11institutetext: Leibniz-Institut für Astrophysik Potsdam (AIP), An der
Sternwarte 16, 14482 Potsdam, Germany
11email<EMAIL_ADDRESS>22institutetext: European Southern Observatory,
Karl-Schwarzschild-Str. 2, 85748 Garching, Germany 33institutetext: Centre for
Space Physics, North-West University, Mahikeng 2735, South Africa
44institutetext: Jeremiah Horrocks Institute, University of Central
Lancashire, Preston PR1 2HE, UK 55institutetext: European Southern
Observatory, Alonso de Cordova 3107, Vitacura, Santiago, Chile
66institutetext: NASA Ames Research Center, Moffett Field, CA 94035, USA

# Testing pulsation diagnostics in the rapidly oscillating magnetic Ap star
$\upgamma$ Equ using near-infrared CRIRES$+$ observations

S. P. Järvinen 11 S. Hubrig 11 B. Wolff 22 D. W. Kurtz 3344 G. Mathys 55 S. D.
Chojnowski 66 M. Schöller 22 I. Ilyin 11

(Received 30 December 2023 / Accepted 16 January 2024)

###### Abstract

Context.  Pulsations of rapidly oscillating Ap stars and their interaction
with the stellar magnetic field have not been studied in the near-infrared
(near-IR) region despite the benefits these observations offer compared to
visual wavelengths. The main advantage of the near-IR is the quadratic
dependence of the Zeeman effect on the wavelength, as opposed to the linear
dependence of the Doppler effect.

Aims.  To test pulsation diagnostics of roAp stars in the near-IR, we aim to
investigate the pulsation behaviour of one of the brightest magnetic roAp
stars, $\upgamma$ Equ, which possesses a strong surface magnetic field of the
order of several kilogauss and exhibits magnetically split spectral lines in
its spectra.

Methods.  Two magnetically split spectral lines belonging to different
elements, the triplet Fe i at 1563.63 nm and the pseudo-doublet Ce iii at
1629.2 nm, were recorded with CRIRES$+$ over about one hour in the $H$ band
with the aim of understanding the character of the line profile variability
and the pulsation behaviour of the magnetic field modulus.

Results.  The profile shapes of both studied magnetically split spectral lines
vary in a rather complex manner probably due to a significant decrease in the
strength of the longitudinal field component and an increase in the strength
of the transverse field components over the last decade. A mean magnetic field
modulus of 3.9 kG was determined for the Zeeman triplet Fe i at 1563.63 nm,
whereas for the pseudo-doublet Ce iii at 1629.2 nm we observe a much lower
value of only about 2.9 kG. For comparison, a mean field modulus of 3.4 kG was
determined using the Zeeman doublet Fe ii at 6249.25 in optical PEPSI spectra
recorded just about two weeks before the CRIRES$+$ observations. Different
effects that may lead to the differences in the field modulus values are
discussed. Our measurements of the mean magnetic field modulus using the line
profiles recorded in different pulsational phase bins suggest a field modulus
variability of 32 G for the Zeeman triplet Fe i at 1563.63 nm and 102 G for
the pseudo-doublet Ce iii at 1629.2 nm.

###### Key Words.:

Stars: individual: $\upgamma$ Equ – Stars: magnetic field – Techniques:
spectroscopic

##  1 Introduction

The rapidly oscillating Ap (roAp) main-sequence stars belong to a sub-group of
H-core-burning SrCrEu peculiar A stars with $T_{\rm eff}$ in the range of
about 6600 K to 8500 K. They are usually strongly magnetic and pulsate in
high-overtone, low-degree, non-radial p-modes with periods in the range of
about 5 to 24 min. They have been observed intensively photometrically since
their discovery by Kurtz (1978), who detected a 12 min pulsation period in
Przybylski’s star (HD 101065) using ground-based photometric observations.
Frequency analyses of the light curves of the roAp stars have yielded rich
asteroseismic information on the degrees of the pulsation modes, the
distortion of the modes from normal modes, the magnetic geometries, and the
interaction of the pulsation with the magnetic field. Whereas several
thousands of Ap stars are currently known (e.g. Renson & Manfroid 2009), the
number of identified roAp stars is just over 100 (e.g. Smalley et al. 2015;
Holdsworth et al. 2021, 2024).

The main property of roAp stars is that they are oblique pulsators, and the
strength and geometry of their global magnetic fields constrain the pulsation
modes. Kurtz (1982) suggested that the pulsation mode axis is aligned with the
magnetic axis, which is itself inclined to the rotation axis so that the
observer sees the pulsation modes from an aspect that varies with rotation.
The roAp stars show not only photometric variability, but also rapid radial
velocity (RV) variations with the same mode frequencies as those obtained from
photometry. The detected pulsation amplitudes of radial velocities depend on
the element used: the lines of rare earth elements (REEs) and the H$\alpha$
core show the highest amplitudes, whereas iron-peak elements with the higher
formation depth in the atmosphere usually show low RV variations (e.g. Kurtz
et al. 2003, and references therein). This is an indication that the pulsation
amplitudes are a function of atmospheric depth.

Assuming that the pulsations of roAp stars are governed by their magnetic
fields, several authors obtained spectral time series in circular polarised
light in a visual spectral region to study magnetic field variations over the
pulsation cycle in a number of stars. Hubrig et al. (2004) measured the
magnetic field variability over the pulsation cycles in six roAp stars using
low-resolution FOcal Reducer low dispersion Spectrograph (FORS 1; Appenzeller
et al. 1998) spectral time series; however, only one roAp star, Przybylski’s
star (HD 101065), showed a signal for magnetic variability with a frequency of
1.365 mHz and an amplitude of $39\pm 12$ G. Leone & Kurtz (2003) used high-
resolution circular polarisation observations of individual lines in the
spectra of the ultra-slowly rotating roAp star $\upgamma$ Equ (HD 201601) and
reported a clear detection of the mean longitudinal field variability of up to
$240\pm 37$ G in the Nd iii line at 5845.07 Å over the pulsation cycle. This
discovery was, however, questioned by Kochukhov et al. (2004), who obtained an
upper limit of 40–60 G using 13 Nd iii lines.

Mathys et al. (2005, 2007) studied the pulsational behaviour of the individual
$\uppi$ and $\upsigma$ components using the Eu ii line at 6437 Å in the
spectroscopic time series of the strongly magnetic roAp star HD 166473, which
exhibits radial velocity variations due to pulsation with three frequencies:
1.833, 1.886, and 1.928 mHz (Kurtz et al. 2003). The results of these studies
hinted at the occurrence of variations of the mean magnetic field modulus with
the pulsation frequency 1.928 mHz and an amplitude of $21\pm 5$ G.

Importantly, more recent studies of the oblique pulsator model showed that the
mode geometries in roAp stars are complex and should be considered as a
function of atmospheric height. Using high-resolution time-series
spectroscopy, it became possible to identify the pulsation geometry not only
over the stellar surface but also vertically in the stellar atmosphere. As an
example, high-speed spectroscopy of the roAp star HD 99563 to study the
pulsation amplitude and phase behaviour of elements in its stratified
atmosphere over the 2.91-d rotation cycle was carried out by Freyhammer et al.
(2009). The authors identified spectral features related to patches in the
surface distribution of chemical elements and studied the pulsation amplitudes
and phases as the patches moved across the stellar disc. The detected
variability had been consistent with a distorted non-radial-dipole pulsation
mode.

Hubrig et al. (2021) took a step further in the analysis of pulsations of roAp
stars and investigated the usefulness of linear polarisation as an enhanced
pulsation diagnostic. While the circular polarisation depends upon the
magnetic vector projection on to the line of sight, the linear polarisation
depends upon the magnetic vector projected on to the plane perpendicular to
the line of sight. The authors reported on a possible detection of pulsational
variability of the transversal field component in Fe, Nd, and Eu lines. The
detected variability had been considered as due to the impact of pulsations on
the transverse magnetic field, causing changes in the obliquity angles of
magnetic force lines.

To better understand the interaction between the pulsation and the stellar
magnetic field and to test the diagnostic potential of near-IR observations,
we recently carried out high-resolution spectroscopical time series of one of
the brightest roAp stars, $\upgamma$ Equ ($m_{v}=4.7$), over about one hour in
the $H$ band using the upgraded CRyogenic InfraRed Echelle Spectrograph
(CRIRES$+$; Dorn et al. 2023). As of today, pulsations of roAp stars and their
interaction with the stellar magnetic field have not been studied in the near-
IR region despite the numerous advantages such observations offer compared to
visual wavelengths. The main advantage of the near-IR is the quadratic
wavelength dependence of the Zeeman effect, as opposed to the linear
dependence of the Doppler effect. This opens the possibility, at a given
magnetic field strength, of observing resolved lines in faster rotating stars
(e.g. Chojnowski et al. 2019) or, conversely, at a given projected equatorial
velocity, of resolving Zeeman split comments in more weakly magnetic stars.
Furthermore, performing a near-IR study permits the exploration of a different
range of photospheric depths, thus providing more insight into the depth
dependence of pulsation. As the detection of the variation of the magnetic
field modulus over the pulsation period is yet to be confirmed by independent
observations, high-resolution spectroscopic time series obtained in the near-
IR allowing the analysis of the pulsational behaviour of the individual
$\uppi$ and $\upsigma$ components is especially important.

$\upgamma$ Equ is distinguished among the roAp stars by very sharp lines
caused by the extremely slow rotation, a rather strong mean longitudinal
magnetic field varying from about 630 G to $-1180$ G (e.g. Savanov et al.
2018), and a magnetic field modulus of 4 kG measured using magnetically split
spectral lines resolved in the visible (e.g. Mathys 2017). The most recent
estimate of the rotation period for this star gives a lower limit of 95 yr
(Bychkov et al. 2016; Savanov et al. 2018). Furthermore, $\upgamma$ Equ
exhibits a rather high amplitude of pulsational line profile radial velocities
in the visual exceeding 1000 m s-1 in individual REE spectral lines and has
well-studied pulsation characteristics, with the highest amplitude frequency
corresponding to a period of about 12 minutes.

In the following we discuss the available observational material and the
analysis of the pulsational behaviour of the $\uppi$ and $\upsigma$ components
observed in the selected magnetically split spectral lines in the high-
resolution spectroscopic CRIRES$+$ time series, and we discuss the potential
usefulness of multi-wavelength observations for the improvement of our
understanding of the pulsational properties of roAp stars.

##  2 Spectroscopic and photometric observations

To study the pulsation and its interaction with the magnetic field in the roAp
star $\upgamma$ Equ, we obtained, on 2022 September 28, high-resolution
spectroscopic CRIRES$+$ time series over 56 min in the $H$ band (from 1438 to
1765 nm), sampling several lines belonging to iron-peak and rare-earth
elements. The line identification in the $H$ band was previously carried out
by Chojnowski et al. (2019). Using the nodding cycle ABBA, we obtained 35
spectra with the narrowest available slit of 0.2 ″, resulting in a spectral
resolution of about 100 000. Within the ABBA nodding cadence, the first
exposure is recorded at the position A; then, it is followed by two exposures
at the position B, and the fourth exposure is obtained moving back to the
position
A111https://www.eso.org/sci/facilities/paranal/instruments/crires/doc/CRIRES_User_Manual_P113.1.pdf.
The purpose of nodding -moving the telescope to the positions A and B along
the direction of the slit- is to remove sky emission, detector dark current
and glow, and some ghosts. With the exposure time of individual frames of 10 s
between the nodding and the full individual sequence, ABBA accounted for about
1.5 min. The data were reduced using the CR2RES pipeline recipes. As both ABBA
and AB/BA nodding cycles are available, the reduction was carried out for both
cycles. For the AB/BA nodding cycles, we obtained 70 spectra with the
individual AB/BA cycle length of about 50 s. The typical signal-to-noise ratio
($S/N$) for the spectra obtained with the ABBA cycle is 212, whereas the $S/N$
for the individual AB/BA cycles is 153.

Figure 1:  Zeeman doublet Fe ii at 6149.25 Å with a Landé factor $g_{\rm
eff}=1.35$ used for measuring the mean magnetic field modulus in observations
obtained in different years using different instruments. The corresponding
Zeeman pattern is displayed above the line profiles, with the $\uppi$
components shown above the horizontal line and the $\upsigma$ components
below. The lengths of the vertical bars are proportional to their relative
strength.

Previous studies of Ap stars revealed that their magnetic fields cover the
whole stellar surface and have a large-scale structure often resembling a
single dipole whose axis is inclined with respect to the stellar rotation
axis. Because the measured magnetic field strength depends on the viewing
angle of the observer, that is, on the rotation phase of the star, it is
important for our analysis that we know the strength of the mean longitudinal
magnetic field and the mean magnetic field modulus at the time of
observations. According to Hubrig et al. (2021), the mean longitudinal
magnetic field was gradually decreasing from $-$922 G in 2011 to $-$572 G in
2020, and the mean magnetic field modulus was gradually decreasing from 3.80
kG in 2011 to 3.58 kG in 2017. Our most recent
observations222https://doi.org/10.17876/data/2024_1 of $\upgamma$ Equ with the
Potsdam Echelle Polarimetric and Spectroscopic Instrument (PEPSI; Strassmeier
et al. 2015) installed at the $2\times 8.4$ m Large Binocular Telescope (LBT)
were obtained on 2022 September 15, just thirteen days before the CRIRES$+$
observations. They show an even lower field modulus of 3.36 kG, confirming the
decreasing trend for the field strength. In Fig. 1, we present magnetically
split components of the Zeeman doublet Fe ii at 6149.25 Å used for the
measurements of the mean magnetic field modulus in different years using
different instruments. The PEPSI observations were made using the cross-
dispersers CD II, covering $4236-4770$ Å, and CD IV, covering $5391-6289$ Å
with a spectral resolution of $R\sim 130\,000$ corresponding to 0.06 Å at 7600
Å. For more details on observations with PEPSI and data reduction, we refer
the reader to Strassmeier et al. (2023) and Hubrig et al. (2021).

Since mean longitudinal magnetic field measurements of $\upgamma$ Equ after
2020 are not available in the literature, we relied on the work of Savanov et
al. (2018), for example, who discussed the variability of the longitudinal
field phase curve using a periodogram analysis. According to this work, the
strength of the field had to be in the range of $-$200 – $-$300 G at the time
of the CRIRES+ observations.

Figure 2:  Frequency spectrum of $\upgamma$ Equ based on Sector 55 TESS
photometry.

To be able to study the magnetic field variability over the pulsational cycle,
we need to assign the pulsational phase to each of our spectra. Because
$\upgamma$ Equ is known to have multiple pulsation periods near 12 min (Kurtz
1983; Libbrecht 1988; Martinez et al. 1996; Gruberbauer et al. 2008), we used
more recent Transiting Exoplanet Survey Satellite (_TESS_ ; Ricker et al.
2015) observations to identify the frequency with the highest amplitude at a
time closest to the spectroscopic observations.

$\upgamma$ Equ was observed by _TESS_ in Sector 55 from 2022 August 5 to 2022
September 1. The analysis of these data is illustrated in Fig. 2, showing
three main frequencies. The highest peak with an amplitude of 0.315 mmag is at
$115.5654\pm 0.0001$ d-1 (1.3386 mHz; or $P=747.0404\,{\rm s}=12.45067$ min).
The other two peaks with amplitudes of 0.127 mmag and 0.042 mmag are at
118.0236 d-1 (1.3660 mHz) and 113.2873 d-1 (1.3112 mHz), respectively.

These three frequencies are in complete agreement with the frequencies derived
independently by Holdsworth et al. (2024). As those authors noted, we also see
the harmonic of the highest amplitude peak and one combination frequency.
However, it is notable that the frequencies previously derived for $\upgamma$
Equ in other studies (Kurtz 1983; Libbrecht 1988; Martinez et al. 1996;
Gruberbauer et al. 2008) range from $113-123$ d-1 and in general do not
coincide with the frequencies derived from the TESS data. Possible
explanations are that the star shows mode changes, that different modes are
detected at different atmospheric depths (since the studies were both
spectroscopic and photometric with different filters), or that there have been
problems with aliasing in the ground-based data, which present significant
time gaps. Importantly, the frequencies derived from the TESS data do not
suffer significant aliasing, and the pulsation amplitude is stable over the
27-d time span of the Sector 55 data. Since the spectroscopic data were taken
only a month after the TESS data, we assume that this mode stability continued
over this short time span. Because of the possibility of some pulsation
amplitude change, however, we cannot directly relate the photometric pulsation
amplitude to the amplitude of the magnetic variations over the pulsation
cycle. That requires simultaneous observations.

The frequency separations of the three peaks seen in Fig. 2 are 27 $\upmu$Hz,
which is plausibly half the large separation; hence, we suggest that the
observed modes are $\ell=2,1,2$ alternating-degree modes. We note that the
modulation timescale for the outer mode frequencies to beat with the main mode
frequency is about 10 hr; hence, these frequencies are not resolved in the 56
min of the CRIRES$+$ time series presented in this paper.

##  3 Analysis of CRIRES$+$ spectroscopic time series

In the recorded CRIRES$+$ spectroscopic time series, two almost blend-free
magnetically split lines, the Zeeman triplet Fe i at 1563.63 nm and the
pseudo-doublet line Ce iii at 1629.20 nm, appear to be most suitable to study
the pulsational variability of the mean magnetic field modulus. Their
identification is based on the work of Chojnowski et al. (2019). Zeeman
patterns for both lines and the corresponding effective Landé factors are
presented in Fig. 3.

Figure 3:  Zeeman patterns of Fe i triplet and Ce iii pseudo-doublet with
corresponding Landé factors $g_{\rm eff}$. Each Zeeman component is
represented by a vertical bar whose length is proportional to its relative
strength. The $\uppi$ components are presented above the horizontal wavelength
axis, and the $\upsigma_{\pm}$ components are shown below it. The unit length
of the wavelength axis is $\Delta\lambda_{B},$ defined as the wavelength shift
from the line centre of the $\upsigma$ components in a normal Zeeman triplet
with a Landé factor of $g_{\rm eff}$=1.  Figure 4:  Mean Ce iii 1629.2 nm line
profiles in their original shape (left) and corrected for telluric
contributions (right), calculated for both sequences ABBA and AB/BA.

While the spectral region around the Fe i 1563.63 nm line is perfectly clean
and is not affected by telluric lines, the Ce iii 1629.2 nm line shows a small
telluric contribution on the red side of the line profile. It was removed
using the ESO SkyCalc based on the Cerro Paranal Sky Model (Noll et al. 2012;
Jones et al. 2013). The original and the mean Ce iii 1629.2 nm corrected for
the telluric contribution profiles for both sequences ABBA and AB/BA are
presented in Fig. 4.

Figure 5:  Individual line profiles for Fe i 1563.63 nm (left) and for Ce iii
1629.2 nm (right) after telluric correction. The time increases from bottom to
top.  Figure 6:  Comparison of line profiles of Zeeman triplet Fe i 1563.63 nm
between 2011 May, 2011 June, 2021 October, and 2022 September. The two spectra
from 2011 were recorded with CRIRES, whereas for the observations in 2021 and
2022 CRIRES$+$ was used. The overplotted profiles are presented at the top.
Figure 7:  Line profiles of two Zeeman triplets and one pseudo-triplet
observed with PEPSI in the visual spectral region in 2020 and 2022. The shape
of the line profiles and the separation between the $\uppi$ and $\upsigma$
components are completely different compared to the appearance of the Zeeman
triplet Fe i 1563.63 nm observed on 2022 September 28 with CRIRES$+$. Zeeman
patterns for the spectral lines are presented at the top of each panel.

Individual line profiles’ time series obtained over 56 min using the ABBA
sequence are presented in Fig. 5. Surprisingly, in contrast to the high-
resolution CRIRES (Kaeufl et al. 2003) and CRIRES$+$ observations of the Fe i
1563.63 nm line obtained in previous years, the line profile shape recorded in
our observations in 2022 shows a significant change, displaying rather flat
$\upsigma$ components and not very clear separation between them and the
$\uppi$ component. While the presence of the $\upsigma$ component on the red
side of the line profile can still be perceived, the position of the blue-side
$\upsigma$ component is not clearly defined. In Fig. 6 we present very
different shapes of the line profiles of the Zeeman triplet Fe i 1563.63 nm
observed in different years using CRIRES and CRIRES$+$. Observations on 2011
May 19 and 2011 June 16 were obtained with an exposure time of 450 s. For the
observations on 2021 Oct 29, four frames with 20 s of exposure time each were
obtained using one nodding cycle. It is possible that the ‘filling’ between
the $\uppi$ and $\upsigma$ components that appeared in 2021–2022, while not
present in 2011, is due to blending with a line of another element whose
wavelength coincides with that of the Fe i 1563.63 nm line. The $\sim$10 years
elapsed from 2011 to 2021 may represent about one tenth of the rotation
period, which may be enough for an overabundance spot of some element to have
appeared on the visible stellar hemisphere as a result of stellar rotation.
However, the measured equivalent width of this line is the same in 2021–-2022
as it was in 2011.

As discussed in Sect. 2, the magnetic field strength in $\upgamma$ Equ is
gradually decreasing and will probably approach the rotation phase of the best
visibility of the magnetic equator in a few years, where the longitudinal
magnetic field component is expected to be weaker and the transverse field
component becomes the strongest. For a transverse field, the $\uppi$
components become, to the first order, twice as strong as the $\upsigma_{+}$
and the $\upsigma_{-}$ components. Interestingly, in contrast with the
observations in the near-IR, the profiles of the Zeeman triplet lines recorded
by us in the visual wavelength region in 2020 and 2022 using PEPSI show much
clearer $\upsigma$ components, which can be more easily used for the
measurement of the magnetic field modulus. The different appearance of the
triplets can probably be explained by different iron vertical stratification
between the visual and near-IR regions. There may also be a significant
difference in the optical depth from which the continuum is issued between the
visible and the IR, and, as a consequence, there could be different
(de)saturations of the split line components between the two wavelength
regions. This alone may potentially result in differently looking triplets. In
Fig. 7, we present line profiles of two Zeeman triplets and one Zeeman pseudo-
triplet observed with PEPSI in the visual spectral region.

Figure 8:  Phase-binned mean line profiles of Fe i line (left) and Ce iii line
(right) calculated using the ABBA sequence. For each bin, profiles are plotted
with different colours and shifted upward for better visibility. The number of
spectra within each bin is indicated close to each profile. At the top, we
show the overplotted profiles and the corresponding error bars.

As shown in Fig. 5, several absorption bumps appear from time to time not only
on both sides of the $\uppi$ component in the line profile of Fe i 1563.63 nm,
but also in the core of the Ce iii 1629.2 nm line. Since the $S/N$ of the
individual spectrum is rather low, it is not clear whether the observed
variability is intrinsic or is caused by the noise. To investigate the nature
of these bumps in more detail, we determined, for each individual spectrum
calculated in the ABBA sequence, the corresponding pulsation phase assuming
the pulsational period of 12.45067 min as determined from the TESS
observations. To increase the $S/N$, the spectra have been binned with the
phase bin of 0.1. The number of individual spectra in each bin varies from two
to five, allowing us to increase the $S/N$ by a factor of 1.4 to 2.2,
respectively. The plots for the mean spectra obtained within each bin are
displayed in Fig. 8 together with the information on the number of spectra
included in each bin. While no radial velocity variability is detected in the
overplotted line profiles, we observe that the depth of the $\uppi$ component
of the Fe i 1563.63 nm line is slightly variable; moreover, the fractions of
the profile around the $\uppi$ component show variable absorption bumps. A
comparison of the variability of the profiles that have the largest numbers of
individual spectra per bin and therefore a higher $S/N$ (phases 0.0–0.1,
0.1–0.2, 0.2–0.3, 0.3–0.4, 0.7–0.8) shows that absorption bumps become
stronger in the red part of the profile in the phase bins 0.0–0.1, 0.2–0.3,
and 0.7–0.8. In the case of the Ce iii line, the separation between the split
components becomes much more clear in several phase bins, the clearest case
being the phase bin 0.4–0.5.

Figure 9:  Dynamical spectrum of whole Fe i 1563.63 nm line. The bluest colour
is for the deepest absorption at the $\uppi$ component, whereas red is at the
continuum level. _Left:_ Profiles reduced using the ABBA sequence. _Right:_
Profiles reduced using the AB/BA sequence.

Figure 10:  As Fig. 9, but $\uppi$ component has been masked out in order to
show the $\upsigma$ components better. _Left:_ Profiles reduced using ABBA
sequence. _Right:_ Profiles reduced using AB/BA sequence.

Figure 11:  Dynamical spectrum of line core of Ce iii 1629.2 nm. The bluest
colour is for the deepest absorption, whereas red is for the shallower
absorption. Black is used for phases not covered by the observations. _Left:_
Profiles reduced using ABBA sequence. _Right:_ Profiles reduced using AB/BA
sequence.

To delve deeper into the character of the line profile variability, in Figs.
9–11 we present dynamical spectra for the observed profiles of Fe i 1563.63 nm
and the Ce iii 1629.2 nm lines phased with the pulsation period and using both
the ABBA and the AB/BA sequences. The blue-green colour in Fig. 9 constructed
for the Fe i 1563.63 nm line clearly shows the variability of the $\upsigma$
component on the red side; it changes intensity and becomes significantly
broader in some phases. The $\upsigma$ component on the blue side almost
disappears in some phases, that is, between the phases 0.0–0.3. This behaviour
is confirmed in the dynamical spectra presented in Fig. 10, where we mask out
the $\uppi$ component to achieve a better contrast. Interestingly, absorption
bumps discovered in the flat blue part of the profile appear shifted and
stronger close to the pulsation phase 0.20, whereas the absorption bump on the
red side becomes strongest in the pulsation phase around 0.7–0.8. In the plot
corresponding to the AB/BA sequence, a sort of arc appears in the phases
0.80–0.90. The inspection of the dynamical spectra in Fig. 11 calculated for
the Ce iii 1629.2 nm line shows that this line is also variable where the
depth of the right component of the split profile almost disappears in the
pulsation phases between 0.25–0.30 and 0.85–0.90.

##  4 Discussion

Figure 12:  Zeeman triplet Fe i 1563.63 nm observed in phases 0.3–0.4 and
0.7–0.8. The black lines present the observed profiles and the red lines are
used to show the triple-Gaussian fits.

Figure 13:  Zeeman pseudo-doublet Ce iii 1629.2 nm observed in phases 0.3–0.4
and 0.5–0.6. The black lines present the observed profiles and the red lines
are used to show the double-Gaussian fits.

Our study of the pulsation behaviour of the magnetic roAp star $\upgamma$ Equ
in the near-IR using CRIRES$+$ time series over 56 min reveals a distinct
variability of the magnetically split lines Fe i 1563.63 nm and Ce iii 1629.2
nm. The character of this variability appears, however, rather complex,
probably due to a significant decrease in the strength of the longitudinal
field component and an increase in the strength of the transverse field
component over the last decade. The reason for the observed absorption bumps
around the $\uppi$ component in the Zeeman triplet Fe i 1563.63 nm is not
clear. We simply speculate that they are possibly related to the near-surface
convection in the presence of the stratification of iron.

In contrast to the older high-resolution CRIRES (Kaeufl et al. 2003)
observations, the line profile shape of the Zeeman triplet Fe i 1563.63 nm
recorded in our observations in 2022 displays rather flat $\upsigma$
components with unclear separation between them and the $\uppi$ component.
Assuming that the outer absorption dips on both sides of the $\uppi$ component
in the line profile of the Zeeman triplet Fe i 1563.63 nm present the
$\upsigma$ components, we tried to measure the mean magnetic field modulus
using the line profiles recorded in the pulsational phase bins 0.3–0.4 and
0.7–0.8. The results of our measurements with a triple-Gaussian fit are
presented in Fig. 12. Interestingly, the measured modulus strengths of 3.929
kG in the phase bin 0.3–0.4 and 3.897 kG in the phase bin 0.7–0.8 are not
identical, implying that the mean magnetic field modulus probably varies over
the pulsation period with an amplitude of $32\pm 7$ G. On the other hand, the
much stronger magnetic field modulus measured in the optical region
contradicts the one reported for the IR (see Sect. 2 and Fig. 1). We measured
a much lower field modulus of 3.36 kG on 2022 September 15 using the Zeeman
doublet Fe ii at 6149.25 Å. Such a difference is not unexpected, for various
reasons. First, the mean magnetic field modulus is the average of the field
modulus over the visible stellar hemisphere, averaged by the local emergent
line intensity. Because the limb darkening is not the same in the visible and
in the IR, the line intensity distribution over the stellar disk is different
between the two wavelength regions, which leads to a different weighting of
the contributions of the various parts of the stellar surface in the magnetic
field averaging process. Furthermore, the continuum optical depth of the line-
forming layer is not the same in the IR as in the visible. Accordingly, the
saturation degree of the individual $\uppi$ and $\upsigma$ components of lines
with anomalous Zeeman patterns may be different, which affects the
interpretation of the observed line profiles in terms of magnetic field
strength. Finally, while vertical gradients of the magnetic field strength may
not be required to justify the difference in the field modulus values derived
from the consideration of diagnostic lines in different wavelength ranges,
their existence and their potential effect on the magnetic field measurements
are plausible.

Much lower magnetic field modulus values are determined using the Zeeman
pseudo-doublet Ce iii 1629.2 nm. In the 0.3–0.4 phase bin, we measure 2.954
kG, whereas the value 2.852 kG is determined for the spectra in the 0.5–0.6
phase bin. These results presented in Fig. 13 imply that the mean magnetic
field modulus probably varies over the pulsation period with the amplitude of
$102\pm 14$ G. Since iron is reported to be gravitationally settled in the
stellar atmospheres of roAp stars compared to the rare earth elements that are
usually concentrated in the upper atmospheric layers (e.g. Nesvacil et al.
2013), it is expected that the lines of different elements trace the magnetic
field strength and the pulsation amplitudes differently. However, because of
the extremely abnormal chemical composition of the atmospheres of magnetic
roAp stars, the calculation of the vertical element abundance distribution is
a very complex process requiring the fitting of spectroscopic, photometric,
and magnetic data using realistic self-consistent atmospheric models. There is
also a number of previous studies indicating the possible presence of vertical
magnetic field gradients in roAp stars (e.g. Nesvacil et al. 2004; Hubrig et
al. 2018), with the mean magnetic field and longitudinal field strengths
changing with atmospheric depth. The significant increase of the magnetic
field in the deeper atmospheric layers would explain the difference in our
measurements of the magnetic field modulus using the iron Zeeman triplet and
the Ce Zeeman pseudo-doublet.

Even though the combination of a vertical gradient of the magnetic field and
of stratification can plausibly contribute to the observed difference between
the magnetic field strengths derived from consideration of the Fe i and Ce iii
lines, there is also another effect that must definitely play a role. Because
the Zeeman pattern of the Fe i line at 1563.63 nm is a pure triplet, the
determination of the mean magnetic field modulus from the separation of the
split components is almost approximation-free. In particular, it does not
depend on radiative transfer effects. By contrast, to determine $\langle
B\rangle$ from the splitting of the line Ce iii 1629.2 nm, whose Zeeman
pattern is a pseudo-doublet, one must use the weak-line approximation. By
doing so, one ignores saturation effects. The outer $\uppi$ components have a
much higher relative strength than their inner counterparts. Therefore, the
outer $\uppi$ components will saturate first. As a result of radiative
transfer, the ratio of the depths of the inner-to-outer $\uppi$ components
will become greater than the ratio of their strengths. As a result, the
wavelength separation of the blue and red line components will be less than it
would be for the Fe i line, for which radiative transfer effects are
negligible. Thus, a smaller value of the mean magnetic field modulus will be
derived. The effect is similar to what we observe. One way to achieve a better
assessment of the importance of this effect (which definitely must be present)
would be to observe a diagnostic line with a pseudo-doublet (or pseudo-
triplet) pattern for which the radiative transfer effects should lead to an
increase in the derived value of $\langle B\rangle$. Unfortunately, no
suitable line in the wavelength range covered by our spectra was observed.

On the other hand, another effect that may also lead to differences in the
values of $\langle B\rangle$ derived from lines of different ions are
horizontal inhomogeneities, which may lead to the different sampling of the
local magnetic field strength. Currently, we do not have enough information to
assess the significance of this effect, but its potential contribution is
still worth mentioning.

The results of our observations suggest that near-IR observations present an
important diagnostic potential in the studies of pulsations in magnetic roAp
stars using spectral lines with different formation heights. Future near-IR
studies of roAp stars with strong mean longitudinal magnetic fields (observed
close to the best visibility of their magnetic poles) in the near-IR using
long, high-resolution spectroscopic time series and combined with detailed
modelling will be worthwhile in the improvement of our understanding of the
concerned physics.

###### Acknowledgements.

We thank the anonymous referee for their useful comments. This work is based
on observations made with ESO telescopes at the La Silla Paranal Observatory
under programme IDs 087.C-0124(A), 0108.D-0659(A), and 0109.D-0659(A) publicly
available via ESO Archive. This work is partially based on observations
carried out with the PEPSI spectropolarimeter. PEPSI was made possible by
funding through the State of Brandenburg (MWFK) and the German Federal
Ministry of Education and Research (BMBF) through their Verbundforschung
grants 05AL2BA1/3 and 05A08BAC. LBT Corporation partners are the University of
Arizona on behalf of the Arizona university system; Istituto Nazionale di
Astrofisica, Italy; LBT Beteiligungsgesellschaft, Germany, representing the
Max-Planck Society, the Leibniz-Institute for Astrophysics Potsdam (AIP), and
Heidelberg University; the Ohio State University; and the Research
Corporation, on behalf of the University of Notre Dame, the University of
Minnesota, and the University of Virginia. This paper includes data collected
by the TESS mission. Funding for the TESS mission is provided by the NASA
Science Mission Directorate.

## References

  * Appenzeller et al. (1998) Appenzeller, I., Fricke, K., Fürtig, W., et al. 1998, The Messenger, 94, 1
  * Bychkov et al. (2016) Bychkov, V. D., Bychkova, L. V., & Madej, J. 2016, MNRAS, 455, 2567
  * Chojnowski et al. (2019) Chojnowski, S. D., Hubrig, S., Hasselquist, S., et al. 2019, ApJ, 873, L5
  * Dorn et al. (2023) Dorn, R. J., Bristow, P., Smoker, J. V., et al. 2023, A&A, 671, A24
  * Freyhammer et al. (2009) Freyhammer, L. M., Kurtz, D. W., Elkin, V. G., et al. 2009, MNRAS, 396, 325
  * Gruberbauer et al. (2008) Gruberbauer, M., Saio, H., Huber, D., et al. 2008, A&A, 480, 223
  * Holdsworth et al. (2021) Holdsworth, D. L., Cunha, M. S., Kurtz, D. W., et al. 2021, MNRAS, 506, 1073
  * Holdsworth et al. (2024) Holdsworth, D. L., Cunha, M. S., Lares-Martiz, M., et al. 2024, MNRAS, 527, 9548
  * Hubrig et al. (2021) Hubrig, S., Järvinen, S. P., Ilyin, I., Strassmeier, K. G., & Schöller, M. 2021, MNRAS, 508, L17
  * Hubrig et al. (2018) Hubrig, S., Järvinen, S. P., Madej, J., et al. 2018, MNRAS, 477, 3791
  * Hubrig et al. (2004) Hubrig, S., Kurtz, D. W., Bagnulo, S., et al. 2004, A&A, 415, 661
  * Jones et al. (2013) Jones, A., Noll, S., Kausch, W., Szyszka, C., & Kimeswenger, S. 2013, A&A, 560, A91
  * Kaeufl et al. (2003) Kaeufl, H.-U., Moorwood, A. F. M., & Pirard, J.-F. 2003, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 4843, Polarimetry in Astronomy, ed. S. Fineschi, 223–232
  * Kochukhov et al. (2004) Kochukhov, O., Ryabchikova, T., & Piskunov, N. 2004, A&A, 415, L13
  * Kurtz (1978) Kurtz, D. W. 1978, Information Bulletin on Variable Stars, 1436, 1
  * Kurtz (1982) Kurtz, D. W. 1982, MNRAS, 200, 807
  * Kurtz (1983) Kurtz, D. W. 1983, MNRAS, 202, 1
  * Kurtz et al. (2003) Kurtz, D. W., Elkin, V. G., & Mathys, G. 2003, MNRAS, 343, L5
  * Leone & Kurtz (2003) Leone, F. & Kurtz, D. W. 2003, A&A, 407, L67
  * Libbrecht (1988) Libbrecht, K. G. 1988, ApJ, 330, L51
  * Martinez et al. (1996) Martinez, P., Weiss, W. W., Nelson, M. J., et al. 1996, MNRAS, 282, 243
  * Mathys (2017) Mathys, G. 2017, A&A, 601, A14
  * Mathys et al. (2005) Mathys, G., Kurtz, D. W., & Elkin, V. G. 2005, in EAS Publications Series, Vol. 17, EAS Publications Series, ed. G. Alecian, O. Richard, & S. Vauclair, 113–118
  * Mathys et al. (2007) Mathys, G., Kurtz, D. W., & Elkin, V. G. 2007, MNRAS, 380, 181
  * Nesvacil et al. (2004) Nesvacil, N., Hubrig, S., & Jehin, E. 2004, A&A, 422, L51
  * Nesvacil et al. (2013) Nesvacil, N., Shulyak, D., Ryabchikova, T. A., et al. 2013, A&A, 552, A28
  * Noll et al. (2012) Noll, S., Kausch, W., Barden, M., et al. 2012, A&A, 543, A92
  * Renson & Manfroid (2009) Renson, P. & Manfroid, J. 2009, A&A, 498, 961
  * Ricker et al. (2015) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003
  * Savanov et al. (2018) Savanov, I. S., Romanyuk, I. I., & Dmitrienko, E. S. 2018, Astrophysical Bulletin, 73, 463
  * Smalley et al. (2015) Smalley, B., Niemczura, E., Murphy, S. J., et al. 2015, MNRAS, 452, 3334
  * Strassmeier et al. (2023) Strassmeier, K. G., Carroll, T. A., & Ilyin, I. V. 2023, A&A, 674, A118
  * Strassmeier et al. (2015) Strassmeier, K. G., Ilyin, I., Järvinen, A., et al. 2015, Astronomische Nachrichten, 336, 324