PULSATION PERIOD VARIATIONS IN THE RRc LYRAE STAR KIC 5520878

The Kepler data were downloaded from the Mikulski Archive for Space Telescopes (MAST) in FITS format and all files were merged with TopCat (Bristol University). The data come with adjustments for barycentering (and the corrections applied), and flux data are available as raw simple aperture photometry (called SAP by the Kepler team). There is also a precleaned version available called PDCSAP, which includes corrections for drift, discontinuities and events such as cosmic rays. The Kepler Data Characteristics Handbook (Christiansen & Jenkins 2011) points out that this cleaning process "[does not] perfectly preserve general stellar variability" and investigators should use their own cleaning routines where needed.

For the period length analysis explained in the next sections, we have tried both SAP and PDCSAP data, as well as applied our own correction obtained from nearby reference stars. All three data sets gave identical results. This is due to the corrections being much smaller than intrinsic stellar variations: while instrumental amplitude drifts are in the range of a few percent over the course of months, the star's amplitude changes by ∼40% during its 0.269 days (6.5 hr) cycle. For a detailed study of best-practice detrending in RR Lyrae stars, we refer to Benkő et al. (2014a).

Most commonly, Fourier decomposition is used to determine periodicity. Other methods are phase dispersion minimization (PDM), fitting polynomials, or smoothing for peak/low detection. The Fourier transformation (FT) and its relatives (such as the least-squares spectral analysis used by Lomb–Scargle for reduction of noise caused by large gaps) have their strength in showing the total spectrum. FT, however, cannot deliver clear details about trends or the evolution of periodicity. This is also true for PDM, which does not identify pure phase variations since it uses the whole light curve to obtain the solution (Stellingwerf et al. 2013).

The focus of our study was pulsation period length and its trends. We have, therefore, tried different methods to detect individual peaks (and lows for comparison).

In the short cadence data (one-minute integrations), we found that many methods work equally well. As the average period length of KIC 5520878 is ∼0.269 days, we have ∼387 data points in one cycle. First of all, we tried eye-balling the time of the peaks and noted approximate times. As there is some jitter on the top of most peaks, we then employed a centered moving-average (trying different lengths) to smooth this out, and calculated the maxima of this smoothed curve. Afterward, we fitted nth order polynomials, trying different numbers of terms. While this is computationally more expensive, it virtually gave the same results as eye-balling and smoothing. Deviations between the methods are on the order of a few data points (minutes). We judge these approaches to be robust and equally suitable. Finally, we opted to proceed with the peak times derived with the smoothing approach as this is least susceptible for errors.

In the long cadence data (30 minutes), there are only ∼13 data points in one cycle (Figure 1). Simply using the bin with the highest luminosity thus introduces large timing errors. We tried fitting templates to the curve, but found this very difficult due to considerable change in the shape of the light curve from cycle to cycle. This stems from two main effects: amplitude variations of ∼3% cycle-to-cycle, and the seemingly random occurrence of a bump during luminosity increase, when luminosity rises steeply and then takes a short break before reaching its maximum. We found fitting polynomials to be more efficient, and tested the results for varying number of terms and number of data points involved. The benchmark we employed was the short cadence data, which gave us 1810 cycles for comparison with the long cadence data. We found the best result to be a fifth-order-polynomial fitted to the seven highest data points of each cycle, with new parameters for each cycle. This gave an average deviation of only 4.2 minutes, when compared to the short cadence data. Figure 1 shows a typical fit for short and long cadence data.

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During the 4 yr of Kepler data, 5417 pulsation periods passed. We obtained the lengths of 4707 of these periods (87%), while 707 cycles (13%) were lost due to spacecraft downtimes and data glitches. Out of the 4707 periods we obtained, we had short cadence data for 1810 periods (38%). For the other 62% of pulsations, we used long cadence data.

The errors given by the Kepler team for short integrations (1 minute) for KIC 5520878 are ⩽35e⁻s⁻¹ for the flux (that is 0.1% of an average flux of 35,000e⁻ s⁻¹). These errors are smaller than the point size of Figure 2, which shows the complete data set. We have applied corrections for instrumental drift by matching to nearby stable stars. Some instrumental residuals on the level of a few percent remain, which are very hard to clean out in such a strongly variable star. As explained above, these variations are irrelevant for our period length determination.

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KIC 5520878 has been found to be metal rich. When expressed as [Fe/H], which gives the logarithm of the ratio of a star's iron abundance compared to that of our Sun, KIC 5520878 has a ratio of [Fe/H] = −0.36 ± 0.06, which is about half that of our Sun. This information comes from a spectroscopic study with the Keck-I 10 m HIRES echelle spectrograph for the 41 RR Lyrae in the Kepler field of view (Nemec et al. 2013). The study also derived temperatures (KIC 5520878: 7250 K, the second hottest in the sample) and radial velocity of −0.70 ± 0.29 km s⁻¹.

When compared to the other RR Lyrae from Kepler, or more generally in our solar neighborhood, it becomes clear that metal-rich RR Lyrae are rare. Most (95%) RR Lyrae seem to be old (>10 Gyr; Lee 1992), metal-poor stars with high radial velocity (Layden 1995). They belong to the halo population of the galaxy, orbiting around the galactic center with average velocities of ∼230 km s⁻¹.

There seems to exist, however, a second population of RR Lyrae. Their high metallicity is a strong indicator of their younger age, as heavy elements were not common in the early universe. Small radial velocities point toward them comoving with the galactic rotation and, thus, belonging to the younger population of the galactic disk, sometimes divided into the very young thin disk and the somewhat older thick disk. Our candidate star, RRc KIC 5520878, seems to be one of these rarer objects, as shown in Figure 3.

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RR Lyrae stars are part of our cosmic distance ladder, due to the peculiar fact that their absolute luminosity is simply related to their period, and hence if one measures the period one gets the light output. By observing the apparent brightness, one can then derive the distance. The empirical relation is 5log₁₀D = m − M − 10 where m is the apparent magnitude and M the absolute magnitude. If we take Kp = 14.214 for KIC 5520878 and M = 0.75 for RR Lyrae stars (Layden 1995), the distance calculates to D = 4.9 kpc (or ∼16,000 LY), that is ∼16% of the diameter of our galaxy—the star is not quite in our neighborhood.

Taking an empirical relation of pulsation period P and radius R of RR Lyrae stars (Marconi et al. 2005) of log₁₀R = 0.90(± 0.03) + 0.65(± 0.03)log₁₀P, one can derive R = 3.4 R_SUN for KIC 5520878. The inclination angle i is not derivable with the data available. Thus, a corresponding minimum rotation period would be 33.8 ± 1 days at i = 90, or 47.8 days at i = 45. The minimum rotation period of 33.8 days is equivalent to ∼126 cycles (of 0.269 days length), so that the rotation period is much slower than the pulsation period.

KIC 5520878 has been studied in detail (Moskalik 2014). In a Fourier analysis, a secondary frequency of f_X = 5.879 day⁻¹ (P_X = 0.17 days) was found. The author calculates a ratio of P_X/P₁ = 0.632, a rare but known ratio among RR Lyrae stars. This rareness is probably caused by instrumental bias, as these secondary frequencies are low in amplitude, and were only detected with the rise of high-quality time-series photometry (e.g., OGLE) and millimag precision space photometry. Indeed, the ∼0.6 period ratio was detected in six out of six randomly selected RRc stars observed with space telescopes (Moskalik 2014; Szabó et al. 2014).

In addition, significant subharmonics were found with the strongest being f_Z = 2.937 day⁻¹(P_Z = 0.34 days), that is at 1/2f_X. Their presence is described as "a characteristic signature of a period doubling of the secondary mode, f_X (...). Its origin can be traced to a half-integer resonance between the pulsation modes (Moskalik & Buchler 1990)." The author also compares several RRc stars and finds "the richest harvest of low amplitude modes (...) in KIC 5520878, where 15 such oscillations are detected. Based on the period ratios, one of these modes can be identified with the second radial overtone, but all others must be nonradial. Interestingly, several modes have frequencies significantly lower than the radial fundamental mode. This implies that these nonradial oscillations are not p-modes, but must be of either gravity or mixed mode character." These other frequencies are usually said to be nonradial, but could also be explained by "pure radial pulsation combinations of the frequencies of radial fundamental and overtone modes" (Benkő et al. 2014b). However, this explanation might only apply to fundamental-mode (RRab) stars. Since RRab and RRc stars pulsate in different modes, period ratios in one class might tell little about the period ratios in the other class. The ∼0.6 period ratio is close to the fundamental mode-second overtone period ratio, but there are no radial modes that fall to this value against the first overtone, and thus the P_X mode is commonly identified as nonradial.

In Section 5.1, we explore the Fourier spectrum in more detail.

Unusual frequencies might be contributed by an unresolved companion with a different variability. To check this, we have obtained observations down to a magnitude of 19.6 with a privately owned telescope and a clear filter. Using a resolution of 0.9 px arcsec⁻¹, it can be easily seen that a faint (18.6 mag) companion is present at a distance of ≈7 arcsec, as shown in Figure 4. The standard Kepler aperture mask fully contains this companion; however, the magnitude difference between this uncataloged companion and KIC 5520878 is larger than 4.4 mag, so that the light contribution into our Kepler photometry is very small. At a distance of 4.9 kpc, the projected distance of this companion is ≈0.15 parsec, resulting in a potential circular orbit of ≈4 × 10⁶ yr. The most likely case is, however, that this star is simply an unrelated, faint background (or foreground) star with negligible impact to Kepler photometry.

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A second argument can be made against blending: with Fourier transforms, linear combinations of frequencies can be found. These are intrinsic behaviors of one star's interior, and cannot be produced by two separate stars.

We have used all short cadence data and created a folded light curve, as shown in Figure 5. It clearly displays the two periods present: the strong main pulsation P₁, and the weaker secondary pulsation P_X.

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With Kepler data, a previously unknown effect in RR Lyrae was discovered, named "period doubling" (Kolenberg et al. 2010). This effect manifests itself in an amplitude variation of up to 10% of every second cycle. It was found in several Blazhko–RRab stars, but its occurrence rate is yet unclear. Our candidate star, KIC 5520878, does not show the period doubling effect with respect to the main pulsation mode. Judging from Figure 6, which shows a typical sample of amplitude fluctuations, the variations are more complex. However, as will be explained below, the star does exhibit a strong and complex period variation in the secondary pulsation P_X.

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Regarding the Blazhko effect, a long-term variation best visible in an amplitude-over-time graph, we can see no clear evidence for it being present in KIC 5520878. Figure 2 shows the complete data set, which features some amplitude variations over time, but not of the typical sinusoidal Blazhko style.

The average period length is constant at P₁ = 0.269d when averaged on a timescale of more than a few days. On a shorter timescale, larger variations can be seen. Peak-to-peak, the variation is in the range [0.24 days to 0.30 days], low-to-low in the range [0.26 days to 0.28 days]. For any given cycle, the qualitative result of measuring through the lows and through the peaks is equivalent, meaning that a short (long) period is measured to be short (long), no matter whether peaks or lows are used. As the peak-to-peak method gives larger nominal differences (and thus better signal-to-noise), we focus on the peak times from here on.

The variation seems to fluctuate in cycles of variable length. We estimate maxima at BJD ∼ 164, 317, 533, 746, 964, 1328, and 1503. Their separation is then (in days): 153, 216, 213, 218, 364, and 175 (Figure 7). As some of these are near one Earth (Kepler) year, or half of it, we have to carefully judge the data regarding barycentering, which has been performed by the Kepler team. We argue against a barycentering error for three reasons: first, the differences caused by barycentering between two subsequent cycles are negligible (<0.5 s), while the cycle has an average length of ∼387 minutes. Second, the cycles are of varying lengths, while a year is not. Third, the large Kepler community would most likely already have found such a major error. We therefore judge the long-term variation of period lengths to be a real phenomenon.

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In Section 5, we will explain how the period length variations originate from the presence of a second pulsation mode.

The period variation of RR Lyrae, the prototype star, has been measured at ∼0.53% (Stellingwerf et al. 2013). This is low compared to ∼22% for KIC 5520878. The low value of RR Lyrae is reproducible with our method of individual peak detection. The authors estimate a scatter of ∼0.01 days caused by measurement errors. We have furthermore looked at several other RR Lyrae stars in the Kepler field of view, and found others (V445 = KIC 6186029; KIC 8832417) that also show large period variations. For KIC 6186029 the variations are 0.49 days to 0.53 days, which is ∼8%. It has been analyzed in a paper whose title summarizes the findings as "Two Blazhko modulations, a nonradial mode, possible triple mode RR Lyrae pulsation and more" (Guggenberger 2013).

KIC 5520878 shows a significant (p > 99.9%) relation of longer periods having higher flux. In a recent paper, a counter clockwise looping evolution between amplitude and period in the prototype RR Lyrae was detected. This looping, thus far only found in RRab–Blazhko stars, cannot be reproduced in KIC 5520878 (Stellingwerf et al. 2013).

The period lengths of the 4707 cycles that we measured are not just Gaussian distributed, and the distribution changes over time considerably. Figure 8 shows their change over time. For every given time, we can see a regime of shorter and a regime of longer period lengths. For the times of higher variance, these regimes are clearly split in two, with no periods of average length (forbidden zone). During times of lower variance, no clear split can be seen.

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It has been proposed that a sufficiently advanced civilization may employ Cepheid variable stars as beacons to transmit all-call information throughout the galaxy and beyond (Learned et al. 2008). The idea is that Cepheids and RR Lyrae are unstable oscillators and tickling them with a neutrino beam at the right time could cause them to trigger early, and hence jog the otherwise very regular phase of their expansion and contraction. The authors proposition was to search for signs of phase modulation (in the regime of short pulse duration) and patterns, which could be indicative of intentional signaling. Such phase modulation would be reflected in shorter and longer pulsation periods. We showed that the histograms do indeed contain two humps of shorter and longer periods. In case of an artificial cause for this, we would expect some sort of further indication. As a first step, we have applied a statistical autocorrelation test to the sequence of period lengths. This will be discussed below. Furthermore, we have assigned "one" to the short interval and "zero" to the long interval, producing a binary sequence. This series has been analyzed regarding its properties. It indeed shows some interesting features, including nonrandomness and the same autocorrelations. We have, however, found no indication of anything "intelligent" in the bitstream, and encourage the reader to have a look himself or herself, if interested. All data are available online (see Appendix B). A positive finding of a potential message, which we did not find here, could be a well-known sequence such as prime numbers, Fibonacci, or the like.

We have also compared several other RR Lyrae stars with Kepler data for their pulsation period distributions, and created a Kernel density estimate (Rosenblatt 1956; Parzen 1962) plots (Figure 9). We found no second case with two peaks, though some other RRc stars come close.

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As explained in Section 3, through Fourier analysis two pulsation periods can be detected, P₁ = 0.269 days and P_X = 0.17 days. While it is clear from Figure 1 that the light curve is not a sine curve, for simplicity we have taken it to be the sum of two sine curves in the following model. We used two sine curves of periods P₁ and P_X, with relative strengths of 100% and 20%, as shown in Figure 11 (left panel). When added up (right panel), period variations can easily be seen as indicated by the gray vertical lines.

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We then ran this simulated curve for 100 cycles, and compared the peak times with the real data. The comparison is shown in Figure 12—this simple model resembles the observational data to some extent. The histogram (Figure 13 bottom right) also shows the two humps. When certain ratios of P₁ and P_X are chosen, e.g., P_X/P₁ = 0.632 ± 0.001, it can also be shown that the two simultaneous sines create specific autocorrelation distributions. Let us use a simple model equation of f(x) = int(xR) − xR, where int denotes the rounding to the next integer and R the period ratio. The function then gives values of close to 0 or close to 0.5 for odd numbers, while for even numbers it gives other values. When renormalized to the usual ACF range of [−1, +1], this gives a high absolute autocorrelation for odd numbers and less so for even numbers.

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We have compared our observational data to this model, and calculated all ACFs (observed) minus ACFs (calculated). The result is an ACF distribution in which primes are still prominent, but not significantly so (p = 91.35%). We judge this to be the explanation of the prime mystery. We have also checked the Fourier decomposition for the corresponding amplitudes, and find that P_X alone only has an amplitude of 5.1% compared to P₁, or 2% after prewhitening. However, the P_X mode is distributed into multiple peaks, harmonics, and linear combinations. In a simultaneous fit to all frequencies, we find a total of 12.7% for all these peaks that are connected with P_X (see Table 1). This is in the required range of the model described above.

Table 1. Strongest Fourier Frequencies

Namea	Frequency	Period	Amplitudeb	Amplitude
		(days)	(e−s−1)	(percent of f1)
f1	3.71515	0.26917	4990	100.0
2f1	7.43029	0.13458	549	11.0
3f1	11.14544	0.08972	308	6.2
fX	5.87884c	0.17010	255	5.1
4f1	14.86058	0.06729	145	2.9
f1 + fX	9.59399	0.10423	119	2.4
fZ = 0.4996fX	2.93760	0.34041	83	1.7
2f1 + fX	13.30913	0.07514	37	0.7
fX − f1	2.16370	0.46217	36	0.7
$f_{Z^{\prime }}=1.499f_{X}$	8.81238	0.11348	27	0.5
$f_{1}+f_{Z^{\prime }}$	12.52753	0.07982	23	0.5
f1 − fZ	0.77755	1.28609	13	0.2
2fX	11.75768	0.08505	10	0.2
$2f_{1}+f_{Z^{\prime }}$	16.24267	0.06157	10	0.2
f1 + fZ	6.65274	0.15031	10	0.2
f1 + 2fX	15.47283	0.06463	7	0.1
2f1 + 2fX	19.18797	0.05212	5	0.1
All secondary pulsations			634	12.7

Notes. ^aFrequencies are taken from Moskalik (2014). ^bIn a simultaneous fit of all frequencies. ^cWe get P_X/P₁ = 0.63195208.

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We first grossly simplify the star into two regions, a stiff dense inner one and a relaxed rarefied outer one. We model these regions with two masses and two springs connected to a fixed wall, as in Figure 14. For equilibrium coordinates x_n[t], the equations of motion are

$$\begin{eqnarray} m_1 \ddot{x}_1 &= -k_1 x_1 - k_2 \left(x_1 - x_2 \right), \end{eqnarray} \tag{ 1a }$$

$$\begin{eqnarray} m_2 \ddot{x}_2 &= - k_2 \left(x_2 - x_1 \right). \end{eqnarray} \tag{ 1b }$$

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We fix initial conditions x₁[0] = 1, x₂[0] = 2, $\dot{x}_1[0] = 0$, $\dot{x}_2[0] = 0$ and vary the stiffness and inertia parameters. For k₁ = 9, m₁ = 8m, k₂ = 1, and m₂ = m, Equation (1) has the solution

$$\begin{eqnarray} x_1 &= +\frac{1}{3} \cos \left[ \omega _{+} t \right] + \frac{2}{3} \cos \left[ \omega _{-} t \right], \end{eqnarray} \tag{ 2a }$$

$$\begin{eqnarray} x_2 &= -\frac{2}{3} \cos \left[ \omega _{+} t \right] + \frac{8}{3} \cos \left[ \omega _{-} t \right], \end{eqnarray} \tag{ 2b }$$

where the frequency quotient

$$\begin{equation} \frac{ \omega _{+} }{ \omega _{-} } = \frac{ \sqrt{3/2\, m} }{ \sqrt{3/4\, m} } =\sqrt{2} \approx \frac{1}{0.707} \end{equation} \tag{ 3 }$$

is Pythagoras' constant, the first number to be proven irrational. For k₁ = 4, m₁ = 4m, k₂ = 4/5, and m₂ = m, Equation (1) has the solution

$$\begin{eqnarray} x_1 &= \frac{5-\sqrt{5}}{10} \cos \left[ \Omega _{+} t \right] + \frac{5+\sqrt{5}}{10} \cos \left[ \Omega _{-} t \right], \end{eqnarray} \tag{ 4a }$$

$$\begin{eqnarray} x_2 &= \frac{5-3 \sqrt{5}}{5} \cos \left[ \Omega _{+} t \right] + \frac{ 5+3 \sqrt{5} }{5} \cos \left[ \Omega _{-} t \right], \end{eqnarray} \tag{ 4b }$$

where the frequency quotient

$$\begin{equation} \frac{ \Omega _{+} }{ \Omega _{-} } = \frac{ \sqrt{ (\sqrt{5} + 1 ) / \sqrt{5}\, m } }{ \sqrt{ (\sqrt{5} - 1 ) / \sqrt{5}\, m } } = \frac{ 1 + \sqrt{5} }{2} \approx \frac{1}{0.618} \end{equation} \tag{ 5 }$$

is the golden ratio, the most irrational number (having the slowest continued fraction expansion convergence of any irrational number). In each case, the free mass parameter m allows the individual frequencies to be tuned or scaled to any desired value.

We identify the star's radius with the outer mass position, r = r_e + x₂, where r_e is the equilibrium radius. If the star's electromagnetic flux is proportional to its surface area, then

$$\begin{equation} F = k(r_e + x_2)^2. \end{equation} \tag{ 6 }$$

(Squaring generates double, sum, and difference frequencies, including those in the original ratios.) For appropriate initial conditions, Figure 15 graphs the Equations (4)–(6) stellar flux as a function of time and is in good agreement with the Kepler data.

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A star balances pressure outward versus gravity inward. Assuming spherical symmetry for simplicity, the radial force balance on a shell of radius r and mass m surrounding a core of mass M is

$$\begin{equation} m \ddot{r} = f_r = 4 \pi r^2 P - \frac{G M m}{r^2}, \end{equation} \tag{ 7 }$$

where the volume

$$\begin{equation} \frac{V}{V_0} = \left(\frac{r}{r_0} \right)^3 \end{equation} \tag{ 8 }$$

and the pressure

$$\begin{equation} \frac{P}{P_0} = \left(\frac{V_0}{V} \right)^\gamma. \end{equation} \tag{ 9 }$$

Assuming adiabatic compression and expansion and including only translational degrees of freedom, the index γ = 5/3, and the radial force

$$\begin{equation} f_r = f_0 \left(\frac{r_e}{r} \right)^2 \left(\frac{r_e}{r} - 1 \right), \end{equation} \tag{ 10 }$$

where r_e is the equilibrium radius, so that f_r[r_e] = 0 and f_r[0.68r_e] ≈ f₀. The corresponding potential energy

$$\begin{equation} U = U_e \left(\frac{r_e}{r} \right) \left(2 - \frac{r_e}{r} \right), \end{equation} \tag{ 11 }$$

where U_e = −f₀r_e/2 is the equilibrium potential energy. The apparent brightness or flux of a star of temperature T at a distance d is proportional to the star's radius squared,

$$\begin{equation} F = \frac{L}{4\pi d^2} = \frac{4 \pi r^2 (\sigma T^4 ) }{4\pi d^2} = \left(\frac{r}{d} \right)^2 (\sigma T^4 ) = k r^2, \end{equation} \tag{ 12 }$$

where k is nearly constant if the star's luminosity L depends only weakly on its temperature T. (In contrast, Bryant 2014 relates a pulsating star's luminosity to the velocity of its surface.)

The resulting flux is periodic but nonsinusoidal. The curvature at the maxima is less than the curvature at the minima due to the difference between the repulsive and attractive contributions to the potential, but the contraction and expansion about the extrema are symmetric. To distinguish contraction from expansion, we introduce a velocity-dependent parameter step

$$\begin{equation} f_0 = \left\lbrace \begin{array}{@{}ll}f_1, & \dot{r} < 0, \\ f_2, &\dot{r} \ge 0. \end{array} \right. \end{equation} \tag{ 13 }$$

For specific parameters, the left plot of Figure 16 graphs the resulting light curve, which asymmetrizes the solution with respect to the extrema. To add a bump to the expansion, we introduce velocity- and position-dependent parameter steps

$$\begin{equation} p = \left\lbrace \begin{array}{@{}ll}p_1, & \dot{r} < 0, \\ p_2, &\dot{r} \ge 0, r < r_C, \\ p_3, &\dot{r} \ge 0, r \ge r_C, \end{array} \right. \end{equation} \tag{ 14 }$$

where p is f₀, r_e, or m. The right plot of Figure 16 graphs the corresponding light curve. This model captures most of the features of Figure 5.

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From a dynamical perspective, simple models with a minimum of relevant stellar physics are sufficient to reproduce essential features of the KIC 5520878 light curve. The masses versus spring example demonstrates that a lumped model can readily be adjusted to exhibit a golden ratio frequency content similar to that of the light curve. The pressure versus gravity example demonstrates that an adiabatic ideal gas model can readily be modified to exhibit the asymmetry of the light curve. Nothing exotic is necessary.