Terra's sorta Second moon (Cruithne)

[NePhilim]

I am looking for some info on moving "Cruithne" from the NEA.ssc file to the solarsys.ssc file under Sol/Terra on mine yours is prolly Sol/Earth. but i need to update all that orbit stuff but i don't know how or can't be bothered to learn. Either way is there a util that can calculate this 4 me also i want to list J002E3 this way.

[selden]

http://www.lepp.cornell.edu/~seb/celest ... meris.html

might help.

[t00fri]

...yours is prolly Sol/Earth. but i need to update all that orbit stuff but i don't know how or can't be bothered to learn. Either way is there a util that can calculate this 4 me...

http://www.lepp.cornell.edu/~seb/celestia/transforming_ephemeris.html

might help.

I am afraid this is prolly also too much work 4 him

Fridger

PS:
Life is getting increasingly complicated 4 us non-Americans.
I wonder whether I should start communicating with our friend in my native German dialect 4 a change? As a theoretically minded person, I immediately ask myself what is 4 + 4?

[selden]

I finally remembered what Cruithne is, and that it has a particularly difficult orbit. My page which describes how to translate between the different standard Keplerian orbital specifications won't work for it.

Unfortunately, Celestia doesn't have a native representation for horseshoe orbits, so only an xyz trajectory can be used in Celestia v1.5.1. (An xyzv trajectory could be used in v1.6.) While xyz trajectories can be obtained from Horizons, learning to translate that format into what Celestia needs will require some study by the novice.

[t00fri]

I finally remembered what Cruithne is, and that it has a particularly difficult orbit. My page which describes how to translate between the different standard Keplerian orbital specifications won't work for it.

Selden,

why do you call Cruithne's orbit difficult?? It seems to me that in the rest system of Sol we are dealing with a canonical Keplerian orbit around Sol, just like (approximately) for Earth. Only after a change of restframe: Sol at rest -> Earth at rest, Cruithne's orbit appears complicated (bean shaped). But this is quite normal. Jupiter's orbit also looks "crazy" in the Earth@rest frame... Of course there is the gravitational disturbance
of Cruithne's orbit by Earth's field, which will make Cruithne's Keplerian parameters "effective", i.e. time dependent...

We are all hoping that after version 1.6.0, Celestia will at last be able to display orbits correctly in various reference frames...

Fridger

PS:

could it be that the anonymous author of the Wiki article
http://en.wikipedia.org/wiki/3753_Cruithne
used Celestia for displaying the orbits, without crediting Celestia?

[Cham]

This animation is interesting :
http://upload.wikimedia.org/wikipedia/c ... _Earth.gif

I'm wondering if I could build a CMOD file to add the yellow orbit in Celestia, using Mathematica.

[t00fri]

This animation is interesting :
http://upload.wikimedia.org/wikipedia/c ... _Earth.gif

I'm wondering if I could build a CMOD file to add the yellow orbit in Celestia, using Mathematica.

Martin,

this beige, bean shaped trajectory in the animation is just the proper Cruithne orbit in the rest frame of Earth! Ptolemaeus would have looked at Cruithne's orbit this way

I thought that meanwhile I have convinced everybody that unlike present Celestia rendering, orbits are frame dependent. Therefore, I assumed that it is definitely on the todo list for after version 1.6 that we will at last implement frame dependent orbit rendering. This should make your proposed add-on superfluous and address the problem generally as it should be done...

Fridger

[selden]

Fridger,

The orbit is not Keplerian.

Whenever the asteroid is near the Earth, it moves either to an interior orbit (slightly smaller than the Earth's) when the Earth approaches from behind the asteroid and slows it, or the asteroid moves to an exterior orbit when it approaches the Earth from behind and is accelerated.

Several moons of Saturn have similar resonances.

[t00fri]

Fridger,

The orbit is not Keplerian.

Whenever the asteroid is near the Earth, it moves either to an interior orbit (slightly smaller than the Earth's) when the Earth approaches from behind the asteroid and slows it, or the asteroid moves to an exterior orbit when it approaches the Earth from behind and is accelerated.

Several moons of Saturn have similar resonances.

Sure, that is what I meant when referring to the influence of Earth's field on Cruithne's orbit, if Earth happens to be near. Yet it seems sensible to me to try and understand the orbit by means of sequential approximations as we theorists do it all the time

After all, the standard parameters for a normal elliptical orbit around the sun are given on this Wiki page. And there it is also said that

Cruithne is in a normal elliptic orbit around the Sun.

So what's the problem apart from some isolated disturbances happening if Earth is close?? This sort of gravitational disturbance happens at SOME level for ALL solar system bodies. That's why eventually we consider VSOP87 etc. But we know that normal elliptical orbits around the Sun are certainly good enough to understand semi-quantitatively what is going on...


Fridger

[selden]

Cruithne is in a normal elliptic orbit around the Sun.

That statement is, of course, an over simplification. I would not call "normal" an orbit which changes its semi-major axis periodically.

[t00fri]

Cruithne is in a normal elliptic orbit around the Sun.

That statement is, of course, an over simplification. I would not call "normal" an orbit which changes its semi-major axis periodically.

Selden,

of course, but physics consists mostly of "oversimplifications" . The art is usually to do these oversimplifications in the right way.

I suspect that the given parameters for the elliptical orbit refer to an average orbit that is not so bad after all. What criterion of precision are you applying? You wrote that some Saturnian moons show similar resonances. What kind of orbits does Celestia use for them?

Fridger

[selden]

Unfortunately, some simplifications eliminate the interesting cases. For example, Celestia uses simple Keplerian ellipses for the moons Janus and Epimetheus, which can't show them exchanging orbital elements.

[t00fri]

Unfortunately, some simplifications eliminate the interesting cases. For example, Celestia uses simple Keplerian ellipses for the moons Janus and Epimetheus, which can't show them exchanging orbital elements.

Selden,

yes I agree, there are some very interesting common phenomena that seem to be quite analogous physicswise, in terms of the circular restricted three-body problem.
On the one hand among Saturn and Janus <->Epimetheus and on the other hand the orbital dynamics of Cruithne as seen from Earth.

Still it would also be worthwhile to elaborate a bit at which scale relative to a normal elliptical orbit description for the bodies involved, these interesting phenomena are taking place.

For example, what would be the angular size of these orbital exchange effects as seen from Earth? Or in other words, how much different in angle would a simple elliptical orbit description of Cruithne look from Earth. From the nice animation on the respective Wiki page, it is clear that a bean-shaped orbit of Cruithne in the Earth rest frame is bound to result also in an elliptical orbit description.

Fridger

[Enio]

And if the asteroid had the same mass of our Moon? Would it still have the same strange orbit or would be attracted to Earth, becoming a true second moon?

[Cham]

I thought that meanwhile I have convinced everybody that unlike present Celestia rendering, orbits are frame dependent. Therefore, I assumed that it is definitely on the todo list for after version 1.6 that we will at last implement frame dependent orbit rendering. This should make your proposed add-on superfluous and address the problem generally as it should be done...

Unfortunately, this desirable feature will not be implemented until a VERY long time. I don't expect this to come in Celestia before 2010 !

I was able to setup some Mathematica code to build a CMOD representation of the orbits, properly sized and oriented in Celestia. Here's the case of Earth and Cruithne (the yellow curves are actually a CMOD model superposed on Celestia's orbits) :

ok.jpg

I now have to defined the proper relation in the object's position on its orbit, and substract both trajectories to obtain that famous yellow cuve shown on the wiki animation.

[t00fri]

Very well, Martin. How about transforming Earth to rest, such as to recover the "bean shaped" Cruithne orbit!?

Good luck,
Fridger

[Cham]

Very well, Martin. How about transforming Earth to rest, such as to recover the "bean shaped" Cruithne orbit!?

Good luck,
Fridger

This is exactly what I'm trying to do. I now need to define the right position of each body on its orbit. This is where the "trouble" is.

[t00fri]

Very well, Martin. How about transforming Earth to rest, such as to recover the "bean shaped" Cruithne orbit!?

Good luck,
Fridger

This is exactly what I'm trying to do. I now need to define the right position of each body on its orbit. This is where the "trouble" is.

How about determining the Galilei transformation matrix [tex]\Lambda(t)[/tex] by the requirement
[tex]\vec{x}(t)^T_{Earth} \,\Lambda(t) =\{0,0,0\}[/tex]

and then transforming the positional vector of Cruithne as follows:
[tex]\vec{x}(t)_{bean\ shape} = \vec{x}(t)^T_{Cruithne} \, \Lambda(t)[/tex]

Plot it! That's all...

Fridger

[Cham]

My idea was simply to substract both trajectories, as this :

[tex]{\bf r}_{Cruithne relative to Earth} = {\bf r}_{Cruithne relative to Sol} - {\bf r}_{Earth relative to Sol}[/tex]

However, to plot the trajectory, I need to define the time variable.

[Cham]

Here's what I got. This shape isn't right. I made a crude calculation just as a test, but there are some parameters that I don't understand (EarthMeanLongitude and CruithneMeanAnomaly at CruithneEpoch) :

peanut.jpg

The relative trajectory is defined as

[tex]{\bf r}(t)_{Earth-Cruithne} = {\bf r}(t)_{Sol-Cruithne} - {\bf r}(t)_{Sol-Earth} + {\bf r}(0)_{Sol-Earth}[/tex]