^{1}

^{2}

^{3}

The first photoelectric light curve analysis of the TT And system in a broad band filter (400 - 700 nm) was carried out using the PHOEBE program. The absolute dimensions of the system are determined and its evolution is discussed. The most distinct feature of the system is a comparatively very low mass of the cooler component
* i.e.* M
_{2} = 0.26M⊙. Moreover, using the Observed-minus-Calculated data, the period changes of the system are studied, which reveal the presence of a third body orbiting the system with a period P
_{3} = 63.89 yr, superimposed on which is a secular period increase, which is ascribed to mass transfer with a rate
−7.60 × 10
^{−12 }M⊙ yr
^{−1} from the cooler secondary component.

The regular Algol-type (also called semi-detached) binaries are stars whose less massive component (secondary) fills its Roche lobe. One of their main characteristics is mass transfer from the Roche lobe filling secondary to the primary forming a circumprimary rotating accretion disk, due to transfer of high angular momentum material by the donor star. The magnetic activities of the secondary, X-ray flares, and variations of light, are the phenomena which make the Algols an interesting field for researchers. The mass transfer is due to the Roche lobe filling of one or both components in a binary system. In a close binary system the more massive star expands during its evolution and at some moment fills up its Roche lobe. Then a rapid mass transfer takes place on the thermal time scale [

Close binary systems usually show orbital period variations in different ways. The apparent sinusoidal variations in the period are because of the presence of additional third or fourth bodies or the precession of the binary orbit due to tidal and rotational effects in its own plane. Another cause of alternating changes in rapidly rotating and possessing outer large convection zone is exchange and distribution of angular momentum between the star’s rotation and their mutual orbit [

TT And (GSC 3623.02323, RA = 23^{h}13^{min}23^{sec}, Dec = 46˚08'48") is a typical short period semi detached (Algol-type) eclipsing binary composed of A_{0}-A_{2} + [G7IV] stars with orbital period of 2.675d. The system was discovered in 1907, its variability was announced by Ceraski in 1913 [

Erdem et al 2007 have studied period variations in five eclipsing binaries i.e. TT And, V342 Aql, RW Cap, BZ Cas and TW Lac [_{3}) = 0.101 ± 0.021M_{⊙}. The semi-ma- jor axis of the third-body orbit around the mass center would be accreted, when relative orbital inclination of the three-body system is equal to the orbital inclination of TT And. This increment shows that the third body revolves far beyond the outer Lagrangian points of TT And, and its orbit should be stable [

The rest of the present paper is organized as follows. The analysis of light curve of TT And is described in section 2 and the analysis of O-C curve is presented in section 3. In section 4 we summarize our findings and discuss evolutionary status of the system their implications.

In the previous section, we pointed out that studies of the TT And system don’t cover all important details. So we purpose to study the properties of this system by using available photometric data. The photometric data used in this study were obtained from the Super WASP (Wide Angle Search for Planets) project which ran between 2006-2008 in a broadband filter with a passband of 400 - 700 nm (for details see [

More than 5800 data points were used, few of them were omitted due to large scattering. However to carry on the LC analysis since the spectral type was reported to be A_{0}-A_{2} ( [_{1} = 9900K, and the appropriate gravity darkening (g_{i}) and Bolometric Albedo (A_{i}) coefficients are selected according to the spectral types of the primary and Algol-type binaries, i.e., g_{1} = 1.0, A_{1} = 1.0 and g_{2} = 0.32, A_{2} = 0.5. The limb darkening coefficients are read from Van Hamme (1993) tables automatically by the PHOEBE program [_{2}/M_{1}). For this purpose a grid of q values (0.1, 0.2, ...) were selected, then for each value of the q the main parameters of the system i.e. Ω_{1} the non-dimensional surface potential of the primary, T_{2} effective temperature of the secondary star, i, inclination of the orbital plan, were adjusted so that to minimize the χ^{2} value and best fit of synthetic to observed points via visual inspections. Then we have plotted χ^{2}/q in ^{2} as the best estimate of q = 0.110 value and used this as initial q-value. This value of q is well agreed with the value of q obtained from the empirical relation: q = 10^{(logL1−logL2)/3.664}.^{ }

Moreover, to estimate the absolute dimensions of the system we have taken the value of the semi-major axis i.e. a = 11.6R_{⊙} from Erdem et al. (2007) [

[_{1} as fixed parameters, the other binary main parameters Ω_{1}, T_{2}, i, eccentricity e, and L_{1}, the monochromatic luminosity of the primary component were set as free parameters. Since the period analysis of the system indicated a third body orbiting around the system, therefore, in addition to the free parameters just mentioned, l_{3}, the luminosity of the third component was also taken as a free parameter. The free parameters were adjusted sequentially by trail and error method so that to minimize the χ^{2} and reduce corrections to the parameters errors. Also best fit of the observatioal points to theoretical LC was inspected visually for each run of the program. We have illustrated the results of analysis in

In order to study period variations of the system, the observed minus calculated (O-C) values were collected from different sources mainly from the updated O-C web page of Czech Astronomical Society. Then with help of the following linear Ephemeris all the O-C points were converted to a common Epoch.

Then we have plotted the O-C residuals against Epoch cycles (E) in the ^{2} through least squares method, where, a = −0.0024 ± 0.006, b = −1.08557 × 10^{−}^{5} ± 6.19475 × 10^{−7}, c = 2.47595 × 10^{−9} ± 2.31675 × 10^{−10}.

Parameter | Values for the Semidetached Sol. |
---|---|

i (^{◦}) T_{1} (K) T_{2} (K) Ω_{1} Ω_{2} = Ω_{in} q L_{1}= (L_{1} + L_{2}) L_{2}= (L_{1} + L_{2}) r_{1} (pole) r_{1} (side) r_{1} (point) r_{1} (back) r_{2} (pole) r_{2} (side) r_{2} (back) r_{2} (point) X_{1} X_{2} | 85.15 ± 0.02 9900 4530 ± 25 5.000 ± 0.014 1.990 0.110 ± 0.001 0.958 ± 0.05 0.042 0.2044 0.2054 0.2057 0.2056 0.1940 0.2017 0.2323 0.2471 0.610 0.610 |

Parameter | Semidetached Sol. |
---|---|

Period (d) A/R_{⊙} M_{1}/M_{⊙} M_{2}/M_{⊙ } Ṝ_{1}/R_{⊙} Ṝ_{2}/R_{⊙} L_{1}/L_{⊙} L_{2}/L_{⊙} M_{1,bol} (mag) M_{2,bol} (mag) | 2.76 (adopted) 11.6 (adopted) 2.47 ± 0.1 0.26 ± 0.03 2.286 ±0.08 2.441 2.930 ± 0.08 0.128 3.90 0.17 |

The residuals between the fitted parabola (i.e. continuous curve) and O-C normal points are displayed in

where, y_{0} = 0.015 ± 0.004 d, A = 0.050 ± 0.004 d, ω = 0.00024π ± 0.00001 d^{−1}, ϕ_{0} = 1.476 ± 0.111 rad, and corresponding period P_{3} = 63.89 yr and illustrated in the

From the light curve, the physical situation can be directly deduced. The EA light curves meaningfully divided in two groups, EAD and EAS. The EAS light curve, with its deep eclipse of an early type star, normally provides a clear indication of Algol evolution.

Reference to _{1} point. Using the Equation

critical radius the fill out factor of secondary comes to be ~100%., however, the hotter A star has partially filled its Roche lobe. Low mass of the secondary and high temperature of the primary A-type star may suggest that the system is at the case B mass transfer evolutionary state. In the Figures 7-9 we have determined the positions of the individual components on the H-R, M-R and M-L diagrams.

As evident from these Figs. both of the components are evolved away from the Main Sequence (MS). This situation for the secondary (cooler component) is expected, however the hot primary shows a bit more evolved on the H-R and M-L diagrams, this may be due to inaccuracies in mass determination of the individual component. Therefore spectroscopic mass determination is need and the masses determined should be used with due caution.

The position of the secondary low mass on the M-R diagram is typical for low mass star and consistent with the departure of low mass stars from the mass radius relation.

Referring to ^{−12} M_{⊙}/yr (assuming conservative mass transfer). After subtraction of the mass transfer effect (i.e. parabolic change) there is still a significant sinusoidal variation in the residual points. These residuals were depicted in the _{3} = 63.89 yr, the mass and orbital radius of the third body can be estimated as follow: Assuming a third body with circular orbit and coplanar with the system, then we may estimate the radius of the orbit and a lower limit to the mass of the possible third body using the equations 4 below, by putting the orbital inclination of the presumed third body i_{3} = 90^{◦} and using the amplitude (A), from the Equation (3), we get (see Mayer 1990) [_{12} sini_{3} = A × c

where the quantities

m_{1} = 2.476 M_{⊙}, mass of the hotter component (primary), m_{2} = 0.272 M_{⊙}, mass of the cooler component (secondary) e = 0, orbital eccentricity of the third body orbit based on these values the estimated mass of the possible third body m_{3} ≃ 1.39 ± 0.50 M_{⊙} and its orbital radius a_{3} ≃ 8.69 ± 0.54 AU.

The mass and period of the presumed third body found are in rough agreement with those of Erdem et al. (2007) [_{⊙}.

TT And is an Algol type system. The eclipsing pair consists of a hot A-type star as primary and the cool secondary is a low mass K-type subgiant star. The period analysis reveals a mass transfer with a rate of dm/dt = 7.6 × 10^{−12} M_{⊙}/yr and a third body orbiting the system with period P_{3} = 63.89 yr and mass m_{3} = 1.4 M_{⊙}. Hence the system is a triplet system.

We have used data from the WASP public archive in this research. The WASP consortium comprises of the University of Cambridge, Keele University, University of Leicester, The Open University, The Queens University Belfast, St. Andrews University and the Isaac Newton Group. Funding for WASP comes from the consortium universities and from the UKs Science and Technology Facilities Council.

Manzoori, D., Abbasvand, S. and Abbasvand, V. (2017) The Light Curve and the Orbital Period Variations of Binary System TT Andromedae. International Journal of Astronomy and Astrophysics, 7, 112-123. https://doi.org/10.4236/ijaa.2017.72009