Mass Transfer in Close Binary Star Systems

Isaac SchultzME 547 Dr. Derek Tretheway

June 4, 2015

The Algol Paradox

• Binary star systems were observed in which less massive star was further developed than more massive star• The more mass a star has, the quicker it will evolve

through its life cycle• This observation presented a serious problem to

scientists for many years• 19th century French astronomer Eduard Roche

developed a model describing the mass transfer between binary stars

More massive, younger star

Less massive, older star

The Algol Paradox- Solved

• Roche found that the reason the less massive star was younger than its partner was that it had originally been more massive before mass transfer

• Mass transfer between two stars in a binary system allows for the originally more massive star to become less massive than its partner

• Roche developed a model for this unique mass transfer process starting from first principles

Close Binary Systems & Roche Lobes• Any binary system in which the mass ratio of the

stars will allow for mass transfer at some point in their lifetime is known as a “close binary system”• Mass transfer in close binaries is mainly governed

by gravity• At some point in a stars life as it fuses through its

fuel, pressure from the core will cause matter towards the surface of the star to be pushed into the surrounding space• When mass from one star gets close enough to

its partner star it will be pulled towards it, facilitating mass transfer

Assumptions in Modeling Roche Lobe Overflow

• Many assumptions need to be made in order to have any hope at modeling the processes involved in mass transfer in close binary systems

• Spherical orbits, the existence of a critical Roche lobe and Lagrangian points, conservation of mass and angular momentum

The Roche Model for Close Binary Systems

• Consider a test particle between two stars, with a coordinate system rotating with stars orbit

• The potential between the stars includes the gravitational potentials from each star in addition to the Coriolis force to account for any motion of the particle relative to the rotating coordinate system

Roche Equipotentials

• By setting the Roche potential equal to a constant, equipotential lines can be drawn• When these equipotential lines meet, mass

transfer can occur between the stars• Whichever potential value results in

intersecting equipotentials is defined as the critical Roche radius• The size of the non-spherical Roche lobe is

conveniently defined by a “Roche lobe radius”

Critical Roche Surface and first two

Lagrangian points

Critical Lagrangian Points in the Roche Potential• Mass transfer will occur at critical points in the

Roche radius, which occur at the solutions to

• Critical points in the Roche potential are called Lagrangian points and represent unstable equilibrium points in the space between the stars

• Any mass located near a Lagrangian point is subject to being transported between stars or being ejected from the system

Stable equilibrium points

Unstable equilibrium points

Predicting the Roche Radius

• Kobal tabulated the critical radius of some close binary systems in a 1959 journal article• Pacyznski 1971 came up with a model to predict the

critical radius based on the mass ratio

• Eggelton 1983 used a numerical method to arrive at a similar result to Pacyznski

Time scale of mass transfer between close binary stars

• Paczynski used 24 model close binary stars to determine time scales of the mass transfer in this process• Initially, mass transfer between the stars is

rapid and takes place on the Kelvin-Helmholtz time scale

• Kelvin-Helmholtz time scale represents the time it takes for a star to radiate away a large amount of its energy (≈30 million years for our sun)

What happens next?

• After mass transfer begins, the size of the Roche lobe will change due to the changing mass of the stars

• If the star looses mass too quickly, its Roche lobe will shrink faster than the physical star and the star may disintegrate

• If the star looses mass slowly, it will shrink down to a smaller size than its Roche lobe, ending mass transfer

• In order for steady mass transfer to occur the mass the size of the star must remain equal to its Roche lobe

Mass Transfer in Close Binary Star Systems

Isaac SchultzME 547 Dr. Derek Tretheway

June 4, 2015