?Source of Io's tidal heating?

What is the primary cause of tidal heating of Jupiter's innermost moon
Io? The moon itself always has the same face toward Jupiter (tidally
locked). That seems to leave only the slight eccentricity (0.0041) of its
orbit around Jupiter as the source...where the maximum tidal stretching
at perigee is alternated with somewhat less stretching at apogee (?).
Is that enough to generate most of Io's internal heat? Or is something
else going on?

Gene

(Gene Partlow) writes:

What is the primary cause of tidal heating of Jupiter's innermost moon
Io? The moon itself always has the same face toward Jupiter (tidally
locked). That seems to leave only the slight eccentricity (0.0041) of its
orbit around Jupiter as the source...where the maximum tidal stretching
at perigee is alternated with somewhat less stretching at apogee (?).
Is that enough to generate most of Io's internal heat?

Yes. If you work out the magnitudes of the tidal stresses Io experiences,
you will find they are =ENORMOUS= in comparison to the tides experienced
by Earth. The "height" of Io's tidal bulges rises and falls by about
_100 meters_ (300 ft) during the course of its 1.8 day orbit !!!
(By contrast, the combined solar and lunar tidal stresses in the
Earth's crust only cause it to rise and fall by a mere _8 inches_...)

Also, the fact that Io's orbital eccentricity is non-zero means that
in addition to the periodic "stretching" it experiences, the longitude
of the tidal stress pattern also "librates" back and forth through a
small angle...


Or is something else going on?

Well, you should also take into account the fact that Io's orbital
eccentricity is driven by its mutual orbital resonance with Europa
and Ganymede, which is why its orbital eccentricity hasn't decayed, yet...

Also, each time Io experiences a conjunction with Europa or Ganymede,
it experiences a brief, sharp, but non-trivial impulsive tidal-stress
from that moon. Even though the tidal stresses exerted by the other moons
are not as large as those exerted by Jupiter, the sharp, impulse-like
nature of the Europan and Ganymedean tides causes them to be _very_
efficient at heating Io...

(In one of Isaac Asimov's collections of factual essays, he works out the
tidal stresses exerted on Io by Jupiter and the other galilean moons;
unfortunately, I can't tell you which collection it is right now, because
my own copy appears to be in storage... :-(


-- Gordon D. Pusch

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(Gordon D. Pusch) wrote in message

Yes. If you work out the magnitudes of the tidal stresses Io experiences,
you will find they are =ENORMOUS= in comparison to the tides experienced
by Earth. The "height" of Io's tidal bulges rises and falls by about
_100 meters_ (300 ft) during the course of its 1.8 day orbit !!!
(By contrast, the combined solar and lunar tidal stresses in the
Earth's crust only cause it to rise and fall by a mere _8 inches_...)

OK...thanks. This makes sense in principle, though I have no means of
doing the next step & calculating the heat generated. But that's OK.

Also, the fact that Io's orbital eccentricity is non-zero means that
in addition to the periodic "stretching" it experiences, the longitude
of the tidal stress pattern also "librates" back and forth through a
small angle...

OK...I've squinted my eyes at this & strained mightily, but I'm
unclear...Do you
mean that since the object in question is in a non-uniform g-field (as
opposed
to a perfectly uniform, and unrealistic, one, that the orbital
eccentricity causes the
'radial vectors' of the g-field passing thru Io to change slightly in
angularity?



Or is something else going on?

Well, you should also take into account the fact that Io's orbital
eccentricity is driven by its mutual orbital resonance with Europa
and Ganymede, which is why its orbital eccentricity hasn't decayed, yet...

Thanks...I did not know this.

Also, each time Io experiences a conjunction with Europa or Ganymede,
it experiences a brief, sharp, but non-trivial impulsive tidal-stress
from that moon. Even though the tidal stresses exerted by the other moons
are not as large as those exerted by Jupiter, the sharp, impulse-like
nature of the Europan and Ganymedean tides causes them to be _very_
efficient at heating Io...
So these sudden hiccups in heat generated don't have much time to fade
out via dissipative processes? And local heat buildup is facilitated.
Has this
actually been observed as effects in volcanoes, say?

cheers,
Gene

(Gene Partlow) writes:

(Gordon D. Pusch) wrote in message

Yes. If you work out the magnitudes of the tidal stresses Io experiences,
you will find they are =ENORMOUS= in comparison to the tides experienced
by Earth. The "height" of Io's tidal bulges rises and falls by about
_100 meters_ (300 ft) during the course of its 1.8 day orbit !!!
(By contrast, the combined solar and lunar tidal stresses in the
Earth's crust only cause it to rise and fall by a mere _8 inches_...)

OK...thanks. This makes sense in principle, though I have no means of
doing the next step & calculating the heat generated. But that's OK.

I haven't figured out a simple method of getting a back of the envelope
estimate, either, beyond very general considerations such as that,
to the extent that Io can be treated as a viscous liquid body,
to a first approximation, the local density of tidal heat generation
should proportional to the square of the local strain rate, and that,
since tidal strain is proportional to radius, squaring and integrating
the strain rate over volume suggests that the tidal heating rate should
scale as the fifth power of a Moon's radius. However, a websearch
suggests that the mean tidal heating rate is on the order of 4x10^14 Watts,
(roughly twice the total heat-production rate of the Earth, squeezed
through a surface area roughly an order of magnitude smaller), and that
the average heat flux at its surface is between 2 and 3 Watts/m^2.
It turns out that these numbers do not quite balance (the tidal heating
rate is smaller than the average surface heat flux by about a factor of two),
so either there is another source of heat that has not yet been accounted
for, or perhaps Io might have passed through some sort of resonance within
a recent geological epoch that heated it at an enhanced rate.

The following webpage, which appears to be a homework solution for a
graduate-level astrophysics course attempts to estimate Io's heating rate:
http://www.lpl.arizona.edu/grad/classes/Hubbard_505/Section_3/apr_23.htm.


Also, the fact that Io's orbital eccentricity is non-zero means that
in addition to the periodic "stretching" it experiences, the longitude
of the tidal stress pattern also "librates" back and forth through a
small angle...

OK...I've squinted my eyes at this & strained mightily, but I'm
unclear...Do you mean that since the object in question is in a
non-uniform g-field (as opposed to a perfectly uniform, and unrealistic,
one, that the orbital eccentricity causes the 'radial vectors'
of the g-field passing thru Io to change slightly in angularity?

No it's something much simpler that that: By Kepler's Second Law
(the "Equal Area" law), when Io is farther from Jupiter, its orbital
angular velocity is smaller, while when it's closer to Jupiter,
its orbital angular velocity is faster. By contrast, its _rotational_
angular velocity is more nearly constant (modulo tidal torques).
Hence, when Io is closer to Jupiter, it is revolving around Jupiter
slightly faster than it rotates, so the sub-jovian point drifts "east,"
while when it's farther from Jupiter, it is revolving around Jupiter
slightly slower than it rotates, so the sub-jovian point drifts "west."
This slight apparent "wobbling" back and forth of a satellite's
sub-primary point is called "libration," and it was first observed for
the Earth's own Moon --- it's one of the reasons we were able to map
slightly more than one lunar hemisphere via terrestrial observations.


Also, each time Io experiences a conjunction with Europa or Ganymede,
it experiences a brief, sharp, but non-trivial impulsive tidal-stress
from that moon. Even though the tidal stresses exerted by the other moons
are not as large as those exerted by Jupiter, the sharp, impulse-like
nature of the Europan and Ganymedean tides causes them to be _very_
efficient at heating Io...

So these sudden hiccups in heat generated don't have much time to fade
out via dissipative processes? And local heat buildup is facilitated.
Has this actually been observed as effects in volcanoes, say?

Dunno. I haven't seen it mentioned in the published literature, but
planetary astronomy isn't my personal bag, so it would be quite possible
that I might have missed a paper on it...

One consideration is that Io's heat capacity is immense, and the timescale
required for it to come to thermal equilibrium is probably on the order of
a geological epoch, whereas the heat-pulses occur on the order of every
couple terrestrial days, so it's quite possible that they are simply
"averaged out." However, even if they are not, it occurs to me that there
might also be practical problems with attempting to observe this effect,
in that unless Galileo or the Hubble just happened to be appropriately
positioned (and not otherwise occupied during the few hours a conjunction
happened to be occurring), it might be difficult to continuously monitor
Io's volcanic activity sufficiently continuously to see if its activity
becomes measurably more intense during conjunctions...


-- Gordon D. Pusch

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I managed to locate a formula for the average tidal-heating power in
http://lasp.colorado.edu/icymoons/europaclass/Obrien.pdf; in slightly
modified notation, it is:

P_tidal = (21/2) * G*M_P^2 * (k_2/Q) * (R^5 * e^2)/(a^6 * T)

where `G' is the gravitational constant, `M_P' is the mass of the primary,
`k_2' is a dimensionless ratio called the "Love Number" that apparently
measures the ratio of the actual tide to the theoretical "static" tide,
http://scienceworld.wolfram.com/physics/LoveNumber.html, `Q' is the
"dissipation parameter" of the satellite (apparently analogous to the
`Q' of a resonant circuit), `R' is the radius of the satellite, `a' is
the semimajor axis of its orbit, `e' is its orbital eccentricity, and
`T' is its orbital period.

Hence, while it appears I was accidentally correct about the R^5 scaling,
the tidal-heating power only scales as 1/T instead of 1/T^2 as I would have
expected from the fact that the local heat-generation rate should scale as
the square of the local strain-rate, so apparently there is an additional
factor of the period ... :-(


-- Gordon D. Pusch

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