## Monday, April 29, 2013

### Solutions to Bessel Function Problems

We now look at the solutions to the last group of math problems:

1) Compute the Bessel functions for Jo(x)  and J1(x) with x = 1 and then compare with the values obtained from the graph shown at top.

Solution:

Use the truncated series:

Jo(x) = 1 - x2/ 22 (1!)2 + x4/ 24 (2!)2 - x6/ 26 (3!)2 +…..

Then for x = 1:

Jo(1) = 1 – (1)2/ 22 (1!)2 + (1)4/ 24 (2!)2 – (1)6/ 26 (3!)2 +…

Jo(1) =  1 – ¼   +  1/ 64  -  1/2304  =  0.765

Compare to graphical value: Jo(1) =   0.8

Next:

J1(x) = x/ 2 - x3/ 23 ·1! 2! + x5/ 25 ·2!3! - x7/ 27 ·3!4! - .....

Then for x = 1:

J1(1) = 1/ 2 – (1)3/ 23 ·1! 2! + (1)5/ 25 ·2!3! – (1)7/ 27 ·3!4! - .....

J1(1) =  ½  -  1/16  +  1/ 384  - 1/ 18432  =   0.44

Compare to graphical value: J1(1) =   0.45

2) Find the twist in a solar loop (take it to be a magnetic tube) if: B q (r)= 0. 1T and B z (r) = 0.2T. Take the radius of the tube to be r = 104  km and the length L =108  m. Is the tube kink unstable or not? (Kink instability is said to obtain when: T(r) >2 p

Solution: the “twist” is defined:

T(r)  =  (L B q(r))/  (r  B z (r))

Where: B q(r) = 0.1 Tesla and B z (r)  = 0.2 Tesla

Also: r =  104  km  = 107  m   and L = 108  m

Therefore: B q(r))/  (B z (r))  = (0.1)/ (0.2) = 0.5  and L/r = (108)/ (107) = 10

So:

T(r)  = (L/r) (0.5)  =   10 (0.5) = 2.0

The tube is not kink unstable since that requires: T(r) > 2p  = 6.28

3) Compute the intensity for the azimuthal magnetic field component (i.e. B q (r)   ) of a large sunspot, if its equilibrium magnetic field Bo = 0.01 T and the value of  J1(ar) conforms to a = 0.4 and r = 40.

Solution: By definition: B q (r)  =  Bo J1(ar)

If a = 0.4 and r = 40 then ar = (0.4)(40) = 16

From the graph: J1(ar) = J1(16))  » 0.17

Therefore:

B q (r)  =  Bo J1(ar)  = 0.01T (0.17) =  0.0017 Tesla