luminosity-calibration

luminosity-calibration-tutorial

Tutorial on use of parallaxes in luminosity calibration; accompanies the Gaia DR2 paper on the use of parallaxes (Luri et al. 2018, A&A Special Issue for Gaia DR2).

Code and notebooks in this tutorial by Anthony G.A. Brown [email protected]

This tutorial expands on the luminosity inference example discussed in chapter 16 in 'Astrometry for Astrophysics', 2013, edited by W.F. van Altena (Cambridge University Press). It can also be seen as a simplified version of Hakwins et al. 2017.

Any errors in this tutorial are of course mine. Comments are welcome

Notebooks

The following python notebooks are included with this tutorial:

• Luminosity calibration with a distance prior. This is the main tutorial explaining how to infer from a set of measured parallaxes and apparent magnitudes of stars, their mean absolute magnitude and the standard deviation around the mean. In particular this tutorial is intended to demonstrate that negative and low-quality parallaxes can be used without any problem in the analysis. The four notebooks below provide supporting material.
• Simulated parallax surveys. This notebook explains in more detail how the simulated parallax survey was created which is used in the luminosity calibration tutorial. It also demonstrates the effect of selecting stars with high-quality parallaxes (introducing a bias on the inferred absolute magnitude) and the effect of the survey magnitude limit (including the mathematics of predicting the apparent magnitude distribution).
• Handling data truncation. A simplified demonstration of how to handle data truncation (i.e. selection functions) in a Bayesian analysis. It is meant to provide more insight into the treatment of the survey magnitude limit in the luminosity calibration problem.
• What's with the negative parallaxes? A simplified explanation on how negative parallaxes arise in astrometric surveys. The aim is to show why negative parallaxes are perfectly legitimate outcomes of the astrometric measurement process.
• Distribution of quantities calculated from parallax data. Provides complementary demonstrations to section 3 of Luri et al. (2018) as to why calculating distances or tangential velocities of stars from parallaxes through the naive inversion of the latter leads to problems. This is done by contrasting the naive inversion of parallaxes to the inference approach taken in Bailer-Jones (2015).

Prerequisites

This tutorial depends on the following Python packages (besides scipy, numpy, matplotlib):

• PyStan
• scikit-learn
• astropy
• corner.py
• PyGaia
• daft (Optional)

Python modules

The notebooks rely on the following python modules which are provided as separate code:

• Code to style the plots: plotstyles.py, distinct_colours.py.
• Version of astroML's convert_to_stdev which is robust to NaNs: mcmcplotting.py.
• Robust statistics (median etc): robuststats.py.
• Tools for compiling Stan models: stantools.py.
• Fit TGAS parallax error distribution: fit-tgas-parallax-errors.py.
• Parallax survey simulation modules, including plotting summary of survey: parallaxsurveys.py.
• Luminosity calibration code: luminosity_inference_distance_prior.py, which runs the Stan models for the luminosity inference problem. This code is intended for running the notebook examples from the command line. The parallax survey simulation parameters can be set with the command line arguments.

Stan models

The following Stan models are included for use in the notebooks or with the command line version of the luminosity inference problem:

• truncated_data.stan. Simplified demonstration of the treatment of truncated data.
• luminosity_inference_volume_complete_distance_prior.stan. Luminosity inference for a volume complete parallax survey.
• luminosity_inference_distance_prior.stan. Luminosity inference for a magnitude limited parallax survey.

TGAS parallax error model

The parallax survey simulation module contains a Hipparcos and a TGAS error model. For TGAS the distribution of parallax errors is assumed to be independent of magnitude. The probability density of the parallax error distribution is obtained as a kernel density estimate from all parallax errors in the TGAS catalogue. One can use the fit-tgas-parallax-errors.py Python code to try to fit this distribution with an inverse Gamma distribution. The necessary data can be extracted from the Gaia archive with the following query (store as "TGAS-allPlxErrorsVsGmag.fits", note the FITS format):

select parallax_error, phot_g_mean_mag from gaiadr1.tgas_source

The survey simulation code makes use directly of the KDE mentioned above.

Subfolder contents

pgm : Probabilistic graphical models for the tutorial