臺灣博碩士論文加值系統

摘要
本論文，研究弱磁場對緻密星（中子星；夸克星）的影響。我們探討不同磁場強度(γ= B/Bcn =0， 0.01， 0.05， 0.1， 0.2)對緻密星的穩定性和結構影響的可能性。首先，討論星球結構的三個基本方程式：質量方程式、ＴＯＶ方程式、狀態方程式，並採用 Euler-MacLaurin expansion 在弱磁場極限下的修正項，以修正狀態方程式。利用數值積分法分析緻密星的 Chandrasekhar 極限，在沒有磁場情況下，且以Ideal Fermi gas描述中子物質，中子星的Chandrasekhar 極限(M0)約為0.66倍太陽質量，而當磁場強度γ=0.01， 0.05， 0.1， 0.2，得到中子星的Chandrasekhar 極限(MB)分別約為0.66， 0.69， 0.79， 1.22倍太陽質量。中子
星的Chandrasekhar 極限隨磁場增強的增加率約為
， ( γ ≤ 0.2 ) 。
在沒有磁場情況下，夸克星的Chandrasekhar 極限(M0)約為1.07倍太陽質量，而當磁場強度γ= 0.01， 0.05， 0.1， 0.2，得到夸克星的Chandrasekhar 極限(MB)分別約為1.07，1.09，1.16，2.15倍太陽質量。夸克星的Chandrasekhar 極限隨磁場增強的增加率約為
， ( γ ≤ 0.2 ) 。
所以，當磁場強度γ＜0.1對緻密星的Chandrasekhar 極限影響不明顯。比較緻密星在沒有磁場和有磁場的Gibbs 自由能，具有磁場的緻密星具有較大的穩定性。

Abstract
In this thesis, we investigate the influence of weak magnetic fields on dense stars (neutron stars & quark stars). We explore the possibility that such dense stars have weak magnetic fields (γ= B/Bcn =0, 0.01, 0.05, 0.1, 0.2) and how their stability and structure are affected. We study properties of a degenerate ideal neutron star and strange star with and without the weak magnetic fields. First, we discuss three fundamental equations: the mass equation; the Tolman-Oppenheimer-Volkoff equation; and the equa-tion of state (EOS). We then apply an Euler-MacLaurin expansion (Ker-nan, Starkman, & Vachaspati 1996) [1] in the weak magnetic field limit to modify the EOS. We use the numerical integration method to obtain the Chandrasekhar limit of the dense stars. The Chandrasekhar limit (M0) of the neutron star is around 0.66 times the solar mass in the absence of a magnetic field. As the strengths of magnetic field varies with γ= 0.01, 0.05, 0.1, 0.2, the Chandrasekhar limit (MB) of neutron stars ranges from 0.66, 0.69, 0.79, 1.22 times the solar mass, respectively. The rate of in-crease for the maximum neutron star mass is given approximately by
, ( γ ≤ 0.2 ).
The Chandrasekhar limit (M0) of the quark star is around 1.07 times the solar mass in the absence of a magnetic field. As the strengths of mag-netic field varies with γ= 0.01, 0.05, 0.1, 0.2, the Chandrasekhar limit (MB) of quark stars ranges from 1.07, 1.09, 1.16, 2.15 times the solar mass, re-spectively. The rate of increase for the maximum quark star mass is esti-mated to be
, ( γ ≤ 0.2 ).
The neutron stars and the quark stars with relatively weak magnetic fields (γ< 0.1) hardly change the Chandrasekhar limit. Gibb’s free energy may be used as a criterion to determine which case the stars will be more sta-ble. The Gibb's free energy for both neutron stars and quark stars with magnetic fields is always lower, so they are more stable.

Contents
1 Introduction 4
1-1 Dense stars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1-2 Magnetic fields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..6
2 Equation of state (EOS) for an ideal Fermi gas 8
2-1 Equation of state for an ideal Fermi gas without a magnetic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2-2 The modification of the equation of state for an ideal Fermi gas in a weak magnetic field. .. . . . . . . . . . . . . . . . . . . . . . . . . . .10
3 Equation of state for quark matter 14
3-1 Equation of state for quark matter with weak magnetic fields
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
4 Calculation and Results 17
4-1 The main program and calculation . . . . . . . . . . . . . . . . . . . . .17
4-2 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5 Summary 22
6 References 28

[1] Kernan, P. J., Starkman, G. D., & Vachuspati, T. 1996, Phys. Rev. D, 54,7207
[2] Hwang , W-Y. P., Z. Phys. A-Hadroons and Nuclei 338, 223-232 (1991)
[3] Witten, E.: 1984 Phys. Rev. D30, 272
[4] Ginzburg, V. L., 1964, Sov. Phys. Dok1., 9, 329
[5] Woltjer, L. 1964 ApJ, 140, 1309
[6] Hewish, A., Bell, S. J., Pilkington, J. D., Scott, P. F., & Collins, R. A. 1968, Na-ture 217, 709
[7] Gold, T. 1968, Nature, 218, 731
[8] Thompson, C., & Duncan, R. C. 1993, ApJ, 408, 194
[9] Ruderman, M. 1980, J. Phys. Suppl., 41, C2-125
[10] Galloway, D. J., Proctor, M. R. E., & Weiss, N. O. 1977, Nature, 266, 686
[11] Shapiro, S. L., & Teukolsky, A. A. 1983, Black Holes, White Dwarfs, and Neu-tron Stars (New York : Wiley)
[12] Lerche, I., & Schramm, D. N. 1977, ApJ, 216, 881
[13] Bocquet, M., Bonazzola, S., Gourgoulhon, E., & Novak, J. 1995, A&A, 301, 757
[14] Landau, L. D., &Lifshitz, E. M. 1938, Statistical Mechanics
[15] Blandford, R. D. & Hernquist, L. 1982, J. Phys. C: Solid State Phys., 15, 6233
[16] Suh, I.-S., & Mathews, G. J. 2000a, ApJ, 530, 949
[17] Suh, I.-S., & Mathews, G. J. 2001, ApJ, 546, 1126
[18] Hwang, W-Y. P., 1986 Program: tov.for