The radius of a black hole is the distance from the black hole's center at which the escape speed is the speed of light. a) What is the radius of a black hole with a mass twice that of the Sun? b) At what radius from the center of the black hole in part (a) would the orbital speed be equal to the speed of light? c) What is the radius of a black hole with the same mass as that of the Earth?

Escape velocity is a fundamental concept in astrophysics, particularly when studying celestial objects such as black holes. It is defined as the speed needed for an object to break free from the gravitational pull of a celestial body without further propulsion.

Imagine you’re throwing a ball upwards; the escape velocity is that initial throw speed needed for the ball to avoid falling back to Earth and instead escape into space. This speed depends on the mass of the celestial body and the distance from its center.

In the context of black holes, the escape velocity at the event horizon, known as the Schwarzschild radius, equals the speed of light. This characteristic makes black holes appear 'black' as no light can escape from within this radius. The exercise provided illustrates how to calculate the Schwarzschild radius using the escape velocity formula and the given black hole mass.

The mass of a black hole is a critical factor in determining its gravitational pull and, consequently, the Schwarzschild radius. Black hole mass is typically expressed in multiples of the Sun's mass because the Sun is a well-known reference point.

The exercise showcased the calculation of the Schwarzschild radius for a black hole with a mass twice that of the Sun. This doubled mass greatly affects the Schwarzschild radius, showcasing the direct relationship between a black hole's mass and its gravitational effects.

To understand this further, remember that the heavier the black hole, the larger the radius at which its escape velocity would equal the speed of light. This concept is essential for comprehending the immense gravitational influence black holes have in their vicinity.

The gravitational constant, denoted by the symbol 'G', is one of the fundamental constants in physics. It quantifies the strength of gravity between two objects and is used in the universal law of gravitation as well as in the equations governing black holes.

With a value of approximately 6.674 x 10^-11 m³ kg^-1 s^-2, the gravitational constant G directly influences the escape velocity and Schwarzschild radius calculations. In the steps provided for the exercises, G is a vital coefficient in determining the exact size of the Schwarzschild radius for a given black hole mass. It’s interesting to note that G is the same everywhere in the universe, making it a constant not just in value, but also in its cosmic ubiquity.

The invariable nature of G allows us to make precise calculations about the properties of black holes and other astronomical phenomena regardless of where they are in the universe.