# Long Term Evolution of Double Planet System in Close Orbit

So, I have a double-planet system in a solar system and I'd like to analyze the long-term evolution of these two planets. In particular, I'd like to know:

1. Is this setup unrealistic in some way? The expectation is that this solar system exists in it's current state after about 1 to 2 billion years of stellar evolution.
2. How would the system be most likely to change in the future?

The double planet in question sits around 1.84 AU from the parent star. One is an ocean planet of .803 Earth masses with a radius of 5721 km and the other is a Venusian world of 1.21 Earth masses with a radius of 6976 km. They orbit a barycenter in a 47.65 hour period with a SMA of 84250 km between them. The system eccentricity is .011.

Edit: All rotations are prograde.

Now, with objects of similar mass like this, it is my understanding that they will move towards being tidally locked. However, this is not necessarily an automatic process. So, I have these bodies currently set up as not being tidally locked. The ocean world has a rotational period of 15.3 hours and the Venusian world has a rotational period of 45.7 hours. (Part of what I would like to know is whether this state of the system is reasonable. I would also like to hear what initial circumstances would be necessary for these planets to arrive in this pattern.)

As for tidal dynamics, I've calculated the tides on the ocean world to average around 3 km in height, so some friction is likely to be caused there, however only 3% of the surface of the planet is land, so I feel like this friction might be lessened overall. The Venusian world has no water, but a very active mantle, so there might be tidal forces impacting it as well.

So, what I'm looking for is how would these forces impact the evolution of these two planets. They would move towards being tidally locked, but what would be the main driving force making them do so and on what time frames would it take place? What would the stable-state of this system look like?

Potentially relevant information: The central star is an F-type main sequence star, at 1.421 stellar masses and a luminosity 3.42 times that of the sun. In the inner system (Around .07 AU out) there is a hot Jupiter of about 2/3 Jupiter masses. In the outer system (around 13 AU out) there is another gas giant, this one at around 2.5 Jupiter masses.

Three possibilities: There are three main avenues that I can see.

1. Rotation to Heat - The two planets could both slow down due to tidal acceleration, the energy being converted to heat through friction, until their rotational periods match.
2. Potential to Heat - Potential energy could be translated into internal heating making them drift together until their orbital period is fast enough that their rotational periods can easily match it. This would also be likely combined with option 1.
3. Rotation to Potential - We could have a repeat of the Earth-Moon system where rotational energy is translated into potential energy, making them drift apart as they spin down.

Which of these three is most likely? Would there perhaps be a mixture of the three or a fourth option I've not yet considered?

## Edit 1 - Final Semimajor Axis

With the information provided by Logan R Kearsley below, I've been able to make another pass at this. I've calculated the angular momentum of the system to be $$3.87273\ast10^{35}$$ $$\frac{m^2kg}{s}$$. Then I took the equation Logan provided and the equation

$$r_1=\frac{a}{1+\frac{1.21}{0.803}}$$

Which gives $$r_1$$ (the distance from the larger body to the barycenter) given $$a$$ (the semimajor axis between the bodies). Using these two together I was able to find that the semimajor axis of the final system would be $$2135020+1416880=3551900$$ $$km$$. I'm currently unsure if this would be close enough for the bodies to stay in each other's influence.

## Edit 2 - Time Period

Alright, using the equation on the Wikipedia page Logan pointed out in the comments below, estimating $$Q$$ as 100 and $$k_2$$ as .3 (close to that of Earth), I arrived at an estimate of about 400 years for the ocean planet to become tidally locked to the Venusian world and about 250 years for the Venusian world to become locked the the ocean world. Obviously, even if these estimates were off by a factor of 1000, they would become locked to each other very quickly on a geological time scale. This is less than desirable, so I might have to go back to the drawing board for this pair of planets.

Is this setup unrealistic in some way? The expectation is that this solar system exists in its current state after about 1 to 2 billion years of stellar evolution.

Sorta-kinda. That's long enough that you might expect enough time to have passed for tidal locking to have occurred. But there's plenty of things that could've messed that up. After all, you may observe that the Earth is not yet locked to the Moon.

How would the system be most likely to change in the future?

That depends on one extra bit of information you did not supply: what direction is each planet rotating in, compared to their orbital direction?

For simplicity, I will assume that both planets are rotating prograde, and that their mutual orbit is also prograde with respect to their solar orbit.

Since both worlds' rotational periods are shorter than their orbital period, they will both be slowed by tidal interactions and transfer spin angular momentum into orbital angular momentum.

The ocean world will experience stronger tidal dissipation than the Venusian world, both because it will experience more fluid friction and because its spin rate is more out of sync to start with, so you should expect its spin rate to reduce more quickly than that of the Venusian world as they move farther and farther apart from each other. The exact rates are pretty much impossible to calculate from first principles; equations for tidal locking timescales tend to have errors of more than a factor of 10, since the results depend strongly on internal structural details of the objects, so either one may end up locked to the other first. The eccentricity is already very low, so its evolution will likely depend more on interactions with other bodies in the system; as is, it will be only weakly suppressed by tidal evolution, and it's entirely plausible that it not change significantly over the life of the system.

While the spin momentum of the planets is converted into orbital momentum (just as in the Earth-Moon system), spin energy will also be converted into heat. That will affect the smaller ocean world more strongly than the larger Venusian world, but on the smaller world more of the tidal heat will be dumped straight into the oceans, while pretty much all of the (lesser) energy dissipation in the Venusian world will go into the mantle.

To figure out what possible states thee system might assume in the future, we need to know what the total angular momentum is, as that is the only close-to-conserved property. (Energy will be dissipated as heat, so we can't rely on that, and angular momentum can be exchanged with other bodies in the system, but we can ignore that as a first approximation.)

The moment of inertia of a sphere (i.e., a planet or approximately uniform density) is $$\frac{2}{5}Ma^2$$. Thus, the total angular momentum of the system is $$\frac{2}{5}(M_1r_1^2\omega_1 + M_2r_2^2\omega_2) + \frac{1}{2}(M_1a_1^2 + M_2a_2^2)\omega_o$$ -- the total spin momentum of each planet, plus the total orbital momentum of each planet.

You haven't provided the radii of the two planets, so I'll assume that they both have the same average density as Earth (which is rather high, just FYI; lowering that might be a bit more realistic) and scale accordingly. That gives us radii of 5928.2km and 6796.4km for the smaller and larger worlds respectively. The distances from each world to the barycenter are 50642km and 33608km respectively.

Filling in the numbers, we get a total angular momentum for the system of and the total angular momentum of the system is $$2.58913624e34 m^2kg/s$$.

In the final equilibrium state, all of the omegas will be equal, so we can rewrite the total angular momentum as $$L = \omega(\frac{2}{5}(M_1r_1^2 + M_2r_2^2) + \frac{1}{2}(M_1a_1^2 + M_2a_2^2))$$ Meanwhile, omega is constrained by the laws of gravity to $$\omega = \frac{1}{2\pi}\sqrt{\frac{G(M_1+M_2)}{(a_1+a_2)^3}}$$

Performing a substitution, we get $$L = (\frac{1}{2\pi}\sqrt{\frac{G(M_1+M_2)}{(a_1+a_2)^3}})(\frac{2}{5}(M_1r_1^2 + M_2r_2^2) + \frac{1}{2}(M_1a_1^2 + M_2a_2^2))$$

If you can solve that for $$a_1+a_2$$, you can plug in the known masses, radii, and total angular momentum to get the final semimajor axis between the worlds, and then plug that back in to figure out the orbital period.

• Apologies on forgetting the radii. They are 5721 km for the ocean world and 6976 km for the Venusian world (a lot of iron ended up in the core of the ocean world). I'll edit those into the post in just a moment. With those in mind I get an angular momentum of $3.87273\ast 10^{35}$ $m^2kg/s$. (Also, would it be possible to get a rough estimate to how long it would take for tidal locking to occur? Are we talking 1 billion years? 100 thousand years?) – Rithaniel Jun 15 '20 at 23:36
• @Rithaniel en.wikipedia.org/wiki/Tidal_locking#Timescale provides formulas for the timescale, but as explained there they depend on factors which in practice just have to be measured--specifically, the Love number $k_2$ (i.e., a measure of rigidity) and the dissipation quality factor $Q$. For physically plausible values of those parameters, you're looking at anywhere from a few hundred million to 10 billion years. – Logan R. Kearsley Jun 16 '20 at 16:12
• Alright, thank you very much for the quick help. I should be able to finish up this system with this information. (I'm surprised I hadn't yet found that Wikipedia page, but it's exactly what I was looking for.) – Rithaniel Jun 16 '20 at 21:13
• Alright, using the equation linked, estimating $Q$ as 100 and $k_2$ as .3 (close to that of Earth), the estimate is about 400 years for the ocean planet to become tidally locked to the Venusian world, which, even if this is off by a factor of 1000, means that this would happen very quickly on a geological time scale. – Rithaniel Jun 16 '20 at 23:23