Constraining Nuclear Matter Equation of States with Rotating Neutron Stars

INPC2025 Constraining of Nuclear Matter Equation of States with Rotating Neutron Stars Kwon Hyukjin Sekizawa Kazuyuki Institute of Science, Tokyo 1

Outline . Introduction . Equation of States . Models of Neutron Star Structure. . Results . Summary & Results 2

Introduction Neutron Stars are Rotating 346Hz P (Period) P dot diagram Constraining works with M-R relation usually based on TOV equation Millisecond Pulsar 205Hz 12714km 12756km Equator is longer than polar!! 3

Outline . Introduction . Equation of States . Models of Neutron Star Structure. . Results . Summary & Results 4

Equation of States Neutron Star Equation of States We use BPS, BBP EoS for Crust Inner crust Nuclei + n + ? ? < 1014?/??3 Outer crust Nuclei + ? 106?/??3< ? < 1011?/??3 Density We use Skyrme interaction in this work Uniform nuclear matter Outer core n + p + ? + ? ? < 1015?/??3 We do not consider inner core Inner core , , (?) Quark Matter (?) ~0.5 km ~0.3 km ~10 km 5

Outline . Introduction . Equation of States . Models of Neutron Star Structure. . Results . Summary 6

Models of Neutron Star Structure Non-Rotating (Spherical Symmetry) Rotating (Axis Symmetry) Gravity Gravity ? = ? + ? 2? ? ? = ? Pressure Centrifugal Force Gravity Pressure Gravity Pressure Pressure 1D ODE 2D PDE Difficult!! 2 = 4??? Centrifugal Force Poisson Equation Hachisu Self Consistent Field (HSCF) Method [1986] ? 1?? + 2??? = ? Integral Form of Equilibrium Condition This calculation is performed self-consistently! ? ? ? ? ?3? = ? Poisson Equation 7

Models of Neutron Star Structure ???= ??? 1 Tolman-Oppenheimer-Volkoff (TOV) Equation 2????= 8????? Einstein Equation TOV equation 1 ?2? 2?2???? 1 + ?2?= 8??? 1 ?2? 2?2???? + 1 ?2?= 8??? ? 2???2? + ???2 ?????? +1 ? + ? [?? ? + 4???3?] ? ? 2?? ? ?? ? ?? = ???? ??? = 8??? Komatsu Eriguchi Hachisu (KEH) Method [1989] Integral form of Metric Potentials Einstein Equation 1 2? ? = 1 1 4?? ? ?? ? 2??? ,? ?? ?? 2 ???/2= ???,? +1 ? +2 ? ? ? 0 1 0 ? ?? 1 ? ?? 1 ?2?? ?2?? ???/2= ???,? 2? 1 2?? ? ?? ?? ?? ? 2sin? ??? ,? log ? ? ?sin?? = 2 0 0 ??(? 2?)/2= ???,? ?? ? 2? 1 1 4?? 2? ? ?? ? 3sin2? cos? ???,? ?? ?? ?sin?cos?? = 2 ? ? 0 0 0 Metric potentials can be computed in an integral form, and the equilibrium configuration is obtained using a self consistent field method 8

Outline . Introduction . Equation of States . Models of Neutron Star Structure. . Results . Summary 9

Results Rotational Effects of Neutron Stars Rotating Effects from KEH Methods Rp Re Sperical Shape (R?= R?) Rp Re Oblated Shape (2R?= R?) As the rotational speed increases, the M-R curve and the radius ratio change 10

Results Internal Structure of Neutron Stars Low Central Density Decrease Crust Radius High Central Density For a deformed (rotating) neutron star, the radius refers to the equatorial radius. 11

Results Observational Constraints PSR J0740+6620 (346Hz) All units are ? KEH TOV ? 0.012 0.010 0.006 0.009 0.007 ?max 2.295 2.192 1.618 1.875 2.062 ?max 2.284 2.182 1.612 1.866 2.055 BSk24 SkI4 SkM* SkTa SLy4 The maximum mass slightly increases due to rotation, but the change remains within 1%, indicating a minor effect. Observational Constraints Reference: Cromartie, H. T. Relativistic Shapiro delay measurements of an extremely massive millisecond pulsar , Nat. Astron. (2019) 12

Results Observational Constraints PSR J0030+0451 (205Hz) All units are ? KEH TOV R 0.076 0.078 0.044 0.090 0.057 ?1.4 12.674 12.694 12.634 12.575 11.770 ?1.4 12.598 12.616 10.590 12.485 11.713 BSk24 SkI4 SkM* SkTa SLy4 The radii of 1.4 solar mass neutron stars increased in all cases, but the increase remained within 1% Observational Constraints Reference: Miller, M.C. et al. PSR J0030+0451 Mass and Radius from NICER Data and Implications for the Properties of Neutron Star Matter , Astrophys. J. Lett. (2019) 13

Results Observational Constraints PSR J1748+2246ad (716Hz) Keplerian frequency maximum rotation rate a neutron star can sustain before mass at the equator becomes unbound due to centrifugal force. Lattimer et al. Neutron stars were found to form well below the 716 Hz constraint curve proposed by Lattimer, indicating that a more conservative rotational limit may be necessary. Observational Constraints Reference: Lattimer, J. M. and Prakash, M. The Physics of Neutron Stars , Science. (2004) 14

Outline . Introduction . Equation of States . Models of Neutron Star Structure. . Results . Summary & Future Works 15

Summary & Future Works Summary Rotation alters the internal structure of neutron stars, and thus rotational effects should be taken into account. For spin frequencies below 500 Hz, the differences from the non-rotating case are minor typically within 1%. Nonetheless, considering rotation is preferable for accurate constraints. Moreover, at frequencies exceeding 500 Hz, rotation induces significant structural changes, making its inclusion essential. Therefore, future observational constraints should also incorporate rotational effects where relevant. Future Works 1) Nuclear Matter Properties 2) Make more conservative 716 Hz constraints with various EoS Mass or Radius change ?0 (Incompressibility) ?0 (Skewness) ? (Slope) New 716Hz Constraints! 16

INPC2025 Thank you Questions and Discussion Get in touch Name : Kwon Hyukjin E-mail : Kwon.h.ab@m.titech.ac.jp Office : Main building 199b 17