Course:ASTR508/Assignments

Assignment 1 [Due 21 September]

Part 1

Using the estimate of the central temperature of the Sun that I derived in class and the equation for radiative energy transport, estimate the luminosity of the Sun if the opacity is given by electron scattering 0.4 cm²g^-1 or by free-free opacity using the mean density and the central temperature of the Sun. Here is the formula for the absorption coefficient in c.g.s. units and its relationship to the free-free opacity $\kappa _{ff}$ :

$\alpha _{ff}=\kappa _{ff}\rho =1.7\times 10^{-25}T^{-7/2}n_{e}n_{i}g_{R}$

Take $Z=g_{R}=1$ . $n_{e}$ and $n_{i}$ are the number densities of electrons and ions respectively. $Z$ is the atomic number and $g_{R}$ is the Rosseland mean Gaunt factor.

Part 2

Complete the MESA exercises at the end of chapter one of Stellar Astrophysics .

The easiest way to get the code for the exercises is to download the repository on GitHub at [1].

Also please download [2] to help you get started with MESA>

Assignment 2 [Due 2 October]

Part 1

Using make_zams a starting point, generate a CMD in MV vs B-V for a solar metallicity ZAMS. If you assume that stars stay the same colour and luminosity on the MS (not strictly true), what does the CMD look like at an age of 10 Myr, 100 Myr, 1 Gyr and 10 Gyr? Using your solar metallcity ZAMS at 100 Myr figure out the distance to the Pleiades and at 10 Gyr for 47 Tuc. You will have to edit the specification file. Use the value of log(tnuc) to estimate which stars are alive after a given time. That is, for 1 Gyr only plot stars with log(tnuc)>9.

You will find the table below useful to convert luminosity and effective temperature to colour.

# SpTyp  MV   BC   MBol   B-V   L/Lsun    Teff  Mass
#  1     2    3     4      5      6        7      8 
  M6  16.5   4.3 12.20  1.7  1.0568e-03   2600  0.17
  M4  12.7   2.7 10.00  1.6  8.0168e-03   3200  0.25
  M2  11.2   1.7  9.50  1.5  1.2706e-02   3400  0.3
  M0   8.9   1.2  7.70  1.41 6.6681e-02   3800  0.35
  K7   8.3   1.0  7.30  1.32 9.6383e-02   4000  0.47
  K5   7.5   0.6  6.90  1.08 1.3932e-01   4350  0.69
  K0   5.9   0.4  5.50  0.84 5.0582e-01   5250  0.78
  G5  4.93   0.2  4.73  0.69 1.0280e+00   5700  0.93
 Sun  4.83  0.07  4.76  0.65 1.0000e+00   5780  1
  G0   4.2   0.2  4.00  0.59 2.0137e+00   6000  1.10
  F5   3.3   0.1  3.20  0.41 4.2073e+00   6500  1.3
  F0   2.4   0.1  2.30  0.32 9.6383e+00   7300  1.7
  A5   1.8   0.1  1.70  0.19 1.6749e+01   7800  2.1
  A0   0.8   0.3  0.50  0.0  5.0582e+01   9400  3.2
  B8 -0.25   0.8 -1.05 -0.11 2.1086e+02  11600  4
  B6  -1.0   1.2 -2.20 -0.14 6.0814e+02  14000  5
  B3  -1.4   1.6 -3.00 -0.18 1.2706e+03  18750  11
  B0  -3.7   3.0 -6.70 -0.30 3.8371e+04  30500  18
  O8  -4.3   3.3 -7.60 -0.321 8.7902e+04  35000  20
  O5  -5.2   3.8 -9.00 -0.322 3.1915e+05  41000  35
  O3  -5.8   4.0 -9.80 -0.323 6.6681e+05  44500  40

To log out from your computer but still have your job run, you will need to disown the job. Here is how. Type "jobs" at the prompt. Find the job in question, and type "disown %1" if it is job number 1. If you type "jobs" again you should find that the job is no longer on the list. If you run "top", you will see the job on the top of the list and you can kill it if you want by typing in the job number.

We have developed a more sophisticated tool to convert the MESA output files to the observable fluxes. It is called paintisochrone.py You can download it at

gcwd.tar.gz

with instructions and lots of ancillary files so that you can try it out.

Part 2

Calculate the function $\theta (\xi )$ for various values of $n$ between 0 and 5. Some important ones in addition to the ones here are $n=3$ and $n=3/2$ .

Assignment 3 [Due 6 October]

Part 1

Let's build upon the solar models that you performed in Week 2. Run the a one-solar-mass model using 1M_pre_ms_to_wd. Idenify the various regions on the H-R diagram for the main sequence, giant branch, horizontal branch and asymptotic giant branch and plot the profiles of density, temperature, energy generation as a function of radius for each stage. Also plot temperature against density on a log-log plot to identify the convective and radiative regions.

Repeat the analysis for a 0.3 solar-mass star and a 3 solar-mass star.

Part 2

Exercise 10.2 from Stellar Astrophysics Notes

Assignment 4 [Due 13 October]

Exercise 11.1 from Stellar Astrophysics Notes

Exercise 11.3 from Stellar Astrophysics Notes

Assignment 5 [Due 20 October]

Following the derivation in class, calculate the pressure at the surface of an isothermal core as a function of the radius of the core including the effects of electron degeneracy. Use units of the envelope pressure times $M_{c}/M$ . That is on the vertical axis you will have

${\frac {M_{c}}{M}}{\frac {P_{c}}{P_{e}}}$

where

$P_{e}={\frac {81}{4\pi }}{\frac {1}{G^{3}M^{2}}}\left({\frac {kT_{c}}{\mu _{e}m_{H}}}\right)^{4}$

and on the horizontal axis you will have $R_{c}$ . Try several values of $M$ .

Hints: First choose $M_{c}$ , $T_{c}$ (say $5\times 10^{7}\mathrm {K}$ ) and plot as a function of $R_{c}$ . What is $P_{c}/P_{e}$ in equilibrium? How can you figure out $M$ ? How can you fix $M$ ?

Assignment 6 [Due 27 October]

You will use the inlist called astero_adipls for this assignment. This model evolves a 1.2 solar mass star from the ZAMS until the hydrogen is nearly exhausted in the centre. I would like you to plot the sound speed (profile column csound) and Brunt frequency (brunt_N2) as a function of radius for the initial stage (ZAMS) , halfway through the evolution and at the final stage. Identify the convectively unstable regions on the plots. Plot the Lamb frequency for l=1,2,3. Identify the turning points of p-mode with ω=3000μHz for different values of angular momentum. What about 10,000μHz? Similarly what are the turning points of a 2000μHz g-mode?

Plot the structure of the first several modes (l=0, n=0 to 6) for the final model. The results are in the file ttt.adipls.prt.

Assignment 7 [Due 3 November]

You will create a 7 solar-mass star from the 7M_prems_to_AGB test suite

First, recall the various regions on the H-R diagram that you identified in the assignment for the solar mass star. Plot the solar mass star and the 7 solar mass star on the same H-R diagram. How does the behaviour of the two models differ?

Plot the profiles of density, temperature, energy generation (dividing into the various reactions using the burn columns of the profile files) as a function of enclosed mass for the main sequence of the 7 solar-mass star and pick two later stages that you find interesting. Make a plot showing the composition of the different regions with the enclosed mass as the x-coordinate.

Plot temperature against density on a log-log plot to identify the convective and radiative regions. Plot neutrino luminosity as a function of time and compare with the one-solar-mass model (from the history file).

Assignment 8 [Due 10 November]

For your assignment I would like you to focus on the following two inlists:

wd_c_core_ignition (only about 3 minutes to run),
split_burn_big_net_30M (only about 8 minutes to run).

For both runs I want you to add the dynamic timescale and the nuclear timescale to the history file. For the wd_c_core_ignition run only, I want you to add the various non-nuclear neutrino processes to the profile file.

Now how do you do this? In each of the directories you will have to edit files called history_columns.list and profile_columns.list. The wd_c_core_ignition by default only gives a few profiles. We want more so change profile_interval to 10 from 50 in the inlist file. You will have to make a few changes to split_burn_big_net_30M to get it to run to collapse. Hint: look at the stopping conditions.

For all of the plots I would like for you to use the enclosed mass as the abscissa.

For the the wd_c_core_ignition run,

I would like for you to plot the neutrino losses through the star for the final profile for each of the processes.
I would also like you to plot the various nuclear reaction rates for both the final profile and the second to last profile.
Which nuclear reaction do you think is driving the instability? Are you surprised?

For the split_burn_big_net_30M I would like you to plot the various nuclear reaction rates, the composition and the specific entropy of the material as a function of the enclosed mass for the initial profile, one in the middle and the final profile.

How much time has elapsed between the beginning of silicon burning and the collapse of the core? To get the core to collapse, you will probably have to change the inlist. Right now, it will stop after only a few models.
Do you notice a mass at which the specific entropy abruptly rises near the centre? What is the significance of this mass?
Can you infer from the specific entropy curve what is happening in the different regions? Hint: compare it with the nuclear burning profiles.

Assignment 9 [Due 17 November]

List and identify with a phrase or two 5 effects that modify the so-called Chandrasekhar Mass from the oft quoted value of 1.45 solar masses. Which effects cause increases and which cause decreases and which one or two are most important within the context of iron cores in massive stars? Which are important for accreting white dwarfs?
What two instabilities lead to the dynamic implosion of the iron core?
What two effects lead to the failure of the "prompt shock" in Type II supernovae?
Consider a sphere of initial radius $5\times 10^{8}$ cm and mass 1.4 solar masses, initially heated 10¹⁰K Assuming constant density, total ionization, Y_e = 0.5, homologous expansion ( $v\propto r$ ), and opacity due to electron scattering, calculate the radius the expanding sphere would have when it first became optically thin. (aside: the gas does not really remain totally ionized but the Doppler broadened forest of iron lines in a SN I provides an opacity comparable to electron scattering, use κ_e = 0.2 cm²g^-1 throughout) If the expansion were adiabatic as well as homologous what would be the temperature of the radiation at this point (assume that radiation entropy is separately conserved). Obviously this is too cold because you have left out the effect of radioactive decay.
For this same sphere, estimate the radius and time when the diffusion time for the radiation would be equal to the expansion time scale for the density. (hint: first show that for homologous expansion, $v\propto r$ , that $\rho /{\dot {\rho }}$ is just the elapsed time divided by 3) for an assumed surface expansion velocity of 10⁹ cm s^-1. If the supernova radiates 10⁴³erg s^-1 as a blackbody (not true!) what would be the effective temperature of the photosphere (assume for example that the photosphere lies at 1/2 the just computed for the edge of the sphere). Compare your answers to the characteristics of Type I supernovae.
Distinguish the terms "deflagration" and "detonation" in the context of the exploding white dwarf model for a Type I supernova. Which is the currently favoured mechanism for propagating the runaway? Why? What sets the peak value of temperature that is experienced behind the flame front? Describe what happens to each of the following (increase or decrease) as one crosses a a) detonation or b) deflagration front: density, pressure.

Assignment 10 [Due 24 November]

For elements with atomic numbers less than that of calcium, the most abundant isotope of each element with an even number of protons has Z=N, e.g. ⁴He, ¹²C, ¹⁶O, ⁴⁰Ca, ... and those with odd proton numbers, N, Na, Al ... have Z nearly equal to N. When one goes to heavier nuclei however there is a surplus of neutrons in the most abundant isotopes; iron-56 has 26 protons and 30 neutrons. Explain both these trends why light nuclei have Z about equal to N while heavy nuclei have Z less than N.
Why is combination of a single neutron and a single proton stable but two protons is not?
Calculate the energy released in erg/g when a composition of pure helium burns to pure carbon-12 and to 50/50 carbon-12 and oxygen 16. What is the energy released when each of these mixtures is burned to Nickel-56? In both cases how much nickel has to be made to produce 10⁵¹ erg?
The neutron capture cross sections at 30 keV for the stable isotopes of barium are ¹³⁰Ba, 715 mb ¹³²Ba, 447mb, ¹³⁴Ba, 221 mb ¹³⁵Ba, 457 mb, ¹³⁶Ba, 69 mb, ¹³⁷Ba, 57 mb, and ¹³⁸Ba, 3.9 mb. The s-only isotopes of barium are 134 and 136 and the nuclear charge is 56. a) Why is the cross section of ¹³⁵Ba greater than that of ¹³⁴Ba or ¹³⁶Ba? Why is the cross section of ¹³⁸Ba so small? What do you expect for the solar ratio of the abundance of ¹³⁴Ba to that of ¹³⁶Ba? Your discussion should at least mention why reactions with large releases of energy have large cross-sections.

Assignment 11 [Due 1 December]

Run the inlist from the test_suite called c13_pocket to develop a Carbon-13 pocket in the AGB star. Plot the abundances profile at the star and the end of the run as a function of mass enclosed (H, He, C, N, O) as a function of mass enclosed. Plot the key burning rates as a function of mass enclosed. Focussing in on the region where the Carbon-13 abundance is greater than 10^-4, plot the abundances of the same species as before.
Plot the abundance of Carbon-13 as a function of the cumulative mass of Carbon-13 through the pocket. Note the abundance at the 10th, 30th, 50th, 70th and 90th percentile. This divides the pocket into five zones with different typical amounts of Carbon-13.
Take the mass fraction of iron to be 10^-3 throughout. Consider that every carbon 13 nucleus will generate a neutron through ¹³C(α,n)¹⁶O, how many neutrons are generated per iron nucleus in the zones noted in the previous part.
Assuming that the number of neutrons added to a given iron nucleus is distributed as a Poissonian, what is the resulting distribution of nuclei as a function of atomic mass for each zone? What is the total abundance pattern? How does this agree with the observed abundances of elements beyond iron?

Assignment 12 [Due 8 December]

Choose one of the directories from the test_suite that you haven't done before. Perform the simulations and prepare a lab report to present your results.