GJ 1002b used to be resonant with c

Now we've got some basic Gliese-1002 parameters, and since my maths aren't professional-tier, I'm wondering if I can just cheat. Like: who else has done this stuff before me. It's worked so far on Venus/Earth cyclers; and on magnetising Ganymede.

I started with that 1979 classic "Melting of Io by Tidal Dissipation", which show us the stakes of the game. This assumed Sinclair 1975 on the triple-Laplacian around Jupiter and, it turns out, Uranus. Now: over GJ 1002, we only have the two planets - the astronomers had the 'scope only for long enough to find the 20-day rockin'. In fact we're here precisely to see if there's that third, 40 days or maybe 30 days (inclined?) or I dunno, 80/3 days.

Sinclair linked in his turn to Greenberg. Good news: Titan/Hyperion, a two-body resonance. Bad news: they run 4:3, and Hyperion's mass might as well be zero. In fact now that Cassini's seen Hyperion, we know that the thing tumbles chaotically. More relevant to us is Sinclair 1972 on Enceladus:Dione and Mimas:Tethys, both 2:1.

Sinclair posted an equation for the change in mean-motion over time: dn/dt = -27n²m/4Q a⁵. Take planet b as it's closer so hits harder. n_b = 34.79572°/day. m and a are ratios, mass and radius respectively. m_b = 2.7031e-5... divided by sin i. We know even less of the radius although I'll assume a Venerean density as to force a radius about Earth's. Since 0.137 R_☉ = 14.933 R_⊕ we'll set a=0.067.

So, er... Q. That's the tidal-dissipation function on the star. I don't think we've studied the tides on that star yet.

Luckily we have similar for Io upon Jupiter. This argues that the tides which Europa and Ganymede raise upon Jupiter are strong enough to pull, in turn, Io backward. Eventually Io's mean-motion increases as the other two decreases. The planets are in resonance now but, eventually, Io will break from it.

Taking that to GJ 1002, the star unlike ten-hour Jupiter is rotating many days behind its satellites, and most of us assume this system is coplanar. The star's tides should be pulling b inward to a faster spiral - and a shorter day. b's own tides should lower it too, as for Phobos. The Deimos of the system, which is c, takes that energy, to rise and to slow. That is what we see: b's day is already shorter than half c's. 2:1 resonants are supposed to be unstable anyway on their own; Kirkwood Gap, anyone? I wonder how long it's been drifting?

Last night I'd intuited another planet to clean up the Laplacian. Interjecting (4:3:2) a 13.8-day planet c would do it, pushing their c to d. The new c would be 0.055 AU, so an Earth mass (maximum) would reach 0.001 to L1/L2. This is, inclination-willing, within our bounds. 4:3:2 is not the problem; we allow it. The problem of course is that we saw the actual 20.2 day c, and didn't see this 13.8 day c-postulate. Also they say that 4:3:2 implies a giant planet or at least a lot of Kepler-90g/h which, for GJ 1002, we should have detected since 1995. Anyway this wasn't even my initial thought last night so I shan't think further of it today.

More serious would be something with longer-period so harder-to-see, n_b - 3n_c + 2n_d = 0. So, n_d in °/day is 16.14287057256621 / 2 = 9.3264. This would be a 38.6 day orbit; a little inbound of double c's. Mind, the tides this sucker would raise on its star would simply add to those of c, pulling b even further away. But maybe not just yet.

Either way that's yet another constraint on anything beneath b. Nothing below b owns momentum enough to raise b's orbit. In fact it's likely b and c both have driven everything down there into a fiery grave by now.