# Delta Cephei and the Size of the Universe

You're driving along a road in the dark, and coming toward you is the single headlight of a motorbike. With just the one bulb, it is very hard to estimate how far away the bike is. Is it a very bright light far away, or a less bright light closer to you? This is the problem that faces astronomers as they look out at the universe. The stars appear as points of light. Are they dim stars relatively close to us, or bright stars very far away?

For the very closest stars, such as Sirius, alpha Centauri and Procyon, we can use the 'method of parallax', which we'll explain in a minute. For most stars, however, there was no way of telling until the astronomer Henrietta Swan Leavitt made an important discovery about a special type of variable star. This Entry will explain about these stars, the method and the various people involved along the way.

### The Method of Parallax

Hold your index finger out in front of you, between you and the computer screen. Shut your right eye and note which part of the screen the finger is in front of. Now swap eyes, and you will notice the finger is covering a different part of the screen. The finger appears to have moved because your two eyes are observing it from slightly different positions. This effect is called 'parallax'. If you measured the angles between your eyes and the finger very carefully, and the distance between your eyes, you could calculate the distance to your finger.

We can do the same with a close star. If we observe the star in spring, say, and then in autumn, we'll be viewing it from two different locations because the Earth has done half a revolution around the Sun in that time. The star will appear to move by a tiny amount against the background of much more distant stars. Knowing the angle by which the star appears to move (the parallax) and the distance the Earth has moved, we can calculate how far away the star is.

If the star were to move by an angle of one degree against the background, we would say it had a parallax of one degree. In fact the angles involved are much smaller - a degree is divided into 60 minutes1 and a minute is divided into 60 seconds2. The closest star has a parallax of less than one second. The further away the star is, the less it will appear to move against the background of more distant stars, so the smaller the parallax will be, making it harder to measure. This method is really only good for the closest stars.

### John Goodricke

In the distant past, it was thought that the stars were constant. There are bright stars and dim stars, but each one would stay the same brightness. Then in 1638, astronomers discovered that the star omicron Ceti is not constant. Regularly, over a period of 11 months, it goes from being so dim as to be invisible, up to being a moderately bright star, and then fading back to invisibility again. This behaviour was considered so amazing that they renamed the star 'Mira', meaning 'Amazing'. Soon after this, in 1669, another variable star was discovered. The star Algol (beta Persei) was found to fade slightly every few days. The situation stayed like that for another century, until the arrival of John Goodricke.

John Goodricke was born in 1764 in The Netherlands to English parents. The family soon moved back to England and settled in York, where John grew up. He was profoundly deaf, and as a result, he never learned to speak - such a person was known in England until very recently as 'deaf and dumb'. This was no comment on their intelligence; 'dumb' just meant 'unable to speak'. Goodricke was sent to a special school for the deaf and was well educated.

Goodricke became friends with his neighbour Edward Pigott, who had a small observatory in his house. He became interested in astronomy and carried out a series of detailed observations of Algol, comparing it to the other stars near it to estimate its brightness. He discovered that the fading of the star was regular: the star normally shines at magnitude 2.1, but every 2 days, 20 hours and 49 minutes, it dims for a duration of 10 hours to magnitude 3.4 (bigger magnitude numbers indicate dimmer stars). Goodricke proposed that Algol must in fact be a binary star - two stars orbiting around a common point. If one star is bright and the other dimmer, then when the dimmer one is in front of the brighter one, it will block the light of the brighter star and the total brightness will diminish. Most of the time, we are seeing the two stars together. He was subsequently shown to be correct in this; stars of this sort are called 'eclipsing binaries'. He presented his results to the Royal Society in 1783 and received a medal for it.

Both Goodricke and Pigott continued to look for variable stars. In the following year, 1784, Goodricke discovered one, delta Cephei, and soon after that Pigott found another, eta Aquilae. These also changed brightness regularly and quickly over a few days like Algol, rather than over nearly a year like Mira, but they could not be explained as eclipsing binaries. They appeared to be a new type of star which actually changes brightness itself, rapidly and regularly. Because delta Cephei was the first such star discovered, the whole class of stars is named after it. Stars like this are called 'Cepheid variables' or just Cepheids for short.

Sadly, John Goodricke didn't live to enjoy his fame. He died from pneumonia in 1786 at the age of 21. He had been elected a fellow of the Royal Society just four days earlier for his work but had not yet received the news.

Recent studies have shown that Goodricke made his observations from the Treasurer's House in York, just next to York Minster.

### Delta Cephei

Delta Cephei is the fourth brightest star in the constellation of Cepheus the King. This is a fairly dim constellation in the part of the sky above the North Pole. Goodricke discovered that the brightness of the star varies over a period of 5 days and 9 hours. It gets steadily brighter for about 1.5 days, then it dims steadily over the next 3 days back to its original brightness. He could not think of any explanation for this behaviour but recognised that it could not be explained by two components orbiting around a common point. The star itself must be changing in brightness.

In fact the explanation for this changing brightness was not discovered until much later, in the late 19th Century. Delta Cephei is a yellow giant star. It is a very big star, with a mass more than four times that of the Sun and a radius of more than 40 times that of the Sun. It has finished burning all the hydrogen in its core and is now fusing helium into more complex elements - this reaction gives off a lot more energy than hydrogen fusion of normal stars, so the star is very hot, making it very bright. Although delta Cephei looks much the same as the other stars around it in the Cepheus constellation, it is actually much brighter and much further away - it gives off approximately 500 times as much light as our Sun.

The changing brightness is caused by a process which is called the κ-mechanism3. The following is a simplified explanation of this process, but it will give a flavour of what is happening. The gas helium in the atmosphere of the star comes in two forms that concern us here: singly ionised helium, with one electron, which is transparent to light; and doubly ionised helium, with no electrons, which is opaque. If singly ionised helium is heated, it will lose its electron and become doubly ionised and opaque; on the other hand, if doubly ionised helium cools, it will gain an electron and become singly ionised and transparent. Density also increases the opacity of the gas.

• When the star is at its smallest, the light flowing out from the centre, where the nuclear reaction is taking place, heats up the helium, causing it to become doubly ionised. This makes it opaque, so it absorbs more light and gets still hotter. The pressure of the light from the core pushes it outwards and the star expands.

• As the star expands, it becomes less dense, and its increased surface area allows it to cool. Eventually the star becomes so big that it cools down and the helium turns back into its singly ionised form, which is transparent. This allows the light flowing outwards from the core to travel through it uninterrupted, so the pressure supporting the expanded star isn't there anymore, and it collapses back down to its smaller size.

This cycle continues over and over with an extremely regular period. In the case of delta Cephei, the pattern repeats every 5 days 8 hours and 48 minutes.

### Henrietta Leavitt

Henrietta Swan Leavitt was born in Massachusetts, USA, on 4 July, 1868. She first attended Oberlin College and then the Society for the Collegiate Instruction of Women, which later became Radcliffe College for Women, a branch of Harvard University4. She graduated in 1892. This was not a good time for women looking for employment in the sciences. Leavitt was interested in astronomy, so in 1895 she chose to work in the Harvard Observatory without pay. It was another seven years before she was considered to have proved herself, and in 1902 she was offered fully paid employment at the observatory. She worked on a project led by Prof Edward Pickering to measure the brightness of every star in the sky. In 1907 she was put in charge of a team to use photographic techniques to measure the brightness of stars. She became an expert on such techniques and made a particular study of variable stars, discovering 2,400 of them in her lifetime - this amounted to more than half the known variable stars at the time.

Leavitt became interested in the Cepheid variables, stars that pulsate in the same way as delta Cephei. She examined the Small Magellanic Cloud, a misty area in the southern skies. To the naked eye, the Cloud looks like a faint patch of white light in the sky, similar to the Milky Way. In a telescope, it is revealed to be a giant cluster of stars. It was not known at the time how far away this cluster is from us, or even whether it is a part of our galaxy or outside of it5. The Small Magellanic Cloud is very far south and is not visible from Harvard, but the observatory owned a smaller observatory in Arequipa, Peru, where they had positioned the Metcalf telescope, a 10-inch refractor. They had done an extensive photographic study of the Cloud. Leavitt studied the photographs in detail and managed to identify 25 separate Cepheid variables in the Cloud, with periods ranging from 1.2 days up to 127 days, averaging 5 days.

Leavitt discovered that there was a marked connection between the brightness of the Cepheid and its period of variability - the brighter the star, the longer it took to pulsate. If these were just stars randomly dotted around the sky, this would not be particularly interesting, because the brighter stars could be brighter because they were closer to us. But all these stars were in the Cloud, and were therefore all the same distance from us - in the same way that New York and London are approximately the same distance from Saturn. This was a momentous discovery because it meant that these were actual differences in the light output of the stars. Given the period of pulsation, which is easily measurable, she could calculate how bright the stars were. This result was published in 1912.

The community of astronomers immediately realised the significance of the discovery. As light travels through space, its intensity decreases with distance by the inverse square law. If something is twice as far away, it will be one quarter as bright. If it is three times as far away, it will be one ninth as bright. So if we know how bright a star actually is, and how bright it appears from our viewpoint on Earth, we can work out how far away it is. Astronomy finally had a way of measuring the distance of stars further away than the very closest ones.

The method actually tells us how bright a Cepheid is compared with other Cepheids, and hence how far it is away compared with other Cepheids. But we need to know the distance of at least one Cepheid to start the whole process off. This was the weak point in the procedure - at the time, the parallax method was not accurate enough to measure the distance to any Cepheid, as even the closest one is quite far away. Other methods which were not particularly accurate were used to estimate the distances of the nearest Cepheids, and these were then used to calibrate the general method.

Henrietta Leavitt continued working in the observatory until she died of cancer on 12 December, 1921, at the age of 53.

### Measuring the Galaxy - Harlow Shapley

Cepheids are known as 'standard candles' because they are sources of light for which we can calculate their actual brightness, allowing us to measure their distance. It wasn't long before they were used to make the first measurements of the size of our galaxy. Up to the early 20th Century, astronomers had no idea how big the galaxy was. Estimates varied widely, with some as small as 10,000 light years across, with the Sun at the centre of the galaxy. It was also thought by many that the visible galaxy was the whole of the universe - there was nothing outside of the galaxy, and the various blobs and spiral-shaped whirls visible in the sky (known as nebulae) were thought to be clouds of dust or small clusters of stars within our galaxy.

Enter Harlow Shapley. Born in 1885 in Nashville, Missouri, USA, Shapley did badly in school and dropped out, then decided to become a journalist. Since he needed a good education for this, he went back to high school, did the entire six-year course in two years and graduated from school the best in his year. He then went to enrol in a course in journalism in the University of Missouri, but there were administrative problems and the course was postponed for a year. To kill time, Shapley opened the University Prospectus at A and decided to do the first course. He said he couldn't pronounce Archaeology so he did Astronomy. Thus was a great astronomy career born.

When Leavitt's discovery became known, Shapley decided to use it to measure the size of the galaxy. Dotted around the galaxy are tight balls of stars known as globular clusters. While normal clusters might contain 50 to 100 stars, a globular cluster can have up to 150,000, all very close together. Among these Shapley was able to identify some Cepheid variables, enabling him to estimate the distances of these clusters. He discovered that the galaxy was much bigger than had been thought. The distance-measuring method was still in its infancy and his estimate, of 300,000 light years across, was too big by a factor of 3, but this was still a remarkable result. Astronomy went from a position of having no idea at all to a definite, though inaccurate, estimate.

Shapley also showed that the Sun was not in the middle of the galaxy but in a rather insignificant position about half way from the centre to the edge. The Earth had already been demoted from the position of centre of the universe, now our Sun suffered the same fate.

### Measuring the Universe - Edwin Hubble

The Earth and the Sun were no longer anything special, but our galaxy must be the centre of the universe, right? Shapley thought so, but another astronomer, Edwin Hubble, took up the challenge of measuring the universe and proving him wrong.

You may be familiar with Hubble's name, as the Hubble Space Telescope was named in his honour. Edwin Hubble, born in 1889, was good at everything as a child. He studied law at his father's request, but never liked it, and after his father's death changed to astronomy. He started work at the Mount Wilson Observatory in California in 1925 just as the new 100-inch Hooker telescope was completed. This was the biggest telescope in the world at the time. Hubble managed to identify Cepheids in the Andromeda Nebula and the Triangulum Nebula, as they were then known, which allowed him to measure their distances. He proved that both of these were way beyond our galaxy, and were in fact themselves galaxies. Suddenly our galaxy, now known as the Milky Way Galaxy, was no longer the centre of the universe but just one in a group of many. We now know that there are billions of galaxies in our universe.

### Refining the Method

There are a few problems with using Cepheid variables to measure distances. Astronomers are constantly studying these to make the method more accurate.

• The inverse square law for light mentioned above only applies to a perfect vacuum. But space is not a perfect vacuum - the gaps between stars are occupied by hydrogen gas. It is very, very thinly spread, but it is there, and it absorbs some light, so we don't get as much light from the stars as we would expect. This makes them appear to be slightly further away than our calculations would show. An extra factor has to be included in the formulas to allow for this 'extinction' effect.

• The basic assumption of the Cepheid variable method is that all such stars behave in the same way wherever they are in the universe. After Hubble's measurement of the distance to the Andromeda and Triangulum galaxies, it was discovered that there are in fact two types of Cepheid variable, and they behave differently to each other. Observations of Cepheids in distant galaxies need to take this into account.

• As already mentioned, we need to measure the actual distance of at least one Cepheid by other means in order to calibrate the method. This brings us back to delta Cephei, which is the closest of the Cepheids to us. As observational techniques have become more refined, it has been possible to use the parallax method to measure the distance of this star. This has been done a number of times - each time a new, more accurate figure becomes available, the distances to everything else in the universe must be revised accordingly, since they are all based on it. The latest figures from the Hipparcos satellite6, using the parallax method, show it to be 887 light years away, with a possible error of 3%.

### Conclusion

Our conception of the universe we live in is constantly changing as astronomers measure and remeasure it, and examine it in more and more detail. The universe is all the time being shown to be bigger and more full of wonder than we could ever have imagined. We must thank John Goodricke and Henrietta Swan Leavitt for providing us with the method of measuring the universe and making sense of it all.

Image credit: NASA
1These minutes are sometimes called 'arcminutes' to distinguish them from minutes of time.2Sometimes called 'arcseconds'.3That's a Greek letter kappa rather than a k.4Other famous women who attended Radcliffe include Margaret Atwood, Benazir Bhutto, Helen Keller, Ursula Le Guin, Bonnie Raitt and Gertrude Stein.5We now know that it is a mini-galaxy outside our own.6Hipparcos is a space telescope specifically designed to measure the positions of stars.