Orbital Mechanics — Circles, Ellipses & Gravity
Orbital Mechanics
Once a rocket reaches space, math takes over completely. Every satellite, space station, and interplanetary probe follows paths dictated by equations discovered by Kepler and Newton centuries ago.
Circular Orbital Velocity
For a circular orbit at altitude $h$ above Earth:
$$v_{\text{orbit}} = \sqrt{\frac{GM}{R + h}}$$
where $G = 6.674 \times 10^{-11}$ N·m²/kg² (gravitational constant), $M = 5.972 \times 10^{24}$ kg (Earth's mass), and $R = 6{,}371$ km (Earth's radius).
Notice: orbital velocity depends on a square root — the same function you study in algebra. As altitude increases, speed decreases, which is why the Moon orbits more slowly than the International Space Station.
The ISS orbits at about 408 km altitude. How fast does it travel?
$$v = \sqrt{\frac{(6.674 \times 10^{-11})(5.972 \times 10^{24})}{6{,}371{,}000 + 408{,}000}}$$
$$v = \sqrt{\frac{3.986 \times 10^{14}}{6{,}779{,}000}} \approx \sqrt{5.879 \times 10^7} \approx 7{,}668 \text{ m/s}$$
That's about 27,600 km/h — fast enough to circle the Earth every 90 minutes!
Kepler's Laws — Geometry in the Sky
Johannes Kepler discovered that planets move in ellipses, not circles. An ellipse is described by:
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$This is a quadratic equation — the same conic sections you study in algebra II. The orbits of every planet, asteroid, and comet are conic sections: ellipses, parabolas, or hyperbolas.
The Hohmann Transfer — Algebra Saves Fuel
To travel from one orbit to another, spacecraft use a Hohmann transfer — an elliptical path that touches both orbits. The required velocity changes are:
$$\Delta v_1 = \sqrt{\frac{GM}{r_1}} \left(\sqrt{\frac{2r_2}{r_1 + r_2}} - 1\right)$$This equation is pure algebra and square roots — exactly the math you're learning in this course. NASA engineers use it every day.
Square roots, quadratics, and ellipses aren't abstract — they're the literal shape of every orbit in the solar system. Learning these concepts means learning the language of space travel.