Rotation period for (fictional and real) planets

I'm having trouble in inferring orbital periods for planets accounting Tidal lock time. Is there a way to "easily" infer (totally putative) rotation periods?

Plus, using following modified formula for "planet-satellite" system, (sorry for "sketchy" appearance):

t= 6 (a⁶)Rμ/m_planetm_star²

Masses are expressed in Kg, distances in meters
μ is rigidity modulus for "satellite" (3e10 for rocky planets and 4e09 for icy ones)

I find earth-sized planets (0.8 - 2 M_Earth) within HZ of stars (0.7-0.9 AUs and mass range 0.8-1.1 M_Sun) being tidally locked after only 2 or 5 Gyrs.

For example I find a hypothetical 0.99 M_Earth planet around 18 Scorpii (1 M_Sun) at 0.8AUs, getting tidally locked after only 1.6 billion years. Whilst if I assumed being located at 0.96 AUs, it gets locked to star at 4.7 billion years, roughly host star's age.
Is there anything wrong in my calculations?

Or else it sounds like most of putative Earth-like planets in the HZ of extrasolar Suns to be tidally locked as all the M-dwarf epistellar worlds.

The truth is that your formula is entirely wrong, no wonder you have troubles with it. Let me explain why.

First, you probably made mistakes when you modified the formula (why ?), and even more mistakes if it comes from Wikipedia.

Second, let's do it with 2 examples with Earth and your planet. With Earth, we have:
a = 150 billion meters
R = 6378000 meters (the radius of the planet in meters, missing in your description)
µ = 3x10^10
planet's mass = 5.97x10^24 kg
star's mass = 1.99x10^30 kg

Let's do the calculation, we have
t = 6*(150,000,000,000^6)*(6378000)*(3x10^10)/(5.97x10^24)*(1.99x10^30)² = 0.553
If t is in years, then the Earth would be tidally locked only 6 months after the Sun's birth. If t is in billion years, we have 553 million years. In both cases, the Earth would be already tidally locked and there would be no life on it.

Now, take your planet, we have:
a = 120 billion meters
R = assuming that it's the same as the Earth
µ = 3x10^10
planet's mass = 5.9103x10^24 kg
star's mass = 1.99x10^30 kg

Doing the same calculation, we have 53.46 days if t is in years and 146.463 million years if t is in billion years. You can see that the results are very bad.

In fact, an Earth-like planet on an Earth orbit will never be tidally locked, or a very long time after the death of the planet.

To get a much better idea, you should use the following formula:
Tidal lock distance = 0.0483((T.M²)/rho)^(1/6)
where Tidal lock distance is pretty much what it says (in AUs), T is the age of the star in years, M is the mass of the star in solar masses and rho is the density of the planet in tons/dm³.
Use it with T as the only unknown value.

Your planet will be tidally locked if the Tidal lock distance is the same or is greater than the semi-major of your planet. Doing the calculations, we see that your planet will be tidally locked 114 billion years after the star's birth (and, of course, a very long time after the death of the planet).
We get 434.77 billion years with the Earth.

Now, your Earth-like planets can live free of tidal lock.

If you look at the Wikipedia page formulae, notice the constant at the end of the "simplified" formula is 10¹⁰ years (i.e. 10 billion years, not 1 billion)

Then again for the Earth-Sun system, that one and the previous one give rather different results, with the first one I get a timescale of about 70 billion years until tidal locking.

Would be nice to get some proper references in here though.

Lazarus wrote:

If you look at the Wikipedia page formulae, notice the constant at the end of the "simplified" formula is 10¹⁰ years (i.e. 10 billion years, not 1 billion)

I wrote:

Let's do the calculation, we have
t = 6*(150,000,000,000^6)*(6378000)*(3x10^10)/(5.97x10^24)*(1.99x10^30)² = 0.553

I wrote:

Doing the same calculation, we have 53.46 days if t is in years and 146.463 million years if t is in billion years.

With the constant, I have 5.53 billion years and 1.46463 billion years respectively.

Lazarus wrote:

Then again for the Earth-Sun system, that one and the previous one give rather different results, with the first one I get a timescale of about 70 billion years until tidal locking.

I made the calculations again but I don't find your timescale. Could you explain how you get it ?

Okay, I understand, but with this difference, the simplified formula is rather useless.

Hey, that's unfair, it's only 1 order of magnitude out. This is astrophysics we're talking about here!

Thanks for useful corrections, I was aware there was something wrong in the formula, but then what's wrong in the formula I provided in main thread?

Which is correct formula for tidal lock timespan?

Plus I still need to know an easy way to "infer" rotation period for planet. Or simply should I fix it arbtrarily?

Edasich wrote:

Thanks for useful corrections, I was aware there was something wrong in the formula, but then what's wrong in the formula I provided in main thread?

The mistake is that the specific terms have been wiped out, which give different results.

Edasich wrote:

Which is correct formula for tidal lock timespan?

Actually, there's no correct formula. The most correct ones are the complete formula and the one I gave in the second post.

Edasich wrote:

Plus I still need to know an easy way to "infer" rotation period for planet. Or simply should I fix it arbtrarily?

If your planet is tidally locked, the rotation period is the same as the orbital one.
If your planet is not tidally locked, you can fix it arbitrarily. Remember that rotation periods of free-spinning planets vary over time. When the dinosaurs were roaming our planet, the rotation period was only 18 hours. It's just a question of tidal forces.

18 hours for the Mesozoic? That's a bit extreme... this paper (pdf) suggests that kind of rotation period roughly 2.45 Gyr ago, way back in the Proterozoic.

Note that Sedna's formula makes its own assumptions about the values of Q and k₂, which in general are very poorly known (even for the planets in our own solar system). The uncertainty in these parameters can lead to huge changes in the resultant timescales: Sedna's formula is in a sense no better or worse than the Wikipedia "simplified" one, it just takes different values for the constants, perhaps making it more useful for different types of systems.

Lazarus wrote:

18 hours for the Mesozoic? That's a bit extreme... this paper (pdf) suggests that kind of rotation period roughly 2.45 Gyr ago, way back in the Proterozoic.

Now you get why Dinosaurs went extinct: the Earth braked and Dinos violently smashed each other dying. *lol*

Lazarus wrote:

18 hours for the Mesozoic? That's a bit extreme... this paper (pdf) suggests that kind of rotation period roughly 2.45 Gyr ago, way back in the Proterozoic.

I always thought this was true, now an old truth is wiped out.

Lazarus wrote:

Note that Sedna's formula makes its own assumptions about the values of Q and k₂, which in general are very poorly known (even for the planets in our own solar system). The uncertainty in these parameters can lead to huge changes in the resultant timescales: Sedna's formula is in a sense no better or worse than the Wikipedia "simplified" one, it just takes different values for the constants, perhaps making it more useful for different types of systems.

That's why I don't smash my head with it...

Edasich wrote:

Now you get why Dinosaurs went extinct: the Earth braked and Dinos violently smashed each other dying. *lol*

This funny story is always better than an asteroid, so we cannot be afraid about that threat.