# Estimation of physical conditions in PDRs Bayesian approach with spatial regularization

Pierre Palud, Franck Le Petit, Pierre Chainais, Emeric E. Bron,
Pierre-Antoine Thouvenin

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Pierre Palud, Franck Le Petit, Pierre Chainais, Emeric E. Bron, Pierre-Antoine Thouvenin. Estimation of physical conditions in PDRs Bayesian approach with spatial regularization. Multi-line Diagnostics of the Interstellar Medium, Apr 2022, Nice, France. . [hal-03953044](https://hal.science/hal-03953044)

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Submitted on 23 Jan 2023

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Distributed under a Creative Commons Attribution 4.0 International License
# Estimation of physical conditions in PDRs
## Bayesian approach with spatial regularization

<figure><img src="image_2.png" /></figure>

Pierre Palud¹,², F. Le Petit¹, P. Chainais²,
E. Bron¹, P.-A. Thouvenin², ORIONB consortium
¹LERMA, Obs. de Paris, PSL Research Univ., CNRS, Meudon
²Univ. Lille, CNRS, Centrale Lille, UMR 9189 – CRISTAL, France

<figure><img src="image_3.png" /></figure>

## Aims and contributions

New large hyper-spectral surveys in radio-astronomy: game changer for study of star formation, feedback mechanisms and chemistry of interstellar gas. ⇒ now possible to observe full molecular clouds (10 pc size) at dense core scale (<0.1 pc) spatial resolution. For instance, the IRAM-30m Large Program «Orion B» [1] provides a 10⁶ pixel observation map, covering ~250 pc² of the Orion B cloud with emission of dozens of molecules.

**Goal** To estimate maps of thermal pressure $P_{th}$, UV radiation intensity $G_0$, visual extinction $A_V$, scaling factor $\kappa$ and the associated uncertainties.

**Challenge** Variability of the SNR: the brightest regions can be well constrained, but the regions with low SNR lead to degenerate solutions.

**Approach**
* Mixture of both additive and multiplicative noises + limited detectability ⇒ more realistic observation model
* Spatial regularization ⇒ exploitation of the information contained in neighbouring pixels.
* Bayesian approach ⇒ access to accurate credibility intervals.

## Problem formulation

→ Meudon PDR code [2]: solves radiative transfer, chemistry, thermal balance on stationary 1D plane-parallel model.

<figure><img src="image_4.png" /></figure>

→ Assumption: physics in PDR = interpolation of a grid of simulations of Meudon PDR code. Defines function $f$ s.t. for one pixel $n$:

<figure><img src="image_5.png" /></figure>

→ observation model involves detectability limit $\omega$, gaussian noise $\varepsilon_{n,l}^{(a)}$ and lognormal noise $\varepsilon_{n,l}^{(m)}$

$$y_{n,l} = \begin{cases} \omega & \text{if } \varepsilon_{n,l}^{(m)} f_{\ell}(x_n) + \varepsilon_{n,l}^{(a)} \le \omega \\ \varepsilon_{n,l}^{(m)} f_{\ell}(x_n) + \varepsilon_{n,l}^{(a)} & \text{else} \end{cases}$$

⇒ defines a likelihood function $\pi(y|x)$.

## Application on toy dataset (100 pixels)

<figure><img src="image_6.png" /><figcaption>Synthetic observations: 10 excited lines of CO (from J=4-3 to 13-12)</figcaption></figure>

<figure><img src="image_7.png" /></figure>

## Bayesian approach

→ State-of-the-art in millimeter/IR astronomy estimations: Maximum Likelihood Estimates (MLE) ⇒ very sensitive to noise

→ For robust estimators: spatial regularization prior $\pi(x)$ (norm of image gradient or laplacian)

→ Posterior probability density function:

<figure><img src="image_8.png" /></figure>

→ To derive estimators and credibility intervals from posterior distribution: **sample** from it with Monte Carlo Markov Chain (MCMC).

→ Our sampler mixes two kernels: one identifies local minima, one efficiently explores them.

→ Our ponctual estimator: MMSE (Minimum Mean Squared Error) = mean of posterior.

## NGC 7023 (1 pixel)

<figure><img src="image_9.png" /></figure>

→ Estimation from 10 lines of CO and 2 lines of CI.

→ In contrast with [3], $G_0$ and $A_V$ estimated along with $\kappa$ and $P_{th}$.

→ Two regions of acceptable solutions identified.

<table><caption>Summary for main mode</caption><thead><tr><th></th><th>MLE [3]</th><th>MMSE</th><th>2.5%</th><th>97.5%</th></tr></thead><tbody><tr><td>κ</td><td>0.7</td><td>4.8</td><td>2.5</td><td>8.6</td></tr><tr><td>P<sub>th</sub> (×10<sup>8</sup>)</td><td>1</td><td>1.1</td><td>1.0</td><td>1.3</td></tr><tr><td>G<sub>0</sub> (×10<sup>3</sup>)</td><td>2.6</td><td>1.6</td><td>1.0</td><td>2.6</td></tr><tr><td>A<sub>V</sub></td><td>10</td><td>3.1</td><td>1.6</td><td>30.5</td></tr></tbody></table>

## References

[1] Pety et al., *The anatomy of the Orion B Giant Molecular Cloud: A local template for studies of nearby galaxies*, 2016

[2] Le Petit et al., *A model for atomic and molecular interstellar gas in the Meudon PDR code*, 2009

[3] Joblin et al., *Structure of photodissociation fronts in star-forming regions revealed by Herschel observations of high-J CO emission lines*, 2018

## Conclusions

### Achieved:

* ✓ Spatial regularization ⇒ more robust estimations of maps of parameters,

* ✓ Bayesian approach ⇒ more complete description than ponctual estimators.

### Future Work:

* → Application to other environments than PDR (shocks, dark clouds, etc.),

* → Code acceleration to scale to ~10⁶ pixel observations (e.g., Orion B IRAM Large Programm).