All about orbits!

What does the Sun's "motion" look like from Earth?


If you follow the "motion" of the Sun throughout the year:

The analemma - the "apparent" path of the Sun around the Earth




So, it was believed that the Sun, stars, and planets ALL revolved around the Earth?  But there was a hitch:

How did they explain the peculiar motion of Mars, etc. - where it seemed to go "backwards" (retrograde) every few months?  The epicycle model - the OLD (and very wrong) way that orbits were conceived:


This wrong theory lasted for nearly 2000 years - until....


Johannes Kepler, 1571-1630



Kepler's laws of planetary motion


Note that these laws apply equally well to all orbiting bodies (moons, satellites, comets, etc.)

1. Planets take elliptical orbits, with the Sun at one focus. (If we were talking about satellites, the central gravitating body, such as the Earth, would be at one focus.) Nothing is at the other focus. Recall that a circle is the special case of the ellipse, wherein the two focal points are coincident. Some bodies, such as the Moon, take nearly circular orbits - that is, the eccentricity is very small.



2. The Area Law. Planets "sweep out" equal areas in equal times. See the applets for pictorial clarification. This means that in any 30 day period, a planet will sweep out a sector of space - the area of this sector is the same, regardless of the 30 day period. A major result of this is that the planet travels fastest when near the Sun.




3. The Harmonic Law. Consider the semi-major axis of a planet's orbit around the Sun - that's half the longest diameter of its orbit. This distance (a) is proportional to the period of time (P) to go around the Sun in a very peculiar fashion:

a^3 = P^2

That is to say, the semi-major axis CUBED (to the third power) is equal to the period (time) SQUARED. This assumes that we choose convenient units:

- the unit of a is the Astronomical Unit (AU), equal to the semi-major axis of Earth's orbit (approximately the average distance between Earth and Sun). This is around 150 million km or around 93 million miles

- the unit of time is the (Earth) year

The image below sums this up nicely (I hope):





Example problem:  Consider an asteroid with a semi-major axis of orbit (a) of 4 AU. We can quickly calculate that its period (P) of orbit is 8 years (since 4 cubed equals 8 squared).

Likewise for Pluto: a = 40 AU.  P works out to be around 250 years.

Note that for the equation to be an equality, the units MUST be AU and Earth years.


Cool:



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