Bessel Function Problem Solutions

 1) Compute the Bessel functions for J_o(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:

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

Then for x = 1:

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

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

Compare to graphical value: J_o(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) A large sunspot has an equilibrium magnetic field  B_o=  0.01T.  Find its azimuthal magnetic field if  x = 2, for the associated Bessel Function, J₁(x).

Solution: By definition: B _q (r)  =  B_o J₁(x)

If x  = 2   we compute J₁(x)  from the series::

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

Then for x = 2:

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

J₁(1) =   1 -  1/2   +  1/ 12  - 1/ 144  =   0.576

Then: B _q (r)  =  B_o J₁(x) =   0.01 T( 0.576) =   0.006 T

3) 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