1. Some Later Stages of Low- Mass Stars
Dead Stars
Some Later Stages of Low- Mass Stars
Eventually, hydrogen at the core gets completely used up
Core is now pure 4He “ash”
It continues burning 1H  4He in a thin shell
This star becomes a red giant
After a while, the 4He becomes so hot, it can undergo nuclear fusion
And shortly thereafter, it can add one more 4He to make 16O
In the Sun, this is the last nuclear process that occurs
The Sun eventually loses almost all its 4He and 1H and ends up as a burnt-out carbon/oxygen star
No longer able to produce new energy it loses heat slowly forever
It becomes a white dwarf
3 4He  12C + 
12C + 4He  16O + 

2. White Dwarfs White dwarfs are made primarily of 12C mixed with 16O
They are no longer undergoing nuclear fusion
Therefore, they are not in a balance of energy lost vs. energy generated
They are not held up by ideal gas pressure
Instead, the Pauli exclusion principle creates pressure
No two particles of the same type can be in the same quantum state
Therefore, as you add more and more to a small volume, they must go into higher energy states
Each type of fermion creates its own degeneracy pressure
Degeneracy pressure exists even at zero temperature
It has nothing to do with interactions between particles

3. Degeneracy Pressure (1)
We will do an order of magnitude calculation to find the degeneracy pressure
Similar to what you did in a homework problem
Consider first a single particle in a box of size V = L3
The energy of a particle in such a box is given by
The minimum energy is
Write in terms of volume
Pressure is the derivative of energy with respect to volume
Because of Pauli exclusion principle, each particle needs its own volume (sort of)
The total volume is
Therefore we have L

4. Degeneracy Pressure (2)
Actual computation is more complicated
Take into account two spin states
Particles actually share a volume
Make assumption of many particles
Correct formula works out to
Note that calculation is independent of temperature
Cooling the star doesn’t change degeneracy pressure
Effect is inversely proportional to mass
Electrons contribute much more than protons, neutrons, or nuclei
This effect has nothing to do with particle interactions
Even neutrinos would have degeneracy pressure

5. Radius vs. Mass for White Dwarfs (1)
Order of magnitude estimate for size of white dwarf
Ignore factors of 2, , etc.
As we did before, assume a star of radius R and mass M
Divide the star in half
Each half has mass ½ M ~ M and are separated by a distance R
The force between the two halves is about
Pressure is force over area
Area of circle is R2 ~ R2
This must match degeneracy pressure
Number density is number of particles over volume
Volume ~ R3
Number of particles is mass divided by mass per particle R R

6. Radius vs. Mass for White Dwarfs (2)
M = mass of star
m = mass of particle
causing pressure
 = mass per degen-
erate particle
Equate these two expressions
Solve for R
For a white dwarf, the electron causes the pressure
 is the mass per electron
Partly 12C, which would have
Partly 16O, which would have
Clearly,
Write the mass in terms of the Sun’s mass

7. Radius vs. Mass for White Dwarfs (3)
If you do all the differential equations properly, you actually get
Compare, REarth = 6370 km
Note that the more massive the white dwarf, the smaller the star
This is still not accurate, because for masses > 0.5 MSun, the electrons become relativisitic
This causes a faster reduction in the size
In fact for the Chandrasekhar mass, 1.4 MSun, the star becomes unstable and undergoes catastrophic collapse

8. Radius vs. Mass for White Dwarfs (4)
By the time the Sun dies, it will have lost a significant fraction of its mass
Mf = 0.6 Msun
Most white dwarfs are in the range 0.5 – 1.3 MSun

9. White Dwarf Cooling and the H-R Diagram
White dwarfs have a relatively small range of radii and masses
M = 0.6 – 1.3 Msun
R = 3,000 – 10,000 km
As they age, they cool, but their size doesn’t change
Their luminosity drops as they age
They end up in a small strip in lower left side of H-R diagram
Young, hot ones near the top
Old, cool ones near the bottom

10. Later Stages of High-Mass Stars
Stars lighter than about 8 solar masses eventually shed most of their mass and end as carbon/oxygen white dwarfs with mass < 1.4 Msun
Stars heavier than this instead go through a large variety of additional fusion processes, ending with 56Fe
They then undergo catastrophic core collapse
The 56Fe disintegrates back into protons and neutrons
The degeneracy pressure from the electrons is going crazy
Electrons have enormous energy because of their low mass
The electrons are then absorbed by electron capture to make neutrons
With the electrons gone, the collapse speeds up
But eventually, the neutrons stop the collapse
The energy released by this fall explodes the rest of the star
A type Ia supernova
But the ball of neutrons – a neutron star - remains + + p+ n0 e- 

11. Radius vs. Mass for Neutron Stars (1)
We can – approximately – use the same formulas we did for white dwarfs
For the neutron star, the particle causing pressure is the neutron
For a neutron star, this is also , the mass per neutron
As before, write it in terms of the Sun’s mass
Assuming our formula is right, we then find
M = mass of star
m = mass of particle
causing pressure
 = mass per degen-
erate particle

12. Radius vs. Mass for Neutron Stars (2)
This value is not actually right, because:
We didn’t actually solve the differential equations
This increases the radius a bit
The neutrons are strongly interacting, so ignoring interactions is wrong
The neutrons are relativistic
Makes the neutron star unstable if mass gets too large
The gravity is so strong that you have to take general relativity into account
See next section of course
Because of all these effects, there is once again a maximum mass for neutron stars
The actual limit is not known very well
Current estimates put it in the range 2.5 – 3.0 MSun

13. Radius vs. Mass for Neutron Stars (3)5 10 15 20
Stars in the range 8 – 30 Msun end up as neutron stars
Final neutron star mass 1.4 – 3.0 Msun
More massive stars end as black holes (next section)