Glossary of documentation terms

Akaike Information Criterion

a common metric for model selection that prevents overfitting of data by penalizing models with higher numbers of parameters (\(k\))

definition:

\[\mathrm{AIC} = 2k - 2\mathrm{ln}(\hat{L})\]

asteroseismology: the study of oscillations in stars
ACF
autocorrelation function: in this context it is a small range of frequencies in the power spectrum surrounding the solar-like oscillations, then the power array is correlated (or convolved) with a copy of the power array. This is a helpful diagnostic tool for quantitatively confirming the p-mode oscillations, since they have regular spacings in the frequency domain and therefore should create strong peaks at integer and half integer harmonics of \(\Delta\nu\)
background: this basically means any other noise structures present in the power spectrum that are not due to solar-like oscillations. This is traditionally parametrized as:

\[B(\nu) = W + \sum_{i=0}^{n} \frac{4\sigma_{i}^{2}\tau_{i}}{1 + (2\pi\nu\tau_{i})^{2} + (2\pi\nu\tau_{i})^{4}}\]

BCPS
background-corrected power spectrum: the power spectrum after removing the best-fit stellar background model. In general, this step removes any slopes in power spectra due to correlated red-noise properties

Note

A background-corrected power spectrum (BCPS) is an umbrella term that has the same meanings as a background-divided power spectrum (BDPS) and a background-subtracted power spectrum (BSPS). Thus it is best *to avoid* this phrase if at all possible since it does not specify how the power spectrum has been modified.

BDPS

background-divided power spectrum

the power spectrum divided by the best-fit stellar background model. Using this method for data analysis is great for first detecting and identifying any solar-like oscillations since it will make the power excess due to stellar oscillations appear higher signal-to-noise

BSPS

background-subtracted power spectrum

the best-fit stellar background model is subtracted from the power spectrum. While this method appears to give a lower signal-to-noise detection, the amplitudes measured through this analysis are physically-motivated and correct (i.e. can be compared with other literature values)

BIC

Bayesian Information Criterion

a common metric for model selection

cadence

the median absolute difference between consecutive time series observations

variable: \(\Delta t\)
units: \(\rm s\)
definition:

critically-sampled power spectrum

when the frequency resolution of the power spectrum is exactly equal to the inverse of the total duration of the time series data it was calculated from

ED

echelle diagram

a diagnostic tool to confirm that dnu is correct. This is done by folding the power spectrum (FPS) using dnu (you can think of it as the PS modulo the spacing) – which if the large frequency separation is correct – the different oscillation modes will form straight ridges. Fun fact: the word ‘echelle’ is actually French for ladder

FFT

fast fourier transform

a method used in signal analysis to determine the most dominant periodicities present in a light curve

FPS

folded power spectrum

the power spectrum folded (or stacked) at some frequency, which is typically done with the large frequency separation to construct an echelle diagram

numax

frequency of maximum power

the frequency corresponding to maximum power, which is roughly the center of the Gaussian-like envelope of oscillations

variable: \(\nu_{\mathrm{max}}\)

units: \(\rm \mu Hz\)

scales with evolutionary state, logg, acoustic cutoff

frequency resolution

the resolution of a power spectrum is set by the total length of the time series \((\Delta T^{-1})\)

FWHM

full-width half maximum

for a Gaussian-like distribution, the full-width at half maximum (or full-width half max) is approximately equal to \(\pm 1\sigma\)

global properties

in asteroseismology, the global asteroseismic parameters or properties refer to \(\nu_{\mathrm{max}}\) (numax) and \(\Delta\nu\) (dnu)

granulation

the smallest (i.e. quickest) scale of convective processes

Harvey-like component

Harvey-like model

named after the person who first person who discovered the relation – and found it did a good job characterizing granulation amplitudes and time scales in the Sun

Kepler artefact

Kepler short-cadence artefact in the power spectrum from a short-cadence light curve occurring at the nyquist frequency for long-cadence (i.e. ~270muHz)

Kepler legacy sample

a sample of well-studied Kepler stars exhibiting solar-like oscillations (cite Lund+2014)

dnu

large frequency separation

the so-called large frequency separation is the inverse of twice the sound travel time between the center of the star and the surface. Even more generally, this is the comb pattern or regular spacing observed for solar-like oscillations. It is exactly equal to the frequency spacing between modes with the same spherical degree and consecutive :term:`radial order`s.

variable: \(\Delta\nu\)

units: \(\rm \mu Hz\)

definition:

\[\Delta\nu = \bigg[2 \int_{0}^{R} \frac{\mathrm{d}r}{c}\bigg]^{-1} \propto \bar{\rho}\]

light curve

the measure of an object’s brightness with time

mesogranulation

the intermediate scale of convection

mixed modes

in special circumstances, pressure (or p-) modes couple with gravity (or g-) modes and make the spectrum of a solar-like oscillator much more difficult to interpret – in particular, for measuring the large frequency separation

notching

a process used to mitigate features in the frequency domain (e.g., mixed modes) by setting specific values to the minimum power in the array

nyquist frequency

the highest frequency that can be sampled, which is set by the cadence of observations (\(\Delta t\))

variable: \(\rm \nu_{nyq}\)

units: \(\rm \mu Hz\)

definition:

\[\mathrm{\nu_{nyq}} = \frac{1}{2 \Delta t}\]

Note

Kepler example

Kepler short-cadence data has a cadence, \(\Delta t \sim 60 \mathrm{s}\). Therefore, the nyquist frequency for short-cadence Kepler data is:

\[\mathrm{\nu_{nyq}} = \frac{1}{2\cdot60\,\mathrm{s}} \times \frac{10^{6}\,\mu\mathrm{Hz}}{1\,\mathrm{Hz}} \approx 8333 \,\mu\mathrm{Hz}\]

oversampled power spectrum: if the resolution of the power spectrum is greater than 1/T
p-mode oscillations
solar-like oscillations: implied in the name, these oscillations are driven by the same mechanism as that observed in the Sun, which is due to turbulent, near-surface convection. They are also sometimes referred to as p-mode oscillations, after the pressure-driven (or acoustic sound) waves that are resonating in the stellar cavity.

power excess: the region in the power spectrum believed to show solar-like oscillations is typically characterized by a Gaussian-like envelope of oscillations, \(G(\nu)\)

\[G(\nu) = A_{\mathrm{osc}} \,\mathrm{exp} \bigg[ - \frac{(\nu-\nu_{\mathrm{max}})^{2}}{2\sigma_{\mathrm{osc}}^{2}} \bigg]\]

PSD

power spectral density

when the power of a frequency spectrum is normalized s.t. it satisfies Parseval’s theorem (which is just a fancy way of saying that the fourier transform is unitary)

unit: \(\rm ppm^{2} \,\, \mu Hz^{-1}\)

PS

power spectrum

any object that varies in time also has a corresponding frequency (or power) spectrum, which is computed by taking the fast fourier transform of the light curve. A general model to describe characteristics of a power spectrum is generalized by the equation below, where \(W\) is a constant (frequency-independent) noise term, primarily due to photon noise. \(B\) and \(G\) correspond to the background and Gaussian-like power excess components, respectively. Finally, \(R\) corresponds to the response function, or the attenuation of signals due to time-averaged observations.

\[P(\nu) = W + R(\nu) [B(\nu) + G(\nu)]\]

scaling relations: empirical relations for fundamental stellar properties that are scaled with respect to the Sun, since it is the star we know best. In asteroseismology, the most common relations combine global asteroseismic parameters with spectroscopic effective temperatures to derive stellar masses and radii:

\[\frac{R_{\star}}{R_{\odot}} = \bigg( \frac{\nu_{\mathrm{max}}}{\nu_{\mathrm{max,\odot}}} \bigg) \bigg( \frac{\Delta\nu}{\Delta\nu_{\odot}} \bigg)^{-2} \bigg( \frac{T_{\mathrm{eff}}}{T_{\mathrm{eff,\odot}}} \bigg)^{1/2}\]

\[\frac{M_{\star}}{M_{\odot}} = \bigg( \frac{\nu_{\mathrm{max}}}{\nu_{\mathrm{max,\odot}}} \bigg)^{3} \bigg( \frac{\Delta\nu}{\Delta\nu_{\odot}} \bigg)^{-4} \bigg( \frac{T_{\mathrm{eff}}}{T_{\mathrm{eff,\odot}}} \bigg)^{3/2}\]

whiten
whitening: a process to remove undesired artefacts or effects present in a frequency spectrum by taking that frequency region and replacing it with simulated white noise. This is typically done for subiants with mixed modes in order to better estimate dnu. This can also help mitigate the short-cadence Kepler artefact.