Time-related binary system characteristics such as orbital period, its rate of change, apsidal motion, and variable light-time delay due to a third body, are measured in two ways that can be mutually complementary. The older way is via eclipse timings, while ephemerides by simultaneous whole light and velocity curve analysis have appeared recently. Each has its advantages, for example, eclipse timings typically cover relatively long time spans while whole curves often have densely packed data within specific intervals and allow access to systemic properties that carry additional timing information. Synthesis of the two information sources can be realized in a one step process that combines several data types, with automated weighting based on their standard deviations. Simultaneous light-velocity-timing solutions treat parameters of apsidal motion and the light-time effect coherently with those of period and period change, allow the phenomena to interact iteratively, and produce parameter standard errors based on the quantity and precision of the curves and timings. The logic and mathematics of the unification algorithm are given, including computation of theoretical conjunction times as needed for generation of eclipse timing residuals. Automated determination of eclipse type, recovery from inaccurate starting ephemerides, and automated data weighting are also covered. Computational examples are given for three timing-related cases-steady period change (XY Bootis), apsidal motion (V526 Sagittarii), and the light-time effect due to a binary's reflex motion in a triple system (AR Aurigae). Solutions for all combinations of radial velocity, light curve, and eclipse timing input show consistent results, with a few minor exceptions.

Among the most accurately measured astrophysical quantities
are binary system orbit periods (P), which are most commonly
found from eclipse timings, with a reference epoch (t0) and
often also a rate of change (dP/dt) to complete an ephemeris.
The exquisitely precise periods allow detection of minute period
changes, leading to mass exchange and angular momentum
exchange estimates that are indispensable for understanding binary
system structure and evolution. We shall refer to ephemerides
based solely on eclipse timings as performed in the traditional
way. Whole light curves (lcs) have recently entered the picture
and have several advantages. For example, whole lc solutions
can utilize information from ellipsoidal (tidal) variation and the
reflection effect in addition to eclipses, and can work even in
the absence of eclipses if tides and/or reflection are sufficiently
large, thus extending ephemeris statistics to low inclinations. An
alternative of long standing, and often the only realistic
alternative in the absence of eclipses, is radial velocity (RV), although
most RV curves lack sharply defined timing ticks. Apart from
binaries with little or no photometric data, or no significant
photometric variation, the main usefulness of RV curves for
ephemerides is in filling gaps where there are no lcs or eclipse
timings, and there they can be very helpful. Eclipses are good
timing ticks, with data records that may extend over decades
or even centuries. Whole lcs are clearly the best ephemeris
resources for newly discovered binaries that lack a history of
eclipse timings. Light curve surveys such as KEPLER4 and
COROT

torically known binary might have a modest number of eclipse
timings and a few cycles of lcs and/or RV curves. Whole lc
analyses are usually tied to full physical models, so departures
from symmetry due to effects such as magnetic cool spots,
accretion hot spots, and eccentric orbits may be taken into account.
A practical point is that whole lcs usually contain far more
points, and thus more information or weight, than RV curves
or the lc segments typically used for eclipse timings. Multiple
data combinations and solution options can show where the best
ephemeris information resides in a given case, through standard
error estimates or numerical experiments. Helpful EB timing
catalogs are provided by

The idea of simultaneous light and RV curve s

2001) cover many of these contributions and still others might
have been covered

Unification needs to be done in a logically consistent way
with proper weighting across the several data types, which
here is accomplished iteratively within the overall process (see
Section 5.2). Applications to real and synthetic binaries are
described in Sections 5.3 and 6?8, including comparisons of
results from the three data types in all combinations. However,
these experiments are only a small start on those needed to
gain experience with unification. Our examples are not chosen
for having exceptionally good data or timewise coverage but
to exhibit particular features such as a large and steady rate of
period change, apsidal motion, or the light-time effect. Timings
can guide lc and RV results by leading the way to more accurate
6 According to its user?s guide, the FOTEL program by

The differential corrections (DC) solution algorithm applied
here is fast, usually converges well, and generates standard
errors based on quantity and precision of the various data
sources. Its basics are c

Conjunction10 times would be fundamental time markers if they could be measured, but the required geometric resolution does not come from lcs or any present observational alternative. Accordingly, observers adopt times of minimum light in eclipses as proxies for conjunction times. For most EBs, times of eclipse minima are very good such proxies since lc asymmetries due to orbital eccentricity, surface brightness irregularities, and surface geometry are usually quite small. Actually, nearly all measured timings are derived by estimating the time coordinate of an idealized vertical symmetry axis (of an eclipse lc) by various analytic or graphical means. That time then serves as a proxy for the time of minimum, which itself is a proxy for conjunction time (a proxy for a proxy). A direct strategy is to adopt conjunction as the theoretical time marker since there is no issue concerning exactly what it means, and that is what we do. Any rare cases where the observational proxies are judged to need asymmetry corrections can be handled by applying such corrections. None of our demonstration binaries have that problem. If asymmetric lc data for a questionable observed timing are at hand, they can simply be entered into the unified ephemeris algorithm as a lc and the previously derived eclipse timing ignored.

For a given binary, each data type or type combination can
give an ephemeris for its covered time span if that span is
sufficiently long. P and dP/dt may change significantly over
years and shorter times, so perspective is gained by evaluation
of a variety of combinations. A unified solution algorithm can
be well suited to that strategy. Note that ephemeris changes
can significantly affect other results. Treatment of a long
data record as one large dataset (as here) may not be the
best path to parameters (ephemeris or non-ephemeris) when
non-steady period changes are evident or suspected. Better
may be timewise separation into eras of coherent behavior,
still via unified solutions, although that option may not often
be realistic, given existing data records. The exercises of
Sections 6?8 are not intended as optimal from a standpoint
of results, but as a way to gain strategic perspective on the
interactions between ephemeris parameters and other system
parameters. One can ask if inclusion of a second period
9 Algorithms other than DC, such as Steepest Descent and Simplex, can be
applied to whole curve ephemeris solutions, as in

The reader may notice that we avoid the widely used terms ?O-C diagram? and ?O-C analysis,? which we consider uninformative. While it may be clear that O means ?observed? and C means ?computed,? O-C diagrams and analyses occur in all physical sciences in virtually limitless contexts, most being unrelated to eclipse timings. Accordingly, terms such as ?eclipse timing residual? (or just ?timing residual? for short) and ?timing diagram? are much more usefully descriptive, especially in connection with areas such as stellar structure and evolution where there are many applications for ephemeris results and many other kinds of residuals.

Three EBs are run through the DC algorithm, each being a typical example of a timing-related phenomenon (steady period change; apsidal motion; variable light time). An all-data solution (RV, lc, timings) begins each experiment, followed by solutions for the six possible data type subsets. Evaluated quantities are binary or triple system parameters that can reasonably be determined for each example and timing-related parameters that are directly relevant to each case. Starting parameter estimates for the subsets were dithered around the all-data values by amounts of the order of 5? .

One may ask why so many data type combinations are processed here for each binary, when all-data solutions utilize the full set of input information and logically give a better hold on parameters than any data subset. The reason is that although our examples have reasonably good RVs, lcs, and eclipse timings, that will not be true for most binaries of potential interest. The primary purpose here, after specification of unification?s core ideas, logic, and mathematics, is to explore realistic possibilities, including examples for which only one or two of the data types exist. It is not expected that typical future applications will solve multiple subsets of available data, but will simply include whatever useful datasets exist. Further experience along these lines will come with such examples, but the idea for the moment is to explore consequences of data type restrictions via intentional data deletions.

The key points of interest for this paper?s numerical results concern intercomparisons of the seven parameter combinations for each binary in regard to P0, dP/dt, d?/dt , and 3b light time. Are results entirely consistent among the seven data combinations, do they show rampant disagreements for one, two, or all three of the binaries, or does reality lie somewhere between these extremes? Although the DC computer program has been thoroughly checked for errors in several ways, mistakes are always possible. Data shortcomings are also possible. For example, RV defects might have only subtle effects in an all-data solution where they are outweighed by more numerous lc and timing data, but stand out in an RV-only solution. Such problems will eventually be sorted out as new observations come along. In overview, the data subset solutions are mostly consistent with the all-data solutions within ranges expected from the formal uncertainties, while the subset uncertainties are typically larger than the all-data uncertainties.

4. EPHEMERIS AND CONJUNCTION COMPUTATIONS Traditional eclipse timing analysis is phenomenological rather than astrophysical, as the computed (i.e., theoretical) times of minima are not from binary star theory but from whole cycle advances in a phase to time formula. Although the events are known to be due to eclipses, the procedure can be (and almost always is) applied without use of that knowledge. The data could be analyzed in the traditional way by someone who did not know they were EB observations, with no effect on the results. Asymmetric eclipses might need timing corrections based on an EB model, but published examples where such corrections were applied are essentially non-existent. The theoretical times in the development below are star?star conjunction times generated from two body orbital motion, with possible apsidal motion and a 3b light-time effect. We designate superior conjunctions of component 1 as type I and its inferior conjunctions as type II, with the star labels (1 and 2) having been assigned. A common practice is to refer to primary and secondary (rather than I and II) minima, although that usage has led to occasional confusion since the designated primary star may be the one covered at secondary (less deep) eclipse.

Our star?star conjunction phases correspond to theoretical
conjunction times that are differenced with minimum times
to produce timing residuals for DC solutions. The phases are
understood to include whole cycles, so their conceptual range
is unlimited, rather than the often seen 0?1 range.11 An eclipse
minimum usually can be assumed to coincide closely with a
conjunction, although very small corrections might usefully be
applied to timings of significantly asymmetric eclipses. A theory
of such asymmetry corrections is specific to any applied scheme
for estimation (not utilization) of minimum times and is beyond
the scope of this paper. That is to say, we assume that the input
timings are published or otherwise known and that they closely
correspond to conjunction times. The orbit period parameter
is denoted by P0, with subscript 0 on any quantity referring
to its value at the ephemeris reference time, t0. For a circular
orbit, one can generate a phase, ?min, from an observed time of
light minimum, tmin, given momentary estimates of ephemeris
parameters t0, P0, and dP/dt. Then simply round ?min to an
integer phase for a computed ?I estimate and add 0.5 if it is a
?II estimate. Phase to time conversion then gives a theoretical
minimum time. The phase from time equation and its inverse
(time from phase) are, respectively

P0 dP/dt

P0
dP/dt
+ t0,
(1)
(2)
and
11 For time to phase and phase to time mathematics and a brief historical
account see

Phase in the Wilson?Devinney (W?D) binary model (

3?

Mconj = Econj ? e sin Econj, and with

?conj2 = 2 ? ?, respectively. Kepler?s equation, applied at conjunction, Econj = 2 arctan 1 ? e 1/2 1 + e tan ?conj 2 , then gives the mean anomaly of superior or inferior conjunction of star 1, Mconj, which is phase-like except for its angular unit (radians) and its starting point at periastron. The relative conjunction phase is given by ??conj =

2?

Here ??conj is the part of ?conj that lies within a cycle (i.e., without the whole cycle part). These equations result in phased curves that are referenced to a definite direction (the periastron direction for ? = ?/2) rather than to the changing periastron direction that goes with apsidal motion. Accordingly, modeled eclipses move in the same way in phase as those of a real binary as ? changes, instead of having only secondary eclipse move relative to primary eclipse as seen in many publications. For any ?, the equations give respective I and II ??conjs of 0.0 and 0.5 in the limiting case of vanishing eccentricity. Computed conjunction times are formed in the same way as for circular orbits except that the shifts are not, in general, the half-cycle shift of the circular II case or the null shift of the circular I case, but are given by Equations (3)?(7) for momentarily applicable e and ?. Note that individual cycles are not necessarily bounded by primary conjunction times, as in the circular case, but by times corresponding to integer phase.

A previously determined eclipse type can be adopted for each
timing if there is no doubt as to its correctness, but an automated
way to distinguish superior from inferior conjunctions can be
very helpful. Whether a minimum is type I or II can be decided
by reducing its phase to the range 0?1 and comparing the result
with the phases of type I and II conjunctions that follow from
12 These mathematical relations have been reprinted in the textbook by

A reasonably correct ephemeris (perhaps including dP/dt and a light-time term) is needed, otherwise an anomaly will become obvious in the run of residuals as time extends well beyond the reference time. A trend to more positive or negative residuals will appear until there is a large abrupt jump, which is the signature of an ephemeris with insufficient validity range. Some timing residuals, especially ones far from the reference time, can then be off by orders of magnitude, and perhaps even correspond to the wrong eclipse (type I versus type II). A solution could get into permanent trouble if there were no escape process. This danger is especially serious for pure timing solutions since neither RVs nor lcs can then help to straighten out the problem. We find that convergence failure occurs only when a substantial fraction of the minima are incorrectly evaluated as to eclipse type. The remedy is to have DC ignore minima when the type computed from the ephemeris disagrees with the designated type. The revised ephemeris from the next iteration should have a greater applicable range, perhaps including all observed minima. This ?clip and solve? process may be repeated until all observed timings have small residuals, and has been automated for our solutions. At succeeding iterations, the remaining timings improve the initial ephemeris enough so that fewer timings need be ignored in the following iteration, then still fewer in the next, until all the timings are utilized. Occasional disagreements due to wrong I versus II assignments (as opposed to an inaccurate ephemeris) will be obvious from residuals that continue to be off by much of a cycle.

A phenomenon that can mimic period variation when only
partial timing information is at hand is the 3b light-time effect,
while another is apsidal motion. A 3b light-time effect has been
in the W?D model for several years

Apsidal motion is often incorrectly called orbit precession, most commonly in regard to the planet Mercury?s historically famous advance of perihelion, but apsidal motion really is orbit rotation. Of course precession involves two planes whereas apsidal motion involves only one. Apsidal motion can be included in solutions via parameter d?/dt , the time derivative of the argument of periastron.

Once the data have been collected into computer files, processing time is short, so all data type combinations can easily be done and results intercompared. Typical future applications may do only all-data solutions, but the option to input several datatype combinations is there. Of course a dataset can be broken into subsets if it shows essentially discontinuous timewise behavior. However, unification offers advantages even with input of eclipse timings only, as it deals coherently with an ephemeris (t0, P, and dP/dt), with apsidal motion (d?/dt ), and with the six parameters of the light-time effect, so a solution for the three phenomena allows the full parameter set to interact.

Input of many timings does not significantly increase DC run time because the conjunction and ephemeris computations of Section 4 and the corresponding partial derivatives, ?t /?t0, ?t /?P , ?t /?(dP/dt), needed by DC have simple analytic expressions. RV points take as long as lc points individually but are usually less numerous by factors of the order of 10?1000.

Simultane

Regardless of applied solution algorithms, true parameter
values for most astrophysical systems are unknown and estimated
only indirectly, so the only way to know if DC recovers correct
numbers is through solutions of synthetic data generated from
known input parameters by a computational model.
Accordingly, synthetic data with simulated noise were generated and
run through the solution process by both authors so as to gain
experience and compare parameter results from the DC program
with values assigned in RV, lc, and timing data generation. Some
of the early experiments uncovered problems that required
fixing but the fixes did succeed, as shown by good convergence in
13 For download of the W?D 2013 version (FORTRAN programs,
documentation, sample input data), go to the anonymous FTP site
ftp.astro.ufl.edu, then to subdirectory pub/wilson/lcdc2013.
subsequent experiments to answers within expected statistical
ranges corresponding to computed standard errors.
6. NEARLY STEADY PERIOD CHANGE: XY BOOTIS
Unification of timing information has partly been stimulated
by the general problem of W Ursae Majoris (W UMa) type
binaries, where matter exchange internal to a common
envelope leads to orbital period behavior that may be approximately
steady for decades but more commonly is erratic. These
variations are one of the few diagnostics of the inner workings of
common convective envelopes, and good statistics are needed if
their messages are to be understood. XY Boo is among a small
minority of W UMa type EBs in the KKN and Gateway timing
diagram catalogs with comparatively uncomplicated parabolas
that indicate an approximately constant rate of period change
over many years. Such behavior is a likely indicator of steady
mass transfer, which can be a means to understand structure
and interactions within W UMa common envelopes. RV curves

The Awadalla & Yamasaki (1984, A&Y) data have many repeating points, with as many as eight consecutive identical -0.1

0 g a m B ijk 0.1 d n e n n iB 0.2 0.3 -0.4 -0.3 g am -0.2 B r e l ikn -0.1 W phase magnitude entries. As no explanation for this unusual circumstance is given by A&Y, we tried to learn the reason from the authors and did receive thoughtful replies from Professor Yamasaki, but without essential success in the end. We felt that the observations should be included in our solutions despite this apparent problem since they are from a time with no other lcs and no RVs. Accordingly, single points were made from the repeating points by averaging their times. The resulting data points agree in form with Binnendijk?s data in B, although not in V at some phases, with both maxima higher than Binnendijk?s and )c 300 e s / m (k 200 s e iit lco 100 e V l ida 0 a R h itc -100 d li H & -200 n a e L cM -300 -0.2 0 0.2

0.4 phase 0.6 0.8 some disagreement in the form of the V eclipses. Details can be seen in Figures 1?3. Such epoch to epoch mismatches are seen in many W UMa lcs and usually ascribed to magnetic spot activity. A solution without the A&Y curves was run but changes from the all-data solution are small, so that solution is not tabulated. Notably the standard error for dP/dt is sensibly smaller with the A&Y observations included, which underscores the importance of filling timewise gaps. The Winkler B and V points that supposedly are at HJD 2,442,557.6711 are out of time sequence and were not entered into our solutions since a misprint is likely. Binnendijk and A&Y used the same comparison star, although Winkler had another one. The differing comparison stars are not a significant issue, as all the lcs were entered separately and therefore have individually adjusted magnitude zero points. McLean & Hilditch have +211 ± 7 km s?1 for XY Boo?s systemic RV (V? ), which is an obvious misprint, as can be seen in their plot and our Figure 4. We find a value of about +14 km s?1 for V? .

Although mass ratios often can be accurately determined from
lcs of overcontact binaries that have total-annular eclipses, they
are not well determined photometrically at low inclinations.
Actually, a dramatic drop in mass ratio accuracy occurs in going
from total-annular to partial eclipses, as shown by solutions of
synthetic W UMa type EBs that have known mass ratios

Table 3 allows comparisons for consistency of the seven combinations of the three input data types. Table 4 lists auxiliary and absolute parameters for the all-data solution. Star 1 is the larger and hotter one that is eclipsed around phase zero. The timings have a significant influence on the ephemeris, as expected from their much longer time base and the likelihood of ephemeris changes over those 53 yr. Only a few timings are prior to the whole curve data, as shown by Figure 5, so just one or two bad ones from those times could be influential. The case to case ephemeris differences are not really large, but are outside the range expected from the standard errors. The timing diagrams show essentially steady period change except for three points, two of which (at JDs 2,444,691.3860 and 2,444,716.3900) lie off the otherwise well established curve and may be mistakes. The earliest points (JDs 2,435,622.4578 and 2,435,627.4596) lack neighbors, so whether they show a real departure from steady behavior is not clear. The oscillation from about JD 2,448,000 to 2,455,000 covers only one cycle so there is no indication of periodicity. Redistribution of matter within the common envelope is a likely cause of the oscillation and would be quite ordinary in a W UMa type binary.

The RV parameters a and V? may seem to disagree among
our five solutions that involve velocities, but they agree within
expectations based on the statistical uncertainties. The

Julian Date - 2,400,000 50000 55000 40000 45000

Julian Date - 2,400,000 50000 55000 9 d ? n 0 a ) 1 ) 1 2 ly 1 (1 × 10 n 8 2 8 7 2 O .. .. 27. .. 1 . 00 595 .02 .. .. .. .. .. .. .l isem . . 66 . 227. .. .0±570 0±. . . . . . tlea T 65 30. 279 lre .53 .1 reT 0 ( 9 ? 0 ) 1 f 2 o 9 9 9 2 ( ×8 5 3 8 2 3 V lnyO ... ... .130 .88 .0001 ... .0000 98859 .0307 .0001 .0001 .0003 .0001 .0002 .0002 llB?.no a i lc ± ± ± ±05 ± ± ± ± ± ± ± re tu .25 .267 7125 1652 307. .4917 .0683 .0083 .0683 .0083 .0683 .0083 teohv tlsao 66 61 .2 .3 0 0 0 0 0 0 a 0 on l-d T B se

a Brackets indicate equal-volume radii, with r = R/a. b The fill-out fraction in terms of the common-surface and critical potentials at the L1 and L2 Lagrangian points, (?L1 ? ?)/(?L1 ? ?L2), is 0.425.

XY Boo RV data have been published. Although neither the RV nor lc data alone allow accurate determination of the mass ratio, the standard errors suggest that the two sets acting together may give at least somewhat reliable results. Of course, an attempt to solve for P0 or dP/dt over the 0.2 days of RV data would be meaningless, so only a, V? , mass ratio, and a reference epoch are in the RV-only solution. However, a solution that adjusts the reference time for a short stretch of data (but not P0 or dP/dt) can give the analog of an eclipse timing without an eclipse.

7. APSIDAL MOTION: V526 SAGITTARII

V526 Sgr is a well-known eccentric-orbit (e ? 0.22) detached
EB with spectral types B9.5 and A2 in a 1.919 day orbit
that rotates with an apsidal period of 155 yr. The many
eclipse timings dating back to 1895 are well distributed in
time

Solution experiments were similar to those for XY Boo, with
results in Table 6. Masses and radii for the all-data solution
are in Table 7. Individual eclipse timing weights are assigned
inversely proportional to the square of the errors listed in the
two cited timing papers (Table 5). All solutions that include lcs
adjusted the rotation parameters F1 and F2, which are rati

Solutions that include third light are needed to fit multiple lcs
well, confirming the presence of extra light from stars B and C in

Our experience with the solutions is similar to that for
XY Boo. Convergence is excellent for data sets that include
eclipse timings, with ephemeris and apsidal motion parameters
retrieved quickly as part of the solution. Convergence was
aided by the method of multiple subsets

Time Range

(JD) 2,437,548 to 2,438,297 2,447,378 to 2,448,071 2,445,273 to 2,449,489 2,413,552 to 2,453,125 , t lyn 044 (511 9× 050 0 550 ilg 0 h O . . . . . . . 00 116 .07 .. .. .. .. .. .. .. .. .. .. .0 .3 .0 rd se .. .. .. .. .. .. .. .0 4 0 . . . . . . . . . . 0 0 0 ih m i T lc 1 .11 1 .022 .014 00290 ()33 01? 004 .004 .004 .004 .003 .003 .003 .003 .14 .25 .000 .76 .016 rfno 7 2 5 3 6 2 3 3 4 4 m ly 1 0 1 0 0 6 .0 46 10 .0 0 0 0 0 0 0 0 0 0 0 0 0 e nO .902 ... ± ± ± ±973 8 × ± ± ± ± ± ± ± ± ± ± ± ± ± tak 0 1 ± ± ±

K 2 1 .225 211 .08 0 4 1 0 , 0 1 s i , 3 7 30 ) 2 4 0 3 2 3 2 3 5 0 lc .0096 .807 .101 01 .0022 .0053 .0050 .0000 (0443 01?01 .0040 .0040 .0040 .0040 .0030 .0030 .0030 .0030 .105 .108 .0000 .080 .0017 raeo iren d 0 w b an ± ± ± ± ± ± ± ±21 × ± ± ± ± ± ± ± ± ± ± ± ± ± t m 1 4 7 1 0 4 4 3 1 6 1 6 9 2 8 4 5 3 5 2 5 nd nu RV .0289 .0107 .982 856 .661 .634 .3079 .6343 9119. 018. .6090 .7017 .6090 .7017 .1007 .0074 .1011 .0017 .30 .16 .2119 .4199 .228 seco teh 1 ? 0 0 s t l u s 6 e e R l r b g a S T 6 s 2 e 5 9.8 10.0 e d u it gn 10.2 a M Note. a Brackets indicate equal-volume radii, with r = R/a. consistent among the data combinations and among the band to band third light results.14

8. LIGHT-TIME EFFECT: AR AURIGAE

AR Aur is a detached EB consisting of two late-B stars in a
4.134 day period circular orbit. A discussion of its dimensions
14 The luminosity ratios, L1/(L1 + L2), are identical for the two B curves and
for the two V curves although the curves were entered separately. The reason is
that radiative behavior in the model is completely specified by the band
designation, and is therefore identical for the Lacy and O?Connell lcs in a
given band. Of course, most parameters (i, m2/m1, etc.) are required to be
identical for all bands. The corresponding third light parameters are not
identical, although similar (Lacy vs. O?Connell), because 3 is expressed
relative to the scaling of individual curves. Similar remarks pertain to the XY
Boo and AR Aur solutions.
and evolutionary state is in Nordstr o¨m & Johansen (1994).
Eclipse timings clearly indicate a 3b light-time effect

Figure 2 of

The

Table 8

AR Aur Data Sets

Sets U, B, V lcs u, v, b, y lcs

RVs RVs RVs RVs

RVs Eclipse timings

Time Range

(HJD) 2438057?2440248 2439096?2439933 2427701?2428116 2427901?2428451 2439795?2445040 2453671?2454091 2452516?2454876 2427887?2455640

Number 271, 686, 688 176, 181, 181, 154 10, 10 35, 29 6, 6 16, 16 85, 85 71 0.0 0.2 0.4 0.6 0.8

1.0
PHASE
a star?s face by its eclipsing companion, thereby reducing the
usual RV averaging. DC quickly converged to definite
subsynchronous F1 and F2 values of approximately 0.3 and 0.2,
presumably based mainly on the (rather small) RM effects.
Complete results for the all-data solution with F1, F2 adjusted are
in Table 10. Computed RV curves for this asynchronous
solution are shown along with the observations in Figure 10, with
residuals in Figure 11. Eight v sin i rotational velocities from
spectral line broadening are compiled in Table 11. In aggregate,
these values, in particular the precise

Figures 9 and 10, and the RV residuals in Figure 11, reveal another puzzling discrepancy. A number of secondary RVs close to primary eclipse, and two primary RVs near secondary eclipse, are well off the solution curve with residuals from 2 to 5 km s?1, while residuals outside eclipse are typically a factor of 10 smaller. A possible explanation is suggested by AR Aur?s similar component spectral types. For simplicity, assume the component spectra to be identical except for a scale factor. At some eclipse phases, the differential Doppler shift between components will be appreciable but not large enough for line separation, and produce asymmetric blended lines. Since the two spectra have the same form, many lines will be distorted in much the same way so that even small asymmetries combine coherently, giving larger line asymmetries in the combined spectra than expected in a non-coherent example where some asymmetries ?go left? while others ?go right.? Such an effect is not part of the physical W?D model so we deleted the primary RVs at JD 2454810.6607 and JD 2454814.6692 and secondary RVs at JDs 2454775.4904, 2454779.5550, 2454779.5766, 2454783.7022, 2454783.7416, and 2454812.6515 from the DC input. -1.0 e -0.5 d u itn 0.0 g a lM 0.5 a it re 1.0 n e iffD 1.5 0.010 0.008 ) 0.006 l(d 0.004 dua 0.002 i se 0.000 gR -0.002 inm -0.004 iT -0.006 -0.008 -0.010 -0.2 0.0 0.4

0.6

Weighting for AR Aur requires more explanation than for our
other binaries because of multiple concatenated RV curves in the
input stream. RVs by

Analysis results for synchronous rotation are in Table 12,
with representative solution curves in Figures 9?12. For each of
the data set combinations, parameters that can be extracted are
those with attached standard errors. The vector length reduction
scheme of recent DC versions and the Levenberg?Marquardt

Although AR Aur?s rotation is puzzling in line broadening
versus RV measures, its solutions show no notable
inconsistencies in the inner binary parameters or the 3b parameters
except for the 3b eccentricity, which does jump around among
the seven solutions by more than its standard errors predict.
Accordingly, the AR Aur solutions can be described as nearly
mutually consistent, although not entirely so. The 3b parameters
of the all-data solution in Table 12 yield a light-time effect for
the EB of semi-amplitude 0.007619 ± 0.000031 days (11
minutes) and a third-body mass of 0.5122 ± 0.0087M , assuming
EB and third-body coplanar orbits. Figure 13 shows the wave
that is seen when the light-time effect is not in the model, and
also the essentially flat distribution of residuals that remains
when the light-time effect is modeled. Table 15 compares the
3b parameters of this paper (based on the all-data solution) with
those of

2 2 2 2 2 2 9 .0 .0 .0 5 .00 (1 0 .0 .0 .0 .0 .0 .0 .0 2 1 5 4 3 ? 0 0 0 0 0 0 0 lynO .5184 ... 0±8±0±0±562 0±456 1×0±0±0±0±0±0±0±.4390 6658 .1458 .262 .69 888 8 0 1 4 4 2 0 1 c s l u o n o r h 2 c 1 n e y l S b ( aT tls a R (k eg (K m P a V i T ? ? m t0 P0 dP L1 L1 L1 L1 L1 L1 L1 a3b P3 i3b e3b ?3b T03b Na a ( ? (d 2 1 2 /2 (Ha r

) L L L L L L L ) ) r to ?s + + + + + + + ) sy ) ) D o m1 JD (ady td/ (/L1 (/L1 (/L1 (/L1 (/L1 (/L1 (/L1 (R (adb (egd (edg (JH .tseF itrend o P 0 1 11 .3 9 o e 7 5 h 0 4 .4 22 860 d t 3 8 0

se s 4 a se 4 2 b e

h d t e t n

e p r o a d p a

n s i 2 a r 4 w e

b
0 4
4 n u
2 tio gfi
u t
l n
o a
s c
0 0
9 9 2 ) 5 5 4 5 5 5 0
8 3 2 7 1 0 0 8 0 6 6 6 6 6 6 6
lc .00 .50 .30 0 .60 .70 .00 .00 (95 1?0 .00 .00 .00 .00 .00 .00 .00 .0
o d
ta 14 67 11 54 45 014 0025 (2159 21×045 045 044 045 044 045 041 7 32 treu and
f r
a .0 .0 .0 .0 .0 .0 .0 6 .2 .0 .0 .0 .0 .0 .00 .00 .79 53 .15 .0 .5 0 ra ta
D 0 0 0 9 0 0 0 0 5 0 0 0 0 0 0 0 5 7 e s
ll 6 p ts
0
1
?
) 0
Parameter
Note. a Brackets indicate equal-volume radii. Relative radius, r, is R/a.
third body light would be part of the observed light. However,
a main sequence star of mass ?0.51 M would be an
earlyM star and have an absolute magnitude MV of, say, 8.8

9. ASSESSMENT OF RESULTS AND PROSPECTS

Results for XY Boo, V526 Sgr, and AR Aur (respectively, exemplifying steady period change, apsidal motion, and 3b light time) are mainly consistent among all-data sets and their six data subsets, with differences of up to a few ? as expected in statistical problems. Exceptions are in XY Boo?s RV-based and RV-plus-timing-based mass ratio and in its purely RV-based a and V? , where the outcomes are understandable in terms of the small number of points in the only published RV observations. However, even for XY Boo, overall consistency of the larger subsets and the all-data set is good.

Most of the work on a unified ephemeris is in collecting RV, lc, and timing data and concatenating them into a DC input file. With that stage completed, solution times (say for 10?20 iterations) are typically less than a second for timingsonly solutions, or tens of minutes if a few thousand light and/ or velocity observations are included. A huge body of binary timing observations appears each year in explicit eclipse timings as well as the implicit timing information in whole light and RV curves. The COROT and KEPLER missions are responsible for a major jump in production, while many well equipped amateurs supplement the work of professionals on both timings and whole curves. Optimal analysis of all these observations is a job for many persons, working efficiently. Although we have specified essential logical and mathematical relations for unified ephemeris solutions, we are well aware that release of the corresponding computer programs is a practical necessity for speeding applications, at least in the near term. Accordingly, public release as a version of the W?D program has high priority and mainly awaits completion of suitable documentation.

Extensive use was made of the NASA ADS Query, SIMBAD,
KKN, and O-C Gateway internet sites. The unpublished AR Aur
RVs by

PARTIAL DERIVATIVES FOR DIFFERENTIAL CORRECTIONS SOLUTIONS OF ECLIPSE TIMINGS The partial derivatives are ? tconj

? e ? tconj ? ?0

? tconj ? ?conj = ? ?conj ? e = Pconj

a Time of periastron passage preceding the time of conjunction T0 of the all-data solution in Table 12.
b Semi-amplitude of the light-time variation of the EB with respect to the triple system barycenter.
c Third body mass for co-planar EB and third body orbits (i3b = 88?. 517). The Albayrak et al. value and its error
are based on Albayrak et al.?s mass function and error, and on the absolute masses M1 = 2.480 ± 0.098 M ,
M2 = 2.294 ± 0.093 M by

Derivative ? ?conj/? (d?/dt ) is evaluated by writing ? = ?0 + (t ? t0) dt

, ? =

.

The first factor on the right is found numerically by running through Equations (3)?(7) twice, first with ?conj = ?0 + (tconj ? t0) dt

, and then with the advanced ?conj slightly incremented. The second factor on the right of Equation (A5) is found by differentiating Equation (A6) with respect to d?/dt , to get ? ?conj ? d? dt

= tconj ? t0.

Analytic forms exist for these three ?conj derivatives but numerical evaluation gives better than needed accuracy and simplifies the programming. (A4) (A5) (A6)