<p>What you want to do is use the relation between Mass, Radius, and Density. The proper expression should be</p>

<p>$$\rho_c = \frac{M_c}{(4/3) \pi R_c^3}$$</p>

<p>Your three Jeans conditions are $M_c &gt; M_J$, $R_c &gt; R_J$, and $\rho_c &gt; \rho_j$. By using the relation above, you can transform any of the three conditions into one of the others.</p>

<p>For example, going from $M_c &gt; M_J \rightarrow R_c &gt; R_J$ is done as follows.</p>

<p>$$M_c = \frac{4}{3}\pi R_c^3 \rho_c &gt; (\frac{5kT}{G \mu m_H})^{3/2} (\frac{3}{4 \pi \rho_c}) ^{1/2}$$</p>

<p>Now, solving for $R_c$ gives</p>

<p>$$R_c^3 &gt; (\frac{5kT}{G \mu m_H})^{3/2} (\frac{3}{4 \pi \rho_c}) ^{3/2}$$</p>

<p>$$R_c &gt; (\frac{15kT}{4 \pi G \mu m_H \rho_c})^{1/2} \equiv R_J$$</p>

<p>The other transformations follow in a similar manner.</p>

CC BY-SA 3.0
2016-06-15T16:29:23.430Z
Thank you zephyr for your very well presented answer! Now I understand. Thank you so much!
DeltaCentauri
CC BY-SA 3.0
2016-06-13T18:14:47.910Z
2016-06-13T18:14:47.910Z
zephyr
<p>The gravitational collapse of a gas cloud can be described by the Jeans Criterion for mass, radius and density of the gas cloud, which is (c stands for cloud): </p>

<p>$$M_J = (\frac{5kT}{G \mu m_H})^{3/2} (\frac{3}{4 \pi \rho_c}) ^{1/2} $$</p>

<p>$$R_J = (\frac{15kT}{4 \pi G \mu m_H \rho_c})^{1/2} $$</p>

<p>$$\rho_J = (\frac{3}{4 \pi M_c^2}) (\frac{5kT}{G \mu m_H}) ^{3}$$</p>

<p>I wanted to show that they are equivalent and simply used </p>

<p>$$\rho = \frac{m}{V} $$</p>

<p>on the criteria above an rearranged the equation. </p>

<p><strong>However this doesn't work. What am I doing wrong?</strong> </p>

<p>Thank you for your help! </p>

CC BY-SA 3.0
2016-06-13T07:50:06.583Z
The Jeans mass is certainly the mass inside a sphere of radius half the Jeans length. $M_c$ only appears in one equation, so it cannot be derived from the other two?
ProfRob
<star><star-formation><stellar-astrophysics><gas>
How to show that the Jeans Criterion for Mass, Radius and Density are equivalent?
DeltaCentauri
# How to show that the Jeans Criterion for Mass, Radius and Density are equivalent?

(Asked by: username_0 on 2016-06-12)

The gravitational collapse of a gas cloud can be described by the Jeans Criterion for mass, radius and density of the gas cloud, which is (c stands for cloud):

$$M\_J = (\frac{5kT}{G \mu m\_H})^{3/2} (\frac{3}{4 \pi \rho\_c}) ^{1/2} $$

$$R\_J = (\frac{15kT}{4 \pi G \mu m\_H \rho\_c})^{1/2} $$

$$\rho\_J = (\frac{3}{4 \pi M\_c^2}) (\frac{5kT}{G \mu m\_H}) ^{3}$$

I wanted to show that they are equivalent and simply used

$$\rho = \frac{m}{V} $$

on the criteria above an rearranged the equation.

**However this doesn't work. What am I doing wrong?**

Thank you for your help!

**Tags:** star, star-formation, stellar-astrophysics, gas
> **Comments:**
> **username_1**: The Jeans mass is certainly the mass inside a sphere of radius half the Jeans length. $M_c$ only appears in one equation, so it cannot be derived from the other two?  

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## 1 answers

### Answer by username_2 on 2016-06-13. Score: 1 | Accepted answer

What you want to do is use the relation between Mass, Radius, and Density. The proper expression should be

$$\rho\_c = \frac{M\_c}{(4/3) \pi R\_c^3}$$

Your three Jeans conditions are $M\_c > M\_J$, $R\_c > R\_J$, and $\rho\_c > \rho\_j$. By using the relation above, you can transform any of the three conditions into one of the others.

For example, going from $M\_c > M\_J \rightarrow R\_c > R\_J$ is done as follows.

$$M\_c = \frac{4}{3}\pi R\_c^3 \rho\_c > (\frac{5kT}{G \mu m\_H})^{3/2} (\frac{3}{4 \pi \rho\_c}) ^{1/2}$$

Now, solving for $R\_c$ gives

$$R\_c^3 > (\frac{5kT}{G \mu m\_H})^{3/2} (\frac{3}{4 \pi \rho\_c}) ^{3/2}$$

$$R\_c > (\frac{15kT}{4 \pi G \mu m\_H \rho\_c})^{1/2} \equiv R\_J$$

The other transformations follow in a similar manner.

> **Comments:**
> **username_0**: Thank you zephyr for your very well presented answer! Now I understand. Thank you so much!  

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