Question

Calculate the Schwarzschild radius of a black hole with the mass of a) the Sun. b) a proton. How does this result compare with the size scale \(10^{-15} \mathrm{~m}\) usually associated with a proton?

Short answer

Answer: The Schwarzschild radius of a black hole with the mass of the Sun is approximately \(2.954 \times 10^3 \mathrm{~m}\), and the Schwarzschild radius of a black hole with the mass of a proton is approximately \(2.478 \times 10^{-54} \mathrm{~m}\). Comparing these values to the size scale usually associated with a proton (\(10^{-15} \mathrm{~m}\)), it shows that a black hole with the mass of a proton would have a Schwarzschild radius much smaller than the proton's size scale.

Step-by-step solution

Define the Given Values and Constants

First, let us define the given values for the mass of the Sun and a proton, and the constants: - Mass of the Sun, \(M_\odot = 1.989 × 10^{30} \mathrm{~kg}\) - Mass of a proton, \(M_p = 1.673 × 10^{-27} \mathrm{~kg}\) - Gravitational constant, \(G = 6.674 × 10^{-11} m^3 kg^{-1} s^{-2}\) - Speed of light, \(c = 2.998 × 10^8 \mathrm{~m/s}\)

Calculate the Schwarzschild Radius of the Sun

Now, we will calculate the Schwarzschild radius of a black hole with the mass of the Sun using the formula: \(R_{s_\odot} = \frac{2GM_\odot}{c^2}\) Plug in the values we defined earlier: \(R_{s_\odot} = \frac{2 × 6.674 × 10^{-11} \mathrm{m^3 kg^{-1} s^{-2}} × 1.989 × 10^{30} \mathrm{~kg}}{(2.998 × 10^8 \mathrm{~m/s})^2}\) After evaluating the expression, we get: \(R_{s_\odot} \approx 2.954 \times 10^3 \mathrm{~m}\) So, the Schwarzschild radius of a black hole with the mass of the Sun is approximately \(2.954 \times 10^3 \mathrm{~m}\).

Calculate the Schwarzschild Radius of a Proton

Now, we will calculate the Schwarzschild radius of a black hole with the mass of a proton using the formula: \(R_{s_p} = \frac{2GM_p}{c^2}\) Plug in the values we defined earlier: \(R_{s_p} = \frac{2 × 6.674 × 10^{-11} \mathrm{m^3 kg^{-1} s^{-2}} × 1.673 × 10^{-27} \mathrm{~kg}}{(2.998 × 10^8 \mathrm{~m/s})^2}\) After evaluating the expression, we get: \(R_{s_p} \approx 2.478 × 10^{-54} \mathrm{~m}\) So, the Schwarzschild radius of a black hole with the mass of a proton is approximately \(2.478 \times 10^{-54} \mathrm{~m}\).

Compare the Schwarzschild Radius with the Size Scale Associated with a Proton

Now, let's compare the calculated Schwarzschild radius of a proton-sized black hole with the size scale usually associated with a proton, which is \(10^{-15} \mathrm{~m}\). The Schwarzschild radius we calculated is much smaller than the size scale associated with a proton: \(R_{s_p} \approx 2.478 \times 10^{-54} \mathrm{~m} << 10^{-15} \mathrm{~m}\) This result shows that a black hole with the mass of a proton would have a Schwarzschild radius much smaller than the size scale usually associated with a proton.

Key concepts

Black Hole Physics

The concept of a black hole is one of the most fascinating subjects in the field of astrophysics. At its core, a black hole is a region in space where the pull of gravity is so strong that nothing, not even light, can escape from it. This intense gravitational attraction is due to the presence of an extremely dense mass concentrated in a small area.

A critical concept when discussing black holes is the Schwarzschild radius, also known as the event horizon. It represents the boundary around a black hole beyond which no information can return to the outside universe. The Schwarzschild radius is determined by the mass of the black hole and is directly proportional to it. Using the known values of constants like the speed of light and the gravitational constant, scientists can calculate this radius and gain insights into the properties of a black hole.

To truly understand the magnitude of a black hole's gravity, consider that the Schwarzschild radius for a black hole with the mass of the Sun is about 3 kilometers. If the Sun were to be compressed to within this radius, the escape velocity at its surface would be equal to the speed of light, thus transforming it into a black hole.

Gravitational Constant

The gravitational constant, denoted as \(G\), is a key value in the calculation of the Schwarzschild radius. It appears in Newton's law of universal gravitation as well as in Einstein’s theory of general relativity. The gravitational constant represents the force of attraction between two masses that are one meter apart.

The value of \(G\) is approximately \(6.674 \times 10^{-11} m^3 kg^{-1} s^{-2}\). This constant is crucial for calculations related to gravity, including the dynamics of planetary orbits, the behavior of objects in free-fall on Earth, and the properties of black holes. When we plug this constant into the formula for the Schwarzschild radius, it helps us comprehend just how the mass of an object affects the extent of its gravitational pull, indicating the size the mass must be confined to for it to become a black hole.

Mass of the Sun

The Sun, with a mass of approximately \(1.989 \times 10^{30} kg\), is the most massive object in our solar system, accounting for about 99.86% of the system's total mass. This immense mass influences the orbit of planets, comets, and asteroids through its gravity.

Using the aforementioned Schwarzschild radius formula, if the Sun were to be compressed below approximately 3 kilometers, it would become a black hole. This radius is minuscule compared to the Sun's current radius of about 696,340 kilometers. The concept is theoretical and helps to illustrate the scale of forces at play within black holes of different masses. This comparison between the Schwarzschild radius and the actual size of the Sun conveys the extraordinary density that a black hole represents.

Mass of a Proton

The mass of a proton is about \(1.673 \times 10^{-27} kg\). In the context of the universe, protons are extremely small; however, they have a significant impact on the scale of atoms and molecules. A comparison between the Schwarzschild radius of a proton-sized black hole and the size of an actual proton illustrates fascinating aspects of scale in the universe.

As calculated, the Schwarzschild radius for a black hole with the mass of a proton is incredibly small, at about \(2.478 \times 10^{-54} m\). In contrast, the size of a proton is about \(10^{-15} m\), which is many orders of magnitude larger. This result is hypothetical and serves to emphasize that, in our current understanding of physics, black holes are expected to form only from significantly larger masses, as the conditions to compress a proton to within its Schwarzschild radius are not met in nature. These calculations allow students to appreciate the extent of densities and sizes we consider when exploring the cosmos.