HD 60431, the CP2 star with the shortest rotational period

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)¹¹1https://gcpd.physics.muni.cz/. This provides initial estimates of the effective temperature and surface gravity of our target star. From Strömgren photometry, we derive a temperature of $T[u-b]=13\,280\pm 270$ K and a surface gravity estimate of $\log g=4.53\pm 0.13$, while Geneva indices provide a hotter temperature of $T_{\textrm{XY}}=14\,200\pm 400$ K and $\log g=4.10\pm 0.10$.

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 = $12\,550\pm 850$ K and T_CP = $12\,990\pm 400$ K from the Strömgren and Geneva photometric data sets, respectively.

From a fit to the observed hydrogen line profiles (cf. Section 3), we derive $T_{\mathrm{eff}}=13\,000\pm 300$ K, $\log g=4.1\pm 0.1$, and a projected rotational velocity value of ${\varv}\sin i=190\pm 20\,\text{km}\,\text{s}^{-1}$. Given the peculiar composition of the star and the effects this may have on the observed colours as well as the fact that the hydrogen line profile is usually the best indicator of effective temperature in CP2 stars (Gray & Corbally, 2009)²²2 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 $T_{\mathrm{eff}}$ estimates for these objects., we have 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 Tool³³3https://github.com/Johaney-s/StIFT, 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 et al., 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 $L/L_{\odot}$ = $1.995\pm 0.032$). 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] = 0.014⁴⁴4The chosen [Z] value corresponds to solar metallicity. As described in the following subsection, HD 60431 is most likely a member of NGC 2547, for which Baratella et al. (2020) derive an average $[Fe/H]=0.10\pm 0.01$ from the analysis of two members, which is slightly higher than solar. The corresponding shift on the HR diagram, however, is no larger than the error bars on $T_{\mathrm{eff}}$ and $L$ (see e.g. Fig. 15 of Bressan et al. (2012)). (cf. Netopil et al., 2016) and employing the isochrones of Bressan et al. (2012), we obtain a radius of $R=1.97\pm 0.11$ $\mathrm{R}_{\odot}$, a mass of $M=3.1\pm 0.1$ $\mathrm{M}_{\odot}$, a surface gravity of $\log g=4.34\pm 0.14$, and an age of about 10 Myr, which indicates that the star is situated on the zero-age main sequence (ZAMS).

An overview over the astrophysical parameters derived by the different methods is presented in Table 1.

Cantat-Gaudin et al. (2019) investigated the young Vela OB2 association on the basis of $Gaia$ DR2 data (Gaia Collaboration et al., 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 to 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 $\gamma$ Velorum cluster (Prisinzano et al., 2016). The topology of the region is quite complex; there are a dozen of 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 Section 2.

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 CFHT⁵⁵5https://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/cfht/. The spectrum covers the wavelength range 3 690 – 10 480 Å, with an average SNR of 120 at 5 500 Å. 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 IRAF⁶⁶6https://github.com/iraf-community/iraf (National Optical Astronomy Observatories, 1999). We utilised a template spectrum synthesised with photometrically derived atmospheric parameters to reconstruct the continuum around the broad structures like hydrogen lines extending on multiple echelle orders. The average of the four subexposures trimmed to 3 740 – 8 000 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 $\leq$ $\varphi$ $\leq$ 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 device⁷⁷7https://github.com/ich-heisse-eugene/PyYAP. 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. 0

The fact that the star has been studied in the framework of the MiMeS survey and no results of magnetic measurements have been published so far 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 ’non-magnetic’ code Synth3 (Kochukhov, 2007, 2012) that uses model atmospheres computed with Atlas9 (Kurucz, 2005, 2017) and line lists extracted from the VALD database (Ryabchikova et al., 2015).

From a simultaneous fit to the six observed hydrogen lines, we derive the effective temperature, surface gravity, and projected rotational velocity values presented in Section 2 ($T_{\mathrm{eff}}=13\,000\pm 300$ K, $\log g=4.1\pm 0.1$, ${\varv}\sin i=190\pm 20\,\text{km}\,\text{s}^{-1}$). 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 the 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 4 075 – 7 990 Å, we simultaneously fitted 19 partially overlapping spectral regions rich in lines (Figure 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 4 471 Å and 5 876 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 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 we may here have an intermediate case between the magnetic He-rich 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 (cf. Table 3) can be explained following the non-LTE approach (Korotin & Ryabchikova, 2018), which is also required to account for the observed depth of the O iii 7771-5 Å triplet. Considering the derived oxygen excess of $\Delta\log\varepsilon=+2$ and the values of the non-LTE correction published by Takeda & Honda (2016), we expect that the real abundance of O in HD 60431 is close to solar.

Ca is most prominent in the H and K Ca ii lines, from 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 noteworthy 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 have been 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 a separate modelling and 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 Sec. 3.3) on the derived abundances, we synthesised a spectrum of a star with the same physical parameters and a dipolar magnetic field of $B_{\mathrm{d}}$ = 1 kG, using the code SynthMag written by Kochukhov (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 accordance with the literature, HD 60431 is a highly peculiar late B-type CP2 star. Besides significant overabundances of Mg, Si, Fe, Cr, and Ti, we find a strong excess 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, 1975; Maitzen, 1976; Adelman, 1979; Kudryavtsev et al., 2006; Hümmerich et al., 2020). This flux depression is detected in the Geneva photometric system through the photometric parameters $\Delta(V1-G)$ (Hauck, 1978) and $Z$ (Cramer & Maeder, 1979). From the average Geneva colors of HD 60431 taken from Rufener (1989) and the definitions of these parameters, we obtain $\Delta(V1-G)=0.072$ and $Z=-0.092$. As normal stars have $\Delta(V1-G)\simeq-0.005$ and $Z\simeq 0.00$, 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 $T_{\mathrm{eff}}$, it achieves maximum strength at around 12 000 – 13 000 K, just at the effective temperature of HD 60431.

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 $V_{\mathrm{r}}$ 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 $V_{\mathrm{r}}$ 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 has been pointed 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 $\langle B_{z}\rangle=270\pm 150$ G.

Despite the fact that 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 that under the assumption of a dipolar field, HD 60431 has a weak magnetic field of the order of 1 kG on the pole, which is, however, not conclusive and needs to be verified by additional studies.

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, 2015), which contain a total of 8 911 measurements collected in 2019 and 2021 (sectors 7, 8, 34, and 35).

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); 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 $P=0\aas@@fstack{d}475\,518$ (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. 6).

While CP2 stars may show very different light curves at different wavelengths, it has been shown by Mikulášek et al. (2004); Mikulášek et al. (2008a) that similar light curve patterns in different passbands are commonly observed in these objects. Obviously, in these stars, only one dominant mechanism is responsible for the redistribution of the flux in the spectrum. All the light curves $F(\lambda,\varphi)$ of HD 60431 can be satisfactorily approximated by the simple model:

where $A$($\lambda$) is the effective semiamplitude of the light curve in the individual photometric passbands, $B(\varphi,\@vec{\alpha})$ is the normalised periodic function of the phase function $\vartheta$. In this case, we define $B(\varphi,\@vec{\alpha})$ as a linear combination of nine simple mutually orthonormal periodic functions, of which five are symmetric and four antisymmetric functions to the phase $\varphi=0$, which describes the eight independent parameters forming the alpha vector $\@vec{\alpha}$. More details are given in Appendix A.

The phase function $\vartheta(t)=E+\varphi$ ($E$ is the epoch and $\varphi$ the common phase) and its inverse function $\mathit{\Theta}(\vartheta)$ 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 $\mathit{\Theta}(\vartheta)$, we can calculate the moments of the zero phases JD${}_{\mathrm{min}}(E)=T(E)$, where $E$ is the epoch.

Since the first period determination by North et al. (1988), HD 60431 has been considered as 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 $P=0\aas@@fstack{d}46566$ 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 twice longer rotation period could not be excluded on the basis of the available data, and called for additional observations to further investigate this matter.

HD 98000 (TIC 82196009) was observed in Sector 10 by TESS for a duration of $\sim$26 days. The ultra-precise TESS data unambiguously show that, in contrast to the findings from the ground-based observations, HD 98000 indeed shows a double-wave light curve (Fig. 9). Obviously, therefore, the period solution of Hümmerich et al. (2017) represents only half the true value. From an analysis of TESS data, we derive a rotational period of $P=0\aas@@fstack{d}93130(3)$. HD 60431, therefore, remains the CP2 star with the shortest known rotational period.

We currently know of nine magnetic chemically peculiar stars, including HD 60431, that have shown changes of their rotational periods (Mikulášek et al., 2021). These objects form a very heterogeneous group, with some members showing increasing or decreasing periods as well as stars with more complex period behaviour. With an observed value of $\dot{P}=2.36(19)\times 10^{-10}$, HD 60431 exhibits the smallest rate of period change among these objects. At the same time, however, the uncertainty with which $\dot{P}$ is known ($\delta\dot{P}=1.9\times 10^{-11}=8\times 10^{-7}$ 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, $\sigma$ Ori E and CU Vir (see Pyper & Adelman, 2020; Mikulášek, 2016). The small uncertainty is due to the very short rotation period of $P=0\aas@@fstack{d}475\,5$, the time span of the observations, the relatively large amplitude of the light curve that exhibits a well defined narrow minimum (see Fig. 6), and, finally, the quality of the employed photometry. Such small value of $\dot{P}$ would be hardly 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 of the rotational period by mere conservation of angular momentum).

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 $\mathit{\Omega}$. This isobar area is an axisymmetric spheroid characterised by the equatorial and polar radii $R_{\mathrm{eq}}$, $R_{\mathrm{p}}$ and flattening $f$, $f=R_{\mathrm{eq}}/R_{\mathrm{p}}$. After some algebra, we obtain the basic relation between the radii from Eqs. (10) and (12) of Zahn et al. (2010), as follows:

where $G$ is the gravitational constant and $\mathit{\Omega}_{\mathrm{crit}}$ the maximum angular velocity $(\mathit{\Omega}\leq\mathit{\Omega}_{\mathrm{crit}})$, beyond which the stellar body is disrupted. $V_{\mathrm{eq}}$ is the equatorial velocity, $V_{\mathrm{eq}}=R_{\mathrm{eq}}\,\mathit{\Omega}$, while $V_{\mathrm{crit}}$ is the maximum allowed equatorial velocity ($V_{\mathrm{eq}}\leq V_{\mathrm{crit}}$). $(a_{\mathrm{cfg}}/g)_{\mathrm{eq}}$ is the ratio of the centrifugal and gravitational accelerations at the equator. It follows from Eq. (14) that the maximum flattening $f_{\rm{crit}}$ of a rotating body is $f_{\rm{crit}}=\textstyle{\frac{3}{2}}$, because $a_{\mathrm{cfg}}=g$ in this case. Critical quantities for the given mass $M$ and polar radius $R_{\mathrm{p}}$ are as follows (with $\rho\equiv R_{\mathrm{eq}}^{\mathrm{crit}}$):

Applying relations (14) and (15) to the case of HD 60431, we assume a mass of $M=3.1\,\mathrm{M}_{\odot}$, $\mathit{\Omega}=1.5294\times 10^{-4}$ s^-1, and $R_{\mathrm{p}}\doteq R=1.97\,\mathrm{R}_{\odot}$. After several iterations of these semianalytic relations between $R_{\rm{eq}}$ and $R_{\rm{p}}$ (Ekström et al., 2008), we estimate the equatorial radius at $R_{\mathrm{eq}}=2.16\,\mathrm{R}_{\odot}$. Bearing in mind that the rotational period of HD 60431 is the smallest among all known CP2 stars, the derived flattening of $f=1.096$ appears surprisingly small at first glance. That is because the flattening $f$ and the ratio of the equatorial centrifugal to gravitational accelerations ($(a_{\mathrm{cfg}}/g)_{\mathrm{eq}}=0.19$ for HD 60431) of a rotating star critically depend on its radius ($\sim R^{3}$). As our target star is nearly on the ZAMS (cf. Section 2), its polar radius is the smallest possible for its mass.

The same conclusion follows from a comparison of the real equatorial velocity $V_{\mathrm{eq}}=50.6\,R_{\mathrm{eq}}/P_{1}=230\,\text{km}\,\text{s}^{-1}$ and the maximum allowed equatorial velocity $V_{\mathrm{crit}}=450\,\text{km}\,\text{s}^{-1}$.⁹⁹9It is worth noting that 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 $V_{\mathrm{crit}}=460\,\text{km}\,\text{s}^{-1}$. The true rotational velocity $V_{\mathrm{eq}}$ is about half the critical one.

We can also evaluate the quantity $R_{\mathrm{eq}}\sin i={\varv}\sin i\ P_{1}/50.6=1.79\pm 0.19\,\mathrm{R}_{\odot}$, determining the lower limit of the stellar equatorial radius $R_{\mathrm{eq}}$. We use the simplistic approximation $R_{\rm{eq}}\approx\langle R\rangle$ according to the relatively small ratio $f=1.096$. Assuming that $R_{\mathrm{eq}}=2.16\,\mathrm{R}_{\odot}$, we derive $\sin i=0.83(9)$ which indicates an inclination angle of $i=56^{\circ}\pm 9^{\circ}$.

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, 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 charaterized by decreasing amplitude towards longer wavelengths (see Figs. 5 and 6).

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 – 0.040 and 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.

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 $\mathrm{M}_{\odot}$, according to Ekström et al., 2012). Despite this short time, radiative diffusion has obviously efficiently established a peculiar atmospheric composition, in agreement with findings for other young stars (Romanyuk et al., 2020, 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, 2021b). 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, 2021b), thus $\mathit{\Omega}(t)=\mathit{\Omega}_{0}=$ const. As the equatorial radius $R_{\mathrm{eq}}(t)$ increases during the main-sequence evolution, the stellar equatorial velocity $V_{\rm{eq}}(t)=\mathit{\Omega}_{0}\,R_{\mathrm{eq}}(t)$ may approach the critical rotation limit $V_{\rm{crit}}$ (see Eq. 15), which results in the loss of the peculiar atmospheric abundance pattern.

The subsequent spinning up of stars during the main-sequence evolution could be at the root of why so few ultra short-period magnetic CP stars are observed.