Many body effects in the line radiative transfer equation

TL;DR

This work derives a generalized many-body radiative transfer equation by treating the gas as an interacting ensemble of moving two-level emitters via Dicke's Hamiltonian, incorporating Doppler and collisional decorrelation. The canonical radiative transfer equation is recovered in the limit of negligible coherent optical depth $\tau_{ν}^{\mathrm{coh}}$, while nonzero coherence yields a multiplicative correction that sums to $e^{-\tau_{ν}^{\mathrm{coh}}}$, modifying line profiles and emergent intensities. The theory predicts distinctive signatures such as double-peaked profiles and reduced effective optical depth in Doppler-broadened lines, with concrete implications for HI $21$ cm and CO rotational lines, and proposes a laboratory experiment (OCS rovibrational line) to detect many-body effects. These results imply potential biases in standard gas-property inferences if many-body correlations are ignored and offer a practical route to constrain molecular parameters via MBRT observations. The framework thereby connects fundamental many-body physics with practical radiative-transfer modeling in astrophysical and laboratory settings.

Abstract

The radiative transfer equation for spectral lines from an extended gas is derived from first principles, treating the gas as a system of many atoms/molecules rather than isolated ones. Line broadening effects are assumed to be dominated by particle motions (Doppler effect), but collisional broadening effects are included in the impact approximation. We retrieve the canonical radiative transfer equation under the condition that the optical depth over a coherence length, defined as the transition-levels lifetime times the speed of light, is much lower than unity. For other cases, the line radiative transfer equation contains a correction factor whose magnitude depends exponentially on a quantity that we call the coherent optical depth. We compute that many-body effects affect line radiative transfer of strongly emitting and astronomically ubiquitous radio- and submillimeter lines, such as the HI 21 cm line, and rotational transitions of the main isotopologue of CO. These results imply that care must be taken when interpreting observations relying on these lines, as many-body effects can significantly alter emergent line profiles and bias inferred physical conditions of the emitting gas. Finally, we propose a simple laboratory experiment that would reveal many-body effects in the transfer of radiation, which could furthermore offer a cost-effective means of constraining fundamental molecular parameters.

Figures (4)

• Figure 1: Interpretation of the coherent optical depth, $\tau_{\nu}^{\mathrm{coh}}$. The factor $\tau_{\nu}^{\mathrm{coh}} \sim [\lambda^2\ell_{\mathrm{coh}} \mathcal{N} (n_- - n_+)] A_0/\Delta \nu$ can be understood as the number of molecules that interact coherently with the radiation field. Many-body interactions occur within a volume, $\lambda^2\ell_{\mathrm{coh}}$, set by the radiation wave cross-section and coherence length. The particle density that interacts coherently is, $\mathcal{N}(n_- - n_+)A_0 / \Delta \nu$, and is selected within a small slice of order, $A_0 / \Delta \nu$, of the velocity distribution.
• Figure 2: Emergent spectra derived from the solution to the many-body radiative transfer equation, for the special case $\tau_{\nu}^{\mathrm{\mathrm{coh}}}>\tau_{\nu}$. We plot profiles due to dominant (a) absorption, and (b) emission, for different line-center optical depths, $\tau_0$. In dashed lines, we also plot the solution of the canonical radiative transfer equation.
• Figure 3: Emergent spectra derived from the solution to the many-body radiative transfer equation, for the case $\tau_{\nu}^{\mathrm{\mathrm{coh}}}<\tau_{\nu}$. We plot profiles due to dominant (a) absorption, and (b) emission, for coherent optical depths at line center, $\tau_0^{\mathrm{coh}}=3$, and for different optical depths at line center $\tau_{0}$. In dashed lines, we also plot the solution of the canonical radiative transfer equation.
• Figure 4: Adjusted line profiles due to many-body effects. When no many-body effects are present, the integrated line profile is normalized to unity. With increasing coherent optical depth, the magnitude of the integrated line profile progressively diminishes.