Characteristic point

The basic integrals along a magnetic field line for the standard pulsar wind are (O78; A79)
    
In the above formulae, subscript ``p'' and ``'' denote the poloidal and toroidal components respectively, is the Lorentz factor, , F, G and are integral constants along a field (or poloidal) line, and the other quantities have their usual meanings. Integrals (1) to (4) are in order the generalised Ferraro isorotation law (the integral of the induction equation), mass flux conservation, the angular momentum conservation and finally the energy conservation. Setting , these equations reduce to those derived earlier by Mestel (1968), except the difference in (4) due to neglecting the pressure and gravity terms. Most important is the Lorentz factor , which characterises the relativistic nature of a pulsar wind. It can be expressed by
 
where
 
It can been seen that there is a unique singularity at in (5), though the regularisation at the Alfvénic point has been made. Note that (5) is a complete expression as the RHS has no dependence on ( is independent of ). The quantity is thus a function of integral constants, the proper density and also the distance. The toroidal velocity may be expressed as
 
Note that (7) is not a complete form because the numerator is a function of which is subject to its singular nature. It turns out that once is regularised at , i.e., (5), in (7) is automatically singular free at the pure Alfvénic point (see LM94). Nevertheless, once
 
which is the definition of the so-called pure Alfvénic point, we must require
 

One can in principle express (5) in a quadratic form as adopted in A95:
 
where new variables
 
Bearing in mind, (10) is not a complete quadratic expression as and R are not independent. However, this does not affect having two solutions of :
 
A95 discussed two solutions and , and found that and behave well at infinity and within respectively. Thus the continuity of a smooth solution requires a transition from to (outwards) at a point . This point has been shown by A95 occurring at and we may call it a characteristic point. Two extra conditions from (12) for the characteristic point are readily obtained:
  
Alternatively, (13) and (14) define the characteristic point. This leads to
 
It is important to realize that A95 does not give the correct form (15) but instead. By (15), we obtain an equivalent relation
 
With the use of (4), (16) becomes
 
According to A79, (17) defines the intermediate point.

For the pure Alfvénic point, one can derive a different relation as compared to (15). Expressing (11) into
 
we find that the pure Alfvénic point, as defined by (8), corresponds to
 
The difference between (19) and (15) shows that the intermediate point is independent of the pure Alfvénic point. We thus conclude that the quadratic analysis of A95 has no relevance to the pure Alfvénic point.

© Copyright Astronomical Society of Australia 1997