Kepler's three laws say orbits are ellipses, an orbiting body moves faster when closer and slower when farther, and the square of the period grows with the cube of the orbit's size.
Johannes Kepler worked out three rules for how planets move round the Sun. He had worked as an assistant to the Danish astronomer Tycho Brahe, whose careful naked-eye records gave him the data. He published the first two laws in Astronomia Nova in 1609 and the third in Harmonices Mundi in 1619.
The same three laws describe satellites going round Earth.
Kepler did not know why these rules held. Isaac Newton later explained them with his theory of universal gravitation.
ISRO's LVM3 put Chandrayaan-3 into an elliptic parking orbit about 170 km high at its lowest point and 36,500 km at its highest.
NASA's planet examples show the third law too: Mercury, closest to the Sun, takes only 88 days per orbit; Earth takes 365 days; Saturn takes 10,759 days.
Watch the Altitude card change slightly as the station goes round — its orbit is not a perfect circle, and its speed shifts with it. Open the ISS tracker
Newton's form of the third law, for a small body orbiting a large one:
T² = (4π² / GM) a³
where a is the semi-major axis. Strictly, GM should be G(M + m); it matters only when the orbiting body is not tiny.
Test on the Moon. a = 384,400 km. With Earth's GM alone (398,600 km³/s²): a³ = 5.680 × 10¹⁶, divided by GM = 1.425 × 10¹¹, square root = 377,500 s, times 2π = 2.372 × 10⁶ s = 27.45 days. Adding the Moon's GM (4,900) makes the total 403,500; the period shrinks by √(398,600/403,500) = 0.9939 to 27.28 days. The measured value is 27.32 days.
Second law as conservation of angular momentum. Equal areas in equal times means r × v⊥ is constant. At perigee and apogee the velocity is perpendicular to r, so r_p v_p = r_a v_a. For Chandrayaan-3's parking orbit (r_p = 6,548 km, r_a = 42,878 km), the vis-viva equation gives v_p ≈ 10.28 km/s and v_a ≈ 1.57 km/s. Check: 6,548 × 10.28 ≈ 67,300 and 42,878 × 1.57 ≈ 67,300.
First law and eccentricity. For an ellipse, r_p = a(1 − e) and r_a = a(1 + e), so e = (r_a − r_p) / (r_a + r_p). For the parking orbit: e = 36,330 ÷ 49,426 ≈ 0.735 — strongly stretched. The ISS has e very close to 0.
Third law across satellites. Because T² ∝ a³, the ratio T²/a³ is the same for every Earth satellite: 4π²/GM ≈ 9.90 × 10⁻⁵ s²/km³. For the ISS: 5,553² ÷ 6,778³ = 3.084 × 10⁷ ÷ 3.114 × 10¹¹ ≈ 9.90 × 10⁻⁵. For geostationary: 86,164² ÷ 42,164³ ≈ 9.90 × 10⁻⁵.
A line joining the satellite to Earth sweeps equal areas in equal times. So the satellite moves fastest at perigee and slowest at apogee.
T ∝ a^(3/2), so the ratio is 1 : 4^(3/2) = 1 : 8.
Isaac Newton, through his theory of universal gravitation.