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Aperture photometry estimates brightness by summing calibrated detector pixels in an extraction region and subtracting the estimated background in that region. For $n$ equally weighted pixels, let $S$ be their total accumulated signal in Electrons, $b$ the estimated background Electrons per pixel over the same exposure, and $\Delta t$ the exposure duration. Then $\hat{r}=(S-nb)/\Delta t$ is a background-subtracted electron rate. Kepler applied aperture photometry to calibrated pixels with a fitted sky background. ^[ref].

This measurement trades source collection against background: enlarging the extraction region can retain more starlight but includes more sky brightness and neighbouring sources. Photon shot noise, read noise, and uncertainty in background subtraction contribute to its uncertainty. Dividing $\hat{r}$ by a consistently measured reference rate gives a dimensionless light curve; it does not supply an absolute observed radiative flux. If contaminating light remains, the reference includes it and the inferred exoplanet transit depth suffers transit dilution.

In the first toy figure, the source's reference rate is $100,000$ electrons per second, the summed sky contribution is $4,000$ electrons per second, and a known multiplicative response is $g(t)=1+0.0003t$ with $t$ in hours from mid-transit. The artificial exoplanet transit depth is $0.010$, its total duration four hours, and its flat-bottom duration $2.8$ hours; the straight ingress and egress are a toy approximation, not an exact exoplanet transit photometry model. Each synthetic sample averages a $0.5$-hour exposure and adds seeded normal distribution noise with relative standard deviation $0.0006$. Subtracting the known sky rate and dividing by the known response and reference rate recovers the dip. Real observations require estimating those corrections and carrying their uncertainty.

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.

Astrophysics uses physics to explain stars, planets, galaxies, and their electromagnetic radiation; cosmology extends the questions to the universe as a whole. Much of its evidence arrives as light. The challenge is to infer the source's physical properties from what reaches an observer: a bright star might be nearby, unusually luminous, or both. Stellar parallax and observed radiative flux make this distinction quantitative.

This is an inverse problem. A forward problem predicts observations from a source model and the intervening radiative transfer; the inverse problem works backwards from observations to possible models. A spectral line can constrain chemical elements, temperature, and motion through the Doppler effect, but its strength also depends on the stellar atmosphere. Combining different measurements is therefore more informative than interpreting brightness alone. The attraction is that a tiny change in the direction of light can supply a distance, turning received radiative flux into intrinsic luminosity. ^[ref].

The fractional decrease in observed radiative flux during an exoplanet transit, relative to the out-of-transit baseline. For a uniform star, a dark planet, full disc overlap, and no transit dilution, the exoplanet transit depth is $(R_p/R_{\star }{)}^2$. For a nonuniform star it depends on the blocked specific intensity as well as the radius ratio.

Exoplanet transit photometry measures the decrease in a star's observed radiative flux during an exoplanet transit. Define $F_0$ as the unobscured star's observed radiative flux and $F(t)$ as the value at time $t$. For a dark, opaque exoplanet crossing a uniformly bright circular stellar disc, a fully superimposed planetary disc blocks exactly the fractionHere $R_p$ and $R_{\star }$ are the planet and star radii. This exoplanet transit depth determines a radius ratio, $R_p/R_{\star }=\sqrt{\delta }$; an absolute planet radius requires an independently estimated stellar radius. For $R_p/R_{\star }=0.10$, $\delta =0.010$: a one-percent dip, with $F_{\mathrm{full}}/F_0=0.990$. Halving the radius ratio to $0.05$ reduces the exoplanet transit depth to $0.0025$, a quarter as much.

For the central exoplanet transit illustrated below, the transit impact parameter is $b=0$. Write $k=R_p/R_{\star }<1$ and let $z$ be the projected centre-to-centre separation in units of $R_{\star }$. The light curve is $F/F_0=1-A(z)/\pi$, where $A(z)$ is the overlap area of circles of radii $1$ and $k$. There is no overlap when $z\ge 1+k$; the planetary disc is fully inside when $z\le 1-k$, giving $A=\pi k^2$. Between those limits, the overlap area grows or shrinks continuously. Thus even a uniform star has curved ingress and egress, with a flat bottom only during full overlap. The illustration uses approximately constant transverse velocity over the short crossing, so $z=|ut|/R_{\star }$ with mid-transit at $t=0$. Analytic models also handle limb darkening. ^[ref].

The simple exoplanet transit depth formula assumes constant stellar brightness, no limb darkening or starspots, negligible light from the exoplanet, and no transit dilution by other light. It describes full overlap in a non-grazing exoplanet transit, requiring $b<1-k$ for a full-overlap interval. A grazing exoplanet transit never hides the whole planetary disc against the star, and limb darkening weights different blocked areas differently. Repeating dips are valuable evidence, but eclipsing binary stars and starspots can imitate parts of the signal, so confirmation and joint modelling matter.

One observed radiative flux cannot determine both luminosity and Euclidean distance: for every $k>0$, replacing $(L,d)$ by $(k^{2}L,kd)$ leaves $F$ unchanged. In particular, $4\times 10^{26}\mathrm{W}$ at $40$ parsecs produces the same observed radiative flux as $10^{26}\mathrm{W}$ at $20$ parsecs. This is a concrete failure of identifiability in an inverse problem.

An independent stellar parallax fixes the distance and breaks this degeneracy; alternatively, a calibrated standard candle supplies the luminosity. Interstellar extinction can further reduce the observed radiative flux, and a photometric passband measures only a selected spectral range. At cosmological distances, use luminosity distance in the bolometric relation. Each inference must specify which radiative flux was measured and what propagation assumptions connect it to luminosity.

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.

For a circular stellar disk whose specific intensity is independent of azimuth, its mean specific intensity over projected area isThe polar coordinates element is $d{A}_{\mathrm{proj}}=2\pi rdr$ after angular integration, and $rdr=-R_{\star }^2\mu d\mu$. Substitution of the linear limb-darkening law givesFor a distant observer at distance $D\gg R_{\star }$ receiving light normally to the detecting plane, the observed radiative flux isIt is spectral irradiance, with units $\mathrm{W}{\mathrm{m}}^{-2}\mathrm{H}{\mathrm{z}}^{-1}$ when $I_{\nu }$ is per hertz. The local outward radiative flux per actual surface area is instead $F_{\nu }=2\pi {\int }_0^1{I}_{\nu }(\mu )\mu d\mu =\pi {\overline{I}}_{\nu }$, a spectral radiant exitance. In a spherical model with the same stellar atmosphere everywhere, the luminosity is $L_{\nu }=4\pi R_{\star }^2{F}_{\nu }$ and $f_{\nu }=L_{\nu }/(4\pi D^2)$, consistent with the inverse-square law. Compare disks at fixed $I_c$ or fixed total observed radiative flux explicitly: increasing $u$ lowers the total in the first comparison, while the second requires a larger $I_c$.

Luminosity is emitted radiative power: the energy carried away by electromagnetic radiation per unit time, measured in watts. Bolometric luminosity includes all wavelengths; a measurement through one photometric passband captures only part of it. Spectral luminosity records how that power is distributed over frequency. Received observed radiative flux has different units, watts per square metre, because it measures what crosses an observer's collecting area. ^[ref].

For example, an ideal source with luminosity $10^{26}\mathrm{W}$ emits $10^{26}\mathrm{J}$ each second regardless of whether it is viewed from $20$ or $40$ parsecs. Its observed radiative flux changes with distance, while its luminosity remains fixed. Combining stellar parallax with observed radiative flux therefore lets us compare the actual outputs of stars rather than their apparent brightnesses.

Methods of detecting exoplanets infer or resolve an exoplanet through its effects on the host star's light or through its own light. The observable determines what can be learned:

An exoplanet secondary eclipse measures the loss of the exoplanet's light when it passes behind its star, complementing exoplanet transit photometry. A detection method also selects which systems are easiest to find: deeper exoplanet transits favour larger radius ratios, and larger stellar radial-velocity semi-amplitudes favour more massive companions, shorter orbital periods, and larger $\sin i$. These preferences must be accounted for when inferring the underlying exoplanet population.

For a steady point-like astronomical source with isotropy of emission, in transparent space with negligible relativistic or cosmological effects, conservation of energy spreads its luminosity over a sphere of surface area $4\pi d^2$. Its bolometric observed radiative flux isThis inverse-square law reflects geometric dilution: the same emitted power crosses increasingly large spheres. It does not require light to lose energy on the journey. For a fixed luminosity, doubling Euclidean distance quarters the observed radiative flux. ^[ref].

With $L_0=10^{26}\mathrm{W}$ and $d_0=20$ parsecs, the result is $F_0\simeq 2.09\times 10^{-11}\mathrm{W}{\mathrm{m}}^{-2}$. At $40$ parsecs, the same source gives $F_0/4\simeq 5.22\times 10^{-12}\mathrm{W}{\mathrm{m}}^{-2}$. An ideal collecting area of $1{\mathrm{m}}^2$ at $20$ parsecs intercepts approximately $2.09\times 10^{-11}\mathrm{W}$, before detector losses.

Radiative flux is the first angular moment $F_{\nu }=\int I_{\nu }\cos \theta d\Omega$ of specific intensity and measures net radiant power crossing unit area. Integrating over frequency gives bolometric radiative flux. The receiving surface's orientation matters through $\cos \theta$; observed radiative flux from a compact distant source conventionally refers to an area perpendicular to its line of sight.

Constant extra light makes an exoplanet transit appear shallower. If the host star supplies out-of-transit observed radiative flux $F_{\star }$ and an unresolved source supplies $F_c$, let $D=F_c/F_{\star }$. Then ${\delta }_{\mathrm{obs}}={\delta }_{\mathrm{host}}/(1+D)$. For $D=1$, a true exoplanet transit depth of $0.010$ appears as $0.005$; ignoring transit dilution gives a radius ratio $\sqrt{0.005}\simeq 0.07071$ instead of $0.10$, under the same uniform-disc assumptions.