Connecting the formation of stars and planets. II: coupling the angular momentum of stars with the angular momentum of planets
L. M. Flor-Torres, R. Coziol, K.-P. Schröder, D. Jack, J. H. M. M. Schmitt
aa r X i v : . [ a s t r o - ph . S R ] J a n Manuscript for
Revista Mexicana de Astronom´ıa y Astrof´ısica (2007)
CONNECTING THE FORMATION OF STARS ANDPLANETS. II: COUPLING THE ANGULARMOMENTUM OF STARS WITH THE ANGULARMOMENTUM OF PLANETS
L. M. Flor-Torres, R. Coziol, K.-P. Schr¨oder, D. Jack, and J. H. M. M.Schmitt, Draft version: January 29, 2021
RESUMENUna muestra de 46 estrellas, hu´espedes de exoplanetas, esta usada para lab´usqueda de una conexi´on entre su proceso de formaci´on y el proceso deformaci´on de los planetas que las orbitan. Separando nuestra muestra en dos,estrellas hu´espedes de exoplanetas de baja masa (LME) y de alta masa (HME),encontramos que las HMEs rotan preferiblemente alrededor de estrellas conalta masa y con mayor rotaci´on que las estrellas de las LMEs. Tambi´en seencontr´o que las HMEs tienen un momento angular orbital m´as alto que lasLMEs y que perdieron m´as momento angular durante su migracion. Nuestrosresultados son consistente con un modelo donde estrellas m´as masivas con altarotacion forman discos protoplanetarios m´as masivos que tambien rotan masr´apido, y que ademas son mas eficientes en disipar el momento angular de susplanetas. ABSTRACTA sample of 46 stars, host of exoplanets, is used to search for a connectionbetween their formation process and the formation of the planets rotatingaround them. Separating our sample in two, stars hosting high-mass exoplan-ets (HMEs) and low-mass exoplanets (LMEs), we found the former to be moremassive and to rotate faster than the latter. We also found the HMEs to havehigher orbital angular momentum than the LMEs and to have lost more angu-lar momentum through migration. These results are consistent with the viewthat the more massive the star and higher its rotation, the more massive wasits protoplanetarys disk and rotation, and the more efficient the extraction ofangular momentum from the planets.
Key Words: stars: fundamental parameters — stars: rotation — stars: for-mation — stars: planetary systems1. INTRODUCTIONThe discovery of gas giant planets rotating very close to theirs stars (hotJupiter, or HJs) has forced us to reconsider our model for the formation ofplanets around low mass stars by including in an ad hoc way large scale Departamento de Astronom´ıa, Universidad de Guanajuato, Guanajuato, Gto, M´exico. Hamburger Sternwarte, Universit¨at Hamburg, Hamburg, Germany. ⊙ , or the maximum mass model with a mass above 0.5 M ⊙ (Armitage 2010).There are few clues which could help us determining which path the PPD ofthe solar system followed (and strong difficulties compared to direct observa-tions of PPD; see Fig. 2 and discussion in Raymond & Morbidelli 2020). Oneis the total mass of the planets, which represents only 0.1% the mass of theSun. This implies the solar system PPD have lost an important amount ofits mass after the formation of the planets. Another clue is that 99% of theangular momentum of the solar system is located in the planets, suggestingthat the initial angular momentum of the PPD might have been conserved inthe planets. However, this is obviously not the case when large scale migrationoccurs, so what was the difference?If the initial angular momentum of the PPD passes to the planets, thenone could use the orbital rotation momentum in exoplanetary systems to testdifferent scenarios connecting the formation of the planets to the formation oftheir stars. For example, how is the angular momentum of the PPD coupledto the angular momentum of the stars? Since large scale migration representsa loss of angular momentum of the planets (at least by a factor 10), what wasthe initial angular momentum of the PPD when it formed and how does thisEVMEXAA MAIN JOURNAL DEMO DOCUMENT 3compared to the initial mass of the PPD? Does this influence the masses ofthe planets and their migration?The answers are not trivial, considering thatthe physics involved is still not fully understood.In particular, we know that the angular momentum is not conserved dur-ing the formation of stars. This is obvious when one compares how fast theSun rotates with how fast its rotation should have been assuming the angularmomentum of the collapsing molecular cloud where it formed was conserved.Actually, working the math (a basic problem, but quite instructive; see coursenotes by Alexander 2017), the Sun effective angular momentum, j ⊙ = J ⊙ /M ⊙ ,is ∼ times lower than expected. Intriguingly, j ⊙ is also 10 lower than theangular momentum of its breaking point, j b , the point where the centripetalforce becomes stronger than gravity (McKee & Ostriker 2007). If that was nottrue, then no stars whatever massive would be able to form. In fact, observa-tions revealed that, in general, the angular momentum of stars with spectraltype O5 to A5 trace a power law, J ∝ M α , with α ∼
2, with typical J ∗ valuesthat are exactly ten times lower than their breaking point. How universalis this “law”and how stars with different masses get to it, however, is unex-plained (Wesson 1979; Brosche 1980; Carrasco et al. 1982; God lowski et al.2003). To complicate the matter, it is clear now that lower mass stars, laterthan A5, do not follow this law, their spin going down exponentially (cf. Fig.6in Paper I). For low-mass stars, McNally (1965), Kraft (1967) and Kawaler(1987) suggested a steeper power law, J ∝ M . , which suggests they loosean extra amount of angular momentum as their mass goes down. What isinteresting is that low-mass stars are also those that form PPDs and planets,which had led some researchers to speculate there could be a link between thetwo.To explain how low-mass stars loose their angular momentum, differentmechanisms are considered. The most probable is stellar wind (Schatzman1962; Mestel 1968), which is related to the convective envelopes of these stars.This is how low-mass stars would differ from massive ones. However, whetherthis mechanism is sufficient to explain the break in the J − M relation is notobvious, because it ignores the possible influence of the PPD (the formation ofa PPD seems crucial; see de la Reza & Pinz´on 2004). This is what the mag-netic braking model takes into account (Wolff & Simon 1997; Tassoul 2000;Uzdensky et al. 2002). Being bombarded by cosmic rays and UV radiationfrom ambient stars, the matter in a molecular cloud is not neutral, and thuspermeable to magnetic fields. This allows ambipolar diffusion (the separationof negative and positive charges) to reduce the magnetic flux, allowing thecloud to collapse. Consequently, a diluted field follows the matter throughthe accretion disk to the star forming its magnetic field (McKee & Ostriker2007). This also implies that the accretion disk (or PPD) stays connected tothe star through its magnetic field as long as it exists, that is, a period thatalthough brief includes the complete phase of planet formation and migration.According to the model of disk-locking, a gap opens between the star and thedisk at a distance R t from the star, and matter falling between R t and the FLOR-TORRES ET AL.radius of corotation, R co (where the Keplerian angular rotation rate of thePPD equals that of the star), follow the magnetic field to the poles of thestar creating a jet that transports the angular momentum out. In particular,this mechanism was shown to explain why the classic T-Tauri rotates moreslowly than the weak T-Tauri (Ray 2012). How this magnetic coupling couldinfluence the planets and their migrations, on the other hand, is still an openquestion (Matt & Pudritz 2004; Fleck 2008; Champion 2019).To investigate further these problems, we started a new observationalproject to observe host stars of exoplanets using the 1.2 m robotic telescopeTIGRE, which is installed near our department at the La Luz Observatory (incentral Mexico). In paper I we explained how we succeeded in determiningin an effective and homogeneous manner the physical characteristics ( T eff ,log g , [M/H], [Fe/H], and V sin i ) of a initial sample of 46 bright stars using iSpec (Blanco-Cuaresma 2014, 2019). In this accompanying article, we nowexplore the possible links between the physical characteristics of these 46 starsand the physical characteristics of their planets, in order to gain new insightabout a connection between the formation of stars and their planets.2. SAMPLE OF EXOPLANETSOur observed sample consists of 46 stars host of 59 exoplanets, whichwere selected from the revised compendium of confirmed exoplanets in theExoplanet Orbit Database . In Table 1 the dominant planet (col. 2) in eachstellar system is identified by the same number (col. 1) which was used inpaper I to identify their stars. In col. 3 and 4 we repeat the magnitude anddistance of the host stars as they appeared in Table 1 of paper I. This isfollowed by the main properties of the planets as reported in the exoplanetOrbit Database: mass (col. 5), radius (col. 6), period (col. 7), major semi-axis (col. 8) and eccentricity (col. 9). Note that an eccentricity of zero couldmean the actual eccentricity is not known. The last two columns identifythe detection method and the distinction between high mass and low massexoplanet (as explained below).Among our sample, one exoplanet, M Jup (Spiegel et al. 2011;Burgasser 2008). Note that Hn-Peg b was detected in imaging (identified asIm in col. 11), and its huge distance from its host star (col. 8) is more typicalof BDs than of exoplanets (the other exoplanet detected in imaging is 34,HD 115383 b, at 43.5 AU from it stars, but with a mass of only 4 M Jup ).Another exoplanet whose mass is close to the lower mass limit for BDs is http: //exoplanets.org/ EVMEXAA MAIN JOURNAL DEMO DOCUMENT 5TABLE 1
PHYSICAL PARAMETERS THE PLANETS IN OUR SAMPLE.
Id. M p R p Period a p e p Detection Planetary
Planet (V) (pc) (M jup ) (R jup ) (days) (AU )
Method type − −
795 0 Im BD40 WASP-8 b 9.9 90.2 2.24 1.04 8.16 0.08 0.31 Tr HME41 WASP-69 b 9.9 50.0 0.26 1.06 3.87 0.05 0.00 Tr LME42 HAT-P-34 b 10.4 251.1 3.33 1.11 5.45 0.07 0.44 Tr HME43 HAT-P-1 b 9.9 159.7 0.53 1.32 4.47 0.06 0.00 Tr LME44 WASP-94 A b 10.1 212.5 0.45 1.72 3.95 0.06 0.00 Tr LME45 WASP-111 b 10.3 300.5 1.85 1.44 2.31 0.04 0.00 Tr HME46 HAT-P-8 b 10.4 212.8 1.34 1.50 3.08 0.04 0.00 Tr HMEAn * in front of the name of the planet identified multiple planetary systems.
FLOR-TORRES ET AL.Another point of importance that can be noted in Table 1 is the fact thatwe have only 14 exoplanets detected in RV. This is most probably due to thefact that we required the radius of the planets to be known, which is easierto determine by the Tr method. Curiously, only three of the 14 exoplanetsdetected in RV show the trend to be located farther from their stars thanthe Tr exoplanets, as is observed in the literature (and as is obvious in theExoplanet Orbit Database). Two are close to the ice line and thus are morewarm than hot, and one with 3.86 M Jup is at the same distance as Jupiterfrom the Sun. This implies that any bias introduced by the different detectionmethods, RV vs. Tr, cannot be explored thoroughly in our present analysis.Although the variety of the characteristics of exoplanets cannot be ad-dressed with our present sample, we can however separate our sample of ex-oplanets in two based on their masses. For our analysis, this distinction isimportant in order to test how the mass of the exoplanet is related to themass of its host star. To use a mass limit that has a physical meaning wechoose 1 . M Jup , which is the mass above which self-gravity in a planet be-comes stronger than the electromagnetic interactions (Padmanabhan 1993;Flor-Torres et al. 2016, see demonstration in Appendix A). Using this limitwe separated our sample in 22 high mass exoplanets (HMEs) and 23 lowmass exoplanets (LMEs). This classification is included in Table 1 in the lastcolumn.According to Fortney et al. (2007) the mass-radius relation of exoplanetsshow a trend for exoplanets above 1 . M Jup to have a constant radius (seealso Fortney et al. 2010). In fact, modelizations of exoplanet structures (e.g.,Baraffe et al. 1998, 2003, 2008) predicts an inflection point in the mass-radiusrelation where the radius starts to decrease instead of increasing as the massincreases. Actually, this is what we observe in brown dwarfs (BDs). Physicallytherefore, this inflection should be located near 1 . M Jup , where self-gravitybecomes stronger than the electromagnetic interaction and the object startto collapse (which is the case of BDs). However, if massive HJs have moremassive envelopes of liquid metallic hydrogen (LMH) than observed in Jupiter(as suggested by JUNO; Guillot et al. 2018; Kaspi et al. 2018; Iess et al. 2018;Adriani et al. 2018), their structures might resist gravity (at least for a while),due to the liquid state being incompressible, pushing the collapse of the ra-dius to slightly higher masses (models and observations verifying this pre-diction can be found in Hubbard et al. 1997; Dalladay-Simpson et al. 2016;Flor-Torres et al. 2016). This possibility, however, is still controversial, andwe only use the mass limit in our analysis to separate our sample of exoplan-ets according to their masses. On the other hand, the possibility of massiveLMH envelopes in the HMEs might have some importance, since these exo-planets would be expected to have higher magnetic fields than the LMEs ,which, consequently, could affect their interactions with the PPD and nearbyhost stars (this fits the case of HD 80606 b, a HME studied by de Wit et al.2016), leading possibly to different migration behaviors. Likewise, we mightalso consider the possibility of inflated radii in the LMEs, since they are soEVMEXAA MAIN JOURNAL DEMO DOCUMENT 7close to their stars, although what could be the effect of inflated radius onmigration is less obvious (one possibility is that they circularize more rapidly).
M(M
Jup ) R ( R J up ) . . . . . . . M J up M J up BDHMEsLMEs
Fig. 1. The M-R diagram for exoplanets in the Exoplanet Orbit Database. Thevertical line at 1 . M Jup corresponds to the mass criterion we used to separate theexoplanets in LMEs and HMEs. The other vertical lime is the lower mass limit forthe BDs, 13 M Jup . To test further our distinction in mass, we trace in Fig. 1 the mass-radiusrelation of 346 exoplanets from the Exoplanet Orbit Database. Based onvisual inspection it is not clear due to inflated radii how to separate the HJs.One quantitative criterion is the mass-radius relation. In figure 1, we tracedthree M-R relations, adopting 1 . M Jup to distinguish LMEs form HMEs.Below this limit we find the relation:ln R LME = (0 . ± . M LME + (0 . ± . r = 0 .
75, whichimplies that the radius within this range of masses continually increases withthe mass. This relation is also fully consistent with what was previouslyreported by Valencia et al. (2006) and Chen & Kipping (2017). Above thelower mass limit for BDs we find the relation:ln R BD = ( − . ± . M BD + (0 . ± . r = 0 .
18, butwhich is sufficient to indicate that the trend is for the radius to decrease with FLOR-TORRES ET AL. T eff (K) V s i n i ( k m / s ) HMEsLMEsBDSun
Fig. 2.
Star rotational velocity vs. temperature, distinguishing between stars hostingHMEs and LMEs. The position of the Sun is included as well as the star with a BDas companion. the mass. This is as expected for objects where self-gravity is stronger thanthe electromagnetic repulsion (BDs not having enough mass to ignite fusionin their core cannot avoid the effect of gravitational collapse). Finally, inbetween these two mass limits defining the HMEs, we obtain the relation:ln R HME = ( − . ± . M HME + (0 . ± . r = 0 .
02, consistent with no correlation. This range of mass, therefore,is fully consistent with HJs near the inflection point, extending this regionover a decade in mass (as previously noted by Hatzes & Rauer 2015). For ouranalysis, these different M-R relations are sufficient to justify our separationbetween LMEs and HMEs (note that this physical distinction criterion wasnever used previously in the literature; the only study which uses a limit closeto ours is Sousa et al. 2011).3. CONNECTING THE STARS TO THEIR PLANETSIn paper I, we verified that the rotational velocity of a star, V sin i , de-creases with the temperature, T eff . In Fig. 2, we reproduced this graphic,but this time distinguishing between stars hosting LMEs and HMEs. We ob-serve a clear trend for the HMEs to be found around hotter and faster rotatorstars than the LMEs. Considering the small number of stars in our sample,we need to check whether this result is physical or due to an observationalEVMEXAA MAIN JOURNAL DEMO DOCUMENT 9TABLE 2 STATISTICAL TESTSHME LME MW sign. diff.Parameter median mean median mean p-value V sin i .
47 10 .
94 4 .
11 4 .
62 0 . T eff < . M ∗ .
22 1 .
21 1 .
11 1 . < . F e/H ] 0 .
21 0 .
19 0 .
15 0 .
16 0 . J ∗ .
83 14 . .
17 7 .
33 0 . J p .
17 4 .
93 1 .
04 1 . < . bias. For example, one could suggest that HMEs are easier to detect thanLMEs through the RV than Tr method around hotter (more massive) andfaster rotator stars. To check for observational biases, we trace first in Fig. 3athe distributions of the absolute V magnitude for the stars hosting LMEs andHMEs. We do find a trend for the stars hosting HMEs to be more luminousthan the stars hosting LMEs. This is confirmed by a non-parametric Mann-Whitney test with a p-value of 0.0007 (Dalgaard 2008). However, in Fig. 3bwe also show that the reason why the stars with HMEs are more luminous isbecause they are located farther out, and this is independent of the detectionmethod. Consequently, the trend for the HMEs to be found around hotterand faster rotator stars than the LMEs does not depend on the method ofdetection, but is a real physical difference. Considering the mass-temperaturerelation on the main sequence, this suggests that the more massive exoplanetsin our sample rotate around more massive stars.In Fig. 4 we compare the physical characteristics of the stars that hostHMEs with those that host LMEs. Fig. 4a shows a clear trend for rotationalvelocity to be higher in the stars hosting HMEs than LMEs. In Table 2we give the results of non-parametric, Mann-Whitney (MW) tests. The useof non-parametric tests is justified by the fact that we did not find normaldistributions for our data (established by running 3 different normality tests).In col. 6 we give the p-values of the tests at a level of confidence of 95%, incol. 7, the significance levels (low, *, medium, ** and high, ***) and in col. 8the acceptance or not of the differences observed. Note that a non-parametrictest compares the ranks of the data around the medians, not the means. In thecase of V sin i , the MW test confirms the difference of medians at a relativelyhigh level of confidence (the stars 22, 33 and 36 were not considered due totheir high uncertainties). In Fig. 4b we see the same trend for T eff on average,also with a significant difference in median in Table 2.In Figure 4c the difference in mass is obvious, and confirmed by the MW0 FLOR-TORRES ET AL. HME LME2468 V abs a) RV(HME) RV(LME) Tr(HME) Tr(LME)0100200300400 d i s t an c e ( p c ) b) Fig. 3. a) Absolute magnitude in V distinguishing stars hosting HMEs and LMEs;b) Distance of the stars also distinguishing by the detection method, radial velocity,RV, or transit, Tr. The bars correspond to the medians and interquartile ranges.
EVMEXAA MAIN JOURNAL DEMO DOCUMENT 11
HME LME V s i n i ( k m / s ) HME LME0.60.81.01.21.41.6 M * ( M ! ) HME LME0204060 J * x ( k g m s - ) HME LME5000550060006500 T e ff ( K ) HME LME-0.4-0.20.00.20.40.6 F e / H HME LME0246810 J p x ( k g m s - ) a) b)c) d)e) f) Fig. 4.
Comparing the physical characteristics of the stars and planet hosting HMEsand LMEs. In each graph the medians and interquartile ranges are drawn overthe data. a) rotational velocity, b) effective temperature, c) mass of the star, d)metallicity, e) the angular momentum of the star, f ) the angular momentum of theorbit of the dominant planet. V sin i ). Considering Eq. 2 inpaper I for the angular momentum of the star, the statistical test confirmsthat the HMEs rotate around more massive and faster rotator stars than theLMEs. Considering the difference in angular momentum of the exoplanets, J p , this result seems to support the hypothesis that more massive planetsform in more massive PPDs with higher angular momentum. The question,then, is how does this difference affects the migration process of the differentexoplanets?In Fig. 5a, following Berget & Durrance (2010), we trace the specific an-gular momentum of the stars, j ∗ = J ∗ /M ∗ , as a function of their masses,distinguishing between stars hosting HMEs and LMEs. The angular momen-tum of all the host stars in our sample fall well below the theoretical relationproposed by McNally (1965). There are is clear distinction between the LMEsand HMEs, except for the differences encountered in Fig. 4 and confirmed inTable 2, and there is no evidence the stars follow a j − M relation.On the other hand, when we compare in Figure 5b the angular momentumfor the systems, j sys = j ∗ + j p , we do see a difference between systems withLMEs and HMEs. But this is expected, since, by definition, the HMEs havinghigher masses naturally have a higher contribution to j sys . However, anddespite being more massive than Jupiter, very few of the HME systems havea value of j sys comparable to the solar system. Obviously, this is becauseof the large scale migration they suffered. To illustrate this point, we tracedin Figure 5c the possible “initial” angular momentum the system could havehad assuming the exoplanets formed at the same distance as Jupiter (5 AU).Comparing with the positions in Figure 5b, the HMEs would have lost onaverage 89% of their initial momentum, compared to 86% for the LMEs.Those losses are enormous. Considering the lost of angular momentum ofthe stars and planets, it might be consequently difficult to expect a couplingbetween J p and J ∗ or even a j − M relation.As a final test, we have calculated the Pearson correlation matrices (alsoexplained in Dalgaard 2008) for the systems with HMEs and LMEs. Theresults can be found in Table 3 and Table 4. Since the correlation matrices aresymmetrical we show only the lower diagonal of each, showing first the matrixEVMEXAA MAIN JOURNAL DEMO DOCUMENT 13 −0.4 0.0 0.2 0.4 log (M ) [M O ] l og ( J / M ) [ m ² / s ] ***** * * * * * * * * * * . . . . . G0 F5F0 A5 A0 (a) −0.4 0.0 0.2 0.4 log (M ) [M O ] l og ( J sys t e m M sys t e m ) [ m ² / s ] * * * * * . . . . . G0 F5F0 A5 A0 (b) −0.4 0.0 0.2 0.4 log (M ) [M O ] l og ( J sys t e m _5 A U M sys t e m ) [ m ² / s ] * * * * * . . . . . G0 F5F0 A5 A0 (c)
Fig. 5.
In (a) Specific angular momentum of the host stars as function of theirmasses. The symbols are as in Fig. 2. The solid black line is the relation proposedby McNally (1965) for low-mass stars (spectral types A5 to G0) and the dottedline is the extension of the relation suggested for massive stars. The black trian-gle represents the Sun; (b) specific angular momentum for the planetary systems,with the inverted black triangle representing the Solar system; (c) original angularmomentum assuming the planets formed at 5 AU.
PEARSON CORRELATION MATRIX FOR THE HME SYSTEMS
HME T eff log g [ Fe/H ] V sin i R ∗ M ∗ J ∗ M p R p a p e p log g Fe/H ] 0.6794 0.1519 V sin i R ∗ M ∗ < < J ∗ < M p R p a p e p J p < < log g [ Fe/H ] V sin i R ∗ M ∗ J ∗ M p R p a p -0.4539 0.5021 -0.5267 -0.5546 0.5544 e p J p -0.4305 0.9393 0.7272 0.4811 TABLE 4
PEARSON CORRELATION MATRIX FOR THE LME SYSTEMS
LME T eff log g [ F e/H ] V sin i R ∗ M ∗ J ∗ M p R p a p e p log g [ F e/H ] 0.3461 0.2153 V sin i R ∗ < < M ∗ < < < J ∗ < M p R p < a p e p J p < g -0.6992[ F e/H ] V sin i R ∗ M ∗ J ∗ M p R p a p -0.4521 e p -0.4418 J p EVMEXAA MAIN JOURNAL DEMO DOCUMENT 15with the p-values(with alpha ∼ . V sin i , T eff and log g . This suggests that the more massive the starthe higher its rotation.One difference between the two systems is that although there are nocorrelations of the temperature with the surface gravity in the HMEs, thereis one anticorrelations in the LMEs. Due to the smallness of our samples, itis difficult to make sense of this difference physically. However, we note thatlog g in the LMEs is also well correlated with the radius, the mass and angularmomentum (but not V sin i itself), something that is not seen in the HMEs.In Fig. 2 wee see that the dispersion of V sin i decreases at low temperature,which implies that the bi-exponential relation of V sin i as a function of T eff and log g becomes tighter, suggesting that the behavior of log g becomes moreordered, which could explain the anti correlations with T eff and correlationswith M ∗ and R ∗ ( and thus also with J ∗ ).It is remarkable to note that [ F e/H ] in both systems is not correlated withany of the other parameters, either related to the stars or the planets. For thestars, the most obvious correlations in both systems are between M ∗ and R ∗ ,or V sin i and J ∗ . Note also that although V sin i shows no correlation with M ∗ and R ∗ in the HMEs it is correlated with M ∗ in the LMEs. This mightalso explain why R ∗ is not correlated with J ∗ in the HMEs, while it is in theLMEs.In the case of the planets, the most important result of this analysis isthe (almost complete) absence of correlations between the parameters of theplanets and the parameters of the stars (most obvious in the LMEs). Theonly parameter that show some correlations with the star parameters is thesemi-axis, a p , of the orbit of the planets in the HMEs. This might suggests adifference in terms of circularization. Two results also seem important. Thefirst is that R p is only correlated to M p in in the HMEs, which is consistentwith the fact that the radius of the HMEs is constant. The second is thatthere is no correlation between J p and J ∗ , which, consistent with the behaviorobserved in Figure 5, and which could suggest there is no dynamical couplingbetween the two. This, probably, is due to important losses of angular mo-mentum during the formation of the stars and migrations of the planets.4. DISCUSSIONAlthough our sample is small, we do distinguish a connection betweenthe exoplanets and their stars: massive exoplanets tend to form around more6 FLOR-TORRES ET AL.massive stars, these stars being hotter (thus brighter) and rotating faster thanless massive stars. When we compare the spin of the stars with the angularmomentum of the orbits of the exoplanets, we found that in the HME systemsboth the stars and planets rotate faster than in the LME systems. This isconsistent with the idea that massive stars formed more massive PPDs, whichrotate faster, explaining why the planets forming in these PPDs are seen alsoto rotate faster.When we compare the effective angular momentum of the stars (Figure 5a)we found no evidence that they follow a j − M relation, in particular, likethe one suggested by McNally (1965), or that the angular momentum of thesystems (Figure 5b) follow such relation. Furthermore, there is correlation be-tween J p and J ∗ , which suggests that there is no dynamical coupling betweenthe two. This is probably due to the important losses of angular momentumof the stars during its formation (by a factor 10 ) and of the planets duringtheir migrations (higher than 80% their possible initial values). For the plan-ets, their final angular momentum depend on their masses where its migrationends, a p . Assuming, consequently that they all form more or less at the samedistance (farther than the ice line in their systems) and end their migration atthe same distance from their stars (which seems to be the case, as we shownin Figure 6), the HMEs would have lost a slightly higher amount of angularmomentum than the LMEs. Within the scenario suggested above, this mightsuggest that more massive PPDs are more efficient in dissipating the angularmomentum of their planets.One thing seems difficult to understand, however, which is consideringHMEs are more massive and lost more angular momentum during their mi-gration, why did they end their migration at almost exactly the same distancefrom their stars as the less massive LMEs. In Fig. 6 the accumulation weperceive between 0.04 and 0.05 AU is consistent with the well known phe-nomenon called the three-days pile-up (3 days is equivalent to slightly higherthan 0.04 AU, assuming Kepler orbits), which is supposed to be an artifactdue to selection effects of ground-based transit surveys (Gaudi 2005). How-ever, Dawson & Johnson (2018) in their review about migration suggestedthis could be physical, and some authors did propose different physical expla-nations (see Fleck 2008, and references therein). The model of Fleck (2008)is particularly interesting because it tries to solve the problem using the samestructure of the PPD that many authors believe explains how the stars loosetheir angular momentum during the T-Tauri phase, that is, magnetic brak-ing. In the Appendix B we did some calculations which show that where theplanets in our sample end their migration could be close to the co-rotationradius, the region of the PPD where the disk turns at the same velocity asthe star. But does this imply that disk migration is more probable thanhigh-eccentricity tidal migration?According to the theory of high-eccentricity tidal migration, one expectsa strong dependence of the tidal evolution timescale on the final location ofEVMEXAA MAIN JOURNAL DEMO DOCUMENT 17 + + + + + Only Dominant planet a (AU) J p ( k g m / s ) HME − exoplanet.orgLME − exoplanet.orgHME − This workLME − This work
Fig. 6.
Angular momentum of the exoplanets in our sample with respect to theirdistances from their stars. For comparison the exoplanets in our initial sample arealso shown in light gray. the orbit of the planets, a final (e.g., Eggleton 1998):˙ a ∝ a final (4)This implies that since the HMEs and LMEs have the same a final ∼ .
04 AUthey should also have the same tidal evolution timescale, and thus no differencewould be expected comparing their eccentricities. What stops the planetmigration in the high-eccentricity tidal migration model is the circularizationof the orbit through tidal interactions with the central stars. Assuming sametidal evolution timescale, therefore, we would expect the eccentricities for theHMEs and LMEs to be all close to zero (Bolmont et al. 2011; Remus et al.2012). Note that using our data to test whether the HMEs and LMEs havesimilar distributions in eccentricity is possibly difficult, because we are notcertain an eccentricity of zero is physical or not (meaning an absence of data).For the HME sample, on 22 exoplanets we count 7 (32%) with zero eccentricity,while in the LMEs out of 23, 10 (43%) have zero eccentricity. The differenceseems marginal. For the remaining planets with non zero eccentricities (15HMEs and 13 LMEs) we compare in Fig. 7 their distributions. There is aweak difference, with a median (mean) of e p = 0 .
19 ( e p = 0 .
22) for the HMEscompared, to e p = 0 .
04 ( e p = 0 .
10) for the HMEs and LMEs. A MW testyields a p-value = 0.0257, which suggests a difference at the lowest significancelevel. Therefore, there seems to be a trend for the HMEs to have on average ahigher eccentricity than the LMEs. The HMEs possibly reacted more slowly8 FLOR-TORRES ET AL.
HME LME0.00.20.40.60.8 e p Fig. 7.
Eccentricities of the exoplanets in our sample, distinguishing between HMEsand LMEs. to circularization than the LMEs (de Wit et al. 2016), suggesting possiblydifferent structures due to their higher masses (Flor-Torres et al. 2016).5. CONCLUSIONSBased on an homogeneous sample of 46 stars observed with TIGRE andanalysed using iSpec we started a project to better understand the connectionbetween the formation of the stars and their planets. Our main goal is tocheck is there could be a coupling between the angular momentum of theplanets and their host stars. Here are our conclusions.There is a connection between the stars and their exoplanets, which passesby their PPDs. Massive stars rotating faster than low-mass stars, had moremassive PPDs with higher angular momentum, explaining why they formedmore massive planets rotating faster around their stars. However, in termsof stellar spins and planets orbit angular momentum, we find that both thestars and their planets have lost a huge amount of angular momentum (bymore than 80% in the case of the planets), a phenomenon which could havepossibly erased any correlations expected between the two. The fact that allthe planets in our sample stop their migration at the same distance from theirstars irrespective of their masses, might favor the views that the process ofmigration is due to the interactions of the planets with their PPDs and thatmassive PPDs dissipates more angular momentum than lower mass PPD.Consistent with this last conclusion, HMEs might have different structuresthan LMEs which made them more resilient to circularization.We like to thank an anonymous referee for his careful revision of our resultsand for his comments and suggestions that helped us improved our work. L. M.F. T. thanks S. Blanco-Cuaresma for discussions and support with iSpec . SheEVMEXAA MAIN JOURNAL DEMO DOCUMENT 19also thanks CONACyT for a scholar grant (CVU 555458) and travel support(bilateral Conacyt-DFG projects 192334, 207772, and 278156), as well as sup-ports given by the University of Guanajuato for conference participation andinternational collaborations (DAIP, and Campus Guanajuato). This researchhas made use of the Exoplanet Orbit Database, the Exoplanet Data Explorer(exoplanets.org Han et al. 2014), exoplanets.eu (Schneider et al. 2011) andthe NASA’s Astro-physics Data System.APPENDICESA. CALCULATING THE SEFL-GRAVITATING MASS LIMITAccording to Padmanabhan (1993) the formation of structures with dif-ferent masses and sizes involves a balance between two forces, gravity, F g andelectromagnetic, F e . For the interaction between two protons, we get: F e F g = κ e e /r Gm p /r = κ e e Gm p = (cid:18) κ e e ¯ hc (cid:19) (cid:18) ¯ hcGm p (cid:19) (A5)where κ e and G are the electromagnetic and gravitational constants, that fixthe intensity of the forces, e and m p the charges and masses interacting ( m p is the mass of a proton) and r the distance between the sources (the laws haveexactly the same mathematical form). Introducing the reduced Plank constant¯ h = h/ π and the velocity of light c (two important constant in physics), thefirst term on the right is the fine structure constant α ∼ . × − , and thesecond term is the equivalent for gravity α G ∼ . × − . This impliesthat: αα G ∼ . × (A6)This results leads to the well-known hierarchical problem: there is no con-sensus in physics why the electromagnetic force should be stronger than thegravitational force in such extreme. The reason might have to do with thesources of the forces, in particular, the fact that in electromagnetic there aretwo types of charge interacting, while there is only one type of mass. The im-portant consequence for the formation of large-scale structures is that whilein the electromagnetic interaction the trend to minimize the potential energyreduces the total charge, massive object being neutral, the same trend in grav-ity is for the gravitational field to increase with the mass. Therefore, smallstructures tend to be dominated by the electromagnetic force, while largestructures are dominated by gravity. Based on this physical fact, Padman-abhan calculated that there is a critical mass, M c , above which the force ofgravity becomes more important than the electromagnetic force. The pointof importance for planet formation is that this critical mass turned out to becomparable to the mass of a gas giant planet.To realize that, it suffices to compare the energy of ligation of a structurewith its gravitational potential energy. Since the escape energy for an electronis: E ≈ α m e c ≃ . × − J (A7)0 FLOR-TORRES ET AL.a spherical body formed of N atoms would have an energy of ligation E = N × E . On the other hand, its mass would be M = N m p and its volumewould be 4 / πR = N × / πa , where a ≃ . × − m, is the Bohrradius. From this we deduce that its radius woud be R ∼ N / a , and itsgravitational potential: E G ≃ GM R ≃ N / Gm p a = N / α G m e αc (A8)Now, the condition for a stable object to form under gravity is E ≥ E G , whilefor E < E G the object would collapse under its own weight. In equilibriumwe would thus have: N / α G m e αc = N α m e c (A9)This yields to a maximum number of atoms, N max ∼ ( α/α G ) / ∼ . × ,and a critical mass: M c = N max × m p ∼ . × kg ∼ . M J (A10)This suggests that the inflection point we observe in the mass-radius relationof exoplanets could be the critical point where an object becomes unstableunder self-gravity.B. AT WHAT DISTANCE IS THE COROTATION RADIUS, R CO In their review about the migration of planets, Dawson & Johnson (2018)claimed that the distribution of exoplanets around their host stars is consis-tent with the co-rotation radius, R co , which is part of the magnetic structureconnecting the star to its PPD. In the model presented by Matt & Pudritz(2004), the authors discussed some specific aspects of this magnetic structurethat could contribute in stopping planet migration (see their Figure 3 andexplanations therein). Following this model, the star and disk rotate at differ-ent angular speeds except at R co , where the magnetic field becomes twistedazimuthally by differential rotation, triggering magnetic forces. These forceswould act to restore the dipole configuration conveying torques between thestar and the disk. As a planet approaches R co , therefore, these torques wouldtransfer angular momentum from the star to the planet stopping its migra-tion (Fleck 2008). One important aspect of this model is that the dumpingwould only work up to a distance R out ∼ . R co , and thus one would expectthe planets to pile-up over this region, that is, between R out and R co . Thequestion then is what is the value of R co ?One way to determine this value is to assume that when the wind of anewly formed star starts evaporating the PPD, R out −→ R co and the mag-netic pressure at R co balances the gas pressure due to the wind, P B = P g .Assuming, that the intensity of the magnetic field decreases as the cube ofthe distance, we get, B ∼ B s R ∗ /r , where B s is the intensity of the magneticEVMEXAA MAIN JOURNAL DEMO DOCUMENT 21field at the surface of the star, R ∗ is the radius of the star, and µ is thepermeability of the vacuum. This yields that: P B = B µ = B s R ∗ µ r (B11)For the gas pressure we used the expression for “Ram pressure”: P g = nmv (B12)where v is the velocity of the wind and nm is its load, that is, the amount ofmass transported by the wind. This load then can be expressed in terms ofthe flux of matter, ˙ M , as: nm = ˙ M πr v (B13)Assuming equality at r = R co , we get: R co = 2 πµ B s R ∗ ˙ M v (B14)which for a star like the Sun ( B s ∼ − T, ˙ M ∼ . × − M ⊙ / yr and v = 2 . × − m/s) yields R co ∼ . × m or 0 .
045 AU. Accordingto this model, therefore, we could expect to see a real pile up of exoplanetsindependent of their masses (and losses of angular momentum) between 0 . − .
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