搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

大质量转动星族Ⅲ恒星中心氢氦燃烧阶段演化

詹琼 宋汉峰 戚世涛 曲新玥 韩卓 钟文丽

引用本文:
Citation:

大质量转动星族Ⅲ恒星中心氢氦燃烧阶段演化

詹琼, 宋汉峰, 戚世涛, 曲新玥, 韩卓, 钟文丽
cstr: 32037.14.aps.74.20250704

Study of the evolutionary stage of H and He burning in the center of massive rotating Population Ⅲ stars

ZHAN Qiong, SONG Hanfeng, QI Shitao, QU Xinyue, HAN Zhuo, ZHONG Wenli
cstr: 32037.14.aps.74.20250704
Article Text (iFLYTEK Translation)
PDF
HTML
导出引用
  • 转动是影响恒星结构和演化的一个非常重要的物理因素. 本文根据零金属丰度恒星演化模型, 研究了转动如何影响第一代(星族Ⅲ)大质量恒星中心氢和氦燃烧阶段的演化. 结果发现转动在此阶段演化过程中最主要的影响是提高恒星光度并降低表面温度. 光度的增大是由于转动混合导致对流核心增大, 而表面温度的下降则是由氢燃烧壳层产生的能量驱动的恒星半径膨胀引起的. 星族Ⅲ恒星的子午环流非常弱, 剪切湍流较强. 因此星族Ⅲ恒星传输角动量的效率非常低而混合化学元素的效率较高. 转动混合可促进氮元素的表面增丰, 然而, 在氦燃烧初期, 转动混合可能因其引发的能量产生机制变化和恒星结构变化而抑制该过程. 转动混合(剪切湍流)作用将包层的氢燃料带入燃烧的核心, 扩大核心区尺寸从而提升产能率并延长其主序寿命, 最终导致恒星光度增强. 在氦燃烧阶段氢燃烧壳层会影响氦核的大小和燃烧强度. 氦核的增长会反过来也影响氢燃烧壳层的尺寸和燃烧强度. 因此, 转动对星族Ⅲ恒星的演化产生至关重要的影响.
    The effects of rotation on the evolution of Population Ⅲ (Pop Ⅲ) stars in the burning stages of core H and He are investigated. Due to their zero-metallicity nature, these stars are initially unable to burn hydrogen through the CNO cycle (Here, C, N, and O stand for carbon, nitrogen, and oxygen, respectively). And without this crucial energy supply, they experience a contraction phase during the early main sequence (MS). The lack of CNO elements not only affects the central region of the star but also leads to energy increase (due to triggering of the CNO cycle) in the stellar envelope due to the outward diffusion of He-burning products. Therefore, rotational mixing has a unique effect on these stars.Rotation significantly affects the observable properties of Pop Ⅲ stars through two main effects. One is that rotational mixing brings additional fuel into the nuclear burning core, which increases the luminosity as well as the stellar lifetimes, and the other is that rotational mixing brings He-burning products from the core to the H-burning shell during later evolutionary phases. This will change the temperature distribution, and may lead to significant expansion in some models, depending on the relative core size. The relative core size is crucial here, because the contribution of the outer shell and the core to the total energy produced tells us about the structure of the star and dominant factors in the evolution of the surface properties.Despite weaker meridional currents in Pop Ⅲ stars, angular momentum can accumulate at the surface in fast-rotating massive models because of their negligible mass loss through radiative winds. This spin-up causes the models with an initial mass of 40M, an initial velocity of υini = 400 km/s, and a metallicity of Z = 10–4 to reach critical rotation during the MS, resulting in increased mass loss.Rotational mixing strongly affects metal enrichment, but unlike stars with high metallicity, it cannot consistently enhance metal production. Rotation leads to an early enhancement of CNO in the H shell during He burning, which may hinder metal enrichment. This effect also occurs during the core He-burning phase. In these cases, the convection caused by the CNO enhancement in the H shell will lead to the retraction of the He-burning core. As the core grows, the speed at which the H shell moves outwards is faster than the speed at which the He-burning products can be expelled from the core through rotational mixing, therefore hindering the interaction of these products with the H-burning shell, which is necessary for metal enrichment. H-He shell interactions after core He burning play a crucial role in metal production, where the rotation may enhance enrichment. This highlights the complexity in the metal enrichment processes of these models. A detailed understanding of the interior structure is therefore required to accurately predict metal yields.
      通信作者: 宋汉峰, hfsong@gzu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 12173010, 11863003)和项目“恒星内部结构和微观物理过程”资助的课题.
      Corresponding author: SONG Hanfeng, hfsong@gzu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12173010, 11863003) and the Internal Structure of Stars and Microscopic Physical Processes, China.
    [1]

    Savage B D, Sembach K R 1996 Annu. Rev. Astron. Astrophys. 34 279Google Scholar

    [2]

    Draine B T 2003 Annu. Rev. Astron. Astrophys. 41 241Google Scholar

    [3]

    Jenkins E B 2009 Astrophys. J. 700 1299Google Scholar

    [4]

    Meynet G, Georgy C, Hirschi R, Maeder A, Massey P, Przybilla N, Nieva M F 2011 Bull. R. Soc. Sci. Liège. 80 266Google Scholar

    [5]

    Cui Z, Wang Z J, Zhu C H 2018 Publ. Astron. Soc. Pac. 130 084202Google Scholar

    [6]

    彭卫国, 宋汉锋, 詹琼, 吴兴华, 景江红 2019 物理学报 68 219701Google Scholar

    Peng W G, Song H F, Zhan Q, Wu X H, Jing J H 2019 Acta Phys. Sin. 68 219701Google Scholar

    [7]

    Wu F W, Song H F, Li Q L, He Y, Qu X Y, Han Z 2024 Chin. Phys. Lett. 41 089701Google Scholar

    [8]

    Marigo P, Chiosi C, Kudritzki R P 2003 Astron. Astrophys. 399 617Google Scholar

    [9]

    Marigo P, Girardi L, Chiosi C, Wood P R 2001 Astron. Astrophys. 371 152Google Scholar

    [10]

    Ekström S, Meynet G, Chiappini C, Hirschi R, Maeder A 2008 Astron. Astrophys. 489 685Google Scholar

    [11]

    Yoon S C, Dierks A, Langer N 2012 Astron. Astrophys. 542 A113Google Scholar

    [12]

    Ekström S, Georgy C, Eggenberger P, Meynet G, Mowlavi N, Wyttenbach A, Granada A, Decressin T, Hirschi R, Frischknecht U, Charbonnel C, Maeder A 2012 Astron. Astrophys. 537 A146Google Scholar

    [13]

    Georgy C, Ekström S, Eggenberger P, Meynet G, Haemmerlé L, Maeder A, Granada A, Groh J H, Hirschi R, Mowlavi N, Yusof N, Charbonnel C, Decressin T, Barblan F 2013 Astron. Astrophys. 558 A103Google Scholar

    [14]

    Groh J H, Ekström S, Georgy C, Meynet G, Choplin A, Eggenberger P, Hirschi R, Maeder A, Murphy L J, Boian I, Farrell E J 2019 Astron. Astrophys. 627 A24Google Scholar

    [15]

    Bromm V, Kudritzki R P, Loeb A 2001 Astrophys. J. 552 464Google Scholar

    [16]

    Abel T, Bryan G L, Norman M L 2002 Sci. 295 93Google Scholar

    [17]

    Heger A, Woosley S E 2002 Astrophys. J. 567 532Google Scholar

    [18]

    Brott I, de Mink S E, Cantiello M, Langer N, de Koter A, Evans C J, Hunter I, Trundle C, Vink J S 2011 Astron. Astrophys. 530 A115Google Scholar

    [19]

    Stacy A, Bromm V, Loeb A 2011 Mon. Not. R. Astron. Soc. 413 543Google Scholar

    [20]

    Stacy A, Greif T H, Klessen R S, Bromm V, Loeb A 2013 Mon. Not. R. Astron. Soc. 431 1470Google Scholar

    [21]

    Hirano S, Bromm V 2018 Mon. Not. R. Astron. Soc. 476 3964Google Scholar

    [22]

    Murphy L J, Groh J H, Ekström S, Meynet G, Pezzotti C, Georgy C, Choplin A, Eggenberger P, Farrell E, Haemmerlé L, Hirschi R, Maeder A, Martinet S 2021 Mon. Not. R. Astron. Soc. 501 2745Google Scholar

    [23]

    Sibony Y, Liu B, Simmonds C, Meynet G, Bromm V 2022 Astron. Astrophys. 666 A199Google Scholar

    [24]

    Aryan A, Pandey S B, Gupta R, Ror A K 2023 Mon. Not. R. Astron. Soc. 521 L17Google Scholar

    [25]

    Tsiatsiou S, Sibony Y, Nandal D, Sciarini L, Hirai Y, Ekström S, Farrell E, Murphy L, Choplin A, Hirschi R, Chiappini C, Liu B, Bromm V, Groh J, Meynet G 2024 Astron. Astrophys. 687 A307Google Scholar

    [26]

    Zahn J P 1992 Astron. Astrophys. 265 115Google Scholar

    [27]

    Maeder A, Zahn J P 1998 Astron. Astrophys. 334 1000Google Scholar

    [28]

    Kippenhahn R, Thomas H C 1969 Mitt. Astron. Ges. 27 168Google Scholar

    [29]

    Maeder A, Meynet G 2004 Proceedings of IAU Symposium Cancun, Yucatan, Mexico, November 11-15, 2002 p500

    [30]

    Kaehler H 1986 Astron. Astrophys. 157 329Google Scholar

    [31]

    Maeder A, Meynet G 2012 Reviews of Modern Physics. 84 25Google Scholar

    [32]

    Heger A, Langer N, Woosley S E 2000 Astrophys. J. 528 368Google Scholar

    [33]

    Maeder A 1995 Astron. Astrophys. 299 84Google Scholar

    [34]

    Maeder A 1997 Astron. Astrophys. 321 134Google Scholar

    [35]

    Vink J S, de Koter A, Lamers H J G L M 2001 Astron. Astrophys. 369 574Google Scholar

    [36]

    Iglesias C A, Rogers F J 1996 Astrophys. J. 464 943Google Scholar

    [37]

    Ferguson J W, Alexander D R, Allard F, Hauschildt P H 2001 Astrophys. J. 557 798Google Scholar

    [38]

    Asplund M, Grevesse N, Sauval A J 2005 Astronomical Society of the Pacific Conference Series Austin, Texas, June 17-19, 2004 p25

  • 图 1  (a) 初始转速为$ {v_{{\text{ini}}}} $= 0, 200, 400 km/s的不同金属丰度的40M恒星模型的角速度比率$ \varOmega /{\varOmega _{{\text{crit}}}} $演化情况, 绿色虚线表示模型达到临界转速的位置; (b) 初始转速$ {v_{{\text{ini}}}} $= 0, 200, 400 km/s的不同金属丰度的40M恒星模型的表面赤道速度的演化情况; (c) 金属丰度Z = 10–4的40M恒星的子午环流的垂直速度在中心氢燃烧阶段的分布轮廓图

    Fig. 1.  (a) The evolution of $ \varOmega /{\varOmega _{{\text{crit}}}} $ for 40M stellar models with different metallicities, and initial rotation speeds of 0, 200, and 400 km/s, the green dashed line in the figure indicates the position where the model reaches the critical rotation speed; (b) the evolution of surface equatorial velocities for 40M stellar models with different metallicities and initial rotation speeds of 0, 200, and 400 km/s; (c) the profile for the vertical components of meridional circulation in a 40M star with metallicity Z = 10–4 during the central H burning.

    图 2  (a) 金属丰度Z = 0, 40M恒星在不同初始转动速度下的星风损失随时间演化; (b) 金属丰度Z = 10–4, 40M恒星在不同初始转动速度下的星风损失随时间演化

    Fig. 2.  (a) The evolution of stellar wind mass loss with time for 40M with Z = 0 metallicity under different initial rotation speeds; (b) the evolution of stellar wind mass loss with time for 40M with Z = 10–4 metallicity under different initial rotation speeds.

    图 3  (a) 无转动情况下金属丰度Z = 0和Z = 10–4的大质量恒星在赫罗图中的演化; (b) 40M恒星在不同转动速度和不同金属丰度下的赫罗图的演化

    Fig. 3.  (a) The evolution of massive stars with Z = 0 and Z = 10–4 metallicities in the HR diagram under non-rotation conditions; (b) the evolution of HR diagrams for 40M stars under different rotation speeds and metallicities.

    图 4  (a) 金属丰度Z = 0, 40M恒星在不同初始转动速度下的对流核随时间演化; (b) 金属丰度Z = 10–4, 40M恒星在不同初始转动速度下的对流核随时间演化

    Fig. 4.  (a) The evolution of convection cores with time for 40M with 0 metallicity under different initial rotation speeds; (b) the evolution of convection cores with time for 40M with 10–4 metallicity under different initial rotation speeds.

    图 5  (a) 金属丰度Z = 0的40M恒星在不同初始转动速度下表面14N随时间的演化; (b) 金属丰度Z = 10–4的40M恒星在不同初始转动速度下表面14N随时间的演化

    Fig. 5.  (a) The evolution of surface N-14 abundance with time for 40M with 0 metallicity under different initial rotation speeds; (b) the evolution of surface 14N abundance with time for 40M with 10–4 metallicity under different initial rotation speeds.

    图 6  (a) 40M恒星在不同初始转动速度和金属丰度下中心温度随时间的演化; (b) 40M恒星在不同初始转动速度和金属丰度下中心密度随时间的演化

    Fig. 6.  (a) The evolution of central temperature with time for 40M stars under different initial rotation and metallicities; (b) the evolution of central density with time for 40M stars under different initial rotation and metallicities.

    图 7  40M的转动恒星金属丰度分别为Z = 0和Z = 10–4的模型在氦燃烧阶段3个不同时刻(以中心氦含量Yc为标识)的能量产生率分布图, 图中绿色(黑色)实线分别代表氦(氢)燃烧产生的能量, 红色虚线表示对应光度对总光度的贡献比例(数值标注于右侧纵轴), 对流区域用灰色阴影区域表示 (a1), (a2) Yc = 0.8; (b1), (b2) Yc = 0.5; (c1), (c2) Yc = 0.2

    Fig. 7.  Energy production capacity distribution diagrams at three different moments during the He-burning core phase for rotating 40M stellar models with metallicities of 0 and 10–4, the green (or black) solid line represent the energy generated by He (or H) burning, respectively, the red dashed line indicates the contribution ratio of the corresponding luminosity to the total luminosity(the numerical values are labeled on the right vertical axis), convective regions are indicated by the grey shaded areas: (a1), (a2) Yc = 0.8; (b1), (b2) Yc = 0.5; (c1), (c2) Yc = 0.2.

    图 8  初始转动速度为400 km/s的40M恒星模型在金属丰度为Z = 0和Z = 10–4下氦燃烧阶段3个不同时刻(以中心氦含量Yc为标识)的元素丰度分布, 丰度曲线展示了恒星从中心到表面的化学元素分布, 对流区域用灰色阴影区域表示 (a1), (a2) Yc = 0.8; (b1), (b2) Yc = 0.5; (c1), (c2) Yc = 0.2

    Fig. 8.  Elemental abundance distributions at three different moments during the He-burning phase for 40M stellar models with an initial rotation speed of 400 km/s, under two metallicities cases Z = 0 and Z = 10–4, the curves in the figure show the chemical element distribution from the center to the surface of the star, convective regions are indicated by the grey shaded areas: (a1), (a2) Yc = 0.8; (b1), (b2) Yc = 0.5; (c1), (c2) Yc = 0.2.

  • [1]

    Savage B D, Sembach K R 1996 Annu. Rev. Astron. Astrophys. 34 279Google Scholar

    [2]

    Draine B T 2003 Annu. Rev. Astron. Astrophys. 41 241Google Scholar

    [3]

    Jenkins E B 2009 Astrophys. J. 700 1299Google Scholar

    [4]

    Meynet G, Georgy C, Hirschi R, Maeder A, Massey P, Przybilla N, Nieva M F 2011 Bull. R. Soc. Sci. Liège. 80 266Google Scholar

    [5]

    Cui Z, Wang Z J, Zhu C H 2018 Publ. Astron. Soc. Pac. 130 084202Google Scholar

    [6]

    彭卫国, 宋汉锋, 詹琼, 吴兴华, 景江红 2019 物理学报 68 219701Google Scholar

    Peng W G, Song H F, Zhan Q, Wu X H, Jing J H 2019 Acta Phys. Sin. 68 219701Google Scholar

    [7]

    Wu F W, Song H F, Li Q L, He Y, Qu X Y, Han Z 2024 Chin. Phys. Lett. 41 089701Google Scholar

    [8]

    Marigo P, Chiosi C, Kudritzki R P 2003 Astron. Astrophys. 399 617Google Scholar

    [9]

    Marigo P, Girardi L, Chiosi C, Wood P R 2001 Astron. Astrophys. 371 152Google Scholar

    [10]

    Ekström S, Meynet G, Chiappini C, Hirschi R, Maeder A 2008 Astron. Astrophys. 489 685Google Scholar

    [11]

    Yoon S C, Dierks A, Langer N 2012 Astron. Astrophys. 542 A113Google Scholar

    [12]

    Ekström S, Georgy C, Eggenberger P, Meynet G, Mowlavi N, Wyttenbach A, Granada A, Decressin T, Hirschi R, Frischknecht U, Charbonnel C, Maeder A 2012 Astron. Astrophys. 537 A146Google Scholar

    [13]

    Georgy C, Ekström S, Eggenberger P, Meynet G, Haemmerlé L, Maeder A, Granada A, Groh J H, Hirschi R, Mowlavi N, Yusof N, Charbonnel C, Decressin T, Barblan F 2013 Astron. Astrophys. 558 A103Google Scholar

    [14]

    Groh J H, Ekström S, Georgy C, Meynet G, Choplin A, Eggenberger P, Hirschi R, Maeder A, Murphy L J, Boian I, Farrell E J 2019 Astron. Astrophys. 627 A24Google Scholar

    [15]

    Bromm V, Kudritzki R P, Loeb A 2001 Astrophys. J. 552 464Google Scholar

    [16]

    Abel T, Bryan G L, Norman M L 2002 Sci. 295 93Google Scholar

    [17]

    Heger A, Woosley S E 2002 Astrophys. J. 567 532Google Scholar

    [18]

    Brott I, de Mink S E, Cantiello M, Langer N, de Koter A, Evans C J, Hunter I, Trundle C, Vink J S 2011 Astron. Astrophys. 530 A115Google Scholar

    [19]

    Stacy A, Bromm V, Loeb A 2011 Mon. Not. R. Astron. Soc. 413 543Google Scholar

    [20]

    Stacy A, Greif T H, Klessen R S, Bromm V, Loeb A 2013 Mon. Not. R. Astron. Soc. 431 1470Google Scholar

    [21]

    Hirano S, Bromm V 2018 Mon. Not. R. Astron. Soc. 476 3964Google Scholar

    [22]

    Murphy L J, Groh J H, Ekström S, Meynet G, Pezzotti C, Georgy C, Choplin A, Eggenberger P, Farrell E, Haemmerlé L, Hirschi R, Maeder A, Martinet S 2021 Mon. Not. R. Astron. Soc. 501 2745Google Scholar

    [23]

    Sibony Y, Liu B, Simmonds C, Meynet G, Bromm V 2022 Astron. Astrophys. 666 A199Google Scholar

    [24]

    Aryan A, Pandey S B, Gupta R, Ror A K 2023 Mon. Not. R. Astron. Soc. 521 L17Google Scholar

    [25]

    Tsiatsiou S, Sibony Y, Nandal D, Sciarini L, Hirai Y, Ekström S, Farrell E, Murphy L, Choplin A, Hirschi R, Chiappini C, Liu B, Bromm V, Groh J, Meynet G 2024 Astron. Astrophys. 687 A307Google Scholar

    [26]

    Zahn J P 1992 Astron. Astrophys. 265 115Google Scholar

    [27]

    Maeder A, Zahn J P 1998 Astron. Astrophys. 334 1000Google Scholar

    [28]

    Kippenhahn R, Thomas H C 1969 Mitt. Astron. Ges. 27 168Google Scholar

    [29]

    Maeder A, Meynet G 2004 Proceedings of IAU Symposium Cancun, Yucatan, Mexico, November 11-15, 2002 p500

    [30]

    Kaehler H 1986 Astron. Astrophys. 157 329Google Scholar

    [31]

    Maeder A, Meynet G 2012 Reviews of Modern Physics. 84 25Google Scholar

    [32]

    Heger A, Langer N, Woosley S E 2000 Astrophys. J. 528 368Google Scholar

    [33]

    Maeder A 1995 Astron. Astrophys. 299 84Google Scholar

    [34]

    Maeder A 1997 Astron. Astrophys. 321 134Google Scholar

    [35]

    Vink J S, de Koter A, Lamers H J G L M 2001 Astron. Astrophys. 369 574Google Scholar

    [36]

    Iglesias C A, Rogers F J 1996 Astrophys. J. 464 943Google Scholar

    [37]

    Ferguson J W, Alexander D R, Allard F, Hauschildt P H 2001 Astrophys. J. 557 798Google Scholar

    [38]

    Asplund M, Grevesse N, Sauval A J 2005 Astronomical Society of the Pacific Conference Series Austin, Texas, June 17-19, 2004 p25

  • [1] 李辰恺, 朱金龙. 高压调控过渡金属硫族化合物及异质结构的光电性质. 物理学报, 2025, 74(17): 176802. doi: 10.7498/aps.74.20250498
    [2] 赵诗艺, 刘承志, 黄修林, 王夷博, 许妍. 强磁场对中子星转动惯量与表面引力红移的影响. 物理学报, 2021, 70(22): 222601. doi: 10.7498/aps.70.20211051
    [3] 刘慧莹, 王树申, 林恒福. 单层haeckelites结构Ⅲ族金属硫族化合物MX (M = Al, Ga, In; X = S, Se, Te). 物理学报, 2020, 69(14): 146802. doi: 10.7498/aps.69.20191955
    [4] 彭卫国, 宋汉峰, 詹琼, 吴兴华, 景江红. 大质量转动沃尔夫-拉叶星的形成及内部核合成研究. 物理学报, 2019, 68(21): 219701. doi: 10.7498/aps.68.20191040
    [5] 周愈之. 过渡金属硫族化合物柔性基底体系的模型与应用. 物理学报, 2018, 67(21): 218102. doi: 10.7498/aps.67.20181571
    [6] 李志, 宋汉峰, 彭卫国, 王靖洲, 詹琼. 转动双星同步和轨道圆化的物理过程研究. 物理学报, 2018, 67(19): 199701. doi: 10.7498/aps.67.20181056
    [7] 邰丽婷, 宋汉峰, 王江涛. 临界转动恒星Achernar的斜压结构与引力昏暗的精细研究. 物理学报, 2016, 65(4): 049701. doi: 10.7498/aps.65.049701
    [8] 詹琼, 宋汉峰, 邰丽婷, 王江涛. 转动潮汐变形双星理论模型研究. 物理学报, 2015, 64(8): 089701. doi: 10.7498/aps.64.089701
    [9] 王瑨, 李春梅, 敖靖, 李凤, 陈志谦. IVB族过渡金属氮化物弹性与光学性质研究. 物理学报, 2013, 62(8): 087102. doi: 10.7498/aps.62.087102
    [10] 宋汉峰, 王靖洲, 李云. 辐射压对非同步转动双星系统洛希势函数的影响. 物理学报, 2013, 62(5): 059701. doi: 10.7498/aps.62.059701
    [11] 李海滨, 王博华, 张志强, 刘爽, 李延树. 一类非线性相对转动系统的组合共振分岔与混沌. 物理学报, 2012, 61(9): 094501. doi: 10.7498/aps.61.094501
    [12] 付宏洋, 文德华, 燕晶. 考虑非牛顿引力下的快速转动混合星性质. 物理学报, 2012, 61(20): 209701. doi: 10.7498/aps.61.209701
    [13] 蔡元学, 掌蕴东, 党博石, 吴昊, 王金芳, 袁萍. 基于Ⅲ-Ⅴ与Ⅱ-Ⅵ族半导体材料色散特性的高灵敏度慢光干涉仪. 物理学报, 2011, 60(4): 040701. doi: 10.7498/aps.60.040701
    [14] 王勇, 郭永新, 吕群松, 刘畅. 非完整映射理论与刚体定点转动的几何描述. 物理学报, 2009, 58(8): 5142-5149. doi: 10.7498/aps.58.5142
    [15] 时培明, 蒋金水, 刘彬. 耦合相对转动非线性动力系统的稳定性与近似解. 物理学报, 2009, 58(4): 2147-2154. doi: 10.7498/aps.58.2147
    [16] 汪 华, 刘世林, 刘 杰, 王凤燕, 姜 波, 杨学明. N2O+离子A2Σ+电子态高振动能级的转动结构分析. 物理学报, 2008, 57(2): 796-802. doi: 10.7498/aps.57.796
    [17] 石筑一, 张春梅, 童 红, 赵行知, 倪绍勇. 102Ru核振动到转动演化的微观研究. 物理学报, 2008, 57(3): 1564-1568. doi: 10.7498/aps.57.1564
    [18] 方建会, 赵嵩卿. 相对论性转动变质量系统的Lie对称性与守恒量. 物理学报, 2001, 50(3): 390-393. doi: 10.7498/aps.50.390
    [19] 阎宏, 常哲, 郭汉英. q变形转动振子模型(Ⅰ)——q振子与双原子分子振动谱. 物理学报, 1991, 40(9): 1377-1387. doi: 10.7498/aps.40.1377
    [20] 顾本源, 董碧珍, 郑师海, 杨国桢. 空间平移不变和角度受限下转动不变的特征识别与滤波器的设计. 物理学报, 1985, 34(6): 760-765. doi: 10.7498/aps.34.760
计量
  • 文章访问数:  396
  • PDF下载量:  4
  • 被引次数: 0
出版历程
  • 收稿日期:  2025-05-30
  • 修回日期:  2025-07-07
  • 上网日期:  2025-07-17
  • 刊出日期:  2025-09-05

/

返回文章
返回