Evolutionary Trends
1. H-R Diagrams and Isochrones
How do we track/follow the evolution of a star? Basic answer – it isn’t possible. Stars
evolve very slowly so it is not possible to see significant changes in any stars during your
life. There are occasional sporadic changes, like supernovae, but those are rare. Stars
will take millions or billions of years to change their observable characteristics. So how
is it possible to observe any evolutionary changes? It is possible to do this by looking at
many stars at once, particularly those with a common origin. Typically one would look at
the characteristics of a cluster of stars.
The observable characteristics (surface temperature and luminosity) of stars in clusters
have a very specific arrangements on the H-R diagram based upon several characteristics
including the age of the cluster, and the composition of the stars. Since cluster stars have
a common origin in time and space, it is reasonable to assign all stars in the cluster with
the same age and composition. A typical H-R diagram for a cluster, or in this case a
Color-Magnitude diagram is shown here. The other convenient aspect is that stars in
clusters are all at about the same distance from the earth, so you don’t even have to
bother with getting absolute magnitudes or luminosities – apparent magnitudes will work
just fine. Another thing that is the same for most clusters is the reddening, unless the
cluster is found in a region of extensive star formation where dust levels could vary
slightly. However that isn’t a
common problem.
The observed characteristics can
then be compared to the results of
computer models, which can be
generated for stars over a wide
range of stellar masses. Individula
models aren’t sufficient for a
cluster, since there will be a range
of stars of different masses. Also
since stars of different masses have
different evolutionary paths,
models have to be compiled for a
wide range of times to show how
far stars have evolved from the
Main Sequence. The combined
evolutionary characteristics of
many stars with a common age and
composition result in an
isochrones, which shows the
distribution of the stars on the
color-magnitude diagram. Here
they are shown as the solid lines in the diagram for different cluster ages.
Notes 4 - 1
To make things a bit more complicated, astronomers may not know the composition of
the stars in the cluster, or they may want to study many different clusters, so isochrones
are usually constructed for quite a few different metallicities as well as ages. The
generated isochrones can then be matched to the observed distribution of stars in an H-R
(or Color-Magnitude) diagram to try and match ages and metallicities.
Since there is still controversy about various aspects of the computer models that go into
these results, not all authors will have the same isochrones or evolutionary tracks even for
the same ages and compositions. Models are continually being revised and updated to
account for effects such as convection, rotation, mass loss, etc., that are not always easy
to model. And of course when you are comparing isochrones to observational data, there
is the influence that dust has on the color, and magnitudes of the observed stars, so
corrections are needed for those effects as well.
In a few cases it is possible to compare stellar models to individual stars. This may be
possible for stars in binary systems where stellar mass data is available or for other stars
where distance and luminosity information can be determined. So whether it is using
observations of clusters to double check the quality of the stellar models or individual
stars, we can now surmise what might be going on inside of stars (what is buried below
the surface) based upon the results of the computer models.
2. Stellar Cores – Pressure-Density
How do the cores of stars change as they evolve?
The best way to characterize the changes in the core is by tracking the changes in the
temperature (T), density ( ), and composition (X,Y,Z) . Actually composition is the least
important of these, since most other stellar characteristics have a much stronger
dependence on T and .
In order to follow the values for Tcore and core we’ll use a graph of the temperature and
density. Actually this plot uses the log values since the values for temperature and
density can be quite varied when you look at all of the different stars out there.
The Log (Tcore) –Log ( core) diagram shown below also indicates transitions between
areas where various equations of state dominate, and these typically just depend only on
the values of Log (Tcore) –Log ( core). The temperature and density both increase in the
normal sense (to the right and upwards in the graph).
Notes 4 - 2
While most of the boundaries are
pretty well defined, the boundary
between ideal and radiation
pressure is not really correct.
Radiation pressure and gas
pressure can exist together, not as
an option – so there isn’t a point
where one replaces the other.
Typically only when radiation
pressure becomes very
overwhelming is it possible to
ignore the contribution of gas
pressure. So the boundary marked
on the graph is the location where
Prad=10 Pgas. The other boundaries
are found by equating the pressure
relationships and solving the
formulae so that they are only
functions of density and
temperature. So those boundaries mark locations where the dominant form of the gas
transitions from one form to another. In reality you will not have exactly everything
being in one form, there will tend to be some material that is in a slightly different form –
but the most likely form can be found using the diagram.
The core of the Sun on this graph would be found in the ideal gas realm, as would all
Main Sequence stars. This means the ideal gas law dominates in the core – it may not be
the only thing happening there but it is the most important.
Another useful thing to check out is
where the various fusion processes
occur since fusion is a function of
density and temperature – here is a
map showing where those events
occur. The lines shown indicate the
lowest possible values for the
reactions to occur.
You notice that there isn’t much
fusion extending into the regions of
degenerate material. This is because
such material is not very stable, and
any reactions that do happen are
pretty short lived.
The iron photodisintegration is the
break down of iron down to helium
Notes 4 - 3
due to extreme temperatures. The strip shown here is where there would be an equal
number of iron and helium nuclei present, indicating an equilibrium location for the
process. Stars in various phases of evolution and fusion would lie above and/or to the
right of the various thresholds that are shown here. No fusion occurs in the lower left
corner.
Logically you would expect stars on the density-temperature graph to move towards the
upper right corner as they evolve, and the various fusion cycles would continue over
time. But how do stars really evolve? While we could find out using a super computer,
we’ll instead use the polytropic models.
You have a relationship from polytropic models for pressure
Pc
(4 )1/ 3 BnGM 2 / 3
4/3
c
3-9
And the ideal gas law
T
Pgas
2-9
These can be combined to get a relationship between , T, M in the form
c
Tc3
M2
Or taking the log of both sides you get
Log(
c
) 3Log(Tc ) 2Log(M )
4-1
When you graph this on the temperature
density plot, you end up with a straight
line from lower left to upper right for
individual stars. This indicates that
stars tend to stay within the domain of
the ideal gas law even as their core
density and temperature increase as they
evolve.
We do know that stars do get into the
other regions of the graph over time,
particularly at later phases of their
evolution – typically a star’s core would
become degenerate over time.
Equation 4-1 shows that the distribution
of stars in the ideal gas domain is
according to mass – they are spaced out
uniformly. However, once a star gets a degenerate core, the density doesn’t depend on
temperature anymore – it is only a function of the mass.
Notes 4 - 4
The evolution line then becomes horizontal. In this area
c
M2
The paths shown in the degenerate areas can only go so high (in terms of mass) since the
Chandrasekhar limit will set the upper limit to the values that will be attained for the
mass.
So theoretically a high mass star goes straight and a low mass star gets curved over into
the degenerate regions as they evolve.
The temperature-density plot shows us some limits for masses such as
1. There is a lower limit for fusion to occur
2. Highest mass possible for a stable star. Any higher, and radiation pressure would
destroy it. This ends up being more than 100 solar mass
3. Different stars will reach different fusion stages depending upon their masses
Do stars actually follow the paths described by polytropes in the Log (Tcore) –Log ( core)
graph? Of course not! There are all sorts of things that can alter the evolution of a star –
1. Mass loss
2. Much more complex equations of state (not just ideal gas)
3. Different fusion processes may occur at the same time, or there may be starting
and stopping of fusion which makes the paths much more complex.
4. Eruptive events like helium flashes
In spite of the limitations, the temperature-density graph can illustrate the changes in the
core of a star over its life. Here is the typical life of such a star –
First there is the formation of the star, which would actually be somewhere off the graph
in the lower left corner. This phase would have the temperatures, pressures and densities
at very low levels.
As the star contracts, its central temperature and density would gradually increase up
along the lines indicating constant mass on the graph until it reaches a fusion threshold.
The first one is hydrogen fusion, and depending on the stars mass, it would either make
use of the P-P cycle or the CNO cycle as the dominant Main Sequence fusion process.
The star then sort of stays in that location on the graph for a while…….
Of course if the mass isn’t big enough to reach the hydrogen fusion limit, it would just
keep contracting slowly while it keeps on cooling. This would be a failed star. Typically
this is the fate for stars with masses less than approximately 0.1 M.
When hydrogen runs out, the core will contract and become hotter and denser (so it
would move up and right). The star keeps going in this direction until it gets to the next
fusion stage (helium fusion). If it doesn’t get up to that threshold, it may turn over
Notes 4 - 5
towards the degenerate area. Once helium fusion does start, it will again pause on the
graph for a long time and keep fusing along happily.
Carbon fusion would start at a point close to the boundary between the degeneracy limit
and the ideal gas law. Theoretically this should result in a fusion event that would be
rather catastrophic – carbon detonation. The odds of this actually happening are
extremely unlikely since a star would have to maintain a constant mass during its entire
life, and it must have a very precise mass (exactly equal to the Chandrasekhar mass). So
carbon ignition isn’t always to dramatic.
Depending on the stars mass the successive fusion processes will start up. However there
is a point where fusion does hit a wall – that of iron fusion (photodisintegration). This
would result in a major release of energy that will basically end the star’s life.
So you actually could have several options – though not all are realistic or likely.
3. H-R Diagram Relations
There are some fun relations that we can derive for stellar structures which allow us to
approximate their physical characteristics on the Main Sequence.
For example we have the following
Conservation of Mass formula
dm
4 r2
dr
2-1
Thermal Equilibrium
dF
dr
4 r2 q
2-2
Hydrostatic Equilibrium
dP
dr
GM
r2
2-3
Radiative Transfer
dT
dr
3 F
4acT 3 4 r 2
2-17
Energy Production
q q0 T n
2-18
And the ideal gas law (for stars on the Main Sequence)
Pgas
T
2-9
Notes 4 - 6
Let’s re-arrange things by switching the values from dr to dm. That isn’t such a big thing
and basically involves substituting equation 2-1 into the others to get the following (in
terms of dm)
dr
1
dm 4 r 2
dF
q0 T n
dm
dP
Gm
dm
4 r4
dT
3
F
3
dM
4ac T 4 r 2
2
Let’s make this easy by getting rid of the differential term (those “d” parts of the above
formulae). There are various ways to do this, such as using dimensionless substitutions,
and redefining the variables, but the end result is the same – simplified formulae.
You end up with the following relations –
P
T
F
F
GM 2
R4
M
R3
P
4-2
4-3
4-4
ac T 4 R 4
M
q0 T n M
4-5
4-6
Okay, what can we learn from this?
If you combine 4-2, 4-3, and 4-4, and solve for T you get a relation for T in terms of M
and R.
This can then be put into 4-5 to get
F
M3
4-7
Or put more simply, on the Main Sequence, mass is related to flux (luminosity) by this
power law – which is sort of what we observe (go back to the first set of notes).
Eddington’s Standard Model also gives this same relationship, but this method just
provides us another way of seeing physical reality in some approximations of complex
formulae.
Notes 4 - 7
We can also combine 4-2, 4-3, 4-4, 4-6 and 4-7 to get a relation between radius and mass
for various values of n (which is the power value for the temperature in the fusion
relation)
R
M
n 1
n 3
4-8
What are the relations for different values of n? For CNO cycles, n=16, so the radius is
almost proportional to the mass (mass raised to a value that is nearly equal to 1). For n=4
(the value of the PP reaction) we have radius related to mass to the 3/7 power, so that the
radius will increase with increasing mass. This is the case for a non-degenerate material.
One more check on reality. We have a basic relation between L, T, R (equation 1-4).
Let’s use some of the new relations and see how that ends up on the HR diagram. Use
equations 4-7 and 4-8 to get rid of the radius in equation 1-4. This will result in the
following ugly beast –
1
L
2( n 1)
3( n 3)
Teff4
4-9
And for Main Sequence using the PP reaction, we have n=4, so
L0.71
Teff4
L Teff5.6
Log( L) 5.6Log(T ) c
And for those using the CNO reaction we have n=16, and you get
Log(L) 8.4Log(T ) c
Do we see this sort of relationship between surface temperature and luminosity on the
Main Sequence? Yes we do – there is a curve or bend in the shape of the Main Sequence
(actually quite a bit of a curve, but the above relationships follow it fairly well.)
Notes 4 - 8
And if you combine 4-2, 4-3 and 4-4, you get the relationship that shows how
temperature varies as M/R. This can be combined with 4-8 to give a relationship that
indicates that temperature varies as M4/(n+3).
With n=4 (PP reaction), that gives us temperature varying with M4/7
We know that stars have to reach a certain temperature for fusion to occur (about 4
million K), so this relation can tell us the minimum mass needed for fusion to occur. Just
scale the mass and temperature to the Sun’s central temperature and mass to get
Tcore
Tcore, sun
M
M sun
4
7
For Tcore = 15 million K, this gives a minimum mass needed for hydrogen fusion being
0.1 M. And using 4-7 this gives a minimum luminosity of 0.001 L. We’ve basically
determined the bottom of the Main Sequence in terms of mass and luminosity.
Log Rho
Now all of this discussion has made use of some major simplifications, such as ideal gas
laws, polytropic models, constancy of physical parameters, etc. How do the really
complex stellar models (the ones with all of the bells and whistles) behave in the
temperature-density plot? Below is a sample of some models showing the evolution
from the Main Sequence, through the various fusion stages and finally to the end when
the core stops
fusing and
8
becomes
degenerate. The
7
lines are ordered
according to the
6
masses of the
stars, but they
clearly don’t
5
follow the same
path that is
4
defined by
assuming that
3
the ideal gas law
is appropriate
inside of all
2
stars, or the path
that is found by
1
using a
polytropic
0
model….but
7
7.2
7.4
7.6
7.8
8
8.2
8.4
8.6
8.8
9
that’s the way it
Log T
works.
Notes 4 - 9
Overall the paths predicted by the various assumptions and simplifications aren’t close to
the complex paths, but they do show the various trends that are expected. So you
shouldn’t complain…..
Notes 4 - 10