amoeba2

Automated Molecular Excitation Bayesian line-fitting Algorithm

amoeba2 is a Bayesian model of the 1612, 1665, 1667, and 1720 MHz hyperfine transitions of OH written in the bayes_spec spectral line modeling framework. amoeba2 is inspired by AMOEBA and Petzler et al. (2021).

Read below to get started, and check out the tutorials here: https://amoeba2.readthedocs.io

• Installation
  • Basic Installation
  • Development Installation
• Notes on Physics & Radiative Transfer
• Models
  • AbsorptionModel
    • mainline_pos_tau
  • EmissionAbsorptionModel
    • mainline_pos_tau
  • ordered
• Syntax & Examples
• Issues and Contributing
• License and Copyright

Installation

Basic Installation

Install with pip in a conda virtual environment:

conda create --name amoeba2 -c conda-forge pymc pytensor pip
conda activate amoeba2
pip install amoeba2

Development Installation

Alternatively, download and unpack the latest release, or fork the repository and contribute to the development of amoeba2!

Install in a conda virtual environment:

cd /path/to/amoeba2
conda env create -f environment.yml
conda activate amoeba2-dev
pip install -e .

Notes on Physics & Radiative Transfer

All models in amoeba2 apply the same physics and equations of radiative transfer.

The transition optical depth is taken from Magnum & Shirley (2015) equation 29. The excitation temperature is allowed to vary between transitions (a non-LTE assumption) and clouds. The excitation temperatures of the 1612, 1665, and 1667 MHz transitions are free, whereas that of the 1720 MHz transition is derived from the excitation temperature sum rule.

The radiative transfer is calculated explicitly assuming an off-source background temperature bg_temp (see below) similar to Magnum & Shirley (2015) equation 23. By default, the clouds are ordered from nearest to farthest, so optical depth effects (i.e., self-absorption) may be present.

Notably, since these are forward models, we do not make assumptions regarding the optical depth or the Rayleigh-Jeans limit. These effects are predicted by the model. There is one exception: the ordered argument, described below.

Models

The models provided by amoeba2 are implemented in the bayes_spec framework. bayes_spec assumes that the source of spectral line emission can be decomposed into a series of "clouds", each of which is defined by a set of model parameters. Here we define the models available in amoeba2.

AbsorptionModel

AbsorptionModel is a model that predicts the OH hyperfine absorption (1-exp(-tau)) spectra. The SpecData keys for this model must be "absorption_1612", "absorption_1665", "absorption_1667", and "absorption_1720". The following diagram demonstrates the relationship between the free parameters (empty ellipses), deterministic quantities (rectangles), model predictions (filled ellipses), and observations (filled, round rectangles). Many of the parameters are internally normalized (and thus have names like _norm). The subsequent tables describe the model parameters in more detail.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
tau	Line-center optical depth	``	$\tau \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.1, 0.1]
log10_depth	log10 line-of-sight depth	pc	$\log_{10} d \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.0, 0.25]
log10_Tkin	log10 kinetic temperature	K	$\log_{10} T_K \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[2.0, 1.0]
velocity	Velocity	km s-1	$V \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.0, 10.0]

Hyper Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
log10_nth_fwhm_1pc	Non-thermal broadening at 1 pc	km s-1	$\log_{10}\Delta V_{\rm 1 pc} \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.2, 0.1]
log10_depth_nth_fwhm_power	Non-thermal broadening vs. depth power law index	``	$\alpha \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.4, 0.1]
baseline_coeffs	Normalized polynomial baseline coefficients	``	$\beta_i \sim {\rm Normal}(mu=0, \sigma=p_i)$	[1.0]*(baseline_degree + 1)

mainline_pos_tau

An additional parameter to AbsorptionModel is mainline_pos_tau. If True, then the mainline (1665 MHz and 1667 MHz) optical depths are required to be positive by changing the prior distribution as follows.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
tau	Line-center optical depth	``	$\tau \sim {\rm HalfNormal}(\sigma=p_1)$	[0.1, 0.1]

EmissionAbsorptionModel

EmissionAbsorptionModel is a more physically motivated model that also predicts the brightness temperature spectra assuming a given background source brightness temperature (where bg_temp is in K and is supplied during model initialization; EmissionAbsorptionModel(bg_temp=3.77) is the default). The SpecData keys for this model must be "absorption_1612", "absorption_1665", "absorption_1667", "absorption_1720", "emission_1612", "emission_1665", "emission_1667", and "emission_1720". The following diagram demonstrates the model, and the subsequent table describe the additional model parameters.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
log10_N0	log10 column density in lowest energy state	cm-2	$\log_{10} N_0 \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[13.0, 1.0]
log_boltz_factor	log Boltzmann factor (-h*freq/(k*Tex))	``	$\ln B \sim {\rm Normal}(\mu=p_0, \sigma=p_1)	[-0.1, 0.1]
log10_depth	log10 line-of-sight depth	pc	$\log_{10} d \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.0, 0.25]
log10_Tkin	log10 kinetic temperature	K	$\log_{10} T_K \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[2.0, 1.0]
velocity	Velocity	km s-1	$V \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.0, 10.0]

Hyper Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
log10_nth_fwhm_1pc	Non-thermal broadening at 1 pc	km s-1	$\log_{10}\Delta V_{\rm 1 pc} \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.2, 0.1]
log10_depth_nth_fwhm_power	Non-thermal broadening vs. depth power law index	``	$\alpha \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.4, 0.1]
baseline_coeffs	Normalized polynomial baseline coefficients	``	$\beta_i \sim {\rm Normal}(mu=0, \sigma=p_i)$	[1.0]*(baseline_degree + 1)

mainline_pos_tau

An additional parameter to EmissionAbsorptionModel is mainline_pos_tau. If True, then the mainline (1665 MHz and 1667 MHz) optical depths are required to be positive by changing the prior distribution as follows.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
log_boltz_factor	log Boltzmann factor (h*freq/(k*Tex))	``	$\ln B \sim {\rm HalfNormal}(\sigma=p_1)	[0.1, 0.1]

ordered

An additional parameter to set_priors for both the AbsorptionModel and EmissionAbsorptionModel is ordered. By default, this parameter is False, in which case the order of the clouds is from nearest to farthest. Sampling from these models can be challenging due to the labeling degeneracy: if the order of clouds does not matter (i.e., the emission is optically thin), then each Markov chain could decide on a different, equally-valid order of clouds.

If we assume that the emission is optically thin, then we can set ordered=True, in which case the order of clouds is restricted to be increasing with velocity. This assumption can drastically improve sampling efficiency. When ordered=True, the velocity prior is defined differently:

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
velocity	Velocity	km s-1	$V_i \sim p_0 + \sum_0^{i-1} V_i + {\rm Gamma}(\alpha=2, \beta=1.0/p_1)$	[0.0, 1.0]

Syntax & Examples

See the various tutorial notebooks under docs/source/notebooks. Tutorials and the full API are available here: https://amoeba2.readthedocs.io.

Issues and Contributing

Anyone is welcome to submit issues or contribute to the development of this software via Github.

License and Copyright

Copyright(C) 2024 by Trey V. Wenger; [email protected]. This code is licensed under MIT license (see LICENSE for details)