1. Stars
Measuring Stars
For any nearby star, we can pretty easily measure the brightness
How much power arrives at Earth from the star
By a variety of methods, we can often determine the distance
Combining this, we can figure out the luminosity L
How bright the star really is
We can get a good estimate of the surface temperature T from the spectrum
Bluer stars are hotter; redder stars are cooler
We can also get a good idea of the composition from the spectrum
¾ hydrogen, ¼ helium, 1-3% other stuff
Most stars are too small to resolve
No direct measurement of radius R
Indirectly deduce radius from formula

2. “Oh Be A Fine Girl, Kiss Me.”
Spectral Type
The following are all equivalent information:
The surface temperature of a star
The color of the star
The spectral type of the star
From hottest to coldest, OBAFGKM
Subdivided 0-9, with 0 the hottest
Sun is a G2 star
The spectral type is easy to determine
Why I hate astronomers
“Oh Be A Fine Girl, Kiss Me.”

3. Spectral Type

4. Intrinsic Properties of Stars
To describe stars, we want to talk about intrinsic properties
Luminosity
Composition
Temperature
Composition is almost always the same
Mass is difficult to measure
Radius can be deduced from Luminosity and Temperature
Radius
Mass
Temperature and Luminosity

5. The Hertzsprung-Russell Diagram
A plot of temperature vs. luminosity
Hot on left, cold on right
Luminous at top, dim at bottom
Stars fall into categories:
The Main Sequence contains about 90% of the bright stars
The Giants are rare but very bright
The Supergiants are very rare but extremely bright
The White Dwarfs are not uncommon but very dim

6. Main Sequence Stars
Main Sequence stars have different sizes, masses, and luminosities
But spectral class determines everything else
This diagram shows correct relative sizes and approximate colors of stars
But not correct relative luminosities

7. Main Sequence Stars Stars in the main sequence are similar to the Sun
At their core, they are fusing hydrogen to make helium
Other stars are doing different things
Pre-main sequence stars are not yet hot enough to fuse hydrogen
Red giants have run out of hydrogen at their core, and are fusing it in a small shell surrounding the core
Horizontal branch stars are fusing helium at their core to carbon and oxygen
White dwarfs are burnt-out carbon/oxygen
We will focus on main sequence stars
And will talk a little about pre-main sequence stars
And I have time, will talk about white dwarfs and neutron stars

8. Stars Are in Balance
Stars spend almost all their lives in a steady state
Pressure vs. gravity
Gravity wants them to shrink
Pressure wants them to expand
If a star gets out of gravitational balance, it adjusts to compensate
If gravity is winning, it will shrink until pressure rises to compensate
If pressure is winning, it will expand until pressure drops
Luminosity vs. nuclear energy generation
Luminosity causes them to lose power
Nuclear fusion replaces it
If a star gets out of energy equilibrium, it adjusts to compensate
If too little power, shrinks and heats up to increase power
If too much power, expands and cools to decrease power

9. Building a Simple Model for Stars
We would like to build an order of magnitude model for main sequence stars
Approximation 1: They are spherical
Pretty good approximation for realistic stars
Approximation 2: The nuclear energy generation rate is infinitely sensitive to temperature
At 10 million K, nuclear energy generation is very slow
At 20 million K, nuclear energy generation is very fast
Therefore, all stars have central temperature Tc ~ 15 million K
Approximation 3: We can approximate all derivatives as differences
Terrible approximation
Correct way to do it is with differential equations
Approximation 4: All factors of 2, , etc., are too small to worry about

10. Sources of Pressure There are three sources of pressure in stars
Degeneracy pressure caused by the Pauli Exclusion principle
Not important for most main sequence stars
Ideal gas pressure
You probably learned a version of this formula in high school chemistry
n is the number of independent particles per unit volume
Because the core is so hot, atoms are completely ionized
When computing n, add together nuclei and all electrons
This is the main source of pressure in the Sun
Radiation pressure is from the electromagnetic radiation
This is more important at higher temperatures
Major source of pressure in high mass stars

11. Radiation Pressure
Imagine a region filled with black-body radiation with a mirror on one side
If the mirror weren’t there, energy would be flowing out at a rate
Consider a single photon bouncing off
Each photon that attempts to leave has momentum
By bouncing off the mirror, it transfers momentum
Collectively, the photons produce a pressure
This works out to

12. Radius-Temperature-Mass Relation (1)
A star is a sphere of mass M and radius R
Divide it in half
Both pieces have mass ½M ~ M
Separated by a distance R
The gravitational force between these two pieces is
The area separating these two halves is R2 ~ R2
The pressure is force over area
This must be matched by the ideal gas pressure
Use central temperature for T
The number density is the number of particles divided by the volume
The volume of a sphere of radius R is about R3
The number of particles is the mass M divided by the mass per particle 
Equate this to the pressure found above
Solve for R R R

13. Radius-Temperature-Mass Relation (2)
The star is made primarily of 1H with mass 1u
But each atom has proton + electron = 2 particles
Therefore
So, how well did we do?
Try the formula out for the Sun
Far better than we deserved to do! R

14. Pre Main-Sequence Stars
Note that for fixed M, the product RTc will be constant
Pre-main sequence stars are not producing energy in their interiors
They lose energy over time
They will consequently shrink
But this relationship will apply at all times
Since M is constant, Tc must rise as R decreases
Eventually, the star will reach a temperature hot enough for fusion to start
10 – 20 million K
At this point, the star becomes a main sequence star

15. Radiative Transport (1)
For the Sun, the energy mostly flows via radiation
Consider a region of slowly changing temperature
There are many electrons which can scatter or absorb photons
It is opaque
The photons scatter and/or are absorbed many times
Because of these many scatterings, the photons will be almost thermal
They will typically go a short distance l before being scattered or absorbed
Consider an imaginary barrier perpendicular to the temperature gradient
The light moving right came from a distance l to the left
The light moving left came from a distance l to the right
The difference between these is the net flux

16. Radiative Transport (2)
The distance l is very small compared to the scale of the star, so we can approximate this as a derivative:
We therefore have
The mean free path l was how far it goes between scatterings
The more stuff there is to scatter from, the smaller it will be
Therefore, l is inversely proportional to the density

17. Mass-Luminosity Relationship
Radiation moves out of the star because it is hotter at the center than the outside
The temperature at the center is roughly constant Tc, and by comparison, temperature on the surface is 0
Multiply by the area of the star to get luminosity
A = R2  R2
Density is mass over volume
Volume proportional to R3
Radius is proportional to mass
Central temperature is roughly constant
By comparing to the Sun, we therefore have

18. Mass-Temperature Relationship
The luminosity is also related to the surface temperature and radius
And radius is proportional to mass
Equating these two expressions, we have
And therefore have
Surface temperature rises slowly as a function of mass
Luminosity rises quickly as a function of mass
The different stars on the main sequence correspond to different masses

19. The Main Sequence High mass stars are hot and very luminous
Intermediate mass stars are in between
Low mass stars are cool and very dim

20. A Dose of Reality How did I lie to thee? Let me count the ways
The nuclear generation rate is not infinitely sensitive to temperature
Higher mass must have slightly higher central temperature
This changes a little
For 0.43 – 2 Msun, the relationship is more like
For high mass stars, the pp-chain is not the most important fusion
There is another method called the CNO cycle
This is very sensitive temperature
So our approximations work a little better
For very high mass stars, radiation pressure dominates
Entire arguments fall apart
For very low mass stars, convection throughout the star, not radiative transport