In previous posts, I have run a constant photon-to-nucleon ratio and a constant entropy network calculation. In this post I will compare such calculations. To do this, I will run an r-process calculation with a hydrodynamic trajectory I previously defined at constant entropy and photon-to-nucleon ratio.

I will run my calculation at the initial entropy per nucleon of *s/k = 250* for both calculations. Since my thermodynamic trajectory starts at a mass density of *1.5 x 10 ^{6}* g/cc at time zero, I will invert the entropy to get the temperature. I change into my

**cd nucnet-tools-code/examples/thermo**

**./examples_make**

I will run the calculation with an initial electron-to-nucleon ratio *Y _{e} = 0.35*; thus, I type

**./compute_temperature_from_entropy ../../data_pub/my_net.xml 1.5e6 250 out.xml 0.35**

This returns

Temperature (K) = 1.02462e+10 Density = 1.5e6 g/cc, Ye = 0.35

I need a starting temperature of *T _{9} = 10.2462* to give an entropy per nucleon of

I next change into *my_examples/network* and compile the constant entropy network code that does not update the abundances in the entropy inversion root-finding steps. I type

**export NNT_USE_SPARSKIT2=1**

**make clean**

**make run_constant_entropy**

When the code has compiled, I create the zone input file *../../data_pub/zone.xmi* to read

<zone_data> <zone> <optional_properties> <property name="tend">1.e6</property> <property name="tau_0">0.035</property> <property name="tau_1">1.</property> <property name="munuekT">-inf</property> <property name="t9">10.2462</property> <property name="steps">20</property> <property name="time">0</property> <property name="dt">1.e-15</property> <property name="rho_0">1.4985e6</property> <property name="rho_1">1.5e3</property> <property name="entropy per nucleon">250</property> <property name="Ye">0.35</property> <property name="iterative solver method">gmres</property> <property name="t9 for iterative solver">2.</property> </optional_properties> <mass_fractions/> </zone> </zone_data>

I run the calculation by typing

**./run_constant_entropy ../../data_pub/my_net.xml ../../data_pub/zone.xml my_output1.xml "[z <= 90]"**

This runs a constant entropy network calculation with data in my_net.xml for a network including nuclei up to *Z = 90*. The calculation uses the Krylov-space solver gmres and employs the previously defined trajectory.

Once the constant entropy network calculation is done, I can run a constant photon-to-nucleon ratio with the same starting conditions. To do this, I select out the first zone from *my_output1.xml* to be the input for this new calculation. I type

**xsltproc --param zone_xpath "//zone[position()=1]" http://nucnet-tools.sourceforge.net/xsl_pub/2016-03-05/select_zone.xsl my_output1.xml > ../../data_pub/zone2.xml**

which creates *../../data_pub/zone2.xml*. I next compile the single-zone code. I use a new version that outputs the XML only at the end of the calculation. I download run_single_zone.cpp to the *my_examples/network* directory and type

**make run_single_zone**

I then run the calculation by typing

**./run_single_zone ../../data_pub/my_net.xml ../../data_pub/zone2.xml my_output2.xml "[z <= 90]"**

This code runs the calculation with *ρ ∝ T ^{3}* that I previously ran but with different starting conditions on the temperature and abundances. In particular, the abundances start in the nuclear statistical equilibrium distribution appropriate for the input conditions.

Once this calcultion is finished, I can compare the photon-to-nucleon ratio in the two calculations. I retrieve the time, temperature, and density by typing

**cd ../../examples/analysis**

**./examples_make**

**cd ../../my_examples/network**

**../../examples/analysis/print_properties my_output1.xml time t9 rho > props1.txt**

**../../examples/analysis/print_properties my_output2.xml time t9 rho > props2.txt**

The photon-to-nucleon ratio *φ* is the number density of photons divided by the number density of nucleons (*ρ N _{A}*, where ρ is the mass density and

Here *k* is Boltzmann's constant, *ℏ* is Planck's constant divided by *2π* and *c* is the speed of light in vacuum. *ζ(3)* is the Riemann zeta of argument 3. The key result is that *φ ∝ T ^{3} / ρ*.

By plotting the appropriate combination of columns in *props1.txt* and *props2.txt*, I can make the following plot showing *φ* relative to *φ _{i}*, the initial

Of course *φ* is constant in the constant photon-to-nucleon ratio calculation. In the constant entropy calculation, on the other hand, the photon-to-nucleon ratio increases to a factor of about 2.75 times its initial value. This is due to the transfer of entropy from the electron-positron pairs and nucleons to the photons.

The entropy density in a gas of relativistic particles is given by

where the radiation constant is

*g* is the multiplicity factor. For photons it is *g = 2*. I may next find the entropy per nucleon (in units of *k*) as

I now relate the entropy per nucleon in relativistic particles to the photon-to-nucleon ratio *φ* as

When relativistic particles dominate the entropy, as is certainly the case for *s/k = 250*, the entropy is proportional to the multiplicity factor times the photon-to-nucleon ratio. The multiplicity factor itself is given by

The photons in the system are bosons. The electrons and positrons are fermions. The factor 7/8 for the fermions comes from the different integrand for fermions compared to bosons. Early in the constant entropy calculation, the electrons and positrons are abundant and act as nearly relativistic particles. The multiplicity factor is thus *g = 2 + 7 x 2 x 2 / 8 = 11/2*, since the *g* for electrons and positrons is *g = 2s + 1 = 2*, where *s = 1/2* is the electron or positron spin. Later in the calculation, after the electron-positron pairs have completely annihilated and transferred all their entropy to the photons, *g = 2*. Because the quantity *gφ* is constant for the relativistic particles, the *φ* at the end of the calculation is *11/4* times larger than at the beginning. This explains the rise of *φ* to 2.75 times its initial value in the constant entropy calculation. The later slight decline in *φ* is due to the transfer of entropy into the now non-relativistic baryons and residual electrons at late times. Of course in the constant *φ* calculation, we are not accounting for the transfer of entropy from electron-positron pairs to the photons.

Because the photon-to-nucleon ratio increases from its inital value in the constant entropy calculation, the temperature at a given time (and density) is larger than in the constant *φ* calculation. By plotting the appropriate columns in the files *props1.txt* and *props2.txt*, I can make this figure:

At late times, the temperature in the constant entropy calculation at a given time is *(11/4) ^{1/3}* times larger than in the constant

I now consider the abundances. I retrieve the abundances versus mass number by typing

**../../examples/analysis/print_abundances_vs_nucleon_number my_output1.xml a "[last()]" > ya1.txt**

**../../examples/analysis/print_abundances_vs_nucleon_number my_output2.xml a "[last()]" > ya2.txt**

The XPath expression *[last()]* selects out the last timestep in the calculation. By plotting column 2 versus column 1 in these files, I get this figure:

The abundance of the heaviest species is higher in the constant entropy calculation. The higher temperature has impeded the assembly of heavy species. I confirm this computing the number of heavy (*Z > 2*) nuclei by typing

**../../examples/analysis/compute_abundance_moment_in_zones my_output1.xml "[z > 2]" z 0 "[last()]"**

which returns

188 0.00356726

and

**../../examples/analysis/compute_abundance_moment_in_zones my_output2.xml "[z > 2]" z 0 "[last()]"**

which returns

193 0.00411635

The constant entropy calculation assembled only 86% as many heavy nuclei as the constant *φ* calculation. The constant entropy calculation thus has more neutrons per seed nucleus and the average nucleus can thus capture neutrons to higher mass.

The famous *(11/4) ^{1/3}* temperature factor seen in this calculation is that relevant for the cosmic neutrino background. Neutrinos are expected to have decoupled from the rest of the particles before the electron-positron pairs annihilated in the early Universe. This means that we expect the relic neutrinos to have a temperature of about

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