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Caverns in Binary Systems


We follow the approach used by Lipunov and Prokhorov (1984)[110]. Suppose that a NS having a rotational magnetic  luminosity tex2html_wrap_inline11095 forms a binary system (with semi-major axis a) with a normal star losing matter in the form of stellar wind (with a loss rate tex2html_wrap_inline11099 ).

Let us find the Schwartzman radius  in accordance with the definition and by equating the plasma pressure tex2html_wrap_inline9471 to the electromagnetic pressure tex2html_wrap_inline9469 :


where R is the distance from the NS. The Schwartzman radius  is (see Lipunov and Prokhorov, 1984[110]; Harding, 1991[64]):


When making estimates, it is convenient to use the fact that the outflow rate, luminosity and the stellar wind velocity of hot stars are connected through the empirical relation (see, for example, Barlow and Cohen, 1977[8])


where tex2html_wrap_inline11109 -0.4. This leads to the approximation


The luminosity of normal stars whose mass varies from a few times to several tens of times the mass of the Sun lies in the interval between tex2html_wrap_inline11111 and tex2html_wrap_inline9158 - tex2html_wrap_inline11115 erg stex2html_wrap_inline8853 . The rotational magnetic luminosity  of NS also varies in the same wide range. For ``aged'' radiopulsars, tex2html_wrap_inline11119 erg stex2html_wrap_inline8853 , while for young pulsars tex2html_wrap_inline11095 may be as high as tex2html_wrap_inline9158 - tex2html_wrap_inline11127 erg stex2html_wrap_inline8853 . Equation (10.4.3) shows that entirely different situations may arise in different binary systems at various stages of evolution. For example, tex2html_wrap_inline11131 or tex2html_wrap_inline11133 . This means that while deriving the formula for a cavern in the general case, we must take into account the proximity of the normal star.

Figure 28: Schematic representation of a pulsar in a pair with a normal star. 

Let us calculate the shape of a cavern  surrounding a NS, assuming that the relativistic wind ejected by the star is spherically symmetric. We also assume that tex2html_wrap_inline11135 . At the cavern boundary (Figure 28), the following equilibrium condition must be satisfied:


where tex2html_wrap_inline9378 is the distance from the normal star, V is the volume of the cavern, and tex2html_wrap_inline11141 is the gas pressure. The second term on the right-hand side of the equilibrium equation (10.4.4) corresponds to the contribution of the pressure of the magnetic dipole radiation which is reflected back by the cavern ``walls'' and is accumulated inside it. The factor tex2html_wrap_inline9046 is zero for an open cavern and approximately 1 for a closed cavern. The static gas pressure tex2html_wrap_inline11147 can be presented in the form


For an isothermal stellar wind  (as a rule, this is the situation commonly encountered), n=0. We use dimensionless variables:


In this case, equation (10.4.4) becomes


Let us consider the case of an isothermal stellar wind (n=0) with no magnetic dipole radiation accumulated (tex2html_wrap_inline11153):


It follows from the symmetry considerations that at the anterior point of the cavern (i.e. the point closest to the normal star), tex2html_wrap_inline11155 . This leads to the following expression for the distance from the anterior point:


Similarly, we can determine the distance tex2html_wrap_inline11157 from the posterior point of the cavern. We get


If tex2html_wrap_inline11159 , the cavern is closed. The value of k is of the order of the ratio of the square of the velocity of sound to the outflow rate of the stellar wind, so that in real situations tex2html_wrap_inline11163 - tex2html_wrap_inline9467 . Hence closed caverns must be quite compact: tex2html_wrap_inline11167 - tex2html_wrap_inline11169 . In most cases, the assumption k=0 may serve as a good approximation.

Only the electromagnetic waves with a frequency higher than the plasma frequency ( tex2html_wrap_inline11173 tex2html_wrap_inline8845 tex2html_wrap_inline11177 Hz) may propagate in a plasma. The characteristic density in the stellar wind is estimated from the continuity condition


where tex2html_wrap_inline11179 . Thus, the magnetic dipole radiation  should be reflected from the walls. Suppose that the reflection coefficient tex2html_wrap_inline11181 . In this case, the above analysis will remain valid if we redefine the Schwartzman radius as


This gives


It can be seen that the size of the cavern has increased. As tex2html_wrap_inline11183 , the cavern cannot remain stationary. It is either always open or pulsating quasi-periodically. In the latter case, the closed cavern expands right up to the maximum value tex2html_wrap_inline11185 :


The shape of the closed cavern is shown in Figure 29.

Figure 29: The cavern around a pulsar in a binary system (Lipunov and Prokhorov, 1984).[110]  

next up previous contents index
Next: The Effect of Relativistic Up: Ejectors in Massive Binary Previous: Why do Radiopulsars not

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