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<!DOCTYPE html>
<html lang="en" xmlns="http://www.w3.org/1999/xhtml">
<head>
<title>Surveying the Solar System</title>
<meta name="description"
content="How Kepler surveyed the solar system.">
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<script>MathJax = {tex: { inlineMath: [["$", "$"], ["\\(", "\\)"]] }};
</script>
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id="MathJax-script">
</script>
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box-sizing: border-box;
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<body>
<div class="sidenav">
<a href="#topdiv">Top</a>
<a href="#planet-clock">Planet Clock</a>
<a href="#orbit-view">Orbit View</a>
<a href="#earth-period">Earth Period</a>
<a href="#mars-period">Mars Period</a>
<a href="#survey-orbits">Survey Orbits</a>
<a href="#mars-inclination">Mars Inclination</a>
<a href="#first-two-laws">First Two Laws</a>
<a href="#third-law">Third Law</a>
</div>
<div id="topdiv" class="textcolumn">
<h1 style="text-align: center;">
<a href="index.html">Surveying the Solar System</a></h1>
<section id="intro">
<h2>Introduction</h2>
<p>We remember Johannes Kepler for his three laws of planetary
motion, but those were just his encore. The really dazzling show
was how Kepler applied surveying methods to astronomical data, for
the first time calculating the relative distances to the planets.
Astronomers had been studying essentially the same data for at
least two millennia without figuring out how to compute the orbits
of the planets in three dimensions. Little more than a half
century after Kepler, his laws of planetary motion became the
cornerstone for Newton's theory of universal gravitation, which
launched modern physics. The Copernican revolution in astronomy,
as completed by Kepler's laws, thereby became the seed for the
subsequent industrial and scientific revolutions that have
completely rebuilt human society.</p>
<p>Specifically, Kepler analyzed the naked-eye observations that
Tycho Brahe made a few decades before the invention of the
telescope. Tycho's data consist of measurements of the positions
of the Sun and planets relative to the fixed stars. His
measurements cover about twenty years with an accuracy of about
two minutes of arc - that is about a fifteenth the diameter of the
Sun or moon, or a half millimeter at a distance of a meter. You
cannot do any better without a telescope.</p>
<p>In order to walk through what Kepler did step by step, you
need to open a second browser tab containing an ephemeris
of planetary data and a series of graphical calculation tools.
That tab has very little explanation of how to use the tools there,
so you will want to switch back and forth between the two tabs,
reading the explanations here, and actually performing the
calculations there. Open the second tab now by
<a href="planetclock.html" target="_blank">clicking this link</a>,
then return to this tab for more instructions.</p>
<p>The sections in this tab (see the left sidebar) correspond to the
main menu items (under the drop down button on the right side) of
the second tab. Each section represents a step that Kepler took
to understand the motions of the planets.</p>
</section>
<section id="planet-clock">
<h2>Planet Clock</h2>
<section id="coords">
<h3>Ecliptic Coordinates</h3>
<p>Before you can start, you need to understand one of the
coordinate systems astronomers use to locate the Sun and planets
in space. J2000 ecliptic coordinates consist of a latitude and
longitude like the coordinates used for points on the surface of
the Earth. The difference is that ecliptic latitude is the angle
from the plane of the Earth's orbit around the Sun instead of the
Earth's equator. Ecliptic longitude is the angle around the
perpendicular to Earth's orbital plane instead of Earth's spin
axis. The ecliptic longitude of the Sun tells which constellation
of the Zodiac it is in - each sign represents 30 degrees of
ecliptic longitude; people have been using ecliptic coordinates
since ancient times. (Conventionally, Aries is 0 to 30 degrees
ecliptic longitude, Taurus 30 to 60, and so on.)</p>
<p>The axis of the Earth precesses slowly relative to the fixed
constellations of the stars, about one degree every 72 years, and
the plane of the Earth's orbit is also changing, though even more
slowly. The "J2000" designation means to use ecliptic coordinates
as the Earth's spin and orbital planes were at noon universal time
on January 1, 2000. Thus, J2000 ecliptic coordinates represent a
direction in space that is fixed relative to the stars (actually
relative to a few hundred even more distant quasars). The J2000
coordinates of the Sun or a planet describe exactly where it
appears in the constellations of fixed stars as seen from
Earth.</p>
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<p>The figures above illustrate the relationship between the "planet
clock" diagram used in the second tab and the actual geometry of
the sky. The globe on the left shows the ecliptic coordinate
system lines of latutude and longitude in gray. The Earth is the
blue dot at the center. The gray ring at the center of the black
ribbon is the ecliptic equator; the Sun will always be found on
this ring. The parallel gray rings above and below the ecliptic
equator are ±5° ecliptic longitude (the black ribbon itself
extends to ±9°). The "planet clock" diagram twists and flattens
this black ribbon into the equatorial plane as shown in the planar
diagram on the right. Since the planets are always found within
the black ribbon, their positions can always be plotted in the
planar diagram.</p>
<p>The globe on the left also indicates the relationship between the
ecliptic coordinate system and the equatorial latitude and
longitude used in terrestrial navigation (as well as in astronomy,
where they are called declination and right ascension,
respectively). The three dark blue lines show the perpendicular
to the ecliptic plane and the lines through the positions of the
Sun at the equinoxes and the solstices; the three lines are
mutually perpendicular. The light blue circle is the ordinary
equator (that is, the intersection of Earth's equatorial plane
with this very large celestial globe). The gray ecliptic equator
is tilted about 23.5° from the ordinary equator. The nearly
vertical light blue line is the Earth's axis, perpendicular to the
ordinary equator. The third light blue line and the dark blue
line of equinoxes both lie in the plane of the ordinary equator
and are perpendicular to Earth's axis. Finally, the tropic and
arctic circle lines are shown in very faint light blue. Note that
the tropic circles are just tangent to the ecliptic equator at the
solstices, while the ecliptic pole lies on the arctic circle. The
north ecliptic pole is about halfway between Polaris and Vega if
you want to find it in the northern sky.</p>
</section>
<section id="radec">
<h3>Playing with the planet clock</h3>
<p>The planet clock reproduces the motion of the Sun and planets
through the constellations quite accurately for any date from 5000
years in the past to 1000 years in the future (3000 BCE to 3000
CE). Thus, you have at your fingertips a distillation of every
planetary observation in recorded history up to Tycho, the last
naked-eye astronomer. (If you want to see the data Kepler worked
from, run the clock from about 1575 to 1595.) This starting point
completely skips over the enormous amount of labor required to
reduce raw observations of the sky to a few pairs of ecliptic
coordinates and a time, but you should spend at least a few
moments appreciating the skill, effort, and persistence of those
observers. Unlike them, you can simply dial in a time and see
where the all the planets appeared. To a certain extent, this was
Kepler's situation as well, since he began with Tycho's data. The
challenge you face here is to make sense of what you see.</p>
<p>The planet clock does remind you of the fundamental problem of
astronomy: You cannot see the stars or planets in daylight. In
fact, it is hard to get good observations within an hour of
sunrise or sunset, which is very roughly indicated by the sky blue
sector of the planet clock - any time a planet appears within that
sector, you will have no accurate data about its position. The
Sun itself is the exception to this basic rule, but since the
stars are never visible at the same time, it is quite difficult to
accurately measure where the Sun is with respect to the
constellations. (The solution requires an accurate clock, another
facet of the long story of how all the data represented by the
planet clock were collected.) Since the Sun moves acrosss the sky
east to west daily, planets to its east - that is counterclockwise
on the planet clock - will be visible for only a short time after
sunset before they set themselves. Similarly, when a planet is
just west - clockwise - of the Sun, it will be visible for only a
short time after it rises before sunrise.</p>
<p>Let's quickly mention the controls on the planet clock to be sure
you have discovered them all:
<ul>
<li>Drag the Sun to change the time (date).</li>
<li>Press and hold one of the yellow triangles below the clock
to move time forward or backward, slowly or quickly depending
on which triangle.</li>
<li>Click the year button to jump to an arbitrary year.</li>
<li>Click a planet or its name in the legend to toggle a radial
line from the Earth at the center of the clock to that planet.
This lets you see the angle between the Sun and planet, which
will be important.</li>
<li>Click the button at the upper left to show or hide the Moon.
The motion of the Moon is more complicated than the planets;
its position shown on the clock is accurate for only a few
centuries around the present. Mostly you should leave it
off.</li>
</ul>
</p>
<p>The first thing you notice is that Venus and Mercury always appear
near the Sun, alternating between the morning side and the evening
side. Mars, Jupiter, and Saturn, on the other hand loop all the
way around the sky. When a planet is in exactly the same
direction as the Sun we say it is at conjunction. Notice that we
can never measure conjunctions accurately. When a planet is
exactly opposite the Sun in the sky - when it crosses the faint
yellow line on the planet clock - we say it is at opposition.
Venus or Mercury is never at opposition. Opposition observations
will be indispensible for working out planetary orbits; since
ancient times astronomers have taken special care to accurately
measure the time of opposition.</p>
<p>Notice the strong asymmetry in the motion of Venus and Mercury:
it takes far longer for them to move from the morning to the
evening side of the Sun than to move back from evening to morning.
When you see their motion on the planet clock, with months of
actual observations reduced to seconds, this asymmetry is even
more striking. You very strongly get the impression that Venus
and Mercury are moving around the Sun, and that we are viewing
that circular motion nearly edge on. The morning-to-evening leg
of their orbit is when they are chasing the Sun, while the
evening-to-morning leg is when - from our vantage point - they are
moving opposite to the Sun. If their orbits around the Sun are in
the same direction as the motion of the Sun through the
constellations (counterclockwise viewed from the north), they must
be passing behind the Sun going from morning to evening, then in
front of the Sun going from evening to morning. The slow
morning-to-evening crossing is called superior conjunction, while
the rapid evening-to-morning crossing is called inferior
conjunction.</p>
<p>Mars, Jupiter, and Saturn generally move through the
constellations in the same direction as the Sun - west-to-east -
but more slowly. However, as they approach opposition, they
reverse direction, moving east-to-west for a time near opposition.
This is called retrograde motion. Notice that Venus and Mercury
also move retrograde near inferior conjunction, so all the planets
have a retrograde portion to their motion. Perhaps the easiest
way to visualize why the outer planets go retrograde near
opposition is to imagine the path Earth would move through the
stars as viewed from Venus: Since the direction of Earth from
Venus is exactly the opposite to the direction of Venus from
Earth, Earth appears to move in exactly the same pattern viewed
from Venus as we see Venus move from Earth - namely it spends most
of its time in prograde motion, with a quicker retrograde part.
But what we see as an inferior conjunction of Venus in the middle
of the retrograde section would appear as an opposition of Earth
as seen from Venus. The retrograde motion of Mars, Jupiter, and
Saturn near opposition has exactly the same cause.</p>
<p>If you are studying the motion of one planet, it helps to toggle
on its radial line (and toggle off all the others). You can then
press and hold the "play" triangle to watch the motion.
Personally, I find the impression that Venus and Mercury are
moving roughly in a circle about the Sun to be quite strong. I
find it much harder to directly see that Mars, Jupiter, and Saturn
are also moving around the Sun, just in a circle larger instead of
smaller than our distance to the Sun. For me, seeing those larger
circles requires some reasoning, like imagining how Earth would
appear to move from Venus.</p>
</section>
</section>
<section id="orbit-view">
<h2>Orbit View</h2>
<p>After you become familiar with the planetary motions shown on the
planet clock, advance to the "Orbit View" using the main menu
button on the right side of the page. This shows you the final
result you will be pursuing - this is the answer sheet. The
radial line for Mars will automatically turn on initially, but you
can toggle any of the radial lines using the legend in the middle
of the planet clock as before. The orbit view on the right shows
you a theoretical view from a point beyond the solar system in the
direction of the north ecliptic pole. The Orbit View panel has
some yellow buttons along its top edge: The triangles in the left
corner zoom in so you can see the orbits of Mercury, Venus, and
Mars, or out so you can see the orbits of Jupiter and Saturn.</p>
<p>You can see that the direction from Earth to any planet and from
Earth to the Sun in the Orbit View always exactly match the
directions in the Planet Clock. Of course, every viable theory of
planetary motion must satisfy this condition first and foremost.
You can switch among three models using the button at the upper
right, labeled "heliocentric", "geocentric", and "epicycles".
They could have been labeled "Copernican" (or simply "modern"),
"Tychonic", and "Ptolemaic" after their best known proponents.</p>
<p>The relative positions of the Earth, Sun, and all the planets are
identical in all three cases; the only differences are which
object you imagine to be at the center, and what geometrical lines
and shapes you imagine describing their positions. No
observations of the planets can possibly distinguish among the
three models, since again, they all produce identical relative
positions of the planets. Our modern bias in favor of the
heliocentric model stems from Newton's theory of universal gravity
rather than from any astronomical observations.</p>
<p>The shape of the blue Earth orbit around the Sun in the
heliocentric model is identical to the shape of the gold orbit of
the Sun around the Earth in the geocentric and epicycles models,
except it is rotated 180 degrees (since the direction of the Earth
from the Sun is 180 degrees from the direction of the Sun from the
Earth). The only difference between the geocentric and epicycles
models is that the large, slow orbits of Mars, Jupiter, and Saturn
are centered near the Earth instead of the Sun, while a copy of
the gold Sun orbit is displaced so it becomes an epicycle whose
center lies near the planetary orbit with the planet on that
epicycle. (Mathematically, the order in which you add the two
roughly circular motions does not matter. I have to admit that I
can more easily visualize an epicyclic motion when the radius of
the epicycle is smaller than the radius of the main cycle,
although the extra copies of the Sun's orbit seem very
strange.)</p>
<p>The important part of the heliocentric hypothesis has nothing to
do with whether the Sun is at the center. You must have noticed
that the motion of the Sun through the constellations is far, far
simpler than the motion of any planet. The important part of the
hypothesis is that, relative to the Sun, the motion of all the
planets is just as simple. Relative to the Sun, there is never
any retrograde motion. Most importantly, relative to the Sun, the
motion of all the planets is periodic, repeating exactly the same
path every orbit around the Sun.</p>
</section>
<section id="earth-period">
<h2>Earth Period</h2>
<p>The point of the Earth Period section is to explore the
extraordinary accuracy with which the Sun repeats its motion each
year. You can use this display to calculate the length of the
sidereal year to within about one thousandth of a day - a little
more than a minute - using ten years of measurements of the
position of the Sun. A sidereal year is the length of time
required for the Earth to complete one orbit around the Sun (or
vice versa). Despite what everyone is taught in school, this is
slightly different than the tropical year we use for our calendar
(because unlike the sidereal year it tracks the seasons), owing to
the precession of the equinoxes, as Hipparchus discovered more
than 2100 years ago.</p>
<p>When you switch to the Earth Period section, a new day counter
display appears in the middle of the Planet Clock, plus a Reset
button. Clicking the Reset button resets the day count to zero at
the current date on the clock. As you move the Sun to change the
date, the day counter increments and decrements accordingly
(except jumping using the year button starts over). When you
click Reset, the Earth period plot automatically plots the number
of days elapsed versus the change in ecliptic longitude of the Sun
measured in revolutions for ten years beginning at the date you
clicked Reset. Clicking the yellow zoom buttons in the upper left
corner zooms in or out on this data. Dragging the Sun around
after you have clicked Reset will move the Sun marker on the plot
to show which point corresponds to which date.</p>
<p>Each yellow box along the diagonal shows one year or Earth
period. One year always represents exactly one revolution, so the
horizontal dimension of the boxes is fixed. The vertical
dimension of the yellow boxes is equal to the number of days per
sidereal year, which you can adjust using the yellow slider at the
right. Your current estimate is displayed above the plot. The
yellow line is the diagonal of the boxes. You find the length of
the year year by adjusting the number of days until the blue data
line matches the yellow diagonal as best you can.</p>
<p>When you zoom in the first time, the ten revolutions are stacked
on top of one another by sliding the yellow boxes and their
contents together. Any error in your estimated period (year)
shows up as the blue data lines not coninciding from one year to
the next. At this magnification, you can adjust the year to
within a couple of tenths of a day. But the Sun moves around the
ecliptic about a degree per day, which means you have only found
the year to within about a tenth of a degree. Recall that Tycho's
naked-eye measurements of the Sun and planets are good to within a
thirtieth of a degree, which over a ten year period ought to allow
you to determine the sidereal year to a few thousandths of a
day.</p>
<p>To improve your Earth period estimate, click the zoom in button a
second time. This does not zoom the horizontal axis any further,
but instead of plotting the measured number of elapsed days
vertically, it plots the small difference between the blue data
and the yellow diagonal line, that is the difference between the
time it actually requires for the Sun to reach a given ecliptic
longitude and the time it would reach that longitude if it moved
at a constant rate. You can now see that the Sun does not move
around the ecliptic at a perfectly constant rate: Its true
position deviates by about four days or 1% of a year from a
perfectly constant speed - so that the blue data line seems to be
a straight line at the other two zoom levels. This view has
effectively zoomed in by a factor of about 100 from the previous
vertical scale. You can see that even these small
non-uniformities in speed repeat from one year to the next with
extraordinary precision. And at this zoom level, you can
determine the length of the year with an accuracy approaching one
thousandth of a day. <em>(This accuracy may be higher than the
JPL approximate position model supports - actual variations in
year lengths can be as much as 20 minutes or 0.01 day? Is this a
problem with the difference between EM barycenter and Earth
center?)</em></p>
<p>Incidentally, the variation in the speed of the Sun around the
ecliptic is large enough to have been noticed since ancient times.
For example, the time from the spring equinox to the fall equinox
is about 5 days longer than the time from the fall equinox back to
the spring equinox - northern hemisphere summer is a few days
longer than northern hemisphere winter.</p>
<p>The most important lesson here is qualitative: The position of
the Sun among the constellations repeats each year with incredible
precision. By naked eye observations it is impossible to detect
any year-to-year variations at all. What is this exact
periodicity telling you? Although you cannot directly measure the
distance to the Sun (at least by naked eye observations), it is
hard to imagine that distance is not also varying periodically
with comparable precision. If both distance and direction vary
periodically, the Sun must be moving around the Earth (or
vice-versa) on a closed path in three dimensions. That is, after
exactly one year, the Earth and Sun will return to exactly the
same relative positions in three dimensions as they have now.</p>
<p>The motion of the other planets is clearly not periodic with
respect to the Earth. The hypothesis you will test is
that <em>the motion of every planet is periodic with respect to
the Sun</em>, repeating the same closed path around the Sun over
and over. If true, their motion with resepct to the Earth will be
a compound of two periodic motions; when the two periods are
incommensurate this compound motion can result in the aperiodic
wanderings of the planets you observe. Historically, this special
feature of the Sun was conflated with the idea that the Sun was at
the center while the Earth moves. Logically, whether the Earth or
Sun or neither is "stationary" is independent of whether the
planets move in periodic closed orbits around the Sun - and the
latter is the only hypothesis Kepler actually used to work out all
the relative positions.</p>
<p>In this section, you have determined the period of Earth's orbit
around the Sun. To test whether the other planets follow simple
periodic orbits around the Sun, you must somehow determine
accurate periods for planets other than Earth. Mars is the
easiest choice, and on Tycho's advice, Kepler chose to focus on
Mars.</p>
</section>
<section id="mars-period">
<h2>Mars Period</h2>
<p>Although you can measure the direction of the Earth-planet line
(the radial line on the Planet Clock), there is no way to measure
the direction of the Sun-planet line. This makes it much harder
to figure out the period of a planet than it was to figure out the
period of Earth. However, there is one situation when
you <em>do</em> know the direction of the Sun-planet line: when
the Earth-Sun and Earth-planet lines are co-linear. Except for
the non-zero ecliptic latitude of the planet, the Sun, Earth, and
planet are lined up at conjunctions or oppositions, and since you
cannot accurately measure exactly when a conjunction occurs, you
are left with oppositions.</p>
<p>The fact that the planet does not have zero ecliptic latitude at
opposition means that the three co-linear points at opposition are
the Sun, the Earth, and the point on the planet's orbit projected
into the plane of the ecliptic. However, this projected point
clearly moves around the Sun with the same period as the planet
itself. Furthermore, if you can find the position of the
projected point at any time (relative to the position of Earth),
you can use its measured ecliptic latitude to find its coordinate
perpendicular to the ecliptic. So you can ignore the ecliptic
latitude and consider only the plane geometry problem of figuring
out the motion of the projection of the planet into the
ecliptic.</p>
<p>When you switch to the Mars Period section, you see a display and
controls very similar to the Earth Period section. Clicking the
Reset button now collects and plots twenty years of data for the
time it takes Mars to reach a given ecliptic longitude. (Recall
that Kepler worked with twenty years of Tycho's data.) These data
appear as a series of light red S-shaped sections near the
diagonal of the plot. The gaps are when Mars is near conjunction
- too close to the Sun to be seen. The dark red dots are the
points at opposition, which lie near the center of each retrograde
section - the kink in the S-shape when ecliptic longitude is
decreasing with increasing time. There is also a light red dot
which moves as you drag the Sun around the clock to show which
point on the data curve corresponds to a particular date on the
clock.</p>
<p>Once again, you see the yellow boxes and the yellow line along
their diagonals. The boxes are exactly one revolution longitude
in width again, and again you can adjust their height using the
yellow slider along the right side of the plot. This time, of
course, you are adjusting your estimate of the number of days it
takes Mars to go once around the Sun. The data is drawn in faint
red because almost all of it is useless - it tells you only about
the direction of Mars as seen from Earth, while you want to know
how long it takes Mars to move around one revolution as seen from
the Sun. The ten or eleven dark red dots at the oppositions are
the only useful data here, because only at those times do you know
the exact direction of Mars (really its projection) as seen from
the Sun. You want to adjust the yellow line to pass through all
the dark red dots as best you can.</p>
<p>The second zoom level once again stacks the yellow boxes on top
of one another, so you can see one hypothesized period of the
motion of Mars around the Sun. As expected, the light red data
lines of ecliptic longitude as viewed from Earth do not come close
to lining up - the motion of Mars is certainly not periodic when
viewed from Earth. The dark red opposition dots have a chance of
being periodic, but in twenty years you only get to see a single
hypothesized period, so you cannot really be sure. However, you
can get a pretty accurate estimate of what the period of Mars must
be <em>if</em> its motion around the Sun is indeed periodic. To
aid you, a dark red line connecting the oppositions has been added
to the plot. This is just an arbitrary smooth curve - in fact a
cubic spline, although Kepler might have used a French curve if he
ever plotted his oppositions by hand. This dark red curve
represents the ecliptic longitude of Mars as seen from the Sun,
but again only the only points on the curve you have actually
measured are the dark red dots. By lining up the ends of this
curve (the curve terminates at the first and last oppositions in
the twenty year data set) you can estimate Mars's period to within
about one day.</p>
<p>For the Earth, the deviation of its longitude as a function of
time from a straight line was very subtle. For Mars, the
opposition points very obviously do not lie on a straight line.
The third zoom level again shows the deviation of from the
straight yellow line. (That line has been adjusted to pass
through the first opposition point so it no longer lies exactly
along the yellow box diagonals. It is now only parallel to the
box diagonal.) At the third zoom level you can see that Mars must
speed up and slow down much more than Earth as it orbits - the
deviation from the uniform rate of the yellow line is over 40
days. You can also see that the arbitrary dark red line
connecting the dots does not line up exactly where it overlaps.
This is because only the opposition points themselves are real
data. The result is that you cannot estimate Mars period nearly
as accurately as Earth's period from just these opposition data -
you can only be sure to within about 0.1 day. This is easily good
enough for a first cut at Kepler's surveying technique.</p>
</section>
<section id="survey-orbits">
<h2>Survey Orbits</h2>
<p>You are now set to experience one of the greatest Aha! moments in
the history of science: Kepler's ingenious technique for surveying
the solar system. Ordinary terrestrial surveying relies on fixed
landmarks, perhaps mountaintops or trees or steeples. By
measuring the apparent directions of these landmarks from
different vantage points, the surveyor can use triangulation to
locate both these vantage points and the relative positions of the
landmarks on a map. Because the Sun and planets are constantly
moving around, there are no fixed landmarks in the sky, and
applying surveying techniques to the sky seemed hopeless to all
the great astronomers and geometers from ancient times until the
year 1600.</p>
<p>Kepler's great insight is that the hypothesis planetary motion is
periodic relative to the Sun provides you with the missing fixed
landmarks: If the planet - Mars in this case - returns to exactly
the same place relative to the Sun after one period, you can use
that point as a fixed landmark. When you make an observation of
Mars exactly one or two or any number of periods before or after
an initial observation, it will have returned to exactly where it
was for your first observation. Used in this way, Mars can play
exactly the same role as the reliable fixed mountaintop or tree or
steeple for the terrestrial surveyor. A surveyor needs at least
two landmarks, but for the astronomer the Sun can play the role of
that second landmark: The Sun and Mars are your two landmarks, and
the Earth is your vantage point.</p>
<p>Notice that you are testing a hypothesis: You merely suspect that
planets move in periodic orbits with respect to the Sun, and you
have realized that <em>if</em> this is correct, <em>then</em> you
would be able to use a planet as a landmark and survey the sky.
All of the work you are about to do is speculative - it may turn
out that your observations of the directions of Mars viewed from
Earth are inconsistent with this hypothesis. You risk only a few
minutes to find out because the compute engine behind your browser
is so powerful; Kepler risked decades of his life to work through
the surveying calculations required to check the periodic orbit
hypothesis.</p>
<section id="select-opposition">
<h3>Select an opposition</h3>
<p>You cannot choose an arbitrary point on Mars orbit for your
landmark. Only an opposition point can serve as your first
landmark (or second landmark if the Sun is the first). So your
first task is to choose which opposition you will use as your
reference landmark. Click on the corresponding yellow circle to
make your choice. When you make your selection, the Planet
Clock on the left side will advance to the opposition you have
selected. You can come back later and choose a different one if
you wish - each opposition will produce a different set of
points around the orbit you are surveying.</p>
<p>The right side plot shows the same "overhead" view of the
ecliptic plane as in the Orbit View section. Here it has become
the surveyor's map view. Once you have selected an opposition,
a black line appears between the Sun and the position of Mars at
that opposition. You know the direction of this line segment,
but not its length in kilometers. However, because you
hypothesize that Mars returns to exactly that point every
period, you are free to use its length as the unit by which you
will measure all distances in your map of the ecliptic.</p>
<p>This map is drawn from a heliocentric perspective with the Sun
always at the center. This is the easiest analogy with
surveying, with the Sun and reference Mars point serving as
fixed landmarks. From a geocentric perspective, you have to
think of the reference Sun-Mars line as a rigid object which is
moving to different positions while not changing its length or
orientation, which is not quite the mindset of a surveyor, even
though the geometry is identical. Kepler of course worked from
the heliocentric perspective adopted here.</p>
<p>Click the yellow Next button at the upper right to proceed to
the next step.</p>
</section>
<section id="earth-orbit">
<h3>Survey Earth's orbit</h3>
<p>When you step forward or back one Mars period, Mars returns to
the reference point, but Earth moves to a new point. From each
of these new vantage points, you can use the measured longitudes
of the Sun and Mars to triangulate and map the location of your
vantage point - the Earth. When you click the yellow -M or +M
buttons at the lower right, the Planet clock will move to the
corresponding date one Mars period earlier or later. Also, the
Earth-Sun and Earth-Mars lines for that date will be
highlighted. The directions of these lines in the right map
panel exactly match the directions of the Earth-Sun and
Earth-Mars radial lines in the left clock panel by construction:
The blue Earth point in the ecliptic map is placed at the unique
point from which those measured directions pass through the Sun
and Mars reference points. The first fruit of your survey is to
be able to mark these eight or nine points on Earth's orbit -
you now know exactly where the Earth must have been relative to
the Sun on those dates, not just its direction. Notice that you
know the Earth-Sun distance on those dates measured not in
kilometers, but in units of the length of the black Sun-Mars
reference line.</p>
<p>The ±M buttons will not step beyond the twenty year span you
specified back in the Mars Period section, restricting you to
about the same amount of data Kepler worked with.</p>
<p>As you step around Earth's orbit in Mars period steps, notice
that one (or rarely two) points are missing roughly opposite to
your selected reference opposition. These missing points are so
close to conjunction that Mars could not be seen from Earth.
The light blue shaded circle around the sun in the right panel
roughly subtends the same 15 degrees from Earth's orbit as the
sky blue sector of the Planet Clock. If the Earth-Mars line
intersects this light blue circle you will have no data, so you
cannot determine where the Earth is on that date. Note also
that you cannot determine the position of Earth on the date of
the reference opposition itself, because triangulation does not
work when the three points are co-linear or nearly so.</p>
<p>Finally, notice that the positions of these points on Earth's
orbit depend on your estimate of the period of Mars. If your
Mars period is wrong, Mars will have moved slightly from its
reference position, in addition to the Earth-Sun and Earth-Mars
directions changing slightly relative to where they were after a
true period. The yellow slider used to adjust your estimated
Mars period has reappeared on the right side, along with the
display of your current estimate. By dragging the slider, you
see the effect of an error in Mars period on the positions of
the points on Earth's orbit that you calculate. An error of
half a day distorts the orbital points into an obvious
non-closed spiral. You need to know the period of Mars quite
accurately before you can apply Kepler's surveying
technique.</p>
<p>You still have not surveyed any new points on Mars orbit.
Furthermore, all practical surveying requires multiple redundant
measurements, known as closures, which you use to check and
reduce the inevitable errors in your angle measurements. Click
the yellow next button at the upper right to proceed to that
final phase of Kepler's surveying program.</p>
</section>
<section id="mars-orbit">
<h3>Survey Mars's orbit</h3>
<p>You know the period of Earth's orbit as well as the period of
Mars's orbit. From every one of the eight or nine points you
found on Earth's orbit, you can step forward or back one Earth
period by clicking on the yellow +E or -E buttons at the lower
left. The Earth - always your vantage point - returns to
exactly its previous position, but now Mars advances to a new
position. In fact, no matter which date Mars was at its
reference position, one Earth period after that date it will
reach exactly the same point in its orbit. Therefore, by
stepping an Earth period from all of the points you found where
Mars was at its reference position, you find over half a dozen
dates on which Mars was at this new position and Earth was at a
known point on its orbit. Again, you can step forward and back
by Mars years to step through these new dates. Stepping either
Earth or Mars periods moves the Planet Clock and highlights the
Earth-Sun and Earth-Mars lines for that particular date, so you
can see that the two directions are exactly the same on the
right map panel as on the left clock panel.</p>
<p>The intersection of any two of the red Earth-Mars lines
determines the new position of Mars, but you have many such line
pairs to choose from. Stunningly, all the red lines very nearly
intersect at a single common point! This is the moment that you
realize you really are on the right track. If you move the
slider at the right to change the estimated Mars period, you can
see that the lines all intersect at a single common point for
only one value of Mars's period - if you get the period wrong,
the lines do not intersect. If you could zoom in around the
intersection point, you could more accurately judge the best
value for the Mars period - the one for which all the red lines
converge to a single point. Instead of a zoom capability, this
map panel provides a statistical summary of how nearly the red
lines intersect at a single point, which is described below.</p>
<p>When you step to a new point on Mars's orbit you have not
visited before, the map panel highlights all the blue points
on Earth's orbit from which you could observe Mars at that
position within your twenty year span of data. Most of these
match the Earth points at the previous step (since the Earth
always comes back to the same place after a step of an Earth
period). However, if you look closely, you will see that one
Earth point may disappear - which happens when the step would
take you out of the twenty year window. More interestingly, one
or two Earth points may be added. New Earth points may appear
either because at the previous step that point falls outside
the twenty year data window, or because at the previous step
that point was too near conjunction for any observation to be
made.</p>
<p>When a new point on Earth's orbit has never had its position
fixed before, it cannot be used to determine the position of the
new point on Mars's orbit. Instead, after you use all the other
Earth points to find the new point on Mars's orbit, you can use
the observation at the new Earth point to determine its position
on the map. Such new points have not only the red Earth-Mars
line drawn, but also a faint gold Earth-Sun line showing the
angle used to fix that new Earth point. This is the same
procedure you used in the previous step to fix the original set
of points on Earth's orbit, but using a different point on
Mars's orbit for your second landmark.</p>
<p>You can keep adding points indefinitely to the orbits of both
Mars and Earth by stepping forward and back by Earth and Mars
periods, respectively, while keeping within the twenty year data
span. Here the maximum number of Mars period steps away from
your selected opposition has been limited to ten, and the number
of Earth period steps has been limited so that there are always
at least four points on Earth's orbit for each point on Mars's
orbit. The total number of points on Earth's orbit constructed
here is thus always 21 (the original opposition plus 10 Mars
period steps in either direction), shown as faint blue dots.
The total number of points on Mars's orbit is about 40, shown as
faint red dots. The total number of observation dates is about
350, all within the twenty year period you chose back in the
Mars Period section.</p>
<p>In lieu of that zoomed view, the map panel displays the angular
errors among the intersection points in yellow above the current
Mars period estimate in the upper right corner. You first need
to know exactly how the red points on the map were computed.</p>
<p>Each blue point on Earth's orbit has been computed from a
single observation date from the direction to the Sun and the
direction to a previously computed point on Mars's orbit. This
procedure is not unique; it is just one choice among many since
you have observations from that same point on Earth's orbit on
several other dates separated by Earth period steps. In the
case of a new point on Mars's orbit, a more complicated
procedure is chosen here that treats all of the observations you
made when Mars was at the new point equally: Each observation is
the direction of a line from Earth to Mars, and we assume that
the position of Earth at the time of each observation has been
previously established. In general these Earth-Mars lines
do not all intersect at a single point (as you can see by
adjusting the estimated Mars period away from its true value).
<p>If you pick any arbitrary point P, the Earth-P line will lie at
some angle from your observed Earth-Mars line. You would like
to find a single point P for which this angle is zero for every
one of your observations, but if the Earth-Mars lines do not
exactly converge to a single point there will be no such P.
However, you can find a point P which is the best compromise.
The figure of merit statisticians prefer is that you find the
point P which minimizes the root mean square angle (that is, the
square root of the average of the squares) between P and all of
your observations of Mars at that point in its orbit. This
turns out to be easy to compute for any set of two or more lines
even in three dimensions. Thus, except for the original
reference opposition point, all the red points on Mars's orbit
plotted on this map minimize the sum of squares of the errors in
angle among all of the observations made of Mars when it was at
that point (all the different Mars period steps). This
calculation has been carried out in three dimensions using both
the measured ecliptic longitude and latitude of Mars, producing
not only Mars's position in the ecliptic plane shown in this map
panel, but also its coordinate out of the plane.</p>
<p>Since each individual point on Mars's orbit (except the
reference opposition point) has been computed by minimizing the
RMS angular error between the computed points and our
observations, a good way to measure the consistency of the
entire data set is by computing the residual angular errors of
all roughly 350 observations that contributed to these 40 points
on Mars's orbit. That total RMS error is the yellow number just
above the current estimated Mars period in the upper right
corner of the map panel. The number above that is the largest
angular error betwen an observed position of Mars and the
position of the best fit point as plotted. You can now adjust
the slider to minimize these errors as best you can. You will
find that you can get the RMS error down below 0.01° and the
maximum error down to about 0.02°. Recall that Tycho's ecliptic
longitude data is accurate to about 0.03°, so this is roughly
the level of accuracy Kepler achieved with his original survey
of Mars's orbit.</p>
<p>Although this Survey Orbits section began with your estimate of
Mars's period from the Mars Period section, your estimate of
Earth's period from the Earth Period section has been ignored.
Instead, an Earth period of 365.25636 days has been assumed
throughout this section. The rationale is that estimating the
exact period of Mars from ten or eleven opposition points is far
less accurate than estimating the period of Earth from ten years
worth of potentially very finely spaced mesurements. In
practice, twenty years of Mars observations really does allow
you to nail down the period of Mars a lot more accurately than
just the ten oppositions it provides.</p>
</section>
</section>
<section id="mars-inclination">
<h2>Mars Inclination</h2>
<p>The procedure for computing points on Mars's orbit in the Survey
Orbits section was to compute the point most consistent with all
of the sight lines measured from different points on Earth's
orbit. This produced not only projected positions in the ecliptic
plane, but also the best fit positions of Mars in three
dimensions. In this section you will look at Mars's orbit out of