The Virial Theorem

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The Virial Theorem
For a collection of objects (could be stars,
atoms, galaxies, etc.) bound together by a force
that is proportional to 1/r2, the total kinetic
energy of the bound system is one half of the
absolute value of the binding energy (electrical
or gravitational).
Examples
Planetary orbit
Multiple planets around the sun, or stars in a
globular cluster
All the planets, stars, and satellites in the
Milky Way galaxy
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The Virial Theorem

• Other examples
• The internal temperatures of stars supported by
  ideal gas pressure
• Electrons in atoms

• Exceptions
• Forces that are not 1/r2 (e.g., the strong
  force)
• Gases that are degenerate (as in white dwarfs)
• Particles that move near the speed of light (as
  in radiation) (it will turn out there that
  KEPE, not PE/2)
• Systems that have not relaxed into a steady
  state equilibrium (e.g., galaxies passing in
  the night)

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• The Virial Theorem is sometimes a useful tool for
• estimating mass in astronomy.
• Consider a globular cluster of stars all bound
  together by gravity.
• The radius of this cluster is determined to be 10
  pc and the typical
• (average) velocity of a member is 10 km/s.
  Estimate the mass of the
• cluster.
• The total mass of the cluster, M, is the sum of
  all the stars masses,
• mi. The total gravitational potential energy is
  approximately the binding energy of a sphere
  with radius r.

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Stellar temperature Total thermal energy in
star ½ total gravitational potential energy
NA is the number of particles in a gram. For
hydrogen, NA 6.02 x 1023
For the sun this gives about 5 million K (this
is the approximate average temperature, not
the central temperature. It is also an
underestimate because the sun does not have
constant density and W is greater)
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Another way of expressing the Virial Theorem is
that the total energy of a bound system is equal
to the negative of its total kinetic energy (if
the total energy were zero, the system would fly
apart) That is PE - 2 KE
total PE KE - 2KE KE
- KE We shall now apply this to atoms
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Rutherford Atom (1911)
e
r
classically, any value of v or r is allowed.
Ze
Protons in nucleus. Electrons orbit like planets.
The neutron was not discovered until 1932
(Chadwick)
BUT,
As the electron moves in its orbit it is
accelerated, and therefore emits radiation.
Because energy is being radiated, the total
energy of the system must decrease. This means v2
must increase and r must get smaller. But
smaller r and larger v also imply greater
acceleration and radiation. In approximately
10-6 s the electron spirals into the nucleus.
Goodbye universe
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The solution lies in the wave-like property of
the electron and of all matter
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Ground state of the hydrogen atom Neils Bohr
(1913)
(lowest possible energy state)
Must fit the wavelength of the electron inside a
circle of radius r, the average distance between
the electron and the proton.
r
p
Note that PE goes as 1/r and KE goes as 1/r2
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For a single electron bound to a single
proton, i.e., hydrogen.
Energy
r
ro
At ro
Energy would have to be provided to the electron
to make it move any closer to the proton, more
energy than e2/r can give.
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For atoms with a single electron H, He, etc.
For arbitrary nuclear charge, Z, and excited
level, n
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E.g.,
Lines that start or end on n1 are called the
Lyman series
Lines that start or end on n2 are called the
Balmer series
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Adjusting the energy of each state in hydrogen by
adding 13.6 eV (so that the ground state becomes
zero), one gets a diagram where the energies of
the transitions can be read off easily.
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Absorption Line Spectrum
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How are excited states populated?

• Absorb a photon of the right energy
• Collisions
• Ionization - recombination

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He II strong, He I increasing from O4 to O9 H
prominent
He I lines dominate H increasing in strength
http//nedwww.ipac.caltech.edu/level5/Gray/Gray_co
ntents.html
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H lines reach maximum strength. Ca II growing. Fe
II, Si II, Mg II reach
H lines start to decrease in strength. Ca II
strong. Fe I growing in strength. Mg II
decreasing.
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Ca II lines strongest, H lines weak, neutral
metal lines strong. G-band of CH is strong.
H lines weak. Lines of neutral metals present but
weakening. Major characteristic is bands from
molecules like TiO and MgH
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Luminosity Classes
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These stars all have essentially the same
temperature and are spectral type M2. However,
based upon the strength of the Ca I 4227 A line,
one can distinguish red supergiants (Class I)
from ordinary red giants (Class III), and main
sequence stars (Class V).
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At other temperatures other lines are used to
distinguish luminosity class. Here, for F0, the
chief diagnostics are blends of Fe II and Ti II
lines at 4172-4178 A and similar blends at
4395-4400, 4417, and 4444 A.