A CLOSER LOOK AT THE FLUCTUATIONS IN THE BRIGHTNESS OF SN 2009IP DURING ITS LATE 2012 ERUPTION

Images of SN 2009ip were obtained in the V, Rc, and Ic bands using four telescopes: the University of Illinois Springfield (UIS) Barber Observatory 20 inch telescope with an Apogee U42 CCD camera using a back-illuminated E2 V CCD42-40 chip (Pleasant Plains, IL), a 16 inch f/6.8 reflecting telescope with a Finger Lakes Instruments CCD camera using a Kodak 16803 chip operated by Franz-Josef Hambsch at the Remote Observatory Atacama Desert (Hambsch 2012), the Perth Exoplanet Survey Telescope (PEST) ⁶ a 12 inch Meade LX200 Schmidt–Cassegrain Telescope (SCT) with an SBIG ST-8MXE CCD camera using a Kodak 0400 chip operated by TG Tan (Perth, Australia), and a C11 SCT with a ATIK 320 E mono CCD camera using a Sony ICX274 chip operated by Ivan Curtis (Adelaide, Australia). The Barber Observatory imaged all bands with 600 s exposures. Hambsch obtained 240 s exposures in the V and Ic bands in pairs.

Curtis and Tan took a series of high-cadence 120 s R-band images over two to five hours each night they observed (see Table 1). Analysis of those images showed no significant trend or significant short-term brightness variation over the course of each observing session. The photometry from individual images in the high-cadence series were averaged together as single magnitude reported for the night.

The brightness was measured using circular apertures adjusted for seeing conditions and sky background annuli around each aperture. The aperture size and sky annuli were held constant when reducing a continuous series of images from a single telescope on the same night. Twenty comparison stars within 10 arcmin of the target were selected from the AAVSO Photometric All-Sky Survey ⁷ (Henden et al. 2012). The Barber and Hambsch images were reduced using an unweighted average of results from all the comparison stars. The Tan and Curtis images covered a smaller field and were reduced using an unweighted average of a sub-set of these comparisons. A small correction was applied to Curtis' photometry to account for any systematic offsets introduced by the subset of comparisons used. When pairs of images were taken by Hambsch at roughly the same time, photometry was measured from the individual images and then averaged together. Statistical errors in the photometry for individual images were typically 0.05 mag or less. Photometry taken by different telescopes on the same night compare favorably with each other within the statistical errors. Finally, the photometry was corrected to the system of Pastorello et al. (2013) using the corrections of: dV = +0.063, dRc = +0.046, and dIc = +0.023 (A. Pastorello, 2012 personal communication). There were no additional transformations for color or other factors applied to the measured magnitudes.

The Rc and Ic magnitudes are published in their entirety in Margutti et al. (2014). The V magnitudes prior to t = +10 days were previously published in Pastorello et al. (2013). The later V magnitudes are in Table 2.

SWIFT UVOT data in the b, u, w1, m1, and w2 bands are taken from Margutti et al. (2014), where they are described in detail. The data were analyzed following the prescription of Brown et al. (2009) with apertures optimized to maximize signal-to-noise and limit contamination from nearby stars. The magnitudes are on the UVOT photometric system (Poole et al. 2008) and revised to the Vega zero points of Breeveld et al. (2011). The u and b data are converted to equivalent Johnson U and B magnitudes by Margutti et al. (2014). They note that this is a minor correction that is not responsible for the bumps in the light curve.

Our measurements begin on t = −9.16 days very near to the start of the rise in brightness toward the 2012-B peak (Pastorello et al. 2013) . The SN brightness increased about 4 mag in 12 days in V, Rc, and Ic, reaching a peak V = 13.80 ± 0.05 on t = +3 days (MJD 56206.04), which is consistent with observations reported by Prieto et al. (2013).

The post-peak bolometric flux of most SNeIIn can be modeled by a power-law function of flux with respect to time (Svirski et al. 2012; Moriya et al. 2013). Bolometric magnitude is directly proportional to the logarithm of bolometric flux. Therefore, a power law in flux-time is linear when transformed to a function of bolometric magnitude with respect to the logarithm of time. For a sample whose duration spans a few decades of time, bolometric magnitude is nearly linear with respect to time.

The dominant trend in the decline in magnitude from the peak is almost linear with respect to time within the first 60 days (see Section 3.2). Fitting a purely linear function, we obtained slopes of 0.060 ± 0.005 mag/day in V, 0.050 ± 0.001 mag/day in Rc, and 0.094 ± 0.026 mag/day in Ic. The slopes differ due to different bolometric corrections in each filter. The rapid rise and rates of decline, within 60 days of the peak, are consistent with other Type IIn light curves (Kiewe et al. 2012; Taddia et al. 2013).

The observed brightness after the peak contains significant fluctuations superimposed on a downward trend that are easily identified in all eight bands and the bolometric luminosity calculated by Margutti et al. (2014). Each telescope observed these fluctuations independently, eliminating the concern that it could be an effect attributable to a single telescope or observing location. The record of the data presented here is also corroborated by the independent observations of Fraser et al. (2013) and Graham et al. (2014). Margutti et al. (2014) notes the regular placement of "peaks" in the general trend and that an analysis of the Rc-band photometry prior to the 2012-B event reveals some power in periods between 20–40 days.

To analyze this structure, we subtracted the peak and linear decline from data in each band. The best theoretical models of the trend after the peak are power-law functions of bolometric flux with respect to time (e.g., Svirski et al. 2012; Moriya et al. 2013), which are linearized as functions of bolometric magnitude with respect to the logarithm of time. Moriya et al. (2013) explicitly state that their model should not be applied when the CSM is optically thick in the early stages of the Type IIn. They also assume the power law governing the radial distribution of circumstellar material s<3. Figure 1 shows fits of the Moriyaʼs model to the bolometric luminosity curve published by Margutti et al. (2014). The power law of the best-fit (α = −2.98) is much steeper than any of the examples given by Moriya et al. (2013) (SN 2005ip, SN 2006jd, and SN 2010jl) and in all allowed self-similar cases for the density slope in the outer envelope of the progenitor yields a power law s = 5 for the CSM. This result is outside the bounds of the modelʼs assumptions and implies that the data are not consistent with the Type IIn model of Moriya et al. (2013).

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A more complicated model (i.e., the sum of two power laws in flux-time) is difficult to linearize and fit to the data using a least squares method. A second degree polynomial of magnitude with respect to time is a reasonable low order approximation to the physical models. Margutti et al. (2014) found significant transitions in the thermal continuum at t = +16 and t = +70 days. The light curves in each band visibly change character at t ∼ +10 days and t ∼ +70–75 days. Graham et al. (2014) confirms a change in the color evolution of the light curve at the same times. To accommodate this, our fit employed two second degree polynomials bracketed by these transitions (see the example in Figure 2). The first was fit to the data around the peak for −8 < t < +10 days. The second was fit to the decline between +10 > t > +75 days. The coefficients for these fits are recorded in Table 3. The small χ² for these fits (Table 3) indicate that there is a high probability that they are consistent with the trend in the data.

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Table 3. Parameters For Light Curve Fits

	a1 (mag)	b1 × 102	c1 × 103		$({{\chi }^{2}})/N$	a2 (mag)	b2 × 102	c2 × 103		$({{\chi }^{2}})/N$	a3 (mag)	b3 × 102		$({{\chi }^{2}})/N$
Filter	(mag)	(mag/day)	(mag/day2)	N	x100	(mag)	(mag/day)	(mag/day2)	N	x100	(mag)	(mag/log(day))	N	x100
Ic	13.81 ± 0.02	−5.5 ± 0.5	3.5 ± 0.5	18	0.48	13.02 ± 0.14	5.5 ± 1.1	a	34	1.54	10.71 ± 0.17	1.14 ± 0.05	35	2.12
Rc	13.76 ± 0.01	−5.2 ± 0.2	6.2 ± 0.4	5	0.02	13.22 ± 0.05	4.8 ± 0.2	a	24	0.92	10.80 ± 0.26	1.17 ± 0.08	24	3.43
V	13.86 ± 0.02	−6.6 ± 0.3	7.9 ± 0.7	16	0.35	13.36 ± 0.07	5.3 ± 0.5	a	55	2.07	11.33 ± 0.21	1.13 ± 0.06	56	12.07
B	13.90 ± 0.02	−6.4 ± 0.3	8.7 ± 0.7	25	0.52	13.62 ± 0.19	4.5 ± 1.0	0.3 ± 0.1	29	3.96	7.07 ± 0.58	2.53 ± 0.16	30	21.19
U	12.91 ± 0.02	−5.5 ± 0.3	8.1 ± 0.5	35	0.45	12.15 ± 0.15	9.3 ± 0.9	a	37	4.41	5.10 ± 0.43	3.03 ± 0.12	38	19.40
w1	12.25 ± 0.02	−3.3 ± 0.4	9.5 ± 0.8	13	0.39	11.48 ± 0.17	15.0 ± 0.9	−0.6 ± 0.1	28	2.61	3.70 ± 0.37	3.53 ± 0.10	29	7.29
m2	12.09 ± 0.03	−2.7 ± 0.6	9.6 ± 1.0	9	0.60	10.97 ± 0.13	19.6 ± 0.6	−1.0 ± 0.1	35	3.39	3.05 ± 0.24	3.86 ± 0.07	36	5.03
w2	12.18 ± 0.02	−1.8 ± 0.3	10.3 ± 0.6	23	0.38	11.01 ± 0.15	21.7 ± 0.9	−1.3 ± 0.1	37	3.92	3.62 ± 0.17	3.80 ± 0.05	38	2.77
Mbol	−19.34 ± 0.01	−2.5 ± 0.2	10.5 ± 0.5	28	0.23	−19.70 ± 0.08	10.6 ± 0.5	−0.6 ± 0.1	37	1.99	−24.07 ± 0.28	2.11 ± 0.08	35	98.08

Note. These are the best-fit parameters to the measured brightness in each band for the time periods −8 < t < +10 days (brightness = a₁+ b₁t+c₁t²) and +10 < t < 75 days (brightness = a₂+ b₂t+c₂t²). For comparison, the fit parameters and χ² for brightness = a₃+b₃Log(t) to +10 < t < 75 days are also included.

^a The fit for this term was consistent with zero.

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For the purpose of comparison with our approach, we performed fits to the light curves in the interval of +10 > t > +75 days to a linear function of magnitude with respect to the logarithm of time. The χ² for all but one those fits (Table 3) demonstrate that they do not fit the data as well as the second degree polynomial. In all cases, the function with respect to log(t) is a poor fit to the data at the extreme ends of the range (illustrated in Figure 2). Note that the bolometric data were also better fit by our approach. The one exception is the data for the w2 filter which is fit slightly better by the linear function with respect to log(t) than the second degree polynomial.

We acknowledge that there is an unquantifiable (but likely small) risk that the second order polynomial over-subtracts the underlying trend. A systematic influence of this type should increase red (Brownian) noise and reduce the significance of the bumps and dips. If this effect is present, there is also some risk that the relative heights of the peaks in different bandpasses could be effected. However, it should have negligible effect on the spacing and frequency of the peaks.

The use of a relatively low order polynomial also risks under-subtraction of the underlying trend. As with the risk of over-subtraction, this may have increased the Brownian noise and possibly over emphasized the bumps in the decline. To test this, similar fits were made to SN IIn light curves from Taddia et al. (2013). Subtraction of the fits generated using the same method from those light curves revealed no significant fluctuations at the one sigma level or greater relative to the quoted measurement errors in the individual data points. If present, under-subtraction could also influence the relative heights of the peaks in each bandpass, but should have a negligible effect on the spacing and frequency of the peaks.

The data with the modeled trends subtracted are plotted in Figure 3. The detrended data has a series of peaks and troughs with an amplitude of 0.10–0.40 mag in each band. There are three larger dips followed by rises at approximately t = +5 ± 15 days, t = +20 ± 40 days, and t = +45 ± 65 days. These fluctuations roughly match transition points in the evolution of the thermal continuum as modeled by Margutti et al. (2014) (see their Figure 8).

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There was no expectation of detecting changes on the timescale of hours, but Tan and Curtis were eager to test for the unexpected and/or provide a useful null result. They made high-cadence observations with a series of 120 s exposures spanning 2–5 hours over 26 separate nights (in 3 circumstances both observing simultaneously on the same night; see Table 1). A few of the observations (particularly the first observations by Tan during the steep rise prior to peak brightness) have a small but significant linear trend over the course of the night. The observations revealed no significant fluctuations in the brightness of 2009ip on the timescale of hours. The standard deviations (σ) of measurements for each observing session are small. There were a few instances where a series of continuous points fell more than two sigma from the average. Careful inspection revealed that these were caused by interference from an uncorrected hot pixel on the CCD. Excepting those instances, there were no sudden flares or drops in the brightness of the target greater than 2σ from the mean over two or more consecutive measures.

Figure 4 shows how the fluctuations evolve relative to each band over time. All the bands start out fluctuating in phase and at roughly the same amplitude. The bump at t ∼ +10 days is the first point where the bands separate, with the shorter wavelengths peaking at a higher amplitude. At that point, the UV bands (w1, w2, and m2) fall out of phase with the optical bands (Ic, Rc, V, B, and U). The UV leads the optical into the dip at t ∼ +25 days and lags the optical by about one day coming out of the dip. This lag was noted by Fraser et al. (2013), but, in contrast to their findings, we observe no lag in U relative to V and the amplitude measured from the low point at t ∼ +25 days to the peak at t ∼ +32 days differs significantly depending on wavelength. At t ∼ +45 days, the U and B bands fall out of phase with the V band, lagging that band by about 1 day and matching phase with the UV fluctuations. The peak at t ∼ +60 days is brightest in the U band and slightly fainter in the B band, with the V and UV bands having roughly the same lower amplitude.

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These fluctuations are likely to be driven by continuum changes rather than by variations in the emission lines. A change in flux of 0.1 mag in the filters used equals a change of 500 Å–2000 Å or more in equivalent width. Adopting the distance modulus and bolometric correction for SN 2009ip used by Margutti et al. (2014), the peaks represent an increase in total luminosity of approximately 10⁴² erg s⁻¹. The energy of a single peak (from the trend-subtracted data) integrated over time is approximately 10⁴⁷ ergs. No fluctuations matching these characteristics were reported in the spectrum of SN 2009ip (Mauerhan et al. 2013; Pastorello et al. 2013; Margutti et al. 2014; Graham et al. 2014).

A peak or dip in brightness driven by a change in size of the continuum-emitting region should affect the brightness in all filters equally. However, Figure 4 reveals relative differences in color in each of the major peaks. A relative heating of the continuum-emitting region should cause shorter wavelengths to brighten relative to longer wavelengths while a relative cooling should cause longer wavelength to brighten relative to shorter wavelengths.

In order to properly interpret the data, recall that we have subtracted the underlying trend so that differences in brightness between bands measure relative color. An increase in brightness in shorter wavelength bands relative to longer wavelengths reveal the color becoming bluer, which can be interpreted as a wavelength-dependent change in opacity (i.e., line blanketing) or relative heating. Line blanketing can have the effect of reducing through increased opacity and enhancing through re-emission the brightness in specific filters leaving others unaffected. Relative heating or cooling will proportionally effect all the filters according to the shift in peak of the blackbody continuum. The analysis in Margutti et al. (2014) determined that while these fluctuations were occurring, the continuum-emitting region was expanding and cooling. In the context of a blackbody continuum, a local peak in brightness can represent heating relative to the subtracted trend in declining temperature, but the absolute temperature of each subsequent peak declines as time progresses. Two peaks may have the same amplitude, revealing comparable heating, but the later one has a cooler absolute temperature.

For the first peak at t ∼ +8 ± 15 days and the following dip between t ∼ +20 ± 30 days the UV bands have a greater amplitude than the visual. This could correspond to heating of the continuum-emitting region to produce the peak and then cooling to produce the dip. According to Margutti et al. (2014), this peak occurs when the effective temperature of the thermal continuum is between 15,000 K to 11,000 K.

The second major peak between $t\sim +30\pm 40$ is different than the first. All bands reach the same amplitude, but the UV bands lag the optical bands by ∼1 day. This does not fit the expectation for a change driven by the shift in wavelength peak of a blackbody continuum. The lag in UV brightness is better understood in the context of line blanketing which simultaneously reduces brightness though increased opacity in one wavelength range and raises the brightness through re-emission in another. Rising out of the minimum, significant line blanketing could simultaneously lower the UV flux and enhance the visual bands through re-emission. This roughly corresponds to the explanation put forward by Fraser et al. (2013) for the lag between bands in this peak. After the peak, the ionization of the continuum-emitting region changes, eliminating the line blanketing and causing the UV to be brighter throughout the decline. According to Margutti et al. (2014), this peak occurs when the effective temperature of the thermal continuum is between 9500 K to 8000 K.

The third major peak shows a lag in both the UV and the blue optical bands. This peak is brightest in the U and B bands and less prominent in the longer wavelength optical and UV. The presence of the Balmer continuum/decrement in the U and B bands along with their divergent behavior relative to the other bands alludes to a transition in the ionization state of hydrogen. According to Margutti et al. (2014), this peak occurs when the effective temperature of the thermal continuum is between 6500 K to 5500 K. This peak occurs at a lower absolute temperature than the first two peaks. There is probably some combination of line blanketing and temperature change at work. Spectra from this time hint at an increase in structure at wavelengths shorter than 4000 Å along with a significant increase in Balmer (Margutti et al. 2014) and Paschen (Fraser et al. 2013) emission.

The placement of the three major peaks and several smaller peaks imply a possible regularity in the fluctuations. Kashi et al. (2013) note the major bumps occur roughly 24 days apart and attributes this to binary interaction. The behavior prior to t = +8 days is also very regular. It should also be noted that the fluctuations after the eruption are consistent with those detected prior to the eruption as part of a binary or single star model. Flickering (fluctuations on the order of 3 mag in brightness over ∼16 days) was observed in earlier eruptions of SN 2009ip (Smith et al. 2010; Foley et al. 2011). Margutti et al. (2014) also found periods of fluctuation between 20-40 days in the R band prior to the 2012-A eruption.

There is already evidence for regularity in the fluctuations. The power spectrum analysis that follows is performed to corroborate and quantify what is already supported by that evidence. Without corroborating evidence, the results of our power spectrum analysis should be considered with caution. One objection may be that the data set is short and spans an insufficient length of time to yield meaningful results. To mitigate this concern, we ignore the lowest frequencies in the power spectrum that correspond to periods more than one-third the total time spanned by the data. We also avoid tools like WOSA (Welch-Overlapped-Segment-Averaging; see Schultz & Strattegger 2002) that subdivide the full data set into smaller segments for analysis.

Another concern is Brownian (red) noise. See the caveats and concerns outlined at the end of Section 3.2. The process to subtract the underlying trend is certain to introduce noise heavily weighted toward low frequencies. To answer those concerns, we have applied a tested and peer-reviewed algorithm to estimate the confidence level of the peaks relative to Brownian noise (see below). A third concern is white noise unequally affecting individual data points. To address this, we have modeled their influence with a Monte–Carlo simulation. There may still be doubts about the validity of the power spectrum analysis results on their own, but, to the extent that they corroborate other observations, they are compelling.

We analyzed the fluctuations using a date compensated discrete Fourier transform (Ferraz-Mello 1981) in the AAVSO software package VStar version 2.16.1. ⁸ The analysis was performed on the detrended data (Figure 3). We will focus primarily on the power spectrum of the bolometric magnitude (bottom panel in Figure 3). That spectrum has notable peaks at 0.043 day⁻¹ (period ∼23 days), 0.079 day⁻¹ (period ∼13 days), and 0.132 day⁻¹ (period ∼ 8 days). The frequencies of the peaks are almost integer multiples of each other. There are also hints of weaker adjacent peaks at 0.058 day⁻¹, 0.095 day⁻¹, and 0.145 day⁻¹. The most significant peaks in the M_bol power spectrum are in the power spectra for each individual photometric band and the power spectrum of the bolometric magnitude data without the trend subtracted (dashed line, bottom panel). V, Rc, and Ic data from Fraser et al. (2013) processed in the same way independently support these results.

The process by which the underlying trend was subtracted is likely to introduce Brownian noise into the data. This is evident in Figure 3 as a continuous decrease in spectral amplitude with increasing frequency. Similar noise dominates plaeoclimate data (Hasselmann 1976) and can be modeled using a first-order autoregressive (AR1) process. Schultz & Mudelsee (2002) provide an algorithm to model this noise in an unevenly spaced time series and simulate confidence levels in order to determine the significance of peaks in the power spectrum. Figure 6 shows the results of applying their code to our data using N = 10⁴ for the Monte–Carlo simulation. Three major peaks and one weaker adjacent peak are above the 99% confidence level in the REDFIT simulation. Peaks above a 99% confidence level have less than a 1% chance of being a false alarm. If our sample is dominated by Brownian noise, this simulation implies that several of the observed peaks in the power spectrum are not a product of AR1-type Brownian noise.

However, it is unclear to what extent Brownian noise may dominate this data (a key assumption of the REDFIT analysis). The white noise from photometric errors could play a more significant role. Irregular spacing of the points in time combined with white noise has a non-linear influence on the power spectrum. We modeled this with a Monte–Carlo simulation of 1000 perturbed data sets generated and run through the same REDFIT analysis. In each simulated set, the bolometric magnitude was perturbed by an amount randomly selected from a normal distribution with a standard deviation equal to the photometric error. The times were perturbed in a similar manner with a standard deviation of 0.05 days. ⁹

The results of the Monte–Carlo simulation are presented in Figure 7. The the peaks for average outcome in the simulation (dashed white line) are at a lower confidence level than the results presented in Figure 5. This indicates that the white noise has a greater influence on the power spectrum than the Brownian noise. In more than half of the simulated power spectra, the three highest peaks have a confidence level of 80% or greater (false alarm rate of 20% or less). As a result of this simulation, we have confidence in the first two peaks. In more than 99% of the simulated power spectra, the first peak (0.043 day⁻¹) was higher than the modeled AR1 noise level. More than 75% of the simulated power spectra had a REDFIT false alarm rate less than 20% for this peak. The second peak (0.079 day⁻¹) was above the modeled AR1 noise level in more than 95% of the simulations and more than 80% of the simulated power spectra had a REDFIT false alarm rate less than 20% for this peak. Conservatively speaking, this implies a better than 1.5σ detection for both peaks. Combined with our other findings, it appears likely that the first two peaks (and likely more) are an actual signal and not a false alarm. Our confidence in several of the less significant peaks is bolstered by their status as sub-harmonics of more significant frequency peaks.

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