Question

A newly discovered planet orbits a distant star with the same mass as the Sun at an average distance of \(112 \mathrm{mil}\) lion kilometers. Find the planet's orbital period.

Short answer

The planet's orbital period is approximately 1 year.

Step-by-step solution

Understand Kepler's Third Law

Kepler's Third Law states that the square of the period of orbit (T²) of a planet is directly proportional to the cube of the semi-major axis of its orbit (a³), when orbiting the Sun-like star. Mathematically, it can be represented as:\[ T^2 = rac{4π^2a^3}{G M} \]where:\\( T \) = period of the orbit (in seconds),\\( a \) = average distance from the star (in meters),\\( G \) = gravitational constant \(6.674 imes 10^{-11} m^3 kg^{-1} s^{-2}\),\\( M \) = mass of the star \((1.989 imes 10^{30} ext{ kg})\).

Convert the Distance to Meters

The average distance of the planet from the star is given in kilometers, so we need to convert it to meters. Since 1 kilometer is 1000 meters:\[ 112,000,000 ext{ km} = 112,000,000 imes 1,000 = 1.12 imes 10^{11} ext{ m} \]

Substitute Values into Kepler's Law

Now, substitute the values into the rearranged formula to find the orbital period:\[ T^2 = rac{4π^2 (1.12 imes 10^{11})^3}{6.674 imes 10^{-11} imes 1.989 imes 10^{30}} \]

Calculate the Orbital Period

This calculation involves several steps: first, compute \((1.12 imes 10^{11})^3\) which gives the volume of the orbit, then multiply by \(4π^2\), and finally divide by the product of \(G\) and \(M\):\[ T^2 = rac{4 × (3.14159)^2 × (1.12 × 10^{11})^3}{(6.674 × 10^{-11}) × (1.989 × 10^{30})} \]Solve the equation to find \(T^2\), then take the square root to find \(T\) (in seconds).

Convert Seconds into Earth Years

The final value of \(T\) will be in seconds, so convert it into Earth years by dividing by the number of seconds in a year (approximately 31,536,000 seconds/year). Compute:\[ T ext{ (years)} = rac{T ext{ (seconds)}}{31,536,000} \] to find the orbital period in Earth years.

Key concepts

Orbital Mechanics

Orbital mechanics is a fascinating area of physics that explains the movement of planets and other celestial objects in space. At its core, it uses the principles of Newton's laws of motion and gravity to describe how objects like planets, moons, and satellites follow specific paths, called orbits, around larger bodies such as stars or planets. This branch of mechanics is crucial in understanding how these objects behave in space, predicting their future positions, and planning space missions. One of the critical aspects of orbital mechanics is Kepler's Third Law, which helps us relate the time it takes a planet to orbit a star to its distance from that star. According to this law, the square of a planet's orbital period is proportional to the cube of the semi-major axis of its orbit. This relationship allows us to predict the orbital period of any planet if we know the average distance of that planet from the star it orbits. In our exercise, we use Kepler's Third Law to find the orbital period of a newly discovered planet orbiting a star similar in mass to the Sun. By converting the planet's average distance from kilometers to meters, we substitute this value, along with known constants like the gravitational constant and the star's mass, into the formula to solve for the orbital period. This principle demonstrates the powerful predictive capability of orbital mechanics in understanding celestial movements.

Astrophysics

Astrophysics is the branch of astronomy that applies the principles of physics and chemistry to understand how stars, galaxies, and the universe behave. It involves studying the physical properties of celestial objects and the interactions between them. Astrophysicists use mathematical models and theories to delve deeper into phenomena such as black holes, neutron stars, and planetary systems. In the context of our exercise, astrophysics plays an integral role. Kepler's Third Law, which underpins the calculation of the planet's orbital period, is part of a broader set of laws and theories that explain the behavior of celestial bodies. The fact that a planet's orbital characteristics can be calculated and understood through the mass of its star and its distance exemplifies the profound impact of physics on our comprehension of the universe. The use of universal constants, such as the gravitational constant and the mass of celestial bodies like stars, showcases how astrophysics bridges the gap between individual celestial movements and the large-scale behavior of celestial systems. Each calculation adds to our broader understanding of the universe's mechanics, offering insights into both our solar system and other planetary systems beyond.

Planetary Science

Planetary science is an exciting field that focuses on the study of planets, including those in our solar system and beyond. This field encompasses a wide range of scientific disciplines, such as geology, geography, and atmospheric science, to explore the composition, structure, and dynamics of planets and their moons. Through planetary science, scientists seek to understand the formation and evolution of planets. They study the materials that make up planets and explore atmospheric conditions, climate, and potential habitability. In the exercise, the calculation of a planet's orbital period around a distant star helps planetary scientists infer vital information about the planet's year and its possible conditions. Investigating such characteristics is crucial for exploring topics like climate change on Earth and identifying potentially habitable planets in other solar systems. By understanding a planet’s orbit and its dynamics, planetary scientists can make predictions about seasonal changes, climate, and more extensive processes within the planet's atmosphere and surface. Ultimately, planetary science is an essential field for expanding our understanding of our universe and its diverse celestial phenomena.