Binary Stars Companion: Classification, Angular Scales, and Archive Reconnaissance
BinaryStarsCompanionClassificationAngularScalesAndArchiveReconnaissance
2026-10-04 04:50:21
2026-10-04 04:50:21
Example
Binary Stars as Physical Laboratories
binary stars
worked examples
visual binaries
spectroscopic binaries
eclipsing binaries
angular separation
projected separation
parallax
SIMBAD
VizieR
astroquery
data provenance
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\section*{Binary Stars Companion: Classification, Angular Scales, and Archive Reconnaissance}
This companion applies the introductory ideas from BIN01 in two ways:
\begin{enumerate}
\item complete conceptual and numerical worked problems,
\item a first reproducible public-archive exercise using SIMBAD and VizieR.
\end{enumerate}
The archive lab deliberately remains small. The goal is to learn how to identify an object, inspect provenance, and make a simple physical calculation before moving to larger survey datasets.
\begin{center}
\includegraphics[width=0.9\textwidth,keepaspectratio]{BIN01E1_fig01_archive_to_physics_strategy.png}
{\small \PMlinkescapetext{Figure 1. The introductory workflow moves from object identity to catalog values, provenance checks, simple physical calculations, and a saved frozen extract that can be compared with future live queries.}}
\end{center}
\section{Part I: problems}
\subsection*{Problem 1: physical pair or optical double}
Two stars appear 3 arcsec apart.
Star A has parallax
\begin{equation}
20.1\pm0.2\ {\rm mas}
\end{equation}
and proper motion
\begin{equation}
(55,-40)\ {\rm mas/yr}.
\end{equation}
Star B has parallax
\begin{equation}
20.3\pm0.3\ {\rm mas}
\end{equation}
and proper motion
\begin{equation}
(53,-41)\ {\rm mas/yr}.
\end{equation}
Does this prove the pair is gravitationally bound? What does it suggest?
\subsection*{Problem 2: projected separation}
A resolved pair has angular separation
\begin{equation}
\theta=0.80\ {\rm arcsec}
\end{equation}
at distance
\begin{equation}
d=25\ {\rm pc}.
\end{equation}
Find its projected physical separation.
\subsection*{Problem 3: parallax distance}
A catalog gives parallax
\begin{equation}
p=50\ {\rm mas}.
\end{equation}
Find the approximate distance in parsecs.
\subsection*{Problem 4: total dynamical mass}
A binary has physical relative semimajor axis
\begin{equation}
a=5\ {\rm AU}
\end{equation}
and period
\begin{equation}
P=10\ {\rm yr}.
\end{equation}
Estimate
\begin{equation}
M_1+M_2
\end{equation}
in solar masses.
\subsection*{Problem 5: center of mass mass ratio}
The barycentric semimajor axes are
\begin{align}
a_1&=2.4\ {\rm AU},\\
a_2&=1.2\ {\rm AU}.
\end{align}
Find
\begin{equation}
\frac{M_1}{M_2}.
\end{equation}
Which component is more massive?
\subsection*{Problem 6: SB2 mass ratio}
A double-lined spectroscopic binary has radial-velocity semi-amplitudes
\begin{align}
K_1&=60\ {\rm km/s},\\
K_2&=90\ {\rm km/s}.
\end{align}
Find
\begin{equation}
\frac{M_1}{M_2}.
\end{equation}
\subsection*{Problem 7: eclipse probability}
Estimate the geometric eclipse probability for a circular binary with
\begin{equation}
R_1+R_2=0.012\ {\rm AU}
\end{equation}
and
\begin{equation}
a=0.080\ {\rm AU}.
\end{equation}
\subsection*{Problem 8: observational classification}
Classify each detection signature. More than one classification may apply.
\begin{enumerate}
\item[(a)] Two images are resolved and their position angle changes over decades.
\item[(b)] One set of spectral lines shifts periodically with a 12-day period.
\item[(c)] Two sets of spectral lines move in opposite radial-velocity phase.
\item[(d)] The flux drops twice per orbit.
\item[(e)] The photocenter traces a periodic ellipse, but the companion is not directly resolved.
\end{enumerate}
\subsection*{Problem 9: SIMBAD frozen extract}
The file \PMlinktofile{BIN01E1_frozen_simbad_bright_binaries.csv}{BIN01E1_frozen_simbad_bright_binaries.csv}
\begin{equation}
\texttt{BIN01E1\_frozen\_simbad\_bright\_binaries.csv}
\end{equation}
contains a small educational extract of SIMBAD basic-data values for Sirius and Algol retrieved for this lesson.
Using the frozen parallax values:
\begin{enumerate}
\item estimate the distance to Sirius,
\item estimate the distance to Algol,
\item state why these values should not automatically be described as the best modern distances to the systems.
\end{enumerate}
\subsection*{Problem 10: angular resolution at two distances}
An instrument resolves pairs wider than
\begin{equation}
0.10\ {\rm arcsec}.
\end{equation}
What projected separation does this correspond to at:
\begin{enumerate}
\item[(a)] 10 pc,
\item[(b)] 100 pc?
\end{enumerate}
Explain the selection effect.
\subsection*{Problem 11: archive choice}
Choose SIMBAD or VizieR as the better starting point for each task:
\begin{enumerate}
\item[(a)] Resolve aliases for a named bright binary.
\item[(b)] Find published catalogs of Algol-type eclipsing binaries.
\item[(c)] Read basic literature-linked measurements for one named star.
\item[(d)] Retrieve rows from a specific published binary catalog for a statistical sample.
\end{enumerate}
\subsection*{Problem 12: provenance checklist}
A student reports:
\begin{quote}
The parallax of the star is 36.27 mas.
\end{quote}
List at least five additional pieces of information needed before the statement is fully reproducible and scientifically interpretable.
\section{Part II: complete solutions}
\subsection*{Solution 1}
The parallaxes agree within the uncertainties and the proper motions are very similar.
This is strong evidence that the pair may be physically associated.
It does not by itself prove that the pair is gravitationally bound. A complete argument could include relative orbital motion, radial velocities, a statistical chance-alignment analysis, or a dynamical solution.
\subsection*{Solution 2}
Use
\begin{equation}
s_\perp({\rm AU})
\approx
\theta({\rm arcsec})d({\rm pc}).
\end{equation}
Then
\begin{align}
s_\perp
&\approx
(0.80)(25)\\
&=20\ {\rm AU}.
\end{align}
Therefore
\begin{equation}
\boxed{s_\perp\approx20\ {\rm AU}.}
\end{equation}
This is a projected instantaneous separation, not automatically the semimajor axis.
\subsection*{Solution 3}
\begin{align}
d
&\approx
\frac{1000}{p({\rm mas})}\\
&=
\frac{1000}{50}\\
&=20\ {\rm pc}.
\end{align}
Thus
\begin{equation}
\boxed{d\approx20\ {\rm pc}.}
\end{equation}
\subsection*{Solution 4}
In AU, years, and solar masses,
\begin{align}
\frac{M_1+M_2}{M_\odot}
&\approx
\frac{a^3}{P^2}\\
&=
\frac{5^3}{10^2}\\
&=
\frac{125}{100}\\
&=1.25.
\end{align}
Therefore
\begin{equation}
\boxed{M_1+M_2\approx1.25M_\odot.}
\end{equation}
\subsection*{Solution 5}
From
\begin{equation}
M_1a_1=M_2a_2,
\end{equation}
\begin{equation}
\frac{M_1}{M_2}
=
\frac{a_2}{a_1}
=
\frac{1.2}{2.4}
=
0.50.
\end{equation}
Thus
\begin{equation}
\boxed{M_1/M_2=0.50.}
\end{equation}
Component 2 is more massive because it has the smaller barycentric orbit.
\subsection*{Solution 6}
For an SB2,
\begin{equation}
\frac{M_1}{M_2}
=
\frac{K_2}{K_1}.
\end{equation}
Therefore
\begin{equation}
\boxed{
\frac{M_1}{M_2}
=
\frac{90}{60}
=
1.5.
}
\end{equation}
Component 1 is more massive.
\subsection*{Solution 7}
Use the circular-orbit approximation
\begin{equation}
P_{\rm eclipse}
\sim
\frac{R_1+R_2}{a}.
\end{equation}
Then
\begin{align}
P_{\rm eclipse}
&\sim
\frac{0.012}{0.080}\\
&=0.15.
\end{align}
Therefore
\begin{equation}
\boxed{P_{\rm eclipse}\sim15\%.}
\end{equation}
\subsection*{Solution 8}
\begin{enumerate}
\item[(a)] Visual binary.
\item[(b)] Single-lined spectroscopic binary, SB1.
\item[(c)] Double-lined spectroscopic binary, SB2.
\item[(d)] Eclipsing binary.
\item[(e)] Astrometric binary.
\end{enumerate}
A real system can satisfy more than one of these at once.
\subsection*{Solution 9}
For Sirius, the frozen extract contains
\begin{equation}
p=379.21\ {\rm mas}.
\end{equation}
Thus
\begin{align}
d
&\approx
\frac{1000}{379.21}\\
&\approx
2.64\ {\rm pc}.
\end{align}
For Algol, the extract contains
\begin{equation}
p=36.27\ {\rm mas}.
\end{equation}
Thus
\begin{align}
d
&\approx
\frac{1000}{36.27}\\
&\approx
27.6\ {\rm pc}.
\end{align}
Therefore
\begin{equation}
\boxed{d_{\rm Sirius}\approx2.64\ {\rm pc}}
\end{equation}
and
\begin{equation}
\boxed{d_{\rm Algol}\approx27.6\ {\rm pc}.}
\end{equation}
These should not automatically be called the best modern distances because SIMBAD is a heterogeneous literature meta-compilation. The frozen values in this lesson come from the specific references displayed in the SIMBAD basic-data records. A precision distance analysis should identify the desired modern astrometric source, understand its treatment of multiplicity, and propagate the measurement uncertainties and systematics.
\subsection*{Solution 10}
At 10 pc,
\begin{equation}
\boxed{s_\perp\approx(0.10)(10)=1\ {\rm AU}.}
\end{equation}
At 100 pc,
\begin{equation}
\boxed{s_\perp\approx(0.10)(100)=10\ {\rm AU}.}
\end{equation}
The same angular-resolution limit removes progressively tighter physical binaries from the resolved sample as distance increases.
\subsection*{Solution 11}
\begin{enumerate}
\item[(a)] SIMBAD.
\item[(b)] VizieR.
\item[(c)] SIMBAD.
\item[(d)] VizieR.
\end{enumerate}
SIMBAD is optimized for object identity, cross-identification, linked measurements, and bibliography. VizieR is optimized for published catalogs and tabular datasets.
\subsection*{Solution 12}
Useful provenance includes:
\begin{itemize}
\item object identifier or component identifier,
\item database or archive,
\item catalog or originating reference,
\item uncertainty,
\item units,
\item measurement epoch when relevant,
\item quality flag or data-quality note,
\item retrieval date,
\item exact query or script.
\end{itemize}
\section{Part III: public-data lab}
\subsection{Lab A: inspect the frozen extract}
Open
\begin{equation}
\texttt{BIN01E1\_frozen\_simbad\_bright\_binaries.csv}.
\end{equation}
The extract contains the SIMBAD values used in Solution 9 plus basic object information needed to reproduce the educational calculation.
The extract is intentionally tiny. Its purpose is not to form a statistical sample.
Its purpose is to make the worked example reproducible if the live database changes.
\subsection{Lab B: rerun the live SIMBAD query}
Run \PMlinktofile{BIN01E1_simbad_vizier_reconnaissance.py}{ BIN01E1_simbad_vizier_reconnaissance.py}
\begin{equation}
\texttt{python BIN01E1\_simbad\_vizier\_reconnaissance.py}.
\end{equation}
The script queries Sirius and Algol and prints a compact table.
Compare the live values with the frozen CSV.
If they differ, do not immediately treat either one as an error. Instead ask:
\begin{itemize}
\item Was the SIMBAD preferred measurement updated?
\item Did the API return a different field or reference?
\item Is the live value from a newer data source?
\item Is the object identifier resolving to the same physical system or component?
\end{itemize}
\subsection{Lab C: use VizieR for catalog discovery}
The same script queries VizieR around Algol in the catalog
\begin{equation}
\texttt{V/115/algols},
\end{equation}
a published catalog of semi-detached eclipsing binaries available through VizieR.
The purpose at this stage is not to adopt every returned parameter.
Instead:
\begin{enumerate}
\item inspect the table name and catalog identifier,
\item print the returned column names,
\item identify units and descriptions,
\item find the original publication reference,
\item decide which columns could become inputs to a later physical model.
\end{enumerate}
\subsection{Lab D: provenance record}
Create a short text file or notebook cell recording:
\begin{itemize}
\item target name,
\item resolved identifier,
\item coordinates,
\item archive or catalog,
\item catalog version or identifier,
\item retrieval UTC time,
\item query radius,
\item relevant columns,
\item units,
\item source reference.
\end{itemize}
This provenance record is as important as the numerical result.
\section{Part IV: extension exercises}
\begin{enumerate}
\item Replace Algol with another bright binary and repeat the SIMBAD reconnaissance.
\item Search VizieR for a catalog of eclipsing binaries and compare its classification fields with SIMBAD object types.
\item Select two parallax values with uncertainties and propagate first-order distance uncertainties using $d=1000/p$ only in the high signal-to-noise regime.
\item Given a live parallax and an assumed angular separation, calculate a projected physical separation and clearly label it as projected rather than orbital semimajor axis.
\item Record one example where two archives provide different-looking values because they serve different scientific purposes.
\end{enumerate}
\section{Summary}
The first data-analysis lesson is not to download the largest possible table.
It is to establish the identity of the object, understand what a catalog value means, and preserve enough provenance to reproduce the calculation.
For introductory binary-star work:
\begin{equation}
\boxed{
\text{SIMBAD}
\rightarrow
\text{object reconnaissance}
}
\end{equation}
and
\begin{equation}
\boxed{
\text{VizieR}
\rightarrow
\text{published catalog data}.
}
\end{equation}
The numerical tools from BIN01 then convert angles, parallaxes, periods, and orbital scales into physically meaningful quantities.
\section*{References}
\begin{thebibliography}{9}
\bibitem{CarrollOstlie2017}
B. W. Carroll and D. A. Ostlie,
\textit{An Introduction to Modern Astrophysics}, 2nd ed.,
Cambridge University Press, 2017.
\bibitem{Wenger2000}
M. Wenger et al.,
The SIMBAD astronomical database,
\textit{Astronomy and Astrophysics Supplement Series}, 143, 9--22, 2000.
\bibitem{Ochsenbein2000}
F. Ochsenbein, P. Bauer, and J. Marcout,
The VizieR database of astronomical catalogues,
\textit{Astronomy and Astrophysics Supplement Series}, 143, 23--32, 2000.
\end{thebibliography}