The ALMA survey to Resolve exoKuiper belt Substructures (ARKS) - III. The vertical structure of debris disks

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Volume 705 (January 2026)

A&A, 705 (2026) A197

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Table 1

Summary of the density functional forms used in the parametric modeling.

Radial Form	Surface Density Profile	Free Parameters
Double Power Law (DPL)	$Σ (r) = 10^{Σ_{c}} {[{(\frac{r}{R_{c}})}^{- α_{in} γ} + {(\frac{r}{R_{c}})}^{- α_{out} γ}]}^{- 1 / γ}$ ${\rm{\Sigma }}(r) = {10^{{{\rm{\Sigma }}_{\rm{c}}}}}{\left[ {{{\left( {{r \over {{R_{\rm{c}}}}}} \right)}^{ - {\alpha _{{\rm{in}}}}\gamma }} + {{\left( {{r \over {{R_{\rm{c}}}}}} \right)}^{ - {\alpha _{{\rm{out}}}}\gamma }}} \right]^{ - 1/\gamma }}$	R_c, α_in, α_out, γ, Σ_c
Double Gaussian (DG)	$Σ (r) = 10^{Σ_{c}} [C_{1} \exp [\frac{- {(r - R_{1})}^{2}}{2 σ_{12}}] + (1 - C_{1}) \exp [\frac{- {(r - R_{2})}^{2}}{2 σ_{22}}]]$ ${\rm{\Sigma }}(r) = {10^{{{\rm{\Sigma }}_{\rm{c}}}}}\left[ {{C_1}\exp \left[ {{{ - {{\left( {r - {R_1}} \right)}^2}} \over {2\sigma _1^2}}} \right] + \left( {1 - {C_1}} \right)\exp \left[ {{{ - {{\left( {r - {R_2}} \right)}^2}} \over {2\sigma _2^2}}} \right]} \right]$	R₁, R₂, σ₁, σ₂, C₁, Σ_c
Error Function + Power Law (Erf)	$Σ (r) = 10^{Σ_{c}} [1 - erf (\frac{R_{c} - r}{\sqrt{2} σ_{in} R_{c}})] {(\frac{r}{R_{c}})}^{- α_{out}}$ ${\rm{\Sigma }}(r) = {10^{{{\rm{\Sigma }}_{\rm{c}}}}}\left[ {1 - {\rm{erf}}\left( {{{{R_{\rm{c}}} - r} \over {\sqrt 2 {\sigma _{{\rm{in}}}}{R_{\rm{c}}}}}} \right)} \right]{\left( {{r \over {{R_{\rm{c}}}}}} \right)^{ - {\alpha _{{\rm{out}}}}}}$	R_c, σ_in, α_out, Σ_c
Asymmetric Gaussian (AG)	$Σ (r) = {_{10^{Σ_{c}} exp [- \frac{{(r - R_{c})}^{2}}{2 σ_{out}^{2}}] r \geq R_{c}}^{10^{Σ_{c}} exp [- \frac{{(r - R_{c})}^{2}}{2 σ_{in}^{2}}] if r < R_{c}}$ $\Sigma (r) = \left\{ {_{{{10}^{{\Sigma _{\rm{c}}}}}{\rm{exp}}\,\left[ { - {{{{(r - {R_c})}^2}} \over {2\sigma _{{\rm{out}}}^2}}} \right]\,\,\,\,{\rm{if }}r\, \ge \,{R_{\rm{c}}}}^{{{10}^{{\Sigma _{\rm{c}}}}}{\rm{exp}}\,\left[ { - {{{{(r - {R_c})}^2}} \over {2\sigma _{{\rm{in}}}^2}}} \right]\,\,\,\,\,\,{\rm{if }}r\, < \,{R_{\rm{c}}}}} \right.$	R_c, σ_in, σ_out, Σ_c
Triple Power Law (TPL)	$\begin{array}{r} Σ (r) = 10^{Σ_{c}} {(\frac{R_{in}}{R_{out}})}^{- α_{mid}} {[{(\frac{r}{R_{in}})}^{- α_{in} γ_{in}} + {(\frac{r}{R_{in}})}^{- α_{mid} γ_{in}}]}^{- 1 / γ_{in}} \\ \times {[{(\frac{r}{R_{out}})}^{- α_{mid} γ_{out}} + {(\frac{r}{R_{out}})}^{- α_{out} γ_{out}}]}^{- 1 / γ_{out}} \end{array}$ $\matrix{ \hfill {{\rm{\Sigma }}(r) = {{10}^{{{\rm{\Sigma }}_{\rm{c}}}}}{{\left( {{{{R_{{\rm{in}}}}} \over {{R_{{\rm{out}}}}}}} \right)}^{ - {\alpha _{{\rm{mid}}}}}}{{\left[ {{{\left( {{r \over {{R_{{\rm{in}}}}}}} \right)}^{ - {\alpha _{{\rm{in}}}}{\gamma _{{\rm{in}}}}}} + {{\left( {{r \over {{R_{{\rm{in}}}}}}} \right)}^{ - {\alpha _{{\rm{mid}}}}{\gamma _{{\rm{in}}}}}}} \right]}^{ - 1/{\gamma _{{\rm{in}}}}}}} \cr \hfill { \times {{\left[ {{{\left( {{r \over {{R_{{\rm{out}}}}}}} \right)}^{ - {\alpha _{{\rm{mid}}}}{\gamma _{{\rm{out}}}}}} + {{\left( {{r \over {{R_{{\rm{out}}}}}}} \right)}^{ - {\alpha _{{\rm{out}}}}{\gamma _{{\rm{out}}}}}}} \right]}^{ - 1/{\gamma _{{\rm{out}}}}}}} \cr }$	R_in, R_out, α_in, α_mid, α_out, γ_in, γ_out, Σ_c
Gaussian (G)	$Σ (r) = 10^{Σ_{c}} \exp [- \frac{{(r - R)}^{2}}{2 σ^{2}}]$ ${\rm{\Sigma }}(r) = {10^{{{\rm{\Sigma }}_{\rm{c}}}}}\exp \left[ { - {{{{(r - R)}^2}} \over {2{\sigma ^2}}}} \right]$	R, σ, Σ_c

Vertical Form	Density Profile	Free Parameters

Gaussian (vG)	$ρ (r, z) = Σ (r) \times [\exp [- 12 {(\frac{z}{H_{σ} (r)})}^{2}] / \sqrt{2 π} H_{σ} (r)]$ $\rho (r,z) = {\rm{\Sigma }}(r) \times \left[ {\exp \left[ { - {1 \over 2}{{\left( {{z \over {{H_\sigma }(r)}}} \right)}^2}} \right]/\sqrt {2\pi } {H_\sigma }(r)} \right]$	h_HWHM
Exponential (vE)	$ρ (r, z) = Σ (r) \times [\exp [- 12 {(\frac{\| z \|}{H_{σ} (r)})}^{ζ}] / \sqrt{2 π} H_{σ} (r)]$ $\rho (r,z) = {\rm{\Sigma }}(r) \times \left[ {\exp \left[ { - {1 \over 2}{{\left( {{{\left\| z \right\|} \over {{H_\sigma }(r)}}} \right)}^\zeta }} \right]/\sqrt {2\pi } {H_\sigma }(r)} \right]$	h_HWHM, ζ
Double Gaussian (vDG)	$ρ (r, z) = Σ (r) \times [\frac{C_{vert} \exp [- 12 {(z / H_{σ 1} (r))}^{2}]}{\sqrt{2 π} H_{σ 1} (r)} + \frac{(1 - C_{vert}) \exp [- 12 {(z / H_{σ 2} (r))}^{2}]}{\sqrt{2 π} H_{σ 2} (r)}]$ $\rho (r,z) = {\rm{\Sigma }}(r) \times \left[ {{{{C_{{\rm{vert}}}}\exp \left[ { - {1 \over 2}{{\left( {z/{H_{\sigma 1}}(r)} \right)}^2}} \right]} \over {\sqrt {2\pi } {H_{\sigma 1}}(r)}} + {{\left( {1 - {C_{{\rm{vert}}}}} \right)\exp \left[ { - {1 \over 2}{{\left( {z/{H_{\sigma 2}}(r)} \right)}^2}} \right]} \over {\sqrt {2\pi } {H_{\sigma 2}}(r)}}} \right]$	h_HWHM1, h_HWHM2, C_vert
Lorentzian (vL)	$\begin{array}{l} ρ (r, z) = Σ (r) \times [\frac{C_{vert} \exp [- 12 {(z / H_{σ 1} (r))}^{2}]}{\sqrt{2 π} H_{σ 1} (r)} + \frac{(1 - C_{vert}) \exp [- 12 {(z / H_{σ 2} (r))}^{2}]}{\sqrt{2 π} H_{σ 2} (r)}] \\ ρ (r, z) = Σ (r) \times \frac{1}{π} \frac{H_{HWHM} (r)}{z^{2} + H_{HWHM} {(r)}^{2}} \end{array}$ $\eqalign{ & \rho (r,z) = {\rm{\Sigma }}(r) \times \left[ {{{{C_{{\rm{vert}}}}\exp \left[ { - {1 \over 2}{{\left( {z/{H_{\sigma 1}}(r)} \right)}^2}} \right]} \over {\sqrt {2\pi } {H_{\sigma 1}}(r)}} + {{\left( {1 - {C_{{\rm{vert}}}}} \right)\exp \left[ { - {1 \over 2}{{\left( {z/{H_{\sigma 2}}(r)} \right)}^2}} \right]} \over {\sqrt {2\pi } {H_{\sigma 2}}(r)}}} \right] \cr & \rho (r,z) = {\rm{\Sigma }}(r) \times {1 \over \pi }{{{H_{{\rm{HWHM}}}}(r)} \over {{z^2} + {H_{{\rm{HWHM}}}}{{(r)}^2}}} \cr}$	h_HWHM

Notes. Top: Radial forms and free parameters. Bottom: Vertical forms and free parameters. For all radial DPL models, we fix γ = 2. We note that while the vertical aspect ratio free parameters are all listed as h_HWHM, some vertical forms are more conveniently expressed in terms of H_σ. The conversion to h_HWHM is described by Equations (2) and (1). In addition to the parameters shown in the table, all models fit for the disk inclination, PA, and stellar flux, as well as RA and declination offsets.

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