I Comparison of tidal forces acting on the Moon vs Enceladus

[Canis Lupus]

I have heard about a moon Enceladus. Which is powered by tidal force. I suppose this force press back and forth on the moon and friction in the core causes heat. I hope I'm right :-)
Now tidal force $F_t=\frac{2GMmr}{R^3}$, where $G$ is Gravitation constant, $M$ is mass of planet causes gravitational field, $m$ and $r$ is mass and radius of body where we looking for tidal force and $R$ is distance of both objects.
I took data from Wikipedia about Enceladus, Saturn, Moon and Earth and put it to this formula ($M_S$~$100M_E$,
$m_{enc}$~$m_m/1000$, $r_{enc}$~$r_m/10$, $R_{enc-S}$~$R_{m-E}$). I find out that force acting on Moon is 100 times stronger. My question is why our Moon is cold and on Enceladus is warm water? Or ok, probably there are other factors (radioactive decay in core of Enceladus and on Moon it is not) but I would expect something more than cold stone :-)
Can anyone explain it?

Vrbic, If the calculations are still astray you might want to consider this paper https://www.nature.com/articles/srep37740

It's possible that the same process which is the subject of the paper is occurring on Enceladus (and a few other masses in the solar system).

It's against the mainstream, I know, but is peer reviewed. One of the paper's implications is that the present definition of a planet is incorrect concerning the lack of a nuclear reaction, unless you exempt Earth from being a planet, which would seem highly irregular.

I cannot say with technical expertise whether I think the paper is correct, anymore than I can help with the details of your calculations as others far better qualified than me have attempted to do thus far, but I would be less than honest if I stated I believe the paper is incorrect.

Hopefully, it may prove helpful with you calculations if you find it impossible to resolve your calculations using reasonable assumptions.

 

[Charles Link]

Not sure how that is different from the mechanism discussed in posts #8, #10, ...

The OP may have already done a similar derivation, and/or used the results of this type of derivation, but if I may, I will show you how I like to derive the tidal forces: There is a difference between the centripetal force that is necessary to create circular motion around the larger body for the parts of the moon that are away from the center and the gravitational force from the larger body at that point. This difference will be the tidal force. To write this out algebraically $F_t=\frac{mv^2}{(r+r_o)}-\frac{GMm}{(r+r_o)^2}$. $\\$ Now $v=\frac{2 \pi(r+r_o)}{T}$. Also, we have $T^2=\frac{4 \pi^2 r_o^3}{GM}$ by Kepler's 3rd law.($T$ is the same for every point on the orbiting moon). $\\$ A little algebra gives $F_t=GMm( \frac{1}{r_o^2}+\frac{r}{r_o^3}-\frac{1}{r_o^2} \frac{1}{(1+\frac{r}{r_o})^2}) \approx 3\, GMm \frac{r}{r_o^3}$. $\\$ Here $r_o$ is the distance from the larger body to the center of the moon, and $r$ is the difference in distance (e.g. extra distance from $r_o$) to a point on the moon that is not at its center. $\\$ This difference in forces is made up to some extent by the moon's own gravity, but if the interior of the moon is part liquid, some motion due to unbalanced forces could certainly result. $\\$ Additional note: Other than the factor of 3 vs. 2, this is in agreement with the equation that the OP @Vrbic presented in post #1.

 

[haruspex]

I can't realize that a sledge has different friction when I go up or down.

The direction of the friction reverses. How different can you get?
(Oh, and it does not depend on speed.)

 

[Vrbic]

Vrbic, If the calculations are still astray you might want to consider this paper https://www.nature.com/articles/srep37740

It's possible that the same process which is the subject of the paper is occurring on Enceladus (and a few other masses in the solar system).

It's against the mainstream, I know, but is peer reviewed. One of the paper's implications is that the present definition of a planet is incorrect concerning the lack of a nuclear reaction, unless you exempt Earth from being a planet, which would seem highly irregular.

I cannot say with technical expertise whether I think the paper is correct, anymore than I can help with the details of your calculations as others far better qualified than me have attempted to do thus far, but I would be less than honest if I stated I believe the paper is incorrect.

Hopefully, it may prove helpful with you calculations if you find it impossible to resolve your calculations using reasonable assumptions.

Interesting paper, but I'm not specialist on nuclear reactions. I have read just abstract but:
1) Because of expectations of high pressures and temperatures in this paper, I'm not sure whether it is applicable on small moon.
2) Peculiarity for me is the catalysis by pions. As far as I know pions exist only in very high pressures and temperaturs only in cores of massive neutron stars (or some colliders).

But as I said I'm not expert. For me was most interisting question why some objects has a potential of tidal heating and some not. So I tried to understand a process of tidal heating.

 

[Charles Link]

One of the factors in tidal heating would seem to be the rotation of the moon. (This was probably previously mentioned.) If the moon is always facing the same way as the Earth's moon with period of rotation equal to period of revolution, the tidal forces=difference in the two forces that I mentioned in post #52, would reach an equilibrium and there were be little motion in the interior. Alternatively, if the moon were rotating, this equilibrium could not occur because the tidal forces, as calculated by the equation $F_t=3 \, GMm \frac{r}{r_o^3}$, at each location keep changing. $\\$ If the heating that occurred were sufficient to cause a molten state, it would allow for increased tidal action and heating inside the moon.

 

[haruspex]

One of the factors in tidal heating would seem to be the rotation of the moon. (This was probably previously mentioned.) If the moon is always facing the same way as the Earth's moon with period of rotation equal to period of revolution, the tidal forces=difference in the two forces that I mentioned in post #52, would reach an equilibrium and there were be little motion in the interior. Alternatively, if the moon were rotating, this equilibrium could not occur because the tidal forces, as calculated by the equation $F_t=3 \, GMm \frac{r}{r_o^3}$, at each location keep changing. $\\$ If the heating that occurred were sufficient to cause a molten state, it would allow for increased tidal action and heating inside the moon.

According to post #3, Enceladus is rotationally locked. See also post #5.

 

[Charles Link]

According to post #3, Enceladus is rotationally locked. See also post #5.

A google of the moon Enceladus and tidal heating that I just did shows Wikipedia has an article that explains the tidal heating of Enceladus apparently comes from an eccentric resonance orbit with another moon. Looks like it's not as simple as what I proposed in post #55. $\\$ Editing: Yes, and I see you covered some of that already in post#5, etc.