A star can vary strongly while a long exposure records only a small change. The detector collects light throughout an interval, so the reported brightness can average together bright and faint phases. A small measured amplitude does not always mean a small intrinsic variation.
Finite exposure averaging changes the signal before the observing schedule samples it. This guide derives the effect for a sinusoidal variable, calculates a complete cancellation and shows when an amplitude correction becomes unstable. The examples are invented and use an ideal linear detector measuring flux, rather than astronomical magnitudes.
THE GOLD STANDARD OF ASTRONOMY
Choose the question you need to answer
Understand the window · Chapters 1–2. Separate timing quantities and derive the response.
Calculate and interpret · Chapters 3–5. Work attenuation, correction and model limits.
Practise and continue · Chapters 6–8. Change the period and check the evidence.
Open the chapter contents · Try the transfer checks · The Gold Standard of Astronomy
Chapter contents
Understand the window · Chapters 1–2
Calculate and interpret · Chapters 3–5
Practise and continue · Chapters 6–8
Imagine averaging the height of a moving wave over a time interval. A brief interval gives something close to the height at its midpoint. An interval covering a crest and a trough can cancel their departures. A camera measuring a variable source encounters the same mathematical issue when its output is proportional to collected photons.
Exposure duration tells us how long one measurement integrates. Cadence describes the spacing between successive measurements. A 30-second exposure repeated every five minutes has a 30-second integration window and a five-minute cadence. Neither number replaces the other.
The daily-alias guide asks whether different periods agree at sampled times. Here we ask what one finite measurement does to a signal even when its period is known. More frequent visits do not turn a long integration into an instantaneous reading.
Assume a stable sinusoid, constant mean flux, uniform sensitivity within each exposure and no saturation, blending or background error. These assumptions isolate the window’s effect. A real instrument may require a different response model.
Write instantaneous flux as F(t) = F₀[1 + a cos(2πt/P)]. Here F₀ is mean flux, P is period and a is fractional semi-amplitude. A semi-amplitude of 0.10 means excursions of 10% above and below the mean: a 20% peak-to-peak range relative to F₀.
Let an exposure of duration Δ be centred at tₘ. Recorded mean flux is the integral of F(t) from tₘ − Δ/2 to tₘ + Δ/2, divided by Δ. Raw photon counts grow with exposure time, whereas the mean flux of a constant source should stay comparable.
Integration gives F̄(tₘ) = F₀[1 + aη cos(2πtₘ/P)], where η = sin(πΔ/P)/(πΔ/P). Angles are in radians. This is the sinc response of a rectangular window: sinc(x) means sin(x)/x here, with limiting value 1 at zero.
For 0 < Δ/P < 1, η is positive and observed semi-amplitude is aη. More generally the amplitude is |aη|; a negative η represents a phase reversal. Check a calculator’s sinc convention before using it.
Chaplin and colleagues discuss finite-integration attenuation in stellar oscillation measurements. The derivation here isolates that measurement effect in a deliberately simpler, independently constructed model.
Take P = 10 minutes and a = 0.10. With Δ = 1 minute, η ≈ 0.9836. Observed fractional semi-amplitude is about 0.09836, or 9.836%. Most of the variation survives.
With Δ = 5 minutes, η = sin(π/2)/(π/2) = 2/π ≈ 0.6366. Semi-amplitude falls to 6.366% although the stipulated star is unchanged. At a midpoint aligned with a maximum, the average flux is approximately 1.06366 F₀ rather than the instantaneous 1.10 F₀.
With Δ = 10 minutes, η = 0. Every exposure averages one whole cycle and returns F₀, regardless of starting phase. A noiseless sequence therefore appears constant despite intrinsic 10% semi-amplitude.
The zero is exact for the stated signal and window. Period drift, additional frequencies or nonuniform sensitivity change the result. The counterexample is still useful: absence of variation in an averaged record does not, by itself, establish a constant source.
| Duration | Δ/P | Response η | Observed semi-amplitude |
|---|---|---|---|
| 1 minute | 0.10 | 0.9836 | 9.836% |
| 5 minutes | 0.50 | 0.6366 | 6.366% |
| 10 minutes | 1.00 | 0 | 0% |
If period and window are established and |η| is sufficiently far from zero, the model suggests a = a_obs/|η|. For observed semi-amplitude 0.030 and η = 0.6366, the corrected estimate is approximately 0.0471, or 4.71%.
Division also magnifies uncertainty. If the standard uncertainty of a_obs is 0.002 and η is treated as exact, its contribution to uncertainty in a is 0.002/0.6366 ≈ 0.00314. Uncertain period or window parameters introduce additional contributions.
Near a response zero, the correction becomes unstable. If η = 0.02, a signal uncertainty of 0.002 becomes 0.10 after division. At η = 0, division is undefined: these averages contain no amplitude information about the stipulated sinusoid. Decimal places cannot restore that information.
The useful next measurement changes the window. Shorter integration may preserve variation better, but collects fewer photons per exposure and can face readout and background limitations. Choose settings from the signal, noise and scientific question together.
A sinusoid has one frequency. A sharp eclipse, flare or transit-like dip has a different time pattern. Averaging can round its edges and change measured depth or duration in ways that one sinusoidal amplitude factor does not describe.
For those events, integrate a proposed light curve through the actual exposure windows and compare those predictions with observations. An instantaneous model evaluated only at midpoints can misrepresent a rapid event. The transit-candidate guide addresses the additional question of what caused the dip.
Flux and magnitude are also different. Magnitude is a logarithmic transformation of flux. Averaging magnitudes generally differs from transforming an averaged flux. The present equations concern a linear flux record.
Retain start and end times, timestamp convention, calibration and uncertainties. If a pipeline combines shorter exposures with intervening gaps, the combined window may differ from one continuous rectangular exposure. A nominal cadence alone cannot supply the missing information.
Transfer check: change the period
Keep Δ = 5 minutes but change P to 20 minutes. Now Δ/P = 0.25 and η ≈ 0.9003. A 10% intrinsic semi-amplitude appears as about 9.003%. The same integration attenuates the slower variable less.
Transfer check: keep the ratio
Compare Δ = 2 minutes, P = 8 minutes with Δ = 5 minutes, P = 20 minutes. Both have Δ/P = 0.25 and equal η. Absolute durations differ; this ideal fractional response depends on their ratio.
Transfer check: change the starting times
A two-minute sinusoid is measured through exact two-minute exposures. Will irregular exposure starts restore its amplitude? No. Each exposure still averages a complete cycle. Start times can address some sampling ambiguities, but this integration zero requires a different duration.
Practical evidence check
From an unfamiliar observing log, identify duration and cadence separately. Calculate Δ/P for proposed periods, predict attenuation and list which assumptions must hold before correction. Compare integrated model predictions with the data before interpreting a small amplitude.
Does longer exposure always give better evidence?
It may improve photon statistics while suppressing rapid changes. Estimating a constant mean and measuring a fast oscillation require different trade-offs.
Can amplitude be negative?
Amplitude is conventionally nonnegative. A negative response factor is represented by a phase reversal; retain that phase information while using its absolute value for amplitude.
Is averaging the same problem as a daily alias?
No. Aliasing concerns indistinguishable sampled patterns. Integration transforms each measurement before sampling. Both can affect one dataset.
Does a flat light curve rule out a variable source?
Only within the sensitivity, windows and models tested. The whole-cycle average supplies an explicit counterexample to a blanket claim.
A defensible amplitude claim connects a proposed flux history, the measurement window and recorded uncertainty. It distinguishes intrinsic variability from variability that the instrument averages away.
Explain the ten-minute example to a partner and let them change the exposure. If they can predict the cancellation before using the formula, the correction has become an understandable measurement model. Continue through the related guides to connect that model with sampling and physical interpretation.
Subject and learning map
The Gold Standard of Astronomy
Education: learning that transfers into life
How X Works: connected subject hub
Timing and physical interpretation
Daily aliases and choosing when to observe again
Vetting a transit candidate: competing explanations
Asteroseismology: reading stellar vibrations
Astrophysics: connect radiation with physical inference
Authoritative sources
Sources checked on 21 October 2026. Numerical cases and practice tasks are original teaching examples, not observations from a published experiment or official statistics.
Chaplin et al.: Super-Nyquist asteroseismology of solar-like oscillators — Primary research on stellar oscillations, sampling and finite-integration attenuation; the numerical cases here are independent teaching examples.