The secular equation

Taking into account all these processes mentioned before, the evo-lution of the semi-major axis is given by (see Zahn 1966; Alexander et al. 1976;Zahn 1977,1989;Livio 1982;Livio & Soker 1984;Villaver

& Livio 2009;Mustill & Villaver 2012;Villaver et al. 2014)

a˙ a

=−M˙_?+ ˙M_pl M?+M_pl

| {z }

Term 1

− 2

M_plv_pl[Ffri+Fgra]

| {z }

Term 2

− a˙

a

t

| {z }

Term 3

, (2.6)

where M? is the stellar mass, M˙_? = −M˙_loss, M˙_loss being the mass loss rate (here given as a positive quantity), Mpl and M˙pl are the planetary mass and the rate of change in the planetary mass,v_plis the velocity of the planet,F_fri andF_gra are respectively the frictional and gravitational drag forces, ( ˙a/a)_t is the term that takes into account the effects due to the tidal forces. We derive Eq. (2.6) and then we analyze its terms individually.

The equations of the motion of an object through the atmosphere of its companion has deeply studied by Alexander et al. (1976). The evolution of the separation when a perturbing force (F) is acting, as drag forces, is given by

¨r=−µ

r³r+ 1

M_plF, (2.7)

whereµ=G(M?+Mpl) =GM with Gis the gravitational constant, andrthe orbital radius vector from the star to the planet. The orbital velocity r˙ has components

Vr= ^dr_dt = µ

p

¹₂

esinf , (2.8)

V_φ= ^(µp)

1 2

r , (2.9)

where p = a 1−e²

and f is the orbital true anomaly, that is the angle between the direction of periapsis and the current position of the planet (see Fig. 2.3).

The stellar wind has a velocity v_wind with the components:

v_wind,r= v_s(r) , (2.10)

v_wind,φ= λV_φ , (2.11)

where λ is the ratio between the transverse velocities of the matter around the planet and the orbital motion at any point.

We can write the perturbing force Fas

F=−Q(r, w)w, (2.12)

where w is equal tor˙−v_wind that corresponds to the velocity of the planet relative to the matter.

Figure 2.3: The true anomaly of point P corresponds to the angle f. The point C and F are the center and the focus of the ellipse, respectively. The segment CZ is the semi-major axis a and e is the eccentricity. Theβ angle is the eccentric anomaly. The orange line is the orbit of the object P.

The equation (2.7) becomes

¨ r=−µ

r³r− 1 Mpl

Q( ˙r−v_wind). (2.13) Writing (2.13) and the cross-product ofr, we have

¨

r×r=−µ

r³r×r− 1

M_plQ( ˙r×r−v_wind×r). (2.14) Using the following relations, valid for an unperturbed Keplerian orbit,

¨

r×r= _dt^d ( ˙r×r) = _dt^d

(µp)¹² bz

, (2.15)

˙

r×r= (µp)¹²bz , (2.16)

r×r= 0 , (2.17)

wherebz is an unit vector normal to the orbit, we find that Eq. (2.14) becomes

d dt

(µp)¹² bz

= −_M¹

plQ

(µp)¹²bz−λ(µp)¹²bz

, (2.18) d

dt

(M p)¹² bz

= −_M¹

plQ(1−λ) (M p)¹²bz , (2.19)

that, resolving the derivative at the first member and doing few

In a similar way, taking Eq. (2.13) and this time the dot product ofr˙ we can find the energy equation. Starting from

¨ Making use of the equation for undisturbed orbit

r= p

1 +ecosf, (2.34)

the equations (2.22) and (2.33), the “osculating condition” (that says the velocity in the disturbed and undisturbed orbits be the same), we have the equation for the semi-major axis can be written as

1−e²

Using the conditions introduced in this section we find that Eq. (2.35) can be easily expressed

Now, we examine the question of tidal stability. Different works have been done to understand the effects produced by tidal forces (Zahn 1977; Alexander et al. 1976; Zahn 1989; Livio 1982; Livio &

Soker 1984). Unfortunately, the physics behind the tidal dissipation is relatively complex that some questions remain open. For instance, when a planet has to be swallowed by host star. Nevertheless, much recent progress has been made on star-planet tides during the RG and AGB phase (Villaver & Livio 2009; Mustill & Villaver 2012; Ogilvie 2014; Villaver et al. 2014). Zahn (1977) analyzed the physical pro-cesses responsible for the tidal torque. He distinguished two parts of the tidal field: the equilibrium tide and the dynamical tide.

The equilibrium tide corresponds to a hydrostatic adjustment of the stellar structure due to the tidal forces raised by the companion, accompanied by a displacement of the fluid needed to bring about the distortion. For this tide, the dissipation mechanism, identified byZahn (1966) as the “turbulent friction”, is the interaction between the con-vective fluxes and the tidal flow.

The dynamical tide is, instead, the response that takes into account the elastic properties of the star and arises from stellar oscillations.

The stars, that possess a convective core and a radiative envelope, studied by Zahn (1977), are in particular subjected to this type of tide. The dissipative mechanism, presenting for the dynamical tide, is

the departure from the adiabaticity of the forced oscillation, produced by “radiative damping” (Cowling 1941;Zahn 1975).

Zahn(1977) evaluated carefully the efficiencies related to these dis-sipation processes. The emerged results show a greater impact for the tides in binary systems where we have stars characterized by an outer convective zone. Therefore, for the RGs, where the external region is a massive convective envelope, the turbulent friction represents the most efficient mechanism to the tidal dissipation (Zahn 1966,1977;Villaver

& Livio 2009).

For that reason, in the cases studied here, we adopt the formalism ofZahn(1977) (Eq. (2.37)), taking into account only the turbulent vis-cous dissipation in the stellar convective envelope. This phenomenon in the convective envelope produces a change on both an orbital semi-major axis and eccentricity. A geometrical representation is given in Fig. 2.4, where we show a star-planet system with spherical polar co-ordinates (r,θ,φ).

Figure 2.4: Geometry of a star-planet system. Ω_? and ω_pl are the stellar and orbital angular velocity respectively. a is the semi-major axis and iis the orbital inclination. The tidal bulge and the spherical polar coordinates are shown (r,θ,φ).

Now, we want to address the following question:

Do we have to consider the tidal bulges induced in the star by the planet, or those ones produced on the planet by the star? Or both?

We suppose only those ones trigger by the planet/brown dwarf, and not vice versa. This point was studied by Mustill & Villaver (2012) and Villaver et al.(2014). They obtained, considering many orders of

magnitude of the tidal quality factor¹ the planetary tides are negligi-ble. Therefore, only stellar tides matter has a large effect.

Concerning the tides, we study a star-planet system taking into account:

• the tides, that can change the orbital configuration, are only equilibrium tides;

• the tidal bulges are only produced on the star by the planet;

• Zahn’s approach is more precise and powerful method for stars characterized by an outer convective envelope during the RG stage.

The secular equations governs the exchanges of angular momentum and energy in a binary system. The secular equation for the semi-major axis, derived byZahn(1977) is

( ˙a/a)_t= χ τ

Menv

M_? q(1 +q) R?

a 8

Ω?

ω_pl −1

, (2.37) and it will be described in Sect.2.1.4. From Eq. (2.36) and (2.37), we obtain the equation of the orbital evolution (2.6).

In the following sections, we describe the terms in Eq. (2.6) and we give some values of each terms focusing on the case of the 15 MJ mass planet orbiting a 2 M star with solar metallicity (Z = 0.014), at an initial distance of 1 au (fiducial model) (see Fig.2.5 for the evolution of the planetary orbital radius). For stars with different masses, if not explicitly indicated, the metallicity corresponds to the solar one.