AMSAT-NA Keplerian Elements Tutorial

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                     AMSAT-NA Keplerian Elements Tutorial
                                       
   This tutorial is based on the documentation provided with
   InstantTrack, written by Franklin Antonio, N6NKF.
   
   Satellite Orbital Elements are numbers that tell us the orbit of each
   satellite. Elements for common satellites are distributed through
   amateur radio bulletin boards, and other means.
   
   Entering satellite elements is easy. Understanding them is a bit more
   difficult. I have tried to make this tutorial as easy to read as
   possible.
   
The Seven (or Eight) Keplerian Elements

   Seven numbers are required to define a satellite orbit. This set of
   seven numbers is called the satellite orbital elements, or sometimes
   "Keplerian" elements (after Johann Kepler [1571-1630]), or just
   elements. These numbers define an ellipse, orient it about the earth,
   and place the satellite on the ellipse at a particular time. In the
   Keplerian model, satellites orbit in an ellipse of constant shape and
   orientation.
   
   The real world is slightly more complex than the Keplerian model, and
   tracking programs compensate for this by introducing minor corrections
   to the Keplerian model. These corrections are known as perturbations.
   The perturbations that amateur tracking programs know about are due to
   the lumpiness of the earth's gravitational field (which luckily you
   don't have to specify), and the "drag" on the satellite due to
   atmosphere. Drag becomes an optional eighth orbital element.
   
   Orbital elements remain a mystery to most people. This is due I think
   first to the aversion many people (including me) have to thinking in
   three dimensions, and second to the horrible names the ancient
   astronomers gave these seven simple numbers and a few related
   concepts. To make matters worse, sometimes several different names are
   used to specify the same number. Vocabulary is the hardest part of
   celestial mechanics!
   
   The basic orbital elements are...
   
    1. Epoch
    2. Orbital Inclination
    3. Right Ascension of Ascending Node (R.A.A.N.)
    4. Argument of Perigee
    5. Eccentricity
    6. Mean Motion
    7. Mean Anomaly
    8. Drag (optional)
       
   The following definitions are intended to be easy to understand. More
   rigorous definitions can be found in almost any book on the subject.
   I've used aka as an abbreviation for "also known as" in the following
   text.
   
  Epoch
  
   [aka "Epoch Time" or "T0"]
   
   A set of orbital elements is a snapshot, at a particular time, of the
   orbit of a satellite. Epoch is simply a number which specifies the
   time at which the snapshot was taken.
   
  Orbital Inclination
  
   [aka "Inclination" or "I0"]
   
   The orbit ellipse lies in a plane known as the orbital plane. The
   orbital plane always goes through the center of the earth, but may be
   tilted any angle relative to the equator. Inclination is the angle
   between the orbital plane and the equatorial plane. By convention,
   inclination is a number between 0 and 180 degrees.
   
   Some vocabulary: Orbits with inclination near 0 degrees are called
   equatorial orbits (because the satellite stays nearly over the
   equator). Orbits with inclination near 90 degrees are called polar
   (because the satellite crosses over the north and south poles). The
   intersection of the equatorial plane and the orbital plane is a line
   which is called the line of nodes. More about that later.
   
  Right Ascension of Ascending Node
  
   [aka "RAAN" or "RA of Node" or "O0", and occasionally called
   "Longitude of Ascending Node"]
   
   RAAN wins the prize for most horribly named orbital element. Two
   numbers orient the orbital plane in space. The first number was
   Inclination. This is the second. After we've specified inclination,
   there are still an infinite number of orbital planes possible. The
   line of nodes can poke out the anywhere along the equator. If we
   specify where along the equator the line of nodes pokes out, we will
   have the orbital plane fully specified. The line of nodes pokes out
   two places, of course. We only need to specify one of them. One is
   called the ascending node (where the satellite crosses the equator
   going from south to north). The other is called the descending node
   (where the satellite crosses the equator going from north to south).
   By convention, we specify the location of the ascending node.
   
   Now, the earth is spinning. This means that we can't use the common
   latitude/longitude coordinate system to specify where the line of
   nodes points. Instead, we use an astronomical coordinate system, known
   as the right ascension / declination coordinate system, which does not
   spin with the earth. Right ascension is another fancy word for an
   angle, in this case, an angle measured in the equatorial plane from a
   reference point in the sky where right ascension is defined to be
   zero. Astronomers call this point the vernal equinox.
   
   Finally, "right ascension of ascending node" is an angle, measured at
   the center of the earth, from the vernal equinox to the ascending
   node.
   
   I know this is getting complicated. Here's an example. Draw a line
   from the center of the earth to the point where our satellite crosses
   the equator (going from south to north). If this line points directly
   at the vernal equinox, then RAAN = 0 degrees.
   
   By convention, RAAN is a number in the range 0 to 360 degrees.
   
   I used the term "vernal equinox" above without really defining it. If
   you can tolerate a minor digression, I'll do that now. Teachers have
   told children for years that the vernal equinox is "the place in the
   sky where the sun rises on the first day of Spring". This is a
   horrible definition. Most teachers, and students, have no idea what
   the first day of spring is (except a date on a calendar), and no idea
   why the sun should be in the same place in the sky on that date every
   year.
   
   You now have enough astronomy vocabulary to get a better definition.
   Consider the orbit of the sun around the earth. I know in school they
   told you the earth orbits around the sun, but the math is equally
   valid either way, and it suits our needs at this instant to think of
   the sun orbiting the earth. The orbit of the sun has an inclination of
   about 23.5 degrees. (Astronomers don't usually call this 23.5 degree
   angle an 'inclination', by the way. They use an infinitely more
   obscure name: The Obliquity of The Ecliptic.) The orbit of the sun is
   divided (by humans) into four equally sized portions called seasons.
   The one called Spring begins when the sun pops up past the equator. In
   other words, the first day of Spring is the day that the sun crosses
   through the equatorial plane going from South to North. We have a name
   for that! It's the ascending node of the Sun's orbit. So finally, the
   vernal equinox is nothing more than the ascending node of the Sun's
   orbit. The Sun's orbit has RAAN = 0 simply because we've defined the
   Sun's ascending node as the place from which all ascending nodes are
   measured. The RAAN of your satellite's orbit is just the angle
   (measured at the center of the earth) between the place the Sun's
   orbit pops up past the equator, and the place your satellite's orbit
   pops up past the equator.
   
  Argument of Perigee
  
   [aka "ARGP" or "W0"]
   
   Argument is yet another fancy word for angle. Now that we've oriented
   the orbital plane in space, we need to orient the orbit ellipse in the
   orbital plane. We do this by specifying a single angle known as
   argument of perigee.
   
   A few words about elliptical orbits... The point where the satellite
   is closest to the earth is called perigee, although it's sometimes
   called periapsis or perifocus. We'll call it perigee. The point where
   the satellite is farthest from earth is called apogee (aka apoapsis,
   or apifocus). If we draw a line from perigee to apogee, this line is
   called the line-of-apsides. (Apsides is, of course, the plural of
   apsis.) I know, this is getting complicated again. Sometimes the
   line-of-apsides is called the major-axis of the ellipse. It's just a
   line drawn through the ellipse the "long way".
   
   The line-of-apsides passes through the center of the earth. We've
   already identified another line passing through the center of the
   earth: the line of nodes. The angle between these two lines is called
   the argument of perigee. Where any two lines intersect, they form two
   complimentary angles, so to be specific, we say that argument of
   perigee is the angle (measured at the center of the earth) from the
   ascending node to perigee.
   
   Example: When ARGP = 0, the perigee occurs at the same place as the
   ascending node. That means that the satellite would be closest to
   earth just as it rises up over the equator. When ARGP = 180 degrees,
   apogee would occur at the same place as the ascending node. That means
   that the satellite would be farthest from earth just as it rises up
   over the equator.
   
   By convention, ARGP is an angle between 0 and 360 degrees.
   
  Eccentricity
  
   [aka "ecce" or "E0" or "e"]
   
   This one is simple. In the Keplerian orbit model, the satellite orbit
   is an ellipse. Eccentricity tells us the "shape" of the ellipse. When
   e=0, the ellipse is a circle. When e is very near 1, the ellipse is
   very long and skinny.
   
   (To be precise, the Keplerian orbit is a conic section, which can be
   either an ellipse, which includes circles, a parabola, a hyperbola, or
   a straight line! But here, we are only interested in elliptical
   orbits. The other kinds of orbits are not used for satellites, at
   least not on purpose, and tracking programs typically aren't
   programmed to handle them.) For our purposes, eccentricity must be in
   the range 0 <= e < 1.
   
  Mean Motion
  
   [aka "N0"] (related to "orbit period" and "semimajor-axis")
   
   So far we've nailed down the orientation of the orbital plane, the
   orientation of the orbit ellipse in the orbital plane, and the shape
   of the orbit ellipse. Now we need to know the "size" of the orbit
   ellipse. In other words, how far away is the satellite?
   
   Kepler's third law of orbital motion gives us a precise relationship
   between the speed of the satellite and its distance from the earth.
   Satellites that are close to the earth orbit very quickly. Satellites
   far away orbit slowly. This means that we could accomplish the same
   thing by specifying either the speed at which the satellite is moving,
   or its distance from the earth!
   
   Satellites in circular orbits travel at a constant speed. Simple. We
   just specify that speed, and we're done. Satellites in non-circular
   (i.e., eccentricity > 0) orbits move faster when they are closer to
   the earth, and slower when they are farther away. The common practice
   is to average the speed. You could call this number "average speed",
   but astronomers call it the "Mean Motion". Mean Motion is usually
   given in units of revolutions per day.
   
   In this context, a revolution or period is defined as the time from
   one perigee to the next.
   
   Sometimes "orbit period" is specified as an orbital element instead of
   Mean Motion. Period is simply the reciprocal of Mean Motion. A
   satellite with a Mean Motion of 2 revs per day, for example, has a
   period of 12 hours.
   
   Sometimes semi-major-axis (SMA) is specified instead of Mean Motion.
   SMA is one-half the length (measured the long way) of the orbit
   ellipse, and is directly related to mean motion by a simple equation.
   
   Typically, satellites have Mean Motions in the range of 1 rev/day to
   about 16 rev/day.
   
  Mean Anomaly
  
   [aka "M0" or "MA" or "Phase"]
   
   Now that we have the size, shape, and orientation of the orbit firmly
   established, the only thing left to do is specify where exactly the
   satellite is on this orbit ellipse at some particular time. Our very
   first orbital element (Epoch) specified a particular time, so all we
   need to do now is specify where, on the ellipse, our satellite was
   exactly at the Epoch time.
   
   Anomaly is yet another astronomer-word for angle. Mean anomaly is
   simply an angle that marches uniformly in time from 0 to 360 degrees
   during one revolution. It is defined to be 0 degrees at perigee, and
   therefore is 180 degrees at apogee.
   
   If you had a satellite in a circular orbit (therefore moving at
   constant speed) and you stood in the center of the earth and measured
   this angle from perigee, you would point directly at the satellite.
   Satellites in non-circular orbits move at a non-constant speed, so
   this simple relation doesn't hold. This relation does hold for two
   important points on the orbit, however, no matter what the
   eccentricity. Perigee always occurs at MA = 0, and apogee always
   occurs at MA = 180 degrees.
   
   It has become common practice with radio amateur satellites to use
   Mean Anomaly to schedule satellite operations. Satellites commonly
   change modes or turn on or off at specific places in their orbits,
   specified by Mean Anomaly. Unfortunately, when used this way, it is
   common to specify MA in units of 256ths of a circle instead of
   degrees! Some tracking programs use the term "phase" when they display
   MA in these units. It is still specified in degrees, between 0 and
   360, when entered as an orbital element.
   
   Example: Suppose Oscar-99 has a period of 12 hours, and is turned off
   from Phase 240 to 16. That means it's off for 32 ticks of phase. There
   are 256 of these ticks in the entire 12 hour orbit, so it's off for
   (32/256)x12hrs = 1.5 hours. Note that the off time is centered on
   perigee. Satellites in highly eccentric orbits are often turned off
   near perigee when they're moving the fastest, and therefore difficult
   to use.
   
  Drag
  
   [aka "N1"]
   
   Drag caused by the earth's atmosphere causes satellites to spiral
   downward. As they spiral downward, they speed up. The Drag orbital
   element simply tells us the rate at which Mean Motion is changing due
   to drag or other related effects. Precisely, Drag is one half the
   first time derivative of Mean Motion.
   
   Its units are revolutions per day per day. It is typically a very
   small number. Common values for low-earth-orbiting satellites are on
   the order of 10^-4. Common values for high-orbiting satellites are on
   the order of 10^-7 or smaller.
   
   Occasionally, published orbital elements for a high-orbiting satellite
   will show a negative Drag! At first, this may seem absurd. Drag due to
   friction with the earth's atmosphere can only make a satellite spiral
   downward, never upward.
   
   There are several potential reasons for negative drag. First, the
   measurement which produced the orbital elements may have been in
   error. It is common to estimate orbital elements from a small number
   of observations made over a short period of time. With such
   measurements, it is extremely difficult to estimate Drag. Very
   ordinary small errors in measurement can produce a small negative
   drag.
   
   The second potential cause for a negative drag in published elements
   is a little more complex. A satellite is subject to many forces
   besides the two we have discussed so far (earth's gravity, and
   atmospheric drag). Some of these forces (for example gravity of the
   sun and moon) may act together to cause a satellite to be pulled
   upward by a very slight amount. This can happen if the Sun and Moon
   are aligned with the satellite's orbit in a particular way. If the
   orbit is measured when this is happening, a small negative Drag term
   may actually provide the best possible 'fit' to the actual satellite
   motion over a *short* period of time.
   
   You typically want a set of orbital elements to estimate the position
   of a satellite reasonably well for as long as possible, often several
   months. Negative Drag never accurately reflects what's happening over
   a long period of time. Some programs will accept negative values for
   Drag, but I don't approve of them. Feel free to substitute zero in
   place of any published negative Drag value.
   
Other Satellite Parameters

   All the satellite parameters described below are optional. They allow
   tracking programs to provide more information that may be useful or
   fun.
   
  Epoch Rev
  
   [aka "Revolution Number at Epoch"]
   
   This tells the tracking program how many times the satellite has
   orbited from the time it was launched until the time specified by
   "Epoch". Epoch Rev is used to calculate the revolution number
   displayed by the tracking program. Don't be surprised if you find that
   orbital element sets which come from NASA have incorrect values for
   Epoch Rev. The folks who compute satellite orbits don't tend to pay a
   great deal of attention to this number! At the time of this writing
   [1989], elements from NASA have an incorrect Epoch Rev for Oscar-10
   and Oscar-13. Unless you use the revolution number for your own
   bookeeping purposes, you needn't worry about the accuracy of Epoch
   Rev.
   
  Attitude
  
   [aka "Bahn Coordinates"]
   
   The spacecraft attitude is a measure of how the satellite is oriented
   in space. Hopefully, it is oriented so that its antennas point toward
   you! There are several orientation schemes used in satellites. The
   Bahn coordinates apply only to spacecraft which are spin-stablized.
   Spin-stabilized satellites maintain a constant inertial orientation,
   i.e., its antennas point a fixed direction in space (examples:
   Oscar-10, Oscar-13).
   
   The Bahn coordinates consist of two angles, often called Bahn Latitude
   and Bahn Longitude. These are published from time to time for the
   elliptical-orbit amateur radio satellites in various amateur satellite
   publications. Ideally, these numbers remain constant except when the
   spacecraft controllers are re-orienting the spacecraft. In practice,
   they drift slowly.
   
   For highly elliptical orbits (Oscar-10, Oscar-13, etc.) these numbers
   are usually in the vicinity of: 0,180. This means that the antennas
   point directly toward earth when the satellite is at apogee.
   
   These two numbers describe a direction in a spherical coordinate
   system, just as geographic latitude and longitude describe a direction
   from the center of the earth. In this case, however, the primary axis
   is along the vector from the satellite to the center of the earth when
   the satellite is at perigee.
   
   An excellent description of Bahn coordinates can be found in Phil
   Karn's "Bahn Coordinates Guide".
   
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