Download Introduction to Galaxies and Cosmology Exercises 2

Introduction to Galaxies and Cosmology
Exercises 2
In this exercise session we will go learn how to solve problems relating the
light coming from galaxies and how the stellar physics affect the output.
Also, some problems relating to active galactic nuclei and galaxy clusters
are included. We will also work with the theoretical framework of cosmology,
the Friedmann-Lemaı̂tre-Robinson-Walker equations and learn how to use
them to relate observational quantities to the theory. The exercises will
touch upon parts from chapters 2–6 in J&L, so make sure that you have
read through those chapters before working with the exercises.
• Galaxies and surface brightness
• Star forming galaxies and galaxy evolution
• AGN, black holes and galaxy clusters
• Basic cosmological theory
Notes:
A bolometric magnitude contains radiation emitted at all wavelengths. Invariably they refer to absolute magnitudes and the related luminosities are
thus the total luminosities of the objects. The bolometric absolute magnitude of the Sun corresponding to L⊙ as tabulated is 4.75 (by definition).
For the location of the event horizon of a black hole, refer to the collection
of formulae.
Galaxies and surface brightness
1. A flat, two-dimensional disk galaxy has an exponentially declining
r
surface brightness distribution given by I(r) = I0 e− a , where a
represents the scale length. Derive an approximate expression for
the total luminosity of the disk.
2. (Challenging) A galaxy with the same surface brightness distribution as in the previous exercise has a scale length of 5 kpc and
a rest-frame B-band luminosity of LB = 1010 L⊙,B .
a) Derive an expression for r as a function of a, I(r) (in units of
L⊙ /pc2 ) and L.
b) Derive a general expression which converts surface brightness,
µ, in units of (mag/arcsec2 ) to I (L⊙ /pc2 ) and thereby show that
surface brightness is independent of distance.
Hint: Surface brightness is related to magnitude by the equation:
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µ = m + 2.5 log A,
where the area A is expressed in units of square arcseconds.
Star forming galaxies and galaxy evolution
3. The colour-magnitude diagrams of globular clusters can be used to
estimate their ages. By including information on the metallicity in
the cluster the age can be further constrained (the metal content
in stars also affect their colours and evolution). The following
expression relates the luminosity at the turn-off point to the age
and metallicity of globular clusters (from Iben & Renzini, 1984):
log LT O ≈ 0.019(log Z)2 + 0.065 log Z + 0.41Y − 1.18 log t9
+1.25 − 0.028(log Z)2 − 0.27 log Z − 1.07Y, (1)
with LT O in units of L⊙ and t9 the age in units of 109 years. In this
equation Z is the “metal content”, this is basically the abundance
of all elements other than hydrogen and helium divided by the
hydrogen abundance (e.g for the Sun Z⊙ = 0.02). Y is the helium
abundance divided by the hydrogen abundance (e.g. for the Sun
Y⊙ = 0.28).
a) Estimate the age of the M3 cluster using equation 1 and figure 3
(left panel). The distance to M3 is 10.4 kpc and you can assume
that the bolometric correction of the turn off point is 0.05 and
that AV = 0. The metal and helium content in M3 is given by:
Z = 0.001, Y = 0.24.
b) Estimate the age of the Pleiades cluster using equation 1 and
figure 3 (right panel). The distance to the Pleiades is 135 pc
and you can assume that the bolometric correction of the turn off
point is 0.15 and that AV = 0. The metal and helium content in
the Pleiades is given by: Z = 0.02, Y = 0.28.
4. The two plots in figure 2 shows the model photometric evolution
of a 106 M⊙ stellar population.
a) Use this model to predict what would be the absolute V-band
magnitude and B-V colour of a 1Gyr old galaxy with a total stellar
mass of 1010 M⊙ (Use the solid-line model prediction).
b) If 106 M⊙ of 107 yr old stars would be added to the galaxy in
the example above, what would its absolute magnitude and colour
then be?
5. Assume that a galaxy is formed by gravitational collapse and that
it contains a mass Mgas of gas. Also assume that half of this mass
is converted into stars (M⋆ = 0.5Mgas ).
a) The absolute bolometric magnitude for the galaxy is Mbol =
2
−21, what is the bolometric luminosity of the galaxy (answer in
units of L⊙ )?
b) If we assume that the stars in the galaxy are all formed over a
short time scale (a few million years), the Initial Mass Function
(IMF, see lecture notes for more info) together with some simple
scaling relations can be used to calculate the approximate total
luminosity and stellar mass (M⋆ ) of the single stellar population
that constitutes this galaxy. The Salpeter IMF gives the number
of stars formed at a certain mass:
dN = ξ(M)dM = ξ0 M−2.35 dM,
(2)
with M the mass in units of solar masses. This function can be
integrated to find the number of stars between two masses. With
Mhigh as the mass upper limit and Mlow as the mass lower limit
we obtain:
Z
Z Mhigh
N=
dN =
ξ0 M−2.35 dM.
(3)
Mlow
The total luminosity and total mass between the upper and lower
mass limits can be found by integrating the moment equations:
Z
Z Mhigh
L(M) · ξ0 M−2.35 dM
(4)
Ltot =
L dN =
M⋆,tot =
Z
Mlow
Z Mhigh
M dN =
Mlow
M · ξ0 M−2.35 dM,
(5)
all luminosities and masses are in solar units (L⊙ and M⊙ ). The
luminosity of main sequence stars can be approximated by a simple scaling law: L/L⊙ = (M/M⊙ )3.5 . You may assume that the
galaxy has just finished forming stars (i.e. all stars are main sequence stars) and that the upper/lower mass limits are given by
Mlow = 0.1M⊙ and Mhigh = 100M⊙ .
b) What fraction of stars in the galaxy have masses higher than
10M⊙ ?
c) What fraction of the luminosity from the galaxy is emitted
from these massive stars?
d) Using the luminosity derived in a), find the mass of the original
gas cloud, Mgas .
AGN, black holes, and galaxy clusters
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6. If the Sun suddenly collapsed to a black hole,
a) at what distance from the solar centre would the event horizon
lie?
b) What would the period of the Earth’s revolution around the
Sun be?
7. For every mass m which is swallowed by a black hole (via an accretion disk, say), an amount of energy νmc2 is liberated, where
ν is the efficiency of the process. A value of ν = 0.1 is realistic.
At what rate Ṁ would a supermassive black hole have to swallow mass to produce the luminosity L = 1040 W = 2.5 · 1013 L⊙ .
Convert your answer to solar masses per year.
8. A quasar has a total luminosity of 1012 L⊙ .
a) If 10% of the restmass can be converted to energy, at what rate
does it accrete gas?
b) What is the mass of the black hole if one assume it to radiate
at 10% of its Eddinton luminosity?
c) What is the Schwarzschild radius of the black hole?
9. (Challenging) A quasar emits two radiating clouds in our general direction at 13/14 the speed of light. They are first observed
when they appear to have been first produced at the central powerhouse, and are subsequently observed to move apparently outward in opposite directions from the centre of the quasar image.
Fourteen years after the quasar actually emitted the clouds, they
are in reality twelve ly. closer to us than the quasar itself.
a) How many years, on earth, elapse between our first observations
of the clouds and when we observe them at the second position?
b) At what speed do they appear to us to be separating?
10. In a gravitationally bound system the virial theorem relates the
mean gravitational (potential) energy to the mean kinetic energy.
This means that if we have information on the size of a cluster
as well as the velocities of galaxies in the cluster we can use the
virial theorem to find the total mass of the cluster. By making
some assumptions and approximations we arrive at:
Mcl =
σ 2 re
,
G
(6)
where Mcl is the dynamical mass of the cluster, σ 2 is the mean of
the squared peculiar velocities of the galaxies in the cluster (i.e.
the squared standard deviation of velocity) and re is the efficient
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radius (can be obtained by studying the radial surface density
profile of galaxies in the cluster).
In table 1 the ten brightest galaxies in the Fornax cluster is listed
with magnitudes and radial velocities. The efficient radius for the
Fornax cluster can be assumed to be 1 Mpc. Using the galaxies
listed in the table, calculate the mass of the cluster.
11. A galaxy cluster is observed to contain 1000 galaxies, within a
radius of 3 Mpc from the centre. The effective radius of the cluster is 2.5 Mpc. Spectral analysis show the cluster to have a mean
redshift of z = 0.03 with a standard deviation of σz = 0.004.
H0 = 70km/s/Mpc.
a) Estimate the dynamical mass of the cluster.
b) Estimate the mass of all the galaxies taken together by assuming that they follow a luminosity function: dN ∝ L−2 dL between
MV = −12 and -22, and that they have an average mass to light
ratio of M/L = 10 in solar units. (The absolute magnitude of the
sun is MV = 4.8)
c) What would the dark matter fraction of this cluster be?
Basic cosmological theory
12. Assuming an Einstein–de Sitter universe (k = 0, Λ = 0, p = 0),
derive an expression for how the scale factor R varies with time
13. a) Assuming a de Sitter (empty) universe (k = 0, p = 0, ρ = 0),
derive an expression for how the scale factor R varies with time.
b) During the inflationary phase of the universe (here assumed to
last between t ∼ 10−36 s and t ∼ 10−34 s) the universe expanded
enormously. The cosmology was dominated by the cosmological
constant and the de Sitter model is a good approximation. If you
assume that the universe grew by a factor 1050 during this period,
what was the value of Λ?
c) Compare this to Λ0 using ΩΛ = 0.7 and equation 5.30 in Jones
& Lambourne.
14. (Challenging) Show that if Λ = 0 and H0 = 75 km/s/Mpc, the
maximum age of the universe is 13 Gyrs.
15. The Friedmann equation, together with the relation ρR3 = constant, may be used to show:
2ṘR̈ = −
8πG
(RṘρ − 2RṘρΛ ).
3
5
(7)
a) Use this, and the definitions given in J & L section 5.4 for H(t),
q(t) and ρcrit (t) to show that at any time t
q(t) =
Ωm (t)
− ΩΛ (t)
2
(8)
b) What is the value of q0 for the currently favoured cosmology
(ΛCDM cosmology)?
(J&L ex. 5.10)
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N
E
b = 22"
a = 50"
Figure 1: Disk galaxy taken from the digitized sky survey. Negative intensity
scaling. North is up, east is left. The angular size of the image is 100 × 100
arcseconds.
Figure 2: Galaxy evolution plots
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Figure 3: Colour-magnitude diagrams for the two stellar clusters M3 (right)
and Pleiades (left). The M3 diagram is from Renzini & Pecci, 1988.
Table 1: The ten visually brightest galaxies in the Fornax cluster
Mag.
9.4
10.2
10.6
10.9
11.3
11.7
11.8
11.8
11.8
11.9
vrad (km/s)
1786
1638
1430
1944
1827
1405
1884
1454
1363
1386
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Answers, Exercises 2
1. L = 2πI0 a2
2. a) r = −aln
2πa2 I(r)
L
−0.4(µ−M⊙ −21.572)
b) I[L⊙/pc2 ] = 10
= 4.254 · 108 10µ−M⊙
3. Note that, due to the logarithmic nature of the magnitude diagrams, the ages are uncertain and depends critically on the value
read off the diagrams.
a) tM 3 = 12.4 ± 2 Gyrs (but note that the higher bound is limited
by the age of the universe) using VT O = 19.0.
b) tP leiades = 1.9 ± 0.5 Gyrs, using VT O = 6.0 (note that the
equation is not valid for luminosities higher than about 20L⊙ ,
corresponding to an age of 20 Myrs).
4. a) MV = −21, B − V = 0.22
b) MV = −21.02, MB = −20.80
5. a) L = 2.0 · 101 0L⊙
b) 0.19%
c) 99.3%
d) 2.5 · 107 M⊙
6. a) Rs = 2.95 km
b) The Newtonian law of gravity is still valid. Far from the event
horizon, the gravitational field of a black hole is indistinguishable
from that of any sperically symmetric object of the same mass
7. Ṁ = 17.7M⊙ /yr
8. a) Ṁacc = 0.68M⊙ /yr
b) MBH = 3 · 108 M⊙
c) Rs = 8.9 · 1011 m ≈ 1 lighthour
9. a) 2 years
b) v = 5 · c (apparent superluminal motion)
10. Mcl = 2.76 · 1013 M⊙
11. a) Mdyn = 2.5 · 1015 M⊙
b) Mlum = 4.8 · 1011 M⊙
c) Mdark /Mlum = 5200
√ 2/3
12. R = 3H03 Ω0
t2/3
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√Λ
13. a) R = Ri e 3 t
b) Λinfl = 4.1 · 1072
c) Λ0 = 1.4 · 10−52 ,
Λinfl
Λ0
∼ 10124
14. –
15. a) –
b) q0 = −0.55
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