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. (b) What is the mass of this black hole, assuming circular orbits? Express your answer in kilograms and as a multiple of our sun's mass.

Verified step by step guidance

1. Identify the given values: orbital period (T) = 27 hours, orbital speed (v) = 30,000 km/s. Convert the orbital period from hours to seconds for consistency in units.
2. Use the formula for the circumference of a circle to find the radius (r) of the orbit: Circumference = 2\pi r. Since the object completes one orbit in one period, the distance traveled in one period is the circumference. Thus, 2\pi r = v \times T.
3. Apply Newton's law of universal gravitation and the formula for centripetal force to relate the mass of the black hole (M) to the radius of the orbit. The gravitational force provides the necessary centripetal force for circular motion: F_{gravity} = F_{centripetal}, which gives \frac{GMm}{r^2} = \frac{mv^2}{r}, where G is the gravitational constant and m is the mass of the orbiting object.
4. Simplify the equation to solve for the mass of the black hole (M): M = \frac{rv^2}{G}. Substitute the expression for r from step 2 into this equation.
5. Convert the mass of the black hole from kilograms to solar masses by dividing the mass in kilograms by the mass of the sun (approximately 2 \times 10^{30} kg).

Key Concepts

Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation states that every mass attracts every other mass in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This principle is essential for understanding how objects, such as clumps of matter orbiting a black hole, are influenced by gravitational forces.

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, directed towards the center of the circle. In the context of the black hole, the gravitational force exerted by the black hole provides the necessary centripetal force to keep the orbiting matter in its circular path, allowing us to relate the orbital speed and radius to the mass of the black hole.

Mass of the Sun

The mass of the Sun is a standard reference point in astrophysics, approximately 1.989 x 10^30 kilograms. When calculating the mass of astronomical objects like black holes, it is common to express their mass as a multiple of the Sun's mass, facilitating easier comparisons and understanding of their scale relative to our solar system.