Introduction

The discovery of rapidly rotating neutron stars with a magnetic field G or T in the late 60s changed our view about stellar degenerate matter, and revealed the existence of an astrophysical laboratory for electrodynamics. A rapidly rotating conducting neutron star with a strong magnetic field will induce significant electric fields across the magnetic field lines. For a typical rotation rate (e.g. 0.03 s for the Crab pulsar) and the typical magnetic field strength shown above, the electric field ( m) has a component along a field line and is strong enough to pull the charged particles out of the stellar surface to form a pulsar magnetosphere, despite huge surface gravity (Goldreich & Julian 1969; Mestel 1971). Such a magnetosphere is responsible for generating spin-modulated radio emission which led to the discovery of these pulsing neutron stars - pulsars. Strong centrifugal effect, due to the rapid stellar spin, becomes dominant at large distance, and a pulsar wind inevitablely developes (cf. e.g. Mestel & Shibata 1994, and Michel 1969; Goldreich & Julian 1970). Understanding these magnetospheres and associated pulsar winds has been one of the important frontiers of theoretical research.

Comparing to an outflow from a main sequence star, the pulsar wind is different as the special relativity effect plays an important role. The pressure and even the gravitational forces acting on the wind plasma, however, may not be as important as the electromagnetic and centrifugal forces. Theoretical modelling of the wind thus often neglects the thermal and gravity effects, and such a wind is usually called a gravity-free, cold relativistic wind (see Michel 1969). An analytic approach usually assumes mass conservation and an ideal plasma (dissipation-free). The former is probably the crudest one among all, as the plasma may be generated due to electron-positron pair production along curved field lines in the the magnetosphere (e.g. Sturrock 1971; Ruderman & Sutherland 1975), indicating that the inner boundary conditions for a wind may not be well-defined. But nevertheless, one may ignore this complication and assume that ideal MHD equations are applicable to the whole domain. Assuming a steady state and axisymmetry, the ideal MHD equations for a pulsar wind can be integrated along a poloidal field line. Although the transfield equation (e.g., Okamoto 1975; Okamoto 1978, thereafter O78; Ardavan 1979, thereafter A79) when considering the three dimensional magnetic field structure cannot be simply integrated, a solution for a cold, gravity-free relativistic pulsar wind derived only along a poloidal field line, the ``standard solution'', is of particular theoretical importance, as it is the simplest and has perhaps inherited the most important physics for a realistic pulsar wind.

The singular nature is one of the important issue for a standard wind. The wind solutions, which relate wind quantities to integral constants, are singular for a physical quantity, and some work (see below) even indicates the ``multi-singular'' nature. It is therefore an important matter whether a singular feature is physical or unphysical. The alternative way to phrase the question is whether the singularity is significant or not. We say that a singularity is significant when it physically constrains the flow parameters, and we say it is spurious when it may only arise from a special mathematical form, without constraining the flows. O78 found two mathematical singularities associated with the relativistic ``splitting'' of the conventional Alfvénic point, i.e., the Alfvénic point and the pure-Alfvénic. O78 argued that the Alfvénic point may be more significant than the pure-Alfvénic point, though the later singularity might still have physical significance. It was suggested by Ardavan (1979; A79) that in addtion to the previous two important singularities, there existed another equal important singularity, the intermediate point, though O78 did not argue for the existence of the intermediate one. The analysis of A79 involves with a selection of different variables other than the Lorentz factor , leading to different ``singularities''. However, other derivations (e.g., Michel 1969; Goldreich & Julian 1970; Kennel, Fujimura, & Okamoto 1983; Camenzind 1986) do not indicate the existence of the pure Alfvénic point and also the intermediate one. Thus some confusion and inconclusiveness remained in the literature about the singular nature of the standard pulsar wind. The complication is entirely due to special relativistic effects, as in the classic stellar wind domain all the proposed three singularities merge into one, the Alfvénic point. Resolving the multi-singular nature of a pulsar wind is clearly not trivial, as this would answer the question whether the special relativistic effects impose more constraints on the wind flow than for the nonrelativistic case. Li & Melrose (1994, thereafter LM94) studied the problem and have shown that only the Alfvénic point is genuine, and once the equations are regularised at the Alfvénic point, other singularities disappear and therefore they are not genuine. The argument is consistent with the physical nature - wave-singularity analogy (see LM94). The unequivocal implication is that the relativistic effect does not yield additional constraints on the wind solution.

However, in a recent paper, Ardavan (1995; A95) reiterated his early argument (A79), and therefore criticised LM94. A95 emphasises that a singularity does not necessarily arise from a ratio of variables as defined at a point in which the denominator goes to zero (and so the numerator must also go to zero). A95, following A79, used a quadratic form of certain variables and discussed the determinant in relation to the two branch solutions. A79 and A95 argued that different singularities might arise by choosing different variables, and these singularities all had the same physical significance. A95 concluded that LM94 considered only a part of the linearised version of singular analysis, and the definition of a singularity could be more general, for instance with the use of a quadratic form.

In fact, the conclusion of A95 is the fundamental argument which we believe may have caused confusion in the literature, since it emphasises the importance of a variation of mathematical form rather than the physical nature, i.e., whether it helps to constrain the flow or not. In this paper we attempt to clarify this matter. We first summarise and study the argument of A95 in section 2, where we show that the characteristic point which A95 argued is the intermediate point but not the pure Alfvénic point, and then demonstrate in section 3 that the intermediate point is not genuine, and finally conclude in section 4. (We use the symbols adopted in A95 simply for an easy comparison.)

© Copyright Astronomical Society of Australia 1997