Issue 
A&A
Volume 589, May 2016



Article Number  A100  
Number of page(s)  10  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201527526  
Published online  20 April 2016 
A study to improve the past orbit of comet C/1917 F1 (Mellish) on the basis of its observed meteor showers
^{1} Astronomical Institute, Slovak Academy of Science, 05960 Tatranská Lomnica, Slovakia
email: ne@ta3.sk
^{2} IMCCE, 77 Av. Denfert Rochereau, 75014 Paris, France
email: Jeremie.Vaubaillon@obspm.fr
^{3} Astronomical Institute, Slovak Academy of Science, Dúbravská cesta 9, 84504 Bratislava, Slovakia
email: astromia@savba.sk
Received: 8 October 2015
Accepted: 8 February 2016
Context. Periodic comets are known to be the parent bodies of meteoroid streams. The stream of a given comet can split into several filaments. These can be observed in the Earth’s atmosphere as more than just a single meteor shower. One such comet is C/1917 F1 (Mellish), which associates at least two, possibly four, meteor showers that have been recorded in the meteor databases. In a recent study, the dynamical evolution of the C/1917 F1’s theoretical stream was followed by only considering the gravitational perturbations. The properties of individual filaments of this stream, corresponding to the appropriate meteor showers, were not predicted perfectly.
Aims. To reach better agreement between theory and observation, we repeatedly model the theoretical stream of C/1917 F1. In the modeling, we also include the PoyntingRobertson drag acting on meteoroids and assume an action of the nongravitational effects on the parent comet dynamics. If success was achieved, the modeling could become a tool that would enable us to recover the past orbital history of the parent comet.
Methods. Considering the nominal orbit, as well as several cloned orbits, of the comet C/1917 F1, we modeled its theoretical streams. The modeling was performed for several past perihelion passages. Each modeled stream consists of 10 000 test particles that are influenced by the PoyntingRobertson drag of various strengths.
Results. We achieve a partial improvement in the prediction of the properties of all four meteor showers. The PoyntingRobertson drag helps to improve the match between the theory and observation of three of the four showers. However, when considering the nominal orbit of the parent comet, a perfect match seems to be impossible. A close match in the case of the most problematic shower is achieved using a cloned orbit, but this is not applicable to reality because the simultaneous predictions of the properties of the other three showers fail.
Key words: comets: individual: C/1917 F1 (Mellish) / meteorites, meteors, meteoroids
© ESO, 2016
1. Introduction
The meteoroid stream of periodic comet C/1917 F1 (Mellish) was recently investigated in the paper by Neslušan & Hajduková (2014; NH14, hereinafter). It was found that the stream of this comet crosses the Earth’s orbit in four filaments, which were denoted (and this denotation is also used in the current paper) as F1, F2, F3, and F4. As a result, four meteor showers associated with C/1917 F1 were predicted. One of these showers, the December Monocerotids (No. 19 in the IAU MDC list of established showers; Jopek & Kaňuchová 2014)^{1}, corresponding to F3, clearly exists. Its relationship to the comet C/1917 F1 has been confirmed, before NH14, by several authors (e.g., McCrosky & Posen 1961; Lindblad 1971; Kresáková 1974; Drummond 1981; OlssonSteel 1987).
Identification of another shower, April ρCygnids (No. 348), with the corresponding predicted filament F1, is not very obvious. However, there is at least a strong suspicion that a reliable identification will be possible after a larger set of data becomes available. NH14 used, in general, the photographic IAU MDC (Neslušan et al. 2014), SonotaCo video (SonotaCo 2009), and radiometeor (Hawkins 1963; Sekanina & Southworth 1975; Lindblad 2003, priv. comm.) databases in their shower identification. With the angular distance of mean radiant from the Sun in the activity maximum lower than 70°, ρCygnids were only, however, found in the video data. A similar, nondefinitive conclusion about the April ρCygnids, selected from several catalogs, and their relationship to C/1917 F1 was also made by Hajduková et al. (2015).
Of another two predicted showers, one (F2) is a daytime shower, which seems to be present in the SonotaCo video and radiometeor databases. Currently, no corresponding shower can be found in the IAU MDC list of all showers. In the SonotaCo video data, there is also an indication of a shower that resembles the last predicted filament, F4.
Not only the unclear identifications of filaments F1, F2, and F4, but also an imperfect match between the predicted filament F3 and the December Monocerotids demand improvement in any modeling of the creation and subsequent dynamical evolution of the C/1917 F1’s stream. In NH14, the influence of the PoyntingRobertson (PR) drag was not considered in the dynamics of modeled stream particles, so we consider its influence in this work.
As we show in the following, the PR drag alone does not, however, yield a perfect match, either. In the first part of our modeling, we assumed the past evolution of the nominal osculation orbit of C/1917 F1 only because of the gravitational perturbations of planets. We note that this orbit was determined by Asklöf (1932). It is known with relatively high precision. Asklöf gave three sets of orbital elements determined for the different assumptions and selections of observational data. The largest differences of corresponding values in these three sets are only 0.001627 d in the time of perihelion passage, 14.9′′ in the argument of perihelion, 21.2′′ in longitude of ascending node, and 5′′ in the inclination. The orbital elements for the equinox 2000.0, given on the JPLbrowser^{2} are q = 0.190186 AU, e = 0.993121, ω = 121.3190°, Ω = 88.6683°, i = 32.6828°, and T = 2421329.6751 (April 11.1751, 1917) for epoch JDT = 2 421 333.5.
In the past, this orbit could be significantly influenced and, therefore, modified by nongravitational effects. The currently observed stream could be and, most likely, was formed when the comet moved in a slightly different orbit than it moved at its last return to the perihelion.
To model a theoretical stream and follow its dynamical evolution, we use the same procedure as Neslušan and Hajduková in NH14 (see Sect. 2 in their paper). The procedure starts with the integration of the parentbody orbit backward for time t_{ev}. Then, the nearest perihelion is found and a set of 10 000 test particles (TPs), representing the meteoroids, are assumed to be ejected from the parent nucleus with a small velocity in all directions. Finally, the orbits of the TPs are integrated forward, up to the present. The TPs in the orbits that approach the Earth’s orbit within 0.05 AU at the present time are selected and used to predict the properties of relevant showers.
Our theoretical predictions are evaluated by comparing them to the actually observed meteors. In this comparison, we use the photographic IAU MDC (Neslušan et al. 2014), SonotaCo video (SonotaCo 2009), and radiometeor (Hawkins 1963; Sekanina & Southworth 1975; Lindblad 2003, priv. comm.) databases. In contrast to NH14, we also use the EDMOND video database (Kornoš et al. 2014).
Fig. 1
Positions of radiants of the particles representing the modeled stream of comet C/1917 F1 in orbits, along which they approach the Earth’s orbit within 0.05 AU. The positions are shown for the stream not influenced by the PR drag, which was modeled in time 50 kyr in the past. The Hammer projection of the equatorial coordinate frame is used. The green full circles (red crosses) show the positions of the radiants of observed meteors recorded in the SonotaCovideo (photographic IAU MDC) database. The black empty squares show the positions of the radiants of modeled particles. The sinusoidlike curve illustrates the ecliptic. The positions are shown in two variants. In plot a), the radiants of observed meteors are covered by those of modeled particles; therefore, the latter are seen transparently. In contrast, the radiants of modeled particles are covered by those of observed meteors in plot b). The positions of the radiants of observed meteors are seen transparently here. The positions of observed radiants are identical in all presented figures. 
In this research note, we describe our attempt to find the appropriate strength of the PR drag influencing the dynamical evolution of the C/1917 F1 stream and causing four meteor showers in the Earth’s atmosphere. The strength of the drag is given with the help of parameter β, which is the ratio of the acceleration due to the PR drag and gravitational acceleration of the Sun. Likewise, because of nongravitational effects in general, we try to find the appropriate modification of the orbit of this parent body in the time when the meteoroids of its stream were released. A more detailed description of the problem solved is given in the next section.
This work aims not only to better predict the properties of the showers associated with the parent comet, but also to point out the possibility of an eventual recovery of the past orbital history of the comet by studying its observed meteor showers. If our work or other future work of this kind is successful, then the meteor databases will become a tool for tracing the past dynamical history (caused by the nongravitational effects) of those parent bodies that associate more than a single meteor shower.
2. Description of problem
We first recall the differences between the predicted and observed filaments of the meteoroid stream of comet C/1917 F1 (Mellish), which have already been found in NH14. In their work, the authors exclusively considered the gravitational perturbations of planets on the meteoroid particles.
These differences are discussed in terms of the positions of the radiant areas of individual filaments from F1 to F4. We note that the members of all four filaments were selected from the observational databases considered, using the “breakpoint method” (Neslušan et al. 1995, 2013).
Filaments F1, F2, and F4 can be separated well only in the SonotaCo catalog, and filament F2 also seems to be present in the radio data. The break point is not very apparent when the EDMOND video data are used to separate F1, F2, and F4. Concerning the F3 filament, very similar mean shower characteristics are obtained from both the SonotaCo and EDMOND catalogs, and this rich filament is present in the photographic IAU MDC and radiometeor data, of course.
The positions of the observed, as well as the predicted, radiants are shown in Fig. 1 for the model with the evolutionary time t_{ev} = 50 kyr. From the observed meteors, the photographic and SonotaCo video meteors are shown. Since the radiant area of filament F3 separated from the EDMOND catalog is almost identical to the area separated from the SonotaCo database, the EDMOND video meteors are not plotted in Fig. 1 or in the other figures. The predicted radiant positions represent all TPs approaching the Earth’s orbit within 0.05 AU at present.
We can see that the radiant area of filament F1, which can hypothetically be identified to the April ρCygnids (No. 348) as suggested in NH14, is almost predicted correctly. (There is, nevertheless, a “tail” of radiants in the southeast direction.) The radiant area of filament F2 is shifted toward a larger right ascension and, significantly, toward a higher declination. In addition, in filament F4, a low abundance of meteors is predicted, which does not match the observed radiant area perfectly. The predicted radiant area of filament F3, identified with the December Monocerotids (No. 19), more or less matches its video observed counterpart, although it is shifted slightly southward.
The differences between the prediction and observation described above could occur because of the nongravitational effects, which were not taken into account in NH14. The most relevant nongravitational effects are the PR drag and the rocket effect on the nucleus of the parent comet. The first effect influences the orbits of meteoroids. The second changes the orbit of the parent; therefore, the particles could be released from the parent’s nucleus in a different position and/or a different time than implied by the wellknown nominal orbit of the C/1917 F1. In the following, we trace the action of these effects in the past.
3. Nominal orbit and PR drag
3.1. An initial discussion
The orbital period of C/1917 F1 is about 145 years, so its stream has needed a relatively long time to spread along the whole orbit. This and the relatively short perihelion distance, 0.19 AU, of its orbit imply that it is reasonable to also consider the influence of solar radiation and solar wind on the dynamics of especially smaller particles in the meteoroid stream of this comet. Therefore, we add the acceleration due to the PR drag to the gravitational acceleration of the TPs. To see the influence of this nongravitational force, we constructed the stream models mostly for t_{ev} = 5, 10, 20, and 50 kyr and followed the dynamics of several streams for each t_{ev}, with different values of parameter β, which characterizes the strength of the PR drag.
Fig. 2
Positions of radiants of the modeled particles for different evolutionary times. The positions of radiants are shown in the same way as those in Fig. 1a. The PR drag influenced the motion of modeled particles in this case. The strength of the drag characterized by parameter β was 0.001. The streams with the radiants in plots a)–f) were modeled for times 5, 10, 20, 30, 50, and 70 kyr in the past, respectively. 
Fig. 3
Radiant positions of the modeled particles for different strength of the PR drag. The positions of radiants are shown in the same way as those in Fig. 1a. The PR drag influenced the motion of modeled particles in this case. The strength of the drag characterized with parameter β was 0.0001, 0.001, 0.002, 0.003, 0.005, and 0.007 at the streams with the radiants in plot a)–f), respectively. All the streams were modeled for time 50 kyr in the past. 
In the following, we give a description of the achieved results based on a comparison of predicted and observed radiant areas of individual filaments. The results were obtained after modeling theoretical streams, starting from the nominal orbit of C/1917 F1, and for various values of PR drag parameter β. Specifically, the radial, a_{r}, and transversal, a_{t}, components of the acceleration due to the drag were calculated as a_{r} = β(GM_{⊙}/r^{2})(1−2v_{r}/c) and a_{t} = −β(GM_{⊙}/r^{2})(v_{t}/c), where G is the gravitational constant, r the heliocentric distance of TP, v_{r} and v_{t} are the radial and transversal component of the TP velocity vector, and c is the speed of light. The transversal component of the acceleration is perpendicular to the radial component and lies in the actual orbital plane of the TP. It is oriented in the direction of the TP velocity. The third component of the PRdrag acceleration, which is perpendicular to both radial and transversal components, is considered to be zero.
3.2. An influence of different evolutionary times
The distributions of the current TP radiants in the streams modeled from the nominal orbit of C/1917 F1 integrated backward to times of 5, 10, 20, 30, 50, and 70 kyr before the present, for β = 0.001, are shown in Fig. 2. The radiants of corresponding shower meteors are also shown in this and in all the other figures for a comparison. This investigation of the influence of evolutionary period on the structure of stream confirms the expected behavior. After the short periods, t_{ev} = 5 and 10 kyr, only filament F3, which is situated in an orbital corridor around the orbit of the parent comet, appears.
Increasing evolutionary time and thus a longer action of some systematic planetary perturbations enables moving a significant part of the TPs also into other orbital corridors. For t_{ev} = 20 kyr, filaments F1 and F2 also occur. Modeled filament F2 is, however, far from its observed counterpart. The modeled radiant area tends to agree more with that of observed radiants with the increasing t_{ev}. However, F2 starts to disappear in t_{ev} = 70 kyr when the best match between the radiants of modeled and observed meteoroids is reached. In addition, modeled filaments F1 and F3 become overabundant for the long evolutionary times, 50 kyr and 70 kyr.
3.3. An influence of different strengths of PR drag
If we fix the evolutionary period, e.g. t_{ev} = 50 kyr, and consider various values of parameter β, we can map the influence of the strength of the PR drag on the evolution of stream structure. The distributions of radiants in this case are presented in Fig. 3. Specifically, we show there the radiant distributions for β equal to 0.0001, 0.001, 0.002, 0.003, 0.005, and 0.007.
The dependence on the value of β is weaker than on t_{ev}. One can “tune” the position of radiant areas of the filaments and their relative abundances. Nevertheless, a simultaneous perfect match of all mutually corresponding theoretical and observed filaments hardly appears to be achieved. The best match is seen for β ranging from ~0.001 to ~0.003. Unfortunately, the radiant areas of modeled filaments F2 and F4 are still discernibly displaced from their observed counterparts, and modeled filaments F1 and F3 are much more abundant than observed.
A more detailed description of each modeled stream is given in the following sections.
3.4. Streams modeled 5 kyr before the present
In the stream modeled for t_{ev} = 5 kyr, only the TPs in the F3 filament approach the Earth’s orbit. In more detail, the F3 filament occurred for β = 0.001 (Fig. 2a) and 0.002 (not shown in a figure). For significantly higher β values, all TPs are deflected from the orbit of our planet and, therefore, from the collision course.
3.5. Streams modeled 10 kyr before the present
Almost the same result is obtained for t_{ev} = 10 kyr: only the F3 filament occurs for β = 0.001 (Fig. 2b) and 0.002. For β = 0.003, this filament consists of only a few TPs and completely disappears for a higher β value.
3.6. Streams modeled 20 kyr before the present
The F1 filament occurs, together with F3, for t_{ev} = 20 kyr and β = 0.0002. If the higher value of β = 0.001 is considered, F2 also occurs, but it is largely shifted northward (Fig. 2c). In addition, an unknown, small, compact filament with the radiant area near F3, but more southeastward and below the equator, occurs. Its mean argument of perihelion, longitude of ascending node, and inclination are ω = 124.5°, Ω = 80.4°, and i = 35.2°, respectively. These elements are similar to their DecemberMonocerotid counterparts (cf. ω = 126.7°, Ω = 80.1°, and i = 35.9°). However, the mean perihelion distance, radiant position, and geocentric velocity are different from those of the densest core of the December Monocerotids (cf. q = 0.217 AU, α = 127.6°, δ = −5.1°, and V_{g} = 33.5 km s^{1} and q = 0.189 AU, α = 102.0°, δ = 7.8°, and V_{g} = 41.9 km s^{1}; so while the first radiant area is situated below, the second radiant area is situated above the ecliptic). This filament is not indicated in the databases of the observed meteors. For β = 0.005 and 0.01, all TPs are deflected from the vicinity of the Earth’s orbit and, thus, no filament occurs.
3.7. Streams modeled 30 kyr before the present
The redundant filament, mentioned in the previous section, does not occur for t_{ev} = 30 kyr. Otherwise, the result of the simulations for this t_{ev} is similar to that for t_{ev} = 20 kyr. For the stream influenced with the PR drag characterized with β = 0.001, the distribution of radiants can be seen in Fig. 2d.
3.8. Streams modeled 50 kyr before the present
3.8.1. β = 0.0001
Filament F4 starts to occur in the models constructed for t_{ev} ≥ 50 kyr. Specifically, there are a few meteors of this filament in the case of t_{ev} = 50 kyr and β = 0.0001 (Fig. 3a). In this model, filament F1 is, however, overabundant, and the main accumulation of filament F2 is shifted northnortheastward of the observed radiant area. The prediction does not match the observed area of F3 perfectly either.
3.8.2. β = 0.001, and 0.0014
For β = 0.001 (Fig. 3b), a slightly better agreement than for β = 0.0001 can be seen between the predicted and observed radiant areas of all four filaments. The prediction considering β = 0.0014 (not shown) is almost the same as for β = 0.001, but the overabundance in F1 is not as high, and the agreement in the case of filament F2 is slightly better.
3.8.3. β = 0.002
Although still far from perfect, the best prediction perhaps occurs for t_{ev} = 50 kyr and β = 0.002 (Fig. 3c). The agreement between the theory and observation is quite good, practically sufficient, in the case of filaments F1 and F4. However, it is still not significantly better than for β = 0.0014 for filaments F2 and F3.
3.8.4. β = 0.003 and 0.005
For β = 0.003 (Fig. 3d), the quality of the prediction is similar to one for the previous value of β, but filament F4 is richer and F3 is slightly more shifted westward. The agreement is then worse for even higher β values. For β = 0.005 (Fig. 3e), the westward shift of F3 is a little larger than for 0.003, and F1 becomes less numerous in comparison with its observed counterpart. A certain improvement occurs for F2, but it is still not sufficient. The predicted radiant area of F4 practically matches its observed counterpart, but another small, compact filament occurs in α ~ 160° and δ ~ −50° (in the southeastward direction from F4).
3.8.5. β = 0.007 and 0.01
For β = 0.007 (Fig. 3f), filament F3 is not numerous enough, and its predicted radiant area is completely outside its observed counterpart (but in touch with it). F1 and F4 are also less numerous, and F2 is still displaced. The redundant “southern” filament becomes more numerous than for β = 0.005. Finally, the stream is deflected from the collision course with the Earth, when even higher β values are assumed. For β = 0.01, there are only three TPs in the orbits passing the orbit of the Earth within 0.05 AU.
3.9. Streams modeled 70 kyr before the present
3.9.1. β = 0.0005
Time t_{ev} = 70 kyr seems to be too long. The match is not as good as for t_{ev} = 50 kyr. Considering β = 0.0005, filament F1 is much more dispersed than the observed radiant area. F3 is also largely dispersed. A southward excess of the radiant area occurs with some radiants even below the equator. F4 is quite well populated but apparently shifted westward. The predicted interval of the declination of radiant area of filament F2 agrees with the corresponding observed interval, but it is still situated a little to the side and is clearly less numerous than the observed shower.
3.9.2. β = 0.001 and 0.002
For β = 0.001, we obtain similar results to the previous case, but with another diffuse filament having its radiant area near the equator and southsouthwestward of F1 (Fig. 2f). If β is increased to the value of 0.002, we obtain a better match for filament F1, but filament F2 is smaller and does not match as well.
3.9.3. β = 0.004
Another increase in β, to 0.004, brings some improvement in comparison to 0.002: filament F1 fits the observations better (though still not perfectly), F2 becomes highly abundant but still discernibly separated from its observed counterpart and F3 is situated only northward of the equator, as it should be (the large dispersion disappeared), but its radiant area is still large and in a position apart from its observed counterpart. The redundant filament in the southsouthwestward of F1 is much less abundant than for lower βvalues, but still discernible. In this case, the predicted radiant area of filament F4 matches the observed one quite well.
3.10. Question on November Orionids
Vereš et al. (2011) suggest that comet C/1917 F1 is also the parent body of another established meteor shower, the November Orionids (No. 250), the mean orbit of which is similar to that of the December Monocerotids. In our models, where we also considered the PR drag, we repeatedly searched for the theoretical November Orionids. However, no indication of such a shower occurred in our improved modeling either, which accords with the conclusion already made in NH14 when the PR drag was ignored.
4. Cloning the nominal orbit
As seen in Figs. 2 and 3, which were constructed to show some aspects of the results described in the previous section, the PR drag alone does not, unfortunately, improve the agreement between the theory and observation enough. The discrepancy can, however, also occur if the nongravitational effects, like the rocket effect, affect the orbit of the parent comet. Since we considered its nominal orbit, the site of the release of meteoroid particles from the cometary nucleus could be different from the site (perihelion) considered at the modeling.
4.1. The cloning procedure
In the following part of the simulations, we create the clones of the nominal orbit of C/1917 F1. To obtain the cloned orbits, we modify some nominal orbital elements slightly. Specifically, we vary the eccentricity e, the argument of perihelion ω, the longitude of ascending node Ω, and the inclination i. Because of computational limitations, it is impossible to consider a large number of clones. To reduce this number to a necessary minimum, we do not vary the perihelion distance, q, and the time of perihelion passage, T. Past studies of major meteoroid streams yielded a quite large dispersion of eccentricity and semimajor axis of the stream members, but a significantly smaller variation of perihelion distance, so a change in q can be expected to be lower than those of the other orbital elements, so we ignore it. We do not expect any change in T greater than a few days because of the observational constraint. During such a short time, the main perturbing planets, especially Jupiter, do not change their positions much; therefore, a shift in T most likely does not change the planetary perturbations calculated with respect to T of the nominal orbit much.
We consider three values of the varied elements. Except for the nominal value, the low incremented and decremented values are also considered as the elements of clones. Specifically, the increments or decrements are chosen to be Δe = 0.005, Δi = 1.8°, and Δω = ΔΩ = 3.5°. Because of some preliminary experiments, indicating the cloned orbits with a smaller inclination as more appropriate, we do not consider the incremented value of inclination, but only the values of i−2Δi, i−Δi, and i.
Considering the orbit of each clone as the parent body of its own meteoroid stream, we create the model, for t_{ev} = 50 kyr (a “standard model”) with no PR drag, for each such parent and follow its evolution until the present. Using the TPs, which are in orbits passing around the orbit of the Earth within 0.05 AU at present, we again predict the properties of individual filaments and evaluate the agreement between the mean orbit of each of the four predicted filaments and the corresponding observed mean orbit. The evaluation was done with the help of the SouthworthHawkins (1963) Ddiscriminant. The D values are given in Table 1.
Values of the SouthworthHawkins Ddiscriminant, D_{j}, between the mean observed and modeled orbits, where j denotes the four assumed filaments of the meteoroid stream of C/1917 F1.
With respect to the observational aspect of twocore filament F3 (there are two break points; see Fig. 6 in NH14), we also consider two variants, F3a and F3b, of this filament. The agreement is, subsequently, evaluated for these variants separately. A general agreement is then given as the average, ⟨ D ⟩ _{a}, of the D values for all four filaments (variant F3a is here taken into account). In Table 1, the clones are arranged by increasing ⟨ D ⟩ _{a}value. We can see that there is a better match between the theory and observation for only four clones, those with the serial numbers 4, 3, 17, and 8.
4.2. Models for the cloned orbits No. 4 and 18 and PR drag
The best agreement occurred in the case of the cloned orbit with serial number 4 (clone No. 4). Since it is still imperfect, we again consider the influence of the PR drag. In more detail, we use the orbit of clone No. 4 to model the streams for t_{ev} = 10, 20, and 50 kyr and several values of parameter β.
Fig. 4
Radiant positions of the particles modeled using a cloned orbit of the comet. The positions of radiants are shown in the same way as those in Fig. 1a. However, the initial orbit to model the stream was the cloned orbit No. 4 and PR drag influenced the motion of modeled particles in this case. In plot a), the stream was modeled in time 10 kyr before the present and parameter β = 0.002. In plot b), the stream was modeled in time 20 kyr before the present and β = 0.009. 
4.2.1. Models from Clone No. 4 for t_{ev} = 10 kyr
No progress is achieved for t_{ev} = 10 kyr. If we assume β = 0.001, only filament F3 occurs. Increasing β to 0.002 (Fig. 4a), there is a good match of F3, and filament F2 also occurs in the prediction. Its radiant area is, however, displaced from its observed counterpart. No F1 or F4 is present. For β = 0.003, filament F3 is almost “extinct”. F2 becomes quite numerous, but still separated from the observed radiant area. F1 and F4 are still missing. For higher βvalues, the prediction fails even more. Specifically, only F2, with the radiant area aside the observed one, is predicted for β = 0.005. Then, this filament becomes more compact and less numerous for β = 0.007. Finally, for β = 0.009, filaments F1 to F4 disappear completely. Instead of these, a new, redundant, compact filament in α ~ 150° and δ ~ −40° is predicted.
4.2.2. Models from Clone No. 4 for t_{ev} = 20 kyr
In the models for t_{ev} = 20 kyr, we assume a sequence of β values ranging from 0.001 to 0.009 with the step of 0.002 and β = 0.013. Filament 4 is not predicted in any of these models. However, we achieved a partial progress in predicting the most problematic, likely, filament F2. A good agreement between its predicted and observed radiant areas is obtained for β = 0.009 (Fig. 4b), as well as 0.007.
Unfortunately, there is a poor match of filaments F1 and F3. F1 is not numerous enough, and the radiant area of F3 is shifted westward. In addition, a redundant shower with a striplike radiant area having the mean α ~ 160°, and δ< −50° is predicted in all models for t_{ev} = 20 kyr.
4.2.3. Models from Clone No. 4 for t_{ev} = 50 kyr
Models based on the cloned orbit No. 4 for t_{ev} = 50 kyr provide a prediction of all four filaments, but the prediction of hardly predictable filament F2 is always worse than for t_{ev} = 20 kyr and β = 0.009. Specifically, we considered the values β = 0.001, 0.005, and 0.009 for t_{ev} = 50 kyr. For β = 0.001, modeled filaments F1 and F3 are shifted toward higher right ascension; F2 is shifted, a little, toward lower right ascension and higher declination; and F4 consists of only two TPs. There are quite a large number of TPs in the region with the declination δ< −60°.
For β = 0.005, we can state a little better match of F1 and F3 with observation than in the previous case. However, the match is worse for F2. The radiant area of filament F4 is, here, clearly separated from its observed counterpart. It is shifted toward a lower declination. And, there is even a higher concentration of the TPs in region δ< −70°.
The prediction failed for β = 0.009. Only a few TPs represent each of the four filaments. Again, a high concentration of TPs in region δ< −70° is predicted.
4.2.4. Models from Clone No. 18
Seeing most of the results described above, it seems that the largest problem is to achieve a good match for filament F2. The PR drag is able to reduce the declination of radiant of F2 members, but the right ascension typically remains too large. In the next step, we try to model the stream using the cloned orbit No. 18 with one of the lowest values of this parameter in the “standard stream”. Some cloned orbits yielded an even smaller right ascension, but their “standard stream” did not consist of all four observed filaments, in contrast to the cloned orbit No. 18.
Starting from the latter, we simulated several streams for t_{ev} = 10, 20, and 50 kyr and a few values of parameter β. Unfortunately, the attempt to progress with the help of this clone completely failed. No good match was achieved, even in the case of only a single filament. Concerning filament F2, the difference between the predicted and the observed mean inclination of the radiant area appeared to be larger than in the case of nominal orbit.
5. Search for a convergence of the orbits of real meteors
5.1. The principle and method
If a swarm of particles is released from the surface of a parent body, they start orbiting the Sun along very similar orbits. With elapsing time, this similarity decreases, and the orbits become more and more dispersed. If we knew the exact orbits of a set of meteoroids from a stream, we could reconstruct their evolution by integrating these orbits back in time. In such a backward integration, the dispersion of orbits should decrease to reach the original state immediately after the particle ejection.
Fig. 5
Radiant positions of the particles modeled using an orbit obtained by backward integration of observed meteors. The positions of radiants are shown in the same way as those in Fig. 1a. However, the initial orbit to model the stream was that of video meteor SNTC118490, which was obtained via numerical integration of the current orbit for time 30 kyr before the present in this case. The PR drag influencing the motion of modeled particles characterized with β = 0.003 was considered, too. 
In reality, we only know the meteoroid orbits with a certain precision. The uncertainty in their determination means that their backward integration does not result in a smaller, but rather in a gradually larger dispersion. As a result, we usually cannot reveal the actual orbit of the parent body in the time of meteoroid production on the basis of the known orbits of these meteoroids.
Nevertheless, if we know a relatively large number of the orbits of meteoroids in a given stream, there is still a chance that their backward integration will reveal a convergence of at least a fraction of them. Such a convergence could perhaps indicate the parameters of the orbit of the parent in the time of meteoroid production. Considering this possibility, we integrate the sets of known orbits from the photographic and video databases used, which correspond to the filaments F1−F4 of the C/1917 F1 stream, and search to see if an orbit from one filament became similar in 10, 20, 30, or 50 kyr in the past to an orbit (or orbits) from another filament (or filaments). Expressed in more detail, we perform several backward integrations, each for another PR parameter β. Values of β equal to 0, 0.0005, 0.001, 0.005, 0.009, and 0.013 are considered for t_{ev} = 10, 20, and 30 kyr. With respect to the results obtained in these simulations, we considered values β = 0.00001, 0.003, 0.005, and 0.007 for t_{ev} = 50 kyr.
The similarity of the orbits from different filaments in the given past is evaluated with the help of the SouthworthHawkins (1963) Ddiscriminant. A given pair of orbits was regarded as similar if D< 0.2.
5.2. Implied convergence
For t_{ev} = 50 kyr, no single filament currently crossing the Earth’s orbit was predicted. For β = 0.007, the aphelion of the parent meteoroid has increased above 1000 AU so far, therefore any observable meteoroid stream could be expected, so we did not create any model in this case.
A certain improvement could be stated for t_{ev} = 10 and 20 kyr. At any rate, the best convergence occurred for β = 0.005 in time 30 kyr before the present. In this model, the orbit of video meteor No. 118490 from filament F1 became similar, in 30 kyr, to that of video meteor No. 49303 from filament F2. The D value between the orbits of this pair was 0.140. At the same time, the orbit of No. 118490 from F1 became similar to that of video meteor No. 112248 from filament F3. This time, D = 0.099. The sum of these D values was the lowest of all found similarities and equal to 0.239. The elements of these similar orbits are given in Table 2.
Largest convergence of orbits of real meteors found from three filaments (fils. F1 to F3) of the C/1917 F1 stream.
Considering the orbits of meteors from three filaments, which were similar 30 kyr ago, we can suppose that (i) the meteors were released from the parent around this time; and (ii) the orbit of the parent body was very similar to the orbits of these three meteors at that time. As a result, we approximatively identify the orbit of the parent to that of meteor No. 118490. In other words, we assumed that the orbit of C/1917 F1 had the elements given in the first line in Table 2 in a time of about 30 kyr in the past.
5.3. Stream modeling and prediction of filaments
Using the orbit mentioned at the end of the previous subsection, we integrated the position of meteor No. 118490 to the perihelion passed in the time closest to 30 kyr and then modeled the theoretical stream. Subsequently, we followed the evolution of stream particles up to the present and predicted the mean dynamical properties of filaments F1 to F4.
In the first model of this sequence, we assumed β = 0.005. At the end, it was found that the radiants of meteors approaching the Earth’s orbit were dispersed over the whole sky. This was the effect of an ejection speed that was too high, which was implicitly assumed. We note that, in the procedure used, the ejection speed of each TP from the parent surface is assumed to be equal to 0.001 v_{p}, where v_{p} is the heliocentric velocity of the parent in its perihelion. However, in this case, the parent body occurred in the orbit with an extremely small perihelion distance (q = 0.01586 AU) 30 kyr ago. Such a small perihelion distance implied the high ejection speed of 333 m s^{1}, so we performed a new modeling with the ejection speed reduced by a factor of four. At the same time, we assumed a lower strength of the PR drag with β = 0.003.
It appears that the video meteor No. 118490 moved in a very special orbit in the context of parent body of a meteoroid stream. Reducing the ejection velocity had practically no effect. Also in this simulation, the TPs approached the Earth’s orbit from almost all directions (see Fig. 5). The hypothetical parent body considered here is an example of such a parent, which supplies the particles immediately to the sporadic background, without the stage of a compact stream. This type of stream is, however, beyond the scope of our current study.
6. Conclusion
Our attempt to improve the prediction of the properties of meteor showers associated with comet C/1917 F1 by modeling its stream and considering the PR drag, as well as nongravitational effects, only resulted in partial success. When the nominal orbit of the comet is considered, the PR drag clearly helps to improve the prediction in the case of filaments F1, F3, and F4. The prediction for F2 is also better than that without the PR drag, but far from perfect.
We found a cloned orbit, which resulted in a practically perfect prediction of the properties of problematic filament F2. This orbit could be the actual orbit of C/1917 F1 in the appropriate past period owing to unknown nongravitational effects. Unfortunately, the simultaneous perfect match of the other three filaments does not occur in the modeling concerned. In reality, no explanation for this specific case can be made.
Our study illustrates the different ages of particles in various filaments, i.e. the different ages of meteoroids in the various showers originating in the same parent body. While the members of December Monocerotids corresponding to filament F3 can be observed soon after their release from the parent comet, most probably much earlier than after 5 kyr, the April ρCygnids corresponding to filament F1 can occur in the vicinity of Earth’s orbit and collide with this planet not earlier than about 20 kyr. According to our simulations, their typical age is at least about another 10 kyr longer. The particles belonging to filament F4, which correspond to a still unknown shower, can collide with the Earth only after about 50 kyr. Maybe the meteoroids released from C/1917 F1 can hardly survive such a long time in interplanetary space, and this could be the reason the real counterpart of filament F4 is observed to be relatively less numerous.
We have to admit that we investigated only a relatively small set of a huge number of possible combinations of modified orbits, each expanded into an even larger number when a variety of strengths of the PR drag is assumed. A much more robust modeling could possibly reveal a convergence in cloning at a narrow interval of PR drag strength in the future. If such a success was
achieved, the studies of meteoroid streams could become a tool for recovering the history of the orbital evolution of their parent bodies owing to the action of nongravitational effects.
Acknowledgments
J. Vaubaillon is grateful to the CINES staff for the use of the occigen machine on which heavy calculations were performed. L. Neslušan and M. Hajduková acknowledge the support from the Project ITMS No. 26220120009, based on the supporting operational Research and development program financed by the European Regional Development Fund. Their work was also supported by the Slovak Research and Development Agency under contract No. APVV051712 (M.H.) and by the VEGA – the Slovak Grant Agency for Science, grant No. 2/0031/14 (L.N.).
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All Tables
Values of the SouthworthHawkins Ddiscriminant, D_{j}, between the mean observed and modeled orbits, where j denotes the four assumed filaments of the meteoroid stream of C/1917 F1.
Largest convergence of orbits of real meteors found from three filaments (fils. F1 to F3) of the C/1917 F1 stream.
All Figures
Fig. 1
Positions of radiants of the particles representing the modeled stream of comet C/1917 F1 in orbits, along which they approach the Earth’s orbit within 0.05 AU. The positions are shown for the stream not influenced by the PR drag, which was modeled in time 50 kyr in the past. The Hammer projection of the equatorial coordinate frame is used. The green full circles (red crosses) show the positions of the radiants of observed meteors recorded in the SonotaCovideo (photographic IAU MDC) database. The black empty squares show the positions of the radiants of modeled particles. The sinusoidlike curve illustrates the ecliptic. The positions are shown in two variants. In plot a), the radiants of observed meteors are covered by those of modeled particles; therefore, the latter are seen transparently. In contrast, the radiants of modeled particles are covered by those of observed meteors in plot b). The positions of the radiants of observed meteors are seen transparently here. The positions of observed radiants are identical in all presented figures. 

In the text 
Fig. 2
Positions of radiants of the modeled particles for different evolutionary times. The positions of radiants are shown in the same way as those in Fig. 1a. The PR drag influenced the motion of modeled particles in this case. The strength of the drag characterized by parameter β was 0.001. The streams with the radiants in plots a)–f) were modeled for times 5, 10, 20, 30, 50, and 70 kyr in the past, respectively. 

In the text 
Fig. 3
Radiant positions of the modeled particles for different strength of the PR drag. The positions of radiants are shown in the same way as those in Fig. 1a. The PR drag influenced the motion of modeled particles in this case. The strength of the drag characterized with parameter β was 0.0001, 0.001, 0.002, 0.003, 0.005, and 0.007 at the streams with the radiants in plot a)–f), respectively. All the streams were modeled for time 50 kyr in the past. 

In the text 
Fig. 4
Radiant positions of the particles modeled using a cloned orbit of the comet. The positions of radiants are shown in the same way as those in Fig. 1a. However, the initial orbit to model the stream was the cloned orbit No. 4 and PR drag influenced the motion of modeled particles in this case. In plot a), the stream was modeled in time 10 kyr before the present and parameter β = 0.002. In plot b), the stream was modeled in time 20 kyr before the present and β = 0.009. 

In the text 
Fig. 5
Radiant positions of the particles modeled using an orbit obtained by backward integration of observed meteors. The positions of radiants are shown in the same way as those in Fig. 1a. However, the initial orbit to model the stream was that of video meteor SNTC118490, which was obtained via numerical integration of the current orbit for time 30 kyr before the present in this case. The PR drag influencing the motion of modeled particles characterized with β = 0.003 was considered, too. 

In the text 
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