Problem 68 - Precalculus: Functions and Graphs

Question

Exer. \(67-68\) : The planets move around the sun in elliptical orbits. Given the semimajor axis a in millions of kilometers and eccentricity \(e,\) graph the orbit for the planet. Center the major axis on the \(x\) -axis, and plot the location of the sum at one focus. $$\text { Pluto's path } a=5913, \quad e=0.249$$

Step-by-Step Solution

Answer

Plot Pluto's orbit using ellipse parameters: semimajor axis = 5913, semiminor axis = 5724.35, sun at focus (1471.737, 0).

Step 1: Understand the Components of an Ellipse

The given parameters are the semimajor axis \(a\) and the eccentricity \(e\). The semimajor axis is half the longest diameter of the ellipse, and the eccentricity quantifies the deviation of the ellipse from being circular. The center of the ellipse is located at the origin \((0,0)\) and the major axis lies along the \(x\)-axis.

Step 2: Calculate the Semiminor Axis

For any ellipse, the relationship between the semimajor axis \(a\), the semiminor axis \(b\), and the eccentricity \(e\) is given by \(b = a\sqrt{1-e^2}\). Using the values \(a = 5913\) and \(e = 0.249\), we calculate:\[b = 5913 \cdot \sqrt{1 - (0.249)^2} \approx 5913 \cdot \sqrt{1 - 0.062001} \approx 5913 \cdot \sqrt{0.937999}\approx 5913 \cdot 0.9685 \approx 5724.35\]

Step 3: Identify the Foci of the Ellipse

The distance of each focus from the center of the ellipse is given by \(c = ae\). Calculate this using \(a = 5913\) and \(e = 0.249\):\[c = 5913 \cdot 0.249 = 1471.737\]The foci are located at \((+c, 0)\) and \((-c, 0)\) along the major axis.

Step 4: Position the Sun at One Focus of the Orbit

The sun is located at one of the foci of the ellipse. Therefore, based on our calculations, we set the sun at the coordinates \((1471.737, 0)\) or \((-1471.737, 0)\) on the \(x\)-axis.

Step 5: Graph the Elliptical Orbit

To graph the orbit, sketch an ellipse centered at the origin. The semimajor axis of length 5913 extends along the \(x\)-axis and the semiminor axis of length 5724.35 extends along the \(y\)-axis. Position the sun at one of the foci calculated in the previous step. Ensure the ellipse's width is shorter in the \(y\)-axis due to the smaller semiminor axis.

Key Concepts

Semimajor Axis

The semimajor axis is a fundamental part of understanding elliptical orbits. It represents the longest radius of the ellipse, essentially half of the ellipse's longest diameter. In astronomical terms, it provides an average distance from a planet to its star (like the sun).
To visualize it, imagine slicing an ellipse lengthwise down the middle; the semimajor axis is that long slice. Since it is an axis, it runs from the center of the ellipse to its edge, along the major diameter.

• For the orbit of Pluto, the given semimajor axis is 5913 million kilometers. This is used as a scale for its orbit, showing how far the planet is from the Sun on average.
• The semimajor axis helps define the shape and extent of the orbit, playing a crucial role in calculating the orbit's overall dimensions.

Understanding the semimajor axis is essential for graphing and analyzing elliptical paths in celestial mechanics.

Eccentricity

Eccentricity is a measure of how much an ellipse differs from being a perfect circle. It can have values from 0 to 1, with 0 being a circle and values closer to 1 indicating more elongated ellipses.
For instance, Pluto's orbit has an eccentricity of 0.249, indicating that its path is noticeably elliptical, rather than circular. Behind this number is a straightforward computation that helps define the shape of the orbit.
To calculate the eccentricity, you can use the relationship with the semimajor axis (\(a\)) and the distance between the two foci (\(c\)): \(e = \frac{c}{a}\). Once you know the eccentricity, you can also easily find the semiminor axis (\(b\)) through \(b = a \sqrt{1 - e^2}\).

• Lower eccentricity (close to 0) results in a more circular shape.
• Higher eccentricity (approaching 1) leads to a more stretched ellipse.

Eccentricity is crucial in understanding the distortion and the nature of orbits in our solar system.

Ellipse Foci

Foci are two fixed points on the interior of an ellipse used in the formal definition of the curve. Each point along an ellipse's perimeter has a total distance to these two foci points that remains constant. This unique property defines the shape and structure of an ellipse.
For Pluto's orbit, we calculated its foci as the points \((1471.737, 0)\) and \((-1471.737, 0)\). Here, the Sun resides at one of these points, which distinguishes a center of gravitational attraction for Pluto.
To find these foci, use the formula that relates the semimajor axis (\(a\)), eccentricity (\(e\)), and the distance (\(c\)) between the center and each focus:\(c = ae\). In this way, the placement of foci suggests how off-center the ellipse can be, helping determine how Pluto—or any celestial body—moves over time.

• Foci determine the layout and positioning of the ellipse within a coordinate system.
• An astronomical body orbits around one focal point, generally the one containing a massive object like the sun.

Understanding the role of foci is key to comprehending the behavior of celestial orbits and their resulting phenomena.