*Show that the position of a planet in its elliptical orbit can be expressed, using a frame with \(x\)-axis in the direction of perihelion (point of closest approach to the Sun), in terms of an angular parameter \(\psi\) by \(x=a(\cos \psi-e), y=b \sin \psi\). (See Problem B.1. In the literature, \(\psi\) is sometimes called the eccentric anomaly, while the polar angle \(\theta\) is the true anomaly.) Show that \(r=a(1-e \cos \psi)\), and that the time from perihelion is given by \(t=(\tau / 2 \pi)(\psi-e \sin \psi)\) (Kepler's equation).

An elliptical orbit is the path traced by an object moving around a central body, typically a star or planet, under the influence of gravity. This type of orbit is characterized by its elongated shape, similar to that of an oval. The two main points defining an ellipse are the foci, one of which is the central celestial body, such as the Sun in the case of planets in our solar system.

The mathematics behind the elliptical orbit can be derived from the fundamental principles of orbital mechanics and the laws of motion and gravity established by Sir Isaac Newton and refined by Johannes Kepler. The eccentricity (\(e\)) of the orbit provides a measure of its deviation from a perfect circle (\(e = 0\) for a circular orbit) and determines the shape of the ellipse. A higher eccentricity indicates a more elongated orbit.

The eccentric anomaly, denoted as \(\psi\), is a crucial concept in understanding orbital movement. It is an angular parameter used to describe the position of a body moving along an elliptical path.

Imagine an auxiliary circle overlaid on the ellipse, sharing the same center. The eccentric anomaly is the angle from the center of the ellipse to a point on this circle, taken from the major axis. This point on the auxiliary circle projects onto the actual position of the body on the ellipse. The eccentricity and geometry of the ellipse govern how the eccentric anomaly relates to the planet's actual position in its orbit.

The significance of the eccentric anomaly comes into play when trying to calculate the position and speed of a body along an ellipse, where simple circular motion equations do not suffice.

True anomaly is another angular parameter, represented by \(\theta\), which complements the eccentric anomaly in describing a planet's position in its orbit. It measures the angle between the direction of the perihelion (the point of closest approach) of the orbit and the current position of the planet, measured at the focal point of the ellipse where the central body is located.

This value changes more rapidly when the planet is close to the perihelion compared to when it is near the aphelion (the farthest point from the central body) due to the varied speed induced by the gravitational pull. The correlation between the true anomaly (\(\theta\)) and the eccentric anomaly (\(\psi\)) is vital for precise calculations in orbital mechanics.

Orbital mechanics, also known as celestial mechanics, is the field of study that deals with the motions of celestial bodies. This discipline applies the laws of physics, specifically Newton's laws of motion and universal gravitation, to predict and understand the orbits of planets, moons, satellites, and other objects in space.

Kepler's laws of planetary motion are fundamental to orbital mechanics. These laws describe the motion of planets around the Sun in three rules: planets move in elliptical orbits with the Sun at one focus, they sweep out equal areas in equal times, and there is a relationship between the orbital period and the semi-major axis of an ellipse.

Using these principles, along with a deep understanding of forces, vectors, and mathematical equations, scientists can calculate the trajectories, velocities, and many other aspects of moving bodies in space. This is essential not only for understanding our universe but also for practical applications like satellite launches and space exploration missions.