Stars & the Cosmos

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

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.

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

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.

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 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 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 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 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 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 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 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.

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.