E2 Celestial Motion and Orbital Mechanics

Topic

Newton's law of universal gravitation, together with Kepler's laws, describes the motion of planets, satellites, comets, and binary stars. The gravitational force exerted between two bodies increases with the product of their masses and decreases rapidly—in inverse proportion to the square of the distance separating them—as the bodies move apart.

Kepler's first law establishes that the orbits of these bodies are elliptical; the second law states that an orbiting body sweeps out equal areas in equal times, meaning its speed increases as it approaches the central body and decreases as it moves away, thereby conserving its angular momentum throughout the orbit.

Kepler's third law relates the orbital period to the size of the orbit itself: the larger the orbit's semi-major axis, the longer the orbital period; conversely, for a given orbit, the greater the mass of the central body, the shorter the period.

Circular orbital velocity is the speed that exactly balances centripetal force with the gravitational attraction of the central body; this velocity decreases as the orbit becomes larger.

Escape velocity, by contrast, is the minimum threshold speed required for a body to completely leave the central body's gravitational field: below this threshold, the body remains bound to its orbit, whereas above it, the body escapes permanently.

A negative total orbital energy signifies this bound state: as long as the energy remains negative, the body stays in orbit around the central body; however, if the energy were to become zero or positive, the body would escape, following an open trajectory rather than a closed orbit. Thus, from the universal gravitation with which this document begins to the orbital energy—determining whether a body remains bound or escapes—with which it concludes, and encompassing Kepler’s three laws, circular orbital velocity, escape velocity, celestial motion, and orbital mechanics, the fate of any body in space—whether to remain in orbit or leave it forever—is entirely dictated by the nature of its own energy.

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