text:
\section{Introduction}
\label{sec:INTRO}

The recent identification of massive black-hole binaries with masses up to $80\,M_\odot$ as powerful sources of gravitational waves \citep{abb2:16,abb1:16} opened the first direct view on the end products of the evolution of the most massive stars. To be able to exploit this new information it is necessary to model stellar populations that are young enough to contain Very Massive Stars (VMS) with masses $\gtrsim$\,$100\,M_\odot$. Because of the steep increase of stellar luminosities with increasing mass, VMS in this regime have luminosities well in excess of $10^6\,L_\odot$. The resulting high $L/M$ ratios $\gtrsim 10^4\,L_\odot/M_\odot$ are bringing such VMS close to the Eddington limit where the radiative acceleration $a_\mathrm{rad}$ equals the gravitational acceleration $g$.

Due to the resulting low effective surface gravities VMS are subject to physical effects affecting their outer envelope structure and stellar winds. The envelopes of VMS are predicted to be subject to the envelope inflation effect, which leads to the formation of radially extended low-density envelopes \citep{ish1:99,pet1:06}. In the limit of low convective efficiency inflated envelopes become unstable above a specific $L/M$ ratio because of a lack of hydrostatic envelope solutions \citep{gra1:12}. This leads to numerical problems in stellar evolution models, which are often tackled by an artificial increase of the convective efficiency \citep[cf.][]{pax1:13}. As a result the inflation effect is suppressed in most evolutionary models. Relaxing this assumption leads to the formation of relatively cool stars with inflated envelopes at the top of the main sequence \citep{gra1:12,koe1:15,san1:15}.

One key question addressed in this work is whether the existence of the inflation effect can be corroborated by observations. This is particularly interesting because the inflation effect may be related to the still enigmatic group of luminous blue variables (LBVs) and their S-Dor type (radius) variability on timescales of months to years \citep[cf.][]{gra1:12,gra1:20}.

The enhanced mass loss of VMS, and its transition from optically thin O-star winds to optically thick Wolf-Rayet winds have been investigated by numerical wind models \citep{gra1:08,vin1:11} that are supported by systematic empirical studies of samples of VMS \citep{gra1:11,bes1:14}. \citet{gra1:11} pointed out that the underlying reason for the enhanced mass loss of WR stars is their proximity to the Eddington limit, and not their surface chemistry as usually assumed in stellar evolution models. The enhanced mass loss is the reason why VMS often appear as emission-line stars, namely as hydrogen-rich Wolf-Rayet (WNh) stars, even during their main-sequence evolution \citep{ham1:06,sch1:08,sch1:09,cro1:10}. Due to their extreme luminosities the spectral signatures of such very massive WNh stars are detectable in young starbursts even at cosmological distances \citep{wof1:14,gra1:15}.

The winds of WNh stars are often confused with those of classical Wolf-Rayet (WR) stars. Classical WR stars are massive stars near the end of their evolution, predominantly in the phase of core He-burning. These stars have lost their hydrogen-rich envelopes during their previous evolution and have much lower masses. They reach high Eddington factors due to the high luminosity of their He-burning cores. Classical WR stars have different physical and wind properties than the more massive core hydrogen-burning WNh stars, but they are usually treated with the same mass-loss prescriptions in evolutionary models.

In this work we construct evolutionary models that take the enhanced mass-loss rates and the envelope inflation of VMS into account. We compute model grids for massive main-sequence stars up to $500\,M_\odot$ and compare the results with the so far best studied sample of VMS obtained in the VLT-Flames Tarantula Survey \citep[VFTS;][]{eva1:11}. The VFTS is a near-complete spectroscopic survey of $\approx$\,$1000$ hot luminous stars in 30\,Dor in the LMC, the closest analogue of a young starburst in our neighbourhood, with a well-controlled brightness cut-off. All single stars in the VFTS sample have been analysed individually, and their star-formation history (SFH) and rotational-velocity distribution have been investigated in detail \citep{sch2:18,sch4:18,ram1:13,duf1:13}. This makes it an ideal target for detailed population synthesis modelling, enabling quantitative predictions of the expected numbers of stars in specific evolutionary phases.

In the following we describe our evolutionary models in Sect.\,\ref{sec:MESA}, and the population synthesis in Sect.\,\ref{sec:POP}. In Sect.\,\ref{sec:30DOR} we compare our models with the observed population of VMS in 30\,Dor, and in Sect.\,\ref{sec:DISC} we discuss the results. The most important conclusions are summarised in Sect.\,\ref{sec:CON}.

\section{Stellar evolution models}
\label{sec:MESA}

We use the Modules for Experiments in Stellar Astrophysics (MESA) version { 11701} \citep{pax1:11,pax1:13,pax1:15,pax1:18,pax1:19} to compute single-star evolutionary models covering the main-sequence phase from zero to 99\% hydrogen exhaustion in the stellar core. The same input physics as in \citet{sch1:18,sch1:19} are used, except for the treatment of convection in the outer stellar envelope and stellar-wind mass loss (see below). In particular, we adopt a wind-scaling as a function of $X_{\rm Fe}/X_{{\rm Fe}, \odot}$, as described by \citeauthor{sch1:19} We use the MESA default chemical composition from \citet{gre1:98} scaled to an initial metallicity of $Z=0.008$ (compared to the corresponding solar metallicity of $Z_\odot = 0.0169$) to match the metallicity, and in particular $X_{\rm Fe}$, in the LMC.

Because of our interest in the envelope inflation effect we employ standard mixing-length theory (MLT) instead of the modified treatment of superadiabatic convection in radiation-dominated regions \citep[MLT++,][]{pax1:13} that is used by default in MESA models. Furthermore, we introduce a new treatment of stellar-wind mass loss to meet the requirements for VMS near the Eddington limit. In the following sections these new mass-loss relations are described.

\subsection{The winds of OB stars}
\label{sec:OB}

The most widely used mass-loss prescription for OB stars is the relation by
\citet{vin1:00,vin1:01}. These theoretical mass-loss estimates are based on a
Monte-Carlo approach that makes use of the first-order Sobolev
approximation. In the Sobolev approximation spectral lines are assumed to be
infinitely narrow, but their flux-mean opacity is high due to the effective
line broadening through Doppler shifts in the wind. Consequently, the
radiative acceleration depends predominantly on the velocity gradient
$\partial\varv/\partial r$, leading to the same wind physics as in \citet[][CAK]{cas1:75} who described the first closed theory for the formation of optically thin OB-star winds.

Notably, the Sobolev approximation does not apply in the limit of large wind
optical depths where the photon mean-free path becomes small, and Doppler
shifts become inefficient \citep[cf.\ the discussion in][]{gra1:17}. In this
work we therefore only apply the \citeauthor{vin1:01} relation in the
optically thin regime. Furthermore, we change the absolute value and the slope
of $\dot{M}$ as a function of luminosity $L$, to match the average observed mass-loss
rates of O\,stars in 30\,Dor as in \citet[][]{bes1:14}.

For the absolute calibration of our mass-loss relations with empirical results in Sect.\,\ref{sec:MDOT30} we adopt a wind clumping factor of $D=10$. This means that we assume that the wind consists of dense clumps with a density that is a factor $D$ higher than the mean density, while the inter-clump medium is void. Clumping constitutes the main uncertainty in the empirical mass-loss rates of OB and WR stars. It increases the line strengths of recombination lines, and therefore reduces the empirical mass-loss rates derived from this type of lines by a factor $\sqrt{D}$ \citep{hil1:91,ham1:98}.

For the most massive O and WNh stars in 30\,Dor \citet{bes1:14} found clumping
factors of the order of $D=10$, based on the relative strengths of the
electron-scattering wings of strong recombination lines
\citep[cf.\ also][]{vin1:12}. For O stars with weaker winds even higher
clumping factors up to $D \approx 100$ have been discussed, based on the
strength of non-saturated UV resonance lines \citep{ful1:06}. However, taking into account
the influence of optically thick clumps on the UV line profiles
\citet{osk1:07,sun1:14} derived more moderate values of the order of 10. Here
we adopt $D=10$ for all stars in our sample, resulting in a reduction of the
empirical mass-loss rates by a factor 3.3 compared to un-clumped winds.

Our comparison with observations in Sect.\,\ref{sec:MDOT30} leads to a
considerable reduction of O-star mass-loss rates, and a shallower slope as a
function of $L$. For the slope we adopt the value of 1.45 derived by
\citet{bes1:14} and replace it in Eq.\,24 of \citet{vin1:01} instead of the
original value of 2.194. This substitution already lowers the mass-loss rates in the VFTS O-star range
considerably. To match the observations, we further apply a constant factor of
0.8 to the altered mass-loss relation. For the dependence of $\dot{M}$ on the
rotational velocity we use the standard relation implemented in MESA
\citep[Eq.\,26 in][]{pax1:13}

\subsection{The winds of WNh stars}
\label{sec:GAMMA}

The most massive and luminous main-sequence stars are thought to appear spectroscopically as WNh stars \citep{gra1:08,cro1:10}. WNh stars are therefore the extension of the O-star regime towards higher masses and luminosities. Previous implementations of wind mass loss in stellar evolution models do not take the transition from the optically thin winds of OB stars to the optically thick winds of WNh stars into account. Instead, they extrapolate the \citeauthor{vin1:00} relations to higher luminosities. { Only when the surface composition becomes substantially deficient in hydrogen,} the mass-loss relations switch to Wolf-Rayet type mass loss, where no distinction between the mass-loss of very massive WNh stars, and the mass loss of classical core He-burning WR stars is made. { For instance, in the default 'DUTCH' MESA wind-scheme the empirical mass-loss relation from \citet{nug1:00} is used for hot stars ($T_{\rm eff} > 10$\,kK) with hydrogen surface mass fractions $X_{\rm s} < 0.4$, otherwise the \citeauthor{vin1:00} relations are used.}

In this work we follow previous theoretical \citep{gra1:08,vin1:11} and empirical \citep{gra1:11,bes1:14} studies of the mass loss properties of VMS that suggest that the proximity to the Eddington limit determines the formation of optically thick WR-type winds. According to these studies WR-type winds show  a distinct, strong dependence on the proximity to the Eddington limit, which is most easily expressed via a dependence on the classical Eddington factor
\begin{equation}
 \Gamma_{\rm e}=\frac{\chi_{\rm e}L}{4\pi c GM}
  = 10^{-4.813} \times \left( 1 + X_{\rm s} \right) \frac{L/L_\odot}{M/M_\odot}.
\label{eq:GAMMA_E}
\end{equation}
Here we define $\Gamma_{\rm e}$ as the ratio between the radiative acceleration on free electrons (with the electron-scattering opacity $\chi_{\rm e}$ for a fully ionised plasma) and the gravitational acceleration of the star. Because $\chi_{\rm e}$ depends only on the number of free electrons per gram, $\Gamma_{\rm e}$ depends only on fundamental stellar parameters, namely the hydrogen surface abundance $X_{\rm s}$ and the $L/M$ ratio. In this picture the mass loss of VMS, which are still in the core hydrogen burning phase, differs from the mass loss of classical core He-burning Wolf-Rayet stars because of their different $L/M$ ratios, effective temperatures, and chemical compositions.

In the following we concentrate on VMS at LMC metallicity. We employ the empirical mass-loss relation for VMS in 30\,Dor from \citet{bes1:14} who derived a relation of the form
\begin{equation}
\label{eq:BEST}
\log(\dot{M}_{\rm thick}) = 5.22 \times \Gamma_{\rm eff} - 0.5 \times \log(D)
  -2.6,
\end{equation}
for the optically-thick winds of WNh stars. Here $D$ denotes the wind clumping factor (cf.\ Sect.\,\ref{sec:MDOT30}), and $\Gamma_{\rm eff}$ the effective Eddington factor including centrifugal acceleration \citep[cf.\ the discussions in][]{gra1:08,gra1:15}. As $\Gamma_{\rm eff}$ varies over the stellar surface we use an intermediate value which lies between the value at the poles, where the centrifugal acceleration is zero, and the equator where $g_{\rm rot} = \Omega^2R$
\begin{equation}
 \Gamma_{\rm eff}= \Gamma_{\rm e} + 0.5 \times \Gamma_{\rm rot}
  \equiv \Gamma_{\rm e} + 0.5 \times \frac{\Omega^2R^3}{GM}.
\end{equation}
{ As a result, our mass-loss prescription is sensitive to a combination of the proximity to the Eddington limit and the critical rotation rate similar to the $\Gamma\Omega$-limit discussed by \citep{mae1:00,lan1:97}.}

As a criterion to switch between the regimes of WR-type winds and optically thin OB-star winds we evaluate their sonic-point optical depth $\tau_{\rm s}$. This approach is motivated by the theory of optically thick winds, where the properties of winds with large $\tau_{\rm s}$ are largely constrained by the conditions at the sonic point and $\tau_{\rm s}$ itself \citep{nug1:02,gra1:17}. As described in Sect.\,\ref{sec:OB}, the different wind physics for the optically thin winds of OB stars (with small values of $\tau_{\rm s}$) leads to lower mass-loss rates.

To decide whether a stellar model with given surface values of $M, L, R, \Omega$ and $X_{\rm s}$ can develop a WR-type wind we estimate the value of $\tau_{\rm s}$ that its wind {\em would have} if it had an optically thick wind. To this purpose we compute $\dot{M}_{\rm thick}$ from Eq.\,\ref{eq:BEST} and estimate the terminal wind velocity $\varv_\infty$ using the empirical relation from \citep[][]{lam1:95} with $\varv_\infty/\varv_{\rm esc}^{\rm eff} = 2.51$ \citep[cf.\ also][]{bes1:14}. For given $\dot{M}$ and $\varv_\infty$ we estimate $\tau_{\rm s}$ using a relation between $\tau_{\rm s}$ and the wind efficiency factor $\eta \equiv \dot{M}\varv_\infty / (L/c)$ from \citet{gra1:17}. $\eta$ denotes the ratio between the mechanical wind momentum and the momentum of the radiation field driving the wind. It is related to $\tau_{\rm s}$ via the number of absorption and re-emission processes needed to drive the wind. \citet{gra1:17} derived an approximate relation of the form
\begin{equation}
\label{eq:TAUS}
    \tau_{\rm s} \approx \frac{\dot{M}\varv_\infty}{L/c}\left(1-\frac{\varv_{\rm esc}^2}{\varv_\infty^2}\right).
\end{equation}

In our models we use $\tau_{\rm s}$ from Eq.\,\ref{eq:TAUS} as a criterion for the onset of WR-type mass loss. For cases where $\tau_{\rm s} < \tau_1$ we use the mass-loss prescriptions for optically thin winds from Sect.\,\ref{sec:OB}, and for $\tau_{\rm s} > \tau_2$ we use the $\Gamma$-dependent mass-loss relation Eq.\,\ref{eq:BEST}. For $\tau_1 < \tau_{\rm s} < \tau_2$ we interpolate between both cases. As demonstrated in Sect.\,\ref{sec:POP} we can reproduce the observed mass-loss rates for VMS in 30\,Dor from \citet{bes1:14} very well with values of $\tau_1=2/3$ and $\tau_2=1$.

\section{Population synthesis models}
\label{sec:POP}

In this section we describe the population synthesis models that we use to compare the models from Sect.\,\ref{sec:MESA} with observations.

\subsection{Stellar evolution models}
\label{sec:STD}

Each of our population synthesis models is based on a set of stellar evolution models for massive stars with initial masses in the range of 9-500\,$M_\odot$ and initial rotational velocities between 0 and 90\,\% of the critical rotation rate.  Typically, each model grid consists of models with 18 different masses and 11 different rotational velocities, that is, a total of 198 models. As start models we use rotating, chemically homogeneous models in thermal equilibrium, with a default initial composition with $Z = 0.008$ (cf.\ Sect.\,\ref{sec:MESA}). The models cover the main-sequence phase up to 99\% hydrogen exhaustion.

\subsection{Population synthesis}

The goal of our population synthesis models is to compute the fraction of stars in a specific stellar population that show one or more observables $o$ in a given parameter range $o_i$. The population is defined through a given initial mass function $\xi(M) = \frac{{\rm d}N}{{\rm d}M}$, initial rotational velocity distribution $P(\varv_{\rm ini})=\frac{{\rm d}N}{{\rm d}\varv_{\rm ini}}$, the star-formation history  $S(t)=\frac{{\rm d}N}{{\rm d}t}$, and the total number of stars $N$.

Our models provide stellar parameters $p$ as a function of time $t$ for each evolutionary track with a given initial mass $M$ and initial fraction of the critical rotation rate $\omega \equiv \Omega/\Omega_{\rm crit}$. We further determine the main-sequence lifetime $T_{\rm MS}$ for each evolutionary track and define the relative age $t_{\rm r} \equiv t/T_{\rm MS}$, so that the $p$ are given as a function of the form $p(t_{\rm r}, M, \omega)$ on the model grid. To compute the $o_i$ we transform the $p(t_{\rm r}, M, \omega)$ onto a homogeneous grid that is logarithmic in $M$ and linear in $\omega$ and $t_{\rm r}$, through linear interpolation between evolutionary tracks. In this step we typically use 100 grid points in $M$ and $t_{\rm r}$, and 64 grid points in $\omega$, where $t_{\rm r}$ lies between 0 and 1, $M$ between 9 and $500\,M_\odot$, and $\omega$ between 0 and 0.9. Based on the $p(t_{\rm r}, M, \omega)$ we then compute any observable $o(t_{\rm r}, M, \omega)$ of interest on the same grid.

To compute the probability distribution of the different $o_i$ for the adopted stellar population we need to compute a weight function $W(t_{\rm r}, M, \omega)$ that takes the star formation history $S(t)$ and the initial distributions of $M$ and $\omega$ into account. The probability $P_i$ to find an observable $o$ in a given parameter range $o_i$ is then
\[
P_i = \frac{ \sum_{t_{\rm r}, M, \omega} W(t_{\rm r}, M, \omega) \left|_{o(t_{\rm r}, M, \omega) \in o_i}\right. }
{ \sum_{t_{\rm r}, M, \omega} W(t_{\rm r}, M, \omega) },
\]
where the summation is a sum over all grid points, and the weight $W(t_{\rm r}, M, \omega)$ takes the initial distribution of $M$ and $\omega$ and the sampling of $t_{\rm r}$, $M$ and $\omega$ into account. We write  $W(t_{\rm r}, M, \omega)$ in the form
\[
W(t_{\rm r}, M, \omega) = W_t(t_{\rm r}) \times W_M(M) \times W_\omega(\omega).
\]
and compute $W_t$, $W_M$ and $W_\omega$ in the following way.

As we use linear time steps $\Delta t_r$, $W_t$ is given by the product of the star formation history $S(t)$ and the derivative ${\rm d}t/{\rm d}t_r=T_{\rm MS}$
\[
W_t = S(t) \times T_{\rm MS}(M,\omega),
\]
with $t=t_{\rm r} \times T_{\rm MS}(M, \omega)$.

For $M$ we use a logarithmic grid with ${\rm d}M/{\rm d}\log(M) = M$, and adopt
an initial mass function $\xi(M) \propto M^{-\gamma}$, so that
\[
W_M = \xi(M) \times \frac{{\rm d}M}{{\rm d}\log(M)} \propto M^{1-\gamma}.
\]

For $\omega$ we use a linear grid, but the initial rotational distribution $P(\varv_{\rm ini})$ is given as a function of the rotational velocity, so that
\[
W_\omega = P(\varv_{\rm ini}) \times \frac{{\rm d} \varv_{\rm ini}}{{\rm d} \omega}.
\]
Here we compute ${{\rm d} \varv_{\rm ini}}/{{\rm d} \omega}$ numerically, where we define $\varv_{\rm ini}(M,\omega) \equiv \varv_{\rm rot}(t_r=0.01, M, \omega)$ to avoid problems due to the rapid adjustment phase of the internal rotation profile at the start of each evolutionary sequence.

\section{Very massive stars in 30\,Dor}
\label{sec:30DOR}

In this section we compare the results of our population synthesis with the empirically derived properties of the stellar population in 30\,Dor. The empirical data consist of the sub-sample of single main-sequence stars from the VLT-Flames Tarantula Survey \citep[VFTS;][]{eva1:11} as compiled by \citet{sch2:18}. In particular, the empirical data are based on detailed spectroscopic analyses of the most massive stars in 30\,Dor from \citet{bes1:14}, O-type giants and supergiants from \citet{ram1:17}, O-type dwarfs from \citet{sab1:17}, B-type supergiants from \citet{mce1:15} and B-dwarfs from \citet{sch2:18}.

For the population synthesis we used the star-formation history $S(t)$ and initial mass function (IMF) $\xi(M)$ for 30\,Dor from \citet{sch2:18}, together with the rotational velocity distribution $P(\varv)$ for single O stars from \citet{ram1:13}. Analogous to the work of \citeauthor{sch2:18} the present work thus focuses on single stars with a luminosity $\gtrsim$\,$10^{4.5} L_\odot$, well above the VFTS brightness cut-off. This means that only stars with masses $\gtrsim$\,15\,$M_\odot$ and ages $\lesssim$\,12\,Myr are considered in the present analysis.

We further employed evolutionary models for single stars as described in Sect.\,\ref{sec:MESA}, tailored to match the empirical properties of the massive single-star population in 30\,Dor. Because of our focus on the physics of very massive stars these models are designated as VMS models in the remainder of this work. We compare these models with empirical data from VFTS and models based on alternative input physics and/or common standard assumptions.

\subsection{Mass distribution}

\label{sec:NvsL}

\begin{figure}
    \centering
    \includegraphics[scale=0.53]{NvsL.pdf}
    \caption{{ Observed luminosity distribution of single stars in 30\,Dor compared to theoretical predictions. The histogram shows} the observed numbers of single stars in the VFTS sample as function of luminosity (blue) compared to the theoretical distribution resulting from our VMS population synthesis { models} described in Sect.\,\ref{sec:30DOR} (green). Both distributions are plotted transparent so that overlaps appear in dark green.}
    \label{fig:NvsL}
\end{figure}

An important input parameter for our population synthesis models is the exponent $\gamma$ of the IMF with $\xi(M)\propto M^{-\gamma}$. In Fig.\,\ref{fig:NvsL} we compare the present-day luminosity distribution resulting from our population synthesis with the observed distribution in the VFTS sample. In this plot we normalised the synthetic distribution to match the total number of 288 observed, putatively single stars above $\log(L/L_\odot) = 4.5$. We further adjusted $\gamma$ to match the number of 20 observed VMS in the range $6.0 < \log(L/L_\odot) < 6.6$ (roughly corresponding to a mass range of 70--200\,$M_\odot$). For our VMS models this was achieved using a value of $\gamma=1.977$, slightly higher than the value of $\gamma=1.90_{-0.26}^{+0.37}$ derived by \citet{sch2:18}. Using alternative models employing the { DUTCH MESA wind scheme (cf.\ Sect\,\ref{sec:GAMMA})} we obtain $\gamma = 1.935$, in agreement with \citeauthor{sch2:18}\ Altogether our models confirm a top-heavy IMF in comparison to the standard \citet{sal1:55} IMF with $\gamma=2.35$, but with a moderate impact of the adopted mass-loss prescription on the present stellar masses and therefore on $\gamma$. In the remainder of this work we use $\gamma=1.977$ to achieve consistency between our models and observations.

Notably, there are no stars with luminosities higher than $\log(L/L_\odot) =6.6$ in the VFTS sample while our VMS models predict 4.7 stars above this luminosity. This upper luminosity limit corresponds to an upper initial-mass limit of $M_{\rm ini}\approx 200\,M_\odot$. We will discuss the relevance of this upper mass limit in Sect.\,\ref{sec:ulim}. For the investigation of the physical properties of VMS in the remainder of this work, we consider the consistency between observed and predicted stellar numbers below $200\,M_\odot$ as satisfactory.

\subsection{Mass loss rates}
\label{sec:MDOT30}

\begin{figure}[tbp!]
    \centering
    \includegraphics[scale=0.45]{DMOM_VMS.pdf}
    \caption{Wind-momentum -- luminosity relation for O and WNh stars in 30\,Dor. Different symbols indicate empirical results for the most massive stars in 30\,Dor from \citet{bes1:14} in blue, O-type giants and supergiants from \citet{ram1:17} Tables C.4 (dark green) and C.5 (light green), and O-type dwarfs from \citet{sab1:17} Tables A.1 (red) and A.2 (pink). Upper limits are indicated in grey. Evolutionary tracks are indicated by coloured lines analogous to Fig.\,\ref{fig:HRD} but restricted to the O\,star range. Predicted absolute numbers of stars according to our population synthesis models are mapped and colour coded as indicated by the colour bar on the right.}
    \label{fig:DMOM_VMS}
\end{figure}

\begin{figure}[tbp!]
    \centering
    \includegraphics[scale=0.45]{DMOM_Dutch.pdf}
    \caption{Wind-momentum -- luminosity relation as in Fig.\,\ref{fig:DMOM_VMS} compared with population synthesis models based on standard MESA models using the DUTCH wind scheme.}
    \label{fig:DMOM_DUTCH}
\end{figure}

\begin{figure}[tbp!]
    \centering
    \includegraphics[scale=0.45]{GAMMA_VMS.pdf}
    \caption{{ Mass loss as function of Eddington factor.}  Mass-loss rates $\dot{M}$ as a function of the estimated classical Eddington factor for chemically homogeneous stars $\Gamma_{\rm e}^{\rm hom}$ (see text) analogous to Fig.\,\ref{fig:DMOM_VMS}. The  $\dot{M}(\Gamma_{\rm e})$ relation for R\,136 from \citet{bes1:20} is indicated by the dashed curve.}
    \label{fig:GAMMA_VMS}
\end{figure}

To test the validity of the mass-loss prescriptions employed in our models we compare the mass-loss relations resulting from our population synthesis models directly with empirical data. In Figs.\,\ref{fig:DMOM_VMS} and \ref{fig:DMOM_DUTCH} the resulting modified wind momenta $D_{\rm mom} = \dot{M}\varv_\infty\sqrt{R/R_\odot}$ are shown as a function of luminosity $L$. According to the CAK theory for optically thin winds, OB stars are expected to follow a tight wind-momentum luminosity relation \citep[WLR;][]{kud1:99}, largely independent of secondary parameters such as mass or radius.

In Figs.\,\ref{fig:DMOM_VMS} and \ref{fig:DMOM_DUTCH} we indicate the expected numbers of stars in this diagram according to our population synthesis models by colour maps. The synthetic $D_{\rm mom}$ were computed based on mass-loss rates $\dot{M}$, radii $R$ and stellar masses $M$ from the underlying stellar evolution models. The $\varv_\infty$ were computed from the resulting effective escape velocities using a ratio of $\varv_\infty/\varv_{\rm esc}^{\rm eff} = 2.51$ as described in Sect.\,\ref{sec:GAMMA}. The empirical wind momenta were compiled from \citet{ram1:17,sab1:17,bes1:14} and cover the complete sample of single O and WNh stars in the VFTS, i.e., they include all apparently single main-sequence stars with effective temperatures $> 29$\,kK. The same temperature cut-off has been applied to the theoretical results.

Our VMS models in Fig.\,\ref{fig:DMOM_VMS} show a good agreement with observations. In particular, the location of the transition from O-star winds to the enhanced mass-loss in the WNh phase is reproduced well. On the other hand, the WLR resulting from the oft-used DUTCH wind scheme in Fig.\,\ref{fig:DMOM_DUTCH} matches the WNh stars well, but over-estimates the mass-loss rates of the O stars in the VFTS sample. Also, the slope of the WLR appears to be too steep in the O-star range. For both wind schemes it is noticeable that the synthetic WLRs are much narrower than the observed one. In particular for O stars the observed WLR shows a large spread in $D_{\rm mom}$, where dwarfs tend to have lower wind momenta than giants and supergiants by up to a factor 10. As we will show later, this may hint on substantially reduced mass-loss rates for younger O stars.

A novelty in our VMS models is the treatment of the transition between the regimes of optically thin O-star winds and optically thick WNh winds. This transition occurs qualitatively differently in the VMS and DUTCH approaches. In the VMS wind scheme in Fig.\,\ref{fig:DMOM_VMS} stars with luminosities above $\log(L/L_\odot)\approx 5.5$ reach high enough Eddington factors at some point during their main-sequence evolution to perform a quick transition to enhanced WR-type mass-loss rates. In the WNh regime the evolution then proceeds at nearly constant luminosity (cf.\ Fig.\,\ref{fig:HRD}). On the other hand, the DUTCH models in Fig.\,\ref{fig:DMOM_DUTCH} show no marked transition between the two regimes but change their behaviour at some point during the WNh stage, when they reach a helium surface mass fraction of $Y_{\rm s}=0.4$. Before this point they evolve continuously with increasing luminosity from the O-star regime into the WNh stage. After they reached $Y_{\rm s}=0.4$ their mass loss rates increase, and their evolution continues with decreasing luminosity.

As noted earlier, the absolute mass-loss rates in the WNh range are reproduced well by both mass-loss prescriptions. One notable outlier in Figs.\,\ref{fig:DMOM_VMS} and \ref{fig:DMOM_DUTCH} is VFTS\,108, which has a WR-type mass-loss rate despite of its low luminosity of $\log(L/L_\odot)=5.7$. This star is highly enriched in helium ($Y_{\rm s}\approx0.8$) and most likely already in the core helium-burning phase. \citet{bes1:14} demonstrated that this star matches the adopted $\dot{M}(\Gamma_{\rm eff})$ relation (Eq.\,\ref{eq:BEST}) precisely, if a lower mass, corresponding to the core helium-burning case, is adopted.

In Fig.\,\ref{fig:GAMMA_VMS} we put the $\Gamma_{\rm e}$-dependent mass-loss prescription (Eq.\,\ref{eq:BEST}) in our VMS models to a direct test. As $\Gamma_{\rm e}$ is not directly observable, we estimate the Eddington factor $\Gamma_{\rm e}^{\rm hom}(L, X_{\rm s})$ for the chemically-homogeneous case instead, where we compute $\Gamma_{\rm e}^{\rm hom}(L, X_{\rm s})$ from Eq.\,\ref{eq:GAMMA_E} using the mass estimate $M_{\rm hom}(L, X_{\rm s})$ for a chemically-homogeneous star from \citet[][]{gra1:11}. Notably, $\Gamma_{\rm e}^{\rm hom}(L, X_{\rm s})$ depends only on observable quantities. In reality, most stars are of course not chemically homogeneous, but have a chemical profile with an increasing mean molecular weight towards their centre. For fixed values of $M$ and $X_{\rm s}$ this results in a higher luminosity than for a chemically homogeneous star. The other way round, for given $L$ and $X_{\rm s}$, $M_{\rm hom}(L, X_{\rm s})$ constitutes a strict upper limit to the real mass of a star, and $\Gamma_{\rm e}^{\rm hom}$ a lower limit to $\Gamma_{\rm e}$.

In Fig.\,\ref{fig:GAMMA_VMS} we plot the synthetic and observed mass-loss rates $\dot{M}$ for our O/WNh sample as a function of  $\Gamma_{\rm e}^{\rm hom}$. To allow for a direct comparison between theory and observations, the $\Gamma_{\rm e}^{\rm hom}(L, X_{\rm s})$ are computed in an identical manner for models and observations. Analogous to the WLR in Fig.\,\ref{fig:DMOM_VMS} the theoretical mass-loss rates increase rapidly as soon as the formation of optically thick winds becomes possible. The transition between O and WR-type winds happens at $\log(\Gamma_{\rm e}^{\rm hom}) \approx -0.5$ and continues with a steeper slope for high $\log(\Gamma_{\rm e}^{\rm hom})$, in good agreement with the observations. As our $\Gamma_{\rm e}$-dependent mass-loss prescription (Eq.\,\ref{eq:BEST}) was initially derived from the same data under the assumption that $\Gamma_{\rm e} = \Gamma_{\rm e}^{\rm hom}$ \citep[cf.\ ][]{bes1:14} this can be seen as a confirmation that most WNh stars in our sample are in fact evolving chemically homogeneously.

Again, the overall agreement between models and observations in the WNh stage (with $\log(\dot{M}/\,M_\odot\,\mbox{yr}^{-1}) \gtrsim -5.2$) is very good, except for a short phase directly after the transition. In this short phase the surface is still more hydrogen-enriched than the stellar core, and thus $\Gamma_{\rm e} > \Gamma_{\rm e}^{\rm hom}$. However, as soon as the enhanced mass loss in the WNh phase sets in, the H-rich surface layers are quickly removed, and the stars become nearly chemically homogeneous.

While the O-star mass-loss rates in Fig.\,\ref{fig:GAMMA_VMS} (with $\log(\dot{M}/\,M_\odot\,\mbox{yr}^{-1}) \lesssim -5.2$) show a good agreement with observations for $\Gamma_{\rm e}^{\rm hom} \gtrsim -1$, the agreement gets worse for lower values of $\Gamma_{\rm e}^{\rm hom}$. For the O dwarfs in this regime \citet{sab1:17} derived upper limits for $\dot{M}$ which are indicated in grey. For O (super)giants \citet{ram1:17} provided explicit values for $\dot{M}$ that tend to be higher than those in our models. Notably, \citeauthor{ram1:17} did not provide absolute values for $D_{\rm mom}$ for the same objects (cf.\ Figs.\,\ref{fig:DMOM_VMS} and \ref{fig:DMOM_DUTCH}) because they considered the results as uncertain (H.\,Sana \& A.\,de\,Koter private communication). Depending on whether the mass-loss rates in this parameter range can be believed or not the differences between O dwarfs and (super)giants, which were already apparent in Fig.\,\ref{fig:DMOM_VMS}, may be even more pronounced for these objects. This is further supported by recent results of \citet{bes1:20,bes2:20} who investigated the mass-loss properties of the most massive stars in the R\,136 cluster in the centre of 30\,Dor. These authors determined a very young age of 1--2\,Myr for R\,136 and derived an $\dot{M}(\Gamma_{\rm e})$ relation which lies up to two orders of magnitude below the values derived for O stars the VFTS sample as indicated by the dashed curve in Fig.\,\ref{fig:GAMMA_VMS}.

\subsection{Location in the Hertzsprung-Russell Diagram}
\label{sec:HRD}

\begin{figure}[tbp!]
    \centering
    \includegraphics[scale=0.45,trim = 1cm 0. 0. 0.]{HRD.pdf}
    \caption{Theoretical Hertzsprung-Russell diagram for { VMS models with standard MLT}. Coloured areas indicate the expected number of stars according to our population synthesis models for 30\,Dor (cf.\ Sect.\,\ref{sec:30DOR}). Evolutionary tracks for $\Omega/\Omega_{\rm crit} = 0.3$ are indicated, labelled with ZAMS masses in $M_\odot$.}
    \label{fig:HRD}
\end{figure}

\begin{figure}[tbp!]
    \centering
    \includegraphics[scale=0.45,trim = 1cm 0. 0. 0.]{OBS.pdf}
    \caption{Observed Hertzsprung-Russell diagram for 30\,Dor. Coloured areas indicate the observed number of stars, evolutionary tracks are the same as in Fig.\,\ref{fig:HRD}.}
    \label{fig:OBS}
\end{figure}

\begin{figure}[tbp!]
    \centering
    \includegraphics[scale=0.45,trim = 1cm 0. 0. 0.]{HRD_MLT++.pdf}
    \caption{{ Influence of envelope inflation. Theoretical Hertzsprung-Russell diagram for models with reduced envelope inflation effect through an} artificially enhanced convective efficiency (MLT++; MESA option {\tt okay\_to\_reduce\_gradT\_excess = .true.}). }
    \label{fig:MLT++}
\end{figure}

In Fig.\,\ref{fig:HRD} we show the theoretical HRD for 30\,Dor including a colour coded map of the expected numbers of stars according to our population synthesis models. For comparison, we show the observed HRD in Fig.\,\ref{fig:OBS} with the observed number of stars colour coded in the same way. Generally, the observed HRD positions of stars above $\log(L/L_\odot) \approx 5$ are well reproduced by our VMS models. The lack of evolved stars below this luminosity in Fig.\,\ref{fig:HRD} is caused by the maximum age of 12\,Myr adopted in our population synthesis models. Beyond this, our VMS models reproduce the observed bending of the zero-age main sequence (ZAMS) for the highest masses, and the lack of stars with cooler temperatures than $\log(T_{\rm eff}/{\rm K})\approx 4.5$ above $\log(L/L_\odot) \approx 6$ very well.

The bending of the ZAMS towards cooler temperatures at the top of the main sequence is a consequence of the envelope inflation effect. This is demonstrated in Fig.\,\ref{fig:MLT++} where the inflation effect has been suppressed by artificially increasing the convective efficiency in radiation-dominated stellar envelopes (MESA option {\tt okay\_to\_reduce\_gradT\_excess = .true.}). As a consequence, the most massive stars above $\approx 80\,M_\odot$ become much hotter, and evolve towards even hotter temperatures during their main-sequence evolution. The reason is that stars in this regime become chemically homogeneous and thus more and more compact throughout their evolution. Only when standard MLT is used (Fig.\,\ref{fig:HRD}) the inflation effect compensates this effect and brings the effective temperatures of stars in this mass range in agreement with observations.

\begin{figure}[t!]
    \centering
    \includegraphics[scale=0.45,trim = 1cm 0. 0. 0.]{UHRD.pdf}
    \caption{{ Effective temperature distribution.} Theoretical Hertzsprung-Russell diagram for the upper main sequence analogous to Fig.\,\ref{fig:HRD} but as a function of the effective temperature relative to the ZAMS temperature.}
    \label{fig:UHRD}
\end{figure}

\begin{figure}[t!]
    \centering
    \includegraphics[scale=0.45,trim = 1cm 0. 0. 0.]{UOBS.pdf}
    \caption{Observed Hertzsprung-Russell diagram analogous to Fig.\,\ref{fig:UHRD}.}
    \label{fig:UOBS}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.45,trim = 1cm 0. 0. 0.]{UHRD_OB.pdf}
    \caption{{ Influence of mass loss.} Theoretical Hertzsprung-Russell diagram analogous to Fig.\,\ref{fig:UHRD} but adopting a pure OB-type mass-loss prescription without enhanced $\Gamma$-dependent mass loss.}
    \label{fig:UHRD_OB}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[scale=0.45,trim = 1cm 0. 0. 0.]{UHRD_DUTCH.pdf}
    \caption{{ Influence of mass loss.} Theoretical Hertzsprung-Russell diagram analogous to Fig.\,\ref{fig:UHRD} but adopting the DUTCH MESA wind scheme.}
    \label{fig:UHRD_DUTCH}
\end{figure}

In Figs.\,\ref{fig:UHRD} and \ref{fig:UOBS} we focus on the distribution of stars in the upper HRD, relative to the ZAMS. To demonstrate how the predicted HRD positions depend on mass loss we show models with alternative mass-loss prescriptions in Figs.\,\ref{fig:UHRD_OB} and \ref{fig:UHRD_DUTCH}. In Fig.\,\ref{fig:UHRD_OB} the onset of the enhanced mass loss of WNh stars has been completely omitted while otherwise the same wind scheme as in our VMS models was used. Consequently, the mass loss of the most massive stars is reduced, and the outer H-rich layers are removed much more slowly, leading to higher hydrogen surface mass fractions $X_{\rm s}$ and therefore higher values of $\Gamma_{\rm e}$ (cf.\ Eq.\,\ref{eq:GAMMA_E}). Due to the strong dependence of the inflation effect on $\Gamma_{\rm e}$ \citep[cf.][Eqs.\ 20, 28, 29]{gra1:12}, this leads to an enhanced envelope inflation and correspondingly cooler stellar temperatures.

The models in Fig.\,\ref{fig:UHRD_DUTCH} are computed using the standard DUTCH wind scheme with much higher mass-loss rates than our VMS models. Because of the higher mass loss the most massive stars become more compact and less massive, and therefore hotter and less luminous. The qualitatively correct reproduction of the observed HRD positions of the most massive stars with $\log(L/L_\odot) \gtrsim 6$ by our VMS models can therefore be seen as an indirect confirmation of our new mass-loss prescription, and in particular of the adopted clumping factor $D$ in Eq.\,\ref{eq:BEST}.

For stars with $\log(L/L_\odot) \lesssim 6$ our models predict very extended envelopes and correspondingly cool stellar temperatures towards the end of the main-sequence evolution, irrespective of the mixing approach (cf.\ Figs.\,\ref{fig:HRD} and \ref{fig:MLT++}). In the observed HRD (Fig.\,\ref{fig:UOBS}) this region is populated by B-supergiants whose evolutionary origin is still unclear \citep[e.g.,][]{vin1:10,mce1:15}. In Fig.\,\ref{fig:UOBS} there are only few objects of this type that are clearly located in the Hertzsprung-gap, that is, outside the expected parameter range for main-sequence stars.

\begin{figure}
    \centering
    \includegraphics[scale=0.53]{NvsT.pdf}
    \caption{{ Effective temperature distribution.} Histogram of the observed numbers of single stars in the VFTS sample with $5 < \log(L/L_\odot) < 6$ as a function of their distance from the ZAMS ($\log(T_{\rm eff}/T_{\rm ZAMS})$) (transparent blue) compared with the theoretical distribution from our VMS population synthesis models (transparent green).}
    \label{fig:NvsT}
\end{figure}

To quantify the contribution of the envelope inflation effect we compare in Fig.\,\ref{fig:NvsT} the distribution of stars between $5 < \log(L/L_\odot) < 6$ as a function of their distance from the ZAMS ($\log(T_{\rm eff}/T_{\rm ZAMS})$) with observations. Our models predict 22.4 stars with $\log(T_{\rm eff}/T_{\rm ZAMS}) < -0.2$ compared to 30 stars that are observed. This means that we expect the majority of stars in this regime to be evolved main-sequence stars with inflated envelopes. Setting the maximum effective temperature for stars to be classified as B-type stars to 29\,kK, we predict 22.4 stars in this phase to appear as B supergiants, while 34 stars are observed. Again, according to our models the majority of B supergiants between $5 < \log(L/L_\odot) < 6$ are evolved main-sequence stars.

While the overall agreement between theory and observation in Fig.\,\ref{fig:NvsT} appears satisfactory, it is notable that there is a small systematic shift in $T_{\rm eff}/T_{\rm ZAMS}$ between the observed and predicted median locations of the main sequence, by 0.025\,dex. In principle, this could be an indication that the predicted core masses in our models are too small, similar to what has been claimed by \citet{tka1:20}. For this reason, we varied the overshooting parameter analogous to \citeauthor{tka1:20} as well as the parameters for rotational mixing in our models but did not achieve a satisfactory agreement between theory and observations. Also, the relatively high oxygen abundance in our models \citep[based on the solar composition from][]{gre1:98} does already bring our models towards slightly lower effective temperatures than what would result from more recent estimates \citep[such as][]{asp1:09}. Therefore, it remains unclear whether the observed shift between theory and observations is due to deficiencies in our models or systematic errors in the spectroscopic analyses of the VFTS sample.

\subsection{Helium surface enrichment}

\label{sec:HE}

\begin{figure}[t]
    \centering \includegraphics[scale=0.45]{XHE.pdf}
    \caption{{ Helium surface enrichment through mass loss.} Helium surface mass fraction $Y$ as a function of the estimated inverse mass-loss timescale $\dot{M}/M_{\rm hom}$ for stars with $T_{\rm eff} > 30$\,kK and $\log(L/L_\odot)>5$, compared with VFTS O and WNh stars labelled as in Fig.\,\ref{fig:DMOM_VMS}. The colour scale indicates the number of objects predicted by our population synthesis. Evolutionary tracks from Fig.\,\ref{fig:HRD} are indicated by coloured lines.}
    \label{fig:XHE}
\end{figure}

In Fig.\,\ref{fig:XHE} we show a comparison between the observed and predicted
helium surface mass fractions $Y_{\rm s}$ for stars with $T_{\rm eff} >
29$\,kK, i.e., for O-type stars and WNh stars in the VFTS, as a function of
their estimated inverse mass-loss timescale $\dot{M}/M_{\rm hom}$. Again, for
a meaningful comparison between models and observations, the mass estimate
$M_{\rm hom}(L,X)$ from \citet{gra1:11} has been computed in an identical manner for models and observations.

Our population synthesis models show two regions with substantially enhanced values of $Y_{\rm s}$ in Fig.\,\ref{fig:XHE}. The WNh stars with $\log{\dot{M}/M_{\rm hom}} \gtrsim -7.5$ follow a clear trend of increasing $Y_{\rm s}$ for increasing $\dot{M}/M_{\rm hom}$ \citep[cf.\ also][]{bes1:14,bes2:20}. The reason is that the enhanced mass loss in the WNh regime efficiently removes the hydrogen rich outer layers, leading to quasi chemically homogeneous evolution with a strong dependence of $\dot{M}$ on $\Gamma_{\rm e}$, and therefore on $Y_{\rm s}$.

\begin{figure}[t]
    \centering
    \includegraphics[scale=0.45]{ROT.pdf}
    \caption{{ Helium surface enrichment through rotation.} Helium surface mass fraction $Y_{\rm s}$ as a function of rotational velocity for { O}\,stars with $T_{\rm eff} > 29$\,kK and $5.0 < \log(L/L_\odot) < 5.9$, compared with observed $Y_{\rm s}$ and $\varv \sin i$  of He-enriched VFTS O stars (with $Y_{\rm s}>0.35$) labelled as in Fig.\,\ref{fig:DMOM_VMS}; stars with $\log(L/L_\odot)<5.0$ are plotted in grey. The colour scale indicates the number of objects predicted by our population synthesis models. Evolutionary tracks for an initial mass of 30\,$M_\odot$ are indicated by coloured lines. Step functions indicate the initial (blue) and present (orange) total number distributions in arbitrary units, as well as the present distribution of He-enriched stars with $Y_{\rm s}>0.35$ (red).}
    \label{fig:ROT}
\end{figure}

In the O-star range, with lower values of $\dot{M}/M_{\rm hom}$, our models
predict substantial He enrichment for initially fast rotating stars. This is
demonstrated in Fig.\,\ref{fig:ROT} where we show the He enrichment for O
stars with $5 < \log(L/L_\odot) < 5.9$ (i.e., excluding WNh stars) as a
function of rotational velocity. According to our models, fast rotating stars
in this regime develop strong winds { due to their proximity to the
$\Gamma\Omega$-limit}. This leads to a phase of enhanced mass loss which removes
angular momentum until the rotational velocity falls below a value of
typically $300$\,km/s. As a consequence, there is a peak in the rotational
velocity distribution for O stars at $\varv_{\rm rot} \approx 300$\,km/s
(cf.\ the orange step function in Fig.\,\ref{fig:ROT}). Notably, \citet{ram1:13} found a similar feature in the observed rotational velocity distribution of the VFTS O\,star sample, however, at a higher velocity of $\approx 400$\,km/s.

According to our models the rotational braking associated with this process
induces substantial mixing of helium-rich material to the surface, with a
maximum in the number distribution of He-enriched stars at $\approx 300$\,km/s
(cf.\ the red step function in Fig.\,\ref{fig:ROT}). The same objects
constitute the group of helium enriched stars with $\log{\dot{M}/M_{\rm hom}}
\lesssim -7.5$ in Fig.\,\ref{fig:XHE}. According to our models we expect 14.6 O\,stars with $Y_{\rm s} > 0.35$ in the luminosity range $5.0 < \log(L/L_\odot) < 5.9$ in the VFTS sample which have been enriched by rotational mixing. This compares to 7 observed objects in the same regime which are indicated by coloured symbols in Fig.\,\ref{fig:ROT}. For lower luminosities we expect 0.7 objects, compared to 5 observed objects which are indicated by grey symbols in Fig.\,\ref{fig:ROT}.

Notably, VFTS\,285, the star with the highest value of $\varv \sin i = 600$\,km/s in our sample and $Y_{\rm s} = 0.36$ belongs to the group of low-luminosity stars for which no He-enhancement is predicted (VFTS\,285 is not visible in Fig.\,\ref{fig:ROT}). In both, the high- and low-luminosity regime, we expect the vast majority of He-enriched objects near a rotational velocity of $\approx 300$\,km/s. Due to the effect of random inclination angles the expected average value of $\varv \sin i$ is $\pi/4$ \citep[cf.][]{bro1:11}, with a decreasing distribution function for small values of $\sin i$. We therefore expect the distribution of He-enriched stars to peak near 235\,km/s, while the observed $\varv \sin i$ distribution appears evenly distributed, with a potential peak near $100$\,km/s. This means that neither the predicted numbers of rotationally enriched stars, nor their distribution agrees with observations. The implementation of rotational mixing and/or rotationally induced mass loss in our models therefore seems to leave space for improvements (cf.\ the discussion in Sect.\,\ref{sec:HE_disc}).

\section{Discussion}
\label{sec:DISC}

In the following we discuss the implications of our results for the physics and evolution of the most massive stars (Sect.\,\ref{sec:WNh_disc}), their upper mass limit (Sect.\,\ref{sec:ulim}), the evolution of OB stars (Sect.\,\ref{sec:OB_disc}), and the mechanisms for surface He-enrichment in the upper HRD (Sect.\,\ref{sec:HE_disc}).

\subsection{WNh stars: the most massive stars}

\label{sec:WNh_disc}

According to our analysis, at LMC metallicity, stars in excess of $\approx 80\,M_\odot$ enter the WNh stage at some point during their main-sequence evolution. This means that they reach high enough Eddington factors to initiate optically thick winds, so that they appear as WR-type emission-line stars. Because they still have hydrogen on their surface, they are traditionally classified as H-rich WN stars, or WNh stars. As shown in Sect.\,\ref{sec:MDOT30}, the absolute mass-loss rates, as well as the onset of WR-type mass-loss in this regime, are well described by our VMS models employing a $\Gamma_{\rm e}$-dependent mass-loss relation.

Because of their enhanced mass loss, the outer radiative layers of WNh stars are quickly removed so that they develop a quasi chemically homogeneous structure, with hydrogen surface abundances starting at near-solar values and almost reaching hydrogen depletion at the end of their main-sequence evolution. This is in stark contrast to the traditional treatment of WR-type mass loss in stellar evolution models, where enhanced mass loss only sets in beyond a limit of $Y_{\rm s} > 0.4$.  In Sect.\,\ref{sec:HE} we have shown that our VMS models describe the resulting He-enrichment through the enhanced mass loss in the WNh stage well, in good agreement with observations.

Because of their proximity to the Eddington limit, WNh stars are subject to the envelope inflation effect. Therefore, their HRD positions are shifted towards cooler temperatures. For masses $\gtrsim 80\,M_\odot$ the ZAMS is bent towards cooler temperatures due to this effect. As we have shown in Sect.\,\ref{sec:HRD}, our VMS models with standard MLT reproduce this effect very well, in agreement with the observed HRD positions. In particular, there appears to be no need to invoke a clumped envelope structure to explain their radii as it has been proposed for the inflated envelopes of classical WR stars \citep{gra1:12,gra2:13}.

A further consequence of the inflation effect is that WNh stars evolve towards cooler temperatures, i.e., towards the right side in the HRD, although they are close to chemical homogeneity. Again, the HRD positions following from our VMS models agree with observations. Models that employ the oft-used modified treatment of super-adiabatic convection in radiation-dominated regions \citep[MLT++,][]{pax1:13} clearly fail to reproduce the observed HRD positions of WNh stars. They show much hotter temperatures than observed and evolve towards the left side in the HRD (cf.\,Figs.\ \ref{fig:OBS} and \ref{fig:MLT++}).

The general agreement of our models with the observed population of WNh stars in 30\,Dor in terms of numbers, HRD positions, mass loss and surface enrichment strongly support their evolutionary status as VMS in the phase of core H-burning.

\subsection{A variable upper stellar mass limit}

\label{sec:ulim}

The first estimate of an upper limit for the masses of stars of $\approx$\,$150\,M_\odot$ was determined by \citet{fig1:05}, based on photometric data of the Arches cluster in the Galactic Centre, one of the youngest and densest clusters in the Galaxy. Later \citet{cro1:10} analysed the brightest WNh stars in R\,136, the central cluster of 30\,Dor, and found luminosities up to $\log(L/L_\odot) = 6.94\pm0.1$, corresponding to initial masses up to $300\,M_\odot$. Because of its high stellar density R136 has not been included in the VFTS sample but \citet{bes2:20} analysed the 55 brightest stars in R\,136 based on new HST spectroscopy of \citet{cro1:16} and revised the upper luminosity limit to $\log(L/L_\odot) = 6.8\pm0.1$, corresponding to an upper mass limit of $\approx 250\,M_\odot$.

According to our present models the brightest WNh stars evolve with near-constant luminosity, implying slightly higher initial masses than the ones derived by \citeauthor{bes2:20}\ As a consequence, a value of $\log(L/L_\odot) = 6.8$ corresponds almost precisely to $M_{\rm ini} = 300\,M_\odot$. As we have shown in Sect.\,\ref{sec:NvsL}, the VFTS sample displays an upper luminosity cut-off at a slightly lower value of $\log(L/L_\odot)\approx 6.6$ (cf.\ Fig.\,\ref{fig:NvsL}), corresponding to an initial mass of $\approx 200\,M_\odot$. According to our population synthesis models we would expect $4.7$ stars above $\log(L/L_\odot)=6.6$ and 1.8 stars above $\log(L/L_\odot)=6.8$ without assuming an upper mass limit. If the R\,136 upper mass limit would be universal, we would therefore miss 3.1 stars with $6.6 < \log(L/L_\odot) < 6.8$ in the VFTS sample. Given that there are 20 stars with $\log(L/L_\odot)>6.0$, the probability to miss 3 stars with $6.6 < \log(L/L_\odot) < 6.8$ by chance would be $(17/20)^{20} = 0.039$. If there were no uncertainties in the individual luminosity measurements this would indicate a significant difference of the upper mass limits in R\,136 and the remainder of 30\,Dor.

Now the relevant most massive stars in both samples are spectroscopically remarkably similar objects of subtype WN5h which have been analysed by \citet{bes1:14} for the VFTS sample and \citet{bes2:20} for R\,136. In both works individual uncertainties of $\pm 0.1$\,dex in $\log(L)$ were estimated which are of course significant compared to the difference in the maximum luminosities of 0.2\,dex. However, we think that the {\em relative} uncertainties in the derived $\log(L)$ may be smaller than the individual errors because of the similarity of the objects, and because they have been analysed by the same person using the same method of analysis. Moreover, the brightest star in the VFTS sample, VFTS\,1025 alias R\,136c is in fact a member of R\,136 with a luminosity of $\log(L/L_\odot) = 6.58$. The second brightest star, VFTS\,682 has a much lower luminosity of $\log(L/L_\odot) = 6.51$, and has been discussed as a slow runaway from R\,136 \citep{ren1:19}. In any case the VFTS population excluding R\,136c clearly displays lower luminosities than $\log(L/L_\odot) = 6.6$ compared to R\,136 with its brightest member R\,136a1 with $\log(L/L_\odot) = 6.78$.

As \citet{bes2:20} report 3 stars with $6.6 < \log(L/L_\odot) < 6.8$ in R\,136, in agreement with expectations for a top-heavy IMF, dynamical mass segregation cannot account for the lack of the same number of stars in the VFTS sample. Instead, the higher mass limit for R\,136 could imply that R\,136 initially formed stars with higher masses, depending on the local conditions in the star-forming region. This could mean that the stars in 30\,Dor and R\,136 already formed in a mass-segregated way \citep[cf.][]{pav1:19}. Alternatively, the most massive stars in R\,136 could be the result of successive stellar mergers due to the high stellar density in the cluster \citep[e.g.][]{por1:04}.

\subsection{OB stars}
\label{sec:OB_disc}

\subsubsection{Mass-loss rates}

For the mass-loss rates of OB stars we employed a modified form of the \citeauthor{vin1:01} recipe adjusted to match the empirical mass-loss rates in the VFTS sample with an adopted clumping factor of $D=10$. To this purpose we adopted a shallower dependence on $L$ from \citet{bes1:14} and used overall reduced mass-loss rates. In Sect.\,\ref{sec:MDOT30} we have shown that this mass-loss relation does indeed match the average O\,star mass-loss rates in the VFTS sample, however, it fails to reproduce their observed wide spread. In particular, the O\,dwarfs analysed by \citet{sab1:17} show systematically lower wind momenta and mass-loss rates than the O\,(super)giants analysed by \citet{ram1:17}. This trend of reduced mass-loss rates for younger objects becomes even more prominent when the VFTS results are compared with the mass-loss relation from \citet{bes1:20} for O and WNh stars in R\,136. The stars in R\,136 are only about 2\,Myrs old, and therefore even younger than most of the stars in the VFTS sample. Notably, the mass-loss rates for O\,stars in R\,136 are up to two orders of magnitudes lower than those found in the VFTS sample (cf.\ Fig.\,\ref{fig:GAMMA_VMS}), possibly indicating a very strong trend of reduced mass-loss rates for young O dwarfs.

A reduction of O-star mass-loss rates is also reported in recent theoretical works that employ more elaborate techniques to compute the radiative force than the oft-used first-order Sobolev approximation as in \citeauthor{vin1:01}\ Employing higher-order Sobolev models \citet{owo1:99} reported a drop in the line source function near the sonic point due to rapid changes in the velocity gradient. \citet{gra4:03} reported a similar effect in non-LTE models employing a co-moving frame (CMF) radiative transfer causing a local drop in the radiative force near the sonic point. More recent mass-loss predictions for OB stars employing higher-order Sobolev or CMF techniques by \citet{luc1:07,krt1:17,sun1:19} report similar effects and predict much lower mass-loss rates than those of \citeauthor{vin1:00}\ One could speculate that young stars with high $\log g$ are more susceptible to this effect because a high $\log g$ implies stronger changes in the velocity gradient. Physical effects occurring in more evolved stars that are not included in current wind models, such as turbulence caused by sub-surface convection \citep{can1:09}, could help to overcome the associated local drop in the radiative force for these objects, and help to bring their mass-loss rates closer to the values of \citeauthor{vin1:00} as it is observed for the O supergiants in the VFTS sample \citep[cf.][]{ram1:17}.

\subsubsection{HRD positions}

In agreement with previous studies by \citet{koe1:15, san1:15} we find that luminous OB stars are subject to the inflation effect, albeit less dependent on the detailed treatment of convection than for the more luminous WNh stars. As a consequence, such objects appear as B\,supergiants during their main-sequence evolution, in regions of the HRD that previously may have been attributed to the Hertzsprung-gap. In Sect.\,\ref{sec:HRD} we have shown that, according to our population synthesis models, 22 out of the 34 bright B\,supergiants in the VFTS sample are expected to be inflated main-sequence stars. However, there are some (at least 4) objects in the VFTS sample whose HRD positions cannot be explained by our models, so that alternative explanations such as blue loops or binary interaction may still be needed to explain the complete B\,supergiant population.

Notably, our population synthesis models are also able to reproduce the lack of B\,supergiants with high luminosities ($\log(L/L_\odot) \gtrsim 6$) in the VFTS sample. In Sect.\,\ref{sec:HRD} we identified the removal of H-rich layers due to the onset of enhanced WR-type mass loss as the reason for this effect.  The successful reproduction of the VFTS population in the HRD is therefore also an indirect confirmation of the absolute mass-loss rates and clumping factors adopted in our models.

Previous works such as \citet{gra1:12,gra1:18} associated stars with inflated envelopes with the S\,Doradus type variability of Luminous Blue Variables (LBVs). The fact that we expect about 20 objects in the VFTS sample to be in this phase, but none of the observed stars shows LBV-type behaviour, implies that envelope inflation is not {\em generally} associated with the LBV phenomenon, in particular not in the H-burning phase. This could still mean that the inflation effect is associated with LBVs, but under more rare conditions such as fast rotation, or in very late (pre-SN) evolutionary stages \citep[cf.][]{gro1:06,gra1:12,gro1:13}.

Finally, a clumped density structure inside inflated stellar envelopes has been proposed by \citet{gra1:12,gra2:13} to explain the observed radii of classical core He-burning WR stars. Our present analysis shows that there is no need to invoke similar clumped sub-surface regions to explain the radii of B supergiants through the inflation effect.

\subsection{Surface helium enrichment through mass loss and rotation}
\label{sec:HE_disc}

As outlined in Sect.\,\ref{sec:WNh_disc} the surface He-enrichment through the strong mass loss of WNh stars is well reproduced by our models. On the other hand, the He-enrichment through rotational mixing appears to be over-predicted by our models. According to our models we expect to find 15 O stars with $Y_{\rm s} > 0.35$ in the VFTS sample that are He-enriched through rotational mixing. These objects are expected to have high rotational velocities ($\approx 300$\,km/s) and high luminosities $5.0 < \log(L/L_\odot) < 5.9$. In reality, we find 12 He-enriched objects that are concentrated at low values of $\varv \sin i$ of $\approx 100$\,km/s, five of which are located at luminosities of $\log(L/L_\odot) < 5.0$. Therefore, the observed sample appears to be largely incompatible with our model predictions, implying a much lower number of truly rotationally enriched O stars.

This result implies that either the internal rotational mixing is over-predicted in our models, or that the rotationally enhanced mass loss, which induces mixing, is too high. A large part of the 15 He-enriched stars in the VFTS sample could therefore be false detections (such as un-identified multiple systems) or products of alternative processes such as binary interaction or mergers \citep[cf.][]{dem1:14}. Whether the observed discrepancy can be explained through changes in internal mixing parameters, or the previously discussed lower mass-loss rates for young O stars, remains to be shown in future works.

\section{Conclusions}
\label{sec:CON}

The implementation of new mass-loss relations in combination with standard-MLT envelope physics in our present evolutionary models leads to a successful reproduction of the properties of the most massive stars ($\gtrsim 80$--$100\,M_\odot$) in 30\,Dor, including their HRD positions, mass-loss rates, and surface He-enrichment. In particular the transition between optically thin O-star winds and the optically thick winds of WNh stars is well reproduced.

For the less massive O\,stars (with $\approx25$--$100\,M_\odot$) we find much more diverse mass-loss properties than previously thought, with indications for substantially reduced mass-loss rates for young O dwarfs. Also, the surface helium enrichment in the O-star range does not appear to be well understood, as we find no clear indications for rotational mixing of helium in contrast to our model predictions. Both phenomena may be related as reduced mass-loss rates for young fast-rotating stars could lead to less rotational braking and mixing.

According to our models the distribution of stellar luminosities in 30\,Dor is in good agreement with the top-heavy IMF from \citet{sch2:18}, with a very modest dependence on the adopted mass-loss prescription. However, the upper luminosity cut-off in the greater 30\,Dor region is lower than for its central cluster R\,136, implying an upper initial mass limit of $\approx 200\,M_\odot$ compared to $\approx 300\,M_\odot$ in R\,136. This could be an indication of a general dependence of the upper stellar mass limit on the conditions in the star-forming environment.

Finally, the observed number distribution of stars as a function of effective temperature supports the existence of inflated stellar envelopes that are theoretically predicted to form when main-sequence stars reach high L/M ratios. This puts the majority of bright B\,supergiants in the end phase of core hydrogen burning where they display cool temperatures due to their large, inflated radii.

\begin{acknowledgements}
  { GG thanks the Deutsche Forschunsgemeinschaft (DFG) for financial support under grant No. GR 1717/5-1, the Stellar Evolution group at the Argelander Institute for Astronomy in Bonn for sharing their expertise in stellar modelling and providing computational resources, and the anonymous referee for reviewing this work.}
\end{acknowledgements}

redpajama_set_name: RedPajamaArXiv


Discussion in 'VOLuminous' started by countvolcula, Oct 3, 2011.
From what I have gathered in the seedy corners of the recruiting forums, we should receive good news on Thursday. O'Brien should give his pledge to play for UT.
You knew to start a thread about it.
Yeah - things are sounding pretty good. It could obviously go against us still, but I feel a lot better than I did three weeks ago. OCI has said the staff has felt confident all along - sounds like the staff may have snagged him a while back, fought of Michigan over the late summer while O'Brien dealt with the home-state school, and ultimately hung onto him...at least that's how I hope it goes down.
Glad to have you aboard. Hopefully Danny will follow.
Dooley has to go to Michigan to get a DT. Can't get one from a football state in the South. Don't be surprised when this guy busts.
Fulmer was really killing it with DT's the last five years or so.
Where was our best one from... Hawaii?
Let me know when Dooley lands an Albert Haynesworth, John Henderson type DT even once, let alone repeatedly.
I don't understand your picture.
I guess we won't have to worry about Dooley winning more than Phil did with those two.
Dooley has only had one full recruiting cycle under his belt at UT.
Ask Phil to explain it to you.


text:
Monica Agosta: Love at First Light
The best way to describe Monica Agosta's photography is love at first sight - whether it was the first time she saw an Ansel Adams image or the excitement she felt when she received her first camera for free after mailing in several Bazooka gum wrappers at age eight - love is infused in her artistry.
Agosta worked as a photo assistant for the Associated Press in Manhattan during the news service's 150th anniversary in 1998. She also served as a photo assistant at Gannett's Journal News, during it's "birth" that same year, and had some of her work published weekly in the gardening section of the paper. In 1991, she interned at Eyewitness News, WABC-TV on the assignment desk and learned the video production process, skills that added to the depth of her photography.
Agosta received her degree in Radio/Television production at the State University of New York at New Paltz in 1991
© 2020 Monicamera
Monica Agosta Photography

redpajama_set_name: RedPajamaCommonCrawl


text:
Student maintenance grant to be protected in Budget 2015
The student maintenance grant is set to be protected from further cutbacks in tomorrow's Budget, Trinity News can reveal. It is understood that the back-to-education allowance, designed to help single parents, the disabled and unemployed back into education, will also be ring-fenced.
Overall, spending on education is set to increase for the first time since 2010, with the departments of health and social protection also set to see their budgets increase.
Previously, Minister for Education and Skills, Jan O'Sullivan, had been non-committal about maintaining student supports at their current levels, telling the Dáil only last Tuesday that "changes to any public service, including the student grant scheme, cannot be ruled out."
However, it is likely that the student contribution charge will still be increased by €250 in Budget 2015, despite strong opposition from a number of backbench Labour TDs. While both Fine Gael and Labour pledged at the last election not to increase the registration fee from its then annual rate of €2,000 – Labour even pledged to return them to the €1,500 level last paid in 2010 – Minister O'Sullivan has been evasive about the charges' future.
A face-to-face poll of 130 students carried out by Trinity News last week revealed deep-seated opposition to any further increases in the charge. 37% of respondents judged the charge's likely rise to €3,000 to be the "correct amount" but 60% thought the figure already "too high".
The Union of Students in Ireland (USI) has been campaigning to protect student supports in recent months and several thousand students from across Ireland attended its Wednesday rally to demonstrate against further cuts to the education sector.
Speaking to this paper last night, USI president, Laura Harmon, said that the union is "cautiously optimistic about the protection of students' supports in Tuesday's Budget. Students' unions across the country have been campaigning on this issue since the summer and we hope that work will pay off."
The proposed increases to certain departments' budgets comes in the wake of somewhat more upbeat economic news than in recent months. Unemployment has dropped steadily since last October's Budget, falling from 12.3% to 11.1%. This is its lowest level since February 2009, slightly below the Eurozone average of 11.4%, although youth unemployment remains stubbornly high at 25.2%, a drop of a mere 0.6% in a year.
Growth in the economy has also picked up with Irish Gross Domestic Product (GDP) expanding 7.7% in the second quarter of 2014, the largest single increase since 2007. As a consequence, the budget deficit has fallen to 3.7% of GDP and is set to fall to 2.4% next year. The extra spending is likely to push the deficit back up to 2.7% but this remains crucially below the 3% maximum allowed under European Union law.


RAW NITRO SHOCK ABSORBERS WILL OUTLAST MANY OF ITS WELL KNOWN RIVALS AT A COMPETITIVE PRICE.
RAW SHOCK ABSORBERS COME WITH A 2 YEAR 40000KM WARRANTY.
This product was added to our catalog on Monday, 14. November 2011.


text:
TIPS & EXPERIENCES
Best day-trips from Bangkok
Ana Lorens
Asia / Thailand
The capital of Thailand, Bangkok, is located right in the Ecuador of the country. It is a clear turning point between the paradisiacal beaches of the south and the lush…
The most beautiful temples in Bangkok
Did you know that there are more than 40,000 Buddhist temples in Thailand? Many of these temples are located in Bangkok, the capital. For this reason, if you are visiting…
One-day-trip to the West Bank
Asia / Palestine
Doing a trip to the West Bank means soaking up its history, returning a few years to the past and being wide-eyed all the time. It is, without a doubt,…
Practical Guide to travel to Israel: What you need to know
Asia / Israel
Israel is a country with enormous cultural wealth and one of the most prominent destinations in the Middle East. I suppose that if you have are reading this publication it…
What and where to eat in Israel: Traditional cuisine
Asia / Gastronomy / Israel
Israel is a very special country. It is well-known for many reasons, one of them is its delicious typical food, influenced by other styles of cuisine, such as Jewish and…
Israel Travel Budget: 10-day-trip
Israel is reputed to be a very expensive destination, which puts a lot of people back when it comes to organizing a trip around the country. The truth is that…
FOLLOW MY TRIPS ON INSTAGRAM
Home sweet home 🏡 . 🇪🇸 Se acabaron los di
Starting 2021 in this beautiful town in the north
Charming village in the Valencian Community 🌸 .
Bits of my lovely hometown ✨ . 🇪🇸 Qué bie
It's nice to be back home 💖! . 🇪🇸 Feliz
(English below) Exprimiendo al max los últimos d
This site uses cookies to improve your browsing experience, do you accept them? Settings and policy (read more)OK
Cookies policy and privacy
Read full cookie policy
Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that guarantee basic functionalities and security features of the website. These cookies do not store any personal information.
Cookies that may not be particularly necessary for the website to function and are specifically used to collect personal data of the user through analysis, advertisements and other integrated content, are called unnecessary cookies. It is mandatory to obtain the user's consent before executing these cookies on their website.


text:
namespace gloperate_qtquick
{

QmlScriptContext::QmlScriptContext(QmlEngine * engine)
: m_engine(engine)
{
}

QmlScriptContext::~QmlScriptContext()
{
}

void QmlScriptContext::initialize(cppexpose::ScriptContext * scriptContext)
{
    m_scriptContext = scriptContext;
}

void QmlScriptContext::addGlobalObject(cppexpose::Object * obj)
{
    m_engine->addGlobalObject(obj);
}

void QmlScriptContext::removeGlobalObject(cppexpose::Object * obj)
{
    m_engine->removeGlobalObject(obj);
}

cppexpose::Variant QmlScriptContext::evaluate(const std::string & code)
{
    QQmlExpression expr(m_engine->rootContext(), nullptr, QString::fromStdString(code));

    return m_engine->fromQVariant(
        expr.evaluate()
    );
}

} // namespace gloperate_qtquick

redpajama_set_name: RedPajamaGithub


text:
/* */
'use strict';

var acorn = require('acorn');
var walk = require('acorn/dist/walk');

// polyfill for https://github.com/marijnh/acorn/pull/231
walk.base.ExportNamedDeclaration = walk.base.ExportDefaultDeclaration = function (node, st, c) {
  return c(node.declaration, st);
};
walk.base.ImportDefaultSpecifier = walk.base.ImportNamespaceSpecifier = function () {};

function isScope(node) {
  return node.type === 'FunctionExpression' || node.type === 'FunctionDeclaration' || node.type === 'Program';
}
function isBlockScope(node) {
  return node.type === 'BlockStatement' || isScope(node);
}

function declaresArguments(node) {
  return node.type === 'FunctionExpression' || node.type === 'FunctionDeclaration' || node.type === 'ArrowFunction';
}
function declaresThis(node) {
  return node.type === 'FunctionExpression' || node.type === 'FunctionDeclaration';
}

function reallyParse(source) {
  try {
    return acorn.parse(source, {
      ecmaVersion: 6,
      allowReturnOutsideFunction: true,
      sourceType: 'module'
    });
  } catch (ex) {
    if (ex.name !== 'SyntaxError') {
      throw ex;
    }
    try {
      return acorn.parse(source, {
        ecmaVersion: 6,
        allowReturnOutsideFunction: true
      });
    } catch (ex) {
      if (ex.name !== 'SyntaxError') {
        throw ex;
      }
      return acorn.parse(source, {
        ecmaVersion: 5,
        allowReturnOutsideFunction: true
      });
    }
  }
}
module.exports = findGlobals;
module.exports.parse = reallyParse;
function findGlobals(source) {
  var globals = [];
  var ast = typeof source === 'string' ?
    ast = reallyParse(source) :
    source;
  if (!(ast && typeof ast === 'object' && ast.type === 'Program')) {
    throw new TypeError('Source must be either a string of JavaScript or an acorn AST');
  }
  var declareFunction = function (node) {
    var fn = node;
    fn.locals = fn.locals || {};
    node.params.forEach(function (node) {
      fn.locals[node.name] = true;
    });
    if (node.id) {
      fn.locals[node.id.name] = true;
    }
  }
  walk.ancestor(ast, {
    'VariableDeclaration': function (node, parents) {
      var parent = null;
      for (var i = parents.length - 1; i >= 0 && parent === null; i--) {
        if (node.kind === 'var' ? isScope(parents[i]) : isBlockScope(parents[i])) {
          parent = parents[i];
        }
      }
      parent.locals = parent.locals || {};
      node.declarations.forEach(function (declaration) {
        parent.locals[declaration.id.name] = true;
      });
    },
    'FunctionDeclaration': function (node, parents) {
      var parent = null;
      for (var i = parents.length - 2; i >= 0 && parent === null; i--) {
        if (isScope(parents[i])) {
          parent = parents[i];
        }
      }
      parent.locals = parent.locals || {};
      parent.locals[node.id.name] = true;
      declareFunction(node);
    },
    'Function': declareFunction,
    'TryStatement': function (node) {
      node.handler.body.locals = node.handler.body.locals || {};
      node.handler.body.locals[node.handler.param.name] = true;
    },
    'ImportDefaultSpecifier': function (node) {
      if (node.local.type === 'Identifier') {
        ast.locals = ast.locals || {};
        ast.locals[node.local.name] = true;
      }
    },
    'ImportSpecifier': function (node) {
      var id = node.local ? node.local : node.imported;
      if (id.type === 'Identifier') {
        ast.locals = ast.locals || {};
        ast.locals[id.name] = true;
      }
    },
    'ImportNamespaceSpecifier': function (node) {
      if (node.local.type === 'Identifier') {
        ast.locals = ast.locals || {};
        ast.locals[node.local.name] = true;
      }
    }
  });
  walk.ancestor(ast, {
    'Identifier': function (node, parents) {
      var name = node.name;
      if (name === 'undefined') return;
      for (var i = 0; i < parents.length; i++) {
        if (name === 'arguments' && declaresArguments(parents[i])) {
          return;
        }
        if (parents[i].locals && name in parents[i].locals) {
          return;
        }
      }
      node.parents = parents;
      globals.push(node);
    },
    ThisExpression: function (node, parents) {
      for (var i = 0; i < parents.length; i++) {
        if (declaresThis(parents[i])) {
          return;
        }
      }
      node.parents = parents;
      globals.push(node);
    }
  });
  var groupedGlobals = {};
  globals.forEach(function (node) {
    groupedGlobals[node.name] = (groupedGlobals[node.name] || []);
    groupedGlobals[node.name].push(node);
  });
  return Object.keys(groupedGlobals).sort().map(function (name) {
    return {name: name, nodes: groupedGlobals[name]};
  });
}


text:
Klaus-Dieter Kurrat, född den 16 januari 1955 i Nauen, Brandenburg, är en östtysk friidrottare inom kortdistanslöpning.

Han tog OS-silver på 100 meters-stafetten vid friidrottstävlingarna 1976 i Montréal.

Referenser

Källor
 Denna artikel är helt eller delvis översatt från motsvarande artikel på engelska wikipedia.

Externa länkar
Sports-reference.com

Personer från Nauen
Östtyska kortdistanslöpare
Östtyska olympiska silvermedaljörer
Olympiska silvermedaljörer 1976
Tävlande vid olympiska sommarspelen 1976 från Östtyskland
Tävlande i friidrott vid olympiska sommarspelen 1976
Tävlande vid olympiska sommarspelen 1980 från Östtyskland
Tävlande i friidrott vid olympiska sommarspelen 1980
Män
Födda 1955
Levande personer

redpajama_set_name: RedPajamaWikipedia


text:
NIS President Clarifies Disgaea 5 Comment, was Merely a Playful Joke
by Brandon Orselli on December 10, 2014 at 10:18 PM, EST
Last night we published a story regarding Nippon Ichi Software and their latest title, Disgaea 5: Alliance of Vengeance. The article focused on what seemed to be a depressing comment made by the company president, Sohei Niikawa.
Nippon Ichi representatives reached out to me and clarified that his comment in the 20th anniversary Playstation edition of Dengeki Playstation was merely in jest, and that he was simply trying to put an emphasis on their sales projections.
As we know, Poe's Law makes it very difficult to read the subtleties of human speech. To be clear, the spirit of the statement and the translation was meant to be as such:
"If Disgaea 5 does not sell as planned, I, as the producer, might need to leave the Japanese gaming industry. (LOL)"
In a response to Niche Gamer, Niikawa-san elaborated on the comment, and their goals with Disgaea 5:
"This comment was simply self-deprecating humor and in no way did it mean or imply that Nippon Ichi Software as a company is going bankrupt or is in financial difficulties. I was merely trying to use humor in order to get the point across that we are committed to selling 150,000 copies of Disgaea 5."
So there you have it. The Disgaea developer doesn't seem to be going anywhere.
dengeki playstation, disgaea 5: alliance of vengeance, Nippon Ichi Software
Pdugna
*whew* definitely buying it though, I can finally play one of the console versions of Disgaea for ridiculous hours with remote play :D.
Pyrotek85
Good to hear, I'd be more than a little sad if they went out of business.
Siveon
Well, now I won't feel guilty for not picking it up.
But guys, seriously, Z.H.P2. Needs to happen.
For the love of…
At least they'll be fine that I won't have to buy the game out of guilt. T_T
Wahh…I can keep my Axel now.
Get them to deal with NISA and I'll buy 10 copies
Ok, good. Fucking text not imparting tone, lol.
Lordlundar
Ugh, corporate representatives REALLY need to learn to watch what they say. I've read about a similar statement causing a very profitable company's shares to tank in the span of a few hours that it never recovered from.
It shouldn't be like that but unfortunately you're right. If I were running a company I'd be very careful what I or my employees said in public, it's easy for it to be misinterpreted and the damage can get out of hand fast.
NuclearKangaroo
good to see they are not in a dire situation, id still want them to consider PC releases
Landale
It's also a matter of translation not just what they themselves said.
Nonscpo
Well sir you can make it up to me by localizing Dangan Ronpa Another Episode, Im sure I can forgive you then for that terrible joke. Oh my how offended I am……lol!
Matt McCoy
Nuuuuu~ Nikawa-sama you baka!! >////<
tetrisdork
Cool that it isn't dire. They still need to embrace the PC, though.
Eritach
I'm relieved. Thank goodness we have you to confirm things like this for us.
Marlon Hernandez
Hey, the important thing is that sempaii noticed you.
TheCynicalReaper
I'm glad, and THANK YOU for clearing this up for us, Niche
That said, I wish they'd finally give a damn and do something about NISA already…
Satori Mask
So uh, the source in the original article was some guy on NeoGAF, which isn't exactly the most reliable source. I feel that this miscommunication could have been avoided if someone on the staff had found the original comments and translated it for themselves.
(This seems to be where the mix-up was first cleared up, it's the last post on the bottom: http://nisamerica.com/forum/viewtopic.php?f=9&t=15580&start=60#p836347)
landlock
150,000 copies on the PS4 in Japan? good luck with that Disgaea 4 only made it to 79,000 in the first week and that was on the PS3 with a huge userbase.
Fenrir007
Good! Now I don't have to feel guilty over not buying it because of NISA.
Ncrdrg
Well, that's a relief. I'm glad it was just some dumb fuck at NeoGAF who misinterpreted the comment.
Sylveria Shini
I just hope their American branch gets their shit together. After Mugen Souls Z being basically unplayable for almost 2 months, The Witch and the Hundred Knight bug, the Disgaea D2 bug which I, tahnkfully, never ran in to, their slapdash approach to localizing is starting to harm the brand.
Good that their situation isn't dire, but if they put their game on platforms people actually own, maybe they would buy it? (Pc, hint hint.)
I'm pretty sure they've said similar things before. Those guys are staying around forever!
Game Guy
Great, they still need to flush NISA down the toilet though
Tyrannikos
Cool, but can we still get rid of NISA?
LurkerJK
hopefully senpai also noticed all those ppl screaming for PC ports
Hydroninja9
Agreed. Its fairly cheap to port to pc and they may get more money on pc than on PS4.
Yamato-san
Indeed. I said it in the previous thread, but even if there isn't quite as much at stake here, I still think NIS is making a mistake by jumping ship to the PS4 too soon. Come to think of it, they made similar mistakes with putting early titles on the PS3 (back when it still got flack for its "five-hundred-and-ninety-nine US dollars" price tag), as well as the PSP and Vita. I'm not sure why NIS is so blindly devoted to Sony, but they really should start considering sticking to an older system with a higher install-base just a little longer (especially if it's during a time when they really are on the brink of bankruptcy).
Though, ironically, it's usually NIS's titles that actually make a Sony system start seeming worthwhile to me. I might've said this before, but I usually wait until I pick up a PlayStation (usually after the next model combined with a price-drop), and NIS titles are often among the first in line to what eventually builds up to be a promising library.
MusouTensei
Glad to read that, would suck if NIS would go away :<
Marburg-chan
I guess this is what happens when you use neogaf as a source :^)


I was making my usual blogging rounds yesterday and happened upon a Happy 1st Birthday blog post at one of my favortie blogs Delighted Momma. I started reminiscing about how my own blog began and realized that I too was coming up on my 1 year mark.
I could have sworn I started blogging in March but sadly I realized that my first post was actualy on February 26th of last year.
I'm not a big fan of lengthy posts but if you are interested in reading about why I started my blog click below!
It took some time for me to remember why I really started blogging. It wasn't for a particular reason but rather a culmination of things.
I graduated from the interior design program at BYU-Idaho in April of 2009. Upon graduating and completing a 2nd internship at a cabinetry design company, I was torn between wanting to be a "career woman" or start a family (my husband and I had been married for a little over 3 years by this time). I realize, like many of you, I could have done both, but in the end, I felt that it was important for me to be a "stay at home mom."
Around the time my son turned 1, I began to feel like I was becoming disconnected with the design world. On top of that I was upset with myself for putting so much time and effort into earning a 4 year design degree and basically doing nothing with it. Also, I had just moved to a small town where everything was new and I didn't know a single soul.
I decided that blogging would be a great way for me to stay connected with the design world and was a place where I could share my own design projects. It also gave me something exciting to work towards and sort of filled the lonely gap that had resulted from moving to a new town and having a husband that worked long hours.
Although the original plan for my blog was to be centered around design, I quickly realized that with everything else going on in my life design posts alone was something I couldn't be consistent with. So, along with design and diy projects, I started blogging about everything else that I do on a regular basis, like baking, thrifted and vintage finds, and of course the "mom stuff."
I like your honesty hence why i follow your blog. I have learnt a few design tips from you and will be redecorating my daughter's bedroom soon.
Yay, congrats to you! I think my 1-year was sometime in Feb too but I haven't checked. Funny that I started writing around my son's 1st birthday too. I think your blog is fabulous!
Well I'm glad you took up blogging because I really look forward to reading your content. Happy belated blog birthday!
I am SO happy you started your blog for a few reasons...1) i know how crazy talented you and Abe are so now with all your tutorials I can copy all your ideas:) 2) We miss you and guys and it's fun being able to see what you guys are up too!
I adore your blog. In an overwhelming list of blogs to "catch up on" that I never do...I can never skip on yours. It's really my favorite. Thanks for all the inspiration!
Congrats on your first birthday - belated, too :) I really love the way your blog is with all its amazing mix of topics ...from design to daily stuff.
Since a long time, even before I started blogging myself, your blog is one of my daily reads. Love to drop by here.
Thanks for starting and continuing writing. Thanks for your inspiration!
Happy Blog Birthday! I love your blog. It's the best.
Happy belated bloggy birthday! I am such a fan of your blog. I still remember the first post that I saw that blew me away--it was a DIY cake decorating post :) keep it up Momma!


Description: Prestigious Westridge Gated Community. Panoramic City Lights View Home. Quiet Cul-De-Sac Location. Cathedral Ceiling Living Room. 3 Car Attached Garage. Main Floor Bedroom. Island Kitchen. Freshly Painted Interior. Three Bedrooms Upstairs with a Loft that can be converted to with Bedroom. Hardwood Flooring. Spacious Low Maintenance Backyard. Conveniently Located Close to Golf Course, Restaurants, Parks and Shopping Centers.


I believe this issue highlights the disparate treatment of women in a world run mostly by men. Child care responsibilities still fall heavily on the shoulders of mothers, especially the working moms who struggle to make ends meet. The least that the City can do is provide a safe, secure and private area for mothers who need to pump breast milk.
As Mayor, I will require an assessment be done of each municipal building to determine where an appropriate space can be set up to accommodate mothers who pump breast milk. This would apply to employees and customers. The findings and recommendations would be included in the City's capital plan and budget.
A parent friendly work environment is vital to any workforce and my administration will provide all necessary accommodations.
Yes, this is not a controversial issue, it just needs the will of the leadership. Like many of the questions here, this should have happened in our city long ago.
This is a basic human accommodation that a civil society should make to a mother who needs to pump breast milk in a dignified and respectful manner.
We will provide the facilities for breastfeeding parents. One would hope that there would not be a need for special training on sensitivity towards breastfeeding parents. People will understand the rules, rights, and expectations of the spaces to accommodate these individuals. I have done this before and it did not require special training. As the husband of a wife who breastfed at least two of my children while working, there did not seem to be a need for training.


PPC or Pay per Click is the most popular online advertising model. The main reason is that its pretty efficient, but the marketers love it also because its pretty simple. Basically, PPC is an advertising structure in which the advertiser pays for each ad click made via the website of the publisher. How much they pay depends on the deal between the advertiser and the publisher. PPC is the model used in Google AdWords campaigns. The average cost per click in this program is between 1 and 2 dollars, although it all depends on the keyword, that is, its search volume and competition. The most expensive keywords can reach the price of over 50 dollars per click!


The preference mostly includes the sort of sauce you serve with it, but it's one of the greatest comfort foods you will find. Some pasta dishes are complex to create and the cooking time is extended. But if you're searching for a recipe that is quick to make but will still have the taste of a complex dish, these pasta recipes are definitely a must-try!
Cook pasta to al dente according to package directions. Put garlic and olive oil in a pan and cook over medium high heat, stirring until garlic is golden.
Combine garlic, olive oil, sugar, soy sauce and red pepper flakes in a large bowl. Add shrimp to the bowl and toss to coat with mixture. Cover and simmer for at least 1 hour. Cook the fish and olive oil mix in a pan over medium high heat until shrimp is cooked through. Add the pasta to the pan, season with pepper and salt and stir fry until heated through. Serve garnished with onions and parmesan.
These pasta recipes will certainly accelerate your cooking game!


Revulsion is released!steemCreated with Sketch.
The game has been ready to release for quite some time now so I finally just decided to release it. I feel really anxious because I have never released a game like this before. I did not raise the price at this time as I had planned I would. But it will be interesting to see how things play out regardless. The game is also at a 25% discount.
Enjoy everyone! I hope you have a good time with the game!
Congratulations, mate. That's fantastic news. Hope it all goes well. Keep us in the loop.
Thanks! I just released another update with info.


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Senscot spent a couple of days at the Gathering this week – mainly through our involvement in the work of the Health SEN and Roundtable. We participated in the 'World Café' Discussions – what's hot in health and social care & what it means for the third sector – with particular focus on 'Joint Strategic Commissioning'. The event also saw the Youth Employment Minister, Angela Constance announce the extension of the Community Jobs Scotland Programme. An extra £5.5m will create a further 1000 jobs as well as extending eligibility to 24 year olds. See, www.se-networks.net/shownotice.php?articleid=930.
Next Tuesday, 5th March, Edinburgh SEN (ESEN) will be hosting the second annual Social Enterprise Edinburgh (SEE) Summit at the Grassmarket Community Project. This year's Summit starts with an afternoon seminar on 'Effective Customer Engagement' and ends with a networking reception. Keynote speaker is Mark Bradley of The Fan Experience. The summit is supported by the City of Edinburgh Council – and already over 30 local SEs have signed up. If you'd like to attend, see www.se-networks.net/shownotice.php?articleid=899.
Mentioned a couple of weeks ago that West Lothian SEN is to receive £50k from WL Council for 2013/14. See press release, www.se-networks.net/shownotice.php?articleid=933.
This week, Dumfries and Galloway (D&GSEN) is also asking those SEs in the Dumfries and Galloway to take 5 minutes to fill in the linked form. See, www.surveymonkey.com/s/6DNDTM2.
Congratulations to our colleagues at Social Enterprise Scotland (SES) this week launched their new TV station – Social Enterprise Scotland TV. The new service is an online channel devoted to social enterprise – containing short films from a wide variety of social enterprises across Scotland. If you've got a short film you'd like included, contact news@socialenterprisescotland.org.uk . For more, see http://bit.ly/Y5ZHSy.
Moray SEN member, Down to Earth Solutions, supports self-build as a way to own an affordable eco-home of your own. They are running a survey (10 mins max) looking at issues like: What do you want in a home? What matters most? Why do you think it's so difficult to get on to the housing ladder? Do you actually want to? There's also a chance to WIN a £50 gift voucher or one of two £20 iTunes vouchers. See, www.surveymonkey.com/s/ahomeofmyown.
Three weeks to go till the Exchange and Early Bird rates are still available to SEN members – £50 plus VAT – as well as bursary places for smaller SEs – £20 per delegate. For eligibility, see www.surveymonkey.com/s/WRXCVVK . NB: only one bursary place per organisation will be considered. See full programme, www.socialenterpriseexchange.com/programme/.
TENDERS: Social Enterprise Hotel in Inverness and Counselling Service for Employees of NHS Greater Glasgow and Clyde Health Board. For more details, see www.readyforbusiness.org. Join the Linked-In group at http://tinyurl.com/7ltz7oz and follow on Twitter https://twitter.com/RdyForBusiness.
PROGRAMME: The Social Enterprise Academy is holding a fully funded Intensive Social Enterprise Planning Programme in Glasgow starting from 6th March. This intensive business planning programme is an essential 4-module programme for individuals or third sector organisations looking to establish a sustainable and thriving social enterprise. See, www.se-networks.net/shownotice.php?articleid=929.
OPEN DAY: Remake Scrapstore are holding an open day on Sunday 24 March from 2-5pm. Join them for cake and a look around the new premises. There will be a special goody bag for the first 10 to join the Remake Scrapstore which officially opens on 1st April. See www.se-networks.net/shownotice.php?articleid=928.
FREE CONSULTANCY: The Scottish Government will be running an 'analytical consultancy' scheme to support third sector organisations. As part of a programme of CPD for analytical staff in the Scottish Government, they want to identify very short, sharp projects where statistical or analytical skills could make a difference to your organisation. See, www.se-networks.net/shownotice.php?articleid=931.
FOR HIRE: Creative Stirling CIC is a community interest company based in Stirling's Old Town Jail. As a not-for-profit, volunteer organisation they have focused on developing a creative hub with space and facilities for hire to support creative and cultural industries, as well as the broader community. See, www.se-networks.net/shownotice.php?articleid=932.
Download full case study, see www.se-networks.net/cultural/mediaco-op.pdf.


For people to engage with your brand, they need to feel a connection. The foundation of this is a brand story.
Connections are what creates strong loyalty with your customers, which ultimately grows into a long-term relationship with your business and brand. One way to build this connection is through storytelling marketing, more commonly known as brand storytelling.
Many websites we review as part of the competitive studies we carry out for our clients suffer from the same problem. They have fallen foul of the 'build and they will come' syndrome, a misconception that some website owners have; that once their website has been created their work is complete and that customers will flock to their online web presence. But consider this, how many times have you browsed online and come across a website and wondered why you had never seen it before?