Problem 55 - Precalculus

Question

The planets travel in elliptical orbits with the sun at one focus. Assume that the focus is at the pole, the major axis lies on the polar axis, and the length of the major axis is \(2 a\) (see figure). Show that the polar equation of the orbit is \(r=a\left(1-e^{2}\right) /(1-e \cos \theta),\) where \(e\) is the eccentricity.

Step-by-Step Solution

Answer

The polar equation of the orbit is \(r=a(1-e^2) / (1-e\cos \theta)\)

Step 1: Identify the Constants

Identify the given information: the major axis of the ellipse is of length \(2a\) and the eccentricity is \(e\). The sun, at one focus of the ellipse, is taken as the pole of the polar coordinate system.

Step 2: Apply the Property of Ellipse

For any point P (r, \(\theta\)) on the ellipse, by the property of an ellipse, the sum of distances from the two foci (S and S') to that point P is constant and equal to the major axis. In terms of r, this can be expressed as: \(r + r' = 2a\). The distance r is the distance from the pole (S) to the point P, and r' from S' to the point P.

Step 3: Relate Distances with Eccentricity

The eccentricity of the ellipse is defined as the ratio of the distance between the foci to the length of the major axis. This can be expressed as \(e = SS' / 2a\). With the sun at the origin, the distance SS' is also \(2ae\). Therefore, the distance from S' to a point P can also be expressed in terms of r and \(\theta\) as: \(r' = 2ae + r \cos \theta\).

Step 4: Substitution into Original Equation

Substitute \(r'\) from Step 3 into \(r + r' = 2a\) from Step 2 to obtain: \(r + 2ae + r \cos \theta = 2a\). Simplifying this equation leads to: \(r = a(1 - e\cos \theta) / (1 - e^2)\)

Key Concepts

Eccentricity

Eccentricity is a fundamental concept when discussing ellipses and their shapes. It is a parameter that measures how much an ellipse deviates from being a perfect circle. The eccentricity is represented by the symbol \( e \).
The value of \( e \) ranges between 0 and 1 for an ellipse. When \( e = 0 \), the ellipse is actually a circle because the two foci are at the same point. As \( e \) approaches 1, the ellipse becomes more stretched and elongated.Understanding eccentricity can help describe different types of orbits and shapes:

• If \( e = 0 \), it is a circle.
• If \( 0 < e < 1 \), it is an ellipse.
• An eccentricity of exactly 1 or greater would describe a parabolic or hyperbolic trajectory, not an ellipse.

In the context of planetary motion, eccentricity helps determine how elliptical the orbit of a planet is.
This is crucial for studying how planets and other celestial bodies move around stars.

Ellipse Properties

An ellipse is a closed curve, which can be thought of as a stretched circle. It has several unique properties that make it interesting in geometry and astronomy.
In mathematics, an ellipse is defined as the set of points in a plane where the sum of the distances to two fixed points, called foci, is constant.Key properties of an ellipse include:

• Major Axis: This is the longest diameter of the ellipse, stretching from one side of the ellipse to the other through both foci. Its length is usually represented as \( 2a \).
• Minor Axis: This is the shortest diameter, perpendicular to the major axis.
• Foci: These are the two special points along the major axis. In the context of an orbital path, one focus is occupied by the central body (like the sun).

The properties of an ellipse are vital when it comes to understanding orbits, as many celestial orbits are elliptical with the central body at one focus.

Polar Coordinates

Polar coordinates provide an alternative way of describing the position of a point. Instead of using \( x \) and \( y \) like in Cartesian coordinates, polar coordinates use the distance from a reference point and an angle from a reference direction.For an ellipse in a polar coordinate system:

• The pole is often considered one of the foci of the ellipse, particularly when modeling orbits.
• \( r \) represents the radial distance from the pole to a point on the ellipse.
• \( \theta \) is the angle formed with a reference direction, typically the positive x-axis.

This coordinate system is especially handy when dealing with problems involving rotational symmetry, like planetary orbits.
In our exercise, we express the orbit of a planet as \( r = a(1 - e\cos \theta) / (1 - e^2) \) using polar coordinates.

Orbital Motion

Orbital motion refers to the path an object, such as a planet, takes as it moves around another object, such as a star. Most orbital paths are ellipses, with the more massive body located at one of the foci of the ellipse.
Orbits are governed by gravitational forces, described by Kepler's laws of planetary motion, which explain how planets and other celestial objects move through space. Key factors in orbital motion include:

• The shape of the orbit, usually elliptical, which depends on eccentricity.
• The speed of the object in orbit, which changes as it moves closer or farther from the central body.
• The position of the object in its orbit, which can be defined using polar coordinates.

Understanding orbital motion requires a good grasp of ellipses and how they function in a sea of gravitational forces.
This knowledge is crucial for predicting not just the movement of planets but also of satellites and other objects in space.