COMET PATHS IN THE SKY -------------------- John Pazmino NYSkies Astronomy Inc www.nyskies.org firstname.lastname@example.org 2009 July 21 initial 2013 August 15 current
Introduction ---------- At the NYSkies Astronomy Seminar of 2009 February 19, the group discussed the path of a comet in the sky as seen from Earth. Many examples from current comets were presented in a handout. Of special interst was comet Lulin, at the time in the sky within binocular range. It was trending close to the ecliptic in a direction against that of the planets. It turns out that there is no good treatment of the geocentric motion of a comet in the litterature for home astronomers. They are given only comet's orbit around the Sun. But that is the view in an inertial frame fixed against the stars. It is not even the view from the Sun himself. What happens when the observer is on the Earth, herself in orbit around the Sun?
Planetary motion -------------- As seen from Earth the planets travel west to east thru the zodiac, roughly parallel to the ecliptic. A planet's location is approximated by its ecliptic longitude. There are minor deviations north or south in ecliptic latitude. The 16 degree width of the zodiac contains just about the maximum latitude displacement from the ecliptic that a planet can do. The planets are continuously observable, save for brief intervals when they are angularly near the Sun. Even then, their motions are smooth enough to anticipate when they emerge again into darker sky away from the Sun. Their behavior could be monitored for patterns and cycles to be compared with records from earlier times. We are able to figure out the kinetics of the planets, including the Sun, and predict with good confidence where they will be in the future. It was this ability to foretell with complete assurance the future position and motion of the planets that lifted astronomy to the highest level of professions. Other forecasting schemes, for weather, crop yield, animal hunts, disease, all fell apart. They still do.
Distances ------- With Earth as a fixed place in the cosmos, there is no good way to fathom the distances to the planets. Only the Moon showed a parallax when seen from different places on Earth. She would occult a certain star over one town but miss the star from an other. The other planets showed no parallax, as if they were of indefinite remoteness from us. An attempt to rank the planets outward from Earth was based on their angular speed. The Moon cruised thru the zodiac fastest of all, so it must be the closest planet. This was confirmed by seeing the Moon always pass in front of other planets, never behind them, during an occultation. Mercury and Venus presented a problem. Over long spans of time, their speed averaged to be the same as the Sun's, But they could not conceptually be at the same distance from us, else they would interfere and collide with the Sun. Each planet must stay in its own lane or orbit. Mercury scoots around a mean location centered on the Sun in about 3-1/2 months. Venus wanders around the Sun's location in about 1-1/2 years. Plausibly, Mercury was the next planet beyond the Moon, then Venus, then Sun. This seems 'logical' but some early astronomers put Venus, Mercury, Sun as the ranking above the Moon. Mars, Jupiter, Saturn are more well behaved. They move slower from Mars. Jupiter, and Saturn. These planets were ranked above the Sun in that order. Farther than Saturn was the realm of the fixed stars. There never was a concept of 'solar system' until Copernicus. The behavior of each planet was modelled separately from other planets, with no attempt to seek a correlations among them. Not even the presence of the Sun was a major factor in the motion of each planet raised the idea of a linked set of planets.
Comets ---- There was one major exception that refused to yield to the methods of analysis that worked so well for the planets. This was the comets. They appeared frequently enough to arouse concern and seemed to be residents of the celestial realm. There were other temporary erratic visitors in the sky. They were shortlived, not widely observed, and appeared to be local like clouds. Such include bolides, aurorae, solar halos, and meteor showers. Comets, on the other hand, were seen from wide regions and persisted for weeks or months. During its apparition a comet attracted major public attention. Astronomers were called to explain them. Before the telescope only brighter comets were observed. Dim ones were missed. Counting only the bright comets, we could expect one every decade with their frequency quite erratic. Some decades had several visitors; others, none at all. A really brilliant comet came around once or twice in a lifetime.
Apparition -------- A comet rised into view by gradually brightening into bare-eye range, At first it was a weak misty spot where no previous star was noticed. It gradually growed in prominence and resolved into a more definite shape. The comet after swelling in prominence typicly sprouted a tail of luminous material. The gradual crescence from an obscure spot to a showy body covering tens of degrees of sky gave the illusion of an approach to Earth from an indefinitely great distance. The point of first visibility of a comet was any where among the stars, not confined to the zodiac like the planets. It can come into view from high ecliptic latitudes and from any ecliptic longitude. This made it tough to anticipate a comet. It sneaked up without warning and may have been in the sky for a few days before astronomers caught notice of it. In somw cases, the astronomer was alerted to the comet by the public seeing it first. After a period of weeks or more the comet contracted in size, the tail shrank, the central body dimmed. The apparition faded gradually from view, as if reproaching back to some remote realm of the heavens. The comet receded out of sight in any place, in any latitude and longitude. There was no way to infer the recession point from the accesion spot. In many instances the two endpoints were within a constellation or two of each other. But there erre enough exceptions, with widely separated endpoints to ruin general rules about endpoints.
Movement ------ The comet moved thru the stars in a path utterly without rhyme or reason discernible by the early astronomer. So irregular was the path that even after tracking the comet for several days, the location on the very next day could not be reliably forecast. The path may have loops, zigzags, speed runs, halts, reversals, spirals. The comet may wander far from the zodiac, into constellations no planet can possibly migrate thru. Or it can hug the ecliptic like a planet, There was usually no correlation bewtween the aspect of the comet and either its location or movement among the stars. Changes in size, shape, texture, features of the comet seemed to have no effect on the body's future position or motion. Attempts to llnk various aspects of the comet to the comet's future behavior enjoyed zero percent of success. This was one gross embarrassment to the stronomer, now confronted with a celestial apparition well beyond the ordinary, and extraordinary, efforts to understand. If the comet was witnessed by the general public with no calming explanation from the astronomers, the comet was treated with fear and dread. A coemt was a bad star, a dis-aster.
Apprehension ---------- We can easily appreciate the apprehension we once had upon seeing a comet. It did not fit into the neat order of the world, had a threatening hostile guise, seemed to wander thru the stars with its own mind and will. A comet sure did look like it could incinerate the Earth, spread poison and disease, collide, terminate humanity in all sorts of horrible ways. For foretelling the fate of a person or nation, a comet had no playbook. The astronomer, who often was also an astrologer, made up a fable to suit the current social and political ambience. To be safe, the interpretation was usually one of calamity of the kind which by prior experience would come against the people anyway, comet or not. A comet was not a pleasnat aster.It was a fearsome disaster. Yes, this is where that word really comes from! In spite of the general dread of comets, astronomers gradually over the ages recognized some interesting features about them. In general, the comet was more lustrous when angularly near to the Sun and was weakest far from him. The tail always pointed away from the Sun, never toward or broadside to him. Comets presented no parallax, meaning they were far beyond the sphere of the Moon. Comets typicly first came into view at dawn or dusk, rarely in full night.
Newton's comet ------------ The behavior of a comet in the sky remained a mystery until the Newton cracked the problem. He showed that the Comet of 1680 moved around the Sun under the force of gravity. What's more, its path was one of the conic sections allowed by a gravity field, specificly a parabola. He could suss this out by employing the heliocentric model of the solar system, with Earth in her own orbit around the Sun between Venus and Mars. This system, the Copernicus system, was still in great debate in the 1600s. Many astronomers clang to the Ptolemaeus scheme, with the Earth at the center of the world and every thing else circulating around her. Newton unravelled the motion of a comet by subtracting out the motion of Earth. The cockeyed path in the sky was a combination of the comet's orbit around the Sun and our view of it from the moving Earth. By removing the component of the Earth's changing place relative to the comet, the comet became merely an other gravity-driven orbiting body, not an orb animated by a crazed spirit. The analysis of Comet 1680 was a conclusive validity for the theory of gravitation. For his work with the planets, arguments were raised that Newton fudged his numbers to fit already kknown history for these bodies. The theory was merely an other kinetic model that best fit the millennia of data built up for the planets. In fact, in some colleges, you could choose to study astronomy by the Copernicus or Ptolemaeus model. The comet had no history. It was a brand new visitor, never before seen by humanity. Yet it yielded so well to the tools of gravity. Being able to correctly foretell its future place among the stars by mathematical methods validated Newton's theory of gravity. Ptolemaeus methods were useless and hopeless for describing comet motions. Newton's study of Comet 1680 also confirmed the suspicion that the luster of a comet depended on its remoteness from the Earth or Sun. This comet first appeared to sight at a far distance along the inward arm of its newly laid-down parabola and vanished from sight at a comparable distance away on the outward arm of the parabola.
Conic sections ------------ Newton's gravity theory puts a body in motion around the Sun along a conic section. This is the cut plane of a solid cone with a circle abse under various slices. The already well-known maths of conic sectins was the same as that for the gravity-governed trajectory of a particle around a large central mass. A slice of the cone perpendicular to its axis makes a circle. An inclined cut of less than the apex angle makes an ellipse. Cutting the cone parallel to its side, equal to its apex angle, creates an open- ended section, the parabola. The hyperbola is formed by slicing the cone within its apex angle up to parallel to its axis. A body orbiting the Sun follows one of these paths. While the circle and parabola have only one shape, differing only in size, the ellipse and hyperbola come in both different shape and size. In Newton's day only ellipse orbits were known, those for the planets and Moon. (The Sun was deplanetized by Copernicus!) The 1680 comet added the first physical example of a parabola orbit.
Chaos theory ---------- The apparently irrational path of a comet in the sky as seen from Earth is one application of chaos theory. Under one scenario of chaos, this seemingly self-motivated action of a comet comes from the combination of smooth, regular, well-behaved motions. Unless these underlying movements are known, the comet's behavior is unpredictable and nondeterministic. The comet's motion in the sky results from the combination of two very simple components. One is the parabola or ellipse orbit of the comet; the other, the nearly-circular ellipse of Earth. It is the line of sight from Earth to comet that swings around so wildly across the sky along a choatic path. The comet looks like it's under the helm of a drunk devil. This is one of the lesser-appreciated facets of Newton's work, showing that comets are driven by physical laws in well-ordered trajectories. He cured humankind from the irratonal dread of comets.
Observing comets -------------- Even after the telescope was applied to comets, these orbs were within view only while close to the Sun. The typical comet did not enter telescope range until it crossed into the asteroid belt. This is by distance in any direction from Sun, not just in the ecliptic plane. It was lost from sight when it retreated to the same order of distance from the Sun. We simply didn't see comets farther away than that. It wsn't until the 1980s, with new large telescopes plus advances in digital imnging, that comets could be found while really far from the Sun, out to Jupiter's distance and beyond. They also could be followed back out to that distance. The first long-range discovery of a comet was comet Kohoutek in 1973, found about 5AU from the Sun. This was acheived by the visual comet discovery methods of the day. It must surely be extra luminous to be seen so far away. This prompted predictions of a monster fiery orb in the sky when it got near to the Sun and Earth. As fate fell out, comet Kohoutek fizzled out -- it became comet Ko-hoax-tek -- and ended up as a dim comet requiring binoculars to find. The most celebrated early accomplishemnt of the new digital techniques was the recovery of Halley's comet in 1982.
Periodic comets ------------- Halley in 1704 noticed that a certain set of comets seen in the prior few hundred years had very similar orbits. He was studying theser comets using the new methods of Newton's gravity theory. He worked out that these comets were really the same body in a closed orbit that brought it to the solar vicinity repeatedly. We saw it only when near the Sun. It was well out of sight in the remoter parts of its orbit, misleading astronomers to think each visit was from a new comet. He predicted that the comet will return in 1759, long after he passed on. The comet was recovered. It followed the orbit laid out by Halley. This, now Halley's comet, was the first comet discovered in an ellipse orbit. The period of this orbit, which reached from about 1/2 AU from the Sun at its nearest point to the orbit of Neptune (then not yet known), is about 76 years. Halley's comet was observed on all following returns, the last in 1986. Modern study of old comets reveals that Halley's comet was seen on every return back to the 200s BC. On each of these early instances, a stretch to fit two within a single human lifetime, it was treated as a new visitor. By summer 2009 we know about 250 periodic comets, with periods from 3-1/3 years, for Encke's comet, to a thousand years. They are given a serial number in the order that their periods are confirmed. Halley's comet is periodic comet #1.
Inclination --------- The orbit of a comet, like that of any other gravity-driven orb, is in a plane fixed in space passing thru the Sun. I neglect for this article the perturbations by planets. These can and do swing comets into new orbits. It's a generalization of Kepler's Law of Planes, stated at first only for the planets. While a good approximation for home stronomers, it is not quite exact. The orbit plane intersects the barycenter of the solar system. This point happens to be within the globe of the Sun about 3/4 radius from his center. The orbit plane cuts the ecliptic plane in a line thru the Sun. Where the comet crosses this line while moving south to north is the ascending node. The opposite point, north to south, is the descending node. This line, the line of nodes, is stable in space. The inclination of the orbit is the angle at the ascending node between the forward direction of the comet and the forward, prograde, direction along the ecliptic. This can be from 0 thru 180 degrees. Inclinations between 0 and 90 are for comets in direct or prograde motion around the Sun; 90 to 180 degrees, retrograde motion. Comets may have any inclination. Planet inclinations are only a few degrees at most, forcing them to circulate within the zodiac. Comets, with arbitrary inclinations, may roam thruout the heavens.
Excentricity ---------- There is only one shape of a circle or parabola. They differ only in size. An ellipse can have various shape and size. The measure of the shape of an allipse is its excentricity, a term held over from Prolemaeus astronomy. In the Ptolenaeus scheme a planet moves in a circular path around the Earth. To account for part of its irregular path in the sky, the circle was displaced off center from the Earth. The planet moved angularly faster, from Earth's eye, in the nearer part of the orbit and slower when in the farther section. The excentricity was given by the off-center distance of the circular orbit divided by the radius of the orbit. Because in the Earth centered cosmos there is no way to rank the planet orbits in linear distance from Earth, the radius for each planet was taken as unity (or 60 in the older base-60 maths). The excentricity was a small fraction of unity only because planet orbits are in fact nearly circular. On a scale of 1 meter radius, the Sun's orbit is excentric from Earth by 16.7 millimeters. It takes a skilled eye to discern that such a large circle isn't quite centered! Yet, the Greeks determined it from monitoring the angular motion of the Sun during the year. Excentricity is nowadays denoted by 'e' or 'epsilon'. It is easy to understand from the pin-&-string method of drawing an ellipse. Two pins are pressed into the drawing board and a loop of string is laid over them. The pencil runs in the loop, keeping it tebsed, and traces out the ellipse. The line joining the pins and extended to cut the ellipse id the major axis. The pins are the foci, one is a focus. The center of the ellipse is the midpoint of the major axis and each half of the major axis is the semimajor axis. Perpendicular to the major axis thru the center is the minor axis. This has only minor use in astronomy. The excentricity is the distance of one focus from the center of the ellipse divided by the semimajor axis, a definition more in line with geomtery and other maths. Because the focus is inside the ellipse, the ratio of its distance from the center over the semimajor axis is less than one. The excentricity ranges from 0 up to, but not equal to, 1. Excentricity is sometimes stated as an angle. The excentricity range of 0 to 1 is the range for the sine of an angle. The arcsine of the axcentricity is the angle sometimes cited in its place. That is, an excenticity of 0.2 is also one of 11.5370 degree because the sine of this angle is 0.2. On this scheme an ellipse excentricity ranges from 0 up to 90 degrees. Geometricly this angle stands at one end of the minor axis with arms to the center and one focus. In this diagram A is one end of the major axis; F, one focus; C, the center; E one end of the minor axis. The ellipse, not drawn, arcs from A thru E in one quadrant, then coninues thru the other ends of the major and minor axos to rejoin a.
+E / | / | +------+-----+---~ A F | C ~ The line EF is equal in length to the semimajor azis AC, by the properties of ellipses. It is unity. FC is the excentricity of the ellipse vecause EC/AC = FC/1 = FC. In the right triangle FCE EC is the side opposite angle E. Sine(E) = EC/EF. Because FE = AC, we have sine(E) = EC/AC = EC/1 = EC. This is why excentricity is sometimes cited as an angle, E = arcsine(EC). When the excentricity is zero, both foci are at the center of the ellipse. The curve traced by the pencil becomes a circle. When the excentricity is unity, the major axis is infinitely long. The distance from one focus to the center, at infinity, is the same as the distance from one apse to that center. EF = EC. As an angle, the excentricity is arcsin(EF/FC) = arcsin(1) = 90 degrees. The loop of string is infinitely long, it never closes around, and the curve is open-ended. This is the parabola.
Sun at focus ---------- In a gravity-driven orbit, the Sun sits at one of the two foci, the other being vacant. Kepler figured this out in his first law of planetary motion. If the orbit was a circle, the Sun would still be at one focus and the other, coincident, focus is still empty. We speak of Kepler's three laws of planet motion. In point of history, there are FOUR laws. To keep the numbering intact, the fourth law is #0. It is split off of the old law #1. Some authors handle this split by calling the laws #1a and #1b. This keeps the 'three' laws. Law #0 or #1a is the Law of Planes discussed above. The orbit is a plane stable against the stars and passing thru the Sun. While this sounds simple today it was a massive step forward in thinking in the early 1600s. For the first time humankind had to realize that motion in space could not be banked off of the Earth in an absolute sense. Becuase the Earth in the heliocentric system moves, Kepler needed a new 'ground' to bank planet motions against. He considered the sphere of fixed stars as the alternative, since until his era there was no motion ever noticed among the stars. They formed a grid or lattice on which to hang other celestial motions. Three centuries later this concept of an external absolute frame of reference would be the foundation of Einstein's work. Law #1 or #1b is the 'Law of Ellipses'. In the orbit plane the planet moves in an ellipse with the Sun at one focus. Newton extended this principle to other conic sections. A comet can move in a parabopla with the Sun at its accessible focus.
Circle ---- When the both foci coincide at the center of the ellispe the excentricity is zero and the ellipse rounds into a circle. No comet or other natural body has a truly circular orbit. Venus has a closely circle orbit [in the heliocentric solar system] with excentricity of 0.007. In a scale drawing of Venus's orbit with semimajor axis of one meter, the each focus is displaced from the center by only 7mm. It is the small excentricities of the planet orbits that allowed early astronomers to lay them out as true circles displaced slightly off center from the Earth, and later from the Sun. The great accomplishment of Kepler was recognizing that the off- centered orbits of Copernicus were NOT circles, but ellipses. The circle is a bit squahsed, enough to deviate the planet from where the Copernicus model put it in the sky. He was lucky to pick Mars as his target. Mars has a large excentricity so the deviation of the ellipse from an off-centered circle showed up better. If he chose Venus to start with, he would likely have failed to distinguish a circle from an ellipse.
Parabola ------ It is not so easy to locate the foci of a parabola but you can visualize them by the mental experiment with an elongated ellipse. If you make the major axis extremely long, the ratio (focus / semimajor axis) tends to unity. The excentricity of a parabola is one. One focus stays to hand, with the SUn occupying it. The other one, empty, removes to infinity. It's easy to see why for a long time comets were believed to have parabola orbits. A comet was visible only near the Sun, along only part of its real ellipse. If the comet had a period of more than a few hundred years, the visible part of the ellipse looks closely like the vertex of a parabola. We had too short a span of the path to trace the arms to a substantial distance and see if they remain separate in a parabola or close up into an ellipse. New comets, with no prior history and not in an obvious ellipse orbit, are handled by parabolae at first. As observations accumulate the path may be refined to an ellipse, indicating a comet of some long period. Or it may remain a parabola as a one-time visitor. The endpoints of the parabola on the sky are the same. The comet appears from one point among the stars and vanishes into that same point. This is in the ideal case of observing the comet regardless of its brightness. In actuality, the endpoints are separate because we discover the comet already well along toward the Sun. We later lose sight of it while still within the solar system. By prolonging the in and out arms we find the endpoints are coincident.
Hyperbola ------- A few comets have hyperbola orbits. The excentricity has no easy geomteric definition, but it is greater than one. Of the comets so far on the books, the excentricities of hyperbola members are only a smidgeon over one. This suggests an inaccurately worked up path due to observational deficiency. There seems to be no strong theory for hyperbola comets. Those for parabola comets appear to fit just as well to the known hyperbola ones. Once a comet approaches from, say, a thousand AU, it's merely dropping to the Sun by freefall. We don't believe any are propelled toward the Sun that would give them excess speed above freefall and put them into a strongly hyperbolic trajectory. A parabola has arms that tend toward the same vanishing point in deep space. This is why for long period or real parabola comets the endpoints of their path in the sky are the same, A hyperbola has arms that diverge at a nonzero angle. The endpoint of one arm is displaced on the sky from that of the other. A comet is a strong hyperbola, excentricity well above unity, the endpoints are widely separate places in space. Hyperbolae come in pairs or napes. For orbits, only one nape counts; the other is disregarded. The pair is a mathematical construction based on the idea of a cone having two parts touching at their common apex. The conic section to make a hyperbola must cut both parts, making tow disconnected curves.
Perihelion -------- So far we studied only the shape of an orbit. The size of an orbit is a bit more tricky. For a circle or ellipse the size is easiest cited by the radius or semimajor axis. For the parabola the semimajor axis is infinitely long. In all three cases the Sun is at one of the foci and one end, apex, vertex, of the curve, an apse, is closest to the Sun at that focus. This apse is the perihelion of the orbit. The size of an orbit in the general case is commonly given by the distance from the Sun to the perihelion. It is, for the ellipse with the semimahor axis as unity, equal to (1 - excentricity). The rest of the major axis, from Sun to the opposite apse, is the aphelion distance. For the ellipse it equals (1 + excentricity)_ A parabola has no finite aphelion becuase the orbit does not close and there is no opposite apse. Perihelion is symbolized by q; aphelion, Q. Sometimes the farthest distance from the Sun is spelled apohelion, (a-po-HEH-lee-yonn). When spelled aphelion, it is sounded 'app-HEH-lee-yonn', not 'a-FEH-lee- yonn' with 'ph' sounding like 'f'. The perihelion POINT is often denoted by 'pi'. Perihwlion also can mean the TIME when the comet crosses the perihelion point. It is denoted by 'T'. We can have a perihelion of so many AU distance and a perihelion of a certain moment and the 3D spatial location of the perihelion point. Context sorts out the meanings.
Law of Areas ---------- We examined the first two laws of Kepler. These are #0 or #1a, Law of Planes, and #1 or #1b, Law of Ellipses. After Newton, the laws were generalized to accommodate other possible orbits under gravity. Law of Ellipses is updated to a Law od Conics to state that the orbit is a conic section with the Sun art one focus. The remaining two laws are essentially unchanged. They work for the expanded range of orbits. They are the Law of Areas (#2) and Law of Periods (#3). The Law od Areas states that the body sweeps, or paints, equal areas of the interior of its orbit for equal spans of time. This at first is tough to understand. Think of an obviously excentric ellipse orbit and join its orb to the Sun by an elastic line. This line is the radius vector, equal to the instant distance of the body from the Sun. If the orbit was a circle the body travels at a constant speed and moves over the same arc any where around the orbit for a given span of time. The arc traversed per unit time is the same all around the circle. The radii vectores are always the same, the radius of the circle. The area of the pie-shaped sector bounded by each pair of radii vectores is found by geometry. The ingredients for the area formula of a sector are the arc and radius. But these are the same for all the sectors. The rate of sweep over the whole orbit is the area of the circle divided by the orbit period
(areal speed) = (orbit area) / (orbit period) = (radius ^ 2 * pi) / (orbit period)
In an ellipse, the planet ranges near and far from the Sun as it rounds the orbit. The radius vector is longest at aphelion; shortest, perihelion. The rate of area coverage is, in spite of the changing shape of the wedge around the orbit, still the area of the entire orbit divided by the orbit period.
(areal speed) = (orbit area) / (orbit period) = ((smaj axis) * (smin axis) * pi) / period
In both the ellipse and circle, for any span less than a full period, the area sweeped out is proportional to that span. Note the formula for the area of an ellipse. If the orbit is a circle, the semimajor and semiminor axes are the same, the circle's radius. The formula becomes area = radius*radius*pi or radius^2*pi. The area painted for a run of the planet over a given span of timee is that of the sector between the start and end of the span. For small spans, a percent or less of the body's period, the wedge may be treated has part of a circle with radius equal to the average of the two radii vectores bounding the sector. Now the orb is near perihelion. The radius vector is shorter, making the sector shorter. To keep the area the same as before, the arc of the sector along the orbit lengthens. This means the orb travels farther along the orbit in the time span. The greater speed makes up for the shortened radius vector and the sector area remains constant. Near apohelion the radius vector is longer, for a taller sector. The body can keep the wedge area constant by traveling over a lesser arc. It travels at slower speed. The visible effect of the Law of Areas is that the angular (linear along the orbit) speed of the orb varies with distance from the Sun. It's faster for lesser distances; slower, greater.
A missed discovery ---------------- It is a weird bit of astronomy history that Ptolemaeus came close to discovering this law! He knew that the planets ran faster and slower in accordance with their distance from Earth. He assigned the planet to run at a constant angular speed around not the Earth but a point displaced from Earth by the same amount as the orbit's center, but in the opposite direction. In modern terms this is the EMPTY focus. This preserved constant circular motion, a prime axiom in Greek philosophy, yet allowed for APPARENT varying speed of the planet. Copernicus did the same trick with his orbits, now around the Sun, but still circular. For small excentricities, those among the planets, the Ptolemaeus method is an amazingly good fit, well within the observation errors of the pretelescope era. If Ptolemaeus went the extra meter he would have seen that the sector areas around the orbit were equal for equal increments of time. He, and Copernicus, didn't.
computing planets --------------- The Law of Areas simplified the chore of calculating where a planet is in its orbit. Pick an increment, say 1% of the planet's orbital period. Start at aphelion and step the planet one increment. Since the areal speed is known from the diemsnions and period of the orbit, the area of a sector sweeped out during each increment is just 1% of this total area. With the sector area and length of the radius vector, averaged between the start and end of the time span, compute the arc needed to enclose this sector area. Move to the next interval around the orbit. Because the radii vectores are now a bit shorter, being away from aphelion, they have to be spaced a bit farther apart along the orbit. The planet moves a bit faster to ocver this extra spacing for the second step. Procede around the orbit thru perihelion, where the sector arc is the longest and the radii vectores are farthest apart. You can skip the other half of the orbit because it is a pure reflection of the half you went thru already. In this way you plot the planet's place around the orbit, 100 of them for 1% increments of the planet's year. This is a permanent chart. You do this work only once. Let's say that for a certain date you observe the planet at a certain place in its orbit. Note where it is on your chart. Now you want to know where the planet is so many days later, which translates to so many percent later of the planet's period or year. Just count off that many percent, that many steps, from the initial position! That's where the planet will be. No mess. No fuss. Kepler freed astronomers from the miserable task of massive maths to compute the positions of planets. The Law of Areas works for all conic sections as orbits, with some adjustment of the maths for a parabola. A parabola has no finite period or closed area. They are infinite. In the part of the orbit near the Sun, within the zone of a comet's visibility, the Law of Areas is essentially the same as for ellipses. Astronomers plotted charts of positions for parabolae of assorted perihelion distances. When a new comet comes along, get a fix on where the comet is on one of the orbit charts. Then procede as you do for a planet. Such charts were in wide use in the 1700s-mid 1900s to work up ephemerides and plot the path among the stars. In the early years of the space age, such charts (and for ellipses) were published to quickly track artificial satellites and ballistic missiles.
Law of Periods ------------ The fourth law is the Law of Periods, still numbered '#3". Kepler having finished developing the first three laws still had a massive problem. So far each law applied to each planet on its own with no relation to other planets. How are the planets linked together within the solar system? Why does Mercury zoom around the Sun in 88 days while Saturn takes 29 years to complete one lap of its own orbit? Kepler tried to explain the bond of the planets with the Sun by magnetism, studied by Wlliam Gilbert in Kepler's day. Is the Sun magnetic? Well, it does have those spots. Are they magnetic poles that as they move around the Sun drag the planets with them? The Earth, by Gilbert's work, is a big magnet.COuldn't it be attracted by magnets in the Sun. Eventually, after many years, Kepler hit on the unifying principle. The period, year, of a planet was a simple function of its distance, semimajor axis, from the Sun. In maths:
(period) ^ 2 = (K) * (semimajor axis) ^ 3
This works for any consistent set of units, like days and kilometers. The proportion canstant K fixes them up. If we take the Earth's semimajor axis as unity and the Earth's period as unity, K becomes one and the Law of Periods is a trivial exercise. Jupiter's orbit is 5.2 times larger than Earth's. His period is
(period) ^ 2 = (K) * (semimajor axis) ^ 3 = (1) * (semimajor axis) ^ 3 = (semimajor axis) ^ 3
(period) = (semimajor axis) ^ (3/2) = 5.2 ^ (3/2) = 11.86
Jupiter's year is 11.86 Earth years. This matches just about exactly the known period of the planet. The law works for all ellipses, even for comets. In fact, in the specs for a comet the period is often missed out because it can be calculated from the semimajor axis. The period for a paravola or hyperbola comet is infinite. After ronerounding of perihelion, it takes infinite time until the next rounding. The comet is a one-time visitor. The law as Kepler stated it is not complete. It works to a good degree because, unknwon to Kepler, the Sun is so overwhelmingly more massive than any of the planets. Newton showed that the Law of Periods is
(period) ^ 2 = (K) * (Sun mass + planet mass) * (semimajor axis) ^ 3
Only because the planet mass is so small against the Sun that the sum (Sun + planet) is always very close to just the Sun's mass and is rolled into the K constant. On the other hand, for binsary stars, where the two stars are of the same order of mass, the full Newton statement must be applied.
Inertial view ----------- Virtually all discussion of comet orbits deal only with the view of the comet path from an inertial frame. The observer sits away from the Sun and is stable against the stars, as if he was in a stadium seat looking at the solar system. In this frame the planets go thru their nitid orbits around the Sun according to Kepler and Newton. Comets course thru the solar system in their ellipse or parabola orbits, coming in from any direction and passing perihelion at assorted disances. This viewpoint has the advantage of showing best the trajectory of a spaceprobe passing from Earth to a comet. It is also the best for taking off the comet's orbit elements, such as perihelion location, inclnation, ascending node, The measurements can be taken directly from the diagram, to the extent the diagram is faithfully plotted with a large enough scale.
Heliocentric view --------------- To the observer on the Sun a comet runs in a 'stright line', a great circle, in the local sky. Foran ellipse this greatcircleis a closedcircuit around the sky and thecometruns in it coninuously in one direction. For a prabola the great circle is cut at the far end of the major axis, opposite the perihelion point. The comet emerges from one side of the cut, rounds the sky,and retreats into the other side of the cut. It does not jump across to the first side to begin a second lap of the sky. From Kepler's Law of Planes a comet's orbit is a plane passing thru the Sun. By standing on the Sun we see that orbit edgeon as a great circle in the Sun's sky. This great circle is stable in the stars. It can be plotted on a Sun starmap, like the ecliptic on an Earth starchart. The comet moves in this circle always in one direction but with varying angular speed. The speed is greatest at the perihelion point and least at the aphelion point of an ellipse or zero at the infinitely far off outer end of a parabola. The place and speed of the comet can be mapped out once at given intervals of time. This plot is good for every return of a periodic comet. The plot is symmetrical across the major axis joining the perihelion and foci. The only difference is the zero date, like that of perihelion pass. Once zeroed, all other dates fall in line and the place and speed of the comet can be quickly and easily figured out for a given round. The diagram here suggests such a plot.
* * * * * * * -7-6 -5 -4 -3 -2 * -1 * 0 +1 +2 * +3 +4 +5 +6+7 +-+--+---+----+-----+------+------+------+-----+----+---+--+--+-+- * * * * * ** * * * * * * *
The '*' are stars near the comet's path. The ticks are at equal intervals of time, say 30 days, starting from 0 at perihelion. If for instance the perihelion is on 2009 March 9, then the ticks correspond to roughly one month steps earlier and later than that. See how the distance between ticks is greater near perihelion, indicating a faster movement in the 30-day span. The ticks close up away from perihelion as the speed decreases. For all of the comets a starchart compiled for an observer on the Sun would have great circles criss-crossing the constellations at every inclination to a reference plane like the solar equator and cutting it in every longitude. Each path has its own timeline of position and motion for its comet.
Hwliocentric rectilinear coordinates ---------------------------------- When computing a comet's trajectory seen from Earth, a geocentric station is used. There is no need for topocentric calculations because a comet is typicly among the planets, with a negligible parallax. Only if the comet comes really close, within a hundred lunar distances, do we factor in the situation of the observer at the Earth's surface. To obtain the place of the comet relative to Earth we subtract the Earth's location relative to the Sun from the comet's position, also with respect to the Sun. This is vastly easier to do with coordinate math rather than by classical orbital descriptions. For this task we set up at the Sun a rectilinear coordinate scheme of XYZ axes. the X-axis aims toward the vernal equinox; Y-axis, summer solstice; Z-axis, north ecliptic pole. Any point in the solar system can be represented by its XYZ coordinate in this system, with units of Earth semimajor axis. This is the 'astronomical unit', AU, equal to about 149,600,000km. The diagram here clarifies the scheme. Picture the X and Y axes in the paper plane and the Z axis orthogonal to it.
+Z NEP -Y | -X Win Sol | Aut Eqx \ | / Sun / | \ / SEP \ / -Z \ Ver Eqx Sum Sol +X +Y
In the bad old days we used worksheets into which we plugged the orbital parameters and the date and stepped thru the instrcutions on the form to generate the XYZ coordinates. Before calculettes this was an allday chore using slide ruler, trig and log tables, and drawing tools. For the Earth we did not have to compute the coords. We used the tables of heliocentric coordinates from the Astronomical Almanac of the US Naval Obsrvatory for 0h UT of each day of the year. Only a few home astronomers got their own copy of the Almanac. We others used the copy at the a large public or college library.
Computation --------- First for both bodies the XYZ coordinates are computed. I leave out the computation because nowadays many computer ephemeris programs can generate these parameters from the orbit elements and range of dates. Starting with the comet's coords, conetX, cometY, conmetZ, and those for Earth as EarthX, EarthY, EarthZ
(cometX wrt Earth) = (cometX wrt Sun) - (EarthX wrt Sun) (cometY wrt Earth) = (cometY wrt Sun) - (EarthY wrt Sun) (cometZ wrt Earth) = (cometZ wrt Sun) - (EarthZ wrt Sun)
Let's work up the position of a madeup comet for 2009 March 9, 0h UT. A comet has a position of, making up numbers, X = +4.126AU, Y = +2.953AU, Z = -1.585AU, all in AU
From the Astronomical Almanac tables we read out Earth's X = -0.973; Y = +0.183, Z = +0.079. Some tables give the SUN'S coord, not Earth's. To get Earth's, negate the signum of the solar values. Plus values become minus; minus, plus. Please read the instructions for the table! We may also use the USNO website for obtaining this data in the lack of the paper book. We collect the values here:
body | X | Y | Z | AU distance ------+--------+--------+--------+--------------- comet | +4.126 | +2.953 | -1.585 | 5.316 from Sun Earth | -0.976 | +0.183 | +0.079 | 0.996 from Sun ------+--------+--------+--------+----------------- diff | +5.102 | +2.770 | -1.664 | 6.039 from Earth
The apparent deviation of Earth from the ecliptic, the Z value of +0.079, is due to the secular variation of the ecliptic inclination. This is quite small and for home astronomy it is usually ignored. The distance values are obtained from the Pythagoras formula;
dist = sqrt((X ^ 2) + (Y ^ 2) + (Z ^ 2))
Mind that the distance of the comet from Earth is NOT the difference of the Sun distances! Figure the comet-Earth distance by plugging the 'diff' XYZ values into the Pythagoras formula. 'diff' is the coordinates of the comet relative to Earth. They have to be converted to the traditional RA and dec.
geocentric coordinates -------------------- Theconversion of XYZlocation to RA & dec is a 2-stage process. The XYZ values are based on the ecliptic, so the first conversion is to ecliptic lat & lon. These are next transformed into RA & dec. To get the lat-lon from XYZ coords, solve these formulae
(ecl lon) = atn(Y / X), then rectify for quadrant
(ecl lat) = atn((Z) / (srt(X^2 + Y^2))
Make a sketch of the ecliptic with the XYZ numbers on it and see where the arctangent falls. Add/subtract 90 or 180 degrees to rectify it into the proper quadrant. This is a process you MUST deliberately go thru to get the proper quadrant for the longitude. For our comet these formulae yield, waypoint steps omitted,
ecl lon = 25.4966 deg ecl lat = -15.9937 deg
The details for transfroming from ecliptic to equatorial coords are left out, but they amount to a rotation of axes. Some astronomers handle this chore with matrix maths. Many astronomy programs have a coordiante sonversion by keying in one set and reading out the others. Here are the results:
rt asc = 01h 58m decl = -05d 03m dist = 6.039 AU
Our comet is in Cetus, near the middle of the triangle of alpha Piscium, omicron Ceti (Mira), and zeta Ceti. It on March 9 is low in west-southwest in dusk. Unless the comet is really lustrous, it likely will not be easily observable. There's no use for extreme precision here. Coords rounded to the minute are entirely sufficient.
Retrograde loops -------------- If we do this computation for a run of days, a chore taking by hand a whole weekend, we obtain points to plot the path of the comet on a starchart. This path can be bizarre, to be gentle. From the two smooth and regular XYZ coord for comet and Earth the difference between the two sets can be insanely erratic. There are some intersting generalities. On the whole a comet with a low inclination, near 0 or 180 degrees, has a sky path resembling a planet's, staying close to the ecliptic. The motion may be prograde (inclination near 0) or retrograde (near 180). Comet Lulin in 2009 is a good example of a comet with a retrograde movement. Such a comet executes retrograde loops. The comet slows, stops, reveres diretion, slows, stops, resumes forward motion. This effect is wntirely an artifact of the Earth's motion around the Sun. The full explanation is exactly that of a planet's retrograde loop. When the inclination increases, the comet's loops stretch out to resemble those of an asteroid. The arc in the stars is more sinuous than a doubling over into a loop. For a high elciptic latitude the loop becomes a sprial or overlapping series of circles.The loop is seen more faceon away from the ecliptic.
Distance from Earth ----------------- A comet moves rapidly thru the inner solar system, traversing the region within the orbit of Mars in only a couple months. When it swings near to Earth, its angular speed on the sky can be several degrees per day. The path during this time is straight or gently curved. When the comet is remote from Earth, there is a strange spiral or oscillating pattern in its path. This was noticed by the early astronomers, who thought that someehow a comet actually span out of some source, maybe one of the planets?, and scooted scross the sky. It then retreated by a spiral into its hutch. This is nothing more than the parallactic oscillation, a form of retrograde loop, where the comet is moving straight toward or away from Earth. The openness of the spiral is a function of ecliptic latitude.Thisenear thepoles are roughly circular. Near the ecliptic they collapse into back-&-forth seinging. With little transverse motion across the sky, the retrograde loops overlap to form a spiral. These spirals exist for all long period and one-time comets, due to their remoteness from Earth, but are often missed from sight by the comet's weakness at large distances from Earth and Sun.
Direction of tail --------------- The usual textbook explains that a comet's tail points away from the Sun. The means that when a comet is approaching the Sun the tail lags the head, altho it does not trail straight 'behind' it as if it was brushed back by a slipstream. After perihelion, when the comet is running out from the Sun, the tail leads the head. Not directly in 'front' like a headlight beam, but definitely it precedes the head. There is a wind in space, the solar wind heaved out from the Sun. This wind pushes the comet's tail away from the Sun. An other force is electrostatic repulsion of ionized gases from the comet. The electric charge of the solar wind repels the comet's tail for tens and hundreds of millions of kilometers from the comet's head. Some comets have two tails, one of ions; the other, dust. The ion tail is usually a jet radially outward from the Sun while the dust tail is more brushy and bushy and can be curved by inertial lag. The best time to appreciate the behavior of the tail is from a comet in the west in dusk or in the east in dawn. You then correlate the Sun's position below the horizon with that of the tail. A comet seen in a night sky aims its tail away from Earth, because Earth is then between th comet and the Sun. The tail may be hidden behind the head or showing only a bit around one edge of the head. Comet Holmes in 2007 was such a comet, seen in the full night sky with its tail peeking around the head. With the RA & dec of the comet we can computer the direction the tail aims away from the head. We need also the RA & dec of the Sun for 2009 March 9 0h UT. They come from just about any astronommy ephemeris program. The tail points toward the bearing, or position angle, p[[psotetp that the Sun holds as seen from the comet. The simplest way to find this is to use the angular distance function of the astronomy program. Key in the omet's RA & dec as the'from' or 'start' point. Then key in the Sun's RA & dec for the 'to' or 'end' point. The nnswer is the angle and bearing of the Sun from the comet. The direction of the tail is 180 degrees from this bearing. For our comet we have --------------------------- coord | comet | Sun -------+----------+--------- rt asc | 01h 58m | 23h 18m decl | -05d 03m | -04d 32m
separation = 39d 51m bearing = 270d 45m from comet to Sun bearing = 90d 45m tail direction from comet
The calculated bearing is the direction FROM the comet TO the Sun. The tail points opposite to this, or into 90d 45m. Recall that bearing, also called position angle, runs from 0 deg at celestial north, thru 90 deg at east; 180, south; 270, west. An easy mistake is to compute the comet's bearing from the Sun, likely by swopping the 'from' and 'to' entries. This sun-to-comet is NOT merely the reverse of the comet-to-Sun bearing, as it would be for plane geometry. Make sure you're going from comet to Sun. As it turns out astronomy programs usually offer only the angular separation but not the bearing from the one point to the other. If this is yoursituation, you have to apply the spherical trig method, found in a math book or website. For routine comet observing you may want to codify the trig method into a BASIC or Excel application for future use. You MUST do a sanity check on the position angle and be prepared to rectify it into the proper quadrant. A sketch of the Sun-comet location will be a huge help to see what massaging of the bearing is needed.
Conclusion -------- The appearance of comets in the sky is a good opportunity for the home astronomer to appreciate the work earlier astronomers went thru to build predictions of position and motion. While today there are comet tracking programs that cut thru the maths, it is a healthy exercise to go thru the maths once by hand (with calculette and graph paper) to better understand the geometry and physics involved.