Corotation RadiusEdit
Corotation radius is a concept that appears in multiple contexts within astrophysics, most prominently in the dynamics of rotating galaxies and in magnetized accretion disks. It marks the location where the angular pattern speed of a non-axisymmetric structure equals the orbital angular velocity of the surrounding material. In galaxies, this often means the radius where a bar or spiral pattern rotates in step with the stars and gas in the disk. In accretion systems around compact objects, a related idea identifies where the disk’s orbital motion is synchronized with the rotation of the central object, setting up important torque regimes and resonant interactions. The corotation radius, and its related resonances, help govern angular momentum transfer, star formation activity, and the long-term evolution of the system.
Definition and mathematical basis
In a rotating disk with a pattern that has a constant angular pattern speed Ω_p, the corotation radius R_cr is defined by the condition Ω(R_cr) = Ω_p, where Ω(R) is the local orbital angular velocity of material at radius R. For nearly circular motion in a galaxy, Ω(R) can be written as V(R)/R, with V(R) the rotation curve. Thus, R_cr solves V(R_cr) / R_cr = Ω_p. When the rotation curve is flat (V(R) ≈ V0 at large R), this reduces to R_cr ≈ V0 / Ω_p.
A closely related concept appears in accretion disks around magnetized stars. There, the corotation radius R_co is defined by the equality of the star’s rotation rate Ω* and the Keplerian angular velocity of the disk at that radius: Ω_K(R_co) = Ω*, where Ω_K(R) = sqrt(GM / R^3) for a central mass M. The radius R_co then marks where the disk material orbits in step with the star’s rotation, influencing whether material is spun up, accreted, or expelled by magnetic torques.
In galactic dynamics, corotation is part of a broader network of resonances. The pattern speed Ω_p connects with local orbital and epicyclic frequencies (Ω and κ) to define resonances such as the inner Lindblad resonance (ILR) at Ω_p = Ω − κ/2 and the outer Lindblad resonance (OLR) at Ω_p = Ω + κ/2. These resonances organize the distribution and evolution of stars and gas, shaping features like bars, rings, and spiral arms. See Lindblad resonance for a detailed treatment.
Corotation in galaxies
Bars, spirals, and pattern speeds
Many disk galaxies host non-axisymmetric patterns such as bars or spiral arms. The rate at which these patterns rotate, the pattern speed Ω_p, determines where corotation lies and how it interacts with the surrounding disk. If the corotation radius lies near the end of a bar, the bar is often described as “fast,” while a corotation well beyond the bar’s end characterizes a slower pattern. Observationally, measuring Ω_p and R_cr requires kinematic data and careful modeling; methods include the Tremaine-Weinberg method and dynamical modeling of gas and stellar motions.
The relationship between R_cr and the length of a bar has implications for the transfer of angular momentum between the bar and the disk. When bars extend close to their corotation, angular momentum exchange tends to be efficient, which can influence the bar’s longevity and growth. In contrast, patterns with corotation far from the bar’s end may behave differently, implying a spectrum of evolutionary paths for barred galaxies.
Observational evidence and debates
A central question in the study of barred galaxies is whether bars are long-lived or transient features. The corotation radius features prominently in this debate because its location relative to the bar constrains models of bar formation and persistence. Some observations suggest bars grow and slow their pattern speeds as they exchange angular momentum with the surrounding disk and halo, potentially moving corotation outward over time. Others argue for more rapid formation and dissolution scenarios, with corotation dynamics playing a different role.
Disagreements in the literature often hinge on how exactly pattern speeds are measured and how representative a given galaxy is of the broader population. The use of resonant radii, including corotation, as diagnostic tools remains powerful, but interpretations can vary with assumptions about the mass distribution, dark matter halo, and the temporal stability of the pattern. See pattern speed and rotation curve for foundational concepts that underpin these discussions.
Corotation in accretion disks and magnetospheres
In systems where a magnetized central object governs the inner disk dynamics, the corotation radius is a key boundary. If the inner disk material spins up to the star’s rotation rate at R_co, magnetic torques can couple the star and disk in ways that affect spin evolution and accretion efficiency. When the disk’s inner edge lies inside R_co, material can be channeled toward the star, potentially spinning it up; when it lies outside, magnetic torques can act to spin the star down or to eject material in a propeller-like regime. The concept of R_co thus informs models of torque balance, accretion regimes, and the spin evolution of objects such as neutron stars and white dwarfs.
Methods and phenomena linked to corotation
- Measurement of pattern speeds via kinematic tracers and luminosity-weighted velocity fields.
- Identification of resonance locations (ILR, OLR) through global rotation curves and epicyclic analysis.
- Modeling of bar formation, maintenance, and dissolution in terms of angular momentum exchange across corotation.
- Application of the corotation concept to accretion physics and magnetospheric interactions in compact-object systems.
Key terms tied to corotation radius include rotation curve, pattern speed, Lindblad resonance, bar (galaxy), and spiral arm.