Astronomy · Why stellar light changes

Eclipses, rotation and surface features

Separate an observed brightness recurrence from an orbital or rotation period, using binary geometry, surface patterns and a two-component synthetic signal.

Continue in the approved chapter order. Necessary vocabulary, quantities and assumptions are explained in the lesson.

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After reading, you can

Explain why an eclipse and a rotating surface pattern can change unresolved stellar light.
Calculate how pattern symmetry can produce two brightness cycles per supplied rotation.
Identify additional observations needed to distinguish orbital motion, rotation and intrinsic variability.

The source need not pulsate to appear variable

An unresolved star can change brightness because its appearance changes with viewing geometry. A companion may pass in front of an emitting surface, or rotation may carry a darker or brighter surface region through the visible hemisphere. The measured light is a sum over the visible system through an observing band.

A starspot is a region with different physical conditions and lower surface brightness than its surroundings in the relevant comparison. It still emits light. Its contribution depends on contrast, projected area and location; a brightness deficit does not directly measure its area without those assumptions.

An eclipse changes how much of each component is visible. Both stars in a binary orbit their common centre of mass. A model in which one dark disc crosses a fixed luminous disc can illustrate some geometry, but it is not a complete description of every stellar binary.

Photometry therefore first establishes a change in the received band light. Calling a recurrence a rotation period or an orbital period is a further inference about what produces that light.

Two eclipses can have different depths

In an eclipsing binary, one component passes in front of the other at one conjunction and behind it at the other when the orientation permits both events. Conjunction describes alignment along the line of sight; it does not imply that the stellar surfaces physically collide.

The eclipse depth depends on the overlapping projected areas and their surface brightness, along with any additional light in the aperture. Limb darkening—the change of apparent brightness across a stellar disc—can affect the shape. A grazing overlap and a total eclipse supply different geometric information.

Two similar eclipses can tempt a period finder to select half the orbital period. Unequal depths can reveal an alternating pattern, but depth alone still does not supply both masses or the full three-dimensional orbit. Spectroscopic velocities add information about orbital motion.

Other binary effects, including changing projected shapes of tidally distorted stars, can modulate the light between eclipses. A periodic curve is not sufficient evidence that every minimum is an eclipse. Its complete shape and independent observations must be explained by the proposed model.

A supplied rotation can produce a shorter brightness recurrence

For an original synthetic signal, stipulate a physical rotation hypothesis of 12 days. Let the band flux follow F(t)/Fref = 1 − 0.02 cos(4πt/12 days) − b cos(2πt/12 days), where Fref is a fixed positive reference flux and b is dimensionless. Time is inserted in days so both cosine arguments are dimensionless radians.

This is a Fourier surrogate for testing symmetry, not a recovered spot map or an eclipse simulation. Set b = 0 for a symmetric pattern. Its flux repeats after 6 days even though the supplied physical rotation is 12 days: the visible pattern repeats twice per stipulated turn.

Now set b = 0.004. The additional component distinguishes the two halves of the turn. At 0 days the normalized flux is 0.976; at 6 days it is 0.984. Their difference is 0.008, while the complete asymmetric signal repeats after 12 days.

The table gives selected exact-model samples rounded for display. The 3-day and 9-day entries are not the exact maxima of the asymmetric curve; adding the slower cosine slightly shifts its peaks. The small vertical range in the accompanying diagram is explicitly a zoom, with zero flux outside the plotted range.

Original continuous signals under a supplied 12-day rotation hypothesis. These rounded model samples are not observations or a unique surface map. Fref is the same fixed positive reference for both signals.
Time (days)	F/Fref, b = 0	F/Fref, b = 0.004
0	0.980000000	0.976000000
1.5	1.000000000	0.997171573
3	1.020000000	1.020000000
4.5	1.000000000	1.002828427
6	0.980000000	0.984000000
7.5	1.000000000	1.002828427
9	1.020000000	1.020000000
10.5	1.000000000	0.997171573
12	0.980000000	0.976000000

Recurrence and a rotation hypothesis

Scroll sideways to see the full diagram. Its explanation is below.

Original synthetic analytic signals, not measured light curves or a reconstructed stellar surface. A physical rotation period of 12 days is supplied as a hypothesis. In one fixed passband the model is F(t)/Fref = 1 - 0.02 cos(4 pi t / 12 days) - b cos(2 pi t / 12 days), with t in days and a fixed positive reference flux Fref. The blue solid curve and filled circles use b = 0; the orange dashed curve and hollow diamonds use b = 0.004. Both curves have the same time and flux scales. The horizontal linear axis is time from 0 to 12 days. The vertical linear axis is explicitly zoomed to relative flux 0.97 through 1.03, not zero based; zero flux lies outside the plot. A grey dotted line marks F/Fref = 1. The blue signal repeats after 6 days although the supplied rotation hypothesis is 12 days. The orange signal has a 12-day repeat and unequal alternating minima: 0.976 at 0 days, 0.984 at 6 days, and 0.976 at 12 days. Their difference is 0.008 in relative flux. The nine table rows are time in days, supplied rotation phase t/12 days, blue flux, orange flux: 0, 0, 0.980, 0.976; 1.5, 0.125, 1.000, 0.997171573; 3, 0.25, 1.020, 1.020; 4.5, 0.375, 1.000, 1.002828427; 6, 0.5, 0.980, 0.984; 7.5, 0.625, 1.000, 1.002828427; 9, 0.75, 1.020, 1.020; 10.5, 0.875, 1.000, 0.997171573; 12, 1, 0.980, 0.976. Decimal precision reports model arithmetic, not measurement uncertainty. Dense analytic curves include the slightly shifted orange maxima, so the 3-day and 9-day rows are not labelled exact maxima. The asymmetric analytic maximum is 1.0201 at approximately 3.095533 days and 8.904467 days. Symmetry of an observed signal alone would not determine a unique physical rotation, surface-feature map, emitting area or cause. This is not an eclipse model or an observing-cadence alias.

Features evolve while the star rotates

McQuillan, Mazeh and Aigrain discuss the half-period ambiguity caused by active regions on opposite hemispheres. An autocorrelation compares a light curve with time-shifted copies of itself: peaks indicate repeated patterns at particular time lags. Such a peak describes self-similarity in the signal before a physical rotation is inferred.

Real surface patterns can grow, decay and migrate. Different latitudes can rotate at different angular rates, so a feature-tracking period need not represent one universal rotation rate for the entire star. The observed modulation also depends on inclination and where the features lie.

A change in amplitude can reflect changing feature contrast or geometry, rather than a comparably large change in the star’s total luminosity. Conversely, a weak modulation does not prove an unspotted surface: the visible configuration can hide or cancel a signal.

This pattern symmetry is distinct from cadence aliasing. The synthetic symmetric function really repeats every 6 days as a continuous signal; no observations were removed to create that recurrence. A real data set can have both physical pattern symmetry and additional aliases from its observing window.

Several causes can coexist in one system

Useful tests include the alternation of eclipse depths, recurrence across successive cycles, band-dependent changes and spectral velocity or line-profile measurements. The interpretation should predict these jointly rather than treating any one resemblance as a guaranteed diagnosis.

Pietrzyński and collaborators provide a concrete example of coexistence in their 2010 study of OGLE-LMC-CEP0227. A pulsating classical Cepheid belongs to an eclipsing binary, and the analysis separates orbital and pulsational contributions using photometry and spectra. The lesson here is the separation of mechanisms, rather than a claim that one label excludes the others.

Folding observations on a trial period—placing times at their fractional cycle phases—can reveal a repeatable shape. It can also hide differences between alternating cycles or erase evolution. Compare the unfolded sequence and the observing intervals before accepting a compact folded display.

A defensible report distinguishes the measured brightness recurrence, the chosen orbital or rotation model, and the quantities inferred within it. That distinction allows new velocities, extra cycles or better surface modelling to resolve an ambiguity without changing the original measurements.

Worked examples

Check the function after half a supplied turn

Why does the b = 0 signal repeat after 6 days?

Advancing t by 6 days adds 4π × 6/12 = 2π to its cosine argument.
A cosine is unchanged by a 2π advance.
The supplied 12-day physical turn therefore contains two identical photometric repeats in this symmetric surrogate.

Its continuous brightness recurrence is 6 days; the 12-day physical rotation is a stipulated hypothesis, not a uniquely recovered quantity.

Test whether the two half-turns remain identical

What difference distinguishes the b = 0.004 signal at 0 and 6 days?

At 0 days both cosines equal 1, giving 1 − 0.02 − 0.004 = 0.976.
At 6 days the first cosine equals 1 and the second equals −1, giving 1 − 0.02 + 0.004 = 0.984.
Subtract the two normalized fluxes: 0.984 − 0.976 = 0.008.

The difference is 0.008Fref in flux. It breaks the half-turn symmetry in this model without specifying a unique surface configuration.

Try the reasoning

What is the smallest positive repeat interval of the supplied b = 0 signal?
Show the answer and reasoning
6 days.
Only the cosine with angular frequency 4π per 12 days remains.
Should a starspot automatically be modelled as a hole emitting no light?
Show the answer and reasoning
No.
Its band-dependent brightness contrast must be specified.
Are the tabulated 3-day and 9-day values necessarily the exact maxima of the asymmetric signal?
Show the answer and reasoning
No.
The added slower cosine changes the slope at those times and shifts the peaks.
Does evidence for stellar pulsation rule out an eclipsing binary in the same unresolved system?
Show the answer and reasoning
No.
Pulsational and orbital changes can coexist and require a joint model.

What to carry forward

An observed light-curve repeat is distinct from an inferred physical rotation or orbit.
Surface symmetry can create two brightness cycles per turn even in a continuously observed signal.
Photometry, spectra and evolving geometry together help distinguish mechanisms that may coexist.

References

Teaching resources and research supporting this lesson’s explanations and examples.