Quark Hybrid Equations of State for Astrophysical Simulations
 |
We present finite temperature quark-hadron hybrid equations of state for applications in astrophysical
simulations of e.g. core-collapse supernovae and
neutron star binary mergers. The details to the equations of state as well their implementation
in core-collapse supernova simulations can be found in [1,2,3].
The phase transition to quark matter is implemented via the Gibbs construction to the hadronic Shen et al. equation of state [4]. In addition, we extend the Shen et al. table by three values for the temperature, namely T~125 MeV, ~160 MeV, ~200 MeV, and ten points in density ρb.
The final ranges for the temperature T, the proton fraction Yp, and the baryon mass density ρb are:
- Temperature T [MeV] :
-0.1 ≤ log10( T ) ≤ 2.3; number of points: 34 ( points 1-31 as in Shen et al., 32-34 added); mesh of log10( T ) ~ 0.1
- Proton fraction Yp :
-2.0 ≤ log10( Yp ) ≤ -0.25; number of points: 71; mesh of log10( Yp ) ~ 0.025
- Baryon number density ρb [g/cm3]:
5.1 ≤ log10( ρb ) ≤ 16.4; number of points: 114 ( 1-104 as in Shen et al., 105-114 added); mesh of log10( ρb ) ~ 0.1
The constructed tables have the same structure as the Shen et al. table. However, now with 34 temperature blocks, divided into 71 blocks of proton fraction, which are composed of 114 rows for the equation of state at given baryon mass density. The tables have 18 columns, whereas entries 1-17 are identical to the original table and the 18th column gives the quark fraction χ. The latter is zero in the pure hadronic phase, 1 for pure quark matter, and has a value of 0 < χ < 1 for the quark-hadron mixed phase. Consequently, the hybrid tables consist of the following entries:
| Entry number |
|
|
Quantity for given temperature T |
|
|
|
| 1 |
|
|
Log of baryon mass density |
|
|
log10(ρb) [g/cm3] |
| 2 |
|
|
Baryon number density |
|
|
nb [1/fm3] |
| 3 |
|
|
Log of proton fraction |
|
|
log10(Yp) |
| 4 |
|
|
Proton fraction |
|
|
Yp |
| 5 |
|
|
Free energy per baryon |
|
|
F=(ε - T)/nb - mfree [MeV] |
| 6 |
|
|
Internal energy per baryon |
|
|
Eint=ε/nb - matomic [MeV] |
| 7 |
|
|
Entropy per baryon |
|
|
S=s/nb [kb] |
| 8 |
|
|
Mass number of heavy nucleus1,2 |
|
|
A |
| 9 |
|
|
Charge number of heavy nucleus1,2 |
|
|
Z |
| 10 |
|
|
Effective mass1 |
|
|
m* [MeV] |
| 11 |
|
|
Free neutron fraction3 |
|
|
Xn |
| 12 |
|
|
Free proton fraction4 |
|
|
Xp |
| 13 |
|
|
Alpha particle fraction1,2 |
|
|
Xa |
| 14 |
|
|
Heavy nucleus fraction1,2 |
|
|
XA |
| 15 |
|
|
Pressure |
|
|
p [MeV/fm3] |
| 16 |
|
|
Chemical potential of neutrons |
|
|
μn=μb - mfree [MeV] |
| 17 |
|
|
Chemical potential of protons |
|
|
μp=μb + μc - mfree [MeV] |
| 18 |
|
|
Quark fraction |
|
|
χ |
1 set to zero in the mixed phase
2 set to zero in the pure quark phase
3 set to 1-Yp in the mixed and pure quark phase
4 set to Yp in the mixed and pure quark phase
ε = energy density [MeV/fm3]
mfree=938.0 MeV,
matomic=931.49432 MeV
Note that for the three last temperature blocks (T ~ 120 MeV, ~ 160 MeV, ~ 200 MeV) we filled the table with pure quark matter for densities starting from nb=0.01 1/fm3. For lower densities the table entries 5-18 are filled with zeros.
As given in the manual of the Shen et al. equation of state, we tested the hybrid equations of state for thermodynamic consistency, consistency in
fractions, and the relation between the free energy F, the internal energy Eint, and the entropy S by:
F + p/nb = μn (1 - Yp) + μpYp
1 = Xn + Xp + Xa + XA
F = Eint - TS + mu - mfree
Refering to the calculated consistencies as:
Consistency1 = abs[ (F + p/nb - μn (1 - Yp) - μpYp) / (μn (1 - Yp) + μpYp) ]
Consistency2 = abs[ Xn + Xp + Xa + XA -1.0 ]
Consistency3 = abs[ ( F - ( Eint - TS + mu - mfree)) / F]
these can be found here for B1/4=165 MeV, whereas the consistencies for all hybrid equations of state are very similar. By its definition and regarding its construction for the presence in quark matter, Consistency2 is fulfilled to a high accuracy. For temperatures T ≤ 100 MeV, the two consistency relations give Consistency1 ≤ 10-7 and Consistency3 ≤ 10-6. For higher temperatures the upper limits of both increase to 0.001 for Consistency1 and 10-4 for Consistency3, respectively, whereas the latter has peaks up to ~ 0.01 for very high and low proton fractions.
The tables are free for download:
Bag constant B1/4=162 MeV, strong coupling constant as=0.0:
Currently unavailable
Bag constant B1/4=165 MeV, strong coupling constant as=0.0:
Download (ca. 13MB)
Bag constant B1/4=155 MeV, strong coupling constant as=0.3:
Download (ca. 13MB)
For the read-in of the tables:
Each temperature block begins with:
cccccccccccccccccccccccccc
Temperature= .....
blank line
Each proton fraction block ends with a blank line
The format for the temperature entry is (in fortran):
"format(a14,1pe12.6)"
The format for one table row is:
"format(1pE13.6,17(1x,1pE13.6))"
References:
[1] Signals of the QCD Phase Transition in Core-Collapse Supernovae
Sagert, I.; Fischer, T.; Hempel, M.; Pagliara, G.; Schaffner-Bielich, J.; Mezzacappa, A.; Thielemann, F.-K.; Liebendörfer, M.
Physical Review Letters, vol. 102, Issue 8, 2009
[2] Strange quark matter in explosive astrophysical systems
Sagert, Irina; Fischer, T.; Hempel, M.; Pagliara, G.; Schaffner-Bielich, J.; Thielemann, F.-K.; Liebendörfer, M.
Journal of Physics G: Nuclear and Particle Physics, Volume 37, Issue 9, pp. 094064 (2010)
[3] The revival of an explosion mechanism of massive stars - the quark hadron phase transition during the early post bounce phase of core collapse supernovae
Fischer, T.; Sagert, I.; Pagliara, G.; Hempel, M.; Schaffner-Bielich, J.; Rauscher, T.; Thielemann, F. -K.; Käppeli, R.; Martínez-Pinedo, G.; Liebendörfer, M.
eprint arXiv:1011.3409
[4] Relativistic Equation of State of Nuclear Matter for Supernova Explosion
Shen, H.; Toki, H.; Oyamatsu, K.; Sumiyoshi, K.
Progress of Theoretical Physics, Vol. 100, No. 5, pp. 1013-1031, 1998
Irina Sagert
Last modified: Tue Feb 28 16:30:30 EST 2012