bayes_cn_hfs

A Bayesian CN Hyperfine Spectroscopy Model

bayes_cn_hfs implements models to infer the physics of the interstellar medium from hyperfine spectroscopy observations of CN as well as the carbon isotopic ratio from observations of CN and $^{13}$CN.

• Installation
  • Basic Installation
  • Development installation
• Notes on Physics & Radiative Transfer
• Models
  • Model Notes
  • CNModel
  • CNRatioModel
  • ordered
• Syntax & Examples
• Issues and Contributing
• License and Copyright

Installation

Basic Installation

Install with pip in a conda virtual environment:

conda create --name bayes_cn_hfs -c conda-forge pymc pip
conda activate bayes_cn_hfs
pip install bayes_cn_hfs

Development installation

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

Install in a conda virtual environment:

conda env create -f environment.yml
conda activate bayes_cn_hfs-dev
pip install -e .

Notes on Physics & Radiative Transfer

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

The transition optical depth and source function are taken from Magnum & Shirley (2015) section 2 and 3.

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. We do not assume the Rayleigh-Jeans limit; the source radiation temperature is predicted explicitly and can account for observation effects (i.e., the models can predict brightness temperature ($T_B$) or corrected antenna temperature ($T_A^*$).

Models can assume local thermodynamic equilibrium (LTE). Under this assumption, the excitation temperature of all transitions is fixed at the kinetic temperature of the cloud.

Non-LTE effects are modeled by considering the column densities of all states and self-consistently solving for the excitation temperature of each transition. We can assume a common excitation temperature (CTEX) across all transitions, or we can allow for hyperfine anomalies by allowing the state column densities to deviate from the LTE values.

For the CNRatioModel, we can (1) assume LTE for both CN and $^{13}$CN, (2) do not assume CTEX for CN, but assume CTEX for $^{13}$CN at the average CN excitation temperature, or (3) do not assume CTEX for either species, but assume $^{13}$CN hyperfine anomalies are similar to those of CN.

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

Models

The models provided by bayes_cn_hfs 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 bayes_cn_hfs.

Model Notes

1. Non-thermal broadening is only considered when prior_fwhm_nonthermal is non-zero. By default, non-thermal broadening is not considered.
2. The velocity of a cloud can be challenging to identify when spectral lines are narrow and widely separated. We overcome this limitation by modeling the line profiles as a "pseudo-Voight" profile, which is a linear combination of a Gaussian and Lorentzian profile. The parameter fwhm_L is a latent hyper-parameter (shared among all clouds) that characterizes the width of the Lorentzian part of the line profile. When fwhm_L is zero, the line is perfectly Gaussian. This parameter produces line profile wings that may not be physical but nonetheless enable the optimization algorithms (i.e, MCMC) to converge more reliably and efficiently. Model solutions with non-zero fwhm_L should be scrutinized carefully.
3. By default, the spectral RMS noise is not inferred, rather it is taken from the noise attribute of the passed SpecData datasets. If prior_rms is not None, then the spectral RMS noise of each dataset is inferred.
4. Hyperfine anomalies are treated as deviations from the LTE densities of each state. The value passed to prior_log10_Tex sets the average excitation temperature, log10_Tex_ul, and statistical weights of every transition, LTE_weights (i.e., the fraction of molecules in each state). Deviations from these weights are modeled as a Dirichlet distribution with a concentration parameter LTE_weights/LTE_precision, where LTE_precision is a cloud parameter that describes the scatter in state weights around the LTE values. A small LTE_precision implies a large concentration around LTE_weights such that the cloud is in LTE. A large LTE_precision value indicates deviations from LTE.
5. For the CNRatioModel, the $^{13}$CN excitation conditions are either (1) assumed constant across transitions with value log10_Tex_ul when assume_CTEX_13CN=True or (2) assumed to be similar to the excitation conditions of CN, with LTE deviations characterized by the same LTE_precision parameter.

CNModel

The basic model is CNModel, a general purpose model for modelling hyperfine spectroscopic observations of CN or $^{13}$CN. The model assumes that the emission can be explained by the radiative transfer of emission through a series of isothermal, homogeneous clouds as well as a polynomial spectral baseline. 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}
log10_N	Total column density across all upper and lower states	cm-2	$\log_{10}N \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[13.5, 1.0]
log10_Tkin	Kinetic temperature	K	$\log_{10}T_K \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[1.0, 0.5]
velocity	Velocity (same reference frame as data)	km s-1	$V \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.0, 10.0]
fwhm_nonthermal	Non-thermal FWHM line width	km s-1	$\Delta V_{\rm nt} \sim {\rm HalfNormal}(\sigma=p)$	0.0
log10_Tex	Average excitation temperature	K	$\log_{10}T_{{\rm ex}, ul} \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[1.0, 0.5]
LTE_precision	LTE precision	``	$1/a_{\rm LTE} \sim {\rm Gamma}(\alpha=1, \beta=p)$	100.0

Hyper Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
fwhm_L	Lorentzian FWHM line width	km s-1	$\Delta V_{L} \sim {\rm HalfNormal}(\sigma=p)$	1.0
rms	Spectral rms noise	K	${\rm rms} \sim {\rm HalfNormal}(\sigma=p)$	None
baseline_coeffs	Normalized polynomial baseline coefficients	``	$\beta_i \sim {\rm Normal}(\mu=0.0, \sigma=p_i)$	[1.0]*baseline_degree

CNRatioModel

bayes_cn_hfs also implements CNRatioModel, a model to infer the $^{12}{\rm C}/^{13}{\rm C}$ isotopic ratio from hyperfine observations of ${\rm CN}$ and $^{13}{\rm CN}$. Both species are assumed to be described by the same physical conditions (velocity, non-thermal line width, etc.) and different assumptions about the excitation conditions can be made.

Cloud Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
log10_N_12CN	Total ${\rm CN}$ column density across upper and lower states	cm-2	$\log_{10}N_{\rm CN} \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[13.5, 1.0]
ratio_12C_13C	$^{12}{\rm C}/^{13}{\rm C}$ abundance ratio by number	``	$^{12}{\rm C}/^{13}{\rm C} \sim {\rm Gamma}(\mu=p_0, \sigma=p_1)$	[75.0, 25.0]
log10_Tkin	Kinetic temperature	K	$\log_{10}T_K \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[1.0, 0.5]
velocity	Velocity (same reference frame as data)	km s-1	$V \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[0.0, 10.0]
fwhm_nonthermal	Non-thermal FWHM line width	km s-1	$\Delta V_{\rm nt} \sim {\rm HalfNormal}(\sigma=p)$	0.0
log10_Tex	Average excitation temperature	K	$\log_{10}T_{{\rm ex}, ul} \sim {\rm Normal}(\mu=p_0, \sigma=p_1)$	[1.0, 0.5]
LTE_precision	LTE precision	``	$1/a_{\rm LTE} \sim {\rm Gamma}(\alpha=1, \beta=p)$	100.0

Hyper Parameter variable	Parameter	Units	Prior, where ($p_0, p_1, \dots$) = prior_{variable}	Default prior_{variable}
fwhm_L	Lorentzian FWHM line width	km s-1	$\Delta V_{L} \sim {\rm HalfNormal}(\sigma=p)$	1.0
rms	Spectral rms noise	K	${\rm rms} \sim {\rm HalfNormal}(\sigma=p)$	None
baseline_coeffs	Normalized polynomial baseline coefficients	``	$\beta_i \sim {\rm Normal}(\mu=0.0, \sigma=p_i)$	[1.0]*baseline_degree

ordered

An additional parameter to set_priors for these models 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 (same reference frame as data)	km s-1	$V_i \sim p_0 + \sum_0^{i-1} V_i + {\rm Gamma}(\alpha=2, \beta=p_1)$	[0.0, 10.0]

Syntax & Examples

See the various tutorial notebooks under docs/source/notebooks. Tutorials and the full API are available here: https://bayes-cn-hfs.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

This code is licensed under MIT license (see LICENSE for details)