Detectors and CCDs

A detector is the last optical element in the chain and the first quantitative one: it turns a stream of photons that has crossed light-years and a telescope aperture into a number a computer can store. For a century that conversion was done by the eye and the photographic plate, which throw away roughly ninety-nine of every hundred photons and respond to brightness in a way no one could reliably calibrate. The charge-coupled device — a grid of metal-oxide-semiconductor capacitors etched into a silicon wafer — changed astronomy by counting almost every photon and reporting the count linearly. When light above the silicon band gap ( $E_g \approx 1.1\,\mathrm{eV}$ , wavelengths shortward of $\sim 1.1\,\mu\mathrm{m}$ ) is absorbed, it lifts an electron across the gap and leaves a hole; applied gate voltages trap the electrons in a potential well, one per pixel, so the charge grows in strict proportion to the photons caught. This article is the instrument-physics hinge of the observational chapter. It takes the flux and magnitude definitions from Luminosity and Radiant Flux and The Magnitude System and asks the practical question they leave open: given a real detector with finite efficiency and unavoidable noise, how faint an object can you measure, and how well? The answer is the CCD equation, which balances the Poisson graininess of source and sky against the amplifier's read noise and the detector's thermal dark current. That single relation sets the signal-to-noise ratio, converts it into a magnitude error through $\sigma_m \approx 1.0857/(S/N)$ , and fixes the limiting depth of every image on which the Cosmic Distance Ladder is built. Everything downstream — colours, standard candles, redshifts, cosmological surveys — inherits its precision from the arithmetic on this page.

Key takeaways

A charge-coupled device is a silicon pixel array: photons above the $E_g \approx 1.1\,\mathrm{eV}$ band gap free photoelectrons that collect in per-pixel wells, giving charge proportional to detected photons.
Quantum efficiency reaches $80\text{–}90\%$ — versus $\sim 1\%$ for the photographic plate — and the response is linear to a full well of $\sim 10^5$ electrons.
The signal-to-noise ratio follows the CCD equation, which adds source Poisson, sky, dark-current and read noise in quadrature; independent variances add, standard deviations do not.
At fixed exposure the bright (source-limited) regime gives $S/N \propto \sqrt{N_\star}$ , the faint (sky-limited) regime $S/N \propto N_\star$ — but with time both scale as $\sqrt{t}$ .
A magnitude error is just an inverse $S/N$ : $\sigma_m \approx 1.0857/(S/N)$ , so $S/N = 100$ buys $0.011\,\mathrm{mag}$ photometry.
Calibration frames — bias, dark, flat — strip the instrument's electronic and optical fingerprints before any count becomes a physical flux.
Key idea: a CCD counts photoelectrons per pixel with high quantum efficiency and linearity, and the CCD equation — balancing Poisson, sky, read and dark-current noise — fixes the faintest object measurable in a given exposure and its magnitude error.

The core insight

A CCD counts photoelectrons in each pixel with high quantum efficiency and linearity; the achievable signal-to-noise ratio — governed by the CCD equation balancing Poisson, sky, read and dark-current noise — sets the faintest object measurable in a given exposure.

Where this lives today

The CCD equation is the daily arithmetic of every observatory and the design driver of every survey. Wide-field mosaics — the Sloan Digital Sky Survey, the Dark Energy Survey, and now the Vera C. Rubin Observatory with its $3.2\,\mathrm{Gpix}$ camera — plan cadence and depth directly from the sky-limited $\sqrt{t}$ scaling, reaching $r \approx 27$ over thousands of square degrees. Space missions push read noise and dark current to near zero: Gaia centroids stars to microarcsecond precision, delivering parallaxes for over a billion stars and re-anchoring the distance ladder. Precision photometry of Type Ia supernovae, at the few-percent level $S/N \sim 100$ affords, turned the CCD into a cosmological instrument and revealed cosmic acceleration. At the frontier, skipper CCDs read the same pixel hundreds of times to average read noise below one electron — the CCD equation taken to its limit.

The physical picture

Two things happen inside a CCD, and the two figures below capture them. First, detection: a photon disappears and an electron appears, but only if the photon carried more than the silicon band-gap energy — the photoelectric effect inside a semiconductor, which fixes both the reddest wavelength the chip can see and the fundamental unit of measurement, one photoelectron. Second, accumulation: each pixel is a tiny electrostatic bucket held by the gate voltages, filling over an exposure with photoelectrons in proportion to the light that landed on it, up to a full-well capacity of order $10^5$ electrons beyond which it overflows into its neighbours.

Figure

A photon with energy hν above the silicon gap E_g = 1.1 eV lifts a valence electron to the conduction band, leaving a hole and fixing the red cutoff λ_max = hc/E_g ≈ 1.1 μm; each pixel is a potential well that stores the freed charge in proportion to the photons detected, up to a full-well capacity of order 10⁵ electrons

The virtue of the CCD is not just that the well fills, but that it fills linearly and for almost every photon: quantum efficiency of $80\text{–}90\%$ wastes little light, and the count is a faithful ruler of flux over four or five decades. But the ruler has a graininess. Even a steady source delivers photons at random times, so the count fluctuates by $\sqrt{N}$ — irreducible Poisson noise. On top sits the sky, itself a photon source; the dark current, which cooling suppresses; and the read noise, a fixed electronic penalty paid once each readout. The second figure is the diagram a detector engineer actually measures — the photon transfer curve — showing which noise term rules at each signal level.

Figure

The photon transfer curve for a CCD with read noise σ_read = 5 e⁻ and 1% flat-field residuals: total noise σ = √(σ_read² + S + (0.01 S)²) is flat at low signal (read-noise limited, slope 0), rises as √S once shot noise dominates near S ≈ 25 e⁻ (slope 1/2), then steepens to slope 1 as flat-field residuals take over approaching the full well of 10⁵ e⁻

The core equations

The reddest light a silicon CCD can register is set by the band gap: a photon must carry at least $E_g$ to free an electron, so wavelengths longer than $hc/E_g$ pass through undetected — which is why silicon runs out in the near-infrared and redder detectors use narrower-gap materials.

E_g \approx 1.1\,\mathrm{eV} \quad\Rightarrow\quad \lambda_{\max} = \frac{hc}{E_g} \approx 1.1\,\mu\mathrm{m} \qquad (1)

where: $E_g$ — silicon band-gap energy (minimum to create an electron–hole pair); $h$ — Planck constant; $c$ — speed of light; $\lambda_{\max}$ — longest wavelength a silicon CCD detects. Reading it: with $hc = 1240\,\mathrm{eV\,nm}$ , dividing by $1.1\,\mathrm{eV}$ gives $1127\,\mathrm{nm} \approx 1.1\,\mu\mathrm{m}$ . Not every photon that lands is counted; the fraction producing a collected photoelectron is the quantum efficiency, a wavelength-dependent number peaking near $90\%$ for a back-illuminated chip and falling toward the ultraviolet and near-infrared edges.

\mathrm{QE}(\lambda) = \frac{N_{e^-}}{N_{\gamma}} \qquad (2)

where: $N_{e^-}$ — photoelectrons detected; $N_{\gamma}$ — photons incident on the pixel; $\mathrm{QE}$ — quantum efficiency, wavelength-dependent, $\lesssim 1$ . Reading it: a plate at $\mathrm{QE}\sim 1\%$ needs roughly ninety times the exposure of a CCD at $\mathrm{QE}\sim 90\%$ for the same electrons. The chip does not report electrons directly. An amplifier and analog-to-digital converter turn the charge into integer counts (ADU), and the gain is the factor you must apply before any Poisson statistics are valid — counts are not electrons.

N_{e^-} = g \, N_{\mathrm{ADU}} \qquad (3)

where: $N_{\mathrm{ADU}}$ — digital output in ADU (counts); $g$ — gain, electrons per ADU; $N_{e^-}$ — number of photoelectrons. Reading it: with $g = 2\,\mathrm{e^-/ADU}$ , a pixel reading $5000$ counts holds $10^4$ electrons — and it is $10^4$ , not $5000$ , whose square root is the Poisson noise. The heart of the article is the CCD equation: the signal-to-noise ratio of an aperture measurement is the source counts divided by the quadrature sum of every independent noise — the source's own Poisson term plus the per-pixel sky, dark current and read noise summed over the aperture.

\frac{S}{N} = \frac{N_\star}{\sqrt{\,N_\star + n_{\mathrm{pix}}\left(N_S + N_D + \sigma_{\mathrm{read}}^2\right)}} \qquad (4)

where: $N_\star$ — total source photoelectrons in the aperture; $n_{\mathrm{pix}}$ — pixels in the aperture; $N_S$ — sky electrons per pixel; $N_D$ — dark-current electrons per pixel; $\sigma_{\mathrm{read}}$ — read noise per pixel (rms). Reading it: the numerator is the signal, and each term under the root is a variance, so the three backgrounds enter as $\sigma^2$ times the pixels they occupy. The counts are not free parameters — each accumulates linearly with collecting area, exposure time and efficiency. The source scales with the star's photon flux; the sky, a surface brightness, also scales with a pixel's solid angle.

N_\star = \mathcal{F}_\star\, A\, t\, \mathrm{QE}, \qquad N_S = \mathcal{F}_S\, A\, t\, \mathrm{QE}\, \Omega_{\mathrm{pix}} \qquad (5)

where: $\mathcal{F}_\star$ — source photon flux in the band; $\mathcal{F}_S$ — sky flux per unit solid angle; $A$ — collecting area; $t$ — exposure time; $\Omega_{\mathrm{pix}}$ — solid angle of one pixel. Reading it: both source and sky carry a factor $t$ , so sky noise grows as $\sqrt{t}$ while signal grows as $t$ — the origin of the $\sqrt{t}$ improvement of $S/N$ . Dark current is charge that appears with no light — electrons thermally excited across the same gap that photons exploit. Its steep temperature dependence is why astronomical CCDs are cryogenically cooled.

N_D = D\, t, \qquad D \propto T^{3/2}\, e^{-E_g/2kT} \qquad (6)

where: $D$ — dark-current rate (electrons per pixel per second); $T$ — detector temperature; $k$ — Boltzmann constant; $t$ — exposure time. Reading it: cooling from $293\,\mathrm{K}$ to $173\,\mathrm{K}$ ( $-100\,°\mathrm{C}$ ) drops the Boltzmann factor by about seven orders of magnitude, so cooled chips integrate for an hour with negligible dark noise. Once you have the signal-to-noise ratio, the photometric error follows: a magnitude is a logarithm of flux, and a fractional flux error is just $1/(S/N)$ .

\sigma_m = \frac{2.5}{\ln 10}\,\frac{1}{S/N} \approx \frac{1.0857}{S/N} \qquad (7)

where: $\sigma_m$ — uncertainty in the measured magnitude; $S/N$ — signal-to-noise ratio of the flux. Reading it: $S/N = 100$ gives $\sigma_m \approx 0.011\,\mathrm{mag}$ , $S/N = 10$ gives $\approx 0.11\,\mathrm{mag}$ — a handy rule of thumb. Finally, no raw count is usable until the instrument's fingerprints are removed. The calibration model states how a raw pixel value is built from the bias offset, the dark charge and the flat-field-scaled sum of sky and source — and how to invert it.

C_{\mathrm{raw}} = C_{\mathrm{bias}} + C_{\mathrm{dark}} + \mathrm{flat} \times (C_{\mathrm{sky}} + C_{\mathrm{source}}) \qquad (8)

where: $C_{\mathrm{raw}}$ — raw counts in a pixel; $C_{\mathrm{bias}}$ — electronic bias offset; $C_{\mathrm{dark}}$ — dark contribution over the exposure; $\mathrm{flat}$ — normalised flat-field response. Reading it: reduction runs the equation backwards — subtract bias, subtract dark, divide by the flat — before a count is a flux.

Derivation AThe CCD equation from independent variances

The goal is to build equation (4) from first principles of counting statistics, so its structure is transparent rather than memorised. We want to see precisely why the source term appears once but each background term is multiplied by the number of aperture pixels, why read noise enters as a variance though it is quoted as a standard deviation, and why every contribution sits under a single square root. Getting this right separates a defensible error bar from a fabricated one.

Step 1. Treat every detected electron as a Poisson random variable. For a Poisson process the variance equals the mean, so the source contributes $\sigma_\star^2 = N_\star$ , the sky $\sigma_S^2 = N_S$ per pixel, and the dark current $\sigma_D^2 = N_D$ per pixel. Why: photon arrivals and thermal-generation events are independent in time — the defining property of a Poisson process — whose signature is that noise is the square root of the count.

Step 2. Add the read noise, which is not Poisson. The amplifier and digitiser inject a fixed Gaussian uncertainty $\sigma_{\mathrm{read}}$ each time a pixel is read, independent of its charge; it enters as a variance $\sigma_{\mathrm{read}}^2$ per pixel. Why: read noise is an electronic measurement error, not a counting fluctuation, so it does not grow with signal — which is why it dominates only when the signal is small.

Step 3. Sum the variances over the aperture. The source contributes $N_\star$ once, but the sky, dark and read terms afflict every one of the $n_{\mathrm{pix}}$ pixels, so they multiply by $n_{\mathrm{pix}}$ : the total variance is $\sigma_{\mathrm{tot}}^2 = N_\star + n_{\mathrm{pix}}(N_S + N_D + \sigma_{\mathrm{read}}^2)$ . Why: independent variances add, and the backgrounds are independent from pixel to pixel, so more pixels means more accumulated background noise — the reason apertures should not be oversized.

Step 4. Form the ratio. The signal is $N_\star$ and the noise is $\sigma_{\mathrm{tot}} = \sqrt{\sigma_{\mathrm{tot}}^2}$ , so dividing gives $S/N = N_\star/\sqrt{N_\star + n_{\mathrm{pix}}(N_S + N_D + \sigma_{\mathrm{read}}^2)}$ , equation (4). Why: the single square root is the whole point — standard deviations combine as the root of summed variances, never linearly.

Go deeper · Derivation B · 1 minDepth in the sky-limited regime

Derivation BDepth in the sky-limited regime

The goal is to find how the limiting magnitude of an image improves with exposure time in the regime that governs all deep imaging, where the sky dominates the noise. This dictates observing strategy on every survey telescope: how long to stare for a target depth, and how brutally the returns diminish. We also correct a common slip — the belief that faint-source $S/N$ grows linearly with time — by tracking the fact that the sky itself accumulates during the exposure.

Step 1. Drop the subdominant terms. When the sky is bright per pixel and swamps dark current and read noise, the CCD equation collapses to $S/N \approx N_\star/\sqrt{n_{\mathrm{pix}} N_S}$ . Why: in a long exposure of a faint object the sky Poisson noise is the only term left standing, so keeping the others is false precision.

Step 2. Insert the time dependence. Both counts accumulate linearly, $N_\star = R_\star t$ and $N_S = R_S t$ , so $S/N = R_\star t / \sqrt{n_{\mathrm{pix}} R_S t} = (R_\star/\sqrt{n_{\mathrm{pix}} R_S})\,\sqrt{t} \propto \sqrt{t}$ . Why: this is the correction — because the sky variance grows with $t$ too, the ratio grows only as $\sqrt{t}$ , not as $t$ ; quadrupling the exposure only doubles the signal-to-noise.

Step 3. Solve for the faintest source at a fixed threshold $(S/N)_{\min}$ . The limiting rate is $R_{\star,\mathrm{lim}} = (S/N)_{\min}\sqrt{n_{\mathrm{pix}} R_S / t} \propto t^{-1/2}$ , and converting to a magnitude with zero point $m_{\mathrm{zp}}$ gives $m_{\mathrm{lim}} = m_{\mathrm{zp}} - 2.5\log_{10} R_{\star,\mathrm{lim}} = \mathrm{const} + 1.25\log_{10} t$ . Why: the limiting flux falls as $t^{-1/2}$ , and through the logarithmic magnitude scale that becomes a depth growing as $1.25\log_{10} t$ .

Step 4. Read off the payoff. Each doubling of exposure deepens the image by $\Delta m = 1.25\log_{10} 2 \approx 0.38\,\mathrm{mag}$ , so one full magnitude fainter costs a factor $10^{1/1.25} \approx 6.3$ in time. Why: this is the notorious slow deepening of deep imaging — and it is $0.38\,\mathrm{mag}$ per doubling, not the $0.75\,\mathrm{mag}$ that applies only if the noise were fixed (the read-noise-limited case).

Go deeper · Derivation C · 1 minMagnitude error from signal-to-noise

Derivation CMagnitude error from signal-to-noise

The goal is to derive the universal relation $\sigma_m \approx 1.0857/(S/N)$ that lets an observer read a photometric error straight off a signal-to-noise ratio, and to see why the constant is $2.5/\ln 10$ . This is the bridge between the instrument physics of this article and the magnitude scale of its neighbours, used in nearly every photometry paper written.

Step 1. Start from the definition of magnitude, $m = -2.5\log_{10} F + \mathrm{const}$ , with $F$ the measured flux and the constant the zero point. Why: the magnitude system is logarithmic in flux, so a flux uncertainty must be propagated through a logarithm, not added directly.

Step 2. Differentiate: $\mathrm{d}m = -\dfrac{2.5}{\ln 10}\,\dfrac{\mathrm{d}F}{F}$ , using $\mathrm{d}(\log_{10} F) = \mathrm{d}F/(F\ln 10)$ . Why: the factor $1/\ln 10$ converts natural to base-10 logarithms, and is precisely what makes the constant $2.5/\ln 10 = 1.0857$ rather than $2.5$ .

Step 3. Identify the fractional flux error with the inverse signal-to-noise, $\mathrm{d}F/F = 1/(S/N)$ , since $S/N$ is by definition the signal over its standard deviation. Substituting gives $\sigma_m \approx 1.0857/(S/N)$ . Why: this closes the loop — the CCD equation delivers $S/N$ , and one line of calculus turns it into a magnitude error, so $S/N = 100$ is $0.011\,\mathrm{mag}$ and $S/N = 5$ (a common detection threshold) a coarse $0.22\,\mathrm{mag}$ .

Limiting cases

The CCD equation is opaque as written but simple once you ask which single noise term dominates. The three regimes below cover essentially all of practice, and knowing which you are in tells you at once how to improve a measurement.

Case	Result	Meaning
Source-limited (bright)	$S/N \approx \sqrt{N_\star}$	When the source outshines all backgrounds and read noise, $S/N$ grows as the square root of collected photons; doubling the exposure gains a factor $\sqrt{2}$ .
Sky-limited (faint)	$S/N \approx \dfrac{N_\star}{\sqrt{n_{\mathrm{pix}}\,N_S}}$	For faint sources on bright sky, at fixed exposure $S/N \propto N_\star$ , but with time still $S/N \propto \sqrt{t}$ ; a small aperture and dark site are decisive, and depth improves only $\approx 0.38\,\mathrm{mag}$ per doubling.
Read-noise-limited (short exposures)	$S/N \approx \dfrac{N_\star}{\sqrt{n_{\mathrm{pix}}}\,\sigma_{\mathrm{read}}}$	For faint sources in short exposures under dark sky the fixed read noise rules; paid once per readout, so one long exposure beats many short, and depth can improve up to $\approx 0.75\,\mathrm{mag}$ per doubling.

Worked examples

1 · The full CCD equation (fully worked)

A star deposits $N_\star = 10{,}000$ photoelectrons in an aperture of $n_{\mathrm{pix}} = 20$ pixels. The sky contributes $N_S = 200\,\mathrm{e^-}$ per pixel, dark current $N_D = 10\,\mathrm{e^-}$ per pixel, and read noise $\sigma_{\mathrm{read}} = 5\,\mathrm{e^-}$ per pixel. Find the signal-to-noise ratio and magnitude error, and compare with the naive bright-star estimate.

Step 1. Assemble the per-pixel background variance: $N_S + N_D + \sigma_{\mathrm{read}}^2 = 200 + 10 + 25 = 235\,\mathrm{e^-}$ — read noise entered as its square, $5^2 = 25$ .

Step 2. Multiply by the aperture pixels and add the source Poisson term: $N_\star + n_{\mathrm{pix}}(235) = 10{,}000 + 20 \times 235 = 14{,}700\,\mathrm{e^-}$ , so the total noise is $\sqrt{14{,}700} = 121.2\,\mathrm{e^-}$ .

Step 3. Divide signal by noise: $S/N = 10{,}000 / 121.2 = \mathbf{82.5}$ , and $\sigma_m = 1.0857/82.5 = \mathbf{0.0132\;mag}$ .

Step 4. Compare with the naive estimate. Ignoring all backgrounds would give $S/N \approx \sqrt{N_\star} = 100$ and $\sigma_m \approx 0.011\,\mathrm{mag}$ . The backgrounds have degraded the real $S/N$ from $100$ to $82.5$ — an $18\%$ hit — through the $4{,}700$ electrons of background variance the twenty pixels contribute. The honest answer is S/N = 82.5, σ_m = 0.013 mag, not the source-limited 100.

2 · Exposure time and survey depth

An imaging survey reaches $S/N = 20$ on a faint galaxy in a $300\,\mathrm{s}$ sky-limited exposure. (a) How long must you integrate to reach $S/N = 100$ ? (b) How much extra time buys one magnitude of extra depth?

Step 1. In the sky-limited regime $S/N \propto \sqrt{t}$ , so the time ratio is the square of the $S/N$ ratio: $t_2/t_1 = (100/20)^2 = 25$ , i.e. $25 \times 300\,\mathrm{s} = 7{,}500\,\mathrm{s} \approx \mathbf{2.1\;hours}$ .

Step 2. For depth, $m_{\mathrm{lim}} = \mathrm{const} + 1.25\log_{10} t$ . One magnitude deeper needs $1.25\log_{10}(t_2/t_1) = 1$ , so $t_2/t_1 = 10^{0.8} = \mathbf{6.3\times}$ .

Step 3. Read the lesson. This brutal $t^{-1/2}$ scaling is why survey depth is planned in fractions of a magnitude and why aperture (collecting area $A$ ) is worth more than exposure — doubling $A$ shifts depth as effectively as quadrupling $t$ . To go from S/N 20 to 100 costs 25× the time; to go one magnitude deeper costs 6.3×.

3 · One long exposure versus many short ones

You need $3{,}000\,\mathrm{s}$ total on a read-noise-limited target ( $\sigma_{\mathrm{read}} = 6\,\mathrm{e^-}$ , $n_{\mathrm{pix}} = 9$ ). Compare one $3{,}000\,\mathrm{s}$ frame with ten $300\,\mathrm{s}$ frames for accumulated read noise.

Step 1. Read noise is charged once per readout. The single frame pays a variance $n_{\mathrm{pix}}\sigma_{\mathrm{read}}^2 = 9 \times 36 = 324\,\mathrm{e^-}$ , so $\sqrt{324} = 18\,\mathrm{e^-}$ total.

Step 2. Ten frames pay it ten times: $10 \times 324 = 3{,}240\,\mathrm{e^-}$ , so $\sqrt{3{,}240} = 57\,\mathrm{e^-}$ — larger by exactly $\sqrt{10} = 3.16$ .

Step 3. The collected signal is identical, so the single long exposure gives $3.16\times$ better read-noise-limited $S/N$ . Fewer, longer exposures win when read noise dominates — bounded in practice by saturation, cosmic rays and tracking, so observers split the difference.

Physics of the portrait

The CCD is a rare case where the decisive instrument and the physics behind it arrived almost together, and where the invention earned a Nobel Prize four decades on. The portrait is the device itself: a slab of silicon that finally let astronomers count photons.

Origins · 1 mincharge-coupled device and how this was discovered

charge-coupled device

the silicon photon-counter of Boyle and Smith, 1969 · astronomical detector physics

The charge-coupled device was conceived in a single afternoon in October 1969 at Bell Laboratories, when Willard Boyle and George Smith sketched a way to shuffle packets of charge along a chain of MOS capacitors by clocking their gate voltages in sequence — the "charge coupling" that names the device. They meant it as a memory and a bucket-brigade delay line, not a camera; that the same silicon which stores charge also generates it photoelectrically, and so the array could be an imager, followed almost at once. By the mid-1970s astronomers had grasped what the plate and photomultiplier could never offer: a detector counting eight or nine of every ten photons, linear over four decades, its output already digitised. The first astronomical CCD images — a crude picture of Uranus in 1976, a few hundred pixels on a side — reached fainter than plates in a fraction of the time, and the technology scaled relentlessly to the 3.2-gigapixel focal plane of the Vera Rubin Observatory. Boyle and Smith shared the 2009 Nobel Prize in Physics for handing science an electronic eye that transformed astronomy, microscopy and everyday photography alike.

History

1969

— Boyle and Smith conceive the charge-coupled device at Bell Labs

1970

— Bell Labs demonstrates the first working CCD shift register

1976

— first astronomical CCD image, of Uranus, on a ground-based telescope

1990

— the Hubble Space Telescope flies WFPC CCD cameras above the atmosphere

2009

— Boyle and Smith share the Nobel Prize in Physics for the CCD

Going further

Texts. Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., §6.1 (detectors and the CCD equation) · Howell, Handbook of CCD Astronomy, 2nd ed. (the definitive practical treatment) · Chromey, To Measure the Sky, 2nd ed., Ch. 8 (detectors) · Binney & Tremaine, Galactic Dynamics, 2nd ed. (observational appendices) · Dodelson, Modern Cosmology (survey observational methods) · Ryden, Introduction to Cosmology, 2nd ed.

Provenance. A verifier should recompute: (1) the silicon cutoff $\lambda_{\max} = hc/E_g$ with $E_g \approx 1.1\,\mathrm{eV}$ giving $1127\,\mathrm{nm} \approx 1.1\,\mu\mathrm{m}$ ; (2) the CCD equation by summing Poisson and read-noise variances in quadrature (Howell §4, Carroll & Ostlie §6.1); (3) the magnitude-error relation $\sigma_m = (2.5/\ln 10)/(S/N) \approx 1.0857/(S/N)$ by differentiating $m = -2.5\log_{10} F$ ; (4) the sky-limited depth scaling $m_{\mathrm{lim}} = \mathrm{const} + 1.25\log_{10} t$ , i.e. $0.38\,\mathrm{mag}$ per doubling — corrected here from the base article's erroneous $0.75\,\mathrm{mag}$ , which applies only to the read-noise-limited case; (5) the worked $S/N = 82.5$ from $N_\star = 10^4$ , $n_{\mathrm{pix}} = 20$ , $N_S = 200$ , $N_D = 10$ , $\sigma_{\mathrm{read}} = 5$ . Standard values (QE $\sim 80\text{–}90\%$ , plate QE $\sim 1\%$ , full well $\sim 10^5\,\mathrm{e^-}$ , cooling to $\sim -100\,°\mathrm{C}$ ) are textbook.

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

Ask the tutor: Want to work the CCD equation for your own telescope, sky brightness and exposure time to predict the signal-to-noise ratio and limiting magnitude?

Why must a CCD be cooled to around −100 °C?

Because dark current — electrons thermally excited across the silicon band gap with no light present — scales roughly as $T^{3/2}e^{-E_g/2kT}$ , savage in temperature. Dropping from $293\,\mathrm{K}$ to about $173\,\mathrm{K}$ cuts the Boltzmann factor by some seven orders of magnitude, from thousands of electrons per pixel per second to a small fraction of one. Otherwise those thermal electrons pile up in a long exposure, adding Poisson noise and eating into the full well. Cooling is a prerequisite for deep imaging, usually by liquid nitrogen or a closed-cycle cryocooler.

What is the difference between read noise and dark current?

They are physically unrelated and behave oppositely with time. Dark current is real charge — thermally generated electrons that accumulate in the well, growing linearly with exposure, so its Poisson noise grows as $\sqrt{t}$ . Read noise is not charge at all; it is the electronic uncertainty the amplifier and digitiser add each time a pixel is measured, independent of exposure and paid once per readout. You fight dark current by cooling the chip, and read noise by reading less often (longer exposures) or by averaging repeated non-destructive reads, as skipper CCDs do. In the CCD equation dark current appears as $N_D$ and read noise as $\sigma_{\mathrm{read}}^2$ , both per pixel.

Why do the noise terms add in quadrature, and why must I convert counts to electrons first?

Adding in quadrature is the statement that independent random variables combine by summing variances, not standard deviations: if two noises are uncorrelated, their fluctuations partly cancel, and the correct total is $\sqrt{\sigma_1^2 + \sigma_2^2}$ , always less than $\sigma_1 + \sigma_2$ . Adding noises linearly assumes they always push the same way, which independent sources do not. The electron conversion is a separate but equally strict point: Poisson statistics say the variance equals the mean in electrons, the things being counted. A pixel reading $5000$ ADU at gain $g = 2$ holds $10^4$ electrons, and its Poisson noise is $\sqrt{10^4} = 100\,\mathrm{e^-}$ , not $\sqrt{5000}$ — take the square root of counts and you get a meaningless number.

Is one long exposure always better than several short ones?

For read noise, yes; for everything else, not necessarily. A single frame pays the read-noise penalty once, while $M$ frames pay it $M$ times, so when read noise dominates the long exposure wins by $\sqrt{M}$ in signal-to-noise. But real exposures are capped: bright stars saturate and bleed at full well, cosmic rays accumulate at roughly one per few minutes per square centimetre, and tracking errors smear long integrations. The usual compromise is to split a deep observation into sub-exposures long enough to stay sky-limited yet short enough to avoid saturation and to let cosmic rays be rejected by combining frames — "dithering" and median-stacking.

If a silicon CCD stops at 1.1 μm, how do we observe redder light?

By changing the detector material. The $1.1\,\mu\mathrm{m}$ cutoff is set entirely by silicon's $1.1\,\mathrm{eV}$ band gap: a redder photon cannot free an electron. To see the infrared, astronomers use narrower-gap semiconductors — mercury cadmium telluride (HgCdTe) arrays, whose gap tunes by composition out to $5\,\mu\mathrm{m}$ and beyond, or indium antimonide and doped silicon for the mid-infrared. These are not CCDs in the charge-shuffling sense but hybrid arrays bonded to a separate silicon readout. The same noise budget applies, but infrared detectors face far higher thermal backgrounds and must be cooled harder, near liquid-nitrogen or even liquid-helium temperatures.

Why do I need a flat field if the CCD is linear?

Linearity and uniformity are different properties. Linearity means each pixel's response is proportional to the light it receives; a flat field corrects the fact that different pixels have different proportionality constants. Pixel-to-pixel quantum-efficiency variations, dust shadows and vignetting all mean a uniform sky produces a non-uniform image, often at the several-percent level. A flat-field frame — an exposure of a uniformly illuminated twilight sky or dome screen — measures that response map, and dividing by the normalised flat removes it. Without it, photometry carries a position-dependent systematic that no signal-to-noise can beat down, because it is not random.