Sky Mechanics Lab

Speed

Shaded wedges each cover the same amount of time (1/12 of an orbit). Dashed line: the orbit predicted from energy alone. Amber arrow: gravity. Blue arrow: velocity.

Why planets move the way they do

Star mass

Launch speed at 1 AU

What's happening now

Energy budget (per kg of planet)

Why it works

The planet is always falling toward the star. It misses because it is also moving sideways. If the sideways speed is exactly right, the fall curves at the same rate as the orbit and the distance stays fixed: a circle. Too slow and it falls inward, speeding up as it drops. Too fast and it climbs outward, slowing down. Gravity's pull is 1/r², so the path closes into an ellipse, as Kepler found from Tycho Brahe's data in 1609.

First law: orbits are ellipses with the star at one focus (not the center).
Second law: equal areas in equal times. Angular momentum is conserved, so when the planet is close it must move faster.
Third law: P² = a³ / M, with P in years, a in AU, M in solar masses.

The total energy (moving energy plus negative gravitational energy) decides everything. Below zero the planet is bound and must return. At or above zero it escapes. That happens at √2 ≈ 1.414 times circular speed.

Speed

Viewed from above the orbit. Earth is toward the bottom of the screen. Strip: star 1's spectrum lines with the Doppler shift magnified. Graph: star 1's velocity toward or away from us over two orbits.

How we find planets we can't see

Star 1 mass

Companion mass

Separation

Orbit tilt seen from Earth

What's happening now

Why it works

Gravity pulls both ways. A planet doesn't orbit its star's center; both circle their shared center of mass, the barycenter. A heavy star sits close to it and moves in a tiny circle. A light planet swings in a wide one.

d★ = a · m₂ / (m₁ + m₂)

We can't see the star move sideways, but we can measure its speed toward or away from us. Light from a star moving away is stretched to longer, redder wavelengths. Light from a star moving toward us is squeezed bluer. The size of the shift gives the speed:

Δλ / λ = v / c

A regular back-and-forth shift reveals an unseen companion. Its period gives the orbit size (Kepler's third law), and its amplitude gives the companion's mass. 51 Pegasi b, the first planet found around a Sun-like star (1995, Nobel Prize 2019), was found this way. A tilted orbit shows less of the motion along our line of sight, so the measured mass is a minimum (m·sin i).

Limb darkening

Graph: the star's total brightness as the planet crosses (a light curve). Faint line is the full prediction, dot is right now.

Planets revealed by a shadow

Star radius

Planet radius

Path offset from star's center

What's happening now

Why it works

We can't see the planet, but we can see the star get slightly dimmer when the planet blocks part of its disk. The fraction of light lost is roughly the fraction of the disk covered:

depth ≈ (R_planet / R_star)²

The light curve's shape tells more. Its sloped edges (ingress and egress) last as long as the planet takes to cross the star's edge, which gives its size and path. The flat-ish bottom is curved because stars are dimmer at their edges. That's limb darkening: near the edge we look through cooler upper layers. Repeating dips give the orbital period.

The orbit must be lined up edge-on from our viewpoint for a transit to happen at all. For an Earth-Sun copy, the odds are about R★/a ≈ 0.5%. Space telescopes like Kepler and TESS beat those odds by watching hundreds of thousands of stars at once, and found thousands of planets this way. Small stars help: TRAPPIST-1's Earth-size planets dim it by about 0.5%, versus 0.008% for Earth crossing the Sun.

Life stage

Hertzsprung–Russell diagram: brightness (up) against surface temperature (hot on the left). Gray band is the main sequence, where stars spend about 90% of their lives. The path is schematic and its timing is not to scale.

Why mass decides a star's fate

Birth mass

What's happening now

Why it works

A star is a tug-of-war. Gravity squeezes inward. Pressure from hot gas, heated by fusion in the core, pushes out. While hydrogen lasts, the two balance (hydrostatic equilibrium) and the star stays steady on the main sequence.

A heavier star squeezes harder, so its core runs hotter, and fusion speeds up very steeply with temperature. Brightness rises roughly as M^3.5, but fuel only rises as M:

lifetime ∝ M / L ∝ M⁻²·⁵ → 10× the mass ≈ 1/300th the life

Color is temperature. A hotter surface peaks at shorter wavelengths (Wien's law: λ_peak = 2.9 mm·K / T). Cool stars glow red, the Sun peaks in green-yellow and looks white, and the hottest stars look blue.

The ending depends on the core left behind. Below about 1.4 solar masses (the Chandrasekhar limit), electron pressure holds it up as a white dwarf. Above that, it collapses to a neutron star. Above roughly 2–3 solar masses, nothing known can stop the collapse, and it becomes a black hole.

Show twin Speed

Solid bodies and their hollow twins start almost identically. Graph: how far apart the two runs are (log scale) as time passes.

When prediction breaks down

Twin's starting nudge

What's happening now

Why it works

Two bodies under gravity always follow neat, repeating ellipses, which Newton solved exactly. Add a third and there is no general formula. Henri Poincaré showed this in 1890, and the discovery became the seed of chaos theory.

Each body's pull on the others keeps changing as they move. Close encounters act like slingshots, and a tiny difference in where a body arrives sends it out on a very different path. In chaotic systems the difference grows exponentially: it doubles every fixed stretch of time until the two runs share nothing.

error(t) ≈ error(0) · e^(λt)

That is why the twin's nudge barely matters. A billionth or a hundredth only shifts when the runs split apart, not whether they do. Some configurations, like the figure-eight (found in 1993) and planets on well-spaced orbits, stay orderly for long times. Our Solar System is mildly chaotic too: planet positions can't be predicted more than about 5–10 million years ahead.