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\begin{document}

\title{SN 2011dh - Understanding a Type IIb Supernova and its progenitor star.}

\author{M.~Ergon\inst{\ref{inst1}} \and J.~Sollerman\inst{\ref{inst1}} \and M.~Fraser\inst{\ref{inst2}} \and A.~Pastorello\inst{\ref{inst3}} \and S.~Taubenberger\inst{\ref{inst4}} \and N.~Elias-Rosa\inst{\ref{inst5}} \and M.~Bersten\inst{\ref{inst6}} \and A.~Jerkstrand\inst{\ref{inst2}} \and S.~Benetti\inst{\ref{inst3}} \and M.T.~Botticella\inst{\ref{inst7}} \and C.~Fransson\inst{\ref{inst1}} \and A.~Harutyunyan\inst{\ref{inst8}} \and R.~Kotak\inst{\ref{inst2}} \and S.~Smartt\inst{\ref{inst2}} \and S.~Valenti\inst{\ref{inst3}} \and F.~Bufano\inst{\ref{inst9},\ref{inst10}} \and E.~Cappellaro\inst{\ref{inst3}} \and M.~Fiaschi\inst{\ref{inst3}} \and A.~Howell\inst{\ref{inst11}} \and E.~Kankare\inst{\ref{inst12}} \and L.~Magill\inst{\ref{inst2},\ref{inst13}} \and S.~Mattila\inst{\ref{inst12}} \and R.~Naves\inst{\ref{inst15}} \and P.~Ochner\inst{\ref{inst3}} \and J.~Ruiz\inst{\ref{inst14}} \and K.~Smith\inst{\ref{inst2}} \and L.~Tomasella\inst{\ref{inst3}} \and  M.~Turatto\inst{\ref{inst3}}}

\institute{The Oskar Klein Centre, Department of Astronomy, AlbaNova, Stockholm University, 106 91 Stockholm, Sweden 
\label{inst1}
\and Astrophysics Research Center, School of Mathematics and Physics, Queens University Belfast, Belfast, BT7 1NN, UK 
\label{inst2}
\and INAF, Osservatorio Astronomico di Padova, vicolo dell'Osservatorio n. 5, 35122 Padua, Italy 
\label{inst3}
\and Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85741 Garching, Germany 
\label{inst4}
\and Institut de Ci\`{e}ncies de l’Espai (IEEC-CSIC), Facultat de Ci\`{e}ncies, Campus UAB, E-08193 Bellaterra, Spain.
\label{inst5}
\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{inst6}
\and INAF-Osservatorio Astronomico di Capodimonte, Salita Moiariello, 16  80131 Napoli, Italy
\label{inst7}
\and Fundaci\'{o}n Galileo Galilei-INAF, Telescopio Nazionale Galileo, Rambla Jos\'{e} Ana Fern\'{a}ndez P\'{e}rez 7, 38712 Bre\~{n}a Baja, TF - Spain
\label{inst8}
\and INAF, Osservatorio Astrofisico di Catania, Via Santa Sofia, I-95123, Catania, Italy
\label{inst9}
\and Departamento de Ciencias Fisicas, Universidad Andres Bello, Av. Republica 252, Santiago, Chile
\label{inst10}
\and Las Cumbres Observatory Global Telescope Network, 6740 Cortona Dr., Suite 102, Goleta, CA 93117 
\label{inst11}
\and Tuorla Observatory, Department of Physics and Astronomy, University of Turku, V\"ais\"al\"antie 20, FI-21500 Piikki\"o, Finland
\label{inst12}
\and Isaac Newton Group, Apartado 321, E-38700 Santa Cruz de La Palma, Spain
\label{inst13}
\and Observatorio Montcabrer, C Jaume Balmes 24, Cabrils, Spain
\label{inst15}
\and Observatorio de C\'{a}ntabria, Ctra. de Rocamundo s/n, Valderredible, Cantabria, Spain
\label{inst14}}

\date{Submitted to Astronomy and Astrophysics}

\abstract{
We present optical and near-infrared (NIR) photometry and spectroscopy of the Type IIb supernova (SN) 2011dh spanning $\sim$2 years and modelling of the bolometric lightcurve using hydrodynamical modelling for diffusion phase (3-40 days) and early tail (40-100 days) and steady-state NLTE modelling for the late tail (100-750 days). The fitting procedure for the hydrodynamical modelling is quantified by the use of a model grid spanning a large volume of paramater space which also allow us to determine the sensitivity of the results to errors in the observed quantities as well as a doing a consistent comparison of the derived quantities to SNe 1993J and 2008ax. The results are in good agreement with our results from \citep{Ber12} and for SNe 1993J and 2008ax we find similar values although the mass of \element[ ][56]{Ni} is sensitive to the adopted extinction and extinction. We also extend the temporal coverage of the model grid to 300 days using a correction for the flux outside the observed wavelength range determined with steady-state NLTE modelling. Applying this to SN 2011dh we find results in good agreement with those based on the 3-100 days bolometric lightcurve. Steady-state NLTE modelling of the 100-750 days bolometric lightcurve using a restricted set of ejecta models gives a reasonable fit to observations up to $\sim$400 days whereas the subsequent evolution cannot be reproduced. Time-dependent NLTE modelling shows that in this late phase freeze-out in the helium zone becomes important and a steady-state assumption is no longer valid. We find an excess in the MIR and NIR as compared to model photometry which could be explained by a modest amount of dust ($\tau=0.2$) being formed in the ejecta, although the evolution after $\sim$400 days is not well reproduced. This amount of dust is also needed to reproduce the change in optical and near-infrared decline rates seen between $\sim$100 and $\sim$200 days.
%This is a caveat for the identification of the yellow supergiant coincident with SN 2011dh as the progenitor star which will then have a 20 percent chance of being hidden behind a dust cloud.
%We also provide a detailed analysis and comparison of the observations to SNe 1993J and 2008ax, review our results obtained sofar and discuss, in some detail, the physics giving rise to the observed photometric and spectral evolution using our optimal model. 
Comparisons of SNe 2011dh, 1993J and 2008ax show strong similarities and many of the differences could be explained by different masses and radii of the hydrogen rich envelope. The initial masses of $\lesssim$15 M$_\odot$ found by steady-state NLTE modelling of nebular spectra in \citet{Jer13} and hydrodynamical modelling of the bolometric lightcurve in this paper suggests that all of these SNe originates from binary systems, which have been previously established for SN 1993J.}

\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{Erg13b}{E13}
\defcitealias{Ber12}{B12}
\defcitealias{Arc11}{A11}
\defcitealias{Mau11}{M11}
\defcitealias{Tsv12}{T12}
\defcitealias{Vin12}{V12}
\defcitealias{Sch98}{S98}
\defcitealias{Sch11}{SF11}
\defcitealias{Mun97}{MZ97}
\defcitealias{Poz12}{P12}
\defcitealias{Val11}{V11}
\defcitealias{Bes12}{BM12}
\defcitealias{Pas09}{P09}
\defcitealias{Val11}{V11}
\defcitealias{Cho11}{C11}
\defcitealias{Mar13}{M13}
\defcitealias{Dyk13b}{D13}
\defcitealias{Sah13}{SA13}
\defcitealias{Shi13}{SH13}
\defcitealias{Jer13}{J13}

\section{Introduction}

Type IIb supernovae (SNe) are observationally characterized by a transition from Type II (with hydrogen lines) at early times to Type Ib (without hydrogen lines but with helium lines) at later times. The physical interpretation is that these SNe arise from stars that have lost most of their hydrogen envelop, either through stellar winds or interaction with a binary companion. Which of these production channels are dominating is still debated but for SN 1993J, the prime example of such an SNe, a companion star was detected by direct observations \citep{Mau04}. The evolution of this binary system has been successfully modelled \citep{Pod93,Sta09} and it is widely accepted that the companion was reponsible for the removal of the hydrogen envelope. Bright, nearby Type IIb SNe as 1993J, 2008ax and the recent 2011dh are essential to improve our understanding of this class. Observations of the progenitor star in pre-explosion images, a search for the companion star when the SN has faded and multi-method modelling of high quality data all provide important clues to the nature of Type IIb SNe and their progenitor stars.

%To enhance our understanding, high quality, multi-wavelength monotoring of nearby Type IIb SNe is essential. Such SNe also provide the opportunity to observe the progenitor star in pre-explosion images and search for a companion star when the SN has faded. Except for SN 1993J, SN 2008ax and the recent SN 2011dh are good examples and, as discussed in this paper, detailed multi-method modelling of these SNe provide important clues to the nature of Type IIb SNe and their progenitor stars.

In this paper we present our extensive optical and near-infrared (NIR) dataset, covering nearly two years, we have obtained for SN 2011dh. The first 100 days of this dataset have earlier been presented in \citet[herafter \citetalias{Erg13b}]{Erg13b}. Detailed hydrodynamical modelling of the SN using those data were presented in \citet[hereafter \citetalias{Ber12}]{Ber12} and steady-state NLTE spectral modelling using late time data in \citet[hereafter \citetalias{Jer13}]{Jer13}. Identification and analysis of the plausible progenitor star have been presented in \citet[hereafter \citetalias{Mau11}]{Mau11} and confirmation of the progenitor identification through its disappearence in \citetalias{Erg13b}.

SN 2011dh was discovered on 2011 May 31.893 UT \citep{Gri11} in the nearby galaxy M51 at a distance of 7.8 Mpc \citepalias{Erg13b}. The SN has been extensively monitored from X-ray to radio wavelengths by several teams. Most observations cover the 3-100 days period but late time data have been published in \citet{Tsv12}, \citet{Dyk13b}, \citet{Sah13}, \citet{Shi13} and \citet{Hel13}. As in \citetalias{Erg13b} we will focus on the UV to MIR emission. The explosion epoch, the distance to M51 and the interstellar line-of-sight extinction towards the SN used in this paper are all adopted from \citetalias{Erg13b}.

The nature of the progenitor star has been a key issue since the identification of a yellow supergiant in pre-explosion images coincident with the SN (\citetalias{Mau11}; \citealt{Dyk11}). Recent progress in modelling of the SN (\citetalias{Ber12}; \citetalias{Jer13}; \citealt{Shi13}) and the disappearence of the progenitor candidate (\citetalias{Erg13b}; \citealt{Dyk13b}) strengthens the hypothesis that the progenitor was a yellow supergiant of moderate mass, as was originally proposed in \citetalias{Mau11}. In this paper we present further modelling in support of this hypothesis. As shown in \citet{Ben12} a binary interaction scenario that reproduces the observed and modelled properties of the yellow supergiant is possible. HST observations that could detect or set useful constaints on the presence of a companion star are scheduled for Cycle 21.

The paper is organized as follows. In Sect.~\ref{s_observations} we present the observations and describe the reduction and calibration procedures and in Sect.~\ref{s_analysis} we present an observational analysis and comparison of the observations to SNe 1993J and 2008ax. In Sect.~\ref{s_modelling} we model the bolometric lightcurve and in Sect.~\ref{s_discussion} we discuss the interpretation of the observations given our optimal model and discuss and review the results obtained so far and our current understanding of Type IIb SNe. Finally, we conclude and summarize the paper in Sect.~\ref{s_conclusions}. In Appendix \ref{a_hydro_modelling} we provide details on the hydrodynamical modelling.

\section{Observations}
\label{s_observations}

\subsection{Software}
\label{s_software}

As in \citetalias{Erg13b} two different software packages have been used for 2-D reductions, measurements and calibrations of the data. The {\sc iraf} based {\sc quba} pipeline \citep[hereafter \citetalias{Val11}]{Val11} and another {\sc iraf} based package which we will refer to as the {\sc sne} pipeline. This package has been developed with the particular aim to provide the high level of automation needed for large sets of data.

\subsection{Imaging}
\label{s_obs_image}

An extensive campaign of optical and NIR imaging was initiated for SN 2011dh shortly after discovery using a multitude of different instruments. The observations for during the first 100 days have been described in \citetalias{Erg13b}. The late time data have been obtained with the Liverpool Telescope (LT), the Nordic Optical Telescope (NOT), Telescopio Nazionale (TNG), the Calar Alto 3.5m and 2.2m telescopes, the Asiago 67/92cm Schmidt and 1.82m Copernico telescopes, the William Herschel Telescope (WHT), the Albanova Telescope (AT) and the United Kingdom Infrared Telescope (UKIRT). The major contributors were the NOT, the Asiago 67/92cm Schmidt and the AT for the optical observations and the the Calar-Alto 3.5m, the WHT and the UKIRT for the NIR observations. The late time dataset includes 61 epochs of optical imaging and 9 epochs of NIR imaging which, together with the early time observations, gives a total of 146 epochs of optical imaging and 32 epochs of NIR imaging.

\subsubsection{Reductions and calibration}
\label{s_obs_image_red_cal}

The optical and NIR raw data have been reduced with the {\sc quba} and {\sc sne} packages respectively as described in \citetalias{Erg13b}, except for the optical LT and NIR UKIRT data for which reductions provided by the automated telescope pipelines have been used. Photometry was performed with the {\sc sne} pipeline as described in \citetalias{Erg13b}. Comparison to photometry on template subtracted images shows that the background contamination is negligible before $\sim$300 days after which we have used photometry on template subtracted images. The optical and NIR photometry was calibrated to the Johnson-Cousins (JC), Sloan Digital Sky Survey (SDSS) and 2 Micron All Sky Survey (2MASS) systems as described in \citetalias{Erg13b} where we also discuss the related uncertainties. The photometry was transformed to the standard systems using S-corrections \citep{Str02} except for the JC $U$ and SDSS $u$ bands which were transformed using the linear colour-terms tabulated in \citetalias{Erg13b}. As discussed in \citetalias{Erg13b} we find the calibration to be accurate to within five percent in all bands for the first 100 days. The accuray of the late time photometry depends critically on the accuracy of the S-corrections. The late time JC and SDSS phototry were mainly obtained with the NOT but comparison between S-corrected NOT, LT and Calar-Alto 2.2m JC and SDSS observations at $\sim$300 days show differences at the 5 percent level suggesting that this precision is maintained. We note that at this phase S-corrections are abolutely neccesary, for example the difference between the NOT and Calar-Alto $I$ band observations are almost one magnitude at $\sim$300 days if these are not applied, mainly because of the strong \ion{Ca}{ii} 7291,7323 and 8498,8542,8662 \AA~lines. The late time 2MASS photometry was obtained with a number of different telescopes and although the sampling is sparse the shape of the lightcurves suggest that errors in the S-corrections are modest. Note that we have used JC-like $UBVRI$ filters and SDSS-like $gz$ filters at NOT whereas we have used JC-like $BV$ filters and SDSS-like $ugriz$ filters at LT and FTN. The JC-like $URI$ and SDSS-like $uri$ photometry were then tied to both the JC and SDSS systems to produce full sets of JC and SDSS photometry.

\subsubsection{Space Telescope Observations}
\label{s_obs_image_space}

We have also performed photometry on the Spitzer 3.6 and 4.5 $\mu$m imaging\footnote{Obtained through the DDT program by G. Helou.} and the SWIFT optical and UV imaging. For the Spitzer imaging we performed aperture photometry using the {\sc sne} pipeline as described in \citetalias{Erg13b} to calculate magnitudes in the natural (energy flux based) Vega system of IRAC as well as JC $UBV$ using S-corrections. All Spitzer images were template subtracted as described in \citetalias{Erg13b}. Optical and UV SWIFT magnitudes were published in \citetalias{Erg13b} where we also provide details on the reductions.

\subsubsection{Results}
\label{s_obs_image_results}

The S-corrected optical and NIR magnitudes and their corresponding errors are listed in Tables \ref{t_jc}, \ref{t_sloan} and \ref{t_nir} and the Spitzer 3.6 and 4.5 $\mu$m magnitudes and their corresponding errors in Table~\ref{t_mir}. For simplicity we also include the magnitudes for the first 100 days already published in \citetalias{Erg13b}. All magnitudes including the SWIFT magnitudes published in \citetalias{Erg13b} are shown in Fig.~\ref{f_uv_opt_nir_mir} which also shows cubic spline fits using 3-5 point knot separation, error weighting and a 5 percent error floor. The standard deviation around the fitted splines is less than 5 percent and mostly less than a few percent except for the SWIFT $UVM2$ band for which the standard deviation is between 5 and 10 percent on the tail. All calculations in Sect. \ref{s_analysis}, including the bolometric lightcurve, are based on these spline fits. In these calculations the errors have been estimated as the standard deviation around the fitted splines and then propagated.

\begin{figure*}[tb]
\includegraphics[width=1.0\textwidth,angle=0]{figs/sn2011dh-UV-opt-NIR-MIR-app.pdf}
\caption{Photometric evolution of SN 2011dh in the UV, optical, NIR and MIR. For clarity each band has been shifted in magnitude. Each lightcurve has been annotated with the name of the band and the shift applied. We also show the S-corrected SWIFT JC photometry (crosses) and cubic spline fits (solid lines).}
\label{f_uv_opt_nir_mir}
\end{figure*}

\subsection{Spectroscopy}
\label{s_obs_spec}

An extensive campaign of optical and NIR spectroscopic observations was initiated for SN 2011dh shortly after discovery with data obtained from a multitude of telescopes. The observation during the first 100 days have been described in \citetalias{Erg13b}. The late time data have been obtained with the NOT, the TNG, the WHT, the Calar Alto 2.2m telescope, the Asiago 1.22m Galileo telescopes and the Gran Telescopio Canarias (GTC). The major contributors were the NOT, the WHT and the GTC. Details of all spectroscopic observations, the telescope and instrument used, epoch and instrument characteristics are given in Table~\ref{t_speclog}. The late time dataset includes 18 optical spectra obtained at 13 epochs and 2 NIR spectra obtained at 2 epochs which, together with the early time observations, gives a total of 73 optical spectra obtained at 39 epochs and 20 NIR spectra obtained at 12 epochs.

\subsubsection{Reductions and calibration}
\label{s_obs_spec_red_cal}

The optical and NIR raw data were reduced and the flux extracted using the {\sc quba} and {\sc sne} pipelines respectively as described in \citetalias{Erg13b}. The optical and NIR spectra were flux and wavelength calibrated using the {\sc quba} and {\sc sne} pipelines respectively as described in \citetalias{Erg13b}. The absolute flux scale of all spectra has been calibrated against interpolated photometry using a least square fit to all bands for which the mean energy wavelength is at least half an equivalent width within the spectral range.

\subsubsection{Results}
\label{s_obs_spec_results}

All reduced, extracted and calibrated spectra will be made available for download from the Weizmann Interactive Supernova data REPository\footnote{http://www.weizmann.ac.il/astrophysics/wiserep/} (WISeREP) \citep{Yar12}.  Figure~\ref{f_spec_evo_opt_NIR_trad} shows the sequence of observed spectra where those obtained on the same night using the same telescope and instrument have been combined. For clarity, and as is motivated by the frequent sampling of spectra, all subsequent figures in this and the following sections are based on time-interpolations of the spectral sequence as described in \citetalias{Erg13b}. To further visualize the evolution, the spectra have been aligned to a time axis at the right border of the panels. Interpolated spectra were also used in the calculations of the bolometric lightcurve (Sect.~\ref{s_bol_lightcurve}) and S-corrections. Figure~\ref{f_spec_evo_opt_NIR} shows the interpolated optical and NIR spectral evolution of SN 2011dh for days 5$-$425 with a 20-day sampling. All spectra in this and subsequent figures spectra have been corrected for redshift and interstellar extinction.

\begin{figure*}[tb]
\includegraphics[width=1.0\textwidth,angle=0]{figs/sn2011dh-spec-evo-opt-NIR.pdf}
\caption{Optical and NIR (interpolated) spectral evolution for SN 2011dh for days 5$-$425 with a 20-day sampling. Telluric absorption bands are marked with a $\oplus$ symbol in the optical and shown as grey regions in the NIR.}
\label{f_spec_evo_opt_NIR}
\end{figure*}

\begin{figure*}[p]
\includegraphics[width=1.0\textwidth,angle=0]{figs/sn2011dh-spec-evo-opt-NIR-trad.pdf}
\caption{Sequence of the observed late-time (100-415 days) spectra for SN 2011dh. Spectra obtained on the same night using the same telescope and instrument have been combined and each spectra have been labelled with the phase of the SN. Telluric absorption bands are marked with a $\oplus$ symbol in the optical and shown as grey regions in the NIR.}
\label{f_spec_evo_opt_NIR_trad}
\end{figure*}

\section{Analysis}
\label{s_analysis}

In this section we provide an observational analysis and comparison to the Type IIb SNe 1993J and 2008ax. Beside SN 2011dh these are the best monitored Type IIb SNe sofar which motivates a detailed comparison. Both occured in nearby galaxies, have progenitor detections and well constrained explosion epochs. As discussed in \citetalias{Erg13b} the systematic errors stemming from the uncertainty in distance and extinction are large for all three SNe which should be kept in mind when absolute quantities are compared. For SNe 1993J and 2008ax we adopt the same values and error bars for the distance and extinction as in \citetalias{Erg13b}. The references for the photometric and spectroscopic data of SNe 1993J and 2008ax used in the comparison are the same as specified in \citetalias{Erg13b}.

\subsection{Photometric evolution}
\label{s_analysis_phot}

Absolute magnitudes were calculated as in \citetalias{Erg13b}. In Fig. \ref{f_opt_comp} we show absolute optical and NIR magnitudes for SN 2011dh as compared to SNe 1993J and 2008ax and in Table \ref{t_lc_char_comp} we tabulate the time and absolute magnitude of the maximum, and the tail decline rates at 100, 200 and 300 days. Most striking is the similarity between the three SNe except for a shift, which as discussed in \citetalias{Erg13b} could be explained by an error in the adopted extinction. Another difference is the early evolution, where the lightcurves of SN 1993J show a decline before $\sim$10 days. Such a decline is also seen in the lightcurves of SN 2011dh before $\sim$3 days (not shown here but see \citetalias{Arc11} and \citetalias{Tsv12}) whereas there is only a hint of this behavior in the lightcurves of SN 2008ax. Except for this, the behavior is most similar showing a strong increase in luminosity from day 3 to the peak followed by a decrease down to a tail with a roughly linear decline rate. The maximum occurs at increasingly later times for redder bands. The drop from the maximum down to the tail is more pronounced for bluer bands and is not seen for bands redder than $I$. 

Except for the early tail the optical decline rates are fairly constant for all three SNe, being roughly twice the decay rate of \element[ ][56]{Co}. For SN 2001dh all decline rates except for the $K$ band tend to decrease at later times, which also seems to be the case for SNe 1993J and 2008ax. The $J$ and $H$ band decline rates are considerably higher than the optical at 100 days for all three SNe, being three or even four times the decay rate of \element[ ][56]{Co}. For SNe 2011dh and 1993J these have decreased considerably at 200 days and for SN 2011dh these are lower than the optical at 300 days. The $K$ band decline rate is a considerably lower than the $J$ and $H$ band decline rates for SNe 2011dh and 1993J at 100 days but is considerably higher than the $J$ and $H$ band decline rates at 300 days for SN 2011dh. Between 450 and 700 days the optical lightcurves of SN 2011dh flattens and the decline rates decrease considerably, approaching the decay rate of \element[ ][56]{Co}.

Contrary to SNe 1993J and 2008ax, SN 2011dh was also monitored in UV and MIR. As seen in Fig. \ref{f_uv_opt_nir_mir} the Swift $UVM2$ band shows a quite different evolution than the other bands with a monotonic strong decrease between 3 and 80 days. The Spitzer 3.6 and 4.5 $\mu$m bands follows a similar evolution as the other bands for the first 100 days although, as discussed in \citetalias{Erg13b}, the 4.5 $\mu$m shows a considerably slower decline than the 3.6 $\mu$m and the NIR bands. Later on, a strong excess in both the Spitzer bands develops between 100 and 250 days. Between 250 and 400 days the decline rates are fairly constant and similar to the optical bands but between 400 and 700 days the Spitzer lightcurves flatten and the decline rates decrease considerably in a way similar to the optical bands.

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-opt-NIR-comp.pdf}
\caption{Photometric evolution of SN 2011dh in the optical and NIR as compared to SNe 1993J and 2008ax. For clarity each band has been shifted in magnitude. Each lightcurve has been annotated with the name of the band and the shift applied.}
\label{f_opt_comp}
\end{figure}

\begin{table*}[tb]
\caption{Times and absolute magnitudes of the maximum and tail decline rates at 100, 200, 300 and 400 days for SN 2011dh as compared to SNe 1993J and 2008ax as measured from cubic spline fits.}
\begin{center}
\include{sn2011dh-lc-char-comp-table}
\end{center}
\label{t_lc_char_comp}
\end{table*}

\subsection{Colour evolution and blackbody fits}
\label{s_analysis_colour}

Figure~\ref{f_colour_evo_comp} shows the intrinsic $U$-$V$, $B$-$V$, $V$-$R$, $V$-$I$, $V$-$J$, $V$-$H$ and $V$-$K$ colour evolution of SN 2011dh as compared to SNe 1993J and 2008ax. This is essentlially the lighturves normalized to the $V$ band and is a way of visualising the distribution of the flux within the SED. The similarity between the three SNe is striking except for a shift, which could be explained by an error in the adopted extinction \citepalias{Erg13b}. For SN 2011dh we see an initial quite strong blueward trend in the $V$-$R$ to $V$-$K$ colours reaching a minimum at $\sim$10 days which is not reflected in the $U$-$V$ and $B$-$V$ colours. SN 1993J shows a similar behaviour although the trend starts later (at the luminosity minimum) and the colour minimum is reached later (at $\sim$20 days). Before $\sim$10 days all the colours of SN 1993J show a redward trend. For SN 2008ax there is no NIR coverage before $\sim$10 days but all other colours shows the blueward trend. For SN 2011dh the blueward trend is followed by a redward trend reaching a maximum at $\sim$40 days. Both SNe 1993J and 2008ax show a very similar behavior.

For SN 2011dh the evolution after the colour maximum is slow and quite diverse among the colours. This is not surprising as in this phase the SED becomes more and more dominated by lines and the colours mainly reflect the evolution of these. Until $\sim$150 days the evolution is fairly consistent though and all colours except $V$-$I$ shows blueward trends of varying strength. After $\sim$150 days the $U$-$B$ and $B$-$V$ colours shows weak blue trends, the $V$-$R$ colour shows a redward trend reaching a colour maximum at $\sim$350 days and then a blueward trend, the $V$-$I$ colour stays nearly constant until $\sim$350 days and then shows a blueward trend, the $V$-$J$ and $V$-$K$ colours show redward trends and the $V$-$K$ colour shows a redward trend reaching a colour maximum at $\sim$250 days and then a blueward trend. Both SNe 1993J and 2008ax show quite similar behavior in all colours. Notable exceptions are the $V$-$I$ colour for 2008ax which is not constant and the $V$-$J$ to $V$-$K$ colours for SN 1993J which do not show the redward trends seen in SN 2011dh.

In \citetalias{Erg13b} we discussed the blackbody temperature and radius for SN 2011dh as inferred from fits the $V$, $I$, $J$, $H$ and $K$ bands. Such fits only have physical meaning during the early phase when the approximation of a blackbody emitting surface is valid. In \citetalias{Erg13b} we did not show a comparison with SNe 1993J and 2008ax though. Fig.~\ref{f_bb_T_evo_comp} show the blackbody temperature for the first 100 days for SN 2011dh as compared to SNe 1993J and 2008ax as well as the systemetic errors arising from the errors in extinction. Apparently SN 2008ax is much hotter than SNe 2011dh and 1993J for the adopted extinction.

\begin{figure}
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-colour-evo-comp.pdf}
\caption{$U$-$V$, $B$-$V$, $V$-$R$, $V$-$I$, $V$-$J$, $V$-$H$ and $V$-$K$ intrinsic colour evolution for SN 2011dh (black dots) as compared to SNe 1993J (red triangles) and 2008ax (blue squares.}
\label{f_colour_evo_comp}
\end{figure}

\begin{figure}
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-bb-T-evo-comp.pdf}
\caption{Evolution of the blackbody temperature for SN 2011dh (black dots) as inferred from fits to the $V$, $I$, $J$, $H$ and $K$ bands compared to SNe 1993J (red crosses) and 2008ax (blue pluses). We also show the systematic errors arising from the errors in extinction (dashed lines).}
\label{f_bb_T_evo_comp}
\end{figure}

%\begin{figure}
%\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-bb-R-evo.pdf}
%\caption{Evolution of blackbody radius for SN 2011dh as inferred from fits to the $V$, $I$, $J$, $H$ and $K$ bands for the adopted extinction. The radius corresponding to the P-Cygni minimum of the \ion{Fe}{ii} 5169 \AA~line is also shown (dashed line).}
%\label{f_bb_R_evo}
%\end{figure}

\subsection {Bolometric evolution}
\label{s_bol_lightcurve}

As in \citetalias{Erg13b} we have used a combination of the spectroscopic and photometric methods, applied to wavelength regions with and without spectral information respectively, when calculating the bolometric lightcurves. The details of these methods have been described in \citetalias{Erg13b}.

Figure.~\ref{f_UK_bol_comp} shows the U to K (3300-24000 \AA) bolometric lightcurve as calculated with the photometric method for SN 2011dh as compared to SNe 1993J and 2008ax for day 0-500. For SN 2011dh the U to K bolometric lightcurve shows a strong increase in luminosity from 3 days to peak luminosity at 20.45 days followed by a decrease down to a tail with a roughly linear decline rate. The decline rate on the tail is 0.020 mag day$^{-1}$ at 100 and 200 days and decrease to 0.016 and 0.015 mag day$^{-1}$ at 300 and 400 days, respectively. There is however a significant increase in the decline rate between 150 and 200 days that breaks this pattern. The data from the first three days published in \citetalias{Arc11} and \citetalias{Tsv12} are insufficient to construct a bolometric lightcurve but it is clear that this phase corresponds to a strong decline of the bolometric luminosity. The U to K bolometric lightcurves of SNe 1993J and 2008ax both have larger luminosities as well as larger peak-to-tail luminosity ratios. As discussed in \citetalias{Erg13b} both these differences could be explained by an error in the extinction. Given this and the caveat that SNe 1993J and 2008ax are not covered in NIR after $\sim$250 and $\sim$150 days respectively, their U to K bolometric lightcurves are remarkably similar to the one of SN 2011dh. There are however some differences. For SN 2008ax the intial decline is not seen although there is a hint in the early $U$ band observations. For SN 1993J the initial decline is slower and lasts for $\sim$10 days and the decline rates at 200 and 300 are increasingly lower. Neither SN 1993J nor SN 2008ax shows the increase in decline rate at 150-200 days.

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-bol-opt-NIR-comp.pdf}
\caption{Pseudo-bolometric U to K lightcurve for SN 2011dh (black circles and solid line) calculated with the photometric method as compared to SNe 1993J (red crosses and solid line) and 2008ax (blue pluses and solid line). The upper and lower error bars for the systematic error arising from extinction and distance (dashed lines) are also shown.}
\label{f_UK_bol_comp}
\end{figure}

\begin{table*}[tb]
\caption{Times and luminosity of the maximum and tail decline rates at 100, 200, 300 and 400 days for the U to K bolometric lightcurve of SN 2011dh as compared to SNe 1993J and 2008ax.}
\begin{center}
\include{sn2011dh-bol-lc-char-comp-table}
\end{center}
\label{t_bol_lc_char_comp}
\end{table*}

The 3-732 days UV to MIR (1900-50000 \AA) pseudo-bolometric lightcurve as calculated with the combined spectroscopic and photometric methods is shown in Figure.~\ref{f_UV_MIR_bol} and the 3-300 days period (for which we have full UV to MIR coverage) listed in Table~\ref{t_UV_MIR_bol} for reference. As expected the UV to MIR lightcurve is very similar to the U to K lightcurve. The decline rates at 100, 200, 300 and 400 days are 0.021, 0.022, 0.015 and 0.016 mag day$^{-1}$ but the increase in decline rate between 150 and 200 days is not as pronounced. Given the caveat that the NIR coverage ends at $\sim$350 days and the sampling is sparse and the measurement errors large after $\sim$500 days the UV to MIR lightcurve show a significant flattening after $\sim$500 days where the decline rate decrease to a value slightly lower than the decay rate of \element[ ][56]{Co}. 

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-UV-MIR-bol.pdf}
\caption{Pseudo-bolometric UV to MIR lightcurve for SN 2011dh calculated with the spectroscopic (black circles and solid line) and photometric (red circles) method. The upper and lower error bars for the systematic error arising from extinction and distance (black dashed lines) and the radioactive decay chain luminosity of 0.075 M$_\odot$ of \element[ ][56]{Ni} (black dotted line) are also shown.}
\label{f_UV_MIR_bol}
\end{figure}

Figure.~\ref{f_bol_frac} shows the fractional UV (1900-3300 \AA), optical (3300-10000) \AA), NIR (10000-24000 \AA) and MIR (24000-50000 \AA) luminosities for SN 2011dh for day 0-750. The optical flux dominates and varies between $\sim$60 and $\sim$80 percent. The NIR fraction varies between $\sim$15 and $\sim$30 percent, the MIR fraction between a few and $\sim$25 percent and the UV fraction between $\sim$1 and $\sim$15 percent. Between $\sim$3 and $\sim$10 days (corresponding to the colour minimum) we see a strong increase in the optical fraction and a strong decrease in the UV fraction whereas the NIR fraction stays roughly constent. Between $\sim$10 and $\sim$40 days (corresponding to the colour maximum) we see a strong decrease in the optical fraction, a continued strong decrease in the UV fraction and a strong increase in the NIR and MIR fractions. Between $\sim$40 and $\sim$100 days we see an increase in the optical fraction and a decrease in the NIR fraction wheras the UV and MIR fractions stay roughly constant. Between $\sim$100 and $\sim$200 days we see a decrease in the optical and NIR fractions and a strong increase in the MIR fraction. The subsequent evolution becomes quite uncertain after $\sim$350 days where the NIR coverage ends and $\sim$500 days where the sampling and measurement errors becomes worse, but the optical, NIR and MIR fractions seems to stay roughly constant during this period.

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-bol-fraction.pdf}
\caption{Fractional UV (black dots), optical (blue dots), NIR (green dots) and MIR (red dots) luminosity for SN 2011dh. The fractional Rayleigh-Jeans luminosity redwards of 4.5 $\mu$m (red dashed line) is also shown for comparison.}
\label{f_bol_frac}
\end{figure}

%Figure~\ref{f_sed_evo} shows the evolution of the SED as calculated with the photometric method overplotted with the blackbody fits discussed in Sect.~\ref{s_analysis_colour} as well as the observed spectra interpolated as described in Sect.~\ref{s_obs_spec_results}. The strong blueward slope in the UV region (except for the first few days) suggests that the flux bluewards of the $UVM2$ band is negligible. The flux redwards of 4.5 $\mu$m could be approximated with a Rayleigh-Jeans tail or a model spectrum. As shown in Fig. \ref{f_bol_frac} the fractional Rayleigh-Jeans luminosity redwards of 4.5 $\mu$m is at the percent level. Note again the excess at 4.5 $\mu$m that develops between 50 and 100 days. Whereas the other bands redward of $V$ are well approximated by the blackbody fits the flux at 4.5 $\mu$m is a factor of $\sim$5 in excess at 100 days. Note also the strong reduction of the flux as compared to the fitted blackbodies in bands blueward of $V$ between 10 and 30 days (Sect.~\ref{s_analysis_spec}).

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-sed-evo.pdf}
\caption{The evolution of the SED as calculated with the photometric method (black dots and dashed lines) overplotted with the blackbody fits discussed in Sect.~\ref{s_analysis_colour} (black dotted lines) as well as the observed spectra interpolated as described in Sect. \ref{s_obs_spec_results} (red solid lines).}
\label{f_sed_evo}
\end{figure}

\subsection{Spectroscopic evolution}
\label{s_analysis_spec}

\subsubsection{Hydrogen lines}
\label{s_analysis_spec_H}

The right panel of Fig.~\ref{f_spec_evo_He_Ha} shows the (interpolated) spectral evolution centred on the H$\alpha$ line. The early evolution of the hydrogen lines were discussed in \citetalias{Erg13b}. Initially these are strong but then soon fade away although the H$\alpha$ persists in absorption until $\sim$80 days. There is a dip in the [\ion{O}{i}] 6300/6364 \AA~line profile seen after $\sim$150 days that corresponds well to the early time H$\alpha$ absorption minimum at $\sim$11000 km s$^{-1}$. However, as discussed in Sect.~\ref{s_analysis_spec_small_scale} this feature repeats in a number of other line profiles and is rather due to clumping/assymetries in the ejecta. Using the steady-state NLTE modelling we find no detectable absorption in H$\alpha$ at this velocity (!Confirm, is this really true? What happens if we introduce a density peak near this velocity like is produced by our hydrodynamical modelling and the hydrodynamical modelling of SN 1993J!). After $\sim$200 days there is also an emerging emission feature near the rest wavlength of H$\alpha$. As discussed in \citetalias{Jer13} the steady-state NLTE modeling find this feature to arise mainly from the [\ion{N}{ii}] 6548 \AA~line although emission from H$\alpha$ contributes a minor fraction of the flux declining with time.

\subsubsection{Helium lines}
\label{s_analysis_spec_He}

The left and middle panels of Fig.~\ref{f_spec_evo_He_Ha} shows the (interpolated) spectral evolution centred on the \ion{He}{i} 10830 \AA~and \ion{He}{i} 20581 \AA~lines, respectively. The early evolution of the helium lines were discussed in \citetalias{Erg13b}. The helium lines emerge between $\sim$10 and $\sim$15 days and, except for the \ion{He}{i} 10830 \AA~and \ion{He}{i} 20581 \AA~lines, fades away or dissapears due to blending with other lines towards $\sim$100 days. The \ion{He}{i} 10830 \AA~and \ion{He}{i} 20581 \AA~lines however, stays strong until our last spectrum at $\sim$200 days, both in emission and absorption.

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-spec-evo-He-Ha.pdf}
\caption{Closeup of (interpolated) spectral evolution centred on the \ion{He}{i} 10830 \AA~(left panel), \ion{He}{i} 20581 \AA~(middle panel) and H$\alpha$ (right panel) lines.}
\label{f_spec_evo_He_Ha}
\end{figure}

\subsubsection{Oxygen lines}
\label{s_analysis_spec_O}

Fig.~\ref{f_spec_evo_O} shows the (interpolated) spectral evolution centred on the [\ion{O}{i}] 6300,6364 \AA, \ion{O}{i} 7774 \AA~and \ion{O}{i} 11300 \AA~lines. As discussed in \citetalias{Erg13b} the \ion{O}{i} 5577, 7774, 9263, 11300, and 13164 \AA~lines emerge between $\sim$25 and $\sim$50 days. After $\sim$100 days the [\ion{O}{i}] 6300,6364 \AA~line emerge as well and stays strong until our last spectrum at $\sim$400 days. The \ion{O}{i} 5577 dissappears after $\sim$300 days, the \ion{O}{i} 7774 starts to decline after $\sim$200 days but is still visible in our last optical spectrum at $\sim$400 days wheras the \ion{O}{i} 11300, and 13164 \AA~lines seems to be still rising in our last NIR spectrum at $\sim$200 days.

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-spec-evo-O.pdf}
\caption{Closeup of (interpolated) spectral evolution centred on the [\ion{O}{i}] 6300,6364 \AA~(left panel), \ion{O}{i} 7774 \AA~(middle panel) and \ion{O}{i} 11300 \AA~(right panel) lines.}
\label{f_spec_evo_O}
\end{figure}

\subsubsection{Sodium lines}
\label{s_analysis_spec_Na}

The right panel of Fig.~\ref{f_spec_evo_Mg_Na} shows the (interpolated) spectral evolution centred on the \ion{Na}{i} 5890,5896 \AA~line. The \ion{Na}{i} 5890,5896 \AA~line is hard to observationally disentangle from the \ion{He}{i} 5876 line. As discussed in \citetalias{Erg13b} this feature is likely arise mainly from the \ion{He}{i} 5876 \AA~line during the first 30 days. After $\sim$100 days the steady-state NLTE modelling find emission to arise mainly from \ion{Na}{i} 5890,5896 \AA~line. The feature stays strong, both in emission and absorption, until our last spectrum at $\sim$400 days.

\subsubsection{Magnesium lines}
\label{s_analysis_spec_Mg}

The left and middle panels of Fig.~\ref{f_spec_evo_Mg_Na} shows the (interpolated) spectral evolution centred on the [\ion{Mg}{i}] 4571 \AA~and \ion{Mg}{i} 15040 \AA~lines, respectively. The [\ion{Mg}{i}] 4571 \AA line emerges at $\sim$150 days and stays strong until our last optical spectrum at $\sim$400 days. As discussed in \citetalias{Jer13} there is a significant blueshift of the peak at $\sim$150 days gradually decreasing towards $\sim$400 days. As discussed in \citetalias{Erg13b} the \ion{Mg}{i} 15040 \AA~line emerge at $\sim$40 days and stays strong until our last NIR spectrum at $\sim$200 days.

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-spec-evo-Mg-Na.pdf}
\caption{Closeup of (interpolated) spectral evolution centred on the [\ion{Mg}{i}] 4571 \AA~(left panel), \ion{Mg}{i} 14050 \AA~(middle panel) and \ion{Na}{i} 5890,5896 \AA~(right panel) lines.}
\label{f_spec_evo_Mg_Na}
\end{figure}

\subsubsection{Calcium lines}
\label{s_analysis_spec_Ca}

Fig.~\ref{f_spec_evo_Ca} shows the (interpolated) spectral evolution centred on the \ion{Ca}{ii} 3934,3968 \AA, [\ion{Ca}{ii}] 7291,7323 \AA~and \ion{Ca}{ii} 8498,8542,8662 \AA~lines. As discussed in \citetalias{Erg13b} the \ion{Ca}{ii} 3934,3968 \AA~and \ion{Ca}{ii} 8498,8542,8662 are both present initially in strong P-Cygni profiles whereas the [\ion{Ca}{ii}] 7291,7323 \AA~line emerges at $\sim$100 days. The \ion{Ca}{ii} 8498,8542,8662 \AA~lines disappears in absorption at $\sim$100 days, and the \ion{Ca}{ii} 8498,8542 \AA~lines in emission at $\sim$300 days. After $\sim$100 days the \ion{Ca}{ii} 8662 \AA~line declines but stays rather strong until our last optical spectrum at $\sim$400 days. The \ion{Ca}{ii} 3934,3968 \AA~line stays strong in absorption and the [\ion{Ca}{ii}] 7291,7323 \AA~line in emission until our last optical spectrum at $\sim$400 days.

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-spec-evo-Ca.pdf}
\caption{Closeup of (interpolated) spectral evolution centred on the \ion{Ca}{ii} 3934/3968 \AA~(left panel), [\ion{Ca}{ii}] 7291/7323 \AA~(middle panel) and \ion{Ca}{ii} 8498/8542/8662 \AA~(right panel) lines.}
\label{f_spec_evo_Ca}
\end{figure}

\subsubsection{Iron group lines}
\label{s_analysis_spec_Fe}

Fig.~\ref{f_spec_evo_Ca} shows the (interpolated) spectral evolution centred on the [\ion{Fe}{ii}] 7155 \AA, [\ion{Fe}{ii}] 16400 \AA~and [\ion{Co}{ii} 10190,10248,10283 \AA~lines. As discussed in \citetalias{Erg13b} the \ion{Fe}{ii} 4233, 4549, 4584, 4924, 5018, 5169 and 5317 \AA~lines are all present initially but fade away or dissappear due to blending after $\sim$50 days. As also discussed in \citetalias{Erg13b} numerous \ion{Fe}{ii} lines also contributes to the reduction in flux bluewards $\sim$5000, most pronounced between $\sim$10 and $\sim$30 days. Similarily, the steady-state NLTE modeling find numerous lines from \ion{Fe}{ii} and \ion{Fe}{i} to be the main source of the quasi-continuum bluewards $\sim$6000 \AA~emerging at $\sim$100 days. The isolated iron lines visible after $\sim$100 days are the [\ion{Fe}{ii}] 7155, 12600 and 16400 \AA~lines, the former emerging at $\sim$150 days and the latter two seen in our last NIR spectrum at $\sim$200 days. The [\ion{Fe}{ii}] 7155 \AA~line persists until our last optical spectrum at $\sim$400 days. The 

The [\ion{Co}{ii}] 9338,9344 line may emerge as early as $\sim$50 days, athough blending from \ion{O}{i} 9220 on the blue side complicates the identification, and persists until our last spectrum covering this region at $\sim$300 days whereas the [\ion{Co}{ii}] 10190,10248,10283 \AA~lines are seen in our last NIR spectrum at $\sim$300 days.

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-spec-evo-Fe-Co.pdf}
\caption{Closeup of (interpolated) spectral evolution centred on the [\ion{Fe}{ii}] 7155 \AA~(left panel), [\ion{FeII}{ii}] 16400 \AA~(middle panel) and [\ion{Co}{ii}] 10190,10248,10283 \AA~(right panel) lines.}
\label{f_spec_evo_Fe_Co}
\end{figure}

\subsubsection{Line velocities and fluxes}

Fig.~\ref{f_line_vel_evo} shows line maxima and the blue and red side velocities at which 90 percent of the line flux is enclosed as measured by a simple automated algorithm for the [\ion{O}{i}] 6300 \ion{Mg}{i} 4571, \ion{Na}{i} 5890 [\ion{Ca}{ii}] 7291, \ion{Ca}{ii} 8662 and the [\ion{Fe}{ii}] 7155 \AA~lines. The continuum level was determined in a way similar to what was done in \citetalias{Jer13} by a linear interpolation between the minimum flux levels on the blue and red sides within a region set to $\pm{6000}$ km s$^{-1}$ for the [\ion{O}{i}] 6300, \ion{Na}{i} 5890 and [\ion{Ca}{ii}] 7291 \AA~lines, $\pm{5000}$ km s$^{-1}$ for the \ion{Mg}{i} 4571 line $\pm{10000}$ km s$^{-1}$ for the \ion{Ca}{ii} 8662 line and $\pm{2000}$ km s$^{-1}$ for the [\ion{Fe}{ii}] 7155 \AA~line. At 100 days the maxima of the [\ion{O}{i}] 6300 and \ion{O}{i} 5577 \AA~lines show a blue-shift of $\sim$1000 km s$^{-1}$ decreasing towards 200 days and at later epochs the maximum of the [\ion{O}{i}] 6300 line lies close to the rest wavelength. At 200 days the \ion{Mg}{i} 4571 shows a similar blue-shift decreasing towards $\sim$300 days and in our last spectrum at $\sim$400 days the maximina lies close to the rest wavelength. In \citetalias{Jer13} we discuss these blue-shifts in detail and find them to be caused by line blocking in the core obscuring emission from the receding (red) side of the SN. No such blue shifts are seen at similar epochs for the oxygen and magnesium lines located in the infrared (not shown in the figure) suffering from less line blocking in support of this conclusion. After $\sim$200 days the \ion{Ca}{ii} 8662 shows a red-shift increasing to $\sim$1000 km s$^{-1}$ towards $\sim$400 days. 

As compared to the line maximum 90 precent of the flux lies within +$\sim$4000-$\sim$3000 km s$^{-1}$ for the [\ion{O}{i}] 6300, \ion{Na}{i} 5890 and [\ion{Ca}{ii}] 7291 \AA~lines, within $\pm{\sim}$2500 km s$^{-1}$ for the \ion{Mg}{i} 4571 \AA~line, within $\pm{\sim}$1500 km s$^{-1}$ for the [\ion{Fe}{ii}] 7155 \AA~line and within $\pm{\sim}$6000 km s$^{-1}$ for the \ion{Ca}{ii} 8662 \AA~line.

Fig.~\ref{f_line_flux_evo} shows evolution of the line flux normalized to the bolometric U to K flux as measured by a simple automated algorithm for the [\ion{O}{i}] 6300,6364, \ion{Mg}{i} 4571, \ion{Na}{i} 5890,5896, [\ion{Ca}{ii}] 7291,7323, \ion{Ca}{ii} 8498,8542,8662 and the [\ion{Fe}{ii}] 7155 \AA~lines. The continuum level was determined as described above. 


\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-line-vel-evo.pdf}
\caption{Line maxima (solid lines) and the blue and red side velocities (dashed lines) at which 90 percent of the line flux is enclosed for the [\ion{O}{i}] 6300/6364 (black dots), \ion{Mg}{i} 4571 (blue diamonds), \ion{Na}{i} 5890 (green squares), [\ion{Ca}{ii}] 7291 (red downward triangles), \ion{Ca}{ii} 8662 (yellow uppward triangles) and the [\ion{Fe}{ii}] 7155 (black crosses) \AA~lines.}
\label{f_line_vel_evo}
\end{figure}

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-line-flux-evo.pdf}
\caption{Evolution of line fluxes normalized to the bolometric U to K flux for the [\ion{O}{i}] 6300/6364 (black dots), \ion{Mg}{i} 4571 (blue diamonds), \ion{Na}{i} 5890 (green squares), [\ion{Ca}{ii}] 7291 (red downward triangles), \ion{Ca}{ii} 8662 (yellow uppward triangles) and the [\ion{Fe}{ii}] 7155 (black crosses) \AA~lines.}
\label{f_line_flux_evo}
\end{figure}

\subsubsection{Small scale fluctuations}
\label{s_analysis_spec_small_scale}

Figure.~\ref{f_spec_line_profiles_comp} shows small scale fluctuations in the [\ion{O}{i}] 6300,6364, \ion{O}{i} 5577, \ion{O}{i} 7774, \ion{Mg}{i} 4571 and \ion{Na}{i} 5890,5896 \AA~lines at 202 and 300 days. A 1000 km s$^{-1}$ box average of the line profile was repeatedly (3 times) subtracted to enhance the smal scale fluctuations. In the upper left panel we show a comparison of the [\ion{O}{i}] 6300 \AA~line profiles at 202 and 300 days. These are very similar and there is not much evolution of the small scale fluctuations in the line profile during this period. We identify 9 features marked A-H with an FWHM between 300 and 600 km s$^{-1}$ present at both epochs. 

However, features G and H interpreted as belonging to the [\ion{O}{i}] 6364 line match very well with the E and F features interpreted as belonging to the [\ion{O}{i}] 6300 \AA~line profile so these are likely to be repetitions. We have corrected the [\ion{O}{i}] 6300 \AA~line profile for the contamination from the [\ion{O}{i}] 6364 \AA~line assuming that the [\ion{O}{i}] 6364 \AA~line dominates the region redwards 3000 km s$^{-1}$ and the [\ion{O}{i}] 6300 \AA~line dominates the region bluewards -500 km s$^{-1}$. Minimizing the difference between the observed flux and $F_{6300}(v) + F_{6364}(v-\Delta v) = (F_{blue}(v) + R F_{red}(v-\Delta v)) + (F_{blue}(v) + R F_{red}(v-\Delta v))/R$ where R is the [\ion{O}{i}] 6300/6364 line ratio and $\Delta v$ the [\ion{O}{i}] 6300/6463 \AA~velocity difference gives a very good fit and line ratios of 2.3 and 2.6 at 202 and 300 days respectively. These line ratios are in reasonable agreement with the 3.0 expected for optically thin emission.

In the upper right panel we show the [\ion{O}{i}] 6300 \AA~line profile corrected by subtraction of the [\ion{O}{i}] 6364 \AA~line profile estimated from the fit. In the lower left panel we show a comparison of the corrected [\ion{O}{i}] 6300 \AA~line profile and the \ion{Mg}{i} 4571 \AA~line profile at 300 days. All features except B are clearly identified and the agreement is very good. In the lower right panel we show a comparison of the corrected [\ion{O}{i}] 6300 line profile and the \ion{O}{i} 5577, \ion{O}{i} 7774, and \ion{Na}{i} 5890/5896 \AA~line profiles at 202 days. The E and F features are clearly identified in all of these line profiles but none of the other features are seen. Since the E and F features are also the strongest it is not clear if the abscence of the other features is real or if the other features are just too faint to be seen.

\citet{Mat02} presented a similar investigation of the small scale fluctuations in the line profiles of SN 1993J. They found a good agreement between the [\ion{O}{i}] 6300, 5577 and 7774 \AA~line profiles which is in agreement with our results for SN 2011dh if we assume that the fainter features are hidden in the noise of the  [\ion{O}{i}] 5577 and 7774 \AA~line profiles. However, they did not find a good agreement between the [\ion{O}{i}] 6300 and \ion{Mg}{i} 4571 line profiles which is a bit surprising since we find an excellent agreement between these line profiles. One possible explanation could be that the [\ion{O}{i}] 6300 is dominated by flux from the O/Ne/Mg zone for SN 2011dh but not for SN 1993J as we expect the \ion{Mg}{i} 4571 \AA~line emerge from this zone wheras the  [\ion{O}{i}] 6300 \AA~line could also have contributions from the O/Si/S and O/C zones. The contributions to different lines from different zones for SN 2011dh are discussed in detail in \citet{Jer13}.

The small scale fluctuations observed provide strong evidence for a clumpy ejecta as have been previously demonstrated for SN 1993J \citep{Mat02} and 1987A \citep{Sta91,Chu94}. \citet{Mat02} used the statistical model for a clumpy ejecta by \citet{Chu94} to estimate a filling factor of $\sim$0.06 for oxygen zone material distributed within a sphere with 3800 km s$^{-1}$ radius. Using their estimated typical clump size of 300 km s$^{-1}$ this corresponds to $\sim$900 clumps. The model requires the radius of the sphere containing the clumps, the typical size of the clumps and the RMS of relative flux fluctuations in lines originating from the clumps. In the case of SN 2011dh we estimate the radius of the sphere containing the bulk of the O/Ne/Mg zone material to $\sim$3500 km s$^{-1}$ from the widths of the [\ion{O}{i}] 6300 \AA~and \ion{Mg}{i} 4571 \AA~lines. For SN 1987A a typical cloud size of 120 km s$^{-1}$ was estimated from the power spectrum of the [\ion{O}{i}] 6300 \AA~line by \citet{Sta91} using high-resolution spectroscopy but it is not clear how this was done by \citet{Mat02} although they had acces to high-resolution spectroscopy. As we have do not have access to high-resolution spectroscopy for SN 2011dh we can only estimate an upper limit on the typical cloud size taken to be 300 km $s^{-1}$, the smallest size of the features seen. The RMS of the flux fluctuations was calculated to $\sim$0.09 in the inner part of the sphere from the [\ion{O}{i}] 6300 \AA~line profile. Using these estimates and applying \citet[eq. 11]{Chu94} we get an upper limit on the filling factor of O/Ne/Mg zone material within the sphere of $\sim$0.07 and a lower limit on the number of O/Ne/Mg zone clumps of $\sim$900. These values are in good agreement with the values estimated by \citet{Mat02} for the clumping of oxygen zone material in SN 1993J.

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-spec-line-profile-comp.pdf}
\caption{Comparison of box average subtracted line profiles. The upper left panel shows the [\ion{O}{i}] 6300 \AA~line profile at 202 (red) and 300 (black) days. The upper right panel shows the corrected [\ion{O}{i}] 6300 \AA~line profile at 202 (red) and 300 (black) days. The lower left panel shows the corrected [\ion{O}{i}] 6300 \AA~line profile (black) and the \ion{Mg}{i} 4571 (red) \AA~line profile at 300 days. The lower right panel shows the corrected [\ion{O}{i}] 6364 \AA~line profile (black) and the \ion{O}{i} 5577 (red), \ion{O}{i} 7774 (green), and \ion{Na}{i} 5890/5896 (blue) \AA~line profiles at 202 days}
\label{f_spec_line_profiles_comp}
\end{figure}

\subsubsection{Comparison to SNe 1993J and 2008ax}
\label{s_analysis_spec_comp}

The early evolution was briefly compared in \citetalias{Erg13b} with focus on the hydrogen and helium lines. In this section we focus on the evolution after $\sim$100 days. Fig.~\ref{f_spec_evo_comp_lines} shows the (interpolated) spectral evolution centred on the [\ion{O}{i}] 6300 \AA, \ion{O}{i} 7774 \AA, \ion{He}{i} 10830 and 20581 \AA, \ion{Na}{i} 5890,5896 \AA, \ion{Mg}{i} 4571 \AA, [\ion{Fe}{ii}] 7155 and 16436 \AA, [\ion{Ca}{ii}] 7291 \AA~and \ion{Ca}{ii} 8662 \AA~lines for SN 2011dh as compared to SNe 1993J and 2008ax.

The double peaked profiles of the [\ion{O}{i}] 6300,6364 \AA~and the \ion{He}{i} 10830 and 20581 \AA~lines for SN 2008ax have been previously discussed in \citet{Tau11} and \citet{Mau10}. \citet{Mau10} used a steady-state NLTE code and a torus like distribution of helium to reproduce the \ion{He}{i} 10830 and 20581 \AA~lines and suggested that scattering by H$\alpha$ could be responsible for the double peaked profile of the [\ion{O}{i}] 6300,6364 \AA~line. However, a number of other lines show similar double peaked or flat topped line profiles, so a further investigation is motivated. 

Fig.~\ref{f_2008ax_spec_comp_lines} shows the [\ion{O}{i}] 7774 \AA, \ion{O}{i} 5577 \AA, \ion{Na}{i} 5890,5896 \AA, \ion{He}{i} 10830 and 20581 \AA, \ion{Mg}{i} 4571 and 15040 \AA, [\ion{Ca}{ii}] 7291 \AA~and \ion{Ca}{ii} 8662 \AA~lines as compared to the [\ion{O}{i}] 6300 \AA~line for SN 2008ax at 131 and 280 days. The \ion{Na}{i} 5890,5896 \AA~line shows a profile most similar to the [\ion{O}{i}] 6300,6364] \AA¸~line at both 131 and 280 days. The [\ion{O}{i}] 7774 \AA~line also shows a similar profile at 280 days whereas the line profile at 131 days is flat topped and has a flux deficit on the red side. The \ion{Mg}{i} 4571 and 15040 \AA~lines shows flat topped profiles fairly similar to the [\ion{O}{i}] 6300,6364 \AA~line but without double peaks at 131 days whereas the \ion{Mg}{i} 4571 has a flux deficit on the red side at 280 days and the single peak is coincident with the blue peak of the [\ion{O}{i}] 6300,6364] \AA~line. The \ion{O}{i} 5577 \AA~line has a flux deficit on the red side and the single peak is coincident with the blue peak of the [\ion{O}{i}] 6300,6364 \AA~line. The [\ion{Ca}{ii}] 7291 \AA~and \ion{Ca}{ii} 8662 \AA~lines shows a quite different behaviour and both have single peaked line profiles centred at their rest wavelengths. Finally, as is also shown in Fig.~\ref{f_2008ax_spec_comp_lines} the profiles of the [\ion{Fe}{ii}] 7155 and 163426 \AA~lines are clearly shifted to the blue peaking at $\sim$1000 and extending to about $\sim$5000 km s$^{-1}$ with little emission on the red side

The good agreement between the [\ion{O}{i}] 6300,6364 \AA~and the \ion{Na}{i} 5890,5896 \AA~lines as well as the \ion{O}{i} 7774 \AA~line at 280 days and the similar but broader double peaked structure seen in the \ion{He}{i} 10830 and 20581 \AA~lines suggests that scattering in H$\alpha$ is not responsibe for the double peaked structure of the [\ion{O}{i}] 6300,6364 \AA~line. For SN 2011dh we don't find any significant scattering in H$\alpha$ at late times with the steady-state NLTE modeling. It is clear that most of the lines, except for the [\ion{Ca}{ii}] 7291 \AA~and \ion{Ca}{ii} 8662 \AA~lines shows either double peaked and/or flat topped profiles. This could be explained by a torus like distribution of the material but a shell-like structure with some amount of bipolar assymetry is maybe a more natural explanation. A non-mixed ejecta preserving the onion-like structure of the nuclear burning zones would result in flat-topped line profiles except for lines arising from the central Fe/Co zone. The centrally peaked [\ion{Ca}{ii}] 7291 \AA~and \ion{Ca}{ii} 8662 \AA~lines could be explained if these are formed by syntesized calcium in the Si/S zone mixed into the central core. An onion-like ejecta structure with some amount of bipolar assymetry, a core of mixed Fe/Co and Si/S material, the oxygen zone material residing outside $\sim$2000 km s$^{-1}$, the helium zone material outside $\sim$3000 km s$^{-1}$ and the Fe/Co zone material assymetrically distributed towards the observer could explain the shape of most line profiles. The rather large volyme occupied by the core could be explained by expansion because of heating from the radioactive decay of \element[ ][56]{Ni}.

There is a clear difference between the shape of the SN 2011dh and 2008ax line profiles. The flat-topped and double peaked profiles seen in SN 2008ax are not clearly seen in SN 2011dh although the [\ion{O}{i}] 6300,6364 \AA~line has a weak double peaked shape to some extent repeated in other lines (Sect. \ref{s_analysis_spec_small_scale}). This difference could be explained by a different viewing angle if the structure of the ejecta is torus-like or has some amount of bipolar assymetry or there could be an intrinsic difference between the SNe. The line profiles of SN 1993J are somewhat intermediate in shape and have flatter line profiles as compared to SN 2001dh except for the [\ion{Ca}{ii}] 7291 \AA~and \ion{Ca}{ii} 8662 \AA~lines which have similar centrally peaked line profiles for all three SNe.

%Leaving the significance of the double peaks for a while and focusing on the flat topped nature of the line profiles we also find the \ion{Mg}{i} 4571 and 15040 \AA~lines at 131 days to be similar. Flat topped line profiles are easiest explained by optically thin emission from a shell-like structure and adding some amount of bipolar assymetry could explain the double peaks without the need of a torus-like distribution. 

%One caveat with a shell-like structure is that it is does not explain the single blue-shifted peaks coincident with the blue peak of the [\ion{O}{i}] 6300,6364 \AA~line seen in the \ion{O}{i} 5577 \AA~line at 131 days and in the \ion{Mg}{i} 4571 \AA~line at 280 days. Neither does it explain the shape of the [\ion{Ca}{ii}] 7291 \AA~and \ion{Ca}{ii} 8662 \AA~lines with a single peak at their rest wavelengths. This could be explained however, if these are formed by syntesized calcium in the Si/S zone residing within the shell like structure. Finally, as is also shown in Fig.~\ref{f_2008ax_spec_comp_lines} the profiles of the [\ion{Fe}{ii}] 7155 and 163426 \AA~lines are clearly shifted to the blue peaking at $\sim$1000 and extending to about $\sim$5000 km s$^{-1}$ with little emission on the red side suggesting a highly assymetric distribution of the iron (and thus the ejected \element[ ][56]{Ni}).

%In summary a structure with mixed Fe/Co and Si/S material, surrounded by oxygen zone material in turn surrounded by helium zone material could explain most of the observed line profiles provided that the oxygen zone material is not mixed inwards within $\sim$2000 km s$^{-1}$ and the helium zone material not inwards within $\sim$3000 km s$^{-1}$ and the Fe/Co zone material is assymetrically distributed towards the observer. The rather large volyme occupied by the two innermost zones could be explained by expansion of these zones because of heating from the radioactive decay of \element[ ][56]{Ni}. The double peaks seen in the [\ion{O}{i}] 6300,6364, \ion{O}{i} 7774 \AA~line, \ion{Na}{i} 5890,5896 \AA~and the \ion{He}{i} 10830 and 20581 \AA~lines could be explained by some amount of bipolar assymetry.

\begin{figure*}[tb]
\includegraphics[width=1.0\textwidth,angle=0]{figs/sn2011dh-spec-evo-comp-lines.pdf}
\caption{Closeup of (interpolated) spectral evolution centred on the [\ion{O}{i}] 6300 \AA, [\ion{O}{i}] 7774 \AA, \ion{He}{i} 10830 \AA, \ion{He}{i} 20581 \AA, \ion{Na}{i} 5890,5896 \AA, \ion{Mg}{i} 4571 \AA, [\ion{Fe}{ii}] 7155 \AA, [\ion{Fe}{ii}] 16436 \AA, [\ion{Ca}{ii} 7291 \AA and \ion{Ca}{ii} 8662 \AA~lines for SN 2011dh as compared to SNe 1993J and 2008ax.}
\label{f_spec_evo_comp_lines}
\end{figure*}

\begin{figure*}[tb]
\includegraphics[width=1.0\textwidth,angle=0]{figs/sn2008ax-spec-comp-lines.pdf}
\caption{Comparison of the [\ion{O}{i}] 7774 \AA, \ion{O}{i} 5577 \AA, \ion{Na}{i} 5890,5896 \AA, \ion{He}{i} 10830 \AA, \ion{He}{i} 20581 \AA, [\ion{Ca}{ii} 7291 \AA and \ion{Ca}{ii} 8662 \AA~lines to the [\ion{O}{i}] 6300 \AA for SN 2008ax at 131 and 280 days.}
\label{f_2008ax_spec_comp_lines}
\end{figure*}

\section{Modelling}
\label{s_modelling}

In this section we use the steady-state NLTE code used in \citetalias{Jer13} and described in \citet{Jer11,Jer12} and {\sc hde}, a new hydrodynamical code similar to the one used in \citetalias{Ber12} and described in Appendix \ref{a_hydro_modelling}, to model the bolometric lightcurve. In Sect.~\ref{s_modelling_day_100_750} we discuss modelling of the 100-750 days bolometric and photometric lightcurves with the steady-state NLTE code and in Sect.~\ref{s_modelling_dust}, \ref{s_modelling_molecules} and \ref{s_modelling_timedep} we discuss the effects on these lightcurves by dust, molecules and time dependenent processes respectively. In Sect.~\ref{s_modelling_day_0_100} we make a quantitative fit of the 0-100 days bolometric lightcurves of SNe 2011dh, 1993J and 2008ax using a model grid spanning a large volume of parameter space constructed with the hydrodynamical {\sc hde} code. In Sect.~\ref{s_modelling_day_0_300} we extend the temporal coverage of this model grid to 300 days and use a correction for the flux within the observed wavelength range determined by the steady-state NLTE modelling to fit the observed 3-300 days $U$ to 4.5 $\mu$m pseuso-bolometric lightcurve of SN 2011dh.

\subsection{Modelling of the bolometric lightcurve day 100-750}
 \label{s_modelling_day_100_750}

In this section we compare synthetic pseudo-bolometric lightcurves for the J13 ejecta models to the observed pseudo-bolometric lightcurves. For day 1-100 we use the {\sc hde} code setup to run in homologous mode. The temperature profile for the initial model is taken to be that of the optimal model found in Sect.~\ref{s_modelling_day_0_100}. In the optimal model homology is reached and the thermal explosion energy gets exhausted at $\sim$3 days so these assumptions are not critical for the subsequent evolution. For day 100-700 we use the same NLTE code as in \citetalias{Jer13}. The parameters of the ejecta models are given in table 2 in \citetalias{Jer13}.

Figures~\ref{f_bol_opt_mir_mcomp} and \ref{f_bol_opt_mir_mcomp_100d} show a comparison of the $U$ to 4.5 $\mu$m (3300-50000 \AA) pseudo-bolometric lightcurves for day 0-750 and day 0-100 respectively calculated from the observed magnitudes and synthetic magnitudes obtained from NLTE spectra (day 100-700) for the J13 ejecta models. In these figures we also plot the bolometric lightcurve as calculated with the {\sc hde} code (day 1-100) for the J13 ejecta models. The 10-15 percent drop in luminosity seen at 100 days in the model lightcurves is caused by the fraction of the luminosity falling outside the $U$ to 4.5 $\mu$m wavelength range. Figures~\ref{f_bol_opt_mcomp} and \ref{f_bol_opt_nir_mcomp} show the same comparison for the $U$ to $z$ (3300-10000 \AA) and $U$ to $K$ (3300-24000 \AA) pseudo-bolometric lightcurves. These figures do not show the bolometric lightcurves calculated with the {\sc hde} code as the larger fraction of the luminosity falling outside the wavelength range would make such a comparison misleading.

Figure~\ref{f_opt_mir_frac} shows the fraction of the bolometric luminosity within the $U$ to 4.5$\mu$m (3300-50000) \AA~range between 100 and 750 days for the \citetalias{Jer13} ejecta models.

%Filling factors for each zones.
%Mixing of each zone.
%Locally absorbed or free-streaming positrons.

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-bol-opt-MIR-model-comp.pdf}
\caption{U to 4.5 $\mu$m pseudo-bolometric lightcurves for the 43 (red short-dashed line), 45 (green short-dashed line), 46 (blue short-dashed line), 47 (yellow short-dashed line), 49 (red solid line), 50 (green solid line), 51 (blue solid line), 52 (yellow solid line), 54 (red long-dashed line) and 55 (blue long-dashed line) J13 models as compared to the observed U to 4.5 $\mu$m bolomectric lightcurve. The lightcurves shown in this figure and Fig. \ref{f_bol_opt_nir_mcomp}, \ref{f_bol_opt_mcomp} and \ref{f_opt_nir_mir_mcomp} have been normalized to the radioactive decay chain luminosity of 0.075 M$_{\odot}$ of \element[ ][56]{Ni}.}
\label{f_bol_opt_mir_mcomp}
\end{figure}

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-bol-opt-MIR-100d-model-comp.pdf}
\caption{U to 4.5 $\mu$m bolometric lightcurves for the J13 models as compared to the observed U to 4.5 $\mu$m bolomectric lightcurve for the first 100 days. The lightcurves are displayed as in Fig. \ref{f_bol_opt_mir_mcomp}.}
\label{f_bol_opt_mir_mcomp_100d}
\end{figure}

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-bol-opt-NIR-model-comp.pdf}
\caption{U to K bolometric lightcurves for the J13 models as compared to the observed U to K bolomectric lightcurve. The lightcurves are displayed as in Fig. \ref{f_bol_opt_mir_mcomp}.}
\label{f_bol_opt_nir_mcomp}
\end{figure}

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-bol-opt-model-comp.pdf}
\caption{U to z bolometric lightcurves the J13 models as compared to the observed U to z bolometric lightcurve. The lightcurves are displayed as in Fig. \ref{f_bol_opt_mir_mcomp}.}
\label{f_bol_opt_mcomp}
\end{figure}

\begin{figure*}[tb]
\includegraphics[width=1.0\textwidth,angle=0]{figs/sn2011dh-opt-NIR-MIR-model-comp.pdf}
\caption{Optical, NIR and MIR magnitudes for the J13 models as compared to the observed magnitudes. The lightcurves are displayed as in Fig. \ref{f_bol_opt_mcomp}.}
\label{f_opt_nir_mir_mcomp}
\end{figure*}

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-opt-MIR-fraction-model.pdf}
\caption{Fractional U to 4.5 $\mu$m bolometric luminosity for the J13 models.}
\label{f_opt_mir_frac}
\end{figure}

\subsection{Dust}
\label{s_modelling_dust}

As discussed in Sect.~\ref{s_bol_lightcurve} and seen in Fig.~\ref{f_bol_frac} there is a strong increase in the fractional MIR luminosity between $\sim$100 and $\sim$200 days and a corresponding decrease in the fractional optical and NIR luminosities. As also discussed in Sect.~\ref{s_bol_lightcurve} and seen in figures \ref{f_UK_bol_comp} and \ref{f_bol_opt_nir_mcomp} there is an significant increase in the decline rates of the $U$ to $K$ pseudo-bolometric lightcurve between $\sim$100 and $\sim$200 days, even more pronounced in the $U$ to $z$ pseudo-bolometric lightcurve (Fig.~\ref{f_bol_opt_mcomp}) but less so in the $U$ to 4.5 $\mu$m pseudo-bolometric lightcurve (Fig.~\ref{f_bol_opt_mir_mcomp}). This means that some process moving flux from the optical and NIR to the MIR is active during this period. One such process is dust formation in the ejecta that would absorb the still quite hot radiation from the SN and re-emit it at a much lower temperature. However, a change of the lines dominating the cooling of the ejecta in some zone could also cause a similar effect. As discussed in Sect.~\ref{s_bol_lightcurve} an increasing excess in the MIR during this period is also seen in SN 1993J but the corresponding increase in the deline rates of the $U$ to $K$ pseudo-bolometric lightcurve is not seen which makes the interpretation less clear. However, circum stellar medium (CSM) interaction that would affect the lightcurve in the opposite way could be important for SN 1993J already at this early phase.

\subsection{Molecules}
\label{s_modelling_molecules}

Figure. \ref{f_spec_CO_flux} shows the integrated continuum subtracted flux between 22750 and 24350 \AA~were we expect CO overtone band emission. The continuum was estimated as a linear interpolation between the averaged 22700-22800 \AA~and 24300-24400 \AA~regions. The integrated continuum subtracted flux was 2.2$\times$10$^{-14}$ and 5.7$\times$10$^{-15}$ erg s$^{-1}$ cm$^{-2}$ at 88 and 206 days respectively. 

The total flux in the 4.5 $\mu$m band, calculated using the zeropoint flux and the equivalent width of the band was 4.9$\times$10$^{-13}$ and 1.7$\times$10$^{-13}$ erg s$^{-1}$ cm$^{-2}$ at 88 and 206 days respectively. Note that the value at 206 days lies in the gap of the Spitzer observations and has been linearly interpolated between 85 and 251 days. If all of the flux in the 4.5 $\mu$m band was due to CO fundamental band emission this would correspond to fundamental to overtone band flux ratios of $\sim$20 and $\sim$30 at 85 and 206 days respectively.

\begin{figure}[tb]
\includegraphics[width=0.48\textwidth,angle=0]{figs/sn2011dh-spec-CO-flux.pdf}
\caption{Continuum subtracted CO overtone region at 88 days (blue) and 206 days (red). The flux at 206 days have been scaled with the ratio of the measured total fluxes in the region.}
\label{f_spec_CO_flux}
\end{figure}

\subsection{Time dependent effects}
\label{s_modelling_timedep}

\subsection{Modeling of the bolometric lightcurve day 0-100}
\label{s_modelling_day_0_100}

In this section we use the {\sc hde} code to construct a large model grid allowing us to find the parameters giving the best fit to the bolometric lightcurve and the photospheric velocities in a more quantitative way than in \citetalias{Ber12}. This allows us to refine the disussion of the sensitivity of the derived quantities to errors in the observed quantites initiated in \citetalias{Erg13b}. It also allows us to fit the bolometric lightcurves of SNe 1993 and 2008ax using the same model grid and refine the discussion of the nature of their progenitors initiated in \citetalias{Erg13b}. The parameters varied are the mass of helium core M$_{\mathrm{He}}$, the explosion energy E, the mass of ejected \element[ ][56]{Ni} M$_{\mathrm{Ni}}$ and the distribution of it. The mass fraction of the \element[ ][56]{Ni} was assumed to be a linearly declining function of mass becoming zero at some fraction of the total mass Mix$_{\mathrm{Ni}}$. The parameter space spanned was M$_{\mathrm{He}}$=2.5-6.5 M$_\odot$, E=0.5-2.0$\times$10$^{51}$ erg, M$_{\mathrm{Ni}}$=0.02-0.2 and Mix$_{\mathrm{Ni}}$=0.5-1.0 using a 10$\times$10$\times$10$\times$10 grid. We have found this resolution to be sufficient to safely interpolate intermediate values. The fitting is done by minimizing the square of the relative residuals giving equal weight to the diffusion phase lightcurve (5-40 days), the tail lightcurve (40-100 days) and the early photospheric velocity evolution (5-40 days). As stellar models we have used the STARS models, removing the hydrogen envelope and re-calculating the density profile using the constraints from hydrostatic and thermal equilibrium as described in \citetalias{Ber12} (! Fix this !). As the {\sc hde} code does not include a network of nuclear reactions the explosive nucleosynthesis as a function of mass and explosion energy have been taken from Woosley et al. (XXXX) (! Fix this !) and linearly interpolated.

Fig.~\ref{f_mg_fit_comp} shows the model and observed lightcurve and photospheric velocity evolution for the optimal models of SNe 2011dh, 2008ax and 1993J as well as contour plots of the error in the fit as a function of mass and energy. Table \ref{t_mg_fit_comp} shows the mass, explosion energy, mass of ejected \element[ ][56]{Ni} and the distribution of it for the optimal models and the corresponding errors. The errors were calculated as the square root of the sum of the squared errors resulting from the errors in distance and extinction and a systematic error in the photospheric velocities. The derived parameters for SN 2011dh are in good agreement with the results in \citetalias{Ber12}. The mass and explosion energy derived for SNe 1993J and 2008ax are very similar to what is derived for SN 2011dh wheras the mass of ejected \element[ ][56]{Ni} differs significantly. The \element[ ][56]{Ni} is distributed far out in the ejecta for all three SNe. We note that the velocity evolution of SN 2008ax is not well fitted which could be explained by a worse correspondance between the absorption miniumum of \ion{Fe}{ii} 5169 \AA~and the photosphere as compared to SNe 2011dh and 1993J. The countor plots show that, as expected and discussed in \citetalias{Erg13b}, there is as a strong degenerecy in mass and explosion energy if the fitting is done using the bolometric lightcurve alone. As seen the constraint from the photospheric velocity evolution decrease this degeneracy significantly and the fit becomes quite robust. This means however that we expect the results to be quite sensitive to errors in the photospheric velocities. 
Fig.~\ref{f_mg_fit_sens} shows the sensitivity of the derived quantities on errors in the distance, extinction and a systemtic error in the photospheric velocity. For the mass and explosion energy the dependence on the distance and extinction is weak although a higher extinction or larger distance tends to lower the mass. The dependence on the photospheric velocity on the other hand is quite strong. For the mass of ejected \element[ ][56]{Ni} the sensitivity on the distance and extinction is strong whereas the dependence on the photospheric velocity is weak. The dependencies of the derived quantities on the distance and extinction are in agreement with the qualitative discussion in \citetalias{Erg13b} but our model grid now makes it possible to quantify these as well as the dependencies on the photospheric velocity. In general we see that an error in the distance and extinction mainly corresponds to an error in the mass of ejected \element[ ][56]{Ni} whereas an error in the photospheric velocity mainly corresponds to an error in the mass and explosion energy. In Fig~\ref{f_mg_fit_sens} we also show the dependencies as expected from approximate considerations. As discussed in \citetalias{Erg13b} we expect the mass of ejected \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 $10^{A_{V}/2.5}$. As the diffusion time is proportional to E/M$^{3}$ and the velocity squared to E/M we expect the mass to be proportional to the velocity and the energy to be proportional to the velocity cubed. As seen in the figure these approximate scalings are well followed by the model.

To calculate the error bars for the derived quantities in Table \ref{t_mg_fit_comp} we have assumed a systematic error in the photosperic velocity of 15 percent. As discussed in \citetalias{Erg13b} the photospheric radius as measured from the absorption minimum of the \ion{Fe}{ii} 5169 \AA~line could be overestimated by as much as $\sim$30 percent for SN 2011dh if we treat the thermalization radius as estimated from blackbody fits to the photometry as a lower limit. Such an error would ruin our lower error bars on the mass and explosion energy wheras the upper error bars would remain unchanged. However, such a $\sim$30 percent overestimate corresponds to an dilution factor (ratio between thermalization and photospheric radii) of 1.0 which is not particularly likely. Actually, the average observed dilution factor between 5 and 40 days of $\sim$0.75 equals the average model dilution factor (Sect.~\ref{s_physics}) between 5 and 25 days (where the thermalization radius disappears in the modelling). Although this argument is only indicative and the observed dilution factor is not known with better accuray than the distance we find the 15 percent error used in the calculation reasonable.

We find a similar good agreement between average observed and model dilution factors for SN 1993J but as discussed in (Sect.~\ref{s_physics}) for SN 2008ax the average observed and model dilution factors is $\sim$0.5 and $\sim$0.80 respectively and the observed blackbody temperature is also much higher than the model thermalization thermperature. This is a clear indication that the adopted extinction is overestimated although the comparison has a number of caveats and other explanations are possible. As discussed above the derived mass and explosion energy are weakly dependent on the adopted extinction and the significant uncertainly in this quantity for SN 2008ax is included in the calculation of the error bars in Table \ref{t_mg_fit_comp}. To get a better correspondance between observed and model dilution factors and temperatures the extinction for SN 2008ax would need to be revised towards the lower error limit which would correspond to an increase of the derived mass to $\sim$3.5 M$_\sun$ (!verify this!). The derived mass of the ejected \element[ ][56]{Ni} on the other hand would be very sensitive to such a revision and reduced with a factor of $\sim$2 (!verify this!) which, as discussed in \citetalias{Erg13b}, makes some sense as we find the other explosion parameters (mass and explosion energy) to be similar to those of SNe 2011dh.

\begin{figure}[tb]
\includegraphics[width=0.5\textwidth,angle=0]{figs/sn2011dh-mg-fit-comp.pdf}
\caption{Bolometric lightcurve (right panels) and photospheric velocity evolution (middle left panels) for the optimal models as compared to the observed U to K pseudo-bolometric lightcurves and the velocity evolution of the absorption minimum of \ion{Fe}{ii} 5169 \AA~for SNe 2011dh (top panels), 2008ax (middle panels) and 1993J (bottom panels). In the middle right and right panels we show contour plots of the error in the fits as a function of mass and explosion energy for the case where the photospheric velocities were used and not used respectively.}
\label{f_mg_fit_comp}
\end{figure}

\begin{figure}[tb]
\includegraphics[width=0.5\textwidth,angle=0]{figs/sn2011dh-mg-fit-sens.pdf}
\caption{Sensitivity of the derived explosion energy (left panels), mass (middle left panels), mass of ejected \element[ ][56]{Ni} (middle right panels) and the distribution of it (right panels) on an error in the distance (upper panels), extinction (middle panels) and photospheric velocities (lower panels).}
\label{f_mg_fit_sens}
\end{figure}

\begin{table*}[tb]
\caption{Helum core mass, explosion energy, mass of the ejected \element[ ][56]{Ni} and the distribution of it for the optimal models of SNe 2011dh, 1993J and 2008ax.}
\begin{center}
\include{sn2011dh-mg-fit-comp-table}
\end{center}
\label{t_mg_fit_comp}
\end{table*}

\subsection{Modeling of the bolometric lightcurve day 0-300}
\label{s_modelling_day_0_300}

In this section we extend the temporal coverage of the model grid described in Sect.~\ref{s_modelling_day_0_100} to 300 days (which is the period for which we have full $U$ to 4.5 $\mu$m coverage) and make a fit of the observed $U$ to 4.5 $\mu$m pseudo-bolometric lightcurve for SN 2011dh to a model grid in a way similar to what was done in Sect.~\ref{s_modelling_day_0_100}. The method used will be to calculate the bolometric lightcurve after 100 days using the {\sc hde} code assuming homologous expansion and that all energy deposited by the radioactive decay chains is instantly emitted as radiation. However, to compare with the pseudo-bolometric lightcurve after 100 days we also include a correction for the flux falling outside the $U$ to 4.5 $\mu$m wavelength range determined with the steady-state NLTE code. Fig.~\ref{f_bol_opt_mir_omcomp} shows the bolometric lightcurve for the optimal bare helium core model for SN 2011dh as compared to the observed $U$ to 4.5 $\mu$m pseudo-bolometric lightcurve, calculated as described and multiplied with the fraction of bolometric flux within the $U$ to 4.5 $\mu$m wavelength range for the optimal NLTE model after 100 days.

Although Fig.~\ref{f_bol_opt_mir_omcomp} is quite convincing a quantative fit as in Sect.~\ref{s_modelling_day_0_100} is needed to determine the error sensitivity and degeneracy of the solution. In anology to the procedure described in Sect.~\ref{s_modelling_day_0_100} the fitting is done by minimizing the square of the relative residuals giving equal weight to the diffusion phase lightcurve (5-40 days), the early tail lightcurve (40-100 days), the late tail lightcurve (100-300 days) and the early photospheric velocity evolution (5-40 days). This weighting scheme give less weight to the photospheric velocities but this is also motivated as we have additional information about the lightcurve.

The correction for the flux falling outside the $U$ to 4.5 $\mu$m wavelength range was determined by evolving a strongly restricted set of the ejecta models with the steady-state NLTE modelling. In most of the parameter space spanned this correction did not vary much and the number of ejecta models were chosen as small as possible to get a reasonable precision ($\sim$10 percent) using linear interpolations. The further restrict the number of ejecta models we excluded those for which the average bolometric luminosity was more than 25 percent below the average observed $U$ to 4.5 $\mu$m luminosity using the lower error bars for the distance and extinction as we know these would never make a good fit.

The method has its limitations and all the free parameters of the steady-state NLTE modelling which are not possible to map from the hydrodynamical modelling, as the degree of macroscopic mixing, the fraction of the energy going into molecule cooling in the O/C and O/Si zones and the amount of dust, have to be assigned some values. Here we have chosen to give these parameters the same values as for our optimal steady-state NLTE model (Sect.~\ref{s_modelling_day_100_750}). On the other hand, as seen in Fig.~\ref{f_opt_mir_frac} the fractional flux within $U$ to 4.5 $\mu$m does not vary much between the \citetalias{Jer13} ejecta models during the first 300 days and as the optical depth to the $\gamma$-rays (and thus the deposited energy) depends on the mass as $M^{2}/E$ we don't expect the derived mass to be very sensitive to changes in this fraction.

%In this section we attempt to model the complete bolometric lightcurve using hydromodelling for day 0-100 and NLTE modeling for day 100-700. Hydro modelling of day 0-100 have been presented in \citetalias{Ber12} but the behaviour of the late tail imposes further constraints on the mass, energy and mass and distribution of \element[ ][56]{Ni}. For the hydrodynamical modelling we use the {\sc hde} code described in detail in Appendix \ref{a_hydro_modelling}. For the NLTE modeling we use the same code as in in \citet{Jer11,Jer12,Jer13}. As we also want to investigate in more detail, the sensitivity of the results to changes in the observed parameters, as well as using a more quantitative estimate of the best fit, we first need to construct a grid of stellar models. To achive this we use the  We vary the explosion energy between 0.5 and 2.0 B, and the mixing of \element[ ][56]{Ni} between .. and ... . The amount of \element[ ][56]{Ni} is assumed to correspond to a pure scaling of the lightcurve but to confirm this a series of models were made for the optimal choice of the other parameters and this parameter was refitted.

\begin{figure}[tb]
\includegraphics[width=0.5\textwidth,angle=0]{figs/sn2011dh-bol-opt-MIR-optimal-model-comp.pdf}
\caption{Bolometric lightcurve for the optimal bare helium core model for SN 2011dh (red solid line) as compared to the observed U to 4.5 $\mu$m pseudo-bolometric lightcurve for the first 300 days (black dots and solid line). After 100 days the optimal bare helium core model flux have been multplied with the fraction of bolometric flux within the U to 4.5 $\mu$m wavelength range for the optimal NLTE model.}
\label{f_bol_opt_mir_omcomp}
\end{figure}

\section{Discussion}
\label{s_discussion}

\subsection{Physical interpretation of photometric and spectral evolution}
\label{s_physics}

In this section we discuss how the photometric and spectroscopic evolution could be understood given our optimal model, a 3.5 M$_\odot$ helium core with a thin 270 R$_\odot$ envelope exploded with an energy of 0.6$\times$10$^{51}$ erg, ejecting 0.075 M$_\odot$ of \element[ ][56]{Ni}. When the shock reached the hydrogen rich envelope at $\sim$100 seconds the explosion energy is roughly equi-partioned between kinetic and thermal energy and the shock speed is $\sim$20000 km s$^{-1}$. In the nearly constant density envelope the shock is decelerated to $\sim$10000 km s$^{-1}$ and when the radiation breaks out from the shock at $\sim$10000 seconds almost all of the thermal energy deposited in the helium core has been cold away by expansion. About on tenth of the explosion energy is deposited in the hydrogen envelope but even here equpartition is not reached and at shock breakout only a small fraction of the explosion energy is in the form of thermal energy. Fig. \ref{f_shock_breakout} shows the density and temperature profiles at shock breakout. Because of the decelaration of the shock the hydrogen envelope has been strongly compressed and the temperature is high. In the three days that follows the hydrogen envelope will accelerate and expand and the temperature and luminoisity at the photosphere, which are initially very high, will decrease rapidly because of expansion cooling and the short diffusion time for the radiation. This is the cooling phase seen in the early photometry published in \citet{Arc11} and \citet{Tsv12}. 

The photosphere reaches the helium core at 4.5 days, after which the position of the photosphere is determined by the ionization front of helium slowly moving inwards in mass coordinates but outwards in radial coordinates. This period corresponds to the diffusion of thermal energy deposited in the ejecta by the radioactive decay chain of \element[ ][56]{Ni}. Defining the thermalization radius as $\sqrt {3 \tau_{abs} \tau_{tot}}=2/3$ \citep{Ens92} this 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 cease to exist. In Fig. \ref{f_bb_T_evo_model_comp} we show the dilution factor (ration of thermalization and photospheric radii) for the optimal bare helium core models for SNe 2011dh, 1993J and 2008ax compared to the observed dilution factor as estimated from blackbody fits to the photometry and the absorption minimum of the \ion{Fe}{ii} 5169 \AA~line. We also show the thermalization temperature compared to the observed temperature as estimated from blackbody fits to the photometry.  Before the temperature peak of SNe 2011dh and 1993j the observed temperature is lower and the observed dilution factor higher than in the model, possibly because of the presence of the hydrogen envelope, but after this the agreement is good. The model temperatures and dilution factors for all SNe are fairly constant at $\sim$9000 K and $\sim$0.8 respectively. The observed temperature of SN 2008ax on the other hand is much higher and the observed dilution factor much lower. In the modelling the thermalization temperature is mainly determined by the ionization temperature of helium and to reach a temperature above 15000 K at the center of the ionization front in a pure helium envelope requires a density higher than 10$^{-9}$ g cm$^{-1}$ which is a factor of 100 above the canonical 10$^{-11}$ g cm$^{-1}$. Although there is a number of caveats, as the variation of the thermalization radius and thus the temperature with wavelength \citep{Des05} and the relation between this temperature and a blackbody fit to the photometry, we find the discrepancey intriguing. One possible explanation is that the extinction adopted for SN 2008ax is overestimated but other explanations as differences in density and composition might be possible as well.

\begin{figure}[tb]
\includegraphics[width=0.5\textwidth,angle=0]{figs/sn2011dh-shock-breakout.pdf}
\caption{Density (left panel) and temperature (right panel) profiles of the optimal model for SN 2011dh at shock breakout.}
\label{f_shock_breakout}
\end{figure}

\begin{figure}[tb]
\includegraphics[width=0.5\textwidth,angle=0]{figs/sn2011dh-bb-RT-evo-model-comp.pdf}
\caption{Left panel: Observed blackbody temperature for SNe 2011dh (black dots), 1993J (red triangles) and 2008ax (blue squares) compared to the thermalization temperature as estimated from the hydrodynamical modelling (solid lines). Right panel: Observed dilution factor for SN 2011dh (black dots), 1993J (red trangles) and 2008ax (blue squares) compared to the dilution factor as estimated from the hydrodynamical modelling (solid lines).}
\label{f_bb_T_evo_model_comp}
\end{figure}

\subsection{The nature of the progenitor star}

In \citetalias{Mau11}, \citetalias{Ber12}, \citetalias{Erg13b}, \citetalias{Jer13} and this paper we have investigated the nature of the progenitor star using a number of different and, at least partially, independent methods. In \citetalias{Mau11} we analysed direct observations of the star by comparison of the observed magnitudes to predictions from stellar athmosphere and evolutionary models. The best match was found to be a yellow supergiant with an initial mass of 13$\pm{3}$ M$_\odot$ and a radius of $\sim$270 R$_\odot$. In \citetalias{Erg13b} we presented observations of the disappearence of this star thus confirming that it was the progenitor of SN 2011dh. 

In \citetalias{Ber12} we presented hydrodynamical modeling, which given the refinements in \citetalias{Erg13b}, shows that a star with a helium core of 3-4 $M_\odot$ and a thin (0.1 M$_\odot$) and extended (200-300 R$_\odot$) envelope, exploded with an energy of 0.6-1.0$\times$10$^{51}$ erg and ejecting 0.05-0.10 M$_\odot$ of \element[ ][56]{Ni} mixed out to a velocity of $\sim$9000 km s$^{-1}$ well reproduce the observed bolometric lightcurve for the first 100 days. In this paper we arrive at similar best fit values using hydrodynamical modeling to scan a large volume of parameter space where we have also extended the temporal coverage to 300 days by the use of a correction for the flux within the observed wavelength range determined with steady-state NLTE modelling.

In \citetalias{Jer13} we presented steady-state NLTE modelling of nebular spectra showing that a star with an initial mass of $\sim$12 M$_\odot$ best reproduce the observed spectral evolution with particular focus on the [\ion{O}{i}] 6300/6364 lines which are very sensitive to the initial mass of the star. In this paper we find the optimal model from \citetalias{Jer13} to give a reasonable fit to the observed $U$ to 4.5 $\mu$m and $U$ to $z$ pseudo-bolometric lightcurves between $\sim$100 and $\sim$450 days and to the observed $U$ to $K$ pseudo-bolometric lightcurve until the NIR coverage ends at $\sim$300 days. After $\sim$450 days we cannot reproduce the observed evolution of the pseudo-bolometric lightcurves but it is clear from modelling using a time-dependent NLTE code \citep{Koz92,Koz98a,Koz98b} that in this phase freeze-out in the helium zone is important and the assumption of steady-state is no longer valid.

 In \citetalias{Erg13b} we estimated a hydrogen mass of 0.01-0.04 M$_\odot$ using a Monte-Carlo atmosphere code in agreement with the 0.024 M$_{\odot}$ estimated by \citet{Arc11} using a similar but more advanced code that included a NLTE treatment of hydrogen and helium. This hydrogen mass is consistent with the \citetalias{Ber12} ejecta model and we also find the interface between the helium core and hydrogen interface to be located at a velocity consistent with this model.

As proposed in \citetalias{Erg13b} and shown in this paper the sensitivity of the mass and explosion energy derived from the hydrodynamical modelling to the errors in distance and extinction is weak and merely effects the derived mass of ejected \element[ ][56]{Ni}. Although we have not scanned parameter space with the steady-state NLTE code as this would be too computationally intensive a similar conclusion is likely to hold for the initial mass as estimated from nebular spectra with steady-state NLTE code. The fitting procedure for the hydrodymanical modelling used in this paper also allows us to quantify the sensitivity to errors in the photospheric velocities and including the errors in all the observables the upper bound on the initial mass is found to be $\sim$15 M$_\odot$.

In all the results obtained with the different methods are consistent and, even given the caveats of each individual method, it is likely that the progenitor star is of moderate initial mass (10-15 M$_\odot$) and has a thin extended hydrogen rich envelope of which most must have been lost either through stellar winds or interaction with a binary companion. The moderate mass suggests that interaction with a binary is needed as stellar winds of stars in this mass range are not strong enough to expell the hydrogen envelope before core-collapse. As we show in \citetalias{Jer13} and in this paper, using steady-state NLTE modelling of nebular spectra and hydrodynamical modelling of the bolometric lightcurve, SNe 2008ax and 1993J are likely to be of similar initial mass and have similar explosion energy as SN 2011dh although the mass of the ejected \element[ ][56]{Ni} may differ significantly depending on the distance and extinction adopted. In particular the upper bound on the initial mass for all three SNe is found to be $\sim$15 M$_\odot$. By the same reason as discussed above this suggests that interaction with a binary have taken place and removed most of the hydrogen envelope of their progenitor stars. In the case of SN 1993J this conclusion is supported by direct observations of the binary companion \citep{Mau04}. Observations that could detect or set useful constraints on the presence of a companion star for SN 2011dh are schedulded for Cycle 21 at HST whereas similar observations for SN 2008ax would not be feasible because of the larger distance. Clearly there is growing evidence that the main production channel for Type IIb SNe are stars stripped on their hydrogen envelope by interaction with a binary companion. Modelling of the nebular spectra and hydrodynamical modelling of the bolometric lightcurves for a larger sample of Type IIb SNe could provide further evidence for this hypothesis.

\section{Conclusions}
\label{s_conclusions}

We present nealy two years of optical and NIR photometric and spectroscopic observations for the Type IIb SN 2011dh. Together with SWIFT UV and Spitzer MIR data we build a UV to MIR pseudo-bolometrci lightcurve covering $\sim$700 days, although the photometric coverage ends at $\sim$100 days in UV and at $\sim$350 days in NIR. The spectral coverage ends at $\sim$200 days in NIR and $\sim$450 days in the optical.

%The late time lightcurve offers a number of challenges like molecule emission, dust absorption and emission, cooling processes and recombination freeze out, all sensitive to the density and mixing of the different ejecta zones. 
We have used a steady-state NLTE code \citep{Jer11,Jer12,Jer13} to find a 12 M$_\odot$ ejecta model that reproduce the $U$ to 4.5 $\mu$m and $U$ to $z$ pseudo-bolometric lightcurves until $\sim$450 days and the $U$ to $K$ pseudo-bolometric lightcurve until $\sim$300 days (where the NIR coverage ends) reasonably well. However, we are unable to reproduce the evolution after $\sim$450 days when a considerable flattening is seen in all lightcurves. Modelling with a time-dependent NLTE code \citep{Koz92,Koz98a,Koz98b} show that in this late phase freeze-out in the helium zone becomes important and therefore the assumption of steady-state is not valid.

We have used the hydrodynamical {\sc hde} code to build a model grid for the early (3-100 days) lightcurve spanning a large volume of parameter space. This allows a more quantitatve fitting procedure than was used in B12 as well as a consistent modelling of SNe 2011dh, 1993J and 2008ax taking into account the significant errors in distance and extinction. The results for SN 2011dh are in good agreement with those found in \citetalias{Ber12}. For SNe 1993J and 2008ax we find values of the mass and explosion energy similar to those of SN 2011dh whereas the mass of \element[ ][56]{Ni} depends sensitively on the adopted distance and extinction. The mass and explosion energy on the other hand are unsensitive to changes in the distance and extinction which is an important quality of the modelling.

We have also constructed an extended model grid with temporal coverage up to 300 days by assuming instant emission of the energy deposited by the radioactive decay chains after 100 days. To compare with the observed $U$ to 4.5 $\mu$m pseudo-bolometric lightcurve we have determined the fractional flux within this wavelength range with the steady-state NLTE code. This allows us to combine the power of the hydrodynamical and steady-state NLTE modelling and the use of a quantiative fitting procedure. Applying this to the observed 3-300 days $U$ to 4.5 $\mu$m pseudo-bolometric lightcurve of SN 2011dh we find best fit values for the mass, explosion energy, mass of ejected \element[ ][56]{Ni} and the distribution of it in good agreement with those based on the 3-100 days bolometric lightcurve.

We find an excess in the MIR and NIR as compared to steady-state NLTE model photometry developing between $\sim$200 and $\sim$300 days, during which an increase in the optical and NIR tail decline rates is also observed. This behaviour could be reproduced by the steady-state NLTE modelling if a modest amount of dust ($\tau=0.2$) is being continiously formed in the ejecta during this period, although the photomeric evolution in the MIR is not well reproduced after $\sim$400 days.

However, as discussed in \citetalias{Erg13b} an excess is developing in the 4.5 $\mu$m already during the first 100 days. This is unlikely to be caused by dust forming in the ejcta but CO fundamental band emission or a thermal dust echo as proposed by \citet{Hel13} are possible explanations. We detect CO first overtone band emission in NIR spectroscopy at $\sim$100 and $\sim$200 days so there is certainly a contribution to the observed 4.5 $\mu$m band flux from CO fundamental band emission at these epochs and probably at earlier and later epochs as well. 

In general the physical explanation for the photometric evolution in the MIR bands could be quite complex and might involve components from CO emission, SiO emission, ejecta and circumstellar dust emission for which, at the best, only a simple and approximate treatment is included in the steady-state NLTE modelling.

%Finally, it is allways worth listening to slave wispering in your ear - Ceasar, in the end you will all be wrong.

\section{Acknowledgements}

\begin{table*}[p]
\caption{Optical colour-corrected JC $U$ and S-corrected JC $BVRI$ magnitudes for SN 2011dh. Errors are given in parentheses. For simplicity data for the first 100 days already published in \citetalias{Erg13b} are included.}
\begin{center}
\include{sn2011dh-jc-table-1}
\end{center}
\label{t_jc}
\end{table*}

\setcounter{table}{2}
\begin{table*}[p]
\caption{Continued.}
\begin{center}
\include{sn2011dh-jc-table-2}
\end{center}
\end{table*}

\setcounter{table}{2}
\begin{table*}[p]
\caption{Continued.}
\begin{center}
\include{sn2011dh-jc-table-3}
\end{center}
\end{table*}

\begin{table*}[p]
\caption{Optical colour-corrected SDSS $u$ and S-corrected SDSS $griz$ magnitudes for SN 2011dh. Errors are given in parentheses. For simplicity data for the first 100 days already published in \citetalias{Erg13b} are included.}
\begin{center}
\include{sn2011dh-sloan-table-1}
\end{center}
\label{t_sloan}
\end{table*}

\setcounter{table}{4}
\begin{table*}[p]
\caption{Continued.}
\begin{center}
\include{sn2011dh-sloan-table-2}
\end{center}
\label{t_sloan}
\end{table*}

\begin{table*}[p]
\caption{NIR S-corrected 2MASS $JHK$ magnitudes for SN 2011dh. Errors are given in parentheses. For simplicity data for the first 100 days already published in \citetalias{Erg13b} are included.}
\begin{center}
\include{sn2011dh-2mass-table}
\end{center}
\label{t_nir}
\end{table*}

\begin{table*}[p]
\caption{MIR Spitzer 3.6 $\mu$m and 4.5 $\mu$m magnitudes for SN 2011dh. Errors are given in parentheses. For simplicity data for the first 100 days already published in \citetalias{Erg13b} are included.}
\begin{center}
\include{sn2011dh-spitzer-table}
\end{center}
\label{t_mir}
\end{table*}

\begin{table*}[p]
\caption{List of late-time (100-415 days) optical and NIR spectroscopic observations.}
\begin{center}
\scalebox{1.0}{
\begin{tabular}{l l l l l l l}
\hline\hline \\ [-1.5ex]
JD (+2400000) & Phase & Grism & Range & Resolution & Resolution & Telescope (Instrument)\\ [0.5ex]
(d) & (d) & & (\AA) & & (\AA) &\\ [0.5ex]
\hline \\ [-1.5ex]
55821.33 & 108.33 & b200 & 3300-8700 & ... & 12.0 & CA-2.2m (CAFOS) \\
55821.33 & 108.33 & r200 & 6300-10500 & ... & 12.0 & CA-2.2m (CAFOS) \\
55828.35 & 115.35 & R300B & 3200-5300 & ... & 4.1 & WHT (ISIS) \\
55828.35 & 115.35 & R158R & 5300-10000 & ... & 7.7 & WHT (ISIS) \\
55864.65 & 151.65 & ? & ?-? & ? & ? & AS-1.22m (AFOSC) \\
55867.71 & 154.71 & ? & ?-? & ... & ? & CA-2.2m (CAFOS) \\
55893.76 & 180.76 & Grism 3 & 3200-6700 & 345 & 12.4 & NOT (ALFOSC) \\
55897.76 & 184.76 & Grism 5 & 5000-10250 & 415 & 16.8 & NOT (ALFOSC) \\
55911.20 & 198.20 & zJ & 8900-15100 & 700 & ... & WHT (LIRIS) \\
55914.70 & 201.70 & R300B & 3200-5300 & ... & 4.1 & WHT (ISIS) \\
55914.70 & 201.70 & R158R & 5300-10000 & ... & 7.7 & WHT (ISIS) \\
55918.69 & 205.69 & HK & 14000-25000 & 333 & ... & TNG (NICS) \\
55951.64 & 238.64 & Grism 4 & 3200-9100 & 355 & 16.2 & NOT (ALFOSC) \\
55998.68 & 285.68 & r200 & 6300-10500 & ... & 12.0 & CA-2.2m (CAFOS) \\
56005.63 & 292.63 & Grism 4 & 3200-9100 & 355 & 16.2 & NOT (ALFOSC) \\
56013.14 & 300.14 & R600B & ?-? & ... & ? & WHT (ISIS) \\
56013.14 & 300.14 & R316R & ?-? & ... & ? & WHT (ISIS) \\
56071.56 & 358.56 & ? & ?-? & ? & ? & GTC (OSIRIS) \\
56072.61 & 359.61 & ? & ?-? & ? & ? & GTC (OSIRIS) \\
56128.47 & 415.47 & ? & ?-? & ? & ? & GTC (OSIRIS) \\

\hline
\end{tabular}}
\end{center}
\end{table*}

\begin{table*}[p]
\caption{Pseudo-bolometric 3-300 days UV to MIR lightcurve for SN 2011dh calculated from spectroscopic and photometric data with a 1-day sampling between 3 and 50 days and a 5-day sampling between 50 and 300 days. Random errors are given in the first parentheses and systematic lower and upper errors (arising from the distance and extinction) respectively in the second parentheses.}
\begin{center}
\include{sn2011dh-UV-MIR-bol-table}
\end{center}
\label{t_UV_MIR_bol}
\end{table*}

\appendix

\section{Hydrodynamical modelling}
\label{a_hydro_modelling}

For the hydrodynamical modelling done in this paper we use the {\sc hde} code, similar in most aspects to the code used in \citetalias{Ber12} implementing the method described in \citet{Fal77} and \citet{Ber11}. The hydrodynamical conservation equations for mass, momentum and energy \citep[eqs. 1-4]{Fal77} are solved by a finite difference scheme similar to the one described by \citet[eqs A1-A12]{Fal77} assuming that the radiative flux is given by the diffusion approximation \citep[eq. 5]{Fal77}. This is motivated in the optically thick regime but not in the optically thin regime where the radiation field is decoupled from the matter. In the optically thin regime we use a flux limiter following the prescription given by \citet{Ber11}, being essentially a transformation of the radiation field from the optically thick to the optically thin limit. To handle strong velocity gradients (shocks) we use an artificial viscosity following the prescription by \citet{Neu50}. The opacity is calculated from the OPAL opacity tables \citep{Igl96} complemented with the low temperature opacities given by \citet{Ale94}. These opacity tables are calculated for a non-expanding medium and therefore the bound-bound opacity, which is strongly dependent on the velocity field, is not applicable to SNe. To handle this we use, as discussed in \citet{Ber11}, a minimum value of the opacity called opacity floor. The value of this floor is set to $0.01$ cm$^2$ gram$^{-1}$ in the hydrogen envelope and $0.025$ cm$^2$ gram$^{-1}$ in the helium core following \citetalias{Ber12} who calibrated these values by comparison to the STELLA hydrodynamical code \citep{Bli98}. 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}. 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 gray opacities, luminosities and decay times as in  \citet{Jer11,Jer12} and the heating rate then fed into the energy equation. The code may also be run in homolgous mode where the dynamics have been switched off and the energy equation is solved given the constraint of homologous expansion.

The momentum equation was written in explicit finite difference form as

\begin{equation}
{{\Delta v^{n}_{k}} \over {\Delta t^{n}}}=-4\pi(r^{n}_{k})^{2}{{(\Delta P^{n}_{k}+ \Delta Q^{n}_{k})} \over {\Delta m_{k}}}-{{G m_{k}} \over {(r^{n}_{k})^{2}}}
\label{eq_1}
\end{equation}

where

\begin{equation}
\begin{split}
&\Delta v^{n}_{k}=v^{n+1/2}_{k}-v^{n-1/2}_{k}\\
&\Delta P^{n}_{k}=P^{n}_{k-1/2}-P^{n}_{k+1/2}\\
&\Delta Q^{n}_{k}=Q^{n}_{k+1/2}-Q^{n}_{k-1/2}
\end{split}
\end{equation}

and solved for $v^{n+1/2}_{k}$. The energy equation was written in implicit finite difference form as

\begin{equation}
{{\Delta E^{n+1/2}_{k+1/2}} \over {\Delta t^{n+1/2}}}=\epsilon^{n+1/2}_{k+1/2}-(P^{n+1/2}_{k+1/2}+Q^{n+1/2}_{k+1/2}){{\Delta V^{n+1/2}_{k+1/2}} \over {\Delta t^{n+1/2}}}-{{\Delta L^{n+\theta}_{k+1/2}} \over {\Delta m_{k+1/2}}}
\label{eq_1}
\end{equation}

where 

\begin{equation}
\begin{split}
&\Delta E^{n+1/2}_{k+1/2}=E^{n+1}_{k+1/2}-E^{n}_{k+1/2}\\
&\Delta V^{n+1/2}_{k+1/2}=V^{n+1}_{k+1/2}-V^{n}_{k+1/2}\\
&\Delta L^{n+\theta}_{k+1/2} = \theta (L^{n+1}_{k+1}-L^{n+1}_{k}) + (1-\theta) (L^{n}_{k+1}-L^{n}_{k})\\
&\epsilon^{n+1/2}_{k+1/2}=(\epsilon^{n+1}_{k+1/2}+\epsilon^{n}_{k+1/2})/2\\
&P^{n+1/2}_{k+1/2}=(P^{n+1}_{k+1/2}+P^{n}_{k+1/2})/2\\
&Q^{n+1/2}_{k+1/2}=(Q^{n+1}_{k+1/2}+Q^{n}_{k+1/2})/2
\end{split}
\end{equation}

and solved for $T^{n+1}_{k-1/2}$, $T^{n+1}_{k+1/2}$ and $T^{n+1}_{k+3/2}$. This was achived by defining the quantity

\begin{equation}
D^{n+1/2}_{k+1/12}=\Delta E^{n+1/2}_{k+1/2}+(P^{n+1/2}_{k+1/2}+Q^{n+1/2}_{k+1/2})\Delta V^{n+1/2}_{k+1/2}+\Delta L^{n+\theta}_{k+1/2} {{\Delta t^{n+1/2}} \over {\Delta m_{k+1/2}}}
\label{eq_1}
\end{equation}

and finding the zero of this function by iteratively solving

\begin{equation}
{{\partial D^{n+1/2}_{k+1/2}} \over {\partial T^{n+1}_{k-1/2}}} \delta T^{n+1}_{k-1/2} + {{\partial D^{n+1/2}_{k+1/2}} \over {\partial T^{n+1}_{k+1/2}}} \delta T^{n+1}_{k+1/2} + {{\partial D^{n+1/2}_{k+1/2}} \over {\partial T^{n+1}_{k+3/2}}} \delta T^{n+1}_{k+3/2} = D^{n+1/2}_{k+1/2}
\label{eq_2}
\end{equation}

for the temperature corrections $\delta T^{n+1}_{k-1/2}$, $\delta T^{n+1}_{k+1/2}$ and $\delta T^{n+1}_{k+3/2}$. The coefficients of this equation system constitutes a tridiagonal matrix which could be inverted by the use of standard methods.

\bibliographystyle{aa}
\bibliography{sn2011dh}

\label{lastpage}

\end{document}
