I should explain what I've been doing. For the past several weeks I've been reading quite a lot of papers on modal anaylsis by going through their derivations of the modal equation (specifically Goldreich/Tremaine 1979, Val-Borro et al. 2008, Laughin/Bodenheimer 1994,...and more). We mainly stayed with Goldreich/Tremaine (GT79) derivation where the modal equation is...

...where...

Phi_1 is the total perturbed potential, phi_1 can be thought of as the perturbed gravitational potential (self-gravity, external potential, etc.) and eta is the perturbed enthalpy.

Now, with radiation pressure, the modal equation becomes...

Sorry if it's too small...I can't seem to make it bigger. Essentially, this equation is a second order ODE of the form...

where...

Note that the term I is also in terms of eta so we believe this equation can be thought of as a third order (homogeneous?) equation. And obviously the coefficients we're dealing are non-constant with derivatives all over the place.

So far, I tried to code this type of ODE (assuming the coefficients are constant) via leapfrog and I have something like this...

...like a decaying free-vibration...which is expected. Also remember that equations like these typically have two solutions (if you solve the characeristic equation you would a few roots - real or imagninary).

So now, I'm trying to figure out how to code this equation with all the non-constant coefficients included.

with non-constant coefficients it's almost the same (except they are almost singular!) - the method's the same. Don't start integrating from zero, I'd set the staring point at 1/4 or so from the place where tau_0 goes up sharply.

ReplyDeleteyour eqs. are nicely visible, no worries. (you need to click on them).

ReplyDeletewhen you solve your eqs, can you display real and imaginary components of your solution overplotted?

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