This activity guide walks students through simulating planetary orbits to discover that they are ellipses with the Sun at one focus.

You’ve correctly completed the activity! Let’s walk through the solution and explanation step-by-step to reinforce your understanding.

---

## 🌟 Activity Summary: Kepler’s First Law — The Shape of Planetary Orbits

This activity uses a simulation (Gizmo) to explore how planets move around the Sun. You’re investigating the *shape* of orbits and where the Sun is located relative to that shape.

---

## 🔍 Step-by-Step Breakdown

✔ 1. Sketch the Orbit

- You placed the planet at r = –3.00i AU → meaning it’s 3 AU to the left of the Sun (on the negative x-axis).
- You set velocity to v = –8.0j km/s → meaning it’s moving straight downward (southward) at 8 km/s.
- When you ran the simulation, the planet followed a curved path — an ellipse — returning to its starting point after one revolution.

> 💡 *Why an ellipse?*
> Gravity pulls the planet toward the Sun, changing its direction continuously. With this initial position and velocity, the planet doesn’t have enough speed to escape (which would make a parabola/hyperbola), nor too little (which would make it crash into the Sun). So it settles into a closed elliptical orbit.

---

✔ 2. Identify the Shape: Ellipse & Foci

- An ellipse is a stretched circle with two special points called foci (plural of focus).
- A key property: For any point on the ellipse, the sum of distances to the two foci is constant.
> Example: `a₁ + a₂ = b₁ + b₂`

- When you turned on “Show foci and center”:
- The Sun is located at one of the foci — not the center!
- This is a crucial insight from Kepler’s First Law.

> 🎯 Answer: “It’s right at one of the foci points.”

---

✔ 3. Experiment: Try Other Initial Conditions

You tested different starting positions and velocities.

#### A. What do you notice about the orbits?
> ✔ “They are ellipses.”

Even when you change the initial position or velocity (as long as it’s bound — not escaping), the orbit remains an ellipse. Sometimes it’s very circular (nearly a perfect circle), sometimes very elongated — but still an ellipse.

#### B. What do you notice about the position of the Sun?
> ✔ “There is always a foci point on it.”

No matter what combination you try, the Sun always sits at one focus of the ellipse. This is true for all planets in our solar system — and for any object orbiting under gravity in a bound orbit.

---

## 📜 Kepler’s First Law — Official Statement

> “The orbit of every planet is an ellipse with the Sun at one of the two foci.”

This was revolutionary in the 1600s — before Kepler, people believed orbits were perfect circles centered on the Sun (or Earth, depending on the model). Kepler’s discovery, based on Tycho Brahe’s precise observations, changed astronomy forever.

---

## 🧠 Why This Matters

Understanding orbital shapes helps us predict:

- Where a planet will be at any time
- How fast it moves (Kepler’s Second Law — faster near perihelion, slower near aphelion)
- How long it takes to orbit (Kepler’s Third Law — relates period to semi-major axis)

Also, this applies beyond planets — moons, comets, satellites, even binary stars follow similar rules!

---

## 🖋️ Final Answer Recap (As You Wrote):

1. Orbit shape: Ellipse
2. Sun’s position: At one focus
3. Experiment Observations:
- A. All orbits are ellipses
- B. Sun is always at a focus

✔ You’ve successfully demonstrated Kepler’s First Law!

---

Let me know if you want to explore Kepler’s Second or Third Law next — or dive into the math behind orbital mechanics! 🚀