from ztf_summerschool import source_lightcurve, barycenter_times %matplotlib inline

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1 In [2]: # imports import numpy as np import matplotlib.pyplot as plt import astropy.coordinates as coords import astropy.units as u from astropy.time import Time from astroml.time_series import \ lomb_scargle, lomb_scargle_bootstrap from ztf_summerschool import source_lightcurve, barycenter_times %matplotlib inline Hands-On Exercise 3: Period Finding One of the fundamental tasks of time-domain astronomy is determining if a source is periodic, and if so, measuring the period. Period measurements are a vital first step for more detailed scientific study, which may include source classification (e.g., RR Lyrae, W Uma), lightcurve modeling (binaries), or luminosity estimation (Cepheids). Binary stars in particular have lightcurves which may show a wide variety of shapes, depending on the nature of the stars and the system geometry. In this workbook we will develop a basic toolset for the generic problem of finding periodic sources. by Eric Bellm ( ) Let's use the relative-photometry corrected light curves we built in Exercise 2. We'll use the utility function source_lightcurve to load the columns MJD, magnitude, and magnitude error. Note that we will use days as our time coordinate throughout the homework. In [3]: # point to our previously-saved data reference_catalog = '../data/ptf_refims_files/ptf_d022683_f02_c06_u _ p12_sexcat.ctlg' outfile = reference_catalog.split('/')[-1].replace('ctlg','shlv') We'll start by loading the data from our favorite star, which has coordinates α J2000, δ J2000 = ( , ). In [4]: ra_fav, dec_fav = ( , ) mjds, mags, magerrs = source_lightcurve('../data/'+outfile, ra_fav, dec_fav) Exercise 0: Barycentering Our times are Modified Julian Date on earth. We need to correct them for Earth's motion around the sun (this is called heliocentering or barycentering). How big is the error if we do not make this correction?

2 In [4]: # CALCULATE AN ANSWER HERE import astropy.constants as const (const.au/const.c).to(u.min) Out[4]: min We have provided a script to barycenter the data--note that it assumes that the data come from the P48. Use the bjds (barycentered modified julian date) variable through the remainder of this notebook. In [5]: bjds = barycenter_times(mjds,ra_fav,dec_fav) In [6]: bjds Out[6]: array([ , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ]) Optional exercise: plot a histogram of the time differences between the barycentered and non-barycentered data. In [12]: s = np.argsort(bjds) bjds = bjds[s] mags = mags[s] magerrs = magerrs[s] Exercise 1: Getting started plotting Complete this function for plotting the lightcurve: In [7]: # define plot function def plot_data(mjd, mag, magerr): # COMPLETE THIS LINE plt.errorbar(mjd, mag, yerr=magerr, # COMPLETE THIS LINE fmt = '_', capsize=0) plt.xlabel('date (MJD)') plt.ylabel('magnitude') plt.gca().invert_yaxis()

3 In [8]: # run plot function plot_data(bjds, mags, magerrs) /opt/local/library/frameworks/python.framework/versions/2.7/lib/python2.7/si te-packages/matplotlib/figure.py:1653: UserWarning: This figure includes Axe s that are not compatible with tight_layout, so its results might be incorre ct. warnings.warn("this figure includes Axes that are not "

4 In [8]: # documentation for the lomb_scargle function help(lomb_scargle) Help on built-in function lomb_scargle in module astroml_addons.periodogram: lomb_scargle(...) (Generalized) Lomb-Scargle Periodogram with Floating Mean lse ue Parameters t : array_like sequence of times y : array_like sequence of observations dy : array_like sequence of observational errors omega : array_like frequencies at which to evaluate p(omega) generalized : bool if True (default) use generalized lomb-scargle method otherwise, use classic lomb-scargle. subtract_mean : bool if True (default) subtract the sample mean from the data before computing the periodogram. Only referenced if generalized is False. significance : None or float or ndarray if specified, then this is a list of significances to compute for the results. Returns p : array_like Lomb-Scargle power associated with each frequency omega z : array_like if significance is specified, this gives the levels corresponding to the desired significance (using the Scargle 1982 formalism) Notes The algorithm is based on reference [1]_. The result for generalized=fa is given by equation 4 of this work, while the result for generalized=tr is given by equation 20. t Note that the normalization used in this reference is different from tha used in other places in the literature (e.g. [2]_). For a discussion of normalization and false-alarm probability, see [1]_. To recover the normalization used in Scargle [3]_, the results should be multiplied by (N - 1) / 2 where N is the number of data points. References [1] M. Zechmeister and M. Kurster, A&A 496, (2009).. [2] W. Press et al, Numerical Recipies in C (2002).. [3] Scargle, J.D. 1982, ApJ 263:

5 Exercise 2: Determining the frequency grid One of the challenges of using the LS periodogram is determining the appropriate frequency grid to search. We have to select the minimum and maximum frequencies as well as the bin size. If we don't include the true frequency in our search range, we can't find the period! If the bins are too coarse, true peaks may be lost. If the bins are too fine, the periodogram becomes very slow to compute. The first question to ask is what range of frequencies our data is sensitive to. Exercise 2.1 What is the smallest angular frequency ω min our data is sensitive to? (Hint: smallest frequency => largest time) In [9]: freq_min = 2*np.pi / (bjds[-1]-bjds[0]) # COMPLETE print 'The minimum frequency our data is sensitive to is {:.3f} radian/days, corresponding to a period of {:.3f} days'.format(freq_min, 2*np.pi/freq_min) The minimum frequency our data is sensitive to is radian/days, corresp onding to a period of days Exercise 2.2 Determining the highest frequency we are sensitive to turns out to be complicated. if Δt is the difference between consecutive observations, π/median( Δt) is a good starting point, although in practice we may be sensitive to frequencies even higher than 2π/min( Δt) depending on the details of the sampling. What is the largest angular frequency ω max our data is sensitive to? In [10]: freq_max = np.pi / np.median(bjds[1:]-bjds[:-1]) # COMPLETE print 'The maximum frequency our data is sensitive to is APPROXIMATELY {:.3f} radian/days, corresponding to a period of {:.3f} days'.format(freq_max, 2*np.pi/freq_max) The maximum frequency our data is sensitive to is APPROXIMATELY radia n/days, corresponding to a period of days Exercise 2.3 We need enough bins to resolve the periodogram peaks, which have ( /lomb_scargle.html) frequency width Δf 2π/( t max t min ) = ω min. If we want to have 5 samples of Δf, how many bins will be in our periodogram? Is this computationally feasible? In [11]: n_bins = np.round(5 * (freq_max-freq_min)/freq_min) # COMPLETE print n_bins

6 Exercise 2.4 Let's wrap this work up in a convenience function that takes as input a list of observation times and returns a frequency grid with decent defaults. In [12]: # define frequency function def frequency_grid(times): freq_min = 2*np.pi / (times[-1]-times[0]) # COMPLETE freq_max = np.pi / np.median(times[1:]-times[:-1]) # COMPLETE n_bins = np.floor((freq_max-freq_min)/freq_min * 5.) # COMPLETE print 'Using {} bins'.format(n_bins) return np.linspace(freq_min, freq_max, n_bins) In [13]: # run frequency function omegas = frequency_grid(bjds) Using bins In some cases you'll want to generate the frequency grid by hand, either to extend to higher frequencies (shorter periods) than found by default, to avoid generating too many bins, or to get a more precise estimate of the period. In that case use the following code. We'll use a large fixed number of bins to smoothly sample the periodogram as we zoom in. In [14]: # provided alternate frequency function def alt_frequency_grid(pmin, Pmax, n_bins = 5000): """Generate an angular frequency grid between Pmin and Pmax (assumed to b e in days)""" freq_max = 2*np.pi / Pmin freq_min = 2*np.pi / Pmax return np.linspace(freq_min, freq_max, n_bins) Exercise 3: Computing the Periodogram Calculate the LS periodiogram and plot the power.

7 In [15]: # calculate and plot LS periodogram P_LS = lomb_scargle(bjds, mags, magerrs, omegas) # COMPLETE plt.plot(omegas, P_LS) plt.xlabel('$\omega$') plt.ylabel('$p_{ls}$') Out[15]: <matplotlib.text.text at 0x110483a90> In [16]: # provided: define function to find best period def LS_peak_to_period(omegas, P_LS): """find the highest peak in the LS periodogram and return the correspondi ng period.""" max_freq = omegas[np.argmax(p_ls)] return 2*np.pi/max_freq In [17]: # run function to find best period best_period = LS_peak_to_period(omegas, P_LS) print "Best period: {} days".format(best_period) Best period: days Exercise 4: Phase Calculation Complete this function that returns the phase of an observation (in the range 0-1) given its period. For simplicity set the zero of the phase to be the time of the initial observation. Hint: Consider the python modulus operator, %. Add a keyword that allows your function to have an optional user-settable time of zero phase. In [18]: # define function to phase lightcurves def phase(time, period, t0 = None): """ Given an input array of times and a period, return the corresponding phase.""" if t0 is None: t0 = time[0] return ((time - t0)/period) % 1 # COMPLETE

8 Exercise 5: Phase Plotting Plot the phased lightcurve at the best-fit period. In [19]: # define function to plot phased lc def plot_phased_lc(mjds, mags, magerrs, period, t0=none): phases = phase(mjds, period, t0=t0) # COMPLETE plt.errorbar(phases,mags,yerr=magerrs, #COMPLETE fmt = '_', capsize=0) plt.xlabel('phase') plt.ylabel('magnitude') plt.gca().invert_yaxis() In [20]: # run function to plot phased lc plot_phased_lc(bjds, mags, magerrs, best_period) How does that look? Do you think you are close to the right period? Try re-running your analysis using the alt_frequency_grid command, searching a narrower period range around the best-fit period.

9 In [23]: # COMPLETE omegas = alt_frequency_grid(.6,.65) P_LS = lomb_scargle(bjds, mags, magerrs, omegas) plt.plot(omegas, P_LS) plt.xlabel('$\omega$') plt.ylabel('$p_{ls}$') Out[23]: <matplotlib.text.text at 0x108c4d850> In [24]: # COMPLETE best_period = LS_peak_to_period(omegas, P_LS) print "Best period: {} days".format(best_period) plot_phased_lc(bjds, mags, magerrs, best_period) Best period: days

10 [Challenge] Exercise 6: Calculating significance of the period detection Real data may have aliases--frequency components that appear because of the sampling of the data, such as once per night. Bootstrap significance tests, which shuffle the data values around but keep the times the same, can help rule these out. Calculate the chance probability of finding a LS peak higher than the observed value in random data observed at the specified intervals: use lomb_scargle_bootstrap and np.percentile to find the 95 and 99 percent significance levels and plot them over the LS power. In [78]: Out[78]: D = lomb_scargle_bootstrap(mjds, mags, magerrs, omegas, generalized=true, N_bootstraps=1000) # COMPLETE sig99, sig95 = np.percentile(d, [99, 95]) # COMPLETE plt.plot(omegas, P_LS) plt.plot([omegas[0],omegas[-1]], sig99*np.ones(2),'--') plt.plot([omegas[0],omegas[-1]], sig95*np.ones(2),'--') plt.xlabel('$\omega$') plt.ylabel('$p_{ls}$') <matplotlib.text.text at 0x1111dced0> [Challenge] Exercise 7: Find periods of other sources Now try finding the periods of these sources: , [Challenge] Exercise 9: gatspy Try using the gatspy ( package to search for periods. It uses a slightly different interface but has several nice features, such as automatic zooming on candidate frequency peaks.

11 In [49]: import gatspy ls = gatspy.periodic.lombscarglefast() ls.optimizer.period_range = (0.2,1.2) # we have to subtract the t0 time so the model plotting has the correct phase origin ls.fit(bjds-bjds[0],mags,magerrs) gatspy_period = ls.best_period print gatspy_period plot_phased_lc(bjds, mags, magerrs, gatspy_period) p = np.linspace(0,gatspy_period,100) plt.plot(p/gatspy_period,ls.predict(p,period=gatspy_period)) - Computing periods at 1358 steps Out[49]: [<matplotlib.lines.line2d at 0x1110c2b10>] [Challenge] Exercise 10: Alternate Algorithms Lomb-Scargle is equivalent to fitting a sinusoid to the phased data, but many kinds of variable stars do not have phased lightcurves that are well-represented by a sinusoid. Other algorithms, such as those that attempt to minimize the dispersion within phase bins over a grid of trial phases, may be more successful in such cases. See Graham et al (2013) ( for a review.

12 In [50]: ss = gatspy.periodic.supersmoother(fit_period=true) ss.optimizer.period_range = (0.2, 1.2) ss.fit(bjds-bjds[0],mags,magerrs) gatspy_period = ss.best_period print gatspy_period plot_phased_lc(bjds, mags, magerrs, gatspy_period) p = np.linspace(0,gatspy_period,100) plt.plot(p/gatspy_period,ss.predict(p,period=gatspy_period)) - Computing periods at 1358 steps Out[50]: [<matplotlib.lines.line2d at 0x110d332d0>] [Challenge] Exercise 10: Multi-harmonic fitting Both AstroML and gatspy include code for including multiple Fourier components in the fit, which can better fit lightcurves that don't have a simple sinusoidal shape (like RR Lyrae).

13 In [76]: from astroml.time_series import multiterm_periodogram omegas = alt_frequency_grid(.2,1.2) P_mt = multiterm_periodogram(bjds, mags, magerrs, omegas) #COMPLETE plt.plot(omegas, P_mt) plt.xlabel('$\omega$') plt.ylabel('$p_{mt}$') Out[76]: <matplotlib.text.text at 0x110f6e210> In [77]: best_period = LS_peak_to_period(omegas, P_mt) # COMPLETE print "Best period: {} days".format(best_period) plot_phased_lc(bjds, mags, magerrs, best_period) Best period: days

14 In [75]: ls = gatspy.periodic.lombscargle(nterms=4) ls.optimizer.period_range = (0.2,1.2) ls.fit(bjds-bjds[0],mags,magerrs) gatspy_period = ls.best_period print gatspy_period plot_phased_lc(bjds, mags, magerrs, gatspy_period) p = np.linspace(0,gatspy_period,100) plt.plot(p/gatspy_period,ls.predict(p,period=gatspy_period)) - Computing periods at 1358 steps Out[75]: [<matplotlib.lines.line2d at 0x111440d10>] [Challenge] Exercise 8: Compute all periods This is a big one: can you compute periods for all of the sources in our database with showing evidence of variability? How will you compute variablity? How can you tell which sources are likely to have good periods? In [43]: # open the stored data import shelve import astropy shelf = shelve.open('../data/'+outfile) all_mags = shelf['mags'] all_mjds = shelf['mjds'] all_errs = shelf['magerrs'] all_coords = shelf['ref_coords'] shelf.close()

15 In [64]: # loop over stars variable_inds = [] best_periods = [] best_power = [] chisq_min = 5 with astropy.utils.console.progressbar(all_mags.shape[0],ipython_widget=true) as bar: for i in range(all_mags.shape[0]): # make sure there's real data wgood = (all_mags[i,:].mask == False) n_obs = np.sum(wgood) if n_obs < 40: continue # find variable stars using chi-squared. If the data points are cons istent with a line, reduced chisq ~ 1 chisq = np.sum((all_mags[i,:]-np.mean(all_mags[i,:]))**2./all_errs[i, :]**2./(n_obs - 1.)) if chisq < chisq_min: continue variable_inds.append(i) bjds = barycenter_times(all_mjds[wgood],all_coords[i].ra.degree,all_c oords[i].dec.degree) ls = gatspy.periodic.lombscarglefast() ls.optimizer.period_range = (0.2,1.2) ls.fit(bjds-bjds[0],all_mags[i,:][wgood],all_errs[i,:][wgood]) best_periods.append(ls.best_period) best_power.append(float(ls.periodogram(ls.best_period))) bar.update()

16 - Computing periods at 1357 steps - Estimated peak width = Computing periods at 1357 steps - Computing periods at 1358 steps - Computing periods at 1358 steps - Computing periods at 1358 steps - Computing periods at 1358 steps - Computing periods at 1358 steps - Computing periods at 1358 steps - Computing periods at 1358 steps

17 In [65]: # cut out the NaNs: w = np.isfinite(np.array(best_power)) best_power = np.array(best_power)[w] best_periods = np.array(best_periods)[w] variable_inds = np.array(variable_inds)[w] # sort by most power in the LS peak s = np.argsort(best_power) # reverse s = s[::-1]

18 In [66]: iis = variable_inds[s][:10] periods = best_periods[s][:10] for i,period in zip(iis,periods): # make sure there's real data wgood = (all_mags[i,:].mask == False) print all_coords[i].ra.degree,all_coords[i].dec.degree bjds = barycenter_times(all_mjds[wgood],all_coords[i].ra.degree,all_coord s[i].dec.degree) ls = gatspy.periodic.lombscarglefast() ls.optimizer.period_range = (0.2,2) ls.fit(bjds-bjds[0],all_mags[i,:][wgood],all_errs[i,:][wgood]) plt.figure() plot_phased_lc(bjds, all_mags[i,:][wgood], all_errs[i,:][wgood], period) p = np.linspace(0,period,100) plt.plot(p/period,ls.predict(p,period=period))

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22 Other effects to consider Many eclipsing binaries have primary and secondary eclipses, often with comparable depths. The period found by LS (which fits a single sinusoid) will thus often be only half of the true period. Plotting the phased lightcurve at double the LS period is often the easiest way to determine the true period. References and Further Reading Scargle, J. 1982, ApJ 263, 835 ( Zechmeister, M., and Kürster, M. 2009, A&A 496, 577 ( Graham, M. et al. 2013, MNRAS 434, 3423 ( Statistics, Data Mining, and Machine Learning in Astronomy ( (Ivezic, Connolly, VanderPlas, & Gray) gatspy documentation (