The HR Diagram
•
Stellar lives, as we have been discussing, are stories about how they generate energy and how that energy competes with other processes in the star (notably gravity) to give the star its physical properties.
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As a consequence, as a star ages, its physical character slowly changes, manifesting itself from an observational viewpoint as changes in luminosity and temperature .
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You should recall that the HR Diagram is a plot of luminosity (or brightness) versus temperature (or color). As such, the location where a star falls on the HR Diagram is an indicator of its evolutionary state.
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We saw this on homework, when we calculated the Kelvin-Helmholtz lifetime for stars of different mass on the main sequence. The result, shown in the figure below, was that low-mass main sequence stars had much longer ∆ t kh
, and high-mass main sequence stars had much shorter ∆ t kh
. We know the Kelvin-Helmholtz is not a good estimator of stellar lifetime because it gives too short a life , but it is based on sound physical arguments, and thus provides a reasonable tool to help build our intuition about stellar lives
•
We can take the Kelvin Helmholtz result now in hand with our understanding of the nuclear processing of hydrogen into helium. Massive stars on the upper end of the main sequence are highly luminous, so they are putting out more energy than low mass companions. If they
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are putting out more energy, the production rate must be higher, so the rate if hydrogen consumption must be higher, and they will run out of fuel sooner.
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In this way, the stars are nuclear laboratories for testing our understanding of nuclear reactions.
Handwaving – Early lives
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Stars form from the collapse of molecular clouds into protostars .
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The star collapses until it reaches hydrostatic equilibrium. Once it reaches that point, the effective temperature of the surface temperature stabilizes, but the star continues to contract on Kelvin-Helmholtz timescales. They are not hot enough to ionize hydrogen yet, but their luminosity decreases as they contract, meaning stars in this phase move along almost vertical lines in the HR Diagram (constant temperature, decreasing luminosity).
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It takes about 10 million years for Sun-like stars to have core temperatures capable of sustaining fusion, and possibly another 20-30 million years before the nuclear burning stabilizes against gravity. Once this happens, the star stabilizes its home on the HR Diagram for the next 10 billion years (or so).
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The part of the main sequence where stars stabilize is usually called the ZAMS — Zero
Age Main Sequence
The Simple Truth of the Main Sequence
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The main sequence is where stars spend most of their lives. Its where the burn hydrogen into helium. This is the story of life on the main sequence:
⊲ Star gets up in the morning and burns hydrogen into helium.
⊲ Star gets up in the morning and burns hydrogen into helium.
⊲ Star gets up in the morning and burns hydrogen into helium.
⊲ Star gets up in the morning and burns hydrogen into helium.
⊲ Star gets up in the morning and burns hydrogen into helium.
⊲ Star gets up in the morning and burns hydrogen into helium.
⊲ Star gets up in the morning and burns hydrogen into helium.
Stellar Old Age ( Grumpy Old Stars , great movie title)
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The star is burning hydrogen in its core; the output of that process is energy (keeping the star shining) and helium ash . This helium is not participating in the fusion process (the temperature is not high enough to fuse helium atoms), hence it is referred to as “ash”
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The consequence of this is that at the core of the the star there builds up a core of non-burning helium.
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Eventually, so much of the hydrogen is used up that fusion cannot continue. When this happens, the radiative pressure decreases, so gravity begins to contract the core.
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The core is contracting, converting gravitational potential energy U
E
K
E
, and the gas heats up .
into kinetic energy
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While the temperature increase is not enough to support pp chain fusion in the core, it is enough to support this process in a thick envelope over the core of (mostly) helium ash.
Red giant branch
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This is all going on down deep in the core, where we can’t see anything. What’s going on at the surface, the parts of the star we can observe ?
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The contraction and the new energy generation means the outer layers of the star become hotter, so the outer layers expand. The outer layers of the star to expand and cool, increasing the luminosity but decreasing the effective surface temperature.
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The star has expanded, the temperature has decreased, but overall the luminosity increases slightly .
◮ On the HR Diagram, this makes a main sequence star move slightly up (brighter) and to the right (cooler) .
•
Low to medium mass stars ( M .
10 M
⊙
) in this phase are called red giant branch stars .
The track looks much like a reversed Hayashi track.
Core evolution
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The core at this time is isothermal . Since there is no fusion in the core at this point, for the core to maintain hydrostatic equilibrium, the pressure must be increasing , which translates into a continuous increase in density toward the center of the star.
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As the hydrogen envelope burning continues, the amount of helium ash increases and the mass of the core increases. How big can the core get? The core can grow in size until it can no longer provide enough pressure to support the material above it. The maximum mass of the isothermal core M ic in proportion to the total mass of the star
Sch¨onberg-Chandrasekhar limit:
M is given by the
M ic
M
2
≃
0 .
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µ env
µ ic where µ is the mean molecular weight of the gas in the indicated areas. For ionized gases
(the case we have in stars) the mean molecular weight µ i
( i is for “ionized”) is defined by
1
µ i
≃
2 X +
3
4
Y +
1 + z
A i
Z
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where
X
Y
Z
≡
≡
≡ total mass of hydrogen total mass of gas total mass of helium total mass of gas total mass of metals total mass of gas
The quantities z and A collectively characterize atoms heavier than helium; A is the atomic weight (i.e. for a given element, A j
= m j
/m
H
), and 1 + z is the total number of particles for a completely ionized atom — the nucleus and the z associated electrons.
•
Generally A j
≃
2 z j for any species (isotopic oddities non withstanding), and for atoms very large compared to helium 1 + z j
≃ z j
, thus
1 + z
A i
≃
1
2
EXAMPLE [BOB 13.1.1]
If a star is formed with the initial composition X = 0 .
68 , Y = 0 .
30 and Z = 0 .
02 , and if all the hydrogen in the core has been converted to helium, what is the maximum mass of an isothermal core?
The envelope has approximately the initial stellar abundances so µ env
µ ic
≃
1 .
34 (note the core is Y = 0 .
98 and Z = 0 .
02). Thus:
≃
0 .
627 and
M ic
M
≃
0 .
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The isothermal core will collapse if the core mass exceeds 8% of the star’s total mass.
Degeneracy Pressure
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Is it possible to exceed the Sch¨onberg-Chandrasekhar limit? Of course — if you have quantum mechanics to help you!
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To not collapse, the core must have an additional source of pressure, beyond the standard thermodynamic pressure we’ve been talking about. As the density in the core increases, the ionized gas behaves less and less like an ideal gas.
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The electrons are forced to occupy all the available energy states. Because they are fermions, the Pauli exclusion principle applies. The occupation of all the lowest quantum energy states produces a pressure — the gas is degenerate .
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The completely degenerate electron gas has a pressure given by
P deg
= κρ 5 / 3
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Does this look familiar to you? The degenerate electron gas equation of state is a polytrope with n = 1 .
5!
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Degeneracy pressure is particularly important in stellar remnants, and we will come back to this soon.
Helium Burning
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As we noted above, if the star is not heavy enough, the contraction of the core heats the center of the star until hydrogen burning starts again. Generally this occurs above the core where the temperature is climbing but the helium content is lower. The result is a core surrounded by a hydrogen-burning shell out to the radius where the temperature and pressure drop below the requirements to fuse hydrogen.
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If the star is heavy enough, the temperature can climb to the point where helium can fuse into heavier elements. As the core contracts, the temperatures climb and when they reach
∼
3
×
10 8 K, then the triple-alpha process can occur:
4
8
He + 4
Be + 4
He
He
→
→
8 Be + γ
12 C + γ
•
If the core were acting like an ideal gas, then energy released by the helium fusion would cause the pressure to increase. The pressure increase makes the gas to expand relieving the pressure and reducing the reaction rate. This would be a regulating feedback on the helium burning.
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But remember that in increasing pressure on the core has made the gas degenerate . In degenerate gases, the pressure does not depend on temperature ! This means the core never
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expands, and the reactions can’t be slowed. Eventually the reactions run away and there is a rapid conversion of helium to carbon known as the helium flash . An enormous amount of energy ( L
∼
10 11 L
⊙
) is released in just a few seconds.
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The energy goes into “lifting” the degeneracy, then into thermal energy to expand the core, decreasing the temperature and pressure and regulating the helium burning.
Horizontal Branch
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During the time that helium burning is stable, energy is being pushed into the atmosphere of the star, but other competing structural processes (contraction, convection) are also causing rapid and dynamic evolution of the interior.
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Many stars in this phase undergo a variety of variations in temperature, luminosity and radius. The general trend however is for the star to evolve to higher effective surface temperatures along a relatively horizontal track in the HR diagram.
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Stars in this phase are called horizontal branch stars ; the phase lasts until the helium in the core is exhausted
Asymptotic giant branch
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As the helium burns, the by product is non-burning carbon ash
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Over time, a non-burning core of carbon ash builds up, eventually leading to a state where the core is surrounded by an envelope of burning helium.
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Sound familiar? This is the same game as before, with helium playing the role of fuel.
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Once again, as fusion shuts down in the core, Kelvin-Helmholtz contraction begins, heating the star. The contraction and the heating means the outer layers of the star become hotter, so the outer layers expand. The outer layers of the star to expand and cool, increasing the luminosity but decreasing the effective surface temperature.
◮ On the HR Diagram, this makes a star move up (brighter) and to the right (cooler) .
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Stars in this phase are called asymptotic giant branch stars , and its steeply vertical profile again looks like a reversal of a Hayashi track.
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If you imagine an extension of the red giant branch to higher luminosity, the asymptotic giant branch asymptotically approaches it. It took them a while to figure out a name.
:-)
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The late part of the AGB phase is called the thermal pulse phase . The temperature increases enough in surrounding layers to ignite hydrogen burning. The helium shells are depleting their fuel, and quasi-periodically shut down, intermittently flashing alive again as the hydrogen shells dump enough helium ash into them to get the helium going again.
•
With all that is going on, AGB stars are known to lose mass a prodigious rates, as high as
˙
∼
10 − 4 M
⊙ yr − 1
•
The story of what comes next is a strong function of the stellar mass — stars evolve past the AGB in different ways depending on whether their mass is less than or greater than
∼
8 M
⊙
. At this mass, the evolution is dictated by whether or not new phases of nuclear burning begin in the core.
ASIDE: Cluster Evolution
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Let’s return to where we started this conversation: the lifetime of a star, and the HR
Diagram
•
As time goes on, the brightest, most luminous stars evolve more quickly, as we would expect from both our Kelvin-Helmholtz and nuclear lifetime calculations.
•
What this means in the context of our current discussion is that the upper end of the main sequence evolves into the red giant phase first .
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Imagine a cluster of stars born together and populating the main sequence. Initially, the entire main sequence of stars is filled from the lower to the upper ends. This is how a young open cluster might appear, such as the Pleiades, whose HR Diagram is shown below. By contract, for an older cluster, the most massive stars have evolved to become red giants, and so the main sequence of the HR Diagram has turned off , as is the case for the globular cluster, M3, also shown below.
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The turn-off point in a cluster is a measure of the age of the cluster! It represents the age of the oldest star that has had enough time to burn its primary fuel and evolve to become a red giant.
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We will come back to this point when we talk about cosmology and the age of the globular clusters
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