11institutetext:  Charles University, Faculty of Mathematics and Physics,
Astronomical Institute, V Holešovičkách 2, CZ-180 00 Praha 8-Trója, Czech
Republic 22institutetext: Uniwersytet Wrocławski, Instytut Astronomiczny,
Kopernika 11, 51-622 Wrocław, Poland 33institutetext: Hvar Observatory,
Faculty of Geodesy, Zagreb University, Kačicéva 26, 10000 Zagreb, Croatia
44institutetext: Czech Academy of Sciences, Astronomical Institute, CZ-25165
Ondřejov, Czech Republic 55institutetext: Université de Montréal, Département
de physique, C.P.6128, Succursale center-Ville, Montréal, Québec, H3C 3J7,
Canada 66institutetext: Department of Physics, Mount Allison University,
Sackville, NB, E4L1E6, Canada 77institutetext: Department of Physics and Space
Science, Royal Military College of Canada, Kingston, Ontario K7K 7B4, Canada
88institutetext: Institut für Astro- und Teilchenphysik, Universität
Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria 99institutetext:
Department of Electronics, Electrical Engineering and Microelectronics,
Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland.
1010institutetext: University of Vienna, Institute for Astrophysics,
Türkenschanzstraße 17, 1180 Vienna, Austria

#  Spectrum of the secondary component and new orbital elements of the massive
triple star $\delta$ Ori A ††thanks:  Based on spectroscopic CCD observations
with a coudé spectrograph attached to the 2m reflector of the Astronomical
Institute AS ČR at Ondřejov, archival Haute Provence and ESO La Silla spectra,
ground-based $U\\!B{}V$ photometry from Hvar, and data collected by the BRITE
Constellation satellite mission, designed, built, launched, operated, and
supported by the Austrian Research Promotion Agency (FFG), the University of
Vienna, the Technical University of Graz, the University of Innsbruck, the
Canadian Space Agency (CSA), the University of Toronto Institute for Aerospace
Studies (UTIAS), the Foundation for Polish Science & Technology (FNiTP MNiSW),
and National Science Centre (NCN). ,††thanks: Tables 14 and 15 are also
available in electronic form at the CDS via anonymous ftp to
cdsarc.cds.unistra.fr (130.79.128.5) or via https://cdsarc.cds.unistra.fr/…+A/

A. Oplištilová 11 P. Mayer Pavel Mayer passed away on November 7, 2018.11 P.
Harmanec 11 M. Brož 11 A. Pigulski 22 H. Božić 33 P. Zasche 11 M. Šlechta 44
H. Pablo 55 P. A. Kołaczek-Szymański 22 A. F. J. Moffat 55 C. C. Lovekin 66 G.
A. Wade 77 K. Zwintz 88 A. Popowicz 99 W. W. Weiss 1010<EMAIL_ADDRESS>
(Received )

$\delta$ Orionis is the closest massive multiple stellar system and one of the
brightest members of the Orion OB association. The primary (Aa1) is a unique
evolved O star. In this work, we applied a two-step disentangling method to a
series of spectra in the blue region (430 to 450 nm), and we detected spectral
lines of the secondary (Aa2). For the first time, we were able to constrain
the orbit of the tertiary (Ab) – to $55\,450\,{\rm d}$ or 152 yr – using
variable $\gamma$ velocities and new speckle interferometric measurements,
which have been published in the Washington Double Star Catalogue. In
addition, the Gaia DR3 parallax of the faint component (Ca+Cb) constrains the
distance of the system to $(381\pm 8)\,{\rm pc}$, which is just in the centre
of the Orion OB1b association, at $(382\pm 1)\,{\rm pc}$. Consequently, we
found that the component masses according to the three-body model are $17.8$,
$8.5$, and $8.7$ M⊙, for Aa1, Aa2, and Ab, respectively, with the
uncertainties of the order of $1$ M⊙. We used new photometry from the BRITE
satellites together with astrometry, radial velocities, eclipse timings,
eclipse duration, spectral line profiles, and spectral energy distribution to
refine radiative properties. The components, classified as O9.5 II + B2 V + B0
IV, have radii of $13.1$, $4.1$, and $12.0\,{\rm R}_{\odot}$, which means that
$\delta$ Ori A is a pre-mass-transfer object. The frequency of $0.478\,$cycles
per day, known from the Fourier analysis of the residual light curve and X-ray
observations, was identified as the rotation frequency of the tertiary.
$\delta$ Ori could be related to other bright stars in Orion, in particular,
$\zeta$ Ori, which has a similar architecture, or $\varepsilon$ Ori, which is
a single supergiant, and possibly a post-mass-transfer object.

###### Key Words.:

Stars: close – Stars: massive – Stars: binaries: eclipsing – Stars:
fundamental parameters – Stars: individual: $\delta$ Ori – Techniques:
spectroscopic

††offprints: A.O.,

##  1 Introduction

The bright star $\delta$ Ori (HR 1852, HD 36486, HIP 25930, ADS 4134) is a
multiple stellar system consisting of six components: Aa1, Aa2, Ab, B, Ca, and
Cb, more specifically, the eclipsing binary Aa1+Aa2, the interferometric
binary (Aa1+Aa2)+Ab, the faint visual companion B, and the spectroscopic
binary Ca+Cb (see Fig. 1). Their properties can be summarised as follows:

  * •

Aa1+Aa2 ($V_{\mathrm{Aa1}}=2.55\,{\rm mag}$, $V_{\mathrm{Aa2}}\simeq 5.5\,{\rm
mag}$)111$\alpha_{\mathrm{J2000}}=5^{\mathrm{h}}\,32^{\mathrm{m}}\,0.398^{\mathrm{s}}$
and $\delta_{\mathrm{J2000}}=-00^{\circ}\,17^{\prime}\,56.69^{\prime\prime}$
is a detached eclipsing binary with a negligible mass transfer, the orbital
period $P_{1}=5.732436\,{\rm d}$ (Mayer et al. 2010), a slightly eccentric
orbit ($0.08$), and apsidal motion ($1.45^{\circ}\,{\rm yr}^{-1}$) (Pablo et
al. 2015).

  * •

Ab ($V_{\mathrm{Ab}}=3.7\,{\rm mag}$) is a nearby companion, which forms an
interferometric pair with Aa1+Aa2. It was discovered by Heintz (1980),
confirmed by speckle interferometry (Mason et al. 1999) and by Hipparcos
astrometry (Perryman & ESA 1997) of the (Aa1+Aa2)+Ab system. The corresponding
orbital period $P_{2}$ must be of the order of tens of thousands of days.

  * •

B ($V_{\mathrm{B}}\simeq 14\,{\rm mag}$) is a very faint distant
companion222$\alpha_{\mathrm{J2000}}=5^{\mathrm{h}}\,31^{\mathrm{m}}\,58.745^{\mathrm{s}}$
and $\delta_{\mathrm{J2000}}=-00^{\circ}\,18^{\prime}\,18.65^{\prime\prime}$
that is probably not associated with the system. Assuming that the component
is a main-sequence star, its absolute magnitude of 6.7 mag corresponds to the
spectral type K.

  * •

Ca+Cb ($V_{\mathrm{Ca+Cb}}=6.85\,{\rm mag}$) is another distant
companion333$\alpha_{\mathrm{J2000}}=5^{\mathrm{h}}\,32^{\mathrm{m}}\,00.406^{\mathrm{s}}$
and $\delta_{\mathrm{J2000}}=-00^{\circ}\,17^{\prime}\,04.38^{\prime\prime}$
to (Aa1+Aa2)+Ab that is a spectroscopic, non-eclipsing binary with a period of
$29.96\,{\rm d}$ and of spectral types B3 V + A0 V (Leone et al. 2010).

In the present paper, we focus on the triple sub-system $\delta$ Ori
(Aa1+Aa2)+Ab, with $V_{\mathrm{Aa1+Aa2+Ab}}=2.223\,{\rm mag}$ (from the
differential photometry at the Hvar Observatory),
$\alpha_{\mathrm{J2000}}=5^{\mathrm{h}}\,32^{\mathrm{m}}\,00.400^{\mathrm{s}}$,
and $\delta_{\mathrm{J2000}}=-00^{\circ}\,17^{\prime}\,56.74^{\prime\prime}$.
Hereinafter, the parameters corresponding to the inner orbit Aa1+Aa2 and to
the outer orbit (Aa1+Aa2)+Ab are denoted by indices 1 and 2, respectively. The
parameters of the components Aa1, Aa2, and Ab are denoted by indices 1, 2, and
3, respectively.

Many researchers have studied the system since the end of the 19th century.
For a detailed summary of the early investigation of $\delta$ Ori, we refer
readers to our earlier study of the system (Mayer et al. 2010). As far as
studies of the 21st century are concerned, Harvin et al. (2002) carried out a
tomographic separation of the ultraviolet and optical spectra into two systems
of spectral lines, interpreted them as the lines of the primary and secondary
of the eclipsing subsystem, and concluded that the components have
unexpectedly low masses ($m_{1}=11.2\,{\rm M}_{\odot}$ and $m_{2}=5.6\,{\rm
M}_{\odot}$). However, Mayer et al. (2010), showed that the optical spectra
are dominated by the spectral lines of the O9.5 II primary (Aa1; Walborn 1972)
and the similarly hot tertiary (Ab), and that the system has normal masses for
O and early-B stars (Harmanec 1988). The previous solution of the light curves
(LCs) led Mayer et al. (2010) to the conclusion that the faint secondary (Aa2)
contributes only a few percent to the total flux. Although they carried out
disentangling of the spectra, they were unable to find its spectral lines
convincingly, and could only rely on an indirect estimate of the mass ratio
$m_{2}/m_{1}$.

Five in-depth studies of $\delta$ Ori were published in 2015 (the first four
are a series): Corcoran et al. (2015) presented an overview of deep Chandra
HETGS X-ray observations that covered nearly the entire binary (Aa1+Aa2)
orbit. The observed X-ray emission was dominated by wind shocks from the
primary (Aa1). Nichols et al. (2015) discussed the time-resolved and phase-
resolved variability seen in the Chandra spectra. For the first time, they
found phase-dependent variability in the X-ray emission line widths. They
identified two periods in the total X-ray flux: $4.76\pm 0.30$ and $2.04\pm
0.05$ days. Pablo et al. (2015) carried out a detailed analysis of space-based
photometry from Microvariability and Oscillations of STars (MOST) and
simultaneously secured ground-based spectroscopy in the residuals of the
orbital LC, with periods ranging from 0.7 to 29 days. Shenar et al. (2015)
carried out a multi-wavelength non-local thermodynamic equilibrium (NLTE)
analysis of spectra. The determined parameters led to a O9.5 II, B1 V, and B0
IV spectral classification for Aa1, Aa2, and Ab, respectively, with evolved
primary (Aa1) and tertiary (Ab) components. They also found wind-driven mass
loss by the Aa1 component at $4\cdot 10^{-7}\,{\rm M}_{\odot}\,{\rm yr}^{-1}$.
Richardson et al. (2015) used cross-correlation of the ultraviolet spectra
from HST to obtain stellar parameter estimates for the primary, secondary, and
the tertiary that was angularly resolved in the observations.

In this work, we continue our earlier analysis (Harmanec et al. 2013), which
was devoted to the detection of very weak He i 6678 Å lines of the secondary
in the red spectral region. Hereinafter, we focus on the blue spectral region.
This study was also motivated by the tentative evidence of the secondary
reported by Richardson et al. (2015), namely in the ultraviolet region,
observed by the Hubble Space Telescope (Space Telescope Imaging Spectrograph).

However, a robust detection of the secondary (Aa2) spectrum is still lacking.
Now, we have a larger set of spectra in the blue part of the optical spectrum
and procedures to successfully detect the secondary’s spectrum. Moreover, new
Gaia DR3 parallax measurements have been published. This provides the
possibility to estimate the distance of bright stars, saturated in the Gaia
images, from the measured distances of their faint companions. We also have
new high-resolution astrometric measurements at our disposal, which enables us
to constrain the long-period orbit of (Aa1+Aa2)+Ab.

Figure 1: Scheme of the multiple system $\delta$ Ori (HD 36486, ADS 4134,
Mintaka). Orbital periods were taken from ∗ Leone et al. (2010), ∗∗ Mayer et
al. (2010), and ∗∗∗ this paper.

##  2 Observational data

In this section, only the spectroscopic and photometric data sets are
described as these data sets are new and fundamental to our analysis. Details
of other data sets (astrometry, spectral energy distribution SED, speckle
interferometry, etc.) are described in the following sections (Sects. 4, 8,
9).

###  2.1 Spectroscopy

We used digital spectra covering the blue spectral region secured at the coudé
focus of the Ondřejov 2m reflector (Škoda et al. 2002). We supplemented these
data sets with spectra from the public archives of the ELODIE echelle
spectrograph (Moultaka et al. 2004) at the Haute Provence Observatory, and the
FEROS echelle spectrograph (Kaufer et al. 1999) at the ESO La Silla
Observatory. The journal of the observations is presented in Table 1 (see
Table 14 for more details). The coverage of orbital phase $\varphi_{1}$ is
illustrated in Fig. 2. The short period $P_{1}$ of 5.732436 d is well covered.
The mean signal-to-noise ratio (S/N) is 208.5 (S/N values of individual
spectra are given in Tables 14 and 15), which was sufficient for spectral
disentangling. We normalised the spectra using polynomials of degree at least
4, with the program
reSPEFO2444https://astro.troja.mff.cuni.cz/projects/respefo/ (written by Adam
Harmanec).

Figure 2: Coverage of the orbital period $P_{1}$ by blue spectra. Phases are
determined with respect to time $T_{0}={\rm HJD}\,2454002.8735$ (time of
periastron passage determined by KOREL) for the eclipsing binary. Table 1:
Journal of digital spectra covering the blue spectral region.

Time interval | No. of | Detector | Resolution
---|---|---|---
$\mathrm{[HJD-2400000]}$ | spectra |  | [Å]
50031.68–50435.40 | 4 | ELODIE | 0.05
54136.58–54953.46 | 6 | FEROS | 0.03
55836.57–58405.57 | 65 | Site-5 CCD | 0.13

555For more details, see Tables 14 and 15.

###  2.2 Photometry

We used space-based photometric data from instruments on board the BRITE
(BRIght Target Explorer; Pablo et al. 2016) and the MOST (Carroll et al. 1998)
satellites and ground-based photometric data obtained at the Hvar Observatory
with the 0.65m telescope. The time coverage is illustrated in Fig. 3. We did
not use the saturated photometry from the Transiting Exoplanet Survey
Satellite (TESS).

Each BRITE nanosatellite hosts a telescope, which has a 3 cm aperture. The
BTr, BHr, and UBr satellites are equipped with a red filter (with effective
wavelength 620 nm); BAb and BLb have a blue filter (420 nm). We have
eliminated instrumental effects from the raw BRITE data by removing outliers
and worst orbits, and by decorrelations. For more information on BRITE data
processing, see Pigulski (2018).

The MOST passband covers the visible range of the spectrum (350–750 nm). The
satellite performs high-precision optical photometry of single bright stars.
It is equipped with a Maksutov telescope with an aperture of 15 cm and a
custom broadband filter. It can point with an error of less than 1 arcsec.
Other information can be found in Table 2.

The $\delta$ Ori LC from MOST continuously covers 3 weeks of observation.
During calibration, we numerically shifted the measured magnitude to the $V$
magnitude from the differential photometry at the Hvar Observatory. Then, we
constructed normal points by centring the errors on the satellite orbital
periods from Table 2, omitting the points with larger than the average
uncertainty (0.5 mmag).

The Cassegrain 0.65m $f/11$ telescope at the Hvar observatory is equipped with
a photoelectric detector (Božić 1998). This telescope was constructed at the
Ondřejov Observatory of the Czechoslovak Academy of Sciences and brought to
the Hvar Observatory at the beginning of 1972. A monitoring programme of
bright variable stars has continued until today. The Hvar all-sky photometry
provides accurate UBVR magnitudes in the Johnson system. For $\delta$ Ori A,
we used UBV differential magnitudes obtained between October 2006 and October
2008 and UBVR between January 2019 and March 2021.

Table 2: Information on satellites.

Satellite | Height | Inclination | Period
---|---|---|---
| [km] |  [∘] | [d]
MOST | 825–840 | 98.7 | 0.07042
UBr (UniBRITE) | 775–790 | 98.6 | 0.06972
BAb (BRITE-Austria) | 775–790 | 98.6 | 0.06972
BLb (BRITE-Lem) | 600–890 | 97.7 | 0.06917
BTr (BRITE-Toronto) | 620–643 | 97.9 | 0.06819
BHr (BRITE-Heweliusz) | 612–640 | 98.0 | 0.06743

666Sources Pablo et al. (2016), Carroll et al. (1998), Webb et al. (2006).

Figure 3: Photometric data from MOST and BRITE displayed with respect to time.
The BRITE data covers six consecutive seasons between 2013 and 2021.

##  3 Parallax and distance of $\delta$ Ori

In Gaia DR3 (Gaia Collaboration et al. 2016, 2021; Vallenari 2022), the
parallaxes of the faint components of bright stars in the Orion OB1
association were measured (see Table 3). The parallax of $\delta$ Ori Ca+Cb,
$\pi=(2.6244\pm 0.0538)\,{\rm mas}$ implies a distance $d=(381\pm 8)\,{\rm
pc}$ and a distance modulus $\mu=(7.90\pm 0.04)\,{\rm mag}$. Hereinafter, we
assume that the components (Aa1+Aa2)+Ab as well as Ca+Cb are located at the
same distance. Statistically, they are located close to each other. The number
of stars brighter than Ca+Cb ($6.62\,{\rm mag}$) is limited, there is only 15
of them within $7200^{\prime\prime}$. Given the separation of
$52^{\prime\prime}$, the probability that stars are physically unrelated is
low, $p<10^{-3}$.

To the contrary, $\delta$ Ori B, which is also a formal member of the multiple
visual system ADS 4134, is located at a substantially smaller distance (by
almost 100 pc). It is therefore not physically related to $\delta$ Ori A.
Either way, it is too faint (14 mag) to affect our results.

The Orion OB1 stellar association is usually divided into four subgroups,
OB1a, OB1b, OB1c, and OB1d (Brown et al. 1994). The system $\delta$ Ori
belongs to OB1b. We used the distances of 131 members from the Gaia DR3
catalogue and estimated the median distance to be $(382\pm 1)\,{\rm pc}$,
using a cumulative distribution function that is sensitive to the local number
density of stars (see Fig. 4). We obtained the same distance as the distance
of $\delta$ Ori Ca+Cb, within the respective intervals. We consider this to be
an independent estimate for the $\delta$ Ori A system since massive stars are
often located in the centre of the given association.

Other bright stars in the Orion belt are also located at very similar
distances (Table 3). For instance, the faint components of $\zeta$ Ori C,
$\sigma$ Ori C, D, and E all have precise parallaxes. Moreover, the single
star $\varepsilon$ Ori has a similar spectroscopic distance modulus. Again,
this is an independent confirmation for the $\delta$ Ori system.

For comparison, the dispersion of distance in the radial direction (1$\sigma$)
of the OB1b subgroup is only 15 pc, as seen in Fig. 4, while the angular
dispersion (1$\sigma$) is about $0.5^{\circ}$, which corresponds to 3 pc, at
the distance of 382 pc. In other words, $1^{\prime}$ corresponds to
$0.11\,{\rm pc}$, and $1^{\prime\prime}$ to $0.0018\,{\rm pc}$; this is a
range of separations for the faint components discussed above.

The age of the OB1b association is estimated between 4 and 5 Myr (Maucó et al.
2018).777Some of the outliers seen in Fig. 4 might actually be former members
of the OB1b association. If they were ejected at the typical speed of
$10\,{\rm km}\,{\rm s}^{-1}$, they may travel $50\,{\rm pc}$ or $7.5^{\circ}$
in the radial or tangential directions. The same is true for $\delta$ Ori B.
The OB1a subgroup (north-west) is older and at a smaller distance (by
approximately 37 pc), while the OB1c and OB1d subgroups (north-east),
including the Trapezium, are younger and at larger distances.

Table 3: Information about bright stars and their companions in Orion.

HD | Name | $V$ | Spectral | $A_{V}$ | $V_{0}$ | $M_{V}$ | $V_{0}\\!-\\!M_{V}$ | Gaia DR3 parallax | Notes
---|---|---|---|---|---|---|---|---|---
|  | [mag] | type | [mag] | [mag] |  |  | [mas] |
36486 |  $\delta$ Ori | 2.22* | O9.5 II | 0.13 | 2.09 | $-5.81$ | 7.90 |  | OB1b association, multiple
37128 |  $\varepsilon$ Ori | 1.68* | B0 Ia | 0.14 | 1.54 |  $-6.25$** | 7.79 |  | OB1b, single, variable 0.05 mag
37742 |  $\zeta$ Ori | 1.75* | O9.5 Ib | 0.17 | 1.58 |  $-6.28$** | 7.92 |  | OB1b, multiple
37468 |  $\sigma$ Ori | 3.82* | O9.5 V | 0.17 | 3.65 | $-4.14$ | 7.79 |  | OB1b, multiple
37043 |  $\iota$ Ori | 2.75 | O8.5 III | 0.09 | 2.66 | $-5.13$ | 7.79 |  | OB1d (Trapezium), multiple
36486 Aa1 |  $\delta$ Ori Aa1 | 2.55 | O9.5 II |  | 2.42 |  $-5.7$** | 8.12 |  | cf. this work
36486 Aa2 |  $\delta$ Ori Aa2 | 5.5? | B2 V |  | 5.4? |  $-2.5$? |  |  |  $0.00052^{\prime\prime}$ from Aa1, Shenar et al. (2015)
36486 Ab |  $\delta$ Ori Ab | 3.83 | B0 IV |  | 3.70 |  $-4.0$*** | 7.70 |  | $0.32^{\prime\prime}$
36486 B |  $\delta$ Ori B | 14.0 | K? |  | 13.9 | $+6.6$ |  | $3.5002\pm 0.0119$ |  $33^{\prime\prime}$, UCAC3 180-24383
36485 Ca |  $\delta$ Ori Ca | 6.62 | B3 V |  | 6.49 |  $-1.6$*** | 8.09 | $2.6244\pm 0.0538$ |  $52^{\prime\prime}$, helium star, Leone et al. (2010)
36485 Cb |  $\delta$ Ori Cb | 9.8? | A0 V |  | 9.7? |  $+1.8$? |  |  |  $0.0012^{\prime\prime}$ from Ca
37742 Aa |  $\zeta$ Ori Aa | 2.1 | O9.5 Ib |  |  |  |  |  | Hummel et al. (2000)
37742 Ab |  $\zeta$ Ori Ab | 4.3 | B0.5 IV |  |  |  |  |  | $0.042^{\prime\prime}$
37743 |  $\zeta$ Ori B | 4.0 | B0 III |  |  |  |  |  | $2.4^{\prime\prime}$
37742 C |  $\zeta$ Ori C | 9.54 | A? |  |  |  |  | $2.5876\pm 0.0387$ | $57^{\prime\prime}$
37468 Aa |  $\sigma$ Ori Aa | 4.61 | O9.5 V |  |  |  |  |  | Simón-Díaz et al. (2015)
37468 Ab |  $\sigma$ Ori Ab | 5.20 | B0.5 V |  |  |  |  |  | $0.00042^{\prime\prime}$
37468 B |  $\sigma$ Ori B | 5.31 | B? |  |  |  |  |  | $0.25^{\prime\prime}$
37468 C |  $\sigma$ Ori C | 8.79 | B0.5 V |  |  |  |  | $2.4720\pm 0.0292$ | $11^{\prime\prime}$
37468 D |  $\sigma$ Ori D | 6.62 | B2 V |  |  |  |  | $2.4744\pm 0.0621$ | $13^{\prime\prime}$
37468 E |  $\sigma$ Ori E | 6.66 | B2 V |  |  |  |  | $2.3077\pm 0.0646$ |  $42^{\prime\prime}$, helium star
37043 Aa1 |  $\iota$ Ori Aa1 | 2.8? | O8.5 III |  |  |  |  |  | Bagnuolo et al. (2001)
37043 Aa2 |  $\iota$ Ori Aa2 |  | B0.8 III |  |  |  |  |  |  $0.0015^{\prime\prime}$, eccentric
37043 Ab |  $\iota$ Ori Ab |  | B2 IV |  |  |  |  |  | $0.15^{\prime\prime}$
37043 B |  $\iota$ Ori B | 7.00 | B8 III |  |  |  |  | $2.7869\pm 0.0476$ | $11^{\prime\prime}$
37043 C |  $\iota$ Ori C | 9.76 | A0 V |  |  |  |  | $2.6057\pm 0.0241$ |  $49^{\prime\prime}$, Parenago (1954), Brun 731

888 * Hvar all-sky photometry; ** Martins et al. (2005); *** Schmidt-Kaler
(1982); spectral types from Maíz Apellániz et al. (2019); Burssens et al.
(2020) and absorption from Lallement et al. (2019). The visual magnitude
including absorption is denoted by $V$; the absorption, by $A_{V}$; the visual
magnitude without absorption, by $V_{0}$; and the distance modulus computed
from visual magnitude, by $M_{V}$.

Figure 4: Sorted distances of objects from the Orion OB1b (blue) and OB1a
(grey) associations (Brown et al. 1994). Parallax data were taken from the
Gaia DR3 catalogue (Vallenari 2022). The median values of distance for OB1b
and OB1a are $(382\pm 1)\,{\rm pc}$ (brown) and $(345\pm 1\,{\rm pc})$
(black), respectively. The distance of $\delta$ Ori C (red), $(381\pm 8)\,{\rm
pc}$, is very close to the median distance of the OB1b distribution.

##  4 Visual orbit of (Aa1+Aa2)+Ab

To determine the parameters of the long-period ($P_{2}$) orbit of the
(Aa1+Aa2)+Ab system, we constructed a simplified two-body model in the Xitau
program (Brož 2017; Brož et al. 2021, 2022a,
2022b)999http://sirrah.troja.mff.cuni.cz/~mira/xitau/. We used astrometric
data from the Washington Double Star (WDS) Catalogue (Mason et al. 2001). If
uncertainties were not available, we assumed the uncertainties of the
separation $\rho$ and the position angle measured from $v$, +DE direction as
follows: $\sigma_{\theta}=1.0^{\circ}$, $\sigma_{\rho}=0.01\,\mathrm{mas}$, or
$\sigma_{\theta}=0.2^{\circ}$, $\sigma_{\rho}=0.005\,\mathrm{mas}$ for
measurements before and after 2013, respectively. After removing 8 outliers
due to poor resolution, incorrect plate scale, or calibration (from
1879.12101010 Niesten (1904) reported micrométriques measures by 15cm Merz
refractor, with two position angles $162^{\circ}$ and a note ‘en contact’.
However, it is unlikely that it corresponds to $\delta$ Ori Ab, because its
separation at that epoch was only $0.11^{\prime\prime}$; separations in other
binaries were $1^{\prime\prime}$ or more. This observation is not compatible
with our model, which indicates $\theta\doteq 127^{\circ}$.  , 1978.10,
1979.06, 1980.02, 1981.01, 1985.74, 1995.05, 1999.78), we used $N=74$ data
points (both $\rho$, $\theta$).

Another data set incorporated into the model included values of the systemic
velocities $\gamma_{1}$ of $\delta$ Ori (Harvey et al. 1987; Harvin et al.
2002), which vary between approximately $12$ and $23\,{\rm km}\,{\rm s}^{-1}$
(see Table 5). This should correspond to the radial velocity of the (Aa1+Aa2)
component. We did not take into account data points with possible systematic
errors in $\gamma_{1}$, that is, blending with Ab (1910, 1948), low amplitude
of RV curve $K_{1}$ (1951, 1969, 1981), and different $\gamma_{1}$ for Aa2
(1987, 1997). In some cases, also the RV of Ab was measured.

In total, we had $N=88$ data points and $M=8$ free parameters, which means
$N-M=80$ degrees of freedom. The model resulted in the best-fit with
$\chi^{2}=95$, with contributions $\chi_{\rm SKY}^{2}=60$ for astrometry and
$\chi_{\rm RV}^{2}=35$ for RVs. Although $\chi^{2}>N-M$, the fit is still
acceptable. The RV amplitude is in agreement, as well as directly measured RV
values of Ab, which is lower than $\gamma_{1}$.

The resulting parameters and parameters that were fixed are shown in Table 4.
We fixed the mass of the (Aa1+Aa2) components based on the phoebe2 model
(Sect. 7) and the distance $d$ based on the parallax (Sect. 3). The orbit is
illustrated in Fig. 5. The fit of RVs is shown in Fig. 6.

We estimated the uncertainties of parameters using $\chi^{2}$ mapping and
verified the MCMC method. According to the $\chi^{2}$ statistics, a 1$\sigma$
level corresponds to $\chi^{2}\doteq 101$. Since the distance was fixed, the
uncertainties are relatively small ($1\%$ for the period, $10\%$ for the
mass). We determined the mass of the Ab component to be $11.0$ M⊙. Therefore,
the total mass of the (Aa1+Aa2)+Ab system is around $37.5$ M⊙.

#### Mirror solution.

We are aware of the existence of a mirror solution, with the opposite sign of
inclination $i_{2}$. It exhibits higher total mass (up to $52\,{\rm
M}_{\odot}$), higher eccentricity ($0.95$), shorter period ($40000\,{\rm d}$),
closer periastron passage. The RV curve of (Aa1+Aa2) component is also
opposite, with a ‘spike’ due to the eccentricity. According to our throughout
testing, it always has a worse best-fit $\chi^{2}$, especially the
$\chi^{2}_{\rm rv}$ contribution. Moreover, in the mirror solution, the RVs of
Ab are larger than $\gamma_{1}$, which is incorrect. A more complex model is
needed to test other constraints (see Sect. 9).

Table 4: Parameters of the orbit (Aa1+Aa2)+Ab, for the model with unreduced
$\chi^{2}=95$.

Parameter | Value | Unit
---|---|---
$T_{0}$ |  2458773.1886 $\pm$ | 0.1 | HJD
$(m_{1}+m_{2})$ f |  26.5 $\pm$ | 2.0 |  M⊙
$m_{\mathrm{3}}$ |  11.102 $\pm$ | 1.2 |  M⊙
$P_{2}$ |  53839 $\pm$ | 550 | d
$e_{2}$ |  0.5886 $\pm$ | 0.016 | 1
$i_{2}$ |  104.710 $\pm$ | 0.4 | ∘
$\Omega_{2}$ |  122.416 $\pm$ | 0.5 | ∘
$\varpi_{2}$ |  258.944 $\pm$ | 2.0 | ∘
$\lambda_{2}$ |  134.307 $\pm$ | 1.0 | ∘
$\gamma_{1}$ |  19.1 $\pm$ | 0.5 | ${\rm km}\,{\rm s}^{-1}$
$d$ f |  382 $\pm$ | 8 | pc

111111 $T_{0}$ denotes the time of periastron passage; $(m_{1}+m_{2})$, the
mass of Aa1+Aa2 (the primary in this model); $m_{3}$, the mass of the Ab
component (the secondary in this model); $P_{2}$, the orbital period of Aa and
Ab; $e_{2}$, the eccentricity; $i_{2}$, the inclination; $\Omega_{2}$, the
longitude of the ascending node; $\varpi_{2}$, the longitude of periastron;
$\lambda_{2}$, the true longitude; $\gamma_{1}$, the systemic velocity
attributed to Aa1+Aa2; and $d$, distance. f indicates the respective parameter
was fixed.

Figure 5:  Orbit of 2-body model. Top: Orbit of (Aa1+Aa2)+Ab components in the
($u$-$v$) plane (green), calculated using the two-body model and the simplex
method. Observations are shown with blue symbols and uncertainty ellipses
(orange). The residuals are plotted in red; the value of $\chi^{2}=95$. The
astrometry used for the fit is from the WDS. The radial velocities of Aa1,
from Table 5 were also used for the fit, extending the time span to 44 000
days. Bottom: Detail of the observed arc. The most precise astrometric
measurements from 2013 and 2019 constrain the orbital period $P_{2}$.

Figure 6:  Synthetic RVs of the Aa1+Aa2 (green) and Ab (black) components,
relative to the barycentre of the (Aa1+Aa2)+Ab system. We used a simplified
two-body model and fitted data from Table 5, plotted with blue symbols. The
residuals are plotted in red; the value of $\chi^{2}=95$. The last most
precise point is from this work.  Table 5:  Observed values of $\gamma_{1}$
velocity of the Aa1+Aa2 components. It is variable due to the presence of the
third (Ab) component. In some cases, also the RV of Ab was measured. If the
reference is not provided, the value is taken from the list of Harvey et al.
(1987) or Mayer et al. (2010), where more information about RV observations is
provided (in their App. A).

$T$ (TDB) | $T_{\mathrm{Bessel}}$ | RV | Observatory or Ref.
---|---|---|---
[JD] | [BY] |  [${\rm km}\,{\rm s}^{-1}$] |
2415793.70 | 1902.1174 | $23.5\pm 1.4$ | Potsdam
2418981.13 | 1910.8443 | $14.3\pm 1.8$ | Allegheny
2420024.23 | 1913.7002 | $20.4\pm 0.6$ | Ann Arbor
2422391.71 | 1920.1822 | $18.6\pm 2.1$ | Vienna
2428382.90 | 1936.5855 | $12.5\pm 0.9$ | Yerkes
2432499.19 | 1947.8555 | $12.4\pm 1.4$ | McDonald
2433656.03∗ | 1951.0229 | $13.6\pm 1.9$ | Heidelberg
2440410.36∗ | 1969.5156 | $15.9\pm 2.1$ | Kodaikanal
2444922.09∗ | 1981.8683 | $16.6\pm 3.0$ | Kavalur
2445139.84 | 1982.4645 | $20.4\pm 0.7$ | IUE
2446865.19∗ | 1987.1883 | $27.5\pm 0.7$ | Harvin et al. (2002)
2450535.00∗ | 1997.2359 | $24.0\pm 3.0$ | Harvin et al. (2002)
2454125.40 | 2007.0661 | $21.7\pm 0.5$ | Mayer et al. (2010)
2457040.9380 | 2015.0486 | $21.1\pm 1.6$ | Richardson et al. (2015)
2457121.0974 | 2015.2681 | $21.4\pm 0.1$ | this work
Ab component: |  |  |
2457022.1757 | 2014.9972 | $19.1\pm 2.7$ | Richardson et al. (2015)
2457064.0764 | 2015.1120 | $15.9\pm 2.9$ | Richardson et al. (2015)
2457121.0974 | 2015.2681 | $12.0\pm 3.0$ | this work

121212∗ denotes the data that were not included in the fit due to systematic
errors (see text).

##  5 Spectral disentangling of residuals

After obtaining the reliable value of the long-period ($P_{2}$) of $\delta$
Ori Aa and Ab, we searched for the secondary’s lines in the spectra. Our
experience with the disentangling technique is that the result is often
sensitive to the choice of initial values of the parameters. This is
understandable since the $\chi^{2}$ sum based on all data points of all
spectra is a complicated function of the orbital elements, and it is easy to
end up in a local minimum.

Moreover, the rotationally broadened spectral lines of the primary (Aa1) and
tertiary (Ab) blend with each other at all orbital phases and altogether
dominate the spectrum. Consequently, the contribution of the faint secondary
(Aa2) to the $\chi^{2}$ sum is almost comparable to the noise. The mass ratio
$q_{1}$ of the Aa2 and Aa1 components is therefore poorly constrained.
Nevertheless, the lines of the secondary can be detected in the residuals by a
procedure called a two-step disentangling.

To disentangle the spectra, we used the KOREL program developed by Hadrava
(1995, 1997, 2004, 2005). Rebinning of the spectra to a linear scale in RV,
needed as input for KOREL, was carried out using the HEC35D program written by
P.H. 131313The program HEC35D with User’s Manual is available at
http://astro.troja.mff.cuni.cz/hec/HEC35 . The relative fluxes for the new
wavelength points were derived using the INTEP program (Hill 1982), which is a
modification of the Hermite interpolation formula. It is possible to choose
both boundaries of the desired spectral region, and the program smoothly
interpolates the rebinned spectra with the continuum values of 1.0 at both
edges.

To account for the variable quality of the individual spectra, we measured
their S/N ratios in the line-free regions and assigned each spectrum a weight
$w$ according to the formula:

$w={(\mathrm{S/N})^{2}\over{(\mathrm{S/N})_{\rm mean}^{2}}},$ (1)

where $(\mathrm{S/N})_{\rm mean}$ denotes the root mean square of S/N ratio of
all spectra.

Fitting with KOREL was performed with the following measurement equation:

${\cal F}I_{i}(y,t)=\sum_{j}s_{ij}\,{\cal F}I_{ij}\cdot\exp({\rm
i}yv_{ij})\quad\hbox{for }\forall i\,,$ (2)

where $j$ denotes the component; $i$, the spectrum; $I$, the normalised
intensity; ${\cal F}I$, its Fourier transform; $y$, the Fourier-transformed
quantity $x\equiv\ln\lambda/\lambda_{0}$, related to the wavelength $\lambda$;
$s_{ij}$, the intensity factors (constant or variable); $v_{ij}$, the radial
velocity.

#### Two-step method.

We used the period derived by Mayer et al. (2010), pericentre rate derived by
Pablo et al. (2015), and parameters from Table 4 as the initial conditions.
With the method of spectral disentangling, we needed to detect a line spectrum
of the secondary in the blue spectral region 4275–4509 Å.

In the first step, we fitted the orbit of the close pair (Aa1+Aa2) and
converged $q_{1}$, $e_{1}$, $\omega_{1}$, $K_{1}$, $T_{0}$, while $P_{1}$,
$\dot{\omega}$ were fixed as well as the outer orbit (Sect. 4). We set the
same and constant intensity of lines of Aa1 and Ab ($s_{1}=1$, $s_{3}=1$)
constant and assigned zero intensity to Aa2 ($s_{2}=0$). The result of the
first step was the disentangled spectra of only the primary (Aa1) and tertiary
(Ab), and the residuals for all individual spectra after disentangling
(O$-$C).

In the second step, we added a value of 1.0 to the residuals and reran KOREL
on this ‘residual’ data set. Now, the intensity factors of Aa1, Ab were zero
($s_{1}=0$, $s_{3}=0$) and the intensity factor of Aa2 was constant $s_{2}=1$.
We fitted the spectrum of the Aa2 component by converging the mass ratio $q$
and fixing $T_{0}$, $e_{1}$, $\omega_{1}$, $K_{1}$, $P_{1}$, and
$\dot{\omega}_{1}$. We successfully detected the desired spectrum of the Aa2
component. The determined parameters are summarised in Table 6. This method
gave higher $e_{1}$, lower $q_{1}$, precise $K_{1}$ and $\omega_{1}$, which
were well constrained. All disentangled spectra in these two steps have a flat
continuum, not wavy. To confirm the detection, we created a pseudo-$\chi^{2}$
map (see Fig. 7).141414 We had some success using only a one-step method and
setting Aa1, Aa2 to have fixed intensity factors $s_{1}=1$, $s_{2}=1$ and Ab
to have free intensity factor $s_{3}$. Parameters $T_{0}$, $e_{1}$,
$\omega_{1}$, $K_{1}$, $q_{1}$ were converged and $P_{1}$, $\dot{\omega_{1}}$
were fixed. This setting prevents fluctuations related to variable $s_{1}$,
$s_{2}$ factors, making the model much more stable. Otherwise, the continuum
is wavy when $s_{1}$ is free. The results from this approach are
$e_{1}=0.0804$, $\omega_{1}=153.9^{\circ}$, $K_{1}=108.3\,{\rm km}\,{\rm
s}^{-1}$, $q_{1}=0.4893$; uncertainties as in Table 6.

#### Three-step method.

A more precise orbital solution can be obtained by using KOREL in a sequence
(three-step method). We started the process by fitting the primary and
tertiary (Aa1+Ab) with variable intensities $s_{1}$, $s_{3}$. We fixed $P_{1}$
and $\dot{\omega_{1}}$ of the close orbit and converged $T_{0}$, $e_{1}$,
$\omega_{1}$, $K_{1}$. The outer parameters of the orbit were fixed.

We continued by fitting Aa1, Ab with the constant $s$-factors and Aa2 with the
variable one. Except for $q_{1}$, all parameters were fixed. Finally, we found
the solution for all three components with constant $s$-factors, free $T_{0}$,
$e_{1}$, $\omega_{1}$, $K_{1}$, $q_{1}$, and fixed $P_{1}$,
$\dot{\omega_{1}}$.

The resulting $s_{1}(t)$ is variable with time and should correspond to the
LC; however, the amplitude of the eclipses (without reflection) is too low
(0.04 mag) to be seen. The resulting value of $q_{1}=0.4517$ is higher,
compared to the two-step disentangling, while $e_{1}=0.0761$ is close to that
found from the LC, and $K_{2}=239.7\,{\rm km}\,{\rm s}^{-1}$.

In the three-step method, which we considered to be more reliable, we also
computed the radial velocities of all three components (see Tables 14 and 15).
We estimated the uncertainties as a standard deviation weighted by S/N.

Table 6: Solution to the disentangling of 75 blue spectra in KOREL. We prefer
the solution from the three-step method (bold).

Parameter | Two-step | Three-step | $\sigma$
---|---|---|---
$P_{\mathrm{anom,1}}\,[{\rm d}]$ |  5.732821 (Mayer et al. 2010)
$\dot{\omega}_{1}\,[^{\circ}\,{\rm y}^{-1}]$ |  1.45 (Pablo et al. 2015)
$T_{0}$ | 2454002.8737 | 2454002.8735 | 0.02
$e_{1}$ | 0.0833 | 0.0761 | 0.01
$\omega_{1}\,[^{\circ}]$ | 153.6 | 153.5 | 3.0
$K_{1}\,[{\rm km}\,{\rm s}^{-1}]$ | 110.1 | 108.3 | 0.5
$q_{1}$ | 0.3996 | 0.4517 | 0.02
$\sigma_{s_{1}}$ | 0 | 0.1 | –
$\sigma_{s_{2}}$ | 0 | 0.4! | –
$\sigma_{s_{3}}$ | 0 | 0.1 | –
$K_{2}\,[{\rm km}\,{\rm s}^{-1}]$ | 295 | 239 | 10
pseudo-$\chi^{2}$ | 197588 | 172575 | –

151515 The anomalistic period $P_{\mathrm{anom,1}}$ and the pericentre rate
$\dot{\omega}_{1}$ were fixed. Free parameters were the time of periastron
passage $T_{0}$, eccentricity $e_{1}$, argument of periastron $\omega_{1}$,
semi-amplitude of the primary $K_{1}$, mass ratio $q_{1}$, standard deviations
of the intensity factors for the primary, secondary, and tertiary,
$\sigma_{s_{1}}$, $\sigma_{s_{2}}$, $\sigma_{s_{3}}$, respectively. The
dependent parameter is the semi-amplitude of the secondary $K_{2}$. The models
are quantified by pseudo-$\chi^{2}$ in Fourier space.

Figure 7:  Plot of the pseudo-$\chi^{2}$ in Fourier space vs. the mass ratio
$q_{1}=m_{2}/m_{1}$. This is related to the two-step disentangling method, to
its second step, when the signal of the secondary component (Aa2) was sought
for in the residuals.

##  6 Atmospheric parameters of Aa1, Aa2, and Ab

We used the disentangled blue spectra to estimate the atmospheric parameters,
namely, $T_{\rm eff}$, log $g$, $v\sin i$, and the relative luminosities of
the three components using the program PYTERPOL (Nemravová et al.
2016)161616https://github.com/miroslavbroz/pyterpol3. The program uses the
simplex minimisation technique to fit the synthetic spectra to the observed
ones. As model spectra, OSTAR and BSTAR grids (Lanz & Hubený 2003, 2007) were
used.

The results are summarised in Table 7, where the uncertainties were estimated
from several independent trials. The fitted spectral line profiles of all
components are shown in Fig. 8. Most of the lines are fitted reasonably well,
except for He I 4471\. The value of $\log g$ was determined primarily from the
H$\gamma$ wings. The metallicity $Z$ is not well constrained.

The sum of relative luminosities that were fitted independently (0.692 + 0.035
+ 0.194 = 0.921) is close to 1, which is an independent verification of the
correctness of the KOREL disentangling. The effective temperatures agree with
the spectral classifications of Aa1 (O9.5 II) and Ab (B0 IV), although the
effective temperature of the Aa2 component is significantly lower (around
25000 K) than that of the other components, corresponding to B1 V, according
to Harmanec (1988) calibrations. The metallicities were fixed to the solar
value since they are not well constrained by the blue spectra containing only
one strong magnesium line. The values from Table 7 are the initial parameters
for the phoebe2 model.

Table 7: Atmospheric parameters derived with PYTERPOL from the blue spectral
region 4271–4513 Å, with the mean resolution of 0.0144 Å.

Parameter | Aa1 | Aa2 | Ab
---|---|---|---
$T_{\rm eff}$[K] | 31400(1000) | 25442(1500) | 30250(1000)
29500 | 25600 | 28400
log $g$ [cgs] | 3.55(5) | 3.48(6) | 3.64(5)
3.37 | 3.9 | 3.5
$v\sin i$ [km s-1] | 114(20) | 89(15) | 216(25)
130 | 150 | 220
$L_{R}$ | 0.692(34) | 0.035(15) | 0.194(10)
0.707 | 0.059 | 0.234
$\chi^{2}_{R}$ | 2.562 | 1.769 | 0.2384

171717 $T_{\rm eff}$ denotes the effective temperature; log $g$, logarithm of
surface gravity; $v\sin i$, projected rotational velocity; $Z$, metallicity;
$L_{R}$, relative luminosity; $\chi^{2}_{R}$, the reduced value of $\chi^{2}$
(divided by the degrees of freedom), and f indicates the fixed parameter. For
comparison with previous results, the values from Shenar et al. (2015) are
shown in grey. In the case of Aa1 and Ab components, the results are usually
in agreement within the uncertainties. More significant differences are for
the Aa2 component; however, our values for the secondary are constrained by
the disentangled spectra and mass ratio from KOREL. The uncertainties of the
parameters are given in concise form in brackets.

Figure 8:  Comparison of the disentangled spectra (blue) of the Aa1, Aa2, and
Ab components with the best-fit synthetic spectra (orange) found by PYTERPOL.
The range from 428 to 449 nm was used for disentangling. The flux is
normalised to the local continuum. The small panels show the residuals (red).
The relative luminosities of the Aa1 and Ab components significantly exceed
that of the Aa2 component (see also Table 7).

##  7 Orbit of eclipsing binary Aa1+Aa2

For the eclipsing binary Aa1+Aa2, we solved the inverse problem using PHOEBE2
(Conroy et al. 2020), obtaining a more precise model than with PHOEBE1 (Prša &
Zwitter 2005) in our preliminary analysis (Oplištilová et al. 2020). The
initial values of parameters for PHOEBE2 were inferred from the analysis
performed with PHOEBE1. We had three photometric data sets available for
analysis, SMEI, MOST, and BRITE (Oplištilová et al. 2020), but we preferred to
use only BRITE data to have a homogeneous data set spanning nine seasons. MOST
was used in Sect. 9. We did not use SMEI data since they suffer from a
contamination problem because stellar images in SMEI image have angular sizes
of the order of 1 degree.

PHOEBE2181818http://phoebe-project.org, a Python module, is software for
modelling eclipsing binaries. To achieve the smallest possible discretisation
error, the software uses a mesh of triangular elements. Each element of the
mesh is assigned local properties (e.g. temperature, intensity), and the
eclipse algorithm determines which elements are visible, which are partially
visible, and which are not visible at all. The total flux is obtained by
integrating over all visible elements.

We implemented a custom object-oriented Python wrapper to construct a model of
the eclipsing binary and combine different data types. Each model was
quantified by the $\chi^{2}$ value. First, we fitted the stellar parameters
using the simplex method (Nelder & Mead 1965) or the supblex method (Rowan
1990). Second, we used the Markov chain Monte Carlo method (MCMC; Robert &
Casella 2011; Tierney 1994), which was originally invented by Slanisław Ulam
alongside with the atomic bomb. This method uses a sequence of random samples
and provides a straightforward algorithm for the numerical estimation of
parameters and their uncertainties. In other words, it describes the topology
of the parameter space in the vicinity of the local/global minimum. The MCMC
method was run using the API OpenMP (application program interface), which
allows our code to run in parallel on multiple CPUs.

The MCMC method relies on Bayes’ Theorem, which relates four probabilities as
follows:

$P(\bm{\Theta}_{\mathrm{M}}|\bm{D})=\frac{P(\bm{D}|\bm{\Theta}_{\mathrm{M}})P(\bm{\Theta}_{\mathrm{M}})}{P(\bm{D})},$
(3)

where $\bm{D}$ denotes the vector of data; $\bm{\Theta}_{\mathrm{M}}$, the
vector of parameters of our model; $P(\bm{D})$, the probability of obtaining
the data (normalisation); $P(\bm{\Theta}_{\mathrm{M}})$, the prior, a priori
knowledge of parameters (we used uniform, uninformative priors);
$P(\bm{D}|\bm{\Theta}_{\mathrm{M}})$, the likelihood function, which is
equivalent to the forward model or $\chi^{2}$; and
$P(\bm{\Theta}_{\mathrm{M}}|\bm{D})$, the posterior distribution, which
quantifies our belief in the parameters after combining our prior distribution
with the current observations and normalising by the overall evidence.

The input data for the script are the RV curves of the primary and secondary,
and the LCs in the blue and red filters. The synthetic fluxes were normalised
by two free parameters $S_{\mathrm{red}}$ and $S_{\mathrm{blue}}$ satisfying:

$F_{c,\mathrm{norm}}=S_{\mathrm{c}}\cdot\frac{F_{c,i}}{\mathrm{max}(F_{\mathrm{c},i})},$
(4)

where $c$ denotes the colour of the filter (blue or red), and $i$ is the point
number.

We set the algorithm parameters as follows: for the spatial discretisation, we
used 1500 triangles covering the surface of the primary and 500 triangles for
the small secondary surface. As a sampler, we used emcee (Foreman-Mackey et
al. 2013) with 30 walkers and 2000 iterations. After some initial tests, we
set the number of initial steps (burn-in) to 300. These are not taken into
account as they are irrelevant and randomly distributed within the prior. The
program ran on 30 CPUs.

In our modelling, we fixed the orbital sidereal period to $5.732436\,{\rm d}$
following Mayer et al. (2010), the pericentre rate
$\dot{\omega}=1.45^{\circ}\,{\rm y}^{-1}$ (Pablo et al. 2015), and in some
models also the effective temperature of the primary
$T_{\mathrm{1}}=31\,000\,{\rm K}$, and the third light, additional to the
components Aa1 and Aa2, $l_{3}=0.26685$ calculated from Table 7 (Sect. 6).

We used the following parameters in our model:

  * •

atmosphere, black-body (approximation),

  * •

limb darkening, linear,

  * •

limb darkening coefficients, interpolated based on van Hamme (1993),

  * •

gravity brightening, 1.0 (corresponding to the $\beta$ coefficient for gravity
darkening corrections),

  * •

reflection and heating fraction, 1.0,

  * •

distortion method, Roche,

  * •

irradiation method, Wilson (1990), Wilson’s original reflection effect scheme
incorporates all irradiation effects, including reflection and redistribution,

  * •

radial velocity method, flux-weighted (i.e. radial velocities are determined
by the radial velocity of each element of visible surface, weighted by its
intensity). Consequently, the RV curve includes the Rossiter-McLaughlin
effect.

We did not take into account either the effects of light travel time or
gravitational redshift. This setting of the phoebe2 model is used for all
models in Sects. 7.1 and 7.2.

###  7.1 Model for season 2016

We had the BRITE LCs from 9 seasons at our disposal (Table 8). First, we
selected well-covered season 2016 and fitted several models with some
parameters free or fixed, namely the effective temperature of the primary and
the third light (for both blue and red filters). The results are presented in
Table 9. We prefer the model with the fixed effective temperature of the
primary, which also has the lowest value of $\chi^{2}$. The data, the model,
and the residuals are shown in Fig. 10.

Then, we used the MCMC method to estimate the uncertainties of the parameters.
Figs. 11 and 12 show the corner plot and the paths of walkers. In particular,
masses $M_{\mathrm{i}}$ and radii $R_{\mathrm{i}}$ show strong positive
correlations. In contrast, the inclination $i_{1}$ and $R_{\mathrm{i}}$ show
negative correlations due to geometrical reasons. In binaries, the sum of
masses is inversely proportional to the third power of $\sin i$; thus, $i_{1}$
and $m_{\mathrm{i}}$ show negative correlations. The value of the systemic
velocity $\gamma$ is a little problematic as the value of $21.96\,{\rm
km}\,{\rm s}^{-1}$ was assumed and subtracted, then our model drifted to about
$-2.5\,{\rm km}\,{\rm s}^{-1}$, so that the resulting value is $18.5\,{\rm
km}\,{\rm s}^{-1}$.

The detached binary system Aa1+Aa2 is shown in Fig. 9. In addition, we derived
several parameters from the nominal phoebe2 model ($\chi^{2}=604$); see Table
10. We estimated the synthetic apparent brightness of $\delta$ Ori A as
follows. The passband flux in Johnson V at the observer location is (in $[{\rm
W}\,{\rm m^{-2}}]$):

$\Phi_{\mathrm{V}}=\Delta_{\rm eff}\kern 0.3pt\omega\\!\sum_{k}I_{\lambda
k}\kern 0.2ptS_{\\!k\,}\mu_{k}\,\eta_{k}\,,$ (5)

where $\Delta_{\rm eff}\,[{\rm m}^{-1}]$ stands for the effective wavelength
range; $\omega=1\,{\rm m}^{2}/d^{2}\,[{\rm sr}^{-1}]$, for the solid angle;
$d$, for the distance of the system; $\sum_{k}$, for the summation over the
triangular elements (grid); $I_{\lambda}\,[{\rm W}\,{\rm m}^{-3}\,{\rm
sr}^{-1}]$, the monochromatic intensity on the stellar surface; $S\,[{\rm
m}^{2}]$, the surface area of the element; $\mu=\cos\theta$, where $\theta$ is
the angle between the normal and the line of sight; $\eta$, the visibility in
the range from 0 (visible element) to 1 (hidden or eclipsed element).

We assumed the monochromatic calibration flux of Bessell (2000):

$\Phi_{\lambda,{\rm cal}}=\Phi_{\nu,{\rm cal}}{c\over\lambda_{\rm
eff}^{2}}=0.03669877\,{\rm W}\,{\rm m}^{-3}$ (6)

and the Johnson V passband flux:

$\Phi_{\mathrm{V,cal}}=\int_{\lambda}f\Phi_{\lambda,{\rm cal}}{\rm
d}\lambda\doteq\Delta_{\rm eff}\Phi_{\lambda,{\rm cal}}\doteq 3.119396\cdot
10^{-9}\,{\rm W}\,{\rm m}^{-2}\,,$ (7)

where $f(\lambda)$ denotes the filter transmission;
$\lambda_{\mathrm{eff}}=545$ nm, the effective wavelength; and
$\Delta_{\mathrm{eff}}=85$ nm, the effective range.

The apparent magnitude $V_{0}$ (without absorption) of the primary Aa1 and Aa2
is then:

$V_{0}=0\,{\rm[mag]}-2.5\log\frac{\Phi_{\mathrm{V}}}{\Phi_{\mathrm{V,cal}}}.$
(8)

For the tertiary, we used the value of the third light:

$\Phi_{\rm V,2}=(\Phi_{\rm V,1}+\Phi_{\rm V,2})\,l_{\rm 3V}\,.$ (9)

Comparing $V_{0}$ of the Aa1 + Aa2 + Ab system with Table 3, we get synthetic
values $2.65+5.91+4.02=2.34\,{\rm mag}$ and observed values
$2.42+5.4+3.70=2.08\,{\rm mag}$. Thus, the total synthetic magnitude $V_{0}$
is about $0.26\,{\rm mag}$ fainter than observed. This result is acceptable,
especially because the phoebe2 model was constrained only by the relative
BRITE photometry (see also Sects. 8, 9). In other words, the result can be
considered to be an independent confirmation of the distance.

Figure 9:  Mesh plot of $\delta$ Ori Aa1+Aa2 binary from the phoebe2 model
(with $\chi^{2}=604$). This is a $(u,v)$ plane projection with scale in R⊙, at
phase $\varphi=0.75$. The grey scale corresponds to the monochromatic
intensity $I_{\lambda}\,[{\rm W}\,{\rm m}^{-3}\,{\rm sr}^{-1}]$ for the
effective wavelength of the BRITE blue filter (420 nm).

Figure 10:  Comparison of observations and the phoebe2 model of $\delta$ Ori
with $\chi^{2}=604$. The values of the effective temperature of the primary
$T_{1}$ and the third light $l_{3}$ were fixed. The upper panel shows the
phased LCs in the blue and red BRITE filters. The lower panel shows the RV
curves for the primary Aa1 ( green) and the secondary Aa2 ( purple). The grey
points correspond to our model, the red lines to the residuals, or
contributions to $\chi^{2}$.

Figure 11:  Corner plot (a full covariance matrix) for the $\delta$ Ori model,
as derived by the MCMC analysis. The model is the same as in Fig. 10. Each
diagonal panel shows a 1-D histogram (posterior distribution) for one
parameter (explained in Table 9). Each sub-diagonal panel shows a 2-D
histogram, the isolines corresponding to the confidence intervals, and the
correlations between parameters.

Figure 12:  Parameter values versus iterations during the MCMC analysis (as in Fig. 11). We used 30 walkers (one walker corresponds to one colour) for the computation and 300 steps for the burn-in. For most parameters, the distribution of walkers is already stationary. The systemic velocity $\gamma$ is stationary only after 1000 iterations. The value of $21.96\,{\rm km}\,{\rm s}^{-1}$ should be added to $\gamma$.  Table 8:  Time intervals of the BRITE LCs (red and blue filters) when $\delta$ Ori was observed. The whole time span was divided into 9 seasons. Time is given in ${\rm HJD}-2400000$. Season 2019 denoted by ∗ was omitted since red-filter data were not available. In total, BRITE nanosatellites measured 499 656 raw data points. The numbers of points are shown for each season and filter. Season | Blue filter | $N_{\rm B}$ | Red filter | $N_{\rm R}$
---|---|---|---|---
2013 | $56628\mbox{--}56734$ | 22822 | $56603\mbox{--}56733$ | 33357
2014 | $56926\mbox{--}57098$ | 38126 | $56924\mbox{--}57095$ | 61672
2015 | $57370\mbox{--}57434$ | 6472 | $57374\mbox{--}57443$ | 9665
2016 | $57645\mbox{--}57810$ | 22800 | $57729\mbox{--}57734$ | 22996
2017 | $58011\mbox{--}58178$ | 17055 | $57645\mbox{--}58178$ | 12812
2018 | $58375\mbox{--}58561$ | 61777 | $58430\mbox{--}58556$ | 28001
2019∗ | $58745\mbox{--}58917$ | 10291 | $-$ | 39888
2020 | $59112\mbox{--}59256$ | 4727 | $59129\mbox{--}59286$ | 44131
2021 | $59469\mbox{--}59638$ | 5065 | $59504\mbox{--}59646$ | 58109
Table 9:  Results of fitting three phoebe2 models for $\delta$ Ori Aa1+Aa2. LC
from season 2016 and all RVs were used to constrain the models. We fixed or
released effective temperature $T_{1}$ of the primary and the third light
$l_{3}$. We prefer the model with fixed $T_{1}$ and $l_{3}$ (bold). The
uncertainty $\sigma$ is estimated the same for all models. The following
numbers of data points were used: $N_{\mathrm{total}}=321$,
$N_{\mathrm{LCB}}=N_{\mathrm{LCR}}=100$, $N_{\mathrm{RV1}}=71$, and
$N_{\mathrm{RV2}}=50$.

Parameter | Fixed $T_{\mathrm{1}}$ |  Fixed $T_{\mathrm{1}}$, free $l_{3}$ | $\sigma$ | Unit
---|---|---|---|---
$T_{0}$ | 0.8303 | 0.8277 | 0.0030 |  HJD∗
$T_{\mathrm{1}}$ | 31000 | 31000 | 300 | K
$T_{\mathrm{2}}$ | 22709 | 22778 | 300 | K
$R_{\mathrm{1}}$ | 13.27 | 13.01 | 0.55 |  R⊙
$R_{\mathrm{2}}$ | 3.70 | 3.66 | 0.35 |  R⊙
$i_{1}$ | 77.671 | 78.45 | 1.1 | ∘
$S_{\mathrm{B}}$ | 1.0079 | 1.0236 | 0.0008 | 1
$S_{\mathrm{R}}$ | 1.0084 | 1.0250 | 0.0009 | 1
$m_{\mathrm{1}}$ | 18.07 | 17.39 | 1.5 |  M⊙
$m_{\mathrm{2}}$ | 8.47 | 8.26 | 0.4 |  M⊙
$e_{1}$ | 0.0825 | 0.0826 | 0.0031 |
$\omega_{1}$ | 129.98 | 137.50 | 2.5 | ∘
$\dot{\omega_{1}}$ |  1.45f |  1.45f |  1.45f |  ∘ yr-1
$\gamma$ | 19.43 | 21.96 | 0.5 |  km s-1
$l_{3\mathrm{B}}$ | 0.26685 | 0.308 | 0.05 |
$l_{3\mathrm{R}}$ | 0.26685 | 0.316 | 0.05 |
$\chi^{2}_{\mathrm{sum}}$ | 604 | 490 |  |
$\chi^{2}_{\mathrm{LCB}}$ | 145 | 136 |  |
$\chi^{2}_{\mathrm{LCR}}$ | 113 | 113 |  |
$\chi^{2}_{\mathrm{RV1}}$ | 171 | 216 |  |
$\chi^{2}_{\mathrm{RV2}}$ | 70 | 72 |  |

191919 $T_{0}$ denotes the time of periastron passage (∗ means HJD$-2457733$);
$T_{\mathrm{1}}$ and $T_{\mathrm{2}}$, the effective temperatures of the
primary and secondary, respectively; $R_{\mathrm{1}}$ and $R_{\mathrm{2}}$,
the radii of the primary and secondary, respectively; $i$, inclination;
$S_{\mathrm{B}}$ and $S_{\mathrm{R}}$, the coefficients adjusting
normalisation of the flux defined in Eq. (4); $m_{\mathrm{1}}$ and
$m_{\mathrm{2}}$, the masses of the primary and secondary, respectively;
$e_{1}$, eccentricity; $\omega_{1}$, the argument of periastron; $\gamma$, the
systemic velocity; $l_{3\mathrm{R}}$ and $l_{3\mathrm{B}}$, the third light in
the blue and red filters, respectively; $\chi^{2}_{\mathrm{sum}}$, the total
value of $\chi^{2}$; $\chi^{2}_{\mathrm{LCB}}$, $\chi^{2}_{\mathrm{LCR}}$,
$\chi^{2}_{\mathrm{RV1}}$, and $\chi^{2}_{\mathrm{RV2}}$, the contributions to
$\chi^{2}$ from the LCs in the red and blue filters and radial velocities of
the primary and secondary; $N_{\mathrm{total}}$, the total number of data
points; $N_{\mathrm{LCB}}$, $N_{\mathrm{LCR}}$, $N_{\mathrm{RV1}}$, and
$N_{\mathrm{RV2}}$, the corresponding numbers of data points. Uncertainties
were estimated from Figs. 12 and 13.

Table 10:  Derived parameters corresponding to the $\delta$ Ori Aa1+Aa2 model
with $\chi^{2}=604$.

Derived parameter | Value | Unit
---|---|---
$\log g_{1}$ | $3.46\pm 0.01$ | cgs
$\log g_{2}$ | $4.27\pm 0.01$ | cgs
$\Phi_{V,1}$ | $2.714\cdot 10^{-10}$ | ${\rm W}\,{\rm m}^{-2}$
$\Phi_{V,2}$ | $1.342\cdot 10^{-11}$ | ${\rm W}\,{\rm m}^{-2}$
$\Phi_{V,1}+\Phi_{V,2}+\Phi_{V,3}$ | $3.618\cdot 10^{-10}$ | ${\rm W}\,{\rm m}^{-2}$
$V_{0,1}$ | 2.65 | mag
$V_{0,2}$ | 5.91 | mag
$V_{0,1}+V_{0,2}+V_{0,3}$ | 2.34 | mag

202020 $\log g$ denotes the logarithm of surface gravity, $\Phi_{V}$ the
passband flux at Earth, $V_{0}$ the corresponding apparent magnitude (without
absorption, at a distance of 382 pc).

###  7.2 Model for all observing seasons

Then, we made fits for all seasons. We kept the effective temperature of the
primary $T_{1}$ and the third light $l_{3}$ fixed (as for the preferred
model). The detailed results are presented in Table 16. We took season 2016 as
a reference. In Fig. 13, we show the deviations from the solution for season
2016.

In season 2018, the red filter measurements had greater uncertainties,
however, it did not lead to any artefacts. We also omitted season 2019 since
only measurements in the blue filter are available. We cannot confirm that
variations of the parameters are intrinsic. Since the masses of the components
must be constant, the mean values over all seasons should be preferred. The
variations are most likely due to the oscillations.

Figure 13:  Relative differences of parameters derived for $\delta$ Ori for
eight seasons (Table 16). In all models, the effective temperature $T_{1}$ of
the primary (Aa1) and the third light $l_{3}$ were fixed. Season 2016 was
taken as a reference. Explanations of the parameters can be found in Table 9.

##  8 Spectral energy distribution (SED)

The absolute flux is an additional observational constraint. In the case of
$\delta$ Ori, the absorption is low because the star is not located behind the
Orion molecular clouds. According to the reddening maps of Lallement et al.
(2019)212121https://stilism.obspm.fr/, $E(B-V)=0.042\,{\rm mag}$, with a
substantial scatter of individual samples (due to the Orion clouds).
Therefore, the total absorption $A_{V}\doteq 3.1\,E(B-V)=0.130\,{\rm mag}$, if
extinction is not anomalous. There are not enough line-of-sight samples in the
maps of Green et al. (2019)222222http://argonaut.skymaps.info/.

From the photometric catalogues in VizieR tool (Allen et al. 2014), we took
the standard Johnson system photometry (Ducati 2002) and measurements from
Hipparcos (Anderson & Francis 2012), Gaia DR3 (Gaia Collaboration 2020), 2MASS
(Cutri et al. 2003), WISE (Cutri et al. 2012), MSX (Egan et al. 2003), Akari
(Ishihara et al. 2010), and IRAS (Neugebauer et al. 1984). The data covered
the spectral range from $0.35$ to $100\,\mu{\rm m}$.

We have removed clear outliers, multiple entries, and points without
uncertainties. The IRAS 60 and $100\,\mu{\rm m}$ measurements show an excess,
probably due to the far-infrared emission behind $\delta$ Ori; thus, they have
been removed too. In the end, our photometric data set contained 31 data
points (see Fig. 15).

For the Hvar photometry (Božić 1998), we performed a removal of eclipse phases
(around 0.0, 0.45, 1.0), and computed average values at the maximum light. In
this case, the absolute photometry is more reliable. The magnitudes
transformed to the Johnson system are as follows: $U=0.940\,{\rm mag}$,
$B=1.977\,{\rm mag}$, $V=2.221\,{\rm mag}$, $R=2.334\,{\rm mag}$, with
uncertainties less than $0.010\,{\rm mag}$. The comparison star used was HD
36591 (HR 1861): $V=5.341\,{\rm mag}$, $B-V=-0.190\,{\rm mag}$,
$U-B=-0.927\,{\rm mag}$, $V-R=-0.050\,{\rm mag}$. In order to compare with the
absolute flux, we used the calibrations from Bessell (2000) (see also Fig.
15).

##  9 Three-body model with all observables

In order to account for additional observables in Xitau, we replaced the two-
body model (Aa1+Aa2)+Ab with a three-body model Aa1+Aa2+Ab. Thus, the
equations of motion were:

$\@vec{f}_{i}=\sum_{j\neq i}{Gm_{j}\over
r_{ij}^{3}}\@vec{r}_{ij}+\@vec{f}_{\rm oblat}+\@vec{f}_{\rm ppn}\quad\hbox{for
}\forall\,i\,,$ (10)

where the first term is Newtonian gravitational interactions; the second,
oblateness; and the third, relativistic effects. This model includes all
relevant $N$-body perturbations (e.g. the radial velocities with respect to
the common centre of mass, the light-time effect, precession of $\Omega_{1}$,
$\varpi_{1}$, $\Omega_{2}$, $\varpi_{2}$, variation, evection; see also
Appendix A), even though some of them are of minor importance for $\delta$
Ori. We included the oblateness of the bodies, parametrised with the Love
number $k_{\rm F10}\simeq 0.015$ (Fabrycky 2010)232323 In the Fabrycky (2010)
model, only the radial force component is included. In the multipole model
(Brož et al. 2021), containing all components, the value of
$J_{2}=-C_{20}\simeq 1.5\cdot 10^{-4}$, or equivalently
$k_{2}=J_{2}(\Omega_{0}/n_{0})^{2}\simeq 2.5\cdot 10^{-3}$. , which results in
the observed value of precession $\dot{\omega}\simeq 1.45^{\circ}\,{\rm
y}^{-1}$. Finally, we also included the parametrised post-Newtonian (PPN)
approximation of relativistic effects (Standish & Williams 2006; Brož et al.
2022b) since the stars are both massive and close. The motion was integrated
numerically using a Bulirsch–Stoer integrator, with a precision parameter
$\varepsilon=10^{-8}$, and output every $0.2\,{\rm d}$ (plus exact times of
observations).

Our model was constrained by astrometry (as in Sect. 4), RVs of all components
(Aa1, Aa2, Ab), eclipse timings, eclipse duration, LCs, synthetic spectra, and
the SED. Individual contributions to the $\chi^{2}$ metrics were multiplied by
weights:

(12)

where subscripts denote the data sets mentioned above, respectively. We used
$w_{\rm\scaleto{\rm SKY}{4pt}}=w_{\rm\scaleto{\rm ECL}{4pt}}=10$, due to the
limited number of points, and $w_{\rm\scaleto{\rm
SYN}{4pt}}=w_{\rm\scaleto{\rm SED}{4pt}}=0.1$, due to the remaining
systematics of rectification of spectra and absolute flux measurements.

Our model had 27 free parameters (see Table 11). The osculating elements are
referenced to the epoch 2458773.188651 (TDB), corresponding to the most
precise speckle interferometry measurement. They are defined in the Jacobi
coordinates, suitable for a system with hierarchical geometry. In this
particular case, the distance $d_{\rm pc}$ (Sect. 3) was fixed.

We used the MOST LC (Pablo et al. 2015) to derive three times of the primary
eclipse timings: 2456283.521, 2456289.277, and 2456294.994 (TDB, barycentric).
Additional timings were obtained from the TESS (Ricker et al. 2014):
2458473.344, 2458479.080, 2458484.830 (TDB, barycentric). Due to large-
amplitude oscillations, the uncertainty is degraded to $0.005\,{\rm d}$. The
primary eclipse duration is $0.667\,{\rm d}$, with an uncertainty of
$0.010\,{\rm d}$, again due to the oscillations. We used a simplified eclipse
algorithm for spherical stars.

At the same time, we computed the synthetic LC with the modified version of
the Wilson-Devinney program (Wilson & Devinney 1971; Wilson 1979; Van Hamme &
Wilson 2007; Wilson et al. 2010; Brož 2017); similarly as in Sect. 7. In this
case, however, the instantaneous true phase and distance were determined by
the N-body integration. The third light is no longer an independent parameter;
instead, it is determined by the third component ($m_{3}$, $T_{3}$, $\log
g_{3}$). This allowed us to constrain our model by eclipse depths. Other
improvements included: a correction of computations for highly eccentric
binaries, precise computations of the Roche potential from the volume-
equivalent radius (Leahy & Leahy 2015), and more photometric filters (Prša &
Zwitter 2005), including MOST. As the oscillations were not accounted for in
the synthetic LC, uncertainties of 0.01 mag were assigned to all data points
(see Fig. 21). The observed spectra cover the blue region (430 to 450 nm). The
synthetic spectra were interpolated by Pyterpol (Nemravová et al. 2016) from
the BSTAR and OSTAR grids (Lanz & Hubený 2003, 2007).

We used the Planck (black body) approximation for the whole range of
wavelengths, or absolute synthetic spectra for the limited range of the
respective grids. The fit was performed with the simplex algorithm (see Fig.
14). We consider the best-fit model to be a compromise because it exhibits a
tension between i) the synthetic spectra (in particular, $\log g_{2}$ or
$R_{2}$) and the duration of eclipses, ii) the minima timings, RVs, and
oblateness (see also Fig. 22). The best-fit parameters are summarised in Table
11, and the derived parameters in Table 12. Uncertainties were estimated by
the $\chi^{2}$ mapping and by the MCMC method. Actually, for the Aa1, Aa2
components, they seem to be comparable to the phoebe2 model (Sect. 7), but
here we use a different and more extensive set of observations, in order to
constrain all components at the same time.

The observed and synthetic SEDs are compared in Fig. 15. Even though the
corresponding contribution $\chi^{2}_{\rm\scaleto{\rm LC}{4pt}}$ is larger
than the number of data points $N_{\rm\scaleto{\rm SED}{4pt}}$, we consider
the fit to be acceptable as there are several multiple (but independent)
measurements of the same band that are not consistent with each other. At the
same time, there is no systematic offset of the SED. In other words, our model
provides independent confirmation of the parallax distance.

All blue spectra are shown in Fig. 16. There were remaining systematics
between the observations and the model related to the rectification procedure,
especially close to the He I 4387 line. While the H$\gamma$ was fitted without
problems, the synthetic He I 4471 line was much shallower than the observed
one. These problematic regions were removed from the fitting. These spectra
constrain not only the RVs but also the relative luminosities $L$, $\log g$,
or radii $R$ of all components.

Contributions of individual components are demonstrated in Fig. 17. Indeed,
the secondary (Aa2) is faint (relative $L_{2}=0.038$). Unfortunately, its
contribution is comparable to the systematics mentioned above. Consequently,
some of the parameters are not very stable (in particular, $\log g_{2}$,
$v_{\rm rot2}$). Nevertheless, our fitting in the direct space is independent
and complementary to the fitting in Fourier space (Sect. 5). Moreover, the
secondary is constrained by other observables (e.g. eclipse duration, eclipse
depth, RVs of the primary, the 3rd-body orbit, or the total mass
$m_{1}+m_{2}+m_{3}$).

#### Mirror solution.

Eventually, we explored the mirror solution (Sec. 4). We fixed the total mass
$m_{1}+m_{2}+m_{3}=52.0\,{\rm M}_{\odot}$ and performed a similar testing as
in Fig. 22. The overall best-fit has $\chi^{2}=25468$, which is worse than the
nominal model. It exhibits a strong tension between the synthetic spectra and
the SED. Especially the H$\gamma$ line profiles are fitted poorly
($\chi^{2}_{\rm syn}=69441$ vs. $44795$). This is directly related to the
$\log g_{3}$ value, which is very low (around 3.2) according to our model, as
well as modelling of the disentangled spectrum of the tertiary component
(Sects. 6). Consequently, we exclude the mirror solution and prefer the
nominal model presented above.

Table 11:  Free parameters of the three-body model of $\delta$ Ori. The best-
fit model has non-reduced $\chi^{2}=23739$.

Component | Parameter | Value | Unit | $\sigma$
---|---|---|---|---
Aa+Ab | $m_{\rm tot}$ | 35.108 |  M⊙ | 2.0
Aa1+Aa2 | $q_{1}$ | 0.4785 | 1 | 0.03
Aa+Ab | $q_{2}$ | 0.3338 | 1 | 0.03
Aa1+Aa2 | $P_{1}$ | 5.733121 | d | 0.000001
| $\log e_{1}$ | $-1.089$ | 1 | 0.01
| $i_{1}$ | 79.124 | deg | 1.0
| $\Omega_{1}$ | 224.294 | deg | 1.0
| $\varpi_{1}$ | 43.451 | deg | 1.0
| $\lambda_{1}$ | 72.695 | deg | 1.0
Aa+Ab | $P_{2}$ | 55453 | d | 1000.0
| $\log e_{2}$ | $-0.248$ | 1 | 0.01
| $i_{2}$ | 105.170 | deg | 1.0
| $\Omega_{2}$ | 122.735 | deg | 1.0
| $\varpi_{2}$ | 261.255 | deg | 1.0
| $\lambda_{2}$ | 133.391 | deg | 1.0
Aa1 | $T_{1}$ | 31385 | K | 1000
Aa2 | $T_{2}$ | 24515 | K | 1000
Ab | $T_{3}$ | 27906 | K | 1000
Aa1 | $\log g_{1}$ | 3.452 | cgs | 0.1
Aa2 | $\log g_{2}$ | 4.128 | cgs | 0.1
Ab | $\log g_{3}$ | 3.220 | cgs | 0.1
Aa1 | $v_{\rm rot1}$ | 114 | ${\rm km}\,{\rm s}^{-1}$ | 10
Aa2 | $v_{\rm rot2}$ | 184 | ${\rm km}\,{\rm s}^{-1}$ | 100
Ab | $v_{\rm rot3}$ | 279 | ${\rm km}\,{\rm s}^{-1}$ | 10
Aa1 | $C_{20}$ | $-9.2\times 10^{-4}$ | 1 | $0.1\times 10^{-4}$
| $z_{0}$ | 2.221 | mag | 0.01
| $\gamma$ | 18.455 | ${\rm km}\,{\rm s}^{-1}$ | 1
| $d_{\rm pc}$ |  382f | pc | $-$
| $N_{\rm\scaleto{\rm SKY}{4pt}}$ | 74 |  |
| $N_{\rm\scaleto{\rm RV}{4pt}}$ | 105 |  |
| $N_{\rm\scaleto{\rm ETV}{4pt}}$ | 6 |  |
| $N_{\rm\scaleto{\rm ECL}{4pt}}$ | 1 |  |
| $N_{\rm\scaleto{\rm LC}{4pt}}$ | 22136 |  |
| $N_{\rm\scaleto{\rm SYN}{4pt}}$ | 76794 |  |
| $N_{\rm\scaleto{\rm SED}{4pt}}$ | 31 |  |
| $N$ | 147147 |  |
| $\chi^{2}_{\rm\scaleto{\rm SKY}{4pt}}$ | 59 |  |
| $\chi^{2}_{\rm\scaleto{\rm RV}{4pt}}$ | 2991 |  |
| $\chi^{2}_{\rm\scaleto{\rm ETV}{4pt}}$ | 82 |  |
| $\chi^{2}_{\rm\scaleto{\rm ECL}{4pt}}$ | 10 |  |
| $\chi^{2}_{\rm\scaleto{\rm LC}{4pt}}$ | 13704 |  |
| $\chi^{2}_{\rm\scaleto{\rm SYN}{4pt}}$ | 44795 |  |
| $\chi^{2}_{\rm\scaleto{\rm SED}{4pt}}$ | 17872 |  |
| $\chi^{2}$ | 23739 |  |

242424 $m_{\rm tot}$ denotes the total mass; $q_{1}=m_{2}/m_{1}$,
$q_{2}=m_{3}/(m_{1}+m_{2})$ the respective mass ratios; $P$, osclulating
period; $e$, eccentricity; $i$, inclination; $\Omega$, longitude of node;
$\varpi$, longitude of pericentre; $\lambda$, true longitude; $T$, effective
temperature; $g$, gravitational acceleration; $v_{\rm rot}$, rotational
velocity; $C_{20}$, quadrupole moment; $z_{\rm 0}$, magnitude zero point;
$\gamma$, systemic velocity; and $d_{\rm pc}$, distance. f indicates the
respective parameter was fixed. Orbital elements are osculating for the epoch
$T_{0}=2458773.188651$ (TDB).

Table 12:  Derived parameters of the three-body model of $\delta$ Ori.

Component | Parameter | Value | Unit | $\sigma$
---|---|---|---|---
Aa1 | $m_{1}$ | 17.803 |  M⊙ | 1.0
Aa2 | $m_{2}$ | 8.518 |  M⊙ | 1.0
Ab | $m_{3}$ | 8.787 |  M⊙ | 1.0
Aa1+Aa2 | $a_{1}$ | 40.099 |  R⊙ | 1.0
Aa+Ab | $a_{2}$ | 20038 |  R⊙ | 100
Aa1+Aa2 | $e_{1}$ | 0.081 | 1 | 0.02
Aa+Ab | $e_{2}$ | 0.565 | 1 | 0.01
Aa1 | $R_{1}$ | 13.119 |  R⊙ | 1.0
Aa2 | $R_{2}$ | 4.168 |  R⊙ | 0.5
Ab | $R_{3}$ | 12.045 |  R⊙ | 1.0
Aa1 | $L_{1}$ | 0.563 | 1 | 0.1
Aa2 | $L_{2}$ | 0.038 | 1 | 0.01
Ab | $L_{3}$ | 0.397 | 1 | 0.1

252525 $m$ denotes the mass; $a$, the semi-major axis; $e$, the eccentricity;
$R$, the stellar radius; $L$, the relative luminosity (in V).

Figure 14:  Example of a convergence of the three-body model of $\delta$ Ori.
The individual contributions to $\chi^{2}$ correspond to astrometry (SKY),
radial velocities (RV), eclipse timings (ETV), eclipse duration (ECL),
synthetic spectra (SYN), and the spectral energy distribution (SED). The total
$\chi^{2}$ is summed with non-unit weights (see values in the main text).

Figure 15:  Model from SED data. Top: Comparison of the observed (blue) and
synthetic (orange) SED of $\delta$ Ori. The residuals are plotted in red. The
wavelength range is from $350\,{\rm nm}$ (ultraviolet) to $25\,\mu{\rm m}$
(far-infrared). Bottom: The same for the limited wavelength range of synthetic
spectra. The Hvar differential UBVR photometry with removed eclipses is
plotted in green. Figure 16:  Comparison of the observed (blue) and synthetic
(orange) rectified spectra. The residuals are plotted in red. All components
Aa1, Aa2, Ab of $\delta$ Ori contribute to the total flux. The wavelength
range included the spectral lines: H$\gamma$ 4341, He I 4378, and numerous
weaker lines.  Figure 17:  Same as Fig. 16, with the distinguished
contributions of individual components Aa1, Aa2, Ab of $\delta$ Ori. The
wavelength range is limited to 430 to 440 nm. Only the sixth spectrum,
2455836.582700 is shown.

##  10 Pulsations

After obtaining a well-constrained model, we analysed the residual MOST LC
(Fig. 18), in order to address the large-amplitude oscillations (BRITE data
were not used because of instrumental issues). Our analysis is similar to that
in Pablo et al. (2015), albeit it is different in several aspects: i) our
model (from Sect. 9) is constrained by all observables, ii) not only did we
subtract the synthetic LC but we also removed the eclipse intervals (both
primary and secondary), to suppress the binary signal. Consequently, the
remaining frequencies should be preferentially related to rotation or
pulsations, even though ‘gaps’ also create spurious signals.

We used the Period04 program (Lenz & Breger 2004) to compute the Fourier
spectrum (Fig. 19), subtract the dominant term (prewhitening), recompute the
spectrum again, and repeat these steps ten times. Our result is shown in Table
13.

For reference, the minimum frequency (also spacing) is given by the time span
of observations, $f_{\Delta}=1/\Delta=0.045\,{\rm c}\,{\rm d}^{-1}$, which
means that frequencies differing by $f_{\Delta}$ are certainly possible to
distinguish. The maximum frequency (also Nyquist) is given by the sampling,
$f_{\rm Ny}=1/(2\delta)\doteq 1000\,{\rm c}\,{\rm d}^{-1}$. The orbital
frequency of the binary Aa1+Aa2 is $f_{\rm orb}=1/P_{1}=0.174\,{\rm c}\,{\rm
d}^{-1}$.

In the MOST data, the first two frequencies are $f_{\rm 1}=0.218\,{\rm
c}\,{\rm d}^{-1}$, $f_{\rm 2}=0.478\,{\rm c}\,{\rm d}^{-1}$. These correspond
to the Chandra data (Nichols et al. 2015), $f_{\rm X1}=0.210\,{\rm c}\,{\rm
d}^{-1}$, $f_{\rm X2}=0.490\,{\rm c}\,{\rm d}^{-1}$, within uncertainties. The
first one was identified as the rotation frequency of the primary (Aa1) by
Nichols et al. (2015). Here, we interpret the second one as the rotation of
the tertiary (Ab). This is confirmed by the rotational broadening. For
parameters from Tables 11, 12, $f_{\rm rot3}=v_{\rm rot3}/(2\pi R_{3}\sin
i_{2})=0.474\,{\rm c}\,{\rm d}^{-1}$, where we assumed an alignment.

Interestingly, $f_{\rm rot1}=v_{\rm rot1}/(2\pi R_{1}\sin i_{1})=0.174\,{\rm
c}\,{\rm d}^{-1}$ derived from rotational broadening is not equal to $f_{1}$,
which differs by $f_{\Delta}$. It corresponds to the orbital frequency, which
would indicate a synchronous binary.

Alternatively, when we assume an asynchronous primary, the synchronicity is
$F_{1}=f_{1}/f_{\rm orb}=1.250$. For comparison, the periastron-synchronised
value is $F_{1}=[(1+e_{1})/(1-e_{1})^{3}]^{1/2}=1.181$. In both cases, it
means a minor modification of the LC.

The remaining frequencies ($f_{3}$, $f_{4}$, $f_{10}$) are likely associated
with pulsations, namely low-order modes $\ell=0,1,2$, or $3$, typical for
$\beta$ Cep or SPB stars (Paxton et al. 2015). They can be present either on
the primary (Aa1), or the tertiary (Ab), which contributes up to 40% of the
light.

Table 13:  Fourier analysis of the residual MOST LC (from Fig. 18). The
synthetic LC was subtracted and eclipses were removed to suppress the binary
signal. The first two frequencies are then identified as the rotation
frequency of the primary (Aa1; Nichols et al. 2015) and the tertiary (Ab; this
work).

| Frequency | Period | Amplitude | Notes
---|---|---|---|---
|  [${\rm c}\,{\rm d}^{-1}$] |  [${\rm d}$] | [mmag] |
$f_{1}$ | 0.218 | 4.572 | 6.37 | rotation of Aa1
$f_{2}$ | 0.478 | 2.090 | 3.39 | rotation of Ab
$f_{3}$ | 0.396 | 2.522 | 4.04 | pulsation?
$f_{4}$ | 0.929 | 1.075 | 3.05 | pulsation?
$f_{5}$ | 0.168 | 5.931 | 2.95 | orbital (Aa1+Aa2)
$f_{6}$ | 1.168 | 0.855 | 1.95 |  orbital$\,+\,1\,{\rm c}\,{\rm d}^{-1}$
$f_{7}$ | 0.975 | 1.025 | 1.96 |  approx. $1\,{\rm c}\,{\rm d}^{-1}$
$f_{8}$ | 0.346 | 2.887 | 2.54 |  $2f_{\rm orb}$?
$f_{9}$ | 0.086 | 11.551 | 1.67 |  $0.5f_{\rm orb}$?
$f_{10}$ | 2.214 | 0.451 | 1.29 | pulsation?

262626 The uncertainty is determined by the time span, $f_{\Delta}=0.045\,{\rm
c}\,{\rm d}^{-1}$. For reference, the orbital frequency of the Aa1+Aa2 binary
$f_{\rm orb}=0.174\,{\rm c}\,{\rm d}^{-1}$, and the orbital frequency of the
satellite $f_{\rm most}=14.19\,{\rm c}\,{\rm d}^{-1}$.

Figure 18:  Residuals of the MOST LC (red), and the Fourier series up to the
order of $\ell=10$ (black; Table 13). Individual orders as they are summed up
are shown in grey. The times of primary (P) and secondary (S) eclipses are
indicated on top.  Figure 19:  Periodogram of the residual MOST LC with the 10
principal frequencies indicated on top. The first periodogram is shown; it is
subsequently modified by subtraction (prewhitening). The broad peak between 10
to $20\,{\rm c}\,{\rm d}^{-1}$ corresponds to the satellite frequency $f_{\rm
most}$ and its combinations with $f_{1}$, etc.

##  11 Conclusions

In this paper, we studied the triple star Aa1+Aa2+Ab in the multiple system
$\delta$ Ori. The close eclipsing binary Aa1+Aa2 contains an O star, is non-
interacting, and has a negligible mass transfer. Consequently, it represents a
target suitable for defining the intrinsic parameters of evolved O stars. Our
main results are as follows:

  1. 1.

The distance of the system was estimated from the new Gaia DR3 parallax of the
faint component $\delta$ Ori (Ca+Cb).

  2. 2.

The outer orbit (Aa1+Aa2)+Ab was constrained by new speckle interferometric
measurements from the WDS (the period of approximately 152 years and
eccentricity $0.58$) and by $\gamma$ velocities.

  3. 3.

The secondary (Aa2) spectrum in the blue spectral region was detected by the
two- and three-step disentangling.

  4. 4.

The RV curve of the secondary was obtained by cross-correlation with a
disentangled template spectrum.

  5. 5.

The two-body model of the eclipsing binary was constructed in PHOEBE2.

  6. 6.

The three-body model in Xitau was constrained by all observables.

Compared to previous studies, we obtained significantly lower masses than
Pablo et al. (2015), in their low-mass model ($23.81+8.54$) M⊙. In contrast to
the study of Shenar et al. (2015), where models were calculated for two
distances, 212 and 380 pc, we adopted the latter. Our results give lower radii
of Aa1, Aa2; Shenar et al. (2015) have [($16.5\pm 1)+(6.5\pm 2$)] R⊙.
Nevertheless, the radius of Ab is in agreement with Shenar et al. (2015) who
reported $(10.4\pm 2)\,$R⊙. We also obtained low $\log g_{3}$ similarly as
Shenar et al. (2015).

#### Hertzprung–Russel diagram.

Given the spectral types of Aa1 + Aa2 + Ab, O9.5 II + B2 V + B0 IV (Pablo et
al. 2015 and this work), the primary has evolved from the main sequence;
however, it has not reached the overflow yet. The Hertzsprung–Russel diagram
(Fig. 20) with the positions of Aa1, Aa2, Ab components indicates an
interesting problem — the Ab component is very offset from a normal position.
This offset is either related to its $\log g_{3}$ value (3.2) or to its
$m_{3}$ value ($8.7\,{\rm M}_{\odot}$). However, it is not easy to modify
these values, because they are well constrained by observations. To put Ab on
the evolutionary track, either $\log g_{3}\simeq 3.7$, or $m_{3}\simeq
18\,{\rm M}_{\odot}$. If all components were normal, the sum of masses should
be about $(24+18+10)\,{\rm M}_{\odot}=52\,{\rm M}_{\odot}$. Interestingly,
this is similar to the mirror solution, which was excluded (see the discussion
in Sects. 4 and 9). Consequently, we are left with an unusual stellar
component. Actually, it is not unusual — see for example $\delta$ Ori Ca, or
$\sigma$ Ori E which are both helium-rich, with H$\alpha$ emission (Tab. 3).
Detailed stellar-evolution models with possible mass transfer between (some
of) the components shall be computed in future work. Additionally, long-
baseline optical or near-infrared interferometry may be able to measure
precisely the angular diameters of the component stars (e.g. (Shabun et al.
2008)) and the separation of the inner orbit in the sky, giving direct
constraints on the size of the orbit, helping to resolve any discrepancies in
the masses measured between this study and other similar studies of $\delta$
Ori.

A comparison of $\delta$ Ori with other bright stars in the Orion belt (see
Table 3) shows that $\sigma$ Ori has a similar architecture
(((Aa+Ab)+B)+C)+D+E, and even a very similar angular scale (Simón-Díaz et al.
2015). All of its components seem to be less evolved.272727$\iota$ Ori is
located farther away, in the Trapezium region. On the other hand, $\zeta$ Ori
exhibits an angular scale about 10 times larger and has the primary evolved in
an O supergiant (Hummel et al. 2000). In this sense, $\varepsilon$ Ori, which
seems to be a single variable B supergiant (Puebla et al. 2016), may represent
an even more evolved object.

Given the fact that all these stars ($\delta$, $\varepsilon$, $\zeta$,
$\sigma$) are the most massive within the Orion OB1b association, they might
have encountered and perturbed (destabilised) each other. Again, a possible
convergence of their proper motions will be analysed in future work.

Figure 20:  Hertzsprung–Russel diagram with the positions of the Aa1, Aa2, and
Ab components of $\delta$ Ori and evolutionary tracks from Paxton et al.
(2015). Numbers next to the main sequence indicate the theoretical masses,
small numbers the theoretical gravitational acceleration $\log g$. According
to the 3-body model, the masses are $17.1$, $8.5$, $8.7\,{\rm M}_{\odot}$. For
the tertiary (Ab), it is in disagreement with the evolutionary track, but in
agreement with the value of $\log g_{3}$ inferred from normalised spectra
(H$\gamma$). The instability region of pulsations ($\beta$ Cep type, order
$\ell=0$) is indicated as grey area. Other modes ($\ell=1,2,3$) can be found
in a very similar region (Paxton et al. 2015). Both Aa1, Ab components are
located here, and they can exhibit photometric variability attributed to
pulsations.

###### Acknowledgements.

This research was supported by the grants P209/10/0715, GA15-02112S, and
GA19-01995S of the Czech Science Foundation. M.B. was supported by the Czech
Science Foundation grant GA21-11058S. We acknowledge the support from the
Research Programmes MSM0021620860 “Physical study of objects and processes in
the solar system and in astrophysics” of the Ministry of Education of the
Czech Republic and the grant AV0Z10030501 of the Academy of Sciences of the
Czech Republic. Thanks to Dr. Rachel Matson for providing the data from
Washington Double Star Catalogue. H.B. acknowledges financial support from the
Croatian Science Foundation under the project 6212 “Solar and Stellar
Variability”. We gratefully acknowledge the use of the program KOREL written
by Petr Hadrava, PYTERPOL by Jana Nemravová, and reSPEFO2 by Adam Harmanec.
This research has made use of the AMBER data reduction package of the Jean-
Marie Mariotti Center282828http://www.jmmc.fr/amberdrs. Our thanks are due to
Š. Dvořáková, D. Korčáková, J. Kubát, J. Nemravová, M. Oksala, P. Škoda, and
V. Votruba who obtained some of the Ondřejov blue spectra used in this study.
We also acknowledge the use of the electronic database from CDS Strasbourg and
the electronic bibliography maintained by the NASA/ADS system. CCL and GAW
acknowledge Discovery Grant support by the Natural Science and Engineering
Research Council (NSERC) of Canada. Last but not least, we would like to thank
our anonymous referee for the constructive useful comments that helped to
improve the paper.

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##  Appendix A Secular rates for $\delta$ Ori A

We used the standard Gauss equations to estimate the secular rates for the
$\delta$ Ori A system. A perturbation due to the quadrupole moment
$J_{2}\equiv-C_{20}$ of the primary induces a precession of the argument of
pericentre:

$\dot{\omega}_{1}=+3n_{1}J_{2}\left({R_{1}\over
a_{1}}\right)^{2}{5\cos^{2}\tilde{i}_{1}-1\over 4\eta_{1}^{4}}\,,$ (84)

and the longitude of the node:

$\dot{\Omega}_{1}=-{3\over 2}n_{1}J_{2}\left({R_{1}\over
a_{1}}\right)^{2}{\cos\tilde{i}_{1}\over\eta_{1}^{4}}\,,$ (85)

where $n_{1}=\sqrt{G(m_{1}{+}m_{2})/a_{1}^{3}}$,
$\eta_{1}\equiv\sqrt{1-e_{1}^{2}}$, and $\tilde{i}_{1}$ is the inclination
with respect to the primary equator. Otherwise, the notation is the same as in
Table 11. For $\tilde{i}_{1}\to 0$, $e_{1}\to 0$, $J_{2}>0$, we would have
$\dot{\omega}_{1}>0$, $\dot{\Omega}_{1}<0$,
$\dot{\omega}_{1}=-2\dot{\Omega}_{1}$.

The quadrupole moment is determined by the internal structure:

$J_{2}=-{1\over
m_{1}R_{1}^{2}}\int_{V}\rho|\@vec{r}|^{2}P_{2}(\cos\theta)\,{\rm d}V\,,$ (86)

where the Legendre polynomial $P_{2}(x)={1\over 2}(3x^{2}-1)$. For a
homogeneous body, it would be related to the ellipticity (Fitzpatrick
2012)292929$r(\vartheta)=R(1-2/3\varepsilon_{0}P_{2}(\cos\vartheta))$:

$J_{2}=-{2\over 5}\varepsilon_{0}\,.$ (87)

For a rotating body, it is related to the Love number303030approximately,
$0.02$ for the Sun; $0.3$ for the Earth, Jupiter, or an M-dwarf; up to $0.75$
for incompressible fluid:

$J_{2}\simeq k_{2}\left({\Omega_{0}\over n_{0}}\right)^{2},$ (88)

where $\Omega_{0}$ is the angular rotation frequency; $n_{0}$, the mean motion
at the surface. Assuming $J_{2}=1.8\cdot 10^{-4}$ results in
$\dot{\omega}=1.47\,{\rm deg}\,{\rm y}^{-1}$, $\dot{\Omega}=-0.73\,{\rm
deg}\,{\rm y}^{-1}$, where the node circulates with respect to the primary
equator, but it only librates with respect to the observer plane.

For the binary Aa1+Aa2 (acting on a mass-less particle), the equations should
be modified as follows:

$\dot{\omega}_{2}=+3n_{2}J_{2}\left({a_{1}\over
a_{2}}\right)^{2}{5\cos^{2}\tilde{i}_{2}-1\over 4\eta_{2}^{4}}\,,$ (89)
$\dot{\Omega}_{2}=-{3\over 2}n_{2}J_{2}\left({a_{1}\over
a_{2}}\right)^{2}{\cos\tilde{i}_{2}\over\eta_{2}^{4}}\,,$ (90)

where the inclination $\tilde{i}_{2}$ is with respect to the binary. The
effective quadrupole moment is:

$J_{2}={1\over 2}{m_{1}m_{2}\over(m_{1}+m_{2})^{2}}\,,$ (91)

because the respective radius ($|\@vec{r}|=a_{1}$) is the same as the
reference radius ($R=a_{1}$). Given that $J_{2}=0.109$, and the ratio of
$a_{1}/a_{2}=0.002$, the precession rates should be of the order of
$10^{-6}\,{\rm deg}\,{\rm y}^{-1}$.

However, for the massive triple system (Aa1+Aa2)+Ab, we used the theory of
Breiter & Vokrouhlický (2015); for the longitudes of pericentre and node:

$\displaystyle\dot{\varpi}_{2}$ $\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!$
$\displaystyle{3\over 8}{n_{2}\over\eta_{2}^{3}}{m_{3}\over
m_{1}\\!+\\!m_{2}\\!+\\!m_{3}}{n_{2}\over n_{1}}\gamma\cdot$ (92)
$\displaystyle\\!\\!\\!\\!\cdot\\!\\!\\!\\!$
$\displaystyle\left[3\cos^{2}\\!J-1-{\gamma\sin J\sin 2J\over 1+\gamma\cos
J+\\!\sqrt{1+\gamma^{2}+2\gamma\cos J}}\right],$ $\dot{\Omega}_{2}={3\over
4}{n_{2}\over\eta_{2}^{3}}{m_{3}\over
m_{1}\\!+\\!m_{2}\\!+\\!m_{3}}{n_{2}\over n_{1}}\cos
J\sqrt{1+\gamma^{2}+2\gamma\cos J}\,,$ (93)

where $m_{1}^{\prime}\\!=m_{1}m_{2}/(m_{1}{+}m_{2})$,
$m_{2}^{\prime}\\!=(m_{1}{+}m_{2})\,m_{3}/(m_{1}{+}m_{2}{+}m_{3})$ denote the
reduced masses; $L_{1}=m_{1}^{\prime}n_{1}a_{1}^{2}$,
$L_{2}=m_{2}^{\prime}n_{2}a_{2}^{2}$, the angular momenta; $\cos J=\cos
i_{1}\cos i_{2}+\sin i_{1}\sin i_{2}\cos(\Omega_{1}-\Omega_{2})$, the mutual
inclination; and $\gamma=L_{1}/(L_{2}\eta_{2})$. Assuming parameters from
Table 11, $\dot{\varpi}_{2}=3.4\cdot 10^{-6}\,{\rm deg}\,{\rm y}^{-1}$,
$\dot{\Omega}_{2}=8.5\cdot 10^{-5}\,{\rm deg}\,{\rm y}^{-1}$, which is
negligible on the observational time span. All these effects were nevertheless
included in our numerical N-body model (Sect. 9).

##  Appendix B Supplementary figures and tables

In Tables 14 and 15, we present more details on the spectral data sets
discussed in Sect. 2.1. In Table 16, we report parameters derived for the 8
seasons observed by the BRITE satellites. The LC from our three-body model
(Sect. 9) is shown in Fig. 21. Individual contributions to $\chi^{2}$ computed
for an extensive set models of $\delta$ Ori is shown in Fig. 22.

Table 14:  Details on spectra from the coudé focus of the Ondřejov 2m reflector in the blue region. The RVs were determined during the spectral disentangling (three-step method) in KOREL.  $T$ | Exposure time | S/N | Heliocentric correction |  RV1 | $\sigma_{\mathrm{RV}_{1}}$ |  RV2 | $\sigma_{\mathrm{RV}_{2}}$ |  RV3 | $\sigma_{\mathrm{RV}_{3}}$
---|---|---|---|---|---|---|---|---|---
[$\mathrm{HJD-2400000}$] | [s] |  |  [km s-1] |  [km s-1] |  [km s-1] |  [km s-1] |  [km s-1] |  [km s-1] |  [km s-1]
55836.5692 | 1113 | 235.3 | $26.2335$ | $-13.03$ | 1.73 | $101.44$ | 1.48 | 17.25 | 1.72
55836.5827 | 1131 | 268.3 | $26.2097$ | $-13.29$ | 1.52 | $100.76$ | 1.30 | 16.46 | 1.51
55836.5963 | 1129 | 217.3 | $26.1849$ | $-13.97$ | 1.87 | $104.81$ | 1.60 | 17.91 | 1.86
55836.6137 | 1250 | 238.0 | $26.1517$ | $-14.86$ | 1.71 | $108.90$ | 1.46 | 17.19 | 1.70
55836.6287 | 1265 | 236.2 | $26.1222$ | $-17.11$ | 1.72 | $115.37$ | 1.47 | 17.70 | 1.71
55836.6434 | 1174 | 254.2 | $26.0929$ | $-19.88$ | 1.60 | $113.68$ | 1.37 | 15.16 | 1.59
55837.5877 | 1124 | 214.3 | $26.0588$ | $-94.63$ | 1.90 | $278.78$ | 1.63 | 16.35 | 1.88
55837.6016 | 1134 | 239.5 | $26.0328$ | $-96.07$ | 1.70 | $280.05$ | 1.45 | 13.32 | 1.69
55837.6152 | 1150 | 232.7 | $26.0064$ | $-95.62$ | 1.75 | $279.59$ | 1.50 | 14.24 | 1.74
55837.6294 | 1218 | 237.1 | $25.9784$ | $-98.09$ | 1.71 | $278.48$ | 1.47 | 13.54 | 1.70
55871.5969 | 1999 | 217.1 | $16.9755$ | $-79.45$ | 1.87 | $251.90$ | 1.60 | 17.42 | 1.86
55893.4601 | 1662 | 226.5 | $7.9171$ | $36.33$ | 1.79 | $-31.61$ | 1.54 | 17.48 | 1.78
55893.4820 | 2003 | 225.8 | $7.8667$ | $36.04$ | 1.80 | $-25.90$ | 1.54 | 11.27 | 1.79
55953.4009 | 1916 | 224.9 | $-19.2387$ | $-11.07$ | 1.81 | $91.43$ | 1.55 | 15.22 | 1.80
55953.4226 | 1766 | 243.8 | $-19.2836$ | $-9.87$ | 1.67 | $87.03$ | 1.43 | 12.58 | 1.66
55953.4508 | 1599 | 250.2 | $-19.3381$ | $-7.24$ | 1.62 | $79.34$ | 1.39 | 14.11 | 1.61
55953.4692 | 1503 | 252.0 | $-19.3704$ | $-5.12$ | 1.61 | $75.37$ | 1.38 | 14.21 | 1.60
55953.4871 | 1468 | 216.3 | $-19.3992$ | $-1.59$ | 1.88 | $71.30$ | 1.61 | 14.07 | 1.87
55956.3498 | 3561 | 230.1 | $-20.1601$ | $58.08$ | 1.77 | $-72.93$ | 1.51 | 11.98 | 1.75
55959.2469 | 2134 | 230.6 | $-20.9128$ | $4.46$ | 1.76 | $68.90$ | 1.51 | 14.61 | 1.75
55977.3276 | 2165 | 209.0 | $-25.6205$ | $86.76$ | 1.95 | $-121.98$ | 1.67 | 14.70 | 1.93
55990.3318 | 4872 | 206.6 | $-27.3138$ | $96.47$ | 1.97 | $-152.94$ | 1.69 | 14.40 | 1.95
55991.2779 | 1466 | 258.1 | $-27.2883$ | $-5.95$ | 1.58 | $77.34$ | 1.35 | 14.43 | 1.56
55992.2949 | 1415 | 199.3 | $-27.3830$ | $-93.85$ | 2.04 | $278.69$ | 1.75 | 18.81 | 2.03
56003.3328 | 1200 | 186.8 | $-27.6165$ | $-72.08$ | 2.18 | $231.46$ | 1.86 | 12.62 | 2.16
56011.3498 | 1361 | 142.0 | $-27.1206$ | $54.90$ | 2.86 | $-52.75$ | 2.45 | 17.89 | 2.84
56167.6213 | 2054 | 154.9 | $26.1472$ | $118.69$ | 2.62 | $-195.80$ | 2.25 | 15.19 | 2.61
56241.4709 | 2405 | 199.5 | $15.4295$ | $112.55$ | 1.86 | $-178.56$ | 1.59 | 6.27 | 1.85
56257.4224 | 2287 | 219.6 | $8.5701$ | $13.12$ | 1.85 | $42.02$ | 1.59 | 16.01 | 1.84
56257.6273 | 3324 | 227.7 | $8.1164$ | $35.28$ | 1.79 | $-5.17$ | 1.53 | 13.21 | 1.77
56330.3789 | 1399 | 245.5 | $-22.8962$ | $-91.20$ | 1.66 | $271.45$ | 1.42 | 16.21 | 1.64
56354.3038 | 3112 | 241.5 | $-27.1587$ | $-50.33$ | 1.54 | $180.31$ | 1.32 | 5.28 | 1.53
56357.3761 | 1138 | 170.5 | $-27.5049$ | $88.62$ | 2.38 | $-124.65$ | 2.04 | 15.73 | 2.37
56596.5631 | 1415 | 212.4 | $19.1204$ | $98.83$ | 1.91 | $-147.64$ | 1.64 | 17.80 | 1.90
56608.6423 | 2196 | 181.0 | $14.3991$ | $121.15$ | 2.25 | $-201.59$ | 1.92 | 13.34 | 2.23
56609.4312 | 3840 | 166.5 | $14.4441$ | $93.94$ | 2.44 | $-153.01$ | 2.09 | 19.46 | 2.43
56621.6375 | 6298 | 264.2 | $8.7784$ | $22.16$ | 1.54 | $14.69$ | 1.32 | 14.10 | 1.53
56629.6451 | 5211 | 232.3 | $5.0123$ | $-30.90$ | 1.75 | $135.90$ | 1.50 | 15.84 | 1.74
56642.5644 | 2340 | 236.0 | $-1.1527$ | $103.79$ | 1.72 | $-163.06$ | 1.48 | 14.29 | 1.71
56643.3817 | 2365 | 218.8 | $-1.2259$ | $121.57$ | 1.86 | $-196.79$ | 1.59 | 12.93 | 1.85
56643.6078 | 4554 | 246.3 | $-1.7159$ | $110.68$ | 1.51 | $-198.65$ | 1.29 | 19.58 | 1.50
56666.3166 | 2968 | 197.2 | $-11.9535$ | $120.06$ | 2.06 | $-196.17$ | 1.77 | 16.15 | 2.05
56666.4940 | 5013 | 237.6 | $-12.3482$ | $120.37$ | 1.56 | $-187.88$ | 1.34 | 8.33 | 1.55
56704.3508 | 2793 | 279.8 | $-24.9170$ | $-15.22$ | 1.45 | $90.44$ | 1.25 | 13.73 | 1.44
56714.2672 | 2292 | 260.6 | $-26.4964$ | $-77.91$ | 1.56 | $246.02$ | 1.34 | 16.17 | 1.55
56719.3096 | 3214 | 240.8 | $-27.1475$ | $-1.73$ | 1.69 | $79.65$ | 1.45 | 18.73 | 1.68
56721.3496 | 2993 | 256.5 | $-27.3779$ | $-32.32$ | 1.59 | $140.09$ | 1.36 | 16.81 | 1.57
56737.3202 | 629 | 226.3 | $-27.4373$ | $-93.55$ | 1.80 | $265.23$ | 1.54 | 15.84 | 1.78
56738.3020 | 2404 | 184.0 | $-27.3532$ | $-58.28$ | 2.21 | $194.34$ | 1.89 | 15.57 | 2.19
56746.2894 | 1392 | 199.8 | $-26.5682$ | $120.01$ | 2.03 | $-201.76$ | 1.74 | 16.48 | 2.02
56928.5832 | 1652 | 221.9 | $26.6856$ | $59.66$ | 1.83 | $-64.14$ | 1.57 | 14.96 | 1.82
57105.2900 | 1986 | 205.6 | $-27.2198$ | $-45.65$ | 1.98 | $170.70$ | 1.69 | 12.24 | 1.96
57106.2877 | 565 | 118.5 | $-27.1303$ | $60.16$ | 3.43 | $-62.99$ | 2.94 | 14.03 | 3.41
57106.3182 | 1784 | 166.9 | $-27.1696$ | $63.36$ | 2.44 | $-69.25$ | 2.09 | 16.40 | 2.42
57116.3066 | 1651 | 156.0 | $-25.8607$ | $-88.80$ | 2.61 | $254.62$ | 2.23 | 11.35 | 2.59
57128.2762 | 1434 | 158.1 | $-23.3452$ | $-39.86$ | 2.57 | $158.43$ | 2.20 | 14.13 | 2.55
57297.5482 | 3202 | 187.7 | $26.2886$ | $83.80$ | 2.17 | $-113.55$ | 1.86 | 20.79 | 2.15
57364.5793 | 2224 | 219.8 | $2.9803$ | $85.92$ | 1.85 | $-124.80$ | 1.58 | 14.50 | 1.84
57445.3477 | 3194 | 218.4 | $-26.6982$ | $114.68$ | 1.86 | $-185.19$ | 1.60 | 16.60 | 1.85
57464.3540 | 6102 | 211.3 | $-27.6372$ | $18.76$ | 1.92 | $28.03$ | 1.65 | 15.26 | 1.91
58389.5221 | 2398 | 176.3 | $26.7622$ | $-29.27$ | 2.31 | $132.85$ | 1.98 | 16.06 | 2.29
58390.6224 | 161 | 79.9 | $26.4953$ | $81.98$ | 5.09 | $-114.83$ | 4.36 | 15.47 | 5.05
58390.6416 | 3001 | 174.5 | $26.4581$ | $85.89$ | 2.33 | $-118.96$ | 2.00 | 17.81 | 2.31
58402.5225 | 2556 | 171.8 | $24.7883$ | $108.15$ | 2.37 | $-177.53$ | 2.03 | 16.77 | 2.35
58405.5661 | 2701 | 193.3 | $24.0578$ | $-89.81$ | 1.92 | $273.70$ | 1.65 | 6.62 | 1.91
Table 15: Details on ELODIE and FEROS spectra in the blue region. ELODIE is at the upper part of the table, FEROS at the lower part. The RVs were determined during the spectral disentangling (three-step method) in KOREL.  $T$ | Exposure time | S/N | Heliocentric correction |  RV1 | $\sigma_{\mathrm{RV}_{1}}$ |  RV2 | $\sigma_{\mathrm{RV}_{2}}$ |  RV3 | $\sigma_{\mathrm{RV}_{3}}$
---|---|---|---|---|---|---|---|---|---
[$\mathrm{HJD-2400000}$] | [s] |  |  [km s-1] |  [km s-1] |  [km s-1] |  [km s-1] |  [km s-1] |  [km s-1] |  [km s-1]
50033.5779 | 187.13 | 86.9 | $14.76$ | $126.27$ | 4.27 | $-206.96$ | 3.66 | $18.15$ | 4.24
50033.5920 | 333.78 | 99.6 | $14.72$ | $123.38$ | 3.73 | $-206.12$ | 3.19 | $16.55$ | 3.70
50435.4013 | 120.50 | 102.9 | $-1.87$ | $112.98$ | 3.61 | $-184.64$ | 3.09 | $20.89$ | 3.58
54136.5830 | 30 | 295.3 | $-22.15$ | $22.27$ | 1.26 | $4.33$ | 1.08 | $14.51$ | 1.25
54809.7226 | 20 | 216.8 | $2.19$ | $76.14$ | 1.71 | $-101.90$ | 1.47 | $15.43$ | 1.70
54809.7238 | 20 | 213.4 | $2.19$ | $76.02$ | 1.74 | $-101.64$ | 1.49 | $15.43$ | 1.73
54809.7255 | 20 | 219.8 | $2.19$ | $72.08$ | 1.69 | $-100.56$ | 1.45 | $12.72$ | 1.68
54953.4599 | 30 | 232.0 | $-18.33$ | $27.82$ | 1.60 | $6.80$ | 1.37 | $15.67$ | 1.59
54953.4615 | 30 | 216.8 | $-18.33$ | $27.35$ | 1.71 | $10.92$ | 1.47 | $10.76$ | 1.70
Table 16:  Results of eight phoebe2 models of $\delta$ Ori. LCs from eight
individual seasons S. (Table 8) and all RVs were used to constrain the models.
We assumed a fixed value of the temperature $T_{1}$ and the third light
$l_{3}$. The uncertainties are the same as in Table 9.

Parameters | S. 2013 | S. 2014 | S. 2015 | S. 2016 | S. 2017 | S. 2018 | S. 2020 | S. 2021 | All seasons
---|---|---|---|---|---|---|---|---|---
$T_{0}$ [HJD] |  733.8355∗ |  733.8355∗ |  733.8412∗ |  773.3830∗ |  733.8412∗ |  733.8359∗ |  733.8343∗ |  733.8447∗ |  733.8340∗
$T_{\mathrm{1}}$ [K] |  31000f |  31000f |  31000f |  31000f |  31000f |  31000f |  31000f |  31000f |  31000f
$T_{\mathrm{2}}$ [K] | 23477 | 23371 | 23055 | 22709 | 23055 | 23455 | 22577 | 21825 | 22940
$R_{\mathrm{1}}$ [R⊙] | 12.96 | 12.79 | 12.78 | 13.27 | 12.78 | 12.82 | 12.95 | 12.61 | 12.87
$R_{\mathrm{2}}$ [R⊙] | 3.69 | 3.47 | 3.54 | 3.70 | 3.54 | 3.43 | 3.75 | 3.39 | 3.56
$i_{1}$ [∘] | 79.21 | 79.48 | 79.46 | 77.67 | 79.46 | 79.53 | 78.85 | 79.55 | 79.15
$S_{\mathrm{B}}$ | 1.00631 | 1.00489 | 1.00556 | 1.00786 | 1.00872 | 1.02264 | 1.00430 | 1.00455 | var
$S_{\mathrm{R}}$ | 1.00375 | 1.00355 | 1.00872 | 1.00841 | 1.00361 | 1.02431 | 1.00342 | 1.00640 | var
$m_{\mathrm{1}}$ [M⊙] | 18.06 | 18.08 | 18.08 | 18.07 | 18.08 | 17.22 | 18.05 | 18.08 | 17.97
$m_{\mathrm{2}}$ [M⊙] | 8.40 | 8.38 | 8.42 | 8.47 | 8.42 | 8.06 | 8.38 | 8.34 | 8.35
$e_{1}$ | 0.0843 | 0.0845 | 0.0900 | 0.0824 | 0.0903 | 0.0881 | 0.0861 | 0.0851 | 0.0864
$\omega_{1}$ [∘] | 133.5 | 132.9 | 131.0 | 130.0 | 131.0 | 129.0 | 134.0 | 135.91 | 129.86
$\dot{\omega_{1}}$ [∘ yr-1] |  1.45f |  1.45f |  1.45f |  1.45f |  1.45f |  1.45f |  1.45f |  1.45f | -
$\gamma$ [km s-1] | 19.27 | 19.02 | 19.33 | 19.78 | 19.33 | 19.71 | 18.93 | 19.52 | 19.36
$l_{3\mathrm{B}}$ |  0.273f |  0.273f |  0.273f |  0.273f |  0.273f |  0.273f |  0.273f |  0.273f | -
$l_{3\mathrm{R}}$ |  0.273f |  0.273f |  0.273f |  0.273f |  0.273f |  0.273f |  0.273f |  0.273f | -
$\chi^{2}_{\mathrm{sum}}$ | 661 | 683 | 1051 | 604 | 1003 | 877 | 1310 | 1176 |
$\chi^{2}_{\mathrm{LCB}}$ | 195 | 174 | 598 | 153 | 280 | 392 | 895 | 61 |
$\chi^{2}_{\mathrm{LCR}}$ | 201 | 254 | 180 | 200 | 463 | 217 | 160 | 284 |
$\chi^{2}_{\mathrm{RV1}}$ | 197 | 188 | 206 | 184 | 192 | 200 | 186 | 207 |
$\chi^{2}_{\mathrm{RV2}}$ | 68 | 66 | 67 | 67 | 68 | 68 | 69 | 77 |

313131 ∗ $-2457000\,$HJD. The explanation of variables is the same as in Table
9. f indicates the respective parameter was fixed. var denotes variable values
for each season (they were between 1.004 and 1.010). For each season, the
numbers of data points were: $N_{\mathrm{total}}=321$,
$N_{\mathrm{LCB}}=N_{\mathrm{LCR}}=100$, $N_{\mathrm{RV1}}=71$, and
$N_{\mathrm{RV2}}=50$.

Figure 21:  Comparison of the observed MOST (blue) and synthetic (orange)
light curve. The residuals are plotted in red. Apart from the eclipses, the
light curve contains large-amplitude oscillations (not included in our model);
uncertainties 0.01 mag were thus assigned to all data points.  Figure 22:
Contributions to $\chi^{2}$ for a set 100 best-fit models of $\delta$ Ori.
Individual contributions are shown in the panels (from top left): astrometry
(SKY), RV, eclipse timings (ETV), eclipse duration (ECL), normalised spectra
(SYN), SED, light curve (LC), and total. Every simplex was initialised with a
different combination of the gravitational accelerations $\log g_{1}$, $\log
g_{3}$, which were kept fixed to obtain a regular grid. All other parameters
were free. The number of convergence steps was limited to 2000, consequently,
200000 models were computed in total. Axes correspond to the values of $\log
g_{1}$, $\log g_{3}$, colours to $\chi^{2}$ (see also tiny numbers). The
colour scale was adjusted as follows: cyan the very best fit for a given data
set, blue acceptable fits (${<}3.0$ min $\chi^{2}$), orange poor fits
(${\geq}3.0$ min $\chi^{2}$). The factor was 1.5 for the SKY, total; and 30.0
for the ECL data set. ‘Forbidden regions’ can be seen, in particular, large
$\log g_{1}$, $\log g_{3}$ due to the SYN, SED, or large $\log g_{1}$ due to
the ECL, LC. The weighted very best fit is denoted by the red circle.