THE CHANGE OF THE ORBITAL PERIODS ACROSS ERUPTIONS AND THE EJECTED MASS FOR RECURRENT NOVAE CI AQUILAE AND U SCORPII

A simplistic way to determine the time of minima is to take the time of the faintest point in the light curve. But this method is poor because ordinary Poisson fluctuations throughout a fairly flat-bottomed minimum will cause the faintest point to be displaced from the true minimum by up to the half-duration of the minimum. So we should have a method to determine the minima times that somehow averages over many points near the minimum. Another possible method is to bisect the times when the ingress and egress pass through some given magnitude. The Poisson fluctuations and binning can be largely eliminated by fitting straight lines to the appropriate portions of the ingress and egress. This method has the advantage of using a fast changing part of the light curve, so that the times can be accurately determined. But this method requires that the light curve be symmetric around the time of minimum, and this is wrong at least far from the minimum. If we picked some bright magnitude then the systematic errors will be large and if we pick a magnitude near the minimum then the measurement uncertainties along the relatively flat portions of the light curve will be large. So we need a method that concentrates near the minimum (so as to have minimal effect of the asymmetries in the light curve), gets above the minimum (to where the rate of change of the light curve is substantial), and uses many points (to minimize Poisson variations).

I determine the time of minimum light by fitting a parabola to the near-minimum portions of the eclipse light curve. The minima of any function will be well approximated by a parabola over a sufficiently small range, so the idea is to perform the fit over a sufficiently small range such that the asymmetric terms are negligible. (If I try to fit higher order terms, then the minimum time is sensitive to the order and to the chosen time interval for the fit. With variable coverage in time and with the eclipse shapes changing, the use of higher order terms only adds large systematic uncertainty.) This method has the advantages of being applicable to any light curve that covers the time of the minimum, of being insensitive to the high-frequency noise from Poisson fluctuations, and of being insensitive to asymmetries in the light curve.

In practice, I determine the best-fit parabola by the usual chi-square minimization. That is, a parabolic model with three parameters (the time of minimum, the faintest magnitude, and the curvature) is varied until the chi-square comparison between the model and the observations is best. The data points for inclusion in this fit are determined by the times over which the model is fainter than some cutoff magnitude. This cutoff is usually taken to be 0.1–0.2 mag brighter than the minimum, so that enough time is included for the curvature around the minimum to dominate and yet not so long that any asymmetries will be apparent. Once a good minimum is achieved, then the points included in the fit are frozen and the fits are optimized again, with this being required so as to avoid effects where the chi-square is changing due to differing points being included in the fit. The 1σ uncertainty on the time of minimum is found by varying the time until the chi-square has risen by unity from its minimum value.

In practice, I find that many of the eclipses have best fits for which the reduced chi-square is substantially smaller than unity. This is seen only in light curves where the quoted error for each point is dominated by the arbitrary systematic error that I added in. (For well-exposed images, the statistical uncertainties, as reported by IRAF, are always unrealistically small, so I have added a typical systematic error of 0.015 mag in quadrature as an effort to make the quoted error bars more realistic. This value is based on the typical scatter for standard stars in a plot of the reduced magnitude versus airmass, with the ultimate cause of this scatter being unknown.) This problem of a low reduced chi-square is fully consistent with the real systematic uncertainty being more like 0.010 mag for many of the observations. With the problem coming from a sometimes slightly conservative choice in the level of systematic errors, the quoted error bars on the minimum time will be slightly too large for some eclipses.

The parabola fitting method has a free parameter of the cutoff magnitude. This cutoff should not be too close to the minimum or else there will be too few points with too little curvature to determine the time of minimum well. Nor should this cutoff be too near the out-of-eclipse level or else the asymmetries will make large systematic uncertainties in the derived time of minimum. Within some middle range, how sensitive is the derived minimum time to the choice of the cutoff magnitude? To test this, I have taken a typical light curve (for the 2009 May 1 eclipse of CI Aql) with coverage throughout the eclipse. For this case, my normal analysis had used a cutoff of V = 16.55 mag and found a minimum magnitude of 16.735 and a minimum time of JD 2454953.7887 ± 0.0003. For cutoff magnitudes of 16.70, 16.65, 16.60, 16.55, 16.50, 16.45, and 16.40, the fractional part of the best-fit Julian Date is 0.7890 ± 0.0009, 0.7889 ± 0.005, 0.7887 ± 0.0003, 0.7887 ± 0.0003, 0.7886 ± 0.0002, 0.7887 ± 0.0002, and 0.7885 ± 0.0001, respectively. As expected, the statistical uncertainty gets smaller as the cutoff is raised. (The systematic uncertainty from the light curve asymmetry will get larger as the cutoff is raised.) Importantly, the variations in the derived minimum time due to changing the cutoff are always small compared to my final error bars, which is to say that the best-fit eclipse time is insensitive to the choice of the cutoff for any reasonable range.

As a further test of the parabola fitting method as well as of the asymmetry in the light curve, I have performed the bisector method for various well-observed eclipses. In the same case as in the previous paragraph (for the 2009 May 1 eclipse of CI Aql), I have fitted line segments to portions of the ingress and egress. With these fits, I get the time when the light curve passes through various fiducial magnitudes, and these times are then bisected to get a minimum time. For fiducial magnitudes of 16.70, 16.65, 16.60, 16.55, 16.50, 16.45, and 16.40, the fractional part of the Julian Date for the minimum is 0.7889, 0.7885, 0.7886, 0.7884, 0.7882, and 0.7880. We see a systematic but small shift as the fiducial magnitude is raised, and this is a demonstration of the level of change caused by the asymmetries in the light curve. For the middle range of fiducial magnitudes, the changes are always small compared to the final quoted uncertainties. From the above, I conclude that the parabola fitting method is robust and produces accurate (or slightly too large) error bars.

A number of eclipses were covered with fast time series photometry, but only a portion of the eclipse was covered. This was usually due to clouds or twilight intervening. Without substantial coverage on both sides of the minimum, the parabolic fitting procedure does not work reliably. Nevertheless, these data still provide strict constraints on the time of the minima. The general procedure that I will use to derive the minima times will be to determine the average offsets between when the eclipse light curve passes through some fiducial magnitude as based on light curves with full coverage (see the previous subsection) and then apply these offsets to the partial-coverage light curves (this subsection).

From Figures 4 and 7, it looks like the shape of the eclipse light curve is nearly constant. (This is not perfect, as both RNe show some variations from orbit to orbit.) When all the light curves are shifted to a common minimum, the light curves fall on top of each other with only moderate scatter. So, for example, for CI Aql, when the ingress fades through a magnitude 0.2 mag brighter than the minimum that time is 0.0216 days before the minimum on average, and when the egress brightens through the same magnitude that time is 0.0203 days after the maximum on average. This can be placed into an equation as

$$\begin{equation} T_{{\rm min}} = T_m + \Delta T, \end{equation} \tag{ 1 }$$

where ΔT is the offset in time from when the brightness passes through the fiducial magnitude (T_m), with this value a function of m_fid, the branch (ingress or egress), and the star. I will take the fiducial magnitude to be 0.1, 0.2, 0.3, and 0.4 mag brighter than the observed minimum magnitude (m_min). This simple offset in time can often be improved by taking account of the slope of the light curve as it passes through the fiducial magnitude. That is, if the eclipse light curve is "flatter" (with a weaker parabolic term), then the slope will be shallower and the time offset will be greater. This relation can be quantified as

$$\begin{equation} T_{{\rm min}} = T_m + (A + S_m B), \end{equation} \tag{ 2 }$$

where T_min is the derived time of minimum, T_m is the time when the eclipse light curve passes through some fiducial magnitude m_fid, A and B are two constants that depend on m_fid, and S_m is the observed slope of the light curve (in magnitudes per day) when it passes through m_fid. T_m and S_m are both calculated from the light curve by fitting a simple line to the magnitudes within a small range centered on m_fid. In general, Equation (2) is better than Equation (1), other than the case where the fiducial magnitude is too close to the minimum so that the measurement uncertainties dominate. Also, for the case of U Sco in the B and V bands, I have too few fully covered light curves to measure A and B, so I can only use Equation (1).

The constants in Equations (1) and (2) can be derived from the well-observed eclipses, for both ingress and egress, for fiducial magnitudes 0.1, 0.2, 0.3, and 0.4 mag brighter than the eclipse minimum. For CI Aql, the constants have been derived from 19 eclipses with good coverage from 2002 to 2008. For U Sco, the constants have been derived for 3, 1, and 17 eclipses with good coverage from 1994 to 2009. The average values of ΔT, A, and B are reported in Table 6 for both branches for both RN for all four fiducial magnitudes. From these eclipses, I have also calculated the rms scatter in the deviation between the derived minima times versus those derived by parabola fitting, σ, with these being the accuracy in the derived eclipse minimum time for any one measure.

If the light curve includes a recognizable minimum, then the fiducial magnitude can be determined, the observed light curve will yield the time and slope for when the RN passes through this fiducial magnitude, and then Equation (1) or Equation (2) plus the values in Table 6 can be used to derive the minimum time and its 1σ uncertainty.

Among the eclipse light curves with partial coverage (so that a parabola fit is not reasonable), many do cover the minimum, so that the minimum magnitude can be measured to better than 0.01 mag. For these, the fiducial magnitudes are known to good accuracy and the 1σ uncertainties in Table 6 represent the entire error bars. For many partial light curves, the minimum is not covered, so the minimum magnitude is not known and the fiducial magnitudes have corresponding uncertainty. So, for example, if we only observed a descending branch, then we could assume that the minimum equals the average minimum, take the fiducial magnitude to be some value, extract a time and a slope from the light curve, and then deduce the time of minimum. Alternatively, if we take the minimum to be fainter than average, then our fiducial magnitude would be fainter, the time when the ingress crosses that fiducial magnitude would be later, and the derived minimum time would be later. Therefore, when the minimum magnitude is not observed, an additional uncertainty will arise from not knowing the correct fiducial magnitudes. For cases where the minimum is not directly observed, the best that I can do is simply to take the average for that RN and filter. For CI Aql in the V band, this is 16.74 ± 0.05. For U Sco, the average m_min is 19.98 ± 0.14, 18.96 ± 0.05, and 18.18 ± 0.07 in the B, V, and I bands, respectively. I have taken account for this by the propagation of errors, where the uncertainty in the minimum magnitude is taken to be the rms scatter of the minima.

For each light curve, I measure the times and slopes for as many of the fiducial magnitudes as possible (for 0.1, 0.2, 0.3, and 0.4 mag above minimum for ingress or egress), and derive a time of minimum for each. If one whole branch plus the minimum is observed, then we will have four derived times, each accurate to 0.0019–0.0066 days (see Table 6), then these times can be combined as a weighted average.

As a test for the procedure described in this subsection, I have applied it to the 10 well-observed CI Aql light curves from 2009. These 10 eclipses have accurate minima times from parabola fitting. The deviations of the derived minima times from the procedure in this section versus the accurate times from parabola fitting will provide a good measure of the real accuracy of the procedure. For the case where the minimum magnitude is known from the observed minimum (as when one of the branches plus the minimum is observed), I have 61 measures of the deviations, each from a particular eclipse and fiducial magnitude. The rms of the deviations is 0.0024 days. For the case where the average minimum magnitude (16.71 ± 0.07) is adopted, the rms of the deviations is 0.0036 days. For the case where I take weighted means of minimum time estimates from multiple fiducial magnitudes, I find that the rms deviations are 0.0022 days for when the minimum magnitude is known and 0.0027 days for when the average case is adopted for the minimum magnitude. These improvements are smaller than expected if the individual measures are independent. I take this to mean that the various measures along each branch are not independent, with the implication that one fiducial magnitude is almost as good as averaging the results from up and down the light curve. Another implication is that the formal error bars obtained by the weighted average from multiple measures will always be too optimistic. With this experience, I will adopt 1σ uncertainties of 0.0022 for the case where the minimum magnitude is well measured and 0.0027 days for the case where it is not measured.

Certainly the best way to get an eclipse time is to have full coverage across a minimum, or at least with substantial portions of the ingress or egress. Unfortunately, many of the key old eclipses were only observed with one image. For CI Aql, we have all of the pre-eruption data as these single images (either from the Harvard plates or from RoboScope). There is still good information as to the time of the eclipse minimum, even though the accuracy will be poorer than the situations with time series. So, for example, if an old plate shows the RN at the magnitude usually taken as the eclipse minimum, then the middle time for that plate will be the best estimate of the minimum time with some to-be-determined error bar. Or if an old CCD image shows the RN halfway between the usual minimum and maximum levels, then the time of minimum will be roughly the middle exposure time plus-or-minus a quarter of the eclipse duration. In most cases, the ambiguity of knowing whether the image is during the ingress or egress will be resolved by comparison with other plates similar in time. With this, the many pre-eruption RoboScope images of CI Aql can be used to determine the eclipse times with fairly good accuracy, and the few Harvard plates showing CI Aql in eclipse are so far back in time that the long lever arm will finely constrain the O − C diagram despite moderately large uncertainties in the eclipse times.

For the RoboScope data on CI Aql (Mennickent & Honeycutt 1995), the magnitudes are almost all once per night. The folded light curve (see Figure 1) easily resolves whether the image is on the descending or rising branch. The estimate of the time of minimum is as easy as interpolating the observed magnitude in the CI Aql 1991–1996 template (Table 5), and adding the corresponding time offset to the observed time. I have selected all images with phase within 0.068 of the eclipse by the original Minnickent & Honeycutt ephemeris. The uncertainty in how much brighter this measured magnitude is above the minimum will be the quoted photometric uncertainty (typically 0.02–0.04 mag) added in quadrature with the vertical scatter in the eclipse light curves (0.05 mag, compare with Figure 4). This uncertainty is then propagated through the eclipse light curve template to get the uncertainty in the time of the minimum. The resultant error bars range from 0.0040 to 0.0117 days (with one at 0.0263 days) depending on the observed magnitude.

For the Harvard plates, the typical exposure time is 1 hr, and this smears out the recorded light curve (Schaefer & Patterson 1983). CI Aql was positively detected on 12 plates in quiescence, and one plate had a very faint limit. (This is in addition to the 20 plates at Harvard which recorded CI Aql in eruption, see Schaefer 2010.) The typical B-band magnitude at maximum light is 17.0–17.2 mag. So the two plates taken with B > 17.5 (plate MF10536) and with B = 17.75 (plate A17852) are certainly near the eclipse minimum. As such, the middle time for the plate exposure is taken as the time of the eclipse. The uncertainty in this estimate will be comparable to the half-exposure time (0.02 days) plus the half-duration of minimum (∼0.01 days), which I will take to be 0.03 days.

After the nova shell becomes optically thin (at the end of the initial fast fall in the light curve), the binary system and its eclipses are visible. The RNe can be well observed during the tail of its eruption, partly because observers are interested in the eruption to motivate sustained photometry and partly because the star is brighter than in quiescence so it is easier to observe. This photometry might be either a fast time series by a single observer that happens to cover the time of an eclipse or as single magnitudes taken on many nights (possibly by many independent observers) folded on the period to provide a phased light curve. A problem with combining magnitudes over many nights is that the uneclipsed nova light is fading, so any folded light curve must be detrended first. A problem with combining magnitudes from many observers is that often each observer will have some small offset so as to convert their quoted magnitudes onto some standard magnitude system. These problems are simple to overcome, yet they will nonetheless lead to some increased scatter in any combined light curve.

For the case of the early tail in the 2000 eruption of CI Aql, the folded and detrended light curve (Figure 8) shows significant and substantial roughly sinusoidal modulation on the orbital period. For days 40–200 after the peak, we see a well-defined sinusoidal envelope roughly half a magnitude in width that contains most of the points. The best-fit sinusoidal amplitude is 0.16 ± 0.02 mag and the phase of minimum is −0.037 ± 0.017. That is, the minimum in the light curve comes 0.0229 ± 0.0105 days early in comparison with the post-eruption ephemeris, and this gives a minimum epoch of HJD 2451792.7070±0.0105. The scatter about the best-fit sine wave is much larger than the observational uncertainty (typically 0.15 mag for visual observations), and I have to add in quadrature a value of 0.26 mag so as to get a reduced chi-square equal to unity. The cause of this larger scatter is undoubtedly a combination of imperfect detrending and intrinsic variations of CI Aql that are on faster timescales than the trend line can take out. The light curve does not show any primary eclipse. I interpret this as being caused by an emission region that is substantially larger than the binary orbit (so any eclipse of its center would cause only a small drop in light) and that is transparent enough so that the inner regions can be seen (with the irradiation of the inner hemisphere on the companion providing the modulation with the orbital period). That is, the outer ejected shell has already faded to insignificance in the continuum light, the luminous and extended envelope would provide most of the light, the binary would primarily be visible due to the hot illuminated hemisphere of the companion star, and the accretion disk is still disrupted by the nova wind that is creating the extended envelope.

Zoom In Zoom Out Reset image size

I have observed two fast time series of CI Aql in its late decline phase (2001 August; see Figure 2). Both are well-observed minima, for which I used the parabola fitting method to find the minimum time. As the light curve had not yet returned to the quiescent level, these two times will have to be treated as eruption times, with potential systematic offsets when compared to the true quiescent conditions.

The 1945 eruption of U Sco was discovered as part of an exhaustive search through the Harvard archival plates, with the (perhaps surprising) result that a full eruption was well covered on 37 plates (Schaefer 2010). Until the 2010 eruption, these plates provided the only late coverage of the U Sco light curve, with 20 magnitudes covering the time from 33 to 47 days after the peak. (The exact HJD of the plates has been confirmed from the original logbooks as well as the calculated values on the plate envelopes.) This late coverage is during a time when eclipses are prominent (Schaefer et al. 2010a), so this provides an opportunity to seek an eclipse time in the tail of the 1945 eruption. With the B-band magnitudes reported in Schaefer (2010), I have determined a trend line, with this being subtracted out. The folded detrended light curve (Figure 9) shows an obvious eclipse, and we can even see the slow ingress known from the 2010 eruption at the same time after peak (Schaefer et al. 2011). The points in the minimum and along the egress are all from one night (1945 July 2). I have performed a chi-square fit to compare the measured magnitudes to the light curve template derived for the same time after peak from the 2010 eruption (Schaefer et al. 2011). The measurement errors for the magnitudes are 0.15 mag, and to this I have added 0.10 mag in quadrature so as to represent the orbit-to-orbit scatter known from the 2010 eruption light curve. The only free parameter in this fit is the phase of the minimum. The best-fit phase of minimum corresponds to HJD 2431639.300. The 1σ uncertainty is ±0.009 days.

Zoom In Zoom Out Reset image size

The 1987 eruption of U Sco (Schaefer 2010) has few magnitudes on the plateau phase (during which eclipses would be visible). These points happen to include no values with phase between 0.78 and 1.08, so no eclipses could have been detected.

The 1999 eruption of U Sco has many magnitudes after the fast early decline (Schaefer 2010). Several eclipses are visible in the detrended light curve. One eclipse at 19 days after the peak has already been reported by Matsumoto et al. (2003), as will be discussed in the next section. Another eclipse is visible 25 days after the peak, but only ingress is recorded. With the minimum magnitude uncertain by perhaps even half a magnitude (due to questions on subtracting the trend as well as the depth of the eclipse at that epoch), the uncertainty in the derived time of minimum (cf. Section 4.2) will be much too large to be of any use. The only other confident eclipse is 30 days after the peak. Here, the light curve is seen to be rising, with the faintest magnitude being 1.21 mag fainter than the accurately known trend line, so the first observations are certainly very close in time to the minimum. I take the time of minimum to be the time of this first observation, and give it an uncertainty appropriate for the sampling and the observed duration of minima from 2010. Thus, I get the minimum time to be HJD 2451265.3060±0.0100.

Eclipse times in quiescence have already been published for both CI Aql and U Sco in Schaefer (1990) and Schaefer & Ringwald (1995), with all of these times now being superseded by the analysis in this paper. Twenty well-observed eclipse times in the tail of the 2010 eruption of U Sco are reported in Schaefer et al. (2011), and these are adopted here. The only other eclipse times in the literature are few in number, isolated in measure, and all in the tail of eruption light curves. Due to the shifts of the eclipse times during the eruptions, these published times cannot be combined with eclipse times from quiescence so as to derive any orbital period change. Without the realization of the time shifts, earlier papers (Matsumoto et al. 2003; Lederle & Kimeswenger 2003) had claimed to measure period changes, and these are completely erroneous.

For CI Aql, Matsumoto et al. (2001) present a large number of magnitudes in many filters, and the orbital modulation can be clearly seen. However, they do not present any minimum times, nor do they give their magnitudes so others can derive minima times, and their plotted light curves are not adequate to determine the minimum times with any accuracy. Their folded light curve from some unstated time during the 300 day long plateau shows an amplitude of 0.6 mag, with an apparent shallow eclipse. They present a new ephemeris for the eclipses, but it is never stated whether the original Mennickent & Honeycutt (1995) data was also used for the equation. The only apparently useful time is their specified epoch for their new ephemeris, HJD 2451701.2086±0.0089, which presumably represents an observed minimum time. However, this time is just 29 days after peak at a time when no eclipses were visible. As such, I cannot accept their epoch as a reliable observed eclipse time, even though they must have many good eclipse times that are unreported.

For CI Aql, Lederle & Kimeswenger (2003) present several long time series taken in 2001 and 2002, as taken with the Innsbruck 60 cm telescope plus a CCD camera with a V filter. They also present a nice and detailed physical model that reproduces their light curves and derives the parameters of the CI Aql system. But they neither present any individual minima times, nor do they present any photometry data, while their plotted light curves are inadequate to get any accurate minima times. The only useful time that I can pull out of their paper is their specified epoch for their new ephemeris, Julian date 2452081.5022 ± 0.0046, which presumably has a heliocentric correction, and which presumably represents an observed minimum time.

For U Sco, Thoroughgood et al. (2001) identified an eclipse as based on seven flux density measures from spectra during the 1999 eruption. The eclipse is at a time 51 days after the peak, which corresponds to the middle of the second plateau phase (Schaefer et al. 2010a). They fitted a parabola to the full minimum and reported the heliocentric Julian date of the minimum as 2451286.2143 ± 0.0050.

For U Sco, Kato (2001) measured 22 magnitudes over 0.14 days (early in the 1987 eruption) to find random fluctuations, for which he picked out 1 to identify as an eclipse minimum. But this claimed minimum is not different from other minima in the light curve, so its identification as an eclipse is poor. Also, this time series was taken just seven days after the peak, at a time when we know that the nova shell is optically thick and eclipses are not visible (Schaefer et al. 2011). So I reject this time as a valid eclipse time. Kato (2001) makes a further claim to have recognized an eclipse in a folded light curve based on data from the VSOLJ. However, the claimed eclipse has much too short a duration to be the real eclipse. And these observations were collected from 5 to 11 days after the peak, so they cannot represent eclipses. Again, I reject the times from this claim.

For U Sco, Matsumoto et al. (2003) present many series of fast photometry throughout the 1999 eruption. They claim to identify a secondary eclipse at a time six days after the peak. However, I only see a usual fluctuation for which they converted a marginally significant inflection in the decline into an apparent minimum by subtracting out an unrealistically steep trend. In any case, we now know that the nova shell was optically thick at six days after the peak, so no eclipse could be visible (Schaefer et al. 2011). Therefore, I reject this claimed secondary minimum time. Matsumoto et al. (2003) also report the time of one primary minimum, based on piecing together an egress and the next ingress. Their reported time is HJD 2451254.211, which is 19 days after the peak, on the first plateau. No error bar in the eclipse time is quoted, so I will adopt ±0.01 days as appropriate for their sampling.

With these observations and analysis procedures, I have tabulated all known eclipse times for both CI Aql and U Sco. These are presented with identical formats in Table 7 for CI Aql and Table 8 for U Sco. The first column gives the UT date of the observed minimum. The second column gives the Observatory and/or telescope used to make the observations, or alternatively the data collecting organization and the observer. The third column lists the heliocentric Julian Date (HJD) of the observed time of minimum light (T_obs) and its 1σ uncertainty. The fourth column gives a running integer, N, which keeps track of the cycle count as used in the ephemeris equations. The last column gives the O − C value (in units of days) for the observed HJD versus the model time as based on the best-fit post-eruption linear ephemeris.

I have 45 post-eruption eclipse times from 2001 November to 2009 September (see Table 7). The O − C diagram is plotted in Figure 10.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For the many highly accurate eclipse times, we see a scatter that is significantly larger than any smooth curve we could draw. This scatter cannot be due to measurement errors, nor to jitter of the position of the stars in their orbit. The likely cause of the scatter is changes in the brightness of the accretion disk across the duration of the eclipse minimum. The backside of the accretion disk is still visible during mid-eclipse (Hachisu et al. 2003; Lederle & Kimeswenger 2003) and that the disk brightness changes on all timescales (see Figures 3 and 4). So, when the unocculted portion of the disk changes in brightness over time, the ingress or egress light curve will be distorted from its average shape, and this distortion will result in small shifts in the fitted time of minimum light. This is an easy and expected explanation for intrinsic jitter in the apparent eclipse minimum times.

With this intrinsic scatter, the real uncertainty of each minimum time for indicating some fiducial orbital phase will be larger than the quoted measurement error bars in Table 7. The intrinsic scatter will be some constant throughout the post-eruption interval, roughly given as the rms scatter of the best-measured points around the best-fit smooth curve. The total uncertainty in the fiducial orbital phase will be the addition in quadrature of the measurement error and the intrinsic scatter. Formally, I will derive the intrinsic scatter as that value which produces the best-fit-reduced chi-square equal to unity for the 2001–2009 interval.

For the model where the O − C curve is linear, I have used a chi-square fitting routine to derive the best-fit period and epoch. For this fit, I have chosen the fiducial epoch to be near the peak date for the eruption. I find an epoch of E₀ = 2451669.0575 ± 0.0002 and P_post = 0.61836051 ± 0.00000006. With an intrinsic scatter of 0.00136 days (1.9 minutes), I get a fit with a chi-square of 43.0 for 43 degrees of freedom. So the scatter caused by the disk variations is less than 2 minutes, while the orbital period is known to 0.0043 s accuracy. This period has the fractional accuracy of 81 parts per billion.

The O − C curves can be readily interpreted if the model is linear. So I should choose some linear model, and I chose to use this best-fit post-eruption linear model as my fiducial linear model. So all my O − C values and plots are constructed with

$$\begin{equation} T_{{\rm model}}=2451669.0575 + N \times 0.61836051, \end{equation} \tag{ 3 }$$

as in Table 7 and Figure 10. N is an integer that counts the orbital cycles from the original epoch.

The system parameters along with Equation (B9) (in Appendix B) can be used to place an upper limit on $\dot{P}$. From Hachisu et al. (2003), M_comp = 1.7^+0.3 _{− 0.7} M_☉, M_WD = 1.2 ± 0.2 M_☉, and $\dot{M} \sim 1.0 \times 10^{-7}$ M_☉ yr⁻¹. From Lederle & Kimeswenger (2003), M_comp = 1.5 M_☉, M_WD = 1.2 M_☉, and $\dot{M}$ is 2.5 × 10⁻⁸ M_☉ yr⁻¹ in 1991–1996 and 5.5 × 10⁻⁸ M_☉ yr⁻¹ in 2002. We also have the more restrictive constraint that q ≲ 5/6. With these inputs, the upper limit on $\dot{P}$ varies from +1 to +9 times 10⁻¹¹ days cycle⁻¹. With this, I can only constrain

$$\begin{equation} \dot{P} < +0.9 \times 10^{-10} \end{equation} \tag{ 4 }$$

days cycle⁻¹.

The 45 post-eruption eclipse times can be fit to Equation (B7) in Appendix B, with no ambiguity relating to period changes related to eruptions. I find the best fit for E₀ = 2451669.0563, P₀ = 0.61836136 days, and $\dot{P}$= −24×10⁻¹¹ days cycle⁻¹. This gives a chi-square of 42.0. The 1σ range (i.e., the range over which the chi-square is within 1.0 of the minimum) is

$$\begin{equation} -52 \times 10^{-11} < \dot{P} < 0 \end{equation} \tag{ 5 }$$

in units of days cycle⁻¹. With this, $\dot{P}$ is likely negative, and this means that the $\dot{J}$ term in Equation (12) must be substantial and will dominate over the effects of the steady mass transfer.

The constraint arising from the 1926 and 1935 eclipse will depend on the derived ΔP value, which will come from Section 6.4. In all, the best constraints on $\dot{P}$ are those given in Equation (4), while the best estimate is $\dot{P}$= −24×10⁻¹¹ days cycle⁻¹.

The post-eruption ephemeris can be extrapolated to the tail of the eruption with high accuracy, where the acceptable range of $\dot{P}$ makes for no significant change. I have four eclipse times for the tail of the 2000 eruption. The three times on the late tail are consistent with the post-eruption ephemeris to within the intrinsic scatter. But the one eclipse time in the early tail comes 0.022 days (32 minutes or 0.037 in phase) early. This offset is large, but the uncertainty is not small because no eclipse is visible (see Figure 8). The chi-square for a sine wave fit to the data from 40 to 200 days after peak with a zero offset is 4.9 worse than the best fit, indicating that the existence of the offset is significant at the 2.2σ level.

If this offset is taken at face value, then the eclipse minima occur before the time expected from the quiescent minima. This shift is too large to be caused by any real change in the orbital period, so it must be caused simply by a shift in the center of light from eruption to quiescence. During the eruption, the dominant light sources will be centered on the WD and centered on the illuminated hemisphere of the companion, both of which will make for minima at the time of conjunction between the two stars. During quiescence, the dominant light source will be the hot spot in the accretion disk, with this being offset from the line connecting the centers of the two stars. The hot spot will be centrally eclipsed at a time roughly 0.03 in phase after the time of the conjunction (based on the geometry in Lederle & Kimeswenger 2003). So we have a consistent explanation for why the eclipse times are systematically offset from quiescence to the eruption.

The eclipse times from 1991 to 1996 can now be used to measure ΔP across the 2000 eruption. For now, I will not use the 1926 and 1935 eclipses in the chi-square calculation. To get a good idea of the situation, let me start by adopting the case that best fits the post-eruption times (P_post = 0.61836136 days, E₀ = 2451669.0563, and $\dot{P}=-24 \times 10^{-11}$ days cycle⁻¹). When ΔP = 0, the chi-square is 65.6 for the 74 non-eruption eclipse times from 1991 to 2009. If ΔP is allowed to vary freely, the chi-square reaches a minimum of 63.8 at ΔP = +5 × 10⁻⁷ days. The 1σ range (where the chi-square is within unity of 63.8) is ±3 × 10⁻⁷ days. The positive sign for ΔP means that the P_post > P_pre, which is to say that the period increases across the eruption.

Next, we can look at the fits for the range of $\dot{P}$ allowed by the post-eruption times. One extreme of the allowed range has $\dot{P}=0$, for which P_post = 0.61836051 days and E₀ = 2451669.0575. For this case, with no period change across the eruption, the chi-square is 77.1 (for 74 non-eruption eclipse times from 1991 to 2009). When the ΔP is allowed to vary, the chi-square reaches a minimum at 64.8 for a period change of −12 × 10⁻⁷ days. The 1σ error bar is ±0.3 × 10⁻⁶ days. For the maximal allowed steady period change ($\dot{P}=-52\times 10^{-11}$ days cycle⁻¹ with P_post = 0.61836231 days and E₀ = 2451669.0550) and zero sudden period change, the chi-square is 112.5. When ΔP is allowed to vary, the chi-square reaches a minimum at 64.8 for a period change of (+ 23 ± 3) × 10⁻⁷ days. Characteristically, the best-fit value for ΔP is correlated with $\dot{P}$, such that the more curved the O − C curve (with the parabola concave down) the more positive the required period change. For the minimal curvature (i.e., flat with $\dot{P}=0$), the period must have decreased across the 2000 eruption (i.e., ΔP < 0), while for the maximal curvature the period must have increased across the eruption.

As the fits go from minimal to optimal to maximal $\dot{P}$ (as based on the post-eruption times alone), the minimum chi-square goes from 64.8 to 63.8 to 64.8. With this, we see that we have closely mapped out the 1σ range as based on the inclusion of the 1991–1996 eclipse times. That is, all the 1991–2009 data is best described by a model with $\dot{P}=-24 \times 10^{-11}$ days cycle⁻¹, with the 1σ range from zero to −2.6 × 10⁻¹⁰ days cycle⁻¹. The best fit has ΔP = +5 × 10⁻⁷ days, with a 1σ range from around −12 to +23 times 10⁻⁷ days.

Finally, we can now do a fit to all 76 non-eruption eclipse times from 1926 to 2009. This is different from the results in the previous section only in that the two eclipses from 1926 and 1935 are included. These old eclipses have substantial error bars in their timing, but they are around 80 years ago, so their long lever arm in time will provide strong constraints. An ambiguity in this fit is whether CI Aql had any undiscovered eruptions between the years 1941 and 2000. Schaefer (2010) puts forth the evidence (based on quiescent magnitudes and discovery probabilities) that one or two eruptions were missed, although zero missed eruptions is certainly possible.

To start this, let us consider the result for the best fit from the 1991 to 2009 times (ΔP = +5 × 10⁻⁷ days, $\dot{P}=-24 \times 10^{-11}$ days cycle⁻¹, P_post = 0.61836136 days, and E₀ = 2451669.0563). With these values, no adjustments, and no undiscovered eruptions, the chi-square for all 76 non-eruption eclipse times is 79.4. The O − C curve passes somewhat below the 1935 point. If I add two eruptions (in 1960 and 1980) with the same ΔP value for all eruptions, then the fit is improved to a chi-square of 72.4. If further the ΔP value is allowed to vary, then a value of ΔP = +9.5 × 10⁻⁷ days produces a good chi-square of 67.2. Alternatively, if $\dot{P}$ is allowed to vary, then the minimum chi-square is 68.5 for $\dot{P}=-22 \times 10^{-11}$ days cycle⁻¹ (still with ΔP = +5 × 10⁻⁷ days).

For the best minimal curvature case ($\dot{P}=0$), with no change from the best fit from 1991 to 2009 and eruptions in 1960 and 1980, we have a good minimum chi-square of 66.1 with no adjustments. If I allow the period change across the eruption to vary, we get a minimum chi-square of 65.6 for ΔP = −13 × 10⁻⁷ days. If I do not allow for any undiscovered eruptions, then the chi-square minimum is poor at 73.8 for ΔP = −18 × 10⁻⁷ days.

For the best maximal curvature case ($\dot{P}=-52\times 10^{-11}$ days cycle⁻¹), with no changes from the best fit of 1991–2009 and eruptions in 1960 and 1980, the chi-square is bad at 129.3 (the O − C curve passing far below the 1935 eclipse). By adjusting only the ΔP value, the best chi-square equals the poor 85.6 for ΔP = +35 × 10⁻⁷ days. If I do not include undiscovered eruptions, the best chi-squares are always very bad because the O − C curve falls far below the 1935 point. With this, the inclusion of the two very old eclipse times places a constraint that the $\dot{P}$ value must be substantially closer to flat than the maximal curvature value.

For the general case, we should try to minimize the chi-square by simultaneously allowing ΔP, $\dot{P}$, and the number of missed eruptions to vary. For this, the period just after the 2000 eruption (P_post and the epoch (E₀)) will be held constant to the values obtained for the 2001–2009 data alone. (This is because the statistical weight of these many and accurate times will dominate over all the other parameters.) I will further assume that $\dot{P}$ remains constant throughout and that ΔP is the same for all eruptions. With this, the global best fit (for all 76 non-eruption eclipse times) has a chi-square of 64.6 with ΔP = −3.7 × 10⁻⁷ days, $\dot{P}=-12\times 10^{-11}$ days cycle⁻¹, P_post = 0.61836092 days, E₀ = 2451669.0570, all with an adopted intrinsic scatter of 0.00136 days and no undiscovered eruptions. This is my best result for the period change of CI Aql across eruptions, and hence the main goal of the CI Aql program.

The 1σ region of parameter space can be determined by seeking those values for which the chi-square is within unity of the global minimum (i.e., ⩽65.6). With this, the allowed range for $\dot{P}$ is −2 to −20 times 10⁻¹¹ days cycle⁻¹, over which ΔP varies from −11 to +6 times 10⁻⁷ days. Over this same range, the E₀ varies from 2451669.0574 to 2451669.0565 and P_post varies from 0.61836060 to 0.61836122 days.

All this work has largely been aimed at getting the one number for CI Aql of ΔP = −0.00000037 days. The 1σ error range is −3.7^+9.2 _{− 7.3} × 10⁻⁷ days. This measure has an accuracy of close to 0.00000085/P days = 1.4 × 10⁻⁶, or 1.4 parts per million. This period change across the 2000 eruption is very small and consistent with zero. The slightly preferred negative value of ΔP implies that P_post < P_pre, which is to say that the orbital period of CI Aql decreased across the 2000 eruption.

The dominant uncertainty comes from not knowing the exact curvature of the O − C curve between eruptions. That is, we can have the long-term shape roughly reproduced with concave-up parabolas plus period decreases through each eruption or by concave-down parabolas plus period increases through each eruption. At this point all old archival sources have been exhaustively examined, so the only way to improve the measure of ΔP is to better constrain the $\dot{P}$ measure (by accumulating more eclipse timings).

The global best fit is displayed as the thick curved line in Figures 10–12. To be explicit, the equation for this best-fit ephemeris is

$$\begin{eqnarray} T_{{\rm eph}}&=&2451669.0570 + 0.61836092 N\nonumber\\ && - 0.5 N^2 12\times 10^{-11}, N>0, \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} T_{{\rm eph}}&=&2451669.0570 + 0.61836129 N \nonumber\\ &&- 0.5 N^2 12\times 10^{-11}, -35045<N<0, \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} T_{{\rm eph}}&=&2451669.0570 + 0.61836587 (N+35045)\nonumber\\ && - 0.5 (N+35045)^2 12\times 10^{-11}, N<-35045. \end{eqnarray} \tag{ 8 }$$

The first part covers the time after the 2000 eruption, the second part from 1941 to 2000, and the third part from 1917 to 1941. This equation is applicable only when CI Aql is in quiescence.

Zoom In Zoom Out Reset image size

To illustrate the extreme acceptable solutions, Figures 10–12 also display the model for the $\dot{P}=0$, as well as for the largest acceptable curvature. To be explicit, the zero curvature model has parameters $\dot{P}=0$, ΔP = −13 × 10⁻⁷ days, P_post = 0.61836051 days, E₀ = 2451669.0575, for eruptions in 1917, 1941, 1960, 1980, and 2000. The largest acceptable curvature model has $\dot{P}=-20\times 10^{-11}$ days cycle⁻¹, ΔP = +5.5 × 10⁻⁷ days, P_post = 0.61836122 days, E₀ = 2451669.0565, for eruptions in 1917, 1941, 1960, 1980, and 2000.

In all, I now have a good O − C curve for CI Aql from 1926 to 2009, and from this I get a confident value for the period change across the 2000 eruption (−3.7^+9.2 _{− 7.3} × 10⁻⁷ days). While the best estimate is for a negative ΔP (the period decreasing across the eruption), the error bars allow for a zero or positive value.

I have 29 eclipse times from 2001 May to 2009 July (see Table 8). The O − C curve is plotted in Figure 13. This shows the best-measured period as well as the curvature caused by any $\dot{P}$ term, all with no complication of period changes across eruptions.

Zoom In Zoom Out Reset image size

Again, we see that the scatter of the best-measured eclipse times around any smooth curve is substantially larger than the quoted error bars. I take this to mean that U Sco has some intrinsic scatter in the stability of its eclipse times, likely associated with the ubiquitous variability of the disk which will tilt the light curve outside of the mid-eclipse interval. To account for this intrinsic variability, I will add the quoted measurement uncertainty (see Table 8) in quadrature with some constant intrinsic uncertainty so as to get the total uncertainty. This total uncertainty is what will be used in chi-square fits to the O − C curve. I evaluate the intrinsic uncertainty as that value which yields a reduced chi-square equal to unity in the best line fit for the 2001–2009 interval. I find that the intrinsic error has its 1σ equal to 0.00242 days (3.5 minutes). With this comes the realization that there is little utility in measuring the eclipse times to better than roughly one-minute accuracy, also that the bigger telescopes (with better photometric accuracy and higher time resolution) are not needed.

With the intrinsic error of 0.00242 days added in, the best-fit line for the 29 eclipse times (hence 27 degrees of freedom) produces a chi-square of 27.0. This best fit has the post-eruption period equal to P_post = 1.23054695 days and the heliocentric Julian date of the epoch equal to E₀ = 2451234.5387. This epoch was chosen as being at the time of the start of the 1999 eruption, when the system lost most of the ejected mass. To be explicit, the linear model

$$\begin{equation} T_{{\rm model}}=2451234.5387 + N \times 1.23054695 \end{equation} \tag{ 9 }$$

will be used for all calculations of O − C for U Sco.

The O − C curve displayed in Figure 13 does not show any obvious curvature. Nevertheless, the orbital period of U Sco in quiescence must change at some level, if only from conservative mass transfer. The O − C curve for 2001–2009 can be used to derive the best-fit $\dot{P}$ with no complication from the sudden period change across eruptions.

The best-fit parabola to the 29 eclipse times from 2001 to 2009 has a chi-square of 26.75, which is only slightly better than a straight line. This immediately tells us that $\dot{P}$ is near zero and could either be positive or negative. This best-fit parabola is with $\dot{P}=-110\times 10^{-11}$ days cycle⁻¹, P_post = 1.23054915 days, and E₀ = 2451234.5369. At the 1σ level, the smallest acceptable value of $\dot{P}$ is −330 × 10⁻¹¹ days cycle⁻¹, with P_post = 1.23055339 days and E₀ = 2451234.5335. At the 1σ level, the largest acceptable value of $\dot{P}$ is +110 × 10⁻¹¹ days cycle⁻¹, with P_post = 1.23054488 days and E₀ = 2451234.5405. So, we can express $\dot{P}$ as (−110 ± 220) × 10⁻¹¹ days cycle⁻¹.

For conservative mass transfer, the limit on $\dot{P}$ is given by Equation (B9) and depends on the masses of the two stars and the accretion rate. Hachisu et al. (2000a) have analyzed the 1999 eruption and conclude that M_WD = 1.37 ± 0.01 M_☉, M_comp = 1.5 M_☉ (with 0.8–2.0 M_☉ being acceptable), and $\dot{M}\sim 2.5 \times 10^{-7}$ M_☉ yr⁻¹. Fortunately, U Sco is a double-lined spectroscopic and eclipsing binary, and Thoroughgood et al. (2001) have measured a radial velocity curve for both stars, concluding that M_comp = 0.88 ± 0.17 M_☉. With this, the theoretical value for no angular momentum loss is $\dot{P}= +126^{+85}_{-57} \times 10^{-11}$ days cycle⁻¹. Pushing all the values to their extremes and realizing that we only have a limit (because the angular momentum losses are unknown), all I can get is $\dot{P}< +211 \times 10^{-11}$ days cycle⁻¹.

Both ranges are comparable in size, similar in magnitude, and one is contained entirely within the other. The best-fit curvature term has $\dot{P} = (-110\pm 220)\times 10^{-11}$ days cycle⁻¹. I will treat the steady period change as a free parameter within the range −330 to +110 times 10⁻¹¹ days cycle⁻¹.

The O − C diagram for the 2001–2009 interval is well determined, and we can extrapolate the ephemeris to the times of the 1999 and 2010 eruptions with negligible error. With the best fit from the previous section, I have

$$\begin{eqnarray} T_{{\rm eph}}&=&2451234.5369 + 1.23054915 N\nonumber\\ &&- 0.5 N^2 110\times 10^{-11}, N>0. \end{eqnarray} \tag{ 10 }$$

We see from Figure 13 that the uncertainties in the O − C for the ephemeris are around ±0.0015, depending on the curvature. With this uncertainty in mind, Equation (10) is a reasonable representation, and thus the O − C values in Table 8 are good for the eclipse times in the tails of the 1999 and 2010 eruptions. With this, I have plotted the O − C values for the eclipse times during the tails of the eruptions in Figure 14. We see a strong roughly linear trend, where the eclipses within 30 days of the peak (i.e., on the long plateau) are early by up to 0.013 days (19 minutes), while the eclipses late in the tail are late by up to the same amount. This trend is highly significant (although with substantial scatter) and is much larger than the uncertainty associated with the exact degree of curvature.

Zoom In Zoom Out Reset image size

The orbital period of U Sco cannot change fast enough to explain the variations seen in Figure 14. So the explanation undoubtedly arises from a shift of the center of light in the U Sco system as the eruption progresses. This is an expected result of changes in the relative brightness of the emission regions centered on the WD and those off the line connecting the stellar centers (like from the hot spot and structure in the accreting material). I would hope that someone can construct a detailed model of the U Sco eruption that reproduces and explains the variations shown in Figure 14.

The large variations of eclipse times from the quiescent ephemeris means that these times should not be combined with the quiescent eclipse times. The deviations from zero in Figure 14 are greatly larger than those in Figure 13. So any naive combination of quiescence and eruption times will be dominated by the offsets during the eruption, and any derived period changes (e.g., Matsumoto et al. 2003) must certainly be wrong.

With the base period for the O − C curve and the observed constraints on the $\dot{P}$, I will now use all the 46 non-eruption eclipse times from 1987 to 2009 to determine the sudden period change across the 1999 eruption. The O − C diagram for these eclipse times is in Figure 15. For this, I will use a chi-square analysis on the eclipse times when compared against a model with a constant $\dot{P}$ throughout and a sharp ΔP at the time of the 1999 eruption. The $\dot{P}$ value will be kept inside the 1σ constraints from Section 7.2. The total uncertainty in the eclipse times will be the quoted measurement uncertainty (from Table 8) added in quadrature with 0.00242 days (for the systematic jitter in the eclipse times).

Zoom In Zoom Out Reset image size

An early report on the ΔP value from my eclipse times (Martin et al. 2011) quotes a negative value, with this being hard to understand because ΔP_drag must be negligibly small. For this early report, I had not appreciated the effects of allowing for a non-zero $\dot{P}$. That is, I had taken the O − C curve in Figure 15, and fitted straight line segments (assuming $\dot{P}=0$) with a break in 1999, and seen that the pre-eruption points fall significantly below the O − C = 0 axis, and then concluded that ΔP must be significantly negative. However, with a substantial concave-down parabola shape between eruptions, we can get a positive ΔP. With this, the sign of ΔP is ambiguous. As such, the mechanism proposed by Martin et al. (2011) is now required not, it is certainly operating at some level, and it might yet be the dominating effect.

Let me report the chi-squares for the 1989–2009 eclipse times for the best and extreme fits as derived from the 2001 to 2009 data alone (Section 7.2). The best fit (with $\dot{P}=-110\times 10^{-11}$ days cycle⁻¹) has a chi-square equal to 60.9 (for 42 degrees of freedom). For one extreme allowed curvature (with $\dot{P}=+110\times 10^{-11}$ days cycle⁻¹), the chi-square is 64.9. For the other extreme allowed curvature (with $\dot{P}=-330\times 10^{-11}$ days cycle⁻¹), the chi-square is 60.4.

The overall best fit for 1989–2009 is for $\dot{P}=-250\times 10^{-11}$ days cycle⁻¹, with a chi-square of 60.2. For this best fit, ΔP = +43 × 10⁻⁷ days. Again, the best fits have ΔP and $\dot{P}$ strongly correlated, so that as $\dot{P}$ gets increasingly negative, the ΔP value gets increasingly positive. The 1σ range for ΔP (i.e., the values over which the chi-square can be less than 61.2) will correspond to a range of $\dot{P}$. For the most positive $\dot{P}$ (−80 × 10⁻¹¹ days cycle⁻¹), the chi-square is 61.2 for ΔP = −24 × 10⁻⁷ days. For the most negative $\dot{P}$ (−430 × 10⁻¹¹ days cycle⁻¹), the chi-square is 61.2 for ΔP = +114 × 10⁻⁷ days. Figure 15 displays these three fits as curves, with the best-fit curve as the thick line.

These fits from 1989 to 2009 represent my best answer for ΔP across the eruption. I find that ΔP = (+ 43 ± 69) × 10⁻⁷ days as $\dot{P}=(-250 \pm 170) \times 10^{-11}$ days cycle⁻¹. The best ΔP value is positive, but the value is small and the uncertainty is such that it might be zero or even negative. The $\dot{P}$ value is negative, which implies that the $\dot{J}$ term is significant.

The eclipse in the tail of the 1945 eruption (see Figure 9) provides a good time for the O − C diagram (see Figure 16). The light curve might be poorly sampled by the standards of my modern time series photometry, but the data are more than good enough to show the existence of the eclipse and to determine the time of minimum to ±0.009 days. The eclipse is in the tail of an eruption, so it should suffer some systematic offset relative to an ephemeris for eclipses during quiescence. The eclipse minimum is from plates taken 34 days after the 1945 peak, so from Figure 14, we see that the offset should be near zero. The uncertainty in this zero offset is around ±0.005 days, with this being around half the size of the 1σ error bar for the eclipse time. Thus, to all needed accuracy, the quoted eclipse time can be directly compared with the ephemeris from quiescence.

Figure 16 shows the back extrapolation of the best-fit ephemeris as based on the 1989–2009 non-eruption eclipse times. The predicted eclipse time for 1945 is earlier than the observed eclipse time by roughly 0.2 days (around 5 hr). The early predicted time (with a phase of 0.84) is completely rejected by the 1945 light curve (Figure 9). The 1945 eclipse time disagrees with the 1989–2009 best fit at close to the 3σ confidence level.

Zoom In Zoom Out Reset image size

To within the errors, the 1945 point has a near-zero O − C value, which implies that the long-term average orbital period (from 1945 to 2009) is nearly the same as the orbital period from 2001 to 2009 (that interval used to select the period for constructing the O − C diagram). That is, the period from 2005 is close to the overall period going back in time. With the analysis from Section 5.3, this means that the steady period change between eruption must closely balance out the abrupt period change for each eruption. The period change throughout an inter-eruption interval with ΔN cycles will be $\Delta N \dot{P}$. This must be balanced by the period change across any one eruption, so $\Delta P = -\Delta N \dot{P}$. We expect the $\dot{P}$ to change, so a better notation would be $\langle \dot{P} \rangle$ to indicate the long-term average of the period change. Thus, $\langle \dot{P} \rangle =-\Delta P/\Delta N$. For the average inter-eruption interval of near 10 years, ΔN will be close to 3000. For our best fit ΔP = +43 × 10⁻⁷ days, we get $\langle \dot{P} \rangle =-140\times 10^{-11}$ days cycle⁻¹. This long-term period change is nearly identical with the best-fit curvature for the 2001–2009 eclipse times alone (Section 7.2).

We now have the case that the long-term average period change nearly equals the best-fit period change from 2001 to 2009, and is somewhat different from the best-fit period change from 1989 to 2009. This merely indicates, with no surprise, that the $\dot{P}$ changes over the years. Schaefer et al. (2010a) calculate a quantity proportional to the average $\dot{M}$ (as based on the B-band flux) and demonstrate variations by a factor of 2.0. From Equation (B8), these changes in $\dot{M}$ will translate into corresponding changes in $\dot{P}$. With this, we expect that curvatures will vary from 1945 to 2009, even though the ΔP values could remain constant. Figure 17 shows the difference between a steady $\dot{P}$ case and a variable $\dot{P}$ case. We do not have enough information to measure both $\dot{M}$ and $\dot{J}$ throughout the various inter-eruption intervals, so we cannot construct any approximate O − C curve back to 1945. Nevertheless, the required $\langle \dot{P} \rangle$ (for the observed ΔP from 1999) is the same as the observed $\dot{P}$ from 2001 to 2009, so the 1945 eclipse time is plausible for the expected situation. In all, the 1945 eclipse time is consistent with the idea that $\dot{P}$ varies somewhat as expected, yet the ΔP value is constant.

Zoom In Zoom Out Reset image size

I have a good measure of the orbital period change across the 1999 eruption. (The measure of ΔP for the 2000 eruption of CI Aql is better because there are more points and the intrinsic timing jitter is smaller.) My best-fit value and the 1σ uncertainty is ΔP = (+ 43 ± 69) × 10⁻⁷ days. Note that the value is within 1σ of zero.

The ΔP value depends primarily on the curvature in the O − C diagram. The best-fit and extreme values of ΔP correspond to the best-fit and extreme values of $\dot{P}=(-250 \pm 170) \times 10^{-11}$ days cycle⁻¹. In retrospect, this degeneracy is caused by the lack of eclipses from 1997 May to 2001 May, which is the interval where the models disagree the most, as can be seen in Figure 15. We do have three eclipses at the time of the 1999 eruption, but these have relatively large error bars on the eclipse times and they have offsets from the quiescent ephemeris. Correcting for the offsets as well observed in the 2010 eruption and taking a weighted average, I get an O − C value of −0.0067 ± 0.0041. This could be plotted in Figure 15. This composite O − C curve point is close to the curve with the maximal ΔP, however its error bars include the best-fit curve, and even the other extreme curve is not confidently rejected. The post-eruption gap will not be repeated for the 2010 eruption, yet a good measure must await the full analysis that would independently yield a ΔP across the 2010 eruption. In all, at this time, I know of no way to break the degeneracy between ΔP and $\dot{P}$ across the 1999 eruption.

To be explicit, let me quote the full best-fit ephemeris for the entire 1989–2009 interval:

$$\begin{eqnarray} T_{{\rm eph}}&=&2451234.5348 + 1.23055183 N \nonumber\\ &&- 0.5 N^2 250\times 10^{-11}, 0<N<3200, \end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray} T_{{\rm eph}}&=&2451234.5348 + 1.2054753 N \nonumber\\ &&- 0.5 N^2 250\times 10^{-11}, -3600<N<0. \end{eqnarray} \tag{ 12 }$$

This best-fit ephemeris is given as the thick curve in Figures 13, 15, and 16. During the tail of an eruption, there are offsets from this ephemeris by up to 0.010 days due to shifts in the center of light within the binary system. Extensions before 1987 are substantially uncertain due to the inevitable secular variations in $\dot{P}$.

In all, I have a confident measure of the period change across the 1999 eruption of U Sco, with the value being (+ 43 ± 69) × 10⁻⁷ days. This was the observational goal of my program started 24 years ago.

The basic formula for determining M_ejecta is Equation (B6). Here, I will collect the various needed system parameters. Hachisu et al. (2003) and Lederle & Kimeswenger (2003) both give the mass of the WD as 1.2 M_☉, although the mass cannot be much smaller for the system to have a recurrence timescale known to be as small as 23.6 years. Hachisu et al. (2003) place only a weak constraint on the mass of the companion star to be greater than 1.0 M_☉, while we have a strong constraint that q ≲ 5/6. M_comp = 1.0 M_☉ is consistent with all the constraints. Hachisu et al. (2003) give a = 4.25 R_☉ and R_comp = 1.69 R_☉, so that β = 0.04. There is no reason to expect that the mass ejected from the system will have a high or low specific angular momentum, so I take α = 1. With this, the best value of A from Equation (B2) is 1.02, while plausible variations in the input parameters allow A to vary over a range with ±0.06. The expansion velocity of the shell is 2000–2500 km s⁻¹ (Kiss et al. 2001). The orbital velocity is 230 km s⁻¹. The drag term in the square bracket of Equation (B6) is negligibly small at −0.016.

With the drag term being negligible, Equation (B6) becomes M_ejecta = M_WD(ΔP/P)/A. The measure of ΔP = −3.7^+9.2 _{− 7.3} × 10⁻⁷ days has its 1σ upper limit at ΔP < +5.5 × 10⁻⁷. With the best values and the 1σ upper limit on ΔP, I find that M_ejecta < 1.0 × 10⁻⁶ M_☉. This limit is insensitive to the adopted stellar masses and sizes. The 3σ upper limit (with ΔP < 24 × 10⁻⁷ days) is M_ejecta < 4.5 × 10⁻⁶ M_☉.

For U Sco, the mass of the WD must be near the Chandrasekhar mass, with Hachisu et al. (2000b) giving M_WD = 1.37 ± 0.01 M_☉. The companion star was measured by Thoroughgood et al. (2001) to have M_comp = 0.88 ± 0.17 M_☉, and this is consistent with the requirement that q ≲ 5/6. Hachisu et al. (2000b) give a = 6.87 R_☉ and R_comp = 2.66 R_☉, so that β = 0.04. These values give A = 1.11 ± 0.05 with the uncertainty dominated by M_comp. The expansion velocity of U Sco is quite high, being close to 5000 km s⁻¹ (Munari et al. 1999). The orbital velocity is 195 km s⁻¹. The drag term in the square bracket of Equation (B6) is −0.007 and completely negligible.

My measured value of ΔP is (+ 43 ± 69) × 10⁻⁷ days. With this, I find the best value to be M_ejecta = 43 × 10⁻⁷ M_☉. (The equality of M_ejecta and ΔP in these units is coincidental.) The 1σ upper limit is 110 × 10⁻⁷ M_☉, while the 3σ upper limit is 250 × 10⁻⁷ M_☉.