text:
\section{\label{sec:level1}INTRODUCTION \protect\\ }
Geometrically frustrated magnetic systems have been studied extensively over the last three decades \citep{Gardner2010}. The frustration between magnetic moments at corner sharing triangular or tetrahedral units can give rise to exotic ground states such as spin liquid \citep{Molavian2007}, ordered and disordered spin ice \citep{Mirebeau2005,Harris1997}, spin glass freezing \citep{Gingras1997} and order by disorder \citep{Savary2012} to name a few. The chromium spinels with formulae ACr$_2$X$_4$ are a family of such compounds where magnetic chromium atoms sit on corner sharing tetrahedra allowing various novel states, including zero energy excitation mode in spin liquid phase, heavy fermionic behavior, zero field and field induced transitions \citep{Lee2010}.

\begin{figure}[]
\includegraphics[width=85.5mm]{XRD.pdf
\caption{\label{synchrotron} The figure shows the fit of the synchrotron powder x-ray diffraction data taken at 300 K and 12 K.  There is a small shift in the intensities and peak positions between the two temperatures, but no different peaks occur at the different temperatures. Inset shows the zoomed view of the [200] peak, characteristic of the breathing pyrochlore structure, presents evidence for a small shift in 2$\theta$ between the two scans. }
\end{figure}
The presence of two different types of non-magnetic elements at the A site leads to the formation of compounds with formulae AA'B$_4$X$_8$ and may lead to A-site ordering, which can allow the formation of
larger and smaller tetrahedra. This occurs by displacement of the octahedral B-site cation and ordering of X anions \citep{Talanov2020}. Numerous mixed A-site pyrochlores have been synthesized in the past with some examples of the A-site ordering \citep[][]{Duda2008, Plumier1989, Plumier1977, Brasen1975, Wilkinson1976, Palmer1999, Lutz1989, Snyder2001, Snyder2000, Aminov2012}.   Since the type of interaction between nearest neighbor chromium atoms is distance sensitive, changing from strongly antiferromagnetic in oxide spinel to ferromagentic in selenium spinel \citep{Yaresko2008}, the magnetic Cr$^{+3}$ atoms at the edges of larger and smaller tetrahedra have different interaction (J and J'). This structure with alternating array of small and large tetrahera is called the Breathing Pyrochlore structure. Along with nearest neighbor interactions, further nearest neighbor interactions also play an important role in deciding the ground state \citep{Pokharel2020a, Ghosh2019}. This unique combination of magnetic frustration along with bond alternation in the breathing pyrochlores allow them to host exotic phases  including classical spin nematic transition in LiGa$_{0.95}$In$_{0.05}$Cr$_4$O$_8$ \citep{Wawrzynczak2017},  cluster frustration and negative thermal expansion in LiGaCr$_4$S$_8$ \citep{Pokharel2020a, Pokharel2018}, spin gap transition and magnetostructural transition in LiFeCr$_4$S$_8$ \citep{Saha2017}. These are a few examples of the diverse set of properties \citep{Reig-I-Plessis2021, Kalvius2008, Okamoto2015, Okamoto2013, Tanaka2018, Lee2021} exhibited by the breathing pyrochlores.

Breathing degrees of freedom can be introduced in a regular spinel structure by substitution of A-site cation with nonequivalent cations. This reduces the symmetry of the lattice, converting centrosymmetric F$\bar{d}$3m (227) structure into non-centrosymmetric F$\bar{4}$3m (216), which belongs to the same group. This allows additional reflections like (200), (420), (600) etc.  in the diffraction pattern, which are forbidden otherwise.

We study the CuAlCr$_4$S$_8$ system that was first synthesized by H. L. Pinch et al. \citep{Pinch1970} in 1970, reporting the A-site ordering in the normal pyrochlore structure. We have  successfully synthesized CuAlCr$_4$S$_8$ in breathing pyrochlore structure. In this structure, non-magnetic Cu$^{+1}$ and Al$^{+3}$ ions  occupy the tetrahedral sites, whereas the Cr$^{+3}$ ions populate the octahedral sites, forming an array of corner sharing tetrahedra in the pyrochlore lattice. The Cr$^{+3}$ ion in Cr breathing pyrochlores adopts spin 3/2 state due to half-filled t$_{2g}$ orbital, which leads to the formation of an ideal spin 3/2 system with strong geometric frustration.

The Heisenberg spin Hamiltonian for a breathing pyrochlore can be written as \[H = J\sum_{ij}S_iS_j\] here J $\epsilon$ \{J, J$'$, J$_{nnn}$\}.  J and J$'$ are exchange coupling constants for small and large tertaherdra, respectively.  J$_{nnn}$ is interaction parameter between next nearest neighboring spins. The breathing factor (B$_f$ = J/J$'$) can be an important parameter to tune different ground states.

\section{EXPERIMENTAL DETAILS}
The polycrystalline sample of CuAlCr$_4$S$_8$ was synthesized by conventional solid-state synthesis method. We used high purity copper, aluminium, chromium and sulfur powders from Alfa Aesar in stoichiometric ratios and ground them into a pellet, which was sealed in a quartz tube under 10 mTorr of argon pressure. We heated the sealed tube at 1000  $^{\circ}$ C in a box furnace for 48 hours. This process was repeated at 1100 $^{\circ}$ C once more with intermediate grinding to obtain the pure sample.

\begin{figure}[t]
\includegraphics[width=85.5mm]{LatticeParameters.pdf
\caption{\label{synchrotron} The figure shows the temperature dependence of the unit cell parameter obtained from the Rietveld refinement. The inset of the figure shows the breathing pyrochlore lattice formed by Cr atoms where the smaller and larger tetrahedra are distinguished through different colored bonds.}
\end{figure}

We performed preliminary room temperature powder X-ray diffraction measurements using a Panalytical X'pert diffraction instrument, which uses copper source with a wavelength of 1.54056 \AA. Rietveld refinement of the XRD pattern was performed using FullProf Suite. The sample contained 4.3 \% weakly ferromagnetic Cr$_3$S$_4$ \citep{Tazuke1979}, which does not interfere with our measurements except for magnetic susceptibility measurements in low fields. Magnetization measurements were performed using a Quantum Design SQUID with an RSO insert. We utilized a GC10/3 helium gas pressure cell from the Institute of High
Pressure Physics, Polish Academy of Sciences, inserted into
the MPMS to allow for magnetization measurements under
pressure up to 600 MPa, above 2 K. We performed specific heat measurements with a Quantum Design 9 T PPMS.
Synchrotron x-ray diffraction measurements were carried out on a powder sample of CuAlCr$_4$S$_8$ using the QM2 beamline at the Cornell High Energy Synchrotron Source (CHESS).  Measurements were performed using x-rays with an incident energy of 29.20 keV ($\lambda$ = 0.42460 \AA ~), and a Pilatus 6M area detector.  The sample was loaded into a borosilicate glass capillary (0.5 mm diameter), and cooled with a liquid helium cryostream.  Measurements were carried out from base temperature (T = 12 K) up to 300 K.  The zero and longitudinal field muon spin relaxation$\slash$rotation ($\mu$SR) measurements were performed at the M20 beamline at TRIUMF Vancouver using a He4 cryostate. The $\mu$SR data was fitted with the help of the musrfit software \citep{Suter2012}.
\begin{table}[
\caption{\label{tab:table1}%
The crystallographic parameters of CuAlCr$_4$S$_8$ determined by Rietveld refinement of the synchrotron XRD data at 300 K. The space group is F$\bar{4}$3m. The lattice parameters are a = b = c = 9.94112(4) \AA\ and B is the thermal factor.
}
\begin{ruledtabular}
\begin{tabular}{c c c c c}
\textrm{ }&
\textrm{ }&
\textrm{x=y=z} &
\textrm{B(\AA$^2$ )}& \textrm{Occupancy}\\
\colrule
 Cu1  &    4a &  0.75000(0)   &  0.451(58) &  0.994 \\
 Al1  &    4d &  0.00000(0) &  0.451(58) &  0.994 \\
 Al2  &    4a &  0.75000(0)  &  0.656(21) &  0.006 \\
 Cu2  &    4d &  0.00000(0)  &  0.656(21) &  0.006 \\
 Cr  &   16e &   0.37901(7)  &  0.431(11) &  1 \\
 S1   &   16e &  0.13212(5)  &  0.255(28) &  1 \\
 S2   &   16e &  0.61693(9)  &   0.431(27) &  1 \\

\end{tabular}
\end{ruledtabular}
\end{table}

\begin{table}[
\caption{\label{tab:table2}%
The crystallographic parameters of CuAlCr$_4$S$_8$ determined by Rietveld refinement of the synchrotron XRD data at 12 K. The space group is F$\bar{4}$3m. The lattice parameters are a = b = c = 9.92774(3) \AA\ and B is the thermal factor.
}
\begin{ruledtabular}
\begin{tabular}{c c c c c}
\textrm{ }&
\textrm{ }&
\textrm{x=y=z} &
\textrm{B(\AA$^2$ )}& \textrm{Occupancy}\\
\colrule
 Cu1  &    4a &  0.75000(0)   &  0.111(52) &  0.994 \\
 Al1  &    4d &  0.00000(0)  &  0.111(52) &  0.994 \\
 Al2  &    4a &  0.75000(0)  &  0.108(17) &  0.006 \\
 Cu2  &    4d &  0.00000(0)  &  0.108(17) &  0.006 \\
 Cr  &   16e &   0.37899(6)  &  0.130(10) &  1 \\
 S1   &   16e &  0.13217(5)  &  0.017(25) &  1 \\
 S2   &   16e &  0.61696(8)  &  0.129(24) &  1 \\

\end{tabular}
\end{ruledtabular}
\end{table}

\section{RESULTS AND DISCUSSION}
\paragraph*{Synchrotron X-ray diffraction}
The powder X-ray diffraction data taken on the polycrystalline samples of CuAlCr$_4$S$_8$ is shown in Figure \ref{synchrotron}. Rietveld refinement of the XRD data ensures good agreement with the breathing pyrochlore structure with space group F$\bar{4}$3m. This crystal structure carries magnetic Cr$^{+3}$ atoms at the corners of two unequal sized tetrahedra, unlike the normal pyrochlore (space group F$\bar{d}$3m) structure. The two Cr-Cr distances are 3.4020(14) and 3.6275(14) for this system. The ratio of the sizes of the two tetrahedra is called the breathing ratio, B$_r$, which is 1.0663(8) for CuAlCr$_4$S$_8$. This value is comparable to other sulfide breathing pyrochlores like CuInCr$_4$S$_8$ (1.06) \cite{Okamoto2018}, LiGaCr$_4$S$_8$ (1.077) \cite{Pokharel2018} and so on, however, it is much less then the rare earth compound Ba$_3$Yb$_2$Zn$_5$O$_{11}$ (1.90) \citep{Haku2016}, where the tetrahedra are decoupled.

\begin{figure}[b]
\includegraphics[width=85.5mm]{Magnetisation.pdf
\caption{\label{suscept}[Top]The dc magnetic susceptibility measurements on CuAlCr$_4$S$_8$. A sharp dip  at 20 K in the magnetic susceptibility, $\chi$, represents the antiferromagnetic transition. [Bottom] The inverse susceptibility measured in 0.1 T field and its Curie-Weiss fit in the temperature range 70 K-100 K. [Inset] The M-H at various temperatures around magnetic transition are plotted. }
\end{figure}

CuAlCr$_4$S$_8$ retains the room temperature crystal structure on cooling down to 12 K. The crystallographic parameters at 300 K and 12 K are listed in Tables \ref{tab:table1} and \ref{tab:table2}, respectively. The lattice parameters as a function of temperature, obtained from the Rietveld refinement, are shown in Figure: \ref{synchrotron}. It shows positive thermal expansion coefficient with a concave upward temperature dependence, unlike LiInCr$_4$S$_8$ and LiGaCr$_4$S$_8$, which show concave downward temperature dependence with positive thermal expansion and negative thermal expansion, respectively \citep{Okamoto2018}. This signifies the complex nature of the correlation between chemical composition and the structure of the breathing pyrochlore magnets.

\begin{figure}[b]
\includegraphics[width=85.5mm]{PressureSusceptibility.pdf
\caption{\label{PressureSusceptibility} The graph shows the magnetic susceptibility plots measured with 0.1 T applied field under different pressures. [Inset] Shows the plot of midpoints of the magnetic transition and Curie-Weiss temperature against applied pressure}
\end{figure}

\paragraph*{Magnetization}
The dc magnetic susceptibility data in Figure \ref{suscept} shows a sharp anomaly at 20 K. We performed low temperature synchrotron XRD measurements, which found no evidence of the structure distortions associated the magnetic susceptibility anomaly. The negative Curie-Weiss temperature obtained from Curie-Weiss fit of the magnetic susceptibility establishes the dominant antiferromagnetic interaction in this system.   The antiferromagnetic ordering temperature, T$_N$, of CuAlCr$_4$S$_8$ is comparable to the other antiferromagnetic samples of this sulfide family like CuInCr$_4$S$_8$ (28 K), and  LiGaCr$_4$S$_8$ (10.3 K)   \citep{Okamoto2018}.
The magnetic transition shifts to a higher temperature on application of hydro-static pressure as shown in the Figure \ref{PressureSusceptibility}. The shift in transition displays an almost linear dependence on pressure, with 600 MPa inducing a 3.2 K increase in the N\'eel temperature as shown in Figure \ref{PressureSusceptibility}[Inset].   This may be due to changing volume of the crystalline lattice induced by hydro-static pressure, which generates stronger interactions among the magnetic spins, allowing for higher magnetic ordering transition temperature. Magnetic susceptibility data were fitted with the Curie-Weiss equation in the range 70 K to 300 K as shown in Figure \ref{suscept} (b).
\begin{equation}
\label{CWeq}
\chi(T) = \frac{C}{T-\theta_{CW}}
\end{equation}
where $C = N_Ag^2\mu_B^{2}S(S+1)/3K_B$. The value of Curie-Weiss constant (C = 1.553(6)) obtained from the fits of the ambient pressure data corresponds to an effective magnetic moment ($\mu_{eff}$) of 3.525(7)*$\mu_B$, which is lower than the other sulfide breathing pyrochlores. This value of $\mu_{eff}$ yields a g = 1.83 for S = 3/2. The Curie-Weiss temperature ($\theta_{CW}$) for CuAlCr$_4$S$_8$ is -68.7(9) K. The smaller value of the N\'eel temperature in comparison to Weiss temperature signifies the suppressed antiferromagnetic (AFM) long-range interactions due to frustration. The frustration factor (T$_f$ = $\theta_{CW}/T_N$) for CuAlCr$_4$S$_8$ is 3.43(7). Curie-Weiss fit of the 600 MPa data exhibits the same Curie-Weiss constant (within the error limit) as ambient pressure data implying the unchanged effective moment. However, Curie-Weiss temperature  shifts to a lower temperature on application of pressure, the temperature dependence is plotted in \ref{PressureSusceptibility} [Inset]. Moreover, the frustration factor, the ratio of $\theta_{CW}$ and $T_N$, decreases by 2.9 \% at 600 MPa pressure.
\begin{figure}[b]
\includegraphics[width=85.5mm]{HeatCapacity.pdf
\caption{\label{HeatCap} [Top] The figure shows that the sample heat capacity and the slope of the magnetic susceptibility peak at the same temperature. [Bottom] The expanded view of zero field heat capacity to elucidate the near T$_N$ ranges. The total entropy (magnetic + non-magnetic) and the maximum entropy of a spin 3/2 system are shown here for comparison.}
\end{figure}
\paragraph*{Heat Capacity}
Heat capacity measurements performed on a polycrystalline sample of CuAlCr$_4$S$_8$ are shown in Figure: \ref{HeatCap}. There is a sharp peak in heat capacity data at 20 K,  which is largely independent of the applied field. The bottom panel of the figure depicts the total entropy as a function of temperature. Due to the absence of the non-magnetic analog of this sample, we could not subtract the lattice contribution from the entropy. The system is not able to recover the full entropy of a spin 3/2 magnetic system until 40 K, despite not correcting for the lattice contribution. The temperature derivative of magnetic susceptibility manifests the same behavior as specific heat near the magnetic transition Figure: \ref{HeatCap} [Top], which in principle looks antiferromagnetic; however, a detailed neutron diffraction experiment is desired to definitely determine the nature of this transition.

\begin{figure}[b]
\includegraphics[width=85.5mm]{ZFmuSR.pdf
\caption{\label{ZF_musr}  The zero field (ZF) $\mu$SR data taken on polycrystalline sample of CuAlCr$_4$S$_8$. (a) The asymmetry spectra taken at various temperatures above and below the T$_N$. The relaxation rates (b) $\lambda_1$ and (c) $\lambda_2$, and (d) stretched exponential factor, $\beta$  as obtained from the fits of the asymmetry are plotted against temperature. All three parameter shows an abrupt change of relaxation rates as we cool down the sample below its ordering temperature. The relaxation rate, $\lambda_2$, is too fast below 20 K and contained to be less than 200 $\mu s^{-1}$ and therefore poorly determined.}
\end{figure}

\paragraph*{Zero-field (ZF)/ longitudinal field (LF) muon spin relaxation/rotation ($\mu$SR)}
Figure \ref{ZF_musr} (a) shows zero field $\mu$SR spectra taken at temperatures between 2 K to 100 K. The sample relaxation rate starts increasing as we cool down from 100 K and major part of the signal relaxes out on going through the magnetic transition. We do not see any spontaneous oscillations in the signal in the ordered state, such as was seen with LiGaCr$_4$O$_8$ \citep{Lee2016} and ZnCr$_2$O$_4$ \citep{Marshall2002}. The fast relaxing part of the signal is fitted with an exponentially relaxing function, whereas the 1/3rd tail is fit with a stretched exponential function as shown in equation \ref{asym}
\begin{equation}
\label{asym}
A = \frac{A_0}{3}{e^{-(\lambda_1t)}}^{\beta}+\frac{2A_0}{3}e^{-\lambda_2t}
\end{equation}
 where A is total asymmetry and $A_0$ is  initial asymmetry. $\lambda_1$ and $\lambda_2$ are fast and slow relaxation rates, respectively. $\beta$ is the exponent of the stretched exponential factor fitting to the 1/3rd tail of the signal. In a polycrystalline sample, the spins are pointing in random directions, 1/3 component of the signal, also called 1/3rd tail, describes the asymmetry due to the spins parallel to initial muon spin while the remaining 2/3rd part describes the spins perpendicular to initial muon spin direction. The monotonically increasing relaxation rates upon cooling signifies the slowing down of thermal fluctuations. Upon cooling below 20 K, the asymmetry signal starts to decay rapidly. Thus, the relaxation rates rise sharply, indicating the quasi-static magnetic ordering at 20 K. The absence of the oscillating component below the transition  means that the field distribution is very broad. The fact that we have 1/3rd and 2/3rd components means that the entire sample undergoes spin freezing. The internal field below T$_N$ = 20 K is very large and as such we are unable to resolve the time dependence of the muon polarization, either to detect oscillations as would be expected in an ordered state, or a non-oscillating but rapidly relaxing signal as would be expected in a glassy state.

We also performed longitudinal field (LF) $\mu$SR measurements. In longitudinal field $\mu$SR experiment, a field is applied parallel to the initial muon spin and this field cants the magnetic moments along the magnetic field, making the perpendicular field smaller, and effective parallel field larger, thereby shifting the 1/3rd tail upward for the polycrystalline samples. This process brings some of the fast relaxing asymmetry signal in the $\mu$SR time window. Figure \ref{LFmusr} shows the asymmetry spectra of the sample under different applied fields. The longitudinal field of 4kG shifts the $\mu$SR spectra up, however, it is not enough to bring the entire fast relaxing component of the signal in the $\mu$SR time window. This implies that the characteristic internal fields are quasi-static and are of the order of few thousand Gauss.

\begin{figure}[t]
\includegraphics[width=85.5mm]{LFmuSR.pdf
\caption{\label{LFmusr}  Longitudinal field (LF) $\mu$SR data taken on polycrystalline sample of CuAlCr$_4$S$_8$ under different fields shows an upward shift of the 1/3rd tail.}
\end{figure}
\section{Conclusion}
We successfully synthesized a new breathing pyrochlore sample CuAlCr$_4$S$_8$, and investigated the structural and magnetic properties. The specific heat data exhibits a sharp peak at 20 K, which is supported by an abrupt dip in the magnetic susceptibility. We observe a linear increase in magnetic transition at a rate of 0.53 K/kbar of applied helium pressure. The close proximity of breathing ratio, and $\theta_{CW}$ of CuAlCr$_4$S$_8$ to CuInCr$_4$S$_8$ may allow it to have similar complex nearest neighbor magnetic interactions, switching from antiferromagnetic in smaller tetrahedra to ferromagnetic in larger tertahedra; however, \textit{ab initio} calculations and neutron scattering experiments are required for more accurate determination of the exchange interaction parameters and magnetic ground state, respectively.

\section{Acknowledgements}
We would like to thank Sarah Dunsiger for her assistance during the $\mu$SR measurements. Work at McMaster was supported by the Natural Sciences and Engineering Research Council of Canada.
Portions of this work are based upon research conducted at the Center for High Energy X-ray Sciences (CHEXS), which is supported by the National Science Foundation under award DMR-1829070. This research was undertaken thanks, in part, to funding from the Max Planck-UBC-UTokyo Center for Quantum Materials and the Canada First Research Excellence Fund, Quantum Materials and Future Technologies Program.

\nocite{*}


text:
\section{Introduction}\label{sec:intro}

In a binary or multiple system with small Doppler shifts compared to the spectral lines widths, the spectra of the individual stellar components must be separated to have access to the profiles and line fluxes of each stellar component. Several algorithms have been developed with that aim, such as the tomographic method of~\cite{BagnuoloJr1991}, the discrete Fourier transform of~\cite{Hadrava1995} and the shift-and-add method of \citet[hereafter MME98]{Marchenko1998}. The MME98 algorithm can be used either on its own, when the radial velocity curve is already known, or in conjunction with a method to determine the Doppler shifts as described by~\cite{Gonzalez2006}. It seems however that all these algorithms suffer from some drawback, either being quite demanding in computer time or producing spurious artefacts in the reconstructed line profiles. Reviews of disentangling methods have been written by~\cite{Hadrava2009} and~\cite{Pavlovski2010}.

Ideally a disentangling algorithm should be efficient (fast and easy to implement) and reliable (reproducing exactly the spectrum of the individual stellar components). To achieve this objective the best route is to choose an algorithm known for its versatility and ease of use, and improve its reliability. A good candidate is the shift-and-add method of MME98. It has been applied successfully over a hundred times: WR141~\citep{Marchenko1998}, $\gamma^2$ Vel~\citep{Eversberg1999}, GP Cep~\citep{Demers2002}, DQ Vel~\citep{Barria2013}, WR62a~\citep{Collado2013}, WR21a~\citep{Tramper2016}, WR148~\citep{Munoz2016}, SMC AB6~\citep{Shenar2018}, among others.

However it is well documented that this algorithm does not always reproduce correctly the wings of broad spectral lines such as the Balmer lines: see e.g. the cases of HD 150136~\citep{Mahy2012}, HD 194649~\citep{Mahy2015}, HD 149404~\citep{Raucq2016}; and can even generate artefacts: see for instance HD 38282~\citep{Schnurr2009}, 9~Sgr~\citep{Rauw2012} and HD 165052~\citep{Ferrero2013}. If these discrepancies could be resolved, this algorithm would meet our goals of efficiency and reliability.

In this paper we examine the causes of the deficiencies of the MME98 algorithm (\S~\ref{sec:analysis}) and propose a novel, highly reliable and efficient disentangling package (\S~\ref{sec:algorithm}). We then apply this new package to the binary 9~Sgr and compare our disentangled results to previous attempts (\S~\ref{sec:sgr}).

\section{Analysis of the shift-and-add Method}\label{sec:analysis}

When a disentangling algorithm is applied to observed spectra it is not possible to know if the individual spectra have been reproduced exactly, as there is no direct access to them. Hence our strategy will be to create two synthetic absorption profiles meant to represent the individual stellar components $A$ and $B$ of a binary system, shift them by their respective radial velocity and add them to simulate a composite spectrum. We then apply our algorithm to a series of such spectra and compare the output to the original profiles.

In a first experiment we choose simple geometric shapes for our synthetic absorption profiles, so the differences between original and disentangled profiles will stand out: a rectangle having a full width half maximum (FWHM) of $1000$ km\,s$^{-1}$ for star $A$ and a triangle with FWHM of $500$ km\,s$^{-1}$ for star $B$. This ensures that the two profiles have equal line fluxes (Figure~\ref{fig:StarA0B0}). Both profiles are sampled every $10$ km\,s$^{-1}$. The continuum is set to zero. We then construct ten composite spectra by applying to both stars radial velocities ranging from $-50$ to $50$ km\,s$^{-1}$, in equal steps of $10$ km\,s$^{-1}$ (omitting the value 0).

\begin{figure}
\centerline{\includegraphics[width=1\columnwidth]{StarA0B0}}
\caption{Synthetic line profiles of individual stellar components before blending. The rectangular shape is that of star $A$ (continuous line), the triangle of star $B$ (dotted line).\label{fig:StarA0B0}}
\end{figure}

The first step of the first iteration of the shift-and-add method aims to reinforce the features of star $A$ in the spectrum and to minimize those of star $B$. For this, each of the $S_{k}$ composite spectra taken at times $t_{k}|^{N}_{k=1}$, is corrected for the radial velocity of star $A$ ($\upsilon_{a,k}$). They are then co-added:

\begin{eqnarray}
A_{1}(\lambda)=\frac{1}{N}\sum^{N}_{k=1}\left[S_{k}(\lambda+\upsilon_{a,k})\right]
\label{eq:starA1}
\end{eqnarray}

As a result, the flux from the line of star $B$ is spread out over a range of wavelengths and diluted. This is taken as the first approximation of star $A$.

In the second step, this approximation is subtracted from each of the composite spectra, which are then corrected for the radial velocity of $B$ ($\upsilon_{b,k}$) and co-added to provide the first approximation of star $B$:

\begin{eqnarray}
B_{1}(\lambda)=\frac{1}{N}\sum^{N}_{k=1}\left[S_{k}(\lambda+\upsilon_{b,k})-A_{1}(\lambda-\upsilon_{a,k}+\upsilon_{b,k})\right]
\label{eq:starB1}
\end{eqnarray}

This process is repeated in the following iterations of the algorithm ($i=2,3,4,...,n$):

\begin{eqnarray}
A_{i}(\lambda)=\frac{1}{N}\sum^{N}_{k=1}\left[S_{k}(\lambda+\upsilon_{a,k})-B_{i-1}(\lambda-\upsilon_{b,k}+\upsilon_{a,k})\right]
\label{eq:starAi}
\end{eqnarray}

\begin{eqnarray}
B_{i}(\lambda)=\frac{1}{N}\sum^{N}_{k=1}\left[S_{k}(\lambda+\upsilon_{b,k})-A_{i}(\lambda-\upsilon_{a,k}+\upsilon_{b,k})\right]
\label{eq:starBi}
\end{eqnarray}

In Figure~\ref{fig:Star_1_20_40_60_80_100_200_2000} we can see that already at the first iteration ($i=1$) the output of the shift-and-add algorithm suffers from two kind of distortions with respect to the original, true profiles: an increase in the line flux of star $A$ and a decrease in that of star $B$; spurious absorption wings for star $A$ and spurious emission wings for star $B$. These wings are symmetrically located on each side of the absorption feature, both roughly at the same distance from the rest wavelength. This characteristic is conserved throughout the iterations. As illustrated in Figure~\ref{fig:Star_1_20_40_60_80_100_200_2000}, the flux of these wings increases during the first few iterations, and then slowly begins to decrease as it is spread out further and further along the velocity axis. After about 100 iterations the difference between two successive profiles becomes imperceptible. This leads to the erroneous impression that convergence has been reached.

\begin{figure*}
\centerline{\includegraphics[width=2\columnwidth]{Star0_1_25_50_75_100}}
\caption{Output of the MME98 algorithm
for the input shown in Figure~\ref{fig:StarA0B0}, after one, 25, 50, 75 and 100 iterations. The last two are very similar and could be interpreted wrongly as nearing  the end of the convergence process.
Left: Star $A$. Right: Star $B$.
\label{fig:Star_1_20_40_60_80_100_200_2000}}
\end{figure*}

We have repeated the tests described in the beginning of this section, now adding Gaussian noise with a signal-to-noise ratio of 50 in the composite spectra, and random uncertainties within 1, 2, and 3 $\sigma$ in the radial velocities. This changes neither the characteristics of the spurious wings nor the convergence speed of the algorithm.

The two effects (spurious wings and incorrect line fluxes) can readily be explained. Indeed the first step of the algorithm consists of shifting the original composite spectra and add them all. This is taken as the first approximation of the spectrum of star $A$. However it should be emphasised that this step spreads the flux from star $B$ over a range of wavelengths, but does not subtract it. It is clear then that, in the given wavelength window, the total flux from star $B$ (integrated over the whole profile) is assigned entirely to star $A$:

\begin{eqnarray}
F(A_1) = F(A_0) + F(B_0)
\end{eqnarray}
where the subindex $0$ refers to the original spectra and the subindex $1$ to the first approximation.

In the second step of the first iteration, the original composite spectra are shifted and centered around the rest wavelength of star $B$, then the first approximation of star $A$ is subtracted, with the aim of enhancing the profile of star $B$. However in this operation it is the total line flux from stars $A$ and $B$ that is actually subtracted, leaving the reconstructed line profile of star $B$ with a total flux equal to zero:

\begin{eqnarray}
 F(B_1) = F(A_0) + F(B_0) - F(A_1) = 0
\end{eqnarray}

At each iteration, the profile of the line core of star $A$ slightly improves and the spurious absorption wings are increasingly spread out on both sides (Figure~\ref{fig:Star_1_20_40_60_80_100_200_2000}). However the integrated flux of the line of star $A$ is unchanged, and remains equal to the total from both stars. For star $B$ too the profile becomes more similar to the true one, but its total line flux remains zero, as the flux in the absorption core is equal to that of the spurious emission wings.

After 100 iterations the algorithm seems to converge but the flux of the line of star $A$ is still much too strong and conspicuous wings are seen in the reconstructed spectra of both stars. Naturally, the stronger the flux in the line of star $B$ compared to star $A$, the stronger these effects. In this simple case, the correct profile with the correct flux is reconstructed after 2,000 iterations.

We also tested MME98 with simple Gaussian absorption profiles, meant to simulate roughly the He\,{\sc ii} 4542 \AA\ line of the individual components of the massive binary 9~Sgr. We assign to the line in star $B$ (the primary) half the line flux of star $A$ (Figure~\ref{fig:StarA0A5B0B5Gauss}), as suggested by Figure 3 of~\cite{Rauw2012}. We then created 51 blends of these synthetic profiles using the values of the radial velocities derived by~\cite{Rauw2012} for the 9~Sgr system. In this test, the number of wavelength steps is 2,000. Because of this increase (compared to 400 for our geometric profiles), the MME98 algorithm now requires 10,000 iterations to reconstruct the original profiles with the correct flux. This takes 35 hours on a computer with an 8-core processor at 4.20 GHz and 24 GB of RAM. And this is for a narrow spectral window covering only one line!

\begin{figure*}
\centerline{\includegraphics[width=2\columnwidth]{StarA0A5B0B5Gauss}}
\caption{When applied to Gaussian profiles (dotted black lines), the MME98 algorithm generates strong spurious wings and discrepant line fluxes.
The continuous blue and broken red lines correspond to 100 and 500 iterations respectively. Left panel, star $A$; right panel, star $B$.
\label{fig:StarA0A5B0B5Gauss}}
\end{figure*}

That algorithm is therefore prohibitively slow: waiting for full convergence is practically impossible in real cases, where the spectral windows include more than one spectral line and some 12,000 wavelengths steps (as in our Figure~\ref{fig:Sgr} below) or more. We can safely conclude that most authors using MME98 never let it reach full convergence and may even believe that convergence has already been reached earlier. This illusion of convergence is due to the fact that its speed decreases exponentially (Figure~\ref{fig:convergencia}). As a consequence, there is no guarantee to obtain a faithful reconstruction by simply requesting that the difference between two successive approximations drop below the noise level or other seemingly reasonable convergence criterion. Only in experiments with synthetic input spectra can we assess the reliability of the reconstructed spectra, as we can compare them with the original profiles.

\begin{figure*}
\centerline{\includegraphics[width=1.9\columnwidth]{convergencia}}
\caption{Root mean square of the difference between two successive reconstructions, over 10,000 iterations of the MME98 algorithm applied to composite spectra of the Gaussian profiles of Figure~\ref{fig:StarA0A5B0B5Gauss}.
\label{fig:convergencia}}
\end{figure*}

If the MME98 algorithm is stopped before full convergence, for example after 100 or 500 iterations, the disentangled profiles are similar to those obtained from the analysis of the observed spectra of 9~Sgr by~\cite{Rauw2012}, but markedly different from the original, true profiles (Figure~\ref{fig:StarA0A5B0B5Gauss}): compared with the input profiles, the reconstructed absorption line profile of star $A$ has a much larger flux and much broader absorption wings, whilst star $B$ exhibits a fainter absorption core, as well as prominent emission wings. None of these characteristics were present in the input profiles.

Adding noise to the spectra or random uncertainties to the radial velocities changes neither the speed of convergence of the MME98 algorithm nor the shape of the spurious wings. In fact, from our experiments with spectral windows of different sizes, we note that the speed of convergence depends only on the number of wavelength steps.

The limitations of the MME98 algorithm as remarked upon in the literature (see \S~\ref{sec:intro}) can be explained by these two effects: incorrect line fluxes and spurious wings. They are caused by its inability to redistribute the line flux between the two stellar components. These discrepancies are concerning, because exact line fluxes are needed to determine spectral types and, via fitting with model spectra, fundamental stellar parameters such as temperatures. Correct wings are required to characterise the surface gravity and the stellar wind.

\section{The QER20 package} \label{sec:algorithm}

We have created the QER20 package in response to the need for a highly efficient and reliable disentangling method. That is, a method that reconstructs line profiles with correct fluxes and without artefacts, and in a reasonable amount of computing time.

The disentangling methods that are most simple to use are those that work in the wavelength domain, since they do not require any space transformation. For this reason, our QER20 package is designed to work in the wavelength space, thus ensuring its efficiency.

To achieve the reliability of QER20, we have introduced a new perspective to this kind of disentangling methods: we treat the integrated spectral line flux as a free parameter. This new approach greatly accelerates the speed of convergence and thus guarantees that the output profiles are correctly reconstructed. It is worth mentioning that considering the line flux as a freely adjustable parameter during the whole process does not generate a loss of information. This is because at each iteration QER20 adjusts the reconstructed spectra to the original observations according to (~\ref{eq:starAi}) and (~\ref{eq:starBi}).

We conducted numerous experiments to probe different flux modification strategies. We describe hereafter the one that turns out to be the most efficient and reliable.

We consider first as a way of example, the case of absorption lines. Our code starts by executing the first iteration of the original MME98 method as formalised in (~\ref{eq:starA1}) and (~\ref{eq:starB1}). Next, QER20 looks for spurious emission wings in the reconstructed profile of star $B$. We do not attempt to work on the wings of the reconstructed profile of star $A$, because both its spurious wings and the real profile are in absorption, and it is not possible to separate them a priori.

To determine if emission wings in the spectrum of star $B$ are spurious, QER20 verifies that in this spectral window no emission is present in the observations. As a further test, QER20 evaluates whether these wings are symmetrically located with respect to the central wavelength of the absorption trough, since such symmetry is typical of artefacts produced by (~\ref{eq:starA1}) and (~\ref{eq:starB1}). When a spurious emission wing is identified, QER20 measures its flux and allows it to flow between the two stars during the second step (~\ref{eq:starB1}) and the two steps (~\ref{eq:starAi}) and (~\ref{eq:starBi}) of each iteration. As a result, the convergence speed dramatically accelerates and yet the correct profiles are produced, as will be shown below. This is because in our strategy, the line fluxes can be freely redistributed between the two stars.

If the lines in the observed composite spectra are in emission, our QER20 package proceeds analogously, with the spurious wings of star $B$ now in absorption. QER20 deals slightly differently if the line to disentangle is in absorption, but there is a true emission line at the position of one of the spurious emission wings, or vice-versa. (This is checked in the observed composite spectra). In that case, the flux correction strategy is applied only to the other spurious wing.

Figure~\ref{fig:StarA0A30B0B30} illustrates the case for rectangular and triangular profiles. The convergence is reached after less than 30 iterations and the true profiles are correctly reconstructed. This result contrasts sharply with the 2,000 iterations required by the MME algorithm. In the case of Gaussian profiles, our QER20 algorithm produces the correct profiles without the strong wings and with the correct line fluxes after 30 iterations (Figure~\ref{fig:StarA0A4B0B4Gauss}), while MME requires 10,000.

\begin{figure*}
\centerline{\includegraphics[width=2\columnwidth]{StarA0A30B0B30}}
\caption{Output of QER20 package after 30 iterations (continuous line) compared with the original profile (dotted line). Left panel: star $A$. Right panel: star $B$.
\label{fig:StarA0A30B0B30}}
\end{figure*}

\begin{figure*}
\centerline{\includegraphics[width=2\columnwidth]{StarA0A4B0B4Gauss}}
\caption{When applied to Gaussian profiles (dotted black lines), QER20 package produces the correct profiles after 30 iterations (continuous red lines). Left panel, star $A$; right panel, star $B$.
\label{fig:StarA0A4B0B4Gauss}}
\end{figure*}

As in \S~\ref{sec:analysis}, we also performed tests where Gaussian noise is added to the blends (of both geometric and Gaussian profiles) with a signal-to-noise ratio of 50, as well as random uncertainties in the radial velocities within 1, 2, and 3 $\sigma$. Under these testing conditions, the convergence speed is unchanged and the correct profiles and fluxes are obtained again, thus confirming the efficiency and the reliability of our approach.

We then apply QER20 to data simulating real conditions more closely: noisy spectra with radial velocities which are unevenly spaced and incommensurate to the wavelength step of the spectra.

To do so we take as input the output of our disentangled spectra of 9~Sgr based on observations by~\cite{Rauw2012}, as described in the next section (\S~\ref{sec:sgr}). We construct composite spectra adding random errors within 3 $\sigma$ to the radial velocities derived by~\cite{Rauw2012}, and we add Gaussian white noise with a SNR of 50 to simulate the observed spectra of this binary. The latter step is needed because in the disentangled spectra the noise has been significantly reduced due to the averaging inherent to the method.

After only 40 iterations our QER20 algorithm reproduces exactly the input spectra, whilst the MME98 algorithm distorts the broader lines (Balmer lines and several  He {\sc ii} lines), adding spurious absorption wings in $A$ and emission wings in $B$ (see Figure~\ref{fig:OriVsQEVsMME}).

\begin{figure*}
\centerline{\includegraphics[width=1.8\columnwidth]{OriVsQEVsMME}}
\caption{Comparison of the output of the MME98 after 100 iterations and QER20 after 40 iterations, for spectra simulating real conditions. Black: original spectra. Blue: spectra reconstructed with MME98. Red: spectra reconstructed with QER20. The spectra of star $B$ have been shifted vertically for clarity.
\label{fig:OriVsQEVsMME}}
\end{figure*}

These experiments show that our new disentangling package QER20 can reproduce exactly the line profiles and fluxes of the individual spectra, and does it in a very small number of iterations, thus fulfilling our aim of reliability and efficiency.

\section{Disentangling the Spectra of 9 Sgr}\label{sec:sgr}

Being now rather confident that our QER20 package produces disentangled spectra quite similar to those of the true individual stars, we apply it to the observations of 9~Sgr used in~\cite{Rauw2012}. This set of 51 spectra was taken with the FEROS spectrograph at La Silla Observatory with a spectral resolving power of 48\,000. The non-thermal radio emitter 9~Sgr is a massive binary consisting of two early-type main-sequence O stars orbiting each other with a period of 9.1 year and an eccentricity of 0.7 \citep{Rauw2012,Rauw2016}.

For the sake of comparison we first disentangle these spectra with the MME98 algorithm around the wavelength of the He\,{\sc ii} 4542 \AA\ line. The output after 100 iterations is shown in Figure~\ref{fig:StarA40B40Sgr} (dotted blue lines). The most striking features are the broad wings, in absorption in star A (left panel, dotted blue line) and in emission in star $B$ (right panel), as also seen in Figure 3 of \citet{Rauw2012}. The absence of emission in the raw data led these authors to suspect that these wings were artefacts. We confirm these suspicions since the spurious wings show high symmetry with respect to the central wavelength of the intense absorption lines.

\begin{figure*}\centering
\centerline{\includegraphics[width=2\columnwidth]{StarA40B40Sgr}}
\caption{Reconstruction of the He\,{\sc ii} 4542 \AA\ line of 9~Sgr with our QER20 package (continuous red lines) after 40 iterations. Left panel, star $A$; right panel, star $B$. For comparison, the output of the MME98 algorithm after 100 iterations
is shown in dotted blue lines.
\label{fig:StarA40B40Sgr}}
\end{figure*}

Our QER20 package converges after only 40 iterations. The results are also displayed in Figure~\ref{fig:StarA40B40Sgr} (continuous red lines). The spurious emission and absorption wings have disappeared. We also note that the absorption line flux has significantly changed. This quantity is now larger for star $B$ whilst the opposite was the case with the MME98 algorithm.

These differences between the output of that algorithm and our package are similar to those of our previous experiments. Because the comparison with the original profiles enabled us to confirm the exactness of the profiles reconstructed with our algorithm, we believe that we can also be fully confident in our new disentangled spectra of 9~Sgr.

The spectrum of 9~Sgr reconstructed with our QER20 package is shown in Figure~\ref{fig:Sgr} (covering most of the 4000 -- 5880 \AA\ range) and Figure~\ref{fig:HalphaHbeta} (H\,$\alpha$ and H\,$\beta$ lines). The output of the MME98 algorithm is also plotted for comparison.

Table~\ref{tab:flux} gives the fluxes of the broadest spectral lines (Balmer lines and some He~{\sc i} and He~{\sc ii} lines). They have been measured with respect to the combined continua, as the respective contribution of each stellar component to the continuum is unknown (because disentangling techniques do not yield the stellar flux ratio). Compared to the output of the MME98 algorithm, the fluxes of the tabulated lines obtained with our package are stronger in the reconstructed spectrum of the primary star (treated as star $B$), and weaker in the secondary (star $A$). The differences are quite significant for the broadest lines because they are severely blended and thus particularly difficult to disentangle with the MME98 algorithm.

We stress that our new package applies the flux corrections only to lines of the spectrum of star $B$ that are considered spurious, i.e. where pairs of roughly symmetric emission wings flank an absorption line. Other lines in its spectrum, such as the C\,{\sc iii} 5696 \AA\ line (Figure~\ref{fig:Sgr}), have been treated as in MME98.

\begin{figure*}
\centerline{\includegraphics[width=1.65\columnwidth]{Sgr}}
\caption{Disentangled spectra of 9~Sgr in the 4000 - 4360, 4450 - 4750, 5170 - 5470 and 5580 - 5880 \AA\ regions. The output of the MME98 algorithm after 100 iterations is represented with dotted blue lines, our package after 40 iterations with black continuous lines. The spectrum of the primary (star $B$) has been shifted vertically for clarity.\label{fig:Sgr}}
\end{figure*}

\begin{figure*}\centering
\centerline{\includegraphics[width=2\columnwidth]{HalphaHbeta}}
\caption{Disentangled spectra of 9~Sgr in the H\,$\alpha$, H\,$\beta$ regions. The output of the MME98 algorithm after 100 iterations is represented with dotted blue lines, our package after 40 iterations with black continuous lines. The spectrum of the primary (star $B$) has been shifted vertically for clarity. Several narrow nebular emission lines from the Lagoon Nebula are seen in the H\,$\alpha$ spectral region.
\label{fig:HalphaHbeta}}
\end{figure*}

\begin{center}
\begin{table*}[t]
\caption{Absorption line fluxes for 9~S\lowercase{gr}.
\label{tab:flux}}
\centering
\begin{tabular*}{500pt}{@{\extracolsep\fill}lcD{.}{.}{3}D{.}{.}{3}cD{.}{.}{3}D{.}{.}{3}D{.}{.}{3}@{\extracolsep\fill}}
\toprule
\multicolumn{2}{c}{}
			& \multicolumn{3}{@{}c@{}}{\textbf{Secondary}}
			& \multicolumn{3}{@{}c@{}}{\textbf{Primary}}\\
  \cmidrule(r){3-5}\cmidrule(l){6-8}
  \textbf{$\lambda$} & \textbf{ID} & \multicolumn{1}{@{}l@{}}{\textbf{MME98}} & \multicolumn{1}{@{}l@{}}{\textbf{This work}} & \textbf{$\Delta$ (\%)} & \multicolumn{1}{@{}l@{}}{\textbf{MME98}} & \multicolumn{1}{@{}l@{}}{\textbf{This work}} & \multicolumn{1}{@{}l@{}}{\textbf{$\Delta$ (\%)}} \\
			\midrule
 4026  & He {\sc i}    & 56.71   & 29.43  & -48.1   & 18.9   & 26.1   & 38.2\\
 4102  & H\,$\delta$   & 257.11  & 169.78 & -34.0   & 71.0   & 98.74  & 39.1\\
 4200  & He {\sc ii}   & 72.41   & 37.26  & -48.5   & 27.25  & 36.62  & 34.4\\
 4341  & H\,$\gamma$   & 306.10  & 186.34 & -39.1   & 80.48  & 114.0  & 47.1\\
 4471  & He {\sc i}    & 14.29   & 13.79  & -3.5    & 6.50   & 6.89   & 6.0 \\
 4542  & He {\sc ii}   & 99.27   & 45.33  & -54.3   & 43.83  & 58.48  & 33.4\\
 4686	 & He {\sc ii}   & 84.82   & 52.31  & -38.3   & 46.84  & 57.85  & 23.5\\
 4862  & H\,$\beta$    & 336.07  & 210.10 & -37.5   & 103.25 & 128.74 & 24.7\\
 5412  & He {\sc ii}   & 149.14  & 77.14  & -48.3   & 67.16  & 84.73  & 26.2\\
 5876	 & He {\sc i}    & 31.89   & 26.69  & -16.3   & 14.63  & 19.86  & 35.7\\
 6563  & H\,$\alpha$   & 282.33  & 213.59 & -24.4   & 111.55 & 116.31 & 4.3\\
\bottomrule
\end{tabular*}
\end{table*}
\end{center}

\section{Discussion and conclusions}\label{sec:conclusions}

It is quite common for massive stars to have their spectral lines significantly broadened by fast rotation and strong winds. It can then be challenging to disentangle their spectra if they belong to binary systems with low orbital velocity. We have started a series of studies with the aim to better understand the behavior of disentangling techniques. To do so, we work with blends of known individual spectra, to which the output of the algorithm can be compared.

In this work we focus on the efficiency and reliability of the disentangling process. We have shown that the convergence of the MME98 method is in some cases too slow, and that it does not always reproduce the line fluxes and the profiles correctly. This leads to potentially misleading conclusions about the stellar type, temperature, gravity, wind, and other physical properties of the stars in the binary systems under study. Once we identified the causes of the deficiencies of this method, we were able to design a reliable and efficient method, implemented in the QER20 package, which reconstructs the individual spectra without artefacts and without error in the line fluxes, and does so in a very short computational time.

As mentioned earlier, the speed of convergence of the MME98 algorithm decreases exponentially. In very simple cases and in conditions far from real (such as our experiments with rectangular and triangular profiles), the MME98 method eventually converges to correct profiles after thousands of iterations. However, in real conditions the convergence can be so slow as to become computationally out of reach. Unfortunately, if full convergence is not reached, the MME98 output yields incorrect fluxes and incorrect wing profiles. This raises concerns about the use of the MME algorithm, either in its original form or as modified by~\cite{Gonzalez2006}.

The disentangled spectra of the stellar components of the O+O binary 9 Sgr that we obtained with the QER20 package are sufficiently different from those reconstructed by~\cite{Rauw2012} with the method of MME98 to warrant a new appraisal of their spectral types. These authors used a qualitative method of classification, namely the visual comparison with published spectra of single O stars, which has the advantage of involving many spectral lines. They assigned a subtype O\,3.5V ((f +)) to the primary (our star $B$) and O\,5-5.5V ((f)) to the secondary (star $A$).

The quantitative classification criteria of~\cite{Conti1971} are based on the ratio of the flux of only two lines, the He~{\sc i} 4471 \AA\ and He~{\sc ii} 4542 \AA\ lines \citep[see also][]{Martins2018}. Applied to the output of the MME98 algorithm, these criteria yield O\,4V ((f +)) for the primary and O4\,V ((f )) for the secondary. However when we substitute to these line fluxes the output of our QER20 package, the subtypes we derive  are O\,3V ((f +)) and O\,5V ((f)), respectively, and are in better agreement with the qualitative classification of~\cite{Rauw2012}.

In the case of 9~Sgr, the effect of the deficiencies of the MME98 algorithm on the classification is not severe, because the two stellar components are of similar subtype, their spectral lines have roughly the same flux and thus their ratio is not much affected. However, when a binary is composed of massive stars with different temperatures, the errors will increase. To check this, we have run tests with composite spectra of Gaussian profiles for different cases: O3+O7, O3+O9, O5+O9, etc. We note that the spectral classification of the MME98 output can be off by up to two subtypes, compared to the synthetic input. This is turn will distort the interpretation of the spectra and of the nature of the binary system, as this could mean an error of some 5000~K for the temperature and of 20 $M_ \odot$ for the masses estimated from the spectral types ~\citep{Martins2005}. In contrast, when the QER20 package is applied to those same composite spectra, there are no differences between the classification of the disentangled spectra and the synthetic input.

We also observe that the MME98 algorithm tends to shift the subtype of one star toward that of the other. Thus it is difficult to distinguish the case of two similar stellar components from the case where they are made similar by the disentangling method.

Our strategy of taking blends of known individual spectra as input to the codes has enabled us to rigorously analyse disentangling techniques, uncover errors and propose remedies. The success of our new QER20 package encourages us to extend the present work and to study other effects, such as that of variable line profiles.

\section*{Acknowledgments}

We are grateful for financial support from the  \fundingAgency{CONACyT-DFG} bilateral project No. \fundingNumber{278156} and from the University of Guanajuato under the Apoyo a Profesores program. We thank the referee for constructive comments that have helped to improve the paper. This research has made use of NASA's Astrophysics Data System Bibliographic Services.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

This paper deals with the combined analysis of two data sources: First the AlgorithmWatch data-set on search results for political queries during the German Federal Elections in 2017 \cite{zweig2018} and second data from a long-term data crawl of search engine query suggestions. This enables us to examine result lists and query suggestions for the same set of politics related queries over a 2 month period.

Query suggestions comprise a set of interactive techniques to assist users with the formulation of their query. As Kelly et al. \cite{kelly_comparison_2009} point out they seem particularly important in cases where users are searching for topics on which they have little knowledge or familiarity and in which they are in a so called ``anomalous state of knowledge'' \cite{belkin_ask_1982}. In the context of political opinion formation little is known about the influence of query suggestions in driving the users information behaviour. Regarding search results, the ``search engine manipulation effect'' was measured in several experimental setups \cite{epstein_search_2015}. It claims that biased search rankings can shift the voting preferences of undecided voters. However, the paper is criticized for presenting exaggerated effect sizes \cite{zweig2017}.

In this paper we (1) present a data set with query suggestions crawled from different web search engines to complement the AlgorithmWatch data. As a first approach we (2) compare the aggregated \emph{ranking stability} of search results and query suggestions for $16$ search queries over a $2$ month observation period. Here, stability refers to the degree of similarity of successive lists over time. We suspect that external events both influence the composition of query suggestions and search results and the goal is to better understand how these two search engine features interact with each other through the search behaviour of the users. \footnote{E.g. an external event such as the marriage of a politician, may change his or her result set since websites dealing with this event become more prominent. In addition, search engine users will possibly also search for the event in connection with the politician's name which influences the composition of the query suggestions.} In a last step, we (3) shortly discuss further research topics.

\section{Data Sets}

Our data-set consists of query suggestions (e.g. autocompletions) to complement the search engine results from the AlgorithmWatch data. For most queries an exact match can be obtained.\footnote{\label{fn:missing}The term ``cdu" is missing in our data-set. The search-terms ``Bündniss90/Die Grünen" and ``die linke" from the AlgorithmWatch data are compared to the search suggestions for the queries ``grüne'' and ``dielinke''.} For both data-sets we consider queries submitted between August 4st and September 30th, 2017. We programmed a web client which made calls to Google's query suggestion API \footnote{\url{http://clients1.google.de/complete/search}}. The crawling ran usually twice a day around 05:00 and 17:00 CET and produced an outcome of $401$ unique suggestion terms during the observation period. A sample of the data set can be seen in table \ref{tab:dataset}. The data is published under a Creative Commons licence \cite{bonart2018}.

\begin{table}[ht]
\centering
\begin{tabular}{ccccc}
  \toprule
 source	& queryterm	& date	& suggestterm	& position\\
 \midrule
google & Alexander Gauland& 2017-08-04 05:30:51 & twitter & 0\\
google & Alexander Gauland& 2017-08-04 05:30:51 & itate & 1\\
google & Alexander Gauland& 2017-08-04 05:30:51 & kontakt & 2\\
google & Alexander Gauland& 2017-08-04 05:30:51 & dorothea gauland & 3\\
google & Alexander Gauland& 2017-08-04 05:30:51 & boateng & 4\\
google & Alexander Gauland& 2017-08-04 05:30:51 & krawatte & 5\\
google & Alexander Gauland& 2017-08-04 05:30:51 & carola hein & 6\\
google & Alexander Gauland& 2017-08-04 05:30:51 & ehefrau & 7\\
google & Alexander Gauland& 2017-08-04 05:30:51 & youtube & 8\\
google & Alexander Gauland& 2017-08-04T05:30:51 & islam & 9\\
$\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ & $\cdots$ \\
\bottomrule
\end{tabular}
\caption{Illustration of the query suggestions data-set.}
\label{tab:dataset}
\end{table}

For the AlgorithmWatch data we restrict the analysis to organic search results for requests submitted from within Germany using a German keyboard layout. We also cleaned the data from errors as described in \cite{zweig2018}. This results in \numprint{896333} unique result lists from \numprint{1441430} single requests.

While the comparison of different rankings from query suggestions is straightforward this is not true for the search results. Although there is only a small effect of search personalization in the data the ranking usually slightly differs between search engine users. However, the public data-set does not provide any persistent user/plugin ids which prevents the comparison of result lists on an individual level over time. As a workaround, for each time-point we construct an aggregated ``typical'' search result list. This is done by calculating the average position of each URL which is present in more than a third of the requests for this time-point and query.

\section{First insights and discussion}

Typically, result and suggestion lists of search engines are incomplete, top-weighted and indefinite. The importance of the items decreases with the depth which justifies a truncation of the list. As a measure of stability we therefore rely on the so called rank-biased overlap (RBO) statistic. It measures the similarity of two indefinite possibly non-conjoint lists and properly incorporates the characteristics of indefinite rankings \cite{webber2010}.

The RBO statistic can take values between $0$ indicating no similarity and $1$ for similar lists. The degree of top-weightiness can be controlled by a parameter $p$. For $p \rightarrow 0$ the first item in both lists are assigned all the weight while for $p \rightarrow 1$ all list items become equally important. For the analysis we choose $p = 0.85$ as this puts almost all the weight ($93\%$) on the first $10$ list items and results in an expected evaluation depth of $6.7$ list items. For each query and time-point we compare the ordered lists of aggregated search results and query suggestions with the preceding list (figure \ref{fig:rbo1}) and with the list from the earliest time-point in the sample (figure \ref{fig:rbo2}). To smooth the time series, a moving average with a window of three days is applied to the RBO values.

\begin{figure}[p]
 \figuretitle{Stability of search results and suggestions over time: Successive comparison}
\centering
\includegraphics[width=\textwidth]{rbo_moving.pdf}
\caption{For each keyword the search results (6 per day) and suggestions (2 per day) from each time-point are compared to the result from the \emph{previous} time-point. The rank biased overlap measure with $p = 0.85$ is used. A moving average with a window size of $n = 18$ and $n = 6$ (e.g. 3 days) is applied to smooth the time series. The horizontal gray line marks a RBO of $0.5$. \footnotesize{*see footnote \ref{fn:missing}.}}
\label{fig:rbo1}
\end{figure}

\begin{figure}[p]
 \figuretitle{Stability of search results and suggestions over time: Fixed comparison}
\centering
\includegraphics[width=\textwidth]{rbo_fixed.pdf}
\caption{For each keyword the search results and suggestions from each time-point are compared to the result from the \emph{earliest} time-point. The RBO measure with $p = 0.85$ is used. A moving average with a window size of 3 days is applied to smooth the time series. The horizontal gray line marks a RBO of $0.5$. \footnotesize{*see footnote \ref{fn:missing}.}}
\label{fig:rbo2}
\end{figure}

The figures show that the degree of stability for the query suggestions and the result sets mainly follow broad similar trends. Obviously, the composition of the lists changes during the two month observation period but the stability seems relatively high: Comparing the last result list with the first one yields a minimum RBO of around $0.5$ for most queries. For some queries, the query suggestions seem to be less stable.

This initial analysis only gives a broad overview over the general trend and can shed light to some interesting patterns in the data (e.g. the negative peak in figure \ref{fig:rbo1} for both search results and query suggestions for the term ``CSU'' around the date of the federal election). To better understand the role of specific events and how search results and query suggestions influence each other a more detailed look is required. For this it is important to study the actual, possibly dynamic, \emph{semantic} similarities between the data-sets over time.  It may be beneficial to include the top news headlines which appear in the organic search result lists to capture the influence of specific important events on the composition of result and suggestion lists.

In a political context it can also be interesting to ask whether query suggestions do lead to new and more diverse information sources. It could be possible that search suggestions for a specific term mainly resemble the topics and trending events listed on the term's result page. If this is the case, the effect of query suggestions on a user's process of political opinion formation could be neglected regardless whether he or she makes use of this feature or not.

\section*{Acknowledgments}
\label{sec:Acknowledgments}

This research was supported by the Digital Society research program funded by the Ministry of Culture and Science of the German State of North Rhine-Westphalia.

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\bibliographystyle{splncs03}