Cartesian vs Keplerian Elements

there are many different ways to describe with satellites orbit about a planet however you are most likely to encounter one of two methods Cartesian elements and Keplerian elements the algorithms to convert between these representations are well-documented so we won't be covering them here instead we'll go through a brief description of each set and the reasons why you want to use one form over another let's say you want to plot out the ballistic trajectory of a cannonball you take out your trusty graph paper and draw two axes the x axis represents the downrange distance of the Cannonball while the y axis represents its height you'll probably place your cannon at point zero zero and give the cannonball an initial velocity which has values in the X and y directions assuming that there is no air resistance the only force acting on the Cannonball is a constant downward acceleration of 9.81 m/s^2 at any point in time you can characterize the current state of a cannonball by recording the x and y components of its position and velocity these state vectors which are also called the ephemeris are a simple form of Cartesian elements in order to remember what the term Cartesian means I prefer to think of the term cartography which reminds me of maps which then reminds me of grids this is a little bit of a flub since the Cartesian elements are actually named after the French philosopher and mathematician Rene Descartes while it isn't exactly correct to use the term cartography in this sense it's still a good way to remember what Cartesian means when describing the Cartesian elements for a satellite you will need to expand it out into three dimensions the X's are typically placed at the center of the earth with the z axis the new dimension pointing to the North Pole in order to describe the Cartesian and femoris for satellite you will need to know the x y and z position as well as the x y&z velocity just like the cannonball example you don't need to record the acceleration due to gravity this can be calculated using the position of the satellite the cap layering elements are a completely different way of describing a satellite orbit these are named after Johann Kepler and are also frequently called the classical orbital elements the way that I remember what these elements are is to think that it's almost like sounding out the vowels in the alphabet ayyy I oh oh and new a is the semi-major axis this describes the size of the orbit for a circular orbit the semi-major axis is the radius of the orbit for an elliptical orbit the semi-major axis is the average of the radius at perigee and the radius at Apogee he is the eccentricity of low orbit and eccentricity of zero is a circular orbit and elliptical orbit has an eccentricity greater than zero but less than one a parabolic orbit has needs intricity equal to 1 while hyperbolic orbits all have eccentricity z-- greater than 1 I is the inclination of the orbit this parameter describes the tilt of the orbital plane when I satellite orbits the earth its orbit is always balanced about the center of the earth it's not possible for the satellite to travel on a halo pattern as depicted here instead if the orbit is tilted it will spend a certain amount of time above and below the equatorial plane when a satellite has zero inclination this means that the orbital plane is aligned with the equatorial plane when a satellite has 90 degree inclination this means that the orbital plane is perpendicular to the equatorial plane this is also called a polar orbit since the satellite travels directly over the North and South Poles Oh or capital Omega is the right ascension of the ascending node for an inclined orbit there are two points when the satellite crosses the equatorial plane when the satellite crosses the plane and then is moving upward this is called the ascending node when it crosses the plane and is moving downward this is called the descending node when you draw a line from the center of the earth to the ascending node this is called the line of nodes the right ascension in the ascending node is the angle from the coordinate system x-axis to the line of nodes this one may be a little confusing so you may want to pause the video to study this picture some more the next parameter o is little Omega naught W this one is called the argument of perigee remember that perigee means the point at which the satellite is closest to the earth the argument of perigee is the angle from the line of nodes to the satellite position vector at perigee again you may want to pause the video to study this picture some more the final parameter is new or the true anomaly since new looks so much like V which can be confused with the velocity of the vehicle you will frequently see it written using another letter I use the letter F a top-down view of the orbit shows the true anomaly more clearly the true anomaly is the angle from the position at perigee to the actual position of the vehicle at that moment in time okay now that we've described Cartesian and Kepler in elements and what situations would you use them here are the Cartesian and Kepler in ephemerides for a GPS satellite they both represent the same instant in time by looking at the first three Keplerian elements you can get a pretty decent idea of what the orbit looks like the radius of the earth is 6378 kilometers so the semi-major axis is about four times the size of the earth by looking at the eccentricity we can tell that the orbit is nearly circular and we can see that the inclination is nearly 55 degrees using this information we can draw the general shape of the orbit in addition you can add the argument of perigee in the true anomaly to get an approximate location of the satellite as you can see here it's very easy to get good idea of the shape of the orbit from Colleran elements it's not quite so intuitive with the Cartesian elements however the Cartesian elements are very useful when propagating the orbit by using small step sizes you can use a numerical integrator to determine the new position velocity of the satellite given the current position velocity and acceleration with this method you can also add in accelerations from other forces such as gravitational effects from the Sun and Moon and solar radiation pressure depending on the fidelity of your force models you can get some very accurate simulations using this method this is not quite so easily done when using Keplerian elements thanks for tuning into orbit nerd where we make astrodynamics easier you

Channel: OrbitNerd

Views: 19,491

Keywords: Cartesian Elements, Keplerian Elements, Classical Orbital Elements, orbit nerd

Id: A-wAiFFSY6o

Length: 7min 16sec (436 seconds)

Published: Thu Aug 22 2013