Effects of finite resistivity

The winding of the magnetic field lines will inevitably force regions of oppositely directed fields into mutual proximity, and thus, possibly, bring about the onset of fast reconnection and accelerate Ohmic dissipation [effects of the finite conductivity of the plasma, Eq. (5.18)].

From the simple geometry of the frozen magnetic field, we see that the distance D.r between two initially remote points that are brought close to each other by the winding of the field line to which both are pinned is

D.r:::::: ~~:l?rr.

^(5.31)

Then, the time scale ^TR at which the field line will reconnect is inversely propor- tional to the reconnection speed which is a fraction of the Alfven speed ^VR = f3vA ((3 ~ 0.1)

(5.32)

As a result of the reconnection, the field which was originally ordered on a large scale ( ~ r_{0 )} is being fragmented into isolated "islands" or "cells," thus reduc- ing its ability to transport angular momentum. However, as long as TR. is larger than the characteristic time scale of field amplification TB

=

B¢/(dB¢/dt) (B¢/ Br

)/lronj or ^I

^{[see Eq. (5.20)], B¢ will} continue to grow. In other words if

TR ³1 m

1

1³ 1 Br 1 3

3 r I

ani

- --v 2 ^X 10 - - - ^{X - - -}

>

1

TB -

/3

Bo B¢

no or ...,

^(5.33)

[where we have again used Eq. (5.19) and r

=

r0x

=

3.2 x 10⁶cm m¹1³x], the magnetic cells will, on average, gain in length along the ¢; direction despite being occasionally broken up by the reconnection.

The reconnection by itself does not cause a significant loss of the magnetic flux. Nevertheless, the reduction of the relevant length scale due to the winding of the field lines will accelerate the familiar Ohmic dissipation which will proceed on the timescale TD = (6.r)²

/ry.

^{Then, from Eqs.} (5.19) and (5.31), we obtain the condition for a continuing amplification of the magnetic field, despite Ohmic dissipation

--:::::: 1.4 ^X 10⁷m²1^{3 -} x²- - - ~ 1.

To I Br ¹

3

r I

ani

TB B¢

no or

^(5.34)

The effect of reconnection in the early phase of accretion ( 1/2

<

x

<

1) might be to stop the amplification of the magnetic field before condition ( 5.22) is satisfied.

Otherwise, as shown in the previous subsection, the field will keep the rotation essentially rigid, therefore removing the danger of an all-out reconnection.

The borderline case occurs when TR/TB ::::::: 1 at the moment the ratio B¢/ Br reaches the value (5.22). This implies [see Eq. (5.33)] that an effective braking, for

1/2

<

x

<

1, can be achieved despite reconnection if

(TR) ^~

³

X 1010~

B6/

>

1,

TB _e.b

/3

^{m14 3}

where we have used

18nj or ^{I -;;_} 2n

0

jr, njn

0 -;;_ 4. Therefore, if 1 815/14

m

<

mR -:::::: 200/3₃₁₁₄ 0

(5.35)

( 5.36) the differential rotation will be braked before the field is fragmented by reconnec- tion. Of course, this assumes that the amplification of the field is efficient enough [condition (5.28)] to give rise to substantial magnetic torques, capable of braking the rotation.

Similarly, at the point of effective braking [Eq. (5.22)], Eq. (5.34) yields (5.37) leading, therefore, to the upper mass limit for negligible dissipation at effective braking

3B18/13

m

<

mo = 2 x 10 0 . (5.38)

If condition (5.38) is satisfied, most of the magnetic flux will be preserved despite reconnection. This raises an interesting possibility that the magnetic field fragmented at x :::::: 1 might still achieve ratio B¢/ Br sufficient for effective braking later in the course of accretion. More specifically, for x

<

1/2

(TR) _ ₉ x 109 2_

B6 2_

(no)

3

.!_I ani

TB e.b. - (3 m¹⁴1³ X 6 0 00 Or ¹ (5.39)

where we have, again, substituted B<i>/ Br from the relation (5.22) and assumed a rough conservation of the flux (B:::::: B₀jx²) which is necessary for Eqs. (5.22) and (5.33) to be valid. If we also assume that, after the reconnection at x :::::: 1, the rotation is virtually uninhibited by any angular momentum transport, n:::::: flo/x 2 (though the magnetic "cells" produced by reconnection do indeed provide some

"friction" [13]), thus minimizing (5.39), we obtain ( TR) = 2 X 10102_

BJ 2__

TB e.b. (3 m14/3 x2 (5.40)

Thus if the accretion is capable of generating strong enough magnetic fields [see Eqs. (5.28) and (5.30)], the reconnection will lose its ability to disrupt the achievement of IB<Pf Br le.b. at the distance (in units ro)

1 BS/2 Xres :::::: 1 X 105 m___!}__/ ,

y p m 7 3

from the black hole for m

<

mA, m

<

mo but m

>

mR.

(5.41)

The angular velocity that has increased during the inflow from 1 to Xres, flres ::::::

flo/x~es> will remain roughly constant inward from Xres to, say, the accretion radius

X a = ra/r₀ ::::::- 1 x 10-⁷ m²1³, where the effective transport of angular momentum

should presumably cease due to the increased radial velocity of the inflow. The requirement that the corresponding specific angular momentum l should be smaller than the last stable orbit angular momentum ltso

ltso

nresra2 (3m 17/3

--=----=---"' 1 x 10-²¹- - -

<

1

2.../31/;GM/c '"" 1/;

Bg '

^(5.42)

translates into a new reconnection limit

(5.43) thus depending only weakly on the details of the mechanism of reconnection (factor

{3),

or the angular momentum of the black hole (factor

1/J).

We note in passing that for x

<

1/2 the danger of Ohmic dissipation is di- minished with decreasing x [see Eq. (5.34)]. Indeed, the ratio

(To/TB)e.b.

actually increases inward from x ~ 1,

( 5.44) [compare with Eq. (5.37)]. Therefore, the condition (5.38) which ensures that the flux is not dissipated near x = 1 guarantees its approximate conservation during the subsequent inflow.

Finally, we conclude that if all three conditions (5.28), (5.38) and (5.43) are satisfied, the magnetic field will be able to preserve a roughly spherical accretion despite the reconnection and Ohmic dissipation.

For B0 greater than about 5 Gauss, we see that the only relevant upper mass limit is (5.28), which is imposed only by the limited ability of the flow to amplify the seed magnetic field through an ideal MHD dynamo action. Therefore, if we want the magnetic braking of the differential rotation to be successful throughout the relevant range of mass

(Mv <

M

< M+),

we would need to have B₀ ;(:. 5 x 10⁵ Gauss.