在经典理论中，不考虑磁场和星体旋转的白矮星的质量上限为1.44M⊙，它一般被称为“钱德拉塞卡质量极限”。当白矮星通过吸积或双星合并使自身质量超过该极限时，会因为内部的电子简并压无法平衡自引力而收缩，随之带来的物质密度增大与核心温度升高将引发不可控制的核聚变反应，产生能量巨大的Ia型超新星爆发。现在之所以将Ia型超新星作为宇宙学距离测量的所谓“标准烛光”，是因为理论上我们期待大部分Ia型超新星都具有相同的本征光度，而这一假定的规律与“钱德拉塞卡质量极限”有着密不可分的联系。然而过去二十年里，数个超高光度Ia型超新星的发现，暗示了质量超越“钱德拉塞卡质量极限”的白矮星的存在。为合理解释超大质量白矮星这一物理可能性，许多研究者将目光投向了磁白矮星的理论模型研究。在本论文的研究中，我们假设磁白矮星是准球对称且完全冷却的，我们考虑广义相对论效应，但忽略星体旋转、电子间的相互作用和因磁场而生的朗道量子化。磁白矮星内部磁场的几何构型还是未知的，我们尝试假定稳定的白矮星内部磁场是紊乱的，为此我们引入了一个横向分量远远大于径向分量，在同一层薄球壳内强度与方向均随机变化的磁场，并取其系综空间方均根值作为磁场强度近似代表值。给定相关边界条件，我们通过径向数值积分求解含有磁场修正的Tolman-Oppenheimer-Volkoff（TOV）方程，从而获得了带有随机横向磁场的白矮星的质量-半径关系等信息。然后，我们对此类型磁白矮星的数值解进行了广义相对论磁流体径向脉动稳定性分析，找到了磁白矮星的磁流体动力学稳定性临界点。我们发现，紊乱横向磁场的存在会使得白矮星相比于无磁场时具有更高的质量上限，且其对应的半径也更大。一个内部磁场强度峰值为3.46×10^14G的磁白矮星的质量上限为1.6507M⊙，这已经明显超过了白矮星的“钱德拉塞卡质量极限”。由于磁场越强，磁白矮星的最大质量也越大，所以当磁场强度非常极端时，白矮星的质量上限甚至可以达到24.6472M⊙，此时的磁场强度峰值为6.38×10^15G。我们还将紊乱横向磁场模型引入到了褐矮星质量上限的研究中。褐矮星的质量上限等于氢燃烧最小质量，这也是物理上褐矮星与主序恒星的质量分界线。通过对完全冷却的磁褐矮星结构进行计算和稳定性分析，我们发现，较弱的紊乱横向磁场会使得褐矮星最大质量比无磁场时稍小，但较大的磁场会使得褐矮星质量上限大大增加，甚至达到无磁场时质量上限的五倍之多。一般而言，磁褐矮星会因磁场的存在而较易被探测到。
According to the classical theory, the upper mass limit of white dwarfs without magnetic fields and rotations is 1.44M⊙, which is commonly referred to as the "Chandrasekhar mass limit". If a white dwarf exceeds this limit by accreting matter or binary merger, it will collapse as its electron degeneracy pressure cannot counterbalance the self-gravity. The resulting increase in mass density and core temperature will ignite a runaway nuclear fusion reaction, leading to a type Ia supernova explosion with enormous energy output. The reason why type Ia supernovae are used as so-called “standard candles” for cosmological distance measurement is that they are expected to have a uniform intrinsic luminosity, which is inextricably related to the "Chandrasekhar mass limit". However, the observation of several super-luminous type Ia supernovae in the past two decades has implied the existence of white dwarfs with masses beyond the “Chandrasekhar mass limit”. To explain the physical possibility of supermassive white dwarfs, many researchers have focused on the theoretical models of magnetized white dwarfs. In this paper, we assume magnetized white dwarfs are quasi-spherically symmetric and fully cooled. We consider the effect of general relativity, but neglect the rotation, electron interaction and Landau quantization due to magnetic fields. The geometric configuration of the internal magnetic fields of magnetized white dwarfs remains unknown, and we presume that magnetic fields inside stable white dwarfs are turbulent. Hence, we introduce a magnetic field with a transverse component much larger than the radial component, whose strength and direction vary randomly within each thin spherical shell, and use its ensemble space root mean square value as an approximation of the magnetic field strength. We solve the magnetic field modified Tolman-Oppenheimer-Volkoff (TOV) equations with given boundary conditions by radial numerical integration to obtain the mass-radius relation and other information for white dwarfs with random transverse magnetic fields. We also analyze the general relativistic magnetohydrodynamics radial pulsation stability of this type of magnetized white dwarfs, and find the critical point of magnetohydrodynamics stability of magnetized white dwarfs. We find that random transverse magnetic fields increase the upper mass limit and the corresponding radius of white dwarfs compared to those without magnetic fields. The maximum mass of a magnetized white dwarf with a peak internal magnetic field strength of 3.46 × 10^14G is 1.6507M⊙, which significantly exceeds the “Chandrasekhar mass limit” of white dwarfs. The stronger the magnetic field, the higher the maximum mass of magnetized white dwarfs, so the upper mass limit of white dwarfs is very astonishing when the magnetic field strength is extreme. The upper mass limit of a white dwarfs with a peak magnetic field strength of 6.38 × 10^15G can reach 24.6472M⊙ We also introduce the random transverse magnetic field model in our study of the upper mass limit of brown dwarfs. The upper mass limit of brown dwarfs is equal to the hydrogen burning minimum mass, which is also the physical dividing line between brown dwarfs and main-sequence stars. By calculating the structures of fully cooled magnetized brown dwarfs and analyzing their stability, we find that relatively weak random transverse magnetic fields make the maximum mass of brown dwarfs slightly smaller than that of nonmagnetic brown dwarfs, but relatively strong magnetic fields greatly increase the upper mass limit of brown dwarfs, even reaching five times the nonmagnetic limit. In general, magnetic fields enhance the detectability of brown dwarfs.
