Relation Between Gravitational Mass and Baryonic Mass for Non-Rotating and Rapidly Rotating Neutron Stars

With a selected sample of neutron star (NS) equations of state (EOSs) that are consistent with the current observations and have a range of maximum masses, we investigate the relations between NS gravitational mass Mg and baryonic mass Mb, and the relations between the maximum NS mass supported through uniform rotation (Mmax) and that of nonrotating NSs (MTOV). We find that for an EOS-independent quadratic, universal transformation formula (Mb=Mg+A×M2g)(Mb=Mg+A×Mg2), the best-fit A value is 0.080 for non-rotating NSs, 0.064 for maximally rotating NSs, and 0.073 when NSs with arbitrary rotation are considered. The residual error of the transformation is ∼ 0.1M⊙ for non-spin or maximum-spin, but is as large as ∼ 0.2M⊙ for all spins. For different EOSs, we find that the parameter A for non-rotating NSs is proportional to R−11.4R1.4−1 (where R1.4 is NS radius for 1.4M⊙ in units of km). For a particular EOS, if one adopts the best-fit parameters for different spin periods, the residual error of the transformation is smaller, which is of the order of 0.01M⊙ for the quadratic form and less than 0.01M⊙ for the cubic form ((Mb=Mg+A1×M2g+A2×M3g)(Mb=Mg+A1×Mg2+A2×Mg3)). We also find a very tight and general correlation between the normalized mass gain due to spin Δm = (Mmax − MTOV)/MTOV and the spin period normalized to the Keplerian period PP, i.e., log10Δm=(−2.74±0.05)log10P+log10(0.20±0.01)log10Δm=(−2.74±0.05)log10P+log10(0.20±0.01), which is independent of EOS models. These empirical relations are helpful to study NS-NS mergers with a long-lived NS merger product using multi-messenger data. The application of our results to GW170817 is discussed

(E)-4-[4-(Dimethyl­amino)benzyl­ideneamino]benzonitrile

The mol­ecule of the title compound, C16H15N3, displays a trans configuration with respect to the C=N double bond. The mol­ecule is not planar, the dihedral angle between the benzene rings being 57.83 (9)°. The crystal packing is stabilized only by van der Waals inter­actions

(E)-3-[(4-Diethyl­amino-2-hydroxy­benzyl­idene)amino]benzonitrile

The mol­ecule of the title compound, C18H19N3O, displays a trans configuration with respect to the C=N double bond. There is a strong intra­molecular O—H⋯N hydrogen-bonding inter­action between the hydr­oxy group and imine N atom. The dihedral angle between the aromatic rings is 30.35 (8)°. The crystal packing is stabilized by O—H⋯N links

1-(4-tert-Butyl­benz­yl)-2-(4-tert-butyl­phen­yl)-1H-benzimidazole

In the mol­ecule of the title compound, C28H32N2, the benzimidazole ring system is almost planar [maximum deviation = 0.0221 (15) Å] and forms dihedral angles of 85.86 (4) and 32.09 (6)° with the benzene rings. In the crystal structure, mol­ecules are linked into chains running parallel to the a axis by inter­molecular C—H⋯N hydrogen bonds. The methyl groups of a tert-butyl group are rotationally disordered over two positions with refined site-occupancy factors of 0.636 (4) and 0.364 (4)

(E)-3-(2-Hydr­oxy-3-methoxy­benzyl­idene­amino)benzonitrile

The mol­ecule of the title compound, C15H12N2O2, displays a trans configuration with respect to the C=N double bond. The dihedral angle between the two benzene rings is 30.46 (14)°. A strong intra­molecular O—H⋯O hydrogen bond stabilizes the mol­ecular structure

(Z)-2-[(2-Hydr­oxy-1-naphth­yl)methyl­eneamino]benzonitrile

The title compound, C18H12N2O, crystallizes in a phenol–imine tautomeric form with a Z conformation for the imine functionality. The dihedral angle between the aromatic rings is 8.98 (9)°. A strong intra­molecular O—H⋯N hydrogen-bond inter­action between the hydroxyl group and imine N atom occurs

(E)-2-[(3,5-Di-tert-butyl-2-hydroxy­benzyl­idene)amino]benzonitrile

The asymmetric unit of the title compound, C22H26N2O, contains three crystallographically independent mol­ecules, in which the aromatic rings are oriented at dihedral angles of 21.74 (5), 27.59 (5) and 27.87 (5)°. Intra­molecular O—H⋯N hydrogen bonds result in the formation of planar six-membered rings, and these are nearly coplanar with the adjacent rings. In the crystal structure, π–π contacts between the benzene rings [centroid–centroid distances = 3.989 (2), 3.802 (1) and 3.882 (1) Å] may stabilize the structure

(E)-3-(2-Hydr­oxy-4-methoxy­benzyl­idene­amino)benzonitrile

In the mol­ecule of the title compound, C15H12N2O2, the aromatic rings are oriented at a dihedral angle of 28.11 (3)°. Intra­molecular O—H⋯N hydrogen bonding results in the formation of a planar six-membered ring, which is nearly coplanar with the adjacent ring at a dihedral angle of 1.53 (3)°. In the crystal structure, π–π contacts between the benzene rings [centroid–centroid distance = 3.841 (1) Å] may stabilize the structure