In the <razor-thin disk approximation>, the three-dimensional <Poisson equation> is
$$
\nabla^2\Phi_d=4\pi G\Sigma(x,y)\delta(z).
$$
For a horizontal <Fourier mode> with <wavevector> $\mathbf k$ and $k=|\mathbf k|>0$, the <Fourier transform> of this equation is
$$
\left(\frac{d^2}{dz^2}-k^2\right)\widehat\Phi_d
=4\pi G\widehat\Sigma\,\delta(z).
$$
The solution that decays away from the disk is
$$
\widehat\Phi_d(\mathbf k,z)
=-\frac{2\pi G}{k}\widehat\Sigma(\mathbf k)e^{-k|z|}.
$$
Consequently the required <razor-thin disk Poisson kernel> in the midplane is
$$
\boxed{\widehat\Phi_{d,m}(\mathbf k)
=-\frac{2\pi G}{k}\widehat\Sigma(\mathbf k)}.
$$
The spatially uniform $k=0$ background is excluded from this local perturbation formula.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 1 / a / Solution

The razor-thin disk approximation replaces a disk's vertical density profile by $\rho(x,y,z)=\Sigma(x,y)\delta(z)$. It is useful for horizontal dynamics on scales much longer than the disk thickness.


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