Magnetic field of gas giant exoplanets and its influence on the retention of their exomoons

We study the magnetic and tidal interactions of a gas-giant exoplanet with its host star and with its exomoons, and focus on their retention. We briefly revisit the scaling law for planetary dynamo in terms of its mass, radius and luminosity. Based on the virial theorem, we construct an evolution law for planetary magnetic field and find that its initial entropy is important for the field evolution of a high-mass planet. We estimate the magnetic torques on orbit arising from the star-planet and planet-moon magnetic interactions, and find that it can compensate tidal torques and bypass frequency valleys where dynamical-tide response is ineffective. For exomoon's retention we consider two situations. In the presence of a circumplanetary disk (CPD), by comparison between CPD's inner and outer radii, we find that planets with too strong magnetic fields or too small distance from its host star tend not to host exomoons. During the subsequent CPD-free evolution, we find, by comparison between planet's spindown and moon's migration timescales, that hot Jupiters with periods of several days are unlikely to retain large exomoons, albeit they could be surrounded by rings from the debris of tidally disrupted moons. In contrast, moons, if formed around warm or cold Jupiters, can be preserved. Finally, we estimate the radio power and flux density due to the star-planet and planet-moon magnetic interactions and give the upper limit of detection distance by FAST.


INTRODUCTION
Planetary or stellar magnetic fields influence the structure of surrounding circumstellar and circumplanetary disks during their infancy, their rates of mass and angular momentum loss throughout their life span, and their dynamical interaction with their companions.These fields are generated by the dynamo action in their convective regions.Thermal convection drives the motion of conducting fluid which interacts with the magnetic field lines to induce new lines to offset magnetic diffusion, i.e. the dynamo action.The dynamo model was initially proposed by Larmor (1919) to interpret the solar magnetic fields.This model was later applied to geomagnetism by Bullard & Gellman (1954).The dynamo theory, especially how turbulent motion generates magnetic fields, was discussed in detail by Moffatt (1978) and Krause & Rädler (1980).These theoretical constructs were followed by several numerical simulations of kinematic dynamos (Gubbins 1972;Galloway & Corresponding author: Xing Wei xingwei@bnu.edu.cnProctor 1992; Wei et al. 2012;Wei 2014).In these kinematic dynamo simulations, the growth of magnetic field is computed for a given (assumed) fluid velocity in the absence of the back reaction of magnetic field on fluid motion.The first three-dimensional (3D) self-sustained dynamos were numerically simulated by Glatzmaier & Roberts (1995).This simulation includes the magnetic feedback on the flow.Subsequently, several follow-up 3D numerical dynamo simulations were carried out (Rotvig & Jones 2002;Christensen & Aubert 2006;Wei 2018).
Due to the enormous computational challenges in the range of spacial and temporal scales, it is not possible to carry out simulations with appropriate magnitude for the relevant physical quantities.In all the simulations, some gross approximations have to be introduced.They are usually carried out with some manageable, but unrealistic, adopted model parameters.For example, the real Ekman number (ratio of rotational timescale to viscous timescale) in the Earth's core is at the order of 10 −15 .But, the smallest Ekman number that can be achieved in the most recent geodynamo simulations is at the order of 10 −7 .For Jovian dynamo, in addition to Jupiter's fast rotation and small viscosity, the electric conductivity in its interior varies over orders of magnitude.All of these properties add to the technical difficulties for the numerical studies of planetary dynamo.In order to link the results obtained with artificial numerical models to actual planetary contexts, it is necessary to find a scaling law (Christensen & Aubert 2006;Jones 2014).Such extrapolation has been applied to the planets in the present-day solar system (Christensen et al. 2009;Davidson 2013) and the evolution of exoplanets (Reiners & Christensen 2010).A summary of some scaling laws can be found in Christensen (2010).On the other hand, the internal structure of planets, especially gas giant planets, has evolved considerably since the epoch of their formation, and the initial entropy is found to have an impact on the structure evolution of a highmass planet (Marley et al. 2007), which might influence the field evolution.In this paper we derive a scaling law and an evolution law of planetary magnetic field in a more rigorous way and then apply it to young planets.
The star-planet or planet-moon magnetic interaction plays an important role in the orbital dynamics, especially at the young age when stellar and planetary fields are strong (Zarka 2007;Laine et al. 2008;Strugarek et al. 2017).With the estimated magnetic fields, we can interpret moon's orbital migration in a circumplanetary disk and find the possibility for the retention of exomoons around the different host planets.We can also estimate the radio emissions due to the star-planet or planet-moon magnetic interaction.
The paper is structured as follows.In §2 we derive the scaling law for planetary dynamo ( §2.1), find the evolution law and estimate the magnetic field of young gas giant planets with different mass( §2.2).In §3 we discuss the various magnetic interactions in a star-planet or planet-moon system ( §3.1), summarize all the magnetic and tidal torques ( §3.2), and compare their relative strength ( §3.3).In §4 we apply the tidal and magnetic torques to the retention of exomoons in a close system by comparison between the inner and outer radii of circumplanetary disk ( §4.1) and between planet's spindown and moon's migration timescales ( §4.2).In §5 we estimate the radio power and flux density due to star-planet and planet-moon magnetic interactions and give the upper limit of detection distance by FAST.In §6 some discussions are given.

MAGNETIC FIELD OF A GAS GIANT PLANET
In this section we discuss the planetary dynamo and planetary field evolution.

Planetary field dynamo
In this subsection we estimate the planetary magnetic field in terms of mass, radius and luminosity.Here we briefly summarize the results and the details can be found in Appendix A. In the first place, we derive the total energy equation to find the power balance, i.e., buoyancy power ∼ ohmic dissipation rate.Next by the two-scale analysis on the length scale of magnetic field, we derive the energy equipartition, i.e., magnetic energy ∼ buoyancy energy.According to the standard mixing length theory, buoyancy energy ∼ kinetic energy.Eventually, we estimate the convective velocity and mixing length by the Lane-Emden equation and find the magnetic energy as a function of mass, radius and luminosity ⟨B 2 /µ⟩ ≈ 0.115M 1/3 R −7/3 L 2/3 . (1) Magnetic energy arises from thermal convection on short timescale and thermal convection arises from gravitational contraction on long timescale.Gravitational contraction leads to luminosity (see the next section).Equation (1) reveals the relation between magnetic energy and luminosity.This estimation yields Jupiter's volume-averaged field ⟨B⟩ ≈ 70 gauss, where Jupiter's intrinsic luminosity is 4.60 × 10 24 erg/s (Li et al. 2018).The dynamo simulations suggest that ⟨B⟩ ≈ 7B dip where B dip is dipole field at surface of dynamo region (Christensen et al. 2009).Therefore, our estimation is in good agreement with Jupiter's present dipole field ≈ 10 gauss.The estimation ( 1) is what we will use in the next section.

Planetary field evolution
In this subsection we study the evolution of magnetic field of gas giant planets with different mass.We apply the virial theorem to the evolution of gas giant planet.For a polytropic gas, the relation of gravitational potential energy and internal energy is E g = −(3/n)E i (Chandrasekhar 1939).With a non-uniform density profile, the gravitational potential energy reads In a planet in the absence of nuclear reaction, its luminosity is the rate of total energy Then we simply assume that the evolution of planet radius follows a power law R ∝ t −γ , and inserting it into the luminosity expression we readily obtain L ∝ M 2 t γ−1 .Usually γ is a small number, e.g., for 1 M J γ ≈ 0.044 fitted from the numerical results in Marley et al. (2007), and thus L ∝ M 2 t −1 which is in good agreement with the numerical simulations (Burrows et al. 1997).
Inserting the evolution of radius and of luminosity into the estimation (1), we obtain the evolution of magnetic field ⟨B⟩ ∝ M 5/6 t (3/2)γ−1/3 ∝ M 5/6 t −0.267 (2) where γ ≈ 0.044 of a low-mass planet is used.Nowadays Jupiter's surface dipole field at 4.6 Gyr is about 10 gauss at poles.We then use the evolution law (2) to extrapolate Jupiter's surface dipole field at the two young ages, one at 2 Myr when the protoplanetary disk has a significant impact on dynamics and the other at 10 Myr when the protoplanetary disk almost disappears.The results are B dip ≈ 80 gauss at 2 Myr and B dip ≈ 50 gauss at 10 Myr.
Our evolution law shows that magnetic field depends on mass.Moreover, different mass leads to different tracks of planetary evolution (Marley et al. 2007;Fortney & Nettelmann 2009), and consequently different field strength.On the other hand, although the initial condition for planetary evolution is eventually forgotten, the planetary evolution at young age greatly depends on the initial condition, namely whether a hot start with a high entropy or a cold start with a low entropy due to the different core-accretion formation models (in the former heat release by accretion shock is transferred to planet interior whereas in the latter heat is radiated away).
For 1 M J the two initial conditions lead to almost the same evolution tracks.But for higher mass the evolution tracks with a hot versus cold start differ more.Therefore, the dipole field and dipole moment of young planets are primarily determined by mass and the initial condition for evolution.Since the initial entropy does not influence the evolution track of a low-mass planet, we use 1 M J as reference to estimate magnetic field of 1 M J at different age, and then use (1) with the numerical results in Marley et al. (2007) to estimate magnetic field of high-mass planets.
Table 1 shows the magnetic field of planets with 1 M J , 4 M J and 10 M J at 2 Myr, 10 Myr and 5 Gyr for the two initial conditions (we do not need to list the two initial conditions for 1 M J because they yield the same results).Clearly, a larger mass and the hot start correspond to stronger magnetic field.At 2 Myr, a 10 M J planet has a magnetic field as high as ∼400 gauss with the hot start.But with the cold start, it is ∼100 gauss.Moreover, magnetic field decays much sharper with the hot start than with the cold start.Reiners & Christensen (2010) uses the evolution model with the hot start results in (Burrows et al. 1997) to estimate dipole field and their results are comparable to ours.

TIDAL AND MAGNETIC INTERACTIONS IN STAR-PLANET AND PLANET-MOON SYSTEMS
In this section we discuss the various magnetic interactions and torques, and then list all the tidal and magnetic torques in the star-planet and planet-moon systems.

Star-planet or planet-moon magnetic interactions
In this subsection we estimate the magnetic torque on orbit arising from star-planet or planet-moon magnetic interaction.The star-planet magnetic interaction consists of three types (Zarka 2007): the interaction of unmagnetized stellar wind and planetary field, the interaction of stellar dipole field and planetary dipole field (i.e., the dipole-dipole interaction) (Strugarek et al. 2015;Strugarek 2016;Strugarek et al. 2017) and the interaction of stellar field and orbital motion of unmagnetized planet (i.e., the unipolar induction) (Goldreich & Lynden-Bell 1969;Laine et al. 2008;Laine & Lin 2012).In addition to these three types, the interaction of the misaligned field and rotation of host star also leads to the time-dependent magnetic interaction with its planet (Laine et al. 2008), which is relatively weak and will not be studied in this paper.The planet-moon magnetic interaction consists of two types: the dipole-dipole interaction (e.g., Jupiter-Ganymede) and the unipolar interaction (e.g., Jupiter-Io).

Star-planet dipole-dipole interaction
We now consider the star-planet magnetic torque for the first two interactions, namely the interaction of unmagnetized stellar wind and planetary field and the dipole-dipole interaction.We use a unified empirical formula for the both magnetic torques (Strugarek et al. 2015), Γ ≈ C d • a • A obs • P t , where C d is the drag coefficient, a is the orbital radius, A obs = πR 2 obs is the obstacle area, and P t is the stellar total pressure at the obstacle radius R obs .The drag coefficient is expressed as a (Zarka 2007), where M a is the Alfvén Mach number M a = v/v a .Here v is the relative velocity v = |ω p − Ω ⋆ |a where ω p is planet's orbital frequency and Ω ⋆ star's rotation frequency, and v a = B ⋆ (R ⋆ /a) 3 / µm p N is the Alfvén velocity where N is the plasma number density, m p the proton mass and µ the magnetic permeability.
For an illustrative estimate, we take the typical values for a close-in star-planet system, i.e., a=0.05 au, M ⋆ = M ⊙ and the rotation period (2π/Ω ⋆ ) of a young star about 3 days.We then readily obtain v = |ω p − Ω ⋆ |a ≈ 50 km/s.We take stellar dipole field B ⋆ ≈ 1500 gauss (Yu et al. 2017) and R ⋆ ≈ 2R ⊙ .The mass loss rate Ṁ⋆ = 4πa 2 N m p v evolves roughly as t −2 (Wood et al. 2002(Wood et al. , 2005)), e.g., the solar mass loss rate at present is ∼ 10 12 g/s.We assume that the wind velocity keeps about 400 km/s (this assumption is valid in Parker's model) so that density also evolves as t −2 and scales with distance as a −2 .The density can be estimated as N = 5(t/4.6Gyr)−2 (a p /1au) −2 cm −3 .
(3) At present N ≈ 5 cm −3 at 1 au, and density can be estimated from Ṁ⋆ scaled to an young age, say, 5 Myr, N ≈ 1.7 × 10 9 cm −3 at 0.05 au.As a result v a ≈ 500 km/s, M a ≈ 0.1 and C d ≈ 0.1.The obstacle radius is determined by the balance between P t and the planetary magnetic pressure, P t ≈ B p (R p /R obs ) 3 2 /µ.P t is the sum of the stellarwind thermal pressure, the stellar-wind kinetic pressure P k = N m p v 2 and the stellar magnetic pressure P m = B ⋆ (R ⋆ /a) 3 2 /µ.In the first interaction, namely the interaction of unmagnetized stellar wind and planetary field, P m ≈ 0, such that we choose P t ≈ P k .In the second interaction, namely the dipoledipole interaction, the numerical result (Strugarek 2016) shows magnetic pressure dominates over kinetic pressure, such that we choose P t ≈ P m .At young age of a star-planet system, stellar field ≈ 1500 gauss and planetary field ≈ 100 gauss as estimated in the last subsection, and thus P m ≫ P k , namely the dipoledipole interaction dominates over stellar-wind-planetfield interaction.For the dipole-dipole interaction, by B ⋆ (R ⋆ /a) 3 2 /µ ≈ B p (R p /R obs ) 3 2 /µ to derive R obs ≈ R p (B p /B ⋆ ) 1/3 (a/R ⋆ ).We insert the typical values: a = 0.05 au, R ⋆ = 2R ⊙ , B ⋆ ≈ 1500 gauss and B p ≈ 100 gauss, and then obtain the obstacle radius R obs ≈ 2.2R p .Consequently, we derive the magnetic torque due to the dipole-dipole interaction Here the subscript ⋆ denotes star (primary) and p planet (secondary), and the superscript m magnetic torque.

Star-planet unipolar interaction
We next consider the star-planet magnetic torque in the third interaction, namely the unipolar induction.The electric field induced by the planet's orbital motion on the stellar field is E = vB ⋆ (R ⋆ /a) 3 where v is the relative velocity as before, the voltage imposed on the planet is V = E • 2R obs = 2vB ⋆ (R ⋆ /a) 3 R obs , and then the electric current is I = 2vB ⋆ (R ⋆ /a) 3 R obs /Λ where Λ is the resistance.The Lorentz force can be calculated as I • B ⋆ (R ⋆ /a) 3 • 2R obs , and by R obs ≈ R p (B p /B ⋆ ) 1/3 (a/R ⋆ ) the magnetic torque due to unipo-lar induction reads (5) The torque Γ m ⋆p in both Eqs (4) and ( 5) is in the direction of the planet's motion relative to the star.In the case that the stellar rotation is in the same direction as the orbital motion, outside the corotation radius magnetic torque induces an outward orbital migration, whereas inside the corotation radius it induces an inward migration.In the case that the stellar rotation and orbital motion are opposite to each other, no matter what the direction of relative velocity is, magnetic torque always retards orbital motion and induces an inward migration.Consequently, magnetic torque and tidal torque are always in the same direction with respect to orbital migration.
In the star-planet unipolar interaction, the resistance is higher at the field footpoint of planetary plasma envelope than planetary interior (Laine & Lin 2012).According to the Alfvén wing model (Neubauer 1980;Zarka 2007;Strugarek 2016), the resistance of Alfvén wing at magnetosphere reads Λ 1 ≈ µv 1 + M 2 a /M a ≈ µv/C d where the expression of a is employed (de Colle et al in preparation).Alternatively, the upper limit of winding angle ∼ 1 also yields a comparable resistance Λ 1 ≈ 0.25v/(10 5 m/s) ohm (Lai 2012).When the star-planet system approaches a synchronous state, i.e., v → 0, Λ 1 tends to vanish and hence the circuit will be established through the planet's atmosphere instead of through Alfvén (Laine & Lin 2012).The resistance of planet's atmosphere arising from Na + and K + is about Λ 2 ≈ 0.1 ohm (Laine et al. 2008;French et al. 2012).We can write Λ = Λ 1 + Λ 2 to take into account both the synchronous and asynchronous states.
At an asynchronous state, inserting Λ ≈ µv/C d into (Eq.5) we obtain the torque magnitude Γ obs • P t under the circumstance P t ≈ P m .The reason is that the dipole-dipole interaction is caused by magnetic reconnection and the Alfvén wing in unipolar interaction is also induced by the same physical mechanism.This is also shown in Zarka (2007) that the two magnetic dissipations are comparable to each other.

Planet-moon and star-moon unipolar interactions
In the planet-moon interaction, when the moon is unmagnetized, e.g., Jupiter-Io, R obs for the star-planet unipolar induction (5) should be replaced with the physical size of exomoon, i.e., moon's radius R m .The resistance Λ p,m = 1/(σ p,m R p,m ) arises from a series circuit through the moon itself and across the footprint of the planetary field.Magnitude of moon's intrinsic conductivity σ m depends on its composition and structure, which varies over many order of magnitude between volatile ices, condensed or molten silicates, and iron.Many known super Earths have density comparable to that of the Earth and surface temperature above the melting point of silicates.If the surface of exomoons around hot Jupiters (with orbits similar to these closein super Earths) is heated to the same temperature and is covered with magma ocean, their conductivity may be very large.But around warm and cold Jupiters, icy moons' intrinsic conductivity may be much smaller.
In highly conductive moons with negligible magnetic diffusivity, the field distortion induced by the planetmoon differential rotation is quickly amplified.Eventually, the induced field diffuses through the surrounding plasma in the Alfvén-wing wake of the moon's orbit ( §3.1.2).Around Jupiter, Io's effective conductivity is caused by the plasma torus around its orbit (Neubauer 1980).Hot Jupiters and their moons are also embedded in the plasma from their host stars' wind (Zarka 2007;Strugarek 2016).In view of these possibilities and uncertainties, we adopt Io's conductivity σ m ≈ 10 −8 ohm −1 cm −1 and the typical radius R m ≈ 3 × 10 8 cm so that moon's effective resistance is 1/(σ m R m ) ≈ 1 ohm, and the resistance at the foot of the planetary field is also around 1 ohm (Goldreich & Lynden-Bell 1969).Our adopted fiducial value may be an overestimate for icy moons' effective σ m and therefore enhance the probability of their retention around warm and cold Jupiters.We set Λ = 1/(σ m R m ) in ( 5), and consequently the magnetic torque due to the planetmoon unipolar induction reads where ω m is moon's orbital frequency and a m its orbital semi-major axis.The subscript p denotes planet (primary) and m moon (secondary), and the superscript m magnetic torque.
In addition to the planet-moon unipolar interaction, star and moon can also induce unipolar interaction.In star's frame of reference, the electric field induced by star-moon unipolar interaction is E = (v p + v m ) × B) where v p is planet's velocity relative to star, v m is moon's velocity relative to planet, and B is star's magnetic field at planet (precisely speaking at moon but the upper limit of planet-moon distance is 10% of starplanet distance, see §4.1).The Lorentz force is then The force arm is r p + r m where r p is the position vector of planet relative to star and r m the position vector of moon relative to planet.We now calculate the torque What we are concerned with is the moon's orbital evolution around planet, and therefore, the first term r p × v p arising from the star-planet interaction should be subtracted, the second and third terms proportional to cos(ω m − ω p )t vanish when averaged over the moon's orbit around planet, and eventually only the last term r m × v m contributes to the moon's orbital evolution.We can readily write the torque of star-moon unipolar interaction Compared to the torque of planet-moon unipolar interaction ( 6), the difference lies in magnetic field, i.e., planetary field at moon in the former whereas stellar field at planet in the latter.Both Γ m pm (6) and Γ m ⋆m (7) are in the direction of the moon's motion relative to the planet.Our estimation shows that for a hot-Jupiter system the star-moon torque is stronger than the planet-moon torque whereas for a cold-Jupiter system the former decaying as a −6 p is far weaker than the latter.

Combination of tidal and magnetic torques
In a two-body system, e.g., star-planet or planet-moon system, both tidal and magnetic torques can transfer angular momentum between the orbit and the spins.We will summarize all the torques in this subsection.Figure 1 shows the sketch of tidal and magnetic torques in a star-planet-moon system.The arrows depicts the primary and secondary, i.e., arrows pointing from secondary to primary.Here the symbols are defined as follows: the first subscript denotes the primary on which the torque is exerted, the second subscript denotes the secondary which exerts the torque ('⋆' for star, 'p' for planet and 'm' for moon), and the superscript denotes the type of torque ('t' for tidal torque and 'm' for magnetic torque).

The Q-value and tidal time lag for equilibrium and dynamical tides
The tidal torque due to the large-scale equilibrium tide in convection zone is proportional to the tidal frequency . Schematic illustration of tidal and magnetic torques in a star-planet-moon system.The first subscript denotes the primary, the second subscript the secondary, and the superscript the type of torque (tidal or magnetic).
ω, i.e., the orbital frequency relative to the primary's spin frequency, and also proportional to turbulent viscosity ν t in primary's convection zone.The combined effect of tidal frequency and turbulent viscosity on tidal torque is modelled with the tidal quality factor where denotes the integral over an orbital cycle, E 0 is the tidal deformation energy, namely the product of tidal force ), Ė is tidal dissipation rate, P ≃ 1/ω is tidal period, and Γ t is tidal torque.Thus, the tidal torque is Γ t ≃ GM 2 2 R 5 /(a 6 Q).Usually Q can be written as Q ≃ 1/ωτ where τ = ν t R 1 /GM 1 is the tidal time lag (Hut 1981;Eggleton et al. 1998).In the weak friction theory, turbulent viscosity ν t or tidal time lag τ is independent of tidal frequency ω such that tidal torque Γ t is proportional to tidal frequency ω; in Zahn's theory (Zahn 1977), turbulent viscosity is inversely proportional to tidal frequency such that tidal torque is independent of tidal frequency; in Goldreich's theory (Goldreich & Nicholson 1977) turbulent viscosity is inversely proportional to the square of tidal frequency such that tidal torque is inversely proportional to tidal frequency.The most recent numerical simulations support Goldreich's theory (Duguid et al. 2020;Vidal & Barker 2020).
In addition to equilibrium tide, dynamical tide (e.g., inertial waves in convection zone or internal gravity waves in radiation zone) also induces tidal dissipation and hence tidal torque.The tidal torque induced by dynamical tide, especially by inertial waves in convection zone, dominates over that induced by equilibrium tide (Ogilvie & Lin 2004;Ogilvie 2014).The tidal torque induced by inertial wave is strongly frequency-dependent because inertial wave has a very dense spectrum (Ogilvie & Lin 2004;Ogilvie 2014).To avoid this complexity, tidal torque induced by dynamical tide is usually cal-culated with its frequency-average and also modelled with Q number (Ogilvie 2014;Strugarek et al. 2017).However, tidal torque will sharply decay to zero when tidal frequency approaches zero at a synchronous state (Ogilvie & Lin 2004, 2007).In order to take into account the synchronous state we calculate the tidal torque induced by dynamical tide with tidal time lag τ instead of tidal Q, The superscript t denotes tidal torque.The observations of close-in binary circularization (Mathieu 1994), exoplanet's (Penev et al. 2018) and Galilean moons' migration (Yoder & Peale 1981;Lainey et al. 2009) suggest that star's or planet's tidal Q is about Q ∼ 10 5−6 .The tidal frequency is about ω ≃ 10 −5 s −1 , and hence tidal time lag of dynamical tide can be estimated τ ∼ 1−10 s.

Bypassing the valleys of dynamical tides
As discussed in the last subsection §3.1, the magnetic torque, in addition to tidal torque, can cause angular momentum transfer between the host star and its planet or between the host planet and its moon.For a more complete view, we introduce a fiducial model for a young (a few Myr old) planetary system with the typical parameters listed in Table 2 (the details of calculation will be shown in the next section).For this model, we evaluate all the eight torques in a star-planet-moon system (first row in Table 3) as well the corresponding spin-orbit interchange due to the angular momentum transfer by each torque (second row).The third row shows the values of the torques at an asynchronous state.We consider only a circular orbit such that there is no interchange of spin and eccentricity.If the moon is un-magnetized, the last torque Γ m mp induced by planet on moon's magnetic field would be irrelevant.
Furthermore, as discussed in §3.1, magnetic torque is proportional to tidal frequency.That is, contrary to the torque due to dynamical tides associated with inertial waves, magnetic torque varies smoothly with tidal frequency.Therefore, during the orbital evolution when tidal frequency varies, magnetic torque smooths the total torque in the sense that it compensates tidal torques and bypasses frequency valleys where dynamical-tide response is ineffective at those particular tidal frequencies.
Figure 2 shows the illustration of this smoothing effect of magnetic torque on dynamical tide.In the Figure the equivalent tidal Q number due to magnetic torque is estimated with Q ≃ E 0 /Γ m where E 0 is tidal deformation energy and Γ m is magnetic torque.It should be noted that the equivalent magnetic Q ∝ a −3 (E 0 ∝ a −6 and Γ m ∝ a −3 ), such that magnetic Q in Figure 2 2. The typical parameters of a young star-planet-moon system.In this fiducial model, the plasma number density N = 5(t/4.6Gyr)−2 (ap/1au) −2 cm −3 .The coefficient of planet's moment of inertia αp = 0.1 and moon's conductivity σm = 10 −8 ohm −1 cm −1 .10 32 1.7 × 10 33 2.1 × 10 34 2.4 × 10 31 7.8 × 10 25 3.1 × 10 25 1.8 × 10 30 N/A 2.2 × 10 26 Table 3.The first row shows all the torques in the star-planet-moon system.The second row shows the corresponding spin-orbit interchange.The third row shows their values (erg) at an asynchronous state with the typical parameters listed in Table 2.
asymmetric about the tidal frequency ω = 0 (in Figure adapted from Ogilvie & Lin (2007) tidal frequency is denoted by Doppler-shifted ω rather than ω).We can then infer such a situation: when a planet migrates inward, say, in a disk due to the angular momentum transfer between the planet and the disk, tidal torque becomes very weak at some orbital frequencies.However, magnetic torque continues to induce the inward migration until the planet plunges into the star.

A brief synopsis of tidal and magnetic torques
To wrap up this subsection, we summarize the expressions of all the torques already discussed: The positive sign means that orbital motion is faster than spin so that the torque spins up the primary body denoted by the first subscript; and vice versa, the negative sign corresponds to the spindown of the primary body.The typical values of the torques are calculated with the parameters in Table 2 and listed in the third row of Table 3.The tidal torque Γ t mp raised by planet on moon depends on moon's tidal quality factor Q m = 1/(ω m − Ω m )τ m , assumed to be 67, i.e., Io's Q (Lainey et al. 2009).Γ t mp is estimated to be stronger than the tidal torque Γ t pm or magnetic torque Γ m pm raised by moon on planet.However, since moon's mass is very small, its spin and orbit will quickly reach a synchronous state under such a strong torque, therefore, Γ t mp will soon vanish, and moon's orbital migration will then be controlled by Γ t pm and Γ m pm .and the horizontal axis denotes the tidal frequency (i.e., the difference between orbital and stellar spin frequencies).The dashed black lines denote the results of equilibrium tide.The two dashed color lines show the magnetic torques of the system of a solar-like star with a hot Jupiter: red denotes stellar surface field 1000 gauss and planetary surface field 100 gauss (young planetary system) and blue denotes stellar surface field 10 gauss and planetary surface field 10 gauss (mature planetary system).Magnetic torque compensates the minuscule tidal torque at some frequencies, and higher fields correspond to a stronger magnetic torque.

Relative strength of magnetic versus tidal torques for planets and moons
In an asynchronous state, Γ t p⋆ ≈ GM 2 ⋆ R 5 p /(a 6 p Q p ) and Γ t ⋆p ≈ GM 2 p R 5 ⋆ /(a 6 p Q ⋆ ).Suppose that the two tidal Q's are comparable, and hence the ratio is Γ t p⋆ /Γ t ⋆p ≈ (M ⋆ /M p ) 2 (R p /R ⋆ ) 5 ≈ 10.Consequently, Γ t p⋆ leads to the synchronization of the planet's spin before it undergoes significant migration.As Ω p approaches ω p , Γ t ⋆p dominates over the diminishing but finite Γ t p⋆ and the planet migrates with an evolving ω p and Ω ⋆ while a state of near Ω p − ω p synchronism is being maintained.In addition, Γ m ⋆p (which is far stronger than Γ m p⋆ ) also contributes to the planet's migration.Similarly, Γ t pm and Γ m pm leads to the moon's near spin-orbit (Ω m − ω m ) synchronism.The leading torques for moon's migration are tidal torque Γ t pm and magnetic torques Γ m pm (Γ m ⋆m is important for the close star-planet distance but decays quickly as a −6 p ).Based on the model parameters for a fiducial young system (Table 2), we now compare the relative strength of the torques for planet's or moon's migration.Figure 3 shows the tidal and magnetic migration timescales for planet's migration (left panel) and for moon's migration (right panel).We firstly focus on planet's migration (left panel).The comparison between the two tidal torques Γ t ⋆p and Γ t p⋆ shows that in an asynchronous state Γ t p⋆ is stronger than Γ t ⋆p by one order, which is consistent with our analysis.The comparison between the tidal torque Γ t ⋆p and magnetic torque Γ m ⋆p shows that the magnetic torque always wins out the tidal torque both inside and outside corotation radius.The magnetic torque Γ m ⋆p wins out Γ t p⋆ when planet's orbital period becomes longer than 6 days, because the former decays as a −2 p whereas the latter a −6 p .However, Γ m ⋆p decreases as the stellar field decays and radius contracts on a timescale of a few 10 7 yr.Around mature main sequence stars (with a life span of a few Gyrs), neither the magnetic nor tidal torque can significantly influence the orbits of planets with period ≳ 10 days.
Next we move to moon's migration (right panel).Outside the corotation radius, the magnetic torque Γ m pm wins out the tidal torque Γ t pm if the moon is located outside 10 R J .However, the migration timescale is already longer than 1 Gyr.Over such a long timescale, Γ m pm would be much reduced as B p declines and R p contracts.Thus, magnetic field is unlikely to have a dominant and significant influence on moon's migration.

RETENTION OF EXOMOONS
Recently, exomoons are possibly found in the circumplanetary disk of, for example, PDS70c (Benisty et al. 2021), and the retention of exomoons has now been studied, e.g., (Tokadjian & Piro 2020).In this section, we will study the circumplanetary disk and the timescales of planet-moon dynamics, and then investigate under what situations the exomoon can retain.

Inner and outer radii of Circumplanetary disks
The radius of circumplanetary disk cannot exceed the planetary Hill radius, otherwise gas would be caught by host star (Martin & Lubow 2011) where a p is the planetary orbital semi-major axis.Hill radius r H is considered to be the outer radius of circumplanetary disk.At the innermost circumplanetary disk, planet's magnetic field is so strong that it can truncate the disk.The magnetic truncation radius, which is considered to be the inner radius of circumplanetary disk, can be estimated by the balance between magnetic pressure and kinetic pressure B p (R p /r T ) 3 2 /(2µ) ≈ ρv 2 (or equivalently the balance between Alfvén speed and orbital speed) where density ρ is estimated with the accretion rate of circumplanetary disk ρ = Ṁp /(4πr 2 T v) and velocity v follows Keplerian motion v = (GM p /r T ) 1/2 .The magnetic truncation radius can be estimated (10) (Koenigl 1991).Inserting the typical values M p = M J , R p = 2R J , Ṁp = 10 −7 M J /yr and B p = 100 gauss, we obtain r T ≈ 3R p ≈ 6R J , which is exactly the Io's position.If spin-orbit synchronism can be established at r T , Jupiter would spin with a period 3-4 times longer than its present-day value, and the subsequent (a factor of 2) contraction would increase its spin rate to its present-day value.The magnetic truncation radius r T gives the lower bound of the inner radius of circumplanetary disk (Koenigl 1991;Batygin 2018), whereas the Hill radius gives the upper bound of the outer radius (Lin & Pringle 1976;Papaloizou & Pringle 1977;Machida et al. 2008;Fung et al. 2015;Li et al. 2021).
Compared to r H ≈ 0.1a p , r T ≈ 6R J is not much less than r H for a hot Jupiter with a p , say, 0.05 au.Since r T ∝ B 4/7 p , if the young planetary field reaches, say, 300 gauss, then r T > r H such that the circumplanetary disk does not exist and hence the exomoon cannot be born.Figure 4 shows the schematic configuration of star-planet-moon system, where moon lies in circumplanetary disk between inner radius r T and outer radius r H . Figure 5 shows the contours of ratio of the two radii r H /r T versus planet's orbital period and magnetic field.The area to the right of the red dashed line with the ratio greater than 1 is favorable for moon's retention.We draw a conclusion that a planet with too strong magnetic fields or too short distance from its host star tends not to have exomoons.When a gas giant planet is born, it spins with an angular frequency comparable to the Keplerian frequency at r T .On the Kelvin-Helmholtz thermal timescale of Gyr, it spins up due to its contraction.However, at its young age a close-in planet spins down due to the tidal torque Γ t p⋆ raised by its host star.During the planet's spin-down, the corotation radius of any circumplanetary satellites moves outward.The top panel of Figure 6 shows planet's spin evolution.In stage 1 ○ shortly after planet is born, planet inflates slightly; in stage 2 ○ planet contracts such that it spins up; in stage 3 ○ planet spins down due to star-planet tidal interaction; and in stage 4

Evolution of planet's corotation radius and moon's migration
○ planet inward migrates and spins up or outward and down.The stage 3 ○ when a close-in planet spins down due to star-planet interaction and corotation radius moves out is what we are concerned with.But the spin of a long-period (with τ cor > τ ⋆ ) the planet does not spin down significantly throughout its host star's life span and its evolution terminates at the end of stage 2 ○.In the meanwhile, a moon located outside the corotation radius also migrates outward due to the tidal and magnetic torques Γ t pm , Γ m pm and Γ m ⋆m .If the corotation radius moves faster than moon's outward migration, the moon will be located inside the corotation radius, migrate inward and eventually plunge into the planet to form a planetary ring (Kanodia et al. 2022).The bottom panel of Figure 6 shows moon's migration and planet's corotation radius evolution.Solid curves represents the three possible moon migration tracks, and dashed curve planet's corotation radius evoltuion corresponding to its spin evolution in the top panel.
In the first place we estimate the planet's spin frequency Ω p when it forms.We assume that the planet's corotation radius r cor is located at its magnetic truncation radius r T (Batygin 2018;Ginzburg & Chiang 2020;Hasegawa et al. 2021).Using the expression of r T in the last subsection, we readily derive the planet's spin frequency Ω p = (GM p /r 3 T ) 1/2 ≈ 0.1(GM p /R 3 p ) 1/2 , which is 10% of the break-up frequency and consistent with the observations (Bryan et al. 2020).With the typical values, the planet's spin period is about 40 hours.
In addition to the spin-up on the Kelvin-Helmholtz timescale due to contraction, a hot Jupiter can spin down on a much shorter timescale induced by the tidal torque Γ t p⋆ exerted on the planet's host star.With the planet spinning down, the corotation radius of satellites moves outward on a timescale where α p ≈ 0.1 is the coefficient of planet's moment of inertia and the coefficient 1.5 arises from Kepler's third law Ω p = ω m ∝ a −1.5 m .An exomoon outside/inside its host planet's corotation radius migrates outward/inward due to the tidal torque Γ t pm and the magnetic torque Γ m pm (these two torques are comparable) as well as Γ m ⋆m (this torque is important for a hot-Jupiter system).The moon's migration timescale is estimated as At the end of stage 2 ○, the moon would be retained with an expanding orbit if its a m > r cor .In the limit a m < r cor , the moon's would decay into the planet if τ cor > τ mig , but it could be retained, at least until the end of stage 3 ○ if τ cor < τ mig .Thereafter, the moon would still plunge into the planet if a m < r cor and τ mig ≤ τ ⋆ before the host star evolves off the main sequence on a timescale τ ⋆ .Otherwise it would be retained.
Since τ cor ∝ a 6 p and τ mig ∝ a 6.5 m (roughly), large a p and small a m favour the retention of exomoon, and Figure 7 verifies this survival preference.The left panel shows the contours of ratio τ cor /τ mig versus planet's and moon's orbital periods.At the end of stage 2 ○, the moon is assumed to be outside corotation radius of its rapidly spinning host planet.In domain A with τ cor /τ mig > 1, the moon's orbit expands prior to the planet's spin down.This region is favorable for the moon's retention.In domain B τ mig < τ ⋆ ∼ 10 Gyr and in domain C τ mig > τ ⋆ .Since τ cor < τ mig in both domains B and C, they contain moons which undergo orbital decay after r cor is reduced interior to a m .Moreover, moons would plunge into their host planet before their host star evolves off the main sequence in domain B whereas they would be retained in domain C. Thus, exomoons tend to be retained around cold but not hot Jupiters, unless the moon's migration timescale is longer than the age of planetary system.Although tidal torque Γ t pm ∝ M 2 m , magnetic torques Γ m pm and Γ m ⋆m ∝ M m (magnetic torques ∝ R 3 m and moon's density is roughly 3 g/cm 3 ).The moon's orbital angular momentum ∝ M m so that the two migration timescales have different scaling laws with respect to moon's mass, i.e., tidal migration timescale ∝ M −1 m but magnetic migration timescale is independent of M m , and therefore a moon with larger mass corresponds to smaller migration  2).Domain A denotes the ratio τcor/τmig > 1, favourable for moon's survival.Domain B denotes τcor/τmig < 1 with short τmig < 10 Gyr, unfavourable for moon's survival.Domain C denotes τcor/τmig < 1 with long τmig > 10 Gyr (longer than the age of planetary system), favorable for moon's survival.In order for a moon to be bound to the planet, ωp < ωm such that top-left half domain D is a zone of avoidance.Right panel: Investigation of moon's mass Mm = 10 25 g (black), 10 26 g (red), and 10 27 g (blue).In all cases Mp = MJ and density of the moon is taken to be 3 g/cm 3 .Solid curves denote ratio τcor/τmig = 1, on the right of which moon can retain.Above dahsed lines τmig > 10 Gyr and moon can retain.
timescale.The right panel of Figure 7 shows moon's retention for different moon's masses.A moon with larger mass is more likely to be found around a planet with shorter distance from its host star.Finally, in order for moons to be bound to host planets, they must be within the planetary Hill radius, i.e. ω p < ω m , such that the top-left area separated by the dotted line is a zone of avoidance.If the planets' spin frequency Ω p is synchronized with their orbital frequency ω p at the end of stage 3 ○, their moons' a m < r H = r cor .Planets' subsequent inward migration would further reduce their moons' survivable probability.Although some host planets may continue to spin down with outward migration, a m < r cor (Fig. 6) and their moons' orbit would not expand during stage 4 ○.

Numerical calculation of spin-orbit evolution
We have estimated the timescales of planet's spindown and moon's migration.To investigate more rigorously we solve numerically planet's orbital and spin equations coupled with its moon's orbital equation in the circular and coplanar limit.A more general treatment is presented in Appendix B.
Based on the above consideration, the evolution equations for a p , Ω p and a m become In the calculation of the magnetic torque, we adopt the evolution of planetary field B p ∝ t −0.267 (Eq.2) and assume the plasma density N ∝ t −2 a −2 p (Eq. 3).The numerical integration of the secular equations (13) starts at the initial time when the planet has already formed through any mechanism: in-situ core accretion (Bodenheimer et al. 2000) for a cold Jupiter, or high-e migration (Rasio & Ford 1996;Wu & Murray 2003) or disk migration (Lin et al. 1996) for a hot Jupiter.This initial time is estimated at 2 Myr (see Table 1).The initial a p is set such that the initial 2π/ω p is, respectively, 4 days, 10 days and 100 days.The initial Ω p is set to be 40 hours.The initial a m is set to be 6R J which is slightly outside corotation radius r cor or 10R J which is far outside r cor .The other parameters are listed in Table 2.
Figure 8 shows the evolution of r cor (solid) and a m (dashed and dotted) with different initial a p and a m .We firstly focus on the dashed lines with initial a m = 6R J , sightly outside r cor .These values of a m and M m are comparable to those of Io.When a planet is initially The initial planet's orbital period at 4 days (black), 10 days (red) and 100 days (blue).Horizontal thin dotted line denotes moon's Roche limit.
close to its host star with an orbital period 2π/ω p = 4 days, the tidal torque Γ t p⋆ raised by its host star is so strong that planet spins down very quickly.As the moon's orbit is engulfed by the expanding r cor within 100 years, it migrates inward until it overflows its Roche lobe (horizontal think dotted line).With an initial 2π/ω p = 10 days, moon's r cor crosses its a m within 2000 years.For a distant planet with 2π/ω p = 100 days, planet's spin down is so slow that moon's r cor remains inside a m until around 10 8 years.At Jupiter's semimajor axis, a p ≃ 5.2 au, the planet's spin retains its value at the end of stage 2 ○.Since a m > r cor , the moon migrates outward slightly.We then move to the dotted lines with initial a m = 10R J which is far outside r T (Eq.10).Moon with initial 2π/ω p = 4 days has an initial a m > r cor .After 4000 years, r cor expands outside a m and the moon migrated inwards until it overflows its Roche radius after 3 × 10 7 years.With the other two initial 2π/ω p 's (10 and 100 moon is retained for at least 10 8 years.Clearly, larger distance between moon and corotation radius prolongs the retention of the moon.In summary, the numerical calculations of the secular equations ( 13) are consistent with the estimations of timescales in §4.2.

RADIO EMISSIONS
The star-planet or planet-moon magnetic interaction induces the radio emissions that can be detected by the radio telescopes, e.g.LOFAR, VLT, FAST, etc. (Zarka 2007;Zarka et al. 2019;Vidotto & Donati 2017;Kavanagh et al. 2022).Recently, the detection of radio emission from the YZ Ceti system has been reported by Pineda & Villadsen (2023).Since it contains several previously known super Earth(Astudillo-Defru et al. 2017), these signals may be associated with the starplanet magnetic interaction (Trigilio et al. 2023).
The power of radio emissions due to the dipole-dipole or unipolar interaction can be estimated by the radiomagnetic Bode's law (Zarka 2007) where v ⋆p,pm = (n p,m − Ω ⋆,p )a p,m is the relative velocity, n p,m is the orbital frequency of the planet for the star-planet interaction or of the moon for the planetmoon interaction, Ω ⋆,p is the host's rotational frequency, and a p,m is the orbital semi-major axis, B ⋆p,pm = B ⋆,p (R ⋆,p /a p,m ) 3 is the local magnetic field of the star/planet at the location of the planet/moon, R ⋆p,pm is the obstacle radius with for the star-planet interaction or R pm ≈ R m for the planet-moon interaction.The dimensionless efficiency coefficient η is about 2 × 10 −3 according to the observations in the Jovian system (Zarka 2007).We take the typical values of a young stellar system to estimate the radio power of the star-planet and planet-moon interactions.As before, the parameters we use are listed in Table 2 for a star-planet-moon system.The estimations show that the radio power of star-planet magnetic interaction is 2.1 × 10 26 erg/s and that of planet-moon magnetic interaction is 2.3 × 10 22 erg/s.For more mature systems with much weaker B ⋆ and B p (Table 2, these powers are much reduced. The averaged flux density (power per unit area per unit frequency) is (Kavanagh et al. 2022) where d is the distance of the planetary system from the Earth, Θ is the solid angle of the radio emission (1.58 sr for Jupiter-Io interaction), ∆ν is the waveband of the radio telescope.We use this formula to estimate the average flux density for different sources and different telescopes.The energy spectrum of cyclotron radiation peaks at Larmor frequency and its double frequency.Suppose that the radio telescope wavebands to detect the star-planet and planet-moon unipolar interactions focus near ∼ 100 MHz (LOFAR at 10 ∼ 240 MHz and the future FAST at 70 ∼ 200 MHz), and that the distance from the earth is 1 kpc, then the averaged flux density of the star-planet interaction is F ≈ 14 mJy and that of the planet-moon interaction is F ≈ 1.5 × 10 −3 mJy.We can then estimate the upper limit on detectable distance for the radio telescopes.For example, the detection sensitivity of FAST is about 1 mJy and it can detect radio signals from star-planet magnetic interaction out to ∼ 4 kpc, and the planet-moon radio signals can be detected within ∼ 40 pc.It is possible to detect star-planet interaction well beyond the distances (∼ 10 2 pc) of the closest Scorpius-Centaurus OB association (Blaauw 1964) and the well-studied Taurus and Orion star formation regions.But radio emission from planet-moons magnetic interaction may be marginally detectable, at best, for only super-Jupiter young (≲ 10Myr) planets (Table 1).
What we considered in the above is a young starplanet or planet-moon system.For a mature star-planet or planet-moon system, the stellar field drops from ∼ 1000 gauss to ∼ 1 gauss and the planetary field from ∼ 100 gauss to ∼ 10 gauss, and the upper limit of detection distance which is almost proportional to magnetic field (Eqs.14 and 15) will be ∼ 13 pc for a mature star-planet system or ∼ 4 pc for a mature planet-moon system.For a mature star-planet-moon system, the radio emission from the planet-moon interaction is comparable to that from the star-planet interaction.

SUMMARY AND DISCUSSIONS
In this paper we derive the scaling law for planet magnetic field through MHD dynamo equations.The scaling law depends on mass, radius and luminosity but it is insensitive to rotation.Then we combine this scaling law with the virial theorem and the planetary evolution track to find the evolution law of magnetic field of the young Jupiter and of the exoplanets with different mass.Next we derive the magnetic torques induced by dipole-dipole and unipolar interactions for star-planet, planet-moon and star-moon systems.As proposed in Lin et al. (1996), a strong field in stellar magnetosphere can terminate the migration of short-period exoplanet, and similarly, such a strong field of young Jupiter can terminate the migration of Io at its present location.
We then investigate the possibility to find exomoons.By comparing the inner and outer radii of circumplanetary disk, we find that too strong planetary fields or too short star-planet distance do not favor the retention of exomoons.By comparing the planet's spindown and moon's migration timescales and numerically computing the equations of dynamics, we find that large exomoons are vulnerable to merge with their mature hot-Jupiter planets because the planets' spin and corotation radius evolve on shorter timescale than that of the exomoon's orbital migration.Nevertheless, the debris of tidally disrupted exomoons can lead to planetary rings which may be potentially detectable.Observational inferences of such rings can provide supportive evidences for disk over high-e migration scenario for hot Jupiters' origin since dynamical instabilities during the highly-eccentric close stellar encounters are likely to effectively dislodge most of the exomoons from the gravitational confinement of their host planets.
Provided they have similar densities, the magnetic torques Γ m pm and Γ m ⋆m increases with moons' mass.Moreover, due to their much weaker tidal torque Γ t pm , small exomoons (sub-Moon) have longer migration timescale, especially at relatively large distances (≳ 10R J ) from their less massive (sub-Jupiter) host planets (with relatively weak fields, Eqs. 2,6,8,& 12).Although they could survive around hot Jupiters if their τ m is longer than the age of the planetary systems, small moons are much less observationally conspicuous.In contrast, the survivability of potentially detectable (i.e.relatively large, super-lunar-size) exomoons are much higher around warm or cold Jupiters because their spins do not have time to become tidally synchronized with their orbits around their host stars.But the transit-detection probability also decreases with the host planets decrease with their semi-major axis.
Finally, we estimate the power and flux density of radio emissions due to the star-planet and planet-moon magnetic interactions, and estimate the upper limit of distance to be detectable by FAST.
This study is primarily based on a simple prescription for the planets' magnetic field during the evolution of planetary radius and luminosity ( §2.2).In general, magnetic field may also influences the planetary evolution to some extent.The interaction of zonal wind (or thermal current) in planetary atmosphere and planetary poloidal field can induce a strong electric current penetrating into the planetary interior.The ohmic dissipation associated with this electric current may be sufficiently strong to lead to the planetary inflation to compensate the gravitational contraction.This mechanism has been suggested to be an explanation for the anomalous large radii of hot Jupiters, and it was firstly studied in a kinematic regime with an externally prescribed field (Batygin & Stevenson 2010) and then in a dynamical regime in which the Lorentz force is considered to be a drag on fluid motion (Batygin et al. 2011).This hypothesis has been challenged by follow-up studies (Huang & Cumming 2012;Wu & Lithwick 2013) and numerical simulations (Rogers & Komacek 2014).However, a Bayesian analysis of observational data (Thorngren & Fortney 2018) shows that ohmic dissipation can be attributed as the culprit of the inflation of hot Jupiters with high mass.In the present context, the coupling between the magnetic field and the thermal current and its effect on the planetary inflation are not directly related to dynamo that we study in this paper.Dynamo is a self-excited field but not a prescribed one, and its energy source is the convective heat flux in the interior.Ohmic dissipation releases heat but it is negligible compared to the convective heat flux.From the point view of the second law of thermodynamics, the convective heat flux can do work to cause fluid motion and generate magnetic field, but ohmic dissipation only releases useless heat that cannot do work on fluid to generate magnetic field.Therefore, the effect of magnetic field on the surface flow and thermal current is unlikely to have a significant impact on the evolution of planetary magnetic field.
On the other hand, the planetary inner core may have an impact on dynamo because the existence of a large core alters the thermal structure of a planet.At the young age of our solar system, the collisions may be frequent due to the high eccentricity of planetesimals arising from the sweeping secular resonance (Zheng et al. 2017a,b), such that a relatively large dilute core can form in Jupiter (Liu et al. 2019).The effect of a large core on the evolution of planetary magnetic field needs to be further studied.
With respect to application to the orbital dynamics, we consider only the circumstance of a co-planar circular orbit.The more complicated orbital dynamics about the semi-major axis, eccentricity and inclination evolution coupled with the magnetic field evolution will be further studied (see Appendix B). and magnetic diffusion, i.e., the former takes place on planetary radius R while the latter on l B , and readily obtain l B /l ≈ (η/vl) 1/2 = Rm −1/2 where magnetic Reynolds number Rm = vl/η is defined on the mixing length l.This twoscale analysis is necessary for such a dissipative system (another well-known example is the boundary-layer analysis in fluid mechanics).Otherwise, if we admitted that the magnetic induction and the magnetic diffusion take place on the same scale then we would find that the magnetic Reynolds number is of order of unity at which dynamo cannot be driven.Inserting Ampere's law J ≈ B/(µl B ) with the estimation l B ≈ (ηl/v) 1/2 into the balance ⟨δρ gv⟩ ≈ ⟨J 2 /σ⟩, we arrive at ⟨B 2 /µ⟩ ≈ ⟨δρ gl⟩, which indicates the equipartition between magnetic energy and buoyancy energy.In the mixing length theory, buoyancy energy is comparable to kinetic energy, i.e. δρ gl ≈ ρv 2 , and thus we obtain ⟨B 2 /µ⟩ ≈ ⟨ρv 2 ⟩.
To estimate turbulent velocity v we introduce heat flux F = ρc p δT v = c p T δρ v ≈ c p T ρv 3 /(gl) where δT /T = −δρ/ρ for ideal gas at constant pressure and the mixing length theory δρ gl ≈ ρv 2 are employed, and then we obtain the estimation of turbulent velocity v ≈ [F gl/(ρc p T )] 1/3 .In stellar convection zone the mixing length l ≈ c p T /g such that v ≈ (F/ρ) 1/3 .This estimation about turbulent velocity is numerically validated by Chan & Sofia (1996) and Cai (2014).In solar convection zone the overall turbulent velocity is estimated around 30 m/s which is consistent with the asteroseismological observation (Hanasoge et al. 2012).Inserting the estimation of v into the balance ⟨B 2 /µ⟩ ≈ ⟨ρv 2 ⟩ we obtain the estimation of magnetic energy ⟨B 2 /µ⟩ ≈ ⟨ρ 1/3 F 2/3 ⟩.However, in planetary interior, due to the small size of planet we replace l with min(l, d), where d is the depth of planetary convection zone, such that the estimation of magnetic energy becomes ⟨B 2 /µ⟩ ≈ ρ 1/3 F 2/3 [min(l, d))/(c p T /g)] 2/3 where the factor min(l, d)/(c p T /g) is dimensionless.Clearly, this scaling law is independent of rotation since Coriolis force does not enter the energy equation that we used to derive this scaling law.Although rotation is not the driving mechanism it plays an important role in field geometry (Christensen 2010).Usually faster rotation leads to more dipolar field on a larger scale due to the columnar structure of fluid flow.In some scaling laws, field strength depends on rotation, e.g.Stevenson (1979), Davidson (2013) and Wei (2022).In the anisotropic rotating turbulence, the mixing length theory, which states the balance between buoyancy force and inertial force, cannot hold but Coriolis force enters the force balance, such that the field strength will eventually depend on rotation (Wei 2022).This inference is valid for stellar convection zone and observations show that stellar fields indeed depend on rotation (Wright et al. 2011;Vidotto et al. 2014).Planet is much smaller than star and the mixing length is usually taken to be the depth of planetary convection zone.Therefore, the rotational effect on the mixing length theory is negligible for planetary dynamo.
Next we consider the planetary internal structure.We normalize the density ρ with the mean density ρ m and the heat flux F with the surface heat flux F s such that the estimation can be written as ⟨B 2 /µ⟩ ≈ βρ is a structure factor.We simply assume F/F s ≈ (R/r) 2 in gas giant planet.Moreover, for a gas giant planet, the polytropic model can well describe its internal structure with the polytropic index n ranging from 1 of partial degeneracy at low density to 1.5 of complete degeneracy at high density (Lissauer & de Pater 2013).Using Jupiter's parameters we solve the Lane-Emden equation to find that β ≈ 1.12 for n = 1 and β ≈ 0.85 for n = 1.5, so that β ≈ 1.We rewrite ⟨B 2 /µ⟩ ≈ ρ 1/3 m F 2/3 s with ρ m = 3M/(4πR 3 ) where M is planet mass and F s = L/(4πR 2 ) where L is intrinsic luminosity (i.e., luminosity due to internal heat) then we are led to ⟨B 2 /µ⟩ ≈ 0.115M 1/3 R −7/3 L 2/3 .(A2)

B. DISCUSSIONS ON SPIN AND ORBITAL ANGULAR MOMENTUM IN STAR-PLANET-MOON SYSTEM
In this section, we systematically construct quantitative relations between several processes which can potentially lead to star's, planet's, and moon's orbital and spin evolution (in §B.1).In order to highlight the dominant mechanisms, some of these effects (such as planetary spin-down due to mass loss, tidally induced eccentricity excitation, and secular interaction) are neglected in this paper ( §B.2, §B.3, & §B.4).With these approximations, we derive the moon's spin ( §B.5) and compute its orbital evolution ( §4.3).B.1.Transfer in the star-planet-moon systems.
Total angular momentum for star-planet-moon systems is are the angular momentum of the star-planet's and planet-moon's orbits, and α ⋆,p,m are, separated by comma, the star's, planet's, and moon's spin angular momentum and coefficient of moment of inertia respectively.The subscripts with commas separate, in corresponding orders, components for star (⋆), planet (p), moon (m), star-planet (⋆p) and planet-moon (pm) systems respectively.Stellar wind Ṡw ⋆ and planetary evaporation Ṡw p carried by their mass losses ( Ṁ w ⋆ and Ṁ w p ) lead to net losses of J tot ⋆pm .Stars' and planets' contraction ( Ṙ⋆ and Ṙp ) changes their Ω ⋆ and Ω p but not their S ⋆ and S p .For infant planets/moons, the effect of their migration through the protostellar / circumeplanetary disks can also be included in Ṡw p and Ṡw m .Due to the combination of stellar/planetary wind and tidal/magnetic torque, the spin angular momenta evolve at rates, Ṡ⋆,p,m = Γ where the plus and minus signs for Γ sec ⋆pm refer to the secular transfer of orbital angular momentum between the starplanet and planet-moon systems respectively.With the neglect of the second-order terms Γ t,m ⋆m and Γ t,m m⋆ , the total angular momentum budget of the system changes at a rate In Eq (B9), the effect of angular momentum and mass loss from stars and planets is included in Ṡw ⋆ and Ṡw p through Ṡ⋆ and Ṡp (Eq.B6), Ṁ⋆ and Ṁp .After the initial T Tauri phase, host stars' mass and radius evolution ( Ṁ⋆ /M ⋆ and Ṙ⋆ /R ⋆ ) is negligible.But, the loss of angular momentum is strongly enhanced by magnetic braking (Mestel 1968).
An empirical formula for the observed rotational velocity ⟨V r sin i⟩ (where i is the inclination between the spin axis and the line of sight) of G and K stars with age τ ⋆ suggests Ω ⋆ R ⋆ ≃ V 0 (τ 0 /τ ⋆ ) 0.5 where V 0 ≃ 4km s −1 and τ 0 ≃ 1 Gyr (Skumanich 1972).The corresponding spin-down timescale For young systems, this loss timescale may be considerable shorter than the tidal circulation timescale (Dobbs-Dixon et al. 2004).But for mature stars with slow Ω ⋆ and vanishing oblateness, angular momentum loss carried by stellar wind becomes negligible.Photo-evaporation of close-in planets can also lead to significant fractional loss of super-Earths' atmosphere (Owen & Wu 2013, 2017;Fulton et al. 2017).However, this photo-evaporation process is unlikely to be effective for Jupiter-mass planets.Under some circumstances (such as high-e migration), strong tidal (Gu et al. 2003(Gu et al. , 2004) ) or ohmic (Laine et al. 2008;Laine & Lin 2012;Hou & Wei 2022) dissipation of host stars' gravitational and magnetic perturbation may lead to intense heating, runaway inflation, and mass loses.The presence of planets' magnetic field may enhance the their angular moment loss.Moons engulfed in such out-flowing envelopes would endure hydrodynamic drag and undergo orbital decay.Such effects are beyond the scope of the present investigation.In this paper, we assume Ṡw p = Ṁp = 0.

B.3. Eccentricity evolution
In Eqs.(B9) and (B10), ȧp,m , ėp,m , and Ω⋆,p,m are separately treated through L's and Ṡ's respectively.Tidal dissipation inside rapidly/slowly spinning stars can excite/damp their planets' eccentricity (Goldreich & Soter 1966;Dobbs-Dixon et al. 2004).Likewise tidal dissipation inside rapidly/slowly spinning planets can also excite/damp their moons' eccentricity.In systems where 1) all spin vectors of the host stars, planets, and moons are aligned with the planets' and moons' orbital angular momentum vectors and 2) a m << r H (so that the stars' secular perturbation on the moons' orbits can be neglected), the governing equations for the eccentricity evolution become ėp = g ⋆p + g p⋆ and ėm ≃ g pm + g mp where g p⋆,⋆p =27k .
These equations indicate that around rapidly spinning stars (with Ω ⋆ or Ω p >> ω p ), tidal torque can excite the planets' eccentricity.Similarly around rapidly spinning planets (with Ω p or Ω m >> ω m ), tidal torque can excite the moons' eccentricity.However, stellar wind leads to the rapid spindown of young solar type stars ( §B.2).For hot Jupiters, the dominance Γ t p⋆ over Γ t ⋆p also leads to synchronization between the planets' spin with their orbit, i.e.Ω p ∼ ω p ( §3.3).Thus, tidal interaction between stars and their planets usually damps their eccentricity (Dobbs-Dixon et al. 2004).Indeed, nearly all the hot Jupiters and most of the warm Jupiters have circular orbits or very small eccentricities.Moreover, since the moons' a m < r H , their ω m is greater than their planets' ω p and therefore Ω p (Ω p with respect to hot Jupiters).Tidal interaction between planets and their moons also usually damps the moons' eccentricity.In this paper, we consider the limiting cases e p << 1 and e m << 1 so that the second term on the left hand side of Eq (B10) is negligible.

B.4. Secular interaction
In Eq (B10), we also consider the dissipationless gravitation interaction between the stars, planets, and moons (included in Γ sec ⋆pm through L⋆p,pm , Eq. B7).Although the Laplace-Runge-Lanz vector (eccentricity, periapsis longitude and ascending node) of stars, planets, and moons generally modulate on a secular timescale, 3-body (star-planet-moon) gravitational interaction does not lead to a net angular momentum and energy exchanges over multiple liberation or circulation cycles (Murray & Dermott 1999).Gravity associated with their axisymmetric natal disks, general relativity, and rotationally-flatten oblateness of their host stars/planets also do not lead to any net changes in planet and moons' angular momentum, albeit they lead to precession of planet's and moon's ascending nodes and periapsis longitudes.
However, during the depletion of their natal disks or the spin down of their host stars and planets (due to Ṡw ⋆ and Ṡw p ), these systems may encounter secular resonances with net angular momentum (i.e.finite Lsec ⋆pm /J tot ⋆pm ) and negligible energy (i.e.conserved a p and a m ) exchanges between the star-planet and planet-moon's orbits (Nagasawa et al. 2003;Nagasawa & Lin 2005).Moreover, when the planet's/moon's apsidal and nodal precession frequency match with their mean motion (ω p and ω m ), their eccentricity and inclination may also be excited through the evection and eviction resonances in coplanar and inclined systems respectively (Touma & Wisdom 1998).Together with the stellar, planetary, and moons' tidal torque, these physical processes may have played a role in the dynamical history of the Sun-Earth-Moon system (Zahnle et al. 2015;Ćuk et al. 2016).They are also likely to disrupt satellite systems formed in the already confined birth-domains ( §4.1) around short-period planets which may have migrated to their present-day location.In this paper, we neglect these dynamical effects.

B.5. The rate of change in the semi-major axis
In Eqs (B9) and (B10), we have listed all the contributing factors which may affect the spin-orbit evolution of a star-planet-moon system, under the influence of tidal, magnetic, and secular interactions as well as angular momentum and mass loss, and stellar and planetary contraction.Some of these contributing factors, such as secular interaction, stellar and planetary winds, are important during the system's formation and infancy epochs.While these effects will be explored in subsequent studies, this paper focuses on the long-term survival of exomoons on the orbital period of their host planets.The above discussions indicate that most these effects have negligible contribution in mature (Gyr old) systems.
Based on the above discussions, we neglect, for mature star-planet-moon system, contributions from Ṡsec ⋆pm , Ṡw ⋆,p , Ṁ⋆,p,m , Ṙ⋆,p,m , e p,m , so that Eqs (B9) and (B10) reduce to We focus on the orbital evolution under the assumption that the eccentricities of the moon's orbit around the planet and the planet's orbit around the star are quickly damped by Γ t mp or Γ m mp and Γ t p⋆ or Γ m p⋆ respectively.The effects introduced by the initial stellar and planetary spin with eccentricity and inclination in young systems will be analyzed in a follow-up investigation.

Figure 2 .
Figure2.Illustration of the smoothing effect of magnetic torque on dynamical tide.Black curves are adapted fromOgilvie & Lin (2007): stellar tidal dissipation versus tidal frequency, the vertical axis Q ′ ≈ Q/k denotes tidal dissipation (where the apsidal motion constant k ≈ 0.035 for the Sun), and the horizontal axis denotes the tidal frequency (i.e., the difference between orbital and stellar spin frequencies).The dashed black lines denote the results of equilibrium tide.The two dashed color lines show the magnetic torques of the system of a solar-like star with a hot Jupiter: red denotes stellar surface field 1000 gauss and planetary surface field 100 gauss (young planetary system) and blue denotes stellar surface field 10 gauss and planetary surface field 10 gauss (mature planetary system).Magnetic torque compensates the minuscule tidal torque at some frequencies, and higher fields correspond to a stronger magnetic torque.

Figure 3 .
Figure 3. Left panel: planet's migration timescales versus planet's orbital period due to different torques and planet's spin periods (40 hours and 3 days).Star's spin period is 3 days.Right panel: moon's migration timescales versus moon's semi-major axis due to different torques and planet's spin periods.

Figure 4 .
Figure 4. Sketch of inner and outer radii of circumplanetary disk (grey area).

Figure 5 .Figure 6 .
Figure5.Contours for the ratio rH /rT of the two radii of circumplanetary disk.The red dashed line denotes the ratio rH /rT = 1.The area to the right of the red dashed line with rH /rT > 1 is favourable for moon's survival and the area to the left with rH /rT < 1 is unfavourable for moon's survival.

Figure 7 .
Figure 7. Left panel: contours for log(τcor/τmig) (Eqs.11 and 12).Moon's and planet's mass are Mm = 10 26 g and Mp = MJ respectively (Table2).Domain A denotes the ratio τcor/τmig > 1, favourable for moon's survival.Domain B denotes τcor/τmig < 1 with short τmig < 10 Gyr, unfavourable for moon's survival.Domain C denotes τcor/τmig < 1 with long τmig > 10 Gyr (longer than the age of planetary system), favorable for moon's survival.In order for a moon to be bound to the planet, ωp < ωm such that top-left half domain D is a zone of avoidance.Right panel: Investigation of moon's mass Mm = 10 25 g (black), 10 26 g (red), and 10 27 g (blue).In all cases Mp = MJ and density of the moon is taken to be 3 g/cm 3 .Solid curves denote ratio τcor/τmig = 1, on the right of which moon can retain.Above dahsed lines τmig > 10 Gyr and moon can retain.

)
Neglecting structure changes (i.e.assuming α⋆ = αp = αm = 0), the spin angular momentum, mass, and radius changes of the stars, planets, and moons in Equation (B5) lead to spin rate evolving at rates Ω⋆Spin and mass loss

Table 1 .
Magnetic field (gauss) for different mass and age with different initial condition.