Analysis of exoplanetary spectra with a multi-fractal and multi-wavelength approach

Sahil Agarwal,1, ∗ Fabio Del Sordo,2, 3, 4, † and J. S. Wettlaufer1, 4, 5, 6, ‡

1Program in Applied Mathematics, Yale University, New Haven, USA2Department of Astronomy, Yale University, New Haven, USA

3Departments of Geology & Geophysics, Yale University, New Haven, USA4Nordita, Royal Institute of Technology and Stockholm University, SE-10691 Stockholm, Sweden

5Departments of Geology & Geophysics, Mathematics and Physics, Yale University, New Haven, USA6Mathematical Institute, University of Oxford, Oxford, UK

(Dated: May 25, 2016)

I. INTRODUCTION

The last three decades have seen the birth of exoplan-etary science. Thousands of planets have been detectedorbiting stas other than the Sun, thanks to various tech-niques which include Doppler measurements, transit pho-tometry, microlensing, and direct imaging. Whilst thecombination of the aforementioned approaches have pro-vided insight on planets of different size and orbital pe-riod, the detection of Earth analogs is still an open prob-lem. Terrestrial-like planets could currently be detectedonly with transit photometry or Doppler measurements.Nonetheless, both these methods are affected from thepresence of instrumental and astrophysical noise whosemagnitude could easily overcome that of the signal weare looking for.

Here we propose a new approach to the analysis ofspectral data to identify the timescales that characterizea planetary system.

II. DATA

A. HD 189733b

• HARPS: Reduced 1-D spectra data from theHigh Accuracy Radial velocity Planet Searcher(HARPS) at the ESO La Silla 3.6m tele-scope for the planet HD 189733b from programs072.C-0488(E), 079.C-0127(A) and 079.C-0828(A)from the European Southern Observatory (ESO)archive. These data cover four nights with tempo-ral resolution ranging from approximately each ob-servation every 5 minutes to 10 minutes. The datafrom the fourth night has been removed from thisanalysis as it was affected by bad weather [1, 2].Each night is treated as an independent data setfor robustness. The resolution of these spectra isreduced from Picometer to Angstrom by using awindow length of 100 Picometer and taking themean over that window.

∗ sahil.agarwal@yale.edu† fabio.delsordo@yale.edu‡ john.wettlaufer@yale.edu

TABLE I. HARPS Data Analyzed for HD 189733b

Observation

DateProgram ID

Night 1 2006-09-07 072.C-0488(E)

Night 2 2007-07-19 079.C-0828(A)

Night 3 2007-08-28 079.C-0127(A)

• Spitzer: To the Basic Calibrated Data (.bcd) files,from the Spitzer pipeline version 18.18.0, we useoptimal extraction tool on the SPICE software thatemploys Optimal Extraction Algorithm by [3] toobtain the reduced 1-D spectra from the observedimages.

B. α Centauri b

The reduced 1-D spectra covering 652 nights from 2008to 2015 are obtained, from the High Accuracy Radialvelocity Planet Searcher (HARPS) at the ESO La Silla3.6m telescope. Only those nights are considered whichhave at least 10 observations. Again, the resolution ofthese spectra is reduced from Picometer to Angstrom byusing a window length of 100 Picometer and taking themean over that window.

C. Simulations

Using the SOAP2.0 tool, we obtain the simulated spec-tra for a planet orbiting a star with an orbital period of26 time units. These spectra are stacked together to giveus 8 orbital period worth of data. White Gaussian Noisewith a specific Signal to Noise Ratio (SNR) is then addedto obtain data sets with different SNR to study the effectof noise of analysis.

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TABLE II. Spitzer Data Analyzed for HD 189733b

AOR

Key

Observation

Date

Wavelength

Range (µm)

18245632 2006-10-21 7.4-14.0

20645376 2006-11-21 7.4-14.0

23437824 2008-05-24 7.4-14.0

23438080 2008-05-26 7.4-14.0

23438336 2008-06-02 7.4-14.0

23438592 2008-05-31 7.4-14.0

23438848 2007-10-31 7.4-14.0

23439104 2007-11-02 7.4-14.0

23439360 2007-06-26 7.4-14.0

23439616 2007-06-22 7.4-14.0

23440384 2008-06-09 5.0-7.5

23440640 2008-06-04 5.0-7.5

23440896 2007-12-07 5.0-7.5

23441152 2007-11-06 5.0-7.5

23441408 2007-11-11 5.0-7.5

23441664 2007-11-09 5.0-7.5

23441920 2007-11-24 5.0-7.5

23442176 2007-11-15 5.0-7.5

23439872 2007-11-04 13.9-21.3

23440128 2007-06-17 13.9-21.3

23442432 2007-12-10 19.9-39.9

23442688 2007-06-20 19.9-39.9

III. METHOD

A. Multi-fractal Temporally Weighted DetrendedFluctuation Analysis

There are four stages in the implementation of MF-TW-DFA [4]. First, one constructs a non-stationary pro-file Y (i) of the original time series Xi, which is the cu-mulative sum

Y (i) ≡i∑

k=1

(Xk − Xk

), where i = 1, ..., N. (1)

Second, the profile is divided into Ns = int(N/s) seg-ments of equal length s that do not overlap. Exceptingrare circumstances, the original time series is not an ex-act multiple of s leaving excess segments of Y (i). Theseare dealt with by repeating the procedure from the endof the profile and returning to the beginning and hencecreating 2Ns segments.

Here, rather than using nth order yν(i)’s to estimateY (i) within a fixed window (as is done in regular MF −DFA), without reference to points in the profile outsidethat window, a moving window which is smaller thans but determined by distance between points is used toconstruct a point by point approximation yν(i) to theprofile. Thus, we compute the variance up (ν = 1, ..., Ns)

and down (ν = Ns + 1, ..., 2Ns) the profile as

Var(ν, s) ≡1

s

s∑i=1

{Y ([ν − 1]s+ i)− y([ν − 1]s+ i)}2

for ν = 1, ..., Ns, and

Var(ν, s) ≡1

s

s∑i=1

{Y (N − [ν −Ns]s+ i)−

y(N − [ν −Ns]s+ i)}2

for ν = Ns + 1, ..., 2Ns. (2)

Therefore we replace the global linear regression of fit-ting the polynomial yν(i) to the data, with a weighted lo-cal estimate yν(i) determined by the proximity of pointsj to the point i in the time series such that |i− j| ≤ s. Alarger (or smaller) weight wij is given to yν(i) accordingto whether |i− j| is small (large) [4].

Finally, the generalized fluctuation function is formedas

Fq(s) ≡

[1

2Ns

2Ns∑ν=1

{Var(ν, s)}q/2]1/q

, (3)

and the principal tool is to examine how Fq(s) dependson the choice of time segment s for a given order q of themoment taken. The scaling of Fq(s) is characterized bya generalized Hurst exponent h(q) viz.,

Fq(s) ∝ sh(q). (4)

When the time series is monofractal then h(q) is inde-pendent of q and is thus equivalent to the classical Hurstexponent H. For the case of q = 2, MF-DFA and DFAare equivalent [5]. Hence, a time series with long-termpersistence has h(2) = 1 − γ/2 for 0 < γ < 1. However,short-term correlated data, decaying faster than 1/s, hasγ > 1 and finite Ts leading to a change at s = s?, andasymptotic behavior defined by h(2) = 1/2. Finally, theconnection between h(2) and the decay of the power spec-trum S(f) ∝ f−β , with frequency f is h(2) = (1 + β)/2[e.g., 6]. Therefore, one sees that for classical white noise,β = 0 and hence h(2) = 1/2, whereas for Brownian orred noise β = 2 and h(2) = 3/2.

IV. DISCUSSION

A. HD 189733b

1. HARPS

Having the 1-D spectra for the HARPS instrument,we obtain the time series for each of the wavelengthsfor each night separately. Using the Multi-fractal Tem-porally Weighted Detrended Fluctuation Analysis (MF-TW-DFA) described above, we analyze all these time se-ries. The second moment of these fluctuation functions

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for data from these time series are shown in Figures 1, 2and 3.

FIG. 1. The second moment of the fluctuation functionsare shown for all the wavelengths for Night 1 (Resolution≈ 10 minutes), with the red pentagram marking the crossovertimescale as ≈ 80 minutes. The slope after this crossover is0.5, signifying the presence of white noise dynamics on timescales above this crossover. The straight blue line has a slopeof 0.5.

FIG. 2. The second moment of the fluctuation functionsare shown for all the wavelengths for Night 2 (Resolution≈ 5.5 minutes), with the red pentagram marking the crossovertimescale as ≈ 82.5 minutes. The slope after this crossover is0.5, signifying the presence of white noise dynamics on timescales above this crossover. The straight blue line has a slopeof 0.5.

These fluctuation functions bring to notice a couple ofpoints. Firstly, almost all of these curves are parallel toeach other signifying the robustness of the effect of the

FIG. 3. The second moment of the fluctuation functionsare shown for all the wavelengths for Night 3 (Resolution≈ 5.5 minutes), with the red pentagram marking the crossovertimescale as ≈ 82.5 minutes. The slope after this crossover is0.5, signifying the presence of white noise dynamics on timescales above this crossover. The straight blue line has a slopeof 0.5.

FIG. 4. The crossover times plotted for all the wavelengths,for all three nights for HD 189733b. Only the ≈ 80minutestimescale is present for all three nights.

transit of the planet on all the wavelengths. Secondly,only a few wavelengths show the departure of the fluc-tuation function curves from all the rest which may bedue to atmospheric interference from the earth or the ex-oplanet in study. Thirdly, we see a robust timescale of≈ 80minutes for all the nights and all the wavelengths.And finally, the dynamics for all wavelengths exhibitwhite noise structure after this ≈ 80minutes scale . InFigure 4 we plot all the crossovers in the fluctuation func-

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tions for the three nights for all the wavelengths. Thisplot shows the robustness of the ≈ 80minutes time scalewhich is present for all nights, whereas other times scalesare only present for one night. Hence, these other timescales may correspond to the noise from the Earth’s at-mosphere, that of the exoplanet or even stellar activity.These time scales would be studied at a later stage.

To understand this time scale we study the followingparameters of the planet HD 189733b:

•(ρpρs

)2= 0.02391± 0.00007, where ρp is the radius

of the planet orbiting the star of radius ρs, [7]

• τtr = 0.07527±0.00037days ≈ 108.4minutes, whereτtr is the transit duration of the planet in front ofthe star, calculated as the duration between firstcontact to the last between the planet and the star,[1].

FIG. 5. A schematic showing the planet transiting it’s parentstar.

Due to the resolution of the data for each night, aswell as the short length of these time series, we are notable to observe the shortest and the longest time scales(Figure 5) in these datasets. The 80 minutes time scalewould thus correspond to the duration between the firstand last events when the whole planet is in front of thestar. We calculate this time as follows:

τtr2ρs + 4ρp

=τf2ρs

(5)

Using Eqn. 7, we obtain τf , the total duration whenthe whole planet is in front of the star as,

τf =τtr

1 + 2(ρpρs

) (6)

From the values for the planet HD 189733b, τf ≈0.0575days = 82.7865minutes.

To understand how degrading the data from picome-ter to Angstrom affects the analysis, we also analyze thedata for Night 1, with original resolution. The fluctua-tion function is plotted in Figure 6. We see that onlythe shorter wavelengths at small time scales show noisybehaviour. The structure at longer time scales is un-changed, and there by justifying our method.

FIG. 6. The second moment of the fluctuation functions areshown for Night 1 (Resolution ≈ 10 minutes), with originalresolution for wavelengths. The straight blue line has a slopeof 0.5.

2. Spitzer

An issue with the HARPS spectrograph is that it isEarth based, which puts a limitation on the durationof the observations, as the stars can only be observedat night. Also these is telluric contamination due tothe Earth’s atmosphere as is evident from the shortertimescales obtained in the HARPS data. To bypass theselimitations, we start to look at the data from the Spitzermission, which observed HD189733b for 22 nights, look-ing at the secondary eclipses of the planet, i.e. when theplanet is behind the star. Although the data from Spitzeris low resolution in the wavelength space, the time res-olution as well as the total duration is good enough towarrant its use. We show the fluctuation function forNight 2 (Table II) in Figure 7. An immediate conclusionis presence of high amount of noise in the data. Presently,the raw data are processed to remove the noisy charac-teristics from the data, but we assume that this noise isa source of information as well [8]. To quantize the cal-culated crossovers, we plot these crossovers as a functionof wavelength in Figure 8.

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There are four prominent timescales in this data, asfollows:

• τ12 = 12.7744 ± 3.4860 minutes

• 55.0966 ± 5.8851 minutes

• τ23 = 93.3848 ± 4.5972 minutes

• τ14 = 118.7942 ± 4.3432 minutes

where the uncertainty is 1 standard deviation around themean.

Firstly, the 55.0966 ± 5.8851 minutes timescales is thepointing wobble in the Infrared Array Camera of theSpitzer telescope, which has been known to be presentdue to the heater which keeps the battery in the tele-scope running properly and efficiently [9, 10].

The other three timescales are related to the tran-sit of the planet behind the star, i.e. the secondaryeclipse. The necessary and sufficient condition for thesetimescales to represent the transit (Figure 5) is

τ14 = τ23 + 2× τ12 (7)

As is evident, this Equation 7 is satisfied for the abovetimescales.

Using these time scales, we can then estimate the ra-tio of the radius of the planet to that of the star usingEquation 6, where τtr = τ14, and τf = τ23.

From Equation 6,

ρpρs

=1

2

(τtrτf− 1

)(8)

Substituting the values for τtr and τf , we obtain theratio of the areas of the disk of the planet to that of the

star as(ρpρs

)2= 0.0205± 0.0112.

We show that without the use of any fitting of modelparameters such as epoch of mid-transit, orbital period,fractional flux deficit, total duration of transit, impactparameter of planet’s path across the stellar disc, transitdepth, shape parameter for transit, transit ingress/egresstimes and many more [11, 12], we can now calculate mostof these from the time scales alone [13].

In Figures 9 and 10 we show the crossovers from allthe available datasets, as described in Table II. Figure10 includes those data sets as well that are not used inother studies due to low signal to noise ratio, or otherissues [14]. Moreover, we see the clear emergence of thesignificant time scales discussed above from all these datasets. Another point to note is the amount of noise in thesedata sets, which as we will see below can be a source ofinformation on robust estimation of these time scales.

FIG. 7. The second moment of the fluctuation functions areshown for all the wavelengths for Night 2 (Resolution ≈ 1.2minutes), with the red pentagrams marking the crossovertimescales. The straight blue line has a slope of 0.5.

0 20 40 60 80 100 120

Wavelength Index

0

20

40

60

80

100

120

140

160

180M

inute

sAOR - 20645376

FIG. 8. The crossover times plotted for all the wavelengths,for the second night for HD 189733b. All four significanttimescales are robustly extracted using our method.

B. SOAP Simulated Data

The study till now has focussed on analysis data ob-served during a primary or secondary eclipse of the ex-oplanet. But these measurements are not always avail-able. Rather the most commonly observed data are thespectrum of the stars around which we are looking forexoplanets. This is due to the motion of the star itselfaround the center of mass of the star-planet system, if theplanet exists. At present technological advancements areonly able to capture the red and blue shifts of the spec-trum when the motion of the star is significant enough,

6

FIG. 9. The crossover times plotted for all the wavelengths,for those nights which have high SNR [14] for HD 189733b.All four significant timescales are robustly extracted using ourmethod.

FIG. 10. The crossover times plotted for all the wavelengths,for all nights for HD 189733b. All four significant timescalesare robustly extracted using our method.

i.e. when a Jupiter-sized planet orbits very closely it’sparent star. Moreover, these data are contaminated withall sorts of noise, such as instrumental noise, atmosphericnoise, stellar noise, absorption lines, etc. Presently, theobservations are fit to a radial velocity curve, which mayor may not be robust and has been known to have ahigh false detection rate. We analyze the data from theSOAP simulations to first study the effect of noise as wellas the robust estimation of crossovers/timescales whichwe know to be present in the data set.

The original (No Noise) data describe the shifts in thespectrum of the star with a radial velocity of amplitude

40 ms−1. We add Gaussian white noise to the spectra ateach time instant for a specific SNR, and thus obtain 10such data sets for SNR 1, 10, 50, 100, 150, 200, 250, 500and 1000. Subfigures in Figure 12 show the extractedtime scales for the different noise cases.

To obtain these crossovers, we calculate the change inslope of the fluctuation functions for each wavelength,based on a threshold value, i.e. a crossover is calculatedif the change in slope of the curve is more than the setthreshold, Cth. Plots in Figure 12 use a threshold valueCth = 0.08. Using this value, we are able to extract theexact timescale for the orbital period of the exoplanet.Even for SNR = 150, the methodology is efficient androbust against noise and we get the correct timescale. Asthe signal quality is further decreased, we see the noisestarting affect this calculated timescale with crossoversscattering above the timescale of what should have beenactually calculated. Further decrease in the signal qualitymakes the timescales disappear altogether. This is dueto domination of the signal with white noise and the dataessentially behaves as a white noise signal and hence has aslope of 0.5, signifying white noise dynamics. In an effortto capture the timescales, even when the signal quality isnot that good, we decrease the threshold to Cth = 0.01.Even this threshold is able to capture the orbital period,but not exactly, i.e. it captures a time scale of 31 timeunits. It also captures other multiple time scales, whichmay be erroneous. The important effect here is how noiseaffects these timescales. As the SNR is decreased, wesee, as before, the time scales remain robust. But ondecreasing it further, the captured time scales are nowthe harmonics of the original ones, showing the effect ofnoise on these timescales, which may be due to some sortof stochastic resonance happening between the threshold,radial velocity measurements and noise. This shows howimportant it is to get the value of threshold right.

In actual data, one cannot calculate the this thresholdfor all the wavelengths separately and for each night, andhence one value is chosen in accordance with the observednoise characteristics. Also, presently, periodograms andfitting on sine curves is done to model the radial velocitycurves from the observations. The above analysis showshow noise can lead to spurious estimation of orbital pe-riods or the folding period of the observations and henceessentially result in spurious detection of exoplanets.

C. α Centauri b

We also study the spectral measurements for the starα Centauri, claimed to have an exoplanet orbiting it.Figure 11 shows the crossovers from all the data setsavailable on the HARPS database for all the wavelengths.There is no robust timescale that is present in all the datasets and hence no timescale above would be related to theexoplanet said to be orbiting this star.

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FIG. 11. The crossover times plotted for all the wavelengths,for all nights for α Centauri b.

V. CONCLUSION

APPENDIX

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[2] Wyttenbach, A., Ehrenreich, D., Lovis, C., Udry, S., andPepe, F., A&A 577, A62 (2015).

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[8] S. Agarwal and J. S. Wettlaufer, Physics Letters A 380,142 (2016).

[9] C. J. Grillmair, D. Charbonneau, A. Burrows, L. Armus,J. Stauffer, V. Meadows, J. V. Cleve, and D. Levine,The Astrophysical Journal Letters 658, L115 (2007).

[10] Spitzer, Modification to observed pointing wobble duringstaring observations, Tech. Rep. (Caltech, 2010).

[11] A. Collier Cameron, D. M. Wilson, R. G. West, L. Hebb,X. B. Wang, S. Aigrain, F. Bouchy, D. J. Christian, W. I.Clarkson, B. Enoch, M. Esposito, E. Guenther, C. A.Haswell, G. Hebrard, C. Hellier, K. Horne, J. Irwin, S. R.Kane, B. Loeillet, T. A. Lister, P. Maxted, M. Mayor,C. Moutou, N. Parley, D. Pollacco, F. Pont, D. Queloz,R. Ryans, I. Skillen, R. A. Street, S. Udry, and P. J.Wheatley, Monthly Notices of the Royal AstronomicalSociety 380, 1230 (2007).

[12] T. D. Morton, S. T. Bryson, J. L. Coughlin, J. F. Rowe,G. Ravichandran, E. A. Petigura, M. R. Haas, and N. M.Batalha, The Astrophysical Journal 822, 86 (2016).

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8

(a) (b)

(c) (d)

(e) (f)

9

(g) (h)

(i) (j)

FIG. 12. The crossover times plotted for all the wavelengths, with Cth = 0.08, for the simulated SOAP spectra with differentSNR. (a)No Noise, (b)SNR = 1000, (c)SNR = 500, (d)SNR = 250, (e)SNR = 200, (f)SNR = 150, (g)SNR = 100, (g)SNR =50, (i)SNR = 10, (j)SNR = 1

10

(a) (b)

(c) (d)

(e) (f)

11

(g) (h)

(i) (j)

FIG. 13. The crossover times plotted for all the wavelengths, with Cth = 0.01, for the simulated SOAP spectra with differentSNR. (a)No Noise, (b)SNR = 1000, (c)SNR = 500, (d)SNR = 250, (e)SNR = 200, (f)SNR = 150, (g)SNR = 100, (g)SNR =50, (i)SNR = 10, (j)SNR = 1