Interstellar Cloud collapse

[CAF123]

Homework Statement

An interstellar cloud, made up of an ideal gas, collapses with its radius decreasing as $$R = 10^{13} \left(\frac{-t}{216}\right)^{2/3} \text{m}$$ with $t$ measured in years. The time $t$ is taken to be zero at zero radius so that $t$ is always negative.

The cloud collapses isothermally at 10K until its radius reaches 10¹³m. It then becomes opaque so that from then on, the collapse takes place adiabatically and reversibly. How many years does it take for the temperature to rise by 800K measured from the time the cloud reaches a radius of 10¹³m.

Homework Equations

The one in question, First Law of Thermodynamics, Adiabatic expansion

The Attempt at a Solution

The question has not specified how they define 'radius', but I assumed a spherical cloud. Considering the end of the isothermal phase, start of the adiabatic phase, and the end of the adiabatic phase, the following holds: $$\frac{T_i}{P_i^{1-1/\gamma}} = \frac{T_f}{P_f^{1-1/\gamma}},$$ Reexpressing gives: $$T_i^{1/\gamma} V_i^{1-1/\gamma} = T_f^{1/\gamma} V_f^{1-1/\gamma},$$ where $T_i = 10, V_i = 4/3 \pi (10^{13})^3, T_f = 810K, V_f = 4/3 \pi R^3$ When I solve for t, I obtain the incorrect answer.

Perhaps my assumption of the spherical shape of the cloud is incorrect?

 

[voko]

What answer did you obtain, and what is the correct answer?

 

[CAF123]

Hi voko,

What answer did you obtain, and what is the correct answer?

I obtained approx t = 57 years which means t ≈ 21 years (57 - 36) in answering the given question. (36 yrs the amount of time for R = 10¹³ ). The correct result is t = 208 yrs.

 

[voko]

What do you use for $gamma$ and why?

 

[CAF123]

What do you use for $gamma$ and why?

Oops, $\gamma$ was given in the question to be 5/3. Sorry about that.

 

[D H]

The correct answer is indeed 208.

How did you get 36 years as the time it takes to collapse to R=10¹³, and how did you get 57 years for the time it takes to collapse to the stage where T=810K? Both of those numbers are wrong. They don't even have the correct sign.

 

[D H]

In fact, looking at it as the time it takes to collapse to R=10¹³ is wrong. t=0 is the time at which R=0.

Start with R=10¹³ meters. Given that $R(t)=10^{13}\left(\frac {-t} {216} \right)^{2/3}\,\text{m}$, what is the value of t that yields R=10¹³ meters?

 

[CAF123]

Start with R=10¹³ meters. Given that $R(t)=10^{13}\left(\frac {-t} {216} \right)^{2/3}\,\text{m}$, what is the value of t that yields R=10¹³ meters?

That would be t ≈ 10.9 yrs. I made an error previously.

 

[voko]

That would be t ≈ 10.9 yrs. I made an error previously.

How is that possible if $R(t)=10^{13}\left(\frac {-t} {216} \right)^{2/3} = 10^{13}$?

 

[CAF123]

How is that possible if $R(t)=10^{13}\left(\frac {-t} {216} \right)^{2/3} = 10^{13}$?

It's not , I made another clumsy error - rather t = -216 yrs.

 

[D H]

Much better. Now how about T=810K? What do you get for the radius, and hence for time?

 

[voko]

Do you get $t_f / t_i = (T_i / T_f)^{3/4}$?

 

[voko]

Hmm. That ratio gives me the correct answer, and, frankly, I do not see why it is incorrect.

 

[D H]

It is correct. I wouldn't solve it that way, though. Personal preference.

 

[CAF123]

Much better. Now how about T=810K? What do you get for the radius, and hence for time?

My simplified expression was: $$\left(\frac{T_i}{T_f}\right)^{1/\gamma} = \left(\frac{-t}{216}\right)^{4/5},$$ which gives t = -8 yrs. Relative to the state of R=0, 216 yrs ago the cloud was at R = 10¹³m and temperature T = 10K. 8 years ago, it was at a temperature of 810K and R ≈ 10¹²m. So it takes (216 - 8) = 208 yrs to reach the latter state from the former state.

Do you get $t_f / t_i = (T_i / T_f)^{3/4}$?

I obtained a different expression. How did you get yours? Edit: Actually, they are the same.

I have some questions about the underlying physics of this process that I would like to ask later.

Thanks.

 

[CAF123]

Just want to check a few conceptual things:
In the isothermal phase, dU = 0 so that dQ = -dW. So the heat that enters the cloud (which is regarded as the system) is then released as work done by the cloud.

In the adiabatic phase, dU = dW, so the work done on the cloud by environment (in compressing it) is increasing its temperature.

Is this fine?

Also, why is it that $\gamma = 5/3$?

 

[voko]

Is it the cloud that does the work, or is the work being performed on it, and by what?

The heat capacity ratio is $\gamma = 5/3$ for monatomic ideal gas. This is related to the degrees of freedom of a gas molecule, which is 3 for monatomic gases.

 

[CAF123]

Is it the cloud that does the work, or is the work being performed on it, and by what?

Both the cloud and the environment do work because of NIII, yes? W_{system on surr} = - W_{surr on system}. I suppose an external agent in the environment that could do work on the cloud would be varying pressure.

 

[voko]

W_{system on surr} = - W_{surr on system}.

That is certainly true. However, in thermodynamics we usually distinguish between work on and work by based on the sign of work.

I suppose an external agent in the environment that could do work on the cloud would be varying pressure.

What external agent is at play here? The cloud is in the interstellar space, which we could consider perfectly void for this problem.

 

[CAF123]

What external agent is at play here? The cloud is in the interstellar space, which we could consider perfectly void for this problem.

Hmm, I am not sure what a possible source could be then to explain the compression of the cloud. Or maybe you could argue that the compression is caused by internal effects and there is no need for any external agent to do the compressing.

 

[voko]

What internal effect could that be?

 

[CAF123]

What internal effect could that be?

I wouldn't be sure of the details. But given that the cloud is interstellar, (i.e no external influences) the effect must be internal.

Alternatively, perhaps you could argue that the process described is simply not realistic?

 

[voko]

I wouldn't be sure of the details. But given that the cloud is interstellar, (i.e no external influences) the effect must be internal.

What kind of interaction is present with any sort of matter?

Alternatively, perhaps you could argue that the process described is simply not realistic?

Well, superficially this does seem to resemble some early stages of star evolution. But I am not sure about the time scales involved. I do not know in what context you were given the problem, though.

 

[D H]

Just want to check a few conceptual things:
In the isothermal phase, dU = 0 so that dQ = -dW. So the heat that enters the cloud (which is regarded as the system) is then released as work done by the cloud.

Not quite. In the initial stages of the collapse is slow and the gas cloud is transparent. Sans radiation, the collapse would cause the temperature to rise. That it can radiate means the temperature remains fairly constant and fairly low (but higher than the background 2.7 kelvin).

In the adiabatic phase, dU = dW, so the work done on the cloud by environment (in compressing it) is increasing its temperature.

The environment is not doing a thing to the gas cloud. The collapse is internally driven. It's gravity.

As the collapse continues, the density and temperature rise, particularly in the center. The cloud becomes opaque, starting in the center but the opaque region grows with time. The opaque cloud does not interact much at all with the environment, not even radiationally.

Also, why is it that $\gamma = 5/3$?

The adiabatic index γ is the ratio of heat capacity at constant pressure to that at constant volume. This is 5/3 for an ideal monatomic gas, 7/5 for an ideal diatomic gas. At low temperatures, molecular hydrogen acts more like a monatomic gas than a diatomic gas. One of the *many* simplifications in this model is the use of a constant value of γ.

 

[CAF123]

What kind of interaction is present with any sort of matter?

If the body has mass m, then it will experience a gravitational attraction. If I understand D H 's post, then the source of the compression is the gravitational collapse of the gas cloud under its own weight.

I do not know in what context you were given the problem, though.

A problem from a book on thermal physics and in a chapter dealing with the first law of thermodynamics. I am self studying at the moment.

 

[CAF123]

Not quite. In the initial stages of the collapse is slow and the gas cloud is transparent. Sans radiation, the collapse would cause the temperature to rise. That it can radiate means the temperature remains fairly constant and fairly low (but higher than the background 2.7 kelvin).

So maybe I should recast the condition as dW = -dQ. As the cloud is gravitationally collapsing, this will tend to increase its temperature. But it radiates as it does so, and hence there is no net effect on the temperature. (isothermal phase)

As the collapse continues, the density and temperature rise, particularly in the center. The cloud becomes opaque, starting in the center but the opaque region grows with time. The opaque cloud does not interact much at all with the environment, not even radiationally.

Okay, so in the adiabatic phase, the gravitational collapse does increase the temperature.

 

[D H]

So maybe I should recast the condition as dW = -dQ. As the cloud is gravitationally collapsing, this will tend to increase its temperature. But it radiates as it does so, and hence there is no net effect on the temperature. (isothermal phase)

There is no work here. The problem is that the simplistic form of the first law of thermodynamics you learned ignores thermal radiation.

 

[CAF123]

There is no work here. The problem is that the simplistic form of the first law of thermodynamics you learned ignores thermal radiation.

Why is there not work done by gravity as the cloud collapses?

 

[D H]

Because it's self gravitation. Work in thermodynamics is that done on the system by the external environment. (Alternatively, it's work done by the system on the external environment, depending on which sign convention one uses.) Gravitation is not doing work on the external environment. The only thing the external environment does is act as a place into which the collapsing cloud can emit electromagnetic energy.