Problem 37 - Precalculus: Functions and Graphs

Question

Kepler's first law asserts that planets travel in elliptical orbits with the sun at one focus. To find an equation of an orbit, place the pole \(O\) at the center of the sun and the polar axis along the major axis of the ellipse (see the figure). (a) Show that an equation of the orbit is $$r=\frac{\left(1-e^{2}\right) a}{1-e \cos \theta}$$ where \(e\) is the eccentricity and \(2 a\) is the length of the major axis. (b) The perihelion distance \(r_{\text {per }}\) and aphelion distance \(r_{\text {aph }}\) are defined as the minimum and maximum distances, respectively, of a planet from the sun. Show that \(r_{\text {per }}=a(1-e) \quad\) and \(\quad r_{\text {aph }}=a(1+e)\) (IMAGE CAN NOT COPY)

Step-by-Step Solution

Answer

The orbit equation is derived: \( r = \frac{(1-e^2)a}{1-e\cos\theta} \), and distances are \( r_{\text{per}} = a(1-e) \) and \( r_{\text{aph}} = a(1+e) \).

Step 1: Understanding ellipse properties

Kepler's first law indicates that a planet's orbit around the sun is an ellipse with the sun at one focus. The general form of an ellipse in the polar coordinate system with the pole at one focus is \( r = \frac{l}{1 + e \cos \theta} \), where \(e\) is the eccentricity and \(l\) is the semi-latus rectum. The major axis length is \(2a\).

Step 2: Adjust ellipse equation to given form

We want to show \( r = \frac{(1-e^2)a}{1-e\cos\theta} \). Since the semi-latus rectum \( l = a(1-e^2) \) for an ellipse, we substitute this into the ellipse equation. Therefore, \( r = \frac{a(1-e^2)}{1-e\cos\theta} \), which is the given form.

Step 3: Perihelion distance calculation

To find the perihelion distance \(r_{\text{per}}\), set \(\theta = 0\) because this is when the planet is closest to the sun. Substitute into the orbit equation: \( r_{\text{per}} = \frac{a(1-e^2)}{1-e\cdot1} = a(1-e) \).

Step 4: Aphelion distance calculation

To find the aphelion distance \(r_{\text{aph}}\), set \(\theta = \pi\) because this is when the planet is furthest from the sun. Substitute into the orbit equation: \( r_{\text{aph}} = \frac{a(1-e^2)}{1-e\cdot(-1)} = a(1+e) \).

Key Concepts

Elliptical Orbits

Kepler's First Law of planetary motion reveals that planets orbit the sun in paths shaped like ellipses, rather than perfect circles. An ellipse can be visualized as a squashed or elongated circle. The sun sits at one of the two foci of this ellipse. This means that as a planet moves along its orbit, its distance from the sun changes continuously.

Think of an ellipse as a shape that has a major and a minor axis. The major axis is the longest diameter that stretches from one end of the ellipse through its center to the other end. The length of this major axis is denoted as "2a," where "a" is called the semi-major axis. The polar coordinate form of the orbit, with the sun at one focus, is given by the equation:

• \( r = \frac{l}{1 + e \cos \theta} \)

Here, \(r\) is the distance from the sun to the planet and \(\theta\) is the angle concerning the polar axis. This formula provides crucial insights into how planets travel in an elliptical orbit around the sun.

Eccentricity

Eccentricity is a crucial factor when understanding elliptical orbits. It quantitatively describes how stretched out an ellipse is. It is represented by the letter "e" and always has a value between 0 and 1. An eccentricity of 0 implies a perfect circle (though circles are not practical representations in orbital terms), while values closer to 1 indicate more elongated ellipses.

Elliptical orbits with lower eccentricity appear more round, whereas orbits with higher eccentricity are distinctly oval-shaped. The formula for eccentricity in the context of an orbit can be derived from the polar representation of an ellipse. It can be calculated through certain parameters such as the perihelion and aphelion distances or using the equation:

• \( e = \sqrt{1-\left(\frac{b}{a}\right)^2} \)

where "b" is the semi-minor axis. Understanding eccentricity allows us to appreciate how varied the paths of planets can be as they journey around the sun.

Perihelion and Aphelion Distances

The terms "perihelion" and "aphelion" describe the nearest and farthest points of a planet's orbit from the sun. These points are directly influenced by the orbit's eccentricity.

**Perihelion Distance:**At perihelion, a planet is closest to the sun. This distance can be calculated using the formula:

• \( r_{\text{per}} = a(1-e) \)

where "a" is the semi-major axis and "e" is the eccentricity.

**Aphelion Distance:**At aphelion, the planet is farthest from the sun. The formula for the aphelion distance is:

• \( r_{\text{aph}} = a(1+e) \)

The difference in these distances due to eccentricity highlights the non-circular nature of planetary orbits. By comprehending perihelion and aphelion, students can better grasp how eccentricity impacts the varying distances a planet experiences from the sun during its orbital path.