Separating competing causes of variability

Begin with what actually changed

A brightness curve can be periodic, irregular, asymmetric or punctuated by brief events. These descriptions summarize the observed signal. Explaining the signal requires a model of the source, its surroundings, the viewing geometry and the observing system.

The previous lessons introduced pulsation, rotation, eclipses, magnetic flares and accretion-related outbursts. An observation can support more than one candidate explanation. It can also contain several physical contributions at once.

AAVSO’s variable-type conventions make a useful distinction: an OR notation records alternative classifications, while an AND notation records different variability types seen together. Uncertainty about which cause is present is logically different from evidence for two simultaneous causes.

Preserve the measured band fluxes, observation times, uncertainties and calibration before compressing them into a class label. A revised classification should improve the explanation of the evidence, not silently replace that evidence.

An amplitude pattern can change without either component changing

For an original noiseless model, define fractional band-flux variation s(t) = 0.01 cos(2πf₁t) + 0.01 cos(2πf₂t), with f₁ = 1.01 cycles per day and f₂ = 0.99 cycles per day. The corresponding normalized total flux is 1 + s(t), which remains positive. The two components have fixed amplitude, frequency and relative phase.

Using cos a + cos b = 2 cos[(a − b)/2] cos[(a + b)/2], write the same function as s(t) = 0.02 cos[2π(0.01 day⁻¹)t] cos[2π(1.00 day⁻¹)t]. The fast cosine is a carrier; its coefficient A_signed(t) = 0.02 cos[2π(0.01 day⁻¹)t] is a signed envelope.

The visible amplitude envelope is |A_signed(t)|. It first reaches zero at 25 days and repeats every 50 days, the reciprocal of the frequency separation 0.02 cycles per day. The signed envelope repeats after 100 days. At 50 days its sign reverses, which is equivalent to a phase reversal of the carrier.

These statements describe a continuous function, before any sampling or noise is introduced. The changing envelope is interference between two fixed components, often called beating. It does not require a changing amplitude of either constituent mode.

Keep three different time descriptions separate

In this specially chosen example the full s(t) also repeats after 100 days. Advancing by 100 days adds 101 cycles to the first component and 99 to the second. Both return to the same phase; no smaller positive advance is a common integer number of their cycles.

At 0 days the variation is +0.02, whereas at 50 days it is −0.02. Those values distinguish the 50-day amplitude-envelope recurrence from a repeat of the complete waveform. At 25 days the signed envelope is zero, so s is zero regardless of the carrier phase at that instant.

The selected samples below do not define a new interpolation rule. The displayed continuous formula defines the entire signal. At 12.5 and 37.5 days, the carrier is −1, which reverses the sign of the signed envelope in s(t).

A single changing stellar parameter could produce other amplitude changes, but this constructed counterexample shows why an envelope alone does not establish that explanation. A good fit with two sinusoids also does not uniquely identify pulsation: the physical interpretation still needs evidence beyond algebra.

Different physical explanations predict more than brightness alone

For a candidate pulsation model, examine frequencies, phase relations and spectral velocities or line-profile changes consistent with moving layers. For a candidate binary, compare eclipse geometry and the orbital velocities of its components. Neither test should ignore additional surface or pulsational contributions.

For rotation, compare repeated pattern shapes across successive cycles and test the persistence of surface features. A spot pattern may evolve; differential rotation and viewing inclination affect its interpretation. Surface symmetry can make the brightness recurrence shorter than a turn even with ideal continuous observations.

For eruptive behaviour, compare the event’s spectral and temporal evolution across bands. Accretion and changing circumstellar extinction can coexist in young stars. An abrupt brightening need not be a flare, and a colour change alone need not give a unique temperature or extinction.

OGLE classification studies and the Cepheid binary analysed by Pietrzyński and collaborators show the value of combining diagnostics. The latter system needs both orbital and pulsational contributions. An inference can become stronger through joint constraints without pretending that every star belongs to a single exclusive mechanism.

Use independent evidence and preserve unresolved alternatives

Check whether the variation follows the sky source or the detector position, whether comparison sources show related changes, and whether calibration or image-subtraction residuals can explain it. Revisit the time convention, observing window and finite exposures before interpreting a new period or phase shift.

Evidence derived from the same images or reference calibration can be correlated. A light-curve shape statistic, a periodogram and a classifier score are not automatically three independent confirmations. Conditional statistical support depends on the actual data and model, rather than how many summary numbers are reported.

Prefer a follow-up observation where credible alternatives predict different results. It might sample an alternating minimum, obtain a spectrum at a discriminating phase, resolve a blend or measure another band with suitable calibration. Additional points at uninformative times can leave the same ambiguity intact.

Report the established observations, the supported mechanism or combination, and the remaining degeneracy separately. Where evidence is insufficient, a bounded uncertain classification is a useful scientific result. It preserves a clear next test instead of turning a plausible resemblance into a permanent explanation.

Worked examples

Try the reasoning

1. Must either 0.01-amplitude component grow when the combined envelope grows?

   Show the answer and reasoning

   No.

   Their relative phase changes while both component amplitudes remain fixed.

2. What is s(t) at 25 days in the exact supplied model?

   Show the answer and reasoning

   0.

   The signed envelope is zero at that instant.

3. Is “pulsation or eclipse” the same claim as “pulsation and eclipse”?

   Show the answer and reasoning

   No.

   The first leaves alternative causes unresolved; the second asserts coexisting contributions.

4. Do three statistics from the same calibrated light curve automatically constitute independent evidence?

   Show the answer and reasoning

   No.

   Shared measurements and calibration can correlate their information and errors.

What to carry forward

• Variability descriptions, physical alternatives and simultaneous mechanisms are different claims.
• Two fixed-frequency signals can create a changing envelope without changing either component amplitude.
• Discriminating follow-up and an explicit observing model help resolve causes more than repeated labels or correlated statistics.