Hi friends,

Following is the code I was writing to simulate the trajectory of sun, earth and moon...
The trajectory of sun and earth works out just fine, but there is some problem with the moon's trajectory... kindly have a look at the code and help me out.

The code is self-explanatory to some extent, but ask to clear any doubts you have about my presumptions, etc.

Please help me out with this... 

% ------PartI: The simulation of sun earth moon system-------
clear;
clc;

Me=5.972E24;                 % Mass of earth
Ms= 1.988E30;                % Mass of sun
R= 1.495E11;                 % Orbital radius of earth
r= 3.85E8;                   % Orbital radius of moon
G= 6.67E-11;                 % Gravitational Constant
T=3.155E7;                   % orbital time of earth
w= sqrt(G*(Me+Ms))/R;        % Angular velocity
de= Ms*R/(Ms+Me);            % Distance between earth and centre of mass
ds= Me*R/(Ms+Me);            % Distance between sun and centre of mass
vv=1.018E3;                  % average velocity of moon

% Propagation of time
t=linspace(0,80,100);

% Equation for coordinates of sun and earth
xs=ds*cos(w.*t);
ys=ds*sin(w.*t);
xe=-de*cos(w.*t);
ye=-de*sin(w.*t);

% Plotting of trajectory of sun and earth

plot(xs,ys,'yo');
hold on;
plot(xe,ye,'go');
grid on;

% -----Part II: Simulation of trajectory of moon around earth and sun-----

% global x y;
x(size(t))=0;
y(size(t))=0;
a= sqrt(((x - xs).^2) + ((y - ys).^2));
b= sqrt(((x - xe).^2) + ((y - ye).^2));
q= ((G*Ms.*(x-xs))./power(a,3) + (G*Me.*(x-xe))./power(b,3));
w= ((G*Ms.*(y-ys))./power(a,3) + (G*Me.*(y-ye))./power(b,3));
X=dsolve('D2x+q=0','x(0)= -(de+r)','Dx(0)=0');
Y=dsolve('D2y+w=0','y(0)=0','Dy(0)=vv');
x=subs(X);
y=subs(Y);
plot(x,y,'k');

-------------------------------------------------------------------------------------------------

If you can give me some interesting ideas about presenting it better, please feel free to add it as well.

On Dec 28, 12:06=A0pm, "Nishant Gupta" <im_nisha...@yahoo.co.in> wrote:
> Hi friends,
>
> Following is the code I was writing to simulate the trajectory of sun, ea=
rth and moon...
> The trajectory of sun and earth works out just fine, but there is some pr=
oblem with the moon's trajectory... kindly have a look at the code and help=
 me out.
>
> The code is self-explanatory to some extent, but ask to clear any doubts =
you have about my presumptions, etc.
>
> Please help me out with this...
>
> % ------PartI: The simulation of sun earth moon system-------
> clear;
> clc;
>
> Me=3D5.972E24; =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 % Mass of earth
> Ms=3D 1.988E30; =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0% Mass of sun
> R=3D 1.495E11; =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 % Orbital radius of earth
> r=3D 3.85E8; =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 % Orbital radius of moon
> G=3D 6.67E-11; =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 % Gravitational Constant
> T=3D3.155E7; =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 % orbital time of earth
> w=3D sqrt(G*(Me+Ms))/R; =A0 =A0 =A0 =A0% Angular velocity
> de=3D Ms*R/(Ms+Me); =A0 =A0 =A0 =A0 =A0 =A0% Distance between earth and c=
entre of mass
> ds=3D Me*R/(Ms+Me); =A0 =A0 =A0 =A0 =A0 =A0% Distance between sun and cen=
tre of mass
> vv=3D1.018E3; =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0% average velocity of mo=
on
>
> % Propagation of time
> t=3Dlinspace(0,80,100);
>
> % Equation for coordinates of sun and earth
> xs=3Dds*cos(w.*t);
> ys=3Dds*sin(w.*t);
> xe=3D-de*cos(w.*t);
> ye=3D-de*sin(w.*t);
>
> % Plotting of trajectory of sun and earth
>
> plot(xs,ys,'yo');
> hold on;
> plot(xe,ye,'go');
> grid on;
>
> % -----Part II: Simulation of trajectory of moon around earth and sun----=
-
>
> % global x y;
> x(size(t))=3D0;
> y(size(t))=3D0;
> a=3D sqrt(((x - xs).^2) + ((y - ys).^2));
> b=3D sqrt(((x - xe).^2) + ((y - ye).^2));
> q=3D ((G*Ms.*(x-xs))./power(a,3) + (G*Me.*(x-xe))./power(b,3));
> w=3D ((G*Ms.*(y-ys))./power(a,3) + (G*Me.*(y-ye))./power(b,3));

Why are you reusing the vaiable w?

> X=3Ddsolve('D2x+q=3D0','x(0)=3D -(de+r)','Dx(0)=3D0');
> Y=3Ddsolve('D2y+w=3D0','y(0)=3D0','Dy(0)=3Dvv');

>> X

X =3D

-1/2*q*t^2-de-r

>> Y

Y =3D

-1/2*w*t^2+vv*t

dsolve is not used properly. it is treating q and w as
constants.

Need to interface double and sym variables correctly.

> x=3Dsubs(X);

??? Error using =3D=3D> sym/subs
Elements of the substitution cell array must be of the same size.

> y=3Dsubs(Y);
> plot(x,y,'k');
>
> -------------------------------------------------------------------------=
--=AD----------------------
>
> If you can give me some interesting ideas about presenting it better, ple=
ase feel free to add it as well.

Don't use yellow as a plot color.

Hope this helps.

Greg

Instead of avoiding the use of yellow as the color representing the sun, I 
would recommend changing the background color of the entire axes to black 
instead of white.  The yellow sun will appear much brighter against the 
black background than it does against white.   This change will have the 
added benefit that it gives the illusion of trajectories as lines against 
the backdrop of outer space.

To accomplish this effect, please try the following:

1.  After the line

        % Plotting of trajectory of sun and earth

and before the line

        plot(xs,ys,'yo');

please add the following lines of code:

        fig = figure;
        ax = axes;

2.  After the very last call to the 'plot' function, add the following line 
of code:

        set(ax,'color','black');


3.  Change the color of the moon's trajectory from 'black' to something 
lighter, maybe 'cyan':

        plot(x,y,'cyan');


HTH.


Rick



"Greg Heath" <heath@alumni.brown.edu> wrote in message 
news:a98fb7e0-b480-4807-8636-e0ca31a8dd38@s5g2000yqm.googlegroups.com...
> On Dec 28, 12:06 pm, "Nishant Gupta" <im_nisha...@yahoo.co.in> wrote:
>> Hi friends,
>>
>> Following is the code I was writing to simulate the trajectory of sun, 
>> earth and moon...
>> The trajectory of sun and earth works out just fine, but there is some 
>> problem with the moon's trajectory... kindly have a look at the code and 
>> help me out.
>>
>> The code is self-explanatory to some extent, but ask to clear any doubts 
>> you have about my presumptions, etc.
>>
>> Please help me out with this...
>>
>> % ------PartI: The simulation of sun earth moon system-------
>> clear;
>> clc;
>>
>> Me=5.972E24;                 % Mass of earth
>> Ms= 1.988E30;                % Mass of sun
>> R= 1.495E11;                 % Orbital radius of earth
>> r= 3.85E8;                   % Orbital radius of moon
>> G= 6.67E-11;                 % Gravitational Constant
>> T=3.155E7;                   % orbital time of earth
>> w= sqrt(G*(Me+Ms))/R;        % Angular velocity
>> de= Ms*R/(Ms+Me);            % Distance between earth and centre of mass
>> ds= Me*R/(Ms+Me);            % Distance between sun and centre of mass
>> vv=1.018E3;                  % average velocity of moon
>>
>> % Propagation of time
>> t=linspace(0,80,100);
>>
>> % Equation for coordinates of sun and earth
>> xs=ds*cos(w.*t);
>> ys=ds*sin(w.*t);
>> xe=-de*cos(w.*t);
>> ye=-de*sin(w.*t);
>>
>> % Plotting of trajectory of sun and earth
>>
>> plot(xs,ys,'yo');
>> hold on;
>> plot(xe,ye,'go');
>> grid on;
>>
>> % -----Part II: Simulation of trajectory of moon around earth and 
>> sun-----
>>
>> % global x y;
>> x(size(t))=0;
>> y(size(t))=0;
>> a= sqrt(((x - xs).^2) + ((y - ys).^2));
>> b= sqrt(((x - xe).^2) + ((y - ye).^2));
>> q= ((G*Ms.*(x-xs))./power(a,3) + (G*Me.*(x-xe))./power(b,3));
>> w= ((G*Ms.*(y-ys))./power(a,3) + (G*Me.*(y-ye))./power(b,3));
>
> Why are you reusing the vaiable w?
>
>> X=dsolve('D2x+q=0','x(0)= -(de+r)','Dx(0)=0');
>> Y=dsolve('D2y+w=0','y(0)=0','Dy(0)=vv');
>
>>> X
>
> X =
>
> -1/2*q*t^2-de-r
>
>>> Y
>
> Y =
>
> -1/2*w*t^2+vv*t
>
> dsolve is not used properly. it is treating q and w as
> constants.
>
> Need to interface double and sym variables correctly.
>
>> x=subs(X);
>
> ??? Error using ==> sym/subs
> Elements of the substitution cell array must be of the same size.
>
>> y=subs(Y);
>> plot(x,y,'k');
>>
>> ---------------------------------------------------------------------------�----------------------
>>
>> If you can give me some interesting ideas about presenting it better, 
>> please feel free to add it as well.
>
> Don't use yellow as a plot color.
>
> Hope this helps.
>
> Greg
> 

Also, I would change the time vector so that the time increment is exactly 
0.8 seconds (right now it is 0.8081... = 80/99).

Please use either:

    dt = 0.8;
    StopTime = 80;
    t = linspace(0,StopTime-dt,100);

OR:

    N = 100;
    StopTime = 80;
    t = linspace(0,StopTime,N+1);

depending on whether you want exactly 100 points (first case) or to stop the 
simulation at exactly 80 seconds (second case).

HTH.

Rick



"Nishant Gupta" <im_nishantg@yahoo.co.in> wrote in message 
news:ifd5dt$i47$1@fred.mathworks.com...
> Hi friends,
>
> Following is the code I was writing to simulate the trajectory of sun, 
> earth and moon...
> The trajectory of sun and earth works out just fine, but there is some 
> problem with the moon's trajectory... kindly have a look at the code and 
> help me out.
>
> The code is self-explanatory to some extent, but ask to clear any doubts 
> you have about my presumptions, etc.
>
> Please help me out with this...
> % ------PartI: The simulation of sun earth moon system-------
> clear;
> clc;
>
> Me=5.972E24;                 % Mass of earth
> Ms= 1.988E30;                % Mass of sun
> R= 1.495E11;                 % Orbital radius of earth
> r= 3.85E8;                   % Orbital radius of moon
> G= 6.67E-11;                 % Gravitational Constant
> T=3.155E7;                   % orbital time of earth
> w= sqrt(G*(Me+Ms))/R;        % Angular velocity
> de= Ms*R/(Ms+Me);            % Distance between earth and centre of mass
> ds= Me*R/(Ms+Me);            % Distance between sun and centre of mass
> vv=1.018E3;                  % average velocity of moon
>
> % Propagation of time
> t=linspace(0,80,100);
>
> % Equation for coordinates of sun and earth
> xs=ds*cos(w.*t);
> ys=ds*sin(w.*t);
> xe=-de*cos(w.*t);
> ye=-de*sin(w.*t);
>
> % Plotting of trajectory of sun and earth
>
> plot(xs,ys,'yo');
> hold on;
> plot(xe,ye,'go');
> grid on;
>
> % -----Part II: Simulation of trajectory of moon around earth and sun-----
>
> % global x y;
> x(size(t))=0;
> y(size(t))=0;
> a= sqrt(((x - xs).^2) + ((y - ys).^2));
> b= sqrt(((x - xe).^2) + ((y - ye).^2));
> q= ((G*Ms.*(x-xs))./power(a,3) + (G*Me.*(x-xe))./power(b,3));
> w= ((G*Ms.*(y-ys))./power(a,3) + (G*Me.*(y-ye))./power(b,3));
> X=dsolve('D2x+q=0','x(0)= -(de+r)','Dx(0)=0');
> Y=dsolve('D2y+w=0','y(0)=0','Dy(0)=vv');
> x=subs(X);
> y=subs(Y);
> plot(x,y,'k');
>
> -------------------------------------------------------------------------------------------------
>
> If you can give me some interesting ideas about presenting it better, 
> please feel free to add it as well. 

Dear Greg Heath <heath@alumni.brown.edu>,

Thanks a lot for your swift and prompt input. 

I don't see where I reused w as variable twice... I presume vv (double v) for velocity of moon looked like w...

I am still stuck at the problem of using sym/subs formulae and making the matrix dimensions equal.

I presume there are following possible problem in the code:

 1)Because the differential equation of x and y contains both the components-> x and y .... so, one have to try the system of equation thing...

 2) another thing could be that when I preallocate space for x and y (in line 30-33), all the following equations (a,b,q,w) gets equated by 0 and whole system get destroyed (for moon's location), and if I don't preallocate, MATLAB shows error of matrices not matching... one way to overcome it might be to try out the loop, where the value of last x and y can be used in the next...  not sure whether this will work or not...

thanks again. appreciate your help.

Hi Rick Rosson <rrosson@mathworks.com>,

Thanks for such a fast reply.
I was trying the black background with 'whitebg' command, but that wasn't desirable. Your approach in this regard was better.

As for the moon.... I am still losing it to infinity...
Some tips in that regard would be very helpful.

Thank you
Nishant

Hi Greg Heath <heath@alumni.brown.edu>

What exactly am I doing wrong with the dsolve method
or is there a better approach than dsolve method?...

On 28/12/10 3:20 PM, Nishant Gupta wrote:
> Hi Greg Heath <heath@alumni.brown.edu>
>
> What exactly am I doing wrong with the dsolve method
> or is there a better approach than dsolve method?...

Hint:

Another poster wrote

dsolve('Dy = sqrt(4*x + 2*y -1)','x')


To do this in Maple directly, I had to write,

dsolve(Diff(y(x),x) = sqrt(4*x + 2*y(x) - 1))

Note what I had to do to y on the right hand side to prevent it from 
being treated as a constant relative to x.

The function 'dsolve' is used to solve differential equations 
_symbollically_, whereas I think you are actually trying to simulate the 
trajectory _numerically_.  If that is the case, I would recommend that you 
use one of MATLAB's ode solvers instead of the function 'dsolve'.  The best 
place to start is to type

     doc ode45

at the MATLAB command prompt, which will take you to the function reference 
page for the entire suite of ode solvers.  I am not sure which is the best 
solver to use for this particular problem, but it is usually helpful to 
start with 'ode45' and see what happens.

Please give it a try, and if you get stuck, you may want to post a new 
message to the Newsgroup with specific questions relating to the ode 
solvers.

HTH.

Rick




"Nishant Gupta" <im_nishantg@yahoo.co.in> wrote in message 
news:ifd5dt$i47$1@fred.mathworks.com...
> Hi friends,
>
> Following is the code I was writing to simulate the trajectory of sun, 
> earth and moon...
> The trajectory of sun and earth works out just fine, but there is some 
> problem with the moon's trajectory... kindly have a look at the code and 
> help me out.
>
> The code is self-explanatory to some extent, but ask to clear any doubts 
> you have about my presumptions, etc.
>
> Please help me out with this...
> % ------PartI: The simulation of sun earth moon system-------
> clear;
> clc;
>
> Me=5.972E24;                 % Mass of earth
> Ms= 1.988E30;                % Mass of sun
> R= 1.495E11;                 % Orbital radius of earth
> r= 3.85E8;                   % Orbital radius of moon
> G= 6.67E-11;                 % Gravitational Constant
> T=3.155E7;                   % orbital time of earth
> w= sqrt(G*(Me+Ms))/R;        % Angular velocity
> de= Ms*R/(Ms+Me);            % Distance between earth and centre of mass
> ds= Me*R/(Ms+Me);            % Distance between sun and centre of mass
> vv=1.018E3;                  % average velocity of moon
>
> % Propagation of time
> t=linspace(0,80,100);
>
> % Equation for coordinates of sun and earth
> xs=ds*cos(w.*t);
> ys=ds*sin(w.*t);
> xe=-de*cos(w.*t);
> ye=-de*sin(w.*t);
>
> % Plotting of trajectory of sun and earth
>
> plot(xs,ys,'yo');
> hold on;
> plot(xe,ye,'go');
> grid on;
>
> % -----Part II: Simulation of trajectory of moon around earth and sun-----
>
> % global x y;
> x(size(t))=0;
> y(size(t))=0;
> a= sqrt(((x - xs).^2) + ((y - ys).^2));
> b= sqrt(((x - xe).^2) + ((y - ye).^2));
> q= ((G*Ms.*(x-xs))./power(a,3) + (G*Me.*(x-xe))./power(b,3));
> w= ((G*Ms.*(y-ys))./power(a,3) + (G*Me.*(y-ye))./power(b,3));
> X=dsolve('D2x+q=0','x(0)= -(de+r)','Dx(0)=0');
> Y=dsolve('D2y+w=0','y(0)=0','Dy(0)=vv');
> x=subs(X);
> y=subs(Y);
> plot(x,y,'k');
>
> -------------------------------------------------------------------------------------------------
>
> If you can give me some interesting ideas about presenting it better, 
> please feel free to add it as well. 

On Dec 28, 4:04=A0pm, "Nishant Gupta" <im_nisha...@yahoo.co.in> wrote:
> Dear Greg Heath <he...@alumni.brown.edu>,
>
> Thanks a lot for your swift and prompt input.
>
> I don't see where I reused w as variable twice... I presume vv (double v)=
 for velocity of moon looked like w...

xs=3Dds*cos(w.*t);
ys=3Dds*sin(w.*t);
xe=3D-de*cos(w.*t);
ye=3D-de*sin(w.*t);


Hope this helps.

Greg

On Dec 28, 4:20=A0pm, "Nishant Gupta" <im_nisha...@yahoo.co.in> wrote:
> Hi Greg Heath <he...@alumni.brown.edu>
>
> What exactly am I doing wrong with the dsolve method
> or is there a better approach than dsolve method?...

When I simulated the two-body problem,
I just solved Newton's equations using
finite differences.

Greg

On Dec 29, 10:30=A0am, Greg Heath <he...@alumni.brown.edu> wrote:
> On Dec 28, 4:04=A0pm, "Nishant Gupta" <im_nisha...@yahoo.co.in> wrote:
>
> > Dear Greg Heath <he...@alumni.brown.edu>,
>
> > Thanks a lot for your swift and prompt input.
>
> > I don't see where I reused w as variable twice... I presume vv (double =
v) for velocity of moon looked like w...
>
> xs=3Dds*cos(w.*t);
> ys=3Dds*sin(w.*t);
> xe=3D-de*cos(w.*t);
> ye=3D-de*sin(w.*t);

I am not familiar with this characterization.

This assumes that in the center of mass coordinate
system both sun and earth rotate in circles instead
of ellipses. Furthermore, the rotation rates are
equal.

How accurate is that?

Is using ellipses that much more difficult?

Hope this helps.

Greg

"Rick Rosson" <rrosson@mathworks.com> wrote in message <ifdsp3$1d2$1@fred.mathworks.com>...
> 
> The function 'dsolve' is used to solve differential equations 
> _symbollically_, whereas I think you are actually trying to simulate the 
> trajectory _numerically_.  If that is the case, I would recommend that you 
> use one of MATLAB's ode solvers instead of the function 'dsolve'.  The best 
> place to start is to type
> 
>      doc ode45
> 
> at the MATLAB command prompt, which will take you to the function reference 
> page for the entire suite of ode solvers.  I am not sure which is the best 
> solver to use for this particular problem, but it is usually helpful to 
> start with 'ode45' and see what happens.
> 
> Please give it a try, and if you get stuck, you may want to post a new 
> message to the Newsgroup with specific questions relating to the ode 
> solvers.
> 
> HTH.
> 
> Rick
----------------------------------------------------

Hi Rick,

I tried ode45... and followed the following help for converting second order differential equation into first order and then solving it.
http://www.mathworks.com/support/tech-notes/1500/1510.html 

But the complexity of the equation itself is not allowing me to break it into pure form, having separate x(t) and y(t) (see the denominator of q and w, which has component of x and y in there).

Unless I can somehow separate the function x and y, the ode45 method won't give me much of a result...

Nishant

Hii Greg Heath <heath@alumni.brown.edu>,

Yeah , I see the mistake now. Thank you pointing it out. But it didn't still change anything in the trajectory.
I am using circle for simplicity.... if I am able to solve the equations and plot the moon, I might upgrade the trajectory to ellipse...

Thank You
Nishant

On Dec 29, 6:56=A0pm, Greg Heath <he...@alumni.brown.edu> wrote:
> On Dec 29, 10:30=A0am, Greg Heath <he...@alumni.brown.edu> wrote:
>
> > On Dec 28, 4:04=A0pm, "Nishant Gupta" <im_nisha...@yahoo.co.in> wrote:
>
> > > Dear Greg Heath <he...@alumni.brown.edu>,
>
> > > Thanks a lot for your swift and prompt input.
>
> > > I don't see where I reused w as variable twice... I presume vv (doubl=
e v) for velocity of moon looked like w...
>
> > xs=3Dds*cos(w.*t);
> > ys=3Dds*sin(w.*t);
> > xe=3D-de*cos(w.*t);
> > ye=3D-de*sin(w.*t);
>
> I am not familiar with this characterization.

Well, now you are.

> This assumes that in the center of mass coordinate
> system both sun and earth rotate in circles instead
> of ellipses. Furthermore, the rotation rates are
> equal.
>
> How accurate is that?

As accurate as can be for objects that can be
modelled as point masses.

> Is using ellipses that much more difficult?

No need. Circles are correct.

Hope this helps.

Greg

On Dec 30, 6:24=A0am, "Nishant Gupta" <im_nisha...@yahoo.co.in> wrote:
> Hii Greg Heath <he...@alumni.brown.edu>,
>
> Yeah , I see the mistake now. Thank you pointing it out. But it didn't st=
ill change anything in the trajectory.
> I am using circle for simplicity.... if I am able to solve the equations =
and plot the moon, I might upgrade the trajectory to ellipse...
>

No need. In the COM frame the 2-body sun-earth
trajectories are circles.

Sorry for my blunder.

Greg

Greg Heath <heath@alumni.brown.edu> wrote in message <1ad2c8f7-5fb9-4d0f-9f8f-bf60525ea24d@k22g2000yqh.googlegroups.com>...
> On Dec 30, 6:24 am, "Nishant Gupta" <im_nisha...@yahoo.co.in> wrote:
> > Hii Greg Heath <he...@alumni.brown.edu>,
> >
> > Yeah , I see the mistake now. Thank you pointing it out. But it didn't still change anything in the trajectory.
> > I am using circle for simplicity.... if I am able to solve the equations and plot the moon, I might upgrade the trajectory to ellipse...
> >
> 
> No need. In the COM frame the 2-body sun-earth
> trajectories are circles.
> 
> Sorry for my blunder.
> 
> Greg

Neither the original problem not the OP's approach to solving it interests me  much. However you were correct in your first impression. Consider the highly eccentric elliptical orbits of some comets (of negligible mass compared to the Sun); these cannot be represented as circular paths in any frame of reference.

On Dec 30, 10:49=A0am, "Mark Shore" <msh...@magmageosciences.ca> wrote:
> Greg Heath <he...@alumni.brown.edu> wrote in message <1ad2c8f7-5fb9-4d0f-=
9f8f-bf60525ea...@k22g2000yqh.googlegroups.com>...
> > On Dec 30, 6:24=A0am, "Nishant Gupta" <im_nisha...@yahoo.co.in> wrote:
> > > Hii Greg Heath <he...@alumni.brown.edu>,
>
> > > Yeah , I see the mistake now. Thank you pointing it out. But it didn'=
t still change anything in the trajectory.
> > > I am using circle for simplicity.... if I am able to solve the equati=
ons and plot the moon, I might upgrade the trajectory to ellipse...
>
> > No need. In the COM frame the 2-body sun-earth
> > trajectories are circles.
>
> > Sorry for my blunder.
>
> > Greg
>
> Neither the original problem not the OP's approach to solving it interest=
s me =A0much. However you were correct in your first impression. Consider t=
he highly eccentric elliptical orbits of some comets (of negligible mass co=
mpared to the Sun); these cannot be represented as circular paths in any fr=
ame of reference.

That makes sense. Now, I have to back up and find out
how my thought processes conspired to deceive me.

Thanks.

Greg