Problem 94 - College Algebra with Modeling and Visualization

Question

Halley's comet travels in an elliptical orbit with \(a=17.95\) and \(b=4.44\) and passes by Earth roughly every 76 years. Note that each unit represents one astronomical unit, or 93 million miles. The comet most recently passed by Earth in February 1986 (Source: M. Zeilik, Introductory Astronomy and Astrophysics.) (a) Write an equation for this orbit, centered at \((0,0)\) with major axis on the \(x\) -axis. (b) If the sun lies (at the focus) on the positive \(x\) -axis, approximate its coordinates. (c) Determine the maximum and minimum distances between Halley's comet and the sun.

Step-by-Step Solution

Answer

(a) \( \frac{x^2}{17.95^2} + \frac{y^2}{4.44^2} = 1 \). (b) Approximately \((17.41, 0)\). (c) Maximum distance: 17.95 AU; minimum distance: 0.54 AU.

Step 1: Equation of the Ellipse

The equation for an ellipse centered at the origin with the major axis along the x-axis (horizontal) can be expressed as \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where \(a\) is the semi-major axis and \(b\) is the semi-minor axis. Substituting the given values of \(a = 17.95\) and \(b = 4.44\), the equation of the ellipse becomes:\[ \frac{x^2}{17.95^2} + \frac{y^2}{4.44^2} = 1 \]

Step 2: Focus of the Ellipse

The coordinates of the foci for an ellipse centered at the origin with the major axis on the x-axis are \((\pm c, 0)\), where\[ c = \sqrt{a^2 - b^2} \]Using the given \(a\) and \(b\):\[ c = \sqrt{17.95^2 - 4.44^2} \approx \sqrt{322.6025 - 19.7136} \approx \sqrt{302.8889} \approx 17.41 \]Thus, the coordinates of the sun (focus) on the positive x-axis are approximately \((17.41, 0)\).

Step 3: Maximum and Minimum Distances

The maximum distance between the comet and the sun is at the vertex on the positive x-axis, which is \(a\) or 17.95.The minimum distance is at the vertex on the negative x-axis, which is the distance from the center of the ellipse to the focus, or \(a - c\):\[ a - c = 17.95 - 17.41 = 0.54 \]Hence, the maximum distance is 17.95 AU, and the minimum distance is 0.54 AU.

Key Concepts

Ellipse

An ellipse is a geometric shape that looks like an elongated circle. It is defined as a set of points where the sum of the distances from two fixed points, called the foci, is constant. The major axis is the longest diameter of the ellipse, and the minor axis is the shortest.
Ellipses have different mathematical properties and are represented by an equation of the form: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] where:

• \(a\) represents the semi-major axis length
• \(b\) represents the semi-minor axis length

For Halley's comet, the values \(a = 17.95\) and \(b = 4.44\) signify its elliptical path around the sun. The distance between the foci is determined by calculating \(c\), where \(c = \sqrt{a^2 - b^2}\). For Halley’s comet, this calculation is necessary to understand the position of the comet's orbit in space relative to the sun.

Orbital Mechanics

Orbital mechanics is the study of the motions of objects in space, influenced by gravitational forces. It's a fundamental aspect of understanding how celestial bodies move and interact.
When dealing with comets, like Halley's comet, which orbits the sun in an elliptical path, understanding orbital mechanics allows us to predict its behavior and trajectory. The calculation of Halley's elliptical orbit includes understanding its semi-major and semi-minor axes and how these determine the shape and eccentricity of the orbit.
Studying these mechanics involves calculating the positions of celestial bodies at any point in time and how gravity influences their paths. This is crucial when determining when the comet will return to Earth, as it does every 76 years. The more predictable the orbit, the better we can understand potential interactions with other cosmic bodies.

Astronomy

Astronomy is the scientific study of celestial objects like stars, planets, comets, and galaxies. It seeks to explain their origin, evolution, and eventual fate through observation and theory.
Halley's comet is a well-known subject in astronomy, often observed due to its periodic close encounters with Earth.
This celestial phenomenon provides vital clues about the composition of the early solar system. Studying Halley's orbit around the sun helps astronomers understand more about the solar system's dynamics and the gravitational forces at work. Exploration of such bodies deepens our knowledge about the formation of planets and other celestial phenomena, significantly contributing to the understanding of the cosmos.

Distance Calculation

Accurate distance calculation in space is crucial for understanding celestial movements. In the context of Halley's comet, we calculate distances using astronomical units (AU), where 1 AU represents the average distance from the Earth to the Sun, about 93 million miles.
For Halley’s comet, the maximum distance from the sun represents the semi-major axis, which is \(a = 17.95 \) AU. Conversely, the minimum distance is the difference between the semi-major axis and \(c\), the focal distance, calculated as \(a - c\). Here, this results in a minimum distance of 0.54 AU.
The calculated distances help predict the comet’s closest and farthest points from the sun during its orbit, which is essential for long-term predictions and observations. Such calculations are key to understanding the interaction between celestial bodies in our solar system.