[Paper Review] Tidal radii of main sequence stars -- I. Physical tidal radius, semi-analytic model and their implications

Taeho Ryu, Julian H. Krolik+2 more|arXiv (Cornell University)|Jul 18, 2019

Astrophysical Phenomena and Observations7 references4 citations

TL;DR

This paper presents a semi-analytic model using MESA stellar evolution and general relativistic dynamics to compute the physical tidal radius 𝒫ₜ for main sequence stars disrupted by a 10⁶M⊙ black hole. It finds 𝒫ₜ ≈ 27 times the black hole's gravitational radius and shows that 𝒫ₜ is nearly constant across stellar masses, leading to a weaker-than-expected dependence of full disruption rates on stellar mass compared to standard estimates.

ABSTRACT

A star is tidally disrupted by a supermassive black hole when their separation is shorter than the "tidal radius". This quantity is often estimated on an order-of-magnitude basis without reference to the star's internal structure. Using MESA models for main sequence stars and fully general relativistic dynamics, we find the physical tidal radius for complete disruption $\cal{R}_t$ for a $10^6M_\odot$ black hole (BH). We find that across a factor $\sim20$ in stellar mass $M_*$, i.e., $0.15M_{\odot}\leq M_*\leq3M_\odot$, $\cal{R}_t\sim27 imes$(BH's gravitational radius). When comparing $\cal{R}_t$ with the commonly used order-of-magnitude estimate $r_t$, we find that $\cal{R}_t\sim1.05-1.45r_t$ for $0.15M_\odot\leq M_*\leq0.5M_\odot$, but between $0.5 M_\odot$ and $1 M_\odot$, $\cal{R}_t$ drops to $\sim 0.45r_t$, and it remains at this value up to $10 M_\odot$. The near-constancy of $\cal{R}_t$ implies a weaker dependence of the full disruption rate on $M_*$ than when predicted with $r_t$. The characteristic energy width of the debris $ΔE$ ranges from $\sim1.2Δ\cal{E}$ for low-mass stars to $\sim 0.35Δ\cal{E}$ for higher-mass stars, where $Δ\cal{E}=GM_{ m BH}R_*/\cal{R}_t^{2}$. We present analytic fits for the $M_*$ dependence of $\cal{R}_t$ and $ΔE$; these fits lead to analytic expressions for the time of peak mass fallback rate and the maximal mass fallback rate. Our results also bear on the fraction of events leading to fast or slow circularization, as well as on the character of the tidal event occurring when the remnant of a partial disruption returns to the black hole. Using a semi-analytic model, we show that $\cal{R}_t$ is primarily determined by the star's central density rather than its mean density. For high-mass stars, the full disruption rate is roughly 1/4 the partial disruption rate, while this ratio is close to unity for low-mass stars.

Motivation & Objective

• To accurately compute the physical tidal radius 𝒫ₜ for main sequence stars disrupted by a 10⁶M⊙ supermassive black hole, accounting for internal stellar structure.
• To compare the physical tidal radius 𝒫ₜ with the conventional order-of-magnitude estimate 𝑟ₜ and assess the implications for disruption rates.
• To derive analytic fits for 𝒫ₜ and the energy width of debris 𝚫𝐸 as functions of stellar mass, enabling predictions of fallback times and peak rates.
• To investigate the role of central density in determining 𝒫ₜ and the fraction of events leading to fast or slow circularization.
• To examine the dynamics of partial disruption remnants returning to the black hole and their tidal event characteristics.

Proposed method

• Using MESA stellar evolution models to generate realistic internal structures for main sequence stars across 0.15M⊙ ≤ M⋆ ≤ 3M⊙.
• Applying fully general relativistic dynamics to compute the tidal disruption radius 𝒫ₜ where the star is fully disrupted.
• Defining the characteristic energy width of debris as 𝚫𝐸 = 𝐺𝑀_𝐵𝐻𝑅_*/𝒫ₜ², with 𝚫ℰ as a reference energy scale.
• Deriving analytic fits for 𝒫ₜ and 𝚫𝐸 as functions of stellar mass, enabling analytical predictions of mass fallback evolution.
• Using a semi-analytic model to show that 𝒫ₜ is primarily governed by central density rather than mean density.
• Comparing full and partial disruption rates to assess the dependence on stellar mass and the likelihood of remnant re-encounter events.

Experimental results

Research questions

• RQ1 How does the physical tidal radius 𝒫ₜ for main sequence stars vary with stellar mass when computed using realistic stellar structure models?
• RQ2 What is the quantitative relationship between the physical tidal radius 𝒫ₜ and the standard order-of-magnitude estimate 𝑟ₜ across different stellar masses?
• RQ3 How does the energy width of the debris 𝚫𝐸 depend on stellar mass, and what are the implications for the mass fallback rate light curves?
• RQ4 To what extent is 𝒫ₜ determined by central density versus mean density in the star?
• RQ5 What fraction of tidal disruption events lead to fast or slow circularization, and how does this depend on stellar mass?

Key findings

• The physical tidal radius 𝒫ₜ is approximately 27 times the gravitational radius of a 10⁶M⊙ black hole across 0.15M⊙ ≤ M⋆ ≤ 3M⊙.
• For low-mass stars (0.15M⊙ ≤ M⋆ ≤ 0.5M⊙), 𝒫ₜ ≈ 1.05–1.45𝑟ₜ, but for stars with M⋆ ≥ 0.5M⊙, 𝒫ₜ drops to ∼0.45𝑟ₜ and remains nearly constant up to 10M⊙.
• The energy width of the debris 𝚫𝐸 ranges from ∼1.2Δℰ for low-mass stars to ∼0.35Δℰ for higher-mass stars, where Δℰ = 𝐺𝑀_𝐵𝐻𝑅_*/𝒫ₜ².
• The full disruption rate depends weakly on stellar mass due to the near-constancy of 𝒫ₜ, with the full-to-partial disruption rate ratio being ∼1/4 for high-mass stars and close to unity for low-mass stars.
• The physical tidal radius 𝒫ₜ is primarily determined by the star’s central density rather than its mean density.
• Analytic fits for 𝒫ₜ and 𝚫𝐸 enable analytical predictions of the time of peak mass fallback rate and the maximal fallback rate as functions of stellar mass.