METHODS OF NUMERICAL SOLUTION

2 Equations of magnetosphere models

2.2 METHODS OF NUMERICAL SOLUTION

ratio dZA('if;)/dZD(1f;) should be very large because of the large conductivity of the plasma making up the disk; equation (2 .l Ob) then implies that

QF ~

QD, i.e.

the field lines are frozen into the disk.

The current potential !('if;) is determined by the equations

(2.12)

for field lines threading the hole and the disk, respectively.

In accordance with the above guidelines, the field-line angular velocity

QF

for field lines threading the hole will be chosen to equal oH / 2 in all models con-

structed here. The currents will be taken to have the functional form defined by

the slow-rotation limit of equations (2 . 12) , i.e . the derivatives dZH I d'if; and

dZA / d'if; will be figured using the 1f of the pre-spinup, Schwarzs child-spacetime

solution. Since the ratio dZA ('if;)/ dZH ('if;) is assumed to be approximately unity,

the acceleration region will be taken to have an effective surfac e resistivity of

order unity; the impedance zA used in equation (2.12) to determine the current

on field lines threading the disk will therefore be defined by an equation of the

form dZA

Cl(

de I sine. The disk will be taken to have infinite conductivity so that

the magnetic field lines are rigidly fro zen into the disk and must rotate with its

angular velocity.

r• = ^r/

^{M ,}

^{a*= a/}

^{M , rzi*}

^{= f:j/}

^{M ,}

^t*=t/M

I

c.>*=c.>M

I

if*=OFu

1V1 ₁

n11 * =

^QH

^{M = a* I 2Mr * . r: = r +I M =}

¹

^{+ (}

^{1 -}

^{a *}

^{2) 11 2 ,}

^(2.13)

B* =BM

^I

E* =EM

^I

1/J* = 1/J/ M . I*= I .

For a black hole of mass

M ""

10

⁸ M_{0 ,}

surrounded by a magnetic field of strength

B "" 10

⁴

G, and with magnetospheric currents of order 10

¹⁸

amperes. the dimen-

sionless quantities B*, 1/J* and /*will all have magnitudes of order 10-

⁸.

If one reformulates all of the equations in section 2.1 in terms of the rescaled quanti- ties (including expressing the '1 operator in terms of r*-derivatives). all equa- tions precisely retain their original forms except for the replac e ment of non- starred by starred quantities . Thus, in the interest of notational simplicity, the stars will be dropped and all quantities will henceforth be understood to be dimensionless.

Equation (2 .5) is elliptic everywhere except on the locus ("'1:rF) 2 = ^{1 or}

(oF - c.>)

²1:j2

/ o.²

= 1 ("velocity-of-light surface"), where it becomes a first-order equation. It is thus amenable to solution by finite-difference, point-iterative relaxation methods on a grid (see e.g. Ames 1977) . The grid chosen was an r-e

coordinate grid with constant stepsizes h,. and

he

in the respective directions.

The computational molecule is shown in Fig. 2.

The gradient operators in equa tion (2.5) are covariant derivatives in abso- lute three-dimensional space. The first term

in

(2.5) may be expressed in terms of ordinary derivatives as

v . lrxv1f] =

^_1_

_vg ^{(Fag Xig}

^{ij ,,,} _'!',) ^{. )} _.i ^.

(2.14)

dimensional metric (2. 1) . The differential equation (2.5) m ay thus be written as a difference equation on the grid of Fig. 2

in

the following form :

-µ1-µ,z

I

[ '1/11 -21/lo +1f;3 '1/11-1/13

e o _(e

-,u.1+Jl-zx~)o

+ (e -µ1+µ,zXc:i),r Io

r;:,o h,.2 2h,.

+

(eµ1-µ,zx~)o

'1f 4 -21/10+1/12 +

(eµ¹-µ,zX~).e

lo ^'1f 4 -'1/12

J ^{(2 .15)}

h~ 2he

(OF -c.J)o I r

^{-2µ -2µ,z}

+ 167T: !(1/lo)

d!

I = ⁰

dOF

e

¹

i o 2 e Io

+ d1f; 'li'ol 4h,.2 (1/11--1/13) +

_4h~

( 1/14 -'1/12) 2

cxo

cxo~o

d'l/J

'iflo

where the subscripts 0-4 refer to the points labeled in Fig. 2 .

Solving this equation for 'I/Jo yields a prescription for an iterative relaxation scheme;

in

updating the array, the new value of 1f; at the point 0 is calculated as a function of the old valµes of

'1/J

at the surrounding points . The parti cular method used was a technique known as successive over-relaxation (SOR) with Gauss-Seidel iteration (see e.g. Ames 1977) . Jn this method, the prospective correction 1/lo(new) -

'1/Jo(old)

is multiplied by a factor {3 ( 1 :::: {3 < 2) before being added to 1/lo(old.)· The relaxation parameter {3 may be chosen to optimize the con- vergence rate : if /.... is the asymptotic ratio of maximum corrections on succes- sive iterations when {3 = 1, then the optimal value of {3 is ^f3soR = 2/ (1 + YT="'X);

this value of {3 produces a convergence ratio (ratio of success ive m aximum corrections) of f3soR - 1.

One point of difficulty with the solution is that, as m entioned previously, equation (2 .5) becomes a first-order equation on the velocity-of-light surface (vf')

²

= 1.

If

0

~ fJF ~ QH

(which is the condition for ene rgy extracti on), the re

'

order to avoid superluminal motion; and an outer surface, corresponding to the familiar pulsar light cylinder, beyond which particles locked to the field lines must slide outward. For a nonrotating hole and field, the inner light surface is coincident with the horizon and the outer one is at infinity.

At the light surfaces, equation (2.5) just becomes a Neumann-type boundary condition

1vx1 ~+ ^{(OF - c.>) doF}

^(V1f;)2

+

16rr2

I df =

0

an

^{C( d}

^1/1

^{cxrzi2 d}

^{1/1 . (2.16)}

where B1{11 an denotes the normal derivative of

1{I

at the light surface .

Considerable difficulty was encountered in finding a difference equation

which was stable at the light surface. Several schemes which were tried before

equation (2.15) was selected were found to produce divergenc es if the outer light

cylinder was within the region of integ ration. Explicit attempts to model the

Neumann boundary condition (2 .16) produced more success, but the best

results were obtained using the differencing scheme embodied in equation

(2.15). Even this method exhibited noticeable oscillations in the solution values

at lattice points which happened to lie very near the light surface (much nearer

than the stepsizes h,. and

h_{8 );}

this is due to the fact that, in solving equation

(2.15) for 1{1

_0,

the quantity X. which vanishes on the the light surface, occurs in

the denominator. Oscillations may occur if a particular iteration moves the

light surface across a lattice point lying near it; the next iteration will then

apply a correction of the opposite sign which will tend to move the light surface

back across the lattice point. But the oscillations were usually small enough

that equation (2.15) converged to a solution for the rotation parameter a less

than about 0. 75, where the criterion for successful convergence was defined by

In situations where a force-free gap was assumed to exist between the hor- izon and the inner edge of the disk, the boundary condition (2.Bd) was enforced by resetting the boundary values of 'if; on the gap after each iteration in such a way as to require the field to pass through the gap vertically. After each itera- tion, the magnetic field would generally come out with a small but nonzero radial component at the gap; condition (2.Bd) was applied by changing the values of 'if; on the gap so that it once more represented a vertical field at the boundary.

In general, this required adding or removing some flux from the horizon; this was accomplished by refiguring the horizon field using (2 .Be) and the newly cal- culated value of the total flux threading the horizon.

3 Specific models