The Kelvin-Helmholtz timescale

[physiks]

The basic idea is that from the Virial Theorem, if the gravitational potential energy of the sun decreases by a certain amount X, it gains thermal energy equal to half of this, 0.5X, and radiates away this same amount of energy too, 0.5X. This is from the relations
E=U+T and 2U+T=0 (energy E, PE U, KE T).

Then this is used to find the time the sun could continue to radiate at its current luminosity for. The problem is everywhere I read about this, the total PE of the sun is calculated in terms of its current state. They all say that the sun can continue contracting until its PE becomes zero (they say all the PE is 'radiated away'), so if we say half of the current PE is radiated away from the above, we can easily do the calculation.

However surely the PE of the sun is currently negative, and as it contracts, its PE gets more and more negative (and goes to negative infinity which obviously means my argument is flawed). If this is the case, why is the above method acceptable, because it's PE doesn't 'radiate away' and go to zero - it get's more negative. Another issue with my thinking is that because my PE is going to negative infinity, the star can continue to radiate forever at this luminosity by contracting - I'm obviously being very silly but can't figure out why!

Thanks for any help :)

 

[mfb]

To get a "timescale", you can calculate how long the sun could have been contracting in that way, as this corresponds to a finite energy difference.

In Newtonian mechanics, the sun could continue to radiate forever. General relativity sets a limit on that, but even that is extremely long.

 

[fayled]

Edit: probably wrong

 

[physiks]

To get a "timescale", you can calculate how long the sun could have been contracting in that way, as this corresponds to a finite energy difference.

In Newtonian mechanics, the sun could continue to radiate forever. General relativity sets a limit on that, but even that is extremely long.

But all of the sources say that the sun 'radiates away' its current PE. But it is getting more and more negative. Sorry I don't quite understand your point. All I can possibly think of is that we are saying the star starts off from zero PE and gets to it's current PE and finding the time for that to occur at it's current luminosity. However why would that make sense, as why not choose any change in PE and find a time. Surely it should be the time for it to contract down to zero size.

Section 1.1.2 here is useful to explain my problem
http://www.maths.qmul.ac.uk/~svv/MTH725U/Lecture1.htm
I'm fine with everything they say and do until they equate the GPE to the luminosity * time. They say themselves the GPE gets more and more negative through time.

 

[mfb]

But all of the sources say that the sun 'radiates away' its current PE.

Just to make that clear: we discuss a historic, wrong model. The sun is not actually using potential energy to radiate.

In this model, it radiates away the positive energy relative to some lower value it will have afterwards. The definition of "zero energy" is arbitrary, so absolut values do not matter.

Surely it should be the time for it to contract down to zero size.

That does not exist.

I'm fine with everything they say and do until they equate the GPE to the luminosity * time. They say themselves the GPE gets more and more negative through time.

And GPE and radiated energy are on the same side of the equation of energy conservation.

 

[physiks]

In this model, it radiates away the positive energy relative to some lower value it will have afterwards. The definition of "zero energy" is arbitrary, so absolut values do not matter.

But doesn't this mean I can choose any 'lower value' I like to do the calculation?

 

[mfb]

If you want to find the time the sun could radiate in this model until it reaches radius X, or if you want to compare two arbitrary states: yes.

 

[physiks]

If you want to find the time the sun could radiate in this model until it reaches radius X, or if you want to compare two arbitrary states: yes.

Would radius X then correspond to the 'lower value' of PE?

I'm just very confused.

So my understanding is:
We find the current PE of the sun. We then say the PE of the sun decreases from 0 to its current value, and from this we can find the energy it radiates (half of the GPE change). From this we find the time this can occur for if it maintains its current luminosity.

But the whole idea of the problem is to find the time the sun can maintain its current luminosity for in total, not just by contracting from one radius to another. Why are we then using the PE from one radius to another.

 

[mfb]

Would radius X then correspond to the 'lower value' of PE?

Which lower value?

I'm just very confused.

And I don't understand why.

Potential energy decreases, radiation is emitted, sun heats up, energy is conserved.
Forget about absolute values for the potential energy. Just look at differences.

But the whole idea of the problem is to find the time the sun can maintain its current luminosity for in total, not just by contracting from one radius to another. Why are we then using the PE from one radius to another.

At least the historic approach was to find the approximate age of the sun - the time the sun could have done this process in the past to reach its current radius today.

 

[physiks]

Which lower value?
And I don't understand why.

Potential energy decreases, radiation is emitted, sun heats up, energy is conserved.
Forget about absolute values for the potential energy. Just look at differences.

At least the historic approach was to find the approximate age of the sun - the time the sun could have done this process in the past to reach its current radius today.

So is the Kelvin-Helmholtz timescale

the time the sun could have radiated at its current luminosity in the past to reach it's current state today, via contraction

or

the time the sun could radiate for in the future from its current state today until it dies?

 

[mfb]

The past. See every google hit for "Kelvin-Helmholtz timescale".

 

[physiks]

The past. See every google hit for "Kelvin-Helmholtz timescale".

Right, that sorts that then. My lecture material wasn't so clear and was worded in such a way to make me think the future, and I didn't see anything to sway me from my misunderstanding whilst looking on the internet, so I'll look again and probably realize now.

Thanks!