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					J. Undergrad. Sci. 3: 113-117 (Fall 1996)                                                                                                                            Astronomy

Rotation Periods and Relative Ages of Solar-Type Stars
CONNIE ING                                                                                         Isochrones are theoretical plots of various-aged stars
                                                                                             on axes of log Teff (effective temperature) and δMbol (bolom-
                                                                                             etric luminosity). Also, Ca II H (396.8 nm) and K (383.4 nm)
The relationship between stellar rotation periods (Prot)                                     chromospheric emissions are empirically linked to ages
and relative ages of stars in a sample of Sun-like stars                                     (Soderblom et al. 1991). By fitting Strömgren uvby-Hβ pho-
was examined, then compared to other age determina-                                          tometry data to theoretical isochrones, isochrone ages may
tion methods. The majority of stars exhibited increas-                                       be compared to chromospheric emission ages of stars.1
ing Prot as a function of increasing age, as expected.                                             The metallicity of a star, a measure of the abundance
The accuracy of the analysis was confirmed when the                                          of certain metals in a star’s photosphere, also serves as an
stars reflected the characteristics of the Vaughan-                                          indicator of age. For example, since the Sun is still burning
Preston gap, which was discovered in 1983. However,                                          hydrogen in its core, it is too “young” to produce heavy met-
the discovery of significant anomalies gave several im-                                      als. Metals like calcium, iron, lithium, and magnesium, ob-
plications. First, data collection methods at Mount Wil-                                     served on the stellar surface, can be assumed to be primor-
son Observatory need to be refined; quality and quan-                                        dial. The same is true for stars of similar mass to the Sun.
tity of observations need to be increased. Second, up to                                     Therefore, the relative abundance of primordial metals re-
a third of the stars could be in their Maunder Minimum                                       maining in the surface can be compared to other stellar in-
activity minimums, which occur on centuries-long                                             dices, such as the CRV (Soon et al. 1993), to estimate ages
timescales and occupy one-third of the lifetime of a star.                                   of stars. However, uncertainties in this model exist because
This fact gives a possible error of up to 33.3%. Another                                     of unknown rates of supernovae events, which might con-
source of error is the possibility of active region growth                                   taminate the stars’ original degree of metallicity.
and decay, which produces abnormally high rates of                                                 Length of stellar activity cycles Pcyc may be correlated
rotation. Third and final, the model for predicting age                                      with age. Stellar activity cycles are decades-long variations
based on rotation period of a star should not depend on                                      in stars’ magnetic fields, which in turn are powered by the
its spectral type (or mass). Previous formulae utilized                                      stellar dynamo. The stellar dynamo is perpetuated by inter-
convective turnover time (a function of spectral type)                                       actions between the convective zone and surface differen-
as a coefficient in the algorithm to determine age; the                                      tial rotation. The Pcyc of solar-type stars is about 11 years
first results of this study indicate that rotation periods                                   (Baliunas 1991), while the period of time required for the
and relative ages of solar-type stars are related, but in-                                   magnetic field to switch poles is two times that, or about 22
dependent of spectral type.                                                                  years. As stars age, activity cycles lengthen (Baliunas and
                                                                                             Soon 1995).
                                                                                                   A comparison of age determination models of stars is
                                 Introduction                                                necessary to confirm the validity of these models. Stellar
                                                                                             ages thus inferred may provide information on the lower limit
      Stellar age, rotation period (Prot), and activity fluxes (S)                           of the age of the universe through an estimation of main
are intimately linked in lower main sequence (LMS) stars                                     sequence lifetime (Deliyannis et al. 1989). In addition, com-
such as the Sun (Radick et al. 1990). Comparing ages in-                                     parison of ages of LMS stars will add to the study of the
ferred from activity fluxes to those inferred from rotation                                  relationship between surface magnetic activity, rotation and
periods will determine the validity of stars’ estimated ages,                                age that all stars on the main sequence share. However, a
as well as determine the validity of each of these age deter-                                detailed discussion of these topics is too broad for the scope
mination methods. This study presents first results of the                                   of this research.
increase in rotation period as a star matures on the lower
main sequence. However, notable inconsistencies in plot-                                                                          Methods
ted trends implicate that basing a formula for Prot on color
index is questionable.                                                                             Olin Wilson of the Mount Wilson Observatory (MWO)
      Currently, other projects in the stellar rotation field are being                      in California first began measuring stellar activity cycles in
conducted. At the Von Vleck Observatory, the Orion Monitor-                                  1966. In 1980, the HK Project, a program of nightly obser-
ing Update, started in 1990, is a program to measure the rota-                               vations, was launched to investigate the solar-terrestrial
tion periods of 700 stars with a 24-inch telescope. Seventy-                                 connection, further the study of stellar activity cycles, and to
five of the 700 stars measured exhibit a Prot with a False Alarm                             focus attention on detecting rotation periods. This is an on-
Probability (FAP) of less than 1%. The Spotted Stars III Project                             going project in which each of its 1200 LMS stars is sched-
began in March of 1995 with the goal of investigating Prot of                                uled for several observations per week. Three measure-
stars in clusters. The slow-rotating stars in α Per and the Pleia-                           ments of that same star are typically made per night.
des are constantly examined for new periods. The NGC 6475                                          Data presented here were obtained either from spec-
and IC4665 clusters are also being studied.                                                  trophotometers on the 60-inch (1.5 m) reflector or 100-inch
      To accurately measure the ages of stars, the consis-                                   (2.5 m) Hooker reflector at MWO (Soon et al. 1993), which
tencies of the various age-determination methods must be                                     utilize the Coude and Cassegrain telescope configurations.
gauged. Current methods include the examination of char-                                     The data comes in the form of time versus mean activity
acteristics such as isochrones, chromospheric emissions,                                     level. Ca II flux (from the stars’ chromospheric emission) is
metallicities, and period of activity cycle Pcyc.                                            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.
114                                                       Journal of Undergraduate Sciences                                           Astronomy

Figure 1. (A) An example of a time series plot of HD 82885 data (top). (B) A   Figure 2. A comparison of observed rotation period versus calculated
Fourier transform periodogram of HD 82885 data (bottom).                       rotation period. The Pcalc = Prot line is plotted.

Julian calendar system. Magnetic heating occurs in the                         as candidates for designation of a rotation period. Rota-
chromospheres of stars; starspots are greatest where mag-                      tion periods of less than five days could be spurious be-
netic fields are greatest. Hence, as starspots or inhomoge-                    cause measurements were made two to three days apart.
neities travel across the surface of a star, Ca II flux will re-               Rotation periods longer than 100 days are beyond the sea-
flect that movement (Donahue 1993). Ca II flux is thus used                    sonal window of the observations. When periods fell within
to observe rotation periods.                                                   a range of 10 days or less, the mean was calculated and
      The search for rotational signals was performed on a                     the standard deviation utilized in place of δProt . A file sum-
set of 102 LMS stars taken from the MWO sample. Criteria                       marizing detected Prot, δProt, and FAP was created for each
for these stars’ selection included a B - V color index similar                star.
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                                                    Results
over 3000, due to the rolling inclusion of stars in the HK
project. To begin the search, data points were organized                              Of a total of 102 stars, only 46 had been observed
into seasons, with a season defined as an interval contain-                    with enough data to span one to three seasons. Six stars
ing at least 30 data points and separated from other inter-                    had only 3-10 points of data, and thus did not qualify for
vals by at least 80 days.                                                      even a single season. However, fifty-one out of 102 stars
      Determining Prot required various techniques. The pri-                   exhibited a measurable P rot. Thirty of these stars exhibited
mary means of obtaining Prot was the use of the program “j,”                   FAPs of less than 1.0 x 10-3, which indicates a fairly accu-
created by R. A. Donahue in 1993. J plots activity-versus-                     rate period.
time observations (activity,2 S, is measured by monitoring                            Among the 51 stars with observable seasons, there were
the emission core fluxes within Ca II H and K resonance                        29 (56.7%) with periods detected in one to three seasons.
lines from LMS stars). An example of a time series plot of                     Eighty-two percent of these stars had detectable periods in
Ca II flux is shown in Figure 1A.                                              at least 50% of their seasons. These statistics may be mis-
      J performs Fourier transform periodograms of data                        leading when examined out of context. Half of the stars have
points. J is specifically designed to handle data series with                  only one or two observable seasons. Therefore, finding a
uneven rates of sampling (Horne and Baliunas 1986). A                          period in one season inflates period percentages. However,
sample periodogram of star HD 82885 is shown in Figure                         the determined periods should be given considerable weight
1B. At least one periodogram was computed for each ob-                         because they are relatively more consistent than their flat
serving season; filtering was necessary for transforms with                    counterparts. Additionally, gaps in data and indeterminate
second peaks. Gaps in the data were filled by executing j                      rotation periods are common in the field of rotation study
manually on selected time intervals.                                           (Radick et al. 1990).
      Another function of j that was utilized is its output of                        Thirty stars in the main sample did not receive a Prot
False Alarm Probability (FAP) of periods detected.3 FAP—                       designation. Data for 26 of these stars were so sparse that
the chance in percent that a measured periodic peak will                       a rating of “NMI,” or Need More Information, was given. Four-
arise from purely random (Gaussian) noise—serves as an                         teen stars in the sample earned a rating of “NDP,” or No
indicator of quality of period detection. It is desirable to look              Detectable Period. This occurred when stars showed con-
for FAPs close to zero when searching for a final rotation                     flicting Prot over various seasons, causing any Prot obtained
period. The stars were ranked on a negative log FAP scale,                     to be unreliable. Only nine stars were flat, meaning that they
which was labeled as FApH because of the similarity to the                     exhibited no discernible Prot at all, and instead yielded static
pH scale.                                                                      time series plots.
      Each star received at least five comprehensive reviews                          Sixteen stars were found to have equivocal rotation
of its detected period. Observed periods of less than 5 days                   periods. These periods featured high False Alarm Probabili-
and greater than 100 days were generally not considered                        ties, indicating they were not likely to be accurate.
115                                                           Journal of Undergraduate Sciences                                                     Astronomy

Figure 3. Spectral type, or mass versus the residuals. The difference               Figure 4. Spectral type versus period of rotation. Note the increasing scatter
between P calc and Pobs is minimal, but note the lower left corner region of slow   of Prot as B - V decreases. Again, correlation with the Vaughan-Preston Gap
rotators. This region mirrors the observed 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
Data Sample. In order to evaluate the data sample in the                            rotation period. To begin, smaller intrinsic spread in Prot is
context of previously-published hypotheses, calculated ro-                          characteristic of older LMS stars, but not young ones. This
tation periods, or Pcalc, were obtained for each star. These                        observation is apparent in Figures 3 and 4. The residuals of
Pcalc are a function of chromospheric activity level as well as                     Pcalc - Prot are plotted against B - V in Figure 3. A concentra-
B - V spectral type. The formula for the calculation of rota-                       tion of data points falls on the line Pcalc - Prot = 0, reinforcing
tion periods was derived from Noyes’ algorithm for deter-                           the consistency between Prot and Pcalc. A number of stars at
mining Rossby number from mean stellar activity. A more                             the lower left corner of the graph have low B - V values and
detailed description is given by Noyes et al. (1984).                               negative residuals. They are rotating slower than predicted,
      These calculated rotation periods were compared against                       indicating a potential Maunder Minimum.
the observed rotation periods of the sample field stars in Fig-                           Figure 4 is a plot of B - V versus Prot . As B - V decreases,
ure 2. The relationship between these two values is close to                        the amount of scatter along the Prot axis increases. At high
one-to-one; over half of the stars fall within 10 days of the line                  values of B - V (0.70-0.80), Prot is confined to a relatively
Pobs = Pcalc. However, significant outliers on either side of the x                 narrow range of 0-20 days. As B - V values approach 0.55
= y line prevent definitive predictions. About ten stars rotate                     from 0.70, scatter approaches a range of 0-85 days. Since
faster than predicted, which can be attributed to abnormally                        the increasing scatter is a function of decreasing age, Prot is
high activity levels, or active region growth and decay (ARGD),                     more useful for identifying relative ages of older stars.
an observed phenomenon (Donahue 1993). About eight stars                                  These results are likely to be quite accurate because
rotate slower than predicted; these could be experiencing ab-                       they reflect the findings of Noyes et al. (1984) in the Vaughan-
normally low chromospheric emission levels, or a Maunder                            Preston gap. This is a relation in which younger stars tend
Minimum4 (Baliunas and Soon 1995).                                                  to be more unpredictable in their rotation periods and activ-
      Two overall trends, amplified and elegantly expressed                         ity levels—as compared to older stars. They have a wider
in Figure 2, are apparent in the rest of the Figures 3-6. First,                    spread along their curve than older stars do in an R’HK ver-
the majority of the sample of field stars behaves as expected                       sus Prot plot.
when plotted against axes such as B - V and activity: older                               An additional implication is that rotation period is inde-
stars spin more slowly. The B - V versus residuals plot in                          pendent of spectral type. These results conflict with Noyes’
Figure 3 is nearly level at residuals = 0. Figure 4 shows that                      determination of Prot. This determination may be derived from
Prot tends to decrease as B - V increases. In Figure 5, most                        Noyes’ algorithm linking stars’ dynamo field generation and
of the F and G stars have Prot under 20 days, which is con-                         Rossby number, or Prot / τc.5 Tau equals convective turnover
sistent with that of the Sun, whose rotation period is about                        time, which is a function of mass. Mass, however, is a func-
27 days. Finally, Figure 6 shows a power-law relationship                           tion of spectral type. Hence, this study would implicate elimi-
between activity (age) and rotation period.                                         nating τc from future predictions of Prot.
      The second trend is that there are outliers which devi-                             An illustration of the trends described above is provided
ate significantly from mean relations. These stars either spin                      in Figure 5, a histogram of rotation periods for the sample.
slowly and have high activity levels, or vice versa—estab-                          At lower rotation periods, a high number of stars is found.
lished trends conflict. Additionally, Pobs deviates from Pcalc.                     As the graph moves toward longer Prot, a tail in the graph
There are outliers with Pobs > 60 days in Figure 2, as well as                      becomes evident. This tail is tapered, meaning that fewer
lower Pobs than Pcalc. A region of stars exhibiting high Pobs for                   stars exhibit high rotation periods. The percentage of slow
low Pcalc is found in Figure 3. Figures 4 and 5 show a signifi-                     rotators approaches 50%. A possible explanation is that as
cant portion of the sample with abnormally indeterminate                            stars age, the increasing numbers of mechanisms for angu-
Prot as a function of age. In Figure 6, two regions are detached                    lar momentum transport cause a larger spread in Prot
from the log R’HK versus log Prot power-law relationship.                           (Donahue 1993).
116                                              Journal of Undergraduate Sciences                                               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         pool of 51 stars is undesirable, staggering the stars in
rotation period may be confirmed in Figure 6. The activity          batches of 10, 15, or 25, for alternating years is an alterna-
index, S, has been expressed in log R’HK units6 to provide a        tive. For example, a study of surface differential rotation
measure of magnetic heating (Noyes et al. 1984). Log R’HK,          would require heavy observation of the stars with good pe-
which is chromospheric emission minus any possible photo-           riods, while a study searching for a broad range of periods
spheric contribution, is plotted against log Prot. A strong dis-    would entail scrutiny of the less well-observed stars.
tribution of stars falls along a line of negative slope (with             Examining the 102 stars by hand is a rate-limiting step
power ~ 1), which illustrates a possible power-law relation-        of the data processing. Computer routines can be devel-
ship between activity (age) and Prot . If the power law is close    oped to refine the discrimination with which automated
to 1, it suggests that R’HK α P -1.                                 periodogram analysis and selection of interval are com-
     Three stars are found above the range of the main              pleted. The main drawback to automated data processing
graph, but they fall on a line of parallel slope. These stars       is the high degree of experience and judgment needed to
rotate slowly (indicating older solar-type stars) but emit high     select an appropriate interval for analysis; improved pro-
levels of Ca II (indicating younger solar-type stars)—per-          grams, however, could alleviate the problem and save time.
haps these are the RS-CVn stars. An even more prominent
cluster of data points falls below the main relation. These                                      Conclusions
are older stars with uncharacteristically small rotation peri-
ods (see Figure 3). This phenomenon deserves further in-                  The relationship between stellar rotation periods (Prot)
vestigation. These stars are unusual because their activity         and relative ages of stars in a sample of Sun-like stars was
levels are low (indicating older solar-type stars), yet they        examined, using the empirical link between activity level and
spin quickly (a characteristic of younger solar-type stars).        age to calibrate the ages. Data from the last 15 years of
                                                                    observation in Mount Wilson Observatory’s HK Project were
Errors and Improvements. As with any astrophysics re-               used. This consisted of time-series plots of Ca II H and K
search project, the accuracy of measurements made is lim-           chromospheric emissions separated by at least 80 days.
ited by several factors. These factors include the potential        Fourier transform periodograms of the data were performed
inclination angles of the stars to the line of sight of the ob-     to determine Prot for each star; each of the 102 stars re-
server. An active region (or zone of high convective turn-          ceived at least five comprehensive reviews.
over rate) occurring before or after another one on the vis-              Fifty-one stars were found to have measurable rotation
ible disk may also interfere with data, as might the differ-        periods—a significant improvement over past studies, where
ence in Prot at different latitudes of the stars (Morfill et al.    Prot of stars were reported in batches of only 15 or 20. The
1991). Differential rotation adds uncertainty to the observed       majority of these stars exhibited increasing Prot as a function
Prot but may contribute to the process of age determination         of increasing age. However, anomalies were discovered as
(Donahue 1993).                                                     well: outliers of older stars that rotated too quickly, and
      Although Prot and Pcalc agree for most stars with Prot < 20   younger stars that rotated too slowly.
days, the |Pcalc - Prot | can be large for some slowly rotating           The implications of these first results are fivefold. First,
stars. This implies an error in either Pcalc, Prot, or both. In-    younger stars rotate faster than older stars, as has been
creasing the quantity and quality of observations will illumi-      predicted previously. Second, the anomalies are most likely
nate the source of these errors.                                    the result of unrefined data collection methods at Mount
      To obtain more accurate measurements of rotation pe-          Wilson Observatory. A recommendation would be to increase
riods, an increased frequency of observations is needed.            the quantity, quality, and regularity of observations—this will
The data sample may be reduced to the 51 best stars or the          decrease the False Alarm Probabilities of the rotation peri-
51 least well-observed stars. Then, a commitment to ob-             ods. Third, the anomalies could also signify stars in their
serve the stars every night must be made. If abandoning a           activity maximums or Maunder Minimums, which are low
117                                                   Journal of Undergraduate Sciences                                             Astronomy

points in their centuries-scale activity flux. Fourth, these re-          (5) log(P / τ) = f(<R’ HK>) = 0.324 - 0.400y - 0.283y2 - 1.325y3, where
sults reflect a Vaughan-Preston gap—a relation in which                   α = 1.9, and y = log τc(2)( B - V).
younger stars exhibit wider spread in Prot than older stars do            (6) Net chromospheric emissions (in log R’HK) have been empiri-
in an H-K flux ( R’HK ) versus Prot plot. This not only supports          cally linked to ages of stars (Soderblom et al. 1991, Donahue 1993).
the Vaughan-Preston relation, but also supports the accu-
racy of the rotation periods found.                                                                   References
      Fifth, and perhaps most significant, this study finds no
correlation between a star’s B - V and its rotation period. Previ-        Baliunas, S. 1991. “The past, present and future of solar magne-
ous studies (e.g., Noyes et al. 1984) have fitted functions to            tism: stellar magnetic activity.” In: The Sun in Time (eds. Sonett, C.
calculate age from Rossby number, a coefficient equal to Prot /           P., M. S. Giampapa, and M. S. Matthews) (Tucson, AZ: University
τc. Tau, or convective turnover time, is a function of B - V spec-        of Arizona Press): 334-67.
tral type. When the residuals between calculated rotation peri-           Baliunas, S., R. A. Donahue, W. H. Soon, et al. 1995. “Chromo-
ods (calculated using a derivative of the Noyes formula) and              spheric variations in main sequence stars. II.” Ap. J. 438: 269-87.
observed rotation periods were plotted against B - V, large scat-
ter occurred. This indicated that rotation periods are indepen-           Baliunas, S., and W. Soon. 1995. “Are variations in the length of
dent of spectral type. That is to say, rotation periods of stars of       the activity cycle related to changes in brightness in solar-type
particular spectral types exhibit relative rates of decline of ro-        stars?” Ap. J. 450: 896-901.
tation period with age. Although one may certainly correlate              Crawford, D. L. 1975. “Empirical calibrations of the uvbyβ systems.
spectral type with age, it is not to be done in a model that incor-       I. The F-type stars.” Astron. J. 80: 955-71.
porates both B - V and color.
      The study of the rotation periods of lower main sequence            Dearborn, D. S. 1991. “Standard solar models.” In: The Sun in Time
stars will afford us greater insight into the life cycles of these        (eds. Sonett, C. P., M. S. Giampapa, and M. S. Matthews) (Tucson,
stars and their evolution on and off the main sequence. Iden-             AZ: University of Arizona Press): 457-90.
tifying the characteristics of these stars will also isolate con-         Deliyannis, C. P., P. Demarque, and M. H. Pinsonneault. 1989. “The
ditions for the origin of life on planets orbiting these stars.           ages of globular cluster stars: effects of rotation on pre-main se-
Comparing the various age determination methods of stars                  quence, main sequence, and turnoff evolution.” Ap. J. 347: L73-
available (the next step in this study would be to compare                L76.
these results against metallicities, isochrone ages, activity
                                                                          Donahue, R. A. 1993. “Surface differential rotation in a sample of
cycle and long-term Ca variability) will enable us to improve
                                                                          cool dwarf stars.” Ph.D. thesis (New Mexico State University): 1-
these methods. Ultimately, one could take a “snapshot in                  205.
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               Duncan, D. K. 1984. “A sample of solar-type stars of known age.”
the Universe could be verified.                                           Astron. J. 89: 515-22.

                                                                          Eaton, N. L., W. Herbst, and L. A. Hillenbrand. 1995. “Rotation
                       Acknowledgments                                    periods and variability of stars in the Trapezium Cluster.” Astron. J.
                                                                          110: 1735-48.
    The efforts of Dr. Sallie Baliunas, Dr. Willie Soon, Dr.
Bob Donahue, Beth Wicklund, and Eric Ford are appreci-                    Horne, J. H., and S. L. Baliunas. 1986. “A prescription for period
                                                                          analysis of unevenly sampled time series.” Ap. J. 302: 757-63.
ated. Additional thanks go to Sara Seager, Ms. Deeley, and
Dr. Yvonne Pendleton, who is at NASA-Ames Research                        Morfill, G., H. Sheingraber, and W. Voges. 1991. “Sunspot number
Center. I would also like to thank the Center for Excellence              variations: stochastic or chaotic.” In: The Sun in Time (eds. Sonett,
in Education and Research Science Institute for generous                  C. P., M. S. Giampapa, and M. S. Matthews) (Tucson, AZ: Univer-
sponsorship of this research.                                             sity of Arizona Press): 123-49.

                                                                          Noyes, R. W., L. Hartmann, S. L. Baliunas, et al. 1984. “Rotation,
                             Endnotes                                     convection, and magnetic activity in lower main-sequence stars.”
                                                                          Ap. J. 279: 763-77.
(1) First, the absolute magnitude, or M v, of the sample solar-type
stars (lower main sequence, or LMS) is plotted against their Teff .       Radick, R. R., B. A. Skiff, and G. W. Lockwood. 1990. “The activity,
Second, δMv, the difference between Mv and zero-age main se-              variability, and rotation of lower main sequence members of the
quence (ZAMS), is calculated and plotted against Teff . The accu-         Coma star cluster.” Ap. J. 353: 524-32.
racy of isochrone ages for LMS stars are tested by analyzing clus-
ters of stars—those in the Hyades and M67, for example, should            Soderblom, D. R., D. K. Duncan, and R. H. Johnson. 1991. “The
have the same isochrones within each cluster.                             chromospheric emission-age relation for stars of the lower main
                                                                          sequence and its implications for the star formation rate.” Ap. J.
(2) S = α [(H + K) / (V + R)], where H and K are the Ca II passband       375: 722-31.
counts and V and R are the violet and red continuum band counts.
α is a calibration factor called lamp correction that changes nightly.    Soon, W. H., Q. Zhang, S. L. Baliunas, et al. 1993. “The Mount
                                                                          Wilson Observatory metallicity index, CRV : comparison with other
(3) FAP = 100[1 - (1 - EXP(-z / ρ2) Ni], where ρ2 equals the total data   photometric systems.” Ap. J. 416: 787-805.
variance and Ni equals the independent frequencies, estimated us-
                                                                          Walter, F., and D. Barry. 1991. “Pre- and main sequence evolution
ing Horne and Baliunas’ technique (1986).
                                                                          of solar activity.” In: The Sun in Time (eds. Sonett, C. P., M. S.
(4) In the 17th century, the Earth experienced a Little Ice Age. French   Giampapa, and M. S. Matthews) (Tucson, AZ: University of Ari-
paintings from this era depict frozen rivers that never freeze over       zona Press): 283-97.
today. During this same time, Maunder observed abnormally low
energy output from the Sun (and established a basis for the solar-
terrestrial connection). Hence, error mean S is up to 33.3%, be-
cause stars spend one-third of their lifetimes in Maunder Minimums.

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