<p>The question <a href="https://astronomy.stackexchange.com/q/20077/7982">Why are there no ISM clouds with temperatures between 100 and 6,000K?</a> has an accepted answer of the form:</p>

<blockquote>
  <p>Let $n$, $T$, and $x_i$ be the number density of hydrogen, the temperature of the gas, and $n_i/n$, where $n_i$ is the number density of the $i$th component of the interstellar medium. We can then write the criteria for thermal equilibrium as
  $$n^2\Lambda(n,T,x_i)-n\Gamma(n,T,x_i)\equiv n^2\mathcal{L}=0$$
  where $\Lambda$ and $\Gamma$ and the heating and cooling functions, respectively, and $\mathcal{L}$ is defined through those functions and $n$. If the equilibrium is unstable,
  $$\left(\frac{\partial\mathcal{L}}{\partial S}\right)&lt;0$$
  for entropy $S$.  </p>
  
  <p>This leads to different instability conditions, termed the <em>isochoric</em> and <em>isobaric</em> instabilities (<a href="http://adsabs.harvard.edu/cgi-bin/bib_query?1965ApJ...142..531F" rel="nofollow noreferrer">Field (1965)</a>, $\text{Eq } 4a,4b$). These can be determined from the temperature, pressure, and density of the gas (also assuming that the gas can be approximated as an ideal gas).</p>
</blockquote>

<p>which is not helping me <em>understand why there aren't any</em> beyond saying simply "because they are unstable."</p>

<p>I think the key is to understand what the "different instability conditions, termed the <em>isochoric</em> and <em>isobaric</em> instabilities" refer to - what those terms mean.</p>

<p>Is it possible to provide an approximate explanation of these two instability conditions in words that would not presuppose the reader has substantial depth in statistical physics? </p>

<hr>

<p>There may in fact be an explanation in <a href="https://physics.stackexchange.com/a/310190/83380">this answer</a>, but I need something unwound a bit further. Also possible help in the several answers to <a href="https://astronomy.stackexchange.com/q/15086/7982">this question</a>.</p>

<p><strong>edit:</strong> I've looked at <a href="http://w.astro.berkeley.edu/~ay216/05/NOTES/Lecture10.pdf" rel="nofollow noreferrer">this</a> and <a href="http://crescent.astro.illinois.edu/yhchu/ISM/Lectures/ISM_L14_Pogge.pdf" rel="nofollow noreferrer">this</a> as well, and I know there is a region when the line labeled "heating rate" has a negative slope, but I don't understand what this actually means. Is there something like a gravitational effect (triggered by density) or a phase change or chemical reaction that 'triggers' this instability? </p>

CC BY-SA 3.0
2017-02-17T08:32:31.050Z
The Physics stack exchange answer is your answer.
ProfRob
CC BY-SA 3.0
2017-02-17T09:16:12.653Z
@RobJeffries  I can't find any instance of either *isochoric* or *isobaric* there. I've also made an edit to the question.
uhoh
CC BY-SA 3.0
2017-02-17T14:54:17.120Z
I've edited my answer to explain a bit more what I mean by the terms. I'm sorry about that; I'm taking a thermal physics course and I'm used to automatically using them to describe processes.
HDE 226868
CC BY-SA 3.0
2017-02-17T15:19:21.420Z
@HDE226868 I know exactly what you mean. The additional explanation in the edit is helpful, and helps explain what the terms represent in this context. So I'm going to vote to close this question as a duplicate and point to the original answer. Thanks!
uhoh
<interstellar-medium><thermodynamics>
Explanation of the isochoric and isobaric instability conditions of interstellar medium gas clouds
uhoh
# Explanation of the isochoric and isobaric instability conditions of interstellar medium gas clouds

(Asked by: username_0 on 2017-02-17)

The question [Why are there no ISM clouds with temperatures between 100 and 6,000K?](https://astronomy.stackexchange.com/q/20077/7982) has an accepted answer of the form:

> Let $n$, $T$, and $x\_i$ be the number density of hydrogen, the temperature of the gas, and $n\_i/n$, where $n\_i$ is the number density of the $i$th component of the interstellar medium. We can then write the criteria for thermal equilibrium as
> $$n^2\Lambda(n,T,x\_i)-n\Gamma(n,T,x\_i)\equiv n^2\mathcal{L}=0$$
> where $\Lambda$ and $\Gamma$ and the heating and cooling functions, respectively, and $\mathcal{L}$ is defined through those functions and $n$. If the equilibrium is unstable,
> $$\left(\frac{\partial\mathcal{L}}{\partial S}\right)<0$$
> for entropy $S$.
>
> This leads to different instability conditions, termed the *isochoric* and *isobaric* instabilities ([Field (1965)](http://adsabs.harvard.edu/cgi-bin/bib_query?1965ApJ...142..531F), $\text{Eq } 4a,4b$). These can be determined from the temperature, pressure, and density of the gas (also assuming that the gas can be approximated as an ideal gas).

which is not helping me *understand why there aren't any* beyond saying simply "because they are unstable."

I think the key is to understand what the "different instability conditions, termed the *isochoric* and *isobaric* instabilities" refer to - what those terms mean.

Is it possible to provide an approximate explanation of these two instability conditions in words that would not presuppose the reader has substantial depth in statistical physics?

---

There may in fact be an explanation in [this answer](https://physics.stackexchange.com/a/310190/83380), but I need something unwound a bit further. Also possible help in the several answers to [this question](https://astronomy.stackexchange.com/q/15086/7982).

**edit:** I've looked at [this](http://w.astro.berkeley.edu/~ay216/05/NOTES/Lecture10.pdf) and [this](http://crescent.astro.illinois.edu/yhchu/ISM/Lectures/ISM_L14_Pogge.pdf) as well, and I know there is a region when the line labeled "heating rate" has a negative slope, but I don't understand what this actually means. Is there something like a gravitational effect (triggered by density) or a phase change or chemical reaction that 'triggers' this instability?

**Tags:** interstellar-medium, thermodynamics

> **Comments:**
> **username_1**: The Physics stack exchange answer is your answer.  
> **username_0**: @RobJeffries  I can't find any instance of either *isochoric* or *isobaric* there. I've also made an edit to the question.  
> **username_2**: I've edited my answer to explain a bit more what I mean by the terms. I'm sorry about that; I'm taking a thermal physics course and I'm used to automatically using them to describe processes.  
> **username_0**: @username_2 I know exactly what you mean. The additional explanation in the edit is helpful, and helps explain what the terms represent in this context. So I'm going to vote to close this question as a duplicate and point to the original answer. Thanks!  

---

## 0 answers

