Self-consistent Emission-line Diagnostics with Integrated Extinction Correction

Abstract We present a closed-form, self-consistent spectroscopic framework in which extinction correction is treated as an integral part of the inference rather than as a separate preprocessing step. From emission-line measurements alone, the method simultaneously solves for the electron density ( n e ), electron temperature ( T e ), extinction index ( c (H β )), and total-to-selective extinction ratio ( R V ) within an adopted parametric family of extinction curves. The solution is obtained by minimizing the variance among c (H β ) surfaces in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>log</mml:mi> <mml:mspace width="0.25em"/> <mml:msub> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> </mml:mrow> </mml:msub> </mml:math> – <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>log</mml:mi> <mml:mspace width="0.25em"/> <mml:msub> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> </mml:mrow> </mml:msub> </mml:math> plane constructed from multiple line-ratio pairs, with R V treated as a free parameter, allowing the extinction law to be constrained directly from the data rather than fixed a priori. A simple rank condition, requiring n ≥ k + 4 lines for k ionic species, ensures the identifiability of unique solutions, while diagnostics based on the c (H β )-variance surface provide inexpensive quality control metrics. Noise-free tests recover input parameters exactly at grid points and to interpolation-limited accuracy off grid, validating the analytic formulation and its numerical implementation. When measurement uncertainty is introduced, a clear hierarchy of robustness emerges: plasma parameters remain accurate to ≲10% over wide ranges of line measurement precision, whereas extinction parameters require subpercent accuracy. For Balmer-line diagnostics, this implies a practical uncertainty threshold of ∼0.1%–0.2% per line for stable four-parameter recovery, while ( n e , T e ) alone remain informative at the 1%–2% level. This behavior follows from the geometric conditioning of the objective function surface, where noise flattens constrained directions. The framework clarifies how diagnostic geometry and diversity govern the solution stability in extinction-aware plasma diagnostics, from nebulae to integrated galaxy spectra.