**3. Considerations on energy from tidal heating**

For cryovolcanism, an indispensable prerequisite must be an energy source. In principle, energy could be gained from accretion and contraction during the formation *Cryovolcanism in the Solar System and beyond: Considerations on Energy Sources… DOI: http://dx.doi.org/10.5772/intechopen.105067*

of the planetary object. This process is among other parameters depending on the size of the object (with *<sup>R</sup>* being the radius of the object, roughly � *<sup>R</sup>*<sup>3</sup> ). As radioactive material is incorporated along with this process, equivalent considerations may be done here. Higher volume-to-surface ratios (� *R*) minimize cooling effects and allow longer stable heating from inside. Rearrangement of material (e.g., impacts as in LHB, Theia-Gaia events, seeding with <sup>26</sup>*Al*) may change the occurrence, intensity, and also duration of volcanic active phases; inhomogeneities in deposition of material may give rise to local volcanism. The starting composition of radioactive material during formation may differ along with, for example, age of the stellar population. These processes will be complete in the very early phase of a stellar system (roughly 0.5 Gy after formation) and any volcanic activity based on this will evolve based on the then built-up conditions for heating and cooling. Models over several Gys imply significant effects for the heating of liquid volatiles in bigger objects of several hundreds of km radius [96].

This "standard" energy production process might not work in smaller objects where other heating sources are required, for instance, tidal heating, a process occurring in planetary systems with masses closely associated and thus impacting each other. The general principles for tidal heating may be considered as based on many more parameters as for accretion/radioactivity. Aspects of volume-to-surface ratios (� *R*) are the same as for accretion and radioactivity, many other parameters differ.

The tidal acceleration *A* acting on an object's surface is

$$A = \frac{GM}{r^2} \left( \frac{\mathbf{1}}{\left(\mathbf{1} \pm \frac{R}{r}\right)^2} - \mathbf{1} \right),\tag{1}$$

*G* as gravitational constant, *M* as mass of the influencing object, *R* as radius of the influenced object, and *r* as distance between the objects. This can be approximated by a Taylor series expansion to

$$A = \mp 2GM \frac{R}{r^3}.\tag{2}$$

Therefore, the tidal force will go with � *R* (for details and elaborated calculations see [95]). The energy transfer and average dissipation rate gets based on even more parameters and may mostly be assumed by � *<sup>R</sup>*<sup>5</sup> [97–103].

$$
\dot{E} = -\frac{21}{2} \frac{k\_2}{Q} \frac{n^5 R^5}{G^\*} \varepsilon^2,\tag{3}
$$

*E*\_ as rate for tidal energy dissipating, *G*<sup>∗</sup> as gravitational constant, *k*<sup>2</sup> as Love number, and *Q* dissipation function of the satellite. *<sup>k</sup>*<sup>2</sup> *<sup>Q</sup>* is telling how "effectively" energy is transferred on the satellite and how this leads to heating. Models with *k*<sup>2</sup> are mainly used, but also models with "higher" Love numbers as *k*3, *k*4, or *k*<sup>6</sup> may be considered reasonable for special systems [97, 104–106].

*Q* is in the range from 10 to 500 are found for the terrestrial planets and satellites of the major planets. On the other hand, *Q* for the major planets is always larger than <sup>6</sup> � <sup>10</sup><sup>4</sup> [106].

Trying to figure out further principles for tidal heating we may approach this by considering when tidal heating may really be minimized.

A body that is tidally locked on an orbit with eccentricity *e* ¼ 0 will not have any type of tidal energy dissipating. Locking will occur in ranges of

$$t\_{\rm lock} = \frac{\alpha a^6 I Q}{\Im G \,^\* m\_p \,^2 R^5 k\_2},\tag{4}$$

*G*<sup>∗</sup> , *k*2, *Q*, *R* as above, *ω* as initial spin rate, *a* for the semi-major axis of the orbit of the satellite around the planet/partner, *ms* as mass of the satellite, *mp* as mass of the planet/partner, and *I* as momentum of inertia [107] (see pages 169–170 of this article; Formula (9) is quoted here, which comes from ref. [108]), with *I*≈0*:*4*msR*<sup>2</sup> :

$$t\_{\rm lock} \approx \frac{\alpha a^6 \mathbf{0}.4 m\_s R^2 Q}{\Im G^\* m\_p 2^6 k\_2} = \frac{\mathbf{0}.4 \alpha}{\Im G^\*} \left. \frac{Q}{k\_2} \right. \frac{a^6 m\_s}{m\_p 2^3 R^3} . \tag{5}$$

With *ms* <sup>¼</sup> <sup>4</sup>*<sup>π</sup>* <sup>3</sup> *<sup>ρ</sup>R*<sup>3</sup> and *<sup>ρ</sup>* as density of the satellite:

$$t\_{\rm lock} \approx \frac{1.6\pi\alpha}{\Re G^\*} \,\frac{Q}{k\_2} \rho \frac{a^6}{m\_p^2}. \tag{6}$$

Apart from *ω* resulting from the formation process, *<sup>Q</sup> <sup>k</sup>*<sup>2</sup> and *ρ*, as parameters for interior composition and "behavior" in heating, *mp* and especially *a* seem to strongly influence the period in which tidal heating may be possible.

The moon Io is actually tidally locked and would be on a far bigger orbit with eccentricity *e* ¼ 0 and so no volcanism at all would occur, if its accompanying moons would not have been influencing it and are distracting it from a round orbit [109, 110].

But *a* may also change over longer periods "on its own" and may so become important regarding the period for tidal heating and so volcanism. This results from an effect of energy transfer by tidal forces beyond heating, yielding a change of orbital velocity because of tidal acceleration or tidal deceleration.

As the energy transfer resulting in heating is not the only effect, tidal acceleration and also tidal deceleration may occur and by changes in velocity, change the orbit of the objects. For tidal acceleration this will bring objects to farther orbits, moving them out of the possible zone for tidal heating, for tidal deceleration, this will lower the orbits and so either crushing the objects when crossing the Roche limit or crashing them on the body which they are orbiting, as it is assumed for Triton [111–113]. These effects have also an impact via changes in the semi-major axis *a* on *t*lock . The changes by tidal acceleration/deceleration are still tiny in our system and so changes in *t*lock maybe on larger scales [111–113].

All these aspects make it obvious how variable volcanism based on tidal heating may be. The discovery of so powered cryovolcanism on the moons Enceladus and also Triton has been quite surprising and many proofs or hints for active or inactive volcanism, of any kind, may have still not been found in the region of the asteroid belt and beyond. A general overview of both silicate volcanism and cryovolcanism is given in **Figure 2**. All sketches of phases given may be powered by both accretion and radioactivity or by tidal heating. Especially if objects are big or young enough, we may also consider overlap of both power types. Known objects in our own system cover only some of these sketches, but still, we do not have proof of volcanism on all objects being considered and, as discussed, some may be cryovolcanic worlds but may have yet not been even put on a list of assumed objects.

*Cryovolcanism in the Solar System and beyond: Considerations on Energy Sources… DOI: http://dx.doi.org/10.5772/intechopen.105067*

#### **Figure 2.**

*Schematic overview of general types of volcanism (1–3) and how silicate and cryovolcanism are linked (2). Remnants of both silicate and cryovolcanism as signs of inactive volcanism in (4). Earth is a known example of silicate volcanism powered by accretion and radioactivity, as well as Io is also known example of silicate volcanism powered by tidal heating, may be both sketched in (1). Both known icy moons with cryovolcanism powered by tidal heating, Enceladus, and also triton may be found in (2) or in some parts may be in (3). Inactive remnants (4) as discussed may be found on many objects, for example, Vesta or the moon.*

Considering this, we may, when looking out of our own solar system, get aware of how problematic identifications of volcanic worlds may get in these faraway systems. Also, some aspects may get stronger influence. Many systems with close orbits, favoring stronger tidal forces, especially around K- and M-stars, have been found and modeled (e.g., [114–118]). But many parameters of these systems being necessary for modeling are barely known and may need even stronger efforts in measuring and obtaining them. First attempts in reconsidering some constraints of these models have been done (as in e.g., [95]) and first assumptions based on reduced parameter sets for the evaluation of state and kind of volcanic worlds have been made. The approach aims at assessing the potential for volcanic worlds on easier than other observable parameters and has been verified in our own system, yielding all known and many assumed volcanic objects, plus hints for further bodies harboring volcanoes. Thus, it may be considered as a pre-scan before deeper and more intensive modeling. The first application in the system of TRAPPIST-1 gave rise to a higher volcanic potential on all planets, not only by forces of the central star but also by mutual tidal influences of the orbiting bodies [95].
