1. Final End of Low Mass Stars
Todays Lecture:
Final End of Low Mass Stars
• Planetary Nebulae
• White Dwarf Stars
• Electron Degeneracy Pressure
• White Dwarf Mass Limit
Homework 5: Due Classtime, Tuesday, March 11
Help Session: Wed., March 12, 5:00-6:30, LGRT 1033
Midterm Exam, in class Thursday, March 13
Reading for Today: Chapter
Reading for Next Lecture: Chapter 11

2. Mass Loss From Stars
During the red giant and AGB phases, stars develop winds that remove some of the star's outer envelope. At the end of the AGB stage, the outer envelope of the star is ejected into space, leaving behind the core of the star.

3. Post AGB Stars or Proto-Planetary Nebulae
The image to the left captures a star at the end of its AGB phase. We see the extended wind material surrounding this star. The star itself is hidden from view this wind material.

4. Planetary Nebulae
The outer envelope expands and escapes the gravitational pull of the star forming a planetary nebula containing the remaining outer envelope of the star.
The remaining core forms a white dwarf star supported by electron degeneracy pressure.
M < 0.4 M⊙ He-rich white dwarf M⊙ < M < 4 M⊙ - C/O-rich white dwarf star M⊙ < M < 8 M⊙ O/Ne/Mg-rich white dwarf star

5. Planetary Nebulae
The envelope of the star is ionized by the ultraviolet radiation from the white dwarf core. Planetary nebulae are emission line nebulae (hot, tenuous gas).

6. Planetary Nebulae
A few examples with very symmetric expanding shells of ionized gas surrounding a white dwarf star.

7. Planetary Nebulae
Sometimes these nebula can be complex, as is the case shown here. Evidence that the current ejection of the outer envelope is running into the material expelled earlier by the star's winds.

8. Planetary Nebulae
Some show an hour glass shape, maybe due to a rapidly expanding shell interacting with slower moving material concentrated around the equator that constricts the flow.

9. The planetary nebula gas disperses into space (recycling material for the next generation of stars) and fades from view in tens of thousands of years leaving behind the core of the star which will become a white dwarf star.

10. Life Cycle of a Star Like the Sun:
The path in the H-R diagram is shown to the right, while the time in each stage of evolution is given below:
(1) Main Sequence – 1010 yrs
(2) Red Giant Star – 109 yrs
(3) Horizontal Branch (Yellow giant star) – 108 yrs
(4) Asymptotic Giant Branch (AGB) – 107 yrs
(5) Planetary Nebulae/White dwarf – 104 yrs
(6) Cooling white dwarf - > 109 yrs

11. Example of White Dwarf Star is Sirius B
Sirius A is a main sequence A star and Sirius B is a white dwarf star, L = L⊙, T = 24,800 K, R = R⊙
From orbital data can derive mass of Sirius B of 0.97 M⊙
Density = mass/volume = 2.2 x 106 gm cm-3.
A cubic centimeter has a mass of 2200 kg !!!!!

12. Electron Degeneracy Pressure
The white dwarf star is the inert remnant core of the low mass star, composed of atomic nuclei and electrons. As we will see, these stars are supported by electron degeneracy pressure.
In general, the gas in stars behaves like an ideal gas, and the pressure results from the random motion of the gas particles – their thermal motions.
P = n k T
However in addition to thermal motions, the electrons in the gas can have motions due quantum mechanical affects:
There are two important principles, the Pauli Exclusion Principle and the Heisenberg Uncertainty Principle, which gives rise to these affects.

13. Pauli Exclusion Principle:
This principle states that no two electrons can have exactly the same quantum mechanical state. Applied to free electrons in the white dwarf star, “state” means the particles position and momentum.
If electrons are packed tightly together, they must have larger and larger momentum to be in different quantum states.
The electrons therefore have motions that are independent of temperature and arise due to quantum mechanics. These motions give rise to a pressure we call the electron degeneracy pressure.
For free electrons, knowledge of the quantum mechanical state is restricted by the Heisenberg Uncertainty Principle:

14. Heisenberg Uncertainty Principle:
It is impossible to define the position and momentum of a particle to an accuracy better than h/2π, where h is Planck's constant (consequence of the wave nature of particles).
Thus, (Δx)(Δpx) > ℏ, where ℏ = h/2π.
This implies that for electrons that are very closely packed together, and thus have less uncertainty in their position, must have a higher uncertainty in their momentum, and therefore a higher average momentum.
Therefore the value of px can be roughly approximated by Δpx. Therefore:
px ~ ℏ/Δx

15. If the number density of electrons is ne (the number of electrons per unit volume), then each electron is confined to a volume given roughly by: V ~ ne-1.
The average separation of particles:
Δx ~ V1/3 ~ ne-1/3.
Therefore the momentum is given by:
px ~ ℏ/Δx ~ ℏ ne1/3.
Since vx = px/me , then vx ~ ℏ ne1/3 me-1
For a typical density in a white dwarf star (1x106 gm cm-3), the electron number density is 3x1029 cm-3. The velocity is therefore is 8x109 cm s-1 or 25% of the speed of light !!

16. In an earlier lecture we discussed pressure, and derived the following expression for gas pressure:
P = n vx px = n vx2 m
We can use this expression to determine the pressure exerted by the motion of these “degenerate” electrons”:
Pe ~ ne vx2 me ~ ℏ2 ne5/3 me-1
If we do this calculation properly, we get the following expression for the electron degeneracy pressure:
Pe = ℏ2 ne5/3 me-1
Close to our crude approximation.

17. The white dwarf stars have a density structure to produce an net outward electron degeneracy pressure force that balances gravity.
White dwarf stars are in hydrostatic equilibrium.
Since electron degeneracy pressure has nothing to do with temperature, the thermal energy can be radiated away and the white dwarf star can cool, but this will not affect the hydrostatic balance.
To produce an equivalent thermal pressure the stars would have to maintain a central temperature in excess of a billion degrees.

18. Mass-Radius Relation for White Dwarf Stars
A white dwarf star is made up of atomic nuclei (charge Z, number density ni and atomic number A) and electrons (number density ne). Thus: ne = Z ni.
The mass density is given: ρ = me ne + mp np + mn nn
ρ = Z ni me + Z ni mp + (A-Z) ni mn
ρ = Z ni (me + mp) + (A-Z) ni mn
The mass of the neutron and proton are roughly the same and if we ignore the electron mass, then: ρ ~ A ni mp = A/Z ne mp
Therefore we can rewrite the electron degeneracy pressure as:
Pe ~ ℏ2/me (Z/A)5/3 (ρ/mp)5/3

19. For stars in hydrostatic equilibrium, we showed earlier that the central pressure of a star is approximately:
Pc ~ GM2/R4.
If we assume that this is provided by electron degeneracy pressure, we find for white dwarf stars:
G M2/R4 ~ ℏ2/me (Z/A)5/3 (ρ/mp)5/3
We can crudely approximate: ρ ~ M/R3, and we find:
G M2/R4 ~ ℏ2/me (Z/A)5/3 (1/mp)5/3 (M5/3 /R5)
multiplying by R5 and dividing by G M2 gives the following:
R ~ ℏ2/(G me mp5/3) (Z/A)5/3 M-1/3

20. So the radius is: R ~ ℏ2/(G me mp5/3 ) (Z/A)5/3 M-1/3
Therefore R  M-1/3, note that as mass increases, the radius of a white dwarf decreases !!!!
As the mass increases, the radius of the star must decrease to produce the larger density needed to produce a larger pressure to resist gravity.
If you solve carefully for the equilibrium configuration of a white dwarf star supported by electron degeneracy pressure, find the following mass-radius relation:
R = 4.5 ℏ2/(G me mp5/3 ) (Z/A)5/3 M-1/3
Note for C and O rich white dwarf stars, Z/A = The homework has you compute the radius for a 0.5 solar mass white dwarf star.

21. Mass-Radius Relation for White Dwarf Stars
The plot below shows the mass-radius relation for white dwarf stars supported by electron degeneracy pressure.

22. Relativistic Electrons
As white dwarf stars becomes more massive, their radius shrinks, and density increases. The velocity of the electrons approach the speed of light. As before the momentum of the electrons is still given by: px ~ ℏ ne1/3.
However we cannot assume that vx = px/me.
Remember the pressure is given by: P = n vx px, and therefore the relativistic electron degeneracy pressure is:
Pe ~ vx ℏ ne4/3
As the electrons start to approach the speed of light, the pressure does not increase as fast with increasing density as for non-relativistic electrons.

23. Relativistic electrons provide less pressure support for a given density, so the density must be larger and the radius smaller than we estimated from the non-relativistic expression. The relativistic mass-radius relation is shown below:

24. White Dwarf Star Mass Limit
The maximum degeneracy pressure is when vx ~ c. The ultrarelativistic degeneracy pressure is given by:
Pe ~ c ℏ ne4/3 , or Pe ~ c ℏ (Z/A)4/3 (ρ/mp)4/3
We can set ultra-relativistic electron degeneracy pressure equal to central pressure for hydrostatic support
G M2/ R4 ~ c ℏ (Z/A)4/3 (ρ/mp)4/3 ~ c ℏ (Z/A)4/3 (M/R3mp)4/3
Thus we find: M ~ [c ℏ /(Gmp4/3)]3/2 (Z/A)2
The radius drops out and because this was the maxiumum pressure, this mass is the maximum mass of a white dwarf star.

25. Chandrasekhar first showed that if the mass were sufficiently large there is no stable solution, the radius of the star would formally shrink to zero !!!
A more accurate expression for this mass (assuming Z/A = 0.5) is given by:
Mch = [c ℏ /(Gmp4/3)]3/2 (Z/A)2 ~ 2.9x1033 gm, or
Mch = 1.44 solar masses
We call this the Chandrasekhar mass.
Electron degeneracy pressure cannot stop gravity if mass exceeds this limit, and the object must collapse.

26. Electron degeneracy pressure does not depend on temperature, so even if the white dwarf star cools, electron degeneracy pressure still supports white dwarf against gravity. As the star cools, its luminosity plummets and eventually the white dwarf fades from view (cooling time > 109 years).

27. White Dwarf star in a binary star system.
We will discuss white dwarf stars with close binary companions after spring break.
White Dwarf Star