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1. What are the wavelengths for the Balmer
lines (n→2 transitions)? What are the
wavelengths of the Lyman (n→1) transitions
in the limit n→∞?
 1
1 
  91.16nm 2  2 
 n2 n1 
1
1
H
 1 1
 91.16 2  2   656.3 nm
2 3 


H
1 1
 91.16 2  2   486.2 nm
2 4 


1
1
Balmer limit
 1 1
 91.16 2    364.64 nm
2 


This means any photon with <364.64 can
completely remove an electron in the n=2 level
from the atom.
1
Lyman limit
1 1
 91.16 2    91.16 nm
1  


A photon with wavelength less than this can ionize
H no matter what level the electron is initially in.
2.
Doppler shifts
a) Typical stars in the solar neighbourhood have
velocities ~30 km/s. What is the size of their
Doppler shift?
This is much less than the speed of light, so
  v r / c
 0.0001
 0.05nm
at visible wavelengths.
At visible wavelengths, l~500 nm, so the
wavelength shift is 0.04 nm. To resolve this
wavelength difference, a grating must have

nN 
 12500

b) Extragalactic objects (mostly galaxies and
quasars) are strongly redshifted due to the
expansion of the Universe. The most distant
object currently known is quasar SDSS1148+5251,
with z=6.42. Since z is not small, we have to use
the full expression:
vr ( z  1) 2  1

c ( z  1) 2  1
7.42 2 1

7.42 2  1
 0.964
c) Barnard’s star has a very high proper motion
of ~88 km/s. The spectrum of Barnard’s star
shows the H absorption line at a wavelength of
656.044 nm. What is its full space velocity?
In the laboratory:
4 c  1
1 



4  2
2 
e  n2 n1 
 656.3 nm

3
1
1
 1 1
 91.176 2  2  nm
2 3 


So the radial velocity is:
vr  c
c


656.044  656.3
656.3
 117 km/s
And the full space velocity is:
v  vr2  v 2pec  146 km/s
3. Pulsars are rapidly spinning neutron stars
which beam light in opposite directions. They
have huge magnetic fields of 108 – 1012 gauss.
How large is the Zeeman splitting?
 
eB
2me c
1.60 10
19


C 1010 G

2 9.1110 31 kg 2.998 108 m/s



 9.32 1011 Hz
The frequency of visible light is about 6x1014 Hz.
So even in this strong magnetic field the Zeeman
splitting is only about a 0.1% effect.
4. The star Rigel has a spectral type B8Ia and
a magnitude V=0.14. What is its distance?
From the Hipparcos HR diagram, this spectral class
corresponds to:
M V  6.5  0.5
So
Dstar  10
 10
 m M  1
5
0.14 6.5
1
5
 213  50 pc