Comets move around the sun in very elliptical orbits. At its...

Comets move around the sun in very elliptical orbits. At its closet approach, in $1986,$ Comet Halley was $8.79 \times 10^{7} \mathrm{km}$ from the sun and moving with a speed of $54.6 \mathrm{km} / \mathrm{s}$. What was the comet's speed when it crossed Neptune's orbit in $2006 ?$

Key Concepts

Elliptical Orbits

Elliptical orbits describe the paths that objects follow under the influence of a central gravitational force, such as a comet orbiting around the sun. These orbits are not circular; rather, they have two focal points, with the central body usually located at one of them. The shape of the ellipse causes the orbital speed of the object to vary, with the object moving faster when it is closer to the focus (perihelion) and slower when it is farther away (aphelion).

Conservation of Angular Momentum

The conservation of angular momentum is a key concept in orbital mechanics that states that in an isolated system with no external torques, the angular momentum remains constant. For an object in an elliptical orbit, this means that as the distance from the central body changes, the speed of the object adjusts to keep the product of the distance and the perpendicular component of the velocity constant. This principle explains why objects not only change speed but maintain a specific relationship between their velocity and distance from the focus of the orbit.

Vis-Viva Equation

The vis-viva equation provides a direct relationship between an orbiting object’s speed, its distance from the central body, and the properties of the orbit. Specifically, it relates the orbital speed to the distance from the central mass and the semi-major axis of the orbit. This equation is a fundamental tool in celestial mechanics for calculating orbital velocities at different points along an elliptical orbit, thereby encapsulating both gravitational influences and orbital geometry.

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