Kepler's Three Laws of Planetary Motion

    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.

    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.

    "Stargazers" contains more material than can ever be covered in a regular class. Still, teachers need a wider knowledge, allowing them to pick and choose material according to circumstances, and to mention odd tidbits without detailed discussion, just to create interest.

    And some very lucky teachers may sometimes find in class a kid or two who really want to find out more. Such students can be directed here to satisfy their interest.

    This overview focuses 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
       (also stated as-- ...of the "semi-major axis" of the orbital ellipse, half the sum of smallest and greatest distances 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.

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's actually the earth that is turning) and the planets were moving on their own "crystal spheres" with variable speed. 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. The Earth then sometimes overtakes the slower planets more distant from the Sun, making their positions among stars move backwards (for a while). The orbits of Venus and Mercury are inside that of Earth, so they are never seen far from the Sun (e.g. at midnight).

    I hope you that describing those features--the "pancake" of the ecliptic, the reversed ("retrograde") motion, Venus always close to the Sun--will help students 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, seems to bounce back and forth across the position of the Sun, and so does Mercury--but since it is much closer to the Sun, you may only see it whe it is 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 may be "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 brought the era of the scientific revolution and of technology. 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 .

Johannes  Kepler

    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 http://www.phy6.org/outreach/books/Tycho.htm 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 its 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).

    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."

    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
http://www.phy6.org/stargaze/Sgravity.htm )

--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.
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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

    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.)
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    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.
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    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 the details are beyond the scope of this review.

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

(also stated as-- ...of the "semi-major axis" of the orbital ellipse, half the sum of smallest and greatest distances 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

    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,