# The impact of tidal migration of hot Jupiters on the rotation of Sun-like
main-sequence stars

Shuai-Shuai Guo
Yunnan Observatories, Chinese Academy of Sciences, P.O. Box 110, Kunming
650011, People’s Republic of China
School of Astronomy and Space Science, University of Chinese Academy of
Sciences, Beijing, People’s Republic of China
Key Laboratory for the Structure and Evolution of Celestial Objects, CAS,
Kunming 650011, People’s Republic of China
<EMAIL_ADDRESS>
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alternative address)— _not_ for acknowledging funding agencies.

###### Abstract

The tidal interactions of planets affect the stellar evolutionary status and
the constraint of their physical parameters by gyrochronology. In this work,
we incorporate the tidal interaction and magnetic braking of the stellar wind
into MESA and calculate a large grid of 25000 models, covering planets with
masses of 0.1-13.0 $M_{\mathrm{J}}$ with different orbital distances that
orbit late-type stars of different metallicities. We also explore the effect
of different stellar initial rotations on the tidal interactions. Our results
show that in the case of tidal inward migration, the stellar rotation periods
are always lower than that of the star without planet before the planet is
engulfed and the difference in the rotation period of its host star always
increases with time. After the planet is engulfed, the stellar rotation
periods are still lower than that of star without planet, but the difference
of periods can be quickly eliminated if the star has a thick convective
envelope(smaller mass and larger metallicity), regardless of the mass of the
planet and the initial rotation period of the star. In the case of stars with
thinner convective envelopes(larger mass and smaller metallicity), the stars
will be spined up and remain the faster rotation in a long time. Meanwhile,
the planet is easily swallowed and the period differences are large if the
initial rotation period of its host star is higher. Final, we also study the
evolution of WASP-19 and estimate the range of tidal quality parameter
$Q^{\prime}_{*}=(4.6\pm 0.9)\times 10^{6}$ and initial semi-major axis as
$(0.035\pm 0.004)$ au.

_Keywords_ planet-star interactions – stars: rotation– stars: low-mass –
stars: metallicities – planetary engulfment event

##  1 Introduction

Up to now, more than 5000 exoplanets have been confirmed since the discovery
of the first exoplanet 51Pegasi b (Mayor and Queloz, 1995). The Kepler, CoRoT,
TESS and ground-based telescopes have enabled astronomers to have a deeper
insight into the exoplanet systems. A large number of close-in hot Jupiters (P
$<$ 10 days) around the low-mass stars have been found (Buchhave et al., 2012;
Fressin et al., 2013; Weiss et al., 2018; Zhou et al., 2019). According to the
current theory of planet formation, it is difficult to accumulate materials
required for planet formation due to such a short distance. These planets are
unlikely to form at the current location (Lin et al., 1996). Thus, in the
early phase of planet formation, hot Jupiter are thought to be migrated to
about 0.1 au due to the drag of the gas nebula in the protoplanetary disk (Lin
et al., 1996; Trilling et al., 1998; Ward, 1997; Papaloizou, 2007). After the
dissipation of protoplanetary disk, the tidal interactions between close-in
planets and their host stars become important, which results in two important
consequences. First, the eccentricities of the hot Jupiters tend to be low
(Winn and Fabrycky, 2015), and the interactions between star and planet also
change the orbital semi-major aixs $a$ (Jackson et al., 2008; Levrard et al.,
2009; Guo, 2010). Second, growing evidences suggest that hot Jupiters play an
important role in modifying the stellar rotation. For example, Alves et al.
(2010) first discovered that the rotations of stars with planets are faster
than that without planets. Recently, Tejada Arevalo et al. (2021) also found
that the host stars of hot Jupiters have faster rotations by analysing the
results of the California Kepler Survey (CKS; (Petigura et al., 2017)) and the
SWEET-Cat catalog (Santos et al., 2013).

The rotation of stars has been studied for more than a century, and such study
has been rapidly developed in recent decades. For stars with masses of 0.3
$M_{\odot}$ $\leq$ M $\leq$ 1.3 $M_{\odot}$, they are consisted of a central
radiative core and a convective envelope. When the mass of the star is less
than 0.3 $M_{\odot}$, the star is fully convective. The convective envelopes
of these low-mass stars contain surface magnetic fields that interact with the
stellar wind to extract angular momentum from the stars. As a result of the
loss of angular momentum, the rotation of the star decreases with time. This
is often called magnetic braking (Kraft, 1967; Skumanich, 1972; Kawaler,
1988). Many models have been developed (Matt et al., 2015; van Saders and
Pinsonneault, 2013; Garraffo et al., 2018), for instance, Amard and Matt
(2020) discussed the evolution of the rotation period of Solar-like stars with
different metallicities, and found that metal-poor stars tend to rotate
faster. As an application of magnetic braking, one can determine the age of
the star through the evolutionary characteristics of its rotation period
because the stellar rotation period will converge to a certain value with the
evolution owing to the magnetic braking. This method is named gyrochronology
(Barnes, 2003, 2007, 2010; Mamajek and Hillenbrand, 2008). The accuracy of the
age obtained by this method is strongly dependent on the completeness of the
stellar model. Although the rotation periods of some open clusters have been
measured ( Praesepe: 670 Myr (Douglas et al., 2017); the Hyades: 30 Myr
(Douglas et al., 2019); NGC 6811: 1.0 Gyr Curtis et al. (2019) and NGC 752:
1.4 Gyr (Agüeros et al., 2018) ), Curtis et al. (2019) found that for low-mass
stars the magnetic braking models tend to predict too much angular momentum
loss so that the inferred ages are likely to be incorrect. In fact, the
orbital decay of planet can rise the rotation of host star (Carone and
Pätzold, 2007). Brown (2014) compared the ages of the host stars by using the
isochrone and gyrochronology on a sample of 68 planet-hosting stars and found
that the ages obtained by gyrochronology are smaller than that obtained by
isochrone, which can be an indirect evidence that star-planet interaction
speeds up the stellar rotation. Furthermore, by including the tidal
interaction, Gallet and Delorme (2019) and Gallet (2020)corrected the ages of
gyrochronology for some stars with planet, which provides a standalone tool
based on tidal-chronology to estimate the age of a massive close-in planetary
system. In addition, the modification of the stellar rotation produced by the
planetary tides is also related to the metallicity of the star. Bolmont et al.
(2017) and Oetjens et al. (2020) show that for 1.0 $M_{\odot}$ stars,
planetary engulfment events are more likely to occur on metal-poor stars.
Thus, the faster rotation of metal-poor stars can be attributed to the
increase of the stellar angular momentum due to planetary engulfment. In fact,
the interaction between the planet and its host star is affected by many
factors. Therefore, our aim is to study the influence of those factors on the
stellar rotation. We calculate a large sample, in which we explore the
influences of free parameters ($M_{*}$, $[Fe/H]$, $M_{\mathrm{pl}}$,
$a_{\mathrm{ini}}$, $Q^{\prime}_{*}$) on the rotation of host star. In
addition, we also check the effect of stellar tides on the rotation before and
after engulfment for different star-planet systems in details.

In this paper, the structure is as follows. In Section 2, we describe the
model of star-planet interaction. In Section 3, we have discussed how
metallicity influences the evolution of the interaction model. In Section 4,
we discussed the influences of 6 parameters on the variation of rotation
periods of star with age and the evolution of WASP-19. In Section 5, we
discussed the limitations of our model. In Section 6, we make conclusions
about this work.

##  2 Star-planet interaction model in MESA

In this section, we study the exchange of angular momentum between the
planetary orbit and the host star’s rotation. The angular momentum of planet
itself is neglected because it is small compared to that of the former. In
fact, the timescale of the exchange of angular momentum is comparable with the
evolutionary timescale of the star. We thus implement the processes of the
exchange of angular momentum into the stellar evolution code MESA, version
11554. So far, MESA has been widely used in the field of stellar and planetary
evolution (Paxton et al., 2011, 2013, 2015, 2018, 2019). For the stellar
evolution, we consider the phase of main sequence for the stars with masses of
0.8-1.3 $M_{\odot}$. We adopt the protosolar abundances of Asplund et al.
(2009) as a reference for all metallicities, and here we use Table 1 of Amard
and Matt (2020) as the input for the metallicity in our model. The stellar
metallicity [Fe/H] is in the range of -0.5 to +0.5 dex. For stellar element
abundances, we use solar metallicities Z = 0.0134 as the benchmark. We set
0.6-16 days as the initial rotation periods of the star. In order to simulate
the change of the periods of the star and planetary orbit caused by the tidal
friction, we only consider massive planets with a mass range of 0.1-13.0
$M_{J}$. For planets with small orbital distances (a <0.1 au), the influence
of tidal force is prominent (Tejada Arevalo et al., 2021). Therefore, the
initial orbital distance is set to be 0.01-0.10 au. In calculating the tidal
dissipation rate (Goldreich, 1963), the tidal quality parameter
$Q^{\prime}_{*}$ of the star is a very important parameter. A smaller
$Q^{\prime}_{*}$ value means a larger tidal dissipation rate, and vice versa.
However, $Q^{\prime}_{*}$ is still an indeterminate quantity from both
theoretical and observational sides. Patra et al. (2020) used the transit-
timing method to detect the tidal quality parameter $Q^{\prime}_{*}$ for 12
close-in hot Jupiter systems. Their results show that the values of
$Q^{\prime}_{*}$ are larger than 105. Hamer and Schlaufman (2019) constrain
the value of $Q^{\prime}_{*}$ empirically and suggest that $Q^{\prime}_{*}$
varies from $10^{5}$ to $10^{9}$. Thus, we also explore a same range of the
tidal quality parameter $Q^{\prime}_{*}$ ($10^{5}$ \- $10^{9}$). We summarize
the values that we have adopted in Table 1.

###  2.1 Stellar rotation and wind braking model

We implement the magnetic braking model of Matt et al. (2015) into the
rotation scheme provided in MESA. We simulate the entire evolution of stars
from pre-main sequence(PMS) to end of the main sequence(MS). In the PMS phase,
the angular momentum is transferred between the stars and their protostellar
disks and the star maintains a constant rotation rate in the disk locking
time. In this paper, the disk locking time scale for all models is 13 Myr
(Gossage et al., 2021). During this period, there is no exchange of angular
momentum with the gas around the star. Thus, the stellar angular momentum is
assumed to be constant. After the disk is dissipated, the magnetic braking
results in the loss of the angular momentum in stars. We use magnetic braking
model of Matt et al. (2015) scheme as following:

$\frac{\mathrm{d}L_{wind}}{\mathrm{d}t}=-T_{0}(\frac{\tau_{\mathrm{cz}}}{\tau_{\mathrm{cz\odot}}})^{p}(\frac{\Omega_{*}}{\Omega_{\odot}})^{p+1}(unsaturated),$
(1)

$\frac{\mathrm{d}L_{\mathrm{wind}}}{\mathrm{d}t}=-T_{0}{\chi}^{p}(\frac{\Omega_{*}}{\Omega_{\odot}})(saturated),$
(2)

$T_{0}=K(\frac{R_{*}}{R_{\odot}})^{3.1}(\frac{M_{*}}{M_{\odot}})^{0.5}{\gamma}^{2m},$
(3)

$\gamma=\sqrt{1+(u/0.072)^{2}},$ (4)
$u=\frac{\Omega_{*}}{\Omega_{\mathrm{crit}}},\Omega_{\mathrm{crit}}=\sqrt{\frac{GM_{*}}{{R_{*}}^{3}}},$
(5)

where $L_{\mathrm{wind}}$ is the angular momentum lost by the magnetic stellar
wind, $R_{*}$, $\Omega_{*}$, and $\Omega_{\mathrm{crit}}$ are the stellar
radius, angular rotation rate, and critical angular rotation rate,
respectively. The constants $K$, $m$, $p$, and $\chi$ are free parameters. The
saturated and unsaturated states in Equations (1) and (2) represent two
different states of stellar magnetic activity, both of which have a strong
correlation with the Rossby number
$R_{\mathrm{o}}=\frac{2\pi}{\Omega_{\mathrm{crit}}\tau_{\mathrm{cz}}}$. The
convective turnover timescale $\tau_{\mathrm{cz}}$ can be expressed as
(Gossage et al., 2021):

$\tau_{\mathrm{cz}}(r)=\alpha_{\mathrm{MLT}}H_{\mathrm{P}}(r)/v_{\mathrm{c}}(r),$
(6)

where $H_{\mathrm{P}}(r)$ is the scale height, $v_{\mathrm{c}}(r)$ is the
convective velocity at radius $r$, and $\alpha_{\mathrm{MLT}}$ is the
convective mixing length. In Equation (6), $\tau_{\mathrm{cz}}$ is the
turnover timescale of the convective zone, which is defined in the location
where $r$ = $r_{\mathrm{BCZ}}$ \+ $0.5H_{\mathrm{P}}(r)$, and
$r_{\mathrm{BCZ}}$ is the radius of the bottom of the outer convection zone.
The values of $\alpha_{\mathrm{MLT}}$ is 1.82.

$R_{\mathrm{o}}$ $\leq$ $R_{\mathrm{osat}}$ is the saturation region, and vice
versa. The parameter $\chi$ = $R_{\mathrm{o}}$ /$R_{\mathrm{osat}}$ relies on
the critical value $R_{\mathrm{osat}}=0.14$ (Wright et al., 2018). The solar
$R_{\mathrm{o}}$ is around 2 (See et al., 2016). Thus we take $\chi$ = 14. The
constant $m$ is set to be 0.22. In order to reproduce the current rotation
period of the Sun, the values of $K$ and $p$ are 1.2 $\times$ $10^{30}$ and
2.6, respectively.

###  2.2 Orbital evolution and angular momentum transfer

For the sake of simplicity, we only consider the circular orbits, i.e. the
eccentricity is zero. We assume that the star is a spherically symmetrical and
uniformly rotating object, and the planet is a mass point. For an isolated
star-planet system, the total angular momentum is not conservative and can be
reduced by the magnetic wind. We have the following expression:

$\frac{\mathrm{d}L_{*}}{\mathrm{d}t}=\frac{\mathrm{d}L_{\mathrm{wind}}}{\mathrm{d}t}-\frac{\mathrm{d}L_{\mathrm{orb}}}{\mathrm{d}t}-\frac{\mathrm{d}L_{\mathrm{sp}}}{\mathrm{d}t},$
(7)

$L_{*}=I_{*}\Omega_{*},$ (8)
$L_{\mathrm{sp}}=I_{\mathrm{sp}}\Omega_{\mathrm{sp}},$ (9)
$L_{\mathrm{orb}}=\frac{M_{*}M_{\mathrm{pl}}}{M_{*}+M_{\mathrm{pl}}}na^{2},$
(10) $n=\sqrt{\frac{G(M_{*}+M_{\mathrm{pl}})}{{a}^{3}}},$ (11)

where $L_{*}$, $L_{\mathrm{sp}}$ and $L_{\mathrm{orb}}$ are the stellar,
planetary and orbital angular momentum, respectively. $I_{*}$ and
$I_{\mathrm{sp}}$ are the stellar and planetary moment of inertia. $n$ is the
orbital angular frequency. $a$ is the semi-major axis, and $M_{\mathrm{pl}}$
is the planetary mass. Since the rotational angular momentum of the planet is
negligible compared to its orbital angular momentum, we assume
$L_{\mathrm{sp}}$ = 0. Combining Equations (7)-(11), we obtain the expression
of the time derivative of the stellar angular rotation rate:

$\frac{\mathrm{d}\Omega_{*}}{\mathrm{d}t}=\frac{1}{I_{*}}(\frac{\mathrm{d}L_{\mathrm{wind}}}{\mathrm{d}t}-\Omega_{*}\frac{\mathrm{d}{I_{*}}}{\mathrm{d}t}-\frac{1}{2}M_{\mathrm{pl}}M_{*}\sqrt{\frac{G}{a(M_{*}+M_{\mathrm{pl}})}}\frac{\mathrm{d}a}{\mathrm{d}t}).$
(12)

From Equation (1) of Penev et al. (2012), we obtain the evolution of the semi-
major axis $a$ (Goldreich, 1963; Kaula, 1968; Jackson et al., 2008):

$\frac{\mathrm{d}a}{\mathrm{d}t}=sign(\Omega_{*}-n)\frac{9}{2}\sqrt{\frac{G}{aM_{*}}}(\frac{R_{*}}{a})^{5}\frac{M_{\mathrm{pl}}}{Q^{\prime}_{*}},$
(13)

where $sign(\Omega_{*}-n)$ takes the value of 1 when the star spins faster
than the planet and -1 when the reverse is true. We use the method of Oetjens
et al. (2020) to set the Roche limit at 1.44 times the radius of the star,
that is, when the semi-major axis $a$ = 1.44 $R_{*}$, the planet is destroyed.
However, it should be noted that we adopt the Roche limit as 1.44 times the
stellar radius, consistent with the approach of Oetjens et al. (2020). When
the planet reaches the Roche limit, the transfer of orbital angular momentum
to the star stops, and the remaining orbital angular momentum is removed from
the system. At this point, the rotational velocity of the star when the planet
is consumed represents a lower limit.

Table 1:  The parameters adopted in this work Parameter |  |  |  |  |  |  |
---|---|---|---|---|---|---|---
$M_{*}$ ($M_{\odot}$) | 0.8 | 0.9 | 1.0 | 1.1 | 1.2 | 1.3 | –
$[Fe/H]$ (dex) | -0.5 | -0.3 | 0.0 | +0.3 | +0.5 | – | –
$M_{\mathrm{p}}$ ($M_{\mathrm{J}}$) | 0.1 | 0.5 | 1.0 | 4.0 | 8.0 | 13.0 | –
$P_{\mathrm{rot,ini}}$ (days) | 0.6 | 3.0 | 8.0 | 16.0 | – | – | –
$Q_{*}$ | $10^{5}$ | $10^{6}$ | $10^{7}$ | $10^{8}$ | $10^{9}$ | – | –
$a_{\mathrm{ini}}$ (au) | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.10

##  3 How metallicity affect the evolution of star-planet systems

(a)

(b)

Figure 1: In panel (a), the top panel shows the relationship between the
evolution time and rotation period for individual stars and star-planet
systems. The middle panel shows the relationship between the evolution time
and planetary orbital period. The bottom panel shows the relationship between
the evolution time and orbital angular momentum $L_{\mathrm{orb}}$. Solid and
dashed lines represent stars with and without planets at different
metallicities, respectively. Here, the mass of the star is 1.0 $M_{\odot}$,
and the metallicity is represented by the red, blue, yellow, and green lines
for [Fe/H] = -1.0, -0.5, 0.0, and +0.5 dex, respectively. The planetary mass
is 1.0 $M_{\mathrm{J}}$, the initial semi-major axis is 0.03 au, the initial
rotation period of the star is 8.0 days, and the tidal quality parameter
$log_{{10}}\mathcal{Q^{\prime}}{{}_{*}}$ is 7. In panel (b), the top panel
shows the evolution of $\delta P$ as a function of time. The middle panel
shows the relationship between the evolution time and stellar spin angular
momentum $L_{*}$. The bottom panel shows the relationship between the
evolution time and magnetic braking angular momentum loss $\lvert
L_{\mathrm{wind}}\rvert$. Here, $\delta P$ represents the difference in
rotation periods of stars with and without planets, quantifying the influence
of the planet on stellar rotation.

In this section, we have extensively discussed the impact of different
metallicities on the tidal interactions between solar-mass stars and close-in
hot Jupiters. This analysis aims to provide a comprehensive understanding of
how various physical quantities evolve in our model. Figure 1 (a) top depicts
the influence of star-planet interactions on stellar rotation for stars with
different metallicities. These stars without planets show the increasing trend
for their rotation periods due to the magnetic braking. In fact, the
metallicity dramatically influences the spin down of the stars. The spin down
of metal-rich stars are much faster than that of metal-poor stars because of
their deeper convective zones. The evolutionary path of the rotation period is
affected when a planet with the mass of Jupiter is included. For metal-rich
stars with and without planets, the divergence of periods occurs at about 1.0
Gyr. In the case, the orbital decay of the close-in planets transfers angular
momentum to the stars so that the rotation of stars are accelerated until the
planets spiral down to the stars. Subsequently, the periods of the stars
increase again. In contrast, metal-poor stars with solar masses have a larger
radius and stronger tidal influences (Equation 13), which are more likely to
result in an early planetary engulfment, in the medium panel of Figure 1 (a),
stars with lower metallicities have faster planetary orbit decay and are more
likely to experience early planetary engulfment, as predicted by Oetjens et
al. (2020). In the bottom panel of Figure 1 (a), as planets migrate inward,
their orbital angular momentum is transferred to the star and gradually
decreases over time. Additionally, for larger radius stars, their Roche limit
is further from the star, so the transfer of orbital angular momentum to the
star is slightly less for larger radius stars. We also note that the
modification of the stellar rotation period by the planet is reduced with the
decrease of the metallicity. The difference of the rotation periods of the
stars with and without planets, $\delta P$, is given by:

$\delta P=P_{\mathrm{without}}-P_{\mathrm{with}},$ (14)

where $P_{\mathrm{without}}$ and $P_{\mathrm{with}}$ are the rotation periods
of the stars without and with a planet. Figure 1 (b) top shows the difference
of the rotation period. One can see that the planets have more effect on the
rotation of metal-rich stars than that of metal-poor stars. This is because
the metal-poor stars have thinner convective envelopes, which results in much
weaker magnetic braking and thus week effects of the planetary angular
momentum transfer on the stellar rotation. In the medium and bottom panels of
Figure 1 (b), we can see that for stars with planets, the rotational angular
momentum decreases rapidly for all stars except those with [Fe/H] = -1.0 dex
within 2.0 Gyr. This phenomenon is mainly due to two reasons: firstly, the
initial rotation speed of the star is faster, resulting in stronger magnetic
braking and more severe loss of angular momentum; secondly, in the early
stages, the planet is far from the star, and the tidal force is weak. Thus,
the angular momentum transferred to the star is smaller than the angular
momentum lost due to magnetic braking, resulting in a slower rotation of the
star. As the evolution progresses, the wind torque decreases while the tidal
torque increases until the tidal torque exceeds the wind torque at around 2.0
Gyr, causing an increase in the star’s rotation. After the planet is engulfed,
only magnetic braking can cause angular momentum loss, and tidal torque
disappears. The star with [Fe/H] = -1.0 dex has a very thin convective
envelope, which results in almost no loss of angular momentum. This system can
be approximately treated as conserving angular momentum, whereby the stellar
angular momentum increases due to the transfer of orbital angular momentum
before the planet is engulfed. After the planet is engulfed, the stellar
angular momentum remains constant.

##  4 The variation of stellar rotation periods of star with age

Our goal is to investigate the influence of tidal interaction on the rotation
of star and orbit of planet at different metallicities. Thus we compare the
evolution of rotation period of the star for the cases with and without a
planet. In the above model, there are six free parameters, namely, the initial
rotation period of the star, the initial orbital distance, the masses of the
star and planet, the metallicity, and tidal quality parameter
$Q^{\prime}_{*}$.

In this section, we will discuss how the stellar rotations are affected by the
tidal interaction. We define the difference of age $\delta Age$ as:

$\delta Age=Age_{\mathrm{now}}-Age_{\mathrm{planet-engulf}},$ (15)

where $Age_{\mathrm{now}}$ is the current age of the star. We choose 0.5, 1.0,
2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0 and 10.0 Gyr from the evolutionary
models as the comparison samples. $Age_{\mathrm{planet-engulf}}$ is the time
when the planet is engulfed. For each age interval, $\delta$Age = 0 is the
moment that the planet has just been engulfed. At the same time, all angular
momentum of planet is transferred to the host star. In addition, the stellar
$\delta$P reached the maximum value in the entire evolution stage (see Figure
1 (b) top) at this moment. The differences of stellar periods are showed in
Figures 2 and 3. The red dashed lines in each panel denote the engulfment
moment, and the left and right sides of the lines are in the evolution stage
before and after the planetary engulfment.

###  4.1 The effects of initial semimajor axis

Figure 2: The relationship between $\delta Age$ and $\delta P$ of a star with
a mass of 1.0 $M_{\odot}$, tidal quality parameter
$log_{{}_{10}}\mathcal{Q^{\prime}}_{{}_{*}}$ = 7, initial semi-major axis of
0.02, 0.05, 0.10 au, respectively. and planetary mass of 1.0, 8.0, 13.0
$M_{\mathrm{J}}$, respectively. The red dashed lines denote the time of
engulfment. The left and right points of the red dashed line means that the
planets are alive or engulfed. The circles, triangles and stars represent the
stars with metallicity [Fe/H] = -0.5, 0, +0.5 dex, respectively; the solid
lines, dashed lines, dot-dashed lines, and dotted lines respectively represent
the initial rotation period of the stars of 16.0, 8.0, 3.0, 0.6 days. Colorbar
represents the relative rate of change of stellar rotation period, i.e.
$\delta P[\%P_{\mathrm{rot,without}}]=\delta P/P_{\mathrm{without}}$.

First, it can be seen from Figure 2 that for the initial semi-major axis of
0.02 au, a planet with a mass larger than 1.0 MJ can be engulfed before 0.5
Gyr. Interestingly, $\delta P$ is almost independent of the mass of the
planet. After the engulfment ($\delta$ Age = 0 denotes the time of
engulfment.), $\delta P$ decreases quickly with the evolution owing to the
magnetic braking. In fact, the process of the magnetic braking is proportional
to the angular frequency of the stellar rotation. When the planet is swallowed
up, the angular frequency is enhanced rapidly. As a consequence, the stellar
rotation period drops faster if the mass of planet is lager. Therefore, the
difference of the $\delta P$ caused by different planetary masses is quickly
eliminated. Moreover, it is worth noting that for stars with an initial
rotation period of 16.0 days and a stellar metallicity of -0.5 dex, the spin-
up caused by planetary engulfment has a great influence during the whole main
sequence, with a period change rate always exceeding 20$\%$. In order to
better explain this phenomenon, we introduce panels (a) and (b) in Figure 5.
It can be seen from the Figure 5 that when the planets of 1.0 $M_{J}$, 8.0
$M_{J}$ and 13.0 $M_{J}$ are engulfed, the the angular frequency of stars with
different metallicities converges uniformly within 2.0 Gyr. The angular
frequency of the stellar rotation has a steep increase when the planet is
engulfed, but this increased angular momentum will have an uncontrolled loss.
The greater the mass of the planet, the more severe the uncontrolled loss.
$\delta P$ will soon be eliminated by the magnetic braking effect. At the same
time, comparing the different metallicities, it can be seen that the angular
frequency of rotation of metal-poor stars is more difficult to return to the
state of stars without planets in the main sequence stage compared to the
solar metallicity stars after planets is engulfed. This means that if you
consider the situation where stars are polluted by planets, due to the
uncertainty of the initial rotation period, the gyrochronology still requires
caution for metal-poor stars. For the case of an initial semi-major axis of
0.05 au, only the 13.0 $M_{\mathrm{J}}$ planet is engulfed during the main
sequence stage of the star. However, for the case of 0.10 au, due to the
sufficient distance which weakens tidal interactions, the orbital decay is
slow, and all planetary masses survive. We can clearly observe that when the
planet is engulfed and the age is fixed, a larger planetary mass corresponds
to a larger $\delta P$.

###  4.2 The effects of planetary mass

We can also see that for the case of an initial semi-major axis of 0.02 au, as
discussed in Section 4.1, due to the close proximity of the planets to the
host star, all planetary masses are engulfed within 1.0 Gyr. For semi-major
axis is 0.05 au, With the evolution of time, we find that $\delta P$ has an
overall rise. This is due to the transfer of angular momentum caused by the
inward migration of the planet to the star, and it can be seen from the
magnetic braking formula that under certain conditions, faster stellar
rotation will bring about greater angular momentum loss, but the increased
angular momentum of the star is always greater than the lost angular momentum,
which makes $\delta P$ always increased. And from the simulation results of
the planet with a mass of 13.0 $M_{J}$, it’s obvious that when the planet is
engulfed, $\delta P$ tends to peak, because at this time the planet transfers
all the angular momentum to the star. At the same time, the relationship
between metallicity and $\delta P$ also shows obvious regularity, that is, the
higher the metallicity, the greater the $\delta P$. However, there is little
correlation between $\delta P[\%P_{rot,without}]$ and the metallicity of
stars. Here we define $\delta P[\%P_{rot,without}]=\delta P/P_{without}$.
Except for metal-poor stars with slow initial rotation, different initial
rotation periods seem to have little effect on the stellar $\delta P$ and
$\delta P[\%P_{rot,without}]$. For the case where the initial semi-major axis
is 0.10 au, it is common for systems with larger planetary masses that the
influence of planets on the rotation of stars is greater. For the system with
a planetary mass of 1.0 $M_{J}$ until the end of the main sequence, $\delta P$
is only 0.08 days, which is almost negligible. In the case of more massive
planets, $\delta P$ can attain 10.0 days. However, the $\delta
P[\%P_{rot,without}]$ of most samples are smaller than 20%.

###  4.3 The effects of stellar mass

Figure 3: The relationship between $\delta Age$ and $\delta P$ of the stellar
mass of 0.9, 1.1, 1.3 $M_{\odot}$. Tidal quality parameter
$log_{{}_{10}}\mathcal{Q^{\prime}}_{{}_{*}}$ of 5, 7, 9, initial semi-major
axis of 0.05 au, and planetary mass of 1.0 $M_{\mathrm{J}}$. Other parameters
are the same as in Figure 2.

(a)$M_{*}$ = 1.0 $M_{\odot}$,$[Fe/H]$ = -0.5 dex,$a_{\mathrm{ini}}$ = 0.02 au

(b)$M_{*}$ = 1.0 $M_{\odot}$,$[Fe/H]$ = +0.5 dex,$a_{\mathrm{ini}}$ = 0.02 au

(c)$M_{*}$ = 1.3 $M_{\odot}$,$[Fe/H]$ = -0.5 dex,$a_{\mathrm{ini}}$ = 0.02 au

(d)$M_{*}$ = 1.3 $M_{\odot}$,$[Fe/H]$ = +0.5 dex,$a_{\mathrm{ini}}$ = 0.02 au

Figure 4: Evolution of the rotation of star as a function of time. The mass of
the star is 1.0 $M_{\odot}$ and 1.3 $M_{\odot}$, respectively. The metallicity
[Fe/H] is -0.5 dex and +0.5 dex, respectively. The initial semi-major axis is
0.02 au, and the mass of the planet is 1.0 $M_{\mathrm{J}}$. The tidal quality
parameter $log_{{}_{10}}\mathcal{Q^{\prime}}_{{}_{*}}$ = 7. The dashed line in
the figure is the model without planets, and the solid line is the model with
planets. Red, green, blue, and yellow indicate the initial rotation period of
the star is 0.6, 3.0, 8.0, and 16.0 days, respectively.

(a)$M_{*}$ = 1.0 $M_{\odot}$,$[Fe/H]$ = -0.5 dex,$a_{\mathrm{ini}}$ = 0.02 au

(b)$M_{*}$ = 1.0 $M_{\odot}$,$[Fe/H]$ = +0.5 dex,$a_{\mathrm{ini}}$ = 0.02 au

(c)$M_{*}$ = 1.0 $M_{\odot}$,$[Fe/H]$ = -0.5 dex,$a_{\mathrm{ini}}$ = 0.03 au

(d)$M_{*}$ = 1.0 $M_{\odot}$,$[Fe/H]$ = +0.5 dex,$a_{\mathrm{ini}}$ = 0.03 au

(e)$M_{*}$ = 1.3 $M_{\odot}$,$[Fe/H]$ = -0.5 dex,$a_{\mathrm{ini}}$ = 0.02 au

(f)$M_{*}$ = 1.3 $M_{\odot}$,$[Fe/H]$ = +0.5 dex,$a_{\mathrm{ini}}$ = 0.02 au

(g)$M_{*}$ = 1.3 $M_{\odot}$,$[Fe/H]$ = -0.5 dex,$a_{\mathrm{ini}}$ = 0.03 au

(h)$M_{*}$ = 1.3 $M_{\odot}$,$[Fe/H]$ = +0.5 dex,$a_{\mathrm{ini}}$ = 0.03 au

Figure 5: Evolution of the rotation of star as a function of time. The mass of
the star is 1.0 $M_{\odot}$ and 1.3 $M_{\odot}$, the metal abundance [Fe/H] is
-0.5 dex and +0.5 dex, respectively. The initial semi-major axis is 0.02 and
0.03 au, respectively. The tidal quality parameter
$log_{{}_{10}}\mathcal{Q^{\prime}}_{{}_{*}}$ = 7. And the initial rotation
period of the star is 16.0 days. The dotted lines are the models without
planets and the solid lines are the models with planets. Red, green, and blue
indicate the planets with masses of 1.0, 8.0, and 13.0 $M_{\mathrm{J}}$,
respectively.

(a)$M_{*}$ = 1.0 $M_{\odot}$,$[Fe/H]$ = -0.5 dex,$a_{\mathrm{ini}}$ = 0.02 au

(b)$M_{*}$ = 1.0 $M_{\odot}$,$[Fe/H]$ = +0.5 dex,$a_{\mathrm{ini}}$ = 0.02 au

(c)$M_{*}$ = 1.3 $M_{\odot}$,$[Fe/H]$ = -0.5 dex,$a_{\mathrm{ini}}$ = 0.02 au

(d)$M_{*}$ = 1.3 $M_{\odot}$,$[Fe/H]$ = +0.5 dex,$a_{\mathrm{ini}}$ = 0.02 au

Figure 6: Evolution of the rotation of star as a function of time. The mass of
the star is 1.0 $M_{\odot}$ and 1.3 $M_{\odot}$, respectively. The metallicity
[Fe/H] is -0.5 dex and +0.5 dex, respectively. The initial semi-major axis is
0.02 au, and the mass of the planet is 1.0 $M_{\mathrm{J}}$. The initial
rotation period of the star is 16.0 days. The dashed line in the figure is the
model without planets, and the solid line is the model with planets. Red,
green and blue indicate the tidal quality parameter
$log_{{}_{10}}\mathcal{Q^{\prime}}_{{}_{*}}$ is 5, 7 and 9, respectively.

We show the variations of stellar periods for the case of 1.0 MJ and initial
orbital distance of 0.05 au in Figure 3. The engulfment of planet only occurs
at $log_{{}_{10}}\mathcal{Q^{\prime}}_{{}_{*}}$ = 5. The panel (a) of Figure 3
shows that for the 0.9 $M_{\odot}$, the higher the metallicity, the greater
the $\delta P$. This is because the planets transfer angular momentum to
compensate the rapid loss of angular momentum for the stars with low
metallicities (thin convective envelopes). Thus, $\delta P$ is small. However,
for the stars with high metallicities (thick convective envelopes), the
angular momentum provided by the planets is insufficient. In this situation,
$\delta P$ is large. The effect of metallicity on the stellar periods shows
the opposite trend in high mass stars. As shown in panels (d) and (g), $\delta
P$ is large in the cases of low metallicity. In fact, the convective envelopes
of 1.1 and 1.3 $M_{\odot}$ stars are very thin in the case of low metallicity.
The angular momentum provided by the planet can be remaind after the planet is
engulfed. For the cases of high metallicity, those angular momentum will be
rapidly lost owing to the magnetic braking. Thus, $\delta P$ is smaller than
that of low metallicity. The influence of planets on the rotation period of
stars is slight ($P_{\mathrm{rot,ini}}$ $<$ 4.0 days) or even negligible
during the entire stellar main sequence for the cases of
$log_{{}_{10}}\mathcal{Q^{\prime}}_{{}_{*}}$ = 7 and
$log_{{}_{10}}\mathcal{Q^{\prime}}_{{}_{*}}$ = 9 because of the weak
interaction (The middle and right panels of Figure 3).

We compare star-planet systems with an initial stellar mass of 1.1 $M_{\odot}$
and 1.3 $M_{\odot}$. It was found that the size of $\delta P$ is different
from that of low-mass stars, but it is closely related to the initial stellar
rotation period. When the metallicity is smaller and the initial stellar
rotation period is larger, stars tend to have larger $\delta P$. And in
Figures 3 (d) and (g), there are two phenomena for 1.1 $M_{\odot}$ metal-poor
stars and 1.3 $M_{\odot}$ metal-poor and solar metal-rich stars: One is that
for systems with an initial rotation period of less than 3.0 days, the planets
have survived and $\delta P$ is reduced and negative, and the angular momentum
is transferred from the star to the planetary orbit. At this time, $\Omega$
$\geq$ n and the planet is always in tidal migration outward. Another
phenomenon is that after the planet is engulfed, $\delta P$ increases for the
initial stellar rotation period of 16.0 days and 8.0 days. This is opposite
for the stars with smaller masses or higher metallicities, because after the
planet is engulfed, the star only loses angular momentum by magnetic braking,
and there is no angular momentum transfer between the planet’s orbit and the
star. From Figures 4 (c) and (d), it can be seen that stars with a mass of 1.3
$M_{\odot}$ and a metallicity of -0.5 dex will not experience significant
angular momentum loss until the end of the main sequence under any initial
rotation period, and the increase in stellar rotation caused by the planet
being engulfed will also continue almost lossless until the end of the main
sequence. In Figures 4 (c), we do not have the portion corresponding to
$P_{rot}$ = 0.6 days. This is because for a star with a metallicity of [Fe/H]
= -0.5 dex and a mass of 1.3 $M_{\odot}$, the rapid rotation model with a
period of 0.6 days exceeds the critical rotational velocity of the star due to
its initial larger radius. Therefore, MESA is unable to generate such models.
In the case of stellar metallicity of +0.5 dex, except for the case where the
stellar initial rotation is too slow. When the planet is engulfed, during the
main sequence, for the stars with different initial periods, the influence of
planetary engulfment on the rotation of stars will still be eliminated. We can
summarize the following rules from the 1.3 $M_{\odot}$ star in Figure 5: (i)
Metal-rich stars have thicker convective envelopes, and for longer orbital
migrations, slower planetary engulfment will also bring about slower $\delta
P$ elimination. (ii) Since the convective envelopes of metal-poor stars are
almost negligible, magnetic braking will hardly cause angular momentum loss,
and the speed of planetary engulfment does not affect $\delta P$.

###  4.4 The effects of tidal quality parameter

In Section 4.3, we extensively discuss the evolution of $\delta P$ for
different stellar masses under the condition of
$log_{10}\mathcal{Q^{\prime}}_{{}_{*}}$ = 5. Throughout the main sequence
stage, planets survive under the cases of
$log{10}\mathcal{Q^{\prime}}_{{}_{*}}$ = 7 and
$log{10}\mathcal{Q^{\prime}}_{{}_{*}}$ = 9, with the influence of stellar-
planet interactions on the stellar rotation period being weak ($\delta P$ $<$
4.0 days) or even negligible (as shown in the middle and right panels of
Figure 3). Furthermore, although these two tidal dissipation factors have
different values for $\delta P$, their trends are almost identical. Finally,
we can see from Figure 6 that the tidal quality parameter $Q^{\prime}_{*}$
will have a great impact on the timescale of planetary migration, and with
$log_{{}_{10}}\mathcal{Q^{\prime}}_{{}_{*}}$ = 9, Jupiter-mass planets have
survived during the main sequence. Comparing the two different metallicity
situations when the mass of the star is 1.0 $M_{\odot}$, we can see that
metal-rich stars will still experience faster stellar rotation angle frequency
loss after the planet is swallowed and stellar rotation angle frequency
converge consistently during the main sequence. For stars with a mass of 1.3
$M_{\odot}$ and metallicity of -0.5 dex, the spin-up of the star caused by the
planet being swallowed will also continue almost lossless until the end of the
main sequence.

###  4.5 The evolution of WASP-19

In this section, we will use an interaction model to simulate the possible
evolutionary trajectory of WASP-19, the parameters of the WASP-19 we used are
shown in Table 2. We use the observed age, current stellar rotation period,
and current orbital semimajor axis to constrain the initial semimajor axis and
tidal quality parameter $Q^{\prime}_{*}$ of the system. We selected this star-
planet system for three main reasons: (i) The stellar mass of this system is
small and has a thick convective envelope. In our previous discussions, we
know that for low-mass metal-rich stars, magnetic braking produces stronger
angular momentum loss in the case of a thicker convective envelope. When the
evolution timescale is long, the difference in initial stellar rotation
periods will be quickly smoothed out over time. The uncertainty in the initial
rotation period will not have a significant impact on the evolution of this
system. (ii) The observation error of the stellar rotation period in this
system is small, and the interaction model can better constrain the
parameters. (iii) Recent studies on Wasp-19 have measured the orbital decay of
the system and calculated the $Q^{\prime}_{*}$ value (Patra et al., 2020;
Petrucci et al., 2020; Rosário et al., 2022), this is significant for our
results and provides valuable reference.

Table 2:  Parameters of the WASP-19 planetary system (Cortés-Zuleta et al., 2020) planet WASP-19 | star WASP-19
---|---
Mass:1.154 $M_{\mathrm{J}}$ | Mass:0.965 $M_{\odot}$
Semi major axis:0.01652 au | Age: 2.9-10.5 billion years
– | Metallicity([Fe/H]):0.04 dex
– | Rotation period:10.3-10.7 days

From Figure 7 (a), we can see that in the late stage when the planet is close
enough to the star, the angular momentum loss caused by the wind torque is
much less than the angular momentum gain caused by tidal torque. And the
star’s rotation rate increases. Therefore, the size of the current stellar
rotation period is more dependent on the tidal dissipation rate, and the
observation gives a small error in the stellar rotation period of WASP-19.
When the planet mass and orbital semimajor axis are fixed, the tidal quality
parameter $Q^{\prime}_{*}$ determines the tidal dissipation rate, which we can
more easily constrain. Under our model assumptions, $Q^{\prime}_{*}$ is a
constant, so given a certain $Q^{\prime}_{*}$, we also obtained the range of
initial orbital semimajor axis. In the end, we determined that
$Q^{\prime}_{*}=(4.6\pm 0.9)\times 10^{6}$ and the initial orbital semimajor
axis is $(0.035\pm 0.004)$ au. The results given by Petrucci et al. (2020) and
Rosário et al. (2022) are $Q^{\prime}_{*}>(1.23\pm 0.231)\times 10^{6}$ and
$Q^{\prime}_{*}>(1.26\pm 0.1)\times 10^{6}$ respectively, and they provided
the minimum limit of $Q^{\prime}_{*}$ for this system in their paper. Our
calculated results are within the possible range and thus have a certain
degree of credibility. For the rotation-age relationship of the star, we
demonstrate in Figure 7 (b) the impact of planets on the age estimation of the
star. From the figure, we can see that for the simulation results of the lower
age limit of observation, the age error caused by the gyrochronology method
begins to appear after 1.0 Gyr. For the simulation results of the upper age
limit of observation, the age error caused by the gyrochronology method begins
to appear after 3.0 Gyr. Furthermore, as the planet orbit decays, the age
error becomes larger. At the current orbital position, the age error reaches
its maximum, and without considering the existence of planets, the
gyrochronology method can estimate the age of the star as only a few hundred
million years. In extreme cases, such as the green dashed line in the figure,
where $\delta t_{\mathrm{gyro}}$ exceeds 10 Gyr, the gyrochronology method can
estimate the age of the star with a current age of 10.5 Gyr as only 220
million years.

(a)

(b)

Figure 7: Panel (a) depicts the evolution of the planetary orbital period and
stellar rotation period in the WASP-19 system. The black horizontal dotted
line represents the observed value of the current orbital period, and the red
rectangle represents the range of observed values for the current age and
rotation period of the star. The colored dashed and solid lines represent the
evolution curves of the planetary orbital period and stellar rotation period,
respectively. The parameters of the red curve are ($P_{\mathrm{rot,ini}}$ =
0.8 days, $Q^{\prime}_{*}=5.5\times 10^{6}$, $a_{\mathrm{ini}}$ = 0.039 au),
the parameters of the green curve are ($P_{\mathrm{rot,ini}}$ = 8
days,$Q^{\prime}_{*}=5.5\times 10^{6}$, $a_{\mathrm{ini}}$ = 0.039 au), the
parameters of the blue curve are ($P_{\mathrm{rot,ini}}$ = 0.8 days,
$Q^{\prime}_{*}=3.7\times 10^{6}$, $a_{\mathrm{ini}}$ = 0.031 au), and the
parameters of the yellow curve are ($P_{\mathrm{rot,ini}}$ = 8
days,$Q^{\prime}_{*}=3.7\times 10^{6}$, $a_{\mathrm{ini}}$ = 0.031 au). It can
be clearly seen from the figure that the differences caused by the initial
stellar rotation period can be almost eliminated after 1.0 Gyr, and the
insensitivity of the initial stellar rotation period can more accurately
predict the initial semi-major axis of the orbit and $Q^{\prime}_{*}$ of this
star from a theoretical perspective. Panel (b) depicts the evolution of
$\delta t_{\mathrm{gyro}}$ in the WASP-19 system. Here, $\delta
t_{\mathrm{gyro}}=t_{\mathrm{without}}-t_{\mathrm{with}}$, where
$t_{\mathrm{without}}$ represents the age of the star obtained using the
gyrochronology method without taking into account the presence of the planet,
and $t_{\mathrm{with}}$ represents the age obtained with the planet. The
dashed lines show the evolutionary trajectory of the star’s age and $\delta
t_{\mathrm{gyro}}$ under different initial parameter conditions.

##  5 Discussion

Our results show the influence of different metallicities on stellar rotation
periods and planetary engulfment timescales under star-planet interactions.

At present, the loss mechanism of stellar angular momentum has various braking
models and reproduces the rotation distribution of some star clusters very
well. However, with the emergence of more and more observational evidences,
some current braking models still have some limitations. For example, Gossage
et al. (2021) implemented the braking model of Matt et al. (2015) and Garraffo
et al. (2018) in MESA for the first time and found that the rotation period of
stars with an age $\geqslant$ 1.0 Gyr is often overestimated. Curtis et al.
(2019) believed that at 1.0 Gyr, there is a stagnation phase of angular
momentum loss for stars with masses $\leq$ 1.0 $M_{\odot}$ and found that
gyrochronology models tend to predict too much angular momentum loss. It may
underestimate the timing of the impact on the stellar rotation period after a
planetary engulfment event.

Our model employs parameterized tidal quality parameter, which may not always
be accurate. The advantage of using parameterized tidal dissipation factors is
that we can systematically analyze the correlations between the various
parameters and tidal quality parameter. Recently, some works tend to use a
self-consistent modeled tidal quality parameter $Q^{\prime}_{*}$ (Bolmont et
al., 2017; Benbakoura et al., 2019; Lazovik, 2021). However, modeled tidal
quality parameter also bring a problem, as the results of Lazovik (2021)
indicate that the efficiency of dynamical tidal dissipation is stronger in
metal-rich stars. This does not support the viewpoint proposed by Lazovik
(2021) that the trend of hot Jupiter occurrence rate with metallicity
variation is due to the fact that metal-poor stars more frequently devour
planets than metal-rich ones. Lazovik (2021) points out in the paper that this
difference can be explained by different techniques of calculating tidal
quality parameter. The impact of self-consistent modeled tidal quality
parameter $Q^{\prime}_{*}$ goes beyond the scope of our article, and we will
discuss it more deeply in our next work.

##  6 Conclusions

We implement the magnetic wind braking model of Matt et al. (2015) into MESA
and further incorporate star-planet tidal interaction. To do this, we focus on
the main sequence stars with the spectral types of F, G and K. We calculate a
large samples and focus on the effect of the tidal interaction on the stellar
spin variation $\delta P$. Our simulations show that for any star-planet
system with orbital decay, $\delta P$ always increases with evolution time
before the planet is engulfed. We find that $\delta P$ is weakly affected by
the initial rotation period of the stars for less massive stars with higher
metallicities, but is greater for stars with higher masses and lower
metallicities. Such behaviors are also appear in the stars with higher initial
rotation periods. For a star with a smaller mass and a higher metallicity, the
changes in the rotation period of the star caused by the engulf of planet will
be eliminated rapidly after a planetary is engulfed. On the contrary, stars
with higher masses and lower metallicities tend to produce more prominant
period changes for larger initial stellar rotation period and planetary
masses, and the impact of planetary engulfment lasts longer.

We also found that the convective envelope is thin enough that the effects of
the initial rotation of the star and the mass of the planet are large and not
negligible, and vice versa. When the planet is engulfed later, the transfer of
angular momentum is mild, and the loss of angular momentum by the stellar
magnetic braking is also relatively mild, and the influence of the planetary
engulfment is eliminated more slowly. For stars with thin convective
envelopes, the early angular momentum loss of fast initial rotation stars is
severe and the change of the rotation period of the stars is easy to be
eliminated. However, the slower initial rotation of the star has weaker
magnetic braking, which tends to produce larger changes in the rotation period
of the star.

Finally we also simulated star-planet systems: WASP-19. For WASP-19 system, it
is a G-type star with a relatively thick convection zone. And system evolution
is insensitive to different initial stellar rotation periods. We estimated
system’s tidal quality parameter $Q^{\prime}_{*}=(4.6\pm 0.9)\times 10^{6}$
and initial orbital semi-major axis as $(0.035\pm 0.004)$ au. We also
discussed how the close-in hot Jupiter affects the star in WASP-19 and how
much error the gyrochronology method may produce. We found that in extreme
cases, the actual age of a star over 10 billion years old may be estimated as
less than 220 million years, which could result in a very old star being
incorrectly estimated as very young. Therefore, for systems with close-in hot
Jupiters, caution is needed when using the gyrochronology method.

##  Appendix A The rotation period of the Sun

In this section, we first test our model by comparing it with the rotation
period of the Sun. In this situation, the tide produced by the planet is
ignored. Figure 8 shows the rotation period as a function of evolution time
for the stars with solar mass and metallicity for different initial rotation
periods. It is clear that, as the stars evolve, the stellar rotation periods
gradually converge to 25.4 days regardless of the initial period. When the age
of the star exceeds 1.0 Gyr, such a behavior shows the possibility of applying
gyrochronology to estimate the ages of stars if the stellar rotation periods
are measured accurately. For isolated stars of F,G, K and M spectral types,
the method of gyrochronology can estimate an age with an accuracy of 10
percent, which is much better than that estimated by the isochrone line with
an average error of 50$\%$. (Mamajek and Hillenbrand, 2008; Collier Cameron et
al., 2009; Delorme et al., 2011). Figure 8 also shows the credibility of our
model.

Figure 8: Evolution of the rotation period as a function of time for the model
of solar mass and metallicity. The initial rotation period of each model are
0.6, 3, 8, 16 days respectively. The pentagram sign represents the current
position of the Sun.

.

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