TriPoD: Tri-Population size distributions for Dust evolution - Coagulation in vertically integrated hydrodynamic simulations of protoplanetary disks

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Volume 691 (November 2024)

A&A, 691 (2024) A45

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Table A.1

Lists of symbols

General symbols

Symbol	Description
a	particle size
ȧ	dust growth rate
a_dıift	drift-limited particle size
a_drift-frag	drift-fragmentation-limited particle size
a_frag	fragmentation-limited particle size
a_{mon ,} a_gr	monomer and grown dust size (`two-pop-py`)
a_turb-frag	turbulent-fragmentation-limited particle size
c_s	soundspeed
D	dust diffusivity
ƒ_fric	aerodynamic friction force
f_m	fudge factor (`two-pop-py`)
G	gravitational constant
H	gas scale height
H_d	dust scale height
k_B	Boltzmann constant
L_*	stellar luminosity
m	particle mass
m_p	proton mass
M_*	stellar mass
M_disk	disk mass
M_⊙	solar mass
n(a), n(m)	number density size/mass distribution
𝒩	size ratio for drift-induced collisions
R	stellocentric cylindrical radius
R_c	characteristic disk radius
Re	Reynolds number
P	gas pressure
q	dust size distribution power-law exponent
q_m	dust mass distribution power-law exponent
St	Stokes number
t_fric	stopping time
t_grow	dust growth time scale
T	gas temperature
$T$	viscous stress tensor
$\bar{v}$ $\bar v$	average dust velocity (`two-pop-py`)
v_d-g,R	radial relative velocity of dust and gas
v_frag	dust fragmentation velocity
v_g,v_d	gas and dust velocity
v_K	Keplerian
	velocity
v_mon, v_gr	monomer and grown dust vel. (`two-pop-py`)
∆v	relative particle velocity (source in subscript)
z	cylindrical distance from the disk midplane

α	turbulence parameter
β_∑	double-log. column density gradient
β_T	double-log. temperature gradient
γ	abs. value of the double-log. pressure gradient
δ	dust diffusion parameter (here δ=α)
ɛ	dust-to-gas ratio
λ_mfp	gas molecule mean free path
µ	mean molecular weight
v	coagulation kernel index
v_turb,v_mol	turbulent and molecular viscosity
ξ	fragment size distribition
ρ_g,ρ_d	gas and dust volume density
ρ_g,mid, ρ_d,mid	gas and dust midplane volume density
ρ_m	material density of dust
$σ_{H_{2}}$ ${\sigma _{{{\rm{H}}_2}}}$	hydrogen molecule collision cross section
σ_SB	Stefan-Boltzmann constant
Ω, Ω_K	angular frequency, Keplerian frequency

Symbols specific to `TriPoD`

Symbol	Description
ȧ_max	growth rate of max. size
a₀, a₁	mass-averaged particle sizes of populations
a_min,a_max	min. and max. size of distribution
a_int	$\sqrt{a_{\min} a_{\max}}$ $\sqrt {{a_{\min }}{a_{\max }}}$
a_lim	lower limit for size reduction in gaps
ƒ_crit	crit. mass depletion coefficient for shrinking
ƒ_drift	drift velocity calibration factor
ƒ_∆v	collision speed parameter
$F_{Σ_{0}, i + 1 / 2}^{drift}$ $F_{{{\rm{\Sigma }}_0},i + 1/2}^{{\rm{drift}}}$	dust flux due to radial drift (small pop.)
$F_{Σ_{1}, i + 1 / 2}^{drift}$ $F_{{{\rm{\Sigma }}_1},i + 1/2}^{{\rm{drift}}}$	dust flux due to radial drift (large pop.)
$F_{a_{\max} Σ_{1}, i + 1 / 2}^{drift}$ $F_{{a_{\max }}{{\rm{\Sigma }}_1},i + 1/2}^{{\rm{drift}}}$	flux of the density-weighted maximum particle size due to radial drift
$F_{Σ_{0}, i + 1 / 2}^{diff}$ $F_{{{\rm{\Sigma }}_0},i + 1/2}^{{\rm{diff}}}$	dust flux due to radial diffusion (small pop.)
$F_{Σ_{1}, i + 1 / 2}^{diff}$ $F_{{{\rm{\Sigma }}_1},i + 1/2}^{{\rm{diff}}}$	dust flux due to radial diffusion (large pop.)
$F_{a_{\max} Σ_{1}, i + 1 / 2}^{diff}$ $F_{{a_{\max }}{{\rm{\Sigma }}_1},i + 1/2}^{{\rm{diff}}}$	flux of the density-weighted maximum particle size due to radial diffusion
$F_{Σ_{0 / 1}, i \pm 1 / 2}$ ${F_{{{\rm{\Sigma }}_{0/1}},i \pm 1/2}}$	total flux of the dust fluids
$F_{a_{\max} Σ_{1}, i \pm 1 / 2}$ ${F_{{a_{\max }}{{\rm{\Sigma }}_1},i \pm 1/2}}$	total flux of the density-weighted maximum particle size
ℱ	size distribution calibration function
$\tilde{ℱ}$ $\mathop {\cal F}\limits^$	vert. int. size distribution calibration function
H₀, H_ı	dust scale height of populations
m₀, m₁ N	particle masses corresponding to a₀ and a₁ number of bins of the reconstructed size distr.
p_frag	fragmentation/growth transition function
p_sweep	l − p_frag
p_turb	turb.-dom./drift-dom. transition function
p_drift	l - p_turb
p_turb.1	turb.1/turb.2 transition function
p_turb.2	l − p_turb.1
q_frag	general fragmentation power-law exp.
q_turb.1	fragmentation power-law exp. in turb. 1 regime
q_turb.2	fragmentation power-law exp. in turb. 2 regime
q_drift-frag	drift-fragmentation power-law exp.
q_sweep	non-equilibrium power-law exp.
q_turb-frag	turbulent-fragmentation power-law exp.
s	steepness parameter of the transition from growth to fragmentation
$v_{Σ_{0 / 1}}$ ${v_{{{\rm{\Sigma }}_{0/1}}}}$	drift flux velocity for populations
∆v₀₁	rel. vel. between grains of sizes a₀ and a₁
∆v₁₁	rel. vel. between grains of sizes a_l and ƒ_∆va_l
∆v_max	rel. vel. between grains of sizes a_max and ƒ_∆va_max

ɛ₀,ɛ₁	vert. integrated dust-to-gas ratios of population
ɛ_tot	total dust-to-gas ratio
λ_lim	flux limiter
${\dot{ρ}}_{0 \to 1}$ ${\dot \rho _{0 \to 1}}$	“sweep-up” rate
${\dot{ρ}}_{1 \to 0}$ ${\dot \rho _{1 \to 0}}$	“fragmentation” rate
σ_oı	coll. cross sec. of grains of size a₀ and a_l
σ₁₁	coll. cross sec. of grains of size a₁ and ƒ_∆va_l
∑₀,∑₁	dust column densities of populations
${\dot{Σ}}_{0 \to 1}$ ${{\rm{\dot \Sigma }}_{0 \to 1}}$	vertically integrated “sweep-up”rate
${\dot{Σ}}_{1 \to 0}$ ${{\rm{\dot \Sigma }}_{1 \to 0}}$	vertically integrated “fragmentation”rate
τ_depletion	dust depletion timescale

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