Kepler's Laws: Quick Reference

Kepler's 1st Law - Planets orbit the Sun in paths shaped like ellipses, which are like stretched-out circles. - The Sun isn't exactly at the center; instead, it's at one of two special points inside the ellipse called foci. The other focus is just empty space. - Think of "eccentricity" as how "squashed" or "stretched" an orbit is. A perfectly circular orbit has an eccentricity of 0, while orbits closer to 1 are very elongated.

Kepler's 2nd Law - Equal Areas in Equal Times: Imagine a line connecting a planet to the Sun. As the planet moves, this line sweeps out an area, like a slice of pie. Kepler's 2nd Law says that in the same amount of time, the planet will sweep out the exact same amount of area, no matter where it is in its orbit. - What this means: Planets move faster when they are closer to the Sun (this closest point is called perihelion) and slower when they are farther away (this farthest point is called aphelion). This way, the "pie slices" always have the same area for the same time.

Kepler's 3rd Law - This law connects how long a planet takes to go around the Sun (its orbital period) to how far away it is from the Sun (its average distance). - Simply put: Planets that orbit closer to the Sun have shorter periods (they finish a revolution faster) and generally move faster. Planets that orbit farther away have longer periods and move slower. - The math behind it: The square of a planet's orbital period ($T$) is directly proportional to the cube of its average distance from the Sun ($a$).

$T^2 imes a^3$

• We can also say that for any planet in our solar system, this relationship is constant:
  $\frac{T^2}{a^3} = \text{constant}$

• For example, Mercury, being closest, is the fastest planet (around $47.9\ \text{km/s}$), while Neptune, the farthest, is the slowest (around $5.5\ \text{km/s}$).

Eccentricity (quick reminder) - Eccentricity is just a number that tells you how much an orbit deviates from a perfect circle. The closer it is to 0, the more circular; the closer it is to 1, the more stretched.

Quick Practice & Takeaways - Q: Does Earth travel faster during US winter or US summer? A: Earth travels faster during US winter (around January), because this is when Earth is closest to the Sun (perihelion), as explained by Kepler's 2nd Law. - Q: Does Earth sweep out a larger "pie slice" of area in July or October, assuming the same time interval? A: According to Kepler's 2nd Law, Earth sweeps out the same area in any equal time interval, regardless of whether it's July or October. However, because it moves faster in October (closer to perihelion), that "slice" will be shorter and wider, while in July it will be longer and skinnier.