Steps to Calculate Planetary Orbits Using Elliptical Conic Sections (Algebra 2).

📚 Understanding Planetary Orbits and Ellipses

Planetary orbits aren't perfect circles; they're ellipses! An ellipse is like a squashed circle, defined by two points called foci (plural of focus). The Sun sits at one of these foci. Let's explore how to describe these orbits mathematically.

📝 Learning Objectives

• 🌍 Define the key properties of an ellipse: foci, major axis, minor axis, and eccentricity.
• 🔢 Apply the equation of an ellipse to model planetary orbits.
• 📐 Calculate orbital parameters such as semi-major axis and eccentricity from given data.
• 🔭 Understand Kepler's First Law of Planetary Motion.

⚗️ Materials Needed

• 📏 Ruler or straightedge
• ✏️ Pencil
• ➗ Calculator
• 📜 Graph paper (optional)

🧠 Warm-up (5 mins)

Review conic sections. Specifically, discuss the standard form equation of an ellipse centered at the origin:

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Where 'a' is the semi-major axis and 'b' is the semi-minor axis.

🧭 Main Instruction

1. 🌌 Kepler's First Law

   • 🍎 State Kepler's First Law: Planets move in elliptical orbits with the Sun at one focus.
   • 🛰️ Explain how this differs from earlier circular orbit models.

2. 📊 Ellipse Properties

   • 📍 Define foci, major axis, semi-major axis (a), minor axis, semi-minor axis (b), and center.
   • ✏️ Draw an ellipse and label its parts.
   • 💡 Understand the relationship: $c^2 = a^2 - b^2$, where 'c' is the distance from the center to each focus.

3. 🧮 Eccentricity (e)

   • ➗ Define eccentricity: $e = \frac{c}{a}$.
   • 🌡️ Explain that $0 < e < 1$ for an ellipse.
   • 🪐 Relate eccentricity to the shape of the ellipse (e.g., e close to 0 is nearly circular, e close to 1 is highly elongated).

4. 📐 Calculating Orbital Parameters

   • 🔍 Example: A planet's orbit has a semi-major axis $a = 1.5 \times 10^8$ km and a distance from the center to a focus of $c = 1.45 \times 10^8$ km. Calculate the eccentricity.
   • ✏️ Solution: $e = \frac{c}{a} = \frac{1.45 \times 10^8}{1.5 \times 10^8} = 0.967$

5. 🛰️ Applying the Ellipse Equation

   • 🌍 Discuss how to orient the ellipse in a coordinate system.
   • 📝 Use the standard form equation to represent the orbit: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

🧪 Assessment

Solve these problems:

1. A planet has a semi-major axis of $2 \times 10^8$ km and an eccentricity of 0.9. Calculate the distance 'c' from the center to a focus.

2. An asteroid's orbit has $a = 3 \times 10^8$ km and $b = 1 \times 10^8$ km. Find the eccentricity.

3. If a comet has an eccentricity of 0.99 and a semi-major axis of $5 \times 10^9$ km, what is the distance from the center to the focus?