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Systemic - Characterizing Extrasolar Planetary Systems

Angular Power Spectra

29 Dec 2014, 19:06 UTC
Angular Power Spectra
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It’s worth a scramble to get a window seat on a Hawaiian inter-island flight. The views are full of craggy green cliffs, porcelain ocean, and wispy masses of fog and cloud. Sometimes, several islands are visible at once, and it’s not hard to imagine that the archipelago might extend over the entire globe.
That would be a very different planet, and, in fact, a world covered by hotspot volcanoes might have a surface elevation profile somewhat reminiscent of the WMAP image of the temperature fluctuations in the cosmic microwave background. The WMAP image brings to mind a planet covered in Hawaiian islands.

Any distribution, \(f(\theta,\phi)\), on the surface of a sphere, be it of temperature, or elevation, or the density of IP addresses, can be expressed as a weighted sum of spherical harmonics
$$f(\theta,\phi)=\sum_{l,m} a_{l,m} Y(\theta,\phi)_{l}^{m}\, ,$$
where the coefficients corresponding to the individual weights, \(a_{l,m}\) are given by
$$a_{l,m}=\int_{\Omega}f(\theta,\phi)Y(\theta,\phi)_{l}^{m \star}d\Omega\, ,$$
and the power, \(C_{l}\) at angular scale \(l\) is
$$C_{l}=\frac{1}{2l+1}\sum_{m=-l}^{l}a_{l,m} {a_{l,m}}^{\star}\, .$$
The power spectrum of the CMB anisotropies peaks at \(l\sim 200\), which corresponds to an angular scale on the sky of \(\Delta \theta \sim 1^{\circ}\), which is very close to the solid angle subtended by ...

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