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 \(38.07 \mathrm{AU}\). What is the eccentricity?

Short answer

Eccentricity is approximately 0.244, with a Cartesian equation: \(\frac{x^2}{19.63^2} + \frac{y^2}{19.035^2} = 1\).

Step-by-step solution

Understand the geometric properties

Pluto's orbit is an ellipse. The major axis is the longest diameter of the ellipse, and the minor axis is the shortest diameter. Given lengths of both these axes help us describe the orbit's size and shape.

Determine semi-major and semi-minor axes

The semi-major axis (\(a\)) is half the length of the major axis, and the semi-minor axis (\(b\)) is half the length of the minor axis. Given:\[a = \frac{39.26}{2} = 19.63 \text{ AU}\]\[b = \frac{38.07}{2} = 19.035 \text{ AU}\]

Calculate the eccentricity

The eccentricity (\(e\)) of an ellipse is calculated using the formula:\[e = \sqrt{1 - \left(\frac{b^2}{a^2}\right)}\]Substitute with the calculated values:\[e = \sqrt{1 - \left(\frac{19.035^2}{19.63^2}\right)}\]

Simplify and find eccentricity

Simplifying inside the square root:\[19.035^2 = 362.333225, \quad 19.63^2 = 385.2769\]\[\frac{b^2}{a^2} = \frac{362.333225}{385.2769} \approx 0.9404\]Now, calculate the eccentricity:\[e = \sqrt{1 - 0.9404} = \sqrt{0.0596} \approx 0.244\]

Find the Cartesian Equation

The standard form of the Cartesian equation of an ellipse centered at the origin with the major axis along the x-axis is:\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]Substitute the values of \(a=19.63\) AU and \(b=19.035\) AU:\[\frac{x^2}{19.63^2} + \frac{y^2}{19.035^2} = 1\]

Key concepts

Eccentricity

Eccentricity is a measure that indicates how much an orbit deviates from being a perfect circle. In celestial mechanics, it defines the shape of an elliptical path. For an orbit:

• An eccentricity ( \( e \) ) of 0 indicates a perfect circle.
• As the eccentricity increases closer to 1, the orbit becomes more elongated.
• Orbits with \( e < 1 \) are elliptical.

In the context of Pluto, its eccentricity can be calculated using the formula: \[ e = \sqrt{1 - \left(\frac{b^2}{a^2}\right)} \]where \( b \) and \( a \) are the semi-minor and semi-major axes, respectively. For Pluto's orbit, this results in an eccentricity of approximately 0.244. This value represents a notable deviation from a circular orbit, showcasing how unique Pluto’s path around the sun is.
This orbit shows Pluto's trajectory is somewhat eccentric, resulting in variations in its orbital speed and distance from the Sun.

Ellipse Equation

The equation of an ellipse is instrumental in defining the path of celestial bodies like planets. A typical Cartesian equation for an ellipse centered at the origin is:\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]where \( a \) is the length of the semi-major axis, and \( b \) is the length of the semi-minor axis.In Pluto’s case:

• \( a = 19.63 \) Astronomical Units (AU)
• \( b = 19.035 \) AU

Plugging these into the standard form gives us:\[ \frac{x^2}{19.63^2} + \frac{y^2}{19.035^2} = 1 \]This ellipse formula provides a complete mathematical representation of the possible locations of Pluto in its orbit, depicting how it travels around the Sun across two principal directions.

Astronomical Units

An Astronomical Unit, or AU, is a standard unit of measurement in astronomy. It represents the average distance between the Earth and the Sun, approximately 149.6 million kilometers. Used primarily in measuring astronomical distances within our solar system, it provides a relatable scale for understanding various orbital paths.When considering the elliptical orbits of celestial bodies like Pluto, using astronomical units simplifies the handling of vast distances. For Pluto:

• The semi-major axis is \( 19.63 \) AU
• The semi-minor axis is \( 19.035 \) AU

The use of AU is particularly helpful in astronomy because it allows scientists and students to discuss distances without resorting to extremely large numbers, thereby enhancing clearer communication and comprehension of orbital mechanics.