The ALMA survey to Resolve exoKuiper belt Substructures (ARKS) - II. The radial structure of debris discs

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Volume 705 (January 2026)

A&A, 705 (2026) A196

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Table 1

Definitions of functional forms for radial parametric modelling.

Function	Surface density profile	Parameters
Double power law	$Σ (r) = {[{(\frac{r}{R_{c}})}^{- α_{in} γ} + {(\frac{r}{R_{c}})}^{- α_{out} γ}]}^{- 1 / γ}$ $\Sigma(r) = \left[\left(\dfrac{r}{R_{\rm{c}}}\right)^{-\alpha_{\text{in}} \gamma} + \left(\dfrac{r}{R_{\rm{c}}}\right)^{-\alpha_{\text{out}} \gamma}\right]^{-1/\gamma}$	$R_{c}, α_{in}, α_{out}, γ = 2$ $R_{\rm{c}}, \alpha_{\text{in}}, \alpha_{\text{out}}, \gamma=2$
Triple power law	$\begin{aligned} Σ (r) = {(\frac{R_{in}}{R_{out}})}^{- α_{mid}} {[{(\frac{r}{R_{in}})}^{- α_{in} γ_{in}} + {(\frac{r}{R_{in}})}^{- α_{mid} γ_{in}}]}^{- 1 / γ_{in}} Σ (r) = \times {[{(\frac{r}{R_{out}})}^{- α_{mid} γ_{out}} + {(\frac{r}{R_{out}})}^{- α_{out} γ_{out}}]}^{- 1 / γ_{out}} \end{aligned}$ $\makecell[tl]{$\Sigma(r) = \left(\dfrac{R_{\text{in}}}{R_{\text{out}}}\right)^{-\alpha_{\text{mid}}} \left[\left(\dfrac{r}{R_{\text{in}}}\right)^{-\alpha_{\text{in}} \gamma_{\text{in}}} + \left(\dfrac{r}{R_{\text{in}}}\right)^{-\alpha_{\text{mid}} \gamma_{\text{in}}}\right]^{-1/\gamma_{\text{in}}}$ \\[12pt] $\textcolor{white}{\Sigma(r) =} \times \left[\left(\dfrac{r}{R_{\text{out}}}\right)^{-\alpha_{\text{mid}} \gamma_{\text{out}}} + \left(\dfrac{r}{R_{\text{out}}}\right)^{-\alpha_{\text{out}} \gamma_{\text{out}}}\right]^{-1/\gamma_{\text{out}}}$}$	R_in, R_out, α_in, α_mid, α_out, γ_in=2, γ_out=2
Power law + error function	$Σ (r) = [1 - erf (\frac{R_{c} - r}{\sqrt{2} σ_{in} R_{c}})] {(\frac{r}{R_{c}})}^{- α_{out}}$ $\Sigma(r) = \left[1-\text{erf}\left(\dfrac{R_{\rm{c}}-r}{\sqrt{2} \sigma_{\text{in}} R_{\rm{c}}}\right)\right] \left(\dfrac{r}{R_{\rm{c}}}\right)^{-\alpha_{\text{out}}}$	R_c, σ_in, α_out
Gaussian	$Σ (r) = \exp [- \frac{(r - R)^{2}}{2 σ^{2}}]$ $\Sigma(r) = \exp\left[-\dfrac{(r-R)^2}{2 \sigma^2}\right]$	R, σ
Asymmetric Gaussian	$\begin{aligned} \begin{array}{l} Σ (r) = {\begin{cases} \exp [- \frac{(r - R_{c})^{2}}{2 σ_{in}^{2}}] & if r < R_{c} \\ \exp [- \frac{(r - R_{c})^{2}}{2 σ_{out}^{2}}] & if r \geq R_{c} \end{cases} \end{array} \end{aligned}$ $\begin{array}{l} \Sigma(r) = \begin{cases} \exp\left[-\dfrac{(r-R_{\rm{c}})^2}{2 \sigma_{\text{in}}^2}\right] & \text{if } r<R_{\rm{c}}\\ \exp\left[-\dfrac{(r-R_{\rm{c}})^2}{2 \sigma_{\text{out}}^2}\right] & \text{if } r \geq R_{\rm{c}} \end{cases} \end{array}$	R_c, σ_in, σ_out
Double Gaussian	$\begin{aligned} Σ (r) = C \times \exp [- \frac{(r - R_{1})^{2}}{2 σ_{1}^{2}}] \\ Σ (r) = + (1 - C) \times \exp [- \frac{(r - R_{2})^{2}}{2 σ_{2}^{2}}] \end{aligned}$ $\makecell[tl]{ $\Sigma(r) = C \times \exp\left[-\dfrac{(r-R_1)^2}{2 \sigma_1^2}\right]$\\[12pt] $\textcolor{white}{\Sigma(r) =} + (1-C) \times \exp\left[-\dfrac{(r-R_2)^2}{2 \sigma_2^2}\right]$ }$	R₁, R₂, σ₁, σ₂, C
Triple Gaussian	$\begin{aligned} Σ (r) = C_{1} \times \exp [- \frac{(r - R_{1})^{2}}{2 σ_{1}^{2}}] \\ Σ (r) = + C_{2} \times \exp [- \frac{(r - R_{2})^{2}}{2 σ_{2}^{2}}] \\ Σ (r) = + (1 - C_{1} - C_{2}) \times \exp [- \frac{(r - R_{3})^{2}}{2 σ_{3}^{2}}] \end{aligned}$ $\makecell[tl]{ $\Sigma(r) = C_1 \times \exp\left[-\dfrac{(r-R_1)^2}{2 \sigma_1^2}\right]$ \\[12pt] $\textcolor{white}{\Sigma(r) =} + C_2 \times \exp\left[-\dfrac{(r-R_2)^2}{2 \sigma_2^2}\right]$ \\[12pt] $\textcolor{white}{\Sigma(r) =} + (1-C_1-C_2) \times \exp\left[-\dfrac{(r-R_3)^2}{2 \sigma_3^2}\right]$ }$	R₁, R₂, R₃, σ₁, σ₂, σ₃, C₁, C₂
Gaussian + double power law	$\begin{aligned} Σ (r) = C_{1} \times \exp [- \frac{(r - R)^{2}}{2 σ^{2}}] \\ Σ (r) = + (1 - C_{1}) \times {[{(\frac{r}{R_{c}})}^{- α_{in} γ} + {(\frac{r}{R_{c}})}^{- α_{out} γ}]}^{- 1 / γ} \end{aligned}$ $\makecell[tl]{ $\Sigma(r) = C_1\times\exp\left[-\dfrac{(r-R)^2}{2 \sigma^2}\right]$ \\[12pt] $\textcolor{white}{\Sigma(r) =} + (1-C_1)\times \left[\left(\dfrac{r}{R_{\rm{c}}}\right)^{-\alpha_{\text{in}} \gamma} + \left(\dfrac{r}{R_{\rm{c}}}\right)^{-\alpha_{\text{out}} \gamma}\right]^{-1/\gamma}$ }$	R, σ, C₁, R_c, α_in, α_out, γ=2

Notes. All surface density profiles are multiplied by 10^>∑_c, the surface density normalisation. We fixed certain parameters at the values noted here to minimise degeneracies; exceptions for specific targets are stated explicitly.

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