BC AST 2008 Lecs21-22Post.ppt...
Transcript of BC AST 2008 Lecs21-22Post.ppt...
Stellar Astrophysics - BC 2008
Homework Assignment 4
CO2 Problems 12.3, 12.7, 12.13, 12.15, and 12.18
Due date: Wednesday 19 March, 13:00
Stellar Astrophysics - BC 2008
Stellar Evolution
The question “What are the stars?” provoked the following
answers (amongst others) when we first met:
The stars are:
“Large balls of hot gas”
“The fabric of the universe”
“Bodies that maintain equilibrium between
gravity
and nuclear reactions”
Stellar Astrophysics - BC 2008
Stellar Evolution
What have we learned?
HOT
Atmosphere
(thin !)
We are indeed dealing
with hot (spectroscopy!)
gaseous (spectroscopy)
balls (gravity)...
that are opaque (radiation
transfer theory) with a very
thin atmosphere as outer
coating (spectroscopy/rad.
transfer theory)
Stellar Astrophysics - BC 2008
Stellar Evolution
What have we learned?
HOT
Atmosphere
(thin !)
Inside, energy is being transported
either by photons taking a random walk
(ideal gas theory, atomic physics) out-
ward, or, if is high enough, by large-
scale motion of atoms and plasma
(convection theory).
Increases in accompany PIZ’s (i.e.
temperature zones). We can regard the
stellar interior as a sequence of spherical
shells that have individual values of T, P,
, and composition.
Stellar Astrophysics - BC 2008
Stellar Evolution
What have we learned?
Inside, energy is being transported
either by photons taking a random walk
(ideal gas theory, atomic physics) out-
ward, or, if is high enough, by large-
scale motion of atoms and plasma
(convection theory).
Increases in accompany PIZ’s (i.e.
temperature zones). We can regard the
stellar interior as a sequence of spherical
shells that have individual values of T, P,
, and composition.
Stellar Astrophysics - BC 2008
Stellar Evolution
What have we learned?
In the core, quantum tunneling allows
fusion of hydrogen nuclei to occur at
rate sufficiently high that the pressure
gradient can maintain hydrostatic equi-
librium with gravity.
Since the feasible fusion reactions all
have a strong temperature dependence,
the fusion zones have sharp edges.
Stellar Astrophysics - BC 2008
Stellar Evolution
What have we learned?
HOT
Atmosphere
(thin !)
At an optical depth of 2/3, we see the
opaque ball’s surface (photosphere)
as a good approximation to a black body
with temperature Te ;
the layer above the photosphere commu-
nicates information about T, Pe and com-
position to us, via the spectral signature
imposed on the BB flux coming form the
photosphere.
Stellar Astrophysics - BC 2008
Stellar Evolution
These “thin-shelled balls
of hot gas” are gene-
rated by the collapse
and fragmentation of a
Giant Molecular Cloud
(GMC) or a dense core.
What have we learned?
Stellar Astrophysics - BC 2008
Stellar Evolution
What have we learned?
The Jeans approach applied to GMC’s leads to the formation of
stellar clusters and the ZAMS
Stellar Astrophysics - BC 2008
Stellar Evolution
The following diagram, from a classic paper by Icko Iben, beautifully shows how
the physical parameters in a star “settle down” as the star “reaches” the ZAMS:
Icko Iben (jr), ApJ (1965) 141, 993
log Te
QRC
log R
log ( c / ave)
log L
Stellar Astrophysics - BC 2008
Stellar Evolution
We have now characterised what stars are, which processes
occur in them, how they form, and what physical characteristics
they are born with. We now turn our attention to their further
evolution:
Icko Iben (jr), ApJ (1965) 141, 993
Stellar Astrophysics - BC 2008
“The science of stellar evolution describes how the observable
properties of stars may sensibly change as time passes”
- Donald Clayton
Stellar Evolution
Which clues do we have to work with ?
Stellar Astrophysics - BC 2008
Stellar Evolution
years 10 6
85 <solar
M
Most stars of a given class more or
less maintain their luminosity
Assumption of hydrogen fusion as
the primary source of luminous
radiation in stellar interiors
years 10 14
07.0solar
M
Stellar Astrophysics - BC 2008
Stellar Evolution
What have we learned?
How can we explain the Post-ZAMS HRD and the observed
content of the galaxies ?
Stellar populations, HII regions, Globular clusters, Open clusters, Dust lanes
Stellar Astrophysics - BC 2008
Stellar Evolution
http://s
eds.o
rg
ww
w.a
str
onom
ynote
s.c
om
Population I and Population II stars
bluer,
more “metals”
in galactic disk
redder,
“metal-poor”
in galactic halo
and bulge
H II region
IPH
AS
Surv
ey, Jonath
an Irw
in,
Institu
te o
f A
str
onom
y, C
am
bridge
Open cluster (Pop I)
Globular cluster
(Pop II)H
ubble
Space T
el
Stellar Astrophysics - BC 2008
Stellar Evolution
http://o
utr
each.a
tnf.csiro.a
u
Every open cluster displays a distinct distribution of
stars on the HRD
Stellar Astrophysics - BC 2008
Stellar Evolution
ww
w.a
str
o.c
olu
mbia
.edu
“ZAMS fitting” allows us to situate individual clusters
in a common framework:
Stellar Astrophysics - BC 2008
Stellar Evolution
outr
each.a
tnf.csiro.a
u
ZAMS fitting produces a “snapshot of cluster ages”
Stellar Astrophysics - BC 2008
The dependent variables here are P, and Mr . In order to
have a solvable system of differential equations, we need one
more equation relating two of these variables, for instance P
as a function of :
This is called the Barotropic Case.
We have obtained mathematical descriptions of gradients and
conservation laws inside static stars. Two of them look like
this:
2r
MG
dr
dPr= )4(
2r
dr
dMr =
Stellar Evolution
)(PP =
Stellar Astrophysics - BC 2008
These three equations can be solved to provide the pressure,
density and mass distribution, as functions of radial distance
from the centre of the star.
2r
MG
dr
dPr= )4(
2r
dr
dMr =
Stellar Evolution
)(PP =
Stellar Astrophysics - BC 2008
If the pressure depends not only on density, but also on tempe-
rature: (as in an ideal gas), we have the
Non-barotropic Case.
Since there are now four dependent variables, we require
another differential equation, describing the temperature
gradient.
These three equations can be solved to provide the pressure,
density and mass distribution, as functions of radial distance
from the centre of the star.
2r
MG
dr
dPr= )4(
2r
dr
dMr =
Stellar Evolution
),( TPP =
Stellar Astrophysics - BC 2008
The choice of temperature gradient depends on the mode of
energy transport , which in turn depends partly on the mode of
nuclear energy generation. The inclusion of all these factors
greatly complicates the computation of stellar structure (i.e. P, T,
and Mr as functions of r).
However, we have two equations for the temperature gradient:
2r
MG
dr
dPr= )4(
2r
dr
dMr =),( TPP =
Stellar Evolution
2344
3
r
L
Tacdr
dTr=
2
H11|
r
GM
k
m
dr
dTr
ad
μ= ;
Stellar Astrophysics - BC 2008
We will solve the Barotropic Case for a few special conditions, as
an example of the general approach to fully determining the stellar
structure.
The equation that we will derive is called
the Lane-Emden equation,
and it looks like this:
n
n
nD
d
dD
d
d=2
2
1
Stellar Evolution
CONCEPT 30
Stellar Astrophysics - BC 2008
Derivation of the Lane-Emden equation:
Start with the equation for hydrostatic equilibrium:
2r
MG
dr
dPr=
)4(2
rdr
dMr =
Re-arrange, then take the
derivative on both sides: dr
dMG
dr
dPr
dr
dr=
2
Stellar Evolution
Then use the second of our
structure equations to simplify (1):
- (1)
Stellar Astrophysics - BC 2008
Derivation of the Lane-Emden equation:
Gdr
dPr
dr
d
r4
12
2=Which gives us
Stellar Evolution
KP =
We have eliminated Mr , and are left with P and . Another
equation relating these two physical variables will allow us to
proceed. We have seen that an ideal gas under adiabatic
conditions obeys the rule:
where = 5/3
A system that obeys this relationship in general, is called a
PolytropeCONCEPT 31
Stellar Astrophysics - BC 2008
Substituting for P , our differential equation becomes:
Gdr
Kdr
dr
d
r4
)(1 2
2=
n
n 1+
Stellar Evolution
Derivation of the Lane-Emden equation:
i.e.G
dr
dr
dr
d
r
K4
2
2
2=
We now define the polytropic index n as follows:
Stellar Astrophysics - BC 2008
Substituting for :
Gdr
dr
dr
d
r
K
n
n nn4
1 /)1(2
2=
+
[ ]nnc
rDr )()(
1)(0 rDn
Stellar Evolution
Derivation of the Lane-Emden equation:
Introducing the dimensionless parameter Dn through:
where
.
,
Stellar Astrophysics - BC 2008
our equation
=dr
dnr
dr
d
dr
dr
dr
dn
n
/121)/1(2
Gdr
dr
dr
d
r
K
n
n nn4
1 /)1(2
2=
+
Stellar Evolution
Derivation of the Lane-Emden equation:
is reduced to a simpler form:
Let’s first tackle the derivative itself:
=dr
dDnr
dr
dnn
c
/12
Stellar Astrophysics - BC 2008
Gdr
dr
dr
d
r
K
n
n nn4
1 /)1(2
2=
+
n
n
n
nn
cD
dr
dDr
dr
d
rG
Kn =+
2
2
/)1(1
4)1(
Stellar Evolution
Derivation of the Lane-Emden equation:
Substituting in, and collecting constants, our differential equation
is reduced to
Stellar Astrophysics - BC 2008
Gdr
dr
dr
d
r
K
n
n nn4
1 /)1(2
2=
+
n
n
n
nn
cD
dr
dDr
dr
d
rG
Kn =+
2
2
/)1(1
4)1(
Stellar Evolution
Derivation of the Lane-Emden equation:
Substituting in, and collecting constants, our differential equation
is reduced to
this is dimensionless
this is dimensionless
So, this is dimensionless too
Stellar Astrophysics - BC 2008
n
n
n
nn
cD
dr
dDr
dr
d
rG
Kn =+
2
2
/)1(1
4)1(
Stellar Evolution
Derivation of the Lane-Emden equation:
Let’s call this factor ( n)2 . Clearly, n is then a parameter of length.
If we now introduce a second dimensionless parameter by defining
this is dimensionless
this is dimensionless
2
2
2
1
r
n=
Stellar Astrophysics - BC 2008
n
n
n
nn
cD
dr
dDr
dr
d
rG
Kn =+
2
2
/)1(1
4)1(
Stellar Evolution
Derivation of the Lane-Emden equation:
Then
is reduced to
n
n
nD
d
dD
d
d=2
2
1
The Lane-Emden equation.
Stellar Astrophysics - BC 2008
Solution of the Lane-Emden equation:
This is a second-order ordinary differential equation. In order to
fully solve it, we need two boundary conditions (BC’s).
BC 1: Let when , i.e. when the density
drops to zero (in practice, when it drops to a sufficiently low
value)
Stellar Evolution
1= 0)( =
nD
BC 2 requires some talking first:
Stellar Astrophysics - BC 2008
Let represent an infinitesimally small radial distance from the
stellar centre. Then, the mass contained inside a sphere with this
radius is:
2r
MG
dr
dPr=
3
3
4=
rM
Stellar Evolution
Solution of the Lane-Emden equation:
If hydrostatic equilibrium holds in this small volume, we can write
0 as 03
4 2= G
dr
dP
, and
Stellar Astrophysics - BC 2008
Solution of the Lane-Emden equation:
0=d
ndD
0=
0dr
dP
Consequently, it is convenient to choose our second boundary
condition as
Stellar Evolution
when
Since we are dealing with polytropes, if
then as well.0dr
d
.
As a final measure, we normalise Dn so that c is the central density:
1)0( =n
D .
Stellar Astrophysics - BC 2008
Stellar Evolution
The Lane-Emden equation applied to polytropic spheres provides
a useful practice ground for the solution of the full set of stellar
structure equations for physically realistic stellar models.
Let’s play a bit...
Stellar Astrophysics - BC 2008
Stellar Evolution
The Lane-Emden equation applied to polytropic spheres provides
a useful practice ground for the solution of the full set of stellar
structure equations for physically realistic stellar models.
Let’s play a bit... Let n = 0:
Cd
dD
dd
dDd
Dd
dD
d
d
+=
=
==
3
3
102
202
0
002
2
,
,11
where our BC’s
make C = 0
Stellar Astrophysics - BC 2008
Stellar Evolution
So:
CD
d
dD
+=
=
20
0
6
1)(
,3
1
and the application of our BC’s and the normalisation we chose, deliver
Stellar Astrophysics - BC 2008
Stellar Evolution
61)(
2
0 =D
Two other choices of n that deliver simple solutions,
are n = 1 and n = 5:
[ ] 2/125
1
3/1)(
;sin
)(
+=
=
D
D
Stellar Astrophysics - BC 2008
Stellar Evolution
61)(
2
0 =D
These three solutions are illustrated in the diagram:
[ ] 2/125
1
3/1)(
;sin
)(
+=
=
D
D
Stellar Astrophysics - BC 2008
Stellar Evolution
The two values of n that are relevant to stellar structure are
n = 1.5 and n = 3.
The first of these corresponds to an adiabatic ideal gas.
The second corresponds to the Eddington standard model and
describes a star in radiative equilibrium:
say ,Pm
kTP
H
gμ
==
fraction of total pressure contributed by the gas
Stellar Astrophysics - BC 2008
Stellar Evolution
say ,Pm
kTP
H
gμ
==
PaTPr
)1(3
1 4==The contribution from radiation pressure is
3/4
4
,)1(3
1 μ
KP
Pk
mPa
H
=
=
Eliminating T from the equation for gas pressure, we find
(i.e. n = 3)
Stellar Astrophysics - BC 2008
As mentioned earlier, this is the full set of equations describing
stellar structure in the spherically symmetric approximation:
2r
MG
dr
dPr= )4(
2r
dr
dMr =
),( TPP =
Stellar Evolution
2344
3
r
L
Tacdr
dTr=
2
H11|
r
GM
k
m
dr
dTr
ad
μ= or
)4(2
rdr
dLr =; ;
and
There are five dependent variables appearing in these four
equations. The required fifth equation is the equation of state
(EOS), usually expressed as
Stellar Astrophysics - BC 2008
Stellar Evolution
As a star fuses hydrogen in its core, the value of the mean
molecular weight and, consequently, the values of opacity and
reaction rate, are forced to change on a continuous basis.
Therefore, the solution to the equations of stellar structure must
also change on a continuous basis.
Stellar evolution is mapped out theoretically by solving these
equations in discrete time steps, feeding the changes occurring in
each step into the equations for the following step.
The first (or at least best-known) detailed solution of this kind was
done by Icko Iben (jr) in a series of seminal papers in the 1960’s.
Stellar Astrophysics - BC 2008
Stellar Evolution
I. I
be
n (
jr),
An
n.R
ev.
Astr
on
. A
str
op
hys.
(19
67
) 5
71
Stellar Astrophysics - BC 2008
Stellar Evolution
ww
w.a
str
o.c
olu
mbia
.edu
“ZAMS fitting” allows us to situate individual clusters
in a common framework: