Scientific Calculators & Simulation Platform

A star's intrinsic brightness — the apparent magnitude it would have at a standard distance of 10 parsecs. Removes distance so stars can be compared on equal footing.

How large an object of known size appears at a given distance. The Moon and the Sun span almost exactly the same half-degree from Earth — the coincidence that makes total solar eclipses possible.

The finest detail a telescope can resolve, set by diffraction at its aperture — the Rayleigh criterion. Bigger apertures see finer detail; a 100 mm telescope resolves about 1.4 arcseconds in green light.

The angle on the sky between two positions given by their right ascension and declination — the great-circle distance between two points on the celestial sphere.

The power radiated per square metre by a blackbody at a given temperature. The Sun's photosphere emits about 63 megawatts per square metre.

The speed of a body on a circular orbit at a given distance from a central mass. The Earth circles the Sun at about 29.8 km/s.

The difference between apparent and absolute magnitude, which encodes distance. A modulus of five corresponds to 100 parsecs; each five magnitudes multiply the distance tenfold.

The orbital distance at which a planet receives the same starlight per square metre as Earth does from the Sun — a first anchor for the habitable zone. It scales with the square root of the star's luminosity.

The minimum speed an object needs to break free of a body's gravity, ignoring drag. Set by the body's mass and radius alone.

The angular extent of sky a camera frames, from the sensor size and focal length. A shorter focal length or a larger sensor takes in more sky.

The radius within which a body's gravity dominates over the larger body it orbits — the region where its moons can hold. Earth's Hill sphere reaches about 1.5 million kilometres.

The distance to a galaxy from its recession velocity and the Hubble constant, by Hubble's law. Because the measured value of H₀ is itself contested — the Hubble tension — it is left as an input rather than fixed.

How much sky each camera pixel covers, from the pixel size and focal length. Matching the image scale to the seeing (roughly 1–2″ per pixel) is the key to sharp astrophotography.

A rule-of-thumb estimate of the faintest star an aperture can show under dark skies. Approximate — real limits depend on sky brightness, magnification, and the observer — so it is offered as a guide, not a guarantee.

Roughly how long a star burns hydrogen in its core, scaled from the Sun's ~10 billion years. Massive stars are prodigal — a two-solar-mass star lasts under two billion years — while red dwarfs last far longer than the present age of the universe.

The steep relation between a main-sequence star's mass and its luminosity: a star twice the Sun's mass shines roughly eleven times as bright. An approximation (exponent ~3.5) valid across the middle main sequence.

The average density of a body from its mass and radius — a clue to its composition. Earth's ~5510 kg/m³ points to a rock-and-iron world; the giant planets are far less dense.

Kepler's third law: the time to complete one orbit, from the semi-major axis and the central mass. A planet at 1 AU around the Sun takes one year.

The most direct distance measurement: a star's distance in parsecs is the reciprocal of its annual parallax in arcseconds. A parallax of one arcsecond defines one parsec — but no star is that close.

The best signal-to-noise achievable when the only noise is the Poisson statistics of the photons themselves — the shot-noise limit. To double the signal-to-noise you must collect four times as many photons.

The temperature a planet settles at from the balance of starlight absorbed and heat radiated, before any greenhouse warming. Earth's is about 255 K (−18 °C); its atmosphere lifts the surface to habitable warmth.

The recession velocity implied by a small cosmological redshift, v ≈ cz. This linear form holds only for low redshift; at large z the full relativistic and cosmological treatment is required and this approximation overstates the speed.

The distance within which a fluid satellite held together only by gravity is pulled apart by tides. Inside the Earth–Moon fluid Roche limit — about 18,000 km — a Moon-like body could not survive.

The radius of the event horizon of a non-rotating black hole of a given mass — the size to which that mass must be compressed to become one. The Sun's is just under three kilometres.

A star's total power output, from its radius and surface temperature by the Stefan–Boltzmann law. Doubling the temperature raises luminosity sixteenfold.

The gravitational acceleration at a body's surface, from its mass and radius. Earth's is about 9.8 m/s².

How often two orbiting bodies return to the same relative alignment — the interval between successive oppositions, say. Earth and Mars line up about every 2.14 years.

The magnification of a telescope–eyepiece pair — the ratio of their focal lengths. Useful magnification is capped by the aperture and the atmosphere, not by the eyepiece alone.

The geometric chance that a planet's orbit is aligned edge-on enough for it to transit its star as seen from Earth. Only about one in 215 for an Earth-like orbit — which is why transit surveys must watch so many stars at once.

The wavelength at which a blackbody radiates most intensely, by Wien's displacement law. The Sun peaks in green light at about 502 nm; hotter stars peak bluer, cooler stars redder.