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Appendix B: Disc structure

B.1. Hydrostatic equilibrium

At stationary equilibrium (), gas velocities v_gr, v_gθ, v_gz and the gas density ρ_g 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 e_z: (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.

B.2. Azimuthal velocity

The radial component of the Euler equation is given by: (B.10)where ρ_g 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 c_s =  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)

B.3. Radial profiles of surface density and temperature

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 ρ_g 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 ρ_g, is given by: (B.29)which ensures that: (B.30)and, using Eq. (B.22), we find: (B.31)

Appendix C: Dimensionless quantities and equations of motion

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, s_opt,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, s_opt,0, for the Epstein and the three Stokes drag regimes. The dimensionless equations of motion for a dust grain are then:

Appendix D: Lemma for the different expansions

Lemma: Let x be either r or θ and i the order of the perturbative expansion. If:

• , and

• can be written as a function of R () with of the expansion in η₀,

then, is of order .

Proof: (D.1)

Appendix E: Epstein regime: perturbation analysis

• 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)

• Order : Applying the lemma in this order of expansion and noting that (E.3)Eq. (11) becomes (E.4)Solving the linear system Eq. (E.4) for provides (E.5)

In addition to the expression of given in Sect. 3, we also note that (E.6)which provides with Eq. (15) (E.7)

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.

• Order : (F.3)

• Order : (F.4)

Finally, we obtain expressions for and : We now compare the NSH86 expansion at R^pS₀ ≪ 1 (A-mode) and the W77 expansion at η₀ ≪ 1.

• NSH86: From Eqs. (16) and Eq. (E.7):

• W77 small grains: From Eq. (F.5):

Clearly, Eqs. (F.6) and (F.7) are identical, demonstrating that the theories of W77 and NSH86 are consistent. We also note that if the simplification of Eq. (15) is not performed, the two W77 expansions directly appear as the expansion of NSH86 in or .

Now, in the case of large grains, we perform a perturbative expansion of the radial equation of motion with respect to while assuming that S₀R^p ≫ 1, and verify that taking the limit at small η₀ provides the expression found for the B-mode in the NSH86 expansion.

Taking the same precautions as for the previous expansions, we write: (F.8) Following the same method as for the small grain sizes expansion, we obtain:

• Order : (F.9) It this order of expansion, the azimuthal velocity of the grain is the standard Keplerian velocity.

• Order : (F.10)

The expansion at higher order is more complicated and will not be used for further developments. At the order , we have for and : (F.11) We now compare the expressions provided by the NSH86 expansion at (B-mode) and the W77 expansion at η₀ ≪ 1 for the radial velocity:

• NSH86: (F.12)

• W77 large grains: (F.13)

Once again, we show that the W77 and NSH86 theories are consistent.

Appendix G: Asymptotic radial behaviour of a single grain

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.

Thus, from Eq. (34), we see that the time for a grain starting at R = 1 to reach some final radius R_f is minimized for an optimal grain size S_m,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), .

  • T_m reaches a minimal value for a size S_m given by (G.6)Therefore (G.7)S_m is of order unity and corresponds to an optimal size of migration. Values of S_m and in the (p,q) plane are shown in Fig. G.2. When S ≃ S_m, both the A- and B-modes contribute in an optimal way to the grains radial motion.

  In this case, grains can be efficiently accreted by the central star if . Thus, they can not contribute to the formation of pre-planetesimals. This process is called the radial-drift barrier for planet formation.

© ESO, 2012