Skylab, the largest spacecraft ever to fall back to Earth,...

Skylab, the largest spacecraft ever to fall back to Earth, met its fiery end on July 11, 1979, after flying directly over Everett, Washington, on its last orbit. On the news that night before the crash, broadcaster Walter Cronkite, in his rich baritone voice, said, "NASA says there is $a$ little chance it will land in a populated area." (Italics added.) After a commercial he immediately corrected himself, saying, "I meant to say 'there is little chance' Skylab will hit a populated area." In fact, it landed primarily in the Indian Ocean off the west coast of Australia, though several pieces were recovered near the town of Esperance, Australia, which sent the U.S. State Department a $\$ 400$ bill for littering. The cause of Skylab's crash was the friction it experienced in the upper reaches of Earth's atmosphere. As the radius of Skylab's orbit decreased, did its speed increase, decrease, or stay the same?

Key Concepts

Orbital Mechanics

Orbital mechanics is the study of the motions of objects in space under the influence of gravitational forces. It explains how the properties of an orbit, such as radius and speed, are related by fundamental laws of physics and how changes in these properties affect an object's path around a massive body.

Circular Orbit Velocity Relation

For objects in stable circular orbits, the orbital speed is determined by the relationship v = ?(GM/r), where G is the gravitational constant, M is the mass of the central body, and r is the orbital radius. This equation shows that if an object moves to a lower circular orbit (a smaller r), its orbital speed must increase in order to maintain a balance between the gravitational pull and the required centripetal force.

Atmospheric Drag

Atmospheric drag refers to the resistance an object encounters as it moves through a planet's atmosphere. For spacecraft in low Earth orbit, drag results in energy loss, causing the orbit to decay (i.e., the orbital radius decreases) and influencing the dynamics of the orbiting object.

Orbital Decay

Orbital decay is the process by which an orbit shrinks over time due to forces such as atmospheric drag. As the orbit decays, the object's altitude decreases and, according to orbital mechanics, this decrease in orbital radius leads to an increase in orbital speed, even though energy is being lost to drag.

Transcript

00:01 So this is a question involving an object orbiting the earth, this was the skylab.

00:08 So if you have an object orbiting the earth, we can derive a formula for the speed at which it is orbiting as a function of the radius of its orbit.

00:21 So we're gonna, by saying radius, i'm assuming a circular orbit, but it turns out that the formula applies for an elliptical orbit as well, as long as you use for that distance, the radius of the orbit, use the average distance that the satellite is from the planet.

00:36 We'll proceed as follows.

00:38 The force that is causing the satellite to orbit, say the planet earth, is the force of gravity.

00:47 And we know the force of gravity is given to us by this formula.

00:52 We have gravitational constant, universal gravitational constant, times the mass of the planet, in this case, it would be the earth, times the mass of the satellite, divided by the radius of the orbit squared.

01:12 So as we derive it here, we're gonna call it the radius, because we're thinking of it being the radius of the orbit.

01:20 But as i said, you can use the same formula that we're gonna derive, just replace r with the average distance of the satellite from the center of the earth.

01:33 Okay, that's the force.

01:35 And because it's going in a circular orbit, we know that the force must equal the mass times the centripetal acceleration formula.

01:45 In other words, this formula here, this force, must equal the mass of the satellite times the speed of the satellite squared divided by the radius, because that is the acceleration.

01:58 V squared divided by r is the acceleration of something that is going at a uniform speed in a circular path...

Please give Ace some feedback