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Stellar Structure
Section 6: Introduction to Stellar Evolution
Lecture 14 – Main-sequence stellar structure:
… mass dependence of energy generation,
opacity, convection zones, density profile
… mass limits
… effects of composition changes
Post-main-sequence evolution:
… calculations and observational tests
Mass dependence of opacity and
energy generation
• Central (and mean) temperature of a MS star increases with
mass, though not strongly (T ~ M/R, R ~ M0.6-0.8)
• Bound-free opacity (Kramers’) decreases with temperature, while
electron-scattering stays constant, thus becoming relatively more
important
• Energy generation increases with temperature – and CNO cycle
much more sensitive than pp chain, so becomes more important
• Roughly speaking:
 M > 1.5 M: CNO cycle, Thomson scattering dominate
 M < 1.5 M: pp chain, bound-free absorption dominate
• Transition actually occurs gradually, at slightly different masses
Mass dependence of convection
• CNO cycle has much stronger T-dependence than pp chain, so
central temperature gradient steeper in more massive stars
• Steep gradient unstable to convection → convective core
(radiative core in less massive stars with pp chain)
• Less massive stars have cooler surfaces => ionisation zones in
surface regions → convective envelopes (more massive stars
ionised right to surface, so have radiative envelopes)
• Thus convective envelopes are deep at low mass, and shrink to
nothing as mass increases, while convective cores grow with
mass, from zero at about 1.1 M (Handout 11, top)
• Note strong mass-dependence of the concentration of mass to
the centre (50% of R contains ~95% M at Sun, <50% at 0.4 M)
Mass dependence of central
conditions
• Seen already that T generally increases slowly with mass –
Handout 11 (foot) shows the detail – note the dramatic change
of T with density for masses around the Sun
• Density decreases as mass increases (contributing to decrease
of bound-free opacity) – but note almost constant central
density (~100 g cm-3) over mass range 0.3 to 1.3 M
• Ratio of central to mean density at Sun ~100
• Radiation pressure increases gradually towards higher mass,
and degeneracy towards lower mass
Mass limits
(see http://www.sheffield.ac.uk/mediacentre/2010/1713.html
for details of R136a1)
• Lower mass limit for MS: ~0.08 M, caused by T being too low at
centre for H fusion (some D fusion pre-main-sequence)
• Less massive stars simply cool slowly; seen as brown dwarfs
(with degenerate cores, and surface molecules, such as
methane); below ~17 Jupiter masses, usually classed as planets
• High mass limit less clear: cores of 50-100 M have luminosities
large enough for radiation pressure to stop further accretion, so
this often taken as upper limit (also: pulsationally unstable)
• Recent VLT observations suggest more massive stars exist:
 eclipsing binary NGC 3603-A1,
masses 116 and 89 M - consistent
Sun
 R136a1, mass ~265 M, birth mass ~320 M !
R136a
Summary of basic picture
• For solar composition, model HR diagram agrees satisfactorily
with observations, remembering that models are zero age and
observed stars have range of ages – see blackboard sketch
• Changes in assumed composition of models cause small shifts
in ZAMS, but little change in shape – see blackboard sketch,
which implies giants cannot remain well-mixed as they evolve
• Theoretical M-L relation agrees reasonably with observation
(Handout 2), despite small number of well-determined masses
• Switchover from convective to radiative envelopes seems to
occur at about the right effective temperature, if we are
interpreting spectra correctly
Post-main-sequence evolution
• Better understood than pre-MS evolution
• Pioneering calculations in 1950s and 1960s by 3 main groups:
 Icko Iben (USA)
 Rudolf Kippenhahn and collaborators (Germany)
 Bohdan Paczynski and collaborators (Poland)
• Results agreed well, despite 3 independent computer codes
• Many other groups now active, from Switzerland to Japan to
USA – results differ from each other only in detail: slightly
different assumptions about equation of state, opacity, nuclear
reaction rates, treatment of convection etc
Observational tests give additional
confidence
• Many observational tests available for MS stars => firm
foundation for post-MS studies
• MS lifetime >> pre-MS timescale => much more data available
(even though star formation studies now observation-led)
• Post-MS timescale also nuclear (except for a few phases) – so
again much more data than for pre-MS studies
• Two kinds of observational constraint
 Statistical studies of large numbers of field stars (problem:
selection effects, e.g. more luminous stars dominate sample)
 Look at star clusters: stars all at ~same distance, and
probably all of ~same age. See next lecture