Download Rotation Periods and Relative Ages of Solar-Type Stars

J. Undergrad. Sci. 3: 113-117 (Fall 1996)
Astronomy
Rotation Periods and Relative Ages of Solar-Type Stars
CONNIE ING
The relationship between stellar rotation periods (Prot)
and relative ages of stars in a sample of Sun-like stars
was examined, then compared to other age determination methods. The majority of stars exhibited increasing Prot as a function of increasing age, as expected.
The accuracy of the analysis was confirmed when the
stars reflected the characteristics of the VaughanPreston gap, which was discovered in 1983. However,
the discovery of significant anomalies gave several implications. First, data collection methods at Mount Wilson Observatory need to be refined; quality and quantity of observations need to be increased. Second, up to
a third of the stars could be in their Maunder Minimum
activity minimums, which occur on centuries-long
timescales and occupy one-third of the lifetime of a star.
This fact gives a possible error of up to 33.3%. Another
source of error is the possibility of active region growth
and decay, which produces abnormally high rates of
rotation. Third and final, the model for predicting age
based on rotation period of a star should not depend on
its spectral type (or mass). Previous formulae utilized
convective turnover time (a function of spectral type)
as a coefficient in the algorithm to determine age; the
first results of this study indicate that rotation periods
and relative ages of solar-type stars are related, but independent of spectral type.
Introduction
Stellar age, rotation period (Prot), and activity fluxes (S)
are intimately linked in lower main sequence (LMS) stars
such as the Sun (Radick et al. 1990). Comparing ages inferred from activity fluxes to those inferred from rotation
periods will determine the validity of stars’ estimated ages,
as well as determine the validity of each of these age determination methods. This study presents first results of the
increase in rotation period as a star matures on the lower
main sequence. However, notable inconsistencies in plotted trends implicate that basing a formula for Prot on color
index is questionable.
Currently, other projects in the stellar rotation field are being
conducted. At the Von Vleck Observatory, the Orion Monitoring Update, started in 1990, is a program to measure the rotation periods of 700 stars with a 24-inch telescope. Seventyfive of the 700 stars measured exhibit a Prot with a False Alarm
Probability (FAP) of less than 1%. The Spotted Stars III Project
began in March of 1995 with the goal of investigating Prot of
stars in clusters. The slow-rotating stars in α Per and the Pleiades are constantly examined for new periods. The NGC 6475
and IC4665 clusters are also being studied.
To accurately measure the ages of stars, the consistencies of the various age-determination methods must be
gauged. Current methods include the examination of characteristics such as isochrones, chromospheric emissions,
metallicities, and period of activity cycle Pcyc.
Isochrones are theoretical plots of various-aged stars
on axes of log Teff (effective temperature) and δMbol (bolometric luminosity). Also, Ca II H (396.8 nm) and K (383.4 nm)
chromospheric emissions are empirically linked to ages
(Soderblom et al. 1991). By fitting Strömgren uvby-Hβ photometry data to theoretical isochrones, isochrone ages may
be compared to chromospheric emission ages of stars.1
The metallicity of a star, a measure of the abundance
of certain metals in a star’s photosphere, also serves as an
indicator of age. For example, since the Sun is still burning
hydrogen in its core, it is too “young” to produce heavy metals. Metals like calcium, iron, lithium, and magnesium, observed on the stellar surface, can be assumed to be primordial. The same is true for stars of similar mass to the Sun.
Therefore, the relative abundance of primordial metals remaining in the surface can be compared to other stellar indices, such as the CRV (Soon et al. 1993), to estimate ages
of stars. However, uncertainties in this model exist because
of unknown rates of supernovae events, which might contaminate the stars’ original degree of metallicity.
Length of stellar activity cycles Pcyc may be correlated
with age. Stellar activity cycles are decades-long variations
in stars’ magnetic fields, which in turn are powered by the
stellar dynamo. The stellar dynamo is perpetuated by interactions between the convective zone and surface differential rotation. The Pcyc of solar-type stars is about 11 years
(Baliunas 1991), while the period of time required for the
magnetic field to switch poles is two times that, or about 22
years. As stars age, activity cycles lengthen (Baliunas and
Soon 1995).
A comparison of age determination models of stars is
necessary to confirm the validity of these models. Stellar
ages thus inferred may provide information on the lower limit
of the age of the universe through an estimation of main
sequence lifetime (Deliyannis et al. 1989). In addition, comparison of ages of LMS stars will add to the study of the
relationship between surface magnetic activity, rotation and
age that all stars on the main sequence share. However, a
detailed discussion of these topics is too broad for the scope
of this research.
Methods
Olin Wilson of the Mount Wilson Observatory (MWO)
in California first began measuring stellar activity cycles in
1966. In 1980, the HK Project, a program of nightly observations, was launched to investigate the solar-terrestrial
connection, further the study of stellar activity cycles, and to
focus attention on detecting rotation periods. This is an ongoing project in which each of its 1200 LMS stars is scheduled for several observations per week. Three measurements of that same star are typically made per night.
Data presented here were obtained either from spectrophotometers on the 60-inch (1.5 m) reflector or 100-inch
(2.5 m) Hooker reflector at MWO (Soon et al. 1993), which
utilize the Coude and Cassegrain telescope configurations.
The data comes in the form of time versus mean activity
level. Ca II flux (from the stars’ chromospheric emission) is
measured and recorded with the date of observation in the
CONNIE ING is a first-year student attending Yale College. She completed the research described in this article under the supervision of Drs. Sallie L. Baliunas, Willie H. Soon, and
Robert A. Donahue at the Harvard-Smithsonian Center for Astrophysics. Ms. Ing was honored as a Westinghouse Science Talent Search finalist for this work, and she was recognized
as the California State Fair Science Student of the Year.
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Astronomy
Figure 1. (A) An example of a time series plot of HD 82885 data (top). (B) A
Fourier transform periodogram of HD 82885 data (bottom).
Figure 2. A comparison of observed rotation period versus calculated
rotation period. The Pcalc = Prot line is plotted.
Julian calendar system. Magnetic heating occurs in the
chromospheres of stars; starspots are greatest where magnetic fields are greatest. Hence, as starspots or inhomogeneities travel across the surface of a star, Ca II flux will reflect that movement (Donahue 1993). Ca II flux is thus used
to observe rotation periods.
The search for rotational signals was performed on a
set of 102 LMS stars taken from the MWO sample. Criteria
for these stars’ selection included a B - V color index similar
to that of the sun’s (i.e., 0.55 < B - V < 0.76). The number of
experimental data points each star had varied from 50 to
over 3000, due to the rolling inclusion of stars in the HK
project. To begin the search, data points were organized
into seasons, with a season defined as an interval containing at least 30 data points and separated from other intervals by at least 80 days.
Determining Prot required various techniques. The primary means of obtaining Prot was the use of the program “j,”
created by R. A. Donahue in 1993. J plots activity-versustime observations (activity,2 S, is measured by monitoring
the emission core fluxes within Ca II H and K resonance
lines from LMS stars). An example of a time series plot of
Ca II flux is shown in Figure 1A.
J performs Fourier transform periodograms of data
points. J is specifically designed to handle data series with
uneven rates of sampling (Horne and Baliunas 1986). A
sample periodogram of star HD 82885 is shown in Figure
1B. At least one periodogram was computed for each observing season; filtering was necessary for transforms with
second peaks. Gaps in the data were filled by executing j
manually on selected time intervals.
Another function of j that was utilized is its output of
False Alarm Probability (FAP) of periods detected.3 FAP—
the chance in percent that a measured periodic peak will
arise from purely random (Gaussian) noise—serves as an
indicator of quality of period detection. It is desirable to look
for FAPs close to zero when searching for a final rotation
period. The stars were ranked on a negative log FAP scale,
which was labeled as FApH because of the similarity to the
pH scale.
Each star received at least five comprehensive reviews
of its detected period. Observed periods of less than 5 days
and greater than 100 days were generally not considered
as candidates for designation of a rotation period. Rotation periods of less than five days could be spurious because measurements were made two to three days apart.
Rotation periods longer than 100 days are beyond the seasonal window of the observations. When periods fell within
a range of 10 days or less, the mean was calculated and
the standard deviation utilized in place of δProt . A file summarizing detected Prot, δProt, and FAP was created for each
star.
Results
Of a total of 102 stars, only 46 had been observed
with enough data to span one to three seasons. Six stars
had only 3-10 points of data, and thus did not qualify for
even a single season. However, fifty-one out of 102 stars
exhibited a measurable P rot. Thirty of these stars exhibited
FAPs of less than 1.0 x 10-3, which indicates a fairly accurate period.
Among the 51 stars with observable seasons, there were
29 (56.7%) with periods detected in one to three seasons.
Eighty-two percent of these stars had detectable periods in
at least 50% of their seasons. These statistics may be misleading when examined out of context. Half of the stars have
only one or two observable seasons. Therefore, finding a
period in one season inflates period percentages. However,
the determined periods should be given considerable weight
because they are relatively more consistent than their flat
counterparts. Additionally, gaps in data and indeterminate
rotation periods are common in the field of rotation study
(Radick et al. 1990).
Thirty stars in the main sample did not receive a Prot
designation. Data for 26 of these stars were so sparse that
a rating of “NMI,” or Need More Information, was given. Fourteen stars in the sample earned a rating of “NDP,” or No
Detectable Period. This occurred when stars showed conflicting Prot over various seasons, causing any Prot obtained
to be unreliable. Only nine stars were flat, meaning that they
exhibited no discernible Prot at all, and instead yielded static
time series plots.
Sixteen stars were found to have equivocal rotation
periods. These periods featured high False Alarm Probabilities, indicating they were not likely to be accurate.
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Astronomy
Figure 3. Spectral type, or mass versus the residuals. The difference
between P calc and Pobs is minimal, but note the lower left corner region of slow
rotators. This region mirrors the observed Vaughan-Preston Gap.
Figure 4. Spectral type versus period of rotation. Note the increasing scatter
of Prot as B - V decreases. Again, correlation with the Vaughan-Preston Gap
provides support for the accuracy of the findings.
Discussion
Age versus Rotation. Several conclusions may be reached
concerning the relationship between a star’s age and its
rotation period. To begin, smaller intrinsic spread in Prot is
characteristic of older LMS stars, but not young ones. This
observation is apparent in Figures 3 and 4. The residuals of
Pcalc - Prot are plotted against B - V in Figure 3. A concentration of data points falls on the line Pcalc - Prot = 0, reinforcing
the consistency between Prot and Pcalc. A number of stars at
the lower left corner of the graph have low B - V values and
negative residuals. They are rotating slower than predicted,
indicating a potential Maunder Minimum.
Figure 4 is a plot of B - V versus Prot . As B - V decreases,
the amount of scatter along the Prot axis increases. At high
values of B - V (0.70-0.80), Prot is confined to a relatively
narrow range of 0-20 days. As B - V values approach 0.55
from 0.70, scatter approaches a range of 0-85 days. Since
the increasing scatter is a function of decreasing age, Prot is
more useful for identifying relative ages of older stars.
These results are likely to be quite accurate because
they reflect the findings of Noyes et al. (1984) in the VaughanPreston gap. This is a relation in which younger stars tend
to be more unpredictable in their rotation periods and activity levels—as compared to older stars. They have a wider
spread along their curve than older stars do in an R’HK versus Prot plot.
An additional implication is that rotation period is independent of spectral type. These results conflict with Noyes’
determination of Prot. This determination may be derived from
Noyes’ algorithm linking stars’ dynamo field generation and
Rossby number, or Prot / τc.5 Tau equals convective turnover
time, which is a function of mass. Mass, however, is a function of spectral type. Hence, this study would implicate eliminating τc from future predictions of Prot.
An illustration of the trends described above is provided
in Figure 5, a histogram of rotation periods for the sample.
At lower rotation periods, a high number of stars is found.
As the graph moves toward longer Prot, a tail in the graph
becomes evident. This tail is tapered, meaning that fewer
stars exhibit high rotation periods. The percentage of slow
rotators approaches 50%. A possible explanation is that as
stars age, the increasing numbers of mechanisms for angular momentum transport cause a larger spread in Prot
(Donahue 1993).
Data Sample. In order to evaluate the data sample in the
context of previously-published hypotheses, calculated rotation periods, or Pcalc, were obtained for each star. These
Pcalc are a function of chromospheric activity level as well as
B - V spectral type. The formula for the calculation of rotation periods was derived from Noyes’ algorithm for determining Rossby number from mean stellar activity. A more
detailed description is given by Noyes et al. (1984).
These calculated rotation periods were compared against
the observed rotation periods of the sample field stars in Figure 2. The relationship between these two values is close to
one-to-one; over half of the stars fall within 10 days of the line
Pobs = Pcalc. However, significant outliers on either side of the x
= y line prevent definitive predictions. About ten stars rotate
faster than predicted, which can be attributed to abnormally
high activity levels, or active region growth and decay (ARGD),
an observed phenomenon (Donahue 1993). About eight stars
rotate slower than predicted; these could be experiencing abnormally low chromospheric emission levels, or a Maunder
Minimum4 (Baliunas and Soon 1995).
Two overall trends, amplified and elegantly expressed
in Figure 2, are apparent in the rest of the Figures 3-6. First,
the majority of the sample of field stars behaves as expected
when plotted against axes such as B - V and activity: older
stars spin more slowly. The B - V versus residuals plot in
Figure 3 is nearly level at residuals = 0. Figure 4 shows that
Prot tends to decrease as B - V increases. In Figure 5, most
of the F and G stars have Prot under 20 days, which is consistent with that of the Sun, whose rotation period is about
27 days. Finally, Figure 6 shows a power-law relationship
between activity (age) and rotation period.
The second trend is that there are outliers which deviate significantly from mean relations. These stars either spin
slowly and have high activity levels, or vice versa—established trends conflict. Additionally, Pobs deviates from Pcalc.
There are outliers with Pobs > 60 days in Figure 2, as well as
lower Pobs than Pcalc. A region of stars exhibiting high Pobs for
low Pcalc is found in Figure 3. Figures 4 and 5 show a significant portion of the sample with abnormally indeterminate
Prot as a function of age. In Figure 6, two regions are detached
from the log R’HK versus log Prot power-law relationship.
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Astronomy
Figure 5. Histogram. Number of stars per Prot.
Figure 6. Activity, interpreted as age, versus log Prot. Note the power-law
relationship between age and rotation.
Finally, the relationship between a star’s age and its
rotation period may be confirmed in Figure 6. The activity
index, S, has been expressed in log R’HK units6 to provide a
measure of magnetic heating (Noyes et al. 1984). Log R’HK,
which is chromospheric emission minus any possible photospheric contribution, is plotted against log Prot. A strong distribution of stars falls along a line of negative slope (with
power ~ 1), which illustrates a possible power-law relationship between activity (age) and Prot . If the power law is close
to 1, it suggests that R’HK α P -1.
Three stars are found above the range of the main
graph, but they fall on a line of parallel slope. These stars
rotate slowly (indicating older solar-type stars) but emit high
levels of Ca II (indicating younger solar-type stars)—perhaps these are the RS-CVn stars. An even more prominent
cluster of data points falls below the main relation. These
are older stars with uncharacteristically small rotation periods (see Figure 3). This phenomenon deserves further investigation. These stars are unusual because their activity
levels are low (indicating older solar-type stars), yet they
spin quickly (a characteristic of younger solar-type stars).
pool of 51 stars is undesirable, staggering the stars in
batches of 10, 15, or 25, for alternating years is an alternative. For example, a study of surface differential rotation
would require heavy observation of the stars with good periods, while a study searching for a broad range of periods
would entail scrutiny of the less well-observed stars.
Examining the 102 stars by hand is a rate-limiting step
of the data processing. Computer routines can be developed to refine the discrimination with which automated
periodogram analysis and selection of interval are completed. The main drawback to automated data processing
is the high degree of experience and judgment needed to
select an appropriate interval for analysis; improved programs, however, could alleviate the problem and save time.
Errors and Improvements. As with any astrophysics research project, the accuracy of measurements made is limited by several factors. These factors include the potential
inclination angles of the stars to the line of sight of the observer. An active region (or zone of high convective turnover rate) occurring before or after another one on the visible disk may also interfere with data, as might the difference in Prot at different latitudes of the stars (Morfill et al.
1991). Differential rotation adds uncertainty to the observed
Prot but may contribute to the process of age determination
(Donahue 1993).
Although Prot and Pcalc agree for most stars with Prot < 20
days, the |Pcalc - Prot | can be large for some slowly rotating
stars. This implies an error in either Pcalc, Prot, or both. Increasing the quantity and quality of observations will illuminate the source of these errors.
To obtain more accurate measurements of rotation periods, an increased frequency of observations is needed.
The data sample may be reduced to the 51 best stars or the
51 least well-observed stars. Then, a commitment to observe the stars every night must be made. If abandoning a
Conclusions
The relationship between stellar rotation periods (Prot)
and relative ages of stars in a sample of Sun-like stars was
examined, using the empirical link between activity level and
age to calibrate the ages. Data from the last 15 years of
observation in Mount Wilson Observatory’s HK Project were
used. This consisted of time-series plots of Ca II H and K
chromospheric emissions separated by at least 80 days.
Fourier transform periodograms of the data were performed
to determine Prot for each star; each of the 102 stars received at least five comprehensive reviews.
Fifty-one stars were found to have measurable rotation
periods—a significant improvement over past studies, where
Prot of stars were reported in batches of only 15 or 20. The
majority of these stars exhibited increasing Prot as a function
of increasing age. However, anomalies were discovered as
well: outliers of older stars that rotated too quickly, and
younger stars that rotated too slowly.
The implications of these first results are fivefold. First,
younger stars rotate faster than older stars, as has been
predicted previously. Second, the anomalies are most likely
the result of unrefined data collection methods at Mount
Wilson Observatory. A recommendation would be to increase
the quantity, quality, and regularity of observations—this will
decrease the False Alarm Probabilities of the rotation periods. Third, the anomalies could also signify stars in their
activity maximums or Maunder Minimums, which are low
Journal of Undergraduate Sciences
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points in their centuries-scale activity flux. Fourth, these results reflect a Vaughan-Preston gap—a relation in which
younger stars exhibit wider spread in Prot than older stars do
in an H-K flux ( R’HK ) versus Prot plot. This not only supports
the Vaughan-Preston relation, but also supports the accuracy of the rotation periods found.
Fifth, and perhaps most significant, this study finds no
correlation between a star’s B - V and its rotation period. Previous studies (e.g., Noyes et al. 1984) have fitted functions to
calculate age from Rossby number, a coefficient equal to Prot /
τc. Tau, or convective turnover time, is a function of B - V spectral type. When the residuals between calculated rotation periods (calculated using a derivative of the Noyes formula) and
observed rotation periods were plotted against B - V, large scatter occurred. This indicated that rotation periods are independent of spectral type. That is to say, rotation periods of stars of
particular spectral types exhibit relative rates of decline of rotation period with age. Although one may certainly correlate
spectral type with age, it is not to be done in a model that incorporates both B - V and color.
The study of the rotation periods of lower main sequence
stars will afford us greater insight into the life cycles of these
stars and their evolution on and off the main sequence. Identifying the characteristics of these stars will also isolate conditions for the origin of life on planets orbiting these stars.
Comparing the various age determination methods of stars
available (the next step in this study would be to compare
these results against metallicities, isochrone ages, activity
cycle and long-term Ca variability) will enable us to improve
these methods. Ultimately, one could take a “snapshot in
time” of an LMS star and be able to identify it as relatively
older or younger than the Sun. A lower limit for the age of
the Universe could be verified.
Acknowledgments
The efforts of Dr. Sallie Baliunas, Dr. Willie Soon, Dr.
Bob Donahue, Beth Wicklund, and Eric Ford are appreciated. Additional thanks go to Sara Seager, Ms. Deeley, and
Dr. Yvonne Pendleton, who is at NASA-Ames Research
Center. I would also like to thank the Center for Excellence
in Education and Research Science Institute for generous
sponsorship of this research.
Endnotes
(1) First, the absolute magnitude, or M v, of the sample solar-type
stars (lower main sequence, or LMS) is plotted against their Teff .
Second, δMv, the difference between Mv and zero-age main sequence (ZAMS), is calculated and plotted against Teff . The accuracy of isochrone ages for LMS stars are tested by analyzing clusters of stars—those in the Hyades and M67, for example, should
have the same isochrones within each cluster.
(2) S = α [(H + K) / (V + R)], where H and K are the Ca II passband
counts and V and R are the violet and red continuum band counts.
α is a calibration factor called lamp correction that changes nightly.
(3) FAP = 100[1 - (1 - EXP(-z / ρ2) Ni], where ρ2 equals the total data
variance and Ni equals the independent frequencies, estimated using Horne and Baliunas’ technique (1986).
(4) In the 17th century, the Earth experienced a Little Ice Age. French
paintings from this era depict frozen rivers that never freeze over
today. During this same time, Maunder observed abnormally low
energy output from the Sun (and established a basis for the solarterrestrial connection). Hence, error mean S is up to 33.3%, because stars spend one-third of their lifetimes in Maunder Minimums.
Astronomy
(5) log(P / τ) = f(<R’ HK>) = 0.324 - 0.400y - 0.283y2 - 1.325y3, where
α = 1.9, and y = log τc(2)( B - V).
(6) Net chromospheric emissions (in log R’HK) have been empirically linked to ages of stars (Soderblom et al. 1991, Donahue 1993).
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