Doppler models - Revision history

NebarnixWikiSysop: /* Circular Case */

2017-04-29T22:36:11Z

Circular Case

NebarnixWikiSysop: /* Polar Spherical Case */

2017-04-29T18:05:13Z

Polar Spherical Case

NebarnixWikiSysop: /* What is a Doppler Shift?= */

2017-04-29T18:04:38Z

What is a Doppler Shift?=

NebarnixWikiSysop at 18:04, 29 April 2017

2017-04-29T18:04:10Z

NebarnixWikiSysop at 17:59, 29 April 2017

2017-04-29T17:59:10Z

NebarnixWikiSysop at 17:47, 29 April 2017

2017-04-29T17:47:33Z

NebarnixWikiSysop: Added animations and totally redid matlab code. Added hidden code places to help confusion factor.

2017-04-29T17:45:04Z

Added animations and totally redid matlab code. Added hidden code places to help confusion factor.

Show changes

NebarnixWikiSysop at 17:26, 29 April 2017

2017-04-29T17:26:20Z

NebarnixWikiSysop at 15:47, 28 April 2017

2017-04-28T15:47:29Z

NebarnixWikiSysop: /* Variable Inclination Spherical Case with Rotating Earth */

2016-05-01T03:51:56Z

Variable Inclination Spherical Case with Rotating Earth

@@ Line 227: / Line 227: @@
 [[File:RaceCar web.gif|400px]][[File:RaceCar2 web.gif|400px]] <br />
 The case for the viewer outside the track. Notice how the relative velocity will always reach the speed of the car, because the car is, for an instant, 'pointed' directly at the observer (A tangent intersects the observer)
 [[File:RaceCarPit Web.gif|400px]][[File:RaceCarPit2 web.gif|400px]]<br />
 The case for the viewer inside the track. Note that the relative velocity is always less than the speed of the car, because the car is never 'pointed' at the observer.

@@ Line 494: / Line 494: @@
 </div>
-==Variable Inclination Spherical Case with Rotating Earth==
+=Variable Inclination Spherical Case with Rotating Earth=
 Whittling away errors, we have now exposed the next problem. Orbits have inclinations, which means that the earth's angular velocity vector and the satellite's motion vector are no longer orthagonal. During ascending nodes and descending nodes the velocity of the earth slightly adds and subtracts from the relative velocity, and thus each node will have a slightly different shape based on this larger or smaller velocity.

@@ Line 10: / Line 10: @@
 <blockquote>Christian Andreas Doppler (/ˈdɒplər/; German: [ˈdɔplɐ]; 29 November 1803 – 17 March 1853) was an Austrian mathematician and physicist. He is celebrated for his principle — known as the Doppler effect — that the observed frequency of a wave depends on the relative speed of the source and the observer. He used this concept to explain the color of binary stars.</blockquote>
-==What is a Doppler Shift?===
+==What is a Doppler Shift?==
 We should all be familiar with the sound of a passing car, airplane, or even more dramatic, passing ambulance. The sound changes as the object rushes by us. The pitch becomes lower, sometimes dramatically depending on how close we are and how fast (and in which direction) the object is moving. We can model this using math!
@@ Line 17: / Line 17: @@
 Doppler shifting is caused by the compression of the waves emitted from a source. For audio waves it is the physical objecting 'catching up' with waves emitted. Sound waves always travel at a constant velocity (speed of sound), so you can 'catch up' with the waves moving in front of, and 'leave behind' the waves emitted backwards. As the space betweeen the waves changes, their frequency changes because frequency is defined by the number of changes of the wave per unit time. More changes per unit distance means you hear the vibrations faster, less changes per unit time means you hear the vibrations slower. Wikipedia has several articles to make this more understandable. For electromagnetic waves, it is actually time compression of the source, but it can be modeled as the same effect with the same equations.
 Project Desert Tortoise will uses these models to perform geolocation of radio transmitters based their change in frequency as heard by the passing spacecraft. This also could be used to determine your location based on the changing frequency of a passing spacecraft, which is exactly how it was done before GPS came along. GPS is a much better solution, but has its own set of challenges (it won't work on Mars, for example).
 =Linear Case=

@@ Line 1: / Line 1: @@
 =Latest Doppler Fit Model for ARGOS-2 Data=
 http://wiki.nebarnix.com/wiki/File:Doppler3DRotationInclination3FitandWebephem.m
 =Understanding Doppler Shifts=
 ==Who is Doppler?==

@@ Line 5: / Line 5: @@
 '''page in work'''
-Doppler shifting is caused by compression of the waves emitted from a source. For audio waves it is the physical objecting 'catching up' with waves emitted at a constant velocity (speed of sound) in front and 'leaving behind' the waves emitted backwards. Wikipedia has several articles to make this more understandable. For electromagnetic waves, it is actually time compression of the source, but it can be modeled as the same effect with the same equations.
+==Who is Doppler?==
-We need to model the Doppler shifts of a radio signal in order to match real world data to an actual orbit (or position along a ground track of a known orbit).
+==What is a Doppler Shift?===

@@ Line 10: / Line 10: @@
-==Linear Case==
+=Linear Case=
 Let's start simple. We have a road and there is an observer (A) standing at some distance d from this road. A car (B) is traveling at a constant velocity v. Let us assume that it is emitting engine noise at frequency ftx. The road is straight, and so we will use Cartesian (x and y) coordinates for this example.
@@ Line 209: / Line 209: @@
 <br />
-==Circular Case==
+=Circular Case=
 It is clear that this picture does not work for a satellite. A satellite is not traveling along a linear path, instead, a circle. Let's follow the same idea but close the track from a country road into a racetrack where the cars are traveling in a circle.
@@ Line 353: / Line 353: @@
 </div>
-==Polar Spherical Case==
+=Polar Spherical Case=
-The circular case models what happens when a satellite passes DIRECTLY over you, but cannot account for what happens when there is an offset to one side of the pass. We can only model this case by moving forward to spherical coordinates. Things get a little messy here, but we will use the exact same approaches we used before -- the equations are just a little longer because we have to account for the new unit vector Phi. We will use the standard where phi represents rotation along the equator and theta represents the angle between the north pole and the vector while r is again the radius.
+The circular case models what happens when a satellite passes DIRECTLY over you, but cannot account for what happens when there is an offset to one side of the pass. We can only model this case by moving forward to spherical coordinates. Things get a little messy here, but we will use the exact same approaches we used before -- the equations are just a little longer because we have to account for the new third axis, represented by the unit vector Phi. We will use the standard where phi represents rotation along the equator and theta represents the angle between the north pole and the vector while r is again the radius.
-This case works quite well, and in fact is almost enough to solve our problem. We can model a 90 degree inclination orbit which is representative of any other circular orbit. We can probably even use this model to solve for geo-locations within about 250km which isn't bad for a 'rough guess'.
+This case models how a satellite travels quite well, and in fact is almost enough to solve our problem. We could model a 90 degree inclination orbit, which is representative of any other circular orbit passing directly overhead. We can probably even use this model to solve for geo-locations within about 250km which isn't bad for a 'rough guess'.
-What is that 250km all about? Well unfortunately this is the distance that a point will move along the equator during a 15 minute satellite pass (assuming a sat altitude of 850km). The effect will get better at the pole and worse at the equator, but we really can't ignore this.
+What is that 250km all about? Well unfortunately this is the distance that a point will move along the equator during a 15 minute satellite pass (assuming a sat altitude of 850km). The effect will get better at the pole and worse at the equator, but we really can't ignore this basic fact: the earth turns under the satellites!
 <div class="toccolours mw-collapsible mw-collapsed">

@@ Line 12: / Line 12: @@
 ==Linear Case==
-Let's start simple. We have a road and there is an observer (A) standing at some distance d from this road. A car (B) is traveling at a constant velocity v. Let us assume that it is emitting engine noise at frequency ftx. The road is straight, and so we will use Cartesian coordinates for this example.
+Let's start simple. We have a road and there is an observer (A) standing at some distance d from this road. A car (B) is traveling at a constant velocity v. Let us assume that it is emitting engine noise at frequency ftx. The road is straight, and so we will use Cartesian (x and y) coordinates for this example.
 <html>
@@ Line 72: / Line 72: @@
 <img src="https://upload.wikimedia.org/math/7/1/5/715e445551d99e52e53ba1e8cdc0aba5.png"><br />
 </html>
 ==Circular Case==

← Older revision		Revision as of 15:47, 28 April 2017
Line 78:		Line 78:

	We can use the same variables, but now we can reference everyone against the center of the track instead of using the observer as the origin. The same car B is traveling at the same constant velocity v on a track with a radius Rb while our observer A is at distance Ra from the center of the track, and an angle Atheta. We are doing away with the above notation because it takes forever and Matlab is so nice! We can still use Wolfram Alpha for calculus and plotting. Note that from here on out we will switch from audio to radio waves (from speed of sound in air to speed of light) and we will be using the expanded doppler formula because satellites move REALLY fast.		We can use the same variables, but now we can reference everyone against the center of the track instead of using the observer as the origin. The same car B is traveling at the same constant velocity v on a track with a radius Rb while our observer A is at distance Ra from the center of the track, and an angle Atheta. We are doing away with the above notation because it takes forever and Matlab is so nice! We can still use Wolfram Alpha for calculus and plotting. Note that from here on out we will switch from audio to radio waves (from speed of sound in air to speed of light) and we will be using the expanded doppler formula because satellites move REALLY fast.
		+
		+	<br />
		+	[[File:RaceCar web.gif\|400px]][[File:RaceCar2 web.gif\|400px]] <br />
		+	The case for the viewer outside the track. Notice how the relative velocity will always reach the speed of the car, because the car is, for an instant, 'pointed' directly at the observer (A tangent intersects the observer)
		+	[[File:RaceCarPit Web.gif\|400px]][[File:RaceCarPit2 web.gif\|400px]]<br />
		+	The case for the viewer inside the track. Note that the relative velocity is always less than the speed of the car, because the car is never 'pointed' at the observer.
		+	<br />

	<pre>		<pre>

← Older revision		Revision as of 03:51, 1 May 2016
Line 251:		Line 251:

	%Vt = (ObsRadSatRad(sin(ObsPhi - atan2(-sin(SatPhase + SatAngularVelt)cos(SatInc), cos(SatPhase + SatAngularVelt)) + ObsAngularVelt)sin(pi/2 - atan2(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc), (SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))sin(ObsTheta)(ObsAngularVel + ((real(cos(SatPhase + SatAngularVelt)) + imag(sin(SatPhase + SatAngularVelt)cos(SatInc)))^2((imag(SatAngularVelsin(SatPhase + SatAngularVelt)) + real(SatAngularVelcos(SatPhase + SatAngularVelt)cos(SatInc)))/(real(cos(SatPhase + SatAngularVelt)) + imag(sin(SatPhase + SatAngularVelt)cos(SatInc))) - ((real(SatAngularVelsin(SatPhase + SatAngularVelt)) - imag(SatAngularVelcos(SatPhase + SatAngularVelt)cos(SatInc)))(imag(cos(SatPhase + SatAngularVelt)) - real(sin(SatPhase + SatAngularVelt)cos(SatInc))))/(real(cos(SatPhase + SatAngularVelt)) + imag(sin(SatPhase + SatAngularVelt)cos(SatInc)))^2))/((imag(cos(SatPhase + SatAngularVelt)) - real(sin(SatPhase + SatAngularVelt)cos(SatInc)))^2 + (real(cos(SatPhase + SatAngularVelt)) + imag(sin(SatPhase + SatAngularVelt)cos(SatInc)))^2)) - (sin(pi/2 - atan2(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc), (SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))cos(ObsTheta)(imag(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) - real((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))^2((imag((2SatRad^2SatAngularVelcos(SatPhase + SatAngularVelt)sin(SatPhase + SatAngularVelt) - 2SatRad^2SatAngularVelcos(SatPhase + SatAngularVelt)sin(SatPhase + SatAngularVelt)cos(SatInc)^2)/(SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2))/2 - real(SatRadSatAngularVelcos(SatPhase + SatAngularVelt)sin(SatInc)))/(imag(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) - real((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2))) + ((real(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) + imag((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))(real((2SatRad^2SatAngularVelcos(SatPhase + SatAngularVelt)sin(SatPhase + SatAngularVelt) - 2SatRad^2SatAngularVelcos(SatPhase + SatAngularVelt)sin(SatPhase + SatAngularVelt)cos(SatInc)^2)/(SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2))/2 + imag(SatRadSatAngularVelcos(SatPhase + SatAngularVelt)sin(SatInc))))/(imag(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) - real((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))^2))/((real(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) + imag((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))^2 + (imag(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) - real((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))^2) + (cos(pi/2 - atan2(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc), (SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))sin(ObsTheta)cos(ObsPhi - atan2(-sin(SatPhase + SatAngularVelt)cos(SatInc), cos(SatPhase + SatAngularVelt)) + ObsAngularVelt)(imag(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) - real((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))^2((imag((2SatRad^2SatAngularVelcos(SatPhase + SatAngularVelt)sin(SatPhase + SatAngularVelt) - 2SatRad^2SatAngularVelcos(SatPhase + SatAngularVelt)sin(SatPhase + SatAngularVelt)cos(SatInc)^2)/(SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2))/2 - real(SatRadSatAngularVelcos(SatPhase + SatAngularVelt)sin(SatInc)))/(imag(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) - real((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2))) + ((real(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) + imag((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))(real((2SatRad^2SatAngularVelcos(SatPhase + SatAngularVelt)sin(SatPhase + SatAngularVelt) - 2SatRad^2SatAngularVelcos(SatPhase + SatAngularVelt)sin(SatPhase + SatAngularVelt)cos(SatInc)^2)/(SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2))/2 + imag(SatRadSatAngularVelcos(SatPhase + SatAngularVelt)sin(SatInc))))/(imag(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) - real((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))^2))/((real(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) + imag((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))^2 + (imag(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) - real((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))^2)))/(ObsRad^2 + SatRad^2 - 2ObsRadSatRad(cos(pi/2 - atan2(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc), (SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))cos(ObsTheta) + sin(pi/2 - atan2(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc), (SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))sin(ObsTheta)cos(ObsPhi - atan2(-sin(SatPhase + SatAngularVelt)cos(SatInc), cos(SatPhase + SatAngularVelt)) + ObsAngularVelt)))^(1/2)</pre>		%Vt = (ObsRadSatRad(sin(ObsPhi - atan2(-sin(SatPhase + SatAngularVelt)cos(SatInc), cos(SatPhase + SatAngularVelt)) + ObsAngularVelt)sin(pi/2 - atan2(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc), (SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))sin(ObsTheta)(ObsAngularVel + ((real(cos(SatPhase + SatAngularVelt)) + imag(sin(SatPhase + SatAngularVelt)cos(SatInc)))^2((imag(SatAngularVelsin(SatPhase + SatAngularVelt)) + real(SatAngularVelcos(SatPhase + SatAngularVelt)cos(SatInc)))/(real(cos(SatPhase + SatAngularVelt)) + imag(sin(SatPhase + SatAngularVelt)cos(SatInc))) - ((real(SatAngularVelsin(SatPhase + SatAngularVelt)) - imag(SatAngularVelcos(SatPhase + SatAngularVelt)cos(SatInc)))(imag(cos(SatPhase + SatAngularVelt)) - real(sin(SatPhase + SatAngularVelt)cos(SatInc))))/(real(cos(SatPhase + SatAngularVelt)) + imag(sin(SatPhase + SatAngularVelt)cos(SatInc)))^2))/((imag(cos(SatPhase + SatAngularVelt)) - real(sin(SatPhase + SatAngularVelt)cos(SatInc)))^2 + (real(cos(SatPhase + SatAngularVelt)) + imag(sin(SatPhase + SatAngularVelt)cos(SatInc)))^2)) - (sin(pi/2 - atan2(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc), (SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))cos(ObsTheta)(imag(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) - real((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))^2((imag((2SatRad^2SatAngularVelcos(SatPhase + SatAngularVelt)sin(SatPhase + SatAngularVelt) - 2SatRad^2SatAngularVelcos(SatPhase + SatAngularVelt)sin(SatPhase + SatAngularVelt)cos(SatInc)^2)/(SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2))/2 - real(SatRadSatAngularVelcos(SatPhase + SatAngularVelt)sin(SatInc)))/(imag(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) - real((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2))) + ((real(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) + imag((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))(real((2SatRad^2SatAngularVelcos(SatPhase + SatAngularVelt)sin(SatPhase + SatAngularVelt) - 2SatRad^2SatAngularVelcos(SatPhase + SatAngularVelt)sin(SatPhase + SatAngularVelt)cos(SatInc)^2)/(SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2))/2 + imag(SatRadSatAngularVelcos(SatPhase + SatAngularVelt)sin(SatInc))))/(imag(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) - real((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))^2))/((real(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) + imag((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))^2 + (imag(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) - real((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))^2) + (cos(pi/2 - atan2(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc), (SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))sin(ObsTheta)cos(ObsPhi - atan2(-sin(SatPhase + SatAngularVelt)cos(SatInc), cos(SatPhase + SatAngularVelt)) + ObsAngularVelt)(imag(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) - real((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))^2((imag((2SatRad^2SatAngularVelcos(SatPhase + SatAngularVelt)sin(SatPhase + SatAngularVelt) - 2SatRad^2SatAngularVelcos(SatPhase + SatAngularVelt)sin(SatPhase + SatAngularVelt)cos(SatInc)^2)/(SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2))/2 - real(SatRadSatAngularVelcos(SatPhase + SatAngularVelt)sin(SatInc)))/(imag(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) - real((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2))) + ((real(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) + imag((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))(real((2SatRad^2SatAngularVelcos(SatPhase + SatAngularVelt)sin(SatPhase + SatAngularVelt) - 2SatRad^2SatAngularVelcos(SatPhase + SatAngularVelt)sin(SatPhase + SatAngularVelt)cos(SatInc)^2)/(SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2))/2 + imag(SatRadSatAngularVelcos(SatPhase + SatAngularVelt)sin(SatInc))))/(imag(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) - real((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))^2))/((real(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) + imag((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))^2 + (imag(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc)) - real((SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))^2)))/(ObsRad^2 + SatRad^2 - 2ObsRadSatRad(cos(pi/2 - atan2(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc), (SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))cos(ObsTheta) + sin(pi/2 - atan2(SatRadsin(SatPhase + SatAngularVelt)sin(SatInc), (SatRad^2cos(SatPhase + SatAngularVelt)^2 + SatRad^2sin(SatPhase + SatAngularVelt)^2cos(SatInc)^2)^(1/2)))sin(ObsTheta)cos(ObsPhi - atan2(-sin(SatPhase + SatAngularVelt)cos(SatInc), cos(SatPhase + SatAngularVelt)) + ObsAngularVelt)))^(1/2)</pre>
		+
		+	=== Errors ===
		+	The model is a circular orbit, not an elliptical one. This has various implications for the velocity and altitude differences from reality. An analysis of the ground-track error was performed. The following plot tracks the error as the model diverges from ephemris data for a random pass of NOAA-18. Max error (for this pass) was only 4.8km after 15 minutes (max pass duration). Running a rough analysis pass, then following it up with a re-initialization of the model using the ephemris parameters at the point of minimum relative velocity should spread the error to the horizons and minimize the solution error.
		+
		+	[[File:ModelvsEphemErrorinGroundKm2.png\|800px]]