A slow spin to win: The gradual kinematic evolution across metallicities of the proto-Galaxy to the high-α disc

Open Access

Table 1

Model parameters, their priors, and their functional forms for the three different models presented in this work.

Prior/functional form
Parameter	Frozen means model on [M/H]−v_ϕ plane	Evolving means model on [M/H]−v_ϕ plane	Evolving means model on [M/H]–η plane
$\vec{w_{k n o t s}}$ $\overrightarrow{w_{{knots}}}$	Dirichlet(0.5 2)	Dirichlet(0.5 2)	Dirichlet(0.5 2)
σ_{mean, GP}	−	𝒰(50 km s⁻¹, 200 km s⁻¹)	𝒰(0.1, 0.9)
ℓ_GP	−	𝒰(0, 1)	𝒰(0, 1)
α_GP	−	𝒰(0, 4)	𝒰(0, 4)
$\vec{μ_{1, k n o t s}}$ $\overrightarrow{\mu_{1, {knots}}}$	−	𝒩(80 km s⁻¹, cov_{spin up})	𝒩(0.3, cov_{spin up})
σ_{1, knots}	−	ℋ𝒩(150 km s⁻¹)	ℋ𝒩(0.3)
σ_{2, knots}	−	𝒯𝒩(100 km s⁻¹, 50 km s⁻¹.; 50 km s⁻¹, 150 km s⁻¹)	𝒯𝒩(0.4, 0.2; 0.4, 0.7)
w	$S_{i n t e r p} ([M / H] ∣ \vec{{k n o t s}_{w}}, \vec{w_{k n o t s}})$ $\mathcal{S}_{{interp}}\left([M / H] \mid \overrightarrow{{knots}_{w}}, \overrightarrow{w_{{knots}}}\right)$	$S_{i n t e r p} ([M / H] ∣ \vec{{k n o t s}_{w}}, \vec{w_{k n o t s}})$ $\mathcal{S}_{{interp}}\left([M / H] \mid \overrightarrow{{knots}_{w}}, \overrightarrow{w_{{knots}}}\right)$	$S_{i n t e r p} ([M / H] ∣ \vec{{k n o t s}_{w}}, \vec{w_{k n o t s}})$ $\mathcal{S}_{{interp}}\left([M / H] \mid \overrightarrow{{knots}_{w}}, \overrightarrow{w_{{knots}}}\right)$
μ₁	𝒩(0 km s⁻¹, 150 km s⁻¹)	$S_{i n t e r p} ([M / H] ∣ \vec{{k n o t s}_{μ}}, \vec{μ_{1, k n o t s}})$ $\mathcal{S}_{{interp}}\left([M / H] \mid \overrightarrow{{knots}_{\mu}}, \overrightarrow{\mu_{1, {knots}}}\right)$	$S_{i n t e r p} ([M / H] ∣ \vec{{k n o t s}_{μ}}, \vec{μ_{1, k n o t s}})$ $\mathcal{S}_{{interp}}\left([M / H] \mid \overrightarrow{{knots}_{\mu}}, \overrightarrow{\mu_{1, {knots}}}\right)$
σ₁	${log}_{10 \frac{σ_{1}}{{kms}^{- 1}}} \sim U (0, 2.5)$ $\log _{10 \frac{\sigma_{1}}{\mathrm{kms}^{-1}}} \sim \mathcal{U}(0, 2.5)$	𝒮_interp([M/H] \| ${\vec{k n o t s}}_{μ}, \vec{σ_{1, k n o t s}}$ $\overrightarrow{knots}_{\mu}, \overrightarrow{\sigma_{1, {knots}}}$ )	𝒮_interp([M/H] \| ${\vec{k n o t s}}_{μ}, \vec{σ_{1, k n o t s}})$ $\overrightarrow{knots}_{\mu}, \overrightarrow{\sigma_{1, {knots}}})$
μ₂	𝒩(0 km s⁻¹, 150 km s⁻¹)	Fixed, 0 km s⁻¹	Fixed, 0
σ₂	${log}_{10} \frac{σ_{2}}{{kms}^{- 1}} \sim U (0, 2.5)$ $\log _{10} \frac{\sigma_{2}}{\mathrm{kms}^{-1}} \sim \mathcal{U}(0,2.5)$	$S_{i n t e r p} ([M / H] ∣ \vec{{k n o t s}_{μ}}, \vec{σ_{2, k n o t s}})$ $\mathcal{S}_{{interp}}\left([M / H] \mid \overrightarrow{{knots}_{\mu}}, \overrightarrow{\sigma_{2, {knots}}}\right)$	$S_{i n t e r p} ([M / H] ∣ \vec{{k n o t s}_{μ}}, \vec{σ_{2, k n o t s}})$ $\mathcal{S}_{{interp}}\left([M / H] \mid \overrightarrow{{knots}_{\mu}}, \overrightarrow{\sigma_{2, {knots}}}\right)$
	w · 𝒩(μ₁, σ₁)+	w · 𝒩(μ₁, σ₁)+	w · ℱ𝒩(μ₁, σ₁;−1, 1)+
v_ϕ or η	(1-w) · 𝒩(μ₂, σ₂)	(1-w) · 𝒩(μ₂, σ₂)	(1-w) · ℱ 𝒩(μ₂, σ₂;−1, 1)

Notes. Abbreviations: ℋ𝒩= half-normal distribution, 𝒯𝒩= truncated normal distribution, 𝒮_interp= spline interpolation, ℱ𝒩= folded normal distribution. The parameters of 𝒯𝒩 and ℱ𝒩 are (μ, σ; min, max). The subscript 1 and 2 stands for the disc-like and halo-like components, respectively, with μ and σ as the Gaussian mean and standard deviations in v_ϕ or η, conditioned on the value of [M/H]. w stands for the relative contribution of the disc-like component with the relative contribution of the halo-like component defined as (1−w). knot s_w and knot s_μ stands for the [M/H] spline knots location for the relative contribution parameter (w) and the velocity/circularity means and standard deviations (μ, σ) respectively. The graphical representation of the models are shown in Figure 5.

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