Download Lecture 6: Main Sequence Stars

VI.
Main sequence stars
h"p://sgoodwin.staff.shef.ac.uk/phy111.html 0. The main sequence
We saw that most stars lie on the main sequence, and
there is a mass-luminosity relationship
but there are also giants and white dwarfs…
Why are most stars on the MS, but some not? Why are
there giants and white dwarfs?
What is happening in a star on the main sequence?
1. Energy generation
We can age the Earth to about 4.5 Gyr from radiometric
dating – so the Sun must be at least this old as well.
The only process that could keep the Sun producing 3.8 x
1026 J s-1 for 4.5 Gyr (that’s 5x1041 J so far!) is nuclear
fusion.
We convert mass to energy from E=mc2 – and c is big, so a
small amount of mass turns into a very large amount of
energy.
We saw the Sun is about 75% H and 25% He – the easiest
way to make energy with this mixture is fusing H to He.
1. Energy generation
To fuse H to He we need very high temperatures (T) and
pressures (P).
High T means the nuclei move very fast, and high P means
they meet each other often. The higher the T and P, the
more reactions there are and the more energy is
generated.
The central T and P depend on stellar mass – the more
massive a star the higher the T and P.
If the mass is less than about 0.1M, the central T and P
never get high enough to fuse H.
1. The pp-chain
The way most stars fuse H to He is via the pp-chain. The
first step is to make deuterium (a proton+neutron)
1H + 1H  2H + e+ + ν + energy
where e+ is a positron and ν is a neutrino (the positron
annihilates with an electron and makes energy).
Then another 1H is added to make 3He
2H + 1H  3He + energy
Then two 3He meet to make 4He and 2 protons
3He + 3He  4He + 21H + energy
1. The pp-chain
There are slightly different ways of doing this, but this is the
most common in the Sun.
In massive (ie. hotter) stars there is the CNO cycle that
involves C, N and O in the fusion process – but we’ll ignore
that.
The final 4He nucleus has 0.7% less mass than the 4 1H
nuclei – this is released as energy: about 4x10-12 J.
So to make the ~4x1026 J the Sun releases every second
needs about 1038 reactions per second! 1. Solar neutrinos
We are confident this is what is happening in the Sun
because we can estimate the number of neutrinos that
must be being produced every second (1038-ish).
Neutrinos hardly interact with anything – most pass right
through the Sun and Earth and never notice them. But a
tiny fraction do interact with large underground detectors in
exactly the right numbers.
[You might read about the Solar Neutrino Problem – we see
only 1/3rd of the neutrinos we first expected, this is because
neutrinos ‘change flavour’ as they travel – it isn’t a problem
anymore.] 2. Lifetimes of stars
We can estimate the length of time the Sun can continue
producing energy this way.
The core where reactions happen is about 10% of the mass
of the Sun = (0.1)x(2x1030 kg). We can convert about 0.7%
of this mass into energy which from E=mc2 is
1x1044 J = (0.007 x 2x1029 kg)(3x108 m s-1)2
Divide by the luminosity 4x1026 W and we get a main
sequence lifetime of 4x1017 s, or about 10 billion years
(10Gyr).
2. Lifetimes of stars
More massive stars have more fuel, but burn it much more
rapidly. Low-mass stars burn at a very slow rate.
A 10 M star has L=3000 L – so 10x more fuel, but burnt
3000x faster  lifetime of about 50 Myr.
A 0.1 M star has L=3x10-4 L – 10x less fuel, but burnt
3000x slower  lifetime of 30 trillion years (30 000 Gyr).
So main sequence lifetimes are a very strong function of
mass.
3. Stellar structure
Stars have the particular radii they do because they are in
‘hydrostatic equilibrium’.
Gravity attempts to make a star smaller, pressure attempts
to make it larger. The vast majority of the time these two
forces are in equilibrium and the star remains stable.
Pressure is mainly due to the thermal energy of the gas/
plasma (there can be radiation pressure as well).
[Because of something called the virial theorem the
balance is always such that the kinetic energy in the
pressure is exactly half the potential energy due to gravity.]
3. Stellar structure
The structure of stars is dominated by having an energy
source at their centres.
Stars must radiate away energy at the rate they produce it
(otherwise they would shrink or grow as their thermal
energy changed).
So the energy from the core must pass through the star in
one of two ways:
Radiation
Convection
3. Stellar structure
How energy is transported depends on the opacity of the
gas – a measure of how easily radiation can pass through
something:
Glass and air have very low opacity.
Wood and steel have very high opacity.
It does depend on wavelength – our atmospheric windows
depend on the opacity of the atmosphere – it doesn’t let
much UV or sub-mm through, but lots of visible or radio.
3. Stellar structure
What sets opacity can be horribly complicated, but stars
can generally be divided into two zones – the core and
envelope. How heat is transported depends on mass (it
depends a lot on the rate of energy generation)
3. Stellar structure
A star like the Sun has a radiative core and a convective
envelope – we see the convection cells on the surface of
the Sun. Sunspots are cooler regions where magnetic
fields inhibit the convection and don’t allow the cooled
material to fall back down.
Summary
Main sequence stars generate their energy through HHe fusion in their cores. For most stars this is via the pp-­‐chain. Stellar lifeCmes are a strong funcCon of mass (because luminosity is). Massive stars live only Myr, Solar-­‐type stars about 10 Gyr, and low-­‐mass stars for trillions of years. The structure of stars is set by a balance between (thermal) pressure and gravity, and how the star is able to transport energy (radiaCve or convecCve). Key points
To describe the basics of nuclear fusion and the pp-­‐chain. To be able to esCmate the lifeCme of a star of a given mass. Understand that structure depends on energy generaCon and energy transport. Quickies
What is the main sequence lifeCme of a 5Msun star? Roughly what fracCon of a star’s mass is turned into energy on the main sequence? Very simply, what is the structure of a Solar mass star? Very simply, what is opacity? What fracCon of the mass is lost in going from 4 protons to a 4He nucleus? Notes
Opacity is very important, but in reality really quite complex. Opacity depends on composiCon, temperature, and density and will change with the wavelength of the light trying to pass through the material. It turns-­‐out that opacity is a very strong funcCon of temperature. At low temperatures everything is neutral (or even molecular) and the main source of opacity are heavy elements (more electrons, more energy levels, and so more absorpCon). As the temperature increases, heavy elements are ionised and create free electrons. These electrons can bind with H to make H-­‐ ions which have a very high opacity (you destroy them with a photon and then they recombine) – the more H-­‐ the higher the opacity and opacity increases roughly as T4. But at about 50000K H-­‐ starts being destroyed and so the opacity rapidly decreases with increasing temperature (about T-­‐4). Thus a mix of thermodynamics and radiaCon transport together with ionisaCon and quantum mechanics determine what happens at different layers in a star – high opacity=convecCon, and low opacity=radiaCon.