Caroline Herschel, sister of the astronomer and telescope maker William Herschel discovered eight comets. Could the orbit of a comet be determined at that time, and if so, how?
The same way we do it now. After all, it was Kepler who figured out the basic math behind how objects orbit. And it doesn’t take supercomputers to figure out an orbit based upon astronomical observations, just a fair amount of really precise calculation. Which you then refine as more observational data comes in.
Thanks.
You can calculate the comet’s position over time, but you can’t view it six months apart to determine parrallax, can you?
A better question is, “What observations does one make of the comet to apply Kepler’s laws?”
In addition to its position in the sky, you can look at the brightness and apparent length of the ‘tail’ (when it is close enogh to the Sun to be observed) to gauge the proximity. It only takes a relatively few observations across a span of a few weeks to get a reasonable approximation of the orbital elements, and from there is is easy enough to predict where it should be in the sky after it passes perihelion. It isn’t necessary (or frequently even possible) to get a good estimate of distance by parallax alone since both the planet and comet are in hogh relative motion to the Sun; you’d already need to have a good idea of the state vector at the observed positions, and by the time you’ve gotten good estimates for that you can already figure relative distance and ephemeris of the comet.
Stranger
An orbit has six degrees of freedom, so you can in principle completely determine an orbit from three measurements of position in the sky (each of which has two coordinates). The algebra can get a bit hairy, but it is still only algebra.
Of course, it’s better to use more than three observations. In that case, the problem is overdetermined, and you’re going to need to use statistics in addition to algebra to solve for all of the orbital parameters. Gauss worked out how to do this (in fact, I’m pretty sure that it was his work with orbit determination that led him to study the statistical distribution that now bears his name).
I thought it was 7 degrees of separation? Don’t all comets have a connexion to Kevin Bacon?
More seriously, getting back to the OP, wasn’t that exactly what Edmund Halley did: work out that some comets have recurring orbits, based on observations using 17th century astronomy and algebra?
In case there’s an actual question that goes with that joke, six degrees of freedom means, basically, that you can describe an orbit using six numbers (though there are many possible choices of sets of six numbers to use). Most commonly, you have the semimajor axis (a sort of average distance from the Sun) to describe the size of the orbit, the eccentricity to describe the shape, three Euler angles to describe the orientation of that shape, and a phase to say where on that path the planet is at some reference time. Alternately, you could, for instance, pick some reference time and specify the planet’s position (three components) and velocity (three components) at that reference time, or take the standard orbital elements and combine them in various ways that eliminate the awkwardness that arises when some of them are zero or close to it. You could even take the three sky-position measurements you took, and call those your degrees of freedom, though they’re really not very useful in that form.
EDIT: Haley had the luxury of historical records to see that the comet was returning to begin with. When you’ve got a comet showing up in your sky in one year, and a comet 76 years before that, and one 76 years before that, and so on, it doesn’t take much math to determine that it’s going to show up again 76 years after the most recent sighting. What’s more impressive, is to take a few observations of a comet from a single orbit, and use those observations to predict when it’ll appear next.
Thanks!
Chronos has the correct answer. Though, to be pedantic, I think that all of Caroline Herschel’s comet discoveries and the orbital determinations thereof were prior to Gauss’s treatment of the problem.
For people could laboriously hack out approximate solutions to cometary orbits without the Gaussian exact version.
William taught Caroline mathematics. I wonder if he or she calculated the orbits of her comets.
Not seven? The Wikipedia entry says that “Sometimes the epoch is considered a “seventh” orbital parameter, rather than part of the reference frame.”
I always thought that it was seven, because you need two different x,y,z positions, plus the time elapsed between them (the Gauss problem.)
Although error estimation in prediction from discrete measurements predated Gauss by almost two decades (the first real efforts at probability analysis can be traced back to Antoine Gombaud, Chevalier de Méré who was interested in likelihood of different results in games of chance and estimations of “truth” and engaged Pierre de Fermat and Blaise Pascal), Gauss was in fact the first person to apply the normal distribution to continuous measurements to address the problem of measurement error and uncertainty in celestial observations, using a linear least squares estimation to reduce to the average of the observations for orbital determination, which is the same method used today for first order evaluation of orbital elements when the observations are relatively close together. For methods where observations are at distant angular arguments, the Double-r iteration or Gooding’s Method are used, and for solutions where the range can be directly determined by laser or range-rate estimation, trilateralzation methods are used. Modern satellite orbital determination uses the Gibbs method as laid out in Bate, Mueller, and White, and in making near-real time estimates from rapid sequential observations the Herrick-Gibbs method or deviations therefrom is what is used.
Gaussian error estimation doesn’t account for other perturbations that deviate from Keplerian orbits (which occur to some extent in all celestial objects) but for gross determination of orbital elements it only requires a few observations that are reasonably distributed around a segment of the orbit. However, even with an overdetermined system it is possible to get an approximate answer without resorting specifically to a least squares reduction, just by calculating a series of orbits from different sets of three observations and seeing which combination falls closest to prediction the largest number of other observations, which is conceptually the method used to determine distributions for orbital or trajectory perturbations and atmospheric or solar particle drag.
Here is a fairly cogent and reasonably brief description of the history of how the normal (Gaussian) distribution was developed and applied to astronomy.
Stranger
There are six orbital arguments. The epoch is basically part of the definition of the reference frame, and so is essentially arbitrary, e.g. you can start measuring the position of the orbit as reference at any point and you’ll get a consistent set of elements.
Stranger
There probably was a time, close to twenty years ago, where I would have understood all of that and agreed. I probably used those same methods myself. That part of my memory, however, has apparently rotted away (though I do remember that my orbital dynamics class was quite a lot of fun, even by the standard of other physics classes).
If I ask you “Where was the comet on January 1, 2000”, and you give me three coordinates, and I then ask “Where was the comet on January 1, 2005”, and you give me three coordinates for that, too, then I have all the information I need to determine the orbit, even though you have given me only six numbers. I will, of course, need to use the five-year difference between those numbers in my calculations, and will need to use the dates themselves to determine the phase parameter, but those are numbers I came up with, not you, and not the comet. I could have chosen any two dates I liked to ask about (though, of course, the noise would be badly amplified if I chose dates too close together).
My degrees are in electronics. My interest in astronomy comes from being a kid during the Gemini and Apollo programs. Excuse my question:
How did Caroline Herschel know that she viewed the same comet five years apart? It is not as though they have license plates.
Orbital mechanics is amazingly fun, and like Mandarin or chess, if you don’t us it in a long time most of the knowledge fades like a tan in a Minnesota winter. Although I didn’t really need them to respond to this thread, I pulled out Vallado and a couple of William Wiesel’s texts (BMW is curiously missing; need to figure out if I loaned it to someone) and spent a pleasant couple of hours paging through just to see what all I’ve forgotten (a lot). I don’t really get to do any trajectory work anymore (and most of what I did was for unsolicited proposals) but it is enormous fun, especially trying to find optimal methods to find low energy intercepts or injection maneuvers.
Stranger
Yes! I was able to get about halfway through Fundamentals of Astrodynamics, by Bate, Mueller, and White.
Some of the equations cannot be “solved” and must be worked by numerical methods, in which you propose a guess for the first approximation, refine that, refine again and again, until the process converges on the correct answer.
(I do not know if Gauss used that kind of approach to solve for the orbit of Ceres.)
Heh, that was the course I audited my junior year, even after I had given up on being a physics and astronomy major; it was fun! (For, I suppose, very large parameters of “fun”, many would say)