# Cepheid variable period magnitude relationship questions

### PHY / The Cepheid Period-Luminosity Relation

There is a relation between pulsation period and luminosity. Cepheid Variable Stars & Distance Determination. Discovery of the Period - Luminosity Relationship; Calculating Distances Using Cepheids; Standard Candles. The relationship is given approximately by where m is the apparent magnitude and the distance d is in parsecs. (One parsec Find the absolute magnitudeMv and the period of this Cepheid variable. Solutions for Problems in Chapter

Such period-luminosity relationships are invaluable to astronomers as they are a vital method in calculating distances within and beyond our galaxy. Discovery of the Period - Luminosity Relationship Credit: Eventually she classified 47 of these in the two clouds as Cepheid variables and noticed that those with longer periods were brighter than the shorter-period ones. She correctly inferred that as the stars were in the same distant clouds they were all at much the same relative distance from us.

Any difference in apparent magnitude was therefore related to a difference in absolute magnitude. When she plotted her results for the two clouds she noted that they formed distinct relationships between brightness and period.

### Cepheid Variable Stars & Distance

Her plot showed what is now known as the period-luminosity relationship; cepheids with longer periods are intrinsically more luminous than those with shorter periods. The Danish astronomer, Ejnar Hertzsprung quickly realised the significance of this discovery. By measuring the period of a Cepheid from its light curve, the distance to that Cepheid could be determined. He used his data on nearby Cepheids to calculate the distance to the Cepheids in the SMC as 37, light years away.

From this he could infer the distance to globular cluster too distant to have visible Cepheids and realised that these clusters were all essentially the same size and luminosity.

By mapping the distribution and distance of globular clusters he was able to deduce the size of our galaxy, the Milky Way. Using these he determined that their distances wereandlight years respectively.

He thus established conclusively that these "spiral nebulae" were in fact other galaxies and not part of our Milky Way. This was a momentous discovery and dramatically expanded the scale of he known Universe. Hubble later went on to observe the redshift of galaxies and propose that this was due to their recession velocity, with more distant galaxies moving away at a higher speed than nearby ones.

## Cepheid Variable Stars & Distance Determination

This relationship is now called Hubble's Law and is interpreted to mean that the Universe is expanding. Period-luminosity relationship for Cepheids and RR Lyrae stars. Let us now see how this relationship can be used to determine the distance to a Cepheid. Photometric observations, be they naked-eye estimates, photographic plates, or photoelectric CCD images provide the apparent magnitude values for the Cepheid. Plotting apparent magnitude values from observations at different times results in a light curve such as that below for a Cepheid in the LMC.

From the light curve and the photometric data, two values can be determined; the average apparent magnitude, m, of the star and its period in days. In the example above the Cepheid has a mean apparent magnitude of Knowing the period of the Cepheid we can now determine its mean absolute magnitude, M, by interpolating on the period-luminosity plot. The one shown below is based on Cepheids within the Milky Way.

Astrophysics - Distance - Cepheid variables (1/2) - (IB Physics, GCSE, A level, AP)

Along with the temperature changes their radii also change during each pulsation e. The brightness changes are more pronounced at shorter wavelengths. Pulsations in an overtone higher than first are rare but interesting. Stars pulsating in an overtone are more luminous and larger than a fundamental mode pulsator with the same period. When the helium core ignites in an IMS, it may execute a blue loop and crosses the instability strip again, once while evolving to high temperatures and again evolving back towards the asymptotic giant branch.

The duration and even existence of blue loops is very sensitive to the mass, metallicity, and helium abundance of the star. In some cases, stars may cross the instability strip for a fourth and fifth time when helium shell burning starts.

More massive and hotter stars develop into more luminous Cepheids with longer periods, although it is expected that young stars within our own galaxy, at near solar metallicity, will generally lose sufficient mass by the time they first reach the instability strip that they will have periods of 50 days or less.

Very massive stars never cool sufficiently to reach the instability strip and do not ever become Cepheids. At low metallicity, for example in the Magellanic Clouds, stars can retain more mass and become more luminous Cepheids with longer periods.

This is due to the phase difference between the radius and temperature variations and is considered characteristic of a fundamental mode pulsator, the most common type of type I Cepheid.

In some cases the smooth pseudo-sinusoidal light curve shows a "bump", a brief slowing of the decline or even a small rise in brightness, thought to be due to a resonance between the fundamental and second overtone. The bump is most commonly seen on the descending branch for stars with periods around 6 days e.

As the period increases, the location of the bump moves closer to the maximum and may cause a double maximum, or become indistinguishable from the primary maximum, for stars having periods around 10 days e. At longer periods the bump can be seen on the ascending branch of the light curve e.

X Cygnibut for period longer than 20 days the resonance disappears.