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Find Analytic Solutions to an Eigenvalue Problem

Whitworth's (1981) Isothermal Free-Energy Surface
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Note from J. E. Tohline to Students with Good Mathematical Skills: This is one of a set of well-defined research problems that are being posed, in the context of this online H_Book, as challenges to young, applied mathematicians. The astronomy community's understanding of the Structure, Stability, and Dynamics of stars and galaxies would be strengthened if we had, in hand, a closed-form analytic solution to the problem being posed here. A solution can be obtained numerically with relative ease, but here the challenge is to find a closed-form analytic solution. As is true with most meaningful scientific research projects, it is not at all clear whether this problem has such a solution. In my judgment, however, it seems plausible that a closed-form solution can be discovered and such a solution would be of sufficient interest to the astronomical community that it would likely be publishable in a professional astronomy or physics journal. At the very least, this project offers an opportunity for a graduate student, an undergraduate, or even a talented high-school student (perhaps in connection with a mathematics science fair project?) to hone her/his research skills in applied mathematics. Also, I would be thrilled to include a solution to this problem — along with full credit to the solution's author — as a chapter in this online H_Book. Having retired from LSU, I am not in a position to financially support or formally advise students who are in pursuit of a higher-education degree. I would nevertheless be interested in sharing my expertise — and, perhaps, developing a collaborative relationship — with any individual who is interested in pursuing an answer to the mathematical research problem that is being posed here.

The Challenge

Formally, this is an eigenvalue problem.

Find one or more analytic expression(s) for the (eigen)function, ~\mathcal{G}_\sigma(x) — and, simultaneously, the unknown value of the (eigen)frequency, ~\sigma — that satisfies the following 2nd-order, ordinary differential equation:


(x^2\sin x ) \frac{d^2\mathcal{G}_\sigma}{dx^2} + 2 \biggl[ x \sin x +  x^2 \cos x \biggr]  \frac{d\mathcal{G}_\sigma}{dx} + 
\biggl[ \sigma^2 x^3  - 2\alpha ( \sin x - x\cos x ) \biggr]  \mathcal{G}_\sigma = 0 \, ,


where, ~\alpha is a known constant. The desired functional solution is subject to the following two boundary conditions: ~\mathcal{G}_\sigma = 0 at ~x = 0; and ~d\ln\mathcal{G}_\sigma/d\ln x = (\pi^2 \sigma^2/2 - \alpha) at ~x = \pi. Note that, in the context of astrophysical discussions, the interval of ~x that is of particular interest is 0 \le x \le \pi.

Context

The challenge posed above is one of a set of closely related eigenvalue problems that arise in the context of the study of the pulsating stars and the governing 2nd-order ODE is often referred to as the Linear Adiabatic Wave Equation (LAWE). In the most general context, the LAWE takes the form,

~\biggl[P \biggr]\frac{d^2\mathcal{G}_\sigma}{dx^2} + \biggl[\frac{4P}{x} 
+ P^' \biggr]\frac{d\mathcal{G}_\sigma}{dx} + \biggl[ \sigma^2 \rho + \frac{\alpha P^'}{x} \biggr]\mathcal{G}_\sigma

~=

~0 \, ,

where, ~P and ~\rho are both functions of ~x that have different prescriptions for each specified astrophysics problem — see the table of examples presented below — and primes denote differentiation with respect to ~x. The symmetries associated with this broad set of eigenvalue problems can perhaps be better appreciated by rearranging terms in the LAWE to obtain,

~- \sigma^2  \mathcal{G}_\sigma

~=

~\frac{P}{\rho} \biggl[ \frac{4\mathcal{G}_\sigma^'}{x}+ \mathcal{G}_\sigma^{' '} \biggr]  
+ \frac{P^'}{\rho} \biggl[ \frac{\alpha \mathcal{G}_\sigma}{x} + \mathcal{G}_\sigma^'\biggr] \, ,

or, equivalently,

~- \sigma^2  \rho \mathcal{G}_\sigma

~=

~ \frac{P}{x^4} \frac{d}{dx}\biggl( x^4 \mathcal{G}_\sigma^' \biggr)  
+ \frac{P^'}{x^\alpha}\frac{d}{dx}\biggl(x^\alpha \mathcal{G}_\sigma\biggr) \, ,

or, equivalently,

~\frac{d}{dx}\biggl(x^4 P \mathcal{G}_\sigma^'\biggr) + \biggl[ \biggl( \sigma^2 + \frac{\alpha P^'}{x\rho} \biggr) x^4 \rho 
\biggr] \mathcal{G}_\sigma

~=

~ 0 \, .


Properties of Analytically Defined Astrophysical Structures
Model ~\rho(x) ~P(x) ~P^'(x)
Uniform-density ~1 ~1 - x^2 ~-2x
Linear ~1-x ~(1-x)^2(1 + 2x - \tfrac{9}{5}x^2) ~-\tfrac{12}{5}x(1-x)(4-3x)
Parabolic ~1-x^2 ~(1-x^2)^2(1  - \tfrac{1}{2} x^2) ~-x(1-x^2)(5-3x^2)
~n=1 Polytrope ~\frac{\sin }{ x} ~\biggl[\frac{\sin x}{x}\biggr]^2 ~\frac{2}{x} \biggl[ \cos x - \frac{\sin x}{x} \biggr]
\frac{\sin x}{x}

Drawing the expressions for ~\rho(x), ~P(x), and ~P^'(x) from the last row of this table and plugging them into this generic form of the LAWE leads to the specific statement of the astrophysically motivated eigenfunction problem presented above — inside the blue-framed box. As is discussed in the subsection that follows, an analogous eigenvalue problem whose analytic solution is known comes from plugging expressions presented in the first row of this table into the generic form of the LAWE.

Analogous Problem with Known Analytic Solutions

Here is an analogous problem whose analytic solution is known. Anyone interested in tackling the challenge, provided above, should study — and even extend — the known set of solutions of this analogous problem. This exercise should provide at least partial preparation for addressing the above challenge.

Statement of the Problem

As above, the task here is to find one or more analytic expression(s) for the (eigen)function, ~\mathcal{F}_\sigma(x) — and, simultaneously, the unknown value of the (eigen)frequency, ~\sigma — that satisfies the following 2nd-order, ordinary differential equation:


(1 - x^2) \frac{d^2 \mathcal{F}_\sigma}{dx^2} + \frac{4}{x}\biggl[1 -  \frac{3}{2}x^2 \biggr] \frac{d\mathcal{F}_\sigma}{dx} +  \biggl[3\sigma^2 - 2 \alpha \biggr]  \mathcal{F}_\sigma  = 0 .

Try a Polynomial Expression for the Eigenfunction

Let's guess that the proper eigenfunction is a polynomial expression in ~x. Specifically, let's try a solution of the form,

\mathcal{F}_\sigma = a + bx + cx^2 + dx^3 + fx^4 + gx^5 + \cdots

truncated at progressively higher- and higher-order terms.

Lowest-order mode (Mode 0)

Try,

\mathcal{F} = a \, ,

in which case,

~\frac{d\mathcal{F}}{dx}

~=

~0 \, ,

and,

~\frac{d^2\mathcal{F}}{dx^2}

~=

~0 \, .

So, the governing 2nd-order ODE reduces to,

~(3\sigma^2 - 2 \alpha )  a

~=

~0 \, ,

which will be satisfied as long as, ~\sigma = (2\alpha/3)^{1/2} \, . We conclude, therefore, that the eigenvector defining the lowest-order (the simplest) solution to the governing ODE has an eigenfunction given by,

~\mathcal{F}_0

~=

~a = \mathrm{constant} \, ,

with a corresponding eigenfrequency whose value is,

~\sigma_0

~=

~\biggl(\frac{2\alpha}{3} \biggr)^{1/2} \, .

Second Guess

Try,

\mathcal{F} = a + bx \, ,

in which case,

~\frac{d\mathcal{F}}{dx}

~=

~b \, ,

and,

~\frac{d^2\mathcal{F}}{dx^2}

~=

~0 \, .

Plugging this trial eigenfunction into the governing 2nd-order ODE gives,

~0

~=

~\frac{4}{x}\biggl[1 -  \frac{3}{2}x^2 \biggr] b +  \biggl[3\sigma^2 - 2 \alpha \biggr]  (a + bx)

 

~=

~\frac{4b}{x} + (3\sigma^2 - 2 \alpha -6) bx +  (3\sigma^2 - 2 \alpha ) a \, .

But this expression can be satisfied for all values of ~x only if ~b = 0, in which case the trial eigenfunction reduces to the earlier solution, ~\mathcal{F}_0. We conclude, therefore, that our "second guess" does not generate a new solution to this eigenfunction problem.

Third Guess

Try,

\mathcal{F} = a + cx^2\, ,

in which case,

~\frac{d\mathcal{F}}{dx}

~=

~2cx \, ,

and,

~\frac{d^2\mathcal{F}}{dx^2}

~=

~2c \, .

Plugging this trial eigenfunction into the governing 2nd-order ODE therefore gives,

~0

~=

~2c(1 - x^2) + 8c(1 -  \frac{3}{2}x^2 ) +  (3\sigma^2 - 2 \alpha )  (a + cx^2)

 

~=

~ \biggl[10c + (3\sigma^2 - 2 \alpha )a\biggr] + \biggl[ (3\sigma^2 - 2 \alpha )  -14 \biggr]cx^2 \, .

This relation will be satisfied for all values of ~x if both expressions inside the square brackets are simultaneously zero, that is, if,

~10c + (3\sigma^2 - 2 \alpha )a

~=

~0 \, ,

and, simultaneously,

~(3\sigma^2 - 2 \alpha )  -14

~=

~0 \, .



  • Mode 1:
x_1 = a + b\chi_0^2, in which case,


\frac{dx}{d\chi_0} = 2b\chi_0; ~~~~ \frac{d^2 x}{d\chi_0^2} =  2b;


\frac{1}{(1 - \chi_0^2)}  \biggl\{ 2b (1 - \chi_0^2) + 8b \biggl[1 -  \frac{3}{2}\chi_0^2 \biggr]  +  A_1 \biggl(1 + \frac{b}{a}\chi_0^2 \biggr) \biggr\} = 0 ,

where,


A_1 \equiv \frac{a}{\gamma_\mathrm{g}}\biggl[ \biggl( \frac{3}{2\pi G\rho_c} \biggr) \omega_1^2+ 2(4 - 3\gamma_\mathrm{g}) \biggr] .

Therefore,


(A_1 + 10b) + \biggl[ \biggl(\frac{b}{a}\biggr) A_1 - 14b \biggr] \chi_0^2  = 0 ,


\Rightarrow ~~~~~ A_1  = - 10b ~~~~~\mathrm{and} ~~~~~ A_1 = 14a


\Rightarrow ~~~~~ \frac{b}{a} = -\frac{7}{5}  ~~~~~\mathrm{and} ~~~~~ \frac{A_1}{a} = 14 = \frac{1}{\gamma_\mathrm{g}}\biggl[ \biggl( \frac{3}{2\pi G\rho_c} \biggr) \omega_1^2+ 2(4 - 3\gamma_\mathrm{g}) \biggr] .

Hence,


\biggl( \frac{3}{2\pi G\rho_c} \biggr) \omega_1^2 = 20\gamma_\mathrm{g}  -8


\Rightarrow ~~~~~ \omega_1^2 = \frac{2}{3}\biggl( 4\pi G\rho_c \biggr) (5\gamma_\mathrm{g}  -2)

and, to within an arbitrary normalization factor,


x_1 = 1 - \frac{7}{5}\chi_0^2 .


Astrophysical Context

A star that is composed entirely of a gas for which the pressure varies as the square of the gas density will have an equilibrium structure that is defined by, what astronomers refer to as, an ~n=1 polytrope. Inside such an equilibrium structure, the density of the gas will vary with radial position according to the expression,

~\frac{\rho}{\rho_c} = \frac{\sin x}{x} \, ,

where, ~\rho_c is the density at the center of the star, and,

~x

~\equiv

~\pi\biggl(\frac{r}{R}\biggr) \, ,

where, ~R is the radius of the equilibrium star. Notice that, according to this expression, the density will drop to zero when ~r = R, in which case, ~x = \pi. If a star of this type is nudged out of equilibrium — for example, squeezed slightly — in such a way that it maintains its spherical symmetry, the star will begin to undergo periodic, radial oscillations about its original equilibrium radius. The 2nd-order ODE whose solution is being sought in the above challenge is the equation that describes the behavior of these oscillations. In particular, the function,

~\mathcal{G}_\sigma(x) \equiv \frac{\delta x}{x}

describes the relative amplitude of the oscillation as a function of position, ~x, within the star, and ~\sigma gives the frequency of the oscillation.


Related Discussions


Whitworth's (1981) Isothermal Free-Energy Surface

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