## Research topics

Some results obtained in the main research topics are described in this section. All these topics, mostly included in celestial mechanics and astrometry fields, are anyhow related to the study of double and multiple stellar systems:

##### Perturbed N-body problem

- N-parametric canonical perturbation method
- Orbital stability of stellar and planetary systems
- Binary systems with mass loss by
*periastron effect*

##### Determination of fundamental stellar parameters

##### Astronomical observations

### Perturbed N-body problem

#### N-parametric canonical perturbation method

The need for solving nonlinear differential equations arises frequently in the study of many dynamical problems. Since, in general, the exact solution cannot be found by means of classical integration methods, it is necessary to use alternative analytical or numerical methods. The perturbation methods, like the one that here is designed, based on asymptotic developments of the equations of motion in terms of one or more small parameters belong to the first class.

Therefore, this work deal with perturbation methods for nonlinear dynamical systems based on the so-called theory of Lie transforms. The first methods of this type, referring to the case of Hamiltonian systems depending on one small parameter, were given by Hori (1966) and Deprit (1969). Although these methods are equivalent (Campbell & Jefferys 1970; 1970; Henrard & Roels 1974), they are not identical. Later, Kamel (1970) and Henrard (1970) generalized them to arbitrary systems of differential equations. Ribera (1981) and Abad & Ribera (1984) obtained a Lie transform method applicable to Hamiltonian systems depending on two small parameters. This was used by Prieto & Docobo (1997a, 1997b) to analytically integrate the two-body problem with slowly decreasing mass. Varadi (1985) also obtained a two-parameter method based on differential geometry, as in Henrard & Roels (1974). This was subsequently extended to the case of three parameters by Ahmed (1993). Recently, a three-parameter canonical method (Andrade 2002) has been applied to the integration of the two-body problem with mass loss depending both on the time and the distance of the bodies.

Despite these partial results, there are a considerable number of interesting problems in different fields in which it is necessary to consider a large number of small parameters and their associated perturbative expansions. To our knowledge, no generalization of the Hori-Deprit canonical method has so far been presented in the case of N parameters, with N arbitrary. That was precisely the main purpose of this investigation, besides the integration of the classical Gyldén–Meŝĉerskij problem —that of the relative motion of two stars that are losing mass over time— when, in addition, the primary's oblateness, as well as relativistic effects, is taken into account. Moreover, speed and accuracy comparisons between this analytical method and a numerical one are accomplished.

M. Andrade

N-parametric canonical perturbation method based on Lie transforms

The Astronomical Journal, 136(3), 1030-1038 (2008). [Download pdf]

#### Orbital stability of stellar and planetary systems

Several perturbations can produce secular variations in the orbital elements of components in stellar or planetary triple (and multiple) systems affecting inevitably its orbital stability. The most interesting ones, taking into account its abundance, are the stellar (hierarchical) triple systems and the stellar binary systems with planets in S-type orbits when considering different perturbations: one component's oblateness, relativistic effects of the gravitational field, fast rotation, mass loss/transfer, accretion disk...

One of the targets is to define general stability criteria what allow us to determine stability/unstability regions and to make predictions about the long-term evolution of these systems.

In a wider interpretation of the orbital stability concept we must consider the determination of the habitability regions, above all for terrestrial-type planets or, even, for satellites of giant planets.

M. Andrade & J.A. Docobo

Long-term stability for the Bb planetary-like object in the triple stellar system Gl 22

XI Jornadas de Trabajo en Mecánica Celeste. Ezcaray (La Rioja), June 25-27 (2008).

#### Binary systems with mass loss by *periastron effect*

Dynamics of binary systems with mass loss depending on time and *periastron effect* —that is, a hypothetical increase in mass loss during the periastron passage— is analyzed by means of analytical and numerical methods. In this way, we study variations of the orbital elements depending on three small parameters that describe the mass loss depending on time (α_{1}, α_{2}) and by *periastron effect* (β). We show that the last one causes secular variations in some orbital elements: eccentricity, semimajor axis and period. Finally, it is suggested that certain anomalous behaviours observed in the orbital motion of some close binaries could be explained taking into account the *periastron effect*.

M. Andrade & J.A. Docobo

Orbital dynamics analysis of binary systems in mass-loss scenarios

Revista Mexicana de Astronomía y Astrofísica (SC), 15, 223-225 (2003). [Download pdf]

M. Andrade & J.A. Docobo

Periastron effect enhancement by Kozai resonance: the BU 1099 AB system

X Workshop on Celestial Mechanics. Barcelona, September 5-7 (2007).

### Determination of fundamental stellar parameters

#### Calculation of binary system orbits

Binary systems are very important in astronomy because the calculation of their orbits is the only direct method to obtain stellar masses, and from them, indirectly, other fundamental parameters. Moreover, these values are essential to establish the empirical mass-luminosity relation (MLR), that itself allows us to estimate masses of individual stars as well.

- WDS 00243+5201 / HU 506 / ADS 328

Orbital elements - WDS 00516+2237 / A 1808 / ADS 701

Orbital elements - WDS 00568+6022 / BAG 10Aa / ADS 784

Orbital elements - WDS 01409+1117 / A 2320 / ADS 1321

Orbital elements - WDS 02278+0426 / A 2329 / ADS 1865

Orbital elements Paper: 2007RMxAA..43..237A

- WDS 02396-1152 / FIN 312

Orbital elements - WDS 04515-3454 / FIN 320

Elementos orbitais - WDS 05117+0031 / HU 33 / ADS 3767

Orbital elements - WDS 13320-6519 / FIN 369

Orbital elements - WDS 14369+4813 / A 347 / ADS 9324

Orbital elements Paper: 2004AJ....127.1181D

- WDS 16115+0943 / FIN 354

Orbital elements - WDS 16198+2647 / A 225 / ADS 10007

Orbital elements Paper: 2008AJ....135.1803D

- WDS 18208+7120 / STT 353AB / ADS 11311

Orbital elements - WDS 19351+5038 / HU 679 / ADS 12656

Orbital elements - WDS 20374+7536 / HEI 7

Orbital elements - WDS 21158-5316 / FIN 329

Elementos orbitais - WDS 21287+7034 / LAB 6 / ADS 15032

Orbital elements - WDS 21477-3054 / FIN 330AB

Elementos orbitais (I) Elementos orbitais (II) - WDS 21597+4907 / HU 774 / ADS 15530

Fax +34 982 285 926

>Orbital elements Paper: 2004AJ....127.1181D

- WDS 23186+6807 / STF 3001AB / ADS 16666

Orbital elements Paper: 2003AJ..126..1522A

M. Andrade

Orbit, masses and spectral analysis of th visual binary A 2329

Revista Mexicana de Astronomía y Astrofísica, 43, 237-242 (2007). [Download pdf]

#### Computation of stellar masses and parallaxes

The observation of stellar binary systems with accurate techniques allows us to obtain reliable measurements of their coordinates (position angle, θ; and separation, ρ). From a set of those large enough it would be possible to calculate accurately the orbit of the system.

Taking into account the obtained orbital elements and, on occasion, additional data from the spectrum, it is possible to calculate directly and very accurately either the parallax —or equivalently the distance to the system— or the mass.

J.A. Docobo & M. Andrade

A methodology for the description of multiple stellar systems with spectroscopic subcomponents

The Astrophysical Journal, 652, 681-695 (2006). [Download pdf]

J.A. Docobo, V.S. Tamazian, M. Andrade, N.D. Melikian & A.A. Karapetian

Refining the parallax in visual double stars using orbital and spectral data: application to the system of the giants, A 1808

The Astronomical Journal, 136, 890-894 (2008). [Download pdf]

### Astronomical observations

#### Optical *speckle* interferometry of binary systems

It is well known that speckle interferometry (Labeyrie 1970) is one of the most effective techniques for high-angular-resolution observations of binary stars. The high-quality astrometric information obtained at different wavelengths and close to the diffraction limit of the telescopes is of vital importance in the determination of orbits and dynamical parameters of binary and multiple stars.

The investigation group of the Astronomical Observatory R.M. Aller (USC) has an optical *speckle* ICCD camera which has been used in several obsevational runs in different observatories since 1999.

The camera was attached to the 3.5-m telescope at Calar Alto Observatory (CAHA, Almeria) in the last of these observational runs (July 2005). Fifty stars with separations between 0.″058 e 2.″1 were observed, which allowed to obtain high-quality optical speckle data in binaries with separations close to the diffraction limit of the telescope.

J.A. Docobo, V.S. Tamazian, M. Andrade, J.F. Ling, Y.Y. Balega, Lahulla & A.F. Maximov

First results of the optical speckle interferometry with the 3.5 m telescope at Calar Alto (Spain): measurements and orbits of visual binaries

The Astronomical Journal, 135, 1803-1809 (2008). [Download pdf]

#### Astrometric detection of extrasolar planets

Hierarchical triple system Gl 22 consists of three red dwarfs Aa, Ab and B. The orbital period of the inner orbit (pair Aa-Ab) is 15.64 yr, whereas that of the outer one (B relative to the mass center of Aa-Ab) is 223.4 yr. Both orbits are coplanar.

When determining the outer orbit, a weak sinusoidal pattern in the apparent motion of the component B has been noticed. It can be attributed to either a very unusual distribution of observational residuals or an unseen fourth body in the system. In the latter case, the star B would consist of the components Ba and Bb (extrasolar planet).

Under assumption of Bb to be a very low-mass object (~16 M_{J}) on a circular orbit around Ba with a period of around 15 years, semimajor axis 0.″35 and coplanar with other two orbits, the observational residuals of the outer orbit are improved. In such case, the component Ba would be moving relative to the mass center of the virtual pair Ba-Bb on an orbit with a semimajor axis of 0.″03.

These motions are illustrated in the last figure on which blue line corresponds to the orbit of the Ba-Bb mass center relative to that of Aa-Ab and the red one shows the motion of the component Ba affected by the virtual component Bb. Similar to all visual, photographic and CCD observations, a single speckle measurement marked as “*speckle* (LC)” had initially been showing the position of B relative to the light center of Aa-Ab. For the orbits calculation, all such measurements have been reduced to the mass center of Aa-Ab.

J.A. Docobo, V.S. Tamazian, Y.Y. Balega, M. Andrade, D. Schertl, G. Weigelt, P. Campo & M. Palacios

A methodology for studying physical and dynamical properties of multiple stars. Application to the system of red dwarfs Gl 22

Astronomy & Astrophysics, 478, 187-191 (2008). [Download pdf]

M. Andrade e J.A. Docobo

On the dynamical stability of the very low-mass object Gliese 22 Bb

Icarus, 215(2), 712-720 (2011). [Baixar pdf]