list of equation other than $ $

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Hasan al Rasyid 3 years ago
parent 23733c0984
commit c8f9322f80
  1. 94
      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
1. a factor containing local timescales only,
2. 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.
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.
<!--
@ -318,11 +345,10 @@ Lp. & Miejscowość
We will now write down the sign (and therefore stability)
determining parts of the left-hand sides of the inequalities
(\ref{ZSDynSta}), (\ref{ZSSecSta}) and (\ref{ZSVibSta}) and thereby
obtain \emph{stability equations of state}.
obtain *stability equations of state*.
The sign determining part of inequality (\ref{ZSDynSta}) is
$3\Gamma_1 - 4$ and it reduces to the
criterion for dynamical stability
$3\Gamma_1 - 4$ and it reduces to the criterion for dynamical stability
\begin{equation}
\Gamma_1 > \frac{4}{3}\,\cdot
\end{equation}
@ -344,7 +370,7 @@ Lp. & Miejscowość
we find the sign determining terms in inequalities (\ref{ZSSecSta})
and (\ref{ZSVibSta}) respectively and obtain the following form
of the criteria for dynamical, secular and vibrational
\emph{stability}, respectively:
*stability*, respectively:
\begin{eqnarray}
3 \Gamma_1 - 4 =: S_{\mathrm{dyn}} > & 0 & \label{DynSta} \\
\frac{ 1- 3/4 \chi^{}_\rho }{ \chi^{}_T } ( \kappa^{}_T - 4 )
@ -403,7 +429,7 @@ create label for \ref{FigVibStab} using #FigVibStab
indicated by vibrational instability. These regions
of instability are much larger in extent and degree of
instability than the second He ionization zone
that drives the Cephe{\"\i}d pulsations.
that drives the Cephe\text{\"\i}d pulsations.
\end{enumerate}

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