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Libration on tidelocked planets (Read 8079 times)
EDG
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oh, crumbs!!!

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Re: Libration on tidelocked planets
Reply #15 - 09/16/07 at 17:57:21
 
huh. that's handy. When did amazon start doing that?!
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(formerly known as Mal)
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Tony
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Re: Libration on tidelocked planets
Reply #16 - 09/16/07 at 20:12:50
 
I don't know.  I didn't feel like looking for my copy of SSD, so I just Googled for "synchronous rotation empty focus" and that popped up.  I was surprised too.  Grin
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frankuitaalst
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Re: Libration on tidelocked planets
Reply #17 - 09/19/07 at 13:02:37
 
It seems to be possible to derive a formula , altough approximative to second order ...(M&D pg 44)
Using the formulas of M&D in the pages 38-44 one can derive the "delta" angle if the anomaly (f) or the eccentric anomly (E) is known  
Derivation is shown in annex .  
 
This formula works under the assumption that the planet always is directed to the empty focus , so the angle under which the sun is seen must be "delta" .  
Plugging this values into excel one can hardly distinguish the "delta" from a sine for eccentricities up to 0.25 ( Pluto ) .  
( Help : isn't it possible to reduce the size of the picture here?)  
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frankuitaalst
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Re: Libration on tidelocked planets
Reply #18 - 09/21/07 at 10:04:11
 
Applying the above formulas gives the following plot ...
In abscis the E - value ( eccentric anomaly in radians) , in ordinate the longitude in degrees under which the sun is seen from the eccentric moving tital locked planet .  
The plot was generated for ascending values of eccentricity starting with 0.0 ....to 0.9 with steps of 0.1.  
The last picture shows the plot for e=0.95.
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Longitude_vs_ecc.gif
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Tony
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Re: Libration on tidelocked planets
Reply #19 - 09/21/07 at 10:20:51
 
That's interesting how it deviates from a pure sine wave.
 
Is this mean longitude or true longitude?
 
What does the y-axis represent?  Is that time?  Does this represent 1 complete orbit?
 
0.0 to 0.9 step 0.1 = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9
I can see 10 frames in the animation.  Where does 0.95 come from?
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frankuitaalst
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Re: Libration on tidelocked planets
Reply #20 - 09/21/07 at 10:25:06
 
Quote from Tony on 09/21/07 at 10:20:51:
That's interesting how it deviates from a pure sine wave.
Is this mean longitude or true longitude?
What does the y-axis represent?  Is that time?  Does this represent 1 complete orbit?
0.0 to 0.9 step 0.1 = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9
I can see 10 frames in the animation.  Where does 0.95 come from?

 
The longitude is the angle of view ( see delta) in the formula , or the angle as seen from the planet between the focus and the empty focus .  
This is the y-axis , in degrees .  
I have 10 values :  0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 0.95.  
The x-axis is the eccentric anomaly , somewhat more than one orbit ( one orbit = 6.28 )  
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Re: Libration on tidelocked planets
Reply #21 - 10/10/07 at 14:42:19
 
Quote from Tony on 09/16/07 at 14:13:55:
Quote from frankuitaalst on 09/16/07 at 13:36:17:
You're right !
Is really amazing ! Is this geometrically correct if you draw lines to the focus ?  

I printed the picture and tried that.  It's close but not perfect.  But I manually pieced together the image, so I wouldn't expect it to be perfect.

Quoting myself  Roll Eyes
 
It turns out it is not quite perfect as Grant pointed out in the BAUT thread: http://www.bautforum.com/questions-answers/64727-tidelocked-planets-libration-cr eating-day-night-cycles.html#post1084211 .  I made a new image to verify.  I used Gravity Simulator to take a screen shot every 1/12th of an orbital period for an e=0.8 orbit.  Then I photoshopped in the two-tone balls, rotating them 30 degrees each time, and drawing a line perpendicular to their hemisphere dividing line.  They don't all point to the empty focus, but they're close.
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frankuitaalst
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Re: Libration on tidelocked planets
Reply #22 - 10/10/07 at 14:51:27
 
Yes , the statement  that one side is always directed towards the empty focus is true as a first order approximation ( Murray&Dermot says ).  
 If also the higher order of e ( ie e^2...) are taken into consideration the statement is not true anymore .  
That's a pitty .  Sad
In the other case it would have as a consquence that the position of a planet could be calculated pure from geometrics .  
BTW : the evolution I pictured in the previous post is also of first order ( ie . +/-exact for small e ).  
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