1
Q
red giants and supergiants
A
late stages of of stars >0.5 solar mass
the only stars that can be directly imaged due to great size
red giants have convective envelopes leading to a pattern of hotter and cooler patches
2
Q
planetary nebulae
A
at end of lives for 1-8 solar mass stars, stars eject their envelopes
envelopes are enriched with the products of nuclear burning, leading to rich spectra
3
Q
planetary nebulae - reason for the complex, non-spherical shapes
A
not clear
could be related to magnetic fields, rotation, presence of disk
4
Q
what is at the centre of a planetary nebulae
A
a stellar remnant
a white dwarf with radius 10^3-10^4 km which starts off very hot and cools over time
5
Q
white dwarfs
A
around 1solar mass (mass of core once outer layers blown off)
supported by electron degeneracy pressure
6
Q
observing white dwarfs
A
older white dwarfs can be seen in the optical (have cooled down enough to emit in optical and they have shed their planetary nebulae
7
Q
properties of neutron stars
A
radius 10km
1-2 solar mass
100s of revs per sec
surface T =10^6K so visible in x-rays of a few keV
supported by neutron degeneracy pressure
strong B fields (10^15 ish times Earth’s)
formed in supernova explosions, the younger ones are embedded in supernova remnants
8
Q
the Crab Nebula
A
an expanding supernova remnant with a neutron star at its centre
9
Q
what ends up as a black hole
A
some fraction of stars with a very high ZAMS mass will end up as black holes
10
Q
black holes happen when
A
neutron degeneracy pressure is insufficient to support the mass of the stellar remnant and no further equilibrium is possible
11
Q
‘stellar mass’ black holes are detected by
A
deducing their presence by orbital properties of their visible companions in a binary
emission from their surrounding (x-ray binaries)
gravitational waves produced when they collide
12
Q
late MS evolution
Hydrogen burning in core –>
A
increase in core u continues
13
Q
late MS evolution
pressure in core decreases –>
A
core contracts
14
Q
towards the end of the MS lifetime, as u rises quickly…
A
the thermostat effect reduces
so energy generation rate and L_nuc increase quickly (cannot be controlled by thermostat effect)
so the core contracts and the rest of star expands
15
Q
late MS evolution
the increased energy flow L(r) means that
A
the internal energy of the envelope increases
16
Q
late MS evolution
the increased U of the envelope, and the core contraction both lead to
A
dramatic envelope expansion
17
Q
late MS evolution
the star develops a
A
hot, compact, H-burning core surrounded by an extended envelope with a low surface temp
(from L=4pir^2 sigma T^4 and L increases, r super increases so T decreases)
18
Q
stars leaving the MS may undergo oscillations
whether or not this happens depends on
A
the distribution of temperature in their envelopes (everything non-core)
19
Q
oscillating stars lie on the
A
instability strip
20
Q
oscillations require layers which are
A
fairly deep in the envelope where either H or He is partially ionised
21
Q
oscillations
in partially ionised zones, the plasma can
A
absorb energy without significant temp increase
22
Q
oscillations
opacity given by Kramers opacity = k0p^2T^-3.5 and if T is approx const,
A
the opacity depends mostly on p
this leads to the k-mechanism
23
Q
the k mechanism
pulsations require a
A
driving force (radiation pressure) and a restoring force (gravity)
24
Q
the k mechanism
the driving force is strongest when
A
the envelope is contracting, due to increased opacity
this provides the outward radiation force to halt contraction and produce expansion
25
steps in the k-mechanism
envelope contracts
density, opacity increase (const T due to ionisation increase)
increased radiation force
envelope expands
density, opacity decrease (const T due to ionisation increase)
decreased radiation force (gravity wins)
so envelope contracts again and repeat
26
why do MS stars not produce large oscillations
the layers of partial ionisation are too close to the surface
27
pulsation period is on the scale of
free-fall timescale
28
when does degeneracy pressure happen?
When you squish things really close together
29
degeneracy pressure
at a number density of n per m^3, the spacing Δx between particles is
Δx = 1/n^1/3
since cube has side length n^1/3
30
degeneracy pressure
by the Heisenberg uncertainty principle, each particle has momentum
p= h bar / Δx
= h bar n^1/3 in terms of number density
= h bar (p/mp)^1/3 in terms of mass density
31
degeneracy pressure
non-relativistic electrons have energies
Ee = p^2 / 2me
= hbar^2 / 2me (p/mp)^2/3 (from subbing in expression for momentum)
and equivalent for protons, neutrons
32
degeneracy pressure
in general, the pressure P is related to the energy density nE (n=number density)
we write
P=BnE
where B is a number of order 1 ("general thermodynamic relation)
33
using P=BnE
for electrons, we have the non-relativistic degeneracy pressure
Pe = Bnr/2 hbar^2/me (p/mp)^5/3
from subbing in expression for energy
34
degeneracy pressure in the relativistic case
have E=pc
so E=h bar c (p/mp)^1/3
giving Pe=Br/2 hbar c (p/mp)^4/3
35
difference in expression for Pe between non-relativistic and relativistic case
for NR - power of 5/3
for R - power of 4/3
36
stars with M < 0.5 solar masses never develop a high enough core temp to
ignite He fusion in their cores
37
low mass - straight to white dwarf
the core contracts (on the K-H timescale) and what increases?
both its temperature and density
38
low mass - straight to white dwarf
electron degeneracy pressure sets in at a temperature that is
too low for He ignition
so no further energy source so becomes a white dwarf and cools slowly
39
white dwarfs
even the H burning MS phase for such low mass stars is calculated to last
longer than the age of the universe
(ie doesn't exist yet)
40
white dwarfs
if H burning MS phase for these stars lasts longer than the age of the universe, how do we see them?
we detect helium WDs but they occur almost exclusively in binaries
thought that a companion strips off the outer H layers of the star before it becomes a WD, revealing He-rich material
41
0.5 to 8 solar mass stars
these stars evolve to become red giants and eventually
expel their envelope in a planetary nebula, leaving a white dwarf remnant
42
before high mass stars expel their envelope, they undergo
a complex series of core and shell burning, which depends on their mass
43
high mass stars
first: hydrogen shell burning
temperature in the centre is high enough for H burning but all the H has been consumed
there is still a shell where there is some H, and temp is high enough for H to He fusion
44
high mass stars
hydrogen shell burning
since there is no source of L in the core (Lcore=0) we have
dT/dr =0
temp gradient is zero
ie an isothermal core
45
high mass stars - core contraction
the core, which is approx isothermal is supported by
the pressure of the inert helium
46
what does inert mean?
not burning
47
high mass stars - core contraction
maximum mass of the core
for a star of total mass M there is a limit to the mass of the core that can be supported by the pressure of inert He against the gravitational force of the overlying envelope
the Schonberg-Chandrasekhar limit
48
in the the Schonberg-Chandrasekhar limit, what are u_env and u_core
the mean molecular mass of the envelope and core
49
high mass stars - core contraction
continued H shell burning adds He to the core mass
once the SC limit is exceeded...
the core begins to contract which may lead to He burning
(depending on the mass of the star)
50
high mass stars - evolutionary path in the HE diagram
in a 5 solar mass star, burning is as follows:
- core burning on MS
- core H burning stops, core contracts so energy is released (L and Teff increase) and envelope expands
- shell H burning, shell narrows as burning proceeds
- envelope above the H-burning shell cools, opacity increases and envelope becomes convective, starting near surface and moving inwards
- takes only a million years for 5 solar mass so not many stars seen - the Hertzsprung gap
- opacity near surface decreases and L increases
51
high mass - He ignition
if the core mass exceeds the Schonberg-Chandrasekhar mass...
the core contracts, heats and becomes denser
depending on core mass, it may become hot enough for He burning
52
high mass - He ignition
the core contracts and heats and the triple alpha process starts when
T around 10^8K
53
high mass - He ignition
the core density also increases; the core may...
become degenerate before triple alpha ignites
(this is the case if the star's mass in < 2 solar masses)
54
if triple alpha ignites in a degenerate core, this results in a
Helium flash
55
simulations of a helium flash show
essentially the whole core ignites simultaneously and Tcore rises to around 2x10^8K
L can reach 10^11 Lsun for a few seconds (short lived so hard to see and also diffused by envelope)
56
high mass stars - the He flash
what happens for an ideal gas
no He flash
when triple alpha starts, T increases so P increases
increasing P expands the core so T decreases
reaction is stabilised
no runaway
57
High mass stars - the He flash
what happens for degenerate gas (non-relativistic)
P does not depend on T so gas does not expand to stabilise reaction
T increases, triple alpha rate increases
a degenerate electron gas has a very high thermal conductivity so the whole core reaches high T - runaway reaction
T gets so high that electron degeneracy is broken; gas behaves like ideal gas
58
expression for gas pressure
Pgas = p kB T / u mH
59
expression for degenerate (non-relativistic) pressure
Pdeg,nr = Knr n^5/3
60
for He flash
what does 'T gets so high that electron degeneracy is broken' actually mean?
gas pressure > degenerate pressure
p kB T / u mH > Knr n^5/3
so gas behaves like an ideal gas
61
what is the triple alpha process
He burns to C via the triple alpha process
He4 +He4 + He4 --> C12
62
triple alpha temperature dependence
the triple alpha process has an extremely strong temperature dependance
ε prop to T^30
63
eventually, the core He runs out leaving
an inert C core, surrounded by a He burning shell and a H burning shell
64
high mass stars - stages of shell burning
1. H core burning (main sequence)
2. H shell burning
3. He core burning and H shell burning ("He main sequence")
4. He shell burning and H shell burning
65
what is the Helium main sequence
burning He to C or O in a stars core
66
why is the helium main sequence much shorter than the hydrogen main sequence
1.there is less energy available (the energy released per unit mass burning He to C is only 10% of that released burning H to He)
2. the energy generation rate is very high because of the T^30 dependence
67
the helium MS for a 5 solar mass star lasts around
10% of the hydrogen MS lifetime
68
what happens to u as the core He burns
u is increasing
increases to a point where the core starts to contract
69
when u increases enough to a point where the core starts to contract, this is accompanied by
the outwards expansion and cooling of the envelope
70
envelope expansion means that Ω increases (becomes less negative) so
U decreases and the stat becomes less luminous
the core He is eventually exhausted
71
after the core He is exhausted, the inert C core contracts and
He shell burning stars
the He shell burning forces the overlying H-burning shell to expand, quenching H burning (temporarily)
72
the red giant branch consists of stars which are
undergoing only shell H burning
73
the horizontal branch corresponds to stars on
the helium-burning main sequence, with a H-burning shell
74
the asymptotic giant branch corresponds to
He shell burning and H shell burning, with an inert C/O core
75
stars on the RGB and AGB have
thick convective envelopes (low Teff means high opacity), hence the appearance of stars such as Betelgeuse
76
order of the layers for for a 5 solar mass star at the onset of He core burning
(from centre)
He core
H-burning shell
H,He envelope
77
order of the layers for for a 5 solar mass star at the early AGB
(from centre)
CO core
He burning shell
He
H burning shell
H,He envelope
78
later AGB evolution: thermal pulses
eventually the He burning shell thins and
burning stops
but H shell burning continues
79
later AGB evolution: thermal pulses
the H burning shell dumps He onto the He layer below, the resulting compression means
that the He layer becomes partly degenerate at its base, while T increases
80
later AGB evolution - thermal pulses
that the He layer becomes partly degenerate at its base, while T increases
this leads to
re-ignition of He burning at the base of the shell, in a He flash (like the earlier core flash)
81
later AGB evolution - thermal pulses
the He flash causes both the He and H burning shells to
expand outwards, cooling and quenching the burning
82
later AGB evolution - thermal pulses
He and H burning shells expand outwards, cooling and quenching the burning
but gravitational contraction means
the H shell source recovers
the process repeats on a timescale of a few 1000 years (for 5 solar mass stars) to 100,000 years (for 1 solar mass stars)
83
typically how many thermal pulses occur
tens of pulses
(all according to models though)
84
the increases in luminosity and radius on the RGB and AGB leads to
strong stellar winds
the driving mechanism is not well understood but high mass-loss rates have been observed
85
for AGB stars, the envelope is lost very rapidly compared to the other evolutionary timescales, leaving behind a
white dwarf core of C/O and an ejected planetary nebula
86
fusion up to Fe is
exothermic
87
one proposed way to build elements heavier than Fe is by
neutron capture
88
neutron capture can happen in two ways called the
rapid and slow processes (r and s)
the s process is believed to occur in AGB stars
89
two ways of displaying the neutron capture process
1. neutrons captured to create heavier (and more unstable) isotopes of Fe. Fe59 eventually beta decays to Co59 before it can capture another neutron
2. the same process but in the neutron-proton plane (along x in neutron number then during beta decay moves back on in neutron and up one in proton (y-axis) )
90
in neutron capture, the neutron source is thought to be
C13 (alpha,n) O16
occurring during thermal pulses
91
after the AGB phase, the CO core contracts and heats up to T around 10^5 K. Inevitably, the density becomes high enough for at least part of this remnant to become degenerate
for masses < or = 8 solar masses
the remnant is supported by electron degeneracy pressure
it reaches a max temp then cools off gradually to become a white dwarf
92
for masses around 8 solar mass, depending on how much mass has been lost in the wind, the remnant core mass
may be too large for support by electron degeneracy pressure
the core collapses catastrophically and temp increases to ignite C and O burning
CO flash
core collapse supernova
93
a white dwarf is a polytrope in HE supported by electron degeneracy pressure
so it can be described using the
Lane-Emden equation introduced in SSE1
pc = central density
p=pc theta^n
ξ is the radius variable such that r=aξ where a is const
94
the polytropic index in the case of non-relativistic degeneracy pressure is n=3/2 so M prop to R^-3
this says that
as the mass of a white dwarf increases, its radius decreases
this is due to the behaviour of degenerate matter
95
high mass white dwarf
at higher masses and densities, particles are more strongly confined and their momentum becomes relativistic
so now
this is a poltrope with n=3
so M prop to R^(3-n)/(1-n)
mass of a white dwarf supported by relativistic degeneracy pressure does not depend on radius!
96
high mass white dwarf
as M does not depend on R, this means
there can only be one value of mass for a stable star (in HE) supported by relativistic electron degeneracy pressure
this is the Chandraskhar limiting mass
97
value of the Chandraskhar limiting mass
1.46 solar masses
98
as the white dwarf mass increases, the slope of the M-R relationship
increases as relativistic effects become more important, up to M_CH
thereafter, no stable solution exists
99
for stars with M>8 solar masses
evolution up to the HB stage
is the same as for less massive stars
the AGB and post-AGB evolution differ
100
for stars with M>8 solar masses
as He shell burning proceeds on the AGB in massive stars, it adds
enough mass to the C/O core to ignite C burning in the core (at around 6x10^8K)
details on the nuclear reactions that follow depend strongly on the star's mass
101
stars with M>8 solar masses
overview of the nuclear reactions that follow C burning in the core
carbon burning produces O, Ne, Na, Mg
after C burning stops, O burning starts producing a new core composition, dominated by Si
Si burning then produces Ni/Fe
(each shell has an inner burning layer and an outer inert layer)
102
burning stages in massive stars
later stages of nuclear burning proceed very fast
eg for a 25 solar mass star, last 3 stages are on order of 1 year, half a year, 1 day
103
after Fe56, reaction become
endothermic and core fusion stops
104
after core fusion stops, the iron core cannot produce energy by further nuclear reactions
it is supported primarily by
electron degeneracy pressure
105
the Fe core mass is continuously increasing due to Si burning in the overlying shell
once it exceeds M_CH...
The Fe core begins to collapse
106
the core, initially composed of He and with a radius of a few 1000km, collapses within around
1 second
the core temperature rises and its composition changes dramatically
107
photodisintegration and inverse beta decay
at temps of a few x10^9K, the gamma ray photons produced in nuclear reactions have
sufficient energy to destroy heavy nuclei
this endothermic reaction is called photodisintegration
108
particularly important reaction in photodisintegration are
Fe56 + gamma --> 13 He4 + 4n
He4 + gamma --> 2p + 2n
109
in about 0.1s, photodisintegration destroys
the heavy elements in the core
110
photodisintegration
as T increases still further, to 10^10K, the electrons that were producing degeneracy pressure are
captured by the protons in inverse beta decay, producing neutrons
p + e- --> n + ve
111
several factors conspire to produce an accelerating collapse:
1. photodisintegration is endothermic, so core ion temp and pressure reduce
2. e- are captured in inverse B-decay, so core e- degeneracy pressure is reduced
3. neutrinos from inverse B-decay stream out taking energy with them, further reducing the core pressure
4. photodisintegration and inverse B-decay both absorb some grav PE that would otherwise be turned into heat
112
the core collapse proceeds differently in different regions:
the inner part of the core
collapses homologously (meaning speed prop to radius)
113
the core collapse proceeds differently in different regions:
the outer core
collapses above the inner part, in freefall, taking milliseconds
density increases so much that neutrinos start to become trapped in inner core
114
the core collapses to about 10km radius, and a density of about 10^17-18 kg/m^3 (the density of nuclear matter)
at this density...
neutrons become degenerate, forming a nearly rigid core
115
overview of steps of core collapse
1. shells of elements undergo fusion, leading to Fe core.
2. at M_CH, the core starts to collapse rapdily
3. inner core solidifies into neutrons, outer core still collapsing
4. bounce-wave forms leading to an outward propagating shock front
thereafter:
5. the shock starts to slow as it runs into overlying material
6. neutrinos produced near the centre of the collapsed core re-energise shock
116
most gravitational PE lost during core collapse is initially
converted into the KE of neutrinos
these are temporarily trapped behind the shock in the high density of the collapsing core
117
as the shock wave propagates into less dense regions, the neutrinos can
stream out, leading to a neutrino burst
(20 neutrinos were observed in 2 Cherenkov detectors, simultaneous with the light signal in SN1987A, confirming this theory)
neutrinos also drive convection behind the shock which helps power it
118
the shock wave drives off the exterior of the star in a supernova explosion
in a core-collapse supernova, the energy comes from
gravitational PE of collapsing core (SN types Ib, Ic, and II)
119
energy of a core collapse supernova is Ep = GM_NS^2 / R_NS
for a neutron star mass of around 1 solar mass and radius 10km, this gives
around 10^46 J
120
the remaining core, composed of neutrons, is called
a neutron star
121
a neutron star is supported by
degeneracy pressure of the very dense neutrons
122
observed/inferred properties of neutron stars
radius = 10km
mass = 1-2 solar masses
Teff = 10^6 K
density = approx nuclear density (10^17-18 kg/m^3)
interior structure of a neutron star is an active topic of research
123
at very high densities, the neutron momentum increases (Heisenberg Uncertainity Principle) to the point that
the neutrons become relativistic
124
for a gamma=4/3 equation of state, no stable solution is possible and the remnant...
collapses directly to a black hole
this occurs at the Tolman-Oppenheimer-Volkoff mass limit
125
theory puts the TOV remnant mass between
2.2 and 2.9 solar masses
observations range from 2.14 to 2.74 solar masses
126
the centrifugal force in a rapidly spinning NS provides an additional stabilising force against gravity, which may account for
the range of NS masses
(M_CH has a value by TOV limit has a range)
127
evolutionary endpoint for 0.5 solar mass star
star's MS lifetime is longer than the age of the universe
128
evolutionary endpoint for 0.5-8 solar masses
WD remnant and planetary nebula
129
evolutionary endpoint for around 8 solar masses
may ignite core C/O burning leading to a core collapse supernova
130
evolutionary endpoint for between 8 and 20 solar masses
core collapse supernova and neutron star remnant
131
evolutionary endpoint for > 20 solar masses
black hole remnant
132
A4 Stellar Structure & Evolution
flashcards
Decks in class (10)
# Cards
-
Lecture 1: Foundations & HR Diagrams34
-
Lecture 2: Mass, Timescales & Equilibrium49
-
Lecture 3: Dynamics, Star Formation & Ignition47
-
Lecture 4: Core Conditions & Limits on Stellar Masses31
-
Lecture 5: Central Pressure, Simple Models & Energy Generation34
-
Lecture 6: Polytropes & the Lane-Emden Equation23
-
Lecture 7: Mass-Radius from Polytropes & Radiative Transport Basics2
-
SSE2 - Pre-MS Evolution107
-
SSE2 - Life on the MS99
-
SSE2 - Post MS Evolution132
