1. PHYS Astronomy
The second exam will be next Monday, November 9 at the regular class time. It is closed book but you may bring in one 8 1/2 X 11 inch “cheat sheet” with writing on both sides. It will cover everything I have covered in class up through last Wednesday’s class (excluding what I covered on the first exam – i.e., starting with the class on 9/28/15). It will not cover anything from today’s or Wednesday’s class. The homework solutions up to assignment #8 are already on line. The solutions to the homeworks will be on line after Wednesday’s class

2. The Source of Stellar Energy
PHYS Astronomy
The Source of Stellar Energy
Recall from our discussion of the sun:
Stars produce energy by nuclear fusion of hydrogen into helium.
In the sun, this happens primarily through the proton-proton (PP) chain

3. Highly temperature dependent
PHYS Astronomy
The CNO Cycle
In stars slightly more massive than the sun, a more powerful energy generation mechanism than the PP chain takes over:
The CNO Cycle.
Highly temperature dependent

4. Energy Transport Structure
PHYS Astronomy
Energy Transport Structure
Inner convective, outer radiative zone
Inner radiative, outer convective zone
Convection
Depends on Opacity - measure of the transparency of a gas
depends on the gas temperature, density, and composition
depends on how easily the gas can move, i.e. its viscosity and any forces (such as gravity) which tend to resist the convective motion. can only operate if it transports more heat than radiation. T
When opacity is high/low (and radiation is inefficient), convection takes over.
CNO cycle dominant
PP chain dominant

5. Summary: Stellar Structure
PHYS Astronomy
Summary: Stellar Structure
Convective Core, radiative envelope;
Energy generation through CNO Cycle
Sun
Mass
Radiative Core, convective envelope;
Energy generation through PP Cycle

6. Convection and the CNO Cycle
As a result of the extreme temperature dependence of CNO burning, those stars that are dominated by CNO fusion have very large values of L/4πr2 in the core. This results in convective instability and convective energy transport is extremely efficient.
• Because of the extreme temperature sensitivity of CNO burning, nuclear reactions in high mass stars are generally confined to a very small region, much smaller than the size of the convective core.

7. Schwarzschild criterion
PHYS Astronomy
Schwarzschild criterion
The conditions under which a region of a star is unstable to convection is expresses by the Schwarzschild criterion:
where g is the gravitational acceleration, and Cp is the heat capacity.
A parcel of gas that rises slightly will find itself in an environment of lower pressure than the one it came from. As a result, the parcel will expand and cool. If the rising parcel cools to a lower temperature than its new surroundings, so that it has a higher density than the surrounding gas, then its lack of buoyancy will cause it to sink back to where it came from. However, if the temperature gradient is steep enough (i. e. the temperature changes rapidly with distance from the center of the star), or if the gas has a very high heat capacity (i. e. its temperature changes relatively slowly as it expands) then the rising parcel of gas will remain warmer and less dense than its new surroundings even after expanding and cooling. Its buoyancy will then cause it to continue to rise. The region of the star in which this happens is the convection zone.

8. Convection and Size
•In stars more than 1.3 times the mass of the Sun, the nuclear fusion of hydrogen into helium occurs via CNO cycle instead of the proton-proton chain. The CNO process is very temperature sensitive, so the core is very hot but the temperature falls off rapidly. Therefore, the core region forms a convection zone that uniformly mixes the hydrogen fuel with the helium product. The core convection zone of these stars is overlaid by a radiation zone that is in thermal equilibrium and undergoes little or no mixing.
In stars of less than about 10 solar masses, the outer envelope of the star contains a region where partial ionization of hydrogen and helium raises the heat capacity. The relatively low temperature in this region simultaneously causes the opacity due to heavier elements to be high enough to produce a steep temperature gradient. This combination of circumstances produces an outer convection zone, the top of which is visible in the sun as solar granulation. Low mass main sequences of stars, such as red dwarfs are convective throughout and do not contain a radiation zone.
As the stellar mass increases, so does the size of the convective core (due again to the large increase in ε with temperature). Supermassive stars with M≈100M would be entirely convective.

9. Low-Mass Star Evolution
PHYS Astronomy
Low-Mass Star Evolution
H-burning continues in a shell around the core, and as T increases, the CNO process can occur in the shell
As CNO T17 energy generation is concentrated in the regions of highest T and highest H content (in shell T ~ 20 x106 K):
Subgiant
branch
This high T causes high P outside the core and the H envelope expands.
- expansion becomes more pronounced when >10% of the stellar mass in the He core.
This early expansion terminates the main-sequence lifetime
Luminosity remains approximately constant - Teff must decrease, star moves right along the red subgiant branch.

10. Solar Composition Change Theoretical estimates of the changes in a Sun-like star’s composition. Hydrogen and helium abundances are shown (a) at birth, just as the star arrives on the main sequence; (b) after five billion years; and (c) after 10 billion years. At stage (b) only about five percent of the star’s total mass has been converted from hydrogen to helium. The change speeds up as the nuclear burning rate increases with time.

11. Expansion onto the Giant Branch
PHYS Astronomy
The shell source slowly burns, moving through the star, as the He core grows.
- the star expands and its surface cools during the phase of an inactive He core and a H-burning shell
But the star cannot expand and cool indefinitely.
When the temperature of the outer layers reach <5000 K the envelopes become fully convective. This enables greater luminosity to be carried by the outer layers and hence quickly forces the star (< ~8 M) almost vertically in the HR diagram - the star becomes a red giant
The Sun will expand beyond Earth’s orbit.

12. Degenerate Matter Star expands but He core continues to contract
PHYS Astronomy
Star expands but He core continues to contract
- core is not producing energy - temperature
not high enough for He fusion
- not enough thermal pressure to resist and balance gravity
- continues to contract until it becomes degenerate
Gas so dense, lower energy levels filled - only two electrons per level- Pauli Exclusion Principle
- max occupancy of phase space volume of phase space cell dxdydzdpxdpydpz=h3
- so in [p,p+dp] 4dpdV/h3 cells each with max occupancy of 2e- (spin ,)
Resists compression - electrons can not be packed arbitrarily close together and have small energies - independent of temperature
Temperature defined mainly by the energy distribution of the heavy particles (He nuclei) - gravitational collapse is resisted by electron degeneracy pressure.

13. “Triple-Alpha Process”
PHYS Astronomy
Red Giant Evolution
H-burning shell keeps dumping He onto the core.
He-core gets denser and hotter until the next stage of nuclear burning can begin in the core:
4 H → He He
He fusion through the
“Triple-Alpha Process”
4He + 4He  8Be + 
8Be + 4He  12C + 

14. Helium Fusion
PHYS Astronomy
For T~108K, triple- reactions start in the very dense core. They generate energy, heating core, and KE of He nuclei increases, increasing the energy production. Energy generation and heating under degenerate conditions - core doesn’t expand, reactions go faster, generates more energy, raises temperature, reactions go faster - leads to runway - the He Flash - core can generate more energy than an entire galaxy. Stars more than ~3 M don’t become degenerate - start burning He without He flash.

15. Red Giant Evolution (3 solar-mass star)
PHYS Astronomy
Red Giant Evolution (3 solar-mass star)
Core T rises until core no longer degenerate - pressure from He fusion causes core to expand - absorbs energy used to support outer envelope
Outer layers contract - surface heats up, luminosity decreases
Star stabilizes - He burning period similar to main phase H burning - luminosity stabilizes but surfaces heats up
As a byproduct, some O is produced:
12C + 4He  16O + 
Inert C and O core created - He/H burning shells - outer layers expand again - surface T decreases
Inactive He
C, O

16. The Asymptotic Giant Branch
PHYS Astronomy
The Asymptotic Giant Branch
The core will soon consist only of C+O, and in a similar way to before, the CO-core grows while a He-burning shell source develops.
These two shell sources force expansion of the envelop and the star evolves up the red giant branch a second time - this is called the asymptotic giant branch.
For high metallicity stars, the AGB coincides closely with the first RGB.
For globulars (typical heavy element composition 100 times lower than solar) they appear separated.

17. PHYS Astronomy
Red Dwarfs
Stars with less than ~ 0.4 solar masses are completely convective.
Mass
Consume H slowly - long lifetimes - longer than the age of the universe
 Hydrogen and helium remain well mixed throughout the entire star.
 No phase of shell “burning” with expansion to giant.
Star not hot enough to ignite He burning.

18. Sunlike stars (~ 0.4 – 4 solar masses) develop a helium core.
PHYS Astronomy
Sunlike Stars
Sunlike stars (~ 0.4 – 4 solar masses) develop a helium core.
Mass
 Expansion to red giant during H burning shell phase
 Ignition of He burning in the He core
 Formation of a degenerate C,O core

19. Evolution of Massive Stars (> 8 M)
PHYS Astronomy
Evolution of Massive Stars (> 8 M)
The electrons in their cores do not become degenerate until the final burning stages, when iron core is reached.
Mass-loss plays an important role in the entire evolution
The luminosity remains approximately constant in spite of internal changes. The track on the HRD is therefore horizontal - star becomes a supergiant.
For a 15-25M star there is a gradual redwards movement. But for higher mass (or stars with different initial compositions) the star moves back and forth between low and high effective temperatures as it burns successively higher mass elements

20. From He Burning to Core-Collapse
PHYS Astronomy
From He Burning to Core-Collapse
The He burning core is surrounded by a H-burning shell
The triple- process liberates less energy per unit mass than for H-burning (~10%). Hence the lifetime is shorter, again around 10%
There is no He-flash as densities in the He-core are not high enough for electron degeneracy.
Star has a core of 12C and 16O, surrounded by He and H burning shells.
The core will again contract and the temperature will rise, allowing C and O burning to Mg and Si.
This process continues, with increasing Z, building up heavier and heavier elements until the iron group elements of Ni, Fe and Co are formed. The core is surrounded by a series of shells at lower T, and lower 

21. Major Nuclear Burning Processes
PHYS Astronomy
Major Nuclear Burning Processes
Common feature is release of energy by consumption of nuclear fuel. Rates of energy release vary enormously.
Nuclear Fuel
Process
Tthreshold
106K
Products
Energy per nucleon (Mev) H PP ~4 He
6.55
CNO 15
6.25 3
100
C,O
0.61 C
C+C
600
O,Ne,Ma,Mg
0.54 Ne
Ne+Ne
1000
O,Mg
~0.4 O
O+O
1500
S,P,Si
~0.3 Si
Alpha
3000
Co,Fe,Ni
<0.18

22. Fusion Into Heavier Elements
PHYS Astronomy
Fusion Into Heavier Elements
Fusion into heavier elements than C, O:
requires very high temperatures; occurs only in very massive stars (more than 8 solar masses).

23. The Life “Clock” of a Massive Star (> 8 Msun)
PHYS Astronomy
The Life “Clock” of a Massive Star (> 8 Msun)
Let’s compress a massive star’s life into one day…
H  He 12 11 1
Life on the Main Sequence
+ Expansion to Red Giant: 22 h, 24 min.
H burning 10 2 9 3 4 8 7 5 6
H  He
He  C, O 12 11 1 10 2
He burning:
(Red Giant Phase) 1 h, 35 min, 53 s 9 3 4 8 7 5 6

24. The Life “Clock” of a Massive Star (2)
PHYS Astronomy
The Life “Clock” of a Massive Star (2)
He  C, O
H  He 12 11 1
C  Ne, Na, Mg, O 10 2 9 3 4 8
C burning:
6.99 s 7 5 6
C  Ne, Na, Mg, O
H  He
Ne  O, Mg
He  C, O
Ne burning:
6 ms
23:59:59.996

25. The Life “Clock” of a Massive Star
PHYS Astronomy
The Life “Clock” of a Massive Star
C  Ne, Na, Mg, O
H  He
Ne  O, Mg
He  C, O
O  Si, S, P
O burning:
3.97 ms
23:59:
C  Ne, Na, Mg, O
H  He
Ne  O, Mg
He  C, O
O  Si, S, P
Si  Fe, Co, Ni
Si burning:
0.03 ms
The final 0.03 msec.

26. Summary of Post Main-Sequence Evolution of Stars
PHYS Astronomy
Summary of Post Main-Sequence Evolution of Stars
Supernova
Evolution of Msun stars is still uncertain.
Fusion proceeds; formation of Fe core.
Mass loss in stellar winds may reduce them all to < 4 Msun stars.
M > 8 Msun
Fusion stops at formation of C,O core.
M < 4 Msun
Red dwarfs: He burning never ignites
M < 0.4 Msun

27. Mass Loss From Stars
PHYS Astronomy
Stars like our sun are constantly losing mass in a stellar wind ( solar wind).
The more massive the star, the stronger its stellar wind - our Sun loses mass at M per billion years
Large radiation pressure of red giant drives mass-loss - particles absorb photons from radiation field - accelerated out of the gravitational potential well.
- observations of red giants and supergiants are in the range 10-9 to 10-4 M yr-1
WR 124

28. Mass-Loss from High Mass Stars
PHYS Astronomy
Mass-Loss from High Mass Stars
Large amount of evidence now that high mass stars loose mass through a strong stellar wind.
The winds are driven by radiation pressure - UV photons from a hot, very luminous star absorbed by the optically thick outer atmosphere layers.
atmosphere is optically thick at the wavelength of many strong UV transitions (resonance transitions) of lines of Fe, O, Si, C (and others).
photons absorbed, imparting momentum to the gas and driving an outward wind.
measured in O and B-stars, and leads to (terminal) wind velocities of up to 4000 km/s and mass-loss rates of up to 5 x 10-5 M yr-1
Huge effect on massive stellar evolution - the outer layers are effectively stripped off the star.

29. Mass-loss is generally classified into two types of wind:
PHYS Astronomy
Mass-loss is generally classified into two types of wind:
1. Stellar wind: described by empirical formula, linking mass, radius, luminosity with simple relation and a constant from observations. Typical wind rates are of order 10-6 M yr-1
2.A superwind: a stronger wind, leading to stellar ejecta observable in shell surrounding central star
- periodic eruptions called thermal pulses can eject large amounts of mass
- triple  process very temperature sensitive ( T30)
- makes He-fusion cell very sensitive to temperature - unstable leads to sudden eruptions that lift outer layers of star and drives gas from surface
Mass loss can be as much as 10-5 M yr M star can reduce mass to
3 M in just 500,000 years
Existence of superwind suggested by two independent variables
- the high density observed within the observed shells in stellar ejecta
- relative paucity of very bright stars on the AGB - expected compared to observed is >10
So superwind causes envelope ejection - cores evolve into C-O white dwarfs. - Core mass at tip of AGB ~0.6 M - close to mass of most white dwarfs.

30. PHYS Astronomy
Age and Metallicity
Two of the fundamental properties of stars that we can measure – age and chemical composition
Composition parameterised with
X,Y,Z  mass fraction of H, He and all other elements
e.g. X = ; Y = ; Z = 0.017
Note – Z often referred to as metallicity
Would like to studies stars of same age and chemical composition – to keep these parameters constant and determine how models reproduce the other observables

31. Evidence for Stellar Evolution: Star Clusters
PHYS Astronomy
We observe star clusters
Stars all at about the same distance
Dynamically bound
Same age
Same chemical composition
Can contain 103 –106 stars
Have wide range of masses - can be used test our understanding of stellar evolution.
NGC3603 from Hubble Space Telescope

32. HR Diagram of a Star Cluster
PHYS Astronomy
HR Diagram of a Star Cluster
Stellar clusters large group of stars born at same time, age of cluster will show on HR-diagram as the upper end, or turn-off of the main-sequence.
We can use this as a tool (clock) for measuring age of star clusters. Stars with lifetimes less than cluster age, have left main sequence. Stars with main-sequence lifetimes longer than age, still dwell on main-sequence.

33. Estimating the Age of a Cluster
PHYS Astronomy
The lower on the MS the turn-off point, the older the cluster.

34. Example: HR diagram of the Globular Star Cluster M 55
PHYS Astronomy
Example: HR diagram of the Globular Star Cluster M 55
Horizontal
branch
High-mass stars evolved onto the giant branch
Turn-off point
Low-mass stars still on the main sequence

35. Star Clusters 47 Tuc – Globular cluster NGC3293 - Open cluster
PHYS Astronomy
47 Tuc – Globular cluster
NGC Open cluster
stars
About 25 pc in diameter
Open, transparent appearance - stars are not crowded together
stars
pc in diameter
Nearly spherical and much closer together than open clusters

36. Globular Cluster HRD Open Cluster HRD
PHYS Astronomy
Globular Cluster HRD
Open Cluster HRD
MS turn off point varies massively, faintest is consistent with globulars
Maximum luminosity of stars can get to Mv-10
Very massive stars found in these clusters
MS turn-off points in similar position. Giant branch joining MS
Horizontal branch from giant branch to above the MS turn-off point
Horizontal branch often populated only with variable RR Lyrae stars

37. Open vs. Globular Clusters
PHYS Astronomy
Open vs. Globular Clusters
Differences are interpreted due to age – open clusters lie in the disk of the Milky Way and have large range of ages. The Globulars are all ancient, with the oldest tracing the earliest stages of the formation of Milky Way (~ 12 109 yrs)
Globular clusters are old and metal poor - produce a horizontal branch
Mass-loss dependent on metallic content - drives bluewards evolution
Open clusters are metal rich - produce a red clump
More metal rich stars appear towards red
Clump stars  extreme red end of HB

38. PHYS Astronomy

39. Modelling star clusters
PHYS Astronomy
Best way to check stellar evolutionary calculations is to compare calculated and observed tracks. But can’t observe stars as they evolve - need to use star clusters.
Isochrones:
A curve which traces the properties of stars as a function of mass for a given age.
Be clear about the difference with an evolutionary track - which shows the properties of a star as a function of age for a fixed mass.
Isochrones are particularly useful for star clusters - all stars born at the same time with the same composition. Consider stars of different masses but with the same age . Make a plot of Log(L/L) vs. LogTeff for an age of 1Gyr. The result is an isochrone.
Important - When we observe a cluster, we are seeing a “freeze-frame” picture at a particular age. We see how stars of different masses have evolved up to that fixed age (this is not equivalent to an evolutionary track).

40. Pleiades young open cluster Age~ 100Myrs NGC6231 young cluster
PHYS Astronomy
Pleiades young open cluster
Age~ 100Myrs
NGC6231 young cluster
Age~ 6Myrs
47 Tuc : globular cluster. Age= 8-10Gyrs
NGC188: old open cluster . Age= 7Gyrs