Eclipsing binaries (EBs) measure distance without need or use for nearby similar objects, with many applications over recent decades. EBs are now considered the most reliable and accurate distance indicators for the very important lower rungs in the cosmic distance ladder, within the Local Group of Galaxies. Among several EB distance algorithms, direct comparison of observed and theoretical fluxes is particularly straightforward, although it requires absolute flux calibrations for which only a modest number of publications exist. Here, we measure UBV RI and uvby flux calibrations and calibration ratios from astronomical objects in ways not previously tried, specifically for EBs, single stars within 80 pc, and the Sun. All of the single stars are below about 6500 K temperature. Interstellar extinction is avoided by a restriction to nearby targets. Some photometric band calibrations in the literature are accurate enough for very good EB distance determinations if star temperatures are accurately known, especially considering that estimated distance has only a square-root dependence on calibration constant, but accurate bandto-band calibration ratios are keys to the combined temperature-distance problem. Band-independent canceling factors (star radii and distances) allow calibration ratio measurement with enhanced accuracy, compared to individual band calibrations. A physical EB model with embedded stellar atmosphere emission optimally matches theory to observations for the binaries. Single star candidates are identified as reliably single if their radial velocity variations are below 100 m s?1. For the most part, we find good agreement with some of the previous calibrations and the ratios are improved.

1. ECLIPSING BINARY DISTANCE ACCURACY
Eclipsing binaries (EBs) allow accurate distance
measurement to objects as remote as the relevant observations can
be made, which today is within the Local Group of Galaxies

Therefore, continued optimization of EB distance
measurement is important to the wider distance problem. Simulations
show that, given observations of typical good accuracy, EB
distances are good to a few percent or better in the ideal case that the
4 An important literature on luminosity measurement for Cepheids based on
interferometry has developed, mostly over the last decade

The basic ideas of EB distance measurement go back at least
to

1.1. Capsule Summary of Direct Distance Estimation
The procedure of interest here is that of Wilson (2007, 2008),
called Direct Distance Estimation (DDE). DDE generates model
fluxes in standard physical units (specifically erg s?1 cm?3), with
full incorporation of all ordinary close binary effects. Given a
few reliable calibration numbers, DDE can lead to accurate and
easy distance estimates of known uncertainty. They are accurate
because spherical star approximations are entirely avoided, easy
because distance is an ordinary solution parameter, and of
known uncertainty because they are accompanied by standard
errors. The DDE algorithm has been inserted into the 2010
version5 of the Wilson?Devinney (hereafter WD) EB light
curve and RV analysis program

A brief summary of EB distance estimation via the DDE
algorithm will clarify its use for flux calibration measurements,
which are basically inverted DDE applications (Cband from
distance d rather than d from Cband). In a given standard photometric
band, one has d = d(EB parameters, Cband), where some of the
EB parameters are derivable mainly from light curves, some
mainly from RVs, some from spectral classification or
spectral modeling, and perhaps others from miscellaneous further
information. DDE inputs a mix of observed light curves
(perhaps multi-band) and RV curves, a Cband for each light curve,
and adopted parameters related to phenomena such as gravity
brightening and the reflection effect. It then converts the light
curve observations to absolute fluxes via the Cband?s, compares
the fluxes with model fluxes computed from the EB parameters,
and eventually arrives at a solution for designated parameters,
including d, by the least-squares criterion. Bandpass emission is
computed locally on each star

EB component and both components (1T and 2T solutions) and noted that 2T temperature determinations require light curves in two bands and are sensitive to Cband ratios, although derived temperatures are unaffected if the Cband?s are changed together so as to keep a fixed ratio (Section 2.1). Accordingly, accurate Cband ratios are crucial to success of 2T solutions, which offer an alternative to spectroscopic temperatures for favorable examples. Temperatures based on spectral classification are subjective and almost always refer to an unknown orbital phase, as observation times of the relevant spectra are seldom reported.7 Other problems with spectroscopic temperatures are that spectral types are discontinuous and coarsely spaced and that separation of T1 from T2 in binaries may be difficult spectroscopically. 1.3. Effective Utilization of Flux Calibrations to Find

Eclipsing Binary Distances

Taken together, these points about prior establishment of Cband ratios mean that instead of N unrelated calibrations for N bands, accurate Cband determination for one band can set good calibrations for other bands if the ratios are known. Given correct temperatures, distance depends only weakly on Cband (by a factor of ?1/Cband), but a wrong Cband ratio mimics a wrong (band-toband) flux ratio so as to give wrong temperatures in 2T solutions, while accurate temperatures are needed for accurate distance. Accordingly, accurate Cband ratios are the keys to temperature estimation without spectra8 with full utilization of light curve temperature information.

After refinement of the Cband ratios, distances can be found routinely in subsequent 2T solutions without knowledge of spectral types if interstellar extinction is measurable or negligible. Overall, a typical 1? uncertainty in T from spectral type is of the order of several percent. An alternative for temperature estimates is color index, but color index calibrated for single stars may not apply reliably to unresolved binaries, most of which are two-temperature sources. A further difficulty requiring proper inversion of binary effects is that color indices saturate (approach fixed values) for very hot stars, thereby losing much of their usefulness as temperature indicators. Although spectral type will sometimes be the best option for temperature estimation and ordinary color index may be satisfactory for some EBs, simultaneous light-velocity EB solutions in two bands can be a third way. Such two-band solutions effectively rely on color index to set temperatures, but with thorough modeling of its variation with phase due to EB effects, two-temperature effects, and their interaction. The subjectivity of spectral classification is then bypassed. 2. FLUX CALIBRATIONS FROM ECLIPSING BINARIES

AND SINGLE STARS

Confidence in calibrations increases when several classes of
objects yield accordant results. The calibrations found in this
paper have in common that all refer to the photometric response
functions adopted in the WD program but, as explained below,
specifics of their measurement differ for EBs, single stars, and
the particular case of the Sun. A given star system has only
one real distance so, if all Cband?s for a given object are found
7 One might think that time is routinely published for nearly all observations
but often the time goes entirely unreported?even the year. The importance of
an observation time may become apparent only long after publication.
8 Unknown amounts of interstellar extinction can undermine T and d
estimates, although our work in progress

SEDs, then averages the fluxes. Although our procedure works in combination with geometric and model stellar atmosphere parameters and therefore is less straightforward than the traditional one, it becomes absolute in its own way and thus gives new results.

Band calibration for a single spherical star is given in terms of apparent magnitude (m) and computed surface flux (f), in the absence of interstellar extinction, by (1) together, percent Cband error due to distance error is the same for all bands and measured Cband ratios are therefore distance independent. This circumstance is important for an EB, where individually estimated Cband?s depend on the (inverse) square of the assumed distance.

The relative importance of various kinds of errors (stellar
emission, photometric transformations, flux calibration
constants, etc.) for EB distance measurement is difficult to assess.
However, stellar atmosphere emission is probably correct within
a few percent when averaged over a band while transformations
by observers to standard photometric systems should also be
accurate within a few percent, and better for uncomplicated
objects.

Single stars of definite brightness and temperature?if reliably known to be single?can supply useful Cband?s in their own way, as described in Section 2.2. The number of suitable single star targets greatly exceeds that of EBs, so averages from hundreds or even thousands of stars can produce rather accurate Cband?s and Cband ratios.

2.1. Eclipsing Binaries: Flux Calibration by Inversion of

Direct Distance Estimation

DDE can find not only d as a function of Cband but can run inversely, estimating Cband?s where distance is known. If d is fixed in separate (say) B and V solutions, in accordance with a given EB having only one distance, the derived CB /CV ratio will be independent of d and should be accurate if the B ? V color index is accurate and interstellar extinction is negligible. The reason is that a Cband measure depends strictly on adopted d according to the inverse square law of flux dilution, such that any error due to d cancels exactly in the CB /CV ratio. This exact cancellation (with other parameters fixed) has been checked by DDE simulations. Temperature dependence remains, but ordinarily will go in the same sense in the two bands so as to be far less important in the ratio than in the individual Cband?s. Coherence is therefore improved in simultaneous solutions that measure calibrations in two or more bands. Cband ratios of thus enhanced accuracy lead to much improved 2T solutions, as confirmed by further simulations. In subsequent applications to measure distance, errors in Cband ratios are equivalent to flux ratio errors that correspond to temperature errors, while temperatures are correlated with distance, so accurate Cband ratios are crucial to reliability of temperature and distance.

Our single star Cband estimation scheme differs from calibrations in the previous literature so as to check earlier results independently, while following the essential information usage of DDE solutions. In other words, the plan is to have a consistent line of attack for EBs and single stars that does not simply repeat previous work. We operate with band-integrated absolute model stellar atmosphere fluxes to find Cband?s for individual stars and then average the Cband?s for many stars. Typical earlier work generates band-integrated fluxes over absolute empirical Cband = 100.4 mfband

R and d are the same for all bands so the R/d factor cancels exactly in band-to-band ratios. The radius and star mass (M) also enter the computation through the effect of surface gravity, g = GM/R2, on theoretical stellar atmospheres, but only slightly since the computed SEDs depend mainly on T and only weakly on g. Taking the B, V ratio, for example, one finds

CV CB = 100.4(B?V ) fB (T , log g, [M/H]) fV (T , log g, [M/H])

(2) where of course T, log g, and chemical composition [M/H] are all band independent. Since distance cancels exactly in Cband ratios, the only reason to prefer nearby stars for the ratios is to be sure of negligible extinction.

The main selection requirement is that the stars be effectively single, as judged by having only the very small or undetectable velocity variations found in some exoplanet search targets.

That is, they should not have companions of stellar mass (and
luminosity), although they may have planets. The input data on
T, log g, [M/H], R, and d are taken from Tables 8 and 9 of
the exoplanet paper by

All of our utilized single stars are within about 80 pc, as seen
in Figure 13 of VF and therefore will have negligible, or at
least unimportant, interstellar extinction. Any star with a double
or variable star label or that appears in the Ninth Catalog of
Spectroscopic Binary Orbits

Valenti & Fischer computed R/R from (L/L )1/2/(T /T )2,
where the L?s are bolometric and based on distances from
three parallax catalogs

Clues to the main error sources are in Figures 1?10, which show logarithmic Cband?s and Cband ratios versus temperature for hundreds of single stars (see Table 1), as computed from Equations (1) and (2). The abscissa and ordinate scales have been chosen to facilitate inter-comparison of noise patterns, with the plots having identical temperature scales and all the logarithmic Cband and Cband ratio scales having the same spacings. A given percent noise corresponds to the same scatterband width for all the plots as a consequence of the logarithmic scales with strictly fixed intervals. The noise is illustrated by many figures because visual comparison conveys a sense of accuracy and precision much more readily than reading 4800 5200 6000

6400 0.25 0.20 of tabulated numbers. One sees immediately that there are (currently unexplained) asymmetries in the histograms and dependences on temperature, while the Sun?s locations show essential agreement (or in a few cases slight disagreement) with the single star results. It may seem unnecessary to include figures for Cband ratios that are not formally independent of other plotted ratios, such as Cv /Cb when there are figures for Cu/Cv and Cu/Cb. However, the extra figures are helpful in showing where random and systematic effects lie (are they mainly in Cu, Cb, or Cv ?).

We find Cband ratios that differ from star to star, with strongly band-dependent standard deviations that range from about 2% in Cb/Cy to about 10% in Cu/Cy and CU /CV . Because Cband ratios are nearly independent of R and d, as shown by Equation (2), possible sources for most of the scatter in the ratios are the observed color indices and estimated temperatures, both of which produce larger scatter at shorter wavelengths.9 In a simple numerical experiment, input temperatures were altered 9 Earth atmosphere extinction (in optical bands) increases steeply toward shorter wavelengths and temperature affects stellar emission more strongly toward shorter wavelengths. by a few hundred Kelvins to see the relative changes in CU and CV , with resulting CU changes being about 1.4 times larger than CV changes, percent wise. Since (percent) CU ? ?s are about 5 times larger than CV ? ?s, any errors in estimated temperature cannot contribute more than a small part of the scatter in CU . By process of elimination, the only noise source that can account for most of the scatter at the shorter wavelengths is photometry.

Naturally, averages based on hundreds of stars will greatly reduce accidental error.10

As shown by Equation (1), derived calibrations are based
on measured star magnitudes (m) that implicitly depend on true
instrumental response functions and on theoretical fluxes (f) that
depend on adopted response functions. The adopted functions
are proxies for the realistically unknowable true functions that
can differ significantly among data sets. Some published Cband?s
differ from those of this paper mainly due to adopted response
functions. For example, our several estimates of CB in this and
later sections cluster around 0.62 erg s?1 cm?3, while Johnson
(1965a) and

6400
respectively, but the discrepancy is understandable in that our
adopted B response (

2.3. The Sun and Interferometrically Observed Stars

In the case of negligible extinction, individual Cband?s could be measured accurately from standard magnitudes of a single star if R/d were somehow known with very small uncertainty, although the Sun may be the only star to meet that requirement.

The main issue for the Sun is that its enormous brightness makes
standard magnitudes difficult to measure. While recognizing
that potential problem, we ran through the computation with
input of solar UBV RIK apparent magnitudes from

The numbers for the Sun agree very closely with the single
star averages of Table 1, although they are from completely
independent data. Table 3 has no ? ?s because we do not know
the uncertainties of the solar magnitudes but, judging from the
Cband agreements with the single star means, the Sun magnitudes
in

The (R/d)2 factor in Equation (1) suggests that interferometry
may be useful for individual Cband measures by providing
angular radii, although interferometry will not help for Cband
ratios since R/d is absent from Equation (2). We examined the
CADARS stellar interferometry catalog by

We did not check the entire catalog of over 13,000 measures, but none of the nearby and clearly single stars such as Vega, Sirius A, ? Ceti, and Eridani that we checked gave CB?s or CV ?s via Equation (1) in good agreement with values from other kinds of data. One would expect this result from scatter among individual interferometric R/d?s that suggests 1? uncertainties of order 5% in R/d (so 10% in [R/d]2) in the better examples. However, prospects for interferometric Cband measurement, for example, by averaging results for many stars, should be examined more thoroughly. 4800 5200 6000

6400 5600 Teff (K) -0.50 -0.45 log10 CV -0.40 -0.35

A single star with known or negligible interstellar extinction can offer physical and computational simplicity for Cband ratio estimation, only requiring a good color index in the relevant bands, approximately correct log g, and approximately correct [M/H]. These are entered into a short program based on Equation (2) that directly calculates a Cband ratio without iteration. The advantage of single star targets is that Cband ratio results are independent of distance and not very sensitive to mass, radius, or composition. The down side is that each star must be effectively single, while it can be difficult to establish that no significantly luminous companion has affected the results (done here by requiring very small to nil RV variation). Similarly, an EB candidate ideally should lack a significantly luminous companion, although third light is an ordinary EB parameter that is routinely evaluated in light curve solutions. Also, companions to EBs can be detected by use of eclipses as timing ticks that quantify orbital motion about the barycenter, although that option is observationally intensive. Masses, radii, and consequently also log g?s are typically measured with good accuracy in EB solutions.

The most common reason for the shortfall of EB candidates, notwithstanding hundreds of nominal EBs in the solar neighborhood, is that the requisite observations have not been made. Also, many supposed EBs that have been reasonably well observed turn out to be ellipsoidal variables or have such shallow partial eclipses as not to be very useful for EB analysis. Many potentially good targets have good light curves but no RVs or RVs but no light curves, while others have good observations of both kinds but with RVs for only one component. Light curves of still others are not on standard photometric systems and not transformable to standard systems for one or more of several reasons. Although standard magnitudes of a comparison star are usually known and ostensibly standard filters may have been used, it may not be clear whether the differential light curves were properly transformed to standard systems. That problem may not be significant if the EB and its comparison star are nearly the same color and the EB?s color is nearly constant, or if the instrumental and standard systems are very close to one another, but often the realistic situation cannot be ascertained 5600

Teff (K)
4800
5200
6000
6400
from an observational paper. Standardization uncertainties have
little impact on traditional non-absolute EB solutions, but do
impact absolute solutions, as they correspond to physical flux
errors. The many light curves that are reported and perhaps
shown graphically in papers, but not published in useful form,
detract further from the candidate list. Finally, we avoided EBs
with substantial third light. In our experience, third light of the
order of a few percent of system light does not degrade
essential results significantly, but amounts of the order of 20% or
more may do so. On the positive side, comparison star standard
magnitudes are now commonly available thanks to networks of
standard stars, e.g.,

The Differential Corrections (DC) algorithm equates a set of residuals, ?f = fO ? fC , for an observable quantity f (one residual for each observation), to a sum of what might be called partial residuals or perhaps individual parameter residuals. One presumes that the partial residuals would add to the overall residual in f if observational error were absent and the linear approximation were entirely adequate. Subscripts O and C, respectively, mean ?observed? and ?computed? (from a model). In this context, the linear approximation means that only the first derivative term in the Taylor series of corrections is applied for each parameter, so iteration is required. Accordingly, the equation of condition for a least-squares application to n parameters (pi , i = 1, . . . , n) can be written ?f = ? ?f ?p1 ? ?f ?p2 ?p1 + ?p2 + . . . + ?pn,

(3) ? ?f ?pn where the ?p?s are differences between momentarily adopted and least-squares p?s. Often one sees the partial derivatives written in terms of only fC rather than fO ?fC because the fO?s are constants in most applications (i.e., the observations are definite and not subject to variation, thus not contributing to the derivatives). However, the partial derivatives needed for Cband adjustment differ from the other DC partials in that they come from variation of fO rather than fC. Given that fO = 10?0.4 m × Cband, with m being stellar magnitude, the 4800 5200 6000

6400 4800 5200 6000

6400 5600 Teff (K) -0.45

log10 CV -0.50 -0.40 -0.35

Cband derivative is just ? ?f/? Cband = ?10?0.4 m if expressed in magnitude or ? ?f/? Cband = ?fO (m)/fO (m = 0) if expressed in light (i.e., flux at Earth). Usually an implicit sign reversal is incorporated into the derivatives because one wants ?p to represent a parameter correction rather than a parameter error. There is another reversal because of the minus sign attached to fC, so the two sign reversals effectively leave a derivative?s overall sign unchanged. For Cband derivatives, however, we differentiate fO rather than fC so there is only one sign reversal, thus reversing the overall sign of ? ?f/? Cband.

The EB-based calibrations of this paper are from
leastsquares solutions that require prior temperature estimates for
one component and solve for the other temperature (and, of
course, for Cband). In all cases, the prior-estimated temperature
was that of star 1 (T1) while the solution temperature was T2.
For binaries whose spectral-type information was not considered
fully reliable, the prior T1?s were based on B ? V outside eclipse.
A practical problem arises when a T1 estimate is made from a
published color index that pertains to an entire EB?how are
the components? flux contributions to be separated? The issue
of which star is hotter will usually be clear from eclipse depths,
but an intuitive guess of T1 based on a T versus B ? V relation
for single stars may be inaccurate. A partial accommodation
adopted here is to tabulate EB solutions for several input T1?s so
as to gauge how strongly the Cband?s and Cband ratios depend on
T1, but naturally one would like the middle of the T1 range
to be close to correct. Accordingly, prior T1?s were based
on a consensus of published temperature calibrations, along
with computational logic to account for the EBs being
twotemperature sources. The relation between observable light ( )
and B ? V for a pair of simple spherical stars may be written as
( 1 + 2)B
( 1 + 2)V
= 100.4(KBV ?(B?V )),
(4)
where KBV is the constant required by the condition that
B ? V = 0 for an average A0 V star, according to photometric
convention. By numerical experiment with

6400
5600
Teff (K)
-1.10
satisfies Equation (4) for main-sequence star data. The
temperature data upon which KBV is based are from Table 15.7 of

(5) Typically, three to five iterations find a definite T1 within a pre-set tolerance that can be very small.

Curve-dependent weighting can be important in simultaneous
least-squares solutions of multiple curves. The DC program
follows rules in

4. CALIBRATION EB STRATEGIES, TARGETS, AND

RESULTS

The EB protocol for Cband solutions was in two steps, with a non-absolute solution for most parameters followed by an absolute solution for only T2, the calibrations, and third lights.

It was the same for all binaries of this paper, except that most solutions did not need third light, and is justified as follows. To begin, orbit size has no effect on theoretical light curve fluxes ( ) in non-absolute computations, as the intent is to represent arbitrarily scaled observations, so semimajor axis length (a) is purely an RV parameter in that case (? /? a = 0). Accordingly, orbit size cannot be found from arbitrarily scaled light curves alone. However, the situation is entirely different for absolute light curve solutions. With other parameters fixed, including relative star sizes r1,2 = R1,2/a, observable flux scales with 4800 5200 6000

6400 5600 Teff (K) a2 since star size scales with a, so parameter a is derivable from one or more absolute light curves, given d. Note that the influence of d is not minor?with all else ideal, a 10% wrong distance will produce a 10% error in orbit size and ? 20% Cband error. The point is that one should not try to find parameter a from an absolute light curve solution unless there is no other way or unless d is somehow known with unusually good (basically unprecedented) accuracy. Derived Cband?s depend strongly on a and d, so simultaneous (RV, absolute light) solutions will produce compromise calibrations based on both kinds of data, with wrong d leading to wrong a, which leads in turn to wrong calibrations. Realistic situations can be made worse by greatly unequal sizes and precisions of typical RV and light curve data sets, with light curves usually winning easily on both quantity and quality. Having 50 times as many light curve data points as RV points is not unusual. To deal with this circumstance, we begin by finding most parameters, including a, in a traditional (non-absolute) solution so as to ensure that ?light curve a?s? do not influence derived Cband?s.

The idea is to evaluate most parameters from a non-absolute solution, leaving only T2, Cband?s, and 3?s for the absolute solution. Our experience is that T2 changes only slightly between the non-absolute and absolute solutions. Parameter 3 is readjusted in the absolute solution because the flux unit differs between the non-absolute and absolute cases. Experiments show that the absolute light-RV solutions for fixed d produce very nearly the same Cband?s whether carried out simultaneously or in each band separately, so they were done simultaneously.

EBs chosen here for calibration analyses have good RV and light curve observations on standard photometric systems, circumstances well suited to strong solutions (mainly reasonably deep eclipses), and absence of strongly irregular behavior. The discussion of common shortcomings in Section 2.4 is in no way meant to imply lack of other good candidates, and interested persons can certainly add meaningfully to EB-based Cband?s. Although good traditional (i.e., non-absolute) solutions for all seven of this paper?s ?calibration? EBs are in the literature, our full parameter results are tabulated so as to place the derived calibrations in proper context. The numbers are in solution Tables 4?10 for TZ Men, V1130 Tau, TY Pyx, V505 Per,

CrA, BG Ind, and WW Aur, respectively. The distances in all these tables are fixed input taken from the revised HIP catalog 4800 5200 6000

6400 4800 5200 6000

6400
5600
Teff (K)

4.1. TZ Mensae

TZ Mensae is a well-detached B9 IV-V EB with V ? 6.2 mag,
d ? 110 pc, and a slightly eccentric, nearly edge-on orbit of
period 8.57 days. A fixed input temperature of T1 = 10,437 K
was estimated for our calibration solutions via the scheme of
Section 3.2, based on Equations (4) and (5). Metallicity, [M/H],
was assumed to be zero. Although there is no evidence for
measurable period variation or apsidal motion, it was necessary
to include the period and reference epoch in the solution
rather than adopt one from the literature because published TZ
Men ephemerides are specifically for the primary or secondary
eclipse. All versions of the WD program, including DDE, work
with ephemerides that are not specifically related to eclipses or
to actual conjunctions but to conjunctions that occur when the
orbital major axes are most nearly along the line of sight, with
star 1 away from the observer (i.e., for argument of periastron
?1 = ?/2 rad). The reason is to have computed eclipses make
phase excursions due to apsidal motion in the same way as real
eclipses (one goes left while the other goes right, and then they
reverse motions), rather than have one eclipse at fixed phase
while the other moves. A consequence for TZ Men of these
contrasting concepts is that our reference time, T0, differs by
more than an hour from the one by

The

4.2. V1130 Tauri

A traditional analysis of V1130 Tauri, an EB with V =
6.6 mag and orbit period 0.80 days, has just appeared

Notes. RVs from

0
0.2
0.4
0.6
0.8
-0.2
0
0.2
0.4
0.6
0.8
9.1
ed 9.2
u
it
ng 9.3
a
um9.4
Notes. RVs and uvby light curves are from

4.3. TY Pyxidis

TY Pyxidis is an RS CVn-type binary11 of spectral-type G5V
and orbit period 3.20 d. Although nearby with a good HIP
distance of 56 pc, TY Pyx may not seem a good candidate for
flux calibration due to erratic behavior, considering its surface
brightness variations and consequent light curve irregularities
caused by chromospheric and magnetic spot activity. It has
a moderately large literature as a soft X-ray source

Since the number of well-observed EBs that are near enough
11 RS CVn?s are understood to have strong dynamo action, driven by
convective motions in fast rotating envelopes. See

One cannot expect to infer details of the spot distribution, so the spotted regions were represented by just one circular spot whose effective temperature supposedly approximates an average over the actual regions. Better light curve fits surely would be possible with two or more spotted areas, but our objective is estimation of calibration constants rather than a definitive general solution.

Spot longitude and the dimensionless spot temperature factor were adjusted via least squares, while spot co-latitude (85?) and angular radius (20?) were not. -0.2

Standard uvby magnitude data for the reference comparison
star, HD 76483, are from Table 1 of

Conversion of the differential photometry in

5. CONSISTENCY AND ERROR PROPAGATION

Band-to-band calibration ratios from EBs and single stars are independent of assumed distance while the individual Cband?s from EB?s are sensitive to assumed distance. We do not claim that our calibrations are more accurate than earlier ones, but only that they were derived differently. We do believe that our Cband ratios are improved in accuracy, as shown by their agreement for single stars, EBs, and the Sun. Some users of this work may want to condition previously published or otherwise independently determined Cband?s to agree with ratios from this paper by making use of Cband averages in setting overall calibration levels. Cband?s drawn from the literature might be acceptably accurate for 1T solutions while their ratios (which require very good accuracy when applied to 2T solutions) might produce somewhat wrong temperatures and therefore wrong distances.

A reasonable strategy for distance applications is to adopt the new band-to-band Cband ratios while at least approximately preserving old Cband levels. For example, one could simply retain the calibration in one judiciously chosen band and revise the others according to the new ratios. However, exact preservation of both the ratio and mean value conditions is easy enough. First note that the geometric mean of Cband?s is preferable to their arithmetic mean due to possible order of magnitude differences among bands. In the Str o¨mgren systems, for example, let the ?old? (i.e., from literature) calibration constants be Cu, Cv , Cb, Cy, and let the newly measured Cbnaenwd ratios be Ruv , Rvb, and Rby . Then new calibrations that have the same geometric mean as the old ones and also agree with newly specified ratios are

2 3 = (CuCv CbCy )/ Ruv RvbRby Cnew

y Cbnew = Rby Cynew, Cvnew = RvbCbnew, Cunew = Ruv Cvnew. 4.4. V505 Persei

V505 Persei (V ? 6.86 mag) is a detached, double-lined
binary with two nearly identical F5V components in a 4.22 d
orbit.

T1 (6512 K) and metallicity ([M/H] = ?0.12) are from

Coronae Australis is a well-known W UMa-type OC binary
with a rather extreme mass ratio (M2/M1 ? 0.12) and light
curve asymmetries that indicate varying degrees of surface
activity (magnetic spots) that affect the quality of fit. The HIP

4.6. BG Indi

BG Indi is a sixth magnitude EB of spectral-type F3V in
a 1.46 d orbit, as determined by

4.7. WW Aurigae

WW Aurigae (V ? 5.8 mag) is a well detached, double-lined
EB with a 2.53 days orbital period, consisting of two nearly
equal metallic-line A-type stars. A fairly detailed analysis in

(6) Cband dependence on star temperature was explored computationally via triplets of EB solutions that allow later interpolation or extrapolation among three T1?s. A similar strategy is not needed for adopted distance because of the exact dependence of Cband on d given by Cband = C0 (d0/d)2. Error estimates in Cband due to 1? errors in T and in d are computed from ?Cband · ?T and

?T ?C?bdand · ?d , respectively. The total 1? Cband errors in our EB calibration solution tables are the root-mean-square sums of these two terms since T and d come from mutually independent data.

Table 11 summarizes mean Cband?s and Cband ratios for our seven calibration EBs. Means of uvby calibration results in Table 12 were calculated in another way?among photometric bands for each of five EBs and for the single star data. Although these numbers are not applicable to absolute solutions, they give an impression of object to object consistency and point to a possible problem with WW Aur being a metallic-line star. The Cband?s naturally jump around since they respond to the input HIP distance, with d errors producing d2 errors in the Cband?s.

However, the ratios are unaffected by distance and should agree closely. Indeed, the single star and EB ratios easily agree to better than 1%, except for WW Aur which differs from the others by about 3%. An explanation for WW Aur in terms of interstellar extinction is possible but unlikely, as its distance is only ?80 pc. A more likely reason is that WW Aur is well known as a strong metallic-line star, while the analysis program?s Kurucz atmospheres are for normal stars. As we lack metallic-line SEDs at present, the best strategy may be to avoid use of the WW Aur results for distance estimation.

Notes. a Weighted mean and standard error of the mean, with weights inversely proportional to individual mean errors in Tables 4, 5, 6, 9, and 10. b Same as Column 4 for the four EBs with WW Aur excluded (see Section 5). Notes. The Cuvby?s are geometric means of Cu, Cv, Cb, and Cy. The Cuvby ratios are geometric means of Cu/Cv, Cu/Cb, and Cu/Cy . Parameter T0 (HJD) P (d) a/R V? (km s?1) i (deg) T1 (K) T2 (K) ?1 ?2 q = M2/M1 d (pc) dHIP (pc) 3B 3V 3RC 3IC 3u 3v 3b 3y 3/ totalB 3/ totalV 3/ totalRC 3/ totalIC 3/ totalu 3/ totalv 3/ totalb 3/ totaly

V505 Per

Notes. The 3 unit is 10?6 erg s?1 cm?3, with total meaning total system light at phase 0p.25. The standard errors on fractional third light ( 3/ total) assume negligible errors in total. as well as uvby constants for WZ Oph, are from Table 1. WZ Oph also needed calibrations for Cousins RC and IC, which are from Bessell (1979, after conversion of units).

V505 Per?one of our ?calibration stars??is used here in
a 2T temperature?distance solution to check agreement with
spectral temperatures. The applied Cband?s are not from the
calibration solution but from our single star means, so the
overall process is not circular. Our distance solutions were in
two bands (B and V), in accord with the T?d theorem

WZ Oph was observed on 28 nights in 2008 May and June with the 0.4 m Schmidt?Cassegrain telescope at Sonoita Research Observatory (SRO) in Sonoita, AZ. A Santa Barbara 12 Perhaps worth noting is that the original HIP distance by Perryman et al.

(1997) is 66.6 ± 3.9 pc, in agreement with DDE. -150 -0.2 0

Instrument Group STL-1001XE CCD camera with Johnson B and V and Cousins RC and IC filters took 751 B, V , RC , IC image sets that were bias/dark subtracted and flat-fielded with the Image Reduction and Analysis Facility (IRAF) of the National Optical Astronomy Observatories. Differential photometry was then performed with TYC 977-116-1 as the comparison (C) star and TYC 976-1177-1 as a check (K) star.

The C ? K differential magnitudes had a standard deviation of about 0.01 mag over the entire observing interval. The Tycho comparison star observations yielded B ?V = 0.55±0.04 while SRO observations over 13 photometric nights, with Landolt standards, yielded B ? V = 0.54 ± 0.02. Observations of WZ Oph from SRO on those 13 nights yielded B ? V = 0.54 ± 0.05.

The identical colors of the comparison star and WZ Oph, along with the lack of variability in the comparison star, confirm its suitability for the differential photometry. The differential magnitudes were then transformed to the standard B, V, RC, and IC systems with the transformation coefficients for the SRO instrumentation. Estimated standard deviations are 0.025 mag in B, 0.012 mag in V, 0.017 mag in RC, and 0.018 mag in IC. Because of the good color match and the proximity of the comparison star to WZ Oph, no corrections for differential atmospheric extinction were necessary.

A comprehensive discussion of WZ Oph with a full traditional
light/RV analysis is in

IC (mag)
(This table is available in its entirety in a machine-readable form in the online
journal. A portion is shown here for guidance regarding its form and content.)
observations through the DDE algorithm to check whether our
new Cband ratios lead to temperatures that agree with spectral
information in the literature. WZ Oph is well detached with
orbit period 4.18 d and main-sequence components of nearly
equal mass and early G spectral type. It poses an interesting
situation with regard to the T?d theorem?s advice that two
bands (not one, not three or more) should be entered into a
2T absolute solution. The interesting point is that we have the
unusual number of eight bands (u, v, b, y, B, V , RC , and IC )
and would like to utilize all, but solutions of the 28
twoband combinations and subsequent inter-comparisons would be
tedious. However, the sole reason not to process three or more
light curves together is to avoid the non-Gaussian residuals that
go with overall misfits that are likely to arise from errors in
the calibrative data. Because of the possibility that there may
be no misfits (and curiosity about how large they might be),
the eight bands were entered together, with the unexpected
result that all bands were matched rather well?there was no
obvious non-Gaussian problem. Accordingly, the Str o¨mgren
u, v, b, y?Johnson B, V ?Cousins RC , IC solution, with RVs
from

The uvby curves are from

8.5 s e tdu 9 u n g a m ph 9.5 O Z W 10

9 9.5 s e d tu 10 i n g a hm10.5 p O Z W 11

11.5 -0.2 0 Figure 16. WZ Oph observed (dots) and model (lines) light curves for the eight-band distance solution (left panel, bottom to top: B, V, RC, and IC; right panel, bottom to top: u, v, b, and y). Note that agreement is good not only in form, but also in absolute level for all bands, which can happen only if the input calibration ratios are accurate. 6. FINAL COMMENTS

To underscore the aims, usefulness, and some active problems of Cband measurement by the ways of this paper, we emphasize several key points.

This work was supported by U.S. National Science
Foundation grant 0307561. Extensive use was made of the NASA ADS
and Simbad databases, and we also consulted the eclipse
timing diagrams and associated World Wide Web site by