Question

For the following exercises, 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 \((A U)\). $$ \text { Halley's Comet: length of major axis }=35.88 \text { , eccentricity }=0.967 $$

Short answer

The polar equation is \( r = \frac{1.164652}{1+0.967 \cos{\theta}} \).

Step-by-step solution

Understand the Polar Equation for an Ellipse

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

Calculate the Semi-Major Axis

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

Substitute Values into the Polar Equation

Substitute the values \( a = 17.94 \) and \( e = 0.967 \) into the polar equation: \[ r = \frac{17.94(1-0.967^2)}{1+0.967 \cos{\theta}} \].

Simplify the Equation

Calculate \( 1 - 0.967^2 \): \( 1 - 0.935089 \approx 0.064911 \). So the equation becomes: \[ r = \frac{17.94 \times 0.064911}{1+0.967 \cos{\theta}} \]. Then calculate \( 17.94 \times 0.064911 \approx 1.164652 \), so: \[ r = \frac{1.164652}{1+0.967 \cos{\theta}} \].

Key concepts

Eccentricity

Eccentricity is a crucial term when discussing celestial orbits, especially when describing their shapes. It is a dimensionless parameter that determines the degree of deviation of an orbit from being circular. In simpler terms, it gives you an idea of how much an ellipse (the general shape of orbits) is stretched. For example:

• An orbit with an eccentricity of 0 is perfectly circular.
• An orbit with an eccentricity between 0 and 1 is elliptical.
• A value of 1 or greater signifies a parabolic or hyperbolic trajectory, not a closed orbit.

For Halley's Comet, the eccentricity is given as 0.967, which indicates a highly elongated elliptical orbit. This means that its path takes a very elongated shape, coming close to the Sun, then swinging far out into space before returning. Understanding eccentricity helps predict the path of objects like comets and planets.

Major Axis

The major axis is the longest diameter of an ellipse, running through the center and the two foci. Think of it as stretching across the body of an elliptical orbit from one edge to the other. It's the ``backbone'' of the ellipse, defining its maximum dimension. In celestial orbits, the length of the major axis is directly linked to the size of the orbit. This axis is important because it helps us understand the geometric shape and size of an orbit.

In the case of Halley's Comet, the length of the major axis is 35.88 astronomical units (AU). This means that the overall width of the comet's orbit, measuring from one side of the ellipse to the other, is 35.88 AU. The major axis is vital for calculating the semi-major axis, a key value used in the polar equation of an ellipse.

Halley's Comet

Halley's Comet is one of the most well-known celestial bodies due to its periodic return to the inner solar system approximately every 76 years. It is a prime example of a comet with a highly elliptical orbit. The comet is named after Edmund Halley, who calculated its orbit and correctly predicted its return. Its orbit is very elongated, represented by its high eccentricity value of 0.967.

Because the comet has such an elongated path, it spends a significant amount of time in the outer solar system, only briefly coming close to the Sun and Earth. Halley's Comet's orbit can be modeled using the polar coordinates, taking into account the length of its major axis and its eccentricity, which allows us to determine its position at any given time. Observations of Halley's Comet provide vital insights into the behavior and composition of comets in general.

Semi-Major Axis

The semi-major axis is half the length of the major axis. In other words, it extends from the center to the furthest point along the major axis of the ellipse. This measurement is crucial in celestial mechanics as it helps define the size of an orbit.

For Halley's Comet, the semi-major axis is calculated by dividing the major axis by 2, resulting in a value of 17.94 AU. This value represents a vital component of the polar equation of an ellipse, allowing for calculations of the orbit's path: \[ r = \frac{a(1-e^2)}{1+e \cos{\theta}} \]where \( a \) is the semi-major axis and \( e \) is eccentricity. By substituting the semi-major axis into the polar equation, one can predict the comet's distance from the Sun depending on its angular position. This eventually assists astronomers in predicting the movement of Halley's Comet and its future appearances.