(9a) May Earth be Revolving around the Sun?

Aristarchus of Samos, an early Greek astronomer (about 310 to 230 BC), was the first to suggest that the Earth revolved around the Sun, rather than the other way around. He gave the first estimate of the distance of the Moon (section (8c)), and it was his careful observation of a lunar eclipse--pin-pointing the Sun's position on the opposite side of the sky--that enabled Hipparchus, 169 years later, to deduce the precession of the equinoxes).

Except for one calculation--an estimate of the distance and size of the Sun--no work of Aristarchus has survived. However, one could guess why he believed that the Sun, not the Earth, was the central body around which the other one revolved. His calculation suggested that the Sun was much bigger than the Earth--a watermelon, compared to a peach--and it seemed unlikely that the larger body would orbit one so much smaller.

Here we will develop a line of reasoning somewhat like the one Aristarchus used (for his actual calculation, see reference at the end). Aristarchus started from an observation of a lunar eclipse (section (8c)). At such a time the Moon moves through the Earth's shadow, and what Aristarchus saw convinced him that the shadow was about twice as wide as the Moon. Suppose the width of the shadow was also the width of the Earth (actually it is less--see below). Then the diameter of the Moon would be half the Earth's.

Aristarchus next tried to observe exactly when half the moon was sunlit. For this to happen, the angle Earth-Moon-Sun (angle EMS in the drawing here) must be exactly 90 degrees.

Knowing the Sun's motion across the sky, Aristarchus could also locate the point P in the sky, on the Moon's orbit (near the ecliptic), which was exactly 90 degrees from the direction of the Sun as seen from Earth. If the Sun were very, very far away, the half-moon would also be on this line, at a position like M' (drawn with a different distance scale, for clarity).

Aristarchus estimated, however, that the direction to the half-Moon made a small angle a with the direction to P, about 1/30 of a right angle or 3 degrees.

As the drawing shows, the angle ESM (Earth-Sun-Moon) then also equals 3 degrees. If R_s is the Sun's distance and R_m the Moon's, a full 360° circle around the Sun at the Earth's distance has length of 2pR_s (p = 3.14159...). The distance Rm = EM is then about as long as an arc of that circle, covering only 3° or 1/120 of the full circle. It follows that

R_m = 2pR_s/120 ~ R_s/19
Therefore
R_s/R_m ~ 19

making the Sun (according to Aristarchus) 19 times more distant than the Moon. Since the two have very nearly the same size in the sky, even though one of them is 19 times more distant, the Sun must also be 19 times larger in diameter than the Moon.

If now the Moon's diameter is half the size of the Earth's, the Sun must be 19/2 or nearly 10 times wider than the Earth. The effect described in the figures of section (8c) modifies this argument somewhat (details here), making the Earth 3 times wider than the Moon, not twice. If Aristarchus had observed correctly, that would make the Sun's diameter 19/3 times--a bit more than 6 times--that the Earth.

Actually, he had not! His method does not really work, because in actuality the position of the half-Moonis very close to the line OP. The angle p, far from being 3 degrees, is actually so small that Aristarchus could never have measured it, especially without a telescope. The actual distance to the Sun is about 400 times that of the Moon, not 19 times, and the Sun's diameter is similarly about 400 times the Moon's and more than 100 times the Earth's

But it makes no difference. The main conclusion, that the Sun is vastly bigger than Earth, still holds. Aristarchus could just as well have said that the angle p was at most 3 degrees, in which case the Sun was at least 19 times more distant than the Moon, and its size at least 19/3 times that of Earth. In fact he did say so--but he also claimed it was less than 43/6 times larger than the Earth (Greeks used simple fractions--they knew nothing about decimals), which was widely off the mark. But it makes no difference: as long as the Sun is much bigger than the Earth, it makes more sense that it, rather than the Earth, is at the center.

Good logic, but few accepted it, not even Hipparchus and Ptolemy. In fact, the opposite argument was made: if the Earth orbited the Sun, it would be on opposite sides of the Sun every 6 months. If that distance was as big as Aristarchus claimed it to be, would not the positions of the stars differ when viewed from two spots so far apart? We now know the answer: the stars are so far from us, that even with the two points 20 times further apart than Aristarchus had claimed, the best telescopes can barely observe the shift of the stars. It took nearly 18 centuries before the ideas of Aristarchus were revived by Copernicus.

Postscript

In the year 1600, William Gilbert, physician to Britain's Queen Elizabeth I and the first investigator of magnetism, published De Magnete ("On the Magnet" in Latin, in which the book was written). That book marks the end of medieval thought, built largely on citations of ancient authors, and the start of modern science based on experiments. (For a large web site containing two reviews of the book and the history of the Earth's magnetism from Gilbert to our time, see here.)

Gilbert was a staunch supporter of Copernicus (see section #9c), but it is interesting to note that he still quotes the result of Aristarchus (Gilbert's "Book 6", section 2, about 2/3 through the section), writing "The Sun in its greatest eccentricity has a distance of 1142 semi-diameters of the Earth." Aristarchus estimated the Sun's distance to be slightly below 20 times that of the Moon, which was at a distance of about 60 Earth radii, and 20x60 = 1200, close to Gilbert's figure. In 1800 years, no one had checked that result!

The introduction to Gilbert' book was written by Edward Wright, who used that value to derive the velocity of the Sun, if it were to circle the Earth every 24 hours, ariving at a speed so high that he considered it impossible:

primum mobile

This is the same argument Aristarchus could have made, and probably did.

For advanced students:

Note: The actual calculation by Aristarchus was more complex and less transparent. See A. Pannekoek: A History of Astronomy, Interscience, 1961, p. 118-120 and Appendix A.

While the method used by Aristarchus to estimate the Earth-Sun distance does not give the correct value--the Sun is too far--a variation of it was used by a Danish student, successfully, to estimate the distance to Saturn, in terms of the Sun-Earth distance.

Saturn has a well-knowen system of rings around its equator. Using a good telescope, one can observe the planet's shadow on the rings, and note its position. Viewing the rings as a round dial, one notes the position of the edge of the shadow relative to the point where the rings are lined up with the center of the planet, as seen from Earth.

That gives the (small) angle between the Sun-Saturn line and the Earth-Saturn line. Knowing the positions in the sky of Saturn and of the Sun gives another angle of the Earth-Sun-Saturn triangle. Regarding the Earth-Sun distance as one "astronomical unit" (AU), one can now calculate the Earth-Saturn distance (and the Sun-Saturn distance) in astronomical units. For details, see:
http://www.amtsgym-sdbg.dk/as/AOL-SAT/SATURN.HTM

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(9c) Discovery of the Solar System

Early History, False Leads

As noted earlier, Aristarchus of Samos proposed that the Earth revolved around the Sun, but the idea was rejected by later Greek astronomers, in particular by Hipparchus. Ptolemy, living in Egypt in the 2nd century AD, expressed the consensus when he argued that all fixed stars were on some distant sphere which rotated around the Earth. Ptolemy tried to assemble and write down all that was known in his day about the heavens, and his influence was great, extending (as noted below) even to the 1600s.

Yet anything that moved across the celestial sphere--Sun, Moon and planets--had to be able somehow to slip around it. The planets, in particular, were hard to understand, because their motion was not simple.

Schematic drawing of the apparent reversal of motion (retrograde motion) observed with Mars.
Positions 1...7 of the Earth correspond to positions 1...7 of Mars, which moves more slowly. As the Earth overtakes Mars (positions 4 and 5) the planet's position in the sky moves backwards.

Venus and Mercury moved back and forth across the position of the Sun, sometimes rising before the Sun as morning stars, sometimes setting after it as evening stars, but never appearing in the midnight sky. Mars, Jupiter and Saturn, on the other hand, did not follow the Sun in the sky. They all tended to move in the same direction around the ecliptic, but now and then they would stop, move backwards for a while ("retrograde motion"), and then resume their usual motion.

here

Ptolemy tried to explain all that, using a theory which started with Hipparchus. The Sun and Moon, he claimed, obviously moved around Earth. To the Greek, the circle represented perfection, and Ptolemy assumed these bodies moved in circles too. Since the motion was not exactly uniform, he assumed that the center of these circles was some distance away from the Earth.

While the Sun moved around Earth, Venus and Mercury obviously moved around it, on circles of their own. But what about Mars, Jupiter and Saturn? Cleverly, Ptolemy proposed that like Venus and Mercury, each of them also rotated around a point in the sky that orbited around Earth like the Sun, except that those points were empty. The backtracking of the planets now looked similar to the backtracking of Venus and Mercury. The center carrying each of those planets accounted for the planet's regular motion, but to this the planet's own motion around that center had to be added, and sometimes the sum of the two made the planet appear (for a while) to advance backwards.

This "explanation" left open the question what the planets, Sun and Moon were. But worse, it was also inaccurate. As the positions of the planets were measured more and more accurately, additional corrections had to be introduced.

Yet Ptolemy's view of the solar system dominated European astronomy for over 1000 years. One reason was that astronomy almost stopped in its tracks during the decline and fall of the Roman Empire and during the "dark ages" that followed. The study of the heavens continued in the Arab world, under Arab rulers, but of all the achievements of Arab astronomers, the one which exerted the greatest influence was the preservation and translation of Ptolemy's books.

Nature

Copernicus (1473-1543)

The full story of the discovery of the solar system--of Copernicus, Galileo, Tycho and Kepler--is long and would take this discussion too far afield. Excellent books exist on the subject--for instance, "The Sleepwalkers" by Arthur Koestler.

Nicolaus Copernicus

Nicholaus Copernicus (the Latin version of Koppernigk) was a Polish church official whose passion was astronomy, and who actually performed some observations. By that time, all sorts of corrections had to be made to fit the motion of the planets to Ptolemy's ideas. Copernicus proposed an alternative theory--that the Earth was a planet orbiting the Sun, and that all planets moved in circles, one inside the other. Mercury and Venus had the smallest circles, smaller than that of the Earth, and therefore their position in the sky was always near the Sun's. That made it easy to estimate their distances from the Sun in terms of the Earth-Sun distance. Mars, Jupiter and Saturn moved in bigger circles, and they moved more slowly, so that whenever the Earth overtook them, they seemed to move backwards.

Copernicus was quite cautious in voicing his theory: not only did it deny that the Earth was the center of the universe, but it, too, did not fully describe the motion of the planets. Some corrections were still needed. Being associated with the church (as practically all European scholars were in those days), Copernicus had to abide by a rigid discipline, and he therefore hedged his ideas and only published them at the end of his life. Because of his caution, many church scholars indeed viewed his theory as a possible alternative to Ptolemy's.

Galileo Galilei (1546-1642)

Many books and plays exist on the life of Galilei, the Italian scholar who laid the foundation to the discipline known for many years as "natural philosophy," now called physics.

He was the first to observe the planets through a telescope, and what he saw convinced him that Copernicus was right. How his agressive defense of the Copernican theory turned the Catholic church against him and cost him his freedom is a fascinating story, but it goes beyond our scope here.

Galileo did not invent the telescope; that was done by lensmakers in Holland and elsewhere (eyeglasses had been in use for centuries). Unlike later astronomical telescopes, which turn the picture upside down, the first version worked the way opera glasses do, combining two lenses of different types. Opera glasses magnify about 2-3 times: Galileo pushed the technology to its limits, magnifying his view 8-fold and in a later instrument 33 times.

That was the instrument with which, in 1609-10, Galileo made his revolutionary discoveries. He observed the Moon and saw a world with mountains and "seas," and risking blindness (since the Sun should never be looked at through a telescope) he also observed sunspots. When he turned his telescope to the planet Jupiter, he saw four moons orbiting around it, all practically in the same plane, close to the ecliptic (and therefore, they and the planet all seem to lie on the same straight line; you can get the same view through good binoculars or any telescope), very much like a miniature version of the kind of solar system proposed by Copernicus.

And when he looked at Venus, he saw its visible shape changing like that of the moon, becoming a crescent when Venus was between us and the Sun, a time when most of its sunlit half faced away from Earth. Galileo was persecuted for advocating the world view of Copernicus, but his observations, which were soon confirmed by other astronomers, convinced all scholars that this was indeed the way the Sun, Earth, Moon and the planets were related.

Exploring Further:

Site about Ptolemy.

Site on Galileo.

Very detailed and long site on the history of astronomy.

Arthur Koestler, The Sleepwalkers, Arkana reprint edition 1990

A concise review, by Owen Gingerich, of "De Revolutionibus Orbium Celesium " (On the Revolutions of Heavenly Spheres) by Nicolaus Copernicus, appeared in the journal Nature--vol. 391, p. 140, 8 January 1998. A 1992 translation of the book by Rossen (452 pages) is available from Willmann-Bell ($39.95)--see http://www.willbell.com. That publisher is also offering other writings by Copernicus, "Galileo at Work" by Drake and various books on the history of astronomy.

Laura Fermi and Gilberto Bernadini: Galileo and the Scientific Revolution, Basic Books 1962

The History and Practice of Ancient Astronomy by James Evans, Oxford Univ. Press, 1998; reviewed by J.D. North, "Nature ", vol. 398, p. 385, 1 April 1999

To buy a bumper sticker "Living on Earth is Expensive but it Includes a Free Trip Around the Sun" go to item #591 under "Science and Education Bumper Stickers" at this web site.

Next Stop: #10 Kepler and his Laws

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(10) Kepler and His Laws

Tycho Brahe (1546-1601)

Tycho was a Danish nobleman interested in astronomy. In 1572 a "new star" (in today's language, a nova) appeared in the sky, not far from Polaris, outshining all others. Tycho carefully measured its position, then measured it again 12 hours later, when the the rotation of the Earth had moved the observing point to the other side of the Earth. Such a move was already known to shift the position of the Moon in the sky, helping astronomers estimate its distance. The position of the "new star" did not change, suggesting it was much more distant than the Moon (click here for the rest of the story).

Drawing of Tycho Brahe.

This event so impressed young Tycho that he resolved to devote himself to astronomy. The king of Denmark supported him and gave him the island of Hven to build an observatory, with the taxes of the island providing him with the funding. The telescope had not yet been invented, and all measurements were done by eye, aided by sights (similar to those used on guns) which could be slid around circles, marked in degrees. Tycho extended such methods to their ultimate limit, the resolution of the human eye, and his star charts were far more accurate than any earlier ones. He even measured and took into account the very slight shift of star positions near the horizon, due to the bending of light in the Earth's atmosphere, similar to its bending in glass or water. And his observations of the planets became the most stringent test of the theories of Copernicus and Ptolemy.

Concerning those theories, Tycho believed that all planets revolved around the Sun, but the Sun circled Earth. That view might have suited Denmark's Protestant church, for Martin Luther, founder of the Protestant doctrine, had rejected the views of Copernicus (who lived at the same time). Tycho's manners, however, were arrogant, and the residents of Hven complained about him, so that after the death of the king who was Tycho's patron, Tycho was forced to leave Denmark.

Johannes
Kepler

He settled in 1599 in Prague--now the Czech capital, then the site of the court of the German emperor Rudolf--and there he became court astronomer. It was in Prague, too, where a German astronomer named Johannes Kepler was hired by Tycho to carry out his calculations. When in 1601 Tycho suddenly died, it was Kepler who continued his work.

(A few more notes and links about Tycho.)

Johannes Kepler (1571-1630)

Kepler had studied astronomy long before he met Tycho: he favored the Copernican world-view and corresponded with Galileo.

Tycho's observations included some very accurate measurements of the position of the planet Mars, which did not agree with either Ptolemy or Copernicus. When Tycho died, Kepler got hold of those observations and tried to puzzle them out. In 1609, the same magic year when Galileo first turned his telescope towards the heavens, Kepler caught a glimpse of what he thought might be the answer. That was when he published his first two laws of planetary motion:

Planets move along ellipses, with the Sun at one focus.
The line from the Sun to the planet
covers equal areas in equal times.

Each of these statement requires some explanation.

Ellipses!

The ellipse, the shape of a flattened circle, was well known to the ancient Greeks. It belonged to the family of "conic sections," of curves produced by the intersections of a plane and a cone.

The curves generated as
"conic sections" when flat
planes are cut across a cone.

As the drawing above on the left shows, when that plane is...

--perpendicular to the axis of the cone, the result is a circle.

--moderately inclined, an ellipse.

--inclined so much that it is parallel to one side of the cone, a parabola.

--inclined even more, a hyperbola.

All these intersections are easily produced by a flashlight in a moderately dark room (drawing below). The flashlight creates a cone of light and when that cone hits a wall, the shape produced is a conic section--the intersection of the cone of light with the flat wall.

The Third Law

The axis of the flashlight is also the axis of the cone of light. Aim the beam perpendicular to the wall to get a circle of light. Slant the beam: an ellipse. Slant further, to where the closing point of the ellipse is very, very far: a parabola. Slant even more, to where the two edges of the patch of light not only fail to meet again, but seem to head in completely different directions: a hyperbola.

After Tycho's death, Kepler became the court astronomer, although the superstitious emperor was more interested in astrology than in the structure of the solar system. In 1619 Kepler published his third law: the square of the orbital period T is proportional to the cube of the mean distance a from the Sun (half the sum of greatest and smallest distances). In formula form

T²= k a³

with k some constant number, the same for all planets. Suppose we measure all distances in "astronomical units" or AUs, with 1 AU the mean distance between the Earth and the Sun. Then if a = 1 AU, T is one year, and k with these units just equals 1, i.e. T²= a³. Applying now the formula to any other planet, if T is known from the observations of many years, the planet's a, its mean distance from the Sun, is readily derived. Finding the value of 1 AU in miles or kilometers, that is, finding the actual scale of the solar system, is not easy. Our best values nowadays are the ones provided by space-age tools, by radar-ranging of Venus and by planetary space probes; to a good approximation, 1 AU = 150 000 000 km.

Kepler's 3rd Law T in years, a in astronomical units; then T² = a³ Discrepancies are from limited accuracy
Planet	Period T	Dist. a fr. Sun	T²	a³
Mercury	0.241	0.387	0.05808	0.05796
Venus	0.616	0.723	0.37946	0.37793
Earth	1	1	1	1
Mars	1.88	1.524	3.5344	3.5396
Jupiter	11.9	5.203	141.61	140.85
Saturn	29.5	9.539	870.25	867.98
Uranus	84.0	19.191	7056	7068
Neptune	165.0	30.071	27225	27192
Pluto	248.0	39.457	61504	61429

Not only were Kepler's laws confirmed and explained by later scientists, but they apply to any orbital system of two bodies--even artificial satellites in orbit around the Earth. The constant k' for artificial satellites differs from k obtained for planets (but is the same for any satellite). By Kepler's formula

T = SQRT (k' a³)

where SQRT stands for "square root of" (the world-wide web does not offer more specific symbols). If T is measured in seconds and a in Earth radii (1 R_E = 6371 km = 3960 miles)

T = 5063 SQRT (a³)

More will be said about Kepler's first two laws in the next two sections.

Kepler's later years were not too happy. His patron, Emperor Rudolf, died in 1612, and although Kepler retained his post as court mathematician and continued to produce important work, his life was increasingly disrupted by war. That was the 30 years' war, a bitter religious battle which pitted Protestants against Catholics; it began in Prague in 1618 and engulfed all of Kepler's part of Europe.

Postscript: The remanant of Tycho's supernova is still visible. In August 1999 it was one of the first targets of NASA's orbiting x-ray telescope "Chandra," resulting in the picture on the right; the small bright spot at the center of the cloud could possibly be the left-over star (see also here). You might compare this to the picture (in visible light) of an older supernova remnant. For more about the "Chandra," observatory, see its web site (click here to return to the top of the page).

Exploring Further:

A site about Tycho Brahe, illustrated, here

Detailed site about Kepler

Picture and links concerning Kepler

Willman-Bell, publishers of astronomy books (http://www.willbell.com) is offering "The Lord of Uraniborg" by Thoren, a biography of Brahe, and "Kepler" by Casper. See also the end of the preceding section, concerning "The Sleepwalkers" by Koestler.

For Teachers: Lesson plan on Kepler's laws of planetary motion.

Next Stop: #10a. The Scale of the Solar System

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The Scale of the Solar System

(10a) The Scale of the Solar System

Kepler's laws agree with all observed planetary motions, and by the table in the previous section, they give the correct proportions of all planetary orbits. If the mean distance of Earth from the Sun is 1 AU ("Astronomical Unit"), then that of Venus is 0.723 AU, of Mercury 0.387 AU and that of Mars is 1.524 AU. But how much is that in kilometers, or miles? In other words--what are the actual dimensions, not just their proportions?

Remember how Hipparchus estimated the distance of the Moon? In a solar eclipse which was total in one location, at another location about 1000 kilometers away, only 80% of the Sun was covered. The body blocking the Sun--the Moon--was close enough that moving an observer by about 1000 kilometers shifted its apparent position in the sky by 1/5 the apparent size of the Sun, or about 0.1 degree.

Tycho still accepted the erroneous estimate by Aristarchus of the Sun's distance, 20 times smaller than the actual one (see section about Aristarchus). Since the Sun's distance sets the scale of the entire solar system, Tycho believed Mars was close enough for its apparent position in the sky to be shifted measurably as the Earth's rotation carried an observer from one side of the globe to the other. Actually, the solar system is much bigger, and the shift was too small to be seen by Tycho's pre-telescope equipment. The story (which has additional twists) is told in "Tycho and the ton of gold" by Owen Gingerich in "Nature", vol 403, p. 251, 20 January 2000.

If we know the proportions of all the orbits in the solar system, measuring just one actual distance in kilometers gives the scale of all orbits around the Sun. Kepler suggested measuring the distance to the planet Mercury when it passed in front of the Sun, but (as Halley noted) Venus is closer and offers a better choice. Now and then Venus passes in front of the Sun, and a telescope observing the Sun (by projecting its image, or using a dark filter) sees its dark disk on the bright solar background. By comparing where on the Sun's disk is the crossing seen from two far-apart points on Earth, and comparing the times Venus was observed crossing the edge of the Sun, one can calculate the distance to Venus and from it the scale of the solar system.

Unfortunately, this never happened during Halley's lifetime. "Transits of Venus" occur in pairs, more than a century apart. One occured in 1639--too early. The next ones did not take place until 1761 and 1769, and astrtonomers were prepared for them. One of the goals of the famous expedition by Captain James Cook to the Pacific Ocean was to observe the transit from a point far from other observers.

No transits of Venus occured in the 20th century, but the next one is due on June 8, 2004. Things being the way they are, you may very well have an opportunity to watch it over the world-wide web. To get ready, read the book June 8, 2004: Venus in Transit by Eli Ma'or, Princeton University Press, 2000, 186 pp., $22.95 (reviewed by Don Fernie in "Nature" vol 406, p. 562, 10 August 2000).

Later astronomers realized that some asteroids passed quite close to Earth. Today we worry about any actually hitting us--but their discovery also made some astronomers happy. Because of their nearness, their distance could be measured much more accurately and it gave a much better estimate of the AU. Still later the giant radio telescope whose (fixed) dish is nestled in a valley near Arecibo, Puerto Rico, was used as a radar to bounce signals off the planet Venus, and timing their "echo" gave an even more accurate estimate of the AU. Today, of course, we also can use the orbital mechanics of space probes, tracked by radio as they pass near major planets.

Next Stop: #11. Graphs and Ellipses

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(11) Graphs and Ellipses

The laws of orbital motion are mathematical, and one cannot explore them without some mathematics. The math used here is rather elementary; if you need a refresher click here. Otherwise, you can just skip the equations and follow the narrative.

The Mathematical Description of a Curve

As already noted, the cartesian system labels any point in a plane (e. g. on a flat sheet of paper) by a pair of numbers (x,y), its distance from two perpendicular axes. These numbers are known as the "coordinates" of the point.

A line in the plane--straight or curved--contains many points, each with its own (x,y) coordinates. Often there exists a formula ("equation") which connects x and y: for instance, straight lines have a relationship

y = ax + b

where any pair of numbers (a,b), positive, negative or zero, gives some straight line. The plot of a line given by one of such relationship (or indeed by any relationship--even pure observation, e.g. temperature against time--is known as a graph. More complicated relationships give graphs that are curves: for instance

y = ax²

gives a parabola, with a any number. Usually (though not always) y is isolated, so that the formula has the form

y = f(x)

where f(x) stands for "any expression involving x" or in mathematical terms, a "function of x." The curves drawn here are the straight line y = -(2/3)x + 2 and the parabola y = x². A list of some of their points follows.

Straight line:

x	-1	0	1	2	3	4
y	8/3	2	4/3	2/3	0	-2/3

Parabola:

x	-2	-1.5	-1	-0.5	0	0.5	1	1.5	2
y	4	2.25	1	0.25	0	0.25	1	2.25	4

The Equation of a Circle

In the vast majority of formula-generated graphs, the formula is given in the form

y = f(x)

Such a form makes it very easy to find points of the graph. All you have to do is choose x, calculate f(x) (= some given expression involving x) and out comes the corresponding value of y.

However, any equation involving x and y can be used as the property shared by all points of the graph. The main difference is that with more complicated equations, after x is chosen, finding the corresponding y requires extra work, (and sometimes it is easier to choose y and find x).

Perhaps the best-known graph of this kind is a circle of radius R, whose equation is

x² + y² = R²

Draw a circle of radius R centered at the origin O of a system of (x,y) axes . Given any point P on the circle with specified values of (x,y), draw a perpendicular line from P to point A on the x-axis. Then

x = OA y = AP R = OP

Here x and/or y may be negative, if they are to the left of the y-axis or below the x-axis, but regardless of sign, x² and y² are both always positive. Since the triangle OAP has a 90° angle in it, by the theorem of Pythagoras, for any choice of P, the relation below always holds:

OA² + AP² = OP²

Since this can also be written

x² + y² = R²

The equation of the circle is satisfied by any point located on it. For instance, if the graph is defined by the equation:

x² + y² = 25

this equation is satisfied by all the points listed below:

x	5	4	3	0	-3	-4	-5	-4	-3	0	3	4	( 5 )
y	0	3	4	5	4	3	0	-3	-4	-5	-4	-3	( 0 )

The Equation of an Ellipse

The equation of the circle still expresses the same relation if both its sides are divided by R²:

(x²/R²) + (y²/R²) = 1

The equation of an ellipse is a small modification of this:

(x²/a²) + (y²/b²) = 1

where (a,b) are two given numbers, for example (8,4). What would such a graph look like? Near the x axis, y is very small and the equation comes close to

(x²/a²) = 1

From which

x² = a² and hence x = a or x = -a (sometimes combined into x = ±a)

The graph in that neighborhood therefore resembles the section of a circle of radius a, whose equation

(x²/a²) + (y²/a²) = 1

also comes close to x² = a² in this region. In exactly the same way you can show that near the y-axis, where x is small, the graph cuts the axis at y=±b and its shape there resembles that of a circle of radius b.

An example

Let us draw the ellipse

(x²/64) + (y²/16) = 1

We already know that it cuts the axes at x=±8 and at y=±4. Let us now add a few points:

(1) Choose y = 2 . Then from the equation

(x²/64) + (4/16) = 1

Substract 1/4 from both sides

(x²/64) =3/4

Take square roots (marked here by the letter SQRT) and retain only 3-4 figures:

x/8 = SQRT(3)/SQRT(4) = 1.732/2 = 0.866

from which x = 6.93 within resonable accuracy.

(2) Choose y = 3 . Then from the equation

(x²/64) + (9/16) = 1

Substract 9/16 from both sides

(x²/64) =7/16

Take square roots (to an accuracy of 3-4 figures):

x/8 = SQRT(7)/SQRT(16) = 2.6457/4 = 0.6674

from which, approximately, x = 5.29

Again, either sign can be attached to x and y. We get 12 points, enough for a crude graph:

x	8	6.93	5.29	0	-5.29	-6.93	-8	--6.93	-5.29	0	5.29	6.93	( 8 )
y	0	2	3	4	3	3	0	-2	-3	-4	-3	-2	( 0 )

A Different View of the Ellipse

The collection of all points for
which R1 + R2 has the same value
is an ellipse

The ellipse was already familiar to ancient Greek scientists (who fell under the term "philosophers", lovers of wisdom), but they defined it differently. To them the ellipse was the collection of all points (in a flat plane) for which the sum of the distances R₁ + R₂ from two given points was the same (see drawing).

It was a natural extension of the definition of a circle, which is the collection of all points at the same distance (the radius R) from one given point (the center). One point defines a circle, two define an ellipse.

The two points were called the foci of the ellipse (each one, a focus), and they are important here, because Kepler found that the Sun always occupied a focus of the orbital ellipse, not (as one might perhaps think) the center--i.e. the origin, when the ellipse is given by its equation in a system of perpendicular (x,y) axis.

(x²/a²) + (y²/b²) = 1

The foci are always located on the longer of the two symmetry axes of the ellipse--the (x, y) axes when the above equation is used--the major axis of the ellipse. If a is larger than b, the majore axis lies along the x-axis, and we leave it to you to show that in such a case

R₁ + R₂ = 2a

[Hint: Make a sketch of the ellipse and the axes which define it, mark one of the points at which it crosses the x-axis, and examine R₁ and R₂ of that point]. The value of a in an elliptic orbit is known in astronomy as the semi-major axis and it is regarded as one of the six orbital elements which define the motion according to Kepler's laws.

Whispers in the US Capitol

The foci of an ellipse have an interesting property. An ellipsoid of revolution is the 3-dimensional figure obtained by rotating an ellipse around one of its axes. If a hollow ellipsoid of this shape is made and its inner surface is silvered to act as a mirror, then if a source of light is placed at one focus, all of its rays will converge at the other one. Even if only part of the ellipsoid is silvered, all light hitting that part will still be concentrated at the other focus.

Sound waves can behave like light, too. The chamber (=room) in the US capitol in Washington in which the House of Representatives used to meet has a ceiling shaped like a quadrant (half of a half) of an ellipsoid, with its foci near the floor. This was done for architectural reasons, nearly 200 years ago, but it also enabled a person at one focus to overhear anything spoken at the other focus, even whispers. Supposedly, Daniel Webster sat at one such spot and made good use of its special nature.

Today the House of Representatives has many more members, uses a much bigger chamber and its old meeting room is a museum displaying statues of distinguished Americans.

Every year many thousands of visitors are guided through that chamber. At one time during their visit, they gather at or near one focus (identified by a brass marker on the floor), to catch whispers by their guide who stands at the other one. By the way, a well-known painting of that chamber, with faces of its occupants clearly identifiable, was made by Samuel Morse, the artist who also invented the electric telegraph. A copy of that painting and its story are also shown in the room; the original is in the Corcoran gallery in Washington.

Next Stop, for those
familar with trigonometry-- #11a Ellipses and Kepler's First Law

Trigonometry--what is it good for?

Otherwise, next stop is: #12 Kepler's Second Law

(11a) Ellipses and Kepler's First Law

As already noted, other ways exist for labeling points in the plane. For instance, a point P may be labeled by its distance r from a central point O ("origin") and the angle f (or Greek f) which the line OP makes with some standard direction. Such "polar coordinates" (drawing on the left, below) are the ones best suited for describing planetary motion.

The Ellipse in Polar Coordinates

Again, if all the values of (r,f) of a curve are related by some equation which can be symbolically written

r = r(f)

then the function r(f) is said to be the equation of the line, in polar coordinates. The simplest function is a constant number a, giving the line

r = a

The value of r equals a for any value of f. That gives a circle around the origin, its radius equal to a, shown in the drawing on the right above.

The Ellipse

Consider next the curve whose equation is

r = a(1- e²)/(1+ e cos f)

where the eccentricity e is a number between 0 and 1. If e = 0, this is clearly the circle encountered earlier.

How about other values? The function cos f represents a wave-like behavior (picture below), and as f goes through a full circle, it goes down, from +1 to 0, then -1, then up again to 0 and +1. The denominator also rises and falls as a wave, and it is smallest when cos f = -1 . Here is the table of the main values (360 is in parentheses, because it represents the same direction as 0 degrees):

f degrees	0	90	180	270	(360)
cos f	1	0	-1	0	1
1 + cos f	1 + e	1	1 - e	1	1 + e

As long as e is less than 1, the denominator is always positive. It is never zero, so that for any f one can name, one can always find a suitable r. In other words, the curve goes completely around the origin, it is closed.

The expression (1 - e²) can be factored--that is, written as two expressions multiplied by each other ("the product of two expressions"). As discussed in the section on algebraic identities

1 - e² = (1 - e)(1 + e)

At some of the points on the above table, either (1 - e) or (1 + e) cancels the denominator, giving:

f degrees	0	90	180	270	(360)
r	a(1 - e)	a(1 - e²)	a(1 + e)	a(1 - e²)	a(1 - e)

The distance of the line from the origin thus fluctuates between a(1 - e) and a(1 + e), and the result is a flattened circle or ellipse; the point O (the origin) is its focus. All planetary orbits resemble ellipses, each with its own value of e or eccentricity: the smaller e is, the closer the shape to a circle. The Earth's orbit is very close to a circle, with e = 0.0068, and other major planets (except for Pluto) have comparable eccentricities: if you saw a scale drawing of that orbit on a sheet of paper, your eye would not be able to tell it apart from a circle. The orbit of Comet Halley, on the other hand, has e quite close to 1.

The collection of all points for
which R1 + R2 has the same value
is an ellipse

As mentioned in the preceding section a second focus O' can be drawn symmetrical to O, and the ellipse can be defined (its original definition, in fact) as the collection of points for which the sum R₁+R₂ of their distances from O and O' is always the same

The longest dimension of the ellipse, its width AB along the line connecting the two foci, is its "major axis." Suppose (R₁,R₂) are the distances of A from the foci O and O'. Then R₁ = OA =a(1 - e) is the smallest distance of the ellipse from O, R₂ = O'A = OB (by symmetry) is the largest and therefore equals a(1 + e). But, OA + OB = AB, hence

AB = a(1 - e) + a(1 + e) = 2a

Hence the quantity a in the equation of the ellipse is known as semi-major axis. We can now state Kepler's law more precisely as "the square of the orbital period T is proportional to the cube of the semi-major axis a of its orbit around the Sun."

The two quantities (a,e) completely define the ellipse. When that ellipse is the orbit of a planet or a satellite, they form two of the six orbital elements which define the state of the orbiting body. A third element, the mean anomaly M, specifies the position of the planet or satellite along the orbit, and the remaining three define the orientation of the orbit in 3-dimensional space. More about the orbital elements can be found in section (12a)

Refining the First Law

Kepler's first law was:

"The orbit of a planet is an ellipse, with the Sun at one focus"

Actually, this is not 100% accurate. Imagine the planet magically getting heavier and heavier, while the Sun is getting lighter and lighter. At some point both of them would be equally heavy: could we then say which is orbiting around which?

To be fully accurate, the first law should have placed the focus of the orbital ellipse at the center of gravity of the planet-Sun system. (The center of gravity will be defined later, but intuitively, if the masses are very unequal, as with a planet and the Sun, it lies close to the center of the heavier object.) Because the Sun is so much heavier than the Mars, the effect on the orbit of Mars (which Kepler studied) was too small to be noted by him. Nevertheless, the Sun also moves in response to motions of its planets, and motions of this type have become an important tool in the search for planets outside the solar system.

An Earth-sized planet orbiting a distant star would be far too dim to be seen with any earthly telescope, especially against the glare of its sun, the star itself. However, as the planet goes around the orbit, its star also moves in a mirror image orbit around the common center of gravity. It is a much smaller orbit and a much slower motion, because the center of gravity is very close to the center of that star (in the Earth-Sun system, it is inside the Sun), but it can still be detected by subtle variations of the starlight. Recently a few such planets have been found, but most are Jupiter-sized and none seems suitable for life. The search, however, goes on, and the most recent discovery (15 April 1999) has been the system of upsilon Andromeda which appears to contain at least 3 planets. For another recent discovery associated with a distant planet, click here.

Bodies in the solar system can also move in other conic sections, in parabolas or hyperbolas, whose equations resemble that of an ellipse, but have e equal to 1 or larger. Properly speaking, these orbits do not have a "semi-major axis," because they are not limited in size like ellipses but extend to infinity. Instead, we must write

r = p/(1 + e cos f)

which doe not raise any problems at e=1. Such bodies are not bound to the Sun, but are free to escape it. The denominator in the equation of the trajectory then becomes zero for some values of f, making its r infinite, and as the moving body approaches those values, it moves further and further away, with no limit. Comets in general have an eccentricity e close to 1, suggesting they have come from the very distant fringes of the solar system. The space probe Voyager 2 has e > 1 and is on its way out of the solar system, never to return.

Next Stop: #12 Kepler's Second Law

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(12) Kepler's Second Law

The Law

The ellipse traced by a planet around the Sun has a symmetric shape, but the motion is not symmetric.

Think of a stone thrown upwards: as it rises it loses speed, then for an instant, at the top of the trajectory, it moves very slowly, and finally it comes down, gathering speed again. The motion of a planet around the Sun or of a scientific satellite around Earth follows different equations (though a connection exists), but in many respect it resembles that of the stone.

That is most evident if the orbit is elongated, that is, its eccentricity is not far from 1. As the planet or satellite rises in its orbit, it slows down, then as it returns, it speeds up again, moving at its fastest during its closest approach. That point of the orbit is called perihelion for a planet ("helios" is the Sun) and perigee for an Earth satellite ("gee" from "geo", denoting Earth-related).

After studying actual observations, mainly of Mars, Kepler proposed the following prescription for predicting the speeding-up and the slowing-down. Let a line ("radius vector") be drawn from the center of the Sun to the planet (or from the center of Earth to the satellite). Kepler's law states:

"The radius vector sweeps equal areas in equal times"

Illustrating Kepler's 2nd law:
segments AB and CD take
equal times to cover.

As an example, let the drawing on the right represent the orbit of an Earth satellite, and let AB and CD be the portions of the orbit covered in 3 hours near apogee and near perigee, respectively. If then O is the Earth's center, the shaded areas OAB and OCD are equal. What it means, obviously, is that CD is much longer than AB, because near perigee the satellite moves much faster and it covers a much greater distance in 3 hours.

Energy

Energy may be loosely defined as anything that can make a machine move. The forms of energy which power our machines are usually electricity or heat; light is another form, converted into electricity by the solar cells which power most satellites.

Gravity can also provide energy. The wheels of grandfather clocks are turned by weights which gradually descend to the bottom of the clock, at which point they have to be cranked up again, or else the clock stops. Thomas Jefferson, at his home near Charlottesville, Virginia, had a clock whose weights (hanging on the side of the room) were cannonballs strung on a rope, and to give the clock a 7-day range, a hole was cut in the floor allowing the balls to descend to the basement.

When a weight or cannonball is raised against the force of gravity, it has potential energy--energy by virtue of its position, proportional to the height to which it was raised. If the weight is dropped, it loses height and potential energy, but gains speed and kinetic energy, the energy due to speed of motion. Kinetic energy can be converted back to potential, as happens to a roller coaster after it passes the bottom of a dip and climbs up again.

A similar change occurs when a stone is thrown upwards with some velocity v. If its mass is m (mass will be defined later, for now view it as something related to weight), its kinetic energy can be shown to be

1/2 mv²

As it rises, v and the kinetic energy decrease, but this is matched by the growth of the potential energy

h m g

where h is the height in meters and g is a constant measuring the strength of the force of gravity: if m is in kilograms, h in meters and v in meters-per-second (written m/sec; walking speed is about 1-2m/sec), g is about 9.81.

The sum of the two is the total energy E and stays constant:

E = 1/2 mv² + h m g = constant

As the stone rises, the kinetic part of its energy gets smaller and smaller, becoming zero when it reaches its highest point, where for a brief instant v = 0. On the downward trip, the opposite changes take place. In a later section we will come back to that formula and to the concept of energy.

For a satellite of mass m orbiting Earth (or for a planet around the Sun) a similar formula exists:

E = 1/2 mv² - k m/r = constant

Here k is some other constant--actually, related to g, because both constants reflect the strength of the Earth's gravity (the exact value is k = gR², where R is the radius of the Earth, in meters). Don't let the minus sign confuse you: as the satellite rises, r increases, k m/r becomes smaller, but -k m/r becomes bigger, it is less negative than near Earth. This equation shows why the satellite's speed decreases as it moves away and grows as it comes back.

Suppose the satellite has just enough velocity to escape Earth's gravity altogether (the "escape velocity" V). Then far from Earth, where k m/r is close to zero, its kinetic energy would also be exhausted, that is, v = 0. Since the sum E is the same everywhere, this suggests that for the space probe which just barely escapes the Earth's gravity, E=0. From that

V² = 2k/R = 2 g R

With g = 9.81 and R =6 371 000 meter on finds V to be about 11200 m/sec.

The Mean Anomaly

Earlier it was stated that a third orbital element is needed to specify where in its orbit is the satellite located. Since the equation of the orbital ellipse is

r = a(1 - e²)/(1 + e cos f)

each value of the angle f --called the "true anomaly"--specifies a position along the orbit. One could therefore use the true anomaly as third orbital element.

The true anomaly f varies periodically around the orbit, quickly near perigee and slowly near apogee. Kepler's second law tells us everything about this variation and should allow us to obtain a formula that gives the way f varies with time t. Unfortunately, no neat way exists for expressing that formula.

The simplest way of expressing f is to use two auxiliary angles, which like f increase by 360 degrees each orbit, the "eccentric anomaly" E (the letter here has nothing to do with energy) and the "mean anomaly" M; an equation then exists connecting f and E, and another that connects E and M. The great virtue of M is that it grows in proportion to the time t:

M = M(0) + nt

where M(0) is the value of M when t = 0 and n is a constant (related to the constant appearing in Kepler's third law). The mean anomaly is what is counted as the third orbital element.

If one wishes to predict the position of a satellite in its orbit at some time t, assuming the elliptical motion of Kepler's laws is good enough for that prediction (neglecting the pull of the moon, friction of the upper atmosphere etc. ) the first step is to derive M from the above formula. Then E is derived from E, and finally f from E, tasks which electronic computers handle quite easily (though at one time, those calculations were done on paper, not nearly as quickly or easily). The formula for r then gives the position of the satellite in its orbit; all that the computation requires are the elements a, e and M(0), the mean anomaly at t=0.

Below, a drawing of the orbit of Mars, from Kepler's writings

More Nitty Gritty: #12a How Orbits are Calculated

Next Regular Stop: #13 The Way things Fall (starting Newtonian mechanics)

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(12a) How Orbital Motion is Calculated

This section is on a higher level than the rest, and is primarily meant for advanced users, who may wonder how orbital motion is actually derived. For others it provides a peek at the complexity of orbital calculations, or it may be skipped.

As stated earlier, the motion of a satellite (or of a planet) in its elliptical orbit is given by 3 "orbital elements":

(1) The semi-major axis a, half the greatest width of the orbital ellipse, which gives the size of the orbit.

(2) The eccentricity e, a number from 0 to 1, giving the shape of the orbit. For a circle e = 0, larger values give progressively more flattened circles, up to e = 1 where the ellipse stretches to infinity and becomes a parabola. The orbits of all large planets are rather close to circles: that of Earth, for instance, has e=0.0068

(3) The mean anomaly M, an angle growing at a steady rate, increasing by 360 degrees each orbit

M = M(0) + 360°(t/T)

where M(0) is the value of M at time t=0 and T is the orbital period. Given those numbers, M is readily calculated for any time t.

However, the actual position of the satellite is given by the true anomaly f. In polar coordinates (r,f) describing the satellite's motion in its orbital plane, f is the polar angle. The equation of the orbit is

r = a(1 - e²)/(1 + e cos f)

The angle f also grows by 360^o each full orbit, but not at all uniformly. By Kepler's law of areas, it grows rapidly near perigee (point closest to Earth) but slowly near apogee (most distant point).

The information needed to derive f for any time t is contained in the law of areas, but the actual calculation is not easy. The process involves an auxiliary angle, the eccentric anomaly E which like f and M grows by 360^o each orbit. At perigee, all three anomalies equal zero.

The drawing on the right gives a geometric construction of those angles (no, don't try to puzzle out the details). The orbital ellipse is enclosed in a circle of radius a, and given a position P of the satellite, a corresponding point Q on the circle can be drawn, sharing the same line perpendicular to the ellipse's axis. Then E is the angle between the long axis of the ellipse and the line drawn from the center of the circle to Q ("eccentric" might mean here "from the center").

Kepler's Equation

Suppose the elements a, e and M(0) at time t=0 are given, and we need to find the value of f at some different time t. With f known, the above equation gives r, and (r, f) together pin-point the satellite's position in its orbital plane. The first step is to derive

M = M(0) + 360°(t/T)

We assume the period T is known (this requires the 3rd law and is discussed for circular orbits in sections 20 and 20a). It can then be shown that the angle E satisfies "Kepler's equation"

M = E - (180°/p)e sinE

where p = 3.14159256... is the ratio between the circumference of a circle and its diameter. How did that number suddenly crop up, you may ask? The fact is, the division of the circle into 360 degrees may be convenient to use (we inherited it from the ancient Babylonians) but the number 360 has no particular place in mathematics. It is probably related to the number of days in a year. The "natural" division of angles which arises in calculus and other branches of math is into radians, with 360 degrees equal to 2p = 6.2831... radians (making each radian equal to about 57.3 degrees). With angles measured in radians, Kepler's equation simplifies to

M = E - e sinE

No matter which form is used, mathematics knows no formula which gives E in terms of M. However, solutions can often be approximated to any degree of accuracy by iteration--by starting with an approximate solution, then improving it again and again by an appropriate procedure ("algorithm"--more about that word, here). If the eccentricity e is not too big--the ellipse not much different from a circle--then M and E are not too different. So an initial guess

E' = M

may not be too far off. Putting this guess into the term sinE gives an improved guess E"

E" = M + (180°/p)e sinE'

One can now insert E" in the sinE term and get an even closer guess, and so on and so forth... until the first (say) ten decimal digits of the value of E no longer change, at which point we may decide we have E to sufficient accuracy and stop the process. Computer handle such a process of continuous improvement ("iteration of the solution"--one form of an algorithm) very rapidly, and other methods also exist, with sufficient speed even when e is not very small.

Given E, a number of formulas will give the true anomaly f. For instance, one can first derive

r = a(1 - e cos E)

And then cos f can be found from

r = a(1 - e²)/(1 + e cos f)

and sin f follows from cos f. All this is easily and automatically computed nowadays, but must have been a real hassle in the days before computers.

The Orbit in Space

The 3 remaining orbital elements are all angles giving the position of the orbit in 3 dimensions. They are described below, but their actual use belongs to a university course in orbital mechanics and will be omitted. The angles:

Inclination i.
The argument of perigee w (small Greek omega).
The longitude of the ascending node W (capital omega).

To orient the orbit in 3 dimensions requires a reference plane and a reference direction. For satellite orbits, the reference plane--the horizontal plane in the drawing--is usually the Earth's equatorial plane (sometimes it is the plane of the ecliptic). The reference direction in either case is the direction from the center of the Earth to the vernal equinox (which belongs to both above planes). We will call it the x direction, since that is its role in (x,y,z) coordinates used in orbital calculations.

Two non-parallel planes always intersect along a line--the way the plane of a door intersects the plane of the wall along the door's hinge. The orbital plane and the equatorial plane (used for reference) do so too, and their intersection is called the line of nodes N. Let the origin O of our coordinates be the center of the Earth, which is also the focus of the ellipse; this point belongs to both the equatorial plane and the orbital plane, and is therefore also on their intersection line N (drawing). Then...

The inclination i is the opening angle of the "hinge" along N. It is best defined by erecting at O lines perpendicular to each plane and measuring the angle between them (drawing).
The angle W is measured in the equatorial plane between N and the reference direction x. One can imagine rotating the "hinge" N around point O, without changing the inclination: the orbital plane then covers all possible values of W.
But what is this "ascending node" bit? The above definition contains some ambiguity: N defines two lines coming out of O, in opposite directions. From which of them should W be measured? To resolve this we note that the plane of the equator divides space into two parts, one north of it and one south of it. Specifying "the ascending node" selects the branch on which the satellite crosses as it enters the northern half-space, rather than the one crossed when leaving it.
Finally, w is the angle measured in the orbital plane between N and the direction from O to the perigee point P. If perigee lies on the "hinge", on the side of positive x, then w = 0; rotating the orbit by 90^o until the line OP is perpendicular to N gives w = 90^o, rotating it further until it reaches the negative side of the x-direction gives w = 180^o.

Suppose you have the orbital elements of some satellite, e.g. the space shuttle (you can often get them off the world-wide web). The first three (a, e, M), with M given at some particular time, enable you to calculate where the satellite will be at any time in its orbit. With (i, w, W) you can then find where it would be in the sky.

problem 8

Next Regular Stop: #13 The Way things Fall (starting Newtonian mechanics).

Author and curator: David P. Stern, u5dps@lepvax.gsfc.nasa.gov
Last updated 3 April 1999