I didn't notice Grant's post #9 until Ken quoted it. I guess that's because it came while I was typing my reply. I appreciate you taking the time to look up the paper.
I'd have to agree with you that if the orbits were being constantly circularized that the graph doesn't have the resolution to show it. In fact, in my astronomy class we were talking about this. Graphs are often tiny because the author has to pay by the page to have his work submitted. I'm not sure if this works for the Letters to Nature or only journal entries.
But if Uranus & Neptune were immersed in a disk thick enough to smooth out the rough edges, often exceeding 1 AU, then it seems to me that they should be doing some migrating during this period.
In my simulations, Uranus and Neptune are not just affecting each others' eccentricity, but they're affecting each others' semi-major axes as well. In some runs of my simulation, it only takes two conjunctions to make Uranus & Neptune switch places. You could still have migrating semi-major axes while retaining a relatively circular orbit. But in the author's diagram, their semi-major axes hold steady during the first 100,000 years.
In my screenshots above, I simply show the results after the first two conjunctions. But if I let it run for thousands of years, Uranus and Neptune will have an encounter close enough to eject one of them, and send the other one on a Jupiter-crossing orbit with a perihelion near the Sun. Even if a planetesimal disk could circularize this orbit, it would be far from the original orbit.
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Originally Posted by Ken G
So the Neptune/Uranus conjuctions must not do that much since they can't make both planets migrate outward (perhaps it helps separate the two orbits, though) whereas the resonant interactions with Saturn and Jupiter can cause an outward spiral even in the presence of the circularization.
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If I understand the paper correctly, Jupiter and Saturn are not causing the outard spiral. Jupiter and Saturn, while teamed up in the 2:1 resonance are only exciting the eccentricities of Uranus and Neptune, causing their aphelions to wander out into the planetesimal disk. It is there that the migrating happens.
I've found something else that confuses me about the author's model.
Jupiter and Saturn perturb each other and do not stay at the 2:1 resonant positions for any extended period of time. If I place them perfectly at 2:1, after their first conjunction, they're not at 2:1 anymore. They're close, but close isn't good enough as it causes the longitude of their conjunction to drift. It's important for his model that the longitude of conjunction remains stationary, so the pertabutions to Uranus and Neptune build upon each other, rather than average out.
In my simulation, with Jupiter and Saturn placed perfectly in 2:1, ( 5AU & 7.9370388140957004 AU) their resonance jumps to 2.01 : 1 after one conjunction, and after 10 conjunctions, it is 2.03 : 1, and the longitude of conjunction has drifted ~30 degrees.