Binary TTL

A binary in time of way of light (TTL), of English timing, light-time effect or light-travel time (LTT) binary, is a binary star or a multiple star of which the luminosity varies in a periodic way and in which the orbital movement is highlighted by the change of this period. The principle is that of a “clock in orbit”, that it is a variable star, a Binaire with eclipses or a Pulsar, whose advance or delay announces that the object approaches or moves away from the observer.

History

The 19^ème century successively saw the discovery of binary by their movement on the sky (binary visual), then by their variation speed or position on the sky (binary astrometrical), then their variation speed on the line of sight (binary spectroscopic). In the space positions and speeds, it thus missed the variation of distance along the line of sight like indication of binarity. Although much more difficult to establish (for example because the variation of induced Parallaxe is so weak that it is drowned in the errors of measurement), it is by the means of the finitude speed of light that was going to prove to be possible in certain cases.

The Binaire with eclipses Algol is currently known like a triple system, the couple with eclipses being respective masses 3,6 and 0,8 solar Masse, orbited with one 680 days period by a companion of 1,6 solar mass. The first suggestion of the presence of a third body was made by Chandler (1892), 2 years after the description of ALGOL like spectroscopic Binaire, with like proof the effect of time of way of light. Actually, the period indicated by Chandler (130 years) was incorrect, and it is the variation the radial speed which was used by Belopolsky in 1906 to highlight the reality of a third body with one period much close to the fact that one knows now. Nevertheless, the effect of time of light was indeed present with the 680 days period (Eggen 1948).

To highlight this effect of TTL, a usual method is to study the dates of the minima of the curve of light of a Binaire to eclipses: these dates of minima should represent a periodic signal. One thus represents according to time the residues between the dates observed of the minima and the dates envisaged by éphéméride (difference named O-C for “observed less calculated”), and these residues should have a sinusoidal variation if they represent the gravitational influence of another body. The problem, it is that the variation of these O-C with time can be explained for other reasons why the presence of another body, for example of the cycles of magnetic activity, variable losses of Angular momentum, a rotation of the line of the Apside S, etc In fact, in spite of two centuries of observations by tens of astronomers, Algol is still not well-known.

Since, several of stars have a suspected or proven effect TTL, with the number of which AH Cephei, Algol, AM Leonis, AR Aurigae, WITH Serpentis, CM Draconis, R Canis Majoris, RT Persei, YOU Ursa Majoris, U Ophiuchi, UV Leonis, V471 Tauri, etc the number is relatively low because the effect requires, to be discovered, to have simultaneously of a triple star with an eclipse or a very regular and double variable star. The great statements of the sky which are currently made or in the future should nevertheless increase this number.

Lastly, and it is not least success of this method, it should be recalled that the first planetary system known was detected by measuring times of arrival of the signals of a pulsar (Wolszczan & Frail 1992).

Theory and application

Equations of the movement

If one writes the variation of position of the primary education (transmitting object) due to his orbital movement around the barycentre, the difference in time of arrival in second is written (Irwin 1952):

\ tau = \ frac {K} {\ sqrt {1-e^2 \ cos^2 \ Omega}} \ left \ cos \ naked} \ sin (\ naked + \ Omega) + E \ sin \ Omega \ right] avec

K = 499 a_1 \ sin I \ sqrt {1-e^2 \ cos^2 \ Omega}

where:

* K = (τ_max-τ_min) /2 = “semi-amplitude” of the effect of time of light in S.

* a₁ = Equatorial radius of the orbit of the primary education around the barycentre in astronomical Unit = 1.49597873011×10¹¹m = 499.0047915433 S of time of light.

* E = eccentricity of the orbit.

* ν = True anomaly, function of the time passed since the date T of the passage to the Pericenter, the orbital Period, and the eccentricity.

*ω = angle enters the node and the Périastre.

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

Function of mass

By using the third law of Képler with the usual units, the time of way of light makes it possible to know the function of mass defined in solar Masse by:

\ frac {M_2^3 \ sin^3 I} {(M_1 + M_2) ^2} = \ frac {1} {499^3 (1-e^2 \ cos^2 \ Omega) ^ {3/2}} \ frac {K^3} {P^2}

variables of the member of left being unknown while the member of right-hand side is obtained by the analysis of the curve of the O-C according to the time, and where:

* P = orbital period in Year.

* M₁ = mass of the “primary education” in solar Mass. If the “primary elections” are binary with eclipse, it is about the sum of the masses of its two components.

* M₂ = mass of the “secondary” in solar Mass. If the “primary elections” are binary with eclipse, it is about the mass of the third body.

Fundamental parameters

To reach the mass of the disturbing body, it is necessary either to make simplifying assumptions, or to have additional details:

If the primary elections are binary with eclipses, and thus which it is most probably spectroscopic Binaire also, the total mass of this binary can already be known. As regards the slope, and for want of anything better, one can make the assumption that the orbits are coplanar to have an estimate of the mass of the third body. If not one has access only to the minimal mass of the invisible body.

If the spectroscopic analysis indicates the orbital movement, the presence of the body is confirmed and the precision of the elements of the orbit is improved, but the slope remains unknown.
On the other hand, if it is the Astrométrie which confirms duplicity, the indetermination of the slope can be raised. It is the case for example for the Binaire with eclipses R Canis Majoris, suspectée as astrometrical Binaire with acceleration in the Catalog Hipparcos; the combination of times O-C (image opposite) and of the astrometrical data suggests that the primary education couple of solar mass 1.24 Masse is orbited by an object of solar mass 0.34 Masse with one 93 years period (Ribas and Al 2002).
Enfin, the optimal one occurs if the Interférométrie makes it possible to solve the components, because the masses, even the luminosities, can then be obtained. It is the case for example for β Cephei (Pigulski & Boratyn 1992).

Detectability

Plus the secondary object is massive, more it is detectable easily.
For masses given, effect TTL is proportional to P ^2/3, making easier detection the binary ones at great orbital period, and this, more especially as measurements of the luminosity of variable stars are classically made since more than one century.
the maximum effect is obtained when I =90°, the normal in the plan of the orbit being perpendicular to the line of sight.

Instruments of observation

Great statements of the sky providing of the curves of variable star light (e.g. the experiments EROS, MACHO, OGLE, DUET, or Hipparcos)

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