• The plot shows the time in days required for the Sun to move to each point around the ecliptic. It covers ten complete revolutions (sidereal years).
• To make this plot you need one measurement of ecliptic longitude and the exact time of that observation every five days or so, for a total of about 730 observations.
• The accuracy achieved by Tycho Brahe for this kind of naked-eye measurement was about 10 seconds in time and 1/30 degree in angle, and Kepler required this level of accuracy to do his surveying.
• The blue line passes through these data points.
• The first observation is plotted as 0 days and 0 revs.
• The gold circle is at the location of the date currently shown on the planet clock, so by running the planet clock forward or back you can see which point on the blue line corresponds to which observation date.
• Each yellow box shows the data collected in one year. The width of the boxes is always exactly one revolution. The height of the boxes is your current estimate of that period.
• Your goal is to measure as accurately as possible the number of days required for the Sun to complete one revolution.
  • Adjust your estimate by dragging the slider on the right edge of the panel up or down.
• The yellow line is the diagonal of all the boxes (which are stacked corner to corner), so you want to adjust your period estimate to make the straight yellow line coincide with the blue data line as best you can.
  • The yellow diagonal line is called the “mean Sun” - a theoretical point in the sky which moves along the same path as the Sun but at a constant rate.
• To be able to make a more accurate estimate, click the lower triangle button to zoom in. This will make ten copies of the data, and shift each of the ten copies so a different one of the boxes overlays the box in the lower left corner.
  • The x-axis now shows only one revolution and the y-axis shows only one year.
  • Drag the slider up or down to an incorrect year estimate to see how the blue lines fail to repeat if your period estimate is incorrect.
  • Run the planet clock to see how the Sun marker follows the blue lines from one year to the next. After you are satisfied you understand this stacked plot of your ten years of data, you can again adjust your year estimate as accurately as possible.
• A more accurate way to present this kind of data is to think about the difference between the actual position of the Sun (blue line) and the mean Sun (yellow line). Click the zoom in button a second time to see that picture.
  • The mean Sun depends on your best estimate of the period - you can change it with the slider.
  • Although it is the mean Sun line that is changing, at this highest zoom level it is the blue true Sun lines which move because you are plotting the difference.
• The deviation from the mean Sun line at the highest zoom level clearly shows the Sun does not move at a constant rate around the ecliptic, but speeds up and slows down in a regular, repeatable way.
  • The deviation is about 4 days over the course of the year.
  • You can refine your period estimate to within about 0.001 day (with ten years of data) - about a minute and a half.
  • Notice the precision with which the Sun repeats its motion every year.

1. Use planet clock controls to set starting date.
2. Click Reset button (in planet clock dial) to collect ten years of Sun position data.
3. Click lower triangle (in upper left corner) to zoom in plot.
4. Click upper triangle (in upper left corner) to zoom out plot.
5. Drag slider (along right edge) up or down to change your estimate of Earth's orbital period.

• The most striking thing about the motion of the Sun is how precisely it repeats every year. Here you can check that the Sun returns to exactly the same place in the zodiac every 365.256 days - to within 0.001 day (a minute and a half)!
• No natural phenomenon on Earth recurs nearly that precisely - in fact, until the last century or two, no human contrivances had that kind of precision.
• What is perhaps even more impressive - and important - is that this motion is not quite uniform. The Sun is speeding up and slowing down as it moves through the constellations in exactly the same pattern over and over again.
• The combination of exact periodicity and non-uniformity hints at some great underlying regularity. Newton finally explained this ancient mystery by recognizing the law of universal gravity, but that is a story for another place. Here you will focus on simply describing this periodic motion in as simple, accurate, and general a way as possible.
• Since the apparent diameter of the Sun in the sky remains constant, at least as far as you can tell using naked-eye observations, you know that the distance from the Earth to the Sun must change very little, if at all, during the course of the year.
  • However, remember how nearly straight the blue data line appeared on the two lower magnification levels in the right panel. Is it possible that the distance to the Sun is changing by a similar tiny amount as it moves around the sky?
• If the distance to the Sun is changing slightly during the year, the precise periodicity of its motion around the sky strongly suggests that any distance changes are also periodic, repeating exactly every year.
  • In other words, periodicty in the motion around the sky strongly suggests that the Sun is moving around the Earth (or vice versa) along a closed path in space, like a train around a fixed track on an unvarying schedule.

What do you mean when you say that the planets all move around the Sun and that the Earth is the third planet? Besides giving a vague description of the geometry of the solar system, you must be saying that if you could view the solar system from the Sun, the planets would all move around the zodiac in a similar manner. You mean that the planets are somehow alike in their motion around the Sun. And if the Earth is just another planet (in this respect), you are saying you know one example of whatever this general kind of motion may be: You know exactly how the Earth appears to move around the zodiac as viewed from the Sun, because it is the same as the motion of the Sun you have measured from Earth, only rotated 180°.

Thus, from the striking periodicity of the Sun's motion, the Earth pretty certainly moves in a closed path around the Sun at a not-quite-uniform but precisely repeating rate. And if the motion of the other planets is somehow similar to Earth's, surely this must be the general rule: The planets all move in periodic closed orbits around the Sun. Another way to think about it is that, when viewed from any planet, the Sun will exhibit the same perfectly periodic motion around the zodiac that you see from Earth.

But this periodic closed orbit hypothesis is an idea you can test by carefully studying your planetary position data! If it is true, then any planet returns to exactly the same point on its orbit after exactly one of its periods. That implies a certain relationship between observations made on dates spaced by one period in time, and you can check whether your observations are, in fact, so related. Like any hypothesis, checking this is not mathematical proof that the hypothesis is true, but it is an extremely strong argument in its favor.

• The diagram shows three different pictures or paradigms for the solar system: heliocentric, geocentric, and epicycles.
  • In all cases, you are viewing the ecliptic plane from a position far from the solar system in the direction of the north ecliptic pole.
  • The three pictures are indistinguishable, in the sense that the relative positions of the Sun and planets are exactly the same in all three pictures. There is no observational way to distinguish among them without reference to bodies outside the solar system.
  • The Sun-Earth and any Earth-planet lines you have toggled on in the planet clock are highlighted. Note that these lines in the diagram on the right remain exactly parallel to their counterparts in the planet clock on the left: These paradigms all model the motions you see in the sky perfectly as you move the planet clock around.
  • None is “right” or “wrong” - the question is usefulness, not correctness.
• The heliocentric picture is the modern accepted model of the solar system, as advocated by Copernicus and surveyed by Kepler. The Sun is at the center, and the planets move around it in closed orbits.
• The geocentric picture was proposed by Tycho. It is identical to the heliocentric picture, except the Earth is translated to remain stationary at the center, forcing the Sun to move around it. All of the closed planetary orbits remain centered on the now moving Sun. The gold orbit of the Sun around the Earth is identical to the blue orbit of the Earth around the Sun in the heliocentric picture, except it is rotated 180°.
• The epicycles picture is a modifed version of the ancient Greek solar system described in Ptolemy's Almagest but dating from much earlier. The modification here is to set the arbitary scale factors and change the epicycle shapes slightly (from circles to ellipses), so that this epicycle picture is geometrically identical to the geocentric picture. However, the model retains all of the key geometrical features of Greek epicycle models.
  • For Mercury, Venus, and the Sun, the epicycles picture is identical to the geocentric picture. The Sun serves as the center of these two epicycles.
  • For Mars, Jupiter, and Saturn, the planetary orbit path has been moved from the Sun to the Earth. A new radial line is drawn from the Earth to a point on this Earth-centered planet orbit, and a copy of the gold Sun orbit is centered at the end of this radial line. Each planet now moves around a small circle - its epicycle, a displaced copy of the Sun's orbit - centered on a larger circle centered on the Earth.
  • A faint new point and line appear on the planet clock for Mars, Jupiter, or Saturn (when you toggle on its radial line). This is the direction from the Earth to the imaginary point at the center of the epicycle (the gold copy of the Sun's orbit).
  • If you run the clock forward, you will see that the outer planets wag back and forth around this epicycle center point in exactly the same way that Mercury and Venus wag back and forth around the Sun: The epicycle center moves around the zodiac smoothly; the planet slowly overtakes it prograde, then quickly crosses back over it retrograde.
  • The epicycles paradigm was designed to make the motion of the outer planets resemble that of the inner planets in this wagging motion.

1. Click upper triangle (in upper left corner) to zoom out view.
2. Click lower triangle (in upper left corner) to zoom in view.
3. Click heliocentric/geocentric/epicycles button to change pictures.

Zoom in all the way to watch the inner planets in the heliocentric picture. Zoom out one step to watch the inner planets in the geocentric or epicycles picture. Zoom out a second step to watch Jupiter (in any picture). Zoom out all the way to watch Saturn.

When you adopt the heliocentric picture, you are likely to guess that the motion of the planets as viewed from the Sun has the same crucial property as the motion of the Sun as viewed from Earth: that every planet, like Earth, travels periodically along a closed path around the Sun. If this hypothesis is true, then you can survey the solar system to find the shapes of all the orbits and how each planet speeds up and slows down as it moves around its orbit. The heliocentric picture strongly suggests a surveying model you can test.

Of course, whatever test you devise will also confirm the other two pictures, since they are geometrically equivalent. However, by demoting the Earth from its central position and supposing it to behave like just another planet, you've already suggested that the orbits of the other planets should be periodic closed loops around the Sun. When the Earth was different from the other planets, the assumption that their orbits around the Sun resemble the Sun's orbit around the Earth in this way feels like a separate hypothesis.

A second reason to believe that the Sun should be at the center is that you know it is far, far larger than the Earth or any other planet. The planets always appear as starlike points, while you can see the disk of the Sun in the sky.

You might be attracted to the geocentric picture because it makes the relationship between the radial lines you actually observe on the planet clock and the radial lines in the map of the ecliptic plane really obvious. Wherever you are, you are always at the point from which all your sight lines emanate. Putting the Earth at the center plays into that perceptual truth.

Tycho had an interesting technical reason for believing that the Earth is stationary, while the Sun and planets orbit it: If the Earth moves around the Sun, then by observing the positions of the stars very carefully, you should be able to detect slight changes in the shapes of the constellations as we view them from different perspectives. Despite his best efforts, however, he could not detect any such stellar parallaxes. He calculated that the stars must therefore be at least several thousand times farther away than the Sun. He reasoned that this vast difference in scale between the distance to the planets and the distance to the stars was even less likely than the Earth being stationary. But the universe holds many surprises.

The epicycles picture was the backbone of astronomy for over two millenia, accepted by nearly all the most brilliant ancient Greek and medieval Islamic astronomers. You see here that their model can be adjusted to be equivalent to our modern heliocentric picture, so they certainly were not completely on the wrong track. It was a very successful descriptive model even in its imperfect forms, and it's hard to argue with success.

The original geometric attraction to the epicycle picture is that Mercury and Venus are pretty clearly moving around the Sun as the Sun moves around the zodiac. That is, they are moving around a circle whose center is in turn moving around us. This kind of motion is much easier to visualize when the radius of the epicycle is less than the radius of its center.

The geocentric picture features epicycle descriptions of Mars, Jupiter, and Saturn, but the epicycle radius is larger than the radius its center moves around (the Sun, as for Mercury and Venus). You can get exactly the same motion by swapping the big circle and the small circle, so that the small one - in this case the orbit of the Sun - becomes the epicycle, while the large one becomes the path for its center.

Watching the outer planets wag back and forth around their ghostly epicycle centers on the planet clock may help you to understand what the ancients saw in epicycles. Remember that the Sun is always below the horizon when you can see any planet - so in a way it is an imaginary point as well.

The ancient Greeks noticed that you can rescale the center point and orbit of any planet, keeping the Earth fixed at the center of this scaling operation, without changing any directions as observed from Earth:

Note that in such a scaled orbit, the center of the epicycle is at an imaginary point in empty space that lies along the Earth-Sun line. Thus, in the ancient epicycles models, the center of the epicycle was always at an imaginary point, not just for outer planet orbits as in the modified epicycles picture drawn in the right panel of this section.

Unlike the translations of the heliocentric, geocentric, and epicycles paradigms drawn here, you can apply a different scaling to the orbit of each individual planet. Like the translations, you cannot disprove such a model by observations of the directions to the planets, since by construction those directions remain fixed. The question was not definitively settled until Galileo observed the phases of Venus through his telescope, proving that its prograde conjunctions are behind the Sun and its retrograde conjunctions in front of the Sun. By naked eye, there is no way to tell. Since the primary reason anyone wanted to independently scale the planetary orbits was the belief that nothing could move from in front of the Sun to behind it (penetrating its "sphere"), the idea of individually scaled orbits died with Galileo.

• You need to use at least one planet as a fixed landmark in order to survey the solar system.
• The only way to do that is if the planet returns periodically to the same position. If it does, you need to know the period of its orbit around the Sun in order to know when it has returned to the place you choose for your landmark.
• Mars is by far the best landmark choice, so here you will explore how to adapt the technique you used to determine the orbital period of Earth to the case of Mars.
  • The plot in the right panel here behaves analogously to the right panel in the Earth Period section, with all the same controls.
  • Here, when you click the Reset button on the clock, you collect twenty years of virtual position data for Mars, with observations spaced ten days apart. Kepler used twenty years of Mars data from Tycho for his survey.
• The Mars data of time (in days) versus longitude (in revs) appears as a series of light red S-curves along the diagonal of the plot.
  • The kinks in the S-curves are the retrograde parts of Mars's orbit, and gaps between them are the times when Mars is too close to the Sun to be able to make any observations.
  • When you move the planet clock in the left panel, a faint red dot traces the time-longitude data along the plot in the right panel, so you can see which date corresponds to each data point.
• Unfortunately, these data record the position of Mars on the zodiac from our point of view on Earth. In order to determine the period of Mars's orbit around the Sun, you need to know where Mars appears as viewed from the Sun, so you can judge how long it takes to go around exactly once.
• Fortunately, on each S-curve there is exactly one point where you do know where Mars appears when viewed from the Sun: When Mars is at opposition, we on Earth are seeing it at exactly the same ecliptic longitude as it would have as viewed from the Sun.
  • These opposition points are marked in dark red on the plot.
  • Of all the data you collected over that twenty year span, only these nine or ten opposition points are of any use for making an accurate estimate of the period of Mars.
  • In practice, the opposition points are interpolated between two actual observations. They are treated here as if they were points you actually observed.
• The yellow boxes along the diagonal again have a width of exactly one revolution, and a height of one estimated Mars period, which you can adjust using the slider on the right edge.
  • The yellow line again has the slope of the diagonal, but instead of actually being the diagonal, it is shifted to pass through the first opposition point.
  • You want to adjust the slope of this yellow line - the period of Mars - to pass through all the opposition points as nearly as possible.
• When you zoom in, the first step once again copies and shifts all the year boxes to stack them on top of the first box at the lower left.
• Since only the opposition points are of interest, they are connected by an arbitrary smooth curve (a natural cubic spline here, but other choices work just as well).
  • Although in reality each of these points is separated by over a full period from the previous one, there really is a continuous orbital path of time versus ecliptic longitude as viewed from the Sun - something like the dark red curve - which passes though each of the opposition points.
  • Twenty years of data was not an arbitrary choice: In twenty years the oppositions progress around the zodiac just slightly more than once.
  • This gives you a small overlap section in the red curve connecting them.
  • By adjusting the estimated Mars period until the overlapping sections match as well as possible, you can get a pretty accurate estimate of Mars's period, within perhaps half a day or a bit better.
• Unlike the case for Earth's orbit, the red curve that constitutes your best guess at how Mars moves around the Sun differs quite markedly from the straight yellow line.
  • Mars speeds up and slows down as it moves around the Sun far more than Earth.
  • This exaggerated variation from uniform motion is another reason why Mars is the best landmark choice.
• When you zoom in a second time, the plot again switches to show you the difference between the red curve and yellow line on the vertical axis.
  • You can make your final adjustment here to find the period of Mars to within a few hundredths of a day.
  • Unlike for the Earth, the overlap sections at this zoom level usually will not match very well.
  • This is not because the orbit of Mars is less periodic than Earth. It is because your data is much sparser - you have only nine or ten points on Mars's orbit, while you had 75 on Earth's.
  • Remember that the red curve has an arbitrary shape - it could be any smooth curve passing through the red opposition points. The actual time-longitude curve would match perfectly.
• Because of the arbitrariness of the red curve, your estimate of Mars's period will be less accurate than your estimate of Earth's period. Once you begin surveying, you will be able to tighten up this estimate. For now it is easily good enough to begin surveying.

1. Use planet clock controls to set starting date.
2. Click Reset button (in planet clock dial) to collect twenty years of Sun position data.
3. Click lower triangle (in upper left corner) to zoom in plot.
4. Click upper triangle (in upper left corner) to zoom out plot.
5. Drag slider (along right edge) up or down to change your estimate of Mars's orbital period.

• Each planet has its own set of challenges. With twenty years of data, you can use this same opposition technique to make a pretty accurate estimate of the period of Jupiter. To use it for Saturn, you would need over thirty years of data.
• Since they are never at opposition, you can never be sure of the direction of Mercury or Venus when viewed from the Sun.
  • Your best alternative is probably to estimate the time of inferior conjunction - the retrograde conjunction - as best you can.
  • The planet is not visible for the shortest period of time near inferior conjunction, so your interpolated time of conjunction will be more accurate than for superior conjunction.
  • To partly compensate for the inaccuracy of the time of inferior conjunction compared to opposition, twenty (Earth) years of data gives you many more revolutions of Mercury or Venus than you had of Mars.
• In practice, however, the best approach is to use the orbits of Mars and Earth you will generate in your upcoming survey to derive Kepler's first two laws.
  • These laws reduce all orbits to just a few parameters.
  • You can then use the position data you actually have (as opposed to interpolated inferior conjunctions) to find the parameter values which best fit your observations.
  • You will use this technique in the Third Law section.

• The diagram is the same map of points in the ecliptic plane as in the Orbit View section.
  • There, the points on the orbits were provided for you by magic.
  • Here, you will actually do the work to survey a large set of points on the orbits of the Earth and Mars.
• This work is divided into three steps:
  1. Select one of the nine or ten oppositions you found in the Mars Period section to serve as a reference landmark point on Mars's orbit.
  2. Starting from this reference point, you will step forward or backward by multiples of your estimated Mars period. This allows you to survey the positions of several points on Earth's orbit.
  3. Starting from each of these different points on Earth's orbit, you will step forward or backward by multiples of Earth's period. This allows you to survey several points on Mars's orbit.
• Move forward through these steps by clicking the Next button, back by clicking the Prev button.
  • Unlike the period determination, this procedure eventually makes full use of all of your twenty years of Mars observations, not just nine or ten opposition points.
• When you select an oppostion in step 1, the radial line from the Sun to Mars at that opposition is highlighted in black. (It is really from the Sun to the projection of Mars into the ecliptic plane.)
  • You know the direction - that is, ecliptic longitude - of this line segment, but you do not know its length.
  • However, you do not know any distances in the ecliptic plane. Therefore, you are free to use this black segment as a reference ruler you will use to measure distances in the solar system.
  • The actual dimensions of the solar system in kilometers were unknown until long after Kepler.
• In step 2, you use this reference segment to survey eight or nine points on Earth's orbit.
  • These are drawn with yellow Earth-Sun lines and red Earth-Mars lines.
  • As you click the ±M buttons to step your observation time forward or back one Mars year at a time, the planet clock will automatically move to the time of that observation, and the yellow and red lines will be highlighted on the right panel.
  • The point on Earth's orbit has been placed where it must be to make the highlighted yellow and blue lines on the right panel exactly match the directions of the yellow and red lines on the planet clock.
  • This is precisely how surveyors triangulate two known landmarks - the Sun and your reference opposition point on Mars's orbit in this case - to chart their own position.
  • Notice that one or two points are missing on this ring of points on Earth's orbit. These points are too close to conjunction to be able to see Mars. You will find their positions later when you have other points on Mars's orbit.
• When you advance to step 3, you will be able to step not only by Mars periods, but also by Earth periods.
  • The diagram simultaneously steps all of the observations you made of Mars at the reference point forward or backward by one Earth year.
  • Since Mars always moves to the same place after any fixed time step, you will be looking at Mars when it is at a single new place from eight or nine different viewpoints.
  • If your hypothesis that Mars is moving periodically relative to the Sun is correct, all eight or nine of the sight lines on these dates should converge at a single point - that new point on Mars's orbit.
  • Click the ±E button and see what happens.
  • Aha! Look at that!
  • Step back and forth with the ±M buttons to move the planet clock to over a half dozen different observation dates and check that all those red Mars sight lines really do converge at a single point.
  • Drag the slider on the right edge away from the true Mars period. The observation sight lines no longer converge properly - you need to be able to step exactly one Mars period to get fixed surveying landmarks and make the scheme work.
• As you step forward and back Earth years, you slowly lose points on Earth's orbit because they will fall outside your twenty year data window. You also occasionally gain a new point as it moves into your twenty year window.
  • Completely new points on Earth's orbit are marked by a faint yellow Earth-Sun line in addition to the red Earth-Mars line.
  • These new points are located by triangulation using the point where all the sight lines from existing points on Earth's orbit converge.
  • Hence as you add new points on Mars's orbit you also pick up new points on Earth's orbit.
• Since the periods of Earth and Mars are incommensurate, you could continue this process indefinitely.
  • The more periods you step away from your reference opposition date, the more the error due to any inaccuracy in your Mars or Earth period grows.
  • Since the uncertainty in your Mars period is greater, this calculator limits you to ten Mars period steps away from your reference opposition in either direction.
  • This arbitrary limit always produces 21 points around Earth's orbit and about 40 points around Mars's orbit.
  • Because you have collected observations of several different points on Mars's orbit from each point on Earth's orbit, and you have viewed each point on Mars's orbit from several different points on Earth's orbit, you have used observations made on a total of about 320 dates in order to survey these 60 odd orbital points.
  • The diagram plots all the surveyed points on both orbits as faint dots.

• When you observe a new point on Mars's orbit from several different points on Earth's orbit (spaced in time by multiples of Mars's period), you add to your chart of points by plotting the common intersection of the direction of Mars you observed at each of those times, starting from the position of the Earth points you previously got by triangulating a previous point on Mars's orbit.
  • However, those lines will never exactly intersect at a common point; in fact, if you use the actual direction in 3D - latitude as well as longitude - no two lines will exactly intersect at all. The lines will always be skew.
  • If the common point of intersection is only approximate, how do you compute exactly where to plot it on your chart of Mars's orbit?
• The modern approach to this kind of problem is called least squares fitting.
  • A least squares fit is a computationally convenient way to define the best compromise point: The lines may not all intersect at any point, but there will always be a point which lies at the minimum RMS (root mean square) distance from all the lines.
  • However, in this case, you are measuring a direction - angles of longitude and latitude - so you really ought to find the point that minimizes the RMS angle by which the line from Earth to this point deviates from your measured direction.
  • The calculator that generates this diagram does just that: Each new point on Mars's orbit is placed at the position that minimizes the RMS angular deviation from all of your measurements of Mars at that point (from previously charted points on Earth's orbit).
• Since both latitude and longitude measurements are used, this gives a position of Mars in 3D, even though this diagram only shows the projection of that point into the plane of the ecliptic. The full 3D position will be shown in the Mars Inclination section.
• The numbers above your current estimated Mars period are two measures of the total angular deviations of all 40 of these fitted points on Mars's orbit, using all 320 direction measurements:
  • The bottom number is the RMS angular deviation of all of your measured directions from the direction of the charted Mars point from the charted Earth point.
  • The top number is the largest angular deviation of any of the 320 observations from its corresponding Mars position on the chart.
  • Using these numbers, you can make a final adjustment to your estimated period of Mars: Move the slider until you minimize one or the other of these two numbers - they will both minimize for nearly the same period.
• After this final period adjustment, you can usually drive the maximum angular error down to about 0.02° and the RMS angular error to about 0.01°.
  • Tycho, the last and greatest naked-eye astronomer in history, measured the positions of planets with an accuracy of about 0.03° - about 1/15 the disk of the Sun or Moon, and Kepler used his data. (Someone with acute vision cannot distinguish two marks separated by less than about half that, or half a millimeter viewed at a distance of a meter.)
  • Notice that the groups of lines converging on every point around Mars orbit converge most accurately for a single value of the period of Mars.
• (Any error in the period of Earth would lead to a similar failure of your survey lines to converge. For simplicity, this calculator uses the exact value of 365.25636 days rather than the value you found in the Earth Period section. As noted, it is much easier to determine the period of Earth than the period of any other planet.)

1. Click on opposition to select reference opposition.
2. Click Next button at upper right to move to the next step.
3. Click ±M buttons at lower right to step Mars periods.
4. Click ±E buttons at lower left to step Earth periods.
5. Click Prev button at upper left to return to the previous step.
6. Drag slider (along right edge) up or down to change your estimate of Mars's orbital period.

• You have discovered a very well hidden regularity in twenty years of painstakingly measured postions of the Sun and Mars.
  • The greatest geniuses of astronomy pored over data very similar to this for two millenia without finding these relationships among the positions.
  • The striking geometrical relationships are buried two levels deep - you first need to find a set of points on Earth's orbit by stepping by Mars years from an opposition, and then find a new point on Mars's orbit by stepping by an Earth year, and only then do you get to see the incredible accuracy with which these several sight lines converge at a single point.
• On the other hand, these steps are routine for terrestrial surveyors.
  • The trick that you need to wait a certain fixed period of time for your landmarks or viewpoints to return to their positions has no real analog in ordinary surveying.
  • Kepler is the one who first figured out how to put the two ideas together.
• You cannot doubt that your survey points represent an accurate map of the positions of the Sun, Mars, and Earth, any more than you can doubt a terrestrial map made by surveyors.
  • In order to make this map, you had to assume that the planets periodically return to the same positions, each after its own fixed period.
  • You then adjusted the periods precisely in order to make your surveying techiniques self-consistent. Is this not circular reasoning?
  • Yes, but by adjusting only one number (two if you count the period of Earth), you were able to exactly fit over 320 independent observations of the directions of the Sun and Mars.
  • Therefore, it is very hard to doubt that the hypothesis of periodic orbits is correct, in addition to the geometry of your survey points.
• Furthermore, you can repeat this survey for any twenty year data set you please throughout the ages, or more easily using a different opposition in the same twenty year window for the starting reference point. The specific points you get around the orbits will be different, but the convergence of the sight lines on all the different days will be just as accurate.

You have proven that Earth and Mars move as if they followed repeating closed orbits around the Sun. Of course, we now routinely measure the tiny discrepancies in the orbits of the planets from this rule. Though the rule may not be exact, nevertheless it is very, very accurate.

Would the astonishing consistency of Kepler's surveying technique have convinced ancient Greek astronomers that the planets orbited the Sun rather than moving along scaled orbits whose centers merely aligned with the direction of the Sun? They may well have acknowledged only the periodicity of the motion and clung to their scaled orbits. At the same time, they would certainly have pounced on Kepler's surveyed orbital shapes and recognized them as the true shapes of their deferents and epicycles.

Your first order of business is to check that the other planets do indeed repeat closed orbits. Are you the kind of person who hopes that the answer is yes? Or no? Is your work done if the answer is yes?

• Initially, the right panel shows the same chart of the orbits of Mars and Earth as the third step of the Survey Orbits section. However, the points you charted on Mars's orbit are now plotted in 3D, as you can see by dragging the sliders on the bottom and right edges.
  • You can see that your surveyed points all lie in a single plane containing the Sun, but tilted at a small angle away from the ecliptic.
  • Because this orbital inclination of Mars is so small, you should magnify the z-coordinates by clicking the 10× button so you can clearly see the tilt.
  • Be sure to click on the point on Mars's orbit where the red sight lines converge to turn them off as you familiarize yourself with this diagram.
• The faint blue line through the Sun is the intersection of the best fit plane to your Mars survey points with the ecliptic plane, which is called the line of nodes.
  • The ecliptic longitude of this line of nodes (specifically the ascending node) is the number at the top right of the diagram.
  • The angle between this best fit plane and the ecliptic plane, the orbital inclination of Mars, is the number under the ascending node longitude.
  • Click the Reset button at the lower left to return to the map of the ecliptic plane. Now drag the bottom slider until the blue node line is vertical (you can use the stationary tick mark at the top of the diagram to align it exactly).
  • Drag the right slider all the way to the top to get the view along the line of nodes, exactly edge on to both the ecliptic and the plane of Mars's orbit.
  • Go back to 1× z-magnification to see how small the tilt really is.
• The ring of faint red dots are the projections of your dark red surveyed points into the ecliptic plane. Viewed from Earth, these points have the same ecliptic longitude as Mars, but zero ecliptic latitude.
• You can click on any of the dark red points to toggle on or off red sight lines for all of the observations you made of Mars when it was at that position through your twenty years of data.
  • The corresponding blue points on Earth's orbit will highlight.
  • When you view any set of these sight lines from different perspectives using the sliders, you can see that they all converge to a single point in 3D, not just in the plane of the ecliptic.
• Finally, as a curiosity, if you jump to the year -3000 and survey the Earth and Mars orbits then, the Mars Inclination section will show you something very interesting:
  • In addition to the plane of Mars's orbit being inclined a couple of degrees, the plane of Earth's orbit is also inclined a fraction of a degree.
  • This is because the J2000 coordinate system adopted here puts zero ecliptic latitude in the plane of the Earth's orbit in the year 2000, and the Earth's orbital plane really does change ever so slowly relative to the fixed stars.
  • This is not to be confused with the much quicker (but still slow) precession of the equinoxes that shifts the calendar - unlike equinoctial precession it has nothing to do with Earth's spin axis.

1. Drag bottom slider to rotate ecliptic plane around Sun.
2. Drag right edge slider to rotate view out of ecliptic plane.
3. Click one of the magnify z buttons at upper left to select z magnification.
4. Click any red point on Mars orbit to toggle on or off sight lines from Earth orbit points.
5. Click Home button at lower left to restore original orientation.

The fact that all the Earth-Mars sight lines emanating from your surveyed positions of Earth at the dates you found by stepping multiples of Mars years all converge at a single point in three dimensions is very striking. Kepler's survey technique works in three dimensions, as it must to be successful.

As if all these convergences were not remarkable enough, you should also reflect on the fact that the survey points all lie exactly in a single plane containing the Sun. You did not assume this - the only thing you assumed to make your survey was that the orbits of Earth and Mars were periodic. You already knew that Earth's orbit was confined to a plane containing the Sun, because that is how you defined the plane of the ecliptic. But you now know that not only is Mars's orbit as periodic as Earth's, but that it too is confined to a plane containing the Sun.

It seems as if Mars, and probably the other planets, completely ignore one another - only the Sun seems to exert any influence over the motion of the planets. Heliocentrism is not about what body lies at the the center so much as what body dominates the dynamics. The answer is apparently that every body except the Sun is negligible.

• The points in the ring are the ones you found in the Survey Orbits section, plotted in the plane of the orbit. In the case of Mars, this is tilted about 2° from the J2000 ecliptic plane you have used for orbit plots until now.
• Drag the three round buttons around to adjust the perihelion (point nearest the Sun), aphelion (point farthest from Sun), and the line of apses (radial line through the Sun connecting perihelion and aphelion).
• The numbers at the bottom left show the perihelion and aphelion of your current fit. The numbers at the bottom right are the three conventional orbital parameters: The a is the semi-major axis (average of perihelion and aphelion). The e is eccentricity (displacement of center from Sun as a fraction of a, equal to the difference between perihelion and aphelion divided by their sum). The angle is the counterclockwise angle of perihelion from the x-axis.
• Your goal is to adjust the three orbital parameters to fit your survey points as closely as possible. One systematic procedure is:
  1. If all points lie outside your current orbit fit, drag the perihelion button outward until your fit just begins to touch points on the orbit. Otherwise, drag the aphelion button inward (this will change perihelion too once they become equal) until all points just barely lie outside it.
  2. In either case, drag the angle button around to align the perihelion with the points which are just touching your fit.
  3. Drag the aphelion button outward until your fit just touches the points nearest aphelion.
  4. Alternate among the three round buttons, moving them to minimize the error number in the upper left corner.
• The error number in the upper left corner is the RMS (root mean square) distance of your orbit fit from your surveyed points. The units are the same as for the perihelion and aphelion numbers at the lower left, namely AU (astronomical units, the a value for Earth).
• Click the Ellipse/Circle button to fit an eccentric circle (simply a circle whose center is displaced from the Sun) instead of an ellipse with one focus at the Sun.
• Click the Mars/Earth button to fit Earth's orbit instead of Mars's orbit.
• Click the Coarse/Vernier button when you want to make fine adjustments to your orbital fit parameters. This does two things: First, you have to move your mouse 20 times farther to change the parameter by a given amount. Second, a set of "ghost" points become visible, which are plotted at a distance from your orbit fit that is 20 times the distance of the actual solid survey points. If you can get all the ghost points to coincide with the actual points you have a perfect fit.
• Click the Auto button if you get frustrated trying to make very small adjustments even on Vernier. You may be able to get a slightly better fit after clicking Auto, but Auto will get you almost all the way to the minimum error fit.

• In the left panel, you have parametrized the shape of the orbits of Mars and Earth - Kepler's first law. In the right panel, you will characterize how the planet speeds up and slows down as it moves around the shape you have fit - Kepler's second law.
• The easiest way to describe the position of the planet on its orbit is to specify its angle around the Sun from some standard point, usually taken to be its perihelion point.
• You saw in the Earth Period section that this angle changes with time at an almost, but not quite, uniform rate. After you have fit Earth's orbit in the left panel, note the direction of the line of apses you have set. Return to the Earth Period section and zoom in twice, then drag the Sun marker on the clock around to the steepest downward point on the sinusoidal curve - it will be in early January (if you are within a few hundred years of the present). Notice that the angle of the Sun hand on the clock matches the perihelion angle you set in your fit in this section! This means the Sun is moving around fastest at perihelion and slowest at aphelion.
• Drag the triangle button on the left panel around to highlight the area enclosed by the line from the Sun to the perihelion point, your orbit fit, and the line from the curent point on your orbit back to the Sun.
• This will show a white dot on the graph in the right panel, which shows you the area of the highlighted area in the left panel (as a fraction of the total area of the elliptical or circular orbit).
• Since the radius is smaller near perihelion, a given change in angle there produces a smaller change in area than elsewhere. Kepler noticed that meant if the area, instead of angle, were changing at a constant rate, the point would move faster near perihelion and slower near aphelion, as you have seen it does.
• Therefore, instead of time as a function of angle, the plot in the right panel shows time as a function of area. The white line is a straight line going from zero days at area zero to one period (of Mars or Earth) at at area one.
• Each orbital point is plotted at the fit area at the same angle as the surveyed point in the left panel, and the time of those observations modulo one period, shifted in time so that the reference opposition you chose in the Survey Orbits section lies exactly on the white line.
• Click the zoom in button to plot the difference between the survey point times and the white line instead of the times themselves. You can click zoom in a second time for an even higher magnification of the vertical axis.
• At the first zoom level, play around with the fit adjustment buttons in the left panel to see how they affect the area-time relationship.
• Recall that the standard ruler in your orbit survey was the distance from the Sun to Mars at your reference opposition. However, there is no reason you need to call this distance 1.0 unit. The standard choice is to call 1.0 unit Earth's a value, which is called the astronomical unit, or AU. To accomplish this, your reference distance has been given a numerical value equal to the ephemeris Sun-Mars distance at the time of your reference opposition. If that had not been done, you would have simply adjusted your units so your fitted a value for Earth's orbit was exactly 1.0 unit.

In the left panel:

1. Drag round button nearest the Sun around the Sun to change perihelion angle of fit.
2. Drag round button nearest the orbit on the side with two round buttons radially to change the perihelion distance of the fit.
3. Drag round button opposite the two round buttons radially to change the aphelion distance of the fit.
4. Click Ellipse/Circle button to fit either an ellipse with one focus at the Sun or an eccentric circle.
5. Click Mars/Earth button to fit either the points you surveyed on Mars's or earth's orbit.
6. Click Coarse/Vernier button to change the sensitivity of all the drag buttons.
7. Click Auto button to set fit parameters to ephemeris values.
8. Drag triangle button around Sun to show area used by the Area-Time relation plot.

In the right panel:

1. Click on triangular buttons in the upper left corner to zoom the plot out or in.

• An ellipse with the Sun at one focus certainly describes the surveyed orbit points for both Mars and the Earth extremely precisely - within about 0.0003 AU.
• The ancient guess for orbit shapes was an eccentric circle. For the case of Earth's orbit, you have seen that the difference between an eccentric circle and an ellipse with the Sun at one focus is completely negligible. By naked eye observations, it is impossible to see the difference - the two models have the same shape.
• As you see here, even for Mars it is very difficult to see the difference between an ellipse and an eccentric circle. The difference between the major and minor axes of the best fit ellipse is about one part in 230. You can see it easily only with the 20 times magnified ghost points in Vernier mode. (Toggle Ellipse/Circle in Vernier mode after Auto.)
• For Mars, the best fit eccentric circle has RMS error of about 0.0023 AU, about seven or eight times larger than the RMS error of the best fit ellipse.
• Thus for Mars, an ellipse is clearly a somewhat better fit than an eccentric circle. However, you need to survey the position of Mars to within a couple of thousandths of its distance to the Sun in order to be able to make that judgement.
• Kepler's first law therefore rests on the very slight deviations of Mars's orbit from an eccentric circle. Until you can verify it with a more eccentric orbit, it is a tentative conclusion.
• Mercury's orbit deviates from an eccentric circle by about 2%, largest of all the planets. Perhaps the first really obviously elliptic orbit was Halley's comet, discovered about 100 years after Kepler published his first and second laws. (Kepler saw this comet in 1607, but did not realize it was orbiting according to his brand new laws.)

• You must have been struck by the accuracy with which the area-time survey points lie along the straight line when you adjust your orbital shape model to be near best fit.
• At best fit, the deviations from a perfectly straight line are of order hundredths of a day for Mars as well as Earth, out of hundreds of days. In order words, the law that the radius from the Sun to a planet sweeps out equal areas in equal times in correct to better than a part in ten thousand. This is at least as good as the accuracy of your survey, or of the fit to an ellipse.
• Note the effects of moving your model parameters away from the best fit orbital parameters, or switching from an ellipse fit to an eccentric circle fit.
• Kepler's second law that planets sweep out equal areas in equal times is extremely convincing, much more so than his first law.
• For both laws, you can very confidently assert that the motions of Mars and Earth are consistent with Kepler's laws to very high accuracy. (Measuring something with an accuracy of one part in ten thousand is like measuring a one meter distance to within a tenth of a millimeter.)

• In the case of Jupiter, using twenty years of data you could survey its orbit using the same techniques you used for Mars, then directly fit its orbit and check the equal area law as you did for Mars. Saturn would require about thirty years of data, and the survey method you used for Mars does not work for Mercury and Venus, since they have no oppositions.
• The best way to test laws like these is to assume them to be true, and work out the consequences. That strategy was a spectacular success in the case of the law that the motion of planets is periodic with respect to the Sun, as you found when you assumed it to survey the orbits of Earth and Mars.
• If you assume Kepler's first two laws are true, the motion of each planet can be described by only six numbers: The ecliptic longitude of its ascending node and its inclination (which together determine its orbital plane), its semi-major axis, eccentricity, and angle of perihelion (which determine the shape and orientation of its ellipse), and its period. (A seventh number is its position at some initial time, which we can take as given.)
• In the next section, you will find these six orbital parameters by trial and error, fitting the observed planet positions in the sky.

• One at a time, select a planet (by clicking on its name in the legend) to trace its path around the sky for a few years - one year for Mercury, two for Venus and Mars, three for Jupiter, and five for Saturn. The starting date is the same as you used in your orbit survey, or the current date in the Planet Clock section if you haven't done the orbit survey.
• The ten dots on each orbit are simulated observations, roughly equally spaced in date over the time span shown above the planet legend. None of these observations is near conjunction (when observations are impossible), so there are some gaps in dates.
• The solid curve is the path predicted by a model in which the planet (and earth) follow an elliptical orbit with the Sun at one focus, sweeping out equal areas in equal times. (That is, according to Kepler's first two laws.)
• The six orbital parameters for the planet appear above the slider. Click on a parameter name and use the slider in the middle to adjust its value in order to change the orbit ellipse and period.
• The solid model curve will change slightly as you change the orbital parameters. In order to make those changes more easily visible, a ghost point will appear near each of the ten observation points. These ghost points are 20 times farther from the model curve than the corresponding true observation point, so you can more easily see that the obseravtion points do not lie exactly on the model curve.
• The error number above the orbit parameters is the maximum distance in degrees on the sky among the ten observations points from its predicted position along the model curve on the corresponding date. Recall that Tycho's observations are accurate to about 0.03° - the best you can do by naked eye.
• In addition to the six parameters listed, the model curve has exactly the ecliptic longitude measured on the first observation date. (This sets the initial position of the planet on its ellipse.)
• The initial parameter values match the actual orbit over the observation interval as closely as possible (or nearly so). As you are adjusting parameters, you can reset all parameters to these initial values by clicking on the highlighted planet name in the legend below.

• The graph in the right panel shows the relationship between orbital period $T$ and the semi-major axis of the orbit $a$. Kepler's third law is that the ratio of $a^3$ to $T^2$ is the same for every planet. If you always measure $a$ in astronomical units (AU) and time in Earth years (yr), this ratio is exactly 1.0. (Note that these are sidereal years of 365.25636 days, not the tropical years of our Gregorian calendar or of Earth seasons.)
• The vertical position of the bar above each planet shows the ratio according to the scale at the left. The numerical value of the ratio is printed below the planet name.
• To show how accurately the power of $a$ must be exactly $3$ (assuming the power of $T$ is exactly $2$), adjust the power using the slider on the right edge.
• To show how changes in model period or semi-major axis affect the ratio, adjust those parameters in the left parameter.

In the left panel:

1. Click on a planet name in legend to display the model for that planet. Click on the highlighted planet name to reset all the model parameters to their initial values.
2. Click on a parameter name above to connect that parameter value to the slider in the middle.
3. Use the slider in the middle to adjust the selected parameter value.

In the right panel:

1. Use slider on the right edge to adjust power $p$ of $a$ in the displayed ratio $a^p/T^2.$ (This is not really an adjustable parameter: $p=3$ is the correct value.)

• For Mercury, Venus, and Mars, you see that $T^2=a^3$ to better than one part in 10,000 - a very convincing result.
• For Jupiter and Saturn, the ratio can be off by a part or two in 1000, depending on what time interval you choose to look at. This is still easily accurate enough to be a useful way to estimate period or orbit size if you know only one.
• The orbit fits you are using to check the cube-square law here depend on the assumption that the orbits are ellipses with the Sun at one focus and that the planet sweeps out equal areas in equal time as it moves around. That is, Kepler's laws are somehow interconnected.
• At least in the case of Jupiter and Saturn, you see hints that $a^3$ is not precisely proportional to $T^2$. In other words, although these laws of planetary motion are very accurate, they may not be quite exact.

• Kepler's laws profoundly change the way you survey an orbit in the solar system.
• Using only terrestrial surveying techniques (plus the period stepping trick), you had to make multiple observations to find each individual point on Earth's or Mars's orbit. Tracing a whole orbit required hundreds of observations.
• Using Kepler's laws, you can get similar quality orbital shapes from only a handful (ten here) of observations.
• This compression comes from the fact that a complete description of the orbit requires only six numbers (seven if you include the ecliptic longitude at the initial time). Generally only a few measurements pin down all six numbers, enabling you to compute points on the orbit very distant from any of your observations.
• After Kepler, using his complete orbital surveying technique became unnecessary, except as an infrequent and independent check. Parameter fitting similar to what you have done in this section remains the standard orbital surveying technique.
• Once you have established the cube-square law, the number of parameters you need to fit drops by one more - you no longer need to fit both period and semi-major axis.

The epicycles of ancient Greek astronomy, like Kepler's laws, reduce planetary orbits to only a handful of parameters. What they lack is the geometrical basis Kepler provided using his survey technique: The Greek epicycles were derived solely by a fitting process little different from what you just did in the Third Law section. Kepler, on the other hand, fit his ellipse and equal area laws to the points he had surveyed on the orbits of Earth and Mars.

While this was a great leap for astronomy, other thinkers would have taken little notice. After all, the motion of the planets has no consequences for or relation to more down-to-earth problems. Or does it?

In 1687, at the end of the same century that began with Kepler's laws, Isaac Newton published his three laws of motion. Newton's laws lay out our modern understanding of the relationship between forces and motion, and provide a general framework for studying how things move.

When Newton applied his laws of motion to the planets, he realized that Kepler's laws are equivalent to the statement that the planets all accelerate directly toward the Sun at a rate inversely proportional to the square of their distance from it. Newton knew that the four Galilean moons of Jupiter also obey Kepler's laws, differing from the planets only in the constant of proportionality between acceleration and the inverse square of distance. When he compared the acceleration of our Moon to the acceleration of gravity here on Earth, Newton made perhaps the single greatest scientific discovery of all time, his law of universal gravitation: Gravity is a property of all matter, which not only holds us to the surface of the Earth, but also holds the planets in orbit around the Sun.

The influence of Newton's work as a trigger for the scientific and industrial revolution cannot be overstated. And that excitement, in large measure, was caused by the neat way Newton explained Kepler's planetary motion laws - most importantly, by connecting them to the familiar force of gravity on Earth. And Kepler's laws are based on surveyor's geometry and the hypothesis of periodic orbits around the Sun - the beginning of the story of modern science.