1
Q
Virial Theorem
A
relating potential energy Ω and total kinetic energy U in a self-gravitating sphere in hydrostatic equilibrium
Ω+2U=0
2
Q
Jeans mass
A
the maximum mass of gas that is stable against gravitational contraction
(ie a mass with M>MJ will contract under its own gravity)
3
Q
Freefall timescale
A
timescale on which a gas sphere collapses if there is no support against gravity (also, time to adjust to a dynamical perturbation)
in early stages of a star - only gravity
Tff = (1/pG)^1/2
4
Q
Kelvin-Helmholtz timescale
A
timescale if radiation powered by release of gravitational potential (also, time taken to adjust to a thermal perturbation)
TKH = alpha GM^2/RL
alpha = constant of order 1
5
Q
nuclear timescale
A
how long a star powered by the p-p chain can radiate
tnuc = fE fM Mc^2/L
fE = fraction of rest-mass energy converted
fm = fraction of stellar mass involved
fEfM = 0.0007 for p-p
6
Q
nuclear reactions - reaction rate per unit mass for p-p chain and CNO cycle is
A
ε prop to pT^n
7
Q
types of opacity
A
electron scattering
free-free
bound-free
bound-bound
8
Q
Hydrostatic equilibrium
A
dP/dr = -Gmp/r^2 = -pg
9
Q
gas pressure
A
Pgas = p kB T / u mH
10
Q
radiation pressure
A
Prad=1/3 aT^4
11
Q
degeneracy pressure
A
non relativistic:
Pdeg, nr = Knr n^5/3
relativistic:
Pdeg, r = Kr n^4/3
12
Q
adiabatic process relationship
A
PV^gamma = constant
P prop to p^gamma
where gamma= Cp/Cv
13
Q
ways to move energy around
A
radiative energy transport
convective energy transport
14
Q
Peliades HR diagram shows only main sequence - what does this suggest
A
young
not yet evolved off the MS
15
Q
how do we know stars evolve?
A
- change is inevitable: the finite energy source of a star must run out
- Different HR diagrams =different evolutionary stages
- theoretical predictions and calculations of stellar evolution are compared with observations
16
Q
evolutionary tracks
A
the paths that stars are predicted to take through the HR diagram as they evolve
17
Q
isochrons have constant
A
time
18
Q
Big picture of pre MS evolution
A
- A cloud of molecular gas and dust starts to contract under self-gravity
- in the initial collapse the gas is in freefall and approx isothermal
- as density increases, cloud fragments into smaller clumps
- contraction becomes adiabatic when density and opacity high enough
- Fragmentation stops, approx HE established and protostar contracts of KH timescale
6.high opacity means protostar initially fully convective - if protostar massive enough, T increases enough for opacity to drop and radiative interior develops
- core T high enough for nuclear fusion to start
19
Q
dense clouds of molecular gas and dust are identified by
A
their absorption in the visible and emission in the infrared-mm range
molecules are present, since clouds are at low temperatures (below molecular dissociation temps)
20
Q
interstellar molecules and dust
A
more than 200 molecules, including some complex ones, are observed in the interstellar medium, primarily using IR to mm spectroscopy
21
Q
molecules have rotational and vibrational transitions giving rise to
A
densely-packed sequences of lines
22
Q
interstellar molecules and dust in the optical
A
stars against a dark structure, and a bright HII region
23
Q
interstellar molecules and dust in the infrared
A
emission from extended molecular cloud - both gas and dust - showing filamentary structure and some protostellar objects
24
Q
protostars accrete mass from
A
their host molecular clouds
and shrink under their own gravity
25
protostars - initially, the cloud is in
free-fall as there is insufficient pressure to establish HE
26
evidence for flows associated with collapse comes from
mm molecular spectral lines showing slightly redshifted absorption
27
pre main sequence sources are low mass objects, bright in the optical and observed to lie
above the theoretical zero-age main sequence (ZAMS), so are not yet fusing H-He
28
hot and luminous, high mass O,B stars on a HR diagram
already evolved into the MS
(live fast, die young)
29
at low T, protostars have a higher luminosity than
for the same T as ZAMS
implies radius bigger than for MS star at a given T
as collapsing towards MS, radius shrinks
30
in the pre-MS phase, signs of protoplanetary disks
1. protoplanetary ionised disks in Orion. in some cases these surround glowing protostars
2. Protoplanetary disk observed by ALMA showing dark rings thought to be caused by protoplanets clearing their orbits
31
T Tauri stars
- <10^7 years and <3Msun
- pre MS - still undergoing grav. contraction
- found only in nebulae or very young clusters
- more luminous than MS stars of similar spectral types
- 50% are surrounded by disks of gas and dust
- winds and jets are confined mostly to the axis perpendicular to these disks
32
T Tauri - why are winds and jets confined mostly to the axis perpendicular to the dust and gas disks
the gas and dust disks are funneling
so stops outwards and outflows are perpendicular
driven outwards by radiation/thermal pressure
33
Herbig Ae/Be stars are differentiated from T Tauri by
MASS
34
Herbig Ae/Be stars
-Pre-MS high mass counterparts of T Tauri (3-8Msun)
-<10^7 years, spectral type A/B
-located to the right of MS
-many newly formed stars eject gas before they become stable
-if a bi-polar outflow is visible: Herbig-Haro object
-A disk of gas and dust, rotating and accreting onto the star, prevents equitorial winds
35
consider a gas cloud with mean density p contracting under self-gravity
initially, there is insufficient...
gas pressure to resist gravity, and the collapse is freefall
36
tff does not depend directly on
the total mass or radius of the cloud
37
small and large clouds of the same mean density have the same
freefall timescale
38
tff changes throughout
contraction and at different locations
(volume changing but mass staying same so mean density changing)
centre also contracts faster than outside
39
tff is really fast compared to
kelvin-helmholtz
if outward pressure removed, sun would have tff of 1/2 an hour
sun tkh=10 million years
40
u in jeans mass expression
mean mass per particle in amu
41
a gas cloud is gravitationally unstable and will collapse further under its own self gravity if
M>Mj
42
If M< or = Mj, the object is
stable and in approx HE
43
with typical molecular cloud properties, Mj is of the order of
10^3 solar masses
ie much bigger than typical stellar masses
ie something must have happened between this collapse and formation of stars we see today - fragmentation
44
we have Mj approx (T^3/p)^1/2
as the cloud shrinks...
p increases
however, T is constant at the beginning (isothermal contraction) then increases (adiabatic contraction)
45
Fragmentation
during isothermal contraction
Mj decreases so smaller and smaller sub-masses can become gravitationally unstable and collapse
46
Fragmentation
during adiabatic contraction
Mj can increase, which halts further contraction and fragmentation
47
Fragmentation
Mj also depends on u but why do we ignore this?
only varies by a factor of 4
(mostly hydrogen u=2 and when ionised protons and electrons u=1/2 - factor of 4 difference)
48
fragmentation eventually stops when
reaches the adiabatic phase and stability is achieved
49
overview of isothermal side of fragmentation flow chart
start with gaseous cloud under collapse
T const, p increased so new Mj'
50
overview of adiabatic side of fragmentation flow chart
adiabatic so T increases, p increases and new Mj
Mj'>Mj
if Mj'M then have reached stability
51
during the early freefall T remains
roughly constant
isothermal contraction
52
isothermal contraction happens for two reasons:
1. the cloud initially has low density and low opacity, so excess energy can be radiated easily
2. even as p and k increase, the temp is kept constant for a while by dissociation and ionisation of the gas. This is the phase change which absorbs energy
53
the latent heat of dissociation/ionisation acts as
a 'thermostat'
adding in energy without changing the temperature
54
the increased number of free electrons also increases
opacity
during this time, u also changes as the number of particles increases
(before mass over 2 particles, after have same mass over 4 particles)
55
adiabatic contraction during freefall
as the hydrogen dissociation and ionisation increases...
the thermostat effect reduces and eventually disappears
the resulting increase in T (and p) leads to strong increase in the opacity and the amount of radiation trapped
at this stage the contraction becomes adiabatic
56
adiabatic contraction during freefall
with T,p and hence P increasing, the pressure gradient stars to
support the star against gravity, and the cloud moves towards HE
57
evolutionary track on HR diagram during freefall
initial free fall
approx isothermal contraction
R and L increasing
58
evolutionary track on HR diagram during freefall
Teff is
the temperature at the 'surface' ie where the optical depth approx 1
(L=4pi R^2 sigma Teff^4)
59
evolutionary track on HR diagram during freefall
as more material accretes,
the tau=1 radius increases so the tau=1 surface area increases
60
evolutionary track on HR diagram during freefall
Initially L increases while Teff...
stays approx constant
61
evolutionary track on HR diagram during freefall
when dissociation/ionisation is complete, contraction becomes
adiabatic and temperature (including Teff) increases
62
evolutionary track on HR diagram during freefall
A short burst of deuterium burning starts in the core, when it stops...
L decreases sharply
the protostar is now in H.E at the start of its Hayashi track
63
The temperature increase during adiabatic contraction can result in the Jeans mass
increasing as density increases
64
showing Jeans mass increasing as density increases during adiabatic contraction
adiabatic relation: Pg prop to p^gamma
therefore pKbT/uM_H prop to p^gamma
and T prop to p^gamma-1
then sub this into Jeans mass equation
65
jeans mass during adiabatic contraction
if gamma >4/3 then Mj
depends on a positive power of p
therefore Mj increases as density increases (opposite to isothermal contraction) and fragmentation stops
66
stability in terms of p and T
how to get to expression for pj
rearrange the jeans mass equation to get an expression for critical density
67
if p >pj
gravitational forces dominate and the cloud contracts
68
if p
gas pressure forces are greater than the gravitational forces and the cloud expands
69
for a given M, u there is a line T prop to pj^1/3 which
divides regimes of contraction and expansion
points on this line represent stable conditions
70
T prop to p^gamma-1
if gamma < or = 4/3
the log T-log p track of a contracting protostar has gradient <1/3 and will never cross the T prop to pj^1/3 line of stability
71
T prop to p^gamma-1
if gamma > 4/3
gradient >1/3 and eventually crosses the line of stability so contraction halts
72
T prop to p^gamma-1
gamma < or =4/3 during a phase change as
energy input results in increased dissociation or ionisation, not increased volume so
gamma = cp/cv =approx 1
or if moleuclar =s+2/s
73
when does fragmentation stop
once in the adiabatic regime with gamma=5/3 (ie ionisation complete)
74
formation of a protostar
density is highest in the
centre (gravity)
so the opacity, temperature and pressure can also be expected to be highest here
an outward pressure gradient is thus established and approx HE can be assumed
75
equation of HE
dP/dr = -Gmp/r^2 = -pg
76
we now have a protostar, which continues to contract slowly on the kelvin-helmholtz timescale
HE is a good approx because
kelvin helmholtz timescale >> free fall timescale
77
note as the protostar is relatively cool and has high opacity, it is
fully convective for at least some of its pre-MS evolution (duration of the convection stage depends on mass)
78
we can estimate the interior temperature at which HE is established using
the virial theorem
omega +2U = 0
79
at the beginning of the adiabatic phase the gas is fully ionised
assuming pure H the total number of particles N=NH+Ne = 2NH (NH is no of hydrogen atoms before ionisation)
then internal energy is
U=3/2 N kB T
=3 NH kB T
=3 M/mH kB T
80
during the isothermal phase of freefall, most of the released gravitational PE leads to
ionisation and dissociation (with a small amount radiated which we neglect)
81
assuming a cloud of hydrogen contracting from R1 to R2
Ω(R1)-Ω(R2)=
N_H2εd +N_Hεi.H
εd is the H2 dissociation energy
εi is the ionisation energy
82
RHS of the equation for interior temperature when HE is established (N_H2εd +N_Hεi.H) is
the difference in potential energy associated with this change
83
what assumption do we make for interior temperature when HE is established
assume final radius R2 at which HE sets in is << R1
and recall that Ω is negative
84
after assumptions, N_H2εd +N_Hεi.H is approx
-Ω(R2)
85
inputting N_H2εd +N_Hεi.H=-Ω(R2) and U=3M/mH kB T into the virial theorem and using NH2=M/2mH
then rearrange for T gives
T=1/kB(εd/12 +εi/6) = 3 x10^4 K
86
what does T=1/kB(εd/12 +εi/6) = 3 x10^4 K tell us
thereafter, the evolution of the protostar towards the MS is governed to a great extent by its opacity, which determines whether energy transport is convective or radiative
87
forbidden zone on HR diagram for convective objects
there is a forbidden zone on the right of the HR diagram where HE is not possible for fully convective objects
88
Teff cannot increase above a few 1000 K in star's surface/subsurface because of
opacity from H- (H atom with a second loosely-bound electron)
H- really good at absorbing photons
89
Hayashi track
A fully convective protostar follows a track of roughly constant Teff to the left of the forbidden zone, called a Hayashi track
90
the reason for a roughly constant Teff on the Hayashi track is that
H- opacity depends strongly on T (k approx T^10)
so a small perturbation in T gives big spike in opacity so radiation cannot get through. This increases T and P so outer layers expand and cool
Teff then reduces again (negative feedback loop)
91
Hayahi tracks - convective transport
Since Teff is kept roughly constant, what happens to L and R
L decreases as R decreases
92
Hayahi tracks - convective transport
low mass stars
have long Hayashi tracks
93
Hayahi tracks - convective transport
very low mass stars (less than half a solar mass)
fully convective throughout their entire pre-MS lifetimes
there central temp does not get high enough to develop a radiative core (|dT/dr| always above critical value for convection)
so very low mass stars arrive on the MS fully convective
94
Henyey tracks - radiative transport
inexorable increase in internal KE and T due to contraction means:
H- ions are destroyed and Teff can rise
in the central region of stars with M>0.5 solar masses, |dT/dr| drops below critical value for convection
energy transport becomes radiative
increasing Teff and constant L characterises the Henyey track
95
Henyey tracks - radiative transport
Luminosity
Luminosity roughly constant since the L=M^3/k relationship for radiative transport (SSE1) is independent of how energy is generated
also true for conversion of gravitational PE
96
at the end of the Hayshi and Henyey tracks, there is still some gravitational contribution to luminosity but the first steps of nuclear burning start
if core T is high enough, then first
12/6 C --> 14/7 N burning starts, leading to a small deviation in the Henyey track
97
12/6 C --> 14/7 N burning starts, leading to a small deviation in the Henyey track
high local L(r) makes core expand a little
expansion means grav contribution to L decreases, and Teff decreases a little
Henyey track turns down/right slightly
P-P and CNO cycle starts and the star arrives on the MS
98
12/6 C --> 14/7N burning only last a short time and finally star settles on the MS
What happens with stars with M<0.5 solar masses
do not get hot enough for 12/6C -->14/7N burning and do not show this behaviour
99
the time from the onset of the fully convective protostar to MS arrival depends strongly on
the mass
eg a 15 solar mass star arrives on the MS in around 10^4 years but a 1 solar mass star takes 5x10^7 years
these lifetimes are calculated form numerical models of contraction
100
if the mass is too low (ie<0.08 solar masses) what happens to the protstar
it does not become a main sequence star
brown dwarf
101
brown dwarf
before the core reaches the temp necessary for the p-p chain...
degeneracy pressure halts gravitational contraction
can burn deuterium (if M>0.013 solar mass) and lithium (if M>0.06 solar mass)
surface temps between 200 and 1200 K
102
brown dwarf
once nuclear burning stops what happens to the brown dwarf
they will contract and gravitationally cool
sometimes called 'missing link' between MS stars and gas giant planets
103
the birth function
assume that the number of stars born at a given time within a given volume is a function only of mass (and not for eg chemical composition)
this function is the birth function ɸ(M)
104
birth function
the number dN of stars born in mass range M to M+dM is
dN=ɸ(M)dm
where ɸ(M) is the birth function
105
initial mass function
the total of mass locked in mass range M to M+dM is M dN and is defined by the initial mass function
MdN = ξ(M) dM
ξ(M) is determined empirically
ξ(M)=ξ0M^-1.35
106
initial mass function
plot of Logξ(M) against Log(M/Msolar)
straight line with slope of -1.35
the IMF rolls over at both high and low masses, at approx values predicted as limits in SSE1
At low masses, stars are harder to detect (being fainter) so this could also account for some of the roll over
107
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
-
Key Links
-
Subjects
-
Company
-
Find Us
-
