# Stellar and accretion disk parameters of the close binary HD 50526

Rosales, J. A. Department of Physics, North-West University, Private Bag
X2046, Mmabatho 2735, South Africa.  Departamento de Astronomía, Universidad
de Concepción, Casilla 160-C, Concepción, Chile.  Main Astronomical
Observatory, National Academy of Sciences of Ukraine, 27 Akademika Zabolotnoho
St, 03680 Kyiv, Ukraine.  Mennickent, R. E  Departamento de Astronomía,
Universidad de Concepción, Casilla 160-C, Concepción, Chile.  Djurašević, G
Astronomical Observatory, Volgina 7, 11060 Belgrade 38, Serbia.  Isaac Newton
Institute of Chile, Yugoslavia Branch, 11060 Belgrade, Serbia.  Schleicher, D.
R. G  Departamento de Astronomía, Universidad de Concepción, Casilla 160-C,
Concepción, Chile.  Zharikov, S  Instituto de Astronomía, Universidad Nacional
Autónoma de México, Apartado Postal 877, Ensenada, Baja California, 22800
México.  Araya, I  Centro de Investigación DAiTA Lab, Facultad de Estudios
Interdisciplinarios, Universidad Mayor, Chile  Celedón, L  Instituto de Física
y Astronomía, Facultad de Ciencias, Universidad de Valparaíso, Chile.  Curé, M
Instituto de Física y Astronomía, Facultad de Ciencias, Universidad de
Valparaíso, Chile.

(Received XX; Revised YY; Accepted ZZ)

###### Abstract

We present a photometric and spectroscopic study of HD 50526, an ellipsoidal
binary member of the group Double Periodic Variable stars. Performing data-
mining in photometric surveys and conducting new spectroscopic observations
with several spectrographs during 2008 to 2015, we obtained orbital and
stellar parameters of the system. The radial velocities were analyzed with the
genetic PIKAIA algorithm, whereas Doppler tomography maps for the H$\alpha$
and H$\beta$ lines were constructed with the Total Variation Minimization
code. An optimized simplex-algorithm was used to solve the inverse-problem
adjusting the light curve with the best stellar parameters for the system. We
find an orbital period of $6\fd 701\pm 0\fd 001$ and a long photometric cycle
of $191\pm 2~{}\mathrm{d}$. We detected the spectral features of the coldest
star, and modeled it with a $\log{g}=2.79\pm 0.02~{}\mathrm{dex}$ giant of
mass $1.13\pm 0.02~{}\mathrm{M_{\odot}}$ and effective temperature $10500\pm
125~{}\mathrm{K}$. In addition, we determine a mass ratio $q=0.206\pm 0.033$
and that the hot star is a B-type dwarf of mass $5.48\pm
0.02~{}\mathrm{M_{\odot}}$. The $V$-band orbital light curve can be modeled
including the presence of an accretion disk around the hotter star. This fills
the Roche lobe of the hotter star, and has a radius $14.74\pm
0.02~{}\mathrm{R_{\odot}}$ and temperature at the outer edge 9400 K. Two
bright spots located in the disk account for the global morphology of the
light curve. The Doppler tomography maps of H$\alpha$ and H$\beta$, reveal
complex structures of mass fluxes in the system.

binaries: close - binaries: eclipsing - binaries: spectroscopic - Stars:
early-type - Stars: mass-loss

††journal: ApJ

##  1 Introduction

Since the discovery of the first Double Periodic Variable stars by Mennickent
et al. (2003), in the Large Magellanic Cloud (LMC) and the Small Magellanic
Cloud (SMC), we have learn that there is a new kind of semi-detached, mass-
transferring binary stars showing two closely linked photometric variations.
These systems show an enigmatic long period on average 33 times longer than
the orbital period (Mennickent et al., 2016; Mennickent, 2017; Poleski et al.,
2010; Pawlak et al., 2013). But an interesting and remarkable property of the
DPVs is the constancy of their orbital periods, which usually does not occur
in the algols undergoing Roche Lobe Overflow mass transfer (Garrido et al.,
2013). To date, it is suspected that some interacting binary systems show
variations of the wind generated in the stream-disk impact region (Mennickent
et al., 2016; van Rensbergen et al., 2008), e.g., the interacting binary V393
Scorpii studied by Mennickent et al. (2012) shows evidence of a cyclically
variable bipolar wind. The prototype $\beta$ Lyrae also is a DPV, and shows
evidence of a jet emanating from the the accretion disk in a process far to be
well understood (Harmanec et al., 1996).

Recently, the nature of the second photometric period in these stars was
associated to a mechanism based on cycles of a magnetic dynamo in the donor
star (Schleicher & Mennickent, 2017; Mennickent et al., 2018). Also, it is
believed that the changes in the DPVs orbital light curves could be related to
changes in disk size/temperature and spot temperature/position (Garcés L. et
al., 2018). Understanding the mechanism associated to the DPV phenomenon will
be of fundamental importance for the study of the mass transfer in
semidetached binaries related to algols. Also it can provide information about
the stellar magnetic dynamos, stellar densities, tidal friction strength and
wind processes involved in these systems, processes that have been considered
in stellar population synthesis by de Mink et al. (2014) and also studied by
Deschamps et al. (2015).

The interacting binary HD 50526 (ASAS ID $065402+0648.8$,
$\alpha_{2000}=06:54:02.0$, $\delta_{2000}=06:48:47.9$, $V=8.23\pm
0.01~{}\mathrm{mag}$, $B-V=0.08~{}\mathrm{mag}$, spectral type
B9)111http://simbad.u-strasbg.fr/simbad, is a system characterized by a long
photometric cycle of 189$\fd$5 in the ASAS
222http://www.astrouw.edu.pl/asas/?page=acvs catalogue (Pojmanski, 1997) and
shows a short period of 6$\fd$7007, as published in
VSX333https://www.aavso.org/vsx/. The distance based on the
GAIA444http://gea.esac.esa.int/archive/ DR2 parallax is 1247 [+50 -47] pc
(Bailer-Jones et al., 2018). Since very few DPVs have been studied
spectroscopically, this object requires a detailed study, specially aimed to
determine its orbital and stellar parameters, and elucidate its enigmatic long
period. In this paper we shed light on the physical parameters of this
interesting system.

In Section 2 we present a detailed photometric analysis of ASAS photometry for
HD 50526 finding ephemeris for the orbital and long cycles. In Section 3 we
present a summary of our spectroscopic data and our methods of data reduction.
In Section 4 we use the method of spectral disentangling to separate the
stellar components and analyze them individually. We also obtain the orbital
parameters of the system using a code based in a genetic algorithm and
constrain the physical parameters of the donor through a minimization of the
deviation between theoretical normalized spectra and the observed one. In this
section we also calculate residual emission spectra and construct Doppler
tomography maps for the H$\alpha$ and H$\beta$ lines. In Section 5 we model
the light curve finding additional constrains for the stellar and orbital
parameters and in Section 6 we discuss our results. Finally, the main results
of our research are summarized in Section 7.

##  2 Photometric Ephemeris

The ASAS Photometric Catalog is maintained separately for each observed field,
so for some stars independent data sets of measurements are available, their
mean magnitudes may slightly differ. In addition, these data are labeled as A
for the best data, B for mean data, C corresponds to A and B data and finally
D are the worst data. The catalog calculates the magnitudes using different
apertures, wherein the photometric data are labeled from Mag0 to Mag4 with
steps of 1, these have size of 2 pixels for the smallest to 6 pixel for the
largest aperture. The smallest aperture is appropriate to the weakest stars,
while the largest apertures are to the brightest stars. So to get rid of the
saturated measurements, as an example it is optimal to compare Mag0 and Mag4
and if the difference between these is greater than 0.05, it is considered as
saturation. Therefore, with this information we performed a photometric
analysis of the ASAS light curve considering the 269 better-quality data
points labeled as A-type by ASAS. The data set was analyzed with the Phase
Dispersion Minimization algorithm PDM-IRAF (Stellingwerf, 1978) 555IRAF is
distributed by National Optical Astronomy Observatories, which are operated by
Association of Universities for Research in Astronomy, Inc., under cooperative
agreement with the National Science Foundation, revealing an orbital period
$P_{\mathrm{o}}=6\fd 701\pm 0\fd 001$. We also determined the epoch of minimum
$\mathrm{HJD}=2453096.5463$. The data set also revealed a long period
$P_{\mathrm{l}}=191\pm 2~{}\mathrm{d}$ with a maximum at
$\mathrm{HJD}=2452942.84607$, presenting a mean magnitude of the system of
$V=8.296\pm 0.044~{}\mathrm{mag}$ and variations from
$V_{\mathrm{max}}=8.148\pm 0.049~{}\mathrm{mag}$ to $V_{\mathrm{min}}=8.615\pm
0.045~{}\mathrm{mag}$ at the visual.

Table 1: Summary of ephemeris for HD 50526. We give the orbital period ($P_{\mathrm{o}}$) and epoch for the minimum brightness ($\mathrm{T_{0}}$). The initial and final HJDs of the photometric time series (minus -2450000) are given by $\mathrm{T_{i}}$ and $\mathrm{T_{f}}$. N indicates the number of data points. We give the mean $V$ value along with the standard deviation in parenthesis. Source | Po (d) | Pl (d) | T0 (mino) | T0 (maxl) |  $V$ (mag) | N | Ti | Tf
---|---|---|---|---|---|---|---|---
ASAS | 6.701(1) | 191(2) | 3096.54630 | 2942.84607 | 8.296(44) | 269 | 2549.84189 | 5162.78517

Once obtained both periods, we disentangled the light curves using a code
specially designed for this purpose by late Zbigniew Kołaczkowski. Briefly,
the code adjusts a Fourier series consisting of the fundamental frequency plus
their harmonics to the orbital signal previous computed. Later, it removes the
signal from the original time series allowing us to obtain the long
periodicity and the orbital one in two separated light curves (Fig. 1). This
code is described in more detail by Mennickent et al. (2012). The process
reveals an orbital modulation typical of an ellipsoidal DPV and a longer cycle
characterized by a quasi-sinusoidal variability as usual in others double
periodic variables. Both cycles are of comparable amplitude, contrary to the
observed in some eclipsing DPVs like V495 Cen, where the orbital variability
dominates (Rosales Guzmán et al., 2018). We have determined the following
ephemeris for the light curves and these will be used for the analysis in the
rest of the paper:

$\mathrm{HJD_{min,orbital}}=2453096.54630+6.701(1)\times E,$ (1)
$\mathrm{HJD_{max,long}}=2452942.84607+191(2)\times E,$ (2)

wherein HJD corresponds to Heliocentric Julian Date labeled with their
respective sub-indexes, i.e. for the minimum orbital cycle (min, orbital) and
for the maximum long cycle (max, long), while $E$ is an entire number
reflecting the number of elapsed cycles. As we will show later, the main
eclipse occurs when the more massive and hotter star is eclipsed. We will call
primary star to this star, while the less massive and coldest is the secondary
star. Since we find evidence of mass transfer from the cold to the hot star,
we also name them donor and gainer, respectively.

Figure 1: ASAS $V$-band phased light curves after disentangling. Orbital (up)
and long cycle (down) phases are shown. Phases were calculated according to
times of light curve minimum and maxima respectively, as given by equations
(1) and (2).

##  3 Spectroscopic observations

We have collected a series of 138 spectra uniformly distributed on orbital
phase, of high/medium spectral resolution with different spectrographs during
the years 2008 and 2015. We started the year 2008 using the
CORALIE666https://www.eso.org/public/chile/teles-instr/lasilla/swiss/coralie/
spectrograph at the ESO La Silla Observatory, obtaining 37 spectra of signal
to noise ratio $(\mathrm{SNR})\approx 58$ with a resolving power
$\mathrm{R}\sim 60000$ (Table 2). Further, during the years 2009, 2013 and
part of the year 2015, we used the
ECHELLE777http://www.astrossp.unam.mx/oanspm/ spectrograph in the San Pedro
Mártir Observatory obtaining 49 spectra of $\mathrm{SNR}\approx 124$ with
$\mathrm{R}\sim 18000$ (Table 3). In addition, between years 2012 to 2013 we
used CHIRON888http://www.ctio.noao.edu/noao/node/847 spectrograph in Cerro
Tololo Interamerican Observatory obtaining a sample of 52 spectra of
$\mathrm{SNR}\approx 47$ (slicer mode) with $\mathrm{R}\sim 80000$ (Table 3).
The covered spectral region were 3875-6895 Å (CORALIE), 3520-7605 Å (ECHELLE)
and 4580-8780 Å (CHIRON); this provided us a broad spectral range for studying
the variability of the spectral features of the stellar components.

The correction of every spectra with flat, bias, and the wavelength
calibration were done with IRAF, and the spectra were normalized to the
continuum. The velocities are expressed in the heliocentric rest of frame. The
signal to noise ratio ($S/N$) of every spectrum was computed over a region of
the continuum depleted of spectral lines. We have not sky subtracted neither
flux calibrated our spectra, but this does not affect the strength of
measurements neither the radial velocities (RVs). Our observations cover
$100\%$ of the orbital cycle, while for the long-term periodicity the coverage
is about $90\%$.

Table 2: Summary of spectroscopic observation using the CORALIE spectrograph
mounted on EULER Telescope. The HJD at mid-exposure for the first spectrum
series is given, the orbital ($\phi_{\mathrm{o}}$) long cycle
($\phi_{\mathrm{l}}$) phases were calculated using the eq. 1 and eq. 2
respectively. These spectra have a spectral resolution of $\mathrm{R\sim
60000}$.

UT-date | exptime (s) | HJD | $\phi_{\mathrm{o}}$ | $\phi_{\mathrm{l}}$ | $S/N$
---|---|---|---|---|---
2008-04-05 | 600 | 2454562.31964120 | 0.7395 | 0.4700 | 46.14
2008-04-05 | 600 | 2454562.32746528 | 0.7407 | 0.4700 | 49.06
2008-04-05 | 900 | 2454562.33877315 | 0.7423 | 0.4701 | 62.89
2008-04-05 | 900 | 2454562.35005787 | 0.7440 | 0.4701 | 58.76
2008-04-06 | 1200 | 2454563.32796296 | 0.8900 | 0.4752 | 62.15
2008-04-06 | 1200 | 2454563.34273148 | 0.8922 | 0.4753 | 68.61
2008-04-06 | 1200 | 2454563.35752315 | 0.8944 | 0.4754 | 69.80
2008-04-07 | 900 | 2454564.31726852 | 0.0376 | 0.4804 | 51.70
2008-04-07 | 900 | 2454564.32856481 | 0.0393 | 0.4805 | 53.90
2008-04-07 | 900 | 2454564.34033565 | 0.0410 | 0.4805 | 51.28
2008-04-08 | 900 | 2454565.32734954 | 0.1883 | 0.4857 | 60.72
2008-04-08 | 900 | 2454565.33864583 | 0.1900 | 0.4858 | 65.20
2008-04-08 | 900 | 2454565.34995370 | 0.1917 | 0.4858 | 64.36
2008-04-15 | 900 | 2454937.46894670 | 0.7236 | 0.4320 | 61.03
2008-04-15 | 900 | 2454937.48075231 | 0.7253 | 0.4321 | 68.15
2008-04-15 | 900 | 2454937.49209491 | 0.7270 | 0.4321 | 67.71
2008-04-16 | 900 | 2454937.50343750 | 0.7287 | 0.4322 | 73.32
2008-04-16 | 1200 | 2454938.47243056 | 0.8733 | 0.4373 | 71.75
2008-04-16 | 1200 | 2454938.48769676 | 0.8756 | 0.4373 | 71.36
2008-04-17 | 1200 | 2454938.50247685 | 0.8778 | 0.4374 | 65.43
2008-04-17 | 1200 | 2454938.51725694 | 0.8800 | 0.4375 | 58.55
2008-04-17 | 1200 | 2454938.53203704 | 0.8822 | 0.4376 | 59.53
2008-05-16 | 900 | 2454968.45359954 | 0.3475 | 0.5941 | 53.94
2008-05-16 | 900 | 2454968.46481481 | 0.3491 | 0.5941 | 56.56
2008-05-22 | 900 | 2454609.32633102 | 0.7544 | 0.7158 | 47.55
2008-05-22 | 900 | 2454609.33790509 | 0.7561 | 0.7159 | 43.55
2008-05-23 | 900 | 2454610.29773148 | 0.8993 | 0.7209 | 56.35
2008-05-23 | 900 | 2454610.30902778 | 0.9010 | 0.7209 | 49.34
2008-05-23 | 900 | 2454610.32034722 | 0.9027 | 0.7210 | 42.64
2008-10-02 | 999 | 2454741.67127315 | 0.5044 | 0.4080 | 57.29
2008-10-02 | 999 | 2454741.68372685 | 0.5063 | 0.4080 | 48.16
2008-10-03 | 1200 | 2454742.68658565 | 0.6559 | 0.4133 | 32.20
2008-10-04 | 1200 | 2454743.66797454 | 0.8024 | 0.4184 | 61.84
2008-10-04 | 1200 | 2454743.68747685 | 0.8053 | 0.4185 | 54.12
2008-12-19 | 900 | 2454819.69328714 | 0.1477 | 0.8160 | 62.10
2008-12-20 | 900 | 2454820.68160635 | 0.2952 | 0.8212 | 63.67
2008-12-22 | 900 | 2454822.69305410 | 0.5954 | 0.8317 | 64.95

Table 3: Summary of spectroscopic observation using the ECHELLE spectrograph
mounted on the San Pedro Mártir Telescope. The HJD at mid-exposure for the
first spectrum series is given, the orbital ($\phi_{\mathrm{o}}$) and long
cycle ($\phi_{\mathrm{l}}$) phases were calculated using eq. 1 and eq. 2
respectively with a spectral resolution of $\mathrm{R\sim 18000}$.

UT-date | exptime (s) | HJD | $\phi_{\mathrm{o}}$ | $\phi_{\mathrm{l}}$ | $S/N$
---|---|---|---|---|---
2009-12-06 | 1200 | 2455171.91678429 | 0.7106 | 0.6582 | 143.05
2009-12-06 | 1200 | 2455171.93134509 | 0.7127 | 0.6583 | 178.73
2009-12-06 | 1200 | 2455171.94591749 | 0.7149 | 0.6584 | 131.58
2009-12-07 | 1200 | 2455172.98771418 | 0.8704 | 0.6638 | 30.89
2009-12-07 | 1200 | 2455173.00228659 | 0.8726 | 0.6639 | 52.55
2009-11-04 | 1200 | 2455140.04944751 | 0.9550 | 0.4915 | 50.68
2009-11-05 | 1200 | 2455141.04128960 | 0.1030 | 0.4967 | 135.79
2009-11-07 | 1200 | 2455142.98124607 | 0.3925 | 0.5069 | 157.92
2009-11-08 | 1200 | 2455143.94206029 | 0.5359 | 0.5119 | 142.01
2009-11-09 | 900 | 2455144.87802708 | 0.6755 | 0.5168 | 89.46
2009-11-09 | 900 | 2455144.88912728 | 0.6772 | 0.5168 | 116.88
2009-11-09 | 900 | 2455144.90022751 | 0.6788 | 0.5169 | 135.28
2009-11-10 | 900 | 2455146.01491087 | 0.8452 | 0.5227 | 129.14
2009-11-10 | 900 | 2455146.02601127 | 0.8468 | 0.5228 | 170.71
2009-11-10 | 900 | 2455146.03710007 | 0.8485 | 0.5228 | 134.24
2011-01-25 | 900 | 2455586.76571973 | 0.6191 | 0.8279 | 122.56
2011-01-26 | 900 | 2455587.76936629 | 0.7689 | 0.8331 | 88.93
2011-01-27 | 900 | 2455588.69713235 | 0.9073 | 0.8380 | 101.30
2011-01-28 | 900 | 2455589.70166633 | 0.0572 | 0.8432 | 103.31
2012-02-11 | 1200 | 2455968.82382656 | 0.6342 | 0.8261 | 186.58
2012-02-12 | 1200 | 2455969.88029338 | 0.7918 | 0.8316 | 152.52
2012-02-13 | 1200 | 2455970.75980939 | 0.9231 | 0.8362 | 175.94
2012-02-14 | 1200 | 2455971.78222342 | 0.0756 | 0.8416 | 207.17
2012-02-15 | 1200 | 2455972.87665882 | 0.2390 | 0.8473 | 13.43
2012-02-18 | 1200 | 2455975.70836929 | 0.6616 | 0.8621 | 120.36
2012-02-18 | 1200 | 2455975.79925421 | 0.6751 | 0.8626 | 154.01
2012-02-18 | 1200 | 2455975.86411098 | 0.6848 | 0.8629 | 124.37
2012-02-19 | 1200 | 2455976.61805307 | 0.7973 | 0.8668 | 182.46
2012-02-19 | 1200 | 2455976.78931463 | 0.8229 | 0.8677 | 129.46
2012-02-20 | 1200 | 2455977.64297014 | 0.9503 | 0.8722 | 174.55
2012-02-20 | 1200 | 2455977.73496695 | 0.9640 | 0.8727 | 144.43
2012-02-21 | 1200 | 2455978.75980151 | 0.1169 | 0.8780 | 73.49
2012-02-21 | 1200 | 2455978.77386302 | 0.1190 | 0.8781 | 51.67
2012-02-22 | 1200 | 2455979.70671364 | 0.2582 | 0.8830 | 195.94
2012-02-22 | 1200 | 2455979.72076366 | 0.2603 | 0.8831 | 175.80
2012-02-23 | 1200 | 2455980.70032159 | 0.4065 | 0.8882 | 138.41
2012-11-13 | 1200 | 2456245.02839128 | 0.8526 | 0.2706 | 215.02
2012-11-14 | 1200 | 2456245.91058693 | 0.9842 | 0.2753 | 182.54
2012-11-18 | 1200 | 2456250.00101000 | 0.5946 | 0.2967 | 227.72
2012-11-19 | 1200 | 2456250.87730000 | 0.7254 | 0.3012 | 119.71
2013-01-28 | 1200 | 2456320.72825435 | 0.1494 | 0.6666 | 183.37
2013-01-28 | 1200 | 2456320.74231642 | 0.1515 | 0.6666 | 158.04
2013-01-28 | 1200 | 2456320.75639001 | 0.1536 | 0.6667 | 144.43
2013-01-30 | 1200 | 2456322.73900591 | 0.4494 | 0.6771 | 130.41
2013-01-30 | 1200 | 2456322.75306786 | 0.4515 | 0.6772 | 132.03
2013-01-30 | 1200 | 2456322.76714142 | 0.4536 | 0.6772 | 139.45
2015-01-13 | 1200 | 2457035.95402000 | 0.8836 | 0.4072 | 215.92
2015-01-13 | 1200 | 2457035.96816000 | 0.8857 | 0.4073 | 159.34
2015-01-13 | 1200 | 2457035.98227000 | 0.8878 | 0.4074 | 144.10

Table 4: Summary of spectroscopic observation using the CHIRON spectrograph.
The HJD at mid-exposure for the first spectrum series is given, the orbital
($\phi_{\mathrm{o}}$) and long cycle ($\phi_{\mathrm{l}}$) phases were
calculated using the eq. 1 and eq. 2 respectively with a spectral resolution
of $\mathrm{R\sim 80000}$ (Slicer mode).

UT-date | exptime (s) | HJD | $\phi_{\mathrm{o}}$ | $\phi_{\mathrm{l}}$ | $S/N$
---|---|---|---|---|---
2012-11-05 | 1200 | 2456236.80526037 | 0.6254 | 0.2276 | 64.30
2012-11-08 | 1200 | 2456239.79422590 | 0.0715 | 0.2433 | 58.38
2012-11-08 | 1200 | 2456239.80832071 | 0.0736 | 0.2433 | 62.83
2012-11-08 | 1200 | 2456239.82241444 | 0.0757 | 0.2434 | 61.62
2012-11-11 | 1200 | 2456242.82871200 | 0.5243 | 0.2591 | 66.10
2012-11-11 | 1200 | 2456242.84280691 | 0.5264 | 0.2592 | 63.63
2012-11-11 | 1200 | 2456242.85690172 | 0.5285 | 0.2593 | 60.01
2012-11-14 | 1200 | 2456245.78951280 | 0.9662 | 0.2746 | 73.99
2012-11-14 | 1200 | 2456245.80360880 | 0.9683 | 0.2747 | 69.66
2012-11-14 | 1200 | 2456245.81770132 | 0.9704 | 0.2748 | 47.09
2012-11-20 | 1200 | 2456251.73843209 | 0.8539 | 0.3057 | 73.50
2012-11-20 | 1200 | 2456251.75252803 | 0.8560 | 0.3058 | 68.08
2012-11-20 | 1200 | 2456251.76662151 | 0.8581 | 0.3059 | 73.05
2012-11-25 | 1200 | 2456256.83286117 | 0.6142 | 0.3324 | 62.18
2012-11-28 | 1200 | 2456259.70042938 | 0.0421 | 0.3474 | 54.80
2012-11-28 | 1200 | 2456259.71452406 | 0.0442 | 0.3475 | 50.92
2012-11-28 | 1200 | 2456259.72861750 | 0.0463 | 0.3475 | 48.31
2012-12-04 | 1200 | 2456265.78505374 | 0.9501 | 0.3792 | 63.94
2012-12-04 | 1200 | 2456265.79914814 | 0.9522 | 0.3793 | 57.33
2012-12-04 | 1200 | 2456265.81324131 | 0.9543 | 0.3794 | 54.12
2013-11-04 | 1200 | 2456600.84049212 | 0.9509 | 0.1316 | 41.22
2013-11-04 | 1200 | 2456600.85227210 | 0.9527 | 0.1316 | 29.08
2013-11-04 | 1200 | 2456600.86405207 | 0.9544 | 0.1317 | 26.02
2013-11-06 | 1200 | 2456602.82314142 | 0.2468 | 0.1419 | 53.63
2013-11-06 | 1200 | 2456602.83723646 | 0.2489 | 0.1420 | 49.99
2013-11-06 | 1200 | 2456602.85133027 | 0.2510 | 0.1421 | 49.40
2013-11-08 | 1200 | 2456604.78689240 | 0.5399 | 0.1522 | 56.14
2013-11-08 | 1200 | 2456604.80098707 | 0.5420 | 0.1523 | 57.10
2013-11-08 | 1200 | 2456604.81508074 | 0.5441 | 0.1524 | 57.05
2013-11-12 | 1200 | 2456608.84935936 | 0.1461 | 0.1735 | 49.36
2013-11-12 | 1200 | 2456608.86345529 | 0.1482 | 0.1735 | 49.10
2013-11-12 | 1200 | 2456608.87754884 | 0.1503 | 0.1736 | 51.28
2013-11-22 | 1200 | 2456618.78744321 | 0.6292 | 0.2254 | 49.51
2013-11-22 | 1200 | 2456618.80153801 | 0.6313 | 0.2255 | 38.32
2013-11-22 | 1200 | 2456618.81563157 | 0.6334 | 0.2256 | 46.79
2013-11-17 | 1200 | 2456613.74164809 | 0.8762 | 0.1990 | 50.55
2013-11-17 | 1200 | 2456613.75574294 | 0.8783 | 0.1991 | 50.10
2013-11-17 | 1200 | 2456613.76983652 | 0.8804 | 0.1992 | 47.54
2013-11-18 | 1200 | 2456614.79142646 | 0.0328 | 0.2045 | 48.91
2013-11-18 | 1200 | 2456614.80552017 | 0.0350 | 0.2046 | 53.69
2013-11-18 | 1200 | 2456614.81961616 | 0.0371 | 0.2047 | 49.32
2013-11-20 | 1200 | 2456616.80744009 | 0.3337 | 0.2151 | 54.65
2013-11-20 | 1200 | 2456616.82153479 | 0.3358 | 0.2152 | 49.41
2013-11-20 | 1200 | 2456616.83562947 | 0.3379 | 0.2152 | 50.11
2013-11-24 | 1200 | 2456620.71242225 | 0.9164 | 0.2355 | 23.26
2013-11-24 | 1200 | 2456620.72651699 | 0.9186 | 0.2356 | 27.04
2013-11-24 | 1200 | 2456620.74061160 | 0.9207 | 0.2356 | 24.93
2013-11-26 | 1200 | 2456622.78521955 | 0.2258 | 0.2463 | 45.55
2013-11-26 | 1200 | 2456622.81340771 | 0.2300 | 0.2465 | 47.72
2013-12-16 | 1200 | 2456642.74995625 | 0.2051 | 0.3508 | 22.40
2013-12-16 | 1200 | 2456642.76405042 | 0.2072 | 0.3508 | 25.21
2013-12-16 | 1200 | 2456642.77814343 | 0.2093 | 0.3509 | 26.70

##  4 Spectroscopic analysis

In this section we provide the analysis of our spectra, revealing constrains
on the system parameters and evidencing the presence of circumstellar matter
in the system.

###  4.1 Radial velocities for the donor

Close binary star orbits can be characterized through the study of the radial
velocities of their stellar components. We selected some characteristic lines
that represent the movement of both components. The radial velocity of the
donor was obtained by Gaussian fit on a set of H$\alpha$ lines from spectra
obtained with the CORALIE spectrograph. In addition, the velocities were
corrected to the absolute heliocentric system and the orbital parameters for
HD 50526 were obtained using the genetic algorithm PIKAIA999
http://www.hao.ucar.edu/modeling/pikaia/pikaia.php of public domain developed
by Charbonneau (1995). This code produces a series of theoretical velocities
and find the best parameters through the minimization of the function $\chi
2$, defined as:

$\chi 2(P_{\mathrm{o}},\tau,\omega,e,K_{2},\gamma)$

$=\frac{1}{\mathrm{N}-6}\sum_{\mathrm{j=1}}^{\mathrm{N}}\left(\frac{V_{\mathrm{j}}^{\mathrm{obs}}-V(t_{\mathrm{j}};P_{\mathrm{o}},\tau,\omega,e,K_{2},\gamma)}{\sigma_{\mathrm{j}}}\right)^{2},$
(3)

where the parameter $\mathrm{N}$ represents the number of observations,
$V_{\mathrm{j}}^{\mathrm{obs}}$ is the radial velocity observed in the data
set and $V(t_{\mathrm{j}};P_{\mathrm{o}},\omega,e,K_{2},\gamma)$ is the radial
velocity at the time $t_{\mathrm{j}}$. $P_{\mathrm{o}}$ is the orbital period,
$\tau$ the time of passage per the periastron, $\omega$ the periastron
longitude, $e$ the orbital eccentricity, $K_{2}$ the half-amplitude of the
radial velocities of the donor, and finally $\gamma$ the velocity of the
center of mass of the system. According to the equation 2.45 given by Hilditch
(2001) the theoretical radial velocity is given by :

$V(t)=\gamma+K_{2}(\cos(\omega+\theta(t))+e{\cos(\omega)}),$ (4)

where the angular parameter called true anomaly $\theta$ is obtained solving
the relationship between the true and eccentric anomalies through the
following equation:

$\tan\left(\frac{\theta}{2}\right)=\sqrt{\frac{1+e}{1-e}}\tan\left(\frac{E}{2}\right),$
(5)

wherein $E$ corresponds to the eccentric anomaly and it is given by the
equation 2.35 of Hilditch (2001):

$E-e\sin(E)=\frac{2\pi}{P_{\mathrm{o}}}(t-\tau),$ (6)

In order to solve the equation 5, it is necessary to solve the equation 6, but
since it has not an analytic solution it must be solved numerically through an
iterative method. After obtaining a solution for the true anomaly it is
possible to obtain a solution for the equation 4 of the theoretical radial
velocity $V(t_{\mathrm{j}};P_{\mathrm{o}},\omega,e,K_{2},\gamma)$. Through a
Monte Carlo simulation we have estimated the error, perturbing the best
fitting solution obtained with the PIKAIA code and computing a $\chi 2$ for
these perturbed solutions. As criterion for deciding when to accept an
elliptical ($p<0.05$) or circular ($p\geq 0.05$) orbit, we have implemented
the statistical test $p_{1}$ of Lucy & Sweeney (1971) :

$p_{1}=\left(\frac{\sum(o-c)^{2}_{\mathrm{ecc}}}{\sum(o-c)^{2}_{\mathrm{circ}}}\right)^{\mathrm{(n-m)/2}},$
(7)

the subindex _ecc_ corresponds to the $\chi 2$ obtained from the fit of an
elliptical orbit, while _circ_ is the $\chi 2$ from a circular orbit, _n_ is
the total number of observational radial velocities and _m_ is the number of
free parameters for the fit of an elliptical orbit. Finally, we obtained the
orbital parameters with its respective errors that are given in Table 5 with a
value $p_{1}=0.0015$ compatible with an elliptical orbit, as is indicated in
the Fig. 2. In this figure the solid black and the dashed gray lines
correspond to $1\sigma$ and $2\sigma$ around the best solution marked with a
red dot, respectively. It has been pointed out that gas stream and
circumstellar matter can distort spectroscopic features in semidetached
interacting binaries, producing skewed RVs and artificial small eccentricities
(Lucy, 2005). For a non interacting binary with the stellar and orbital
parameters of HD 50526 dynamical tides should have circularized the orbit and
synchronized the rotational periods (Zahn, 1975, 1977). This should imply that
the observed small eccentricity might be spurious. In addition, we find a
small $2\sigma$ difference between the spectroscopic and photometric period.
However, due to the larger density of data and time baseline of the
photometric time series we trust more in the period obtained from photometry.

The radial velocities for the donor and gainer stars were measured through
Gaussian fits in the CORALIE spectra, that turned to be the best spectra for
this task. These velocities are listed in Tables 6 and 7. The radial
velocities (RV) of the gainer star for He I 4471.477 Å was fitted with a sine
function of amplitude $31.8\pm 5.0~{}\mathrm{km\,s^{-1}}$ and zero point
$0.0\pm 3.4~{}\mathrm{km\,s^{-1}}$. The RV of the donor measured with the
H$\alpha$ line, was fitted with an amplitude of $154.0\pm
2.3~{}\mathrm{km\,s^{-1}}$ and a zero point $0.0\pm
1.8~{}\mathrm{km\,s^{-1}}$. Both solutions assume a non-circular orbit and
yield a mass ratio $q\equiv K_{1}/K_{2}=0.206\pm 0.033$ (Fig. 3).

Figure 2: (Up) The orbital period as a function of the eccentricity calculated by Monte Carlo simulations. The solid black line corresponds to $1\sigma$ isophote, the dashed gray line to $2\sigma$ isophote, and the red dot shows the minimum $\chi 2$. (Down) The contour corresponds to the $\Delta\chi 2$ of 2 degrees of freedom that includes 68.3% of probability, i.e. $\Delta\chi 2=2.30$. Figure 3: Radial velocities of He I 4471.477 and H$\alpha$ lines for the primary component (black dots) and for the secondary component (yellow dots) respectively. All measurements were through Gaussian fits. The best fitting functions are also shown. Table 5: Orbital elements for the donor of HD 50526 obtained through minimization of $\chi 2$ given by equation (1). The value $\tau^{*}=\tau-2450000$ is given and the maximum and minimum are one isophote $1\sigma$. $\begin{array}[]{l r r r}\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\mathrm{Parameter}&\mathrm{Best~{}value}&\mathrm{Low~{}limit}&\mathrm{Upper~{}limit}\\\ \hline\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr P_{\mathrm{o}}~{}(\mathrm{d})&{6.7048}&{6.7034}&{6.7063}\\\ \tau^{*}&{566.392}&{566.348}&{566.436}\\\ \emph{e}&{0.04974}&{0.02200}&{0.07600}\\\ \omega~{}(\mathrm{rad})&{0.76636}&{0.72639}&{0.80639}\\\ K_{2}~{}($\,km\,s$^{-1}$$)&{153.008}&{149.158}&{156.958}\\\ \gamma~{}($\,km\,s$^{-1}$$)&{-5.16994}&{-7.93987}&{-2.33987}\\\ \hline\cr\end{array}$ Table 6: Radial velocities of the donor using the H$\alpha$ line in CORALIE spectra. Typical error is 1.9 km s-1 and HJD’ stands for HJD-2454000. HJD’ | $\phi_{o}$ | RV (km s-1) | HJD’ | $\phi_{o}$ | RV (km s-1) | HJD’ | $\phi_{o}$ | RV (km s-1)
---|---|---|---|---|---|---|---|---
564.31726852 | 0.0376 | 42.3 | 937.46894670 | 0.7236 | -152.8 | 938.50247685 | 0.8778 | -111.2
564.32856481 | 0.0393 | 43.6 | 937.48075231 | 0.7253 | -152.6 | 938.51725694 | 0.8800 | -106.3
564.34033565 | 0.0410 | 46.9 | 937.49209491 | 0.7270 | -150.3 | 938.53203704 | 0.8822 | -105.5
819.69328714 | 0.1477 | 97.1 | 937.50343750 | 0.7287 | -155.2 | 563.32796296 | 0.8900 | -82.7
565.32734954 | 0.1883 | 148.7 | 562.31964120 | 0.7395 | -157.6 | 563.34273148 | 0.8922 | -80.0
565.33864583 | 0.1900 | 166.0 | 562.32746528 | 0.7407 | -150.5 | 563.35752315 | 0.8944 | -78.7
565.34995370 | 0.1917 | 151.1 | 562.33877315 | 0.7423 | -153.7 | 610.29773148 | 0.8993 | -70.5
820.68160635 | 0.2952 | 148.4 | 562.35005787 | 0.7440 | -153.3 | 610.30902778 | 0.9010 | -72.7
968.45359954 | 0.3475 | 127.6 | 609.32633102 | 0.7544 | -150.4 | 610.32034722 | 0.9027 | -68.3
968.46481481 | 0.3491 | 126.0 | 609.33790509 | 0.7561 | -154.0 |  |  |
741.67127315 | 0.5044 | -27.9 | 743.66797454 | 0.8024 | -141.2 |  |  |
741.68372685 | 0.5063 | -28.3 | 743.68747685 | 0.8053 | -132.5 |  |  |
822.69305410 | 0.5954 | -87.3 | 938.47243056 | 0.8733 | -113.0 |  |  |
742.68658565 | 0.6559 | -137.1 | 938.48769676 | 0.8756 | -109.9 |  |  |
Table 7: Radial velocities of the gainer and their respective errors, using the He I 4471.477 Å line in CORALIE spectra. Typical error is 3.5 km s-1 and HJD’ stands for HJD-2454000. HJD’ | $\phi_{o}$ | RV (km s-1) | HJD’ | $\phi_{o}$ | RV (km s-1) | HJD’ | $\phi_{o}$ | RV (km s-1)
---|---|---|---|---|---|---|---|---
819.69328714 | 0.1477 | -31.7 | 742.68658565 | 0.6559 | 52.3 | 938.50247685 | 0.8778 | 30.9
565.32734954 | 0.1883 | -23.4 | 937.46894670 | 0.7236 | 39.1 | 938.51725694 | 0.8800 | 13.6
565.33864583 | 0.1900 | -32.5 | 937.49209491 | 0.7270 | 10.8 | 938.53203704 | 0.8822 | 10.7
820.68160635 | 0.2952 | -25.0 | 937.50343750 | 0.7287 | 40.6 | 563.32796296 | 0.8900 | 2.7
968.45359954 | 0.3475 | -21.6 | 562.35005787 | 0.7440 | 20.9 | 563.34273148 | 0.8922 | 23.8
968.46481481 | 0.3491 | -18.1 | 609.32633102 | 0.7544 | 35.9 | 563.35752315 | 0.8944 | 13.5
741.67127315 | 0.5044 | 9.3 | 743.66797454 | 0.8024 | 12.6 | 610.29773148 | 0.8993 | 8.4
741.68372685 | 0.5063 | 17.6 | 938.47243056 | 0.8733 | -47.3 | 610.30902778 | 0.9010 | 14.1
822.69305410 | 0.5954 | 23.4 | 938.48769676 | 0.8756 | 17.0 |  |  |

###  4.2 Spectral disentangling

Since we have independently distinguished the absorption lines of each
component using the CORALIE spectra, we performed an iterative method of
spectral disentangling proposed by González & Levato (2006), which is quite
effective in separating the absorption lines of the stellar components. This
method uses alternately the spectrum of one component to calculate the
spectrum of the other one, hence eliminating gradually the spectral features
of one stellar component until the convergence is assured and virtually
disappears the flux contribution of its companion. In the process we used the
theoretical radial velocities obtained from the sinusoidal fits in the
previous section as input parameters until the seventh iteration for both
components, obtaining successfully clean average spectra for both stars. The
disentangling process for the gainer star reveals a double emission line in
H$\alpha$, confirming the presence of circumstellar matter around the hot
component. Furthermore, as the emission is double, an accretion disc is in
principle inferred, in agreement with a semidetached algol mass-transferring
binary (Fig. 4).

Figure 4: Disentangled spectra of the gainer star around the H$\alpha$ line.
Data of three different spectrographs are independently shown.

###  4.3 Determination of the donor physical parameters

The determination of physical parameters of the donor is of fundamental
importance to constrain the size of the system. We have constructed a grid of
synthetic theoretical spectra, modeling stellar atmospheres with the code
SPECTRUM 101010http://www.appstate.edu/~grayro/spectrum/spectrum.html (Gray &
Corbally, 1994; Gray et al., 2001) and using a grid of model atmospheres given
by ATLAS9111111http://wwwuser.oats.inaf.it/castelli/grids.html (Castelli, &
Kurucz, 2003). The grid was constructed in Local Thermodynamic Equilibrium
(LTE) with different free parameters such as the effective temperature, which
has two groups which vary $4000\leq T_{\mathrm{2}}<11000~{}\mathrm{K}$ with
steps of 250 K and $11000\leq T_{\mathrm{2}}\leq 20000~{}\mathrm{K}$ with
steps of 1000 K. The surface gravities vary from 0.0 to 5.0 dex with steps of
0.5 dex, the micro-turbulent velocity which is defined as the scale of
turbulence in the stellar atmosphere in which the size of a turbulent cell is
smaller than the mean free path, has unique options of 0.0 and 2.0
$\mathrm{km\,s}^{-1}$, $v\sin{i}$ varies from 0 to 150 $\mathrm{km\,s}^{-1}$
with step of 10 $\mathrm{km\,s}^{-1}$ and the macro-turbulence velocity from 0
to 10 $\mathrm{km\,s}^{-1}$ with step 1 $\mathrm{km\,s}^{-1}$. This parameter
is defined as the scale of turbulence in the stellar atmosphere in which the
size of the turbulence cell is greater than the mean free path of the photon.
We also consider a veiling factor from 0.10 to 0.90 with step of 0.1
(dimensionless), this is defined as a constant of proportionality between the
theoretical spectrum and the observed one, needed to account for the light
contribution of the hot star veiling the absorption lines of the more evolved
and cold star. Finally, a fixed mixing length parameter $l/\mathrm{H}=1.25$
was the default for the grid, this parameter corresponds when the blobs of
convected fluid travel a distance “$l$” from their position of equilibrium and
then disrupt and disperse into the surroundings.

The implemented analysis was based on a chi-square optimization algorithm
$\chi
2=\sum_{i=\lambda_{0}}^{\lambda_{\mathrm{f}}}(O_{\mathrm{i}}-E_{\mathrm{i}})^{2}/E_{\mathrm{i}}$,
this consists in the minimization of the deviation between the theoretical
normalized spectrum ($E_{\mathrm{i}}$) and the observed average spectrum of
the donor star already disentangled and corrected by a veiling factor on the
analyzed spectral range ($O_{\mathrm{i}}$), and the lines were analyzed at the
rest frame. The implemented method yielded the best result with a temperature
of $T_{2}=10500\pm 125~{}\mathrm{K}$, $\log{\mathrm{g}}_{2}=3.0\pm
0.5~{}\mathrm{dex}$, with fixed $v_{\mathrm{mic}}=0.0~{}\mathrm{km\,s^{-1}}$,
$v_{\mathrm{mac}}=1.0~{}\mathrm{km\,s^{-1}}$, $v\sin{i}=70\pm
20~{}\mathrm{km\,s^{-1}}$, veiling factor $\eta=0.5\pm 0.05$ and a
$l/\mathrm{H}=1.25$ (Fig. 5). The comparison between the observed and the
theoretical spectrum shows a good match in the 4000-4400 Å region as well as
in the broader region of 4500-4600 Å (Fig. 6 and Fig. 7).

An inspection of the behaviour of $\chi 2$ versus $v\sin{i}$ shows that the
projected rotational velocity is loosely constrained. Actually, $\chi 2$
changes very little in the range 50-150 $\mathrm{km\,s}^{-1}$. If we assume
synchronous rotation for the donor, because of the close binary nature (Zahn,
1975, 1977), and considering that the expected rotational velocity can be
calculated from equation 3.9 of Eggleton (2006),

$\frac{v_{\mathrm{rot}}\sin{i}}{K_{2}}=\frac{R_{\mathrm{L}}}{a}\approx(1+q)\frac{0.49q^{2/3}}{0.6q^{2/3}+\ln(1+q^{1/3})},$
(8)

where $R_{\mathrm{L}}$ is the volume Roche Lobe radius of the secondary, we
get $v\sin{i}$ = 47 $\mathrm{km\,s}^{-1}$, using $q$ = 0.206 and
$K_{2}=154~{}\mathrm{km\,s^{-1}}$. This is just in the lower limit of our
determined value for the donor projected rotational velocity. Therefore, our
analysis is consistent with a rotationally synchronized donor star. We notice
that the approximation in Eq. 8 is accurate to 1% for all $q$ values
(Eggleton, 2006).

We calculated $R_{\mathrm{L}}/a=0.455\pm 0.130$ using Eq. 8. Assuming $R_{2}$
= $R_{\mathrm{L}}$ we have quantified the mean density $\bar{\rho}$ (g cm-3)
of the donor star. Using $\rho=3M_{2}/4\pi R^{3}_{\mathrm{2}}$,
$M_{\mathrm{2}}=Mq/(1+q)$ and the Kepler’s third law we get:

$\bar{\rho}_{2}=\frac{3q}{(1+q)}\frac{1}{(R_{\mathrm{2}}/a)^{3}}\frac{\pi}{GP^{2}},$
(9)

and we obtain a highly affected mean density of $\bar{\rho}_{2}=0.0008\pm
0.0004~{}\mathrm{g\,cm^{-3}}$, which is characteristic of evolved stars,
possibly caused by the loss mass during the mass transfer process.

###  4.4 On the gainer star

A similar procedure as described in the previous section was performed for the
gainer star. The spectrum of the gainer is characterized by weak H$\alpha$
emission and weak H$\beta$, H$\gamma$ and H$\delta$ absorptions, along with
broad He I lines, sometimes surrounded by emission flanks. This suggest that
the spectrum is a blend of the contribution of the gainer and the accretion
disk. We choose a spectral range with only helium lines to search for the best
spectral model, and run the $v\sin{i}$ parameter between 10 and
$400~{}\mathrm{km\,s^{-1}}$. We find a best model with
$T_{1}=16000~{}\mathrm{K}$, $\log{g}\geq 3.0~{}\mathrm{dex}$,
$v_{\mathrm{mic}}=0.0~{}\mathrm{km\,s^{-1}}$,
$v\sin{i}=130~{}\mathrm{km\,s^{-1}}$,
$v_{\mathrm{mac}}=1~{}\mathrm{km\,s^{-1}}$, veiling factor $\eta=0.2$ and
$\chi 2=2.215$ (Fig. 8). This suggests that the gainer is a B-type star with
spectral type around B 4. We notice that the low surface gravity found could
be due to the influence of line emission of circumstellar material affecting
the width of the line profiles.

Figure 5: Chi-square analysis to find the best theoretical normalized spectrum
with respect to the average observed spectrum, computed with six free degrees
obtained for the cold star, and with all parameters at their optimized values.
The best model is obtained at $\chi 2=14.197$ and is represented by a red dot
at $T=10500~{}\mathrm{K}$ and a $\log{g}=3.0~{}\mathrm{dex}$. Figure 6:
Comparison between the best theoretical model obtained for the cold star (red
line) and the disentangled average observed spectrum (black line). Figure 7:
Detailed comparison between the disentangled observed (black line) and
theoretical (red line) secondary spectrum. Figure 8: Disentangled CORALIE
spectrum for the gainer star, showing broad helium lines surrounded by
emission shoulders and the best model.

###  4.5 The H$\alpha$ and H$\beta$ emission line profiles

We have performed a complementary analysis to the H${\alpha}$ and H$\beta$
profiles removing the donor and gainer theoretical contributions. We observe
H${\alpha}$ and H$\beta$ emission at all phases, being in H$\alpha$ broad and
usually double (Fig. 9). This is consistent with an origin in an accretion
disk. The profile shapes show variable asymmetry. The H$\alpha$ equivalent
width - measured before gainer subtraction - changes through the orbital cycle
but most notably during the long cycle, being lower (eventually due to more
emission) near the long-cycle maximum (Fig. 10).

We measured the RV of the H$\alpha$ residual emissions using fits with
Gaussian functions, but the H$\beta$ emission was quite irregular and in this
case we measured the RVs using a fit only considering the upper portion of the
profiles. The velocities show large scatter; the averages are
$\mathrm{\overline{RV}}_{\alpha}=30.6\pm 26.7~{}\mathrm{km\,s^{-1}}$ and
$\mathrm{\overline{RV}}_{\beta}=-42.3\pm 85.2~{}\mathrm{km\,s^{-1}}$. While
the H$\alpha$ velocities are consistent with no orbital variability, the
H$\beta$ velocities seems to roughly follow the motion of the donor but with
large scatter (Fig. 11).

Figure 9: Residual H$\alpha$ (up) and H$\beta$ (down) emission in CORALIE
spectra, smoothed by a factor 50, during a complete orbital cycle. The orbital
phases are displayed. Figure 10: Equivalent width of the H$\alpha$ profile of
the gainer star for several orbital (Top) and long cycle phases (Bottom)
without the donor contribution, using CORALIE (black dots), Echelle (blue
dots) and CHIRON (orange dots) spectrographs. The red continuum line is a
fitted 1st order polynomial to the EWs with a confidence interval region of
$95\%$. Figure 11: Radial velocities of the residual emission along with the
theoretical fit to the RVs of the donor.

Figure 12: Doppler maps of disentangled spectra: donor H$\alpha$ absorption
(left) and H$\alpha$ and H$\beta$ emission (donor and gainer subtracted,
center and right). Flux 1 indicates the continuum level. The Roche lobes of
the donor and gainer are plotted projected in the orbital plane, along with
the theoretical path followed by the gas stream.

###  4.6 Doppler tomography

Doppler tomography is an image reconstruction technique introduced by Marsh &
Horne (1988) that has been widely used in the study of cataclysmic variables
and more recently, in the study of algol systems (Richards, 2004). The
technique consists of the reconstruction of the system emissivity in the
velocity space from spectra taken along the orbital cycle.

We constructed Doppler maps for the H$\alpha$ and H$\beta$ lines using 25
CHIRON spectra, 37 CORALIE spectra and 15 SPM spectra. These were selected as
those spectra with highest S/N sampling well the orbital cycle and we used the
disentangled spectra, alternatively removing the contribution of the donor and
gainer star. The Doppler maps were constructed by means of the Total Variation
Minimization (DTTVM) code that also deals with absorption lines; this code was
presented by Uemura et al. (2015).

We find that the H$\alpha$ donor spectra reveal an absorption line centered in
the cold star, as expected (Fig. 12, left panel). In addition, the residual
emission is observed inside the Roche lobe of the gainer, much clearly in the
H$\beta$ line without further structure, while the H$\alpha$ line shows
additional high-velocity emission structures in the lower right quadrant and a
diffuse and thick donuts-shaped emission structure centered at the low-left
side of the gainer (Fig. 12, central and right panels). An additional mild
emission is observed in the upper left quadrant around the track of the gas
stream in the H$\alpha$ map. These emissions are reminiscent of that expected
from an optically thin accretion disk observed at mid latitudes. However, the
presence of structured regions of higher emissivity and the localized H$\beta$
emission inside the Roche lobe of the gainer probably reflects the complex
geometrical and physical structure of the circumstellar material surrounding
the gainer. This material might even not to be restricted to the orbital
plane, breaking the assumptions of the Doppler tomography reconstruction.
Eventual mass outflows would also influence the general appearance of the
Doppler maps.

##  5 Light-curve model and system parameters

In this section we determine the stellar and system parameters that best fit
the orbital light curve.

###  5.1 The fitting procedure

We fit the orbital light curve with a code described by Djurašević (1992). The
algorithm applies the inverse-problem solving method based on the simplex
algorithm, considering a binary system with an optically thick disc
surrounding the more massive star. The system is assumed semidetached, with
the less massive star filling its Roche lobe. We used the Nelder-Mead simplex
algorithm (Press et al., 1992) with optimizations described by Dennis and
Torczon (1991). While the direct problem comprises the calculation of the
light curve from model parameters given a priori, the inverse problem is the
process of finding the set of parameters that will optimally fit the synthetic
light curve to the observations.

The theoretical model considers a hot spot located on the edge of the disc, in
the place where the gas stream from the donor falls encountering the disc.
This active region is described by the ratio of the hot spot temperature and
the unperturbed local disc temperature and the angular dimension and longitude
of the spot. An additional, bright spot, with similar parameters, is also
included in the outer disc edge, following results of hydro-dynamical
simulations of gas interchanged among close binary systems (e.g. Heemskerk,
1994; Kaigorodov, Bisikalo, & Kurbatov, 2017). The disk is assumed in physical
contact with the gainer and is characterized by its radius Rd, outer and inner
edge thicknesses and its temperature:

$T(r)=T_{\mathrm{d}}\left(\frac{T_{\mathrm{d}}}{r}\right)^{a_{\mathrm{T}}}$
(10)

where $T_{\mathrm{d}}$ is the disk temperature at its outer edge
($r=R_{\mathrm{d}}$) and $a_{\mathrm{T}}$ is the temperature exponent
($a_{\mathrm{T}}\leq 0.75$). The value of exponent $a_{\mathrm{T}}$ shows how
close is the radial temperature profile to the steady-state configuration
($a_{\mathrm{T}}=0.75$). The radial dependencies of the accretion disk
temperature shows that the surface of the disk is hotter in the inner regions,
and that disk gets cold as one moves away from the center. The model and code
have been widely used during our recent research of intermediate-mass
interacting binaries (Mennickent & Djurašević, 2013; Mennickent et al., 2015;
Rosales Guzmán et al., 2018). Based on our results of previous sections, we
fixed the mass ratio to $q=0.206$ and the temperature of the cold and less
massive star to $T_{2}=10500~{}\mathrm{K}$.

Table 8: Results of the analysis of DPV HD 50526 V-filter light-curve obtained
by solving the inverse problem for the Roche model with an accretion disk
around the more-massive (hotter) gainer in the critical non-synchronous
rotation regime.

$\begin{array}[]{llll}\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr{\rm
Quantity}&&{\rm Quantity}&\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr n&265&\cal
M_{\rm{}_{1}}{[\cal M_{\odot}]}&5.48\pm 0.02\\\
{\rm\Sigma(O-C)^{2}}&0.0565&\cal M_{\rm{}_{2}}{[\cal M_{\odot}]}&1.13\pm
0.02\\\ {\rm\sigma_{rms}}&0.0146&\cal R_{\rm{}_{1}}{\rm[R_{\odot}]}&3.57\pm
0.03\\\ i{\rm[^{\circ}]}&61.28\pm 0.2&\cal
R_{\rm{}_{2}}{\rm[R_{\odot}]}&7.09\pm 0.01\\\ {\rm F_{d}}&0.998\pm 0.2&{\rm
log}\ g_{\rm{}_{1}}&4.07\pm 0.02\\\ {\rm T_{d}}[{\rm K}]&9401\pm 200&{\rm
log}\ g_{\rm{}_{2}}&2.79\pm 0.02\\\ {\rm d_{e}}[a_{\rm orb}]&0.152\pm
0.01&M^{\rm h}_{\rm bol}&-2.40\pm 0.26\\\ {\rm d_{c}}[a_{\rm orb}]&0.102\pm
0.02&M^{\rm 2}_{\rm bol}&-2.06\pm 0.02\\\ {\rm a_{T}}&0.41\pm 0.05&a_{\rm
orb}{\rm[R_{\odot}]}&28.04\pm 0.02\\\ {\rm f_{1}}&14.71\pm 0.5&\cal{R}_{\rm
d}{\rm[R_{\odot}]}&14.74\pm 0.02\\\ {\rm
F_{1}}&1.000&\rm{d_{e}}{\rm[R_{\odot}]}&4.26\pm 0.02\\\ {\rm T_{1}}[{\rm
K}]&16000&\rm{d_{c}}{\rm[R_{\odot}]}&2.87\pm 0.03\\\ {\rm T_{2}}[{\rm
K}]&10500&\\\ {\rm A_{hs}=T_{hs}/T_{d}}&1.36\pm 0.03&\\\
{\rm\theta_{hs}}{\rm[^{\circ}]}&19.8\pm 2.0&\\\
{\rm\lambda_{hs}}{\rm[^{\circ}]}&343.1\pm 3.0&\\\
{\rm\theta_{rad}}{\rm[^{\circ}]}&-16.2\pm 2.0&\\\ {\rm
A_{bs}=T_{bs}/T_{d}}&1.18\pm 0.03&\\\ {\rm\theta_{bs}}{\rm[^{\circ}]}&46.6\pm
4.0&\\\ {\rm\lambda_{bs}}{\rm[^{\circ}]}&129.5\pm 5.0&\\\ {\Omega_{\rm
1}}&9.796\pm 0.09&\\\ {\Omega_{\rm 2}}&2.248&\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\end{array}$

FIXED PARAMETERS: $q={\cal M}_{\rm 2}/{\cal M}_{\rm 1}=0.206$ \- mass ratio of
the components, ${\rm F_{1}}=R_{1}/R_{zc}=1$ \- filling factor for the
critical Roche lobe of the hotter, more-massive gainer (ratio of the stellar
polar radius to the critical Roche lobe radius along z-axis for a star in
synchronous rotation regime), ${\rm T_{1}=16000K}$, ${\rm T_{2}=10500K}$ \-
temperature of the more massive (hotter) gainer and the less-massive (coldest)
donor, ${\rm F_{2}}=1.0$ \- filling factor for the critical Roche lobe of the
donor, $f{\rm{}_{1,2}}=1.00$ \- non-synchronous rotation coefficients of the
system components, ${\rm\beta_{1}=0.25}$, ${\rm\beta_{2}=0.25}$ \- gravity-
darkening coefficients of the components, ${\rm A_{1}=1.0}$, ${\rm A_{2}=1.0}$
\- albedo coefficients of the components.

Note: $n$ \- number of observations, ${\rm\Sigma(O-C)^{2}}$ \- final sum of
squares of residuals between observed (LCO) and synthetic (LCC) light-curves,
${\rm\sigma_{rms}}$ \- root-mean-square of the residuals, $i$ \- orbit
inclination (in arc degrees), ${\rm F_{d}=R_{d}/R_{yc}}$ \- disk dimension
factor (the ratio of the disk radius to the critical Roche lobe radius along
y-axis), ${\rm T_{d}}$ \- disk-edge temperature, $\rm{d_{e}}$, $\rm{d_{c}}$, -
disk thicknesses (at the edge and at the center of the disk, respectively) in
the units of the distance between the components, $a_{\rm T}$ \- alpha disk
temperature distribution coefficient, $f{\rm{}_{h}}$ \- non-synchronous
rotation coefficient of the more massive gainer (in the synchronous rotation
regime), ${\rm A_{hs}=T_{hs}/T_{d}}$ \- hot spot temperature coefficient,
${\rm\theta_{hs}}$ and ${\rm\lambda_{hs}}$ \- spot angular dimension and
longitude (in arc degrees), ${\rm\theta_{rad}}$ \- angle between the line
perpendicular to the local disk edge surface and the direction of the hot-spot
maximum radiation, ${\rm A_{bs}=T_{bs}/T_{d}}$ \- bright spot temperature
coefficient, ${\rm\theta_{bs}}$ and ${\rm\lambda_{bs}}$ \- bright spot angular
dimension and longitude (in arc degrees), ${\Omega_{\rm 1,2}}$ \-
dimensionless surface potentials of the hotter gainer and coldest donor, $\cal
M_{\rm{}_{1,2}}{[\cal M_{\odot}]}$, $\cal R_{\rm{}_{1,2}}{\rm[R_{\odot}]}$ \-
stellar masses and mean radii of stars in solar units, ${\rm log}\
g_{\rm{}_{1,2}}$ \- logarithm (base 10) of the system components effective
gravity, $M^{\rm{1,2}}_{\rm bol}$ \- absolute stellar bolometric magnitudes,
$a_{\rm orb}$ ${\rm[R_{\odot}]}$, $\cal{R}_{\rm d}{\rm[R_{\odot}]}$,
$\rm{d_{e}}{\rm[R_{\odot}]}$, $\rm{d_{c}}{\rm[R_{\odot}]}$ \- orbital semi-
major axis, disk radius and disk thicknesses at its edge and center,
respectively, given in solar units.

Figure 13: Observed (LCO) and synthetic (LCC) light-curves of HD 50526
obtained by analyzing photometric observations; final O-C residuals between
the observed and synthetic light curves; fluxes of donor, gainer and of the
accretion disc, normalized to the donor flux at phase 0.25; the views of the
model at orbital phases 0.05, 0.30, and 0.80, obtained with parameters
estimated by the light curve analysis.

###  5.2 The best light-curve model

The obtained results for the system are summarized in Fig. 13 and presented in
Table 1. The best model shows an orbital inclination $61.\\!\\!^{\circ}28\pm
0.\\!\\!^{\circ}2$ degree. The stars have masses $5.48\pm 0.02$ and $1.13\pm
0.02~{}\mathrm{M_{\odot}}$, radii $3.57\pm 0.03~{}\mathrm{R_{\odot}}$ and
$7.09\pm 0.01~{}\mathrm{R_{\odot}}$ and surface gravities $\log{g}=4.07\pm
0.02\mathrm{dex}$ and $\log{g}=2.79\pm 0.02~{}\mathrm{dex}$. Considering their
temperatures and surface gravities they correspond to a pair B5 V + A0 III.
The B type star fits relatively well the mass-radius expectations for main
sequence stars but the A type star turns to be slightly bigger than expected,
which causes that its surface gravity and mean density appears affected and
look more evolved than it appears to be.

The model also contains an accretion disc, concave and optically thick around
the hot star with a radius of $R_{\mathrm{d}}=14.74\pm
0.02~{}\mathrm{R_{\odot}}$, which is 4.1 times larger than the star in its
center ($R_{\mathrm{h}}=3.57\pm 0.031~{}\mathrm{R_{\odot}}$), with a central
thickness $d_{\mathrm{c}}=2.87\pm 0.032~{}\mathrm{R_{\odot}}$ and an edge
thickness $d_{\mathrm{e}}=4.26\pm 0.02~{}\mathrm{R_{\odot}}$. The gainer star
is partly hidden by the disc whose temperature in the external border is
$T_{\mathrm{d}}=9401\pm 200~{}\mathrm{K}$. The disk presents a hot spot on the
its outer edge with a temperature of $T_{\mathrm{hs}}=12785\pm
392~{}\mathrm{K}$ and a bright spot temperature of $T_{\mathrm{bs}}=11093\pm
368~{}\mathrm{K}$, placed almost in the opposite location of the disk.

The disk dominates the light of the system at the $V$-band, contributing 67%
to the total flux at orbital phase 0.25, whereas the donor contributes 25% and
the gainer the remaining 8%. Let’s remember that the disk luminosity is not
accretion-driven in double periodic variables, being the disk mainly heated by
high-energy radiative flux from the hotter star (Mennickent et al., 2016).

##  6 Discussion

HD 50526 turns to be a typical DPV, consisting of a B dwarf + A giant
semidetached pair, and with evidence of circumstellar matter around the hotter
star. While the continuum light curve is well represented including an
optically thick accretion disk around the gainer, the line emission Doppler
maps reflect the complex dynamics of gaseous mass fluxes.

Spectroscopic studies of non-eclipsing double periodic variables are scarce.
If the region causing the long photometric cycle is located somewhere in the
orbital plane, it should be differently visible in eclipsing and non-eclipsing
systems. In these later, the projected disk surface is larger and the observer
should dispose of a better view of the disk features. It makes sense then to
compare HD 50526 with HD 170582, two DPVs seen under intermediate latitudes.
Data for these two systems are listed in Table 9. Both systems are binaries
consisting of a B dwarf and a A giant, observed at inclination slightly larger
than 60$\fdg$0.

The disk contributes 67% to the flux at the $V$-band in HD 50526, while it
does only 30% in HD 170582. This is partly due to bigger secondary star in
this system, characterized also by a larger orbital separation and longer
photometric periods. Both orbital separation and donor radius are about twice
larger in HD 170582. While the gainer is about 2kK hotter in HD 170582, the
donor is about 2kK coldest. Interestingly, the full amplitude of variability
for the long cycle is 4 times smaller in HD 170582. In this binary the hot and
bright spots are hotter than the surrounding disk compared with those of HD
50526. However, in terms of absolute temperatures, HD 50526 have hotter disk
spots than HD 170582, 12785 K (hs) and 11093 K (bs) versus 8980 K and 7899 K,
respectively. The disk outer edge temperature is about 9400 K for HD 50526 and
5400 K for HD 170582. We notice that the large amplitude long cycle is
observed in the system with hotter disk spots. Another peculiar
characteristics of HD 170582 is the presence of a Helium 5875 absorption
feature arising from near the hot spot, not detected in HD 50526. The presence
of Helium absorption suggest high temperature gas around the hot spot,
something not observed in the shorter period system. This is in apparent
conflict with the temperatures derived from the light curve model, indicating
hotter spots in the shorter period system. At present we have not an
explanation for this discrepancy.

We observe that differences exist in the two systems under comparison, and
conjecture that not only orbital and stellar parameters determine the observed
behaviour, but probably other variables like the mass transfer rate.

In semidetached Algol, mass from the coldest (donor) star flows through the
inner Lagrangian point onto the hotter (gainer) star forming a gas stream. If
the gainer is large enough, the stream hits the star forming a hot shock
region on its surface. If the gainer is small enough then the stream turns
around the star hitting itself and eventually forming an accretion disk when
spreading by the effect of the viscosity. This was investigated by Lubow & Shu
(1975) who determined this gainer critical radius $r_{\mathrm{c}}$ as a
function of the mass ratio. Actually, due to the finite width of the stream, a
range of radii marks the transition boundary between impact systems and disk
systems. On the other hand the disk, if formed, cannot extend beyond the
region when the tidal forces impede its stability, this maximum radius
$r_{\mathrm{tidal}}$ is often also parameterized in terms of the system mass
ratio.

Impact and disk systems can be studied in a diagram of fractional radius (the
radius relative to the orbital separation) and mass ratio. We notice that the
fractional radius of the hotter star $R_{1}/a_{\mathrm{orb}}=0.127\pm 0.001$
along with the mass ratio $q=0.206$ place HD 50526 into the area of tangential
stream impact where most DPVs are found (Figure 14). This means that if mass
transfer occurs due to Roche lobe overflow, the gas stream hits tangentially
the more massive star transferring efficiently the angular momentum and
eventually spinning-up the gainer star. The fact that most DPVs are found in
this area was noticed by Mennickent et al. (2016). The HD 50526 disk, on the
other hand, has a fractional radius $R_{d}/a_{\mathrm{orb}}=0.526\pm 0.001$
and is located very near the tidal radius limit, where the influence of tidal
forces should disperse the matter particles of the disk limiting its size
(Fig. 14). This means that the radial extension of the disk is large, since
almost fills the Roche lobe of the gainer and material in the outer edge could
be subject to tidal forces and flow around the system in non-Keplerian orbits.
We notice that not all disks of DPVs reach the tidal limit. Since most radii
have been determined for the optically thick disk contributing to the
continuum flux and modeling the light curve, still is possible that optically
thin parts contributing to line emission extend beyond the measured radius.
This could explain the complex morphology of the Doppler maps in HD 50526.

We observe a clustering of DPVs around $q$= 0.16-0.3 (Fig. 14). Only the
relatively massive UU Cas (total mass $\sim$ 26 M⊙) have a larger $q$
$\approx$ 0.54. For this system, a long cycle of $\sim$ 270 d was detected in
the $I_{c}$-band that needs to be ratified for further photometric campaigns
(Mennickent et al., 2020).

It has been argued that DPV disk luminosities are not accretion-driven but
supported by heating from the central star (Mennickent et al., 2016). Can we
test if this statement is applicable for HD 50526? Assuming that the disk is
an optically thick, geometrically thin steady-state accretion disk, its
temperature is directly related to its mass transfer rate ($\dot{M}$). We use
canonical formulae to determine $\dot{M}=2.44(2)\times
10^{-6}~{}\mathrm{M_{\odot}\,yr^{-1}}$ (e.g. Frank, King, & Raine, 2002, Eq.
5.43). If the accretion process is conservative, then an increase in the
orbital period of 2.98 s/yr is expected (e.g. Hilditch, 2001, Eq. 4.56).
Considering the error in the orbital period $\epsilon_{P}=0.001~{}\mathrm{d}$
and the time baseline of the photometric time series $\Delta
T=2613~{}\mathrm{d}$, we expect at most a drift in the period of
$2\times\epsilon_{P}/\Delta T=24~{}\mathrm{s\,yr^{-1}}$. As we see, more
photometric data is needed to constrain further the orbital period and test if
the disk has the aforementioned mass transfer rate. Since there are no
evidence of significant mass outflows like P-Cygni emission profiles, highly
shifted lines in radial velocity, strong systemic reddening or a circumstellar
nebula, it is possible that the mass transfer in the system is near
conservative.

All these arguments must be evaluated in the context of accretion-decretion
disks expected in some DPVs. Since the tangential impact should accelerate the
star until critical rotation, we might have a disk with an inner boundary
transporting angular momentum and matter outside and the outer boundary
transporting mass inside. This kind of relatively massive, self-gravitating
accretion-decretion disks are starting to be studied by Wilson (2018, 2020)
and its importance in the field of algols, double periodic variables and other
close binaries remains to be established by future investigations.

Table 9: Comparison between the HD 50526 data obtained in this paper and those
of HD 170582 obtained by Mennickent et al. (2015) in the synchronous case.
$\Delta V$ is the long-cycle full amplitude and $f_{r}$ the fractional disk
contribution to the $V$ band flux at $\Phi_{\mathrm{o}}=0.25$.
$\begin{array}[]{@{\extracolsep{+0.0mm}}lrr@{}}\hline\cr\vskip 3.0pt plus
1.0pt minus 1.0pt\cr{\rm Quantity}&{\rm HD\,50526}&{\rm HD\,170582}\\\ \vskip
3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr{\rm P_{o}}[{\rm days}]&6.701&16.871\\\ {\rm P_{l}}[{\rm
days}]&190&587\\\ {\rm P_{l}/P_{o}}&28.4&34.8\\\ i{\rm[^{\circ}]}&61.3&67.4\\\
\Delta V&0.20&0.05\\\ a_{\rm orb}{\rm[R_{\odot}]}&28.0&61.2\\\ {\rm
T_{d}}[{\rm K}]&9401&5410\\\ {\rm T_{h}}[{\rm K}]&16000&18000\\\ {\rm
T_{c}}[{\rm K}]&10500&8000\\\ \cal{R}_{\rm d}{\rm[R_{\odot}]}&14.7&20.8\\\
f_{r}&0.67&0.30\\\ {\rm A_{hs}=T_{hs}/T_{d}}&1.36&1.66\\\
{\rm\lambda_{hs}}{\rm[^{\circ}]}&343.1&336.6\\\ {\rm
A_{bs}=T_{bs}/T_{d}}&1.18&1.46\\\
{\rm\lambda_{bs}}{\rm[^{\circ}]}&129.5&134.8\\\ \cal M_{\rm{}_{h}}{[\cal
M_{\odot}]}&5.5&9.0\\\ \cal M_{\rm{}_{c}}{[\cal M_{\odot}]}&1.1&1.9\\\ \cal
R_{\rm{}_{h}}{\rm[R_{\odot}]}&3.6&5.5\\\ \cal
R_{\rm{}_{c}}{\rm[R_{\odot}]}&7.1&15.6\\\ {\rm log}\ g_{\rm{}_{h}}&4.1&3.9\\\
{\rm log}\ g_{\rm{}_{c}}&2.8&2.3\\\ \hline\cr\vskip 3.0pt plus 1.0pt minus
1.0pt\cr\end{array}$
Table 10: Primary star and disk radius relative to the orbital separation and
mass ratio for double periodic variables, along with their errors. iDPV stands
for OGLE 05155332-6925581 and OGLE157529 for OGLE-BLG-ECL-157529. Data are
from Mennickent et al. (2016) and references therein and from Mennickent et
al. (2020) and Gorda (2017) (UU Cas), Mennickent et al. (2020) (OGLE-BLG-
ECL-157529, two epochs ”a” and ”b” are given) and Rosales Guzmán et al. (2018)
(V495 Cen).
$\begin{array}[]{@{\extracolsep{+0.0mm}}lcccccc@{}}\hline\cr\vskip 3.0pt plus
1.0pt minus 1.0pt\cr{\rm System}&{\rm R_{d}/a}&{\rm eR_{d}/a}&{\rm
R_{1}/a}&{\rm eR_{1}/a}&q&eq\\\ \vskip 3.0pt plus 1.0pt minus
1.0pt\cr\hline\cr\vskip 3.0pt plus 1.0pt minus 1.0pt\cr\hline\cr{\rm
V393\,Sco}&0.276&0.012&0.103&0.007&0.256&0.030\\\ {\rm
DQ\,Vel}&0.434&0.014&0.121&0.008&0.301&0.030\\\ {\rm
AU\,Mon}&0.302&0.017&0.121&0.013&0.171&0.030\\\ {\rm
iDPV}&0.401&0.020&0.159&0.008&0.209&0.020\\\
{\rm\beta\,Lyrae}&0.484&0.008&0.103&0.004&0.226&0.020\\\ {\rm
HD\,170582}&0.340&0.006&0.090&0.004&0.210&0.010\\\ {\rm
V495\,Cen}&0.486&0.017&0.054&0.003&0.158&0.008\\\ {\rm
DPV\,065}&0.501&0.010&0.176&0.007&0.203&0.008\\\ {\rm
HD\,50526}&0.526&0.001&0.127&0.001&0.206&0.033\\\ {\rm
UU\,Cas}&0.398&0.027&0.134&0.002&0.537&0.138\\\ {\rm
OGLE\,157529a}&0.424&0.005&0.069&0.003&0.220&0.010\\\ {\rm
OGLE\,157529b}&0.514&0.005&0.069&0.003&0.220&0.010\\\ \vskip 3.0pt plus 1.0pt
minus 1.0pt\cr\hline\cr\end{array}$ Figure 14:  The fractional radius,
relative to the orbital separation, for primaries and disks in algols with and
without disks. Red circles and triangles show primaries and disks of double
periodic variables. Semidetached algol primaries from Dervişoǧlu, Tout, &
Ibanoǧlu (2010) are also shown as black points. Below the circularization
radius shown by the solid black line, a disk should be formed and below the
dashed line, a disk might be formed. The tidal radius indicates the maximum
possible disk extension (upper dashed line). Data are from Table 10.

##  7 Conclusions

In this work we have studied the double periodic variable HD 50526 using
survey photometry and new spectroscopic data obtained in optical wavelengths
with several spectrographs at high and medium resolution. We summarize our
main conclusions as follows :

  * •

From the photometric study of ASAS data, we determined an orbital period of
$P_{\mathrm{o}}=6\fd 701\pm 0\fd 001~{}\mathrm{d}$ and a long period of
$P_{\mathrm{l}}=191\pm 2~{}\mathrm{d}$, both results more reliable than the
obtained from spectroscopic analysis due to the larger density of data and
time baseline of the photometric time series.

  * •

We performed a radial velocity study using the PIKAIA code resulting in the
best orbital parameters given in Table 5. In particular, we obtain a half-
amplitude of radial velocity for the cold star
$K_{2}=153~{}\mathrm{km\,s^{-1}}$ and a systemic velocity
$\gamma=-5.17~{}\mathrm{km\,s^{-1}}$.

  * •

Sharp lines of the cold stellar component dominate the spectrum. A comparison
with a grid of synthetic stellar spectrum indicates a temperature
$T_{\mathrm{2}}=10500\pm 125~{}\mathrm{K}$, surface gravity $\log{g}=3.0\pm
0.25~{}\mathrm{dex}$, and $v\sin{i}=70\pm 20~{}\mathrm{km\,s^{-1}}$.

  * •

Double emission flanks surrounding the H$\alpha$ cold star absorption lines
are detected, suggesting the existence of an accretion disk in the system.

  * •

We find, through light curve modeling, a complete solution for the system
including stellar and disk parameters that are listed in Table 8. We find a
binary consisting of a pair of B5 V + A0 III stars. The cold star fills its
Roche lobe and transfer mass onto the hot star, forming and accretion disk of
radius $14.7~{}\mathrm{R_{\odot}}$ and outer edge temperature 9400 K.

  * •

According to the model, the accretion disk has two bright and hot active
regions, 1.36 and 1.18 times hotter than the surrounding disk, and located 343
and 47 degrees apart from a line connecting the stellar centers as measured in
the orbital motion direction.

  * •

We construct Doppler maps for the H$\alpha$ and H$\beta$ lines, finding
additional and complementary evidence of circumstellar emitting material
located around the gainer star, consistent with the existence of the accretion
disk.

  * •

The long-cycle variability, although unknown in its nature, could be related
to the existence of mass flows in the systems as shown in the Doppler maps and
general spectroscopic variability.

## Acknowledgements

Part of the observations presented in this paper were conducted by the late
Zbigniew Kołaczkowski during his postdoctoral stay in the University of
Concepción, Chile. We thanks Sebastián Otero for useful insights on
photometric data surveys. This publication makes use of VOSA, developed under
the Spanish Virtual Observatory project supported from the Spanish MICINN
through grant AyA2011-24052. J.R. and R.E.M. gratefully acknowledges support
from the Chilean BASAL Centro de Excelencia en Astrofísica y Tecnologías
Afines (CATA) grant PFB-06/2007, AFB-170002 and support by VRID-Enlace
218.016.004-1.0. This research was funded in part by a scholarship from
Faculty of Physical Sciences and Mathematics of the Universidad de Concepción
(UdeC). R.E.M. and D.R.G.S. thanks for funding through FONDECYT regular,
project codes 1201280 and 1190621, and through the “Concurso Proyectos
Internacionales de Investigacion, Convocatoria 2015”, project code
PII20150171. G. D. gratefully acknowledge the financial support of the
Ministry of Education, Science and Technological Development of the Republic
of Serbia through contract No. 451-03-68/2020-14/200002. S. Z. acknowledges
PAPIIT-DGAPA-UNAM grant IN102120. I.A. and M.C. acknowledges support from
FONDECYT project Nº 1190485. I. A. is also grateful for the support from
FONDECYT project Nº 11190147. Also this work has made use of data from the
European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),
processed by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

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# WEcharge: democratizing EV charging infrastructure

Md Umar Hashmi1  Mohammad Meraj Alam2  Ony Lalaina Valerie Ramarozatovo3
Mohammad Shadab Alam4<EMAIL_ADDRESS>1KU Leuven & EnergyVille,
Belgium 2FlexThor BV, Belgium 3Independent Consultant, France 4University of
Hasselt, Belgium

###### Abstract

The sustainable growth of EVs will have to be met with proportional growth in
EV charging infrastructure. With limited urban spaces to place new charging
stations, shrinking profitability, privately owned charging facilities need to
be shared. WEcharge will allow privately owned charging infrastructure to be
shared with public EV owners using a business model. We propose a resource
matching algorithm that takes into account incoming EV preferences, hard
constraints for such EV, and provides the best suited resource for charging.
We demonstrate the applicability of the matching model by showing a realistic
case study with a Nissan Leaf 40 kW EV and 25 company and publicly owned
charging stations (DC fast charger, AC rapid charger, level 1 and level 2
charger) in Hasselt, Belgium. The case study shows that consumer preferences
will govern resource matching.

###### keywords:

Electric vehicle, charging infrastructure, matching model, business model

##  1 Introduction

Electric vehicle (EV) share in transportation is expected to grow in the
coming decades. According to BNEF report, every sixth car sold in the world
will be an EV by 2025 and the global sales can hit approximately 17 million
bloomberg . Although, a slow growth in the first quarter of 2020 and 2019 has
been observed, even then the overall market of EV sales rose to 26 percent and
overall by 44 percent in Europe in 2021 mckinsey . Moreover, US Energy
Information Administration (EIA) predicts that by 2050, EV fleet will compose
34% in OECD countries and 28% of total light duty vehicles in non-OECD
countries euobservatory . This projection is from EV consisting of a meagre
0.7% of the total light duty vehicles in the year 2020 to 31% globally by 2050
eiaprojects . One of the main reasons behind this rapid growth and market
acceptability of EVs has been the steep decline in lithium battery prices over
the last decade, with a mean change of 19% year-on-year, as shown in Figure 1
(a) and (b).

The unprecedented growth of EV is helping the market and the governments to
achieve the zero emissions objective. However. with such an unparalleled
growth of EVs, the stress on existing EV charging infrastructure is growing as
well, leading to an increase in government and company owned investments in
installing new charging infrastructure hall2017emerging . Figure 2 (a) and (b)
shows the growth of EV charging infrastructure in Europe and on average EV per
public charging facility. As EVs become more popular, more public charging
stations need to be installed which is expensive, time-consuming, permit
required and grid connections upgrades. According to a study done by NREL, the
US will need to build 50000 DC fast chargers and 1.2 million level 2 fast
chargers for 35 million EVs by 2030 wood2017national .

Many EV owners have installed EV chargers at their homes which they use to
charge their EV’s generally after the office hours (in the evening and at
night when the network tariff is low) and during the weekends. Since these
privately-owned chargers are available only for private use, they remain non-
utilized from 16 to 18 hours per day on an average (60% – 70% of duration)
virta . The reason behind this non-participation of privately-owned chargers
with normal public is non-existence of any digital platform at this moment
where the homeowners can host their privately-owned chargers on rent and
monetize their energy asset when not in use. There is a clear gap between the
private charger owner and public EV owner, where a business landscape exist to
connect both players together.

This paper discusses the pain point of EV owners when travelling to a
destination with incomplete real-time information about availability of public
charging stations during peak-hours, under-utilized privately-owned chargers,
and presents an innovative business model between the privately-owned charger
owner and EV owner to fill this gap in section 2. In section 3, an adaptive
infrastructure matching algorithm is proposed to match charging stations with
incoming EVs. Moreover, a case study on matching mechanism has been
demonstrated with a 40 kW Nissan Leaf and 25 charger stations in a city of
Belgium, Hasselt, in section 4. Section 5 discusses the conclusions of the
paper.

Figure 1: (a) Lithium battery pack price (b) Mean change in battery pack
prices bloomberg  Figure 2: (a) Growth of EV charging infrastructure in Europe
(b) Average EV per public charging facility euobservatory .

##  2 Business model

Presently, an EV owner has incomplete real-time information about any EV
charging station infrastructure such as (i) if a resource occupied or idle,
(ii) charger and socket type, (iii) location, (iv) charging tariffs and
penalties, etc. When EV owners travel to other destination, they charge their
EV’s using charging stations owned by the government or charging stations
companies. According to a survey, there is a waiting time of 10 minutes to 15
minutes for EV owners in real-time to find a publicly available charging spots
during the peak-hours while many private charging stations owned by
homeowners, SME’s and private companies stand idle and underutilized evcharg .

The goal of WEcharge platform is to reduce and evenly distribute (increasing
utilization factor of underutilized resources) the stress on the EV charging
station infrastructure by bringing the homeowner and private companies owned
charging station into the mainstream through a privately-owned charging
station-as-a-service business model. Using our platform, not only the public
charging stations owned by the government will be listed but also the charging
stations owned by any private companies and homeowners can be listed along
with their characteristics as

  * •

real-time and future availability,

  * •

charging tariff and mode of charging,

  * •

charger type, socket, number of phases, and AC/DC,

  * •

electrical properties, voltage, power rating, amperage,

  * •

geographical coordinates,

  * •

manufacturer and mode of payment.

Private charging stations owned by homeowners, SME’s and private companies can
participate in this business model to decrease their charger non-utilization
time and earn remuneration to decrease the payback of their installation.
Moreover, a service to reserve a charging spot in advance or pre-book any
charging stations before commencing any travel is facilitated. Additionally,
the platform notifies EV owner in case of penalty for not removing EV after
its fully charged.

###  2.1 Customer segment

The WEcharge rent-a-charging spot service directly focuses on the following
customer segments, where the EV owner receives the ease of charging facility
at the tip of his finger and the other hand charger owner earns money on their
under-utilized privately-owned chargers. The EV charging infrastructure
includes

  * •

charging stations privately owned by privately,

  * •

charging stations owned by SME’s and companies,

  * •

and public/municipal owned charging stations.

###  2.2 Revenue model and partnerships

The platform services are a subscription based model that includes services
like pre-book charging stations, EV battery monitoring, real-time pricing,
price forecasting and charger-as-a-service participation. The customers have
the freedom to choose pay-as-you-go model with higher subscription fee as the
services increases. There are potential partnerships as shown in Fig. 4 listed
as

  * •

DSO and TSO for improving grid stability,

  * •

EV charger OEMs for up-to-date standards and protocols,

  * •

Charger installers for upgrading charging infrastructure,

  * •

Municipal/government for policies and regulation updates.

Figure 3: WEcharge business model Figure 4: WEcharge business partnerships

##  3 WEcharge: adaptive resource matching

The increase in EVs will make it more challenging to install new charging
infrastructure due to space constraint, and shrinking profitability flowbank .
This may bottleneck future growth of EVs forbes9 . The increased investment
could be avoided by facilitating private EV charging infrastructure to share
their resources for charging incoming EVs. This will reduce stress on public
and company owned charging resources, thus avoiding a probable bottleneck in
the growth of EV. Similar to Uber, which matches taxi driver to interested
rider, our proposed solution, WEcharge, matches incoming EVs to a pool of
public and private charging infrastructures based on (a) charging preference,
(b) EV make, (c) geographical proximity, (d) travel time to charging facility,
(e) remaining state-of-charge (SoC) of EV battery, (f) cost of charging, (g)
reduce incoming EV waiting period. Additional criteria for matching can be
added for considering consumer preferences, local distribution network
congestion issues, renewable energy integration, driving range, health of EV
batteries hall2017emerging  etc.

WEcharge will work autonomously and does not require human intervention.
However, in case the private charging infrastructure is going to be not
available, the owner can manually opt out. This will result in dropping of
this resource from WEcharge platform. This resource must be locally activated
to be shown in WEcharge. The skeleton of the matching model is shown in Fig.
5. In order to utilize the WEcharge platform, in step (J1) it gathers
attributes of the EV chargers, in step (J2) the incoming EV provides its
charging preferences as shown in Fig. 6, and the matching algorithm is
executed in (J3) step to find the best suited EV charger meeting consumer
needs. The bottleneck for WEcharge is the lack of universal EV charging
infrastructure: this could stress resources more than others, depending on the
growth of certain types of EVs. For example, if Tesla EVs compose of 30% total
EVs, however, Tesla compatible chargers compose of 40% then on average all
Tesla compatible charging infrastructure will be underutilized.

In Belgium, new charging infrastructure placement requires a feasibility study
conducted by an electric utility for analysing the impact of charging stations
on the distribution network. It is probable that some location with higher
probabilities of voltage and thermal congestion may limit such a placement.
However, such nodes will be highly effective in mitigating probable congestion
issues on such and neighbourhood nodes. Incoming EVs which are flexible in
charging profile could significantly contribute in realization of active
distribution network.

Figure 5: Matching incoming EV to charging infrastructure.

Proposed matching models have the following properties:

  * •

the democratization of charging infrastructure so as anyone can participate.
WEcharge encourages privately-owned EV charging infrastructure to participate
in resource sharing thus avoids public and corporate investment in new
charging infrastructures, and

  * •

matches incoming EV with charging infrastructure while considering various
charging modes and preferences.

###  3.1 Resource matching algorithm

The resource matching algorithm classifies the vehicle parameters and user
inputs into hard and soft constraints. The hard limitations need to be
satisfied and soft limitations should be optimized to find the best
alternative.

The hard constraints include

  * •

Vehicle charger type and socket compatibility,

  * •

Will the remaining SOC in the electric vehicle be able to reach the charging
facility or not, and

  * •

Availability of charging infrastructure at the present time or for pre-
booking.

The soft constraints include

  * •

Geographical distance denotes the distance between vehicle and the charging
infrastructure,

  * •

Duration of charging at the rated charging rate,

  * •

Waiting time: is assumed to be inversely proportional to the number of
chargers at the charging station,

  * •

Cost of charging: assuming EV is charged from zero SoC.

Consider there are $N$ charging infrastructures which satisfies the hard
constraints. An option $O_{i}$ for resource $i$ is given as

$O_{i}=\\{d_{i},t_{i},s_{i},c_{i}\\},$ (1)

where $d_{i}$ is the geographical distance, $t_{i}$ is the charging duration,
$s_{i}$ denotes waiting time, and $c_{i}$ is the cost of charging. Note, in
this paper we present a simple matching model. Several additional
considerations can also be included.

The performance metric is denoted as

$\begin{split}P_{i}&=\Big{(}\frac{w_{1}}{\bar{w}}\frac{d_{i}}{\max_{i\in\\{1,..,N\\}}d_{i}}+\frac{w_{2}}{\bar{w}}\frac{t_{i}}{\max_{i\in\\{1,..,N\\}}t_{i}}\\\
&+\frac{w_{3}}{\bar{w}}\frac{s_{i}}{\max_{i\in\\{1,..,N\\}}s_{i}}+\frac{w_{4}}{\bar{w}}\frac{c_{i}}{\max_{i\in\\{1,..,N\\}}c_{i}}\Big{)},\end{split}$
(2)

The matching algorithm returns the best suited charging infrastructure, given
as

$i^{*}=\arg\min_{i}P_{i},$ (3)

where $w_{1},w_{2},w_{3},w_{4}$ denotes the weights for features
$d_{i},t_{i},s_{i},c_{i}$ respectively. These weights are input by a user
looking for matching infrastructure. $\bar{w}$ denotes the mean weights, given
as $\bar{w}=\frac{1}{4}\sum_{k=\\{1,2,3,4\\}}w_{k}.$

Figure 6: The incoming EV charging preference is shown. These weights are used
for solving (3).

In (3), the weights and features are normalized for avoiding the skewness
caused by magnitude difference of the feature parameters. For all the
parameters in $O_{i}$ needs to minimized, therefore, (3) is a sum of product
of normalized weights and normalized features to be minimized. In case of a
parameter which needs to be maximized, such as utility of EV owner, then it
should be subtracted in (3).

Note that one charging infrastructure can be considered more than an option
based on number of modes of charging. Some of the EV charging modes could be

  * •

Only charging,

  * •

Vehicle to grid mode of charging:

    * –

Providing services to DSO by performing voltage regulation, congestion
management, peak reduction etc.,

    * –

Providing services to TSO by performing ancillary services, frequency
regulation etc.,

    * –

Performing energy arbitrage or power quality improvement based on billing
rules of the retailer.

The details of these modes of charging are not included in this paper. Since
the proposed matching algorithm is computationally fast, therefore, real-time
adaptive mapping will be easily implemented. This feature will be desired by
moving EVs searching for an electric vehicle charging point.

In the next section, we detail a case study showing how this algorithm can be
applied for real world mapping.

##  4 Case-study

The case study demonstrates the matching mechanism proposed in this paper.
Table 1 lists the details of the incoming EV. We consider EV charging matching
of a Nissan Leaf (2018) 40 kW located at point $x$, shown in Fig. 6. Based on
the remaining SoC of the battery, suppose 25 charging stations are present, as
shown in Fig. 6. Note that except for charger 15 in Fig. 6, all other chargers
are compatible to Nissan Leaf, implying hard constraints are satisfied.

Table 1: Incoming EV details podpoint ; mobilityhouse ; evbox  Model | Nissan Leaf 2018
---|---
Battery capacity | 40kWh
Total electrical range | 378 km
Plug Type | Type 2
Domestic, 10 A, 1-phase | 2.3 kW | 17 hours
1-Phase, 16A | 3.7 kW | 11 hours
1-Phase, 32A | 7.4 kW | 6 hours
3-Phase, 16A | 11 kW | 10 hours
3-Phase, 32A | 22 kW | 6 hours
DC Fast Charger | 50 kW | 45 minutes
Figure 7: Map of Hasselt in Belgium google ; chargemap , with EV charging
infrastructure marked (non-exhaustive list). The location of incoming EV is
marked with $x$.

Table 2 summarizes the performance metrics based on four parameters described
in (3). The performance metrics are calculated for two scenarios:

  * •

S1: Equal weights; $w_{1}=w_{2}=w_{3}=w_{4}=0.25$,

  * •

S2: Unequal weights; $w_{1}=w_{3}=0.4$, and $w_{2}=w_{4}=0.1$.

Fig. 8 shows the performance metric for all charging infrastructure satisfying
the hard constraints. For scenario S1 the first EV station is best suited and
for S2 EV charging station id 12 should be selected. Clearly, we can observe
that incoming EV charging preferences will govern the selection of charging
infrastructure.

Table 2: Performance metrics for vehicle at point $x$ |  |  |  | Norm. | S1 | S2
---|---|---|---|---|---|---
id | Wait | Norm. | Norm. | charge | Equal | Varying
| time | distance | cost | time | weight | weight
1 | 0.17 | 0.696 | 1 | 0.068 | 0.483 | 0.452
2 | 0.33 | 0.717 | 1 | 0.068 | 0.530 | 0.527
3 | 0.5 | 0.783 | 0.722 | 0.545 | 0.638 | 0.640
4 | 0.5 | 1 | 0.722 | 0.545 | 0.692 | 0.727
5 | 0.5 | 0.674 | 0.722 | 0.545 | 0.610 | 0.596
6 | 0.5 | 0.891 | 0.722 | 0.545 | 0.665 | 0.683
7 | 0.08 | 0.652 | 0.722 | 0.545 | 0.501 | 0.421
8 | 0.5 | 0.239 | 0.722 | 0.545 | 0.502 | 0.422
9 | 0.17 | 0.652 | 0.722 | 0.545 | 0.522 | 0.454
10 | 0.5 | 0.543 | 0.722 | 0.545 | 0.578 | 0.544
11 | 0.5 | 0.652 | 0.722 | 0.545 | 0.605 | 0.588
12 | 0.5 | 0.174 | 0.722 | 0.545 | 0.485 | 0.396
13 | 0.5 | 0.109 | 0.722 | 0.909 | 0.560 | 0.407
14 | 0.5 | 0.217 | 0.722 | 0.909 | 0.587 | 0.450
16 | 0.5 | 0.348 | 0.722 | 0.909 | 0.620 | 0.502
17 | 0.5 | 0.652 | 0.722 | 0.909 | 0.696 | 0.624
18 | 0.5 | 0.348 | 0.722 | 0.909 | 0.620 | 0.502
19 | 0.5 | 0.522 | 0.722 | 0.909 | 0.663 | 0.572
20 | 0.5 | 0.609 | 0.722 | 0.909 | 0.685 | 0.607
21 | 0.5 | 0.478 | 0.741 | 0.909 | 0.657 | 0.556
22 | 0.5 | 0.457 | 0.722 | 0.909 | 0.647 | 0.546
23 | 0.5 | 0.696 | 0.722 | 0.545 | 0.616 | 0.605
24 | 0.5 | 0.348 | 0.722 | 1 | 0.643 | 0.511
25 | 1 | 0.717 | 0.722 | 1 | 0.860 | 0.859
Figure 8: Performance metrics

##  5 Conclusion

Democratization of EV infrastructure is a crucial next step to ensure
sustained growth of new EVs. We propose a business model, WEcharge, which
could allow privately owned EV charging infrastructure to share their
resource. This will reduce congestion and waiting time for company and
publicly owned charging infrastructure. WEcharge will also assist in mapping
incoming EVs to their best suited EV charging infrastructure. We demonstrate
the applicability of this matching algorithm using a realistic case study.
This case study consider Renault Leaf at the city centre in Hasselt in
Belgium. There are 25 options for charging. Our matching algorithm provides
the best suited charging infrastructure based on consumer preferences.

In future work, we will extend this matching models to consider more real-time
parameters and various modes of charging while providing services to DSOs
and/or TSO.

## References

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