Luminosity Converter

Absolute magnitude M is the apparent magnitude a star would have at a standard distance of 10 parsecs. The relation to luminosity: M = M_sun − 2.5 × log₁₀(L/L_sun), where M_sun = 4.83 for the visual band. A star 100× more luminous than the Sun has M = 4.83 − 5 = −0.17. The distance modulus is μ = m − M = 5 × log₁₀(d/10pc), connecting apparent magnitude m, absolute magnitude M, and distance d.

Gamma ray bursts (GRBs) are the most luminous transient events in the universe. Typical peak luminosities are 10⁵¹ erg/s = 10⁴⁴ W ≈ 2.6 × 10¹⁷ L☉ — briefly outshining entire galaxies. A long GRB (duration >2 s) releases about 10⁵¹ erg total in gamma rays. This represents the complete gravitational collapse energy of a stellar core, mostly carried by neutrinos (10⁵³ erg) with a small fraction in gamma rays via relativistic jets.

The Jansky (1 Jy = 10⁻²⁶ W/m²/Hz) is the unit of spectral flux density — the power received per unit collecting area per unit frequency bandwidth. It is named after Karl Jansky, who discovered cosmic radio emission in 1932. Typical flux densities at radio wavelengths: Cassiopeia A (3C461) ≈ 3700 Jy at 100 MHz; Crab Nebula ≈ 1040 Jy at 100 MHz; faintest radio sources ~μJy. One Jy = 10⁻²⁶ W/m²/Hz = 10⁻²³ erg/s/cm²/Hz.

Luminosities relative to the Sun: Proxima Centauri 0.00155 L☉; Sun 1 L☉; Sirius 25 L☉; Rigel ~1.2 × 10⁵ L☉; Betelgeuse ~1.5 × 10⁵ L☉; R136a1 (most luminous known) ~8.7 × 10⁶ L☉. Main sequence luminosity scales approximately as L ∝ M⁴ (mass-luminosity relation) for stars of mass 0.5-5 M☉, steepening to L ∝ M² for more massive stars.

The Milky Way has total luminosity ~2 × 10¹⁰ L☉ (total, all wavelengths) with about 100-400 billion stars. The Andromeda Galaxy is slightly more luminous. Dwarf galaxies: 10⁵-10⁸ L☉. Giant ellipticals (cD galaxies in cluster centers): up to 10¹³ L☉. Quasars (active galactic nuclei): 10⁴⁶-10⁴⁸ erg/s = 10³⁹-10⁴¹ W = 10¹³-10¹⁵ L☉ — outshining their host galaxies by 100-fold.

Luminous efficacy (lm/W): incandescent bulb ~15 lm/W; halogen ~20 lm/W; fluorescent ~60-80 lm/W; white LED ~100-200 lm/W. A 100W incandescent provides ~1500 lm; a 10W LED also provides ~1500 lm but uses 10× less energy. The theoretical maximum for monochromatic 555 nm light is 683 lm/W; the white light maximum (under photopic vision) is about 347 lm/W for an optimal spectral distribution.

Apparent brightness (flux) is measured photometrically. Combined with the distance (from parallax, Cepheid period-luminosity relation, type Ia supernovae, or other standard candles), luminosity is computed via L = 4πd²F. Bolometric luminosity (all wavelengths) requires corrections for atmospheric absorption and the spectral energy distribution beyond optical bands. The bolometric correction varies from negligible for G stars to large for very hot (UV-bright) or cool (IR-bright) stars.

The Eddington luminosity L_Edd = 4πGMm_p c/σ_T ≈ 1.26 × 10³¹ W × (M/M☉) is the luminosity at which radiation pressure balances gravity for hydrogen plasma. For the Sun: L_Edd ≈ 3.28 × 10⁴ L☉ — the Sun is well below its Eddington limit. For accreting black holes and neutron stars, exceeding the Eddington limit drives strong outflows. Hyperluminous X-ray sources exceed the Eddington limit by factors of 10-100, possibly via anisotropic emission or photon trapping.

The CMB energy density is about 4.17 × 10⁻¹⁴ J/m³, corresponding to a blackbody at 2.725 K. The total power intercepted by a 1 m² sphere from all directions is about 3.14 × 10⁻⁶ W/m². This contributes about 3.13 × 10⁻⁶ W/m² to the cosmological energy density, or about 5 × 10⁻⁵ W per steradian per square metre of sky at the CMB peak frequency of ~160 GHz.