • In the Schwarzschild radius formula, \( G \) helps determine the radius of a black hole for a given mass.
  • It is a universal constant, meaning it remains the same throughout the universe, no matter where you are.
  • Without \( G \), we wouldn't be able to compute celestial mechanics accurately and consistently.
Speed of Light
The speed of light in a vacuum, represented by \( c \), is a fundamental constant in physics, fundamental as it becomes a limit for the speed at which information or matter can travel. Its value is about \( 3 \times 10^8 \, \text{m/s} \).

Understanding the speed of light is crucial because:
  • It plays a pivotal role in the theory of relativity, especially in connecting time and space.
  • In the context of calculating the Schwarzschild radius, \( c^2 \) acts as the necessary factor that balances gravity in the formula \( r_s = \frac{2GM}{c^2} \).
  • It is a constant figure that helps unify concepts across different domains of physics, ensuring consistency in calculations.
The speed of light isn't just about the speed of photons, but also about understanding the universe's fundamental structure, as it maxes out the interaction speed of all physical events.
Planetary Mass
Planetary mass is a critical parameter in understanding celestial objects, like planets and stars. It refers to the total amount of mass contained in a given planet. When discussing the Schwarzschild radius, the planetary mass is essential in determining the potential size of a black hole formed from such a mass.

For our particular exercise, Saturn's mass, \( M = 5.683 \times 10^{26} \, \text{kg} \), serves as the benchmark for calculations.

Key reasons why planetary mass is important:
  • It allows us to predict the gravitational influence of the planet on other bodies, thereby affecting their orbits.
  • In the Schwarzschild scenario, planetary mass determines the radius of the event horizon of a black hole, giving insights into the gravitational pull and escape velocity.
  • Understanding mass helps in comparing planets with one another, understanding their formation, and studying their potential to support various phenomena, like atmospheric conditions or rings.
Thus, planetary mass is not just a number鈥攊t's a fundamental element that shapes our understanding of planetary science, gravitational interactions, and cosmic occurrences.

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Most popular questions from this chapter

Use the Starry Night Enthusiast \({ }^{\mathrm{TM}}\) program to examine X-ray images of galaxies with supermassive black holes at their centers. Open the Options pane and expand the Deep Space layer. Select Chandra Images and deselect all of the other options in this layer. Use the Find pane and Zoom controls to examine each of the following galaxies: (i) \(\mathrm{NGC} \mathrm{4261 \text {;(ii)VirgoA(M87); }}\) (iii) M31. Open the Options pane again and select Messier Objects and deselect Chandra Images and compare the visual images of Virgo A (M87) and M31. Suggest why supermassive black holes were discovered in these galaxies only after relatively recent advances were made in telescope and detector technology.

Search the World Wide Web for information about supernova SN 2006aj, which was associated with gamma-ray burster GRB 060218. In what ways were this supernova and gamma-ray burster unusual? Are the observations of these objects consistent with the collapsar model?

Search the World Wide Web for information about the intermediate-mass black hole candidate in M82. Is this still thought to be an intermediate-mass black hole? What new evidence has been used to either support or oppose the idea that this object is an intermediate-mass black hole?

. What is the law of cosmic censorship?

What is the Schwarzschild radius of a black hole whose mass is that of (a) the Earth, (b) the Sun, (c) the supermassive black hole in NGC 4261 (Section 22-5)? In each case, also calculate what the density would be if the matter were spread uniformly throughout the volume of the event horizon.
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