Minneapolis, February 6, 2012
Dear Students,
unfortunately I had some emergency surgery last Friday when I was on
travel. I am currently at home and will not be able to hold the
office hours today. I start having some fever, so there is a chance I
have to get back into the ER if this gets worse, in which case I may
not hold the class tomorrow.
As for the HW, the program that does the integration should not be
long, so if you can print it out, that would be nice (I trust it runs)
- but if not, and I would like me to help, an electronic version may
be useful (...). FORTRAN90 would be best - for you to use - as MESA
is also written in that (I, personally, am more an expert in F77).
Python is also a nice tool; my version of the LE integrator is F90
code called by Python using f2py and transferring data in common
blocks, which runs about 150x faster than plain python/numpy.
If you can determine the radius, this is the first step.
If you can get from that P, T, rho as a function of r and plot this
(assuming ideal gas - pure fully ionized 1H - with mean molecular
weight 0.5) that would be nice.
Sightly more challenging might be to obtain the quantities as a
function of mass coordinate - I think we have not discussed this as
part of the solution on Thursday.
Finally - and I do not know whether this works easily, I have never
tried, so this is left for the ambitious - if you can compute the
effective T and from that luminosity of the star. Here we would
assume just electron (Thompson) scattering and find what T is at an
optical depth of tau = 2/3, as a crude approximation. You would
likely use a plane-parallel approximation for the "atmosphere".
as a reminder, the other parameters were
n = 3
central density 0.1 g/cm**3
mass = 1 M_sun
So, yes, if you can submit
1) notes on the reformulation to 1st order ODE
this may include derivation of the 2nd derivative of
theta a 0 (which my code seems to need,
but I may be incompetent)
2) derivation for the other quantities
3) a program to integrate the LE
4) plots as mentioned above
5) optionally the T/L as well, if this can be done.
There is a chance I many not be in this week, so there is no point to
make you work hard till Thursday if I then don't pick up the work,
hence I am fine to collect the work next Tuesday (Feb 14).
Just to clarify, and since one student has asked, yes, the sun does
have a much higher central density, and its polytropic index is > 3 on
average. What we look at would be a model of the sun as it is still
forming and fully convective, before it contracts to star in thermal
equilibrium and starts burning in the center.
In case I cannot come in for class, I would like to ask you to review
the remainder of the chapter on the Lane Emden Equation in KW, then
the derivation of the Chandraskhar mass based on that. The next item
I had planned was to have a closer look at burning stages, starting
with the overview in KW, the pp chains and the CNO cycle.
There is a very general formulation on nuclear burning processes for
reaction networks that I would like to go over as well, unfortunately
the only place to find that (to my knowledge) is in my thesis. I have
programmed lots on nuclear networks, and I think this is useful - you
have to remember only one formula for all cases - the only
disadvantage is that I do not know a good mathematical formalism to
write it, so it looks long and ugly, but after looking at it many
times over many years it seems finally clear (it's not that bad). As
a parallel thread to studying the relevant process we will look at an
example how to solve them. Maybe Charles can show a simple case for
an implicit solver that he has just implemented in case I cannot come.
Best wishes,
Alexander