The compact neutron star in 4U 1746-37 revisited: Reassessing the mass and radius

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Volume 710 (June 2026)

A&A, 710 (2026) A204

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Table A.1.

Scale height, density, and temperature parameters for disk models

	z₀ (cm)	ρ₀ (g cm⁻³)	T (K)
Case 1^a	$7.8 \times 10^{5} (\frac{\dot{M}}{0.25 {\dot{M}}_{Edd}}) \times [1 - {(\frac{R}{R_{0}})}^{- 1 / 2}]$ $Mathematical equation: $ 7.8 \times 10^5 \ (\frac{\dot{M}}{0.25 \dot{M}_{\mathrm{Edd}}}) \times [1-(\frac{R}{R_0})^{-1/2}] $$	$9.9 \times 10^{- 5} {(\frac{α_{v}}{0.1})}^{- 1} {(\frac{\dot{M}}{0.25 {\dot{M}}_{Edd}})}^{- 2}$ $Mathematical equation: $ 9.9 \times 10^{-5} \ (\frac{\alpha_v}{0.1})^{-1} (\frac{\dot{M}}{0.25 \dot{M}_{\mathrm{Edd}}})^{-2} $$	$3.3 \times 10^{7} {(\frac{α_{v}}{0.1})}^{- 1 / 4} \times {(\frac{M}{1.4 M_{⊙}})}^{1 / 8} {(\frac{R}{R_{0}})}^{- 3 / 8}$ $Mathematical equation: $ 3.3 \times 10^7 \ (\frac{\alpha_v}{0.1})^{-1/4} \times (\frac{M}{1.4M_{{\odot}}})^{1/8} (\frac{R}{R_0})^{-3/8} $$
		$\times {(\frac{M}{1.4 M_{⊙}})}^{- 1 / 2} {(\frac{R}{R_{0}})}^{3 / 2} {[1 - {(\frac{R}{R_{0}})}^{- 1 / 2}]}^{- 2}$ $Mathematical equation: $ \times (\frac{M}{1.4M_{{\odot}}})^{-1/2} (\frac{R}{R_0})^{3/2} [1-(\frac{R}{R_0})^{-1/2}]^{-2} $$
Case 2^b	$1.4 \times 10^{4} {(\frac{α_{v}}{0.1})}^{- 0.1} {(\frac{\dot{M}}{0.25 {\dot{M}}_{Edd}})}^{1 / 5}$ $Mathematical equation: $ 1.4 \times 10^4 \ (\frac{\alpha_v}{0.1})^{-0.1} (\frac{\dot{M}}{0.25 \dot{M}_{\mathrm{Edd}}})^{1/5} $$	$16 {(\frac{α_{v}}{0.1})}^{- 0.7} {(\frac{\dot{M}}{0.25 {\dot{M}}_{Edd}})}^{0.4}$ $Mathematical equation: $ 16\ (\frac{\alpha_v}{0.1})^{-0.7} (\frac{\dot{M}}{0.25 \dot{M}_{\mathrm{Edd}}})^{0.4} $$	$2.4 \times 10^{8} {(\frac{α_{v}}{0.1})}^{- 0.2} {(\frac{\dot{M}}{0.25 {\dot{M}}_{Edd}})}^{0.4}$ $Mathematical equation: $ 2.4 \times 10^8 \ (\frac{\alpha_v}{0.1})^{-0.2} (\frac{\dot{M}}{0.25 \dot{M}_{\mathrm{Edd}}})^{0.4} $$
	$\times {(\frac{M}{1.4 M_{⊙}})}^{- 0.35} {(\frac{R}{R_{0}})}^{1.05} {[1 - {(\frac{R}{R_{0}})}^{- 1 / 2}]}^{1 / 5}$ $Mathematical equation: $ \times (\frac{M}{1.4M_{{\odot}}})^{-0.35} (\frac{R}{R_0})^{1.05} [1-(\frac{R}{R_0})^{-1/2}]^{1/5} $$	$\times {(\frac{M}{1.4 M_{⊙}})}^{0.55} {(\frac{R}{R_{0}})}^{- 33 / 20} {[1 - {(\frac{R}{R_{0}})}^{- 1 / 2}]}^{0.4}$ $Mathematical equation: $ \times (\frac{M}{1.4M_{{\odot}}})^{0.55} (\frac{R}{R_0})^{-33/20} [1-(\frac{R}{R_0})^{-1/2}]^{0.4} $$	$\times {(\frac{M}{1.4 M_{⊙}})}^{0.3} {(\frac{R}{R_{0}})}^{- 0.9} {[1 - {(\frac{R}{R_{0}})}^{- 1 / 2}]}^{0.4}$ $Mathematical equation: $ \times (\frac{M}{1.4M_{{\odot}}})^{0.3} (\frac{R}{R_0})^{-0.9} [1-(\frac{R}{R_0})^{-1/2}]^{0.4} $$
Case 3^c	$8.2 \times 10^{3} {(\frac{α_{v}}{0.1})}^{- 0.1} {(\frac{\dot{M}}{0.25 {\dot{M}}_{Edd}})}^{0.15}$ $Mathematical equation: $ 8.2 \times 10^3 \ (\frac{\alpha_v}{0.1})^{-0.1} (\frac{\dot{M}}{0.25 \dot{M}_{\mathrm{Edd}}})^{0.15} $$	$84 {(\frac{α_{v}}{0.1})}^{- 0.7} {(\frac{\dot{M}}{0.25 {\dot{M}}_{Edd}})}^{0.55}$ $Mathematical equation: $ 84 \ (\frac{\alpha_v}{0.1})^{-0.7} (\frac{\dot{M}}{0.25 \dot{M}_{\mathrm{Edd}}})^{0.55} $$	$7.9 \times 10^{7} {(\frac{α_{v}}{0.1})}^{- 0.2} {(\frac{\dot{M}}{0.25 {\dot{M}}_{Edd}})}^{0.3}$ $Mathematical equation: $ 7.9 \times 10^7 \ (\frac{\alpha_v}{0.1})^{-0.2} (\frac{\dot{M}}{0.25 \dot{M}_{\mathrm{Edd}}})^{0.3} $$
	$\times {(\frac{M}{1.4 M_{⊙}})}^{- 3 / 8} {(\frac{R}{R_{0}})}^{9 / 8} {[1 - {(\frac{R}{R_{0}})}^{- 1 / 2}]}^{0.15}$ $Mathematical equation: $ \times (\frac{M}{1.4M_{{\odot}}})^{-3/8} (\frac{R}{R_0})^{9/8} [1-(\frac{R}{R_0})^{-1/2}]^{0.15} $$	$\times {(\frac{M}{1.4 M_{⊙}})}^{5 / 8} {(\frac{R}{R_{0}})}^{- 15 / 8} {[1 - {(\frac{R}{R_{0}})}^{- 1 / 2}]}^{0.55}$ $Mathematical equation: $ \times (\frac{M}{1.4M_{{\odot}}})^{5/8} (\frac{R}{R_0})^{-15/8} [1-(\frac{R}{R_0})^{-1/2}]^{0.55} $$	$\times {(\frac{M}{1.4 M_{⊙}})}^{1 / 4} {(\frac{R}{R_{0}})}^{- 3 / 4} {[1 - {(\frac{R}{R_{0}})}^{- 1 / 2}]}^{0.3}$ $Mathematical equation: $ \times (\frac{M}{1.4M_{{\odot}}})^{1/4} (\frac{R}{R_0})^{-3/4} [1-(\frac{R}{R_0})^{-1/2}]^{0.3} $$
Case 4^d	$6.7 \times 10^{5} (\frac{R}{R_{0}})$ $Mathematical equation: $ 6.7 \times 10^{5} \ (\frac{R}{R_0}) $$	$1.1 \times 10^{- 4} {(\frac{α_{v}}{0.1})}^{- 0.1} (\frac{\dot{M}}{0.25 {\dot{M}}_{Edd}})$ $Mathematical equation: $ 1.1 \times 10^{-4} \ (\frac{\alpha_v}{0.1})^{-0.1} (\frac{\dot{M}}{0.25 \dot{M}_{\mathrm{Edd}}}) $$	$3.2 \times 10^{11} (\frac{M}{1.4 M_{⊙}}) {(\frac{R}{R_{0}})}^{- 1}$ $Mathematical equation: $ 3.2 \times 10^{11}\ (\frac{M}{1.4M_{{\odot}}}) (\frac{R}{R_0})^{-1} $$
		$\times {(\frac{M}{1.4 M_{⊙}})}^{- 1 / 2} {(\frac{R}{R_{0}})}^{- 3 / 2}$ $Mathematical equation: $ \times (\frac{M}{1.4M_{{\odot}}})^{-1/2} (\frac{R}{R_0})^{-3/2} $$

^a Thin-disk model from Shakura & Sunyaev (1973), Region a): radiation pressure dominated, Thomson scattering opacity P_r ≫ P_g and σ_T ≫ σ_ff.^b Thick-disk model from Shakura & Sunyaev (1973), Region b): gas pressure dominated, Thomson scattering opacity P_r ≪ P_g and σ_T ≫ σ_ff.^c Thick-disk model from Shakura & Sunyaev (1973), Region c): gas pressure dominated, free-free opacity P_r ≪ P_g and σ_T ≪ σ_ff.^d Advection-dominated accretion flow (ADAF) from (Narayan & Yi 1994) with no radiative cooling (f = 1) and γ = 4/3.

Notes. Parameters are expressed in terms of accretion rate (Ṁ), central mass (M), radius (R), and viscosity (α_v). Scale height z₀ represents half-thickness, ρ₀ is volume mass density, and T is temperature. For thick-disk models (Cases 2-4), temperature is derived from sound speed; for the thin-disk model (Case 1), it is derived from energy density. The Eddington accretion rate Ṁ_Edd and reference radius R₀ are treated as constants independent of stellar mass M for consistency across models.

Notes. Usage in main analysis: in the geometric analysis presented in Sect. 2, the thin-disk model corresponds to the standard Shakura-Sunyaev disk (Case 1), while the thick-disk model is based on the advection-dominated accretion flow (ADAF) solution (Case 4).

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