University of Ljubljana
Faculty of Mathematics and Physics
Department of physics
Seminar Ia - 1st year, 2nd cycle
Big Bang Nucleosynthesis
Author: Mitja Fridman
Mentor: dr. Jernej Fesel Kamenik
Ljubljana, january 2015
Abstract
In this seminar, the Big Bang Nucleosynthesis or Primordial Nucleosynthesis, is
presented. First, the nuclear statistical equilibrium is introduced to set the initial
values of all nuclear densities and to understand the conditions before nucleosynthesis actualy began. The production of light elements is then described in three steps.
A short study of the predicted and observed abundances of the most abundant
species follows. At the end, a description of how to use primordial nucleosynthesis
as a probe of conditions in the early Universe, is considered.
Contents
1 Introduction
1
2 Nuclear statistical equlibrium
2
3 Production of
3.1 Step 1 . .
3.2 Step 2 . .
3.3 Step 3 . .
3
4
5
6
the light
. . . . . .
. . . . . .
. . . . . .
elements
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Primordial abundances
4.1 Predictions . . . . . . . . . . . .
4.2 Observations . . . . . . . . . . . .
4.2.1 Deuterium . . . . . . . . .
4.2.2 Helium-3 . . . . . . . . . .
4.2.3 Lithium-7 . . . . . . . . .
4.2.4 Helium-4 . . . . . . . . . .
4.3 Confrontation between theory and
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observation
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7
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5 Primordial nucleosynthesis as a probe of conditions
10
6 Conclusion
11
1
Introduction
Nucleosynthesis is a proces in which new atomic nuclei are created from pre-existing
nucleons, which are primarily protons and neutrons. About three minutes after the Big
Bang, the first nuclei were formed through a process called Big Bang Nucleosynthesis. At
that time hydrogen and helium were formed and became the building blocks of the first
stars. This process is also responsible for the hydrogen/helium ratio of the cosmos, which
is observed today [1].
Big Bang Nucleosynthesis (also known as primordial nucleosynthesis) describes the
production of nuclei, heavier than the lightest isotope of hydrogen (1 H with a single
proton as a nucleus), in the early phases of the Universe [1].
A paper, published in April 1948, named “The Origin of Chemical Elements”, was
the first to discuss the formation of light elements in the early Universe, or Big Bang
Nucleosynthesis [2]. It was writen by Ralph Alpher and later became his PhD dissertation
under the guidance of his advisor George Gamow [2]. Gamow humorously added the name
of his friend, physicist Hans Bethe, to this paper, so the authors would be writen as Alpher,
Bethe and Gamow like the first three Greek leters α, β and γ. The paper was later known
as “αβγ paper ”. But this addition of an eminent physicist to the paper overshadowed the
discovery of Alpher, a graduate student, who actualy did all the work by him self [3].
1
2
Nuclear statistical equlibrium
In the early stages of evolution of the Universe, before the era of nucleosynthesis, it was
dense and hot enough for photons, electrons, positrons, neutrinos and nucleons, to be in
kinetic and chemical equilibrium due to high (weak and electromagnetic) interaction rates.
Even if the state of the Universe after inflation was not in equilibrium, it thermalizes
very fast and gets in to equilibribrium state, because of the weak and electromagnetic
interactions. Equilibrium does not mean that the thermodynamical parameters are not
changing, but that the time evolution is adiabatic and that the thermal distributions
of individual species have an equilibrium form at a given temperature of the plasma.
Strong and gravitational interactions can be neglected, because they have freezed out in
the earlier stages of the evolution of the Universe. In particular, the initial values of all
nuclear densities are set by Nuclear Statisical Equilibrium (NSE) [4].
To understand primordial nucleosynthesis, we will first have to consider the consequences of NSE between the light nuclear species. While in kinetic equilibrium, the
number density of a nonrelativistic species A(Z), with mass number A and charge Z is
given by
µA − mA
mA T
exp
,
(2.1)
nA = gA
2π
T
where µA is the chemical potential of the species, T is the sorounding temperature, mA
is the mass of the species and gA is the number of the deegres of freedom for the species.
The assumption of a nonrelativistic species is valid, because the velocities of nuclei are
not comparable with the speed of light even at temperatures as high as 10 MeV. While in
chemical equilibrium, the chemical potential of the species A(Z) is related to the chemical
potentials of the neutron and proton by
µA = Zµp + (A − Z)µn .
(2.2)
Equation (2.1) also holds for the neutron and proton. With this in mind, we can express
the part exp (µA /T ) in terms of the neutron and proton number densities
µ Zµp + (A − Z)µn
A
= exp
exp
T
T
3A/2
Zmp + (A − Z)mn
2π
Z A−Z
−A
,
(2.3)
= np nn
2 exp
mN T
T
where mN ' mp ' mn ' mA /A is the nucleon mass. Considering the binding energy of
a nucleon WbA = Zmp + (A − Z)mn − mA , we can write the species number density (2.1),
using equation (2.3) as [5]
3(A−1)/2
2π
WbA
3/2 −A
Z A−Z
nA = gA A 2
np nn exp
.
(2.4)
mN T
T
Because of the expansion of the Universe, the particle number densities decrease as
R−3 (R3 is the comoving
volume) and it is therefore useful to use the total nucleon density,
P
nN = nn + np + i (AnA )i , as an initial quantity. Mass fraction, contributed by nuclear
species A(Z), must therefore be considered,
nA A X
XA =
,
Xi = 1.
(2.5)
nN
i
2
Considering this definition we see that in NSE the mass fraction (or abundance) of nuclear
species A(Z) is given by
(A−1) (1−A)/2 (3A−6)/2
XA = gA [ζ(3)
π
2
]A
5/2
T
mN
3(A−1)/2
WbA
(A−1) Z (A−Z)
η
Xp Xn
exp
, (2.6)
T
where η = nN /nγ is the present baryon-to-photon ratio and ζ(3) is the Riemann zeta
function, which comes from the solution of the number density integral over the thermal
distribution of particles
Z
Z ∞
gA
(E 2 − m2A )1/2
gA
3
nA =
EdE =
f
(p)d
p
=
(2π)3
2π 2 mA exp [(E − µ)/T ] ± 1
(ζ(3)/π 2 )gA T,
BE distribution
=
(2.7)
(3/4)(ζ(3)/π 2 )gA T, FD distribution,
where f (p) = [exp [(E − µ)/T ] ± 1]−1 are the Bose-Einstein (for −1) and Fermi-Dirac
(for +1) distributions. The most significant fact to primordial nucleosynthesis is that the
Universe is “hot” (η 1, which means the entropy per baryon is very high) [5].
3
Production of the light elements
Before the description of the production of the light elements, we first have to consider the
initial conditions from which the production started. The neutron to proton ratio is very
important in the outcome of primordial nucleosynthesis, because nearly all the neutrons in
the Universe incorporate into 4 He. The weak interactions maintain the balance between
neutrons and protons [5]:
n ←→ p + e− + νe ,
νe + n ←→ p + e− ,
e+ + n ←→ p + νe .
(3.1)
Figure 1: Feynman diagram of neutron decay (left) and n − p weak reactions (right)
3
Considering the chemical equilibrium and assuming that the lepton number densities
are small ( 1), we get the neutron to proton ratio as
n
Q
= exp −
,
(3.2)
p EQ
T
where Q = mn − mp = 1.293 MeV. Figure 2 shows the neutron to proton ratio depending
on temperature T [5].
Figure 2: The dashed line shows the equilibrium ratio and the solid line shows the actual ratio through time (temperature of the Universe, presented in this plot, is always a
function of time) [5]
Comparing the rates for the weak interactions Γ that interconvert neutrons and protons
with the expansion rate of the Universe H, gives [5]
Γ
∼
H
T
0.8 MeV
3
,
(3.3)
for T ≥ me . For temperatures greater than about 0.8 MeV the neutron to proton ratio is
equal to its equilibrium value. If the temperatures are much greater then an MeV, then
Xn ' Xp and the rates for the nuclear reactions are more rapid than the expansion rate
of the Universe [5].
The production of the light elements is described within three steps. The steps represent different times (epochs) and temperatures in the evolution of the early Universe.
3.1
Step 1
The age of the Universe was t = 10−2 seconds and the temperature was T = 10 MeV.
At this time the Universe was dominated by radiation in terms of energy density. The
4
relativistc degrees of freedom are: e± , γ and 3 neutrino species. Every weak rate is much
larger than the expansion rate H, so that (n/p) = (n/p)EQ ' 1. NSE then holds for
the light elements, but they have small abundances because η is very small. η should
be nearly constat from the beginning of primordial nucleosynthesis to the presen day,
because most of the baryons formed in the era of baryogenesis which took place before
the nucleosythesis era, so for t → 0, follows η → 0. For example, with η = 10−9 the
abundances are
Xn , Xp = 0.5,
3/2
2.22
T
η exp
' 6 × 10−12 ,
X2 = 4.1
mN
TM eV
3
T
7.72
2
X3 = 7.2
η exp
' 2 × 10−23 ,
mN
TM eV
9/2
T
28.3
3
X4 = 7.1
' 2 × 10−34 ,
η exp
mN
TM eV
33/2
T
92.2
11
X12 = 79
η exp
' 2 × 10−126 ,
mN
TM eV
(3.4)
where TM eV = T /M eV , Xn + Xp + X2 + X3 + X4 + X12 = 1 and indices n, p, 2, 3, 4, 12
represent neutrons, protons, deuterons, 3 He nuclei, 4 He nuclei and 12 C nuclei respectively
[5].
3.2
Step 2
The age of the Universe was t ' 1 second and the temperature was T = TF ' 1 MeV.
Before this epoch, all the neutrino species decoupled from the plasma and the e± pairs
annihilate. With annihilation they transfer their entropy to the photons and raise the
photon temperature relative to the neutrino temperature. The weak interactions which
interconvert neutrons and protons freeze out at about this time. This happens because
the weak rates Γ become smaller than the expansion rate of the Universe H. When the
freeze out happens, the neutron to proton ratio is approximately given by its equilibrium
value,
Q
1
n
= exp −
' .
(3.5)
p f reeze out
TF
6
After the freeze out the neutron to proton ratio slowly decreases because of the occasional
weak interactions (dominated by free neutron decays) and does not remain constant as it
can be seen on figure 2. The deviation of the neutron to proton ratio from its equilibrium
value becomes significant when the nucleosynthesis begins. Here, the light nuclear species
are still in NSE, but have very small abundances [5]
Xn ' 1/7,
X2 ' 10−12 ,
X4 ' 10−28 ,
Xp ' 6/7,
X3 ' 10−23 ,
X12 ' 10−108 .
5
(3.6)
3.3
Step 3
The age of the Universe was t = 1 to 3 minutes and the temperature was T = 0.3 to
0.1 MeV. At this time the e± pairs have completely disappeared and transfered their
entropy to the photons. If the freeze out would not happen, the neutron to proton ratio
would have its equilibrium value (n/p)EQ = 1/74 at T = 0.3 MeV [5]. The reactions (3.7)
which take place are schematicaly shown in figure 2 [6].
1.
n
2
1
3. H + H
5. 2 H + 2 H
7. 3 H + 4 He
9. 3 He + 2 H
11. 7 Li + 1 H
→
→
→
→
→
→
H + e− + νe ,
3
He + γ,
3
H + 1 H,
7
Li + γ,
4
He + 1 H,
4
He + 4 He,
1
1
2.
H+n↔
2
4.
H + 2H →
6. 2 H + 3 H →
8. 3 He + n →
10. 3 He + 4 He →
7
12.
Be + n →
2
H + γ,
He + n,
4
He + n,
3
H + 1 H,
7
Be + γ,
7
Li + 1 H.
3
(3.7)
Figure 3: Scheme of reaction products [4]
At this same temperature the NSE value of the mass fraction of 4 He approaches its
limit which is also observed today. Before that, the actual amount of 4 He is smaller than
its NSE value. This happens because the rates for the processes that produce 4 He are
not fast enogh for the increasing NSE “demand” for 4 He. There are two reasons why the
reaction rates are not fast enough:
1. The abundances of D, 3 He and 3 H begin to exceed their NSE values, but are still
very small, Xi = 10−12 , 2×10−19 , 5×10−19 , respectively. Because of this the number
densities of these species are small.
2. At this time also the Coulomb barrier suppression is becoming significant. The
Coulomb barrier is the energy barrier, which comes from the electrostatic interaction
between two nuclei. The two nuclei must overcome this barrier for the nuclear
6
reaction to take place. The barrier “height” depends on the temperature. The
lower the temperature, the higher the barrier, so this suppresses the reactions and
must be taken into account. The thermal average of the barrier penetration factor
is
#
"
1/3
2A (Z1 Z2 )2/3
,
(3.8)
hσ|v|i ∝ exp −
1/3
TM eV
where A = A1 A2 /(A1 + A2 ) and hσ|v|i is the thermally averaged cross section
multiplied with relative velocity. Until the abundances of D, 3 He and 3 H become the
same order of magnitude, these reactions do not produce sufficient 4 He to maintain
its NSE abundance. When these abundances build up, all the available neutrons
bound into 4 He.
The 4 He synthesis begins at about T ' 0.1 MeV. Also some 7 Li is sythesized (7 Li/
H' 10−10 to 10−9 ). This trace amount of 7 Li is a very valuable probe of primordial
nucleosynthesis. Some amounts of D and 3 He do not react, bacause the rates for the
reactions that burn them to 4 He, become small and the reactions freeze out. These rates
are proportional to η, so the amounts of D and 3 He should decrease with increasing η [5].
4
Primordial abundances
As mentioned in the introduction, the primordial nucleosynthesis dates back to Gamow
in 1946. Not long before the discovery of the cosmic microwave background radiation
(CMBR), an estimation of the amount of 4 He, synthesized in the early stages of a hot big
bang, was made. After that, a lot of independent nucleosynthesis calculations were made.
The calculations were mostly made with computer programs. The theoretical framework
for predicting primordial abundances of elements, from which the calculations were made,
is the NSE, which is explained in chapter 2.
The observational methods are mainly spectroscopic, but others, such as analyzing
meteorites, are also used to derive the primordial abundances of elements.
4.1
Predictions
The NSE is the first and only theoretical model for primordial nucleosynthesis, because it
is very accurate and therefore there is no need for a different approach. The abundances
are sensitive to the input nuclear physics data. The relevant cross sections are, in practice,
known accurately enough so that the theoretical uncertanties are irrelevant. The predicted
4
He abundance depends only on the weak reaction rates which determine the freeze out
neutron to proton ratio [5].
Figure 4 displays how the abundances of light elements evolve through time. The
sensitivity of the abundances to the free cosmological parameter η and the two physical parameters g∗ (which is the total number of effectively massless degrees of freedom)
and τ1/2 (n) (which is the half life of the neutron τ1/2 (n) = (10.3 ± 0.2)min), should be
considered before comparing the predicted abundances to the observed abundances [5].
• τ1/2 (n): All the weak rates are given by Γ ∝ G2F (1 + 3gA2 ), or Γ ∝ T 5 /τ1/2 (n). If
the neutron half life would increase, the weak rates that interconvert neutrons and
protons would decrease, so that the freeze out of the neutron to proton ratio would
7
occur at a higher temperature TF ∝ τ1/2 (n)1/3 and the value of (n/p) would be
larger. Because the final 4 He abundance depends on the value of (n/p)f reeze out , an
increase in τ1/2 (n) leads to an increase in the predicted 4 He abundance. Also the
abundances of other light elements change, but are not of great interest.
1/2
• g∗ : The expansion rate is given by H ∝ g∗ T 2 . If g∗ would increase, the expansion
rate of the Universe would be larger at the same temperature. This also leads to an
1/6
earlier freeze out of the neutron to proton ratio TF ∝ g∗ .
• η: The abundances of species A(Z) are proportional to XA ∝ η A−1 while in NSE. If
η is larger, the abundances of D, 3 He and 3 H build up earlier, so the 4 He synthesis
begins earlier, when the neutron to proton ratio is larger, which results in a greater
abundance of 4 He. At the time 4 He synthesis begins significantly (T ' 0.1MeV), the
neutron to proton ratio is slowly decreasing and the sensitivity of 4 He production
to η is small.
Figure 4: Time evolution of light elements [7]
4.2
Observations
The predicted primordial abundances have well defined and small theoretical uncertainties, while the observed primordial abundances are less certain and large. The goal is
to measure the primordial cosmic abundances which synthesized at a time before other
astrophysical processes (stellar production, destruction and similar processes) became important. What can be measured are present day abundances in selected astrophysical sites
8
[5]. Models for post-nucleosynthesis evolution of species (production and destruction in
stellar and other processes) are used to derive the following primordial abundances from
observations.
4.2.1
Deuterium
The D abundance has been measured in studies of the solar system, local interstellar
medium (ISM) and deuterated molecules in the ISM. Determinations from the solar system are based on the measurements of the abundances of deuterated molecules in the
atmosphere of Jupiter (D/H' (1 − 4) × 10−5 ) and considering the pre-solar D/H ratio
from data of meteorites and Sun on the abundance of 3 He (D/H' (1.5 − 2.9) × 10−5 ).
These determinations match with a pre-solar value of D/H' (2 ± 1) × 10−5 . The average
ISM value (D/H' 2 × 10−5 ) has been derived from studies of UV absorption of the local
ISM. These determinations present only the lower bound of the abundance of D, because
of its production and desruction nature [5].
4.2.2
Helium-3
The 3 He abundance has also been measured in studies of the solar system and by observations of the 3 He+ line in galactic HII (notation for once ionized hydrogen) regions. In
the oldest meteorites and carbonaceous chondrites, the abundance of 3 He has been determined to be 3 He/H = 1.2 ± 0.4 × 10−5 . It is believed that these objects were formed at
about the same time as the solar system and therefore they provide a sample of pre-solar
material. From the solar wind, the abundance of 3 He has been determined, by analizing
gas rich meteorites, lunar soil and the foil on the surface of the moon. These measurements represent the pre-solar sum of D and 3 He, because D is burned to 3 He when the Sun
approaches the main sequence. The determinations of D+3 He match with the pre-solar
((D + 3 He)/H ' (3.6 ± 0.6) × 10−5 ) [5].
4.2.3
Lithium-7
The abundance of 7 Li, first derived from pre-solar meteorites, the local ISM and Pop I
stars (7 Li/H ' 10−9 ) is about a factor of 10 greater than the predicted by primordial
nucleosynthesis. It is not possible to accurately measure the primordial abundance of
7
Li, because it is also produced by cosmic ray spallation and some stellar processes and
is easily destroyed in environments where T ≥ 2 × 106 K. 7 Li was later observed in the
atmospheres of unevolved halo and old disk stars with low metallicity and masses in the
range M = 0.6 − 1.1 M . A correlation between the 7 Li abundance and the mass of
the star was noticed. With decreasing mass of a star, the 7 Li abundance decreases and
the other way around, but there is an uper limit for the more massive stars which is
called a plateu. From this plateu a primordial abundance of 7 Li was derived (7 Li/H '
(4.8 ± 0.4) × 10−10 ). This abundance represents the pre-Pop II 7 Li abundance and if the
Pop III stars’ destruction /production is neglected, it represents the actual primordial
abundance [5, 4].
4.2.4
Helium-4
The 4 He abundances were observed in ISM and intergalactic medium (IGM) and derived
from recombination lines in HII regions. Because 4 He is also synthesized in stars, some of
9
the 4 He is certainly not primordial. The mass fraction of the primordial 4 He (4 He/H) is
usualy marked with Yp . There is a correlation between the stellar 4 He abundance Y and
the metallicity Z. The lower Z is, the lower Y is [5].
The most reliable way to estimate Yp is to focus on the 4 He determinations for objects
with low metallicity. The metal-poor galaxies (Z ≤ Z /5) have been studied to get a
weighted average of the primordial abundance Yp ' 0.250 ± 0.003 [5, 4].
4.3
Confrontation between theory and observation
The only predicted isotopes which were produced in significant amounts during the time of
primordial nucleosynthesis are D, 3 He, 4 He and 7 Li. The predicted primordial abundances
match very well with the observed primordial abundances. This provides strong evidence
that the standard model cosmology (SMC) is valid at times as early as 10−2 s after the
big bang [5].
2
H/H
He/H
7
Li/H
Yp
3
predicted
2.6 × 10−5
1.03 × 10−5
4.6 × 10−10
0.248
observed
2.0 × 10−5
1.2 × 10−5
4.8 × 10−10
0.250
Table 1: Comparison of predicted and observed abundances of primordial species
The baryonic matter can be studied very well in laboratories on Earth and observed in
the Universe in several ways, so there is much information about it to construct a working
theory about it’s origin. Unlike dark matter, which, for now, can only be observed by it’s
gravitational influence on baryonic matter and can’t be studied like baryonic matter yet,
so there is no fundamental way to construct a working theory for it’s origin yet.
5
Primordial nucleosynthesis as a probe of conditions
If we accept the validity of the SMC model, because of its success, primordial nucleosynthesis can be used as a probe of conditions in the early Universe and with it, of cosmology
and particle physics. The baryon density, derived from studies of primordial abundances,
is [5]
4 × 10−10 ≤ η ≤ 7 × 10−10 .
(5.1)
As mentioned before, one of the reasons for increased 4 He is the freeze out of the
neutron to proton ratio, which occurs at a temperature of ' 0.8 MeV. At this time,
the relativistic degrees of freedom are γ, 3 species of neutrinos (both ν and ν) and e±
pairs. The mass fraction of 4 He, Yp increases with increasing of η, τ1/2 (n) and g∗ (T ) as
mentioned in chapter 4.1 [5].
By measuring the width of the neutral Z 0 weak boson, the neutrino species number
can be determined. A neutrino flavour, which is lighter than mZ /2, contributes around
190 MeV to the width of the Z 0 . Today it is known that there are Nν = 3 neutrino
10
species. The Big Bang Nucleosynthesis predicted an upper bound to the number of the
neutrino species, before they were experimentaly known [5, 4].
The process of primordial nucleosynthesis also predicts the relative baryon density
ΩB = ρB /ρC (where ρB is the baryon density and ρC is the critical density of the Universe).
ΩB is a different version of η. The results show that there must be much more matter in
the Universe which can not be seen - dark matter. The determined relative density of all
matter is ΩM ' 0.315, from which the density of baryonic matter is only ΩB ' 0.049 [8].
6
Conclusion
The NSE gives the number densities of the primordial species and their time (temperature)
dependence while in kinetic and chemical equilibrium. Also mass fractions, which are
most interesting in this study, of the primordial elements can be derived from NSE. The
time evolution in NSE, the time of the freeze out and the time evoution after the freeze
out is described with a three step process of light element production. The primordial
abundances change their time dependence because the reaction rates Γ become smaller
than the expansion rate of the Universe H. The theoretical description gives a good
insight of how sensitive to the input parameters g∗ , η and τ1/2 (n) the abundances are.
The observational data later confirmed the theoretical abundances.
Primordial nucleosynthesis is the earliest and the most strict test of SMC and a very
important probe of cosmology and particle physics. The results show that theory and
observation are consistent and that the SMC is a valid description of the Universe at least
back to times as early as 10−2 s after the big bang and temperatures as high as 10 MeV.
As a probe, primordial nucleosynthesis provides a very good determination of ΩB , a strict
limit to Nν and some other particle physics and cosmology constraints.
References
[1] Big Bang Nucleosynthesis, accessible at http://www.sciencedaily.com/articles/
b/big bang nucleosynthesis.htm (25.11.2014)
[2] R. Alpher, H. Bethe, G. Gamow, The Origin of Chemical Elements, Physical Review,
73 (7), 803-804 DOI: 10.1103/PhysRev.73.803, 1948
[3] S. Singh, Veliki pok (Big Bang) (Učila International, Tržič, 2008)
[4] F. Iocco, G. Mangano, G. Miele, O. Pisanti, P. D. Serpico, Primordial Nucleosynthesis: from precision cosmology to fundamental physics, arXiv:0809.0631v2, 2009
[5] E. W. Kolb, M. S. Turner, The Early Universe, (Addison-Wesley publishing company,
Illinois, 1989)
[6] J. Brorsson, J. Jacobsson, A. Johansson, Big Bang Nucleosynthesis, accessible at
http://fy.chalmers.se/subatom/nt/index.php?page=bachelor (4.1.2015)
[7] M. Rieke, The Hot Big Bang, accesible at http://ircamera.as.arizona.edu/astr 250/Lectures
(6.1.2015)
[8] A. Gomboc, Lectures: Astronomy II, summer semester 2013
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