Question

Mass of the Central Black Hole. Suppose you observed a star orbiting the galactic center at a speed of \(1000 \mathrm{km} / \mathrm{s}\) in a circular orbit with a radius of 20 light-days. Calculate the mass of the object that the star was orbiting.

Short answer

The black hole's mass is approximately \(4.2 \times 10^7 \) solar masses.

Step-by-step solution

Understand the context

We're tasked with calculating the mass of a black hole based on the orbital velocity and radius of a star orbiting it. This is a problem involving gravitational dynamics and requires using Newton's version of Kepler's Third Law.

Convert radius to meters

Convert 20 light-days into meters. A single light-day is the distance light travels in one day:\[ 1 \text{ light-day} = c \times 86400 \text{ s} \] where \( c \approx 3 \times 10^8 \text{ m/s} \) is the speed of light.So:\[ 20 \text{ light-days} = 20 \times 3 \times 10^8 \times 86400 \text{ m} \]

Use the formula for gravitational force

Use the orbit equation:\[ v^2 = \frac{GM}{r} \]where:- \( v = 1000 \text{ km/s} = 1 \times 10^6 \text{ m/s} \) (converted to meters per second),- \( r \) is the orbital radius in meters,- \( G = 6.674 \times 10^{-11} \text{ m}^3\text{kg}^{-1}\text{s}^{-2} \) is the gravitational constant.Rearrange to solve for \( M \):\[ M = \frac{v^2 \times r}{G} \]

Calculate the mass of the black hole

Substitute the known values into the rearranged formula:\[ M = \frac{(1 \times 10^6 \text{ m/s})^2 \times (20 \times 3 \times 10^8 \times 86400 \text{ m})}{6.674 \times 10^{-11} \text{ m}^3\text{kg}^{-1}\text{s}^{-2}} \]Calculate the result to find the mass of the black hole.

Key concepts

Gravitational Dynamics

Gravitational dynamics is a critical concept in understanding the behaviors of celestial bodies like stars and black holes. At its core, it describes the interaction between objects with mass through the force of gravity. Gravity is what keeps planets in orbit around stars and governs the motion of galaxies and clusters. It works according to Newton's universal law of gravitation.

In the context of calculating a black hole's mass, gravitational dynamics comes into play when analyzing how a star orbits the black hole. The star's velocity and the distance from the black hole provide essential clues. By observing these parameters, scientists can use gravitational dynamics to infer the immense mass of hidden objects purely through their indirect gravitational effects. It's fascinating how through such calculations, scientists are able to determine the characteristics of objects millions of light-years away.

In summary, gravitational dynamics is about understanding how massive objects influence one another and allows us to solve problems like determining the mass of a black hole simply by looking at the motion of nearby stars.

Orbital Mechanics

Orbital mechanics is the study of how celestial objects move under the influence of gravity, primarily serving to explain and predict the motion of planets, moons, and other astronomical objects. It involves physics and mathematics to understand orbits of planets, moons, satellites, and other celestial bodies.

The motion of an object in orbit can be defined by several elements: speed, radius of its orbit, and the mass of the bodies involved. This subject is vital to comprehend how distances, like the 20 light-days radius mentioned in the problem, convert into more comprehensible units like meters, which are crucial for calculations.

In our task of finding the mass of the black hole, orbital mechanics tells us that the star's velocity and distance from the black hole (its orbital radius) relate directly to the force of gravity acting upon it. This relationship allows us to calculate the central body's mass using the star's orbital characteristics—something central to both space exploration and astrophysical research.

• Velocity: The speed at which an object orbits is crucial; faster objects usually indicate a stronger gravitational pull exerted by the central mass.
• Radius: This is the distance between the orbiting object and the center mass; greater distance usually weakens the gravitational pull and affects orbital speed.

Orbital mechanics applies physical laws and equations systematically, ensuring accurate calculations in celestial navigation and the discovery of cosmic phenomena.

Kepler's Laws

Kepler's Laws of planetary motion are essential for understanding orbits and dynamic systems in space. They were derived by Johannes Kepler in the early 17th century and provide foundational guidelines for celestial motions.

The exercise mentioned uses Newton's version of Kepler's Third Law. Kepler's Third Law states that the square of an orbiting object's period is proportional to the cube of the semi-major axis of its orbit. In simpler terms, this states a relationship between the time it takes for a body to complete an orbit and its average distance from the main body around which it orbits. Newton generalized Kepler's Third Law to include any two masses in orbit about their common center of mass, which allows us to compute the mass of astronomical entities we cannot directly observe, like black holes:

• It connects the gravitational force to orbital time and distance, serving as a tool to pinpoint masses of celestial bodies based on their influence over others.
• This law is invaluable for determining the mass of an unseen massive object by observing the orbital characteristics of a nearby visible object (e.g., a star).

Employing Newton's refinement of Kepler’s laws equips us with the mathematical framework needed to calculate the mass of a black hole, as demonstrated in the exercise through using the formula for gravitational force.