We used Danby for our text. As the professor explained, “There are two very good textbooks for orbital dynamics. This one costs $25.”.
As for the fun, for one of our projects, we were given a set of prior orbital elements with error bars, plus an incomplete set of high-precision measurements (four distance measurements from fixed points at particular times, I think it was), and had to refine the orbital elements. For the eccentricity, the prior error bars were actually larger than the measurement itself, and only about half the class managed to figure out how to handle that without their codes blowing up (since of course eccentricity can’t go negative).
Five years was just for the sake of example, not anything that Herschel actually did. It’s not like someone would look at a comet once, and then not look at it again until five years later. In practice, I expect that Herschel probably observed her comets continually every clear night until they were lost in the sun’s glare or became too faint.
Is the observation continuous for your example? I was under the impression that it was observing a comet when it is first discovered, is not observable for x years and is again observed on it’s return.
Thanks yet again.
I’ve never read Danby. BMW seems to be the typical text that everyone in industry uses for basic astrodynamics (and is available in a supercheap Dover edition), and you’ll see Vallado on the bookshelf of any professional (or would-be, such as myself) astrodynamicist as it goes deep into actual numerical solution methods as well as extensive theory, but it is so dense that trying to cover all of the material in a lecture class would be three or four semester courses.
Once you’ve made a sufficient number of position measurements (over a span of a few weeks for a distant object like a comet) you can calculate the orbit and generate a table of the ephemeris. When the comet re-appears on its next journey to perihelion it should fall very near that prediction (within measurement error) which gives high confidence that it is the same object. There are also some particular characteristics of comets, like the albedo length of the tail at a given distance from the Sun but that would probably not be discernible by the crude optics of the day. Depending on the position of the Earth relative to the position of the comet at perihelion, it may be observable from the orbit of Saturn to inside the orbit of Mercury either coming or going, which gives plenty of observations to make a good estimate of the orbital parameters and period.
Comet flybys are not discrete events. On each orbit, a comet will approach the Sun, stay relatively close to the Sun for a few months or so, and then recede further away from the Sun. You can and probably will make observations for that entire span of months. If you already have yesterday’s observation, today’s probably won’t be very valuable by itself, since the comet won’t have moved very much over that time, but when you get today’s and tomorrow’s and the day after’s and so on for the entire time the comet is close, you should be able to get a very good calculation indeed.
Just to get this out of the way: the hairy “tails” of comets–by that very word and their pictures since I don’t know how many centuries–can show which way they’re heading? That sounds very atmospheric-effect. But those are meteorites, actually doing that. Historical conflation?
And more generally, this makes me think of a problem when other aspects of 3-D reference are missing: a streak may be x-miles long seen full on planar, but if the streak is oriented towards or away from you, and no diminution or other parameter is changed, the streak will be optically shorter.
Did I just discover perspective, made rigorous in art 1435, and is the basics of astronomical interpretation since forever?
A comet has different kinds of tails, which will point in slightly different directions, but both will mostly point away from the Sun (and hence, will lead the comet when it’s moving away from the Sun).
And comet tails are of course subject to perspective foreshortening, but there are multiple ways to correct for that.
To elaborate, a comet has a tail of neutral dust and gasses that points strait backwards away from the sun (blown by the solar wind) and an ion tail that also points backwards, but curves along the magnetic field lines of the sun.
@Darren Garrison: IMO “backwards” is a notso hotso word choice in this case.
Unlike a meteor trail which records the path of passage, the comet tail mostly “blows” away from the Sun. Which is where the comet was on the “descent” towards the Sun, but is where the comet will be during the “ascent” away from the Sun.
So either backwards or forwards depending on whether it’s on the inbound or outbound leg to/from the Sun.
I know you know the difference, but some folks read more literally and literarily than others.
Even in the 18th century, there was teaching someone mathematics and there was teaching someone mathematics.
Neither William nor Caroline, for all their talents and brilliance, were the ones trying to calculate orbits. That was just not their thing. William never tried to calculate the orbit of what became the planet Uranus, etc.
Looking over the multiple editions of her memoirs and letters, I’m really not sure she ever remotely expressed an interest in the calculation of orbits. Simply discovering a comet was the achievement.
That is damned odd. I would think that knowing when it would be visible again would be important for them. Surely something was mentioned when they gave the dinner party in the telescope tube.
Keep in mind that most comets are either one-and-done, or very long period. When it’ll be visible again is a lot less interesting when the answer is “4000 years from now”, or “never”. Halley’s is very much the exception, which is why it’s so famous.
Besides which, even if they didn’t calculate the orbits, somebody else would have. Yeah, it’s a lot more work without computers, but hey, that’s what grad students are for.
My Grandfather had the chance to see Halley’s comet twice. When he was a kid, and when I was a kid, but G-d hates me, and there is always cloud cover when I might see a comet, save for the one Mrs. Plant (v.3.0) and I drove around North Little Rock to see.
Hyakutake and Hale-Bopp were both easily-enough visible, and visible for a long enough time, that there was no need to miss either. Both were bright enough to be seen even from within a city, and both stayed at that brightness for over a month. Both were also a heck of a lot more impressive than Halley 1986, and Hale-Bopp just might have been more impressive than Halley 1910 (accounts of which I suspect to have been rather exaggerated).
I can recall Comet Ikeya-Seki. One of my earliest memories is getting up early one morning to see it in the Eastern sky before sunrise. It was amazingly bright, as I recall.
Possibly not much of a concern when you’re doing vast regular sweeps of the sky looking for new comets anyway.
Though there was a case where her brother extrapolated a position from foreign reports and she tried in vain to find the comet.
It’s perhaps worth describing how she was reporting her finds. Her two main points of contact were Nevil Maskelyne, the Astronomer Royal, and Joseph Banks, the President of the Royal Society. In the latter case, I can’t see Banks having done anything with the news other than pass it on and I suspect there was a sense that telling him was to make it all “official”. Maskelyne however was making much more active use of the news and you get complaints from him to her that the post is taking too long and that her reported positions are too vague. Leading to the occasion when she rides alone through the night from Slough to Greenwich to tell him that she’s discovered another. That said, I suspect his enthusiasm for quick and accurate reports is less that he wants a good first data point for any calculation and more that he wants to minimise the bother in him finding it himself. Once he’s located it, building up a good series of observations from Greenwich is straightforward.
To take the obvious contemporary, Messier’s observations of comets were being turned into orbits by Bochart de Saron (well, at least until the guillotine intervened in 1794).
In terms of what was being calculated, the big useful approximation was to assume a parabolic orbit. Even for a periodic comet, that’s likely to be quite a good fit for the observed portion of the orbit. Newton had given a somewhat fudged solution to that in Principia and people had improved on that in the intervening century.
They probably spent the time moaning about how George III still thinks he can see Hanover through the telescope.
