Satellites, weightlessness and Kepler's laws - TCE Physics (Tasmania)

What this dot point is asking

This dot point applies gravity and circular motion to real orbits, and explains the often misunderstood idea of weightlessness in space.

Apparent weightlessness

Weight is the gravitational force $W = mg$, and it does not disappear in orbit. At the altitude of the International Space Station, gravity is still about $90$ per cent of its surface value. What changes is the apparent weight, which is the normal force a surface pushes back with.

In orbit the station and everything in it are in continuous free fall toward Earth, accelerating at the local $g$. Because the floor falls with the astronaut, it pushes with zero normal force, and with no normal force there is nothing to sense as weight. This is true weightlessness only in deep space far from any mass; in orbit it is apparent weightlessness caused by free fall.

The same free-fall idea explains the "vomit comet" aircraft used to train astronauts: when the plane flies a parabolic arc and its engines provide only enough thrust to overcome air resistance, everyone inside falls together and feels weightless for around half a minute. No spacecraft is needed; weightlessness is purely a consequence of everything sharing the same gravitational acceleration with no supporting normal force.

Low Earth and geostationary orbits

The orbital speed $v = \sqrt{\dfrac{GM}{r}}$ shows that lower orbits are faster. A low Earth orbit has a period of about $90$ minutes. A geostationary orbit is special: its period is exactly one sidereal day, so the satellite stays above a fixed point on the equator, ideal for communications. Setting $T = 86400\ \text{s}$ in Kepler's third law gives a radius of about $4.2 \times 10^7\ \text{m}$, roughly $36\,000\ \text{km}$ above the surface. Low orbits are used for imaging and the space station because they are close and cheap to reach; geostationary orbits are used for broadcast and weather because the satellite appears fixed in the sky.

Kepler's three laws

Kepler described planetary orbits before Newton explained them:

1. The law of ellipses: each planet moves in an ellipse with the Sun at one focus. For most school problems we approximate orbits as circles.
2. The law of equal areas: a line from the Sun to a planet sweeps out equal areas in equal times, so a planet moves faster when closer to the Sun.
3. The law of periods: the square of the orbital period is proportional to the cube of the orbital radius.

The third law follows directly from equating gravity to the centripetal force:

so $\dfrac{T^2}{r^3}$ is the same constant for every satellite of a given central body, and you can compare two orbits without knowing $G$ or $M$. The second law is a consequence of the conservation of angular momentum, which is why a comet whips quickly past the Sun at perihelion and dawdles at aphelion.

In the exam, when comparing orbits use Kepler's third law as a ratio so the unknown constants drop out. When asked about weightlessness, state clearly that gravity still acts and that the sensation comes from the absence of a normal force during free fall.