Math Revisited: Bessel Functions

Plot of Bessel function of the first kind, J_α(x), for integer orders α = 0, 1, 2.

Among the most important special functions is the Bessel function. In the field of solar physics, for example, it's of inestimable importance in the analysis of solar magnetic fields and their evolution. One very key equation is the Helmholtz, viz.

1/ r  [¶/ ¶r  ( r  ¶/ ¶r)] B  +  (a)² B = 0

where r is the radial coordinate, B  the magnetic field intensity, and a a quantity called the "force free parameter". Then the axially symmetric (i.e.- in cylindrical coordinates r, z, q) Bessel function solutions are

B _z (r)    =   B_o J_o(a r)  

B _q (r)  =  B_o J₁(ar)

where the axial (top) and azimuthal magnetic field components are given, respectively, and  J_o(a r) is a Bessel function of the first kind, order zero and J₁(ar) is a Bessel function of the first kind, order unity. (See graphs at the top for Bessel functions of the orders 0, 1 and 2).

The Bessel functions are mathematically defined (cf. Menzel, 'Mathematical Physics', 1961, p. 204):

J_m (x) = (1/ 2^m m!) x^m [1 -  x ²/ 2² 1! (m + 1)  +  x⁴/ 2⁴2! (m + 1) (m + 2) -  ….(-1)^j x^2j / 2 ^2j j! (m + 1) (m + 2)…(m + j) +  …]
 

which we terminate with second order terms.

For m = 0 and m = 1 forms one gets:

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

and:

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

The equations in B _z (r),  B _q (r),  with the special Bessel functions at root, are critical in describing the respective magnetic fields for a magnetic tube. For a cylindrical magnetic flux tube (such as a sunspot represents viewed in cross-section) the “twist” is defined:

T(r)  =  (L B _q(r))/  (r  B _z (r))

Where L denotes the length of the sunspot-flux tube dipole and r, the radius. If the twist exceeds 2p then the magnetic configuration may be approaching instability and a solar flare.

Problems for the Math Maven:

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.

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 = 10 ⁴  km and the length L = 10 ⁸  m.   Is the tube kink unstable or not? (Kink instability is said to obtain when: T(r) > 2p)

Looking at Bessel Functions & Applications

Plot of Bessel function of the first kind, J_α(x), for integer orders α = 0, 1, 2.

Among the most important special functions is the Bessel function. In the field of solar physics, for example, it's of inestimable importance in the analysis of solar magnetic fields and their evolution. One very key equation is the Helmholtz, viz.

1/ r  [¶/ ¶r  ( r  ¶/ ¶r)] B  +  (a)² B = 0

where r is the radial coordinate, B  the magnetic field intensity, and a a quantity called the "force free parameter". Then the axially symmetric (i.e.- in cylindrical coordinates r, z, q) Bessel function solutions are

B _z (r)    =   B_o J_o(a r)  

B _q (r)  =  B_o J₁(ar)

where the axial (top) and azimuthal magnetic field components are given, respectively, and  J_o(a r) is a Bessel function of the first kind, order zero and J₁(ar) is a Bessel function of the first kind, order unity. (See graphs at the top for Bessel functions of the orders 0, 1 and 2).

The Bessel functions are mathematically defined (cf. Menzel, 'Mathematical Physics', 1961, p. 204):

J_m (x) = (1/ 2^m m!) x^m [1 -  x ²/ 2² 1! (m + 1)  +  x⁴/ 2⁴2! (m + 1) (m + 2) -  ….(-1)^j x^2j / 2 ^2j j! (m + 1) (m + 2)…(m + j) +  …]

which we terminate with second order terms.

For m = 0 and m = 1 forms one gets:

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

and:

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

The equations in B _z (r),  B _q (r),  with the special Bessel functions at root, are critical in describing the respective magnetic fields for a magnetic tube. For a cylindrical magnetic flux tube (such as a sunspot represents viewed in cross-section) the “twist” is defined:

T(r)  =  (L B _q(r))/  (r  B _z (r))

Where L denotes the length of the sunspot-flux tube dipole and r, the radius. If the twist exceeds 2p then the magnetic configuration may be approaching instability and a solar flare.

Problems for the Math Maven:

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.

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 = 10 ⁴  km and the length L = 10 ⁸  m.   Is the tube kink unstable or not? (Kink instability is said to obtain when: T(r) > 2p)

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

The Remarkable World of Special Functions

The world of mathematical special functions is as intriguing and interesting as any domain, say in physics, or the natural world. The trouble is that one usually doesn't appreciate these functions enough while taking Calculus or Linear Algebra or Differential Equations, and one must wait until a research focus brings them to prominence.

Let's take the case of the Bessel functions. In solar physics one very key equation (Helmholtz) for which the (axially symmetric- in cylindrical coordinates r, z, t) Bessel function solution is:

B_z (r) = B_o J_o(a r)

B_t(r)) = B_o J_1(ar)

t = theta

J_0(a r) is a Bessel function of the first kind, order zero and J1 (ar) is a Bessel function of the first kind, order unity. The Bessel functions are defined (cf. Menzel, 'Mathematical Physics', 1961, p. 204):

J_m(x) = (1/ 2^m m!) x^m [1 – x^2/ 2^2 1! (m + 1) + x^4/ 2^4 2! (m +1) (m + 2) -…

-(1)^j x^(2j) / 2^(2j) j! (m + 1) (m +2 )……(m + j) + …]

for m = 0 and m = 1 forms one gets:

J_o(x) = 1 - x^2/ 2^2 (1!)^2 + x^4/ 2^4 (2!)^2 - x^6/ 2^6 (3!)^2 + ......

J_1(x) = x/ 2 - x^3/ 2^3*1! 2! + x^5/ 2^5 *2!3! - x^7/ 2^7 *3!4! - .....


The equations in B_z, B_t, with the special Bessel functions at root, are critical in describing the respective magnetic fields for a magnetic tube.

For a cylindrical magnetic flux tube (such as a sunspot represents viewed in cross-section) the “twist” is defined:

T(r) = (L * B_t(r))/ (r * B_z (r))

Where L denotes the length of the sunspot-flux tube dipole. If the twist value exceeds 2(pi) then the magnetic configuration may be approaching instability and a solar flare.

Then there is the Gamma function (call it 'G') for which:

G(a) = (a - 1 )!

where 'a' is a positive integer.

Thus, for a = 3:

G(3) = (3 - 1)! = 2! = 2*1 = 2

One can also make use of a recursion formula:

G(a + 1) = a G(a)

For example: G(4) = G (3 + 1) = 3 G(3) = 3 (2) = 6

Check this from the earlier formula: G(a) = (a - 1)!

G(4) = (4 - 1)! = 3! = 3*2*1 = 6

Now, there is also the Beta function, call it B(u,v) which can be expressed in terms of the Gamma functions G(u), G(v)

Thus:

B(u,v) = G(u) G(v)/ G(u + v)

Consider the Beta function B(3, 4):

B(3, 4) = G(3) G(4)/ G(3 + 4) = (2) (6)/ G(7) = 12 / G(7)

where: G(7) = (7 - 1)! = 6! = 6*5*4*3*2*1 = 720

so B(3,4) = 12/ 720 = 1/60

Why all the fuss about Beta and Gamma functions here?

Well, if any readers happened to have caught the PBS series 'The Elegant Universe', they'd have seen string theorist Brian Greene scribbling the Euler equation for string theory on a blackboard:

B(p, q) = G (p) G(q)/ G(p + q)

and arriving at the specific string theory form:

Beta([1- alpha(s)][1 - alpha(t)] =

Gamma (1 - alpha(s)) Gamma (1 - alpha(t))/ Gamma(2 - alpha(s) - alpha(t))

Special functions can indeed be loads of fun and have wide applications. In a future foray I'll look at fractional Gamma functions.