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Add non-zero surface term to fix homogeneous model $f$-modes - #33

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rboston628 merged 1 commit into
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uniform-f-mode
Jul 4, 2026
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Add non-zero surface term to fix homogeneous model $f$-modes#33
rboston628 merged 1 commit into
mainfrom
uniform-f-mode

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@rboston628 rboston628 commented Jul 4, 2026

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The uniform density (aka isopycnic) star model of Thrain was used model as a calibration tool, for comparing to the exact Pekeris solutions. However, it had a known flaw, that it consistently produced the wrong value for the $f$-mode. See Issue #16 .

This problem is due to the non-vanishing surface density of the uniform density model. Unlike real stars, and even polytropes, the uniform density star's density does not taper to zero at the surface, but actually has a sharp $\Theta$-function cutoff.

To account for this, Unno et al. mention the possible surface term. It can be seen from the perturbed gravitationl field equation,
$\frac{1}{r^2}\frac{d}{dr} \big(r^2 \frac{d\Delta\Phi}{dr}\big) - \frac{\ell(\ell+1)}{r^2}\Delta\Phi = 4\pi G\Delta\rho.$$
For most starts, $\Delta\rho \rightarrow 0$ at the surface. For the uniform density model, it becomes
$$\Delta\rho = \rho_o \delta(R-r) \xi^r,$$
and integrating
$$r\frac{d\Delta\Phi}{dr} = -(\ell+1) \Delta\Phi + 4\pi G \rho_o \xi^r r$$

In Dziembowski variables, this term becomes
$$y_4 = -(\ell+1) y_3 - Uy_1.$$

Since for all non-uniform stars this final term is zero, we can add this extra term, which will appear only in the case of a star with a surface term.

Now, we get the correct value for the uniform star $f$-mode. Also makes some changes to reduce the compiler warnings.

Closes #16

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rboston628 marked this pull request as ready for review July 4, 2026 14:12
@rboston628 rboston628 changed the title fix issue with surface Add non-zero surface term to fix homogeneous model $f$-modes Jul 4, 2026
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rboston628 merged commit 91bec35 into main Jul 4, 2026
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Homogeneous f-mode is off.

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