3.3 The disc-instability scenario
There is, however, an alternative to the core-accretion scenario that succeeds in predicting the formation of giant planets, including those with Mp > MJup via direct collapse of the gas in the protoplanetary disc. The key idea of this disc-instability scenario is that a sufficiently cold and/or massive disc tends to be gravitationally unstable. Thus, the disc can undergo fragmentation to form gravitationally bound clumps that evolve into giant planets. For fragmentation to occur, the local surface density in the disc needs to be high enough that the self-gravity of the gas and its differential rotation (both drivers of gravitational collapse) are higher than the thermal pressure (which counteracts collapse). These competing effects are nicely summarised by the Toomre criterion, which states that, for a disc to fragment, its Toomre Q parameter must satisfy:
Here, cs is the sound speed (Equation 3), ωK is the Keplerian angular speed (Equation 2) and Σ is the gas surface density.
To understand the significance of the Toomre criterion for exoplanet systems, it is instructive to consider it in the context of a protoplanetary disc similar to the one that formed our Solar System. To that end, it is useful to introduce the concept of the minimum-mass solar nebula. This is a hypothetical protoplanetary disc with a surface density profile Σsn(r) defined as the minimum value of the surface density that a disc would need to have (as a function of radius r from a Sun-like star) to form our Solar System. The composition of this model nebula is derived from the observed mass of heavy elements in the Solar System planets, plus enough hydrogen and helium to mimic the solar composition. The distribution of this model nebula is determined by spreading the mass needed for each planet over an annulus extending over the distance between them. The accepted surface density distribution for the minimum-mass solar nebula is:
The following activity shows how to express Q as a function of the stellar mass, disc mass and the disc aspect ratio, and compares the minimum value of Σ required for fragmentation to that of the minimum-mass solar nebula.
Activity 7
a.Starting from the definition of scale height (Equation 6), demonstrate that for a protoplanetary disc with uniform surface density Σ, the Toomre Q parameter can be written as
Equation label: (Equation 23)where r is the distance from the star, H is the scale height, M* is the mass of the central star and Mdisc is the mass of the disc.
b.Show that for the Toomre criterion to be satisfied, the surface density must obey:
c.Consider a disc with aspect ratio H/r = 0.05 around a solar-type star (M* = 1 M☉). Show that the minimum surface density at r = 1 au for the disc to fragment is roughly two orders of magnitude greater than that of the minimum-mass solar nebula at the same radius.
Comment
a.Using the scale height definition, Equation 22 becomes:
Using (Equation 2), and noting that for a disc with uniform surface density, then Q becomes
as required.
b.Starting from Equation 23 and multiplying by a factor of and the relevant normalised quantities gives
For the Toomre criterion to be satisfied, we need Q < 1, therefore rearranging the equation above:
c.Using the result from part (b), for a solar mass star with a disc whose aspect ratio is 0.05, fragmentation occurs at r = 1 au when Σ > 1.4 × 106 kg m-2. Comparing this to the surface density of the minimum-mass solar nebula at r = 1 au gives
Activity 7(c) showed that the minimum-mass solar nebula does not meet the Toomre criterion for fragmentation at r = 1 au. In fact, the Toomre criterion would only have been met at distances greater than several thousand astronomical units in the case of the minimum-mass solar nebula. So the Solar System planets probably did not form via disc instability.
Computational simulations show that as a disc becomes unstable, due to Q falling below 1, shock waves are generated within the disc. These shock waves follow a spiral pattern and heat up the disc. Since , the net effect of the shocks is for Q to increase again so the disc stabilises. This effect is known as self-regulation and because of this the disc temperature and surface density tend to reach values for which Q ~ 1. Therefore, an additional condition is necessary for fragmentation: the cooling needs to be fast enough to prevent self-regulation. This is known as the cooling criterion and it is satisfied if the cooling time obeys:
The fact that both the conditions in Equation 22 and Equation 24 need to be satisfied for fragmentation effectively limits the mass and semimajor axis of the planets that can form via the disc-instability scenario, as shown in Figure 9.
