--- title: "Non-Uniform Observations and Lomb-Scargle" author: Jed Rembold date: February 25, 2025 slideNumber: true theme: tokyo-night-light highlightjs-theme: tokyo-night-light width: 1920 height: 1080 transition: slide p5js: true --- ## Announcements - Be working on HW3! - Don't forget the check-in form this weekend! - Quiz scores are finished, I just need to get them in the gradebook - No class on Thursday! I am in Pittsburgh. - If you have questions over the next 5 days, message me on Discord. I'll be much slower in responding that per usual, but I'll make sure I get to them each evening ## Recap - Looking at discrete Fourier transforms brings extra effects: - The overall observing window controls the broadness of the found peaks - Smaller windows make for broader peaks - The sampling rate determines how often we get aliased peaks - Finer sampling results in further spaced aliases - It is worth noting that, for a general Discrete Fourier Transform power spectrum, the max frequency reported is before the aliasing would start - When using pure Fourier Transform approaches, you won't need to worry about aliases ## Discussing Today - What is the limit of frequencies we can detect? - What happens when observations are not consistently spaced? - The Lomb-Scargle Periodogram - Period-Folding - Exoplanet Hunting: - Astrometric Method - Doppler Method # Taking it to the Limit ## The Nyquist Limit - Note that if the window of observations gets too small, or the time between observations too large, our Fourier Transform peaks will begin to overlap! - In this case not all of the frequency information can be recovered - This is called the Nyquist Limit, and occurs at a frequency of half the sampling frequency - The FFT algorithm generally measures frequencies _up to_ but not beyond this point, so you shouldn't see aliases in your results, but your results might not capture what you were hoping to see. ## Nyquist Visual ![](../images/python/conv3_annotated.svg) # Variety is the Spice of Life ## Non-uniform Observations ::::::cols ::::col - We don't always get to have perfectly uniform observations! - Non-uniform observations add what will seem to be "noise" to the FFT - Effects of small segments of periodicity between various observations :::: ::::col ![](../images/python/conv4_annotated.svg) :::: :::::: ## Non-uniform FFT - In practice, computing a DFT using a computational FFT can be problematic, since the returned frequencies are computed assuming a uniform spacing - The `d` in `fftfreq(n, d)` - You could super-sample the observations, adding an evenly spaced grid with 0's wherever no data exists - Still might be difficult to get an evenly spaced grid that exactly contains all observations - A better option though is to shift to a slightly different estimator: the _Lomb-Scargle Periodogram_ # Lomb-Scargle Methods ## Lomb-Scargle - Lomb and Scargle considered a more generalized form of the periodogram, with added functions $A$, $B$, and $\tau$, and then showed that you could choose those functions such that: - The periodogram reduces to the classical form for evenly spaced observations - The periodogram's statistics are analytically computable - The periodogram is insensitive to global time-shifts in the data - The same basic insights we had with the classic periodogram will still hold qualitatively with Lomb-Scargle, even if not exactly the same quantitatively - The real price you pay vs the classic periodogram is that you have to deal with aliases ## Lomb-Scargle in Python :::{style='font-size:.9em'} - Scipy's Signals library provides an implementation of the Lomb-Scargle periodogram - `from scipy.signal import lombscargle` - Works with **angular frequencies**: $$\text{Angular frequency } = \omega = \frac{2\pi}{T} = 2\pi f $$ - You need to provide 3 arrays: - The sequences of observation times - The observed signal - The desired angular frequencies to compute the periodogram over ```python power = lombscargle(ts, signal, afreqs, floating_mean=True) ``` ::: ## Lomb-Scargle in R - Install the "lomb" library to get access to nice Lomb-Scargle functions in R - `install.packages("lomb")` - Works with normal frequencies! - You can actually have it work directly in terms of period as well if you like - Need to provide: - Observation times - Observed signals - Starting and stopping frequency/period values ```R model <- lsp(df, from=0.001, to=1) ``` ## Demo - The file [here](../demos/lombscargle_demo.csv) is of a simple sine wave with noise added. - It has been sampled randomly to create a non-uniform sample rate. - Our goal is to use Lomb-Scargle to extract the period. ## Activity! - Provided [here](../demos/lombscargle_activity.csv) is a noisy signal that is a combination of several signals with differing periods - Determine the period of all underlying signals - How did you parse the different aliases? # Forging a Signal ## Phase Folding - Periodograms are excellent for determining the frequency / period of hidden signals, but they don't let you _see_ those signals - Often times want to go one step further and use the found period to "fold" the signal over again on itself - Snippets of the signal that may have been captured by very different observations get properly aligned or stacked - The _phase_ of the signal describes how far is signal is through its period - Calculating phase is a classic example of using the modulo operator: ```python phase = times % period normalize_phase = phase / period ``` ## Visual Phase Folding ::::::{.cols style='align-items: flex-start'} ::::col - Often times you'll see the phase scaled by the period, so that it starts at 0 and ends at 1 - Be careful! Folding at integer multiples of the true period may _look_ clean, but will contain more than a single oscillation - [Notebook Demonstration](../demos/PhaseFoldingDemo.ipynb) (requires the ipywidgets package) :::: ::::col ![Phase Folding Demo](../images/phase_folding.png) :::: :::::: # The Exoplanet Quest ## Why are planets so hard to see? ::::::cols ::::col - _Exoplanets_ are planets orbiting stars that are not our own Sun - Commonly far too tiny to be observed directly - Recall we can't even resolve most **stars** - We must then rely on other, more subtle measures :::: ::::col \begin{tikzpicture}%%width=100% [scale=.9, transform shape] \fill[inner color=yellow, outer color=orange] (0,0) circle (3.5cm); \fill[black] (-1,0) circle (3.5mm) node[above,yshift=3mm] {Jupiter}; \fill[black] (1,0) circle (0.3mm) node[above, yshift=1mm] {Earth}; \end{tikzpicture} :::: :::::: ## Gravity Tugs! ::::::cols ::::col
![](../images/sun_wobble.png){width=70%} :::: ::::col - Planets and their host stars actually orbit the _center of mass_ between the planet and its star - In most system, this point might still be inside the radius of the star, but it is not at the **center** of the star - As the star oscillates then, you get a bit of "wobble" - Most pronounced for massive planets far from their host star :::: :::::: ## The Astrometric Method - Make **extremely** precise measurements of a star's location against its background stars - Need slightly ridiculous levels of precision to do well - More "obvious" planets would have longer orbits, and thus we would need to observe longer to see their effects - These types of measurements are one of the main aims of the [GAIA](https://www.esa.int/Science_Exploration/Space_Science/Gaia) mission - Hangs out at the $L_2$ Lagrange point (same as Webb!) - Needs to know its exact position **to within 150 m** every day! ## {data-background-image=../images/gaia_skymap_dr3.png} ## Wiggle Wiggle - Often, we are not viewing the plane of an exoplanetary system directly from the top - How we see this "wiggle" from Earth depends on how the planets orbit is oriented relative to us - A perfectly "top down" view would have us seeing the planet making little circles - A perfectly "side" view would have us seeing the planet move towards us and away from us on the left and right sides - In general, it is easier for us to detect and measure the forwards and backwards motion, but instruments like Gaia _can_ detect the tinier circular motion for some systems ## Doppler Wiggle - The idea then is to monitor the dominant frequency of light emitted over a period of time - Should result in a sinusoidal curve as the star wiggles - The amount of wiggle will depend on **both** the mass of the orbiting planet **and** our perspective ![](../images/pegasi_51.png) ## Full Example