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The Stopping Radius


Now we consider qualitatively the effect of the electromagnetic field of a magnetic rotator on the accreting plasma. Consider a magnetic rotator with a dipole magnetic moment tex2html_wrap_inline9376 , rotational frequency tex2html_wrap_inline9374 and mass M. At distances tex2html_wrap_inline9413 the surrounding plasma is characterized by the following parameters: density tex2html_wrap_inline9397 , sound velocity tex2html_wrap_inline9399 and/or velocity tex2html_wrap_inline9270 relative to the star. The plasma will tend to accrete on to the star under the action of gravitation. The electromagnetic field, however, will obstruct this process, and the accreting matter will come to a stop at a certain distance.

Basically, two different cases can be considered: (1) when the interaction takes place beyond the light cylinder, tex2html_wrap_inline9455 , and (2) the accreting plasma penetrates the light cylinder tex2html_wrap_inline9457 .

(1) Case (1) tex2html_wrap_inline9455 was first considered by Schwartzman (1970b,c)[175, 174]. In this case the magnetic rotator generates a relativistic wind consisting of a flux of different kinds of electromagnetic waves and relativistic particles. The form in which the major part of the rotational energy of the star is ejected is not important at this stage. What is important is that both relativistic particles and magnetic dipole radiation will transfer their momentum and hence exert pressure on the accreting plasma. Indeed, random magnetic fields are always present in the accreting plasma. The Larmor radius of a particle with energy tex2html_wrap_inline9461 tex2html_wrap_inline9463  eV moving in the lowest interstellar magnetic field tex2html_wrap_inline8945 tex2html_wrap_inline9467  G is much smaller than the characteristic values of radius of interaction, so the relativistic wind will be trapped by the magnetic field of the accreting plasma and thus will transfer its momentum to it.

Thus, a relativistic wind can effectively impede the accretion of matter. A cavern is formed around the magnetic rotator,   and the pressure of the ejected wind tex2html_wrap_inline9469 at its boundary balances the ram pressure of the accreting plasma tex2html_wrap_inline9471 :


This equality defines a characteristic size of the stopping radius, which we call the Schwartzman radius  tex2html_wrap_inline9473 .

(2) The pressure of the accreting plasma is high enough to permit the plasma to enter the light cylinder tex2html_wrap_inline9457 . Since the magnetic field inside the light cylinder decreases as a dipole field, the magnetic pressure is given by


Matching this pressure to the ram pressure of the accreting plasma yields the Alfvén radius  tex2html_wrap_inline9477 .

The magnetic pressure and the pressure of the relativistic wind can be written in the following convenient form:


We introduce a dimensionless factor tex2html_wrap_inline9479 such that the power of the ejected wind is


Assuming tex2html_wrap_inline9481 , we get for tex2html_wrap_inline9483 a continuous function tex2html_wrap_inline9485 whose qualitative behavior is shown in Figure 4.

Figure 4: Internal and external pressure laws.   

The accreting pressure of plasma outside the capture radius  is nearly constant, and hence gravitation does not affect the medium parameters significantly. In contrast, at distances inside the gravitational capture radius tex2html_wrap_inline8865 the matter falls almost freely and exerts pressure on the ``wall'' equal to the dynamical pressure. For spherically symmetric accretion   we obtain


Here we used the continuity equation tex2html_wrap_inline9489 . When presented in this form, the pressure tex2html_wrap_inline9471 is a continuous function of distance (Figure 4).

Summarizing, for the stopping radius  we get


The expressions for the Alfvén radius  are:


and for the Schwartzman radius: 


next up previous contents index
Next: The Stopping Radius in Up: ``Ecology'' of Magnetic Rotators Previous: The Environment of the

Mike E. Prokhorov
Sat Feb 22 18:38:13 MSK 1997