Universal gravitation and satellites

China’s Tiangong space station circles the Earth about every hour and a half at a height of roughly 400 km, while the geostationary satellites of the BeiDou navigation system seem to “hang” above one point of the equator. The same force — universal gravitation — holds all of them; only the radius of the orbit differs. This lesson is the second half of the CSCA syllabus line “Circular motion and universal gravitation”: you will apply F = mv²/r from the lesson “Circular motion” to the force of gravity.

Kepler’s laws

1. First law (orbits): every planet moves on an ellipse with the Sun at one focus.
2. Second law (areas): the line from the Sun to the planet sweeps equal areas in equal times. So the planet is fastest at perihelion (近日点) and slowest at aphelion (远日点).
3. Third law (periods): for all bodies orbiting the same central body the ratio a³/T² is the same.

Newton’s law of universal gravitation

Any two bodies attract each other. By Newton’s third law the forces are equal and opposite: the Earth pulls the Moon exactly as hard as the Moon pulls the Earth. The formula holds for point masses and uniform spheres; for spheres r is the distance between the centres.

g at the surface and at a height; mass and density of a planet

At the surface of a planet the weight of a body equals the gravitational force (we ignore the planet’s rotation): mg = GMm/R². This gives g = GM/R² and the most useful equality in CSCA: GM = gR². Chinese textbooks call it the “golden substitution” (黄金代换) — when a problem gives g and R instead of M and G, replace GM by gR².

Satellites: orbits, cosmic velocities, geostationary satellites

In a circular orbit only gravity acts on the satellite, and it plays the role of the centripetal force: GMm/r² = mv²/r = mω²r = m · 4π²r/T² = ma. Each equality gives one formula, and the satellite’s mass m cancels in all of them. An astronaut on board feels weightless (完全失重) because he and the station “fall” with the same acceleration.

Changing orbits. To move to a higher orbit a spacecraft fires its engine to speed up: now mv²/r is larger than gravity, and the craft rises along an elliptical transfer orbit. At the far point it speeds up again to enter the new circular orbit — where its speed is lower and its period longer than before. To come down, it slows down. At a given point the acceleration a = GM/r² does not depend on which orbit the craft is on, but the speed does. A craft docking with a station starts in a lower orbit and speeds up; speeding up in the station’s own orbit would lift it higher and make it fall behind.

How CSCA asks about this

Gravitation items in CSCA are usually two steps: one equation (GMm/r² = …) and one ratio. You have about 75 seconds per item. The typical patterns:

• Kepler’s laws: ellipse and focus, speed at perihelion/aphelion, a period or radius from a³/T² = k;
• how F changes when a distance or mass changes; who measured G (Cavendish);
• g at a height, g on another planet, a planet’s mass and density (GM = gR², ρ = 3π/(GT²));
• comparing two satellites (v, ω, T, a) — often from a table or a graph;
• the first cosmic velocity, “which statement is correct?” about geostationary satellites, changing orbits and docking.