Ellipses and Elliptical Orbits

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eccentricity, semi-major and semi-minor axes, perigee and apogee, perihelion and aphelion, etc.

          An ellipse is a closed geometrical curve of which the circle is a special case. The Cartesian plane formula for a circle is

        x2+y2=r2 , 

where r is the radius. The ellipse formula is

        (x/a)2+(y/b)2=1 , 

where a and b are, respectively, the semi-major and semi-minor axes (a>b asssumed). If a=b, then the ellipse is circle of radius a. The figure to the right shows an ellipse with its foci and accompanying formulae. A string fixed at two points and held taut with a pen then the pen can be used to trace an ellipse with the two points becoming the foci: the semi-major axis half the string's length. For astronomical orbital purposes, it turns out that the physically important distance is from one focus to the curve, and not from the geometric center to the curve.

          The ECCENTRICITY E of an ellipse (which defined mathematically on the figure above) is loosely speaking a measure of the DEVIATION of the ellipse from circularity. If e=0, the ellipse is a circle. If e=4/5, the ellipse is quite quite elliptical: the semi-minor to semi-major axis ratio is 3/5. If the semi-minor to semi-major axis ratio is 1/10, the e=0.995 approximately. If e=1, then the ellipse has flattened into a line segment if one sends semi-minor axis b to zero and holds the semi-major a axis constant. (You get different answer for e=1, when you allow a and b to go to infinity in just the right way.)

          Beyond the scope of intro astro, there are more ellipse arcana.

          One of Newton's epochal discoveries was that two bodies that can be treated as point masses, isolated from all other bodies in space, will orbit their mutual center of mass in ellipses where the center of mass is at one of the focuses of each of the ellipses: the other focus in each ellipse is just an empty point in space---the center of mass can be just an empty point too, of course.

          Of course, one means ``orbit the center of mass'' in a physical sense. If there is no net outside force on the system, then the CENTER OF MASS IS UNACCELERATED. The BODIES ARE ACCELERATED even if the center of mass is not since they arn't moving in straight lines at constant speed. There is an internal force causing them to move in orbit.

          That internal force is, of course, GRAVITY. Elliptical orbits are a consequence of the INVERSE-SQUARE LAW nature of gravity. The discovery that gravity must cause elliptical orbits for any two-body system was one of Newton's greatest discoveries.

          Another example of ``orbiting'' is the circular motion of a SWIRLED SLING. Here the center of force is relatively unmoving (unaccelerated) hand and the swung object is accelerated by the TENSION force of the string into circular motion. If the tension force vanished, the object would fly off in a straight line if not acted on by gravity. Of course, with a sling the flying off is the whole point.

          If one body in a two-body gravitation system is sufficiently massive, then it is effectively. the center of mass itself. The Sun and a planet form a two-body system with the SUN EFFECTIVELY AT THE CENTER OF MASS. Thus the planet orbits the Sun in an ellipse where the Sun is a focus of the ellipse.

          Now you may say that there are NO TRUE TWO-BODY SYSTEMS because the universe is full of bodies. True, but APPROXIMATE TWO-BODY SYSTEMS are everywhere. The Sun and each planet is approximately a two-body system because the mass of the Sun dominates the interaction of SUN AND PLANET: the other planets are small perturbations because of their small mass and other stars even smaller perturbations because of their great distance. A PLANET AND EACH OF ITS MOONS also constitute an approximate two-body system. Of course, for detailed analysis the perturbations must be considered. But often the two-body approximation gives dominant behavior.

          Part of ASTRO-JARGON are special names for the points of closest and farthest separation of a two-body system. From the Earth these are PERIGEE AND APOGEE: peri meaning something like around, apo something like off, and gee meaning Earth: all derived from the Greek. Perigee and apogee can probably be used generically for two-body systems. But when one body is the Sun one can use PERIHELION AND APHELION. One can also invent variations: periastron and apastron for stars, peri-Jupiter and ap-Jupiter for Jupiter, peri-object and ap-object, etc. Perhaps this carrying astro-jargon too far.

          The SPEED OF A BODY in orbit varies. It is faster when nearer to the center of force and slower when farther from the center of force. The highest speed is at perigee and the lowest, at apogee.

          Below we give a table of planet mean distances (from the Sun), eccentricities, and ecliptic angles. Don't try to memorize these numbers: look at them and think about what they mean.

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Table of Planet Distances, Eccentricities, and Ecliptic Angles

  Planet        Mean Distance (AU)    Eccentricity    Ecliptic Angle (degrees)

  Mercury        0.38710              0.2056           7.00
  Venus          0.72333              0.0068           3.39
  Earth          1.00000              0.0167           0
  Mars           1.52369              0.0933           1.85
  Jupiter        5.20283              0.048            1.31
  Saturn         9.53876              0.056            2.49
  Uranus        19.19139              0.046            0.77
  Neptune       30.06107              0.010            1.77
  Pluto         39.52940              0.248           17.15

          The PLANET ORBITS are close to CIRCULAR: i.e., the eccentricities are small. For example consider the Earth's eccentricity of 0.0167. This means that the Earth is only ever 1.67 % farther from the Sun that its mean distance and only 1.67 % closer to the Sun than its mean distance. Venus has the smallest eccentricity. Pluto and Mercury have the two largest eccentricities by far.

          The ECLIPTIC ANGLE is the angle inclination of the plane of the orbit from ecliptic plane (i.e., the plane of the Earth's orbit). We see that the planets nearly orbit all in the same plane. Pluto is the largest deviator by far. The asteroids orbits are close to the ecliptic too. However, long-period comet orbits can be at any angle relative to the ecliptic (Se-569).

          Thinking of the planet orbits as CIRCULAR is a fine first order approximation. But for detailed predictions one must go to ELLIPTICAL ORBITS and even further to PERTURBED ELLIPTICAL ORBITS. Detailed prediction of angular position on the sky has always been one of the goals of astronomy since ancient times. In fact ASTRONOMICAL ACCURACY is a byword.

          Alas, ASTROPHYSICAL ACCURACY is a byword too: sympathetically it means order of magnitude accuracy; unsympathetically it means ``we don't know what were talking about.''