Theoretical Model of Non-Conservative Mass Transfer with Uniform Mass Accretion Rate in Contact Binary Stars
aa r X i v : . [ a s t r o - ph . S R ] O c t Theoretical Model of Non-Conservative MassTransfer with Uniform Mass Accretion Rate inContact Binary Stars
Prabir Gharami ,Koushik Ghosh , Farook Rahaman Taki Bhabanath High School, P.O.: Taki, North 24 Parganas, Pin-743429 West Bengal,India . e-mail: [email protected] Department of Mathematics, University Institute of Technology, University of BurdwanGolapbag (North), Burdwan-713104 West Bengal, India.e-mail: [email protected] Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal, India.e-mail: [email protected](Submitted on xx.xx.xxxx; Accepted on xx.xx.xxxx)
Abstract.
In contact binaries mass transfer is usually non-conservative which ends into lossof mass as well as angular momentum in the system. In the present work we have presenteda new mathematical model of the non-conservative mass transfer with a uniform mass ac-cretion rate in a contact binary system with lower angular momentum. The model has beendeveloped under the consideration of reverse mass transfer which may occur simultaneouslywith the original mass transfer as a result of the large scale circulations encircling the entiredonor and a significant portion of the gainer. These circulations in contact binaries withlower angular momentum are caused by the overflow of the critical equipotential surface byboth the components of the binary system making the governing system more intricate anduncertain.
Key words:
Contact binary; donor star ; gainer star; non-conservative mass transfer,reversemass transfer
There exists a large number of binary systems in which the component pairsof stars are with very small separations and with orbital periods nearly lessthan 10 years such that mass can transfer from one star to another. This masstransfer can change the structure of both the stars causing subsequent evolu-tion in the system as a whole (Podsiadlowski 2001). These paired systems areknown as contact binaries. Thus mass transfer is a regular event in contactbinary stars. This mass transfer may occur in two ways. First type is conser-vative in which total mass and angular momentum remains unchanged. Thesecond type is non-conservative where some mass is lost during its journeyfrom donor to gainer. Non-conservative mass transfer is of two types: withuniform mass accretion rate (Podsiadlowski et al. 1992; Sepinsky et al. 2006;Van Rensbergen et al. 2010; Davis et al. 2013) and with non-uniform massaccretion rate with respect to time (Stepien and Kiraga 2013; Izzard et al.2013; Gharami et al. 2015).In contact binaries reverse mass transfer may occur simultaneously withthis regular mass transfer making a significant change in the overall dynamicsand evolution (Longair 1994; Nelson and Eggleton 2000; Stepien 2009; Stepienand Kiraga 2013). In case of contact binaries with lower angular momentumthe overflow of the critical equipotential surface by both components driveslarge scale circulations surrounding the entire donor and a huge fraction thegainer. A portion of the mass transported by the donor to the gainer returnsback to the donor by this circulation with the mass flux of the order of 10 − to 10 − M per year (Stepien 2009). Bulgarian Astronomical Journal 21, 2014 P. Gharami, K.Ghosh, F.Rahaman
In this paper we have proposed a theoretical model of non-conservativemass transfer with uniform mass exchange rate with respect to time in contactbinaries with lower angular momentum under the consideration of reverse masstransfer following the argument of Stepien (Stepien 2009) . We have furnisheda numerical model for the presently proposed theory.
We offer the following theoretical model of non-conservative mass transfer withuniform mass accretion rate with respect to time in contact binaries with lowerangular momentum taking into account the reverse mass transfer originatedas a result of large scale circulations encircling the entire donor and a majorportion of the gainer star. We here assume that M is the mass of the gainerand M is the mass of the donor at any time t.˙ M = β ˙ M o ) (1)˙ M o ) = − Aτ M (2)˙ M = − ˙ M o ) + ˙ M i ) (3)˙ M i ) = γ (1 − β ) ˙ M o ) (4)where β characterizes the mass accretion process by the gainer and γ portrays the process of formation of reverse mass jet directed to the donor.(2) comes from the Bernoulli’s law applied in the mass flow through the innerLagrangian point considering adiabatic index γ = assuming the componentstars being with convective envelops. A is a numerical constant lying between1 and 2 and τ is the entire timescale during which the mass transfer is takingplace (Pols 2012). M ( o )2 and M ( i )2 indicate respectively the total outgoing jetof mass as a result of non-conservative mass transfer from donor to gainer andtotal incoming mass flow directed towards the donor due to the reverse masstransfer in time t.Using (2) and (4) in (3) we get,˙ M = − [1 − γ (1 − β )] Aτ M (5)Integrating equation (5) with the initial condition that at t = 0 , M = M , , we get M = M , e − A [1 − γ (1 − β tτ (6)Again using (2) in (1) and on integration with the initial condition that at t = 0 , M = M , we have heoretical Model of Non-Conservative Mass 3 M = M , + β M , − γ (1 − β ) (cid:20) − e − A [1 − γ (1 − β tτ (cid:21) (7)Again using (6) in (2) we get,˙ M o ) = Aτ M , e − A [1 − γ (1 − β tτ (8)Integrating (8) with the initial condition M ( o )2 = M ( o )2 , at t = 0, we get M ( o )2 = M ( o )2 , + M , − γ (1 − β ) (cid:20) − e − A [1 − γ (1 − β tτ (cid:21) (9)Again using (2), we get from (4)˙ M i ) = γ (1 − β ) Aτ M (10)Integrating equation (10) with the initial condition at t = 0 , M ( i )2 , = 0(understandably there should not be any initial jet of reverse flow at the verybeginning instant of mass transfer) we get, M ( i )2 = γ (1 − β )1 − γ (1 − β ) M , (cid:20) − e − A [1 − γ (1 − β tτ (cid:21) (11) Here we produce a numerical example taking the initial masses of the gainerand donor as M , = 9 × (g) and M , = 4 × (g) respectively. For thepresent calculation we take the values of the parameters as A = 1 . γ = 0 . β = 0 .
4. The time scale of the mass exchange is taken as τ = 10 ( years)and the outgoing mass from the donor at initial instant i.e. at time t = 0 istaken as M ( o )2 , = 1 . × (g). Also M ( i )2 , = 0 i.e. at the initial instant noincoming mass jet is taken into account towards the donor. The orbital periodof the system is considered to be less than 100 days so that the gainer can fillthe Roche lobe during its expansion (Monzoori 2011). Here we have producedthe graphs for mass incoming to the donor vs. time (years) (Figure 1), massoutgoing from the donor vs. time (years) (Figure 2), mass of the donor vs.time (years) (Figure 3) and mass of the gainer vs. time (years) (Figure 4).Figure 1, 2 and 4 show increasing profiles while Figure 2 for obvious reasonexhibit decreasing profile. P. Gharami, K.Ghosh, F.Rahaman
The present work proposes a theoretical model of non-conservative mass trans-fer with uniform mass accretion rate with respect to time in contact binarieswith lower angular momentum under the consideration of reverse mass trans-fer based on the proposal of Stepien (2009) which points out a possible largescale circulation generated by the overflow of the critical equipotential surfaceby both components as the possible driving force of this reverse mass transfer.Stepien and Kiraga (2013) proposed an alternative cause of reverse mss trans-fer in contact binaries. They proposed that as the altitude of the equatorialbulge quantifies to a certain percentage of the stellar radius above the surfacehence when the radius of the accretor tends to the size of the Roche lobe bythis quantity the apex of the bulge starts to protrude beyond the inner criticalsurface. As a result, part of the matter flows above the Roche lobe and comesback to the donor. In future, we may work in this aspect of reverse mass trans-fer. Moreover, the present work focuses only on the issue of non-conservativemass transfer with uniform accretion rate. Future work may be carried in thedirection of non-uniform mass accretion rate. There is also a provision to studythe effect of magnetic field in reverse mass transfer. Theoretical study in futurein this direction should bring some interesting results.
Fig. 1.
Mass incoming to the donor. heoretical Model of Non-Conservative Mass 5
Fig. 2.
Mass outgoing from the donor .
Fig. 3.
Mass of the donor .
Fig. 4.
Mass of the gainer.
P. Gharami, K.Ghosh, F.Rahaman
FR gratefully acknowledges support from the Inter-University Centre for As-tronomy and Astrophysics (IUCAA), Pune, India for providing research fa-cility. PG, KG and FR express their sincere gratitude to Prof. Probhas Ray-chaudhuri, Professor (Retd.), Department of Applied Mathematics, Universityof Calcutta, Kolkata, India for a careful reading of the manuscript and for hisvaluable suggestions in order to improve the quality of the present work.