1 The Orbital Elements Overview Orbital elements are a set of six constants that people use to describe the orbit of a satellite and the exact location of the satellite in 3-­‐dimensions with respect to a predetermined reference frame that is inertially fixed. 1) Semi-­‐major axis indicates the size of the orbit, 2) eccentricity indicates the shape of the orbit, 3) inclination indicates the steepness of the orbit, 4) right ascension of the ascending node indicates the position where the satellite would go from the south hemisphere to the north hemisphere, 5) periapsis location indicates the point on the orbit that is closest to Earth, 6) and periapsis passage indicates the exact position of the satellite on the orbit at the given time. *Earth Centered Inertial System *Angular Momentum Vector Orbital Plane *Node Vector (Original image fro http://pages.erau.edu/~ericksol/courses/sp300/sp300_orbits.html ) a, Semi-­‐Major Axis The value of “a” tales us the size of the orbit. Orbits that wraps around the Earth are either circular orbits or elliptical orbits. For a circular orbit, “a” is just the radius of the circle. On the other hand, an elliptical orbit’s “a” is its semi-­‐major radius as shown below. Semi-­‐major radius, a Center (Not the location of the Earth) e, Eccentricity Eccentricity is the magnitude of the eccentricity vector which describes the shape of the orbit, it has no unit. The eccentricity vectors always lay on the major-­‐axis of the orbit and point to the *periapsides of the orbits. For orbits that go around, the value of eccentricity can range from 0 to less then 1, as 0 being perfectly circular and 1 being parabolic which means that the satellite is on a trajectory and would not come back. i, Inclination Inclination is the angle between the angular momentum vector and the Z-­‐axis of the Earth centered inertial system. The value of the inclination suggests the steepness of the orbit and it ranges from 0 degree to 180 degrees. A 90-­‐degree inclination indicates that the orbit is perpendicular to the *equatorial plane, while a 0-­‐degree and a 180-­‐degree inclination indicate that the orbits lay in the equatorial plane. Ω, Right Ascension of the Ascending Node (RAAN) The right ascension of the ascending node is an angle between the X-­‐axis of the Earth centered inertial system and the node vector. If the orbit is tilted, it goes somewhat “diagonally” around the Earth when viewed from the side. Thus there would be two points on the orbit that intersect with the equatorial plane, the *ascending node and the *descending node and they are separated 180 degrees. The RAAN indicates the location of the ascending node relative to the X-­‐axis which points to the first point of the constellation Aries; the angle ranges from 0 degree to 360 degrees. 2 ω, Periapsis Location The *periapsis location is determined by the angle between the node vector and the eccentricity vector. One characteristic these two vectors share is that they both lay on the orbital plane. Thus the angle tells us how far apart on the orbit the periapsis is from the ascending node which were determined from the previous orbital element, RAAN. The measurement of ω can range from 0 degree to 360 degrees. T0, Satellite’s location on the Orbit With the above five orbital elements one can accurately position the orbit into space. One last thing to determine is where is the satellite exactly now. T0 is the periapsis passage in seconds, and it can help we mark the exact location of the satellite once the orbit is in place. The value of T0 can be both positive and negative, with positive meaning the time it takes for the next periapsis passage and negative meaning the time passed since the last periapsis passage. If you find yourself in a situation with a negative value for T0 but you want the time it takes until the next periapsis passage, you can simply add the period of the orbit to the negative value you have. Conclusion The orbital elements give us an insight of the size, the shape, and the position of the orbit, also indicates the exact location of the satellite on the orbit at the given time. Besides directly describe the current position of the satellite in space, the orbital elements also allow engineers to determine the future location of the satellite with more complicated calculation; to take it one step further, one can even judge when and where would be the most efficient for introducing a propulsion in order to perform a desired orbital transfer or maneuver with the orbital elements. 3 4 Glossary 1. Earth Centered Inertial System (ECI) Earth centered inertial system is the right hand oriented system that the orbital elements are based on. The predetermined system is fixed with respect to the stars, which means that the three axes of the system do not rotate with the earth. This is important because orbits also do not rotate with the earth. • Z-­‐axis The Z-­‐axis always points to the North star. Coincidentally, this is also the Earth’s rotational axis; thus even the ECI does not rotate with the Earth, this axis always goes through the North pole. • X-­‐axis The X-­‐axis always points in a direction of the ram, which is Aries. One other characteristic of the X-­‐axis is that it is also the direction from the center of the Earth to the center of the Sun right at the moment of the first day of spring, also known as the vernal equinox. • Y-­‐axis You might wonder why the three axes were introduced in this order. It is because that the Z-­‐axis and the X-­‐axis are sufficient to describe the whole system, the rest (Y-­‐axis) can be determined with the right hand rule. By the right hand rule, Z cross X gives you Y and that is how to find the Y-­‐axis of the ECI. 2. Angular Momentum Vector Angular momentum vector is the product of the satellite’s position vector cross the satellite’s velocity vector. Angular momentum vectors are always perpendicular to the orbital planes. 3. Node Vector Node vector is the product of the Z-­‐axis cross the angular momentum vector. Node vectors always lay on both the equatorial plane and the orbital plane. 4. Periapsis Periapsis is the point of the orbit that is closest to the Earth. When the orbit is elliptical, the location of the Earth would not be at the center of the ellipse, instead, the location would be some where on the semi-­‐major axis. The the plural form of periapsis is periapsides. 5. Equatorial Plane Equatorial Plane is the plane constructed by the 0-­‐degree latitude. 6. Ascending Node Ascending node is the point on the orbital path when the satellite flies from the south hemisphere to the north hemisphere. On the contrary, descending node is where the satellite flies from the north hemisphere to the south hemisphere.