A forma mais comum de representar o momento angular estelar ´e conhecida como momentum angular espec´ıfico, cujo na perspectiva estelar pode ser representado porj? = MJ?
?
(s´ımbolo adotado nesse estudo). Com o vi´es de n˜ao depender da massa estelar, sua medida no SI ´e:m2s−1, onde o expoenteαda lei de kraft (J? ∝M?α) , ´e representado:
j? = J?
M? ∝ M?α
M? ∝M?α−1 (2.23)
Kraft [7], encontrou uma lei de potˆencia,hj?i ∝M?0,57, para estrelas mais massivas que1,5M, retratado pela equac¸˜ao 2.23 . No caso da investigac¸˜ao apenas do momentum angu-lar esteangu-lar (J?), sua relac¸˜ao se tornariaJ? ∝M?1,57, note, que ambas expressam leis equivalentes.
Estimamos o erro bruto emj?como sendo:
σj? =j? s
σJ? J?
2
+ σM?
M? 2
(2.24) O momentum angular espec´ıfico, pode ser evidenciado tamb´em para os planetas (jp = mJp
p) ou at´e mesmojtot = mJtot
p+M?, sendo o ´ultimo uma interac¸˜ao total do sistema (estrela + planetas), deste modo, a representac¸˜ao do momentum angular por unidade de massa se estende para qualquer estudo (estelar, planet´ario ou estrela-planetas) tomando apenas o devido cuidado na troca dos ´ındices e letras de acordo com estudo abordado.
Bergetet al.[49] analisam o momentum angular espec´ıfico estelar para28sistemas extrasolares, encontrando que estrelas hospedeiras (asteriscos) s˜ao inferiores as do estudo de
34
McNally[8] (cruzes)3, como mostra a figura 12. O Sol (caixa inferior) acompanha a regi˜ao ocu-Figura 12 – Momentum angular espec´ıfico versus massa estelar para os dados de Berget &
Durrance (2010) e McNally (1965).
l
sys= L
∗+ L
pM
∗+ m
p(3)
To study the angular momentum distribution within a planetary system, the percentage of the angular momen-tum that resides in the exoplanet is calculated using the following equation:
%L = L
pL
p+ L
∗(4)
2.3. Error Estimates
Errors were propagated through the above calculations to determine the total error for each quantity. Errors were available for most of the parameters in Table 1.
Errors in e and i were not used since they were either not given or were too small to have much effect. Errors in the stellar rotational velocity dominate the other errors in the calculations and are thus listed separately in Table 1.
As previously mentioned, six of the stars in this research had no error listed for V sin i so for those systems no errors are calculated and their errors are unknown.
Stellar rotational velocities can be determined a num-ber of ways, including Doppler broadening of narrow spectral lines, Fourier transform methods, Doppler imag-ing, and interferometry. Our literature search found that Doppler broadening was the technique used for most of the stars in this study. In this method, the width (FWHM) of He I lines in the stellar spectra are measured and, using a relationships between line width and the projected rotational velocity, V sin i can be determined.
With this method, aspects such as gravity, temperature, and turbulence must be accounted for (De Medeiros et al. 2006, Pace & Pasquini, 2004).
3. RESULTS AND DISCUSSIONS
A study performed by McNally (1965) showed that more massive main-sequence stars rotate more rapidly and contain more specific angular momentum than less massive ones. A plot of McNally’s results is shown in Figure 1. Each of the crosses gives the average angu-lar momentum per unit mass for a given spectral class.
There is a definite break in this trend at spectral class A5, where the slope of decreasing specific angular mo-mentum with decreasing mass becomes steeper. The two solid lines are least squares fits to the data above and be-low the break. Also included are the angular momentum per unit mass of the Sun (lower box) and the angular momentum per unit mass of the entire solar system (up-per box). It has been suggested that if the angular mo-mentum of the planets were included in specific angular momentum measurements, the trend in specific angular momentum with system mass might continue with the same slope as for more massive stars and that this might indicate planet formation. It has also been pointed out that it is risky to extrapolate with only one data point.
To compare the exoplanetary systems studied here with this trend in angular momentum, Figure 2 plots the angular momentum per unit mass of the host stars (asterisks) and the angular momentum per unit mass of the complete systems (triangles) as a function of stellar mass. The average angular momentum per unit mass of the various spectral types provided by the data from Mc-Nally (1965) are also shown with crosses and the Sun and
Fig. 1.— Plot of the average angular momentum per unit mass of main-sequence stars for eight spectral classes ranging from O5 in the upper right corner to G0 near the bottom left (McNally, 1965). Also included are the angular momentum per unit mass of the Sun (lower box) and the angular momentum per unit mass of the entire solar system (upper box).
Solar System with boxes. The scale has been expanded somewhat to emphasize the exoplanetary data by leav-ing off the more massive stars in the McNally data. The dashed line is an extension of the slope from the O5 to A5 data that is included to guide the eye. Errors are not shown for these data because they clutter the figure; a typical value is ≈ 0.1 decade.
Fig. 2.— Plot of the angular momentum per unit mass of the host stars (asterisks) and the angular momentum per unit mass of the complete systems (triangles) as a function of stellar mass. The average angular momentum per unit mass of the various spectral types from McNally (1965) are also shown with crosses and the Sun and Solar System with boxes.
It is clear from this plot that the host stars have an-gular momentum per unit mass at or below the slope for stars in this mass range and that the angular mo-mentum per unit mass of the complete systems (star + planets) fall above this slope. It is also clear from this that the angular momentum per unit mass of the com-plete systems fall below the extended slope of the more massive stars. Thus the conjecture that the specific an-gular momentum of the complete system might continue with the same slope as the O5 to A5 data is not sup-ported; however, since the specific angular momentum of a planet scales like l
p∝ √
a, a Jupiter mass planet in a Jupiter like orbit would have ≈ 10 times greater l
pthan the planets in this study. Both radial velocity and transit methods are biased toward planets in small or-bits. It seems reasonable that there could be undetected
Fonte: Retirado de Berget e Durrance [49]
pada pelas estrelas com planetas e o Sistema Solar (caixa superior) sugere uma alta concentrac¸˜ao de momentum angular por unidade de massa, al´em disso,jtot (momentum angular espec´ıfico total do sistema) aparecem na forma de triˆangulos com valores superiores aos dej?(momentum angular espec´ıfico estelar) [49].
3Este resultado ´e o oposto ao apresentado por Alveset al.[4] (ver artigo), no entanto, estes dois estudos abrem uma discuss˜ao entre a diferenc¸a nas taxas de momentum angular em estrelas com e sem planetas. Assim, tomando como base estes dois trabalhos apresentamos no cap´ıtulo 4 uma an´alise envolvendo 355 estrelas da sequˆencia principal com e sem planetas confirmados visando trazer `a tona os resultados apresentados por Bergetet al.[49] e Alveset al.[4].
3 DADOS OBSERVACIONAIS
Nesse cap´ıtulo, apresentamos as principais propriedades f´ısicas presentes nas amos-tras para o estudo do momentum angular estelar, planet´ario e total. Para isso, usamos uma amostragem total de578estrelas em diferentes est´agios evolutivos (355da sequˆencia principal e223do ramo das gigantes e subgigantes (ver tabela 2)).
Tabela 2: Informac¸˜oes para578estrelas em diferentes est´agios evolutivos presente nessa pesquisa
M´etodo Empregado
Classe Estelar VR Trˆansito NoEstrelas
SP 131 224 355
Gigantes 157 0 157
Subgigantes 66 0 66
Total 578
Fonte: Autor.
257de355estrelas s˜ao da sequˆencia principal e tem planetas confirmados e foram catalogados de The Exoplanet Orbit Database1[42] (131 estrelas do tipo espectral F, G e K, na qual102pertencem a sistemas com um planeta (uma estrela + um planeta) e29a sistemas multiplanet´arios (uma estrela + dois ou mais planetas), hospedando cerca161 exoplanetas que foram descobertos pela t´ecnica de velocidade radial) e NASA Exoplanet Archive2[41] (126 estrelas (F, G e K), em que 112 possuem somente um planeta e 14 hospedam mais de um planeta, totalizando141exoplanetas encontrados pelo m´etodo de trˆansito).
Para as98estrelas restantes do total de355, as mesmas n˜ao possuem informac¸˜oes de planetas e foram obtidas de Lanzafameet al.[39] (essa amostra tinha um total de217estrelas, no entanto, utilizamos apenas98devido algumas componentes terem rotac¸˜oes muito elevadas)3. J´a para223estrelas com e sem planetas confirmados dos ramos evolu´ıdos (30subgigantes e56 gigantes com planetas,36subgigantes e101 gigantes sem planetas)4as mesmas foram obtidas de Jofr´eet al.[40]5.
1http://exoplanets.org/
2https://exoplanetarchive.ipac.caltech.edu/
3Lanzafame et al.[39] obteve esses dados de Hartmanet al.[50] que utilizaram instrumentos de buscas por trˆansitos planet´arios para estimarem os per´ıodos de rotac¸˜oes estelares.
4A amostra de Jofr´eet al.[40] foi obtida de uma base de dados que o m´etodo empregado ´e a velocidade radial nas buscas por exoplanetas, s´o que apenas86estrelas da mesma tem planetas confirmados.
5Vale salientar que as amostras utilizadas nessa pesquisa n˜ao est˜ao isentas de planetas, no entanto, n˜ao h´a informac¸˜oes sobre os mesmos ou possuem planetas de baixas massas que ainda n˜ao foram encontrados.