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Celestial Mechanics formulas? (Read 13833 times)
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Re: Celestial Mechanics formulas?
Reply #15 - 12/01/06 at 17:14:40
 
I had a quick go at this problem and gave up half way as it was clear that it was going to turn into a horrible 6th order polynomial in two variables, as frankuitaalst said. My approach was to first find the minimum distance between an ellipse and a point. Then I was going to assume the point was on the second ellipse, and find the overall minimum. But it was just going to turn out horrible. And that's with the ellipses in cartesian coordinates. If you wanted a closed mathematical formula you'd have to change from celestial coords to cartesian, work out the answer, and then convert back to celestial. More horribleness.
 
I would say your best bet is to write a computer program that selects, say, a million points on the first ellipse and a million points on the second ellipse, and work out an approximate minimum from there.
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Re: Celestial Mechanics formulas?
Reply #16 - 12/01/06 at 18:29:20
 
Ok but now can any one tell me how to write the formula for an 2d ellipse in 3d without it being an ellipsoid using the proper transforms.  Because my brain is going WTF  I keep trying to wrap it around the concept and I'm generally wonderful at various maths, but this one has me scratching my brain via my my sinus cavity.
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frankuitaalst
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Re: Celestial Mechanics formulas?
Reply #17 - 12/01/06 at 23:56:57
 
One way one can proceed is as following :  
consider the first ellips in carthesian :  
x2 + y2/b^2= c^2 with z=0
the other ellipse is somewhat similar , but with terms in x^1 and y^1 , maybe also x*y and a term in z.  
The second ellipse also must obey a second :..x+..y+..z = ct.  
Three equations also .  
From this equations one can derive the distance : r2 = distx^2+disty^2+distz^2.  
One can substitute z as it should remain at first order .  
Then differentiate for x1,x2,y1,y2,z1,z2 to get the minimun .  
... and get a "solid" equation set to solve . There are methods to solve this . But writing down the equations is hard , also converting from celestial coordinates to carthesian .  
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Re: Celestial Mechanics formulas?
Reply #18 - 12/05/06 at 16:49:14
 
since you gave me a place to start with it's turning out to be not quite so bad to derive the formula I need. TYVM for the start on it and giving me the direction to go in...I'll post the Spacial equations for an ellipse once I'm done the derivations.
 
Ok so may be I'm not as close as I though... Sad
 
But I have found the Conversion formulae for cart. to polar (3d).
 
X=P sin ϕ cos è
Y=P sin ϕ sin è
Z=P cos ϕ  
 
 
From your equation  
 
e=(1-b^2)^0.5
Which set a Upper limit of 1 on b for real answers
 
Another Known is, When the equation of an ellipse is in the form Ax2+Bxy+Cy2+Dx+Ey+F=0
 
(1-tan^2(è))/(2 tan^2(è)) = (A-C)/B
 
now as for the rest I'm scratching again.
 
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« Last Edit: 12/05/06 at 20:33:37 by abyssoft »  
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frankuitaalst
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Re: Celestial Mechanics formulas?
Reply #19 - 12/16/06 at 09:41:54
 
I think I found the "formula" to calculate the distance between two ellipses . Quite complicated to solve ...
It comes to the point to solve 6 highly nonlinear equations of the second degree...Numerical solving should be done  
First the equations of both ellipses are writen down in general coordinates ;  
ax^2+by^2+c^2+d*xy+e*yz+f*xz+g*x+h*y+i*z=1 ,  for both equations .  
If 9 points of each ellips are given the a....i can be calculated by solving 9 linear equations ... should be possible .  
Then deriving the distance gives 4 more equations , which are function of x1,x2...z1,z2 . So in total there are 6 equations , and 6 unknowns ...Thus solveble , but the equations themselves are products between each unknown , also difficult to solve , even numerically .  
If one feels interested to get a copy of this equations I can send them ...
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Reply #20 - 02/01/07 at 13:54:21
 
http://orbitsimulator.com/gravity/articles/hillsphere.html
 
Example:  To compute the Earth's Hill sphere with respect to the Sun, enter 1 AU for a, 1 M_Earth for m, and 1 M_Sun for M.  Change the output units to AU.  You will see that Earth's Hill sphere is about 0.001 AU.
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frankuitaalst
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Re: Celestial Mechanics formulas?
Reply #21 - 02/04/07 at 06:00:40
 
Re : elliptical distances -formulae .  
Maybe this site can give a key to obtain some formulae :
http://newton.dm.unipi.it/cgi-bin/neodys/neoibo?objects:2004VD17;statpts;gif
The italians build a database of asteroid data . They calculate the MOID ( Miss distance ) or distances between two elliptical orbits ...
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frankuitaalst
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Re: Celestial Mechanics formulas?
Reply #22 - 02/04/07 at 14:09:32
 
RE : distances ellipes .
Here's a link to an article of Mr. Gronchi
http://copernico.dm.unipi.it/~gronchi/PDF/kep_dist.pdf
I'm afraid it is not easy ... cry
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Re: Celestial Mechanics formulas?
Reply #23 - 02/04/07 at 15:43:58
 
TY VM  Grin
 
These were the resources I was needing
 
Your an  Smiley
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frankuitaalst
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Re: Celestial Mechanics formulas?
Reply #24 - 02/04/07 at 22:40:01
 
Quote from abyssoft on 02/04/07 at 15:43:58:

These were the resources I was needing

Do you really want to code "chinese" ?  Grin
Concerning the angel : this must be Prof of Dr Garchi , he's closer to Rome .. Smiley
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Re: Celestial Mechanics formulas?
Reply #25 - 05/16/07 at 12:42:40
 
recently I found this link :  
http://www.cdeagle.com/html/aa_cm.html - closest approach - may work perhaps with planets too.
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