Anyone who has gazed
at the starry night sky knows what marvelous and complex patterns it holds,
and why it has impressed those patterns on our imaginations since the dawn of
human memory. But in addition to the fixed patterns of the stars, there are
the movements of the "wanderers" - the planets, the Moon, and the
Sun - that presented another sort of riddle to us. How do they move, and why?
This lecture is about
that riddle. There was an ancient theory resulting from the work of Claudius
Ptolemy of Alexandria (AD 127-41) that came to be known as the Ptolemaic (or
geocentric) theory. His most important work, that contains detailed studies
of the motions of the Sun, Moon and Planets (around the Earth!) is the Almagest,
which is a treatise in thirteen books. His book was extremely influential (rivaling
Euclid's Elements as a scientific doctrine) for nearly 1400 years.
However, we imagine
today, as Copernicus did, that the planets (including the Earth) move around
the Sun - for some reason. The "fixed" stars form the background for
this dance. They are a fixed canopy against which the planetary motions can
be reckoned. Without the stars, it would be extremely difficult to trace the
tracks of the "wanderers" in the sky.
The first thing to
notice is that the planets (all but one) seem to move in a plane that contains
the Sun. That plane is called the "ecliptic." It is an imaginary
plane in the sky that we can observe, since we move in it also. The planets
move in closed curves around the Sun in that plane, and they all seem to move
in the same direction. These qualitative observations are our starting point,
as they were Kepler's. As we mentioned earlier, Kepler tried to show that these
closed curves were perfect circles with the Sun at the center. We will shortly
see why it follows from Newton's Gravitational law that each of the planets
should move in a plane that contains the Sun. Most of the planets move in the
same plane, the ecliptic, but that is a different story whose explanation belongs
to the origin and history of the Solar System.
Following is a little
more background. John Kepler (1571-1630), was the German mathematician who stated
the three laws of planetary motion that led Newton to formulate his Universal
Law of Gravitation and to invent the Calculus. He used the astronomical observations
of Tycho Brahe. As a Copernican, he rejected the view, held since the time of
Plato, that the Earth was the center of the "world". In 1596 he wrote
Mysterium Cosmographicum, which led to discussions with Galileo and Tycho
Brahe. His Astronomia Nova (1609) contained the first two of what became
Kepler's Three Laws; the third law appeared in 1619 in his Harmonice Mundi
(Harmony of the Worlds). These laws were the result of calculations based on
Brahe's accurate observations, which Kepler published in the Tabulae Rudolphinae
(1627).
Kepler's laws of
planetary motion are three mathematical statements derived from observation,
essentially from Tycho's records, since John Kepler did not have the constitution
for nocturnal observation. While he sought initially simply to confirm the Copernican
view that planets moved in circular orbits around the Sun, he was astonished
(and dismayed) to discover that the planetary paths were not circles, but were
ellipses.
As we discussed in
the Conic Sections part of the Polar Coordinates Section of Satellite
Orbits, an ellipse is a type of "conic section." These curves
were described by the Greek Geometer Apollonius.
Kepler's laws describe
the revolutions of the planets around the sun. The first law states that the
shape of each planet's orbit is an ellipse with the sun at one focus.
The
second law states that if an imaginary line is drawn from the sun to the planet,
the line will sweep out equal areas in space in equal periods of time for all
points in the orbit.
The third law states
that the ratio of the cube of the semimajor axis of the ellipse (i.e., the average
distance of the planet from the sun) to the square of the planet's period (the
time it needs to complete one revolution around the sun) is the same for all
the planets.
As we shall later
see, Newton gave a physical explanation for Kepler's laws based upon both his
laws of motion and on his law of Universal Gravitation. In fact, Newton invented
the calculus as a mathematical system of ideas that could express his explanation.
Galileo, who was
roughly a contemporary of Kepler, began at about the same time a systematic
study of what he called "Natural motions". By a process of abstraction,
he discovered some surprising facts, and we surveyed a few of them in the Pre-calculus
Introduction. When a body falls freely, or when a body is thrown upward
with a given velocity, it moves in such a way that, in equal intervals of
time, there are equal changes in downward component of velocity. Now it
was believed that freely falling bodies were attracted more strongly to Earth,
the heavier they were.
This observation
of Galileo was therefore surprising, and it required two abstractions: one had
to ignore the effects of air resistance to see that it was true. Furthermore,
if gravitational effects were themselves ignored, then another law, Galileo's
law of inertia, was seen to hold. This law stated that, in the absence of
outside influences (like gravity), a body in motion executed a natural motion,
that is: in equal intervals of time, it moved equal distances in any direction.
This was, outside
the domain of methodology, perhaps Galileo's principal contribution to the study
of gravity: the observation that gravity acts directly on the upward velocity
of a moving object, and not, as his predecessors thought, on the position.
It is this concept
that enabled Newton finally to formulate his laws of motion. Galileo saw in
these two laws an analogy. Natural motion, either freely falling, ignoring air
resistance, or motion uninfluenced by the effects of gravity and ignoring air
resistance, was natural or equable. In equal intervals of time, there were,
in the first case, equal changes in downward velocity; there were in the second
case, equal changes in position in any direction.
Why Galileo did not
go farther, is a long and interesting story. He believed in the synthetic mathematics
of the Greeks, and so he chose not to use algebra (in its modern form) in any
of his demonstrations. This rendered his arguments clumsy by modern standards,
but logically impeccable and beautiful by any standards. Isaac Newton, who published
only decades later, did not have these qualms.
Newton used algebra
freely, and extended it in a direction that no doubt Galileo would have felt,
at first sight, was outrageous. He developed the Calculus to articulate the
conceptual scheme that eventually lent coherence and order to Kepler's and Galileo's
seemingly different observations. We should not exclude from this episode in
the history of mathematics the independent discovery of the Calculus by Gottfried
Leibniz, albeit by a different route. For students who have difficulty with
calculus, this may be useful. In spite of its brilliant success in science,
calculus took almost two centuries for Western Europe to assimilate.
Galileo, one of the
giants on whose shoulders Newton stood, and who adumbrated the birth of calculus,
would almost certainly have rejected it. We say almost, because Galileo
was a scientist as well as mathematician. He would have seen its merit from
the vantage point of science. And so we must not speculate how the struggle
between the need for a pure mathematics and the desire for clear scientific
principles would have played out in Galileo's mind. Blaise Pascal once penned
the perspicuous words: "The heart has its reasons that reason can never
know." And we shall never know.
In the 20th Century,
there was a similar story of heart and mind. The Galileo of that story was Albert
Einstein. The Newton were collectively: Niels Bohr, Werner Heisenberg, Paul
Dirac, Max Born and some others. And the story is about Quantum Mechanics. And
it is true that Einstein not only adumbrated Quantum Mechanics, but elevated
it to the rank of serious science with his theory of the photon. But he never
accepted the mathematical interpretation of the wave function that came to be
known as the "Copenhagen Doctrine."
What Galileo did
not do was draw the connection between the motions of the celestial orbs and
the natural motions of freely falling bodies. What he did not see, but his successor
Newton did, was that the planets were falling (just as the cannon ball falls
to Earth) around the Sun. This is the marvelous unification of ideas concerning
gravity. And whatever disputes have arisen concerning priority of ideas, we
owe to Isaac Newton this deep insight into the mechanism of Nature. And we owe
to Galileo the point of departure for this tortuous journey. Gravity acts not
on positions, but on velocities. Finally, we owe
to Einstein the understanding that gravity is essentially Inertia (the Equivalence
Principle), and may be viewed as the geometric fabric of the world, an effect
of the 'curvature' of space-time that somehow is determined (and co-determines)
the distribution of matter and energy in space-time.
We should point out
that Newton never felt that his law of gravitation was an explanation
of gravitational effects, but was only a description of them. Einstein proposed
an explanation with his "Equivalence Principle." Matter curves space-time
(Why? We do not know.) And the curvature of space-time causes the mutual acceleration
of things, because things "want" to travel along straightest possible
paths. This curvature of space-time has much in common with the Curvature of
plane curves that we studied in Curves in Art and Nature.
Now let us return
to the wanderings of the planets. Certain of the planets (Mercury and Venus)
seem to love the Sun. They only appear in the Sky in close proximity to the
Sun. We now understand, from the Copernican viewpoint, that they orbit the Sun
as we do. But if we think of the Sun as being 'down' from us, then they are
farther down than we are, they traverse smaller orbits than we do. We must look
down to see the Sun, and we must also look down to see them.
That is easy to
understand. So we dub these planets 'morning stars' or 'evening stars' because
they become clearly visible in the dawn sky or the dusk sky of evening. But
it is easy to assimilate this to a picture of the Sun and planets moving in
an enormous wheel, turning once a day. These planets of course have an apparent
motion relative to the Sun, and that had to be explained in the Ptolemaic theory
by the invention of 'epicycles' or 'wheels within wheels'.
But when we look
at Mars, we see something very strange. Unlike Mercury and Venus, Mars is not
always visible in the night sky. In fact, it is clearly visible high in the
night sky only about 
of the time, as we will see. And for some of the time when it is visible in
the night sky, it seems to be moving in the wrong direction! What can that mean?
Let us take up the question of visibility first.
Since Mars is 'up'
from us, we have to look 'up' to see it sometimes. We certainly have to look
up (away from the Sun) when we are on the same side of the Sun as Mars is. On
the other hand, when Mars is, roughly speaking, on the other side of the Sun,
then we have to look down to see it -- towards the Sun. There are times (for
example, when it is almost directly opposite us) when Mars is lost in daylight
from a Terrestrial standpoint, and simply cannot be seen in the daytime sky
at all. But most of the time, even when it is on the other side of the Sun,
there are times of day, early in the morning or late in the afternoon, when
Mars can be glimpsed close to the horizon. At those times, we can track its
progress against the canopy of stars that may also be seen. We will see this
clearly in the exploration. But the following pictures illustrate.
In the first picture,
Mars is on the other side of the Sun from Earth. This means that the angle from
Earth to Sun to Mars is larger than 90 degrees. You must look Sunward to see
it. The apparent motion of Mars in the sky in this situation is generally from
West to East as recorded over a period of weeks or months. This is because if
the planets move with this orientation, then Mars moves in the clockwise sense
with respect to the Earth. If the Earth spins on its axis each day in the clockwise
sense, then this motion on its axis causes the Sun to appear to move from East
to West (counter-clockwise) and therefore causes Mars to appear over the weeks
and months to move from West to East.
The next picture
is more interesting. Here, Earth and Mars are on the same side of the Sun. The
angle from Earth to Sun to Mars is less than 90 degrees. It is only possible
to see Mars at night in such a situation. But then, the apparent motion of Mars
in the sky is sometimes from East to West as the Earth "overtakes Mars".
This is because during the time when the angular motion of Earth is greater
than that of Mars, the latter appears to move in the counter-clockwise sense
relative to Earth and against the starry background. This is the same sense
in which the Sun appears to go around the Earth each day.
Question 1: We cannot compare the
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of the two planets a priori, but if they are roughly equal then the principle of reciprocity that we discussed in Conservation of Energy indicates that if the orbits also have roughly the same shape -- Earth and Mars have essentially circular circles, having eccentricities 0.02 and 0.09 respectively -- then Mars should be moving more slowly than the Earth. In fact, we know from observation that Mars' year is roughly 1.88 Earth years. Also, Mars' distance from the Sun is essentially 1.52 Astronomical Units, where the Earth is 1 Astronomical Unit from the Sun by definition. From the first fact alone, you can calculate Mars' angular speed. It is roughly 0.5319 times that of Earth. The Earth is sweeping out angles around the Sun faster. The second fact will enable you also to calculate Mars'
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You can also use
these facts to check Kepler's Third law. Go ahead.
End of Question
But there is more.
Since the Earth is 'overtaking' Mars for a while when we are on the same side
of the Sun, Mars will appear to move from East to West in the sky. Actually,
it will first slow down in its West to East motion and come closer to the Earth.
It will appear to stop, and then move backwards. The backward motion typically
contunues for 9 or 10 weeks.
This was very puzzling
to the ancients. You can see why Copernicus' hypothesis, which was still based
on ideas, not observation as Galileo would teach us to do later, was attractive.
But let us do the experiment for ourselves!
Exploration: A visit to the planetarium
In order to give
an idea just how intricate the nocturnal dance of the planets is, we present
here a model Solar System. Actually it is only a part of the Solar System consisting
of the Sun, and, as we move away from the Sun, the planets: Mercury, Venus,
Earth, and Mars. This Solar System shows the motions of the planets quite accurately.
It is based on Isaac
Newton's picture
of Gravitation. We will explain later in this Chapter the details of the model
we use here. For now, we only want to give an idea of how the planets move in
the sky.
First, there are
three 3-dimensional views of the Solar System. One from outside the system,
another from the Earth during the day, and the third from the Earth at night.
The Solar System view looks at the Sun and planets from outside the ecliptic
(plane of their motion). Later in this Section, we are going to justify, using
calculus, the assumption that each planet separately moves in a plane containing
the Sun. From this view, in particular, you can see the Earth and its relation
with the other planets.
The daytime view
from the Earth is taken within the ecliptic from the Earth. While Mars is in
the daytime sky (most of the time) you will see the Sun, Mercury, Venus and
Mars. The view we take there is directly towards the Sun, which is in the center
of the picture. So, during the day, we are facing the Sun at high noon. The
large yellow sphere is the Sun. The orange ball closest to it is Mercury. Next,
the white cloud-covered planet is Venus. After that, is Earth, of course, and
on the outermost orbit is the red planet, Mars. In the daytime view along the
ecliptic, Mars generally moves from right to left, from West to East.
Now when Mars leaves
the daytime sky (is on the same side of the Sun as we are), the scene changes
to the nighttime view. During the nighttime view, the Sun and inner planets
are removed from the scene and motion is tracked against the canopy of stars.
Mars will jump to its new position and begin to move again. The only object
visible in the sky then is Mars. You will see it change color from red to light
green at the beginning of its retrograde motion, which as we said, only happens
when Mars is in the nighttime sky. At the end of its retrograde motion, it will
change again to red and continue until it enters the daytime sky. At that point,
it will be joined by the Sun and inner planets.
If you check the
Sound ? checkbox before you do the Earth view or Solar System
simulation
you will hear a musical announcement when that happens. If the box is unchecked, there will be a written message saying that "Mars is retrograde (counter-clockwise)!" Of course, the Sun and inner planets will be gone from view at that time. During the retrograde period, Mars will change color to light green. Since it is continually moving, because the camera is tracking the Sun, this is the visual cue that we are overtaking it.
The view from outside
the ecliptic shows the planets moving in their slightly elliptical orbits around
the Sun. In this view, Mars will also change color from red to light green during
the retrograde motion, and you will have the messages: "Mars is retrograde
(counter-clockwise)!" and "Mars is moving clockwise again!"
as well as the musical cues if you check the Sound ? checkbox.
As we said, the view
from the Earth shows them as they would appear at high noon (from a satellite
telescope). We show them when we are watching during the day because we want
to track the movements of Mercury and Venus closely, those being our nearest
neighbors. And we cannot see them at midnight (Why?).
The Sun remains fixed
at the center of our screen in this view, because we want to abstract away the
distracting apparent motion caused by the Earth's rotation. The ancients could
do that also. Obviously the stars in the midnight sky are different from those
that are visible at dawn or dusk. They imagined that this canopy or sphere of
stars rotated once every day around the Earth. But a planet like Mars moved
among them, and they could pinpoint its relative motion with some accuracy.
That is why its retrograde motion in the midnight sky astonished them.
The Earth View of
Mars in the nighttime sky highlights an important problem that the "Ptolemaic"
astronomers had to face. The unsatisfactory resolution of this problem (with
its introduction into an otherwise clean geometric theory of "epicycles",
"deferents", and so on) paved the way for the Copernican view, a view
finally established by Kepler, Galileo, Descartes, and Newton.
Question 2: Why should we expect Mercury and Venus to be in the vicinity of the Sun when we see them in the sky? What is the "Morning Star"? What is the "Evening Star?" and how do you think these names originated? What about Mars? How long on average do you find the retrograde motion to be as you modify the initial positions of the planets?
You may place the
planets at any place on their orbits before you begin viewing them. To do that,
click on one of the sliders:
The angle with the positive x axis will be printed below it as you move it. Whatever positions you set here will be used in your simulations when you press one of the four buttons:
Each time you press
one of the buttons on the left, the simulation will begin again using the slider
angles as initial positions. The velocities are calculated according to a formula
based on Kepler's accurate laws. You may pause the simulation at any time by
pressing the "p" key.
To continue the simulation
in either view (Solar System, or Earth View) press the Continue
button next to that option. This is useful when, for example in Earth View,
you would like to watch the Earth overtaking Mars. As long as you alternate
between Pausing and the Continue buttons (also backing up if desired)
you can watch the story unfold in an understandable fashion. Do not press the
other buttons unless you want to start the simulation again. After you press
the Continue button to pause, you may move the planets around with the
sliders before continuing.
To reverse or back
up the simulation, place a minus sign before the number in the "Step size"
field. This has the effect of reversing the time. To speed it up, increase that
positive number. To slow it down, decrease it.
The time elapsed during your simulation in Earth Days is recorded in the Time in days field. In the Solar System View you may notice interesting "syzygies"or alignments of planets.
If you pay close
attention to the motion of Mars in Earth View, you will also notice peculiar
variations in its apparent speed as it moves across the sky. In fact, if you
look closely at Mars at the extremities of its trek across the sky, just before
it slows down, and backs up, you will see its apparent distance from the Earth
change! You may understand why this is so by toggling back and forth between
the Solar System view, and the Earth View, using the P key to pause, and the
appropriate Continue button to continue.
Question 3: Why does Mars appear to slow down and back up at different times of the year? Explain why (on the modern view) this should happen. The Ancients had Mars moving around the Earth on a circle attached to its orbit in order to "save the phenomena." When Mars is in the daytime sky, there will be times when it is visible around dawn, and other times of the year when it is visible at dusk. What determines this ? Why do those times vary from one year to the next. Try to convince yourself, by making measurements from the Time in days field that Mars is more often in the daytime sky than in the nighttime sky. Why should that be? Do you think the same is true of planets like Jupiter and Saturn?
Question 4: What are the lengths of the solar years of Mercury, Venus, Earth, and Mars ? To find them with some accuracy, set the angles initially to 0.
End of Question
