# Real light curves of FK Comae Berenices: Dead end

L. Jetsu
$DepartmentofPhysics,P.O.Box64,FI-00014UniversityofHelsinki,Finland$ E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Recently, we presented a general model for the light curves of
chromospherically active stars, where the observed light curve is interference
of two real constant period light curves of long-lived starspots. In this
first paper, we make six specific questions which undermine this argument,
because it contradicts the current widely held views about the stellar surface
differential rotation and the starspots. Our aim is to answer these six
questions. We present evidence that the long-lived starspots of our general
model have already been detected in the earlier surface imaging studies. The
Lomb-Scargle power spectrum method analysis of the real and the simulated data
of FK Com reveals that this method fails to detect the two real constant
period light curves of our general model. If our model is valid, this method
gives incompatible period, amplitude and minimum epoch estimates telling
nothing about the real periods, the real amplitudes and the real minimum
epochs of the two real light curves. This would mean that all earlier one-
dimensional period analyses of the light curves of chromospherically active
stars have given spurious results which have been widely and uncritically
accepted since the discovery of the starspots in the year 1947. However, we
arrive at a dead end, because we can not solve the real light curves of FK
Com. In our second paper, we solve these real light curves with a new two-
dimensional period finding method, prove the validity of our general model,
and answer all six questions made in this first paper.

###### keywords:

Methods: statistical – Methods: data analysis – Stars: starspots – Stars:
activity – Stars: individual (FK Comae Berenices, HD117555)

††pubyear: 2018††pagerange: Real light curves of FK Comae Berenices: Dead end
–A

##  1 Introduction

The ancient Egyptian papyrus Cairo 86637, which contains the calendar of lucky
and unlucky days, is the oldest preserved historical document of the discovery
of a variable star, Algol (Porceddu et al., 2008; Jetsu et al., 2013; Jetsu &
Porceddu, 2015; Porceddu et al., 2018). The Sun is also a variable star. The
luminosity changes of Algol can be observed with naked eye, but those of the
Sun can be reliably confirmed only with satellite observations (Willson &
Hudson, 1991; Radick et al., 2018). The oldest preserved drawing of a sunspot
was made by John of Worcester in the year 1128 (Van Helden, 1996). Schwabe
(1844) discovered the eleven years cycle in the number of sunspots. The Zeeman
effect caused by the solar magnetic field was discovered by Hale (1908), who
also found out that this magnetic field is stronger in the sunspots.

Kron (1947) discovered the starspots in the light curves of the eclipsing
binary AR Lacertae. He observed short-term light curve changes “within a few
weeks to a few months”. FK Comae Berenices (HD 117555, FK Com) was among the
first late–type stars where the starspots were also discovered (Chugainov,
1966). This chromosperically active single G4 giant (Strassmeier, 2009) is the
prototype of a class of variable stars, the FK Com class, defined by Bopp &
Stencel (1981) as rapidly rotating single G–K giants. Only a few stars
belonging to this class have been found (Puzin et al., 2014; Howell et al.,
2016; Puzin et al., 2017). The members of this class may represent recently
coalesced W UMa binaries (Webbink, 1976; Bopp & Stencel, 1981; Eggen & Iben,
1989; Welty & Ramsey, 1994).

The starspots on the surface of FK Com seem to concentrate on two long–lived
active longitudes separated by 180 degrees, and undergo abrupt shifts between
these longitudes. These shifts are called the flip–flop events (Jetsu et al.,
1991; Jetsu et al., 1993). This phenomenon has also been observed in the
chromospherically active binaries (e.g. Jetsu, 1996, $\sigma$ Gem). Hackman et
al. (2013) applied the Kuiper method (hereafter the K-method) to the seasonal
light curve minimum epochs of FK Com. This analysis gave the active longitude
period $P_{\mathrm{act}}=2.^{\mathrm{d}}401155\pm 0.^{\mathrm{d}}000092$.
Numerous photometric studies of FK Com have been made (e.g. Korhonen et al.,
2002; Oláh et al., 2006; Panov & Dimitrov, 2007; Hackman et al., 2013). The
starspot distribution of FK Com has also been mapped with the surface imaging
methods (e.g. Korhonen et al., 1999, 2000, 2004, 2007, 2009; Hackman et al.,
2013; Vida et al., 2015).

Jetsu et al. (2017) presented a general light curve model for the
Chromospherically Active Binary Stars (hereafter CABS). They studied the long-
term Mean Light Curve (hereafter MLC) of fourteen CABSs as a function of
orbital phase, and also showed how their new model can be applied to the
photometry of Chromospherically Active Single Star (hereafter CASS), like FK
Com.

Jetsu et al. (2017, JHL = Jetsu, Henry, Lehtinen) made the following argument

-

JHL-argument: “The observed light curves of chromospherically active binary
and single stars are interference of two real constant period light curves of
long-lived starspots. These constant periods are the non-stationary active
longitude period $P_{\mathrm{act}}$ and the stationary rotation period
$P_{\mathrm{rot}}\approx P_{\mathrm{orb}}$.”

At first sight, this JHL-argument would seem to contradict the current
overwhelming observational evidence for stellar Surface Differential Rotation
(hereafter SDR). There are two widely used observational methods for measuring
stellar SDR (Strassmeier, 2009, Sect. 7). In the first method, the rotation
periods $P_{\mathrm{rot}}$ of starspots at different latitudes are measured
from the Surface Images (hereafter the SI-method, e.g. Petit et al., 2004;
Barnes et al., 2005; Collier Cameron, 2007; Strassmeier, 2009; Balona &
Abedigamba, 2016; Kővári et al., 2017). The second method measures the range
of the photometric rotation period $P_{\mathrm{phot}}\approx P_{\mathrm{rot}}$
changes in the Light Curves (hereafter the LC-method, e.g. Hall & Busby, 1990;
Hall, 1991; Reinhold & Reiners, 2013; Reinhold et al., 2013; Reinhold & Gizon,
2015; Lehtinen et al., 2016; Distefano et al., 2016). There are numerous
studies, where the SI-method and the LC-method give different SDR estimates
even for the same individual star. For example, the SI-method study by
Korhonen et al. (2000) indicated “solid body rotation” in FK Com, but
photometric rotation period changes of about 3.1% were measured with the LC-
method by Hackman et al. (2013, $Z\approx 0.0308$).

If the JHL-argument is true, then we must be able to answer at least five of
the six undermining questions made below. The validity our argument does not
depend on the  3rd question. On the contrary, the JHL-argument provides an
answer to this question.

In our Abstract, we refer to these six specific questions:

-

1st question: Why have these two constant periods, the non-stationary active
longitude period $P_{\mathrm{act}}$ and the stationary rotation period
$P_{\mathrm{rot}}\approx P_{\mathrm{orb}}$ of the JHL-argument, not been
detected in the surface images?

-

2nd question: Why have these two constant periods, the non-stationary active
longitude period $P_{\mathrm{act}}$ and the stationary rotation period
$P_{\mathrm{rot}}\approx P_{\mathrm{orb}}$ of the JHL-argument, not been
detected in the light curves? Why are so many different
$P_{\mathrm{rot}}\approx P_{\mathrm{phot}}$ periods observed in the light
curves of the same star, if the photometric data contains only these two
constant periods?

-

3rd question: Why do the surface images and the light curves give different
surface differential rotation estimates even for the same individual star?

-

4th question: Hackman et al. (2013) applied the Kuiper method to the
$t_{\mathrm{min,1}}$ light curve minima of FK Com. Does this kind of an
analysis give an unambiguous estimate for the non-stationary active longitude
period $P_{\mathrm{act}}$ of the JHL-argument?

-

5th question: The Kuiper method detects the non-stationary active longitude
period $P_{\mathrm{act}}$ of the JHL-argument from the seasonal
$t_{\mathrm{min,1}}$ light curve minima of chromospherically active binary
stars (Jetsu et al., 2017) and single stars (Hackman et al., 2013). Why does
this method not detect the stationary rotation period $P_{\mathrm{rot}}\approx
P_{\mathrm{orb}}$ of the JHL-argument? The long-term mean light curves of
chromospherically active binary stars in Jetsu et al. (2017) follow the
stationary rotation period $P_{\mathrm{rot}}\approx P_{\mathrm{orb}}$ of the
JHL-argument. Why do these long-term mean light curves not follow the non-
stationary active longitude period $P_{\mathrm{act}}$ of the JHL-argument?

-

6th question: What explains the observed abrupt flip–flop events in the
chromospherically active stars, if the long-lived starspots rotate with the
two regular constant periods, the non-stationary active longitude period
$P_{\mathrm{act}}$ and the stationary rotation period $P_{\mathrm{rot}}\approx
P_{\mathrm{orb}}$?

In this first paper, we can not answer all these six questions. In our next
second paper, we present a new period analysis method based on the JHL-
argument, apply this method to the photometry of FK Com and give compact
answer to all the above six questions (Jetsu, 2018, hereafter paper ii).

Our Appendix gives all abbreviations used in this paper.

Table 1: Discarded data. Observing time (HJD), magnitude ($V$), telescope (TEL), segment (SEG) and discarding criterion (C) HJD | $V$ | TEL | SEG | C
---|---|---|---|---
2449876.7555 | 8.213 | 2 | – | 1
2449878.6786 | 8.109 | 2 | – | 1
2449882.6775 | 8.192 | 2 | – | 1
2449883.6783 | 8.158 | 2 | – | 1
2449883.7229 | 8.155 | 2 | – | 1
2449886.7003 | 8.309 | 2 | – | 1
2449886.7292 | 8.304 | 2 | – | 1
2449887.6784 | 8.180 | 2 | – | 1
2449888.6833 | 8.203 | 2 | – | 1
2449891.6809 | 8.346 | 2 | – | 1
2449893.6815 | 8.260 | 2 | – | 1
2449894.6718 | 8.250 | 2 | – | 1
2449896.6696 | 8.330 | 2 | – | 1
2449898.6701 | 8.308 | 2 | – | 1
2449900.6693 | 8.217 | 2 | – | 1
2449901.6691 | 8.289 | 2 | – | 1
2449902.6708 | 8.139 | 2 | – | 1
2449903.6705 | 8.318 | 2 | – | 1
2449904.6692 | 8.167 | 2 | – | 1
2449905.6591 | 8.210 | 2 | – | 1
2449906.6593 | 8.217 | 2 | – | 1
2449907.6589 | 8.161 | 2 | – | 1
2449908.6573 | 8.329 | 2 | – | 1
2450130.0134 | 8.094 | 2 | 1 | 5
2450504.8958 | 8.799 | 2 | 2 | 4
2450866.8368 | 8.154 | 1 | 1 | 3
2450934.6557 | 8.264 | 1 | 1 | 4
2450955.7222 | 8.144 | 1 | 1 | 3
2450974.6695 | 8.284 | 1 | 1 | 4
2450979.6702 | 8.322 | 1 | 1 | 4
2450986.7320 | 8.315 | 1 | 1 | 4
2451278.7070 | 8.104 | 1 | 2 | 3
2451278.7273 | 8.120 | 1 | 2 | 3
2451593.8157 | 8.217 | 2 | 5 | 5
2451706.7363 | 8.353 | 1 | 3 | 2
2451731.6859 | 8.195 | 2 | 5 | 2
2452251.0008 | 8.383 | 2 | 7 | 4
2452258.0132 | 8.386 | 2 | 7 | 4
2452348.8437 | 8.190 | 2 | 7 | 5
2452460.7124 | 8.324 | 2 | 7 | 2
2452461.6795 | 8.195 | 2 | 7 | 2
2452790.7993 | 8.248 | 1 | 6 | 5
2452807.6800 | 8.104 | 2 | 8 | 3
2452815.7371 | 8.288 | 1 | 6 | 2
2452824.7039 | 8.179 | 1 | 6 | 2
2452826.6991 | 8.171 | 1 | 6 | 2
2452828.6936 | 8.249 | 1 | 6 | 2
2453046.8944 | 8.013 | 2 | 9 | 5
2453070.8216 | 8.298 | 1 | 7 | 5
2453074.7515 | 7.999 | 2 | 9 | 3
2453139.6703 | 8.088 | 1 | 7 | 2
2453140.7700 | 8.379 | 2 | 9 | 4
2453144.6940 | 8.135 | 1 | 7 | 2
2453182.6611 | 8.060 | 1 | 7 | 2
2453184.6607 | 8.232 | 1 | 7 | 2
2453185.7171 | 8.155 | 1 | 7 | 2
2453709.0464 | 8.256 | 1 | 9 | 2
2453711.0410 | 8.346 | 1 | 9 | 2
2453711.0474 | 8.345 | 1 | 9 | 2
2453922.6616 | 8.243 | 1 | 9 | 2
2453922.6978 | 8.254 | 1 | 9 | 2
2454540.7813 | 8.087 | 1 | 11 | 3
2454807.0399 | 8.220 | 1 | 12 | 2
2454811.0305 | 8.260 | 1 | 12 | 2
2454822.0034 | 8.205 | 1 | 12 | 2
2454822.0349 | 8.254 | 1 | 12 | 2
2454999.7593 | 8.341 | 1 | 12 | 2
2455004.7444 | 8.298 | 1 | 12 | 2

##  2 Data

We analyse the standard Johnson $V$ photometry of FK Com published by Hackman
et al. (2013). These observations were made with the “T7” and “Ph10”
telescopes. We denote these separate samples of observations with TEL=1 (“T7”)
and TEL=2 (“Ph10”). Hackman et al. (2013) stored them as two separate files
into the CDS database. We analyse these two files separately, to avoid bias,
as Hackman et al. (2013) also did. The accuracy of this photometry is between
$0.^{\mathrm{m}}004$ and $0.^{\mathrm{m}}008$ in good photometric nights
(Henry, 1995; Strassmeier et al., 1997). All observations where the standard
deviation of three measurements exceeds $0.^{\mathrm{m}}020$ are automatically
discarded.

We discard $n=68$ observations (Table 1). The number of remaining analysed
observations is $n=3807$.

The isolated first segment of TEL=2 data in Hackman et al. (2013) contains too
few observations for reliable modelling, and it is discarded (Table 1: C=1,
$n=23$).

We divide the remaining data into seasonal segments (Table 2: SEG). These
segment numbers are also used for the rest of the discarded data in Table 1.

Some isolated observations are discarded in the beginning or in the end of a
few segments (Table 1: C=2, $n=24$). These isolated observations would mislead
the second order polynomial fits which are used to eliminate the changes of
the mean brightness within segments (Sect. 3: Eq. 1). The fraction of all
isolated discarded observations, $n=47$, is 1.2% in all data.

We use a sliding window to identify the first group of outliers. The mean
($m$) and the standard deviation ($s$) is computed for all observations that
are within $t\pm 30$ days of each individual observation $V(t)$. We discard
those individual $V(t)$ observations that are below $m-2.5s$ or above
$m+2.5s$. Some observations below the $m-2.5s$ limit may represent photometric
flares (Table 1: C=3, $n=7$). These events have been observed earlier in FK
Com (Jetsu et al., 1993). All observations above the $m+2.5s$ limit are very
probably erroneous (Table 1: C=4, $n=8$).

If two observations made during the same night with the same telescope deviate
more than $0.^{\mathrm{m}}06$, we discard one of them as an outlier. This
second group of outliers contains only $n=6$ observations (Table 1: C=5). The
fraction of all outliers, $n=21$, is 0.5% in all data.

All the above mentioned outliers are identified before our analysis. This
analysis will reveal that there are many other outliers (paper ii: Sects. 3
and 4). However, we do not reject those outliers, because they are identified
only after the analysis.

Table 2: Standard Johnson $V$ photometry of FK Com. Telescope (TEL), segments (SEG), first and last observing time ($t_{1}$ and $t_{n}$), time span and number of observations ($\Delta T$ and $n$) TEL | SEG | $t_{1}$ | $t_{n}$ | $\Delta T$ | $n$
---|---|---|---|---|---
|  | HJD | HJD | d |
1 | 1 | 2450850.875 | 2450997.687 | 146.8 | 221
1 | 2 | 2451153.047 | 2451360.673 | 207.6 | 378
1 | 3 | 2451534.046 | 2451694.803 | 160.8 | 67
1 | 4 | 2451883.044 | 2452078.743 | 195.7 | 88
1 | 5 | 2452248.044 | 2452461.698 | 213.7 | 220
1 | 6 | 2452614.048 | 2452790.765 | 176.7 | 170
1 | 7 | 2452978.048 | 2453144.694 | 166.6 | 127
1 | 8 | 2453343.046 | 2453565.681 | 222.6 | 192
1 | 9 | 2453721.051 | 2453907.661 | 186.6 | 189
1 | 10 | 2454075.045 | 2454285.686 | 210.6 | 181
1 | 11 | 2454440.049 | 2454643.730 | 203.7 | 187
1 | 12 | 2454807.040 | 2455004.744 | 197.7 | 129
1 | 13 | 2455172.044 | 2455297.831 | 125.8 | 111
2 | 1 | 2450085.055 | 2450265.674 | 180.6 | 111
2 | 2 | 2450412.038 | 2450636.678 | 224.6 | 209
2 | 3 | 2450778.042 | 2450997.706 | 219.7 | 205
2 | 4 | 2451144.039 | 2451362.728 | 218.7 | 179
2 | 5 | 2451508.043 | 2451712.687 | 204.6 | 157
2 | 6 | 2451873.042 | 2452089.727 | 216.7 | 193
2 | 7 | 2452242.033 | 2452448.752 | 206.7 | 188
2 | 8 | 2452613.009 | 2452826.679 | 213.7 | 159
2 | 9 | 2452972.029 | 2453194.682 | 222.7 | 154

Hackman et al. (2013) applied the Continuous Period Search method (hereafter
the CPS-method) to this photometry. We refer to the model of this method as
the CPS-model (Lehtinen et al., 2012, their Eq. 3). Hackman et al. (2013)
modelled 1464 subsets of photometry, but they did not publish their numerical
values. Their light curve periods, amplitudes, primary minimum epochs and
secondaryminimum epochs are now published in electronic form 111The full Table
3 is only available via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or
via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/… (Table 3).

Table 3:  CPS-method results from Hackman et al. (2013). The first column gives the telescope (TEL). The remaining columns give the same subset parameters as described in the Appendix of Lehtinen et al. (2011, their Table A.1): first observing time $(t_{1})$, last observing time $(t_{n})$, mean observing time $(\tau)$, statistically independent estimates (IND: 1=Yes, 0=No), number of observations $(n)$, period $(P\pm\sigma_{P})$, peak to peak light curve amplitude $(A\pm\sigma_{A})$, epochs of primary and secondary minima $(t_{\mathrm{min,1}}\pm\sigma_{\mathrm{t_{min,1}}},t_{\mathrm{min,2}}\pm\sigma_{\mathrm{t_{min,2}}})$. The dummy value “-1.000” denotes the cases where no estimate was obtained. The units of Heliocentric Julian Days are HJD-2 400 000. Note that we show only the ten first lines of this table. The full table is published only in electronic form. It contains 1464 lines (TEL=1: 760 lines) and (TEL=2: 704 lines). TEL | $t_{1}$ | $t_{n}$ | $\tau$ | IND | $n$ | $P$ | $\sigma_{P}$ | $A$ | $\sigma_{A}$ | $t_{\mathrm{min,1}}$ | $\sigma_{\mathrm{t_{min,1}}}$ | $t_{\mathrm{min,2}}$ | $\sigma_{\mathrm{t_{min,2}}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| HJD | HJD | HJD |  |  | d | d | mag | mag | HJD | d | HJD | d
1 | 50850.875 | 50874.887 | 50863.555 | 1 | 29 | 2.4129 | 0.0036 | 0.059 | 0.006 | 50851.594 | 0.032 | -1.000 | -1.000
1 | 50851.871 | 50875.883 | 50865.793 | 0 | 28 | 2.4154 | 0.0029 | 0.059 | 0.004 | 50854.043 | 0.034 | -1.000 | -1.000
1 | 50852.867 | 50876.883 | 50867.895 | 0 | 29 | 2.4141 | 0.0036 | 0.051 | 0.004 | 50853.980 | 0.031 | -1.000 | -1.000
1 | 50855.867 | 50878.809 | 50870.176 | 0 | 31 | 2.4196 | 0.0033 | 0.050 | 0.003 | 50856.395 | 0.027 | -1.000 | -1.000
1 | 50856.863 | 50880.867 | 50872.070 | 0 | 32 | 2.4155 | 0.0034 | 0.047 | 0.003 | 50858.812 | 0.029 | -1.000 | -1.000
1 | 50861.852 | 50885.855 | 50876.176 | 0 | 39 | 2.3930 | 0.0037 | 0.044 | 0.002 | 50863.754 | 0.029 | -1.000 | -1.000
1 | 50867.836 | 50891.840 | 50878.883 | 0 | 43 | 2.3923 | 0.0078 | 0.042 | 0.003 | 50868.613 | 0.042 | 50870.176 | 0.060
1 | 50868.828 | 50892.840 | 50880.621 | 0 | 43 | 2.3852 | 0.0051 | 0.041 | 0.003 | 50871.016 | 0.035 | 50870.215 | 0.057
1 | 50872.820 | 50896.828 | 50884.023 | 0 | 49 | 2.3699 | 0.0059 | 0.040 | 0.004 | 50875.008 | 0.041 | 50873.477 | 0.036
1 | 50873.820 | 50897.832 | 50885.551 | 0 | 49 | 2.3732 | 0.0061 | 0.039 | 0.004 | 50875.004 | 0.051 | 50875.844 | 0.037

##  3 Models

Our notation for a $V(t)$ magnitude observation made at an observing time
$t=t_{i}$ is $y_{i}=y(t_{i})$. The time span of these $n$ observations is
$\Delta T=t_{n}-t_{1}.$ We fit a second order polynomial $h(t)$ to the $y_{i}$
data of each individual segment. This polynomial is used to eliminate the
seasonal changes of the mean brightness of FK Com. The analysed data are

$\displaystyle y_{i}^{,}=y(t_{i})-h(t_{i})=y_{i}-h_{i}.$ (1)

We use this second order polynomial $h(t)$ in all segments, because higher
orders tend to fluctuate at the ends of some segments. It would not be
consistent or objective to vary the order of this polynomial in different
segments. However, we will mention if this second order polynomial can, or can
not, reproduce the changes of the mean brightness in individual segments
(paper ii: Sects. 3 and 4).

The abbreviation datax,y is hereafter used for the TEL=x observations in
segment SEG=y. Our notations for the mean and the standard deviation of
$y_{i}^{,}$ are $m_{y^{{}^{\prime}}}$ and $\sigma_{y^{{}^{\prime}}}$. We model
each segment of the $y_{i}^{,}$ data with the two and one period models.

###  3.1 Two period model (2P-model)

The complex 2P-model has two parts. The first part is

$\displaystyle
g_{1}(t)\\!=\\!g_{1}(t,\bar{\beta}_{1})\\!=\\!\sum_{k=1}^{K_{1}}B_{k}\cos(k2\pi
f_{1}t)+C_{k}\sin(k2\pi f_{1}t),$ (2)

where $K_{1}=2$. Higher $K_{1}$ values are unnecessary, because the detection
of three minima in the photometric light curve is practically impossible (e.g.
Lehtinen et al., 2011, their Sect. 3.2). The five free parameters are the
amplitudes $B_{1},B_{2},C_{1}$ and $C_{2}$, and the frequency $f_{1}$. The
frequency units are $[f]={\mathrm{d}}^{-1}$. The vector of free parameters is
$\bar{\beta}_{1}=[B_{1},B_{2},C_{1},C_{2},f_{1}]$. In every segment, we
determine the following parameters of the $g_{1}(t)$ function

$P_{1}=$ period $=f_{1}^{-1}$

$A_{1}=$ peak to peak amplitude

$t_{\mathrm{g1,min,1}}=$ epoch of primary (i.e. deeper) minimum

$t_{\mathrm{g1,min,2}}=$ epoch of secondary minimum (if present)

The units are $[P_{1}]={\mathrm{d}}$, $[A_{1}]={\mathrm{mag}}$ and
$[t_{\mathrm{g1,min,1}}]=$ $[t_{\mathrm{g1,min,2}}]=$
${\mathrm{HJD}}-2~{}400~{}000$. We use the epoch of the first primary and
secondary minimum within each individual segment. The phases of $g_{1}(t)$ are
computed from

$\displaystyle\phi_{1}={\mathrm{FRAC}}[(t-t_{0})/P_{1}],$ (3)

where ${\mathrm{FRAC}}[x]$ removes the integer part of its argument $x$, and
$t_{0}=t_{1}=$ the first observing time of the segment data.

The second part of the 2P-model is

$\displaystyle
g_{2}(t)\\!=\\!g_{2}(t,\bar{\beta}_{2})\\!=\\!\sum_{k=1}^{K_{2}}D_{k}\cos(k2\pi
f_{2}t)+E_{k}\sin(k2\pi f_{2}t),$ (4)

where $K_{2}=2$ and $\bar{\beta}_{2}$ $=[D_{1},D_{2},E_{1},E_{2},f_{2}]$. The
following parameters

$P_{2}=$ period $=f_{2}^{-1}$

$A_{2}=$ peak to peak amplitude

$t_{\mathrm{g2,min,1}}=$ epoch of primary minimum

$t_{\mathrm{g2,min,2}}=$ epoch of secondary minimum (if present)

of the $g_{2}(t)$ function are determined for every segment. The units are the
same as those used for the $g_{1}(t)$ function. We compute the phases of
$g_{2}(t)$ from

$\displaystyle\phi_{2}={\mathrm{FRAC}}[(t-t_{0})/P_{2}],$ (5)

where $t_{0}=t_{1}$, as in Eq. 3.

These two parts are combined in the complex 2P-model

$\displaystyle
g_{\mathrm{C}}(t)\\!=\\!g_{\mathrm{C}}(t,\bar{\beta}_{\mathrm{C}})\\!=\\!g_{1}(t,\bar{\beta}_{1})+g_{2}(t,\bar{\beta}_{2}).$
(6)

This nonlinear model has $p_{\mathrm{\mathrm{C}}}=10$ free parameters
$\bar{\beta}_{\mathrm{C}}=[\bar{\beta}_{1},\bar{\beta}_{2}]$. Note that the
functions $g_{1}(t)$ and $g_{2}(t)$ have unique phases (Eqs. 3 and 5), but no
such phases exist for the $g_{\mathrm{C}}(t)$ function which is constantly
changing in time.

###  3.2 One period model (1P-model)

The simple 1P-model is

$\displaystyle
g_{\mathrm{S}}(t)\\!=\\!g_{\mathrm{S}}(t,\bar{\beta}_{\mathrm{S}})\\!=\\!\sum_{k=1}^{K_{3}}F_{k}\cos(k2\pi
f_{3}t)+G_{k}\sin(k2\pi f_{3}t),$ (7)

where $K_{3}=2$ and
$\bar{\beta}_{\mathrm{S}}=[F_{1},F_{2},G_{1},G_{2},f_{3}]$. There are
$p_{\mathrm{S}}=5$ free parameters in this nonlinear model.

We determine the following parameters

$P_{3}=$ period $=f_{3}^{-1}$

$A_{3}=$ peak to peak amplitude

$t_{\mathrm{S,min,1}}=$ epoch of primary minimum

$t_{\mathrm{S,min,2}}=$ epoch of secondary minimum (if present)

of the $g_{\mathrm{S}}(t)$ function. The units are the same as those of the
$g_{1}(t)$ and $g_{2}(t)$ functions. The phases of $g_{\mathrm{S}}(t)$ are

$\displaystyle\phi_{3}={\mathrm{FRAC}}[(t-t_{0})/P_{3}],$ (8)

where $t_{0}=t_{1}$, as in Eq. 3.

The 1P-method is a “one-dimensional” period finding method because it searches
for only one periodicity in the data. We will consistently use the notations
$f_{3}$, $P_{3}$ and $A_{3}$ also for the frequencies, the periods and the
amplitudes of all other one-dimensional period finding methods that are
mentioned in this study.

###  3.3 Nested models

The complex model of Eq. 6 and the simple model of Eq. 7 are nested. They
represent the same model if

case1: $f_{1}=f_{2}$ in Eqs. 2 and 4

case2: $B_{1}=B_{2}=C_{1}=C_{2}=0$ in Eq. 2

case3: $D_{1}=D_{2}=E_{1}=E_{2}=0$ in Eq. 4

We denote the cases where the complex model approaches the simple model with

$\displaystyle g_{\mathrm{C}}\rightarrow g_{\mathrm{S}}.$ (9)

The 2P-model ”breaks down” when this happens. The alternatives when the
2P-model and the 1P-model are the same model occur in these three cases

$\displaystyle f_{1}-f_{2}$ $\displaystyle\rightarrow$ $\displaystyle 0$ (10)
$\displaystyle A_{1}/A_{2}$ $\displaystyle\rightarrow$ $\displaystyle 0$ (11)
$\displaystyle A_{2}/A_{1}$ $\displaystyle\rightarrow$ $\displaystyle 0.$ (12)

The residuals of the complex model

$\displaystyle\epsilon_{i}=y_{i}^{,}-g_{\mathrm{C}}(t_{i})$ (13)

give the sum of squared residuals
$R_{\mathrm{C}}=\sum_{i=1}^{n}\epsilon_{i}^{2}.$ The simple model
$g_{\mathrm{S}}(t)$ residuals

$\displaystyle\epsilon_{i}=y_{i}^{,}-g_{\mathrm{S}}(t_{i})$ (14)

give $R_{\mathrm{S}}=\sum_{i=1}^{n}\epsilon_{i}^{2}$. We compute the test
statistic

$F=\left({{R_{\mathrm{S}}}\over{R_{\mathrm{C}}}}-1\right)\left({{n-p_{\mathrm{C}}-1}\over{p_{\mathrm{C}}-p_{\mathrm{S}}}}\right).$
(15)

The null hypothesis is

$H_{\mathrm{0}}$: “The complex model $g_{\mathrm{C}}(t)$ does not provide a
significantly better fit than the simple model $g_{\mathrm{S}}(t)$.”

Under $H_{0}$, the test statistic of Eq. 15 has an $F$ distribution with
$(\nu_{1},\nu_{2})$ degrees of freedom, where
$\nu_{1}=p_{\mathrm{C}}-p_{\mathrm{S}}$ and $\nu_{2}=n-p_{\mathrm{C}}$ (Draper
& Smith, 1998). The probability that $F$ reaches or exceeds a fixed level
$F_{0}$ is called the critical level $Q_{F}=P(F\geq F_{0})$. We will reject
$H_{0}$, if and only if

$\displaystyle Q_{F}<\gamma_{F}=0.001,$ (16)

where $\gamma_{F}$ is a pre-assigned significance level. It represents the
probability of falsely rejecting $H_{0}$ when it is in fact true.

##  4 Earlier 2P-model analysis

If the frequencies $f_{1}$ and $f_{2}$ in Eqs. 2 and 4 are unknown free
parameters, the $g_{\mathrm{C}}(t)$ model of Eq. 6 is nonlinear and there is
no unique least squares fit solution. This $g_{\mathrm{C}}(t)$ model becomes
linear, if these $f_{1}$ and $f_{2}$ frequencies are fixed to some known
numerical constant values, like $f_{1}=1/P_{\mathrm{act}}$ and
$f_{2}=1/P_{\mathrm{orb}}$ in Jetsu et al. (2017). In this case, the solutions
for the eight remaining free parameters of a least squares fit, the amplitudes
$B_{1},B_{2},C_{1},C_{2},D_{1},D_{2},E_{1}$ and $E_{2}$ of the
$g_{1}(t,\bar{\beta}_{1})$ and $g_{2}(t,\bar{\beta}_{2})$ functions, are
unambiguous.

Jetsu et al. (2017) studied the long-term photometry of 14 CABSs with the
2P-model. They showed that the long-term MLC of all these CABSs followed the
orbital period $P_{\mathrm{orb}}$ (Jetsu et al., 2017, Figs. 1-14: panels
“b-g”). These CABSs rotate synchronously, $P_{\mathrm{orb}}\approx
P_{\mathrm{rot}}$. This means that the starspot distribution causing the MLC
of these stars is stationary in the rotating reference frame. It was also
shown that the line connecting the centres of the two binary components
intersects the longitudes of this stationary part.

Jetsu et al. (2017) analysed the epochs of the primary minima
$t_{\mathrm{min,1}}$ of the seasonal light curves of CABSs with the Kuiper
method. This analysis revealed the presence of long-lived active longitudes
rotating with a constant period of $P_{\mathrm{act}}$. Due to the inequality
$P_{\mathrm{act}}\neq P_{\mathrm{orb}}$, the seasonal light curve minima
connected to the former period migrate in the rotating reference frame, while
the minima connected to the latter period do not (Jetsu et al., 2017, Figs.
15-27: tilted non-stationary and horizontal stationary lines in panels “a”).

Two segments of the photometry of RS CVn binary $\sigma$ Gem were modelled by
using $f_{1}=1/P_{\mathrm{act}}$ and $f_{2}=1/P_{\mathrm{orb}}$ in Eqs. 2 and
4 (Jetsu et al., 2017, Figs. 30 and 31). It was argued that all observed light
curves of CABSs may be interference of two real light curves with periods of
$P_{\mathrm{act}}$ (non-stationary in rotating reference frame) and
$P_{\mathrm{orb}}\approx P_{\mathrm{rot}}$ (stationary in rotating reference
frame).

Jetsu et al. (2017, Fig. 32) also modelled one segment of FK Com data with the
2P-model. They first fixed $f_{1}=1/P_{\mathrm{act}}$ in Eq. 2. Their
numerical value was $P_{\mathrm{act}}=2.^{\mathrm{d}}401155\pm
0.^{\mathrm{d}}000092$ (Hackman et al., 2013). Then, they tested
$P_{\mathrm{rot}}$ values which were $\pm 15$% at both sides of
$P_{\mathrm{act}}$. For each of these tested $f_{2}=1/P_{\mathrm{rot}}$ in Eq.
4, they computed a test statistic $z_{\mathrm{test}}(f_{2})$ (Jetsu et al.,
2017). For their chosen tested $f_{1}=1/P_{\mathrm{act}}$ and
$f_{2}=1/P_{\mathrm{rot}}$ values, their test statistic $z_{\mathrm{test}}$
was equal to the test statistic that we will introduce later in paper ii
(Sect. 2.1: Eq. 2).

##  5 Measuring surface differential rotation (SDR)

###  5.1 Solar SDR

Howard (1994) discussed the uncertainties in all measurements of solar SDR. He
emphasized that these measurements are averages of many features over
latitude, and he then gave an example of the real  SDR of individual features.
His “sobering reminder” illustrated the solar SDR measurements of 36 708
sunspot groups made at Mount Wilson between the years 1917 and 1985 (Howard,
1994, his Fig. 2). He emphasized that the latitude of any individual sunspot
group did not predict its rotation period.

###  5.2 Stellar SDR

If the individual sunspot groups do not follow the law of solar SDR (see Eq.
22), then why should the individual starspots or starspot groups do so? More
than ten individual SI-method surface temperature maps are available only for
a few stars (see Strassmeier, 2009, Table 2). The most extensive LC-method
studies are typically based on about one hundred individual statistically
independent light curve $P_{\mathrm{phot}}$ values (e.g. Jetsu et al., 1999;
Lehtinen et al., 2016). Our intention is not to undermine the previous SI-
method and LC-method studies, but it is fair to remind about the uncertainties
in these stellar SDR measurements, if they truly rely on the “solar-stellar
connection”.

Table 4: Sample of thirteen CABSs in Jetsu et al. (2017). Rotation period $(P_{\mathrm{rot}}\approx P_{\mathrm{orb}})$, active longitude rotation period $(P_{\mathrm{act}})$, difference $(|P_{\mathrm{rot}}-P_{\mathrm{act}}|)$, differential rotation coefficient (Eq. 18: $|k|$) and angular velocity difference (Eq. 19: $|\Delta\Omega|$). Note that the parameters are given in the order of increasing $|\Delta\Omega|$. Star | $P_{\mathrm{rot}}\approx P_{\mathrm{orb}}$ | $P_{\mathrm{act}}$ | $|P_{\mathrm{rot}}-P_{\mathrm{act}}|$ | $|k|$ | $|\Delta\Omega|$
---|---|---|---|---|---
| $\mathrm{d}$ | $\mathrm{d}$ | $\mathrm{d}$ |  | ${\mathrm{rad~{}d}}^{-1}$
DM UMa | $7.492\pm 0.009$ | $7.4898\pm 0.0008$ | $0.0022\pm 0.0090$ | $0.0003\pm 0.0012$ | $0.0002\pm 0.0010$
HK Lac | $24.4284\pm 0.0005$ | $24.40\pm 0.01$ | $0.028\pm 0.010$ | $0.00116\pm 0.00041$ | $0.00030\pm 0.00011$
V1149 Ori | $53.57465\pm 0.00072$ | $53.14\pm 0.06$ | $0.435\pm 0.060$ | $0.0081\pm 0.0011$ | $0.00096\pm 0.00013$
EL Eri | $48.263\pm 0.206$ | $47.69\pm 0.02$ | $0.57\pm 0.21$ | $0.0119\pm 0.0043$ | $0.00156\pm 0.00056$
BM CVn | $20.6252\pm 0.0018$ | $20.513\pm 0.006$ | $0.1122\pm 0.0063$ | $0.00545\pm 0.00031$ | $0.001666\pm 0.000093$
II Peg | $6.724333\pm 0.000010$ | $6.7119\pm 0.0007$ | $0.01243\pm 0.00070$ | $0.00185\pm 0.00010$ | $0.001731\pm 0.000098$
$\sigma$ Gem | $19.604471\pm 0.000022$ | $19.497\pm 0.005$ | $0.1075\pm 0.0050$ | $0.00550\pm 0.00026$ | $0.001767\pm 0.000083$
HU Vir | $10.387522\pm 0.000031$ | $10.419\pm 0.001$ | $0.0315\pm 0.0010$ | $0.003026\pm 0.000096$ | $0.001827\pm 0.000058$
XX Tri | $23.96924\pm 0.00092$ | $23.77\pm 0.01$ | $0.199\pm 0.010$ | $0.00835\pm 0.00042$ | $0.00220\pm 0.00011$
V1762 Cyg | $28.58973\pm 0.00002$ | $28.17\pm 0.02$ | $0.420\pm 0.020$ | $0.01479\pm 0.00071$ | $0.00327\pm 0.00016$
FG UMa | $21.35957\pm 0.00040$ | $21.12\pm 0.01$ | $0.240\pm 0.010$ | $0.01128\pm 0.00047$ | $0.00334\pm 0.00014$
EI Eri | $1.947227\pm 0.000008$ | $1.9545\pm 0.0008$ | $0.00727\pm 0.00080$ | $0.00373\pm 0.00041$ | $0.0120\pm 0.0013$
V711 Tau | $2.83774\pm 0.00001$ | $2.8924\pm 0.0002$ | $0.05466\pm 0.00020$ | $0.019078\pm 0.000069$ | $0.04184\pm 0.00015$

##  6 SI-method

The compact answer to the  1st question in paper ii (Sect. 12) is based on
results presented in the next Sects. 6.1-6.3.

###  6.1 SI-method parameters

In the SI-method, the aim is to determine the rotation periods of recognizable
starspot distribution patterns at different latitudes. There are numerous
different approaches (see Strassmeier, 2009, 2011; Kochukhov, 2016). For
example, the rotation periods can be estimated directly from the surface
images (e.g. Barnes et al., 2005; Balona & Abedigamba, 2016), or simultaneous
photometry can be used as an additional inversion constraint, or the latitudes
of starspots in the surface images can be compared to the simultaneous light
curves (e.g. Berdyugina et al., 1998; Korhonen et al., 2000; Hackman, 2004;
Korhonen et al., 2007; Cole et al., 2015).

The SI-method inversion represents an “ill posed problem”, because an infinite
number of different solutions can fit the spectroscopic observations. Several
stellar physical parameters are fixed before these inversions, e.g.
inclination, rotation period and local spectral line profiles. This introduces
additional uncertainty to the SI-method results (see Kochukhov, 2016, his Sect
9.2.3.). One of the crucial fixed parameters in these inversions is the
stellar rotation period $P_{\mathrm{rot}}$ value. It is fixed to some constant
numerical value, like $P_{\mathrm{phot}}$ or $P_{\mathrm{orb}}$.

The SI-method results for stellar SDR are usually measured by using the
angular velocity difference

$\displaystyle\Delta\Omega=\Omega_{\mathrm{max}}-\Omega_{\mathrm{min}}={{{2\pi}\over{P_{\mathrm{min}}}}}-{{{2\pi}\over{P_{\mathrm{max}}}}}$
(17)

(e.g. Barnes et al., 2005, their Table 1). For solar SDR, these values are
$\Omega_{\mathrm{max}}=\Omega_{\mathrm{eq}}=2\pi/P_{\mathrm{eq}}$ and
$\Omega_{\mathrm{min}}=\Omega_{\mathrm{pole}}=2\pi/P_{\mathrm{pole}}$, where
$P_{\mathrm{eq}}\approx 25^{\mathrm{d}}$ and $P_{\mathrm{pole}}\approx
35^{\mathrm{d}}$ are the periods at the equator and the pole, i.e. the Sun has
$P_{\mathrm{eq}}<P_{\mathrm{pole}}$.

###  6.2 SI-method general results

In this section, we discuss the  1st question in the general context. Let us
assume that the SI-method would be used to measure SDR from the observed
difference between the periods of non-stationary starspots
($P_{\mathrm{act}}$) and stationary starspots ($P_{\mathrm{rot}}$). We compute
the stellar surface differential coefficient (see Eqs. 22, 24 and 25) values
from

$\displaystyle|k|=|{{P_{\mathrm{act}}-P_{\mathrm{rot}}}\over{(P_{\mathrm{act}}+P_{\mathrm{rot}})/2}}|$
(18)

for the thirteen CABSs where the active longitude period $P_{\mathrm{act}}$
was detected in Jetsu et al. (2017, their Tables 1 and 3). The angular
velocity range is

$\displaystyle|\Delta\Omega|=|{{2\pi}\over{P_{\mathrm{act}}}}-{{2\pi}\over{P_{\mathrm{rot}}}}|.$
(19)

The results are given in the order of increasing $|\Delta\Omega|$ in Table 4.
The lower and upper limits are
$|\Delta\Omega|=0.0002{\mathrm{~{}rad~{}d^{-1}}}$ (DM UMa) and
$|\Delta\Omega|=0.042{\mathrm{~{}rad~{}d^{-1}}}$ (V711 Tau), respectively. Our
lower limit, $|\Delta\Omega|\approx 0$, agrees with the results of the earlier
SI-method studies because weak SDR has been observed in several stars.
However, our $|\Delta\Omega|$ upper limit is about three times smaller than in
Barnes et al. (2005, Table 1: $0.14{\mathrm{~{}rad~{}d^{-1}}}$) and over ten
times smaller than in Balona & Abedigamba (2016, Table 2:
$0.47{\mathrm{~{}rad~{}d^{-1}}}$).

Barnes et al. (2005) and Balona & Abedigamba (2016) give several
$|\Delta\Omega|$ values for the same star. Even if the 2P-model were correct,
the SI-methods are not so accurate that they would always give exactly the
same $|\Delta\Omega|$, or equivalently the same $P_{\mathrm{act}}$ and
$P_{\mathrm{orb}}$, value for the starspots of the same star. This must have
increased the scatter (i.e. the range) of these earlier published upper
$|\Delta\Omega|$ limits. In addition to the previously mentioned inversion
uncertainties, the identification of the same structure in temporally
separated different images of the same star is not always unambiguous (see
Sect 7.4.4: map-incompatibility). Finally, there are several additional
reasons why our $|\Delta\Omega|=0.042{\mathrm{~{}rad~{}d^{-1}}}$ upper limit
is certainly an underestimate, as will be explained later in Sects. 7.7. In
this general context, the earlier published SI-method $|\Delta\Omega|$
estimates of SDR do not contradict the idea that the observed light curves of
CABSs and CASSs could be interference of the two constant period
($P_{\mathrm{act}}$ and $P_{\mathrm{orb}}\approx P_{\mathrm{rot}}$) “real”
light curves of the JHL-argument.

Figure 1: Orbital phases for starspots of II Peg detected with the SI-method.
(a) Larger starspots (heavier crosses) and smaller starspots (lighter crosses)
with the ephemeris of Eq. 20 (Berdyugina et al., 1998, their Fig. 4). The two
highlighted lighter crosses are shifted one orbital round downwards in the
next panel. (b) Same data with the ephemeris of Eq. 21. The horizontal dotted
lines show the orbital phases $\phi_{\mathrm{orb}}=0.25$ and 0.75. Otherwise
as in (a).

###  6.3 SI-method particular results

Here, we discuss the  1st question in the particular context of the SI-method
results presented by Berdyugina et al. (1998, II Peg) and Korhonen et al.
(2000, FK Com).

Berdyugina et al. (1998) published nine surface images of II Peg between the
years 1992 and 1997. Their orbital phase ephemeris was

$\displaystyle{\mathrm{HJD~{}}}2449582.9268~{}+~{}6.724333{\mathrm{E}}.$ (20)

Their zero epoch was equal to that given in Jetsu et al. (2017, “Ac” epoch in
their Table 2). Berdyugina et al. (1998) used the $P_{\mathrm{orb}}$ value of
Eq. 20 in the SI-method inversions. Our Fig. 1a reproduces their phase plot
for the larger and smaller starspots identified with the SI-method. Berdyugina
et al. (1998) concluded that this diagram shows two active longitudes, and
that a flip–flop event occurred in the year 1994.

Jetsu et al. (2017) used the orbital period ephemeris

$\displaystyle{\mathrm{HJD~{}}}2449581.246~{}+~{}6.724333{\mathrm{E}}$ (21)

for II Peg. Their $P_{\mathrm{orb}}$ was the same as in Eq. 20, but their zero
epoch was $\Delta t=P_{\mathrm{orb}}/4\equiv\Delta\phi_{\mathrm{orb}}=0.25$
earlier than in Eq. 20 (Jetsu et al., 2017, “Ab” epoch in their Table 2). One
of their main results was that all photometric minima of II Peg concentrated
close to $\phi_{\mathrm{orb}}=0.25$ between the years 1988 and 1997 (Jetsu et
al., 2017, their Fig. 27a). In other words, the stationary part of the light
curve ($P_{\mathrm{orb}}=P_{\mathrm{rot}}$) dominated during this nine year
time interval which overlaps the four year time interval of the SI-method
images in Berdyugina et al. (1998).

Our Fig. 1b shows the phases of the same starspots as in Fig. 1a, except that
we use the ephemeris of Eq. 21. Unlike Berdyugina et al. (1998), we do not
deliver evidence for a flip–flop event by adding one extra orbital round to
the two first $\phi_{\mathrm{orb}}$ values of smaller starspots (Figs. 1ab:
two highlighted lighter crosses). The starspots of the first image in the year
1992 are exactly at $\phi_{\mathrm{orb}}=0.25$ and $\phi_{\mathrm{orb}}=0.75$,
respectively. All larger starspots are at both sides of phase
$\phi_{\mathrm{orb}}=0.25$. Most of the smaller starspots concentrate close to
$\phi_{\mathrm{orb}}=0.75$. The simultaneous MLC of II Peg also confirms that
the main activity concentrated on the longitude coinciding with the line
connecting the centres of the two binary components (Jetsu et al., 2017, Fig.
14d-e: $\phi_{\mathrm{orb}}=0.25$). Berdyugina et al. (1998) noted that the
mean latitudes of these starspots discovered with the SI-method remained
approximately the same in all images. This is a convincing particular case,
where the SI-method detects a stationary starspot of II Peg rotating with a
constant period $P_{\mathrm{orb}}=P_{\mathrm{rot}}$ during a time interval of
over four years.

Korhonen et al. (2000) published six surface images of FK Com between the
years 1994 and 1997. They used the period
$P_{\mathrm{phot}}=2.^{\mathrm{d}}4002466\pm 0.^{\mathrm{d}}0000056$ in the
SI-method image inversions. Comparison of their six images revealed the
presence of long-lived starspots rotating with a constant period
$P_{\mathrm{SI}}=2.^{\mathrm{d}}4037\pm 0.^{\mathrm{d}}0005$. The dates of
these images overlap the time interval of our photometry between the years
1995 and 1997, except for the year 1994. The thirteen first statistically
independent $t_{\mathrm{min,1}}$ epochs in our Table 3 (IND=1) are between the
years 1995.43 and 1997.41. The K-method analysis of these thirteen epochs
gives $P_{\mathrm{act}}=2.^{\mathrm{d}}4035\pm 0.^{\mathrm{d}}0005$. The
$P_{\mathrm{SI}}-P_{\mathrm{act}}$ difference is only $0.^{\mathrm{d}}0002\pm
0.^{\mathrm{d}}0007$. The latitude range of the starspots detected with the
SI–method was between $45\degr$ and $70\degr$, and the migration of these
starspots followed “solid body rotation”(Korhonen et al., 2000). In this
particular case, the SI-method undoubtedly detects a long-lived non-stationary
starspot of FK Com rotating with a constant period $P_{\mathrm{act}}$.

The starspots detected in the above two particular SI-method studies by
Berdyugina et al. (1998) and Korhonen et al. (2000) were long-lived and
rotated with a constant angular velocity, as predicted by the JHL-argument.
This type of stability is quite common in surface images. For example, the
sequence 37 consecutive surface images during 57 nights show that the
strongest starspot in V711 Tau appears to be nearly stationary in the rotating
reference frame (Strassmeier & Bartus, 2000, their Fig. 8: starspot “A”). Our
answer to the  1st question is that the general and the particular evidence
strongly indicates that the $P_{\mathrm{act}}$ and $P_{\mathrm{rot}}\approx
P_{\mathrm{orb}}$ periods of the JHL-argument have already been detected with
the SI-method.

##  7 LC-method

We give compact answers to the  2nd question and the  3rd question in paper ii
(Sect. 12). These answers are based on the results presented in Sects.
7.3-7.4.1.

###  7.1 LC-method parameters

In the LC-method approach, the most widely used period finding method is the
Lomb-Scargle periodogram (hereafter the LS-method, Lomb, 1976; Scargle, 1982).
It is equivalent to finding the best least squares sinusoidal model for the
data (hereafter LS-model). The LS-method is a one-dimensional period finding
method because it determines only one period value for the data. There are
numerous other one-dimensional period finding methods that can search for more
complex models in the data (e.g. Lehtinen et al., 2011, the CPS-method).

The LC-methods usually use the following approximation for the law of solar
SDR

$\displaystyle P(b)={{P_{\mathrm{eq}}}\over{1-k\sin^{2}b}},$ (22)

where $b$ is the latitude, $P_{\mathrm{eq}}$ is the rotation period at the
equator and $k$ is the SDR coefficient (e.g. Hall & Busby, 1990; Hall, 1991,
solar $k=k_{\odot}=0.186$). A useful parameter is

$\displaystyle h=\sin^{2}(b_{\mathrm{max}})-\sin^{2}(b_{\mathrm{min}}),$ (23)

where $b_{\mathrm{min}}$ and $b_{\mathrm{max}}$ are the mimimum and maximum
latitudes of starspot formation. The largest possible latitude range,
$b_{\mathrm{min}}=0\degr$ and $b_{\mathrm{max}}=\pm 90\degr$, gives $h=1$. All
other latitude ranges give $h<1$. This leads to the relation

$\displaystyle k={\Delta P\over{hP_{\mathrm{eq}}}},$ (24)

where $\Delta
P=|P(b_{\mathrm{max}})-P(b_{\mathrm{min}})|=P_{\mathrm{phot,max}}-P_{\mathrm{phot,min}}$
is the difference between the largest and smallest observed photometric
rotation period $P_{\mathrm{phot}}$ in any particular star. Since $h\leq 1$,
the LC-methods use the lower limit

$\displaystyle|k|\geq{{P_{\mathrm{phot,max}}-P_{\mathrm{phot,min}}}\over{P_{\mathrm{phot,mean}}}}$
(25)

for the absolute value of stellar SDR coefficient, where
$P_{\mathrm{phot,mean}}$ is the mean of all observed values of
$P_{\mathrm{phot}}$. There are four uncertainties in this relation. Firstly,
$P_{\mathrm{phot,mean}}\approx(P_{\mathrm{phot,max}}+P_{\mathrm{phot,min}})/2$
must to be used as an approximate estimate for $P_{\mathrm{eq}}$ of Eq. 24,
because it is not known whether $P_{\mathrm{phot,min}}$ or
$P_{\mathrm{phot,max}}$ represents $P_{\mathrm{eq}}$. Some LC-methods use
$P_{\mathrm{eq}}=P_{\mathrm{phot,max}}$ (e.g. Reinhold et al., 2013).
Secondly, the sign of $k$ is unknown, where $k>0$ represents solar and $k<0$
represents anti-solar SDR. Thirdly, the latitude range of starspot formation,
the value of $h$ in Eqs. 23 and 24, is unknown. Fourthly, it is not known if
the full range of period changes has already been observed, or does the
observed $P_{\mathrm{phot,max}}-P_{\mathrm{phot,min}}$ difference represent an
underestimate of the real range.

Another approach for measuring the range of changes for $n$ photometric
rotation periods $P_{\mathrm{phot,i}}$ (e.g. Lehtinen et al., 2016) is to use
the weighted mean

$\displaystyle
P_{\mathrm{w}}={{\sum_{i=1}^{n}w_{i}P_{\mathrm{phot,i}}}\over{\sum_{i=1}^{n}w_{i}}}$
(26)

and the weighted standard deviation

$\displaystyle\Delta
P_{\mathrm{w}}=\sqrt{{{\sum_{i=1}^{n}w_{i}(P_{\mathrm{phot,i}}-P_{\mathrm{w}})^{2}}\over{\sum_{i=1}^{n}w_{i}}}},$
(27)

where $\sigma_{P_{\mathrm{phot,i}}}$ is the error of $P_{\mathrm{phot,i}}$ and
the weights are $w_{i}=\sigma_{P_{\mathrm{phot,i}}}^{-2}$. Equal weights
$w_{i}=1$ give $P_{\mathrm{w}}=m_{\mathrm{P}}$ and $\Delta
P_{\mathrm{w}}=\sigma_{\mathrm{P}}$, where $m_{\mathrm{P}}$ and
$\sigma_{\mathrm{P}}$ are the mean and standard deviation of all observed
$P_{\mathrm{phot,i}}$ values. In this case, the “three sigma” upper limit for
the $P_{\mathrm{phot,i}}$ changes is

$\displaystyle Z={{6\Delta
P_{\mathrm{w}}}\over{P_{\mathrm{w}}}}={{6\sigma_{\mathrm{P}}}\over{m_{\mathrm{P}}}}.$
(28)

The relation

$\displaystyle|k|\approx Z/h$ (29)

is valid, if $P_{\mathrm{phot,max}}-P_{\mathrm{phot,min}}\approx 6\Delta
P_{\mathrm{w}}$ (Jetsu et al., 2000).

###  7.2 LC-method results

The idea that the value of $P_{\mathrm{phot}}$ is connected to the stellar
rotation period $P_{\mathrm{rot}}$ is logical. In the SDR context, the
observed value of $P_{\mathrm{phot}}$ could tell something about the latitude
of the spot. If the observed $P_{\mathrm{phot}}$ has changed, it is logical to
assume that the latitude of the spot has changed. Thus, the range of
$P_{\mathrm{phot}}$ changes could measure SDR.

Hall (1991) applied the LC-method to the photometry of 277 late-type stars. He
concluded that the stellar SDR correlates strongly with
$P_{\mathrm{phot}}\approx P_{\mathrm{rot}}$. It decreases when
$P_{\mathrm{phot}}$ decreases. He showed that SDR was so weak in some rapidly
rotating stars that it approached solid-body rotation. This general relation
between $P_{\mathrm{phot}}$ and SDR has been amply confirmed by subsequent LC-
method studies of much larger samples (e.g. Reinhold et al., 2013, 24 124
stars). We have also formulated our own LC-methods, and used them to measure
stellar SDR (e.g. Jetsu & Pelt, 1999; Jetsu et al., 1999; Lehtinen et al.,
2011, 2016, the TSPA-method and the CPS-method). Considering all these
findings, our JHL-argument runs into its next problem: what is the answer to
the  2nd question. Note that we have very good reasons for emphasizing the
words many, different, observed and same in this particular question, as will
become evident in the next Sects. 7.3-7.4.1.

Figure 2:  (a) Simulated $y^{*}_{i}$ data of Eq. 33 for equal amplitudes
$a_{1}=a_{2}=0.^{\mathrm{m}}05$ in Eqs. 30 and 31. (b) Periods
$P_{1}=2.^{\mathrm{d}}39313$ and $P_{2}=2.^{\mathrm{d}}40762$ of the 2P-model
are denoted with red and blue circles. The green circles show the periods
$P_{3}$ detected with the LS-method. The large black circles around some green
circles highlight the cases, where the highest $z_{\mathrm{LS}}$ periodogram
value is in the beginning or in the end of the tested period interval. (c)
Amplitudes $A_{1}=2a_{1}$ and $A_{2}=2a_{2}$ of the 2P-model are denoted with
red and blue circles. Note that these red and blue symbols are slightly
shifted apart, because they would otherwise overlap. The green circles show
the peak to peak amplitudes $A_{3}$ of the LS-model sinusoids. The same models
as in "b" are highlighted again. (d) The red and blue circles show the light
curve minimum phases of $g_{1}(t)=s_{1}(t)$ and $g_{2}(t)=s_{2}(t)$ with
$P_{2}$ in Eq. 3. The green circles show the light curve minima of the LS-
method sinusoids. The highlighted models are the same as in "bc".

Figure 3: Simulated $y^{*}_{i}$ data of Eq. 33 for unequal amplitudes
$a_{1}=0.^{\mathrm{m}}025$ in Eq. 30 and $a_{2}=0.^{\mathrm{m}}050$ in 31,
otherwise as in Fig. 2.

###  7.3 Simulated 2P-model data

Here, we assume that the 2P-model is correct and we simulate artificial
photomeric data of two different cases. The model used in creating the
simulated data is the sum of two sinusoidal light curves

$\displaystyle s_{1}(t)$ $\displaystyle=$ $\displaystyle a_{1}\sin{(2\pi
f_{1}t)}$ (30) $\displaystyle s_{2}(t)$ $\displaystyle=$ $\displaystyle
a_{2}\sin{(2\pi f_{2}t)}.$ (31)

This model has $s_{1}(t)=g_{1}(t)$ in Eq. 2, $s_{2}(t)=g_{2}(t)$ in Eq. 4 and
$s(t)=s_{1}(t)+s_{2}(t)=g_{\mathrm{C}}(t)$ in Eq. 6. The peak to peak
amplitudes are $A_{1}=2a_{1}$ and $A_{2}=2a_{2}$. The frequencies are fixed to
$f_{1}=P_{1}^{-1}$ and $f_{2}=P_{2}^{-1}$, where the periods are
$P_{1}=2.^{\mathrm{d}}39313$ and $P_{2}=2.^{\mathrm{d}}40762$. These period
values are taken from the ephemerides of Eqs. 20 and 21 of paper ii. We
compute the lap cycle period

$\displaystyle
P_{\mathrm{lap}}=|P_{1}^{-1}-P_{2}^{-1}|^{-1}=|f_{1}-f_{2}|^{-1}\approx
398^{\mathrm{d}}$ (32)

of these periodicities (Jetsu et al., 2017, their Eq. 4). The interference
pattern of $s(t)=s_{1}(t)+s_{2}(t)$ is repeated regularly after every $\Delta
T=P_{\mathrm{lap}}$. Hence, the time span of the simulated data segment is
fixed to $\Delta T=P_{\mathrm{lap}}$. We use multiples of sidereal day
$P_{\mathrm{sid}}=0.^{\mathrm{d}}99726958$ to create $n^{*}=250$ time points

$\displaystyle t^{*}_{i}=i^{*}P_{\mathrm{sid}}+\delta t^{*}_{i},$

where $i^{*}$ are a random sample of integers $1\leq i^{*}\leq 398$. Any
particular integer $i^{*}$ value is used as many times as the random selection
favours it. The simulated random shifts $\delta t^{*}_{i}$ at both sides of
the mid-nights $i^{*}P_{\mathrm{sid}}$ are evenly distributed between
$-0.^{\mathrm{d}}2$ and $0.^{\mathrm{d}}2$. The simulated data are

$\displaystyle y^{*}_{i}=s(t^{*}_{i})+\epsilon^{*}_{i},$ (33)

where $\epsilon^{*}_{i}$ are a random sample of $n^{*}=250$ residuals drawn
from a Gaussian distribution with a zero mean and a standard deviation of
$0.^{\mathrm{m}}008$.

Inside the simulated segment, we select datasets containing all
$y^{*}_{\mathrm{i}}$ that are within a sliding window of $\Delta
T=30^{\mathrm{d}}$. We apply the LS-method to all those datasets that contain
at least $n=12$ values of $y^{*}_{i}$. Similar sliding window one-dimensional
period analysis has been applied to real photometry, e.g. by Distefano et al.
(2016, the LS-method) or Lehtinen et al. (2016, the CPS-method). The maximum
and minimum values of tested $f_{3}=P_{3}^{-1}$ frequencies of the LS-model
are

$\displaystyle f_{\mathrm{min}}=(1-a)f_{\mathrm{mid}},$ $\displaystyle
f_{\mathrm{max}}=(1+a)f_{\mathrm{mid}},$ (34)

where $P_{\mathrm{mid}}=(P_{1}+P_{2})/2$,
$f_{\mathrm{mid}}=P_{\mathrm{mid}}^{-1}$, and $a=0.03$. This is the $\pm 3$%
range at both sides of $f_{\mathrm{mid}}$.

####  7.3.1 $a_{1}=a_{2}$ simulation (test${}_{a_{1}=a_{2}}$)

In the first case, we use equal amplitudes $a_{1}=a_{2}=0.^{\mathrm{m}}05$ in
Eqs. 30 and 31. We will hereafter refer to these simulations as
test${}_{a_{1}=a_{2}}$. The results are shown in Fig. 2.

The amplitude of the simulated data decreases to zero during the first half of
the segment and then increases back to its original level during the last half
(Fig. 2a). This pattern is repeated during every lap cycle $\Delta
T=P_{\mathrm{lap}}$ (Eq. 32), because the interference of these two sinusoids
is regular. The periods $P_{1}$ and $P_{2}$ of the simulated model do not
change (Fig. 2b: red and blue circles). However, the periods detected with the
LS-method show large variation (Fig. 2b: green circles), especially when the
amplitude of the simulated data approaches zero in the middle of the segment.
There are cases where the highest LS-method periodogram $z_{\mathrm{LS}}$
value is at the end of the tested period interval (Fig. 2b: highlighted green
circles). This is a clear sign of that the tested period interval is too
narrow for this particular period finding method. The amplitudes $A_{1}$ and
$A_{2}$ of the 2P-model remain unchanged (Fig. 2c: red and blue circles). The
peak to peak amplitudes $A_{3}$ of the sinusoidal LS-models vary between
$0.^{\mathrm{m}}0$ and $0.^{\mathrm{m}}2$ (Fig. 2c: green circles). The phases
of the light curve minima of $s_{2}(t)$ are stable with $P_{2}$ in Eq. 3 (Fig.
2d: blue circles), while those of $s_{1}(t)$ show regular linear migration
(Fig. 2d: red circles). Finally, the minima of the sinusoidal models detected
with the LS-method show minor linear changes and an abrupt 0.5 phase shift in
the middle of the segment (Fig. 2d: green circles). This phase shift occurs
when the amplitude $A_{3}$ approaches zero, and this event coincides with the
cases when no periodicity is detected with the LS-method (Figs. 2bcd:
highlighted green circles). We emphasize that this half a rotation phase shift
occurs in every simulated artificial data sample of Eq. 33 when $a_{1}=a_{2}$.

####  7.3.2 $a_{1}<a_{2}$ simulation (test${}_{a_{1}<a_{2}}$)

In the second case, the amplitudes are unequal: $a_{1}=0.^{\mathrm{m}}025$ in
Eq. 30 and $a_{2}=0.^{\mathrm{m}}05$ in Eq. 31. These simulations are
hereafter referred to as test${}_{a_{1}<a_{2}}$.

The amplitude of the simulated data first decreases from $0.^{\mathrm{m}}15$
to $0.^{\mathrm{m}}05$, and then increases back to $0.^{\mathrm{m}}15$ (Fig.
3a). The periods $P_{1}$ and $P_{2}$ do not, of course, change (Fig. 3b: red
and blue circles). There is large variation in the $P_{3}$ periods detected
with the LS-method (Fig. 3b: green circles). This variation is largest in the
middle of the segment when the amplitude is lowest. The amplitudes $A_{1}$ and
$A_{2}$ of the 2P-model do not change (Fig. 3c: red and blue circles). The
peak to peak amplitudes $A_{3}$ of the LS-models vary regularly between
$0.^{\mathrm{m}}05$ and $0.^{\mathrm{m}}15$ (Fig. 3c: green circles). The same
phases of the light curve minima of $s_{1}(t)$ and $s_{2}(t)$ as in the
previous Fig. 2d are shown again in Fig. 3d (red and blue circles). The phases
of the minima of the sinusoidal LS-models fluctuate (Fig. 3d: green circles).
The cases when the LS-method detects no periodicity do not occur as often as
in test${}_{a_{1}=a_{2}}$, because the amplitude does not decrease to zero in
the middle of the segment (Figs. 3bcd: one highlighted circle).

Figure 4: (a) Cumulative distribution functions $F(|k|)$ (red) and $F(Z)$
(blue) in 10 000 simulations of test${}_{a_{1}=a_{2}}$ with $a=0.03$ in Eq. 34
(b) same as in “a” for test${}_{a_{1}<a_{2}}$. (c) $a=0.05$ in Eq. 34,
otherwise as in “a”. (d) $a=0.05$ in Eq. 34, otherwise as in “b”. (e) $a=0.10$
in Eq. 34, otherwise as in “a”. (f) $a=0.10$ in Eq. 34, otherwise as in “b”.

Figure 5: (a) Overlapping data2,3 and data1,1 photometry (red and blue filled
circles). Vertical green lines show the beginning and the end of fully
overlapping observations. (b) Fully overlapping photometry. (c) Red and blue
filled circles denote $P_{3}$ periods detected from the fully overlapping
photometry with the LS-method. The $|k|$ and $Z$ results are given above this
panel.

###  7.4 Incompatibility

The test${}_{a_{1}=a_{2}}$ and test${}_{a_{1}<a_{2}}$ simulations reveal
something that we will hereafter refer to as “incompatibility”. If the JHL-
argument is true, this incompatibility manifests itself in the observed light
curve periods, amplitudes and minima determined with any one-dimensional
period finding method, like e.g. the LS-method or the CPS-method.

The amplitudes $A_{1}$ and $A_{2}$ of the simulated light curves are constant
in the above mentioned test${}_{a_{1}=a_{2}}$ and the test${}_{a_{1}<a_{2}}$
simulations. All incompatibility effects described in the next Sects.
7.4.1-7.4.4 are even stronger in the real data, because these $A_{1}$ and
$A_{2}$ amplitudes are changing. These effects are nicely illustrated in the
periodogram of all data which resembles a patchwork quilt (see paper ii: Fig.
A70 discussed in Sect. 9).

####  7.4.1 Period-incompatibility

The test${}_{a_{1}=a_{2}}$ and test${}_{a_{1}<a_{2}}$ simulations both show
that the LS-method detects spurious $P_{3}$ periods and spurious $P_{3}$
changes although the 2P-model periods $P_{1}$ and $P_{2}$ do not change.

The $P_{3}$ period results in Fig. 2b represent only one particular simulated
test${}_{a_{1}=a_{2}}$ case where $a=0.03$ in Eq. 34, and $a_{1}=a_{2}=0.050$
in Eqs. 30 and 31. We simulated 10 000 such cases having this same $a$,
$a_{1}$ and $a_{2}$ combination. Each case gave us one estimate of $|k|$ (Eq.
25) and $Z$ (Eq. 28). The cumulative distribution functions $F(|k|)$ and
$F(Z)$ of these parameters are shown in Fig. 4a (red and blue continuous
lines). The parameter ranges are approximately $0.035\leq|k|\leq 0.06$ and
$0.03\leq Z\leq 0.07$. A steep rise of $F(|k|)$ occurs at $2a=0.06$. The
probability for $F(|k|\geq 2a|)$ is nearly 0.5. The $|k|$ values can not
exceed this $2a$ limit, because the maxima and the minima of detected $P_{3}$
values have to be within the tested period interval (Eq. 34). In nearly half
of the 10 000 cases, the LS-method detects $P_{3}$ periods over the whole
tested period interval $(1\pm a)P_{\mathrm{mid}}$. The shape of $F(Z)$
resembles that of a Gaussian cumulative distribution function.

The results for 10 000 simulated test${}_{a_{1}<a_{2}}$ cases with $a=0.03$,
$a_{1}=0.025$ and $a_{2}=0.050$ are shown in Fig. 4b. The approximate ranges
are $0.02\leq|k|\leq 0.06$ and $0.02\leq Z\leq 0.05$. The minimum peak to peak
amplitude of the simulated $s(t)$ light curve is $A_{3}=2a_{1}=0.050$ in these
test${}_{a_{1}<a_{2}}$ simulations. Since $A_{3}$ does not decrease to zero,
as in test${}_{a_{1}=a_{2}}$, the LS-method succeeds in detecting
periodicities that are within the tested period interval.

The combination $a=0.05$, $a_{1}=0.050$ and $a_{2}=0.050$ (Fig. 4c) shows what
happens in test${}_{a_{1}=a_{2}}$ with a longer tested $\pm 5$% period
interval. The $|k|$ and $Z$ ranges increase, and the “expected” steep $F(|k|)$
rise occurs at $2a=0.10$. The test${}_{a_{1}<a_{2}}$ combination $a=0.05$,
$a_{1}=0.025$ and $a_{2}=0.050$ shows no steep $F(|k|)$ rise at $|k|=2a$ (Fig.
4d). The results for $a=0.10\equiv\pm 10$% are shown in Figs. 4ef. The LS-
method keeps on finding $P_{3}$ periods over the whole period interval in
test${}_{a_{1}=a_{2}}$ simulations, because the $|k|$ maximum is at $2a=0.20$.
The general result is that the $|k|$ and $Z$ ranges increase when the tested
range, $a$ in Eq. 34, increases.

The simulations shown in Figs. 4a–f also reveal that the relation $|k|\approx
Z/h$ of Eq. 29 is a poor approximation. This is no surprise, because $|k|$ is
computed from two individual $P_{3}$ values, while $Z$ is computed from all
$P_{3}$ values.

If the JHL-argument is correct, our simulations also predict that two
observers observing the light curve of same star during the same time interval
will get different $P_{3}$, $|k|$ and $Z$ values with the LS-method. We show
the temporally overlapping data2,3 and data1,1 photometry in Fig. 5a. The
fully overlapping data are shown in Fig. 5b. We divide these fully overlapping
data into subsets using a sliding window of 30 days, and analyse all subsets
having at least 12 observations with the LS-method (Eq. 34: $a=0.05$). In Fig.
5c, we show the $P_{3}$, $|k|$ and $Z$ results for the fully overlapping data
of segments data2,3 and data1,1. The respective results for all temporally
overlapping pairs of segments are given in Table 5. These results confirm the
above prediction.

If the JHL-argument is true, no-one ever gets the same LS-method analysis
results from the simultaneous photometry of the same star. This general result
applies to all one-dimensional period finding methods. All the above findings
are amazing considering that the underlying 2P-model,
$g_{\mathrm{C}}(t)=s(t)=s_{1}(t)+s_{2}(t)$, does not change at all. Since this
period incompatibility solves the  2nd question, it is easy to the answer the
3rd question (see paper ii: Sect. 12).

Table 5: $|k|$ and $Z$ for fully overlapping real observations. Columns 1 and 4 give the temporally overlapping segments. The other columns give the $|k|$ and $Z$ values for fully overlapping data. | $|k|$ | $Z$ |  | $|k|$ | $Z$
---|---|---|---|---|---
data2,3: | 0.077 | 0.097 |  data1,1: | 0.047 | 0.061
data2,4: | 0.100 | 0.141 |  data1,2: | 0.063 | 0.085
data2,5: | 0.022 | 0.032 |  data1,3: | 0.012 | 0.016
data2,6: | 0.014 | 0.018 |  data1,4: | 0.010 | 0.012
data2,7: | 0.079 | 0.068 |  data1,5: | 0.070 | 0.096
data2,8: | 0.051 | 0.062 |  data1,6: | 0.025 | 0.032
data2,9: | 0.088 | 0.091 |  data1,7: | 0.048 | 0.038

####  7.4.2 Amplitude-incompatibility

The observed light curve amplitudes $A_{3}$ in Figs. 2c and 3c do not
certainly tell anything about the real light curve amplitudes $A_{1}$ and
$A_{2}$ of the 2P-model. We will hereafter refer to this problem as
“amplitude-incompatibility”. It is common for all one-dimensional period
finding methods.

The period analysis of the observed amplitudes $A_{3}$ can sometimes be used
to determine the numerical value for the lap cycle period $P_{\mathrm{lap}}$
of Eq. 32. These $P_{\mathrm{lap}}$ cycles were clearly present in the
observed $A_{3}$ amplitudes of thirteen CABSs (Jetsu et al., 2017, their Figs.
15-27). The numerical $P_{\mathrm{lap}}$ value can also be used to indirectly
estimate other 2P-model parameters (see Sect. 7.7).

####  7.4.3 Minima-incompatibility

The one-dimensional period finding methods also suffer from something that we
will hereafter refer to as “minima-incompatibility”. If we would analyse the
light curve minima epochs of Figs. 2d and 3d with the K-method, we would not
detect the $P_{1}$ period or the $P_{2}$ period. We could detect $P_{2}$ when
$A_{1}=0$ ( case2) or $P_{1}$ when $A_{2}=0$ ( case3). Unfortunately, the
observed light curve does not give us any unambiguous information of when
$A_{1}=0$ or when $A_{2}=0.$ Thus, the next logical step is to begin to search
for the answers to the  4th question, the  5th question and the  6th question.

####  7.4.4 Map-incompatibility

If the JHL-argument is true, the non-stationary and stationary starspots
migrate with different $\Omega$ angular velocities. In this case, the
comparison of surface images from different years can lead to false
identifications of the non-stationary ($P_{\mathrm{act}}$) and the stationary
($P_{\mathrm{orb}}$ or $P_{\mathrm{rot}}$) starpots. In general, there is no
unique period in the $g_{\mathrm{C}}(t)$ model for comparing the longitudinal
difference between two starspots, say A and B, over a longer period of time.
It is also uncertain, if A moves forwards or backwards with respect to B. The
2P-model predicts that one can reliably compare two surface images of the same
star with the same constant period only if the time difference of these
surface images is much shorter than $P_{\mathrm{lap}}/2$. For example, the lap
cycle period of FK Com is only $P_{\mathrm{lap}}=398^{\mathrm{d}}$. This is
comparable to the typical one year gap between observing seasons. We will
hereafter refer to this problem as “map-incompatibility”. This effect
resembles the shuffling of cards. The current order of cards (i.e. longitudes
of starspots A and B) does not give any information about the order before the
shuffling. This effect has certainly been overlooked or underestimated in
earlier SI-method studies, like Washuettl et al. (2009, EI Eri: eleven years),
Lindborg et al. (2011, II Peg: six years) or Hackman et al. (2013, FK Com: 13
years). This map incompatibility must have caused numerous misidentifications
of the rotating structures, because there are no unique phases or longitudes
in the constantly changing 2P-model. Such misidentifications have increased
the range of $|\Delta\Omega|$ estimates, e.g. in Barnes et al. (2005) or
Balona & Abedigamba (2016), which were discussed earlier in Sect 6.

###  7.5 Dead end

Jetsu et al. (2017) assumed that the K-method analysis of the
$t_{\mathrm{min,1}}$ minima epochs of Table 3 would give an unambiguous
$P_{\mathrm{act}}$ value for FK Com (the  4th question). The next step in
their reasoning was to use this $P_{\mathrm{act}}$ value in solving an
unambiguous $P_{\mathrm{rot}}$ value for FK Com. However, we have just shown
that if the $t_{\mathrm{min,1}}$ minima epochs are determined with any one-
dimensional period analysis method, then the minima-incompatibility prevents a
unique $P_{\mathrm{act}}$ solution. Any particular $t_{\mathrm{min,1}}$ value
in our Table 3 may, or may not, be connected to $P_{\mathrm{act}}$ and/or
$P_{\mathrm{rot}}$ (see Sect. 7.7: migration1, migration2 and migration3
alternatives). This problem could be at least partly solved, if we knew the
time intervals when the interference caused by the other periodicity is
absent, and only the $P_{\mathrm{act}}$ or $P_{\mathrm{rot}}$ period totally
dominates the light curves of FK Com ( case2 or  case3). However, we have no
way of knowing when takes place.

We could try to use the known orbital period $P_{\mathrm{orb}}\approx
P_{\mathrm{rot}}$ to solve $P_{\mathrm{act}}$, if FK Com were a binary, like
the 14 CABSs studied in Jetsu et al. (2017). This certainly known
$P_{\mathrm{rot}}$ could then be used to solve $P_{\mathrm{act}}$ in a similar
way, as $P_{\mathrm{rot}}$ was solved by using the presumably known
$P_{\mathrm{act}}$ (Jetsu et al., 2017, their Fig. 32). Unfortunately, we do
not have the “luxury” of knowing the $P_{\mathrm{rot}}$ period of FK Com
before the analysis.

If the JHL-argument is true, we must conclude that all our earlier one-
dimensional period analyses have given spurious results, because these results
suffer from incompatibility. The same applies to all one-dimensional period
finding analyses of starspots done by others during the past 70 years since
Kron (1947) published his first observations of starspots. Due to this
incompatibility, neither one of the two $P_{\mathrm{act}}$ and
$P_{\mathrm{rot}}$ periods of FK Com can be fixed before the analysis. These
two periods must be solved simultaneously. At this point, we have a serious
problem, because our current study of FK Com photometry has arrived at a dead
end: we can not use the $t_{\mathrm{min,1}}$ minima epochs of Table 3 to get
an unambiguous solution for the numerical $P_{\mathrm{rot}}$ or
$P_{\mathrm{act}}$ period values of FK Com (the  4th question and the  5th
question).

###  7.6 Sum of two different frequency sinusoids

In Sect. 7.3, we assumed that the 2P-model is correct and simulated artificial
data with this model. Here, we show how these results could have been
anticipated mathematically, and especially the incompatibility. The
mathematical relations presented in this section explain all graphical results
presented earlier in Figs. 2 and 3.

If $f_{1}\approx f_{2}$, the solution for the sum $s(t)$ of sinusoidal light
curves $s_{1}(t)$ and $s_{2}(t)$ of Eqs. 30 and 31 depends only on the
amplitudes $a_{1}$ and $a_{2}$. The solutions are different for

$\displaystyle a_{1}$ $\displaystyle=$ $\displaystyle a_{2}$ (35)
$\displaystyle a_{1}$ $\displaystyle\neq$ $\displaystyle a_{2}.$ (36)

If Eq. 35 is true, then

$\displaystyle s(t)=s_{1}(t)+s_{2}(t)=a_{a}(t)\sin{(2\pi f_{a}t)},$ (37)

where the amplitude is

$\displaystyle a_{a}(t)=a_{1}\sqrt{2+2\cos{[2\pi(f_{1}-f_{2})t]}}$ (38)

and the frequency is

$\displaystyle f_{a}=(f_{1}+f_{2})/2.$ (39)

This frequency of the sum $s(t)$ remains constant $f=f_{a}$. Hence, the
$P_{3}$ periods detected with the LS-method concentrate between $P_{1}$ and
$P_{2}$ in Fig. 2b. The amplitude $a_{a}(t)$ of Eq. 38 varies between
$a_{\mathrm{a,min}}=0$ and $a_{\mathrm{a,max}}=2a_{1}$ with the lap cycle
period $P_{\mathrm{lap}}$. This regularity is seen in Fig. 2c. Finally, an
abrupt $f_{a}/2$ phase shift of $s(t)$ occurs when $a_{a}(t)$ goes to zero, as
seen in Fig. 2d.

If Eq. 36 is true, we assume that $a_{1}<a_{2}$. In this case, the sum is

$\displaystyle s(t)=s_{1}(t)+s_{2}(t)=a_{b}(t)\sin{[2\pi
f_{2}t+\phi_{b}(t)]},$ (40)

where the amplitude is

$\displaystyle
a_{b}(t)=\sqrt{a_{1}^{2}+a_{2}^{2}+2a_{1}a_{2}\cos{[2\pi(f_{1}-f_{2})t]}}$
(41)

and the phase shift is

$\displaystyle\phi_{b}(t)=\arctan{\left[{{a_{1}\sin{[2\pi(f_{1}-f_{2})t]}}\over{a_{1}\cos{[2\pi(f_{1}-f_{2})t]+a_{2}}}}\right]}.$
(42)

The $[\phi_{b}(t)]$ units are radians.

The amplitude $a_{b}(t)$ of Eq. 41 varies regularly between

$\displaystyle a_{\mathrm{b,min}}=a_{2}-a_{1}$ (43)

and

$\displaystyle a_{\mathrm{b,max}}=a_{1}+a_{2}$ (44)

during the lap cycle period $P_{\mathrm{lap}}$, as seen in Fig. 3c.

The phase modulation $\phi_{b}(t)$ induces frequency changes of $s(t)$ between

$\displaystyle
f_{\mathrm{b,max}}=f_{2}+{{a_{1}(f_{1}-f_{2})}\over{a_{2}+a_{1}}}$ (45)

at the $a_{\mathrm{b,max}}$ epochs, and

$\displaystyle
f_{\mathrm{b,min}}=f_{2}-{{a_{1}(f_{1}-f_{2})}\over{a_{2}+a_{1}}}$ (46)

at the $a_{\mathrm{b,min}}$ epochs. These frequency changes of $s(t)$ at both
sides of $f_{2}$ are repeated regularly during $P_{\mathrm{lap}}$. The
frequency $f_{2}$ of the stronger $s_{2}(t)$ sinusoid $(a_{1}<a_{2})$
dominates these frequency changes of $s(t)$. The $P_{3}$ changes in Fig. 3b
concentrate on the level of the stronger $P_{2}$ periodicity.

The maximum $\phi_{b}(t)$ phase shift is

$\displaystyle\phi_{\mathrm{b,max}}=\arcsin{(a_{1}/a_{2})}$ (47)

to both directions. The $[\phi_{\mathrm{b,max}}]$ units are radians. For
$a_{1}=a_{2}$, the maximum phase shift $\phi_{\mathrm{b,max}}$ is $\pi/2$
radians, or equivalently $(\pi/2)/(2\pi)=1/4$ in dimensionless phase units
with $f_{2}$. Since this shift at $a_{\mathrm{b,min}}$ occurs to both
directions, its full range is

$\displaystyle\Delta\phi_{\mathrm{b}}=[\arcsin{(a_{1}/a_{2})}]/\pi$ (48)

in dimensionless phase units. This gives $\Delta\phi_{\mathrm{b}}=0.5$ when
$a_{1}=a_{2}$. No phase shift can exceed this limit. This abrupt phase shift
occurs also when $a_{1}\neq a_{2}$, but it is smaller. For example, the
$a_{1}=0.^{\mathrm{m}}048$ and $a_{2}=0.^{\mathrm{m}}050$ combination causes
an abrupt $\Delta\phi_{\mathrm{b}}=0.41$ phase shift at $a_{\mathrm{b,min}}$.
The predicted phase shift in Fig. 3d is only $\Delta\phi_{\mathrm{b}}=0.17$.

In conclusion, the observed amplitude, period and mimimum of the sum $s(t)$
fluctuate regularly with the lap period cycle $P_{\mathrm{lap}}$ when the
condition of Eq. 36 is true.

###  7.7 Indirect approach

Now that we know all these different mathematical relations for the sum
$s(t)=s_{1}(t)+s_{2}(t)$ presented in the previous Sect. 7.6, it is still next
to impossible to figure out what happens in some particular sample of
photometry. These mathematical relations do, however, give us some hope in
resolving the $P_{\mathrm{act}}$ and $P_{\mathrm{rot}}$ dilemma of FK Com,
because they open up several indirect approaches. Here, indirect means that we
do not solve the $P_{1}$ and $P_{2}$ periods simultaneously and directly.
These indirect approaches are valid only if the real light curves $g_{1}(t)$
and $g_{2}(t)$ are sinusoids (Eqs. 30 and 31). For example, the relations of
Eqs. 37 and 40 show that the shape of the sum curve $s(t)$ is sinusoidal with
only one minimum and maximum. If this observed $s(t)$ light curve has two
minima, the indirect approaches presented below are no longer valid because at
least one of the real light curves $g_{1}(t)$ and $g_{2}(t)$ can not be a
sinusoid.

It has already been mentioned in Sect. 7.4.2 that the period analysis of the
observed light curve amplitudes $A_{3}$ may reveal the lap cycle period
$P_{\mathrm{lap}}$. The relation of Eq. 32 is easier to manipulate in the
frequency domain, $f_{\mathrm{lap}}=|f_{1}-f_{2}|=\pm(f_{1}-f_{2})$. This
gives

$\displaystyle f_{1}=\pm f_{1}=f_{\mathrm{lap}}\pm f_{2}$ (49)

We can use the equality $f_{1}=\pm f_{1}$, because a negative $f_{1}$ value
makes no sense in this particular context. Assuming that
$f_{2}=P_{\mathrm{orb}}^{-1}\approx P_{\mathrm{rot}}^{-1}$ in CABSs, the above
relation gives two solutions for $f_{1}=P_{\mathrm{act}}^{-1}$ when
$P_{\mathrm{lap}}=f_{\mathrm{lap}}^{-1}$ is known. If the signal of the
$P_{\mathrm{orb}}$ period does not dominate the light curves, the migration of
the light curve minimum epochs is tilted as a function of phase $\phi_{2}$
with $P_{2}=P_{\mathrm{orb}}=f_{2}^{-1}$ (Eq. 5). During these time intervals,
the forwards or backwards migration of these minima reveals if
$P_{\mathrm{act}}^{-1}=f_{1}=f_{\mathrm{lap}}+f_{2}$ or
$P_{\mathrm{act}}^{-1}=f_{1}=f_{\mathrm{lap}}-f_{2}$. Unfortunately, this
relation of Eq. 49 does not help with any CASS, like FK Com.

There are three migration alternatives for the observed
$s(t)=s_{1}(t)+s_{2}(t)$ light curve minima

migration1: If $a_{1}\approx a_{2}$ (Eqs. 30 and 31), the migration is linear
with $(f_{1}+f_{2})/2$ (Eq. 39)

migration2: If $a_{1}<a_{2}$ (Eqs. 30 and 31), the migration is linear with
$f_{2}$ and fluctuating (Eqs. 45 and 46)

migration3: If $a_{1}>a_{2}$ (Eqs. 30 and 31), the migration is linear with
$f_{1}$ and fluctuating (Eqs. 45 and 46)

It is not easy to unambiguously separate these three alternatives from each
other in the real photometric observations. Nevertheless, we know at least
that the stronger signal usually dominates the linear fluctuating migration
(Eqs. 45-46), because it is reasonable to assume that the cases $a_{1}\approx
a_{2}$ occur less frequently than the cases $a_{1}<a_{2}$ or $a_{1}>a_{2}$.
When the observed $s(t)$ light curve minima $t_{\mathrm{min,1}}$ are analysed,
e.g. with the K-method, the result can be $f_{1}$, $f_{2}$ or
$(f_{1}+f_{2})/2$, or none of these. Different migration alternatives of CABSs
were displayed in Jetsu et al. (2017, Figs. 15-27: tilted non-stationary and
horizontal stationary lines in panels “a”). We know now that there are at
least three different migration alternatives even in the most simple case of
two interacting sinusoids $s_{1}(t)$ and $s_{2}(t)$. For example, the minima
of II Peg first concentrated close to $\phi_{\mathrm{orb}}=0.25$ between the
years 1988 and 1997, and then these minima began to wander regularly away from
this phase (Jetsu et al., 2017, their Fig. 27a). All available
$t_{\mathrm{min,1}}$ light curve minima were used to determine the
$P_{\mathrm{act}}$ value of this CABS. However, we know now that the result
for $P_{\mathrm{act}}$ was not correct, because some $t_{\mathrm{min,1}}$
values, like those between the years 1988 and 1997, were certainly connected
$P_{\mathrm{orb}}$, but not to $P_{\mathrm{act}}\neq P_{\mathrm{orb}}$. For
any particular star, the K-method may detect $f_{1}$, $f_{2}$ or
$(f_{1}+f_{2})/2$ depending on which one of the migration1, migration2 or
migration3 modes dominates. The long-term photometry of any CABS or CASS is a
mixture of all these three migration alternatives, because the amplitudes
$A_{1}=2a_{1}$ and $A_{2}=2a_{2}$ can change.

It may be possible to identify the time intervals when one of the above
mentioned three migration modes dominates. For example, it is usually easy to
notice the particular time intervals when $P_{\mathrm{orb}}$ dominates the
$s(t)$ light curve minima of some CABS (Jetsu et al., 2017, e.g. II Peg). It
may be possible to solve $P_{\mathrm{act}}$ by analysing the
$t_{\mathrm{min,1}}$ values of the other remaining time intervals with the
K-method. Finally, the observing seasons displaying a flip–flop event
represent the migration1 case having an exceptional amplitude and periodicity
combination. The detection of the $P_{\mathrm{orb}}$ or $P_{\mathrm{act}}$
periods is impossible during these observing seasons when $A_{1}\approx
A_{2}$, because an infine number of $P_{1}=f_{1}^{-1}$ and $P_{2}=f_{2}^{-1}$
period combinations can induce the observed period
$P_{3}=[(f_{1}+f_{2})/2]^{-1}$.

In Sect. 6.2, we noted that our $|\Delta\Omega|$ values in Table 4 may be
underestimates. The $P_{\mathrm{act}}$ values of some stars published by Jetsu
et al. (2017) are too close to $P_{\mathrm{orb}}$, because they used all
$t_{\mathrm{t_{min,1}}}$ values in their K-method analysis. Had they excluded
the $t_{\mathrm{min,1}}$ values connected to the $P_{\mathrm{orb}}$
periodicity before their K-method analysis, some
$|P_{\mathrm{orb}}-P_{\mathrm{act}}|$ differences would have been larger.
Hence, we have underestimated the $|\Delta\Omega|$ values in our Table 4. In
some cases, the K-method may even have detected only the $P_{\mathrm{orb}}$
periodicity, because $P_{\mathrm{act}}$ never dominated the observed light
curve. This may have happened for example with DM UMa (Jetsu et al., 2017,
Fig. 15a).

Regardless of the minima-incompatibility, the three migration alternatives
migration1, migration2 and migration3 do provide the qualitative answers to
the  4th question, the  5th question and the  6th question, which are given in
paper ii (Sect. 12).

There are other indirect ways to obtain less important additional information.
Firstly, the ratio $a_{1}/a_{2}$ can be estimated from the range of
$\phi_{\mathrm{min,1}}$ fluctuation (Eq. 47). Secondly, if the observed $s(t)$
light curve amplitude $A_{3}$ repeats some regular pattern, the amplitudes
$A_{1}=2a_{1}$ and $A_{2}=2a_{2}$ can be solved from Eqs. 43 and 44. This
gives

$\displaystyle A_{1}$ $\displaystyle=$
$\displaystyle(A_{\mathrm{3,max}}-A_{\mathrm{3,min}})/2$ (50) $\displaystyle
A_{2}$ $\displaystyle=$
$\displaystyle(A_{\mathrm{3,max}}+A_{\mathrm{3,min}})/23$ (51)

where $A_{\mathrm{3,max}}$ and $A_{\mathrm{3,min}}$ are the maximum and
minimum values of the observed amplitude $A_{3}$.

Although the number of possible $g_{1}(t)$ and $g_{2}(t)$ combinations is
infinite, the above indirect approaches give us some hope of detecting the
$P_{\mathrm{act}}$ and $P_{\mathrm{rot}}$ periodicities of FK Com. However,
these indirect approaches rely on the assumption that the observed $s(t)$
light curve is a sum of two sinusoids. Even if this assumption were true, the
interpretation of the observed light curves of FK Com would remain ambiguous,
especially due to incompatibility. We have obtained qualitative answers to the
4th question, the  5th question and the  6th question, but the quantitative
estimates for the numerical values of $P_{\mathrm{act}}$ and
$P_{\mathrm{rot}}$ are still missing. Hence, we are still at the dead
encountered in Sect. 7.5.

##  8 Discussion

Jetsu et al. (2017) presented the general light curve model for CABSs and
CASSs. This model was based on the JHL-argument. We define the complex
2P-model $g_{\mathrm{C}}(t)$ and the simple 1P-model $g_{\mathrm{S}}(t)$ in
Sect. 3 (Eqs. 6 and 7).

The models of one-dimensional period finding methods, like the sinusoidal LS-
model, have unique phases, and the shape of the light curve model does not
change. These models can not describe the light curve of the 2P-model, because
there are no unique phases for presenting the observed $g_{\mathrm{C}}(t)$
light curve (Eq. 6). The shape of this $g_{\mathrm{C}}(t)$ light curve is
constantly changing. Therefore, the $g_{\mathrm{C}}(t)$ light curve can be
studied only as a function of time, not as a function of phase. The only
unique phases are the $\phi_{1}$ and $\phi_{2}$ phases of the real $g_{1}(t)$
and $g_{2}(t)$ light curves (Eqs. 3 and 5).

The complex $g_{\mathrm{C}}(t)$ model and the simple $g_{\mathrm{S}}(t)$ model
are nested (Sect. 3.3). They are the same model when $f_{1}=f_{2}$ ( case1),
and when $g_{1}(t)$ has $A_{1}=0$ ( case2) or $g_{2}(t)$ has $A_{2}=0$ (
case3). If none of these three cases occurs, we can compare the complex
$g_{\mathrm{C}}(t)$ model and the simple $g_{\mathrm{S}}(t)$ model with the
statistical criterion of Eq. 16, and choose which one is the better model for
the data. Such comparisons will be done later in paper ii (Sects. 2-4).

The earlier 2P-model analysis of thirteen CABSs relied on the fixed
$P_{\mathrm{act}}$ and $P_{\mathrm{rot}}\approx P_{\mathrm{orb}}$ values
(Jetsu et al., 2017). One season of FK Com photometry was also studied with
the 2P-model by first solving the fixed $P_{\mathrm{act}}$ value, and then
solving the fixed $P_{\mathrm{rot}}$ value (Jetsu et al., 2017, Sect. 6.2).

In Sects. 5.1-5.2, we draw attention to the “sobering reminder” presented by
Howard (1994), which may mean that all results obtained for the stellar SDR
with the SI-method and the LC-method are not necessarily true, especially if
the solar-stellar connection holds. Nevertheless, those earlier SDR results
certainly contradict the JHL-argument (e.g. Strassmeier, 2009, Sect. 7). For
example, if this argument is true, the earlier SI-method studies should have
detected the two constant periods $P_{\mathrm{act}}$ and
$P_{\mathrm{rot}}\approx P_{\mathrm{orb}}$ of the JHL-argument (the  1st
question). We argue that the parameters $|k|$ (Eq. 18) and $|\Delta\Omega|$
(Eq. 19) of thirteen CABSs in Table 4 support the idea that the earlier SI-
method studies could have already detected these two periodicities. We also
present general and particular evidence for this (Sects. 6.2 and 6.3). Long-
lived starspots rotating with a constant angular velocity have been detected
in many SI-method studies (e.g. Berdyugina et al., 1998; Korhonen et al.,
2000; Strassmeier & Bartus, 2000). These results provide our answer to the
1st question.

The LC-method gives several estimates for stellar SDR, like the $|k|$ or $Z$
parameters (Eqs. 24, 25 and 28). The values of these parameters are computed
from the observed changes of the photometric rotation period under the
assumption that $P_{\mathrm{phot}}\approx P_{\mathrm{rot}}$ (Sect. 7.1).
Numerous LC-method studies have confirmed that the stellar SDR decreases when
$P_{\mathrm{phot}}$ decreases (Sect. 7.2). These $P_{\mathrm{phot}}\approx
P_{\mathrm{rot}}$ changes in the observed light curves of CABSs and CASSs
contradict the JHL-argument (the  2nd question).

The $P_{\mathrm{phot}}\approx P_{\mathrm{rot}}$ values have usually been
determined with the one-dimensional period finding methods, like the LS-
method. We simulate artificial data with the most simple 2P-model: the sum of
two sinusoids (Sect. 7.3). These simulated data are analysed with the LS-
method (Sect. 7.3.1-7.3.2). The results for the light curve periods $(P_{3})$,
amplitudes $(A_{3})$ and minimum epochs $(t_{\mathrm{1,min}})$ reveal an
effect that we refer to as “incompatibility” (Sect. 7.4). The observed values
of all these three parameters change, although the underlying 2P-model does
not change at all (Sects. 7.4.1-7.4.3). Due to this incompatibility effect,
two observers observing the light curve of same star during the same time
interval will get different $P_{3}$, $|k|$ and $Z$ estimates with the LS-
method, or with any other one-dimensional period finding method. The
incompatibility of the $P_{3}$ periods detected with the LS-method, or with
any other one-dimensional period finding method, provides answers to the  2nd
question and the  3rd question. This incompatibility effect does not only
contaminate the LC-method results, because it can also mislead the
identification of starspots in the surface images obtained with the SI-methods
(Sect. 7.4.4).

Due to the minima-incompatibility effect, our study arrives at a dead end in
Sect. 7.5. The K-method analysis of the minimum epochs $t_{\mathrm{min,1}}$
determined with the one-dimensional CPS-method does not give us an unambiguous
$P_{\mathrm{act}}$ value for FK Com. The relation $P_{\mathrm{rot}}\approx
P_{\mathrm{orb}}$ can not be used, because FK Com is not a binary. In other
words, we have no qualitative or quantitative answers for the  4th question,
the  5th question or the  6th question.

The most simple 2P-model, the sum of two sinusoids, is already studied in
Sect. 7.3. We show how the periods, the amplitudes and the minimum phases of
this particular 2P-model can be solved analytically (Sect. 7.6). These
solutions allow us to obtain indirect information of the real light curves
(the two sinusoids) that cause the observed light curve (the sum of two
sinusoids). For example, this indirect approach reveals the three migration
alternatives migration1, migration2 and migration3 (Sect. 7.7). These
migration alternatives provide qualitative answers to the  4th question, the
5th question and the  6th question. For example, perfect flip–flop events
$\Delta\phi_{b}=0.5$ occur when the two real light curves are equal amplitude
sinusoids (Eq. 48).

The above indirect solutions are valid only, if the correct complex
$g_{\mathrm{C}}(t)$ model is a sum of two sinusioids. In this case, the sum
curve $g_{\mathrm{C}}(t)$ is also a sinusoid (Eqs. 37 and 40). For example, if
the observed $g_{\mathrm{C}}(t)$ light curve has two minima, then at least one
of the real $g_{1}(t)$ and $g_{2}(t)$ light curves can not be a sinusoid. In
this case, the analytical relations for a sum of sinusoids given in Sect. 7.6
are no longer valid. Thus, the quantitative answers to the  4th question and
the  5th question are still missing. The only way out of this dead end is to
formulate a new two-dimensional period analysis method, which gives the
unambiguous solutions for all parameters of the 2P-model. This is done in
paper ii.

##  9 Conclusions

In this first paper, we study the real and the simulated photometry of FK Com
to verify validity of the following argument made by Jetsu et al. (2017):

-

JHL-argument: “The observed light curves of chromospherically active binary
and single stars are interference of two real constant period light curves of
long-lived starspots. These constant periods are the non-stationary active
longitude period $P_{\mathrm{act}}$ and the stationary rotation period
$P_{\mathrm{rot}}\approx P_{\mathrm{orb}}$.”

This argument is not supported by the current widely held views of the stellar
surface differential rotation and the starspots (e.g. Strassmeier, 2009, Sect.
7). Therefore, we make six specific questions which undermine this argument
(Sect. 1). Our aim is to answer all these questions.

We present general and particular evidence that the long-lived starspots
predicted by the JHL-argument have already been detected in the earlier SI-
method studies. Our LS-method analysis of the real and the simulated
photometry of FK Com reveals that the results obtained with any one-
dimensional period analysis method suffer from incompatibility. If the JHL-
argument is true, the one-dimensional period finding methods, like the LS-
method, detect spurious period, amplitude and minimum epoch estimates from the
observed light curves. These spurious estimates have nothing to do with the
real periods, the real amplitudes and the real minimum epochs of the two real
light curves of the JHL-argument. This would mean that all earlier one-
dimensional period analysis studies of the light curves of chromospherically
active stars have given spurious results since the starspots were discovered
by Kron (1947). The spurious period changes detected with these LC-methods
would explain why the light curves and the surface images give different
surface SDR estimates even for the same star. We also show that it is possible
to obtain indirect information of the two real light light curves, if the
observed light curve is a sum of two constant period sinusoids.

In conclusion, if the JHL-argument is true, we can already answer the  1st
question, the  2nd question, the  3rd question and the  6th question. We can
also give qualitative answers to the  4th question and the  5th question. Even
if the JHL-argument were true, the quantitative answers to the  4th question
and the  5th question are still missing, because we can not solve the
unambiguous values for the $P_{\mathrm{act}}$ and $P_{\mathrm{rot}}$ periods
of FK Com. Hence, our study arrives at a dead end, because we can not prove
that the long-lived starspots predicted by the JHL-argument do exist in FK
Com. We solve this problem in our second paper, where we formulate a new two-
dimensional period finding method, solve the real light curves of FK Com and
determine its unambiguous $P_{\mathrm{act}}$ and $P_{\mathrm{rot}}$ period
values. The compact answers to all the above mentioned six specific questions
are given in paper ii (Sect. 12).

## Acknowledgements

We thank Thomas Hackman and Ilana Hiilesmaa for their comments of this
manuscript. This work has made use of NASA’s Astrophysics Data System (ADS)
services.

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##  Appendix A Abbreviations

We use the following abbreviations in paper i and paper ii

-

K-method: Kuiper method (paper i: Sect. 1)

-

CABS: Chromospherically Active Binary Star (paper i: Sect. 1)

-

MLC: Mean Light Curve (paper i: Sect. 1)

-

CASS: Chromospherically Active Single Star (paper i: Sect. 1)

-

SDR: Surface Differential Rotation (paper i: Sect. 1)

-

JHL-argument: Jetsu et al. (2017) argument (paper i: Sect. 1)

-

SI-method: Surface Images measure SDR (paper i: Sect. 1)

-

LC-method: Light Curves measure SDR (paper i: Sect. 1)

-

TEL=1: T7 telescope photometry (paper i: Sect. 2)

-

TEL=2: Ph10 telescope photometry (paper i: Sect. 2)

-

SEG: Data segment number (paper i: Sect. 2)

-

CPS-method: Continuous Period Search method (paper i: Sect. 2)

-

CPS-model: Continuous Period Search model (paper i: Sect. 2)

-

datax,y: TEL=x data of SEG=y (paper i: Sect. 3)

-

2P-model: Two Period model (paper i: Sect. 3.1)

-

1P-model: One Period model (paper i: Sect. 3.2)

-

LS-method: Lomb-Scargle method (paper i: Sect. 7)

-

LS-model: Lomb-Scargle model (paper i: Sect. 7)

-

test${}_{a_{1}=a_{2}}$: 2P-model test $A_{1}=A_{2}$ (paper i: Sect. 7.3.1)

-

test${}_{a_{1}<a_{2}}$: 2P-model test $A_{1}<A_{2}$ (paper i: Sect. 7.3.2)

-

2P-method: Two period method (paper ii: Sect. 2.1)

-

1P-method: One period method (paper ii: Sect. 2.2)

-

WK-method: Weighted Kuiper method (paper ii: Sect. 7)

We introduce these abbreviations in the sections given above in the
parenthesis. For easy readability, we do not use these abbreviations in our
six specific questions or in our answers to these six questions, except for
the JHL-argument abbreviation.