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Irish Astr. J., 23(1), 33­36, (1996) A. G. GUNN
RE­INVESTIGATION OF ROTATION­ACTIVITY RELATIONS
USING RADIO LUMINOSITIES
A. G. GUNN
Armagh Observatory, College Hill, Armagh, BT61 9DG, Northern Ireland
agg@star.arm.ac.uk
ABSTRACT. A re­analysis of a comprehensive survey of radio fluxes for chromospherically active binary systems is given.
The correlation between activity and a convection­dependent Rossby number is found to be marginally better than correlations
with simple rotation parameters. It is suggested that the evolved state of RS CVn binaries and the tidal effects on convection
dynamics requires an improved convection­dependent parameter for use in rotation­activity studies.
1. INTRODUCTION
A possible correlation between rotation and chromospheric ac­
tivity was first suggested by Kraft (1967). The hypothesis sug­
gested that late­type stars have magnetic fields generated by
a dynamo mechanism, that field generation depends partly on
the forces produced by rotation and that stellar activity seen
in many wavebands is directly influenced by the presence of
emerging magnetic flux. As well as rotation other parameters
expected to influence magnetic activity include stellar mass,
luminosity class and spectral type which dictate the properties
of the convection zone where the dynamo is rooted. In partic­
ular, the depth of this zone, or its convective turnover time,
is thought to play a role in the field generation. It is expected
that stars of later spectral type (that is, with deeper convection
relative to their radius) will show stronger dynamo behaviour
than stars of earlier spectral type, for a given rotation period.
The theory is at least partly demonstrated for a number of
stellar classes. Relations between coronal emissions and rota­
tion have been reported by Pallavicini et al. (1981) and Schri­
jver, Mewe & Walter (1984) for mostly single stars. Chromo­
spheric relations were also found by Middelkoop (1981) and
Noyes et al. (1984). X­ray observations (Vaiana et al. 1981;
Walter 1981) have also demonstrated the link between stellar
activity, rotation rate and spectral type. Although the mech­
anism is only partly understood its existence is now firmly
established.
Some of the highest levels of stellar activity are seen in
close binary systems consisting of cool stars, i.e. the chromo­
spherically active binaries. The RS CVn binaries (Hall 1976)
are a class of objects normally consisting of at least one com­
ponent of spectral type F­G V­IV and which show the Ca ii
H & K lines in emission outside of eclipse. Common features
include strong coronal X­ray and radio emission, variable Hff
emission, strong UV chromospheric and transition region line
emission, frequent flaring activity and orbital period changes.
Popper & Ulrich (1977) discussed the evolutionary status of
these stars and concluded that the emission characteristics de­
velop as they enter the Hertzsprung gap. More recently Bar­
rado et al. (1994) confirmed the evolved state of RS CVn com­
ponents. In this part of the HR diagram hydrogen shells can
ignite in the stellar interior and envelope expansion can oc­
cur. Hence the convection zone properties can be significantly
altered from main­sequence stars. Metallicity is also important
because the temperature of stars at the base of the giant branch
is very sensitive to this parameter. Since RS CVn binaries are
tidally locked causing rapid rotation and are evolved stars the
convection zone dynamics can be significantly altered from sim­
ilar main­sequence stars. Consequently the activity levels may
undergo some quite rapid variations over a narrow range in
temperature. It has also been suggested that a relevant pa­
rameter for activity studies in binary systems is the ratio of
the stellar to the Roche radius (Young & Koniges 1977).
Many authors have studied the correlation of RS CVn flux
measurements (over a wide range of wavelengths) with rota­
tional parameters; namely the rotational period or rotational
surface velocity. The dynamo theory of activity production sug­
gests that correlations should be more readily apparent using
a parameter which involves the convective properties of the
star and its interaction with the magnetic field. Drake et al.
(1989; hereafter D89) analyzed a comprehensive survey of ra­
dio flux measurements for a large sample of chromospherically
active binaries. These data were originally discussed in terms
of simple rotational parameters and not a convection­related
parameter. The incentive for this work was to re­analyze these
data using the standard Rossby number. If the correlations are
not improved by the use of such a parameter then either the re­
ported dependences are unreal or the simple dynamo theory is
not sufficient to explain the observations. In this paper a brief
summary of dynamo related parameters is given, the use of
empirical turnover times is discussed, the statistical technique
is presented and the results discussed in terms of the dynamo
generated activity in binary stars.
2. CONVECTION, ROTATION AND ACTIVITY
In this analysis the assumption is made that RS CVn binaries
are tidally locked causing orbital and rotational synchronicity
(i.e. Prot = Porb ). The most common quantity used in dynamo
theory investigations of stellar activity is the Rossby number
Ro defined as;
Ro = Prot=Üconv (1)
where Üconv is the convective turnover timescale. Ro is an im­
portant indicator in hydromagnetic dynamo theory and mea­
sures the extent to which rotation can induce both helicity
and differential rotation required for dynamo activity in the
c
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Irish Astr. J., 23(1), 33­36, (1996) ROTATION­ACTIVITY A. G. GUNN
convection zone. Since Üconv is a function of spectral type Ro
characterizes the level of dynamo activity for stars of different
rotation periods. It is expected that magnetic activity should
increase with decreasing Rossby number. Empirically this as­
sertion has been demonstrated by Noyes (1983) who found that
Ca ii emission strength, when normalized to bolometric flux,
was related to the product of angular rotation rate and an ad­
ditional function of spectral type. That function is identified
with the convective turnover timescale.
To form an estimate of Ro for a given star it is necessary
to calculate the convective turnover timescale Üconv . The usual
method involves modelling the stellar convection zone using
standard mixing length theory and stellar structure codes (e.g.
Eggleton 1972). In these models the efficiency of convection is
described by the parameter l, the characteristic mixing length
of the convection flow. In the usual formulation the convective
eddies are assumed to form with length scales of:
l = ffH (2)
where H has been variously defined as a density scale height or
a pressure scale height and ff is the product of mean helicity of
convection and Üconv . It should be noted that Ro scales with ff 5
2
and is therefore heavily dependent on the choice of ff. Calibra­
tion of ff is possible only with helioseimological studies of the
Sun which directly measure the depth of the convection zone
(Gough 1986). Rucinski & Vandenberg (1986) used a value of
ff=1.6 in computing their stellar convective envelopes because
computed isochrones based on this value best reproduce the
observed colour­magnitude diagrams for globular clusters with
wide­ranging age and composition (Vandenberg 1983;1984).
Noyes et al. (1984; hereafter N84) used convective turnover
timescales computed from an empirical fit to the models of
Gilman (1980). Although Gilman's work involved models for
various stellar masses N84 used a standard mass­colour rela­
tionship to find a mean relation between tconv and the (B­
V) colour index. An iterated fitting procedure found that the
observational data were best described with an extrapolated
Gilman model of ff=1.9.
Although the results of N84 favoured ff=1.9, there are in­
herent differences in the formulation of the mixing length the­
ory for various convective models and these themselves create
an uncertainty in the choice of ff. Even so, some progress has
been made in this field by Gilliland (1985) who demonstrated
the variability of convective properties for evolved stars of dif­
ferent masses and showed that different colour­temperature re­
lations require quite different values of ff necessary to repro­
duce the N84 relationship.
Hence the precise way in which Rossby number enters the
physics of dynamo field generation and the uncertainty in the
formulation of convection zone properties and its calibration
complicates the rotation­activity correlation. There are also
problems in using published stellar models. A consistent set
of determinations of Üconv however should yield a qualitative
correlation with activity indicators.
3. EMPIRICAL TURNOVER TIMESCALES
As a first step in the re­analysis of the radio flux measurements
presented by Drake et al. (1989; D89) calculations based on the
empirical relationship between Üconv and the spectral type de­
fined by the (B­V) index can be used. Figure 1 shows several of
these relationships taken mainly from studies of main­sequence
stars of given mass and composition. Although the basic form
of these relations is the same differences in log (Üconv ) of greater
than 0.2 are possible for a single spectral type. As mentioned
above, the curve given by N84 is an interpolated fit which
matches models derived by Gilman (1980) to empirical data
and therefore ignores the range of masses possible within a
main­sequence spectral type. Indeed, the relationship should
only be considered appropriate as a mean description of a small
sample of main­sequence stars. However in the absence of con­
sistent theoretical relationships the polynomial form presented
by N84 can be used to improve the correlations given by D89.
The relationship shows that for (B­V) ! 0.9, Üconv increases
with advancing spectral type, steeply at earlier spectral types
and less steeply at later spectral types. For (B­V) ? 0.9, Üconv
varies little.
Fig. 1. The relationship between convective turnover timescale
Üconv and (B­V) magnitude. The functions plotted are polynomial
fits to either model calculations (Rucinski & Vandenberg (1986);
Stepien (1989); Gilliland (1985)) or interpolated fits using observa­
tional data (Noyes et al. (1984; N84)).
4. EXPECTATION MAXIMIZATION ANALYSIS
The D89 survey consists of radio flux measurements at 6­cm
for a sample of 122 RS CVn and related binary systems with
widely varying orbital periods. They searched for correlations
of the radio flux of these systems with various systemic and
stellar parameters such as rotational periods, surface velocities
and X­ray emission. In this paper a re­analysis of these data
is performed using Rossby numbers calculated using the N84
relationship.
The basic problem in this analysis is how to handle prop­
erly the large percentage (¸ 46%) of cases where the data
represents only an upper limit to the radio emission. Such
detections should not be ignored because they to provide in­
formation about the possible emission from a given system.
However upper limits cannot be regarded as detected points
since this will introduce an unnecessary bias into the sample.
Fortunately, there are advanced statistical techniques appli­
cable to so­called censored data, i.e. data containing numbers
which are upper or lower limits. The method incoorporates
an iterative maximum likelihood (ML) linear regression proce­
dure known as the expectation maximization (EM) algorithm.
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Irish Astr. J., 23(1), 33­36, (1996) ROTATION­ACTIVITY A. G. GUNN
Essentially the ML analysis does a linear least­squares fit to
the data and includes a statistically correct treatment of the
censored points. This field of survival statistics is reviewed by
Miller (1981) and its application to astronomical data analysis
has been presented by Feigelson & Nelson (1985) and Isobe,
Feigelson & Nelson (1986).
To study correlations rather than simply regressions a
measurement of goodness of fit is required. Even if very weak
relations exist between parameters (in that the gradient of the
linear relation is small) the data can still be highly correlated.
Some of the conclusions made by D89 were based only on re­
gression analysis. For example, they conclude that two param­
eters show no significant correlation simply when the gradient
of the regression line is small. Since these regressions were for
log­log relationships the gradients are a measure of the power
of the dependent variable in the regression relationship and
therefore tells us nothing about how well the actual data are
correlated. Fortunately, the usual product moment correlation
coefficient r is a sufficient measure for censored data since the
effect of censoring on this parameter is small. A zero gradi­
ent naturally implies that data are completely uncorrelated.
The correlation coefficient measures the dispersion of the data
around the regression line so r cannot uniquely declare the
correspondence between two parameters. For these reasons an
improvement in a correlation can be defined as an increase in
both the modulus of the ML gradient and r. A reversal of sign
of either parameter is regarded as no significant improvement.
5. RESULTS AND DISCUSSION
Figure 2 shows the radio luminosities against the Rossby num­
bers derived using the (B­V) magnitude and the N84 relation­
ship. Symbols indicate the luminosity class of the stars and
whether these luminosities are upper limits. Also shown are the
maximum likelihood regression lines for all stars in the sample
and also for the subsets of giant, subgiant and dwarf stars. Fig­
ure 3 shows the radio luminosities normalized to the bolomet­
ric luminosities (L6/L bol ) against the Rossby number. From
these diagrams it appears that the relation of radio emission
to rotation becomes steeper as one goes from dwarfs towards
giants. It has been found that the EM algorithm can give sig­
nificantly different results from the standard LS method which
does not take account of censored data. Inspection of these fig­
ures reveals that rotation­activity correlations at radio wave­
lengths are by no means convincing and do not display the
often tight relationships associated with chromospheric activ­
ity indicators.
D89 found no statistically significant correlation of the 6­
cm radio luminosity with either orbital or rotational period but
found a fair degree of correlation between L6 and the surface
rotation velocity vrot . The normalized luminosities were corre­
lated with both the orbital period Porb and the surface rotation
velocity. The relations presented by D89 can be summarized
as follows;
L6 ¸ v 1:0\Sigma0:3
rot (3)
L6=L bol ¸ P \Gamma0:9\Sigma0:2
orb (4)
L6=L bol ¸ v 1:4\Sigma0:3
rot (5)
Separating these systems into three subclasses depending on
the luminosity class of the active star in each system results
Fig. 2. Plot of log Ro against log L 6 for stars in the D89 survey. Ro
values are derived from the (B­V) magnitude using the relationship
of Noyes et al. (1984). Maximum likelihood linear regression fits
to the data are shown. Symbols are squares for giants, circles for
subgiants and triangles for dwarfs. Those points which are upper
limits are plotted as open symbols.
in the dwarfs showing no significant depedence on the rota­
tion parameters while the subgiant and giant systems show
reasonable correlations between their radio emission and their
rotation.
Table 1 shows the results of the regression and correlation
analyses on the data using orbital periods and Rossby numbers.
This table lists the parameters used, the number of stars in the
sample, the subset of the data used (a, g, s or d), the ML linear
regression line gradient a and the product moment correlation
coefficient r. From the regression and correlation results there
appears to be no significant correlation between radio lumi­
nosity and Rossby number or rotation period for dwarf stars.
Correlations are more significant for giants and subgiants and
these appear to be marginally better with Rossby number than
for rotation period. This is to be expected on the basis of the
dynamo theory of stellar activity but is by no means confirmed
by the small increase in correlation for these data. Evidently
the luminosity class is a significant factor in the determina­
tion of active levels or at least its dependence on convection.
The use of a radio luminosity normalized to the bolometric
luminosity also appears to improve the correlations.
This analysis has concerned a single activity indicator
which relates to the large scale coronal activity of the binary
systems. The lack of significant correlations like those found for
lower atmospheric activity indicators may mean other mech­
anisms may be operating to mask the underlying rotation­
activity relation. Also, the use of empirically derived Rossby
numbers may be inappropriate since they are based on fits to a
limited sample of main­sequence stars whilst RS CVn binaries
are known to be slightly evolved. The convective properties
of such stars may differ significantly from their main­sequence
counterparts and may therefore show widely varying actvity
levels for similar rotation periods. It is suggested that the
present definition of Rossby number is an inadequate measure
of the efficiency of magnetic activity for evolved close binary
systems.
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Irish Astr. J., 23(1), 33­36, (1996) ROTATION­ACTIVITY A. G. GUNN
Fig. 3. Plot of log Ro against log L 6 /L bol for stars in the D89 sur­
vey. Maximum likelihood (ML) linear regression fits to the data are
shown. Symbols and abbreviations are the same as those in Figure
2.
6. CONCLUSIONS
It has been demonstrated that the use of an empirically derived
Rossby number does not significantly improve the rotation­
activity correlation for radio luminosity measurements of RS
CVn binaries over that found using the orbital period. Such
correlations at radio wavelengths remain unconvincing. The
results strongly suggest that luminosity class is an important
parameter for magnetic activity generation. To improve our
understanding of activity generation in evolved close binary
stars, additional stellar structure modelling of the convection
zone and a study of the changes in convection dynamics as
stars evolve off the main­sequence are required. Such theoreti­
cal studies may give more reliable estimates of Rossby numbers
for evolved stars (thus improving the correlations) or suggest
a more appropriate parameter to characterize the activity gen­
eration as a function of spectral type, luminosity class and
rotation. Investigation of the effects of binarity and tidal inter­
action on the convection zone dynamics is also required.
In the future it would be instructive to firstly standardize
the stellar and systemic parameters for this sample of stars by
eliminating or re­assessing poorly known parameters. Further
observational work is required to firmly establish the evolu­
tionary status of the RS CVn systems. Once this is done, a
re­analysis of activity indicators from all levels of the chromo­
sphere and corona, and comparison with theoretical results,
will give a better indication of the mechanisms of activity pro­
duction in evolved close binary systems.
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