Abstract
For Cataclysmic Variables (CVs), the record of many eclipse times (as presented in the traditional O − C curves) often displays a small-amplitude fast jitter on timescales of under a few years, with these often being interpreted as physical effects such as from third bodies in the system (including planets), or from azimuthal movements of the Hot Spot around the edge of the accretion disk. Recently, with the new and excellent light curves from spacecraft missions TESS and Kepler, we can measure large numbers of eclipse times down to orbit-by-orbit timescales. For four representative CVs (DQ Her, U Sco, RW Tri, and UX UMa), I find that the O − C jitter is constant from one-orbit-to-the-next out to eclipses separated by many years. This demonstrates that the jitter is not caused by physical effects on the orbit, nor by moving Hot Spots. This result was predicted for the expected jitter in eclipse times arising from the ubiquitous flickering in all CV light curves, where random flickers before/after eclipse minima will shift the measured eclipse times later/earlier than the true time of conjunction.
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1. Jitter in CV O − C Curves
The best observed CVs have hundreds of accurate eclipse timings, which can be plotted on an O − C curve to show variations about some smooth best fit ephemeris (Patterson et al. 2018). The best O − C curves display small variations on timescales of under a few years, with these variations being larger than the nominal measurement errors. Many workers then fit parabolas and sinewaves to this jitter (e.g., Schaefer 2011, 2020; de Miguel et al. 2016). Some workers then interpret the parabolas to derive mass transfer rates, and sinewaves to get orbital periods of third bodies.
A wide variety of models have been proposed to explain this O − C jitter. The usual interpretation is that changes in the O − C reflect changes in the orbital period, and this can arise from third bodies (stars or planets) or from mass transfer. Another means to make small alterations in the orbital period is to have stellar activity cycles make changes in the companion's magnetic field, with such changes being presumably on timescales of a decade or perhaps faster (Applegate & Patterson 1987). Another set of ideas relates the variation in eclipse times to variations in the light distribution within the accretion disk, where the predominant light from the Hot Spot (where the accretion stream impact the outer edge of the accretion disk) moves azimuthally around the disk edge, making for variations in the eclipse profile that produce variations in the eclipse times (de Miguel et al. 2016). For another idea, that I have never seen published anywhere, an inevitable source of jitter must occur from the ubiquitous flickering in the CV light curve, where a flicker soon before the eclipse minimum will make the eclipse time appear later than expected from the ephemeris, and where a flicker soon after the eclipse minimum will make the eclipse time appear earlier than expected.
2. The RMS Scatter In the O − C Curve
The jitter can be quantified by a parameter that I will notate as Δ(N), which is the difference in O − C for pairs of eclipse times separated by N orbital cycles. The O − C is calculated for any smooth ephemeris that reasonably fits the data on long-timescales. For example, the difference in the O − C values for two eclipses separated by three orbital periods (Porbit) will give a value of Δ(3), and we can collect many such values from many eclipse pairs. For a given CV and a given set of eclipse times, we can calculate the RMS scatter of Δ(N), or RMS[Δ(N)]. This tells us the degree of scatter in the O − C curve on the timescale of N-orbits. These RMS values will have a minimum equal to the average measurement error for the set of eclipse times.
Each model for the jitter makes a prediction as to how RMS[Δ(N)] should behave as a function of N. For the jitter from flickering, the jitter will be independent between all eclipses, so RMS[Δ(N)] will be a constant (to within the expected sampling variations) for all N. (The RMS for the N = 0 case will just be 1.4× the average measurement error for a single eclipse time for that data set.) For physical effects where Porbit changes, the RMS[Δ(N)] will start rising when NPorbit is a substantial fraction of orbital period of the third body. For effects controlled by the stellar activity cycle, RMS[Δ(N)] will rise above the flickering minimum for N values corresponding to large fractions of the activity period. For effects arising from motion of the Hot Spot, the azimuthal position cannot change too fast, so the effects in increasing RMS[Δ(N)] can only be apparent on timescales of days to months (e.g., de Miguel et al. 2020). Within this model, we would expect to see RMS[Δ(N)] as a constant (due to flickering) for low N, but then to rise up to another plateau (associated with the range of motion) for N corresponding to some timescale longer than a few days.
So I have a plan to go from CV eclipse times to test all the models for jitter. For this, I have selected out four CVs with excellent sets of eclipse times, with these four spanning the various classes of CVs. Each of these CVs have large numbers of excellent ground based eclipse times. Further, with the launch of the Kepler and TESS spacecraft, I have used their public domain data to derive large numbers of top quality eclipse times for many dozens of consecutive eclipses. Details on the CVs, Porbit in days, and the numbers of eclipses (#) are presented in the Table 1.
Table 1. RMS of Δ(N) in Seconds
| CV | Porbit | Mission | # | 0 | 1 | 2 | 3 | 4 | 5 | 9–11 | 30–40 | 90–110 | 300–400 | 9000–1100 | 3000–4000 | 9000–11,000 | 30,000–40,000 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| DQ Her | 0.194 | nonTESS | 144 | 10 | 37 | ⋯ | ⋯ | 34 | 48 | 47 | 30 | 49 | 60 | 89 | 110 | 124 | 116 |
| TESS | 238 | ⋯ | 23 | 24 | 22 | 23 | 24 | 23 | 24 | 26 | ⋯ | ⋯ | ⋯ | ⋯ | ⋯ | ||
| U Sco | 1.230 | nonKepler | 107 | ⋯ | 840 | 1070 | 1190 | 550 | 650 | 630 | 820 | 610 | 530 | 730 | 1000 | 770 | ⋯ |
| Kepler | 59 | ⋯ | 360 | 350 | 410 | 370 | 360 | 480 | 400 | ⋯ | ⋯ | ⋯ | ⋯ | ⋯ | ⋯ | ||
| RW Tri | 0.232 | nonTESS | 362 | 15 | 110 | 145 | 136 | 136 | 284 | 206 | 123 | 175 | 196 | 127 | 254 | 286 | 1013 |
| TESS | 88 | ⋯ | 53 | 53 | 54 | 48 | 45 | 49 | 46 | ⋯ | ⋯ | ⋯ | ⋯ | ⋯ | ⋯ | ||
| UX UMa | 0.197 | nonTESS | 492 | 44 | 45 | 44 | 47 | 49 | 50 | 66 | 50 | 39 | 56 | 61 | 68 | 128 | 264 |
| TESS | 243 | ⋯ | 23 | 26 | 21 | 22 | 24 | 24 | 24 | 24 | ⋯ | ⋯ | ⋯ | ⋯ | ⋯ |
From the Kepler and TESS light curves, I have derived the eclipse times by multiple methods. These include fitting parabolas to the minima, finding bisectors to lines cutting across the eclipse profiles at various depths, taking the interpolated crossing time at various depths for both the ingresses and egresses, and simply taking the bin of minimum light. The quality of these measures can be quantified as the RMS scatter of the resulting O − C curve, i.e., RMS[Δ(N)]. The worst measures by far are those involving the brighter half of the eclipse, and the time bin of minimum light. The best measures (being roughly equal in quality) are for the fitted parabolas to the minimum light curve and the bisectors cutting across the eclipse profile at roughly 50%–80% of the depth of minimum.
From my sets of eclipse times for the four CVs, I derive RMS[Δ(N)] for a wide range of N. These are tabulated in the Table 1, in units of seconds, and with the column headers giving the range of N.
3. Results
All four CVs show RMS[Δ(N)] to be constant (to within the expected sampling errors) for N = 1 up to large N. That is, the jitter mechanism operates much faster than one orbit, and remains constant up to many orbits.
We know that the ubiquitous flickering jitter will be the same for all N, and this is what we see. We know that real orbital period changes cannot cause changes as fast as N = 1 or N = 100, so this model is eliminated as the source for this plateau in RMS[Δ(N)]. Similarly, the variations that could arise from the mechanism with stellar activity cycles cannot account for the flat curve on timescales faster than a year or so. And the motion of the Hot Spot cannot be on timescales from a few hours up to a few days, so the constant RMS[Δ(N)] must be from some other cause. That is, none of the physics models can account for the constant jitter seen from N = 1 to large-N. Yet the flicker-noise model accurately predicted the constant RMS[Δ(N)].
The other mechanisms could possibly be present in any of the CVs. It is just that the known and required mechanism of flickering dominates for all timescales of less than two years or longer. We do see rises in RMS[Δ(N)] for DQ Her with N > 1000, for RW Tri with N > 15000, and for UX UMa with N > 10,000 or so. Indeed, DQ Her appears to rise to a second, higher, plateau over the range of timescales from 4000 to 40,000 cycles (1.6–16 yr), indicating that some other mechanism has started to dominate, but only over long timescales.
My overall conclusion is that the small-amplitude jitter faster than a few years is certainly caused by the mundane effects of flickering in the CV light curve, and not by any of the other possible mechanisms. Past and future workers must avoid using their physical models to fit O − C curves for which this random jitter dominates.