True Anomaly vs. Mean Anomaly

the true anomaly Ameen anomaly our variables used to describe a satellites position within an orbit before we continue you'll want to make sure you have good understanding of basic orbital shapes and how a satellite moves within an elliptical orbit conveniently clicking on this basic orbit shapes link will take you to a video which explains this topic if you don't know what the true anomaly is or need a refresher on the Keplerian elements you can click on this link go ahead click on the links I'll wait okay now that we're all up to speed let's talk a little more about the true anomaly hopefully the videos you just watched have shown you that a the true anomaly is an angular measure of the vehicles position relative to perigee and B the vehicle doesn't move at a constant speed when in in elliptical orbit the problem you usually encounter with orbits is this we know the satellites current position but where will it be in 1 second or 10 seconds or even one hour since the true anomaly isn't moving at a constant rate this is a tricky problem to solve directly this is where the mean anomaly comes in to save the day it makes solving this problem easy the mean anomaly is based off of the mean motion of the satellite no not that type of mean what I'm talking about is the mean as an average for any circular or elliptical orbits you can calculate what's called the mean motion which is the average motion of the satellite in its orbit this is actually a really simple value to calculate if you know the semi-major axis for the satellite the only other thing you'll need is mu which is the gravitational constant for the earth simply plug in your semi-major axes and you'll get your mean motion in radians per second the really great part about the mean motion is that it is constant if you know your current mean anomaly in your mean motion you can calculate your mean anomaly at any other time by multiplying the amount of time you want to travel by the mean motion and adding it to your current mean anomaly simple right well maybe we'll do an example later on to show how this is done the only problem is that we don't have a way of associating the true anomaly to the mean anomaly how do we link the two this is where another parameter called the eccentric anomaly comes in to help the mean anomaly in saving the day the eccentric anomaly is a helper variable that gets us from the true anomaly to the mean anomaly and back again the mean anomaly is Batman then the eccentric anomaly is Robin let's see how the true anomaly mean anomaly and eccentric anomaly are related to each other first let's draw an elliptical orbit and put in a true anomaly next we draw this thing called auxilary circle around the ellipse then we can draw a line up from the satellites position to the auxilary circle making sure the line is perpendicular to a line drawn from perigee to Apogee using the center of the auxiliary circle as a reference point we can then measure the eccentric anomaly as the angle from perigee to the intersection of the line we just drew with the auxiliary circle the mean and not only is off on its own moving around the auxiliary circle at a constant rate it's also measured from the center of the auxiliary circle maybe we can throw in a flux capacitor and that will clear up the problem clears mud right okay maybe a static picture isn't the best way of showing this hopefully this animation will help the green dot is the mean anomaly the blue dot is the true anomaly and the red dot is the eccentric anomaly note that the true anomaly and the eccentric anomaly are connected by a vertical black line throughout the orbit as you can see they are constantly changing speed as you know from the other videos the true anomaly moves fastest at perigee and slowest at Apogee because it is connected to the true anomaly by that magical black line the eccentric anomaly will move in a similar fashion meanwhile the mean anomaly keeps trucking along at a constant rate note that all the anomalies will arrive at perigee and Apogee at the same time the conversions between true anomaly eccentric anomaly and mean anomaly are all defined in standard texts I've provided a link in the show notes to some scripts that convert between these values now for an example you have an orbit with a semi-major axis of 26,000 500 kilometers and the eccentricity of 0.6 the current true anomaly is 5 degrees you want to know where the satellite will be in 60 seconds and in one hour you also want to know where the satellite was 10 minutes ago how can you do this first you calculate the mean motion then you convert the true anomaly to the mean anomaly using the eccentric anomaly as a go-between the only information you need in order to make this calculation is the true anomaly and the eccentricity of the orbit make sure the true anomaly is in radians and not degrees now that you have the current mean anomaly you can calculate the mean anomaly at each of the desired times using the original mean anomaly the mean motion and the time in seconds for each of the mean anomalies that you just calculated you can then convert them back into true anomalies using the eccentric anomaly these true anomalies will now give you the position of the satellite at the desired times using this method you can do a really simple propagation of a satellite orbit this is also called two-body propagation of the orbit the two bodies in this case are the earth and the satellite it's not something that you would use for a highly accurate orbit propagation but it's good enough for back-of-the-envelope calculations well maybe not an envelope how about a legal pad instead thanks for tuning into orbit nerd where we make astrodynamics easier you

Channel: OrbitNerd

Views: 39,622

Keywords: True Anomaly, Mean Anomaly, Eccentric Anomaly, orbit nerd

Id: cf9Jh44kL20

Length: 6min 19sec (379 seconds)

Published: Thu Sep 26 2013