Products of Binary Evolution I

Notes on: Binary Stars by Onno Pols.

Day 20 - November, 4, 2025

Agenda:

• Reminders - Report Draft - Due: Nov. 11 via email/Slack
• Lecture (25m)
• ICA 18 + Report out - 3/4 Groups - Due: Not for Credit (45m)

Evolution in Close Binary Systems

Close Binary System: containing two stars in which the evolution of either star is strongly influenced by the presence of the other.

Most stars are in binaries (or multiples) and of that, half are close binary systems.

Some Defitions:

• - semi-major axis of the system
• - mass ratio (), where is the primary (or "donor"/more massive) star and is the secondary.

Roche Lobe

The Roche equipotential surfaces

• inner Lagrangian surface,
• contains two Roche lobes, one surrounding each star
  • Region around a star in a binary system within which orbiting material is gravitationally bound to that star.

Example Binary Star Systems

Detached Binary

• In a detached configuration neither star fills its Roche lobe.
• Most stars are this.

Example Binary Star Systems

Semi-Detached Binary

• In a semidetached configuration one star fills its Roche lobe.

Example Binary Star Systems

Contact Binary

• In a contact configuration both stars overfill their Roche lobes.

Credit: Philip D. Hall

Binary Mass Transfer

• When a star "exceeds its Roche lobe", its surface extends out beyond its Roche lobe and the material which lies outside the Roche lobe can "fall off" into the other object's Roche lobe via the first Lagrangian point.

• In binary evolution this is referred to as mass transfer via Roche-lobe overflow.

If mass transfer large enough, accretion disk will form about Star 2!

Let's consider the scenario where the primary (donor or ) will fill it's Roche Lobe at some point. A Semi-Detached Binary Star System.

The First Mass Transfer Phase and its Consequences

Three cases:

• Case A - RLOF occurs during core hydrogen burning. Typically short period binaries.

• Case B - RLOF occurs after hydrogen core exhaustion but before helium ignition.

• Case C - RLOF occurs after helium core exhaustion.

The First Mass Transfer Phase and its Consequences

Example Case A - RLOF occurs during core H-burning. Typically short period binaries.

• Times (t) are given in millions of years.
• For the last four panels, the white circles represent helium-rich material, whereas gray is original hydrogen-rich.

Note the somewhat dubious assumption here: no mass is supposed to have escaped from the system.

The First Mass Transfer Phase and its Consequences

We can define an equivalent radius of a given Roche lobe for a sphere of equivalent volume of the lobe, leading to the following dependence on the seperation and mass ratio:

for .

In the above example, you might expect the now more massive secondary to also undergo mass transfer once filling its Roche lobe.

Conservative Binary Evolution

dynamical time scale - time scale on which a star counteracts a perturbation of its hydrostatic equilibrium.

Given by the ratio of the radius of the star R and the average sound velocity of the stellar matter:

Conservative Binary Evolution

thermal or Kelvin-Helmholtz time scale - time scale on which a star reacts when energy loss and energy production are no longer in equilibrium.

Given by the ratio of the thermal energy content of the star and the luminosity .

Conservative Binary Evolution

nuclear time scale - This is the time scale on which a star uses its nuclear fuel. It is given by the product of the available fusable matter and the fusion energy per unit mass , divided by the stellar luminosity.

Example Case A Binary System - Primary Star ()

• The primary expands on the MS until filling its Roche Lobe (A to B)
• From B to C, a period of rapid mass (thermal-timescale) mass transfer, with a max of as the primary readjusts to reach equilibrium.

Example Case A Binary System - Primary Star ()

• After reaching new equilibrium, mass transfer continues driven by expansion from C to D but now limited by the longer (MS) timescale.

• Note the longer timescale in the right plot.

Example Case A Binary System - () and ()

Q: Which star is the primary based on this plot and why?

• Primary increases until RLOF (A to B), then decreases due to mass transfer/loss (B to C).
  • Secondary (yellow) response is increase in due to the accretion (B to C).

Example Case A Binary System - () and ()

• From C to D, the nuclear timescale limited mass transfer takes place with stars in new equilibrium.
• Net result is that the primary appears over luminous for its mass
  • Secondary has appropriate and for its new mass on the MS.

Example Case A Binary System - HR diagram

Example of the HR diagram for the binary system. Semi-detached, so-called Algol-type binary system.

• Primary increases until RLOF (A to B), then decreases due to mass transfer/loss (B to C).
  • Secondary (yellow) response is increase in due to the accretion (B to C).

In-Class Assignment 18

In class: Work on ICA here with groups per usual. Discuss conceptual questions together and prepare answers to share at the end of class.

• Choose someone that will report out the groups responses ahead of time!

After Class: Due: Not for Credit

Note: The goal of ICAs are to use plots produced in the notebook for discussion and interpretation of results presented in lecture in groups and as a class.