Volume 537, January 2012
|Number of page(s)||19|
|Section||Planets and planetary systems|
|Published online||11 January 2012|
At stationary equilibrium (), gas velocities vgr, vgθ, vgz and the gas density ρg obey mass conservation and the Euler equation: (B.1)The solution of Eq. (B.1) requires: (B.2)which ensures mass conservation. Projecting the Euler equation on ez: (B.3)Assuming that: (B.4)and dividing both sides of Eq. (B.3) by , we have: (B.5)Integrating Eq. (B.5) between 0 and z provides: (B.6)This expression can be simplified by the following approximations:
In the special vertically isothermal case, wherethe sound speed depends only on the radial co-ordinate, Eq. (B.6) simplifies to:(B.7)
Further, assuming a thin disc (), a Taylor series expansion of Eq. (B.7) leads to: (B.8)with: (B.9)which is the classical scale height for vertically isothermal thin discs.
The radial component of the Euler equation is given by: (B.10)where ρg is given by Eq. (B.6). Thus: (B.11)with: (B.12)To simplify Eq. (B.10), we first use the following identity: (B.13)which becomes with Eq. (B.11): (B.14)Noting that f(r,z = 0) = 0 and integrating I1 by parts provides: (B.15)and: (B.16)Then, Eq. (B.10) becomes: (B.17)Noting that : (B.18)and integrating the last term of the right hand side of Eq. (B.18) by parts provides: (B.19)Therefore, Eq. (B.17) reduces to: (B.20)which can be more elegantly written as: (B.21)Thus, the expression of the azimuthal velocity of such a disc can be separated in three terms, called the Keplerian, the pressure gradient and the baroclinic terms respectively. This last term is neglected in most studies. For a three dimensional disc, this term rigorously cancels for cs = constant. In this case, the flow is inviscid and derives from a potential, and isobars and isodensity surfaces coincide: thus, there is no source of vorticity and the azimuthal velocity depends only on the radial coordinate. This terms also cancels out for flat discs in two dimensions. If the disc is vertically isothermal, Eq. (B.21) becomes: (B.22)
In this section, we consider that the disc surface density and the temperature (and thus the sound speed) depend only on the radial coordinate and are given by the following power-law profiles: (B.23)For vertically isothermal thin discs, the vertical density is therefore given by Eq. (B.8) with the scale height given by Eq. (B.9), which can be expressed as: (B.24)with: (B.25)The expression of ρg compatible with the vertical hydrostatic equilibrium and providing the power-law profile set by Eq. (B.23) is written as: (B.26)Indeed: (B.27)Hence, with and , (B.28)which gives the correct surface density profile when integrated with respect to z. With this expression of ρg, is given by: (B.29)which ensures that: (B.30)and, using Eq. (B.22), we find: (B.31)
To highlight the important physical parameters involved, we set and introduce dimensionless quantities given by the following expressions: with: (C.2)The dimensionless parameter η0 gives the order of magnitude of the relative discrepancy between the Keplerian motion and the gas azimuthal velocity. We note that: (C.3)Then, we set and define Writing the coefficient of the drag force of Eq. (5) as: (C.5)and using dimensionless coordinates, we have: (C.6)We also introduce: (C.7)and (C.8)so that Eq. (C.6) becomes: (C.9)Physically, sopt,0 corresponds to the grain size at which the drag stopping time equals the Keplerian time at r0. In Table C.1, we give the expressions of y, λ, sopt,0, for the Epstein and the three Stokes drag regimes. The dimensionless equations of motion for a dust grain are then:
Expressions of the coefficients y, λ, sopt,0 and for different drag regimes.
Lemma: Let x be either r or θ and i the order of the perturbative expansion. If:
can be written as a function of R () with of the expansion in η0,
then, is of order .
Order : At this order of expansion,η0R − q is negligible compared to . Thus, substituting Eq. (13) into Eq. (11) provides (E.1)At this stage, we do not know the order of . We show in the lemma of Appendix D that . Applying this lemma, we see that at the order , taking and (which ensures that ) is a relevant solution for the equations of motion (which corresponds to circular Keplerian motion). Thus, (E.2)
Appendix F: Link with W77’s original derivation
Following W77’s historic reasoning for small grains (see Sect. 3.1), we perform a perturbative expansion of the radial equation of motion with the S0 variable. We verify that taking the limit at small η0 provides the expression found for the A-mode in the NSH86 expansion. (Formally, we will show that ). Hence, we set: (F.1) where we have used for convenience the same formalism as for the expansion in η0 – see Eq. (13) (noting of course that represents different functions). An important point is that the lemma of Appendix D holds when substituting S0 to η0. Therefore, substituting Eq. (F.1) into Eq. (11) provides the equations of motion for different orders of :
Order : Eq. (11) provides expressions for the velocities: (F.2) In this order of expansion, the azimuthal velocity corresponds to the sub-Keplerian velocity of the gas. There is no radial motion.
Now, in the case of large grains, we perform a perturbative expansion of the radial equation of motion with respect to while assuming that S0Rp ≫ 1, and verify that taking the limit at small η0 provides the expression found for the B-mode in the NSH86 expansion.
Noting the position of a grain integrated directly from the equation of motion (Eq. (11)) and the position integrated from the NSH86 approximation (Eq. (34)), we highlight (Fig. G.1) that the discrepancy between the motion from the exact equations and its NSH86 approximation is negligible (the relative error is lower than 10-3 for all the considered sizes). It is therefore justified to use the analytical results derived in Sect. 3 to interpret the grain behaviour.
The discrepancy (bottom panel) between the exact motion (top panel) and its NSH86 approximation (central panel) is negligible. This is illustrated plotting the radial motion of dust grains for S0 = 10-2, η0 = 10-2 and for (solid) and (dashed). Top: , middle: , bottom: relative difference .
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Thus, from Eq. (34), we see that the time for a grain starting at R = 1 to reach some final radius Rf is minimized for an optimal grain size Sm,f given by (G.1)with (G.2)As shown in Eq. (34), the outcome of the grain radial motion depends on the value of :
If : (G.3)For such disc profiles, all grains pile-up and fall onto the central star in an infinite time. Indeed, the surface density profile given by is steep enough to counterbalance the increase of the acceleration due to the pressure gradient. Therefore, grains fall onto the central star in an infinite time, whatever their initial size. Such an evolution happens because the grains always end migrating in the A-mode when they reach the disc’s inner regions. A crucial consequence is that grains are not depleted on the central star and therefore stay in the disc where they can potentially form planet embryos.
If : (G.4)where (G.5)In this case, grains fall onto the central star in a finite time. The surface density profile given by is now too flat to counterbalance the increasing acceleration due to pressure gradient. We note that:
For small sizes(S0 ≪ 1), .
For large sizes (S0 ≫ 1), .
Tm reaches a minimal value for a size Sm given by (G.6)Therefore (G.7)Sm is of order unity and corresponds to an optimal size of migration. Values of Sm and in the (p,q) plane are shown in Fig. G.2. When S ≃ Sm, both the A- and B-modes contribute in an optimal way to the grains radial motion.
© ESO, 2012
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