Moving longitude of conjunction of Jupiter's moons

[Spanier]

Since the longitude, where the conjunctions between Jupiter's inner 3 moons occure propagates in a retrograde orbit, the relation between the orbital period of Europa T₂ and that of Io T₁ is slightly greater than 2. But the relation between the orbital period of Ganymede T₃ and that of Europa T₂ is even bigger.

After a bunch of random assumptions and calculations, I discovered, that the fractions T₂/T₁=A and T₃/T₂=B are in the following relation:

2/(3-A)=B

Is this a happy coincidence or a condition for the resonance?

[Sillychris]

No coincidence, it turns out that due to the close proximity of the orbits there is a resonant system going on. Io does exactly 4 orbits for exactly 2 orbits of Europa for exactly 1 orbit of Ganymede.

[Spanier]

That wasn't my question, I knew that. I'm asking for the B=2/(3-A) connection.

[Sillychris]

Well yeah, but that's where the answer to your question lays:

The orbital resonance of these guys gives a T2/T1=A=2 (Europa's period is twice Io's) and a T3/T2=B=2 (Ganymede's period is twice Europa's) so if you substitute the numbers into your relation B=2/(3-A) you get 2=2/(3-2) ie 2=2

[Spanier]

Well yeah, but that's where the answer to your question lays:

The orbital resonance of these guys gives a T2/T1=A=2 (Europa's period is twice Io's) and a T3/T2=B=2 (Ganymede's period is twice Europa's) so if you substitute the numbers into your relation B=2/(3-A) you get 2=2/(3-2) ie 2=2

No, if you look to Wikipedia, the value isn't excactly 2 but 2.00069... for Io and Europa and 2.0014... for Europa and Ganymede.

[Sillychris]

Interesting, I always thought those numbers were exact. Probably a tidal effect similar to how the moon is gradually escaping Earth.

Anyway, with the new more precise values of A & B your equation doesn't hold.

[Spanier]

Interesting, I always thought those numbers were exact. Probably a tidal effect similar to how the moon is gradually escaping Earth.

Anyway, with the new more precise values of A & B your equation doesn't hold.

The equation does hold, and that is what struggles me. The error is about 10^-7.

[Sillychris]

2/(3-2.00069)=2.001381 does not precisely equal 2.0014, but does approximately. What your relationship is saying is two divided by almost 1 almost equals 2.

The error is 6th order, not 7th and your values are reported to 6th order. I'd say to the accuracy of the reported values, your equation does not hold.

[Spanier]

2/(3-2.00069)=2.001381 does not precisely equal 2.0014, but does approximately. What your relationship is saying is two divided by almost 1 almost equals 2.

The error is 6th order, not 7th and your values are reported to 6th order. I'd say to the accuracy of the reported values, your equation does not hold.

The valueds I wrote here aren't the exact ones, I just wrote them from my memory, and I can't look them up right now. Just check them yourself on Wikipedia (A=~2.007)

[Sillychris]

Equation still doesn't hold.

[Spanier]

Since you are obvisouly not capable of looking up the correct values yourself, I do it for you know.

T₁ = 1.7691378 d

T₂ = 3.551181 d

T₃ = 7.154553 d

A = T₂/T₁ = 2.00729474

B = T₃/T₂ = 2.014696801

2/(3-A) = 2.014696688

So is it coincidence or not. Maybe someone more qualified then SILLYchris can also answer?

Edited April 29, 2014 by Spanier

[Z-Man]

Gee, attitude.

Assuming perfectly circular orbits (*), it's the equation that holds iff every time the outer and middle moon are in conjunction, the inner moon is in the same relative position to them. And overtakes the outer moon three times in between two such events. So yes, it is the equation that determines resonance of that type. More generally, (AB-1)/(B-1) needs to be a rational number, in this case 3. Massage it a bit and you get your form.

*: They aren't, of course, which I guess is why A and B are close to 2.

[Spanier]

The point, where the moons conjunct rotates in space. That does not influence the shape of the orbits in any way.

[Winter Man]

The orbits of the other moons in the system could quite easily account for that discrepancy.

[Spanier]

The orbits of the other moons in the system could quite easily account for that discrepancy.

You mean Callisto? Not excactly. The Laplace resonance is nearly perfect, the libration around 3 of the 4 resonance angles are so narrow, you can say, that there NEVER is a triple conjunction. But the forth is rotating through all angles, which means, conjunction can occure anywhere on Ganymedes orbit.

From Wikipedia:

The longitudes of the Io–Europa and Europa–Ganymede conjunctions change with the same rate, making the triple conjunctions impossible.

Is this the reason for my equation to work?

[Z-Man]

Is this the reason for my equation to work?

That is correct.

[Spanier]

So everytime, 3 bodies are in a 1:2:4 resonance with a non-fixed longitude of conjunction, the orbital period of the third body T₃ = 2T₂/(3-T₂/T₁)

Edited April 29, 2014 by Spanier

[Z-Man]

Well, with a non-fixed conjunction longitude, they are not in a 1:2:4 resonance. They only would look like that in a frame co-rotating with the conjunction. But apart from that nitpick, yes.

[Spanier]

Well, with a non-fixed conjunction longitude, they are not in a 1:2:4 resonance. They only would look like that in a frame co-rotating with the conjunction. But apart from that nitpick, yes.

Well, the Galilean Moons are also said to be in an 1:2:4 resonance, despite their longitude of conjunctions rotates with ~3°/d.

[Z-Man]

Yep, "said to be". I guess they come close enough.

[Yasmy]

I find you relation to be highly unlikely. A first order effect like this shouldn't involve this kind of product: B * (3-A) = 2.

Zeroth order approximations gives perfect keplerian orbits. Should be independent of A or B.

First order effects gives resonances and deviations such as precession of apses. Should depend linearly on A or B.

Second order effects would give wobbles and non-elliptical deviations. Should depend quadratically on A and B, such as the product A*B.

But let's look at what your relation says. First, define the deviations from perfect resonance:

Let A = T2/T1 = 2 + a

Let B = T3/T2 = 2 + b

Now plug these into your relation:

B = 2 / (3 - A)

2 = B * (3 - A) = (2+ * (3-2-a) = 2 - 2a + b - a*b

0 = a*b + 2a - b

This is a very strange relation between a and b. But note that a*b is tiny. Much smaller than a or b. Let's drop it.

2a = b

Now this is much nicer. It still fits your observation, and the relation is much simpler.

Of course this says nothing of causation...

I wouldn't be surprised to find that the system is evolving to enforce that relation, but for now it's idle speculation.

Edited April 29, 2014 by Yasmy

[Spanier]

Ok, after nearly one year, I looked into my old posts, and found this topic still not entirely resolved, so I gave my old calculations some new love.

First of all, I define some variables:

Ä = Time between alignments

α = Angle, by which the alignment moves after Ä

β = 2À+α = 360°+α

η = β/2À = β/360°

Éₓ = 2À/Tₓ = Angular speed of moon x

Tâ‚“ = Orbital period of moon x

We can easily see, that for the moons to align, following equation must be true:

(8À+α)/É₠= (4À+α)/É₂ = (2À+α)/É₃ = Ä

The numbers assigned to the moons is in ascending order in relation to their distance to their parent planet.

We can write this equation also in these forms:

(6À+β)/É₠= (2À+β)/É₂ = β/É₃ = Ä

Tâ‚Â(3+η) = Tâ‚‚(1+η) = T₃η = Ä

From the latter one we find the following relations:

T₂/T₠= (3+η)/(1+η) = A

T₃/T₂ = (1+η)/η = B

Through transforming we now get the following relation:

2/(3-A) = B

or

2Tâ‚ÂTâ‚‚/(3Tâ‚Â-Tâ‚‚) = T₃ â– 

Edited May 10, 2015 by Spanier

[Bill Phil]

The universe isn't static. They're speeding up, and thusly gaining altitude. Due to their different distances the Galilean moons have small differences in their relative orbital ratios.

We actually don't know the exact ratios. They're about 1:2:4, but not exactly.

Interesting fact: Eventually the moons will catch up with Callisto, and it will enter the resonance too.