Figures 3 and 4 shows the general schematic of the geometry of resonant zones in the case of 1:6 (an = 17.17 AU) and 2:13 (an = 18.11 AU) resonances, respectively. Here, we quote an = the “nominal resonance location” (Murray and Dermott 1999, Section 8.4), which is the value of semimajor axis corresponding to a resonant orbital period assuming the most simple case where (which denotes orbital precession) is zero, i.e., as implied by Kepler's third law (Kepler 1609, 1619).
Figure 3. (a,M) space for 1:6 resonance showing regions where particles undergo resonant librations, as a function of initial semimajor axis and mean anomaly at the initial epoch JD 1208880.5 in 1404 B.C.
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In a real case, is never exactly zero, e.g., with the 1:6 resonance (resonant argument as per Equation (5)):
If we assume, as a time average, for resonant libration:
Then, from Equation (7),
where M denotes the mean anomaly.
Simplifying the expression we get
Differentiating Equation (10) on both sides with respect to time and using Equation (8),
Therefore, the actual resonant value of semimajor axis would be slightly greater than the ones given in this section. A much more comprehensive and general description about this subject and its application to meteor streams can be found in Emel'yanenko (2001).
Integrations were performed by taking 7,200 particles, varying the initial semimajor axis from 16.5 to 17.9 in steps of 0.014 AU for 1:6 MMR and from 17.6 AU to 18.6 AU in steps of 0.01 AU for 2:13 MMR, and initial M from 0 to 360 degrees in steps of 5 degrees, keeping all other orbital elements (namely e, i, ω, and Ω) constant. The starting epochs for Figs. 3 and 4 are 1P/Halley's perihelion return times in 1404 B.C. and 240 B.C. respectively. All the particles were integrated for 2,000 yr using the RADAU algorithm with accuracy parameter set to 10−12. Output data were generated for every 10 yr. Resonant particles were identified by employing a simple technique, which looks at the overall range of the resonant arguments (for various combinations allowed by D'Alembert rules, see previous section) of all particles during the whole integration time to check when there is no circulation. Checks on an extensive set of representative particles covering many different libration amplitudes confirm that those particles filtered by this algorithm will have librating resonant arguments, which thereby confirm the presence of respective MMR.
Figures 3 and 4 give a general picture of these resonances: we can visualize 6 or 13 resonant zones spaced in mean anomaly along the whole orbit, each zone consisting of individual librating particles (cf. Emel'yanenko 1988). These zones, or clouds of resonant particles, are preserved for as long as substantial numbers of particles continue to librate. Our test integrations showed that for some particular ejection epochs, the 1:6 MMR is exceptionally effective in retaining the compact dust trail structures for as long as 30,000 yr. However, particles disperse in mean anomaly much faster (in a few thousands of years) in the case of the 2:13 resonance, and do not show such high stability. Typically, the rule of thumb is that the higher the order of resonance (denoted by q, see Equations (2) and (3)), the lower the strength of the resonance.
A necessary condition for a resonant meteor outburst is that the Earth should encounter one of these clusters of resonant particles, i.e., when the Earth misses these clusters, there is no enhancement (which is the common case in most years) in meteor activity, at least due to the MMR mechanism. Of course, there are various other factors like nodal distance, solar longitude, date and time of intersection of the meteoroid with Earth, geocentric velocity, etc., which play a key role in confirming the occurrence and characteristics of a meteor outburst or storm (see 'Specific Calculations' section). It is possible in reality first that many resonant zones would only be partially filled (unlike the uniform pattern shown in Figs. 3 and 4), and second that within a given resonant zone, there is significant fine structure, which could lead to enhanced meteor phenomena if Earth happens to pass exactly through the densest parts.
Similar plots to Figs. 3 and 4 reveal mean anomaly distributions of resonant particles in the long term (a few millennia). The cluster of particles in a single 1:6 resonant zone occupies a much longer part of the orbit (covering 5–6 yr) than the equivalent for a 2:13 resonant zone (only 1–2 yr). This means that for the 1:6 MMR, there can be 5–6 consecutive years of enhanced meteor activity (depending on the exact parts of the resonant stream's fine structure encountered by the Earth), compared with just 1–2 yr of outburst possibilities from 2:13. The 1:6 resonance is also more effective in trapping considerably larger numbers of particles compared with 2:13. Hence, meteor outbursts from the 1:6 resonance would have a higher intensity than those due to the 2:13 resonance in most cases, although dependent again on the fine structure within the respective resonances. As there is a long record of observations for Orionids (Rendtel 2008), the same pattern could be compared and justified. The maximum ZHR in 2007 was about 80 (Arlt et al. 2008), whereas in 1993, it was about 35 (Rendtel and Betlem 1993); cf. normal rates of 20–25.
When the comet is resonant, it would remain in a single resonant zone and populate that particular zone. When the comet goes out of resonance, it would keep traversing between zones and thereby populate different resonant zones gradually, implying that meteor outbursts could come from different zones. One of the main aims of this work was to correlate particular resonant zones with past and present meteor outbursts. Our calculations show that the outbursts during 1436–1440, 1933–1938, and 2006–2010 (see 'Specific Calculations' section) are from the same 1:6 resonant zone, specifically the same one in which the comet librated from 1404 B.C. to 690 B.C. Also, a future meteor outburst in 2070 would occur due to the particles in the same 2:13 resonant zone that caused increased meteor activity in 1916 and 1993. Halley presumably released many meteoroids into the 2:13 resonant zone in which it librated from 240 B.C. to 1700 A.D., but most of these meteoroids do not have the precession rate required for producing Earth-intersecting orbits at the present time. Comparison with resonant zones is an excellent way to match observations with theoretical simulations.
The numbers of particles trapped in other resonances close to this range of semimajor axis (16.5–19 AU) were also checked. For example, the percentage of particles in 1:7 (an = 19.03 AU), 3:17 (an = 16.53 AU), 3:19 (an = 17.80 AU), and 3:20 (an = 18.42 AU) resonances with Jupiter is very low compared to the 1:6 (an = 17.17 AU) and 2:13 (an = 18.11 AU). It should be pointed out that if particle ejection were centered around the resonant value of 1:7 MMR, then there would be substantial amounts of resonant particles, which can cause outbursts, but in reality, the comet is never near this resonant semimajor axis in the time frames that we consider in this work. Also, the ratio of particles trapped in the Saturnian resonances of 2:5 (an =17.56 AU), 3:8 (an = 18.34 AU), 5:12 (an = 17.09 AU), 5:13 (an = 18.03 AU), 7:17 (an = 17.23 AU), 7:18 (an = 17.90 AU), and 8:19 (an = 16.97 AU) are extremely small owing to the overpowering effect of Jupiter's gravity. Hence, significant measurable enhancements in meteor activity can be ruled out from these obscure Jovian and Saturnian resonances. Even if such resonant particles encounter the Earth, it will be almost impossible to distinguish them because of the lack of any sizeable increase in ZHR in any year. Hence, it should be understood that just the mere fact of having some resonant particles intersecting the Earth does not mean an increase from normal meteor rates. The sole criterion depends on how effective that resonance mechanism is in trapping very large numbers of ejected particles and subsequently avoiding close encounters with other planets.
Past observations (Millman 1936; Lovell 1954; Imoto and Hasegawa 1958; Rendtel and Betlem 1993; Dubietis 2003; Rendtel 2007; Trigo-Rodríguez et al. 2007; Arlt et al. 2008; Spurny and Shrbeny 2008; Kero et al. 2011) of Orionids have shown enhanced meteor activity in some particular years. Previous interesting works (Rendtel 2007; Sato and Watanabe 2007) have highlighted the significance of 1:6 MMR in explaining the outburst in 2006. According to Sato and Watanabe (2007), the meteor outburst in 2006 was caused by 1:6 resonant particles ejected from the comet in −1265 (1266 B.C.), −1197 (1198 B.C.) and −910 (911 B.C.). All these ejection years can be directly linked to the time frame in which the comet itself was 1:6 resonant (see previous section). Hence, more meteoroids became trapped in this resonance during this time frame compared with other years when the comet was not resonant.
Calculations were performed on similar lines to Sato and Watanabe (2007). Ejection epochs were set between 1404 B.C. and 1986 A.D. All the ejections were performed by keeping the perihelion distance and other elements as constant and by varying the semimajor axis and eccentricity. In this simple model, ejection was performed at perihelion (M = 0) in the tangential direction. Ejection velocities were set in the range −50 to +50 m s−1, i.e., both behind and ahead of the comet, meaning that over all epochs collectively, the initial orbital periods range from 60 to 88 yr, encompassing all possible 1:6 and 2:13 resonant particles, positive ejection velocity corresponding to larger periods. Radiation pressure and Poynting-Robertson effects were not incorporated into these calculations. Because they span a range of orbital periods, test particles ejected tangentially at perihelion and moving only under gravitational perturbations are able, over the time frames we consider here, to represent the motion of all meteoroids released over the comet's perihelion arc with different velocities and subject to different radiation pressures (Kondrat'eva and Reznikov 1985; Asher and Emel'yanenko 2002).
To confirm the correlation between theory and observations, it is vital to match the time (second half of October) when the meteoroids reach their ascending node, solar longitude (approximately 204–210 degrees) at the node and heliocentric distance of ascending node (by analyzing the difference Δr between heliocentric distances of Earth and ascending node of meteoroid; Earth diameter is about 0.0001 AU). These essential parameters from our simulations (Table 1) can be matched with real observations (listed in past meteor records).
Table 1. Data of dust trails which caused various Orionid outbursts
|Ejection year||Expected peak time (UT)||Solar longitude (J2000.0)||∆r (AU)||Ejection velocity (m s−1)||Period at ejection (years)|
|−1197||1436 Oct 13 01:44||207.551||+0.0012||+13.16||71.76|
|−985||1436 Oct 14 17:40||208.212||+0.0027||−12.79||71.64|
|−910||1437 Oct 14 03:00||208.344||−0.0013||−13.6||71.95|
|−836||1438 Oct 14 13:30||208.522||+0.0010||−22.08||70.12|
|−1265||1439 Oct 14 23:54||208.698||+0.0021||+11.88||70.67|
|−985||1439 Oct 15 00:00||208.702||−0.0001||−14.74||71.23|
|−910||1439 Oct 16 15:03||210.324||−0.0008||−18.07||71.01|
|−910||1440 Oct 13 19:17||208.258||+0.0037||−20.37||70.54|
|−985||1916 Oct 17 07:40||204.771||+0.0002||+10.64||77.22|
|−910||1916 Oct 17 12:57||204.990||+0.0014||+8.79||77.29|
|−1265||1933 Oct 21 02:24||208.170||+0.0011||+15.69||71.54|
|−985||1933 Oct 21 02:52||208.190||+0.0086||−12.18||71.77|
|−1333||1934 Oct 21 12:14||208.321||+0.0044||+9.13||71.82|
|−985||1934 Oct 21 12:28||208.330||+0.0002||−11.16||71.99|
|−985||1935 Oct 21 13:26||208.120||+0.0034||−16.42||71.88|
|−910||1935 Oct 22 05:16||208.771||+0.0087||−14.35||71.79|
|−1265||1936 Oct 21 16:19||208.982||−0.0073||+17.35||71.93|
|−1197||1936 Oct 22 06:28||209.568||+0.0058||+10.77||71.21|
|−1265||1937 Oct 21 20:24||208.890||+0.0067||+17.27||71.91|
|−836||1937 Oct 21 23:02||208.998||+0.0022||−13.68||71.85|
|−1197||1938 Oct 21 21:21||208.675||−0.0091||+7.00||71.37|
|−1265||1938 Oct 22 02:24||208.883||+0.0001||+14.31||71.22|
|−1333||1993 Oct 17 22:48||204.662||+0.0039||+31.88||77.70|
|−985||1993 Oct 18 00:14||204.724||+0.0041||+12.68||77.77|
|−910||1993 Oct 18 02:26||204.774||−0.0017||+8.98||77.34|
|−835||1993 Oct 18 02:40||204.782||+0.0088||+10.45||77.64|
|−1265||2006 Oct 21 02:09||207.452||+0.0005||+11.03||70.48|
|−910||2006 Oct 23 03:38||209.461||−0.0069||−17.73||71.08|
|−1333||2007 Oct 21 18:14||207.858||+0.0043||+8.60||71.70|
|−1265||2007 Oct 22 00:28||208.118||−0.0004||+12.95||70.91|
|−1197||2007 Oct 22 04:36||208.252||+0.0032||+11.25||71.32|
|−985||2007 Oct 22 09:21||208.486||+0.0047||−15.22||71.13|
|−910||2007 Oct 22 10:04||208.513||+0.0002||−15.81||71.48|
|−836||2007 Oct 23 01:12||209.140||−0.0071||−15.76||71.41|
|−1333||2008 Oct 23 05:16||210.050||+0.0073||+6.61||71.25|
|−1265||2008 Oct 20 14:40||207.412||+0.0022||+14.65||71.30|
|−1197||2008 Oct 21 07:28||208.114||−0.0019||+9.30||70.88|
|−985||2009 Oct 21 15:07||208.214||+0.0077||−18.72||70.41|
|−910||2009 Oct 21 19:43||208.367||−0.0008||−17.34||71.16|
|−836||2010 Oct 21 21:36||208.225||−0.0011||−17.88||70.97|
|−1333||2010 Oct 21 23:31||208.302||+0.0056||+6.30||71.18|
|−1265||2010 Oct 22 02:24||208.424||+0.0013||+15.26||71.44|
|−910||2070 Oct 18 19:12||204.770||−0.0029||+9.24||77.41|
|−910||2070 Oct 18 19:26||204.779||+0.0003||+9.80||77.56|
Each entry in Table 1 is a carefully chosen meteoroid, which has the average value out of many candidate particles covering the small range of orbital periods that favors an outburst in the given year. For example, Figs. 5-7 are plots of heliocentric distance of ascending node, solar longitude, and difference in time of nodal crossing of the particles from the time of observed outburst (all three parameters computed at 2007 October 22) versus initial semimajor axis of meteoroids (ejected at −910 return). Heliocentric distance of Earth on 2007 October 22 was 0.995 AU. These results show that the conditions for an outburst to occur (from particles ejected in −910) at the observed time in 2007 are satisfied if initial semimajor axis is around 17.22 AU, but the total suitable range in initial semimajor axis and ejection velocities for meteoroids are 17.20 AU < a < 17.25 AU and −16.53 m s−1 < v < −15.05 m s−1, respectively. The plots are similar for other ejection epoch/outburst year pairs as well.
Our simulations indicate meteor outbursts from 1436 to 1440 A.D. due to 1:6 resonant meteoroids, which were ejected around Halley's −1265, −1197, −985, −910, and −836 returns (details in Table 1). The initial orbital periods of ejected meteoroids, which lead to all five outbursts show that most of them had almost 6 times the Jovian period. There are also historical observational records, which show heightened activity, indicating hundreds of bright meteors, in 1436 and 1439 (Imoto and Hasegawa 1958) and match our theoretical simulations well. We converted the dates from Julian calendar to proleptic Gregorian calendar in order that all dates in Table 1 are referred to a single calendar. No observational records could be traced or identified for 1437, 1438, and 1440 though. Either there were no observations performed in those years (unfavorable lunar phase is a possible explanation only in 1438) or the meteor outbursts would have been insignificant in 1437, 1438, and 1440 compared with the ones in 1436 and 1439. In our simulations, we find that resonant meteoroids ejected with positive ejection velocity (higher orbital period) encountered the Earth in 1436 and 1439. The ones with negative ejection velocity (smaller period) encountered the Earth in 1437, 1438, and 1440. Radiation pressure (not included in our integrations) would always act in the direction that would increase the orbital period of meteoroids, i.e., affects the period in the same sense as positive ejection velocities. In general, we expect the peak of the ejection velocity distribution is close to zero and so the largest number of particles, if affected by radiation pressure, is represented by particles having positive ejection velocities in our gravitational integration model. Radiation pressure is having a detrimental effect (with regard to causing meteor outbursts) when we calculate that negative ejection velocities are required to produce a meteor outburst in a particular year. This can explain why we find this trend in ejection velocities for resonant meteoroids reaching Earth in 1436 and 1439, which caused meteor outbursts (agreeing with past observations as shown in Imoto and Hasegawa 1958) and possibly no (or very low) activity in 1437, 1438, and 1440.
We calculate that the meteor outburst in 1993 was due to 2:13 resonant meteoroids ejected around Halley's −1333, −985, −910, and −835 returns. The outburst (Miskotte 1993; Rendtel and Betlem 1993) occurred when solar longitude was between 204.7° and 204.9°, a notably different time compared with other known outbursts. Our theoretical calculations match this unusually early peaking on 1993 October 18. Our simulations also indicate a meteor outburst in 1916 from the 2:13 resonance and there is a hint of enhanced meteor rates from the past observations in 1916 (Olivier 1921) compared with the adjacent years of 1915 and 1917. For the future, we predict a similar outburst (like in 1993 because favorable ejection velocities are similar in both cases) from the 2:13 resonance mechanism in 2070.
The ejection epochs for 1:6 resonant meteoroids, which caused continuous enhanced activity in 2007, 2008, 2009, and 2010 (Trigo-Rodríguez et al. 2007; Arlt et al. 2008; Kero et al. 2011; IMO Records) are also given in Table 1. These ejection years correspond to the time when the comet itself was 1:6 resonant. Hence, it is obvious that a large number of meteoroids would have been trapped into this resonance during those time frames, which would clearly indicate the reason for high ZHR apart from the contribution due to the inherent geometry (cf. Fig. 3) of these zones. Our simulations match the observed ranges (207–210°) of solar longitude and outburst times (Trigo-Rodríguez et al. 2007; Arlt et al. 2008; Kero et al. 2011; IMO Records) for these outburst years very well. Even though the uncertainties in semimajor axis (to directly compare with theoretical values in our calculations) of observed meteoroids from these highly successful observations are quite high (which is the typical case for all meteor observations, especially when the semimajor axis itself is high), the matching of outburst time frames and solar longitudes from these papers itself is a very effective way of comparing the orbital evolution of resonant meteoroids with real observations. Our results show that 1:6 resonant meteoroids ejected from the resonant comet also caused an enhanced activity from 1933 to 1938, which match old observational records (Millman 1936; Lovell 1954). Most of these meteoroids had positive ejection velocities, which are more favorable for stronger outbursts as discussed above.
Figure 8 clearly shows that meteoroids with initial orbital periods corresponding to 1:6 (around 71 yr) and 2:13 (around 77 yr) resonances have their present ascending nodes near the orbit of the Earth. Hence, it can be concluded that resonance mechanisms (specifically 1:6 and 2:13 MMR in this case) aid these particles to come near the Earth at the present epoch while the nonresonant ones precess away from the Earth's orbit. This is typical of other ejection epochs (as shown in Table 1) as well. The ascending node of Halley during its last apparition (in 1986) was 1.8 AU. One could clearly see that the number of particles trapped in 1:6 resonance is considerably larger than the number trapped in 2:13 resonance. Moreover, in Fig. 8, we notice that particles having orbital periods of almost 5Pj(59.2 yr), 6 Pj (71.1 yr), and 7 Pj (83.0 yr) come near the Earth's orbit, which agrees with earlier calculations performed by Sato and Watanabe (2007).
The typical ZHR for Orionids during nonoutburst years is about 20 (Rendtel and Betlem 1993; Rendtel 2008; IMO Records). From the recorded previous observations, it is seen that the ZHR is about 60 (Rendtel 2007; Kero et al. 2011; IMO Records) due to 1:6 MMR during 2006–2010 and about 35 (Rendtel and Betlem 1993) due to 2:13 MMR in 1993. Using these previous observations and flux, one could actually make a simplistic estimation of the mass delivered to the Earth from the Orionid stream during these outburst years. According to the detailed work of Hughes and McBride (1989), the typical influx rate (at shower maximum perpendicular to the radiant) of Orionids is 1.8 × 10−18 g cm−2 s−1, which in turn (after multiplying with the incident area of Earth) predicts 7 g s−1 during 2006–2010 and 4 g s−1 during 1993. It should be made clear that enhanced ZHR could be quite different in other outburst years (in past as well as future) as it depends on the exact cross section and density distribution of resonant trails intersecting the Earth. Moreover, given the high speed of Orionid meteors and the strong dependence of meteor brightness on velocity, the meteoroidal mass influx to Earth is not dominated by this level (ZHR = a few tens) of Orionid outburst.