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超强磁场下中子星壳层的电导率和磁星环向磁场欧姆衰变

陈建玲 王辉 贾焕玉 马紫微 李永宏 谭俊

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超强磁场下中子星壳层的电导率和磁星环向磁场欧姆衰变

陈建玲, 王辉, 贾焕玉, 马紫微, 李永宏, 谭俊

Conductivity of neutron star crust under superhigh magnetic fields and Ohmic decay of toroidal magnetic field of magnetar

Chen Jian-Ling, Wang Hui, Jia Huan-Yu, Ma Zi-Wei, Li Yong-Hong, Tan Jun
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  • 磁星是指主要由磁场提供辐射能量的一类脉冲星. 部分宁静状态下的磁星X射线有热起源, 对应的温度kT为0.2—0.6 keV (1 eV = 1.602 × 10–19 J), 这比转动供能的脉冲星的典型温度值高很多, 并且可以用黑体谱来拟合. 对磁星的观测和理论研究是当前脉冲星领域一个重要的热点. 结合物态方程, 本文首先计算了在超强磁场下壳层的电导率; 从统计上研究了由于环向磁场衰变, 磁场能释放率与磁星软X射线光度之间的关系. 通过分类和数值拟合, 所得到的新的拟合公式能较好地反映磁星软X射线光度和旋转能损率之间的关系. 研究发现, 对于绝大多数高X射线光度的磁星, 环向磁场欧姆衰变足够提供其观测的软X射线辐射; 对于低X射线光度的暂变磁星, 其软X射线辐射可能来源于旋转能损率、磁层流或粒子星风. 随着对磁星理论和观测研究的深入, 本文模型也会得到进一步的改进, 理论结果将更好地符合磁星的软X射线观测.
    Magnetar is a kind of pulsar powered by magnetic field energy. Part of the X-ray luminosities of magnetars in quiescence have a thermal origin and can be fitted by a blackbody spectrum with temperature kT ~ 0.2-0.6 keV, much higher than the typical values for rotation-powered pulsars. The observation and theoretical study of magnetar are one of hot topics in the field of pulsar research. The activity and emission characteristics of magnetar can be attributed to internal superhigh magnetic field. According to the work of WGW19 and combining with the equation of state, we first calculate the electric conductivity of the crust under a strong magnetic field, and then calculate the toroidal magnetic field decay rate and magnetic energy decay rate by using an eigenvalue equation of toroidal magnetic field decay and considering the effect of general relativity. We reinvestigate the LX-Lrot relationship of 22 magnetars with persistent soft X-ray luminosities and obtain two new fitting formulas on LX-Lrot. We find that for the magnetars with LX < Lrot, the soft X-ray radiations may originate from their rotational energy loss rate, or from magneto-sphere flow and particle wind heating. For the magnetars with LX > Lrot, the Ohmic decay of crustal toroidal magnetic fields can provide their observed isotropic soft X-ray radiation and maintain higher thermal temperature.As for the initial dipole magnetic fields of magnetars, we mainly refer to the rersearch by Viganò et al. (Viganò D, Rea N, Pons J A, Perna R, Aguilera D N, Miralles J A 2013 Mon. Not. R. Astron. Soc. 434 123), because they first proposed the up-dated neutron star magneto-thermal evolution model, which can successfully explain the X-ray radiation and cooling mechanism of young pulsars including magnetars and high-magnetic field pulsars. Objectively speaking, as to the decay of toroidal magnetic fields, there are some differences between our theoretical calculations of magnetic energy release rates and the actual situation of magnetic field decay in magnetars, this is because the estimate of initial dipolar magnetic field, true age and the thickness of inner crust of a magnetar are somewhat uncertain. In addition, due to the interstellar-medium’s absorptions to soft X-ray and the uncertainties of distance estimations, the observed soft X-ray luminosities of magnetars have certain deviations. With the continuous improvement of observation, equipment and methods, as well as the in-depth development of theoretical research, our model will be further improved, and the theoretical results are better accordant with the high-energy observation of magnetars.We also discuss other possible anisotropy origins of soft X-ray fluxes of magnetars, such as the formation of magnetic spots and thermoplastic flow wave heating in the polar cap. Although anisotropic heating mechanisms are different from Ohmic decay, all of them require that there exist strong toroidal magnetic fields inside a magnetar. However, the anisotropic heating mechanisms require higher toroidal multipole fields inside a magnetar (such as magnetic octupole field) and are related to complex Hall drift: these may be our research subjects in the future.
      通信作者: 陈建玲, chenjianling62@163.com
    • 基金项目: 国家自然科学基金(批准号: U1631106, U1431125, 11573059, 11847307, U1831102)、山西省高等学校科技创新项目(批准号: 2019L0863)和运城学院博士启动基金(批准号: YQ-2014013)资助的课题.
      Corresponding author: Chen Jian-Ling, chenjianling62@163.com
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. U1631106, U1431125, 11573059, 11847307, U1831102), the Scientific and Technologial Innovation Programs of Higher Education Institutions in Shanxi, China (Grant No. 2019L0863), and the Scientific Research Project of Yuncheng University, China (Grant No. YQ-2014013).
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  • 图 1  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.

    图 2  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.

    图 3  在无力磁场结构位型下壳层归一化磁场分量${{{B_r}}/{\left( {B\cos \theta } \right)}}$(红线), ${{{B_\theta }}/{\left( {B\sin \theta } \right)}}$(蓝线), 及${{{B_\phi }}/{\left( {B\sin \phi } \right)}}$(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: ${{{B_r}}/{\left( {B\cos \theta } \right)}}$(red line),${{{B_\theta }}/{\left( {B\sin \theta } \right)}}$(blue line), and ${{{B_\phi }}/{\left( {B\sin \phi } \right)}}$ (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.

    图 4  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.

    图 5  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$\sigma = 2.52 \times {10^{24}}\; {{\rm{s}}^{{\rm{ - 1}}}}$$\sigma = 8.75 \times {10^{24}} \;{{\rm{s}}^{{\rm{ - 1}}}}$

    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$\sigma = 2.52 \times {10^{24}}\; {{\rm{s}}^{{\rm{ - 1}}}}$ and $\sigma = 8.75 \times {10^{24}}\; {{\rm{s}}^{{\rm{ - 1}}}}$, respectively.

    图 6  在各向同性加热模型下磁星及相关致密天体$L_{\rm{X}}^\infty $-${L_{{\rm{rot}}}}$的关系图

    Fig. 6.  The $L_{\rm{X}}^\infty $-${L_{{\rm{rot}}}}$ plot for our magnetars and selected objects in isotropic heating models.

    图 7  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.

    表 1  在NL3, GM1和TMA模型下饱和核物质特性.

    Table 1.  Saturation properties of nuclear matter in the parameterizations for NL3, GM1 and TMA models.

    RMF模型${\rho _0}$/fm–3${E_0}$/MeV${K_0}$/MeVm*K′/MeVJ/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
    下载: 导出CSV

    表 2  在TMA模型中磁星的m, R, Rcore/R, μI的部分值

    Table 2.  Partial values of m, R, Rcore/R, μ and I for magnetars in TMA model.

    m/MR/kmRcore/R$\mu $I/g·cm2
    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
    注: *在TMA模型下由物态方程给出的最大中子星质量.
    下载: 导出CSV

    表 3  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)

    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.

    T = 1 × 108 KT = 2 × 108 KT = 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 G4.66 × 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 G4.66 × 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
    下载: 导出CSV

    表 4  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$\mu = 1.676$; 表格上和下半部分分别对应着$\sigma = 8.75 \times {10^{24}}\; {{\rm{s}}^{{\rm{ - 1}}}}$$\sigma = 2.52 \times {10^{24}}\; {{\rm{s}}^{{\rm{ - 1}}}}$)

    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 $\mu = 1.676$, respectively. The top and bottom parts correspond to $\sigma = 8.75 \times {10^{24}}\; {{\rm{s}}^{{\rm{ - 1}}}}$ and $\sigma = 2.52 \times {10^{24}}\; {{\rm{s}}^{{\rm{ - 1}}}}$, respectively.

    $\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
    下载: 导出CSV

    表 5  具有软X射线辐射的22颗磁星的到达时间及其辐射特性

    Table 5.  The persistent timing, ages and emission characteristics for 22 magnetars with observed soft X-ray flux.

    SourceP/s$\dot{ P}$/10–11 s–1${\tau _{\rm{c}}}$/kaAge 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]
    下载: 导出CSV

    表 6  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

    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.

    SourceBp(0)/GPL Ind.$T_{BB}^{\infty} $/keVD/kpc$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.
    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}$0.44(2)53.6(1.2)5.50 × 10–111.89 × 10352.28 × 103687.11 × 103526[50,65]
    SGR1900+143.0 × 10151.9(1)0.47(2)13.0(1.2)4.82 × 10–129.0 × 10342.2(6) × 10367(1)7.8(8) × 103519(2)[50,66]
    SGR1806–203.0 × 10151.6(1)0.55(7)8.8(1.6)1.81 × 10–121.63 × 10353.8(4) × 10367.4(8)8.9(9) × 103526(2)[50,69]
    注: a表示$\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.
    下载: 导出CSV
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出版历程
  • 收稿日期:  2019-05-19
  • 修回日期:  2019-07-12
  • 上网日期:  2019-09-01
  • 刊出日期:  2019-09-20

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