Time-sequence Spectroscopy of Epsilon CrA: The 518 nm Mg i Triplet Region Analyzed with Broadening Functions

The BFs for CrA are very similar to those for AW UMa in that they are dominated by the strong feature of the primary component of the binary system. We show the BFs for four cardinal orbital phases in Figure 2; the results for other observations are tabulated in Table 2. We see the following. (1) The rotational profile of a single, rapidly rotating star with $V\sin i=147$ km s⁻¹ describes the upper part of the profile of the primary component (above the 0.35 of the maximum) very well; the transit eclipse is an obvious exception. (2) The profile shows high-velocity extension wings, which we called the "pedestal" for the case of AW UMa. (3) The surface of the primary component appears to be covered by low-contrast inhomogeneities to a lesser degree than in AW UMa. (4) Obvious signs of distributed matter in the binary are present and may be affecting the primary profile. One such is a conspicuous BF perturbation at the primary (deeper) eclipse during the transit of the secondary over the disk of the primary component (the lowest panel in Figure 2). The perturbation, in the form of a spike, is most likely caused by gaseous matter flowing toward the observer over the surface of the secondary component; this is discussed further in Section 3.2.

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The details in the profiles in Figure 2 are all significant and are not caused by the BF uncertainties, which are very small (Section 2). The errors have a constant distribution along the whole length of a given BF; their size may be estimated directly from the scatter in the baseline, for example from the empty portion at velocities <350 km s⁻¹. It is one of the advantages of the BF approach that all points are treated equivalently and independently by the linear deconvolution process. The spectral resolution of FWHM of 8.5 km s⁻¹ was set at the start of the BF determination and is the only source of correlation between the data points.

The width of the profile of the primary component (Figure 3) changes slightly with the orbital phase. The solid-body rotational profile for the limb darkening coefficient u = 0.5 was fit to the individual profiles above 0.35 of the maximum value to avoid contamination by the high-velocity pedestal. The fits gave the smallest $V\sin i\simeq 143$ km s⁻¹ during the short interval of the secondary (shallower) eclipse, ϕ = 0.5, when the secondary component with all its complications (Sections 3.2–3.3) underwent the occultation by the primary. This interval appears to be longer before the center of the eclipse, starting at ϕ ≃ 0.40 and ending abruptly at ϕ ≃ 0.54. These phases coincide with large perturbations of the primary profile by an unexplained spectral feature discussed further in Section 3.3. The short duration of the totality phases agrees with the orbital inclination angle of i ≃ 72°–73° (Twigg 1979; Wilson & Raichur 2011) estimated from the approximate relations of Mochnacki & Doughty (1972) and our determination of the mass ratio of q_sp = 0.13 (Section 3.2). The median projected equatorial velocity for all phases is $V\sin i=147\,\pm 3$ km s⁻¹, where the assigned uncertainty is an estimate of the scatter at orbital phases away from the extended wings of the eclipses and not perturbed by the gas streams (Figure 2). Of the total of 361 observations, 32 rotational-profile fits were noticeably disturbed by optically thick gas motions in the system and were eliminated from the determination of the orbit of the primary component (marked by red open circles in Figure 3). The rotation rate of $V\sin i=147$ km s⁻¹ is about 80%–85% of the rotation synchronous with the orbital motion of the inner critical equipotential ($178\,\sin i$ km s⁻¹); however, the additional broadening in the pedestal of about 25–35 km s⁻¹ would bring it into approximate agreement with the synchronous rate.

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The radial velocities of the primary component of CrA are listed in Table 3 while the orbital solution based on the centroids of the rotational-profile fits is shown in Figure 4. The orbital fit to these velocities by a sine curve is very well defined, particularly after the strongly deviating measurements close to the phases around 0.58 and 0.87 are removed. The zero-point determined here is used in this paper as the origin of the orbit and orbital-phase counting. The resulting parameters of the spectroscopic orbit for the primary component are listed in Table 4, together with the results for the secondary component discussed in Section 3.2. The orbital period P is for the locally linear ephemeris following the quadratic elements in Section 2.

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Table 3.  The Radial Velocities of the Primary Component of CrA

HJD	Orb	V1	$V\,\sin i$	Key
0.5211	0.760	97.84	149.6	1
0.5263	0.769	97.07	149.3	1
0.5319	0.778	97.07	148.4	1
0.5372	0.787	96.59	147.9	1
0.5426	0.796	96.63	147.8	1

Note. The columns are as follows: HJD—the heliocentric Julian date for the middle of the exposure counted from HJD = 2,458,312; Orb—the same time expressed in units of the orbital period; V₁—the centroid of the rotational velocity fitting profile expressed in km s⁻¹; $V\,\sin i$—the projected rotational velocity determined from the upper 65% of the profile; Key—zero signifies that the observation was not used for determination of the velocity of the primary component.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Table 4.  The Spectroscopic Orbit of CrA

Parameter	Result	Error	Unit
P	0.59145447		day
HJD(pri)	2,458,312.0716	0.0004	day
K1	34.718	0.084	km s−1
K2	267.13	1.37	km s−1
V0	62.541	0.076	km s−1
qsp = M2/M1	0.1300	0.0010
$A\sin i$	3.527	0.016	R⊙
$({M}_{1}+{M}_{2}){\sin }^{3}i$	1.685	0.023	M⊙
${M}_{1}{\sin }^{3}i$	1.491	0.020	M⊙
${M}_{2}{\sin }^{3}i$	0.1938	0.0026	M⊙

Note. The period P was assumed for the epoch of the observations, as described in the text. The time of the deeper (transit) eclipse HJD(pri) is from the velocity curve of the primary component. The radial-velocity amplitude of the secondary component was obtained from measurements of the locations of the profile centroid in 2D images, such as in Figures 5–9, with the value of V₀ assumed to be the same as for the primary component. The errors are formal least-squares errors and do not reflect systematic uncertainties in the data—see the text in Section 3.2 for more details.

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While the parameters of the sine-curve fit to the radial velocities of the primary component have very small errors, the quality of the fit is characterized by a relatively large single-point random error of ±1.44 km s⁻¹. As shown in Figure 4, a large fraction of the uncertainty is due to the systematic deviations reaching −4 to +2 km s⁻¹ close to the eclipses; the scatter at the orbital phases around 0.25 and 0.75 is considerably smaller, ±0.38 km s⁻¹. The deviations before and after the primary eclipse (the transit of the secondary component in front of the primary) are reminiscent of the "Rossiter–McLaughlin effect" effect, which was also noticeable in the AW UMa velocity data. However, observed with the accuracy available here, the effect is not symmetric relative to the eclipse center, implying the existence of gas-flow motions affecting even the central parts of the primary profile.

Figures 5–9 show the 2D images of the BF time variability on the five nights with more than 40 observations of CrA per night (similar, smaller images exist for the three remaining nights). The images were obtained by interpolation into an equal-step grid of orbital phases with a phase step of 0.008 and then—to improve the visibility of the weak features—by subtraction of the rotational-profile fits for the primary component scaled and shifted by the orbital velocity. Figure 10 should be consulted for interpretation of these figures in terms of the sense of rotation and of the orbital phase system. The reader is reminded that a connection of velocities to locations and spatial dimensions is not trivial and requires assumptions.

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A comparison of the 2D images of CrA with those of AW UMa in Paper I shows the primary components of the two binaries to be very similar to each other, but with some differences: (1) the large-scale disturbances visible as gentle, winding waves in the 2D pictures have smaller amplitudes in CrA, with the largest intensity modulation of about 4%, while in AW UMa some of them reached 8%; (2) the small-scale (<1%), high-frequency, regularly spaced disturbances that we called "ripples" appear to be absent in CrA; (3) we do not see isolated spots, which may have been present on the primary of AW UMa.

We conclude that the primary component of CrA is covered by surface inhomogeneities to a lesser degree than AW UMa. This absence of the wavy ripples, so clearly visible in AW UMa, is particularly significant because the BF data are of higher quality (S/N ratio of 120 versus 57 for AW UMa; Section 2) and the overall monitoring time spent on CrA was 3.7 times longer than for AW UMa. However, we should remember that the 3.1 times longer exposure time used for CrA may have led to some smoothing over high-frequency disturbances. The pedestal feature of CrA is very similar to that in AW UMa and remains unexplained.

In contrast to the well understood and basically phase-invariable central profile of the rapidly rotating primary, the secondary component of CrA presents a rather complex picture. Its spectral lines are weak: expressed as integrals of the respective BF features, the spectral lines are 7.97 ± 0.54 times weaker than the lines of the primary component, with the large error in this value reflecting the difficulty of separating the two features. While radial-velocity profiles of the secondary can be defined in individual BFs, they are easier to analyze in the 2D images, such as Figures 5–9 where the complexity of the profiles is fully visible thanks to the BF phase intercorrelation. The images show that the secondary component is to some extent connected to its dominating companion, but that the radial-velocity profiles are different than expected for the Lucy model. The profiles suggest a system of gas motions or surface inhomogeneities rather than an unperturbed, stable photosphere. Spatial localization of these features is generally difficult to establish, but this ambiguity can be sometimes lifted by the eclipses.

A silhouette of the secondary in radial velocities is well defined during the transit (primary, deeper) eclipses (ϕ = 0.0) when the secondary star is projected against its larger companion. We observed transit eclipses on four nights, numbers 12, 13, 16, and 28. The secondary is delineated by two symmetrically located BF features, giving its rotational velocity half-width of 66 ± 2 km s⁻¹. We see a considerable difference in the appearance of the two defining "sides" of the secondary component during the transits (see Figures 5, 6, and 9): The negative-velocity side, at −66 km s⁻¹, relative to the center of mass of the secondary, is always more strongly defined by a spike-like feature in the BFs. A similar, enhanced spectral-line feature was observed in AW UMa, but it is much stronger and better defined in CrA (see the lowest panel of Figure 2). We discuss this feature as a representation of interbinary gas motions further in Section 3.3.

In contrast to the approaching side, the edge of the rotationally receding side is poorly defined during the transit, but its visibility improves after the primary eclipse when the secondary continues at positive velocities relative to the projected center of mass. These out-of-eclipse phases carry information on the varying velocity distribution over the secondary surface at other aspect angles. Of importance is the admittedly faintly defined negative-velocity limit of the secondary component when this star is moving away from the observer (the phase range 0 < ϕ < 0.5; marked by the arrow in Figure 9). It appears that the rotational motion is at least partly confined to the secondary component as if in a local surface motion, without a connection to the velocity field of the primary component. The Lucy model, with its strict solid-body rotation, predicts no separation in the velocities of the two components. We saw a very similar inner edge to the velocities of the secondary component in AW UMa, but it is much better defined in CrA. This implies that in both binaries the secondary component has its own velocity field, independent of the primary component.

While the secondary component of CrA shows a relatively flat profile on the positive-velocity side of the primary, with numerous but barely marked and tenuous structures, it is very different when it reappears after the occultation at ϕ = 0.5. Then, at negative orbital velocities, it shows a more complicated structure dominated by what appears as a "wisp" in the 2D images (discussed further in Section 3.3) and as several delicate, phase-dependent weaker features. The individual profiles at negative relative velocities are dependent on orbital phase, with only weak traces of a tendency for a central depression or meniscus shape, as was suggested for the less extensively observed AW UMa. Generally, however, the secondary of CrA does not seem to show a profile similar to that of an accretion disk, as was suggested in Paper I for AW UMa. We note that we probably see the two binaries at orbital inclination angles differing by about 10° so that phenomena taking place in the plane of the binary orbit may be visible differently.

The orbital radial velocities of the secondary component were determined from the 2D images (Figures 5–9) for 184 epochs as centroids of the profiles of the secondary component. In addition to the orbital motion, we observed variations of the secondary profile in its width and structure, most likely because of the complex gas flows around and over the surface of the secondary component. The secondary profile shows more structure and is narrower after the secondary emerges from behind the primary (ϕ = 0.5) than after the transit over the primary (ϕ = 0.0) when the secondary recedes from the observer.

The individual profiles of the secondary do not appear to be dominated by rotation, as for the primary component, nor do they have shapes expected by the Lucy model. Thus, we cannot directly identify their observed width with $V\sin i;$ we can, however, determine their observed widths from the relatively sharp edges at their BF base. The half-widths of the secondary profiles vary gradually with the orbital phase. The widths can be best established at the phases of the maximum velocity of approach and recession, in addition to the direct radial-velocity imprint at the transit eclipse; the measurements at intermediate phases are more affected by the poor definition of the profile edges and could not be used. The observed half-widths range between 57 km s⁻¹ at the maximum approach of the secondary (ϕ = 0.75) and 89 km s⁻¹ at its recession from the observer (ϕ = 0.25), with the half-width at the transit eclipse (ϕ = 0.0) of 66 km s⁻¹ falling in between. The above numbers carry an uncertainty of about ±3 km s⁻¹. The half-widths at the profile base, interpreted as corresponding to $V\sin i$, are summarized in Table 5, together with the half-widths expected for the rigidly rotating Roche structure; Figure 10 gives a schematic distribution of velocity vectors in the CrA system.

The centroid velocities of the profiles of the secondary component were determined to ±3.5 km s⁻¹, as estimated from repeated measurements. They follow the sine curve, as shown in Figure 4. The orbital parameters given in Table 4 assume that the center-of-mass velocity, V₀, of the secondary is the same as for the primary component. While the uncertainty of K₂ of ±1.37 km s⁻¹ is small, compared with orbital solutions for other W UMa binaries, the single-point error for the secondary, ±14.7 km s⁻¹ is large and reflects systematic trends in the deviations within some of the individual orbits reaching −30 to +40 km s⁻¹. Since neither the Lucy model nor the simplified sine-curve fits describe such deviations, we treat them as contributors to the random errors. As the result, the orbital parameters presented here should be treated as provisional and subject to re-evaluation once a correct model becomes available. We note that a solution for the secondary with the mean systemic velocity V₀ treated as a free parameter differed in the mean velocity by ΔV₀ = −0.94 ± 1.77 km s⁻¹ from that for the primary component, indicating the overall level of systematic uncertainties in our data processing and velocity measurements.

The quantity that should be least affected by the inadequacy of the adopted model should be the mass ratio, q = K₁/K₂. The simulations of light-curve synthesis provide information on the mass ratio that is independent of the spectroscopic result; however, it depends entirely on the adopted Lucy model. The most recent, very accurate photometric value of the mass ratio, q_ph = 0.1244 ± 0.0014 by Wilson & Raichur (2011), is only slightly smaller than our spectroscopic result, q_sp = 0.1300 ± 0.0010; the latter is practically identical with the result by Goecking & Duerbeck (1993), q_sp = 0.128 ± 0.014. Apparently, the systematic difference, q_ph < q_sp, discovered by Niarchos & Duerbeck (1991, Section 5) is very weakly present in CrA, only at the 4σ level, and with error estimates that may be systematically too optimistic.³ Apparently, the similarity of CrA to AW UMa—analyzed in the same way and using the same technique in Paper I—does not extend to the discrepancy in mass ratio. This subject is discussed further in Section 4.1.

The profiles of the secondary component, as shown in Figures 5–9, are fairly complicated yet appear to be surprisingly stable, suggesting a binary-wide velocity field in stationary conditions. However, it is not entirely obvious what is the cause of the details in the profiles. Our BF information-extraction method is not sensitive to the rarefied gas: it can only detect the presence of a dense, high-temperature gas with similar characteristics to those of the photospheric layers of the primary component. If not projected against a photosphere, the gas must be sufficiently dense to accumulate to an optical depth of at least the order of unity along the line of sight to produce an absorption feature detectable by the BF technique. This condition may be fulfilled in localized places close to the secondary component and possibly in the region between the binary components. The radial-velocity measurements give us information about one velocity component of the gas motions only, not about localization of the radiating gas. In addition to the eclipse shadowing effects, a model or plausible assumptions are the only possibilities to visualize the actual patterns of the gas motions.

The BF spike at the negative velocity of the secondary component during its transit was described in Section 3.2 as defining the approaching side of that star (see the lowest panel of Figure 2). It is one of the most prominent manifestations of the interbinary matter, but it is visible only in a narrow range of phases around ϕ = 0.0, becoming strongest at the exact alignment of the stars. Its phase dependence is difficult to measure because the spike is small in the absolute sense, with an integrated intensity at its maximum of less than 1% of that of the integrated profile of the primary component. If strengthening of the spectral lines was coincident with a continuum increase, such a faint feature would be detectable in broadband photometry, possibly combining with other small variations during the eclipse branch. In the case of the current program, it was possible to detect it spectroscopically, mostly thanks to the linear properties of the BF technique and after subtraction of the dominant profile of the primary component. Its width at the base is 25 ± 2 km s⁻¹, as estimated from repeated transit events. We think that the spike is produced by the outer layers of a stream of matter flowing away from the primary component and sliding along the surface of the secondary, directly toward the observer. The outer layers of the flow attain a sufficient optical depth when visible along and into their length; they must be relatively slim and long to produce a strong dependence on the angle.

The radial-velocity profile of the secondary component appears to be more complicated within the orbital phases when the secondary approaches the observer (the phase interval 0.5 < ϕ < 1.0). The most prominent feature visible in the radial-velocity images in Figures 5–9 is called here a "wisp" (see also the third panel of Figure 2 and Figure 11). The wisp emerges from behind the primary component around the orbital phase ϕ ≃ 0.58 and remains visible to ϕ ≃ 0.87. The phases of the appearance and disappearance of the wisp are the only positional indications of the location where the wisp is formed, but these limiting phases are difficult to determine; they are coincident with the intervals when the radial-velocity measurements of the primary component required rejection of several discrepant measurements from the final velocity curve (Figure 4).

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The wisp has very different properties from the brief BF spike at the binary-component alignment producing the transit eclipse: it is not rapidly changing its intensity with the orbital phase; instead, it shows a systematic drift in radial velocities within its visibility phase range, 0.58 < ϕ < 0.87 (the best visible in Figure 6; see also Figure 11). It stays confined to the middle of the profile of the secondary component, suggesting a feature possibly anchored on the stellar disk and apparently confined to one hemisphere since it does not reappear on the positive-velocity side of the system when the secondary component recedes from the observer (ϕ > 0.0). The wisp tends to be wider soon after its emergence from behind the primary component, with a full width at the base of about 35–40 km s⁻¹, then it narrows down to 15–20 km s⁻¹ before its disappearance. Its migration in the radial velocities relative to the center of mass of the secondary component can be approximated by a linear dependence on the orbital phase, Drift = 29.24(±0.64) + 217.50(±6.53) × (ϕ − 0.75) km s⁻¹ (Figure 12), where the single-measurement error of the relation, 3.3 km s⁻¹, is mostly due to measurement uncertainties.

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Assuming the projected rotational velocity of the secondary component of −66 km s⁻¹ and orbit-synchronous rotation, the constant intensity and the apparent radial-velocity drift of the wisp can be explained by a spatially stationary patch located on this component and visible for half of the orbital revolution. This patch of an increased strength of the spectral lines would be then located on the positive Y-axis side in Figure 10, at about 60°–65° away from the line joining the stars. This location would explain why the patch is invisible within the orbital phase range 0 < ϕ < 0.5, as well as the early appearance after mid-occultation at ϕ = 0.58 and an even earlier disappearance before mid-transit at ϕ = 0.87. These numbers assume a moderately low orbital inclination angle of i ≃ 70°, producing a grazing totality eclipse. The wisp, as a positive BF feature with the strength of about 1.5%–2% of the integrated BF intensity of the primary component, is produced by a localized enhancement of the absorption line spectrum. Although it is so weak, its strength is not negligible relative to the secondary component, where it contributes up to 20% of the integrated BF intensity.

The input spectra were individually normalized (to the unit level at the observed continuum) so that without simultaneous photometric observations we can only state that the spectral lines become stronger during the visibility of the wisp. But the mechanism is not known because the lack of independent information on the continuum precludes a proper interpretation of the wisp. We see two possible explanation of the enhanced spectrum producing the wisp.

• 1.  
  The patch of stronger lines is produced by a localized piling-up of the dense gas. Normally, material added on top of the atmosphere would simply shift the location of the surface corresponding to the optical depth of one. But, since we do not know the exact geometry of the gas circulation in the CrA system, this may be a viable solution. The broadband photometric effect of such a gas concentration is expected to be very small at a level of ≤0.1% since the wisp corresponds to a ≤2% modification of the spectrum, whose integrated equivalent width of all spectral lines in our spectral window is about 5.5% relative to the continuum level.
• 2.  
  A spot with a depressed effective temperature. The locally lower temperature would lead to a decrease in the brightness of the star and to a richer and stronger absorption line spectrum. Our spectra were normalized so that information on the stellar brightness was lost. This matter could not be resolved without broadband data, but we have indirect indications (see Section 3.4 and Figure 13) that the light-curve maximum after the secondary eclipse (ϕ ≃ 0.75) was about 2% lower than the first maximum. This would suggest that the dark spot causing the wisp may have indeed influenced the continuum level. Such a spot would most likely be very different from magnetic starspots because the drop in effective temperature must be rather moderate. Usually magnetic spots are dark and it is hard to detect their spectral contribution, whereas here the enhancement in the line spectrum would overcompensate for the decrease in brightness.

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We demonstrated for the case of AW UMa (Paper I) that, with some assumptions, the spectroscopic results can give a direct link to the photometric phase variability of this binary. (1) We utilized the BF normalization of the integrated equivalent width (IEW) of all spectral lines in the spectral window to the IEW of the model spectrum. Building upon the observed approximate constancy of the effective temperature with the orbital phase, (2) we made a strong assumption that the spectral lines of the primary component remain constant with the orbital phase. The results for CrA are shown in Figure 13, while additional explanations of the two assumptions are as follows.

• 1.  
  The BF's for AW UMa and CrA were determined using the same model atmosphere, F2V for the solar abundance. The IEW in AW UMa showed only a very small IEW phase variation of about 2% around the value of 1.10 (Figure 2, Paper I). This agreed with the approximate constancy of the effective temperature and could be interpreted as a slightly later spectral type of AW UMa than of the template. In CrA, the IEW is larger, with a mean value of 1.18 (Figure 13), in agreement with suggestions of an even later spectral type of the binary. The IEW shows about 5% departures from the mean value, reaching 1.23 at the orbital phases around ϕ ≃ 0.75. We note that these are the phases when the secondary component approaches the observer and when the wisp-producing feature (Section 3.3) could influence the continuum spectrum; apparently, the feature produces stronger absorption lines as if the local effective temperature were lower. The IEW for CrA is not as invariable with phase as that for AW UMa; the difference between the binaries may possibly relate to their different orbital inclination angles and the different visibility of their orbital planes, where—according to the model of Stȩpień (Section 4.2)—the energy is transported between the binary components.
• 2.  
  The assumption (2) states that the primary component does not change with the orbital phase and remains a rapidly rotating star describable by a rotational profile; it is oblivious to the existence of the secondary component. In other words, whatever happens in the system in terms of the line-producing photospheric area is due to the secondary and the matter around it. The assumption is made simply for convenience; its correctness is verifiable by the result.

The "spectroscopic light curve" is calculated using the maximum value (S₁) of the rotational profile of the primary component to form a phase dependence of its inverse C/S₁ (see the discussion in Paper I, Section 3). It turns out that such a dependence is a very reasonable approximation to the photometric light curve, as shown in the upper panel of Figure 13, confirming our assumptions. In Paper I, we used for C the normalized integral of the rotational profile, but here we simply use an arbitrary constant adjusted to produce a light curve normalized to unity at the maximum light. Such a light curve is correctly phased and can be compared with the calibrated light curves of Knipe (1967) (shown in Figure 13), Tapia (1969), and Hernández (1972). Only the primary (transit) eclipse comes out too shallow, but this is expected since the assumption of no apparent change to the profile of the primary component breaks down during these phases.

In very simple terms, the "spectroscopic light curve" derived in the above way expresses the amount of the spectral line-producing surface projected onto the sky. Figure 13 tells us that this amount is very close to what can be calculated using the Lucy model. Thus, such a spectroscopically derived light curve does carry the same information about the binary—but admittedly in a more convoluted way—as a standard photometric light curve.

The successful application of our approach may turn out to be restricted to systems with a low mass ratio such as our two targets, AW UMa and CrA, where the primary components strongly dominate in the energy balance. However, thanks to the steep mass–luminosity relation in the lower main sequence, the domination of the primary component may extend to moderate mass ratios. The range of binary parameter for the above approach remains to be determined.

The new, spectral time-monitoring observations of CrA show the binary as a twin to AW UMa analyzed in Paper I. The observations were more extensive (3.1 times longer total exposure) and more accurate (the observational S/N improved by about a factor of two) so that the results carry more weight in a discussion of both binaries. CrA appears to be slightly more massive, as expected for a longer orbital period, but the spectral types are practically identical, although CrA may have more evolved cores judging by its larger luminosity and the slightly lower effective temperature. The two binaries are very similar in having small mass ratios (q_sp = 0.10 and 0.13). Most importantly, they show an identical disagreement with the assumptions of the Lucy model by having an apparently stationary velocity field in the rotating frame of the binary. The gas flows are best visible within the profile of the secondary component and in the volume between the components. We did not see any obvious changes in the flow velocity patterns over 27 orbital periods, so CrA may have more stable internal velocities than AW UMa; the latter, however, was not observed as extensively as CrA. In CrA, a "wisp" (Section 3.3) in the 2D time-sequence images of the surface velocity motions, associated with the secondary component, was seen to be stable over the duration of the observations. Similar but less well-defined velocity features were seen in AW UMa. Such features are most probably produced by gas streams seen along their longer dimensions through accumulated optical depth or they may come from isolated areas of enhanced spectral-line strength of currently unexplained properties.

The general stability of both binaries is confirmed by the moderate systematic variations in period, with dP/dt at a level of a few × 10⁻⁷ day yr⁻¹. The changes in period in these two binaries are of opposite sign, hence they are most likely unrelated to their secular evolution. They may be explainable by a small imbalance in the net mass transfer between the binary components or by external perturbations; we recall that W UMa binaries practically always exist in triple systems: see Tokovinin et al. (2006) and Rucinski et al. (2007). Since large amounts of matter carrying radiation energy appear to circulate between the components (see Section 4.2), the flows must be well balanced in terms of the net amount of matter transferred between the stars.

The primary in both binaries is a rapidly rotating early F-type star with a projected equatorial rotational velocity $V\sin i$ at about 80% of the rate synchronous with the orbital motion (see Section 3.1). The BF profile appears symmetric and well represented by the standard rotational profile. The slower-than-expected rotation suggested for the higher latitudes of the primary may partially result from the anticyclonic gas motion (in the counter-rotating direction) seen in the mass-losing hydrodynamic models of Oka et al. (2002). Gas velocities in such a high-pressure cell are expected to link the high stellar latitudes of the primary to the outflow from the Lagrangian L₁ point. Only a modest reduction of the observed rotational velocity is expected, however, perhaps by a value of the order of the sonic or subsonic photospheric velocity, which for the spectral types of both stars is ≃7 km s⁻¹. The pedestal at the base of the primary rotational profile with the additional broadening of 25–35 km s⁻¹ would bring the observed $V\sin i$ into approximate agreement with the synchronous rate with the orbit. The pedestal contributes moderately to the total depth of the spectral lines, typically at a level of 0.06–0.10 of the integrated rotational profile of the primary. The pedestal likely reflects the presence of the matter in the orbital plane, returning to the primary after its orbit around the secondary component.

The secondary components of W UMa-type binaries are important for our understanding of the unresolved secret of how these binaries function as stable structures. The secondaries of both CrA (Section 3.2) and AW UMa (Paper I) look similar in that they show changes in orbital phase that are repeatable and stable in time, and appear to be related to the phase-dependent visibility of an otherwise stationary velocity field. We identified two features in the somewhat complex radial-velocity picture: (1) a stream directed toward the observer over the secondary surface, accumulating to sufficient optical line depth within a narrow cone of visibility; (2) a region where the absorption line spectrum is for some reason stronger than in the surrounding region; it can be a region of a local, geometrically complex piling-up of the gas, possibly through collision of a direct flow and a circum-secondary flow from the primary, or an area of a slightly lower effective temperature with an enhanced absorption line spectrum. These features may be the sought-after surface imprints of a deeper energy transport from the primary component to the secondary. In spite of the complex radial-velocity picture, the amount of the visible, line-producing photosphere of the secondary and of the matter between the stars approximately agrees with that predicted by the model of Lucy (1968b) based on the Roche model.

Our observational results for AW UMa and CrA relate to two W UMa-type systems with a small mass ratio. Despite their small mass ratios, both binaries have always been considered genuine members of and representatives for the whole class of these objects. We assume that our results indeed apply to the whole class of W UMa-type binaries.

The available radial-velocity data for AW UMa and CrA show that the main assumption of the model of Lucy (1968a, 1968b), the solid-body rotation of the components as a prerequisite for the definition of the Roche-model equipotentials, is not confirmed. A complex system of velocities in the rotating system of coordinates is particularly clearly visible within the phase-dependent radial-velocity profiles of the secondary component. The velocities are moderate, of the order of or less than the rotational velocity of the secondary component, while their observed ranges are of the order of the local sound speed. But their spatial pattern is of great importance because we see circulation fully confined to the secondary component, similar to that expected on a non-contact binary component. The impact of such well organized velocity systems on the assumed W UMa-binary model and on interpretation of light-curve observations is currently unknown and should be investigated.

The solid-body assumption at the time of creation of the Lucy model was simply the easiest to make, although no mechanism had been offered as to why the components would rotate in such a way. This assumption made the Lucy model particularly easy to use as a light-curve modeling tool so that many light-curve solutions have been derived utilizing the model's simplicity. Most of these solutions produced determinations of the photometric mass ratio, q_ph, utilizing the one-to-one mapping between the mass ratio and the relative dimensions of the components, first discussed by Mochnacki & Doughty (1972). While this relationship has been particularly useful for light-curve solutions, it appears that it may require a correction or modification. This had been signaled by Goecking & Duerbeck (1993), was strongly confirmed for AW UMa, and is now indicated for CrA. While the discrepancy between the spectroscopically determined q_sp and photometrically determined q_ph for CrA is indicated at the level of 4σ (with possibly overoptimistic error estimates), the discrepancy (q_sp − q_ph)/q_sp for AW UMa reached the unbelievable relative value of ≥10%! This suggests that either the spectroscopic or the photometric determinations of q may be biased. We point out that unrecognized but omnipresent companions tend to push photometric solutions toward smaller mass ratios and/or smaller orbital inclinations by a change in photometric scale; no equally simple biasing mechanism exists for spectroscopic determinations of the mass ratio executed at an adequate resolution. Whatever the reason, the continuing routine use of the Roche lobe-based photometric model to determine the values of q_ph may give systematically wrong individual results, biasing statistics of the mass ratio, masses, and orbital inclination angles.

The model of Lucy (1968a, 1968b), when it was developed more than half a century ago, offered the first physically acceptable description of W UMa-type binaries. It generated extensive interest, but also considerable controversy after its implied properties were studied in more detail. The main difficulties were related to the internal structure and effective-temperature equalization processes. Further research led to the consideration of the thermal cycles consisting of semi-detached and contact configurations (Flannery 1976; Lucy 1976); the problems did not disappear but were related to the structure during a part of the thermal cycle after re-establishment of the contact. The discussion culminated in detailed studies of the energy transport in contact by Webbink (1976, 1977) and a heated discussion of the concept of a "contact discontinuity" (Shu et al. 1976, 1979; Hazlehurst & Refsdal 1978; Papaloizou & Pringle 1979). The "discontinuity" concept was important because it envisaged the secondary component fully enshrouded in the matter of the primary component. This assumption solved the equality of the effective temperatures while the rest of the observables, particularly the solid-body-rotating equipotential structure, remained as in the original Lucy model. The end to further discussion and similar first-principles considerations came from the realization that—with only photometric data—there exist no observational verification methods for this complex problem. The model concept of Stȩpień (2009) returned to the idea of the gas of the primary component surrounding the secondary, but now not as a static equipotential structure but rather as a stationary flow of optically thick matter confined to the equatorial regions and eventually returning to the primary. In its motion, the flow is controlled by the gravitational pull to the orbital plane and the dispersing tendency of the Coriolis force in the plane. The Stȩpień study specifically addressed the energy transport requirements and did not offer any predictions of the observational consequences of the flow, making his ideas less directly applicable at the present stage than the Lucy model.

Our detection of the organized gas flows in AW UMa and CrA supports the energy transport model of Stȩpień (2009) and the pioneering hydrodynamic calculations of Oka et al. (2002). The essential component of a new vision is a stationary flow that dominates the thermal equilibrium of the binary and which starts and ends on the primary component. The flow submerges the equatorial regions of the secondary in optically thick gas. It leaves the primary at high latitudes to slide down to the low latitudes of the L₁-point region, where it starts its orbit around the secondary component, first curving to one side of the binary (the negative Y-axis side in Figure 10), and then returning to the primary at its low latitudes. When all the matter returns to the primary and there is no net mass transfer between the components, then no changes in orbital period are expected to be observable. The stream at its return may lead to formation of the perturbed region identified between the stars in CrA; it may also excite the standing atmospheric waves visible as ripples in AW UMa. We see only the outermost layers of the moving matter, although the process may involve large amounts of matter carrying substantial energy. Generally, only moderate observable effects are expected, such as those we detected in AW UMa and CrA, but some details are not easily explainable, e.g., the phase-dependent velocity extent of the secondary component.

Two directions can be followed for further research to develop modifications of the current W UMa-binary model: the theoretical one specifically addressing the detailed properties of the Stȩpień energy transfer model, perhaps following the directions established by the pioneering hydrodynamical models of Oka et al. (2002), and the observational one based on high-resolution time-sequence spectroscopy. Both may present considerable difficulties. The former would require extensive use of supercomputers in a simulation incorporating complex physical phenomena, while the observations would be difficult to arrange because they require relatively large amounts of time on moderate- or large-size telescopes equipped with high-efficiency, high-resolution spectrographs. As always, the observations would provide the necessary guidance, but are harder to generalize than theoretical models. The result of the effort is expected to lead to a paradigm shift from an elegant picture of a static, solid-body, Roche-model structure with the inexplicable equality of temperature over its surface, to a more complex vision of a gaseous binary approximately agreeing with the Roche model, with the primary component engulfing the secondary component in its equatorial regions by an optically thick, stationary flow carrying the temperature-equalizing energy.