In last month’s note we saw that a link between C/1661 C1 and C/2002 C1 (Ikeya-Zhang) was, to all intents and purposes, all but certain. We further noted that C/1532 R1 could also have originated from a common parent nucleus that split sometime in the distant past, with the fragment orbits developing rapidly away from each other as a result of the influence of Jupiter. (In much the same way as seems to have happened in the case of 42P/Neujmin 3 and 53P/Van Biesbroeck.) Given that Ikeya-Zhang’s decending node is located so close to the orbit of Jupiter, the opportunity for such fragment-separating encounters is fairly common and so it is unlikely that any splitting scenario put forward at the present time will be unique. In the present note we pursue this further in what may be aptly termed an exercise in orbital gymnastics which the reader may wish to take cum grano salis !

Based on Nakano’s Feb 25 1661-2002 linked solution, the previous calculations indicated that the apparition prior to 1661 occurred in 1273 April/May. Given that a comet is recorded in the Chinese annals as being first sighted on 1273 April 9, it is reasonable to ask whether this is in fact another record of Ikeya-Zhang or merely a coincidence. What leads us to the former opinion is that the details given of the comet’s apparent motion mimic rather well the expected track of Ikeya-Zhang if its perihelion passage were about a month earlier than that indicated by the preliminary calculations.

The comet of 1273 was first seen at the ecliptic longitudes of the Hyades and to the ‘north’ of Auriga. This latter is rather imprecise but still useful for our purposes. Of more use is the statement that the comet subsequently passed from the asterism 28 / Nu / Phi / Theta /15 Uma and then penetrated the ‘ladle’ of the Plough. This piece of information effectively limits an Ikeya-Zhang type orbit to having a perihelion passage time between March 26.5 and 28.5. The record then goes on to say that the comet passed through Bootes and reached the region of Pi Boo. All of this happened, according to Ho’s translation, in 21 days. For our orbit this track would be covered in two months and the end point would be nearer to Eta Boo. To a certain extent, such details are negotiable (due to possible copying errors etc.) but the time period is something of a problem, as is the visibility for the extended period if the comet were indeed Ikeya-Zhang with its current absolute magnitude.

In spite of the possible identification problems we now assume that the comet of 1273 was indeed Ikeya-Zhang and see how it can help us in our quest to reconcile C/1532 R1 with Ikeya-Zhang. We take as starting point the MPEC 2002-F55 orbit solution and then introduce nongravitational terms into the integration to move the previous perihelion time from mid-1662 to 1661 Jan 28.900. This can be achieved by any number of possible pairings of the nongravitational parameters A1 and A2, as shown in the table below which gives sample sequences of perihelion passage times derived by integration from the 2002 epoch.

Although there are many possible combinations of A1 and A2 that will give the desired result as regards 1661, if we want to force a fit to our possible 1273 comet at the same time then the options become rather limited. For instance, if we were to want to fit to, say, T = 1273 Mar. 30 then the orbit prior to 454 would have been hyperbolic due to a close (0.12au) approach to Jupiter.

From the foregoing we have seen that it is quite possible to link 1273-1661-2002 by the introduction of nongravitational acceleration parameters that are deemed to be constant over periods of several centuries. Whilst one might reasonably expect these parameters to change at each perihelion passage (due to different outgassing geometries), assuming constant values is no less legitimate than assuming that they are identically zero at all times! Moreover, the constraints we have used have at least resulted in fairly modest values. Where the problems now arise is what to do to try to generate a 1532 apparition by integrating forward from a chosen time in the past; the idea being to (semi-arbitrarily) select a time of nuclear splitting and use different values of A1 and A2 for the two fragments. The problem we are faced with is that we effectively have three free parameters and only one constraint - that we should end up with a perihelion passage around 1532 Oct 19. To make things simpler we do the following for no apparently obvious reason: we assume that the ratio of A1 to A2 for the second fragment is the same as for the 1273-1661-2002 fragment. There is absolutely no reason why this should be true, but it is as good as any other assumption under the circumstances. This reduces the nongravitational part of the problem to a single parameter but we still have to decide on when we would like the nuclear splitting to occur and trying to second-guess comets is notoriously difficult.

Given that the window of solution for the 1273-1661-2002 linkage is very limited around 450 due to the position of Jupiter, it seems reasonable to assume that the splitting occurred around this time in order to take advantage of the Jovian perturbations about 400 days after perihelion passage. Also, if we assume that the splitting actually occurred at perihelion then the difference in the assumed nongravitational effects will have separated the two fragments by enough to produce useful differential Jovian perturbations when the fragments pass through their descending nodes. Thus, we adopt the middle calculation in the table as being our ‘main’ fragment and assume its A1/A2 ratio with the splitting taken to happen on 451 Oct 23.0 (0.7 days after perihelion passage but chosen in preference as it is a standard 40-day epoch). We now proceed by performing trial integrations using various assumed values of A1 and soon find out that values near 0.0397 are the ones that will generate orbits relevant to a 1532 perihelion passage, with a sequence of perihelia of 879 Oct 2 – 1202 July 17 – 1532 Oct 19. Once again, we must repeat the obligatory disclaimer - there is absolutely no guarantee, even within our rather arbitrary constraints, that this solution is unique. In particular, we should also note that if the foregoing perihelia sequence for the second fragment were correct then we should have seen a return of the 1532 fragment in October 1916.

Finally, for those wishing to plot the tracks using planeteria software the following table gives the osculating elements (J2000) for the 1273, 1532 and 1661 orbits obtained above. I shall leave it to the interested reader to decide whether or not the mooted 1273 / Ikeya-Zhang connection needs to be plied with sodium chloride.

Ó graeme.waddington@bioch.ox.ac.uk