TURUN YLIOPISTO
Turku 2008
TURUN YLIOPISTON JULKAISUJA
ANNALES UNIVERSITATIS TURKUENSIS
SARJA - SER. A I  OSA - TOM. 382
ASTRONOMICA - CHEMICA - PHYSICA - MATHEMATICA
THE COSMIC PRODUCTION
OF HELIUM
by
Luca Casagrande
From the Department of Physics
University of Turku
Turku, Finland
Supervised by
Dr. Chris Flynn and Dr. Laura Portinari
Tuorla Observatory
University of Turku
Turku, Finland
Reviewed by
Dr. Norbert Christlieb
Department of Astronomy and Space Physics
University of Uppsala
Uppsala, Sweden
and
Dr. Sofia Feltzing
Lund Observatory
University of Lund
Lund, Sweden
Opponent
Prof. Bengt Gustafsson
Department of Astronomy and Space Physics
University of Uppsala
Uppsala, Sweden
Cover picture: Luca Casagrande 2008
ISBN 978-951-29-3559-8 (PRINT)
ISBN 978-951-29-3560-4 (PDF)
ISSN 0082-7002
Painosalama Oy - Turku, Finland 2008
Fly me to the moon
And let me play among the stars
Let me see what spring is like
On Jupiter and Mars
From ’Fly Me to the Moon’ (Bart Howard, 1954)
Solution to the cover rebus in bibliography.
Acknowledgments
Many warm and well deserved thanks are due to all the people who contributed to make
my life as a doctoral student so enjoyable over the last four years. These thanks are
intended especially for the benefit of those who are actually not so interested in the rest
of the thesis, but are driven to open these pages, or these three pages only, to know what
I actually did during those years (or if their names are in the list!). To their convenience,
I will sneak some informal tale from behind the scenes.
First, I would like to thank my supervisor, Christopher Flynn, for welcoming me
to Tuorla, his scientific guidance and constant support throughout the entire thesis. He
always made me feel comfortable and fully supported me even when I suffered from
an embarrassing lack of scientific results and grants. Thanks to him, I attended vari-
ous conferences and schools, worked in institutions abroad and networked with other
researchers in the field. The month I spent in Mount Stromlo was not only highly pro-
ductive but also very pleasant thanks to the company and the touristic guidance of Chris
and Leena at various occasions. The student company at the house on the mountain was
unique as well as were those cute wallabies jumping around when walking back from
the observatory in the evening.
Laura Portinari joined Tuorla few months after my arrival and provided invaluable
support to the thesis and helped to build my skills in the field of Galactic Chemical
Evolution (work in progress, though). Her mid-afternoon genuine Italian espresso was
an extraordinary chance to wake up my sleepy mind after the Majatalo lunch and also
provided a moment for bright (from her side) scientific discussions on various topics of
stellar astrophysics and chemical evolution, among others.
Chris & Laura significantly helped me by discussing, structuring and focusing the
results obtained throughout these years. If I did not feel the famous mid-PhD crisis, it is
certainly because of their support and optimism, too.
Leo Girardi: in addition to providing the theoretical stellar tracks computed espe-
cially for this thesis, he efficiently replied to all my questions on stellar modeling, as
did Raul Jimenez thanks to whom I also had a great stay in Philadelphia. Even though
I did not meet them personally, I wish to thank the co-authors of my last paper, Mike
Bessell and Chris Koen for their help in that work. My former undergraduate advisor,
Antonaldo Diaferio, helped me by keeping alive my interests in other exciting fields of
astrophysics during my various visits in Turin. During these years, I also benefited from
enlightening discussions with Johan Holmberg, Jorge Meléndez, Regner Trampedach,
Daisuke Kawata, Sergio Sousa, Chris Thom and Hans Kjeldsen at many times. Even
though most of the time doing research is spent in front of a computer, I could share re-
ally nice moments and learn a lot from the various people I met at schools, conferences
and visits abroad. This is also the place to note that my entire thesis would not exist as
it is now if the formidable 2MASS data had not been publicly available!
I sincerely thank the whole Observatory staff and the other PhD students for an
inspirational environment to do research and study, and in particular Janne–Dr. Alban
for being one of my mentors in having fun and partying outside the working hours, not
to forget Tuorla’s Late Afternoon, starring Stefano Ciprini.
For financial support I have to thank the following contributors: Tuorla Observatory,
the Finnish Graduate School of Space Physics and Astronomy, the Magnus Ehrnrooth
foundation, the Otto A. Malm foundation, the Turku University Foundation and last
but not least the Academy of Finland, through Chris and Laura. Special thanks also
go to Leo Takalo for providing funds to attend the NOT summer school in La Palma.
Observing with the NOT was NOT only highly instructive, but it turned out to be lots of
fun inverting night and day as real astronomers do.
I thank the official reviewers of this thesis for speed reading and useful criticism. I
am greatly honored that Prof. Bengt Gustafsson has agreed to act as my opponent in the
public defense of this dissertation.
I had the fortune to come to Finland and find nice and friendly people here, from the
very beginning, for which I have to thank Paula. Sami, Teemu & Jari soon turned from
merely friendly flatmates to strict advisors to the Finnish lifestyle and properly educated
me to ways of a tough guy from Pohjanmaa. I wonder if they fully succeeded (although
I did my best!), but I really enjoyed their wit and relaxed style. By the Galilean concept
of clear evidence of sense, they indisputably proved that it is also a lot of fun inverting
night and day as non–astronomers do! Along the same lines, I am glad to shortlist Osmo,
Ville, Mikko, Jojo, Kimmo, Annastiina, Peter & Tero.
My Finnish language skills have remained at a steady level throughout these years
and it is still a matter of debate whether they will end up in the big crunch or not. I could
not cope with the many issues (well, except for understanding yet another parking fine)
concerning everyday life without the help of the ihanat people around me. I have been
truly blessed in this respect. I have only good memories of my time in Finland, and I
have to thank Sini for that.
Of course, as a migrant, I had the secret dream of becoming a godfather: I probably
misunderstand something, but I am the happiest person ever to have Giorgia and Feliks
as my godchildren. Although most of the time was actually spent in Finland, the Italian
connection has always been strong. That was certainly because of the good company of
Giacomo, Giulio & Ruggero, besides the other members of Samba Carioca School the
few occasions I got time to play there.
The love with my hometown grew stronger at every return (and every New Year’s
Eve) thanks to the incredible and trustful group of friends I will always count on (e
anche di più!): Alberto, Alessia, Anna, Asso, Bruna, Daniele, Elisa1, Elisa2, Fabio,
Filippo, Giorgio, Nadia, Paolo, Pietchro and Silvana. To those involved in this business,
I already promise my attendance to the next 10–year–meeting, to be held in Mondovì
in April 2017 (Indiscutibilmente Posso Prometterlo Ora), but I look forward to closer
events to enjoy with you!
My interest toward astronomy started when I was just a few years old, from the
wonder of looking at the sky from my family’s summer house and the fear of learning
from my brother that stars are actually much much bigger than the Earth! Everything
started there and then and I consider that as the first proof of support by my family
towards my passion, support that has never been lacking. Ringrazio pertanto i miei
genitori per l’affetto e l’appoggio nelle mie scelte. Senza di loro non sarei arrivato fino
a qui.
Finally, to those who wish to know the tips and tricks of accomplishing a PhD, I
confidently say that all the people mentioned here are part of it and to them I owe my
deepest gratitude.
Kiitos!
Luca Casagrande
Contents
Acknowledgments 4
List of publications 8
1 Introduction 9
2 Big Bang Nucleosynthesis 11
2.1 Historical perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Primordial abundances: a brief overview . . . . . . . . . . . . . . . . . 11
2.3 Beyond the standard model . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Predicted and observed abundances . . . . . . . . . . . . . . . . . . . 17
3 Fine Structure in the Lower Main-Sequence 24
3.1 Quasi–homology relations . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Two recent works on ∆Y/∆Z from low mass stars . . . . . . . . . . . 26
3.3 ∆Y/∆Z in nuce . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 The chemical evolution of helium 29
4.1 The overall picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 ∆Y/∆Z from different stellar populations . . . . . . . . . . . . . . . . 32
4.3 A model for the chemical evolution of the Galaxy . . . . . . . . . . . . 37
4.4 The evolution of helium . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5 Summary of the original publications 46
Bibliography
Original papers 57
51
List of publications
I Accurate fundamental parameters for lower main–sequence stars
Casagrande L., Portinari L. and Flynn C.
MNRAS 373, 13 (2006)
II The helium abundance and ∆Y/∆Z in lower main–sequence stars
Casagrande L., Flynn C., Portinari L., Girardi L. and Jimenez R.
MNRAS 382, 1516 (2007)
III M dwarfs: effective temperatures, radii and metallicities
Casagrande L., Flynn C., Bessell M. and Koen C.
submitted to MNRAS
CHAPTER 1
Introduction
Low mass stars are long–lived objects and can be regarded as snapshots of the stellar
populations formed at different times over the history of our Galaxy, the Milky Way. Not
only these stars retain in their atmospheres a fossil record of the chemical elements in the
interstellar medium at the time of their formation, but also their orbits similarly encode
their dynamical histories. In this thesis we deal with the first of these two aspects. We
characterize the physical and chemical properties of low mass stars in the more general
framework of Galactic Chemical Evolution, with a particular attention to the helium
production.
Traditionally, F and G dwarfs have been preferential targets for studies dealing with
various aspects of the chemical evolution of the Milky Way (e.g. Wallerstein 1962; Pagel
& Patchett 1975; Edvardsson et al. 1993; Wyse & Gilmore 1995; Feltzing, Holmberg
& Hurley 2001; Bensby, Feltzing & Lundström 2003; Nordström et al. 2004; Brewer &
Carney 2006). However, F and G dwarfs are sufficiently massive that some of them have
begun to evolve away from the main-sequence and these evolutionary corrections might
complicate the comparison with chemical and stellar evolutionary models.
It has been long recognized that K dwarfs make a cleaner sample because for these
stars the evolutionary corrections are negligible. K dwarfs are intrinsically fainter than
stars of earlier spectral type, so it has not been until recently that high-quality spectra and
accurate abundance analyses have become common (e.g. Feltzing & Gustafsson 1998;
Allende Prieto et al. 2004). To fully understand the information carried by these stars,
accurate stellar parameters and most notably effective temperatures, are needed. The
determination of effective temperatures and bolometric luminosities for a local sample
of K and G dwarfs has been subject of Paper I of this thesis.
With the availability of better data, K dwarfs have begun to be used for chemical
evolution studies (e.g. Kotoneva et al. 2002). K dwarfs display a similar metallicity dis-
tribution as G dwarfs (e.g. Flynn & Morell 1997), in which most stars have metallicities
around and slightly below the solar value, a feature not expected in the simplest models
of Galactic Chemical Evolution (i.e. the “G dwarf problem”). Although nowadays the
abundance patterns of metals in K dwarfs is relatively straightforward to measure, the
same cannot yet be said for M dwarfs. Recently, major improvements in model atmo-
spheres of very cool stars and the availability of high-quality infrared data have boosted
a number of studies on the subject. Taking advantage of such a favourable conjunction,
the determination of precise physical parameters for M dwarfs has been tackled on Paper
III. The study of very cool dwarfs seems to confirm also the existence of an “M dwarf
10 CHAPTER 1. INTRODUCTION
problem”, but further studies are needed (Bonfils et al. 2005). The definition of an ac-
curate and homogeneous effective temperature scale for G-K-M dwarfs as well as the
technique presented in Paper III for estimating M dwarf metallicities will be major steps
in this direction.
The work presented in Paper I and III is crucial to precisely determine the location
of low mass stars in the HR diagram. With such an accuracy it is possible to address
various questions related to the fine structure of the lower main-sequence, which has
relevance for both chemical evolution and stellar astrophysics, as discussed in Paper II
and III.
Presently, G and K dwarfs are the stellar populations most used for Galactic Chemi-
cal Evolution studies, since their metal abundance pattern provides valuable information
for disentangling the complex puzzle of the Milky Way’s formation. Parallel to the metal
content in a stellar population, also the helium abundance and the differential production
rate of the helium mass fraction Y relative to the metal mass fraction Z (i.e. ∆Y/∆Z)
has profound implications in many areas of astrophysics and cosmology.
The ratio can be used to infer the primordial helium abundance YP by extrapolating
to Z = 0. This technique is usually applied to extragalactic HII regions as discussed in
Chapter 2, where the primordial helium abundance predicted in the standard Big Bang
Nucleosynthesis scenario is also outlined.
The ratio ∆Y/∆Z also governs the stellar clock and thus how long stars live. The
position of not evolved dwarf stars (i.e. those with mass approximately below 0.8M⊙)
on the HR diagram is also uniquely (or almost) determined by their helium and metal
content. The ratio ∆Y/∆Z thus determines the broadening of the lower main-sequence
and it can be understood in simplified terms by means of so-called quasi-homology re-
lations. This is explained in Chapter 3, where quasi-homology stellar models are intro-
duced to provide the theoretical background behind the measurement of ∆Y/∆Z , which
is then the subject of Paper II.
Starting from its primordial abundance, helium is mildly topped up by stars, to-
gether with the production/destruction of other elements. Whereas metals mainly come
from supernovae with high mass progenitors, helium is also ejected into the interstellar
medium by mass loss from intermediate and low mass stars: ∆Y/∆Z can thus be com-
puted from stellar evolutionary theory for a given initial mass function. This is discussed
in Chapter 4 where a chemical evolution model for the Solar Neighbourhood is also pre-
sented. The ratio ∆Y/∆Z predicted by the chemical evolution model is used to validate
the extrapolation technique presented in Chapter 2 and to discuss the results obtained in
Paper II.
Finally in Chapter 5 the results obtained throughout the thesis are summarized. Fu-
ture prospects for projects arising from the current work are also briefly outlined.
CHAPTER 2
Big Bang Nucleosynthesis
“. . . this big bang idea . . . ”
Fred Hoyle on March 28, 1949 – BBC Third Programme
2.1 Historical perspective
The idea of primordial nucleosynthesis dates back to Gamow (1946) and the Big Bang
theory of the Universe pioneered by Gamow, Alpher and Herman in the late 1940s and
early 1950s. One of the first steps in this direction was made in the “αβγ” paper by
Alpher, Bethe & Gamow1 (1948): they supposed that during the first few minutes of
the radiation–dominated Universe, matter was originally present in form of neutrons and
that, after some free decay, protons captured neutrons and successive captures built up
all the elements.
Hayashi first put the theory on a sound physical basis by pointing out that, at the
high densities and temperatures involved, there would be thermal equilibrium between
protons and neutrons at first, followed by a freeze–out, and this intensified the difficulties
already inherent in that theory that the absence of stable nuclei at mass number 5 and
8 would prevent significant nucleosynthesis beyond helium. In the meantime, progress
in the theory of stellar evolution and nucleosynthesis led to comparative neglect of the
Big Bang Nucleosynthesis (BBN) theory, until the detection of the cosmic microwave
background (CMB) radiation (Penzias & Wilson 1965). The existence of the CMB had
been foreseen already by Gamow and proposed on a more solid base by Dicke et al.
(1965) in the Big Bang framework.
The discovery of the CMB marked a major step in favour of the BBN theory, which
was developed subsequently by Peebles, Wagoner, Fowler, Hoyle, Schramm, Steigman
and others. Such a theory intermingles cosmology and particle physics and it has been
very successful in several respects, as we discuss in more details throughout this Chapter.
2.2 Primordial abundances: a brief overview
As the Universe evolved from its early, hot, dense beginnings to its present, cold, dilute
state, it passed through a brief epoch when the temperature and density of its nucleon
1Although Bethe did not take part to this work his name was included by Gamow to complete the “αγ”
sequence.
12 CHAPTER 2. BIG BANG NUCLEOSYNTHESIS
component were such that nuclear reactions building complex nuclei could occur. It
is worthwhile to point out here that throughout the thesis, unless stated, although we
talk about hydrogen (H), deuterium (D), tritium (t), helium (4He) and more complex
elements, we always refer to nuclei.
The nuclear reactions occurring in the early Universe are different from those oc-
curring in stars in several ways: the cosmological density is dominated by radiation,
and the matter density is much lower than in stars. More rigorously, at temperatures2
much higher than 1 MeV the Universe is filled with radiation (photons, e±, neutrinos
of all flavors) along with baryons (nucleons) and dark matter particles. Nuclear and
weak reactions are occurring among the neutrons, protons, e± and neutrinos at rates fast
compared to the Universe expansion rate. The balance between neutrons and protons is
maintained by the weak interactions3
n ←→ p+ e− + νe (2.1)
νe + n ←→ p+ e
− (2.2)
e+ + n ←→ p+ νe. (2.3)
For temperatures of few tens of MeV, protons and neutrons are not relativistic (i.e.
kBT ≪ mpc
2 ∼ mnc
2) and their equilibrium —in the standard scenario without asym-
metry between the number of νe and νe— can be described by the Maxwell–Boltzmann
distribution. DefiningNn (Np) as the number density of neutrons (protons), the neutron–
to–proton ratio is (
Nn
Np
)
∼ exp(−Q/kBT ) (2.4)
where Q = (mn − mp)c2 = 1.293 MeV. When the Universe is hot enough, i.e. for
kBT ≫ Q the ratio given by equation (2.4) is almost one and the number of protons and
neutrons in the Universe practically identical.
While neutrons and protons are interconverting, they are also colliding among them-
selves creating complex nuclides as deuterium n + p ←→ D + γ. However, at early
times, when the density and the average energy of the photons are very high, the newly
formed deuterium is bathed in a background of high–energy gamma rays capable of
photodissociating it. Therefore, deuterium is photodissociated before it can capture a
neutron, a proton or another deuterium to build heavier nuclei. This bottleneck to BBN
persists until the temperature drops sufficiently below the binding energy of deuterium at
about 2.2 MeV. As the Universe expands and cools, the lighter protons are favoured over
the heavier neutrons and the neutron–to–proton ratio decreases, tracking the equilibrium
form in equation (2.4). However, if the equilibrium is to be maintained, the ratio would
2In the rest of the Chapter we talk of temperature in terms of energy, where 1MeV ∼ 1.2 1010 K.
3The reaction (2.1) is the least important for the equilibrium because of the long lifetime of the neutron,
τn = 885.7 ± 0.8 s (Yao et al. 2006) as compared to the Hubble time of the primordial Universe. The
backward reaction is also less likely to occur as it involves a three body process.
2.2. PRIMORDIAL ABUNDANCES: A BRIEF OVERVIEW 13
decrease to a very small value at low temperature. The neutron–to–proton ratio therefore
freezes–out at some characteristic value. As the temperature of the Universe decreases,
other processes occur too. At about 1 MeV, neutrinos decouple since they are no longer
energetic enough for νν −→ e−e+ to happen. Immediately following the neutrino de-
coupling, e+e− −→ 2γ is possible, but not the reverse. As the e± pairs annihilate, they
transfer their entropy to the photons alone, thereby raising the photon temperature rela-
tive to that of the neutrinos. Following the annihilation of the positrons and electrons, a
large number of photons remains. From now on, as the Universe expands, the densities
of baryons and photons in a comoving volume are unchanged with time. It is thus use-
ful, and conventional, to quantify the universal abundance of baryons by comparing the
number density of baryons (nB) to the number density of CMB photons (nγ)
η10 = 10
10ηB ≡ 10
10(nB/nγ) (2.5)
where η10 measured at BBN, at recombination and at present remains the same for the
reason outlined above (e.g. Steigman 2007).
By this time, when the energy of the Universe is somewhat below 1 MeV, the reac-
tions (2.2) and (2.3) cease and the neutron–to–photon ratio is given approximately by its
equilibrium value (
Nn
Np
)
freeze−out
∼ exp(−Q/kBT ) ∼
1
6
. (2.6)
After the freeze–out, the neutron–to–proton ratio does not remain truly constant, but
actually slowly decreases due to occasional weak interactions, eventually dominated by
free neutron decays (2.1). Unless the neutrons are not locked away in nuclei before
∼ τn, their relic abundance would decay to zero. Since the freeze–out point (2.6) occurs
at an age of the Universe of few seconds, there are only a few e-foldings of expansion
available to save the neutrons.
The baryon–to–photon ratio η10 sets the energy at which the photodissociation of
deuterium becomes inefficient, which is at about 0.1 MeV. Before having time to decay,
most of the neutrons end up in 4He nuclei through the following set of reactions
p+ n −→ D+ γ
D+ p −→ 3He + γ
D+D −→ 3He + n
D+D −→ p+ t
3He + n −→ p+ t
t + D −→ 4He + n
3He + D −→ 4He + p (2.7)
while the ratio (2.6) actually decreases to ∼ 1/7 (e.g. Kolb & Turner 1990).
14 CHAPTER 2. BIG BANG NUCLEOSYNTHESIS
It is clear from the above reactions that in the early Universe, the only nuclei pro-
duced in any significant abundance are H and 4He. The latter because it is the most
stable light nucleus (i.e. that with the highest binding energy), and the former because
there are not enough neutrons for all the protons to bind with, and so some protons are
left over. We can easily get an estimate of their relative abundance, normally quoted as
the mass fraction. Since every helium nucleus contains two neutrons, all neutrons end
up in helium and the number density of 4He is NHe = Nn/2. Each 4He nucleus weights
about four proton masses, so the primordial fraction of the total mass in helium, known
as YP is4
YP =
4NHe
Ntot
=
2Nn
Nn +Np
∼ 0.25. (2.8)
This simple treatment tells us that the fraction of matter in the Universe that ends up
into helium is about 25% of the total. A more rigorous and detailed treatment involves
keeping track of the whole network of nuclear reactions rates and the expansion of the
Universe. However, the conclusion remains that about 25% of the Universe’s primordial
composition is in helium, with all the rest in hydrogen. For this reason, it is a common
habit among astronomers to talk of hydrogen mass fraction X, helium mass fraction Y
and metal mass fraction Z , where “metal” is to be understood as all elements heavier
than helium. The condition X + Y + Z = 1 obviously holds.
The full BBN reaction network is considerably more complicated (e.g. Wagoner
1973; Kawano 1992) but allows one to estimate the trace abundances of all the other
nuclei which form in the early Universe. The main reactions occurring during BBN, in
addition to those in (2.7), are
3He + 4He −→ 7Be + γ
t + 4He −→ 7Li + γ
7Be + n −→ 7Li + p
7Li + p −→ 4He + 4He (2.9)
and they are all shown in Figure 2.1.
By the time 4He is produced, Coulomb–barrier suppression is very significant. This
fact, together with the absence of tightly–bound isotopes with mass 5 and 8, prevented
significant nucleosynthesis beyond 4He. The few reactions that manage to bridge the
mass 5 gap lead mainly to mass 7, producing 7Li and 7Be. Later, when the Universe
has cooled further, 7Be captures an electron and decays to 7Li. For the range of η10 of
interest, the BBN predicted abundance of 6Li is more than 3 orders of magnitude below
that of the more tightly bound 7Li. Finally the other gap at mass 8 ensures that there
is no significant production of heavier nuclides, which are in fact produced in stars. It
4Defined this way, YP is not really the mass fraction because this expression adopts precisely 4 for the
4
He to H mass ratio. However, the reader should be warned that YP defined this way is conventionally
referred to by cosmologist as the 4He mass fraction (e.g. Steigman 2007)
2.3. BEYOND THE STANDARD MODEL 15
Figure 2.1: Network diagram of the 12 primary reactions in the processing of the light elements.
The nucleus of deuterium is indicated by d (from Smith, Kawano & Malaney 1993).
is therefore a very good approximation to assume that the primordial mass fraction of
metal is Z = 0.
The primordial nuclear reactor is thus short-lived. As the temperature drops below∼
30 keV, when the Universe is about 20 minutes old, Coulomb barriers abruptly suppress
all nuclear reactions. Afterwards, until the first stars form, no pre–existing, primordial
nuclides are destroyed, except for those like t and 7Be that are unstable and decay to 3He
and 7Li respectively, and no new nuclides are created (e.g. Iocco et al. 2007). In these
20 minutes the BBN has run its course.
2.3 Beyond the standard model
In the previous Section we have presented the standard BBN scenario. Here we briefly
mention possible departures from such a picture and illustrate how in this case the pre-
16 CHAPTER 2. BIG BANG NUCLEOSYNTHESIS
dicted abundances depend on additional parameters. The reader can refer e.g. to Kneller
& Steigman (2004) for further details.
The Hubble parameter, H measures the expansion rate of the Universe. Deviations
from the standard model (H ′) may be parametrized by an expansion rate parameter S ≡
H ′/H or equally by the equivalent number of neutrinos ∆Nν ≡ Nν − 3. ∆Nν and
S are equivalent ways to quantify any deviation from the standard model expansion
rate and ∆Nν is not necessarily related to extra or fewer neutrinos. Changes (from the
standard model) in the expansion rate at BBN would modify the neutron abundance and
the time available for nuclear production/destruction, changing the BBN predicted relic
abundances. For example, in the case of 4He, a faster expansion would provide less time
for neutrons to transform into protons, and the higher neutron fraction would result in an
increase of YP . The primordial mass fraction of 4He is thus an excellent chronometer of
the early Universe expansion rate.
In the currently most popular particle physics models the universal lepton and baryon
asymmetries are comparable, so as to keep the Universe globally neutral. Because any
charged lepton asymmetry is limited by the baryon asymmetry to be very small, any fur-
ther non–negligible lepton asymmetry must be among the neutral lepton, the neutrinos.
Neutrino asymmetries orders of magnitude larger than the baryon asymmetry are in fact
not excluded by any experimental data. Such neutrino asymmetries are quantified by
the parameter ξα (α ≡ e, µ, τ ). Although any neutrino asymmetry increases the energy
density in the neutrinos, resulting in an effective ∆Nν > 0, the range of ξα of interest
to BBN is limited to sufficiently small values that the increase in S due to a non–zero
ξα is negligible. However, a small asymmetry between electron type of neutrinos and
antineutrinos, can have a significant impact on BBN since νe affects the interconversion
of neutrons to protons. Thus, a non–zero ξe results in different numbers of νe and νe,
altering the neutron–to–proton ratio at BBN, therefore changing the yields of the light
nuclides. For example, in the case ξe > 0, the neutron–to–proton ratio increases over its
standard BBN value, leading to an increase in the 4He yield.
Any model beyond the standard BBN can therefore be parametrized essentially as
function of S and ξe. High accuracy measurements of primordial abundances have thus
the possibility to constrain and/or open interesting windows on the physics and the cos-
mology beyond the standard models. We illustrate and briefly discuss some of the most
recent measurements of light elements in the next Section. As regards helium, such al-
ternative models predict variations in its primordial abundance up to a percent or less
and therefore do not provide any viable solution to the low helium abundances presented
in Paper II, which are most likely entirely stemming from inadequacies in extant low
mass, low–metallicity stellar models.
2.4. PREDICTED AND OBSERVED ABUNDANCES 17
2.4 Predicted and observed abundances
Over the years, many researchers have written computer codes to integrate the coupled
set of differential equations that track element production/destruction to solve for the
BBN predicted abundances of the light nuclides.
In the standard BBN scenario with S = 1 (Nν = 3) and ξe = 0 the predicted
primordial abundances depend on only one free parameter, the baryon density η10 as
shown in Figure 2.2. Simple arguments help us to understand this. The reactions burning
D, t and 3He to 4He are fast compared to the early Universe expansion rate, ensuring
that almost all neutrons present at the time are incorporated into 4He once the deuterium
bottleneck is breached. As a result, the 4He primordial abundance is very insensitive to
the baryon abundance. The very slight increase in YP with increasing η10 reflects the fact
that for a higher baryon density BBN begins slightly earlier, when slightly more neutrons
are available. Nuclear reactions burn D, t and 3He to 4He at a rate which increases with
increasing nucleon density, accounting for the decrease in the abundances of D and 3He
(the latter receives a contribution from the β-decay of t) with higher values of η10 (e.g.
Steigman 2007).
The behaviour of 7Li reflects the two pathways to mass 7. At relatively low values
of η10, mass 7 is largely synthesized in form of 7Li by t(α, γ)7Li reactions. As η10
increases 7Li is destroyed more efficiently by collisions with protons. Further increasing
η10, mass 7 is largely synthesized as 7Be via 3He(α, γ)7Be reactions. The end product
of this channel, 7Be is more tightly bound than 7Li and, therefore harder to destroy.
Therefore, as η10 increases the primordial abundance of 7Be increases. Later when the
Universe cools and neutral atoms begin to form, 7Be captures an electron and β-decays
to 7Li. These two paths to mass 7 account for the valley shape of the 7Li abundance
curve in Figure 2.2.
The success of BBN relies on its ability to predict the observed primordial abun-
dances and, conversely, to learn about the cosmological parameters using the same abun-
dances. Now that observations of the CMB and of the distribution of the Large Scale
Structure (LSS) have become sufficiently precise, the range of η10 of interest is restricted
to around 5.7 < η10 < 6.5 (Spergel et al. 2007; Tegmark et al. 2004). Assuming stan-
dard BBN, it is therefore possible to find accurate linear fits to the predicted abundances
as function of η10 only
yD ≡ 10
5(D/H) = 2.68(1 ± 0.03)(6/η10)
1.6 (2.10)
y3 ≡ 10
5(3He/H) = 1.06(1 ± 0.03)(6/η10)
0.6 (2.11)
YP = 0.2483 ± 0.0005 + 0.0016(η10 − 6) (2.12)
yLi ≡ 10
10(7Li/H) = 4.30(1 ± 0.1)(η10/6)
2. (2.13)
The comparison of predictions and observations is far from trivial because of the further
processing of the nuclei since the end of BBN. Deuterium is the baryometer of choice,
18 CHAPTER 2. BIG BANG NUCLEOSYNTHESIS
Figure 2.2: Solid lines: primordial abundances of 4He (mass fraction),D, 3He and 7Li (number
ratios relative to hydrogen) as a function of η10 assuming standard BBN. The dashed lines are
1σ deviations. Some of the observations discussed throughout this chapter are shown with 1σ
boxes. The vertical band represent the WMAP η10 range (from Romano et al. 2003).
because of the strong dependence of its abundance on η10 and because BBN observed
abundance should have only decreased as gas is cycled through stars where deuterium
is burned to heavier, more tightly bound nuclei. As a result, deuterium observed any-
where in the Universe, at any time during its evolution, should provide a lower bound
to its primordial abundance (e.g. Pettini 2006). Further, the deuterium observed in the
2.4. PREDICTED AND OBSERVED ABUNDANCES 19
high redshift, low metallicity QSO absorption line systems should be very nearly pri-
mordial, although hydrogen absorption in a cloud along the same line of sight can bias
measurements toward a much higher deuterium abundance (e.g. Steigman 1994). Al-
though the data is still scarce and the scatter remains large, the measurements appear to
converge to a mean primordial value yD ∼ 2.4 ± 0.4 (e.g. Burles & Tytler 1998; Pettini
& Bowen 2001; O’Meara et al. 2001, 2006) which corresponds (Eq. 2.10) to η10 ∼ 6.4,
in excellent agreement with the value η10 = 6.116 obtained from WMAP (Spergel et
al. 2007). In contrast, the post-BBN evolution of 3He and of 7Li are considerably more
complicated, involving the competition between production, destruction and survival.
As a result, at least so far, the current, locally observed (in the Galaxy) abundances of
these nuclides have been considered of less value than that of deuterium. Indeed, for
the currently accepted range of η10, the primordial abundance of 3He is predicted to lie
in the narrow range 1.0 < y3 < 1.1, in excellent agreement with that inferred from
Galactic observations of distant, metal–poor HII regions (Bania, Rood & Balser 2002).
Thus, 3He provides similar, but less compelling constraints than does deuterium. In the
post BBN Universe, 7Li along with 6Li, 9Be, 10B and 11B, is produced in the Galaxy by
cosmic ray spallation/fusion reactions (e.g. Pagel 1997). Furthermore, observations of
super–lithium–rich red giants provide evidence that at least some stars are net produc-
ers of lithium (e.g. Brown et al. 1989; de la Reza 2000). Therefore, in order to probe
the BBN yield of 7Li, it is necessary to restrict attention to the oldest, most metal–poor
halo stars in the Galaxy. The 7Li abundance measured in the atmosphere of old stars
is a factor of two to four lower than the predicted abundance assuming the currently
accepted value of η10 (Spite & Spite 1982; Ryan, Norris & Beers 1999; Asplund et al.
2006; Bonifacio et al. 2007) although, most notably, the adopted temperature scale (e.g.
Meléndez & Ramírez 2004) and the effect of diffusion (e.g. Korn et al. 2006, 2007)
could help in solving the discrepancy.
Although the main focus of the present thesis has been the study of helium, some
of the results obtained might be of relevance for the determination of other primordial
abundances. The derivation of an accurate effective temperature scale for G-K-M dwarfs
is subject of Paper I and III and the effect of diffusion in altering the position of dwarf
stars in the HR diagram is also briefly discussed in Paper II. The extension of the tem-
perature scale presented in Paper I and III to very metal–poor and earlier spectral type
stars will be valuable for accurate measurements of the primordial lithium abundance in
such a type of stars.
2.4.1 Helium
As gas cycles through generations of stars, hydrogen is burned to 4He and beyond,
increasing the helium abundance above its primordial value YP (Burbidge, Burbidge,
Fowler, Hoyle 1957). As a result, the 4He mass fraction in the Universe at any given
epoch after the Big Bang, Y , contains a contribution from stellar nucleosynthesis, so
that Y > YP .
20 CHAPTER 2. BIG BANG NUCLEOSYNTHESIS
Helium abundances can be estimated in various objects with different techniques,
but direct measurements are still rather challenging. Measurements from the Voyager
and Galileo spacecraft data on Jupiter and Saturn have returned a helium mass fraction
in the range 0.19 − 0.25 (Conrath & Gautier 2000). The value for the interior of the
Sun is based on theoretical models of its evolution, constrained by its known mass, age,
chemical composition (apart from helium itself!) and present day luminosity and radius
(Chapter 3 and Paper II). Helioseismology has led to greatly improved understanding of
the Sun’s internal structure and it also indicates an appreciably lower helium abundance
in the convective envelope, which is attributed to gravitational settling (e.g. Christensen-
Dalsgaard, Proffitt & Thompson 1993; Basu, Pinsonneault & Bahcall 2000). Based on
seismic data, the helium abundance in the solar convective envelope is estimated to range
from Ysurf,⊙ = 0.24 to Ysurf,⊙ = 0.25 (see Basu & Antia 2008 for a review) whereas
the standard solar model predict Y⊙ = 0.2725 and Y⊙ = 0.2600 if the Grevesse &
Sauval (1998) or the Asplund, Grevesse & Sauval (2005) heavy element abundances are
used (Bahcall, Serenelli & Basu 2006). Estimates of helium in the chromosphere have
been made on the basis of hydrogen and helium emission lines from prominences, but
these measurements are quite imprecise5. What it is important from the above discus-
sion is that most of the helium in the Sun has primordial origin and its exact value has
been increased with respect to YP only by few hundreds. The most metal–poor stars in
the solar neighbourhood are thus of particular interest, because their content of heavy
elements is so low that they can be considered to have been born with essentially the
primordial helium abundance. The attempt to determine the helium abundance in the
most metal–poor, non evolved dwarf stars in the Solar Neighbourhood is the subject of
Paper II.
Accurate measurements of helium abundances come from measurements of recom-
bination lines of hydrogen and helium in the emission spectra of planetary nebulae (e.g.
Pottasch, Bernard-Salas & Roellig 2007) and HII regions (e.g. Izotov, Thuan & Stasin´ska
2007; Peimbert, Luridiana & Peimbert 2007), using theoretical effective recombination
coefficients calculated from quantum mechanics. In Galactic HII regions it is also pos-
sible to observe radio recombination lines, which is especially useful when the optical
spectrum is obscured by dust (e.g. Balser 2006). Peimbert & Torres–Peimbert (1974,
1976) outlined a programme whereby measurements in Galactic and extragalactic HII
regions having different Z abundances (represented mainly by oxygen, although some-
times nitrogen is used, too) could be used to plot a regression represented by
Y = YP + Z
∆Y
∆Z
= YP + (O/H)
∆Y
∆(O/H)
. (2.14)
Extrapolation to Z = (O/H) = 0, would give the primordial helium abundance YP ,
while the slope of the regression, ∆Y/∆Z (not necessarily constant) would place a
5It is worth mentioning that helium was first discovered in 1868 by Pierre Janssen and subsequently
confirmed by Norman Lockyer and Edward Frankland by observing the chromosphere of the Sun during a
solar eclipse.
2.4. PREDICTED AND OBSERVED ABUNDANCES 21
Figure 2.3: Helium mass fraction Y vs. oxygen abundancesO/H for 30 metal–poor HII regions,
when the effect of stellar absorption is taken into account. The resulting slope is∆Y/∆(O/H) =
20± 25 which corresponds to ∆Y/∆Z = 1.1± 1.4 (Fukugita & Kawasaki 2006).
significant constraint on stellar nucleosynthesis, by giving the rate at which helium is
freshly synthesised and ejected into the InterStellar Medium (ISM) compared to the
corresponding rate for metals. The application of this idea to objects in our own Galaxy
suffers from the fact that the abundances represented by Z are rather high, so that a large
extrapolation is needed to obtain an estimate of YP . In addition, planetary nebulae need a
small correction for helium dredged up to the surface during prior stellar evolution, while
Galactic HII regions mostly have ionizing stars that are not hot enough to guarantee the
absence of otherwise undetectable neutral helium. Consequently, the best objects in
which to apply this idea are extragalactic HII regions in dwarf irregular galaxies and in
blue compact galaxies, where the abundances of oxygen and other heavy elements are
low and the ionizing stars often very hot. It is worth mentioning that the occurrence of a
complete ionization is however a rather ideal case also for extragalactic HII regions (e.g.
Viegas, Gruenwald & Steigman 2000). The present inventory of such regions studied
for their helium content exceeds 80 (Izotov & Thuan 2004; Izotov, Thuan & Stasin´ska
2007), although often a posteriori selected subsample is used in order to determine YP .
Over the last fifteen years there has been a considerable trend in the estimated value
22 CHAPTER 2. BIG BANG NUCLEOSYNTHESIS
Figure 2.4: Helium mass fraction Y vs. oxygen abundances O/H from the chemical evolution
models presented in Chapter 4. Continuous line refers to model A (built to match the mean
observational constraints in the Solar Neighbourhood) and dotted line to model B (appositely
selected to reach high metallicities at the present time of 13 Gyr). The gradients ∆Y/∆(O/H)
and ∆Y/∆Z are computed for the same range of the data in Figure 2.3. Notice that in the
chemical evolution models YP = 0.24 has been assumed.
of YP , from a “low” primordial helium abundance (YP = 0.228 ± 0.005 Pagel et al.
1992; YP = 0.234 ± 0.002 Olive, Skillman & Steigman 1997; YP = 0.2391 ± 0.0020
Luridiana et al. 2003) to a “high” primordial helium abundance in better agreement with
YP computed assuming standard BBN and η10 from CMB (YP = 0.2421 ± 0.0021 Izo-
tov & Thuan 2004; YP = 0.249 ± 0.009 Olive & Skillman 2004; YP = 0.250 ± 0.004
Fukugita & Kawasaki 2006; YP = 0.2477 ± 0.0029 Peimbert, Luridiana & Peimbert
2007; YP = 0.2516 ± 0.0011 Izotov, Thuan & Stasin´ska 2007). The lesson from the
above determinations is that although recent attempts to determine the primordial abun-
dance of 4He have achieved high precision, their accuracy remains in question. The
latter is limited by the understanding of and ability to account for systematic corrections
and their errors, not by statistical uncertainties. The convergence of the measured pri-
mordial helium abundance to a closer agreement with that predicted assuming standard
BBN and the measured baryon–to–photon ratio hopefully reflects a better understanding
of the complex physics of HII region and not a forcing of the data.
2.4. PREDICTED AND OBSERVED ABUNDANCES 23
The most recent analyses (Olive & Skillman 2004; Fukugita & Kawasaki 2006; Pe-
imbert, Luridiana & Peimbert 2007) fail to find evidence for the anticipated correlation
between the helium and the oxygen abundance (Figure 2.3), calling into question the
model–dependent extrapolations to zero metallicity often employed in the quest for pri-
mordial helium. The theoretical correlation between helium and oxygen abundance is
addressed in Figure 2.4 by means of the chemical evolution model presented in Chapter
4. Interestingly, the very flat evolution of Y compared to oxygen observed by Fukugita &
Kawasaki (2006) is confirmed by our chemical evolution model. Although the observed
correlation is obtained studying HII regions, and our model is best suited to describe the
evolution of the Galactic disc, the predictions are rather insensitive to the detail of the
model calibration, as we discuss in more details in Chapter 4.
According to Steigman (2007) all the analyses presented to date have different sort
of systematic errors. In particular, because hydrogen and helium recombination lines are
used, the observations are blind to any neutral helium or hydrogen, which are anyway
present. Estimates of the ionization correction factor are unfortunately model depen-
dent and possibly also very large (Viegas, Gruenwald & Steigman 2000; Gruenwald,
Steigman & Viegas 2002). The value proposed by Steigman (2007) for the primordial
helium abundance is
YP = 0.240 ± 0.006 (2.15)
where the adopted error is an attempt to account for the systematic as well as the statisti-
cal uncertainties. We adopt this value for the composition of the protogalactic gas cloud
used in the chemical evolution model presented in Chapter 4.
CHAPTER 3
Fine Structure in the Lower
Main-Sequence
“Stars shining bright above you”
Dream a Little Dream of Me – Gus Kahn
3.1 Quasi–homology relations
We have already stated that the main goal of this thesis is to carefully derive physical
parameters for low mass stars in order to gauge insight into the chemical evolution of
helium. Performing such a task using nearby stars has always proved challenging and all
studies to date have exploited the fact that the broadening of the lower main-sequence
with metallicity effectively depends on the helium content (e.g. Faulkner 1967; Ström-
gren, Olsen & Gustafsson 1982; Fernandes, Lebreton & Baglin 1996; Pagel & Portinari
1998; Jimenez et al. 2003). To this purpose, precisely locating low mass stars on the HR
diagram is crucial, as we discuss in what follows.
For stars still on their zero-age main-sequence, it is well known that an increase
of helium (Y ) keeping the metallicity (Z) constant induces an increase of luminosity
(MBol) and effective temperature (Teff ) at a given mass. On the other hand an increase
of Z at fixed Y leads to the opposite effect, i.e. luminosity and effective temperature
decrease (e.g. Fernandes, Lebreton & Baglin 1996). This behaviour is clearly shown in
Figure 3.1 plotting some of the tracks and isochrones used in Paper II. Figure 3.1 also
highlights how the broadening of the lower main-sequence is independent of the age
and how the shift at a fixed mass with increasing age is rather small and into a direction
parallel to the main-sequence.
Such a behaviour is due to opacity and mean molecular weight effects and can be
easily explained in the framework of quasi–homology theory which we briefly review
here. A thorough discussion can be found e.g. in Cox & Giuli (1968). Using the quasi–
homology relations, luminosity and effective temperature of a main-sequence star can
be expressed as
L = ǫ−0.0770 κ
−1.077
0 µ
7.769M5.462 (3.1)
Teff = ǫ
−0.096
0 κ
−0.346
0 µ
2.211M1.327 (3.2)
where µ = (2X + 0.75Y + 0.5Z)−1 is the mean molecular weight (complete ionized
gas), ǫ0 the nuclear energy generation rate, κ0 the opacity and M the stellar mass. By
3.1. QUASI–HOMOLOGY RELATIONS 25
Figure 3.1: Left panel: change in MBol and Teff with increasing Y for a fixed Z . Right panel:
change in MBol and Teff with increasing Z for a fixed Y . Thick lines are stellar tracks at
two given masses (0.7M⊙ and 0.5M⊙) whereas dotted and dashed lines are the isochrones
built interpolating the stellar tracks at the ages of 1 Gyr and 5 Gyr, respectively. Filled (open)
triangles refer to 0.7M⊙ at an age of 1 Gyr (5 Gyr). Filled (open) circles refer to 0.5M⊙ at an
age of 1 Gyr (5 Gyr).
combining equations (3.1) and (3.2) and making some simplifying assumptions on the
form of ǫ0 ∝ X2 (i.e. main source of energy through pp-chain, where X is the hydrogen
mass fraction) and κ0 ∝ (1 +X)(100Z +1) (i.e. opacity dominated by bound-free and
free-free transitions) it is possible to find an analytical formula between the luminosity
and the effective temperature, which depends only on the stellar chemical composition
X,Y,Z . Further assuming that
Y = Y0 + Z
∆Y
∆Z
(3.3)
and since X + Y + Z = X0 + Y0 = 1 it follows that
X = 1− Y − Z = X0 − Z
(
∆Y
∆Z
+ 1
)
(3.4)
and thus and entire family of luminosity–Teff relations can be built by simply scaling the
chemical composition with Z as a function of ∆Y/∆Z . For a given chemical composi-
tion, the resulting luminosity–Teff relation effectively acts like an isochrone
L ∝ (X + 0.4)2.67(100Z + 1)0.455f(Teff). (3.5)
26 CHAPTER 3. FINE STRUCTURE IN THE LOWER MAIN-SEQUENCE
By choosing a reference luminosity–Teff relation is then possible to compute the differ-
ence in luminosity (or effective temperature) for all the other relations as function of Z
and ∆Y/∆Z only (e.g. Pagel & Portinari 1998).
3.2 Two recent works on ∆Y/∆Z from low mass stars
The formalism previously introduced provides a physical ground to understand the broad-
ening of the lower main-sequence in terms of stellar helium and metal content. However,
for a better comparison with the observational data, theoretical isochrones derived from
the latest stellar models are usually used.
Among various works done in the past, two in particular have tackled the ques-
tion in great details, Pagel & Portinari (1998) and Jimenez et al. (2003). By using the
formalism previously introduced, Pagel & Portinari (1998) numerically derived fitting
formulae linking the difference in effective temperature (∆ log Teff ) from a reference
isochrone as function of Z and ∆Y/∆Z using a set of Padova isochrones. The refer-
ence isochrone was chosen to be the solar one. Their sample included about thirty K
dwarfs with Hipparcos parallaxes which were studied in the MV − log Teff plane, us-
ing effective temperatures derived via the InfraRed Flux Method (IRFM) from Alonso,
Arribas & Martínez-Roger (1996). They found ∆Y/∆Z = 3± 2.
Jimenez et al. (2003) used a rather similar approach. Also in this case the sam-
ple was composed by approximately thirty K dwarfs for which accurate parallaxes ob-
tained with the Hipparcos satellite were available. The analysis was performed in the
MV − (B − V ) plane, exploiting the difference in luminosity (∆MV ) with respect to a
reference isochrone, which they chose to be the primordial one. They assumed a value
for the primordial helium Y0 and computed different models for various Z and ∆Y/∆Z .
For both models and sample stars ∆MV was computed with respect to the reference
isochrone. Since all the isochrones with difference helium abundances at a given Z were
computed starting from the same Y0, the effect of different helium enhancements became
relevant only at the highest metallicities which had most of the leverage in determining
the final result ∆Y/∆Z = 2.1 ± 0.4.
3.3 ∆Y/∆Z in nuce
The motivating idea behind this thesis was to determine the broadening of the lower
main-sequence with precision greater than ever before, so as to provide the most strin-
gent determination yet of ∆Y/∆Z from lower main-sequence stars. Collecting a larger
dataset of highly accurate and homogeneous observational data on K dwarfs, together
with the choice of working directly in the MBol − Teff plane (where the effect of the
helium is more pronounced, e.g. Castellani, degl’Innocenti & Marconi 1999) has been
the driving idea of the project and the motivation beyond Paper I. In this paper, we have
3.3. ∆Y/∆Z IN NUCE 27
Figure 3.2: Helium (Y) to metal (Z) enrichment factor obtained in Paper II. For metallicities
slightly below solar (Z⊙ = 0.017) extremely low helium abundances are found.
developed our own version of the IRFM to recover the fundamental stellar parameters
from multi–band photometry.
A new set of low mass stellar tracks, computed especially for us at Padova, for vari-
ous metallicities and helium abundances has been used to derive ∆Y/∆Z from the new
sample (Paper II). Theoretical stellar models have a number of free parameters usually
calibrated on the Sun. One of this parameters is the helium mass fraction, which auto-
matically fixes the hydrogen mass fraction, since that of the metals comes from observa-
tions (although its correct value is currently a matter of debate, e.g. Basu & Antia 2008
for a review). Therefore, when dealing with theoretical stellar models, the abundances
introduced in equations (3.3) and (3.4) must be intended as calibration parameters, the
exact value of which varies depending on the input physics implemented in the adopted
stellar evolutionary code.
For this reason, X0 and Y0 introduced here do not necessarily correspond to the
primordial hydrogen and helium mass fraction presented in the previous Chapter. The
choice of the calibration parameters is simply a zero-point effect, which is irrelevant
when dealing with a differential quantity such as ∆Y/∆Z and thus it justifies both ap-
28 CHAPTER 3. FINE STRUCTURE IN THE LOWER MAIN-SEQUENCE
proaches used in Pagel & Portinari (1998) or Jimenez et al. (2003), who adopted differ-
ent reference isochrones in their studies. In our work (Paper II) we chose to use the solar
isochrone as the reference one, since it is the calibration zero–point of stellar models;
however, comparison with other isochrones sets, including that of Jimenez et al. (2003)
with a fixed primordial isochrone, did not affect our results.
Working in the theoretical HR diagram allowed us to highlight shortcomings in the
low-metallicity theoretical stellar models, which unfortunately precluded us from deriv-
ing ∆Y/∆Z over the entire Z range represented by stars in the Solar Neighbourhood.
Starting from metallicities slightly below solar, we have found that anomalously low
helium abundances (well below the primordial value) are needed to reproduce the data
(Figure 3.2), namely theoretical isochrones are too hot with respect to observations.
Besides accounting for non-local thermodynamic equilibrium (LTE) in deriving metal-
licities, including diffusion in stellar models partly helps to solve the problem. Such a
conclusion can be interpreted as an independent hint for diffusion acting also in nearby
metal–poor field stars, corroborating the evidence already seen in globular clusters (Korn
et al. 2006). Although helping, both non-LTE and diffusion seem not enough to entirely
solve the low helium problem and a mixing–length decreasing with decreasing metallic-
ity has also been proposed in Paper II. Therefore, at present it is not yet possible to derive
∆Y/∆Z from stellar models at low Z . The result obtained in Paper II was thus limited
to stars with metallicities around and above the solar one. However, the technique pre-
sented in Paper I to recover effective temperatures and bolometric luminosities of low
mass stars has been further improved and developed so to be successfully applied also to
M dwarfs (Paper III). The relevance of the results obtained throughout this thesis in the
more general context of studies dealing with Galactic Chemical Evolution are outlined
in the next chapters.
CHAPTER 4
The chemical evolution of helium
“For the things we have to learn before we can do them, we learn by doing them”
Aristotle
4.1 The overall picture
The small ripples observed in the CMB (e.g. Spergel et al. 2003, 2007) are the seeds
of the galaxies which form the LSS of the Universe. In the currently accepted frame-
work, the dynamic of the LSS is dominated by dark matter and dark energy. In this very
schematic picture, the baryons left from the Big Bang in the form of H, D, 3He, 4He and
7Li (see Chapter 2) collapse in the potential well of the dark matter haloes (e.g. White
& Rees 1978) to eventually form the stars and the galaxies we observe today. The actual
process of galaxy formation is considerably more complicated and involves highly non-
linear processes, so that only detailed numerical simulations can tackle them —at least
to some extend. The ultimate goal of a successful theory of galaxy formation is to be
able to understand and describe the formation and the evolution of the various compo-
nents (stellar and non) which characterize different types of galaxies, observed at various
redshifts. In this quest, our Galaxy, the Milky Way, offers a formidable nearby template
where to test and calibrate some of the various ingredients needed for a self-consistent
theory of galaxy formation. The purpose of Galactic Chemical Evolution (GCE) models
is exactly to focus on some (or as many as possible) of those processes responsible for
the chemical enrichment of the Milky Way as deduced from the abundances observed
in stars and gas. GCE models thus specifically deal with the chemistry of the Galaxy,
highly simplifying the complex dynamic involved in its formation.
Figure 4.1 shows a sketch of the Milky Way, seen edge-on. Several components con-
sisting of stars and gas with various properties as regards kinematics and chemical com-
position are present. The stellar populations of these main components, halo, bulge and
disk, have quite different chemical compositions, kinematic and dynamical properties
reflecting different evolutionary histories. Traditionally, two scenarios have competed
to explain the formation of the Milky Way. The first one, proposed by Eggen, Lynden-
Bell & Sandage (1962), describes the rapid monolithic collapse of a protogalactic gas
cloud to form the halo. The Galactic disk wold have subsequently formed as the residual
gas dissipationally collapsed, spun up due to conservation of angular momentum. This
would naturally give rise to two populations of stars: an older, more metal–poor group
found in the halo, and a younger, more metal–rich group orbiting closer to the Galactic
30 CHAPTER 4. THE CHEMICAL EVOLUTION OF HELIUM
Figure 4.1: A sketch of the structure of the Milky Way: the stellar halo, bulge, thick and thin disk
are indicated together with the mean metallicity of the stars in each Galactic component. The
galactocentric distance of the Sun together with the dimension of the optical disk are shown at
the bottom (from Matteucci 2001).
mid–plane. Searle & Zinn (1978) offered an alternative to the monolithic collapse pic-
ture, proposing that the Galaxy was constructed from smaller cloud fragments, in which
stars may have already started forming.
The Galaxy’s true formation history is likely to lie somewhere between the two ex-
tremes of primordial collapse and hierarchical formation, in the more general framework
of ΛCDM structure formation (e.g. Freeman & Bland-Hawthorn 2002; Chiosi 2007).
The recent evidence that the halo of the Milky Way is composed by two populations, the
inner and the outer halo, with distinct kinematic properties and metallicity distribution
functions, further confirms such a conclusion (Carollo et al. 2007).
Independent of the exact details of this picture, galaxy formation is accompanied by
star formation leading to synthesis of heavier elements and modification of primordial
abundances: D is destroyed by stellar activity, 4He is mildly topped up and 3He and
7Li are both created and destroyed by stars (e.g. Chiappini, Renda & Matteucci 2002;
Romano et al. 2003; Carigi & Peimbert 2008).
When stars are formed, there are broadly few things that can happen to the ISM as
a result of their evolution. Galactic Chemical Evolution models deal specifically with
the stellar evolutionary processes so as to explain the chemical composition observed in
4.1. THE OVERALL PICTURE 31
stars and gas in the Galaxy. Stars of different mass ranges give different contribution to
the chemical evolution of the Galaxy, as summarized below. There are still uncertainties
in the exact boundaries of the stellar mass ranges discussed below, also depending on
the stellar metallicity and whether or not stellar models with convective overshooting
are employed. In the present introduction, we aim to give a qualitative overview.
Very low mass stars, withM < 0.8M⊙ have such long lifetimes that they essentially
do nothing: they simply serve to lock up part of the gas and remove it from circulation,
but they remain visible today as archaeological tracers of the composition of the ISM
at the time and place where they formed. These stars can be used to trace the chemical
evolution of the Milky Way. Their characterization is the subject of Paper I and III and
their study by means of theoretical stellar models is presented in Paper II. In particular,
one of the basic assumptions is that the chemical composition in the atmospheric layers
of such stars is representative of the gas out of which the stars once formed, i.e. no
processes substantially altered their chemical composition at the surface with time. Such
an assumption is of paramount importance for chemical evolution studies and it has been
partly discussed also in Paper II.
Low and Intermediate mass stars, approximately from 0.8 to 5 M⊙ —in models
with convective overshooting— undergo various dredge–up and mass loss episodes as
they evolve after the main-sequence and pass through the Red Giant Branch (RGB),
the Horizontal Branch (HB), the Asymptotic Giant Branch (AGB) and the Planetary
Nebula (PN) phases before they end up as CO–white dwarfs. Low mass stars, with mass
below about 2 M⊙ (e.g. Maeder & Meynet 1989; Chiosi, Bertelli & Bressan 1992) ignite
helium in an electron degenerate core (He flash), whereas above this mass intermediate
mass stars avoid the core helium flash and burn He quiescently. Close binaries in this
mass range can undergo more dramatic events as a result of mass transfer. In particular,
they can explode as supernovae Ia which contribute substantially to the nucleosynthesis
of iron and explain the observed evolution of the abundance ratio of α–elements with
respect to [Fe/H] (e.g. Sneden et al. 1979; Matteucci & Greggio 1986; McWilliam
1997).
Quasi massive stars, in the range ∼ 5 − 8M⊙ undergo core C–burning in non de-
generate conditions, but develop highly degenerate O-Ne-Mg–cores. They become dy-
namically unstable and are thought to explode as electron capture supernovae, although
their final state is still a matter of debate. Super–Asymptotic Giant Branch evolution
might alternatively develop in the non–violent formation of a massive white dwarf, or in
a weak explosion at most (Ritossa, Garcia-Berro & Iben 1996; Eldridge & Tout 2004).
Massive stars, in the range ∼ 8 − 120M⊙, have simultaneous occurrence of two
phenomena: the dominant mass loss by stellar wind during the whole evolutionary his-
tory (significant especially above 30M⊙) and the completion of the nuclear sequence
down to the formation of an iron core in presence of strong neutrino cooling. These stars
end up as Type II supernovae or Type Ib,c supernovae if the progenitor suffered from
effective mass loss and lost its H–rich envelope before the explosion. Either a neutron
star or a black hole is left over.
32 CHAPTER 4. THE CHEMICAL EVOLUTION OF HELIUM
Very massive stars i.e. objects more massive than ∼ 120M⊙ are strongly pulsa-
tionally unstable and suffer from violent mass loss. The final outcome is regulated by
the mass of the CO–core: these stars can either collapse to a black hole, suffer from
complete thermonuclear explosion or recover the behaviour of the massive stars above.
The net result of the stellar evolution schematically outlined above is clear: it is
to change the initial Galactic chemical composition. The purpose of GCE models is
to interpret all these processes within the bigger framework of the stellar populations
which compose the Galaxy, so as to account for the abundances observed in stars and
gas. Obviously, the larger the number of observables, the more the framework provided
by GCE can be useful in constraining the history of the Milky Way. In particular, for
obvious reasons, the Solar Neighbourhood is the best observed system.
4.2 ∆Y/∆Z from different stellar populations
In order to trace the history of chemical evolution it is quite important to analyze in
detail the nucleosynthesis of various elements taking place in stars with different masses
and metallicities. Following the classical definition by Tinsley (1980), the stellar yield,
pi(M,Z) of a given element i, is the mass fraction of a star with initial mass M and
metallicity Z that is converted into the element i and returned to the ISM during its
entire lifetime. Stellar yields are thus important for studies of the chemical evolution of
the galaxies.
In this Section we address the topic more specifically, considering the global yields
of a stellar population born with a typical Solar Neighbourhood Initial Mass Function
(IMF). The inclusion of the stellar yields in our chemical evolution model will be the
subject of the next Section. The yields adopted in the present study are those of Portinari,
Chiosi & Bressan (1998) for massive stars (6 < M < 120M⊙) and Marigo (2001) for
intermediate and low mass stars (0.8 < M < 5M⊙). Both Portinari et al. (1998) and
Marigo (2001) stellar yields are based on the Padova evolutionary tracks. The range in
metallicity covers from Z = 0.0004 to Z = 0.05. In Section 4.2.1 we compare the
adopted yields with other sets which have recently appeared in literature and we justify
our choice.
In Figure 4.2 the adopted stellar yields have been properly weighted by a chosen
IMF; we mainly adopt the IMF by Kroupa (1998), and we discuss later the differences
when assuming e.g. the Salpeter (1955) IMF. At low Z , helium is produced mostly by
stars in the mass range 0.8 < M < 10M⊙. The peak of helium production around
2 − 3M⊙ is mostly due to dredge-up episodes during the Thermally Pulsating Asymp-
totic Giant Branch (TP-AGB), whereas the contribution at 4 − 5M⊙ is caused by the
additional occurrence of Hot Bottom Burning (HBB) in addition to dredge-up episodes
during the TP-AGB. In this mass range, the production of helium is more pronounced
at the lowest metallicities when the number of stellar pulses is larger and the TP-AGB
duration longer (e.g. Marigo 2001). More massive stars have minor importance in the
4.2. ∆Y/∆Z FROM DIFFERENT STELLAR POPULATIONS 33
Figure 4.2: Stellar yields in helium and heavy elements, weighted by the IMF (Kroupa) for
models with two different initial metallicities (Z = 0.0004 and Z = 0.02)
production of helium, especially at low Z . With increasing metallicity, however, stel-
lar wind and mass loss in massive stars become important, so that the contribution of
metal–rich massive stars to the production of helium is more relevant. The production
of C also peaks around 2 − 3M⊙, corresponding to the largest number of dredge-up
episodes during the TP-AGB phase, provided that HBB has not operated. The subse-
quent decline towards higher masses is initially due to fewer dredge-up events, and then
to the prevailing effects of HBB. The contribution of massive stars to the enrichment of
C increases with metallicity due to more efficient mass loss. At low metallicities N is
mostly produced during the HBB in intermediate mass stars, while at higher metallici-
ties secondary production of N via the CNO cycle occurs over the whole mass range. O
is mainly produced by massive stars at all metallicities, with a minor contribution from
stars with M < 3.5M⊙ thanks to the dredge-up events during the TP-AGB phase. In
this Figure Fe comes from Type II supernovae; the other main contribution, from Type
Ia supernovae, will be included only in the chemical evolution model.
While the global metallicity Z is mostly produced by massive stars, with low and
intermediate mass stars contributing some 15–40% (at Z=0.02 and Z=0.0004 respec-
tively), the latter give a relevant contribution to helium enrichment: from about 60%
at low Z , to 40% at metallicities around solar. Therefore helium is partly produced on
longer timescales than the metals Z , a behaviour qualitatively reminding the trend of
34 CHAPTER 4. THE CHEMICAL EVOLUTION OF HELIUM
Figure 4.3: ∆Y/∆Xi where Xi is the total metal mass fraction (Z) or the mass fraction of
specific elements (C, N, O, Fe) plotted as function of the metallicity Z covered by the adopted
stellar yields. ∆Y/∆Xi is computed integrating over the entire stellar population weighted by
a given IMF (Salpeter or Kroupa). Notice that Fe is here produced by Type II supernovae only.
The inclusion from supernovae of Type Ia is done in the GCE model presented in the next Section.
[α/Fe] ratios.
Figure 4.3 shows the helium–to–metal enrichment ratio when Z or the mass fraction
of other elements (C, N, O, Fe) is considered. The curves result from the adopted stellar
yields when they are integrated over the entire stellar population (i.e. from 0.1 to 120M⊙
although only masses above 0.8M⊙ contribute) and weighted by two different IMFs,
the Salpeter (1955) and Kroupa (1998). Choosing between different IMFs obviously
changes the global ∆Y/∆Xi as different proportions of massive or low and intermediate
mass stars are involved (e.g. Chiosi & Matteucci 1982).
The global ∆Y/∆Z expected to be produces by a stellar population with the Kroupa
IMF is between 1.8 and 1.4; this is somewhat lower, but comparable to our observational
result of ∆Y/∆Z ≃ 2. With the Salpeter IMF, which has a shallower slope and hence a
smaller percentage of low and intermediate mass stars, the expected ∆Y/∆Z is lower,
between 1.5 and 1. Therefore adopting the Kroupa (1998) IMF for our GCE models
of the Solar Neighbourhood has a better chance to recover the observed value; notice
however that the difference with respect to the Salpeter IMF is at the same level as the
current observational accuracy.
4.2. ∆Y/∆Z FROM DIFFERENT STELLAR POPULATIONS 35
4.2.1 Comparison with other chemical yields
Although other sets of yields have been recently published (e.g. Ventura, D’Antona &
Mazzitelli 2002; Maeder & Meynet 2002; Hirschi, Maeder & Meynet 2005; Karakas &
Lattanzio 2007), these are restricted to mass and metallicity ranges much smaller than
those used in our GCE model, so that to use them one should patch yields computed by
various authors. The occurrence of important nucleosynthesis processes in certain mass
and metallicity ranges depends upon the physics implemented in different stellar codes
(e.g. convection model used, method for determining the convective borders, inclusion
of overshooting). It is therefore obvious that taking yields from different authors can
introduce a source of inaccuracy which is avoided when adopting the complete and ho-
mogeneous set of Padova yields. Nonetheless, we compare here our adopted set with
other recently published, to check if that would make any major difference in our final
results.
Karakas & Lattanzio (2007) have recently computed stellar yields for 74 chemical
species at various metallicities (from Z = 0.0001 to Z = 0.02) in the mass range
1−6M⊙. Stars in this mass range undergo various evolutionary stages, in particular the
TP-AGB phase, whose modeling is still a difficult task, due to both the high complexity
of the physics involved and the remarkable requirement of computing time (e.g. Marigo
2001; Marigo & Girardi 2007; Karakas & Lattanzio 2007). A viable approach is of-
fered by synthetic models, which summarize the results of complete stellar calculations
through simple and practical analytical relations depending upon few parameters. The
parameters used in the synthetic models of Marigo (2001) are calibrated to reproduce
fundamental observables (e.g. the carbon stars luminosity function, the initial-final mass
relation of low and intermediate mass stars) whereas those computed by Karakas & Lat-
tanzio (2007) are not. The main difference is a deeper third dredge-up in the Marigo
(2001) models, so that Karakas & Lattanzio (2007) produce a lower amount of metal, C,
N and particularly O, as well as less helium (see also the detailed comparison presented
in Karakas & Lattanzio 2007). When the yields are integrated over the entire stellar
population from 1 to 6M⊙ and weighted by the Kroupa (1998) IMF, the ∆Y/∆Z from
Karakas & Lattanzio (2007) varies from about 4.5 to 8.7, whereas the yields from Marigo
(2001) predict a shallower slope ranging from 2.8 to 4.6, depending on the metallicity.
For massive stars, a recent set of yields is that available from the Geneva group
(Maeder & Meynet 2002; Hirschi et al. 2005) which also takes into account the effect of
stellar rotation. These yields are presently given for three metallicities (Z = 10−5, 0.004
and 0.02) in the mass range ∼ 10− 60M⊙. A throughout comparison between rotating
and non-rotating models is given in the original papers. Briefly, CNO yields from rotat-
ing models are usually slightly higher than those from non-rotating models whereas the
helium production is rather similar, thus implying only a moderate decrease in ∆Y/∆Z
for the rotating case. With respect to the yields of Portinari et al. (1998), the rotating
yields predict a similar ∆Y/∆Z at Z = 0.004, whereas it decreases to almost half at
solar metallicity. What is important in rotating models is the effect of stellar winds, espe-
36 CHAPTER 4. THE CHEMICAL EVOLUTION OF HELIUM
Figure 4.4: Comparison between the yields from Marigo (2001) and Portinari et al. (1998)
represented by continuous lines (MP) and a patchwork including the yields from Karakas &
Lattanzio (2007), Maeder & Meynet (2002) and Hirschi et al. (2005) instead (KMH, dashed
lines). The comparison is shown for two different metallicities. The upper panels show the helium
(purple) and Z (black) yields, whereas the lower panels display the C (cyan), N (yellow) and
O (blue) yields. Also reported are the helium to various elements enrichment ratios obtained
integrating over the entire stellar population at the metallicity given in each plot. A Kroupa
(1998) IMF is assumed. The peak in each plot is normalized to one and a logarithmic scale is
used for the mass in order to facilitate the comparison.
cially at extremely low metallicities (Z = 10−8 − 10−5) which should be representative
of the —yet unobserved— first stars responsible for the early chemical evolution of the
Galaxy. Whereas the yields (i.e. the total contribution from stellar winds plus supernova
explosions) from rotating and non-rotating models are similar, the efficient mixing in
rotating models enriches considerably the stellar surface which in turn favours a large
mass loss by stellar wind (rich in helium), whereas the supernova contribution is greatly
diminished (Meynet, Ekström & Maeder 2006). Ad hoc scenarios in which the winds of
high mass stars drive the chemical evolution without the contribution in heavy elements
from supernovae are worth investigating to explain e.g. the anomalous abundance pat-
4.3. A MODEL FOR THE CHEMICAL EVOLUTION OF THE GALAXY 37
terns observed in ω Centauri (Maeder & Meynet 2006), but this is clearly outside the
purpose of the present study.
Figure 4.4 shows the stellar yields in helium and heavy elements when a patchwork
including yields from Karakas & Lattanzio (2007) below 6M⊙ and the high mass rotat-
ing yields from Maeder & Meynet (2002) and Hirschi et al. (2005) is used instead of the
yields presented in Section 4.2 and adopted in our GCE model. Despite the differences
in the adopted sets, when the yields are integrated over the entire mass range, the ratios
reported in Figure 4.4 compare quite nicely. Especially for ∆Y/∆Z the difference is
about 30%, substantially at the same level as the current observational accuracy; it is
lower at low Z , higher at high Z for the Padova yields compared to more recent ones.
Such a conclusion ensures that our results do not depend too critically on the specific set
of yields employed. In addition, the fact that the TP-AGB phase in our yields is empir-
ically calibrated is a fundamental additional constraint and since all our adopted yields
are computed using the same Padova stellar evolutionary code, this ensures a greater
homogeneity in our final results.
4.3 A model for the chemical evolution of the Galaxy
We have previously outlined the main stellar populations which compose the Galaxy
and their contribution in driving its chemical evolution. However, to put together a
model for the chemical evolution of the Galaxy, one still needs to combine the following
ingredients.
• The Star Formation Rate (SFR), i.e. a measure of the typical timescale and mass
of gas that goes into stars, usually quantified in terms of M⊙pc−2Gyr−1. Besides
the assumption of a constant SFR, the simplest possible parametrization of the
SFR is to assume that it is just proportional to the surface density of gas.
• The Initial Mass Function (IMF) i.e. the relative number of stars with different
initial mass mM⊙ in a given interval dm.
• The stellar ejecta of different elements, i.e. a budget of the products of stellar
evolution.
Here we briefly summarize the main features of our chemical evolution model, which
we apply to the Solar Neighbourhood. Full details are available in Portinari et al. (1998).
An open model with continuous infall of primordial gas that builds the disk gradually
is adopted. Such a process is supported by dynamical studies (e.g. Larson 1974, 1976;
Carraro, Lia & Chiosi 1998; Sommer–Larsen, Götz & Portinari 2003) and provides
the favoured solution to the “G dwarf problem” (e.g. Lynden-Bell 1975; Tinsley 1980;
Chiosi 1980; Pagel 1989; Matteucci 1991) as shown in Figure 4.5.
The Galactic disk is divided into concentric cylindrical shells, which evolve indepen-
dently; radial flows of gas (Portinari & Chiosi 2000) are neglected in the present work.
38 CHAPTER 4. THE CHEMICAL EVOLUTION OF HELIUM
-2 -1.5 -1 -0.5 0 0.5
0
0.05
0.1
0.15
0.2
0.25
Figure 4.5: Observational data on the local metallicity distribution, in relative number, of long–
lived stars shown together with the prediction of our open model A (see Section 4.4 for further
details).
Each ring consists of a homogeneous mixture of gas and stars, so that for each ring
the only independent variable is the time t (“one zone” model, Talbot & Arnett 1971).
Galactic discs are comfortably described in terms of surface mass density σ(r, t), which
depends both on the galactocentric radius r of the ring and also on time t, since in each
ring the surface density is growing in time due to gradual infall of gas. Since we apply
the model only to the Solar Neighbourhood, in the following we drop the dependence on
r.
4.3. A MODEL FOR THE CHEMICAL EVOLUTION OF THE GALAXY 39
If we indicate with σg(t) the surface gas density, the gas fraction in the ring at any
given time t is the ratio
σg(t)
σ(t)
(4.1)
while the stellar surface density is
σs(t) = σ(t)− σg(t). (4.2)
In closed models the total surface density σ(t) is constant and the other quantities can
be normalized with respect to it. In open models it is suitable to normalize with respect
to the total surface density at the present age of the Galaxy tG i.e. at the final stage of
the model. It is therefore convenient to introduce the normalized surface gas density
G(t) =
σg(t)
σ(tG)
. (4.3)
In each ring the gas is assumed to be chemically homogeneous, and the normalized gas
density for each chemical species i is
Gi(t) = Xi(t)G(t) (4.4)
where Xi is the fractionary mass abundance of species i and
∑
iXi = 1, by definition.
The chemical evolution of the ISM is the evolution of the set of the Gi’s described by
d
dt
Gi(t) = −Xi(t)ψ(t)+
∫ Mu
Ml
ψ(t−τM )RMi(t−τM )φ(M)dM+
[
d
dt
Gi(t)
]
inf
(4.5)
where ψ(t) is the SFR, φ(M) is the IMF, RMi(t) is the mass fraction of a star of mass
M ejected into the ISM in the form of element i, Ml and Mu are the lower and upper
limit for stellar mass respectively and τM is the lifetime of a star of mass M . The first
term on the right hand side represents the depletion of species i from the ISM due to
star formation; the second term represents the amount of species i returned to the ISM
by stellar ejecta and the third term is the contribution of the infalling gas. The main
ingredients are described in more details in the following subsections.
4.3.1 Stellar ejecta
To solve equation (4.5) one needs to calculate
RMi(t) =
EMi(t)
M
(4.6)
where EMt(t) is the amount (in M⊙) of species i which the star expels into the sur-
rounding environment via stellar wind, supernova explosion or planetary nebula. The
stellar ejecta EMi(t), or rather EMi(t, Z) since metallicity dependence is also taken into
40 CHAPTER 4. THE CHEMICAL EVOLUTION OF HELIUM
account are also taken from Portinari et al. (1998) and Marigo (2001). The restitution
fractions depend therefore on metallicity as well.
To follow the temporal behaviour of different chemical species, the instantaneous
recycling approximation is dropped by taking into account the role of finite lifetimes
for stars of different masses. Stellar lifetimes are adopted from the same Padova tracks
which served as the base for the adopted stellar yields and therefore allow to consider
the effects of different metal content not only on the ejecta but also on the lifetimes. The
lifetimes are calculated as the sum of the H-burning and He-burning timescales.
Each star is assumed to expel its ejecta all at once at the end of its lifetime and that the
ejected material is immediately mixed in the ISM, which remains always homogeneous.
This instantaneous mixing approximation is suitable to reproduce the average trends of
abundances observed in the disc, while it cannot model the observed scatter of the data
around the average trend.
Notice that the ejecta of a given star are the sum of its yields plus the original material
which is returned to the ISM without being processed. The effect of stellar ejecta is
thus of fundamental importance in the chemical evolution model, but as long as we are
interested in a differential quantity such as ∆Y/∆Z the stellar yields only provide a
valuable insight, as we have already discussed in Section 4.2.
4.3.2 The infall term
In open models the surface mass density σ(t) increases by slowly accreting gas at a
rate σ˙inf (t) until it reaches the observed present value. An infall rate exponentially
decreasing in time with a timescale τ
σ˙inf (t) = A e
−
t
τ (4.7)
well reproduces the results of dynamical models (e.g. Larson 1976; Carraro et al. 1998;
Sommer–Larsen et al. 2003). A is obtained by integrating over time and by requiring
that at the age tG the observed present surface mass density σ(tG) is matched
A
(
1− e−
tG
τ
)
τ = σ(tG). (4.8)
The contribution of the infalling gas to the evolution of the gas fraction G(t) is[
d
dt
G(t)
]
inf
=
σ˙inf (t)
σ(tG)
(4.9)
and for each single chemical species i[
d
dt
Gi(t)
]
inf
=
σ˙inf (t)Xi,inf
σ(tG)
(4.10)
which gives the third term on the right hand side of Eq. (4.5).
4.3. A MODEL FOR THE CHEMICAL EVOLUTION OF THE GALAXY 41
4.3.3 The star formation rate
The formulation for the SFR introduced by Talbot & Arnett (1975) is adopted
d
dt
σg(r, t) = −ν
[
σ(r, t)σg(r, t)
σ2(r⊙, tG)
]κ−1
σg(r, t). (4.11)
Here, κ (usually ranging from 1 to 2) is the exponent of the Schmidt (1959) law for
star formation, ψ ∝ ρκ; σ(r⊙, tG) is the surface mass density at (r⊙, tG), adopted
as a normalization factor; ν is a free parameter for the star formation efficiency. The
dependence on the surface mass density is introduced because the SFR is related to the
local dynamical timescale, which is shorter where the mass density is larger.
In terms of the formalism introduced in the previous Sections, the adopted SFR
becomes
ψ(r, t) = ν
[
σ(r, t)σ(r, tG)
σ2(r⊙, tG)
]κ−1
Gκ(r, t) (4.12)
and limiting to the solar ring simplifies to
ψ(t) = ν
[
σ(t)
σ(tG)
]κ−1
Gκ(t) (4.13)
where the dependence on r has been dropped and ν is to be fixed so as to reproduce the
features of the Solar Neighbourhood.
4.3.4 The initial mass function
The Kroupa (1998) IMF derived for field stars in the Solar Neighbourhood is adopted1
φ(M)dM ∝M−αdM (4.14)
where
α =


0.5 M ≤ 0.5
1.2 0.5 < M ≤ 1.0
1.7 1 < M < 100
This IMF is steeper than the Salpeter one (φ(M) ∝ M−1.35) in the high mass range
(M > 1M⊙) but it flattens out progressively at low masses (M < 1M⊙). The steep
slope in the high mass range was actually taken after Scalo (1986). In a more recent
determination Kroupa (2001) finds instead a shallower slope of 1.3, close to the Salpeter
value, but the steeper Scalo slope is recovered if unresolved binary systems are taken
into account (Kroupa 2002). Also, steeper slopes are favoured in the average field with
respect to individual star clusters (Kroupa & Weidner 2003).
The adopted IMF is one of the ingredients which compose a GCE model, and we
have already shown in Section 4.2 that the choice between different types of IMF has
little impact for the purpose of the present investigation.
1We adopt an upper mass cutoff of 100M⊙, i.e. no very massive stars are supposed to be present.
42 CHAPTER 4. THE CHEMICAL EVOLUTION OF HELIUM
4.4 The evolution of helium
The model previously presented is used to study the chemical evolution of helium in the
Solar Neighbourhood. Standard observational counterparts for chemical models such as
the (1) current gas fraction, (2) the rate of type I and type II SNe, (3) the age–metallicity
relation, (4) the past and current estimated SFR, (5) the metallicity distribution of long–
lived stars (G dwarf problem) are used to calibrate the free parameters of the model.
0 5 10
0
0.01
0.02
0.03
0.04
t
Figure 4.6: Evolution of the metal mass fraction with time for the two models used to study the
evolution of helium in the Solar Neighbourhood. Details on the two models are discussed in the
text.
In our case the free parameters are the star formation efficiency (ν), the exponent of
the star formation law (κ), the infall timescale (τ ) and the amplitude factor for the SNe
Ia. The other parameters adopted in the model are directly constrained by observational
4.4. THE EVOLUTION OF HELIUM 43
Figure 4.7: The evolution of the helium mass fraction Y as function of the total metal mass
fraction Z . Also shown is the evolution of Y as function of the mass fraction of carbon (C),
oxygen (O), nitrogen (N) and iron (Fe).
determinations: the local surface mass density for the Solar Neighbourhood is chosen to
be σ(r⊙) = 50M⊙/pc2 (Holmberg & Flynn 2000, 2004) and tG = 13 Gyr, consistent
with the WMAP value for the age of the Universe (Spergel et al. 2007). The infalling
material is assumed to be protogalactic gas, with primordial chemical composition X =
0.76, Y = 0.24 and Z = 0 (see also Chapter 2). Notice that the value of YP does not
influence the model predictions on the rate of helium vs. metals production by the stellar
populations, ∆Y/∆Z .
As we discuss in the following, the helium–to–metal enrichment ratio is actually
quite insensitive to the exact calibration of the chemical evolution model, the reason
being that ∆Y/∆Z mostly depends on the adopted stellar yields. The calibration of a
model that satisfactorily matches all the constraints previously mentioned is not straight-
forward. We have selected two models for the chemical evolution of the Solar Neigh-
bourhood. Model A is built to match the observational constraints previously mentioned
whereas model B is built ad hoc to reach a high Z at the present time tG (Figure 4.6)
since our observational constraints in Paper II include highly metal–rich local stars
(Casagrande et al. 2007). In fact model A is calibrated to match the average proper-
ties of the Solar Neighbourhood, and it does not reach a metal mass fraction higher than
Z⊙ ∼ 0.017, as the majority of local stars are of metallicity lower than the Sun (Figure
44 CHAPTER 4. THE CHEMICAL EVOLUTION OF HELIUM
4.5). The difficulty is even more striking considering that such a Z⊙ should have already
been in place 4.6 Gyr ago. Interestingly, the low Z for the Solar Neighbourhood returned
by our GCE model A is in better agreement with the abundances measured in the local
ISM (e.g. André et al. 2003) as well as in nearby B stars (e.g. Gies & Lambert 1992;
Cunha & Lambert 1992; Sofia & Meyer 2001) and it appears also consistent with the
updated solar metallicity (Asplund et al. 2005).
The higher metal mass fraction in model B is obtained by adopting a very short
infall timescale (τ = 2 Gyr) which reduces the amount of pristine gas newly supplied
to the Solar Neighbourhood, whereas increasing the efficiency of the SFR has minor
impact. In order to reach a high metal mass fraction in the Solar Neighbourhood, a
higher efficiency of the the SNe Ia is also assumed in model B. Although model B is
not a realistic description of the chemical evolution of the Solar Neighbourhood, it is
useful to demonstrate that the slope of ∆Y/∆Z is rather insensitive to the details of
the chemical evolution model adopted. The theoretical predictions presented here are
therefore robust.
Figure 4.7 shows the predicted evolution of the helium mass fraction Y as function
of the total metal mass fraction Z and of the mass fraction of other single elements (C,
N, O, Fe). When the relation of Y vs. Z is considered, such a plot can be regarded as the
theoretical counterpart of the extensive observational work of Paper II and reproduced
also in Figure 3.2 where we measured Y and Z from nearby K dwarf stars. The helium–
to–metal enrichment ratio predicted by both chemical evolution models (A and B) is
well approximated by a linear relationship, although that is not necessarily the case
when the abundances of single elements are considered. In both models the slope over
the entire Z range is ∆Y/∆Y = 1.7. A discussion of the slope in the very metal–poor
regime has already been done in Chapter 2, and it is very close to the results found here.
The helium–to–metal enrichment ratio predicted by our chemical evolution models also
agrees with the slope found in Paper II using nearby K dwarfs with metallicity around
and above solar.
For subsolar metallicities, the ∆Y/∆Z found using K dwarfs is considerably dif-
ferent from the theoretical expectation discussed in this Section. At the same time, the
good agreement between the theoretical and the empirical ∆Y/∆Z measured above so-
lar metallicity and the theoretical and the empirical ∆Y/∆(O/H) in low metallicity HII
regions ensures that it is possible to use the GCE model presented in this Chapter to
obtain meaningful conclusions on the cosmic evolution of helium.
We have already clearly stated that we believe the low helium abundances found in
Paper II stem from inadequacies in the extant stellar models for low mass stars. Nonethe-
less, for the sake of completeness, we briefly discuss what this result would imply taken
at face value. In Figure 3.2 the mean slope ∆Y/∆Z = 9.2 below Z = 0.013 would
extrapolate to a primordial helium mass fraction YP = 0.13. From the discussion in
Chapter 2 is obvious that there is overwhelming evidence for a much higher YP ∼ 0.24
so that we do not question the BBN constraint here. The main problem is to find a
suitable astrophysical mechanism responsible for such a large depletion/destruction of
4.4. THE EVOLUTION OF HELIUM 45
helium (of order ∆Y ∼ 0.10) in metal–poor star forming regions of the Galaxy. Be-
sides, it would still be challenging for any GCE model to produce a helium enrichment
rate capable to fully counterbalance the depletion/destruction previously mentioned.
It is interesting to notice that similar, or even larger ∆Y are obtained when theoret-
ical stellar isochrones are used to fit the stellar populations in some globular clusters,
like ω Centauri (Norris 2004) or NGC2808 (D’Antona et al. 2005). In such cases,
the helium abundance for the most metal–poor population is chosen to be primordial
YP ∼ 0.24, thus implying Y ∼ 0.35 − 0.40 for the other populations observed in the
clusters. Nonetheless, from the study of the nearby K dwarfs we have found that in the
metal–poor regime, the observed main-sequence is considerably narrower than predicted
by models with standard ∆Y/∆Z ∼ 2. The only solution to fit the observations with
present models is to assume a much steeper slope for the helium–to–metal enrichment ra-
tio below, say, Z = 0.013. We strongly suspect that the result obtained for the K dwarfs
stems from inaccuracies in the current metal–poor low mass stellar models and it would
be interesting to check weather the extreme values of ∆Y/∆Z deduced by isochrone
fitting the main-sequence of some globular clusters are at least partly the same problem.
The inverted blue and red main-sequence in ω Centauri would still imply a considerable
helium enrichment, but maybe not so high as it is currently believed (∆Y/∆Z ∼ 70
Piotto et al. 2005) and very challenging to explain (e.g. Karakas et al. 2006; Choi & Yi
2008).
CHAPTER 5
Summary of the original publications
“Success is not final, failure is not fatal: it is the courage to continue that counts”
Winston Churchill
This thesis consists of three journal papers. The main goals, methods and results
of each are summarized below. After the summaries, I briefly present prospects for
immediate research arising from the work.
• Paper I: Casagrande L., Portinari L., Flynn C., “Accurate fundamental parameters
for lower main-sequence stars”, 2006, MNRAS, 373, 13-44
• Paper II: Casagrande L., Flynn C., Portinari L., Girardi L., Jimenez R., “The
helium abundance and ∆Y/∆Z in lower main-sequence stars”, 2007, MNRAS,
382, 1516-1540
• Paper III: Casagrande L., Flynn C., Bessell M., Koen C., “M dwarfs: effective
temperatures, radii and metallicities”, submitted to MNRAS, 25 pages
The author’s contribution:
The author performed some of the optical observations presented in Paper I using the
remotely operated KVA telescope in LaPalma. Significant help in using the reduction
software for the photometric observations came from C. Flynn. The development of
the InfraRed Flux Method (Paper I) and of the Multiple Optical Infrared TEchnique
(Paper III) was done by the author as well as most of the analysis and the writing of
all three papers. The near–infrared observations are all from 2MASS. The theoretical
stellar models used in Paper II were provided by L. Girardi and R. Jimenez. Most of the
optical observations presented in Paper III were conducted by C. Koen and for the same
paper M. Bessell provided significant help in discussing the results. L. Portinari and
C. Flynn provided fundamental support for the chemical evolution model of the Solar
Neighbourhood discussed in Chapter 2 – 4.
47
5.0.1 Paper I
As discussed throughout the thesis, low mass stars can be seen as fossils of the Galaxy’s
evolution. Their accurate characterization, both observationally and theoretically, is of
fundamental importance for unveiling the formation and evolution of the Milky Way.
To successfully achieve such a task accurate fundamental physical parameters and thor-
oughly tested theoretical stellar models are needed.
This paper presents accurate BV (RI)CJHKS photometry for a large sample of
nearby G and K dwarfs. The comparison between the observed and synthetic colours has
been used to test the most recent theoretical model atmospheres (ATLAS9 and MARCS)
for these stars. Such a comparison shows a generally good agreement. Special attention
has been paid to the accuracy and homogeneity in generating synthetic colours, with
a special care in discussing the effect of the adopted zero–points and absolute calibra-
tion. The latter point is also crucial for setting the scale of stellar effective temperatures
determined via the IRFM.
The IRFM (Blackwell & Shallis 1977; Blackwell, Shallis & Selby 1979; Blackwell,
Petford & Shallis 1980) is known to be one of the most reliable techniques to derive
fundamental stellar parameters i.e. effective temperatures, bolometric luminosities and
angular diameters. It uses infrared photometry as a sensitive proxy of the stellar effective
temperature and a great deal of observational information to make the derivation of the
stellar parameters almost model independent. In this paper, an implementation of the
IRFM has been done ab initio, so as to keep full control on all possible sources of
uncertainties, in particular because we put these stars to work in Paper II, using them
to track helium production in the Galaxy. Major improvements with respects to similar
work done in the past are the excellent quality and homogeneity of the observational
data (also thanks to a large near–infrared survey as 2MASS), the use of multi–band
photometry covering the wavelength range where these stars emit most of their flux
(∼ 0.4 − 2.2µm) in order to recover their bolometric luminosity, the adoption of the
latest generation of model atmospheres and the most recent zero–points and absolute
calibration for Vega as thoroughly discussed in Appendix A of the paper.
The tightness of the derived colour–temperature, colour–angular diameter and colour–
luminosity relations reflects the high quality of the input data used and the internal ac-
curacy of the work. Because of the adopted 2MASS absolute calibration, the proposed
temperature scale is found to be about 100K hotter than recent analogous determina-
tions in literature, but in very good agreement with various spectroscopic temperature
scales. We thus conclude that Teff determinations for lower main-sequence stars still re-
tain systematics of the order of a few percent. Currently, interferometric measurements
of G and K dwarf angular diameters break only partly this impasse on the tempera-
ture scale. In fact, angular diameters measured via interferometry must be corrected
for limb-darkening, thus introducing model dependence. With respect to classical 1D
model atmospheres, 3D models predict different limb-darkening coefficients, which im-
ply smaller angular diameters and thus hotter effective temperatures (Allende Prieto et
48 CHAPTER 5. SUMMARY OF THE ORIGINAL PUBLICATIONS
al. 2002). The departure from 1D to 3D limb-darkening is more relevant for hotter stars
and decreases considerably going to cooler M dwarfs (Bigot et al. 2006), as discussed
also in Paper III.
5.0.2 Paper II
In this article the fundamental stellar parameters determined in Paper I are used to test
some of the most up–to–date stellar models, newly computed for various metallicities
and helium abundances. The aim is to measure the helium–to–metal enrichment ratio
∆Y/∆Z , a diagnostic of the chemical history of the Solar Neighbourhood as discussed
in Chapter 4.
Taking advantage of the detailed analysis performed in Paper I, we have performed
a direct comparison between stellar models and observations in the MBol − Teff plane,
thus working directly with physical quantities. For the parameter space covered by the
isochrones, the number of stars used, the accuracy and homogeneity of the data —crucial
when it comes to analysing small differential effects in the HR diagram— the analysis
performed in Paper II is the most stringent test on the helium content of lower main-
sequence stars undertaken to date.
Around and above the solar metallicity the helium content scales with the metal
mass fraction more or less consistently with GCE model prediction (Chapter 4), with
∆Y/∆Z ∼ 2± 1 which is also the value usually assumed in the literature by a number
of studies. Such a trend breaks down going to the lowest metallicities, where the broad-
ening of the observed lower main-sequence is considerably narrower than predicted by
theoretical stellar models under the standard assumption ∆Y/∆Z ∼ 2. To fit the posi-
tion of metal–poor stars in the HR diagram, anomalously low helium abundances, well
below the standard BBN predictions, must be used. Possible alternatives to such low
helium abundances are discussed throughout the paper. Including diffusion in stellar
models and accounting for non-LTE effects in deriving metallicities do not go very far to
solve the problem. A viable option is to assume a mixing–length that decreases steadily
with decreasing metallicity.
5.0.3 Paper III
Paper III extends the results of Paper I to M dwarfs. Although M dwarfs make up about
half of the stellar mass of the Galaxy, until now discussions on these objects could just
be largely qualitative because of the intrinsic difficulties in determining their stellar pa-
rameters. Major recent improvements in determining abundances for M dwarfs, in their
theoretical model atmospheres and the availability of very accurate and homogeneous
photometry for them, make it very timely to extend to M dwarfs the work started in
Paper I.
A new technique, the Multiple Optical Infrared TEchnique (MOITE) is introduced.
The MOITE exploits the bolometric to monochromatic flux ratios in different bands
49
(rather than in the infrared only, as the IRFM does) to recover the fundamental stellar
parameters for M dwarfs. A strong sensitivity to the metallicity has been found for the
bolometric to monochromatic flux ratios of cool dwarfs. By exploiting such a features,
the MOITE provides a tool to determine photometrically the metal content in dwarfs of
late spectral type.
The results provided by the MOITE offer thus the interesting opportunity of effec-
tively using M dwarfs as tracers of the Galactic Chemical Evolution. The results obtained
in this paper also have important consequences for the study and modelling of low mass
stars. A comparison with the widely used low mass stellar models of Baraffe et al.
(1998) is made. Thanks to the accuracy and homogeneity of the data, a sharp transition
between K and M dwarfs has been found. Such a feature is not yet reproduced by stellar
models and appears to be related to a sudden increase in the radii of M dwarfs. This
discrepancy between the observed and the theoretical radii for M dwarfs is discussed
in the light of similar results obtained in recent years from much smaller samples of M
dwarfs.
5.0.4 Future prospects
The technique developed in Paper I and III well suits to be applied to other classes of
stars, so as to cover a wider temperature and metallicity range. A better definition of the
effective temperature scale, eventually extending to the extremely metal–poor regime
would be of importance for a variety of studies, from Galactic archaeology to stellar
modelling.
Uncertainties in the Vega zero points and absolute calibration still hinder much of
a progress in reducing the systematic errors that plague the temperature scale. More
angular diameters measurements, corrected using limb-darkening coefficients computed
from 3D model atmospheres would greatly help to firmly establish the temperature scale
in different regions of the HR diagram, finally solving “il dilemma delle calibrazioni as-
trofisiche” (Gustafsson & Gråe-Jørgensen 1985). VLTI angular diameter measurements
for three solar type stars will be soon available as Co-I in the VLTI accepted proposal
081.D-0412.
The technique presented in Paper III for determining the metallicity of M dwarfs
needs to be further tested. New photometric and spectroscopic observations of more
Hipparcos common proper–motion companions with M dwarf secondaries from the list
of Gould & Chanamé (2004) are a simple way forward as metallicities for the M dwarfs
can be deduced from the binary hosts. We could obtain ten times more stars than anal-
ysed in Paper III.
The IRFM and the MOITE are unique tools for extracting stellar parameters and
I will continue updating them to be valuable for the latest data, most notably upcom-
ing photometric surveys such as SkyMapper (Keller et al. 2007). With Mike Bessell,
we are obtaining optical spectrophotometry for few hundreds of stars whose fundamen-
tal parameters have already been determined with the IRFM/MOITE. The data will be
50 CHAPTER 5. SUMMARY OF THE ORIGINAL PUBLICATIONS
similar to those expected from the low-resolution spectrograph on GAIA. The precise
IRFM/MOITE–based data on nearby stars will be used as a template onto which cali-
brate new tools for extracting reliable stellar parameters from such low-resolution spec-
tra, so as to readily identify the best targets for further follow-up observations.
GAIA, as well as other ground based projects such as RAVE, SEGUE and 0Z will
produce a vast amount of data for a billion halo, disc, and bulge stars. Reconstructing
the formation and evolution of the Milky Way in a star–by–star fashion will be a crucial
step towards achieving a successful theory of Galaxy formation. Such data will also
considerably improve our understanding of Galactic Chemical Evolution.
The further use of the chemical evolution model presented in Chapter 4 will be very
valuable for studying the evolution of the helium and other elements in the Solar Neigh-
bourhood. It will be challenging to try to use such approach to independently confirm
the recent update in the solar chemical composition (Asplund et al. 2005).
As a result of such new abundances, the solar model is currently under profound
revision. The results provided by Paper II also suggest that theoretical stellar models for
metal–poor low mass stars need to be improved, possibly accounting for a metallicity
dependent mixing–length. Currently, 3D hydrodynamical models by treating convection
in a fully self-consistent way offer the best option to test any dependence of the mixing–
length with metallicity. Expanding the range within which we will confidently use stellar
models will also push forward the limit of our knowledge on the formation and evolution
of the Galaxy.
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Original papers