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The Masses of Stars
•  Stellar masses can only be determined via the
application of Kepler’s and Newton’s laws, i.e.,
(M1 + M2) P2 = a3
where
  M1 and M2 are the stellar masses (in solar units)
  P is the orbital period (in years)
  a is the semi-major axis of the orbit (in A.U.)
This requires binary stars!
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Binary Stars
Half the stars in the sky are binary stars!
Many stars look like they are
binaries, but are really not:
they are optical doubles.
Fortunately, many other are
true binaries, even if they’re
too close together to be
resolved.
Binary systems are called
different things, depending on
how we perceive them. A
true double star can be called
a visual binary, a spectrum
binary, a spectroscopic binary,
and/or an eclipsing binary,
depending on what we see.
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Visual Binaries
When both stars can be seen, it’s called a Visual Binary.
Castor
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Spectrum Binaries
If the stars are too close together to be seen separately, it is possible
to identify the object as a binary based on its spectrum.
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Spectroscopic Binaries
If the Doppler shift of a star’s
absorption lines changes with
time (redshift, then blueshift,
then redshift, etc.), it’s a
spectroscopic binary.
If one star is much
fainter than the other,
you may not see its lines.
The object is then a
single-line spectroscopic
binary. If both sets of
lines are seen, then it’s
called a double-line
spectroscopic binary.
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Eclipsing Binaries
If two stars eclipse each other while orbiting, it’s an eclipsing binary.
Eclipsing binaries are somewhat rare, since they need to
be seen edge on.
This system is one in which
both stars undergo a total
eclipse.
Frequently, the
eclipses are partial.
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Determining Masses from Binaries
•  The relative speeds of the stars gives you their relative masses.
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Determining Masses from Binaries
•  The relative speeds of the stars gives you their relative masses.
•  The absolute velocities of the stars (times the period) gives
you the circumference of their orbits. From that, you can
derive the orbits’ semi-major axes. In other words,
Circumference = v · t = 2 π a
(at least for circular orbits)
•  From the semi-major axis and the period, you can derive the
total mass of the system through
(M1 + M2) P2 = a3
•  Since you already know the relative masses, you now know
everything!
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Results from Binary Stars Measurements
1)  All stars have masses between 0.1 M and 60 M
2)  Main sequence stars obey a mass-luminosity relation:
the brighter the star, the more massive the star.
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Results from Binary Stars Measurements
1)  All stars have masses between 0.1 M and 60 M
2)  Main sequence stars obey a mass-luminosity relation:
the brighter the star, the more massive the star.
3)  The white dwarf stars
are all less than 1.4 M
4)  There is no pattern to
the masses of red
giants.
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The H-R Diagram
There are patterns in
the HR diagram. Most
stars lie on the main
sequence, and obey a
mass-luminosity relation. (Low mass stars
are faint, high mass
stars are bright.)
But there are huge red
giants that have all
sorts of masses, and
very
small
white
dwarfs that are all less
than 1.4M Why??
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Temperature, Pressure, and Energy
•  Gas Temperature: a measure
of how fast atoms are
moving in random directions
•  Gas Pressure: the force (per
unit area) these atoms put on
their surroundings via their
collisions
Energy, Temperature, and Pressure are related through the
Equation of State: where one goes, the others go!
Energy ↑ ⇔ Temperature ↑ ⇔ Pressure ↑
Energy ↓ ⇔ Temperature ↓ ⇔ Pressure ↓
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Hydrostatic Equilibrium
•  The Sun is very massive, so it has
a lot of gravity. The center of the
Sun is under great pressure!
•  In order to keep from collapsing,
the gas pressure must balance the
pull of gravity. This is called
hydrostatic equilibrium.
•  High pressure means high
temperature: the center of the Sun
is very hot: about 14,000,000° !
•  High temperature means the Sun
produces a lot of energy (through
the blackbody law). This heat
must flow out!
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How Does Heat Get Transported?
•  There are 3 ways to transport heat:
  Conduction:
fast moving electrons hit
slower moving electrons. (This does NOT
happen in most stars.)
  Convection: hot material is mixed into
cooler material. (This only happens in the
outermost layers of some stars like the Sun.)
  Radiation:
blackbody law photons are
emitted, then absorbed by cooler material.
Energy is transported
via the random walk of
photons. In the Sun, it
takes
energy
over
10,000,000 yr to get to
the surface!
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•  As the energy leaks out, the central
temperature of the Sun drops.
•  Lower temperature means lower gas
pressure.
•  The lower gas pressure cannot hold
up against gravity – the Sun shrinks.
•  The added compression puts the
Sun’s center under greater pressure,
so the central temperature increases.
•  The higher temperature produces
higher pressure, which fights off
gravity.
•  The high temperature produces
blackbody photons which leak out…
Gravitational contraction
could keep the Sun shining
for 40,000,000 yrs!
Remember this argument: it
will be critical for you understanding of stellar evolution.
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( very rarely)
quickly
In order to keep up the gas pressure (and
prevent collapse), the center of the Sun
must continually replenish the energy that
is lost. This is done by nuclear fusion (of
hydrogen).
The energy produced
maintains hydrostatic equilibrium.
The sequence of fusions is called the
proton-proton chain.
quickly
Net Result: 4 H → 1 He
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About the Proton-Proton Chain
•  The net result of the proton-proton chain is to turn 4 hydrogen
atoms into 1 helium atom. But there is a mass defect – the 4
hydrogen atoms have 0.7% more mass than the 1 helium atom
(plus the other junk). Where did the missing mass go?
E = m c 2 Energy!!!
•  If the Sun had more mass, it would have more gravity, and its
center would be under greater pressure. The greater the
pressure, the greater the temperature, and the more violent the
nuclear collisions. More fusion would occur, and more
energy would be produced. This explains the main sequence!
•  Fusion only occurs in the core, where the temperature and
density are greatest. The rest of the star just sits there.
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Solar Neutrinos
The energy produced by the proton-proton chain takes over 10
million years to reach the Sun’s surface. But the neutrinos
barely interact with anything, and leave the Sun immediately. It
is thus possible (with great difficulty) to detect these neutrinos.
For many years, these neutrinos weren’t
detected. But we now understand why –
they’re even harder to detect than we
thought!
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The Sun Won’t Shine Forever
Stars spend over 90% of their life on the main sequence,
fusing hydrogen to helium. The Sun has already lived 4.5
billion years. It will live about 5 billion more years.
Stars mostly transport
energy via radiation,
not convection. So
hydrogen from the
outside of the star
does not get mixed
into the core. When
the core’s hydrogen
runs out, bad things
begin to happen.
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