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SPHEROIDAL POPULATED STAR SYSTEMS
Pietro Giannone
Dipartimento di Fisica Universita’ “Sapienza” Roma
SUMMARY
Globular clusters and star evolution
Elliptical galaxies
Population synthesis
Star population
age / metallicity
Open cluster
NGC 3293
Globular
Cluster
M 13
R
Elliptical galaxy
NGC 4374
MOTIVATIONS for their STUDY
Globular clusters are the oldest objects in the Universe
- probes for cosmological issues
(age of the Universe, Big Bang nucleosynthesis)
- protogalactic collapse
Elliptical galaxies are the most populated star systems
Both contribute information on:
- Initial mass function (IMF)
- Star formation rate (SFR)
-
-
Star evolution
Ages
Chemical composition
Stellar populations
Stellar dynamics
Galactic evolution
OBSERVATIONAL DATA
AGB
HB
CM
diagram
RGB
M3
SGB
TO
MS
G6a
INGREDIENTS FOR A STELLAR POPULATION SYNTHESIS
- Birth rate function b(m,t,r,Z)
- Star evolution
- Mass loss from stars
- Model atmospheres (conversions)
- SNe (progenitors, rate, SNRs)
- Nucleosynthesis
- Dynamics
Most common assumptions
-
b(m, t )  (m) (t )
-
( m)  m  x
-
 (t )  g (t ) k
i.e. IMF
with
with
 SFR
1  x  2.7
1 k  2
[Salpeter x=2.35]
SIMPLE STELLAR POPULATION
System of stars with the same age and the same initial chemical composition
Age 15 Gyr
Pop. II
  0 .4
x  2.35 for M  0.4
x 1
6-5
“
M  0 .4
WDs
Simple Star
Population
(SSP)
i.e .
Coeval Stars with
the same initial
chemical
composition
6-7
yr
SSP Model for M3
6-8
°
(A)
INTEGRATED COLOURS AND SPECTRA OF SPHEROIDAL SYSTEMS
Observational data
- Colour-magnitude relation
dMV / d (B V )  0
- Mean metallicity-magnitude relation
d  Z s  / dMV  0
Mass
“
relation
-
104  Z s  0.01
-
[O/Fe]
-
Increasing spread of metallicities with increasing system mass
“
complexity of star populations
and
-
Mg  0
-
Z g  Z sun
in GCs and DSs ,
[ /Fe]
in GCs and DSs ,
[Mg/Fe]
0
 Zs  Zsun  0.02
in
Es
sovrasolar
in Es ,
104 103 M G
in the intergalactic medium in clusters of galaxies
in gEs
EVOLUTIONARY DYNAMICAL MODELS (L. Angeletti, R. Capuzzo, P.G.)
Globular cluster
5105 M sun multi-mass stellar components with star-mass loss
Main results
- increasing core concentration and envelope diluition
- velocity dispersion is isotropic in the core and anisotropic in the envelope
- differential central segregation of star masses
- differential “evaporation” of stars (up to 45 % of the initial mass and
40% of the initial number)
GALACTIC WINDS
- Continuous star formation and star evolution
progressive metal enrichment
overproduction of metals (too redward colours)
galactic wind
- Intracluster gas contains metals
(L. Angeletti & P.G.)
Evolution of spheroidal star systems from globular clusters to elliptical galaxies
(
105
to
1011
solar masses). Galactic wind when residual thermal energy of
Eth (tw )  g (tw )
SNRs reaches the gravitational binding energy
Results
 w  5  555 106 yrs
M g ( w ) / M o  0.34  0.08
as mass is increased
“ “
“
“
M g ( aw ) / M o  0.37  0.26 “ “
“
“
x
Star system
x  r 2  R2
Projection
line of sight
Apparent disk
r = spatial radial distance
 ( R)  2

X
0
 ( x)dx  2
R = projected radial distance

R*
R
r (r )
r 2  R2
dr
In order to determine light and colours at P’(R) we need to know the number of
stars along the line of sight within the system and their specific contributions
ADDITIONAL OBSERVATIONAL DATA FOR ELLIPTICAL GALAXIES
Radial projected profiles of various Johnson/Cousins colour and Lick spectral
indices across galaxy images, through slit or circular apertures
Fig2a
large variations
Projected radial gradients of indices are suggested to stem from spatial
abundance gradients that developed when Es formed through a monolithic
dissipative collapse
a) Dissipative models of galaxy formation can produce metallicity gradients
b) Star formation can proceed near the center for a longer time than farther out
THE R1/4 LAW OF THE PROJECTED SURFACE BRIGHTNESS
Generalization of R1/4 to R1/n with 2  n  8 for the spherical
mass-model derived by deprojection from the surface-brightness profile
R = projected radial distance of the slit position
Re = effective radius corresponding to half of the total light
R 1/ n
)
Re
Surface brightness profile
 ( R )   o  an (
Surface intensity profile
I ( R)  I o exp bn ( R / Re )1/ n

 (r )
In terms of the luminosity density
r (r )
I ( R)  2  R
R*
r 2  R2
dr
By inversion
 (r)  
1


R*
r
dI
dR
dR
R2  r 2
mag/arcsec2

From
 (r ) M

 constant
 (r )
L
 (r ) 
M
 (r )  Mf (n, Re , r )
L
M (r ) 

 (r ) 
r
4r 2  (r )dr
0
GM (r )
G
r
and derivatives

R*
4r (r )dr
gravitational potential
r
d / dr, d 2 / dr2 , d / d , d 2  / d 2
For unit mass at r
binding energy
angular momentum
   (r )  v 2 / 2  0
0  J  rv sin   J r ( )  rv
maximum value
 vr2  vt2 
Models with isotropic velocity dispersion:
Energy distribution function
( ) 
1
2 8


0
d 2
d 2
d
 
For the mass density of stars at r
d ( , J , r )  4
( ) J
r J r2 ( )  J 2
ddJ
Anisotropic models:  vr2  vt2 
Osipkov-Merritt models
J2
q   2 0
2ra
1  ra   = anisotropic radius
(for
ra    q  
i.e. isotropic models)
Distribution function of q’s
( q ) 
where
PROBLEM
1
2 8

q
0
d 2 q
d
d 2
q 
r2
 q (r )  (1  2 )  (r )
ra
to derive the metallicity distribution function, through the
spatial radial mass-density, from the energy distribution
and the angular momenta
(L. Angeletti & P.G.)
R1/ n law,
”Simple model” (SM) , “Concentration model” (CM) , and additions
SM : a one-zone and close-box model with instantaneous gas recycling
Gas is well mixed and its uniform metallicity (by mass) is (with
Z g   p ln g
(for Z  1 ) ,
where
Mean star metallicity
Z g 1  g p
G (t )
g  g (t ) 
Mo =
g ln g
 Z s  p(1 
)
1 g
Zo  0 )
(for Z  1 )
Gas mass
Galaxy mass
p = metal yield
Ms(t) = Mo - G(t) = total mass of the stellar component
(long-living stars and compact stellar remnants)
p = metal yield = fractional mass of the new metals formed in stars and ejected
into the ISM with respect to the total mass “locked up” in stars
p


m*
0. 1
100
mZ (m) (m)dm
0.1
m (m)dm 

100
m f (m) (m)dm
m*
mZ(m) = mass of the new metals ejected by a star with mass m
mf(m) = mass of the “final remnant” of a star with mass m
p will be expressed in units of Zsun= 0.0169
CM : takes into account the gas contraction in the galaxy
In the model
M = Lagrangian mass coordinate (in units of Mo)
- At time ta gas contracts within a decreasing mass coordinate Ma= M(ta) and
forms stars with Ap’s within Ma and Z=Za .
The mass of all stars with Ap’s within Ma (and all Z’s ) is (generalized ansatz)
s( M a )  g
where
ga  g (ta )
From SM + CM :
and
c
a
0  c 1
concentration index
Z a   p ln g a  
Two-parameter ( p and c ) family models
p
ln s(M a )
c
We define
 (M , Z a )
 (M , Z a )
 (M , Z a ) 
 (M , Z M )
= cumulative mass of the stars with Ap’s within
Z  Za
M  M a and
(born until ta )
 (M , Za )   (M , ZM )  s(M )  gM  s(M )  M 1/ c
of the stars with Ap’s within M and
p
Z  Z M   ln s(M )
c
Metallicity distribution function
 ( M , Z a )
( M , Z a ) 
Z a
for the stars with Ap’s within M
= cumulative mass
At r, from
  


rap  J 2  rap   


( , J )
1 / 2
  
M ap  s M rap
(M ap , Z )
The radial profile of index I is
( R)  2

r
R*
r 2  R2
R


Za p ( , J )
dr

 (r )
0
( )d

J r ( )
0
4J
r J r2 ( )  J 2
dJ 
( M ap , Z )I SSP ( Z )dZ
0
Integrated value of index I within a circular concentric aperture with radius
R'
(eventually the galaxy radius)
( R' ) 

R'
0
2R( R)dR
c


Tafig2b


RESULTS
Sample of 11 Es
Ranges of parameters for the best fits:
4n8
an increase of n smoothes variations of the radial gradients
between core and envelope
0.50  c  0.95
gradient slopes increase with increasing c
1.1  p / Z sun  2.2
increasing p moves index (except H ) profiles upwards
1  ra / Reo  
anisotropy produces shallower profiles in the envelopes
We also considered (for Mg2 )
i) Changes of age from 13 to 17 Gyr
ii) Star formation in an initial main burst lasting 2 Gyr and in a delayed minor
episode (1 Gyr long and starting at age 8 Gyr)
iii) A spread of durations for the star formation lasting from 2 to 11 Gyr after
the initial burst
iv) A terminal wind at time tw involving the gas mass 0.05 Mo when
Mw= 0.18Mo and r(Mw) = 0.4 Re(B)
Results of the additional implementations:
i) An increase of age operates like changes of p
ii) Delayed episodes of star formation flatten the index behaviours in the cores
and steepen them in the envelopes
iii) Prolonged phases of star formation emphasize the tendencies mentioned in ii)
iv) A terminal wind flattens the index profiles in the envelopes improving the fit
PROBLEMS
1. Non-solar partition for metals in SSPs
2. Lack of reliable SSPs for Z > 0.05
3. Opacity and surface convection for star models
4. Model atmospheres for log Teff - colours and BC for cool stars
5. Contributions of BHB and PAGB stars to light and colours
6. Contribution of evolved binaries
7. IMF
8. SFR
9. Degeneracy of relation
 Z
10. Non-uniform gas density
11. Instantaneous gas recycling to be replaced by stellar lifetimes
12. Dark matter
CONCLUSIONS
1. Stars in each globular cluster are coeval and were formed with the same initial
chemical composition owing to a prompt wind from the stellar system
SSP
2. Intermediate-mass Es experienced an early wind allowing for a moderate
inhomogeneity of metallicities among stars
3. Star formation was prolonged in gEs leading to a mixture of star metallicities
more stellar populations
4. Es are characterized by different space mass densities
5. Gas distributions (and therefore star formations) differed in Es
6. Mean star metallicities in Es range from solar to sensibly sovrasolar abundances
7. Different metal enrichments in Es evidentiate differences in their evolutions
Proposed scenarios for the formation of Es
1. All luminous Es are coeval and old systems, that formed through a monolithic
dissipative collapse, occurred early in the evolution of the Universe
2. Es formed through a lengthy hierarchical clustering of small objects into larger
ones with star formation extended over a long time
Inside Ma it is assumed that
i) Newly formed stars are distributed radially like the stars born before ta
ii) Stars that form at ta have Ap’s within Ma
Therefore the mass of the stars with Ap’s within Ma ,
Z  Za
and born until ta is
 (M a , Za )  s(M a )  ga  s(M a )  M a1/ c
SM and CM provide explicitly a two-parameter ( p and c ) family of
metallicity distribution functions
Information on ages
Age spread of a few Gyr among the central regions of most Fornax and Virgo Es
Age spread of some Gyr among the innermost regions of the field or Es in small
groups
STAR EVOLUTION
Single stars
Conservation of mass
Hydrostatic equilibrium
Energy balance
Radiative and/or conductive energy transport
Convective energy transport
Criterion for the radiative stability
Equation of state
Opacity
Nuclear energy generation
Quiescent nucleosynthesis
Input data: star mass m and initial chemical composition ( Y , Z )
Mass loss and stellar winds
(solar units)
Eddington luminosity
Reimers formula
with
L  LEd  5104 M
dM
LR
13
 4 10 
dt
M
1
  3
3
Msun yr-1
Cloud fragmentation
Protostar
Gravitational Contraction
Quiescent Nuclear Termofusion
H, He, C, O ……
Fe
Mass loss
Cooling
White Dwarf
Semi-empirical estimate
for M i  8
Explosive Nuclear Termofusion
Mass
 8=
?
>8
w8
M f  0.077M i  0.48
Supernova II
Zn.. Ba.. Pb.. U
Nebular Remnant +
Neutron star (pulsar)
Quiescent nucleosynthesis
3-2
15 Msun
Cosmological Nucleosynthesis
Quiescent Nucleosynthesis
Cosmic
Abundances
J1a
eff
F8b
Evolutionary tracks [ i.e. loci of constant m with L(t) and Teff(t) ]
isochrones [ i.e. loci of constant t with L(m) and Teff(m) ]
AGB
RGB
HB
M3
SGB
TO
MS
MS
SGB +RGB
HB
AGB
G6a
H
H
He
He
He in the core
He in the shell
C+O in the core, H
C+O in a shell,
H
He in the shell
He in a shell
Luminosity functions i.e. frequencies of stars in the various evolutionary stages
N HB  HB

 f (Y , )
R
N RG  RG


Y  (0.370  )logR  (0.186 )
33
55
Yp  0.23  0.24
Fig23c
Y  0.230.26
Y
Y  Yp 
Z  0.228 2.7Z
Z
Globular cluster
M13
Isochrones of
7 , 9 , 12 , and 15
Gyr
GCs are coeval
with  = 13 Gyr
TO
Luminosity functions of pre-white-dwarf stars and white-dwarf stars
at ages: 9.5 Gyr and 12.3 Gyr. Comparison with data for M3
6-6
R
Radial ratio
of projected
numerical
densities of
stars in various
evolutionary
stages
i.e. with
different
masses
segregation
6-2a
R
R
M 15
R
Radiale profiles in the V and B photometric bands from surface
brightness and star counts (solid curves are seeing profiles)
Fig25a/26a
Fig27a
y
Jacobi’s
integral
V 2  2U  C
Hill’s
curves
V 0
VM20
M1
O
M2
x
d
sd
c
W UMa
Roche classification
L10a
M1  M 2

A
P
0.67
13.0
1.5
“
“
6.665+6.665
1.00
12.0
1.3
3.83+9.50
2.48
17.8
2.4
2.90+10.43
3.60
25.4
4.2
8.00+5.33
“
“
(solar masses and radii)
“
(periods in days)
L9a
L2
L4
L5
L6
L8
Case C
 20 30 Termofus.
WR stars
He in
Conservative
evolution
core/shell
large
M  M 1  M 2  const.
h  M 1M 2
Case B2:
 20 30 Termofus.
GA
 const.
M
MS+MS
He in core
Blue
Dwarfs
H in shell
Rapid
transfer
of matter
Mass
Separation
inversion
Case B1
small
 3 Termofus.
H in shell
Slow
transfer
of matter
3
.4-1a
White
Dwarfs
Case A
Termofus.
H in core
Algol
Systems
Stellar
wind
MS+WDHe
Nuclear
termofus.
Transfer
of mass
Losses of
Mass and
Angular
momentum
Novae
Low-mass
X ray
sources
Transfer
of mass
MS + NS
Accretion
Stellar
wind
4-7a
Massive
X ray
sources
MS + WDC
Nuclear
termofus.
Losses of
Mass and
Angular
momentum
MWD  M Ch
WD + WD
SN I
SNR
Gravitational
waves
Coalescence
4-8a