Fragmentation and History of Comet Pair C/1844 Y1 and C/2019 Y4

Inspection of the employed data, reported by the MPC and summarized in Table 1, indicates that Fragments A and B were simultaneously imaged 38 times in the period of 6–18 April, but the results of preliminary tests presented in Table 2 suggest that on a few occasions at least one of the two fragments was either misidentified or its position measured too inaccurately to be of any use in the model computations. Adopting a rejection cutoff of $\pm$1${}^{\prime\prime}\!$.5 for the residuals $O\!-\!C$ (observed minus computed from the model) of A–B in right ascension and declination, the number of acceptable observations has turned out to equal 33.

My efforts were first directed toward the simplest, two-parameter solution, trying to determine the time of separation of Fragment A from Fragment B and the nongravitational acceleration of A relative to B. I began with a separation time of 75 days before perihelion, based on Hui & Ye’s (2020) conclusion that “the comet has disintegrated since 2020 mid-March.” Unfortunately, after a few initial iterations suggesting a meaningless separation time as far back in time as some 600 days (sic) before perihelion, the differential-correction optimization procedure failed to converge. Similarly disappointing were the successive attempts at solutions with more than two parameters.

The overall impression from these runs was that even though the separation time appeared to be indeterminate, the solutions in which the separation was assumed to have taken place much earlier than $\sim$100 days before perihelion (say, prior to February 2020) looked distinctly preferable, offering a lower mean residual and better distribution of individual residuals.

Given that splitting of a comet’s nucleus is known to often coincide with suddenly elevated activity showing up as a brightening in the light curve, the abrupt change in the slope around 22 January, or 130 days before perihelion, in Figure 1 is a strong suspect as a potential candidate for the fragmentation event that gave birth to components A and B. Incorporating this assumption, the one-parameter solution fitted the observations with an unacceptably high mean residual and with the individual residuals displaying strong systematic trends of up to some 3^′′. Solving in addition for the normal component of the separation velocity improved the solution’s quality (BA₅ in Table 3), the systematic trends in the distribution of residuals now reaching not more than about 1${}^{\prime\prime}\!$.

The most satisfactory fit to the data was offered by a three-parameter solution, presented in Table 3 as BA₁. The acceleration slightly exceeds 20 units of 10^-5 the solar gravitational acceleration and the separation velocity stays below 2 m s^-1. The table shows that this solution is distinctly preferable to BA₄, an equivalent solution for an assumed separation time of 75 days before perihelion (mid-March), which requires a high separation velocity of nearly 5 m s^-1 and an acceleration of companion A that is four times higher. Solution BA₁ is also preferable to BA₃, in which the contribution from the acceleration is taken over by the (increased) separation velocity. On the other hand, a four-parameter solution BA₂ that incorporates the radial component of the separation velocity among the parameters is a case of overkill, documented by their unacceptably high errors.

The residuals from the fit of Solution BA₁ of Table 3 to the 38 A–B separations in right ascension and declination are listed in Table 4; of the five rejected data points three are incorrect identities (Table 2) and two are inaccurate data. The peculiar separation trajectory of Fragment A relative to B, plotted in Figure 3, is consistent with a scenario that incorporates the fragmentation event in late January and is, in addition, supported by the nuclear-condensation’s light curve in Figure 1 and by the fact that the nucleus was not seen double during February and especially in March 2020, when the predicted separation distance between the two fragments was much greater than in April (Figure 3).

The key to understanding both the light curve and the detection anomaly is the fundamentally different behavior of the two fragments: while Fragment A was subjected to a high rate of mass loss, Fragment B was of low activity and too faint to detect in any telescope employed in the comet’s observations. The observed light curve between late January and early April was exclusively a product of Fragment A.

Closer inspection of the light curve shows that the constant apparent brightness of the nuclear condensation between late December 2019 and late January 2020 actually meant a decline of activity. If interpreted in terms of the cross-sectional area of dust grains near the nucleus, the drop was from 96 km² on 20 December to 33 km² on 22 January at an assumed geometric albedo of 4 percent and accounting for the phase effect using the Marcus (2007) law. The observed brightness increase from apparent magnitude of 18.8 on 22 January to 15.0 on 25 February, at a fairly constant rate of 0.11 mag per day is equivalent to an exponentially growing cross-sectional area of released dust between the two dates from 33 km² to 367 km². At time $t$ this cross-sectional area equals

where $t_{\rm frg}$ is the presumed fragmentation time, 22 January. As $\exp(34\,\zeta)=367/33=11.1$, one finds $\zeta=0.071$ per day. The comet brightening continuously at the observed rate of 0.11 mag per day and its description via Equation (1) is thus readily interpreted as progressive disintegration of Fragment A, whose cross-sectional area doubled in approximately 10 days.

To properly investigate the crumbling of Fragment A that presumably began on 22 January, one should disregard the presence of dust in the atmosphere at that time and ${\cal A}_{\rm frg}$ in Equation (1) should be the cross-sectional area of the surface of Fragment A. The value of $\zeta$ would then be somewhat higher. For example, taking conservatively the diameter of Fragment A at the time of separation from Fragment B to be 500 meters, one would obtain $\exp(34\,\zeta)=4\times$367/($\pi\!\times\!0.5^{2})\simeq 1870$ or $\zeta=0.22$ and the doubling time a little over 3 days. For a fragment 100 meters across, $\zeta=0.32$ and the doubling time more than 2 days.

Continuing with this line of argumentation, let at time $t>t_{\rm frg}$ the number of pieces of debris of the crumbling fragment be ${\cal N}(t)$ and the mean cross-sectional area of one piece $\langle{\cal A}\rangle$, so that

where $\eta(t)=\exp[\zeta(t\!-\!t_{\rm frg})]$. Expressing a cross-sectional area in terms of a characteristic dimension $\Re(t)$, such as the radius or diameter, one can write the conditions for the mean cross-sectional area and volume at time $t$:

where $\Re_{\rm frg}^{2}$ and $\Re_{\rm frg}^{3}$ refer to Fragment A at the fragmentation time. The ratio,

is a function of the size distribution, $f(\Re)\,d\Re\sim\Re^{-s}d\Re$, of the debris. If the minimum and maximum dimensions at time $t$ are $\Re_{\rm min}(t)$ and $\Re_{\rm max}(t)$, respectively, the mean values are given by

For less steep distributions, for which $s<3$ the mean values are essentially independent of $\Re_{\rm min}$ and the ratio is at time $t$

Comparison of Equations (4) and (6) gives for the dimension of the largest piece of debris of Fragment A at time $t$ in terms of the fragment’s dimension at $t_{\rm frg}$:

Regardless of exact values for $\zeta$ and $s$ (if lower than 3), Equation (7) suggests that months after the separation of A from B, the dimensions of the largest remaining piece of A should be orders of magnitude smaller than the initial size. This is contrary to the results that Ye et al. (2021) obtained from their study of the comet’s debris with the HST in late April. It appears that the process of disintegration of Fragment A was proceeding in a highly irregular manner. The rate of disintegration may have varied strongly with time and/or the process affected only a fraction of the object. In any case, the presence of major anomalies is implied by the dramatic fluctuations in the size distribution of the debris of Fragment A, as documented by Ye et al.

The statistics of the split comets with more than two fragments suggest that, as a rule, the secondary nuclei (or companions) have separated from the principal mass (Sekanina 1982). Less often has a second companion split off instead from the first companion, or, more generally, a companion of a later generation from a companion of an earlier generation. Accordingly, one would expect Fragment C of comet ATLAS to separate, like Fragment A, from B. Yet, it is desirable to test both options. An obvious constraint is that Fragment C could not have broken off from A before A did from B.

Tables 1 and 2 show that the simultaneous astrometric observations of B and C have been less numerous than those of A and C, and both pairs less numerous than the simultaneous astrometry of B and A. Because of these limitations, the rejection cutoff for examining the parent of C was increased from 1${}^{\prime\prime}\!$.5 to 2${}^{\prime\prime}\!$.0. I applied a variety of the fragmentation model’s versions and found that unlike in the case of the motion of Fragment A relative to B, the separation time always converged. However, solutions with higher number of parameters than three consistently failed to converge; the two converging ones for either fragmentation scenario are compared in Table 5.

Solutions BC₁ and BC₂ offer similar separation times, which precede the peak of the light curves in Figure 2 by several days, but Solution BC₁ is preferred because the normal component of the separation velocity from Solution BC₂ is poorly determined and therefore an unnecessary parameter. An unexpected problem is a systematic trend in the residuals in declination, which mars either solution. For Solution BC₁ this undesirable anomaly is apparent from Table 6 and very obvious from the separation trajectory of Fragment C relative to Fragment B displayed in Figure 4.

Contrary to Solutions BC₁ and BC₂, Solutions AC₁ and AC₂ differ from each other considerably. As seen from both the residuals in Table 7 and from the plot of the separation trajectory in Figure 5, the fit to the data by Solution AC₂ can hardly be better. On the other hand, Solution AC₁ has a number of weaknesses. In the adopted overall fragmentation scenario it is in fact meaningless, because it implies that Fragment C had broken off from Fragment A before A separated from B (cf. Section 3.1). In addition, as seen from Table 8, this solution offers acceptable residuals (below 2^′′) from only 20 of the 22 data points and strong systematic trends of about 1^′′ are clealy apparent in the distribution of residuals in right ascension.

The presented evidence suggests rather strongly that it is Fragment A that is the parent to Fragment C. The failure to get an acceptable distribution of residuals on the assumption that the parent was the principal mass B obviously had fundamental roots and was not caused by inaccurate astrometric observations.

The determined fragmentation sequence has implications for the meaning of the observed positions of Fragment C relative to Fragment B. Dynamically, their separations represent the sum of separations of C from A and A from B:

The averaged daily separations of C–B computed in this fashion for the mean times of observation of this pair (cf. Table 6) are listed in Table 9, where the residuals $O\!-\!C$ show rather good accord with the observed separations.

The examination of the origin and motion of Fragment D has followed closely the routine used in the previous section for Fragment C, including the rejection cutoff of 2^′′. Unfortunately, the identity of fragments in some observations was somewhat uncertain. Chronologically, the first such case referred to April 10.81943 (code A77), wherein the MPC assigned Fragment C had to be replaced with D. This improved the fit, even though C had been borderline. A more difficult was the observation on April 11.90888 (code C10), likewise requiring that Fragment C be changed to D, mainly because of the poor fit in the C–A separation. Although C–B and D–B fitted reasonably well, C–B had to be rejected. The observation of April 14.86279 (code 970) was a relatively straightforward case; the position of Fragment B apparently was in error, as the separation D–A fitted well, but A–B and D–B did not at all. On the other hand, the observation of April 14.91613 (code J95) was a major conundrum. The separations C–A and D–A pointed to the need to correct the MPC assigned Fragment D to C, as C–A fitted well but D–A did not. But when I tested the separations C–B and D–B, C was a problem. The separation A–B was acceptable, so it was not clear where the culprit was. Fragment C appeared preferable to D, but the contradiction persisted. The final case was the observation of April 14.91977 (code Z80), where I also replaced Fragment D with C, as the separation C–B offered a better fit than D–B, even though both were acceptable.

The results for Fragment D show similarities with the results for Fragment C, except that the numbers of available (B, D) and (A, D) pairs have now been comparable. The fragmentation models are summarized in Table 10, which suggests — like Table 5 for Fragment C — two similar solutions for the motion of D relative to B and two very different solutions relative to A.

Solution BD₂ again includes a normal component of the separation velocity that is not well determined, so that Solution BD₁ is preferred. Unlike in the case of Solution BC₁, the distribution of residuals offered by Solution BD₁ is rather acceptable, as seen from Table 11 and Figure 6.

Of the two solutions for the motion of Fragment D relative to A, Solution AD₁ is inferior to Solution AD₂ in that it has been unable to fit three of the 20 data points within the rejection cutoff. The full data set, represented by Solution AD₂ with a mean residual of $\pm$0${}^{\prime\prime}\!$.84 (Table 10), has been fitted by Solution AD₁ with a mean residual as high as $\pm$0${}^{\prime\prime}\!$.99. However, unlike Solution AC₁, Solution AD₁ strictly is not meaningless because its separation from Fragment A is found to have taken place within 1$\sigma$ of the time of separation of A from B. Although the distribution of residuals is not bad, the solution is shown above to be weak and the derived separation time is not realistic, being affected by neglect of the normal component of the separation velocity. On the other hand, Solution AD₂ and the BD solutions are in fairly close agreement in that D split off from its parent in mid-, or the second half of, March.

For comparison with Solution BD₁, the distribution of residuals offered by Solution AD₂ is displayed in Table 12 and the separation trajectory of Fragment D relative to A is plotted in Figure 7. The plot indicates that except for three data points at the very beginning of the observed segment of the trajectory, the fit is in fact better than the fit to the motion of Fragment D relative to B exhibited in Figure 6.

The other point to argue is that it is unlikely that both fragments, A and B, should have broken up almost simultaneously, one releasing C, the other D. On the other hand, it should not be surprising that the gradually disintegrating A produced, next to the debris, two subfragments in a row, or perhaps one that almost immediately split into two. In any case, there is some evidence for preferring A to B as the parent to D, but it is tenuous.

As is apparent from Table 12, the separations of (D–A) on April 17.24276 and 18.23496 UT (code F65) left residuals in declination of almost 3^′′ and 4^′′, respectively, from Solution AD₂. On the other hand, the separation of (A–B) left on the 17th an excellent residual in either coordinate from Solution BA₁ (Table 4), indicating that the problem was not with Fragment A. Under these circumstances, one can rule out that the third fragment on the 17th and the second fragment on the 18th were D. Since Fragment C replacing D would have fitted the separation trajectory even worse, the odds were that on the two days one had to do with a new fragment, E.

Furthermore, given that the separation of (D–A) on April 9.80653 UT (code K51) left, from Solution AD₂ in Table 12, a large residual in declination that essentially equaled the rejection cutoff, it is reasonable to suspect that here too the involved fragment may have been E rather than D. Computer runs linking the observations from April 9.8, 17.2, and 18.2 did indeed lead to successful solutions AE₁ and AE₂, displayed in Table 13. The residuals left by the latter solution are listed in Table 14. On account of the extremely small number of observations, either solution is deemed provisional, the second being slightly preferable.

The deletion of the (D–A) separation on April 9.8 from the set in Table 12 had a relatively minor effect on the fragmentation parameters, leading to Solution AD₃ also presented in Table 13; the revised list of residuals is in Table 15. It is noted that the removal of the data point from the 9th had, on the average, a beneficial effect on the residuals from other observations, the one on the 8th in particular. In the following I somewhat reluctantly adopt Fragment A as the most likely parent of D and Solution AD₃ as the latter’s best available fragmentation solution.

The reason for investigating only the scenario in which Fragment E derived from A was the availability of the (E–A) separations on three days. This data set, however humble, compares to a single occasion, on April 17, of the simultaneously observed Fragments B and E. The parent of Fragment E remains unclear and it is merely for the reasons already commented on in Section 3.3 that one can speculate that the most likely candidate for being its parent is Fragment A.

The number of simultaneously observed (C, D) pairs is very limited. From the data in Tables 1 and 2, their total number is assessed at six. I have attempted a fragmentation solution to check whether Fragment D could have separated from Fragment C rather than from A (or B). The result, presented as Solution CD₁ in Table 16, is inconclusive. The acceleration in particular is determined with large uncertainty and no separation velocity could be detected. As shown in Table 17, two of the six observations could not be accommodated by the solution. There is no evidence that would point at a close relationship between Fragments C and D and at their separation from the parent as a single body.

Similar examinations of possible commonalities between Fragments C and E on the one hand and between Fragments D and E on the other hand cannot be undertaken because of the absence of reported simultaneous observations of either pair of these objects.

If the late January sudden change in the light curve of the nuclear condensation signaled a fragmentation event, it unquestionably was a breakup of the parent mass into Fragments A and B. Although the model failed to converge when the time of separation became one of the parameters to solve for, the distribution of residuals from the 33 used separations presented in Table 4 was deteriorating dramatically as the separation time was being moved forward. While the mean residual reached $\pm$0${}^{\prime\prime}\!$.522 for a separation time of 130 days before perihelion (from adopted Solution BA₁ in Table 3), it increased to $\pm$0${}^{\prime\prime}\!$.528 at 100 days before perihelion and to $\pm$0${}^{\prime\prime}\!$.563 at 70 days before perihelion. A formal minimum of $\pm$0${}^{\prime\prime}\!$.518 was reached 650 days before perihelion, an unrealistic solution.

Of the other fragments, the conclusion from the computations in the preceding sections is that separation from Fragment A was very probable for C, perhaps likely for D, and possible for E. Adopting for the three objects the fragmentation solutions of, respectively, AC₂ from Table 5, AD₃ from Table 13, and AE₂ from Table 13, one finds that D and E broke off essentially simultaneouly, while C about a week later, but the C and D events were within 1$\sigma$ of each other. These separation dates were centered on mid-March and approximately coincided with the sudden drop in the comet’s rate of brightening, days before the light curves in Table 2 reached the peak.

Given that the duplicity of the comet’s nucleus was not detected until early April, the breakup in late January meant that one component — the principal mass B, as it turned out — remained unobserved for at least 11 weeks. The disintegrating component A was the only part of the original nucleus — itself a companion — that was under observation continuously from the breakup until late April. In addition, the separation trajectory of Fragment A relative to B in Figure 3 suggests that in mid-March, when only A was seen, the two fragments were nearly 8^′′ apart, while by mid-April, when both A and B were observed, their separation diminished to less than 5^′′. The separation distance had obviously no effect on the detection of the duplicity. Instead, it was low activity of Fragment B prior to April that was keeping its brightness below the detection threshold of the telescopes used. It is likely that the HST would have been able to image the duplicity for most of the time it lasted, beginning not later than early February.

Interestingly, Fragments C, D, and E were all observed within four weeks or so of their presumed release from Fragment A, starting at separations of 10^′′ to 12^′′.

The sublimation-driven nongravitational acceleration $\gamma$ is an important fragmentation parameter, because it is determined by the physical properties of the fragment, just as in Whipple’s (1950) model the nongravitational acceleration of a comet is determined by its physical properties. However, when the fragment’s motion is measured relative to the parent nucleus, the derived value is a differential acceleration. The absolute acceleration of the fragment is the sum of its differential acceleration and the parent’s acceleration.

The acceleration is statistically correlated with the fragment’s lifespan, which is conveniently measured by the endurance, $\cal E$, defined as the difference between the separation time and the time of last observation, reduced to a normalized heliocentric distance of 1 AU, and expressed in equivalent days (or e-days). For comets in nearly-parabolic orbits (Sekanina 1982)

where $q$ is the perihelion distance (in AU) and $u_{\rm frg}$ and $u_{\rm fin}$ are the true anomalies (in radians) at the times of separation and final observation, respectively. Because of effects due to observing conditions, the measured lifespan is in general a lower limit to the true lifespan.

The limited statistics of two dozen data points suggest the existence of three basic categories of cometary fragments: persistent, short-lived, and minor (Sekanina 1982). As already noted in Section 1, the endurance was found to vary with the acceleration as

where $C=800$ e-days for the most massive, persistent fragments, 200 e-days for the short-lived fragments, and 87 e-days for the minor fragments.

From Solutions BA₁, AC₂, AD₃, and AE₂ one finds, in units of 10^-5 the solar gravitational acceleration, the nongravitational accelerations of 21.7 $\pm$ 2.8 for Fragment A relative to B and 161 $\pm$ 30, 135 $\pm$ 32, and 173 $\pm$ 21, respectively, for Fragments C, D, and E relative to A. Fragment A is positively a short-lived fragment, whose acceleration implies an endurance of 58 $\pm$ 3 e-days. The other fragments are apparently short lived as well, in which case the endurances equal 26.2 $\pm$ 2.0 e-days for Fragment C, 28.1 $\pm$ 2.7 e-days for D, and 25.5 $\pm$ 1.2 e-days for E. Should they be minor fragments, their endurances would be too short: 10.8 $\pm$ 0.8 e-days, 11.5 $\pm$ 1.1 e-days, and 10.6 $\pm$ 0.5 e-days, respectively.

These numbers apply on the assumption that the principal mass B was affected by no nongravitational acceleration. As the comet was itself a companion fragment of a larger comet, this is unlikely to be the case. Indeed, the orbit computed by the MPC Staff (2020) indicates that between 20 December 2019 and 3 April 2020 the comet’s motion was affected by a strong nongravitational acceleration. Of course, if the nucleus split into Fragments A and B around 22 January, the MPC elements timewise cover only 31 percent of the motion of the pre-split nucleus and 69 percent of the motion of Fragment A, as Fragment B was not seen until after 3 April. As the MPC Staff determined the orbit using Marsden et al.’s (1973) model with the standard $g(r)$ nongravitational law, it is necessary to convert the integrated effect of the radial sublimation-driven force from the Marsden et al. nomenclature to the style used in this paper.

The fundamental condition to be satisfied is that the dominant nongravitational effect in the radial direction, whose magnitude in the period from 20 December to 3 April was measured in the MPC Staff’s (2020) orbital solution by a parameter $A_{1}=+16.16\times\!10^{-8}$ AU day^-2, be equated with the sum of the radial nongravitational effects, expressed in the style used in this paper, (i) on the pre-split nucleus between 20 December and 22 January, defined by a parameter $\gamma_{\rm par}$; plus (ii) on Fragment A between 22 January and 3 April. The latter part consists of the total radial nongravitaional effect on Fragment B, defined by a parameter $\gamma_{\rm b}$; plus the radial nongravitational effect on Fragment A relative to Fragment B, defined by the tabulated parameter $\gamma_{\rm ba}=21.7$ units of 10^-5 the solar gravitational acceleration. The condition thus reads

where $t_{\rm beg}$, $t_{\rm end}$, and $t_{\rm frg}$ are, respectively, the times of the beginning and the end of the orbital arc, integrated by the MPC Staff (2020), and the time of fragmentation. Expressed in days from perihelion, they are equal to $t_{\rm beg}=-163$ days, $t_{\rm end}=-58$ days, and, as adopted, $t_{\rm frg}=-130$ days; $r_{\oplus}=1$ AU, so that $\gamma_{\rm par}$ and $\gamma_{\rm b}$ refer to 1 AU and, like $\gamma_{\rm ba}$, are in units of 10^-5 the solar gravitational acceleration; at 1 AU from the Sun this unit equals $\Gamma\!_{\odot}=0.593\times\!10^{-5}$ cm s${}^{-2}=0.296\times\!10^{-8}$ AU day^-2. Employing Kepler’s Second Law and elementary operations, and leaving $r_{\oplus}$ out, Equation (11) can be rewritten thus:

where $u_{\rm beg}$, $u_{\rm end}$, and $u_{\rm frg}$ are the true anomalies (in radians) at, respectively, $t_{\rm beg}$, $t_{\rm end}$, and $t_{\rm frg}$; and

The integration over $r$, with $r_{x}=r(t_{x})$, is a parabolic approximation valid on the assumption that $u_{\rm beg}<0$ and $u_{\rm end}<0$.

The condition (12) includes two unknowns: $\gamma_{\rm par}$ and $\gamma_{\rm b}$. Its solution requires an additional condition on the two accelerations. If the parent and Fragment B were about equally active, one would expect that $\gamma_{\rm par}<\gamma_{\rm b}$, the two values the closer to each other the smaller Fragment A relative to B. Because Fragment A was much more active than B, it is likely that in relative terms the parent was more active than Fragment B, in which case $\gamma_{\rm par}$ could equal $\gamma_{\rm b}$ or even exceed it. As a first-guess estimate I assume equality, in which case condition (12) becomes

For the considered MPC orbit, $u_{\rm end}\!-\!u_{\rm beg}=0.2813$ radian, $u_{\rm end}\!-\!u_{\rm frg}=0.2304$ radian, and from Equation (13) $\Im_{\rm beg,end}=0.1167$ AU², so that

More generally, one could assume

where $g>0$ and $g\neq 1$. The solution (14) is now slightly modified,

One can show that if $\gamma_{\rm par}=\gamma_{\rm b}>0$, then $\gamma_{\rm par}>0$ and $\gamma_{\rm b}>0$ for any $g>0$. The only condition for $\gamma_{\rm par}>0$ is a constraint on the ratio of $\gamma_{\rm ba}$ to $A_{1}$,

where the units are, as above, 10^-5 the solar gravitational acceleration for $\gamma_{\rm ba}$ and AU day^-2 for $A_{1}$.

Limited experimentation with the factor $g$ suggests that for values moderately exceeding, or slightly lower than, unity, the accelerations $\gamma_{\rm par}$ and $\gamma_{\rm b}$ do not differ from (15) by more than a few units of 10^-5 the solar gravitational acceleration. As seen, $\gamma_{\rm b}$ is about a factor of 4 to 5 smaller than $\gamma_{\rm ba}$ and does not amount to more than a few percent (well within the uncertainty) of the high nongravitational accelerations that Fragments C, D, and E are subjected to.

Turning back to the lifespan of the fragments, I note that by eliminating the endurance from Equations (9) and (10) it is possible to approximately predict the time of final observation of a fragment, $t_{\rm fin}$, from the model parameters — the time of separation, $t_{\rm frg}$, and the nongravitational acceleration, $\gamma$ — when the fragment’s category (persistent, short-lived, or minor), determining the parameter $C$, is known. The prediction can be tested on the data available from observations. In a parabolic approximation, adequate for this purpose, the procedure includes three steps as follows:

where $t_{\pi}$ is the time of perihelion. As before, the true anomalies are in radians, the distances in AU, and the times in days, the signs giving $t_{\rm fin}$ before/after perihelion.

The predicted dates of last sighting are compared in Table 18 with the reported times. The agreement is fairly good, when appropriate fragment categories are assigned. This is dramatically seen in the case of the principal fragment B, the most massive part of the pre-split comet, a companion to the Great Comet of 1844, as already noted. Let us assume that the two comets separated from their parent at the time of previous perihelion, several thousand years ago. If C/2019 Y4 should have survived as a persistent fragment until the 2020 perihelion, its endurance would have been 506 e-days. From $\gamma_{\rm par}$ in Equation (15) its endurace is expected to equal 420 e-days, last observed just 86 e-days short of the full revolution, a period of time equivalent to $\sim$7 days. Since the endurance varies as time weighted by $r^{-2}$, the ratio 86/7 $\simeq$ 12.3 is, as it should be, about halfway between (1/0.34)${}^{2}\simeq 8.7$, where 0.34 AU was the heliocentric distance on May 24, and (1/0.25)² = 16, where 0.25 AU was the perihelion distance on May 31. If the comet were instead a short-lived fragment, its endurance would have been only a little more than 100 e-days and it would have fragmented and disintegrated at some point after the previous perihelion, on its way to, and long before reaching, aphelion. C/2019 Y4 would never have been discovered.

By contrast, Fragment A was a short-lived fragment, which should have lasted until early to nearly mid-May. Of the few data sources that I inspected, it was the last time seen in the images taken with the HST on 23 April and its lifespan must have been longer than that. The three fragments that separated, presumably from A, around mid-March, were obviously also short lived. They could not have been minor fragments, given that they were observed after 8–13 April.

Even though the definition of the last sighting is not straightforward, the introduction of the endurance is very beneficial in that it offers some insight into the morphology of comet fragmentation products and demonstrates that the sublimation-driven nongravitational acceleration has important physical implications in more than one respect.

The disintegration time of a sizable fragment differs from the disintegration time of a comet as such. The laws of cascading fragmentation imply that a comet’s particulate debris survives and is detectable as a fairly condensed cloud over a period distinctly longer than the individual massive fragments because the debris has a much higher ratio of cross-sectional area to mass. There is nothing controversial, especially among comets of small perihelion distance, about large fragments dissolving before, while the comet’s remains after, perihelion. Indeed, C/2019 Y4 was reported to have been detected by the HI-1 imager of STEREO A as late as 6 June (Knight & Battams 2020), about a week following perihelion.

An important property of the nongravitational acceleration of a fragment is that it serves as a crude size estimate of the object. The acceleration varies as the mass-loss rate of sublimated gas from its surface and inversely as the object’s mass. The mass-loss rate varies as the square of the size, while the mass as the cube of the size. Accordingly, the nongravitational acceleration varies inversely as the size.

One can of course think of variations on the basic sublimation model, including the nuclear shape, rotation, etc. For example, the following empirical relation approximates a link between the acceleration $\gamma$ (again in units of 10^-5 the solar gravitational acceleration) and the fragment’s initial diameter $\cal D$ (in km) implied by a slightly more complex model (Sekanina 1982):

For the pre-split nucleus of C/2019 Y4 this law gives an effective initial diameter of 500 meters, for Fragment A about 100 meters, and for Fragments C, D, and E between 16–20 meters.

The fragmentation sequence of C/2019 Y4, jointly with the peculiar separation trajectory of Fragment A relative to B (Figure 3), are behind the rather intricate motions of the individual fragments relative to one another in projection onto the plane of the sky. The picture is further complicated by their rapid and irregular brightness variations.

The projected relative positions of the five fragments as a function of time are illustrated in Figure 8 by their status on seven selected dates, including the two on which the disintegrating comet was imaged by the HST. A particularly confusing situation is seen to have developed in early April, when the multiplicity was first detected. The three more recently born fragments were moving in a general direction of the line connecting the formerly existing Fragments A and B, not to mention that it was at this very time that B suddenly became bright enough to detect. Subsequently, the high nongravitational accelerations of C, D, and E caused the rapid eastward motions of these fragments away from A and B.

The data on the groups of long-period comets, presented in Table 21 to be discussed in Section 7.2, display some interesting features. The term long period is defined by the range of original orbital periods of the four groups, whose principal fragments are known. According to the catalogue by Marsden & Williams (2008), the periods varied from $\sim$2900 yr for comet Liller to $\sim$14,000 yr for comet Levy. The Great Comet of 1844 and comet Hind had intermediate periods of 7600 yr and 8300 yr, respectively. The typical original orbital period is thus on the order of several thousand years, with a maximum-to-minimum ratio of approximately 5:1.

The appearance of the principal and companion fragments, the order of return to perihelion, and the range of time lags between the principal and companion fragments among these long-period comets vary so strikingly that there must be major reasons for the differences. The principal fragments are intrinsically much brighter than the companions and their light curves are, to the extent known, more or less symmetrical with respect to perihelion. By comparison, the companion fragments usually exhibit a steep drop in brightness after perihelion, if they survive that long. The process of disintegration for at least two — C/1996 Q1 and C/2019 Y4 — began early and their demise may have been essentially completed by the time of perihelion or shortly afterwards.

The principal fragment always returned to perihelion first, a circumstance that has a diagnostic value as proposed below. This rule is known to also apply to the nontidally split comets with persistent companions in the short-period orbits, e.g., 3D/Biela, 73P/Schwassmann- Wachmann, 79P/du Toit-Hartley, 141P/Machholz, etc. (see Marsden & Williams 2008).

The range of time lags of the companion fragments trailing behind the principal fragments at perihelion is astonishing, extending from as little as 0.21 yr for the pair of C/1988 F1 and C/1988 J1 to 175.5 yr for the pair of C/1844 Y1 and C/2019 Y4, nearly three orders of magnitude. Persistent companions of the split comets in short-period orbits, such as 3D, 73P, or 141P, are known to pass perihelion a few hours to about one day after the principal fragment, which is on the order of hundredths of one percent of the orbital period. On the other hand, the lag for C/2019 Y4 equaled fully 2.3 percent of the orbital period of C/1844 Y1, if not more.

When a comet in a long-period orbit splits into two fragments that survive a full revolution about the Sun, the companion fragment arrives at its perihelion at a time different from the perihelion time of the principal fragment. This differential effect could be triggered by (i) the separation velocity, (ii) the sublimation-driven nongravitational acceleration, and/or (iii) the indirect planetary perturbations (even in the absence of either fragment’s close encounter with a planet).

I now investigate the magnitude of the differential effect in the orbital period for each of the three categories of provenance. I assume that the companion separates from the parent in the general proximity or perihelion. Differentiating the expression for the orbital velocity $V_{\rm orb}$ as a function of heliocentric distance $r$ at a fixed location of the fragmentation event, given by $r_{\rm frg}$, the relation between the separation velocity, $V_{\rm sep}(r_{\rm frg})$, measured along the orbital-velocity vector, and the corresponding relative change in the orbital period, $\Delta P/P$, is

where $q$ is the perihelion distance of the parent comet, $e$ its eccentricity, and $k_{0}$ the Gaussian gravitational constant. In a pseudo-parabolic approximation,

where the constant $\kappa$ equals

When $r_{\rm frg}$ is given in AU, $P$ and $\Delta P$ in years, and $V_{\rm sep}$ in m s^-1, the constant is numerically $\kappa=7020$.

The differential effect of a nongravitational acceleration consists in the fact that the fragments orbit the Sun in gravity fields of slightly unequal magnitudes. The effect on the orbital period, $P$, follows from Kepler’s Third Law

where $a$ is the semimajor axis. Differentiating this equation with respect to $k_{0}$, which is no longer a constant, one finds

so that

Here $\gamma$ is the parameter of the nongravitational model applied in Section 3 to investigate the fragmentation of C/2019 Y4; see also Equations (10) and following. The units of $\gamma$ have been $10^{-5}$ the solar gravitational acceleration.

Even though the planets gravitationally affect the motions of comets in all six elements, the perturbations of the reciprocal semimajor axis, $1/a$, have always been of primary interest because they present changes in the orbital energy and are relevant to the issues of comets’ perihelion returns or escape from the Solar System. The indirect perturbations of $1/a$ of long-period comets were investigated extensively by Everhart & Raghavan (1970). Everhart (1968) also undertook a Monte Carlo study of perturbations by a single planet, concluding that their distribution was not Gaussian.

As two long-period fragments are on their way to aphelion, the indirect planetary perturbations essentially cease at heliocentric distances smaller than 50 AU. By that time the separation distance between the two objects has not increased enough to imply a major differential perturbation effect. However, even in the absence of a significant contribution from a separation velocity and/or a nongravitational acceleration, the aphelion distances of the fragments are not identical. It is near aphelion, where their separation distance could increase dramatically, so that the original semimajor axes of the fragments determined from their subsequent near-perihelion arcs could become clearly uneven because of the unequal integrated effects of the indirect planetary perturbations at heliocentric distances smaller than 50 AU.

Changes in the original orbital energy (i.e., its value near aphelion, referred to the barycenter of the Solar System), $\Delta(1/a)_{\rm orig}$, are convertible into changes in the original orbital period, $P_{\rm orig}$. Since Kepler’s Third Law can obviously be written as

with $k_{\rm b}$ being the “barycentric” Gaussian constant, the differentiation with respect to $(1/a)_{\rm orig}$ gives

and, dividing Equation (29) by Equation (28),

where the expression on the right applies only when $P_{\rm orig}$ — and $\Delta P_{\rm orig}$ — are reckoned in years. Instead of differentiation one can work directly with the differences.

I now assess the three proposed sources for $\Delta P$ and $\Delta P_{\rm orig}$. For the sake of simplifying the matters, I assume that the parent comet fragmented at perihelion. The results in Table 21 offer surprisingly robust conclusions about effects of the separation velocity, the nongravitational acceleration, and the indirect planetary perturbations on the returns of the fragments to perihelion.

First of all, it is not accidental that the orbital periods of comets in the groups or pairs are not longer than thousands of years. If they were many tens of thousands of years or longer, the fragments would have been returning over periods in the past well beyond our records of cometary orbits. In fact, Table 21 illustrates that we already begin to have problems of this kind (such as the missing principal fragment of Group E). In this context, it is possible, for example, that the disintegrating comet C/1999 S4 (LINEAR) was a companion to a principal fragment whose orbital period may have been, say, 100,000 yr or so and had returned to perihelion a number of centuries earlier, as I already suggested (Sekanina 2000). C/1999 S4 masqueraded as a potential Oort Cloud comet (see footnote 1 on page 19), but was strongly deficient in carbon monoxide ice (Weaver et al. 2001), which helped reveal its true origin.

Separation velocities can hardly be the source of the investigated effect on companion fragments that are all trailing the principal fragments. On the contrary, one expects that effects of the separation velocity should lead to statistically equal numbers of companion fragments preceding and following the principal fragments. Even though the dataset in Table 21 is very small, the absence of leading companions is striking. Also suspect are the minuscule magnitudes of the derived separation velocities; one would have to assume that all fragmentation events occurred far from the Sun, thereby dramatically increasing the effect of $r_{\rm frg}$ in Equation (23).

Effects of the nongravitational acceleration are consistent with the dominance of trailing companion fragments, but the obvious problem is the gigantic magnitude of the acceleration required (with the exception of C/1988 J1). Objects with such enormous nongravitational efffects could not appear as observable objects and survive the entire revolution about the Sun. There is also no correlation between the likelihood of survival and the magnitude of $\Delta P$ in Table 21. For example, if the nongravitational acceleration governed this relationship, one would expect C/2019 Y1 much more likely to have begun disintegrating before perihelion than C/1996 Q1, contrary to the observed behavior of the two comets. Yet, it should be admitted that the nongravitational acceleration does to a degree contribute to the distribution of perihelion returns among the companion fragments, tilting $\Delta P$ to positive values in Table 21.

This leaves the indirect planetary perturbations as the major source of the effects that influence the semichaotic timing of the companion fragments’ perihelion returns. The three well-established groups/pairs in Table 21 suggest a strong dependence on the perihelion distance. One could also note that this empirical finding fits in with a pair of the Kreutz sungrazers, even though these comets have distinctly shorter orbital periods. While the perihelion returns of the companion fragments of the Liller group lagged the principal fragment, whose perihelion distance amounted to 0.84 AU, by 0.3 percent to 1.1 percent of the principal fragment’s orbital period, C/2019 Y4 trailed C/1844 Y1, whose perihelion distance was 0.3 the value of C/1988 A1, by at least 2.3 percent of its orbital period. By comparison, the Kreutz sungrazer C/1965 S1 (Ikeya-Seki), a less massive sibling of C/1882 R1 (Great September Comet), trailed the latter at perihelion by $>$11 percent of its orbital period.

In any event, the pair of C/1844 Y1 and C/2019 Y4 is an extreme case among the known groups/pairs of long-period comets. If the tabulated original orbital period of 7600 yr for C/1844 Y1 (Marsden et al. 1978) is replaced by T. Kobayashi’s value of 4000 yr (Green 2020a), the perihelion return of C/2019 Y4 would have lagged that of the principal fragment by as much as 4.4 percent of its orbital period, but still by significantly less than in the case of the Kreutz pair. And even though the value of $\Delta P_{\rm orig}$ would change from $-$2756 yr to a positive value of +844 yr, the unpublished 1$\sigma$ uncertainty in the orbital period of C/1844 Y1 derived by Kobayashi is estimated at hardly much less than $\pm$40 percent or about $\pm$1500 yr (given that the orbital solution had a high mean residual of $\pm$5${}^{\prime\prime}\!$.2), making $\Delta P_{\rm orig}$ for C/2019 Y4 essentially indeterminate. Further contributing to the uncertainty is the error of $P_{\rm orig}$ of C/2019 Y4 itself. Because original orbital periods derived from nongravitational solutions are known to be notoriously unreliable (Marsden et al. 1973), the tabulated $P_{\rm orig}$ for C/2019 Y4 was taken from one of the last gravitational-orbit determinations, before introducing the nongravitational terms into the equations of motion became inevitable.