AstroSpeedrun
8.8

Other variable stars and reading light curves

Builds on
Sections 8.4, 8.6 and 8.7
Reading time
25 min
Exercises
3
Olympiad problems
1

In 1596 the Dutch pastor David Fabricius watched a third-magnitude star in Cetus fade over a few weeks until he could no longer see it. Years later it was back. The star, now called Mira, "the wonderful", brightens and fades every 332 days, from about magnitude 3 to magnitude 9 or fainter, so that for months each year it cannot be seen without a telescope. Mira is a cool red giant swelling and shrinking slowly. Today hundreds of thousands of variable stars are known, and the data-analysis round often asks which kind a light curve belongs to, or what it tells us. In this section we learn to read them.

8.8.1Kinds of variable star

Before you readMake a guess first

A light curve shows two dips in each cycle, of nearly equal depth, with the light changing smoothly all the time. What is the star most likely to be?

Variable stars fall into a few families, set by why the light changes.

  1. Pulsating stars swell and shrink, as Section 8.7 describes. Classical Cepheids have periods of 1 to 100 days, RR Lyrae stars less than a day, and the red giant Mira variables 100 to 1000 days with very large ranges in visual light.
  2. Eruptive and explosive stars brighten suddenly. They include novae (Section 8.6), dwarf novae, whose accretion discs flare up every few weeks, young stars still gathering gas, flare stars and supernovae.
  3. Eclipsing binaries change because one star passes in front of the other (Section 8.4). Algol-type stars are steady between eclipses, β Lyrae-type stars change all the time because their stars are stretched into egg shapes, and W Ursae Majoris stars are pairs in contact with periods under a day.
  4. Rotating stars change as dark spots or bright regions turn with them.

Three things identify a light curve. The first is whether it repeats, and with what period. The second is its range in magnitudes. The third is its shape: whether it rises faster than it falls, whether it has one minimum or two in each cycle, and whether it has flat parts.

Figure 8.8Light curves of variable stars
Kind
Period
Range
What to look for
The figure shows a model light curve for each of eight kinds of variable star, with brighter upwards, over about two cycles or over an outburst. The curves are simplified, with typical periods and ranges, and real stars differ in detail. The figure can be set to , , , , , , , or .
Quick checkHow much Mira changes

A Mira variable ranges over 6.0 magnitudes. By what factor does its visual brightness change?

times

8.8.2The mean magnitude

The period-luminosity relation of (8.12) uses the mean brightness of the star, and there are two ways to take a mean. Averaging the magnitudes over a cycle gives one answer. Averaging the flux and turning the result into a magnitude gives another, slightly brighter one, because the magnitude scale squeezes together the bright part of the cycle.

ResultThe intensity-mean magnitude(8.14)
⟨m⟩I=−2.5log⁡(1N∑k=1N10−0.4mk)
⟨m⟩Ithe magnitude of the mean flux over a cycle
mkthe magnitudes at N moments spread evenly over the cycle
For a light curve that rises fast and falls slowly, both means lie below halfway between the brightest and faintest magnitudes, since the star spends longer near minimum. A problem will say which mean it wants, and when it gives only the brightest and faintest magnitudes, halfway between them is the usual estimate.
Example 8.15Two means of a simple light curve

A star spends half its time at magnitude 14.1 and half at 14.8. Find the mean of the magnitudes and the intensity-mean magnitude.

Solution

The mean of the magnitudes is (14.1+14.8)/2=14.45. The fluxes, in units where magnitude 0 has flux 1, are 10−5.64=2.29×10−6 and 10−5.92=1.20×10−6, with mean 1.75×10−6.

AnswerBy (8.14), ⟨m⟩I=−2.5log⁡(1.75×10−6)=14.39, brighter by about 0.05 magnitudes than the mean of the magnitudes.
Example 8.16The true period of a contact binaryfinish it

A star's light curve has minima of almost equal depth, one every 0.18 days, with smooth changes between them. Folding the data on 0.18 days gives a neat curve. What is the true period?

The start

A contact binary has two eclipses in each orbit, one when each star passes in front of the other. When the stars are alike, the two minima look nearly the same, and folding on the time between them lays the primary and secondary minima on top of each other.

Your stepNow decide the period.
days
Common errorsSection 8.8
  1. Halfway between the brightest and faintest magnitudes is not the mean magnitude of a skewed light curve. A star that rises fast and fades slowly spends most of its time on the faint side.
  2. The minima of an eclipsing binary come twice in each orbit. Taking the time between neighbouring minima as the period halves it when the minima look alike.
  3. A large range in visual light, as in Mira, does not mean a large change in total light. Most of the visual change comes from the star's cooling and from molecules forming in its atmosphere.
  4. A repeating light curve need not belong to a pulsating star. The shape tells the difference, since an eclipse gives flat or sharp minima and a pulsating star gives a smooth rise and fall with one maximum in each cycle.
ExercisesSection 8.83 questions
  1. 8.8.1

    A star spends half its time at magnitude 3.0 and half at 4.0. What is its intensity-mean magnitude?

    mag
  2. 8.8.2

    A light curve repeats every 0.6 days, rises by 1 magnitude in about an hour and fades slowly for the rest of the cycle. What kind of star is it?

  3. 8.8.3

    A star stays at a steady brightness except for a 1.3-magnitude dip lasting ten hours, which comes back every 2.9 days, with a tiny dip halfway between. What kind of star is it?

Olympiad problemsPast IOAA problems, restated1 problem
2013A Cepheid in a nearby galaxy10 points

Adapted from IOAA 2013 (Volos), theory problem S4. We have restated it in our own words and kept the original data. The original gave a period-luminosity graph and a light curve. We give the readings from them instead. The solution was last checked on 21 October 2026.

A classical Cepheid in a galaxy of the Local Group has a light curve that varies between visual magnitudes 14.1 and 14.8 with a period of about 11 days. A graph of the period-luminosity relation gives an absolute magnitude of −4.2 for this period.

(a) Estimate the distance of the Cepheid. [8 points]

(b) Revise the estimate for an extinction of A=0.25 magnitudes towards the Cepheid. [2 points]

Answer(a) About 54 kpc. (b) About 48 kpc, closer than before, since part of the faintness is caused by dust. The official solution adds the extinction instead of taking it off, and also misprints the powers of ten, giving 57.5 kpc and then 64.5 kpc. A careful reading of the light curve, which spends longer near minimum, gives a mean nearer 14.5, which moves both answers out by about 2%.