![Page 1: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/1.jpg)
Rotating neutron star
in strong magnetic fields
and the MR relationsCHINATSU WATANABE, KOTA YANASE*,
AND NAOTAKA YOSHINAGA
SAITAMA UNIVERSITY, *TOKYO UNIVERSITY
24th, June XQCD2019
![Page 2: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/2.jpg)
Background and Motivation- Neutron stars (NS)
- Motivation
Formulation- Metric and Hartle equation
- Relativistic Mean Field (RMF) Theory
Results and Summary- Mass-Radius (MR) relations for rotating and magnetic field NS
- Summary
Contents
![Page 3: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/3.jpg)
Background
![Page 4: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/4.jpg)
Neutron Stars (NS)
Mass 1.4 M☉
Radius 10 km
Central density ρ = 10 ρ0
ρ0≃0.15 nucleons/fm3
(normal nuclear density)
The Crab Nebula and pulsar, Chandra X-ray scopeCredit: NASA/CXC/ASU/J. Hester et al.
P. Haensel, A.Y. Potekhin, and D.G. Yakovlev“NEUTRON STARS 1” , ASSL, (2007)
![Page 5: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/5.jpg)
Structure of NS
Supported by neutron degeneracy pressure
Inner crust
Outer core
Inner core
![Page 6: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/6.jpg)
Motivation
![Page 7: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/7.jpg)
2 neutron stars with around twice the solar mass.
B. Demorest et al. Nature 467, (2010) 1081-1083
John Antoniadis et al. Science 340, (2013) 1233232
Two solar mass problem
![Page 8: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/8.jpg)
2 neutron stars with around twice the solar mass.
B. Demorest et al. Nature 467, (2010) 1081-1083
John Antoniadis et al. Science 340, (2013) 1233232
Two solar mass problem
Such a very heavy neutron star gives a strong limit on the equation of state.
![Page 9: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/9.jpg)
NS with strong magnetic field or/and rapid rotation may have such a large mass.
S. A. Olausen and V. M. KaspiAPJ Supplement Series,212:6 (22pp) (2014)
J.Hessels et al.Science Vol.311, Issue5769,pp.1901-1904 (2006)
How can we explain 2M☉ NS ?
We consider magnetic fields or/and rotation.
![Page 10: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/10.jpg)
NS with strong magnetic field or/and rapid rotation may have such a large mass.
S. A. Olausen and V. M. KaspiAPJ Supplement Series,212:6 (22pp) (2014)
J.Hessels et al.Science Vol.311, Issue5769,pp.1901-1904 (2006)
How can we explain 2M☉ NS ?
We consider magnetic fields or/and rotation.
In this study
![Page 11: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/11.jpg)
NS with strong magnetic field or/and rapid rotation may have such a large mass.
S. A. Olausen and V. M. KaspiAPJ Supplement Series,212:6 (22pp) (2014)
J.Hessels et al.Science Vol.311, Issue5769,pp.1901-1904 (2006)
How can we explain 2M☉ NS ?
We consider magnetic fields or/and rotation.
In this study
observation
![Page 12: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/12.jpg)
S. A. Olausen and V. M. KaspiAPJ 212, 22(2014)
Theory of magnetized neutron stars
P-P diagram・
![Page 13: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/13.jpg)
We can limit the radius from the observation.The radii suggested by observational considerations.
About Radius
Mass (M☉) Radius (km) method
0.86−2.42 > 7.6−10.4 Black body from surface
1.2−1.7 < 9.0−13.2 Eddington limit
@ 1.4 > 6.6 The absorption line red shift
@ 1.4 ≲ 13.6 Gravitational waveAnnala et al. (2018)
Waki et al. (1984)
Zamfir et al. (2012)
Guillot et al. (2013)
![Page 14: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/14.jpg)
P. B. Demorest et al.Nature 467, 1081(2010)
Equation of state (EoS)and MR relation
nucleonsnucleons + hyperonsstrange quark matter
Masses do not surpass twice the solar mass, once hyperons are included.
Various kinds of EoSs to have been proposed to describe NS.
![Page 15: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/15.jpg)
How to Judge
equation of state
Using TOV equation
Pressure PEnergy density ε
Number density ρ
Mass MRadius R
MR relationprediction
Judge
stiff
soft
stiff
soft
![Page 16: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/16.jpg)
Formulation
![Page 17: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/17.jpg)
Metric of slowly rotating neutron star considering axial deformation in GR.
Θ
S. Chandrasekhar & John C. Miller, Mon. Not. R. astr. Soc., 167, 63-79 (1974)
![Page 18: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/18.jpg)
Metric of slowly rotating neutron star considering axial deformation in GR.
Θ
S. Chandrasekhar & John C. Miller, Mon. Not. R. astr. Soc., 167, 63-79 (1974)
![Page 19: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/19.jpg)
Metric of slowly rotating neutron star considering axial deformation in GR.
Θ
S. Chandrasekhar & John C. Miller, Mon. Not. R. astr. Soc., 167, 63-79 (1974)
![Page 20: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/20.jpg)
Hartle Equations
To calculate the additional mass for
slowly rotating neutron star.
S. Chandrasekhar & John C. Miller, Mon. Not. R. astr. Soc., 167, 63-79 (1974)
![Page 21: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/21.jpg)
Relativistic Mean Field (RMF) TheoryWe use the following Lagrangian which includes interactions between baryon octet and σ, ω, ρ, σ*,and φ mesons.
![Page 22: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/22.jpg)
N. Takigawa, Nuclear Physics(2013)
Nuclear propertiesBethe-Weizsäcker formula ( 1935 )
Nuclear binding energy can be described roughly by a liquid‐drop model.Radius of nuclei : R=r0A1/3 r0~0.15fm
volume surface coulombsymmetry energy
Pairingenergy
16.2MeV 19.0MeV 0.76MeV
23.5MeV
1MeV
![Page 23: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/23.jpg)
Nuclear properties
Nucleon numberdensity
Binding energy pernucleon
Incompressibility
Symmetry energy ( ) 23.5MeV
𝐿 = 3𝜌0𝑑𝑆
𝑑𝜌𝜌0
𝐾 = 9𝜌02𝑑2𝑆
𝑑𝜌2𝜌0
asym= 𝑆 𝜌0Symmetry-energy slope parameter
incompressibility of symmetry energy
![Page 24: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/24.jpg)
EoS B/AMeV
ρ0fm-3
asym
MeV
KMeV
LMeV
Ksymfm-3
GM1(*1) 16.3 0.153 32.5 300 94.4 18.1
TM1-a(*2) 16.3 0.146 36.9 281.2 111.2 33.8
TM1-b(*2) 16.3 0.146 36.9 281.2 111.2 33.8
TM2-ωρ-a(*2) 16.4 0.146 32.1 281.7 54.8 -70.5
TM2-ωρ-b(*2) 16.4 0.146 32.1 281.7 54.8 -70.5
(*1)N.Glendenning & S.MoszkowskiPhy.Rev.Letter, vol.67, Num.18(1991)
(*2) M.Fortin et al.,Physical Review C95, 065803 (2017)
Properties of various EoSs
![Page 25: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/25.jpg)
ResultsandSummaryCOMPARISON OF VARIOUS EOSS
ROTATION AND MAGNETIC FIELDS
![Page 26: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/26.jpg)
Comparison of rotating NS masses(5 EoSs)
![Page 27: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/27.jpg)
TM1-b and TM2-ωρ-b EoSs give over twice the solar mass.
Comparison of rotating NS masses(5 EoSs)
![Page 28: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/28.jpg)
TM1-b and TM2-ωρ-b EoSs give over twice the solar mass.
Comparison of rotating NS masses(5 EoSs)
![Page 29: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/29.jpg)
For magnetic field, we used following .
(Spherically Symmetric Magnetic Pressure)
Magnetic field B(ρ)
2×1018
20
![Page 30: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/30.jpg)
For magnetic field, we used following .
(Spherically Symmetric Magnetic Pressure)
Magnetic field B(ρ)
R.H.Castro , L.B.Castro & D.P.Menezes, Phy. Rev. C 89, 015805 (2014)
![Page 31: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/31.jpg)
Comparison of magnetizedNS masses (5 EoSs)
![Page 32: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/32.jpg)
Only TM2-ωρ-b EoS gives over twice the solar mass.
Comparison of magnetizedNS masses (5 EoSs)
![Page 33: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/33.jpg)
Comparison of rotating and magnetized NS (5 EoSs)
![Page 34: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/34.jpg)
Comparison of rotating and magnetized NS (5 EoSs)
All 5 EoSs give over twice the solar mass.
![Page 35: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/35.jpg)
EoS Rkm
R(rot)km
R(mag)km
R(rot&mag)km
GM1 13.46 13.34 13.77 13.49
TM1-a 14.07 14.12 14.33 14.26
TM1-b 14.10 14.12 14.33 14.26
TM2-ωρ-a 13.23 13.02 13.47 13.21
TM2-ωρ-b 13.24 13.02 13.47 13.21
Radius for 1.4M☉NS
Observation6.6<R≦13.6 km
Ω=0.03km-1 B0=2×1018G
Annala et al. (2018)
Waki et al. (1984)
![Page 36: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/36.jpg)
EoS Rkm
R(rot)km
R(mag)km
R(rot&mag)km
GM1 13.46 13.34 13.77 13.49
TM1-a 14.07 14.12 14.33 14.26
TM1-b 14.10 14.12 14.33 14.26
TM2-ωρ-a 13.23 13.02 13.47 13.21
TM2-ωρ-b 13.24 13.02 13.47 13.21
Observation6.6<R≦13.6 km
ApproximatelyGM1 &TM2-ωρ-a &TM2-ωρ-bare in the range.
Ω=0.03km-1 B0=2×1018G
Radius for 1.4M☉NS
![Page 37: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/37.jpg)
EoS Rkm
R(rot)km
R(mag)km
R(rot&mag)km
GM1 13.46 13.34 13.77 13.49
TM1-a 14.07 14.12 14.33 14.26
TM1-b 14.10 14.12 14.33 14.26
TM2-ωρ-a 13.23 13.02 13.47 13.21
TM2-ωρ-b 13.24 13.02 13.47 13.21
Observation6.6<R≦13.6 km
ApproximatelyGM1 &TM2-ωρ-a &TM2-ωρ-bare in the range.
EoS LMeV
GM1(*1) 94.4
TM1-a(*2) 111.2
TM1-b(*2) 111.2
TM2-ωρ-a(*2) 54.8
TM2-ωρ-b(*2) 54.8
Ω=0.03km-1 B0=2×1018G
Radius for 1.4M☉NS
![Page 38: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/38.jpg)
Summary
✓ We calculated the mass-radius relations for magnetized and rotating neutron stars using various kinds of EoSs.
✓ We obtained neutron stars with masses more than 2 M☉ both with strong magnetic fields and in rapid rotations for 5 hadronic EoSs.
✓ TM1-a and TM1-b EoSs have larger radius than expected. They are out of bounds.
![Page 39: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/39.jpg)
Future work
• To calculate MR relations for hybrid stars (mixture of quarks and hadrons) under the circumstance of rotation and magnetic fields.
• If the rotational axis and the deformation axis are different, gravitational waves might occur (wobbling motion). We are planning to look into that.
Some part of our work will be published in PTEP soon.
![Page 40: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/40.jpg)
Thank you for your attention.
![Page 41: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/41.jpg)
Back upBACK UP
![Page 42: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/42.jpg)
Including mesons
EoS σ ω ρ σ* φ
GM1 〇 〇 〇 - -
TM1-a 〇 〇 〇 〇 〇
TM1-b 〇 〇 〇 〇 〇
TM2ωρ-a 〇 〇 〇 〇 〇
TM2ωρ-b 〇 〇 〇 〇 〇
![Page 43: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/43.jpg)
![Page 44: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/44.jpg)
Parameters for EoSs
![Page 45: Rotating Neutron stars with Magnetic Fields · Rotating neutron star in strong magnetic fields and the MR relations CHINATSU WATANABE, KOTA YANASE*, AND NAOTAKA YOSHINAGA SAITAMA](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e4e0e61cbf77d5c447fd4/html5/thumbnails/45.jpg)