We can determine luminosities of stars by measuring their distances
(from parallax) and brightnesses, and using the inverse square law.
When parallax is measured using the baseline provided by the Earth orbital
motion around the Sun, we know the distance in terms of AU. How do we know
what 1AU really is in kilometers? Remember, that Kepler's laws helped us relate the periods
of planets in years to distances from the Sun in AUs. So, in order to determine
the absolute distance scale of the Solar System, we need to know at least one
distance between the planets. Historically, Giovanni Cassini in 17th century
measured the distance to Mars by using parallax using the baseline on the Earth surface.
This helped to set the absolute distance scale in the Solar System. Modern methods
include bouncing radar signal off planets and measuring the propagation time delay. 

Back to stars. Enjar Hertzsprung and Princeton's own Henry Norris Russell
independently made the 'Hertzsprung-Russell' (or HR) diagram: plotting
temperature vs. luminosities of stars.  Most stars fall along a main
sequence, hotter stars having higher luminosities.

 Luminosity and temperature are related by:
L = sigma T^4 4 pi R^2
    
So it makes sense that hotter stars have higher luminosities.  There
are hot stars with low luminosities; they must be small; we call these
'white dwarfs'.  Similarly, very luminous red stars must be enormous,
we call them 'red giants' (or for the really big ones, red
supergiants).  

We know the distances and luminosities of the stars. But what are their masses?
We measure the masses by studying binary stars, or stars orbiting each other
(actually, stars in a binary orbit their common center of mass). 
There, if we know any two of the three quantities (period, velocity, separation), 
we can obtain the mass of the stars from using a slightly more sophisitaced 
version of the Kepler's laws. The gravity that is responsible for the stellar orbits
is set by masses of the stars, so by measuring the orbital properties, we can 
reconstruct their masses. 

We discussed that there are 3 types of binaries: visual, eclipsing, and spectroscopic. 
In visual binaries you can see the stars move on the sky, eclipsing binaries show periodic
dimming due to eclipses, and spectroscopic binaries display absorption lines in their
spectra that can be seen periodically moving due to the Doppler effect. Knowing the shift
of the spectral lines from the Doppler effect, allows us to find the velocities of stars. 

When we determine the masses of representative stars, we learn that the main 
sequence is a sequence in mass: high surface temperature, high luminosity stars are
particularly massive. Mass determines everything about the star because the 
size, temperature and luminosity of the star are set by the pull of star's gravity,
which is set by mass. 

Stars form from clouds of gas (molecular Hydrogen), the cloud starts to 
collapse and fragment into pieces. These pieces initially collapse converting 
the gravitaional energy of the gas into heat. Eventually the collapse
is slowed by gas pressure. This contracting "protostar" radiates energy
as a black body, and the energy needed for radiation is coming from the 
contraction. The contraction is halted when the core is hot enough to find a 
stable source of energy at the center -- nuclear fusion. This when the "star"
is turned on, and enters main sequence. 

  What keeps a star shining?  The heat generated as the gas that
  formed the star fell together is not nearly enough to keep a star
  like the Sun shining for billions of years.  

Stars are balls of gas held together by gravity.  The star is in
balance between gravity pulling it together, and the heat pressure
pushing outwards.  The surface of the Sun is at 6000K, but it gets
hotter and denser as you go inward.  The core of the Sun is at 15
million K, and has a density 150 times higher than water.  At this
temperature, the hydrogen and helium is a *plasma*; the electrons are
torn off from the nuclei, and are roaming freely.  Protons don't like
to come close (they are like charges, and therefore repel each other),
but at these high temperatures and pressures, they are whizzing around
so fast that they can collide.  If they get close enough, they stick
(something called the 'strong force' takes over), and in a series of
reactions, protons can become helium nuclei:
    p + p + p + p --> ppnn + two positrons + two neutrinos + gamma rays

The really interesting thing is that the four protons have about 0.7%
more mass than the helium nucleus (the ppnn thing).  Somehow, mass has
been lost in the reaction.  Where did it go?  The answer is that it
got turned into energy.  Einstein in 1905 realized (as he was
developing the Special Theory of Relativity) that mass and energy can
transform into each other in special circumstances.  One of these
special circumstances is that which holds in the interior of the Sun.
The amount of energy equivalent to a given amount of mass is given by
Einstein's famous formula:
           E = m c^2
(c here is the speed of light).  

So this is how stars are powered; these thermonuclear reactions take
place in the core of the Sun, turning mass into energy.  This is a
powerful energy source, enough to fuel the Sun for billions of years. 


  You will be responsible for the material through this lecture for
  the midterm. 

Notes for Lecture 11

© Copyright 2009 Anatoly Spitkovsky