Problem 319 - 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 \((\mathrm{AU})\). Hale-Bopp Comet: length of major axis = 525.91, eccentricity \(=0.995\)

Step-by-Step Solution

Answer

The polar equation is \( r(\theta) = \frac{2.622}{1 + 0.995 \cos(\theta)} \).

Step 1: Define the Polar Equation of an Ellipse

The polar form of the equation for an ellipse is \( r(\theta) = \frac{a(1-e^2)}{1 + e \cos(\theta)} \), where \( a \) is the semi-major axis and \( e \) is the eccentricity.

Step 2: Determine the Semi-Major Axis

The semi-major axis \( a \) is half the length of the major axis. Therefore, \( a = \frac{525.91}{2} = 262.955 \).

Step 3: Substitute Values Into the Polar Equation

Substitute \( a = 262.955 \) and \( e = 0.995 \) into the polar equation to obtain the orbit of Hale-Bopp Comet: \[ r(\theta) = \frac{262.955(1 - (0.995)^2)}{1 + 0.995 \cos(\theta)} \].

Step 4: Simplify the Equation

Simplify the expression for the numerator: \( 1 - (0.995)^2 = 1 - 0.990025 = 0.009975 \). Thus:\[ r(\theta) = \frac{262.955 \times 0.009975}{1 + 0.995 \cos(\theta)} \].

Key Concepts

Major Axis

An ellipse is a fascinating shape often compared to an elongated circle. A key part of understanding an ellipse is the major axis, which is its longest diameter. This straight line stretches from one end of the ellipse, passes through its center, and reaches to the other end. To visualize it, imagine a race track that stretches longer horizontally than it does vertically. The length of the entire track is similar to the major axis.
For an ellipse in polar form, the major axis plays a crucial role. The given problem for the Hale-Bopp comet specifies a major axis length of 525.91 astronomical units. To proceed, it is important to find the semi-major axis, which is exactly half of the major axis. This is because the semi-major axis (\( a \)) is what is used in the polar equation for an ellipse. In this scenario, the semi-major axis is calculated by simply dividing the total length of the major axis by 2. Thus, \( a = \frac{525.91}{2} = 262.955 \) astronomical units.

Eccentricity

Eccentricity is a measure of how much an ellipse deviates from being a perfect circle. A circle can be thought of as a special type of ellipse where the eccentricity equals zero. The more stretched out the circle gets, the closer the eccentricity value will get to one. For the Hale-Bopp comet, the eccentricity is provided as 0.995, indicating a highly elongated orbit. When working with polar equations for ellipses, eccentricity is represented by \( e \), and it influences the shape considerably. A smaller eccentricity results in a more circular shape, while a larger eccentricity leads to a more stretched, elongated form. In the equation of an ellipse's orbit, \( r(\theta) = \frac{a(1-e^2)}{1 + e \cos(\theta)} \), eccentricity affects both the numerator and the denominator. Calculating \( 1 - e^2 \) for Hale-Bopp with \( e = 0.995 \), you get \( 1 - (0.995)^2 = 0.009975 \). This calculation shows how significant the elongation is in determining the precise path of the orbit.

Astronomical Units

Astronomical units (AU) are a convenient way to measure vast distances in space, typically used for distances within our solar system. One AU is about the distance from the Earth to the Sun, approximately 93 million miles or 150 million kilometers. This unit simplifies the complex numbers and makes calculations and comparisons much more manageable.
In the context of the Hale-Bopp comet's orbit, the use of astronomical units helps to provide a relatable context to the enormous scale of space travel. For this case, the length of the major axis—525.91 AU—demonstrates just how far the comet travels in its elongated orbit, when compared to the Earth's relatively circular path around the Sun. By understanding AU, students can better appreciate the magnitude of celestial bodies' orbits and the extensive distances they travel.