Thesis
Reference
Star-planet interactions: planetary orbital evolution and Engulfment
PRIVITERA, Giovanni
Abstract
L'orbite d'une planète peut changer significativement au cours du temps. À cause des forces de marée entre l'étoile et la planète, des échanges entre le moment angulaire orbital de la planète et le moment angulaire rotationnel de l'étoile peuvent survenir. Lorsque l'orbite de la planète est à l'intérieur du rayon de corotation, ces interactions de marées tendent à transférer du moment angulaire de l'orbite de la planète à l'étoile. Cela signifie que la distance entre la planète et l'étoile diminue et que la vitesse de rotation de l'étoile augmente. L'inverse se produit quand le rayon de l'orbite de la planète est plus grand que le rayon de corotation.
Le principal objectif de la thèse était d'identifier, à l'aide de modèles théoriques, des signatures claires d'interaction de marée/d'engloutissement puis de vérifier si ces caractéristiques sont observées.
PRIVITERA, Giovanni. Star-planet interactions: planetary orbital evolution and Engulfment. Thèse de doctorat : Univ. Genève, 2016, no. Sc. 5013
URN : urn:nbn:ch:unige-915140
DOI : 10.13097/archive-ouverte/unige:91514
Available at:
http://archive-ouverte.unige.ch/unige:91514
Disclaimer: layout of this document may differ from the published version.
Star-planet interactions:
Planetary orbital evolution and Engulfment
THÈSE
présentée à la Faculté des sciences de l’Université de Genève
pour obtenir le grade de Docteur ès sciences, mention astronomie et astrophysique
par
Giovanni Privitera
de
Catane (Italie)
Thèse No 5013
GENÈVE
Acknowledgments V
Interactions entre étoiles et planètes: Évolution de l’orbite de la planète et son engloutissement par l’étoile VII
Introduction 1
1 Physics of the evolution of the star 8
1.1 Basic input physics . . . 8
1.1.1 Equations of stellar evolution . . . 8
1.1.2 Nuclear reaction rates . . . 10
1.1.3 Opacities . . . 11
1.1.4 Equation of state . . . 11
1.1.5 Convection and overshooting . . . 11
1.1.6 Mass loss . . . 12
1.1.7 The diffusion equations . . . 12
1.1.8 Transport of angular momentum . . . 15
1.2 Initial conditions of the stars and impact of rotation . . 16
2 Physics of the evolution of the planetary orbit 21 2.1 Physical aspects. . . 22
2.1.1 The secular equation . . . 25
2.1.2 Term 1: Mass variations . . . 31
2.1.3 Term 2: Drag forces . . . 42
2.1.4 Term 3: Tidal forces . . . 45
3 Physics of engulfment 56 3.1 Physics of the engulfment . . . 57
3.1.1 Destruction timescale . . . 58
Contents
3.1.2 Where is the angular momentum deposited? . . . 59 4 Results and comparisons with the observations 65 4.1 Planetary Orbital Evolution . . . 66 4.1.1 Initial conditions considered . . . 66 4.1.2 Planetary orbit evolution . . . 66 4.1.3 Impact of stellar rotation on the conditions lead-
ing to an engulfment . . . 70 4.1.4 Impact of the orbital decay on the stellar rotation 73 4.1.5 Comparisons with observed systems . . . 76 4.2 Planet Engulfment . . . 80
4.2.1 Impact of planet engulfment on the surface ve- locities of red giants . . . 80 4.2.2 Planet engulfment by a 1.5 M star . . . 80 4.2.3 Planet engulfment by stars with masses between
1.6 and 2.5 M . . . 86 4.2.4 How long does a red giant star remain fast ro-
tating after an engulfment? . . . 89 4.2.5 Impact of tidal forces with respect to engulfment 91 4.3 Comparisons with observations . . . 92 4.3.1 Stars with a clear signature of a past interaction 95 4.3.2 Can planet engulfment explain the cases likely
resulting from an interaction? . . . 96 4.3.3 Can the initial conditions be deduced from ob-
servations? . . . 96 4.3.4 The surface compositions after an engulfment . . 97 5 Planetary orbital evolution and planet engulfment:
Study on the effect of the metallicity 104 5.0.5 Planetary orbit evolution . . . 104 5.1 The dissipation region for stars with different metallicity 109 5.2 Impact of planet engulfment on the surface velocities of
red giants . . . 111 5.3 The surface compositions after an engulfment . . . 113 6 Surface magnetic field as a new signature of planet en-
gulfment 117
6.1 Physics of the models . . . 118 6.2 Evolution of the Rossby number. . . 122 6.3 Planet engulfment and surface magnetic field . . . 124
Conclusions 134
engulfment . . . I
Bibliography VIII
Acknowledgments
The project of my thesis started in 2012 at the University of Geneva.
Studying here was for me a wonderful experience that enriched me, both personally and professionally. I worked with nice and incredible people, that love physics and their job. One of this is Prof. Georges Meynet. His passion and experience inspired me and explained how an excellent physicist should be. Thanks a lot Georges for being a mentor. Thanks for the time spent to help me and for your patience (you really had a lot!), for the continuous support of my Ph.D. study and related research, for your motivation and immense knowledge.
Grazie di cuore!
My sincere thanks also goes to Dr. Patrick Eggenberger for innumer- able and interesting discussions had in these years. He spent really a lot of his time following me in this complex project.
Without their passionate participation and input, this thesis could not have been successfully conducted.
A particular thanks to the IRSOL institute. This project, indeed, has been possible thanks the collaboration between the University of Geneva and IRSOL. Grazie mille al Dr. Michele Bianda e al Dr.
Renzo Ramelli per avermi dato la possibilità di svolgere il mio dot- torato su questo interessante progetto e in questo bellissimo paese che è la Svizzera.
I am also grateful to the coauthors of the papers published. In particular, thanks to Prof. Eva Villaver and Prof. Aline Vidotto for the important advices and for your precious contributions.
Thanks also to the other members of the stellar group, among which Dr. Lionel Haemmerlé, Dr. Sylvia Ekström, Dr. Cyril Georgy, Prof. José Groh and Arthur Choplin. Thanks a lot Lionel for the huge help that you gave me!
Vorrei inoltre ringraziare il piccolo gruppo di italiani che ho conosci-
Martina; Rosaria, Marco e Gabriele; Claudia, Bruno e i loro figli Teo e Gaël.
Grazie a tutti i miei amici. Nonostante le migliaia di chilometri che ci separano, so di poter sempre contare su di voi. Vi voglio un mondo di bene.
I would like to thank my fiancée. Grazie mille amore mio per essermi sempre stata vicino in questi anni, per avermi sempre incorag- giato e supportato/sopportato.
Last but not the least, I would like to thank my family: grazie mille ai miei genitori, a mio fratello Antonino e alle mie sorelle Maria, Domenica e Concetta, ai miei cognati e tutti i miei bellissimi nipotini per tutto l’amore e l’affetto che mi avete dato.
Abstract
L’orbite d’une planète peut changer significativement au cours du temps. À cause des forces de marée entre l’étoile et la planète, des échanges entre le moment angulaire orbital de la planète et le moment angulaire rotationnel de l’étoile peuvent survenir. Lorsque l’orbite de la planète est à l’intérieur du rayon de corotation, ces interactions de marées tendent à transférer du moment angulaire de l’orbite de la planète à l’étoile. Cela signifie que la distance entre la planète et l’étoile diminue et que la vitesse de rotation de l’étoile augmente. L’inverse se produit quand le rayon de l’orbite de la planète est plus grand que le rayon de corotation. Le rayon de corotation correspond à la distance à laquelle la période orbitale de la planète est égale à la période de rotation de l’étoile (système synchronisé).
De nombreux travaux ont été effectués dans le passé pour déter- miner l’évolution des planètes dans le système solaire et plus générale- ment des planètes autour d’étoiles de différentes masses initiales et ayant différentes métallicités. Une conclusion intéressante de ces travaux est qu’à cause des interactions de marées, les planètes orbitant suff- isamment proche de leur étoile pendant la Séquence Principale sont englouties par l’étoile hôte quand celle-ci devient une géante rouge.
Quand ce phénomène se produit, les modèles prédisent que des change- ments survenant à la surface de l’étoile peuvent être observés: une aug- mentation rapide et transitoire de la luminosité (Siess & Livio 1999b), une modification de l’abondance du lithium à la surface (Carlberg et al.
2010; Adamów et al. 2012) ou une augmentation de la vitesse de ro- tation à la surface de l’étoile (Siess & Livio 1999a,b; Carlberg et al.
2009, 2010). Il est intéressant de noter que certaines étoiles présen- tent des caractéristiques observationnelles qui pourraient être liées à l’engloutissement d’une planète comme par exemple les quelques pour- cents de géantes rouges ayant une vitesse de rotation élevée (Fekel &
Le principal objectif de la thèse de Giovanni Privitera était d’identifier, à l’aide de modèles théoriques, des signatures claires d’interaction de marée/d’engloutissement puis de vérifier si ces caractéristiques sont observées. Les résultats principaux de cette thèse ont fait l’objet de trois articles publiés dans la revue Astronomy & Astrophysics (revue à comité de lecture). Le premier (Privitera et al. 2015) étudie com- ment l’orbite planétaire évolue sous l’influence des forces de marée.
Nous avons déterminé les conditions initiales (masses de l’étoile et de la planète, distance initiale entre la planète et l’étoile) qui mènent à l’engloutissement de la planète pendant la phase géante rouge. Nous nous sommes concentrés sur le cas de l’engloutissement pendant la phase géante rouge car nous voulions savoir à quel point les quelques pour-cents de géantes rouges à rotation rapide pouvaient être expliqués par l’engloutissement d’une planète. Dans ce premier article, nous avons montré que ces interactions de marée peuvent augmenter sig- nificativement la vitesse de rotation de surface des géantes rouge et les amener à des valeurs compatibles avec celles observées pour les géantes rouges en rotation rapide. Nous mettons l’accent sur le fait que ces modèles sont les premiers où un réel couplage est effectué en- tre le modèle stellaire en rotation et l’orbite de la planète. Les études précédentes étaient basées sur des modèles beaucoup plus simples sup- posant une rotation solide de l’étoile. On sait aujourd’hui par les observations astérosismiques que les géantes rouges ne tournent pas dans leur intérieur en rotation solide (Beck et al. 2012;Mosser et al.
2012;Deheuvels et al. 2012,2014).
L’engloutissement des planètes est le sujet du second article (Priv- itera et al. 2016b). La rotation de surface est encore augmentée lorsque la planète est engloutie par l’étoile. Nous avons montré que l’engloutissement peut mener la surface de l’étoile à des vitesses de rotation non at- teignables sans interaction avec un second corps. En d’autre termes, nous avons montré qu’une étoile simple sans planète et débutant son évolution avec le maximum de moment cinétique ne peut pas atteindre ces grandes vitesses de rotation de surface pendant la phase de géante rouge. Des géantes rouges avec une telle vitesse de rotation à la sur- face ont été observées. Cette étude montre donc que ces étoiles ont du nécessairement interagir avec un compagnon dans le passé.
Nous avons aussi étudié à quel point d’autres signatures observa- tionnelles, telles que des changements de l’abondance de surface du lithium, pourraient survenir d’un engloutissement. L’enrichissement en lithium (Li) de la surface apparait comme un effet possible de l’engloutissement d’une planète, mais le Li est un élément fragile et cette signature n’est pas claire. Par conséquent, une vitesse de rota-
tion à la surface élevée apparait pour le moment comme la meilleure signature de l’engloutissement d’une planète.
Dans le dernier et troisième article (Privitera et al. 2016a), nous avons exploré la possibilité que l’engloutissement d’une planète active un effet dynamo et produise un fort champ magnétique de surface.
À l’aide de relations semi-empiriques liant rotation à la surface, pro- priétés de la zone convective externe et champ magnétiques de sur- face, nous avons pu établir des prédictions quant au champ magné- tique à la surface des géantes rouges ayant englouties une planète.
Nous avons montré que l’engloutissement d’une planète peut produire un fort champ magnétique de surface dans la partie supérieure de la branche des géantes rouges où l’on ne s’attend pas à trouver un champ magnétique de surface pour les étoiles sans planètes. Nous avons donc identifié une nouvelle signature observationnelle de l’engloutissement d’une planète. Il est à noter que ce papier a suscité une demande de temps pour observer le champ magnétique de géantes rouges en rotation rapide.
In the last decades we faced the discovery of an incredible number of planets, but most surprisingly one of the important lessons of the extra-solar planet discoveries is the very large variety of systems en- countered in nature. This has triggered new excitement and interest in this field. Many researchers have tried to answer to questions related to the different structures and characteristics of the planets, their life, their death and their eventual impact on the evolution of their host stars.
The star-planet interaction can happen through several processes.
Among all these possibilities, the tidal forces can play a crucial role.
In some cases, tidal forces between the star and the planet can become so large that the semi-major axis of the orbit decreases, which leads to the engulfment of the planet by the star. When this occurs, models in- dicate that changes at the stellar surface can be observed: a transitory and rapid increase in the luminosity (Siess & Livio 1999b), a change in the surface abundance of lithium (Carlberg et al. 2010; Adamów et al. 2012), or an increase in the surface rotation rate (Siess & Livio 1999a,b;Carlberg et al. 2009,2010). Interestingly, some stars present observed characteristics that could be related to planet engulfments, such as the few percents of red giants (RGs) that are fast rotators (Fekel & Balachandran 1993; Massarotti et al. 2008; Carlberg et al.
2011;Carlberg 2014). Indeed, after the main-sequence phase, the very large expansion of the envelope imposes very low surface rotation veloc-
Figure 1: Evolution of the surface equatorial velocity for our stellar models. No tidal interactions and planet engulfments are considered.
The initial masses and rotations are indicated. The rectangle area indicated at the right bottom corner indicates the region that is zoomed on the Fig. 2.
ities (see Fig.1). Indeed, early survey of projected rotational velocities vsini (Gray 1981, 1982) showed that giants cooler than about 5000 K (log g . 3) are predominantly slow rotators and are characterized by vsini of a few km s−1. However, there exists a small percentage of red giants that present much higher vsini(Fekel & Balachandran 1993;Massarotti et al. 2008;Carlberg et al. 2011;Tayar et al. 2015).
Fig.2, the rectangle area indicated in Fig.1, shows that along the RG branch the surface velocity drops to lower and lower values when the surface gravity decreases. The bulk of red giants observed byCarlberg et al.(2012a)1are characterized by initial masses (estimated from their positions in the HR diagram) between 1.3 and 3 M. They are well framed by our slow and fast rotating models despite the fact that our initial rotations span only a subset of the range of values shown by the progenitors of red giants.
This mainly arises from the fact that the inflation of the star during
1 The other observations showed in the figures, will be better discussed in the following chapters.
dredge-up occurs. Some observations are shown. The dots, squares and diamonds show the observations byCarlberg et al.(2012b),Tayar et al.(2015) and byAurière et al.(2015), respectively. The black ones show the stars with a υsini ∼ 8 km s−1, while the red ones have υsini >8 km s−1. The green pentagons are observed by Lèbre et al.
(2009). The filled blue triangles are observations by Adamów et al.
(2014), the filled blues square is the Li-rich star BD+48 740 discussed by Adamów et al.(2012), and the blue star is the super Li-rich giant HD 107028 studied byAdamów et al.(2015). The yellow hexagons are analyzed byMassarotti et al.(2008).
the RG phase is so large that many different initial surface rotations on the ZAMS converge to similar values at that stage. As we can see, there are some observations of stars characterized by values of rotational velocities six times higher than expected ones. Those objects cannot be explained evoking single stellar models, whatever is the initial rotation velocity.
To explain the high rotation rates of (apparently) single red giants, two kinds of scenarios have been proposed. The first scenario involves a mechanism occurring in the star itself. Simon & Drake(1989) proposed that, at the time of the first dredge-up, the surface could be accelerated
by the transfer through convection of angular momentum from the central fast spinning regions to the surface. Fekel & Balachandran (1993) explained the high rotation and high lithium abundance they observed in their sample of red giants as resulting from such a scenario.
The dredge-up episode would bring to the surface not only angular momentum but also freshly synthesized lithium. It is interesting to underline here that this dredge-up scenario is expected to cause rapid rotation at a particular phase in giant stars evolution, namely when the first dredge-up occurs. In the large sample studied by Carlberg et al. (2011), a clustering of the rapid rotators at this phase (between Teff equal to∼4500 and 5500 K or log Teff between 3.732 and 3.740) is not observed. Moreover, we showed inPrivitera et al.(2015) that the dredge-up actually produces no significant acceleration of the surface and thus cannot be a realistic reason for the high surface rotations that we are discussing here.
A second scenario proposed to explain the high rotation rates in giants involves the swallowing of planet (Peterson et al. 1983; Siess
& Livio 1999a; Livio & Soker 2002; Massarotti et al. 2008; Carlberg et al. 2009). The phenomenon of planet and brown dwarf ingestion was studied theoretically bySandquist et al.(1998,2002) for main-sequence (MS) stars and by Soker et al.(1984);Siess & Livio(1999a,b), and re- cently by Passy et al. (2012) and Staff et al. (2016) for giant stars.
Siess & Livio (1999a,b) studied the accretion of a gaseous planet by a red giant and an asymptotic giant branch star (AGB). They consid- ered cases where the planet is destroyed in the stellar envelope. They focused their study on possible consequences of this engulfment on the luminosity and surface composition of the star. In order to under- stand which condition brings to the planet engulfment and when this phenomenon eventually happens, a detailed analysis of the star-planet interactions is required. Many works have addressed this need, some of them focusing on how a planetary orbit changes under the action of tides between the star and the planet (Livio & Soker 1984;Soker et al.
1984;Sackmann et al. 1993; Rasio et al. 1996;Siess & Livio 1999a,b;
Villaver & Livio 2007;Sato et al. 2008;Villaver & Livio 2009;Carlberg et al. 2009;Nordhaus et al. 2010;Kunitomo et al. 2011;Bear & Soker
initial conditions for an engulfment to occur in terms of the mass of the star, mass of the planet, initial distance between the planet and the star, and of the importance of other physical ingredients, such as the mass loss rates or overshooting. One of these works has specifically studied the possibility that these engulfments are the origin of fast rotating red giants (Carlberg et al. 2009). Although the work ofCarl- berg et al. (2009) provides a detailed and very interesting discussion of the question, it suffers the fact that its conclusions were based on non-rotating stellar models. This limitation was indeed recognized by the authors. In this thesis we present a first attempt to fill this gap. In particular, we include a comprehensive treatment of rotational effects to compute for the first time the simultaneous evolution of the plane- tary orbit and of the resulting internal transport of angular momentum and chemicals in the planet-host star.
We have followed the orbital evolution of a planet accounting for all the main effects impacting the orbit, which are changes of masses of the star and the planet, frictional and gravitational drags and tidal forces. In parallel, we computed the changes of the stellar rotation due to the planet orbital changes. For that reason, we have used rotating stellar models which allow us to follow the angular momentum transport inside the star in a consistent way.
In particular, the present approach will provide us answers to the following points:
• What are the surface rotations of red giants predicted by single star models (stars with no interaction with an additional body)?
A proper comparison between the evolution of the surface ve- locity with and without tides cannot be made with non-rotating stellar models without a priori assumption on the internal dis- tribution of angular momentum. Here, this internal distribution is not a priori imposed, but is computed self-consistently by fol- lowing the changes in the structure of the star, the transport of angular momentum by the shear turbulence and the meridional currents, the impact of mass loss and tides.
• Rotational mixing changes the size of the cores and, among other features, the time of apparition of the external convective zone that is crucial for computing the tidal force according to the formulation by Zahn (1992). It is interesting to see whether these changes are important or not for the computation of the orbital evolution.
• The evolution of the orbit depends on the ratio between the angu- lar velocity of the outer layers of the star and the orbital angular velocity of the planet (see Eq.2.37below); only rotating models can thus account for this effect in a self-consistent manner.
• By how much can the surface rotation increase through a planet engulfment process?
• How long is the period during which a rapid surface velocity can be observed after an engulfment?
• Can the increase of the surface velocity trigger some internal mixing?
• Are there other signatures in addition to fast surface rotation that are linked to a planet engulfment event?
• Can the metallicity of the star have an impact on the planetary orbital evolution and on the formation of fast rotators red giant?
We have also studied to what extent such tidal interactions and en- gulfment processes may trigger a magnetic field and also whether this planet-induced magnetic field might be strong enough to slow down the stellar rotation through the process of wind magnetic braking (ud- Doula & Owocki 2002;ud-Doula et al. 2008).
All these questions are addressed in the present thesis in the follow- ing way: in Chapt. 1, we describe the physics included in our models for the stars. The physics of the orbit and engulfment are explained in Chapt. 2and 3, emphasizing the new aspects of our approach with respect to previous works. Results and comparisons with observations are discussed in Chapt. 4. The variations with different stellar metal- licities are analyzed in Chapt. 5. Planet-induced magnetic fields and
CHAPTER 1
Physics of the evolution of the star
It is important to study the evolution of single rotating stars to reveal, by contrast, the differences that appear when tidal interactions with a planet are occurring. In this chapter we describe the Geneva stellar evolution code, which includes a comprehensive treatment of rotational effects (see Eggenberger et al. 2008, for more details). The ingredients of these models have been computed as in Ekström et al.
(2012). The basic input physics used in the Geneva code is discussed in Sect.1.1. In particular, we will focus on the equations of stellar evo- lution, nuclear reaction rates, opacities, equation of state, convection and overshooting, mass loss rates, diffusion coefficients for rotation and the transport of angular momentum. We will also present the initial conditions of the stars studied in this work where we will show the impact of rotation on their structure and evolution.
1.1 Basic input physics
1.1.1 Equations of stellar evolution
The entire structure of a star and its evolution can be described using four equations. For rotating stellar models, we have to say that spherical symmetry is no longer valid and we cannot derived the ef- fective gravity from a potential. The angular velocity is constant on
isobars (Zahn 1992), when we assume that there is a strong horizontal (along isobars) turbulence. Meynet & Maeder (1997) have described this case, indicated as shellular rotation. In this system, we have that the four structure equations can be written as:
• Hydrostatic equilibrium:
∂P
∂MP =−GMP
4πrP4 fp (1.1)
• Continuity equation:
∂rP
∂MP = 1
4πr2Pρ¯ (1.2)
• Energy conservation:
∂LP
∂MP
=nucl−ν+grav =nucl−ν −cP
∂T
∂t +δ ρ
∂P
∂t (1.3)
• Energy transport equation:
∂ln ¯T
∂MP =−GMP
4πr4Pfpmin
∇ad,∇radfT
fP
(1.4) where ∇ = ∇ad = T¯P δρc¯
P in convective zones, and ∇ = ∇rad =
3 16πacG
κlP
mT¯4 in radiative zones, fP= GM4πrP4
PSP
1 hgihg−1i, fT=
4πr2 P
SP
2 1 hgihg−1i,
hxi is x average on an isobaric surface, x¯ is x average in the volume separating two successive isobars and the indexP refers to the isobar with a pressure equal to P1. Some physical quantities are needed in order to solve these equations. In particular, we have:
• the nuclear reaction rates in order to evaluate nucl and ν (see Sect.1.1.2);
• the opacities to calculate∇rad (see Sect. 1.1.3);
• the equation of state to determine ρ and the needed thermody- namic quantities (see Sect.1.1.4);
1 the other variables have their usual meaning as inMeynet & Maeder(1997)
• a treatment of convection to compute the convective flux and the diffusion (see Sect.1.1.5).
On top of that, the equations of the evolution of chemical elements abundances are to be followed. In the Geneva evolution code, these equations are calculated separately from the structure equations ac- cording to time splitting method.
1.1.2 Nuclear reaction rates
Concerning hydrogen burning, we calculate thepp chains and the CNO tri-cycle in detail. Moreover, the evolution of the main nuclear species is followed explicitly. Concerning helium burning, the following reactions have taken into account:
• the 3α reaction,
• 12C(α,γ)16O(α,γ)20Ne(α,γ)24Mg,
• 13C(α,n) 16O,
• 14N(α,γ) 18F(β,ν) 18O(α,γ)22Ne(α,n) 25Mg,
• 17O(α,n) 20Ne,
• 22Ne(α,γ) 26Mg.
We have determined the system of nuclear reactions and the abun- dances variations for 15 isotopes: H, 3He, 4He, 12C, 13C, 14N, 15N,
16O, 17O, 18O, 20Ne, 22Ne, 24Mg, 25Mg, and 26Mg. We take the val- ues of the following isotopic ratios 3He/He, C/13C, 14N/15N, O/18O,
18O/17O, 21Ne/20Ne, 22Ne/20Ne, 25Mg/24Mg, and 26Mg/24Mg from Maeder(1983) that chose, the ratios between radioastronomical obser- vations and the interstellar material, when it was available. Moreover, the Ne-Na and Mg-Al chains in H-burning regions (see e.g. Meynet &
Maeder 1997) and for the neutron capture reactions during He-burning (see e.g. Meynet & Arnould 2000) can be also accounted for.
Another important task of the Geneva code is to compute stellar models during the advanced stages of evolution (seeHirschi et al. 2004, for more details). The list of elements followed during C-, Ne-, O- and
Si-burnings is then: α, 12C, 16O, 20Ne, 24Mg, 28Si, 32S, 36Ar, 40Ca,
44Ti, 48Cr, 52Fe and56Ni. Concerning Li, it has been followed by an estimate and how it evolves inside the convective envelope during the red giant phase (see Chapt.4and 5).
We generate the nuclear reaction rates using the NetGen tool2. In particular, these are taken mainly from the NACRE database (Angulo et al. 1999). Nevertheless, some of them have been redetermined more recently and updated. More details and an accurate comparison be- tween NACRE values and most recent ones are given byEkström et al.
(2012).
1.1.3 Opacities
Using the OPAL tool3(based onIglesias & Rogers 1996) we are able to generate the opacities. At low temperatures we complement them taking the opacities from Ferguson et al. (2005). The opacity tables are available in the Geneva code for the standard solar abundances of Grevesse & Noels(1993) and Grevesse & Sauval (1998) as well as for the new solar abundances ofAsplund et al. (2004).
1.1.4 Equation of state
We can describe the relation between the three physical parameters (p,T,ρ) using the equation of state. For the stellar models studied in this work, the equation of state is that of a mixture of perfect gas and radiation, and account for partial ionization in the outermost layers, as inSchaller et al.(1992), and for the partial degeneracy in the interior in the advanced stages. This allows us to determine the thermodynamic quantities necessary for the computation of a model (as, for example, δ, ∇ad, cp, ...), but also the third physical parameter from the two others mentioned before.
1.1.5 Convection and overshooting
We determine the convective zones with the Schwarzschild crite- rion. During the H- and He-burning phases, we adopt an overshoot
2 http://www-astro.ulb.ac.be/Netgen/ 3 http://adg.llnl.gov/Research/OPAL/opal.html
parameterdover/HP = 0.10from 1.7Mand above, 0.05 between 1.25 and 1.5 M to determine the extension of the convective core.
Ekström et al.(2012) compared the MS bandwidth of rotating stel- lar models (vini/vcrit = 0.4), computed with the Geneva code, with the ones published in Schaller et al. (1992) (Fig. 1.1). They obtained a slightly narrower MS width thanSchaller et al.(1992), which provided a good fit to the observed MS width (see the discussion in Schaller et al. 1992). For stars at the end of the MS phase will scatter around the limit shown by moderately rotating models, more rapid rotators lying beyond it and slightly enlarging the MS width. The combina- tion of the rotational mixing and of an overshoot of 0.1 reproduces the effects obtained in models with no rotational mixing and a stronger overshoot of 0.2. The rotational mixing, indeed, contributes to mak- ing the convective cores larger at a given evolutionary stage (seeTalon et al. 1997). According to the mixing length theory, the outer convec- tive zone is characterized by the mixing-length parameter (αMLT). For the low-mass stars, we have αMLT≡`/HP = 1.6.
1.1.6 Mass loss
For low mass stars (M.7M) the main sequence evolution is cal- culated at constant mass. On the red giant branch and on the asymp- totic giant branch, the mass loss becomes however non negligible and is taken into account by using the prescription byReimers (1975) (more details will be given in Sect. 2.1.2.1)
M˙loss= 4×10−13ηL?R?/M? M yr−1 , (1.5) with η = 0.5 (Maeder & Meynet 1989a). L?, R?, and M? are the luminosity, the radius, and the mass of the star. The mass loss is one of the parameters that drives the planetary orbital evolution.
1.1.7 The diffusion equations
Many studies on stellar rotation have been developed and analyzed in papers published by our group (the interested reader may refer to these papers to get the full developments Maeder 1999,1997;Maeder
Figure 1.1: Comparison of the MS band between the present rotating models ofEkström et al.(2012) (solid black line) and the non-rotating models ofSchaller et al. (1992) (long-short dashed blue line).
& Meynet 2000;Maeder & Zahn 1998a;Meynet & Maeder 1997,2000).
We summarize here the main aspects.
First of all, we consider the horizontal turbulence4 is very strong (the shellular-rotation hypothesis)5. Due to the horizontal turbulence, we have the following diffusion coefficient (Zahn 1992)
Dh ≈νh= 1
ch r |2V(r)−αU(r)|, (1.6) wherechis a constant (we considered the value1),α= 12dln(rdln2rΩ)¯ ,V(r) andU(r) are the horizontal and vertical component of the meridional circulation velocity, respectively.
The meridional circulation is another important large-scale circu- lation (Eddington 1925; Sweet 1950). This arises according to the von Zeipel theorem (von Zeipel 1924) that establishes a relation be- tween the radiative flux and the effective gravity at a certain colatitude
4 The turbulence along an isobar. 5 We do not have any restoring force in the same direction, as for example the buoyancy force (related to the density gradient).
This one acts in the vertical direction.
on the surface of a rotating star. The vertical component is to first order u(r, θ) = U(r)P2(cosθ), where P2(x) is the second Legendre polynomial. Zahn(1992) andMaeder & Zahn(1998a) determined the formulation of the radial amplitude U(r)
U(r) = P ρgCPT
1
[∇ad− ∇rad+ (ϕ/δ)∇µ]
· L
M? [EΩ? +Eµ] +CP
δ
∂Θ
∂t
, (1.7)
where CP is the specific heat at constant pressure, the adiabatic gra- dient corresponds ∇ad = ρT CP δ
P, M? =M
1−2πgρΩ2
m
, and Θ = ˜ρ/¯ρ indicates the ratio of the variation of the density to the average den- sity on an equipotential (seeMaeder & Zahn 1998a, for details on these expressions).
Deff is the effective diffusion coefficient, that contains the effects of the horizontal diffusion and that of the meridional circulation as (Chaboyer & Zahn 1992)
Deff= 1 30
|r U(r)|2 Dh
. (1.8)
The shear turbulence arises because of the differential rotation.
The Richardson criterion tells us when this shear instability appears.
In particular, it grows between the layers with different rotational ve- locities, when the thermal dissipation reduces the restoring force of the density gradient. Maeder(1997) determined the coefficient of diffusion by shear turbulence Dshear as
Dshear= K
ϕ
δ∇µ+ (∇ad− ∇rad)
·HP
gδ
"
fenerg
9π
32 Ω dln Ω dlnr
2
− ∇0− ∇
#
, (1.9)
where K= 3κρ4acT2C3P is the thermal diffusivity, fenerg indicates the frac- tion of the excess energy in the shear that contributes to mixing, and (∇0− ∇) is the difference between the internal non-adiabatic gradient and the local gradient.
1.1.8 Transport of angular momentum
The transport of angular momentum obeys an advection-diffusion equation, that we can write in Lagrangian coordinates as (Zahn 1992;
Maeder & Zahn 1998b):
ρ d
dt(r2Ω)¯ Mr = 1 5r2
∂
∂r(ρr4ΩU¯ (r)) + 1 r2
∂
∂r(ρDr4∂Ω¯
∂r), (1.10) whereΩ¯ is the mean angular velocity on an isobaric surface, r is the radius, ρ is the density, Mr is the mass inside the radius r, U is the amplitude of the radial component of the meridional circulation and Dis the total diffusion coefficient in the vertical direction, taking into account the various instabilities that transport angular momentum (convection, shears). The divergence of the advected flux of angular momentum is indicated by the first term on the right-hand side of this equation, while the divergence of the diffused flux is indicated by the second one.
The full solution of (1.10) taking into account U(r) and D gives the non-stationary solution of the problem. The expression of U(r) (1.7) involves derivatives up to the third order; (1.10) is thus of the fourth order and implies four boundary conditions. The first boundary conditions impose momentum conservation at convective boundaries (Talon et al. 1997). If we want to take into account the interaction with a planet, the boundary condition imposing momentum conservation at the bottom of the convective envelope has to include the impact of tides owing to the presence of the second body:
∂Ω¯
∂t Z R?
Renv
r4ρdr =−1
5R4envρΩU¯ +FΩ, (1.11) withRenv the radius at the base of the convective envelope. FΩ rep- resents the torque applied at the surface of the star. It is given here by
FΩ= d(Ω?Ice) dt =−1
2Lpl a˙
a
t
, (1.12)
whereΩ? is the angular velocity at the surface of the star and Ice is the moment of inertia of the convective envelope, Lpl is the angular momentum of the planetary orbit, and ( ˙a/a)t is the inverse of the
Figure 1.2: Evolutionary tracks in the Hertzsprung-Russell diagram for rotating models of 1.5, 1.7, 2.0, and 2.5 M. The solid and dashed lines indicate models with Ωini/Ωcrit= 0.1 and0.5, respectively.
timescale for the change of the orbit of the planet resulting from tidal interaction between the star and the planet. The expression of ( ˙a/a)t
is discussed in Chapt. 2. When we compute single stellar models,FΩ is taken equal to 0.
1.2 Initial conditions of the stars and impact of rotation In this work, we have studied different stellar models in mass, ro- tation and metallicity. We considered a range of mass between 1.5 and 2.5 M, with an increment of 0.1. The angular velocity of the stars is 0.1, 0.3, 0.5 and 0.6 Ωcrit. The metallicities are 0.014, 0.02 and 0.03.
Obviously, we do not go into the details of the severals models. We only focus on the impact of rotation on the models used and the main properties that characterize them.
In Fig. 1.2, the evolutionary tracks in the HR diagram of our stel- lar models are shown. The metallicity is Z=0.026. Differences between rotating models for a given mass appear mainly during the MS phase
6 The impact of the metallicities on the star will be analyzed in Chapt.5. Here, we analyze the effects produced by stellar rotation.
Figure 1.3: Top: Evolution of stellar radii for the 2 M models.
Continuous and dotted lines correspond to stars with initial veloci- tiesΩini/Ωcrit = 0.1 and 0.5, respectively. Time 0 corresponds to the end of the MS phase. Centre: Evolution of the total mass of the star (upper red lines) and of the masses of the convective envelopes (lower blue curves) for the same stars as in the left panel. Bottom: Radii at the bottom of the convective envelopes for the same stars as in the left panel.
and during the crossing of the HR gap. Faster rotating models are overluminous at a given mass and the MS phase extends to lower ef- fective temperatures. The widening of the MS is mainly due to the increase of the convective core that is related to the transport by rota- tional mixing of fresh hydrogen fuel in the central layers. The impact on the luminosity results from both the increase of the convective core and the transport of helium and other H-burning products into the radiative zone (e.g. Eggenberger et al. 2010;Maeder & Meynet 2012).
The increase of the convective core also leads to an increase of the MS lifetime. Typically, faster rotating models will reach a given luminosity along the RG branch at an older age than the slower rotating models.
Along the RG branch, at a given effective temperature, the initially fast rotating track is slightly overluminous with respect to the slow one.
This means that the radius of the fast rotating star is also larger than the radius of the slow one. The tidal forces will therefore be stronger around the fast rotating star (the tidal force varies with the stellar radius at the power 8) and we may expect that, everything being equal, when an engulfment occurs, it will occur at an earlier evolutionary stage (typically at smaller luminosities along the RGB) around fast rotating stars than around slow rotating ones7. We note that since rotation increases the MS lifetime, a given evolutionary stage along the RGB occurs at a greater age when rotation increases. We also note that the tip of the RG branch occurs at slightly lower luminosities for the faster rotating models (see Fig. 1.2). This implies that the maximum radius reached by the star decreases when the initial rotation of the star increases8.
The decrease in luminosity at the tip comes from the fact that the mass of the core at He-ignition in the 2 M with Ωini/Ωcrit = 0.5 is smaller than in the same model with Ωini/Ωcrit = 0.1. At first sight, it might appear strange that the core mass is smaller in the faster ro- tating model. Indeed, rotation, in general, makes the masses of the
7 More details will be given in Chapt.4. 8 Therefore, increasing the stellar rota- tion slightly lowers the maximum initial distance between the star and the planet that leads to an engulfment. Indeed, the larger the maximum stellar radius, the stronger the tidal forces and thus the larger the range of initial distances between the star and the planet that leads to an engulfment. This point will be discussed in Chapt.2.
Figure 1.4: Relation between the mass of the He-core at ignition and the initial stellar mass for models with overshooting (solid line) (Maeder & Meynet 1989b).
cores larger. The point to keep in mind here is that we are referring to the mass of the core required to reach a given temperature, which is the temperature for helium ignition. This mass depends on the equa- tion of state. In semi-degenerate conditions, (Fig.1.4), the core mass needed to reach that temperature is larger than in non-degenerate ones (Maeder & Meynet 1989b). For a given initial mass, faster rotation, by increasing the core mass during the core H-burning, makes the he- lium core less sensitive to degeneracy effects. In other words, rotation shifts the mass transition between He-ignition in semi-degenerate and non-degenerate regimes to lower values (see also Fig.1.5).
The evolution of the radius as a function of time along the RG branch in the case of the 2 Mmodels is shown in Fig. 1.3(top panel).
The radius at the tip of the RG branch is lower by about 0.1 dex, therefore by 25% in the case of the faster rotating model. The small bump seen along the curves at time coordinates 0.06 and 0.075 is due to the fact that, when the star climbs the RG branch, the H-burning shell moves outwards in mass and, at a given point, encounters the chemical
Figure 1.5: Relation between the mass of the He-core at ignition and the initial stellar mass for models at different initial rotational veloci- ties. Here, we show the case withΩini/Ωcrit= 0.1, 0.3 and 0.5.
discontinuity left by the convective envelope that also slowly recedes outwards after the first dredge-up. This produces a rapid increase in the abundance of hydrogen in the H-burning shell, a variation in the energy output of this shell, and a change in the structure that produces these bumps.
The central and bottom panels of Fig. 1.3 show how the masses and the radii at the base of the convective envelope vary as a function of time in both rotating models. In the faster rotating model, the convective envelope has a slightly smaller extension in mass and radius than in the slower rotating model. However these changes are minor.
We can conclude that rotation, by changing the transition mass between stars going through a He-flash and those avoiding it, may have a non-negligible impact on the orbital evolution in this mass range.
Outside this mass range, the impact of rotation will be modest.
Physics of the evolution of the planetary orbit
The main goal of this chapter is to explain how the orbit of planet evolves and which parameters can determine the destruction of a planet by own star.
What do we know about orbital evolution? A lot of works has been done to answer to the curiosity about the destiny of our planet and, more in general, solar system. The future evolution of some planets of the solar system seems to be doomed, as, for instance, Mercury and Venus. During the red giant phase, the Sun will increase its radius, swallowing these planets and destroying them inside the convective en- velope. About the Earth,Vila(1984) found it will be swallowed by the Sun before its convective envelope shrinks again. Actually, this sce- nario depends on the maximum radius of the Sun reached at the tip of the red giant phase. Another factor, taken into account bySackmann et al.(1993), is the mass loss of the Sun. Contrary toVila(1984), they obtained that our planet, due to the decrease of the solar mass, would move outward, avoiding its destruction. The following theories have been refined, incorporating other aspects as tidal interactions (seee.g.
Rasio et al. 1996). The tides could change orbital evolution of the Earth, but the impact of these forces is strictly connected to uncon- strained tidal parameters. This complicates the problem not giving a more probable behavior. Later, the gravitational drag, ram pressure,
wind accretion have been added to the orbital evolution simulation (Livio & Soker 1984;Villaver & Livio 2009;Mustill & Villaver 2012).
Up to now, we do not know what is the fate of our planet.
It is clear that understanding deeply a planetary orbit implies de- tailed knowledges of stellar evolution. Kunitomo et al. (2011) and Villaver et al.(2014) studied the orbital evolution and the swallowing of a planet by host stars evolving on the red-giant branch. Kunitomo et al.(2011) found the value of the survival limit, below which planets will be engulfed, is quite sensitive to the stellar masses in the range between 1.7 and 2.1 M. Villaver et al.(2014) showed the effect on or- bital radius of a planet for different mass loss prescriptions, in a stellar mass range noplt so different from the one studied in Kunitomo et al.
(2011). From their models, the mass loss rate has no important influ- ence on the survival limit. Other parameters, as planetary mass, can play a role for the orbital evolution (Villaver et al. 2014). These au- thors studied the dependency of the survival limit considering a planet of mass Neptunian and 1, 2, 5, and 10 MJ. They obtained the survival limit is more sensitive to the mass of the planet than stellar mass and mass loss rate.
In spite of many computations and works on orbital evolution pre- sented in the literature other important aspects of the stellar physics has to be studied and deepened. One of them, that we propose in the present work, is the effects produced by the stellar rotation on the planetary evolution.
2.1 Physical aspects
A lot of factors can change the orbital radius of a planet and each one of them gives a different contribute. Before to describe these fac- tors, we try to image the system that we want to study. We suppose a planetary system characterized by a planet that revolves around its star with a semi-major axis a. For simplicity, we consider a circular orbit (e= 0) and aligned with the equator of the star (orbital incli- nation i = 0◦), and we neglect the rotation of the planet on its axis (Fig. 2.1).
We assume that the planet, with a mass between 1 and 15 MJ, is a
Figure 2.1: A simple scheme of the planetary system taken into ac- count. We have a circular orbit and the equatorial plane coincident with stellar equatorial plane. We neglect the rotation of the planet on its axis.
polytropic gaseous sphere of indexn= 1.5(Siess & Livio 1999a). We use the mass-radius relation byZapolsky & Salpeter(1969)
Rpl≈0.105 2X1/4 1 +X1/2
!
[R], (2.1)
with X = Mpl/(0.0032[M]), where Mpl and Rpl are given in solar mass and solar radius, respectively. In our computations, we have also taken into account the fact that the effective radius may be higher due to the planetary magnetic field (Vidotto et al. 2014). In processes, as the interaction between the planet and the matter around it, the magnetic field surface is several time larger than the real planetary surface. This magnetic surface increases the impact of the frictional force (more details in Sect.2.1.3). We use the magnetic pressure radius given by
Rmp=Rpl
Bpl2 8πρv2wind
!1/6
, (2.2)
whereBpl is the dipole’s magnetic field strength of the planet (Chap- man & Ferraro 1930;Lang 2011),ρthe density of the environment and vwindthe velocity of the stellar wind (more details aboutρare given in Sect. 2.1.2.2.1). The value of the magnetic field, taken in our compu- tations, corresponds to equatorial dipole surface field for Jupiter, with
Bpl=BJ= 4.28 G. In Fig.2.2, we show the magnetic radius per unit of Rpl of a giant planet of mass 1 MJ as a function of the density of the environment with vwind = 5 km s−1, and Bpl = BJ (blue line), Bpl= 2×BJ(green line) andBpl= 10×BJ(red line). During the red giant phase, due to the stellar mass loss, the density goes from 10−20 to 10−14 g cm−3 at 1 au. For Bpl =BJ, we have a variation of Rmp
between 25 and 3 Rpl(2.5 - 0.3 R). With higher value of the magnetic field, for instance one order of magnitude, causes an increase of Rmp
almost 2 times (Rmp ∝ B1/3J ). Instead, during the MS phase, when the stellar winds are very rarefied, we consider a density ρ=nISMmH, withnISM the number density in the interplanetary medium along the orbit of the planet and mHthe mass of a hydrogen atom. In this case, the density is very low (∼10−27g cm−3) andvwindcan be greater than 400 km s−1 bring Rmp to a value above 80 Rpl.
We can also obtain other important parameters (Siess & Livio 1999a)
ρc≈ 8.4 (Mpl) (Rpl)−3 [g cm−3] (2.3) Tc≈ 1.7×107µpl(Mpl) (Rpl)−1 [K] (2.4) Pc≈ 8.7×1015(Mpl)2(Rpl)−4 [dyne cm−2] (2.5) whereρc,TcandPcare the central density, temperature and pressure, and µpl is the planetary mean molecular weight.
The processes, destabilizing the system and modifying the plane- tary orbit, are related to:
• the rate of change in planet mass: a planet, interacting with matter in the environment around, can increase its mass. On the other hand, a planetary object can reduce its mass due to the heating produced by the stellar radiation and the ram pressure exerted by the interplanetary medium;
• the drag forces: the interactions between planet and interplane- tary medium provoke the arising to gravitational and frictional drag forces;
• the stellar parameters: we have stellar radius, mass loss rate, stellar rotation, stellar wind, mass and size of the envelope and
Figure 2.2: Magnetic radius vs density of the environment withvwind= 5 km s−1, Bpl =BJ and a planetary mass 1 MJ. Three values of the magnetic field are taken into account: Bpl = BJ (blue line), Bpl = 2×BJ (green line) andBpl= 10×BJ(red line).
radiation. The factors that can change the stellar evolution, for instance the stellar metallicity, have an impact on the orbital radius;
• the tidal forces: these depend on several parameters among which some stellar parameters and the semi-major axis of the planetary orbit.
2.1.1 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˙?+ ˙Mpl M?+Mpl
| {z }
Term 1
− 2
Mplvpl[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,vplis the velocity of the planet,Ffri andFgra 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
MplF, (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= drdt = µ
p
12
esinf , (2.8)
Vφ= (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 vwind with the components:
vwind,r= vs(r) , (2.10)
vwind,φ= λVφ , (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˙−vwind 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−vwind). (2.13) Writing (2.13) and the cross-product ofr, we have
¨
r×r=−µ
r3r×r− 1
MplQ( ˙r×r−vwind×r). (2.14) Using the following relations, valid for an unperturbed Keplerian orbit,
¨
r×r= dtd ( ˙r×r) = dtd
(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 sim- plifications, we obtain
M p˙ +Mp˙= −M2
plQ(1−λ) (M p) , (2.20) M˙
M +p˙
p = −M2
plQ(1−λ) , (2.21) that brings to
˙ p
p =−M˙ M − 2
Mpl
Q(1−λ). (2.22)
In a similar way, taking Eq. (2.13) and this time the dot product ofr˙ we can find the energy equation. Starting from
¨
r·r˙ =−µ
r3r·r˙− 1
MplQ( ˙r·r˙−vwind·r)˙ . (2.23) and using the following relations
¨
r·r˙ = 12dtd ( ˙r˙r) , (2.24)
˙
r·r˙ = Vr2+Vφ2 =V2 =µ 2r −1a
, (2.25)
r·r˙ = 12dtd (rr) , (2.26)
vwind·r˙ = vsdr
dt +λVφ2 , (2.27)
we obtain
¨
r·r˙= −rµ3r·r˙−M1
plQ( ˙r·r˙−vwind·r)˙ ,(2.28) 1
2 d dt( ˙rr)˙
| {z }
A
= −1 2
µ r3
d dt(rr)
| {z }
B
− 1 Mpl
Q
Vr2+Vφ2−vs
dr dt −λVφ2
| {z }
C
.(2.29)
The terms A, B and C can be written
1 2
d
dt( ˙r˙r) = 12dtd
Vr2+Vφ2
= dtd 12V2
, (2.30)
−1 2
µ r3
d
dt(rr) = −GM
r2 dr dt =dtd
GM
r
−GrM˙ , (2.31)
− 1 Mpl
Q
Vr2+Vφ2−vsdr dt −λVφ2
= − 1
MplQ
V2−vsdr
dt−λGM pr2
. (2.32)
and we finally find d
dt 1
2V2−GM r
=−GM˙ r − 1
MplQ
V2−vs
dr
dt −λGM p r2
. (2.33) Making use of the equation for undisturbed orbit
r= p
1 +ecosf, (2.34)
