Kepler's third law

A thrown stone arcs through the air, but a planet around the Sun follows a closed curve. Why an ellipse, and not a circle or a spiral? Kepler's first law provides the answer, but its usual proof requires heavy computation. This work offers a shorter path — through a single simple ratio of velocities.

In the dark kitchen of the Taurus molecular cloud, 450 light-years away, dozens of planetary systems are being prepared. Each young star is a chef, and its protoplanetary disk is a counter piled with flour-like dust and steam-like gas.

On September 26, 2022, the DART probe slammed into the natural satellite at a speed of 6,6 km/s. A fountain of debris burst into space. Rocks, dust, sand grains — everything erupted in a silent flash. Two years later, a numerical simulation of 20 million trajectories revealed the fate of each particle.

Edge of the spotlight. Remove the dazzling light of the central star, and a theater of shadows emerges. The James Webb coronagraph, like a dense curtain, hid the star TW Hydrae — and on a stage 120 astronomical units long (four times the distance to Neptune), its protoplanetary disk came alive.

26,000 light-years from Earth, in the heart of the Milky Way, star S301 tears through space at 18,000 kilometers per second. It loops around the supermassive black hole Sgr A* on an elongated orbit. At its closest approach, just 280 gravitational radii separate the star from the black hole — roughly the distance from the Sun to Uranus.

Tabby's Star, a beacon in the cosmic sea, has baffled astronomers since 2015. Its light flickered not in the rhythm of planets, but as if scattered by a storm — deep dips, sharp spikes. But amid this chaos, a smooth, nearly perfect dip was found. The eye of the storm.

Every 165 days, the giant planet dives toward its star, nearly grazing its fiery corona, then flies off into the icy void. This is TIC 241249530 b — an exoplanet with a record eccentricity of 0,94.

A tiny satellite, like a swimmer, cuts through invisible layers of gas in the bloated envelope of a red giant. Each orbit excites an acoustic wake — compression waves traveling through plasma.

Among the scorching exoplanets that circle perilously close to their stars lies the secret of their birth. Nearly three decades ago, Michel Mayor and Didier Queloz discovered the first such world, and since then astronomers have been tormented by the question: how does a gas giant end up pressed against…

Seventeen minutes: that’s the heartbeat frequency of AM Canum Venaticorum, setting the rhythm for the entire class of ultracompact binaries. In these systems, two white dwarfs whirl in a death dance, tightening their orbit to the still-inaudible whisper of gravitational waves.

At the center of the Milky Way, shrouded in dust and glare, lurks a supermassive black hole — Sagittarius A*. Around it, like moths around a flame, swarm dozens of guide stars. Two of them, S2 and S38, with eccentric orbits and periods of just a few years, have become the protagonists in testing the fundamental laws of the cosmos.

A quarter-century ago, Michel Mayor and Didier Queloz discovered the first hot Jupiter, and shattered all the rules of planet formation. A gas giant on a days-long orbit couldn't have formed next to the star: it's far too hot and lacks building material. So, it must have been born far away and then migrated.

Two thousand years ago, a brilliant beacon shone in the constellation Eridanus, recorded in the registers of ancient astronomers. Hipparchus in 129 BCE noted it as a first-magnitude star, Ptolemy confirmed its brightness in the Almagest, and the Persian scholar Al-Sufi still saw it among the brightest in 964.

The three-body problem is the eternal stumbling block of celestial mechanics. Even Poincaré realized in the late 19th century that its chaos defies general analytical solutions. In this gravitational madness, islands of regularity appear only rarely — periodic orbits.

Long-period radio transients (LPTs) are a mysterious class discovered only recently. Their pulses last minutes or hours, and their nature was completely unknown. ASKAP J1745-5051, caught by the Australian telescope ASKAP, became the first to get a full explanation.

Some 158 light-years away, around the dim star TOI-6894, a red dwarf, a planet moves leisurely. Its year lasts less than two Earth days, yet it is a cool world with an equilibrium temperature around 420 K, just a bit warmer than a sauna.

Imagine a person in an old photograph standing in a crowd while spinning — guessing their age would be nearly impossible. That's roughly how astronomers determine the ages and masses of distant stars, and they often get it wrong. For stars slightly heavier than the Sun (from 1.4 to 2.5 solar masses), standard methods like chemical-composition 'clocks' work poorly.

In a superconductor, electrons pair up like dancers holding hands. Usually, the distance between them, the pair size, depends on how fast they move and how strongly they are bound. But sometimes the dance floor almost disappears, and there's nowhere to move. This happens in materials with flat electronic bands: electrons nearly freeze, and the usual rules stop working.

The heart doesn't push blood in a steady stream but in pulses. Each beat sends a wave racing through the arteries, like ripples along a taut garden hose. For a long time, it was thought that an animal's metabolic rate depended only on how vessels branch, like pipes in a house. But new work flips that view: what matters isn't the shape of the pipes but the pulse itself.

Every 1151 years, seven planets of the Solar System, from Mercury to Neptune, find themselves at almost the same points in their orbits. Their positions scatter by less than a degree, as if the celestial mechanism is wound to the same tune. This rhythm was discovered by comparing planetary positions over 2600 years using precise tables of celestial motion.

The Kepler telescope spent years watching thousands of stars, spotting planets by the tiny dips in starlight when a planet passes in front of its star. Such a passage, or transit, lets us measure the planet's size but not the shape of its path.

A heavy spinning top launched near a black hole doesn't follow the curve that textbooks draw. Its own rotation and oblateness, like a pumpkin squashed at the poles, make the trajectory wobble. In a new study, this wobbling path has been calculated more precisely than ever before. Usually, a small body in a black hole's field is treated as a point — without size or spin.

When a typical computer searches for a law of nature from data, it tries to guess a complete formula in one go. It's like a cook who, without knowing the recipe, dumps all the ingredients into a pot at once and hopes for soup. Most often, the result is an inedible mess: the formula is long but meaningless, and it fails on new data.

A solar flare is like a short circuit in a giant neon sign. Inside the Sun, magnetic fields twist into loops, like rubber bands, and when they reconnect, energy bursts into space. Part of it goes into hard X-rays, produced by particles accelerated to nearly the speed of light.

At the center of a cosmic dust cloud, astronomers detected an invisible star. It's wrapped in a rotating gas disk. This disk works like a set of scales: the gas speed at different distances from the center reveals the star’s mass. The pattern is simple: the closer to the center, the faster the motion.

Three distant giant planets circle their stars so closely that a year on them lasts just a few Earth days. These worlds are called hot Jupiters, and their orbits are a real puzzle: how did they end up right next to the star if they were born far away, in the cold outskirts of the system? One key clue is the angle at which a planet orbits relative to the star's rotation.

Two translators listen to the same symphony. One hears only the fundamental tone: a clean, predictable melody. The other catches the overtones that color every note.

The three planets in the Kepler-51 system have long intrigued astronomers: they are Saturn-sized but less dense than water. These 'super-puff' worlds, with masses no more than ten Earths, are like cosmic dandelions — massive yet almost weightless. Their orbits are arranged like musicians in an orchestra, each playing its part, beating the rhythm with transits.