The minimum mass ratio of W UMa-type contact binaries—a new calculation
Xu-Dong Zhang
Abstract
Abstract A model is developed to establish the relationship between the critical gyration radius k of the primary component and the mass ratio ( q ) by considering the different dimensionless gyration radii of main-sequence stars with varying masses. The next step involves obtaining the low mass ratio limit ( $${q_{\text{min}}} = 0.038 \sim 0.041$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>q</mml:mi> <mml:mtext>min</mml:mtext> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0.038</mml:mn> <mml:mo>∼</mml:mo> <mml:mn>0.041</mml:mn> </mml:mrow> </mml:math> for overcontact degree $$f = 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ~ 1) of W UMa-type contact binaries. Furthermore, the radial density distributions are estimated within the range of $$0.3 M_{\odot } \sim 4.0 M_{\odot }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0.3</mml:mn> <mml:msub> <mml:mi>M</mml:mi> <mml:mo>⊙</mml:mo> </mml:msub> <mml:mo>∼</mml:mo> <mml:mn>4.0</mml:mn> <mml:msub> <mml:mi>M</mml:mi> <mml:mo>⊙</mml:mo> </mml:msub> </mml:mrow> </mml:math> , based on the mass-radius relationship of main-sequence stars. Subsequently, the physical meaning of the minimum k value is proposed, which leads to an explanation for the cause of the minimum mass ratio. Finally, a stability criterion is proposed, which is based on both the mass ratio ( q ) and the total mass of the two components ( $$M_{tot}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow> <mml:mi>tot</mml:mi> </mml:mrow> </mml:msub> </mml:math> ).