29:62 General Astronomy
First Hour Exam
February 14, 1997
All problems are worth 10 points. Write
legibly, preferably in pen. Good luck and no whining.
(1) A star has a surface temperature of 10000K. Draw an accurate plot
of its continuum spectrum, with a correct labeling (with numbers)
of the wavelength axis.
Answer: The plot should be of radiative flux density versus
wavelength. The spectrum should be a continuum one, and show a maximum
at a wavelength given by 
. Plugging in the
numbers, we have
meters, or 288 nanometers.
(2) Describe how we can determine the temperature of the solar corona. The
more numerical information you can furnish, the better.
Answer: The best indicator is of spectral lines from highly ionized atoms.
Examples would be iron X and Ni XV. These ions have ionization potentials
of the order of 300 to 400 electron volts. There is no way electrons could
collisionally ionize atoms to such a state unless the gas has temperatures of order
electron volts, or
(3) What is the typical speed of an electron in the solar wind?
The solar wind has a temperature of 
K.
The mass of an electron is 
kg.
Answer: We use the relationship between the mean kinetic energy and
the temperature, 
. Rearranging,
the root mean squared speed is
meters/sec.
(4) See the energy level diagram of a hypothetical atom in the Figure 1.
In an experiment, an electron beam (all electrons have the same energy)
is passed through the gas, collisionally
ionizing the atoms. They recombine, and emission lines are produced as
a result. 1 electron volt (eV) = 
Joules.
(a) What must the energy of the ionizing electrons be?
(b) Which of the radiative transitions is in the ultraviolet?
(c) What is the wavelength of this transition or transitions?
Answer: (a) The energy of the electrons must be at least 9 electron
volts, since this is the ionization potential of the atom. (b) Transitions
in the ultraviolet have wavelengths less than 400 nanometers =
meters. The energy of a photon is given by
= 
Joules or 3.10 electron volts. Looking at the energy
level diagram in Figure 1 we see that the following transitions have this
much energy: 
,
and 
. (c) The wavelengths of these transitions are
250nm, 207nm, 177, and 155 nm, respectively.
(5) A star is known to have an absolute magnitude of -2. It is observed with
an apparent magnitude of 4. What is the distance to this star?
Answer: The distance modulus is m-M = 6 magnitudes. Using the
relationship between distance modulus and distance,
where d is the distance in parsecs, we have
parsecs.
(6) Assume an atom has a first ionization potential of 1.0 electron volts.
For what class of stars would you expect to see spectral lines of the
neutral atom? Describe your reasoning and calculations.
Answer: If the first ionization potential is 1.0 electron volts,
the mean energy per particle should be of the order of 0.5 electron volts
to get a significant amount of ionization. Thus
J, or
K. This is lower than the temperature of the sun, and
is characteristic of a spectral class K or M star.
Mathematical Formulae
Note: All units MKS unless otherwise indicated.
Wien's Law: Wavelength at which a blackbody radiator is brightest
where 
(Planck's Constant),
(speed of light), 
.
Kinetic Energy of Thermal Motion:
where m is the mass of the particle (electron, atom, ion, molecule) and
v is its speed.
Relation between Flux and Magnitude:
is the flux (Watts/m 
) from object 1, 
is that from
object 2, 
is the magnitude of object 1, and 
is the magnitude
of object 2. Then,
Energy of a Photon: 
where 
is the wave frequency (units Hertz). Photon energy is often
conveniently expressed in electron volts, 1 eV = 
Joules.