# Frédéric Dahringer

"EAAE Summerschools" Working Group

CLEA - France

#### Abstract

The aim of this activity is to discover that:

- the brightness of the star d Cephei is periodic,

- the variations of the brightness of Cepheid stars depends on their magnitude,

- this property can be used for the determination of the distances of the near galaxies.

#### δ Cephei, the star

- It is a variable star in Cepheus. (fig.1).

- Its magnitude varies with a very regular period (5,36 days) from 3,4 to 4,6.

- Its distance to us is 300 parsecs.

- Its diameter is 60 time bigger than the Sun one, and varies about 8% during a period.

- Its spectral type evolutes from F5, with a surface temperature of 6580 K, to the G2 type so that the temperature of its surface is only 5800 K.

#### A small lexicon

A parsec

It is an unity of distance: the distance at which a star would have an annual parallax of one second 0f arc. That means : from this star the radius of Earth's path around the Sun would be seen with an arc of one second. 1 parsec = 3,08 × 1013 km = 3,26 light-year

The apparent magnitude of a star

The stars appear to us more or less brilliant. They have been classified by the observers , in the old time according to their brightness, first "Greatness" (the most brilliant) to the 6th "Greatness", at the limits of visibility.

In modern time, it came be possible to measure the brightness of a star: it is the quantity of light energy which is received each second of time (the power) en a plane surface, perpendicularly to the corning direction of the light.

This measurable brightness has been connected to the ancient classification with a new word: the apparent magnitude of the star.

If we have two stars with the brightness E1 and E2, the apparent magnitudes will be connected through the relation:

m2 - m1 = -2,5 log (E2 / E1)

The value of the apparent magnitude is rather the same number than the "greatness" if we give to 1 Ursa Minor the magnitude 6,55.

The absolute magnitude

Two stars with the same brightness, and se with the same apparent magnitude, could have the same Luminosity if they are at the same distance of us, or could have different luminosities if their distances are not the same.

The luminosity is the the quantity of energy which is given by the star, each second of time in definite wave-lengthes.

If the distance of the star is d, and its brightness E, then its luminosity is given by the relation:

E = L /4πr2

The energy which is produced each second will be distributed on the surface of the sphere which radius is the distance covered by the light.

The absolute magnitude has been introduced in order to compare the stars each other. In this case we suppose that all the stars are at the same distance of 10 parsecs.

Using the relation (1) we can write the next one; M is the absolute magnitude and D is the distance of 10 parsecs.

m - M = - 2,5 1og ( L /4πr2 : L /4πr2 )

or:

m - M = 5 log d - 5

Spectral type

The stars art classified according to their light spectrum. The absorption rays which are to be seen on a light spectrum give us informations about the cempesitien and the temperature of the star's atmosphere.

The spectrum of F5 type stars have large hydrogen rays and faint calcium rays.

To the opposite the 62 type stars have a spectrum with faint hydrogen rays and large calcium rays.

#### Light variations of δ Cephei

The next data give us the apparent magnitude in visual wave-lengths, me, according to the date of the observations; rather the same time each day (observations of Mr Schweitzer, 1979, in Astronomies , A. Acker et C. Joshed). One compare Cepheid with stars in the neighbourhood which magnitudes are near of the different magnitudes of d Cephei. (Argelander method).

Of course, the observations have be done during the nights and with clear sky so it is not possible to have data exactly for one or two whole periods.

So, we can use the "phases-method" we have the estimate the value of the period and then we calculate how many periods have spend from first observation ~ the very day cant. This number is not integer; the decimal part is the "phase" and will lie the abscises on the diagram.

It will work as if we had the values only for one period.

To avoid too many calculations, we shall use a graphic method. The data will give us several pieces of the curve that we will put together to obtain a periodic curve. It works as if we delete the periods without observations.

The result is rather the same that the vault given in first paragraph.

#### What is a Cepheid star?

The Cepheid stars are at a particular moment of the evolution of stars, between red giant stars and white dwarf.

The fusion of Hydrogen is at its end in the centre of the star an Helium stars to "burn" giving a lot of energy. This energy permits the Hydrogen fusion in a belt around the nuclear of the star. The star become instable and deliver energy, periodically, so that its brightness and diameter have periodic variations.

#### The "Period - apparent magnitude" relation

Henrietta Leawitt's discovery

During 1912, Henrietta Leavitt measured periods and apparent magnitudes of 24 Cepheid stars in the Small Magellanic Cloud.

Light curves of four of them are given on fig 3.

We shall measure the period and the mean apparent magnitude cf these four Cepheid stars and then build the graph

mv = mean apparent magnitude

P = period

using aise the other data given we obtain an affine function:

mv = a log P + b

The sizes of the Small Magellanic Cloud are very small cornpared to its distance te us; so we can consider that all the stars are at the same distance of us.

In consequence, the relation between period and apparent magnitude that we have différence of a constant number (m - M = 5 log d - 5).

This constant number can be obtaintd when we measure the periode of a Cepheid star which distance is known.

in the opposite, if we measure the period of a Ccpheid star, we shall know its absolute magnitude ; comparing it with its apparent magnitude we can calculate its distance.

#### Relation between periode and absolute magnitude

Observing Cepheid stars of our galaxie (the Milky Way) which distances are known, Krafts has obtained, in 1961, the data below. The curve = f (log P) is a straight line nearly parallel to the straight line concerning the Cepheid stars in Small Magellanic Cloud.

The difference between mv and Mv permits us to calculate the distance of the Small Magellanic Cloud.

Mv = 5 log d - 5 = 15,5

So

d = 50 000 parsecs = 1 630 000 light-year

Nowadays we accept the value: 4 200 000 l.y.

#### Conclusion

This discovery cf the relation between period and magnitude has been very important: Henrietta Leawitt is the first famous astronomer women, though she has not been encouraged in her work by her colleagues.

This relation has been used by Hubble to prove that the M 31 Nebula was another galaxy, far from the Milky Way.

However, we can use this method only to measure the nearby galaxies in which we can distinguish Cepheid stars. We have also to increase in the calculation, the effect of light absorption in intergalactic space, which modify the apparent magnitude.

During the last years, this relation has been recalculated with the very accurate data that we have got with the Hipparcos satellite.