Physics Asked on July 18, 2021
I was reading this paper on astro-ph (Sanders et al. 2020, "Models of Distorted and Evolving Dark Matter Haloes"). It’s about modelling the gravitational potential in a cosmological simulation using basis function expansions.
Specifically, I was looking at section 4.2 ("Error Measure") on page 10. In the second paragraph of that section, it says the period of the orbits is "computed by taking a (zero-padded) fast Fourier Transform of the particle’s original trajectory and computing one cycle with respect to the dominant frequency".
Is anyone here familiar with using this sort of methodology to calculate orbital periods? I am trying to do something similar, but I’m confused on how to apply this to a 3D system. I understand for a 1D system, you could just take the Fourier Transform and find the frequency/period associated with the peak. However, no matter what coordinate system I choose, won’t I have three frequencies? (someone I work with referred me to the third chapter from Giacomo Monari’s Ph.D. thesis, but even here, they seem to be working with three angular frequencies: $omega_R$, $omega_{phi}$, and $omega_z$)
I’m working with halo stars, so the stars are not necessarily travelling in orbits along the disk, meaning it doesn’t make sense to just, say, work in cylindrical coordinates and take $omega_{phi}$.
TL,DR: Does anyone know how to estimate the orbital period of stars by somehow taking the Fourier Transform of the trajectory, despite the fact that the system is three dimensional?
These frequencies $omega_R$, $omega_phi$, $omega_z$ would be the same for a Keplerian orbit, but could be quite different otherwise. If the orbit is non-Keplerian then you need to define what is meant by "the orbital period".
As an example, for the Sun, the three periodicities would be about 250 million years in $phi$, about 140 million years in $R$ and about 75 million years in $z$. The orbit is non-Keplerian and isn't closed.
I suppose we could argue that 250 million years is "the orbital period" given that it is the periodicity on which the position of the Sun changes the most and that the amplitudes of the excursions in $R$ and $z$ are comparatively small.
Thus perhaps that is your solution - pick the periodicity with the largest amplitude - a.k.a. the dominant frequency - and define that as "the orbital period".
Answered by ProfRob on July 18, 2021
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