Problem 347 - Calculus Volume 3

Question

Determine the Cartesian equation describing the orbit of Pluto, the most eccentric orbit around the Sun. The length of the major axis is \(39.26 \mathrm{AU}\) and minor axis is 38.07 AU. What is the eccentricity?

Step-by-Step Solution

Answer

Eccentricity is 0.2445 and the Cartesian equation is \( \frac{x^2}{385.2169} + \frac{y^2}{362.3322} = 1 \).

Step 1: Identify the semi-major and semi-minor axes

The length of the major axis is given as 39.26 AU. Therefore, the semi-major axis, denoted as \( a \), is half of this length: \( a = \frac{39.26}{2} = 19.63 \) AU. Similarly, the minor axis is 38.07 AU, making the semi-minor axis \( b = \frac{38.07}{2} = 19.035 \) AU.

Step 2: Calculate the eccentricity

The eccentricity (\( e \)) of an ellipse is given by the formula \( e = \sqrt{1 - \left(\frac{b}{a}\right)^2} \). Substituting \( a = 19.63 \) AU and \( b = 19.035 \) AU into the formula, we get:\[ e = \sqrt{1 - \left(\frac{19.035}{19.63}\right)^2} \]Calculate \( \frac{19.035}{19.63} = 0.9697 \). Hence,\[ e = \sqrt{1 - (0.9697)^2} \approx \sqrt{1 - 0.9402} \approx \sqrt{0.0598} \approx 0.2445 \]

Step 3: Determine the Cartesian equation of the ellipse

The equation of an ellipse centered at the origin with semi-major axis \( a \) along the x-axis and semi-minor axis \( b \) along the y-axis is given by:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]Substitute \( a = 19.63 \) AU and \( b = 19.035 \) AU into this equation:\[ \frac{x^2}{(19.63)^2} + \frac{y^2}{(19.035)^2} = 1 \]Calculating the denominators, we get:\( (19.63)^2 = 385.2169 \) and \( (19.035)^2 = 362.3322 \). Thus, the Cartesian equation becomes:\[ \frac{x^2}{385.2169} + \frac{y^2}{362.3322} = 1 \]

Key Concepts

Eccentricity

Eccentricity is a key measurement in understanding the shape of an ellipse. It tells us how much an ellipse deviates from being a perfect circle. This is particularly important in the context of planetary orbits like that of Pluto. The eccentricity (\( e \)) of an ellipse can be calculated using the formula:

\[ e = \sqrt{1 - \left(\frac{b}{a}\right)^2} \]where \( a \) is the semi-major axis and \( b \) is the semi-minor axis.

• If the eccentricity is 0, the shape is a perfect circle.
• If the eccentricity is closer to 1, the ellipse is elongated.

In the case of Pluto's orbit, having an eccentricity of approximately 0.2445 means it is not a perfect circle, but rather an elongated oval. This gives us valuable insight into the nature of its path around the Sun.

Ellipse

An ellipse is a geometric shape that looks like a flattened circle. It is defined by two main axes: the major and minor axes. Unlike a circle, an ellipse has varying radii of curvature depending on the direction. Understanding the properties of an ellipse is crucial in astronomy, engineering, and physics.

• Major Axis: The longest diameter of the ellipse, passing through both foci.
• Minor Axis: The shortest diameter, perpendicular to the major axis.

In the context of planetary movements, the elliptical nature of orbits means that a planet's distance from the Sun changes as it travels along its path. This is why we use ellipses, rather than circles, to describe most planetary orbits. For Pluto, the exchange in distances results in varying seasons and solar energy intake.

Semi-major Axis

The semi-major axis is one half of the major axis. It extends from the center of the ellipse to the furthest edge. This is perhaps the most important dimension of an ellipse because it greatly influences an object's path and its orbital period when thinking of celestial bodies.

The length of the semi-major axis is used to determine the size of the ellipse and is part of the formula to calculate the eccentricity. For Pluto, this value is 19.63 AU, which partially informs us about the scale of Pluto’s orbit around the Sun. It is essential in calculations involving the orbital velocity and period of an object in space.

Semi-minor Axis

Similarly to the semi-major axis, the semi-minor axis is half of the minor axis. It stretches from the center to the closest edge of the ellipse along the minor axis. Although smaller than the semi-major axis, it is crucial for determining the specific shape of the ellipse.

For an ellipse, knowing both the semi-major and semi-minor axes allows us to calculate the ellipse's eccentricity and write its equation in a Cartesian coordinate system. Pluto's semi-minor axis is 19.035 AU, contributing to its slightly oval-shaped orbit. This axis helps complete the necessary parameters to define Pluto’s unique orbit.