Problem 62 - Precalculus: Functions and Graphs

Question

Mercury's orbit The planet Mercury travels in an elliptical orbit that has eccentricity 0.206 and major axis of length 0.774 AU. Find the maximum and minimum distances between Mercury and the sun.

Step-by-Step Solution

Answer

The maximum distance is 0.466 AU, and the minimum distance is 0.307 AU.

Step 1: Understand the Elliptical Orbit

The orbit of Mercury is an ellipse, characterized by the major axis and eccentricity. In an elliptical orbit, the major axis is the longest diameter that passes through both foci of the ellipse.

Step 2: Define the Eccentricity and Major Axis

Given: Eccentricity (\(e\)) is 0.206, and the major axis (\(a\)) is 0.774 AU. The major axis consists of the sum of the perihelion (closest point to the sun) and aphelion (farthest point from the sun) distances.

Step 3: Calculate the Semi-Major Axis

The semi-major axis (\(a\)) is half of the major axis. Calculate it by dividing the length of the major axis by 2:\[a = \frac{0.774}{2} = 0.387 \text{ AU}.\]

Step 4: Use Eccentricity to Find Minimum and Maximum Distances

The perihelion distance (\(r_{min}\)) is given by: \[r_{min} = a(1 - e).\]The aphelion distance (\(r_{max}\)) is given by:\[r_{max} = a(1 + e).\]

Step 5: Calculate the Perihelion Distance

Substitute the values into the equation for perihelion distance:\[r_{min} = 0.387(1 - 0.206) = 0.387 \times 0.794 = 0.307482 \text{ AU}.\]

Step 6: Calculate the Aphelion Distance

Substitute the values into the equation for aphelion distance:\[r_{max} = 0.387(1 + 0.206) = 0.387 \times 1.206 = 0.466362 \text{ AU}.\]

Key Concepts

Eccentricity

Eccentricity is a measure of how much an ellipse deviates from being a perfect circle.
When you hear about eccentricity in the context of orbits, it helps to understand that it tells us the shape of the orbit.
For an orbit:

• If eccentricity (\(e\)) is 0, the orbit is a perfect circle.
• If \(e\) is between 0 and 1, the orbit is an ellipse.
• As \(e\) approaches 1, the ellipse becomes more elongated.

Mercury has an eccentricity of 0.206, meaning its orbit is not perfectly round. Rather, it is an ellipse with a slightly elongated shape. This value indicates that Mercury's orbit is more elongated when compared with planets like Venus or Earth, which have smaller eccentricities. Understanding eccentricity can help us visualize the type of path a planet takes around the sun, revealing whether the planet moves closer and farther away during its orbit.

Semi-Major Axis

The semi-major axis of an ellipse is one of its most important elements. An ellipse has two axes: the major and minor.
The major axis is the longest diameter across the ellipse, stretching through both foci. The minor axis is the shortest one parallel to the focus points.
The semi-major axis is half of the major axis. It's a line from the center of the ellipse to one end of the major axis.
Imagine the major axis is the measurement taken with a ruler, the semi-major axis is the center-to-object end portion of that measurement:

• In mathematical terms, if the total length of the major axis is \(a\), the semi-major axis is \(\frac{a}{2}\)

For Mercury, this is \(0.387\) AU, calculated from its major axis of \(0.774\) AU.

The semi-major axis is very significant because it helps to determine the size of the orbit. This value is also used in equations to find perihelion and aphelion distances.

Perihelion and Aphelion Distances

In an elliptical orbit, the planet doesn't stay at an even distance from the sun.
Instead, there are two key distances that scientists often focus on:

• Perihelion: This is the closest point to the sun in a planet's orbit. Calculated using the formula \(r_{min} = a(1 - e)\), where \(a\) is the semi-major axis and \(e\) is the eccentricity.
• Aphelion: Conversely, this is the farthest point from the sun. The formula for aphelion distance is \(r_{max} = a(1 + e)\)

By computing these, we know exactly how far Mercury goes in both extremes of its solar orbit.
Using the values given,

• Perihelion Distance: \(r_{min}\) came out to be \(0.307482\) AU, indicating its minimum point is just over \(0.30\) Astronomical Units from the sun,
• Aphelion Distance: \(r_{max}\) calculated to \(0.466362\) AU, meaning Mercury will go to almost \(0.47\) Astronomical Units at its farthest.

Grasping these distances allows us to understand the dynamics of Mercury's motion in space and observe how it interacts with the sun throughout its orbit.