The Small Angle Approx-imation

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The Small Angle Approx-imation
The 2p shows the fact that the natural units for
angle here are radians 2p of them for a full
circle (so 57.3 degrees per radian).
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Measuring distance with triangulation
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Parallax direct distances to the stars
d(parsec) 1/p(arcsec) 1 parsec 3.26
light years
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The Luminosity Law
The intrinsic luminosity of a star depends on its
temperature (because blackbody radiation goes
like T4 per unit area, and its size (to account
for its total area R2)
l2
L2
l1
L1
R2
R1
T1
T2
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Observation of stellar properties

• You can tell something from the brightness and
  color of a star,
• but there are complications.
• The color gives you an approximate temperature.
• It is hard to tell the distance from the
  brightness, because the size is also quite
  variable
• A small blue star can be as luminous as a larger
  red star. A blue star can be the same brightness
  as a red star if they have the same size but the
  red star is much closer.
• If you get the distance from parallax, then you
  can tell the size from the color and apparent
  brightness.
• Since most stars are many light years away, their
  parallaxes are small fractions of an arcsec.
• Observing from space helps with this.

• Their sizes are generally too small to be
  measured directly (much smaller fractions of an
  arcsec).
• Interferometry will help with this.

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The Inverse-Square Law
The apparent brightness of a source goes down
like the square of the distance to the source
(and depends on the intrinsic luminosity of the
source
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Energy Levels in Atoms
We can think of an atom as consisting of a
positive nucleus (protons and neutrons)
surrounded by negative electrons. The electrons
can be thought of as orbiting the nucleus, but
are only allowed in certain orbits (or energy
levels). A photon with exactly the right energy
can excite the electron from one level to
another. The electron will drop back to the
ground state, and emit photons with specific
energies as it does so.
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Energy transitions and photons
The energy of photon that can interact with a
level jump just depends on the energy difference
between the levels. Levels can be skipped.
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Unique Atomic Signatures
Each atom has a specific set of energy levels,
and thus a unique set of photon wavelengths with
which it can interact.
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Astro Quiz
The spectra from two different atoms are
observed. A spectral line in the first atom has
exactly one-half the wavelength of a spectral
line in the second atom. Which of the below is
TRUE?

• The spacing between 2 levels in the first atom
  must be twice the spacing between 2 levels in the
  second atom.
• The spacing between 2 levels in the first atom
  must be half the spacing between 2 levels in the
  second atom.
• The energy of the upper of 2 levels in the first
  atom must be half that of the energy of the upper
  of 2 levels in the second atom.

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Spectrometers
A spectrometer makes an image (usually of its
entrance slit) at every wavelength. A telescope
illuminates the slit with starlight.
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Diffraction gratings
Very fine facets or grooves will act like many
little slits and produce an interference pattern
that spreads wavelengths.
far infrared grating
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Transitions in Hydrogen
Hydrogen is particularly important and
distinctive. The H-alpha line is the one from
levels 2-3 the Balmer series from level 2 is in
the visible. The Lyman series lines are all
ultraviolet.
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The Solar Spectrum
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Stellar Spectral Types
Different stars show different spectral lines, or
different line strengths. They were sorted by
this appearance (Annie Jump Cannon).
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Decoding Stellar Spectra
The reasons why spectra change were a bit subtle,
so the ordering of the letters got scrambled when
the temperature sequence was determined.
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Properties of the Spectral Types
Most stars will be in their stable
hydrogen-burning phase, called the main
sequence. For these, the spectral type has a
simple relation to the stars properties. Note
that luminosity has a large variation.
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The Brightest and Nearest Stars
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Towards the HR diagram
Basketball player
Skinny
Height
Adults
Anorexic
Fat
Sumo wrestler
Children
Weight
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The Hertzsprung-Russell Diagram
Hertzsprung and Russell had the idea of plotting
the luminosity of a star against its spectral
type. This works best for a cluster, where you
know the stars are all at the same distance. Then
apparent brightness vs spectral type is basically
the same as luminosity vs temperature. They found
that stars only appear in certain parts of the
diagram.
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Basics of the HR diagram
Size
Mass
Red dwarfs
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Size/Luminosity
Hot stars are very bright but very rare. They can
affect the light, but not the mass of the Galaxy.
Red supergiants are more common.