Term 1: Mass variations

, (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.

2.1.2 Term 1: Mass variations

Term 1 is characterized by the masses of star and planet and, in particular, by the mass loss rate and the rate of change in planet mass.

We describe in details these last two factors.

1 We can estimate the impact of the tidal forces using a different approach from Zahn(1977). In this method, a parameter, so-called tidal quality factor (Q), is introduced. This term replaces the unknown parameters that characterize the tidal dissipation in Eq. (2.37), simplifying considerably the equation.

Figure 2.5: Evolution of the semi-major axis for the fiducial model (blue line) as a function of time. The green dashed line indicates an unperturbed orbit. The red zone corresponds to the stellar radius.

2.1.2.1 Stellar mass loss rate

A star can be simply represented as a sphere of mass and energy, immersed in the interstellar medium. In this type of configuration, a star is considered an open system. It radiates energy, but it also emits a flux of matter. This flux of matter, generally called stellar wind, has an important role during the stellar evolution. The phenomenon of stellar mass loss has already been studied many times, in different works and for different types of stars. In this section we limit ourselves to explain the effect produced on the planetary orbit and the mass loss rate used in our models.

Intuitively, in simple planetary system composed of a star and a planet, if we decrease the stellar mass, we decrease the gravity force. Because of that, we will have an increase of the planetary orbital radius. We try to go into more detail doing very simple assumptions. An object moving in a gravitational field, as a planet around a star, has an energy characterized by two components. These components are the kinetic

energy (E_K) and potential energy (U) given by

EK= ^Mpl₂^vpl , (2.38)

U = −^GM?_d^Mpl , (2.39)

wheredis the distance between the planet and the star and G is the gravitational constant. The total energy (E_tot) of a planetary orbit is a constant and per unit of mass is given by

E_tot= E_K+U Mpl

= v_pl

2 −GM_?

d =constant, (2.40) In case of a circular orbit, the Kepler’s Third Law tells us

P²= 4π²µ GM?Mpl

a³, (2.41)

whereP is the orbital period of a planet, a the semi-major axis and µ reduced mass that is equal to M?Mpl/(M?+Mpl). Taking into account M_? M_pl and the planetary speed for a circular orbit can simply writtenvpl= 2πa/P, we obtain

E_K

The total energy can be written, using Eq. (2.42), Etot=−1

2 GM?

a . (2.43)

What happens when the mass of the star changes? We consider at timet₀ we have a stellar massM_?,0and a distance between planet and stara0. We suppose that the stellar mass will become smaller at time t₁. The potential energy will decrease and the kinetic energy, at first, remains as it was. This will produce an orbital radius larger, with an additional reduction of the potential energy and also the kinetic energy. This phenomenon will continue until a stable state is reached at time t2, and EK,2, the new kinetic energy, is numerically equal to half the size of U₂, the new potential energy. Numerically speaking, we have (from now on the kinetic and potential energy will be given per unity of mass)

E_K,0 = ¹₂^GM_a^?,0

0 , (2.44)

U₁ = −^GM_a^?,1

0 , (2.45)

and the total energy

that will be in the stable configuration Etot= U2

supposing only a single variation of the stellar mass and therefore M?,2 =M?,1. From Eq. (2.46) and (2.47), we obtain

We easily find from Eq. (2.48) the ratio between a2 anda0

a₂

a₀ = p

2p−1. (2.49)

where p is M?,1/M?,0. In Table 2.1 we show the ratios of a2/a0 at different stellar mass ratios. We can see that, for instance, withp= 0.8 the final orbital radius of a planet will become 1.33 a0. Therefore a decrease of stellar mass corresponds to an increase of the radius of the planetary orbit. If the mass loss would be the most important term for the orbital evolution, the planet will move outward until this process ends.

Table 2.1: Values of the ratio of a2/a0 at different stellar mass ratios.

M?,1/M?,0 0.99 0.98 0.97 0.96 0.95 0.80 0.70 0.60 a2/a0 1.01 1.02 1.03 1.04 1.06 1.33 1.75 3.00

In a more rigorous way,Hadjidemetriou(1963) andOmarov(1962) studied the isotropic mass loss case, in which both star and planet lose mass. They realized the equation

in order to isolate the perturbing term and quantify that. From Eq. (2.51), they obtained in terms of eccentric anomalyβ(see Fig.2.3)

da dt =−a

1 +ecosβ 1−ecosβ

M˙?+ ˙M_pl M_?+M_pl

!

. (2.52)

Knowing what is the impact of the stellar mass loss on the plan-etary orbit, we now describe which rate we considered in the models.

The stellar mass loss rates, M˙_loss, is be given by the Reimers’ law (Reimers 1975)

M˙_loss= 4×10⁻¹³ηL?R?/M? M yr⁻¹, (2.53) L?,R?andM? are the luminosity, the radius and the mass of the star.

The free parameterηgoverns the efficiency of the process. The value of this factor depends on the type of the star we study. For stars of mass in range between 9 and 15 M,η goes from 0.7 to 1.0. With smaller stellar mass the value decreases at 0.6 (for a stellar mass around 7 M) to 0.5 for red giants of lower initial masses (Maeder & Meynet 1989a). We show in Fig. 2.6 mass loss rate vs luminosity for low and intermediate mass stars (fromMaeder & Meynet (1989a)). The stars, taken into account in this work, have a mass below 3 M. For that reason, we choseη= 0.5.

The decrease of the mass of the star due to stellar winds produces a shift of the orbital radius towards larger values (the planet orbital velocity corresponds to a keplerian velocity for a larger radius orbit when the mass of the star decreases). This term does not transfer any angular momentum from the orbit to the star (of course the mass lost by the star modifies its angular momentum but this is not due to the change of the planet orbit but simply due to the mass loss).

The term due to the stellar winds can be written −^M_M^˙?

?, where we neglect the mass of the planet in the denominator, because this mass represents less than one percent (∼0.75%) of the mass of the star (the mass of the planet may increase by accretion but as we shall see this effect remains extremely modest). During the pre-RG phase the stellar wind part is very small and in our stellar evolution code, we neglect it. When the star enters the RG branch regime, stellar winds become

Figure 2.6: The mass loss rate vs luminosity for low and intermediate mass stars on the Hayashi line. FromMaeder & Meynet(1989a).

more important. For a star of 2 M, however, a little mass is lost during the ascent of the RG branch (about 0.07 M), which means a relative decrease of the mass by 3.5%. Considering only this effect, the orbital radius would thus increase by the same relative amount during that same period. This remains modest. Actually, the increase of the radius will be still much less in the present case, since the engulfment occurs before the tip of the RG branch and most of the mass loss occurs at the very end of the ascent of the RG branch. In Fig. 2.7we show the variation of the semi-major axis due to only the term −^M_M^˙?

?

for different stellar masses with Ωini/Ωcrit = 0.1 and the logarithm of the stellar mass loss as a function of time (t_{M S} is the time spent in the Main Sequence). Around the star, a planet of mass 15 MJ with an orbital radius 1 au is taken into account. With the increase of the mass loss, the impact on the orbit is higher (for instance see the case for 1.6 M star in Fig. 2.7). The behavior of the mass loss for 2.4 M is very different from the other cases. This is characterized by the fact the star reaches the He-burning in our computation (for the other stars the computations are stopped at the tip of the RG phase).

Figure 2.7: Variation of the semi-major axis due to only mass loss and the stellar mass loss as a function of time. We show different stars of mass 1.6 (blue line), 1.8 (red line), 2 (magenta line), 2.2 (black line) and 2.4 M (cyan line) and with Ωini/Ωcrit = 0.1. Top: Impact of the mass loss on the orbits of a planet of 15 MJ at 1 au during the stellar evolution. The green dashed line indicates an unperturbed orbit. Bottom: Mass loss as a function of the time.

Stellar winds may affect the orbit through another channel. The wind enhances the density of the circumplanetary environment. This increases the accretion rate of mass onto the planet. This is discussed just below.

2.1.2.2 Rate of change in planet mass

The planetary mass could be subjected to a variation. The term M˙pl is given by (Villaver & Livio 2009)

M˙_pl= ˙M_pl|acc−M˙_pl|ev , (2.54) whereM˙_pl|accandM˙_pl|ev are the accretion rate onto the planet (Bondi

& Hoyle 1944) and the evaporation rate of the planet (Watson et al.

1981;Villaver & Livio 2007). In Eq. (2.54), we suppose the accretion and evaporation processes act simultaneously. This one can happen,

for example, in case the accretion occurs from the downstream side (Livio et al. 1979).

2.1.2.2.1 Mass accreted by the planet

Planets may accrete mass by moving in the interplanetary medium which might become denser when the star has a high mass loss rate and when planet moves inward closer to the star. The accretion rate M˙_pl|acc is given by

M˙_pl|acc=πR²_Aρv_pl , (2.55)

whereR_A= 2GM_pl/v_pl² is the accretion radius,ρ is the density of the environment and G the gravitational constant. The orbital velocity of the planet v_pl is given by p

GM_?/a. When R_A . R_pl (R_pl being the radius of the planet), we have used RARpl instead of R²_A (such a situation happens at short distances). This corresponds to a correction of the geometrical radius by gravitational focusing effects (Villaver &

Livio 2009). The density is computed from the relation ρ= M˙?

4πa²v_wind , (2.56)

with vwind the wind velocity. In case stellar winds are very rarefied (which is the case in our computations before the red giant phase), we consider a density ρ=nISMmH, with nISM the number density in the interplanetary medium along the orbit of the planet andm_H the mass of a hydrogen atom.

We assume that the process of accretion does not change the ve-locity of the planet and keep constant the orbital angular momentum (thus no acceleration or slowing down due to the accretion process and no torque exerted too). In that case, from the constancy of the orbital angular momentum, one obtains that when the mass of the planet increases, the radius of the orbit decreases (and the contrary when the mass of the planet decreases, see Sect. 2.1.2.2.2). As for the stellar wind, this process (and the planet evaporation one discussed in Sect. 2.1.2.2.2) does not produce any exchange between the orbital angular momentum and the angular momentum of the star.

The accretion of mass by the planet is given by Eq. (2.55), with the densityρ equal ton_IPMm_H. During the pre-RG phase, we take for

n_IPM values between 0.001 and 5 cm⁻³. The low value corresponds to the density of the Local Bubble in which the Solar System is immersed (Cox & Reynolds 1987). The high value corresponds, instead, to a typical particle density around the Earth resulting from the solar wind, that is not exactly the same case here where we have a 2 M star.

Indeed, taking the canonical mass loss rate for the Sun, 2 10⁻¹⁴ M

per year, considering a velocity of 500 km s⁻¹, we obtain a density at a distance of 1 au equal to about 5 particles per cm⁻³. In our case, the wind may be stronger, but as we shall see even enhancing the density by large factors, the mass accreted by the planet would remain extremely low, see text for further details.

In the present case we consider a 15 MJplanet. It has according to Eq. (2.1), a radius equal to about 0.1 R. At the ZAMS, the velocity of the planet along its orbit is 42 km s⁻¹, the accretion radius RA is 3 R, thus about 30 times the radius of the planet, and the accretion rate is (see Eq. (2.55), with n_IPM=0.001 cm⁻³) of the order of 10⁻¹⁹ MJper year. With such a rate, the mass of the planet during the MS phase (about 10⁹ years for a 2 M) would increase by 10⁻¹⁰ MJ. This is completely negligible. Even considering a more realistic value for the number density of 5 cm⁻³, the accreted mass during the MS phase would be 5 10⁻⁷ MJ, again an extremely small value.

Let us now turn to the value of the accretion term when the star is losing mass during the RG phase. We use again Eq. (2.55), but with the density given by Eq. (2.56). We shall consider wind velocities between 5 and 50 km s⁻¹ (Reimers 1981). It is surprising to note that these wind velocities are much smaller than the escape velocity of a 2 M along the RG branch which varies between 450 and 70 km s⁻¹ from bottom to top of the RG branch. Actually, it is well known that there is a loose relation between the wind velocity and the escape velocity for RG stars (Reimers 1981). This comes from the way matter is accelerated into the wind. In case the acceleration occurs over large part of the trajectory of the ejected particle above the surface of the star, we can expect indeed a very loose relation between the wind and the escape velocity.

With a wind velocity of 5 km s⁻¹, the number density at 1 au varies

between a few times 10⁵ at the bottom of the RG branch up to 10⁹ cm⁻³ at the tip of the RG branch. If we consider a value of 10⁷ cm⁻³, the accretion rate will be of 10⁻¹⁰ MJ per year. The lifetime of the star during the RG ascent is 0.07 Gyr, so that the mass accreted during that phase will be around 10⁻² MJ. This remains small. We would have used instead, a wind velocity of 50 km s⁻¹, this would have decreased the accreted mass during the RG branch ascent by an order of magnitude. So again here, this effect does appear quite negligible.

In the estimates above, we assumed that the distance between the planet and the star keeps a constant value equal to 1 au. Now in case of engulfment, the distance will decrease and then we can wonder how the accretion rate will evolve. Let us consider here the case where we have that R_Ais larger than R_pl. In this case, one writes

M˙_pl|acc= Mpl

M_?

2 −M˙?

v_wind

! rGM?

a . (2.57)

We see that the accretion rate scales with the inverse of the square root of the distance. This means that passing from a distance equal to 1 au to a distance equal to 0.2 au from the centre of the star will increase the accretion rate by a factor 2.2, all other quantities being kept constant. If the accretion rate at 1 au is around 10⁻¹⁰ MJ per year (see above), at a distance five time smaller, the accretion rate would be about 2 10⁻¹⁰ MJ. So, this remains quite modest. Even if this rate would be considered during the whole 0.07 Gyr of the ascent of the RG branch, this would only increase the mass of the planet by 2%! Actually, as we will see in Sect. 2.1.4 when discussing the tidal term, the time during which the planet has a distance to the centre of the star inferior to 0.6 au is of the order of 6.5×10⁴ yr, so we see that the amount of mass that will be accreted by the planet will be extremely small during the falling of the planet onto the star. For instance, in case of the fiducial model, due to the accretion term, the mass of the planet would increase by only 7×10⁻³MJduring the whole period before the engulfment.

2.1.2.2.2 Evaporation of the planet

As we have already mentioned, planets can be heated by the stellar

radiation. Some atoms can then be lost in the upper atmosphere, when their velocity becomes larger than the escape velocity: this is the thermal evaporation term. Another way of losing mass for the planet is through the ram pressure exerted by the interplanetary medium on the moving planet. Both effects are accounted for in the expression for the evaporation rate, M˙_pl|ev, which is given by (see Jeans 1925;

Villaver & Livio 2007, for more details) M˙_pl|ev=πR²_plm_H V₀

2√

πn_c(1 +E)e^−E , (2.58) wheren_candV₀ are the number density and the velocity of the hydro-gen particles that are able to escape, andE is the escape parameter (see below). For n_c, we use the mean planetary density, which is an upper value for this quantity. V₀ = p

2k_bT_max/m_H, where k_b is the Boltzmann’s constant and Tmax is the maximum value between the equilibrium temperature of the planetT_pl(Villaver & Livio 2007) and the temperature of the shocked gasTshock (Villaver & Livio 2009). Tpl

andT_shock have been taken as follows T_pl= _L?

16πa²σ(1−A)¹₄

, (2.59)

T_shock= ^3m_16k^H

bv_pl² , (2.60)

withσ the Stefan-Boltzmann constant and A the Bondi albedo. The escape parameter is given byE =GM_plm_H/(R_plk_bTmax).

To evaluate the evaporation term, we have to chose the maximum temperature between Tpl and Tshock. We shall consider an Albedo A equal to 0.5 (the exact value of this number has no impact, see below).

It is easy to check that Tshock (of order 10⁵ K) is much larger than T_pl (of order 10³ K), by about two orders of magnitudes all along the evolution of the star until engulfment. ThusT_shock has to be used in Eq. (2.58). We can estimate the radius of the planet using Eq. (2.1) to find a value of about 0.1 R. For n_c, we use the average number density of the planet which is clearly an overestimate. A more realistic value for nc would produce a lower evaporation rate than the value obtained here. The escape parameter E is nearly always larger than 20 except when the distance becomes inferior to about 0.21 au for the case considered here. When E .20, Eq. (2.58) is not valid anymore,

(see the discussion in Villaver & Livio 2007). In case this condition is not fulfilled, we can estimate the evaporation rate as follows (Watson et al. 1981;Baraffe et al. 2004;Villaver & Livio 2007)

M˙_pl|ev = L_XUVR²₁R_pl 4a²GMpl

, (2.61)

where the part of the stellar XUV luminosity transformed in thermal energy is given by LXUV, and R1 is the planetary radius absorbing the XUV radiation.

Thus the shift to Eq. (2.61) occurs only for the very last part of the trajectory. To estimate Eq. (2.61), we assumed that LXUV is equal to L_?, clearly also an overestimate, we also took R₁ equal to R_? the actual stellar radius, again an overestimate. With all these very rough estimates, we obtain that the evaporation rate is at maximum (i.e.

just before engulfment) of the order of 10⁻⁶ MJ per year (at 1 au it would be of the order of 10⁻³¹ MJ per year!). Values of 10⁻⁶ MJ

per year are reached only at short distance (below 0.3 au) and for a small duration (a few thousand years). Thus the mass that will be evaporated remain very modest, a few ten thousands of Jupiter mass.

Moreover, as indicated above, this is an overestimate, thus we can safely conclude that this term has a negligible impact.