• Acta Physica Sinica
  • Vol. 68, Issue 18, 180401-1 (2019)
Jian-Ling Chen1、*, Hui Wang2, Huan-Yu Jia2, Zi-Wei Ma1, Yong-Hong Li1, and Jun Tan3
Author Affiliations
  • 1Department of Physics and Electronic Engineering, Yuncheng University, Yuncheng 044000, China
  • 2School of Physical Science and Technology, Southwest Jiaotong University, Chengdu 610031, China
  • 3Maths and Information Technology School, Yuncheng University, Yuncheng 044000, China
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    DOI: 10.7498/aps.68.20190760 Cite this Article
    Jian-Ling Chen, Hui Wang, Huan-Yu Jia, Zi-Wei Ma, Yong-Hong Li, Jun Tan. Conductivity of neutron star crust under superhigh magnetic fields and Ohmic decay of toroidal magnetic field of magnetar[J]. Acta Physica Sinica, 2019, 68(18): 180401-1 Copy Citation Text show less
    Relationships between mass and radius of neutron stars in NL3, GM1 and TMA model.在NL3, GM1和TMA模型下中子星的质量和半径的关系
    Fig. 1. Relationships between mass and radius of neutron stars in NL3, GM1 and TMA model.在NL3, GM1和TMA模型下中子星的质量和半径的关系
    Relationship of moment of inertial I to mass M and radius R for magnetars in TMA models.在TMA模型中磁星的转动惯量I随质量m和半径R的关系
    Fig. 2. Relationship of moment of inertial I to mass M and radius R for magnetars in TMA models. 在TMA模型中磁星的转动惯量I随质量m和半径R的关系
    Normalized magnetic field components of the crustal confined for the force-free field: (red line),(blue line), and (yellow line) vs. normalized radial coordinate x. Here we assume the parameter μ = 1.676, corresponding to M = 1.45M⊙, R = 11.77 km and I = 1.45 × 1045 g·cm2 in the TMA model.在无力磁场结构位型下壳层归一化磁场分量(红线), (蓝线), 及(黄线)与归一化径向坐标x的关系(选取μ = 1.676, 对应在TMA模型下的M = 1.45M⊙, R = 11.77 km及I = 1.45 × 1045 g·cm2)
    Fig. 3. Normalized magnetic field components of the crustal confined for the force-free field: (red line), (blue line), and (yellow line) vs. normalized radial coordinate x. Here we assume the parameter μ = 1.676, corresponding to M = 1.45M, R = 11.77 km and I = 1.45 × 1045 g·cm2 in the TMA model. 在无力磁场结构位型下壳层归一化磁场分量 (红线), (蓝线), 及 (黄线)与归一化径向坐标x的关系(选取μ = 1.676, 对应在TMA模型下的M = 1.45M, R = 11.77 km及I = 1.45 × 1045 g·cm2)
    Relationship of σ to ρ, Τ and Q in the inner crust for magnetar: (a) The conductivity due to electron-phonon scattering; (b) the conductivity due to electron-impurity scattering. The EOS of BBP model is used.磁星壳层电导率随密度、温度及不纯净度参数的变化 (a)电导率由电子-声子散射主导; (b)电导率由电子-杂质散射主导; 物态方程一律采用BBP 模型
    Fig. 4. Relationship of σ to ρ, Τ and Q in the inner crust for magnetar: (a) The conductivity due to electron-phonon scattering; (b) the conductivity due to electron-impurity scattering. The EOS of BBP model is used. 磁星壳层电导率随密度、温度及不纯净度参数的变化  (a)电导率由电子-声子散射主导; (b)电导率由电子-杂质散射主导; 物态方程一律采用BBP 模型
    Numerical fitting of Ohmic decay for magnetars: (a) The poloidal magnetic field, Bp, as a function of t at x = 1; (b) the toroidal magnetic field, Bt, as a function of t when at x = 1; (c) the poloidal magnetic field decay rate, dBp/dt, as a function of t when at x = 1; (d) the toroidal field decay rate, dBt/dt, as a function of t when at x = 1; (e) the poloidal field energy decay rate, Lp, as a function of t; (f) the toroidal filed energy decay rate, Lt, as a function of t. The red and blue lines in (a)−(f) indicateand , respectively.磁星磁场欧姆衰变的数值模拟 (a) 在x = 1处极向磁场Bp随时间t的变化; (b) 在x = 1处极向磁场Bt随时间t的变化; (c) 在x = 1处极向磁场衰减率dBp/dt, 随时间t的变化; (d) 在x = 1处环向磁场衰减率dBt/dt, 随时间t的变化; (e) 极化磁场的能量衰减率Lp随时间t的变化; (e) 环向磁场的能量衰减率Lt随时间t的变化; 在(a)−(f)图中红色和蓝颜色的线分别表示和
    Fig. 5. Numerical fitting of Ohmic decay for magnetars: (a) The poloidal magnetic field, Bp, as a function of t at x = 1; (b) the toroidal magnetic field, Bt, as a function of t when at x = 1; (c) the poloidal magnetic field decay rate, dBp/dt, as a function of t when at x = 1; (d) the toroidal field decay rate, dBt/dt, as a function of t when at x = 1; (e) the poloidal field energy decay rate, Lp, as a function of t; (f) the toroidal filed energy decay rate, Lt, as a function of t. The red and blue lines in (a)−(f) indicate and , respectively. 磁星磁场欧姆衰变的数值模拟 (a) 在x = 1处极向磁场Bp随时间t的变化; (b) 在x = 1处极向磁场Bt随时间t的变化; (c) 在x = 1处极向磁场衰减率dBp/dt, 随时间t的变化; (d) 在x = 1处环向磁场衰减率dBt/dt, 随时间t的变化; (e) 极化磁场的能量衰减率Lp随时间t的变化; (e) 环向磁场的能量衰减率Lt随时间t的变化; 在(a)−(f)图中红色和蓝颜色的线分别表示 和
    The -plot for our magnetars and selected objects in isotropic heating models.在各向同性加热模型下磁星及相关致密天体-的关系图
    Fig. 6. The - plot for our magnetars and selected objects in isotropic heating models. 在各向同性加热模型下磁星及相关致密天体 - 的关系图
    Fitting relationship between the soft X-ray luminosity and rotational energy loss rate of magnetars in the isotropic heating model.在各向同性加热模型下拟合得到的磁星的旋转能损率与软X射线光度的关系
    Fig. 7. Fitting relationship between the soft X-ray luminosity and rotational energy loss rate of magnetars in the isotropic heating model.在各向同性加热模型下拟合得到的磁星的旋转能损率与软X射线光度的关系
    RMF模型${\rho _0}$/fm–3${E_0}$/MeV ${K_0}$/MeV m* K′/MeV J/MeV ${L_0}$/MeV $K_{{\rm{sym}}}^0$/MeV $Q_{{\rm{sym}}}^0$/MeV $K_{\tau ,V}^0$/MeV
    NL30.148–16.24271.530.60–202.9137.40118.53100.88181.31–698.85
    TMA0.147–16.33318.150.635572.1230.6690.1410.75–108.74–367.99
    GM10.153–16.02300.500.70215.6632.5294.0217.9825.01–478.64
    Table 1.

    Saturation properties of nuclear matter in the parameterizations for NL3, GM1 and TMA models.

    在NL3, GM1和TMA模型下饱和核物质特性.

    m/MR/km Rcore/R$\mu $I/g·cm2
    注: *在TMA模型下由物态方程给出的最大中子星质量.
    1.2011.420.9151.6781.03(1) × 1045
    1.4511.770.9171.6761.47(2) × 1045
    1.7212.050.9191.6751.87(2) × 1045
    2.03*11.250.9141.6792.09(2) × 1045
    Table 2.

    Partial values of m, R, Rcore/R, μ and I for magnetars in TMA model.

    在TMA模型中磁星的m, R, Rcore/R, μI的部分值

    T = 1 × 108 K T = 2 × 108 K T = 3 × 108 K
    $Q = 1$$Q = 5$$Q = 10$$Q = 1$$Q = 5$$Q = 10$$Q = 1$$Q = 5$$Q = 10$
    $\rho $/g·cm–3ZA$\sigma $/1023 s–1$\sigma $/1023 s–1$\sigma $/1023 s–1$\sigma $/1023 s–1$\sigma $/1023 s–1$\sigma $/1023 s–1$\sigma $/1023 s–1$\sigma $/1023 s–1$\sigma $/1023 s–1
    Bp = 5 × 1014 G 4.66 × 1011401270.4552.150.7521.691.150.5910.9980.8210.490
    6.61 × 1011401300.6412.580.8652.241.450.7031.180.9820.592
    8.79 × 1011411340.9283.220.9912.971.540.8221.311.200.702
    1.20 × 1012421371.263.721.153.882.210.9532.081.490.787
    1.47 × 1012421401.974.631.234.892.501.042.431.690.867
    2.00 × 1012431442.624.781.426.313.101.223.182.111.03
    2.67 × 1012441492.675.591.687.823.751.414.082.591.29
    3.51 × 1012451543.426.411.8510.304.521.625.203.141.40
    4.54 × 1012461614.207.262.0815.605.241.896.533.781.65
    6.25 × 1012481705.588.562.3717.506.422.188.604.681.96
    8.38 × 1012491816.959.672.6622.207.462.4910.905.552.23
    1.10 × 1013511938.5811.402.9927.908.752.8113.706.602.55
    1.50 × 10135421110.8012.903.4535.6010.403.2417.307.952.95
    1.99 × 10135723213.0014.903.9543.6012.103.7321.209.373.12
    2.58 × 10136025715.2016.904.4651.2013.804.2224.9010.803.88
    3.44 × 10136529617.7019.705.2259.6016.204.9328.9012.504.53
    4.68 × 10137235420.4023.506.2367.7019.105.8732.6014.605.37
    5.96 × 10137842121.7026.507.0869.0021.106.6333.8015.906.02
    8.01 × 10138954822.1031.208.4869.8023.807.8234.7017.206.95
    9.83 × 101310068323.2035.309.7869.8025.408.8336.0017.507.64
    1.30 × 101412099025.5040.3011.8070.8026.5010.1038.2018.108.20
    Bp = 3 × 1015 G 4.66 × 1011401270.4632.210.7641.701.180.6031.040.8300.505
    6.61 × 1011401300.6492.670.8732.291.500.7211.361.040.605
    8.79 × 1011411340.9433.301.093.051.710.8421.421.290.723
    1.20 × 1012421371.323.771.193.982.321.012.211.590.854
    1.47 × 1012421401.704.761.365.092.841.192.661.870.937
    2.00 × 1012431442.004.851.656.413.291.303.402.281.12
    2.67 × 1012441492.665.661.817.993.791.434.182.651.31
    3.51 × 1012451543.486.491.9111.304.581.645.203.171.45
    4.54 × 1012461614.207.322.1115.805.311.926.563.811.69
    6.25 × 1012481705.588.642.4417.906.492.248.654.741.99
    8.38 × 1012491816.949.742.6923.107.532.5211.205.612.27
    1.10 × 1013511938.5812.003.0628.808.802.8613.806.652.68
    1.50 × 10135421110.9013.203.5035.7010.803.2917.407.982.97
    1.99 × 10135723213.1015.103.9843.7012.603.7721.309.403.45
    2.58 × 10136025715.3017.004.4851.3014.004.2425.0011.103.90
    3.44 × 10136529617.7019.905.2559.7016.404.9528.9012.704.55
    4.68 × 10137235420.5023.706.2567.7019.305.8932.7014.705.38
    5.96 × 10137842121.8026.707.1069.0021.306.6533.8016.006.03
    8.01 × 10138954822.1031.308.4969.8023.907.8334.7017.306.96
    9.83 × 101310068323.2035.409.7970.3025.508.8536.1017.707.65
    1.30 × 101412099025.5040.3011.8070.8025.5010.1028.2018.108.20
    Table 3.

    Partial values of electrical conductivity for different temperatures and impurity parameters in the crust of magnetars. Here we use the equation of station (EOS) of BBP model.

    在不同温度和不同纯净度参数下磁星壳层电导率的部分值(采用BBP模型)

    $\sigma $/s–1t/a ${B_{\rm{p}}}$/G ${{{\rm{d}}B_{\rm{p}}^{}}/{{\rm{d}}t}}$/G·a–1${L_{\rm{p}}}$/erg·s–1${B_{\rm{t}}}$/G ${{{\rm{d}}B_{\rm{t}}^{}}/{{\rm{d}}t}}$/G·a–1${L_{\rm{t}}}$/erg·s–1${L_B}$/erg·s–1
    8.75 × 10245.0 × 1021.995 × 1015–5.92 × 1091.57 × 10341.965 × 1016–5.84 × 10106.28 × 10356.44 × 1035
    2.0 × 1031.981 × 1015–4.65 × 1091.15 × 10341.953 × 1016–4.58 × 10104.59 × 10354.70 × 1035
    2.0 × 1041.954 × 1015–1.37 × 1083.61 × 10331.927 × 1016–1.35 × 10101.44 × 10351.48 × 1035
    2.0 × 1051.844 × 1015–5.91 × 1081.63 × 10331.818 × 1016–5.84 × 10106.52 × 10346.68 × 1034
    2.0 × 1061.373 × 1015–8.61 × 1071.56 × 10321.354 × 1016–8.48 × 1086.24 × 10336.40 × 1033
    2.0 × 1076.865 × 1014–4.36 × 1077.85 × 10316.772 × 1015–4.29 × 1083.14 × 10333.22 × 1033
    2.52 × 10245.0 × 1021.990 × 1015–1.51 × 10103.98 × 10341.96 × 1016–1.49 × 10111.59 × 10361.63 × 1036
    2.0 × 1031.977 × 1015–5.43 × 10101.65 × 10341.95 × 1016–5.36 × 10106.61 × 10356.77 × 1035
    2.0 × 1041.931 × 1015-–1.86 × 1094.74 × 10331.905 × 1016–1.83 × 10101.90 × 10351.94 × 1035
    2.0 × 1051.745 × 1015–7.21 × 1091.69 × 10331.721 × 1016–7.11 × 10106.76 × 10346.93 × 1034
    2.0 × 1068.712 × 1014–3.87 × 1094.46 × 10328.592 × 1015–3.82 × 10101.78 × 10341.83 × 1034
    2.0 × 1072.749 × 1013–1.33 × 1074.82 × 10292.711 × 1014–1.31 × 1081.93 × 10311.98 × 1031
    Table 4. Partial values of Bp, dBp/dt, Lp, Bt, dBt/dt, Lt and LB when Bp(0) = 2.0 × 1015 G. Here we assume a medium-mass magnetar M = 1.45M, R = 11.77 km, Rc = 0.97 km, corresponding to I = 1.47I45 and , respectively. The top and bottom parts correspond to and , respectively. 当Bp(0) = 2.0 × 1015 G时Bp, dBp/dt, Lp, Bt, dBt/dt, LtLB的部分值(假定一个中等质量的磁星M = 1.45M, R = 11.77 km, Rc = 0.98 km, 对应着I = 1.47I45和 ; 表格上和下半部分分别对应着 和 )
    SourceP/s $\dot{ P}$/10–11 s–1${\tau _{\rm{c}}}$/ka Age Est/kaAssocia.Method$L_{\rm{X}}^\infty $/erg·s–1Lrot./erg·s–1Refs.
    SGR 0418+57299.078390.0004(1)36000550SMC磁热模拟9.60 × 10293.1 × 1029[46,48,49]
    1E 2259+5866.979040.04837230.010—20SNR CTB109SNR年龄1.70 × 10347.37 × 1031[5052]
    4U 0142+618.688700.2022(4)68.068.0SMC特征年龄1.05 × 10351.85 × 1032[49,50,53]
    CXOU J16471010.61< 0.04> 420.0> 420Cluster Wdl特征年龄4.50 × 1032< 1.88 × 1031[54,55]
    1E 1048–59376.457872.2504.54.5GSH 288.3–0.5–28特征年龄4.90 × 10344.65 × 1033[5658]
    CXOU J0100438.020391.88(8)6.86.8SMC特征年龄6.50 × 10342.33 × 1033[49,59]
    1RXS J17084911.005021.9455(13)9.09.0MC 13A特征年龄4.20 × 10347.37 × 1032[50,55]
    1E 1841–04511.788984.092(15)4.700.5—1.0SNR Kes73SNR年龄1.84 × 10351.47 × 1033[50,60]
    SGR 0501+45165.762060.594(2)16.004—6SNR HB9SNR年龄8.10 × 10321.85 × 1033[6163]
    SGR 0526–668.054(2)3.8(1)3.4004.8SNR N49SNR年龄1.89 × 10354.22 × 1033[64,65]
    SGR 1900+145.199879.2(4)0.9003.98—7.9Massive star Cluster自行年龄9.00 × 10343.79 × 1034[6668]
    SGR 1806–207.5477349.50000.2400.63—1.0W31, MC13A自行年龄1.63 × 10356.68 × 1034[68,69]
    XTE J1810–1975.540350.777(3)1111W31, MC13A特征年龄4.3 × 10312.93 × 1035[69,70]
    IE 1547–54082.072124.770.690.63SNR G327.24–013SNR年龄1.3 × 10333.11 × 1035[71,72]
    3XXMJ18524611.5587< 0.014> 13005—7SNR Kes 79SNR年龄< 4.0 × 1038< 4.8 × 1038[73,74]
    CXOU J1714053.825356.400.955CTB 37BSNR年龄5.6 × 10346.13 × 1034[45,75]
    SGR 1627–412.594581.9(4)2.25.0SNR G337.0–0.1SNR年龄3.6 × 10335.87 × 1034[76,77]
    Swift J1822–16068.437720.0021(2)63006300HII region特征年龄< 4.0 × 10292.0 × 1030[78,79]
    Swift J1834–08642.48230.796(12)4.960200SNR W41SNR年龄< 8.4 × 10303.1 × 1034[80,81]
    PSR J1622–49504.326(1)1.7(1)4.0≤ 6.0SNR G33.9+0.0SNR年龄4.40 × 10321.18 × 1034[63,82]
    SGR J1745–29003.76361.385(15)4.304.30Galaxy Center特征年龄1.10 × 10321.47 × 1034[83,84]
    PSR J1846–02580.326570.710700.730.9-4.3SNR Kes75SNR年龄1.90 × 10348.10 × 1036[49,85]
    Table 5.

    The persistent timing, ages and emission characteristics for 22 magnetars with observed soft X-ray flux.

    具有软X射线辐射的22颗磁星的到达时间及其辐射特性

    SourceBp(0)/G PL Ind.$T_{BB}^{\infty} $/keV D/kpc $F_{\rm{X}}^\infty $/erg·s–1·cm2$L_{\rm{X}}^\infty $/erg·s–1$L_B^{\rm{a}}$/erg·s–1$\eta _{}^{\rm{a}}$/% $L_B^{\rm{b}}$/erg·s–1$\eta _{}^{\rm{b}}$/% Ref.
    注: a表示 $\sigma = 2.52 \times {10^{24} }\; { {\rm{s} }^{ {\rm{ - 1} } } }$的情况; b表示 $\sigma = 8.75 \times {10^{24} } \;{ {\rm{s} }^{ {\rm{ - 1} } } }$的情况; PL Ind. 表示幂率指数.
    SGR 0418–57293.0 × 10140.302.02.0 × 10–119.60 × 10295.35 × 10320.312.26 × 10320.74[48,49,50]
    1E 2259+5865.0 × 10143.75(4)0.37(1)3.2(2)1.41 × 10–111.70 × 10346.5(1.0) × 103522(6)1.4(3) × 103547(8)[5052]
    CXOU J1647103.0 × 10143.86(22)0.59(6)3.9(7)2.54 × 10–114.50 × 10328.65 × 103393.62 × 103321[50,54,95]
    3XXMJ1852463.0 × 10140.67.11.0 × 10–154.0 × 10333.53 × 10343.11 × 1035[73,74]
    4U 0142+613.0 × 10153.88(1)0.413.6(4)6.97 × 10–111.0 × 10351.14 × 1036154.85 × 103537[50,53,96]
    1E1048–59371.0 × 10153.14(11)0.56(1)9.0(1.7)5.11 × 10–114.90 × 10347.19 × 1035123.08 × 103527[50,57,58]
    CXOU J0100431.0 × 10150.30(2)62.4(1.6)1.40 × 10–116.50 × 10346.82 × 1035163.22 × 103534[50,97]
    IRXS J1708491.0 × 10152.79(1)0.4563.8(5)2.43 × 10–114.20 × 10347.65 × 103593.23 × 103521[50,53,96]
    1E1841–0451.0 × 10151.9(2)0.45(3)8.6(1.1)2.13 × 10–111.84 × 10351.2(2) × 103626(4)5.9(7) × 103546(4)[50,98,99]
    SGR 0526–663.0 × 1015$2.5_{ - 0.12}^{ + 0.11}$0.44(2)53.6(1.2)5.50 × 10–111.89 × 10352.28 × 103687.11 × 103526[50,65]
    SGR1900+143.0 × 10151.9(1)0.47(2)13.0(1.2)4.82 × 10–129.0 × 10342.2(6) × 10367(1)7.8(8) × 103519(2)[50,66]
    SGR1806–203.0 × 10151.6(1)0.55(7)8.8(1.6)1.81 × 10–121.63 × 10353.8(4) × 10367.4(8)8.9(9) × 103526(2)[50,69]
    Table 6.

    The X-ray emission characteristics and magnetic field energy decay rates of 12 magnetars with rotational energy loss rates less than their soft X-ray luminosities.

    12颗旋转能损率远小于软X射线光度的磁星的辐射特性及磁场能衰变率

    Jian-Ling Chen, Hui Wang, Huan-Yu Jia, Zi-Wei Ma, Yong-Hong Li, Jun Tan. Conductivity of neutron star crust under superhigh magnetic fields and Ohmic decay of toroidal magnetic field of magnetar[J]. Acta Physica Sinica, 2019, 68(18): 180401-1
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