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Astrophysics Calculators

Astrophysics

Planetary motion, gravity, orbits

Mostly Kepler's laws with some graphics too!

milkyway

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 $e$

? Semi-major axis: the largest radius from the center of the orbit to the orbital path
? Semi-minor axis: the smallest radius from the center of the orbit to the orbital path
? 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 $\Delta t$. kepler's 2nd law

(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.

? Eccentricity is a measure of how far an orbit deviates from a circle and must be between 0 (circle) and < 1


? Semi-major axis is the longest radius of the orbit and along with eccentricity it determines size of orbit. It is one of two parameters determining orbital period T (Kepler's 3rd law).


? Mass of the orbited object, one of two parameters determining orbital period T (Kepler's 3rd law)



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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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