Absolute magnitude ($M$)

Absolute magnitude $M$, in the stellar convention used here, is the apparent magnitude a source would have at $10$ parsecs, with interstellar extinction removed. The hypothetical observation uses the specified photometric passband and magnitude zero point, preserving the source's emitted spectrum and luminosity. Moving every source to this common distance lets us compare intrinsic brightness without confusing it with distance. ^[ref].

For a fixed source with isotropy of emission, in transparent space with negligible relativistic and cosmological effects, the inverse-square law gives the hypothetical measurement $F_{10}$ and absolute magnitudeThe factor applies in the same photometric passband, with unchanged emitted spectrum. Thus absolute magnitude compares luminosity in that response, rather than automatically the total bolometric luminosity over all wavelengths.

Consider a toy source with $M=0$. Its apparent magnitude is $m=0$ at $10$ parsecs, $m=5$ at $100$ parsecs, and $m=10$ at $1000$ parsecs, neglecting interstellar extinction. Its absolute magnitude remains $M=0$ throughout: moving the observer changes received radiative flux, not emitted luminosity. Another toy source with $M=-5$ has $100$ times its passband output under the same isotropy and calibration assumptions. Negative absolute magnitudes are allowed too; "absolute" names the reference-distance convention, not an absolute value.

A calibrated standard candle reverses this reasoning: its known absolute magnitude and measured apparent magnitude give a distance modulus. This is why estimating intrinsic brightness is so useful for measuring astronomical distances. ^[ref].

The K-correction converts an observed photometric passband to a specified emitted photometric passband, accounting for cosmological redshift and the source spectrum. For a narrow same-band measurement, $K_R=-2.5{\log }_{10}[(1+z)L_{\nu }((1+z){\nu }_R)/L_{\nu }({\nu }_R)]$. Thus its sign depends on the spectrum, not on redshift alone. For $L_{\nu }\propto {\nu }^{\alpha }$, it is $-2.5(1+\alpha ){\log }_{10}(1+z)$. Finite-band corrections use the actual detector response and magnitude zero point.

For emitted spectral power per steradian $P_{\nu }\propto {\nu }^{-\alpha }$, the luminosity distance gives $S_{{\nu }_o}=(1+z)P_{(1+z){\nu }_o}/d_L^2$. Photon energy and arrival-rate redshifts supply two factors of $1+z$, while the observed frequency interval is reduced by one. Substitution of the power-law spectrum gives the displayed K correction. This convention uses spectral flux per unit frequency and power per steradian; using per unit wavelength or total spectral luminosity changes the corresponding factors.

A standard candle is an astronomical source whose calibrated absolute magnitude or luminosity allows its apparent magnitude or flux to determine a distance. A population with unknown common absolute magnitude only determines relative distances without an external luminosity calibration. Type Ia supernova cosmology models this distinction explicitly.

A standardizable candle has a predictable absolute magnitude after correcting measured source properties, rather than identical intrinsic brightness for every object. Type Ia supernova cosmology uses light curve shape and colour for this purpose.

A flux-limited sample preferentially includes intrinsically brighter objects at a given distance. Calibrating standard candles from that selected sample can bias inferred distances.

For a uniform Euclidean population, a complete flux-limited sample weights luminosity by accessible volume $L^{3/2}$. Since $L\propto 10^{-0.4M}$, an intrinsic Gaussian magnitude distribution is multiplied by $e^{-0.6\ln (10)M}$. Completing the square shifts its mean by the displayed amount without changing its variance. Extra distance bounds, extinction, evolution and other selection cuts alter this simple result.

A noise-free flux threshold truncates the distance distribution of objects with exactly identical relevant-band luminosity. It cannot preferentially select intrinsically brighter members when no luminosity scatter exists. A luminosity distribution introduces the accessible-volume weighting $L^{3/2}$ in homogeneous Euclidean space. Measurement scatter across the flux threshold and nonuniform completeness introduce additional selection effects even for identical intrinsic luminosities.

For a flux threshold $f_{\min }$ and isotropic point-source luminosity $L$, the detection radius is $r_{\max }=\sqrt{L/(4\pi f_{\min })}$. In a homogeneous Euclidean population the accessible volume is proportional to $L^{3/2}$, so the selected luminosity distribution is proportional to $\phi (L)L^{3/2}$. This assumes flux selection and inverse-square propagation, not an arbitrary surface-brightness-selected diffuse field.