Problem 318 - Calculus Volume 3

Question

Determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbits of the comets or planets. Distance is given in astronomical units \((\mathrm{AU})\). Halley's Comet: length of major axis \(=35.88\), eccentricity \(=0.967\)

Step-by-Step Solution

Answer

The polar equation is \( r(\theta) = \frac{1.164}{1 + 0.967\cos\theta} \).

Step 1: Understand the Problem

We need to convert information about Halleys' Comet orbit into a polar equation of an ellipse. We're provided the length of the major axis and eccentricity.

Step 2: Compute the Semi-Major Axis

The length of the major axis is given as 35.88 AU. The semi-major axis \(a\) is half of the major axis. Thus, \(a = \frac{35.88}{2} = 17.94\, \mathrm{AU}\).

Step 3: Use Eccentricity Formula

The eccentricity \(e\) is given as 0.967. The equation in polar form is derived from ellipse properties, where \(e = \frac{c}{a}\). Thus, compute \(c = e \cdot a = 0.967 \times 17.94\, \mathrm{AU}\).

Step 4: Calculate Distance to Focus (c)

Calculate \(c = 0.967 \times 17.94 = 17.35\, \mathrm{AU}\), where \(c\) is the distance from the center to a focus.

Step 5: Write Polar Equation

The polar equation for an ellipse with one focus at the origin is given by \(r(\theta) = \frac{a(1-e^2)}{1 + e\cos\theta}\). Substitute \(a\) and \(e\) into the equation: \[ r(\theta) = \frac{17.94(1-0.967^2)}{1 + 0.967\cos\theta} \].

Step 6: Simplify the Expression

Calculate \(1 - 0.967^2 = 1 - 0.935089 = 0.064911\), and substitute to get \[ r(\theta) = \frac{17.94 \times 0.064911}{1 + 0.967\cos\theta} \].

Key Concepts

Polar Coordinates

Polar coordinates offer a different way to represent points on a plane compared to traditional Cartesian coordinates. Instead of using a grid of x and y axes, polar coordinates use a combination of a distance from a fixed point, known as the pole (often corresponding to the origin in Cartesian coordinates), and an angle from a fixed direction, usually the positive x-axis.

The keys to polar coordinates are:

• The radial coordinate (r), which represents the distance from the pole.
• The angular coordinate (θ), which measures the angle from the fixed direction.

This system is particularly useful in astronomy and physics when dealing with orbits, as many natural phenomena are naturally circular or elliptical in pattern. By using polar coordinates, we can more easily describe the paths of objects like planets or comets in their orbital paths.

Ellipse Properties

Ellipses are fundamental shapes in mathematics and astronomy, particularly when describing the orbits of celestial bodies. An ellipse is an elongated circle and has distinct geometric properties.

Key properties of an ellipse include:

• Major Axis: The longest diameter of the ellipse, passing through its foci.
• Minor Axis: The shortest diameter perpendicular to the major axis.
• Foci: Two fixed points on the major axis. The sum of the distances from any point on the ellipse to the foci is constant.

In the context of orbits, the Sun or a planet resides at one of the foci. This makes the path of planets, like Halley's Comet, elliptical around these celestial bodies. By understanding ellipse properties, we can derive equations that describe such orbits accurately.

Astronomical Units

Astronomical units (AU) serve as a standard unit of measurement in astronomy to express distances within our solar system. One AU is defined as the average distance between the Earth and the Sun, approximately 149.6 million kilometers (or about 93 million miles).

Astronomical units are particularly useful because:

• They simplify the large values involved in interplanetary distances.
• They provide a consistent frame of reference when comparing different celestial objects' distances.

In calculations involving orbits, such as Halley's Comet with its major axis measured in AU, using this unit allows for more intuitive understanding and communication of distances within the solar system.

Eccentricity

Eccentricity is a measure of how much an orbit deviates from being circular. It is a dimensionless parameter that defines the shape of the ellipse formed by an orbital path.

The eccentricity (e) of an ellipse is calculated as the ratio of the distance between the foci to the length of the major axis:

• If e = 0, the orbit is a perfect circle.
• If 0 < e < 1, the orbit is elliptical.
• If e = 1, the shape becomes a parabola.

For Halley's Comet, an eccentricity of 0.967 indicates a highly elongated ellipse, meaning its orbit is far from circular. Understanding eccentricity allows astronomers to predict the shape of an orbit and consequently the behavior of celestial bodies within it.