A comparison of pendulum models for large-amplitude longitudinal prominence oscillations

Abstract

Large-amplitude prominence oscillations offer diagnostic information relevant to understanding the magnetic and plasma structure of solar prominences. Accurate prominence seismology requires the use of reliable models. The so-called pendulum model for large-amplitude longitudinal prominence oscillations has demonstrated robustness against observations and numerical simulations. Recent improvements have extended the model to situations with non-uniform gravity, thus leading to corrections that have implications for the inference of the magnetic field strength. In this study we quantify how the different model predictions given by the original and extended pendulum models impact the inference of the minimum magnetic field strength derived from the observed periods of large-amplitude longitudinal prominence oscillations. The analysis we conducted follows a Bayesian approach to solve the inference problem and assess the absolute and relative plausibilities of the two considered models in explaining the observed data, with their uncertainty. We find that the Bayesian solution to the inference problem provides well-constrained posteriors for the minimum magnetic field strength. However, the solutions from each adopted model differ, with differences increasing with the oscillation period. A model comparison analysis results in the extended model being more plausible in the full range of observed periods. However, the magnitude of the Bayes factor is not large enough to determine whether there is positive evidence supporting any of the models. We suggest computing model-averaged posteriors as the most reasonable solution to the inference problem.

Figures (5)

• Figure 1: Top: Surface plots for the period of longitudinal oscillations under model $M_1$ (left, given by Equation [\ref{['pm1']}]) and model $M_2$ (right, given by Equation [\ref{['pm2']}]) as a function of particle density and magnetic field strength. Bottom: Cuts along a given value of magnetic field strength (left, $B=50$ G) and particle density (right, $n= 10^{10}$ cm$^{-3}$). The solutions have been computed on a uniform 2D grid with $N_{n} = N_{B} = 800$ points.
• Figure 2: Marginal and joint posterior distributions for the minimum magnetic field strength $B$ and the plasma particle density $n$ for a LALO with period $P=60\pm10$ min under model $M_1$ given by Equation (\ref{['pm1']}). The left panels are the results obtained with uniform priors, $\mathcal{U}(B \,\mathrm{[G]}; 1,100)$ and $\mathcal{U}(n$ [cm$^{-3}$]; 10$^9$,10$^{11}$) (blue-dotted lines). The right panels are the results obtained with $\mathcal{U}(B \,\mathrm{[G]}; 1,100)$ and a Gaussian prior on particle density, $\mathcal{G}(n$ [cm$^{-3}$]; $\mu_{n},\sigma_{n})$ (red dotted lines), with $\mu_n=10^{10}$ cm$^{-3}$ and $\sigma_n=0.2\mu_n$. The top and middle panels show the marginal posteriors, and the bottom panels show the joint posteriors, with the white line enclosing the 68$\%$ credible interval. For the magnetic field strength, the median and upper bounds at the 68% credible intervals are $B=19^{+5}_{-6}$ G for the uniform priors and $B=8^{+1}_{-1}$ G for the Gaussian prior on density. Solutions computed over a 2D grid with $N_{n} = N_{B} = 800$ points.
• Figure 3: Posterior densities for the minimum magnetic field strength for four values of the period under the original $M_1$ (solid) and the extended $M_2$ (dashed) pendulum models. The top panel displays results obtained with uniform priors, $\mathcal{U}(B \,\mathrm{[G]}; 1,100)$ and $\mathcal{U}(n$ [cm$^{-3}$]; 10$^9$,10$^{11}$). The bottom panel shows the results obtained with $\mathcal{U}(B \,\mathrm{[G]}; 1,100)$ and a Gaussian prior on particle density, $\mathcal{G}(n$ [cm$^{-3}$]; $\mu_{n},\sigma_{n})$, with $\mu_n=10^{10}$ cm$^{-3}$ and $\sigma_n=0.2\mu_n$. The numerical summaries of the posteriors are given in Table \ref{['table:stats']}. A value of $\sigma_P = 10$ min and a 2D grid with $N_{n} = N_{B} = 800$ points were considered.
• Figure 4: Top: Marginal likelihood for models $M_1$ and $M_2$ as a function of the period, with uncertainty, computed using Equation (\ref{['eq:ml']}). Bottom: Bayes factors for the relative plausibility between $M_1$ and $M_2$ as a function of the oscillation period, computed using Equation (\ref{['eq:bf']}). The solid (dotted) line corresponds to the use of uniform (Gaussian) priors. A value of $\sigma_P = 10$ min and a 2D grid with $N_{n} = N_{B} = 800$ points were considered.
• Figure 5: Marginal posteriors under models $M_1$ and $M_2$ and model-averaged posterior for the magnetic field strength for the oscillation with $P= 99$ min analysed by zhang17. A uniform prior, $\mathcal{U}(B \,\mathrm{[G]}; 1,100)$, and a Gaussian density prior, $\mathcal{G}(n$ [cm$^{-3}$]; $\mu_{n},\sigma_{n})$, with $\mu_n=4.25\times10^{10}$ cm$^{-3}$ and $\sigma_n=0.2\mu_n$ have been used. The numerical summaries of the posteriors are given in Table\ref{['table:stats']}. A value of $\sigma_P = 10$ min and a 2D grid with $N_{n} = N_{B} = 800$ points were considered.