Binary astrometrical

A astrometrical Binaire is a binary star whose two components are not solved, duplicity being revealed by the orbital movement Photocentre on the sky. In particular, when the companion is much weaker than primary star, it is the movement reflex of this one which is observed. Precise and very exact astrometrical measurements are necessary to detect these objects, but this method could lead in the future to many detection planets extrasolaires.

History

After having been the first to estimate a stellar Parallax precisely, that of 61 Cygni in 1838, Bessel was also going to discover by chance the first two binary astrometrical. In a letter of August 10th, 1844, Bessel indicated that the own Mouvement of Sirius and Procyon was not constant. After having eliminated various assumptions, he concluded with accuracy in the two cases with the presence from a massive but obscure body orbiting with one period from approximately a half-century, an assumption however perturbing which he justified by: “ the light is not a real property of the mass. The existence of innumerable visible stars does not exclude the existence from innumerable invisible stars ”.

This discovery is not without pointing out to the prediction of Neptune by Urbain the Glassmaker two years later, whose François Arago says that he “ saw the new star at the end of his feather ”. For the binary astrometrical ones, the confirmation required however more time. 7 years had to be waited so that the orbit of Sirius was actually calculated (Peters 1851), the companion of Sirius was seen only in 1862 by Alvan Graham Clark and that of Procyon that in 1896 per John Mr. Schaeberle, transforming blow these binary astrometrical into binary visual. These new companions were besides the white dwarf first known.

This first success nevertheless was not followed of an avalanche of new results. More than one century later, one only counted only 17 binary astrometrical (and 14 suspect cases) (van of Kamp, 1975).

Astrometry requires very exact observations, and can lead in the contrary case to incorrect results. In 1943, K. Strand announced the presence of a planet extrasolaire around the star 61 Cygni. In 1960, S. Lippincott made an identical advertisement for Lalande 21185. In 1963, P. Van de Kamp found a planet massive one 24 years period around the star of Barnard, then indicated in 1978 qu ' it acted of two planets. None of these advertisements was confirmed since and the most probable explanation would be the presence of systematic errors in the observations.

Recent technological developments and to come could nevertheless change gives it, quantitatively and qualitatively. In particular, the Catalog Hipparcos contains approximately 4000 objects suspectés to be the binary astrometrical ones.

Classification

According to the orbital Period, the size of the Equatorial apparent radius (angular), and them precise details of the astrometrical instrument concerned, one can define several categories of binary astrometrical. A more precise instrument or a base of time of longer observation thus modifies this classification. The main part of the categories indicated comes from the Catalog Hipparcos, thanks to its precision and with its number of objects observed.

• the binary orbital ones: the orbital period is of about size of the period of observation of the instrument, and the Orbite can really be calculated.
• the binary ones with acceleration: the orbital period is very long, and only a variation even an inflection of the own Mouvement can be observed.
• binary the stochastic S: the period is short or intermediate and the low amplitude, resulting in a “random” dispersion of appearance of individual measurements. In this case, as in the precedent, only of the complimentary measures allowing to obtain a complete orbit could justify in any rigor the character of binary.

For this, it is necessary to add double stars for which one notes a nonorbital astrometrical movement. In the majority of the cases it can be the binary ones at very long period, but it can sometimes be a question of a nonbinary star couple considering fortuitously on the same line of sight (optical doubles):

• double variables: if one of the components is variable, then the photocentre of the system changes place according to the luminosity of variable star
• double photocentric: if one makes Astrométrie in several spectral bands simultaneously, and if the two components have a different color, the position of the photocentre will vary with the band of observation.

In what follows, one will be interested only in those whose orbit can be highlighted, but without making assumption on the nature of the secondary object, which it is stellar, brown Naine or planet extrasolaire.

Theory and application

Equations of the movement

The photocentre describes a Orbite around the Barycentre which is homothetic in general with that of the most brilliant star but with a Equatorial radius $a\_0$ which can be different in the face. The equatorial variations of position in coordinated on the tangent level of the sky are written:

$\backslash \; left\; \backslash \; \{\backslash \; begin\; \{matrix\}$

\ Delta \ alpha \ cos \ delta = a_0 \ frac {1-e^2} {1+e \ cos \ naked} \ left \ sin \ Omega + \ sin (\ nu+ \ Omega) \ cos \ Omega \ cos I \ right \ \ \ Delta \ delta = a_0 \ frac {1-e^2} {1+e \ cos \ naked} \ left \ cos \ Omega - \ sin (\ nu+ \ Omega) \ sin \ Omega \ cos I \ right \end{matrix}\right. where:

* E = eccentricity of the orbit.

*$\backslash \; nu$ = True anomaly, function of the time passed since the passage to the Pericenter, of the orbital Period, the date of passage to the Pericenter and of the eccentricity.

*$\backslash \; omega$ = angle enters the node and the Périastre.

*$\backslash \; Omega$ = angle of position of the ascending Node.

* I = Slope, angle enters the normal in the plan of the orbit and the line of sight.

Function of mass

Even if one does not see the Orbite of each component, nor the relative orbit of the secondary around the primary education, the third law of Képler in adapted units states nevertheless that:

$(M\_1\; +\; M\_2)\; =\; \backslash \; frac\; \{(\backslash \; varpi\; has)\; ^3\}\; \{P^2\}$

where:

* M₁ = mass of primary star in solar Mass.

* M₂ = mass of the secondary object in solar Mass.

* a₁ = Equatorial radius of the orbit of the primary education around the barycentre in Second of arc.

* a₂ = Equatorial radius of the orbit of the secondary around the barycentre in Second of arc.

* has = a₁ + a₂ = Equatorial radius of the relative orbit in Seconde of arc.

*$\backslash \; varpi$ = annual Parallax in Second of arc

* P = orbital period in Year S.

In addition, by definition of the center of gravity, there is $a\_1\; M\_1\; =\; a\_2\; M\_2$ thus $a\_1\; =\; B\; a$ where the fractional mass of the secondary is noted

$B\; =\; \backslash \; frac\; \{M\_2\}\; \{M\_1\; +\; M\_2\}$

In the same way, if one notes

* L₁ = Luminosité of primary star in solar Luminosité in the spectral Bande observed,

* L₂ = Luminosité of the secondary object with the same units,

* $\backslash \; Delta$m = -2.5 log ( L₂ / L₁ ) the difference of magnitude between the components,

then, the distance $d$ of the photocentre to the primary education is such as $d\; L\_1\; =\; (has\; -\; d)\; L\_2$, = \ beta a is $d\; where\; the\; fractional\; luminosity\; is\; noted$

$\backslash \; beta\; =\; \backslash \; frac\; \{L\_2\}\; \{L\_1\; +\; L\_2\}\; =\; (1+10^\; \{0.4\; \backslash \; Delta\; m\})\; ^\; \{-\; 1\}$

To know this difference in magnitude would give access to the magnitudes of each component, because the magnitude of the unsolved object is already measured. The equatorial radius of the orbit of the photocentre is worth thus

$a\_0\; =\; (B\; -\; \backslash \; beta)\; a$

In general this term is positive, for example when the two components are on the principal sequence, but the contrary sign can also occur in certain cases.

The third law of Képler watch as well as binary astrometrical makes it possible to give access to the function of masses (and luminosities)

$(M\_1\; +\; M\_2)\; \backslash \; cdot\; (B\; -\; \backslash \; beta)\; ^3\; =\; \backslash \; left\; (\backslash \; frac\; \{a\_0\}\; \{\backslash \; varpi\}\; \backslash \; right)\; ^3\; \backslash \; frac\; \{1\}\; \{P^2\}$

where the variables of the member of left are unknown while the member of right-hand side is obtained by the astrometrical analysis.

Masses and luminosities

It is seen that only one equation for the three unknown factors which are the masses and the difference in magnitude informs little about the nature of the involved objects… More, it is necessary either to resort to additional assumptions, or to be in the presence of a spectroscopic Binaire, when it is possible.

• the most favorable case occurs when the object can also be detected like spectroscopic Binaire with two spectra (binary named BS2 ). The union making the force, the astrometrical orbit raises the ambiguity of the Inclinaison which handicaps the spectroscopic orbits, and the masses of each component are then obtained. The function of masses above then gives access to the difference in magnitude. The masses and luminosities of each component are obtained, as in the example illustrated opposite. In this example, the star is detected like binary stochastic by Hipparcos, but the precision of measurements does not make it possible to have an orbit. Also detected like BS2 of period 4,5 years by the Coravel spectrometer, the combination of astrometrical and spectroscopic measurements would indicate that M₁ = 0,7, M₂ = 0,6 mass solar, $\backslash \; Delta$ m = 1,8 magnitudes.
• When the object is known like spectroscopic Binaire with a spectrum only ( SB1 ), all is not lost. In such a case, not only the spectroscopic orbit helps with the determination of the whole of the orbital parameters, but moreover to secondary can be it regarded as much weaker than the primary elections. Consequently, one can consider that $\backslash \; beta\; \backslash \; approx\; 0$ and $a\_0\; \backslash \; approx\; a\_1$. The color and the absolute magnitude of the unsolved object are thus primarily those of the primary education. Using a model, one from of roughly deduced the mass from the primary education, and masses it above secondary from of deduced by the function from mass. Naturally, it is only one approximation.

Detectability

Detection, the confirmation, the precision of the orbit depend on the size of the Equatorial radius of the orbit of the photocentre, or in any case of the relative error on this one. Taking into account the relations above:

• It will be known all the more precisely as the object will be close (large parallax).
• Concerning the masses and luminosities, one notes two extreme cases. When $B\; \backslash \; approx\; \backslash \; beta$, for example for two twin objects, no astrometrical signal can be observed. In the same way, when the secondary companion has a very small mass and a very weak luminosity compared to the primary education, and it is in particular the case for the planets extrasolaires. The capacities of detection by astrometry are optimal when $|B\; \backslash \; beta|$ is maximum.
• For masses and luminosities given, $a\_0\; \backslash \; propto\; P^\; \{2/3\}$, therefore the sensitivity will be better at the orbital long periods, in so far as the base of time of observation is rather long.
• Lastly, contrary to the visual binary for which it is the relative movement of the two components which is measured, the detection of binary astrometrical requires to have a reference mark of very precise reference to minimize the systematic errors, one of the reasons explaining the need for space Astrométrie.

Instruments of observation

• Formerly, the meridian Circles
• the satellite Hipparcos
• the Space telescope Hubble
• In the future, the satellite Gaia,

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