1. 5. The interiors of stars

2. 5. The interiors of stars
5.1 Foreword 5.2 Fundamental equations 5.3 Equation of state 5.4 Material equations 5.5 Stellar models

3. 5.1 Foreword
Analysis of the starlight allows us to determine a variety of quantities that are related to the outer layers of stars, e.g. T, L, chemical composition However, there is no direct way to observe the inner, opaque part of the stars (with a few exceptions; see later).
We can learn about the stellar interiors only by constructing models Such models have to be consistent with the physical laws The predictions of these models can be tested by comparison with the observed properties of individual stars or large groups of stars..
The fundamental physics of the stellar structure was worked out essentially in the first half of the 20th century .*
However, knowledge about the stellar interiors comes mainly from computer models. Computers and sophisticated software codes became available only since the ~1960s.
(*) Fundamental work: „The Internal Constituation of Stars“ by Arthur Eddington (1926)

4. 5.1 Foreword
Fundamental properties of stars which can be determined by observations and compared with model predictions (Summary from Sect. 3):
Stellar property
Determined from
Range of values
Mass (M )
Binary stars
10 … 10
Luminosity (L )
Flux density, distance and bolometric correction
Radius (R )
Interferometry and distance or eclipsing binaries
Effect. temperature T (K)
Spectral type or continuous spectrum
… 10
Heavy element abundance Z
Line spectrum
~0 … 0.03
Age (yr)
HRD
0 … 10
eff -1 2 -4 6 -2 3 5 10 ⊙ -1

5. 5.1 Foreword Main relationships (see Sect. 3): Structure of the HRD
L – T relationship for main sequence
L – M relationship for main sequence
eff -1

6. 5.2 Fundamental equations
The fundamental equations of stellar structure are the equations for
Force balance: hydrostatic equilibrium
Mass conservation
Energy conservation
Energy transoprt

7. 5.2.1 Basic assumptions Spherical symmetry (stars are gaseous spheres)
i.e., only radial dependence
i.e., we neglect deformations caused by fast rotation, nearby companions, and strong magnetic fields
Stars are stationary (steady state)
no explicite time dependence of the physical quantities
i.e., we neglect, for the moment, that a star must change its constituation as it radiates energy and thus must eventually use up their resources … and finally die...
However, usually stars change at an imperceptibly low rate by human standards (though rapid phases occur as well; see later)

8. 5.2.2 Hydrostatic equilibrium
r + dr dF G P r
towards center
Consider an infinitesimal cylindric volume element with the volume dV and mass dm = ρ dV = ρ A dr
Assume that the only forces acting on the cylinder are gravity, dF , and pressure, dF G P
In a static star, the gravitational force is exactly canceled by the outward pressure force
dF = dF G P dP
where dF = P(r) A – P(r+dr) A = – A dr dr P
and dF = G ρ A dr where M = mass inside sphere of radius r M r 2 G dP dr
= – G ρ(r) = – g(r) ρ(r) r M 2
Note that P is the total pressure (gas pressure + radiation pressure + ...)
(5.1)
with g(r) = -G gravitational acceleration at r

9. Mass conservation ≪
Consider an infinitesimally thin concentric, spherical shell of radius r and thickness dr r
The volume of the shell is dr r dM
dV = 4π r dr 2 r
The mass of the shell is therewith
dM = ρ 4π r dr 2 dM dr
= ρ(r) 4π r 2
Rewriting yields the mass conservation equation r
(5.2)
Or, in the integral form:
M = ∫ρ(r') 4π r' dr' 2 r

11. Blackboard 5.2.4 Applications
Application: estimation of central pressure and temperature for star models with constant density
Central pressure
3 G M
8 π R
P(0) ≈ 2 4
(5.3)
Blackboard
Central temperature
(5.4)
G m M
2 k R
T(0) ≈
M: stellar mass, R: radius, m: mean particle mass, k: Boltzmann constant, G: gravitation constant

12. Energy conservation
The contribution to the total luminosity of a star due to stellar material in an infinitesimal mass element dM is dr r dL
dL = ε dM
ε: specific energy production rate [J s kg ] -1 -1 r
The contribution to L from a shell is therewith
dL = ε ρ 4π r dr 2 dL dr
= ε(r) ρ(r) 4π r 2
(5.5)
Rewriting yields the energy conservation equation r
Generally, ε can be due to
- inner energy sources
- gravitational energy (contraction)
- thermal energy (cooling)

14. Blackboard 5.2.5 Energy conservation
Application: Contraction time scale (Kelvin-Helmholtz time scale)
3 G M 2 E
rad
t = ≈
(5.6) KH L
8 R L KH
For the Sun
t = ≈ ≈ yr E
rad L 7
3 G M 2
8 R L ⊙
Blackboard
However, estimated age of rocks from the Earth and the Moon's surface is over yr!* 9
(*) Moon rocks on Earth come from three sources: (a) US Apollo manned lunar missions from (380 kg), (b) Soviet unmanned Luna mission from 1970, (c ) >120 lunar meteorites found on Earth

15. 5.2.6 Energy transport Ways of energy transport:
- radiative transfer (em-abs-em-abs...) … important
- convection (motion of mass packages) … important
- heat conduction (collisions of particles) … negligible for normal stars (but important for WD; see later)

16. 5.2.6 Energy transport (a) Radiative transfer RTE (Eq. 4.09) with
Multiplication with and integration over Ω:
For isotropy:
= 4π/3
= 0
= Φ (from Eq. 4.3) dI dr L
4πr 3 4π κρ
═ ─
(5.7) r

17. Energy transport
On the other hand, the outward directed radiation flux is given by
Φ = π I (Eq. 4.3) and Φ = σ T (Eq. 4.5) + + 4
therewith +
with
which yields
Replacing dI/dr by Eq. (5.7) and rewriting, we find
(5.8)
Remark: - The T gradient is negative because T increases inwards
- If there would be no T gradient, the radiation field would be the same in all direction and the net radiation flux would vanish.

18. Energy transport
(b) Remark: Is the radiation field isotropic in the stellar interior? r
Isotropy:
i.e.,
Consider the Sun at r = R /2 (energy sources inside): ⊙
(a) Net flux density:
(b) Outward flux density:
Φ = |Φ | ─ |Φ | ≈ 0 (compared with Φ ) + -
|Φ | ≈ |Φ | (i.e., isotropy)

19. 5.2.6 Energy transport (c ) Application: Mass-luminosity relation
dT T(0)
Assumptions:
constant T gradient
T(R) T(0)
κ = const., ρ = const.
energy sources strongly concentrated towards center, e.g. ε ~ 1/r
therewith (Eq. 5.5), L ~ r and
dr R
= const =
dT T(R) ─T(0) ≪
dr R
≈ ─
and T ≈ T(0) 1 r
L L
r R =
Inserting in Eq. (5.8):
T(0) L M 1
R R R T
~ ─ ─ 2 3
L M
R T(0) 5
Rewriting:
L M R 4
T(0) ~ 4 M R ~
L M
L ~ M 3
(5.9)
In very good agreement with the observed L-M relation (Sect ) M
On the other hand (Eq. 5.4): R
T(0) ~
Radiative energy transfer must be important!

20. 5.2.6 Energy transport (d) Convective transport
When the radiative transfer is inefficient, the motion of gas parcels sets in to carry the energy more efficiently than radiation.
Hot gas bubbles rise upwards into cooler layers where they loose energy
Cool gas sinks towards hotter layers where it is heated and rises again …
The rising and sinking gas elements transport energy and mix the stellar material
The height of the convection cell is called mixing length (similar to mean free path length in thermodynamics) r
Fig.: Numerical simulations of multi-celled convection (red = hot, blue = cool).

21. 5.2.6 Energy transport (d) Convective transport
Assumptions:
The gas bubble expands adiabatically (i.e. no energy exchange with the surrounding gas until the end of the upward motion)
Pressure equilibrium between the bubble and the surrounding gas
Ideal gas, i.e. P = n k T = ρ k T / m g
Adiabatic gas law:
adiabatic exponent
(*)
Ideal gas equation:
Replacing dρ/dr by (dρ/dr) and rewriting: ad
(5.10)
(*) C : specific heat at constant pressure, C : specific heat at constant volume P V

22. 5.2.6 Energy transport (e) Condition for convection
Final state
Bubble (T,P,ρ)
(b) f
Surrounding gas (T,P,ρ)
A bubble continues to rise if
(s) f
The density after an infinitesimal path dr can be expressed by a Taylor series approxi-mation dr
Bubble (T,P,ρ)
(b) i
Surrounding gas (T,P,ρ)
Initial state
(s) i
Let us assume an initial thermal equilibrium and pressure equilibrium at all times:
Then, the condition for the bubble to rise is:
Now, we express this condition in terms of quantities for the surrounding gas using the adiabatic equation (left hand side) and the ideal gas equation (right hand side):
(*)

23. 5.2.6 Energy transport (e) Condition for convection
For pressure equilibrium, we have
(i.e., the superscript is now redundant)
Therewith, Eq. (*) becomes
Now, we drop the superscripts, since all quantities are related to the surrounding gas:
adiabatic T gradient
actual T gradient
Convection sets in when the radiative T gradient becomes steeper (larger in absolute values) than the adiabatic gradient (i.e. either the radiative gradient is steep or the adiabatic gradient is shallow) or
(5.11)

24. 5.2.7 Summary Fundamental equations of stellar structuredP dr
= – G ρ(r) = – g(r) ρ(r) M 2
(5.1)
Hydrostatic equilibrium: dM r
Mass conservation:
= ρ(r) 4π r 2
(5.2) dr dL
Energy conservation: r
= ε(r) ρ(r) 4π r 2
(5.5) dr
for radiative transport
(5.8)
Energy transport:
for convective transport
(5.10)
with the convection condition:
(5.11)