notation should be interpreted over the natural numbers). In the base case
$k=n$ and, by the truth lemma, there is
$\Theta_{n}\sqsupseteq_{\mathrm{c}}\Gamma_{n}$ such that
$\varphi\in\Theta_{n}$. So, suppose that $(\Theta_{i})_{i=k+1}^{n}$ has been
constructed with the desired properties. Since the canonical model is
forth–down confluent, there is some $\Theta_{k}$ such that
$\Gamma_{k}\sqsubseteq_{\mathrm{c}}\Theta_{k}\preccurlyeq_{\mathrm{c}}\Theta_{k+1}$,
yielding the desired $\Theta_{k}$.

The problem is that the sequence $\Theta=(\Theta_{i})_{i\leq n}$ may not be an
element of $W_{\Sigma}$. We instead choose a suitable subsequence
$\Delta=(\Theta_{j_{i}})_{i\leq m}$, where $j_{k}$ is defined by (forward)
induction on $k$. For the base case, we set $j_{0}=0$, and $m_{0}=0$ (meaning
that $\Delta$ currently has $m_{0}+1$ elements). Now, suppose that $m_{k}$ has
been defined as has been $j_{i}$ for $i\leq m_{k}$, in such a way that
$(\Theta_{j_{i}})_{i\leq m_{k}}\in W_{\Sigma}$ and $(\Gamma_{i})_{i\leq
k}\sqsubseteq_{\Sigma}(\Theta_{j_{i}})_{i\leq m_{k}}$. To define $m_{k+1}\geq
m_{k}$ and $(\Theta_{j_{i}})_{i\leq m_{k+1}}$, we consider two cases. First
assume that there is $\psi\in\Sigma$ such that
$\psi\in\Theta_{k+1}\setminus\Theta_{j_{k}}$. In this case, setting
$m_{k+1}=m_{k}+1$ and $j_{m_{k+1}}=k+1$ we see that $(\Theta_{j_{i}})_{i\leq
m_{k+1}}$ has all desired properties. In particular, the existence of the
formula $\psi$ guarantees that $(\Theta_{j_{i}})_{i\leq m_{k+1}}\in
W_{\Sigma}$. Otherwise, set $m_{k+1}=m_{k}$. In this case we see that
$\Gamma_{k+1}\sqsubseteq_{\mathrm{c}}\Theta_{k+1}\succcurlyeq^{0}_{\Sigma}\Theta_{j_{m_{k}}}$,
so that $\Gamma_{k+1}\sqsubseteq^{0}_{\Sigma}\Theta_{j_{m_{k}}}$ and thus
$(\Gamma_{i})_{i\leq k+1}\sqsubseteq_{\Sigma}(\Theta_{j_{i}})_{i\leq
m_{k+1}}$. That $(\Theta_{j_{i}})_{i\leq m_{k+1}}\in W_{\Sigma}$ follows from
the fact that $m_{k+1}=m_{k}$ and the induction hypothesis. The claim then
follows by setting $m=m_{n}$ and $\Delta=(\Theta_{j_{i}})_{i\leq m}$.

The second claim is proven similarly, but by contraposition. We leave the
details to the reader. ∎

From Lemmas 31 and 33 we immediately obtain the following.

######  Lemma 34.

For $\Gamma\in W_{\Sigma}$ and $\varphi\in\Sigma$, we have
$(\mathcal{M}^{\mathsf{S4I}}_{\Sigma},\Gamma)\models\varphi$ if and only if
$\varphi\in\ell^{+}(\Gamma)$.

######  Lemma 35.

For $\varphi\in\Sigma$, we have $\varphi\in\mathsf{S4I}$ if and only if
$\mathcal{M}^{\mathsf{S4I}}_{\Sigma}\models\varphi$.

###### Proof.

We know that $\varphi\in\mathsf{S4I}$ if and only if
$\mathcal{M}^{\mathsf{S4I}}\models\varphi$. Now, given $\Gamma\in W_{\Sigma}$,
we have $\ell^{+}(\Gamma)=\Theta\cap\Sigma$, for some prime set $\Theta$
containing all derivable formulas, so $\varphi\in\ell^{+}(\Gamma)$. Lemma 34
yields that $\varphi\in\mathsf{S4I}$ implies
$(\mathcal{M}^{\mathsf{S4I}}_{\Sigma},\Gamma)\models\varphi$. Conversely, if
$\varphi\not\in\mathsf{S4I}$ then there is $\Gamma\in W_{\mathrm{c}}$ such
that $\varphi\not\in\Gamma$, which setting $\Gamma^{\prime}\\!=\\!(\Gamma)$ to
be a singleton sequence yields
$(\mathcal{M}^{\mathsf{S4I}}_{\Sigma},\Gamma^{\prime})\not\models\varphi$. ∎

######  Theorem 8.

$\mathsf{S4I}$ has the finite frame property. That is, if a formula $\varphi$
is falsifiable (respectively satisfiable) on a $\mathsf{S4I}$ frame, then
$\varphi$ is falsifiable (respectively satisfiable) on a finite $\mathsf{S4I}$
frame. Hence $\mathsf{S4I}$ is decidable.

###### Proof.

In view of Theorem 4(2), it suffices to show that $\mathsf{S4I}$ has the
shallow, forest-like model property. Fix a formula $\varphi$ and let $\Sigma$
be the set of subformulas of $\varphi$. By Lemmas 30 and 32,
$\mathcal{M}^{\mathsf{S4I}}_{\Sigma}$ is a model based on a shallow, forest-
like $\mathsf{S4I}$ frame, and by Lemma 35,
$\mathcal{M}^{\mathsf{S4I}}_{\Sigma}\models\varphi$ if and only if
$\varphi\in\mathsf{S4I}$, as needed. ∎

##  6 Concluding remarks

We have settled the long-standing problems of the finite frame property for
$\mathsf{CS4}$ and the decidability of $\mathsf{GS4}$ and
${\mathsf{GS4}}^{\mathsf{c}}$. We also introduced a logic closely related to
$\mathsf{IS4}$ that also enjoys the finite frame property. The logics we
considered correspond to classes of models with combinations of the forth–up,
back–up and forth–down confluence properties. There are a handful of other
logics that may be defined in this fashion.

The forth–up confluence property alone is the basis of what we call
_$\Diamond$ -regular_ logics. We have not provided an axiomatisation for the
logic of reflexive transitive $\Diamond$-regular frames, as our completeness
proofs rely on either Axiom 2.1 or Axiom 2.1 being available. Note that this
class does not validate $\Box p\to\Box\Box p$. First results on logics of
frames satisfying only forth–up confluence were recently presented in [34],
where the $\mathsf{K}$ version was axiomatised and shown to be decidable.

Similarly, we may consider the class of forth–down confluent frames. While
Axiom 2.1 holds on frames that are forth–up _and_ forth–down confluent, it is
unclear how the class of forth–down confluence frames may be axiomatised. On
the other hand, it seems that the shallow model construction we have provided
for $\mathsf{S4I}$ should readily adapt to frames satisfying only the forth–up
or the forth–down confluence property, so all that is needed for finite frame
property results in these settings is to find the respective axiomatisations.

In total, there are eight logics that could be obtained from combinations of
the three confluence frame conditions, and eight more for their upward-linear
variants. The study of logics of frames satisfying both forth–up and
forth–down confluence [35] has recently been initiated (in the enriched _bi_
-intuitionistic modal language), although the decidability results in this
paper are not obtained via shallow model properties.

Finally, we mention that Gödel modal logics are of independent interest (see
for example [15, 20]), with the Gödel variants of the modal logics
$\mathsf{K}$ and $\mathsf{S5}$ enjoying the finite model property [21].
However, the decidability of $\mathsf{GS4}$ and ${\mathsf{GS4}}^{\mathsf{c}}$
remained challenging problems, since these logics do not enjoy a finite model
property for real-valued semantics. We have shown how birelational semantics
do not have this issue, and indeed the logics enjoy the finite frame property
for this semantics, and hence are decidable. We believe that this work could
lead to a systematic treatment of other prominent Gödel modal logics whose
decidability was previously unknown, by first providing them with a
birelational interpretation and then establishing the finite frame property
under this semantics.

An interesting question left open is that of the complexity of the two logics
$\mathsf{GS4}$ and ${\mathsf{GS4}}^{\mathsf{c}}$. The exponential bound on the
size of models yields a nexptime upper bound on complexity, but it is possible
that validity is in fact pspace-complete since this is the case of Gödel
linear temporal logic, which is defined by similar frame conditions [23].

## Acknowledgments

David Fernández-Duque and Brett McLean were partially funded by the SNSF–FWO
Lead Agency Grant 200021L_196176 (SNSF)/G0E2121N (FWO).

Martin Diéguez was partially funded by the research project Amorcage/EL4HC of
the program RFI Atlanstic 2020.

## References

  * [1] Rowan Davies and Frank Pfenning.  A modal analysis of staged computation.  J. ACM, 48(3):555–604, may 2001.
  * [2] Vikraman Choudhury and Neel Krishnaswami.  Recovering purity with comonads and capabilities.  Proceedings of the ACM on Programming Languages, 4(ICFP):1–28, 2020\.
  * [3] Kenji Miyamoto and Atsushi Igarashi.  A modal foundation for secure information flow.  In Workshop on Foundations of Computer Security, pages 187–203, 2004.
  * [4] Valentijn Anton Johan Borghuis.  Coming to terms with modal logic: on the interpretation of modalities in typed lambda-calculus.  PhD thesis, Technische Universiteit Eindhoven, 1994.
  * [5] Simone Martini and Andrea Masini.  A Computational Interpretation of Modal Proofs, pages 213–241.  Springer Netherlands, Dordrecht, 1996.
  * [6] Fitch R.B.  Symbolic logic: an introduction.  Ronald Press, 1952.
  * [7] Ranald Clouston.  Fitch-style modal lambda calculi.  In Christel Baier and Ugo Dal Lago, editors, Foundations of Software Science and Computation Structures, pages 258–275, Cham, 2018. Springer International Publishing.
  * [8] Frederic B Fitch.  Intuitionistic modal logic with quantifiers.  Portugaliae mathematica, 7(2):113–118, 1948.
  * [9] Duminda Wijesekera.  Constructive modal logics I.  Annals of Pure and Applied Logic, 50(3):271–301, 1990.
  * [10] A. Simpson.  The proof theory and semantics of intuitionistic modal logic.  PhD thesis, University of Edinburgh, Edinburgh, UK, 1994.
  * [11] Natasha Alechina, Michael Mendler, Valeria de Paiva, and Eike Ritter.  Categorical and Kripke semantics for constructive S4 modal logic.  In Proceedings of 15th International Workshop on Computer Science Logic (CSL 2001), 10th Annual Conference of the EACSL, Paris, France, volume 2142 of Lecture Notes in Computer Science, pages 292–307. Springer, 2001.
  * [12] Marianna Girlando, Roman Kuznets, Sonia Marin, Marianela Morales, and Lutz Straßburger.  Intuitionistic S4 is decidable.  In LICS, pages 1–13, 2023.
  * [13] F. Wolter and M. Zakharyaschev.  The relation between intuitionistic and classical modal logics.  Algebra and Logic, 36(2):73–92, Mar 1997.
  * [14] Frank Wolter and Michael Zakharyaschev.  Intuitionistic modal logic.  In Logic and Foundations of Mathematics: Selected Contributed Papers of the 10th International Congress of Logic, Methodology and Philosophy of Science, Florence, Italy, pages 227–238. Springer, Dordrecht, Netherlands, 1999.
  * [15] X. Caicedo and R. Rodríguez.  Standard Gödel modal logics.  Studia Logica, 94:189–214, 2010.
  * [16] K. Gödel.  Zum intuitionistischen aussagenkalkül.  Anzeiger der Akademie der Wissenschaften in Wien, 69:65–66, 1932\.
  * [17] A. Heyting.  Die formalen Regeln der intuitionistischen Logik.  In Sitzungsberichte der Preussischen Akademie der Wissenschaften, page 42–56. Deutsche Akademie der Wissenschaften zu Berlin, 1930.  Reprint in Logik-Texte: Kommentierte Auswahl zur Geschichte der Modernen Logik, Akademie-Verlag, 1986.
  * [18] Gisèle Fischer Servi.  On modal logic with an intuitionistic base.  Studia Logica, 36(3):141–149, 1977.
  * [19] George Metcalfe and Nicola Olivetti.  Towards a proof theory of Gödel modal logics.  Logical Methods in Computer Science, 7(2), 2011.
  * [20] George Metcalfe and Nicola Olivetti.  Proof systems for a Gödel modal logic.  In Proceedings of the 18th International Conference on Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2009), Oslo, Norway, volume 5607 of Lecture Notes in Computer Science, pages 265–279. Springer, 2009.
  * [21] Xavier Caicedo, George Metcalfe, Ricardo Oscar Rodríguez, and Jonas Rogger.  A finite model property for Gödel modal logics.  In Proceedings of the 20th International Workshop on Logic, Language, Information, and Computation (WoLLIC 2013), Darmstadt, Germany, volume 8071 of Lecture Notes in Computer Science, pages 226–237. Springer, 2013.
  * [22] Petr Hájek.  Metamathematics of Fuzzy Logic.  Springer, Dordrecht, Netherlands, 1998.
  * [23] Juan P. Aguilera, Martín Diéguez, David Fernández-Duque, and Brett McLean.  Time and Gödel: Fuzzy temporal reasoning in PSPACE.  In Proceedings of the 28th Workshop on Logic, Language, Information and Computation - 28th International Conference (WoLLIC 2022), Icşi, Romania, pages 18–35, 2022.
  * [24] Juan P. Aguilera, Martín Diéguez, David Fernández-Duque, and Brett McLean.  A Gödel calculus for linear temporal logic.  In Proceedings of the 19th International Conference on Principles of Knowledge Representation and Reasoning - 19th International Conference (KR 2022), Haifa, Israel, pages 2–11. IJCAI, 2022.
  * [25] Xavier Caicedo and Ricardo Oscar Rodríguez.  Bi-modal Gödel logic over [0, 1]-valued Kripke frames.  Journal of Logic and Computation, 25(1):37–55, 2015.
  * [26] Ricardo Oscar Rodríguez and Amanda Vidal.  Axiomatization of crisp Gödel modal logic.  Studia Logica, 109(2):367–395, 2021.
  * [27] Lutz Schröder and Dirk Pattinson.  Shallow models for non-iterative modal logics.  In KI 2008: Advances in Artificial Intelligence, 31st Annual German Conference on AI, Kaiserslautern, Germany, pages 324–331, Berlin, Heidelberg, 2008. Springer.
  * [28] Philippe Balbiani, Martín Diéguez, and David Fernández-Duque.  Some constructive variants of S4 with the finite model property.  In Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science, (LICS 2021), pages 1–13. IEEE, 2021.
  * [29] Martín Diéguez and David Fernández-Duque.  Decidability for $\mathsf{S4}$ Gödel modal logics.  In María Eugenia Cornejo, István Á. Harmati, László T. Kóczy, and Jesús Medina-Moreno, editors, Computational Intelligence and Mathematics for Tackling Complex Problems 4, pages 1–7. Springer International Publishing, 2023.
  * [30] Gisèle Fischer Servi.  Axiomatizations for some intuitionistic modal logics.  Rendiconti del Seminario Matematico Università e Politecnico di Torino, 42(3):179–194, 1984.
  * [31] Ryuta Arisaka, Anupam Das, and Lutz Straßburger.  On nested sequents for constructive modal logics.  Logical Methods in Computer Science, 11(3), 2015.
  * [32] D. Gabelaia, A. Kurucz, F. Wolter, and M. Zakharyaschev.  Non-primitive recursive decidability of products of modal logics with expanding domains.  Annals of Pure and Applied Logic, 142(1-3):245–268, 2006.
  * [33] P. Balbiani, J. Boudou, M. Diéguez, and D. Fernández-Duque.  Intuitionistic linear temporal logics.  ACM Transactions on Computational Logic, 21(2):14:1–14:32, 2020\.
  * [34] Philippe Balbiani, Han Gao, Çigdem Gencer, and Nicola Olivetti.  A natural intuitionistic modal logic: Axiomatization and bi-nested calculus.  CoRR, abs/2309.06309, 2023.
  * [35] David Fernández-Duque, Brett McLean, and Lukas Zenger.  A Family of Decidable Bi-intuitionistic Modal Logics.  In Proceedings of the 20th International Conference on Principles of Knowledge Representation and Reasoning, pages 262–271, 8 2023\.


# Seismic Signatures of Stellar Magnetic Activity — What Can We Expect from
TESS?

René Kiefer 1,∗, Anne-Marie Broomhall 1, Warrick H. Ball 2,3

###### Abstract

Asteroseismic methods offer a means to investigate stellar activity and
activity cycles as well as to identify those properties of stars which are
crucial for the operation of stellar dynamos. With data from CoRoT and Kepler,
signatures of magnetic activity have been found in the seismic properties of a
few dozen stars. Now, NASA’s Transiting Exoplanet Survey Satellite (TESS)
mission offers the possibility to expand this, so far, rather exclusive group
of stars. This promises to deliver new insight into the parameters that govern
stellar magnetic activity as a function of stellar mass, age, and rotation
rate. We derive a new scaling relation for the amplitude of the activity-
related acoustic (p-mode) frequency shifts that can be expected over a full
stellar cycle. Building on a catalogue of synthetic TESS time series, we use
the shifts obtained from this relation and simulate the yield of detectable
frequency shifts in an extended TESS mission. We find that, according to our
scaling relation, we can expect to find significant p-mode frequency shifts
for a couple hundred main-sequence and early subgiant stars and for a few
thousand late subgiant and low-luminosity red giant stars.

##  1 Keywords:

stellar activity, asteroseismology, stellar activity cycles, TESS, p-mode
frequency shifts

##  2 Introduction

The primary seismic signature of stellar activity and activity cycles is a
systematic change in mode frequencies over timescales associated with activity
cycles, which are typically of the order of months to decades [1, 2, 3].
Measuring this phenomenon for stars is primarily limited by the baseline
length of the observations. For the Sun, however, we have Sun-as-a-star
helioseismic observations lasting for decades, covering several eleven-year
solar activity cycles. From these data it is found that the frequency of low-
degree modes close to the frequency of maximum oscillation power
$\nu_{\textrm{max}}$ change by approximately $0.4\,\mathrm{\mu Hz}$ between
activity minimum and maximum (e.g., [4, 5]). Furthermore, this shift is
strongly correlated with the level of activity along the cycle [4, 6].

The first asteroseismic evidence of variations in stellar activity were found
in an F5 V star observed by CoRoT [7, 8], HD 49933. For this star García et
al. [9] found that not only did the frequencies vary systematically with time
but, as is also observed for the Sun [10, 11, 6], the amplitudes of the modes
varied in anti-phase with the frequencies. In a more detailed analysis,
Salabert et al. [12] found that the observed frequency shifts are larger than
seen in the Sun but the same dependence on mode frequency is observed, i.e.,
the magnitude of the frequency shift increases with mode frequency (e.g., [13,
14]).

Since then examples of systematic variations in the frequencies of stars
observed by Kepler [15, 16] have been found (Salabert et al. [17], Régulo et
al. [18], Kiefer et al. [19], Salabert et al. [20], Santos et al. [21]).
Indeed, Kiefer et al. [19] found that 23 out of the 24 stars in their sample
showed significant and systematic frequency shifts in time, while Santos et
al. [21] found quasi-periodic variations on 60% of their sample of 87 solar-
type stars. The sample of potential candidates for observing asteroseismic
activity cycles is limited by the fact that activity suppresses the amplitude
of acoustic oscillations [22]. Since activity also increases with decreasing
rotation period [e.g., 23, 24, 25, 26] these stars tend to be more active and
consequently are less likely to have high signal-to-noise oscillations. The
dependence of the length of stellar activity cycles on surface rotation rate
or on Rossby number is complex and is actively researched [e.g., 27, 28, 29].
While it has proven to be difficult to observe complete activity cycles with
Kepler, it is still possible to search for systematic variations with time in
the properties of p modes. A concise review on the capabilities and prospects
of asteroseismology regarding the inference on stellar cycles and activity is
also given by Chaplin and Basu [30].

Stellar magnetic activity also manifests in variability of stellar luminosity
[see, e.g., 31, and references therein]. Over the solar cycle the total
irradiance is found to change by about 0.1% in temporal correlation with solar
activity [32]. This slight increase in luminosity is associated with faculae
[32]. In contrast to faculae, spots decrease solar and stellar luminosities
[32, 33]. For the Sun, the increase in luminosity caused by faculae dominates
over the decrease in luminosity brought about by sunspots, hence the overall
increase in total irradiance between cycle minimum and maximum. However, for
other stars it is the decrease in luminosity caused by the presence of spots
that is the dominant source of variability [34]. Magnetic features have
typical life times which are of the order of the rotation period of a star.
This has to be taken into account when a meaningful proxy of stellar magnetic
activity is to be constructed from the variability of the stellar luminosity.
Indeed, Mathur et al. [35] defined such a photometric proxy for stellar
activity with the $S_{\text{ph}}$ index. It is given by taking the mean of the
standard deviations of segments of the time series each having a length 5
times the rotation period of the star (see also [36] and [37]). As long
cadence photometric time series (CoRoT: 512 s, Kepler: 29.4 min, TESS: 30 min)
are sufficient for such studies, the number of available light curves for the
investigation of stellar magnetic activity through photometry is rather large
(CoRoT: $\sim$163 000 targets, Kepler: $\sim$197 000 targets, TESS: $>10^{7}$
targets at end of nominal mission). Studies in which the variability of light
curves is explored regarding signatures of activity or activity cycles
include, but are not limited to [3, 38, 36, 39, 40]. In this article, we do
not consider the variability of light curves, as activity-related changes are
not included in the synthetic time series we use (see [41] and Section 3).

With the launch of NASA’s Transiting Exoplanet Survey Satellite [TESS, 42],
the opportunity of studying seismic signatures of many more stars is at hand.
So far, p-mode frequency shifts have only been detected for a few dozen Kepler
targets and one CoRoT target, and as of yet, the dependence of the amplitude
of the activity-related frequency shifts (which corresponds to the amplitude
of stellar activity cycles) on fundamental stellar parameters is not known.
With the large number of stars that will be observed by TESS, asteroseismology
can become an important additional tool to learn about stellar cycles. As we
will discuss in Section 7, a mission extension of TESS to four or six years is
however necessary for this. This paper is aimed at a prediction of how many
detections of activity-related frequency shifts we can expect in an extended
TESS mission.

##  3 Synthetic TESS sample

To simulate the yield of seismic signatures of activity that can be expected
from TESS, we use the synthetic catalogue of Ball et al. [41], which consists
of light curves that realistically mimic the properties of the TESS targets in
short-cadence data (120 s). The sample consists of 12 731 stars, covering
main-sequence, subgiant, and low-luminosity red giant stars, which were
selected from a synthetic Milky Way population simulated with TRILEGAL [43].
The selection criteria are the same as for the real Asteroseismic Target List
(ATL; see Schofield et al. [44]) of the TESS Asteroseismic Science Consortium
(TASC), which are chosen such as to maximise the yield of detected
oscillations and optimise the coverage of different stages of stellar
evolution. Ball et al. [41] give a detailed description of their methods to
obtain oscillation mode parameters (frequencies, mode damping widths,
amplitudes), noise levels, granulation background, and other aspects of the
simulated time series. We note that the time series of Ball et al. [41] are
provided without noise, which has to be added as
$\mathcal{N}\left(0,\sigma_{\text{noise}}\right)\cdot\sqrt{30}$ to the time
series. Here, $\sigma_{\text{noise}}$ is the noise amplitude as given in the
catalogue of Ball et al. [41] and the factor $\sqrt{30}$ is accounting for the
2-minute sampling of the time series.

Figure 1 shows a Kiel diagram of the synthetic sample with stellar ages given
by the colour of the dots. We separated the sample into main-sequence stars
(in the following called region I) and post-main-sequence stars (region II)
along the Terminal Age Main-Sequence (TAMS), which we defined at a core
hydrogen abundance content of $10^{-5}$. Figure 7 shows the Kiel diagram as in
Figure 1 but colour coded for the two regions. As can be seen in these
figures, the number of stars on the lower main-sequence is rather small. This
is due to the low intrinsic mode amplitudes of solar-like oscillations in cool
dwarf stars (see, e.g., Kjeldsen and Bedding [45]). Thus, these stars have a
low detection probability for their oscillation modes with TESS data [46]. A
large number of targets are young, early subgiants and stars towards the end
of their main-sequence life with $6000\,\mathrm{K}\lesssim
T_{\text{eff}}\lesssim 6900\,\mathrm{K}$. Subgiants ascending to the red giant
branch and low-luminosity red giants have large mode amplitudes and high
detection probabilities, hence, many of the best asteroseismic targets are in
this region of the Kiel diagram.

For a study of the temporal evolution of seismic parameters, it is necessary
that the targets are observed for either a long, uninterrupted period of time
or that they are observed again after some time. During its nominal two year
mission, TESS will observe 13 sectors per hemisphere with an area of four
times $24^{\circ}\times 24^{\circ}$ for a duration of about 27.4 days per
sector. The sectors partly overlap, chiefly in the region of the ecliptic
poles. Thus, some stars will have up to 13 sectors’ worth of data. A detailed
description of the TESS observing strategy can be found in Ricker et al. [42].
The time series of Ball et al. [41] reproduce this observing strategy, i.e.,
their length depends on the position of the respective star in the sky. Here,
we shall assume that the TESS mission is extended after its two year nominal
duration. We further assume that the observation strategy of the nominal
mission will be repeated in possible two year and four year mission
extensions, and thus that the same targets are observed again for the same
number of sectors after two and four years. However, the scaling relations for
p-mode frequency shifts described below are not TESS-specific.

Figure 1: Kiel diagram of the synthetic TESS sample of Ball et al. [41]. The
age of the stars is given by the colour of the dots. The Sun is indicated by
its usual symbol. Evolutionary tracks for stars with masses from 0.8 to 2.0 M⊙
in steps of 0.2 M⊙ are overlaid as solid black lines.

##  4 Reconsidering the scaling relation for p-mode frequency shifts

There are two models for activity related frequency shifts in the literature:
Chaplin et al. [47] simply use the amplitude of the activity cycle $\Delta
R^{\prime}_{\text{HK}}$ (see Equation 1) as a proxy for the frequency shift
and then scale the values such that the solar frequency shift equals the
observed $0.4\,\mathrm{\mu Hz}$. Saar and Brandenburg [48] found that $\Delta
R^{\prime}_{\text{HK}}$ scales with the average logarithmic fraction of the
stellar luminosity that is emitted in the Calcium II H and K spectral lines
$R^{\prime}_{\text{HK}}$:

$\displaystyle\Delta
R^{\prime}_{\text{HK}}\propto{R^{\prime}_{\text{HK}}}^{0.77}.$ (1)

For the second model, Metcalfe et al. [49] find that the frequency shifts
$\delta\nu$ scale as

$\displaystyle\delta\nu\propto\frac{D}{I}\Delta R^{\prime}_{\text{HK}},$ (2)

where $I$ is the inertia of the modes and $D$ is the depth of the perturbation
that causes the mode frequencies to shift. They further assume that the depth
$D$ scales as the pressure scale height $H_{p}$, which is proportional to
$L^{0.25}R^{1.5}M^{-1}$ at the stellar photosphere, and that mode inertia
scales as $MR^{-1}$, where $R$, $M$, and $L$ are stellar radius, mass, and
luminosity respectively. Thus, Metcalfe et al. [49] find

$\displaystyle\delta\nu\propto\frac{R^{2.5}L^{0.25}}{M^{2}}\Delta
R^{\prime}_{\text{HK}}$ (3)

for the frequency shift amplitude over a full activity cycle.

Here we take a new look at the sensitivity of acoustic mode frequencies to
activity. As a starting point we use Equation (1) of Metcalfe et al. [49] to
estimate the frequency shift $\delta\nu$:

$\displaystyle\delta\nu=\frac{\int dVKS}{2I\nu},$ (4)

where $K$ is a kernel for the sensitivity of p modes to perturbations, $S$ is
a source function, and $\nu$ is the frequency of the mode. Following Metcalfe
et al. [49], we use

$\displaystyle K$ $\displaystyle\propto\left|\nabla\cdot\vec{\xi}\right|^{2},$
(5) $\displaystyle S$ $\displaystyle=A\delta(D-D_{c}),$ (6)

where $\vec{\xi}$ is the eigenfunction of the considered mode, $A$ is the
amplitude of the source function, and the $\delta$ function determines the
location of the source $D_{c}$. For now, we set $A=1$ and reintroduce it later
via the $R^{\prime}_{\text{HK}}$ index.

In the following, we concentrate on radial modes, which simplifies the
calculations considerably. The eigenfunctions for radial modes can be
described by

$\displaystyle\vec{\xi}=\xi_{r}(r)\vec{e}_{r},$ (7)

where $\xi_{r}(r)$ is the radial displacement eigenfunction and $\vec{e}_{r}$
is the unit vector in the radial direction. We do not introduce the spherical
harmonic $Y_{0}^{0}$ into the eigenfunction here, as it would only appear as a
constant in the following equations.

With this, the kernel function can be written as

$\displaystyle
K=\left|\frac{2}{r}\xi_{r}(r)+\frac{\partial\xi_{r}(r)}{\partial
r}\right|^{2}.$ (8)

Approximating the derivative $\frac{\partial\xi_{r}(r)}{\partial
r}\approx\frac{\xi_{r}(r)}{r}$ and assuming that the radial position $r$ which
we are considering is close enough to the surface so that the amplitude of the
eigenfunction is similar to that at the photospheric stellar radius $R$ we get

$\displaystyle K\approx\frac{9|\xi_{r}(R)|^{2}}{R^{2}}.$ (9)

Thus we have fixed the source to be at $D_{c}=R$. The mode inertia $I$ can be
approximated as follows:

$\displaystyle I=\int{dV\rho\left|\vec{\xi}\right|^{2}}\approx
R^{3}\overline{\rho}|\xi_{r}(R)|^{2},$ (10)

where we used again the eigenfunction of a radial mode, replaced the radial
integral $\int dr$ by simply multiplying with the photospheric radius $R$, and
replaced the density $\rho$ with the mean density $\overline{\rho}$.

With Equations (8) and (10) we can rewrite Equation (4):

$\displaystyle\delta\nu$
$\displaystyle\approx\frac{\frac{9|\xi_{r}(R)|^{2}}{R^{2}}}{2R^{3}\overline{\rho}|\xi_{r}(R)|^{2}\nu}\int
dV$ $\displaystyle=\text{const}\cdot\frac{R}{M\nu}.$ (11)

Throughout this, we assumed that the source is close enough to the
photospheric radius so that $D_{c}=R$. All constant factors are absorbed into
const. Equation (11) estimates the sensitivity of radial modes to a near-
surface perturbation.

The strength of the perturbation and hence the magnitude of the frequency
shift is imparted by the strength of the magnetic activity on the star for
which a proxy is used. From the synthetic TESS sample, the
$R^{\prime}_{\text{HK}}$ index can be estimated with a relation found by Noyes
et al. [50] for main-sequence stars:

$\displaystyle\log\left(\tau_{c}/P_{\text{rot}}\right)=-\left(0.324-0.400y+0.283y^{2}-1.325y^{3}\right),$
(12)

where $y=\log\left(R^{\prime}_{\text{HK}}\times 10^{5}\right)$, $\tau_{c}$ is
the convective turnover time, and $P_{\text{rot}}$ is the star’s rotation
period. Both these values are taken from the models of Ball et al. [41]. The
(local) convective turnover time $\tau_{c}$ is calculated half a mixing length
$\alpha H_{p}/2$ above the base of the outer convection zone of each star
following Gilliland [51], Landin et al. [52]:

$\displaystyle\tau_{c}=\frac{\alpha H_{p}}{v},$ (13)

where the solar-calibrated mixing length parameter $\alpha=1.957$ is used,
$H_{p}$ is the pressure scale height, and $v$ is the convective velocity.

To take into account that a large fraction of the stars in the synthetic TESS
sample are not on the main-sequence (in contrast to the stars considered in
Saar and Brandenburg [48] and Noyes et al. [50]), we modify the values of
$R^{\prime}_{\text{HK}}$ with three factors: First, we include a factor
$R^{-2}$ to account for the increasing surface area over which magnetic
activity can be distributed, thus affecting the oscillations less. Second,
starting at a Rossby number of $\text{Ro}=\frac{P_{\text{rot}}}{\tau_{c}}=2$,
we divide by $\sqrt{\text{Ro}-1}$. This is to accommodate the results of
Metcalfe and van Saders [26] who found that, as stars reach a critical Rossby
number of $\approx 2$, a transition in the magnetic activity appears to
happen: Magnetic braking weakens, rotation periods increase more slowly, and
cycle periods grow longer and eventually disappear. Finally, we explicitly
include the weakening of magnetic activity with the inverse square root of
stellar age $t_{\text{Age}}$ $\left[\text{Gyrs}\right]$ found by Skumanich
[53]:

$\displaystyle R^{\prime}_{\text{HK,
mod}}=\frac{R^{\prime}_{\text{HK}}}{R^{2}\max\left(\sqrt{\text{Ro}-1},1\right)t_{\text{Age}}^{0.5}}.$
(14)

With this and Equation (11), the expected frequency shift over a complete
stellar activity cycle at the frequency of maximum power $\nu_{\text{max}}$
scales as:

$\displaystyle\delta\nu$
$\displaystyle\propto\frac{R}{M\nu_{\text{max}}}\Delta R^{\prime}_{\text{HK,
mod}}.$ (15)

Figure 2 shows the expected p-mode frequency shifts for the synthetic TESS
sample between the activity minima and maxima of a full stellar activity
cycle. The values are scaled to a solar frequency shift of 0.4 $\mu$Hz at
$\nu_{\text{max}}$. Stellar age is given by the colour of the dots. For the
Sun, we assume $R^{\prime}_{\text{HK}}=-4.901$ (value from [2]) and do not
estimate it with Equation (12). The spike in shifts at 6500 K is caused by a
transition of how rotation rates are modelled by [41], cf. their Section 2.5
and their Figure 2.

Figure 2: Frequency shift amplitudes for full activity cycles for the
synthetic TESS sample calculated with Equation (15). The colour of the dots
indicates stellar age. The Sun is indicated by its usual symbol.

##  5 Measuring $\delta\nu$

Two main methods have been developed to determine the change in p-mode
frequency as a function of time. The first technique uses the cross-
correlation (CC) between power spectra of different epochs of the time series
(see, e.g., [54, 55, 18, 19]). The regions around the frequency of maximum
oscillation amplitude $\nu_{\text{max}}$ of the power spectra are retained and
their cross-correlation function is computed. Typically, only the region in
which p-mode peaks can be identified is retained. As the expected frequency
shifts between two power spectra of any epoch is only of the order of a few
$\mathrm{\mu Hz}$, the complete cross-correlation function does not need to be
calculated: restricting the cross-correlation function to a certain lag regime
considerably speeds up the computation. The central peak of the cross-
correlation function is then fitted to obtain the frequency shift
$\delta\nu_{\text{CC}}$ between the two power spectra. To estimate the
uncertainty $\sigma$ of the measured frequency shift, we employ the resampling
approach described in [18].

The second technique uses the individual mode frequencies (peak-bagging: PB)
obtained at different epochs. Here, the frequency shift
$\delta\nu_{\text{PB}}$, of a particular mode with harmonic degree $\ell$ and
radial order $n$ is given by

$\displaystyle\delta\nu_{\text{PB}}=\nu_{\ell,n}(t)-\overline{\nu_{\ell,n}},$
(16)

where $\nu_{\ell,n}(t)$ is the mode frequency obtained from a time series
observed between $t$ and $t+\delta t$, and $\overline{\nu_{\ell,n}}$ is a
reference frequency for that mode. Both methods (CC and PB) require the data
to be split into time series of length $\delta t$, where $\delta t$ is some
fraction of the full baseline of observations available. Ideally, $\delta t$
will be sufficiently short to resolve changes with time but long enough that
the power spectrum has sufficient resolution and signal-to-noise for the modes
to be observed clearly. See [21] for a detailed description of the peak-
bagging procedure in determining frequency shifts.

Most of the targets in the synthetic TESS sample have only few sectors’ worth
of data (one sector: 7930 stars, two sectors: 2601 stars, three sectors: 781
stars, four sectors or more: 1419; see also Tables 2 and 3). We thus calculate
the power spectra for the complete synthetic time series of each star without
separating them into shorter segments: here, $\delta t$ is equal to the length
of the time series.

In general, the two methods CC and PB are found to produce results in good
agreement with each other but the uncertainties associated with the cross-
correlation technique tend to be systematically larger than the frequency
shifts determined directly from the mode frequencies (see [55, 21] for
comparisons of the frequency shift uncertainty obtained with the two methods).
However, the peak-bagging method requires good resolution and signal-to-noise
for the mode frequencies to be determined accurately and precisely and so
there are instances where the cross-correlation method is preferable. The PB
technique is also much more reliant on input parameters like initial guesses
for mode frequencies and widths than the CC method. We do not apply the PB
method here, but assume that the frequency shifts measured with both methods
are in agreement and that the uncertainties of the PB method are smaller by a
factor of two than those obtained with the CC method.

We implemented the CC technique to measure the frequency shifts in the
following way: the power spectrum is calculated from the synthetic time series
with an oversampling factor of 8. This power spectrum is then shifted in
frequency by the amount that can be expected over the fraction of the stellar
cycle that is covered in the two (four) years that pass between the TESS
nominal mission and the second observation of any given star (see Section 3).
We estimate the length of the activity cycles by

$\displaystyle
P_{\text{cyc}}\left[\text{years}\right]=0.5P_{\text{rot}}\left[\text{days}\right]\cdot\max\left(\sqrt{\text{Ro}-1},1\right).$
(17)

We chose this rather crude estimation as a compromise between the short and
long activity cycle branches that can be found in a
$P_{\text{cyc}}$-$P_{\text{rot}}$ diagram like that of Böhm-Vitense [23] or
Metcalfe and van Saders [26]. The lengthening of activity cycles above
$\text{Ro}\approx 2$ found by Metcalfe and van Saders [26] is again modelled
into this with the factor $\max\left(\sqrt{\text{Ro}-1},1\right)$. Thus, the
input frequency shift is given by

$\displaystyle\delta\nu_{\text{input
shift}}=\delta\nu\cdot\min\left(\frac{2n}{P_{\text{cyc}}},1\right),$ (18)

where the factor 2 accounts for the fact that the full cycle frequency shift
is observed over a half-cycle, between activity minimum and maximum, $n=2$ or
4 is the number of years between observations, depending on the length of the
TESS mission extension, and $\delta\nu$ is calculated with Equation (15).
After shifting the power spectrum, we return to only the natural frequencies
without oversampling. The oversampling was only introduced to be able to shift
by frequencies smaller than the natural frequency resolution, which is
$\approx 0.422\,\mathrm{\mu Hz}$ for stars with only one sector of data.

The regions around the frequency of maximum oscillation amplitude
$\nu_{\text{max}}$ of the unshifted and shifted power spectra are then cross-
correlated. We retain a width of $\pm 5\Delta\nu$ around $\nu_{\text{max}}$,
where $\Delta\nu$ is the large frequency separation. Both $\Delta\nu$ and
$\nu_{\text{max}}$ are given in the catalogue of Ball et al. [41]. To obtain
the frequency shift between the two power spectra, we fit the cross-
correlation function with a Lorentzian profile.

##  6 Influence of assumptions on the predicted shift amplitudes

There are several assumptions made and approximations applied in the
derivation of the scaling relation given by Equation (15). The first
assumption is that the magnitude of activity-related frequency shifts should
be proportional to the strength of stellar magnetic activity. This is
certainly justified by the fact that for the Sun there is a clear positive
correlation between the p-mode frequency shifts and proxies for magnetic
activity as shown by, e.g., [6]. Also, in a follow-up study to [21], in which
they investigate the dependence of the observed frequency shifts on stellar
fundamental parameters, Santos et al. (submit.) find that for a sample of 30
solar-like stars, the amplitude of the frequency shifts decreases with
decreasing $R^{\prime}_{\text{HK}}$. Hence, the $R^{\prime}_{\text{HK}}$
activity index, which is available for a relatively large number of stars
(see, e.g., [1] and the database of [56]) and is employed in the scaling
relation of [55], is a good starting point for stars similar to the Sun. In
principle, spectroscopic activity indices other than $R^{\prime}_{\text{HK}}$
can be used in any of the frequency shift scaling relations we discuss here;
[49], for example, use the Mg II emission index $i_{\text{Mg II}}$. However,
the scaling of the amplitude of the index over a complete activity cycle, as
is done for $R^{\prime}_{\text{HK}}$ in Equation (1), has to be adjusted
accordingly.

To account not only for the strength of the activity, as is done in the
Chaplin et al. scaling, but also for the physics of the oscillation modes,
[49] derived the sensitivity of mode frequencies to near surface
perturbations. We note that the sensitivity of modes we find with Equation
(11) is in fact very similar to that found by [49] and quoted by [57] in their
Equation (14). Inserting the scaling relation of [45] $\nu_{\text{max}}\propto
MR^{-1}T_{\text{eff}}^{-0.5}$ into Equation (11) and using the Stefan-
Boltzmann law $\sqrt{T_{\text{eff}}}\propto\left(R^{-2}L\right)^{1/8}$, gives
$\delta\nu\approx\frac{R^{2.75}L^{0.125}}{M^{2}}$. All quantities are assumed
to be normalised to solar values in this discussion. The difference of our
Equation (11) to the mode sensitivity factor of [49] is thus only a factor
$R^{-0.25}L^{0.125}\propto\sqrt{T_{\text{eff}}}$, which is in the range
0.91–1.14 for the sample of stars investigated here, i.e., is of order unity.
Thus, the approximations we made in the derivation of Equation (11) —
approximation of the radial integral by simply multiplying with the stellar
radius $R$, approximation of the derivatives, localising the perturbation to
the surface — yield essentially the same result as the calculations of [49].

The three scaling relations are most easily distinguished for stars reaching
the lower red giant branch, which is where the disagreement is largest. In
Figures 8 and 9 we show the full-cycle frequency shifts of the synthetic TESS
sample for the Chaplin et al. and Metcalfe et al. scalings, respectively. The
Chaplin et al. [47] relation, simply using the $R^{\prime}_{\text{HK}}$
estimation of [50] for all stars from main-sequence to the lower red giant
branch, gives maximal full-cycle shift values of $\approx 3\,\mathrm{\mu Hz}$.
With estimated cycle lengths of at least 10 years, no shifts larger than
$\approx 1.2\,\mathrm{\mu Hz}$ should be found in TESS data with a two year
mission extension. This value increases to $\approx 5\,\mathrm{\mu Hz}$ for
the new scaling relation presented in this article and to even larger values
for the scaling relation of [49].

As can be seen in Figure 9, the full-cycle frequency shifts obtained by
employing the scaling relation of [49] would lead to very large values of over
50 $\mu$Hz for stars reaching the lower red giant branch. This is why we
looked for reasonable adjustments. As the sample of stars on which [50] based
their results is exclusively comprised of main-sequence stars, we modified the
$R^{\prime}_{\text{HK}}$ index with the factors discussed above. This aims to
capture some of the evolution that can naively be expected in stellar magnetic
activity. Here, the decrease of activity with $t_{\text{Age}}^{-0.5}$ found by
[53] is again extrapolated from the main-sequence up to the lower red giant
branch. The inclusion of this factor must be understood as an attempt to
circumvent our ignorance of how exactly activity evolves after the main-
sequence. It is completely possible that activity does indeed decrease faster
than just proportional to $t_{\text{Age}}^{-0.5}$ or even stops completely at
a certain evolutionary stage. The same applies for the factor
$\max\left(\sqrt{\text{Ro}-1},1\right)$.

Contemporaneous measurements of the $R^{\prime}_{\text{HK}}$ index and
photometric time series for asteroseismology would facilitate a better
differentiation between these scaling relations. If indeed $\delta\nu\propto
A\cdot\left(R^{\prime}_{\text{HK}}\right)^{B}$, with $A,B$ factors dependent
on stellar fundamental parameters regulating mode sensitivity to perturbations
($A$) and the amplitude of activity over a full cycle ($B$), then measurements
of frequency shifts and activity level can be used to learn about mode physics
as well as stellar dynamos. We note that measurements of stellar rotation
periods are essential for the prediction of the $R^{\prime}_{\text{HK}}$ index
with Equation (12) or with the modified expression (14) should direct
spectroscopic measurements of stellar activity not be available.

The estimation of the cycle period as described by Equation (17) is obviously
very crude. If cycle periods are underestimated with this, then the resulting
shifts we can expect to observe decrease and our predictions for $\delta\nu$
are an upper limit in this case. More, reliably measured cycle periods are
needed to replace this estimation with a more rigorous prediction. We remark
however, that Equation (17) does include the latest findings concerning the
evolution of cycle periods by [26] through the lengthening of cycles with the
factor $\max\left(\sqrt{\text{Ro}-1},1\right)$.

Figure 3: As Figure 1 but only stars with significant frequency shifts are
shown. Shifts are measured with the cross-correlation technique and a four
year TESS mission extension is assumed. Note that the colour bar is different
from Figure 1. Table 1: Predicted number of stars with significant frequency
shifts for different spectral types and evolutionary stages. A four year TESS
mission extension is assumed.
Spectral type | Region | Number | CC Detections | CC Fraction [%] | PB Detections | PB Fraction [%]
---|---|---|---|---|---|---
F | I | 6676 | 141 | 2.1 | 673 | 10.1
F | II | 1642 | 24 | 1.5 | 196 | 11.9
G | I | 54 | 30 | 55.6 | 44 | 81.5
G | II | 1628 | 765 | 47.0 | 1092 | 67.1
K | II | 2731 | 1486 | 54.4 | 1799 | 65.9
FG | I | 6730 | 171 | 2.5 | 717 | 10.7
FGK | II | 6001 | 2275 | 37.9 | 3087 | 51.4
total |  | 12731 | 2446 | 19.2 | 3804 | 29.9

##  7 Potential of TESS

We evaluated the frequency shifts obtained from four settings: the TESS
mission is either extended by two years or by four years and the frequency
shifts are determined by the cross-correlation method (CC) or the peak-bagging
method (PB), where the PB shift results are simply taken as those of the CC
method but with half the uncertainty as described in Section 5. The predicted
number of detections of significant frequency shifts for different subsets of
the cohort of TESS ATL short-cadence stars is given in Table 1 for a four year
mission extension and Table 4 for a two year mission extension. We define
significant frequency shifts as those which are at least $1\sigma>0$, and
within $3\sigma$ of the input frequency shift $\delta\nu_{\text{input
shift}}$, where $\sigma$ is the uncertainty of the measured frequency shift.
Tightening the second condition to $2\sigma$ or $1\sigma$ would eliminate some
of the more precisely but less accurately measured frequency shifts.

We separated the spectral types according to their effective temperature with
boundaries at $T_{\text{eff}}=5200\,\mathrm{K}$ between G and K and at
$T_{\text{eff}}=6000\,\mathrm{K}$ between F and G. There are no K-type main-
sequence stars in the sample. The separation into region I and II is discussed
in Section 3. Table 1 gives the number of stars for each subset.

The suppression of p-mode amplitudes by activity has not been taken into
account in our simulations. However, only stars whose oscillations have a low
detection probability and small intrinsic mode amplitudes are likely to be
affected by this. These are mainly late-type dwarf stars whose number in the
mock sample is rather small, thus the overall number of expected detections is
not strongly affected by this. As can be seen in Tables 2 and 3 the fraction
of stars with detectable shifts increases with the length of the time series
as frequency resolution increases. However, even for stars which are observed
for only one sector, it should be possible to detect shifts at a significant
level, especially for stars with long-lived modes.

With the CC method we were able to measure significant frequency shifts in 171
stars in region I and in 2275 stars in region II for a four year TESS mission
extension. The large difference between the two regions reflects the increased
amplitude and longer mode life times of solar-like oscillations for stars in
region II. Also, the much smaller mode widths of cooler stars allow one to
measure even smaller shifts, especially with the CC method. Figure 3 shows a
Kiel diagram of the stars with significant shifts measured with the CC method
for the four year extension. By comparison of Figure 1 with Figure 3 it can be
seen that stars with detectable frequency shifts are largely found at
effective temperatures below $\lesssim$6500 K. Note the different colour
tables used in these two figures. For region I, the fraction of F-type stars
for which we predict significant shifts is only 2.1% (141 out of 6676 stars)
and only 1.5% (24 out of 1642) for region II F stars. This reflects the
predicted smaller full cycle frequency shifts and large mode widths of F
stars. The fraction of stars with predicted detectable frequency shifts
increases for G stars: 55.6% (30 out of 54) for region I and 47.0% (765 out of
1628) for region II. For the G dwarfs of region I this increase is mainly due
to the smaller mode line widths compared to F dwarfs. Furthermore, for region
II G-type stars the detectability of shifts is enhanced by the larger
predicted full cycle shifts and the larger mode amplitudes. A large fraction
of K-type region II stars are predicted to have detectable shifts with 54.4%
(1486 out of 2731). With the smaller uncertainties which we assumed for the PB
method, these numbers increase for all subsets of stars, adding up to a total
of 3804 out of 12731 stars (29.9%) for the complete cohort. Table 4 gives the
predicted number of detections for a 2 year TESS extension.

We also tested for the number of false positives caused by the realisation
noise. For this we set the input shift to zero for all stars and proceeded as
described above. We found no false positives detections with the PB method for
either a two or a four year extension even when ommiting the condition that
the measured shift be within $3\sigma$ of the input shift value.

In Figures 4 and 5 the significant frequency shifts obtained with the cross-
correlation method for a four year TESS extension are plotted for three
subsets of stars (rows from top to bottom: all stars, region I, region II) as
a function of effective temperature (left column of panels) and rotation
period (right column of panels). The colour of the dots gives stellar age in
Figure 4 and stellar mass in Figure 5. We capped the ordinate in these plots
at $5\,\mathrm{\mu Hz}$. Higher frequency shifts tend to have large
uncertainties and including them would only distort the presentation.

Figure 4: Significant frequency shifts of the synthetic TESS sample for: (a)
all stars as a function of effective temperature; (b) all stars as a function
of rotation period; (c) stars in region I as a function of effective
temperature; (d) stars in region I as a function of rotation period; (e) stars
in region II as a function of effective temperature; (f) stars in region II as
a function of rotation period. The frequency shifts are measured with the
cross-correlation method. Here, a four year TESS mission extension is assumed.
The colour of the dots indicates stellar age as given in the colour bar on the
right. Note that the range of the abscissa differs from panel to panel. Figure
5: Same as Figure 4 but colour coded with stellar mass.

By comparing panel (a) of Figure 4 with Figure 2 it can be seen that for older
stars, as stellar cycles grow longer with stellar age, only a fraction of the
predicted shift over a full activity cycle can be detected with data that are
separated by only four years. As expected, younger and faster rotating stars
tend to show larger detectable shifts. The older the stars become (green and
blue data points), the smaller the shifts that can be expected to be observed
after two years become. This can also be seen in panel (b), where the measured
shifts are plotted as a function of rotation period: as stars age, their
rotation period becomes longer, the level of activity decreases, and the
frequency shifts become smaller. Obviously, this by design for our model for
the full-cycle frequency shifts (Equation 15) with the factors
$t_{\text{age}}^{-0.5}$ and the decreasing rotation period with age as it is
modelled in the synthetic TESS sample (cf. Section 2.5 and Figure 2 of [41]).
It is noteworthy that for more evolved stars, which have smaller mode damping
widths, shifts are detected more precisely and smaller shifts are detectable
at a significant level. The lack of stars with a large frequency shift at
short rotation periods in panel (b) of Figures 4 and 5 is due to the fact that
most of these young, faster-rotating stars in the synthetic TESS sample are
hot stars ($T_{\text{eff}}\gtrsim 6500\,\mathrm{K}$) with broader p-mode peaks
for which shifts are harder to measure precisely. Also, the full cycle
frequency shifts of these stars are rather small as can be seen from Figure 2.

The discrete structure in the measured frequency shifts in Figures 4 and 5 is
due to the large number of stars with only one sector of observation (7930
stars). The frequency resolution of one sector’s worth of observation is 0.422
$\mu$Hz. As the cross-crorrelation function is computed from periodograms with
this resolution, the fitted shift value is likely to be a multiple of this
value.

For the stars in region I (panels (c) and (d)) there is no clear tendency in
frequency shift with effective temperature or rotation period. This is partly
due to the way that rotation periods are modelled in the synthetic sample. For
stars with $T_{\text{eff}}>6500\,\mathrm{K}$, most of which are in region I,
the rotation period is drawn from a normal distribution with mean 5 days and a
standard deviation of 2.1 days, i.e., there is no dependency on age or
effective temperature. Overall, only a small percentage of targets with
$T_{\text{eff}}>6500\,\mathrm{K}$ have predicted detections. The same
explanation for this as given above applies. Also, stellar mass introduces a
selection bias here: due to their shorter evolutionary time scales, the more
massive stars in this sample do not have the time to brake their rotation and
thus stars with $M\gtrsim 1.8\,\mathrm{M_{\odot}}$ for which detection of
frequency shifts are predicted, are all at rotation periods
$P_{\text{rot}}\lesssim 20\,\mathrm{d}$ for stars in region I (panel d) and at
$P_{\text{rot}}\lesssim 40\,\mathrm{d}$ for stars in region II (panel f).
Similarly, only lower mass stars evolve slowly enough for rotation period to
increase and the activity to decrease: note the accumulation of old, low-mass
stars at $T_{\text{eff}}\approx 5000\,\mathrm{K}$ in panels (e), which
corresponds to the base of the red giant branch seen in Figure 1.

Figure 6: Left panel: Significant frequency shifts from region I of the
synthetic TESS data as function of effective temperature in red. A four year
extension of the TESS mission is assumed and the error bars of the CC method
are halved to approximate the PB method. Significant frequency shifts found by
[21] are shown in blue. Right panel: Kiel diagram of the stars which are
included in the left panel.

###  7.1 Comparison to real observations

To test the agreement between our predictions and real observations we
concentrate on main-sequence stars as there are currently no measured
frequency shifts of more evolved stars. We use the results of [21] available
online at this link. This is the largest available record on activity related
frequency shifts to date. [21] employ a PB approach on segments of Kepler time
series with a length of 90 days which overlap by 45 days. We calculate the
frequency shift for each star in their sample by taking the difference between
the minimum and maximum value of the mean frequency shift, which combines
their results for modes with harmonic degree $l=0$ and 1.

The left panel of Figure 6 shows the shifts of [21] in blue and the shifts of
the region I stars from this work in red as a function of effective
temperature. As can be seen, our predictions cover essentially the same range
of shifts as the measurements. The baseline of the Kepler data used by [21] is
typically around three years, which is shorter than the baseline we assumed
for our simulations. Thus, when assuming that the Kepler data do not cover a
full activity cycle, extrapolating onto the same baseline would give somewhat
larger shift values for the measured frequency shifts compared to our
predictions. However, our sample is dominated by stars closer to the TAMS as
can be seen in the right hand panel of Figure 6, which shows a Kiel diagram of
the stars of [21] in blue and our region I stars in red. This biases the
predicted shifts more towards lower values, as these stars tend to have slowed
down their rotation compared to their younger counterparts of similar mass and
effective temperature. Restricting our sample further would however leave us
with only a very small number of stars. We will have to wait for data of a
hopefully extended TESS mission to be able to truly test our scaling relation
Equation (15) and those of [47] and [49] on a larger sample of stars.

##  8 Overestimation of global stellar parameters

The full-cycle frequency shifts as shown in Figure 2 have typical values of
$6\,\mathrm{\mu Hz}$ for a low-luminosity red giant. For such a star, let
$\nu_{\text{max}}=324\,\mathrm{\mu Hz}$, $\Delta\nu=23.3\,\mathrm{\mu Hz}$,
$T_{\text{eff}}=5058\,\mathrm{K}$, $M=1.064M_{\odot}$ (target 00197 in the
sample of [41]). Using the scaling relation (see, e.g., Kjeldsen and Bedding
[45])

$\displaystyle M=M_{\odot}\left(\frac{\nu_{\text{max}}}{\nu_{\text{max,
}\odot}}\right)^{3}\left(\frac{\Delta\nu_{\odot}}{\Delta\nu}\right)^{4}\left(\frac{T_{\text{eff}}}{T_{\text{eff,}\odot}}\right)^{\frac{3}{2}},$
(19)

the mass value given above is recovered. The solar reference values are those
used in [46], which are $\nu_{\text{max, }\odot}=3090\,\mathrm{\mu Hz}$,
$\Delta\nu_{\odot}=135.1\,\mathrm{\mu Hz}$,
$T_{\text{eff,}\odot}=5777\,\mathrm{K}$. If $\nu_{\text{max}}$ is shifted by
$6\,\mathrm{\mu Hz}$ to higher frequencies, a mass of
$M_{\text{shift}}=1.12M_{\odot}$ is obtained from the scaling relation Thus,
if $\nu_{\text{max}}$ were measured during times of higher activity, stellar
mass would be overestimated by $0.06\,\mathrm{M_{\odot}}$ with this scaling
relation. The statistical uncertainties of asteroseismic stellar masses
obtained from the standard scaling relations are typically of the order of 10%
[58, 59]. Thus, an activity-related shift to higher frequencies would
introduce a systematic error and lead to an overestimation of mass and radius
when using the standard scaling relations. Indeed, [60] found in their study
of 10 eclipsing binary systems with at least one oscillating red giant that
the dynamical masses were consistently smaller than those obtained with the
asteroseismic scaling relations. The same arguments apply for stellar radii.
We note that for more evolved red giants, a shift to higher frequencies of
only $1\,\mathrm{\mu Hz}$ would lead to an even larger overestimation:
Considering the values of [60] for the star KIC 5786154
($\nu_{\text{max}}=29.75\,\mathrm{\mu Hz}$, $\Delta\nu=3.523\,\mathrm{\mu
Hz}$, $T_{\text{eff}}=4747\,\mathrm{K}$) and using the solar reference values
and scaling relation as given above, we find a mass for this star of
$M=1.44M_{\odot}$. With $\nu_{\text{max}}=30.75\,\mathrm{\mu Hz}$, which
incorporates the frequency shift predicted here, the scaling relation gives
$M=1.58M_{\odot}$. This example shows that it might be necessary to calibrate
the global seismic parameters $\nu_{\text{max}}$ and $\Delta\nu$ for the level
of activity of the star when the scaling relations are to be used for an
estimation of mass and radius of more evolved stars. A paper on activity-
related frequency shifts in evolved Kepler stars is currently in preparation.
As we will show there, frequency shifts of a few tenths of $\mu$Hz are not
uncommon on the red giant branch.

##  9 Discussion and Conclusion

To estimate the yield of detections of activity-related p-mode frequency
shifts in an extended TESS mission, we first derived a scaling relation for
full-cycle shifts (Equation 15) for stars from the main-sequence up to low
luminosity red giants. We used a catalogue of simulated light curves (Ball et
al. [41]) and input frequency shifts which can be expected to occur within the
two (four) years which would pass between a first observation of a given star
and a repeated observation by TESS. We then measured these frequency shifts
with a cross-correlation technique and found that we can expect to find a
couple hundred main-sequence and early subgiant stars and a few thousand more
evolved subgiant and low-luminosity red giant stars with activity-related
p-mode frequency shifts in an extended TESS mission according to our scaling
relation.

Our scaling relation does not account for the varying sensitivity of modes of
different harmonic degree: for the Sun it is observed that modes of higher
harmonic degree experience larger shifts over the solar cycle [e.g., 5]. For
several Kepler stars, larger shifts have been measured for $l=1$ modes than
for $l=0$ modes by [21]. A dedicated study of how the frequency shifts of
different harmonic degrees can influence the estimation of fundamental stellar
parameters is presented in this collection by Pérez Hernández et al. [61].

Up until now there is no confirmed detection of frequency shifts in subgiant
and low-luminosity red giant stars. This may either be because their activity
cycles are too long to be detectable with the available time series of Kepler
and CoRoT, they no longer have activity cycles, or nobody has looked for this
phenomenon in these stars yet. We also note that it is entirely possible that
the scaling relation we have derived here is not valid for stars that have
evolved far off the main-sequence even with the adjustments we have included
to account for suppression of activity, see Section 6.

A dedicated search for activity-related frequency shifts (or frequency offsets
in the case of very long cycles, which could also be attributed to activity
see Section 8), should therefore be carried out with the existing Kepler data
sets. Such results can then also be used to refine the estimates we presented
here. Given our scaling relation, Equation (15) and the large number of
confirmed oscillating subgiants and red giants (e.g., [62, 63]) such a study
could also help to decide between the scaling relations for the p-mode
frequency shifts by Chaplin et al. [47], Metcalfe et al. [49], and the one
presented here.

So far, only a few dozen solar-like stars, most of them Kepler targets, have
measured p-mode frequency shifts. An extended sample of seismic detections of
stellar activity and activity cycles will also be useful to determine the
fundamental parameters which govern stellar dynamos and their evolution as
stars age. [64] have shown that it is possible to measure stellar differential
rotation with asteroseismology. This can be connected with photometric
measurements of activity and the temporal variation of p-mode frequencies to
learn more about the intricate connection of rotation, its change through
stellar evolution, and stellar activity cycles.

Carrying out searches for and investigations of stellar magnetic activity with
seismic measures (mode frequencies, and amplitudes) are greatly enhanced in
their reliability by contemporaneous spectroscopic measurements of magnetic
activity. As, e.g., [65] have shown, it is feasible to employ seismology
together with ground-based observation to study stellar cycles and activity in
detail. Stars near the ecliptic poles, which are observed almost continuously
during TESS’s observation (615 stars in the synthetic sample have 13 sectors
of data; a similar number is expected for the real observations) can be used
for equivalent studies to fill the $P_{\text{rot}}$-$P_{\text{cyc}}$ diagram.
Long-term ground-based spectroscopic measurements of the level of activity of
these targets would be expedient for such studies. The rationale to optimally
select targets for such campaigns was laid out by Karoff et al. [57, 66].
Furthermore, the investigation of stellar variability can strengthen the
seismic detection of magnetic activity as, e.g., [9, 17] have shown.

Our predictions for frequency shifts measured after four years for stars on
the main-sequence and early subgiant stars (region I) agree qualitatively with
those found by [19] and Santos et al. (submit.) for Kepler stars of similar
evolutionary states. The reason the decrease of shifts with decreasing
effective temperature seen by Santos et al. (submit.) is less clear in our
simulations is most likely due to a difference in the underlying sample of
stars. We will have to await TESS data from a hopefully extended mission,
calculate models of the observed stars, measure the frequency shifts, and then
compare predictions from our scaling relation and those of [47] and [49] to
these observed shifts. As also mentioned above, measurements of a
spectroscopic activity proxy like $R^{\prime}_{\text{HK}}$ and reliable
measurements of the stellar rotation periods and age would facilitate this
considerably.

We found that it should be possible to reliably measure even small average
frequency shifts between two TESS observations each with a length of only one
month. This should also enable the search for photometric and asteroseismic
signatures of short activity cycles in those young, fast rotating stars which
have many sectors of data in TESS observations (see [3, 67] for such studies
with Kepler data). Obviously, long and uninterrupted photometric time series
with high stability are optimal for asteroseismology in general and
specifically for the search for seismic signatures of activity cycles. Even
though most of the targets of TESS only have few consecutive sectors of
observation, time series provided by TESS, especially by an extended mission,
can be used for such searches. In the future, PLATO [68] will pick up the
torch, observe two target fields for 2-3 years each, and provide data for
asteroseismology. These data will make another major contribution in the
seismic sounding of stellar activity cycles.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

RK led the work on this article and wrote most of the text. AMB contributed to
the writing of Sections 2, 5, and the overall refinement of the writing. WHB
generated and provided the dataset which is utilised in this article, computed
additional necessary quantities, and helped in the writing of the relevant
sections.

## Funding

This work was funded by STFC consolidated grant ST/L000733/1. WHB acknowledges
support from the UK Science and Technology Facilities Council (STFC).

## Acknowledgements

RK & AMB acknowledge the support of the STFC consolidated grant ST/L000733/1.

## Data Availability Statement

The datasets used for this study can be found in this repository: Link.

## References

  * Baliunas et al. [1995] Baliunas SL, Donahue RA, Soon WH, Horne JH, Frazer J, Woodard-Eklund L, et al.  Chromospheric variations in main-sequence stars.  Astrophys. J. 438 (1995) 269–287.  10.1086/175072.
  * Saar and Brandenburg [1999] Saar SH, Brandenburg A.  Time Evolution of the Magnetic Activity Cycle Period. II. Results for an Expanded Stellar Sample.  Astrophys. J. 524 (1999) 295–310.  10.1086/307794.
  * Vida et al. [2014] Vida K, Oláh K, Szabó R.  Looking for activity cycles in late-type Kepler stars using time-frequency analysis.  Mon. Notices R. Astron. Soc. 441 (2014) 2744–2753.  10.1093/mnras/stu760.
  * Jiménez-Reyes et al. [1998] Jiménez-Reyes SJ, Régulo C, Pallé PL, Roca Cortés T.  Solar activity cycle frequency shifts of low-degree p-modes.  Astron. Astrophys. 329 (1998) 1119–1124.
  * Broomhall [2017] Broomhall AM.  A Helioseismic Perspective on the Depth of the Minimum Between Solar Cycles 23 and 24.  Solar Phys. 292 (2017) 67.  10.1007/s11207-017-1068-5.
  * Broomhall and Nakariakov [2015] Broomhall AM, Nakariakov VM.  A Comparison Between Global Proxies of the Sun’s Magnetic Activity Cycle: Inferences from Helioseismology.  Solar Phys. 290 (2015) 3095–3111.  10.1007/s11207-015-0728-6.
  * Baglin et al. [2006] Baglin A, Auvergne M, Barge P, Deleuil M, Catala C, Michel E, et al.  Scientific Objectives for a Minisat: CoRoT.  Fridlund M, Baglin A, Lochard J, Conroy L, editors, The CoRoT Mission Pre-Launch Status - Stellar Seismology and Planet Finding (2006), ESA Special Publication, vol. 1306, 33.
  * Auvergne et al. [2009] Auvergne M, Bodin P, Boisnard L, Buey JT, Chaintreuil S, Epstein G, et al.  The CoRoT satellite in flight: description and performance.  Astron. Astrophys. 506 (2009) 411–424.  10.1051/0004-6361/200810860.
  * García et al. [2010] García RA, Mathur S, Salabert D, Ballot J, Régulo C, Metcalfe TS, et al.  CoRoT Reveals a Magnetic Activity Cycle in a Sun-Like Star.  Science 329 (2010) 1032.  10.1126/science.1191064.
  * Pallé et al. [1990] Pallé P, Régulo C, Roca Cortés T.  The spectrum of solar p-modes and the solar activity cycle.  Osaki Y, Shibahashi H, editors, Progress of Seismology of the Sun and Stars (Springer Berlin Heidelberg), Lecture Notes in Physics, vol. 367 (1990), 129–134.  10.1007/3-540-53091-6_73.
  * Komm et al. [2000] Komm RW, Howe R, Hill F.  Width and Energy of Solar p-Modes Observed by Global Oscillation Network Group.  Astrophys. J. 543 (2000) 472–485.  10.1086/317101.
  * Salabert et al. [2011] Salabert D, Régulo C, Ballot J, García RA, Mathur S.  About the p-mode frequency shifts in HD 49933.  Astron. Astrophys. 530 (2011) A127.  10.1051/0004-6361/201116633.
  * Libbrecht and Woodard [1990] Libbrecht KG, Woodard MF.  Solar-cycle effects on solar oscillation frequencies.  Nature 345 (1990) 779–782.  10.1038/345779a0.
  * Salabert et al. [2004] Salabert D, Fossat E, Gelly B, Kholikov S, Grec G, Lazrek M, et al.  Solar p modes in 10 years of the IRIS network.  Astron. Astrophys. 413 (2004) 1135–1142.  10.1051/0004-6361:20031541.
  * Borucki et al. [2010] Borucki WJ, Koch D, Basri G, Batalha N, Brown T, Caldwell D, et al.  Kepler planet-detection mission: Introduction and first results.  Science 327 (2010) 977–980.  10.1126/science.1185402.
  * Koch et al. [2010] Koch DG, Borucki WJ, Basri G, Batalha NM, Brown TM, Caldwell D, et al.  Kepler Mission Design, Realized Photometric Performance, and Early Science.  Astrophys. J. Lett. 713 (2010) L79-L86.  10.1088/2041-8205/713/2/L79.
  * Salabert et al. [2016a] Salabert D, Régulo C, García RA, Beck PG, Ballot J, Creevey OL, et al.  Magnetic variability in the young solar analog KIC 10644253. Observations from the Kepler satellite and the HERMES spectrograph.  Astron. Astrophys. 589 (2016a) A118.  10.1051/0004-6361/201527978.
  * Régulo et al. [2016] Régulo C, García RA, Ballot J.  Magnetic activity cycles in solar-like stars: The cross-correlation technique of p-mode frequency shifts.  Astron. Astrophys. 589 (2016) A103.  10.1051/0004-6361/201425408.
  * Kiefer et al. [2017] Kiefer R, Schad A, Davies G, Roth M.  Stellar magnetic activity and variability of oscillation parameters: An investigation of 24 solar-like stars observed by Kepler.  Astron. Astrophys. 598 (2017) A77.  10.1051/0004-6361/201628469.
  * Salabert et al. [2018] Salabert D, Régulo C, Pérez Hernández F, García RA.  Frequency dependence of p-mode frequency shifts induced by magnetic activity in Kepler solar-like stars.  Astron. Astrophys. 611 (2018) A84.  10.1051/0004-6361/201731714.
  * Santos et al. [2018] Santos ÂRG, Campante TL, Chaplin WJ, Cunha MS, Lund MN, Kiefer R, et al.  Signatures of Magnetic Activity in the Seismic Data of Solar-type Stars Observed by Kepler.  Astrophys. J. Suppl. Series 237 (2018) 17.  10.3847/1538-4365/aac9b6.
  * Chaplin et al. [2011] Chaplin WJ, Bedding TR, Bonanno A, Broomhall AM, García RA, Hekker S, et al.  Evidence for the Impact of Stellar Activity on the Detectability of Solar-like Oscillations Observed by Kepler.  Astrophys. J. Lett. 732 (2011) L5.  10.1088/2041-8205/732/1/L5.
  * Böhm-Vitense [2007] Böhm-Vitense E.  Chromospheric Activity in G and K Main-Sequence Stars, and What It Tells Us about Stellar Dynamos.  Astrophys. J. 657 (2007) 486–493.  10.1086/510482.
  * Mamajek and Hillenbrand [2008] Mamajek EE, Hillenbrand LA.  Improved Age Estimation for Solar-Type Dwarfs Using Activity-Rotation Diagnostics.  Astrophys. J. 687 (2008) 1264–1293.  10.1086/591785.
  * Suárez Mascareño et al. [2016] Suárez Mascareño A, Rebolo R, González Hernández JI.  Magnetic cycles and rotation periods of late-type stars from photometric time series.  Astron. Astrophys. 595 (2016) A12.  10.1051/0004-6361/201628586.
  * Metcalfe and van Saders [2017] Metcalfe TS, van Saders J.  Magnetic Evolution and the Disappearance of Sun-Like Activity Cycles.  Solar Phys. 292 (2017) 126.  10.1007/s11207-017-1157-5.
  * Strugarek et al. [2017] Strugarek A, Beaudoin P, Charbonneau P, Brun AS, do Nascimento JD.  Reconciling solar and stellar magnetic cycles with nonlinear dynamo simulations.  Science 357 (2017) 185–187.  10.1126/science.aal3999.
  * Warnecke [2018] Warnecke J.  Dynamo cycles in global convection simulations of solar-like stars.  Astron. Astrophys. 616 (2018) A72.  10.1051/0004-6361/201732413.
  * Strugarek et al. [2018] Strugarek A, Beaudoin P, Charbonneau P, Brun AS.  On the Sensitivity of Magnetic Cycles in Global Simulations of Solar-like Stars.  Astrophys. J. 863 (2018) 35.  10.3847/1538-4357/aacf9e.
  * Chaplin and Basu [2014] Chaplin WJ, Basu S.  Inferences on Stellar Activity and Stellar Cycles from Asteroseismology.  Space Sci. Rev. 186 (2014) 437–456.  10.1007/s11214-014-0090-2.
  * Fabbian et al. [2017] Fabbian D, Simoniello R, Collet R, Criscuoli S, Korhonen H, Krivova NA, et al.  The variability of magnetic activity in solar-type stars.  Astronomische Nachrichten 338 (2017) 753–772.  10.1002/asna.201713403.
  * Fröhlich [2013] Fröhlich C.  Total Solar Irradiance: What Have We Learned from the Last Three Cycles and the Recent Minimum?  Space Sci. Rev. 176 (2013) 237–252.  10.1007/s11214-011-9780-1.
  * Shapiro et al. [2017] Shapiro AI, Solanki SK, Krivova NA, Cameron RH, Yeo KL, Schmutz WK.  The nature of solar brightness variations.  Nature Astronomy 1 (2017) 612–616.  10.1038/s41550-017-0217-y.
  * Shapiro et al. [2014] Shapiro AI, Solanki SK, Krivova NA, Schmutz WK, Ball WT, Knaack R, et al.  Variability of Sun-like stars: reproducing observed photometric trends.  Astron. Astrophys. 569 (2014) A38.  10.1051/0004-6361/201323086.
  * Mathur et al. [2014] Mathur S, Salabert D, García RA, Ceillier T.  Photometric magnetic-activity metrics tested with the Sun: application to Kepler M dwarfs.  Journal of Space Weather and Space Climate 4 (2014) A15.  10.1051/swsc/2014011.
  * Salabert et al. [2016b] Salabert D, García RA, Beck PG, Egeland R, Pallé PL, Mathur S, et al.  Photospheric and chromospheric magnetic activity of seismic solar analogs. Observational inputs on the solar-stellar connection from Kepler and Hermes.  Astron. Astrophys. 596 (2016b) A31.  10.1051/0004-6361/201628583.
  * Salabert et al. [2017] Salabert D, García RA, Jiménez A, Bertello L, Corsaro E, Pallé PL.  Photospheric activity of the Sun with VIRGO and GOLF. Comparison with standard activity proxies.  Astron. Astrophys. 608 (2017) A87.  10.1051/0004-6361/201731560.
  * Ferreira Lopes et al. [2015] Ferreira Lopes CE, Leão IC, de Freitas DB, Canto Martins BL, Catelan M, De Medeiros JR.  Stellar cycles from photometric data: CoRoT stars.  Astron. Astrophys. 583 (2015) A134.  10.1051/0004-6361/201424900.
  * Montet et al. [2017] Montet BT, Tovar G, Foreman-Mackey D.  Long-term Photometric Variability in Kepler Full-frame Images: Magnetic Cycles of Sun-like Stars.  Astrophys. J. 851 (2017) 116.  10.3847/1538-4357/aa9e00.
  * Karoff et al. [2019] Karoff C, Metcalfe TS, Montet BT, Jannsen NE, Santos ARG, Nielsen MB, et al.  Sounding stellar cycles with Kepler - III. Comparative analysis of chromospheric, photometric, and asteroseismic variability.  Mon. Notices R. Astron. Soc. 485 (2019) 5096–5104.  10.1093/mnras/stz782.
  * Ball et al. [2018] Ball WH, Chaplin WJ, Schofield M, Miglio A, Bossini D, Davies GR, et al.  A Synthetic Sample of Short-cadence Solar-like Oscillators for TESS.  Astrophys. J. Suppl. Series 239 (2018) 34.  10.3847/1538-4365/aaedbc.
  * Ricker et al. [2015] Ricker GR, Winn JN, Vanderspek R, Latham DW, Bakos GÁ, Bean JL, et al.  Transiting Exoplanet Survey Satellite (TESS).  Journal of Astronomical Telescopes, Instruments, and Systems 1 (2015) 014003.  10.1117/1.JATIS.1.1.014003.
  * Girardi et al. [2005] Girardi L, Groenewegen MAT, Hatziminaoglou E, da Costa L.  Star counts in the Galaxy. Simulating from very deep to very shallow photometric surveys with the TRILEGAL code.  Astron. Astrophys. 436 (2005) 895–915.  10.1051/0004-6361:20042352.
  * Schofield et al. [2019] Schofield M, Chaplin WJ, Huber D, Campante TL, Davies GR, Miglio A, et al.  The Asteroseismic Target List for Solar-like Oscillators Observed in 2 minute Cadence with the Transiting Exoplanet Survey Satellite.  Astrophys. J. Suppl. Series 241 (2019) 12.  10.3847/1538-4365/ab04f5.
  * Kjeldsen and Bedding [1995] Kjeldsen H, Bedding TR.  Amplitudes of stellar oscillations: the implications for asteroseismology.  Astron. Astrophys. 293 (1995) 87–106.
  * Campante et al. [2016] Campante TL, Schofield M, Kuszlewicz JS, Bouma L, Chaplin WJ, Huber D, et al.  The Asteroseismic Potential of TESS: Exoplanet-host Stars.  Astrophys. J. 830 (2016) 138.  10.3847/0004-637X/830/2/138.
  * Chaplin et al. [2007a] Chaplin WJ, Elsworth Y, Houdek G, New R.  On prospects for sounding activity cycles of Sun-like stars with acoustic modes.  Mon. Notices R. Astron. Soc. 377 (2007a) 17–29.  10.1111/j.1365-2966.2007.11581.x.
  * Saar and Brandenburg [2002] Saar SH, Brandenburg A.  A new look at dynamo cycle amplitudes.  Astronomische Nachrichten 323 (2002) 357–360.  10.1002/1521-3994(200208)323:3/4¡357::AID-ASNA357¿3.0.CO;2-I.
  * Metcalfe et al. [2007] Metcalfe TS, Dziembowski WA, Judge PG, Snow M.  Asteroseismic signatures of stellar magnetic activity cycles.  Mon. Notices R. Astron. Soc. 379 (2007) L16–L20.  10.1111/j.1745-3933.2007.00325.x.
  * Noyes et al. [1984] Noyes RW, Hartmann LW, Baliunas SL, Duncan DK, Vaughan AH.  Rotation, convection, and magnetic activity in lower main-sequence stars.  Astrophys. J. 279 (1984) 763–777.  10.1086/161945.
  * Gilliland [1986] Gilliland RL.  Relation of Chromospheric Activity to Convection, Rotation, and Pre–Main-Sequence Evolution.  Astrophys. J. 300 (1986) 339.  10.1086/163807.
  * Landin et al. [2010] Landin NR, Mendes LTS, Vaz LPR.  Theoretical values of convective turnover times and Rossby numbers for solar-like, pre-main sequence stars.  Astron. Astrophys. 510 (2010) A46.  10.1051/0004-6361/200913015.
  * Skumanich [1972] Skumanich A.  Time Scales for CaII Emission Decay, Rotational Braking, and Lithium Depletion.  Astrophys. J. 171 (1972) 565–567.  10.1086/151310.
  * Palle et al. [1989] Palle PL, Regulo C, Roca Cortes T.  Solar cycle induced variations of the low L solar acoustic spectrum.  Astron. Astrophys. 224 (1989) 253–258.
  * Chaplin et al. [2007b] Chaplin WJ, Elsworth Y, Miller BA, Verner GA, New R.  Solar p-Mode Frequencies over Three Solar Cycles.  Astrophys. J. 659 (2007b) 1749–1760.  10.1086/512543.
  * Egeland [2018] Egeland R.  sunstardb: A Database for the Study of Stellar Magnetism and the Solar-stellar Connection.  Astrophys. J. Suppl. Series 236 (2018) 19.  10.3847/1538-4365/aab771.
  * Karoff et al. [2009] Karoff C, Metcalfe TS, Chaplin WJ, Elsworth Y, Kjeldsen H, Arentoft T, et al.  Sounding stellar cycles with Kepler - I. Strategy for selecting targets.  Mon. Notices R. Astron. Soc. 399 (2009) 914–923.  10.1111/j.1365-2966.2009.15323.x.
  * Chaplin et al. [2014] Chaplin WJ, Basu S, Huber D, Serenelli A, Casagrande L, Silva Aguirre V, et al.  Asteroseismic Fundamental Properties of Solar-type Stars Observed by the NASA Kepler Mission.  Astrophys. J. Suppl. Series 210 (2014) 1.  10.1088/0067-0049/210/1/1.
  * Guggenberger et al. [2016] Guggenberger E, Hekker S, Basu S, Bellinger E.  Significantly improving stellar mass and radius estimates: a new reference function for the $\Delta$$\nu$ scaling relation.  Mon. Notices R. Astron. Soc. 460 (2016) 4277–4281.  10.1093/mnras/stw1326.
  * Gaulme et al. [2016] Gaulme P, McKeever J, Jackiewicz J, Rawls ML, Corsaro E, Mosser B, et al.  Testing the Asteroseismic Scaling Relations for Red Giants with Eclipsing Binaries Observed by Kepler.  Astrophys. J. 832 (2016) 121.  10.3847/0004-637X/832/2/121.
  * Pérez Hernández et al. [2019] Pérez Hernández F, García RA, Mathur S, Santos ARG, Régulo C.  Influence of magnetic activity on the determination of stellar parameters through asteroseismology.  Frontiers in Astronomy and Space Sciences 6 (2019) 41.  10.3389/fspas.2019.00041.
  * Huber et al. [2014] Huber D, Silva Aguirre V, Matthews JM, Pinsonneault MH, Gaidos E, García RA, et al.  Revised Stellar Properties of Kepler Targets for the Quarter 1-16 Transit Detection Run.  Astrophys. J. Suppl. Series 211 (2014) 2.  10.1088/0067-0049/211/1/2.
  * Yu et al. [2018] Yu J, Huber D, Bedding TR, Stello D, Hon M, Murphy SJ, et al.  Asteroseismology of 16,000 Kepler Red Giants: Global Oscillation Parameters, Masses, and Radii.  Astrophys. J. Suppl. Series 236 (2018) 42.  10.3847/1538-4365/aaaf74.
  * Benomar et al. [2018] Benomar O, Bazot M, Nielsen MB, Gizon L, Sekii T, Takata M, et al.  Asteroseismic detection of latitudinal differential rotation in 13 Sun-like stars.  Science 361 (2018) 1231–1234.  10.1126/science.aao6571.
  * Karoff et al. [2018] Karoff C, Metcalfe TS, Santos ÂRG, Montet BT, Isaacson H, Witzke V, et al.  The Influence of Metallicity on Stellar Differential Rotation and Magnetic Activity.  Astrophys. J. 852 (2018) 46.  10.3847/1538-4357/aaa026.
  * Karoff et al. [2013] Karoff C, Metcalfe TS, Chaplin WJ, Frandsen S, Grundahl F, Kjeldsen H, et al.  Sounding stellar cycles with Kepler - II. Ground-based observations.  Mon. Notices R. Astron. Soc. 433 (2013) 3227–3238.  10.1093/mnras/stt964.
  * Reinhold et al. [2017] Reinhold T, Cameron RH, Gizon L.  Evidence for photometric activity cycles in 3203 Kepler stars.  Astron. Astrophys. 603 (2017) A52.  10.1051/0004-6361/201730599.
  * Rauer et al. [2014] Rauer H, Catala C, Aerts C, Appourchaux T, Benz W, Brandeker A, et al.  The PLATO 2.0 mission.  Experimental Astronomy 38 (2014) 249–330.  10.1007/s10686-014-9383-4.

##  Appendix A Additional plots and tables

Figure 7: Kiel diagram of the synthetic TESS sample of Ball et al. [41]. The
Sun is indicated by its usual symbol. Evolutionary tracks for stars with
masses from 0.8 to 2.0 M⊙ in steps of 0.2 M⊙ are overlaid as solid black
lines. Stars on the main-sequence are blue, stars after the Terminal Age Main-
Sequence (TAMS) are red. Stars for which the central hydrogen abundance
$Y_{c}$ is below $10^{-5}$ are here defined to have crossed the TAMS. Figure
8: As Figure 2 but for the full-cycle frequency shifts according to the
scaling relation of [47]. Figure 9: As Figure 1 but for the full-cycle
frequency shifts according to the scaling relation of [49] given in Equation
(3).

Table 2: Number of stars with $N$ consecutive sectors of data, number of
significant shift detections, and fraction of detections of the total number
of stars, for all stars in the sample, the main-sequence stars (region I), and
post-main-sequence stars (region II). Shifts measured with the CC and PB
methods for a two year TESS mission extension.
Sectors $N$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Total number | 7930 | 2601 | 781 | 306 | 9 | 0 | 0 | 40 | 144 | 105 | 114 | 86 | 615
CC Detections | 1101 | 595 | 232 | 98 | 3 | 0 | 0 | 12 | 53 | 38 | 44 | 32 | 141
CC Fraction [%] | 13.9 | 22.9 | 29.7 | 32.0 | 33.3 | 0.0 | 0.0 | 30.0 | 36.8 | 36.2 | 38.6 | 37.2 | 22.9
PB Detections | 1749 | 782 | 284 | 114 | 4 | 0 | 0 | 14 | 56 | 45 | 48 | 37 | 172
PB Fraction [%] | 22.1 | 30.1 | 36.4 | 37.3 | 44.4 | 0.0 | 0.0 | 35.0 | 38.9 | 42.9 | 42.1 | 43.0 | 28.0
Region I | 4251 | 1391 | 393 | 155 | 3 | 0 | 0 | 20 | 69 | 53 | 50 | 42 | 303
CC Detections region I | 57 | 17 | 8 | 2 | 0 | 0 | 0 | 0 | 2 | 3 | 0 | 1 | 4
CC Fraction region I [%] | 1.3 | 1.2 | 2.0 | 1.3 | 0.0 | 0.0 | 0.0 | 0.0 | 2.9 | 5.7 | 0.0 | 2.4 | 1.3
PB Detections region I | 287 | 104 | 30 | 8 | 0 | 0 | 0 | 1 | 2 | 7 | 2 | 3 | 14
PB Fraction region I [%] | 6.8 | 7.5 | 7.6 | 5.2 | 0.0 | 0.0 | 0.0 | 5.0 | 2.9 | 13.2 | 4.0 | 7.1 | 4.6
Region II | 3679 | 1210 | 388 | 151 | 6 | 0 | 0 | 20 | 75 | 52 | 64 | 44 | 312
CC Detections region II | 1044 | 578 | 224 | 96 | 3 | 0 | 0 | 12 | 51 | 35 | 44 | 31 | 137
CC Fraction region II [%] | 28.4 | 47.8 | 57.7 | 63.6 | 50.0 | 0.0 | 0.0 | 60.0 | 68.0 | 67.3 | 68.8 | 70.5 | 43.9
PB Detections region II | 1462 | 678 | 254 | 106 | 4 | 0 | 0 | 13 | 54 | 38 | 46 | 34 | 158
PB Fraction region II [%] | 39.7 | 56.0 | 65.5 | 70.2 | 66.7 | 0.0 | 0.0 | 65.0 | 72.0 | 73.1 | 71.9 | 77.3 | 50.6

Table 3: Number of stars with $N$ consecutive sectors of data, number of
significant shift detections, and fraction of detections of the total number
of stars, for all stars in the sample, the main-sequence stars (region I), and
post-main-sequence stars (region II). Shifts measured with the CC and PB
methods for a four year TESS mission extension.
Sectors $N$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Total number | 7930 | 2601 | 781 | 306 | 9 | 0 | 0 | 40 | 144 | 105 | 114 | 86 | 615
CC Detections | 1139 | 628 | 248 | 102 | 3 | 0 | 0 | 12 | 55 | 39 | 39 | 33 | 148
CC Fraction [%] | 14.4 | 24.1 | 31.8 | 33.3 | 33.3 | 0.0 | 0.0 | 30.0 | 38.2 | 37.1 | 34.2 | 38.4 | 24.1
PB Detections | 2094 | 874 | 311 | 126 | 4 | 0 | 0 | 14 | 65 | 45 | 47 | 40 | 184
PB Fraction [%] | 26.4 | 33.6 | 39.8 | 41.2 | 44.4 | 0.0 | 0.0 | 35.0 | 45.1 | 42.9 | 41.2 | 46.5 | 29.9
Region I | 4251 | 1391 | 393 | 155 | 3 | 0 | 0 | 20 | 69 | 53 | 50 | 42 | 303
CC Detections region I | 108 | 30 | 14 | 6 | 0 | 0 | 0 | 0 | 0 | 3 | 1 | 3 | 6
CC Fraction region I [%] | 2.5 | 2.2 | 3.6 | 3.9 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 5.7 | 2.0 | 7.1 | 2.0
PB Detections region I | 473 | 140 | 42 | 20 | 0 | 0 | 0 | 1 | 5 | 7 | 2 | 4 | 23
PB Fraction region I [%] | 11.1 | 10.1 | 10.7 | 12.9 | 0.0 | 0.0 | 0.0 | 5.0 | 7.2 | 13.2 | 4.0 | 9.5 | 7.6
Region II | 3679 | 1210 | 388 | 151 | 6 | 0 | 0 | 20 | 75 | 52 | 64 | 44 | 312
CC Detections region II | 1031 | 598 | 234 | 96 | 3 | 0 | 0 | 12 | 55 | 36 | 38 | 30 | 142
CC Fraction region II [%] | 28.0 | 49.4 | 60.3 | 63.6 | 50.0 | 0.0 | 0.0 | 60.0 | 73.3 | 69.2 | 59.4 | 68.2 | 45.5
PB Detections region II | 1621 | 734 | 269 | 106 | 4 | 0 | 0 | 13 | 60 | 38 | 45 | 36 | 161
PB Fraction region II [%] | 44.1 | 60.7 | 69.3 | 70.2 | 66.7 | 0.0 | 0.0 | 65.0 | 80.0 | 73.1 | 70.3 | 81.8 | 51.6

Table 4: Detectability of frequency shifts for different samples of stars. A
two year TESS mission extension is assumed.
Spectral type | Region | Number | CC Detections | CC Fraction [%] | PB Detections | PB Fraction [%]
---|---|---|---|---|---|---
F | I | 6676 | 62 | 0.9 | 416 | 6.2
F | II | 1642 | 6 | 0.4 | 49 | 3.0
G | I | 54 | 32 | 59.3 | 42 | 77.8
G | II | 1628 | 775 | 47.6 | 1120 | 68.8
K | II | 2731 | 1474 | 54.0 | 1678 | 61.4
FG | I | 6730 | 94 | 1.4 | 458 | 6.8
FGK | II | 6001 | 2255 | 37.6 | 2847 | 47.4
total |  | 12731 | 2349 | 18.5 | 3305 | 26.0