Pressure gradient in Jeans Lenght

In summary, the conversation is about the demonstration of Jeans instability, which starts with the inequality \rho g \ge \nabla P and then substitutes \nabla P with nkT/R, using the ideal gas law PV = nkT. However, the speaker is confused about the gradient of P, thinking it should be nkT/R^4 instead of nkT/R. This is causing a problem in their understanding of the demonstration. The speaker acknowledges their poor English and notes that \rho g \ge \nabla P is a sufficient condition, not a necessary one.
  • #1
Daaavde
30
0
All the demonstrations on Jeans instability start with:

[itex]\rho g \ge \nabla P[/itex]

Then they substitute [itex]\nabla P[/itex] with [itex]nkT/R[/itex].

But from ideal gas law: [itex]PV = nkT[/itex], so [itex]P = nkT/V = nkT/R^3[/itex]
(I'm not interested in proportionality factors, so let's not bother about [itex]4/3 \pi[/itex], ecc...)

Now, gradient of [itex]P[/itex], to me, means to derivate with respect to [itex]R[/itex].
Hence:

[itex]\nabla P = nkT/R^4[/itex] (again, not interested in proportionality factors)

So, here's the problem:

On demonstrations: [itex]\nabla P = nkT/R[/itex]
To me: [itex]\nabla P = nkT/R^4[/itex]

Obviously, it's plain I'm missing something, still i can't see what it is.

(I apologize for my poor scholastic english, I'm not native speaker)
 
Last edited:
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  • #2
Thank you for your help. P.S.: I'm aware of the fact that \rho g \ge \nabla P is only a sufficient condition and not a necessary one, but this is not the question here.
 

Related to Pressure gradient in Jeans Lenght

1. What is the pressure gradient in Jeans length?

The pressure gradient in Jeans length is a measurement of the change in pressure over a certain distance. It is specifically used in astrophysics to describe the change in pressure within a gas cloud that is undergoing gravitational collapse.

2. How is the pressure gradient in Jeans length calculated?

The pressure gradient in Jeans length can be calculated using the Jeans length formula, which takes into account the density, temperature, and mass of the gas cloud. It can also be calculated using the virial theorem, which relates the kinetic and potential energies of a system in equilibrium.

3. What is the significance of the pressure gradient in Jeans length?

The pressure gradient in Jeans length is significant in understanding the process of star formation. It determines whether a gas cloud is stable or unstable, and if it is unstable, it will collapse under its own gravity to form a protostar.

4. How does the pressure gradient in Jeans length affect the evolution of a gas cloud?

The pressure gradient in Jeans length plays a crucial role in the evolution of a gas cloud. It determines the rate at which the cloud collapses and the formation of stars within it. A higher pressure gradient will result in a faster collapse and a shorter time for star formation.

5. Can the pressure gradient in Jeans length be measured in other astrophysical phenomena?

Yes, the pressure gradient in Jeans length is not only applicable to gas clouds but can also be used to study other astrophysical phenomena, such as galaxy clusters and the interstellar medium. It is a valuable tool for understanding the dynamics of these systems and their evolution over time.

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