Our April 2009 project provides a simple means of measuring what’s called the sidereal day.
We all know that the reason everything in the night sky rises and sets is that the Earth is rotating, giving us day and night, though it looks to us as if the sky is rotating and the Earth is fixed. But there’s another thing that happens — as the Earth goes round the Sun, the view of the stars we get at the same time each night shifts slightly. This is what gives rise to the changing skies from season to season. By careful observation you should be able to measure the changes in just a short time, and find out something about your own powers of observation as well.
The project involves watching a bright star disappear behind a wall or other vertical edge and noting the exact time at which it does so. You need to do this on a few nights. So the first task is to find your star and observation point. We are assuming that you recognise a few constellations so you can pick out individual stars. At this time of year, Leo is an easy constellation to recognise as it’s got a very characteristic shape. Its brightest star, Regulus, is visible through even the worst light pollution and you should be able to pick out one or two of the others as well. This year, Saturn is up there as well, adding an extra star to the pattern. The other advantage of choosing Leo or stars in its neighbourhood is that they will move faster through the sky than those elsewhere, in the same way that a roundabout moves fastest at its edge rather than at its centre. The Pole Star, for example, hardly moves at all over the night as it is close to the sky’s axis of rotation, so it would be a very poor choice. But stars that are due south and on the sky’s equator are the best to choose, as they move fastest. If they are too high in the sky, it will be hard to find a vertical edge for them to disappear behind. You must also start your project fairly late in the evening, so that your chosen star will not move into the evening twilight.
Next, find an object against which you can measure the star’s disappearance, such as the wall of a house or a pole. Ideally it should be to your south, but anywhere from southeast to southwest will do. Beyond that range, stars are moving at an greater angle to the horizontal, so again your measurement gets more difficult. Ideally it should be more than 10 metres away. You will also need a fixed place to stand and something to rest your head against, so that your head is in exactly the same position each time you make your observation. The only apparatus you’ll need is an accurate watch, a torch and a means of making notes. A digital watch is fine as long as it reads out seconds.
So stand in your chosen spot and watch the star disappear behind the object. As it gets nearer, close one eye so that you have only one observation point. Note the exact time, to the nearest second, that the star disappears, and of course write down the date as well. Repeat the observation the next clear night, but get to your position a few minutes earlier this time, and do the same again as often as you can over a period of many days (ideally a month or so), getting there a few minutes earlier each time. You can repeat the experiment with different stars, but make sure your notes are clear so you don’t mix one star up with another. Try it with Saturn as well!
Analysing your results
Books tell us that the stars rise about four minutes earlier each night, so your task is to see how accurately you can measure this. From one night to the next, your observation may be many seconds out. But over seven days, say, the difference should be about 7 × 4 minutes, which is about 28 minutes earlier. So divide the time difference between the number of whole days, and get a different value for each measurement. Even if each observation is a few seconds out, by measuring over a period of time your error is spread over a greater number of days, so the accuracy of your measurement is increased. Actually, the error in any one measurement could go either way, so you are likely to get a spread of values for the same measurement. A better way to look at the results would be to draw a graph of the time difference against the number of days, and find a best-fit straight line to your observations from which you can measure the average time difference per day.
Think about the ways in which you could have made an error, and try to find ways to reduce the error. Are you standing with your head in exactly the same spot each time? How accurate is your watch, and can you make sure each timing is more accurate? Does it matter how far away from you is the object that hides the star? What about the time it takes between seeing the star disappear and your looking at your watch? Does that make any difference to the result?
The solution – and why you might have got it wrong
The actual sidereal day is 23h 56m 4s. You can calculate this figure for yourself without having to make any observations because it’s the same as 365.25/366.25. The Earth makes one more rotation relative to the Sun than relative to the stars each year because it has rotated once in its orbit. Subtracting this from 24 hours gives the difference in rising time between one night and the next – 3m 56s. But you might not have got exactly this figure in your measurements. Why not?
How precise can you be about your observation spot each time? If your position changes by, say, 10 centimetres from one night to the next, the error will depend on how far away is the straight edge behind which the star disappears. If it is 10 metres away, that will make a whopping 2-minute difference in the disappearance time! (To calculate this, we said that 10 centimetres in 10 metres is 1 in 100. The inverse tan of 0.01 is 0.573, so this is the number of degrees that 10 cm subtends at 10 m. The sky ‘turns’ at the rate of 1º in 4 minutes (240 seconds), so multiplying 0.573 by 240 gives 137 seconds.) If the vertical is closer, the error will be even greater. So you need to keep your observation point precise to much less than 1 cm if you can. This is why a really long observation period of a month or so is needed to reduce this daily error.
Your clock might vary by a few seconds from day to day. Even digital watches are not perfect, and you’d need to check the accuracy of your watch with time signals daily in order to be sure how it’s varying. Say your experiment lasted 10 days and over that time your watch lost or gained three seconds. That would make 0.3 second error in your figure.
How accurate is each timing? When you see the star disappear, you have to look at your watch and note the time. How well you can do this depends on your own abilities, so such errors are referred to as your personal error. But it’s likely to be similar each time, and is probably much less than 1 second. So if you do the same thing each time, in theory you should have the same personal error for each measurement so when divided by the number of days over which you do the project, this figure is quite small.
Actually, there is another error which you may have noticed when carrying out the project. Instead of winking out instantly, as you would expect a point source such as a star to do, it may well have faded over a matter of a second or so. How can this be?
The answer is that your own detector – your eye – has a definite size as a result of the diameter of your pupil. This is also a personal error, so again it should be fairly constant for each observation and over a period of time should not make a great difference.
Once you have got your technique right, you could observe Saturn and use the results to plot its movements through the sky. And you’d be following in great footsteps, because until the invention of the telescope, all positional measurements were made in a similar way to this. Once telescopes came into use, it allowed the measurements to be made more accurately, The device used was called a transit telescope, and observers had to time the moment that a star crossed the crosswires of such a telescope. Such telescopes, now highly automated, are still in use, such as the Carlsberg Meridian Telescope on La Palma.
Robin Scagell and Margaret Penston