In 2005 astronomers announced the discovery of a large black hole in the galaxy Markarian 766 having clumps of matter orbiting around once every 27 hours and moving at 30,000 km/s. (c) What is the radius of its event horizon?

Verified step by step guidance

1. Identify the given values: The speed of the orbiting matter (v) is 30,000 km/s, and the orbital period (T) is 27 hours.
2. Convert the orbital period from hours to seconds for consistency in units. Since 1 hour = 3600 seconds, multiply 27 hours by 3600.
3. Use the formula for the circumference of a circle, C = 2\pi r, where r is the radius. The matter completes one orbit in one period, so the circumference can also be expressed as the product of speed and period, C = v \times T.
4. Solve the equation 2\pi r = v \times T for r to find the radius of the orbit.
5. To find the radius of the event horizon, use the Schwarzschild radius formula, R_s = \frac{2GM}{c^2}, where G is the gravitational constant, M is the mass of the black hole, and c is the speed of light. You will need to estimate the mass of the black hole using the orbital radius and speed (using Newton's version of Kepler's third law or other relevant methods).

Key Concepts

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 gravitational pull becomes so strong that escape velocity exceeds the speed of light. For a non-rotating black hole, the radius of the event horizon is known as the Schwarzschild radius, which is directly proportional to the mass of the black hole.

Schwarzschild Radius

The Schwarzschild radius (Rs) is a measure of the size of the event horizon of a black hole. It is calculated using the formula Rs = 2GM/c², where G is the gravitational constant, M is the mass of the black hole, and c is the speed of light. This radius indicates the distance from the center of the black hole to the event horizon, providing a critical parameter for understanding black hole properties.

Orbital Mechanics

Orbital mechanics is the study of the motion of objects in space under the influence of gravitational forces. In the context of black holes, it helps to understand how matter orbits around them. The speed and period of the orbiting clumps of matter can be used to infer the mass of the black hole, which is essential for calculating the Schwarzschild radius and, consequently, the event horizon.