Problem 35 - Algebra 2

Question

SATELLITES For Exercises \(33-35,\) use the following information. Two satellites are placed in orbit about Earth. The equations of the two orbits \(\operatorname{are} \frac{x^{2}}{(300)^{2}}+\frac{y^{2}}{(900)^{2}}=1\) and \(\frac{x^{2}}{(600)^{2}}+\frac{y^{2}}{(690)^{2}}=1,\) where distances are in kilometers and Earth is the center of each curve. Compare the orbits of the two satellites.

Step-by-Step Solution

Answer

The first orbit is more elongated (higher eccentricity) than the second orbit.

Step 1: Recognize the Orbit Types

Both given equations are in the standard form of ellipses centered at the origin. The general form of an ellipse equation is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \( a \) and \( b \) are the lengths of the semi-major and semi-minor axes, respectively.

Step 2: Identify the Ellipses Parameters

For the first satellite's orbit, \( a = 300 \) km and \( b = 900 \) km. For the second satellite's orbit, \( a = 600 \) km and \( b = 690 \) km.

Step 3: Determine the Orientation and Shape

Since \( b > a \) in both equations, both ellipses have their major axis along the \( y \)-axis. The first ellipse is more elongated compared to the second one due to the larger difference between \( b \) and \( a \) values.

Step 4: Calculate the Eccentricity

Eccentricity \( e \) for an ellipse is calculated using the formula \( e = \sqrt{1 - \frac{a^2}{b^2}} \). For the first orbit: \( e_1 = \sqrt{1 - \frac{300^2}{900^2}} = \sqrt{1 - \frac{1}{9}} = \sqrt{\frac{8}{9}} \approx 0.94 \). For the second orbit: \( e_2 = \sqrt{1 - \frac{600^2}{690^2}} = \sqrt{1 - \frac{400}{529}} = \sqrt{\frac{129}{529}} \approx 0.50 \).

Step 5: Compare the Eccentricities

The first orbit has a higher eccentricity (\( \approx 0.94 \)), which means it is more elongated compared to the second orbit, which has an eccentricity of (\( \approx 0.50 \)).

Step 6: Concluding the Comparison

The first satellite's orbit is more elongated because it has a higher eccentricity compared to the second orbit. Thus, the first orbit is more elliptical, while the second one is closer to being circular.

Key Concepts

Eccentricity

Eccentricity is a measure that describes how much an ellipse deviates from being a perfect circle. It ranges from 0 to just under 1, where 0 indicates a perfect circle and values closer to 1 represent more elongated ellipses. The formula to calculate the eccentricity (\(e\)) of an ellipse is:

• \[ e = \sqrt{1 - \frac{a^2}{b^2}} \]

Here, \(a\) is the length of the semi-major axis, and \(b\) is the length of the semi-minor axis. For the given satellite orbits, we calculated \(e_1 \approx 0.94\) for the first orbit, indicating it is quite elongated. Meanwhile, \(e_2 \approx 0.50\) for the second orbit, showing it's more circular by comparison. Thus, eccentricity is crucial in understanding the shape and nature of an orbit.

Ellipse Equation

The ellipse equation is a fundamental component in understanding the geometry involved in the orbits of celestial bodies. An ellipse centered at the origin is generally represented in its standard form as:

• \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

Where:

• \(a\) is the semi-major axis (the longest radius of the ellipse).
• \(b\) is the semi-minor axis (the shortest radius of the ellipse).

This equation shows the relationship between the distances along the x and y axes that maintain the ellipse’s constant width. By examining these, we can tell how stretched or compressed an orbit is. In our example, for the first orbit, \(a = 300\) km and \(b = 900\) km, while for the second, \(a = 600\) km and \(b = 690\) km, helping us visualize the size and orientation of the orbits.

Major Axis

The major axis of an ellipse is the longest diameter and spans across the ellipse passing through the center. It's the line that goes through the farthest points on the ellipse, providing insight into the symmetry and layout of the ellipse.

For an ellipse described by the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the major axis is determined by which is larger: \(a\) or \(b\). Since \(b\) was greater than \(a\) in both given satellite equations, the major axis lies along the \(y\)-axis.

• In the first orbit, the major axis is: \(2b = 1800\) km.
• In the second orbit, the major axis is: \(2b = 1380\) km.

Understanding where the major axis lies helps determine the orbit's orientation and extent.

Minor Axis

The minor axis is the shortest diameter of an ellipse, perpendicular to the major axis and passing through the center as well. It provides a measure of the narrowness of the ellipse.

• In the ellipse equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), the minor axis is determined by the smaller value between \(a\) and \(b\).
• This axis gives the shortest possible distance across the ellipse.

For our satellite orbits, the minor axis is oriented along the x-axis:

• In the first orbit, the minor axis length is: \(2a = 600\) km.
• In the second orbit, the minor axis length is: \(2a = 1200\) km.

The minor axis helps in visualizing how compact or elongated the ellipse is relative to its major axis.