On the Darwin instability effect in binary systems

TL;DR

This paper investigates the Darwin instability in various binary systems using a Regge-like law model, providing new analytical formulas for orbital characteristics.

Contribution

It introduces novel analytical formulas for binary system parameters based on a Regge-like law model, enhancing understanding of Darwin instability effects.

Findings

Derived new formulas for relative distance and orbital period

Applied model to different binary systems including planets and stars

Provided insights into the stability conditions of binary systems

Abstract

The Darwin instability effect in the binary systems (planets, stars, and galaxies) is analyzed within the model based on the Regge-like laws. New analytical formulas are presented for the relative distance between components of the binary and orbital rotation period of the binary.

Equations

$${\mathbf{J}}_{\mathrm{tot}}=\mathbf{L}+{\mathbf{S}}_1+{\mathbf{S}}_2.$$

$$J_{\mathrm{tot}}=\hslash {\left(\frac{M}{m_p}\right)}^{(1+n)/n}$$

$$S_k=\hslash {\left(\frac{M_k}{m_p}\right)}^{(1+n)/n},$$

$$L_{\max }=J_{\mathrm{tot}}+S_1+S_2$$

$$L_{\min }=J_{\mathrm{tot}}-S_1-S_2,$$

$$\frac{S_1+S_2}{L_{\min }}=\frac{(1+\eta {)}^{(1+n)/n}+(1-\eta {)}^{(1+n)/n}}{2^{(1+n)/n}-(1+\eta {)}^{(1+n)/n}-(1-\eta {)}^{(1+n)/n}},$$

$$\frac{S_1+S_2}{L_{\max }}=\frac{(1+\eta {)}^{(1+n)/n}+(1-\eta {)}^{(1+n)/n}}{2^{(1+n)/n}+(1+\eta {)}^{(1+n)/n}+(1-\eta {)}^{(1+n)/n}}.$$

$$\frac{S_1+S_2}{L_{\min }}=\frac{1}{2^{1/n}-1}>1$$

$$\frac{S_1+S_2}{L_{\max }}=\frac{1}{2^{1/n}+1}>\frac{1}{3}.$$

$$\frac{S_1+S_2}{L_{\min }}\to \infty$$

$$\frac{S_1+S_2}{L_{\max }}=\frac{1}{2}.$$

$$R_m$$

$$P_{\mathrm{orb}}$$

$$R_k=g{M}_k^l$$

Full text

Keywords: binary stars, binary galaxies, mass asymmetry

It is now commonly believed that the contact binaries, for example, the W UMa binaries, end their evolution by merging into a single star Cher:2013 . The dissipation of the orbital energy in the initial violent phase of merger resulted in the luminous red nova V1309 Sco observed in 2008. The luminous red novae have recently been identified as a distinct class of stellar transients Kulkarni:2007 . They are characterized by relatively long outbursts with spectral distributions centered in the red, ranging in luminosities between classical novae and supernovae. The observations in the Optical Gravitational Lensing Experiment have revealed a shape of the light curve characteristic for contact binary system with an exponentially decreasing orbital period Tylenda:2011 and confirmed the earlier conjecture of Ref. Soker:2003 that the luminous red novae arise from merging contact binary stars. The spectacular case is KIC 9832227 which was predicted Molnar:2017 to be merge in 2022, enlightening the sky as a red nova. The compact binaries composed of white dwarfs, neutron stars, and black holes eventually merge through gravitational wave emission Abbott:2016a ; Abbott:2016b ; Abbott:2017 .

The details of specific mechanism, which triggers the merger of the contact binary components, are still controversial Sana:2012 ; Kocha:2014 ; Fabrycky:2007 ; Tokovinin:2006 ; Eggleton:2012 ; Pejcha:2017 . There is assumption Eggleton:2012 on the gradual mass transfer from less massive (and hence smaller radius) secondary star to the primary one (heavier star). This mass transfer is driven by the structural change in the secondary star caused by the energy received from the primary star. The contact is sustained by the magnetic braking or thermonuclear evolution. When the mass ratio is extreme enough for the Darwin instability, a merger starts that triggers the outburst in red novae Darwin:1879 ; Tylenda:2011 . The Darwin instability happens when the spin angular momentum of the system is more than one third of the orbital angular momentum. This instability plays a role once the mass ratio becames small enough that the companion star can no longer keep the primary star synchronously rotating via the tidal interaction. For most of the primary massive stars, this occurs at the mass ratio $q=M_{2}/M_{1}<0.1$ Rasio:1995 . There is alternative scenario Stepien:2011 : At first contact in a binary system, a brief, but intense, mass transfer sets in changing originally more massive star into less massive one. This process may oscillate until eventually a stable contact configuration is reached. Also the dynamic mass transfer without the Darwin instability DSouza:2006 and mergers triggered by a tidal runaway based on a non-equilibrium response to tidal dissipation Koenigsberger:2016 have been investigated. Thus, there is a large quest for detailed observational and theoretical investigations Stepien:2011 ; DSouza:2006 ; Koenigsberger:2016 .

As shown in Refs. Murad1 ; Murad2 ; Murad3 , the angular momentum $J$ of many astronomical objects, from planets to clusters of galaxies, and, possibly, the universe as a whole, can be predicted by simple Regge-like laws from the mass $M$ of the object. The Regge-theory proved to be very influential in the development of elementary-particle physics. In Refs. Murad1 ; Murad2 ; Murad3 , the cosmic analog of the Chew-Frautschi plot with two important cosmological Eddington and Chandrasekhar points on it has been constructed. The application of Regge ideas to astrophysics has shown that the spins of planets and stars are well described by the Regge-like law for a sphere ($J\sim M^{4/3}$), while the spins of galaxies and clusters of galaxies obey the Regge-like law for a disk ($J\sim M^{3/2}$) Murad1 ; Murad2 ; Murad3 . In contrast to earlier semi-phenomenological approaches these expressions contain only fundamental constants as the parameters and are independent of any fitted empirical quantities. The aim of the present article is to study the Darwin instability effect in the binary star or galaxy by using the model of Refs. Murad1 ; Murad2 ; Murad3 based on the Regge-theory.

The total angular momentum ${\bf J}_{\rm tot}$ of binary system is the sum of orbital angular momentum ${\bf L}$ and the spins ${\bf S}_{k}$ ($k=$1,2) of the individual components:

[TABLE]

The $J_{\rm tot}$ and $S_{k}$ are expressed using the Regge-like law for stars and planets ($n=3$) or galaxies ($n=2$):

[TABLE]

and

[TABLE]

where $\hbar$, $m_{p}$, $M_{k}$ ($k=$1,2), and $M=M_{1}+M_{2}$ are the Planck constant, masses of proton and astrophysical objects (planets, stars or galaxies), and the total mass of system, respectively. The maximum (the antiparallel orbital and spins angular momenta) and minimum (the parallel orbital and spins angular momenta) orbital angular momenta are

[TABLE]

and

[TABLE]

respectively. Using the mass asymmetry (mass transfer) coordinate $\eta=(M_{1}-M_{2})/M$ instead of masses $M_{1}=\frac{M}{2}(1+\eta)$ and $M_{2}=\frac{M}{2}(1-\eta)$ IJMPE and Eqs. (3)–(5), we derive

[TABLE]

At $\eta=0$, we have

[TABLE]

and

[TABLE]

For the symmetric binary star (planet) and binary galaxy, $(S_{1}+S_{2})/L_{\rm max}\approx$ 0.44 and 0.41, respectively. At $\eta=1$, we have

[TABLE]

and

[TABLE]

As follows from last two expressions, for very asymmetric binaries, the ratios $(S_{1}+S_{2})/L_{\rm max,min}$ almost independent of the value of $n$. According to Ref. Rasio:1995 , the Darwin instability can occur when the binary mass ratio is very small ($q=M_{2}/M_{1}<0.1$) or the mass asymmetry is very large ($\eta=(1-q)/(1+q)>0.82$). As seen in Fig. 1, the ratios $(S_{1}+S_{2})/L_{\rm max}$ and $(S_{1}+S_{2})/L_{\rm min}$ continuously increases with $\eta$ from 0 to 1. Because their absolute values are larger than 1/3, all possible binary stars (planets) or binary galaxies, independently of their mass asymmetry $\eta$, should have the Darwin instability ($S_{1}+S_{2}\geq\frac{1}{3}L$) and, correspondingly, should merge. However, the observations do not support this conclusion which probably means that there is no the Darwin instability effect in such binary systems and, correspondingly, the mechanism of merger has other origin.

Note that in the cases of antiparallel spins with $L_{1}=J_{tot}+S_{1}-S_{2}$ and $L_{2}=J_{tot}-S_{1}+S_{2}$ (Fig. 1), the ratios $|S_{2}-S_{1}|/L_{1}$ and $|S_{1}-S_{2}|/L_{2}$ are larger than $\frac{1}{3}$ for the asymmetric binaries with $|\eta|\geq 0.5$.

As seen in Fig. 2, the dependencies of $L_{\rm max}$, $L_{\rm min}$, $L_{1}$, and $L_{2}$ on mass asymmetry have different behavior. The evolution of system in mass asymmetry (mass transfer) can increase or decrease the orbital angular momentum. For example, at $\eta\to 0$ the binary system has smaller $L=L_{\rm max}$. The observations of the dependence of $L$ on $\eta$ may be useful to distinguish the difference between the orientations of orbital and spins angular momenta.

Employing Eqs. (1)–(3) and results of Refs. nash , we obtain new analytical formulas for the relative distance between the components of the binary

[TABLE]

at $R_{m}>R_{t}=R_{1}+R_{2}$ ($R_{k}$ are the radii of binary components) and

[TABLE]

at $R_{m}\leq R_{t}$, and for the orbital rotation period in the binary

[TABLE]

at $R_{m}>R_{t}$ and

[TABLE]

at $R_{m}<R_{t}$ nash . Here, $G$, $R_{t}=R_{1}+R_{2}$, and $R_{k}$ ($k=1,2$) are the gravitational constant, touching distance, and radius of the component of binary, respectively. The values $\epsilon_{1}=\epsilon_{2}=1$ and $\epsilon_{1}=\epsilon_{2}=-1$ correspond to the cases of antiparallel and parallel orbital and spins angular momenta. The values $\epsilon_{1}=-\epsilon_{2}=1$ and $\epsilon_{1}=-\epsilon_{2}=-1$ correspond to the cases of antiparallel spins. The observational data result in the relationship

[TABLE]

between the radius and mass of the star, where the constants $l=\frac{2}{3}$ and $g=R_{\odot}/M_{\odot}^{2/3}$ ($M_{\odot}$ and $R_{\odot}$ are mass and radius of the Sun) Vasil:2012 and the galaxy, where the constant $l$ depending on mass is in the interval $\left[\frac{2}{5},\frac{2}{3}\right]$ Karachentsev . As seen in Figs. 3–5, at $R_{m}>R_{t}$ ($R_{m}\leq R_{t}$), $R_{m}$ decreases ($R_{t}$ increases) with decreasing $|\eta|$ and, finally, the $P_{orb}$ decreases (increases). At $R_{m}>R_{t}$, the dependence of $P_{orb}$ as a function of mass asymmetry has similar behavior in the cases when the orbital and spin angular momenta are antiparallel and parallel. At $R_{m}\leq R_{t}$, the value of $P_{orb}$ does not depend on orientations of orbital and spin angular momenta.

In conclusion, within the model Murad1 ; Murad2 ; Murad3 based on the Regge-like laws, we have shown that all possible binary stars (planets) or binary galaxies, independently of their mass asymmetry $\eta$, satisfy the Darwin instability condition ($S_{1}+S_{2}\geq\frac{1}{3}L$) which contradicts to the observations. This conclusion is not sensitive to the parameters of model. Therefore, one should search for other mechanism that triggers the merger of the contact binary components.

Employing the Regge-like laws, we have derived the new analytical formulas for the relative distance and orbital rotation period of the binary system, which depend on the fundamental constants $G$, $\hbar$, $m_{p}$, masses of the binary components, and the experimental relation radius-mass. We have predicted that decreasing and increasing periods as functions of mass asymmetry are related, respectively, with the non-overlapping ($R_{m}>R_{t}$) and overlapping ($R_{m}\leq R_{t}$) stage of the binary object.

I Acknowledgements

This work was partially supported by Russian Foundation for Basic Research (Moscow) and DFG (Bonn).