an autodidact meets a dilettante…

‘Rise above yourself and grasp the world’ Archimedes – attribution

Posts Tagged ‘Kepler

the movements of the Earth, the ecliptic, the celestial sphere…

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Why does the Earth spin? Initial conditions plus Newton’s first law is the basic explanation. And from these it should be easy to guess that it’s slowing down as tiny but inexorable forces act upon it, and it will continue to do so unless something unforeseeable happens. The tidal friction caused by the moon, which itself is decreasing over time (or at least I assume so, since the moon is spiralling away from us) is the Earth’s principal brake. Some say that Earth has been gradually slowing down since the last great collision, which created the moon, and which left the planet spinning full circle every six hours, but I think that’s still speculative.

Anyway, we don’t just rotate (in an anti-clockwise direction), we revolve (anti-clockwise) around the sun on a plane tilted at 23.4 degrees from our spin – that’s tilted from the perpendicular. But why? And there’s this thing called precession, right? Spin a top, as I did as a kid, and the most successful spin will have the least precession – the smallest circle (actually a cone) around which the axis of rotation wobbles, but as the top slows that cone will widen until all falls in a heap. In the Earth’s case, it’s most commonly called the precession of the equinoxes, or ‘the wobble’ (maybe).

So the Earth moves in mysterious ways, and I’ve barely begun. It orbits the sun – why? Its orbit is elliptical – why? Its rotational and revolutionary speed vary – why? And what about other movements – the solar system, the galaxy, the universe?

A cool video I’ve been watching tells me something I’d never known or thought of before. We’re all on meridian lines, which pass through us in a north-south direction, from the north pole to the south pole. Lines of longitude. When the sun is at its highest point in the sky, at noon, it’s aligned perfectly with our meridian. The shadow it casts, our shadow, thus points precisely to the north or south pole, depending on the sun’s position north or south of ‘directly overhead’. If the sun is directly overhead, congratulations, you’re on the sub-solar point, and your shadow will disappear beneath your feet, so to speak. Right now the sub-solar point is a circular area in the Atlantic, a little north of the equator, and just touching land in west Africa. I doubt if we ever experience it here in Australia, as it seems to hang close to the equator.

The point to make here is one about time. As there’s a meridian line for just about everyone, it follows that everyone on a different meridian is experiencing a different time. Noon, or any other time, isn’t the same for everyone – but that’s massively inconvenient, so we’ve regularised time via zones, so we can do our business.

Looking again at our rotation, we might think we have it nailed at very close to 24 hours per full rotation, but not quite, for all is relative. The sun, for example, has its movements too, as does everything else. We’ve found that, measured from a distant star, one meridian completes a revolution in 23.9 hours, also known as a sidereal day. Our calendars, though, are based on the solar day. As the Earth turns, it moves forward in its revolution around the sun. So by the time it has turned 360 degrees it needs to spin a little more for the same spot to be facing the sun as was the case 24 hours before. That slightly greater than 360 degree turn is what we call the solar day. From our perspective it seems like an exact 360-degree turn because we’re facing the sun again, exactly as the day before. Or so it seems.

We revolve around the sun in an ellipse. Or not precisely around the sun. Kepler’s first law of planetary motion, presented to the world without fanfare in 1609, had it that all the planets traced an elliptical orbit around a focal line, with the sun as one of its end-points, or foci. And while we’re at it, let’s look at Kepler’s three laws and how they were arrived at. The second law, presented in the same year, states that ‘a line segment joining a planet to the sun will sweep out an equal area over an equal time interval’, and the third law, announced to a largely indifferent world in 1618, is perhaps less linguistically elegant, or at any rate simple: ‘The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.’ I ripped this from Wikipedia, the greatest gift to all dilettantes and autodidacts ever developed.

Kepler’s laws improved on those of Copernicus, but of course they accepted Copernicus’ heliocentric system as the basis. All Kepler really added was the eccentricity of planetary orbits, a minor detail really, but certainly an improvement. His laws weren’t presented as such at the time: they weren’t described as laws until Voltaire’s  publication of Eléments de la philosophie de Newton, no doubt largely the work of his intellectual superior, Emilie du Chatelet.

So, the first two laws. Kepler was given access to some of the detailed astronomical data of his employer Tycho Brahe, who asked him to calculate precisely the orbit of Mars. Tycho apparently withheld the bulk of his observations from Kepler, because he suspected him of being one of those upstart heliocentrists. Kepler wanted, for largely mystical reasons, to define the Mars orbit as a perfect circle, but after years of trying the calculations wouldn’t work out. What he did discover was that, although the orbital path wasn’t circular – the sun was sometimes further away, sometimes closer –  if you drew a line from Mars (or any other planet, including Earth) to the sun, and then another line, say exactly six days later, the triangle created always had the same area, no matter where you were in the orbit. For this to happen, the planet must be moving faster nearer the sun than when further from the sun. This was Kepler’s second law, which helped him to calculate the first. The planets’ orbits appeared to be elliptical. If the sun was offset from the centre of the planetary orbits, but still obviously essential to those orbits, then the offset could be calculated precisely such that all the planetary orbits fitted. And so it was. Most astronomers consider this to be his greatest contribution.

Kepler’s third law, with its interesting mathematical basis, provided the greatest inspiration to Newton:

The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

P2 = a3

I rarely do maths here, but surely this one’s simple enough even for me! The orbital period (p) of a planet is the time it takes to make a complete revolution around the sun. Note that it’s a measure of time, not distance. The semi-major axis of an ellipse is half its longest diameter. In the special case of a circle, it would be the radius. This law enables us, apparently, to determine the distance of planets from the sun, though it appears to entangle time and space. Generally these distances are given in relative terms. with the Earth’s distance from the sun given the value 1 AU (astronomical unit). By that reckoning, the outermost planet, Neptune, has a value of 30.06 AU, approximately, according to one site providing such data. Similarly, we reckon the orbital period in Earth years. Neptune’s orbital period is 164.79 years. So, for Neptune, 164.79² = 30.0611³. Try it on a calculator and you’ll find it doesn’t quite work out, but this may be due to eccentricity of orbits, in time and space. Other sites have different figures. The Kepler equation seems to capture the pattern rather than the precise detail. It’s probable that the publication of logarithmic tables between Kepler’s calculation of the first two laws and the third was vital.

I’m of course no expert on any of this – go to more reputable sites for a more complete story, though you’ll probably find what I found – a fair amount of interesting confusion.

I’ll finish with the ecliptic. The Earth’s orbit sketches out an elliptical plane, which we call the ecliptic. Then again, the ecliptic is also described as the apparent motion of the sun in the sky with respect to the fixed stars – not to be confused with the apparent daily movement caused by Earth’s rotation. In fact Wikipedia describes the ecliptic as ‘the mean plane in the sky that the sun follows in the course of a year’, and Wikipedia is always way more right than I am in these matters, but it’s confusing. The plane can be visualised as stretching out into space, way beyond the actual orbit around the sun and bounded within a celestial sphere, with a ‘celestial equator’, on the same plane as Earth’s equator, also marking a circular section of the sphere at 23.4° from the ecliptic. The north-south celestial axis, an extension of Earth’s axis to the celestial sphere, is again at an angle of 23.4°, on average, from the north-south ecliptic axis, which runs perpendicular to the ecliptic plane.

There’s more, but I’ll stop at this. The ecliptic plane for Earth is an average, as there are always perturbations. The other planets don’t follow this ecliptic precisely, but they’re not too far away, probably as a result of uniforming forces at the creation of the solar system.

 

Written by stewart henderson

September 8, 2018 at 9:43 pm