Kepler's Laws — Orbit Shape and Ellipses (Quick Notes)

Part 1: The Shape of an Orbit

  • Background: Kepler refined Copernicus' heliocentric model with three laws of planetary motion.

  • Objective: Test if orbits are circular or elliptical via simulation.

  • Key terms:

    • Semi-major axis: a

    • Semi-minor axis: b

    • Sun-to-center distance (focus offset): c

  • Circular Orbit:

    • Eccentricity e = 0.0

    • Example: a = 2\text{ AU},\quad b = 2\text{ AU},\quad c = 0\text{ AU}

    • Rule: For a perfect circle, a = b and c = 0.

  • Observations: Circular orbits match Copernicus's predictions. The model helps identify real-world data points for circularity.

Part 2: Making Rules for Elliptical Orbits

  • Concept: Initial velocity (magnitude and direction) determines the elliptical shape of an orbit.

  • Observations:

    • Small initial velocity $\to$ near-circular orbit (low eccentricity).

    • Larger initial velocity $\to$ more elongated ellipses.

    • "Predict Orbit" tool helps forecast orbit shape.

  • Kepler’s 2nd Law (Area Law):

    • Pink/pie-slice lines show swept areas over time.

    • Higher eccentricity: lines bunch closer to the Sun (faster motion near perihelion) and spread farther away (slower motion near aphelion).

    • Takeaway: Line density indicates speed; closer to Sun means faster, farther means slower.

Part 3: Comparing Orbits of Different Sizes

  • Setup: Two planets (e = 0) orbiting a star at different radii.

  • Distance: Closer planet travels a shorter distance (2\pi r) per orbit.

  • Example: If distance ratio is $\approx 2$, the closer red planet completes about 2 orbits for every 1 orbit of the farther blue planet.

  • General Rule: The closer an object is to the star, the faster its orbital speed (for a given central mass). The mass of the orbiting body does not affect orbital speed.

  • Key takeaway: Closer to star $\Rightarrow$ faster orbital speed; farther $\Rightarrow$ slower.

Mars and Other Bodies

  • Mars Orbit:

    • Eccentricity: e \approx 0.093

    • Sun is not at the center (asymmetry).

    • Semi-major axis (a) $\neq$ semi-minor axis (b).

    • Conclusion: Mars' orbit is an ellipse.

  • Other bodies (e.g., Pluto, Uranus): Their orbits are also ellipses.

  • Observations for a circle vs. ellipse:

    • Circle: a = b and c = 0

    • Ellipse: Either a \neq b or c \neq 0

    • Example (e=0.4): a = 2\text{ AU},\quad b = 1.83\text{ AU},\quad c = 0.8\text{ AU} (where c^2 = a^2 - b^2 is consistent).

  • Quick Rule: An orbit is an ellipse if a \neq b or c \neq 0.