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Delta Cephei

What is a Star?

To begin to understand the complexities of variable stars, such as Delta Cephei, we need to understand what a basic star is.

A star is a giant ball of hot glowing gases, and the reactions that occur at the centre of a star provide energy to the star, and this energy is emitted as light and other forms of electromagnetic radiation. It is this energy that we see when we look at the stars at night.

Radial Structure of a Star Like our own sun and planet, stars are made up of layers or radial structures. The innermost of these layers is the core.


The Core

The core is the reaction zone of the star where the nuclear reactions that energise the star take place. The initial reaction is always the fusion of hydrogen to form helium. However as stars evolve, helium begins to accumulate at the centre of the star and causes the hydrogen fusion zone to move out, forming a shell around a small core of helium. If the star becomes massive enough, the pressure and temperature at this centre can be sufficient to cause helium fusion to occur. It is possible for 2 or possibly more reaction shells to exist simultaneously.


Convective and Conductive Zones

Moving out through the star, we come to the convective zone and then the conductive zone. In these sections, nuclear reactions do not take place due to lack of temperature, they are too cool. However, as the their names suggest, energy from the reaction core moves outwards through these areas to the next zone of the star, the photosphere.



Photosphere and Chromosphere


Photosphere and Chromosphere of a Star The photosphere is the visible part of the star that we can see. It is characterised by emission spectra. A rarefied chromosphere, characterised by selective absorption surrounds the photosphere.



The Corona


Corona of a StarThis is the outermost zone of a star. A tenuous, wispy phenomenon that extends for several stellar diameters into space away from the star. Oddly the corona is generally hotter than the surface of the star. It is not known exactly why this is, but the current theory is that the corona is heated by magneto-hydrodynamic waves or other magnet phenomena, much like particles of plasma are heated to millions of degrees by interactions with Jupiter’s magnetosphere.


Main Sequence Stars and Variables


Most “normal” stars, such as our sun, can be put into a general group called “Main sequence stars” in which luminosity increases with surface temperature. This means that the luminosity of a star can increase due to a rise in temperature OR because they are very large OR a combination of the both. We can draw a basic graph called the Hertzsprung-Russell Diagram. This is a graph of absolute magnitude against spectral class. Working independently, Hertzsprung and Russell discovered that for a representative sample of stars, there was a clear relationship between spectral class (which depends on temperature) and absolute magnitude (proportional to luminosity). The basic graph looks like this:


Basic Hertzsprung-Russel Diagram
As you can see, the majority of stars lie close to the main sequence. This implies that this represents some kind of stability and stars spend most of their life here.

Across the whole H-R diagram, however, can be seen a strip called the Instability Strip. This is the area we are most interested in, as our star of study is located within this band.



The Instability Strip


In this strip a star's luminosity does not remain constant but varies and stars that are found in this band are called variable stars. There are many types of variable stars, but we are interested in the ones that are known as the Cepheid class.

The Cepheid class of variable is characterised by a change in brightness of several hundredths to 2 magnitudes, which occur with extreme periodicity of 1 to 135 days.

In terms of stellar evolution, the Cepheid class of star is seen to be luminous white or yellow giants, and may possibly form a link between the red giant and main sequence-type phase.

These stars are commonly of spectral class F at maximum magnitude, and spectral class G to K at minimum magnitude. A common rule is that the later the spectral class, the longer its period.

Stars that begin with 5-20 solar masses pass through the instability strip. Those that become Cepheids generally have a range of temperatures from 5000 to 6000 Kelvin.



The Importance of Cepheid Variable Stars

Except for the sun Cepheid stars are the most important stars in the universe, for an astronomer anyway. This is because they provide a key to measuring cosmological distances.

In 1912, Henrietta Swan Leavitt of Harvard College Observatory catalogued over 1777 variable stars in the neighbouring small Magellanic clouds (S.M.C). Out of these, 25 were found to be of the Cepheid class. This impressive discovery, however, was made even more impressive when Leavitt arranged the Cepheids by period.

She found that by ordering these stars in terms of increasing period, they had simultaneously been ordered by increasing brightness. Therefore she had discovered that the brighter the Cepheid, the longer it took to vary.

This simple relation between the brightness and period of the variable is called the Period-Luminosity relationship.

In 1917, Harlow Shapley extended Leavitt’s work and confirmed that the P-L relationship was true for all Cepheids, and not just ones present in the S.M.C.



The Method for Calculating Cosmic Distances


  1. Identify a star as a Cepheid by studying it's spectra or the shape of it's light curve.
  2. Calculate it's period.
  3. Use the period luminosity relationship to determine the absolute magnitude.
  4. Use the inverse square law to calculate how far a star of that magnitude would have to be moved from the standard distance of 32.6 light years to appear as a star of the apparent magnitude observed.

Discovering Delta Cephei


Delta Cephei was first discovered to be variable in 1784 by a young deaf-mute called John Goodricke. At the time, only 6 variable stars were known, but an English astronomer, Edward Piggot, was convinced that there were more and decided to set out on a campaign to find these distinctive stars. With him in his search was his 17-year old deaf-mute neighbour, John Goodricke. Despite the fact that at that time it was not usual for handicapped people to benefit from an education, Goodricke’s parents had seen to it that he was well educated and he proved to be a bright student who excelled in mathematics and astronomy.

In 1783, Goodricke was honoured by the Royal Society of London for his work on determining the period and cause of variation in the eclipsing variable star Algol. In 1784 he discovered variations in the light from both Beta Lyrae and Delta Cephei. Sadly this young and promising life was extinguished at the age of 21, when he succumbed to pneumonia that he might have caught while observing Delta Cephei in 1786. Piggot went on to discover 2 more variable stars in 1795.

Delta Cephei is the archetype of all Cepheid Variables – the class of stars that have helped to establish the modern day model of the universe.



Structure of Delta Cephei

The structure of Delta Cephei is basically the same as all stars. Like them it is made of concentric zones of hot gases, so why does it pulsate as it does?

All stars must exist in 2 types of equilibrium: Mechanical and Thermal.

Mechanical equilibrium is maintained by the weight of stellar matter, falling towards the central core under the influence of gravity. This force is counteracted with the equal but opposite force of the radial pressure of the compressed stellar gases pushing up against the weight of the matter. Kippenhahn uses a very good analogy of a piston. The weight of the piston due to gravity, is balanced by the compressed gases below it. If the piston were to continue its descent, the gases would become more compressed and push the piston back up to its equilibrium position. If the piston were raised, the pressure beneath it would not be enough to hold the pistons weight, so it would fall back down to its equilibrium position.

Hence the piston, like the star is in a state of equilibrium between the weight of the piston (or stellar matter) against the pressure beneath it.


Thermal equilibrium is maintained through the effects of pressure on both the rate of energy generation in the core, and the opacity of the outer layers. If the star began to cool significantly, the pressure supporting the outer layers would reduce, allowing the star to contract under the influence of gravity. Due to the contraction the pressure on the core is increased, driving the fusion reactions there to proceed more quickly. This increased rate of fusion in turn increases the temperature (thereby stopping the cooling) and the pressure (preventing further contraction). Also as the star contracts, the increase of pressure in the outer layers causes them to become more opaque, meaning they trap more radiation, so this also stops any further cooling.

This is the case inside stable main sequence stars. Yet in variable stars, equilibrium is not constant. It continuously shifts from a maximum to a minimum.

We will go back to Kippenhahn’s analogy of the piston, as it’s a good way of looking at the mechanics of the situation.

If the piston is pushed down from its equilibrium position, and suddenly released, it begins to oscillate. It is pushed upwards by the pressure of the excessively compressed gases beneath it. It is pushed past its resting point due to the inertia of the piston. Now the pressure of the gas is not enough to support the piston, so it falls again, compressing the gas excessively and subsequently being forced up past its rest position again, and the cycle repeats itself.

In an ideal closed system this would continue indefinitely, but energy is continuously being radiated away and so the oscillations should slow down and eventually stop. Perhaps this is what is happening to Polaris at the moment?

However, if the system can be kept energetic, the oscillations can keep up indefinitely. In the example of Delta Cephei, the increase in opacity of the surface layers due to contraction is strong enough to keep enough energy in the star to overcome the damping effects due to radiation of energy, so the star can oscillate.

Energy Convection in Delta Cephei

The cause of the increase in opacity in Delta Cephei was proved in the 1960’s by an American astronomer, John Cox, to be helium gas.

Normally when a star’s helium is compressed, its temperature increases, and the gas becomes more transparent. However, Cox proved that in certain upper layers of the star, compression causes the helium gas to ionise thus becoming more opaque instead of rising in temperature. This compressed helium traps the star’s heat, much like greenhouse gases trap our own heat, which pushes the star’s surface outwards. As the star expands, helium ions recombine with the free electrons and so the gas becomes transparent again releasing all the pent up heat. This causes the star to contract and the cycle to start all over again.

Delta Cephei has the same stellar composition of most ordinary stars - about 90% hydrogen, 9% helium and 1% other materials. These measurements have been made using spectroscopy, in which light from the star is split up into its different wavelengths and then analysed. If certain wavelengths show up, these are unique to certain elements so scientists can work out the actual composition of the object from which the light originated. Spectroscopy is one of the key tools for an astrophysicist, as it can show many things. Mainly it’s used to show the composition of stars, or nebulae, but also it’s been used to show the velocity at which a star pulsates.

In 1894, a Russian astronomer A. A. Belopolsky noticed that the spectral lines from Delta Cephei shifted back and forth with the same 5.366 day period of the star. From the Doppler effect, he was able to translate these shifts in wavelength, to speeds, and was able to draw the velocity curve shown on the next page.
You will see that the light curve and velocity curve are mirror images of each other.

When the star expands...it is brighter.
When the star contracts...it is dimmer.


Another type of variable star, similar in some respects to cepheid variables, are RR Lyrae variables. These variable stars are generally low mass, post helium-flash stars that pass through the lower end of the instability strip. These stars all have periods shorter than 1 day and have roughly the same average brightness as the stars on the horizontal branch.

Below are some interesting graphs that show the typical rate of pulsation for Cepheid variables.


Radial Velocity:

Graph to show the change in Radial Velocity with Time


Surface Temperature:

Graph to show the Change in Surface Temperature with Time


Diameter:

Graph to show the Change in Diameter with Time




Observing Delta Cephei


Another reason that Delta Cephei is such an important star, is not only because of the advances in astronomy it has led to, but also because it is extremely convenient to be observed by anyone.

It is found at a declination of +58 degrees, meaning it is very high in the sky and so is a prime target for Northern Hemisphere observers.

It changes in magnitude from 3.5 to 4.4 – a range that can be followed easily with the naked eye.

Its period of 5.366 days means that you can view the entire light curve in a week – if the weather is favourable!

In addition it has 2 comparison stars next to it that conveniently lie at either end of its magnitude range.

It is possible for anyone to find and investigate this star for themselves and to plot their own light curve to see if they can get results that resemble the classic shark-fin light curve typical of this star and other Cepheid variables.

Below is a light curve a member of our team achieved by observing the star, over a period of several months in 1989, with the naked eye and making estimated guesses at the magnitude. As you can see, it is clear that this is a variable star, though the telltale shark fin like wave is not quite seen. The correct graph should be a rise to maximum magnitude over a period of about 1.5 days and then a slower decline to minimum magnitude over a period of nearly 4 days. The vertical lines are approximations of errors.


Light curve for the Apparent Magnitude of Delta Cephei against Days



The correct graph should look more like this:

Actual Shape for the Apparent Magnitude of Delta Cephei against Days

As you can see, this is a clear shark-fin like shape, with a swift rise to maximum magnitude, and a slow decline to minimum magnitude. It is difficult to achieve perfect results when observing Delta Cephei, but it is far from impossible.

Below is a project guideline to observe Delta Cephei that anyone can use and try out.



Delta Cephei: Monitoring Exercise


Important viewing notes:


Try and find somewhere that is outside a city and away from any lights and night pollution. Good examples are in the country.

When viewing the star, try and view it out of the corner as your eye, as this is where your eye is most sensitive to black and white light.

Make well separated observations throughout the night, but try to forget all previous observations.

If possible, try and view the star when the weather forecast is good for a week, as otherwise there will be gaps in the measurements of the stars magnitude and period.


Finder Chart:


Finder Chart for Delta Cephei

Cephei = Star A
Cephei = Star E

Cephei should be compared to and on the scale shown in next section.


Method of Observation


Instruments are not required to make magnitude estimations. Simply look at Delta Cephei out of the corner of your eye, and compare it’s brightness to its to companion stars.

Each evening one observation should be made, noting the brightness of Delta Cephei on the scale A, B, C, D, E. The time of observation should be recorded.

The scale is as follows:
A = As bright as Star A
B = Almost as bright as Star A
C = In between Star A and Star E
D = Almost as dim as Star E
E = As dim as Star E

You can also go in between this scale if you feel your eyesight is keen enough. The scale is only a rough guide.

The date of the observations must be noted and using the information gathered, draw a light curve and analyse it.




Table:



Record your reading's here

Draw a graph of your data here

It is also possible to take pictures of Delta Cephei and use computer programs to calculate it's magnitude. Below is an image obtained using a 50mm camera lens attached to a CCD camera (Exposure: 0.5s at f2.8: 24th September, 2002 21:47 UT)



Conclusion

Delta Cephei is truly a most amazing star that has had such an important impact on the view of our universe. Its greatest impact has been on how we view the size of the universe and our place in it. Many of the physical parameters we know about stars and galaxies can be determined only if we know their distance from us and by using Cepheid variables we can find these distances. No doubt John Goodricke would have been pleased that his observations of Delta Cephei would ultimately lead to our modern view of the Universe.

Delta Cephei was the beginning of great discoveries, and the cause of many more to come, so it is only fitting it should be in the constellation of the King Cepheus, as it truly is a majestic star.



References


  • Microsoft encarta 98
  • Advanced Physics by Steve Adams & Jonathan Allday
  • http://www.peripatus.gen.nz
  • http://www.stargazing.net
  • http://astro.estec.esa.nl/Hipparcos/education-extra.html