1. Outline – Stellar Evolution
Review Stellar Evolution (Quick)
Compact Objects & White Dwarfs
Lane-Emden Equation
Chandrasekar Limit
Mixing Length Theory

2. A Census of the Stars
Faint, red dwarfs (low mass) are the most common stars.
Bright, hot, blue main-sequence stars (high-mass) are very rare.
Giants and supergiants are extremely rare.

3. Stellar Evolution Sun Mass
High mass star (> 8 M ) burn fuel faster, are brighter, shorter life  SuperNova
Medium mass - (0.4 to 4 M) - burn fuel moderatley, live long
 PN + WD
Low mass - (< 0.4 M) burn fuel slowly - live long long time!
Sun
Mass

4. Energy Transport Structure
Many Open Questions on stellar interior models
A-stars with X-rays; Models for low mass stars (BDs)
O & B-stars with Spots
Inner convective, outer radiative zone
8 Msun
4 Msun
Inner radiative, outer convective zone
CNO cycle dominant
PP chain dominant

5. Lane-Emden Equation Use as a simple model of a star with polytropes in
Use as a simple model of a star with polytropes in
Hydrostatic equilibrium
-or-
Think of it as Poisson’s Eq for a gravitational potential of a Newtonian self gravitating, spherically symmetric, polytropic fluid.

6. Solutions of Lane-Emden equation
for n=0 to 5

7. Mass Functions – The Eddington Solution
Red – standard solar model
Blue = n=3 polytrope
Black – linear
density law
Solar & n=3
Agree very well!

8. Solutions of Lane-Emden equation
n = 0, the density of the solution as a function of radius is constant,
ρ(r) = ρc. This is the solution for a constant density incompressible sphere.
n = 1 to 1.5 approximates a fully convective star, i.e. a very cool late-type
star such as a M, L, or T dwarf.
n = 3 is the Eddington Approximation.
There is no analytical solution for this value of n, but it is useful as it
corresponds to a fully radiative star, which is also a useful approximation
for the Sun.
n > 5, the binding energy is positive, and hence such a polytrope
cannot represent a real star.

9. Analytical solutions of Lane-Emden equation
for n=0,1,5

11. Recall - More massive stars have shorter lifetimes….
The higher a star’s mass, the brighter it is:
40 Msun: ~0.2 million yr!!!
L ~ M3.5
High masses
High-mass stars have much shorter lives than low-mass stars:
Mass
tlife ~ M-2.5
Sun: ~ 10 billion yr.
Low masses
15 Msun: ~ 3 million yr.
3 Msun: ~ 370 million yr.
0.1 Msun: ~ 0.2 trillion yr.

12. Figure 9.25
The mass–luminosity relation shows that the more massive a main-sequence star is, the more luminous it is. The open circles represent white dwarfs, which do not obey the relation. The red line represents the equation L = M 3.5.
Fig. 9-25, p. 198

13. Expansion onto the Giant Branch
Expansion and surface cooling during the phase of an inactive He core and a H- burning shell
Sun will expand beyond Earth’s orbit!

14. “Triple-Alpha Process”
Red Giant Evolution
Core - gravitational contraction makes more heat
H-burning shell keeps dumping He onto the core.
He-core gets smaller & denser & hotter
At shell, density and gravity higher, must burn faster to balance More E, star expands
4 H → He He
GMm
the next stage of nuclear
burning can begin in the core: r
He fusion through the
“Triple-Alpha Process”
4He + 4He  8Be + g
8Be + 4He  12C + g

15. 3 types of Compact Objects
White Dwarfs < 1.4 Msun (Chandrasekar limit)
Neutron Stars > 1.4 Msun < 3 Msun
Black Holes > 3 Msun
White Dwarfs - size of Earth (6000 km radius)
Neutron Stars - size of small city (10 km radius)
Black Holes - smaller* still than a city (< 10 km radius)!
(* really depends on mass)

16. Sizes of Stars & Stellar Remnants (1)
Sizes of Stars
Pauli’s Exclusion Principle:
No 2 electrons can have the same
Quantum mechanical state!
Sun
WD BD
Sizes of Stars & Remnants WD

17. White Dwarfs (2) The more massive a white dwarf, the smaller it is!
The more massive a white dwarf, the smaller it is!
R ~ M-1/3
 Pressure becomes larger, until electron degeneracy pressure can no longer hold up against gravity.
WDs with more than ~ 1.4 solar masses can not exist!
Chandrasekhar Limit = 1.4 Msun

18. The more massive a white dwarf, the smaller it is!
White Dwarfs (3)
The more massive a white dwarf, the smaller it is!
R ~ M-1/3

19. Formation of Compact Objects

20. Summary of Stellar Evolution

21. Mixing Length Theory
The mixing length is conceptually analogous to the concept of mean free path in
thermodynamics: a fluid parcel will conserve its properties for a characteristic
length, , before mixing with the surrounding fluid.
Prandtl described that the mixing length:
“may be considered as the diameter of the masses of fluid moving as a whole
in each individual case; or again, as the distance traversed by a mass of this
type before it becomes blended in with neighbouring masses... “
Prandt 1925