Scalings

March 18, 2020, Revision: 1.1

1 Solving the disk structure by integration

In our approach, we prescribe a heating function, H(z), in the energy equation

\frac{dF}{dz} = H(z)  (1)

which leads to a fixed flux profile, F (z) = F_{∞} f(z), where f(z) goes from 0 (in the midplane) to 1 for z ≥ L. In our approach, the flux at infinity is given in terms of Ṁ via

F_{∞} = H_{0}L = \frac{Σ̇}{3π} (\frac{3}{2}Ω)^{2} = σ_{SB}T_{eff}^{4},  (2)

which thus also determines the effective temperature. Next, we have

\frac{dT}{dz} = −F (z)/K(ρ, T)  (3)

\frac{dP}{dz} = −ρg  (4)

where g = Ω^{2}z is the local gravitational acceleration. Divide by each other, and use P = RT ρ/μ, so

∇_{rad} ≡ \frac{d ln T}{d ln P} = \frac{R}{μ} \frac{F_{∞}}{KΩ^{2}} g(z)  (5)

where

g(z) = f(z)/z = \{ \begin{matrix}1 & (for z ≤ L), \\ 1/z & (for z > L). \end{matrix}  (6)

and

Ṁ = 3π \frac{2F_{∞}}{(\frac{3}{2}Ω)^{2}} \frac{H}{L}  (7)

Furthermore,

F_{∞}f(z) = −\frac{16σ_{SB}T^{3−b}}{3κ_{0}ρ^{1+a}} \frac{dT}{dz}  (8)

1 Next, replacing dT/dz → T/H and T → T_{max} and using (γ − 1)c_{p}T_{max} = c_{s}^{2} = Ω^{2}H^{2} we have

F_{∞}f(z) ∝ \frac{16σ_{SB}H^{7−2b}}{3κ_{0}ρ^{1+a}}  (10)

F_{∞}f(z) ∝ \frac{16σ_{SB}H^{7−2b}}{3κ_{0}(ρH)^{1+a}} H^{1+a}  (11)

F_{∞}f(z) ∝ \frac{16σ_{SB}H^{7−2b}}{3κ_{0}Σ^{1+a}} H^{1+a}  (12)

F_{∞}f(z) ∝ \frac{16σ_{SB}H^{8−2b+a}}{3κ_{0}Σ^{1+a}}  (13)

\frac{4σ_{SB}T^{4−b}}{3κ_{0}Σ^{1+a}} H^{a} = ν_{t}Σ(\frac{3}{2}Ω)^{2}  (14)

2 Optically thin case

In the optically thin case, we have T = const ∝ Ṁ^{1/4}. Furthermore, from Ṁ ∝ ΣH^{2}, and since H^{2} ∝ T we have Ṁ^{3/4} ∝ Σ or Ṁ ∝ Σ^{4/3}.

3 Disk parameters

factor 2?

α_{SS} = Ṁ/3π ΣΩH^{2}  (15)
σ_{SB}T_{eff}^{4} = (Ṁ/3π) (\frac{3}{2}Ω)^{2}  (16)

and

                                                                                         $Header: /var/cvs/brandenb/tex/helena/SSdisk/scaling.tex,v 1.1 2016/08/26 01:17:16 brandenb Exp $

F_{∞}f(z) = − \frac{16σ_{SB}}{3κ_{0}} (\frac{T^{n}}{ρ} )^{1+a} \frac{dT}{dz}  (9)

1The following rewrite may be useful:

where n = (3 − b)/(1 + a) is the polytropic index.

Figure 1: conv0a0bm3p5H8.0  Figure 2: conv0a0bm3p5H7.0

2 3