Problem 4 - Algebra 2

Question

Write an equation for the ellipse that satisfies each set of conditions. At its closest point, Earth is 0.99 astronomical units from the center of the Sun. At its farthest point, Earth is 1.021 astronomical units from the center of the Sun. Write an equation for the orbit of Earth, assuming that the center of the orbit is the origin and the Sun lies on the \(x\)-axis.

Step-by-Step Solution

Answer

Earth's orbit equation is \( \frac{x^2}{1.0055^2} + \frac{y^2}{1.01080975} = 1 \).

Step 1: Understanding the Ellipse Equation

An ellipse centered at the origin has the equation: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \(a\) is the semi-major axis, \(b\) is the semi-minor axis, and \(c\) is the distance from the center of the ellipse to the focus.

Step 2: Assign Values to Conditions

Given that Earth's closest point to the Sun is 0.99 AU, and farthest is 1.021 AU, we can identify the closest point (perihelion) as \(a - c = 0.99\) and the farthest point (aphelion) as \(a + c = 1.021\). Here, \(a\) is half of the major axis (the average of these distances), and \(c\) is the distance from the center to the focus.

Step 3: Calculate Values for a and c

Add the equations \(a - c = 0.99\) and \(a + c = 1.021\) to solve for \(a\):\[ (a - c) + (a + c) = 0.99 + 1.021 \] which simplifies to \[ 2a = 2.011 \]. Dividing by 2 gives \(a = 1.0055\).

Step 4: Calculate c

Subtract the equations \(a - c = 0.99\) from \(a + c = 1.021\) to solve for \(c\):\[ (a + c) - (a - c) = 1.021 - 0.99 \] this simplifies to \[ 2c = 0.031 \]. Dividing by 2 gives \(c = 0.0155\).

Step 5: Calculate b Using the Ellipse Formula

Using the relationship \(c^2 = a^2 - b^2\), substitute the values of \(a\) and \(c\) to solve for \(b^2\):\[ 0.0155^2 = 1.0055^2 - b^2 \] which results in \[ 0.00024025 = 1.01105 - b^2 \]. Solving for \(b^2\) gives \(b^2 = 1.01080975\). Taking the square root, \(b = \sqrt{1.01080975}\).

Step 6: Write the Ellipse Equation

Substitute \(a\) and \(b\) into the standard equation for an ellipse: \( \frac{x^2}{1.0055^2} + \frac{y^2}{1.01080975} = 1 \). This is our equation representing Earth's orbit.

Key Concepts

Astronomical Units

Astronomical units, often abbreviated as AU, are a convenient way to express distances within our solar system. One astronomical unit is roughly the average distance from the Earth to the Sun, approximately 149.6 million kilometers.
This unit provides an easier means of comparing distances without resorting to large numbers. When discussing objects within the solar system, like planets or asteroids, astronomers frequently use AU to simplify calculations and convey information effectively.

• For example, the average distance between the Sun and the Earth is 1 AU.
• When studying the Earth's orbit, determining its perihelion and aphelion points in AU helps astronomers model and understand planetary movement.

In the context of elliptical orbits, knowing these distances in AU allows us to calculate essential properties like the semi-major and semi-minor axes.

Semi-Major Axis

The semi-major axis of an ellipse is a crucial parameter that represents half the longest diameter of the ellipse. It extends from the center to the farthest point on the edge of the ellipse.
In orbital mechanics, it plays a key role in determining the size and shape of an orbit. Using the Earth's orbit as an example, the semi-major axis is approximately 1.0055 AU, which is derived from the average of the perihelion and aphelion distances.

• To find the semi-major axis, add the perihelion and aphelion distances and divide by 2.
• The semi-major axis is not only a measure of size but also directly influences the orbital period of a planet via Kepler's Third Law.

Knowing the semi-major axis allows astronomers to further explore various properties of celestial bodies and predict their behavior within the solar system.

Semi-Minor Axis

The semi-minor axis of an ellipse is the smaller of the two axes and represents half of the shortest diameter. It is perpendicular to the semi-major axis.
In the context of planetary orbits like Earth's, the semi-minor axis helps define the orbit's shape. A smaller semi-minor axis relative to the semi-major axis indicates a more elongated ellipse.

• The semi-minor axis is calculated using the ellipse relationship: \( c^2 = a^2 - b^2 \), where \(c\) is the distance from the center to the focus.
• For the Earth's elliptical orbit around the Sun, the calculated \(b^2\) equals 1.01080975, meaning \(b\) approximately equals 1.0054 AU when solved using the equation.

This axis is essential for determining the total area of the orbit and completing the understanding of the ellipse's geometric properties.

Perihelion and Aphelion

Perihelion and aphelion refer to the points in an object's orbit closest and farthest from the Sun, respectively. These terms are specific to orbits around the Sun—a fact significant in describing Earth's yearly journey.
Earth reaches its perihelion around early January and its aphelion in early July:

• Perihelion distance: 0.99 AU
• Aphelion distance: 1.021 AU

The difference in these distances indicates the non-circular nature of Earth's orbit.
These parameters are critical when calculating the orbital elements, like the semi-major and semi-minor axes, and defining the ellipse mathematically. Understanding perihelion and aphelion provides insight into factors like seasonal changes and variations in solar energy received by Earth. Solving orbital equations also incorporates these values to determine eccentricity.

Eccentricity of Ellipse

The eccentricity of an ellipse is a measure of how much the shape deviates from being a perfect circle. It is a dimensionless number ranging from 0 (a circle) to just below 1 (a very elongated ellipse).
For Earth's orbit, the eccentricity is derived from the relationship between the semi-major axis \(a\) and the distance from the center to the focus \(c\). The formula is:

• \( e = \frac{c}{a} \)

With \( a = 1.0055 \) AU and \( c = 0.0155 \) AU, Earth's orbital eccentricity is approximately 0.0154. A low eccentricity like this means Earth's orbit is nearly circular, stabilizing the planet's climate by maintaining consistent solar energy levels throughout the year. Understanding eccentricity helps predict variations in Earth's climate patterns and provides essential insights into planetary motion and dynamics.