a) Explain what is meant by the 'Boltzmann factor', corresponding to ...

[pic] DEPARTMENT OF
PHYSICS AND ASTRONOMY ASTRONOMY 2 Session 2004-2005 Values of Physical Constants
|Speed of light |c |2.998 ( 108 ms-1 |
|Gravitational constant |G |6.673 ( 10-11 Nm2kg-2 |
|Planck constant |h |6.626 ( 10-34 Js |
|Boltzmann constant |k |1.381 ( 10-23 JK-1 |
|Stefan-Boltzmann |( |5.671 ( 10-8 Wm-2K-4 |
|constant | | |
|Gas constant |R |8.315 Jmol-1K-1 |
|Proton mass |mp |1.673 ( 10-27 kg |
|Electron mass |me |9.109 ( 10-31kg |
|Charge on an electron |e |1.602 ( 10-19 C |
|Thomson cross-section |(T |6.652 ( 10-29 m-2 |
|Astronomical Unit |AU |1.496 ( 1011 m |
|Parsec |pc |3.086 ( 1016 m |
|Light year |Ly |9.461 ( 1015 m |
|Solar mass |M( |1.989 ( 1030 kg |
|Solar radius |R( |6.960 ( 108 m |
|Solar luminosity |L( |3.826 ( 1026 W |
|Earth mass |M( |5.976 ( 1024 kg |
|Earth radius |R( |6.378 ( 106 m |
The collection of tutorial problems listed in this booklet has been
assembled over the past few years. It covers all four courses taught in
Astronomy A2Z, and will generally provide the questions set in the tutorial
assignments. Note, however, that lecturers and course syllabuses change
from year to year. Thus, some of the tutorial problems listed here may
relate to material that is covered slightly differently, in less depth - or
indeed not at all - in this year's lectures. Your lecturers will ensure
that any problems set as tutorial assignment questions are appropriate to
this year's syllabus. Some lecturers may supplement the problems in this handbook with additional
problems and exercises given in their lecture notes, or given out
specifically as assignment questions. Past exam papers also provide a rich
source of questions to attempt; copies of past papers can be obtained from
the University Library, and from the Astronomy Secretary at the end of the
Second Term. You are, of course, encouraged to attempt all of the tutorial problems in
this handbook - not simply those problems set as assignment questions. If
you are unsure if a particular question is appropriate to this year's
syllabus, don't hesitate to ask your lecturers or small group supervisor.
Dr Martin Hendry Astronomy 2Z Class and Lab Head
Theoretical Astrophysics 1.1 The total mass of the visible solar corona is 1016kg comprised almost
entirely of hydrogen. During a solar eclipse a coronal iron line at 500 nm
produces a flux of 10-2 Wm-2 at the earth. If the solar abundance of iron
atoms is 7 x 10-5 by number relative to hydrogen and if 10-4 of all iron
atoms have an electron in the upper level of this transition at coronal
temperatures and densities, calculate the Einstein coefficient A21 for the
transition. State whether the line is allowed or forbidden and calculate
its natural width ((0. [Atomic weight of Fe = 56]
1.2 The luminosity of the Galaxy in the 21cm line of neutral hydrogen is
1029W. If this line were emitted solely by spontaneous downward
transitions with A21 = 3 x 10-15 s-1 calculate:
(a) the natural width of the line
(b) a lower limit to the mass of neutral hydrogen in the Galaxy in solar
masses If the interstellar density and temperature are 106 m-3 and 100K
respectively, calculate:
(c) the thermal line width
(d) the collisional line width for collisions with atoms (take (coll = 10-
20 m2) e) If the Galactic radius and rotation period near the sun are 10kpc and
108 years respectively, estimate very roughly how wide the observed 21cm
line is due to Galactic rotation, assuming the rotation speed, [pic]
constant.
3. Show that the eccentricity e of the ellipse describing the relative
profile (F(??)/F(0)
versus ??/??) of a spectral line for a star of equatorial rotation speed
VEQ seen at inclination i is
[pic]. Hence show that for a star rotating at its centrifugal break up limit (see
A1 Pulsar notes)
this becomes
[pic] where Rs = 2GM/c2 is the Schwarzschild radius of the star.
[pic]
1.4 In an accretion disk, matter at radius r orbits at the Keplerian
speed [pic] where M is the central mass. As the matter spirals slowly
inward it heats up to temperature [pic] at a distance r with T0, r0
constants. Show that
(a) the maximum Doppler shift of light of rest wavelength (0 emitted at r
is
[pic]
where [pic] is the Schwarzschild radius of the central object;
(b) the thermal line width of the line emitted at r is
[pic]
for an atom of mass mA. Hence, by eliminating r, show that the ratio of thermal broadening to
Doppler shift for a line formed at temperature T is
[pic]
1.5 Show that the ratio of (electron) collisional line width to thermal
line width of a line at wavelength (21 from an ion of mass mA is
[pic]
(Note that this is independent of T if (coll is constant.) Calculate this ratio for a line at 500nm from hydrogen atoms in the solar
photosphere if n=1023 m-3 and (coll = 10 -20 m2. At what wavelength range
and for what kind of atoms would you expect collisional broadening to
become more important?
6. Given the Maxwell speed distribution
[pic]
show that
(a) the most probable speed (at which N(v) peaks, i.e. dN(v)/dv=0) is
[pic];
(b) the mean speed [pic] , defined as [pic]is[pic]
and that
(c) the mean kinetic energy, defined as [pic], is [pic].
1.7 Show that the fraction f of the total thermal kinetic energy (not the
fractional number) contained in those particles of a Maxwellian
distribution which have speeds [pic] is given by
[pic]
It is found that during rapid heating of the solar corona (total electron
density [pic]) to a temperature of [pic] in a solar flare, plasma waves are
generated which trap all the heated electrons except those with [pic]
which escape. By approximating the integral above (e.g. graphically)
estimate the total thermal energy of flare electrons which escape in this
way if the total heated volume is 1017 m-3. [n.b.[pic]]
1.8 A globular cluster comprises N stars each of mass M, distributed
through a sphere of radius R, and with a Maxwellian distribution of speeds
at effective temperature T (such that [pic]). Show that the fraction of
stars capable of escape at any instant is approximately
[pic]
If the stars have average mass 1031 kg, and mean speed 300 kms-1, calculate
the 'temperature' involved, and comment on the result.
1.9 A planet has an albedo [pic] at optical wavelengths, and [pic] at
infrared wavelengths, so that its temperature is given by the usual
expression
[pic]
where subscript [pic] denotes the solar value. The spectrum [pic] of
radiation observed from its
sunlit hemisphere comprises two components:
(a) radiation [pic]emitted by the planet, where [pic] is the Planck
function
(b) reflected sunlight [pic] due the small but non-zero albedo. Show that the the reflected and emitted radiation are in the ratio
[pic]
Use the approximation [pic] to show that at long wavelengths the emitted
radiation will exceed the reflected provided
[pic]
(Note: it is this fact that allows radio spectra to be used to measure
[pic])
1.10 The 'colour' of an object around wavelength [pic] may be defined by
the dimensionless index
[pic]
Show that for a black body
[pic]
and also that
[pic].
Draw a graph of [pic] at optical [pic] for [pic] and deduce that spectral
data in the optical are unsuitable for measuring the temperatures of hot
stars.
1.11 The lifetime of an electron in the first and second excited states of
hydrogen is about 10-8 s. Show that the natural width of the H( line is
approximately 4.57 ( 10-5 nm.
1.12 The Sun's photosphere has a temperature of 5770 K. Show that the
Doppler broadening of the H( line from photospheric hydrogen is
approximately 0.043 nm. Taking the number density of hydrogen atoms to be
1.5 ( 1023 m-3, show that the collisional broadening of this line is
comparable to its natural broadening.
1.13 Calculate the power spectra of the following waveforms: a) A pure sinusoid:
[pic] for all t, b) A pulse of duration 2( :
[pic] for |t| 0
[pic] for t < 0, where [pic], [pic] and ( are constants. In each case sketch both the
real part of the waveform in the time domain, and the power spectrum in the
fr