Exploring the probing power of γ Dor’s inertial dip for core magnetism: The case of a toroidal field

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Volume 701 (September 2025)

A&A, 701 (2025) A253

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Table G.1.

Summary of the analytical expressions derived in this work.

		Le_core = Le_env	General Le_core ≠ Le_env
	coupling	$F_{l}^{m} (ν_{M}) \frac{\sqrt{3}}{2} \frac{α}{Λ_{k}^{m} {(ν_{M})}^{2 / 3}} s_{M}^{2 / 3}$ $F_{l}^{m}(\nu_{\mathrm{M}})\dfrac{\sqrt{3}}{2} \dfrac{\alpha}{\Lambda_{k}^{m}(\nu_{\mathrm{M}})^{2/3}}s_{\mathrm{M}}^{2/3}$	$F_{l}^{m} (ν_{M, core}) \frac{\sqrt{3}}{2} \frac{α}{Λ_{k}^{m} {(ν_{M, env})}^{2 / 3}} s_{M, env}^{2 / 3} \frac{s_{M, core}^{2}}{s_{M, env}^{2}} α_{rot}^{2}$ $F_{l}^{m}(\nu_{\mathrm{M, core}}) \dfrac{\sqrt{3}}{2} \dfrac{\alpha}{\Lambda_{k}^{m}(\nu_{\mathrm{M, env}})^{2/3}}s_{\mathrm{M,env}}^{2/3}\dfrac{s_{\mathrm{M, core}}^{2}}{s_{\mathrm{M, env}}^{2}}\alpha_{\mathrm{rot}}^{2}$
	equation	$\times [cot (\frac{π^{2} s_{M}}{Ω Π_{0, M}} - \frac{π}{6}) + \frac{1}{\sqrt{3}}] ≃ ϵ$ $\times \left[\cot\left(\dfrac{\pi^2s_{\mathrm{M}}}{\Omega\Pi_{0,\rm M}}-\dfrac{\pi}{6}\right)+\dfrac{1}{\sqrt{3}}\right] \simeq \epsilon$	$\times [cot (\frac{π^{2} s_{M, env}}{Ω_{env} Π_{0, M}} - \frac{π}{6}) + \frac{1}{\sqrt{3}}] ≃ ϵ$ $\times \left[\cot\left(\dfrac{\pi^2s_{\mathrm{M, env}}}{\Omega_{\mathrm{env}}\Pi_{\mathrm{0,M}}}-\dfrac{\pi}{6}\right)+\dfrac{1}{\sqrt{3}}\right] \simeq \epsilon$

continuous N	V_M/V	${[{(\frac{u^{- 1} (ν_{M})}{ν_{M}})}^{2 / 3}]}_{ν_{M}^{}}$ $\left[\left(\dfrac{u^{-1}(\nu_{\mathrm{M}})}{\nu_{\mathrm{M}}}\right)^{2/3}\right]_{\nu_{\mathrm{M}}^{}}$	$[{(\frac{Λ_{k}^{m} (ν_{M, core})}{Λ_{k}^{m} (u_{env} ° G_{M} ° u_{core}^{- 1} (ν_{M, core}))})}^{2 / 3}$ $\left[\left(\dfrac{\Lambda_{k}^{m}(\nu_{\mathrm{M, core}})}{\Lambda_{k}^{m}(u_{\mathrm{env}}\circ G_{\mathrm{M}} \circ u^{-1}_{\mathrm{core}}(\nu_{\mathrm{M, core}}))}\right)^{2/3}\right.$
			$\times {{(\frac{G_{M} ° u_{core}^{- 1} (ν_{M, core})}{ν_{M, core}})}^{2 / 3} {(\frac{u_{core}^{- 1} (ν_{M, core})}{G_{M} ° u_{core}^{- 1} (ν_{M, core})})}^{2} α_{rot}^{2}]}_{ν_{M, core}^{}}$ $\times \left. \left(\dfrac{G_{\mathrm{M}}\circ u_{\mathrm{core}}^{-1}(\nu_{\mathrm{M, core}})}{\nu_{\mathrm{M, core}}}\right)^{2/3} \left(\dfrac{u_{\mathrm{core}}^{-1}(\nu_{\mathrm{M, core}})}{{G_{\mathrm{M}}\circ u_{\mathrm{core}}^{-1}}(\nu_{\mathrm{M, core}})}\right)^{2}\alpha_{\mathrm{rot}}^{2} \right]_{\nu_{\mathrm{M, core}}^{}}$
	$\frac{1}{Δ P_{M}} - \frac{1}{Π_{0, M}} ≃$ $\frac{1}{\Delta \mathrm{P_{M}}} - \frac{1}{\Pi_{0,\rm M}} \simeq$	$\frac{\frac{Γ_{M}}{π} \frac{d u}{d s_{M}} \|_{{\bar{s}}_{M}}}{{((P_{M} - P_{M}^{}) \frac{d u}{d s_{M}} \|_{\frac{{\bar{s}}_{M} + s_{M}^{}}{2}} + \frac{Γ_{M}}{\sqrt{3}})}^{2} + Γ_{M}^{2}}$ $\dfrac{\frac{\Gamma_{\mathrm{M}}}{\pi}\frac{\mathrm{d}u}{\mathrm{d}s_{\mathrm{M}}}\big\|_{\bar{s}_{\mathrm{M}}}}{\left((\mathrm{P_{M}}-\mathrm{P}_{\mathrm{M}}^{})\frac{\mathrm{d}u}{\mathrm{d}s_{\mathrm{M}}}\big\|_{\frac{\bar{s}_{\mathrm{M}}+s^{}_{\mathrm{M}}}{2}} + \frac{\Gamma_{\mathrm{M}}}{\sqrt{3}}\right)^{2}+\Gamma_{\mathrm{M}}^{2}}$	$\frac{\frac{Γ_{M, \neq}}{π} \frac{d u_{core} ° G_{M}^{- 1}}{d s_{M, env}} \|_{{\bar{s}}_{M, env}}}{{((P_{M} - P_{M}^{}) \frac{d u_{core} ° G_{M}^{- 1}}{d s_{M, env}} \|_{\frac{{\bar{s}}_{M, env} + s_{M, env}^{}}{2}} + \frac{Γ_{M, \neq}}{\sqrt{3}})}^{2} + Γ_{M, \neq}^{2}}$ $\dfrac{\frac{\Gamma_{\mathrm{M, \neq}}}{\pi}\frac{\mathrm{d}u_{\mathrm{core}}\, \circ \, G_{\mathrm{M}}^{-1}}{\mathrm{d}s_{\mathrm{M,env}}}\big\|_{\bar{s}_{\mathrm{M,env}}}}{\left((\mathrm{P_{M}}-\mathrm{P}_{\mathrm{M}}^{})\frac{\mathrm{d}u_{\mathrm{core}}\, \circ \, G_{\mathrm{M}}^{-1}}{\mathrm{d}s_{\mathrm{M, env}}}\big\|_{\frac{\bar{s}_{\mathrm{M,env}}+s^{}_{\mathrm{M,env}}}{2}} + \frac{\Gamma_{\mathrm{M,\neq}}}{\sqrt{3}}\right)^{2}+\Gamma_{\mathrm{M,\neq}}^{2}}$

	coupling	$F_{l}^{m} (ν_{M}) \frac{s_{M}}{2 Λ_{k}^{m} {(ν_{M})}^{1 / 2}}$ $F_{l}^{m}(\nu_{\mathrm{M}}) \dfrac{s_{\mathrm{M}}}{2\Lambda_{k}^{m}(\nu_{\mathrm{M}})^{1/2}}$	$F_{l}^{m} (ν_{M, core}) \frac{s_{M, env}}{2 Λ_{k}^{m} {(ν_{M, env})}^{1 / 2}} \frac{s_{M, core}^{2}}{s_{M, env}^{2}} α_{rot}^{2}$ $F_{l}^{m}(\nu_{\mathrm{M, core}})\dfrac{s_{\mathrm{M,env}}}{2\Lambda_{k}^{m}(\nu_{\mathrm{M, env}})^{1/2}}\dfrac{s_{\mathrm{M, core}}^{2}}{s_{\mathrm{M, env}}^{2}}\alpha_{\mathrm{rot}}^{2}$
	equation	$\times cot (\frac{π^{2} s_{M}}{Ω Π_{0, M}} - \frac{π}{4}) ≃ \tilde{ϵ}$ $\times \cot\left(\dfrac{\pi^2s_{\mathrm{M}}}{\Omega\Pi_{0,\rm M}}-\dfrac{\pi}{4}\right) \simeq \tilde{\epsilon}$	$\times cot (\frac{π^{2} s_{M, env}}{Ω_{env} Π_{0, M}} - \frac{π}{4}) ≃ \tilde{ϵ}$ $\times \cot\left(\dfrac{\pi^2s_{\mathrm{M, env}}}{\Omega_{\mathrm{env}}\Pi_{\mathrm{0,M}}}-\dfrac{\pi}{4}\right) \simeq \tilde{\epsilon}$

discontinuous N,	${\tilde{V}}_{M} / \tilde{V}$ $\tilde{V}_{\mathrm{M}}/\tilde{V}$	${[\frac{u^{- 1} (ν_{M})}{ν_{M}}]}_{ν_{M}^{}}$ $\left[\dfrac{u^{-1}(\nu_{\mathrm{M}})}{\nu_{\mathrm{M}}}\right]_{\nu_{\mathrm{M}}^{}}$	$[{(\frac{Λ_{k}^{m} (ν_{M, core})}{Λ_{k}^{m} (u_{env} ° G_{M} ° u_{core}^{- 1} (ν_{M, core}))})}^{1 / 2} (\frac{G_{M} ° u_{core}^{- 1} (ν_{M, core})}{ν_{M, core}})$ $\left[\left(\dfrac{\Lambda_{k}^{m}(\nu_{\mathrm{M, core}})}{\Lambda_{k}^{m}(u_{\mathrm{env}}\circ G_{\mathrm{M}} \circ u^{-1}_{\mathrm{core}}(\nu_{\mathrm{M, core}}))}\right)^{1/2}\left(\dfrac{G_{\mathrm{M}}\circ u_{\mathrm{core}}^{-1}(\nu_{\mathrm{M, core}})}{\nu_{\mathrm{M,core}}}\right)\right.$
Δρ = 0			$\times {{(\frac{u_{core}^{- 1} (ν_{M, core})}{G_{M} ° u_{core}^{- 1} (ν_{M, core})})}^{2} α_{rot}^{2}]}_{ν_{M, core}^{}}$ $\times \left. \left(\dfrac{u_{\mathrm{core}}^{-1}(\nu_{\mathrm{M, core}})}{G_{\mathrm{M}}\circ u_{\mathrm{core}}^{-1}(\nu_{\mathrm{M, core}})}\right)^{2} \alpha_{\mathrm{rot}}^{2}\right]_{\nu_{\mathrm{M,core}}^{}}$

	$\frac{1}{Δ P_{M}} - \frac{1}{Π_{0, M}} ≃$ $\frac{1}{\Delta \mathrm{P_{M}}} - \frac{1}{\Pi_{0,\rm M}} \simeq$	$\frac{\frac{{\tilde{Γ}}_{M}}{π} \frac{d u}{d s_{M}} \|_{{\bar{s}}_{M}}}{{((P_{M} - P_{M}^{}) \frac{d u}{d s_{M}} \|_{\frac{{\bar{s}}_{M} + s_{M}^{}}{2}})}^{2} + {\tilde{Γ}}_{M}^{2}}$ $\dfrac{\frac{\tilde{\Gamma}_{\mathrm{M}}}{\pi}\frac{\mathrm{d}u}{\mathrm{d}s_{\mathrm{M}}}\big\|_{\bar{s}_{\mathrm{M}}}}{\left((\mathrm{P_{M}}-\mathrm{P}_{\mathrm{M}}^{})\frac{\mathrm{d}u}{\mathrm{d}s_{\mathrm{M}}}\big\|_{\frac{\bar{s}_{\mathrm{M}}+s^{}_{\mathrm{M}}}{2}}\right)^{2}+\tilde{\Gamma}_{\mathrm{M}}^{2}}$	$\frac{\frac{{\tilde{Γ}}_{M, \neq}}{π} \frac{d u_{core} ° G_{M}^{- 1}}{d s_{M, env}} \|_{{\bar{s}}_{M, env}}}{{((P_{M} - P_{M}^{}) \frac{d u_{core} ° G_{M}^{- 1}}{d s_{M, env}} \|_{\frac{{\bar{s}}_{M, env} + s_{M, env}^{}}{2}})}^{2} + {\tilde{Γ}}_{M, \neq}^{2}}$ $\dfrac{\frac{\tilde{\Gamma}_{\mathrm{M, \neq}}}{\pi}\frac{\mathrm{d}u_{\mathrm{core}}\, \circ \, G_{\mathrm{M}}^{-1}}{\mathrm{d}s_{\mathrm{M,env}}}\big\|_{\bar{s}_{\mathrm{M,env}}}}{\left((\mathrm{P_{M}}-\mathrm{P}_{\mathrm{M}}^{})\frac{\mathrm{d}u_{\mathrm{core}}\, \circ \, G_{\mathrm{M}}^{-1}}{\mathrm{d}s_{\mathrm{M, env}}}\big\|_{\frac{\bar{s}_{\mathrm{M, env}}+s^{}_{\mathrm{M,env}}}{2}}\right)^{2}+\tilde{\Gamma}_{\mathrm{M,\neq}}^{2}}$

Notes. Top panels: Continuous treatment of N. Bottom panels: Discontinuous treatment of N. Left panels: Case 1, uniform rotation rate and Alfvén frequency. Right panel: Case 2, bi-layer rotation rate and Alfvén frequency.

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