Problem 38 - Astronomy: At Play in the Cosmos

Question

A black hole has a Schwarzschild radius of \(7 \times 10^{6} \mathrm{km}\). What is its mass, in kilograms?

Step-by-Step Solution

Answer

The mass of the black hole is approximately \(4.72 \times 10^{37}\) kilograms.

Step 1: Understand the Schwarzschild Radius Formula

The Schwarzschild radius is given by the formula \( r_s = \frac{2GM}{c^2} \), where \( r_s \) is the Schwarzschild radius, \( G \) is the gravitational constant \( 6.674 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2} \), \( M \) is the mass of the black hole in kilograms, and \( c \) is the speed of light \( 3 \times 10^8 \text{ m/s} \). We need to solve for \( M \).

Step 2: Rearrange the Formula to Solve for Mass

Rearrange the Schwarzschild radius formula to solve for mass \( M \). We get: \[ M = \frac{r_s \cdot c^2}{2G} \]

Step 3: Convert Schwarzschild Radius to Meters

Convert the Schwarzschild radius from kilometers to meters as the formula requires meters:\( 7 \times 10^6 \text{ km} = 7 \times 10^9 \text{ m} \).

Step 4: Calculate the Mass

Substitute the values into the formula:\[ M = \frac{(7 \times 10^9 \text{ m}) \cdot (3 \times 10^8 \text{ m/s})^2}{2 \times 6.674 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2}} \]Calculate step-by-step:- \((3 \times 10^8)^2 = 9 \times 10^{16} \).- \(7 \times 10^9 \times 9 \times 10^{16} = 63 \times 10^{25} \).- \(2 \times 6.674 \times 10^{-11} = 13.348 \times 10^{-11} \).- Now divide: \( \frac{63 \times 10^{25}}{13.348 \times 10^{-11}} \approx 4.72 \times 10^{37} \text{ kg} \).

Key Concepts

Schwarzschild radius

The Schwarzschild radius is a critical concept when discussing black holes. It defines the size of the event horizon—the point at which no information or matter can escape the black hole's gravity. Named after the German physicist Karl Schwarzschild, this radius represents the boundary beyond which the escape velocity equals the speed of light. If you were to compress a mass into this tiny space, it would form a black hole.The formula used to calculate the Schwarzschild radius is \[ r_s = \frac{2GM}{c^2} \] where:

• \( r_s \) is the Schwarzschild radius
• \( G \) is the gravitational constant
• \( M \) is the mass of the object
• \( c \) is the speed of light

Understanding this radius helps us know the point where light and matter cannot escape a black hole’s pull, hence its significance in astrophysics.

gravitational constant

The gravitational constant, denoted as \( G \), is a fundamental constant in physics that appears in Newton's law of universal gravitation. It is crucial in calculating the force of gravity exerted between two masses.## Nature of Gravitational Constant\( G \) is a universal constant, meaning it is the same throughout the universe, and its value is \( 6.674 \times 10^{-11}\, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \). This constant helps us understand how objects with mass attract each other and determines the strength of this attraction.It plays a key role in calculating not only gravitational forces on Earth but also the forces that govern celestial bodies, such as black holes. This makes \( G \) essential for understanding the dynamics of the universe.

speed of light

The speed of light, denoted by the symbol \( c \), is one of the most fundamental constants in physics. It represents the maximum speed at which all energy, matter, and information in the universe can travel. Its precise value is \( 3 \times 10^8 \text{ m/s} \).### Importance in Physics- The speed of light acts as a speed limit for the entire universe.- It is constant in all inertial frames of reference, unaffected by the motion of the source or observer.### Role in the Schwarzschild Radius FormulaIn the context of black holes, the speed of light is crucial for calculating the Schwarzschild radius. It appears in the denominator of the formula, \[ r_s = \frac{2GM}{c^2} \],and scales the effect of mass and gravity at this boundary. Understanding this constant is critical for anyone studying relativity, quantum mechanics, and astrophysics.