Problem 39 - Astronomy: At Play in the Cosmos

Question

What is the Schwarzschild radius, in meters, of a black hole whose mass is \(4 \times 10^{22} \mathrm{kg}\) ?

Step-by-Step Solution

Answer

The Schwarzschild radius is approximately \( 5.9324 \times 10^{-5} \) meters.

Step 1: Understanding the Schwarzschild Radius

The Schwarzschild radius \( R_s \) is the radius of a sphere such that, if all the mass of an object were to fall within that sphere, the escape speed from the sphere's surface would equal the speed of light. It is given by the formula: \[ R_s = \frac{2GM}{c^2} \] where \( G \) is the gravitational constant \( 6.674 \times 10^{-11} \, \text{m}^3\text{kg}^{-1}\text{s}^{-2} \), \( M \) is the mass of the object, and \( c \) is the speed of light \( 3 \times 10^8 \, \text{m/s} \).

Step 2: Substitute the Given Values

Substitute the values into the Schwarzschild radius formula. Here, \( M = 4 \times 10^{22} \, \text{kg} \). Thus, the equation becomes: \[ R_s = \frac{2 \times 6.674 \times 10^{-11} \times 4 \times 10^{22}}{(3 \times 10^8)^2} \]

Step 3: Calculate the Numerator

Calculate the product in the numerator: \( 2 \times 6.674 \times 10^{-11} \times 4 \times 10^{22} = 5.3392 \times 10^{12} \).

Step 4: Calculate the Denominator

Calculate the square of the speed of light: \( (3 \times 10^8)^2 = 9 \times 10^{16} \).

Step 5: Calculate the Schwarzschild Radius

Divide the numerator by the denominator to determine the Schwarzschild radius: \[ R_s = \frac{5.3392 \times 10^{12}}{9 \times 10^{16}} = 5.9324 \times 10^{-5} \, \text{meters} \].

Step 6: Final Verification

Verify that the units are correct and double-check the arithmetic to confirm that the calculation was performed correctly. The Schwarzschild radius should be expressed in meters.

Key Concepts

Black Hole

A black hole is one of the most fascinating objects in the universe. These are regions in space where the gravitational pull is so extreme that nothing, not even light, can escape from it.
This phenomenon occurs when a massive star collapses in on itself at the end of its life cycle, leading to a concentration of mass in an incredibly small area. This concentration causes the gravitational attraction to become incredibly strong.

• The boundary around a black hole is called the "event horizon," and once anything crosses this boundary, it never comes back out.
• Inside this boundary, physics as we know it breaks down due to the immense density and gravitational forces.

The concept of black holes challenges our understanding of physics and requires us to think beyond conventional realms.

Escape Velocity

Escape velocity is the minimum speed needed for an object to break free from the gravitational pull of a celestial body. On Earth, if you throw a ball fast enough, it'll reach the escape velocity and zoom off into space instead of falling back to the ground.
For any object, including black holes, the escape velocity is determined by the formula: \[ v_e = \sqrt{\frac{2GM}{r}} \]where:

• \(v_e\) is the escape velocity.
• \(G\) is the gravitational constant.
• \(M\) is the mass of the celestial body.
• \(r\) is the distance from the center of the mass.

In the case of a black hole, if the escape velocity at any point is greater than or equal to the speed of light, then nothing can escape, which is precisely what defines the boundary known as the event horizon. Hence, for the Schwarzschild radius, the escape velocity equals the speed of light.

Gravitational Constant

The gravitational constant, denoted by \( G \), is a fundamental constant that plays a key role in the equations governing gravity. It is used in Newton's law of universal gravitation and is vital for calculations involving gravitational force.
The value of \( G \) is approximately \( 6.674 \times 10^{-11} \, \text{m}^3\text{kg}^{-1}\text{s}^{-2} \).
This constant provides the strength by which two masses attract each other at a given distance.

• It's incredibly small, highlighting the relatively weak force of gravity compared to other fundamental forces, like electromagnetism.
• In our solar system, it allows us to calculate the orbits of planets, predict satellite trajectories, and much more.

In the universe of black holes, \( G \) is incorporated into the equation for the Schwarzschild radius, reflecting how gravity plays an essential role in their formation and behavior.

Speed of Light

The speed of light, symbolized by \( c \), is one of the most important constants in physics. It represents the speed at which light travels in a vacuum and is approximately \( 3 \times 10^8 \, \text{meters/second} \).
This speed is not only the speed limit for light; it is also the upper limit for how fast any information or matter can travel through the universe.

• This constant is crucial in E=mc², Einstein's equation that shows the equivalence of mass and energy.
• When we talk about cosmic events, like detecting light from distant galaxies, we're referencing the speed of light.

The speed of light is critical in the context of black holes and the Schwarzschild radius because it sets the threshold for the escape velocity at the event horizon. In simpler terms, for any object or information to escape a black hole, it would need to travel faster than light, a feat that is impossible according to our current understanding of physics.