I’m working on something using MUSES CE, generating a star composed only of protons and neutrons.
Since I need both gravitational and baryonic mass, to obtain the latter I integrate the baryon number density n(R) taken from local_functions_e_#.csv.
The problem I encounter is that the baryonic mass is less than the gravitational mass, so, looking at the theory, it’s incorrect.
The integral has already been checked several times, so its form is correct.
Another check I made was to insert a different equation of state, for which I already knew the baryonic and gravitational mass parameters, from the one generated with MUSES in the QLIMR module. This gave me the opportunity to check whether the problem lies in the construction of the equation of state or in the QLIMR module. What I found is that the gravitational mass is the same within error, but the baryonic mass is underestimated by a factor of 2.2, as is the baryon number density n(R) at the center by a factor of 3, then tending towards the “real” n(R) as the radius increases.
I would like to know how to solve this problem and/or how to interpret this underestimation.
Dear Frederico, you should integrate the baryonic density multiplied by the 1/(1-2M/R)^{-1/2} . This should solve your problem. You can look at section 3.13 of Glendenning’s book Compact Stars.
This isn’t the problem; as I’ve already written, the integration formula has already been checked several times and is correct.
The problem arises from the fact that the QLIMR module underestimates the baryon number density by a factor of 3 in the core, converging (more or less) to the real value toward the crust.
I say this because I made a comparison plot between the baryon number density obtained with MUSES and that obtained with another program.
At the moment, QLIMR does not yet compute the baryon mass, but this will be included in an upcoming update. In QLIMR, the baryon number density ( n(r) ) is obtained by first computing the chemical potential ( \mu(r) ). There was indeed a typo involving a factor of 2 in the expression for ( \mu(r) ).
With the metric conventions used in QLIMR (see the documentation for details), the chemical potential is defined as
[
\mu(r) = \mu_0 \sqrt{1 - \frac{2 M_\star}{R_\star}}, e^{-\nu/2},
]
where ( \mu_0 = 930.34,\mathrm{MeV} ) (following Glendenning).
Once ( \mu(r) ) is computed, the baryon number density is obtained using Gibbs’ relation,
[
n(r) = \frac{p + \varepsilon}{\mu(r)}.
]
Thank you very much for noticing this issue. The confusion arose from the metric convention used in the definition of ( \mu(r) ): the factor of 2 belongs in the exponential ( e^{-\nu/2} ), whereas the current implementation uses ( e^{-\nu} ), which corresponds to Glendenning’s metric convention. I will be updating QLIMR soon with this fix and additional improvements, in case you would like to use it.