Download Low mass stars

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Stars and Stellar Evolution
For instance, the UBV system has about 100 standard stars measured to
about ± 0.01 magnitude. Then if we can calibrate the flux of just one of
these stars, we have calibrated the system. The calibration is usually given
for zero magnitude at each filter; all fluxes are then derived from this base
level. The star usually chosen as the calibration star is Vega.
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Colour index in the BV system. Blackbody curves for 20,000 K and 3000 K,
along with their intensities at B and V wavelengths. Note that B - V is
negative for the hotter star, positive for the cooler one.
The Hertzsprung-Russell Diagram
In 1911, Ejnar Hertzsprung plotted the first such two-dimensional diagram
(absolute magnitude versus spectral type) for observed stars, followed
(independently) in 1913 by Henry Norris Russell.
The simple HR diagram represents one of the great observational syntheses
in astrophysics. Note that any two of luminosity, magnitude, temperature,
and radius could be used, but visual magnitude and temperature are
universally obtained quantities.
An original idea was that a star was born hot (early type) and cooled (late
type).
It’s a particular colour-magnitude diagram.
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Important stars: no obvious pattern…Sirius B, Betelgeuse in opposite
corners:
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Nearby stars: main-sequence appears. Most stars are less luminous and
cooler than the Sun (alpha Centauri, nearest to us and a triple system, is
similar). Note the hot small stars: the white dwarfs.
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Most stars have properties within the shaded region known as the main
sequence. The points plotted here are for stars lying within about 5 pc of
the Sun. The diagonal lines correspond to constant stellar radius, so that
stellar size can be represented on the same diagram as luminosity and
temperature.
The first H-R diagrams considered stars in the solar neighbourhood and
plotted absolute visual magnitude, M, versus spectral type, which is
equivalent to luminosity versus spectral type or luminosity versus
temperature. Note (a) the well-defined main sequence (class V) with everincreasing numbers of stars toward later spectral types and an absence of
spectral classes earlier than A1 (Sirius), (b) the absence of giants and
supergiants (class III and I), and (c) the few white dwarfs at the lower left.
The brightest stars:
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An H-R diagram for the 100 brightest stars in the sky. Such a plot is biased
in favour of the most luminous stars--which appear toward the upper right-
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because we can see them more easily than we can the faintest stars. These
are the GIANTS and SUPERGIANTS
In contrast, the H-R diagram for the brightest stars includes a significant
number of giants and supergiants as well as several early-type mainsequence stars. Here we have made a selection that emphasises very
luminous stars at distances far from the Sun. Note that the H-R diagram of
the nearest stars is most representative of those throughout the Galaxy: the
most common stars are low-luminosity spectral type M.
The most prominent feature of the H-R diagram is the Main Sequence:



Strong correlation between Luminosity and Temperature.
Hotter stars are Brighter than cooler stars along the M-S.
About 85% of nearby stars, including the Sun, are on the M-S.
All other stars differ in size:
Giants & Supergiants:

Very large radius, but same masses as M-S stars
White Dwarfs:

Very compact stars: ~Rearth but with ~0.6 Msun!
Example: Betelgeuse:
• An M2 Iab Supergiant star - Red star in constellation of Orion.
• Very luminous: L ~ 100000 LSUN.
• Cooler than the Sun: T ~ 3300 K (Sun is G2 V MS star, T ~
5800 K.)
From Stefan-Boltzmann Law, (R/RSUN) ~ (TSUN/T)2(L/LSUN)0.5, so
R/RSUN = (5800/3300)2(100000/1)0.5
•
R ~ 977 RSUN
This is about 4.5 AU – Betelgeuse would swallow everything out
until almost the orbit of Jupiter if it was where the Sun is – it is big!
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Stellar luminosity classes:






Ia : Brightest Supergiants
Ib : Less luminous supergiants
II : Bright giants
III : Giants
IV : Subgiants
V : Main-sequence stars
Luminosity Classes
Stellar luminosity classes in the H-R diagram. Note that a star's location
could be specified by its spectral type and luminosity class instead of by its
temperature and luminosity. Giants possess cool low-density photospheres,
hence absorption lines identify them (e.g. narrower lines). After spectral
classification, their distance can be estimated according to their luminosity
class. This is their spectroscopic parallax.
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Magnitude versus Colour
Because stellar colours and spectral types are roughly correlated, we may
construct a plot of absolute magnitude versus colour - called a colour-
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magnitude diagram. The relative ease and convenience with which colour
indices (such as B - V) may be determined for vast numbers of stars
dictates the popularity of colour-magnitude plots. The resulting diagrams
are very similar to the magnitude-spectral type H-R diagrams considered
above.
The Mass-Luminosity Relationship
Mass is the most important physical characteristic of stars - it determines
luminosity, temperature, lifetime etc.
How do we obtain the mass of stars? – use binary star systems and Kepler’s
3rd Law (for visible binaries – for spectroscopic binaries the orbital
inclination needs to be known). Eclipsing binaries are even better! The
method is the same as used to calculate the properties of extrasolar planets.
As we need to know the distance (covered earlier in the course) to
the binary system in order to work out the masses of the stars, we
just have to measure the magnitudes of the stars and then calculate
the luminosity.
This was done for lots of binary star systems, and the resulting data plotted
(Luminosity as a function of Mass).
The resulting correlation is only really valid for Main Sequence stars!
When the observed masses and luminosities for stars in binary systems are
plotted, we obtain the correlation called the mass-luminosity relationship.
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In 1924, Arthur S. Eddington calculated that the mass and luminosity of
normal stars like the Sun are related by
L æ M ö
÷÷
= çç
LQ è M Q ø
a
His first crude theoretical models indicated that α ≈ 3. On a log-log plot,
this gives a straight line with a slope of 3. Main sequence stars do seem to
conform to this relationship, although the exponent varies from α ≈ 3 for
for dim red stars of low mass. From a sample of 126 well-studied binary
systems, we find that the break in slope below this value is 2.26; above it,
3.99. Or :
• The lifetime of a star depends on how quickly it uses up its fuel
and how much fuel there is.
• The rate of burning depends on the central temperature of the
star.
• The central temperature of the star depends on its mass.
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• The luminosity is proportional to the rate of fuel burning, hence
central temperature, hence mass.
For L proportional to Mn ,
α
Mass range (in
Msun)
2.3
M < 0.4
3.9 – 4
0.4 < M < 7
3
7 < M < 25
2.7
25 < M < ~80
Lifetime Mass/Luminosity
Mass-3
-- S
Suunn ccaann bbee ppoow
weerreedd ffoorr 55 bbiillliioonn yyeeaarrss bbyy ccoonnvveerrttiinngg 55%
%
hhyyddrrooggeenn ttoo hheelliiuum
.
m.
-- A
s
t
a
r
1
0
t
i
m
e
A star 10 timess aass m
maassssiivvee aass tthhee S
Suunn hhaass 1100 ttiim
meess
hhyyddrrooggeenn ttoo ppoow
weerr nnuucclleeaarr ffuussiioonn
-- B
Buutt iitt iiss 1100000000 ttiim
meess aass bbrriigghhtt
-- TThheerreeffoorree iitt sshhoouulldd uussee uupp iittss ffuueell 11000000 ttiim
meess m
moorree qquuiicckkllyy
ooff iittss
m
moorree
Massive stars are very short-lived!
If we use the mass-luminosity relation for stars of 0.4MSun and greater,
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or
so a star with 10 x the mass of the Sun will have a main sequence
lifetime of only 10 million yrs!
So we know that O stars, the most massive stars, have main sequence
lifetimes of only a million years so the fact that we see some O stars now
means that star formation is still occurring in the Milky Way.
The more massive stars burn their fuel very rapidly, leading to short
lifetimes………..
Stellar (Main Sequence) Properties With Mass
Mass
40
MSun
17
7
2
1
0.2
Temp
Radius
Luminosity
tMS
habitable zone
35,000 K
18 RSun
320,000 LSun
106 yrs
350-600 AU
21,000
13,500
8,100
5,800
2,600
8
4
2
1
0.32
13,000
630
20
1
0.0079
107
8x107
2x109
1010
5x1011
1-2
0.1-0.2
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In order of spectral type….………..
Spectral Mass Luminosity Temperature
Type
(MSUN) (LSUN)
(K)
Radius
(RSUN)
Mean
Density
(kg m-3)
O5
40
400000
40000
13
25
B0
15
13000
28000
4.9
200
A0
3.5
80
10000
3
200
F0
1.7
6.4
7500
1.5
700
G0
1.1
1.4
6000
1.1
1200
K0
0.8
0.46
5000
0.9
1600
M0
0.5
0.08
3500
0.8
1400
Some stars have still not left the main sequence………
M*/Msun
60
30
10
3
1.5
1
0.1
time (years)
3 million
11 million
32 million
370 million
3 billion
10 billion
1000's billions
Spectral type
O3
O7
B4
A5
F5
G2 (Sun)
M7
tthhee lliiffeettiim
meess ooff ssttaarrss w
wiitthh m
maassss << 00..8855 M
Mssuunn aarree lloonnggeerr tthhaann 1155 bbiilllliioonn
yyeeaarrss ((>> tthhee aaggee ooff tthhee uunniivveerrssee))
Note the M-L law does not apply to highly evolved stars, such as red giants
(with extended atmospheres) and white dwarfs (with degenerate matter.
The ranges. While most
stellar masses lie in the narrow range from 0.085Msun
to 100Msun ,
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stellar luminosities cover the vast span 10-4 ≤ L/LSun ≤ 106.
A useful relationship to give a rule of thumb estimate of a stars surface
temperature is:
æMö
T @ 5870 ç ÷
è M* ø
0.5
Stellar Density
Mean Stellar Density: Mean Density = Mass / Volume
Main Sequence: quite small range of mean densities:



Sun (G2v): ~1.6
g/cc
O5v Star: ~0.005 g/cc
M0v Star: ~5
g/cc
Giants: Low-density stars: ~10-7 g/cc (e.g., K5III)
Supergiants: Very low-density: ~10-9 g/cc (e.g., M2I)
White Dwarfs: High-density stars: ~105 g/cc
For reference, at sea level on Earth, water has a density of 1 g/cc, and air
has a density of ~0.001 g/cc.
Stellar Evolution:
In this section, we explain the HR tracks qualitatively in terms of:
1. The energy source…..chemical, gravitational and nuclear reactions.
We exclude chemical energy (e.g. forest fires) for stars. Gravity (if
contracting) can operate for short periods.
2. Transport from the source to the surface…..conduction, convection
or radiation. We exclude conduction as ineffective.
3. Radiative transfer through the photosphere, as discussed above.
Hydrogen ions can provide the opacity in stars like the Sun.
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The end of Hydrogen burning:

During main sequence lifetime hydrogen burning is confined to the
core. Hydrogen burning converts hydrogen into helium in the core.

Eventually the core hydrogen is exhausted. Energy is then derived
from a hydrogen shell

With no energy production in the core, it contracts to maintain
thermal hydrostatic equilibrium. The collapse of the core will cause
it to heat up.

The hydrogen burning shell dumps further helium onto the core.
Hydrogen burning moves outward.
The core collapses, releasing energy and the star’s envelope expands
and cools – a subgiant branch phase


Over a million years, the core of a Sun sized star decreases to about
1/10 original size.

The core temperature rises from 15 to about 100 million K.
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
The core is composed of helium ‘ash’.

The outer layers of the star become heated by their proximity to the
energy source. The inert hydrogen outside the shell hinders the
movement of the photons.
The energy is then transported by convection. (low temperature, high
opacity, high temperature gradient, just what you need for
convection)
Processed material from the core mixes for the first time with the
envelope - and photosphere. We call this the first dredge-up which
should be visible as an increase in N at the expense of C and O.



The outer layers are not so tightly bound by gravity and will expand
enormously forming a red giant
Why Helium won’t burn yet

Hydrogen, a single proton, has a single electrostatic charge

Helium has two.

Helium nuclei must have a much higher kinetic energy (speed) to get
close enough to bind
Helium burning begins

When the central temperature reaches 100 million K, helium burning
starts.

Two helium nuclei fuse to form an isotope of beryllium.

Beryllium is very unstable. If it is hit by another helium they fuse
into a stable isotope of carbon.

This is known as the triple alpha process.

A high energy gamma ray is released by each reaction
A Star’s Safety-valve

Gravity tries to compress a star

When a perfect gas is compressed its density and temperature
increase.

If a gas heats up its pressure increases.
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
The pressure tries to expand the star.

If a reaction starts to run away, the temperature rises and the star
expands.

This drops the temperature and the reaction is slowed.
Perfect and degenerate

In a low-mass red giant (<3 Msol), the core must undergo
considerable compression to drive the temperature high enough to
start helium burning.

No two identical particles may occupy the same quantum state.

The electrons obey the Pauli exclusion principle (Wolfgang Pauli,
1925) and will not be compressed any further.

The gas is said to be degenerate and is supported by degenerateelectron pressure.

In the highly compressed core, free electrons are so crowded together
that quantum effects must be considered.
Helium flash:

When the temperature in the core reaches that required for helium
fusion, energy begins to be released.

Because the star is supported by electron degenerate pressure, it does
not expand.

(Remember degeneracy is a quantum effect and not influenced by
temperature in the same way.)

Without its safety valve the temperature soars and the fusion process
runs away.

This runaway takes only a few seconds and is called a Helium Flash

It releases a vast quantity of energy which drives the temperature so
high that the gas behaves in an ideal way again: the degeneracy is
‘lifted’.
The Helium Flash is not observable, since the photons produced in
the explosion are trapped in the Hydrogen layers.

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Low mass stars:

After the helium flash, substantial carbon and oxygen ‘ash’ is
dumped at the core.

The core contracts until electron degeneracy again supports the
star.

The temperature reached is enough to start shell helium burning
around the core

Helium shell burning, like the hydrogen shell before it, heats the
outer layers of the star and it expands again to form a red
supergiant.
Low mass planetary nebulae

The helium shell is much thinner than the hydrogen one and is
unable to swell the star to relieve the temperature build up.
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
The process runs away until the helium layer is thick enough to
expand the star thus cooling it.

These helium flashes raise the luminosity from 100 to 100,000 times
that of the Sun.

The flashes can also re-start the hydrogen burning.
Can be so energetic that the outer layers of the star are blown clean
off. The escape velocity from the surface of a star is v esc =
(2GM/R)1/2 .



The expanding shell of ejected gasses is ionized by ultraviolet light
from the hot core left behind. The White Dwarf core has a surface
temperature over 100,000 K. Wein's law for a hot body with this
temperature gives a peak wavelength of 2.9 x 10 -8m, corresponding
to ultraviolet light.
When the electrons recombine with the surrounding ions, they often
enter an excited state and then jump down to the ground state
emitting visible photons. This process is known as fluorescence.
HST images of
Planetary Nebulae
Henize 1357
The Helix
NGC 6543
MyCn18
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Planetary nebulae

Last for around 50,000 years after which it has dispersed and faded
from view.

Accounts for 15% of matter returned to the Inter-Stellar Medium
(ISM) by stars.

The planetary nebula takes ~ 60% of the star with it leaving only the
core.
White dwarfs

< 4 Msol, never produce temperature high enough to ignite carbon
and oxygen.

During this phase, the star moves to the left on the H-R diagram.

The track will sometimes loop corresponding to thermal pulses.

As the ejected nebula fades and the core cools, the stars track turns
sharply downward.

The core becomes more and more compressed as the temperature
drops.

Most of the matter becomes degenerate again and the contraction
halts.

The star is now called a white dwarf - about the same size as the
Earth.

Its density is typically 109 kg/m3.

One teaspoon weighs as much as an elephant (5.5 tons)

Remember that electron degeneracy is a quantum effect. This means
that the more massive a white dwarf, the smaller it becomes.
The end of the road


The Chandrasekhar mass (1.4 Msun) is the largest mass that a white
dwarf can possibly have.
Highly ionized atoms floating in a sea of degenerate electrons.
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
As the star cools, the random motions of the particles slow and the
electric forces between ions line them up in a crystalline lattice.

From this point on the star is ‘solid’

the electrons, though degenerate, may move around the lattice.

The core is similar to copper or silver.

As it cools further it evolves into a cold dark diamond sphere of
carbon and oxygen, about the size of the Earth.
Higher Mass Stars: How far can it go?

For an element to serve as fuel energy must be given off when its
nuclei collide and fuse.

This energy comes from packing together more tightly the neutrons
and protons in the ash nuclei than in the fuel nuclei.

Once iron is reached with 56 protons and neutrons, no further energy
can be extracted by the addition of more.

Iron does not burn.

The fuel layers burn outward dumping more and more iron onto the
core which is supported by degeneracy pressure alone.

Eventually this fails, catastrophically and violently.
**Mass Dictates the Life of a Star** (Russell-Vogt Theorem)





The early stages of evolution for the medium and high mass stars are
very similar to the low mass stars, but they occur faster.
The life of stars of all masses during the main sequence phase is very
similar.
The main difference is that the higher the mass, the more luminous
the star and the shorter the main sequence lifetime.
What happens after the main sequence phase depends on the mass of
the star.
Define the following mass ranges:
o Low Mass Stars: M < 4 MSun
o Medium Mass Stars: 4 MSun < M < 8 MSun
o High Mass Stars: M > 8 MSun
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Medium and High mass stars are not degenerate while red giants.




Low mass stars end up as White Dwarfs composed of mainly Carbon
and Oxygen.
Medium mass stars have higher temperatures in their cores.
The higher T allows fusion reactions creating Oxygen, Neon,
Sodium and Magnesium.
Medium mass stars end up as White Dwarfs composed of the higher
mass elements.
High Mass Stars: Mass > 8 MSun


High mass stars can have many successive stages of fusion of higher
mass element in a core and lighter elements in shells around the core
The general trend is for the star's surface to become cooler and to
become a blue giant and later a red supergiant.

If the mass of the white dwarf in the core is larger than 1.4 M Sun
(called the Chandrasekhar limit) the electrons would have to move
faster than the speed of light in order to create enough degeneracy
pressure to halt the gravitational collapse.

Electrons can't move faster than light, so a white dwarf with M > 1.4
MSun collapses.

Main sequence stars with mass larger than about 8 M Sun eventually
form white dwarf stars with masses larger than the Chandrasekhar
limit and collapse.

This is the beginning of a Core Collapse Supernova also known as a
Type II Supernova
A Supernova in a star with 8 MSun < M < 20 MSun

When the supernova begins the iron core collapses rapidly under
free-fall and becomes denser.

When the density is very high, protons and electrons can combine
together
to
form
neutrons
and
neutrinos:
p + e -> n + 

This reaction is called inverse beta decay.

The neutrinos escape easily since they don't interact well with matter
and carry off energy.
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
The resulting neutron gas collapses until the density is extremely
high.

The core of neutrons held stable by neutron degeneracy pressure is
called a neutron star.

The outer layers collapse and collide with the hard surface of the
newly formed neutron star.

This collision causes a violent rebound and a shock wave.
 This
energy can provide the fuel which allows the endothermic
fusion reactions to create very high mass elements such as Uranium.
The supernovae are responsible for all the elements with masses
larger than iron found on Earth.

Core Collapse Supernovae probably occur about once every 50 years
in our galaxy, but most of them are hidden by the dust of the galaxy.
A Supernova in a star with M > 20 MSun

The evolution of very massive stars is similar with the formation of a
neutron star at the core in a supernova.

However, neutron stars (like white dwarfs) have a maximum mass
near 3 MSun, over which neutron degeneracy pressure can't balance
gravity.

In the very high mass stars, the neutron star goes over the critical
mass, and the neutron star collapses.

No other sources of pressure are available, and the collapsing
material forms a black hole.
A White Dwarf in a Semi-Detached Binary: Type Ia Supernovae

Suppose that a white dwarf is receiving mass from a companion star.

The White dwarf's mass will slowly increase.

If it receives enough mass, the White Dwarf's mass will approach the
Chandrasekhar limit and collapse.

The collapse causes the degenerate Carbon gas in the White dwarf to
begin fusing together explosively.
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This type of supernova is essentially a giant Carbon bomb.
Summary of endpoints of stellar evolution
Type
White Dwarf
Neutron star
Core Mass Main Sequence Mass
Msun
Msun
<1.4
<~6
1.4-~3
~6-~12
Source of pressure
electrons
neutrons
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MDS
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----
Protostar – Young Star evolution is not so well known:
From Protostar to Young Star

Protostars are cool when they begin to shine in the visible so start to
the right of the diagram.

Continued gravitational contraction of the protostar.

Decreasing surface area means a reduction of luminosity
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
Decreasing radius (higher pressure and therefore higher temperature)

Different masses of star will follow different paths to their main
destination on the main sequence.
Protostars shine because they are hotter than their surroundings:


Need an energy source to stay hot, but
Central temperature is too cool for nuclear fusion to ignite
Initial energy source is gravity:
(1) accretion on to ‘core’ + (2) contraction of core.
(1) Accretion: bolometric luminosity of protostar is
·.
GM M
L=
R
Where M*dot* is the mass accretion rate and GM/R is the energy released
per unit mass onto the protostar of (accumulating) mass M and radius R.
(2) Gravitational Contraction (aka, the Kelvin-Helmholz Mechanism):



The Protostar shrinks slowly, releasing gravitational energy
50% goes into photons, and is radiated away as starlight
other 50% goes into heating the Protostar interior
How long does this last?
Kelvin-Helmholz Timescale
To understand how long a Protostar can shine by Gravitational Contraction,
we need to compare two numbers


The Energy Source: (GM2/R)
The Energy Loss Rate: Luminosity (L)
The ratio is the Kelvin-Helmholz Timescale:
The Kelvin-Helmholz timescale is ~30 Myr for a 1 solar mass protostar.
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Consequences:


Shorter K-H time for high-mass protostars
Longer K-H time for low-mass protostars
H-R Diagram of pre-Main Sequence evolution for stars of various masses:
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Note the vertical Hayashi tracks: as a low-mass protostar contracts,
convection transports energy to surface. Opacity at surface determines the
surface temperature. Luminosity falls but temperature is constant.
Later, or for high-mass stars, radiative energy transport becomes effective
– central temperature rises – luminosity increases slightly as surface
temperature rises and contraction continues  Heyney track
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At an age of 1 million years the most massive stars have contracted to the
Main sequence, lived out their hydrogen-burning lifetimes and are evolving
off the Main Sequence. Lower mass stars like the sun are still in the PreMain Sequence phase. The youngest clusters observed in the Milky Way
are estimated to have ages of a few million years.






At 10 million years:
stars of 1 solar mass are still above the Main Sequence, just
beginning nuclear reactions. They will be observed as T-Tauri stars.
Stars with M ~ 20M are just moving off the Main Sequence. Such
clusters will still be associated with regions of gas & dust from
which they formed.
At 100 million years most stars are on or near the Main Sequence,
but stars with M > 5M are now moving off the Main Sequence. The
Pleiades cluster is estimated to have an age of about 100 million
years.
With an age of a billion years:
Cluster stars with masses between 2--3 M are moving off the Main
Sequence. The Main Sequence location at which stars are just
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Astrophysics
Dr Dirk Froebrich
32
beginning to exhaust the hydrogen fuel in their cores and move
toward the Red Giant region is called the Main Sequence Turnoff
The oldest clusters in the Milky Way, the globular clusters, are
estimated to have ages of the order of 10-15 billion years and show
H-R Diagrams like that at the right. Because the globular cluster
stars have very low abundances of the elements heavier than helium
(C,N,O ...) some corrections need to be made to compare their H-R
diagrams to younger clusters with higher abundances.
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Astrophysics
Dr Dirk Froebrich
33
Observed clusters:
After ten million years:
The cooler stars are still collapsing and haven't yet reached their MS stage.
H-R diagram of stars in the cluster NGC 2264

On this diagram the Main Sequence is marked with a red line.

We don't see the hottest, most luminous (type O) MS stars since they
have evolved out of the MS stage already.
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Astrophysics
Dr Dirk Froebrich
34
After one hundred million years

All the O and B MS stars have used up their Hydrogen.

When we look at this cluster's H-R diagram, we shouldn't see O and
B Main Sequence stars.

All of the less massive stars should still be in their MS stage, since
they burn their H slower than the O and B stars.

In this H-R diagram for the Pleiades we don't see any of the hottest
Main Sequence Stars.

The "Turnoff Point" is the hottest temperature MS star which exist in
the cluster.

The temperature can be used to estimate the age of the cluster. The
hotter the turnoff point temperature, the younger the cluster.

The stars found above the red line correspond to the O and B stars
which have left the Main Sequence.
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Astrophysics
Dr Dirk Froebrich
35
After 3 billion years:

After 3 x 109 years, all of the O, B, and A stars have used up their H.

We won't see O, B, A Main Sequence stars when we plot an old
cluster's H-R diagram.

An HR diagram of the Open Cluster M67 is shown below.

Here we see only the cooler main sequence stars.

We also see lots of Red Giants.