Problem 63 - Precalculus

Question

The comet Hale-Bopp has an elliptical orbit with an eccentricity of \(e \approx 0.995 .\) The length of the major axis of the orbit is approximately 500 astronomical units. Find a polar equation for the orbit. How close does the comet come to the sun?

Step-by-Step Solution

Answer

The polar equation for Hale-Bopp's orbit is \(r = \frac{0.995 \cdot 250}{1 + 0.995 \cos \theta}\) and the closest distance it comes to the sun is approximately 49850 astronomical units.

Step 1: Establish the Polar Equation

The polar equation of a conic section is: \(r = \frac{ed}{1 + e \cos \theta}\). The eccentricity \(e\) is given as 0.995 and the semi major axis which is the distance from the sun to the center of the ellipse or distance from the center to the directrix \(d\) is 250. Substituting these values into the polar equation, we get \(r = \frac{0.995 \cdot 250}{1 + 0.995 \cos \theta}\).

Step 2: Find the Closest Distance to the Sun

The minimum value of the polar equation \(r = \frac{0.995 \cdot 250}{1 + 0.995 \cos \theta}\) occurs when \(1 + 0.995 \cos \theta\) is maximum. This happens when \(\cos \theta = -1\). Using this value for \(\cos \theta\) in the equation we get minimum radius \(r = \frac{0.995 \cdot 250}{1 - 0.995} \approx 49850\).

Key Concepts

Eccentricity of an Ellipse

Eccentricity is a parameter associated with every conic section that describes how much the conic deviates from being circular. Specifically, for an ellipse, which is the path our comet Hale-Bopp takes, the eccentricity value (\( e \) ) lies between 0 and 1; where 0 corresponds to a perfect circle and values closer to 1 indicate an elongated ellipse.

In our example, the eccentricity of Hale-Bopp's orbit is given as 0.995, which is very close to 1. This suggests that its orbit is highly elongated, more so than common celestial elliptical paths. The knowledge of the comet's eccentricity can allow us, among other things, to calculate its closest and farthest distances from the Sun during its orbit.

Astronomical Units

An astronomical unit (AU) is a unit of length typically used to measure distances within our solar system. It is approximately equal to the average distance from the Earth to the Sun, roughly 149.6 million kilometers or 92.96 million miles.

In the context of comets, like Hale-Bopp, understanding distances in astronomical units provides an accessible scale for comparison. For instance, Hale-Bopp's major axis being 500 AU means the furthest points of its orbit span a distance 500 times the average distance between the Earth and the Sun, illuminating just how vast its journey through space is.

Polar Coordinates

Polar coordinates offer a way of representing points in a plane using a distance from a reference point, called the pole (similar to the origin in Cartesian coordinates), and an angle from a reference direction, typically the positive x-axis. A point in polar coordinates is denoted by (\( r \theta \) ), where \( r \) is the radius or the distance from the pole, and \theta is the angle in radians.

Polar coordinates are particularly useful in astronomy and navigation because they align well with the circular and elliptical patterns that celestial objects often follow, as well as with directions based on a compass or celestial sphere.

Conic Sections in Polar Form

Conic sections, such as ellipses, parabolas, and hyperbolas, can be elegantly expressed in polar coordinates. This form relates directly to a conic's focus, which for celestial bodies is often a star, like our Sun.

The general polar equation of a conic section is given by \( r = \frac{ed}{1 + e \times \text{cos}(\theta)} \times \theta \) where \( e \) is the eccentricity and \( d \) is the distance from the directrix to the focus. In our example with Hale-Bopp, we use the eccentricity and the semi-major axis (half the length of the major axis) to find the polar equation of its orbit. This equation can be used to calculate significant orbital characteristics, such as the comet's closest approach to the Sun, also known as its perihelion.