HD 60431, the CP2 star with the shortest rotational period Physical parameters and period analysis

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Introduction
About 10% of upper main-sequence stars (spectral types from about early B to early F) show distinct spectral peculiarities indicative of peculiar surface abundances.These are the chemically peculiar (CP) stars, which are subdivided into several subgroups, such as the metallic-line (Am/CP1) stars, the Bp/Ap (CP2) stars, the HgMn (CP3) stars, and the He-peculiar stars (Preston 1974).In general, the observed peculiar abundances are attributed to the processes of atomic diffusion, that is, the interplay between selective radiative levitation and gravitational settling taking place in the calm outer layers of slowly rotating stars (Michaud 1970;Michaud et al. 1981;Richer et al. 2000).
HD 60431, also known as V343 Puppis, which is the focus of the present investigation, belongs to the group of CP2 stars.
These objects are characterised by overabundances of elements such as Si, Cr, Sr, or Eu, and possess globally organised magnetic fields with strengths of up to several tens of kG (Aurière et al. 2007).They show spectroscopic and photometric variability with the rotation period, which is generally of the order of several days, although objects with periods of years or even centuries are known (Renson & Manfroid 2009;Mathys et al. 2020).
HD 60431 was recognised as a CP2 star by Bidelman & MacConnell (1973), who identified Si and Mg peculiarities.North et al. (1988) derived the unusually short photometric period of P = 0. d 475 518(59) d and estimated an equatorial velocity of V eq = 245 km s −1 assuming a radius of R = 2.3 R .As rapid rotation induces meridional circulation that counteracts the diffusion processes, HD 60431 poses a challenge to the atomic diffusion scenario.Although the effects of meridional circulation were described more than half a century ago (Michaud 1970;Watson 1970Watson , 1971)), they have been discussed in a qualitative way.A more quantitative study, such as that presented by Quievy et al. (2009) on meridional circulation in horizontal branch stars, is still lacking for CP2 stars.In this context, HD 60431 may turn out to be a keystone of the understanding of the processes taking place in these objects.
Here we present an investigation of HD 60431 using newly acquired and archival photometric and spectroscopic observations, with the aim of deriving physical parameters and studying its chemical composition and period behaviour.

Physical properties and evolutionary status
Photometric calibrations from Napiwotzki et al. (1993) and Künzli et al. (1997) were applied to the Strömgren (Paunzen 2015) and Geneva photometric indices (Paunzen 2022) listed in the General Catalogue of Photometric Data (Mermilliod et al. 1997) 1 .This provides initial estimates of the effective temperature and surface gravity of our target star.From Strömgren photometry, we derive a temperature of T [u − b] = 13 280 ± 270 K and a surface gravity estimate of log g = 4.53 ± 0.13, while Geneva indices provide a hotter temperature of T XY = 14 200 ± 400 K and log g = 4.10 ± 0.10.
The difference of about 1000 K between the temperature estimates derived from the Strömgren and Geneva indices may be accounted for by the well-known fact that CP2 stars are characterised by a peculiar distribution of the overall stellar flux (e.g.Leckrone 1973; Leckrone et al. 1974;Jamar et al. 1978;Adelman 1975;North 1981;Stigler et al. 2014).Therefore, we employed the temperature domain corrections for CP stars developed by Netopil et al. (2008) and derive corrected effective temperatures of T CP = 12 550 ± 850 K and T CP = 12 990 ± 400 K from the Strömgren and Geneva photometric data sets, respectively.
From a fit to the observed hydrogen line profiles (cf.Sect.3) we derive T eff = 13 000±300 K, log g = 4.1±0.1, and a projected rotational velocity value of v sin i = 190 ± 20 km s −1 .Given the peculiar composition of the star and the effects this may have on the observed colours, and given that the hydrogen line profile is usually the best indicator of effective temperature in CP2 stars (Gray & Corbally 2009)  2 , we adopted the values derived from the hydrogen line profile fitting as the most reliable parameters for further discussion.The temperature value derived in this way agrees very well with the corrected temperature values derived following Netopil et al. (2008).
To estimate mass, radius, and age, we employed the Stellar Isochrone Fitting Tool 3 , which builds on the methodology of Malkov et al. (2010) in estimating the mass, age, radius, and evolutionary phase of a star according to its effective temperature and luminosity as calculated on the basis of Gaia EDR3/DR3 data (Gaia Collaboration 2021).The distance d = 407.0+4.7 −5.0 pc taken from Bailer-Jones et al. (2021) and the Bolometric Correction derived for CP stars by Netopil et al. (2008) were directly transformed into the luminosity (log L/L = 1.995 ± 0.032).
1 https://gcpd.physics.muni.cz/ 2 This is primarily the case for cool CP2 stars; for hotter CP2 stars such as HD 60431, the hydrogen lines are more sensitive to surface gravity.Nevertheless, simultaneous modelling of several hydrogen lines can still provide good T eff estimates for these objects. 3https://github.com/Johaney-s/StIFT
The extinction in this region and distance from the Sun can be neglected, which is supported by the standard dereddening routine in the Strömgren photometric system (Napiwotzki et al. 1993).
The tool automatically searches for similar data in models based on evolutionary tracks and selects the four grid points closest to the input value in the Hertzsprung-Russell diagram.From these grid points, the output parameters are obtained by repetitive linear interpolation.For the error estimation, the program uses the errors of the input parameters and a full statistical Monte-Carlo analysis.Assuming [Z] = 0.014 4 (cf.Netopil et al. 2016 and employing the isochrones of Bressan et al. 2012), we obtain a radius of R = 1.97±0.11R , a mass of M = 3.1±0.1 M , a surface gravity of log g = 4.34 ± 0.14, and an age of about 10 Myr, which indicates that the star is situated on the zero age main sequence (ZAMS).
An overview of the astrophysical parameters derived by the different methods is presented in Table 1.

HD 60431 as a member of the Vela OB2 complex
Cantat-Gaudin et al. ( 2019) investigated the young Vela OB2 association on the basis of Gaia DR2 data (Gaia Collaboration 2018), using proper motions, parallaxes (as a proxy for distance), and photometry to separate the various components of the Vela OB2 complex.This is a young stellar grouping (age between 10 and 30 Myr) located in the direction of the constellations Vela and Puppis, at a distance of 350-400 pc from the Sun.It was initially identified by Kapteyn (1914), who made a first attempt at finding the parallaxes of B-type stars brighter than sixth magnitude.Using Hipparcos data, de Zeeuw et al. (1999) were able to perform a more detailed membership study, which led to the detection of a local group of pre-main-sequence stars (Pozzo et al. 2000) that is referred to as the γ Velorum cluster (Prisinzano et al. 2016).The topology of the region is quite complex; there are many different regions interacting with each other to a certain extent.Most likely, this is the result of various star formation processes triggered by supernovae exploding 30 Myr ago (Cantat-Gaudin et al. 2019).
HD 60431 is located in the Vela OB2 complex.According to Cantat-Gaudin et al. (2019), there is a probability of 99% that the star is a member of the open cluster NGC 2547.This agrees very well with the parameters and the age estimate of about 10 Myr presented in Sect. 2.

Spectroscopic observations and data reduction
For the spectroscopic analysis we utilised an echelle spectrum of HD 60431 collected in 2012 within the framework of the project Magnetism in Massive Stars (MiMeS; programme ID 11BP14, PI Gregg Wade) using the spectropolarimeter ESPaDOnS attached to the Canada-France-Hawaii Telescope (CFHT) in spectropolarimetric mode.The reduced 1D data were extracted from the scientific archive of the CFHT5 .The spectrum covers the wavelength range 3690-10 480 Å, with an average S/N of 120 at 5500 Å.The spectral resolution R is 65 000.The intensity spectrum consists of four subexposures, which were re-normalised separately order-by-order to the continuum level using routines from the astronomical data processing system IRAF6 (National Optical Astronomy Observatories 1999).We utilised a template spectrum synthesised with photometrically derived atmospheric parameters to reconstruct the continuum around the broad structures, for example hydrogen lines extending on multiple echelle orders.The average of the four subexposures trimmed to 3740-8000 Å was used in the subsequent analysis.Furthermore, a circularly polarised spectrum (V Stokes) was employed to check the presence of a longitudinal magnetic field in the star.
The ESPaDOnS combined spectrum was obtained during the narrow phase interval of 0.015 ≤ ϕ ≤ 0.040.To increase phase coverage, an additional spectrum of HD 60431 was taken with the echelle fibre-fed spectrograph MRES installed on the 2.4 m Thai National Telescope at Doi Inthanon (Thailand) in December 2021.The raw observational material was processed in a standardised way using the pipeline PyYAP, which was written in Python by one of the authors (ES) for this particular device7 .For the wavelength calibration we used a ThAr spectrum recorded immediately after the star.After the application of the pipeline, the continuum level was adjusted using IRAF.The extracted and calibrated spectrum covers the wavelength region of 4 200-7 100 Å and has an average resolution of R = 17 000.The spectrum was employed to check the stability of the radial velocity and spectral variability of our target star.A fragment of the combined observational journal is shown in Table 2.

Astrophysical parameters and abundance analysis
The fact that the star has been studied in the framework of the MiMeS survey and no results of magnetic measurements have been published to date suggests that the magnetic field is small, which allows us to neglect its influence in the process of spectrum modelling.We synthesised a spectrum assuming local thermodynamic equilibrium (LTE) with the 'nonmagnetic' code Synth3 (Kochukhov et al. 2007;Kochukhov 2012), which uses model atmospheres computed with Atlas9 (Kurucz 2005(Kurucz , 2017) ) and line lists extracted from the VALD database (Ryabchikova et al. 2015).
From a simultaneous fit to the six observed hydrogen lines, we derived the effective temperature, surface gravity, and projected rotational velocity values presented in Sect. 2 (T eff = 13 000 ± 300 K, log g = 4.1 ± 0.1, v sin i = 190 ± 20 km s −1 ).The corresponding fit is shown in Fig. 1.It is important to note that, for a correct description of the observed level of the continuum, we had to significantly increase the abundances (and thus the number of lines) of most of the chemical elements involved in the peculiar abundance patterns of CP2 stars in our requests to VALD.
Chemical anomalies of SiMg type were initially reported for HD 60431 in the Bidelman & MacConnell (1973) catalogue.Lines of Si and Mg are conspicuous in the observed spectrum, but numerous strong lines of iron are also present in blends with various elements.
Adjusting the abundance of the main contributing elements and ions within the wavelength range of 4075-7990 Å, we simultaneously fitted 19 partially overlapping spectral regions rich in lines (Fig. 2).We found that Si and Mg are overabundant by 1.92 dex and 1.83 dex, respectively; Fe, the third most noticeable contributor, is enhanced by 1.77 dex with respect to the solar abundance.
In addition, several lines of He, O, and Ca were identified in the spectrum.He is represented by at least the He i 4471 Å and 5876 Å lines and, as deduced from an analysis of the blends with significant He contributions, has nearly solar abundance.This result is intriguing because He is generally found to be deficient in late-B type CP2 stars, especially in young ones (Bailey et al. 2014).HD 60431 forms a noteworthy exception.Its comparably high He abundance might be linked to its fast rotation, and here we may have an intermediate case between the magnetic Herich stars, which generally have a spectral type close to B2 (see e.g.Järvinen et al. 2018), and the magnetic CP2 stars, which are generally He-weak.More He abundance determinations in rapidly rotating late-B type CP2 stars are needed to clarify this point.
The observed peculiarly enhanced He i 5876 Å line (Table 3) can be explained following the non-LTE approach (Korotin & Ryabchikova 2018), which is also required to  Calcium is most prominent in the H and K Ca ii lines, of which only the latter line is suitable for modelling.The corresponding Ca ii K 3933 Å line indicates a Ca excess of 1.63 dex.However, the derived value does not take into account the influence of non-LTE effects and may suffer from missing weak blended lines.It is note-worthy that in addition to the broad stellar Ca lines, the observed spectra also contain sharp Ca components of interstellar origin.
The derived abundances of several other elements are reported in the overview provided in Table 3.As these values were derived mostly from modelling weak lines or complex blends, they should be regarded as providing general abundance trends only.The accuracy of the derived abundances depends mainly on the uncertainties of the physical parameters, the continuum normalisation, and the reliable identification of blends.In some cases non-LTE effects require separate modelling, and A159, page 4 of 13 they were not included in the evaluation of errors.In this manner we estimate that the typical error of the individual abundance measurements amounts to 0.20 dex for Mg, Si, and Fe and to 0.30 dex for He and Ti.Less accurate results in Table 3 are identified by a colon (:).
To check for possible effects of a weak magnetic field (see Sect. 3.3) on the derived abundances, we synthesised a spectrum of a star with the same physical parameters and a dipolar magnetic field of B d = 1 kG, using the code SynthMag written by Kochukhov et al. (2007).Although the presence of the magnetic field visibly affects some individual lines, these effects are well contained within the errors.We therefore conclude that the influence of the magnetic field on the derived abundances is negligible.
In summary, our results indicate that, in agreement with the literature, HD 60431 is a highly peculiar late B-type CP2 star.In addition to significant overabundances of Mg, Si, Fe, Cr, and Ti, we find strong excesses of many other elements, including three lanthanides.The classification as a CP2 star is further corroborated by the presence of a conspicuous flux depression at around 5 200 Å, which has been shown to be a characteristic of magnetic chemically peculiar stars (Kodaira 1969;Adelman 1975Adelman , 1979;;Maitzen 1976;Kudryavtsev et al. 2006;Hümmerich et al. 2020).This flux depression is detected in the Geneva photometric system through the photometric parameters ∆(V1 − G), (Hauck 1978) and Z (Cramer & Maeder 1979).From the average Geneva colours of HD 60431 taken from Rufener (1989) and the definitions of these parameters, we obtain ∆(V1 − G) = 0.072 and Z = −0.092.As normal stars have ∆(V1 − G) −0.005 and Z 0.00, we see that HD 60431 has extreme values of both parameters (Z is negative for CP2 stars).While the depth of the depression depends on T eff , it achieves maximum strength at around 12 000-13 000 K, just at the effective temperature of HD 60431.

LSD profiles and magnetic field
The least-squares deconvolution (LSD; Kochukhov et al. 2010) profiles of HD 60431 were derived from the echelle spectra obtained in 2012 and 2021 and listed in Table 2. To extract the averaged profiles we used a mask based on a line list from VALD that represents the physical parameters of a typical CP2 star of temperature type B9.The spectral range and resolution of the two spectra were adjusted to each other.The LSD profiles have variable shapes, which are most likely caused by the presence of abundance spots (Fig. 3).Distinctive spot signatures were found for Si and Fe after adjusting the mask to the observed spectrum.This finding agrees with the expectations for CP2 stars and with the observed rotational modulation in our target star.
The radial velocity V r as measured from the LSD profiles amounts to 18.4 km s −1 in 2012 and 23.2 km s −1 in 2021.The difference in V r can be explained by the variable core, and falls well within the measurement error of 5-8 km s −1 .From the available data, no meaningful conclusion can be drawn about a possible multiplicity of HD 60431.
As we point out above, HD 60431 has been studied in the framework of the MiMeS survey and no results of magnetic measurements have been published.Along with the intensity spectra, we therefore extracted one circularly polarised spectrum from the CFHT archive.To check the presence of a significant longitudinal magnetic field, we analysed the spectropolarimetric data based on the LSD technique described in Donati et al. (1997) and Kochukhov et al. (2010).From the corresponding LSD profiles, in the velocity space from −165 to 223 km s −1 , we measure a mean longitudinal magnetic field of B z = 270 ± 150 G.
A159, page 5 of 13 Even though a typical Zeeman signature is visible in the mean V Stokes profile, we cannot assert that the star is truly magnetic.From a statistical analysis of the signal, we derive a false alarm probability of 0.99.This result implies that the magnetic field of HD 60431 is either very weak or could not be detected because of insufficient data quality or an observation at an inappropriate phase of the magnetic curve (see discussion in Sect.5.4).In summary, our analysis provides evidence, under the assumption of a dipolar field, that HD 60431 has a weak magnetic field of the order of 1 kG on the pole; however, this is not conclusive and needs to be verified by additional studies.The mean LSD profiles used for measuring the longitudinal magnetic field are shown in Fig. 4.

Photometric analysis
This section contains a description of the observed light curves of HD 60431 in the wavelength range 3 400-8 000 Å and an analysis of the period evolution in the last decades.For the latter purpose, we relied on our own methods of phenomenological modelling of light curves or their segments (Mikulášek 2016, and references therein) and their solution by robust regression (Mikulášek et al. 2020).

Observations
Our study is based on diverse data sources.We took into account 357 individual observations in seven bands of the Geneva system (see Table 4), which were obtained using the Swiss telescope at the European Southern Observatory, La Silla (Chile) in 1981-88 and published by North et al. (1988).Furthermore, 667 measurements in the V band were gleaned from the archives of the third phase of the All Sky Automated Survey (ASAS-3; Pojmanski 2002), which cover the period 2000-09 and were divided into two parts: ASAS 3-I and ASAS 3-II (see Table 5).The most accurate information on HD 60431, however, is represented by four data sets obtained by the Transiting Exoplanet Survey Satellite (TESS) (Ricker et al. 2014(Ricker et al. , 2015)), which contain a total of 8 911 measurements collected in 2019 and 2021 (sectors 7, 8, 34, and 35).

Multicolour light curve modelling
According to North et al. (1988), HD 60431 shows a fairly large amplitude in the U band of almost 0.1 mag, while the amplitude in the V band hardly reaches 0.04 mag (see Table 4 and Fig. 5).All colours were shown to vary in phase.
To investigate the relationship of the phased light curves in different passbands quantitatively, we applied the technique of advanced principal component analysis (Mikulášek et al. 2004) to the Geneva photometry of HD 60431.It was found that the light curves phased on the period of P = 0. d 475 518 (North et al. 1988) display only one significant principal component.All other components can be neglected without a deterioration in the accuracy of the fit.This finding also applies to the ASAS-3 and TESS light curves (see Fig. 5).
While CP2 stars may show very different light curves at different wavelengths, it has been shown by Mikulášek et al. (2004Mikulášek et al. ( , 2008a) ) that similar light curve patterns in different passbands are commonly observed in these objects.In these stars only one dominant mechanism is clearly responsible for the redistribution of the flux in the spectrum.All the light curves F(λ, ϕ) of HD 60431 can be satisfactorily approximated by the simple model where A(λ) is the effective semi-amplitude of the light curve in the individual photometric passbands, and B(ϕ, α) is the nor-   4).The mean light curves are defined by the filled dots, which represent normal points, as defined in Appendix B. The model light curves are represented by the solid lines.
malised periodic function of the phase function ϑ.In this case we define B(ϕ, α) as a linear combination of nine simple mutually orthonormal periodic functions, of which five are symmetric and four antisymmetric functions to the phase ϕ = 0, which describes the eight independent parameters forming the alpha vector α.More details are given in Appendix A.

Phase function model
The phase function ϑ(t) = E + ϕ (where E is the epoch and ϕ the common phase) and its inverse function Θ(ϑ) are related to an instantaneous period P(t) at the time t through simple differential equations with a boundary condition (Mikulášek et al. 2008b; Mikulášek 2016):  Using Θ(ϑ), we can calculate the moments of the zero phases JD min (E) = T (E), where E is the epoch.

Linear model solution
If the period of the variable star is constant (P(t) = P 1 ), the solutions of Eqs. ( 2) and ( 3) are trivial ϑ 1 = (t − M 1 )/P 1 ⇒ JD min 1 (E) = M 1 + P 1 E, (4) with only two parameters of the linear ephemeris, P 1 and M 1 .The linear model of the phase function can be introduced into the light curve model of HD 60431 and all free parameters can be found simultaneously by robust regression, which suppresses the influence of any outlying data points (see e.g.Mikulášek et al. 2020).Following this approach, we derived the linear ephemeris M 1 = 2 459 219.151 09(16) and P 1 = 0. d 475 515 90(7), which roughly describes the course of the phase function ϑ(t) and its inverse Θ(ϑ).
However, a detailed analysis indicates that the basic assumption of a constant period P is not fulfilled.This can be demonstrated, for example, by the fact that the linear periods derived from different data sets are not compatible with each other.Omitting the TESS observations, we derive a period of P 1 = 0. d 475 515 456(20), (M 1 = 2 449 538.609 6( 21)).If we then omit the Geneva and ASAS-3 measurements, a period of P 1 = 0. d 475 516 50( 11), (M 1 = 2 459 218.675 57( 16)) is obtained.This can be interpreted as a lengthening of the period by 0.17 ± 0.02 s over a time span of 26.5 years.The same result is derived from an analysis of the (O-C) values for individual groups of observations (see Fig. 6).
We therefore conclude that the period of HD 60431 has not remained constant over the observed time span.The simplest model that adequately describes the observations is provided by the assumption that the period increases linearly with time, so Ṗ > 0.

Quadratic model solution
Assuming that the rate of change in the period length Ṗ remains constant during the whole interval of the observations, we find that the phase function ϑ(t) is expressed by a quadratic polynomial, which we adjust to its orthogonal form: JD min (E) Here the function IP(ϑ) is the integer part of ϑ, while FP(ϑ) denotes the fractional part; P 1 is the linear approximation to the period; M 1 is the time of light minimum near the weighted center of all observations; ϑ 1 (t) is the linear phase function; a 1 , a 2 are the quantities that characterise the distribution of the observations and their quality; P(ϑ 1 ) is a prediction of the instantaneous period at time t = M 1 + P 1 ϑ 1 ; and JD min (E) predicts the Julian date of the light minimum for epoch E.
The times of the light minima for the observational subsets listed in Table 5 and the corresponding O-C diagram (Fig. 6) are visualisations of the model results according to the relations given in Mikulášek et al. (2012).
Ignoring the uncertainties of the predicted quantities, the quadratic ephemeris can be represented in a more elegant form as   To test the reality of the observed period change in HD 60431, we subjected the model calculations to several tests, relying in particular on a bootstrap approach with 5000 cycles, and checked the robustness of the results to estimate the uncertainties.We found a perfect agreement of the results obtained with the above method, and verified that the data variance entirely agrees with the Gaussian ideal (see Fig. 8).In summary, the available evidence proves that the observed period change in HD 60431 is real.

Discussion
5.1.Is HD 60431 the CP2 star with the shortest known rotational period?
Since the first period determination by North et al. (1988), HD 60431 has been considered the CP2 star with the shortest known rotational period.However, this perception has been challenged recently by Hümmerich et al. (2017).Using photometric time series data from ASAS-3, these authors derived a rotational period of P = 0. d 46566 for the Si CP2 star HD 98000, which, if confirmed, would represent the shortest observed rotational period among this group of objects.However, the authors cautioned that the amplitude of the variability is very small and that a rotation period that is twice as long could not be excluded on the basis of the available data, and called for additional observations to investigate this matter further.HD 98000 (TIC 82196009) was observed in Sector 10 by TESS for a duration of ∼26 days.The ultra-precise TESS data unambiguously show, in contrast to the findings from the ground-based observations, that HD 98000 indeed shows a double-wave light curve (Fig. 7).Therefore, the period solution of Hümmerich et al. (2017) represents only half the true value.From an analysis of the TESS data, we derive a rotational period of P = 0. d 93130(3).HD 60431, therefore, remains the CP2 star with the shortest known rotational period.

Period variability
We currently know of nine magnetic chemically peculiar stars, including HD 60431, that have shown changes in their rotational periods (Mikulášek et al. 2021).These objects form a heterogeneous group, with some members showing increasing or decreasing periods, as well as stars with more complex period behaviour.With an observed value of Ṗ = 2.36(19) × 10 −10 , HD 60431 exhibits the smallest rate of period change among these objects.At the same time, however, the uncertainty with which Ṗ is known (δ Ṗ = 1.9 × 10 −11 = 8 × 10 −7 s cycle −1 ) is very small.In this respect, the case of HD 60431 is comparable to the cases of the best-observed magnetic CP stars exhibiting period variations, σ Ori E and CU Vir (see Pyper & Adelman 2020;Mikulášek 2016).The small uncertainty is due to the very short rotation period of P = 0. d 475 5, the time span of the observations, the relatively large amplitude of the light curve that exhibits a well-defined narrow minimum (see Fig. 5), and finally the quality of the employed photometry.Such a small value of Ṗ would hardly be detectable in the vast majority of known CP2 stars, which raises the question of whether the periods of all magnetic CP stars are actually variable to some extent, and on timescales shorter than justifiable by stellar evolution (the moment of inertia varies with evolution on the main sequence, resulting in a change in the rotational period by mere conservation of angular momentum).

Can the period variations be explained in purely kinematic terms?
It is not certain a priori that the secular period change is intrinsic to the star.It might possibly be due to orbital motion in a longperiod binary system, to acceleration linked with the motion within the Vela OB2 association, or even to a radial acceleration due to the change in orientation of the line of sight linked with the proper motion.In this section these possibilities are briefly discussed.
First, there is a link between Ṗ and Vr , where V r is the radial velocity, through the classical Doppler effect formula: The last term of the second equality in Eq. ( 12) disappears because the intrinsic frequency of the star is stable by hypothesis, A159, page 8 of 13 so we are left with Inserting the relevant numbers, we obtain Vr ≈ 5.4 km s −1 per century = 2.2 km s −1 over 40 years.
With a proper motion of 9.38 mas yr −1 , the tangential velocity of HD 60431 is similar to its radial velocity, and the change in orientation of the line of sight is so small (assuming a constant spatial velocity vector relative to the Sun) that the change in radial velocity would amount to no more than a few cm s −1 .Second, the motion of the star within the Vela OB2 association is too slow to provide any significant acceleration (σ V ≈ 4.5 km s −1 ; Melnik & Dambis 2020) since the crossing time exceeds tens of thousands of years (for the Trapezium in Orion, which represents a minimum value, one finds t cross ≈ 13 kyr; Pflamm-Altenburg & Kroupa 2006), a value that greatly exceeds the time base of 40 yr.
Third, assuming that HD 60431 has a 1 M companion in a binary system with a circular orbit and an orbital period of 200 yr, its orbital velocity would amount to about 2 km s −1 .For an orbital inclination close to 90 • and an orbital phase where the radial velocity is close to the systemic value (and hence varies almost linearly), V r will change by about 2.3 km s −1 over 40 years 8 .This is very close to the radial acceleration needed to explain Ṗ.On the other hand, the semi-major axis of the primary orbit would amount to 13.6 au, corresponding to 33 mas on the sky.The Gaia DR3 data cover 34 months (i.e.1.42% of the orbit), which corresponds to a motion of ∼3 mas on the sky, a value easily measurable by Gaia, although this might be interpreted as a mere contribution to the proper motion.However, the 'non-single star' parameter in the DR3 data is null for HD 60431, which casts doubt on the idea that Ṗ is caused by orbital motion.Moreover, the renormalised unit weight error parameter (RUWE = 1.08) indicates that the standard singlestar model fits the Gaia data well; doubt arises only in the case of RUWE > 1.4.

Possible intrinsic cause
An interpretation of the period change observed in magnetic CP stars is far from obvious.In cases of period lengthening, it is expected that this is due to loss of angular momentum linked with mass loss where the stellar wind flows along open magnetic lines.This scenario has been proposed for σ Ori E (Ud-Doula et al. 2009;Townsend et al. 2010).The observed characteristic spin-down time of this 9 M star is τ spin = 1.34 Myr (Townsend et al. 2010), which is consistent with theoretical predictions (Ud-Doula et al. 2009) given a mass loss rate of Ṁ = 2.4 × 10 −9 M yr −1 estimated through modelling (Krtička et al. 2006).
This star, however, is three times more massive than HD 60431 with a luminosity of log(L/L ) = 3.5 (Shultz et al. 2019), while the luminosity of HD 60431 is log(L/L ) = 2.0 (adopting R and T eff from Table 1).A rough extrapolation using Fig. 1 of de Jager et al. (1988) indicates that such a low luminosity suggests a mass loss of ∼10 −12 M yr −1 at most, and this is much higher than the ∼10 −16 value proposed by Babel (1995) 8 Increasing the orbital period to 400 yr would lead to a change in V r of ∼1 km s −1 at most over the same time span of 40 yr.On the other hand, a shorter orbital period could not accommodate the time span of 40 yr during which the radial velocity varies quasi-linearly.on a theoretical basis.Assuming that Eq. 25 of Ud-Doula et al. (2009) is valid for the low effective temperature of HD 60431 (which is admittedly debatable), and inserting the very high value of 10 −12 M yr −1 , we get τ spin ∼ 520 Myr (adopting M = 3 M , R = 2 R , k = 0.1, B p = 1 kG, and v 8 ≡ v ∞ /1000 km s −1 = 1).This is almost two orders of magnitude longer than the observed value of τ spin = 5.5 ± 0.4 Myr.In summary, it seems difficult to explain the observed period lengthening through stellar wind.
Moreover, for a significant fraction of period-changing magnetic CP stars, an acceleration or cyclic changes in the rotation period are observed (see Mikulášek et al. 2021).This can be interpreted within the model of torsional oscillations (Krtička et al. 2017;Takahashi & Langer 2021).This model, however, requires an internal magnetic field with strength equal to the surface field to explain the observed timescales of period variability, which contradicts the current understanding of the internal magnetic field of magnetic CP stars.It is generally admitted that the magnetic field of these objects is a fossil field (Braithwaite & Spruit 2004;Schneider et al. 2019), and that the strength of such a field increases with depth, as observed by Lecoanet et al. (2022), among others.It is not excluded that torsional oscillations may affect only outer layers frozen in the outer magnetic field (Mikulášek et al. 2018), but this idea has not been theoretically studied yet.

Rotational flattening, equatorial velocity, and inclination of the rotational axis
The simplest approximation for the shape of a uniformly rotating single star is provided by the Roche model, which accounts for the centripetal gravitational force and the centrifugal force of rotation.Following this approach, the stellar surface has the shape of an isobar surface expressed by the sum of the gravitational potential approximated by the field of a point mass with mass M of the star and the centrifugal potential of a rigid body rotating with angular velocity Ω.This isobar area is an axisymmetric spheroid characterised by the equatorial and polar radii R eq , R p and flattening f , f = R eq /R p .After some algebra, we obtain the basic relation between the radii from Eqs. ( 10) and ( 12) of Zahn et al. (2010), as where G is the gravitational constant; Ω crit the maximum angular velocity (Ω ≤ Ω crit ) beyond which the stellar body is disrupted; V eq is the equatorial velocity; V eq = R eq Ω, while V crit is the maximum allowed equatorial velocity (V eq ≤ V crit ); and (a cfg /g) eq is the ratio of the centrifugal and gravitational accelerations at the equator.It follows from Eq. ( 14) that the maximum flattening f crit of a rotating body is f crit = 3 2 because a cfg = g in this case.Critical quantities for the given mass M and polar radius R p are calculated as follows (with ρ ≡ R crit eq ): A159, page 9 of 13 Applying relations ( 14) and ( 15) to the case of HD 60431, we assume a mass of M = 3.1 M , Ω = 1.5294 × 10 −4 s −1 , and R p R = 1.97 R .After several iterations of these semi-analytic relations between R eq and R p (Ekström et al. 2008), we estimate the equatorial radius at R eq = 2.16 R .Bearing in mind that the rotational period of HD 60431 is the smallest of all the known CP2 stars, the derived flattening of f = 1.096 appears surprisingly small at first glance.This occurs because the flattening f and the ratio of the equatorial centrifugal to gravitational accelerations ((a cfg /g) eq = 0.19 for HD 60431) of a rotating star critically depend on its radius (∼R 3 ).As our target star is nearly on the ZAMS (see Sect. 2), its polar radius is the smallest possible for its mass.
The same conclusion follows from a comparison of the real equatorial velocity V eq = 50.6R eq /P 1 = 230 km s −1 and the maximum allowed equatorial velocity V crit = 450 km s −1 . 9The true rotational velocity V eq is about half the critical value.
We can also evaluate the quantity R eq sin i = v sin i P 1 /50.6 = 1.79 ± 0.19 R in determining the lower limit of the stellar equatorial radius R eq .We use the simplistic approximation R eq ≈ R according to the relatively low ratio f = 1.096.Assuming that R eq = 2.16 R , we derive sin i = 0.83(9), which indicates an inclination angle of i = 56 • ± 9 • .

Interpretation of the light variability
As shown by Krtička et al. (2007), the main cause of rotationally modulated light changes in chemically peculiar stars is their uneven distribution of overabundant chemical elements across the stellar surface.In the temperature range of HD 60431, the most important element in this respect is Si (Krtička et al. 2009(Krtička et al. , 2012)), which has a number of absorption lines and bound-free transitions in the far-ultraviolet (UV) region that cause a massive redistribution of far-UV flux to near-UV and optical regions.This mechanism, also known as blanketing, affects the entire mentioned spectral region; however, the intensity of the flux redistribution decreases with increasing wavelength (Krtička et al. 2015).This is totally in line with the observed light variability pattern of HD 60431, which is characterised by decreasing amplitude towards longer wavelengths (see Figs. 9 and 5).
Following this scenario, the areas of increased Si abundance appear brighter at optical and infrared wavelengths than those with a lower content of this element.Therefore, during the time of maximum brightness, bright Si spots are expected to cover the major part of the visible stellar hemisphere.In this respect, it is noteworthy that the spectropolarimetric measurement used for the abundance analysis covers the rotational phases 0.015-0.040and was therefore taken almost directly during minimum light.Consequently, the derived overabundance of Si (and perhaps that of Fe, if the surface distribution of both elements agree) may in fact be significantly smaller than the average overabundance.

Development of chemical peculiarity during the main-sequence stage
With a derived age of 10 Myr (Sect.2), HD 60431 is a very young object that has only spent about 1/27 of its expected main-sequence life time (365 Myr for 3.1 M , according to Ekström et al. 2012).Despite this short time, radiative diffu- 9 The much more sophisticated models of rotating stars by Georgy et al. (2013), which account for details of the inner structure, yield almost identical estimates of the astrophysical parameters of HD 60431, resulting in V crit = 460 km s −1 .4).
sion has obviously efficiently established a peculiar atmospheric composition, in agreement with findings for other young stars (Romanyuk et al. 2020(Romanyuk et al. , 2015)).
The subsequent rotational evolution of HD 60431 depends on how angular momentum is transported within the star (Keszthelyi et al. 2020).The non-magnetic evolutionary models of Granada et al. (2013), which use the horizontal diffusion coefficient of Zahn (1992) and the shear diffusion coefficient of Maeder (1997), predict a nearly constant equatorial rotational velocity on the main sequence.Following this scenario, HD 60431 should maintain its rotational speed and possibly also its chemical peculiarity during main-sequence evolution.
On the other hand, for magnetic stars it may be more reasonable to assume solid-body rotation (Takahashi & Langer 2021).Neglecting angular momentum loss due to the magnetised wind, which is a reasonable assumption for a star with a temperature of HD 60431 (Krtička 2014), the rotational period is predicted to remain nearly constant during the main-sequence lifetime (Takahashi & Langer 2021), thus Ω(t) = Ω 0 = const.As the equatorial radius R eq (t) increases during the main-sequence evolution, the stellar equatorial velocity V eq (t) = Ω 0 R eq (t) may approach the critical rotation limit V crit (see Eq. ( 15)), which results in the loss of the peculiar atmospheric abundance pattern.
The subsequent spinning up of stars during the mainsequence evolution could be at the root of why so few ultrashort-period magnetic CP stars are observed.

Conclusions
Using high-resolution spectroscopy and an extensive set of ground-and space-based time series photometry, we carried out a detailed study of the CP2 star HD 60431.Our main results are summarised as follows: 1.With an age of only 10 Myr, an effective temperature T eff = 13 000 ± 300 K, a surface gravity log g = 4.10 ± 0.10, radius R = 1.97±0.09R , and mass M = 3.1±0.1 M , HD 60431 is situated close to the zero age main sequence, and is a member of the open cluster NGC 2547 in the Vela OB2 complex.2. We estimated the flattening of the star at a value of 1.096, which means that the star is nearly spherical.This agrees with the fact that the equatorial velocity V eq = 50.6R/P = 230 km s −1 is half the calculated critical equatorial velocity v crit = 460 km s −1 .The observed quantities of R sin i = 1.79± 0.19 R and sin i = 0.83 ± 0.09 imply an inclination angle of i = 56 • .
A159, page 10 of 13 3. HD 60431 is a classical late B-type CP2 star showing strong overabundances of Mg (1.8 dex), Si (1.9 dex), Ca (1.6 dex), Ti (2.2 dex), and Fe (1.8 dex).We showed that within the errors, a dipolar magnetic field of the order of 1 kG does not significantly affect the abundance determination.4. In agreement with this classification, we find spectroscopic evidence for the presence of chemical surface spots, which are responsible for the observed light changes.Si, in particular, is expected to be a major contributor to the photometric variability.5.No conclusive evidence for the presence of a strong magnetic field was found in the available spectroscopic data.A typical Zeeman signature was found in one ESPaDOnS spectrum, from which we derive a longitudinal magnetic field of B z = 270±150 G.However, from a statistical point of view, this signal is not significant, which implies that the magnetic field is either very weak or could not be detected because of insufficient data quality or unfortunate phase coverage.6.We confirm HD 60431 as the CP2 star with the shortest known rotational period (P = 0. d 4755).We found the available multicolour light curves from 1981-2021 covering the wavelength range 3400-8 000 Å to be constant in shape and amplitude, which made it possible to apply the phenomenological model used, see Eq. ( 1) and Appendix A. 7.After careful treatment of the photometric data, we conclude that the rotational period of HD 60431 is slowly lengthening ( Ṗ = 2.36(19) × 10 −10 = 7.4(6) ms yr −1 ).The best fit to the available data is provided by the following quadratic ephemeris: HJD min (E) = 2 459 212.969 35 + 0. d 475 516 64 E + 5. d 62 × 10 −11 E 2 .8.Although the derived value of Ṗ is the smallest rate of period change known among magnetic chemically peculiar stars, the detection is highly significant.It was made possible thanks to the shortness of the period, the long time span, and the high quality of the time series photometry, and to a light curve with relatively large amplitude and a well-defined narrow minimum.In summary, HD 60431 is an outstanding object that may ultimately turn out to be a keystone of the understanding of CP2 star evolution.Theory needs to explain the establishment and maintenance of chemical peculiarities in such a young and fastrotating object.
The phenomenon of period variability may be inherent to all magnetic chemically peculiar stars.However, only further detailed (and hence time-and resource-consuming) studies will be able to shed more light on this subject.As yet, we can offer no consistent explanation of this phenomenon.

Fig. 2 .
Fig. 2. Sample of spectral regions used for the evaluation of abundances.Red and black lines indicate the observed spectrum and the model fit, respectively.

Fig. 5 .
Fig. 5. Light curves of HD 60431 in [U], B1, [B], B2, V1, [V], V, G, and T (see Table4).The mean light curves are defined by the filled dots, which represent normal points, as defined in Appendix B. The model light curves are represented by the solid lines.

Fig. 6 .
Fig. 6.Virtual O-C lin diagram derived using relations in Mikulášek et al. (2012), see also Table 5.The solid line denotes the model, dashed lines its 1σ uncertainty.

Fig. 7 .
Fig. 7. Phased TESS light curve of HD 98000, which clearly shows a double-wave light curve.The small blue dots denote detrended observations; the large blue dots are normal points representing the mean phased light curve (see Appendix B).

Fig. 8 .
Fig. 8. Histogram of the results of 5000 bootstrap trials for Ṗ proving its non-zero value.

Table 1 .
Astrophysical parameters of HD 60431 derived by the indicated methods.

Table 2 .
Journal of spectroscopic observations used in this work.

Table 3 .
Derived chemical abundances of HD 60431.Notes.The solar abundance values are from Magg et al. (2022; O, Mg, Si, Ca, and Fe) and Asplund et al. (2021; all other elements).∆ log ε is the abundance relative to solar, defined by ∆ log ε = log N/Ntot (N/Ntot) Sun .A colon indicates an uncertain value.

Table 4 .
Photometric bands of the observations used in this study (cf.Figs. 9 and 5).

Table 5 .
(Mikulášek et al. 2012)lášek et al. 2012)for individual groups of photometric data.Notes.Columns: E mean is the mean epoch, HJD min is the corresponding HJD moment of the light minimum, O-C lin and O-C sqrt are the O-C values computed from linear (4) and quadratic (7) ephemerides, δ HJD min is the uncertainty on the minimum moment determination (all in days); σ is the scatter of data (in mmag) around the fitted model curve; N is the number of measurements.