# Doppler models

## Contents

# Latest Doppler Fit Model for ARGOS-2 Data

http://wiki.nebarnix.com/wiki/File:Doppler3DRotationInclination3FitandWebephem.m

# Understanding Doppler Shifts

**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.

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).

## 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.

The distance between them is just the Pythagorean Theorem, which is the distance formula.

Or in the full expanded form

Now what is the relative velocity between these two points? It is simply the rate of change of the distance. If you don't know calculus, this is a GREAT example of just how powerful it can be. Luckily, we can take what is called a derivative -- which is exactly the rate of change of a formula and we can use Wolfram Alpha to do the hard work for us. Seriously. Click the link or type "derivative of sqrt{(A1-(v*t+B1))^2+(A2-B2)^2} with respect to t" into the box on the main page. It will spit the following back at you.

Now that we have the formula for relative velocity of an object moving at a constant speed along the x axis, let's plot an example with the following parameters (you are standing 5 meters from the road and the car moving from -10 to +10 meters along the road at 1 meter per second).

and

Relative Distance Between the car and the observer as a function of time:

http://www.wolframalpha.com/input/?i=plot+sqrt%7B%280-%281*t%2B-10%29%29%5E2%2B%280-5%29%5E2%7D+from+0+to+20

Relative Velocity Between the car and the observer as a function of time:

http://www.wolframalpha.com/input/?i=plot+%28%280-%28-10%29-x%29%29%2Fsqrt%28%280-%28-10%29-x%29%5E2%2B%280-5%29%5E2%29+x+from+0+to+20

We're almost there. The formula for the doppler shift of an emitted frequency is on the wikipedia article where ft is the transmitted frequency and c is the speed of waves in the medium (in this case, the speed of sound in air)

So this makes our final (huff puff!) equation!

Assume engine noise Ftx = 3khz and c = 343 m/sec (at standard conditions)

Plotting this (Wolfram Alpha "plot (2*3000*((0-(-10)-x))/sqrt((0-(-10)-x)^2+(0-5)^2))/343 x from 0 to 20") shows the final received frequency over time! YAY! NEEEEEEEEOoowwwww!!! Except our car is traveling a whopping 1 meter per second, so the doppler is only 15Hz. Feel free to imagine that this is 150Hz and the car is moving at 10m/sec.

http://www.wolframalpha.com/input/?i=plot+%282*3000*%28%280-%28-10%29-x%29%29%2Fsqrt%28%280-%28-10%29-x%29%5E2%2B%280-5%29%5E2%29%29%2F343+x+from+0+to+20

This relationship is linear as long as the car stays below some fraction of the speed of sound (Hopefully!). If we wish to assume that the car can go REALLY fast, we need to use the expanded form of the doppler equation which is left up to the reader and Wolfram Alpha:

## 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.

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.

%% 2D circular Relative Velocity %using kilometers and seconds clear all; ObsAlta = 6378; %sea level SatAlt = 6378+850; %850km orbit OrbitalPeriod = 6120; %102 minutes SatAngularVel = 2*pi/OrbitalPeriod; t=0:OrbitalPeriod/100:2*OrbitalPeriod; %1000 seconds InitialAngle = 0; Vt = -(ObsAlta*SatAlt*SatAngularVel.*sin(InitialAngle+SatAngularVel.*t))./(sqrt(ObsAlta^2+SatAlt^2-2*ObsAlta*SatAlt.*cos(InitialAngle+SatAngularVel.*t))); subplot(3,1,1); plot(t,Vt_den); title(['Distance to Observer: ' num2str(min(Vt_den)) 'km at ' num2str(TcrossingInterp2) 's']); subplot(3,1,2); plot(t,Vt); title('Velocity to Observer');

## 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.

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'.

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.

%% 3D Relative Velocity %using kilometers and seconds %ObsPhi = 20*pi/180; %ObsTheta = pi/2; %ObsRad = 6378; %sea level ObsRad = 6368; ObsTheta = 1.74; ObsPhi = 0.135; SatRad = ObsRad + 854; %sea level + altitude SatPhi = 0; %inclination of 90 SatTheta = pi/2;; %Initial position of 0 (equator) OrbitalPeriod = 6127.2; %102 minutes SatAngularVel = 2*pi/OrbitalPeriod; Fscale = 4.9; Foffset = 2048; t=-.07*OrbitalPeriod:OrbitalPeriod/300:.07*OrbitalPeriod; %1000 seconds Vt_num = (ObsRad*SatRad*SatAngularVel*(sin(ObsTheta)*cos(SatPhi-ObsPhi)*cos(SatAngularVel*t+SatTheta)-cos(ObsTheta)*sin(SatAngularVel*t+SatTheta))); Vt_den = sqrt(ObsRad^2+SatRad^2-2*ObsRad*SatRad*(sin(SatTheta+SatAngularVel*t)*sin(ObsTheta)*cos(SatPhi-ObsPhi)+cos(SatTheta+SatAngularVel*t)*cos(ObsTheta))); Vt = Vt_num ./ Vt_den; subplot(3,1,1); plot(t,Vt_den); title(['Distance to Observer: ' num2str(min(Vt_den)) 'km at ' num2str(TcrossingInterp2) 's']); subplot(3,1,2); plot(t,Vt); title('Velocity to Observer'); %%Doppler c = 299792; %in km per second Ft = 401.65e6; %401.65Mhz uplink freq F =(2*Vt*Ft)./(c-Vt); subplot(3,1,3); plot(t,F,xdata,Fscale*ydata+Foffset,'-o'); title('Doppler Shift of Observer (at 401Mhz)'); max(F)

## Polar Spherical Case with Rotating Earth

This is a simple modification. We can just make phi of the observer also time dependent then re-calculate the derivative. The curve changes shape noticeably, which is expected because 250 miles is no small distance.

%% 3D Relative Velocity with Earth's Rotation %derivative of sqrt(A1^2+B1^2-2*A1*B1*(sin (v*t+A2)*sin (B2)*cos (A3-(v2*t+B3))+cos (v*t+A2)*cos (B2))) with respect to t %(d)/(dt)(sqrt(A1^2+B1^2-2 A1 B1 (sin(v t+A2) sin(B2) cos(A3-(v2 t+B3))+cos(v t+A2) cos(B2)))) = -(A1 B1 (v2 sin(B2) sin(A2+t v) sin(A3-B3-t v2)+v sin(B2) cos(A2+t v) cos(A3-B3-t v2)-v cos(B2) sin(A2+t v)))/sqrt(A1^2-2 A1 B1 (sin(B2) sin(A2+t v) cos(A3-B3-t v2)+cos(B2) cos(A2+t v))+B1^2) %A1 is sat radius => SatRad %v is sat angular velocity = SatAngularVel %A2 is sat theta start = SatTheta %A3 is sat phi => SatPhi %B1 is obs radius => ObsRad %B2 is obs theta => ObsTheta %B3 is obs phi start => ObsPhi %v2 is obs angular velocity => ObsAngularVel ydata = [1506, 1022, 493, -61, -618, -1154, -1648, -2089, -2472, -2798, -3070, -3296, -3482]; xdata = [42.311,72.211,102.4552,132.2255,162.215,192.2178,222.314,252.2178,282.2255,312.2255,342.2255,372.2255,402.2255]; ObsRad = 6368; %center of European pass (50 degrees) ObsTheta = 1.74; ObsPhi = 0.135; SidrealPeriod = 86164; %23 hours 56 minutes and 4 seconds => seconds ObsAngularVel = 2*pi/SidrealPeriod; SatRad = ObsRad + 854; %sea level + altitude SatPhi = 0; %inclination of 90 (really its 98, but we don't realy have a term for this SatTheta = pi/2; %Initial position of 0 (equator) OrbitalPeriod = 6127.2; %102.12 minutes => seconds SatAngularVel = 2*pi/OrbitalPeriod; Fscale = 4.9; Foffset = 2048; t = -0.07*OrbitalPeriod : OrbitalPeriod/1000 : 0.07*OrbitalPeriod; %1000 seconds Vt_num = (SatRad * ObsRad * (ObsAngularVel*sin(ObsTheta).*sin(SatAngularVel*t+SatTheta).*sin(SatPhi-ObsPhi-ObsAngularVel*t)+SatAngularVel*sin(ObsTheta).*cos(SatTheta+SatAngularVel*t).*cos(SatPhi-ObsPhi-t*ObsAngularVel)-SatAngularVel*cos(ObsTheta).*sin(SatTheta+t*SatAngularVel))); Vt_den = sqrt(ObsRad^2 + SatRad^2 - 2*SatRad*ObsRad*(sin(ObsTheta).*sin(SatAngularVel*t+SatTheta).*cos(SatPhi-ObsPhi-ObsAngularVel*t)+cos(ObsTheta).*cos(SatTheta+SatAngularVel*t))); Vt = Vt_num ./ Vt_den; subplot(3,1,1); plot(t,Vt_den); TcrossingInterp2= interp1(Vt,t,0); title(['Distance to Observer: ' num2str(min(Vt_den)) 'km at ' num2str(TcrossingInterp2) 's']); subplot(3,1,2); plot(t,Vt); title('Velocity to Observer'); %%Doppler c = 299792; %in km per second Ft = 401.65e6; %401.65Mhz uplink freq F =(2*Ft*Vt)./(c-Vt); subplot(3,1,3); plot(t,F,xdata,Fscale*ydata+Foffset,'-o'); title('Doppler Shift of Observer (at 401Mhz)'); max(F);

## 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.

It turns out, this isn't so easy. Our models break down when we try to add some sort of tilt to the axis. Never fear, start where you know! I was honestly stumped, so I started making circles and ovals and thinking about what an orbit looks like on the XY, YZ, and XZ planes. I came up with this:

x = (r.*cos(omega.*t+phase)); y = (r.*-cos(inc).*sin(omega.*t+phase)); z = (r.*sin(inc).*sin(omega.*t+phase));

Go ahead, plot this. Its a circular orbit with variable inclination! But now the hard part, we have to transfer this to spherical coordinates to work with the models we want.

If we follow the following transform:

phi = atan2(y,x); theta = acos(sin(z/(r*sin(theta)))); r = r;

we can arrive at the final answer. we can convert acos to arctan using an identity to make matlab happy (it didn't seem to work with acos)

phi = atan2((-cos(inc)*sin(omega*t+phase)),(cos(omega*t+phase))); theta = (pi/2.0)-atan2((r*sin(inc)*sin(omega*t+phase)),sqrt((r.*-cos(inc).*sin(omega.*t+phase)).^2 + (r.*cos(omega.*t+phase)).^2));

That makes the distance formula between the observer and the sat:

%Distance formula in spherical coordinates is %sqrt(r1^2 +r2^2 -2*r1 *r2 *(sin(theta1)*sin(theta2)*cos(phi1-phi2)+cos(theta1)*cos(theta2))) %D(t) = sqrt(SatRad^2+ObsRad^2-2*SatRad*ObsRad*(sin(((pi/2.0)-atan2((SatRad.*sin(SatInc).*sin(SatAngularVel.*t+SatPhase)),sqrt((SatRad.*-cos(SatInc).*sin(SatAngularVel.*t+SatPhase)).^2 + (SatRad.*cos(SatAngularVel.*t+SatPhase)).^2))))*sin(ObsTheta)*cos(atan2((-cos(SatInc).*sin(SatAngularVel.*t+SatPhase)),(cos(SatAngularVel.*t+SatPhase)))-(ObsAngularVel*t+ObsPhi))+cos(((pi/2.0)-atan2((SatRad.*sin(SatInc).*sin(SatAngularVel.*t+SatPhase)),sqrt((SatRad.*-cos(SatInc).*sin(SatAngularVel.*t+SatPhase)).^2 + (SatRad.*cos(SatAngularVel.*t+SatPhase)).^2))))*cos(ObsTheta)))

If we take the derivative (yes, yes, its VERY ugly due to the use of atan2...) then we arrive at the final formula.

%syms t SatRad ObsRad ObsPhi ObsTheta ObsAngularVel SatInc SatAngularVel ObsAngularVel SatPhase %diff(sqrt(SatRad^2+ObsRad^2-2*SatRad*ObsRad*(sin(((pi/2.0)-atan2((SatRad.*sin(SatInc).*sin(SatAngularVel.*t+SatPhase)),sqrt((SatRad.*-cos(SatInc).*sin(SatAngularVel.*t+SatPhase)).^2 + (SatRad.*cos(SatAngularVel.*t+SatPhase)).^2))))*sin(ObsTheta)*cos(atan2((-cos(SatInc).*sin(SatAngularVel.*t+SatPhase)),(cos(SatAngularVel.*t+SatPhase)))-(ObsAngularVel*t+ObsPhi))+cos(((pi/2.0)-atan2((SatRad.*sin(SatInc).*sin(SatAngularVel.*t+SatPhase)),sqrt((SatRad.*-cos(SatInc).*sin(SatAngularVel.*t+SatPhase)).^2 + (SatRad.*cos(SatAngularVel.*t+SatPhase)).^2))))*cos(ObsTheta))),t) %Vt = (ObsRad*SatRad*(sin(ObsPhi - atan2(-sin(SatPhase + SatAngularVel*t)*cos(SatInc), cos(SatPhase + SatAngularVel*t)) + ObsAngularVel*t)*sin(pi/2 - atan2(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc), (SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2)))*sin(ObsTheta)*(ObsAngularVel + ((real(cos(SatPhase + SatAngularVel*t)) + imag(sin(SatPhase + SatAngularVel*t)*cos(SatInc)))^2*((imag(SatAngularVel*sin(SatPhase + SatAngularVel*t)) + real(SatAngularVel*cos(SatPhase + SatAngularVel*t)*cos(SatInc)))/(real(cos(SatPhase + SatAngularVel*t)) + imag(sin(SatPhase + SatAngularVel*t)*cos(SatInc))) - ((real(SatAngularVel*sin(SatPhase + SatAngularVel*t)) - imag(SatAngularVel*cos(SatPhase + SatAngularVel*t)*cos(SatInc)))*(imag(cos(SatPhase + SatAngularVel*t)) - real(sin(SatPhase + SatAngularVel*t)*cos(SatInc))))/(real(cos(SatPhase + SatAngularVel*t)) + imag(sin(SatPhase + SatAngularVel*t)*cos(SatInc)))^2))/((imag(cos(SatPhase + SatAngularVel*t)) - real(sin(SatPhase + SatAngularVel*t)*cos(SatInc)))^2 + (real(cos(SatPhase + SatAngularVel*t)) + imag(sin(SatPhase + SatAngularVel*t)*cos(SatInc)))^2)) - (sin(pi/2 - atan2(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc), (SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2)))*cos(ObsTheta)*(imag(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc)) - real((SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2)))^2*((imag((2*SatRad^2*SatAngularVel*cos(SatPhase + SatAngularVel*t)*sin(SatPhase + SatAngularVel*t) - 2*SatRad^2*SatAngularVel*cos(SatPhase + SatAngularVel*t)*sin(SatPhase + SatAngularVel*t)*cos(SatInc)^2)/(SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2))/2 - real(SatRad*SatAngularVel*cos(SatPhase + SatAngularVel*t)*sin(SatInc)))/(imag(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc)) - real((SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2))) + ((real(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc)) + imag((SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2)))*(real((2*SatRad^2*SatAngularVel*cos(SatPhase + SatAngularVel*t)*sin(SatPhase + SatAngularVel*t) - 2*SatRad^2*SatAngularVel*cos(SatPhase + SatAngularVel*t)*sin(SatPhase + SatAngularVel*t)*cos(SatInc)^2)/(SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2))/2 + imag(SatRad*SatAngularVel*cos(SatPhase + SatAngularVel*t)*sin(SatInc))))/(imag(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc)) - real((SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2)))^2))/((real(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc)) + imag((SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2)))^2 + (imag(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc)) - real((SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2)))^2) + (cos(pi/2 - atan2(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc), (SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2)))*sin(ObsTheta)*cos(ObsPhi - atan2(-sin(SatPhase + SatAngularVel*t)*cos(SatInc), cos(SatPhase + SatAngularVel*t)) + ObsAngularVel*t)*(imag(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc)) - real((SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2)))^2*((imag((2*SatRad^2*SatAngularVel*cos(SatPhase + SatAngularVel*t)*sin(SatPhase + SatAngularVel*t) - 2*SatRad^2*SatAngularVel*cos(SatPhase + SatAngularVel*t)*sin(SatPhase + SatAngularVel*t)*cos(SatInc)^2)/(SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2))/2 - real(SatRad*SatAngularVel*cos(SatPhase + SatAngularVel*t)*sin(SatInc)))/(imag(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc)) - real((SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2))) + ((real(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc)) + imag((SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2)))*(real((2*SatRad^2*SatAngularVel*cos(SatPhase + SatAngularVel*t)*sin(SatPhase + SatAngularVel*t) - 2*SatRad^2*SatAngularVel*cos(SatPhase + SatAngularVel*t)*sin(SatPhase + SatAngularVel*t)*cos(SatInc)^2)/(SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2))/2 + imag(SatRad*SatAngularVel*cos(SatPhase + SatAngularVel*t)*sin(SatInc))))/(imag(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc)) - real((SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2)))^2))/((real(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc)) + imag((SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2)))^2 + (imag(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc)) - real((SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2)))^2)))/(ObsRad^2 + SatRad^2 - 2*ObsRad*SatRad*(cos(pi/2 - atan2(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc), (SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2)))*cos(ObsTheta) + sin(pi/2 - atan2(SatRad*sin(SatPhase + SatAngularVel*t)*sin(SatInc), (SatRad^2*cos(SatPhase + SatAngularVel*t)^2 + SatRad^2*sin(SatPhase + SatAngularVel*t)^2*cos(SatInc)^2)^(1/2)))*sin(ObsTheta)*cos(ObsPhi - atan2(-sin(SatPhase + SatAngularVel*t)*cos(SatInc), cos(SatPhase + SatAngularVel*t)) + ObsAngularVel*t)))^(1/2)

### 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.