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\label{sect:conclusion} A stellar wind from CTTS There is too much parameter degeneracy to turn this argument around and to derive $\dot M$, $v_\infty$ or $P(z)$ from the fact that observe X-ry X-ray and FUV emission a few tens of AU from the star. \citet{2009A&A...493..579G}

\section{Results} \label{sect:results} The last section already showed that for all parameters consistent with the theoretical and observational constraints the stellar wind is enclosed in a finite region by a shock front. This shock front generally reaches a maximum cylindrical radius of only a few AU, but a much larger height above the accretion disk. While a detailed numerical treatment of the post-shock cooling zone is beyond the scope of this work, the shape of the shock front indicates that the post-shock zone will also be rather narrow in $\omega$. The highest post-shock temeratures temperatures are generally reached at the base of the jet when the stellar wind encounters the inner disk rim or at large $z$ when the shock front intersects the jet axis. Thus, the position of the hottest post-shock cooling plasma must be very close to the jet axis. The temperature in our feducial fiducial model stays just below 1~MK -- too little to explain X-ray emission in the jets (Fig.~\ref{fig:result}, solid red line), but small changes in the parameters, well within the observational and theoretical contraints, constraints, are sufficient, are sufficent sufficient to drive the maximal temperatures over 1~MK for a small fraction of the mass loss (other lines in the figure). \citet{2009A&A...493..579G} showed that a small faction, about $10^{-3}$, of the total mass loss rate in the outflow is enough to power the observed X-ray emission at the base of DG~Tau's jet. All but the fiducial scenario in Fig.~\ref{fig:result} have a significant, but small fraction of the stellar wind that gets heated to $>1.5$~MK and thus can easily emit X-rays. In this article we concentrate on the stellar wind mass loss, but in the observations the boundary is not that clear. The slower jet components observed further away from the jet axis carry much of the mass flow \citep{2000ApJ...537L..49B}. Their origin is probably the inner region of the disk and not that star \citep{2003ApJ...590L.107A}. Thus, it is fully consistent that our model predicts a mass loss fraction larger than $10^{-3}$ at X-ray emitting temperatures. If the disk wind dominates the mass loss over the stellar wind, then the fraction of hot gas in the (stellar pluts plus inner disk) jet might still be small.

\subsection{Wind speed} The launching mechanism of the stellar wind in CTTS is uncertain. For a solar-type wind, it is not unreasonable to expect similar wind speeds. The solar wind consists of a slow wind with a typical velocity of 400~km~s$^{-1}$ and a fast wind around 750~km~s$^{-1}$ \citep{2005JGRA..11007109F}. The relative contribution and the launching position of the two types changes over the solar cycles, but the slow wind often emerges from regions near the solar equtor and fast wind is generally associated with coronal holes \citep{1999GeoRL..26.2901G,2003A&A...408.1165B,2009LRSP....6....3C}. Despite these differences, the total energy flux in the solar wind is almost independent of the lattitude, because the slower wind is denser than the faster wind \citep{2012SoPh..279..197L}. In this article, we use $v_\infty=500$~km~s$^{-1}$ as the fiducial outflow velocity and we assume that the wind is accelerated close to the star and has reached $v_\infty$ before it interacts with the shock front. In our model, we use a spherically symmetric stellar wind with a constant velocity. For a solar-type wind this works well to derive the shape of the shock front because eqn.~\ref{eqn:r0} contains the total energy flux $\dot E = \rho v^2_\infty \propto \dot M v_\infty$. However, a lower wind velocity and higher density at the equator would lead to lower post-shock temperatures with a higher emission measure close to the disk plane. \citet{2007IAUS..243..299M} show the stellar winds from CTTS that are launched hot cannot have a total mass loss above $10^{-11}M_\odot\mathrm{ yr}^{-1}$ or the high densities required to reach this mass loss would lead to a catastrophic cooling the wind that should be observable and would probably hinder the launching. Thus, the winds of CTTS are probably more complex than just a scaled up version of the solar wind. Still, the wind speeds observed in the sun provide a reasonable estimate for $v_\infty$. Figure~\ref{fig:v_infty} shows how a large $v_\infty$ and a correspondingly large ram presure of the stellar wind pushes the shock front out to much larger heights a above the disk plane similar to outflows with a larger $\dot M$. Additionally, $v_\infty$ is the single most imporant parameter that controls the maximal post-shock temperatures and the amount of hot plasma that is generated.

\begin{equation} R_0(z) = \sqrt{\frac{\dot{M} v_{\infty}}{4\pi P(z)}}, \end{equation} where $R_0(z)$ is the zylindrical cylindrical radius of the shock front if no further conditions are placed. In the case of young stars, however, there is another constraint. Due to the circum-stellar disk the stellar wind cannot expand freely in the $z=0$ plane; it is constrained to $R_{\textrm{shock}}(0) = R_{\textrm{inner}}$. Thus, we start the integration at $R_{\textrm{inner}} = 0.05$~AU. For general $P(z)$ this ODE needs to be solved numerically\footnote{It is possible to remove all trigonometric functions from eqn.~\ref{eqn:ode} by means of addition formulae, but that introduces singularities into the solution. Thus, we numerically solve the ODE in the form of eqn.~\ref{eqn:ode}.}. The solution to the ODE determines the locations of the shock front. Thus allows us to calculate the pre-shock velocity perpendicular to the shock front using eqn.~\ref{eqn:v0} and the post-shock temperature $T_{\mathrm{post-shock}}$. From eqn.~\ref{eqn:RH3} with negligible pre-shock presure pressure and $v_0=4\;v_1$ we derive: \begin{equation} T_{\mathrm{post-shock}}(z) = \frac{3}{16} \frac{\mu m_{\textrm{H}}}{k} v_0(z)^2,\label{eqn:T} \end{equation} where $m_{\textrm{H}}$ the mass of the hydrogen atom, $k$ the Boltzman Boltzmann constant and $\mu=0.7$ is the mean particle mass for a hot, highly ionized plasma.

\subsection{Disk wind as boundary conditions for a stellar wind} Different models exist to explain wind launching from the stellar surface \citep{1988ApJ...332L..41K,2005ApJ...632L.135M}, the X-point close to the inner disk edge \citep{1994ApJ...429..781S} and magneto-centrifugal launching from the disk \citep{1982MNRAS.199..883B,2005ApJ...630..945A}. It is likely that more than one mechanism contributes to the total outflow from the system. In this case, we expect a contact discontinuity between the different components. Numerically, the magneto-centrifugally accelerated disk wind is probably the best explored component. Magneto-hydrodynamic (MHD) simulations of the disk wind have been performed in 2D \citep{2005ApJ...630..945A}, 2.5D \citep{2011ApJ...728L..11R} or 3D \citep{2006ApJ...653L..33A}, but typically do not resolve the stellar wind, where the magneto-centrifugal launching is not effective. However, they show that the disk wind is collimated close to the axis and that the densities are largest in this region. Furthermore, the inner layers of the outflow close to the jet are within the Alfven surface, the boundary between a magnetically dominated flow and a gas-pressure dominated flow, even at distances of several tens of AU from the central star, in contrast to the outer, less collimated layers of the wind, which leave the magnetically dominated region at a few AU. \citet{2009A&A...502..217M} present analytical and numerical solutions for several scenarios that mix an inner, presumably stellar, wind and an outer disk wind. In contrast to our approach, they impose a smooth transition between stellar wind and disk wind, which allows them to model the entire outflow region numerically. With some time variability in the wind launching their models produce promising knot features in the jet. In the context of our analysis, we note that the presure pressure in their models is magnetically dominated and much higher close to the jet axies axis than at larger radii in apparent contrast to Kompaneet's approximation. However, the an inner jet component as suggested in this article is so narrow that it essentiall essential stays confind confined to the innermost resolution elements. The presure pressure at the jet axis is high initially and reaches a plateau after at $P_\infty$ dropping by one to two orders of magnitude. Below we use simple exponential ($P(z)=P_\infty+P_0\exp\left(-\frac{z}{h}\right)$) or power-law functions for that mimic this behaviour. behavior. Similar profiles for the inner density and presure pressure are seen in simulations by competing groups \citep[e.g.]{2005ApJ...630..945A,Li_Krasnopolsky_Blandford_2006,2008ApJ...678.1109M}. Observations of jets and winds from CTTS indicate that typical temperatures are a few thousand K and typical densities of the range $10^4-10^5 \mathrm{ cm}^{-3}$ \citep[e.g.][]{2000A&A...356L..41L,2007ApJ...657..897K} and we chose the paramters parameters of $P(z)$ to reach presure pressure compatible with the observed densities and temperatures. Figure~\ref{fig:p_ext} shows how different presure pressure profiles influence the shock position. As expected, larger presures pressures force the shock front onto the symetry symmetry axes for smaller $z$ (top row). If the presure pressure profile is almost linear in the region where the shock front hits the symetry symmetry axis, then the angle between the shock front and the jet axis is large, which causes high post-shock temperatures (dotted black and solid red line in the upper row). In contrast, the presure pressure gradient becomes less steep when the shock front bends towards the jet axis again, then it will approach the axis slower. The shock front and the stream lines form a smaller angle and pre-shock speeds and thus the post-shock temperatures are lower (dashed green line in the top row). The solutions shown in the bottom row of the figure are for the same scale heights as those in the upper row, but here we use smaller $P_0$ for scenarios with large $h$, so that the shock front reaches the jet axis at approximately the same $z$. Close to the disk plane the pre-shock speeds differ signifcantly, significantly, but at large $z$ they reach very similar values. However, the scenerios scenarios with the smaller $P_0$ values reach slightly larger radii and the slightly different shape of the shock front leads to more material at high temperature.

\subsection{Mass-loss rates} The measured mass loss rates in the outflows from CTTS vary widely between objetcs objects and even for a single objects very different mass loss rates can be found, depending on the spectral tracers chosen and on the assumptions required to calculate mass loss rates from lines fluxes. One notoriously uncertain variable is the filling factor that describes which fraction of the observed volume is occupied by the hot gas. The innermost, fastest jet component is generally not resolved, so that this question cannot be answered observationally. Also, measurement of the mass loss in a jet are only possible once the outflow can be seen over the much brigther brighter emission of the central star. Typical mass loss rates found in the literature for CTTS outflows are in the range $10^{-8}-10^{-6}M_{\odot}\textrm{ yr}{-1}$ \citep{1999A&A...342..717B,2006A&A...456..189P}. In the specific case of the well-studied jet from DG~Tau \citet{1997A&A...327..671L} calculate the mass loss rate as $6.5\cdot 10^{-6}$~M$_{\odot}$~yr$^{-1}$; \citet{1995ApJ...452..736H} obtain $3\cdot 10^{-7}$~M$_{\odot}$~yr$^{-1}$ and a further out in the jet \citet{2000A&A...356L..41L} find $1.4\cdot 10^{-8}$~M$_{\odot}$~yr$^{-1}$. \citet{2009A&A...493..579G} show that only a small mass loss rate is required to explain the X-ray emission from the jet as shock heating, and it is possible that the optical jet further out entrains some disk wind material, so it might not track the stellar mass loss correctly. Therefore, we conservatively use $1.4\cdot 10^{-8}$~M$_{\odot}$~yr$^{-1}$, a value on the low end of the suggested mass loss rates, as fiducial stellar mass loss in the remainder of the article. Figure~\ref{fig:dot_m} showns shows how a larger mass loss rate and therefore a high density and ram presure i nthe pressure in the stellar wind pushes the shock front out to larger radii and larger heights. The different shape of the shock front also influences the post-shock temperatures. In the high mass loss rate scenario (black dotted line) the shock front reaches its maximum radius at 60~AU and most of the spherically symmetric wind passes the shock front at shallow angles, so this scneriao scenario has the highest fraction of low temperature material.

\section{To-Do} Explain why we do not look at cooling layer as in Koler Kohler et al: For us radiative cooling is important and that cannot be done with analytical solution. Mention that similar observations exisit exist for HD 163296

\section{The model} In this section we develop an analytical model for the interface between the stellar wind and the surrounding disk wind. The presure pressure of the disk wind collimates the stellar wind into a jet. The two flows are separated by a contact discontinuity, whose exact position is given by pressure equilibrium between the outer, disk wind component and the inner, stellar wind component. We assume that the stellar wind is initially emitted radially. As it encounters the contact discontinuity, the velocity component perpendicular to the discontinuity is shocked. Thus, our model needs to distinguish three zones: (i) the cold pre-shock stellar wind, (ii) the host post-shock stellar wind, and (iii) the disk wind. Our goal is to calculate the geometrical shape of the stellar wind shock, since this determines the velocity jump across the shock front and thus the temperature of the post-shock plasma. The Rankine-Hugoniot jump conditions relate the density, velocity and pressure on both sides of a strong shock. For ideal gases and non-oblique shocks the conservation of mass, momentum and energy across the shock can be written as follows \citep[][chap.~7, \S~15]{http://adsabs.harvard.edu/abs/1967pswh.book.....Z}, where the state before the front of the shock front is marked by the index 0, that behind the shock by index 1: \begin{eqnarray}

where $\rho$ denotes the total mass density of the gas and $P$ its pressure. Initially, the stellar wind is relatively cold and thus the thermodynamic pressure can be neglected, setting $P_0=0$. The shock front moves outward, until the pressure of the stellar wind equals the post-shock presure pressure and the contact discontinuity adjusts to equilize equalize the the post-shock presure pressure and the confining external pressure of the disk wind $P(z)$. In our case we are dealing with an oblique shock (Figure~\ref{fig:sketch}). Equations~\ref{RH1} to \ref{eqn:RH3} stay valid if only the velocity component perpendicular to the shock front is taken as $v_0$. Figure~\ref{fig:sketch} shows the geometry of the problem. We use a cylindrical coordinate system with an origin on the central star. We place the $z$-axis along the jet outflow direction and assume rotational symmetry around the jet axis. Thus, the flow can effectively be written in $(z,\omega)$.

\section{Constrains on parameters} In this section we discuss observational and theoretical constrains on boundary conditions and input values for the model, most notably $P(z)$, $\dot M$, $v_\infty$ and $\omega(z=0)$. We vary the paramters parameters individually to show how it affects the solution of the ODE.

\subsection{Starting point of integration} From a mathematical point of view, the starting point of the integration in the plane of the disk can be chosen freely anywhere between $\omega=0$ and $\omega=R_0(z=0)$. Figure~\ref{fig:omega_0} compares differnet different starting point under otherwise equal conditions. For small inital initial radii the ram presure pressure of the stellar wind pushes the shock surface out to larger radii in a comparatively small $\Delta z$. This leads to small pre-shock speeds in this region because the direction of the flow and the shock surface are almost parallel. This region also represents a large fraction of the total mass loss of the stellar wind, because it covers a large angle in the $(z,\omega)$-plane and a large solid angle of the spherical wind emission. Consequently, models with small values for $\omega_0$ show much less material that is heated up high temperatures. Physically, the position of the shock front is restricted by the position of the disk - the shock between the stellar wind and the disk material (in the disk itself or the disk wind) must occour occur within the inner hole of the disk. Fortunately, figure~\ref{fig:omega_0} shows that the two solutions for $\omega_0=0.01$~AU and $0.1$~AU are almost indistinguishable and the extact exact value for this parameter is not important as long as it is small. We use $\omega_0 = 0.01$~AU as the fiducial starting point for the integration.