Orbital Resonance of Jupiter's Moons

[Stub Mandrel]

I understand that Io, Europa and Ganymede have a 1:2:4 ratio of orbital speeds.

I thought this ratio was solidly fixed so you can never have all three moon lined up on the same side of Jupiter. But if you take their quoted orbital periods, Europa and Ganymede have slightly longer orbital periods than twice and four-times Io's, enough that the whole pattern should drift out of alignment at least once a year.

I want to make a 'Jovilabe' that actually shows the positions of the moons accurately. What figures should I use?

[Pippy]

Moons of Jupiter orbital parameters ..

https://en.wikipedia.org/wiki/Moons_of_Jupiter

 

[Stub Mandrel]

9 minutes ago, Pippy said:

Moons of Jupiter orbital parameters ..

https://en.wikipedia.org/wiki/Moons_of_Jupiter

 

Yes, but although it shows the 1:2:4 in brackets the more accurate numbers don't add up...

It seems the numbers are relative to the 'perijove' of Io:

https://en.wikipedia.org/wiki/Orbital_resonance#Laplace_resonance

Although I can't say I understand about what that means in practice...

[Pippy]

A picture / video shows a thousand words ..

If you center your focus at the 12 o'clock and 6 o'clock positions, you'll see that the inner 2 moons have a resonance together with the outer 2 moons having a resonance together with the inner and outer moon having a resonance.

 

edit: oops, this doesn't explain the discrepancy you're asking about

 

[Stub Mandrel]

Sorry, I need to be clearer - I can gear he three moons to have 1:2:4 and they will be right BUT I get the impression that this actually revolves around Jupiter in its own 'frame' but I don't know how to work out what that is...

[SilverAstro]

2 hours ago, Stub Mandrel said:

It seems the numbers are relative to the 'perijove' of Io:

https://en.wikipedia.org/wiki/Orbital_resonance#Laplace_resonance

Oh gosh, it has been a long time, I dunno is the simple answer,  but I guess I would start looking into  "Argument of periapsis"  ('perigee' when I was doing orbital mechanics ! ), and " apsidal precession."

Will you need to construct a 'notional' or virtual body to rotate as does the axis of apsides of IO and ref the gears of the 1:2:4 to that.

Just a random thought whilst I watch railway time on Beeb4

[Putaendo Patrick]

Don't know if any of this will help, it's from The Moons of Planet Jupiter by June Garmon.

 

EUROPA

 

Periapsis: 664 862 km

Apoapsis: 676 938 km

Mean orbit radius: 670 900 km

Eccentricity: 0.009

Orbital period: 3.551 181 d

Average orbital speed: 13.740 km/s

Inclination: 0.470° (to Jupiter's equator)

 

Orbit and rotation

 

Europa orbits Jupiter in just over three and a half days, with an orbital radius of about

670,900 km. With an eccentricity of only 0.009, the orbit itself is nearly circular, and the orbital

inclination relative to the Jovian equatorial plane is small, at 0.470°. Like its fellow Galilean

satellites, Europa is tidally locked to Jupiter, with one hemisphere of the satellite constantly

facing the planet. Because of this, there is a sub-Jovian point on Europa's surface, from which

Jupiter would appear to hang directly overhead. Europa's prime meridian is the line intersecting

this point. Research suggests the tidal locking may not be full, as a non-synchronous rotation has

been proposed: Europa spins faster than it orbits, or at least did so in the past. This suggests an

asymmetry in internal mass distribution and that a layer of subsurface liquid separates the icy

crust from the rocky interior.

The slight eccentricity of Europa's orbit, maintained by the gravitational disturbances from the

other Galileans, causes Europa's sub-Jovian point to oscillate about a mean position. As Europa

comes slightly nearer to Jupiter, the planet's gravitational attraction increases, causing the moon

to elongate towards it. As Europa moves slightly away from Jupiter, the planet's gravitational

force decreases, causing the moon to relax back into a more spherical shape. The orbital

eccentricity of Europa is continuously pumped by its mean-motion resonance with Io. Thus, the

tidal flexing kneads Europa's interior and gives the moon a source of heat, possibly allowing its

ocean to stay liquid while driving subsurface geological processes. The ultimate source of this

energy is Jupiter's rotation, which is tapped by Io through the tides it raises on Jupiter and is

transferred to Europa and Ganymede by the orbital resonance.

 

CALLISTO

 

Periapsis: 1 869 000 km

Apoapsis: 1 897 000 km

Semi-major axis: 1 882 700 km

Eccentricity: 0.007 4

Orbital period: 16.689 018 4 d

Average orbital speed: 8.204 km/s

Inclination 0.192° (to local Laplace planes)

 

Orbit and rotation

 

Callisto is the outermost of the four Galilean moons of Jupiter. It orbits at a distance of

approximately 1 880 000 km (26.3 times the 71 492 km radius of Jupiter itself). This is

significantly larger than the orbital radius—1 070 000 km—of the next-closest Galilean satellite,

Ganymede. As a result of this relatively distant orbit, Callisto does not participate in the mean-

motion resonance—in which the three inner Galilean satellites are locked—and probably never

has.

Like most other regular planetary moons, Callisto's rotation is locked to be synchronous with its

orbit. The length of the Callistoan day, simultaneously its orbital period, is about 16.7 Earth

days. Its orbit is very slightly eccentric and inclined to the Jovian equator, with the eccentricity

and inclination changing quasi-periodically due to solar and planetary gravitational perturbations

on a timescale of centuries. The ranges of change are 0.0072–0.0076 and 0.20–0.60°,

respectively. These orbital variations cause the axial tilt (the angle between rotational and orbital

axes) to vary between 0.4 and 1.6°.

The dynamical isolation of Callisto means that it has never been appreciably tidally heated,

which has had important consequences for its internal structure and evolution. Its distance from

Jupiter also means that the charged-particle flux from the planet's magnetosphere at its surface is

relatively low—about 300 times lower than, for example, that at Europa. Hence, unlike the other

Galilean moons, charged-particle irradiation has had a relatively minor effect on the Callistoan

surface. The radiation level at the surface of Callisto is equivalent to a dose of about 0.01 rem

(0.1 mSv) per day.

 

GANYMEDE

 

Periapsis: 1,069,200 km

Apoapsis: 1,071,600 km

Semi-major axis: 1,070,400 km

Eccentricity: 0.0013

Orbital period: 7.15455296 d

Average orbital speed: 10.880 km/s

Inclination: 0.20° (to Jupiter's equator)

 

Ganymede orbits Jupiter at a distance of 1,070,400 km, third among the Galilean satellites, and

completes a revolution every seven days and three hours. Like most known moons, Ganymede is

tidally locked, with one side of the moon always facing toward the planet. Its orbit is very

slightly eccentric and inclined to the Jovian equator, with the eccentricity and inclination

changing quasi-periodically due to solar and planetary gravitational perturbations on a timescale

of centuries. The ranges of change are 0.0009–0.0022 and 0.05–0.32°, respectively. These orbital

variations cause the axial tilt (the angle between rotational and orbital axes) to vary between 0

and 0.33°.

The Laplace resonances of Ganymede, Europa, and Io

Ganymede participates in orbital resonances with Europa and Io: for every orbit of Ganymede,

Europa orbits twice and Io orbits four times. The superior conjunction between Io and Europa

always occurs when Io is at periapsis and Europa at apoapsis. The superior conjunction between

Europa and Ganymede occurs when Europa is at periapsis. The longitudes of the Io–Europa and

Europa–Ganymede conjunctions change with the same rate, making the triple conjunctions

impossible. Such a complicated resonance is called the Laplace resonance.

The current Laplace resonance is unable to pump the orbital eccentricity of Ganymede to a

higher value. The value of about 0.0013 is probably a remnant from a previous epoch, when such

pumping was possible. The ganymedian orbital eccentricity is somewhat puzzling; if it is not

pumped now it should have decayed long ago due to the tidal dissipation in the interior of

Ganymede. This means that the last episode of the eccentricity excitation happened only several

hundred million years ago. Because the orbital eccentricity of Ganymede is relatively low—

0.0015 on average—the tidal heating of this moon is negligible now. However, in the past

Ganymede may have passed through one or more Laplace-like resonances that were able to

pump the orbital eccentricity to a value as high as 0.01–0.02. This probably caused a significant

tidal heating of the interior of Ganymede; the formation of the grooved terrain may be a result of

one or more heating episodes.

There are two hypotheses for the origin of the Laplace resonance among Io, Europa, and

Ganymede: that it is primordial and has existed from the beginning of the Solar System; or that it

developed after the formation of the Solar System. A possible sequence of events for the latter

scenario is as follows: Io raised tides on Jupiter, causing its orbit to expand until it encountered

the 2:1 resonance with Europa; after that the expansion continued, but some of the angular

moment was transferred to Europa as the resonance caused its orbit to expand as well; the

process continued until Europa encountered the 2:1 resonance with Ganymede. Eventually the

drift rates of conjunctions between all three moons were synchronized and locked in the Laplace

resonance.

 

IO

 

Periapsis: 420,000 km (0.002 807 AU)

Apoapsis: 423,400 km (0.002 830 AU)

Mean orbit radius: 421,700 km (0.002 819 AU)

Eccentricity: 0.0041

Orbital period: 1.769 137 786 d (152 853.504 7 s, 42 h)

Average orbital speed: 17.334 km/s

Inclination: 2.21° (to the ecliptic) 0.05° (to Jupiter's

equator)

 

Orbit and rotation

 

Io orbits Jupiter at a distance of 421,700 km (262,000 mi) from the planet's center and 350,000

km (217,000 mi) from its cloudtops. It is the innermost of the Galilean satellites of Jupiter, its

orbit lying between those of Thebe and Europa. Including Jupiter's inner satellites, Io is the fifth

moon out from Jupiter. It takes 42.5 hours to complete one orbit (fast enough for its motion to be

observed over a single night of observation). Io is in a 2:1 mean-motion orbital resonance with

Europa and a 4:1 mean-motion orbital resonance with Ganymede, completing two orbits of

Jupiter for every one orbit completed by Europa, and four orbits for every one completed by

Ganymede. This resonance helps maintain Io's orbital eccentricity (0.0041), which in turn

provides the primary heating source for its geologic activity. Without this forced eccentricity,

Io's orbit would circularize through tidal dissipation, leading to a geologically less active world.

Like the other Galilean satellites of Jupiter and the Earth's Moon, Io rotates synchronously with

its orbital period, keeping one face nearly pointed toward Jupiter. This synchronicity provides

the definition for Io's longitude system. Io's prime meridian intersects the north and south poles,

and the equator at the sub-Jovian point. The side of Io that always faces Jupiter is known as the

subjovian hemisphere, while the side that always faces away is known as the antijovian

hemisphere. The side of Io that always faces in the direction that the moon travels in its orbit is

known as the leading hemisphere, while the side that always faces in the opposite direction is

known as the trailing hemisphere.

[Stub Mandrel]

My brain now hurts. I think I will keep[ it simple and accept that it may need resetting once a year!

A plan, aimed at keeping errors <<1% is (target figures in brackets):

Rotate Jupiter once every 9.925 hours (9.925).

Jupiter:Io 77:18 = 42.457 hours (42.459 negligible error)

Io: Europa 1:2 = 84.914 hours (85.228 0.4% error)

Europa:Ganymede 1:2 = 169.838 hours (171.109 0.7% error)

Io:Callisto 1849:196 = 400.525 hours (400.536 negligible error).

The ratios may seem weird but they actually translate to two gear pairs per ratio close enough to all run on the same pair of pivots except for one pair which will need to be slightly coarser.

Having thought about it as long as the ratio between Io:Europa equals the ratio Europa: Ganymede then they will keep eh orbital resonance, but as it drifts away from 1:2 the whole thing will rotate ever faster about Jupiter.

Using a ratio of 265:1302 (1:2.007) instead of 1:2 gives almost perfect orbital times for Europa and Ganymede.

I have it all worked out on paper, and it should be perfectly buildable. I just need to invest in some brass, but I could make a plastic prototype!

[Pippy]

Don't forget that resonance itself in effect tunes / cancels out any slow drift away from phase lock that may occur, which basically means any small tendancy for one of moons to drift out of position (out of resonance) will be cancelled out / compensated for by the other moons having a 'tugging' effect on said moon as they pass by one another.

Nothings perfect, the orbits of all the bodies in the system will be ever changing in some small way due to just about anything in the vacinity and beyond (solar wind, other bodies further away etc etc etc). That's why we can't really predict the exact position of any body in the solar system past a number of years into the future, their are countless parameters (how ever small they may seem) that would need to be taken into account, it's very much like trying to predict the weather, to get a far future prediction correct would really require knowing the state of every atom, every ray of sun light etc etc - an impossible task.

[Putaendo Patrick]

2 hours ago, Stub Mandrel said:

My brain now hurts. I think I will keep[ it simple and accept that it may need resetting once a year!

Think I probably need to reset my brain now

[Stub Mandrel]

I realise nothing is simple, but  to sum up:

Checking the numbers the commonly presented animation should be slowly rotating so that after roughly 1843 orbits of Io they get back (approximately) to its starting position.

This can be simulated almost exactly by making Europa orbit 2.07 times as fast as Io and Ganymede orbit 2.007 times as fast as Europa. This maintains the relationship between the orbits but the two outer moons catch up slightly early on each orbit.

Adding this should mean taht my Jovilabe keep good time for several years between tweaks.

[Stub Mandrel]

I have corroboration!

The orbital resonance article on Wikipedia says the drift is -0.7395 degrees/day.

So one full lap takes 486.815 days or 11,683.6 hours

That's 275.173 Io orbits. or an 0.00363 laps per io orbit.

As Europa takes two Io orbits to make one orbit, then the lag per Europa orbit is 0.0072, total 2.0072.

I think using 2.0076 will be ample accuracy for my needs as the drift will only be out by 2% after about five years.

 

[Stub Mandrel]

For the brave, these are my gear calculations:

And proof of concept in Turbocad: