Kepler's Laws

• Although the heliocentric model worked just as well as the geocentric model, to make it work over a millenium Copernicus still had to add epicycles.
  
   
• The German astronomer Johannes Kepler (1571-1630) had a different idea, however.
  
  Kepler didn't believe planetary orbits were necessarily circles, but could instead be other closed curves, such as the ellipse or oval.
  
   
• Recall that a circle is defined as the set of all points that are a constant distance r (the radius) from the center C.
  
   
• An ellipse is a generalization of a circle, involving twopoints F₁ and F₂(each called a focus, and together thefoci) and two distances r₁ and r₂, whose sum is a constant:
  
  r₁ + r₂ =2a
  
  When the foci coincide (coming together at the center), the result is a circle with a radius a.
  
• The constant 2a is equal to the length of the longer or "major" axis, so a is called the semimajor axis.
  
  The semimajor axis therefore describes the overall size of the ellipse.
  
  It can be shown that a is the average distance of the ellipse from one focus.
  
   
• The constant c describes how far each focus is from the center, which determines how elongated the ellipse is (for a given value of a).
  
  However, it is more useful to use the eccentricity:
  
  e = c/a
  Because c is always less than a, the value of e varies between 0 and 1.
  
  When e = 0, c = 0, the foci coincide, and we have a circle.
  
  When e =1, the foci approach the opposite ends of the ellipse; the result is so elongated that, from one focus, both the center and the other focus are infinitely far away, forming a curve called a parabola.
  
   
• Kepler came to work with Tycho in 1600, and the latter's astronomical records provided Kepler with the data he needed to test his hypothesis.
  
  After many years of laborious calculations, Kepler was able to demonstrate what is now known as Kepler's First Law:
  
  Planetary orbits are ellipses, with the Sun at one focus.
   
• Because the Sun is off-center, we can describe two special positions on a planet's orbit, both on the major axis:
  
  The perihelion is the point of closest approach to the Sun; it is a distance a(1 - e) from the Sun.
  
  The aphelion is the point where the planet is farthest from the Sun; it is a distancea(1 + e) from the Sun.
  
  

As can be seen in the table at the right, the eccentricity of the planets' orbits is generally quite small, except for Mercury and Pluto, whose orbits are noticeably elongated. This is why circles initially worked well in describing planetary orbits. The table also shows theorbital inclination, or tilt, of the planets' orbits relative to the ecliptic plane.

Planet Semimajor Axis a Sidereal PeriodP Eccen- tricitye Orbital Inclination Mercury 0.3871 AU 0.2408 y 0.206 7.00° Venus 0.7233 AU 0.6152 y 0.007 3.39° Earth 1.0000 AU 1.0000 y 0.017 0.00° Mars 1.5237 AU 1.8809 y 0.093 1.85° Jupiter 5.2028 AU 11.862 y 0.048 1.31° Saturn 9.5388 AU 29.458 y 0.056 2.49° Uranus 19.1914 AU 84.01 y 0.046 0.77° Neptune 30.0611 AU 164.79 y 0.010 1.77° Pluto 39.5294 AU 248.5 y 0.248 17.15°

The orbital inclination is usually quite small, except for Pluto.
Question: where did we see orbital inclination previously?
Question: why doesn't a planet usually disappear behind the Sun when they are in conjunction?

• Kepler also noticed another characteristic of planetary motion: planets move fastest at perihelion, and slowest at aphelion.
  
  Kepler was able to quantify these varying speeds in what is known as Kepler's Second Law:
  
  Planets sweep out equal areas in equal times.
  
  
• Kepler published his First and Second Laws in 1609 in a book entitled New Astronomy.
  
  
• Ten years later, in 1619, Kepler discovered and published an additional relationship.
  
  Kepler's Third Lawquantifies the observation that more distant orbits have longer periods:
  
  a³ = P²
  
  Here, the semimajor axisa is measured in A.U. and the orbital period P is measured in years.
  
  The graph at the right shows log P vs. log a; the data falls along a straight line, with a slope of 3/2.
  
  
• Kepler also noticed that the Galilean satellites obeyed the Third Law, as can be seen by the same 3/2 slope in the graph at the right.
  
  This implied that Kepler's Third Law was a general principle.
  
  
• Galileo himself refused to accept Kepler's ideas, clinging to the notion that planetary orbits must be circular, though his reasons were based on his studies of motion rather than on tradition.