Bias versus variance when fitting multi-species molecular lines with a non-LTE radiative transfer model - Application to the estimation of the gas temperature and volume density

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Volume 686 (June 2024)

A&A, 686 (2024) A255

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Table 4

Sets of species, the assumed chemistry, the number of available lines, and the number of unknown parameters.

Set of species	Relative abundance	# lines	# unknowns
𝒮 {¹³CO} or 𝒮 {C¹⁸O}	–	2	5
𝒮 {¹³CO, HCO⁺}	free	3	6
𝒮 {¹³CO, HCO⁺; 𝒜}	$\frac{N (^{13} CO)}{N ({HCO}^{+})} = 10^{2.88}$ ${{N\left( {^{13}{\rm{CO}}} \right)} \over {N\left( {{\rm{HC}}{{\rm{O}}^ + }} \right)}} = {10^{2.88}}$	3	5
𝒮 {¹³CO, HCO⁺; 𝒜^★}	$\frac{N N^{13} CO)}{N ({HCO}^{+})} = 10^{3.18}$ ${{\left. {N{N^{13}}{\rm{CO}}} \right)} \over {N\left( {{\rm{HC}}{{\rm{O}}^ + }} \right)}} = {10^{3.18}}$	3	5
𝒮 {¹³CO, C¹⁸O}	free	4	6
𝒮 {¹³CO, C¹⁸O; 𝒜}	$\frac{N (^{13} CO)}{N (C^{18} O)} = 10^{0.90}$ ${{N\left( {^{13}{\rm{CO}}} \right)} \over {N\left( {{{\rm{C}}^{18}}{\rm{O}}} \right)}} = {10^{0.90}}$	4	5
𝒮 {¹³CO, C¹⁸O; 𝒜^★}	$\frac{N (^{13} CO)}{N (C^{18} O)} = 10^{1.20}$ ${{N\left( {^{13}{\rm{CO}}} \right)} \over {N\left( {{{\rm{C}}^{18}}{\rm{O}}} \right)}} = {10^{1.20}}$	4	5
𝒮 {C¹⁸O, H¹³CO⁺}	free	3	6
𝒮 {C¹⁸O, H¹³CO⁺; 𝒜}	$\frac{N (C^{18} O)}{N (H^{13} {CO}^{+})} = 10^{3.78}$ ${{N\left( {{{\rm{C}}^{18}}{\rm{O}}} \right)} \over {N\left( {{{\rm{H}}^{13}}{\rm{C}}{{\rm{O}}^ + }} \right)}} = {10^{3.78}}$	3	5
𝒮 {C¹⁸O, H¹³CO⁺; 𝒜^★}	$\frac{N (C^{18} O)}{N (H^{13} {CO}^{+})} = 10^{3.48}$ ${{N\left( {{{\rm{C}}^{18}}{\rm{O}}} \right)} \over {N\left( {{{\rm{H}}^{13}}{\rm{C}}{{\rm{O}}^ + }} \right)}} = {10^{3.48}}$	3	5

𝒮 {¹²CO, ¹³CO, HCO⁺}	$\frac{N (^{12} CO)}{N (^{13} CO)} = 10^{1.80}$ ${{N\left( {^{12}{\rm{CO}}} \right)} \over {N\left( {^{13}{\rm{CO}}} \right)}} = {10^{1.80}}$	4	6
𝒮 {¹²CO, ¹³CO, C¹⁸O}	$\frac{N (^{12} CO)}{N (^{13} CO)} = 10^{1.80}$ ${{N\left( {^{12}{\rm{CO}}} \right)} \over {N\left( {^{13}{\rm{CO}}} \right)}} = {10^{1.80}}$	5	6

Notes. When more than one species is fitted, we assume that they share the values of ( $n_{H_{2}}$ ${n_{{{\rm{H}}_2}}}$ , T_kin, C_V, and σ_V).

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