Fundamental physical constants required in this
course
a radiation density constant 7.55  10-16 J m-3 K-4
c velocity of light 3.00  108 m s-1
G gravitational constant 6.67  10-11 N m2 kg-2
h Planck’s constant 6.62  10-34 J s
k Boltzmann’s constant 1.38  10-23 J K-1
me mass of electron 9.11  10-31 kg
mH mass of hydrogen atom 1.67  10-27 kg
NA Avogardo’s number 6.02  1023 mol-1
 Stefan Boltzmann constant 5.67  10-8 W m-2 K-4 ( = ac/4)
R gas constant (k/mH) 8.26  103 J K-1 kg-1
e charge of electron 1.60  10-19 C
L luminosity of Sun 3.86  1026 W
M mass of Sun 1.99  1030 kg
Teff, effective temperature of Sun 5780 K
R radius of Sun 6.96  108 m
Parsec unit of distance 3.09  1016 m
Classical Stellar Evolution
Star clusters
4
We observe star clusters
• Stars all at same distance
• Dynamically bound
• Same age
• Same chemical composition
Can contain 103 –106 stars
Goal of this course is to
understand the stellar content of
such clusters
NGC 3766, ESO
The Sun – best studied example
Stellar interiors not directly observable
Neutrinos emitted at core and detectable
Helioseismology - vibrations of solar surface can be used to probe density structure
Must construct models of stellar interiors – predictions of these models are tested
by comparison with observed properties of individual stars
Observable properties of stars
Basic parameters to compare theory and observations:
• Mass (M)
• Luminosity (L)
– The total energy radiated per second i.e. power (in W)
𝐿 = 𝐿 𝜆 𝑑𝜆
∞
0
• Radius (R)
• Effective temperature (Teff)
– The temperature of a black body of the same radius as
the star that would radiate the same amount of energy
L = 4 R2  Teff
 3 independent quantities
Basic definitions
Measured energy flux depends on distance to star
(inverse square law)
𝐹 =
𝐿
4𝜋𝑑2
Hence if d is known then L determined
We can determine distance if we measure parallax
Classical astrometric approach
Now: Gaia
Stellar radii
Angular diameter of sun at distance of 10 pc:
 = 2R/10 pc = 5 10-9 radians = 10-3 arcsec
Compare with Hubble resolution of ~0.05 arcsec
 very difficult to measure R directly
Radii of stars measured with techniques such as
interferometry and eclipsing binaries
JMMC Stellar Diameters Catalogue - JSDC. Version 2:
about 470 000 stars, median error of the diameters is
around 1.5%
Stellar radii
The Hertzsprung - Russell diagram
M, R, L and Teff do not vary
independently
Two major relationships
– L with Teff
– L with M
The first is known as the
Hertzsprung-Russell (HR)
diagram or the
colour-magnitude diagram
Colour and Teff
• Measuring accurate Teff for stars is an intensive task – spectra
needed and model atmospheres
• Magnitudes of stars are measured at different wavelengths
• Colours => Calibration => Teff
• The Asiago Database on Photometric Systems (ADPS) lists
about 200 different systems
Colour and Teff
Various calibrations can be
used to provide the colour
relation:
(B – V) = f(Teff)
Remember that observed
(B - V) must be corrected for
interstellar extinction to
(B - V)0
Absorption = Extinction = Reddening
• AV = k1 E(B-V) = k2 E(V-R) = …
• General extinction because of the ISM characteristics
between the observer and the object
• Differential extinction within one star cluster because
of local environment
• Both types are, in general wavelength dependent
Reasons for the interstellar extinction
• Light scatter at the interstellar dust
• Light absorption => Heating of the ISM
• Depending on the composition and density of the
ISM
• Main contribution due to dust
• Simulations and calculations in Cardelli et al. (1989,
ApJ, 345, 245)
Cardelli et al., 1989, ApJ, 345, 245
Important parameter:
RV = AV/E(B - V)
Normalization factor
Standard value used
is 3.1
Be careful, different
values used!
Depending on the
line of sight
Absolute magnitude and bolometric
magnitude
• Absolute Magnitude M defined as apparent magnitude of a
star if it were placed at a distance of 10 pc
m – M = 5 log(d/10) - 5
where d is in pc
• Magnitudes are measured in some wavelength. To compare
with theory it is more useful to determine bolometric
magnitude Mbol – defined as absolute magnitude that would
be measured by a bolometer sensitive to all wavelengths. We
define the bolometric correction to be
BC = Mbol – MV
Bolometric luminosity is then
Mbol – Mbol, = -2.5 log L/L
Bolometric Correction
BC from Flower, 1996, ApJ, 469, 355
Bolometric Correction
The Hertzsprung - Russell diagram - Gaia
The Hertzsprung - Russell diagram - Gaia
H and He-rich cores
of White Dwarfs
Masses measured in binary systems
Heuristic mass-luminosity relation
L  Ma
Where a = 2 – 5; slope less steep for low and
high mass stars
This implies that the main-sequence (MS) on the
HRD is a function of mass, i.e. from bottom to
top of MS, stars increase in mass
Mass – Luminosity Relation
We must understand the M - L relation and
L - Teff relation theoretically
Models must reproduce observations
Mass – Luminosity Relation
Metallicity - Basics
• Metallicity as [X:Y:Z]
• X = Hydrogen
• Y = Helium
• Z = „the rest“
𝑋 ≡
𝑚 𝐻
𝑀
Y ≡
𝑚 𝐻𝑒
𝑀
Z =
𝑚𝑖
𝑀𝑖>𝐻𝑒 = 1 – X – Y
Metallicity - designations
• In the literature you will find
– [Z]
– [Fe/H]
– [M/H]
– [Element 1 / Element 2]
• Relations for the transformation are necessary
Metallicity - designations
• [dex], e.g. [Fe/H] = -0,5 dex
dex factor dex factor
-2 0,01 0,1 1,26
-1,5 0,03 0,2 1,58
-1 0,10 0,3 2,00
-0,9 0,13 0,4 2,51
-0,8 0,16 0,5 3,16
-0,7 0,20 0,6 3,98
-0,6 0,25 0,7 5,01
-0,5 0,32 0,8 6,31
-0,4 0,40 0,9 7,94
-0,3 0,50 1 10,00
-0,2 0,63 1,5 31,62
-0,1 0,79 2 100,00
Abundance - Sun
• Problems with
–Hydrogen
–Helium
–Elements with only a few lines
–Elements with only weak lines
• LTE versus NLTE (Local Thermodynamic
Equilibrium )
Abundance - Sun
Asplund et al., 2009, Annual Review of
Astronomy & Astrophysics, 47, 481
Abundance - Sun
Asplund et al. (2009)
Abundance - Sun
Asplund et al. (2009)
Isochrones - Metallicity
Different He abundances – [Z] constant
NGC 3766 – Open Cluster M55 – Globular Cluster
Star clusters
33
HRD- Open Clusters
Turn-off
Turn-off
HRD – Globular Clusters
Star Clusters as Laboratories
• In clusters, age and metallicity must be same
for all stars
• Hence differences must be due to masses
• Stellar evolution assumes that the differences
in cluster stars are due only (or mainly) to
initial masses (IMF)
• Cluster HR (or colour-magnitude) diagrams are
quite similar – age determines overall
appearance
Globular vs. Open clusters
Globular Open
• MS turn-off points in similar
position. Giant branch joining
MS
• Horizontal branch from giant
branch to above the MS turn-off
point
• Horizontal branch often
populated only with variable RR
Lyrae stars
• MS turn off point varies
massively, faintest is consistent
with globulars
• Maximum luminosity of stars
can get to Mv  -10 mag
• Very massive stars found in
these clusters
The differences are interpreted due to age – open clusters lie in
the disk of the Milky Way and have large range of ages. The
Globulars the oldest objects known tracing the earliest stages of
the formation of Milky Way (~12 109 yrs)