a) How large is the Schwartzschild radius for a black hole of a million solar masses? b) What about a billion (10°) solar masses? c) How do these sizes compare with the radius of the sun (the actual sun, not the sun as a black hole), and the size of the solar system? Remember you must compare the sizes in kilometers. The radius of the solar system is about 100 AU. 1 AU is 150 million kilometers.
a) How large is the Schwartzschild radius for a black hole of a million solar masses? b) What about a billion (10°) solar masses? c) How do these sizes compare with the radius of the sun (the actual sun, not the sun as a black hole), and the size of the solar system? Remember you must compare the sizes in kilometers. The radius of the solar system is about 100 AU. 1 AU is 150 million kilometers.
Applications and Investigations in Earth Science (9th Edition)
9th Edition
ISBN:9780134746241
Author:Edward J. Tarbuck, Frederick K. Lutgens, Dennis G. Tasa
Publisher:Edward J. Tarbuck, Frederick K. Lutgens, Dennis G. Tasa
Chapter1: The Study Of Minerals
Section: Chapter Questions
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![### The Schwarzschild Radius and Black Holes
A black hole has an event horizon, also known as the Schwarzschild radius, which is proportional to its mass. We can write a handy version of this equation for stars in terms of solar masses as follows:
\[ R = 3M \]
Where \( R \) is measured in kilometers and \( M \) in solar masses. So a 1 solar mass object (the Sun, for example) if turned into a black hole would have a Schwarzschild radius of 3 km. We believe galaxies have supermassive black holes at their centers with masses that can be a million, or even a billion times the mass of the Sun.
**Questions:**
a) How large is the Schwarzschild radius for a black hole of a million solar masses?
b) What about a billion (\( 10^9 \)) solar masses?
c) How do these sizes compare with the radius of the Sun (the actual Sun, not the Sun as a black hole), and the size of the solar system? Remember you must compare the sizes in kilometers. The radius of the solar system is about 100 AU. 1 AU is 150 million kilometers.
### Explanation
- **Schwarzschild Radius Calculation:**
- For a black hole of a million solar masses: \( R = 3 \times 1,000,000 = 3,000,000 \) km
- For a black hole of a billion solar masses: \( R = 3 \times 1,000,000,000 = 3,000,000,000 \) km
- **Comparison:**
- The radius of the Sun is about 696,340 km.
- The size of the solar system is about 100 AU, which is approximately 15 billion km (100 AU \(\times\) 150 million km/AU).
These calculations help us understand the immense scale of black holes compared to familiar astronomical objects.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F812c6a7c-253a-49d0-89c1-1b0910bc9362%2F23e75f0e-4613-466e-adbd-2efd520f4b2d%2F45a3hun_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### The Schwarzschild Radius and Black Holes
A black hole has an event horizon, also known as the Schwarzschild radius, which is proportional to its mass. We can write a handy version of this equation for stars in terms of solar masses as follows:
\[ R = 3M \]
Where \( R \) is measured in kilometers and \( M \) in solar masses. So a 1 solar mass object (the Sun, for example) if turned into a black hole would have a Schwarzschild radius of 3 km. We believe galaxies have supermassive black holes at their centers with masses that can be a million, or even a billion times the mass of the Sun.
**Questions:**
a) How large is the Schwarzschild radius for a black hole of a million solar masses?
b) What about a billion (\( 10^9 \)) solar masses?
c) How do these sizes compare with the radius of the Sun (the actual Sun, not the Sun as a black hole), and the size of the solar system? Remember you must compare the sizes in kilometers. The radius of the solar system is about 100 AU. 1 AU is 150 million kilometers.
### Explanation
- **Schwarzschild Radius Calculation:**
- For a black hole of a million solar masses: \( R = 3 \times 1,000,000 = 3,000,000 \) km
- For a black hole of a billion solar masses: \( R = 3 \times 1,000,000,000 = 3,000,000,000 \) km
- **Comparison:**
- The radius of the Sun is about 696,340 km.
- The size of the solar system is about 100 AU, which is approximately 15 billion km (100 AU \(\times\) 150 million km/AU).
These calculations help us understand the immense scale of black holes compared to familiar astronomical objects.
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