PAIRS AND GROUPS OF GENETICALLY RELATED LONG-PERIOD COMETS AND PROPOSED IDENTITY OF THE MYSTERIOUS LICK OBJECT OF 1921

By coincidence, strong evidence for the existence of pairs of genetically related long-period comets—other than the Kreutz sungrazers—became available several years after the debate ended. The circumstances clearly favored Whipple's (1977) conservative approach.

The first comet pair consisted of C/1988 F1 (Levy) and C/1988 J1 (Shoemaker–Holt). They were discovered within 2 months of each other, and their perihelion times were only 76 days apart. Their osculating orbits, computed by Marsden (1989b) and listed in Table 1, were virtually identical, yet their true orbital period, defined by the original barycentric semimajor axis, was close to 14,000 yr. The striking orbital similarity and a very small temporal separation both suggested that the two comets still had been part of a single body during the previous return to perihelion around 12,000 BCE—and for a long time afterward.

Table 1.  Orbital Elements of Comets C/1988 F1 (Levy) and C/1988 J1 (Shoemaker–Holt) (Equinox J2000.0)

Orbital Element/Quantity	Comet C/1988 F1	Comet C/1988 J1
Osculation epoch (TT)	1987 Nov 21.0	1988 Feb 9.0
Time of perihelion passage tπ (TT)	1987 Nov 29.94718	1988 Feb 14.22162
Argument of perihelion ω (deg)	326.51491	326.51498
Longitude of ascending node Ω (deg)	288.76505	288.76487
Orbit inclination i (deg)	62.80744	62.80658
Perihelion distance q (au)	1.1741762	1.1744657
Orbital eccentricity e	0.9978157	0.9978301
Orbital period P (yr) $\left\{\begin{array}{l}\mathrm{osculation}\\ {\mathrm{original}}^{{\rm{a}}}\end{array}\right.$	12,460 ± 460	12,590 ± 230
	13,960	13,840
Orbital arc covered by observations	1988 Mar 22–1988 Jul 18	1988 May 13–1988 Oct 20
Number of observations employed	30	60
rms residual (arcsec)	±0.7	±0.9
Orbit-quality codeb	2A	2A
Reference	Marsden (1989b)	Marsden (1989b)

Notes.

^aReferring to the barycenter of the solar system. ^bFollowing the classification system introduced by Marsden et al. (1978); the errors of the elements other than the orbital period are unavailable.

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On the other hand, the relative motions of C/1988 F1 and C/1988 J1 are inconsistent with those of fragments of an ordinary split comet except when observed many years after separation. Survival of secondary fragments (or companions) over such long periods of time is untypical—weeks or months are the rule—among the split short-period comets, the most notable exceptions being 3D/Biela (Marsden & Sekanina 1971; Sekanina 1977, 1982) and 73P/Schwassmann–Wachmann (Sekanina 2006).⁴ And the gap of 76 days in the perihelion time, minuscule in terms of the orbital period of C/1988 F1, is unusually protracted when compared to an average split comet, whose fragments are commonly separated by a fraction of a day or a few days at the most.

The motion of a companion relative to the primary is known to be affected by both an orbital-momentum change at breakup (resulting in a nonzero separation velocity) and a continuous outgassing-driven differential nongravitational acceleration after separation (Sekanina 1978, 1982). As the dominant radial component of the nongravitational acceleration causes the less massive—and usually the intrinsically fainter—companion to start trailing the more massive primary in the orbit, the brighter leading fragment represents a signature of the role of the nongravitational forces at work.⁵ We now examine the two mechanisms as potential triggers of the 76-day gap between C/1988 F1 and C/1988 J1.

2.1.1. Effects of a Separation Velocity

Well-determined separation velocities of split comets' fragments are known to range from ∼0.1 to ∼2 m s⁻¹ (e.g., Sekanina 1982). Keeping this in mind, we employ a simple model to demonstrate that the fragmentation event that led to the birth of C/1988 F1 and C/1988 J1 could not have occurred during the previous return to perihelion. We begin by combining the virial theorem with Kepler's third law,

$$\begin{eqnarray}{V}_{{\rm{frg}}} & = & {c}_{0}\sqrt{\displaystyle \frac{2}{{r}_{{\rm{frg}}}}-\displaystyle \frac{1}{a}},\\ P & = & c\;{a}^{\tfrac{3}{2}},\end{eqnarray} \tag{ 1 }$$

where r_frg and V_frg are, respectively, the heliocentric distance and the orbital velocity at the time of fragmentation, a is the semimajor axis of the orbit, P is the orbital period, and c₀ and c are constants. Expressing V_frg in m s⁻¹, r_frg and a in au, and P in years, the values of the constants are ${c}_{0}=2.978\times {10}^{4}$ and c = 1. By differentiating both equations, requiring that ${r}_{{\rm{frg}}}\ll 2a$, and writing an increment ${\rm{\Delta }}(1/a)$ in terms of an increment ${\rm{\Delta }}P$ of the orbital period and 1/a in terms of P, we obtain the following expression for the relationship between ΔV, a separation velocity (in m s⁻¹) in the direction of the orbital-velocity vector, and ΔP:

$$\begin{eqnarray}&&{\rm{\Delta }}V=\displaystyle \frac{{c}_{0}{c}^{\tfrac{2}{3}}\sqrt{2}}{6}\;{r}_{{\rm{frg}}}^{\tfrac{1}{2}}\;{P}^{-\tfrac{5}{3}}{\rm{\Delta }}P=C\;{r}_{{\rm{frg}}}^{\tfrac{1}{2}}\;{P}^{-\tfrac{5}{3}}{\rm{\Delta }}P,\end{eqnarray} \tag{ 2 }$$

where C is a constant whose value is 7019 when ΔP is in years or 19.22 when in days. For the pair of C/1988 F1 and C/1988 J1, we find from Table 1 $P\simeq {\rm{14,000}}$ yr and ${\rm{\Delta }}P=76.3$ days, so that the inferred separation velocity (in m s⁻¹) in the direction of the orbital-velocity vector becomes

$$\begin{eqnarray}&&{\rm{\Delta }}V=0.00018\;{r}_{{\rm{frg}}}^{\tfrac{1}{2}}.\end{eqnarray} \tag{ 3 }$$

Since ${a}_{{\rm{orig}}}\simeq 580$ au, r_frg is to be much smaller than 1160 au and the separation velocity at the time of fragmentation near the previous perihelion is predicted to be ${\rm{\Delta }}V\lt 0.006\;{\rm{m}}$ s⁻¹, orders of magnitude lower than the least separation velocities for the split comets. Only if the separation velocity should be exactly normal to the orbital-velocity vector, rather an absurd premise, might this condition be satisfied. A much more plausible scenario is C/1988 F1 and C/1988 J1 having separated from their common parent fairly recently, when it already was on its way from aphelion to the 1987 perihelion.

This argument is fully supported by a more rigorous treatment of the effects on the companion's perihelion time that are caused by its separation velocity (reckoned relative to the primary) acquired upon the parent comet's fragmentation. Let the primary's perihelion time be t_π, its perihelion distance q, its eccentricity $e\lt 1$, and its parameter $p=q(1+e)$. Furthermore, let ${\boldsymbol{P}}$, ${\boldsymbol{Q}}$, ${\boldsymbol{R}}$ be the unit orbit-orientation vectors with the components ${P}_{{\rm{x}}},\ldots ,\;{R}_{{\rm{z}}}$ in the right-handed ecliptical coordinate system. We wish to determine the temporal gap between the perihelion passages of the companion and the primary,

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{\pi }={t}_{\pi }^{\prime }-{t}_{\pi },\end{eqnarray} \tag{ 4 }$$

where ${t}_{\pi }^{\prime }$ is the companion's perihelion time; a positive ${\rm{\Delta }}{t}_{\pi }$ signifies that the companion trails the primary.

With the planetary perturbations and nongravitational forces ignored, the conditions at the time of fragmentation, t_frg, are described as follows: the comet's heliocentric distance is r_frg, and the position vector's coordinates in the ecliptical system are $({x}_{{\rm{frg}}},{y}_{{\rm{frg}}},{z}_{{\rm{frg}}})$; the primary fragment's true anomaly is ${v}_{{\rm{frg}}}=\mathrm{arccos}[(p/{r}_{{\rm{frg}}}-1)/e]$, its orbital velocity is V_frg, and the components of its velocity vector in the ecliptical system are $({\dot{x}}_{{\rm{frg}}},{\dot{y}}_{{\rm{frg}}},{\dot{z}}_{{\rm{frg}}})$; and the companion's separation velocity relative to the primary is U, its components in the cardinal directions of the RTN coordinate system,⁶ U_R, U_T, and U_N, being related to the ecliptical components U_x, U_y, and U_z by

$$\begin{eqnarray}&&\left[\begin{array}{c}{U}_{{\rm{x}}}\\ {U}_{{\rm{y}}}\\ {U}_{{\rm{z}}}\end{array}\right]=\left[\begin{array}{ccc}{P}_{{\rm{x}}} & {Q}_{{\rm{x}}} & {R}_{{\rm{x}}}\\ {P}_{{\rm{y}}} & {Q}_{{\rm{y}}} & {R}_{{\rm{y}}}\\ {P}_{{\rm{z}}} & {Q}_{{\rm{z}}} & {R}_{{\rm{z}}}\end{array}\right]\cdot \left[\begin{array}{ccc}\mathrm{cos}{v}_{{\rm{frg}}} & -\mathrm{sin}{v}_{{\rm{frg}}} & 0\\ \mathrm{sin}{v}_{{\rm{frg}}} & \mathrm{cos}{v}_{{\rm{frg}}} & 0\\ 0 & 0 & 1\end{array}\right]\cdot \left[\begin{array}{c}{U}_{{\rm{R}}}\\ {U}_{{\rm{T}}}\\ {U}_{{\rm{N}}}\end{array}\right].\end{eqnarray} \tag{ 5 }$$

The ecliptical components of the companion's orbital velocity vector at time t_frg are

$$\begin{eqnarray}&&\left[\begin{array}{c}{\dot{x}}_{{\rm{frg}}}^{\prime }\\ {\dot{y}}_{{\rm{frg}}}^{\prime }\\ {\dot{z}}_{{\rm{frg}}}^{\prime }\end{array}\right]=\left[\begin{array}{c}{\dot{x}}_{{\rm{frg}}}\\ {\dot{y}}_{{\rm{frg}}}\\ {\dot{z}}_{{\rm{frg}}}\end{array}\right]+\left[\begin{array}{c}{U}_{{\rm{x}}}\\ {U}_{{\rm{y}}}\\ {U}_{{\rm{z}}}\end{array}\right],\end{eqnarray} \tag{ 6 }$$

and its orbital velocity ${V^{\prime} }_{{\rm{frg}}}$:

$$\begin{eqnarray}&&{V^{\prime} }_{{\rm{frg}}}=\sqrt{{({\dot{x}}_{{\rm{frg}}}^{\prime })}^{2}+{({\dot{y}}_{{\rm{frg}}}^{\prime })}^{2}+{({\dot{z}}_{{\rm{frg}}}^{\prime })}^{2}}.\end{eqnarray} \tag{ 7 }$$

The companion's orbital eccentricity, e', is equal to

$$\begin{eqnarray}&&{e}^{\prime }=\sqrt{1+{p}^{\prime }\left[{\left(\displaystyle \frac{{V}_{{\rm{frg}}}^{\prime }}{k}\right)}^{2}-\displaystyle \frac{2}{{r}_{{\rm{frg}}}}\right]},\end{eqnarray} \tag{ 8 }$$

where k is the Gaussian gravitational constant,

$$\begin{eqnarray}&&{p}^{\prime }=\displaystyle \frac{{{\mathfrak{R}}}_{{\rm{xy}}}^{2}+{{\mathfrak{R}}}_{{\rm{yz}}}^{2}+{{\mathfrak{R}}}_{{\rm{zx}}}^{2}}{{k}^{2}},\end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray}{{\mathfrak{R}}}_{{\rm{xy}}}=\left|\begin{array}{cc}{x}_{{\rm{frg}}} & {y}_{{\rm{frg}}}\\ {\dot{x}}_{{\rm{frg}}}^{\prime } & {\dot{y}}_{{\rm{frg}}}^{\prime }\end{array}\right|,{{\mathfrak{R}}}_{{\rm{yz}}}=\left|\begin{array}{cc}{y}_{{\rm{frg}}} & {z}_{{\rm{frg}}}\\ {\dot{y}}_{{\rm{frg}}}^{\prime } & {\dot{z}}_{{\rm{frg}}}^{\prime }\end{array}\right|,{{\mathfrak{R}}}_{{\rm{zx}}}=\left|\begin{array}{cc}{z}_{{\rm{frg}}} & {x}_{{\rm{frg}}}\\ {\dot{z}}_{{\rm{frg}}}^{\prime } & {\dot{x}}_{{\rm{frg}}}^{\prime }\end{array}\right|.\end{eqnarray} \tag{ 10 }$$

The companion's perihelion distance is ${q}^{\prime }={p}^{\prime }/(1+{e}^{\prime })$, and its true anomaly ${v}_{\mathrm{frg}}^{\prime }$ at t_frg is computed from

$$\begin{eqnarray}\mathrm{sin}{v}_{{\rm{frg}}}^{\prime } & = & \displaystyle \frac{\sqrt{{p}^{\prime }}}{{{kr}}_{{\rm{frg}}}{e}^{\prime }}({x}_{{\rm{frg}}}{\dot{x}}_{{\rm{frg}}}^{\prime }+{y}_{{\rm{frg}}}{\dot{y}}_{{\rm{frg}}}^{\prime }+{z}_{{\rm{frg}}}{\dot{z}}_{{\rm{frg}}}^{\prime }),\\ \mathrm{cos}{v}_{{\rm{frg}}}^{\prime } & = & \displaystyle \frac{1}{{e}^{\prime }}\left(\displaystyle \frac{{p}^{\prime }}{{r}_{{\rm{frg}}}}-1\right).\end{eqnarray} \tag{ 11 }$$

The eccentric anomaly at t_frg, ${E}_{\mathrm{frg}}^{\prime }$, then comes out to be

$$\begin{eqnarray}&&{E}_{{\rm{frg}}}^{\prime }=2\mathrm{arctan}\left(\sqrt{\displaystyle \frac{1-{e}^{\prime }}{1+{e}^{\prime }}}\mathrm{tan}\tfrac{1}{2}{v}_{{\rm{frg}}}^{\prime }\right)\end{eqnarray} \tag{ 12 }$$

and the companion's perihelion time is finally derived from the relation

$$\begin{eqnarray}&&{t}_{\pi }^{\prime }={t}_{{\rm{frg}}}-\displaystyle \frac{{E}_{{\rm{frg}}}^{\prime }-{e}^{\prime }\mathrm{sin}{E}_{{\rm{frg}}}^{\prime }}{k}{\left(\displaystyle \frac{{q}^{\prime }}{1-{e}^{\prime }}\right)}^{\displaystyle \frac{3}{2}}.\end{eqnarray} \tag{ 13 }$$

Inserting Equation (13) into Equation (4), we obtain the temporal separation between the companion and the primary at perihelion, ${\rm{\Delta }}{t}_{\pi }$, the quantity we were looking for.

We note that, as by-products of the computations, we also determined the effects of the companion's separation velocity on the perihelion distance, ${\rm{\Delta }}q={q}^{\prime }-q$, and eccentricity ${\rm{\Delta }}e={e}^{\prime }-e$. It is similarly possible to derive the separation velocity's effects on the remaining elements—the inclination, the longitude of the ascending node, and the argument of perihelion.

For C/1988 F1 and C/1988 J1 these computations show in Table 2 that a gap of 76 days in the time of arrival at perihelion is achievable with a separation velocity of ∼1 m s⁻¹ in the direction away from the Sun at a heliocentric distance of barely 400 au, less than 700 yr before the 1987 perihelion. The same effect is also achievable with a separation velocity of 1 m s⁻¹ in the direction perpendicular to the radial direction in the orbital plane at a heliocentric distance of less than 900 au some 3000 yr before the 1987 perihelion. Only the component of the separation velocity that is normal to the orbital plane cannot bring about a gap of this magnitude.

Table 2.  Separation Velocity Needed for 76.274-day Gap between Perihelion Arrival Times of C/1988 F1 and C/1988 J1 (Heliocentric Osculating Approximation)

Parent Comet's Fragmentation			Separation Velocity
Distance	Time before Perihelion		Radial,	Transverse,
rfrg (au)	(yr)	(units of P)	UR (m s−1)	UT (m s−1)
200	−227.5	−0.0183	+4.30	+14.95
300	−431.2	−0.0346	+1.80	+7.76
400	−688.3	−0.0552	+0.94	+4.80
500	−1002.5	−0.0805	+0.56	+3.25
600	−1381.6	−0.1109	+0.35	+2.33
700	−1840.0	−0.1477	+0.23	+1.72
800	−2404.1	−0.1929	+0.16	+1.30
900	−3129.0	−0.2511	+0.11	+0.97
1000	−4176.3	−0.3352	+0.067	+0.70
1073.74a	−6230.0	−0.5000	+0.034	+0.43
1000	−8283.7	−0.6648	+0.019	+0.28
900	−9331.0	−0.7489	+0.014	+0.22
800	−10055.9	−0.8071	+0.011	+0.18
700	−10620.0	−0.8523	+0.0094	+0.15
600	−11078.4	−0.8891	+0.0079	+0.13
500	−11457.5	−0.9195	+0.0066	+0.10
400	−11771.7	−0.9448	+0.0055	+0.082
300	−12028.8	−0.9654	+0.0045	+0.061
200	−12232.5	−0.9817	+0.0034	+0.040
100	−12381.4	−0.9937	+0.0023	+0.020

Note.

^aAphelion.

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It is noted in Table 2 that, based on the radial component of the separation velocity, the fragmentation of the parent of C/1988 F1 and C/1988 J1 most probably occurred between 300 and 900 au and not longer than one-quarter of the orbital period before the 1987 perihelion, possibly as recently as the fourteenth century AD. If it had occurred much earlier, the radial component would have to have been more than an order of magnitude smaller than the transverse component, an unlikely scenario.

Since we neglected the planetary perturbations, we checked the accuracy of our results against a rigorous orbit-determination run. We integrated the set of elements of C/1988 F1 from Table 1 back in time to 1300 January 1, when the comet was 402.8 au from the Sun, added exactly 1 m s⁻¹ to the radial component of its orbital velocity, and integrated forward in time to the epoch of 1987 November 21. The effects on the elements are shown in column 2 of Table 3. The perihelion passage came out to take place on 1988 February 18.2664 TT; the computations based on Equations (4)–(13) show a perihelion time of 1988 February 20.1137 TT, differing by ∼2% of the 80.3-day gap and indicating that the effect of planetary perturbations is almost two orders of magnitude smaller than the separation-velocity effect.

Table 3.  Comparison of Effects of Separation Velocity at 402.8 AU from the Sun and Planetary Perturbations on Orbital Elements of Comet C/1988 F1 (Epoch 1987 November 21.0 TT)

Increment	Rigorous	Separation Velocity (m s−1) Alone
in Orbital	Integration
Element	of Orbita	${U}_{{\rm{R}}}\;=\;+1.00$	${U}_{{\rm{T}}}\;=\;+5.02$	${U}_{{\rm{N}}}\;=\;+0.10$
${\rm{\Delta }}{t}_{\pi }$ (days)	+80.3136	+82.1665	+82.1665	+0.0007
Δω(deg)	+0.0004	+0.0029	−0.2186	−0.0125
ΔΩ (deg)	−0.0038	0.0000	0.0000	+0.0272
Δi(deg)	−0.0025	0.0000	0.0000	−0.0444
Δq (au)	+0.000073	+0.000003	+0.106562	+0.000001
Δe	+0.0000142	−0.0000044	−0.0001966	0.0000000

Note.

^aIncrements are the sums of an effect due to a separation velocity of ${U}_{{\rm{R}}}\;=\;+1.00$ m s⁻¹ and the planetary perturbations.

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Table 3 summarizes the test results for the effects of separation velocity not only on the perihelion time but on the other elements as well. We note from columns (2) and (3) that the perihelion time t_π is the only orbital element for which the effect of the radial component of the separation velocity dominates the planetary perturbations. The tabulated values of ΔΩ and Δi are the net effects due to the perturbations, because no change can be triggered by a radial separation velocity in a direction off the orbital plane that these elements reflect.

In the penultimate column of Table 3 we list the orbital increments that would be triggered by a transverse component of the separation velocity generating the same effect in the perihelion time as a 1 m s⁻¹ radial component. Striking are the large changes in ω and q, which are not observed. The last column shows the increments due to a small normal component of the separation velocity. The exceptional similarity of the angular elements and perihelion distance of the orbits of C/1988 F1 and C/1988 J1 (Table 1) suggests that the transverse and normal components of the separation velocity must have been very small compared to the radial component U_R and almost certainly much smaller than 0.1 m s⁻¹.

2.1.2. Effects of a Nongravitational Acceleration

Our reconstruction of the light curves of C/1988 F1 and C/1988 J1 is shown in Figure 1, its caption describing the sources of brightness data used. The plot leaves no doubt that C/1988 F1, the leading fragment, was intrinsically brighter than C/1988 J1 by more than 1 mag; the latter object was also fading at a steeper rate, making the brightness ratio of C/1988 F1 to C/1988 J1 increase with time. Thus, the relative positions of the two fragments in the orbit are consistent with the fainter (and presumably less massive) one having been decelerated relative to the brighter (and presumably more massive) one. Accordingly, the temporal separation of 76 days between both comets could have been affected to a degree by a differential nongravitational acceleration. We now investigate this problem in more detail.

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We begin by assuming that the common parent of comets C/1988 F1 and C/1988 J1 had moved in a gravitational orbit identical with that of C/1988 F1, fragmenting on its way from aphelion to perihelion at a time t_frg, when it was at a heliocentric distance r_frg (Section 2.1.1). If the primary fragment, C/1988 F1, arrived at perihelion at time t_π, the length of the orbital arc that it traveled between t_frg and t_π is equal to

$$\begin{eqnarray}&&{\ell }={\displaystyle \int }_{{t}_{{\rm{frg}}}}^{{t}_{\pi }}{V}_{{\rm{gr}}}(t)\;{dt},\end{eqnarray} \tag{ 14 }$$

where V_gr(t) is the orbital velocity of the primary's gravitational motion at time t. We now assume that the orbital motion of the companion C/1988 J1 has been—ever since the secession from the parent (at a zero separation velocity)—affected by a nongravitational acceleration pointing in a general direction against the orbital motion. During the period of time from t_frg to t_π, the orbital arc traveled by the companion is accordingly shorter by ${\rm{\Delta }}{\ell }$,

$$\begin{eqnarray}&&{\ell }-{\rm{\Delta }}{\ell }={\displaystyle \int }_{{t}_{{\rm{frg}}}}^{{t}_{\pi }}{V}_{{\rm{ng}}}(t)\;{dt},\end{eqnarray} \tag{ 15 }$$

where V_ng(t) is the orbital velocity of the companion's nongravitational motion. But V_ng(t) is the sum of V_gr(t) from Equation (14) and the nongravitational part ${\rm{\Delta }}{V}_{{\rm{ng}}}(t)\lt 0$, which is equal to an effect of a nongravitational acceleration in the direction of the orbital-velocity vector integrated from t_frg to t. Subtracting Equation (15) from Equation (14), introducing a ratio $z=q/r$ (where r is the heliocentric distance at time t) as a new integration variable, and approximating the gravitational orbit by a parabola, we readily find

$$\begin{eqnarray}&&{\rm{\Delta }}{\ell }=-\displaystyle \frac{{q}^{\tfrac{3}{2}}}{k\sqrt{2}}{\displaystyle \int }_{{z}_{{\rm{frg}}}}^{{z}_{\star }}{\rm{\Delta }}{V}_{{\rm{ng}}}{(z){z}^{-\tfrac{5}{2}}(1-z)}^{-\tfrac{1}{2}}\;{dz},\end{eqnarray} \tag{ 16 }$$

where k = 0.0172021 au^3/2 day⁻¹ is the Gaussian gravitational constant, q is again the perihelion distance of the primary's orbit, ${z}_{{\rm{frg}}}=q/{r}_{{\rm{frg}}}$, and ${z}_{\star }=q/{r}_{\star }$, where ${r}_{\star }$ is the heliocentric distance of the companion at the time the primary is at perihelion. The companion still has ${\rm{\Delta }}{t}_{\pi }=76.274$  days to go to its perihelion, and the condition that ${z}_{\star }$ should satisfy is

$$\begin{eqnarray}&&{\rm{\Delta }}{t}_{\pi }=\displaystyle \frac{{q}^{\tfrac{3}{2}}\sqrt{2}}{3k}\;{z}_{\star }^{-\tfrac{3}{2}}(1+2{z}_{\star })\sqrt{1-{z}_{\star }}.\end{eqnarray} \tag{ 17 }$$

Next, we consider separately the radial, j_R, and transverse, j_T, components of the nongravitational acceleration in the orbital plane of the RTN right-handed coordinate system (Section 2.1.1). In the Marsden et al. (1973) Style II formalism, they are written in the following form:

$$\begin{eqnarray}&&\left[\begin{array}{c}{j}_{{\rm{R}}}(r)\\ {j}_{{\rm{T}}}(r)\end{array}\right]=\left[\begin{array}{c}{A}_{1}\\ {A}_{2}\end{array}\right]g(r),\end{eqnarray} \tag{ 18 }$$

where g(r) is the standard empirical nongravitational law that the formalism employs,

$$\begin{eqnarray}&&g{(r)=0.1113(r/{r}_{0})}^{-2.15}{[1+{(r/{r}_{0})}^{5.093}]}^{-4.6142},\end{eqnarray} \tag{ 19 }$$

with a scaling distance ${r}_{0}=2.808\;{\rm{au}}$ appropriate for an acceleration driven by the sublimation of water ice. The law is normalized to $g(1\;{\rm{au}})=1$, so that the parameters A₁ and A₂ are the radial and transverse accelerations at a unit heliocentric distance.

The integrated contribution from the nongravitational acceleration to the magnitude of the orbital-velocity vector, ${\rm{\Delta }}{V}_{{\rm{ng}}}$ (now reckoned positive in the direction opposite the orbital motion), is expressed, in a parabolic approximation, for any time t (${t}_{{\rm{frg}}}\lt t\lt {t}_{\pi }$) along a preperihelion orbital arc by

$$\begin{eqnarray}&&{\rm{\Delta }}{V}_{{\rm{ng}}}(t)={A}_{1}{\displaystyle \int }_{{t}_{{\rm{frg}}}}^{t}g[r(t)]\sqrt{1-z}\;{dt}\end{eqnarray} \tag{ 20 }$$

from the radial component and

$$\begin{eqnarray}&&{\rm{\Delta }}{V}_{{\rm{ng}}}(t)=-{A}_{2}{\displaystyle \int }_{{t}_{{\rm{frg}}}}^{t}g[r(t)]\sqrt{z}\;{dt}\end{eqnarray} \tag{ 21 }$$

from the transverse component. Inserting Equations (20) and (21) into Equation (16) and again replacing time with the ratio z as an integration variable, we find in the case that the effect is due entirely to the radial component of the nongravitational acceleration

$$\begin{eqnarray}&&{\rm{\Delta }}{\ell }=\displaystyle \frac{{A}_{1}{q}^{3}}{2{k}^{2}}{\displaystyle \int }_{{z}_{{\rm{frg}}}}^{{z}_{\star }}{z}^{-\tfrac{5}{2}}{(1-z)}^{-\tfrac{1}{2}}\;{dz}{\displaystyle \int }_{{z}_{{\rm{frg}}}}^{z}g\left(\displaystyle \frac{q}{\zeta }\right){\zeta }^{-\tfrac{5}{2}}\;d\zeta \end{eqnarray} \tag{ 22 }$$

and similarly in the case that it is due entirely to the transverse component

$$\begin{eqnarray}{\rm{\Delta }}{\ell } & = & -\displaystyle \frac{{A}_{2}{q}^{3}}{2{k}^{2}}{\displaystyle \int }_{{z}_{{\rm{frg}}}}^{{z}_{\star }}{z}^{-\tfrac{5}{2}}{(1-z)}^{-\tfrac{1}{2}}\;{dz}\\ & & \times {\displaystyle \int }_{{z}_{{\rm{frg}}}}^{z}g\left(\displaystyle \frac{q}{\zeta }\right){\zeta }^{-2}{(1-\zeta )}^{-\tfrac{1}{2}}\;d\zeta .\end{eqnarray} \tag{ 23 }$$

These expressions determine the parameters A₁ and A₂, since the traveled orbital-arc length ${\rm{\Delta }}{\ell }$ is, in an adequate parabolic approximation, a function of only q and ${z}_{\star }$

$$\begin{eqnarray}{\rm{\Delta }}{\ell } & = & k{\displaystyle \int }_{{t}_{\pi }}^{{t}_{\pi }^{\prime }}\sqrt{\displaystyle \frac{2}{r}}\;{dt}=q{\displaystyle \int }_{{z}_{\star }}^{1}\displaystyle \frac{{dz}}{{z}^{2}\sqrt{1-z}}\\ & = & 2q{\displaystyle \int }_{0}^{\sqrt{{z}_{\star }^{-1}-1}}\sqrt{1+{x}^{2}}\ {dx}\\ & = & q\left(\displaystyle \frac{\sqrt{1-{z}_{\star }}}{{z}_{\star }}+{\mathrm{log}}_{{\rm{e}}}\displaystyle \frac{1+\sqrt{1-{z}_{\star }}}{\sqrt{{z}_{\star }}}\right).\end{eqnarray} \tag{ 24 }$$

With ${\rm{\Delta }}{t}_{\pi }=76.274$ days and $q=1.17418$ au, we find ${z}_{\star }\;=0.70859$ (or ${r}_{\star }=1.65705$ au) from Equation (17) and ${\rm{\Delta }}{\ell }\;=1.60358$ au from Equation (24).

The nongravitational parameters A₁ and A₂ are now functions of only the heliocentric distance at fragmentation, r_frg. There is no measurable contribution from the heliocentric distances much greater than the scaling distance r₀ (Equation (19)), as the nongravitational law g(r) falls off precipitously at those distances. This expectation is fully corroborated by numerical integration of Equations (22) and (23), resulting in the nongravitational parameters ${A}_{1}\;=\;+0.00642$ au day⁻² and ${A}_{2}=-0.00575\;{\rm{au}}$ day⁻² for any ${r}_{{\rm{frg}}}\gtrsim 3.6$ au. For smaller r_frg the magnitudes of A₁ and A₂ are greater still. The inferred minimum values $| {A}_{1}|$ and $| {A}_{2}|$ are found to exceed the gravitational acceleration of the Sun at 1 au by a factor of ∼20 or more and are meaningless. We conclude that the nongravitational effects in the motions of C/1988 F1 and C/1988 J1 cannot account for the gap of ∼76 days between their perihelion times.

These results, too, were confronted with rigorous orbital computations, which showed in the first place that a positive radial nongravitational acceleration (${A}_{1}\gt 0$) would in fact make the companion C/1988 J1 pass through perihelion before the primary C/1988 F1. This is so because by reducing the orbital velocity, this acceleration increases the perihelion distance and significantly shortens the orbital period.

With reference to a negative radial nongravitational acceleration, the rigorous orbital computations suggested that if a fragmentation event occurred, for example, at 402.8 au before perihelion, an acceleration described by ${A}_{1}=-{10}^{-6}$ au day⁻², about 0.3% of the Sun's gravitational acceleration that would delay the perihelion time by merely 0.044 days, should already be judged as unacceptably high in magnitude, because it would—contrary to the trivial differences between the orbital elements of C/1988 F1 and C/1988 J1—increase the argument of perihelion by 0137 and decrease the perihelion distance by 0.0014 au. If nearly comparable with the Sun's gravitational acceleration, these nongravitational effects would make the companion's orbit strongly hyperbolic.

Similarly, the transverse nongravitational acceleration described by ${A}_{2}=-{10}^{-6}$ au day⁻², which would delay the passage through perihelion by 0.24 days, is already unacceptably high in magnitude, because it would increase the argument of perihelion by 0227 and reduce the perihelion distance by 0.0018 au. If nearly comparable with the Sun's gravitational acceleration, these nongravitational effects would transform the companion into a short-period comet of a small perihelion distance.

Another comet that was discovered in early 1988 was C/1988 A1 (Liller), a fairly, but not extraordinarily, active one. We had to wait for more than 8 yr to appreciate its special status. In 1996 August, a newly discovered comet C/1996 Q1 (Tabur) turned out to have an orbit nearly as similar to that of C/1988 A1 as was the orbit of C/1988 J1 to that of C/1988 F1. And just months before this writing, yet another object was discovered, C/2015 F3 (SWAN), also with an orbit nearly identical to those of C/1988 A1 and C/1996 Q1. The sets of orbital elements for the three comets are compared in Table 4. It is unfortunate that their quality is very uneven. The best determined orbit is that of C/1988 A1, which was astrometrically measured both before and after perihelion, and the observed orbital arc was just about 6 months. Although the formal errors of the elements were not—except for the orbital period (or the semimajor axis)—published, their uncertainty is not greater than a few units of the last or penultimate decimal place. The second-best-determined orbit is that of C/2015 F3, whose story of discovery and follow-up observation is peculiar (Green 2015). Although its SWAN Lyα images were detected as early as 2015 March 3, 6 days before perihelion, the astrometric and brightness observations did not commence until March 24, more than 2 weeks after perihelion. The delay notwithstanding, the comet was astrometrically measured over a period of slightly more than 2 months. By contrast, C/1996 Q1, whose orbit is least well determined, was discovered more than 10 weeks before perihelion, but the astrometric observations had to be terminated some 8 weeks later, still before perihelion, because of the loss of the nuclear condensation. The orbital arc's length covered by the astrometry was a few days short of 2 months.

Table 4.  Orbital Elements of Comets C/1988 A1 (Liller), C/1996 Q1 (Tabur), and C/2015 F3 (SWAN) (Equinox J2000.0)

Quantity/Orbital Element	Comet C/1988 A1a	Comet C/1996 Q1	Comet C/2015 F3
Osculation epoch (TT)	1988 Mar 20.0	1996 Nov 13.0b	2015 Feb 27.0
Time of perihelion passage tπ (TT)	1988 Mar 31.11442	1996 Nov 3.53123 ± 0.00274	2015 Mar 9.35613 ± 0.00077
Argument of perihelion ω(deg)	57.38762	57.41221 ± 0.00314	57.56668 ± 0.00121
Longitude of ascending node Ω (deg)	31.51540	31.40011 ± 0.00102	31.63895 ± 0.00042
Orbit inclination i(deg)	73.32239	73.35626 ± 0.00119	73.38679 ± 0.00023
Perihelion distance q (AU)	0.8413332	0.8398052 ± 0.0000139	0.8344515 ± 0.0000121
Orbital eccentricity e	0.9965647	0.9986821 ± 0.0001399	0.9964470 ± 0.0000348
Orbital period (yr) $\left\{\begin{array}{l}\mathrm{osculation}\\ {\mathrm{original}}^{{\rm{c}}}\end{array}\right.$	3832.7 ± 9.4	16,100 ± 2600	3599 ± 53
	2933.0	10,500	3300
Orbital arc covered by observations	1988 Jan 12–1988 Jul 14	1996 Aug 21–1996 Oct 18	2015 Mar 24–2015 May 28
Number of observations employed	100	199	283
rms residual (arcsec)	±0.8	±1.0	±0.8
Orbit-quality coded	1B	2B	2A
Reference	Green (1988)	JPL database (Orbit 15)e	Williams (2015) f

Notes.

^aErrors of the elements other than the orbital period are unavailable. ^bAfter integration of the elements from an initial nonstandard epoch of 1996 September 18.0 TT. ^cReferred to the barycenter of the solar system; rigorous orbit integration gives for t_π at the previous return the date of −944 June 20. ^dFollowing the classification system introduced by Marsden et al. (1978). ^eSee http://ssd.jpl.nasa.gov/sbdb.cgi. ^fErrors of the elements are from the Minor Planet Center's Orbits/Observations Database: http://www.minorplanetcenter.net/db_search.

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In spite of being fragments of a common parent, the three comets had a very different appearance, morphology, and behavior. This is illustrated by both the sample images in Figure 2 and the light curves in Figure 3.

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Comet C/1988 A1, presented in Figure 2 as it looked near its peak intrinsic brightness, displays a coma of at least 3', equivalent to 170,000 km, in diameter and appears to be healthy and active. The light curve in Figure 3 shows that its postperihelion fading was, at least at heliocentric distances smaller than 2 au, substantially less steep than its preperihelion brightening and that the intrinsic brightness peaked some 3 weeks after perihelion. Overall, the light curve was fairly smooth, again especially within 2 au of the Sun.

The behavior of C/1996 Q1 was, in several respects, contrary to that of C/1988 A1. The image of C/1996 Q1 in Figure 2, from November 9, 6 days after perihelion, consists of a headless long narrow tail, without any nuclear condensation, which explains the termination of astrometric observations long before this image was taken. The light curve in Figure 3 is quite erratic, displaying at least three outbursts between 40 and 20 days before perihelion. The first of them is seen to have begun more than 50 days before perihelion, and for more than 10 days in this span of time C/1996 Q1 was intrinsically brighter than C/1988 A1 at the same heliocentric distance (note the shift in the brightness scale in Figure 3). The last of the three outbursts was accompanied by major morphological changes in the comet's head, with the condensation disappearing within a few days following the event's peak. It was at this time, about 16 days before perihelion, that all astrometric observations terminated. The light curve shows a continuous steep decline of brightness ever since the peak of the third outburst. The last confirmed magnitude estimates were obtained about 2 weeks after perihelion, but extremely faint "ghost" images of the comet's residual tail were marginally detected⁷ as late as mid-January of 1997.

Comparison of C/2015 F3 with C/1996 Q1 suggests only one common trait: both comets were fading much more rapidly than C/1988 A1, obviously the primary, most massive fragment of their common parent. Otherwise, C/2015 F3 differed from C/1996 Q1 significantly. First of all, C/2015 F3 survived perihelion seemingly intact. In fact, relative to perihelion, this comet was not even discovered in the SWAN images (Green 2015) by the time C/1996 Q1 had already disintegrated. Second, the light curve of C/2015 F3 is in Figure 3 located above the light curve of C/1996 Q1. Third, no prominent tail was observed to have survived the head of C/2015 F3, as illustrated in Figure 2 by one of the images in its advanced phase of evolution. It was not until some 50 or so days after perihelion that this comet began to lose its nuclear condensation, which resulted in an accelerated fading (Figure 3). And fourth, there is no evidence that this rapid fading was triggered by an outburst, as it was in the case of C/1996 Q1.

2.2.1. Fragmentation of the Parent

With the previous return to perihelion computed to have occurred in 945 BCE (note c in Table 4), we find that both C/1996 Q1 and C/2015 F3 are likely to have separated from C/1988 A1 in their common parent's fragmentation event or events in a general proximity of that perihelion. Equation (2) implies that the gaps of 3139.42 days between the perihelion times of C/1988 A1 and C/1996 Q1 and 9839.24 days between C/1988 A1 and C/2015 F3 require separation velocities (in m s⁻¹) along the orbital-velocity vector of, respectively,

$$\begin{eqnarray}{\rm{\Delta }}V & = & 0.092\sqrt{{r}_{{\rm{frg}}}/q}\ \mathrm{for}\ {\rm{C}}/1996\ {\rm{Q}}1\;\mathrm{versus}\ {\rm{C}}/1988\ {\rm{A}}1,\\ {\rm{\Delta }}V & = & 0.289\sqrt{{r}_{{\rm{frg}}}/q}\ \mathrm{for}\ {\rm{C}}/2015\ {\rm{F}}3\;\mathrm{versus}\ {\rm{C}}/1988\ {\rm{A}}1.\end{eqnarray} \tag{ 25 }$$

Since the direction of the separation velocity vector ${{\boldsymbol{V}}}_{\mathrm{sep}}$ generally differs from the direction of the orbital-velocity vector, ${{\boldsymbol{V}}}_{\mathrm{frg}}$, at the time of fragmentation, it is always V_sep = $| {{\boldsymbol{V}}}_{\mathrm{sep}}| \geqslant {\rm{\Delta }}V$. If θ is an angle between the two vectors, then $\mathrm{cos}\theta ={\rm{\Delta }}V/{V}_{{\rm{sep}}}$ and a statistically averaged angle $\langle \theta \rangle$ is given by integrating over a hemisphere centered on the direction of the orbital-velocity vector,

$$\begin{eqnarray}&&\displaystyle \frac{\pi }{2}\;\langle \mathrm{cos}\theta \rangle ={\displaystyle \int }_{0}^{\tfrac{\pi }{2}}\mathrm{sin}\theta \mathrm{cos}\theta \;d\theta =\displaystyle \frac{1}{2},\end{eqnarray} \tag{ 26 }$$

so that $\langle \mathrm{cos}\theta \rangle =1/\pi$ and a statistically averaged separation velocity $\langle {V}_{{\rm{sep}}}\rangle =\pi {\rm{\Delta }}V$.

Fragmentation events are most likely to occur at, but are not limited to, times not too distant from the perihelion time. Considering, for example, that the parent comet of C/1988 A1, C/1996 Q1, and C/2015 F3 split at a time of up to 1 yr around perihelion, equivalent to heliocentric distances of less than ∼5 au, an overall range of implied velocities ${\rm{\Delta }}V$ for the two companions is from ∼0.1 to ∼0.7 m s⁻¹, and a range for the separation velocities V_sep is from ∼0.3 to ∼2 m s⁻¹, in perfect harmony with expectation (Section 2.1.1).

2.2.2. Comparison of C/1988 A1 and C/1996 Q1 in Terms of Their Physical Properties

The manifest sudden termination of activity of comet C/1996 Q1 about 16 days before perihelion is independently documented by the water production data listed in Table 5. The numbers also show that the production of water from C/1996 Q1 was apparently stalling for at least 10 days before its termination, that 10 days before perihelion the water production of C/1988 A1 was more than 10 times as high as that of C/1996 Q1, and that 43 days after perihelion C/1988 A1 produced about as much water as C/1996 Q1 19 days before perihelion.

Table 5.  Water Production Rates from Comets C/1988 A1 (Liller) and C/1996 Q1 (Tabur)

	Time from	Distance	Production		Active
	Perihelion	from Sun	Rate of H2O	Method	Areab
Comet	(days)	(au)	(105 g s−1)	Useda	(km2)	Reference
C/1988 A1	−10c	0.86	35	OH(radio)	5.5	Crovisier et al. (2002)
	+40	1.12	13.7	OH(UVB)	3.8	A'Hearn et al. (1995)
C/1996 Q1	−30d	1.01	16	OH(radio)	3.5	Crovisier et al. (2002)
	−28	0.99	10	H2O+(red)	2.1	Jockers et al. (1999)
	−26e	0.97	14	OH(radio)	2.8	Crovisier et al. (2002)
	−23f	0.94	11	OH(radio)	2.1	Crovisier et al. (2002)
	−19	0.92	12.6	H(EUV)	2.3	Mäkinen et al. (2001)
	−10g	0.86	<3	OH(radio)	<0.5	Crovisier et al. (2002)

Notes.

^aWater production rate assumed 1.1 times hydroxyl productioni rate; OH(radio) = from observation of hydroxyl emission at 18 cm with a radiotelescope at Nançay; OH(UVB) = from narrowband photometry of hydroxyl emission at 3085 Å; H₂O⁺(red) = from column density maps of ionized water emission at 6153 Å; H(EUV) = from maps of Lyα emission of atomic hydrogen at 1216 Å taken with the SWAN imager on board SOHO. ^bEffective sublimation area, when the Sun is at zenith. ^cAverage from data taken between 1988 March 17 and 24. ^dAverage from data taken between 1996 October 2 and 5. ^eAverage from data taken between 1996 October 7 and 9. ^fAverage from data taken between 1996 October 10 and 12. ^gAverage from data taken between 1996 October 20 and 27.

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The image of C/1996 Q1 in Figure 2 is one of three that Fulle et al. (1998) used to study the comet's dust tail. The employed method (Fulle 1989) allowed them to determine, among others, the production rate of dust as a function of time. For a bulk density of dust particles of 1 g cm⁻³ and a product of their geometric albedo and phase function (with the phase angles of 65°–75° at the imaging times) of 0.02, Fulle et al.'s effort resulted in four similar dust production curves, one of which is plotted in Figure 4. Because the employed method has a tendency to smooth short-term variations, minor outbursts are suppressed. This is the reason why the curve in the figure does not show the steep increase in the amount of dust in the comet's atmosphere around September 14, 50 days before perihelion (an event that was called attention to by Lara et al. 2001), and the terminal outburst 1 month later shows up as a relatively small bulge.

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Comparison with the water production data in Figure 4 suggests a water-to-dust mass production rate ratio of ∼15, so that C/1996 Q1 was an exceptionally dust-poor comet. Integrating the dust production curve from 100 days before perihelion to 25 days after perihelion, Fulle et al. (1998) determined that over this span of time the comet lost $1.2\times {10}^{12}$ g of dust. An exponential extrapolation over the rest of the orbit (outlined by the dashed curve in Figure 4) does not increase the mass to more than 1.4 ×  10¹² g. Assumptions of an overall constant water-to-dust mass production rate ratio (15:1) and of water dominance among volatile species (see Womack & Suswal's (1996) unsuccessful search for carbon monoxide) led to a conservative lower limit of $2\times {10}^{13}$ g for this comet's total mass in the case of complete disintegration. Possible survival of a limited number of boulder-sized fragments cannot significantly affect this estimate.

Fulle et al. (1998) alleged that the nucleus of comet C/1996 Q1 did not fragment and that its activity ceased as a result of mantle formation, seasonal variations, or depletion of volatile species, including water. These authors argued that the fairly high water production rate in the first half of October, a month before perihelion (see Table 5), suggested that the nuclear diameter was greater than 0.7 km. We remark that the implied bulk density would then have been close to 0.1 g cm⁻³. Contrary to Fulle et al. (1998), Wyckoff et al. (2000) concluded from their observations with large telescopes that the nucleus probably began to fragment around October 7, some 4 weeks before perihelion, and that during the next several weeks it completely dissipated.

Because the same method of dust-tail analysis was also applied to C/1988 A1 (Fulle et al. 1992), it is possible to compare the two genetically related comets in terms of dust production. The images of C/1988 A1 selected for this study were taken with a 106 cm Cassegrain telescope at the Hoher List Observatory in 1988 mid-May (Rauer & Jockers 1990), and the adopted product of the geometric albedo and the phase function was 0.06, that is, three times higher than for C/1996 Q1, while the phase angle was 50°. Normalizing to the same geometric albedo, the results show for C/1988 A1 a dust mass loss of 1.4 × 10¹⁴ g integrated between 200 days before perihelion and 40 days after perihelion; the total loss of dust over one revolution about the Sun can be estimated at more than $2\times {10}^{14}$ g. Keeping in mind the factor of three, comparison of the dust production curve with the water production 10 days before perihelion (Table 5) suggests a water-to-dust mass production rate ratio of about 0.2. Similarly, at the time of the postperihelion water production data point in the table, the ratio is about 0.3. Thus, with grains as dark as in C/1996 Q1, C/1988 A1 was a truly dust-rich comet, about 60 times dustier than C/1996 Q1, and its total loss of mass per revolution was near 3 × 10¹⁴ g, some 15 times higher than C/1996 Q1's.

It should be remarked that a strikingly different gas-to-dust production ratio for the two genetically related comets was noticed by Kawakita et al. (1997) after they compared the results of their spectroscopic observations of C/1996 Q1, made between 1996 September 14 and October 16, with the results for C/1988 A1 published earlier by Baratta et al. (1989) and by A'Hearn et al. (1995). Because the data on the water and dust production of C/1996 Q1 (Fulle et al. 1998; Mäkinen et al. 2001; Crovisier et al. 2002) were not yet available, Kawakita et al. could determine this comet's gas-to-dust ratio only very approximately. From a continuum flux at 5200–5300 Å in their spectra they calculated the dust production's Afρ proxy (A'Hearn et al. 1984) and from ${{\rm{C}}}_{2}({\rm{\Delta }}v=0$) band the production rate $Q({{\rm{C}}}_{2})$. They found that in the same range of heliocentric distance the ratio of $Q({{\rm{C}}}_{2})/{\text{}}{Af}\rho$ for C/1996 Q1 was four times greater than for C/1988 A1. Employing Fink & DiSanti's (1990) technique for deriving the water production rate from the intensity of the forbidden [O i]${}^{1}D$ line of atomic oxygen at 6300 Å, Kawakita et al. computed the ratio $Q({{\rm{H}}}_{2}{\rm{O}})/{\text{}}{Af}\rho$ for C/1996 Q1 at the same heliocentric distance at which it was known for C/1988 A1 from A'Hearn et al. (1995) and found that for C/1996 Q1 the ratio came out to be ∼15 times higher; they concluded that the nucleus of the parent comet must have been strongly inhomogeneous.⁸

The issue of whether the uneven dust content in the two comets correlates with the chemical composition of volatile species was addressed by Turner & Smith (1999). Comparing their own results for C/1996 Q1 from 1996 September 19–20 with those by A'Hearn et al. (1995) for C/1988 A1, they showed that the production ratios C₂/CN, C₃/CN, and NH/CN were practically identical. The only exception was the abundance of OH relative to these molecules, which in C/1996 Q1 was an order of magnitude lower. Turner & Smith admitted that their OH results, derived from the observations of the (0–0) band at 3085 Å, may have been burdened by large errors.⁹ Ignoring this discrepancy, the authors arrived at a conclusion that their results are consistent with a homogeneous composition for the ices in the nucleus of the parent comet.

Mineralogical comparison of the dust content in the comets C/1988 A1 and C/1996 Q1 is unfortunately not possible because the necessary data are not available for the first object. Examining the spectrophotometric data for C/1996 Q1 in the thermal infrared between 7.6 and 13.2 μm, Harker et al. (1999) found that a mineralogical model that best fits the spectra obtained on October 8–10, more than 3 weeks before perihelion, is a mix of grains of crystalline olivine 2 μm in diameter and amorphous pyroxene 6 μm in diameter. On the other hand, Turner & Smith (1999) allowed for the possible existence of gradually destroyed grains containing water ice in order to explain the blue color of the continuum in the optical spectrum at small distances from the nucleus on its sunward side.

It is too early to compare C/1988 A1 and C/1996 Q1 with C/2015 F3. However, the very fact that this new member of the group was first spotted owing to its Lyα images of atomic hydrogen but appeared fairly faint and with no prominent dust tail to the ground observers suggests that, like C/1996 Q1, this comet was most probably water-rich and dust-poor.

An important point is raised by Campbell's (1921d) statement that the Lick object was brighter than Venus in the same position and circumstances. Since any small elongation, E, from the Sun is reached by Venus near either the superior or inferior conjunction, we consulted the phase function of the planet to derive its apparent brightness for those two scenarios. With E = 33 and Earth's distance from the Sun of r_⊕ = 1.01387 au at the time, Venus's phase angle at its mean heliocentric distance would have been $4.\!\!{}^\circ 7$ near the superior conjunction and 1753 near the inferior conjunction.

A history of the planet's phase function investigations from a standpoint of The Astronomical Almanac's needs was summarized by Hilton (2005). A recently updated phase curve, the only one that accounts for forward-scattering effects of sulfuric-acid droplets in the planet's upper atmosphere and covers all angles from 2° to 179°, was published by Mallama et al. (2006). Based on their results, we find that at $E=3.\!\!{}^\circ 3$ the V magnitude of Venus is −3.9 and −3.7 near the superior and inferior conjunction, respectively. Incorporating a minor color correction to convert the V magnitude to the visual magnitude (e.g., Howarth & Bailey 1980) and interpreting Campbell et al.'s statement to imply that the Lick object was at least ∼0.5 mag brighter than Venus would have been, we adopted that the object's apparent visual magnitude was about ${H}_{{\rm{app}}}=-4.3$ or possibly brighter.

Campbell et al.'s remarks on the Lick object allow us to examine the constraints on its magnitude H₀, normalized to unit heliocentric and geocentric distances (sometimes called an absolute magnitude), as a function of the unknown heliocentric distance r (in au), on the assumption that the object's brightness varied with heliocentric distance as r⁻ⁿ,

$$\begin{eqnarray}&&{H}_{0}=-4.3-5\mathrm{log}{\rm{\Delta }}-2.5\;n\mathrm{log}r-2.5\mathrm{log}{\rm{\Phi }}(\varphi ),\end{eqnarray} \tag{ 27 }$$

where n is a constant and Δ, φ, and ${\rm{\Phi }}(\varphi )$ are, respectively, the object's geocentric distance (in au), phase angle, and phase function. The distances r and Δ are constrained by the observed elongation angle E,

$$\begin{eqnarray}&&{r}^{2}={r}_{\oplus }^{2}+{{\rm{\Delta }}}^{2}-2{r}_{\oplus }{\rm{\Delta }}\mathrm{cos}E,\end{eqnarray} \tag{ 28 }$$

where ${R}_{\oplus }$ is again Earth's heliocentric distance at the time of Campbell et al.'s observation. Because the phase function depends on the object's unknown dust content and the small elongation implies strong effects of forward scattering by microscopic dust at the object's geocentric distances ${\rm{\Delta }}\lesssim 0.95$ au, we constrained the phase function by employing two extreme scenarios: dust-free and dust-rich. In the first case we put ${\rm{\Phi }}(\varphi )\equiv 1$ at all phase angles, while in the second case we approximated ${\rm{\Phi }}(\varphi )$ with the Henyey–Greenstein law as modified by Marcus (2007) but normalized to a zero phase angle $[{\rm{\Phi }}(0^\circ )=1]$ rather than to 90°.

We varied Δ from 0.001 to 2 au to obtain a plot of the absolute magnitude H₀ against r in Figure 5 for three values of the exponent n; a dust-free case is on the left, a dust-rich case on the right. Although the models differ from one another enormously at ${\rm{\Delta }}\lt 1$ au, some properties of the plotted curves are independent, or nearly independent, of the phase law.

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Rather odd looking in Figure 5 are the magnitude curves for n = 2, typical of dynamically new comets arriving from the Oort Cloud (Whipple 1978), which generally imply an unacceptably high intrinsic brightness of the Lick object. In the dust-free scenario the solutions are way off, while in the dust-rich model the acceptable options imply a very small geocentric distance ${\rm{\Delta }}\lt 0.2$ au. But such scenarios are ruled out by (i) being inconsistent with Campbell et al.'s observation of the object's stellar appearance, (ii) failing to explain why the object was not observed systematically for a longer period of time around August 7 (at an expected rate of motion of up to several degrees per day), and (iii) explicitly contradicting Campbell's failure to find it on the following days at the times of sunrise and/or sunset. It is therefore highly unlikely that the Lick object was a dynamically new comet and it was not near Earth.

Campbell's (1921d) unsuccessful search for the object on August 8–9 (Mount Hamilton time) suggests that by then it may have disintegrated. If it was a member of a comet pair and not a dynamically new comet, it should, like the members of the C/1988 A1 group, belong to Whipple's (1978) class III (with an orbital period on the order of thousands of years), for which Whipple found an average $\langle n\rangle =4.7\pm 0.8$ before perihelion. Not unexpectedly, the preperihelion light curves of C/1988 A1 and C/1996 Q1 (up to the onset of terminal fading) in Figure 3 show that they, too, are fairly consistent with n = 4 or slightly higher. The magnitude curves for n = 4 in Figure 5 are thus expected to approximate class III comets more closely than the other curves. The possibility that the Lick object disintegrated near perihelion, just as did C/1996 Q1, suggests that it could be a companion to the pair's more massive member, which was likely to have arrived at perihelion long before the Lick object. Regardless of the dust content, Figure 5 shows that the minimum on the magnitude curves for n = 4 is ${({H}_{0})}_{{\rm{min}}}\simeq 8$. Accordingly, the Lick object should have had a small perihelion distance, $q\lesssim 0.1$ au, and the small elongation reflected its true proximity to the Sun. Since the minimum absolute magnitude varies with n approximately as ${({H}_{0})}_{{\rm{min}}}=3.09\;n-4.3$, it equals ∼10 for $n=4.7$. We remark that C/1996 Q1 had ${H}_{0}\simeq 6.6$ before fading (Figure 3), being intrinsically about 0.7 mag fainter than C/1998 A1 at ∼1 au from the Sun.

The curve for n = 7 is presented in Figure 5 as an example of an unusually steep light curve—displayed, e.g., by the sungrazing comet C/2011 W3 (Sekanina & Chodas 2012)—and to illustrate a degree of nonlinearity in the plot of the Lick object's H₀ magnitude. Figure 3 shows that the r⁻⁷ law is not consistent with the preperihelion light curve of either C/1988 A1 or C/1996 Q1 and that it could be a plausible match only to the postperihelion light curve of C/2015 F3.

To learn more about the Lick object's possible disintegration, we recall that a failure of long-period comets to survive perihelion was investigated, among others, by Bortle (1991). For such comets with perihelion distances smaller than 0.51 au he found that the chance of disintegration depends on the intrinsic brightness. By fitting the same r⁻⁴ law to the light curves of the numerous comets that he examined, Bortle estimated that there was a 70% probability of disintegration for the comets that were intrinsically fainter than the absolute magnitude ${H}_{10}^{{\rm{surv}}}$ (equivalent in our notation to H₀ for n = 4 and ${\rm{\Phi }}=1$). This survival limit is, according to Bortle, related to the perihelion distance q (in au) by ${H}_{10}^{{\rm{surv}}}=7+6\;q$. It is plotted in Figure 5 against r, thus assuming the equality between the plotted r and the perihelion distance. For $r\gt q$, H₁₀ lies along a horizontal line that passes through the Bortle curve at r = q. The Bortle limit suggests that comets intrinsically brighter than ${H}_{10}=7$ have an increasingly higher probablility to survive, even if their perihelia are within, say, 0.1 au of the Sun.

On certain assumptions, the Bortle limit can readily be extended to photometric laws ${r}^{-n}$ with $n\ne 4$. For example, if an orbital arc covered by the magnitude observations, employed to determine the absolute magnitude, is centered on a heliocentric distance ${r}_{{\rm{aver}}}$ for which $\mathrm{log}{r}_{{\rm{aver}}}=\mathrm{log}\;(q+0.5)$, the survival limit ${H}_{2.5n}^{{\rm{surv}}}$ is related to ${H}_{10}^{{\rm{surv}}}$ by ${H}_{2.5n}^{{\rm{surv}}}={H}_{10}^{{\rm{surv}}}\;+(10-2.5n)\mathrm{log}\;(q+0.5)$. In Figure 5 we display, as a dashed curve, a survival limit ${H}_{17.5}^{{\rm{surv}}}$ for the magnitude curve of n = 7. For perihelion distances smaller than 0.5 au, ${H}_{17.5}^{{\rm{surv}}}$ is fainter than the ${H}_{10}^{{\rm{surv}}}$ and its variation with the perihelion distance is much flatter.

While the minimum on the magnitude curve provides a lower bound to the Lick object's absolute brightness, the Bortle limit offers a sort of an upper bound, if the object's disintegration is considered as very likely. Figure 5 shows that for n = 4 the two bounds almost coincide, leaving a narrow allowed range from 8.1 to 7.3. In the case of a dust-poor comet, such as was C/1996 Q1 (Section 2.2.2), the magnitude curve for n = 4 in Figure 5 suggests that the Lick object's perihelion distance did not exceed 0.07 au, while the same curve in the case of a dust-rich comet allows a perihelion distance of up to 0.20 au. For the unlikely magnitude curves with n = 7 the limits on the perihelion distance would be, respectively, 0.17 and 0.46 au.

Given that the above bounds are not absolute, a search for the Lick object's pair member should allow for some leeway. However, it ought to be pointed out that if the apparent brightness assigned to the object, based on Campbell's comparison to Venus, should be increased to, say, magnitude −4.5 or even −5, the curves in Figure 5 would move up and further reduce a range of conforming perihelion distances, compensating for any intrinsic brightness increase caused by softening the Bortle survival limit's constraint.

There is circumstantial evidence, to be discussed below, that the Lick object's brightness, as observed by Campbell et al. at sunset on August 7, was a result of more complex temporal variations than a simple power law of heliocentric distance can explain. If this indeed was so, our conclusion that, based on Campbell et al.'s account, the object was truly in close proximity of the Sun (and not only in projection onto the plane of the sky) should further be strengthened.

The Lick object's observations by Nelson Day (Markwick 1921), on August 7, 8.5 hr before Campbell et al.'s sighting, by Emmert (1921a, 1921b), on August 6, another 20 hr earlier, and possibly by others, were made with the naked eye in broad daylight, with the Sun a number of degrees above the horizon. This kind of visual observation has both advantages and disadvantages over twilight observations, such as that made by Fellows (1921a, 1921b). An obvious advantage is that the angular distance of the object from the Sun is readily observed. A disadvantage that has implications for the Lick object's physical behavior is a potentially enormous uncertainty in the estimated brightness.

In Section 3 we already hinted that Nelson Day's report of the Lick object's apparent magnitude of −2 at 4° from the Sun on August 7 must be grossly underrated. The issue of daylight visual-magnitude estimates of bright celestial objects was addressed by Bortle (1985, 1997), who presented a formula for a limiting magnitude of an object just detectable with 8 cm binoculars as a function of its elongation from the Sun. At 4° from the Sun one could observe with this instrument an object as faint as −1.3. Furthermore, on a Web site Bortle posted a list of threshold visual magnitudes for daytime observations with the unaided eye.¹³ From this document it follows, for example, that an object of magnitude −4.0 is just visible with the unaided eye if more than 5° from the Sun, while an object of −5.5 or somewhat brighter is rather easily visible when the Sun is blocked. Since, in reference to the Ferndown observation, Markwick (1921) conveyed that the Lick object looked "striking," it must have been at this time quite a bit brighter than magnitude −4, possibly even brighter than −5.

Essentially the same constraint applies to Emmert's sighting on August 6. Emmert estimated the Lick object's brightness by asserting that it "was fully as bright as Venus in twilight at her greatest brilliancy" (Emmert 1921a),¹⁴ while elsewhere he claimed that it "shone altogether too bright for Venus" (Emmert 1921b). The two statements are not incompatible, as the latter refers implicitly to the planet's expected brightness (fainter than −4; Section 4.1) at the time of observation; the former statement implies a magnitude just shy of −5.

We cannot rule out that the Lick object was getting fainter as it was approaching the Sun, which brings to mind the instances of post-outburst terminal fading of disintegrating comets, including C/1996 Q1 (Figure 3). It is possible that both Emmert and Nelson Day saw the Lick object during its final outburst, while Campbell's (1921d) observation was made along the outburst's subsiding branch and his unsuccessful search was conducted on the following days after the event was over.

We were now ready to undertake a search for a comet (or comets), whose orbit could serve as a basis for fitting the limited data on the Lick object, as described in Section 3. Performed in Equinox J2000, the search rested on the following set of diagnostic tests of the evidence:

• 1.  
  The most credible data were provided by Campbell (1921d): the object's position for 1921 August 8.1347 UT, R.A. (J2000) = 9^h275, decl. (J2000) = +15°04', with a probable error of ±10' (see footnote ¹⁰).
• 2.  
  The observation by Nelson Day on 1921 August 7.7806 UT (Markwick 1921), placing the object 4° from the Sun, was judged less reliable than Campbell's and was employed as a second, less diagnostic test, with an estimated uncertainty of ±30'. The somewhat cryptic depiction of the object–Sun orientation (Section 3) suggests, if correctly interpreted, the object's equatorial coordinates of R.A. (J2000) = 9^h296, decl.(J2000) = +16°29', with an estimated uncertainty of ±30' or more.
• 3.  
  Emmert's (1921a, 1921b) observation was judged to furnish information that was much less credible than Campbell's and less credible than Nelson Day's. We used the object's reported angular distance from the Sun, 5° on 1921 August 6.9514 UT, as the only piece of data worth testing; its error was estimated at ±1° at best.To explain the Lick object's sudden appearance, disappearance, and other circumstances, it was judged helpful if a genetically related comet and its orbit satisfied four additional constraints.
• 4.  
  The most likely reason for the object's sudden appearance was thought to be its unfavorable trajectory in the sky during its long approach to perihelion: from behind the Sun, along a narrow orbit that would entail a small elongation from the Sun over an extended period of many weeks before near-perihelion discovery.
• 5.  
  A very small perihelion distance, q < 0.1 au, was preferred because it would most straightforwardly account for the object's brief brightening, thus aiding the argument in the previous point. It would also expose the object to a highly perilous environment near perihelion, thus increasing the chances of its disintegration and thereby its abrupt disappearance (Section 4.1).
• 6.  
  Since in comet pairs and groups the chance of near-perihelion demise was limited to companions (Section 2), which typically arrived at perihelion much later than the primary members, one would expect that the Lick object was a companion to the pair's primary and, accordingly, that the primary would more probably have arrived at perihelion well before 1921.
• 7.  
  Expressed by Equation (2), the correlation between (i) a velocity at which the fragments begin to separate after their parent's breakup near perihelion and (ii) a temporal gap between their observed arrivals to perihelion one revolution later, favors, for a typical gap of dozens of years, fragments of a comet with an orbital period on the order of thousands of years. Thus, such a comet is more likely to be genetically related to the Lick object than a comet with a shorter or longer orbital period; we address the preferred range for our test purposes in Section 5.

Having described the tests and constraints that the comets potentially associated with the Lick object were expected to satisfy, we next compiled an initial list of selected candidates. We allowed a wide range of orbital periods, P, by letting the statistically averaged separation velocity, $\langle {V}_{{\rm{sep}}}\rangle$ (for the definition, see the text below Equation (26)), vary from 0.1 to 3 m s⁻¹; the difference in the orbital period, ΔP, between the primary and the companion, to range from 3 to 300 yr; and the heliocentric distance at fragmentation, r_frg, to span from  0.01 to 10 au. Expressing ΔV from Equation (2) in terms of $\langle {V}_{{\rm{sep}}}\rangle$, the orbital periods range from 100 to 100,000 yr. Requiring conservatively that the perihelion distance not exceed ∼0.25 au, we assembled all comets from Marsden & Williams's (2008) catalog that satisfied these conditions (the orbital period as derived from the original semimajor axis; for C/1769 P1 computed from an osculating value by R. Kracht), and with orbits of adequate precision (a minimum of five decimals in q and three in the angular elements). For the period of 2008–2015 we consulted the JPL Small-Body Database Search Engine.¹⁵

The candidate comets are presented in Table 7. In the Marsden–Williams catalog we found merely 14 objects that satisfied our selection constraints, including C/1917 F1 (Mellish), a parent body of an extensive meteor-stream complex (e.g., Neslušan & Hajduková 2014). We found no candidate comets from 2005 to 2015. Since comets with an orbital period between 100 and 100,000 yr may easily appear among those for which only a parabolic approximation is available, we extended the set of candidates by adding small-perihelion comets with parabolic orbits. Even though we argued that the comets before 1921 should be the more attractive candidates (see point 6 above), we allowed the comets that arrived after 1921 to be subjected to the same tests. We ended up with a total of 50 candidate comets.

Close examination of the history of this comet's orbit determination shows that the study by Holetschek (1891, 1892), whose "best" orbital set was cataloged by Marsden & Williams (2008) and used by us, was preceded by orbital investigations of Vogel (1852a, 1852b), B. Peirce (Kendall 1843), and Pingré (1784).¹⁶ There are very dramatic differences among the computed orbits, which are traced to different interpretations of the crude observations. Although the comet was discovered about a week before perihelion (Kronk 1999, p. 380) and was later observed also on the island of Ternate, Indonesia (Pingré 1784), and in China (Struyck 1740; Pingré 1784; Hasegawa & Nakano 2001), its approximate positions could only be constructed from the accounts reported on four mornings after perihelion by the observers at Pondicherry, then a French territory in India (Richaud 1729), and at Malacca, Malaysia (de Bèze & Comilh 1729).¹⁷ The difficulties encountered in an effort to determine this comet's orbit were befittingly summarized by Plummer (1892), including the intricacies that Holetschek (1891, 1892) got involved with in order to accommodate a constraint based on a remark by de Bèze and Comilh that the comet reached a peak rate of more than 3° per day in apparent motion between 1689 December 14 and 15. Unable to find an orbital solution that would satisfy this constraint, Holetschek himself eventually questioned the report's credibility, considering his preferred orbital solution (based on the adopted positions from December 10, 14, and 23 and on some experimentation with the observation time on the 10th) as a compromise.

Once the condition of a peak rate of apparent motion is dropped, there is no longer any reason to prefer this compromise solution over other available orbital sets. Accordingly, we examined whether or not the orbits by Pingré (1784), Peirce (Kendall 1843; Cooper 1852), and Vogel (1852a, 1852b) and the alternative orbits by Holetschek (1891) himself matched Campbell's position for the Lick object as closely as did the cataloged orbit for C/1689 X1.

The two alternative orbits by Holetschek (1891) that we considered are compared to the cataloged orbit in Table 10. The first set of elements, referred to as "Hol I" in Table 10, was a key solution that Holetschek derived before he began a complicated data manipulation. The second set, referred to as "Hol II" in Table 10, is one of three considered by him—in the course of another effort to accommodate the high rate of apparent motion—to avoid the tendency of the computed path to have strayed to the northern hemisphere before perihelion, and the only of the three to have the perihelion distance smaller than 0.25 au (thus complying with our selection rules). It is somewhat peculiar that Holetschek's orbit in the Marsden–Williams catalog is the only set of elements (among the six we consider for C/1689 X1) that shows the comet's motion to be direct; the rest are all retrograde, with the inclinations between 110° and 150°.

Table 10.  Sets of Parabolic Orbital Elements of C/1689 X1 Derived by J. Holetschek (Equinox J2000)

Quantity/Orbital Element	Catalog	Hol I	Hol II
Perihelion time tπ(1689 Nov UT)	30.659	30.545	28.461
Argument of perihelion ω	78134	163857	143535
Longitude of ascending node Ω	283754	67841	51682
Orbit inclination i	63204	143211	135308
Perihelion distance q (au)	0.06443	0.09281	0.16673

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In Table 11, organized in the same fashion as Table 8, the various orbits for C/1689 X1 are compared in terms of the quality of matching the Lick object's position published by Campbell. The table shows that owing to enormous uncertainties in the orbit determination, the very small offset in Figure 6 and Table 8 from the cataloged set of elements must be judged as purely fortuitous. Accordingly, a genetic relationship between the Lick object and C/1689 X1 is, on account of the orbital uncertainties, entirely indeterminable. Arguments that tests 3 and 4 in Table 9 are unfavorable to the relationship are—while correct—under these circumstances unnecessary.

Table 11.  Orbits of C/1689 X1 Emulating Campbell's Position for the Lick Object

	Offset		Time Diff.	Distance (au) to
Computer/			${t}_{{\rm{obs}}}-{t}_{\pi }$			Phase
Designation	Dist.	P.A.	(days)	Sun	Earth	Angle (deg)
Catalog	55	76°	−1.285	0.1050	0.9263	144.8
Hol I	85.4	56	−0.098	0.0930	0.9596	123.5
Hol II	371.6	50	−0.517	0.1682	0.9257	117.3
Vogel	152.3	272	+0.081	0.0211	1.0060	111.2
Peirce	152.1	282	+0.279	0.0386	1.0493	22.9
Pingré	167.0	270	+0.989	0.0948	1.1065	11.5

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This was an intrinsically bright comet. On the assumption that its brightness, normalized to a unit geocentric distance and corrected for a phase effect according to the law by Marcus (2007), varied with heliocentric distance as r⁻ⁿ, a least-squares solution to several magnitude estimates collected by Holetschek (1913) yields an absolute magnitude of ${H}_{0}=3.70\pm 0.22$ and n = 3.62 ± 0.43. At first sight it is surprising that the comet was not discovered before perihelion, but a straightforward explanation is provided by Figure 7, which shows that the approach to perihelion was from the high southern declinations: the culprit was the lack of comet discoverers in the Southern Hemisphere in the early nineteenth century. For 2 months, from the beginning of 1823 September to the beginning of November, the comet was nearly stationary relative to the Sun, at elongations between 55° and 62° and at heliocentric distances from 2.2 to 1.1 au.

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The comet was poorly placed for observation in an early postperihelion period as well, as it took 3 weeks before its elongation increased to 40°, at which time it was discovered as a naked-eye object¹⁸ independently by several observers.

The most comprehensive orbit for this comet to date was computed by Hnatek (1912) from hundreds of astrometric observations made between 1823 December 30 and 1824 March 31. The perturbations by five planets, Venus through Saturn, were accounted for. An unusual feature of Hnatek's study, marring its scientific value, was that while he computed both the most probable parabola and the most probable ellipse (with an osculation orbital period of 9764 yr) in the process of orbit refinement, he presented only a parabolic solution as the comet's final orbit, dropping the last normal place (because of large residuals) and failing to demonstrate that the error of the eccentricity exceeded its deviation from unity.

Before deciding whether or not to redetermine the orbit with increased attention to its possible ellipticity, we noticed numerous accounts in the literature of the comet's unusual apperance over a period from 1824 January 22 through 31, around the time of Earth's transit across the comet's orbit plane on January 24.0 UT. At least five observers (Hansen 1824; Harding 1824; Olbers 1824b; von Biela 1824; Gambart 1825) reported that the comet displayed, next to its ordinary tail in the antisolar direction, a second extension that pointed sunward and was referred to by both Harding and Olbers as an anomalous tail. Exhibited since by many other comets, most memorably by C/1956 R1 (Arend–Roland) and C/1973 E1 (Kohoutek), this type of comet-dust feature is nowadays customarily known as an antitail, and its dynamics, governed by the conservation-of-orbital-momentum law, and appearance, controlled in large part by conditions of the projection onto the plane of the sky, are well understood (for a review, see Sekanina et al. 2001 and the references therein). The appearance of an antitail is clearly a signature of a comet rich in large-sized dust, but it is not diagnostic of its orbital category. The statistics show that both dynamically new comets, arriving from the Oort Cloud, and dynamically old comets (with the orbital periods on the order of thousands to tens of thousands of years) did show an antitail in the past. Indeed, as Marsden et al. (1973, 1978) showed, both C/1956 R1 and C/1973 E1 were dynamically new comets, as were several additional objects, e.g., C/1895 W1 (Perrine), C/1937 C1 (Whipple), C/1954 O1 (Vozárová), or, quite recently, C/2011 L4 (PANSTARRS), while other antitail-displaying objects, such as C/1844 Y1, C/1961 O1 (Wilson–Hubbard), C/1969 T1 (Tago–Sato–Kosaka), or, recently, C/2009 P1 (Garradd) and C/2014 Q2 (Lovejoy), were not dynamically new comets. As part of a broader antitail investigation, the feature in C/1823 Y1 was investigated by Sekanina (1976) and found to consist of submillimeter-sized and larger dust grains released from the nucleus before perihelion as far from the Sun as Jupiter, if not farther away still.

Given the orbit ambivalence in the antitail statistics, we decided to go ahead with the orbit redetermination, in order to ascertain whether C/1823 Y1 is in compliance with test 7 of Table 9. Employing the code EXORB7 developed by A. Vitagliano, the second author performed the computations, using selected consistent astrometric observations made between 1824 January 2 and March 31, which were re-reduced with the help of comparison-star positions from the Hipparcos and Tycho-2 catalogs. The results are described in the Appendix; here we only remark that the original semimajor axis came out to be ${(1/a)}_{{\rm{orig}}}$ = +0.006077 ± 0.000471 (au)⁻¹, implying a true orbital period of ∼2100 yr and demonstrating that the comet had not arrived from the Oort Cloud.

Having determined that C/1823 Y1 moved in an orbit that did not rule out its potential association with the Lick object (test 7), we next examined the comet's compliance with the other constraints summarized in Table 9. A parameter that is a little outside the proper range is the perihelion distance (test 5). The Lick object's motion along the orbit of C/1823 Y1 should have two implications for the observing geometry at the time of Campbell et al.'s observation, as seen from Table 8: (i) the object would have been only 0.78 au from Earth, and (ii) its apparent brightness should have been strongly affected by forward scattering, if the object was as dust-rich as C/1823 Y1. We reckon that for an r⁻⁴ law of variation, the Lick object would have appeared to Campbell et al. ∼11.5 mag brighter than was its absolute magnitude H₀. With an estimated apparent visual magnitude of −4.3, H₀ ≈ 7.2. With an r⁻⁵ law, the absolute brightness would drop to H₀ ≈ 8.7. If the Lick object was caught by Campbell et al. in the midst of the terminal outburst's subsiding branch, its absolute brightness in quiescent phase could have been fainter still.

While we find that the range of absolute magnitude is plausible for a companion of an intrinsically bright long-period comet, this model for the Lick object has difficulties that involve other tests besides test 5. The emulated trajectory for the period of August 6–8 predicts that the object should have moved from the southwest to the northeast (Figure 7), passing closest to the Sun, at an elongation of 316, on August 7.867 UT, about 2 hr after Nelson Day's observation. As a result, the object's modeled elongation was increasing with time when Campbell et al. detected it. While this by itself is not contradicted by the observation (because the direction of the object's motion relative to the Sun during the several minutes of monitoring at Lick could not be established), the predicted elongation of 32 at the time of the Ferndown observation is not consistent with the observed 4° elongation and its estimated uncertainty of ±30' (test 2 in Table 9).

The Lick object's predicted motion after August 7 suggests that the day-to-day observing conditions were improving at sunset, but remained poor at sunrise. Of the three occasions on which Campbell failed to recover the object, the best opportunity presented itself unquestionably at sunset on August 8, when the predicted location was about 53 from the Sun in a position angle of ∼70°; the object would then still have had some 30 hr to go to perihelion. This hypothesis would thus indicate that the object disintegrated more than 30 hr before perihelion.

Finally, we examine the implications of the nearly 100 yr gap between the perihelion times of C/1823 Y1 and the Lick object for the separation velocity at the time of their presumed common parent's fragmentation. With the orbital period of 2110 ± 260 yr (see Table 13 in the Appendix), the relation between the statistically averaged separation velocity (governed by Equation (26)), $\langle {V}_{{\rm{sep}}}\rangle$ (in m s⁻¹), and the heliocentric distance at separation, r_frg (in au), is

$$\begin{eqnarray}&&\langle {V}_{{\rm{sep}}}\rangle ={6.2}_{-1.1}^{+1.5}\;{r}_{{\rm{frg}}}^{\tfrac{1}{2}}.\end{eqnarray} \tag{ 29 }$$

We note that even if the parent comet split right at perihelion, the required separation velocity would be close to 3 m s⁻¹, already exceeding the plausible upper limit (Section 2.1.1); and at r_frg ≃ 2–3 AU it would reach a completely intolerable magnitude of nearly 10 m s⁻¹. The only way to avoid this contradiction is to assume that the Lick object survived not one but two (or more) revolutions about the Sun, an option that is rather unattractive. Accordingly, it appears that this scenario fails to satisfy test 6.

We summarize by noting that while the hypothesis of a genetic relationship between the Lick object and comet C/1823 Y1 passes tests 1, 3, and 7, it is, strictly, incompatible with the constraints presented by tests 2, 5, 6, and, to some degree, 4.

This is the last of the top three candidates in Table 8 and the most promising one, as argued below. Figure 8 shows that, unlike C/1823 Y1, this comet was approaching the Sun from the high northern declinations; it was discovered by Hind 52 days before perihelion (Bishop & Hind 1847), when 1.49 au from the Sun, as a faint telescopic object. The comet brightened steadily and was detected with the naked eye more than 3 weeks before perihelion (Bond 1847; Schmidt 1847a, 1847b). The discoverer observed the comet telescopically in broad daylight on 1847 March 30, several hours before perihelion, within 2° of the Sun (Hind 1847). As Figure 8 shows, the comet remained close to the Sun during the first few weeks after perihelion. As a result, there were merely three astrometric positions available along the receding branch of the orbit, made on April 22–24. Interestingly, around April 12 the projection conditions were favorable for the appearance of an antitail (Sekanina 1976); unfortunately, the comet was then only 15° from the Sun.

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Employing a total of 160 astrometric observations made between 1847 February 6 and April 24 and accounting for the perturbations by Mercury through Jupiter, Hornstein (1870) derived an elliptical orbit with a period of 10,220 ± 570 yr for the osculation epoch at perihelion, implying a true orbital period (i.e., a time interval between 1847 and the previous perihelion) of 8310 ± 570 yr (Marsden et al. 1978). To find out whether we could rely on Hornstein's computations, we derived two new sets of orbital elements for C/1847 C1 based on the observations made only in London (critical at the beginning of the orbital arc), Berlin (critical at its end), and Vienna. The choice of two different cutoffs for the residuals of rejected observations (±10'' and ±6'') yielded osculation orbital periods of, respectively, 8190 ± 930 yr and 14,100 ± 2300 yr. Interestingly, Hornstein (1854a) estimated, in his earlier study based on 145 astrometric observations from the same period of time, that the orbital period was not shorter than 8000 yr and not longer than 14,000 yr. In that early study Hornstein (1854a, 1854b) established the most probable orbital period of 10,818 yr for the osculation epoch at perihelion; the details of that paper indicate the orbital period's rms error of ±1510 yr. Even though the quality of Hornstein's final set of orbital elements was classified as only category 2B by Marsden et al. (1978), the numbers for the orbital period appear to be consistent. We conclude that Hornstein's results are credible enough to indicate that the comet's true orbital period amounted to about 8000 yr (with an uncertainty of several centuries) and was compatible with the constraints of test 7 in Table 9.

As a companion to C/1847 C1, the Lick object was at the time of Campbell et al.'s observation a little beyond the Sun, so that—unlike in the C/1823 Y1 scenario—its observed brightness was unaffected by forward scattering. On the other hand, the heliocentric distance was now much smaller, which made the object brighter. Accounting for a phase effect (Marcus 2007), we find that the object should have appeared to Campbell et al. about 10.7 mag brighter than was its absolute magnitude with an r⁻⁴ law but 13.6 mag brighter with an r⁻⁵ law. The estimated apparent magnitude of −4.3 implies for the two laws an absolute magnitude of ${H}_{0}\approx 6.4$ and 9.3, respectively. We argue that, again, this is a plausible range of absolute magnitude for a companion of an intrinsically bright long-period comet, with the bright end of the range very close to the absolute magnitude of C/1996 Q1, one of the companions to C/1988 A1 investigated in detail in Section 2.2 (Figure 3).

Because the trajectory of the Lick object emulated by C/1847 C1 runs in Figure 8 in a generally southwestern direction on August 6–8, the predicted motion brought the object almost 2° south of the Sun in close proximity of perihelion late on August 8. Viewed from the Lick Observatory, the object was still more than 2° below the horizon at sunrise and already about 2° below the horizon at sunset on August 8, so in this scenario Campbell's failure to see it was due to unfavorable geometry and not necessarily due to its disintegration. However, by sunrise on August 9 the object swayed already far enough to the west of the Sun that it rose before the Sun, with a net difference of more than 2° in elevation. Hence, Campbell's failure to see the Lick object at sunrise on August 9, the last occasion on which he was searching for it, cannot be explained by unfavorable geometry. However, the constraint on the object's disintegration time is relaxed from ∼6 hr preperihelion to ∼18 hr past perihelion.

A review of the tests in Table 9 shows that they all are satisfied by C/1847 C1. Particularly gratifying is this scenario's correspondence with the evidence, criteria, and requirements in tests 1–4. The Lick object's elongation on approach to the Sun is predicted to have dropped below 50° at a heliocentric distance of more than 2 AU and to continue decreasing with time until after perihelion.

The absolute magnitude of C/1847 C1 has not been well determined. For an r⁻⁴ law, Vsekhsvyatsky (1958) gave ${H}_{10}=6.8$, but the brightness estimates by Schmidt (1847a, 1847b) and by Bond (1847) strongly suggest that between 1.1 and 0.5 au from the Sun the comet brightened less steeply with decreasing heliocentric distance. Correcting a few available magnitude estimates for the phase effect (Marcus 2007) and the geocentric distance, we obtained by least squares ${H}_{0}=5.41\pm 0.13$ and n = 2.02 ± 0.31. We note that Holetschek (1913), assuming a brightness variation with an inverse square heliocentric distance, derived from a single estimate H₀ = 5.7, a value that, like Vseksvyatsky's, was not corrected for a phase effect. Because the phase angle varied between 55° and 90° during the relevant period of time, an average phase correction should make the absolute magnitude brighter by ∼0.4 to ∼0.7 mag, depending on the dust content.

Since Figures 1 and 3 show that companion comets have generally steeper light curves than the primaries, the difference between the derived slope for the light curve of C/1847 C1, as the pair's primary, and the assumed range of slopes (n = 4–5) for the light curve of the Lick object, as the companion to C/1847 C1, is not unrealistic. In summary, we find that even if the Lick object was not in terminal outburst when observed by Campbell et al., its absolute magnitude was probably at least 1 mag—and possibly several magnitudes—fainter than the absolute magnitude of C/1847 C1. If the object was in terminal outburst, its quiescent-phase absolute magnitude was yet another magnitude or so fainter.

The last point of our investigation of C/1847 C1 and the Lick object, as a pair of long-period comets that are likely to have split apart from a common parent near its perihelion some 8310 ± 570 yr before 1847, is their statistically averaged separation velocity $\langle {V}_{{\rm{sep}}}\rangle$ (Section 2.2.1), an issue that we focused on in discussing the fragmentation of long-period comets in Sections 2.1 and 2.2. With a temporal gap of 74.4 yr between the arrivals of C/1847 C1 and the Lick object (Table 9), the separation velocity (in m s⁻¹) is equal to

$$\begin{eqnarray}&&\langle {V}_{{\rm{sep}}}\rangle ={0.48}_{-0.04}^{+0.05}\;{r}_{{\rm{frg}}}^{\tfrac{1}{2}}.\end{eqnarray} \tag{ 30 }$$

If one assumes that the common parent of C/1847 C1 and the Lick object split right at perihelion, the fragments separated at a rate of $\langle {V}_{{\rm{sep}}}\rangle =0.1\;{\rm{m}}$ s⁻¹; if at a heliocentric distance of ${r}_{{\rm{frg}}}\simeq 1$ au, then $\langle {V}_{{\rm{sep}}}\rangle \simeq 0.5\;{\rm{m}}$ s⁻¹; and if at ${r}_{{\rm{frg}}}\simeq 10$ au, then $\langle {V}_{{\rm{sep}}}\rangle \simeq 1.5\;{\rm{m}}$ s⁻¹. This is a typical range of separation velocities as documented by the known split comets (Section 2.1.1; Sekanina 1982).

In spite of the incompleteness of the data on the Lick object, we believe that a fairly compelling case has been presented for its genetic relationship with C/1847 C1, implying that their separation had occurred in the general proximity of the previous perihelion in the seventh millennium BCE. This example demonstrates that the temporal gap between the members of a genetically related pair or group of long-period comets could easily reach many dozens of years and that both the primary and the companion can survive for millennia after the parent comet's splitting. Finally, it is nothing short of remarkable that a premise of the membership in a comet pair can successfully be defended in instances of unidentified interplanetary objects with poorly known motions.