@@ -24,18 +24,6 @@
% -----------
% - Binary engulfment reference: https://arxiv.org/pdf/1002.2216.pdf
% Outline:
% - Intro
% - Data (APOGEE description)
% - Methods
% - Re-measure velocities
% - The Joker / orbit sampling
% - Results
% - Catalog of companions (including samplings)
% - Differences between RGB / RC
% - Other interesting shit
% - Discussion
\include {gitstuff }
\include {preamble }
@@ -68,7 +56,7 @@
\begin {document }\sloppy\sloppypar\raggedbottom\frenchspacing % trust me
\title {Binary companions of red giant stars I: \\
catalog and companions that survive the common envelope phase }
catalog and companions that survive the common envelope}
\author [0000-0003-0872-7098]{Adrian~M.~Price-Whelan}
\affiliation {Department of Astrophysical Sciences,
@@ -147,101 +135,93 @@
\section {Introduction } \label {sec:intro }
Time-domain radial velocity measurements of stars fundamentally contain
information about massive companions: even with two observations of a single
star, a difference in the measured radial velocities implies the existence of at
least one companion.
Of course with just two observations, the orbital properties of the companion(s)
will be very poorly constrained.
Most prior searches for companions using radial velocity data have therefore
restricted their searches to only stars with `` sufficient''
orbital solution is unamibiguous under the ... noise model.
Many of the largest (in number of objects) stellar spectroscopic surveys observe
large samples of stars sparsely, non-uniformly, and imprecisely.
Even if period, generic orbital parameters are poorly known, limits on companion
mass or pericenter
If we want to use a spectroscopic survey to responsibly measure the
binary-star population---that is, if we want to get (say) the mass,
mass ratio, and period distribution of binary-star systems---we face
(at least) two challenges.
The first is how, given a small number of observations of a primary
star (a single-lined spectroscopic binary star), do we reliably obtain posterior
information about the binary-system properties?
After all, if there are only a few radial-velocity measurements made
per star, any observed radial-velocity values will be consistent with
many different combinations of period and amplitude (not to mention
eccentricity and argument of perihelion) for the primary orbit.
In general, the likelihood function, and the posterior pdf under any
reasonable prior pdf, will be highly multimodal in these kinds of
problems, and many of the modes will have comparable integrated
probability density.
We have solved this problem previously, though with limitations (to be
discussed more below), with \thejoker \ (CITE).
\thejoker \ is a Simple Monte Carlo rejection sampler that is
computationally expensive but probabilistically righteous: It delivers
independent (zero-auto-correlations) posterior pdf samples for
single-companion binary model parameters, given any number of
radial-velocity measurements.
The second challenge is how, given all these noisy, multi-modal
posterior pdfs over binary-system properties, many of which will
permit a wide range of qualitatively different system properties, and
many of which will be consistent with no companion at all, can we
constrain parameters of a model of the full binary-system population?
No histogram of best-fit values, nor stack of multi-modal posteriors,
would be anything like a good estimate of the true population
distribution.
We have also solved this problem previously, though in the exoplanet
domain, with hierarchical probabilistic models based on importance
sampling (CITE DFM, and also HOGG).
These hierarchical models derive posterior estimates of the population
parameters that accurately describe the population, given the noisy
information about the individual systems, provided that the noise
model is accurate---that is, provided that the likelihood functions
used in the individual-system inferences are themselves accurate.
Here we mash up these two solutions to these two challenges.
We deliver a catalog of binary-star systems with \RC \ primaries, and also
posterior information about the full population statistics, for at least
a toy model of that full population.
We use all the data---even the non-detections, marginal detections, and
badly multi-modal posterior pdfs---when we infer the population model;
that is, we don't arbitrarily cut to a catalog of `` clean''
For the study of single-line binaries, red-giant stars---and even
better red-clump stars---are ideal.
For one, because they are so luminous, they are unlikely (in general)
to have equally-bright companions, and therefore are well approximated
or fit as single-line objects.
For two, they are standard candles, or can be distance-calibrated.
For three, they have masses that can be estimated spectrally (making
use of dredged-up elements; CITE MARTIG and CITE NESS).
Time-domain radial-velocity measurements of stars contain information about
massive companions: even with two successive observations of a single star, a
difference in the measured radial velocities implies the existence of at least
one companion.
However, with few or imprecise radial-velocity measurements, the orbital
properties of the companion(s) are very poorly constrained.
Most prior searches for companions using survey RV data have therefore
restricted their searches to only sources with many epochs, so that the orbital
solution can be unambiguously determined.
The vast majority of spectroscopic targets with repeat observations in the
largest (by number of objects) stellar spectroscopic surveys are in the opposite
regime: targets are often observed just a few times with sparse, non-uniform
phase coverage.
% With such data, even if the period or other generic orbital parameters
% are very poorly constrained, limits on companion mass or pericentric distance
% of the companion may still XX.
If there are only a few radial-velocity measurements made per star, and the
companion spectrum is not observed, any measured radial-velocity values will be
consistent with many different combinations of (primary) orbital parameters
(period, amplitude, eccentricity, etc.).
To identify companions to the typical star observed in a spectroscopic survey,
we therefore face at least one major challenge: how, given a small number of
observations of a primary star, do we reliably obtain posterior information
about the binary-system properties?
In general, the likelihood function---and the posterior probability distribution
function (pdf) under any reasonable prior pdf---will be highly multimodal, and
many of the modes will have comparable integrated probability density.
For example, with just two radial-velocity measurements, a harmonic series of
period modes will exist in the likelihood function.
We have solved this problem previously, though with limitations (to be discussed
more below), with \thejoker \ (\citealt {Price -Whelan:2017 }).
\thejoker \ is a Monte Carlo rejection sampler that is computationally expensive
but probabilistically righteous:
It delivers independent posterior pdf samples for single-companion binary
orbital parameters, given any number of radial-velocity measurements.
Here we use \thejoker \ to generate posterior pdf samples for \todo {which?} stars
observed by the \apogee \ surveys (see \sectionname ~\ref {sec:data };
\citealt {Majewski:2015 }).
The \apogee \ surveys primarily target red-giant-branch (\RGB ) stars, which are
ideal for the study of single-line binary systems.
For one, because they are so luminous, they are unlikely (in general) to have
equally-bright companions, and their spectra are therefore well-approximated or
fit as single-line objects.
The subset of \RGB \ stars in the `` red clump'' \RC ) are even more powerful as
they are standard candles and have masses that can be estimated using
spectroscopy (using dredged-up elements; \citealt {Martig:2016 ,Ness:2016 }).
With primary-star mass estimates, the binary-orbit fitting will return
$ m \, \sin i$ mass-function
estimates .
For four , the \apogee \ pipelines, and also \thecannon \, produce
detailed abundance estimates for red giants and \RC \ stars.
If there are causal relationships between chemical abundances and
binary companions---and we expect there will be---these should become
visible here.
$ m_ 2 \, \sin i$ estimates of the
mass-function .
Additionally , the \apogee \ pipelines ( \citealt { Garcia - Perez:2016 }) and also
\thecannon \ ( \citealt { Ness:2015 }) produce detailed abundance estimates for \RGB \
and \RC \ stars.
If there are causal relationships between chemical abundances and binary
companions---as are expected---these should be measurable here.
HOGG: Why and how does \apogee \ rock it?
By making cuts on this library of posterior pdf samples (described in detail in
\sectionname ~\ref {sec:whatever }), we deliver a catalog of binary-star systems
from the \apogee \ survey with no cuts or constraints on data quality or volume.
We additionally highlight the subset of this catalog with \RC \ star primaries,
for which we can compute companion mass and pericentric distance.
\section {Data } \label {sec:data }
HOGG: What is up with delivering a catalog (it requires decision-making)?
\todo {HOGG: fill in this ish}
HOGG: Why and how does \apogee \ rock it?
\section {Data }
ASPCAP: \citealt {Garcia -Perez:2016 }
SDSSIV: \citealt {Blanton:2017 }
Overview of what \apogee \ is, and what data we are using.
How we updated the individual-visit radial velocity measurements.
What we did (if anything) with the missing visits.
\section {Method: Detection and orbit fitting }\label {sec:fitting }
\section {Methods }
\subsection {Re-measured radial velocities }\label {sec:rv-remeasure }
\todo {HOGG: fill in this ish}
\subsection {Detection and orbit fitting }\label {sec:fitting }
Our approach here is to proceed in two phases.
In the first phase, we obtain a posterior sampling in binary-system
@@ -278,146 +258,25 @@ \section{Method: Detection and orbit fitting}\label{sec:fitting}
Examples of outputs and etc.
\section {A catalog of red-clump binary systems }
\section {Results }
\subsection {Catalog / posterior samples whatever } \label {sec:full-catalog }
\todo {APW}
\subsection {A catalog of red-clump binary systems } \label {sec:rc-catalog }
\todo {APW}
Thresholding on what now?
Catalog.
Some highlights from this catalog.
\section {Method: Population inferences }\label {sec:popinference }
Now we want to take the individual-star fits---or samplings---that
we made in \sectionname ~\ref {sec:fitting } and, from these, produce
an inference of the properties of the whole \RC \ binary-star population.
Important necessary properties of this inference are the following:
\begin {description }
\item [utilize non-detections] Even stars which are consistent with
zero radial-velocity variation (no binary companion) are relevant to
a populations inference. This is for two reasons: The first is that
the small-mass and long-period ends of the binary population will
imprint only low-amplitude signals in the data. These signals are
present at low signal-to-noise, but not if we remove them because
they don't make some catalog or threshold cut. The second reason
these stars are important is that they constrain the fraction of
stars with no companions. Any populations-inference method we employ
must responsibly harvest the information in these non-detection
stars.
\item [marginalize out individual-star parameters] Every star---even
any star with a high-confidence companion detection---will have a
highly multi-modal likelihood function, with many qualitatively
different companion models that are nonetheless locally
optimal. This means that most posterior samplings (created in
\sectionname ~\ref {sec:fitting }) under our interim prior will show
multiple, qualitatively distinct modes in binary-system parameter
space. Any correct population inference will marginalize out these
extremely non-trivial distributions over individual-star
binary-system parameters.
\end {description }
We can meet these requirements---at least in principle---with a
hierarchical Bayesian inference (hierarchical probablistic model) of
the binary-star population.
In addition to the above requirements, we make the following assumptions
that restrict our attention to a well-defined model space:
\begin {description }
\item [correct samplings] accurate representations of posterior pdf
given prior and likelihood; correct likelihood function. We hereby
take on all the assumptions given in \sectionname ~\ref {sec:fitting },
about noise and kinematics, and so on.
\item [no trinaries] related to the above: no trinaries, quads, or anything
crazier.
\item [sufficent samplings] density and support of samplings
\item [independent stars] no relation between one star and another; each
drawn from the same population; no multiple populations.
\item [masses known] The mass of each primary \RC \ star is known, with
a correctly known posterior pdf, from prior work (CITE NESS).
\item [population parameterization] We adopt the following
parameterization of the population. In what follows, $ \hyperpars $
stand in for the full set of hyper-parameters (population
parameters).
\begin {itemize }
\item Each primary \RC \ star has a probability $ F$
$ 0 <F<1 $ $ F$
in $ \hyperpars $
\item For those with secondary companions, the log-period $ \ln P$
distribution $ p(\ln P\given\hyperpars )$
end-points, or
\begin {eqnarray }
\ln p(\ln P\given\hyperpars ) &=& g_0 + (g_1 - g_0)\, \frac {\ln P - x_0}{x_1 - x_0}
\quad \mbox {for $ x_0 <\ln P<x_1 $
\end {eqnarray }
where $ g_0 , g_1 , x_0 , x_1 $ $ \hyperpars $
Parameters $ g_0 , g_1 $
Parameters $ x_0 , x_1 $
\item Each primary star has a true mass $ M$
noisily) and the companion mass is potentially larger or
smaller (!). The log-mass-ratio $ \ln q$
$ p(\ln q\given\hyperpars )$
\begin {eqnarray }
\ln p(\ln q\given\hyperpars ) &=& h_0 + (h_1 - h_0)\, \frac {\ln q - y_0}{y_1 - y_0}
\quad \mbox {for $ y_0 <\ln q<y_1 $
\end {eqnarray }
where $ h_0 , h_1 , y_0 , y_1 $ $ \hyperpars $
Parameters $ h_0 , h_1 $
Parameters $ y_0 , y_1 $
\item The eccentricity distribution $ p(e\given P,\hyperpars )$
and permitted to vary with period:
\begin {eqnarray }
p(e\given P,\hyperpars ) &=& B(e\given \alpha (P),\beta (P))
\\
\alpha (P) &=& a_0 + (a_1 - a_0)\, \frac {P - x_0}{x_1 - x_0}
\\
\beta (P) &=& b_0 + (b_1 - b_0)\, \frac {P - x_0}{x_1 - x_0}
\quad ,
\end {eqnarray }
where $ B(x\given \alpha ,\beta )$
control parameters $ \alpha ,\beta $ $ a_0 , a_1 , b_0 , b_1 $
hyper-parameters in $ \hyperpars $
Parameters $ a_0 , a_1 $
at the log-period limits, and parameters $ b_0 , b_1 $
beta-distribution beta-parameter at the limits.
\item The inclination $ i$ $ p(i)$
argument-of-perihelion $ \omega $ $ p(\omega )$
orbital-phase $ \phi $
distribution $ p(\phi )$
distributions. They bring no hyper-parameters to $ \hyperpars $
\end {itemize }
\item [hyper-priors] whatevs
\end {description }
\subsection {Differences in companions of RC and RGB stars } \label {sec:rc-rgb }
The binary-system parameters for each individual \RC \ star were
inferred (in \sectionname ~\ref {sec:fitting }) with an interim prior that
is very simple and doesn't represent our true beliefs about the
binary-star population...
In a hierarchical Bayesian inference, we replace the binary-star
system priors used in \sectionname ~\ref {sec:fitting } with a parameterized
model for the binary-star population...
Because all we have is a sampling of each system under the interim prior
$ p(\pars\given H_0 )$
we have to re-weight or importance-sample the individual-star samplings
in this hierarchical inference... Cite Hogg et al; Cite DFM et al.
% The following needs checking for symbol conventions
\begin {eqnarray }
\ln p(y\given\hyperpars ,H_1) &=& \sum _{n=1}^N \ln p(y_n\given\hyperpars ,H_1)
\\
\ln p(y_n\given\hyperpars ,H_1) &=& \ln\left [\frac {1}{K_n}\, \sum _{k=1}^{K_n}\frac {p(\pars _{nk}\given\hyperpars ,H_1)}{p(\pars _{nk}\given H_0)}\right ] + Q_n
\quad ,
\end {eqnarray }
where $ y$ $ y_n$ $ n$
$ \hyperpars $
$ H_1 $
hyper-parameter-controlled hierarchical model,
$ H_0 $ \sectionname ~\ref {sec:fitting },
the $ \pars _{nk}$ $ n$ $ K_n$
the posterior under $ H_0 $
and $ Q_n$
hyper-parameters.
\section {The population of red-clump binary companions }
\todo {APW}
\section {Discussion }