Nothing can escape the event horizon of a black hole, not even light. You can think of the event horizon as being the distance from a black hole at which the escape speed is the speed of light, 3.00 ✕ 10⁸ m/s, making all escape impossible. What is the radius of the event horizon for a black hole with a mass 5.0 times the mass of the sun? This distance is called the Schwarzschild radius.

검증된 단계별 안내

The Schwarzschild radius \( r_s \) 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 black hole, and \( c \) is the speed of light \( 3.00 \times 10^8 \, \text{m/s} \).

The mass of the black hole is given as \( 5.0 \) times the mass of the Sun. The mass of the Sun \( M_\odot \) is approximately \( 1.989 \times 10^{30} \, \text{kg} \). Therefore, the mass of the black hole \( M \) is \( M = 5.0 \times M_\odot = 5.0 \times 1.989 \times 10^{30} \, \text{kg} \).

Substitute the values of \( G \), \( M \), and \( c \) into the Schwarzschild radius formula: \( r_s = \frac{2 \times (6.674 \times 10^{-11}) \times (5.0 \times 1.989 \times 10^{30})}{(3.00 \times 10^8)^2} \).

Simplify the numerator by multiplying \( 2 \), \( G \), and \( M \), and simplify the denominator by squaring \( c \).

Divide the simplified numerator by the simplified denominator to find the Schwarzschild radius \( r_s \). This will give you the radius of the event horizon for the black hole.

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주요 개념

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Event Horizon

The event horizon is the boundary surrounding a black hole beyond which no information or matter can escape. It represents the point at which the escape velocity equals the speed of light, making it impossible for anything, including light, to break free from the gravitational pull of the black hole.

Schwarzschild Radius

The Schwarzschild radius is a specific radius that defines the size of the event horizon for a non-rotating black hole. It is directly proportional to the mass of the black hole and can be calculated using the formula r_s = 2GM/c², where G is the gravitational constant, M is the mass of the black hole, and c is the speed of light.

Escape Velocity

Escape velocity is the minimum speed needed for an object to break free from the gravitational attraction of a celestial body without any additional propulsion. For a black hole, the escape velocity at the event horizon reaches the speed of light, which is why nothing can escape once it crosses this threshold.