Wrong Explanations for the Luminosity of Main-Sequence Stars

[Ken G]

Maybe for NS also?

Yes, true!

Outside the stellar core, shouldn't L(r) remain constant? Otherwise energy would be accumulating or draining from material in some spherical shell

If so, then

4pi r^2 T(r)^4 = L(R) = constant

T ~ r^-0.5

The expression on the left is only true at a surface-- not in an optically thick interior. In the interior, T⁴ has the meaning of an energy density per volume, not a flux per unit area. To connect it with flux, you need to know the optical depth, and divide your expression on the left by the optical depth to that r. You end up with a T⁴ gradient (in tau space rather than real space) that acts a lot like a heat conduction, where the gradient in T⁴ gives you the flux, not T⁴ itself. That's the radiative diffusion physics that gave L ~ R⁴T⁴/M, because optical depth goes like M/R² for a characteristic total value.

 

[TEFLing]

If I try to consider concentric spherical shells, each radiating outwards and inwards, balanced by inwards from the next shell out plus outwards from the next shell in...

Then it seems like dL/dr = constant

E.g. Surface shell radiates one unit of power out and another in...

The next shell inwards radiates two units outwards ( replenishing the surface shell ) and two units inwards...

The third shell radiates three units...

And so on

 

[Ken G]

If I try to consider concentric spherical shells, each radiating outwards and inwards, balanced by inwards from the next shell out plus outwards from the next shell in...

Then it seems like dL/dr = constant

E.g. Surface shell radiates one unit of power out and another in...

The next shell inwards radiates two units outwards ( replenishing the surface shell ) and two units inwards...

The third shell radiates three units...

And so on

That works if you make each shell one photon mean-free-path thick. Then you get the same thing as above, it gives a diffusion process.

 

[TEFLing]

That works if you make each shell one photon mean-free-path thick. Then you get the same thing as above, it gives a diffusion process.

Oh I understand now

dL/d(tau) = constant

If you ignore the surface luminosity as small compared to the core luminosity...

Then

L(r) = L0 (1 - (r/R))

T(r) ~ (1/r^0.5) (1 - (r/R))^0.25

 

[TEFLing]

Oh I understand now

dL/d(tau) = constant

If you ignore the surface luminosity as small compared to the core luminosity...

Then

L(r) = L0 (1 - (r/R))

T(r) ~ (1/r^0.5) (1 - (r/R))^0.25

trying to upload figure from Wolfram Alpha website

 

[Ken G]

Something isn't quite right there, outside the fusion zone it is L itself that is constant.

 

[TEFLing]

L can increase linearly inwards, and still keep every shell in equilibrium

Shell two radiates two units of power outward and inwards (-4), but gets one from above and three from below (+4)

L increases inwards but the star is in equilibrium

 

[Ken G]

L can increase linearly inwards, and still keep every shell in equilibrium

Shell two radiates two units of power outward and inwards (-4), but gets one from above and three from below (+4)

L increases inwards but the star is in equilibrium

That situation does not have an increasing L, it has a nonzero L (it has L=1).

 

[snorkack]

The expression on the left is only true at a surface-- not in an optically thick interior. In the interior, T⁴ has the meaning of an energy density per volume, not a flux per unit area.

Total energy, or energy in form of photons?

To connect it with flux, you need to know the optical depth, and divide your expression on the left by the optical depth to that r. You end up with a T⁴ gradient (in tau space rather than real space) that acts a lot like a heat conduction, where the gradient in T⁴ gives you the flux, not T⁴ itself.

So, if heat conduction is in a narrow range of temperature, then the energy flux
dE/dt*dS=c1*(dT/dr)
If the relevant gradient is T⁴ then its derivative
d(T⁴)=4*T³*dT
and then
dE/dt*dS=c2*4*T³*(dT/dr)
so it matches
c1=c2*4*T³
Thus, conductivity proportional to the cube of the temperature.
Correct?

 

[Ken G]

Total energy, or energy in form of photons?

Photons only.

So, if heat conduction is in a narrow range of temperature, then the energy flux
dE/dt*dS=c1*(dT/dr)
If the relevant gradient is T⁴ then its derivative
d(T⁴)=4*T³*dT
and then
dE/dt*dS=c2*4*T³*(dT/dr)
so it matches
c1=c2*4*T³
Thus, conductivity proportional to the cube of the temperature.
Correct?

Yes. So that's why the cooler interiors of protostars must carry their high luminosities convectively-- effectively lousy radiative heat conductivity, despite the low optical depths. But as they contract and T rises, eventually their radiative heat conductivity rises to the point that radiation can transport their luminosity throughout most of the star, and that's when L saturates at its ~ M³ value.

 

[TEFLing]

Photons only.
Yes. So that's why the cooler interiors of protostars must carry their high luminosities convectively-- effectively lousy radiative heat conductivity, despite the low optical depths. But as they contract and T rises, eventually their radiative heat conductivity rises to the point that radiation can transport their luminosity throughout most of the star, and that's when L saturates at its ~ M³ value.

Is there a radius effect also?

Cooler stars tend to be smaller
And total power transport = energy per time ~ r^2 T^4
So cooler smaller stars have less energy flux per area, and also less / not as much area, a double whammy as it were

 

[TEFLing]

That situation does not have an increasing L, it has a nonzero L (it has L=1).

I want to dispute that

Let me speak in terms of radial step sizes each of dr = one optical depth / MFP

The surface shell is like a hollow shell of onion
It radiates one unit of power = dE/dt outwards and another inwards = -2 units

The next onion shell in radiates two units of power outwards and another two units inwards = -4 units

But its two units outward are absorbed by the optically thick outer surface shell, so that shell stays in equilibrium = 0 = -1 -1 +2

The second shell can stay in equilibrium too, if the third shell inwards radiates 3 units of power outwards ( and another 3 inwards)

Shell         1.       2.        3.      
=====
Losses       -1-1.    -2-2.   -3-3
Gains         +2.     +1+3.   +2+4
--------
Net.           0.       0.      0

 

[Ken G]

Is there a radius effect also?

Cooler stars tend to be smaller
And total power transport = energy per time ~ r^2 T^4

The energy per time isn't R² T⁴, that assumes time ~ R. Actually it's diffusive, so time ~ R/v, where v is not c, it is c/tau, for tau the optical depth. Since tau ~ M/R², it actually works the other way-- larger R allows light to escape faster. So that's where the L ~ R⁴ T⁴ / M comes from.

 

[TEFLing]

...- larger R allows light to escape faster..

I thought that was what I was saying

I guess I'm confused

A diffusion equation with no net diffusion would imply a zero second derivative, yes?

 

[Ken G]

I want to dispute that

The problem with your scenario is not that it isn't in equilibrium, it's that it has a constant luminosity of L=1 everywhere, yet you said it had a radial gradient in L. I'm sure you'll see this-- if L increased between shells, energy could not be conserved, just as a flow of water has to go somewhere. What is increasing inwards is not L, it is the energy density T⁴, and that's completely correct-- your scenario is just what is happening, but it's constant L.

 

[TEFLing]

http://en.wikipedia.org/wiki/Kramers'_opacity_law

Doesn't free free scattering dominate electron scattering given the large coefficients?

 

[Ken G]

I thought that was what I was saying

I guess I'm confused

A diffusion equation with no net diffusion would imply a zero second derivative, yes?

In an idealized plane-parallel geometry, then zero second derivative means it is time steady diffusion, but there is still diffusion. The spherical symmetry changes things a bit.

 

[Ken G]

http://en.wikipedia.org/wiki/Kramers'_opacity_law

Doesn't free free scattering dominate electron scattering given the large coefficients?

Kramers type opacity can certainly enter, especially for cooler stars. For hotter stars, electron scattering is at least close to dominant, if not completely dominant. So no doubt, assuming constant opacity is a bit of a stretch, but it's simple. There is no general expression, because electron scattering has no T dependence if H is ionized, Kramers opacity has a steeply dropping dependence with T, and H-minus opacity, important in the envelopes of very cool stars, actually rises with T at low T because you need some free electrons to make it work. So the opacity is all over the map, and you'd be totally wrong using electron scattering for stars less massive than the Sun, but you can get order-unity kinds of results for most of the main sequence. It's definitely the weakest assumption, some advanced authors try to include some opacity effects but there are subjective choices in the tradeoffs between complexity and accuracy.

 

[TEFLing]

I think I need to clarify vocabulary terms

I want one term for 'shell luminosity ' = 4pi r2 T4
Which is what you would see if you could magically excavate the stellar envelope down to that radius

And I want another word for Transported through luminosity ' which is the net diffusion of energy outwards...
And which is constant outside the core like you said for the energy conservation reason rationale you stated

 

[TEFLing]

I want to try a calculation

From radiative equilibrium

d²L/ds² = 0

Where L = 4 pi r² T⁴ is the shell luminosity, and ds = rho k dr is the differential optical depth.

So

dL/ds = constant

L = L_core - A s

At the surface

L_* = L_core - A rho k R

For ES, k ~ constant, rho R ~ M/R², L_core ~ rho² Tⁿ x V_core ~ M² / R³ Tⁿ

Using the VT, T~ M/R, So

L_* = (T/R) x ( B Tⁿ⁺² - A )

If the LHS ~ 4 pi R² T⁴, and for n=4...

R² T⁴ ~ (T/R) x ( B T⁶ - A )

R³ T³ ~ ( B T⁶ - A )

M³ + A ~ B T⁶

That is vaguely like T ~ sqrtM, which the VT converts to R ~ sqrtM ~ T

As we said above, and implies L ~ M^3

 

[TEFLing]

For Kramers opacity

L_core ~ M² / R³ Tⁿ

rho k r ~ rho² T^-3.5 ~ M² / R⁵ T^-3.5

L_* ~ R² T⁴ ~ M² T²

Cancel out the M², multiply by R⁵, and use VT

M² R³ ~ M² T^n-2 - T^-3.5

Again ignoring the second term on the RHS, and for n=4

T² ~ R³

So

M ~ R^2.5
L ~ R⁸ ~ M^3.2

Which is basically the above also

 

[TEFLing]

L_core ~ M² / R³ Tⁿ

L_* ~ R² T⁴ ~ M² T²

R³ ~ T^n-2

n = 4
R³ ~ T²

n=18
R³ ~ T¹⁶
T ~ R^3/16
M ~ RT ~ R^19/16
L ~ R² T⁴ ~ R^2.75 ~ M^2.3

Seems like all calculation paths lead to the vicinity of L ~ M³

 

[TEFLing]

L ~ M² / R³ Tⁿ
L ~ R² T⁴

Those are the scaling relations I was looking for
They capture the general gist of the balance between core fusion and surface luminosity
And they give informatively useful and accurate results

Precise details are much more complicated

 

[Ken G]

I think I need to clarify vocabulary terms

I want one term for 'shell luminosity ' = 4pi r2 T4
Which is what you would see if you could magically excavate the stellar envelope down to that radius

And I want another word for Transported through luminosity ' which is the net diffusion of energy outwards...
And which is constant outside the core like you said for the energy conservation reason rationale you stated

OK, then all is fine. Note that in standard lexicon, "transported through luminosity" is just "luminosity", and "shell luminosity" is just energy density times surface area, with the appropriate constant. But these are just the ways the words are normally used, when one is deep in the interior of the star it's pretty arbitrary what we might want to call the luminosity, and I agree with you that yours is a useful scenario for understanding how diffusion works.

 

[Ken G]

L ~ M² / R³ Tⁿ
L ~ R² T⁴

Those are the scaling relations I was looking for
They capture the general gist of the balance between core fusion and surface luminosity
And they give informatively useful and accurate results

Precise details are much more complicated

There is indeed some latitude in what you do if you are not exactly solving the equations of stellar structure, giving the basic explanation for L a kind of subjective character. Personally, I'm not in favor of using the core and surface T as though they were the same thing, because they are set by very different physical processes. The core T is set by the need to self-regulate fusion to replace the heat that is diffusing out, and the surface T is set by the need for the blackbody emission at the surface to match that luminosity, so the former is fusion physics and the latter is Stefan-Boltzmann physics, and there is no reason the two should be the same or even proportional. So I wouldn't use L ~ R² T⁴ for anything but a way to infer the surface T. Also, if you use diffusion physics to get L, you never need L ~ M² / R³ Tⁿ, you can just use L ~ R⁴ T⁴ / M from diffusion with constant opacity (or modify that for some other opacity like Kramers, but it won't be more accurate), and T ~ M/R from VT, and get L without even saying whether fusion is happening or not. However, if you want to get T in the core, and R, and then T at the surface, you would need to do what you are doing, since then you do need to know that fusion is happening.

The bottom line is, it is possible to get simple scaling relations to help understand, at a rough level of approximation, all the basic properties of a main-sequence star that is not completely convective and does not have strong radiative forces. These general scaling laws have been around for a long time, and there is really no excuse for any astronomy textbook to claim that high-mass main-sequence stars are more luminous because their stronger gravity compresses the core to higher pressure and temperature. None of the scaling laws have that character, it's really just completely wrong.

 

[TEFLing]

...

Also, if you use diffusion physics to get L, you never need L ~ M² / R³ Tⁿ, you can just use L ~ R⁴ T⁴ / M from diffusion with constant opacity (or modify that for some other opacity like Kramers, but it won't be more accurate), and T ~ M/R from VT, and get L without even saying whether fusion is happening or not...

M² / R³ Tⁿ ~ L ~ R⁴ T⁴ / M

M³ Tⁿ ~ R⁷ T⁴ ~ R³ M⁴

Tⁿ ~ R³ M ~ R⁴ T

T^n-1 ~ R⁴ ~ M/R ~ M/T^(n-1)/4

T^5(n-1)/4 ~ M ~ R⁵

L ~ M^2/5 M^16/5(n-1)

n=4
-----
L ~ M^1.5

Seems like the two equations for L are not compatible...(??)

 

[TEFLing]

I want to ask about the VT...

Isn't the VT based on the IGL and an assumption of HSE?

For, the VT is consistent with the one zone scaling relation form of the HSE equation

dP/dr = -rho g

P/R ~ M² / R⁵

rho T ~ rho g R

T ~ M/RBUT what if pressure support derives from RADIATION pressure, not gas pressure??

Wouldn't wht

dP/dr = -rho g

P/R ~ M² / R⁵

T⁴ ~ M² / R⁴

T ~ sqrt(M) / R

 

[Ken G]

T^n-1 ~ R⁴ ~ M/R ~ M/T^(n-1)/4

Wasn't there a problem in the middle there?

 

[Ken G]

I want to ask about the VT...

Isn't the VT based on the IGL and an assumption of HSE?

Yes exactly.

BUT what if pressure support derives from RADIATION pressure, not gas pressure??

Then you need a different form for the VT, but it's only an issue for the highest mass stars.

T⁴ ~ M² / R⁴

T ~ sqrt(M) / R

Radiation P is not T⁴, it is ~ T⁴/R³. You should ultimately find the Eddington limit: L ~ M, if you take constant opacity.

 

[TEFLing]

If we have scaling relations...

L ~ R² T⁴
L ~ M² Tⁿ / R³
T ~ M/R or M^1/2 / R

Gas support
==========
R² T⁴ ~ (RT)² Tⁿ / R³ ~ Tⁿ⁺² / R

R³ ~ T^n-2

M³ ~ Tⁿ⁺¹

L ~ R² T⁴ ~ T^2(n-2)/3 M^12/(n+1) ~ M^2(n-2)/(n+1) M^12/(n+1) ~ M^(2n+8)/(n+1)

L ~ M^3.2 (n=4)
Radiation support
===============
R² T⁴ ~ (RT)⁴ Tⁿ / R³

R ~ Tⁿ

M^1/2 ~ Tⁿ⁺¹

L ~ R² T⁴ ~ T²ⁿ M^2/(n+1) ~ M^n/(n+1) M^2/(n+1) ~ M^(n+2)/(n+1)

L ~ M^1.1 (n=18)Isn't this qualitatively correct? Doesn't L ~ M for high mass stars?

 

[TEFLing]

Take x2...
========

M² / R³ Tⁿ ~ L ~ R⁴ T⁴ / M

M³ Tⁿ ~ R⁷ T⁴ ~ R³ M⁴

Tⁿ ~ R³ M ~ R⁴ T

T^n-1 ~ R⁴

T ~ R^4/(n-1) ~ M/R

M ~ R^(n+3)/(n-1)

R ~ M^(n-1)/(n+3)

T ~ M^4/(n+3) L ~ R² T⁴ ~ M^{(2n+14)/(n+3)}

n=4
L ~ M^22/7

 

[Ken G]

Radiation support
===============
R² T⁴ ~ (RT)⁴ Tⁿ / R³

R ~ Tⁿ

M^1/2 ~ Tⁿ⁺¹

L ~ R² T⁴ ~ T²ⁿ M^2/(n+1) ~ M^n/(n+1) M^2/(n+1) ~ M^(n+2)/(n+1)

L ~ M^1.1 (n=18)Isn't this qualitatively correct? Doesn't L ~ M for high mass stars?

It is qualitatively correct, but it's kind of a coincidence. What actually happens is L ~ M exactly, if we use constant opacity, and the value of n makes no difference at all. The flaw in this analysis is that it assumes the core T and the surface T are either the same, or at least proportional, and there's no physical basis for that assumption. All the same, since it works out to be approximately true that they are proportional, you can get away with asserting it in the derivation. But doing so puts the cart before the horse-- we know L ~ M for other reasons (the Eddington limit), so your scaling analysis should actually be used to derive the ratio of surface T to interior T. When you find that the two are approximately proportional, you have demonstrated something important, but it's not the same thing being demonstrated as if you just assume that from the start.

 

[Ken G]

Take x2...
========

M² / R³ Tⁿ ~ L ~ R⁴ T⁴ / M

M³ Tⁿ ~ R⁷ T⁴ ~ R³ M⁴

Tⁿ ~ R³ M ~ R⁴ T

T^n-1 ~ R⁴

T ~ R^4/(n-1) ~ M/R

M ~ R^(n+3)/(n-1)

R ~ M^(n-1)/(n+3)

T ~ M^4/(n+3)L ~ R² T⁴ ~ M^{(2n+14)/(n+3)}

n=4
L ~ M^22/7

Yes, that's bang on. So we see we have some subjective freedom as to how we will derive L. We can take an n for nuclear burning, even though that will limit our range of applicability (p-p fusion doesn't dominate past a solar mass), and give exactly the analysis you have. You get a slightly complicated power in the mass-luminosity relation, but it is quite useful all the same, and works fine. Alternatively, we can ignore any need to assert an n for fusion, and just say that fusion induces a thermostatic effect that regulates T to be something like 20 million K. Unfortunately this will not be at all exact, but we do get a very bare bones kind of derivation that gives L ~ M³. A simpler power, simpler physics, but not more justifiable-- just a different set of things to idealize. So it's a subjective choice at this point.

 

[TEFLing]

http://www.physics.rutgers.edu/~pryor/ph442/lecture12.pdf

I found the above link for the index of refraction of a plasma...

In slide 1 I think they mean to say n is IMAGINARY at low frequencies, not negative

I think at the high temperatures inside stars, n~1 is a very accurate approximation

 

[TEFLing]

Yes, that's bang on. So we see we have some subjective freedom as to how we will derive L. We can take an n for nuclear burning, even though that will limit our range of applicability (p-p fusion doesn't dominate past a solar mass), and give exactly the analysis you have. You get a slightly complicated power in the mass-luminosity relation, but it is quite useful all the same, and works fine. Alternatively, we can ignore any need to assert an n for fusion, and just say that fusion induces a thermostatic effect that regulates T to be something like 20 million K. Unfortunately this will not be at all exact, but we do get a very bare bones kind of derivation that gives L ~ M³. A simpler power, simpler physics, but not more justifiable-- just a different set of things to idealize. So it's a subjective choice at this point.

WOW

I think I almost understand what you are saying

Roughly speaking, we start with a diffusion equation for the starlight, which is time invariant, d/dt -> 0

=> constant Laplacian double derivative on the other side...

Implying a constant heat flow

L ~ c l d/dr(4 pi r2 T4)

Where l~1/(rho sigma) is a mean free path measure

So the scaling relation simplification is

L ~ L / ( rho sigma T )

Rho sigma R ~ constant

Which says that if a star becomes more opaque per particle, the column density of particles must decrease to let the same light luminosity out through the plasma to the surface

Assuming ES is dominant, sigma ~ constant, and rho R must be constant too

==============
Using FF opacity

Sigma ~ rho T^-3.5

Rho sigma R ~ rho^2 T^-3.5 R ~ 1

M^2/R^5 ~ T^3.5

-----------------------
Using VT , RT ~ M

M^2 ~ R^1.5 M^3.5

1 ~ (RM)^1.5

M ~ 1/R

(??)-----------------------
Using VT , RT ~ sqrt(M)

M^2/R^5 ~ T^3.5

M^2 ~ R^1.5 M^1.75

M^.25 ~ R^1.5

M ~ R^6

M^2/R^5 ~ T^3.5

M^7/6 ~ T^7/2

M ~ T^3

L ~ R2T4 ~ M^.333 M^1.333 ~ M^5/3