Term 3: Tidal forces

As we have already said, since we are dealing with stars off the main sequence, the expression for the tidal term ( ˙a/a)_t is the one corresponding to the case where the star has a massive convective en-velope. In that case, the waves driven by the tidal forces in the stellar envelope can be very efficiently dissipated. The tidal dissipation mech-anism is due to the equilibrium tide. As a result, angular momentum passes from the planet orbit to the star or the inverse depending on whether the orbital angular velocity of the planet is smaller or larger than the axial angular rotation of the star.

The term 3 is given by (Zahn 1966,1977,1989):

( ˙a/a)_t= χ τ

M_env M?

q(1 +q) R_?

a 8

Ω_? ωpl

−1

, (2.64) with χ a numerical factor taken equal to (P/2τ)² when τ > P/2 in order to consider the only convective cells that give a contribution to

Figure 2.9: Evolution of the radius of the orbit resulting from the action of the stellar winds, forces of friction and gravitational drags only compared to the evolution of the radius when all these three terms are accounted simultaneously and when the tidal term is also accounted for.

viscosity, otherwise we take χ = 1 (Villaver & Livio 2009), M_env is the mass of the convective envelope, q =Mpl/M?, Ω? is the angular velocity at the surface of the star, ω_pl is the orbital angular velocity of the planet. τ is the eddy turnover timescale, which is taken as in Rasio et al. (1996)

τ =

"

Menv(R?−Renv)² 3L_?

#1/3

, (2.65)

withRenv the radius at the base of the convective envelope of the star.

As we shall see, the term 3 is clearly the leading term in Eq. (2.6).

In the present work, we consider the effects of equilibrium tides only during the red giant phase, when a well developed external convective zone is present (see Eq.2.64and the discussion about the derivation of the tidal term above).

We show in Fig.2.11 (top panel) the temporal evolution of a few quantities for a phase comprising the 100 Myr before engulfment. The dashed line, in particular, shows the evolution of the semi-major axis of

the orbit of the planet (D_orb) accounting for all the terms in Eq. (2.6).

We see that the appearance of the external convective envelope (i.e. whenM_env becomes non-zero) is concomitant with an inflation of the star. This is because inflation increases the opacity in the external layers and thus favors convection. This inflation lowers considerably the surface angular velocity of the star pushing out the corotation radius (D_corot), which becomes, from this time on, much larger than the actual orbital radius (D_orb). Indeed, on the ZAMS, the orbital period of the planet (∼0.7 year) is much larger than the rotation period of the star (2.34 days), thus the planet orbits well outside the corotation radius. The corotation radius is 0.0497 au, much smaller than the semi-major axis of the planet orbit. This happens for all the cases studied in the present paper.

Why is it important to know the corotation radius? The corotation radius corresponds to the orbital radius for which the orbital period would be equal to the rotation period of the star. It is given by the following relationDcorot= (GM/Ω²_∗)^1/3. When the actual distance of the planet to the star is inferior to the corotation radius, tidal forces reduce the orbital radius, while when the actual distance of the planet to the star is larger, tidal forces enlarge the orbital radius. This is accounted for in Eq. (2.64) through the term

Ω?

ωpl −1

. The two panels of Fig. 2.10 show the evolution of the corotation radius for various rotating models (without planets). The top panel shows the situation during the MS phase and the crossing of the HR gap, while the bottom panel shows the evolution along the RG branch (and in case of the 2.5 M also during the core He-burning phase).

In order to have an engulfment, a necessary condition (but of course not a sufficient one) is that the actual radius of the planetary orbit is inferior to the corotation radius. For an engulfment to occur, it is in addition needed that the tidal forces are of sufficient amplitude when the orbital radius becomes smaller than the corotation radius. We see that the corotation radius is very small during the MS phase. Even in case tidal dissipation would be efficient at that stage, tidal forces can only decrease the radius of the orbit when the distance is below 0.05-0.15 au if the star has a quite low initial rotation rate (Ω_ini/Ω_crit= 0.1).

Figure 2.10: Top panel: Evolution of the corotation radius (in au) for rotating stellar models with no tidal interactions during the RG branch phase. For the 2.5 M, the core He-burning phase is also shown. The rectangle area indicated at the left bottom corner is the region that is zoomed on the bottom panel. Bottom panel: Same as the top panel, but the MS phase and the crossing of the Hertzsprung-Russel (HR) gap are shown.

If the star is initially rotating rapidly (Ω_ini/Ω_crit >0.6), in case of the 2 M, the corotation radius is inferior to 0.01 and 0.03 au during the MS phase. Along the RG branch, the corotation radius increases a lot as a result of the decrease of the surface rotation rate. The corotation radii are shifted downwards when the initial rotation rate increases, as was the case during the MS phase.

In our computations, the tidal forces will make the orbital radius to decrease as a function of time (inward migration). The falling of the

planet accelerates as a function of time. This is shown in Table2.3.

To better understand what happens, let us adopt for a while a simplified analytic approach. Let us consider the simplified equation (from Eq. (2.64))

a⁷( ˙a)_t =−χ τ

M_env M?

q(1 +q)R⁸_?

| {z }

c

, (2.66)

During a period, sufficiently short before engulfment, the righthand term can be considered constant and in that case, the equation has an analytical solution given by

a(t) = (a⁸₀−8C(t−t₀))^1/8, (2.67) where a₀ is the radius of the orbit at the time t₀. From Fig. 2.11 (top panel), we see that during the last 10⁵ yrs, the characteristics of the star keep nearly constant values. Let us adopt the properties of the star at engulfment for the fiducial model (see Table 2.2). In that case, we obtain the (red) continuous line labeled by a 1 in the bottom panel of Fig. 2.11. We see that even keeping constant the stellar characteristics, there is a strong acceleration of the falling down due to the ever increasing tidal forces when the distance to the star decreases.

The line obtained by the analytic formula begins to overlap the dashed line obtained following the stellar characteristics, but falls down a few thousands years before the dashed line. This corresponds actu-ally only to a slight difference when compared to the duration of the whole process that lasts for 61 Myr; it is however interesting to note that this difference comes from slight changes in the radius during that phase. It happens that during the last 10⁵ yr before engulfment, the radius of the star changes from 44.02 to 44.36 R. Changing the radius of the star from 44.36 to 44.14 R (decrease of ∼0.5%) gives the line 2 and passing from 44.14 to 43.70 (a further decrease of ∼1%) gives the line 3. This high sensitivity reflects the fact that the ratio of the stellar radius to the orbit radius is at the power 8!

This acceleration implies that the timescales in the very last part of the process are much shorter than the typical time steps normally

Figure 2.11: Evolution as a function of time of a few relevant quantities during the last 100 Myr before engulfment for our fiducial model. The horizontal axis is the logarithm of the difference between a reference time (t_ref) taken here equal to 1.16327e+09 years and the age of the star. We chose a reference time slightly above the age of the star at engulfment to avoid to have zero for this difference which would imply an undetermined value when the logarithm is taken. The short-dashed line labeled by Dorbshows the evolution of the semi-major axis of the orbit of the planet in units of au. Superposed to this short-dashed line, are shown three continuous lines (in red) showing the behavior of the semi-major axis of the orbit given by Eq. (2.67) with three different values of the radius of the star. The curve labeled by a 1 is for a radius equal to 44.36 R, the lines labeled by a 2 and 3 are for radii equal to respectively 44.14 and 43.70 R. The dotted line labeled by Dcorot is the corotation radius. The continuous lines (Menv), long-dashed line (R?) show the evolution of the mass (in M) of the external convective envelope and of the radius of the star (in au). The bottom panel is a

Table 2.2: Properties of the star on the ZAMS and at the time of planet engulfment for our fiducial model. “Env." refers to the external convective envelope of the RG star. Furthermore, at the engulfment, we indicates the rotational velocity of the star taking into account the transfer of the orbital angular momentum into the convective envelope.

Otherwise, Veq should be around 0.3 km s⁻¹. AT THE ZAMS

Mass 2 M

Radius 1.529 R

Veq 33.0 km s⁻¹

AT THE ENGULFMENT

Age 1.1632 Gyr

Mass 1.98477 M

Radius 44.36 R

Veq 15.98 km s⁻¹

Ω 5.180E-07 s⁻¹

Luminosity 443.21 L

Mass loss rate -1.9802E-09 M yr⁻¹

Mass (Env.) 1.656 M

Radius (Base of Env.) 0.9037 R

Inertia (Env.) 4.348E+57 cm² g Momentum (Env.) 5.1412E+49 g cm² s⁻¹

used for computing stellar models. During the few hundred years that precedes the engulfment, the radius of the star changes so little that we can also safely use the analytic approach. The timescales indicated in Table 2.3 for the last two lines have actually been obtained using the analytic formula.

There are a few interesting points that we can discuss now, having in hand the behavior of the orbit until the engulfment. One can won-der whether during the falling down of the planet, it is reasonable to assume that the planet always keeps a keplerian velocity? Or in other words, whether the balance between the gravity and the centrifugal ac-celeration has always time to set up? We could imagine for instance a braking so efficient that it would stop the planet on a timescale shorter than the dynamical timescale. In that case, the planet would then free fall on to the star without orbiting around it. Actually this is not

pos-Table 2.3: Duration for passing from a planet orbit equal to R1 to R2

starting at a time about 61 Myr before the engulfment.

R1 [au] 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 R2 [au] 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

∆t[1000 yr] 56442 3679 774 230 65 16 3 0.3

sible because the tides are dissipative processes that have much larger timescales than the dynamical timescale. This can be seen in Fig.2.12 (top panel), where the tidal timescale is compared to the dynamical timescale. We see that there are orders of magnitudes between these two times, indicating that the planet fall onto the star keeping at each instant a Keplerian velocity.

Another point of interest is to look at the relative period variation when the planet is falling down onto the star. This variation is shown in the bottom panel of Fig. 2.12 for the last part of the evolution of the orbit. We see for instance that when the orbital period is 0.1 yr, the relative variation of the period is around 0.0018. This means that the period of 36 days 12.0 hours at a given year will become a period equal to 36 days 10.42 hours the following year (a decrease of ∼1.6 hours). Such a period difference will likely be detectable in the future.

Of course hardly any systems will be observed at a stage so close from being engulfed since the duration of those stages are quite short.

Tidal forces transfer angular momentum from the orbit to the star.

One can wonder whether, well before the engulfment, this transfer may accelerate the rotation at the surface of the star. The answer is yes.

Indeed, the angular momentum in the orbit when the planet is at 1 au is about 1.8×10⁵¹ g cm² s⁻¹, that means around 32 times the angular momentum of the star on the ZAMS. When the planet fall from the distance of 1 au to 0.2 au, it will lose an angular momentum of about 9.98×10⁵⁰g cm² s⁻¹, that means more than half of its initial angular momentum. If we consider that at engulfment the angular momentum of the external convective envelope is ∼0.5×10⁵⁰ g cm² s⁻¹, then we see that, roughly, the angular velocity would increase by about a factor 10, as well as the surface velocity. Of course this

Figure 2.12: Top panel: Comparison between the timescales for the braking of the planets due to tides and the free fall timescale at dif-ferent distance from the star. The timescale for the braking has been estimated by(a/a)˙ t and the free fall timescale byp

a³/GM?. Bottom panel: Relative period variations during the falling down of the planet onto the star as a function of the orbital period. The vertical lines show the periods corresponding to different radii of the orbit. The dash-dot line on the right indicates the period just before engulfment.

Both panels concern the fiducial model at the RG phase.

is a rough estimate, it does not account for the effects of internal transport processes, neither for the change of angular momentum due to mass loss, or for the change of the radius or of the lower limit of the convective envelope, it also does not account for other mechanisms

Figure 2.13: The last part of the evolution of the radius of the orbit is shown here computed from Eq. (2.67) (green dashed-line), from Eq. (2.6), but keeping Ω? constant (cyan dash-dot line), and from Eq. (2.6), assuming that the loss of orbital angular momentum is completely transferred to the star (blue continuous line).

slowing down the planet along its orbit like drag forces that do not produce any transfer of angular momentum to the star. Actually, we saw in the previous sections that the drag forces have a small impact on the orbital evolution compared with the tidal term. Only a small percentage of the total angular momentum is lost in the matter around the planet and is not transferred to the convective envelope of the star (more details will be given in Chapt. 4) .

The simplified Eq. (2.67) corresponds to the approximation Ω?

ωpl −1

∼ −1 in Eq. (2.6). Moreover, when this term is accounted for one should account for the change ofΩ_? resulting from the transfer of angular momentum by tides from the orbit to the star. The impact of these various effects can be seen in the left panel of Fig. 2.13. We see that the impacts are not negligible but remain modest. They will mainly shift the time of the engulfment by about 60 thousand years, but would not change significantly otherwise the overall shape of the falling curve.

From the discussion above, we can deduce that for the case con-sidered here, the most important term in Eq. (2.6) is by far the tidal

term. The impact of the stellar winds and of the friction and drag forces act at the level of the percent when one looks for the relative changes of the orbit. This is quite in agreement with previous works (seee.g. Villaver & Livio 2009;Villaver et al. 2014). The mass of the planet is changed (by accretion or evaporation) by less than the per-cent. This is illustrated in Fig. 2.9. Similar discussions could have been made for the various cases studied here and similar qualitative conclusions would have been deduced, indicating that the tidal term remains the most important one in every cases. The results will show in the Chapt.4

CHAPTER 3

Physics of engulfment

In the previous chapter we studied how the orbit of a planet changes, taking into account the main effects that influencing it. Now we ana-lyze the changes of the stellar rotation due to the orbital decay makes the planet to be engulfed by the star. Rotating stellar models allow to follow in a consistent way the angular momentum transport inside the star. More precisely we want to address the following questions:

• By how much the surface rotation can increase through a planet engulfment process?

• How does this increase depend on the initial mass and rotation of the star, on the mass of the planet and its initial distance to the star?

• When the surface of a star is accelerated either by tidal forces or by planet engulfment, how long is the period during which a rapid surface velocity can be observed? This is an important point to know for checking whether such a process has indeed a chance to leave some observational signatures in red giant stars.

• Can the planet engulfment process be the unique cause for at least some of the observed fast rotating red giants?

• Will this inclusion of angular momentum inside the star change its further evolution? Can for instance the increase of the surface velocity trigger some internal mixing?

• Linked to the above points, are there other expected signatures in addition to fast surface rotation linked to a planet engulfment event?

• Is the observed frequency of fast rotating red giants in agreement with theoretical expectations assuming fast rotating red giants are all produced by planet engulfment?

In the following sections we will discuss the physics included in our models and how we determine where in the star, the planet will be destroyed and deliver its angular momentum and we will present post-engulfment evolution of various stellar models.

3.1 Physics of the engulfment

For a given mass of a star and mass of the planet, we studied in Chapt. 2 the range of initial semi-major axis that leads to a planet engulfment along the red giant branch. First of all, one can wonder if the spiraling planet is swallowed by the star or it is destroyed before to reach that. We know that a planet is subjected to the stellar radiation.

This process works in order to smash the planetary object. We have already seen in Sect. 2.1.2.2.2 the planetary evaporation provoked by the star should not bring to a complete destruction of a planetary object with mass & 1 MJ. The time spent close to the star is short enough to avoid the destruction. In Sect. 2.1.4, we showed that to across the region between 0.3 - 0.2 au, the planet needs only some hundreds years. This time, indeed, is not enough for having a complete disintegration of the planet.

Now, in order to describe what will happen next, we need some prescriptions for the fate of the planet inside the star. According to the work bySoker et al. (1984), planets with masses inferior or equal to about 20 MJwill be destroyed in the star. In the present work, we

consider only planets with masses equal or below 15 MJ, assuming the complete destruction.

In order to be able to compute the impact on the surface velocity of this destruction process we need to know two quantities. The first one is the timescale for the destruction of the planet and the second one is the location of this destruction inside the star. For instance, if the destruction occurs in a very short timescale (much shorter than the evolutionary timescale) then, once engulfment occurs, the whole orbital angular momentum of the planet orbit will be given to the star in one shot. Moreover, if the destruction occurs in the convective envelope, then this angular momentum can be added in a very simple way to the convective envelope (see below).