Photometry and Signal-to-Noise

Photometry is the operational bridge between the abstract quantities of luminosity and radiant flux and the discrete electron counts a real detector delivers. Every measured brightness — a stellar magnitude, a galaxy flux, a supernova light curve — begins as photons striking a charge-coupled device (CCD), being converted to photoelectrons, digitized, and stored as an image of pixel values. The task of photometry is to run this chain backwards: to recover, with a stated uncertainty, the flux of an astronomical source from a noisy, background-contaminated frame. This article is where the magnitude system, filters, and flux meet the hardware — it takes Pogson's relation $m_2-m_1=-2.5\log_{10}(F_2/F_1)$ from the preceding articles and asks what it takes to actually measure $F$ on a real night, and how well. Two ideas organize everything that follows. The first is technique: aperture photometry sums the counts inside a synthetic circle and subtracts a local sky level, while point-spread-function fitting models the instrumental profile of a point source and fits scaled copies of it — robust simplicity versus optimal, deblending precision. The second, and the heart of the article, is noise. Photons arrive as a Poisson process, so a source of $N$ electrons carries shot noise $\sqrt{N}$ ; sky, dark current, and read noise add further variance. Combining them in quadrature gives the CCD equation, which turns into a single number, the signal-to-noise ratio, and hence the magnitude error via $\sigma_m\approx1.0857/\mathrm{SNR}$ . Understanding how SNR grows — as $\sqrt{t}$ when background-limited but linearly when read-limited — is what lets an observer plan an exposure, and what ultimately underwrites every rung of the cosmic distance ladder.

Key takeaways

Photometry converts detector counts into calibrated source flux; aperture photometry sums counts in a circle and subtracts a sky annulus, while PSF fitting models the point-source profile for crowded fields and faint detections.
The precision is set by a noise budget: Poisson shot noise from source and sky photons, plus dark current and read noise, combined in quadrature as the CCD equation.
Signal-to-noise scales as $\mathrm{SNR}\propto\sqrt{t}$ in the source- or sky-limited regime and $\propto t$ when read-noise-limited, which is why long single exposures beat many short ones for faint work.
A magnitude error follows from SNR by $\sigma_m\approx1.0857/\mathrm{SNR}$ , so $\mathrm{SNR}=100$ gives about $0.011$ mag and going one magnitude deeper at fixed SNR costs $2.512^2\approx6.3$ times the exposure.
Instrumental magnitudes are meaningless until tied to a standard system through a zero point, an airmass-dependent extinction term, and a color term.
Key idea: a measured flux is only as good as its noise budget — counts must be converted to electrons, corrected for background, and assigned a Poisson-plus-detector variance through the CCD equation, so that SNR directly fixes the magnitude error via $\sigma_m\approx1.0857/\mathrm{SNR}$ .

The core insight

A measured flux is only as good as its noise budget: recorded counts must be converted to electrons, corrected for background, and assigned a Poisson-plus-detector uncertainty through the CCD equation, so that the signal-to-noise ratio directly determines the magnitude error via $\sigma_m\approx1.0857/\mathrm{SNR}$ .

Where this lives today

The CCD equation is the arithmetic behind every modern survey's exposure-time calculator. The Vera C. Rubin Observatory's Legacy Survey of Space and Time plans a decade of $15$ -second pairs across the southern sky, and it is precisely the read-noise-versus-sky crossover that dictates that cadence — short enough to freeze the sky, long enough that the co-added stacks reach beyond magnitude $27$ by accumulating $\sqrt{t}$ depth over a thousand visits. Gaia has delivered calibrated photometry for nearly two billion stars, its milli-magnitude precision resting on the same $\sigma_m=1.0857/\mathrm{SNR}$ conversion applied to exquisitely characterized detectors. The James Webb Space Telescope pushes faint-source photometry toward AB magnitude $30$ by beating down the background from space and integrating for hours in the source- and read-limited regime where the ground cannot follow. And time-domain astronomy — difference-imaging pipelines that discover supernovae and calibrate Type Ia standard candles, crowded-field PSF fitting that yields cluster color-magnitude diagrams — lives or dies by an honest noise budget. Get the variance wrong and the error bars lie; get it right and your measured flux becomes a rung on the distance ladder.

The physical picture

Think of a photometric measurement as counting raindrops in a bucket during a storm. The source is the rain you care about; the sky background is a steady drizzle falling into every bucket; the read noise is the imprecision of reading the water level off the bucket's scale. To find the source rate you must subtract the drizzle and account for both the randomness of how many drops actually fell — Poisson noise — and the finite resolution of your ruler. A CCD frame is a grid of such buckets, and photometry is the arithmetic that turns the filled grid back into a flux with an error bar. The first figure fixes the geometry and the arithmetic together. A point source lands on the detector as a compact profile — the point-spread function — and aperture photometry draws a circle around it, sums the pixels inside, and estimates the sky from a surrounding annulus of blank pixels. The number that matters is not the sum but its uncertainty, which the noise budget lays bare: for a realistic faint star the background fluctuations swamp the source's own shot noise, so the measurement is "sky-limited" and every trick — a darker site, a larger aperture only up to a point, a tighter PSF — is really a fight against the sky.

Figure

The photometric aperture and sky annulus on a stellar profile, with the worked-example noise budget in electrons: source √30000 ≈ 173, sky √120000 ≈ 346, read √1280 ≈ 36, adding in quadrature to 389 e so the background term dominates and the measurement is sky-limited

The second idea is how that budget evolves as you integrate. Source, sky, and dark counts all grow linearly with exposure time, but read noise is levied once per readout and never grows. So a very short exposure is read-noise-limited and its SNR climbs linearly with time; keep integrating and the accumulating photon shot noise takes over, at which point SNR climbs only as the square root of time. The crossover between these two slopes is the single most important thing an observer plans around: it says how long an exposure must be before the detector stops throwing away signal, and why a survey stacks long sub-exposures rather than many snapshots.

Figure

Signal-to-noise versus exposure time for a star giving 100 e/s over 20 pixels with sky 20 e/s per pixel and read noise 8 e per pixel: below the crossover at t ≈ 2.6 s read noise dominates and SNR ∝ t, above it the background dominates and SNR ∝ √t, passing through SNR ≈ 77 at t = 300 s

The choice between aperture and PSF photometry is a trade between simplicity and information. An aperture throws away the known shape of a point source and weights every pixel equally; PSF fitting exploits the profile, weighting each pixel by how much signal it should contain, which is statistically optimal in the low-SNR and crowded regimes but demands an accurate PSF model. Calibration, finally, is a change of coordinates: instrumental magnitudes live in the detector's own idiosyncratic frame, and zero points, extinction, and color terms rotate and shift them onto the universal standard system so that a magnitude from one telescope on one night can be compared with any other.

The core equations

Photometry starts from the definition of magnitude in terms of flux. Pogson's relation encodes the historical convention that five magnitudes are exactly a factor of one hundred in flux, and its inverse recovers flux from a calibrated magnitude.

m_2-m_1=-2.5\log_{10}\!\left(\frac{F_2}{F_1}\right),\qquad F=F_0\,10^{-0.4\,m} \qquad (1)

where: $m_1,m_2$ — apparent magnitudes; $F_1,F_2$ — corresponding fluxes; $F_0$ — reference flux setting the zero point of the system. Reading: a difference of magnitudes is a logarithmic ratio of fluxes, so the whole enterprise reduces to measuring flux ratios accurately. Measuring that flux begins with a background-subtracted sum of pixel counts inside the chosen aperture. The sky level is estimated from a surrounding annulus and removed pixel by pixel.

C_\star = \sum_{i\in\mathrm{ap}}\left(I_i-\bar{s}\right) \qquad (2)

where: $C_\star$ — background-subtracted source counts in ADU; $I_i$ — raw counts in pixel $i$ ; $\bar{s}$ — mean sky counts per pixel from the annulus; $\mathrm{ap}$ — the set of pixels inside the aperture. Reading: the aperture defines what you call "source" and the annulus what you call "sky"; both choices feed directly into the noise. The central result of the article is the CCD equation, which assembles the signal-to-noise ratio from independent variances. It is the equation an exposure-time calculator solves.

\mathrm{SNR}=\frac{N_\star}{\sqrt{N_\star+n_{\mathrm{pix}}\left(N_S+N_D+R^2\right)}} \qquad (3)

where: $N_\star=g\,C_\star$ — total source electrons; $g$ — gain in electrons per ADU; $N_S$ — sky electrons per pixel; $N_D$ — dark-current electrons per pixel; $R$ — read noise in electrons rms per pixel; $n_{\mathrm{pix}}$ — number of pixels in the aperture. Reading: the numerator is signal, the denominator is the square root of the total variance — source shot noise plus per-pixel background and read variance summed over the aperture. A signal-to-noise ratio converts directly into a magnitude uncertainty by propagating a fractional flux error through Pogson's logarithm.

\sigma_m\approx\frac{2.5}{\ln 10}\,\frac{1}{\mathrm{SNR}}=\frac{1.0857}{\mathrm{SNR}} \qquad (4)

where: $\sigma_m$ — magnitude uncertainty in mag; $\mathrm{SNR}$ — signal-to-noise of the flux. Reading: a per-cent flux error is about $0.011$ mag, so $\mathrm{SNR}=100$ yields roughly one hundredth of a magnitude — a convenient mental conversion. Raw counts become a magnitude on the standard system only after calibration against standard stars, which fixes the zero point, removes atmospheric extinction, and corrects the bandpass mismatch.

m_{\mathrm{std}}=m_{\mathrm{inst}}+Z-k\,X+c\,(\text{color}),\qquad m_{\mathrm{inst}}=-2.5\log_{10}\!\left(\frac{g\,C_\star}{t}\right) \qquad (5)

where: $m_{\mathrm{std}}$ — calibrated magnitude; $m_{\mathrm{inst}}$ — instrumental magnitude from the count rate; $Z$ — nightly zero point; $k$ — first-order extinction coefficient in mag per airmass; $X$ — airmass; $c$ — color-term coefficient; $t$ — exposure time. Reading: the instrument measures a count rate, and three additive corrections carry it onto a system shared across the sky. The airmass itself is fixed by geometry, well approximated by the secant of the zenith angle with a small refractive correction near the horizon.

X(z)\approx\sec z\,\left[1-0.0012\,(\sec^2 z-1)\right] \qquad (6)

where: $X$ — airmass, dimensionless and equal to $1$ at the zenith; $z$ — zenith angle. Reading: observing near the horizon piles up atmosphere ( $X$ grows fast), which is why standard stars are measured across a range of airmass to solve for $k$ .

Derivation ADeriving the CCD equation and its scaling regimes

The goal is to build the signal-to-noise expression of equation (3) from first principles — as a sum of statistically independent variances — and then to read off the three regimes that decide observing strategy: source-limited, sky-limited, and read-noise-limited. We consider a source depositing $N_\star$ electrons into an aperture of $n_{\mathrm{pix}}$ pixels, sitting on a background of $N_S$ sky and $N_D$ dark electrons per pixel, read out with $R$ electrons rms of read noise per pixel, and we track how each contribution grows with exposure time so that the slopes emerge naturally.

Step 1. After background subtraction the signal is simply the source electron count, $S=N_\star=g\,C_\star$ , because the mean sky has already been removed pixel by pixel. Why: the gain converts arbitrary digital counts into physical electrons, and only electrons obey photon statistics — this conversion must happen before any noise is computed.

Step 2. The source photoelectrons are a Poisson process, so their variance equals their mean, $\sigma_\star^2=N_\star$ . Why: independent photon arrivals in a fixed interval are Poisson-distributed, and the defining property of that distribution is variance equal to mean, which is the origin of " $\sqrt{N}$ " shot noise.

Step 3. Each pixel also carries Poisson fluctuations from sky and dark current, and these persist even after the mean sky is subtracted, contributing $\sigma_{\mathrm{bkg}}^2=n_{\mathrm{pix}}(N_S+N_D)$ summed over the aperture. Why: subtracting the mean removes the level but not the randomness — the shot noise of the background photons remains and grows with the number of pixels you sum over, which is why a bloated aperture hurts.

Step 4. Reading the chip out injects a per-pixel Gaussian noise of variance $R^2$ , once per readout, giving $\sigma_R^2=n_{\mathrm{pix}}R^2$ . Why: read noise is set by the output amplifier and digitization, is independent of how many photons landed, and is charged exactly once each time the aperture's pixels are read.

Step 5. Because the noise sources are independent their variances add, and the SNR is signal over total noise, $\mathrm{SNR}=N_\star/\sqrt{N_\star+n_{\mathrm{pix}}(N_S+N_D+R^2)}$ — equation (3). Why: quadrature addition is the statement that uncorrelated errors do not conspire, and it is the single assumption that makes the whole budget tractable.

Step 6. Now let time run: $N_\star$ , $N_S$ , and $N_D$ all scale as $t$ while $R^2$ is constant. When the source dominates, $\mathrm{SNR}\approx\sqrt{N_\star}\propto\sqrt{t}$ ; when the sky dominates, $\mathrm{SNR}\approx N_\star/\sqrt{n_{\mathrm{pix}}N_S}\propto t/\sqrt{t}=\sqrt{t}$ ; when read noise dominates, $\mathrm{SNR}\approx N_\star/(\sqrt{n_{\mathrm{pix}}}\,R)\propto t$ . Why: these three slopes are the practical payoff — they tell you that faint work is a slow √t grind against the sky, and that only very short exposures suffer the linear read-noise penalty that longer integrations escape.

Go deeper · Derivation B · 1 minFrom signal-to-noise to the magnitude error

Derivation BFrom signal-to-noise to the magnitude error

The goal is to justify the compact and universally used relation $\sigma_m\approx1.0857/\mathrm{SNR}$ , showing exactly where the constant $1.0857$ comes from and why a fractional flux error maps onto an absolute magnitude error. This matters because observers quote depths in magnitudes but detectors measure flux, so the conversion between "SNR of a measurement" and "error bar on a catalog magnitude" must be exact and dimensionally clean, and it turns out to depend on nothing but the base of Pogson's logarithm.

Step 1. Start from the magnitude as a logarithm of flux, $m=-2.5\log_{10}F+\text{const}$ , the differential form of Pogson's relation. Why: uncertainty propagation is a statement about small changes, so we need $m$ as a smooth function of $F$ before differentiating.

Step 2. Differentiate, using $\mathrm{d}(\log_{10}F)/\mathrm{d}F=1/(F\ln 10)$ , to get $\mathrm{d}m=-(2.5/\ln 10)\,\mathrm{d}F/F$ . Why: the chain rule converts a change in flux into a change in magnitude, and the awkward factor $2.5/\ln 10$ is simply the price of measuring brightness in a base-10 logarithmic scale.

Step 3. Replace the differentials by uncertainties and identify the signal-to-noise ratio as the inverse fractional flux error, $\mathrm{SNR}\equiv F/\sigma_F$ , so $\sigma_m=(2.5/\ln 10)\,\sigma_F/F=1.0857/\mathrm{SNR}$ , since $2.5/\ln 10=1.08574$ . Why: this is the whole point — the SNR of a flux measurement is exactly the reciprocal fractional error, and multiplying by the fixed constant $1.0857$ turns it straight into a magnitude error with no further inputs. For SNR ≳ 5 the linearization is excellent; only near the detection threshold does the error distribution become noticeably asymmetric.

Go deeper · Derivation C · 1 minThe optimal aperture radius and the aperture correction

Derivation CThe optimal aperture radius and the aperture correction

The goal is to show that for a point source with a known profile there is a finite aperture radius that maximizes SNR — larger is not better — and to derive the aperture correction that recovers the light such an aperture necessarily throws away. This is one of the most consequential practical results in photometry: it explains why measuring apertures are kept surprisingly small, roughly one full-width-at-half-maximum, and why every catalog quietly applies a fixed offset measured on bright stars to its faint-star magnitudes.

Step 1. Let the fraction of source light enclosed within radius $r$ be $\eta(r)$ , so the captured signal is $N_\star(r)=N_{\star,\infty}\,\eta(r)$ , while the number of pixels grows as $n_{\mathrm{pix}}=\pi r^2/p^2$ for pixel scale $p$ . Why: growing the aperture buys more of the source's flux but pays for it with more background-carrying pixels, and the two effects are captured entirely by $\eta(r)$ rising toward one and $n_{\mathrm{pix}}$ rising as $r^2$ .

Step 2. In the sky-limited regime the noise is $\propto\sqrt{n_{\mathrm{pix}}}\propto r$ , so $\mathrm{SNR}(r)\propto\eta(r)/r$ , and maximizing it requires $\mathrm{d}[\eta(r)/r]/\mathrm{d}r=0$ . Why: at small $r$ the enclosed fraction climbs faster than the noise, at large $r$ the noise wins because the profile's wings add pixels but almost no signal — the optimum sits where the two balance, typically near $r\approx0.7$ – $1$ FWHM for a Gaussian core.

Step 3. Because that optimal aperture deliberately misses the profile's wings, the measured magnitude is too faint by $\Delta m_{\mathrm{ap}}=-2.5\log_{10}\eta(r)$ , the aperture correction. Why: one measures $\eta(r)$ on bright, isolated stars where SNR is ample, then applies the same constant offset to faint stars measured in the small aperture — recovering the total flux without ever paying the sky-noise cost of a large aperture.

Limiting cases

The CCD equation collapses into three regimes, each dictating a different observing strategy:

Case	Result	Meaning
Source-limited (bright star)	$\mathrm{SNR}\approx\sqrt{N_\star}=\sqrt{g\,C_\star}$	Source photons dominate; the error is pure shot noise and $\mathrm{SNR}\propto\sqrt{t}$ . Bright-star photometry is limited by how many source photons you collect.
Sky-limited (faint star, long exposure)	$\mathrm{SNR}\approx\dfrac{N_\star}{\sqrt{n_{\mathrm{pix}}\,N_S}}$	Background shot noise summed over the aperture sets the floor; still $\mathrm{SNR}\propto\sqrt{t}$ . Favors dark sites, tight PSFs, and small apertures.
Read-noise-limited (very short exposure)	$\mathrm{SNR}\approx\dfrac{N_\star}{\sqrt{n_{\mathrm{pix}}}\,R}$	Fixed per-readout noise dominates, so $\mathrm{SNR}\propto N_\star\propto t$ . Favors fewer, longer exposures to avoid paying $R$ repeatedly.

Common trapCounts are not electrons, and subtracted sky is not silent

The two most common photometry blunders are statistical. First, do not apply $\sigma=\sqrt{N}$ to counts in ADU: Poisson statistics govern the number of physical electrons, so you must multiply counts by the gain, $N=g\,C$ , before taking a square root — the digital number is electrons scaled by an arbitrary factor and carries no statistics of its own. Second, subtracting the mean sky removes its level but never its noise: each pixel keeps the Poisson fluctuation of the background photons that fell on it, and those fluctuations, summed over the whole aperture, are usually the dominant term for faint sources. "I removed the sky" does not mean "the sky no longer costs me" — it costs you $\sqrt{n_{\mathrm{pix}}N_S}$ every time.

Worked examples

1 · Bright star, source-limited (confidence builder)

A well-exposed star yields $N_\star=10{,}000$ source electrons in an aperture where sky, dark, and read noise are all negligible. Find its SNR and magnitude error.

Step 1. With every background term negligible the CCD equation collapses to the source-limited form $\mathrm{SNR}=\sqrt{N_\star}=\sqrt{10{,}000}=\mathbf{100}$ . The measurement is pure photon shot noise.

Step 2. Convert to a magnitude error with equation (4): $\sigma_m=1.0857/\mathrm{SNR}=1.0857/100=\mathbf{0.011\ mag}$ . So a hundred-to-one measurement is good to about one hundredth of a magnitude — a benchmark worth memorizing. SNR = 100, σ_m ≈ 0.011 mag.

2 · A faint star with a full noise budget

A star delivers a source rate of $100\ \mathrm{e^-/s}$ into an aperture of $n_{\mathrm{pix}}=20$ pixels. The sky contributes $20\ \mathrm{e^-/s}$ per pixel, dark current is negligible, and the read noise is $R=8\ \mathrm{e^-}$ per pixel. What SNR and magnitude error result from a $t=300\ \mathrm{s}$ exposure, and which term limits it?

Step 1. Accumulate the electrons. Source: $N_\star=100\times300=\mathbf{30{,}000\ e^-}$ . Sky per pixel: $N_S=20\times300=6000\ \mathrm{e^-}$ .

Step 2. Build the three variances. Source $\sigma_\star^2=30{,}000$ ; sky $\sigma_S^2=n_{\mathrm{pix}}N_S=20\times6000=\mathbf{120{,}000}$ ; read $\sigma_R^2=n_{\mathrm{pix}}R^2=20\times64=1280$ . The total is $151{,}280$ , so the noise is $\sigma=\sqrt{151{,}280}=\mathbf{389\ e^-}$ .

Step 3. Form the ratio: $\mathrm{SNR}=30{,}000/389=\mathbf{77}$ . Because the sky variance ( $120{,}000$ ) dwarfs the source ( $30{,}000$ ) and read ( $1280$ ) terms, this is a sky-limited measurement — the same conclusion the noise-budget figure draws graphically.

Step 4. Magnitude error: $\sigma_m=1.0857/77=\mathbf{0.014\ mag}$ . A three-hundred-second exposure on this star delivers roughly fourteen-millimagnitude photometry, background-limited. SNR = 77, sky-limited, σ_m ≈ 0.014 mag.

3 · Exposure planning and the cost of one magnitude

A faint star gives $10\ \mathrm{e^-/s}$ against a sky of $40\ \mathrm{e^-/s}$ per pixel over $n_{\mathrm{pix}}=25$ pixels, deep in the sky-limited regime. How long must you integrate for $\mathrm{SNR}=20$ , and how much longer would a star one magnitude fainter need?

Step 1. In the sky-limited limit $\mathrm{SNR}\approx f_\star t/\sqrt{n_{\mathrm{pix}}f_S\,t}=f_\star\sqrt{t}/\sqrt{n_{\mathrm{pix}}f_S}$ , with source rate $f_\star$ and sky rate $f_S$ . Solve for time: $t=\left(\mathrm{SNR}\right)^2 n_{\mathrm{pix}}f_S/f_\star^2$ .

Step 2. Insert the numbers: $t=(20)^2\times25\times40/(10)^2=400\times1000/100=\mathbf{4000\ s}$ , about $1.1$ hours. A full-budget check confirms $\mathrm{SNR}=19.9$ at $4000\ \mathrm{s}$ — the source and read terms are genuinely negligible here.

Step 3. A star one magnitude fainter has $2.512$ times less flux, so $f_\star$ drops by $2.512$ and, at fixed SNR, $t\propto 1/f_\star^2$ grows by $2.512^2=\mathbf{6.31}$ . The required time balloons to $\approx25{,}200\ \mathrm{s}$ , roughly seven hours. Reaching one magnitude deeper at fixed SNR costs about 6.3× the exposure — the single hardest fact of survey economics.

Physics of the portrait

Photometry's history is the story of replacing the human eye, first with a fixed logarithmic scale, then with photocells, and finally with silicon arrays whose noise is understood well enough to write down in one equation.

Origins · 1 minthe CCD equation and how this was discovered

the CCD equation

photometric signal-to-noise, from Pogson's ratios to the silicon detector · observational astrophysics and detector noise

The signal-to-noise equation of modern photometry is the confluence of two very different lineages. One is the scale: Norman Robert Pogson, cataloguing stars at Oxford and later as Government Astronomer at Madras, proposed in 1856 that a difference of five magnitudes be fixed as exactly a factor of one hundred in flux, turning the ancient six-grade eyeball ranking of Hipparchus into a precise logarithmic law, $m_2-m_1=-2.5\log_{10}(F_2/F_1)$ . The other lineage is the statistics of counting: Siméon Denis Poisson's 1837 distribution describes independent rare events, and when photodetection was understood as photon counting it supplied the irreducible " $\sqrt{N}$ " shot noise at the heart of every brightness measurement. For half a century the detector between scale and statistics was the photographic plate — nonlinear, hard to calibrate — until Joel Stebbins pioneered photoelectric photometry around 1910, replacing the eye with a measurable current and finally making error bars meaningful. The decisive hardware arrived in 1969, when Willard Boyle and George Smith invented the charge-coupled device at Bell Labs; its linearity, high quantum efficiency, and characterizable read noise let astronomers write the noise budget down term by term. Peter Stetson's 1987 DAOPHOT package then turned PSF fitting into a reproducible standard for crowded fields. The CCD equation is what remains when all of this is distilled: a scale, a statistic, and a detector, fused into one line.

History

1610

— Galileo's telescopic star counts hint that brightness is a measurable quantity, not merely a rank

1837

— Siméon Denis Poisson publishes the distribution governing photon shot noise

1856

— Norman Pogson fixes five magnitudes as exactly a factor of 100 in flux

1910

— Joel Stebbins pioneers photoelectric photometry, replacing the eye with a measurable current

1969

— Willard Boyle and George Smith invent the charge-coupled device at Bell Labs

1987

— Peter Stetson releases DAOPHOT, standardizing PSF-fitting photometry in crowded fields

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 3 (magnitudes, flux, detectors) · Howell, Handbook of CCD Astronomy, 2nd ed., Ch. 4–5 (the CCD equation and aperture/PSF photometry) · Chromey, To Measure the Sky, 2nd ed. (calibration, extinction, color terms) · Binney & Merrifield, Galactic Astronomy (photometric systems) · Ryden & Peterson, Foundations of Astrophysics (introductory synthesis).

Provenance. Pogson's relation and the flux–magnitude law are standard (Carroll & Ostlie Ch. 3). The CCD equation matches Howell, Handbook of CCD Astronomy 2nd ed. Eq. 4.8; the magnitude-error relation uses the exact constant $2.5/\ln 10=1.08574$ . The three worked examples are directly recomputable: source-limited $\sqrt{10{,}000}=100\Rightarrow0.011$ mag; the full-budget faint star gives $30{,}000/\sqrt{151{,}280}=77.1$ and $\sigma_m=0.014$ mag; the sky-limited planning example gives $t=4000$ s for SNR $20$ and a $2.512^2=6.31$ time penalty per magnitude. The airmass approximation and the optimal-aperture results follow standard practice (Howell Ch. 5). No non-standard numbers are asserted.

Prerequisites. Luminosity and Radiant Flux · The Magnitude System · Photometric Systems and Filters. Leads to. Color Indices and Bolometric Corrections · Standard Candles · The Cosmic Distance Ladder.

Ask the tutor: ask me to size an exposure for any star and sky brightness, or to walk through the CCD equation term by term on your own data.

What is the difference between aperture and PSF photometry, and when do I use each?

Aperture photometry sums all the counts inside a fixed circle and subtracts a sky level measured in a surrounding annulus. It is simple, assumption-free about the source shape, and ideal for bright, isolated stars and galaxies. PSF-fitting photometry instead builds a model of the point-spread function from bright stars and fits scaled, shifted copies of it to the data. Because it weights each pixel by the signal it should contain, it is statistically optimal at low SNR, and because it can fit several overlapping profiles at once it is the only workable method in crowded fields like globular clusters. The rule of thumb: aperture for isolated and bright, PSF for faint and crowded — and PSF whenever you need to deblend.

Why is my measurement sky-limited even after I subtracted the sky?

Because subtracting the sky removes its average level, not its randomness. The background is made of photons that arrive as a Poisson process, so each pixel carries a fluctuation of order $\sqrt{N_S}$ electrons even after the mean is removed. When you sum over the $n_{\mathrm{pix}}$ pixels of your aperture, those independent fluctuations add in quadrature to $\sqrt{n_{\mathrm{pix}}N_S}$ , and for a faint source this term is far larger than the source's own shot noise. "Sky-limited" names exactly this situation. The defenses are physical, not arithmetic: observe from a darker site, use a narrower filter, keep the PSF tight, and choose the smallest aperture that still captures most of the light.

Why not always use a huge aperture to capture all the light?

Because a larger aperture adds background-carrying pixels faster than it adds source flux. The source's light falls off in the wings of the PSF, so beyond about one FWHM each extra ring of pixels contributes almost no new signal but its full share of sky and read noise. In the sky-limited regime the noise grows as the aperture radius while the enclosed signal flattens, so the SNR actually peaks at a modest radius and then declines. The standard solution is to measure in a small, high-SNR aperture and then apply an aperture correction — a fixed offset, measured on bright stars, that accounts for the fraction of light the small aperture deliberately left out.

How does the magnitude error really relate to SNR?

Through one clean line of calculus. Magnitude is $-2.5\log_{10}F$ , so a small fractional change in flux produces a change in magnitude of $\mathrm{d}m=(2.5/\ln 10)\,\mathrm{d}F/F$ . Since the signal-to-noise ratio is by definition the reciprocal of the fractional flux error, $\mathrm{SNR}=F/\sigma_F$ , the magnitude error is $\sigma_m=(2.5/\ln 10)/\mathrm{SNR}=1.0857/\mathrm{SNR}$ . The constant is exact: $2.5/\ln 10=1.08574$ . So SNR $10$ gives $0.11$ mag, SNR $100$ gives $0.011$ mag, and SNR $1000$ gives about a milli-magnitude. The relation is reliable for SNR above about five; nearer the detection limit the flux errors become asymmetric and this linear formula slightly understates the true uncertainty.

What exactly do the zero point, extinction, and color terms do?

They carry a raw instrumental magnitude onto a universal standard system in three steps. The zero point $Z$ is a single additive constant per night and filter that absorbs the telescope's throughput and the detector's gain, anchoring your count rate to the standard scale. The extinction term $-kX$ removes the dimming imposed by the atmosphere, which grows with airmass $X$ ; you solve for $k$ by observing standard stars over a range of altitude. The color term $c\times(\text{color})$ corrects for the fact that your filter's real bandpass never exactly matches the standard one, so redder and bluer stars are converted slightly differently. Together they let magnitudes from different nights, instruments, and sites be compared to milli-magnitude precision.

Why do surveys prefer fewer long exposures over many short ones?

Because read noise is charged once per readout regardless of how long you integrate. If you split a fixed total time into many short frames, you pay $R$ electrons of noise per pixel on every single one, and in the read-limited regime the SNR of each frame grows only linearly, so the penalty is real. One long exposure pays the read-noise toll just once, letting the photon shot noise — which you cannot avoid — dominate the budget. The competing pressures are saturation of bright stars, cosmic-ray hits, and tracking errors, all of which favor shorter frames, so real surveys pick a sub-exposure long enough to become sky-limited but short enough to stay safe, then co-add many of them for depth.