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Message started by Brent on 03/22/05 at 23:27:14

Title: Celestial Mechanics formulas?
Post by Brent on 03/22/05 at 23:27:14

There are some formulas that would be useful for using Gravity Simulator.  Do you know the formulas for velocity for circluar orbit, and for escape velocity?  I'd especially like to know escape since the program seems to compute circular for you.

Title: Re: Celestial Mechanics formulas?
Post by Tony on 03/23/05 at 10:55:36

For circular and escape:
G=6.673*10-11
M = Mass of the object being orbited
d=distance from the center of mass M

http://orbitsimulator.com/gsimyabb/formula001.GIF
http://orbitsimulator.com/gsimyabb/formula002.GIF

Title: Re: Celestial Mechanics formulas?
Post by Tony on 04/12/06 at 00:40:39

Here's a calculator for computing orbital period:
http://orbitsimulator.com/gravity/articles/PeriodCalculator.html

And the same formula re-written to solve for semi-major axis:
http://orbitsimulator.com/gravity/articles/smaCalculator.html

Use this calculator to solve for orbital altitude:
http://orbitsimulator.com/gravity/articles/AltitudeCalculator.html

Title: Re: Celestial Mechanics formulas?
Post by Tony on 11/13/06 at 22:28:47

A little off topic since it's not much use for Gravity Simulator, but here's a formula/calculator for computing the maximum velocity you can achieve from a solar sail.

http://orbitsimulator.com/astrobiology/solarpressure2.html

Title: Re: Celestial Mechanics formulas?
Post by Tony on 11/22/06 at 00:13:01

Compute a star's visual magnitude from a its absolute magnitude and distance:

http://www.orbitsimulator.com/gravity/articles/vmag.html

Try entering 4.8 for the Sun's absolute magnitude, then switch the distance to different planetary distances (use AU) to see how the Sun's compares in brightness between planets.

Title: Re: Celestial Mechanics formulas?
Post by Mal on 11/22/06 at 08:11:51


Tony wrote:
Compute a star's visual magnitude from a its absolute magnitude and distance:

http://www.orbitsimulator.com/gravity/articles/vmag.html

Try entering 4.8 for the Sun's absolute magnitude, then switch the distance to different planetary distances (use AU) to see how the Sun's compares in brightness between planets.


Erm, I'm not seeing a "Calculate" button, I can enter the AbsMag and the distance, but no number appears on the other side of the equals sign. I'm using Firefox 1.5 if that makes a difference. (I tried entering 1 PC and AbsMag 4.82)

Title: Re: Celestial Mechanics formulas?
Post by Tony on 11/22/06 at 10:43:57


Mal wrote:
Erm, I'm not seeing a "Calculate" button, I can enter the AbsMag and the distance, but no number appears on the other side of the equals sign. I'm using Firefox 1.5 if that makes a difference. (I tried entering 1 PC and AbsMag 4.82)

I just tried it in Firefox and you're right.  It works fine in IE.  Firefox also gets the colors wrong.  The answer box should be dark red, and the 2 input boxes should be black.  I'll look into it.  I suspect that IE is more forgiving in javascript syntax errors.

Title: Re: Celestial Mechanics formulas?
Post by Tony on 11/22/06 at 13:01:05

Try it now.  Appearently, IE is not that picky about bad syntax, and Firefox is.  I've cleaned it up so it works in both browsers now.  I'll have to do the same for all the other calculators in this post.

Thanks for pointing this out!

Can anyone tell me if this works in Opera or Netscape, or IE 7?

Title: Re: Celestial Mechanics formulas?
Post by abyssoft on 11/27/06 at 14:48:17

Does any one know the formula for determining the minimum distance of two ellipses in 3d space?

That would help me with the graphs to help show some more info.

Title: Re: Celestial Mechanics formulas?
Post by frankuitaalst on 11/29/06 at 12:52:24

hallo ,
can you explain your question some more ?
what coordinates are you using ?

x/a^2+y/b^2+z/c^2 = 1 ore something else ?

I suppose you have two ellipses of this form and want the determine the distance ?


Title: Re: Celestial Mechanics formulas?
Post by abyssoft on 11/29/06 at 18:25:29

I'm actully not sure..
I am defining the ellipse in astrometric terms
a=semimajor Axis
e=eccentricity
i=inclination angle
O=Longitude of the ascending node

The formula for this in standard algebraic notation is also baffleing me any help would be most appreciated.


Title: Re: Celestial Mechanics formulas?
Post by frankuitaalst on 11/29/06 at 22:46:02

hallo ,
i suppose that the two ellipses share 1 commom focus point .
Is this correct ? This might simplify things

Title: Re: Celestial Mechanics formulas?
Post by abyssoft on 11/30/06 at 05:17:51

Yes both share a common Focus point C which is by astrometric definition located at (0,0,0) in a cartesian system and (0, Sigma=0 deg, Theta=0 deg) in a 3d polar system.

Title: Re: Celestial Mechanics formulas?
Post by frankuitaalst on 12/01/06 at 14:05:38

I'm afraid there is no straightforward formula for this . Ellipses are very hard to deal with . At the first glance it seems one needs to solve a sixth order non linear equation . ...

Title: Re: Celestial Mechanics formulas?
Post by abyssoft on 12/01/06 at 16:42:51

Ty Vm for trying I was going to use the information to track potential approaches of Uranus, Neptune, and Pluto

Title: Re: Celestial Mechanics formulas?
Post by Nexus on 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.

Title: Re: Celestial Mechanics formulas?
Post by abyssoft on 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.

Title: Re: Celestial Mechanics formulas?
Post by frankuitaalst on 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 .

Title: Re: Celestial Mechanics formulas?
Post by abyssoft on 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... :(

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.


Title: Re: Celestial Mechanics formulas?
Post by frankuitaalst on 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 ...

Title: Hill Sphere Calculator
Post by Tony on 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.

Title: Re: Celestial Mechanics formulas?
Post by frankuitaalst on 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 ...

Title: Re: Celestial Mechanics formulas?
Post by frankuitaalst on 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 ... :'(

Title: Re: Celestial Mechanics formulas?
Post by abyssoft on 02/04/07 at 15:43:58

TY VM  ;D

These were the resources I was needing

Your an  [smiley=engel017.gif]

Title: Re: Celestial Mechanics formulas?
Post by frankuitaalst on 02/04/07 at 22:40:01


abyssoft wrote:
These were the resources I was needing

Do you really want to code "chinese" ?  ;D
Concerning the angel : this must be Prof of Dr Garchi , he's closer to Rome .. :)

Title: Re: Celestial Mechanics formulas?
Post by frankuitaalst on 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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