Origin of eclipsing time variations: contributions of different modes of the dynamo-generated magnetic field
Felipe H. Navarrete, Petri J. Käpylä, Dominik R.G. Schleicher, Carolina A. Ortiz, Robi Banerjee
MMNRAS , 1–13 (2021) Preprint 23 February 2021 Compiled using MNRAS L A TEX style file v3.0
Origin of eclipsing time variations: contributions of different modesof the dynamo-generated magnetic field
Felipe H. Navarrete, , ★ Petri J. Käpylä, , Dominik R.G. Schleicher, Carolina A. Ortiz, and Robi Banerjee Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, 21029 Hamburg, Germany Nordita, KTH Royal Institute of Technology and Stockholm University, 10691 Stockholm, Sweden Institut für Astrophysik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany Departamento de Astronomía, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Av. Esteban Iturras/n Barrio Universitario, Casilla 160-C, Chile
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
The possibility to detect circumbinary planets and to study stellar magnetic fields throughbinary stars has sparked an increase in the research activity in this area. In this paper we revisitthe connection between stellar magnetic fields and the gravitational quadrupole moment 𝑄 𝑥𝑥 .We present three magnetohydrodynamical simulations of solar mass stars with rotation periodsof 8.3, 1.2, and 0.8 days and perform a detailed analysis of the magnetic and density fields usinga spherical harmonic decomposition. The extrema of 𝑄 𝑥𝑥 are associated with changes of themagnetic field structure. This is evident in the simulation with a rotation period of 1.2 days. Itsmagnetic field has a much more complex behaviour than other models as the large-scale non-axisymmetric field dominates throughout the simulation and the axisymmetric componentis predominantly hemispheric. This triggers variations in the density field that follow themagnetic field asymmetry with respect to the equator, changing the 𝑧𝑧 component of theinertia tensor, and thus modulating 𝑄 𝑥𝑥 . The magnetic fields of the other two runs are lessvariable in time and more symmetric with respect to the equator such that there are no largevariations in the density, therefore only small variations in 𝑄 𝑥𝑥 are seen. If interpreted via theclassical Applegate mechanism (tidal locking), the quadrupole moment variations obtainedin the simulations are about two orders of magnitude below the observed values. However,if no tidal locking is assumed, our results are compatible with the observed eclipsing timevariations. Key words: magnetohydrodynamics – dynamo – methods: numerical – stars: activity –binaries: eclipsing
Post-common-envelope binaries (PCEBs) are commonly composedof a white dwarf and a low-mass main-sequence star. Observationsof eclipses in these systems reveal deviations from the calculatedeclipsing times in approximately 90% of these systems (Zorotovic& Schreiber 2013), with binary period variations in the order of10 − . . . − modulated over periods of the order of decades.The two main explanations, although not mutually exclusive,are the planetary hypothesis (Brinkworth et al. 2006; Völschowet al. 2014) and the Applegate mechanism (Applegate 1992; Lanzaet al. 1998; Völschow et al. 2018; Lanza 2020). In the planetaryhypothesis, sufficiently massive planets can force the barycenter ofthe binary to change its location as they orbit, which would then ★ E-mail: [email protected] explain the observed-minus-calculated (O-C) diagram of the eclips-ing times. On the other hand, the Applegate mehcanism explainsthe variations via the connection between stellar magnetic fieldsand the gravitational quadrupole moment 𝑄 . The idea behind thismechanism is that when 𝑄 increases, the gravitational field also in-creases. For this to happen, there must be a redistribution of angularmomentum within the star. When angular momentum is carried tothe outer parts of the convective zone (CZ), these will rotate fasterand, overall, the star will become more oblate, which is reflectedby an increase in the gravitational quadrupole moment. As there isno angular momentum exchange between the orbit and the star, theorbital velocity increases and the radius decreases in order to main-tain the angular momentum of the binary. Thus, the orbital periodshortens. In order for this mechanism to work, Applegate (1992) in-voked the presence of a sub-surface magnetic field of ∼
10 kG whichis responsible for redistributing the internal angular momentum ofthe star. © a r X i v : . [ a s t r o - ph . S R ] F e b Felipe H. Navarrete et al.
Confirming the planetary hypothesis requires a detection of theproposed circumbinary bodies in PCEBs by either directly imagingthem, as attempted by Hardy et al. (2015), or via indirect methodssuch as those employed by Vanderbosch et al. (2017). However,these studies did not detect the proposed third body, a brown dwarf,in V471 Tau, which is a PCEB with a Sun-like main-sequence starand a white dwarf, and was the system Paczynski (1976) used todevelop the theory of PCEB formation. Furthermore, direct mod-eling of the Applegate mechanism is also challenging and targetednumerical simulations that may help to understand observationshave been lacking. Navarrete et al. (2020) presented the first self-consistent 3D magnetohydrodynamical (MHD) simulations of stel-lar magneto-convection aimed at this problem, in an attempt toclose this gap. In that study the time evolution of the gravitationalquadrupole moment and its correlation with the stellar magneticfield and rotation was studied using two simulations of a solar massstar with three and twenty times solar rotation, which correspondsto rotation periods of 8.3 and 1.2 days. However, the centrifugalforce, a key ingredient in the original Applegate mechanism, wasnot included in these simulations. Nevertheless, there were still sig-nificant temporal variations of 𝑄 due to the response of the stellarstructure to the dynamo-generated magnetic field.Recently, Lanza (2020) presented an alternative to the Apple-gate mechanism by extending the earlier work by Applegate (1989).He assumed the presence of a persistent non-axisymmetric magneticfield inside the CZ of the main-sequence star that was modeled as asingle flux tube lying at the equatorial plane. The density is smallerwithin the magnetic region in comparison to the rest of the CZ,whose effects were modeled as two point masses lying on a lineperpendicular to the axis of the flux tube at the equator. By furtherassuming that the star is not tidally locked with the primary, thisnon-axisymmetric contribution to the quadrupole moment exerts anextra force onto the companion. Applegate (1989) and Lanza (2020)identified two possible scenarios: the libration model, where theflux tube oscillates around a mean value, and the circulation model,where the axis of the flux tube varies in a monotonic way. Thesemodels reduce the energetic requirements by a factor of 100 to 1000compared to the Applegate mechanism, which is much more re-strictive from an energetic point of view (see e.g. Brinkworth et al.2006; Völschow et al. 2016; Navarrete et al. 2018; Völschow et al.2018). Previous models generally require luminosity variations of ∼
10 per cent, whereas the improved model of Lanza (2020) reducesthe energy requirement by a factor of ∼ ... .The transition to predominantly non-axisymmetric large-scalemagnetic fields in solar-like stars was investigated by Viviani et al.(2018) with the same model as that used by Navarrete et al. (2020).They found that the dominant dynamo mode switches from axi-to non-axisymmetric at roughly three times the solar rotation rate.However, this study showed also that the dominant dynamo modedepends on the resolution of the simulations such that rapidly rotat-ing models at modest resolutions were again more axisymmetric. Inthe present study we revisit both simulations presented in Navarreteet al. (2020) and include one more run to the analysis to explore theimportance of (non-) axisymmetric magnetic fields in the modula-tion of the gravitational quadrupole moment. The model employed here is the same as that described by Käpyläet al. (2013). We solve the compressible MHD equations in a spher-ical shell configuration resembling the solar convection zone with the Pencil Code (Brandenburg et al. 2020). The computationaldomain is given by 0 . 𝑅 ≤ 𝑟 ≤ 𝑅 , 𝜃 ≤ 𝜃 ≤ 𝜋 − 𝜃 , 0 ≤ 𝜙 ≤ 𝜋 ,for the radial, latitude, and longitude coordinates, respectively, andwhere 𝜃 = 𝜋 / To investigate the proposed connection between the non-axisymmetric component of the magnetic field and the fluid den-sity, we perform the same decomposition as in Viviani et al. (2018)for the radial magnetic field at various radial depths. A function 𝑓 = 𝑓 ( 𝜃, 𝜙 ) can be written as 𝑓 ( 𝜃, 𝜙 ) = 𝑙 max ∑︁ 𝑙 = 𝑙 ∑︁ 𝑚 = − 𝑙 ˜ 𝑓 𝑚𝑙 ( 𝜃, 𝜙 ) 𝑌 𝑚𝑙 ( 𝜃, 𝜙 ) , (1)where˜ 𝑓 𝑚𝑙 = ∫ 𝜋 ∫ 𝜋 − 𝜃 𝜃 𝑓 ( 𝜃, 𝜙 ) 𝑌 𝑚 ∗ 𝑙 sin 𝜃 𝑑𝜃 𝑑𝜙. (2)For the radial magnetic field 𝐵 𝑟 ( 𝜃, 𝜙 ) , we impose the condition (seeKrause & Rädler 1980) 𝐵 − 𝑚𝑟,𝑙 = (− ) 𝑚 𝐵 𝑚 ∗ 𝑟,𝑙 , (3)and because the same property applies to the spherical harmonics 𝑌 𝑚𝑙 , we have 𝐵 𝑟 ( 𝜃, 𝜙 ) = 𝑙 max ∑︁ 𝑙 = 𝐵 𝑙,𝑟 𝑌 𝑙 + (cid:32) 𝑙 max ∑︁ 𝑙 = 𝑙 ∑︁ 𝑚 = 𝐵 𝑚𝑙,𝑟 𝑌 𝑟𝑙 (cid:33) . (4)The term containing 𝑙 = Each run is characterized by the Taylor number, Coriolis num-ber, fluid and magnetic Reynolds numbers, and fluid, subgrid-scale(SGS) and magnetic Prandtl numbers. These are defined asTa = (cid:20) Ω ( . 𝑅 ) 𝜈 (cid:21) , Co = Ω 𝑢 rms 𝑘 , (5)Re = 𝑢 rms 𝜈𝑘 , Re M = 𝑢 rms 𝜂𝑘 , (6)Pr = 𝜈𝜒 𝑚 , Pr M = 𝜈𝜂 , Pr SGS = 𝜈𝜒 mSGS , (7)where 𝜈 is the viscosity, 𝑢 rms the root-mean-square velocity, 𝑘 = 𝜋 / . 𝑅 an estimate of the wavenumber of the largest eddies, 𝜂 the magnetic diffusivity and 𝜒 mSGS is the SGS entropy diffusion at 𝑟 = . 𝑅 (cid:12) . http://pencil-code.nordita.org/ MNRAS , 1–13 (2021) ontribution of magnetic field modes to ETVs Figure 1.
Azimuthally averaged radial magnetic field ( 𝐵 𝑟 ) near the surfaceof the domain at 𝑟 = . 𝑅 as a function of latitude and time for Run A(top, 𝑃 rot = . 𝑃 rot = . 𝑃 rot = . 𝑄 𝑥𝑥 for each run is shown from the time periods where thisdiagnostic is available. The color scale of 𝐵 𝑟 in each panel has been clippedat ± We present the results of three runs, labeled as A, B, and C, withrotation periods of 8.3, 1.2 days, and 0.8 days, corresponding to 3,20, and 30 times the solar rotation rate. We summarize input anddiagnostic quantities that characterize each simulation in Table 1.
We begin our analysis by comparing the time-dependent diagnosticsof magnetic fields and the gravitational quadrupole moment. InFigure 1 we show the azimuthally averaged radial magnetic field( 𝐵 𝑟 ) near the surface of the stars at 𝑟 / 𝑅 = .
98 (color contours),along with the evolution of the 𝑥𝑥 component of the gravitationalquadrupole moment 𝑄 (black line) . The latter is defined as 𝑄 𝑖 𝑗 = 𝐼 𝑖 𝑗 − 𝛿 𝑖 𝑗 Tr 𝐼, (8)where 𝐼 𝑖 𝑗 = ∫ 𝜌 ( 𝒙 ) 𝑥 𝑖 𝑥 𝑗 d 𝑉 (9)is the 𝑖 𝑗 component of the inertia tensor expressed in Cartesian co-ordinates and 𝜌 ( 𝒙 ) is the density. In Run A, the variations of 𝑄 𝑥𝑥 are small, of the order of 10 ... × and have a similar period( ∼ 𝑄 𝑥𝑥 that repeats atleast once in the data. Roughly half of the data for Run B, up to ∼ 𝑄 𝑥𝑥 between 𝑡 ∼
30 to 𝑡 ∼
85 years was interpreted as a transientdue to insufficient thermodynamic and magnetic saturation. How-ever, with the longer time series we see that the quadrupole moment Note that the values of 𝑄 𝑥𝑥 for Run A and B differ from Navarreteet al. (2020). This is because in that study, 𝐼 𝑧𝑧 was erroneously calculated.However, this difference does not change their conclusions. is modulated on a timescale of ∼
80 years. It also appears that 𝑄 𝑥𝑥 is roughly correlated to 𝐵 𝑟 : low values of the quadrupole momentapproximately coincide with times when 𝐵 𝑟 is weak on both hemi-spheres ( 𝑡 = . . .
85 and 𝑡 = . . .
160 years, respectively). Thecycle is perhaps starting again at 𝑡 =
160 as the magnetic activ-ity appears to be resuming with a corresponding change in 𝑄 𝑥𝑥 .Its largest variation occurs between 𝑡 = ...
155 years, with amaximum amplitude variation of 10 kg m . It corresponds tothe largest variation of 𝑄 𝑥𝑥 of the three runs. Overall, the gravi-tational quadrupole moment appears to follow the radial magneticfield strength near the surface of the star independently of the hemi-spheric asymmetry. In contrast to the other two runs, diagnosticsfor 𝑄 𝑥𝑥 in Run C are available starting at 𝑡 = 𝑡 ∼
10 yr. The magnetic field back-reacts and re-adjusts the thermo-dynamic quantities, such as density, after which 𝑄 𝑥𝑥 settles to a statewith minimal variations around a mean value of 1 . × kg m .These variations are about two times smaller than in Run A. As seenin Fig. 1, the axisymmetric part of 𝐵 𝑟 is similar in Run A and C, butvery different in Run B. In the first two, 𝐵 𝑟 migrates towards higherlatitudes in a regular fashion and in the latter the the dynamo ismuch more hemispheric such that activity alternates between bothhemispheres seemingly every ∼
50 to 60 years.In Fig. 2 we show instantaneous snapshots of the radial mag-netic field at the surface of the three runs at the times of interest, i.e.maxima (minima) of 𝑄 𝑥𝑥 at top (bottom) row. These correspond to 𝑡 = ,
74 yr for Run A, 𝑡 = ,
155 yr for Run B, 𝑡 = ,
98 yr forRun C, and all of them are at the saturated dynamo regime. In Run A,a predominantly 𝑚 = 𝑚 = 𝑚 = 𝑚 = 𝑄 𝑥𝑥 (middle panel at the lower row), 𝐵 𝑟 is symmetric with respectto the equator with a dominating 𝑚 = 𝑡 ∼
36 yr and 𝑡 ∼
75 yr but at the opposite hemisphere. Run C is similar to Run Bin the sense that the large-scale non-axisymmetric structures span alarge region, but the dominant mode is either 𝑚 = 𝑚 =
2. Thesemodes appear to alternate between the hemispheres but there is vir-tually no variation in the gravitational quadrupole moment in thiscase in contrast to Run B. However, the low-order non-axisymmetricfields in Run C are of the same order of magnitude as the 𝑚 = 𝑚 = 𝑄 .Figures 3 and 4 show the spherical harmonic modes 𝑚 = 𝑚 = 𝑚 = 𝐵 𝑚 = 𝑟 at 𝑡 =
62 yris mostly present at the northern hemisphere with a weak southerncounterpart. At the minimum of 𝑄 𝑥𝑥 the 𝑚 = 𝑚 = 𝑚 = 𝑡 =
62 yr, but at a minima of 𝑄 𝑥𝑥 it increasesin magnitude at regions near the latitudinal boundaries. The caseof Run B is quite different. The first non-axisymmetric mode has a MNRAS , 1–13 (2021)
Felipe H. Navarrete et al.
Figure 2.
Instantaneous radial magnetic field at 𝑟 = . 𝑅 for each run at two times. Top (bottom) row corresponds to a maxima (minima) of 𝑄 𝑥𝑥 . Figure 3.
First non-axisymmetric mode of the radial magnetic field ( 𝐵 𝑚 = 𝑟 ) at 𝑟 = . 𝑅 for each run. Figure 4.
Second non-axisymmetric mode of the radial magnetic field ( 𝐵 𝑚 = 𝑟 ) at 𝑟 = . 𝑅 for each run.MNRAS , 1–13 (2021) ontribution of magnetic field modes to ETVs Table 1.
Simulation parameters. Ω / Ω (cid:12) is the rotation rate in units of mean solar angular velocity. Co is the Coriolis number, Re and Re M are the fluid andmagnetic Reynolds numbers. Δ t is the total simulated time.Run Ω / Ω (cid:12) 𝑃 rot (days) Ta Co Re Re M Pr Pr M Pr SGS Δ t [yr]A 3 8.3 5 . × . × . × Table 2.
Summary of some quantities of interest taken from Fig. 1. 𝑄 𝑥𝑥 is in units of 10 kg m and 𝐵 𝑟 in kG. 𝐸 mag , total is the volume-averagedmagnetic energy in units of 10 J m − 𝑄 max 𝑥𝑥 𝑄 min 𝑥𝑥 𝐵 max 𝑟 𝐵 min 𝑟 𝐸 maxmag , total 𝐸 minmag , total A 2.25 1.94 8.65 -9.70 2.98 1.26B 1.38 0.315 24.7 -19.7 3.98 2.28C 1.80 1.54 16.8 -13.8 2.32 1.30 very similar magnitude at both hemispheres at 𝑡 =
110 yr, but withstrong hemispheric asymmetry. At the minimum ( 𝑡 =
155 yr) the 𝑚 = 𝑚 = 𝑡 =
110 yr as seen in Fig. 2. Thisasymmetry disappears at the minimum of 𝑄 𝑥𝑥 at 𝑡 =
155 yr. Run Cdoes not show as marked differences between the two times, expectfor a mild strengthening (weakening) of the 𝑚 = 𝑚 =
2) mode atthe southern hemisphere.
The variations of the gravitational quadrupole moment are relatedto changes in the mass distribution within the star as can be seenfrom Eqs.(8) and (9). Snapshots of the density from all three runsnear the surface of the star are shown in Fig. 5. As before, the showntimes correspond to a maxima (top row) and a minima (bottom row)of 𝑄 𝑥𝑥 .In Run A there is an overall change in density betweenthe two times. At 𝑡 =
62 yr (top panel) when the gravitationalquadrupole moment is larger there are no noticeable large-scalenon-axisymmetric features, whereas when 𝑄 𝑥𝑥 is at a minimum( 𝑡 =
74 yr), weak non-axisymmetric features appear. This can beseen from the two blue stripes around 𝜃 = ± ◦ where the over-all density decreases with patches of increased (decreased) densityaround 𝜙 = ◦ ( 𝜙 = ◦ ). Closer to the poles and at the equatorthe average density increases but no clear non-axisymmetric fea-tures are present. In Run B we identify a few characteristics. First,when the quadrupole moment is larger at 𝑡 =
110 yr there is aclear asymmetry with respect to the equator, such that the density islarger close to the north pole. As 𝑄 𝑥𝑥 becomes smaller, the asym-metry disappears, and non-axisymmetric structures become visibleat 230 ◦ < 𝜙 < ◦ and 𝜃 = ± ◦ . As the magnetic field changesits configuration from one that is dominated by an 𝑚 = 𝑚 = 𝜌 asym ( 𝑟, 𝜃, 𝜙 ) = 𝜌 ( 𝑟, 𝜃, 𝜙 ) − 𝜌 ( 𝑟, − 𝜃, 𝜙 ) 𝜌 rmsasym = (cid:16) (cid:104) 𝜌 (cid:105) 𝜃 𝜙 (cid:17) / . (11)The time evolution of 𝜌 rmsasym , together with 𝑄 𝑥𝑥 , is shown in Fig. 6.In Run A (top panel) there is an anti-correlation between the two.However, in the case of Run B (middle panel) there is a positivecorrelation between the two but with an apparent time delay. Therms value of 𝜌 rmsasym lags behind 𝑄 𝑥𝑥 by ∼
10 yr for example near theextrema between 80 and 100 yr. In Run C both the variations of thedensity and 𝑄 𝑥𝑥 are weak. It is also less clear whether a correlationbetween the two exists. The variations of 𝜌 rmsasym are between 6 to 10times larger in Run B than in A and C. This is can be an indicationthat the former is in a different regime where the magnetic field ismore strongly coupled to the density field that makes the quadrupolemoment variations larger.Such a connection can be studied by comparing the sphericalharmonic modes of magnetic field (Figures 3 and 4) with the onesof density field shown in Figures 7 and 8. In Run A, the densityvariations of the first non-axisymmetric mode are similar in the twotimes, but the 𝑚 = 𝑚 = 𝑚 = 𝑡 =
110 yr, 𝜌 𝑚 = is larger at the southernhemisphere, whereas 𝜌 𝑚 = is larger at the northern portion of thestar. At 𝑡 =
155 yr density variations become noticeably larger alsoat the northern hemisphere and, in general, increase throughout thewhole region at 𝑟 = . 𝑅 . The non-axisymmetric modes of radialmagnetic field are stronger at regions where the ones of density aresmaller. For example, in Fig. 4 we see that at 𝑡 =
110 yr the 𝑚 = 𝑚 = 𝑄 𝑥𝑥 is reached at 𝑡 =
155 yr, 𝐵 𝑚 = 𝑟 weakens and 𝜌 𝑚 = increases at the southern hemisphere. Thisis because when the magnetic field weakens, the magnetic pressurealso does so, and density must increase in order to keep pressurebalance. No strong equatorial asymmetry of density is observed inRun C. While there are variations in the radial magnetic field mode 𝑚 =
1, this is compensated by the opposite variations of the 𝑚 = 𝐼 𝑧𝑧 . Figure 9 shows the evolutionof the three components of the inertia tensor that contribute to 𝑄 𝑥𝑥 , MNRAS , 1–13 (2021)
Felipe H. Navarrete et al.
Figure 5.
Instantaneous snapshots of density at 𝑟 = . 𝑅 for each Run. Figure 6. 𝑄 𝑥𝑥 (in black) and root mean square value of density 𝜌 asym (inyellow) as a function of time. which is computed as 𝑄 𝑥𝑥 = 𝐼 𝑥𝑥 − (cid:0) 𝐼 𝑥𝑥 + 𝐼 𝑦𝑦 + 𝐼 𝑧𝑧 (cid:1) . (12)In all three simulations 𝐼 𝑧𝑧 is always smaller than the other two components. In Runs A and C all components of 𝐼 𝑖 𝑗 have compa-rable variations, whereas in Run B (middle panel) the variations of 𝑄 𝑥𝑥 are significantly larger. This coincides with larger variationsof 𝜌 𝑚 = , in this run. In Run B a maximum of 𝑄 𝑥𝑥 coincides witha minima of 𝐼 𝑧𝑧 . This corresponds to the star rotating slightly fasterat a maxima (minima) of 𝑄 𝑥𝑥 ( 𝐼 𝑧𝑧 ).To see such differences, we show figures of mean (azimuthallyaveraged) rotational profiles for each Run in Fig. 10 where Ω = Ω + 𝑢 𝜙 𝑟 sin 𝜃 . (13)There are some minor differences in the rotation profiles of Run Aand Run B between a maxima and minima of 𝑄 𝑥𝑥 , while no differ-ences are observed in Run C. Both Run A and Run B have a largerdifference between angular velocities of polar and equatorial regionsat a minima of 𝑄 𝑥𝑥 (lower panels), but and accelerated northernpole is seen at the top panel of Run B. This implies that stars woulddeform and adopt an ellipsoidal shape, adding a further contribu-tion to the quadrupole moment. However, as we have fixed boundaryconditions, we cannot resolve such reaction. The three runs show asolar-like rotation profile with the equatorial region rotating fasterthan the poles as a consequence of Coriolis numbers above the tran-sition region from anti-solar to solar-like differential rotation. Thistransition is found around Co (cid:38) The magnetic energies of the axisymmetric and first two non-axisymmetric modes are shown in Fig. 11. In Run A (top panel)the axisymmetric 𝑚 = 𝑄 𝑥𝑥 . There are short episodes where the 𝑚 = 𝑚 = 𝑄 𝑥𝑥 . In Run A, the 𝑚 = 𝑚 = 𝑄 𝑥𝑥 .This is because, as explained in the previous section, it produces MNRAS , 1–13 (2021) ontribution of magnetic field modes to ETVs Figure 7.
First non-axisymmetric mode of density at 𝑟 = . 𝑅 for each Run. Figure 8.
First non-axisymmetric mode of density at 𝑟 = . 𝑅 for each Run. equatorially asymmetric density fluctuations that modulate the mo-ment of inertia aligned with the rotation axis of the star. The secondnon-axisymmetric mode ( 𝑚 =
2) is as strong as the axisymmetricmode and also correlates with 𝑄 𝑥𝑥 . In Run C all modes have similarenergy levels, with 𝑚 = 𝑄 𝑥𝑥 which can be at-tributed to the fact that the variations of the magnetic field do notresult in significant density perturbations relevant for the quadrupolemoment. Azimuthal dynamo waves (ADW) are magnetic waves that migratein the azimuthal direction and can be prograde or retrograde andtheir period is not the same as the rotation period of the star (seee.g. Krause & Rädler 1980; Cole et al. 2014; Viviani et al. 2018). We take 𝑚 = 𝑚 = ◦ and show the migration ofsuch ADWs in Fig. 12.Run A has an 𝑚 = ∼ 𝑡 =
60 yr.Meanwhile, the 𝑚 = 𝑚 = ∼ ...
100 years. There is an 𝑚 = ∼
40 years of thesimulation. In Run C there are two equally strong ADWs. Bothmigrate in a prograde way with the difference that the period of the 𝑚 = 𝑚 = In Fig. 13 we show the phase of the first two non-axisymmetricspherical harmonic modes of radial magnetic field at the surface ofthe three runs taken at 𝜙 = ◦ . The evolution of the phase of the MNRAS000
40 years of thesimulation. In Run C there are two equally strong ADWs. Bothmigrate in a prograde way with the difference that the period of the 𝑚 = 𝑚 = In Fig. 13 we show the phase of the first two non-axisymmetricspherical harmonic modes of radial magnetic field at the surface ofthe three runs taken at 𝜙 = ◦ . The evolution of the phase of the MNRAS000 , 1–13 (2021)
Felipe H. Navarrete et al.
Figure 9.
Time evolution of the diagonal components of the inertia tensorfor Run A (top), Run B (middle), and Run C (bottom). 𝑚 = 𝑄 𝑥𝑥 . Thephase of 𝐵 𝑚 = 𝑟 does not show any particular evolution in Run C,while for 𝑚 = 𝑄 𝑥𝑥 . The cases of Run A and Run B point to anunderlying relation between a non-axisymmetric dynamo mode andthe gravitational quadrupole moment evolution. The Applegate mechanism (Applegate 1992) is based on the redis-tribution of angular momentum throughout the star due to the cen-trifugal force. More recently, Lanza (2020) presented a new mech-anism where the centrifugal force is no longer needed. In this workthe quadrupole moment is constant in the frame of reference of themagnetically active star due to a time-invariant non-axisymmetricmagnetic field modeled as a single flux tube. The companion starexperiences a non-axisymmetric gravitational quadrupole moment that varies in time due to the assumption that the active star is nottidally locked.In our simulations we compute the gravitational quadrupolemoment in the rotating frame of the reference of the star. The simu-lations thus provide a test whether magnetic activity can relevantlyinfluence the stellar structure. We stress however that the physicalprocess that occurs here is not the classical Applegate mechanism,as it is based on the centrifugal force, which is not included in oursimulations. It will have to be explored in future work whether thecentrifugal force could enhance the quadrupole moment changes ina relevant way. We note also that, even if we were to compute thequadrupole moment from a frame of reference that is oscillatingaround the axis ˆ 𝑥 that originates at the center of the star, pointstoward the companion and rotates together with the star, as in thescenario outlined by Applegate (1989) and Lanza (2020), we wouldstill obtain a time dependent signal. The results described in Sect. 3show that the connection between magnetic fields and gravitationalquadrupole moment is quite complex. It is due to the asymmetryof the magnetic field with respect to the equator rather than due tonon-axisymmetry, which is particularly noticeable in Run B. Thethree simulations we present differ only in the rotation rate of thestar and yet they present different scenarios of quadrupole momentvariations.Simulations of stellar magneto-convection have shown thatdynamo solutions depend mainly on the rotation rate of the star.For example, Viviani et al. (2018) studied the transition from axi-to non-axisymmetric magnetic fields as a function of rotation. Theyfound that there is a transition from the former to the latter at rotationrates greater than Ω ∼ . Ω (cid:12) . However, Viviani et al. (2018) foundthat at sufficiently rapid rotation, the magnetic field returns to apredominantly axisymmetric configuration if the resolution is nothigh enough. A similar sequence is also observed in the simulationsdescribed in this paper. Run A is at a regime where the axisymmetricmode is slightly larger than the first non-axisymmetric mode. Run Bhas a rotation rate 6.7 times the one of Run A and there the 𝑚 = 𝑚 = 𝑚 = 𝑚 =
0. If the resolutionwas to be increased, corresponding to higher Reynolds numbers, it ispossible that a non-axisymmetric solution with stronger quadrupolemoment variations would be recovered.Besides resolution effects, it is possible that Run B is in a pa-rameter regime that is conductive to hemispherical dynamos like thedynamo solutions found by Grote & Busse (2000), Busse (2002),and Käpylä et al. (2010). Brown et al. (2020) reported a cyclic single-hemisphere dynamo in a simulation of a fully convective star, andmentioned that such dynamos are present in other simulations withsimilar parameters. Käpylä (2020) presented a set of simulations offully convective stars where in a single run short periods of hemi-spheric dynamo action was seen, but even in this case the dynamois predominantly present on both hemispheres. It is important to ex-plore this parameter regime in more detail as it will help to addressthe question of whether the eclipsing time variations in PCEBs havea magnetic origin. If this is the main ingredient, however, it wouldimply that PCEBs that show variations in the O-C diagram are inthis particular regime.If we first consider the case where the eclipsing time variationis due to a time-dependent quadrupole moment (classical Applegatemechanism), the expected period variation due to a change of thegravitational quadrupole moment can be computed from (Applegate
MNRAS , 1–13 (2021) ontribution of magnetic field modes to ETVs Figure 10.
Snapshots of mean angular velocity ( Ω ) minus time-averaged angular velocity ( (cid:104) Ω (cid:105) 𝑡 ) normalized to the angular velocity of the frame of reference ( Ω ) for each run. Δ 𝑃𝑃 = − Δ 𝑄 𝑥𝑥 𝑀𝑎 , (14)or Δ 𝑃𝑃 = 𝜋 O − C 𝑃 mod , (15)where Δ 𝑃 / 𝑃 is the amplitude of the orbital period modulation, 𝑀 the stellar mass, and 𝑎 the binary separation, 𝑂 − 𝐶 is the amplitudeof the observed-minus-calculated diagram and 𝑃 mod its modulationperiod. For the target system V471 Tau, the parameters are 𝑀 = . 𝑀 (cid:12) and 𝑎 = . 𝑅 (cid:12) . There are two contributions to the orbitalperiod modulation that individually result in two orbital periodmodulations (Marchioni et al. 2018). These are ( Δ 𝑃 / 𝑃 ) = . × − , (16) ( Δ 𝑃 / 𝑃 ) = . × − . (17)The corresponding quadrupole moment variations are Δ 𝑄 𝑥𝑥, = . × kg m , (18) Δ 𝑄 𝑥𝑥, = . × kg m . (19)For the purpose of comparing with the quadrupole momentvariations in simulations, we recall here that density fluctuationsand the quadrupole moment itself need to be scaled as explained inNavarrete et al. (2020). They scale as Δ 𝜌 ∼ 𝔉 / 𝑟 , (20)where 𝔉 𝑟 is the ratio between the fluxes of the simulation and the target star, i.e. 𝔉 𝑟 = 𝔉 sim 𝔉 ★ , (21)so the quadrupole moment is accordingly scaled as 𝑄 ★ = 𝔉 / 𝑟 𝑄 sim . (22)In Run B the amplitude of the variation is Δ 𝑄 𝑥𝑥 = . × kg m corresponding to Δ 𝑃 / 𝑃 = × − . By adopting amodulation period of 𝑃 mod =
80 yr which corresponds to theperiod of 𝑄 𝑥𝑥 , we have an observed minus calculated value ofO − C = . 𝑂 − 𝐶 amplitude is still four and thirty timessmaller than the values reported by Marchioni et al. (2018). Thereare a few reasons behind this mismatch. First, we are not includingthe centrifugal force in the present runs so density fluctuations weare observing have a magnetic nature and are not produced by theApplegate mechanism. Secondly, the stars we are modelling have aCZ extension of the external 30% of the radius, whereas the main-sequence star in the target system V471 Tau has a mass of 0 . 𝑀 (cid:12) and therefore a slightly more extended CZ. There will more room tohave density fluctuations and thus more run for angular momentumredistribution (Völschow et al. 2018), although we expect this con-tribution to be very small. Lastly, we are imposing sphericity whichis specially important at the surface of the star. A relaxed boundarythat reacts to the physical quantities inside of the star may allowlarger variations of the quadrupole moment, specially if the star caninflate and deflate. MNRAS000
80 yr which corresponds to theperiod of 𝑄 𝑥𝑥 , we have an observed minus calculated value ofO − C = . 𝑂 − 𝐶 amplitude is still four and thirty timessmaller than the values reported by Marchioni et al. (2018). Thereare a few reasons behind this mismatch. First, we are not includingthe centrifugal force in the present runs so density fluctuations weare observing have a magnetic nature and are not produced by theApplegate mechanism. Secondly, the stars we are modelling have aCZ extension of the external 30% of the radius, whereas the main-sequence star in the target system V471 Tau has a mass of 0 . 𝑀 (cid:12) and therefore a slightly more extended CZ. There will more room tohave density fluctuations and thus more run for angular momentumredistribution (Völschow et al. 2018), although we expect this con-tribution to be very small. Lastly, we are imposing sphericity whichis specially important at the surface of the star. A relaxed boundarythat reacts to the physical quantities inside of the star may allowlarger variations of the quadrupole moment, specially if the star caninflate and deflate. MNRAS000 , 1–13 (2021) Felipe H. Navarrete et al.
Figure 11.
Time evolution of the gravitational quadrupole moment (blackline) together with the magnetic energy contained in the axisymmetric mode( 𝑚 =
0, yellow), as well as the first ( 𝑚 =
1, red) and second ( 𝑚 =
2, blue)non -axisymmetric modes for Run A (top), Run B (middle), and Run C(bottom).
In the previous calculation of Δ 𝑃 / 𝑃 following Applegate (1992),we made the implicit assumption of tidal locking and that the stellarrotational axis is perpendicular to the plane of the orbital motion. Inthis scenario, the ˆ 𝑥 axis points towards the companion and thus it willrotate together with the stellar spin. Under those conditions, onlythe 𝑄 𝑥𝑥 component of the gravitational quadrupole moment con-tributes to the modulation of the binary period (Applegate 1992).In contrast, in the scenario put forward by Applegate (1989) andLanza (2020), the star is not yet tidally locked, and its companioneffectively experiences a time-varying quadrupole moment due tothe relative rotation of the magnetically active star. This holds evenif the quadrupole moment in the co-rotating frame of the star wasconstant. This implies then that different components of 𝑄 con-tribute. In the simplified Lanza (2020) scenario, the magnetic fieldis modelled as a permanent single flux tube that lies at the equatorand produces a non-axisymmetric density distribution and thus, apermanent non-axisymmetric gravitational quadrupole moment.While there is a strong non-axisymmetric magnetic field in ourRun B, it is stronger at mid- and at high latitudes rather than at theequator. In our simulations, the choice of the ˆ 𝑥 and ˆ 𝑦 axes in theequatorial plane along which the moments of inertia are calculated is arbitrary, i.e. as the companion star is not being modeled. Oncefixed, we perform rotations about the ˆ 𝑧 axis in steps of 𝜋 /
16 up to 𝜋 ,and then we calculate the two moments of inertia about the rotatedaxes. These axes would correspond to ˆ 𝑠 and ˆ 𝑠 (cid:48) of Lanza (2020).The former is the rotated ˆ 𝑥 axis, and the latter is the rotated ˆ 𝑦 axis.In Lanza (2020) ˆ 𝑠 is chosen to be along the axis of symmetry ofthe magnetic flux tube, which is the only magnetic structure in theconvective zone of the magnetically active star. In our simulationsthe performed rotation is not unique as there is no single radial mag-netic field structure that extends from the bottom to the surface of theconvective zone in our simulations that would otherwise allow us tounequivocally choose ˆ 𝑠 . However, a clear radial magnetic structureat the equator is seen at 𝑡 =
155 yr (see Fig. 14), but magnetic fieldswith different structure and strength dominate at different latitudes.In this configuration, the non-axisymmetric quadrupole moment isdefined as 𝑇 = 𝐼 𝑠 − 𝐼 (cid:48) 𝑠 , where 𝐼 𝑠 and 𝐼 (cid:48) 𝑠 are the moments of inertiaabout the ˆ 𝑠 and ˆ 𝑠 (cid:48) axes. The moment of inertia of the active starabout the spin axis is 𝐼 𝑝 = 𝐼 𝑥𝑥 + 𝐼 𝑦𝑦 . The order of magnitude of theperiod variations can then be estimated as (Eq. 2 of Lanza 2020) 𝑇𝐼 𝑝 ≈ (cid:18) 𝑀 𝑇 𝑚 𝑆 (cid:19) (cid:18) 𝑚𝑎 𝐼 𝑝 (cid:19) (cid:18) 𝑃𝑃 mod (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) Δ 𝑃𝑃 (cid:12)(cid:12)(cid:12)(cid:12) , (23)where 𝑀 𝑇 is the total mass of the binary, 𝑚 𝑆 is the mass of thecompanion, 𝑚 is the reduced mass, 𝑃 is the orbital period, and 𝑃 mod is the modulation period. We take the density fields of Run B at 𝑡 =
110 yr and 𝑡 =
155 yr and compute the two quadrupole moments 𝑇 and 𝐼 𝑝 . By using Eq. (23) and the parameters of V471 Tau (seee.g. Hardy et al. 2015; Vaccaro et al. 2015) we can obtain an orderof magnitude estimate of Δ 𝑃 / 𝑃 . Fig. 15 shows the absolute valueof the amplitude of the orbital period modulation as a function ofseparation angle 𝛼 between ˆ 𝑥 and ˆ 𝑠 for 𝑡 =
110 yr (black dots)and 𝑡 =
155 yr (yellow triangles). | Δ 𝑃 / 𝑃 | ranges between 1 . × − and 1 . × − , which contains the two contributions to theobserved variations as well as their sum (Marchioni et al. 2018).From our simulations we get a value of 𝐼 𝑝 that is of the same orderof magnitude as in Lanza (2020), while 𝑇 is about one order ofmagnitude larger here. It is important to note that we have obtained 𝑇 based on a detailed 3D magneto-hydrodynamical simulation, whileLanza (2020) simply calculated which 𝑇 would be required in orderto explain the observed eclipsing time variations.In general, the gravitational potential felt by the companioncan be written as (Applegate 1992; Lanza 2020) Φ = − 𝐺 𝑀𝑟 − 𝐺 𝑟 ∑︁ 𝑖, 𝑗 𝑄 𝑖 𝑗 𝑥 𝑖 𝑥 𝑗 𝑟 , (24)where 𝐺 is the gravitational constant, 𝑀 the mass of the active star, 𝑟 the distance between the center of the active star and the com-panion, 𝑄 𝑖 𝑗 is the quadrupole moment tensor, and 𝒙 are Cartesiancoordinates. Writing out the summation explicitly and expressing 𝑥 𝑖 and 𝑥 𝑗 in a spherical coordinate system ( 𝑟, 𝜃 (cid:48) , 𝜙 (cid:48) ) with its origin MNRAS , 1–13 (2021) ontribution of magnetic field modes to ETVs Figure 12.
Migration of the 𝑚 = 𝑚 = Figure 13.
Time evolution of the phase of the 𝑚 = 𝑚 = 𝜙 = ◦ for Run A (left panels),Run B (middle panels), and Run C (right panels) in yellow. The black line corresponds to 𝑄 𝑥𝑥 . coinciding with the center of the star, we arrive at Φ = − 𝐺 𝑀𝑟 − 𝐺 𝑟 (cid:26) 𝑄 𝑥𝑥 sin 𝜃 (cid:48) cos 𝜙 (cid:48) + 𝑄 𝑦𝑦 sin 𝜃 (cid:48) sin 𝜙 (cid:48) + 𝑄 𝑧𝑧 cos 𝜃 (cid:48) + (cid:18) 𝑄 𝑥𝑦 sin 𝜃 (cid:48) cos 𝜙 (cid:48) sin 𝜙 (cid:48) + 𝑄 𝑥𝑧 sin 𝜃 (cid:48) cos 𝜃 (cid:48) cos 𝜙 (cid:48) + 𝑄 𝑦𝑧 sin 𝜃 (cid:48) cos 𝜃 (cid:48) sin 𝜙 (cid:48) (cid:19)(cid:27) . (25)The case of 𝜃 (cid:48) = 𝜋 / 𝜙 (cid:48) = Φ = − 𝐺 𝑀𝑟 − 𝐺 𝑟 (cid:18) 𝑄 𝑥𝑥 cos 𝜙 (cid:48) + 𝑄 𝑦𝑦 sin 𝜙 (cid:48) + 𝑄 𝑥𝑦 cos 𝜙 (cid:48) sin 𝜙 (cid:48) (cid:19) . (26)Here the effects of deviations from tidal locking can be modelledby making 𝜙 (cid:48) time-dependent. There are two alternatives, namely 𝜙 (cid:48) = 𝛼 cos ( 𝜔𝑡 ) , (27) 𝜙 (cid:48) = 𝜔𝑡. (28)In the former case, the companion is seen in the frame of referenceof the rotating star as oscillating in the orbital plane with maximumamplitude 𝛼 and angular velocity 𝜔 . In that case, 𝜙 (cid:48) corresponds tothe analogous of the libration model. The latter expression for 𝜙 (cid:48) corresponds to the circulation model presented by Lanza (2020).This would introduce two further contributions to the binary periodvariation that come from 𝑄 𝑦𝑦 and 𝑄 𝑥𝑦 (see Eq. 26). In Run B 𝑄 𝑦𝑦 is, on average, the same as 𝑄 𝑥𝑥 . Meanwhile, 𝑄 𝑥𝑦 is 10 ... times MNRAS000
Time evolution of the phase of the 𝑚 = 𝑚 = 𝜙 = ◦ for Run A (left panels),Run B (middle panels), and Run C (right panels) in yellow. The black line corresponds to 𝑄 𝑥𝑥 . coinciding with the center of the star, we arrive at Φ = − 𝐺 𝑀𝑟 − 𝐺 𝑟 (cid:26) 𝑄 𝑥𝑥 sin 𝜃 (cid:48) cos 𝜙 (cid:48) + 𝑄 𝑦𝑦 sin 𝜃 (cid:48) sin 𝜙 (cid:48) + 𝑄 𝑧𝑧 cos 𝜃 (cid:48) + (cid:18) 𝑄 𝑥𝑦 sin 𝜃 (cid:48) cos 𝜙 (cid:48) sin 𝜙 (cid:48) + 𝑄 𝑥𝑧 sin 𝜃 (cid:48) cos 𝜃 (cid:48) cos 𝜙 (cid:48) + 𝑄 𝑦𝑧 sin 𝜃 (cid:48) cos 𝜃 (cid:48) sin 𝜙 (cid:48) (cid:19)(cid:27) . (25)The case of 𝜃 (cid:48) = 𝜋 / 𝜙 (cid:48) = Φ = − 𝐺 𝑀𝑟 − 𝐺 𝑟 (cid:18) 𝑄 𝑥𝑥 cos 𝜙 (cid:48) + 𝑄 𝑦𝑦 sin 𝜙 (cid:48) + 𝑄 𝑥𝑦 cos 𝜙 (cid:48) sin 𝜙 (cid:48) (cid:19) . (26)Here the effects of deviations from tidal locking can be modelledby making 𝜙 (cid:48) time-dependent. There are two alternatives, namely 𝜙 (cid:48) = 𝛼 cos ( 𝜔𝑡 ) , (27) 𝜙 (cid:48) = 𝜔𝑡. (28)In the former case, the companion is seen in the frame of referenceof the rotating star as oscillating in the orbital plane with maximumamplitude 𝛼 and angular velocity 𝜔 . In that case, 𝜙 (cid:48) corresponds tothe analogous of the libration model. The latter expression for 𝜙 (cid:48) corresponds to the circulation model presented by Lanza (2020).This would introduce two further contributions to the binary periodvariation that come from 𝑄 𝑦𝑦 and 𝑄 𝑥𝑦 (see Eq. 26). In Run B 𝑄 𝑦𝑦 is, on average, the same as 𝑄 𝑥𝑥 . Meanwhile, 𝑄 𝑥𝑦 is 10 ... times MNRAS000 , 1–13 (2021) Felipe H. Navarrete et al.
Figure 14.
Radial magnetic field of Run B at the equator at 𝑡 =
155 yr. Theˆ 𝑥 and ˆ 𝑦 axes lie at 𝜙 = ◦ and 𝜙 = ◦ , respectively. The ˆ 𝑠 and ˆ 𝑠 (cid:48) axes areobtained by performing clockwise rotations. Figure 15.
Absolute value of Δ 𝑃 / 𝑃 as a function of separation angle 𝛼 between ˆ 𝑥 and ˆ 𝑠 for 𝑡 =
110 yr (black dots) and 𝑡 =
155 yr (yellow triangles). smaller so it can be neglected. Thus,
Φ = − 𝐺 𝑀𝑟 − 𝐺 𝑟 (cid:18) 𝑄 𝑥𝑥 cos 𝜙 (cid:48) + 𝑄 𝑦𝑦 sin 𝜙 (cid:48) (cid:19) . (29)In contrast to previous studies, we can directly calculate eachcomponent of the gravitational quadrupole moment from our sim-ulations. In this case it is advantageous to use Eqs. (25) and (26)rather than taking the limit of 𝜙 (cid:48) =
0. However, we would need touse new expressions to derive Δ 𝑃 / 𝑃 considering the libration andcirculation models. Alternatively, it is also possible to try differentvalues of 𝛼 and 𝜔 , and then directly solve Eq. (25) in a two-bodysimulation, which is however beyond the scope of the presentedstudy. The influence of differences between 𝑄 𝑥𝑥 and 𝑄 𝑦𝑦 can bestudied with N-body simulations by prescribing their time evolu-tion and varying their amplitudes. It would be interesting to derivethe parameters that can reproduce the observations and to comparethem with our simulations. We have presented three magnetohydrodynamical simulations ofstellar convection and studied the gravitational quadrupole moment and its connection to dynamo-generated magnetic fields. The anal-ysis is based on a spherical harmonic decomposition of density andmagnetic fields. Our results of Run B ( 𝑃 = . 𝑄 . We also expect to have a further mod-ulation of 𝑄 that comes from the centrifugal force which will beincluded in a future work as it is the responsible of the angularmomentum redistribution in the Applegate mechanism (Applegate1992).When our results are interpreted in the context of the classi-cal Applegate mechanism, i.e. the star is tidally locked, then onlythe 𝑄 𝑥𝑥 component of the quadrupole moment contributes to theperiod variations. In this scenario, we obtain orbital period mod-ulations between one and two orders of magnitude smaller thanobserved in the target system V471 Tau (Marchioni et al. 2018). Inthe context of the models by Applegate (1989) and Lanza (2020),the order of magnitude estimate of the amplitude of the period mod-ulation is 10 − . . . − . This range encompasses the two observedcontributions to the 𝑂 − 𝐶 diagram, as well as their combined effect.It may also be that the observed period variations are a combinationof both, namely, both the axi- and non-axisymmetric quadrupolemoments contribute to them. The implication of the first interpreta-tion is that there must be an hemispheric dynamo with alternatinghemisphere in order to modulate 𝑄 𝑥𝑥 as seen in our simulations.The second interpretation implies that the star is not tidally lockedand that there is a non-axisymmetric magnetic field in the CZ of themagnetically active star.Observational studies suggest that both scenarios discussedabove are plausible. Recently, Klein et al. (2021) reported the re-construction of the surface magnetic field of Proxima Centauri usingZeeman-Doppler Imaging (ZDI). They found that the magnetic fieldis mainly poloidal with a dominant feature that is tilted at 51 ◦ tothe rotation axis (see their Figure 3) with a strength of 135 G, i.e.a field distribution that is asymmetric with respect to the equator.This is a rather weak field so density fluctuations should be smallerthan what we find in our simulations. However, Proxima Centauriis a slowly rotating M5.5 fully-convective star. The magnetic fieldstrength of fully-convective stars increases with rotation until a sat-uration regime is reached, as measured by X-ray emission (see e.g.Wright & Drake 2016), so density variations in magnetically activecomponents of PCEBs are expected to be larger. This might alsobe the case for more massive partially-convective stars as a similarscaling property was recently found (Lehtinen et al. 2020). Study-ing the differences of stellar spots during a minima and maxima ofof 𝑂 − 𝐶 diagrams in PCEBs will provide direct evidence of theconnection between the underlying dynamo and the orbital periodvariations. Furthermore, the determination of tidal synchronizationis equally important, as a deviation from synchronization resultsin a more complex relation between the gravitational quadrupolemoment and eclipsing time variations and potentially larger binaryperiod variations (see Lanza 2020, and also Sect. 4.1 of this paper).Lurie et al. (2017) studied tidal synchronization of F, G, and Kstars in short-period binaries. The authors find 21 eclipsing binariesthat are not synchronized and argue that this could be explainedeither because they are young or have a complex dynamical history. MNRAS , 1–13 (2021) ontribution of magnetic field modes to ETVs Considering the dynamical evolution of post common envelope bi-naries, where the secondary star is engulfed by the companion andspirals inwards toward the core of the more massive star (Paczynski1976), it is conceivable that they fall in this category.
ACKNOWLEDGEMENTS
FHN acknowledges financial support from the DAAD (DeutscherAkademischer Austauschdienst) for his doctoral studies. PJKacknowledges the financial support from the DFG (DeutscheForschungsgemeinschaft) Heisenberg programme grant No. KA4825/2-1. DRGS thanks for funding via Fondecyt regular (projectcode 1201280), ANID Programa de Astronomia Fondo QuimalQUIMAL170001 and the BASAL Centro de Excelencia en As-trofisica y Tecnologias Afines (CATA) grant PFB-06/2007. RBacknowledges support by the DFG under Germany’s ExcellenceStrategy – EXC 2121 "Quantum Universe" – 390833306. RB isalso thankful for funding by the DFG through the projects No. BA3706/14-1, No. BA 3706/15-1, No. BA 3706/17-1, and No. BA3706/18. The simulations were run on the Leftraru/Guacolda su-percomputing cluster hosted by the NLHPC (ECM-02), the Kultruncluster hosted at the Departamento de Astronomía, Universidad deConcepción, and on HLRN-IV under project grant hhp00052.
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