Massive objects beyond the frost line are likely giant planets.

Let's talk about the frost line, which you helpfully specified (3.94 AU). For anyone not familiar with the term, the frost line is the distance beyond which - in the protoplanetary nebula, while the system was still forming - heavy elements we call volatiles could condense into solid grains. These small bits of solid matter could then clump together into larger conglomerates, which in turn made it possible for larger planetesimals to form. This in turn encouraged the formation of giant planets.

The same thing doesn't happen inside the frost line, meaning that smaller protoplanets formed, and their growth was limited. They would eventually become the smaller, less-massive terrestrial planets. There are some exceptions to the rule; Hot Jupiters are giant exoplanets found orbiting extremely close to their parent stars. However, these giant planets likely moved to their present locations via planetary migration, which could have happened in a number of ways:

• Through interactions with the protoplanetary disk
• Through interactions (scattering) with other planets and protoplanets
• Through interactions with gas nearby

The system here has 12 planets - more than any known planetary system. I find it unlikely that any giant planets could have stably migrated inwards without disrupting the system. Assuming giant planets can only form beyond the frost line, then, we conclude that the inner planets (1-7) are terrestrial.

Calculating the frost line

The radius of the frost line changes with time, increasing as the star becomes more luminous. For instance, I believe the Sun's current frost line is around 5 AU, whereas when the Solar System formed, it was around 3 AU. The flux from a star drops off as $r^{-2}$, so I would argue that the relationship between the radius of the frost line and the luminosity of the star should go as $$r_f=r_{f,\odot}\left(\frac{L_*}{L_{\odot}}\right)^{1/2}$$ where $r_{f,\odot}$ is the Solar System's frost line radius, and $L_*$ is the luminosity of the star. I thought I had found a formula of that sort in Hayashi (1981), but looking back, I can't find it.

Now, you've said that $L_*\approx0.67L_{\odot}$, so we find that, at the present time, $r_f\approx2.2\text{ AU}$ - much less than your value of 3.94 AU. Plus, the young star would have been even less luminous than the Sun. A value of perhaps 1.5-1.75 AU is more realistic, meaning that only planets 1-4 would be terrestrial.

Stellar parameters

I also want to do a quick reality check of sorts on your star. Stars are pretty good black bodies, which is a thermodynamic term describing how they absorb and radiate radiation. We can use something called the Stefan-Boltzmann law to calculate the luminosity of a star if we know its radius ($R_*$) and temperature ($T_*$) using the Stefan-Boltzmann constant, $\sigma$: $$L_*=4\pi\sigma R^2T^4$$ Plugging in your parameters for radius and temperature, I find that for your star, $L_*\approx3.88L_{\odot}$ - much larger than you calculated. I would recommend reducing the surface temperature to about 4,000 K, and similarly reducing the radius to achieve the desired luminosity. This would make it a K-type main sequence star.

Orbital resonances

The largest exoplanetary system we know of is Kepler-90, with 8 planets. It's pretty compact, with all 8 planets being within 1.01 AU of the parent star. I'd like to note that some of the planets have fallen into stable orbital resonances. Essentially some of their orbits are in integer ratios of one another. We see the same thing in some of the moons in our Solar System. In the large and complex system you're describing, I'd also expect some of the planets to be in orbital resonances. If you care enough to calculate some stable ones, you can do that (note that not all resonances are stable).

Also, on the subject of Kepler-90, I'd like to mention that the outer two planets are in fact giant planets, and they're likely inside the frost line. That's an interesting twist; perhaps giant planet migration is indeed possible. That said, it's also possible that Kepler-90 was significantly dimmer in its past, or that the migration of these two planets resulted in the scattering of other former planets, now gone. We don't know for sure.

posted almost 6 years ago

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