BM Camelopardalis (=12 Cam) is a chromospherically active binary star with a relatively large orbital eccentricity. Systems with large eccentricities usually rotate pseudosynchronously. However, BM Cam has been a puzzle since its observed rotation rate is virtually equal to its orbital period indicating synchronization. All available photometry data for BM Cam have been collected and analyzed. Two models of modulated ellipticity effect are proposed, one based on equilibrium tidal deformation of the primary star and the other on a dynamical tidal effect. When the starspot variability is removed from the data, the dynamical tidal model was the better approximation to the real physical situation. The analysis indicates that BM Cam is not rotating pseudosynchronously but rotating in virtual synchronism after all

Busby, Michael R.

Hall, Douglas S.

Teke, Mathias

English

NASA Technical Reports Server

N91-28 104
SOLVING THE BM CAMELOPARDALIS PUZZLE
Mathias Teke
Department of Mechanical Engineering
Tennessee State University
Nashville, Tennessee
Michael R. Busby
Center of Excellence in Information Systems
Tennessee State University
Nashville, Tennessee
Douglas S. Hall
Dyer Observatory
Vanderbilt University
Nashville, Tennessee
John Davis
Technical Monitor
NASA/Marshall Space Flight Center
ABSTRACT
BM Camelopardalis (= 12 Cam) is a chromospherically active binary star with a relatively large
orbital eccentricity. Systems with large eccentricities usually rotate pseudosynchronously. However, BM
Cam has been a puzzle since its observed rotation rate is virtually equal to its orbital period indicating
synchronization. All available photometry data for BM Cam have been collected and analyzed. Two
models of a modulated ellipticity effect are proposed, one based on equilibrium tidal deformation of
the primary star and the other on a dynamical tidal effect. When the starspot variability is removed from
the data, the dynamical tidal model was the better approximation to the real physical situation. The
analysis indicates that BM Cam is not rotating pseudosynchrnously but is rotating in virtual synchronism
after all.
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https://ntrs.nasa.gov/search.jsp?R=19910018790 2020-03-19T16:43:18+00:00Z
SOLVING THE BM CAMELOPARDALIS PUZZLE
BM Camelopardalis (= 12 Cam) is an SB1 in which a K1 giant orbits its unseen companion star
in an eccentric 80-day orbit. The K1 giant shows strong emission in its H and K lines of Ca II, so it is
chromospherically active (Abt, Dukes, and Weaver 1969). The photometric variability, produced by
longitudinally concentrated starspot regions, was discovered by Eaton et al. (1980). Hall and Osborn
(1986) analyzed the photometric variability and found two periodicities, 0.3% faster than and 1.0%
slower than the 800.174469 orbital period, presumably caused by two starspot regions at different
latitudes. Hall (1986) calculated that, if the K1 giant is rotating pseudo-synchronously, the photometric
period should be 45 °. 2 -+ 20. 5. He noted that it was a disturbing coincidence to have the observed
rotation rate virtually equal to the orbital period. If the rotation is not pseudosynchronous, then any
period faster than or slower than 45 days would be possible, with no reason for 80 days to be preferred.
This is the BM Cam puzzle.
To solve this puzzle we collected all available photometry. The 1979 and 1980 photometry
discussed by Eaton et al. (1980) was not published but was available in our files. Fernandes (1983)
published some 1983 photometry. We had in our files photometry from the years 1980 through 1985,
obtained by 15 different observers and sent to us for analysis. Photometry obtained between 1983 and
1987 by the 10-inch automatic telescope in Arizona will be published by Boyd, Genet, Busby, Hall, and
Strassmeier (1989). The preliminary analyses by Nelson et al. (1987) and by Strassmeier, Hail, Boyd and
Genet (1989) were based on subsets of these data. We also had access to 1988 photometry obtained by
the Vanderbilt 16-inch automatic telescope on Mt. Hopkins (tiall 1988). The analysis in this paper is
restricted to the V-band data, which was the most extensive. All telescopes except that of Fernandes
used the same comparison star, HR 1688. To compensate, we added --0.M995 tO his differential
magnitudes.
One source of photometric variability which should be exactly in phase with the orbital period is
the ellipticity effect (Morris 1985). That, however, should produce two maxima and two minima during
each orbital cycle and thus show up in a periodogram at P = 40 days. A periodogram of the entire data
set showed the most power around 80 days, very little around 40 days.
In the discussion which follows m = magnitude, e = orbital eccentricity, w = angle of periastron
measured from the ascending node of the spectroscopic primary in the direction of its orbital motion,
M = mean anomaly, v = true anomaly, a = orbital semi-major axis, d = distance between star centers,
and R = stellar radius.
Then it occurred to us that the ellipticity effect should be modulated by the varying star-to-star
separation in this eccentric orbit. The giant star should experience greater tidal deformation at
periastron and less at apastron. This should be a strong effect because it depends on the cube of the
ratio R/d (Russell and Merrill 1952). Ifw is near 90 ° or 270 °, then the light curve should show a maxima
of equal height but minima of unequal depth. Abt, Dukes, and Weaver had found w = 72 °.5, not far
from 90 °, so we formed the tentative hypothesis that this modulated ellipticity effect could explain at
least one of the strong 80-day periodicities.
To quantify this hypothesis, we considered two versions. The first assumes that the long axis of the
tidally distorted star always points towards the other star and that the effect on the light curve, in
magnitude units, is proportional to (R/d) 3. This would correspond to the theory of equilibrium tides.
The second assumes that the long axis of the tidally distorted star rotates uniformly in time, i.e., directly
299
proportionalto the meananomaly,andthat the effecton the light curvevariesuniformlyin time
betweenthetwoextremes,i.e.,betweenthemaximumeffectatd = a(1-e)andtheminimumeffectat
d = a(1+e). Thiswouldcorrespondto thetheoryof dynamicaltides.Bothversionshavethesamecos
2_dependence,where_ istheanglebetweenthelineof sightandthelongaxisof thetidallyelongatcd
star.Thus,in bothversions,theellipticityeffectvanishesat_ = 45°, 135°, 215°, and 225°.
For the first version, the change in magnitude produced by the ellipticity effect is given by
Am = k (d/a) -3 cos2_, (1)
where _ =v + w - 90 ° and the usual equations of the two-body problem can be used to compute d
and v as functions of time.
For the second version, the change in magnitude produced by ellipticity is given by
Am = k (a +fl cos M) cos24, (2)
where now _ = M + w -90 °. In this equation
a = v'2(fp + (3)
fl = v'z (fp - fa) (4)
and
fp = (1 -e) -3 (5)
fa =(1 +e) -3 (6)
Note that, in this second version, there is no need to compute v or d as functions of time.
In both versions the coefficient k would correspond approximately to the coefficient A2 as defined
by Russell and Merrill (1952). In the case of a circular orbit, whcre d = a in equation (1) or e = 0 in
equation (2), one would get k = A2.
To try this hypothesis on BM Cam we proceeded by iteration. The observed magnitudes were
plotted modulo; the known orbital period and means were taken in bins 0.02 phase units wide. There
was considerable dispersion within each bin, of course, because of the other variability present in the
system with supposedly different periodicities. We were encouraged that the resulting mean light curve
had roughly the expected shape: two nearly equal maxima and two quite unequal minima. Then this
mean light curve was subtracted from the observed magnitudes and the residuals examined.
The residuals showed something we recognized as variability produced by starspots. Within each
observing season there was a roughly sinusoidal variation with a period similar to but significantly
different from the orbital period. That period was about 82 d.5 up through 1984.5 and about 81 'J.0 after
that, with a half-cycle phase shift also around 1984.5. The amplitude of this roughly sinusoidal variation
changed dramatically: up to a maximum of 0.M15 at 1981.3, down to a minimum of 0 .M03 at 1984.0, up
to another maximum of 0.M15 at 1985.6, and possibly decreasing thereafter. In addition, the average
light level changed significantly from year to year, covering a range of 0 .m06.
300
The next step of the iteration was to find analytical expressions to approximate this starspot
variability. An assumed sinusoid required as its parameters the mean light level, the epoch of light
minimum, the period, and the amplitude. Additionally, we allowed the mean light level to increase or
decrease linearly with time.
The starspot variability was then removed from the original observed magnitudes with these
analytical expressions. These residuals, which we presume contain only the ellipticity effect, were then
fit in turn with equations (1) and with equation (2). Both fits yielded P, k, e, and w. The period P enters
as a parameter when we convert Julian date into mean anomaly.
Results with equation (2) were superior, in the sense that the sum of the squares of the residuals
reached a much smaller minimum. We take this as indication that the theory in the second version is a
better approximation to the real physical situation. The parameters are presented in Table I with their
formal standard errors and compared to the corresponding elements of the Abt, Dukes, and Weaver
(1969) solution to the spectroscopic orbit.
Table I
Derived Elements
element photometric spectroscopic
P(orb.) 79 d.93 80 d. 174469
+ . 05 + . 000003
0.25 0.35
__+. 03 --.. 02
w 95°.0 720.5
_+5.0 ___3.0
k 0.M018
_+ . 002
The difference between the two determinations of the orbital period amounts to 5 standard errors
and thus poses an apparent conflict. The spectroscopic determination was supposedly uncertain by only
__.0 a.00003, but we can show that it could be significantly in error. The solution of Abt, Dukes, and
Weaver (1969) was based on 13 radial velocities obtained in 1966, 1967, and 1968 plus three radial
velocities obtained long before, in 1916 and i9i7. That long 50-year baseline gave them their high
301
precisionbut dependedcriticallyonproperphasing.Thethreeoldvelocitiesfell coincidentallyatvery
nearlythesamephaseandhadaboutthesamevelocity,namelyVr = -8.km/sec.If oneplacesthemon
thefallingslopeof theradialvelocitycurveratherthantherisingslope,onecouldhavegottenorbital
periodof 80a.04 or 80 d.79. The first of these would differ from our photometric determination by only
2 standard errors.
The difference between the two determinations ofw amounts to 3 standard errors and thus poses
another apparent conflict. There is, however, an easy explanation for this discrepancy, namely apsidal
motion. The 22 °. 5 increase in w between 1967 and 1984 would correspond to an apsidal motion period
of 272 years. Theory shows that the rate of apsidal motion is proportional to the fifth power of the ratio
R/a. Since that ratio must be large in any binary with an observable ellipticity effect, it is expected that
apsidal motion in BM Cam would be relatively rapid, i.e., measurable within a few decades.
The period of the photometric variation produced by the starspots is a measure of the rotation
period of the K1 giant. The fact that it differs only a few percent from the orbital period indicates that
BM Cam is not rotating pseudosynchronously but is rotating in virtual synchronism after all. From this
we might conclude that the theory of pseudosynchronism does not apply when dynamical tides
phase-locked with the orbital period are more important than the equilibrium tides.
This investigation is not quite finished. We need to improve the analytical representation of the
starspot variability by allowing the amplitude of the assumed sinusoidal variation to increase or decrease
linearly with time within each season. We need to repeat the iterative process a few more iterations,
and we need to compute the theoretically expected apsidal motion period, which will require a
determination or estimate of the orbital inclination and the mass ratio, neither of which is known directly
for a single-lined spectroscopic binary nor for a non-eclipsing ellipsoidal variable.
Acknowledgments
The Vanderbilt 16-inch telescope was obtained with NSF research grant AST 84-14594 and the
analysis in this paper was supported by NASA research grant NAG 8-111.
References
Abt, H.A., Dukes, R.J., and Weaver, W.B. 1969, Ap. J. 157, 717.
Boyd, L.J., Genet, R.M., Busby, M.R., Hall, D.S., and Strassmeier, K.G. 1989, Ap. J.
Suppl., in preparation.
Eaton, J.A., Hall, D.S., Henry, G.W., Landis, H.J., McFaul, T.G., and Renner, T.R.
1980, I.B.V.S. No. 1902.
Fernandes, M. 1983, B.A.V. Rundbrief 32, 119.
Hall, D.S. 1986, Ap. J. 309, L83.
302
Hall, D.S. 1988, in Automatic Small Telescopes, edited by D.S. Hayes and R.M, Genet
(Mesa: Fairborn Press), in press.
Hall, D.S. and Osborn, S.A.G. 1986, in Cool Stars, Stellar Systems, and the Sun, edited by
M. Zeilik and D.M. Gibson (Berlin: Springer-Verlag), p.84.
Morris, S.L. 1985, Ap. J. 295, 143.
Nelson, E.R., Boyd, L.J., Genet, R.M., Hall, D.S. 1987 in Cool Stars, Stellar Systems,
and the Sun, edited by J.L. Linsky and R.E. Stencel (Berlin: Springer-Verlag), p. 497.
Russell, H.N. and Merrill, J.E. 1952, Contrb. Princeton Univ. Obsv. No. 26, chapter IV.
Strassmeier, K.G., Hall, D.S., Boyd, L.J., Genet, R.M. 1989, Ap. J. Suppl. 69, 141.
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Solving the BM Camelopardalis puzzle

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