Question

A planet \(P\) revolves around the Sun in a circular orbit, with the Sun at the center, which is coplanar with and concentric to the circular orbit of Earth \(E\) around the Sun. \(P\) and \(E\) revolve in the same direction. The times required for the revolution of \(P\) and \(E\) around the Sun are \(T_{P}\) and \(T_{E}\). Let \(T_{S}\) be the time required for \(P\) to make one revolution around the Sun relative to \(E\) : show that \(1 / T_{S}=1 / T_{E}-1 / T_{P}\). Assume \(T_{P}>T_{E}\).

Short answer

The formula representing the relative time for planet \(P\) to revolve around the sun with respect to Earth is \(1 / T_{S}=1 / T_{E}-1 / T_{P}\).

Step-by-step solution

Understand Revolution Times

Planet \(P\) takes a time duration \(T_{P}\) to complete one revolution around the Sun, while Earth (E) takes time \(T_{E}\) to do the same. There is also a defined relative revolution time of \(P\) with respect to \(E\) which is \(T_{S}\). Since we know \(T_{P}>T_{E}\), it means Planet \(P\) takes more time to revolve around the sun than Earth.

Calculate Average Revolution Time

The mean time taken by \(P\) to make one revolution around the Sun relative to \(E\) is \(T_{S}\). This can be found by taking the average of the time taken by \(P\) and \(E\) to make a revolution around the Sun but due to different periods, the formula is \(T_{S}\) = \(\frac{1}{1/T_{E} - 1/T_{P}}\) .

Present Final Equation

By simply rearranging the formula, you get the final result as \(1 / T_{S}=1 / T_{E}-1 / T_{P}\). This equation shows how the relative time for planet \(P\) to revolve around the sun with respect to Earth can be calculated by knowing their respective revolution times.

Key concepts

Planetary Motion

Planetary motion captures the movement of planets as they orbit around the Sun. This intricate dance is a result of the interplay between gravitational forces and the momentum of the planets. When looking at the example provided, where we have a planet P and Earth E orbiting the Sun, understanding their revolution times is crucial for uncovering the mathematical relationship between their orbits.

Planets follow elliptical orbits, but for simplicity, the problem assumes circular orbits. It's important to realize that the revolution time, denoted by T, is the period it takes for a complete orbit. Earth's revolution time, known as a year, is used as a reference for measuring the relative motion of other planets. The relative revolution time (\(T_{S}\)) is a way to understand the movement of P with respect to E, taking into account the principle that each planet's motion is independent but also interconnected through the central gravitational pull of the Sun.

Circular Orbits

Circular orbits are a simplification often used in exercises to make calculations more straightforward. In reality, planetary orbits are elliptical, but when we approximate them as circles, we assume the distance from the Sun remains constant, and the force of gravity acts as a centripetal force keeping the planet in its path.

The Importance of Orbit Speed and Radius

The speed at which a planet moves along its orbit is determined by the gravitational pull and the radius of the orbit. In a perfectly circular orbit, this speed remains constant. This premise is essential in the exercise, where planet P and Earth E are assumed to be in coplanar and concentric orbits. Understanding circular motion is crucial when interpreting the formula \(T_{S} = \frac{1}{1/T_{E} - 1/T_{P}}\), which represents the relative period of one planet to another. This formula is refined from the complexities of elliptical motion to a simpler expression by assuming circular paths.

Kepler's Laws of Motion

Kepler's laws of motion are the foundation of understanding planetary motion. They explain the relationship between the time a planet takes to orbit the Sun (its period) and the shape and size of its orbit.

Kepler's First Law

States that planets move in elliptical orbits with the Sun at one focus. However, for simplicity, the exercise assumes circular orbits.

Kepler's Second Law

Sometimes called the law of equal areas, it states that a line drawn from a planet to the Sun sweeps out equal areas in equal times, implying that planets move faster when they are closer to the Sun.

Kepler's Third Law

This law provides a direct relationship between the square of a planet's orbital period and the cube of the semi-major axis of its orbit (\(T^2 \propto r^3\)). While not used directly in the exercise's equation, the concept of relative motion and periods discussed can be seen as an extension of Kepler's reasoning into the realm of relative observation, where the reference frame (Earth, in this case) is also in motion. The equation provided in the solution simplifies the concepts derived from Kepler's laws to account for relative motion, offering a practical mathematical tool for relating the motions of two planetary bodies.