Circular Orbital Motion

Draw the free-body diagram of a satellite and there is exactly one arrow: gravity, of size GMm/r², pointing at the planet's centre. There is no separate centripetal force to add beside it and no outward force to balance it; centripetal is the name of a job, and gravity is what does it. Uniform circular motion needs a net inward force of mv²/r, so Newton's second law along the radius is GMm/r² = mv²/r. Nothing acts along the tangent, so the speed stays constant. Notice that m multiplies both sides: the orbit belongs to the radius and the central mass, not to the satellite. A one-gram screw and a station of several hundred tonnes, released side by side at the same r with the same velocity, stay side by side.

Solve the radial law for v: v = √(GM/r). For Earth GM = 3.986 × 10¹⁴ m³ s⁻², and r is measured from Earth's centre, so a satellite 400 km up has r = 6.371 × 10⁶ m + 4.00 × 10⁵ m = 6.771 × 10⁶ m and v = 7.67 km s⁻¹. Put the altitude in place of r and you get 31.6 km s⁻¹, four times faster than any circular orbit around Earth can be — the signal that r was measured from the wrong place. The formula says higher orbits are slower: at the GPS radius of 2.656 × 10⁷ m, about four Earth radii, v = 3.87 km s⁻¹, half the low-orbit speed. The fastest circular orbit skims the surface, r = R, where v = √(g₀R) = 7.91 km s⁻¹ with g₀ = GM/R² = 9.82 m s⁻².

The period is circumference over speed, T = 2πr/v, and substituting v gives T = 2π√(r³/GM), or T² = (4π²/GM) r³. At 400 km that is 5.54 × 10³ s, 92 minutes. Because T² ∝ r³, doubling the radius multiplies the period by 2√2 = 2.83, and the GPS radius, 3.92 times the low orbit's, gives 11.97 h, two orbits per sidereal day. Run the formula backwards for a geostationary satellite, which must turn with the Earth: T = 86 164 s, one rotation relative to the stars, gives r = ∛(GMT²/4π²) = 4.216 × 10⁷ m, about 35 800 km above the equator, at 3.07 km s⁻¹. Using 24 h instead of the sidereal day puts the satellite 77 km too high, and it slowly drifts westward over the ground.

Rearranging T² = 4π²r³/(GM) gives M = 4π²r³/(GT²), and the satellite's mass is nowhere in it: timing any small body on a circular orbit weighs whatever it circles. The Moon has r = 3.844 × 10⁸ m and T = 27.32 days = 2.360 × 10⁶ s, so M = 4π²(3.844 × 10⁸ m)³/[(6.674 × 10⁻¹¹ N m² kg⁻²)(2.360 × 10⁶ s)²] = 6.03 × 10²⁴ kg. That is about 1% more than Earth's accepted 5.97 × 10²⁴ kg, and the gap is itself physics: the Moon has 1.2% of Earth's mass, so Earth also swings round the pair's centre of mass and the formula really returns M + m. For an artificial satellite that correction is below one part in 10¹⁹, which is why the model treats the planet as fixed.

Stand on a bathroom scale in a station 400 km up. Two forces act on you: gravity, mg(r) with g(r) = GM/r² = 8.69 m s⁻², and the scale's push N. Taking the direction toward Earth as positive, Newton's second law reads mg(r) − N = ma. You share the station's orbit, so your acceleration is its orbital acceleration, v²/r = (7.67 × 10³ m s⁻¹)²/(6.771 × 10⁶ m) = 8.69 m s⁻², which is g(r) itself. So N = 0 and the scale reads zero, although gravity pulls a 70 kg astronaut with 609 N, 89% of the 687 N they weigh on the ground. What you feel as weight is the contact force, not gravity. A released pen floats beside you because it shares your r and v, and therefore your path, not because gravity has been switched off.

Each step leaned on an assumption, and each has a limit. Spherical symmetry lets the planet act as a point mass; Earth's equatorial bulge breaks it slightly and makes low orbits precess. A fixed planet needs M ≫ m: the Earth–Moon pair already needs M + m, and two stars of similar mass circle a point between them. No drag: at 400 km the thin upper atmosphere removes energy steadily, so stations need periodic reboosts. And a circle needs exactly the right velocity, tangential and of size √(GM/r). Launch from the same point a little faster or slower and the path becomes an ellipse through that point; ellipses, and the energy it costs to move between orbits, come next.