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Nuclear Astrophysics
Lecture 9 &10
Thurs. Dec. 16, 2010
Prof. Shawn Bishop, Office 2013,
Ex. 12437
[email protected]
1
Alternative Rate Formula(s)
We had from last
lecture:
With:
And:
And (pg 16,
Lecture 6):
And we have approximated the S-factor, at the astrophysical energies, as a simple
constant function
(units of keV barn).
2
We have (again)
Set the argument of the exponential to be
From the definitions of
that:
(dimensionless)
on the previous slide, you can work out
3
A Power Law expression for the Rate
First, an algebraic step of substituting
We substitute this, along with Eeff into
into the equation for
above. After some algebra:
4
Let’s pull out the physics of this complicated formula by trying to write it as something
simpler, like a power law function.
First, equate the last expression for the rate to something like a power law:
Take the natural logarithm of both sides:
Differentiate both sides wrt
:
5
Numerically,
Summarizing then:
Choose a value of temperature
to evaluate the rate at:
Once we have this value for the rate at some temperature you have chosen, to get
the rate at any other temperature, just make the trivial calculation:
6
Finally, recall
7
Summary of Reaction Rate
8
Examples:
The core of our Sun has a central temperature of about 15 million degrees kelvin.
Reaction
Z1
Z2
mu
T6
Eeff
n
p+p
1
1
0.5
15
5.88
3.89
p + 14N
1
7
0.933
15
26.53
19.90
4He
+ 12C
2
6
3
15
56.09
42.81
16O
+ 16O
8
8
8
15
237.44
183.40
Reaction rate becomes more sensitive to temperature as we “burn” heavier
nuclei together.
Such a large sensitivity to the temperature suggests structural changes in star
must occur at some point for certain reactions.
9
Full Reaction Rates
Solar abundances, density = 100 g cm-3:
Rate
p + p reaction
14N + p
12C + a
10
Reaction Rate: Temperature
Dependence
p + p reaction
14N + p
12C + a
11
Energy Generation & Standard Model
Let us consider a representation for the nuclear rate of energy production (in units of
energy per unit time, per unit mass) to be given by the following mathematical form:
Then we can, quite generally, also write:
For a Polytrope model, once the solution to Lane-Emden equation is known, then
we have the following results (for n = 3 Polytrope):
12
Using the equations on previous slide, we can now write the scaled nuclear energy
generation rate in the very compact and simple form:
The luminosity of the star (after settling into equilibrium) comes from the nuclear
furnace in the core. Let’s try to write the luminosity in the following form:
Where we use the mass-averaged
energy generation rate:
We use an averaged energy rate because it is clear that not all mass in the star is
contributing to the nuclear energy rate: only the core of the star is contributing to the
energy generation rate. So, we average over the mass of the star in the hope that the
averaged rate times total mass (equation *) gives a reasonable result.
13
We must now consider the integral:
Our differential mass element is, in the scaled radial coordinate
(refer to Lecture 2):
And, from previous page,
Collecting everything into
the integral:
On pages 29, 30 of Lecture 2 we had:
14
Finally, then, we have that the mass-averaged nuclear energy rate is given by:
15
Lane-Emden Function for n = 3
The integral we must consider is
We are raising the Lane-Emden function to the power of (3u+s).
If this number is large, then we are multiplying by itself many
times. The function is always smaller than one: take multiple
products of it will cause it to become more sharply peaked, and
its tail to grow smaller in magnitude.
We can exploit this feature to make a suitable
approximation of the function in the integrand
above.
16
The power series expansion of
, for the first few terms looks like this:
Let’s use the exponential approximation in the integral for
Mass-averaged
nuclear energy rate:
17
A Polytrope Sun
We had the general energy generation rate expressed as:
For the proton-proton chain in the Sun,
Therefore, we have, in terms of that:
and
(at
Integrating this over the star introduces another
from
integrand of the luminosity function then is something like:
).
. The
Exact e
Gaussian Approx.
18
Structure of Polytrope Sun
Let us define the core to be that volume in which 95% of the nuclear energy is generated.
We find that this occurs around
variable is
.
(next page). Recall the stellar radius in the
So the core occupies about 23% of the total stellar radius.
So the fractional volume occupied by the core is
Mass occupied by the core: From plot on page 16, it is about 33% of stellar mass.
Question for you:
This sun is 1 solar mass, and 1 solar radius in size. Take
Use these numbers and the result on page 11 to determine the mass of hydrogen per
second this Sun burns. Also use: u =2, and s = 4.6
19
Integrated Luminosity
This is 95% at
; this radius corresponds to
1.6/6.89 = 23% of the stellar radius.
Let us take this as being an (arbitrary) criterion for
defining the core.
As a fraction of the star’s volume, the core is only
20
Derivative of Luminosity
21
Mass Interior to Radius
Mass contained in the core
about 36%.
is
So, a star the size of our Sun, in purely
radiative equilibrium (no convection) has
36% of its mass contained in only 1.2% of its
volume!
22
Arbitrary Units
Standard Solar Model: Radial Run of Temperature, Density and Nuclear Energy Rate
Temperature
Density
Pressure
Epsilon
Remember:
23
HERTZSPRUNG-RUSSEL DIAGRAM &
NUCLEAR BURNING
24
Proton-Proton-Chain
p + p  d + e+ + n
p + d  3He + g
86%
3He
14%
+ 3He  4He + 2p
3He
99.7%
PP-I
Qeff= 26.20 MeV
+ 4He  7Be + g
+ e-  7Li + n
7Li + p  2 4He
7Be
0.3%
+ p  8B + g
8B  8Be + e+ + n
7Be
PP-II
2 4He
Qeff= 25.66 MeV
PP-III
Qeff= 19.17 MeV
Netto:
4p  4He + 2e+ + 2n + Qeff
25
CNO Nuclear Cycles
CNO burning cycles for hydrogen
under non-explosive conditions.
Each cycle converts 4 protons into
an alpha particle.
It is these cycles that nuclear
burning enters once protonproton and 4He burning (by the
triple-alpha process) have built up
enough 12C abundance.
Remember: These form cycles
because, at some point along the
path, we encounter a nucleus “Y”
that has a binding energy for alpha
particles that is lower than the
minimum excitation energy of “Y”
when it is made by proton
capture. Example in CNO3:
18O(p,g)19F*  a + 15N
26
The nuclear energy production in Main Sequence stars comes either from the protonproton chain or from the CNO cycle. (C = Carbon, N = Nitrogen, O = Oxygen).
PP-Chain Parameters
CNO-Cycle Parameters
27
From Lecture 2, we derived an expression for the central temperature of a Polytrope star
as,
Central density:
In these formulae, the temperature is in kelvin, and density is grams/cm3.
Numerically, we find that:
28
PP-Chain Nuclear Energy Rate vs Core Temperature
From page 17, we had:
Dropping the “pp” subscripts, and considering this formula at the centre of the star:
(Remembering
Substitute in the previous result for
)
:
29
CNO-Cycle Luminosity vs Core Temperature
From page 17, we had:
Dropping the “CNO” subscripts, and considering this formula at the centre of the star:
Substitute in the previous result for
:
Last lecture, we derived, for Polytrope (n = 3) the result:
Refer to Page 2 to understand that
.
30
The power per unit mass produced in the stellar core is, after all, what is responsible
for the luminosity at the stellar surface. We now try to come up with a generalized
formula for the nuclear energy generation rate for any Main Sequence star.
We do this by performing a mass-averaged energy generation rate. And this massaveraged rate times the stellar mass should, if the model is anywhere close to reality,
give us a quantitative trend that is similar to the observational data.
We now have all the ingredients we need to determine the relationship between surface
temperature (effective temperature) and the stellar luminosity. Start substituting the
results from previous slides. I will do the case for the pp-chain
Multiply and divide by unity:
so that numerator will have
31
Continuing:
Divide both sides by Solar Luminosity,
and
. Then, we have:
, and use
and
pp-chain burning
This result is the relationship between the luminosity of the star in terms of its core
temperature, at the centre of the star, for proton-proton nuclear burning.
Similar steps will lead you to a corresponding formula for luminosity in terms of core
temperature for stars with the nuclear energy generation rate dominated by CNO-Cycle
burning. (And you should do this: you have all the formula from page 27 and 28 to do it).
32
From page 15, Lecture 6, the Main Sequence Mass-Luminosity relationship was derived.
We found that:
And we have just found that the nuclear burning for pp-chains and CNO cycle give:
Solar abundances give, (refer also to page 5):
,
,
. Note: The Eddington Quartic Equation (page 37, Lect. 2/3) lets us solve for
for a chosen stellar mass .
Once we have
temperature
and
we can equate ( ** ) and ( *** ) with ( * ) and get the stellar core
of Main Sequence stars in terms of the stellar mass.
33
Mass-Luminosity: Main Sequence
We use this curve for the left
hand sides of the equations
on the previous page.
Data are from: G. Torres et al., Astron.
Astrophys. Rev. (2009)
34
Main Sequence: Core Temperature vs Mass (for fixed
Luminosity)
PP Chain
CNO Cycle
At
, we find that the pp-chain
and CNO-Cycle give equal core temperature at the same luminosity.
We see that, for any stellar masses
, the CNO
cycle does not require as high a core temperature to produce the
same luminosity as does the pp-chain. Less than this mass, the ppchain can burn at lower temperatures than CNO to produce the
same luminosity.
35
A Consistency Check on the Crossover Temperature
At the crossover temperature, the energy generation rates of both burning cycles must be
equal. So, set
For you to show this result for
from the equations on page 5 and 6
Many terms cancel on both sides, and we require the temperature to be the same. After
cancelling density, mass,
and so on, and solving for the common temperature:
36
Luminosity vs Effective Temperature: Hertzsprung-Russel Diagram
The previous plot tells us that: For a particular value of luminosity, the pp-chain requires a
higher core temperature than does the CNO cycle for stellar masses
.
We cannot, of course, measure the core temperature of stars, so the previous result
would otherwise be only purely theoretical.
However, astronomers do measure the effective surface temperatures of stars. Our Main
Sequence model should be able to at least quantitatively show the trend of stellar surface
temperatures to confirm the previous theoretical result of how the two reaction cycles
depend on core temperature and stellar mass.
In other words: can the stellar surface temperatures be used to definitively determine
where in the Main Sequence stars the pp-chain dominates, and where the CNO-cycle
dominates?
Let’s work toward the answer to this. Let the data tell us how the nuclear reactions in the
core are connected to the temperature and luminosity we observe from the stars.
37
Back on page 28, we had:
Now, the surface temperature of the star is related to its luminosity by the Black Body
relationship,
with Stefan-Boltzmann constant:
This is how we will get the effective temperature related to the nuclear luminosity.
We need to relate the core temperature above to the stellar radius. We can do this with
the Polytrope formalism because
and
Using:
We use this now in the equation up top to express the core temperature in terms of the
stellar radius (and mass). Using
, gives the simple result:
Coupled to each other by
structure constraints
38
We have a first “hint” here of things to come from the last equation:
(Core temp related to radius)
As stellar mass grows, we expect the core temperature to increase, which would seem to
suggest that the pp-burning will dominate over CNO burning. However, as the core
temperature increases, the star’s structure will adopt a new configuration; namely, the
radius will grow (expansion) and it will be the competition of the mass versus the extent of
expansion (growing r) that will regulate the core temperature and, therefore, the type of
nuclear burning happening within the core.
Let us continue. At the stellar surface:
Sub into equation up top:
39
Almost done. On page 32, we derived, for the pp-chain, the following result:
Sub in the last expression for
, and after some algebra, we finally have:
So, we choose a value for m, use EQE to get and use the mass-luminosity function of
Lecture 5 on the LHS. Finally, solve for
as a function of stellar luminosity. Result on
next slide, with Main Sequence data.
40
Main Sequence: Luminosity vs Effective Surface
Temperature
Data tells us the truth: The pp-chain cannot be
the dominant burning source for high mass
stars.
For stars with
, the pp-chain
dominates. Otherwise, it is the CNO cycle!
CNO Cycle
PP Chain
Sun
Data from: G. Torres et al., Astron. Astrophys.
Rev. (2009)
41
Summary: Main Sequence Stars
•
•
•
•
We have learned the mass-luminosity relationship based on the physics of the
stellar structure equations + a “simple” Polytrope model
Using the mass-luminosity relationship (data), and nuclear reaction results (to be
taught in the New Year), we have extended the above result to learn the behaviour
of stellar core temperatures as a function of stellar mass (page 12)
For high mass stars, the pp-chain requires a higher core temp. than the CNO cycle
to produce the same luminosity. (page 12)
Our Polytrope Main Sequence model, while not perfect, agrees with the trend of
the observational Luminosity-Effective Temperature data. The data and model
show that, for those stars with
the luminosity is the result of the CNO
nuclear reaction cycle; pp-chain nuclear reaction cycle is primary stellar energy
source for masses less than
.
42
Hertzprung-Russel Diagram
CNO Cycle
Understanding how stars
become Giants, and the
nuclear processes within,
will be our next step in
the New Year.
PP-Chain
43
Polytrope model, for polytrope index n = 3
APPENDIX: SCALED MAIN
SEQUENCE STRUCTURE PLOTS
44
Integrated Luminosity
Let us take 95% of luminosity as being the (arbitrary)
criterion for defining the core. Then core has a
radius
As a fraction of the star’s volume, the core is only
45
Derivative of Luminosity
46
Mass Interior to Radius
Mass contained in the core
is about 36%.
So, a star the size of our Sun, in purely
radiative equilibrium (no convection) has
36% of its mass contained in only 1.3% of its
volume!
47
Arbitrary Units
Standard Solar Model: Radial Run of Temperature, Density and Nuclear Energy Rate
Temperature
Density
SolarPressure
Model Calculation
Epsilon
48
Nuclear Generation Rate
Using the pp-chain , which is a power law in temperature, with
exponent n = 4.6, we end up with the Polytropic energy generation rate
expressed on in the integral on the LHS below, where is the LaneEmden function of index = 3.
The L-E function, raised to such a high power, can be approximated to
an excellent degree by a Gaussian (RHS), which can then be analytically
integrated.
49