太阳日冕波动加热机制研究
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摘要
近半个多世纪以来，日冕加热问题一直都是日地物理学界关注研究的难题。本论文则主要地研究了Alfven波在日冕加热过程中的作用。我们从两个方面对这个问题进行了探讨：
第一个方面主要地研究了加热日冕的Alfven波源。我们采用Lighthill-Stein理论分析了对流层中的湍流运动对扭转Alfven波的激发作用。就是将MHD方程中的非线性项通过Fourier变换转化为湍动源项，并采用Kolmogroff幂律谱来描述对流层的湍动谱。这样，通过计算可以得到磁通管中被激发的扭转Alfven能谱。当被激发的扭转Alfven波沿着磁通管向上传播到达日冕，其携带的能量就可以加热日冕。
第二个方面是研究冕环中的Alfven波耗散机制。我们在线性共振吸收理论的基础上，对波动耗散过程中的非线性效应做了进一步的研究。通过三波共振模型的分析，可以发现Pump波能被集聚在共振层附近的大幅度背景扰动强烈调制。这个大幅度扰动则是冕环对足点运动造成驱动的共振响应所形成的。由于位于共振层附近的相调层非常薄，那么，对应的位于这个薄层之内的Pump波就会和在这个薄层之外传播的Pump波有强烈的相混作用，从而对共振层附近的Pump波有很强的阻尼作用。
在完成了本论文的理论工作后，我们将这两个部分的理论结合起来，就得到了一个相对完整的日冕加热模型，包含了Alfven波的波源和耗散机制。利用理论结果的计算表明，对流层的湍动对流完全有可能激发足够的波能去补充日冕中等离子体对流和辐射造成的能量损失。将计算得到的激发扭转Alfven能谱做为边界条件代入，可以对扭转Alfven波在冕环中的耗散做一些定量的分析。从我们的计算中得到的Pump波的阻尼长度的量级大概可以和最近的冕环足根加热观测相符合。
本课题相关的论文已经投递和发表在国际主要的学术期刊A&A和ApJ上，得到了国内外同行较好的评价。
As one of the major problems in solar-terrestrial physics, coronal heating has kept to attract physicists' attention for a few decades. In this thesis, we investigate the role of Alfven wave in the coronal heating process. All the work consists of two main parts.
In the first part, turbulent motion in convection zone as source to generate Alfven waves is studied following Lighthill-Stein theory, in which the turbulent source was described by nonlinear terms in MHD equations. Using the technique of Fourier transformation, we can derive the Alfven wave energy spectrum generated from convection turbulence modeled with Kolmogroff-like power spectrum. Then, torsional Alfven waves propagate upwards along the coronal loop and the carried wave energy can make a play in coronal heating.
The second part of our work is devoted to the dissipation mechanism of Alfven wave in coronal loop. On the basis of the linear theory of resonant absorption, nonlinear effect is taken into account in the wave dynamics. With a three wave interaction model, the pump wave mode is found to be strongly modulated by the background fluctuation, accumulated due to Alfven resonance near the resonant layer. And consequently, the phase modulation dissipation arises from the strong phase mixing between the waves inside and outside the modulation layer, and the pump wave is strongly damped near the resonant layer of coronal loop.
After above analytical theories, we combine the two part of work into an integrated model of coronal heating. Evaluation results show that the turbulence in convection zone is capable of generating enough wave energy to supply coronal energy lose due to convection and radiation. Then, the obtained energy spectrum of the generated torsional Alfven wave is considered as boundary condition of coronal loop to calculate the subsequent wave dissipation. More interestingly, the derived damping length scales of the pump wave are consistent with the constraints proposed recently from the observation of localized heating of coronal loops.
The papers related to this thesis were submitted and published in the central scientific journals, such as A&A and ApJ, and were highly evaluated by researchers overseas.
第一个方面主要地研究了加热日冕的Alfven波源。我们采用Lighthill-Stein理论分析了对流层中的湍流运动对扭转Alfven波的激发作用。就是将MHD方程中的非线性项通过Fourier变换转化为湍动源项，并采用Kolmogroff幂律谱来描述对流层的湍动谱。这样，通过计算可以得到磁通管中被激发的扭转Alfven能谱。当被激发的扭转Alfven波沿着磁通管向上传播到达日冕，其携带的能量就可以加热日冕。
第二个方面是研究冕环中的Alfven波耗散机制。我们在线性共振吸收理论的基础上，对波动耗散过程中的非线性效应做了进一步的研究。通过三波共振模型的分析，可以发现Pump波能被集聚在共振层附近的大幅度背景扰动强烈调制。这个大幅度扰动则是冕环对足点运动造成驱动的共振响应所形成的。由于位于共振层附近的相调层非常薄，那么，对应的位于这个薄层之内的Pump波就会和在这个薄层之外传播的Pump波有强烈的相混作用，从而对共振层附近的Pump波有很强的阻尼作用。
在完成了本论文的理论工作后，我们将这两个部分的理论结合起来，就得到了一个相对完整的日冕加热模型，包含了Alfven波的波源和耗散机制。利用理论结果的计算表明，对流层的湍动对流完全有可能激发足够的波能去补充日冕中等离子体对流和辐射造成的能量损失。将计算得到的激发扭转Alfven能谱做为边界条件代入，可以对扭转Alfven波在冕环中的耗散做一些定量的分析。从我们的计算中得到的Pump波的阻尼长度的量级大概可以和最近的冕环足根加热观测相符合。
本课题相关的论文已经投递和发表在国际主要的学术期刊A&A和ApJ上，得到了国内外同行较好的评价。
As one of the major problems in solar-terrestrial physics, coronal heating has kept to attract physicists' attention for a few decades. In this thesis, we investigate the role of Alfven wave in the coronal heating process. All the work consists of two main parts.
In the first part, turbulent motion in convection zone as source to generate Alfven waves is studied following Lighthill-Stein theory, in which the turbulent source was described by nonlinear terms in MHD equations. Using the technique of Fourier transformation, we can derive the Alfven wave energy spectrum generated from convection turbulence modeled with Kolmogroff-like power spectrum. Then, torsional Alfven waves propagate upwards along the coronal loop and the carried wave energy can make a play in coronal heating.
The second part of our work is devoted to the dissipation mechanism of Alfven wave in coronal loop. On the basis of the linear theory of resonant absorption, nonlinear effect is taken into account in the wave dynamics. With a three wave interaction model, the pump wave mode is found to be strongly modulated by the background fluctuation, accumulated due to Alfven resonance near the resonant layer. And consequently, the phase modulation dissipation arises from the strong phase mixing between the waves inside and outside the modulation layer, and the pump wave is strongly damped near the resonant layer of coronal loop.
After above analytical theories, we combine the two part of work into an integrated model of coronal heating. Evaluation results show that the turbulence in convection zone is capable of generating enough wave energy to supply coronal energy lose due to convection and radiation. Then, the obtained energy spectrum of the generated torsional Alfven wave is considered as boundary condition of coronal loop to calculate the subsequent wave dissipation. More interestingly, the derived damping length scales of the pump wave are consistent with the constraints proposed recently from the observation of localized heating of coronal loops.
The papers related to this thesis were submitted and published in the central scientific journals, such as A&A and ApJ, and were highly evaluated by researchers overseas.
引文
[1] 章振大，2000，日冕物理，科学出版社
[2] 林元章，2000，太阳物理导论，科学出版社
[3] Grotrian W. 1939, Naturwiss, 27, 214
[4] Edlen, B. 1941. Ark. Mat., astr.och fys. 28B. No.1
[5] Biermann, L. 1946, Naturwiss, 33, 118
[6] Schatzman, E. 1949, Ann, d' Astrophs. 12, 203
[7] Narain, U., & Ulmschneider, P. 1990, Space Sci. Rev., 54, 377
[8] Narain, U., & Ulmschneider, P. 1996, Space Sci. Rev., 75, 453
[9] Biermann, L. 1948, Z. Astrophys, 25, 161
[10] Schwarzschild, M. 1948, Astrophys_ J., 107, 1
[11] Jordan, S. D., 1993, ApJ, 414,337
[12] Ulmschneider, P., 1970, Solar Phys., 12, 403
[13] Endler, F.& Deubner, F. L., 1983, A&A, 121, 291
[14] Deubner, F. L. 1988, in R. Stalio and L. A. Wilson (eds.), Pulsation and Mass Loss in Stars, Kluwer, 163
[15] Califano, F., Chiuderi, C., & Einaudi, G 1992 Astrophys J 390 560
[16] Uberoi, C. 1990, in Basic Plasma Processes on the Sun, ER Priest, V Krishan Eds., Kluwer, Dordrecht, 239
[17] Porter, L. J., Klimchuk, J. A. &Sturrock, P. A. 1994a, Astrophys. J. 435, 482
[18] Porter, L. J., Klimchuk, J. A. &Sturrock, P. A. 1994b, Astrophys. J. 435, 502
[19] Alfven, H., 1947, MNRAS, 107, 211
[20] Parker, E. N., 1983, ApJ, 264, 642
[21] Parker, E. N., 1972, ApJ, 174, 499
[22] Priest, E. R. & Raadu, M. A., 1975, Solar Phys., 43, 177
[23] Strauss, H. R. 1988, Astrophys. J., 326, 412
[24] Beaufume, P., Coppi, B. & Golub, L., 1992, ApJ, 393, 396
[25] Parker, E. N., 1988, Astrophys. J. 330, 474
[26] Sturruck, P. A. et al., 1990, ApJ, 356, L31
[27] Aschwanden, M. J. 2001, ApJ, 560, 1035
[28] Schrijver, C. J. & Aschwanden, M. J. 2002, \apj, 566, 1147
[29] Lighthill, M. J. 1952, Proc. Roy. Soc. London A, 211, 564
[30] Stein, R. F. 1967, Sol. Phys., 2, 385
[31] Collins, W. 1989a, ApJ, 337, 548
[32] Collins, W. 1989b, ApJ, 343, 499
[33] Collins, W. 1992, ApJ, 384, 319
[34] Musielak, Z. E., Rosner, R., & Ulmschneider, P. 1989, ApJ, 241, 625
[35] Musielak, Z. E., Rosner, R., Gail, H. P.,& Ulmschneider, P. 1995, ApJ, 448, 865
[36] Musielak, Z. E., Ulmschneider, P. 2001, A&A, 370, 541
[37] Lee, J. W. 1993, ApJ, 404, 372
[38] Musielak, Z. E., Rosner, R., & Ulmschneider, P. 2000, ApJ, 541,410
[39] Spruit, H. C. 1981, A&A, 98, 155
[40] Spruit, H. C. 1982, Sol. Phys., 75, 3
[41] Roberts, B., & Ulmschneider, P. 1997, in Solar and Heliospheric Plasma Physics, ed. G. M. Simett, C. E. Alissandrakis, & L. Vlahos (Springer Verlag, Berlin), 75
[42] Ziegler,U.; & Ulmschneider, P. 1997, Astronomy and Astrophysics, v.324, 417-431
[43] Ziegler,U.; & Ulmschneider, P. 1997, Astronomy and Astrophysics, v.327, 854-862
[44] Hollweg, J. V. 1984, Solar Phys., 91, 269
[45] Zhugzhda, Y. D. & Locans, V. 1982, Solar Phys., 76, 77
[46] Hollweg, J. V. 1984, ApJ, 277, 392
[47] Heyvaerts, J., Priest, E. R. 1983, A&A, 117, 220
[48] De Moortel, I., Hood, A.W. & Arber, T.D. 2000, A&A, 354, 334
[49] Ruderman, M. S., Goldstein, M. L., Roberts, D. A., Deane, A., & Ofman, L. 1999, J. Geophys. Res., 104, 17057
[50] Ireland, J. & Priest, E. R. 1997, Solar Phys., 173, 31
[51] Wright, A. N., Allan, W., Elphinstone, R. D. & Cogger, L. L. 1999, J. Geophys. Res., 104, 10159
[52] Tstaronis&Grossmann,1973，Z. Phys. 261, 203
[53] Hen & Hasegawa 1974, Phys. Fluids, 17, 1399
[54] Ionson, J.A., 1978, ApJ, 226, 650.
[55] Davila, J. M., 1987, ApJ, 317, 514
[56] Poedts, S., Goossens, M., & Kerner, W. 1989, Sol. Phys., 123, 83
[57] Goossens, M., Ruderman,M. S., & Hollweg, J. V. 1995, Sol. Phys.,157, 75
[58] Hollweg, J. V. 1997, J. Geophys. Res., 102, 24127
[59] Ruderman, M. S. 1999, ApJ, 521, 851
[60] Ofman, L., Klimchuk, J. A. & Davila, J. M. 1998, ApJ, 493, 474
[61] Belien, A. J. C., Martens, P. C. H., & Keppens, R. 1999, ApJ, 526, 478
[62] Stenflo. J. O. 1994 Solar Magnetic Fields, Kluwer, Dordrecht
[63] Aschwanden, M. J., Nightingale, R. W. & Alexander, D. 2000, ApJ, 541, 1059
[64] Priest, E. R., Foley, C. R., Heyvaerts, J., Arber, T. D., Culhane, J. L., & Acton, L. W. 1998, Nature, 393, 545
[65] Priest, E. R., Foley, C. R., Heyvaerts, J., Arber, T. D., Mackay, D., Culhane, J. L., & Acton, L. W. 2000, ApJ, 539, 1002
[66] Mackay, D. H., Galsgaard, K., Priest, E. R., & Foley, C. R. 2000, Sol. Phys., 193, 93
[67] Aschwanden, M. J., & Nitta, N. 2000, ApJ, 535, L59
Aschwanden, M. J., Schrijver, C. J., & Alexander, D. 2001, ApJ, 550,
[68] 1036
[69] Musielak, Z. E., Rosner, R., Stein, R. F., & Ulmschneider, P. 1994, ApJ, 423, 474
[70] Muller, R., Roudier, Th., Vigneau, J., & Au.ret, H. 1994, A&A, 283, 232
[71] Huang, P., Musielak, Z. E., & Ulmschneider, P. 1995, A&A, 279, 579
[72] Roberts, B., & Ulmschneider, P. 1997, in Solar and Heliospheric Plasma Physics, ed. G. M. Simett, C. E. Alissandrakis, & L. Vlahos (Springer Verlag, Berlin), 75
[73] Nakariakov, V. M., Roberts, B., & Murawski, K. 1997, Sol. Phys., 175, 93
[74] Ulmschneider, P., & Musielak, Z. E. 1998, A&A, 338, 311
[75] Berghmans, D. and De Bruyne, P., 1995, ApJ, 453,495
[76] Sakurai, T., Goossens, M., & Hollweg, J. V. 1991, Solar Phys., 133, 227
[77] Sakurai, T., Goossens, M., & Hollweg, J. V. 1991, Solar Phys., 133, 247
[78] Stenuit, H., Erdelyi, R. & Goossens, M. 1995, Solar Phys., 161, 139.
[79] Goossens, M. & Ruderman, M.S., 1995, Phys. Scripta, T60, 171.
[80] Boris, J.P., 1968, 'Resistive modified normal modes of an inhomogeneous incompressible plasma', Ph.D. Thesis Princeton University, UMI Dissertation Services, Ann Arbor Michigan, p. 172.
[81] Ofman, L. & Davila, J. M. 1995, J. Geophys. Res., 100, 23427
[82] Ofman, L. & Davila, J. M. 1995, ApJL, 456, L123
[83] De Groof, A. & Goossens, M. 2002, A&A, 386, 681
[84] Berghmans, D. & Tirry, W. J., 1997, A&A, 325, 318
[85] Tirry, W. J & Berghmans, D, 1997, A&A, 325, 329
[86] De Groof, A. & Goossens, M. 2000, A&A, 356, 724
[87] De Groof, A., Tirry, W.J. & Goossens, M. 1998, A&A, 335, 329
[88] J. Weiland, H. Wilhelmsson, Coherent Nonlinear Interaction of Waves in Plasmas, Pergamon, Oxford, 1977
[89] Manley, J. M. & Rowe, H. E. 1959, Proc. IER., 47, 2115
[90] Fuchs, V. & Beaudry, G. 1975, J. Math. Phys., 17, 208
[91] David K. Rollins, Bhimsen K. Shivamoggi, 2000, Physics Letters A. 268, 382
[92] Parker, E. N. 1991, ApJ, 376, 355
[2] 林元章，2000，太阳物理导论，科学出版社
[3] Grotrian W. 1939, Naturwiss, 27, 214
[4] Edlen, B. 1941. Ark. Mat., astr.och fys. 28B. No.1
[5] Biermann, L. 1946, Naturwiss, 33, 118
[6] Schatzman, E. 1949, Ann, d' Astrophs. 12, 203
[7] Narain, U., & Ulmschneider, P. 1990, Space Sci. Rev., 54, 377
[8] Narain, U., & Ulmschneider, P. 1996, Space Sci. Rev., 75, 453
[9] Biermann, L. 1948, Z. Astrophys, 25, 161
[10] Schwarzschild, M. 1948, Astrophys_ J., 107, 1
[11] Jordan, S. D., 1993, ApJ, 414,337
[12] Ulmschneider, P., 1970, Solar Phys., 12, 403
[13] Endler, F.& Deubner, F. L., 1983, A&A, 121, 291
[14] Deubner, F. L. 1988, in R. Stalio and L. A. Wilson (eds.), Pulsation and Mass Loss in Stars, Kluwer, 163
[15] Califano, F., Chiuderi, C., & Einaudi, G 1992 Astrophys J 390 560
[16] Uberoi, C. 1990, in Basic Plasma Processes on the Sun, ER Priest, V Krishan Eds., Kluwer, Dordrecht, 239
[17] Porter, L. J., Klimchuk, J. A. &Sturrock, P. A. 1994a, Astrophys. J. 435, 482
[18] Porter, L. J., Klimchuk, J. A. &Sturrock, P. A. 1994b, Astrophys. J. 435, 502
[19] Alfven, H., 1947, MNRAS, 107, 211
[20] Parker, E. N., 1983, ApJ, 264, 642
[21] Parker, E. N., 1972, ApJ, 174, 499
[22] Priest, E. R. & Raadu, M. A., 1975, Solar Phys., 43, 177
[23] Strauss, H. R. 1988, Astrophys. J., 326, 412
[24] Beaufume, P., Coppi, B. & Golub, L., 1992, ApJ, 393, 396
[25] Parker, E. N., 1988, Astrophys. J. 330, 474
[26] Sturruck, P. A. et al., 1990, ApJ, 356, L31
[27] Aschwanden, M. J. 2001, ApJ, 560, 1035
[28] Schrijver, C. J. & Aschwanden, M. J. 2002, \apj, 566, 1147
[29] Lighthill, M. J. 1952, Proc. Roy. Soc. London A, 211, 564
[30] Stein, R. F. 1967, Sol. Phys., 2, 385
[31] Collins, W. 1989a, ApJ, 337, 548
[32] Collins, W. 1989b, ApJ, 343, 499
[33] Collins, W. 1992, ApJ, 384, 319
[34] Musielak, Z. E., Rosner, R., & Ulmschneider, P. 1989, ApJ, 241, 625
[35] Musielak, Z. E., Rosner, R., Gail, H. P.,& Ulmschneider, P. 1995, ApJ, 448, 865
[36] Musielak, Z. E., Ulmschneider, P. 2001, A&A, 370, 541
[37] Lee, J. W. 1993, ApJ, 404, 372
[38] Musielak, Z. E., Rosner, R., & Ulmschneider, P. 2000, ApJ, 541,410
[39] Spruit, H. C. 1981, A&A, 98, 155
[40] Spruit, H. C. 1982, Sol. Phys., 75, 3
[41] Roberts, B., & Ulmschneider, P. 1997, in Solar and Heliospheric Plasma Physics, ed. G. M. Simett, C. E. Alissandrakis, & L. Vlahos (Springer Verlag, Berlin), 75
[42] Ziegler,U.; & Ulmschneider, P. 1997, Astronomy and Astrophysics, v.324, 417-431
[43] Ziegler,U.; & Ulmschneider, P. 1997, Astronomy and Astrophysics, v.327, 854-862
[44] Hollweg, J. V. 1984, Solar Phys., 91, 269
[45] Zhugzhda, Y. D. & Locans, V. 1982, Solar Phys., 76, 77
[46] Hollweg, J. V. 1984, ApJ, 277, 392
[47] Heyvaerts, J., Priest, E. R. 1983, A&A, 117, 220
[48] De Moortel, I., Hood, A.W. & Arber, T.D. 2000, A&A, 354, 334
[49] Ruderman, M. S., Goldstein, M. L., Roberts, D. A., Deane, A., & Ofman, L. 1999, J. Geophys. Res., 104, 17057
[50] Ireland, J. & Priest, E. R. 1997, Solar Phys., 173, 31
[51] Wright, A. N., Allan, W., Elphinstone, R. D. & Cogger, L. L. 1999, J. Geophys. Res., 104, 10159
[52] Tstaronis&Grossmann,1973，Z. Phys. 261, 203
[53] Hen & Hasegawa 1974, Phys. Fluids, 17, 1399
[54] Ionson, J.A., 1978, ApJ, 226, 650.
[55] Davila, J. M., 1987, ApJ, 317, 514
[56] Poedts, S., Goossens, M., & Kerner, W. 1989, Sol. Phys., 123, 83
[57] Goossens, M., Ruderman,M. S., & Hollweg, J. V. 1995, Sol. Phys.,157, 75
[58] Hollweg, J. V. 1997, J. Geophys. Res., 102, 24127
[59] Ruderman, M. S. 1999, ApJ, 521, 851
[60] Ofman, L., Klimchuk, J. A. & Davila, J. M. 1998, ApJ, 493, 474
[61] Belien, A. J. C., Martens, P. C. H., & Keppens, R. 1999, ApJ, 526, 478
[62] Stenflo. J. O. 1994 Solar Magnetic Fields, Kluwer, Dordrecht
[63] Aschwanden, M. J., Nightingale, R. W. & Alexander, D. 2000, ApJ, 541, 1059
[64] Priest, E. R., Foley, C. R., Heyvaerts, J., Arber, T. D., Culhane, J. L., & Acton, L. W. 1998, Nature, 393, 545
[65] Priest, E. R., Foley, C. R., Heyvaerts, J., Arber, T. D., Mackay, D., Culhane, J. L., & Acton, L. W. 2000, ApJ, 539, 1002
[66] Mackay, D. H., Galsgaard, K., Priest, E. R., & Foley, C. R. 2000, Sol. Phys., 193, 93
[67] Aschwanden, M. J., & Nitta, N. 2000, ApJ, 535, L59
Aschwanden, M. J., Schrijver, C. J., & Alexander, D. 2001, ApJ, 550,
[68] 1036
[69] Musielak, Z. E., Rosner, R., Stein, R. F., & Ulmschneider, P. 1994, ApJ, 423, 474
[70] Muller, R., Roudier, Th., Vigneau, J., & Au.ret, H. 1994, A&A, 283, 232
[71] Huang, P., Musielak, Z. E., & Ulmschneider, P. 1995, A&A, 279, 579
[72] Roberts, B., & Ulmschneider, P. 1997, in Solar and Heliospheric Plasma Physics, ed. G. M. Simett, C. E. Alissandrakis, & L. Vlahos (Springer Verlag, Berlin), 75
[73] Nakariakov, V. M., Roberts, B., & Murawski, K. 1997, Sol. Phys., 175, 93
[74] Ulmschneider, P., & Musielak, Z. E. 1998, A&A, 338, 311
[75] Berghmans, D. and De Bruyne, P., 1995, ApJ, 453,495
[76] Sakurai, T., Goossens, M., & Hollweg, J. V. 1991, Solar Phys., 133, 227
[77] Sakurai, T., Goossens, M., & Hollweg, J. V. 1991, Solar Phys., 133, 247
[78] Stenuit, H., Erdelyi, R. & Goossens, M. 1995, Solar Phys., 161, 139.
[79] Goossens, M. & Ruderman, M.S., 1995, Phys. Scripta, T60, 171.
[80] Boris, J.P., 1968, 'Resistive modified normal modes of an inhomogeneous incompressible plasma', Ph.D. Thesis Princeton University, UMI Dissertation Services, Ann Arbor Michigan, p. 172.
[81] Ofman, L. & Davila, J. M. 1995, J. Geophys. Res., 100, 23427
[82] Ofman, L. & Davila, J. M. 1995, ApJL, 456, L123
[83] De Groof, A. & Goossens, M. 2002, A&A, 386, 681
[84] Berghmans, D. & Tirry, W. J., 1997, A&A, 325, 318
[85] Tirry, W. J & Berghmans, D, 1997, A&A, 325, 329
[86] De Groof, A. & Goossens, M. 2000, A&A, 356, 724
[87] De Groof, A., Tirry, W.J. & Goossens, M. 1998, A&A, 335, 329
[88] J. Weiland, H. Wilhelmsson, Coherent Nonlinear Interaction of Waves in Plasmas, Pergamon, Oxford, 1977
[89] Manley, J. M. & Rowe, H. E. 1959, Proc. IER., 47, 2115
[90] Fuchs, V. & Beaudry, G. 1975, J. Math. Phys., 17, 208
[91] David K. Rollins, Bhimsen K. Shivamoggi, 2000, Physics Letters A. 268, 382
[92] Parker, E. N. 1991, ApJ, 376, 355