Question

How do I calculate the tidal heating power for Europa, Callisto, lo, and Ganymede?

Answer 1

   Tidal heating is the increase in internal thermal content of a planet or moon associated with the differential gravitational force (or tide) between two bodies in orbit about their common center of mass.
It is interesting to see moons in our solar system that are so geologically active while our own Moon is not. Due to its size, our Moon has had enough time to radiate its heat into space, causing its interior to cool, prohibiting geological activity. However, Europa is smaller than our Moon (Table1), yet it is still geologically active. The reason for its continued activity is due to tidal heating - a continual flexing and stretching of the Europa caused by the shape of its orbit and gravitational pull from Jupiter, Io, and Ganymede.

The magnitude of gravitational force is defined as:

F = GMm/r²

Where G is the gravitational constant, M is the mass of the more massive body (Jupiter), m is the mass of the less massive body (Europa), and r is the distance of the two bodies. It can be seen that as the distance between the two bodies decreases, the force increases as r², and as the mass of the bodies increase the force also increases.

Tidal force is the differential gravitational force felt across a body. Since the front face of Europa is closer to Jupiter than the far side, it will feel a greater gravitational force. The magnitude of force felt on the side of Europa facing Jupiter is:

F_front = GMm/r²

The force felt on the side facing away from Jupiter is:

F_back = GMm/(r+d)²

where d is the diameter of the moon. The difference in these two forces is the tidal force felt across Europa:

F_tidal = 2GMmd/r³

It can be seen that larger moons have greater tidal forces across them. However, it is not tidal force alone that causes tidal heating within a body. In order for heating to occur, the tidal force must change. This can be imagined if you take a rubber band and stretch it repeatedly. A band that is stretched and left stretched will not generate heat. However, if the band is repeatedly stretched and flexed, heat will be generated. This is becuase stretching and flexing induce frictional heating, similar to rubbing your hands together on a cold day. There is enough tidal heating occurring in Europa to keep its interior warm and the moon to stay geological activity, even though it is smaller than our own Moon.

The change in tidal force (also known as tidal stress) occurs because of the orbits of Io, Europa, and Ganymede. Table 1 shows that the first three Galilean moons are locked in a 4:2:1 orbital resonance. For every orbit Ganymede completes, Europa completes two and Io completes four. This is illustrated in the diagram below (As shown in above Figure). This resonance forces Europa to have an eccentricity of e = 0.01. Eccentricity is a measure of how much an orbit deviates from a perfect circle. A perfect circle has an eccentricity of 0, an elliptical orbit has an eccentricity between 0 < e < 1, e = 1 is a parabolic orbit, and e > 1 is a hyperbolic orbit. Since Europa does not have a perfectly circular orbit, its distance from Jupiter changes. This causes a change in the tidal force on the moon. In addition, Io and Ganymede change their distance with respect to Europa and also introduce a tidal stress.

Europa is locked into a synchronous rotation around Jupiter, just like our Moon is around the Earth. For every revolution Europa makes around Jupiter it also completes one rotation. It has a 3.55 day orbital period and rotation period, so the same side of Europa always faces Jupiter. Thus, is makes one revolution around Jupiter every Europa day. The tidal stress experienced over one orbit is referred to as the "diurnal" stress. The changing tidal force induced by the eccentricity of Europa and tug from the other moons deforms Europa by as much as 3% each Europa day. Moore and Schubert calculate that this should result in a deformation anywhere between 1 m and 30 m depending on if the ice shell is solid down to the silicate layer or if the ice shell is thinner and sits above a liquid water layer. The dissipation of this strain heats the interior of the moon and flexing from the diurnal stress results in the geomorphic features seen across the surface.

Table 1

Satellite	Mass (x 1020)	Radius (km)	Density (g/cm3)	Distance from Parent Body (x 103 km)	Orbital Periods (days)
Io	893.3	1821.3	3.53	421.77	1.77
Europa	480	1565	3.02	671.08	3.55
Ganymede	1482	2634	1.94	1070.4	7.15
Callisto	1076	2403	1.85	1882.8	16.69
Our Moon	734.9	1736	3.34	384.40	27.32

Also,The neighboring Galilean satellites, Ganymede, and Callisto, while quite similar in size and mean density, have traveled very different evolutionary paths. The surface of Callisto, the outermost satellite, appears to have been unmodified by internal processes since shortly after its formation 4.5 billion years ago. Ganymede, on the other hand, shows several distinct phases of surface modification, the most recent occurring perhaps 2 billion years ago . These surficial clues to internal structure were supplemented by the radio science and magnetometer experiments on the Galileo spacecraft, which revealed that the interiors of the two bodies are also very different. Ganymede is a thoroughly differentiated body, having separated into an ice shell about 900 km thick surrounding a rocky mantle and metallic core. The metallic core was furthermore identified as the most plausible site for the generation of Ganymede’s large intrinsic magnetic field which stands off the jovian field in the equatorial regions of the satellite. Callisto, on the other hand, does not appear to have differentiated very much at all, having a moment of inertia which is incompatible with complete separation of rock from ice in its interior . Callisto also lacks an intrinsic magnetic field, but it does respond to the variations of the jovian magnetic field (due to Jupiter’s offset dipole) as if it had a conductivity similar to sea water at a depth of less than about 200 km. In this respect it is very similar to Europa, and the conclusion drawn from this observation is that liquid water exists beneath Callisto’s ice.