A&A 376, 1080-1089 (2001)
DOI: 10.1051/0004-6361:20011007
R. Knaack 1 - M. Fligge 1 - S. K. Solanki2 - Y. C. Unruh3
1 -
Institute of Astronomy, ETH-Zentrum, 8092 Zürich,
Switzerland
2 - Max-Planck-Institut für Aeronomie, 37191 Katlenburg-Lindau, Germany
3 - Institute of Astronomy, University of Vienna, Türkenschanzstr. 17, 1180 Vienna, Austria
Received 10 April 2001 / Accepted 5 July 2001
Abstract
Compared with Sun-like stars, the irradiance variations of the Sun over the solar cycle appear to be relatively small for its average activity level (Lockwood et al. 1992; Radick et al. 1998). It has been proposed that the special position of Earth-based observers in the ecliptic plane may give the impression of a subdued solar photometric variability (Schatten 1993). The aim of the present paper is to examine the influence on irradiance variations of a solar rotation axis inclined towards the observer. A three-component model is used to calculate relative flux variations of a given active-region distribution on the surface of the Sun as a function of inclination and wavelength. Wavelength-dependent intensity spectra are used to describe the contributions of the undisturbed photosphere, sunspots and faculae. The spectra result from models that have successfully been used to reproduce a host of solar data and thus represent realistic estimates of the radiative output from these solar features. We find that an inclined rotation axis increases the total solar irradiance variations maximally by .
The most probable value is approximately
.
This is much less than that suggested by former studies, which were based on simple contrast functions. In the averaged Strömgren filters we estimate a most probable increase of the solar variability of
.
In addition, we estimate the dependence of the flux in the chromospheric Ca II H&K lines on inclination. We find that the average chromospheric activity level depends only slightly on the inclination angle. The chromospheric variability of Sun-like stars, however, is significantly affected. Nonetheless, our results indicate that a different average inclination of stellar rotation axes relative to the observer cannot explain the discrepancy between the brightness variations of the Sun and Sun-like stars.
Key words: Sun: activity - Sun: sunspots - Sun: faculae, plages - Sun: photosphere - stars: activity
The total irradiance of the Sun exhibits a prominent eleven-year cycle with an amplitude of about
(Fröhlich 2000), which is correlated with the appearance and evolution of magnetic active regions on the solar surface. The small imbalance between the excess flux of faculae and the flux deficit of sunspots causes the Sun to be brighter during activity maximum than during activity minimum (Foukal & Lean 1988). Measurements of stellar Ca II H&K flux (used as a proxy of chromospheric magnetic activity) and contemporaneous Strömgren
photometry (at
and
,
respectively) have revealed that the photometric variability of the Sun is subdued by a factor of about 2-3 compared to Sun-like stars of similar magnetic activity (Lockwood et al. 1997; Radick et al. 1998, henceforth referred to as R98).
This gap cannot be explained by increased spectral irradiance variations of the Sun towards shorter wavelengths. Model calculations by Fligge et al. (1998) indicate that the solar spectral variability in the Strömgren filters does not exceed the total irradiance variations by more than 30%.
The special position of a terrestrial observer, who sees the Sun almost equator-on, combined with the fact that solar active regions are confined to low and intermediate heliographic latitudes has been proposed as another explanation for the discrepancy between the Sun and Sun-like stars (Schatten 1993, henceforth referred to as S93). An observer high above the ecliptic plane would see a different distribution of sunspots and faculae on the solar disc since the active regions would appear closer to the limb, which is illustrated in Fig. 1 for two zonal bands.
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Figure 1:
Visibility of two zonal latitude bands between
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This affects the observed solar irradiance variations because white-light faculae have a higher contrast near the limb than at disc centre while the contrast of sunspots changes only slightly with limb distance. Altogether, a net increase of total irradiance variations is expected when viewing the Sun out of the equatorial plane. Therefore, assuming a random distribution of stellar rotation axes with a most probable inclination of
(R98), this effect tends to enhance the photometric variability of Sun-like stars. At the same time we expect the apparent activity level, as judged by the star's Ca II H&K flux, to decrease with decreasing inclination i since the Ca II H&K intensity contrast is roughly independent of limb-distance and the projected area of the active-region bands decreases with i (cf. Fig. 1).
S93 placed active regions as individual dark and bright features, representing sunspots and faculae, on the surface of an artificial Sun in order to investigate the influence of an inclined rotation axis on solar (stellar) irradiance variations. The centre-to-limb variations of the quiet Sun and the active regions were described by quadratic contrast functions adopted from Sofia et al. (1982). The initial variability of the total solar irradiance was set to ,
which yielded a ratio of 5:4 for the contributions from faculae and sunspots. Then, the flux variations for several inclinations were calculated. In the most probable case (i.e. at
), S93 obtained an increase of the total solar irradiance variability by roughly a factor of 3, which would be sufficient to explain the difference between the Sun and Sun-like stars in the sample of Lockwood et al. (1992).
However, R98 repeated the computations using the same contrast functions, but a more appropriate cyclic variation of
(Fröhlich 2000). Moreover, they applied a different ratio of 2:1 for the flux contribution of faculae relative to sunspots and an active-region distribution that was completely smeared out over latitude bands. In contradiction to the results of S93, R98 obtained a most probable increase of a factor of only 1.36, which would make the inclination effect of only marginal importance for explaining the observed difference between the Sun and Sun-like stars.
In the following sections, we provide further evidence that the impact of an inclined rotation axis on total solar irradiance variations is small (even smaller than found by R98) and hence not likely to explain the discrepancy between the Sun and Sun-like stars. We improve on the earlier investigations in two ways. Firstly, we make use of empirical atmospheric models to describe the quiet Sun, sunspot and facular components, which is in contrast to the quadratic contrast functions used by S93 and R98. The models have successfully been used to reproduce measurements of facular contrasts (Unruh et al. 1999) as well as total and spectral irradiance variations (Fligge & Solanki 2000; Fligge et al. 2000a,b). Secondly, we explicitly consider the spectra of each of these components so that we are able to model the fluxes at different wavelengths, e.g. in the Strömgren
filter-bands used for stellar observations.
In addition, we consider the dependence of the Ca II H&K activity level and variability on the inclination angle. A relative shift between the activity levels of the Sun and Sun-like stars could also lead to a seeming discrepancy between their brightness variations since stellar brightness variability depends on the level of activity (R98). We find that the average activity level only marginally depends on i whereas the cyclic Ca II flux variation between activity minimum and maximum is significantly affected.
Our model is described in detail in Sect. 2. In order to verify our model assumptions, we reproduced the results of S93 and R98. This is shown in Sect. 3. New model calculations for the total irradiance of a star identical to the Sun seen at different i as well as the results for selected wavelength regions and Ca II H&K flux are presented in Sect. 4. Finally, the results are summarized and our conclusions are given in Sect. 5.
Our analysis of the inclination effect hence depends on two basic factors. Firstly, the intensity spectra for the individual components and secondly, the geometric configuration of the magnetic features on the solar surface.
The quiet-Sun intensities were obtained with Kurucz' standard solar model atmosphere of
.
The spot intensities (averaged over umbra and penumbra) were calculated from a model atmosphere of
which in turn was interpolated from the Kurucz grid of model atmospheres. This temperature corresponds to a ratio of umbral to penumbral area of approximately 1:3. Model P of Fontenla et al. (1993) was used as a starting point for the facular model. Slightly modified, this model has been remarkably successful in reproducing the spectral irradiance variations obtained by VIRGO (Fligge et al. 2000a; Fligge et al. 2000b). A detailed description of the models and the resulting spectra can be found in Unruh et al. (1999).
The brightness of active regions relative to the undisturbed photosphere is expressed in terms of the contrast
,
which depends on the wavelength range
and on the limb distance parameterized by
,
with
being the angle between the line-of-sight and the normal to the Sun's surface. The contrast is defined by
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Figure 2:
Facular contrast (
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ratio |
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0.36 | 2.06 | 5.7 |
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0.14 | 0.92 | 6.6 |
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0.02 | 0.33 | 16.5 |
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0.01 | 0.13 | 13.0 |
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0.03 | 0.24 | 8.0 |
Sofia et al. (1982) | 0.005 | 0.167 | 33.4 |
Compared to the contrast functions applied by S93 and R98, our spot contrast remains practically independent of
for the total wavelength range (Fig. 2a). Note that foreshortening effects due to the Wilson depression were neglected since we do not model individual active regions but active-region bands which are completely smeared out (cf. Sect. 2.2). We do not expect this simplification to significantly influence the results since further investigations showed that the gradient with respect to
of the spot contrast only marginally influences the relative flux variations. However, even small changes of the facular contrast can have a strong impact on the results (cf. Sect. 4).
Recordings of the total spot area
during solar cycle 22 suggest a value of
at maximum activity (Chapman et al. 1997). Butterfly diagrams for the same period show that sunspots were mainly confined to a band between
and
on both hemispheres. We hence fixed our sunspot bands accordingly and allowed the associated facular bands to range between
and
.
In order to achieve an irradiance increase of
(Fröhlich 2000) between activity minimum and maximum at an inclination
,
the facular area
had to be set to
.
This corresponds to a ratio of
,
which is in good agreement with observations by Chapman et al. (1997).
Disc integration of the intensities over all visible surface elements finally yielded the total flux
of the Sun at activity maximum as a function of i:
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(2) |
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(3) |
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(4) |
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(5) |
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(6) |
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Figure 3:
Dependence of the projected area a on the inclination i of the solar rotation axis for spots (![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Disc-integrated chromospheric Ca II H&K line emission is widely used as a proxy of solar and stellar magnetic activity (Skumanich et al. 1975; Wilson 1978; Schrijver et al. 1989). The underlying assumption is that the summed H&K flux reflects the amount of nonthermal chromospheric heating above faculae, both in active regions and network, which in turn is associated with surface magnetic fields. An instrumental index of the Ca II H&K flux is the S-index, which is proportional to the summed H and K fluxes normalized by the summed fluxes of two nearby continuum bands (see R98 for details).
As discussed in Sect. 4.3, the Ca II flux also depends on inclination. Hence, we investigated S as a function of i. Measurements of Skumanich et al. (1984) indicate that the facular intensity at the solar Ca K line centre darkens the same way as the quiet-Sun intensity with decreasing .
Since our intensity spectra were calculated under the assumption of LTE, they do not describe the Ca II H&K line cores accurately. Thus we adopted
from Skumanich et al. (1984).
Under the explicit assumption that the facular contrast in the Ca II H&K lines remains constant over the disc, i.e.
The Ca II K line profile has been measured by two independent programs conducted at the Sac Peak Evans' Facility and the Kitt Peak McMath-Pierce telescope. We used the daily Sac Peak K-index (
), which can be adjusted to the monthly Kitt Peak K-index (
)
by
(White et al. 1998). The transformation between
and S is given by R98 as
.
We thus obtained
for the minimum in 1986 (which is close to the value of 0.169 suggested by R98) and
for the maximum in 1990. These values imply that
.
The dependence of
on i is presented in Sect. 4.3.
In contrast, our calculations employ intensity spectra for faculae, spots and the quiet Sun as described in Sect. 2.1. Moreover, we explicitly fix the spot area
according to observations and tune the facular area
(and hence the relative flux variation
)
in order to obtain a cyclic variation of
.
Consequently,
and
are outputs of our model and can hence be compared to observations.
The models also differ in the assumed geometric distribution of the active regions. S93 placed individual dark and bright features on the solar surface while R98 used smeared distributions, which is similar to our procedure.
S93 placed the individual groups of spots and faculae within two bands ranging from
to
on each hemisphere. The areas were adjusted to provide a cyclic variation of about
,
which is in reasonable agreement with the results of the uncorrected ERB radiometer on Nimbus 7 (Kyle et al. 1994) but almost a factor of 2 larger than the currently accepted value. Besides, the numbers of S93 imply an (unsigned) ratio of facular-to-spot flux variation of
.
The original results of S93 are shown in Fig. 4a. The relative flux variation
(solid line) starts at
(
)
and increases up to
(
), i.e. by a factor of 6, followed by a pronounced decline. For the statistically most probable case of
,
the irradiance increases by a factor of about 3. The dotted line shows the contribution of spots
and the dashed line the contribution of faculae
.
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Figure 4:
Relative variation
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Our reproduction under the same conditions is shown in Fig. 4b. Instead of our intensity spectra we hence used the polynomials proposed by Sofia et al. (1982), i.e. the same limb-darkening function for the undisturbed photosphere, the same contrast functions for the active regions and finally a similar geometry like the one applied by S93 (although we used smeared latitude bands). It turned out that it was necessary to employ very large facular and spot areas to fulfil the boundary conditions of S93 (i.e.
and
at
), namely
and
.
This is a factor of about 10 larger than observations of the active Sun imply. In agreement with S93, we then obtained a maximal increase between
and
of a factor of 6 while the value for the most probable inclination was about 2. As can be seen, however, there is a substantial difference between the result of S93 and our reproduction regarding the dependence of the flux variation on inclination, which is probably due to the different distributions applied.
In a next step we replaced the limb-darkening function of the quiet Sun and the contrasts for spots and faculae by our intensity spectra. However, the relative flux variations at
were kept, i.e.
and
.
This reduced the necessary total facular and spot areas to
and
,
which is still too high but significantly less than before. The result is shown in Fig. 4c. The variation increases now maximally by a factor of 2.7, in the most probable case only by a factor of 1.3. Our facular and sunspot models hence predict a much smaller inclination effect, even if we employ the cyclic variation of 0.18%.
Again we reproduced their results by employing the same boundary conditions and the same limb-darkening and contrast functions. Because of the similar geometry we expected better accordance this time. Figure 5 shows our reproduction for
and
.
We calculated a most probable variation of
and a maximum variation of
.
This is in good agreement with R98. However, the total facular area of
implies a facular-to-spot ratio of
whereas observations by Chapman et al. (1997) suggest a ratio of
for the solar maximum.
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Figure 5:
Reproduction of the inclination effect using the same
contrast and boundary conditions as R98, i.e.
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The disagreement between the prediction of S93 and the one of R98 indicates that the inclination effect strongly depends on the boundary conditions, i.e. the exact values of
and the ratio
.
Our analysis reduced the free parameters (besides the geometric distribution) to the total sunspot area
and the total irradiance variation
(which directly determine the total facular area
and the ratio
:
). Both parameters are accessible to modern observations. The boundary conditions of S93 required unrealistically large areas for active regions. The values assumed by R98 implied a spot area of
and a ratio of
.
While the Sun may have reached such a large
value at the maximum of cycle 19, the most active cycle on record, the facular-to-spot area ratio of 25 is definitely too large for solar maximum so that the
value itself is too large.
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Figure 6:
Relative variation
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We have obtained a most probable amplitude for the cyclic variation of
and a maximum amplitude of
.
These variations are significantly smaller than the values found by R98, which is mainly due to the different centre-to-limb variation (CLV) of the facular contrast of our model (cf. Fig. 2). Using an alternative facular contrast, R98 found a most probable variation of
,
which is closer to our result and underlines the strong influence of the CLV of the facular contrast. We also investigated the influence of varying the input parameters: changes of
in the spot area or the total irradiance variation (at
)
resulted in a range from
to
for the most probable amplitude and from
to
for the maximum.
Further test calculations showed that changes in the linear spot contrast function hardly affect the gradient of
,
whereas changes in the facular contrast have a significant influence. Nevertheless, it is unlikely that a maximum increase exceeding a factor of 1.7 can be obtained for reasonable values of the facular area
.
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Figure 7:
Relative flux variation
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There is a sharp rise in
for wavelengths smaller than 450 nm due to the increasing facular variability
in the ultraviolet. Unexpectedly,
becomes larger than
for
.
The irradiance variability hence decreases for smaller values of i, which is the opposite to the behaviour at wavelengths longer than
.
This is seen more clearly in Fig. 7b, which shows the change of
from
to
and
,
respectively. The reasons for this behaviour are discussed below in conjunction with Fig. 9.
The passbands of the Strömgren
filters, which are used for the photometric observations of Sun-like stars, are centered at
and
.
For the averaged filters, i.e. (b+y)/2, we estimate a most probable increase of a factor of roughly 1.3 and a maximum increase of about 2.2. However, the facular contrast in these passbands (and hence
and
)
sensitively depends on the exact temperature of the facular model at the relevant atmospheric layers, which is not well determined due to the lack of suitable observations. The values for the variability at the Strömgren wavelengths can therefore only be taken as a first estimate. More reliable estimates must await, for instance, resolved measurements of solar spectral irradiance variations which will allow tighter constraints on the employed facular model.
Finally, we calculated the relative flux variability for the passbands
,
,
and
.
The results are shown in Fig. 9. There is again a decrease in
from
to
for the filters with
(panels a and b).
The reason lies in the CLV of the facular contrast. For
,
the increase of
from disc centre (
)
to the limb (
)
is too small to compensate for the reduction in projected facular area (cf. Table 1 and Fig. 2). This results in a monotonicly falling facular variability
from
(maximal area) to
(minimal area). Since the qualitative behaviour of
does not vary significantly with wavelength, the wavelength dependence of
is mainly dominated by
.
For faculae with a total area of
and
located within latitude bands from
to
,
the projected surface area drops from
at
to
at
,
i.e. by
(cf. Fig. 3). Since the quiet Sun also contributes to the measured Ca II H&K flux, the chromospheric flux is expected to change by a smaller factor with i.
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Figure 8:
S-index vs. i (left axis) calculated with Eq. (8) for the active Sun and
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Figure 8 shows the S-index of the active Sun as a function of i according to Eq. (8). At
,
S is equal to 0.187. As i changes from
to
,
S drops from 0.187 to 0.179, i.e. by
.
From
to
,
the drop is only
.
However, the effect is much larger for
where
is the S-index of the Sun at activity minimum and hence independent of i. From
to
,
decreases by
,
from
to
even by
(i.e. nearly by a factor of 2).
Hence, when comparing stars with similar apparent Ca II H&K fluxes, we might actually look at stars with different true Ca II H&K emissions. This bias, again, is introduced by the different inclination angles of the stellar rotation axes. Although the magnitude of the bias is in the range of only a few percent for the absolute flux, it has a significant impact when the flux variation between activity minimum and maximum is considered. Measurements of the temporal Ca II H&K flux variation thus tend to underestimate the true cyclic chromospheric variability of Sun-like stars.
The i-dependence of
may hence partly be responsible for the dispersion seen in the cyclic chromospheric variation of the sample of Sun-like stars investigated by Lockwood et al. (1992) and R98. However, as explained above, the absolute chromospheric activities are only marginally affected. Therefore, we do not expect that this effect will significantly influence the gap between the broad-band brightness variations exhibited by the Sun and Sun-like stars.
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Figure 9:
Relative flux variation
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model | spots | faculae | ![]() |
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max | prob |
Schatten |
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5 : 4 | ![]() |
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Radick |
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2 : 1 | ![]() |
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present analysis |
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We have investigated the influence of an inclined rotation axis on the total irradiance variations of the Sun, using atmospheric models to calculate the centre-to-limb variation of the quiet photosphere and the contrasts of faculae and sunspots. These models have already been successfully employed to reproduce time series of solar irradiance on time scales ranging from days up to the length of the solar cycle. We have found that the effects of an inclined rotation axis on solar variability have been overestimated. This is summarized in Fig. 10, which compares our results for the total irradiance with the calculations made with similar choices of the parameters as R98 and S93. The results of both studies could be roughly reproduced by our model, but only under assumptions concerning the employed spot and facular areas that are at best only partly consistent with observations. The parameters needed for the reproductions are listed in Table 2.
Based on a model that is consistent with observational constraints, we predict an increase of the total solar irradiance variations of
when decreasing i from
(solar case) to the most probable case of
.
The predicted maximum increase is
when decreasing i from
to
.
Considering the spectral irradiance variations in the Strömgren b & y filters,
we estimate a most probable increase of roughly
for the averaged (b + y)/2 filter. Although the amount of the flux increase depends sensitively on the facular contrast, it appears unlikely that the excessive irradiance variations of other inactive cool stars can be explained by an inclination effect. This is in agreement with the conclusions of R98 and recent observations by Henry (1999).
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Figure 10: Comparison of the inclination effect for the flux integrated over the total wavelength range predicted by our model and under the assumptions of R98 and S93. |
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Moreover, we have found that the Ca II flux variation of Sun-like stars between activity minimum and maximum significantly depends on the inclination of the rotation axis relative to an observer. This result indicates that the chromospheric variability of Sun-like stars may have been systematically underestimated by up to a factor of 2. However, the averaged Ca II flux depends only marginally on the inclination and should be a reliable measure of the chromospheric activity.
Although we have taken care to keep the parameters of our model realistic, constraining them wherever possible by observations, some uncertainties remain. Firstly, the work of Topka et al. (1997) and Ortiz et al. (2000) suggests that a single facular model may not be sufficient to describe the brightness signature of both active region faculae and active network features (whose surface area coverage also changes somewhat over the activity cycle). Secondly, we have only considered G2V stars whereas the sample investigated by R98 exhibits a considerable range in spectral types. Stars with different spectral types exhibit different spectral variability. Also, it is unclear if the ratio between spots and faculae on such stars is exactly the same as on the Sun. Thirdly, we have not taken into account a possible change in the brightness of the poles over the solar cycle. At activity minimum, when the Sun's global magnetic field is closest to a north-south dipole, there is an excess of magnetic flux at the poles. To what extent this influences the irradiance variations at different inclinations is unknown.
Acknowledgements
We would like to thank A. C. Cameron for the use of his Doppler imaging code. Y. C. Unruh would also like to acknowledge support through the Fond zur Förderung der wissenschaftlichen Forschung under grant S7302-AST.