The Ellipse in Space
Have you ever heard of Halley’s Comet?
If not then let me tell you a brief history of the comet. Halley's [HAL-lee]
Comet has been known since at least 240 BC and possibly since 1059 BC. Its most
famous appearance was in 1066 AD when it was seen right before the Battle of
Hastings. It was named after Edmund Halley, who
calculated its orbit. He determined that the comets seen in 1531 and 1607 were
the same object that followed a 76-year orbit. Unfortunately, Halley died in
1742, never living to see his prediction come true when the comet returned on
Christmas Eve 1758. The last sighting was in 1986 and the next sighting will be
in 2062, which poses the question why does it take 76 years to orbit Earth and
why can’t we see it every day like we see the moon? The answer is long and many
books have been written on what I am about to explain in only short detail, but
the answer, believe it or not, lies within one of the conic sections.
The orbits of
the planets are ellipses but the eccentricities (the amount of
"flattening" of the ellipse) are so small for most of the planets
that they look circular at first glance. For most of the planets one must
measure the geometry carefully to determine that they are not circles, but
ellipses of small eccentricity. Pluto and Mercury are exceptions: their orbits
are sufficiently eccentric that they can be seen by inspection to not be
circles.
It mostly
begins with Johannes Kepler and his three laws. For more inspection into these
laws go here,
otherwise stay with me and I’ll give you the short version. Kepler’s first law
states that the orbits of the planets are ellipses, with the Sun at one focus
of the ellipse. As a side note to all of this, scientists call an equation a
"Law" when they discover something that they see as fundamental to
the way the entire universe works and that has never been contradicted by other
empirical evidence that can't be explained. The whole strength behind
Kepler's First Law was that it allowed for the future positions of the planets
to be predicted precisely far into the future, something that had never been
possible before with the previously proposed complicated schemes. The
elliptical orbit idea was so simple while neatly explaining and predicting
planetary motion extremely well that it was seen as a fundamental revolution in
scientific thinking. His laws explained the facts of how scientists knew
the universe behaved. Kepler's First Law illustrated that the Sun is not at the
center of the ellipse, but is instead at one focus (generally there is nothing
at the other focus of the ellipse). The planet then follows the ellipse in its
orbit, which means that the Earth-Sun distance is constantly changing as the
planet goes around its orbit.
Kepler's second law is illustrated in
the preceding figure. The line joining the Sun and planet sweeps out equal
areas in equal times, so the planet moves faster when it is nearer the Sun.
Thus, a planet executes elliptical motion with constantly changing angular
speed as it moves about its orbit. The point of nearest approach of the planet
to the Sun is termed perigee;
the point of greatest separation is termed apogee. Hence, by Kepler's second
law, the planet moves fastest when it is near perigee and slowest when it is
near apogee.
Kepler's Third Law implies that the period
for a planet to orbit the Sun increases rapidly with the radius of its orbit.
Thus, we find that Mercury, the innermost planet, takes only 88 days to orbit
the Sun but the outermost planet (Pluto) requires 248 years to do the same.
Calculations Using Kepler's
Third Law
A convenient unit of measurement for periods is in Earth
years, and a convenient unit of measurement for distances is the average
separation of the Earth from the Sun, which is termed an astronomical unit and
is abbreviated as AU. If these units are used in Kepler's 3rd Law, the denominators
in the preceding equation are numerically equal to unity and it may be written
in the simple form
P (years)2 = R (A.U.s)3
This equation may then be
solved for the period P of the planet, given the length of the semi-major axis,
P (years) = R (A.U.)3/2
or for the length of the
semi-major axis, given the period of the planet,
R (A.U.) = P (Years) 2/3
So based on
Kepler’s three laws we now know that the planets move around the solar system
in elliptical patterns, they move a various speed during a full cycle, and we
can calculate how long it takes a planet to rotate around the sun (in Earth
days or years) based on how far it is from the sun. This is a very rough
estimation and in fact the real equation involves several factors such as the
length of the apogee and perigee combined with what phase the planet is in with
the orbit. But for our investigation it’s a nice estimation just to know.
To end this
long but interesting discussion here’s an Excel spreadsheet to give you
different planetary orbit Durations. Enjoy.
Note: An
astronomical unit AU = 149,600,000
km
92,960,000
miles
490,800,000,000
feet
8.317 light minutes
To give you an idea of what a
light year is check this out: 1 ly (light-year)
= 63,241 AU. That’s fast.