Download Powerpoint of lecture 9

Stellar Structure
Section 4: Structure of Stars
Lecture 9 - Improvement of surface boundary conditions
(part 1)
Definition of optical depth
Simple form of improved surface conditions
Surface boundary conditions and
optical depth (re-cap)
• One obvious better condition is T = Teff at the surface.
(4.53)
• “the surface” (“photosphere”) is the “visible surface” = surface from which
radiation just escapes: photon mean free path is infinite
• Mean free path (m.f.p.) ≡ ‘e-folding distance’ of the radiation
• Define monochromatic m.f.p. by
r  
  ds  1
r
• Integrating over frequency, and taking the frequency-integrated mean free path
to be infinite, we call this integral the optical depth,  :

    ds
• Then:
r
photosphere, or visible surface, is equivalent to  = 1.
Improved surface boundary
conditions
• Need new conditions when zero conditions give a model in
which the radiation escapes from a surface either much hotter
or much cooler than the effective temperature:
=1
=1
T = Teff
• New conditions? See blackboard
T = Teff
Useful approximation to surface
boundary conditions
• Boundary conditions involve an integral – awkward to use.
• Approximate integral by using fact that main contribution comes
from region just above photosphere, where only density
changes rapidly (Sun: e-folding distance ~300 km << RSun).
• Assume every other variable is constant, neglect radiation
pressure, and evaluate the integral (see blackboard) for this
“isothermal atmosphere”.
• Finally leads to simple boundary conditions:
T = Teff and P = g at M = Ms
• Vital to use these for cool giant stars – see Section 6.
(4.60)
Stellar Structure
Section 5: The Physics of Stellar Interiors
Lecture 9 – Limited thermodynamic equilibrium
(part 2)
Composition and molecular weight …
… for fully ionised gas
Deviations from simple pressure laws:
… no changes needed for radiation pressure
… need relativistic and quantum effects for
gas pressure (next lecture)
Limited thermodynamic equilibrium
• Is it true that P,  and  are functions only of , T and
composition?
• Strictly true only for complete thermodynamic equilibrium
• Good approximation provided interactions occur either very fast
or very slowly compared to timescale of problem of interest
• Nuclear interactions – very slow – “never” reach equilibrium
• Atomic interactions – very fast – “instantaneously” reach
equilibrium
• Limited thermodynamic equilibrium:
tnuclear >> tproblem >> tatomic => P, , = P, , (,T,composition)
• Valid for most (not all) problems of interest
Composition and molecular weight:
approximation and definitions
• Pre-main-sequence stars fully convective (Section 6), so expect
‘zero-age’ main-sequence stars to have uniform composition.
• Molecular weight  depends on abundances and on whether
gas is molecular, atomic or ionised (or some combination).
• Outer layers of stars – partially ionised:  very complicated.
• Through bulk of star – gas essentially fully ionised: can make
useful approximation to find expression for .
Define:
X = fraction of material by mass in form of hydrogen
(5.1)
Y = fraction of material by mass in form of helium
(5.2)
Z = fraction of material by mass in form of other elements. (5.3)
• Note that
X + Y + Z = 1.
(5.4)
Composition and molecular weight:
formula for fully ionised gases
•
Number of particles per hydrogen atom mass:
(a) Hydrogen:
2
(2 per mH – 1 proton, 1 electron)
(b) Helium:
¾
(3 per 4 mH – 1 He nucleus, 2 electrons)
(c) “Metals”:
~½
(Z+1 per A mH if fully ionised)
•
Then calculate:
N = total number of particles per unit volume, starting by finding
total number of H atom masses per unit volume =  /mH
•
Do this by adding up the numbers for each of (a) to (c) and using
definitions of X, Y and Z (see blackboard). Finally obtain:
1
N
 
   / mH

3
1
  2 X  Y  Z .
4
2

(5.6)
Value for molecular weight
• X, Y, Z found from observation of surface layers of stars
• Assume they’re the same in the interiors
• Similar for different stars – Handout 4
• Decline towards large mass number consistent with all heavy
elements being formed inside stars – so older stars have fewer
heavy elements
• Taking solar abundances, find:
 ≈ 0.62.
• Only really valid in deep interior, but that is most of mass of
star:
~90% of mass is within ~50% of radius
• Solar convection zone: ~30% of radius, only ~1% of mass
Pressure – do we need to modify our
simple expressions? Prad (and Pgas)
•
•
(a) Radiation pressure – simple expression follows if intensity of
radiation nearly equals Planck function:
1 4
I  B  Prad  aT .
3
Two potentially important deviations:
(i) Anisotropic radiation field: needs tensor pressure, but only
important near surface
Prad << Pgas near surface => tensor effects normally unimportant
(ii) Plasma effects:
EM waves cannot propagate if their frequency is less than the
natural oscillation frequency of the plasma (see blackboard), but:
Prad << Pgas near surface => plasma effects normally unimportant
•
(b) Gas pressure – do need to modify: see next lecture