DYNAMICAL MODEL FOR SPINDOWN OF SOLAR-TYPE STARS
Aditi Sood
1
, Eun-jin Kim
1
, and Rainer Hollerbach
2
1
School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK
2
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
Received 2015 October 11; revised 2016 August 26; accepted 2016 September 10; published 2016 November 21
ABSTRACT
After their formation, stars slow down their rotation rates by the removal of angular momentum from their surfaces,
e.g., via stellar winds. Explaining how this rotation of solar-type stars evolves in time is currently an interesting but
difficult problem in astrophysics. Despite the complexity of the processes involved, a traditional model, where the
removal of angular momentum by magnetic fields is prescribed, has provided a useful framework to understand
observational relations between stellar rotation, age, and magnetic field strength. Here, for the first time, a
spindown model is proposed where loss of angular momentum by magnetic fields evolves dynamically, instead of
being prescibed kinematically. To this end, we evolve the stellar rotation and magnetic field simultaneously over
stellar evolution time by extending our previous work on a dynamo model which incorporates nonlinear feedback
mechanisms on rotation and magnetic fields. We show that our extended model reproduces key observations and is
capable of explaining the presence of the two branches of (fast and slow rotating) stars which have different
relations between rotation rate Ω versus time (age), magnetic field strength
B
∣
∣
versus rotation rate, and frequency of
magnetic field
w
cyc
versus rotation rate. For fast rotating stars we find that: (i) there is an exponential spindown
Wµ
-
e
t
1.35
, with t measured in Gyr; (ii) magnetic activity saturates for higher rotation rate; (iii)
w
µW
cyc
0.83
. For
slow rotating stars we find: (i) a power-law spindown
Wµ
-
t
0.52
; ( ii ) that magnetic activity scales roughly linearly
with rotation rate; (iii)
w
µW
cyc
1.1
6
. The results obtained from our investigations are in good agreement with
observations. The Vaughan–Preston gap is consistently explained in our model by the shortest spindown timescale
in this transition from fast to slow rotators. Our results highlight the importance of self-regulation of magnetic
fields and rotation by direct and indirect interactions involving nonlinear feedback in stellar evolution.
Key words: evolution – stars: activity – stars: magnetic field – stars: rotation – stars: solar-type
1. INTRODUCTION
Spindown of stars is one of the most debated and interesting
issues in astrophysics. Stellar rotation rate is the key parameter
which is believed to affect the spindown process. Spindown is
not only influenced by stellar properties such as mass, radius,
and age, but also depends upon the evolution of stellar
magnetic fields and their interaction with the stellar atmosphere
(Scholz 2008). After their formation from interstellar clouds,
which involves various internal changes, stars undergo
rotational evolution in different stages (Keppens et al. 1995;
Tassoul 2000), briefly summarized as follows. During early
pre-main sequence evolution, the contraction that occurs in the
star along with other various internal structural changes lead it
to spin-up. Also, owing to diverse internal changes, a radiative
core develops which rotates faster than the convective
envelope. Coupling between the radiative core and convective
envelope should be strong enough for angular momentum to be
constantly transferred from the core to the envelope. This
persistent supply of angular momentum reduces the amount of
differential rotation produced in the star. By the time the star
reaches late pre-main sequence or early main sequence,
rotational evolution is modi fied by the stellar wind. Angular
momentum loss via the stellar wind gradually decelerates and
stops the spin-up of the convective envelope toward the end of
late pre-main sequence phase, and causes a fast spindown of
the envelope on the main sequence. The timescale at which the
decoupling of the core and envelope occurs is observed to be
very rapid (Keppens et al. 1995). With increasing rotation, the
timescale for angular momentum loss through the stellar wind
decreases and affects the magnetic field strength.
Consequently, for rapidly rotating stars, the magnetic field
strength does not increase beyond a critical value at a certain
rotation rate and instead becomes independent of rotation no
matter how rapid. When the convection zone spins down
toward the end of the pre-main sequence, the magnetic field
strength is believed to scale linearly with rotation rate in the
case of slow rotating stars.
Based upon the whole spindown process, stars are often
classified into two groups: fast and slow rotating (Saar &
Brandenburg 1999, Brandenburg et al. 1998, Barnes 2003;
Pizzolato et al. 2003; Mamajek & Hillenbrand 2008; Wright
et al. 2011; Vidotto et al. 2014). The existence of two branches
of stars exhibiting different dependences of cyclic variation of
stellar magnetic activity, expressed as cycle period P
cyc
on
rotation period P
rot
, was confirmed by Brandenburg et al.
(1998), and later by Saar & Brandenburg (1999). We note that
a relationship between cycle period and rotation period was
first established by Noyes et al. (1984) as
µPP
n
cyc
ro
t
with
=
n
1.25 0.5
. Brandenburg et al. showed all young, active,
and fast rotating stars lie on one branch, namely the active
branch (A) with scaling exponent n=0.80, while all old,
inactive, and slow rotating stars lie on another branch, namely
the inactive branch (I) with scaling exponent n=1.15
(Charbonneau & Saar 2001; Saar & Brandenburg 2001).
Furthermore, stars on the A branch experience rapid spindown
for which the rotation rate Ω is related to time/age with an
exponential law given as
Wµe
mt
, where m is a negative
constant, and in this case magnetic activity is found to be
saturated, that is, it becomes independent of rotation rate for
rapidly rotating stars. Stars on the I branch undergo a very slow
spindown with a power-law dependence as
Wµ
-
t
12
, known
The Astrophysical Journal, 832:97 (10pp), 2016 December 1 doi:10.3847/0004-637X/832/2/97
© 2016. The American Astronomical Society. All rights reserved.
1
as power-law spindown (Skumanich 1972), and in this case
magnetic activity is thought to scale linearly with rotation rate.
The relationship between magnetic activity and rotation rate is
important for understanding the physical process responsible
for spindown of a star and was first determined by Pallavicini
et al. (1981), while Micela et al. (1985) observed that this
relationship does not hold for rapidly rotating stars. We note
that the regime where magnetic activity increases linearly with
rotation rate is termed the “unsaturated (non-saturated)
regime,” while the regime where magnetic activity becomes
independent of rotation rate is termed the “saturated regime” in
observational studies (e.g., Pizzolato et al. 2003; Mamajek &
Hillenbrand 2008; Wright et al. 2011; Vidotto et al. 2014).
One of the challenges in explaining spindown is the
existence of a gap between the two branches of stars. During
spindown, the star suddenly jumps from the A to the I branch,
creating a gap between the two branches where stars are sparse.
This gap was first observed by Vaughan & Preston (1980) and
is now known as the V–P gap. Various mechanisms have so far
been proposed for this gap, but the underlying physics is still an
open question. Some of the previous suggestions are as
follows. Durney et al. (1981) advocated a change in magnetic
field morphology from complex to simple at the time when
rotation decreases to a certain value. Saar (2002) proposed that
the existence of two distinct branches of stars could be due to
changes in differential rotation, α-effect, and meridional flow
speed (which is proportional to Ω in case of flux transport
models, e.g., see Dikpati & Charbonneau 1999) with stellar
rotation rate. Barnes (2003) studied period–color diagrams of
open clusters and suggested that the transition from the
convective (fast rotators) to the interface sequence (slow
rotators) is due to the shear produced during the decoupling of
the core and envelope. This shear gives rise to large-scale
magnetic fields, and recoupling of the core and convection zone
shifts the star from the convective sequence to the interface
sequence. Structural changes in large-scale magnetic fields
(Donati & Cameron 1997), changes in dynamo action (Böhm-
Vitense 2007), and manifestation of different dynamos for
different stars (Wright et al. 2011) were also proposed as
possible reasons for the V–P gap.
Given the complexity of the spindown problem, which
depends upon various parameters such as rotation rate,
evolution of magnetic fields, and differential rotation, it is not
possible to study a full magnetohydrodynamic model over the
entire spindown timescale (e.g., from
10 to 10
79
yr). There-
fore, various simplified models have been utilized to under-
stand stellar evolution (e.g., Weber & Davis 1967;
Mestel 1968; Mestel & Spruit 1987; Kawaler 1988; Cohen
et al. 2009; Cranmer & Saar 2011; Matt et al. 2012; Garraffo
et al. 2015; Johnstone et al. 2015; Matt et al. 2015). One such
model is the double zone model (DZM) which is based upon
the stellar wind torque law (Weber & Davis 1967; Mestel 1968;
Belcher & MacGregor 1976; Kawaler 1988). The main feature
of this model is the bifurcated expression considered for the
torque acting on the star (depending on the critical rotation rate)
due to its magnetized stellar wind. MacGregor & Brenner
(1991) used this DZM model for coupled (ordinary differential)
equations for the rotation rates of the stellar envelope and
radiative core, where the angular momentum loss is prescribed
according to the relation between rotation and magnetic field
strength. To understand the distribution of stellar rotation at
different ages, Keppens et al. (1995) extended this
parameterized model to describe the evolution of a single star
by taking into account angular momentum exchange, moment
of inertia evolution, and torque exerted on the core and
envelope, which cause changes in angular momentum. Since
then, this model was extended by considering different initial
conditions and tested against various observations in the
spindown process (Krishnamurthi et al. 1997; Irwin & Bouvier
2009; Denissenkov et al. 2010; Leprovost & Kim 2010; Spada
et al. 2011; Reiners & Mohanty 2012; Gallet & Bouvier 2013;
Epstein & Pinsonneault 2014
). Apart from DZM there are other
models such as the symmetrical empirical model (SEM,
Barnes 2010; Barnes & Kim 2010) and the metastable dynamo
model (MDM, Brown 2014). Both SEM and MDM utilize
observational data of two different sequences of stars to fine-
tune their models and thus are descriptive rather than
explanatory. Specifically, SEM uses the different period
evolution of the two sequences (for active and inactive stars)
depending on whether the rotation rate is above/below the
critical value and fits the parameters from period–color
diagrams by obtaining a best fit to the observational data.
Unlike SEM, MDM uses one function for all rotation rates but
two different coupling constants. By fine-tuning the values of
these two coupling constants and the probability for the
transition from small to large couplings, MDM improves the
agreement with observations over SEM. Although it is still
empirical, MDM is remarkable in introducing into a spindown
model a threshold-like behavior with different coupling
constants and their probabilistic nature. Possible mechanisms
for these different coupling constants was later provided, e.g.,
by invoking a change in magnetic complexity ( Garraffo et al.
2015; Réville et al. 2015). Recently, Matt et al. (2015)
proposed a stellar wind torque model which reproduces the
shape of the upper and lower envelopes, which corresponds to
the transition region between the saturated and unsaturated
regimes by explaining the mass-dependence of stellar magnetic
and wind properties.
In this paper, we propose for the first time a dynamical
model of spindown where the loss of angular momentum by
magnetic field is treated dynamically, instead of being
prescribed kinematically. To this end, we evolve the stellar
rotation and magnetic field simultaneously over the stellar
evolution time by extending our previous work (Sood & Kim
2013, 2014) which incorporates nonlinear feedback mechan-
isms on rotation and magnetic fields via α-quenching and
magnetic flux losses as well as mean and fluctuating rotation.
We note that Sood & Kim (2013, 2014)
demonstrated that
nonlinear feedback plays a vital role in the generation and
destruction of magnetic fields as well as self-regulation of the
dynamo. In particular, it was found that a dynamic balance is
required not only in the generation and destruction of magnetic
fields but also in the fluctuating and mean differential rotation
for the working of dynamo near marginal stability; their results
were consistent with observations such as the linear increase in
cycle frequency of the magnetic field with moderate rotation
rates, levelling off of magnetic field strength with sufficiently
large rotation rates, and quenching of shear. We extend this
model to simultaneously evolve rotation and magnetic fields
over the spindown timescale of a star, since their dynamics are
closely linked through angular momentum loss and dynamo.
That is, the angular momentum loss responsible for the
spindown of a star depends upon magnetic fields which in
turn are affected by rotation rates. We show that this model has
2
The Astrophysical Journal, 832:97 (10pp), 2016 December 1 Sood, Kim, & Hollerbach
the capability of explaining the existence of the two branches of
stars, different rotation rate dependence of cycle frequency of
the magnetic fields for these two branches, and the gap between
the two branches, reproducing the main observations. By
extending our previous work, our model is designed in such a
way that it has the essential ingredients mentioned above to
explain the complex process of spindown of solar-type stars,
and to highlight the importance of nonlinear feedback in this
process.
2. THE MODEL
We propose a dynamical model for the evolution of rotation
rate and magnetic field in spindown by extending a previous
nonlinear dynamo model (Weiss et al. 1984; Sood & Kim
2013, 2014). In particular, Sood & Kim (2013, 2014)
incorporated various nonlinear transport coefficients such as
α-quenching and flux losses and took the control parameter D,
known as the dynamo number, to scale with rotation rate as
µW
D
2
. The model equations in dimensionless form are given
as
k
l=
+
-+A
DB
B
BA
2
1
1,1
2
1
2
˙
(∣ ∣ )
[(∣∣)] ()
l=+ - -+
*
Bi wA iAw BB1
1
2
1,2
02
2
˙
() [(∣∣)] ()
n=--
**
wiABAB w
1
2
.3
000
˙( ) ()
n=- -wiABw.4˙()
Here, the poloidal magnetic field is represented by A, the
toroidal magnetic field is given by B, w
0
is the mean differential
rotation, and w is the fluctuating differential rotation; A, B and
w are complex variables whereas w
0
is real. We note that w
0
has zero frequency, and w has twice the frequency of A and B.
The complex conjugates of A and B are denoted by A
*
and B
*
.
In this model, A is generated by B (e.g., α-effect through
helicity) which is assumed to be proportional to the rotation
rate Ω (see Equation (1)). Equation (2) represents the
generation of B by A, where the quenching of the Ω-effect is
incorporated by the total shear
+ w1
0
. The differential rotation
is inhibited by the tension in the magnetic field lines via the
Lorentz force and causes the quenching of the Ω-effect. Due to
back-reaction, the total shear is reduced from 1 to
+<w11
0
as w
0
is always negative and is given by
+=DWWw1
0
. The
generation of w
0
and w is represented by Equations (3) and (4),
respectively.
n
0
and ν represent the viscosity of w
0
and w,
respectively; κ,
l
1
and
l
2
are constant parameters which
represent the strength of nonlinear feedback due to the Lorentz
force by the magnetic field and enhanced magnetic dissipation
(e.g., magnetic flux loss). In particular, κ represents the
efficiency of the quenching of the α-effect while
l
1
and
l
2
represent the efficiency in the poloidal and toroidal magnetic
flux losses, respectively (see Sood & Kim 2013 for full details).
To understand the evolution of rotation rate and magnetic
field in the spindown of solar-type stars, we extend this model
by upgrading Ω from a kinematically prescribed to a dynamic
variable. To this end, we first replace D by the square of the
time-dependent rotation rate Ω(t) in Equation (1):
k
l=
W
+
-+A
B
B
BA
2
1
1,5
2
2
1
2
˙
(∣ ∣ )
[(∣∣)] ()
where Ω is real. Second, we need to include an additional
equation for the evolution of
W t(
)
to model the spindown of a
star by the loss of angular momentum due to magnetic fields.
While the latter depends on many factors such as the mass flux
and geometry and complexity of the magnetic fields (e.g.,
Garraffo et al. 2016), for example, the Alfvén radius over
which it acts as a rotational brake and the latitude at which the
mass release occurs, for simplicity, we incorporate their overall
effects in our dynamical model by the ansatz that the decay rate
of Ω is proportional to the strength of the magnetic fields as
e
e+
W
B
A
1
2
2
2
∣∣
∣∣
with the two tunable parameters
1
and
2
.
Here,
B
∣
∣
represents the strength of the toroidal magnetic field
and
W
A∣∣
is the strength of the poloidal magnetic field in physical
units due to our non-dimensionalization (see Sood & Kim
2013). Our empirical model is thus described by the following
equation for Ω:
eeW=- W-
W
WB
A
.6
1
2
2
2
2
˙
∣∣
∣∣
()
Equation (6) represents the overall spindown of the star as a
whole due to the loss of angular momentum through magnetic
fields. The constant parameters
e
1
and
e
2
represent the
efficiency of angular momentum loss via toroidal and poloidal
magnetic fields, respectively, which are taken to be indepen-
dent in general, given the uncertainty regarding the precise role
of these fields in spindown. Equation (6) is motivated to
capture the key feature of the previous model (e.g., DZM)
where the dependence of the angular momentum loss on Ω is
roughly proportional to
W
3
for slowly rotating stars (below the
critical rotation rate), and to Ω for fast rotating stars (above the
critical rotation rate). Specifically, for fast rotating stars with
the rotation rate above the critical value,
B
∣
∣
and
A
∣
∣
become
independent of Ω, and Equation (6) reduces to
Wµ-
W
˙
,
resulting in the exponential decay of Ω with time. On the other
hand, for slow rotating stars,
W~-W
3
˙
would be reproduced
should the magnetic field increase linearly with Ω as
~WBA,
∣
∣∣∣
(see Section 3.2 for the scaling relation).To
summarize, our extended model consists of Equations (2)–(4)
and (5), (6), where Equations (2)–(4) are the same as in our
previous model, Equation (5) is the modified form of Equation
(1), and (6) is a new equation to model the time evolution of Ω.
This system is investigated taking
n
= 0.5
,
n
= 35.0
0
,
k = 0.025
,
l
= 1.125
1,2
, and
e
=
-
3.5 10
1,2
5
·
. The parameters
ν,
n
0
, κ, and
l
1,2
are much the same as in our previous work
(Sood & Kim 2013, 2014). As can be seen from Equation (6),
the two new parameters
e
1,2
control the rate of the spindown
process. The value
-
3
.5 10
5
·
was chosen as it yields an overall
spindown timescale of several Gyr; larger (smaller) values of
e
1,2
were also investigated, and yielded qualitatively the same
dynamics, simply occurring on shorter (longer) timescales. In
particular, we have checked that qualitatively similar results are
obtained in the limiting cases where
= 0
1
or
= 0
2
.
Correspondingly, the dimensionless timescales are such that
3
The Astrophysical Journal, 832:97 (10pp), 2016 December 1 Sood, Kim, & Hollerbach
the largely completed spindown process translates to the
present-day age of the Sun of 4.5 Gyr.
To model the spindown process, we take the initial value of
Ω to be 30, corresponding to thirty times the present-day solar
rotation, which is
W=1
in our non-dimensionalization. An
initial value of
W=30
is intended to model the rotation rate of
young stars at an age of around ∼10
7
years. In contrast to Ω,
which can only decrease monotonically according to
Equation (6), the initial conditions of the other four variables
are not important, as they can increase as well as decrease, and
turn out to settle into statistically stationary states on
comparatively rapid timescales; that is, transients depending
on the initial conditions of these quantities quickly vanish, and
the subsequent evolution depends only on the initial value
chosen for Ω. Finally, note that because Ω is monotonically
decreasing in time, we can effectively invert the relationship
W t(
)
as
Wt (
)
, and therefore consider all the other variables as
functions of Ω rather than t.
3. RESULTS
3.1.
W
Versus Age Relationship
Figure 1 shows the relationship between Ω and t. A sharp
decrease in Ω can be seen for earlier times, which slows down
as age starts increasing. In the left panel of Figure 2 we fit this
curve using an exponential law, that is,
Wµe
mt
. The best fit,
for stars with rotation periods in the range
P13
rot
, has
=-m 1.35
, corresponding to an e-folding time of 0.74 Gyr. In
the right panel of Figure 2 we fit this curve using power laws,
that is,
Wµt
n
. For larger times we get power-law scalings
Figure 1. Left panel shows Ω as a function of age over the full range [0.03, 4.53] Gyr. As noted in Section 3, we scale our dimensionless time in physical units by the
age of the present-day Sun of 4.5 Gyr, while our dimensionless rotation, represented on the vertical axis, is scaled with thirty times solar rotation to obtain the rotation
period (P
rot
) in days, which is depicted on the right side of the y-axis. The right panel shows a zoomed-in view for age=[0.065155, 0.065185] Gyr, and reveals the
presence of fluctuations superimposed on the general spindown trend.
Figure 2. Left panel shows exponential spindown with
Wµ
-
exp
t1.35
for ages
Î 0.03, 0.7325[]
Gyr depicted in red, while blue represents the trend in semi-
logarithmic scale for age
Î 0.03, 4.53[
]
Gyr. The right panel shows the power-law spindown,
Wµt
n
, with scaling exponent −0.52 for solar-type stars. A gradual
decrease in
n
∣
∣
suggests a drop in the efficiency of angular momentum loss, which seems to align with the suggestion for the reduction in the efficiency of magnetic
braking from recent observations from the Kepler space telescope (e.g., Garraffo et al. 2016).
4
The Astrophysical Journal, 832:97 (10pp), 2016 December 1 Sood, Kim, & Hollerbach
which vary gradually for different rotation rates, that is, n
becomes smaller for smaller rotation as observed in MacGregor
& Brenner (1991). For larger ages (slower rotation rates), the
power-law exponent n is found to be around −0.52 for stars
with rotation periods in the range
<P
2
3 25.65
rot
. For
different rotation rates we summarize the scalings in Table 1.
3.2.
B
∣
∣
Versus
W
Relationship
Magnetic field strength
B
∣
∣
is shown as a function of rotation
rate in Figure 3 ( left panel). The unit of B is normalized by the
strength of the magnetic field in the present-day Sun, which is
roughly of order 10
4
G in the solar tachocline and 3 G in the
atmosphere. Figure 3 exhibits notably different behavior of
B
∣
∣
in the two different rotation rate regimes. For slow rotation
rates, we can clearly see the increasing behavior of
B
∣
∣
with
rotation rate which attains a maximum value at
W»5.8
. For
WÎ 1.17, 5[
]
, the scaling of
B
∣
∣
with respect to Ω is found to
vary between 2.73 and 0.36. We observe an average scaling of
1.47 for
WÎ 1.25, 2[
]
which is close to the observed scaling of
1.38±0.14 (Vidotto et al. 2014). We note that
A
∣
∣
also scales
with Ω similarly to
B
∣
∣
. Interestingly, there is a decrease in
B
∣
∣
which continues up to
W 12.5
. For
W 12.5
, that is, for
very high rotation rates,
B
∣
∣
fluctuates on a very rapid timescale,
but with a cycle-averaged value, depicted in red, that is
essentially independent of Ω. The rapid fluctuations in
B
∣
∣
are
due to the presence of two modes with different frequencies.
The fluctuating behavior of
B
∣
∣
with Ω can be seen in Figure 3
(right panel) for a small cut of
WÎ 23.30, 23.31[
]
. Note how
the system spends more time near the top as opposed to the
bottom, which explains why the cycle-averaged value of
B
∣
∣
(the red curve in the left panel) is higher than the simple
average of the cycle maxima and minima (the highs and lows
of the blue curves). (Observationally this would suggest that
stars might be more likely to be observed close to a peak of
magnetic activity rather than a trough.) Furthermore, we notice
a gap between the two different rotation rate regimes in the
region
WÎ 5.8, 12.5[
]
.
3.3. Power Spectra of B and the
w
cyc
Versus
W
Relationship
To understand how the rapid cycles in
B
∣
∣
gradually evolve as
Ω spins down, we divided the entire time series into discrete
chunks of 0.0106 Gyr, and performed a Fourier transform on
each chunk separately. The precise length of the individual
sections is not important, the only requirements being that it
should be long compared with the fast cycle time, but short
compared with the gradual spindown evolution time. Figure 4
shows Fourier spectra for eight such sections. It is notable that
at earlier times shown in the first and second rows, there are
two main peaks around
w
~ 10
in the spectra, whereas at later
times there is only one in the third and fourth rows, with the
peaks furthermore shifting to lower frequencies. In particular,
in the second and third rows, where time increases from 0.1460
to 0.2831 Gyr, we find the peaks shifting gradually toward
lower frequency as time increases. This behavior continues
until we reach a time of approximately 0.3253 Gyr, beyond
which the multiple peaks of frequency are found to diminish.
This behavior of frequency can be seen in panel 7 of Figure 4
for time ≈[0.3148, 0.3253] Gyr while for time ≈[0.3569,
0.3675] Gyr we find only a single peak of frequency (see
Figure 4 panel 8). The behavior of the power spectra of
B
∣
∣
clearly shows that the second peak of frequency vanishes as
time increases, that is, as the rotation rate decreases. We note
that in addition to the two main peaks at
w
~ 10
or
w
< 10
that
we discussed above, one or two more peaks are also observed
at higher frequency
w
~ 20
in the first and second rows. These
high-frequency modes have much weaker power than the main
peaks and are simply their subharmonics. In the following, we
do not discuss these modes and only focus on the behavior of
the main peaks (e.g., the higher-frequency modes are not
shown in Figure 5).
The gradual transitions in the spectra of
B
∣
∣
are further
illustrated in Figure 5, showing so-called short-time Fourier
transforms. In this technique the signal is again divided into
short chunks, but these now overlap, essentially forming a
moving window, and hence giving an overview of the
continuous evolution of frequencies and amplitudes. Using
this method, the most pronounced frequency of
B
∣
∣
is obtained
in Figure 5 (left panel) where high to low intensity of frequency
is illustrated via bright red to dark blue colors as shown in the
color map. For early times
<t 0.3253 Gyr
, we observe two
curves of frequency of maximum intensity
w
cyc
(depicted in
red) with age in Gyr. The lower curve has a larger amplitude of
frequency than the upper curve. The existence of these two
curves is the manifestation of complex time behavior of fast
rotators and is reminiscent of the complexity of magnetic
topology for active branch stars, discussed in recent papers
(e.g., Matt et al. 2015). Both upper and lower curves show that
the frequency of maximum intensity decreases with age rapidly
until
~t 0.3253 Gyr
when the upper curve disappears while
the lower curve exhibits a change in behavior. This single curve
for
>t 0.3253 Gyr
is interpreted as an inactive branch.
In order to investigate further, we examine the scaling of
w
cyc
by showing the behavior of frequency of maximum intensity
w
cyc
against rotation rate Ω in the right panel in Figure 5.
Again, we notice that for fast rotation rate we have two curves
of frequency for maximum intensity, whereas for slow rotation
rate we have only a single curve. We use a power-law
relationship, that is,
w
µW
p
cyc
with a power-law index p to
obtain the scaling. For the upper curve, we find the value of
~p 0.8
3
for stars with rotation rate
W12.8 30
. On the
other hand, the scaling exponent p of the lower curve varies
with rotation rate, as shown in Table 2. Interestingly, for fast
rotators with
W>1
2
, an average value is
~p 0.9
, which is
close to the observational value for active branch stars (Saar &
Brandenburg 2001); for slow rotators, solar-like stars with
rotation rate in the range
1.17, 3.5
[]
have
~p 1.16
, in good
agreement with observed scaling exponent for solar-type stars
lying on the inactive branch (Saar & Brandenburg 2001).
3.4. Total Shear Versus
W
Relationship
In our dimensionless units, the total shear is given by
+ w1
0
. Figure 6 shows how this total shear changes with
rotation rate Ω.AsΩ increases from
W=1
, the total shear is
seen to decrease by 90% from 1 to 0.1 with increasing Ω. This
Table 1
Power-law Exponent n for Stars with Different Rotation Period in Days
n Ω P
rot
(days)
−1.38
WÎ 3.5, 1.99[
]
8.57–15
−0.97
WÎ 1.99, 1.50[
]
15–20
−0.70
WÎ 1.50, 1.28[
]
20–23.34
−0.52
WÎ 1.28, 1.17[
]
23.34–25.65
5
The Astrophysical Journal, 832:97 (10pp), 2016 December 1 Sood, Kim, & Hollerbach
![Figure 3. Magnetic field strength B∣ ∣ as a function of Ω (left panel) with the cycle-average of B∣ ∣ depicted in red. We note that the value of »B 0.1∣ ∣ at W = 1 corresponds to about 3 G at the star’s surface, whereas the rotation period Prot( ) in days is depicted on the top of the plot. Also, the fluctuating behavior of B∣ ∣ for the high rotation rate regime is shown for W Î 23.30, 23.31[ ] (right panel).](/figures/figure-3-magnetic-field-strength-b-as-a-function-of-o-left-3ohh3ml8.webp)
![Table 3 Power-law Exponent s for Magnetic Activity with Age t of Stars Î 1.066, 4.5[ ] Gyr](/figures/table-3-power-law-exponent-s-for-magnetic-activity-with-age-2og28z1z.webp)
