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tokens: we find κ = 2 − 835.6440/582.143 23 = 0.564 5388. With these values of κ for both stars and a representative range for the neutron star radius (see Section 2.1 ), we can solve for M in equation ( 1 ), including the κ corr term from equation ( 3 ), the rotational correction κ 2 as given by Fig. 2 and a differential rotation term κ 3 consistent with a realistic differential rotation with Δ f ≳ 1 Hz inside the star. In Fig. 3 , we present our results for uniformly and differentially rotating models, and compare with the fit given by equation ( 5 ) of Andersson et al. ( 2014 ). Our results for uniformly rotating stars are ≈ 15 per cent lower than the estimate given by Andersson et al. ( 2014 ), where the fully relativistic r-mode frequency for slowly rotating uniform density stars from Lockitch et al. ( 2003 ) was combined with the second-order rotational corrections for Newtonian polytropes from Lindblom et al. ( 1999 ). As we mentioned in Section 3.3 , the main factor for this difference is our improved calculation of the rotational correction κ 2 (see Fig. 2 ). Differential rotation increases our mass estimate, but the effect is rather small: Δ f = 10 Hz increases the total mass by only ≈0.02 M ⊙ . In the range 11–13 km for the stellar radius, we find from our results 1.41–1.71 M ⊙ for XTE J1751−305 and 1.53–1.87 M ⊙ for 4U 1636−536. We can use the transversal curves in Fig. 3 , which indicate M ( R ) for a choice of three realistic tabulated equations of state (APR4 Akmal, Pandharipande Ravenhall 1998 , SLy Douchin Haensel 2001 and H4 Lackey, Nayyar Owen 2006 ) to gain more insight. Each equation of state crosses our M ( r ) line only once, providing a unique prediction for M and R . However, we can see that H 4 requires R > 13 km for both stars and is therefore less favoured according to current expectations for the neutron star radius. Combining the constraints from both the r-mode frequencies and the APR4 and SLy equations of state, we find a more restricted range of values for M and R : R ≈ 11.5–12 km and M ≈ 1.48–1.56 M ⊙ for XTE J1751−305 and R ≈ 11.4–11.9 km and M ≈ 1.60–1.68 M ⊙ for 4U 1636−536. However, it must be stressed here that these tight constraints are only valid if the frequencies reported in Strohmayer Mahmoodifar ( 2014a , b ) are indeed core r-mode frequencies (see the discussion in Section 1 ). 5 CONCLUSIONS Here, we have explored the consequences for the mass and radius estimates of these two stars if the frequencies detected by Mahmoodifar and Strohmayer really correspond to their r-modes. We used a careful calculation of the r-mode frequency, taking into account relativistic, rotational, and differential rotation corrections, as well as an estimate for the error due to the Cowling approximation, to find improved constraints on the compactness M / R of XTE J1751−305 and 4U 1636−536. Our results yield, for a representative radius of 12 km, masses of 1.55 M ⊙ and 1.70 M ⊙ , respectively. For XTE J1751−305, our value is