Chapter 19: Problem 37
Estimate the Schwarzschild radius for a supermassive black hole with a mass of 26 billion \(M_{\text {Sun. }}\).
Short Answer
Expert verified
The Schwarzschild radius is approximately 7.67 \times 10^{13} meters.
Step by step solution
01
Understand the formula for Schwarzschild radius
The formula for the Schwarzschild radius is given by: \[ R_s = \frac{2GM}{c^2} \] where - \( R_s \) is the Schwarzschild radius, - \( G \) is the gravitational constant \(6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}\), - \( M \) is the mass of the black hole, - \( c \) is the speed of light in a vacuum \(3 \times 10^8 \, \text{m/s} \).
02
Convert mass to kilograms
The mass of the black hole is given in terms of the mass of the Sun. We first need to convert this to kilograms. The mass of the Sun \(M_{\text{Sun}}\) is approximately \(1.989 \times 10^{30} \) kg. Therefore, the mass of the black hole \(M\) is: \[ M = 26 \times 10^9 \times M_{\text{Sun}} = 26 \times 10^9 \times 1.989 \times 10^{30} \text{ kg} = 5.1714 \times 10^{40} \text{kg} \]
03
Substitute values into the formula
Now substitute the values for \( G \), \( M \), and \( c \) into the Schwarzschild radius formula: \[ R_s = \frac{2 \times (6.67430 \times 10^{-11} \text{ m}^3 \text{ kg}^{-1} \text{ s}^{-2}) \times (5.1714 \times 10^{40} \text{ kg})}{(3 \times 10^8 \text{ m/s})^2} \]
04
Calculate the numerator
First, calculate the numerator: \[ 2 \times 6.67430 \times 10^{-11} \times 5.1714 \times 10^{40} = 6.90497204 \times 10^{30} \text{ m}^3 \text{ s}^{-2} \]
05
Calculate the denominator
Next, calculate the denominator: \[ (3 \times 10^8)^2 = 9 \times 10^{16} \text{ m}^2 \text{ s}^{-2} \]
06
Divide numerator by denominator
Now divide the numerator by the denominator to find \( R_s \): \[ R_s = \frac{6.90497204 \times 10^{30}}{9 \times 10^{16}} = 7.67219049 \times 10^{13} \text{ m} \]
07
Round to appropriate significant figures
Round the result to three significant figures: \[ R_s \approx 7.67 \times 10^{13} \text{ m} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
supermassive black hole
A supermassive black hole is a type of black hole that has an extraordinarily large mass, typically millions to billions of times more than our Sun. These colossal objects are often found at the centers of galaxies, including our Milky Way. The immense gravitational pull of a supermassive black hole can have significant effects on the surrounding stars, gas, and dust, often forming an accretion disk that emits intense radiation. Understanding the properties of supermassive black holes, such as their Schwarzschild radius, helps astronomers study how they influence their host galaxies and the universe at large.
gravitational constant
The gravitational constant, denoted as \( G \), is a key constant in Newton's law of universal gravitation. It quantifies the intensity of the gravitational force between two masses. Its value is \( 6.67430 \times 10^{-11} \, \text{m}^3 \, \text{kg}^{-1} \, \text{s}^{-2} \). This constant is essential for calculating gravitational forces in various scenarios, from objects on Earth to celestial bodies in space.
mass conversion
To compute the Schwarzschild radius, it's crucial to convert mass into a standard unit, commonly kilograms (kg). In this exercise, the mass of the supermassive black hole is given in terms of the Sun's mass (\( M_{\text{Sun}} \)), which is approximately \( 1.989 \times 10^{30} \) kg. For a black hole with a mass of 26 billion times the Sun's mass, the conversion to kilograms is: \( M = 26 \times 10^9 \times 1.989 \times 10^{30} \text{ kg} = 5.1714 \times 10^{40} \text{ kg} \).
Schwarzschild radius formula
The Schwarzschild radius is the radius of the event horizon surrounding a black hole. It marks the boundary beyond which nothing, not even light, can escape the black hole's gravitational pull. The formula for calculating the Schwarzschild radius \( R_s \) is: \[ R_s = \frac{2GM}{c^2} \] Here, \( G \) is the gravitational constant, \( M \) is the mass of the black hole in kilograms, and \( c \) is the speed of light in a vacuum (\( 3 \times 10^8 \text{ m/s} \)). By substituting these values into the formula, we can determine the Schwarzschild radius.
significant figures
Significant figures are used in calculations to indicate the precision of a measured or calculated quantity. When performing scientific calculations, the result is often rounded to a certain number of significant figures to reflect the accuracy of the input values. For example, in the calculation of the Schwarzschild radius for a supermassive black hole, the final result was rounded to three significant figures: \( R_s \approx 7.67 \times 10^{13} \text{ m} \). This ensures that the result is presented with an appropriate level of precision, based on the precision of the given data and calculations.