diff --git a/theory/SPH/Flavours/sph_flavours.tex b/theory/SPH/Flavours/sph_flavours.tex index 5fe1277373552d60607671299437a371e068169c..3c80fefb4989505b76cfaf8b38676ac4276b8da8 100644 --- a/theory/SPH/Flavours/sph_flavours.tex +++ b/theory/SPH/Flavours/sph_flavours.tex @@ -36,8 +36,9 @@ and the derivative of its density with respect to $h$: \rho_{\partial h_i} \equiv \dd{\rho}{h}(\vec{x}_i) = \sum_j m_j \dd{W}{h}(\vec{x}_{ij} , h_i). \end{equation} -The gradient terms (``h-terms'') can then be computed from the density -and its derivative: +This corresponds to $x_i = \tilde{x}_i = m_i$, and $y_i =\tilde{y}_i = \rho_i$ +in the \citet{hopkins2013} formalism. The gradient terms (``h-terms'') can +then be computed from the density and its derivative: \begin{equation} f_i \equiv \left(1 + \frac{h_i}{3\rho_i}\rho_{\partial h_i} @@ -300,15 +301,17 @@ smoothing length using: \bar P_{\partial h_i} \equiv \dd{\bar{P}}{h}(\vec{x}_i) = \sum_j m_j \tilde{A_j} \dd{W}{h}(\vec{x}_{ij}), \label{eq:sph:pe:P_dh} \end{equation} +This corresponds to $x_i = m_i \tilde{A}_i$, $\tilde{x}_i = m_i$, $y_i = +\bar{P}_i$, and $\tilde{y}_i = \rho_i$ in the \citet{hopkins2013} formalism. The gradient terms (``h-terms'') are then obtained by combining $\bar -P_{\partial h_i}$ and $\rho_{\partial h_i}$ -(eq. \ref{eq:sph:minimal:rho_dh}): +P_{\partial h_i}$ and $\rho_{\partial h_i}$ (eq. \ref{eq:sph:minimal:rho_dh}): -\begin{equation} - f_i \equiv \left(\frac{h_i}{3\rho_i}\bar P_{\partial +\begin{align} + f_{ij} = & ~ 1 - \tilde{A}_j^{-1} f_i \nonumber \\ + f_i \equiv & \left(\frac{h_i}{3\rho_i}\bar P_{\partial h_i}\right)\left(1 + \frac{h_i}{3\rho_i}\rho_{\partial h_i}\right)^{-1}. -\end{equation} +\end{align} \subsubsection{Hydrodynamical accelerations (\nth{2} neighbour loop)}