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Appendix A: Orbital Elements

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Appendix A: Orbital Elements Appendix A: Orbital Elements Many of the items in this Appendix have been explained elsewhere in the text, but, for clarity, they have been drawn together into this one section. The size and shape of a satellite’s orbit is determined by its speed and mass, and also by the mass of the object it is orbiting. The size, shape and orientation of an orbit can be described by six orbital elements or orbital parameters, known as Keplerian ele- ments or element sets. Different sets of elements can be used for different purposes. The terms used sound complicated, but taken individually each is relatively simple. These explanations have been taken with reference to an Earth-orbiting satellite, however, they can easily be adapted for satellites orbiting other celestial bodies. If all of the elements are known, the exact location of a satellite can be calculated. The Keplerian elements are: Inclination (i) Longitude of the ascending node (W) Argument of periapsis (w) Eccentricity (e) Semi-major axis (a) Anomaly at epoch (v) Time of periapsis or perigee passage (T) may be used instead of the Anomaly at epoch (v) Inclination (i) A satellite will always travel around the centre of the body it is orbiting and therefore it will always cross the equator twice in every orbit, unless, of course, it orbits directly above the equator. It will cut it as it travels from the southern hemisphere to the northern and again when it crosses from the northern to the southern hemi- sphere. The angle it makes when it crosses the equator from the 303 304 It’s ONLY Rocket Science southern hemisphere to the northern hemisphere is called the inclination, and this defines the orientation of the orbit with respect to the equator. Therefore, an orbit that is directly above the equator has an inclination of 0° and one that goes directly over the north and south poles has an inclination of 90°. In Figure 1.2, the orbit shown in red has an inclination of 28°, which is the inclina- tion of most of the USA’s scientific satellites. The yellow orbit shows the path of the International Space Station, which has an inclination of about 52°. Longitude of the Ascending Node (W) The ascending node is where the orbit of a satellite crosses the Earth’s equator when the satellite is travelling from south to north, as shown in Figure A.1. The longitude of the ascending node is defined with reference to the terms celestial sphere, the ecliptic and the First Point of Aries, which therefore need to be defined first: The celestial sphere is the imaginary sphere encompassing the Earth, as can be seen in Figure 6.5. All objects in the sky can be thought FIGURE App A.1 Orbital Elements Appendix A: Orbital Elements 305 of as being on the sphere. The celestial equator and the celestial poles are projected from their corresponding earthly equivalents. Angles measured north or south from the line of the celestial equa- tor are called the declination and are measured in degrees. This is similar to lines of latitude. The longitude equivalent is called the Right Ascension or RA. This is measured in hours eastwards from the First point of Aries, which is given the symbol ^. The Sun does not always appear directly above the equator but instead follows a path called the ecliptic, which takes it above and below it. When its path crosses the equator, it is called an equi- nox. For the northern hemisphere, the Vernal Equinox is usually about March 20. This is when the length of daylight and darkness is exactly equal, and the Sun is directly above the equator. The direction of the Sun from the Earth on this day is known as the First point of Aries. The longitude of the ascending node is the angle from the First Point of Aries to the ascending node, taken anticlockwise if look- ing down from the north, in the plane of the Earth’s equator. Argument of Periapsis or Perigee (w) This defines where the perigee of the orbit is, in respect to the Earth’s surface. This is the angle formed, in the plane of the satel- lite’s orbit, from where the satellite crosses the Earth’s equator at the ascending node, to when it reaches perigee. If the angle is less than 180°, perigee is in the northern hemisphere, larger than 180°, it is in the southern hemisphere. Eccentricity (e) This is the shape of the orbit. Most orbits are not circular but look liked squashed circles, called ellipses. How flat the ellipse looks is called its eccentricity and it is given a value from zero to one. An ellipse with zero eccentricity is a perfect circle and a very flat ellipse has an eccentricity nearing one, as can be seen in Figure 1.3. 306 It’s ONLY Rocket Science Semi-major Axis (a) This defines the size of the orbit. The long axis of an ellipse is called the major axis, the shorter one, the minor axis. In a circle, both the major and minor axes are the same, and are called the diameter. Half of the major axis is called the semi-major axis. This is the equivalent of the radius in a circle. The length of the semi- major axis is used to describe the size of the ellipse. The properties of an ellipse are shown in Figure 4.4. Anomaly at Epoch (v) Anomaly in this sense is an angle. The anomaly at epoch is the angle between where the satellite is now, compared to where it was at perigee. If the satellite is at perigee, the anomaly is zero. At apogee, it is 180°, and back to perigee is 360°. As most orbits are not circular, the measurement of this angle is not easy and so the angle is given in terms of a fraction of an orbit, with one orbit being 360°. Time of Periapsis or Perigee Passage (T) This is simply the time when the satellite was at perigee. This usu- ally comprises the date and time, given in Universal Time or UT. For most purposes, this can be taken to be the same as Greenwich Mean Time (GMT). Appendix B: Coordinate Systems Geocentric Coordinate Systems Usually, the centre of the Earth is used as the origin for the coordinate system for spacecraft in orbit around the Earth. This is called a geo- centric coordinate system and is shown in Figure B.1. A geocentric equatorial coordinate system is a Cartesian system that has the fun- damental plane as the plane of the equator and the x direction towards the Vernal Equinox or First Point of Aries. The z direction is through the North Pole and the y direction makes up the right-handed set. Except for precession, the wobbling motion of the Earth’s axis of rotation, this system is fixed to the stars and the Earth rotates within it. Heliocentric Coordinate Systems For satellites travelling to other places within our solar system, a coordinate system with the origin at the centre of the Sun is used, such as the heliocentric ecliptic coordinate system. This can be seen in Figure B.2. The fundamental plane is the plane of the ecliptic, with the x direction being towards the Vernal Equinox or First Point of Aries at a particular epoch. The z direc- tion is the ecliptical North Pole, and the y direction makes the right-handed set. Perifocal Coordinate System A perifocal coordinate system is often used to describe the motion of a satellite. The fundamental plane is the plane of the satellite’s orbit, and the x-axis points towards the part of the orbit nearest to the body the satellite is orbiting, known as the periapsis. The y-axis is on the orbital plane, and rotated 90° from the x-axis in the direction of the satellite’s motion. The z-axis completes the right-handed set. 307 308 It’s ONLY Rocket Science FIGURE App B.1 Geocentric Equatorial Coordinate System FIGURE App B.2 Heliocentric Ecliptic Coordinate System Appendix C: Web Site Addresses Name Web site Comment The Radio www.amsat.org An educational organi- Amateur Satel- zation with the goal lite Corpora- of fostering amateur tion (AMSAT) radio’s participation in space research and communication. Amateur www.ariss-eu.org Common working Radio on the group of the national International amateur radio and Space Station amateur satellite radio (ARISS) societies involved in amateur radio opera- tions on board the International Space Station. Belgian Work- www.vvs.be Group that coordinate ing Group for and collate amateur’s Satellites data about the flash period of satellites. The web site is in Dutch. Celestrak www.celestrak.com Dr. T.S. Kelso’s web site, which provides a lot of satellite tracking and other information. ESA www.esa.int Web site of the Euro- pean Space Agency. HearSat www.hearsat.org Satellite radio signal monitoring web site. Heavens www.heavens-above.com Provides information Above needed to observe sat- ellites as other space- flight and astronomical information. (continued) 309 310 It’s ONLY Rocket Science Name Web site Comment International www.imo.net Ensures interna- Meteor Organ- tional cooperation ization (IMO) of meteor amateur work. Collects meteor observations by several methods from around the world. ISS Fan Club www.issfanclub.com Provides realtime information about the International Space Station. Most of the content is related to amateur radio. It is not affiliated to any official body or organization. Mike McCants www.io.com/~mmccants/ Mike McCants’ satel- lite tracking web pages.
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