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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].

In Type Ia supernova cosmology, the predicted apparent magnitude depends on the common absolute magnitude $M_0$ and the Hubble constant only through $\beta =M_0-5{\log }_{10}(H_0/H_{\mathrm{ref}})$, after choosing a fixed reference $H_{\mathrm{ref}}>0$. A flat improper prior on $M_0$ makes the likelihood integrated over $M_0$ independent of $H_0$, by translation of the integration variable. If the other prior factors are independent and the posterior is proper, the marginal posterior of $H_0$ equals its prior. An external calibration of $M_0$ can break this lack of identifiability.

Apparent magnitude $m$ is a calibrated description of the observed radiative flux received from an astronomical source. It answers how bright the source appears in a specified photometric passband, rather than how much luminosity the source emits. Define $F_{\mathrm{ref}}>0$ as the measurement assigned zero apparent magnitude by the chosen magnitude zero point. ThenHere $F$ is the positive passband measurement, with the same units and response convention as $F_{\mathrm{ref}}$. For an energy-weighted detector this measures received radiative flux through its response; for a photon-counting detector use the corresponding calibrated count rate for both quantities. Actual photometric passbands include the instrument's response across wavelength, so naming the photometric system is part of reporting an apparent magnitude. ^[ref]. ^[ref].

As toy examples in one photometric passband, $F=F_{\mathrm{ref}}$ gives $m=0$, $F=100F_{\mathrm{ref}}$ gives $m=-5$, and $F=0.01F_{\mathrm{ref}}$ gives $m=+5$. A negative apparent magnitude therefore means more received radiative flux than the reference; it is physically allowed. Zero apparent magnitude does not mean zero radiative flux. The ordinary logarithm formula requires $F>0$.

A small apparent magnitude alone cannot distinguish a nearby faint source from a distant source with large luminosity: this is the flux-distance degeneracy. Stellar parallax supplies independent distance information, while a calibrated standard candle supplies intrinsic brightness. Interstellar extinction also changes the received radiative flux and hence the measured apparent magnitude.

Astronomical magnitude expresses brightness on a logarithmic scale: equal steps in astronomical magnitude represent equal ratios of received radiative flux. The scale runs backwards, so a brighter source has a smaller astronomical magnitude. A difference of five magnitudes means exactly a factor of $100$ in flux; one magnitude means a factor of $100^{1/5}\simeq 2.511886$, rather than exactly $2.5$. This compresses the enormous range of astronomical brightnesses into manageable numbers. ^[ref].

Apparent magnitude describes received brightness, while absolute magnitude compares sources at a shared reference distance. Both require a specified photometric passband and magnitude zero point. A magnitude difference turns a ratio of positive radiative fluxes into a difference of numbers; it does not measure an additive change in emitted luminosity.

The pulsation period of a Cepheid variable is correlated with its absolute magnitude in a specified photometric band. Calibrating the relation with stars of known distance allows an observed period and corrected apparent magnitude to give a distance. Extinction and metallicity corrections matter for a precise cosmic distance ladder.

A Cepheid variable is a pulsating star whose period is tightly related to its luminosity, allowing its distance to be inferred from its measured period and apparent magnitude.

A colour index is the difference of an object's measured apparent magnitudes in two photometric passbands, such as $B-V$. Since magnitude is logarithmic, this difference measures a flux ratio after the respective magnitude zero points are accounted for. It describes spectral shape rather than the object's overall brightness, and changes with both intrinsic radiation properties and interstellar extinction.

The distance modulus compares received brightness with intrinsic brightness through $\mu =m-M$, the difference between apparent magnitude and absolute magnitude in the same photometric passband and magnitude zero point. With no interstellar extinction, fixed emitted spectrum and luminosity, isotropy of emission, and the local inverse-square law, it determines Euclidean distance:The ratio inside the logarithm is dimensionless. To see the factor $5$, use the inverse-square law in the magnitude difference formula:For example, $m=5$ and $M=0$ give $\mu =5$, hence $d=10\mathrm{pc}\times 10=100\mathrm{pc}$. A tenfold increase in distance adds five magnitudes; an increase of one magnitude multiplies distance by $10^{1/5}\simeq 1.584893$. The distance modulus is a logarithmic scale for distance, not a distance in length units. ^[ref].

At $10$ parsecs, $\mu =0$ and $m=M$. Within $10$ parsecs, the distance modulus is negative; for example, $\mu =-5$ gives the positive distance $d=1\mathrm{pc}$. A negative distance modulus does not imply negative distance. At cosmological distances, the inference needs luminosity distance and a K correction; the simple local formula must not silently absorb these effects.

Interstellar extinction increases apparent magnitude. The observed distance modulus then includes both geometric dimming and attenuation; extinction bias in photometric distance explains how to separate them.

Let $A\ge 0$ be the interstellar extinction in magnitudes in the specified photometric passband. Compared with the same unobscured source at the same distance, its radiative flux is multiplied by $10^{-0.4A}$, so its apparent magnitude increases by $A$. Consequently,Here ${\mu }_0$ is the extinction-corrected distance modulus. The correction subtracts $A$ before using the distance formula. Caltech's NED distance-method documentation explicitly distinguishes an apparent distance modulus from an extinction-corrected distance modulus. ^[ref].

Ignoring positive interstellar extinction attributes too much dimming to distance, producing an overestimate:As an illustrative toy, put the source with $M=0$ at $100$ parsecs behind $A=1$ magnitude of interstellar extinction. The measured apparent magnitude is $m=6$. Setting $A=0$ incorrectly gives $d_{\mathrm{inferred}}=10\mathrm{pc}\times 10^{6/5}\simeq 158.489\mathrm{pc}$, a factor $1.584893$ too large. Subtracting the known $A=1$ recovers $100$ parsecs. The diagram's fixed $A=1$ offset is a controlled illustration, not a model of how actual interstellar extinction grows with distance, and none of its values are star catalogue measurements.

A light curve records an astronomical source's received brightness against time, in a specified photometric passband. Its measured form is a time series of timestamps, brightness estimates, and uncertainties; a smooth $F(t)$ is a model of that record. Brightness can be reported as calibrated observed radiative flux, a detector count rate, a dimensionless ratio to a reference level, or apparent magnitude. These quantities require different units and calibration conventions. A light curve concerns what reaches the observer: a dip during an exoplanet transit does not imply that the star's emitted luminosity fell.

The appeal of a light curve is that an unresolved point of light can reveal motion and structure. In exoplanet transit photometry, repeated dips constrain the orbital period; under the uniform-disc, dark-planet, full-overlap assumptions, the exoplanet transit depth is the blocked area fraction $(R_p/R_{\star }{)}^2$. A toy one-percent dip therefore corresponds to a radius ratio of $0.10$. Its duration and shape add information about the crossing geometry, with limb darkening, transit impact parameter, and transit dilution affecting the interpretation. See exoplanet transit photometry for the geometry. The first figure instead follows the measurement: an instrumental trend and sky contribution can obscure a small physical dip.

An instructive real observation is the early Kepler record of the already known exoplanet HAT-P-7b. Welsh and colleagues analysed 33.5 days from 13 May to 15 June 2009, containing 15 complete exoplanet transits at an orbital period of about $2.20473$ days. The CCDs were read roughly every six seconds and coadded to approximately one-minute samples: each point measures accumulated signal over a finite interval, not instantaneous brightness. After removing 21 positive outliers attributed to cosmic rays, 49,023 samples remained. The authors estimated a scatter of about $150$ parts per million per one-minute sample. A gradual $0.13\%$ count decrease, suspected to arise from focus drift with a fixed extraction aperture, was removed with a cubic polynomial fitted while masking the exoplanet transits. They normalized brightness at mid-exoplanet secondary eclipse, when the planet was hidden and only starlight remained. ^[ref].

What does the instrument actually measure? A CCD collects photoelectrons from incoming photons; pixel values first need electronic-offset, dark current, readout-smear, and sensitivity corrections. ^[ref]. Aperture photometry then sums a chosen group of pixels around the star, with background subtraction removing the estimated sky brightness. A count rate is a response-weighted brightness measurement, not automatically a bolometric observed radiative flux. Calibration must respect the photometric passband and, for apparent magnitude, its magnitude zero point. Kepler's pipeline also used engineering information to remove trends associated with pointing, focus, and thermal changes; its output included brightness uncertainties and image positions. ^[ref].

Calibration and interpretation need separate checks. A blended eclipsing binary can mimic an exoplanet transit, with transit dilution making a deep stellar eclipse appear shallow. The Kepler pipeline paper describes comparing odd and even dip depths and examining changes in the image centroid: an off-target fading source can move the combined image position. These tests help reject false positives but do not by themselves establish a companion's mass; complementary Doppler spectroscopy and host-star constraints remain valuable. ^[ref].

Even a real exoplanet's signal can have several contributors. Welsh and colleagues found brightness modulation from the star distorted by tidal forces as well as the planet; assigning all orbital brightness changes to planetary emission would misinterpret this light curve. ^[ref]. Starspots, limb darkening, and transit dilution likewise belong in the relevant physical model. A flexible trend fitted through a dip could absorb that dip; the masked fit in the real example illustrates why the fitting procedure is part of the measurement. Exposure-averaged light curves and phase folding of a light curve make two further limitations explicit.