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. (a) How far are these clumps from the center of the black hole?

Verified step by step guidance

1. Identify the given values: orbital period (T) = 27 hours, velocity of the clumps (v) = 30,000 km/s. Convert the orbital period from hours to seconds for consistency in units, knowing that 1 hour = 3600 seconds.
2. Use the formula for the circumference of a circle to find the radius of the orbit, where circumference C = 2\pi r. Since the clumps complete one orbit in one period, the distance traveled in one period is the circumference.
3. Substitute the velocity and period into the circumference formula to express the radius in terms of the velocity and period: C = v \times T. Solve for r by rearranging the formula: r = \frac{v \times T}{2\pi}.
4. Plug in the values for v and T (converted to seconds) into the formula to calculate the radius r.
5. The calculated radius r will give the distance of the clumps from the center of the black hole in kilometers.

Key Concepts

Gravitational Force

Gravitational force is the attractive force between two masses, described by Newton's law of universal gravitation. It states that the force is proportional to the product of the masses and inversely proportional to the square of the distance between their centers. This concept is crucial for understanding how objects, like clumps of matter, are influenced by the black hole's mass.

Orbital Motion

Orbital motion refers to the movement of an object in a curved path around a central body due to gravitational attraction. The speed and radius of the orbit are related; for a stable orbit, the gravitational force must equal the centripetal force required to keep the object moving in a circle. This principle helps determine the distance of the clumps from the black hole based on their orbital period and speed.

Kepler's Third Law

Kepler's Third Law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. In the context of the black hole, this law can be applied to relate the orbital period of the clumps to their distance from the black hole, allowing for the calculation of how far they are from the center based on their orbital characteristics.