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#10. Kepler & Laws
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#12a. More on 2nd law

Kepler's Three Laws of Planetary Motion

An Overview for Science teachers

By David P. Stern

        Below is a lecture given on March 23, 2005, to science teachers of Anne Arundel County, Maryland. It contains an overview of Kepler's laws with examples, applications, problems and related history, a resource for classroom materials
        It is keyed and linked to appropriate sections of "From Stargazers to Starships." The teachers were also given disks with the web material, allowing it to be accessed off-line.

  Much of this overview is drawn from "From Stargazers to Starships", a detailed course on astronomy, Newtonian mechanics, physics of the Sun and spaceflight. Its home page is and it also includes translations (Spanish and French), a glossary, a timeline, problems, over 150 answers to questions from users and more. It uses algebra and trigonometry (on which a short course is included), stresses conceptual understanding, history, applications and ties to culture and society, and its sections cover a wide range of levels, from middle school to freshman college.

    A quick guide to sections of "Stargazers" related to Kepler's laws can be found in the section "Kepler's Laws". In what follows, those sections will sometimes be referred to by their numbers. You can also reach the complete list of links either from "Site Map" at the top of this page or from "Back to the Home Page" at the end.

    Note that addresses here are abbreviated, because you are already logged onto "Stargazers."
    Thus the home page is Sintro.htm

    The disk has more material than can ever be covered in your class. But you are teachers, and need a wider knowledge--allowing you to pick and choose material according to circumstances, and mention odd tidbits without detailed discussion, just to create interest.

    And if you are very lucky, you may sometimes find in your class a kid or two who really want to find out. Those you can send to here, to the web, for a broader education.

    Let me focus on three items:
---what are Kepler's laws, what do they mean, and why are they important.

The laws were formulated between 1609 to 16l9, and are (as usually stated):

  1. Planets move around the Sun in ellipses, with the Sun at one focus
  2. The line connecting the Sun to a planet sweeps equal areas in equal times.
  3. The square of the orbital period of a planet is proportional to the cube (3rd power) of the mean distance from the Sun
        in (or in other words--of the"semi-major axis" of the ellipse, half the sum of smallest and greatest distance from the Sun)

The Significance of Kepler's Laws

    Kepler's laws describe the motion of planets around the Sun.
Kepler knew 6 planets: Earth, Venus, Mercury, Mars, Jupiter and Saturn.
[IMAGE: The orbit of the Earth around the Sun]
The orbit of the Earth around the Sun.
This is a perspective view, the shape of
the actual orbit is very close to a circle.

    All these (also the Moon) move in nearly the same flat plane (section #2 in "Stargazers"). The solar system is flat like a pancake! The Earth is on the pancake, too, so we see the entire system edge-on--the entire pancake occupies one line (or maybe a narrow strip) cutting across the sky, known as the ecliptic. Every planet, the Moon and Sun too, move along or near the ecliptic. If you see a bunch of bright stars strung out in a line across the sky--with the line perhaps also including the Moon, (whose orbit is also close to that "pancake"), or the place on the horizon where the Sun had just set--you are probably seeing planets.

        Ancient astronomers believed the Earth was the center of the Universe--the stars were on a sphere rotating around it (we now know, it actually was the earth turning) and the planets were moving on their own "crystal spheres" in funny ways. They usually moved in the same direction, but sometimes their motion reversed for a month or two, and no one knew why.

    A Polish clergyman named Nicholas Copernicus figured out by 1543 that those motions made sense if planets moved around the Sun, if the Earth was one of them, and if the more distant ones moved more slowly --so sometimes the Earth overtakes them, and they seem to move backwards for a while. The orbits of Venus and Mercury were inside that of Earth, so they never move far from the Sun. Which is why you never see Venus at midnight!

    I hope you that as you describe those features--the "pancake" of the ecliptic, reversed ("retrograde") motion, Venus always close to the Sun--to help students will get a feeling for the appearance of planets in the sky, as bright stars moving along the same track as Sun and Moon. The 12 constellations along that line are known as the zodiac, a name which should be familiar to those who follow astrology. Venus, the brightest planet, oscillates back and forth across the position of the Sun, and Mercury too--but since it is much closer to the Sun, you may only see it at its most distant from the Sun, and then only shortly after sunset or before sunrise.

    Students will probably have heard or read that the pope and church fought the idea of Copernicus, because in one of the psalms (which are really prayer-poems) the bible says that God "set up the Earth that it will not move" (that was one translation: a more correct one is "will not collapse"). Galileo , an Italian contemporary of Kepler who supported the ideas of Copernicus, was tried by the church for disobedience and was sentenced to house arrest for the rest of his life.

    It was an age when people often followed ancient authors (like the Greek Aristoteles) rather than check out with their own eyes, what Nature was really doing. When people started checking, observing, experimenting and calculating, that became the scientific revolution . Our modern technology is the ultimate result, and Kepler's laws (together with Galileo's work, and that of William Gilbert on magnetism) are important, because they started that revolution .

    Kepler worked with Tycho Brahe, a Danish nobleman who pushed pre-telescope astronomy to its greatest precision, measuring positions of planets as accurately as the eye could make out (Brahe died in 1602 in Prague, now the Czech capital; telescopes started with Galileo around 1609). If you want to read about it, I recommend "Tycho and Kepler" by Kitty Ferguson, reviewed at or at least, read the review. Let me quote from it:

        Religious intolerance was widespread--indeed, events were moving towards the 30 years' war (1618-48), Europe's most destructive religious battle, mirrored by the civil war in Britain. Kepler was forced out of Graz, among all other employees of Protestant colleges in town, after the ruling archduke decreed they must leave the city by nightfall, that same day. It was also an era when Kepler's mother was arrested for witchcraft, when most of his numerous children died in childhood, and when Tycho's marriage was regarded as a second-rate "slegfred" union because his chosen wife was not from the nobility.

    Try to get that across to students, too. 1620 was when the "Pilgrims" landed in Plymouth Rock, fleeing from the outbreak of the religious war which later devastated Europe. Quite possibly it was the memory of such wars that led the US, much later, to decree the separation of church and state. Explain how the development of science and society are often closely related.

Kepler's First Law

    (1) Planets move around the Sun in ellipses, with the Sun at one focus

First explain what an ellipse is: one of the "conic sections, " shapes obtaining by slicing a cone with a flat surface. A flashlight creates a cone of light: aim it at a flat wall and you get a conic section.

    Hit the wall perpendicular. The wall cuts the cone perpendicular to the axis and you get a circle of light.

    Slant the cone relative to the wall: an ellipse. The more you slant, the more far away the ellipse closes.
      The curves generated as
     "conic sections" when flat
    planes are cut across a cone.

    Finally, if the axis of the cone is parallel to the wall, the curve never closes: you get a parabola. Kepler's laws (as we now know them) allow all conic sections, and parabolas are very close to the orbits of nonperiodic comets, which start very far away.

    (Tilt still more and you get hyperbolas--not only don't the trajectories close, but the directions of coming and going make a definite angle).

Ellipses have other properties--they have two special points, "foci", and if you take any two points on the ellipse, the sum of distances (r1+r2) from the two foci is always the same (for that ellipse). The end of section #11 also has a nice story "Whispers in the US Capitol", on how an ellipsoid--the surface created by twirling an ellipse around its axis--can focus sound waves. --------------------------------------

    There is much, much more... but let me just bring up two points. They are good points to raise in class, because they bring together Kepler's work of about 1610 with the latest scientific discoveries of the 21st century.
    First of all, a very famous ellipse is shown below. Its story is told in section #S7-a

    You probably all know our sun is part of a huge disk-shaped collection of stars--about 100 billion at last count--called the galaxy. It's a flat disk, a pancake like the solar system--and here too, we look at that pancake sideways, so it too cuts out view in a narrow strip. In that strip we see a belt of faint stars running all around the globe of the sky, the "Milky Way."

Orbit of star S2

    What holds our galaxy (and more distant ones) together? For a long time it was believed there was a humongous black hole in the middle, but that middle was obscured by dust clouds and hence not easy to observe. Recently, high resolution telescopes sensitive to infra red light were built, which can see though the dust, and they have shown a large concentration of fast moving stars near the center of the galaxy, in orbits which obey Kepler's laws. The web site shows the ellipse of a star orbiting the center once in 15.2 years, and calculations deduce a mass of about 3.7 million suns, give or take 1.5 millions.

        [For astronomers only: the central mass helps keep the galaxy together, but there is a lot of more mass involved, so the rotation of the more extended parts of galaxies does not obey Kepler's 3rd law. In fact, their main parts seem to rotate like solid disks, which is hard to explain unless we assume galaxies contain, in addition to shining stars, a lot of "dark matter" which affects gravity but is invisible. See note and end of #20]

    Second, we said the Earth orbits the Sun (and by the way, the same laws also hold for artificial satellites that orbit Earth). But imagine you could gradually make Earth heavier and heavier, and the Sun at the same time lighter and lighter. What then? At the point where Earth and Sun are equally heavy--which orbits which?

    About 50 years after Kepler Isaac Newton explained Kepler's laws (and in doing so, firmly established the "scientific revolution", from there on). Here is what he did:

    ---First he devised the basic laws of motion--known ever since then as "Newton's 3 laws of motion", and you probably teach them, too.

    ---Second, he gave us the law of universal gravitation--showed that the same force which caused apples and stones to fall down, also held the Moon in its orbit--and therefore, probably, created all orbits in the solar system.

        (For more on that (even that apple), see section #20 )

    --and third, he, proved that if the above two held, Kepler's laws could be derived mathematically...

    ... but with one small change: planets orbited not around the Sun, but around a common center of gravity. While the Earth makes a big circuit each year, the Sun also makes, a very small one, around the Sun-Earth center of gravity.

    (actually, the Sun is also moved by Jupiter, Saturn etc.. and the resultant pattern is complicated.)

    Why is this important? Because it helps us discover if other stars have planets! We cannot see those planets--too dim--but if the star wiggles back and forth in a complicated way, it may be because a planet makes it move so.

    Does it work? Yes and no (end of #11a). Many planets have been discovered this way, but most of them are too close to the star (wiggles on a time scale of weeks) and are very big. To discover Earth-like planets is harder--the wiggle is smaller and we need observe for many years to extract a periodicity of the order of one year. But stay tuned, astronomers are working on it.

Kepler's 2nd law

    (2) The line connecting the Sun to a planet sweeps
            equal areas in equal times.

(That line is sometimes called "the radius vector").
 Illustrating Kepler's 2nd law:
 segments AB and CD take
 equal times to cover.

    An ellipse is symmetric elongated oval, with two foci symmetrically locates towards the "sharper" ends--one focus contains the Sun, the other is empty. (Draw such an ellipse.) If we bring the foci closer and closer, the ellipse appears more and more like a circle, and when they overlap, we do have a circle.

        [ The Earth's orbit, and most planetary orbits, are very close to circles. If one you were shown the Earth's orbit without the Sun at the focus, you would probably not be able to distinguish it from a circle. With the Sun included, though, you might notice it was slightly off-center.]

    The point of Kepler's 2nd law is that, although the orbit is symmetric, the motion is not. A planet speeds up as it approaches the Sun, gets its greatest velocity when passing closest, then slows down again.

    (The star S2 speeds up to 2% of velocity of light when approaching the black hole at the center of our galaxy)

    What happen is best understood in terms of energy. As the planet moves away from the Sun (or the satellite from Earth), it loses energy by overcoming the pull of gravity, and it slows down, like a stone thrown upwards. And like the stone, it regains its energy (completely--no air resistance in space) as it comes back.

    There is an easy exercise here, which is also in section #12A

    Suppose you have a planet whose smallest/greatest distance from the center are (r1, r2)--they are called perihelion and aphelion [ap-helion]) if the center is the Sun, or (perigee, apogee) if the center is the Earth. (Distances are always measured from the center of the bodies, or from centers of gravity)

    Say it is a planet orbiting the Sun. Then --the velocity V1 at perihelion is the fastest one for the orbit. It is therefore the distance covered in one second at perihelion. --the velocity V2 at aphelion is the slowest one for the orbit. It is therefore the distance covered in one second at aphelion.

The area swept by the "radius vector" r during one second after perihelion is a right-angled triangle of base V1, so its area is 0.5 r1 V1

The area swept by the "radius vector" r during one second after aphelion is a right-angled triangle of base V2, so its area is 0.5 r2 V2

By the law of areas, both areas are the same, so

r1 V1   =   r2 V2

Divide both sides by r1V2
and get
V1:V2   =   r2:r1

    If the aphelion r2 is 3 times the distance of perihelion, the velocity V2 there is 3 times slower. (Note: this ratio only works at these two points of the orbit. At other point the velocity and the radius are not perpendicular.)

    When are we closest to the Sun? About January 4th, by about 1.5%, not enough to make the Sun look different.
    Here is a quick way to demonstrate this asymmetry (although you may not have time to cover it in class). Draw an ellipse, with the long axis and a line perpendicular to it through the Sun)
   . It so happens (pure accident) that spring equinox and fall equinox, when day and night are equal, typically March 21, September 22 or 23, fall very close to that perpendicular line.

    Look at the schematic view of the Earth's orbit in section #3. The long axis (as defined above) is the line connecting December-June in that drawing, and the perpendicular line is the one connecting March-September.

    If the orbit were exactly a circle (in which case what we call "long axis would be completely arbitrary, a diameter no different from any other), then by Kepler's 2nd law, the Earth would move at a constant speed and spend equal times in the summer half and the winter half of the year. Actually, it spends about 2 days fewer in the winter half! (Take a calendar and count days from one equinox to the other). That may mean

  • The winter half is shorter, or
  • The Earth moves faster in the winter half

Actually, both conditions hold, if Earth is closest to the Sun around January 4. The "half" of the ellipse (determined by the perpendicular line defined above) which is closer to the Sun is smaller (demonstrate with a drawing of an ellipse that is notably oval), and by Kepler's 2nd law, the Earth moves faster when closer to the Sun.

    The fact the northern hemisphere is closest to the Sun in mid-winter and most distance in mid-summer, moderates the seasons, making them milder.
    In the southern hemisphere, they would be harsher, although the big oceans there moderate this effect.

    But the axis of the Earth moves around a cone, in about 26000 years. In 13,000 years we will be closest to the Sun in midsummer, and climate will get harsher. As described in section 7, this may be one effect tied to the origins of ice ages, but we do not have time for details.

Kepler's 3rd Law

    (3) The square of the orbital period of a planet is proportional
    to the cube of the mean distance from the Sun

    (or in other words--of the"semi-major axis" of the ellipse, half the sum of smallest and greatest distance from the Sun)

    This is a mathematical law, and your students need calculators with square roots, also 3/2 powers and 2/3 powers (and maybe cube roots or 1/3 powers, same thing)..

    If two planets (or two Earth satellites--works the same) have orbital periods T1 and T2 days or years, and mean distances from the Sun (or semi-major axes) A1 and A2 then the formula expressing the 3rd law is

(T1 / T2)2   =   (A1 / A2)3

    Students will ask right away--we can count days to get orbital period T (although it may be tricky, we need subtract the Earth's motion around the Sun)--but how do we know distance A?

    In truth, we don't, but notice only ratios of distances are needed, and units don't affect ratios. For instance, suppose "Planet 2" is the Earth, and all times are in years. Then T2 =1 (year) and we can measure all distances in Astronomical Units (AU) , the mean Sun-Earth distance, so that A2 =1 (AU). The law then becomes, for any other planet,

(T1)2   =   (A1)3
This can be checked, and in section 10 you find the results on a table:

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

    You can see that, even with our limited accuracy, the law holds pretty well. It also shows the greater the distance, the slower the motion, which leads to the overtaking of outer planets by the Earth, making them (for a while) seem to move backwards relative to the fixed stars in the sky. You can prove all this mathematically for circular orbits using Newton's laws (see section #21), but again, I'll skip that.

    In kilometers the astronomical unit is about 150,000,000 km, 400 times the Moon's distance. All sorts of attempts were made to derive it, starting with the ancient Greek Aristarchus (sect. #9a) and they are discussed in sect #10a. It was first done with any accuracy in 1672, and the excitement over the recent "transit of Venus " in front of the Sun was motivated by a proposal made around then by Halley (of comet fame) to use such rare transits (last one was 2004, next in 2012, after that one waits more than a century) to measure the AU. The calculation, not a short one, is in sections #12c to #12e of "Stargazers" . (Some other "methods" were given on the web, involving the transit of Venus but not its duration, and they are fakes.)

    All sorts of problems can be solved with Kepler's 3rd law. Here are a few:

  1. How long does it take to reach Mars, in the most efficient orbit? This is called the "Hohmann Transfer Orbit" (Wolfgang Hohmann, 1925). The spaceship must first get free of Earth (it still orbits the Sun together with Earth, at 30 km/s, at a distance of 1 AU), then it adds speed so that its aphelion (in its orbit around the Sun) just grazes the orbit of Mars, A = 1.524 AU (ignoring ellipticity).
    The Hohmann Transfer Orbit

        For the Hohmann orbit, the smallest distance is 1.00 AU (Earth), the largest one 1.524 AU (Mars), so the semi-major axis is

    A = 0.5(1.00 + 1.524) = 1.262 AU
    A3 = 2.00992 = T2

    The period is the square root T = 1.412 years
    To reach Mars takes just half an orbit or T/2 = 0.7088 years
    It equals about 8.5 months; more details are in section #21b..

  2. How long would it take for a spacecraft from Earth to reach the Sun?
    The Sun is the hardest object in the solar system to reach! It's far easier to escape to interstellar space (yes, those people who speak of hurling nuclear waste into the Sun need to learn astronomy.)

        To reach the Sun directly from Earth, we need shoot the spacecraft free of Earth. It still orbits the Sun with Earth, at 30 km/sec (low Earth orbit only takes 8 km/s), so we need give it an opposing thrust, adding (-30 km/s) to its velocity. It then falls straight into the Sun.

        That orbit is also an ellipse, though a very skinny one. Its total length is 1 (AU), so the semimajor axis is A = 0.5 AU. By the 3rd law, A3 = 0.125 = T2, and taking the square root , T=0.35355 years. We need divide this by 2 (it's a one-way trip!) and multiply by 365.25 to get days. Multiplying:

    T/2 = (0.5) 0.35355 (365.25) = 64.6 days

  3. How far (from the center of Earth) do synchronous satellites orbit? These are (mostly) communication satellites and have a 24 hour period, which helps them hang above the same station? The Moon is at 60 RE (earth radii) away and has a period of T = 27.3217 days (see section 20 on gravitation). The synchronous orbit is circular, so A is also its radius R. We get
    (R/ 60)3   =   R3 / 216,000   =   (1 / 27.3217 days)2  
               =   1/ (27.3217 days)2   =   1 / 746.5753

    R3   =   216,00/746.5753   =   289.32

    This number is between 63 = 216 and 73 = 343, so when the calculator gives R = 6.614 RE. you know you've got it about right.

  4. How far does Halley's comet go?

    Its period is about 75 years, and 752 = 5625. Take the cube root: A = 17.784 AU. That, however is the SEMImajor axis. The length of the entire orbital ellipse is 2A = 35.57 AU. Perihelion is inside the Earth's orbit, less than 1 AU from the Sun, so aphelion is about 35 AU from the Sun--as the table shows, somewhere between Neptune's orbit and Pluto's


If you are a teacher trying to cover Kepler's laws, I hope this quick overview has given you a wide range of tools and insights which may prove useful in the classroom.

Now pass it along! You will find a lot more in the web sites described here or included on your disk.

First among sections on Kepler's laws: #10 Kepler and his Laws

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Author and Curator:   Dr. David P. Stern
     Mail to Dr.Stern:   stargaze["at" symbol] .

Last updated: 3-21-2005