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# Astrophysics Calculators

## Astrophysics

Planetary motion, gravity, orbits

Mostly Kepler's laws with some graphics too!

### Kepler's 1st Law

The orbit of every planet is an ellipse with the Sun at one of the two foci. Your browser does not support HTML5 Canvas. You need a better browser
The orbit size and shape can described by $a$ and $b$, $a$ and $e$, or $b$ and $e$

 $a$ ? Semi-major axis: the largest radius from the center of the orbit to the orbital path $b$ ? Semi-minor axis: the smallest radius from the center of the orbit to the orbital path $e$ ? Eccentricity: how much the orbit deviates from a circle (e=0). Must be less than 1 for an ellipse

### Kepler's 2nd Law

A line joining a planet and the Sun sweeps out equal areas, $A$, during equal intervals of time $\Delta t$.

(not to scale)

### Kepler's 3rd Law

"The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit."

$T=\sqrt{\frac{4\pi^{2} a^{3}}{GM}}$

Where $T$ is the period, $a$ semi-major axis of the orbit, $M$ is the larger mass, and $G$ is the gravitational constant

By definition $T^2$/$a^3$ = 1 yr2 / 1 A.U.3 where A.U. is an astronomical unit = 1.5x1011m = 9.2x107miles.

### Kepler's Laws in action!

Sweeping out equal areas means that the orbiting planet moves faster when it is nearer the star.

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To put eccentricity into perspective:

• Objects with e equal to 0 have circular trajectories
• Objects with e equal to 1 have parabolic trajectories
• Objects with e greater than 1 have hyperbolic trajectories
• Earth currently has e = 0.0167 (nearly circular)
• Mercury has the highest eccentricity (e = 0.2056) of all of the planets in the Solar System
• Pluto has a higher eccentricity (e = 0.248) but it is no longer considered a planet :(

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