MadSci Network: Astronomy

Re: Re: grand tack model, outward why?

Date: Thu Jun 22 01:15:12 2017
Posted By: Vladimir Escalante-Ramirez,
Area of science: Astronomy
ID: 1487830846.As

You are right in that a planet in a protoplanetary disk is supposed to move inward. If the planet is large, it will clear the gas in an annulus as it moves inward. The reason is that the gas particles move in a Keplerian motion. That means that they move in a central field of force according to Kepler's law. In that case bodies in the outer parts are overtaken by bodies in the inner parts. Thus the planet overtakes the gas in the outer parts of the disk. As the planet moves past the gas particles in the outer disk, the planet transfers angular momentum to those particles, and the particles gain momentum and move outward while the planet loses angular momentum and moves inward. At the same time the planet is overtaken by particles in the inner regions. Therefore those particles lose angular momentum and move inward while the planet gains angular momentum and moves outward. The net result is that the gas moves away from the planet leaving a gap in the disk. What is not clear is whether the planet moves inward or outward. That depends on how much gas there is in the inner and outward parts of the disk. Numerical simulations show that the planet moves inward if the disk is more or less uniform.

The Grand Tack model was proposed by Walsh and coworkers in 2011 (“A low mass for Mars from Jupiter’s early gas–driven migration”, Nature, Vol. 475, p. 206). The model assumes that Jupiter formed before Saturn, and cleared the gas in an annulus in the disk as it migrated inward. Jupiter might have migrated inward indefinitely, and there would have been no terrestrial planets, but here comes Saturn to the rescue! Saturn forms in the outer parts of the disk, migrates inward and faster than Jupiter did because of its lower mass. Saturn partially clears the disk in its vicinity and catches up with Jupiter. The two planets then become locked in an orbital resonance. If the orbital period of one planet is P1 and the orbital period of the other planet is P2, the planets become “locked” in a resonance if P1/P2=p/q, where p and q are integers. Right now the two planets are near a resonance of p/q=5/2, but in the past it may have been a resonance of 3/2. When two planets are locked in a resonance, they cannot move independently. The two exchange angular momentum back and forth and maintain a fixed separation. If one loses momentum, the other loses momentum too. The two planets join forces to clear overlapping gaps in the disk with an inner edge at a distance of about 1 AU from the Sun. The gas in the inner part will have an enhanced density to form the terrestrial planets.

Why can Jupiter and Saturn move "outward" in the Grand Tack model? As we said before, when a planet moves in a gas disk, it loses angular momentum when it interacts with the outer parts of the disk, and gains momentum when it interacts with the inner parts of the disk. At the same time it forms a gap in the disk. In the Grand Tack model, however, we have two planets that move in the gap. Jupiter moves in the inner part of the gap while Saturn moves near the outer edge of the gap, and the two planets are locked in a resonance. Therefore both planets move either outward or inward. Jupiter will tend to move outward because it is near the inner edge of the gap while Saturn will tend to move inward because it is near the outer edge of the gap. Which one wins? When a planet exchanges angular momentum with the gas, there is a torque on the planet. A torque is the rate of change of angular momentum with time. It can be shown that the torque in this case is proportional to the square of the planet mass. That means that the torque on Jupiter is about 10 times greater than the torque on Saturn. Therefore the torque on Jupiter wins and the Jupiter-Saturn system moves outward.

The outward movement of Saturn makes the material in the outer disk flow past Saturn, and actually cross the gap, and fall into the inner disk. That gas provides more material to build asteroids and terrestrial planets. The infalling gas increases the angular momentum of Jupiter and Saturn, and makes them migrate further out.

How do we know the orbital resonance is enough to reverse migration? The key to the Grand Tack model is that it needs a lot of gas to flow past Saturn and inward, and we just do not know that because there are a lot of uncertainties in the models. There are additional details in the calculations that I won't mention here. In these types of model it is easier to do a simulation with a computer that calculates how the planets form and interact with the gas, and see what happens after many hours of computations. Afterwards one tries to understand what happened in simpler qualitative terms. You can see a more complete account of how Jupiter and Saturn can migrate outward in Masset and Snellgrove, (“Reversing type II migration: resonance trapping of a lighter giant protoplanet”, Monthly Notices of the Royal Astronomical Society, Vol. 320, Number 4, L55–L59, 2001). There is no need for planetesimals in the Grand Tack model, but planetesimals are another way to move a giant protoplanet outward. If the protoplanet attracts and moves planetesimals to lower orbits, the protoplanet will migrate outward. You can read more on the subject of planet formation and migration in gas disks, and how to estimate torques produced by the gas in chapter 7 of Astrophysics of Planet Formation by Philip J. Armitage (Cambridge University, 2010).

Vladimir Escalante Ramírez
Institute for Radio Astronomy and Astrophysics
Morelia, Michoacán, México

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