Problem 55 - Algebra and Trigonometry Real Mathematics, Real People

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 of a planet is $$r=\frac{\left(1-e^{2}\right) a}{1-e \cos \theta}$$ where \(e\) is the eccentricity.

Step-by-Step Solution

Answer

The polar equation of the orbit of a planet is derived from the definition of an ellipse using polar coordinates and the eccentricity. The equation is \(r=a(1-e^2)/(1-e\cos\theta)\) where \(a\) is the semi-major axis, \(e\) is the eccentricity and \(\theta\) is the angle from the major axis.

Step 1: Definition of Eccentricity

The eccentricity of an ellipse with major and minor axes lengths of \(2a\) and \(2b\) respectively, is given as \(e=\sqrt{1-\left(\frac{b}{a}\right)^2}\). Eccentricity only ranges from 0 to 1 with 0 for a circle and approaches 1 for a highly elongated ellipse.

Step 2: Use Polar Coordinates

In polar coordinates, any point can be described by two values. The distance, \(r\), from the origin (pole) and the angle, \(\theta\), from the polar axis. The distance, \(r\), from any point on the ellipse to the pole is given by \(r=a(1-e^2)/(1-e\cos\theta)\). Here, \(a\) is the semi-major axis, \(\theta\) is the angle made by the line joining the sun and the planet with the major axis, and \(e\) is the eccentricity of the ellipse.

Step 3: Confirmation

Substituting the expression for \(r\) into the polar equation of the ellipse derived in step 2, we have \(r\left[1-e\cos(\theta)\right] = a(1-e^2)\), which can be rearranged to get \(r=a(1-e^2)/(1-e\cos(\theta))\), which is the required expression.

Key Concepts

Eccentricity

Eccentricity is a key concept when dealing with elliptical orbits like planetary paths. It measures how much an ellipse deviates from being circular. The eccentricity, denoted as \( e \), can be defined using the semi-major and semi-minor axes of the ellipse. For an ellipse with semi-major axis length \( a \) and semi-minor axis length \( b \), the formula is:\[ e = \sqrt{1 - \left(\frac{b}{a}\right)^2} \]This value of \( e \) ranges between 0 and 1. An eccentricity of 0 means the shape is a perfect circle. As the eccentricity approaches 1, the ellipse becomes more elongated. This property is vital in astronomy as it helps in understanding the shapes of planetary orbits. Planets with high eccentricity have orbits far from circular, while low eccentricity indicates nearly circular paths.

Polar Coordinates

When discussing orbits, especially celestial ones, polar coordinates offer a convenient way to describe positions. In this system, any point can be represented with two values: the distance \( r \) from a central point (the pole) and the angle \( \theta \) relative to a fixed line (the polar axis). For instance, in planetary motion, the pole is often chosen as one of the foci of the ellipse (in this case, where the sun is placed), and the angle \( \theta \) indicates the direction of the planet with respect to the fixed line, usually the major axis of its orbit. The polar coordinates system simplifies the complex calculations involved in tracking an object's path, crucial for both theoretical simulations and practical applications like space missions.

Ellipse Equation

The ellipse equation in polar coordinates clearly illustrates how planets move in orbits. The equation given for planetary motion is:\[ r = \frac{(1-e^2)a}{1-e\cos\theta} \]Here, \( r \) represents the radius (or distance) from the sun at any point of the orbit, while \( \theta \) is the angle as previously explained. The terms \( a \) is the semi-major axis, and \( e \) is the eccentricity of the orbit.This equation is significant as it provides insight into how the distance between the planet and the sun changes as the planet travels along its orbit. When \( \theta = 0 \), the cosine term is at its maximum, and the distance \( r \) is at one of its minimum or maximum values, defining the perihelion (closest point to the sun) or aphelion (farthest point) of the planet's elliptical path. This mathematical representation is fundamental to the study of celestial mechanics, helping scientists predict planetary positions over time.