AC07 · The times you choose

Observing Cadence and Aliases

Why can the same variable star appear to have a different period when you change the observing times? Regular gaps can make different periods match the same measurements. Moving or spreading the visits changes those competing matches. Brief flashes can fall entirely between exposures, and an alert must wait for a useful exposure to finish and be processed.

Drag the observing window around a clock. Sample a pulsating star and a brief flash with the same camera schedule, compare period aliases, and see when the first alert can arrive.

Saved exposures $24$

Strongest trial period $2.35\,\mathrm d$

Daily window power $1.00$

Flash trigger exposures $0$

Choose when the shutter opens

Window center

3

t_{\rm center}\;[\mathrm h]

Window duration

0.5

T_{\rm window}\;[\mathrm h]

Visit spacing

60

\Delta t\;[\mathrm{min}]

Drag the mint window around the clock. The same schedule samples both sources; changing its times saves fresh synthetic exposures.

Weather, jitter and alert processing

Night-to-night time jitter

0

J\;[\mathrm h]

Cloud-loss probability

0

p_{\rm cloud}

Processing latency

20

t_{\rm process}\;[\mathrm{min}]

Five-minute exposures average the incident source intensity before Gaussian noise. Timing jitter and weather use one fixed schedule seed so their changes can be compared; detector noise has its own exported seed. Period search spans one tenth to five cycles per day on a fixed grid.

Read the cadence course and OGLE replay

Model assumptions

The camera measures two synthetic sources through five-minute rectangular exposures with independent Gaussian noise: a sinusoidal variable and a Gaussian flash. A fixed nightly schedule, optional jitter and weather gaps determine all saved samples. A weighted floating-mean single-harmonic period search analyzes the variable. The clock is an observing scheduler in relative hours, not a sky-visibility or daylight ephemeris; source size and propagation distances are schematic. Alert timing includes exposure completion and a fixed processing delay, relative to an explicitly defined continuous threshold crossing. A fitted peak is not a probability of the correct period. The separate observed OGLE tutorial calibrates full-search false alarms and injection recovery at its actual sample times.

The clock shows a nominal observing window in relative hours; small ticks show actual scheduled visits on the replayed day. The sky is schematic, and the camera screen retains its last acquired intensity.

One variable, chosen observing times

Competing trial frequencies

The sampling window

Did the camera catch the flash?

Gray and muted gold are ideal source references; cyan and bright gold are noisy exposure samples with error bars. Violet uses observing times only, and a red dashed line marks the flash trigger threshold.

The saved schedule missed the flash: no exposure crossed the trigger threshold.

What the fitted period and alert mean

The variable has an injected period of 0.635 days. A single-harmonic weighted floating-mean fit searches all declared trial frequencies; equal daily sampling can give indistinguishable aliases. The browser reports a peak, with no false-alarm probability. The real OGLE tutorial simulates full-search null maxima at exactly its observed times and grid, then separately counts correct, aliased and missed injections.

The transient is a separate Gaussian source. Its trigger uses five times the known per-exposure noise. The reference time is when its ideal continuous intensity first crosses that same threshold, not its peak or an emission start. Noise can cause an earlier trigger. This is a simplified trigger, not a calibrated survey classifier.

Start with the essentials

Focus question: Will adding visits at the same hour resolve a daily alias?
One-sentence intuition: The pattern of sample times determines which trial frequencies compete.

Core mathematical model

Daily aliases

f_{\rm alias}=|f_{\rm true}+k f_s|,\qquad k\in\mathbb Z

For exactly regular samples, adding an integer number of cycles between visits leaves the sampled phases equivalent, with an adjusted phase.

Sampling window

W(f)=\frac1{N^2}\left|\sum_{j=1}^N e^{2\pi i f t_j}\right|^2

This transform uses the actual observing times only. A strong daily window peak predicts competing daily aliases.

A floating-mean period search

y_j=c+a\cos(2\pi f t_j)+b\sin(2\pi f t_j)+\epsilon_j,\quad P(f)=1-\frac{\chi^2(f)}{\chi^2_0}

At each trial frequency fit the offset, cosine and sine together. Power measures improvement over a constant fit, with the same measurement-error weights.

Common difficulties

A peak is not a probability

Typical misconceptionThe tallest peak has a ninety-percent power, so its period is ninety-percent likely to be correct.

Better mental modelPower is a normalized fit improvement. False-alarm calibration asks how often noise gives a peak at least that high somewhere in the full search. Choosing the correct alias needs a separate injection-recovery experiment or additional observations.

Run the experiment

01
Break a daily repetition
Compare One visit each night with Spread visits through the night. Predict whether the period or the star itself will change.
What to observe: The source period stays fixed, but extra phases distinguish the aliases.
02
Move a window across a flash
Choose Miss the short flash, then drag the window toward the sixteen-hour mark.
What to observe: The sampled flash crosses the trigger threshold only when an exposure overlaps it.
03
Wait for the alert
Choose Catch the short flash and add sixty minutes of processing latency in the advanced settings.
What to observe: The physical samples stay at the same times; the first alert moves an hour later.

Observing Cadence and Aliases

The camera chooses which changes survive

Core mathematical model

Daily aliases

Sampling window

A floating-mean period search

Common difficulties

A peak is not a probability

Run the experiment

Break a daily repetition

Move a window across a flash

Wait for the alert

Related experiments and articles

Measuring the sky

Observing cadence and aliases