Astronomy & Celestial Mechanics
the sky, computed
How this site is made

The figure library

Every picture here is generated from code at build time. This page lists the generators, each rendered at its defaults.

No figure on this site is a drawing that was made once and saved. Each one is a function: it takes parameters and returns SVG, so the same generator produces the 27° incline and the 5° incline without either being redrawn.

That is the reason the collection can keep growing without the illustrations drifting apart. A generator is written once, checked once, and every essay that calls it inherits the same line weights, the same colour roles, and the same behaviour in dark mode. There are 24 of them so far.

blackbody

Blackbody curves at 3000, 5800, 10000 KThermal emission against wavelength, each curve scaled to its own peak so the shift can be seen on one plot. The peak moves to shorter wavelengths as the temperature rises, which is why colour is a thermometer.visible050010001500200000.20.40.60.811.2wavelength (nm)3000 K, peak 966 nm5800 K, peak 500 nm10000 K, peak 290 nmeach curve scaled to its own peak

celestial-sphere

The sky from latitude 52°The celestial sphere seen from latitude 52 degrees. The pole stands 52 degrees above the horizon, the celestial equator meets the horizon due east and west, and a star at declination 20 degrees traces the drawn circle once a day. Everything below the horizon is drawn faint.celestial poleobserverzenithnorthsouthcelestial equatorthe daily circle of a star at δ = 20°the pole sits 52° up, because the observer is at latitude 52°faint arcs are below the horizon

conic-family

Every orbit one force allowsCircle, ellipse, parabola and hyperbola, all sharing a focus and a closest approach. The eccentricity alone decides which one a body is on, and whether it returns.common periapsiscircleellipse, e = 0.5ellipse, e = 0.9parabola, e = 1hyperbola, e = 1.5e < 1 returnse ≥ 1 never does

distance-ladder

The distance ladder, and its overlapsThe reach of each distance technique on a logarithmic scale in parsecs. Each rung is calibrated where it overlaps the one below it, so an error low on the ladder propagates all the way to the top.10^-610^-410^-210^010^210^410^610^810^10radar rangingdirect: a timed echoparallaxgeometry, and nothing assumedspectroscopic parallaxassumes a star like the calibratorsCepheid variablesassumes the period–luminosity relation holdsTully–Fisherassumes rotation tracks luminositytype Ia supernovaeassumes a standard explosionHubble's lawassumes the expansion rate is knowndistance (parsecs)each rung is calibrated on the one belowan error at the bottom moves everything above it

eclipse-geometry

Umbra and penumbraThe shadow of a body lit by a source larger than itself. The umbra is a cone of finite length, computed from the two radii and the separation; the penumbra spreads outward and is the region that sees only part of the source.the umbra ends hereumbrapenumbranot to scale: the source is 400 times further away than drawnumbra length = D·r/(R − r) = 150 units for these radii

effective-potential

Effective potential along the line of centres, mass fraction 0.12The combined gravitational and centrifugal potential along the line joining two bodies in the frame that rotates with them. Its three stationary points are the collinear Lagrange points, and all three are maxima along this line.-1.5-1-0.500.511.5-2.5-2-1.5position along the line of centresprimarysecondaryL₁L₂L₃stationary, and unstable — every one of them

ellipse-orbit

An orbit at eccentricity 0.6An orbit of eccentricity 0.6. The primary sits at a focus, offset from the centre by 0.6 of the semi-major axis, and the closest and furthest points differ by a factor of 4.00.empty focusabae = 0.6arperiapsisapoapsis

equal-areas

Equal areas in equal times, at eccentricity 0.65Positions of an orbiting body at equal intervals of time, obtained by solving Kepler's equation. The two shaded sectors span the same interval and enclose the same area — a long thin one at the far end, a short fat one at the close approach.fast hereslow here16 equal intervals of time, one full orbit

escape-velocity

Circular and escape speedOrbital and escape speed against distance, in units of the circular speed at the surface. The escape curve is the circular one multiplied by the square root of two, at every distance without exception.2468101200.511.5distance, in body radiiescape speedcircular speedthe surfacea high orbitgeostationarythe gap is always a factor of √2

gravity-assist

A gravity assist with a 70° turnThe velocity triangle of a flyby. In the planet's frame the spacecraft's speed is unchanged and only its direction turns; adding the planet's own velocity converts that turn into a gain in speed measured from the Sun.the planet's velocityinout70°speed before: 0.505speed after: 0.661gained 0.157, and burned nothingthe circle: constant speed in the planet's frame

hohmann

A Hohmann transfer, 2.6 to 1 in radiusTwo circular orbits and the ellipse that touches both. The first burn raises the far point to the outer orbit; the second, half an orbit later, circularises. The costs are computed from the vis-viva relation.burn 1: +0.202burn 2: +0.158total 0.360 — and no way to spend lesscoast: half an ellipse, 1.21 of an inner-orbit yearspeeds in units of the inner circular speed

hr-diagram

The Hertzsprung–Russell diagramLuminosity against surface temperature, both in solar units and on logarithmic axes, with temperature increasing to the left. The main sequence is computed from the mass–luminosity and mass–radius relations; the dashed diagonals are lines of constant radius.R = 0.01R☉R = 0.1R☉R = R☉R = 10R☉R = 100R☉giantssupergiantswhite dwarfs1M☉3M☉20M☉40M☉the Sun10−410−21102104106surface temperature (K), increasing to the leftluminosity, in solar units

kepler-third

Period against size for the planets, around the SunOrbital period against semi-major axis on logarithmic axes. The line has slope exactly three-halves — the harmonic law — and the measured bodies sit on it.0.321.03.210320.100.321.03.21032100316semi-major axis (AU)MercuryVenusEarthMarsJupiterSaturnUranusNeptuneslope 3/2 — P² ∝ a³period (years)

lagrange-points

The five Lagrange points at mass fraction 0.12The five points at which a small body can keep station with two larger ones. The three on the line of centres are roots of a quintic and are unstable; the two forming equilateral triangles are stable for a sufficiently lopsided mass ratio.L₁L₂L₃L₄L₅L₄ and L₅ are exactlyequilateral with both bodiesthe collinear three areroots of a quintic

magnitude-scale

The magnitude scale, plottedThe logarithm of the received light against magnitude, with the vertical scale left unlabelled because only its slope matters. The relation is a straight line of slope −0.4, which is what makes five magnitudes exactly a hundredfold — and the numbers run backwards, so brighter is smaller.-20-100102030apparent magnitudethe Sunfull MoonVenus at its bestSiriusVega, by definitionthe naked-eye limita good amateur telescopea large ground-based surveythe deepest exposuresbrighter ←→ fainterfive magnitudes = ×100 in light

mass-luminosity

Luminosity against massMain-sequence luminosity against mass, both in solar units, on logarithmic axes. The slope is between three and four across most of the range, so a small spread in mass becomes an enormous spread in output.0.100.321.03.210320.011.0100100001000000mass (solar units)0.2M☉ → 0.01L☉1M☉ → 1L☉5M☉ → 391L☉20M☉ → 50,088L☉dashed: a pure slope of 3.5

orbit-speed

Speed against distance, for orbits of the same periodOrbital speed against distance from the primary. Every orbit with the same semi-major axis follows the same curve; the eccentricity decides only which stretch of it the body uses.00.511.50123distance from the primary (a = 1)e = 0e = 0.4e = 0.8circular speed at r = a

parallax

Parallax for a star at 1.3 parsecsThe same star observed from two ends of a baseline. The angle between the two sight lines is 0.77 arcseconds for a star 1.3 parsecs away — the definition of the parsec is the distance at which it would be exactly one.distant starsJanuaryJuly2 AU1.3 pc2p = 1.54″distance = 1 ÷ parallax in arcsecondsangles hugely exaggerated: the true angle here is a five-thousandth of a degree

phases

Phases are a viewing angle, not a shadowA satellite at eight points of its orbit. Exactly half of it is lit at every one of them; what changes is how much of the lit half faces the centre. Nothing is in shadow except at an eclipse.sunlightnewwaxing crescentfirst quarterwaxing gibbousfullwaning gibbouslast quarterwaning crescentas seen from the centrehalf lit, always

radial-velocity

The wobble of a star, companion at eccentricity 0.4The star's velocity along the line of sight, over two orbits, computed from the companion's orbit. A circular orbit gives a sine wave; an eccentric one gives a skewed curve whose shape encodes the eccentricity.00.511.52-101orbitstoward the observeraway from the observere = 0.4: skewed, and the skew is the measurement

stellar-lifetime

How long a star lasts, against its massMain-sequence lifetime against mass, on logarithmic axes. Fuel grows in proportion to mass and consumption grows as its three-and-a-half power, so the lifetime falls steeply — a star of thirty solar masses lives for a few million years.0.321.03.2103210^610^710^810^910^1010^1110^12mass (solar units)the age of the universe0.5M☉ — 80 Gyr1M☉ — 10 Gyr3M☉ — 458 Myr10M☉ — 23 Myr30M☉ — 1 Myrslope ≈ −2.5

sun-path

The Sun's altitude through the day at latitude 52°Solar altitude against the hour of the day, at one latitude, for the solstices and the equinox. Where a curve crosses zero is sunrise or sunset, and the width between the crossings is the length of the day.05101520-40-200204060hour (local solar time)June solstice — 61° at noon, 16.5 h of dayequinox — 38° at noon, 12.0 h of dayDecember solstice — 15° at noon, 7.5 h of daybelow the horizon

tidal-field

The tidal field is a differenceThe pull of a distant body at each point of a sphere, minus its pull at the sphere's centre. What remains stretches along the line to the source and squeezes across it — two bulges, not one.toward the sourcethe near side is pulled harder than the centrethe far side is pulled less — and so falls behindnot to scale: the source is far outside this frame

two-body-barycentre

Two bodies at a mass ratio of 3 to 1Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 3 to 1, so the heavier body's path is 3 times smaller.barycentrethe line joining them always passes through it