We have calculated the equilibrium properties of a star in a circular, equatorial orbit about a Super-Massive Black Hole (SMBH), when the star fills and overflows its Roche lobe. The mass transfer time scale is anticipated to be long compared with the dynamical time and short compared with the thermal time of the star, so that the entropy as a function of the interior mass is conserved. We have studied how the stellar entropy, pressure, radius, mean density, and orbital angular momentum vary when the star is evolved adiabatically, for a representative set of stars. We have shown that the stellar orbits change with the stellar mean density. Therefore, sun-like stars, upper main sequence stars and red giants will spiral inward and then outward with respect to the hole in this stable mass transfer process, while lower main sequence stars, brown dwarfs and white dwarfs will always spiral outward.
When a star loses mass on a time scale faster than the Kelvin-Helmholtz time (but slower than the dynamical time), as can happen in Extreme Mass-Ratio Inspirals (EMRI), the structure of the star will evolve adiabatically so that the entropy as a function of interior mass S(m) is approximately conserved. This will be true of radiative and convective zones (except near the surface). Webbink (1985) gave a general introduction to binary mass transfer on various time scales. Hjellming & Webbink (1987) used polytropic stellar models to explore the stability of this adiabatic process, and listed several critical mass ratios above which the donor stars are unstable on dynamical time scales. Soberman et al. (1997) further discussed the stability of binary system mass transfer processes on the thermal and dynamical times scales.
The previous works generally assumed comparable masses of the two objects in the binary system, and an unchanging distance between them throughout the process. In this paper, we will use real stellar models to study how stars respond to the loss of mass adiabatically on a time scale slower than the dynamical time scale but still faster than the thermal one. The star’s orbital radius from the hole is no longer held as a constant. The way that the star evolves in such a mass-transfer environment depends upon the mean density of the stripped star as a function of decreasing mass. Nuclear reactions will be shut off as soon as the mass loss starts, as they are highly temperature-sensitive (Woosley et al. 2002) and the central temperature decreases, at least in the cases that concern us here.
In our EMRI binary system, the star is assumed to be in a circular, equatorial orbit about the central massive black hole. The mass transfer starts when the star just fills its Roche lobe, when materials will flow out of the inner Lagrangian point L1. We call this the Roche mass transfer process. During the mass transfer phase, the stellar orbital radius may increase or decrease. However, we assume that the change is slow enough so that the stellar orbit remains circular. If the orbit crosses the black hole’s innermost stable circular orbit (ISCO) before it is tidally consumed, it will plunge into the hole. The ISCO has radius 6Rg for a non-rotating black hole, and 1.237Rg for a spin parameter a = 0.998 black hole (Thorne 1974). Here Rg is the gravitational radius of the hole defined as GMBH/c 2 , where G is the gravitational constant, MBH is the mass of the black hole, and c is the speed of light.
In this paper we consider the evolution of a representative set of stars and planets under these conditions. In section 2 we discuss the radius-mass relation of a star composed of ideal gas when losing mass adiabatically. In section 3 we study the case of a solar type star. In section 4 we investigate lower main sequence stars, upper main sequence stars, red giants, white dwarfs, brown dwarfs and planets. We will illustrate the use of these evolutionary models in upcoming papers.
After this work was completed, our attention was drawn to a recent paper: Ge et al. (2010). Although the end application of their paper is different, some of the calculations overlap (and agree with) those presented below.
For a star composed of ideal gas, we have:
where r is the radius, m is the interior mass, and ρ is the density. This gives
where P is the gas pressure, and G is the gravitational constant.
Ignoring radiative contributions, Coulomb and degeneracy corrections, the entropy per particle will be S/N ∼ Cv ln (P/ρ γ ) with γ = 5/3 for ideal monatomic gas, up to a constant; and we can define an effective entropy S = P 3/5 ρ -1 for such an ideal gas.
Then, for a mass-losing star, with constant local entropy, we have
For a star with a known entropy profile, we can solve for its new equation of state with boundary conditions that r⋆(0) = 0 and P⋆(0) = P0⋆. We solve for the total stellar mass M⋆ and the surface radius R⋆ = r⋆(M⋆), where P⋆(M⋆) = 0. Then the volume and average density of the star in this new model can be obtained.
When the star orbits in a circular equatorial orbit and loses mass under stable evolution during this Roche mass transfer phase, its orbital period will follow PR ∝ ρ-1/2 ⋆ , where PR is the Roche orbital period, and ρ⋆ is the mean density of the stripped star. This formula can be obtained The pressure vs interior mass for a masslosing Sun-like star. The four colored curves represent the four cases when the stellar central pressure decreases to 0.7 (blue), 0.4 (green), and 0.1 (red) from 1 (purple) P 0⊙ . by comparing the tidal force from the hole and the gravitational force from the star for a point mass on the Roche surface, and also using Kepler’s laws in the Newtonian limits. Through simple calculation we can also get the angular momentum of the system L⋆ ∼ m⋆P
⋆ , which should decrease for stability throughout the Roche mass transfer phase through slowly gravitational radiation.
Here we neg
This content is AI-processed based on open access ArXiv data.
