The math behind the wonder.
Useful astronomy relationships, paired with an experiment so you can see what the symbols mean.
Orbit simulator
In astronomical units, solar masses, and Earth years, Kepler’s third law connects the size of an orbit to its period. Increasing eccentricity stretches the orbit without changing its period when the semi-major axis stays fixed.
- Kepler’s third law: P² = a³ / M★ — P in years, a in AU, M★ in solar masses
- Closest and farthest distance: r_min = a(1 − e) · r_max = a(1 + e) — Perihelion and aphelion
- Orbital speed (vis-viva): v = √(GM (2/r − 1/a)) — Fastest at closest approach
Moon phase explorer
The Sun lights half of the Moon. As the Moon orbits Earth, our viewing angle changes, revealing different fractions of that sunlit half. The repeating phase cycle is a synodic month.
- Illuminated fraction: k = (1 − cos(2π × age / 29.53)) / 2 — age in days since new Moon
- Synodic month (phase cycle): 29.53 days — New Moon to new Moon
- Sidereal month (one orbit): 27.32 days — Measured against the stars
Gravity & escape velocity
A planet’s mass increases its pull, while a larger radius puts its surface farther from the center. Your mass stays the same on another world; your weight, a force measured in newtons, changes.
- Surface gravity: g = GM / R² — M = mass, R = radius
- Scaling from Earth: g ≈ 9.82 m/s² × (M / M⊕) ÷ (R / R⊕)² — Masses and radii in Earth units
- Weight: W = m × g — Newtons; your mass m stays the same
- Escape velocity: vₑ = √(2GM / R) — Earth ≈ 11.2 km/s
Light travel time
Even light has a finite speed. A light-year is the distance light travels in a Julian year, not a unit of time. Light travel makes remote observations a view of the past.
- Travel time: t = d / c — c = 299,792.458 km/s
- One astronomical unit: 1 AU ≈ 499 light-seconds — About 8 min 19 s
- One light-year: 1 ly ≈ 9.46 × 10¹² km ≈ 63,241 AU — A distance, not a time
Star color & luminosity
Temperature strongly affects emitted power per unit area. A large, cool star can be more luminous than a small, hot star because it has more radiating surface. Wien’s law relates temperature to the wavelength of peak emission.
- Luminosity (Stefan–Boltzmann): L / L☉ = (R / R☉)² × (T / 5772 K)⁴ — Radius and temperature relative to the Sun
- Wien’s displacement law: λ_peak = 2,897,772 nm·K ÷ T — Hotter stars peak at shorter wavelengths
Telescope calculator
Magnification depends on both telescope and eyepiece. Exit pupil describes the diameter of the light beam leaving the eyepiece. A wider true field lets you see more of the sky at once.
- Magnification: M = telescope focal length ÷ eyepiece focal length — 1000 mm ÷ 25 mm = 40×
- Exit pupil: exit pupil = aperture ÷ M — 100 mm ÷ 40 = 2.5 mm
- True field of view: TFOV ≈ apparent field ÷ M — 60° ÷ 40 = 1.5°
- Dawes’ limit: resolution ≈ 116 ÷ aperture (mm) arcseconds — 116 ÷ 100 mm = 1.16″
- Maximum useful magnification: ≈ 2 × aperture in mm (50× per inch) — 100 mm → about 200×
- Focal ratio: f-number = focal length ÷ aperture — 1000 ÷ 100 = f/10
Exoplanet transit
A transiting planet hides a fraction of its star’s visible disk. Larger planets block more light. The light curve plots the remaining brightness as the planet moves across the star.
- Transit depth: δ ≈ (Rp / R★)² — Planet fully in front of a uniform star
- Planet radius from depth: Rp ≈ R★ × √δ — Invert a measured dip
- Earth–Sun radius ratio: R⊕ / R☉ ≈ 0.00916 — So δ ≈ 0.0084%
Stellar parallax
As Earth moves around the Sun, a nearby star appears to shift against distant background stars. The annual parallax is half of the total angular shift between opposite points in Earth’s orbit.
- Parallax distance: d (parsecs) = 1 ÷ p (arcseconds) — p is the annual parallax
- Parsec in light-years: 1 pc ≈ 3.26 ly
- Parsec in AU: 1 pc ≈ 206,265 AU
Black hole radius
The event horizon is a boundary in spacetime, not a material surface. For a non-rotating, uncharged black hole, its radius grows linearly with mass.
- Schwarzschild radius: rₛ = 2GM / c² — G = 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻², c = 299,792,458 m/s
- Quick estimate: rₛ ≈ 2.95 km × (M / M☉) — Grows in proportion to mass
- Event horizon diameter: D = 2rₛ — Black hole “size”
Stellar brightness
The magnitude scale runs backward: smaller numbers mean brighter objects. It is logarithmic, so a difference of five magnitudes corresponds to a factor of one hundred in brightness.
- Brightness ratio: F_A / F_B = 10^(0.4 × (m_B − m_A)) — Same photometric band
- Magnitude difference: m_B − m_A = 2.5 log₁₀(F_A / F_B)
- One magnitude: ≈ 2.512× in brightness — The fifth root of 100
Constants and units
| Quantity | Value used |
|---|---|
| Speed of light, c | 299,792.458 km/s (exact) |
| Astronomical unit, AU | 149,597,870.7 km (exact) |
| Julian year | 365.25 days |
| Gravitational constant, G | 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² |
| Solar mass | 1.98847 × 10³⁰ kg |
| Solar radius | 695,700 km |
| Mean Earth radius | 6,371 km |
| Earth mass | 5.9722 × 10²⁴ kg |
These educational models use rounded physical parameters. Reference planet tables use equatorial diameters; the gravity model uses mean Earth radius.