Problem 88 - A Graphical Approach to Precalculus with Limits

Question

Solve each application. Neptune and Pluto both have elliptical orbits with the sun at one focus. Neptune's orbit has \(a=30.1\) astronomical units (AU) and eccentricity \(e=0.009,\) whereas Pluto's orbit has \(a=39.4 \mathrm{AU}\) and \(e=0.249 .(1 \mathrm{AU}\) is equal to the average distance from Earth to the sun and is approximately \(149,600,000\) kilometers.) (Source: Zeilik, M., S. Gregory, and E. Smith, Introductory Astronomy and Astrophysics, Saunders College Publishers.) (a) Position the sun at the origin, and determine an equation for each orbit. (b) Graph both equations on the same coordinate axes. Use the window \([-60,60]\) by \([-40,40]\)

Step-by-Step Solution

Answer

The equations are \( \frac{x^2}{30.1^2} + \frac{y^2}{30.096^2} = 1 \) for Neptune and \( \frac{x^2}{39.4^2} + \frac{y^2}{36.91^2} = 1 \) for Pluto.

Step 1: Understand Ellipse Equation Format

The general equation for an ellipse with the center at the origin is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \(a\) and \(b\) are the semi-major and semi-minor axes respectively.

Step 2: Calculate the Semi-Minor Axis for Neptune

For an ellipse, \(b\) can be found using \( b = a \sqrt{1 - e^2} \). For Neptune, \(a = 30.1\) AU and \(e = 0.009\). Substitute these into the formula to find \(b\).

Step 3: Compute Neptune's Semi-Minor Axis

Substitute the given values into the formula: \( b = 30.1 \sqrt{1 - (0.009)^2} = 30.1 \sqrt{1 - 0.000081} = 30.1 \times 0.9999595 \approx 30.096 \).

Step 4: Formulate Neptune's Orbit Equation

Using \(a = 30.1\) and \(b \approx 30.096\), Neptune's orbit becomes: \( \frac{x^2}{30.1^2} + \frac{y^2}{30.096^2} = 1 \).

Step 5: Calculate the Semi-Minor Axis for Pluto

For Pluto, \(a = 39.4\) AU and \(e = 0.249\). Substitute these values into \( b = a \sqrt{1 - e^2} \) to find \(b\).

Step 6: Compute Pluto's Semi-Minor Axis

Substitute to get: \( b = 39.4 \sqrt{1 - (0.249)^2} = 39.4 \sqrt{1 - 0.062001} = 39.4 \times 0.937185 \approx 36.91 \).

Step 7: Formulate Pluto's Orbit Equation

Using \(a = 39.4\) and \(b \approx 36.91\), Pluto's orbit equation becomes: \( \frac{x^2}{39.4^2} + \frac{y^2}{36.91^2} = 1 \).

Step 8: Graph Both Orbits on the Same Axes

Graph the two derived ellipse equations on the coordinate plane, setting the range from \([-60,60]\) for the x-axis and \([-40,40]\) for the y-axis. Ensure each ellipse is centered at the origin with the specified axes lengths.

Key Concepts

Eccentricity

Eccentricity is a crucial concept in understanding the shape of an elliptical orbit. When dealing with planets like Neptune and Pluto, the eccentricity, denoted as \( e \), is a measure of how much an orbit deviates from being circular. It ranges between 0 (a perfect circle) and 1 (a highly elongated ellipse).
- For Neptune, the eccentricity is \( e = 0.009 \). This value indicates that Neptune's orbit is almost a perfect circle because the eccentricity is very close to 0.- In contrast, Pluto has an eccentricity of \( e = 0.249 \), suggesting a more elongated shape compared to Neptune.Understanding eccentricity helps astronomers predict the path a planet will follow around the sun, affecting its distance at various points in its orbit.

Semi-Major and Semi-Minor Axes

In an ellipse, the longest and shortest diameters are known as the semi-major and semi-minor axes, respectively. They represent one half of the major and minor axes. In terms of planetary orbits:- The semi-major axis \( a \) relates to the average distance from the sun to a planet over one orbit period. For example, Neptune's orbit semi-major axis is \( 30.1 \) AU, while Pluto's is \( 39.4 \) AU.- The semi-minor axis \( b \) can be calculated using the equation \( b = a \sqrt{1 - e^2} \).These axes are essential in defining the specific shape and orientation of a planet's orbit. For Neptune and Pluto, calculating these values allows us to visualize and understand their paths around the sun.

Astronomical Units

An Astronomical Unit (AU) is a standard unit of measurement used in astronomy to describe distances within our solar system. One AU is defined as the average distance from the Earth to the Sun, approximately 149,600,000 kilometers.
Using AU simplifies the expression of large distances in the universe, making it easier to comprehend and communicate when describing planetary positions and orbits. For instance:- Neptune's orbit with a semi-major axis of \( 30.1 \) AU suggests that, on average, it is 30.1 times the distance from Earth to the sun.- Pluto, with a semi-major axis of \( 39.4 \) AU, orbits even further away from the sun.This unit provides a more understandable framework for discussing and comparing the vast distances in space.

Ellipse Equation

The equation governing an ellipse is expressed as \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This format allows us to model the elliptical paths of planetary orbits.- \( a \) is the semi-major axis, and \( b \) is the semi-minor axis. These values determine the stretch and shape of the ellipse.- For Neptune, substituting into this equation using \( a = 30.1 \) AU and \( b \approx 30.096 \), we derive the orbit equation of Neptune: \( \frac{x^2}{30.1^2} + \frac{y^2}{30.096^2} = 1 \).- For Pluto, with \( a = 39.4 \) AU and \( b \approx 36.91 \), its equation becomes: \( \frac{x^2}{39.4^2} + \frac{y^2}{36.91^2} = 1 \).These equations form the basis for plotting the ellipses on a graph, helping astronomers and students visualize the orbits of these distant planets.