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    Sextant science: measure the speed of light
    From: Frank Reed CT
    Date: 2004 Dec 31, 21:59 EST
    Since the topic came up a week ago on the list, I thought it might be fun to consider a backyard science project that could be done using a sextant. You're going to measure the speed of light. It's known historically as the "annual aberration of starlight", but it's fundamentally a relativistic effect due to the Earth's motion and the finite speed of light.
    First, I'll describe an extreme hypothetical case. Suppose we could take the whole Earth and accelerate it off towards the belt of Orion at 90% of the speed of light. What would happen to the appearance of the sky? There's the well-known Doppler effect, of course, which would "blue shift" the light of anything we're approaching and "red shift" the light of anything we're flying away from. This means that the stars of Orion would appear even bluer than they are while the stars on the opposite side of the sky, towards Ophiuchus, would appear much redder (actually their spectra would be shifted which would lead to appearance changes depending on the actual distribution of light in the specific star's spectrum, but you get the idea..).
    Meanwhile, the positions of the stars in the sky would also shift. This is "aberration". The stars would shift forward towards the direction of our motion, apparently squeezing the constellations together towards Orion while dragging them away from the opposite side of the sky. At 90% of the speed of light, this change would be quite significant --most of the stars in the sky would appear to be bunched together around a squished Orion. Stars which we see from Earth in the whole hemisphere around Orion (in other words those less than 90 degrees from Orion's belt) would be squeezed into a section of the sky with a radius of only 26 degrees. The measured distance from Polaris to the belt of Orion would be only about 26 degrees. By contrast, the opposite section of the sky would be stretched out by aberration to the point of being almost devoid of stars.
    Now imagine taking our relativistic Earth, bringing it to a stop after a few months (hang on!), and then turning around and accelerating in exactly the opposite direction until we are travelling in the opposite direction at 90% of the speed of light. Everything is reversed, and now the constellations are squeezed together in the direction of Ophiuchus. The measured distance from Polaris to the belt of Orion would now be about 154 degrees. That's a huge difference from before and one that we could detect with no instrumentation at all.
    Let's get back to the real Earth. The velocities involved are much smaller, but the effect is still present. Right at this moment, on New Year's Eve 2004, the Sun is located in the sky in the northern part of the constellation Sagittarius. The Earth is moving in a direction 90 degrees away from that spot in the sky (along the ecliptic), approximately towards the star gamma Virginis, known to the Romans as Porrima. This means that the constellations are all shifted, slightly, towards this star. Three weeks from now, in late January, when you look up in the early morning sky and see Spica, you'll know that we're travelling almost straight towards it, and the constellations are all visually compressed a bit in that direction, squeezed together a bit on the focal point of Spica.
    But what happens in six months? Six months from now, the Earth will have travelled to the opposite side of its orbit and so we will then be moving almost directly away from Spica. The constellations will now be compressed (slightly) towards the constellation Pisces while the stars around Spica will be stretched a bit away from that side of the sky. Every year, this pattern of compression and stretching of the apparent places of the stars in their constellations repeats itself. And it can be measured with a sextant.
    This phenomenon is not as stunningly obvious as the case where we were travelling at 90% of the speed of light, and the stars were shifting back and forth by over 60 degrees in each direction. This annual aberration of the stars' positions amounts to only about 20 arcseconds in each direction semi-annually for a net change in position of about 40 seconds or 0.7 minutes of arc. It's 20 arcseconds because the Earth's speed around the Sun is 30km/sec while the speed of light is 300,000km/sec and for such small relative velocities, the angular shift is simply equal to the ratio of the speeds: 1/10,0000. This is the angle expressed as a pure ratio (a.k.a. radians); as with any angle, this can be converted to seconds of arc by multiplying by 206265 (the number of arcseconds in a unit angle -- or arcseconds in "one radian"). With a properly maintained metal sextant and the proper procedures for clearing a distance across the sky measured between two stars, this is well within the range of detectability.
    The effect of annual aberration is largest when measured between two stars that are about 90 degrees apart in the sky. So let's use Spica and Pollux. In January, we know we're travelling towards Spica while Pollux is "abeam", more or less opposite the Sun's position in the sky. If you get outside during January in the early morning hours, you can measure the distance between these two stars with your sextant. You should also measure the altitudes of the stars (or record the time so that you can calculate them later) but you only need to measure the altitudes to the nearest degree since the altitude corrections for stars are very small. This distance will need to be cleared for the effects of refraction. This calculation is very similar to the methods once used for clearing lunar distance observations, and I have described it before on this list (see "star-star sights" in the list archives for April, 2004). By repeating this observation roughly once a month for as long as both stars can be observed in the sky at night, you will be able to detect a cyclic compression and expansion of the angular distance between them. This is a direct observation of the speed of light (as compared with the Earth's orbital speed). And all it takes is a sextant and a little math.
    This might be the sort of backyard science project that could be proposed to local school groups possibly as a means of getting them interested in the techniques of sextant use and celestial navigation. It's real science, and although it would take the better part of a year to complete the project, it has the advantage of requiring only an hour or so of actual work, once a month.
    42.0N 87.7W, or 41.4N 72.1W.
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