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\begin{document}

\title{Hydrodynamical modelling of Type IIb SNe.}

\author{M.~Ergon\inst{\ref{inst1}} \and F.~Taddia\inst{\ref{inst1}} \and M.~Stritzinger\inst{\ref{inst2}} \and A.~Jerkstrand\inst{\ref{inst3}} \and J,~Sollerman\inst{\ref{inst1}} \and C.~Fransson\inst{\ref{inst1}}}

\institute{The Oskar Klein Centre, Department of Astronomy, AlbaNova, Stockholm University, 106 91 Stockholm, Sweden 
\label{inst1}
\and Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, 8000 Aarhus C, Denmark
\label{inst2}
\and Astrophysics Research Center, School of Mathematics and Physics, Queens University Belfast, Belfast, BT7 1NN, UK 
\label{inst3}
\and Kavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan
\label{inst4}}

\date{Submitted to Astronomy and Astrophysics}

\abstract{
We use HYDE, a new one-dimensional hydrodynamical code, to construct a grid of supernova (SN) models based on solar metallicity bare helium core models evolved to the verge of core-collapse with MESA STAR. This grid is well suited to model Type IIb SNe, which progenitor stars are thought to have lost all but a tiny fraction of their hydrogen envelopes. As previously demonstrated such an envelope only affects the early lightcurve, and the diffusion phase and the early tail phase lightcurves are determined by the helium core. Relatively massive hydrogen envelopes do affect the photospheric velocities during the diffusion phase, however, which could lead to underestimates of the explosion energy. Using an automated procedure we fit the bolometric lightcurves and photospheric velocities of the Carnegie Supernova Project (CSP) and literature samples of Type IIb SNe to the grid of SN models. We find the distribution of initial masses for the combined sample, consisting of Type 17 IIb SNe, to be reasonably well described by a standard Salpeter IMF. The fraction of SNe with an initial mass <15 M$_\odot$ and <20 M$_\odot$ is 50 and 88 percent, respectively, suggesting either the binary channel to dominate the the production of Type IIb SNe or a significant revision of stellar mass loss rates. We find correlations between the explosion energy, initial mass and mass of \element[ ][56]{Ni}, the explosion energy increasing with initial mass and the mass of \element[ ][56]{Ni} increasing with explosion energy. The method used allows us to determine the errors in the model parameters arising from the observed quantities and the degeneracy of the solution. We find an error in the distance and extinction to propagate mainly to the derived mass of \element[ ][56]{Ni}, whereas an error in the photospheric velocity propagates mainly to the derived helium core mass and explosion energy. Fits using the bolometric lightcurve alone are completely degenerate along the M$_{\mathrm{ej}}^{2}$/E=const curve, whereas fits using also the photospheric velocities are quite robust for well-sampled SNe. Finally, we provide a description and tests of the HYDE code, and a discussion of the limitations of the method used.}

\keywords{supernovae: general --- supernovae: individual (SN 2011dh) --- galaxies: individual (M51)}

\titlerunning{SN 2011dh - Understanding a Type IIb SNe.}
\authorrunning{M. Ergon et al.}
\maketitle

\defcitealias{Erg14a}{E14a}
\defcitealias{Erg14b}{E14b}
\defcitealias{Ber12}{B12}
\defcitealias{Jer14}{J14}
\defcitealias{Tad14}{T14}

\section{Introduction}

Type IIb supernovae (SNe) are thought to arise from stars that have lost most of their hydrogen envelopes, either through stellar winds or through Roche-lobe overflow to a binary companion. These SNe are observationally characterized by a transition from Type II (with a hydrogen signature) to Type Ib (without a hydrogen but with a helium signature). Whether binary or single progenitor systems are dominating the production of Type IIb SNe is still debated, but for SN 1993J a companion star have been detected by direct observations \citep{Mau04,Fox14}. Because most of the hydrogen envelope have been lost, whereas the helium core is still intact, we expect these SNe to be well approximated by the explosion of bare helium cores, except during the early cooling phase. This method have been used by \citet{Ber12}, and allows for estimates of the helium core mass, explosion energy and mass of \element[ ][56]{Ni}, whereas the progenitor radius can not be estimated without taking the hydrogen envelope into account. Type IIb SNe have the unique quality to allow an estimate of the helium core mass which, contrary to the ejecta mass, is directly linked to the initial mass of the star. Some parameter studies have been published \citep[e.g.][]{Lym14}, but are all based on approximate lightcurve modelling \citep[e.g.][]{Arn82}. The aim of this paper is to use the new hydrodynamical code HYDE to construct a grid of SN models based on bare helium core models, and use this to estimate the progenitor and SN parameters for the Carnegie Supernova Project (CSP) and literature samples of Type IIb SNe. 

Application of hydrodynamics to SNe lightcurves was introduced in the 70:ths \citep[e.g.][]{Fal77}, and since then a number of codes spanning a wide range of complexity have followed. Some implements more advanced physics, as multi-dimensional \citep[e.g.][]{Mul91} and radiation \citep[e.g.][]{Bli98} hydrodynamics, whereas others are one-dimensional and based on the diffusion approximation \citep[e.g.][]{Ber11}. The different codes all have their different applications and no code is yet capable of modelling a core-collapse (CC) SN consistently, including all the relevant physics. HYDE belongs to the latter category, and like other simplified codes it have the benefit of being fast, which is critical when building model grids covering large volumes of parameter space. The use of model grids to determine the SN and progenitor parameters have been explored before \citep[e.g.][]{Lit83,Lit85} with somewhat mixed results \citep[e.g.][]{Ham03}, but the decreasing computational cost and the increasing amount of data, motivates a renewed interest in this approach. A model grid also allows the degeneracy of the solution and the errors in the SN and progenitor parameters to be estimated.

The paper is organized as follows. In Sect.~\ref{s_hyde} we describe and test the HYDE code, and discuss its limitations. In Sect.~\ref{s_model_grid} we describe the grid of bare helium core and SN models, and discuss the dependence of the observed properties on the SN and progenitor parameters. In Sect.~\ref{s_hydrogen_envelope} we present models with low-mass hydrogen envelopes, and discuss the effects of these on the observed properties. In Sect.~\ref{s_modelling_IIb} we describe our fitting procedure, use the grid of SN models to estimate the progenitor and SN parameters for the CSP and literature samples of Type IIb SNe, and discuss the total sample statistics. The observational details for the CSP and literature samples are given in Appendices~\ref{a_csp_sample} and \ref{a_literature_sample}, respectively. Finally, we conclude and summarize the paper in Sect.~\ref{s_conclusions}. 

\section{The HYDE code}
\label{s_hyde}

HYDE is a one-dimensional Lagrangian hydrodynamical code based on the flux-limited diffusion approximation, following the method described in \citet{Fal77}. The code may also be run in homologous mode, where the dynamics have been switched off and the thermodynamical state is solved for given the constraint of homologous expansion.

\subsection{Hydrodynamics}
\label{s_hyde_hydro}

The hydrodynamical conservation equations for mass, momentum and energy coupled with the diffusion approximation \citep[eqs.~1-4]{Fal77} are solved by a finite difference scheme similar to the one described by \citet[eqs.~A1-A12]{Fal77}. The dynamical state is solved for using a forward difference scheme, whereas the thermodynamical state is solved for using a backward difference scheme. The latter results in a non-linear equation system which is solved using a Newton-Raphson like method. To handle strong velocity gradients (shocks) an artificial viscosity following the prescription by \citet{Neu50} is used, and in the optically thin regime a flux limiter following the prescription given by \citet{Ber11} is used. A nuclear reaction network is not included and the effect of explosive nucleosynthesis is ignored.

\subsection{Opacity}
\label{s_hyde_opacity}

The opacity is interpolated from the OPAL opacity tables \citep{Igl96} complemented with the low temperature opacities given by \citet{Ale94}. In addition we use an opacity floor set to $0.01$ cm$^2$ gram$^{-1}$ in the hydrogen envelope and $0.025$ cm$^2$ gram$^{-1}$ in the helium core, following \citet[private communication]{Ber12}, who calibrated these values by comparison to the STELLA hydrodynamical code \citep{Bli98}.

\subsection{Equation of state}
\label{s_hyde_eos}

The equation of state is taken to be that of a mixture of gas and radiation, and degeneracy pressure is not included. The electron density needed in the equation of state is calculated by solving the Saha equation, using the same atomic data as in \citet{Jer11,Jer12}.

\subsection{Radioactive heating}
\label{s_hyde_radio}

The transfer of the gamma-rays and positrons emitted in the decay chain of \element[ ][56]{Ni} is calculated with a Monte-Carlo method, using the same grey opacities, luminosities and decay times as in \citet{Jer11,Jer12}, and the heating rate then fed into the energy equation.

%\subsection{Limitations}

%The effects of explosive nucleosynthesis are ignored as a nuclear network is not included, and macroscopic mixing is prohibited as the code is one-dimensional. The optically thin regime is not handled correctly as the code is based on the diffusion approximation, but a rough treatment is provided by the flux limiter. The table bound-bound opacity is not correctly calculated as a static medium was assumed, but a rough correction is provided by the use of opacity floor.

\subsection{Tests of the code} 

\begin{figure}[tbp!]
\includegraphics[width=0.5\textwidth,angle=0]{figs/hydro-IIb-b12-model-comp.pdf}
\caption{Bolometric lightcurve for the 4 M$_\odot$ bare helium core model from \citet{Nom88} as modelled with HYDE (black) and the adjusted version of the \citet{Ber12} He4 model presented in \citet{Erg14a} (blue).}
\label{f_b12_model_comp}
\end{figure}

%\begin{figure}[tbp!]
%\includegraphics[width=0.5\textwidth,angle=0]{figs/hydro-IIb-b12-model-comp-early.pdf}
%\caption{The same as Fig.~\ref{f_b12_model_comp} but zoomed in on the early evolution.}
%\label{f_b12_model_comp_early}
%\end{figure}

\begin{figure}[tbp!]
\includegraphics[width=0.5\textwidth,angle=0]{figs/massloss-lightcurve-bol.pdf}
\caption{Progression of model bolometric lightcurves calculated with HYDE for a 15 M$_\odot$ MESA model with the massloss adjusted to yield a final mass of 11.0, 8.0, 6.0, 5.0, 4.8, 4.6, 4.4, 4,2, 4.1, 4.05 and 4.0 M$_\odot$, colour coded from blue (11.0 M$_\odot$) to red (4.0 M$_\odot$). The explosion parameters were E=1.0$\times$10$^{51}$erg, M$_{\mathrm{Ni}}$=0.1 M$_\odot$ and Mix$_{\mathrm{Ni}}$=M$_{\mathrm{He}}$/M (Sect.~\ref{s_model_grid}).}
\label{f_M15_massloss}
\end{figure}

The homologous behaviour have been tested by comparison to analytical solutions by \citet{Ims92}, and the deposition of radioactive decay energy have been tested by comparison to the steady-state NLTE code described in \citet{Jer11,Jer12}. The hydrodynamical behaviour have been tested by comparison to the results presented in \citet{Ber12} using the same 4 M$_\odot$ bare helium core model from \citet{Nom88}. Figure~\ref{f_b12_model_comp} shows a comparison between the lightcurve calculated with HYDE and the lightcurve for the adjusted version of the \citet{Ber12} He4 model presented in \citet{Erg14a}. Both models have the same explosion parameters (E=1.0$\times$10$^{51}$ erg, M$_{\mathrm{NI}}$=0.075 M$_\odot$ and Mix$_{\mathrm{Ni}}$=0.95). Except at $\lesssim$5 days the differences are small, but the early time luminosity is considerably lower. The reason for this difference is unclear and needs further investigation, but as we only make use of the diffusion and tail phase lightcurves in this work, we find the agreement satisfactory. Fiure~\ref{f_M15_massloss} show lightcurves calculated with HYDE for a series of 15 M$_\odot$ MESA models were the massloss was adjusted to yield final masses in the range 11-4 M$_\odot$. The sequence of lightcurves shows the expected transformation from an explosion energy powered Type IIP like lightcurve to a radioactively powered Type Ib like lightcurve.
%, suggesting that the behaviour of the code is healthy. 
Further support comes from the behaviour of observed and physical properties of bare helium core and extended models, discussed in Sects.~\ref{s_dependence}, \ref{s_physics}, \ref{s_hydrogen_effects} and \ref{s_hydrogen_physics}.
%, and the agreement with previous modelling for individual SNe discussed in Sect.~\ref{s_modelling_IIb_lit_results}.
%from the agreement with expectations for the observed properties as a function of the progenitor and SN parameters, discussed in Sect.~\ref{s_dependence}, and the evolution of the physical properties for typical bare helium core and extended models, discussed in Sect.~\ref{s_physics}.

\section{The model grid}
\label{s_model_grid}

\subsection{Progenitor models}

The progenitor models were constructed using MESA STAR \citep{Pax11} by evolving solar metallicity helium cores until the verge of core-collapse. This is similar to what was done in \citet{Nom88}, and relays on the assumption that the hydrogen envelope does not appreciably affect the evolution of the helium core. Evolving a set of solar metallicity 15 M$_\odot$ models, adjusting the mass loss to yield final masses in the range 15-4 M$_\odot$, resulted in helium cores of very similar size and composition, in support of this assumption. The MESA configuration used was the default one, and the evolution were terminated at a central density of 10$^{9.5}$ g cm$^{-3}$, which typically occurred slightly before core-collapse.
%, as indicated by the core infall velocity. 
The evolved models spans M$_{\mathrm{He}}$=4.0-5.0 M$_\odot$ in 0.25 M$_\odot$ steps and M$_{\mathrm{He}}$=5.0-10.0 M$_\odot$ in 0.5 M$_\odot$ steps. Below 4.0 M$_\odot$ the late burning stages ignited off center, which caused convergence problems, and these stellar models were constructed by a scaling of the 4.0 M$_\odot$ density profile.

\subsection{SN models}

As all similar codes, HYDE does not include a treatment of the physics of the core-collapse itself. Instead the outcome of this event is simulated by the injection of thermal energy (thermal bomb) at some location assumed to correspond to the division between the collapsing core and the ejected material. This location is fixed to 1.5 M$_\odot$ in all our models, and the explosion energy (E) is treated as a free parameter. The current version of HYDE does not include a network of nuclear reactions, so the explosive nuclear burning occurring in the iron core and the inner parts of the oxygen zones, synthesizing the radioactive isotopes powering the lightcurve, can not be modelled. Because of this, and the absence of multi-dimensional effects as macroscopic mixing in 1-D (spherical symmetric) modelling, the mass (M$_{\mathrm{Ni}}$) and mixing (Mix$_{\mathrm{Ni}}$) of the \element[ ][56]{Ni} are also treated as free parameters. The mass fraction of \element[ ][56]{Ni} ($\mathrm{X}_{\mathrm{Ni}}$) was assumed to be a linearly declining function of the ejecta mass ($\mathrm{m}_{\mathrm{ej}}$) becoming zero at some fraction (Mix$_{\mathrm{Ni}}$) of the total ejecta mass, expressed as $\mathrm{X}_{\mathrm{Ni}} \propto 1-\mathrm{m}_{\mathrm{ej}}/(\mathrm{Mix}_{\mathrm{Ni}} \mathrm{M}_{\mathrm{ej}}), \mathrm{X}_{\mathrm{Ni}} \geq 0$. Note that this expression allows Mix$_{\mathrm{Ni}}>1$, although the interpretation then becomes less clear. The SN explosion is thus parametrized using three parameters (E, M$_{\mathrm{Ni}}$ and Mix$_{\mathrm{Ni}}$) and the progenitor star using one (M$_{\mathrm{He}}$). The total parameter space spanned is M$_{\mathrm{He}}$=2.5-10 M$_\odot$, E=0.4-6.0$\times$10$^{51}$ erg, M$_{\mathrm{Ni}}$=0.015-0.3 M$_\odot$ and Mix$_{\mathrm{Ni}}$=0.5-1.4 using a 21$\times$24$\times$15$\times$9 grid. We find this resolution to be sufficient to safely interpolate intermediate values.

\subsection{Dependence on progenitor and SN parameters}
\label{s_dependence}

\begin{figure}[tbp!]
\includegraphics[width=0.5\textwidth,angle=0]{figs/hydro-IIb-mg-L.pdf}
\caption{Model bolometric lightcurves for day 1-100 showing the dependence on M$_{\mathrm{He}}$ (2.5-7.0 M$_\odot$; upper left panel), E (0.4-2.2$\times$10$^{51}$ erg; upper right panel), M$_{\mathrm{Ni}}$ (0.05-0.25 M$_\odot$; lower left panel) and Mix$_{\mathrm{Ni}}$ (0.6-1.0; lower right panel). Low to high values are displayed in red to blue colour coding and the values for the parameters not varied are M$_{\mathrm{He}}$=4.0 M$_\odot$, E=1.0$\times$10$^{52}$ erg, M$_{\mathrm{Ni}}$=0.1 M$_\odot$ and Mix$_{\mathrm{Ni}}$=1.0.}
\label{f_mg_L}
\end{figure}

\begin{figure}[tbp!]
\includegraphics[width=0.5\textwidth,angle=0]{figs/hydro-IIb-mg-v-phot.pdf}
\caption{Model photospheric velocities for day 1-100 showing the dependence on M$_{\mathrm{He}}$ (2.5-7.0 M$_\odot$; upper left panel), E (0.4-2.2$\times$10$^{51}$ erg; upper right panel), M$_{\mathrm{Ni}}$ (0.05-0.25 M$_\odot$; lower left panel) and Mix$_{\mathrm{Ni}}$ (0.6-1.0; lower right panel). Low to high values are displayed in red to blue colour coding and the values for the parameters not varied are M$_\mathrm{He}$=4.0 M$_\odot$, E=1.0$\times$10$^{52}$ erg, M$_\mathrm{Ni}$=0.1 M$_\odot$ and Mix$_\mathrm{Ni}$=1.0.}
\label{f_mg_v_phot}
\end{figure}

Fig.~\ref{f_mg_L} shows the dependence of the bolometric lightcurve on M$_{\mathrm{He}}$, E, M$_{\mathrm{Ni}}$ and Mix$_{\mathrm{Ni}}$, varying the parameters for a reference model with M$_{\mathrm{He}}$=4.0 M$_\odot$, E=1.0$\times$10$^{52}$ erg, M$_{\mathrm{Ni}}$=0.1 M$_\odot$ and Mix$_{\mathrm{Ni}}$=1.0. Qualitatively, we expect either an increase of the explosion energy or a decrease of the ejecta mass to decrease the diffusion time for thermal radiation, to decrease the optical depth for the $\gamma$-rays emitted in the decay chain of \element[ ][56]{Ni}, and to increase the expansion velocities. We therefore expect such a change to decrease the time at which peak luminosity occurs, to decrease the luminosity on the tail and to decrease the photospheric velocity. Qualitatively, we also expect the luminosity to scale with the mass of \element[ ][56]{Ni}. As seen in Fig.~\ref{f_mg_L}, all these qualitative dependencies are well followed by the models. Quantifying the dependencies by measuring the time (t$_{\mathrm{m}}$) and photospheric velocity (v$_{\mathrm{m}}$) at maximum luminosity (L$_{\mathrm{m}}$) for Mix$_{\mathrm{Ni}}$=1.0, and fitting a power-law expression to the model grid, we get

%\begin{equation}
%\mathrm{log~t_{m} = 1.17- 0.39~log~E + 0.69~log~M_{ej} + 0.10~log~M_{Ni}}
%\end{equation}

%\begin{equation}
%\mathrm{log~v_{m} = 1.05 + 0.48~log~E - 0.18~log~M_{ej}}
%\end{equation}

%\begin{equation}
%\mathrm{log~L_{m} = 1.40 + 0.19~log~E - 1.38~log~M_{ej} + 0.83~log~M_{Ni}}
%\end{equation}

\begin{equation}
\mathrm{log~t_{m} = 1.14- 0.37~log~E + 0.65~log~M_{ej} + 0.09~log~M_{Ni}}
\label{eq_dep_1}
\end{equation}

\begin{equation}
\mathrm{log~v_{m} = 1.10 + 0.46~log~E - 0.23~log~M_{ej}}
\label{eq_dep_2}
\end{equation}

\begin{equation}
\mathrm{log~L_{m} = 1.34 + 0.19~log~E - 1.04~log~M_{ej} + 0.89~log~M_{Ni}}
\label{eq_dep_3}
\end{equation}

which gives the (average) dependence of the observed quantities on the SN and progenitor parameters for the model grid. Fitting the inverse relations\footnote{Note that the observed quantities are not necessarily independent.}, we get

%\begin{equation}
%\mathrm{log~E = -3.21 + 0.53~log~t_{m} - 0.07~log~L_{m} + 2.55~log~v_{m}}
%\end{equation}

%\begin{equation}
%\mathrm{log~M = -2.65 + 1.40~log~t_{m} - 0.17~log~L_{m} + 1.20~log~v_{m}}
%\end{equation}

%\begin{equation}
%\mathrm{log~M_{Ni} = -5.45 + 2.23~log~t_{m} + 0.92~log~L_{m} + 1.46~log~v_{m}}
%\end{equation}

\begin{equation}
\mathrm{log~E = -3.85 + 0.80~log~t_{m} - 0.11~log~L_{m} + 2.80~log~v_{m}}
\label{eq_dep_4}
\end{equation}

\begin{equation}
\mathrm{log~M = -3.16 + 1.65~log~t_{m} - 0.19~log~L_{m} + 1.40~log~v_{m}}
\label{eq_dep_5}
\end{equation}

\begin{equation}
\mathrm{log~M_{Ni} = -4.32 + 1.75~log~t_{m} + 0.92~log~L_{m} + 0.99~log~v_{m}}
\label{eq_dep_6}
\end{equation}

which gives the (average) dependence of the SN and progenitor parameters on the observed quantities for the model grid. Using the observed values for SN 2011dh we get values for E, M$_{\mathrm{ej}}$ and M$_{\mathrm{Ni}}$ within $\sim$30 percent of those derived by the fitting procedure in Sect.~\ref{s_modelling_IIb}, and a clever parametrization of the model grid could actually be an alternative to this fitting procedure. However, given the dubious results obtained from the model grid fits for Type IIP SNe by \citet{Lit83,Lit85} in e.g. \citet{Ham03}, care has to be taken, and we do not investigate this approach further in this work. The approximate model by \citet{Arn82} is often used to infer the SN and progenitor parameters for stripped envelope SNe \citep[e.g.][]{Lym14}. In this model the diffusion time and the expansion velocity depends on the explosion energy and ejecta mass as $\mathrm{t_{d} \propto (M^{3}/E)^{1/4}}$ and $\mathrm{v \propto (E/M)^{1/2}}$, and inverting these gives $\mathrm{E \propto t_{\mathrm{d}}^2 v^3}$ and $\mathrm{M \propto t_{d}^2 v}$. Comparing to the model grid fits we see that these scalings are qualitatively followed, 
%but the photospheric velocity at maximum is considerably less sensitive to the ejecta mass, and the energy is considerably less sensitive to the time at which maximum luminosity occurs. 
but significant quantitative differences exist, e.g. E is considerably less sensitive to t$_{\mathrm{m}}$.
%but v$_{\mathrm{m}}$ is considerably less sensitive to M$_{\mathrm{ej}}$, and E is considerably less sensitive to t$_{\mathrm{m}}$.
Clearly, t$_{\mathrm{m}}$ and t$_{\mathrm{d}}$ and in particular v and v$_{\mathrm{m}}$ are not identical, but most important is likely the fact that the \citet{Arn82} model assumes a constant opacity, whereas in the hydrodynamical models the (average) opacity is decreasing with time as the helium recombination front recedes through the ejecta (Sect.~\ref{s_physics}). An important consequence of the differences in the scalings is that t$_{\mathrm{m}}$ is independent of the quantity M$^{2}$/E, whereas that t$_{\mathrm{d}}$ is independent of the quantity M$^{3}$/E in the \citet{Arn82} model. This has implications for the degeneracy of the solution, as we will discuss further in Sect.~\ref{s_degeneracy}.

%\subsection{Model physics}
%\label{s_physics}

%\begin{figure}[tbp!]
%\includegraphics[width=0.5\textwidth,angle=0]{figs/sn2011dh-optimal-model-rho-T-early-radius.pdf}
%\caption{Evolution of the temperature (left panel) and density (right panel) profiles between 1 and 200 seconds (shock breakout) in 10 logarithmically spaced intervals for the best-fit model of SN 2011dh.}
%\label{f_opt_model_rho_T_early}
%\end{figure}

%\begin{figure}[tbp!]
%\includegraphics[width=0.5\textwidth,angle=0]{figs/sn2011dh-optimal-model-T-evo.pdf}
%\caption{Evolution of temperature profile for the best-fit model of SN 2011dh. The position of the ionization front of helium (black triangles), the photosphere (red circles) and the thermalization surface (blue squares) have been marked, and each temperature profile annotated with the time since explosion.}
%\label{f_opt_model_T}
%\end{figure}

%\begin{figure}[tbp!]
%\includegraphics[width=0.5\textwidth,angle=0]{figs/sn2011dh-optimal-model-He-ion-evo-1.pdf}
%\caption{Evolution of the helium ionization profile for the best-fit model of SN 2011dh. The thermalization radius (blue squares) have been marked and each ionization profile annotated with the time since explosion.}
%\label{f_opt_model_He_ion}
%\end{figure}

%Here we discuss the physics of our models, exemplified by a typical Type IIb model, taken as the optimal model for SN 2011dh (Sect.~\ref{s_modelling_IIb}). We stress that the early evolution will differ from that of an extended progenitor and would rather correspond to that of a star that have just lost its hydrogen envelope. 

%Fig.~\ref{f_opt_model_rho_T_early} shows the evolution of the density and temperature profiles from the injection of explosion energy until shock breakout, which occurs at $\sim$200 seconds. The shock initially accelerates to a speed of $\sim$10000 km~s$^{-1}$, but decelerates to $\sim$6000 km~s$^{-1}$ after the passage into the helium envelope because of the decreasing density gradient. This gives rise to a reverse shock propagating backwards into the O/C core. In the outermost layers the density gradient increases drastically and the shock accelerates to a speed of $\sim$30000 km~s$^{-1}$ at shock breakout. The thermal and kinetic energy behind the shock is close to equipartition and the temperature high enough for the equation of state to be completely radiation dominated. During the passage of the shock through the star some thermal energy is lost due to expansion but when the radiation breaks out from the shock the explosion energy is still roughly equipartitioned between kinetic and thermal energy (E$_{\mathrm{th}}$/E$_{\mathrm{k}}$=0.35).

%In the few minutes that follows the ejecta expands and the temperature and luminosity at the photosphere decrease rapidly because of diffusion and expansion cooling. At $\sim$700 seconds the outermost parts becomes optically thin and the photosphere starts to recede into the ejecta. At $\sim$3 hours helium starts to recombine and at $\sim$10 hours the helium ionization is overrun by the photosphere. Subsequently the position of the photosphere is determined by the ionization front of helium, slowly moving inwards in mass coordinates but outwards in radial coordinates. Fig.~\ref{f_opt_model_T} shows the evolution of the temperature and helium ionization profile between 1 and 50 and 1 and 25 days respectively, where we have also marked the positions of the photosphere, the thermalization surface and the ionization front of helium. The thermalization surface, here defined as $\sqrt {3 \tau_{\mathrm{abs}} \tau_{\mathrm{tot}}}=2/3$ \citep{Ens92}, is located near the outer edge of the ionization front of helium, and follows the evolution of this until $\sim$25 days, when helium recombines and the thermalization surface ceases to exist. During this period the temperature at the thermalization surface declines only slowly, and is roughly constant at $\sim$9000 K, a few thousand degrees below the temperature at the center of the ionization front.

\subsection{Model physics}
\label{s_physics}

\begin{figure}[tbp!]
\includegraphics[width=0.5\textwidth,angle=0]{figs/bare-helium-core-model-rho-T-early-radius.pdf}
\caption{Evolution of the temperature (left panel) and density (right panel) profiles between 1 and 282 seconds (shock breakout) in 10 logarithmically spaced intervals for the 4 M$_\odot$ helium core model.}
\label{f_opt_model_rho_T_early}
\end{figure}

\begin{figure}[tbp!]
\includegraphics[width=0.5\textwidth,angle=0]{figs/bare-helium-core-model-T-evo.pdf}
\caption{Evolution of temperature profile for the 4 M$_\odot$ helium core model. The position of the recombination front of helium (black triangles), the photosphere (red circles) and the thermalization surface (blue squares) have been marked, and each temperature profile annotated with the time since explosion.}
\label{f_opt_model_T}
\end{figure}

\begin{figure}[tbp!]
\includegraphics[width=0.5\textwidth,angle=0]{figs/bare-helium-core-model-He-ion-evo-1.pdf}
\caption{Evolution of the helium ionization profile the 4 M$_\odot$ helium core model. The thermalization surface (blue squares) have been marked and each ionization profile annotated with the time since explosion.}
\label{f_opt_model_He_ion}
\end{figure}

Here we discuss the physics of our bare helium core models, exemplified by a 4 M$_\odot$ model with explosion parameters E=1.0$\times$10$^{51}$erg, M$_{\mathrm{Ni}}$=0.1 M$_\odot$ and Mix$_{\mathrm{Ni}}$=1.0. We stress that the early evolution differ from that of an extended progenitor, which is discussed in Sect.~\ref{s_hydrogen_physics}.

Fig.~\ref{f_opt_model_rho_T_early} shows the evolution of the density and temperature profiles from the injection of explosion energy until shock breakout, which occurs at $\sim$300 seconds. The shock initially accelerates to a speed of $\sim$10000 km~s$^{-1}$ in the oxygen core, but decelerates to $\sim$6000 km~s$^{-1}$ in the helium envelope, where the density gradient is small. 
%This gives rise to a reverse shock propagating backwards into the oxygen core. 
In the outermost layers the density gradient increases drastically before it levels out in the thin convective envelope, and the shock accelerates to a speed of $\sim$30000 km~s$^{-1}$ at shock breakout. The thermal and kinetic energy behind the shock is close to equipartition and the temperature high enough for the equation of state to be completely radiation dominated. During the passage of the shock through the star some thermal energy is lost due to expansion, in particular during the passage through the thin envelope, and when the radiation breaks out from the shock the thermal fraction of the energy is $\sim$15 percent.

In the few minutes that follows the ejecta expands and the temperature and luminosity at the photosphere decrease rapidly because of diffusion and expansion cooling. At $\sim$500 seconds the outermost parts becomes optically thin and the photosphere starts to recede into the ejecta. At $\sim$3 hours helium starts to recombine and at $\sim$10 hours the recombination front overruns the photosphere. Subsequently the position of the photosphere is determined by the recombination front, slowly moving inwards in mass coordinates but outwards in radial coordinates. Fig.~\ref{f_opt_model_T} shows the evolution of the temperature and the helium ionization profile between 1 and 50 and 1 and 25 days, respectively, where we have also marked the positions of the photosphere, the thermalization surface and the recombination front. The thermalization surface, here defined as $\sqrt {3 \tau_{\mathrm{abs}} \tau_{\mathrm{tot}}}=2/3$ \citep{Ens92}, is located near the outer edge of the recombination front, and follows the evolution of this until $\sim$25 days when the helium has recombined.
%, and the thermalization surface soon ceases to exist. 
During this period the temperature at the thermalization surface is roughly constant, and declines only slowly from $\sim$9000 K to $\sim$8000 K, a few thousand degrees below the temperature at the center of the recomombination front. 

We note that this temperature is in good agreement with the blackbody temperature measured for SN 2011dh in \citetalias{Erg14a}. \citet{Pir14} argue that the blackbody temperature for SN 2011dh and other similar SNe is too low to ionize helium and that their helium envelopes might be effectively transparent, but according to our results this is not the case. Instead, we find that the gradual recombination of the helium envelope is actually what shapes the major part of the diffusion peak lightcurve for a Type IIb SNe.

\subsection{Model limitations}

\begin{figure}[tbp!]
\includegraphics[width=0.5\textwidth,angle=0]{figs/hydro-IIb-kappa-floor.pdf}
\caption{Upper panels: Model bolometric lightcurves (right panel) and photosheric velocities (left panel) for day 1-100 showing the dependence on the opacity floor (0.012-0.048 gram cm$^{2}$). Low to high values are displayed in blue to red colour coding and the model parameters are M$_{\mathrm{He}}$=4.0 M$_\odot$, E=1.0$\times$10$^{52}$ erg, M$_{\mathrm{Ni}}$=0.1 M$_\odot$ and Mix$_{\mathrm{Ni}}$=1.0. Lower panels: Sensitivity of the explosion energy (left panel) and helium core mass (right panel) to the opacity floor, calculated by fitting the model lightcurves and photospheric velocities to the model grid using the procedure described in Sect.~\ref{s_fitting_proc}.}
\label{f_kappa_floor}
\end{figure}

In spite of being superior to the \citet{Arn82} like models used in many other sample studies, the hydrodynamical models used in this study still suffer from a number of limitations. Below we discuss briefly the most important of these and the limitations in our treatment of the opacity in some more detail, as the bolometric lightcurves depends critically on this quantity.

\paragraph{Progenitor models}

The progenitor models only differ in helium core mass, but are all non-rotating and with solar metallicity. This is clearly a limitation, but as the number of SN models would increase drastically we have chosen not to vary these progenitor parameters. The effect of a low-mass hydrogen-envelope, not present in our bare helium core models, is discussed separately in Sect.~\ref{s_hydrogen_envelope}.

\paragraph{Hydrodynamics} As HYDE does not include a nuclear reaction network the effect of the explosive nucleosynthesis in the inner part of the ejecta is not included, and as HYDE is 1-dimensional, macroscopic mixing of the nuclear burning zones is prohibited. The mass-cut is fixed at 1.5 M$_\odot$ and although fallback of material onto the compact remnant is not prohibited, the artificial zero velocity inner boundary condition will cause this material to bounce. To properly handle fallback a pistong-driven explosion without an inner boundary condition on the velocity would be needed. 

\paragraph{Radiative transfer} The optically thin regime is not handled correctly as the code is based on the diffusion approximation. The treatment of the optically thin region is critical when calculating spectra or broad-band photometry, but probably of less importance when calculating the bolometric lightcurve. A correct treatment of the optically thin region could also be important for the radiative acceleration of the outer parts of the ejecta occuring at early times. This could effect the bolometric lightcurve during the cooling phase but probably not later on.

%The opacity tables used (Sect.~\ref{s_hyde_opacity}) are calculated for a static medium and does not take into account the effect of a velocity gradien, which tend to increase the line opacity \citep[e.g.][]{Kar77}.

\paragraph{Opacity}

One major limitation with HYDE is the abscense of a proper treatment of the line (bound-bound) opacity. The opacity tables used are calculated for a static medium and does not take into account the effect of a velocity gradient, which tend to increase the line opacity \citep[e.g.][]{Kar77}. To compensate for this lack of opacity (as well as for other reasons, see \citealp{Ber11}), HYDE makes use of an opacity floor (Sect.~\ref{s_hyde_opacity}). The value of this floor is set to 0.024 gram$^{-1}$ cm$^{2}$, 
%which was calibrated to STELLA \citep{Bli98} by \citet{Ber12} for modelling of SN 2011dh. This 
which 
is much lower than the electron scattering opacity of $\sim$0.2 gram$^{-1}$ cm$^{2}$ for fully ionized helium core material, and only affects the region outside the recombination front (Sect.~\ref{s_physics}). The upper panels of Fig.~\ref{f_kappa_floor} shows the dependence of the model lightcurves and photospheric velocities on the opacity floor for a 4 M$_\odot$ model with explosion parameters E=1.0$\times$10$^{51}$erg, M$_{\mathrm{Ni}}$=0.1 M$_\odot$ and Mix$_{\mathrm{Ni}}$=1.0. These are not particularly sensitive, and a doubling of the opacity floor corresponds to an increase in the photospheric velocities and a shift to later times of the diffusion peak of 10-15 percent. The lower panels of Fig.~\ref{f_kappa_floor} shows the sensitivity of the helium core mass and explosion energy to the opacity floor, calculated by fitting the model lightcurves and photospheric velocities to the model grid using the same procedure as in Sect.~\ref{s_modelling_IIb}. The explosion energy depends strongly on the photospheric velocities (Eqs.~\ref{eq_dep_4}), so this quantity is rather sensitive and almost proportional to the opacity floor, whereas the helium core mass is less affected and a doubling of the opacity floor corresponds to an increase of $\sim$25 percent. A proper investigation of the effects of the opacity floor on our results in Sect.~\ref{s_modelling_IIb} would require a thorough comparison with a code capable of calculating the line opacity correctly, and is outside the scope of this paper. We conclude, however, that the helium core mass is not particurly sensitive to the choice of opacity floor, whereas this choise is more critical with respect to the explosion energy.

\section{The hydrogen envelope}
\label{s_hydrogen_envelope}

As our aim is to use the grid of bare helium core models to fit the bolometric lightcurves and the photospheric velocities of Type IIb SNe, which may have extended low-mass hydrogen envelopes surrounding their helium cores, it is of importance to investigate which effect such envelopes would have on the observed properties, as well as on the results obtained in Sect.~\ref{s_modelling_IIb}. It is also of interest to investigate the physics of such models and compare to the physics of bare helium core models discussed in Sect.~\ref{s_physics}.

\subsection{Effect on the observed properties}
\label{s_hydrogen_effects}

\begin{figure}[tb!]
\includegraphics[width=0.5\textwidth,angle=0]{figs/type-IIb-model.pdf}
\caption{Upper panels: Progression of model bolometric lightcurves (left panel) and photospheric velocities (right panel) calculated with HYDE for a 15 M$_\odot$ MESA model with the massloss adjusted to yield a final mass of 4.2, 4.15, 4.1, 4.05, 4.025 and 4.0 M$_\odot$, colour coded from blue (4.2 M$_\odot$) to red (4.0 M$_\odot$). The explosion parameters were E=1.0$\times$10$^{51}$erg, M$_{\mathrm{Ni}}$=0.1 M$_\odot$ and Mix$_{\mathrm{Ni}}$=M$_{\mathrm{He}}$/M. Lower panels: Sensitivity of the explosion energy (left panel) and helium core mass (right panel) to the mass of the hydrogen envelope, calculated by fitting the model lightcurves and photospheric velocities to the model grid using the procedure described in Sect.~\ref{s_fitting_proc}.}
\label{f_hydrogen}
\end{figure}

%\begin{figure}[tb!]
%\includegraphics[width=0.5\textwidth,angle=0]{figs/type-IIb-model-lightcurve-bol.pdf}
%\caption{Progression of model bolometric lightcurves calculated with HYDE for a 15 M$_\odot$ MESA model with the massloss adjusted to yield a final mass of 4.2, 4.15, 4.1, 4.05, 4.025 and 4.0 M$_\odot$, colour coded from blue (4.2 M$_\odot$) to red (4.0 M$_\odot$). The explosion parameters were E=1.0$\times$10$^{51}$erg, M$_{\mathrm{Ni}}$=0.1 M$_\odot$ and Mix$_{\mathrm{Ni}}$=M$_{\mathrm{He}}$/M.}
%\label{f_hydrogen_bol_lightcurve}
%\end{figure}

%\begin{figure}[tb!]
%\includegraphics[width=0.5\textwidth,angle=0]{figs/type-IIb-model-velocities.pdf}
%\caption{Progression of model photospheric velocities calculated with HYDE for a 15 M$_\odot$ MESA model with the massloss adjusted to yield a final mass of 4.2, 4.15, 4.1, 4.05, 4.025 and 4.0 M$_\odot$, colour coded from blue (4.2 M$_\odot$) to red (4.0 M$_\odot$). The explosion parameters were E=1.0$\times$10$^{51}$erg, M$_{\mathrm{Ni}}$=0.1 M$_\odot$ and Mix$_{\mathrm{Ni}}$=M$_{\mathrm{He}}$/M.}
%\label{f_hydrogen_velocities}
%\end{figure}

%Figs.~\ref{f_hydrogen_bol_lightcurve} and \ref{f_hydrogen_velocities} 
The upper panels of Fig.~\ref{f_hydrogen} 
shows the bolometric lightcurve and photospheric velocities for a sequence of 15 M$_\odot$ MESA models, where the mass loss was adjusted to yield final masses in the range 4.2-4.0 M$_\odot$. Defining the hydrogen envelope to begin where X>0.01, this corresponds to hydrogen envelope masses in the range 0.27-0.07 M$_\odot$. The bolometric lightcurves for all models show an initial decline phase corresponding to the cooling of the thermal explosion energy deposited in the hydrogen envelope, the length of which decreases with decreasing mass of the envelope. The reason for this is twofold, first the thermal energy deposited in the hydrogen envelope decrease with the mass of it, and secondly the radius of the progenitor stars decrease, decreasing the time scale for expansion cooling. Models for Type IIb SNe often have an increased helium abundance in the hydrogen envelope \citep[e.g.][]{Woo94}, because of mixing of helium into the base of the hydrogen envelope. This results in smaller progenitor radii due to decreased opacities, and therefore in shorter durations of the cooling phase. Our models have lower helium abundances as compared \citet{Woo94} and \citet{Shi94}, which should be kept in mind. During most of the cooling phase the photospheric velocities are much higher than those for a bare helium core model, but decrease quickly at the luminosity minimum, after which follow a period when they are significantly lower. The latter effect is larger for models with more massive hydrogen envelopes and is likely caused by deceleration of the helium core. 
%It is of interest to investigate the effects of a hydrogen envelope on the results in Sect.~\ref{s_modelling_IIb}, and as 
As 
suggested by Eqs.~\ref{eq_dep_4}-\ref{eq_dep_5}, and as demonstrated in Sect.~\ref{s_error_sensitivity}, the sensitivity of the estimated explosion energy to an error in the photospheric velocity is high (E $\sim$ v$^{3}$), whereas the sensitivity of the helium core mass is lower.
%As suggested by the dependencies on the photospheric velocity discussed in Sect.~\ref{s_dependence} and as demonstrated in Sect.~\ref{s_error_sensitivity}, the sensitivity of the estimated explosion energy to an error in the photospheric velocity is high (E $\sim$ v$^{3}$), whereas the sensitivity of the helium core mass is lower (M$_{\mathrm{He}}$ $\sim$ v). 
This is quantified by the lower panels of Fig.~\ref{f_hydrogen}, which shows the sensitivity of the estimated explosion energy and helium core mass to the mass of the hydrogen envelope, calculated by fitting the model lightcurves and photospheric velocities to the model grid using the same procedure as in Sect.~\ref{s_modelling_IIb}. If the mass of the hydrogen envelope is larger than $\sim$0.2 M$_\odot$ the explosion energy is significantly underestimated, whereas the helium core mass is less affected. Therefore the presence of a relatively massive hydrogen envelope could have a significant effect on the estimated explosion energy when using bare helium core models.
%The lower panels of Fig.~\ref{f_hydrogen} shows the sensitivity of the estimated explosion energy and helium core mass to the mass of the hydrogen envelope, calculated by fitting the model lightcurves and photospheric velocities to the model grid using the procedure described in Sect.~\ref{s_fitting_proc}. This quantifies the qualitative argument and shows that if the mass of the hydrogen envelope is larger than $\sim$0.2 M$_\odot$ the explosion energy may be significantly underestimated, whereas the helium core mass is less affected.
%Therefore the presence of a relatively massive hydrogen envelope could have a significant effect on the estimated explosion energy when using bare helium core models.
%Otherwise, 
Except for the effect on the photospheric velocities, 
the presence of the hydrogen envelope does not seem to appreciably affect the observed properties after the luminosity minimum.

\subsection{Model physics}
\label{s_hydrogen_physics}

Here we discuss the physics of models with low-mass hydrogen envelopes, exemplified by the 4.05 M$_\odot$ model shown in 
%Figs.~\ref{f_hydrogen_bol_lightcurve} and \ref{f_hydrogen_velocities}. 
Fig.~\ref{f_hydrogen}. 
This model have an hydrogen envelope of 0.17 M$_\odot$, an average hydrogen fraction in the envelope of 0.5, and reach the luminosity minimum at $\sim$11 days, which is similar to, but slightly later than was observed for SN 1993J.

The passage of the shock through the helium core proceeds as described for the bare helium core model, and in the steep density gradient between the helium core and the hydrogen envelope it accelerates to $\sim$20000 km~s$^{-1}$. Once in the hydrogen envelope, where the density is roughly constant, the shock gradually decelerates to $\sim$6000 km~s$^{-1}$, which give rise to a reverse shock propagating backwards into the helium envelope. During the passage of the shock through the hydrogen envelope, the helium core expands and most of the deposited thermal energy is cooled away. At shock breakout the (relatively) cool and expanded helium core is surrounded by the hot and compressed hydrogen envelope, and the subsequent evolution is determined by the expansion and cooling of this envelope. The fraction of the energy deposited in the envelope is about 10 percent, roughly equipartioned into thermal and kinetic energy.

Fig.~\ref{f_hydrogen_rho_T_early} shows the evolution of the density and temperature profiles from shock breakout, which occurs at $\sim$0.3 days, until the luminosity minimum. The profiles are similar to those obtained by modelling of SN 1993J in \citet{Woo94}, \citet{Shi94} and \citet{Bli98}. Initially the hydrogen envelope is opaque and ionized, and the surface luminosity and temperature decreasing by expansion cooling, but at $\sim$4 days the outer parts become optically thin and the photosphere starts to recede into the ejecta. The helium starts to recombine at about the same time, whereas hydrogen stays ionized until $\sim$7 days, and at about $\sim$6 days the photosphere starts to trace the helium recombination front as in the bare helium core models. At the luminosity minimum the hydrogen in the envelope has recombined, the photosphere is located close to the interface between the hydrogen and helium envelope, and the temperature at the thermalization surface is $\sim$9500 K.

\begin{figure}[tbp!]
\includegraphics[width=0.5\textwidth,angle=0]{figs/type-IIb-model-rho-T-early-radius.pdf}
\caption{Evolution of the temperature (left panel) and density (right panel) profiles betwen shock breakout and the luminosity minimum in 10 logarithmically spaced intervals for the 4.05 M$_\odot$ MESA model. The interface between the helium and hydrogen envelopes (dashed line), the photosphere (circles and dashed line) and the helium (downward triangles and dot-dashed line) and hydrogen (upward triangles and dotted line) ionization fronts are also shown.}
\label{f_hydrogen_rho_T_early}
\end{figure}

\section{Model grid fits}
\label{s_modelling_IIb}

Here we use an automated procedure to fit the bolometric lightcurves and photospheric velocities for the CSP and literature samples of Type IIb SNe to those of our grid of SN models. Our method allows us to determine the sensitivity of the derived quantities to errors in the observed quantities, as well as to investigate the degeneracy of the solutions found. As discussed the grid is based on bare helium core models, and given the influence of a low-mass hydrogen envelope on the observed properties (Sect.~\ref{s_hydrogen_effects}), 
%The model grid consist of bare helium core models without a hydrogen envelope (Sect.~\ref{s_model_grid}), and 
the diffusion phase and early tail lightcurves and the diffusion phase photospheric velocities are used to determine the SN and helium core parameters. We also make the assumption, justified for Type IIb SNe, that the helium core is not affected by mass loss. The hydrogen envelope only affects the bolometric lightcurve in the cooling phase, and the parameters of this envelope has to be modelled separately. However, relatively massive hydrogen envelopes do affect the photospheric velocities in the diffusion phase, which might lead to significant underestimates of the explosion energy (Sect.~\ref{s_hydrogen_effects}).

\subsection{The CSP and literature samples}

The Type IIb SNe in the CSP sample consists of SNe 2004ex, 2004ff, 2005Q, 2006T, 2006ba, 2006bf, 2008aq, 2009K, 2009Z and 2009dq \citep{Tad14}, of which SNe 2006bf and 2009dq have been excluded due to bad sampling of the lightcurves. The Type IIb SNe in the literature sample consists of SNe 1993J \citep[e.g.][]{Ric94,Ric96}, 1996cb \citep{Qiu99}, 2003bg \citep{Hamu09}, 2008ax \citep[e.g.][]{Tau11}, 2011dh \citep[e.g.][]{Erg14a}, 2011fu \citep{Kum13}, 2011hs \citep{Buf14}, 2011ei \citep{Mil13} and 2013df \citep{Dyk14}. Out of these SNe 1993J, 2008ax and 2011dh stands out by the quality of the data as well as the hard constraints on the explosion epochs. The details of the observations, the constraints on the explosion epochs and the adopted distances and extincions for the CSP and literature samples are given in Appendices~\ref{a_csp_sample} and \ref{a_literature_sample}, respectively, where we also describe how the photospheric velocities were estimated. The pseudo-bolometric lightcurves were calculated from the photometry using the methods described in \citetalias{Erg14a}, and a UV to MIR bolometric correction determined from SN 2011dh applied.
%, using the data and methods described in \citetalias{Erg14a}. 
The flux falling outside this wavelength range was not corrected for, but given the results from the steady-state NLTE modelling of SN 2011dh presented in \citet{Erg14b}, this correction is likely to be small (<0.15 mag).

\subsection{Fitting procedure}
\label{s_fitting_proc}

The fitting is done by minimization of the square of the relative residuals, giving equal weight to the diffusion phase lightcurve, the tail lightcurve and the diffusion phase photospheric velocity evolution. The division between the diffusion and tail phases is made roughly at the point where the decline rate of the bolometric lightcurve becomes constant. If there is any sign of a cooling phase, the start of the diffusion phase is set to a few days after the rise to peak begins, and otherwise to the first observation. Photospheric velocities above the interface between the helium core and the hydrogen envelope are excluded from the fit. As discussed in \citetalias{Erg14a} this velocity can be estimated from the minimum velocity for the  H$\alpha$ absorption minimum, but is otherwise set to 10000 km~s$^{-1}$. As the explosion epochs in many cases are not well constrained we fit, not only the SN and progenitor parameters (M$_{\mathrm{He}}$, E, M$_{\mathrm{Ni}}$ and Mix$_{\mathrm{Ni}}$), but also the epoch of explosion, which is allowed to vary between the hard limits obtained from detections and non-detections. The errors in the bolometric lightcurves arising from the uncertainties in distance and extinction, and a systematic error in the photospheric velocities, assumed to be 15 percent, were propagated by standard methods.

\subsection{Results for the CSP sample}
\label{s_modelling_IIb_csp_results}

Figure~\ref{f_mg_fit_csp_comp} shows the best-fit model bolometric lightcurve and photospheric velocity evolution compared to the (estimated) observed UV to MIR pseudo-bolometric lightcurve and estimated photosphere velocity evolution for the CSP sample of Type IIb SNe. Table \ref{t_mg_fit_csp_comp} gives the helium core mass, explosion energy, mass and mixing of \element[ ][56]{Ni}, and explosion epoch for the best-fit models and the corresponding errors. Below we review the results and compare to other results obtained in the literature.

\begin{figure*}[tb!]
\includegraphics[width=1.0\textwidth,angle=0]{figs/mg-fit-csp-comp.pdf}
\caption{Bolometric lightcurve (upper panels) and photospheric velocity evolution (lower panels) for the best-fit models as compared to the observed UV to MIR pseudo-bolometric lightcurve and estimated photospheric velocity evolution for the CSP sample of Type IIb SNe. The lower panels shows contour plots of the standard deviation in the fits, normalized to that of the optimal model, projected onto the E-M$_{\mathrm{He}}$ plane. We also show the constraints M$_{\mathrm{ej}}$/E=const (blue) and M$_{\mathrm{ej}}^{2}$/E=const (red) provided by the photospheric velocity evolution and the bolometric lightcurve, respectively.}
\label{f_mg_fit_csp_comp}
\end{figure*}

\begin{table*}[tb!]
\caption{Helium core mass, explosion energy, mass of the \element[ ][56]{Ni} and epoch of explosion for the best-fit models for the CSP sample of Type IIb SNe.}
\begin{center}
\include{mg-fit-csp-comp-table}
\end{center}
\label{t_mg_fit_csp_comp}
\end{table*}

\paragraph{SN 2004ex}

The bolometric lightcurve and photospheric velocity evolution of SN 2004ex is best fit with M$_{\mathrm{He}}$=3.6 M$_\odot$, E=0.65$\times$10$^{51}$ erg and M$_{\mathrm{Ni}}$=0.13 M$_\odot$. The fits to the bolometric lightcurve and the photospheric velocity evolution are both quite good and the solution is well constrained in the E-M$_{\mathrm{He}}$ plane.

\paragraph{SN 2004ff}

The bolometric lightcurve and photospheric velocity evolution of SN 2004ff is best fit with M$_{\mathrm{He}}$=4.6 M$_\odot$, E=1.6$\times$$10^{51}$ erg and M$_{\mathrm{Ni}}$=0.11 M$_\odot$. The fits to the bolometric lightcurve and the photospheric velocity evolution are both quite good and the solution is well constrained in the E-M$_{\mathrm{He}}$ plane. \citet{Lym14} found M$_{\mathrm{ej}}$=1.8 M$_\odot$, E=2.9$\times$$10^{51}$ erg and M$_{\mathrm{Ni}}$=0.22 M$_\odot$ using a fit to approximate \citet{Arn82} models. Within error bars this is consistent with our results except for M$_{\mathrm{Ni}}$, which could be explained by differences in the extinction and the explosion epoch.

\paragraph{SN 2005Q}

The bolometric lightcurve and photospheric velocity evolution of SN 2005Q is best fit with M$_{\mathrm{He}}$=4.5 M$_\odot$, E=1.1$\times$10$^{51}$ erg and M$_{\mathrm{Ni}}$=0.18 M$_\odot$. The fit to the bolometric lightcurve is quite good whereas the fit to the photospheric velocity evolution is worse. Due to the limited rise to peak and tail coverage the constraint from the bolometric lightcurve is weak and the solution is quite degenerate along the M/E=const curve.

\paragraph{SN 2006T}

The bolometric lightcurve and photospheric velocity evolution of SN 2006T is best fit with M$_{\mathrm{He}}$=3.7 M$_\odot$, E=0.70$\times$$10^{51}$ erg and M$_{\mathrm{Ni}}$=0.075 M$_\odot$. The fits to the bolometric lightcurve and the photospheric velocity evolution are both good and the solution is well constrained in the E-M$_{\mathrm{He}}$ plane. \citet{Lym14} found M$_{\mathrm{ej}}$=1.7 M$_\odot$, E=1.2$\times$$10^{51}$ erg and M$_{\mathrm{Ni}}$=0.22 M$_\odot$ using fits to approximate \citet{Arn82} models, which is consistent within error bars with our results.

\paragraph{SN 2006ba}

The bolometric lightcurve and photospheric velocity evolution of SN 2006ba is best fit with M$_{\mathrm{He}}$=4.1 M$_\odot$, E=0.90$\times$$10^{51}$ erg and M$_{\mathrm{Ni}}$=0.088 M$_\odot$. The fits to the bolometric lightcurve and the photospheric velocity evolution are both good (although there is only one photospheric velocity measurement) and the solution is well constrained in the E-M$_{\mathrm{He}}$ plane.

\paragraph{SN 2008aq}

The bolometric lightcurve and photospheric velocity evolution of SN 2008aq is best fit with M$_{\mathrm{He}}$=3.3 M$_\odot$, and E=0.50$\times$$10^{51}$ erg and M$_{\mathrm{Ni}}$=0.050 M$_\odot$. The fits to the bolometric lightcurve and the photospheric velocity evolution are both reasonably good and the solution is well constrained in the E-M$_{\mathrm{He}}$ plane.

\paragraph{SN 2009K}

The bolometric lightcurve and photospheric velocity evolution of SN 2009K is best fit with M$_{\mathrm{He}}$=6.0 M$_\odot$, E=1.1$\times$$10^{51}$ erg and M$_{\mathrm{Ni}}$=0.16 M$_\odot$. The fits to the bolometric lightcurve and the photospheric velocity evolution are both reasonably good. Due to the the missing lightcurve coverage after the peak, the constraint from the bolometric lightcurve is weak, and the solution is quite degenerate along the M/E=const curve.

\paragraph{SN 2009Z}

The bolometric lightcurve and photospheric velocity evolution of SN 2009Z is best fit with M$_{\mathrm{He}}$=4.1 M$_\odot$, E=1.2$\times$$10^{51}$ erg and M$_{\mathrm{Ni}}$=0.22 M$_\odot$. The fits to the bolometric lightcurve and the photospheric velocity evolution are both reasonably good. The solution is reasonably constrained in helium core mass but worse in explosion energy and is a bit degenerate along the M$^{2}$/E=const curve.

\subsection{Results for the literature sample}
\label{s_modelling_IIb_lit_results}

Figure~\ref{f_mg_fit_lit_comp} shows the best-fit model bolometric lightcurve and photospheric velocity evolution compared to the (estimated) observed UV to MIR pseudo-bolometric lightcurve and estimated photospheric velocity evolution for the literature sample of Type IIb SNe. Table \ref{t_mg_fit_lit_comp} gives the helium core mass, explosion energy, mass and mixing of \element[ ][56]{Ni}, and explosion epoch for the best-fit models and the corresponding errors. Below we review the results and compare to other results obtained in the literature.

\begin{figure*}[tb!]
\includegraphics[width=1.0\textwidth,angle=0]{figs/mg-fit-lit-comp.pdf}
\caption{Bolometric lightcurve (upper panels) and photospheric velocity evolution (lower panels) for the best-fit models as compared to the observed UV to MIR pseudo-bolometric lightcurve and estimated photospheric velocity evolution for the literature sample of Type IIb SNe. The lower panels shows contour plots of the standard deviation in the fits, normalized to that of the optimal model, projected onto the E-M$_{\mathrm{He}}$ plane. We also show the constraints M$_{\mathrm{ej}}$/E=const (blue) and M$_{\mathrm{ej}}^{2}$/E=const (red) provided by the photospheric velocity evolution and the bolometric lightcurve, respectively.}
\label{f_mg_fit_lit_comp}
\end{figure*}

\begin{table*}[tb!]
\caption{Helium core mass, explosion energy, mass of the \element[ ][56]{Ni} and epoch of explosion for the best-fit models for the literature sample of Type IIb SNe.}
\begin{center}
\include{mg-fit-lit-comp-table}
\end{center}
\label{t_mg_fit_lit_comp}
\end{table*}

\paragraph{SN 1996cb}

The bolometric lightcurve and photospheric velocity evolution of SN 1996cb is best fit with M$_{\mathrm{He}}$=4.9 M$_\odot$, E=1.4$\times$10$^{51}$ erg and M$_{\mathrm{Ni}}$=0.11 M$_\odot$. The fits to the bolometric lightcurve and the photospheric velocity evolution are both good, but the solution is quite degenerate along the M/E=const curve suggesting that the constraint from the lightcurve is relatively weak. \citet{Lym14} found M$_{\mathrm{ej}}$=2.4 M$_\odot$, E=2.1$\times$$10^{51}$ erg and M$_{\mathrm{Ni}}$=0.12 M$_\odot$ using fits to approximate \citet{Arn82} models, which is consistent within error bars with our results. 

\paragraph{SN 1993J}

The bolometric lightcurve and photospheric velocity evolution of SN 1993J is best fit with M$_{\mathrm{He}}$=3.4 M$_\odot$, E=0.60$\times$10$^{51}$ erg and M$_{\mathrm{Ni}}$=0.10 M$_\odot$. The fits to the bolometric lightcurve and the photospheric velocity evolution are good and the solution is well constrained in the E-M$_{\mathrm{He}}$ plane. \citet{Shi94} found M$_{\mathrm{He}}$=4.0 M$_\odot$, E=1.0-1.2$\times$10$^{51}$ erg and M$_{\mathrm{Ni}}$=0.073 M$_\odot$, and \citet{Woo94} found M$_{\mathrm{He}}$=3.5 M$_\odot$, E=1.5$\times$10$^{51}$ erg and M$_{\mathrm{He}}$=0.073 M$_\odot$ to well reproduce observations. These results are consistent within error bars with ours, except for the explosion energies which are considerably higher. The disagreement is likely caused by the presence of a relatively massive hydrogen envelope (Sect.~\ref{s_hydrogen_envelope}), as suggested by the extent of the cooling phase. The explosion energy deposited in the helium core for the \citet{Shi94} models was $\sim$0.6$\times$10$^{51}$ erg, which is in agreement with our result.

\paragraph{SN 2003bg}

The bolometric lightcurve and photospheric velocity evolution of SN 2003bg is best fit with M$_{\mathrm{He}}$=6.4 M$_\odot$, E=1.6$\times$10$^{51}$ erg and M$_{\mathrm{Ni}}$=0.18 M$_\odot$. The fits to the bolometric lightcurve and the photospheric velocity evolution are acceptable, but the solution is quite degenerate along the M/E=const curve. This suggests that the constraint from the lightcurve is relatively weak, which is not surprising given the sparse sampling during the diffusion peak. \citet{Maz09} find M$_{\mathrm{ej}}$=3.9-4.8 M$_\odot$, E=5$\times$10$^{51}$ erg and M$_{\mathrm{Ni}}$=0.15-0.18 M$_\odot$ to well reproduce observations. This is consistent within error bars with our results, except for the explosion energy which is considerably higher. The disagreement could be caused by the presence of a relatively massive hydrogen envelope (Sect.~\ref{s_hydrogen_envelope}), although there is only a hint of a cooling phase in the earliest observations. According to \citet{Maz09} the kinetic energy of the helium core was $\lesssim$2.5$\times$10$^{51}$, which is in better agreement with our result.

\paragraph{SN 2008ax}

The bolometric lightcurve and photospheric velocity evolution of SN 2008ax is best fit with M$_{\mathrm{He}}$=3.4 M$_\odot$, E=0.65$\times$10$^{51}$ erg and M$_{\mathrm{Ni}}$=0.081 M$_\odot$. The fit to the bolometric lightcurve is good whereas the fit to the photospheric velocity evolution is a bit worse, and the solution is reasonable well constrained in the E-M$_{\mathrm{He}}$ plane. \citet{Tsv09} find M$_{\mathrm{He}}$=3.5 M$_\odot$, E=1.5$\times$10$^{51}$ erg and M$_{\mathrm{Ni}}$=0.11 M$_\odot$ to well reproduce observations using the SN 1993J model 13C from \citet{Woo94}, with M$_{\mathrm{Ni}}$ adjusted to match SN 2008ax. These results are consistent within error bars with ours, except the explosion energy which is condsiderably higher. However, in this case the situation is reversed as compared to SN 1993J. The absence of an extended cooling phase suggests a relatively low-mass hydrogen envelope for SN 2008ax, so the SN 1993J based model used by \citet{Tsv09} could lead to a significant overestimate of the explosion energy (Sect.~\ref{s_hydrogen_envelope}).

\paragraph{SN 2011dh}

The bolometric lightcurve and photospheric velocity evolution of SN 2011dh is best fits with a helium core mass of 3.44 M$_\odot$, and explosion energy of 0.55$\times$10$^{51}$ erg and a mass of \element[ ][56]{Ni} of 0.075 M$_\odot$. The fits to the bolometric lightcurve and the photospheric velocity evolution are good and the solution is well constrained in the E-M$_{\mathrm{He}}$ plane. \citet{Ber12} found M$_\odot$3.3-4.0 M$_\odot$, E=0.6-1.0$\times$10$^{51}$ erg and M$_{\mathrm{Ni}}$=0.05-0.010 M$_\odot$ \citep[adjusted in][]{Erg14a} to well reproduce observations. These results are consistent within error bars with our results, which are also in good agreement with those obtained in \citet{Erg14b} based on the <400 days lightcurve and an extended version of the model grid using a bolometric correction determined with steady-state NLTE modelling.
%The results for SN 2011dh are the same as previously presented in \citetalias{Erg14b}. The fits to the bolometric lightcurve and the photospheric velocity evolution are good and the degeneracy of the solution is well constrained in the E-M$_{\mathrm{He}}$ plane. The results are in good agreement with those presented in \citet{Ber12} obtained with the hydrodynamical code presented by \citet{Ber11} which, as previously discussed, is very similar to HYDE.

\paragraph{SN 2011ei}

The bolometric lightcurve and photospheric velocity evolution of SN 2011ei is best fit with M$_{\mathrm{He}}$=7.4 M$_\odot$, E=3.8$\times$10$^{51}$ erg and M$_{\mathrm{Ni}}$=0.032 M$_\odot$. The fits to the bolometric lightcurve and the photospheric velocity evolution are both good, but due to the limited tail coverage the constraint from the bolometric lightcurve is relatively weak, and the solution quite degenerate along the M/E=const curve. This SN shows the most extreme values for the SN and progenitor parameters in the sample. Whereas the helium core mass and explosion energy are the highest, the mass of \element[ ][56]{Ni} is the lowest. The high helium core mass and explosion energy derived stem mainly from the unusually high photospheric velocities, whereas the time at which peak luminosity occurs is quite typical. The small ejecta mass of 0.3 M$_\odot$ estimated from modelling of nebular spectra in \citet{Mil13} seems to be excluded by our results.

\paragraph{SN 2011fu}

The bolometric lightcurve and photospheric velocity evolution of SN 2011fu is best fit with M$_{\mathrm{He}}$=5.8 M$_\odot$, E=1.8$\times$10$^{51}$ erg and M$_{\mathrm{Ni}}$=0.23 M$_\odot$. The fits to the bolometric lightcurve and the photospheric velocity evolution are reasonably good and the degeneracy of the solution acceptable, although there is clear degeneracy along the M/E=const curve suggesting that the constraint from the lightcurve is relatively weak.

\paragraph{SN 2011hs}

The bolometric lightcurve and photospheric velocity evolution of SN 2011hs is best fit with M$_{\mathrm{He}}$=2.6 M$_\odot$, E=0.45$\times$10$^{51}$ erg and M$_{\mathrm{Ni}}$=0.088 M$_\odot$. The fits to the bolometric lightcurve and the photospheric velocity evolution are good and the degeneracy of the solution is well constrained above in the E-M$_{\mathrm{He}}$ plane. However, the solution lies at the border of the covered parameter space and is not well constrained below. \citet{Buf14} found M$_{\mathrm{He}}$=3.3-4 M$_\odot$, E=0.6-0.9$\times$10$^{51}$ erg and M$_{\mathrm{Ni}}$=0.037-0.040 M$_\odot$ of \element[ ][56]{Ni} to well reproduce the observations. The lower bound on the explosion energy is consistent within error bars with our result, whereas the lower bound on helium core mass and the upper bound on the mass of $^{56}$Ni lies just outside our error bars. It is worth noting that the lower bounds on the explosion energy and helium core mass, which agrees best with our results, corresponds to the case of strong mixing of the $^{56}$Ni, in agreement with our results. The results in  \citet{Buf14} were obtained with the hydrodynamical code presented by \citet{Ber11} which, as previously discussed, is very similar to HYDE.

\paragraph{SN 2013df}

The bolometric lightcurve and photospheric velocity evolution of SN 2011df is best fit with M$_{\mathrm{He}}$=3.8 M$_\odot$, E=0.95$\times$10$^{51}$ erg and M$_{\mathrm{Ni}}$=0.056 M$_\odot$. The fits to the bolometric lightcurve and the photospheric velocity evolution are good and the solution is well constrained in the E-M$_{\mathrm{He}}$ plane. \citet{Mor14} found M$_{\mathrm{ej}}$=0.8-1.4, E=0.4-1.2$\times$10$^{51}$ erg and M$_{\mathrm{Ni}}$=0.1 M$_\odot$ using fits to approximate \citet{Arn82} models. The explosion energy is consistent within error bars with our results, whereas the ejecta mass is below and the mass of $^{56}$Ni above our error bars, the latter partly explained by the difference in the adopted distance.

\subsection{Error sensitivity}
\label{s_error_sensitivity}

\begin{figure}[tb]
\includegraphics[width=0.5\textwidth,angle=0]{figs/hydro-IIb-error-propagation.pdf}
\caption{Sensitivity of the derived explosion energy (upper panels), ejecta mass (middle panels), mass of \element[ ][56]{Ni} (lower panels) to a change in the distance (left panels), extinction (middle panels) and photospheric velocities (right panels). For consistency the changes in all quantities are expressed in magnitudes. The derived quantities for the CSP and literature samples of Type IIb SNe are shown as black dots, power-law fits as red solid lines and the scalings expected from the \citet{Arn82} model as blue solid lines.}
\label{f_mg_error_prop}
\end{figure}

 Fig.~\ref{f_mg_error_prop} shows the sensitivity of the derived quantities to errors in the distance, extinction and a systematic error in the photospheric velocity for the CSP and literature samples of Type IIb SNe. To compare with approximate scalings we have replaced the helium core mass with the ejecta mass. For the ejecta mass and explosion energy the dependence on the distance and extinction is weak, whereas the dependence on the photospheric velocity is strong. For the mass of \element[ ][56]{Ni} the sensitivity on the distance and extinction is strong whereas the dependence on the photospheric velocity is weak. In general we see that an error in the distance and extinction mainly corresponds to an error in the mass of \element[ ][56]{Ni}, whereas an error in the photospheric velocity mainly corresponds to an error in the helium core mass and explosion energy. This behaviour is in agreement with the model grid dependencies discussed in Sect.~\ref{s_dependence} and with the qualitative discussion in \citetalias{Erg14a}, based on the \citet{Arn82} model. In Fig.~\ref{f_mg_error_prop} we show the scalings expected from the \citet{Arn82} model and, as previously found in Sect.~\ref{s_dependence}, these are well followed by the model grid.
%As discussed therein we expect the mass of \element[ ][56]{Ni} to be proportional to the distance and, if we assume that the SED is peaking near the $V$ band and the change in extinction is reasonably small, to be proportional to $10^{A_{V}/2.5}$. From the \citet{Arn82} model also follows that the ejecta mass and explosion energy depend on the expansion velocity as $\mathrm{M \propto v}$ and $\mathrm{E \propto v^3}$ (Sect.~\ref{s_dependencies})
%diffusion time depends on the ejecta mass and explosion energy as M$_{\mathrm{ej}}$$^{3}$/E and the velocity squared as E/M$_{\mathrm{ej}}$. Given this we expect the ejecta mass to be proportional to the velocity and the energy to be proportional to the velocity cubed. As seen in Fig.~\ref{f_mg_error_prop} these approximate scalings are well followed by the model.

\subsection{Degeneracy of the solution}
\label{s_degeneracy}

\begin{figure}[tbp!]
\includegraphics[width=0.5\textwidth,angle=0]{figs/hydro-IIb-degeneracy-1.pdf}
\caption{Model bolometric lightcurve (upper panels) and photospheric velocity evolution (middle panels) as compared to the observed UV to MIR pseudo-bolometric lightcurve and estimated photospheric velocity evolution for SN 2011dh. Models with a normalized standard deviation <2 are shown in red for the cases when the lightcurve and photospheric velocity evolution (left panels), only the lightcurve (middle panels) and only the diffusion phase lightcurve (right panels) where used in the fit. The lower panels shows the corresponding contour plots displayed as in Figs.~\ref{f_mg_fit_csp_comp} and \ref{f_mg_fit_lit_comp}.}
\label{f_mg_deg}
\end{figure}

Fig.~\ref{f_mg_deg} shows the bolometric lightcurve and photospheric velocity evolution for models with a normalized standard deviation <2 as compared to observations for SN 2011dh. The left, middle and right panels show the cases when lightcurve and photospheric velocity evolution, only the lightcurve and only the diffusion phase lightcurve were used in the fit, respectively. The lower panels shows the corresponding contour plots and in the two latter cases, where the photospheric velocity evolution is not used in the fit, the solution is completely degenerate. This is not obvious as the diffusion phase and tail phase lightcurves might provide independent constraints arising from the diffusion time for thermal radiation and the optical depth for $\gamma$-rays, respectively. In \citetalias{Erg14a} we argued that this could be the case, given that the diffusion time and optical depth provide the constraints M$_{\mathrm{ej}}^{3}$/E=const and M$_{\mathrm{ej}}^{2}$/E=const, respectively, in the \citet{Arn82} model. However, as discussed in Sect.~\ref{s_dependence}, this model assumes a constant opacity and the constraint provided by diffusion phase lightcurve is rather M$_{\mathrm{ej}}^{2}$/E=const for the hydrodynamical models. Therefore the diffusion phase and tail phase lightcurves appears to provide similar constraints  
%in the hydrodynamical models, and when only the lightcurve is used
and, as seen in the middle and right panels of Fig.~\ref{f_mg_deg}, 
the degeneracy regions well follows the M$_{\mathrm{ej}}^{2}$/E=const relation. The photospheric velocity evolution is expected to provide a constraint similar to M$_{\mathrm{ej}}$/E=const, which would break the degeneracy and, as seen in the left panels of Fig.~\ref{f_mg_deg}, this is also the case. We have exemplified with SN 2011dh but the conclusion is the same for the other SNe in the CSP and literature samples.

\subsection{Sample statistics}

\begin{figure*}[tb!]
\includegraphics[width=1.0\textwidth,angle=0]{figs/hydro-IIb-sample-stat-1.pdf}
\caption{E versus M$_{\mathrm{He}}$ (left panel), M$_{\mathrm{Ni}}$ versus M$_{\mathrm{He}}$ (middle panel) and M$_{\mathrm{Ni}}$ versus E (right panel) for the CSP (blue) and literature (red) samples of Type IIb SNe. We also show power law fits as black dashed lines.}
\label{f_sample_stat_1}
\end{figure*}

\begin{figure}[tb!]
\includegraphics[width=0.5\textwidth,angle=0]{figs/hydro-IIb-sample-stat-2.pdf}
\caption{Number of SNe with initial mass in the 8-15, 15-20, 20-25 and 25-30 M$_\odot$ bins for the CSP (blue) and literature (red) samples of Type IIb SNe as compared to a standard Salpeter IMF (black).}
\label{f_sample_stat_2}
\end{figure}

No parameter studies specifically aimed at Type IIb SNe have been published, but the stripped-envelope SN parameter study by \citet{Lym14} include all Type IIb SNe from the literature sample except SNe 2011fu, 2011ei and 2013df. On the other hand, their sample includes the Type IIb SN 2004el missing from ours. Their results are based on the approximate \citep{Arn82} model, but are nevertheless interesting to compare with. Fig.~\ref{f_sample_stat_1} shows E versus M$_{\mathrm{He}}$ (left panel), M$_{\mathrm{Ni}}$ versus M$_{\mathrm{He}}$ (middle panel) and M$_{\mathrm{Ni}}$ versus E (right panel) for the combined CSP and literature sample of Type IIb SNe. The caveat that the explosion energy might be underestimated for some SNe with more massive hydrogen envelopes (Sect.~\ref{s_hydrogen_effects}) should be kept in mind.

We find a correlation between E and M$_{\mathrm{He}}$, best fitted with a power law with index 1.8. Some caution is advised as this is similar to the M$_{\mathrm{ej}}^{2}$/E=const degeneracy curve (Sect.~\ref{s_degeneracy}). However, the SNe that obviously belongs to the low and high mass ends, like SN 2011hs and 2003bg, well follows the relation. \citet{Lym14} also find a correlation, and assuming a mass for the compact remnant of 1.5 M$_\odot$, their results for the Type IIb SNe are best fitted with a power law with index of 1.8, in agreement with our results. The correlation found is also in qualitative agreement with the relation between progenitor mass and explosion energy for Type IIP SNe suggested by \citet{Poz13}, which has a power law index of 3. 

We find correlations between M$_{\mathrm{Ni}}$ and E and between M$_{\mathrm{Ni}}$ and M$_{\mathrm{He}}$, best fitted with power laws with indices 0.7 and 1.1, respectively. \citet{Lym14} also find correlations between these quantities, and assuming a mass for the compact remnant of 1.5 M$_\odot$, their results for the Type IIb SNe are best fitted with power laws with indices of 0.89 and 1.36, respectively, in good agreement with our results. \citet{Ham03} find a correlation between M$_{\mathrm{Ni}}$ and E for Type IIP SNe, using relations derived from model grid fits by \citet{Lit83,Lit85}, and their results are best fitted with a power law with index 0.88, in good agreement with our results for Type IIb SNe. The correlations found are also in qualitative agreement with the relations between progenitor mass and M$_{\mathrm{Ni}}$ and expansion velocities and M$_{\mathrm{Ni}}$ for Type IIP SNe found by \citet{Fra11} and \citet{Mag12}, respectively. We note that SN 2011ei is an extreme outlier in the M$_{\mathrm{Ni}}$ versus M$_{\mathrm{He}}$ and M$_{\mathrm{Ni}}$ versus E figures, suggesting a different nature of this SN as compared to the other. 

Fig.\ref{f_sample_stat_2} shows the number of SNe with initial mass in the 8-15, 15-20, 20-25 and 25-30 M$_\odot$ bins for the combined CSP and literature sample of Type IIb SNe as compared to a standard Salpeter IMF. We find 6 SNe instead of the expected 3 SNe in the 15-20 M$_\odot$ bin and no SNe instead of the expected 4 SNe in the >25 M$_\odot$ bins. Except for this there are no significant deviations from a standard Salpeter IMF. We find 50 percent of the SNe in the <15 M$_\odot$ bin and 88 percent of the SNe in the <20 M$_\odot$ bins. The results for the Type IIb SNe from \citet{Lym14} are similar, with 67 percent of the SNe in the <15 M$_\odot$ bin and 100 percent of the SNe in the <20 M$_\odot$ bins. Given the current estimates of stellar mass loss rates (references) the implication of our result is quite clear; either the binary channel is dominating the production of Type IIb SNe or these rates needs a serious revision. The idea that Type IIb SNe arise mainly from the binary channel has been proposed by several authors (references), but the size of the sample and the relatively detailed modelling used strengthens the support for this scenario considerably.

\section{Conclusions}
\label{s_conclusions}

We present HYDE, a new one-dimensional hydrodynamical code and use it to build a grid of SN models based on bare helium core models evolved with MESA STAR. Such a grid is well suited to model the diffusion and early tail phase of Type IIb SNe, as the progenitors of these are thought to have lost all but a tiny fraction of their hydrogen envelopes. We investigate the dependence of the observed quantities on the progenitor and SN parameters, and find these to qualitatively follow expectations from the approximate \citet{Arn82} model. However, significant quantitative differences do exist, likely because of the constant opacity assumed in this model. We also investigate the effects of a low-mass hydrogen envelope on the observed properties, and find these to be negligible after the luminosity minimum, expect for relatively massive hydrogen envelopes, where the photospheric velocities are decreased, likely because of deceleration of the helium core.

We use an automated fitting procedure 
%giving equal weight to the diffusion phase and early tail phase bolometric lightcurve and the photospheric velocities, 
to fit the bolometric lightcurves and photospheric velocities for the CSP and literature samples of Type IIb SNe to this grid of SN models. This allows us to take into account the uncertainties in distance, extinction and photospheric velocities, as well as to investigate the degeneracy of the solutions. We find SN and progenitor parameters in reasonable agreement with previous hydrodynamical modelling for SNe 1993J, 2003bg, 2008ax, 2011dh, 2011hs and 2003bg. However, in the case of 1993J and 2003bg, the derived explosion energies are significantly lower, likely due to the effect of their relatively massive hydrogen envelopes. For SN 2011ei we find an initial mass quite different from that obtained by modelling of nebular spectra in \citet{Mil13}. We find an almost complete degeneracy in the helium core mass and explosion energy along the M$_{\mathrm{ej}}^{2}$/E=const curve if the photospheric velocities are not used in the fit. When these are included the degeneracy is broken and the fit becomes quite robust. We find an error in the distance and extinction to propagate mainly to derived the mass of \element[ ][56]{Ni}, and a systematic error in the photospheric velocity to propagate mainly to the derived helium core mass and explosion energy.

We find correlations between the SN and progenitor parameters, the explosion energy increasing with helium core mass, and the mass of $^{56}$Ni increasing with the explosion energy. The former correlation is best fitted with power-law index of 1.8 and the latter with a power-law index of 1.1, in good agreement with the results in \citet{Lym14}, obtained for a smaller sample using the approximate \citet{Arn82} model. The initial masses of the combined CSP and literature samples follows a standard Salpeter IMF reasonably well, although there is an overpopulation in the 15-20 M$_\odot$ bin and an underpopulation in the >25 M$_\odot$ bins. The fraction of SNe with mass <15 M$_\odot$ and <20 M$_\odot$ are 50 and 88 percent, respectively, in good agreement with the results in \citet{Lym14}. The implication of this result is quite clear; either the binary channel is dominating the production of Type IIb SNe or stellar massloss rates are considerably higher than expected. This conclusion is not new, and the evidence for this have been growing since the discovery of SN 1993J, but the size of the sample and the relatively detailed modelling used strengthen it considerably. 
%Finally, as discussed a number of caveats do exist, but with further development of the method and increasing size of the sample our results could be refined. A population of Type IIb SNe arising from single stars would for example be expected to result in a bump in the IMF at higher masses.

\section{Acknowledgements}

\appendix

\section{The CSP sample of Type IIb SNe}
\label{a_csp_sample}

The Type IIb SNe in the CSP sample consists of SNe 2004ex, 2004ff, 2005Q, 2006T, 2006ba, 2006bf, 2008aq, 2009K, 2009Z and 2009dq. The observational data for these SNe are described in \citet{Tad14} and SNe 2006bf and 2009dq have been excluded due to bad sampling of the lightcurves. 
%The pseudo-bolometric lightcurves were calculated from the photometry using the methods described in \citetalias{Erg14a} and a UV to MIR bolometric correction as determined for SN 2011dh, using the data and methods described in \citetalias{Erg14a}, applied. 
The photospheric velocities were estimated from the absorption minimum of the \ion{Fe}{ii} 5169 \AA~line, in turn measured from the observed spectra as described in \citet{Tad14}. Wherever used, the line-of-site extinction within the Milky way have been adopted from the \citet{Sch98} extinction maps, recalibrated by \citet{Sch11}.

\paragraph{SN 2004ex}

Discovered 2004 Oct 11.34 UT \citep{Jac04} in NGC 182 at an apparent magnitude of 17.7, and the explosion epoch is constrained by non-detections from Oct 6.35 UT (<20.0 mag) and 10.33 (<19.0 mag) \citep{Shi04}. The photometric data covers the $u$ to $H$ bands and the 5-85 days period after discovery. Both the diffusion peak and early tail are well covered. The distance modulus for the NGC 182 was adopted as the median and standard deviation of the literature values given by the NASA/IPAC Extragalactic Database (NED), being 34.83$\pm{0.27}$ mag. The line-of-sight extinction within NGC 182 was estimated to $E$($B$-$V$)$_{\mathrm{H}}$=0.078 mag by comparison of the $V$-$i$ colour at $r$-band maximum with SN 2011dh. Adding the line-of-sight-extinction within the Milky Way ($E$($B$-$V$)$_\mathrm{MW}$=0.021 mag) gives $E$($B$-$V$)=0.099 mag. The pseudo-bolometric UV to MIR bolometric lightcurve was calculated using the $uBVriJH$ bands.

\paragraph{SN 2004ff}

Discovered 2004 Oct 30.40 UT \citep{Pug04} in ESO 552-G40 at an apparent magnitude of 18.0, and the explosion epoch is constrained by a non-detections from 2004 Oct 13.41 (<19.0 mag) and 21.40 UT (<18.0 mag) \citep{Pug04}. The photometric data covers the $u$ to $H$ band and the 5-80 days period after discovery. The rise to peak is not well covered and we have included observations from \citet{Pug04} to extend this coverage. In the absence of literature measurements of the distance to ESO 552-G40 we adopt the Virgo, Great Attractor and Shapley corrected kinematic distance modulus given by NED, being 34.82$\pm{0.15}$ mag. The line-of-sight extinction within ESO 552-G40 was estimated to $E$($B$-$V$)$_{\mathrm{H}}$=0.138 mag by comparison of the $V$-$i$ colour at $r$-band maximum with SN 2011dh. Adding the line-of-sight-extinction within the Milky Way ($E$($B$-$V$)$_\mathrm{MW}$=0.029 mag) gives $E$($B$-$V$)=0.167 mag. The pseudo-bolometric UV to MIR bolometric lightcurve was calculated using the $uBVriJH$ bands.

\paragraph{SN 2005Q}

Discovered 2005 Jan 28.80 UT \citep{Mon05} in ESO 244-G31 at an apparent magnitude of 17.2, and the explosion epoch is constrained by a non-detection from 2004 Dec 30.81 UT (<18.7 mag). The photometric data covers the $u$ to $i$ bands and the 0-50 days period after discovery. The rise to peak and the early tail is not well covered. The distance modulus for ESO 244-G31 was adopted as the median and standard deviation of the literature values given by NED, being 34.83$\pm{0.18}$ mag. The $V$-$i$ colour at $r$-band maximum was bluer than for SN 2011dh, so the line-of-sight extinction within ESO 244-G31 was assumed to be negligible. The line-of-sight-extinction within the Milky Way is $E$($B$-$V$)$_\mathrm{MW}$=0.023 mag. The pseudo-bolometric UV to MIR bolometric lightcurve was calculated using the $uBVri$ bands.

\paragraph{SN 2006T}

Discovered 2006 Jan 30.99 UT \citep{Mon06a} in NGC 3054 at an apparent magnitude of 17.2, and the explosion epoch is constrained by a non-detection from 2006 Jan 16.96 UT (<18.0 mag) \citep{Mon06a}. The photometric data covers the $u$ to $H$ bands and the 0-125 days period after discovery. Both the diffusion peak and the early tail are well covered. The distance modulus for NGC 3054 was adopted as the median and standard deviation of the literature values given by NED, being 32.58$\pm{0.35}$ mag. The $V$-$i$ colour at $r$-band maximum was bluer than for SN 2011dh, so the line-of-sight extinction within NGC 3054 was assumed to be negligible. The line-of-sight-extinction within the Milky Way is $E$($B$-$V$)$_\mathrm{MW}$=0.181 mag. The pseudo-bolometric UV to MIR bolometric lightcurve was calculated using the $uBVriJH$ bands.

\paragraph{SN 2006ba}

Discovered 2006 Mar 19.81 UT \citep{Mon06b} in NGC 2980 at an apparent magnitude of 17.7, and the explosion epoch is constrained by a non-detection from 2006 Feb 5.04 UT (<18.8) \citep{Mon06b}. The photometric data covers the $u$ to $H$ bands and the 5-80 days period after discovery. The rise to peak is not well covered. The distance modulus for NGC 2980 was adopted as the median and standard deviation of the literature values given by NED, being 34.48$\pm{0.26}$ mag. The line-of-sight extinction within NGC 2980 was estimated to $E$($B$-$V$)$_{\mathrm{H}}$=0.212 by comparison of the $V$-$i$ colour at $r$-band maximum with SN 2011dh. Adding the line-of-sight-extinction within the Milky Way ($E$($B$-$V$)$_\mathrm{MW}$=0.046) gives $E$($B$-$V$)=0.258. The pseudo-bolometric UV to MIR bolometric lightcurve was calculated using the $BVriJ$ bands.

\paragraph{SN 2008aq}

Discovered 2008 Feb 27.44 UT \citep{Chu08} in MCG -02-33-20 at an apparent magnitude of 16.3, and the explosion epoch is constrained by a non-detection from 2008 Feb 10.47 UT (<19.1 mag) \citep{Chu08}. The photometric data covers the $u$ to $H$ bands and the 5-120 days period after discovery. The rise to peak is not well covered but we have included observations from \citet{Chu08} and \citet{Bro08} to extend this coverage. The distance modulus for MCG -02-33-20 was adopted as the median and standard deviation of the literature values given by NED, being 32.45$\pm{0.43,0.43}$ mag. The $V$-$i$ colour at $r$-band maximum was bluer than for SN 2011dh so the line-of-sight extinction within MCG -02-33-20 was assumed to be negligible. The line-of-sight-extinction within the Milky Way is $E$($B$-$V$)$_\mathrm{MW}$=0.040 mag. The pseudo-bolometric UV to MIR bolometric lightcurve was calculated using the $uBVriJH$ bands.

\paragraph{SN 2009K}

Discovered 2009 Jan 14.07 UT \citep{Pig09} in NGC 1620 at an apparent magnitude of 14.9, and the explosion epoch is constrained by a non-detection from 2009 Jan 11.08 UT (<18.0 mag) \citep{Pig09}. The photometric data covers the $B$ to $H$ bands and the 0-50 days period after discovery. The early tail is not covered. The distance modulus for NGC 1620 was adopted as the median and standard deviation of the literature values given by NED, being 33.15$\pm{0.22}$ mag. The line-of-sight extinction within NGC 1620 was estimated to $E$($B$-$V$)$_{\mathrm{H}}$=0.057 mag by comparison of the $V$-$i$ colour at $r$-band maximum with SN 2011dh. Adding the line-of-sight-extinction within the Milky Way ($E$($B$-$V$)$_\mathrm{MW}$=0.051 mag) gives $E$($B$-$V$)=0.108 mag. The pseudo-bolometric UV to MIR bolometric lightcurve was calculated using the $uBVri$ bands.

\paragraph{SN 2009Z}

Discovered 2009 Feb 2.53 UT \citep{Gri09} in SDSS J140153.80-012035.5 at an apparent magnitude of 18.1, and the explosion epoch is constrained by a non-detection from 2008 Jun-Jul (<19.4 mag) \citep{Gri09}. The photometric data covers $u$ to $i$ bands and the 5-85 days period after discovery, although additional NIR photometry where obtained at $\sim$400 days. The rise to peak is not well covered but we have included observations from \citet{Gri09} to extend this coverage. In the absence of literature measurements of the distance to SDSS J140153.80-012035.5 we adopt the the Virgo, Great Attractor and Shapley corrected kinematic distance modulus given by NED, being 35.26$\pm{0.15}$ mag. The $V$-$i$ colour at $r$-band maximum was bluer than for SN 2011dh so the line-of-sight extinction within SDSS J140153.80-012035.5 was assumed to be negligible. The line-of-sight-extinction within the Milky Way is $E$($B$-$V$)$_\mathrm{MW}$=0.042 mag. The pseudo-bolometric UV to MIR bolometric lightcurve was calculated using the $uBVri$ bands.

\section{The literature sample of Type IIb SNe}
\label{a_literature_sample}

The Type IIb SNe in the literature sample consists of SNe 1993J \citep[e.g.][]{Ric94,Ric96}, 1996cb \citep{Qiu99}, 2003bg \citep{Hamu09}, 2008ax \citep[e.g.][]{Tau11}, 2011dh \citep[e.g.][]{Erg14a}, 2011fu \citep{Kum13}, 2011hs \citep{Buf14}, 2011ei \citep{Mil13} and 2013df \citep{Dyk14}. 
%Out of these SNe 1993J, 2008ax and 2011dh stands out by the quality of the data as well as the hard constraints on the epoch and the results for these have been previously discussed in \citetalias{Erg14b}. The pseudo-bolometric lightcurves were calculated from the photometry using the methods described in \citetalias{Erg14a} and a UV to MIR bolometric correction as determined for SN 2011dh, using the data and methods described in \citetalias{Erg14a}, applied. 
Wherever used, the line-of-site extinction within the Milky way have been adopted from the \citet{Sch98} extinction maps, recalibrated by \citet{Sch11}.

\paragraph{SN 1996cb}

Discovered 1996 Dec 15.71 UT \citep{Nak96} in NGC 3510 at an apparent magnitude of 16.5, and the explosion epoch is constrained by a non-detection from 1996 Nov 29 UT (<19.0 mag) \citep{Qia96}. The photometric data was taken from \citet{Qiu99} and covers the $B$ to $R$ bands and the 5-160 days period after discovery. To extend the rise to peak coverage we also included observations from \citep{Nak96} and \citep{Qia96}. The distance modulus for NGC 3510 was adopted as the mean and standard deviation of the literature values given by NED being 30.57$\pm{1.03}$. The total line-of-sight extinction was taken as the mean of the upper limit determined by \citet{Qiu99} from comparison to SN 1993J ($E$($B$-$V$)=0.12) and the line-of-sight extinction within the Milky Way ($E$($B$-$V$)=0.12), giving $E$($B$-$V$)=0.073$\pm{0.047}$. The pseudo-bolometric UV to MIR bolometric lightcurve was calculated using the $BVR$ bands. Estimates of the photospheric velocities using SYNOW were adopted from \citet{Den01}. This method has been shown by \citet{Tak12} to give results similar to those obtained from the absorption minimum of \ion{Fe}{ii} 5169 \AA~line for a sample of Type IIP SNe.

\paragraph{SN 2003bg}

Discovered 2003 Feb 25.70 UT \citep{Woo03} in MCG -05-10-15 at an apparent magnitude of 15.0, and the explosion epoch is constrained by a non-detection from 2002 Nov 7.0 UT (<18.0 mag) \citep{Woo03}. The photometric data was taken from \citet{Hamu09} and covers the $B$ to $K$ bands and the 5-325 days period after discovery. The distance modulus for the host galaxy MCG-05-10-015 was adopted from \citet{Kel00}, and the total line-of-sight extinction taken as the mean of the line-of-sight extinction within the Milky Way ($E$($B$-$V$)=0.02) and an assumed upper limit of 0.1 mag additional extinction, giving $E$($B$-$V$)=0.070$\pm{0.05}$. The pseudo-bolometric UV to MIR bolometric lightcurve was calculated using the $BVRIJHK$ bands. Estimates of the photospheric velocities using the Monte Carlo (MC) radiative transfer code by \citet{Maz93,Luc99,Maz00} was adopted from \citet{Maz09}.

\paragraph{SN 2011ei}

Discovered 2011 Jul 25.43 UT \citep{Mar11b} in NGC 6925 at an apparent magnitude of 18.0, and the explosion epoch is constrained by a non-detection from 2011 Jun 23.58 UT (<19.1 mag) \citep{Mar11b}. The photometric data was taken from \citet{Mil13} and covers $U$ to $I$ bands and the 0-50 days period. The distance modulus for the host galaxy NGC 6925 was adopted as the mean and standard deviation of the literature values given by NED being 32.42$\pm{0.27}$. The total line-of-sight extinction was taken as the mean of the upper limit determined by \citet{Mil13} ($E$($B$-$V$)=0.232), who used the equivalent width of the interstellar \ion{Na}{i} D interstellar absorption lines and the relation between this and $E$($B$-$V$) by \citet{Tur03}, and the line-of-sight-extinction within the Milky Way ($E$($B$-$V$)=0.052), giving $E$($B$-$V$)=0.142$\pm{0.09}$. The pseudo-bolometric UV to MIR bolometric lightcurve was calculated using the $UBVRI$ bands. Estimates of the photospheric velocities using SYNOW was taken from \citet{Mil13}.

\paragraph{SN 2011fu}

Discovered 2011 Sep 21.04 UT \citep{Cia11} in UGC 1626 at an apparent magnitude of 15.8, and the explosion epoch is constrained by a non-detection from 2011 Aug 10 UT (<18.8 mag) \citep{Cia11}. The photometric data covers the $U$ to $I$ bands and the 10-175 days period and was taken from \citet{Kum13}. In the absence of literature measurements of the distance to the host galaxy UGC 1626 we adopt the Virgo, Great Attractor and Shapley corrected kinematic distance modulus given by NED, being 34.36 and assume an error in this estimate of 50 percent. The total line-of-sight extinction was adopted from \citet{Kum13}, who used the equivalent width of the interstellar \ion{Na}{i} D interstellar absorption lines and the relation between this and $E$($B$-$V$) by \citet{Mun97} to estimate $E$($B$-$V$)=0.22$\pm{0.11}$. The pseudo-bolometric UV to MIR bolometric lightcurve was calculated using the $UBVRI$ bands. Measurements of the absorption minimum of \ion{Fe}{ii} 5169 \AA~line was taken from \citet{Kum13}. 

\paragraph{SN 2011hs}

Discovered 2011 Nov 12.48 UT \citep{Dre11} in IC 5267 at an apparent magnitude of 15.5, and the explosion epoch is constrained by a non-detection from 2011 Oct 9.574 UT (<18.7 mag) \citep{Dre11}. The photometric data covers the $U$ to $K$ bands and 0-120 days period and was taken from \citet{Buf14}. The distance modulus for the host galaxy IC 5267 was adopted as the mean and standard deviation of the literature values given by NED, being 32.05$\pm{0.41}$. The total line-of-sight extinction was adopted from \citet{Buf14}, who used the equivalent width of the interstellar \ion{Na}{i} D interstellar absorption lines and the relation between this and $E$($B$-$V$) by \citet{Poz12} to estimate $E$($B$-$V$)=0.17$\pm{0.08}$. The pseudo-bolometric UV to MIR bolometric lightcurve was calculated using the $UBVRIJH$ bands. Spectroscopic data was taken from \citet{Buf14}, and the photospheric velocities estimated from the absorption minimum of the \ion{Fe}{ii} 5169 \AA~line, in turn measured with a simple automated algorith described in \citetalias{Erg14a}.

\paragraph{SN 2013df}

Discovered 2013 Jun 7.87 UT \citep{Cia13} in NGC 4414 at an apparent magnitude of ?.?, and the explosion epoch is constrained by a non-detection from 2013 ? ?.? UT (<?.= mag) \citep{Cia13}. The photometric data covers the $B$ to $H$ bands and the 5-65 days period and was taken from \citet{Dyk14}. The distance modulus for the host galaxy NGC 4414 was taken as the Cepheid based measurement by \citet{Fre01}, being 31.10$\pm{0.05}$. The total line-of-sight extinction was adopted from \citet{Dyk14}, which used the equivalent width of the interstellar \ion{Na}{i} D interstellar absorption lines and the relation between this and $E$($B$-$V$) by \citet{Poz12} to estimate $E$($B$-$V$)=0.097$\pm{0.016}$. The pseudo-bolometric UV to MIR bolometric lightcurve was calculated using the $BVRIzJH$ bands. The spectroscopic data was taken from \citet{Dyk14} and the photospheric velocities estimated from the absorption minimum of the \ion{Fe}{ii} 5169 \AA~line in turn measured with a simple automated algorithm described in \citetalias{Erg14a}.

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