Two astronauts are in the same circular orbit of radius R...

Two astronauts are in the same circular orbit of radius $R$ around the earth, $180^{\circ}$ apart. Astronaut A has two cheese sandwiches, while Astronaut B has none. How can A throw a cheese sandwich to B? In terms of the astronaut's period of rotation about the earth, how long does it take the sandwich to arrive at $\mathrm{B}$ ? What is the semi-major axis of the sandwich's orbit? (There are many solutions to this problem, assuming that A can throw the sandwich with arbitrary speeds.)

Key Concepts

Transfer Orbits

A transfer orbit is an elliptical orbital path used to move a spacecraft from one orbit to another. Here, by choosing an appropriate impulse, the thrown object can be set on an elliptical orbit that intersects the circular orbit at a point where astronaut B will be at the same time. The concept is central to orbital rendezvous problems, where timing and the shape of the orbit are adjusted to ensure a meeting.

Orbital Mechanics

Orbital mechanics is the study of the motion of objects in space under the influence of gravity. In the context of this problem, it provides the framework for understanding how an object (in this case, the sandwich) can be made to follow a different orbital path from that of the spacecraft. This includes analyzing how small impulses change an object’s velocity and, consequently, its orbit.

Impulsive Maneuvers

Impulsive maneuvers refer to rapid changes in an object’s velocity (delta?v) that alter its orbit. In problems like this one, a throw (or impulse) given to an object changes its orbital energy and shape, thereby allowing it to enter a transfer trajectory that intersects the orbit of the target (astronaut B). Such maneuvers are idealizations that assume the change in velocity can be applied instantaneously.

Kepler’s Third Law

Kepler’s Third Law relates the orbital period of an object to the semi?major axis of its orbit. Specifically, the square of the period is proportional to the cube of the semi–major axis. In this problem, a designer of the transfer orbit can choose a flight time (for example, one half the orbital period of the original circular orbit) and then use Kepler’s Third Law to determine the semi–major axis of the ellipse that the object must follow in order to meet the target at the correct time.

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