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title	author	email	titleshort	authorshort	processDate	linkDir	appendix	abstractTex	keywords	acknowledgements	include-headers
Paperlighter Template Implementation Example	{number 1} {name Author One} {correspond true} {affiliation My City University} {address Orenomachi, Orenoshi, Orenoken, Japan} {email one@myuni.ac.jp}] [{number 2} {name Author Two} {affiliation My Other City University} {address Hokanomachi, Orenoshi, Orenoken, Japan} {email two@myuni.ac.jp}	xxx@myuni.ac.jp	Paperlighter Example	Author One et.al.	[{received March 11, 2022} {accepted April 11, 2022}]	[Figure Output]	[appendix/1 appendix/2]	\abstract{To investigate the physical nature of the 'nuc-leated instability' of proto giant planets, the stability of layers in static, radiative gas spheres is analysed on the basis of Baker's standard one-zone model.} {To investigate the physical nature of the 'nuc-leated instability' of proto giant planets, the stability of layers in static, radiative gas spheres is analysed on the basis of Baker's standard one-zone model.} {It is shown that stability depends only upon the equations of state, the opacities and the local thermodynamic state in the layer. Stability and instability can therefore be expressed in the form of stability equations of state which are universal for a given composition.} {The stability equations of state are calculated for solar composition and are displayed in the domain $-14 \leq \lg \rho / \mathrm{[g, cm^{-3}]} \leq 0 $, $ 8.8 \leq \lg e / \mathrm{[erg, g^{-1}]} \leq 17.7$. These displays may be used to determine the one-zone stability of layers in stellar or planetary structure models by directly reading off the value of the stability equations for the thermodynamic state of these layers, specified by state quantities as density $\rho$, temperature $T$ or specific internal energy $e$. Regions of instability in the $(\rho,e)$-plane are described and related to the underlying microphysical processes.} {Vibrational instability is found to be a common phenomenon at temperatures lower than the second He ionisation zone. The $\kappa$-mechanism is widespread under 'cool' conditions.}	giant planet formation -- $\kappa$-mechanism -- stability of gas spheres	Part of this work was supported by the German Deutsche Forschungsgemeinschaft, DFG project number Ts 17/2--1.	\newcommand\mC[1]{\multicolumn{1}{c}{#1}} \newcommand\latex{\LaTeX\xspace } \newcommand\v[1]{\verb-#1-}

Introduction

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 the critical mass of the core to be about $12 ,M_\oplus$ ($M_\oplus=5.975 \times 10^{27},\mathrm{g}$ is the Earth mass), which is independent of the outer boundary conditions and therefore independent of the location in the solar nebula. This critical value for the core mass corresponds closely to the cores of today's giant planets.

Although no hydrodynamical study has been available many workers conjectured that a collapse or rapid contraction will ensue after accumulating the critical mass. The main motivation for this article is to investigate the stability of the static envelope at the critical mass. With this aim the local, linear stability of static radiative gas spheres is investigated on the basis of Baker's (\citeyear{mitchell80}) standard one-zone model.

Phenomena similar to the ones described above for giant planet formation have been found in hydrodynamical models concerning star formation where protostellar cores explode (Tscharnuter \citeyear{kearns89}, Balluch \citeyear{MachineLearningI}), whereas earlier studies found quasi-steady collapse flows. The similarities in the (micro)physics, i.e., constitutive relations of protostellar cores and protogiant planets serve as a further motivation for this study.

Markdown

pubsEngine use Markdown text-file as an input to be processed into \latex document. Markdown is a very simplified text markup language. This section would shows several common syntax of Markdown. More detailed syntax can be accessed in the Markdown cheatsheet.

Headings

We use \v{#} on various levels. Headings should be shown as a line that prefixed by \v{#}. (Ex. \v{# The First Level Heading}). Please include space after \v{#}.

Paragraph

blank line stands as a paragraph separator. No indentation for the first line (it only required in multilined lists to show that the next line is part of the item if it is non-blank and indented).

Italic

\v{*} Ex. \v{this text is italic} : this text is italic

Bold

\v{**} Ex. \v{this text is bold} : this text is bold

Bold and Italic

\v{***} Ex. \v{this text is bold and italic} this text is bold and italic

Verbatim

Verbatim should use backticks \v{text}, Ex. this is `the text` this is the text. To include backticks inside the text, use multiple bacticks.

``this is `the text` multiple``

CodeBlocks

Multiple verbatim lines in Markdown can be expressed as a CodeBlock. This particular block is considered as special, due to its popular use in verbatim representation of source-code in the manuscript. Technically, a CodeBlock is a range of lines that surrounded by triple tilde ~~~, for example:

~~~
main :: IO ()
main = putStrLn "we are one"
~~~

will produce:

main :: IO ()
main = putStrLn "we are one"

The abuse of CodeBlocks for another purpose will be discussed in another section (Enhancement by pubsEngine).

Mathematical Equations

By default, we may use $$ environment for mathematical formula. Nonetheless, actually, we can use any \latex scenario, such as:

$$ $$ is equal with \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}

Equations as independent paragraph equation is not numbered, as below.

$$ \nabla \cdot \vec{W} = \sigma W \nonumber $$

And this is another example for inline equation, such as: $$y = 5\cdot x^2$$ You can see that inline equation have automatically numbered.

Tables

Table creation will be described in another section (Enhancement by pubsEngine).

Images

![the caption](Figure/icml_numpapers.eps)
{#fig:FigVibStab width=3.43in height=2.71in}

will produce Figure \ref{fig:FigVibStab}. Make sure that you calculated by yourself the appropriate width and height for the image to fit in a column. Full width image and automatic sized image for single-column will be described at another section (Enhancement by pubsEngine).

{#fig:FigVibStab width=3.43in height=2.71in}

Lists

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Ordered List

For ordered list, we just use numbers. The block should be as an independent paragraph (blank line above and under the block).

1. first in line
2. second
3. and the third

will produce

first in line
second
and the third

Unordered List

For unordered list, use dash sign (-). We also can use asterisk and plus signs ( * +)

- first in line
- second
  should multiline
- and the third

will produce

first in line
second should multiline
and the third

Links

We use [Duck Duck Go](https://duckduckgo.com).

produces: We use Duck Duck Go.

Email link

<myemail@myurl.org>

produces email link:

myemail@myurl.org

Enhancement by pubsEngine

The enhancements provided by pubsEngine spans on various aspects. Most parts of this are expressed in terms of CodeBlocks.

Table

pubsEngine provides its table capability outside the defaults available in Pandoc. Under the table, pubsEngine will use MultiMarkdown to process its table. We can create tables using the following template:

~~~{.multiTable}
|              |          Grouping           ||
|First Header  | Second Header | Third Header |
| ------------ | :-----------: | -----------: |
|Content       |          *Long Cell*        ||
|Content       |   **Cell**    |         Cell |
|New section   |     More      |         Data |
|And more      | With an escaped '\|'        ||
[More complicated table caption.]
~~~

|              |          Grouping           ||
|First Header  | Second Header | Third Header |
| ------------ | :-----------: | -----------: |
|Content       |          *Long Cell*        ||
|Content       |   **Cell**    |         Cell |
|New section   |     More      |         Data |
|And more      | With an escaped '\|'        ||
[More complicated table can be done using multimarkdown in .multiTable Code Block. You have to use this format for all table as default.]

Currently, we cannot create full width two-columns table automatically using above default syntax. For the moment, we should use a complete \latex syntax to fulfill this. The detailed syntax for this purpose will be shown in the Appendix.

Include .md Files

We can create a rather decent categorical separation of our documents by dividing the .md files and gather them using includes.

~~~include
include/addition1.md
~~~

include/addition1.md

Above paragraph was included from include/addition1.md.

Incorporating Diagram

We can include a diagram script, following an Embedded domain-specific Language from Haskell diagrams. We can see the implementation of following CodeBlock as Figure \ref{fig:dia1}.

~~~{#fig:dia1 .diagram}
let x = circle 10
 in x
~~~

let x = circle 10
 in x

Baker's standard one-zone model

\begin{figure*} \centering \includegraphics{Figure/icml_numpapers.eps} \caption{Adiabatic exponent $\Gamma_1$. $\Gamma_1$ is plotted as a function of $\lg$ internal energy $\mathrm{[erg,g^{-1}]}$ and $\lg$ density $\mathrm{[g,cm^{-3}]}$.} \label{fig:FigGam}% \end{figure*}

In this section the one-zone model of \citet{DudaHart2nd}, originally used to study the Cephe\text{"{\i}}d pulsation mechanism, will be briefly reviewed. The resulting stability criteria will be rewritten in terms of local state variables, local timescales and constitutive relations.

\citet{DudaHart2nd} investigates the stability of thin layers in self-gravitating, spherical gas clouds with the following properties:

* hydrostatic equilibrium,
* thermal equilibrium,
* energy transport by grey radiation diffusion.

non-numbered list can be written as above, and shown as:

hydrostatic equilibrium,
thermal equilibrium,
energy transport by grey radiation diffusion.

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 notation:

\noindent 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 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}}} },, \end{equation} Baker's $K$ and $\sigma_0$ have the following form: \begin{eqnarray} \sigma_0 & = & \frac{\pi}{\sqrt{8}} \frac{1}{ \tau_{\mathrm{ff}}} \ K & = & \frac{\sqrt{32}}{\pi} \frac{1}{\delta} \frac{ \tau_{\mathrm{ff}} } { \tau_{\mathrm{co}} },; \end{eqnarray} where $E_{\mathrm{th}} \approx m (P_0/{\rho_0})$ has been used and \begin{equation} \begin{array}{l} \delta = - \left( \frac{ \partial \ln \rho }{ \partial \ln T } \right)P \ e=mc^2 \end{array} \end{equation} is a thermodynamical quantity which is of order $1$ and equal to $1$ for nonreacting mixtures of classical perfect gases. The physical meaning of $\sigma_0$ and $K$ is clearly visible in the equations above. $\sigma_0$ represents a frequency of the order one per free-fall time. $K$ is proportional to the ratio of the free-fall time and the cooling time. Substituting into Baker's criteria, using thermodynamic identities and definitions of thermodynamic quantities, \begin{displaymath} \Gamma_1 = \left( \frac{ \partial \ln P}{ \partial\ln \rho} \right){S} , , ; \chi^{}\rho = \left( \frac{ \partial \ln P}{ \partial\ln \rho} \right){T} , , ; \kappa^{}{P} = \left( \frac{ \partial \ln \kappa}{ \partial\ln P} \right){T} \end{displaymath} \begin{displaymath} \nabla_{\mathrm{ad}} = \left( \frac{ \partial \ln T} { \partial\ln P} \right){S} , , ; \chi^{}T = \left( \frac{ \partial \ln P} { \partial\ln T} \right){\rho} , , ; \kappa^{}{T} = \left( \frac{ \partial \ln \kappa} { \partial\ln T} \right){T} \end{displaymath} one obtains, after some pages of algebra, the conditions for stability given below: \begin{eqnarray} \frac{\pi^2}{8} \frac{1}{\tau{\mathrm{ff}}^2} ( 3 \Gamma_1 - 4 ) & > & 0 \label{ZSDynSta} \ \frac{\pi^2}{\tau_{\mathrm{co}} \tau_{\mathrm{ff}}^2} \Gamma_1 \nabla_{\mathrm{ad}} \left[ \frac{ 1- 3/4 \chi^{}\rho }{ \chi^{}T } ( \kappa^{}T - 4 ) + \kappa^{}P + 1 \right] & > & 0 \label{ZSSecSta} \ \frac{\pi^2}{4} \frac{3}{\tau{ \mathrm{co} } \tau{ \mathrm{ff} }^2 } \Gamma_1^2 , \nabla{\mathrm{ad}} \left[ 4 \nabla{\mathrm{ad}} - ( \nabla_{\mathrm{ad}} \kappa^{}_T + \kappa^{}_P ) - \frac{4}{3 \Gamma_1} \right] & > & 0 \label{ZSVibSta} \end{eqnarray}

For a physical discussion of the stability criteria see \citet{DudaHart2nd} or \citet{anonymous}.

We observe that these criteria for dynamical, secular and vibrational stability, respectively, can be factorized into

a factor containing local timescales only,
a factor containing only constitutive relations and their derivatives.
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.

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 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 \begin{equation} \Gamma_1 > \frac{4}{3},\cdot \end{equation} Stability of the thermodynamical equilibrium demands \begin{equation} \chi^{}\rho > 0, ;; c_v > 0, , \end{equation} and \begin{equation} \chi^{}T > 0 \end{equation} holds for a wide range of physical situations. With \begin{eqnarray} \Gamma_3 - 1 = \frac{P}{\rho T} \frac{\chi^{}T}{c_v}&>&0\ \Gamma_1 = \chi\rho^{} + \chi_T^{} (\Gamma_3 -1)&>&0\ \nabla{\mathrm{ad}} = \frac{\Gamma_3 - 1}{\Gamma_1} &>&0 \end{eqnarray} 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 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 ) + \kappa^{}P + 1 =: S{\mathrm{sec}} > & 0 & \label{SecSta} \ 4 \nabla{\mathrm{ad}} - (\nabla{\mathrm{ad}} \kappa^{}T + \kappa^{}P) - \frac{4}{3 \Gamma_1} =: S{\mathrm{vib}} > & 0,.& \label{VibSta} \end{eqnarray} The constitutive relations are to be evaluated for the unperturbed thermodynamic state (say $(\rho_0, T_0)$) of the zone. We see that the one-zone stability of the layer depends only on the constitutive relations $\Gamma_1$, $\nabla{\mathrm{ad}}$, $\chi_T^{},,\chi\rho^{}$, $\kappa_P^{},,\kappa_T^{}$. These depend only on the unperturbed thermodynamical state of the layer. Therefore the above relations define the one-zone-stability equations of state $S_{\mathrm{dyn}},,S_{\mathrm{sec}}$ and $S_{\mathrm{vib}}$. See == [@fig:FigVibStab] == Fig. \ref{fig:FigVibStab} for a picture of $S_{\mathrm{vib}}$. Regions of secular instability are listed in Table 1.

$Vibrational stability equation of state $S_{\mathrm{vib}}(\lg e, \lg \rho)$. $>0$ means vibrational stability.$ {#fig:FigVibStab width=3.43in height=2.71in}

Conclusions

The conditions for the stability of static, radiative layers in gas spheres, as described by Baker's (\citeyear{DudaHart2nd}) standard one-zone model, can be expressed as stability equations of state. These stability equations of state depend only on the local thermodynamic state of the layer.
If the constitutive relations -- equations of state and Rosseland mean opacities -- are specified, the stability equations of state can be evaluated without specifying properties of the layer.
For solar composition gas the $\kappa$-mechanism is working in the regions of the ice and dust features in the opacities, the $\mathrm{H}_2$ dissociation and the combined H, first He ionization zone, as 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\text{"\i}d pulsations.

21 KiB Raw Blame History