Stars are often found in binaries.  When one
object orbits the other, the central object doesn't sit still.
Indeed, the two objects orbit around their common center of mass.  In
a pair of stars orbiting each other, you can see both stars moving
easily, and use Kepler's Third Law (in a slightly more sophisticated
version than we've seen thus far) to infer the masses of each star. 

  When we do so, we learn that the main sequence is a sequence in
mass: high surface temperature, high luminosity stars are
particularly massive.  This makes sense: the more massive the star,
the greater the internal temperature and pressure needed to hold the
star up against its self-gravity.  The higher the temperature in the
core, the more energetically all those protons are banging into each
other, and the faster the rate at which they fuse to give off energy.  

  With modern technology, it remains impossible to directly see the
light of a planet around another star; the planet is ridiculously
faint and its light is lost in the glare of the *much* brighter star.
This problem is particularly severe for observations using telescopes
from the ground, where turbulence from Earth's atmosphere blurs the
images of stars.  The next generation of space telescopes are being
designed to be able to directly take pictures of planets around other
stars, but it is a real technical challenge.

  The best way, then, to find planets around other stars is to see the
motion of the star due to the orbit of the planet. We see the star
move through the Doppler effect, whereby the exact wavelength lambda
of the absorption lines in a stellar spectrum are shifted by an amount
that depends on the speed v of the star: (lambda - lambda_0)/lambda_0
= v/c Here lambda_0 is the wavelength of the line if the star were to
be at rest (equivalently, the speed we could measure for a given line
in the laboratory) and c is the speed of light.  What counts here is
the component of the motion of the star along the line of sight.  So
as the star moves in its orbit around the center of mass of the
star-planet system, its Doppler shift systematically increases, then
decreases, in a cycle.  Measuring this cycle gives a period of the
orbit, and that, together with the speed from the Doppler effect,
allows one to determine a variety of properties of the star.

  About 300 planets have been discovered this way, the first in 1995.
The planets typically have masses similar to that of Jupiter or
somewhat bigger.  The effect of a wimpy planet the size of the Earth
is too small to be detectable with current technology.

  However, there is another way to find planets.  Consider a planet
orbiting such it passes in front of its parent star once per orbit, as
seen from Earth.  The light from the star will be very slightly dimmed
by the shadow from the planet.  From a big star like Jupiter, the
effect is about a 1% dimunition of light.  There have been about a
dozen planets discovered this way: i.e., carefully measuring the
brightness of a star, and noticing that it periodically dims by about
1%.  NASA has launched a satellite last Friday which has the
sensitivity to see the much more subtle 0.01% effect we'd expect from
an Earth-like planet transiting the star.

Main sequence stars are powered by hydrogen fusing to helium in their
cores, and releasing energy as a consequence.  A star has a finite
lifetime on the main sequence, when all the hydrogen in its core is
burned up.  For a star like the Sun, it takes 10 billion years.  More
massive stars have more hydrogen, but they burn it *much* more
quickly; the net result is that they have substantially shorter main
sequence lifetimes.    Indeed, roughly speaking, the luminosity of a
star is proportional to the mass to the fourth power, and so the
lifetime of a star is proportional to M^{-3}.  A massive O star has a
lifetime of only a few million years, while the lowest-mass M star has
a life of 10 trillion years, much longer than the current age of the
universe.  

  So the main sequence is a sequence in *mass*: the more massive a
  star is, the higher its surface temperature and luminosity.  We know
  this, because we can measure the masses of binary stars directly,
  via our understanding of Newton's law of gravity. 

  At the low-mass end: for stars with mass less than 8% the mass of
  the Sun (or 80 times the mass of Jupiter), the temperatures and
  pressures in the core are not adequate to cause protons to fuse into
  helium nuclei.  Such objects, then, are "failed stars"; their
  internal energy source never turns on.  But such objects can and do
  exist: gravity pulls together a mass of gas to make such an object,
  which will initially be moderately hot (say, 2000 K on its surface)
  just from the conversion of gravitational potential energy to heat.
  So it will radiate as a blackbody, dimly, with most of the energy
  coming out in the infrared.  Such an object (sometimes referred to
  as a 'brown dwarf') has no internal furnace, and so will slowly cool
  off, getting dimmer and dimmer, over billions of years.  Sensitive
  infrared surveys of the sky have found such objects; they represent
  an extension of the familiar O B A F G K M to two cooler classes, L
  and T.  

  We discussed the future of our Sun, and the age of the Sun. We know
  that each thermonuclear reaction of proton fusion converts 0.7% of
  mass in energy. Then the amount of hydrogen the Sun converts per
  second is the luminosity divided by the energy released in each
  interaction, or energy_rel=(delta M) x c^2 x 0.007. Equating this to
  solar luminosity (4x10^26 W), we find that delta M = 700000 tons of
  hydrogen are convered to helium in the core per second. Given that
  only 10\% of the mass of the Sun is in the core, this rate of
  burning will deplete the core of hydrogen in about 10 billion years. 
  Having this lifetime for 1 solar mass star, we can extrapolate it to 
  more massive stars (lifetime proportional to M^(-3)).


Post-main sequence life of low mass stars (0.08Msun C), and oxygen (C+He -> O). 
This vigorous energy release, which lasts for a hundred thousand years, 
causes the outer envelope to swell up and actually decouple, forming 
"planetary nebulae".
Low mass stars do not reach temperatures in the core that would ignite 
oxygen, and the core of the star, consisiting of carbon and oxigen continues
to contract on its way to becoming a white dwarf. 


  After the break, we'll ask the question of what happens after a massive star
  has exhausted all its nuclear fuel.  The answer turns out to be
  rather dramatic...
  

Notes for Lecture 12

© Copyright 2009 Michael A. Strauss and Anatoly Spitkovsky