Question: An astronomer observes a protoplanetary disk rotating uniformly, where a planetesimal orbits along a circular path of radius $ R $. If the arc length between two opposite points on the orbit is $ 120^\circ $ of the circle, what is the ratio of the chord length (distance between these two points) to the radius $ R $?

["Title: Understanding Chord Length in Circular Orbits: A Deep Dive into Protoplanetary Dynamics", "In the study of planetary formation, astronomers often examine protoplanetary disks—vast rotating clouds of gas and dust surrounding young stars—where planetesimals begin their journey from dust grains to planets. One critical observation involves tracking a planetesimal orbiting uniformly along a circular path of radius $ R $. A key geometric problem arises when measuring the distance across the orbit: if the arc subtended by opposite points measures $ 120^\circ $, what is the ratio of the chord length connecting these points to $ R $? This article explores the mathematics behind this astronomical measurement.", "### The Geometry of a Circular Orbit", "A planetesimal moving along a circular orbit traces a full circle of circumference $ 2\pi R $. The orbit is symmetric, and the angular displacement between two diametrically opposed positions is $ 180^\circ $. However, the problem specifies an angular separation of $ 120^\circ $ between these two extreme points—this is significant because it defines a chord connecting these two locations, which carries physical importance in orbital mechanics and observational astronomy.", "### Computing the Chord Length", "Let $ O $ be the center of the circular orbit, and let $ A $ and $ B $ be the two points on the circumference separated by a central angle $ \ heta = 120^\circ = \frac{2\pi}{3} $ radians. The chord $ AB $ connects these two points.", "The length of chord $ AB $ in a circle of radius $ R $ is given by the formula:\n[\n\ ext{Chord length} = 2R \sin\left(\frac{\ heta}{2}\right)\n]\nSubstituting $ \ heta = 120^\circ = \frac{2\pi}{3} $:\n[\n\ ext{Chord length} = 2R \sin\left(\frac{120^\circ}{2}\right) = 2R \sin(60^\circ)\n]\nWe know $ \sin(60^\circ) = \frac{\sqrt{3}}{2} $, so:\n[\n\ ext{Chord length} = 2R \cdot \frac{\sqrt{3}}{2} = R\sqrt{3}\n]", "### Finding the Ratio", "The ratio of chord length to radius $ R $ is:\n[\n\frac{\ ext{Chord length}}{R} = \frac{R\sqrt{3}}{R} = \sqrt{3}\n]", "### Astronomical Significance", "This ratio is not merely a mathematical curiosity—it reflects the spatial scale of growing planetary bodies within protoplanetary disks. Observing and calculating such geometric relations helps astronomers infer mass distribution, orbital stability, and collision risks among forming planetesimals.", "### Conclusion", "When an astronomer observes a planetesimal tracing a circular orbit of radius $ R $, with opposite points separated by $ 120^\circ $, the chord connecting them measures $ R\sqrt{3} $. Thus, the ratio of chord length to radius is $ \sqrt{3} $, a fundamental geometric relationship that bridges observational data and theoretical models in planetary science.", "---", "Keywords: planetesimal, protoplanetary disk, arc length, chord length, circular orbit, orbital mechanics, astronomy, NASA, space science, circular geometry, physics, celestial body, angular distance, radial distance."]