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@ -74,11 +74,8 @@ similarities in the (micro)physics, i.e., constitutive relations of |
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protostellar cores and protogiant planets serve as a further |
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protostellar cores and protogiant planets serve as a further |
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motivation for this study. |
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motivation for this study. |
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# Baker's standard one-zone model |
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# Baker's standard one-zone model |
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<!-- we need figure* for make it full screen, so it must be in latex code as follows |
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<!-- we need figure* for make it full screen, so it must be in latex code as follows |
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--> |
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--> |
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@ -102,6 +99,14 @@ constitutive relations. |
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self-gravitating, |
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self-gravitating, |
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spherical gas clouds with the following properties: |
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spherical gas clouds with the following properties: |
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~~~ |
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* hydrostatic equilibrium, |
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* thermal equilibrium, |
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* energy transport by grey radiation diffusion. |
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~~~ |
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non-numbered list can be written as above, and shown as: |
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* hydrostatic equilibrium, |
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* hydrostatic equilibrium, |
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* thermal equilibrium, |
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* thermal equilibrium, |
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* energy transport by grey radiation diffusion. |
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* energy transport by grey radiation diffusion. |
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@ -109,12 +114,11 @@ spherical gas clouds with the following properties: |
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For equations, we can use several environment: |
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For equations, we can use several environment: |
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~~~ |
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~~~ |
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\begin{equation} |
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$$ $$ is equal with \begin{equation} |
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\begin{eqnarray} |
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\begin{eqnarray} |
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\begin{array} |
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\begin{array} |
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\begin{displaymath} |
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\begin{displaymath} |
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\begin{align} |
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\begin{align} |
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~~~ |
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~~~ |
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This is an example: |
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This is an example: |
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@ -133,6 +137,16 @@ This is an example: |
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\nabla \times \vec{E} &= -\frac{\vec{B}}{t} |
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\nabla \times \vec{E} &= -\frac{\vec{B}}{t} |
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\end{align} |
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\end{align} |
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And this is another example for inline equation, such as: $$y = 5\cdot x^2$$ |
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You can see that inline equation have automatically numbered. |
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Independent paragraph equation is not numbered, as below. |
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$$ |
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\nabla \cdot \vec{W} = \sigma W \nonumber |
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$$ |
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The block should be as an independent paragraph (blank line above and under the block). |
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For the one-zone-model Baker obtains necessary conditions |
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For the one-zone-model Baker obtains necessary conditions |
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for dynamical, secular and vibrational (or pulsational) |
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for dynamical, secular and vibrational (or pulsational) |
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stability (Eqs. (34a,b,c) in Baker \citeyear{DudaHart2nd}). Using Baker's |
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stability (Eqs. (34a,b,c) in Baker \citeyear{DudaHart2nd}). Using Baker's |
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@ -412,25 +426,23 @@ create label for \ref{FigVibStab} using #FigVibStab |
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# Conclusions |
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# Conclusions |
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\begin{enumerate} |
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1. The conditions for the stability of static, radiative |
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\item The conditions for the stability of static, radiative |
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layers in gas spheres, as described by Baker's (\citeyear{DudaHart2nd}) |
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layers in gas spheres, as described by Baker's (\citeyear{DudaHart2nd}) |
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standard one-zone model, can be expressed as stability |
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standard one-zone model, can be expressed as stability |
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equations of state. These stability equations of state depend |
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equations of state. These stability equations of state depend |
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only on the local thermodynamic state of the layer. |
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only on the local thermodynamic state of the layer. |
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2. If the constitutive relations -- equations of state and |
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\item If the constitutive relations -- equations of state and |
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Rosseland mean opacities -- are specified, the stability |
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Rosseland mean opacities -- are specified, the stability |
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equations of state can be evaluated without specifying |
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equations of state can be evaluated without specifying |
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properties of the layer. |
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properties of the layer. |
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3. For solar composition gas the $\kappa$-mechanism is |
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\item For solar composition gas the $\kappa$-mechanism is |
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working in the regions of the ice and dust features |
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working in the regions of the ice and dust features |
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in the opacities, the $\mathrm{H}_2$ dissociation and the |
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in the opacities, the $\mathrm{H}_2$ dissociation and the |
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combined H, first He ionization zone, as |
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combined H, first He ionization zone, as |
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indicated by vibrational instability. These regions |
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indicated by vibrational instability. These regions |
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of instability are much larger in extent and degree of |
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of instability are much larger in extent and degree of |
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instability than the second He ionization zone |
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instability than the second He ionization zone |
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that drives the Cephe\text{\"\i}d pulsations. |
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that drives the Cephe\text{\"\i}d pulsations. |
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\end{enumerate} |
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