A Unified Maxwell-Bloch Framework for Multi-periodic 6.7 GHz Methanol Flaring in G9.62+0.20E

TL;DR

The study analyzes a decade of 6.7 GHz methanol maser monitoring in G9.62+0.20E, confirming the well-known $p_1$ and $p_2$ cycles and uncovering three new periodicities. It demonstrates that all observed flares across multiple velocity channels can be reproduced within a unified Maxwell–Bloch framework operating in the fast-transient superradiance regime, driven by narrow periodic pump pulses, with environmental timescales $T_1$ and $T_2$ remaining broadly consistent. The modelling yields stable physical conditions in the masing region, notably $T_1 \approx 39$–$41$ d, $T_2 \approx 5.0$–$5.4$ d, and inverted-column densities $n_0 L \sim 10^3$–$10^4$ cm$^{-2}$, across five periods. This supports superradiance as a general description for multi-periodic maser flaring and suggests multiple, potentially distinct periodic drivers in a single star-forming region.

Abstract

We analyze a decade of 6.7 GHz methanol monitoring data in G9.62+0.20E, confirming the known periodicities of p1 = 241.3 +/- 2.3 d and p2 = 52.5 +/- 0.3 d, and identifying three new cycles at p3 = 127.0 +/- 1.6 d, p4 = 163.9 +/- 2.9 d, and p5 = 204.1 +/- 1.5 d. The 241.3-d and 204.1-d periods occur in multiple velocity channels, while the others are confined to single components. Despite their diverse morphologies and timescales, all flares can be reproduced within a unified Maxwell-Bloch framework operating in the fast-transient superradiance regime, driven by narrow periodic pump excitations. Model fits yield consistent environmental parameters across periodicities (temperatures, collisional timescales), pointing to broadly uniform physical conditions in the masing region. The discovery of new periodicities and their unified Maxwell-Bloch modeling provide a consistent picture of multi-periodic flaring in G9.62+0.20E and support superradiance as a general framework for maser flaring.

Figures (5)

• Figure 1: Lomb--Scargle periodograms for three selected velocity components: (a)$v_{\mathrm{lsr}} = -0.2$ km s$^{-1}$, (b)$v_{\mathrm{lsr}} = 1.3$ km s$^{-1}$, and (c)$v_{\mathrm{lsr}} = 8.8$ km s$^{-1}$, along with the exact period values derived from the periodograms. For each velocity channel, the most prominent peaks are labelled, together with the first three harmonics of the strongest peak. The dashed grey line shows the significance threshold, defined as the average of the maximum powers from 1000 Lomb--Scargle periodograms generated from simulated light curves with flux values randomly distributed between $-0.3$ and $+0.3$ Jy, representing the observational noise level.
• Figure 2: Phase-folded 6.7 GHz methanol flares in G9.62+0.20E at $v_{\mathrm{lsr}} = -0.2$ km s$^{-1}$. The MBE model fits (solid blue curves) overlaid on the observational data points (black dots) are shown in each panel. The vertical axis represents the integrated flux density, while the horizontal axis shows time in days. Panels (a) to (c) correspond to the average periodicities of $p = 241.3$ d, $163.9$ d, and $127.0$ d, respectively. The MBE fits reproduce the flare profiles of all three cycles using consistent environmental parameters ($T_{1} \simeq 39.9$ d, $T_{2} \simeq 5.3$ d). The pump duration $T_{\mathrm{P}}$, pump amplitudes $\Lambda_0$ and $\Lambda_1$, and the resulting initial inverted column density $n_0L$ are as follows: (a)$T_{\mathrm{P}} = 4.5$ d, $\Lambda_{0} = 3.8 \times 10^{-19}$ cm$^{-3}$ s$^{-1}$, $\Lambda_{1} = 1.3 \times 10^{-19}$ cm$^{-3}$ s$^{-1}$, yielding $n_0L = 1.0\times10^{3}$ cm$^{-2}$; (b)$T_{\mathrm{P}} = 3.8$ d, $\Lambda_{0} = 2.2 \times 10^{-18}$ cm$^{-3}$ s$^{-1}$, $\Lambda_{1} = 1.9 \times 10^{-19}$ cm$^{-3}$ s$^{-1}$, yielding $n_0L = 2.6\times10^{3}$ cm$^{-2}$; (c)$T_{\mathrm{P}} = 4.8$ d, $\Lambda_{0} = 3.0 \times 10^{-18}$ cm$^{-3}$ s$^{-1}$, $\Lambda_{1} = 1.5 \times 10^{-19}$ cm$^{-3}$ s$^{-1}$, yielding $n_0L = 8.0\times10^{3}$ cm$^{-2}$.
• Figure 3: Same as Fig. \ref{['fig:MBE_-0.23']}, but for $v_{\mathrm{lsr}} = 1.3$ km s$^{-1}$. Panels (a) and (b) correspond to average periodicities of $p = 241.3$ d and $p = 204.1$ d, respectively. The fit parameters and the resulting initial inverted column densities are as follows: (a)$T_{\mathrm{P}} = 4.5$ d, $\Lambda_0 = 4.3 \times 10^{-19}$ cm$^{-3}$ s$^{-1}$, $\Lambda_1 = 2.9 \times 10^{-19}$ cm$^{-3}$ s$^{-1}$, $T_1 = 37.8$ d, $T_2 = 5.1$ d, yielding $n_0L = 1.1\times10^{3}$ cm$^{-2}$; (b)$T_{\mathrm{P}} = 2.8$ d, $\Lambda_0 = 3.1 \times 10^{-18}$ cm$^{-3}$ s$^{-1}$, $\Lambda_1 = 2.5 \times 10^{-19}$ cm$^{-3}$ s$^{-1}$, $T_1 = 41.1$ d, $T_2 = 5.2$ d, yielding $n_0L = 8.2\times10^{3}$ cm$^{-2}$.
• Figure 4: Same as Fig. \ref{['fig:MBE_-0.23']}, but for $v_{\mathrm{lsr}} = 8.8$ km s$^{-1}$. Panels (a) and (b) correspond to average periodicities of $p = 241.3$ d and $p = 52.5$ d, respectively. The fit parameters and the resulting initial inverted column densities are as follows: (a)$T_{\mathrm{P}} = 4.5$ d, $\Lambda_0 = 4.7 \times 10^{-19}$ cm$^{-3}$ s$^{-1}$, $\Lambda_1 = 1.5 \times 10^{-18}$ cm$^{-3}$ s$^{-1}$, $T_1 = 35.0$ d, $T_2 = 5.0$ d, yielding $n_0L = 1.1\times10^{3}$ cm$^{-2}$; (b)$T_{\mathrm{P}} = 3.4$ d, $\Lambda_0 = 1.7 \times 10^{-18}$ cm$^{-3}$ s$^{-1}$, $\Lambda_1 = 5.3 \times 10^{-18}$ cm$^{-3}$ s$^{-1}$, $T_1 = 40.0$ d, $T_2 = 5.4$ d, yielding $n_0L = 4.3\times10^{3}$ cm$^{-2}$.
• Figure 5: Effect of pump period on MBE solutions. (a)Top: Flux density from MBE solutions corresponding to the modelling of $p_1$ for $v_{\mathrm{lsr}}=8.8$ km s$^{-1}$, using identical parameters to those in the top panel of Fig. \ref{['fig:MBE_8.76']}. Bottom: The associated population inversion density (vermilion) and periodic pump (blue). (b) Similar to panel (a), but with a pump period of 52.5 d. All other parameters are identical to those of panel (a) in Fig. \ref{['fig:MBE_8.76']}.