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tokens: for σ u and bottom panels for σ l ; left panels for substituting ν + △ν and right panels for substituting ν − △ν into the MHD equations. In order to make a more reasonable estimate of the NS mass, we consider the error of QPOs. After substituting ν upper + △ν or ν upper − △ν into the MHD equations, the mass of the NS in 4U 0614+09 can be estimated as M 2.60 M ⊙ or M 2.50 M ⊙ from the minimum σ l . Similarly, after substituting ν lower + △ν or ν lower − △ν into the MHD equations, the mass of the NS in 4U 0614+09 can be estimated as M 2.50 M ⊙ or M 2.60 M ⊙ from the minimum σ u . The results are shown as panels (a), (b), (c) and (d) of Fig. 8 . Similar to the above process, the mass of the NS in 4U 1636–53 can be estimated as 2.05 M ⊙ M 2.31M ⊙ , 2.00 M ⊙ M 2.31M ⊙ with σ u from the panels of Fig. 9 . In summary, the mass of the NS in 4U 0614+09 is estimated as M 2.60 M ⊙ and that in 4U 1636–53 as 2.00 M ⊙ M 2.31M ⊙ . We do not recalculate the result of 4U 1608–52 because there are very limited results with errors reported. Figure 9. The minimum like-standard deviations σ l and σ u corresponding to the different masses of the NS in 4U 1636–53 when the errors of the observed kHz QPOs are considered. Top panels for σ u and bottom panels for σ l ; left panels for substituting ν + △ν and right panels for substituting ν − △ν into the MHD equations. Figure 10. Mass–radius relationships for various equations of state given in Özel Freire ( 2016 ). The dotted line, twin solid lines and twin dash–dot lines correspond to the masses M = 2.60 M ⊙ , , , respectively, which are obtained from the estimation by the twin kHz QPOs in 4U 0614+09, 4U 1636–53, 4U 1608–52. 5 DISCUSSION As the dividing line between NSs and black holes, EOSs affect the number of observable NSs and their properties, because different EOSs might lead to different mechanisms of supernova explosions (Özel Freire 2016 ). As seen from Fig. 10 , different EOSs have different maximum masses of NSs. Thus a high-mass NS can be used to rule out some EOSs. For 4U 0614+09, M 2.60 M ⊙ , and no constraint can be put on any EOS. For 4U 1636–53, 2.00 M ⊙ M 2.31 M ⊙ rules out EOSs of strange quark matter, which is consistent with the conclusion obtained from the lower limit on the NS radius (11.0 km) that was inferred from observed mHz QPOs modelled as marginally stable burning on the NS surface (Stiele, Yu Kong 2016 ). Güver et al. ( 2010 ) estimated the mass and radius of the NS in 4U 1608–52 when the most probable distance () was measured by the interstellar extinction towards the source. They obtained a mass of M = 1.74 ± 0.14 M ⊙ and a radius of . Steiner, Lattimer Brown ( 2010 ) obtained a larger radius of the NS (about ) when the photospheric radius is much larger than the radius of the NS. Poutanen et al. ( 2014 ) studied 21 photospheric radius expansion bursts from 4U 1608–52 and obtained a NS radius R larger than 12 km for 1.0 M ⊙ M 2.4 M ⊙ , and R restricted to be between 13 and 16 km for