The motions of rockets, satellites, planets and stars all obey a handful of elegant equations. Understanding them turns the night sky and the space race from a spectacle into something you can actually calculate. This guide explores the key ideas, from a falling apple to a black hole.
Gravity: the force behind it all
Newton's law of universal gravitation states that every mass attracts every other with a force that grows with their masses and falls with the square of the distance between them. This single law explains why an apple falls, why the Moon orbits the Earth and why the planets circle the Sun — they are all the same phenomenon at different scales. Because the force weakens with the square of distance, doubling the distance quarters the pull, which is the key to understanding orbits and escape.
Escape velocity
To leave a planet for good, an object must reach escape velocity — the speed at which its kinetic energy exactly overcomes the planet's gravitational pull. It depends on the planet's mass and radius, not on the mass of the object escaping. Earth's is about 11.2 km/s, which is why rockets must be so enormously powerful. The escape velocity calculator works it out for any body, from the Moon to Jupiter.
Staying in orbit
A satellite does not escape; it falls around the planet forever at just the right speed. The clever insight is that an orbit is continuous free-fall: the satellite moves sideways so fast that as it falls, the curved Earth falls away beneath it at the same rate, so it never lands. Orbital velocity is lower than escape velocity and decreases with altitude — too slow and it spirals down, too fast and it flies off. The orbital velocity calculator finds the speed for a given orbit.
Surface gravity and your weight
Gravity at a planet's surface depends on its mass and radius, which is why you would weigh far less on the Moon and far more on Jupiter, even though your mass never changes. The surface gravity calculator gives the gravity for any body, and the weight on other planets calculator shows what you would weigh across the solar system. Density — how tightly mass is packed into a volume — ties size and gravity together, which the planet density calculator explores, revealing for instance that Saturn is so light it would float in water.
The scale of the cosmos
Space is almost incomprehensibly vast, so astronomers measure distance by how long light takes to cross it. Light travels about 300,000 km every second, yet still takes over eight minutes to reach us from the Sun and years from the nearest stars. This means looking outward is also looking back in time: the starlight you see tonight left its source years, centuries or millennia ago. The light travel time calculator makes these distances concrete.
Black holes and the Schwarzschild radius
Here the maths turns strange. If you compressed any mass into a small enough sphere, its escape velocity would exceed the speed of light — and since nothing travels faster than light, not even light could escape. That critical size is the Schwarzschild radius, the 'point of no return' that defines a black hole's event horizon. The Schwarzschild radius calculator computes it; remarkably, for the Earth it is under a centimetre, meaning our whole planet squeezed to marble-size would become a black hole.
Kepler's laws and the dance of the planets
Long before Newton explained why, Johannes Kepler described how the planets move in three laws. Orbits are ellipses, not perfect circles, with the Sun at one focus; a planet sweeps out equal areas in equal times, so it moves faster when closer to the Sun; and the square of a planet's orbital period is proportional to the cube of its average distance, neatly linking how far a planet is to how long its year lasts. These laws let astronomers predict planetary positions with astonishing precision and, turned around, let them weigh distant stars and discover unseen planets from the wobble or dimming they cause. They are a beautiful example of simple maths capturing the motion of whole worlds.
Bringing the universe down to numbers
Finally, telescopes are how we reach across these distances, and even they obey simple maths: magnification is the telescope's focal length divided by the eyepiece's. The telescope magnification calculator connects backyard stargazing to the same physics that launches spacecraft. Taken together, these formulas show that the cosmos, for all its grandeur, runs on a small set of rules you can calculate with — which is perhaps the most astonishing thing about it. The same equations that drop an apple steer a probe to the outer planets.