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PH607/13/A3
UNIVERSITY OF KENT
SCHOOL OF PHYSICAL SCIENCES
PH607: GALAXIES
2013
Assignment 3: Universe
Due back: Never
1. Briefly describe the differences between Fanaroff-Riley Type I and Fanaroff Riley
Type 2 radio galaxies. The same gas physical conditions produce the observed radio
emission. What are these conditions?
2. The Cygnus A radio galaxy has a luminosity of about 1039 W. Given a jet outflow
speed of one-third the speed of light, roughly estimate the minimum mass outflow rate
in solar masses per year.
3. Describe the colour, stellar population and gas content of E, SO and Irr galaxies.
4. Explain the difference between the monolithic and hierarchical theories for the
formation of elliptical galaxies. How do the two theories explain the gas content of
ellipticals?
5. Sketch the galaxy Luminosity Function for all, E/SO, spirals and Irregular galaxies.
State which different types of galaxy dominate by number at low, intermediate and
high luminosities.
6. Assuming the following form of the Schecter function:
(L) dL = *(L/L*) exp{-L/L*}d(L/L*),
Calculate and explain the values of α which would lead to (a) an infinite number of
galaxies and (b) an infinite luminosity in the Universe.
7.
According to General Relativity, the relation between an interval of proper time to
and world time t at a distance R from a spherically symmetric mass M is
2
t o  2GM 
 1 
t 
Rc 2 
1/ 2
.
Two identical clocks are synchronised on the earth, which may be assumed to be in a
zero gravitational field. One of them is then placed on the surface of Jupiter for a
period of 100 years, after which it is returned to the earth undamaged. Calculate the
time discrepancy in minutes between the two clocks.
Take M jupiter  1.90  10 27 kg,
and R jupiter  7.15  10 7 m.
8. The quasar 3C273 showed a blob of radio emission which moved 2.4 milliarcseconds away
from the nucleus in three years. Assuming the motion is in the plane of the sky and using a
proper distance to 3C273 of 640 megaparsecs, show that the motion is superluminal.
9. A photon passing a distance ro from a mass M is bent through an angle
4GM
ro c 2
radians.
Demonstrate that two images will be formed when the light from a source at distance dS
passes a gravitational lens of mass M at a distance dL from the observer, with the angles
subtended with the lens by the two images, 1 and 2, given by
1 2  
4GM d S  d L
.
dS dL
c2
[18 marks]
10. State the specific condition which produces an Einstein ring of light. Write down the
angular radius of the ring.
ANSWERS
1. Fanaroff-Riley Type II radio galaxies possess hot spots at their outer edges (edgebrightened). Type I do not possess distinct hot spots or they are located within the lobe.
The FR Is are generally of lower luminosity and the jet speed is lower. [1 marks].
Wide radio jets leading to plumes are often visible in the FR Is. In the FR Is the jets are
relatively weak.
3
One need relativistic electrons, magnetic field to produce the synchrotron radiation.
2. The mass outflow is given by 2L/vjet2 where L is the power.
This yields 2 1039/1016 kg/s = 2 1023 kg/s.
One solar mass per year is roughly 2 1030/ 3 107 kg/s = 6.7 1022 kg/s.
Therefore, about 3 solar masses per year must be ejected
3.
Ellipticals, E: red, old stellar population, very little gas, hot diffuse gas present.
Sa: reddish, old/intermediate populations, some gas
Irr: blue, intermediate to young stellar population, highest gas content
4. The monolithic theory asserts that ellipticals formed from large gas clouds which collapsed
and cooled. Ellipticals formed in a monolithic collapse, which induced violent relaxation of
the stars, stars are now in an equilibrium state.
The hierarchical theory assumes the formation of elliptical galaxies through the merging of
smaller galaxies.
In the monolithic theory, the gas is lost in the initial phases through the burst of star formation.
In the hierarchical theory, gas is ejected during galactic interactions and mergers
5.
turn over
4
Elliptical dominate the high L end.
Prirals dominate the intermediate luminosities.
Irregulars and dwarf ellipticals dominate the low end.
6. If we integrate the Schechter function, we get the total number of galaxies (per Mpc3), we
find:
N = ∫(L)dL
= * L* (+1)
Where  is the gamma function, (j+1)=j! when j is an integer
If <-1, (+1) is undefined (!), and N is infinite!!
OR
take the low-luminosity approximation for which the exponential is a constant,
leaving N = * L*/(+1)
If  = -1, (+1) N is infinite!!
We can also integrate to find the total luminosity
total lum = ∫ L(L)dL
= * L* (+2), which diverges if  < -2
OR
take the low-luminosity approximation for which the exponential is a constant,
leaving N = * L*/(+2)
If  = -2,total lum. is infinite!!
5
7. The clock delivered to Jupiter will tick slower compared to the one on earth by
a fraction
the subtraction of a dark frame of the same time and
t t o  1  2GM2 
t 
Rc 
1/ 2
Therefore it will display a time of
 t

100 years x  o  1
 t

relative to the earth bound clock.
On approximation to
t o
 GM 
 1  2  , or/and by direct substitution
t
 Rc 
Yields -256 s or -7.6 10-6 years
Which is about 4 minutes slow.
8. 2.4 milliarcseconds in three years is a speed of
(2.4x10-3/3.0x365.25x24x60x60) arcsec per second
= 2.535 x 10-11 arcsec/s
This is 2.535 x 10-11 x 640 x 106 AU/s at a distance of 640Mpc
= 2.4 109 m/s
This is eight times the speed of light.
9. Employ small angle trigonometry throughout.
The light is bent through an angle 
Suppose the source subtends an angle beta with the lens.
Suppose the source appears to subtend an angle theta with the lens.
The light passes by the lenses with impact parameter ro = tan  dL
Then geometry (finding the apparent transverse position of the source from the observer-lens
axis) yields:
dS tan  = (dS - dL)tan  + dS tan 
turn over
6
dS  = (dS - dL)  + dS 
Substituting for  and ro, yields
4GM d S  d L 1
 

c2 d S d L 
This is a quadratic with two roots:  2 
4GM d S  d L
  = 0
c2 d S d L
Substituting (  1 )(   2 )  0 yields the two roots, the product of which is found on
equating coefficients to be the middle term\in the above equation, as required.
10. An Einstein ring is produced when the source lies directly behind the lens.
Putting 

yields 1   2   ,
yields the angular radius of the Einstein ring:


4GM d S  d L
c2 d S d L