WAS COMET C/1945 X1 (DU TOIT) A DWARF, SOHO-LIKE KREUTZ SUNGRAZER?

Since the observed orbital arc is too short to reliably compute the eccentricity, it needs to be determined from other considerations. As already mentioned above, the constraint is here provided by adopting the semimajor axis of the hybrid's osculating orbit, 96.7 AU at the epoch of 1945 December 26.0 TT, for all solutions that follow. We begin with a gravitational orbit to fit all five astrometric positions in Table 4.

Deriving orbits B and C and comparing them with orbit A, Marsden (1989) demonstrated that elliptical solutions fitted the five observations of C/1945 X1 much better than a parabolic approximation. Yet, neither orbit, B or C, provides a satisfactory fit. One possible contributor to this problem could be the constraint introduced to satisfy the prescribed direction of the line of apsides (for which Marsden took ${L}_{\pi }=282\buildrel{\circ}\over{.} 7$, ${B}_{\pi }=\;+35\buildrel{\circ}\over{.} 2$ after conversion to Equinox J2000), the same issue that we encountered with the dwarf sungrazers of the Kreutz system (Paper I; see also Section 1).

Comparison of orbit C, the more realistic of the two sets in terms of the comet's orbital period, with our optimum gravitational fit to the five observations is presented in Table 5. Marsden (1989) residuals from orbit C in this table have a tendency to become more positive toward both ends of the observed arc in R.A. and to get more negative with time in decl. We tried to emulate the residuals from orbit C, employing the set of truncated elements in Marsden (1989) Table V, but succeeded to achieve this within about 4'' only in decl. Although we allowed for the rounding off of the perihelion time, the residuals in R.A. came out systematically more negative by about 12'' and all except the last one were negative. This effect is due apparently to the elements' truncation.

Table 5.  Marsden (1989)'s Orbit C and Our Optimum Gravitational Solution to Fit All Five Boyden Observations of C/1945 X1 (Equinox J2000)

Orbital Element		Orbit C		Best Fit
Perihelion time tπ (1945 TT)		Dec 27.976		Dec 27.982
Argument of perihelion ω		6810		7099
Longitude of ascending node Ω		34554		34950
Orbit inclination i		14160		14191
Perihelion distance $q\left\{\begin{array}{c}(\mathrm{AU})\\ ({R}_{\odot })\end{array}\right.$		0.00699		0.007244
		1.502		1.556
Orbit eccentricity		0.99993010		0.99992506
Semimajor axis (AU)		100		96.67
Orbital period (year)		1000		950.5
Longitude of perihelion Lπ		28270		28314
Latitude of perihelion Bπ		+3520		+3568
Osculation epoch 1945 (TT)		(none)		Dec 11.0
Mean residuala		±1195		±617
Distribution of Residualsa, $O-C$
Time of Observation	Orbit C		Best Fit
1945 (UT)	R.A.	Decl.	R.A.	Decl.
Dec 11.04687	+56	+224	+132	+80
12.04691	+6.1	+5.3	+6.1	+2.5
13.07189	−14.8	−11.8	−6.6	−5.6
14.07000	+10.3	−1.9	+7.8	+8.0
15.06885	+15.6	−10.9	−2.7	−3.2

Note.

^aFrom unweighted observations for orbit C, weighted for Best Fit.

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The residuals from our Best Fit solution, although clearly better than those from orbit C, are not quite satisfactory either, suggesting that the first observation may be inferior. That should not be surprising, considering that the comet was rather far from the plate center on this discovery exposure (Section 3). Unfortunately, the line of apsides deviates from the proper direction by 0571, which is unacceptably large. Although the individual angular elements of the Best Fit solution are burdened with errors of up to several tenths of a degree, the correlation among them draws the uncertainty in the orientation of the line of apsides down to only hundredths of a degree.

Next, we compared Marsden (1989)'s orbit D with the hybrid's orbit from Table 2. As with orbit C, we first tested whether the low-precision orbit D from Marsden's Table 5 is able to reproduce the residuals in his Table 6. Since this exercise did not meet with success, we decided to reconstruct the high-precision version of orbit D, not published by Marsden, by subtracting the residuals in his Table 6 from the observed astrometric positions, retaining his constraint on the reciprocal semimajor axis (0.01016 AU⁻¹). This approach was successful; the resulting orbit D is shown in Table 6 to be exceedingly similar to the hybrid's orbit, the two sets of elements differing mostly in the fourth or higher significant digit. In particular, we note that the lines of apsides agree to better than 001. This correspondence is not fortuitous, as both sets of elements represent perturbed versions of the orbits of two comets observed 83 yr apart, whose motions in space were virtually identical, as already pointed out by Marsden (1967): C/1882 R1, used by us to derive the hybrid's orbit; and C/1965 S1, employed by Marsden (1989) to derive orbit D.

Table 6.  Marsden (1989)'s Orbit D and Hybrid's Orbit (Equinox J2000)

Orbital Element			Orbit D			Hybrid's Orbit
Perihelion time tπ (1945 TT)			Dec 27.9798a			Dec 27.9801
Argument of perihelion ω			678356			678514
Longitude of ascending node Ω			3453571			3453753
Orbit inclination i			1415553			1415607
Perihelion distance $q\left\{\begin{array}{c}(\mathrm{AU})\\ ({R}_{\odot })\end{array}\right.$			0.007129			0.007123
			1.5315			1.5302
Orbit eccentricity			0.99992756			0.99992632
Semimajor axis (AU)			98.42			96.67
Orbital period (year)			976.4			950.5
Longitude of perihelionb, Lπ			28284			28284
Latitude of perihelionb, Bπ			+3516			+3516
Osculation epoch 1945 (TT)			Dec 11.0			Dec 11.0
Mean residualc			$\left\{\begin{array}{c}\pm 7\buildrel{\prime\prime}\over{.} 63({\rm{M}})\\ \pm 8\buildrel{\prime\prime}\over{.} 17({\rm{SK}})\end{array}\right.$			±1375
Distribution of Residualsc, O–C
Time of Observation	Orbit D(M)		Orbit D(SK)		Hybrid's Orbit
1945 (UT)	R.A.	Decl.	R.A.	Decl.	R.A.	Decl.
Dec 11.04687	+108	+51	+123	+66	+264	−63
12.04691	−2.5	−2.8	−1.1	−1.1	+13.7	−12.1
13.07189	−15.2	−11.4	−14.0	−9.6	+1.2	−18.7
14.07000	+4.3	+5.6	+5.3	+7.6	+20.4	0.0
15.06885	+2.8	+2.5	+3.6	+4.5	+18.4	−1.8

Notes.

^aTo fit weighted observations, ${t}_{\pi }=\mathrm{December}\;27.9800$. ^bThese ecliptical oordinates define the reference line of apsides. ^cEntry (M) and columns 2–3 from unweighted observations; entry (SK) and columns 4–5 from weighted observations; residuals from hybrid's orbit are from weighted observations.

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Table 6 also lists three sets of residuals: those left by orbit D from the unweighted observations, in columns 2 and 3 (taken from Marsden 1989's Table 6 and marked M); those left by orbit D from the weighted observations, as derived by us⁹ (marked SK), in columns 4 and 5; and those left by the hybrid's orbit from the weighted observations, in columns 6 and 7.

Even though the residuals in Table 6 are fairly large, the ones left by orbit D are much better than the strongly systematic ones left by the hybrid's orbit. From this comparison, as well as from the difference in the mean residual, we infer that the orbital evolution of comet C/1945 X1, as a fragment of its common parent with C/1882 R1 and C/1965 S1, was apparently more similar to the orbital evolution of the latter than the former. This is an important though tentative conclusion, consistent with the fragmentation hierarchy of the Kreutz system proposed by Sekanina & Chodas (2004). According to their evolutionary model, C/1882 R1, C/1965 S1, and the precursor to C/1945 X1 separated from their common parent, possibly X/1106 C1, at the same time, some 18 days after perihelion. Relative to C/1882 R1, the precursor moved in the same direction as, and with a velocity only about 20% lower than, C/1965 S1 (see also Sekanina & Chodas 2002). The subsequent separation of C/1945 X1 from its precursor around 1700 AD notwithstanding, the comet's orbit in 1945 should indeed resemble the orbit of C/1965 S1 to a greater degree than that of C/1882 R1.

Since orbit D and the hybrid's orbit are so very similar, yet the residuals they leave substantially differ, we tested whether this effect is due to the minor differences in the angular elements or the orbital dimensions. We replaced the semimajor axis 98.4 AU with 96.7 AU and noted that the residuals from orbit D in Table 6 did not change; the new orbit, D', is now t_π = 1945 December 27.9803 TT, $q=0.007126\mathrm{AU}=1.5309$ ${R}_{\odot }$, e = 0.99992630, and for the equinox of J2000, $\omega =67\buildrel{\circ}\over{.} 8377$, ${\rm{\Omega }}=345\buildrel{\circ}\over{.} 3558$, and $i=141\buildrel{\circ}\over{.} 5570$. The residuals in Table 6 reflect thus entirely the differences of up to 002 in the angular elements.

Before turning to nongravitational solutions, we take notice of a possibility that one of the five astrometric observations is inferior and should be discarded before computing any orbit. Accordingly, we are now going to search for solutions that could fit four observations. Such solutions will be referred to as ${\mathcal{G}}\{{\mathfrak{I}}\},$ where ${\mathcal{G}}$ stands for gravitational and $\{{\mathfrak{I}}\}$ is a progression of reference numbers of the observations that such solutions are based on; these numbers are listed in column 1 of Table 4. The Best Fit solution presented in Table 5 can now be referred to as ${\mathcal{G}}\{1,2,3,4,5\}.$ As already mentioned above, the discovery position may be inferior, so that one of the tested four-observation solutions is ${\mathcal{G}}\{2,3,4,5\}.$ The residuals from orbit D in Table 6 suggest that the third observation may be even worse than the first, raising interest in the solution ${\mathcal{G}}\{1,2,4,5\}.$ Finally, the derivation of the cataloged parabolic orbit implied that the fourth observation failed to fit that solution (Marsden 1967), hence a need to test ${\mathcal{G}}\{1,2,3,5\}.$ The results of all three of these solutions are presented in Table 7. No orbits are being computed based on three observations only, as these are regarded for our purposes as meaningless.

Table 7.  Comparison of Three Gravitational Solutions Based on Four Observations of C/1945 X1 (Equinox J2000)

Orbital Element		${\mathcal{G}}\{2,3,4,5\}$		${\mathcal{G}}\{1,2,4,5\}$		${\mathcal{G}}\{1,2,3,5\}$
Perihelion time tπ (1945 TT)		Dec 27.986		Dec 27.979		Dec 27.983
Argument of perihelion ω		6598		6743		7400
Longitude of ascending node Ω		34284		34482		35352
Orbit inclination i		14137		14150		14207
Perihelion distance $q\left\{\begin{array}{c}(\mathrm{AU})\\ ({R}_{\odot })\end{array}\right.$		0.007004		0.007120		0.007415
		1.505		1.530		1.593
Orbit eccentricity		0.99992755		0.99992635		0.99992329
Longitude of perihelion Lπ		28254		28280		28349
Latitude of perihelion Bπ		+3477		+3509		+3622
Reference apsidal line's offseta		0462		0078		1184
Osculation epoch 1945 (TT)		Dec 11.0		Dec 11.0		Dec 11.0
Mean residual (weighted)		±511		±179		±106
Distribution of Residualsb, $O-C$
Time of Observation	${\mathcal{G}}\{2,3,4,5\}$		${\mathcal{G}}\{1,2,4,5\}$		${\mathcal{G}}\{1,2,3,5\}$
1945 (UT)	R.A.	Decl.	R.A.	Decl.	R.A.	Decl.
Dec 11.04687	⋯	⋯	+53	+11	−33	+33
12.04691	+252	+120	−8.2	−7.4	+4.9	+2.7
13.07189	−5.1	−4.3	⋯	⋯	+0.1	−0.8
14.07000	+4.3	+5.5	+0.1	+1.3	⋯	⋯
15.06885	−1.1	−2.3	+0.1	−0.8	−0.2	+0.3

Notes.

^aReference line of apsides is defined by the hybrid's L_π and B_π. ^bFrom weighted observations.

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Since a semimajor axis a between 96.7 and 98.4 AU was shown to make no difference, the four-observation solutions below are subjected to the same constraint as the Best Fit solution in Table 5, namely, a = 96.7 AU at an osculation epoch of 1945 December 26.0 TT. The solutions listed in Table 7 differ from one another considerably. The orbits ${\mathcal{G}}\{2,3,4,5\}$ and ${\mathcal{G}}\{1,2,3,5\}$ are both unacceptable on account of their large offsets from the reference apsidal direction (given by the hybrid's orbit and Marsden's orbit D). The set ${\mathcal{G}}\{2,3,4,5\}$ is in addition handicapped by the very large residuals left by the December 12 observation. Even cursory inspection shows that the orbit ${\mathcal{G}}\{1,2,4,5\}$ is by far the most promising of the three, implying that it is the December 13 observation that is inferior. The offset from the reference line of apsides, slightly less than 01, is reasonably low though not completely satisfactory. It is somewhat surprising that the apparently inferior astrometric position is one of those taken with the Metcalf refractor. We would expect that the discarded data point should be one of the first two observations, made with the Cooke camera. Under the circumstances, the solution ${\mathcal{G}}\{1,2,4,5\}$ can at best be regarded as marginally acceptable to approximate the orbital motion of C/1945 X1.

In any case, it is worth examining whether any meaningful refinement of the orbit, an apsidal line in particular, can be achieved by incorporating a nongravitational acceleration, which, if comparable to those affecting the motions of the dwarf sungrazers, might be detectable over a span of 4 days. Besides, such alternative solutions should prove beneficial to estimating a range of uncertainties in the comet's orbital motion (Section 7).

In order to find out whether there is evidence for nongravitational effects in the motion of comet C/1945 X1 in the meager set of astrometric data available, we begin with a standard formalism of Marsden et al. (1973), based on a water-ice sublimation model. The erosion-driven nongravitational accelerations in the three cardinal directions, i.e., in the radial (away from the Sun), ${\bf{R}}$, transverse, ${\bf{T}}$, and normal, ${\bf{N}}$, directions of a right-handed ${\bf{RTN}}$ coordinate system that is referred to the comet's orbital plane, are in this formalism expressed by

$$\begin{eqnarray}&&\left[\begin{array}{c}{a}_{{\rm{R}}}(r)\\ {a}_{{\rm{T}}}(r)\\ {a}_{{\rm{N}}}(r)\end{array}\right]=\left[\begin{array}{c}{A}_{1}\\ {A}_{2}\\ {A}_{3}\end{array}\right]\cdot {g}_{\mathrm{std}}(r),\end{eqnarray} \tag{ 8 }$$

where ${g}_{\mathrm{std}}(r)$ is the standard law employed by Marsden et al. (1973), approximating the sublimation rate of water-ice from an isothermal spherical nucleus and normalized so that ${g}_{\mathrm{std}}(1\;\mathrm{AU})=1$,

$$\begin{eqnarray}&&{g}_{\mathrm{std}}(r)=\alpha {\left(\displaystyle \frac{r}{{r}_{0}}\right)}^{-m}{\left[1+{\left(\displaystyle \frac{r}{{r}_{0}}\right)}^{n}\right]}^{-k}.\end{eqnarray} \tag{ 9 }$$

Here $m=2.15$, $n=5.093$, nk=23.5, a scaling distance r₀ = 2.808 AU, and a normalization constant $\alpha =0.1113$. The parameters A₁, A₂, A₃ are, respectively, the radial, transverse, and normal components of the nongravitational acceleration at 1 AU from the Sun; their units usually employed in orbital studies are 10⁻⁸ AU day⁻². Since the vector of the acceleration due to the generally sunward-directed sublimation points away from the Sun, physically meaningful values of A₁ should always be positive. The law (9) has over the past 40 yr served admirably in countless orbit-determination applications to comets with perihelia typically several tenths of AU.

The numerical procedure that we apply next follows the method described in Paper I. Briefly, since the direction of the line of apsides was found to be a function of the nongravitational acceleration, a minimum offset from the reference, or nominal, apsidal line (defined here by L_π and B_π for the hybrid's orbit or orbit D; see Table 6) serves as the constraint that determines the nongravitational acceleration's most probable magnitude.

Similarly to our notation for the gravitational runs, we refer to these nongravitational solutions by ${{\mathcal{N}}}_{\mathrm{std}}^{\;({\rm{X}})}\{{\mathfrak{I}}\},$ where $\{{\mathfrak{I}}\}$ is again a progression of reference numbers of the employed observations from Table 4, while ${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{X}})}$ denotes, on the one hand, Marsden et al. (1973)'s formalism of accounting for the nongravitational effect, with the standard law (9) describing the variations with heliocentric distance; and, on the other hand, one of the two versions applied: either solving for the radial component of the acceleration with the parameter A₁ when ${\bf{X}}={\bf{R}}$; or for the normal component with A₃ when ${\bf{X}}={\bf{N}}$. Whenever we attempted to solve for both components, the run aborted. We recall that the radial component (${A}_{1}\gt 0$) dominates the magnitude of the nongravitational acceleration in the motions of the cataloged comets in nearly parabolic orbits (Marsden & Williams 2008), while the normal component (A₃) was found to contribute significantly to the magnitude of the acceleration that affects the motions of the dwarf Kreutz sungrazers (Paper I).

Table 8 summarizes the most important results from the runs based on Marsden et al. (1973)'s standard formalism. Altogether we computed eight such solutions for four different observational sets $\{{\mathfrak{I}}\},$ the same ones as before: ${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{R}})}\{1,2,3,4,5\}$, ${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{R}})}\{2,3,4,5\}$, ${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{R}})}\{1,2,4,5\}$, and ${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{R}})}\{1,2,3,5\}$; and similarly with ${\bf{N}}$ instead of ${\bf{R}}.$ Both the ${\bf{R}}$ and the ${\bf{N}}$ versions of the ${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{X}})}\{1,2,3,4,5\}$ and ${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{X}})}\{1,2,3,5\}$ runs are inferior to their respective gravitational solutions (Tables 5 and 7) in terms of the mean residual: ±642 (${\bf{R}}$) and ±6${}^{\prime\prime }.$68 (${\bf{N}}$) against ±617 in the case of the $\{1,2,3,4,5\}$ solution; and ±303 (${\bf{R}}$) and ±261 (${\bf{N}}$) against ±106 in the case of the {1, 2, 3, 5} solution. In addition, for these two sets of solutions the ${\bf{R}}$-version parameters A₁ come out to be negative and therefore physically meaningless.

Table 8.  Comparison of the Nongravitational Parameters and Test Results for Eight Orbital Solutions Based on Marsden et al.'s Formalism

Nongravitational	Parametera	Apsidal-lineb	Mean residual
Solution	A1 or A3	Offset	(Weighted)
${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{R}})}\{1,2,3,4,5\}$	−0.82  ±  0.17	0116	±642
${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{R}})}\{2,3,4,5\}$	+0.67  ±  0.18	0.118	±5.05
${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{R}})}\{1,2,4,5\}$	+0.113  ±  0.019	0.013	±1.73
${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{R}})}\{1,2,3,5\}$	−1.70  ±  0.40	0.270	±3.03
${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{N}})}\{1,2,3,4,5\}$	−1.87  ±  0.23	0.073	±6.68
${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{N}})}\{2,3,4,5\}$	+1.54  ±  0.30	0.085	±4.88
${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{N}})}\{1,2,4,5\}$	+0.234  ±  0.018	0.006	±1.72
${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{N}})}\{1,2,3,5\}$	−4.18  ±  0.65	0.212	±2.61

Notes.

^aIn units of 10⁻⁵AU day⁻². ^bOffset from direction of the reference line of apsides (Table 6).

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Even though the set $\{2,3,4,5\}$ avoids the misfortunes of the sets $\{1,2,3,4,5\}$ and $\{1,2,3,5\},$ its offsets from the reference line of apsides fail to improve over the corresponding offsets from $\{1,2,3,4,5\};$ in addition, the residuals left by the December 12 observation are comparable to those from the equivalent gravitational solution (Table 7) and unacceptable. In summary, the standard nongravitational solutions based on the set $\{1,2,4,5\}$ (Table 9) are by far the best, in terms of both the offset from the reference apsidal line and the mean residual. Also, both A₁ and A₃ of the $\{1,2,4,5\}$ solutions come out in Table 8 to be almost or just about one order of magnitude smaller than those of the poor solutions.

As the final comments we note that (i) relatively to the gravitational fit in Table 7, both nongravitational solutions based on the set $\{1,2,4,5\}$ improve the mean residual only marginally (from ±179 to ±172/173), but reduce the offset from the reference apsidal line considerably (from 0078 to 0013/0006); (ii) both A₁ and A₃ appear to be fairly well determined (with the relative errors of ±8%–17%) and on the same order of magnitude, ∼10⁻⁶ AU day⁻², as the parameters of the normal component of the nongravitational acceleration of the dwarf sungrazers C/2009 L5 and C/2006 J9 in Table 4 of Paper I; and (iii) these parameters are one order of magnitude greater than the peak values of A₁ for the cataloged comets in nearly parabolic orbits and at least two orders of magnitude greater than the typical values of A₁ for such comets (Marsden & Williams 2008). In summary, there is some—though, owing to the limited data, not overwhelming—evidence that in terms of the nongravitational effects in its orbital motion, C/1945 X1 may share the properties of some dwarf Kreutz sungrazers.

The EXORB7 orbit-determination code employed by the second author allows one to vary arbitrarily all five parameters of the nongravitational law (9)—the exponents m, n, k, the scaling distance r₀, and the normalization constant α. This option facilitates a more robust orbital experimentation, when the standard nongravitational model of Marsden et al. (1973) fails to provide satisfactory results. In Paper I we gained some experience with what we hereafter refer to as a modified nongravitational law ${g}_{\mathrm{mod}}(r;{r}_{0}),$ which is given by Equation (9), but while the three exponents remain unchanged from the standard law, the scaling distance varies broadly, always entailing a change in α as well.

The physics behind this sublimation-law experimentation involves a parallelism between the scaling distance r₀ and the snow line, a boundary of the zone in vacuum (or near vacuum), beyond which it is cold enough for a volatile substance to exist only in the solid phase. Calibrated by water-ice, for which in the isothermal model ${r}_{0}\simeq 2.8$ AU, this distance depends in the first approximation on the effective latent heat of sublimation, L (in cal mol⁻¹), of the sublimating species:

$$\begin{eqnarray}&&{r}_{0}\;\dot{=}{\left(\displaystyle \frac{\mathrm{const}}{L}\right)}^{2},\end{eqnarray} \tag{ 10 }$$

where r₀ is expressed in AU and the constant is equal to 19,100 AU${}^{\frac{1}{2}}$ cal mol⁻¹. Marsden et al. (1973) showed that in a plot of log (normalized sublimation rate) against log (heliocentric distance) the value of r₀ shifts the curve left or right along the axis of heliocentric distance r. Hence, by properly choosing r₀, the erosion-driven nongravitational effects in the orbital motion of any comet can approximately be expressed by a universal curve of log (normalized sublimation rate) against $\mathrm{log}(r/{r}_{0}).$

Highly refractory materials, such as metals or silicates, can sublimate only very close to the Sun, so that their snow line and scaling distance are much smaller than 1 AU. For example, for forsterite, the Mg end-member of the olivine solid solution system, the latent heat of sublimation is 130,000 cal mol⁻¹ (Nagahara et al. 1994), so that its ${r}_{0}\simeq 0.02$ AU.

In limiting cases the empirical Equation (9) offers the following expressions for variations in the sublimation rates (equally applying to the modified and standard laws):

$$\begin{eqnarray}\underset{x\to 0}{\mathrm{lim}}{g}_{\mathrm{mod}}(r;{r}_{0}) & \sim & {r}^{-2.15},\\ \underset{x\to \infty }{\mathrm{lim}}{g}_{\mathrm{mod}}(r;{r}_{0}) & \sim & {r}^{-25.65},\end{eqnarray} \tag{ 11 }$$

where $x=r/{r}_{0}$. The first limit approaches an extreme scenario in which all incident solar energy is spent on sublimation, while the second limit crudely approximates the other extreme, when the energy is spent entirely on heating the object (and increasing its thermal reradiation). The actual limiting expressions are ${r}^{-2}$ for $r/{r}_{0}\to 0$ and

$$\begin{eqnarray}&&{\gamma }_{\mathrm{mod}}(r;{r}_{0})\sim \mathrm{exp}\left(-\displaystyle \frac{L}{{RT}}\right)\ \ \ \mathrm{for}\ r/{r}_{0}\to \infty ,\end{eqnarray} \tag{ 12 }$$

where the sublimation heat L is related to r₀ by Equation (10), R is the gas constant, and T is the temperature at heliocentric distance r. The differences relative to (11) are due to the approximate nature of the law (9), which is responsible for a peculiar feature when applied to the orbital motion of C/1945 X1 (see below). To illustrate it, we begin by introducing a local slope ζ of a modified nongravitational law between r and $r+{\rm{\Delta }}r;$ similarly to (11), it is

$$\begin{eqnarray}&&\underset{{\rm{\Delta }}r\to 0}{\mathrm{lim}}{g}_{\mathrm{mod}}(r;{r}_{0})\sim {r}^{-\zeta },\end{eqnarray} \tag{ 13 }$$

where

$$\begin{eqnarray}&&\zeta (r;{r}_{0})=-\displaystyle \frac{\partial \mathrm{ln}{g}_{\mathrm{mod}}(r;{r}_{0})}{\partial \mathrm{ln}(r/{r}_{0})}=m+\displaystyle \frac{{nk}}{1+{(r/{r}_{0})}^{-n}}.\end{eqnarray} \tag{ 14 }$$

The limits on ζ are those given in (11). Since r₀ is nearly 3 AU in the standard law, the contribution to the integrated nongravitational effect from $r\gg {r}_{0}$ is always negligible and the exact shape of the standard nongravitational law at heliocentric distances $r\gg {r}_{0}$ is unimportant. However, when r₀ in a modified law is considerably smaller than that of the standard law, the power-law approximation to the exponential law for the sublimation rate beyond r₀ could play a major role for comets whose perihelion distances q are much greater than r₀.

To find out at what heliocentric distance ${r}_{*}$ does a modified law begin to deviate from the sublimation law ${\gamma }_{\mathrm{mod}}(r;{r}_{0}),$ we first define the local slope ξ of ${\gamma }_{\mathrm{mod}}(r;{r}_{0})$ similarly to (14):

$$\begin{eqnarray}&&\xi (r;{r}_{0})=-\displaystyle \frac{\partial \mathrm{ln}{\gamma }_{\mathrm{mod}}(r;{r}_{0})}{\partial \mathrm{ln}(r/{r}_{0})}=\displaystyle \frac{L\sqrt{r}}{2{{RT}}_{0}}=\displaystyle \frac{L\sqrt{{r}_{0}}}{2{{RT}}_{0}}\sqrt{\displaystyle \frac{r}{{r}_{0}}},\end{eqnarray} \tag{ 15 }$$

where $T(r)={T}_{0}/\sqrt{r}$ at the heliocentric distances of extremely low sublimation rates; for the isothermal model, ${T}_{0}=278.3$ K. Since, from Equation (10), $L\sqrt{{r}_{0}}$ is a constant and $L\sqrt{{r}_{0}}/{{RT}}_{0}=34.54$, the point of deviation of the modified law ${g}_{\mathrm{mod}}(r;{r}_{0})$ from the exponential sublimation law, given by the condition of equal slopes, $\xi ({r}_{*};{r}_{0})=\zeta ({r}_{*};{r}_{0})$, determines ${r}_{*}$ by comparing (14) with (15). After inserting the numerical values,

$$\begin{eqnarray}&&{r}_{*}=2.12\;{r}_{0}.\end{eqnarray} \tag{ 16 }$$

Less than a millionth part of the Sun's incident energy at the distance ${r}_{*}$ is spent on the sublimation.

Requiring that ${g}_{\mathrm{mod}}({r}_{*};{r}_{0})={\gamma }_{\mathrm{mod}}({r}_{*};{r}_{0})$ and writing the exponential sublimation law in the form

$$\begin{eqnarray}&&{\gamma }_{\mathrm{mod}}(r;{r}_{0})=\beta \;\mathrm{exp}\left[34.54\left(1-\sqrt{\displaystyle \frac{r}{{r}_{0}}}\;\right)\right],\end{eqnarray} \tag{ 17 }$$

we have

$$\begin{eqnarray}&&\beta =0.0267\alpha .\end{eqnarray} \tag{ 18 }$$

The relationship between ${g}_{\mathrm{mod}}(r;{r}_{0})$ and ${\gamma }_{\mathrm{mod}}(r;{r}_{0})$ is displayed in Figure 4. At present, the ${\gamma }_{\mathrm{mod}}(r;{r}_{0})$ law is not incorporated into the orbit-determination code that we employ, as in most cases it would make hardly any difference numerically. However, the approximation of ${\gamma }_{\mathrm{mod}}(r;{r}_{0})$ by ${g}_{\mathrm{mod}}(r;{r}_{0})$ at $r\gt {r}_{*}$ (i) prevents one from testing nongravitational laws with a variable exponent ζ at these heliocentric distances and (ii) yields identical results from all solutions based on the scaling distances r₀ that are by more than 2.12 r₀ smaller than the least heliocentric distance at which the comet is observed, as seen from Figure 4.

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In Paper I we employed the modified-law paradigm to great advantage. We determined for all eight in-depth examined dwarf sungrazers that the astrometric positions were fitted better (and the offsets from the reference line of apsides came out to be smaller) when the nongravitational acceleration affecting their orbital motions was assumed to vary with heliocentric distance r much more steeply than prescribed by the standard law ${g}_{\mathrm{std}}(r).$ The orbital motions of six out of the eight cases were fitted best with ${r}_{0}\lt 0.13$ AU, which corresponds to $L{\rm{\gtrsim }}$ 50,000 cal mol⁻¹, and for none of the eight was r₀ higher than ∼2.1 AU. The heliocentric distances of the dwarf Kreutz sungrazers in Paper I varied between 0.037 and 0.068 AU (or between 8 and 15 ${R}_{\odot }$), so that these objects were considerably closer to the Sun than C/1945 X1 when under observation from Boyden.

For C/1945 X1, the modified-law nongravitational solutions with ${r}_{0}\ll 0.2$ AU fit most satisfactorily the preferred set of four observations, $\{{{\mathfrak{I}}}_{0}\}=\{1,2,4,5\}$. This match is better than that from the standard-law solutions listed in Table 9. We refer to these modified-law solutions as ${{\mathcal{N}}}_{\mathrm{mod}}^{({\rm{X}})}[{r}_{0}]\{{{\mathfrak{I}}}_{0}\},$ where ${\bf{X}}$ has the same meaning as before and r₀ is in AU. Because the comet was not observed closer to the Sun than 0.6 AU, all these solutions are identical, as pointed out more generally under (ii) below Equation (18). We confirm this result by running solutions for r₀ equal to 0.01, 0.05, and 0.10 AU. We then select, as examples, the solutions ${{\mathcal{N}}}_{\mathrm{mod}}^{({\rm{R}})}[0.05]\{{{\mathfrak{I}}}_{0}\}$ and ${{\mathcal{N}}}_{\mathrm{mod}}^{({\rm{N}})}[0.05]\{{{\mathfrak{I}}}_{0}\}$ and present them in Table 10; relative to the results based on the standard law, the offsets from the reference line of apsides now dropped, respectively, from 0013 to 0005 and from 0006 to 0003, while the weighted mean residuals dropped, respectively, from ±173 and ±172 down to ±166 in both ${\bf{R}}$ and ${\bf{N}}$.

Table 9.  Orbital Solutions ${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{R}})}\{1,2,4,5\}$ and ${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{N}})}\{1,2,4,5\}$ (Equinox J2000)

Orbital Element		${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{R}})}\{1,2,4,5\}$		${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{N}})}\{1,2,4,5\}$
Perihelion time tπ (1945 TT)		Dec 27.97928		Dec 27.97938
Argument of perihelion ω		67910		67883
Longitude of ascending node Ω		345437		345412
Orbit inclination i		141572		141565
Perihelion distance $q\left\{\begin{array}{c}(\mathrm{AU})\\ ({R}_{\odot })\end{array}\right.$		0.0071167		0.0071250
		1.5287		1.5307
Orbit eccentricity		0.99992638		0.99992629
Longitude of perihelion Lπ		28282		28283
Latitude of perihelion Bπ		+3516		+3516
Reference apsidal-line offset		00133		00060
Parameter A1(10−5AU day ${}^{-2})$		+0.113a		⋯
Parameter A3(10−5AU day ${}^{-2})$		⋯		+0.234b
Osculation epoch 1945 (TT)		Dec 11.0		Dec 11.0
Mean residual (weighted)		±173		±172
Distribution of Residuals $O-C$
Time of Observation	${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{R}})}\{1,2,4,5\}$		${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{N}})}\{1,2,4,5\}$
1945 (UT)	R.A.	Decl.	R.A.	Decl.
Dec 11.04687	+49	+20	+49	+16
12.04691	−8.2	−7.3	−8.0	−7.2
14.07000	+0.3	+1.0	+0.2	+1.1
15.06885	−0.1	−0.6	0.0	−0.7

Notes.

^aWith a mean error of $\pm 0.019\times {10}^{-5}$ AU day⁻² (Table 8). ^bWith a mean error of $\pm 0.018\times {10}^{-5}$ AU day⁻² (Table 8).

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Table 10.  Orbital Solutions ${{\mathcal{N}}}_{\mathrm{mod}}^{({\rm{R}})}[0.05]\{{{\mathfrak{I}}}_{0}\}$ and ${{\mathcal{N}}}_{\mathrm{mod}}^{({\rm{N}})}\;[0.05]\{{{\mathfrak{I}}}_{0}\}$ (Equinox J2000)

Orbital Element		${{\mathcal{N}}}_{\mathrm{mod}}^{({\rm{R}})}[0.05]\{{{\mathfrak{I}}}_{0}\}$		${{\mathcal{N}}}_{\mathrm{mod}}^{({\rm{N}})}[0.05]\{{{\mathfrak{I}}}_{0}\}$
Perihelion time tπ (1945 TT)		Dec 27.97937		Dec 27.97940
Argument of perihelion ω		67876		67869
Longitude of ascending node Ω		345403		345397
Orbit inclination i		141562		141561
Perihelion distance $q\left\{\begin{array}{c}(\mathrm{AU})\\ ({R}_{\odot })\end{array}\right.$		0.0071256		0.0071279
		1.5308		1.5313
Orbit eccentricity		0.99992629		0.99992626
Longitude of perihelion Lπ		28283		28284
Latitude of perihelion Bπ		+3516		+3516
Reference apsidal-line offset		00055		00034
Parameter A1(10−10AU day ${}^{-2})$		+0.307a		⋯
Parameter A3(10−10AU day ${}^{-2})$		⋯		+0.877b
Osculation epoch 1945 (TT)		Dec 11.0		Dec 11.0
Mean residual (weighted)		±166		±166
Distribution of Residuals O–C
Time of Observation	$\;{{\mathcal{N}}}_{\mathrm{mod}}^{\;({\rm{R}})}[0.05]\{{{\mathfrak{I}}}_{0}\}$		$\;{{\mathcal{N}}}_{\mathrm{mod}}^{({\rm{N}})}[0.05]\{{{\mathfrak{I}}}_{0}\}$
1945 (UT)	R.A.	Decl.	R.A.	Decl.
Dec 11.04687	+47	+20	+48	+17
12.04691	−8.0	−6.9	−7.9	−6.9
14.07000	+0.3	+0.9	+0.3	+1.0
15.06885	−0.1	−0.5	$\ 0.0$	−0.6

Notes.

^aWith a mean error of $\pm 0.021\times {10}^{-10}$ AU day⁻². ^bWith a mean error of $\pm 0.046\times {10}^{-10}$ AU day⁻².

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The nongravitational parameters A₁ and A₃ are now determined with relative errors of only ±5%–7%. Their apparent discrepancy of nearly five orders of magnitude compared to those of the standard law is due entirely to the very different steepness of the two laws' slopes. Since the comet was observed between 0.718 AU from the Sun (on December 11) and 0.598 AU from the Sun (on December 15), the magnitudes of the effective nongravitational accelerations need to be compared in this range of heliocentric distances, rather than at 1 AU from the Sun, to test whether the results from the standard law and the modified law are generally consistent. This test shows that the radial components from the two laws reach the same value, $+0.3\times {10}^{-5}$ AU day⁻², at r = 0.639 AU, while the normal components equally reach $+0.6\times {10}^{-5}$ AU day⁻² at r = 0.648 AU, in either case well inside the observed range of heliocentric distances. The comet's favorable comparison with the dwarf sungrazers, mentioned in Section 6.2, is consistent with the results from the modified-law solutions: C/1945 X1 appears to be subjected to a nongravitational acceleration that is near the lower end of the range of accelerations typical for the dwarf sungrazers.

In summary, we obtain some evidence suggesting that (i) the nongravitational solutions fit the observations of C/1945 X1 better than the gravitational solutions; and (ii) a law with a very steep slope (of the type $\sim {r}^{-25}$) may be superior to the standard law. Over all, however, the promising performance of the modified law has not overturned our conclusion expressed at the end of Section 6.2 that the limited data sample does not make a conclusion about the existence of large nongravitational effects in the motion of C/1945 X1 sufficiently compelling. Every effort should be expended to improve the comet's orbit by extending the observed arc with the help of additional images on Boyden patrol plates.

It may seem to be questionable to use the astrometric data of less than high quality for supporting the type of in-depth orbital analysis of C/1945 X1 that is described in the previous sections. With the observed arc of merely 4 days and the images taken with small cameras of poor spatial resolution and long exposures tracked at sidereal rates, the data may at first sight seem overexploited. However, the addressed objectives are not only to learn as much as possible about the orbital behavior of C/1945 X1 in the context of other Kreutz sungrazers, but, as illustrated in Table 11, also to provide a basis for constraining an ephemeris for the comet's motion both back and forward in time from the observed period, a capability that is amply put to use in our search for more images of the comet, as described in Section 7.

Table 11.  Geocentric Ephemeris for Comet C/1945 X1 from 1945 September 12 to 1946 January 30 (Equinox J2000)

	From Solution ${{\mathcal{N}}}_{\mathrm{mod}}^{({\rm{N}})}[\ 0.05\ ]\{{{\mathfrak{I}}}_{0}\}$						App. Magnitude		Differential Ephemerides from Solutions
Date TT	R.A.	Decl.	Δ	r	Phase Angle	PA(${\boldsymbol{RV}}$)	${r}^{-4}$	${r}^{-7}$	${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{N}})}\{{{\mathfrak{I}}}_{0}\}$		${\mathcal{G}}\{{{\mathfrak{I}}}_{0}\}$		Orbit D'
1945 Sep 12	7h390	−10°00'	2.860	2.466	201	2585	15.7	19.7	−0S.31	+25	−2S.8	+117''	−0S.25	+103
22	7 50 . 1	−11 48	2.592	2.310	22.7	263.5	15.2	19.0	−0.31	+2.7	−2.8	+116	−0.26	+10.3
Oct 2	8 01 . 6	−14 03	2.315	2.148	25.6	268.2	14.6	18.2	−0.32	+2.8	−2.9	+115	−0.27	+10.4
12	8 13 . 7	−16 54	2.031	1.980	28.8	272.8	14.0	17.3	−0.33	+3.0	−3.0	+113	−0.29	+10.4
22	8 27 . 0	−20 36	1.744	1.804	32.5	277.2	13.3	16.3	−0.34	+3.2	−3.1	+110	−0.31	+10.4
Nov 1	8 42 . 3	−25 33	1.458	1.619	37.2	281.5	12.4	15.1	−0.35	+3.4	−3.2	+107	−0.35	+10.3
11	9 02 . 4	−32 30	1.180	1.423	43.5	285.4	11.4	13.6	−0.38	+3.5	−3.3	+94	−0.40	+10.0
21	9 35 . 1	−42 48	0.922	1.213	53.0	286.9	10.2	11.9	−0.42	+3.2	−3.5	+71	−0.53	+8.9
Dec 1	10 54 . 6	−57 49	0.713	0.982	69.1	277.1	8.7	9.7	−0.46	+1.5	−3.3	+30	−0.90	+5.6
6	12 34 . 0	−65 04	0.645	0.855	80.7	256.5	7.9	8.5	−0.36	+0.1	−2.1	+8	−1.22	+1.7
11	15 11 . 8	−65 27	0.617	0.719	94.6	219.8	7.0	7.0	−0.03	+0.1	−0.1	+1	−0.97	−3.4
16	17 12 . 4	−56 11	0.640	0.569	108.9	189.8	6.1	5.3	+0.04	+1.7	−0.3	−4	−0.39	−5.9
1946 Jan 5	18 32 . 1	−42 17	0.662	0.434	126.4	201.3	5.0	3.4	+0.99	−8.7	+9.8	+204	+1.10	+19.9
10	19 14 . 6	−55 37	0.566	0.601	114.8	184.7	6.1	5.5	+2.02	−7.4	+22.1	+219	+2.12	+22.3
20	0 37 . 9	−69 21	0.525	0.882	84.7	104.0	7.6	8.2	+0.66	+17.4	+36.6	+508	+1.78	+44.0
30	3 32 . 8	−48 43	0.664	1.124	60.6	$70.5$	9.1	10.6	−1.47	+4.7	−2.8	+723	−0.87	+55.6

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Our extensive orbit examination in Section 6 allows us now to extrapolate, with a fair degree of confidence, the comet's motion way beyond the period of December 11–15 and to realistically estimate uncertainties of an ephemeris as a function of time. The nominal geocentric ephemeris of the comet for the period of time from 1945 September 12.0 TT to 1946 January 30.0 TT, presented in Table 11, is computed from the nongravitational solution ${{\mathcal{N}}}_{\mathrm{mod}}^{({\rm{N}})}[0.05]\{{{\mathfrak{I}}}_{0}\}$ (whose elements and residuals are in Table 10), but the differential ephemerides, derived from the solution ${{\mathcal{N}}}_{\mathrm{std}}^{({\rm{N}})}\{{{\mathfrak{I}}}_{0}\}$ (elements and residuals in Table 9), from the gravitational solution ${\mathcal{G}}\{{{\mathfrak{I}}}_{0}\}$ (Table 7), and from the orbit ${{\rm{D}}}^{\prime }$ (see the text near Table 6), are also included for comparison. The two numbers listed in Table 11 for each of these three additional solutions are the corrections in R.A. and decl., respectively, that are to be added to the nominal coordinates in columns 2 and 3; note the different units used.

The first six columns of Table 11 are self-explanatory. The seventh column gives the position angle of the prolonged radius vector (direction away from the Sun), and the eighth and ninth columns include a predicted apparent magnitude on two assumptions for the brightness variations: as an inverse fourth and inverse seventh power of heliocentric distance, with no phase effect. Both versions were normalized to the comet's reported magnitude 7 at discovery on December 11, which predicts a magnitude of 9.5 at 1 AU from both the Sun and the Earth for the inverse fourth-power law and 10.6 for the inverse seventh-power law (see Section 7.2).

We find the agreement among the ephemerides based on the orbit ${{\rm{D}}}^{\prime }$ and the two nongravitational solutions quite remarkable. On the other hand, the ephemeris from the gravitational solution based on the same observations as the two nongravitational orbits is very disappointing, differing from the other three by as much as 02 over the period of 4.5 months. We note that the three consistent ephemerides come from the orbital sets that imply that the line of apsides lies within 001 of the reference value, while the ${\mathcal{G}}\{{{\mathfrak{I}}}_{0}\}$ set leaves an offset of nearly 008.

To further appraise the magnitude of ephemeris uncertainties, we completed 200 Monte Carlo runs with pure gravitational solutions fitted to the $\{{{\mathfrak{I}}}_{0}\}$ set of observations, assuming, conservatively, measuring errors of up to ±9${}^{\prime\prime }$ in the December 11–12 positions and up to ±25 in the December 14–15 positions. The standard deviations from the 200 clones vary, in the 1945 September–November period, between 24 and ${3}^{\prime }$ in R.A. and between 1${}^{\prime }.$4 and ${5}^{\prime }$ in decl.; in the 1946 January period, the range is from ${6}^{\prime }$ to nearly ${20}^{\prime }$ in R.A. and from ${8}^{\prime }$ to ${31}^{\prime }$ in decl. On the average the offsets of the ${\mathcal{G}}\{{{\mathfrak{I}}}_{0}\}$-based ephemeris in Table 11 amount to about 40% of these standard deviations. To sum it up, we believe that our nominal ephemeris is accurate, in a worst-case scenario, to a few arcminutes over the entire period of interest to us.

The next step is a search for all patrol plates whose fields cover (or could cover) the comet's ephemeris position. Ideally, the physical size of each plate and its scale provide the angular distance from the center to an edge as a function of a position angle. It is in principle very easy to establish whether the comet's image is or is not located within the plate's limits. In reality, the problem is less straightforward because of a potentially significant error in the reported position of the plate center. Because of this uncertainty (much larger than in the ephemeris), the comet's predicted position may not be within the limits of a plate when it should, or vice versa. The search is further complicated by the comet's unknown brightness, the observing conditions (the sky transparency, the moon interference, the limiting magnitude, etc.), the exposure time, and the distribution of field stars (needed for the astrometry), all of which determine whether the comet's image, even if present on the plate, can in fact be recognized, astrometrically measured and reduced, and photometrically evaluated.

Table 12 presents information on a total of 18 plates that may show an image of C/1945 X1 or its debris, including the unreduced plate AM 25208 from December 14 (Table 1) but not the five from Table 4. Because of the position uncertainties, we include plates that nominally miss the comet's position by up to 25.

These are all patrol plates, with very large fields of view (Section 3). The Ross-Fecker plates (RB) cover a rectangle of 2232 in R.A. and 2790 in decl. (∼600 square degrees), the Cooke plates (AM) 3448 by 4310 (∼1500 square degrees). The column of Table 12, titled Plate, shows whether the comet's predicted position, given by the standard polar coordinates relative to the plate center, is (ON) or is not (OFF) within the limits of the plate. The column Edge indicates the distance of the comet's predicted position from the plate edge. The smaller the number in an ON case, the more difficult it will be to identify and astrometrically evaluate the image because of optical imperfections far from the optical axis of the instrument and a potential lack of appropriate field stars. The smaller the number in an OFF case, the greater is a chance that the image could, after all, appear on the plate because of the positional uncertainties.

We list a total of seven preperihelion plates on which the comet should show up if bright enough and four such plates with the comet's predicted location barely missed; see Section 7.2 for the brightness constraints. After perihelion there are only two potentially useful plates by the end of January to check whether any material part of the head survived; two additional plates just miss the comet's expected position. On one of the plates taken on January 21, we predict that the comet's location is not far from the center. The comet's head should also appear on five plates exposed during February (AM 25269 on the 7th, AM 25272 and AM 25273 on the 8th, RB 14224 on the 19th, and RB 14246 on the 27th), but there is no point in examining these unless the comet shows up, rather unexpectedly, on the January plates.

Finally, there is a group of three plates (footnotes e and f in Table 12) that miss the predicted position of the comet's head by a wide margin but may be relevant to a search for traces of the comet's dust debris (Section 8.2). This group includes the January 8 plate (Table 1), taken with the Ross-Fecker camera specifically for the purpose to recover C/1945 X1 (Section 3). Another plate, taken almost simultaneously with the Metcalf refractor (Table 1), is not listed, because it is deemed useless on account of its small field.

To predict the brightness of a comet is always risky, as has amply been documented by historical examples. For the Kreutz sungrazers, the task is further complicated by apparent differences between the shapes of the preperihelion light curves of a prominent Kreutz comet (observed as a bright object from the ground) and a dwarf sungrazer (observed only from SOHO/STEREO), as suggested by Knight et al. (2010). However, a preperihelion light curve is known for only one prominent sungrazer, C/1965 S1 (Sekanina 2002).¹⁰ It is not advisable to generalize this light curve as a standard attribute of all bright Kreutz sungrazers. This caveat is supported by the perceived differences among the post-perihelion light curves for five prominent Kreutz objects (C/1843 D1, C/1882 R1, C/1963 R1, C/1965 S1, and C/1970 K1), which were fading with heliocentric distance at average rates from ${r}^{-3.3}$ to ${r}^{-5.1}$ (Sekanina 2002).

Knight et al. (2010) concluded that the rate of brightening of the dwarf Kreutz sungrazers observed with the SOHO coronagraphs changes strikingly near 24 ${R}_{\odot }$ (0.11 AU) from the Sun from $\sim {r}^{-7.3}$ to $\sim {r}^{-3.8};$ this slower rate of brightening then holds, according to them, down to 16 ${R}_{\odot }$ (0.075 AU), where the light curve begins to flatten. They estimated that the steep rate is unlikely to persist at heliocentric distances greater than 50 ${R}_{\odot }$ (0.23 AU) and that farther from the Sun the dwarf sungrazers brighten on their way to perihelion by again following an ${r}^{-3.8}$ law. However, Ye et al. (2014) suggested that there is a significant diversity in the preperihelion light curves of the dwarf Kreutz sungrazers. They called attention to either an outburst (whose amplitude exceeded 5 mag) or an extremely steep brightening (${r}^{-11}$ or steeper) of a dwarf sungrazer C/2012 E2 between 1.06 and 0.52 AU from the Sun, while the object followed, on the average, an $\sim {r}^{-4}$ (or flatter) law during its approach to perihelion between at least 0.52 AU (rather than 0.23 AU) and 0.07 AU. Ye et al. also pointed out that another dwarf Kreutz sungrazer, C/2012 U3, was not detected at a heliocentric distance of 1.31 AU before perihelion and was then much fainter than it should have been if it followed the Knight et al. (2010) prediction.

Very illuminating is Ye et al. (2014)'s comparison of the light curves of C/2012 U3 and C/2011 W3 (Lovejoy). Although C/2011 W3 was not a dwarf sungrazer, it brightened according to an ${r}^{-6.9}$ law (Sekanina & Chodas 2012) from 0.75 AU down to 0.34 AU, thus behaving very differently from the prominent sungrazer C/1965 S1 (${r}^{-4.1}$ between 1.02 and 0.03 AU; Sekanina 2002). It appears that there is (i) a distinct possibility that, among the dwarf sungrazers, the steep, ${r}^{-7}$ law applies to heliocentric distances of around 1 AU and possibly even farther away from the Sun; and (ii) at least some of these objects appear to be subjected to outburst-like brightening at moderate distances from the Sun on their way to perihelion.

The potential implications for the brightness evolution of C/1945 X1 are obvious. A major obstacle to accepting this object as a dwarf sungrazer is its considerable brightness reported at the time of discovery. Comparison with C/2011 W3 shows that C/1945 X1 was intrinsically (i.e., after a correction has been applied for the geocentric distance) fully 4.5 mag (!) brighter at the same heliocentric distance. Assuming that the reported estimate is correct, this problem can only be overcome if one accepts that C/1945 X1 experienced a major outburst some time before the discovery, which was thereby substantially facilitated. If the amplitude of the outburst was greater than ∼4.5 mag, and therefore comparable to that of C/2012 E2, C/1945 X1 would have been fainter than C/2011 W3 before the onset of the outburst. From the standpoint of chances that C/1945 X1 could be detected on any of the pre-discovery plates in Table 12, the pivotal parameter is the time when the putative event took place. If this general scenario is correct, the comet's unusual brightness is indeed explained, but there is rather a slim chance of detecting the comet on any of the pre-outburst plates.

An alternative explanation of the reported magnitude of C/1945 X1, if it should be a dwarf sungrazer, is simply a gross overestimation of its brightness by the discoverer. In any case, these considerations show the critical need for a thorough photometric examination of the comet's Boyden images, both those from the period of December 11–15 and any pre-discovery ones.

Keeping all our options open, we illustrate the dramatic differences, over a wide range of heliocentric distances, between the C/1945 X1 brightness predictions based on the ${r}^{-4}$ and ${r}^{-7}$ laws by listing the apparent magnitudes in columns 8–9 of Table 11. One should, however, be able to discriminate between the two laws over a much shorter range of r. In fact, in the known Boyden images taken over the period of December 11–15, the comet should have brightened by 0.7 mag if the ${r}^{-4}$ law applied, but by 1.3 mag if the steeper law was in effect. The ephemeris suggests that C/1945 X1 might have been as bright as magnitude 15 in the second half of September, about 3 months before perihelion. In order that no image be missed, we began our search for plates that fit the predicted position of the comet as early as mid-September, since the Ross-Fecker camera, the more powerful of the two patrol instruments, has nominally a limiting magnitude 15 as well (Section 3).

Based on large numbers of observations, the members of the Kreutz system are usually divided into two categories: prominent (bright and often quite spectacular as seen from the ground) and dwarf (defined in Section 1). This is a very simplistic classification, because it ignores intermediate objects, which occupy the transition between both categories and, albeit scarce, are momentous for a better understanding of the disintegration process of the Kreutz sungrazers.

In an effort to determine where C/1945 X1 is likely to fit in, we selected three "standards" to define a scale for transition objects in the order of increasing similarity to the dwarf members. The standards were chosen to be represented by C/2011 W3, C/1887 B1, and C/2007 L3. Next we describe their diagnostic properties, which will be searched for on the best-suited plates of C/1945 X1 in order to classify the comet on this scale.

C/2011 W3 was the most persistent and nearest the status of a prominent member among the three objects. With a perihelion distance of 1.19 ${R}_{\odot }$, this comet survived intact (without its nucleus disintegrating completely) until 38 ± 5 hr after perihelion, at which time it suddenly fell apart in a terminal outburst, losing suddenly its residual, ∼10¹² g nucleus and head (Sekanina & Chodas 2012). The tail, made up of microscopic dust particles ejected from the nucleus before perihelion, was seen to survive perihelion for at least 24 hr as a syndyname—a locus of dust released at different times but subjected to the same radiation pressure acceleration β—of $\beta =0.6$ the solar gravitational acceleration, typical for dielectric submicron-sized particles. Only grains ejected less than 0.1 days before perihelion, which approached the Sun within 1.8 ${R}_{\odot }$, sublimated away. After perihelion the comet formed a new, spectacular, headless tail, most of which was a product of the event ∼38 hr after perihelion. This tail was a synchrone—a locus of dust grains of different sizes (subjected to a broad range of radiation pressure accelerations), all released at the same time—that was under observation for up to 90 days, until mid-2012 March. Its representative length during most reported sightings again suggested a maximum radiation pressure acceleration of $\beta \simeq 0.6$ the solar gravitational acceleration. Only during the late observations (60–90 days after perihelion) did the visible tail consist of merely micron-sized and larger grains, whose $\beta \ll 0.6$ the solar gravitational acceleration.

C/1887 B1 is the runner-up to C/2011 W3. Available information is rather limited, because this object was discovered only after perihelion as a headless, narrow, ribbon-like tail, again a synchrone. Nonetheless, the systematic variations in this tail's orientation with time suggest that the nucleus survived intact until 5.8 ± 0.8 hr after perihelion (Sekanina 1984), at which point it must have suddenly disintegrated in a terminal outburst similar to that of C/2011 W3. At perihelion the comet was merely 1.04 ${R}_{\odot }$ from the Sun (Sekanina & Chodas 2004). The tail, truly spectacular during the early sightings from 8.5 days after perihelion on, was under observation until 18.5 days after perihelion, and its length implied a maximum radiation pressure acceleration of dust particles that dropped from ∼0.4 the solar gravitational acceleration at discovery to ∼0.05 during the last three reported observations. The minimum particle size in the visible part of the tail was thus increasing with time from less than 1 μm to several microns.

C/2007 L3 is, of the three standards, the most remote from the prominent Kreutz-system members, being in fact a representative of a relative small group of bright dwarf Kreutz sungrazers, which develop long, extremely narrow tails upon their approach to perihelion and which we refer to hereafter as the superdwarfs. Like ordinary dwarf sungrazers, their nuclei fail to survive perihelion, so that they form no post-perihelion tails. We chose C/2007 L3 as a representative primarily because it was imaged by both SOHO and STEREO, unlike most of its peers. The comet had a perihelion distance of 1.530 ${R}_{\odot }$ (Marsden 2008), very close to our best estimate for C/1945 X1, its brightness peaked at magnitude ∼3, and its head disappeared about 80 minutes before perihelion, at a heliocentric distance of 2.75 ${R}_{\odot }$ (Green 2007). The SOHO and STEREO images were studied in considerable detail by Thompson (2009), who closely confirmed the earlier conclusions by Sekanina (2000), based on several objects, that preperihelion images of these tails show that they are synchrones, referring to emission-shutoff times of 26 ± 7 hr before perihelion and to heliocentric distances at shutoff of 23.5 ± 4.5 ${R}_{\odot }$.¹¹ This average agrees remarkably well with Knight et al. (2010)'s distance at which the slope of the light curves suddenly changes from −7.3 to −3.8 (Section 7.2).¹² The tail of C/2007 L3 survived the comet's perihelion, as did the preperihelion tail of C/2011 W3. Thompson examined its remnant and found, unlike Sekanina & Chodas (2012) for C/2011 W3, that the feature remained synchronic to its last examined image, nearly 17 hr after perihelion, but that its position was slightly modified by a loss of angular momentum, possibly due to atmospheric drag from the solar corona. The apparent discrepancy is explained by the fact that the dust emission of C/2011 W3 did not shut off about 1 day before perihelion, but continued. Both Sekanina (2000) and Thompson (2009) detected the maximum radiation pressure accelerations of $\beta \simeq 0.6$ the solar gravitational acceleration. Comet C/2007 L3 was not the only Kreutz superdwarf whose dust tail was observed to survive perihelion. Other examples are C/1979 Q1, the first Solwind comet (Michels et al. 1982), C/1998 K10, one of the dozen comets examined by Sekanina (2000), etc. It is likely that most, if not all, superdwarfs would show their tails surviving perihelion, if their images are closely analyzed. About 10 such comets from 1995–2005 in Knight et al. (2010)'s Table 2 and all 19 sungrazers from 2006–2013 in Sekanina & Kracht (2013)'s Table 1 belong to this subcategory, which is at present estimated to include about three dozen objects.

So where does C/1945 X1 fit in? Employing the constraint that no tail was ever reported with the unaided eye after perihelion, it is certain that this comet was too faint to be placed between C/2011 W3 and C/1887 B1 or even near C/1887 B1. Since a preperihelion tail is always found to survive for only 1 day or so after perihelion, there is no chance of detecting it on any of the plates listed in Table 12. Should we find traces of the comet's synchronic tail, a product of a post-perihelion outburst, on any of these plates, then C/1945 X1 is to be classified on this three-point scale between C/1887 B1 and C/2007 L3, in which case it was a transition object more massive than a superdwarf. If not, it has to be concluded that C/1945 X1 cannot be positioned higher than C/2007 L3, so that, at best, it was a superdwarf.

From these considerations it follows that while a search for, and examination of, the comet's preperihelion images on the plates listed in Table 12 should serve to improve our understanding of the comet's orbital motion and its light curve, a careful search for traces of the comet on, and examination of, the plates taken after perihelion should provide information on the comet's fate and help answer the question of whether C/1945 X1 was a transition object or a dwarf sungrazer; because of the lack of information on the comet's appearance in close proximity of perihelion (Table 12), we cannot distinguish between its classification as a dwarf or a superdwarf.

Table 12.  Additional Boyden Plates with Possible Images of Comet C/1945 X1

		Plate Centerb				Comet's Position (Absolute and Relative)${}^{{\rm{b}},{\rm{c}}}$
Plate Numbera	UT Time at Mid-exposure	R.A.	Decl.	Exp. Time (minute)	Logbook Reference	R.A.	Decl.	Plate	Dist.	P.A.	Edge	Observerd
RB 14049	1945 Sep 19.11997	8h046	−15°17'	60	rb10_148	7h469	−11°13'	ON	591	3126	685	Bl(?)
AM 25110	Oct 10.07683	7 04 . 5	−15 09	85	am45b_0144	8 11 . 4	−16 18	ON	16 . 11	96 . 3	1 . 18	du Toit
RB 14086	17.07399	8 04 . 0	−30 17	120	rb10_132	8 20 . 3	−18 39	ON	12 . 20	18 . 5	2 . 33	J(?)
AM 25132	27.06607	9 03 . 5	−45 24	60	am45b_0148	8 34 . 4	−22 55	OFF	23 . 25	342 . 8	0 . 71	Bl(?)
AM 25145	Nov  5.05881	9 04 . 2	−30 24	90	am45b_0150	8 49 . 7	−28 04	ON	3 . 93	305 . 4	13 . 99	Bester
AM 25150	12.06740	10 04 . 0	−45 29	80	am45b_0152	9 05 . 1	−33 25	ON	16 . 54	311 . 8	4 . 87	J(?)
RB 14121	13.03559	8 03 . 2	−45 17	120	rb10_138	9 07 . 6	−34 16	OFF	16 . 51	53 . 8	2 . 12	Bl(?)
AM 25169	15.06371	10 33 . 7	−60 31	85	am45b_0154	9 13 . 4	−36 11	OFF	27 . 50	323 . 1	0 . 50	J(?)
AM 25187	Dec  7.04693	11 04 . 6	−45 32	90	am45b_0156	13 04 . 0	−65 58	OFF	25 . 94	152 . 4	1 . 49	du Toit
AM 25190	8.01580	10 33 . 7	−60 31	90	am45b_0156	13 34 . 2	−66 28	ON	20 . 50	126 . 1	0 . 63	J(?)
AM 25208	14.03336	16 40 . 9	−70 12	40	am45b_0158	16 34 . 1	−60 38	ON	9 . 60	355 . 0	11 . 94	J(?)
AM 25231e	1946 Jan  5.04600	12 05 . 2	−70 33	90	am45b_0162	18 32 . 4	−42 24	OFF	52 . 65	112 . 7	31 . 37	Bl(?)
RB 14183e	6.04985	12 05 . 2	−60 33	115	rb10_148	18 38 . 1	−44 57	OFF	55 . 57	121 . 9	35 . 99	Bl(?)
RB 14184f	8.07729	17 17 . 5	−47 19	40	rb10_148	18 53 . 5	−50 19	OFF	16 . 02	109 . 7	3 . 88	Bester
AM 25240	21.79113	0 05 . 1	−44 27	90	am45b_0164	1 36 . 3	−66 22	OFF	25 . 13	158 . 5	1 . 89	Britz
AM 25241	21.84446	1 33 . 6	−59 29	60	am45b_0164	1 37 . 8	−66 15	ON	6 . 78	176 . 5	14 . 73	Britz
AM 25244	24.85635	5 25 . 4	−68 55	90	am45b_0164	2 40 . 0	−59 43	OFF	19 . 62	277 . 3	2 . 26	Bl(?)
AM 25260	30.85589	5 03 . 9	−29 52	90	am45b_0166	3 38 . 4	−47 05	ON	23 . 86	217 . 8	2 . 56	Bl(?)

Notes.

^aAM plates taken with the Cooke lens (covering 3448 in R.A. and 4310 in decl.); RB plates with the Ross-Fecker camera (2232 by 2790). ^bEquinox J2000. ^cColumn Plate indicates whether or not the comet's predicted position is within the plate limits; columns Dist. and P.A. show the predicted position relative to the plate center; and in column Edge is the distance of the comet's image from the plate edge (either on or off the plate). ^dThe identities of the observers whose abbreviations are Bl and J could not be determined, major efforts notwithstanding; see also Table 1. ^eThis plate may show a segment of an early post-perihelion tail, if the comet still produced significant amounts of dust (see Figure 6 or 7). ^fThis is the only deliberate attempt at detecting C/1945 X1 after perihelion; unfortunately, the searched position is incorrect (see Figure 5).

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Before we discuss specific search opportunities, we note that the dust released from this comet before perihelion was affected by sublimation less severely than the dust from C/2011 W3. The orbital elements for C/1945 X1's preperihelion ejecta subjected to radiation pressure accelerations $\beta =0.6$ the solar gravitational acceleration are summarized in Table 13. It follows that if this comet's dust had sublimation properties similar to those of the dust from C/2011 W3, only particles ejected within ∼1 hr of perihelion should have sublimated completely, because of a greater perihelion distance of C/1945 X1. A major delay of several days in passing through perihelion is noted as a yet another effect: for example, particles ejected 50 days before the comet's perihelion did not reach their own perihelion until late on January 5.

Table 13.  Orbital Elements of Dust Particles Subjected to $\beta =0.6$ and Ejected from Comet 1945 X1 up to Perihelion

Time of Ejection (days)a	Time of Perihelion (days)a	Perihelion Distance (${R}_{\odot }$)	Orbital Eccentricity
−50	+8.868	3.757	1.035
−40	+7.142	3.749	1.041
−30	+5.410	3.732	1.049
−25	+4.540	3.721	1.055
−20	+3.667	3.706	1.064
−15	+2.788	3.682	1.077
−10	+1.899	3.641	1.101
−7	+1.357	3.596	1.127
−5	+0.989	3.545	1.157
−3	+0.610	3.450	1.217
−2	+0.412	3.356	1.279
−1	+0.203	3.134	1.426
−0.7	+0.136	3.027	1.527
−0.5	+0.089	2.896	1.642
−0.3	+0.041	2.673	1.861
−0.2	+0.017	2.481	2.082
−0.1	−0.004	2.144	2.567
−0.05	−0.010	1.843	3.150
−1h	−0.009	1.779	3.299
0.0	0.000	1.530	4.000

Note.

^aReckoned from the time of the comet's perihelion passage.

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As to the sources of information on comet C/1945 X1 after perihelion, a remote opportunity occurred on 1946 January 3.51 UT, the time of a solar eclipse.¹³ Unfortunately, it was merely partial, of magnitude 0.553, and visible only from Antarctica and surrounding oceanic regions. Even less of the Sun was occulted at the locations of several permanent British scientific stations in Antarctica near the southernmost tip of South America (Operation Tabarin; e.g., Haddelsey 2014). At J. Shanklin's suggestion we checked the UK Met Office Archives in Exeter and, not surprisingly, found no report on any relevant observations at the time.

Back at Boyden, an attempt was made by M. J. Bester on January 8 to recover the comet; the search, as pointed out (Sections 3 and 7.1), was conducted with the Metcalf refractor and the Ross-Fecker piggyback camera. Both exposures pointed in the same direction because they overlapped each other; the Metcalf 45-minute exposure began 20 minutes earlier and was longer by 5 minutes. The pointing of the telescope is, however, a mystery: as performed, it could not accommodate the comet's predicted position within the limits of either plate.

January 8 was the date of publication of Cunningham (1946b)'s ephemeris, but early in the morning Bester could not have been in the possession of the official announcement from Copenhagen. Yet, the telescope was set to point precisely in accord with the ephemeris in decl. and was—most peculiarly—off by exactly 2 hr to the west of the ephemeris place in R.A.. Such a coincidence surely cannot be accidental. We presume that Bester was notified directly by Cunningham ahead of the ephemeris' publication, perhaps by cable, but there was a miscommunication, an error in one coordinate. In any event, the time of Bester's observation was fundamentally incorrect because during the exposures the comet, less than 30° from the Sun, was below and, later, very close to the horizon. Since the comet was almost exactly south of the Sun, it could have been observed either shortly after sunset or, somewhat preferably, just before sunrise. The need to take a fairly long exposure did of course constrain the choice of the observation time. If observed at the correct time with the telescope set to point in accord with Cunningham's ephemeris in both coordinates, the comet's position would have been exposed on both plates, even though only ∼05 from the edge of the Metcalf plate.

Figure 5 displays the projected positions of the comet and the Sun relative to the pointing direction of the plate RB 14184, taken with the Ross-Fecker camera on January 8, 11 days after perihelion. Although dust ejecta are predicted to be all over the plate, their detection, contingent on the time-dependent emission rate, is problematic on account of their generally low surface brightness, except when released during a sharply peaked event, such as a terminal outburst. The curves emanating in the figure from the comet's head and distributed mostly in the lower part of the plate are the near-perihelion synchrones. Their selected range is from 1.2 days before perihelion to 1.6 days after perihelion. Few post-perihelion ejecta project onto the plate, barely touching its southeastern corner, and only a limited fraction of the perihelion ejecta crosses the plate. On the other hand, much of the dust released as far back as ∼40 days before perihelion is projected onto the plate. We emphasized in Section 8.1 that the preperihelion tails of the transition object C/2011 W3 and of C/2007 L3 and many other (if not all) superdwarfs survived perihelion for not much longer than ∼1 day at the most. Such features are too close to the Sun to observe from the ground. It is only for the sake of interest that Figure 5 shows the syndyname $\beta =0.6$, made up of dielectric submicron-sized particles ejected from C/1945 X1 before perihelion, to miss the field of the sky covered by the plate RB 14184, nearly grazing it along the entire northern edge.

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The plate RB 14184 may prove useful. If C/1945 X1 experienced a terminal outburst and disintegrated within an hour or two after perihelion and if the ejecta contained a large abundance of grains several microns in diameter, there is a possibility that this debris could show up as a band of light across the lower left corner of the plate. The phase angles favor a modest forward-scattering effect, being mostly in a range of 100°–110°. Submicron-sized ejecta released from the comet at the same times should miss the plate. Thus, a detection or nondetection of a post-perihelion synchronic tail could provide meaningful constraints on the existence of a terminal outburst and on the time of nucleus disintegration.

We found two more plates that might provide additional limited information on a potential post-perihelion terminal outburst: AM 25231, taken with the Cooke lens in the morning of January 5, and RB 14183, taken with the Ross-Fecker camera 24 hr later. They both are listed in Table 12. The synchrones for the January 5 plate are depicted in Figure 6, those for the January 6 plate in Figure 7. The positions of dust particles relative to the comet's nucleus, determined by a computer code in terms of their polar coordinates, were converted to the polar coordinates of the offsets from the plate center, using a technique described in some detail in the Appendix. An advantage of these plates relative to that on RB 14184 is their exposure times, 115 and 90 minutes, respectively, as opposed to only 40 minutes (Table 12). The January 5 plate might provide useful information only if the terminal outburst took place between, at most, ∼2 and ∼4 hr after perihelion; a synchronic tail containing earlier ejecta would miss this plate, while segments of later synchrones limited to $\beta \gt 0.6$ the solar gravitational acceleration are unlikely to refer to any real tails (Section 8.1). Similarly, the January 6 plate might provide information only on an outburst that occurred between ∼1 and ∼5 hr after perihelion. No forward-scattering effect can be expected in either case, as the phase angles range between about 60° and 80°.

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In the highly unlikely case that a major fragment of the original nucleus survived for weeks after perihelion, the best search opportunity is offered by a Cooke plate AM 25241, taken on January 21, on which the comet is predicted to be less than 7° from the center (Table 12). Less favorable circumstances should accompany a search on a plate AM 25260, taken 9 days later, on which the comet's predicted position is only 25 from the edge. Additional search opportunities are more discouraging still.