text:
\section{Introduction}
Long period variables (LPVs) are easily detectable representatives of the asymptotic giant branch (AGB) phase of the evolution of low- and intermediate mass stars.
The AGB phase is critical for stellar and galactic evolution \citep[e.g.][]{pastorelli_etal_2019, marigo_etal_2017, marigo_15, Bruzual_07, Maraston_etal_06}.
In terms of stellar evolution, high mass-loss rates are responsible for the end of the life of a star.
At the same time, a large fraction of the intermediate to heavy elements are produced and returned to the interstellar medium during the AGB phase \citep[e.g.][]{Ventura_etal_18, Slemer_etal_17, Cristallo_etal_15, karakas_lattanzio_2014}.
Therefore, this phase is of high importance for the chemical evolution of galaxies.

Long period variables are characterised by large amplitude variations in the visual and pulsation periods of about 10 to 1000 days.
The period-luminosity diagram (PLD) has become one of the main tools for studying LPVs and their evolution along the AGB.
\citet{wood_sebo_1996} were the first to detect that semi-regular variables (SRVs) follow an additional P-L relation other than that of Miras. This finding was interpreted, and later confirmed,  as due to pulsations in overtone modes, while Miras pulsate in the fundamental mode \citep[e.g.][]{wood_etal_1999,wood_2000}.
As a consequence of the lack of reliable distances to LPVs in the solar neighbourhood, studies of the PLD have relied on variables in stellar systems, most of all the Magellanic Clouds \citep{feast_et_al_1989, wood_2000, ita_et_al_2004a, fraser_et_al_2005, soszynski_etal_2007}, but also in globular clusters \citep{feast_et_al_2002, lebzelter_wood_2005}, the Galactic
Bulge \citep{wood_bessell_1983, glass_et_al_1995} and some local
group galaxies \citep{whitelock_2012, whitelock_et_al_2013, menzies_et_al_2015}.

Studies of the Magellanic Clouds finally revealed the presence of at least five parallel P-L relations, or sequences \citep{wood_etal_1999,ita_etal_2004}.
Distinct sequences are associated with distinct pulsation modes, with two exceptions: sequences B and C$^{\prime}$ are both due to the same pulsation mode \citep[the first overtone (1O) mode;][]{trabucchi_etal_2017}, while sequence D hosts the so-called long secondary periods (LSPs), which are likely the result of a different kind of variability, but whose origin is still unknown \citep{hinkle_etal_2002, wood_nicholls_2009, saio_etal_2015}.
\citet{trabucchi_etal_2017} argued that when stars pulsating predominantly in the 1O mode are crossing the region between sequences B and C$^{\prime}$, the stars tend to develop a LSP with a larger amplitude than the 1O mode itself.
The LSP is thus more easily detected, resulting in the apparent gap between sequences B and $C^{\prime}$.
Yet, despite pulsating in the same mode, stars associated with sequence $C^{\prime}$ (generally classified as SRVs) have observed properties markedly distinct from those on sequence B \citep[the so-called OGLE\footnote{Optical Gravitational Lensing Experiment} Small Amplitude Red Giants, OSARGs;][]{wray_etal_2004}.
The SRVs not only show larger visual amplitudes, but they also have higher infrared excess and estimated mass-loss rates \citep{mcdonald_trabucchi_2019}.
In this scenario, Mira variables are found to lie on sequence C, associated with pulsation in the fundamental mode and are experiencing the latest stages of the AGB.

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{{Figures/G2MD_apparent.png}}
\caption{\textit{Gaia}\xspace-2MASS diagram of LPVs in the LMC (left) and SMC (right). Boundaries between various groups of stars are defined as in \citet{lebzelter_etal_2018}, except for the redefined boundary between branches (c) and (d) (see Sect.\,\ref{samplechar}).
Boundaries of the SMC account for the different distance with adopted distance moduli $18.49$ mag and $18.96$ mag, respectively \citep{degrijs_etal_2017}.
        }
\label{fig:LMC_SMC_G2MD_apparent}
\end{figure*}

The past 20 years of research in this area revealed the dominant role of large surveys for the detection and study of P-L relations.
Among the ground-based studies, the monitoring of the Magellanic Clouds and the galactic bulge within the OGLE project \citep{soszynski_etal_2009_ogle3lmc,soszynski_etal_2011_ogle3smc,soszynski_etal_2013_ogle3bulge} stands out because of the long time coverage of light changes, the high time resolution and the sky area it comprises.
With the advent of \textit{Gaia}\xspace Data Release 2 (DR2), a database became available which provides a deep all-sky monitoring of LPVs in one wide (\ensuremath{G}\xspace) and two semi-wide (\ensuremath{G_{\rm BP}}\xspace and \ensuremath{G_{\rm RP}}\xspace) optical bands covering bright and faint stars from few mags down to $\ensuremath{G}\xspace=20.7$~mag, and with distances and proper motions as additional information.
The 22 months of data published in DR2 already allow for a study of periods and pulsational behaviour of LPVs \citep{mowlavi_etal_2018_dr2lpv}.

Recently, our group showed \citep{lebzelter_etal_2018} that a combination of Wesenheit functions using \textit{Gaia}\xspace and near-infrared wavelength bands allows us to efficiently separate groups of stars on the AGB according to their mass and surface chemistry (C/O-rich).
To simplify the description, in the following, a diagram plotting 2MASS $\ensuremath{K_{\rm s}}\xspace$ against the above-mentioned difference of Wesenheit indices (see Sect.\,\ref{samplechar} for the definitions of $W_{BP,RP}$ and $W_{J,K}$) is called a \textit{Gaia}\xspace-2MASS diagram.
This diagram allows the possibility to add a mass and dredge-up indicator to the PLD.
A further exploration of the applicability of this diagram to various stellar systems has been presented in \citet{eslab53_Mowlavi_2019}.

Pulsation models of LPVs predict a dependency of the P-L relations on mass \citep{trabucchi_etal_2019} and surface chemistry \citep[e.g.][]{lebzelter_wood_2007}.
The goal of this paper is to extend our understanding of variability on the AGB, particularly for the more massive and thus more luminous part of this evolutionary phase, and in relation to the distinction between C- and O-rich atmospheric chemistry.
We investigate PLDs using \textit{Gaia}\xspace photometry as brightness indicators to add these relations to the suite of tools for the exploration of LPVs.
At the same time, we aim to explore further the strengths of the \textit{Gaia}\xspace-2MASS diagram for the study of evolved stellar populations.

\section{Sample}
\subsection{Selection and characterization of sample stars}\label{samplechar}

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{{Figures/MCs_Gaia_OGLE3_P_hist}.png}
\caption{Left panel: Ratio of the \textit{Gaia}\xspace DR2 period to the primary period from the OGLE-III database for the LMC (blue) and SMC (red), respectively, as well as the corresponding histogram. Right panel: Same, but using the OGLE-III period whose value is closest to the \textit{Gaia}\xspace DR2 period. Above $P_{Gaia}=500$ days, \textit{Gaia}\xspace periods become uncertain owing to the limited time span of the \textit{Gaia}\xspace DR2 data.
        }
\label{Fig:MCs_Gaia_OGLE_periods}
\end{figure*}

The starting point for our sample selection was the \textit{Gaia}\xspace DR2 database of LPVs \citep{mowlavi_etal_2018_dr2lpv} because for these objects the availability of \ensuremath{G_{\rm BP}}\xspace and \ensuremath{G_{\rm RP}}\xspace photometry allows for a distinction according to mass using the \textit{Gaia}\xspace-2MASS diagram and because \textit{Gaia}\xspace also covers the brightest objects in the Magellanic Clouds, which OGLE does not.
We selected stars in the LMC and SMC similar to the selection process in \citet{lebzelter_etal_2018} and \citet{eslab53_Mowlavi_2019} according to their location in the sky, their proper motion, and their parallax.
While we present results for both clouds in this paper, the description and interpretation focusses on the LMC owing to the much lower number of sample stars in the SMC.

The sample inherits the limitations of the first \textit{Gaia}\xspace catalogue of LPVs, which focussed on candidates with variability amplitudes larger than 0.2~mag in \ensuremath{G}\xspace, and prioritised low contamination over completeness, resulting in a catalogue of LPV candidates with a completeness of about 45\% in the Magellanic Clouds.
In addition, the study of the periods in this paper is restricted to the range between 70 and 1000 days, the lower limit resulting from restrictions due to the \textit{Gaia}\xspace scanning law introducing spurious frequencies in the low end, and the upper limit resulting from the 22-month time span of \textit{Gaia}\xspace DR2 \citep[we refer to][for a description of the properties of the catalogue]{mowlavi_etal_2018_dr2lpv}.
While the latter condition would in principle limit the period range to a maximum of $\sim$500~d, we still extend the study up to 1000~d to include LPVs that show variations on that timescale, particularly LSPs forming sequence D in the PLD, keeping in mind that the uncertainties on the periods longer than 500~d can be large.
Finally, we note that the DR2 catalogue of LPVs includes only the variability period of the largest amplitude. Multi-periodicity, while being very common among LPVs, is awaiting additional data in future data releases to be properly characterised.

These limitations, in combination with a higher limiting amplitude, substantially affect the appearance of the PLD, especially when compared with OGLE observations. The lower period limit, in particular, effectively leads to the loss of the low-brightness parts of the P-L sequences C$^{\prime}$ and C (cf. Sect.~\ref{sec:results}). In addition, stars with a 1O mode period transitioning between sequences B and C$^{\prime}$ are likely to develop a more easily detectable LSP on sequence D. When, as in the present case, a single observed period per star is available, this contributes to the apparent depletion of sequence C$^{\prime}$.

We then attributed classifiers for mass (low, intermediate, and high) and chemical type (O-rich or C-rich) to each of our LPVs using the \textit{Gaia}\xspace-2MASS diagram and the definitions for the LMC set up in \citet{lebzelter_etal_2018}.
This is illustrated in the left panel of Fig.~\ref{fig:LMC_SMC_G2MD_apparent}, in which the magnitude in the 2MASS $\ensuremath{K_{\rm s}}\xspace$ band is shown against the difference between the Wesenheit indices constructed with \textit{Gaia}\xspace and 2MASS photometry, respectively,
i.e.\begin{equation}\label{Wrpdef}
    \ensuremath{W_{\rm BP,RP}}\xspace = \ensuremath{G_{\rm RP}}\xspace - 1.3 \left( \ensuremath{G_{\rm BP}}\xspace - \ensuremath{G_{\rm RP}}\xspace \right)
\end{equation}
\begin{equation}\label{WJKdef}
    \ensuremath{W_{\rm J,K_s}}\xspace = \ensuremath{K_{\rm s}}\xspace - 0.686 \left( J - \ensuremath{K_{\rm s}}\xspace \right)
\end{equation}

\citet{lebzelter_etal_2018} identified four branches in the \textit{Gaia}\xspace-2MASS diagram, labelled with letters from (a) to (d).
In this work, we use the same labels as in \citet{lebzelter_etal_2018} to identify the four branches in the \textit{Gaia}\xspace-2MASS diagram.
In addition, we use the label (a-f) to identify the portion of branch (a) populated by RGBs and faint AGBs (i.e. stars below the tip of the RGB), and label (b-x) for the right-most side of branch (b), containing stars identified as
extremely dusty stars.
That group predominantly contains C-stars, which are efficiently producing high opacity dust grains \citep{hoefner_olofsson_2018}, but may also include highly reddened OH-IR stars.
Using the list of OH-IR stars of the LMC recently presented by \citet{goldman_et_al_2017} we find nine OH-IR stars among the oxygen rich stars in our diagram, and six objects in the part denoted as (b-x).
All but one of the latter OH-IR stars are among the reddest objects in (b-x).

We applied one modification to the definitions of the various groups in this diagram compared to \citet{lebzelter_etal_2018}, namely for the boundaries for groups (c) (intermediate-mass) and (d) (high-mass), in which case we replaced the original definitions with, respectively,
\begin{equation}
    K_{\rm s} > 8.74 + (\ensuremath{W_{\rm BP,RP}}\xspace - \ensuremath{W_{\rm J,K_s}}\xspace)\;{\rm and}\;K_{\rm s}\leq10.5+(\ensuremath{W_{\rm BP,RP}}\xspace-\ensuremath{W_{\rm J,K_s}}\xspace)\,;
\end{equation}
\begin{equation}
    K_{\rm s}\leq 8.74 + (\ensuremath{W_{\rm BP,RP}}\xspace - \ensuremath{W_{\rm J,K_s}}\xspace)\,.
\end{equation}
This new boundary between groups (c) and (d) accounts better for the distribution of LPVs in the SMC (right panel of Fig.~\ref{fig:LMC_SMC_G2MD_apparent}) while keeping a good solution for the LMC.
A possible dependency of the boundaries on metallicity still needs to be explored.

The \textit{Gaia}\xspace-2MASS diagram of the SMC shown in Fig.~\ref{fig:LMC_SMC_G2MD_apparent} accounts for the difference in distance with the LMC assuming a distance modulus $18.49\pm0.09$~mag for the LMC and $18.96\pm0.02$ ~mag for the SMC \citep{degrijs_etal_2017}.
It is clear that the chosen values of the distance moduli affect the classification of stars that are located nearby boundaries between groups, thereby introducing a source of error.
This uncertainty overlaps with instrumental uncertainties, which are in the millimagnitude range for the \textit{Gaia}\xspace photometry \citep{evans_etal_2018} and about 0.03\,mag for the 2MASS data.
However, because of the limited sampling of the light curves, deviation from the mean brightness of a star, owing to its variability and cycle-to-cycle fluctuations,  certainly dominates the uncertainty in the two quantities plotted in Fig.~\ref{fig:LMC_SMC_G2MD_apparent}.
Naturally, this leads to misclassified objects.
Additional sources of error stem from the different extinction between the two systems (here neglected) as well as the different magnitude of the RGB tip, whose value was employed by \citet{lebzelter_etal_2018} to discriminate between low-mass O-rich AGBs and the group consisting of RGBs and faint AGBs.
We do not consider this a problem in the present paper since we do not aim to derive statistical quantities from our analysis here.

The bolometric corrections (BCs) for each object in our sample are solely based on \textit{Gaia}\xspace data (\ensuremath{G}\xspace, \ensuremath{G_{\rm BP}}\xspace, \ensuremath{G_{\rm RP}}\xspace).
These corrections differ from those used in DR2 and are described in detail in Appendix~\ref{sec:bc}.
These BCs are used to compute the bolometric magnitudes \ensuremath{m_{\rm bol}}\xspace from the \ensuremath{G}\xspace-band photometry.

\subsection{Comparing OGLE and \textit{Gaia}\xspace periods}
As mentioned above, the OGLE survey provides one of the best catalogues of LPVs in the Magellanic Clouds in terms of completeness and accuracy and
listing multiple periods and amplitudes.
Accordingly, OGLE data have been widely used to study PLDs of LPVs, in particular for the study of \citet{trabucchi_etal_2017}.
For our study of the pulsational properties of LPVs with \textit{Gaia}\xspace data, it seemed self-evident to use OGLE data as a reference to judge the accuracy, in particular, of the \textit{Gaia}\xspace periods and to compare any conclusions deduced from our study to previous works on the PLD.
A first test of that kind was part of the paper by \citet{mowlavi_etal_2018_dr2lpv} indicating a reasonable reproduction of OGLE periods and amplitudes with the help of \textit{Gaia}\xspace data.
We note that there are two advantages of using \textit{Gaia}\xspace data for our study. First, it is possible to access the stellar parameters mass and chemistry (using the \textit{Gaia}\xspace-2MASS diagram) and, second, the data include bright AGB stars in the Magellanic Clouds, which have been cut off due to the upper brightness limit of the OGLE survey. The disadvantages of \textit{Gaia}\xspace
data are the shorter lengths of the time series, lower sampling density of the light curves, and -- for DR2 data -- the availability of only one single period.

\begin{figure}
    \centering
    \includegraphics[width=\columnwidth]{{Figures/RSG_example_plots.png}}
    \caption{Example \textit{Gaia}\xspace light curves of four group [d] stars. Top row: Two representative supergiants. Bottom left: Large amplitude variable, likely a massive AGB star. Bottom right: Incomplete light curve coverage to determine a period reliably.}
    \label{fig:rsg_example}
\end{figure}

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{{Figures/PLD_logP_Ks_LMC_SMC_byGroup_OGLEfit.png}}
\caption{Period-luminosity diagram in the 2MASS $\ensuremath{K_{\rm s}}\xspace$ band of LPVs in the LMC (left) and SMC (right). Stars are colour-coded according to the group they belong to as in Fig.~\ref{fig:LMC_SMC_G2MD_apparent}. Best-fit relations to O-rich (solid lines) and C-rich (dashed lines) are taken from \citet{soszynski_etal_2007}.
        }
\label{fig:PLD_logP_Ks_LMC_SMC}
\end{figure*}

We compare periods between the two surveys in Fig.~\ref{Fig:MCs_Gaia_OGLE_periods}. First, we compare \textit{Gaia}\xspace periods with the primary periods (i.e. those with the largest amplitude in a given star) detected by OGLE\footnote{
    We cross-matched the OGLE and \textit{Gaia}\xspace DR2 catalogues to 1 arcsec of radius and found a match for 5279 of the total 12811 Magellanic Cloud LPVs from the \textit{Gaia}\xspace DR2 data set (the majority of the remaining LPVs are out of the OGLE field of view).
}.
The ratio between the two is plotted against the \textit{Gaia}\xspace period in the left panel of Fig.~\ref{Fig:MCs_Gaia_OGLE_periods}, which also shows the histogram of these ratios.
We find a high rate of agreement; about 70\% of the \textit{Gaia}\xspace DR2 periods in the range 70-500 days, differing by less than 25\% from OGLE primary periods.
At longer periods, the limited length of the \textit{Gaia}\xspace time series introduces a larger period uncertainty.
For a significant fraction of stars, there are evident discrepancies between the two surveys, a direct consequence of multi-periodicity and of the lower period limit of the \textit{Gaia}\xspace catalogue.
Indeed,  a multi-periodic star with a primary period shorter than 60 days is present in the OGLE catalogue, but not in \textit{Gaia}\xspace , which lists one of the  longer periods of the star instead.
This is the case for objects in the upper right corner in the left panel of Fig.~\ref{Fig:MCs_Gaia_OGLE_periods}, for which the \textit{Gaia}\xspace pipeline detected a LSP in place of, for example the 1O mode period found by OGLE.
Similarly, the 1O mode period may be listed in the present catalogue in place of the fundamental mode period or vice versa.
Figure 2 shows candidates with this difference to the OGLE results in the two groups located directly above and below  P$_{Gaia}$/P$_{OGLE}$=1.
A large fraction of objects disappear from these two groups when switching to P$_{Gaia}$/P$_{closest}$, i.e. when using the OGLE period nearest to the \textit{Gaia}\xspace period (right panel of Fig.~\ref{Fig:MCs_Gaia_OGLE_periods}).
In the remaining cases the \textit{Gaia}\xspace period search algorithm seems to have picked the first harmonic of the OGLE period, likely because of the sampling of the light curve by \textit{Gaia}\xspace which is very different from OGLE.
The bias introduced by multi-periodicity can thus be partially accounted for by comparing \textit{Gaia}\xspace periods with the closest period listed for the same star in the OGLE catalogue, rather than the primary OGLE period.

Our comparison proves that within the investigated period range, \textit{Gaia}\xspace periods are in very good agreement with the OGLE periods. Therefore, the use of \textit{Gaia}\xspace light curve data in this study is a valid approach to investigate the PLDs of LPVs. Since we restrict our conclusions on the variability found on sequence D to qualitative statements, accuracy limitations of \textit{Gaia}\xspace data near the long period end play less of a role. The same is true for the bias associated with the lack of multi-periodicity information in the \textit{Gaia}\xspace catalogue.

The high-mass stars and supergiants [group (d)] require a further discussion to verify the quality of the periods for this group.
The red supergiants, as the most luminous objects in the PLD, naturally show, on average, the longest periods and populate a period range where the length of the \textit{Gaia}\xspace DR2 data set is of the same order as the period.
We show in Fig.~\ref{Fig:MCs_Gaia_OGLE_periods}  that periods above 500\,days derived from the \textit{Gaia}\xspace DR2 data exhibit an increasing uncertainty.
However, this finding is based on a general comparison of \textit{Gaia}\xspace and OGLE data.
The OGLE observations are saturated for the supergiants in the Magellanic Clouds and therefore a direct comparison with our group (d) stars is not possible except for those with the lowest brightnesses.
Hoping to find an alternative sample for comparison, we extracted light curve data for our supergiants from the Massive Compact Halo Objects Project (MACHO) database.
However, it turned out that those measurements seem to be affected by saturation effects as well and do not provide conclusive results.
Owing to the lack of usable comparison data we decided to look at the individual light curves to evaluate the quality of the derived periods.
This exercise confirmed the variable nature of our sample supergiants.
Figure \ref{fig:rsg_example} shows four example \textit{Gaia}\xspace light curves from our group (d) stars.
The top row shows two typical supergiant light curves that are well sampled by the \textit{Gaia}\xspace time series.
In the bottom row, the left example shows a case that has very large variability amplitude and a light curve reminiscent of a Mira.
We therefore think that this star is a massive AGB star close to the end of its evolution, which is also supported by the presence of a bump in the light curve before maximum light \citep{mcsaveney2007}.
The very red colour of this object is remarkable considering that in \ensuremath{K_{\rm s}}\xspace the star belongs to the most luminous group, while in \ensuremath{G}\xspace it is of average brightness.
The fourth example is a supergiant with a period that is too long to be fully covered by the time span included in DR2.
Such objects, although they come with a period in the DR2 catalogue, were excluded from our analysis.
We also rejected stars that had badly sampled or irregular light curves.
Excluding those objects we ended up with a total sample of 78 group (d) stars in the LMC and 17 in the SMC, which is in both cases about 45\% of all candidates in this group.

\section{Results}
\label{sec:results}

Relations between period and luminosity are a common tool to study pulsational properties, such as the dominant pulsation mode, of a distinct group of variable stars.
In observational studies of LPVs, the luminosity, a quantity rarely available for large samples of stars, is typically replaced by the absolute $K$-band magnitude or the Wesenheit function \ensuremath{W_{\rm J,K_s}}\xspace.
In our study, we follow this approach using the $P$-\ensuremath{K_{\rm s}}\xspace and $P$-\ensuremath{W_{\rm J,K_s}}\xspace diagrams, and further investigate the PLDs using two luminosity indicators derived from \textit{Gaia}\xspace, namely \ensuremath{W_{\rm BP,RP}}\xspace and \ensuremath{m_{\rm bol}}\xspace.

The study was performed for both the LMC and SMC.
Assuming that all stars within one of the Clouds have the same distance, we did not apply distance corrections to the used magnitudes.
The average line-of-sight depths of the LMC and SMC were estimated by \citet{2009A&A...496..399S} to be about 4~kpc and ~4.5kpc, respectively, i.e. less than 10\% of their distance to the Sun.
These depths would translate into magnitude depths of 0.2~mag.
However, for the SMC significantly larger depths for some directions occur \citep{ripepi_et_al_2017, sun_et_al_2018}.
The line-of-sight depths are expected to add some scatter around the P-L relations, in particular for the SMC.
No reddening corrections were applied either, in accordance with the very low impact of interstellar extinction in the \ensuremath{K_{\rm s}}\xspace\  band for objects outside the disc.
Besides, the small differential interstellar extinction across each Cloud \citep[e.g.][]{milone_etal_2009} leads to small dispersion of the \ensuremath{m_{\rm bol}}\xspace values.

We start this section with a qualitative description in Sect.~\ref{sec:PLDs} of the PLDs using the four luminosity indicators \ensuremath{K_{\rm s}}\xspace, \ensuremath{W_{\rm J,K_s}}\xspace, \ensuremath{W_{\rm BP,RP}}\xspace, and \ensuremath{m_{\rm bol}}\xspace.
The description is then further detailed for intermediate- and high-mass O-rich stars in Sects.~\ref{Sect:IMstars} and \ref{Sect:HMstars}, respectively.
The case of C-stars is tackled in Sect.~\ref{Sect:Cstars}.

\subsection{Period-luminosity diagrams for various luminosity indicators}
\label{sec:PLDs}

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{{Figures/PLD_4x2_Orich.png}}
\caption{Period-luminosity diagrams of \textit{Gaia}\xspace LPV candidates in the LMC (left) and SMC (right), including only stars classified as O-rich, and using different luminosity indicators. From top to bottom,  $\ensuremath{K_{\rm s}}\xspace$ band and $\ensuremath{W_{\rm J,K_s}}\xspace$ index from 2MASS, $\ensuremath{W_{\rm BP,RP}}\xspace$ index from \textit{Gaia}\xspace, and the bolometric magnitude computed as described in Appendix~\ref{sec:bc}. Lines in the top four panels are best fit to the P-L relations of O-rich LPVs from \citet{soszynski_etal_2007}.
        }
\label{fig:PLD_4x2_Orich}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{{Figures/PLD_4x2_Crich.png}}
\caption{Period-luminosity diagrams of \textit{Gaia}\xspace LPV candidates in the LMC (left) and SMC (right), including only stars classified as C-rich (with extreme C-rich stars noted x-C-AGB in the panels), and using different luminosity indicators. From top to bottom, $\ensuremath{K_{\rm s}}\xspace$ band and $\ensuremath{W_{\rm J,K_s}}\xspace$ index from
2MASS, $\ensuremath{W_{\rm BP,RP}}\xspace$ index from \textit{Gaia}\xspace, and the bolometric magnitude computed as described in Appendix~\ref{sec:bc}. Lines in the top four panels are best fit to the P-L relations of C-rich LPVs from \citet{soszynski_etal_2007}.
        }
\label{fig:PLD_4x2_Crich}
\end{figure*}

The P-\ensuremath{K_{\rm s}}\xspace diagram of our complete samples for the two Clouds are shown in Fig.~\ref{fig:PLD_logP_Ks_LMC_SMC}.
The LPVs for which a period has been measured from \textit{Gaia}\xspace DR2 data populate the bright ends of sequences C$^{\prime}$, C, and D, as visible in the figure.
We point out that the \textit{Gaia}\xspace data set extends the OGLE-III observations \citep{soszynski_etal_2009_ogle3lmc,soszynski_etal_2011_ogle3smc} towards higher brightness by almost 2\,mag in \ensuremath{K_{\rm s}}\xspace.
Our study thus fills the gap between low-mass giants and supergiants left in previous studies \citep[see figure 7 of][]{kiss_etal_2006}.

The consistency with the best-fit relations derived by \citet{soszynski_etal_2007}, also shown in Fig.~\ref{fig:PLD_logP_Ks_LMC_SMC}, further testifies the good agreement with OGLE data discussed above.
Sequence D as observed by \textit{Gaia}\xspace is more heavily scattered \citep[cf.][their figure~1]{soszynski_etal_2009_ogle3lmc} and systematically shifted towards shorter periods, a consequence of the degraded precision in the determination of periods longer than $\sim500$ days from DR2 data.
O-rich stars with masses $\lesssim 1.5\,{\rm M}_{\odot}$ [branch (a) in Fig.~\ref{fig:LMC_SMC_G2MD_apparent}] are observed predominantly on sequence D, meaning that their primary variability is associated with a LSP.
This is especially true for stars in the (a-f) group in Fig.~\ref{fig:LMC_SMC_G2MD_apparent}, classified as RGBs or faint AGBs.
These stars usually also pulsate in the 1O mode, but with a lower amplitude than the LSP and a relatively short period \citep[on the left side of sequence C$^{\prime}$, e.g.][]{trabucchi_etal_2017,wood_2015}, so that it is less likely characterised in DR2.

The PLDs defined by the other luminosity indicators, namely \ensuremath{W_{\rm J,K_s}}\xspace, \ensuremath{W_{\rm BP,RP}}\xspace, and \ensuremath{m_{\rm bol}}\xspace, are shown in Fig.~\ref{fig:PLD_4x2_Orich} for O-rich stars, and in Fig.~\ref{fig:PLD_4x2_Crich} for C-rich stars.
In light of the all-sky coverage of the \textit{Gaia}\xspace survey, the use of \ensuremath{W_{\rm BP,RP}}\xspace and \ensuremath{m_{\rm bol}}\xspace, which enables a study of PLDs solely based on data from this spacecraft, provides an important step in the full exploration of the \textit{Gaia}\xspace data set on LPVs.
Figures~\ref{fig:PLD_4x2_Orich} and \ref{fig:PLD_4x2_Crich} are organised in the same way, giving P-\ensuremath{K_{\rm s}}\xspace, P-\ensuremath{W_{\rm J,K_s}}\xspace, P-\ensuremath{W_{\rm BP,RP}}\xspace, and P-\ensuremath{m_{\rm bol}}\xspace from top to bottom, respectively.
The left column in each figure is used for LMC data, the right for SMC data.
The colour-coding of the various groups is explained in the top left corner of each panel.

All four luminosity indicators in Figs.\,\ref{fig:PLD_4x2_Orich} and \ref{fig:PLD_4x2_Crich} lead to a set of clear relations between the indicator and the period.
At the same time we immediately note significant differences between the four kinds of diagrams and between the relations for the various groups of LPVs identified in Fig.~\ref{fig:LMC_SMC_G2MD_apparent}.

For the O-rich stars (C-rich stars are discussed in Sect.~\ref{Sect:Cstars}), we note the following points when comparing the diagrams using the different luminosity indicators (see Fig.~\ref{fig:PLD_4x2_Orich}):
(1)The P-\ensuremath{K_{\rm s}}\xspace- and the P-\ensuremath{W_{\rm J,K_s}}\xspace diagrams show a high similarity as has been outlined in the literature before \citep[e.g.][]{2013ApJ...763..103S}.
(2) In the P-\ensuremath{W_{\rm BP,RP}}\xspace diagram (third row in Fig.~\ref{fig:PLD_4x2_Orich}), sequence C' is narrower and thus seemingly better defined compared to the infrared indicators.
We note that opposite to \ensuremath{W_{\rm J,K_s}}\xspace the quantity \ensuremath{W_{\rm BP,RP}}\xspace consists of photometry averaged over a light cycle.
This may in part be relevant for the smaller scatter.
However, only 5\% of the stars on C' have large amplitudes in \ensuremath{G}\xspace ($>$1 mag), and they are expected to have even smaller amplitudes in \ensuremath{K_{\rm s}}\xspace.
Therefore, the effect of averaging is expected to be comparably small.
This also suggests that differences between the widths of the sequences in various luminosity indicators are not primarily caused by using single or averaged multi-epoch photometry.
(3) The structures seen in the P-\ensuremath{m_{\rm bol}}\xspace diagram (bottom row of Fig.~\ref{fig:PLD_4x2_Orich}) coincide more with the infrared luminosity indicators than with \ensuremath{W_{\rm BP,RP}}\xspace.
While the diagram still facilitates a distinction between various groups, relations, in particular for the low-mass stars, are flatter.
A difference in inclination between the relations for low- and intermediate-mass stars is likely present.
The supergiants and the high-mass AGB stars do not show any distinct sequences in this diagram.
The findings described in this section are applicable to both the LMC and SMC sample, although the SMC sample is less clear because of the lower number of stars.

\subsection{Intermediate-mass AGB stars.}
\label{Sect:IMstars}

A large fraction of the bright AGB stars in the \textit{Gaia}\xspace sample, which \citet{lebzelter_etal_2018} classified as intermediate-mass O-rich AGBs [branch (c)] or as RSGs and massive AGBs [branch (d)], are not present in the OGLE-III database, and are thus of particular interest.
Stars on branch (c) clearly follow three P-L relations, which is consistent with sequences C$^{\prime}$, C, and D. But these are systematically shifted towards shorter periods compared to low-mass O-rich stars in the infrared PLDs (first and
second rows in Fig.~\ref{fig:PLD_4x2_Orich}).
This is a consequence of their higher masses and agrees with the theoretical prediction of \cite{wood_2015}.
This offset is not visible when the $\ensuremath{W_{\rm BP,RP}}\xspace$ index is employed (Fig.~\ref{fig:PLD_4x2_Orich}, third row), and its absence accounts for a much narrower sequence C' compared to \ensuremath{K_{\rm s}}\xspace and \ensuremath{W_{\rm J,K_s}}\xspace.
In the P-\ensuremath{m_{\rm bol}}\xspace diagram, the offset between the relations for low- and intermediate-mass LPVs is clearly detectable only for sequence C'.
Sequence C is very wide for the intermediate-mass stars and the location of the relation is much less defined than for \ensuremath{W_{\rm J,K_s}}\xspace, for example.
However, a smooth transition between low- and intermediate-mass objects is suggested.
We explore the cause of this difference between the various luminosity indicators in Sect. \ref{Discus:PLD_Mass}.

\subsection{High-mass AGB and RSG stars.}
\label{Sect:HMstars}

The LPVs that belong to branch (d) have even higher masses than those on branch (c).
Therefore, we would expect to find an offset for these objects as well.
However, their distribution in the PLD suffers from a larger scatter, and it is less straightforward to identify the P-L relation(s) they belong to.
This is further complicated by the fundamentally different behaviour they exhibit depending on whether visual or near-infrared bands are used to track their luminosity.
Judging by the PLDs in the upper panels of  Fig.~\ref{fig:PLD_4x2_Orich}, where $\ensuremath{K_{\rm s}}\xspace$ and $\ensuremath{W_{\rm J,K_s}}\xspace$ are used, the bulk of stars from branch (d) appear to lie on a prolongation of sequence C$^{\prime}$.
In contrast, when \textit{Gaia}\xspace photometry is used in the PLD (i.e. the $\ensuremath{W_{\rm BP,RP}}\xspace$ index, Fig.~\ref{fig:PLD_4x2_Orich}, third row), most of these bright stars seem to follow the P-L sequence C.

The P-\ensuremath{m_{\rm bol}}\xspace diagram (fourth row in Fig.~\ref{fig:PLD_4x2_Orich}) does not show pronounced indications for a connection of  the high-mass LPVs and red supergiants to any of the relations found from low- and intermediate-mass stars.
The location of these stars in this diagram, however, is found in excellent agreement with the same kind of diagram presented by \citet{wood_etal_1983}; see their figures~6 and~7.
The diagram of \citet{wood_etal_1983} included supergiants not only from the Magellanic Clouds but also from the Galactic field.

There are no high-mass O-rich LPVs that show a variation connected to sequence D.
The existence of such LSPs had been reported by several authors \citep[e.g.][]{kiss_etal_2006}.
However, we determined only a single period and the corresponding sequence D period would likely exceed our upper period cut-off.

\subsection{C-stars}
\label{Sect:Cstars}

The P-L relations of C-rich stars [groups (b) and (b-x)] are shown in Fig.~\ref{fig:PLD_4x2_Crich}.
For these objects, the differences in the PLD arising from using distinct luminosity indicators become even more striking.
When using \ensuremath{W_{\rm BP,RP}}\xspace (third row of Fig.~\ref{fig:PLD_4x2_Crich}), the highly reddened C-stars are shifted away from the bluer C-stars and the M-stars (Fig.~\ref{fig:PLD_logP_Ks_LMC_SMC}) towards lower brightness.
This indicates that the \textit{Gaia}\xspace Wesenheit function \ensuremath{W_{\rm BP,RP}}\xspace does not compensate sufficiently the reddening of large amounts of circumstellar dust affecting \ensuremath{G_{\rm RP}}\xspace.
We do not observe such a difference in \ensuremath{W_{\rm J,K_s}}\xspace (upper panels of Fig.~\ref{fig:PLD_4x2_Crich}), indicating that $\ensuremath{G_{\rm BP}}\xspace-\ensuremath{G_{\rm RP}}\xspace$ gets saturated for large extinction values opposite to $J-K$, where the thick circumstellar shell starts to emit in the infrared adding to the brightness in \ensuremath{K_{\rm s}}\xspace and \ensuremath{W_{\rm J,K_s}}\xspace.
This agrees with results from the preliminary analysis of the \textit{Gaia}\xspace Wesenheit function presented in \citet{lebzelter_etal_2018}.
Since the BC used to compute \ensuremath{m_{\rm bol}}\xspace also depends on $\ensuremath{G_{\rm BP}}\xspace-\ensuremath{G_{\rm RP}}\xspace$, we observe the same effect as for \ensuremath{W_{\rm BP,RP}}\xspace in the $P$-\ensuremath{m_{\rm bol}}\xspace diagram (bottom panels in Fig.~\ref{fig:PLD_4x2_Crich}).

Aside from the offset of the highly reddened C-stars, the P-\ensuremath{W_{\rm BP,RP}}\xspace and P-\ensuremath{m_{\rm bol}}\xspace diagrams show structures very similar to the P-\ensuremath{K_{\rm s}}\xspace and P-\ensuremath{W_{\rm J,K_s}}\xspace diagrams.
Like the O-rich LPVs, the C-rich LPVs [branch (b)] follow all three P-L relations, while the extreme C-rich AGBs [group (b-x)] are likely to follow only sequence C, as is most evident when using $\ensuremath{W_{\rm J,K_s}}\xspace$ in the PLD (Fig.~\ref{fig:PLD_4x2_Crich}, second row).
The few stars of this group that are found on sequences C$^{\prime}$ or D are very close to the separation line between C-rich and extreme C-rich stars in Fig.~\ref{fig:LMC_SMC_G2MD_apparent}.
Therefore, a misclassification cannot be excluded.
Those extreme C-stars below sequence C probably have very thick shells (i.e. high mass-loss rates) and emit mostly in the far-infrared so \ensuremath{K_{\rm s}}\xspace is indeed faint.
Again, a qualitatively similar behaviour is seen in both the LMC and SMC.

It is noteworthy that C-stars forming the P-L sequence with the longest periods, usually dubbed sequence D, is clearly offset to the location of sequence D defined by carbon stars in \citet{soszynski_etal_2007}, shown by the dotted lines in Fig.~\ref{fig:PLD_4x2_Crich}.
We found a smaller shift in the same direction when studying the O-rich stars (Fig.~\ref{fig:PLD_4x2_Orich}), which is likely caused by the limited time span of the \textit{Gaia}\xspace light curves and the upper period limit resulting from it.
This effect is more expressed for C-stars than for O-rich objects, likely because of their on average longer periods compared to M-type stars.
Having a period closer to our period cut-off, there is a higher risk that our period search fails to derive the correct value.

\section{Discussion} \label{Sect:Discussion}

\subsection{Period-luminosity relations at different masses}
\label{Discus:PLD_Mass}

As illustrated in the top two rows of Fig.~\ref{fig:PLD_4x2_Orich} and as pointed out in Sect.\,\ref{Sect:IMstars}, there is a very obvious shift between the P-L relations of low- and intermediate-mass O-rich stars; the latter show a shorter period at a given luminosity. This shift has
previously been noted for sequence C (the Miras) by \citet{feast_et_al_1989} and \citet{hughes_wood_1990}.
As briefly mentioned above, such a shift is expected to occur because a higher mass of the pulsating red giant leads to a shorter period at a given luminosity.
\citet{whitelock_etal_2003} noted that at least some of the stars shifted to a shorter period from sequence C had spectral signatures of hot-bottom burning, consistent with their intermediate-mass assignment.
This is the first time the shift in period with mass has been observed for LPVs pulsating in the 1O.
Because of the conversion of stars in the mass range between groups (a) and (c) into carbon stars, we do not see a smooth transition but rather a break up into two sequences.

Interestingly, the shift disappears in the P-\ensuremath{W_{\rm BP,RP}}\xspace diagram.
To understand this, we have to keep in mind that a star of higher mass also shows a higher effective temperature, which again may affect fluxes in the various photometric bands and accordingly also colours.
We explored the effect of temperature on the various filters with the help of hydrostatic models from \citet{aringer_etal_2016}.
Fig.~\ref{fig:spectrum_and_bands} presents model spectra of O-rich AGB stars of various temperatures, and compares them with the responses of the two \textit{Gaia}\xspace and the two near-infrared passbands used in our analysis.
 Figure~\ref{fig:Tdepend_filters} shows the runs of \ensuremath{K_{\rm s}}\xspace, $J-K_s$, \ensuremath{W_{\rm J,K_s}}\xspace, \ensuremath{G_{\rm BP}}\xspace-\ensuremath{G_{\rm RP}}\xspace, and \ensuremath{W_{\rm BP,RP}}\xspace with temperature;
\ensuremath{K_{\rm s}}\xspace and $J-K_s$ are only weakly dependent on temperature.
As a consequence, \ensuremath{W_{\rm J,K_s}}\xspace behaves very similarly to \ensuremath{K_{\rm s}}\xspace, and any systematic temperature difference between low- and intermediate-mass stars does not become visible in the corresponding PLDs.

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{{Figures/Mspectra_lin.png}}
\caption{Synthetic spectra \citep{aringer_etal_2016} of O-rich stars with $M=2\,\ensuremath{{\rm M}_{\odot}}\xspace$, $\log(g)=0.0$ and different effective temperatures. The black line indicates the model with $T_{\rm eff}=3600\,{\rm K}$.
        }
\label{fig:spectrum_and_bands}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{{Figures/Teff_sensitivity_Gaia_v5.png}}
\caption{Sensitivity on temperature of the various filters and filter combinations used in the present analysis. Values were derived using synthetic spectra of O-rich stars with $M=2\,\ensuremath{{\rm M}_{\odot}}\xspace$, $\log(g)=0.0$ and different effective temperatures (cf.\,Fig.~\ref{fig:spectrum_and_bands}).
        }
\label{fig:Tdepend_filters}
\end{figure*}

In the \ensuremath{G_{\rm BP}}\xspace filter, the absorption bands of TiO make a huge impact on the flux, which therefore is highly dependent on the temperature of the star.
Since the \ensuremath{G_{\rm RP}}\xspace flux is less affected by TiO and VO absorption than the \ensuremath{G_{\rm BP}}\xspace flux, the colour constructed from these two filters shows a high sensitivity on temperature, in particular down to about 3000\,K.
To compute \ensuremath{W_{\rm BP,RP}}\xspace, the temperature-sensitive colour \ensuremath{G_{\rm BP}}\xspace-\ensuremath{G_{\rm RP}}\xspace is subtracted from the \ensuremath{G_{\rm RP}}\xspace, thereby compensating for the flux depression in \ensuremath{G_{\rm RP}}\xspace.
However, we see in Fig.~\ref{fig:Tdepend_filters} that while \ensuremath{G_{\rm RP}}\xspace and \ensuremath{G_{\rm BP}}\xspace-\ensuremath{G_{\rm RP}}\xspace are decreasing in value with temperature, the combined quantity \ensuremath{W_{\rm BP,RP}}\xspace is increasing in value towards higher temperatures because the slope of   the temperature dependency of 1.3(\ensuremath{G_{\rm BP}}\xspace-\ensuremath{G_{\rm RP}}\xspace) is steeper than for \ensuremath{G_{\rm RP}}\xspace.
The factor 1.3 used in the construction of the Wesenheit function obviously leads to an overcompensation of the temperature dependencies of the individual \textit{Gaia}\xspace filters.
Both intermediate-mass and low-mass LPVs are made brighter in \ensuremath{W_{\rm BP,RP}}\xspace, but this occurs to a smaller extent for the former owing to their higher temperature.
The two effects, the offset due to mass (i.e. shorter period due to higher mass at given luminosity) and the effect of temperature on \ensuremath{W_{\rm BP,RP}}\xspace, equal out, and thus sequences C' and C become single sequences more narrow than their near-infrared counterparts.

When interpreting the P-\ensuremath{m_{\rm bol}}\xspace diagram in this context, we have to be careful since the BC is a function of \ensuremath{G_{\rm BP}}\xspace-\ensuremath{G_{\rm RP}}\xspace.
Therefore, our \ensuremath{m_{\rm bol}}\xspace values are expected to exhibit some similarity with the \ensuremath{W_{\rm BP,RP}}\xspace values, although the factors in the colour terms involved are very different.
As a consequence, the P-\ensuremath{m_{\rm bol}}\xspace sequences of low- and intermediate-mass stars do not align as for P-\ensuremath{W_{\rm BP,RP}}\xspace.

After detecting the mass dependency of the P-L relation for O-rich stars in our observations, we decided to investigate the C-rich stars for indications of a mass dependency as well.
With the onset of efficient dust production, circumstellar absorption systematically depresses brightness in all optical and near-infrared filter bands.
Therefore, such an investigation has to be limited to C-stars with low amounts of dust around them, i.e. group (b) stars in the \textit{Gaia}\xspace-2MASS diagram.
There is no doubt that with this approach some of the most massive C-stars are excluded.
Within group (b) in our diagram, a separation of the stars according to mass is not obvious.
Using the brightest C-stars in \ensuremath{K_{\rm s}}\xspace gives a sample that is scattered over the bright part of the PLDs in Fig.~\ref{fig:PLD_4x2_Crich}.
However, selecting the brightest C-stars in the \ensuremath{G}\xspace\  band (bottom panel of Fig.~\ref{fig:massiveCcandidatesG2MD}) gives a sample of stars that form a sequence in the P-\ensuremath{K_{\rm s}}\xspace diagram slightly offset to the bulk of carbon stars on sequence C'
(Fig.~\ref{fig:massiveCcandidatesPeriods}).
This group strongly recalls the behaviour of the intermediate-mass stars in the O-rich case.
For C-stars, the luminosity indicator \ensuremath{W_{\rm J,K_s}}\xspace gives the most narrow sequences.
We suspect, in analogy to the O-rich stars, that the C-stars delineated in Fig.~\ref{fig:massiveCcandidatesPeriods} are more massive than the majority of the group (b) C-stars in our sample, and that \ensuremath{W_{\rm J,K_s}}\xspace, like \ensuremath{W_{\rm BP,RP}}\xspace in the O-rich case, compensates for the temperature and mass dispersion leading to single, narrow sequences C' and C.

\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Figures/MassiveCcanditates_G2MD.png}
\caption{Top panel: Same as Fig.~\ref{fig:LMC_SMC_G2MD_apparent}, but massive C-star candidates are highlighted as black crosses. Bottom panel: Selection of massive C-star candidates in a modified \textit{Gaia}\xspace-2MASS diagram using the \textit{Gaia}\xspace $G$ band on the vertical axis.}
\label{fig:massiveCcandidatesG2MD}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Figures/MassiveCcanditates_Periods.png}
\caption{Same as Fig.~\ref{fig:PLD_4x2_Crich}, with massive C-star candidates highlighted as black crosses.}
\label{fig:massiveCcandidatesPeriods}
\end{figure}

\subsection{Comparison with evolutionary models for low- and intermediate-mass stars}

We compare the observed data of LPVs with the AGB evolutionary tracks computed with the \texttt{COLIBRI} code \citep{marigo_etal_2013} and presented in \citet{pastorelli_etal_2019}.
The models used for the present work cover the mass range between 1 and 5\,\ensuremath{{\rm M}_{\odot}}\xspace for two choices of the initial metallicity, Z=0.006 and Z=0.002.
Along each AGB track periods and amplitude growth rates are calculated using the pulsation models of \citet{trabucchi_etal_2019}.
The results for 1O mode models are shown in Figs.~\ref{fig:PLD_4x2_Orich_wModels} and \ref{fig:PLD_4x2_Crich_wModels}.
For the SMC, the selected tracks refer to  $Z = 0.002$, thus slightly underestimating the average metallicity of this galaxy.
This has to be kept in mind in the comparison of models and observations

The colour coding represents the current mass of the star, hence there are colour changes along the tracks.
Current pulsation models of LPVs suffer from a known tendency to overestimate the period of the fundamental mode, especially at large luminosities \citep[e.g.][]{trabucchi_etal_2017}, hence we preferred to limit the comparison to 1O mode periods, for which we find a good agreement with observations.

Features identified in the four kinds of PLDs are well reproduced by our set of models.
The `mass spread' across individual P-L sequences is clearly visible in the plots of period versus \ensuremath{K_{\rm s}}\xspace and \ensuremath{W_{\rm J,K_s}}\xspace.
When periods are shown against \ensuremath{W_{\rm BP,RP}}\xspace, this effect is suppressed, which is clearly visible for LMC metallicities ($Z\simeq0.006$) and is less obvious in the case of the SMC ($Z\simeq0.002$).
To understand the difference between the two Magellanic Clouds we need to consider that the lower metallicity leads to weaker molecular bands at a given temperature, which again reduces the sensitivity of \ensuremath{G_{\rm BP}}\xspace-\ensuremath{G_{\rm RP}}\xspace on temperature.
In contrast to the more metal-rich case of the LMC, where the colour term in the Wesenheit function compensates for temperature and mass shift, we see --  in the observations and models -- a slight offset between the P-\ensuremath{W_{\rm BP,RP}}\xspace-relations of low- and intermediate-mass stars, respectively.
Overall, the model results agree well with our observational findings, supporting the discussion presented in Sect.\,\ref{Discus:PLD_Mass}.

For the C-rich LPVs, Fig.~\ref{fig:PLD_4x2_Crich_wModels}, the calculated tracks for 1O pulsation show the same behaviour as the observations when going from a P-\ensuremath{K_{\rm s}}\xspace diagram (first row) to a P-\ensuremath{W_{\rm J,K_s}}\xspace diagram (second row), namely a narrowing of sequence C'.
We emphasise that in this diagram the current masses of the model stars are shown, corresponding to main sequence masses between 1.5 and 2.6\,\ensuremath{{\rm M}_{\odot}}\xspace.
We decided to show the current mass since it is relevant for the pulsation properties of the star.
The flattening of the sequences seen in the P--\ensuremath{m_{\rm bol}}\xspace diagram is reproduced by the models as well.

For completeness, results for fundamental mode pulsation of O-rich models are shown in Fig.~\ref{fig:PLD_4x2_Orich_wModelsF} for the two luminosity indicators \ensuremath{W_{\rm J,K_s}}\xspace and \ensuremath{W_{\rm BP,RP}}\xspace.
Keeping in mind the limitations of the pulsation models for predicting fundamental mode periods, an overall agreement with the observations is present.
Periods become visibly too long at bright magnitudes.
The observed narrowing of sequence C is well reproduced by stellar evolution and pulsation models.

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{{Figures/PLD_4x2_Orich_wModels_currentMass.png}}
\caption{Same as Fig.~\ref{fig:PLD_4x2_Orich}, but including AGB models with 1O mode periods. Evolutionary tracks are limited to the portion where ${\rm C/O}<1$ and the 1O mode is expected to be dominant. Tracks are colour coded with current mass, and labelled with their initial mass in the top left panel. They have initial metallicity Z=0.006 (left panels) and Z=0.002 (right panels).
        }
\label{fig:PLD_4x2_Orich_wModels}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{{Figures/PLD_4x2_Crich_wModels_currentMass.png}}
\caption{Same as Fig.~\ref{fig:PLD_4x2_Orich_wModels}, but comparing observed stars on branch (b) with the portions of evolutionary tracks having ${\rm C/O}>1$. The represented tracks roughly correspond to the minimum and maximum initial masses that produce carbon stars pulsating in the 1O mode at the chosen metallicities. For clarity, tracks with an initial mass that is intermediate between these two values have been omitted.
        }
\label{fig:PLD_4x2_Crich_wModels}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{{Figures/PLD_2x2_Orich_wModelsP0_currentMass.png}}
\caption{Same as Fig.~\ref{fig:PLD_4x2_Orich_wModels}, but showing predicted fundamental mode periods along evolutionary tracks.
        }
\label{fig:PLD_4x2_Orich_wModelsF}
\end{figure*}

Figures~\ref{fig:PALD_LMC_SMC_Orich}, \ref{fig:PALD_LMC_SMC_RSG}, and~\ref{fig:PALD_LMC_SMC_Crich} show a variant of the period-amplitude diagram, where the period on the horizontal axis is replaced by the quantity $\log(P)-(W_{\rm JK}-12)/\Delta_{P,W_{JK}}$ \citep{wood_2015}, where $\Delta_{P,W_{JK}}$=-4.444 is assumed to be a representative value for the average slope of the sequences in the $\log(P)$--$W_{JK}$ PLD.
As shown by \citet{trabucchi_etal_2017}, this is a better tool than the usual PLD to discriminate between pulsation in different modes, particularly between fundamental mode pulsation (sequence C) and LSPs (sequence D).
We point out that this study is limited to the period range 70 to 1000~d.

The three distributions visible in Fig.~\ref{fig:PALD_LMC_SMC_Orich} are periods associated with the 1O mode, the fundamental mode, and LSP variability \citep{trabucchi_etal_2017}.
In both the LMC and SMC, the majority of stars classified as RGBs or faint AGBs show variability in a LSP.
Less than 2 percent of LMC stars in this group show variability in the fundamental mode, and only two show variability in the 1O mode, while in the SMC only two stars of this group show
fundamental or 1O variability compared to 91 stars that exhibit a LSP.
We note that such small numbers are highly sensitive to how the boundaries are defined in the \textit{Gaia}\xspace-2MASS diagram and to the chosen values of the distance moduli and the $\ensuremath{K_{\rm s}}\xspace$-band luminosities of the RGB tip in the two systems. Low- and intermediate-mass O-rich AGBs have periods in all three groups.
In contrast, the periods of RSGs and massive AGB stars systematically populate the region associated with 1O mode pulsation (except for a few large amplitude objects that pulsate in the fundamental mode) and exhibit no LSPs (Fig.~\ref{fig:PALD_LMC_SMC_RSG}).  We note, however, that the identification of pulsation modes in this diagram by \citet{trabucchi_etal_2017} is based on AGB pulsation models, and may not be valid for RSG variables (see Sect.\,\ref{Discus:modeRSG}).
Finally, C-rich stars (Fig.~\ref{fig:PALD_LMC_SMC_Crich}) appear in all three groups, but extreme C-rich stars are essentially limited to pulsation in the fundamental mode, which is consistent with the fact that these are evolved objects with high mass-loss rates \citep{2009AJ....137.3139V}.

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{{Figures/PALD_LMC_SMC_byGroup_Orich.png}}
\caption{Projected-period vs. \textit{Gaia}\xspace G-band amplitude (see text) of low- and intermediate-mass O-rich AGBs in the LMC (left) and SMC (right). The green diamonds indicate RGBs and faint AGBs; the blue symbols indicate low-mass O-rich AGBs; and the cyan squares indicate intermediate-mass O-rich AGBs. The dashed lines represent approximate separations between the three distributions corresponding to 1O mode periods, fundamental mode periods, and LSPs.
        }
\label{fig:PALD_LMC_SMC_Orich}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{{Figures/PALD_LMC_SMC_byGroup_RSG.png}}
\caption{Same as Fig.~\ref{fig:PALD_LMC_SMC_Orich}, including the group consisting of RSGs and massive O-rich AGBs. Other O-rich AGB stars are shown as small black symbols, for reference. The dashed lines have the same meaning as in Fig.~\ref{fig:PALD_LMC_SMC_Orich}.
        }
\label{fig:PALD_LMC_SMC_RSG}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{{Figures/PALD_LMC_SMC_byGroup_Crich.png}}
\caption{Same as Fig.~\ref{fig:PALD_LMC_SMC_Orich}, but for C-rich (red) and extreme C-rich (large orange symbols) AGB stars. The dashed lines have the same meaning as in Fig.~\ref{fig:PALD_LMC_SMC_Orich}.
        }
\label{fig:PALD_LMC_SMC_Crich}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[width=\textwidth]{{Figures/PALD_WRP_LMC_SMC_byGroup_RSG.png}}
\caption{Same as Fig.~\ref{fig:PALD_LMC_SMC_RSG}, but the horizontal coordinate is obtained by projecting the period in the $\log(P)$ vs. $\ensuremath{W_{\rm BP,RP}}\xspace$ diagram, allowing us to better distinguish 1O mode periods from fundamental mode ones. The dashed lines have the same meaning as in Fig.~\ref{fig:PALD_LMC_SMC_Orich}.
        }
\label{fig:PALD_WRP_LMC_SMC_RSG}
\end{figure*}

\subsection{Pulsation mode of high-mass LPVs}
\label{Discus:modeRSG}

As noted in Sect.\,\ref{Sect:HMstars} these objects appear connected to different relations depending on the luminosity indicator used
(Fig.~\ref{fig:PLD_4x2_Orich}).
Taking the result from the P-\ensuremath{W_{\rm BP,RP}}\xspace diagram at face value would indicate that the bulk of the supergiants and massive AGBs are pulsating in fundamental mode while the P-\ensuremath{K_{\rm s}}\xspace and the P-\ensuremath{W_{\rm J,K_s}}\xspace diagrams favour a relation to the first overtone pulsation on sequence C'.
In all cases a small fraction of sequence (d) objects in the LMC, of the order of 10\%, seems to be shifted away from the bulk of the stars and possibly pulsate in a different pulsation mode.
In the SMC the number of objects in this group is very small and it is difficult to reach conclusions.

\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics{Figures/PGaia_amplitudes_3x1.png}}
\caption{Period-amplitude diagrams of O-rich stars, colour coded by evolutionary group to which they belong. The green dots indicate faint AGBs and RGBs, the blue dots indicate low-mass AGBs, and the cyan squares indicate intermediate-mass AGBs. The massive AGBs and RSGs from branch (d) are shown as orange/red diamond symbols according to whether they belong to sequence C'/C in the P-\ensuremath{W_{\rm BP,RP}}\xspace diagram.
The three panels present the amplitudes in \ensuremath{G}\xspace, \ensuremath{G_{\rm BP}}\xspace, and \ensuremath{G_{\rm RP}}\xspace, respectively, taken from the magnitude ranges published in \textit{Gaia}\xspace DR2.}
\label{fig:AmplPerMassive}
\end{figure}

To investigate whether there are two groups of LPVs among the high-mass stars that can be distinguished by their pulsation mode, in Fig.~\ref{fig:AmplPerMassive} we compare the period with the amplitudes (taken from the magnitude ranges published in \textit{Gaia}\xspace DR2) in the \ensuremath{G}\xspace, \ensuremath{G_{\rm BP}}\xspace, and \ensuremath{G_{\rm RP}}\xspace bands for our complete sample.
In this diagram, massive AGB stars and RSGs are denoted according to the P-\ensuremath{W_{\rm BP,RP}}\xspace sequences to which they belong.
A difference in the pulsation amplitude may be a hint that different modes are excited.
For instance, there is an obvious gap in the amplitude distribution at approximately 2\,mag for P$>$300\,d.
The large amplitude stars form a band starting at a period of about 150\,d, and all these objects are located on sequence C and are fundamental mode pulsators.

It can be seen that all but three high-mass stars show very similar amplitudes of less than 2\,mag.
This result is independent of the photometric band used.
We conclude that the pulsation amplitudes do not support the presence of two groups with different pulsation modes among the high-mass LPVs.
The three objects with a significantly larger variability amplitude are all found close to sequence C in all luminosity indicators tested and their light curves show bumps in the rising parts of their light curves which are typical for massive Mira-like pulsators (see Fig.\,\ref{fig:rsg_example}).
We therefore conclude that these are massive AGB stars pulsating in fundamental mode.
In the P-\ensuremath{W_{\rm BP,RP}}\xspace diagram they are not distinguishable from the bulk of the stars on that sequence.

\citet{chatys_etal_2019} published a large collection of red supergiants from the galactic field and the LMC with their periods and visual amplitudes.
Like in our study, these authors found light amplitudes to be lower than 1 mag for the vast majority of objects.
Only one star in their list, UZ CMa, which has been classified as a RV Tau variable, shows an amplitude of 2 mag.
Even though the amplitudes reported in \citet{chatys_etal_2019} are in photometric bands different from those of \textit{Gaia}\xspace , we find our results to be in good agreement with that study.

While amplitudes alone do not allow us to draw a conclusion on the dominant pulsation mode, models suggest that higher mass tends to suppress overtone instability \citep[e.g.][their figure~22 and 23]{trabucchi_etal_2019}.
A study of the velocity amplitude of typical representatives from this object class would add an important observational constrain to identify their pulsation modes.
If we assume the fundamental mode to be the dominant mode of pulsation in the red supergiants, models from \citet{wood_etal_1983} give typical masses between 15 and 30\,\ensuremath{{\rm M}_{\odot}}\xspace.

In addition to the paper by \citet{chatys_etal_2019} mentioned above, \citet{ren_etal_2019} recently presented PLDs for supergiants in M31 and M33.
A comparison with pulsation models allowed the latter authors to attribute the supergiants to fundamental or first overtone pulsation.
In Fig.~\ref{fig:M31M33PLD} we plot the period and \ensuremath{K_{\rm s}}\xspace values for supergiants derived by them together with the \textit{Gaia}\xspace DR2 results for the LMC.
It can be seen that the \textit{Gaia}\xspace group (d) stars fall on the relations defined by the M31 and M33 supergiants.
A similar agreement was found with the PLDs presented by \citet{chatys_etal_2019} for galactic and LMC red supergiants.

From this plot, it also becomes clear that the LSPs of red supergiants occur on timescales of a few thousand days and are therefore out of reach for \textit{Gaia}\xspace DR2.
Using the conclusions of \citet{ren_etal_2019} on the pulsation mode of these objects by comparing their location in the P-\ensuremath{K_{\rm s}}\xspace\  plane with pulsation models, we find the bulk of our group (d) stars to be fundamental mode pulsators.
If we want to use the PLD to identify the pulsation mode of a star and if we attribute fundamental mode pulsation to sequence C, we must use the P-\ensuremath{W_{\rm BP,RP}}\xspace\  diagram because it is the only one with the massive stars falling on that sequence.
For other indicators of luminosity such as \ensuremath{K_{\rm s}}\xspace or \ensuremath{W_{\rm J,K_s}}\xspace, the shift likely resulting from the mass and temperature difference relative to low-mass stars is of a size that puts these stars onto sequence C'.
A parallel sequence shifted towards shorter periods, which is seen in our study (Fig.~\ref{fig:M31M33PLD}), in the paper by \citet{ren_etal_2019} and in the work by \citet{chatys_etal_2019}, is likely formed by first overtone pulsators among the massive AGB stars and the supergiants.

\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics{Figures/RSGs_LMC_M31_M33_modGuoLi02.png}}
\caption{P-\ensuremath{K_{\rm s}}\xspace diagram combining our \textit{Gaia}\xspace DR2 results for the LMC with observations of supergiants in M31 and M33 by \citet{ren_etal_2019}. Stars from \citet{ren_etal_2019} are indicated according to the pulsation mode given there, namely open symbols for first overtone pulsators and filled symbols for stars pulsating in fundamental mode. Long secondary periods are delineated with crosses. First overtone (thin lines) and fundamental mode (thick lines) periods from RSG pulsation models by \cite{guo_li_2002} are also shown, for a few different values of mass.}
\label{fig:M31M33PLD}
\end{figure}

\section{Conclusions}
Gaia observations of the LMC and SMC
have allowed the study of the variability of red supergiants and massive AGB stars in
contrast to the extensive previous ground-based observations of OGLE and MACHO in which these
stars saturated the detectors used.
The \textit{Gaia}\xspace-2MASS diagram, constructed for the LMC, provides significant advantages in comparing the AGB and red supergiant populations of different stellar systems.
First of all, the quantity on the x-axis is independent of both distance and (global) reddening, and thus insensitive to the uncertainties in both quantities.
A comparison of the two plots in Fig.~\ref{fig:LMC_SMC_G2MD_apparent} suggests that the \textit{Gaia}\xspace-2MASS diagram is applicable to other galaxies as well.

Our diagram allows us to distinguish between stars of different mass and chemistry.
This paper focussed on the application of this distinction to the P-L relations of LPVs.
The appearance of the P-L relations depends on the spectral range of observations, so that different kinds of information can be derived from the PLD depending on the photometric pass-bands employed in its construction \citep[e.g.][their figure~1]{soszynski_etal_2007}.
In this paper, we considered four such PLDs employing the $\ensuremath{K_{\rm s}}\xspace$-band and $\ensuremath{W_{\rm J,K_s}}\xspace$ index from 2MASS, the $\ensuremath{W_{\rm BP,RP}}\xspace$ index from \textit{Gaia}\xspace DR2 photometry, and the bolometric magnitude \ensuremath{m_{\rm bol}}\xspace, respectively.

We could, for the first time, clearly show the existence of an offset between low- and intermediate-mass, oxygen-rich stars for both fundamental and 1O pulsators and thus confirm predictions from pulsation theory.
The offset is, however, not visible when using the Wesenheit index \ensuremath{W_{\rm BP,RP}}\xspace.
We explain this as the result of a temperature sensitivity inherent in the chosen combination of the \textit{Gaia}\xspace BP and RP filters.
The temperature effect is different for intermediate-mass and low-mass stars, and compensates the offset due to the mass effect.
As a consequence, sequence C' in the P-\ensuremath{W_{\rm BP,RP}}\xspace diagram is found to be the most narrow as compared to the other luminosity indicators investigated in this work.
Therefore, this relation may become the preferred approach to determine distances from the variability behaviour of AGB stars.
In addition, we possibly found a way to identify the most massive C-rich stars in our sample, using an analogy to the O-rich case.
However, this result requires validation by further tests and observational proof.

A detailed comparison with combined stellar evolution and pulsation models are presented, both for the O-rich and  C-rich cases.
We showed that synthetic photometry from these models allows us to reproduce the observed behaviour of first overtone pulsators in all our four luminosity indicators very well.
As pointed out in other papers already, the linear pulsation models used in this work are not capable of reproducing the observations for fundamental mode pulsations.

Our findings for massive AGB stars and red supergiants agree with former studies presented in the literature.
Based on this, we agree with earlier claims that the majority of stars in this group pulsate in fundamental mode.
However, the amplitude of these fundamantal mode pulsating red supergiants never reaches the large values exhibited by intermediate- and low-mass stars during their final pulsating Mira stages.
In contrast to the P-\ensuremath{K_{\rm s}}\xspace diagram and similar to the situation for intermediate-mass stars, the P-\ensuremath{W_{\rm BP,RP}}\xspace\  diagram based on \textit{Gaia}\xspace data shows this group of stars as a smooth extension of sequence C.
A few members of our group (d) seem to pulsate in first overtone mode according to their locations in the various PLDs.
For the supergiants, as for the other mass ranges, results for the two Magellanic Clouds are qualitatively similar.

\begin{acknowledgements}
The authors thank Franz Kerschbaum and Josef Hron for providing access to their photometric data archive used to derive bolometric correction relations. Furthermore, we thank Bernhard Aringer for support with stellar model spectra. MT, PM, and GP acknowledge the support from the ERC Consolidator Grant funding scheme ({\em project STARKEY}, G.A. n. 615604).\\
This work has made use of data from the European Space Agency (ESA) mission {\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia} Data Processing and Analysis Consortium (DPAC, \url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}).
Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\it Gaia} Multilateral Agreement.
This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.
This paper utilises public domain data obtained by the MACHO Project, jointly funded by the US Department of Energy through the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48, by the National Science Foundation through the Center for Particle Astrophysics of the University of California under cooperative agreement AST-8809616, and by the Mount Stromlo and Siding Spring Observatory, part of the Australian National University.
\end{acknowledgements}

\bibliographystyle{aa}

redpajama_set_name: RedPajamaArXiv


5-566 PROGRAM TRACKING AND REPORTING SUBSYSTEM (PTRS) ACTIVITY CODE. 1528 (as appropriate, for renewal, reinstatement, added rating, or upon application).
5-567 OBJECTIVE. This task determines if an applicant is eligible for a Gold Seal Flight Instructor Certificate. Successful completion of this task results in the issuance of a Temporary Airman Certificate for a gold seal flight instructor or the denial of a Gold Seal Flight Instructor Certificate.
A. Purpose of a Gold Seal Flight Instructor Certificate. Flight Instructor Certificates with distinctive gold seals are issued to instructors who currently meet certain qualifications. These certificates are intended to identify those instructors who have high personal qualifications and good records as active flight instructors. The Federal Aviation Administration (FAA) hopes to issue as many gold seal certificates as possible to provide an incentive for flight instructors to improve their qualifications.
B. Renewal and Eligibility. The holder of a Gold Seal Flight Instructor Certificate who applies for renewal does not need to reestablish his or her eligibility for a gold seal.
C. Issuance. If the applicant meets the criteria stated in subparagraph 5-571E, the aviation safety inspector (ASI) may add a gold seal to the instructor's certificate upon renewal or reinstatement, when adding an additional rating, or upon initial application.
5-569 PREREQUISITES AND COORDINATION REQUIREMENTS.
• Qualification as an ASI (Operations) or aviation safety technician (AST).
• Accident Incident Data System (AIDS).
5-570 REFERENCES, FORMS, AND JOB AIDS.
• Advisory Circular (AC) 61-65, Certification: Pilots and Flight and Ground Instructors.
• FAA Form 8060-4, Temporary Airman Certificate (Figure 5-61).
• FAA Form 8710-1, Airman Certificate and/or Rating Application (Figure 5-60).
C. Job Aids. Job Task Analysis (JTA) O3.1.54, Issue a Gold Seal Flight Instructor Certificate.
• A reliable record of flight instruction.
B. The Applicant Arrives for the Appointment.
1) Collect the documents listed above.
2) Open the PTRS Record.
C. Review FAA Form 8710-1.
2) Check the application for accuracy, using the instructions on the form attachment. The applicant is not to complete section II.
3) In section III, Record of Pilot Time, the applicant must list at least the aeronautical experience required for the Airman Certificate and rating sought. Graduates of 14 CFR part 141 pilot schools or 14 CFR part 142 training centers must provide their aeronautical experience in this section of FAA Form 8710-1, even though the graduation certificate is evidence of having completed the course of training. If aeronautical experience has no bearing on the airman certification action being sought, then it is not necessary for an applicant to complete section III. For example, with flight instructor renewal applications, flight instructor reinstatement applications, ground instructor qualification applications, pilot type rating applications, and utilization of a Bilateral Aviation Safety Agreement (BASA), aeronautical experience does not have a bearing on the airman certification action, and thus the applicant would not be required to complete section III. However, all applicants are encouraged to complete section III, because the application remains on file with the FAA and can be used to substantiate past aeronautical experience if a person were to ever lose his or her logbook.
D. Verify the Applicant's Identity. Inspect acceptable forms of identification to verify the applicant's identity (see Volume 5, Chapter 1, Section 3). Compare the identification with the personal information provided on FAA Form 8710-1.
1) If the ASI or AST can verify the applicant's identity, continue the issuing process.
2) If the ASI or AST cannot verify the applicant's identity because of a lack of or inadequate identification, explain what types of identification are acceptable. Advise the applicant to return with appropriate identification to reapply.
• A combination of the above two (two practical tests equal one trained and recommended applicant).
F. Discrepancies. If a discrepancy that cannot be immediately corrected exists in any of the documents, return the application and all submitted documents to the applicant. Inform the applicant why he or she is ineligible and explain how the applicant may correct the discrepancies.
G. The Applicant Meets Requirements. If an applicant meets all of the requirements for a Gold Seal Flight Instructor Certificate, the ASI or AST should issue FAA Form 8060-4 with the appropriate flight instructor ratings.
1) Prepare the certificate in duplicate, per Volume 5, Chapter 1, Section 5.
2) Enter the words "Gold Seal" as shown in Figure 5-61, Sample FAA Form 8060-4, Temporary Airman Certificate—Gold Seal Certificate.
3) Number the certificate with the same certificate number that appears on the applicant's superseded Flight Instructor Certificate.
4) Complete the ASI/AST certification section. Sign the reverse side of FAA Form 8710-1.
5) Forward the completed file to AFS-760.
H. Complete the PTRS Record. Complete the PTRS record in accordance with the PPM.
• Denial of issuance of a Gold Seal Flight Instructor Certificate.
NOTE: If the ASI or AST denies the applicant a Gold Seal Flight Instructor Certificate, record the action in the PTRS.
• Renewal of Flight Instructor Certificate.
• Reinstatement of Flight Instructor Certificate.
• Added rating to Flight Instructor Certificate.
RESERVED. Paragraphs 5-574 through 5-590.


Chris is one of the leading corporate lawyers in Northern Ireland, specialising in high-value and complex M&A and private equity transactions. As head of the Corporate team, he brings significant value by being able to manage multi-faceted projects and resolve issues in order to deliver the right outcome for clients.
In addition, Chris has substantial experience in all corporate matters including MBOs/MBIs, investments, joint ventures, corporate restructuring and shareholder arrangements and exits.
Chris was named Insider Young Dealmaker of the Year in 2015.
He's got a good pragmatic approach. He comes across as a deep thinker and has a strong commercial position.


Beautiful family film with a heartbreaking story. Claude Jarman Jr. (who won a mini Academy Award as outstanding juvenile actor of the year) is wonderful as a little boy being raised in the rough woodlands of Florida by his always optimistic father (Gregory Peck) and tired and worn out mother (Jane Wyman in an outstanding performance). He longs desperately to have a pet animal to take care of, and finally gets his wish when his father shoots a deer and Jarman takes her baby doe home and raises it as his own. This little deer is so adorable it even sleeps in his bed (much to the chagrin of his mother's sense of smell), but come time for the crop yielding and the lovable pet becomes a menace, threatening their survival by not allowing them to keep any of the food they work so hard to grow. The tribulations that these characters endure is presented as harshly as the family scenes are loving, and both combine to make a very satisfying work that is a three-hankie weepie if ever you've seen one. Based on the novel by Marjorie Kinnan Rawlings, and filmed on her actual estate in Florida where she wrote the book (the writing of which is the subject of the film Cross Creek).


A Xinjiang-style döner kebab in Beijing, China. Basically a Turkish döner kebab (土耳其烤肉) covered in spices from Xinjiang Cuisine (新疆菜), including cumin and chilies.
This place specifically sold "土耳其烤肉夹馍" or "Turkish ròu jiā mó". The name ròu jiā mó (肉夾饃) refers to meat sandwiches served on a bun, sometimes affectionately called "Chinese Hamburgers". Probably the most well known ròu jiā mó are from Shaanxi (陕西), served on a specific flakey "Old Tongguan" (#老潼关) bun. This "Turkish" version was just served on a folded steambun.


text:
Trump's trade chief vows to work with Democrats on NAFTA revamp
Jenny Leonard and Erik Wasson, Bloomberg News
Robert Lighthizer Photographer: Andrew Harrer/Bloomberg , Bloomberg
U.S. Trade Representative Robert Lighthizer said he's committed to working with Democrats on the new North American trade agreement, and predicted that the resulting deal will get broad approval in Congress.
"I am aware of specific areas where members have ideas to strengthen the agreement, and we are having constructive discussions," Lighthizer said in his prepared opening remarks at a Senate Finance Committee hearing on trade policies Tuesday. "I continue to believe that the USMCA will win broad support in Congress, as it's designed to do."
He made no direct mention of China trade talks in his prepared remarks, though he's likely to get grilled by lawmakers on that issue above all others.
His boss, President Donald Trump, cheered markets earlier Tuesday by saying he spoke with Chinese counterpart Xi Jinping and they'll have an "extended meeting" later this month, boosting hopes for a breakthrough in stalled negotiations.
The U.S.-Mexico-Canada agreement, which was negotiated over a year and signed by leaders in November, needs approval by legislators to take effect. Trump's surprise proposal this month to slap tariffs on all Mexican imports over border-security concerns threatened to derail the plan, but it's now back on track after the president got a migration deal and dropped his threat.
'Timely Passage'
Lighthizer already has been working closely with Democrats, particularly House Speaker Nancy Pelosi, to resolve concerns about the new accord -- including labor and environmental standards, as well as drug pricing -- before it can come up for a vote. He told senators on Tuesday that he's also spent "countless" hours consulting business, labor and other groups, and said there's wide agreement that Congress should pass the deal.
While business leaders and some in the White House are pushing for a vote before Congress goes on vacation in August, Democrats say they don't want to rush the process.
"Timely passage of implementing legislation for the USMCA will benefit American workers, farmers, ranchers, and businesses throughout the century," Lighthizer said in the remarks.
Trade will likely dominate the agenda when Canadian Prime Minister Justin Trudeau meets with Trump in Washington on Thursday. Canada is proceeding with ratification of the agreement, though time is running out to do so before an election there this fall, with Trudeau trailing in polls.
Trump says he'll meet with Xi at G20; stocks rally
Trump tariffs are set to affect fireworks, and no one's happy about it
Trudeau and Trump to meet this week on trade

redpajama_set_name: RedPajamaCommonCrawl


We are proud to announce that this Jun 1st, 1:00 PM to 4:00 PM we will be hosting an Open House at : 3849 Point McKay Road NW in the Point McKay neighborhood, Calgary. This is an opportunity to visit this excellent Townhouse for sale in beautiful Point McKay.
We are proud to announce that this May 25th, 1:30 PM to 4:00 PM we will be hosting an Open House at : 3849 Point McKay Road NW in the Point McKay neighborhood, Calgary. This is an opportunity to visit this excellent Townhouse for sale in beautiful Point McKay.
Beautifully maintained 2 storey home on 4 acres of land with a South back yard and only a short drive away from all the conveniences of the City!
I just sold this Apartment Condominium at : 211 9 Country Village Bay NE, Calgary, AB Country Hills Village.
Beautifully maintained home on a 15.2m x 33.5m lot with a South facing back yard and mature trees!
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I just sold this Detached at 37 Bridleglen Manor SW, Calgary, AB Bridlewood.


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A guide to help parents and students interested in developing a mentorship. The guidebook will help you answer such questions as: Is a mentoring partnership appropriate? How do I locate a mentor? What kind of relationship is most beneficial? Use as an information guide to help establish, maintain, and conclude a mentorship... (requires Adobe Reader)
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Mentorship: The Essential Guide for Schools and Business by Jill M. Reilly
This book describes a highly successful mentorship program in 'how to' terms that are easily applied and adaptable to a variety of situations.
Mentorships: A Guidebook by The Davidson Institute Team
Designed to help parents and students interested in developing a mentorship. The guidebook will help you answer such questions as: Is a mentoring partnership appropriate? How do I locate a mentor? What kind of relationship is most beneficial? What will the connection entail? Ideally, this guidebook is to be used as an information guide to help establish, maintain, and conclude a mentorship... (requires Adobe Reader)
NASA G.I.R.L.S. Giving Initiative and Relevance to Learning Science
Women@NASA has created a virtual mentoring project that offers a one-of-a-kind experience to middle school girls across the country. Applicants must be U.S. Citizens, grades 5-8 or home-schooled equivalent, have access to the internet and a video chat program, and willing to commit to the full 5-week program...
"QuarkNet brings high school students and teachers to the frontier of the 21st century research that seeks to resolve some of the mysteries about the structure of matter and the fundamental forces of nature." Centers currently in California and Switzerland; 60 centers expected within two years!
QuestLeadership part of the QuestBridge
Free mentorship and leadership-training program for exceptionally bright, low income students with interests in higher education, leadership, and professional careers. QuestLeadership begins with a five-week summer residential program on the Stanford campus for high school students entering their junior and senior years. Its support continues...
TeleMentor Program by Hewlett Packard
Making a difference in math, science, communication skills and career planning
Promote girls' interest and skills in science, technology, and engineering through multi-faceted programs, as well as to develop resources for teachers, role models, families and partners, for Girl Scouts in grades 5-12. Also check out the free Techbridge Family Guide, Science: It's a Familiy Affair (English / Chinese / Spanish) for great projects for families!
Through another's eyes: the pinnacle project by Rena Subotnik, in Gifted Child Today ($)
Seven highly gifted adolescents participated in a weeklong mentorship program designed to cultivate their extraordinary talents. Each of these young scholars worked individually with an established master in his or her field of interest...
WEST (Women Entering and Staying in Tech) Mentorship Project
Enables 1:1 mentorship for women in the early- to middle-stages of their technology careers. Sign up today to be notified of 2015 Application schedule and process...
Young leaders: growing through mentoring by Sandra Manning, in Gifted Child Today ($)
Describes a model for implementing a cross-age mentoring strategy such as the one described above. The goal of this strategy is to provide leadership enhancement opportunities for gifted learners in the general or gifted education classroom. This article also offers a brief review of the literature on leadership and mentoring as they relate to gifted and talented students...
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text:
Route 120 is a state highway in Connecticut, running entirely in the town of Southington. It serves as a more direct connection between the town center of Southington and the city of Meriden.

Route description
Route 120 begins at an intersection with Route 322 in southeastern Southington, just west of the Meriden city line and about  from an interchange with I-691. It heads in a northwest direction, crossing Misery Brook about  later, passing by the St. Thomas Cemetery, then intersecting with Route 364 after another . Route 120 ends at an intersection with Route 10 in the center of town after another . The entire length of Route 120 is two lanes wide and is known as Meriden Avenue. Route 120 is classified as an urban major collector road and carries an average daily traffic volume of 8,900.

History
In the 1920s, the direct Southington-Meriden route was designated as a secondary state highway known as Highway 326. The old highway was renumbered to Route 120 as part of the 1932 state highway renumbering. The route has remained mostly unchanged since then.

Junction list

References

External links

120
Transportation in Hartford County, Connecticut
Southington, Connecticut

redpajama_set_name: RedPajamaWikipedia