THE SOUTHWORTH PLANETARIUM
70 Falmouth Street      Portland, Maine 04103
(207) 780-4249      usm.maine.edu/planet
43.6667° N    70.2667° W
Founded January 1970
2022-2023: XLVI
Sunrise: 6:54 a.m.
Sunset: 4:06 p.m.
Civil twilight begins: 6:21 a.m.
Civil twilight ends: 4:41 p.m.
Sun's host constellation: Ophiuchus
Moon phase: First quarter
Moonrise: 12:45 p.m.
Moonset: 11:41 p.m.
Julian date: 2459914.21
"Cosmic terror appears as an ingredient of the earliest folklore of all
races and is crystallised in the most archaic ballads, chronicles, and
sacred writings." -H.P. Lovecraft

Johannes Kepler (1571–1630) devised what are now considered to be the three
fundamental laws of planetary motion. He discovered these laws through
painstaking analysis of Tycho Brahe’s extensive and meticulous observations
of the planet Mars. Although these three laws serve as the foundation of
planetary astronomy, they have often proven confusing. Today, we'll attempt
to explain these all-important rules despite their perceived complexity.

   1. Every planetary orbit is an ellipse with the Sun at one focus.

Prior to the development of Kepler’s first law, planets were thought to
have traveled along perfectly circular orbits. Kepler realized that these
orbits are elliptical. What’s the difference? To explain, imagine that you
and a friend are trapped within an ellipse. The image above shows it to be
slightly oval-shaped. (Ellipses can actually be circular, oval-shaped, or
shaped like a tautly stretched elastic. See first footnote.) Within the
ellipse are two dots spaced apart along a line connecting the two
diametrically opposed points along the ellipse’s horizontal center. The
dots are the ellipse’s foci and the horizontal line is the major axis. Your
friend stands on one dot and you are stationed on the other. You are now
going to play a game. You each have to walk along a straight line toward
any point on the ellipse. Once you touch the ellipse, you must then walk in
another straight line toward the other person’s dot in such a way so that
your walking distance is less than that of your friend’s. Let’s assume that
despite the deep affection you might harbor for each other, you are
nonetheless fiercely competitive and spend the entire day walking back and
forth. You both find much to your chagrin that you are always tied. The
straight line distance from each focus to any point on the ellipse and then
back to the other will always be the same.

Even if you space the dots far apart, the sum of these distances will
remain the same. The only change will be in the shape of the ellipse. It
will become more elongated. Now, you could also merge the dots so that they
occupy the same point. Now, you have a perfect circle: the distance between
this central dot and any point on the ellipse will be constant.

Kepler’s law places the Sun and one of these foci while the planet travels
along its own elliptical path. This discovery was quite revolutionary
because even the heliocentric (Sun-centered) model proposed by Nicolas
Copernicus retained Ptolemy’s circular orbits. The introduction of the
ellipse greatly improved the accuracy of the predicted planetarpositions.

2. The radius vector connecting the Sun and a planet sweeps out equal areas
in equal intervals of time.

I’ll rephrase it: the closer a planet is to the Sun, the faster it moves.
Let’s now imagine a wire connecting Earth and the Sun. For any interval of
time, the area Earth sweeps out in its orbit is equal to the area it sweeps
out at any other given time. At times, Earth is close to the Sun, at a
position called perihelion and at others it is farther from the Sun, a
position known as aphelion. No matter where Earth is in its orbit, the
amount of space its orbit sweeps out equals that swept out at any other
time provided the time intervals are equal. A planet moves fastest when at
perihelion anslowest when at aphelion.

[Note:Isaac Newton showed that Kepler’s second law was a consequence of the
conservation of angular momentum. The same law that explains how skaters
can move more quickly when they draw in their arms.]

3. The Harmonic Law. (Please bear with me.) The square of a planet’s
orbital period is proportional to the cube of its semimajor axis.

P = the amount of time a planet requires to complete one orbit around the
Sun
a = the planet’s average distance from the Sun, otherwise known as the
semi-major axis (or half the major axis we encountered when discussing
Kepler’s first law)

If we measure the planet’s period (P) in Earth years and the average
distance (a) in Astronomical Units (AU) the proportionality becomes an
equality. In other words, one doesn’t have to insert any other value into
the formula. [Astronomical unit = Earth’s mean distance from theSun,
approximately 93 million miles.]

Just by knowing the amount of time required for a planet to complete one
orbit, we can know its average distance. Let’s use Jupiter as an example:

   - Jupiter’s orbital period: 11.86 Earth years. (11.86)^2 = 140.6. The
   cube root of 140.6 equals 5.2. So, Jupiter’s average distance is 5.2 AU

[We should note that Kepler never knew the actual separation distances
between the Sun and planets. He simply knew the distances in relation to
each other. The first determination of the Astronomical Unit would not be
made until the 1760’s.]

I hope this answer proves helpful and is not a verbose disaster.

*We use eccentricity (e) to measure an ellipse’s departure from
circularity. *

   - an ellipse with e = 0 is a perfect circle
   - e = 1 is a parabola
   - e > 1 is a hyperbola

   Earth’s orbital eccentricity is 0.0167: very slight. If one could draw
   Earth’s orbit, it would appear almost circular. The orbital eccentricities
   of the other major planets save, perhaps, Mercury’s (0.205) are also low
   and would also appear circular (Venus 0.006; Mars 0.093; Jupiter 0.048;
   Saturn 0.054; Uranus 0.047; Neptune 0.008

**These are average values. Planets will move faster when they are at
perihelion (the point of least separation distance between a planet and the
Sun) than they do at aphelion (the point of greatest distance



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