Problem 39 - University Physics with Modern Physics

Question

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 \(30,000 \mathrm{km} / \mathrm{s}\) . (a) How far are these clumps from the center of the black hole? (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. (c) What is the radius of its event horizon?

Step-by-Step Solution

Answer

The clumps are about \(4.63 \times 10^{12} \text{ meters}\) from the black hole. The black hole's mass is \(6.39 \times 10^{31} \text{ kg}\) or \(32 M_{\odot}\). The event horizon radius is \(9.49 \times 10^{4} \text{ meters}\).

Step 1: Identify Given Values

From the problem, we know that the period of orbit of the clumps is \( T = 27 \text{ hours} = 27 \times 3600 \text{ seconds} \) and the orbital speed is \( v = 30,000 \text{ km/s} = 30,000,000 \text{ m/s} \) .

Step 2: Use Circular Motion Formula for Distance

We use the formula for circular motion which relates the orbital speed \(v\), radius \(r\), and period \(T\) as follows: \( v = \frac{2\pi r}{T} \). Rearranging, we find \( r = \frac{vT}{2\pi} \). Substitute the given values to find \( r \).

Step 3: Calculate Radius of Orbit

Substituting the given values into the formula, \( r = \frac{30,000,000 \times 97,200}{2 \pi} \), we calculate \( r \approx 4.63 \times 10^{12} \text{ meters} \).

Step 4: Determine Black Hole Mass using Kepler's Third Law

Kepler's Third Law relates the mass of the central object \( M \), orbital radius \( r \), and period \( T \): \( T^2 = \frac{4 \pi^2 r^3}{GM} \). Solving for \( M \), we have \( M = \frac{4 \pi^2 r^3}{G T^2} \), where \( G = 6.674 \times 10^{-11} \text{ m}^3\text{/kg}\cdot\text{s}^2 \).

Step 5: Calculate Mass of Black Hole

Substitute the values \( T = 97,200 \text{ s} \), \( r \approx 4.63 \times 10^{12} \text{ m} \), and \( G = 6.674 \times 10^{-11} \) to calculate \( M \approx 6.39 \times 10^{31} \text{ kg} \).

Step 6: Express Mass as Multiple of Solar Mass

The mass of the sun is \( M_{\odot} = 1.989 \times 10^{30} \text{ kg} \). The black hole's mass as a multiple of the solar mass is \( M \approx 32 \times M_{\odot} \).

Step 7: Calculate Event Horizon Radius (Schwarzschild Radius)

The event horizon or Schwarzschild radius \( r_s \) is given by \( r_s = \frac{2GM}{c^2} \), where \( c = 3 \times 10^8 \text{ m/s} \) is the speed of light. Substitute \( M \approx 6.39 \times 10^{31} \text{ kg} \) into the formula to find \( r_s \approx 9.49 \times 10^{4} \text{ meters} \).

Key Concepts

Circular Motion

Understanding circular motion is crucial when analyzing the motion of clumps of matter around a black hole. In circular motion, objects travel along a circular path. Here, the clumps orbit around the black hole in a circular trajectory. The speed remains constant but the direction continuously changes, keeping the object in the circle.

Key equations link speed, radius, and time in circular motion. The formula is: \[v = \frac{2\pi r}{T}\]Where,

• \(v\) is the orbital speed,
• \(r\) is the radius of the orbit,
• \(T\) is the period of motion.

If we know the speed and the time it takes for a full orbit, we can calculate the orbital radius of the clumps. Rearranging the formula gives us:\[r = \frac{vT}{2\pi}\]This calculation is essential to understanding their position relative to the black hole, helping us determine the gravitational effects influencing their path.

Kepler's Third Law

Kepler's Third Law is pivotal in determining the mass of celestial objects, such as black holes. The law states a direct relationship between the period of an orbit and the radius of that orbit when the orbit is circular. Specifically, it says:\[T^2 = \frac{4 \pi^2 r^3}{GM}\]Where:

• \(T\) is the orbital period,
• \(r\) is the radius of the orbit,
• \(G\) is the gravitational constant \((6.674 \times 10^{-11} \text{ m}^3/\text{kg}\cdot\text{s}^2)\),
• \(M\) is the mass of the central object, such as a black hole.

By rearranging this equation, the mass \(M\) can be derived, helping quantify a black hole's mass from observed data. This practical use of Kepler's law allows astronomers to estimate the mass of black holes and other massive objects in space.

Event Horizon

The event horizon is a fundamental concept when studying black holes. It represents the boundary around a black hole beyond which nothing can escape, not even light. This stark demarcation is what gives a black hole its 'blackness'.

The radius of this boundary is known as the Schwarzschild radius, calculated using the formula:\[r_s = \frac{2GM}{c^2}\]Here:

• \(G\) is the gravitational constant,
• \(M\) is the mass of the black hole,
• \(c\) is the speed of light \((3 \times 10^8 \text{ m/s})\).

The concept of the event horizon is central to understanding how black holes interact with their surroundings. It essentially marks the point of no return. This boundary helps astronomers predict how nearby objects will behave when close to a black hole, shedding light on some of the universe's most mysterious phenomena.