Orbital Mechanics: Decoding the Dance of Satellites

Examples To completely determine the motion of an earth orbiting satellite we need six numbers. First we need to determine the plane of the ellipse, then the orientation of the ellipse on the plane, finally we need to determine the shape of the ellipse. To start we use a frame of reference. This will be a Cartesian coordinate system (xyz) whose axis agrees with the mean direction of the axis of rotation of the earth. The axis of rotation is not constant, but has a precession once in about 23,000 years. However, in the time frames in which we are interested--several hours to several years--we assume it constant. As -axis, we use a use an axes perpendicular to the axis which points to the sun at the vernal equinox. The is a third perpendicular line to complete a right handed coordinate system. The Earths' equatorial plane is then the -plane of this coordinate system. The plane of motion will be either equal to this plane or intersect it in a line, called the line of nodes. To uniquely determine the plane of the motion, we use two angles: 1. is the angle of right ascension of the line of nodes from the axis, i.e. the angle between the -axis and the line of nodes measures in mathematical positive direction. 2. is the angle between the -axis and the normal of the plane of motion. This uniquely establishes the plane of the motion. The next thing is to establish the direction of the ellipse. This is done by the angle between the line of modes and the perigee of the ellipse. This angle is denoted by . The shape of the orbit is now fixed by its semi-major axis and its eccentricity . Finally, we need to know either the time or the mean or true anomaly to establish the position of the satellite. These orbital elements are illustrated in Figure 3, below1. Figure 3: Graphical representation of orbital elements To measure time, we need a reference time, this is called the Epoch. The original Cartesian coordinate system depends on this reference time as well, since the direction of the and -axes are time dependent. There are two common frames of reference. Most commonly we use the M50 frame of reference, in which the epoch starts with the beginning of 1950. Also in use is the J2K frame of reference, where the epoch is given by the beginning of year 2000. The data for many satellites are available in so called two line element sets at http://celestrak.com/NORAD/elements/. For example one of the GPS satellites (GPS BII-05 (PRN 17)) is given as 1 20361U 89097A 01154.90156813 -.0000008400000-0 00000-0 0 7462 2 20361 56.2556 342.0793 0127851 179.5306 322.3780 2.00562298 74668 We now explain these data care fully. We start with the meaning of the first line: Column Description 01 03-07 Line Number Satellite Number 08 Classification 10-11 Launch year 12-14 Launch number 15-17 Launch piece 19-20 Epoch year 21-32 Epoch day and fraction of a day 34-43 First time derivative of mean motion 45-52 Second time derivative of the mean motion 54-61 drag term 63 Ephemeris type 65-68 Element number 69 Check sum We see that our satellite was launched in launch number 97 of 1989 as piece A of this launch. The time of this data is 01154.90156813 i.e. it is the 154th day of 2001 (i.e. June 3), 21h 38min 15.486432 sec. The next data are of little relevance for us. It is element # 746. The check sum is the sum of all numbers modulo 10 and is used a a control datum whether or not the data was transmitted correctly. More important data are in the second row: Column Description 01 Line number 03-07 Satellite number 09-16 Inclination 18-25 Right ascension of the ascending node 27-33 Eccentricity 35-42 Argument of the perigee 44-51 Mean anomaly 53-63 Mean motion (revolutions per day) 64-68 Revolution number at epoch 69 in degrees (or ) Check sum Therefore in our example, we have: The orbit is at an inclination of 56.2556 degrees, . The eccentricity of the orbit is 0.0127851. The argument of the perigee is . The mean anomaly . The satellite completes 2.00562298 revolutions (orbits) per day, and was in its 7466th revolution at the time given. We observe, that is not given in this set of data. However, from the mean motion and Kepler's third law we can compute . Clearly and This gives a semi-major axis days. Kepler's third law is