From c8f9322f80d4e9756367f04db8aa095a9e650bca Mon Sep 17 00:00:00 2001 From: Hasan al Rasyid Date: Mon, 14 Mar 2022 06:53:20 +0900 Subject: [PATCH] list of equation other than $ $ --- manuscript.md | 102 +++++++++++++++++++++++++++++++------------------- 1 file changed, 64 insertions(+), 38 deletions(-) diff --git a/manuscript.md b/manuscript.md index c530997..ec496f6 100644 --- a/manuscript.md +++ b/manuscript.md @@ -45,7 +45,7 @@ include-headers: | # Introduction -In the \emph{nucleated instability\/} (also called core +In the *nucleated instability* (also called core instability) hypothesis of giant planet formation, a critical mass for static core envelope protoplanets has been found. \citet{langley00} determined @@ -106,18 +106,45 @@ spherical gas clouds with the following properties: * thermal equilibrium, * energy transport by grey radiation diffusion. +For equations, we can use several environment: + +~~~ +\begin{equation} +\begin{eqnarray} +\begin{array} +\begin{displaymath} +\begin{align} + +~~~ + +This is an example: + +~~~ +\begin{align} +\nabla \cdot \vec{E} &= \rho \nonumber \\ +\nabla \cdot \vec{B} &= 0 \nonumber \\ +\nabla \times \vec{E} &= -\frac{\vec{B}}{t} +\end{align} +~~~ + +\begin{align} +\nabla \cdot \vec{E} &= \rho \nonumber \\ +\nabla \cdot \vec{B} &= 0 \nonumber \\ +\nabla \times \vec{E} &= -\frac{\vec{B}}{t} +\end{align} + For the one-zone-model Baker obtains necessary conditions for dynamical, secular and vibrational (or pulsational) -stability (Eqs.\ (34a,\,b,\,c) in Baker \citeyear{DudaHart2nd}). Using Baker's +stability (Eqs. (34a,b,c) in Baker \citeyear{DudaHart2nd}). Using Baker's notation: \noindent - and with the definitions of the \emph{local cooling time\/} + and with the definitions of the *local cooling time* (see == [@fig:FigGam] == Fig. \ref{fig:FigGam}) \begin{equation} \tau_{\mathrm{co}} = \frac{E_{\mathrm{th}}}{L_{r0}} \,, \end{equation} - and the \emph{local free-fall time} + and the *local free-fall time* \begin{equation} \tau_{\mathrm{ff}} = \sqrt{ \frac{3 \pi}{32 G} \frac{4\pi r_0^3}{3 M_{\mathrm{r}}} @@ -164,7 +191,7 @@ notation: { \partial\ln T} \right)_{T} \end{displaymath} one obtains, after some pages of algebra, the conditions for - \emph{stability\/} given + *stability* given below: \begin{eqnarray} \frac{\pi^2}{8} \frac{1}{\tau_{\mathrm{ff}}^2} @@ -195,31 +222,31 @@ notation: We observe that these criteria for dynamical, secular and vibrational stability, respectively, can be factorized into - \begin{enumerate} - \item a factor containing local timescales only, - \item a factor containing only constitutive relations and - their derivatives. - \end{enumerate} - The first factors, depending on only timescales, are positive - by definition. The signs of the left hand sides of the - inequalities (\ref{ZSDynSta}), (\ref{ZSSecSta}) and (\ref{ZSVibSta}) - therefore depend exclusively on the second factors containing - the constitutive relations. Since they depend only - on state variables, the stability criteria themselves are \emph{ - functions of the thermodynamic state in the local zone}. The - one-zone stability can therefore be determined - from a simple equation of state, given for example, as a function - of density and - temperature. Once the microphysics, i.e.\ the thermodynamics - and opacities (see Table \ref{KapSou}), are specified (in practice - by specifying a chemical composition) the one-zone stability can - be inferred if the thermodynamic state is specified. - The zone -- or in - other words the layer -- will be stable or unstable in - whatever object it is imbedded as long as it satisfies the - one-zone-model assumptions. Only the specific growth rates - (depending upon the time scales) will be different for layers - in different objects. + +1. a factor containing local timescales only, +2. a factor containing only constitutive relations and + their derivatives. +3. To make a numered list, make sure that: + 1. it stands on its own paragraph (blank line above and below the list), + 2. the number starts at first column on each line. + +The first factors, depending on only timescales, are positive +by definition. The signs of the left hand sides of the +inequalities (\ref{ZSDynSta}), (\ref{ZSSecSta}) and (\ref{ZSVibSta}) +therefore depend exclusively on the second factors containing +the constitutive relations. Since they depend only +on state variables, the stability criteria themselves are *functions of the thermodynamic state in the local zone*. The +one-zone stability can therefore be determined +from a simple equation of state, given for example, as a function +of density and temperature. Once the microphysics, i.e. the thermodynamics +and opacities (see Table \ref{KapSou}), are specified (in practice +by specifying a chemical composition) the one-zone stability can +be inferred if the thermodynamic state is specified. +The zone -- or in other words the layer -- will be stable or unstable in +whatever object it is imbedded as long as it satisfies the +one-zone-model assumptions. Only the specific growth rates +(depending upon the time scales) will be different for layers +in different objects.