Download Nuclear Astrophysics

Nuclear Astrophysics
Lecture 6
Thurs. Nov. 22, 2010
Prof. Shawn Bishop, Office 2013,
Ex. 12437
[email protected]
1
The 4 Equations of Stellar Structure
Energy generation rate per unit
mass of material
average opacity coefficient
in the material
2
Return to the Standard (Stellar) Model
The stellar gas is a mixture of photons and Ideal particles. Thus,
Total Pressure:
In thermodynamic equilibrium, these two gases have the same temperature. And let
and
Then, we have:
Polytrope of type 3
3
Polytrope Solutions
n=5
n=0
n=1
n=2
n=3
n=4
4
Mass Luminosity Relation
Take the following equations from the 4 Structure Equations:
And use our friend:
Sub this into first equation
above
Small for all but the most massive of stars.
5
Eddington’s Quartic Equation
In 3rd Lecture, pages 13 & 20, it was shown that the stellar mass can be written as,
Product of the root of f
and its 1st derivate at the
root.
For all, but the most massive stars,
is a
small quantity. So, rearrange the above equation to
isolate
and sub result into the previous
equation for to get L(R*)
Parametric plot of
versus
6
After the algebra (you should check, to make sure I’m right  ), we arrive at the Mass
Luminosity Relation for Main Sequence stars!
Numerically:
We now have our first theoretical prediction of the relationship between two observable
properties of stars.
The Luminosity of Main Sequence Stars (H-burning, hydrostatic, up
to ~15 solar masses) should be proportional to the 3rd power of the
stellar mass. (first order result, we can do better).
To do better, we have to deal with that annoying opacity,
.
7
Total Opacity of Solar Composition
Material
Kramer’s Opacity: Varies (crudely) as
and almost as
Empirical relation given as:
This factor of 2 is required to get agreement for
curves with
Thomson’s Opacity: Constant, and has a value
of
The exponent in Kramer’s Opacity is also:
We need to simplify the
Kramer’s opacity so that it is
“averaged” over all of the stars
we are considering in the MassLuminosity relationship.
8
The first step: get a formula that expresses temperature in terms of density. This will give
us a Kramer’s formula that is now only a function of density.
Recall, from page 3, we found (and this is also in Lecture 2/3 pages 36, 37) that:
Get his from squaring and rearranging the
stellar mass formula on page 6.
You have all the ingredients here to now relate temperature to density. The final
result, after doing the algebra:
9
Use this last result in the Kramer’s Opacity formula by replacing :
Now, no two stars are alike, so we have to start doing some reasonable averages of T and .
First thing is to average over the stellar temperature. And remember, the temperature of
the ideal gas and photon gas are the same. However, the photon gas pressure depends only
on the temperature (not on density), so there seems like a good place to start.
Try a volume average of the
radiation temperature
10
From the polytrope formalism, the solution to the Lane-Emden
equation gives the
run of density as a function of radial coordinate, r. For n=3 polytrope, we had (page 29
of Lec. 2/3)
. On previous slide, we had
.
The above integrals can be done numerically (Mathematica), using numerical
Result is:
.
We will need Tc
We now have an “average” temperature (weighted over volume) in terms of central
temperature. Next, we need the density that corresponds with this Tav
Two slides ago (slide 9) we had the following result:
Need to eliminate this
11
And from the Polytrope formalism in Lec. 2/3, we found the following result:
Where,
From the table on page 4.
Finally, for the Sun (a Main Sequence Star),
What have we got now:
and
And we need to complete:
We still need the central temperature, and then we are DONE!
12
The central temperature: From page 9 we can write:
From the last slide, we had:
And:
For a Solar-type star:
and
Assuming
fully ionized
Collecting all the numbers, we finally have:
13
Finally, Kramer’s Opacity becomes simplified to:
Total Opacity:
And, for fully ionized material:
where
And X = 0.71 for Solar.
The total Opacity is now:
And we had for Luminosity:
14
Calling
we finally have the function for Luminosity:
Or, using
With
, as before, given by its Solar value:
Assuming
fully ionized
The function above is parametric in . We work the function by choosing a value for
m, and then solving Eddington’s Quartic equation for , to evaluate the RHS.
What does it look like when plotted against REAL Main
Sequence data??
15
Mass-Luminosity: Main Sequence
Data are from: G. Torres et al., Astron.
Astrophys. Rev. (2009)
16
Thermonuclear Reaction Rate in Stars
THE ROAD TO NUCLEAR REACTION
RATES
17
Some basic kinematics: We have two particles with masses
and
and
with velocities
The velocity of their common centre of mass is:
The velocity of particle 1 relative to the CoM velocity is just:
And v is just the relative velocity between 1 and 2.
Similarly, particle 2 has a velocity relative to CoM velocity:
18
Before the collision, the total incident kinetic energy is:
Using the previous two vector equations, we can substitute in for v1 and v2 in terms of v
and V. (An exercise for you)
The first term is the kinetic energy of the center of mass itself; while the second term is
the kinetic energy of the reduced mass as it moves in the center of mass frame.
19
Nuclear reaction rate: The reaction rate is proportional to the number density of particle
species 1, the flux of particle species 2 that collide with 1, and the reaction cross section.
Flux of N2 as seen by N1 :
Flux of N1 as seen by N2 :
Reaction cross section:
This v is the relative velocity between the two
colliding particles.
Important: this reaction rate formula only holds when the flux of particles has a
mono-energetic velocity distribution of just
20
Inside a star, the particles clearly do not move with a mono-energetic velocity
distribution. Instead, they have their own velocity distributions.
We must generalize the previous rate formula for the stellar environment. From
Lecture 2,3 the particles 1 and 2 will have velocity distributions given by MaxwellBoltzmann distributions. We have the 6-D integral:
The fraction of particles 1 with velocities between
is therefore,
And similarly for particle species 2.
Let’s take a closer look at:
21
From equations on page 4, we can write the argument in [...] in terms of the center of
mass velocity
and relative velocity .
So in terms of the CoM parameters,
The reaction rate now becomes (6-D integral):
And we note:
and
22
We now need to change the differential variables into the new CoM variables.
From page 3, in component form, we have:
Jacobian:
And this is the same for the case of y and z components.
23
The rate integral now becomes:
24
Note: the product N1N2 is the number of unique particle pairs (per unit volume). If it
should happen that 1 and 2 are the same species, then we must make a small
correction to the rate formula to avoid double-counting of particle pairs.
Kronecker delta
25
We can extend the previous result to the case when one of the particles in the entrance
channel is a photon. So reaction is:
The rate:
As before, we generalize this by integrating over the number density distributions: A
Maxwell-Boltzmann for species 1, and for photons we recall from Lecture 2,3 page 18 the
following:
Number of photons per unit volume between
and
:
26
The Einstein postulate of Special Relativity: speed of light is the same in all reference
frames. Therefore, the relative velocity
= N1 because N1 is M-B
And
is the photo-disintegration cross section.
27
Reaction Rate Summary
Reaction rate for charged particles:
Reaction rate for photodisintegration (photon in entrance channel):
28