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Orbits
by Dr Jamie Love
© 1997 - 2005
As in any series of lessons, we start with the simple and move to the complex. You learned about
lunar and planetary motion in previous lessons but I oversimplified some parts of the story in
order to keep the lessons on track. Here I will teach you about some of the complexities of orbits
and in doing so I will clarify some of the (cryptic?) caveats and concerns I included in previous
discussions about orbits and planetary motion.
So far we have been imagining orbits as circular. This is an honest mistake to make. Indeed, you
will often hear professional physicists use expressions like "the Moon circles the Earth" or "the
Earth circles the Sun" and from that you could be forgiven if you have gone through life thinking
that moons and planets follow circular orbits. They do not!
Circles are easy to understand and draw. There is an intuitive belief that the universe should
follow "perfection" and what could be more perfect than a simple, circular orbit? Indeed, this
steadfast belief in a "perfect, simple universe" has been a thorn in the side of science for
centuries (and continues to be a problem even with some of today's most popular theories in
cosmology. But, hey, don't get me started! )
In 1543 a Polish churchman named Mikolaj Kopernik (but known to the outside world by his
Latinized name of "Copernicus") presented the view that the Earth goes around the Sun. Of
course, we now know that he was right but his view was not well accepted at the time. The
Church condemned Kopernik, his theory and his book. The Church was not Kopernik's only
enemy. Circles were too! Kopernik clung to the belief that orbits are circular.
This man was not afraid to move the Earth from the center
of the universe but he was unwilling to accept that the
Earth and the other planets traveled around the Sun in
anything except a perfect circle.
Unfortunately, even Kopernik could not fit his wonderful
ideas to the observations. Careful measurements of the
motion of the planets through our sky did not fit the
motions you should get if planets moved in circles. So, did
Kopernik abandon the idea that planets traveled in a circle?
Well, no!
Instead he proposed that planets circled the Sun while they
also circled, well, nothing! Kopernik believed that the
planets moved in epicycles - small circles whose center
rolls around the circumference of a larger, fixed circle
whose center is the Sun.
Kopernik died believing that planets circled the Sun in complicated epicycles - circles along
circles.
(Epicycles were also used by some folks to explain retrograde motion but that's another story.)
A Danish astronomer named Tycho Brahe thought Kopernik's ideas were a lot of rubbish. Tycho
believed in the good old-fashion "Earth-centered universe" and he set out to prove that Kopernik
was wrong. Using some of the best observational equipment of the time, Brahe produced a star
catalogue of amazing accuracy and then used it to plot the motion of the planets with
unprecedented precision. His measurements proved that the planets do not circle the Sun. Indeed,
Brahe measurements proved that Kopernik's "Earth circles the Sun" idea had to be wrong! Later
in his years Brahe agreed that some of Kopernik's ideas had some merit. Maybe some of the
other planets circled the Sun but certainly not the Earth! When Brahe died in 1601 he went to his
grave with the comfortable thought that he had proven Kopernik wrong and laid to rest this
nonsense about the Earth circling the Sun.
Brahe's assistant, a German mathematician named Johannes Kepler, inherited Brahe's mountain
of data. Kepler analyzed Brahe's observations and came to the conclusion that they could be
explained if the planets orbited the Sun in ellipses! Kepler discovered that Brahe, his former
teacher and employer, was wrong. The Earth actually did orbit the Sun - not in a circle but in an
ellipse. So Kepler discovered that Kopernik was wrong too.
This was Earth shattering. We live in an "imperfect" universe! We are not at its center and
planets do not follow circular perfection in their motion. Kepler took his "ellipse idea" and
developed them into three laws which we now call "Kepler's Laws of Planetary Motion". These
laws (the first two published in 1609) made astronomy more than just an "observational science".
The motion of the planets (and later - the stars and galaxies) could be predicted and calculated
with amazing accuracy thanks to Kepler's Laws!
OK, but what's an "ellipse"?
That depends on how you want to define it.
If you're into the geometry of solids you would define an ellipse as "a conic formed by the
intersection of a plane with a right circular cone, so that the plane is inclined to the axis of that
cone at an angle in excess of half the apex angle of that cone."
Ah, what if you're into something useful?
Then an ellipse is a squashed circle.
The line along its longest diameter is called the major axis and
the line along its shortest dimension is called the minor axis.
They meet at the center and are perpendicular to each other.
Off course a circle doesn't have a major or minor axis (or, if it
does, then they are both equal to each other).
But what really distinguishes an ellipse from a circle, and makes it important, are the foci (or
focus, when discussing just one of them). Both foci are positioned in two special places on the
major axis such that for any point along the ellipse the sum (total) of the distances from that
point to both foci is constant (the same).
Huh? What does that mean?
Hmm, that is a complicated way of saying what I am trying to teach. An image is worth a
thousand words, so lets try drawing it.
Using this diagram as an example, I have marked the two foci as
dots along the major axis and I also have two different points
marked on the ellipse. The lengths of the lines #1 and #2 when
added together give the same total length as when the lengths of
lines #3 and #4 are added together.
Or, to remind you what I just said, for any point along the ellipse
the sum total of the distances from that point to both foci is the
same.
Another interesting feature of an ellipse is that the two foci are always an equal distance from the
center of the ellipse and they are also equal distances from the ends of the major axis. [If you
know some geometry, you'll see that I am repeating the obvious. If you don't understand
geometry, you might want to have a good look at this diagram and try to become comfortable
with it.]
If you imagine this ellipse as a flexible loop you
can imagine that as you squeeze along the major
axis, the two foci will approach each other and the
minor axis will become larger. If you continue to
do that you will "unsquash" the ellipse creating a
circle where the two foci are actually at the center
and the two axes are equal. All this "squashing" of
the loop doesn't change its circumference ("circular
length"), but it causes important changes in the foci
and axes. There are various ways to define these
"squash values" but "eccentricity" is the best one.
We can define eccentricity as the amount of
"circular squashness". [Mathematicians are now
writing me nasty emails insisting that I explain
conics or at least provide a more "rigorous"
definition of eccentricity based upon the axes and
foci. Please understand that I am trying to give my
students a user friendly feel for the ellipse without
the heavy equations.]
A perfect circle has no eccentricity at all. If we were to squash a circle (say by squeezing it from
above and below) we would start to great an ellipse and the more we squashed it the more
elliptical it would become. An ellipse has an eccentricity (abbreviated "e") between zero (0) and
one (1). Only a slight amount of squashing produces only a slightly elliptical shape and it has
low eccentricity. More squashing and we produce a more elliptical shape with more eccentricity.
Yeah, OK. So what does this have to do with orbits? Is Kepler's discovery all that important?
Oh, yes! The motion of planets can be described correctly by knowing that they travel in an
ellipse. We don't have to resort to silly epicycles to explain planetary motion because ellipse
("squashed circles") to the job so well.
Kepler's first law is that planets move in ellipses with the Sun at one focus. The other focus is
empty.
[That's right. The other focus is empty. Neither Jupiter nor any other object is in the other focus.]
That means that every planet has a point in its orbit
when it is closest to the Sun and another point in its
orbit when it is furthest from the Sun.
Once each orbital period, a planet reaches aphelion, a
point in its orbit when it is at it furthest from the Sun
("ap" means "far" and "helion" refers to the Sun). Half
an orbital period later (half that planet's "year") we
find that planet at its closest approach to the Sun - a
position in its orbit called perihelion.
By the way, the aphelion and perihelion have NOTHING to do with the seasons. (It's the tilt of
the Earth that causes seasons and we will come back to that in a few months.) Indeed, folks
living in the Northern Hemisphere are often amazed to learn that we are furthest from the Sun
around the 4th of July and closest around the 4th of January. The orbit of the Earth is not terribly
elliptical - it's just a little bit elliptical. The eccentricity of the Earth's orbit is about 0.017. That
means the differences between the Earth's perihelion and aphelion is not that great.
Here are some numbers to think about.
The Earth's average distance to the Sun is 149.6 million kilometers and astronomers like to call
that distance an astronomical unit, or simply an AU. The AU is commonly used when
describing distances in our Solar System so it's a good idea to become familiar with it. [People
who talk about stars and galaxies often use light-years or parsecs but people who talk about
planets, comets or anything around our star or any other star, commonly use AUs. Oh, by the
way, we'll come back to the definition of "parsecs" in a few months when I teach some of the
techniques used to measure distances in space.]
Anyway, around the 4th of July the Earth will be 152 million kilometers from the Sun. That's
about 1.016 AUs. (I divided the perihelion distance of 152 million by the average distance of
149.6 million to get 1.016 AUs.) In six months time the Earth will be at the opposite end of its
orbit and closest to the Sun. At perihelion the Earth is just 147 million kilometers from the Sun or
0.983 AUs. That means the difference between perihelion and aphelion is only 0.033 AUs and
that small difference does not greatly affect our seasons.
The amount of "ellipsity" of an orbit, called its eccentricity (right?), varies from planet to planet.
Here's a list of each planet's eccentricity (with its maximum and minimum distances from the
Sun in parenthesis). Use this list to get an idea of how the eccentricity affects the orbit's shape.
Mercury = 0.206 (aphelion = 70 millions of kilometers & perihelion = 46 millions of kilometers)
Venus = 0.007 (aphelion = 109 millions of kilometers & perihelion = 107 millions of kilometers)
Earth = 0.017 (aphelion = 152 millions of kilometers & perihelion = 147 millions of kilometers)
Mars = 0.093 (aphelion = 249 millions of kilometers & perihelion = 207 millions of kilometers)
Jupiter = 0.048 (aphelion = 816 millions of kilometers & perihelion = 741 millions of
kilometers)
Saturn = 0.056 (aphelion = 1507 millions of kilometers & perihelion = 1347 millions of
kilometers)
Uranus = 0.047 (aphelion = 3004 millions of kilometers & perihelion = 2735 millions of
kilometers)
Neptune = 0.009 (aphelion = 4537 millions of kilometers & perihelion = 4456 millions of
kilometers)
Pluto = 0.248 (aphelion = 7375 millions of kilometers & perihelion = 4425 millions of
kilometers)
Pluto's orbital eccentricity is so great that it actually crosses the orbit of Neptune! In 1989 Pluto
reached its most recent perihelion and it is now swinging outward from the Sun. In 1999 it
crossed back over Neptune's orbit to again become the most distant (known) planet in our Solar
System. It takes 248 years for Pluto to complete its highly eccentric orbit around the Sun. Since
Pluto's last perihelion was in 1989, we can calculate that it will reach its next aphelion in the year
2113. (That's 124 years, half the time it takes Pluto to complete one orbit, after perihelion.)
As you can see, planets travel in ellipse, not circles. That's Kepler's first law and it is a very
important one to understand because it explains a lot of observations and helps us understand
why planets have aphelions and perihelions.
Is Kepler's second law important?
Oh, yes!
Kepler's second law explains how quickly a planet moves in its orbit at any particular time.
If planets had circular orbits, as Kopernik insisted, they should move at a constant rate around
the Sun. However, planets don't move at a constant rate around the Sun. Observations proved that
(and that is why Kopernik resorted to epicycles).
Because planets have elliptical orbits, there are times when they are closer (perihelion) and other
times when they are further (aphelion) from the Sun's strong gravity. This affects the planet's
speed. Kepler didn't know about Newton or his laws about gravity (they were 80 years in the
future) but Kepler had Brahe's data and a good enough understanding about ellipses to put
together his second law.
Kepler's second law is that the radius vector (the line joining the planet to the Sun) sweeps out
equal areas in equal times. That means a planet moves fastest around perihelion and slowest
around aphelion.
In this diagram I have shown the areas produced by a planet over
three different, but equal, time periods. (Imagine each of these
three samples to be the change in position of a planet after, say,
one week.) Near aphelion the distance between the two positions
is very small, but they are far from the Sun so a long, thin triangle
is produced. At perihelion the planet moves much further along in
the same amount of time but it is closer to the Sun so it produces
a short, fat triangle. I've also included a third time period as a
kind of average.
The areas of these three triangles are the same. (You might not see that as obvious but it can be
proven to be true using the geometry of ellipses.)
Kepler's second law helps us to predict the motion of planets. Because planets change speed
during the course of their orbits, they do not follow simple models of motion. We have to use
more complicated methods that take into account the eccentricity of the planet's orbit and its
exact position along that orbit. Objects with highly eccentric orbits, like comets and some manmade satellites, are strongly influenced by Kepler's second law.
You may recall from your lesson on planetary motion that the synodic period (the time between
successive returns to the same position relative to the Sun as seen from Earth) of Mercury was
supposed to be 116 days. That number is actually an average synodic period that takes into
account the average speed of the Earth and Mercury. Now that you understand Kepler's second
law you will also understand that these kinds of calculations are trickier than we first assumed.
Mercury's synodic period is around 116 days but it varies depending upon the specific position,
and therefore velocity, of both the Earth and Mercury. The same complication is true of all the
planets but it's most obvious with Mercury because it has such a large eccentricity (e = 0.2060).
(Note that the time it takes to COMPLETE its orbit is not changed. The planet just moves more
quickly but then more slowly so it still completes its orbit in the same amount of time. Its
sidereal period is not changed.]
Around January 4th the Earth is moving at its fastest velocity but around July 4th it is moving at
its slowest. Similarly, in 1989 Pluto was at its maximum speed (during its perihelion) and it is
now slowing down as it moves outwards from the Sun.
[Insects move faster when they are warmer and maybe that is why planets do too. WRONG!
Planets change speed as they move in their elliptical orbits, due to the geometry discovered by
Kepler and with Newton's laws of gravity. Temperature has nothing to do with it (but it's a
convenient way to remember it ).]
I bet Kepler's third law is about ellipses and it very important too.
Well, actually his third law has almost nothing to do with ellipses. Indeed, it uses a planet's
average orbit speed to predict its relative position so it would have been possible to understand
Kepler's third law without ever seeing an ellipse. But you're right - it's very important.
Kepler's third law states that the square of the time taken to complete an orbit is proportional to
the cube of its average distance from the Sun.
Huh? What good is this? What are you talking about!?
Yes, it is a little complicated so let me explain. If we measure distances in AUs and periods in
Earth years we have common units of measurements and the math is a little easier for
comparisons. The Sun-Earth distance is 1 AU (by definition) and of course, we have to consider
the planet's sidereal period in the calculations.
So Kepler's third law can be restated as the planet's sidereal period2 = orbital radius3.
Let's use an example.
Mars has an orbit that is 1.52369 AUs and takes 1.88081 years to circle the Sun.
Square the period = 1.880812 = 3.537446256
Cube the distance = 1.52369 3 = 3.53744626
As you can see, the numbers come out the same, 3.53744626.
So what? How is Kepler's third law of any use?
Kepler used his third law to turn observations of orbital periods into calculations of orbital
radius. Here's how.
Consider a planet that had a period of 8 years. 82 = 64. The radius of that planet's orbit, when
cubed, must give a value of 64. We must find the cube root! A cube root is a difficult number to
calculate and most calculators do not even have that function but a good mathematician, like
Kepler, can work it out. The cube root of 64 is 4. (Check that. 43 = 4 x 4 x 4 =64.) So this planet,
with a sidereal period of 8 years, must have an orbit of 4 AUs.
Let's use a real example (but the numbers get harder). Saturn takes 29.4566 (Earth) years to
complete its orbit.
The square of 29.4566 (29.45662) = 867.69128356. Kepler knew that the cube of Saturn's orbital
radius must be a number which, when cubed, equals 867.69128356. In other words, the cube root
of 867.69128356 equals the distance (in AUs) of Saturn from the Sun! The cube root of
867.69128356 is 9.53884 so Saturn must be 9.53884 AUs from the Sun.
Kepler's third law was a kind of "by the way" thingy. Kepler just saw the relationship and told
the world about it. It has nothing to do with ellipses, and the calculations are easiest to do by
assuming a circular orbit anyway!
Kepler was a great mathematician and able to see relationships that others had missed. His
understanding of geometry allowed him to see things Brahe didn't bother to see and things
Kopernik refused to see but Kepler really didn't know why the planets moved the way they do. It
wasn't until Isaac Newton published Principia, 80 years later, that the laws of Kepler started to
make sense. However, even without Newton's help, Kepler was confident that he had found a
better way to describe the Solar System and his ideas changed the way we imagine the universe.
So, now I know all there is to know about orbits! Right?
Well, we haven't gone into the vigorous math that would be useful because we have been trying
to keep it simple. However there is one extra complication you should know. We've been
assuming that planets orbit the Sun in the same flat plane. They don't!
The Earth's orbital plane is defined as having an inclination of zero degrees (0o) and this plane is
referred to as the ecliptic plane or simply the ecliptic. In fact, it's the apparent path of the Sun
through our sky that defines the ecliptic but the apparent path of the Sun is really the effect of the
Earth's path around the Sun. (We just look at it the other way around.) Each planet's orbital
inclination is a comparison of the orbital plane of the planet to the orbital plane of the Earth (the
ecliptic).
Mars'
orbital
plane is at
an
inclinatio
n of 1.8o
with
respect to
the Earth's
orbital
plane.
This
image
illustrates
that angle.
Notice that I have drawn the Earth's plane perfectly horizontal. Mars' orbital plane is "off" the
ecliptic by 1.8o.
(The angle and relative sizes of the orbits are drawn correctly but the size of the Sun and planets
are not to scale.)
Venus'
orbital
plane is at
an
inclinatio
n of 3.4o
so it is
even
farther
from the
ecliptic.
Jupiter's orbital plane is closer to the ecliptic - a mere 1.3o.
Saturn, on the other hand, has an orbit whose plane is at a steeper angle to that of the Earth's -
around 2.5o.
Uranus' orbital plane is only 0.8o, the smallest inclination of any planet. There's no particular
reason for Uranus and Earth to have very similar orbital planes. It just happened that way.
Neptune has an inclination of 1.8o, the same as Mars, but you would be wrong to think that Mars
and Jupiter orbit the Sun in the same plane. You can make an infinite number of planes that are
the same angle to the ecliptic but they won't all be positioned in the same plane. You have to
think (imagine) in three dimensions to understand that. My artistic ability is limited to just two
dimensions. Sorry.
Interestingly, the planets that orbit at the extremes of our Solar System, Mercury and Pluto, also
have the most extreme inclinations and eccentricities.
Pluto's orbital plane is inclined 17.15o from the ecliptic and, you'll recall, Pluto has a very
eccentric orbit too. Some astronomers feel that Pluto's extreme orbital properties, along with its
apparent composition, are evidence that it was not formed like a normal planet. They believe
Pluto is either a lost moon from one of the giant planets or a captured "planet" from beyond the
normal neighborhood.
Mercury's orbit is inclined 7o from the ecliptic and that causes it to be more variable in its
position in our sky than any other visible planet. Indeed, its visibility depends a great deal on
whether you are in the Northern or Southern Hemisphere when Mercury is at its maximum
distance north or south of the ecliptic. For example, if Mercury is at its greatest elongation (and
thus best for viewing because it is far from the Sun) while at its most northern point in its
inclined orbit, observers in the Northern Hemisphere will get a much better view than observers
in the Southern Hemisphere. This is especially true if it is winter in the Northern Hemisphere,
when the Northerners have more nighttime in order to see Mercury.
Clearly a planet's inclination is an important consideration when trying to figure out its position.
This lesson has shown you that planets do not travel in simple circles nor in the same plane.
It is important to appreciate the complexity of elliptical orbits because they are the reason that
planets do not always move at constant speed or at constant distances. Kepler's first and second
laws showed us the true motion of the planets and his third law was an important observation
upon which Newton could expand. All planets, indeed all objects in an orbit, move in ellipses
and follow Kepler's laws. That causes certain events to be different and difficult to understand
(easily) without resorting to the complicated math of elliptical geometry.
Sometimes you hear the statement that "such-and-such-a-planet will be closer to the Earth this
week than it will be for the next 12 years"! That's because the aphelion of one planet and the
perihelion of the other have lined up in a particularly close distance during that opposition.
I told you in a previous lesson that a planet is closest to the Earth at opposition, but I was
oversimplifying the situation. The closet approach occurs around opposition but the specific date
depends on the specifics of the two orbits.
The elliptical shapes of orbits force us to leave behind the simple situations and it forces us to
use more complicated equations to understand planet motion. We will not go into those
equations but you should be aware that they are there and underlie the calculations.
You should also understand that a planet's orbital inclination affects where it will be in the sky. If
we lived in a solar system with all the planets in the same plane we would discover that they all
have the same declinations and they would occult each other during each orbit.
Orbital inclinations and eccentricities add more complexity to our predictions and understanding.
They are particularly important for understanding multiple star systems, occulations and eclipse.
In the next few months I'll teach you about those things, so be sure you understand this lesson
before we move on.