Wrong Explanations for the Luminosity of Main-Sequence Stars

[TEFLing]

...

( T_core / T_surface ) ~ M/R²

...

From VT,

T_core ~ M/R

So

T_surface ~ R

I think that is qualitatively correct, and maybe even semi quantitatively so

 

[Ken G]

From VT,

T_core ~ M/R

So

T_surface ~ R

I think that is qualitatively correct, and maybe even semi quantitatively so

Yes, that logic works for me. It overestimates a little how much the surface T varies, but that's not surprising given that the opacity is assumed constant and convection is neglected. If anyone is just now coming into the thread, notice we are strictly talking about ideal-gas-pressure stars that transport energy by radiative diffusion, so these expressions do not apply to giants or white dwarfs, or even main-sequence stars at very low or very high mass. But it should all give reasonably semi-quantitative results for main-sequence stars between about 0.5 and 50 solar masses.

 

[TEFLing]

In QM, you can calculate the Bohr radius of H by writing the equation for electron energy, and differentiating to find the dx = h/dp which minimizes the total energy

Does something similar work for stars as well?

U ~ -2/5 GM^2/R
K ~ -1/2 U

?

 

[Ken G]

In QM, you can calculate the Bohr radius of H by writing the equation for electron energy, and differentiating to find the dx = h/dp which minimizes the total energy

Does something similar work for stars as well?

U ~ -2/5 GM^2/R
K ~ -1/2 U

?

Only for white dwarfs, which are the stellar equivalent of a multiple-particle Pauli-excluded Bohr atom.

 

[TEFLing]

Maybe for NS also?

I have a question about radiative equilibrium...

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

 

[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.