Destruction timescale

Livio & Soker (1984) studied the evolution of star-planet systems during the red giant phase. They find that the engulfed planets with masses equal or below 15 MJare destroyed after a few thousand years for a star of mass 3 M at AGB stage. This time-scaleτ_dcorresponds to the time necessary to cross the internal region of the star before to reach the zone where the planet is completely disappeared. The time-scale is simply evaluated as the orbital decay rate. It is given by (Siess & Livio 1999a;Livio & Soker 1984)

τ_d= where m?(r) and ρ(r) are the mass and the density at the distance r_?¹. For our fiducial model we obtain a time-scale even smaller (few tens years). This is because the region to cross is smaller than the case studied by Siess & Livio (1999a). Moreover, τ_d is very short with respect to the evolutionary timescale during the red giant phase.

Indeed, the duration of the first ascent of the red giant branch branch (i.e., before the ignition of helium burning in the core) is∼110 Myr for a 2 M star. Therefore, we can consider that the angular momentum

1 r?,m?(r)andρ(r)are given in R, Mand g cm⁻³

that remains in the orbit of the planet at the moment of the engulfment will be delivered to the star in one shot.

3.1.2 Where is the angular momentum deposited?

The location of the planet dissolution depends on the physical mechanism that is responsible for it. We can consider two main mech-anisms: thermal and mechanical destruction of the planet. Thermal dissolution is obtained where the virial temperature of the planet be-comes smaller than the local stellar temperature. This means that from this point on, the thermal kinetic energy of the stellar material is larger than the binding energy of the planet.

Knowing the internal structure of the star, it is possible to compare the local temperature in the star and the virial temperature of the planet (Siess & Livio 1999a)

T_v,pl∼ GMplµplmH

kR_pl , (3.2)

where k is the Boltzmann constant and µ_pl is the mean molecular weight for the planet. Here we consider a mass fraction of hydrogen X= 0.707, a mass fraction of heliumY = 0.273and a mass fraction of the heavy elementsZ = 0.02. With that composition,µ_pl= 0.614. We shall consider different compositions for the planet for what concerns carbon and lithium. These changes have however very limited impacts on the value of µpl and are neglected in the estimate of this quan-tity. The relation 3.2 can be easily found using the Virial Theorem.

Approximating a planet as a gaseous sphere, we can write E_K,pl^tot =−1

2U_pl , (3.3)

where E_K,pl^tot and U_pl are the total kinetic and the potential energy, respectively. We can write Eq. (3.3) as

3

2N kTv,pl= GM_pl² 2Rpl

, (3.4)

where N is the number of particles in the planet. Knowing M_pl/N = µ_plm_H and after few steps we obtain Eq. (3.2) that can be written as

inSiess & Livio (1999a)

T_v,pl∼2.3×10⁶µ_pl Mpl

0.01

0.1 R_pl

[K] . (3.5)

withM_pl and R_pl given inM and R, respectively.

Virial temperatures are typically in the range between hundreds and thousands degrees for planets of mass 1 and 5 MJ and millions of degrees for larger ones.

In Fig.3.1, the regions where the stellar temperature is equal to the virial temperature of planets of various masses are so-called dissolution points and are indicated here. Actually, the lines indicated in Fig. 3.1 are isothermal lines. These lines would not be isothermal lines in case the structure of the planet would change during its journey inside the star. Here it is assumed that during the very short migration time the planet structure does not change significantly. The only change occurs at the very end, when the planet dissolves. We also have indicated by vertical dashed lines the engulfment time of planets of various masses and initial semi-major axis as obtained in Chapt. 2. As seen above, the migration time of the planet is very short, thus the destruction of those planets will occur at the intersection of the isothermal line for the planet mass considered with the corresponding vertical lines indicating the time of engulfment (only a few cases are shown for illustration). A few interesting points can be noted looking at Fig. 3.1:

• When the evolution of the star proceeds, the dissolution point reaches (in general) deeper layers in the stellar interior. This is a consequence of the expansion of the envelope during the red giant phase that produces a lowering of the temperature at a given Lagrangian mass. Only when the star contracts, for example during the bump, the dissolution point shifts for a time outwards. This explains the local minimum that can be seen for instance for the 5 MJ mass planet at a time around 0.08 Gyr for the 2 M model (right panel of Fig.3.1).

• More massive planets go deeper inside the star. This is of course expected, since a more massive planet requires more drastic con-ditions to be destroyed than lighter ones.

• In general, the relative depth of the dissolution is larger in the 2 M than in the 1.5 M. This comes from the fact that more massive stars expand faster than less massive ones.

• Under the assumptions above and in the light of Fig. 3.1, it is reasonable to assume that the planets are destroyed in the convective envelope of the star.

Figure 3.1: Kippenhahn diagrams during the red giants phase for stars of mass 1.5 M in the upper panels and for 2 M in the lower ones with Z=0.014 and withΩini/Ωcrit= 0.1and 0.6 of the critical velocity.

The central radiative zone and the convective envelope are represented respectively by a white area and an area with black dots. The four lines indicate the limit layers that can be reached by the planets of mass 1 MJ (red solid line), 5 MJ (blue dashed line), 10 MJ (green dash-dot line) and 15 MJ (cyan solid line). tmS is the MS lifetime. The vertical dashed-lines labeled A, B and C indicate when the engulfment occurs for respectively the 15 MJ planet at an initial distance of 0.5 au, the 10 MJplanet at an initial distance of 1.0 au and the 1 MJplanet at an initial distance of 1.5 au. All the other cases occur between the lines A and C. Only for the right lower panel we indicate the case C1 that is for a planet of mass 5 MJ, because we do not an engulfment for C case.

Siess & Livio (1999a) also examined the possibility that when the

planet gets closer to the stellar core, tidal effects induce strong distor-tions of the planet that can lead to its total destruction. Using the elongation stress at the centre of the planet as approximated bySoker et al. (1987)

P_stress=ξGρ¯_plR_pl²M

R³ [dyne cm⁻²] . (3.6) where ρ¯_pl is the mean density of the planet, M and R are the mass and the radius of the object causing the destruction of the planet.

The term ξ is a parameter of order unity. The condition to have a complete disintegration of the planet is P_stress > P_c, where P_c is the central pressure of the planet defined in Eq. (2.5). Using this condition, we can find the stellar radius below which the planet is destroyed owing to the tides which is We find in most cases that the planet would be destroyed in the con-vective envelope as is the case when the criterion based on the virial temperature is used. Only in a few cases, the planet would be de-stroyed by tides just below the convective envelope (see Fig. 3.2). In the following we shall consider only the criterion based on the virial temperature and thus assume that all our engulfed planets will deliver their whole angular momentum in the convective envelope. We call L_pl,tot the angular momentum of the planet orbit and L_?(ce) the an-gular momentum of the external convective envelope (ce) of the star just before engulfment. To obtain the new angular velocity of the envelope after engulfment,Ω, we write

Ω(ce) = Ω(ce)

where Ω(ce) is the angular velocity of the convective envelope just before the engulfment. We can find this relation starting from a simple consideration that is

L_?(ce) =L_pl,tot+L_?(ce) , (3.9) where is L_?(ce) is the angular momentum of the convective envelope after the engulfment. After few steps, we obtain Eq. (3.8). In Fig. 3.3,

Figure 3.2: Kippenhahn diagram during the red giants phase for star of mass 2 M as in Fig.3.1. Here, we only show the limit layers for a planet of mass 15 MJ. The cyan and magenta solid lines indicate the layers where the planet should be destroyed using the method of the virial temperature and tides, respectively. We see that the tidal forces should smash the planet in the first layers of the radiative zone.

we show the variation of the surface velocity during the orbital evo-lution with the evoevo-lution of the semi-major axis as a function of the luminosity. This figure is for the fiducial model already introduced at the end of Sect.2.1.1.

Figure 3.3: We show the increase of the rotational velocity V_eq (gray line) of the star for the fiducial model, using Eq. (3.8). We also show the evolution of the semi-major axis (green line) and stellar radius (red line).

More details on the results will be given in Chapt.4.

Results and comparisons with the observations

The results presented in this chapter have been published in the following articles (see on pages 150 and 164):

• Star-planet interactions - I. Stellar rotation and planetary orbits Giovanni Privitera, Georges Meynet, Patrick Eggenberger, Aline A. Vidotto, Eva Villaver and Michele Bianda, A&A 591, A45 (2016)

DOI:http://dx.doi.org/10.1051/0004-6361/201528044

• Star-planet interactions - II. Is planet engulfment the origin of fast rotating red giants?

G. Privitera, G. Meynet, P. Eggenberger, A. A. Vidotto, E.

Villaver, M. Bianda, A&A, Forthcoming article Received: 21 April 2016 / Accepted: 23 June 2016

DOI:http://dx.doi.org/10.1051/0004-6361/201628758

In this chapter, we show the results coming from Chapt.2 and 3 in Sec.4.1and in Sec.4.2, respectively.

4.1 Planetary Orbital Evolution 4.1.1 Initial conditions considered

As we have already mentioned in Chapt.1, we consider stars with initial masses between 1.5 and 2.5 M, in increments of 0.1 M. The initial rotation is equal to Ωini/Ωcrit = 0.1 until 0.6, whereΩini is the initial angular velocity on the ZAMS andΩ_crit the critical angular ve-locity on the ZAMS (i.e. the angular velocity such that the centrifugal acceleration at the equator balances the acceleration due to the gravity at the equator).

The initial mass range considered in this work (M>1.5M) con-tains relatively fast rotators in contrast with lower initial mass stars (M < 1.5 M). Indeed, stars above 1.5 M do not have an enough extended outer convective zone to activate a dynamo during the main sequence, so that unless they host a fossil magnetic field, they do not undergo any significant magnetic braking. Presently only a small frac-tion of these stars (of the order of 5-10%, see the review by Donati &

Landstreet 2009, and references therein) host a surface magnetic field between 300 G and 30 kG. Lower initial mass stars have an extended outer convective zone during the main sequence and thus activate a dy-namo and suffer a strong braking of the surface by magnetized winds.

In this section, we will analyze the results for the cases with metal-licity Z = 0.02. In Chapt. 5, we will discuss models with Z = 0.014 and Z = 0.03. Planets with masses equal to 1, 5, 10 and 15 Jupiter masses (M_J) have been considered. The initial semi-major axes (a₀) have been taken in the range [0.2-4.5] au. The eccentricities of the orbits are fixed to 0. The computations were performed until the tip of the RG branch, except for some stars with higher mass (&2.3 M).

These models ignite helium in a non-degenerate regime, there is there-fore no helium flash. In those case, we pursued the evolution until the early AGB phase.

4.1.2 Planetary orbit evolution

In Fig. 4.1, we compare the evolution of the semi-major axis for planets of 1 MJorbiting around 2 Mstars with an initial slow (_Ω^Ω_critⁱⁿⁱ =

0.1) and rapid rotation (_Ω^Ω_critⁱⁿⁱ = 0.5and _Ω^Ω_critⁱⁿⁱ = 0.6) on the ZAMS. As previously obtained by many authors (see e.g. Kunitomo et al. 2011;

Villaver & Livio 2009; Villaver et al. 2014), and as recalled in the Chapt. 2, the evolution of the orbit for planets that are engulfed is a kind of runaway process. Once the tidal forces begin to play an important role, a rapid decrease of the radius of the orbit is observed as a result of the very strong dependency of the tidal force on the ratio between the stellar radius that is increasing and the semi-major axis that is decreasing (term in (R?/a)⁻⁸ in Eq. (2.37)). Comparing the top, central and bottom panels of Fig. 4.1, we see that the interval of initial distances leading to an engulfment during the RG branch is shallower for the slower rotating models (see the change of the limit between the continuous red and the dashed blue lines). As explained in Sect.1.2, larger the initial rotation rate, lower the minimum initial mass for igniting helium in a non-degenerate core and thus lower the luminosity at the RG tip. A lower luminosity leads to a lower value for the maximum stellar radius, and thus to more restricted conditions for having an engulfment. Figure 4.2 shows the impact of different initial rotation rates for a 2 M on the orbital decay of planets of various masses beginning all their evolution at a distance equal to 0.5 au. The orbit for the 15 MJ planet for instance presents a different evolution around the slow and the fast rotator. The orbital decay occurring around the slow rotating star (magenta dashed-dotted line on the top panel) occurs just during the bump. The decrease of the semi-major axis slows down when the stellar radius decreases since the tidal force depends on (R_?/a)⁻⁸. Around the fast rotating model (see the magenta dashed-dotted line on the central panel), the orbit decay occurs just at the beginning of the bump and is quite rapid.

As mentioned above, this illustrates that stellar rotation changes the structure of the star and modifies thereby the evolution of the orbit. In general, the engulfment occurs at an earlier evolutionary stage when the initial rotation rate increases. If we compare the Figs 4.2 central and bottom panels, we do not note an important variation on the orbits. The impact of the bump in both cases seems to drive the main effect on the orbital evolution.

Figure 4.1: Top panel: Evolution of the semi-major axis of a 1 MJ

planet orbiting a 2 M star computed with an initial rotation on the ZAMS of Ω_ini/Ω_crit = 0.1. The different lines correspond to different initial semi-major axis values. Only the evolutions in the last 50-60 million years are shown. Before that time, the semi-major axis remains constant. The red solid lines represent the planets that will be engulfed before the star has reached the tip of the RG branch; the blue dashed lines represent the planet that will avoid the engulfment during the RG branch ascent. The upper envelope of the green area gives the value of the stellar radius. Central panel: Same as the top panel, but for an initial rotationΩ_ini/Ω_crit = 0.5. Bottom panel: Same as the top panel, but for an initial rotation Ω_ini/Ω_crit= 0.6.

Figure 4.2: Top panel: Evolution of the orbit for planets of masses equal to 1 MJ (blue solid line), 5 MJ (black dotted line), 10 MJ (red dashed line) and 15 MJ (magenta dashed-dotted line) around a 2 M

star with an initial rotation equal to 10% the critical angular velocity.

The initial semi-major axis is equal to 0.5 au. Central panel: Same as the top panel, but for an initial rotation rate equal to 50% the critical angular velocity. Bottom panel: Same as the top panel, but for an initial rotation rate equal to 60% the critical angular velocity.

4.1.3 Impact of stellar rotation on the conditions lead-ing to an engulfment

As already discussed in the literature (see e.g. Villaver & Livio 2009;Villaver et al. 2014), we see that for a given mass of the planet and for given properties of the host star, there exists a maximum ini-tial semi-major axis below which engulfment will occur during the RG phase. These maximal values are given in Table 4.1 and shown in Fig. 4.3. These results illustrate that the conditions for engulfment are more restricted around fast rotating stars, but the effect remains quite modest for the 1.5 and for the models with mass&2.3 M, while it is more important for the intermediate mass cases for the reasons already explained above. In top panel of Fig.4.3, we note that the ef-fects of rotation do not produce overlaps between the curves, indicating that the change due to the initial mass dominates over the change due to rotation. The present results for the evolution of planetary orbits are well in line with previous works (Kunitomo et al. 2011; Villaver

& Livio 2009; Villaver et al. 2014). More specifically, we also obtain that the conditions for engulfment are more favorable for more mas-sive planets and less masmas-sive stars (note that this is because the less massive stars reach larger luminosities at the tip of the RGB). More-over,Kunitomo et al.(2011) found that the orbital radius above which planet engulfment is avoided is quite sensitive to the stellar mass at the transition between those going through a helium flash and those avoiding the helium flash. For the massive stars that avoid the helium flash, the impact of the rotation and the stellar mass is less impor-tant. Qualitatively, this is exactly what we find here (see top panel of Fig. 4.3). The general behavior tell us that with the increase of the initial rotation, the stellar radius, reached at the tip of the red giant phase, will be smaller. This is true until the rotation is high enough to give an extra effect. This effect changes the threshold to go through the helium flash, that will correspond to higher value for the luminosity at the tip (see Sec. 1.2). Such effect explains why for 2.3 M, with the increase of the initial rotational velocity we obtain smaller survival limits (top panel of Fig. 4.3). We have also plotted the semi-major axis above which no engulfment occurs (survival limit)

Table 4.1: Initial semi major axes below which the planet is engulfed during the RG phase for different initial rotational velocities and plan-etary masses. The models with high rotational velocity (0.6), that has been studied, are 4.

Figure 4.3: Variation of the maximum semi-major axis below which engulfment occurs during the RG ascent as a function of the mass of the planet, of the mass of the star and its initial rotation rate. The dashed, short-long dashed and continuous lines are respectively for an initial stellar angular velocity 10%, 30% and 50% of the critical angular velocity. The empty symbols connected by light dashed-dotted lines are the results obtained byVillaver et al.(2014) for planets with masses of 1, 2, 5 and 10 MJorbiting a non-rotating 1.5 Mstar. The triangles are for models using weak mass loss rates during the red giant phase (η= 0.2in Eq. 2.53), the circles are for models with normal red giant mass losses (η= 0.5), and the squares are for models withη = 0.5and overshooting.

predicted by Villaver et al.(2014) for planets with masses between 1, 2, 5 and 10 MJ around a non-rotating 1.5 M computed with differ-ent physical ingredidiffer-ents (see caption). The upper curve from Villaver et al. (2014) was obtained with a smaller mass loss rate during the ascent of the RG branch. Lowering the mass loss rate leads to larger radii at the tip of the RG branch and thus shifts the survival limit to larger values. The two lower curves by Villaver et al. (2014) use the

predicted by Villaver et al.(2014) for planets with masses between 1, 2, 5 and 10 MJ around a non-rotating 1.5 M computed with differ-ent physical ingredidiffer-ents (see caption). The upper curve from Villaver et al. (2014) was obtained with a smaller mass loss rate during the ascent of the RG branch. Lowering the mass loss rate leads to larger radii at the tip of the RG branch and thus shifts the survival limit to larger values. The two lower curves by Villaver et al. (2014) use the