The force-free magnetosphere around an oblique rotator

Autor: Toshio Uchida
Rok vydání: 1998
Předmět:
Zdroj: Monthly Notices of the Royal Astronomical Society. 297:315-322
ISSN: 1365-2966
0035-8711
DOI: 10.1046/j.1365-8711.1998.01528.x
Popis: 1 I N T RO D U C T I O N A rapidly rotating neutron star with a magnetosphere is a generally accepted picture of the radio pulsar (see Michel 1982, 1991 for review). The magnetic field of the pulsar is so strong that the structure of the pulsar magnetosphere is largely decided by the electromagnetic force. This indicates that the force-free condition becomes a good approximation at least in the vicinity of the star. The axisymmetric rotator has already been studied in the forcefree approximation extensively (Goldreich & Julian 1969; Michel 1973; Scharlemann & Wagoner 1973; Okamoto 1974, 1975). Consequently, we have already understood the method of dealing with a stationary axisymmetric force-free electromagnetic field. Further, overall consistency of the theoretical models has been discussed by confronting the resulting electromagnetic structure with the various microscopic pictures of the constituent plasma in the aligned rotator case (see Michel 1982, 1991 for review). In contrast, in spite of the belief that a pulsar is actually an oblique rotator, the theory for the oblique rotator is still in a preliminary level. Although some works have treated this problem (e.g., Mestel 1971, 1973; Endean 1972, 1976), the method of treating the force-free equation in the oblique rotator geometry has never been established. As a result, the relation between the axisymmetric rotator and the oblique rotator has not been fully clarified yet. Recently, we have given a general theory of the force-free electromagnetic field, and also developed a method for treating configurations with symmetry (Uchida 1997a,b). Since there is still much uncertainty about the plasma in the pulsar magnetosphere, it seems appropriate to treat the global electromagnetic field structure of the magnetosphere and the microscopic picture of the plasma separately. In this work we study the former problem, applying the theory presented in our foregoing works. We give a method to treat obliquely rotating force-free electromagnetic fields and clarify their properties. 2 T H E O RY O F T H E F O R C E F R E E E L E C T RO M AG N E T I C F I E L D We apply the force-free approximation to the whole magnetosphere. The force-free approximation is justified when the electromagnetic energy density is much greater than the rest mass of the plasma. Namely, the inertial force acting on the plasma is neglected. However, we do not need any further assumptions about the plasma. The basic equations are the force–force condition FmnJ n 1⁄4 0; ð2:1Þ and Maxwell’s equations (with c 1⁄4 1) =lFmn þ =mFnl þ =nFlm 1⁄4 0; ð2:2Þ =mF lm 1⁄4 4pJ; ð2:3Þ where Fmn is the electromagnetic field and J m is the four-current. The force-free electromagnetic field is a degenerate field, which satisfies Fmn*F mn 1⁄4 2B·E 1⁄4 0 where *F is the dual tensor of Fmn, B is the magnetic field and E is the electric field. Further, the physical force-free electromagnetic fields must satisfy the inequality FmnF mn > 0; ð2:4Þ if the velocity field U satisfying FmnU n 1⁄4 0 exists. A degenerate electromagnetic field is written as Fmn 1⁄4 ∂mf1∂nf2 1 ∂mf2∂nf1; ð2:5Þ in terms of two Euler potentials f1 and f2 (Uchida 1997a). This is equivalent to B 1⁄4 =f1 × =f2; E 1⁄4 1∂tf1=f2 þ ∂tf2=f1: ð2:6Þ In degenerate electrodynamics, an individual magnetic field line is regarded as a string-like entity that has its self-identity (Newcomb 1958). It draws a two-dimensional surface, i.e. world sheet, in the four-space–time. Thus the magnetic field lines introduce a family Mon. Not. R. Astron. Soc. 297, 315–322 (1998) q 1998 RAS *E-mail: uchida@yso.mtk.nao.ac.jp of two-dimensional surfaces in the four-space–time. The surfaces of constant f1 and f2 correspond to them. We call these surfaces the flux surfaces. A vector field tangential to the flux surface is called the generator of the flux surface. By equation (2.5), equation (2.2) is satisfied. Equations (2.1) and (2.3) become a closed equation for the Euler potentials. Since ∂lf1 and ∂lf2 are independent if the electromagnetic field does not vanish, it is decomposed as ∂mfi=n ∂f1∂f2 1 ∂f2∂f1 y 1⁄4 0; i 1⁄4 1; 2: ð2:7Þ These are the basic equations for the Euler potentials. Generally we must solve two non-linear equations for two Euler potentials to decide one force-free electromagnetic field configuration. 3 E L E C T RO M AG N E T I C F I E L D A R O U N D T H E O B L I Q U E RO TAT O R 3.1 The Euler potentials In the magnetosphere around the oblique rotator, the electromagnetic field satisfies the condition L yFmn 1⁄4 0; ð3:1Þ where Ly is the Lie derivative with respect to the vector field y, which is defined by y 1⁄4 ∂t þ Q∂J; ð3:2Þ where Q is the angular velocity of the star. Substituting expression (2.5) into equation (3.1), we have 0 1⁄4∂mðy∂lf1Þ∂nf2 þ ∂mf1∂nðy∂lf2Þ 1 ∂mðy∂lf2Þ∂nf1 1 ∂mf2∂nðy∂lf1Þ: ð3:3Þ As shown by Uchida (1997b), this yields y∂mf1 1⁄4 0; y∂mf2 1⁄4 1kðf1Þ; ð3:4Þ where kðf1Þ is an arbitrary function of f1. Integrating these equations, we have the forms of the Euler potentials as f1 1⁄4 w1ðr; v; JÞ; f2 1⁄4 1kt þ w2ðr; v; JÞ; ð3:5Þ where J 1⁄4 J 1 Qt: ð3:6Þ Equations (3.5) transform the dynamical variables from the Euler potentials to w1 and w2, which satisfy Lyw1 1⁄4 Lyw2 1⁄4 0. We should distinguish the k 1⁄4 0 case. If k Þ 0, we can set k to 1. In fact, we can introduce another Euler potentials of the form f1 1⁄4 kðw1Þdw1; f2 1⁄4 f2=kðw1Þ 1⁄4 1t þ w2=kðw1Þ; ð3:7Þ without a change in the electromagnetic field. Thus we can regard k as 1 or 0 in equation (3.5). In Sections 6 and 7, we use this freedom to shorten equations. In the following subsection only the degeneracy of the electromagnetic field and the condition for the symmetry are assumed. The force-free condition is not used yet. Thus the results are common to all the degenerate electromagnetic fields that satisfy equation (3.1). 3.2 Components of the electromagnetic field From equations (2.5) and (3.5), Fmn becomes Ftr 1⁄4 k∂rw1 þ Q ∂rw1∂Jw2 1 ∂rw2∂Jw1 y
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