Trajectory analysis for the nucleus and dust of comet C/2013~A1 (Siding Spring)
Davide Farnocchia, Steven R. Chesley, Paul W. Chodas, Pasquale Tricarico, Michael S. P. Kelley, Tony L. Farnham
aa r X i v : . [ a s t r o - ph . E P ] A p r Submitted
Trajectory analysis for the nucleus and dust of comet C/2013 A1 (SidingSpring)
Davide Farnocchia ∗ a , Steven R. Chesley a , Paul W. Chodas a , Pasquale Tricarico b , Michael S. P.Kelley c , Tony L. Farnham c ABSTRACT
Comet C/2013 A1 (siding Spring) will experience a high velocity encounter withMars on October 19, 2014 at a distance of 135,000 km ± > Subject headings:
Comets: individual (C/2013 A1); Methods: analytical; Celestial Me-chanics; Radiation: dynamics
1. Introduction
Comet C/2013 A1 (Siding Spring) was discovered on January 2013 at the Siding Spring ob-servatory (McNaught et al. 2013). Shortly after discovery it was clear that C/2013 A1 was headed * [email protected] a Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA b Planetary Science Institute, Tucson, AZ 85719, USA c Department of Astronomy, University of Maryland, College Park, MD 20742, USA
2. Ballistic trajectory
We examined all available ground-based optical astrometry (Right Ascension and Declinationangular pairs) as of March 15, 2014. To remove biased contributions from individual observatorieswe conservatively excluded from the orbital fit batches of more than four observations in the samenight with mean residual larger than 0.5 ′′ , and batches of three or four observations showing meanresidual larger than 1 ′′ . We also adopted the outlier rejection scheme of Carpino et al. (2003)with χ rej = 2. To the remaining 597 optical observations we applied the standard one arcseconddata-weights used for comet astrometry. Figure 1 shows the residuals of C/2013 A1’s observationsagainst our new orbit solution (JPL solution 46).[Figure 1 about here.] 3 –Our force model included solar and planetary perturbations based on JPL’s planetary ephemeridesDE431 , the gravitational attraction due to the 16 most massive bodies in the main asteroid belt,and the Sun relativistic term. No significant nongravitational forces were evident in the astromet-ric data and so the corresponding JPL orbit solution is ballistic, identified as number 46. Table 1contains the orbital elements of the computed solution.[Table 1 about here.]Table 2 provides information on the close encounter between C/2013 A1 and Mars. C/2013 A1passes through the orbital plane of Mars 69 minutes before the close approach epoch, while Marspasses through the orbital plane of C/2013 A1 99 minutes after the close approach. The MinimumOrbit Intersection Distance (MOID) is the minimum distance between the orbit of the comet andthe orbit of Mars (MOID, Gronchi et al. 2007). The MOID points on the two orbits are not on theline of nodes. Mars arrives at the minimum distance point 101 min after the close approach epoch,while C/2013 A1 arrives at the minimum distance point 70 min before the close approach, whichmeans that the comet is 171 min early for the minimum distance encounter.[Table 2 about here.]A standard tool to analyze planetary encounters is the b -plane (Kizner 1961; Valsecchi et al.2003), defined as the plane passing through the center of mass of the planet and normal to the in-bound hyperbolic approach asymptote. The coordinates on the b -plane described in Valsecchi et al.(2003) are oriented such that the projected heliocentric velocity of the planet is along − ζ . There-fore, ζ varies with the time of arrival, i.e., a positive ζ means that the comet arrives late at theencounter while a negative ζ means that the comet arrives early. On the other hand ξ is relatedto the MOID. The b -plane is used on a daily basis for asteroid close approaches to the Earth andcomputing the corresponding impact probabilities (Milani et al. 2005).Figure 2 shows the projection of the 3 σ uncertainty ellipsoid of JPL solution 46 on the b -plane.The projection of the velocity of Mars on this plane is oriented as − ζ , while the Mars-to-Sun vectorprojection is on the left side, at a counterclockwise angle of 186 ◦ with respect to the ξ axis. Thenegative ζ coordinate of the center of the ellipse corresponds to the 171 min time shift betweenMars and C/2013 A1. [Figure 2 about here.] http://ssd.jpl.nasa.gov/?ephemerides
3. Nongravitational perturbations
Comet trajectories can be significantly affected by nongravitational perturbations due tocometary outgassing. We use the Marsden et al. (1973) comet nongravitational model: a NG = g ( r )( A ˆ r + A ˆ t + A ˆ n ) (1)where g ( r ) is a known function of the heliocentric distance r , and A i are free parameters that givethe nongravitational acceleration at 1 au in the radial-transverse-normal reference frame definedby ˆ r , ˆ t , ˆ n .The observational dataset available for C/2013 A1 does not allow us to estimate the nongrav-itational parameters A i . Still, nongravitational accelerations could cause statistically significantdeviations at the close approach epoch. To deal with this problem, we analyzed the propertiesof known nongravitational parameters in the comet catalog. Figure 3 shows the known A and A in the catalog. A values have an order of magnitude similar to that of A . Figure 4 containsscatter plots of nongravitational parameters showing the correlation between these parameters. Forcomets with an orbit similar to that of C/2013 A1, i.e., with large orbital period ( >
60 yr) andhigh eccentricity ( > A are on average ∼ − au/d , but they can be as large as ∼ − . A and A are generally one order of magnitude smaller, i.e., on average they are ∼ − au/d but can be as large as ∼ − au/d . We can see that A is generally one order of magnitudelarger than A and A , which makes sense since the radial component is usually the largest fornongravitational accelerations. [Figure 3 about here.][Figure 4 about here.]According to the properties of the comet population we considered three different scenarios asdescribed in Table 3: the ballistic scenario corresponds to JPL solution 46; the “reference” scenariouses typical values of the nongravitational parameters; the “wide” scenario assumes extreme valuesof the nongravitational parameters. We selected the A uncertainty so that its range would spanfrom 0 au/d to twice the nominal value at 3 σ . For A and A the nominal value is 0 au/d sincethese components can be either positive or negative, while A can only be positive.[Table 3 about here.]Figure 5 shows the position difference among the three scenarios compared to the positionuncertainty of the ballistic solution. The available observations put a strong constraint the tra-jectory of C/2013 A1 for heliocentric distances between 3 au and 8 au from the Sun. Outside ofthis distance range we have no observations and therefore the uncertainty increases. Because of 5 –the fast decay of the g ( r ) function in Eq. (1) the contribution of nongravitational accelerations forlarge heliocentric distances is well within the uncertainty and so the trajectory of C/2013 A1 in thepast is not significantly affected. It is worth pointing out that the function g ( r ) represents watersublimation while distant activity is not driven by water and therefore may be inaccurate a largedistances. However, for such large distances the position uncertainty is large enough to make thispossible discrepancy irrelevant. Finally, for smaller heliocentric distances nongravitational pertur-bations become relevant and can affect the predictions for the Mars encounter, especially in thewide scenario. [Figure 5 about here.]For the three different scenarios, Table 4 gives the close approach information while Fig. 6shows the projection of the orbital uncertainties on the b -plane. The ballistic and reference solutionsprovide very similar predictions, from which we conclude that nongravitational perturbations willnot significantly affect the orbit unless they are larger than expected. The wide solution, which hasto be regarded as an extreme case, produces a significantly different nominal prediction and quite alarge uncertainty. In all three scenarios, the nominal close approach distance is more than 130,000km from Mars and therefore there is no chance of an impact between the nucleus of C/2013 A1and Mars. [Table 4 about here.][Figure 6 about here.]
4. Uncertainty evolution
The predictions and the uncertainty provided so far are based on the optical astrometry avail-able as of March 15, 2014. At the time of submission of this paper (April 2014), comet C/2013 A1was difficult to observe because of the low solar elongation. On June 18, 2014 the solar elongationbecomes larger than 60 ◦ and we therefore expect observations to resume, which will help in furtherconstraining the trajectory of C/2013 A1. To quantify the effect of future optical astrometry, wesimulated geocentric optical observations, with two observations every five nights.Figure 7 shows the evolution of the position uncertainty on the b -plane. The curves representthe semimajor axis of the projection of the 3 σ uncertainty ellipsoid on the b -plane. The ballisticand reference solution curves are close, with an uncertainty that goes from the current 5000 km toless than 1000 km when all the pre-encounter observations are accounted for. The wide solutionhas a much larger uncertainty that decreases to a minimum of about 6000 km.[Figure 7 about here.] 6 –Figure 8 shows the 3 σ uncertainty evolution for the close approach epoch. The ballistic andreference scenarios have a current uncertainty of 3 min and this uncertainty decreases to less than0.2 min right before the close approach. For the wide scenario the uncertainty goes from 45 mindown to 1–2 min. [Figure 8 about here.]As already discussed in Sec. 3, the wide solution produces predictions significantly differentfrom the ballistic and reference solutions. Thus, at some point observations will reveal whether ornot the nongravitational perturbations are behaving as in the wide scenario. Figure 9 shows theuncertainty in A when estimated from the orbital fit as a function of time. When this uncertaintybecomes smaller than a given value of A , the observation dataset reveals such A value if it isreal. By comparing the uncertainty evolution to the nominal values of A assumed for the differentscenarios, we can see that large nongravitational accelerations to the level assumed in the widescenario are detectable about 90 days before the close encounter. On the other hand, the referencesolution becomes distinct from the ballistic solution only a couple of weeks before the encounter.[Figure 9 about here.]Some skilled observers are capable of gathering comet observations even for solar elongationssmaller than 60 ◦ . Therefore, we also simulated observations using 40 ◦ as a lower threshold for thesolar elongation, which makes it possible to collect new observations for C/2013 A1 starting onMay 7, 2014. However, the improvement in the uncertainties discussed above is a factor of 1.3 orless and is therefore not relevant.
5. Dust tail
Though an impact the nucleus of C/2013 A1 on Mars is ruled out, there is a chance that dustparticles in the tail could reach Mars and some of the orbiting spacecrafts. Due to their small size,the motion of dust particles is strongly affected by solar radiation pressure. It is therefore to usethe β parameter (Burns et al. 1979), i.e., the non-dimensional number corresponding to the ratiobetween solar radiation pressure and solar gravity. In terms of physical properties, β is proportionalto the area-to-mass ratio and inversely proportional to both the density and to the radius of theparticle: β = 0 . Qaρ (2)where a is the particle radius in µ m, ρ is the density in g/cc, and Q is the solar radiation pressureefficiency coefficient. 7 –For each ejected particle, the location on the b -plane for the Mars encounter is determined bythe β parameter, the heliocentric distance r at which the particle is ejected (or the ejection epoch),and the ejection velocity ∆ v . Figure 10 shows the typical behavior using as an example β = 0 . v = | ∆ v | = 10 m/s. For each given β we have a curve on the b -plane corresponding to zeroejection velocity. This curve can be parameterized by the heliocentric distance at which the ejectiontakes place. Finally, the ejection velocity ∆ v yields dispersion around the curve: the larger the ∆ v the wider the dispersion. [Figure 10 about here.]The ejection velocity depends on the particle size and density, as well as the heliocentricdistance at which the particle is ejected (Whipple 1951). Since cometary activity is very hard topredict, modeling the ejection velocities is a complicated task and is subject to continuous updatesas additional observations are available. Therefore, we decided to adopt a different approach: forgiven ejection distance r and β parameter we computed the minimum ∆ v required to reach Mars.In mathematical terms we look for the tridimensional ∆ v that is a minimum point of ∆ v = | ∆ v | under the constraint that the particle reaches Mars, i.e., ( ξ, ζ )( r, β, ∆ v ) = (0 , v weare looking for must satisfy the following system of equations: ( ξ, ζ )( r, β, ∆ v ) = (0 , ∂ | ∆ v | ∂ ∆ v = λ ∂ξ∂ ∆ v ( r, β, ∆ v ) + λ ∂ζ∂ ∆ v ( r, β, ∆ v ) (3)where λ and λ are free parameters. To solve this system, we first tested the linearity of ( ξ, ζ ) in∆ v and then linearized system (3) around ∆ v = 0, thus obtaining the following linear system: ( ξ, ζ )( r, β, ∆ v ) = ( ξ, ζ )( r, β,
0) + ∂ ( ξ, ζ )∆ v ( r, β, v = (0 , v = λ ∂ξ∂ ∆ v ( r, β,
0) + λ ∂ζ∂ ∆ v ( r, β, . (4)To compute the required ∆ v , we followed these steps: • We sampled β in log-scale from 10 − to 1 and r from 1.4 au to 30 au; • For each couple ( r, β ) we computed the b -plane coordinates ( ξ, ζ ) obtained without ejectionvelocity as well as a finite difference approximation of the ( ξ, ζ ) partials with respect to ∆ v ; • We solved system (4).We scaled the resulting ∆ v to account for the size of Mars and the 3 σ uncertainty of the particleprojection on the b -plane. For this analysis we used the ballistic solution as reference trajectory. 8 –Figure 11 shows the required ∆ v needed to reach Mars as a function of the heliocentric distanceat which the ejection takes place for different values of β . On the right side of the plot the requiredvelocities are almost the same. This behavior makes sense as the closer we get to Mars the less timeis available for solar radiation pressure to affect the trajectory. Therefore, the required ejectionvelocity is almost independent of the particle size and density. For β = 1 . × − we can seethat the required velocity goes to zero for heliocentric distances around 22.5 au. The reason forthis is that the curve on the b -plane defined by this particular value of β passes through the centerof Mars. Thus, if ejected at the right distance, i.e., 22.5 au, the particle reaches Mars under theaction of solar radiation pressure, with no ejection velocity at all. It is also worth noticing thatthe β = 0 . β , solar radiationpressure is extremely strong and the particle does not even experience the close encounter withMars if ejected too far in advance. [Figure 11 about here.]The results obtained so far can be used to assess the possibility that particles of a given sizecould reach Mars for a given ejection velocity model. For instance, the best fit for the ejectionvelocity according to Farnham et al. (2014) is∆ v = 418 m/s (cid:18) β (cid:19) . (cid:18) r (cid:19) . . (5)As shown in Fig. 12, we can scale the required velocity to β = 1 and make a comparison to thevelocity given by (5). We can see that, according to this ejection velocity model, impacts arepossible only for particles with β ∼ × − or smaller ejected at more than ∼
16 au from the Sun.[Figure 12 about here.]Figure 13 shows a comparison to the ejection velocity model considered by Tricarico et al.(2014): ∆ v = 1 . (cid:18) β . × − (cid:19) . (cid:18) r (cid:19) . (6)In this case impacts are possible only for particles ejected more than 13 au from the Sun and β ∼ − . The figure also makes the comparison for larger ejection velocities (also considered byTricarico et al. 2014): ∆ v = 3 m/s (cid:18) β . × − (cid:19) . (cid:18) r (cid:19) . (7)In this case impacts are possible also for β = 0 .
001 and particles ejected as close as ∼ • millimeter to centimeter dust grains are ejected from the nucleus more than 13 au from theSun; • the ejection velocities are larger than current estimates by a factor >
6. Conclusions
To study the Oct 19, 2014 encounter with Mars, we analyzed the trajectory of comet C/2013 A1(Siding Spring). The ballistic orbit has a closest approach with Mars at 135,000 km ± β parameter, i.e., the ratio between solar radiation pressure and solar gravity. By comparing ourresults to the most updated modeling of dust grain ejection velocities, impacts are possible only for β of the order of 10 − , which, for a density of 1 g/cc, corresponds to millimeter to centimeter particles.However, the particles have to be ejected at more than 13 au, which is generally considered unlikely.See Kelley et al. (2014) for a discussion of the maximum liftable grain size at these distances. Thearrival times of these particles are in an interval of about 20 minutes around the time that Marscrosses the orbit of C/2013 A1, i.e., Oct 19, 2014 at 20:09 TDB. In the unlikely case that ejectionvelocities are larger than currently estimated by a factor >
2, impacts are possible for particleswith β = 0 .
001 that are ejected as close as ∼ Acknowledgments
Part of this research was conducted at the Jet Propulsion Laboratory, California Institute of Tech-nology, under a contract with the National Aeronautics and Space Administration.Copyright 2014, California Institute of Technology.
REFERENCES
Burns, J. A., Lamy, P. L., and Soter, S. (1979). Radiation forces on small particles in the solarsystem.
Icarus , 40:1–48.Carpino, M., Milani, A., and Chesley, S. R. (2003). Error statistics of asteroid optical astrometricobservations.
Icarus , 166:248–270.Farnham et al. (2014). The Activity of Comet C/2013 A1 (Siding Spring).
In preparation .Gronchi, G. F., Tommei, G., and Milani, A. (2007). Mutual geometry of confocal Keplerian orbits:uncertainty of the MOID and search for virtual PHAs. In Valsecchi, G. B., Vokrouhlick´y,D., and Milani, A., editors,
IAU Symposium , volume 236 of
IAU Symposium , pages 3–14.Kelley, M. S. P., Farnham, T. L., Bodewits, D., Tricarico, P., and Farnocchia, D. (2014). A Studyof Dust and Gas at Mars from Comet C/2013 A1 (Siding Spring).
In preparation .Kizner, W. (1961). A Method of Describing Miss Distances for Lunar and Interplanetary Trajec-tories.
Planet. Space Sci. , 7:125–131.Marsden, B. G., Sekanina, Z., and Yeomans, D. K. (1973). Comets and nongravitational forces. V. AJ , 78:211.McNaught, R. H., Sato, H., and Williams, G. V. (2013). Comet C/2013 A1 (Siding Spring). CentralBureau Electronic Telegrams , 3368:1.Milani, A., Chesley, S. R., Sansaturio, M. E., Tommei, G., and Valsecchi, G. B. (2005). Nonlinearimpact monitoring: line of variation searches for impactors.
Icarus , 173:362–384.Moorhead, A. V., Wiegert, P. A., and Cooke, W. J. (2014). The meteoroid fluence at Mars due toComet C/2013 A1 (Siding Spring).
Icarus , 231:13–21. 11 –Tricarico, P., Samarasinha , N. H., Sykes, M. V., Li, J.-Y., Farnham, T. L., Kelley, M. S. P.,Farnocchia, D., Stevenson, R. A., Bauer, J. M., and Lock, R. E. (2014). Delivery of dustgrains from comet C/2013 A1 (Siding Spring) to Mars.
Submitted to ApJ .Valsecchi, G. B., Milani, A., Gronchi, G. F., and Chesley, S. R. (2003). Resonant returns to closeapproaches: Analytical theory.
A&A , 408:1179–1196.Vaubaillon, J., Maquet, L., and Soja, R. (2014). Meteor hurricane at Mars on 2014 October 19from comet C/2013 A1.
MNRAS , 439:3294–3299.Whipple, F. L. (1951). A Comet Model. II. Physical Relations for Comets and Meteors.
ApJ ,113:464.
This preprint was prepared with the AAS L A TEX macros v5.2.
12 –
List of Figures σ uncertainty of JPL solution 46 on the October 2014 b -plane.The nominal prediction for the b -plane coordinates is ( ξ, ζ ) = ( − , , − , b -plane. The minimum distance between the orbits of Mars and C/2013 A1 is ∼ A and A for the comets in the catalog. A is reported in absolute value. Circles correspond to comets with a period larger than60 yr or an eccentricity larger than 0.9. Crosses are for all other comets. The dashedline corresponds to the total magnitude of C/2013 A1. . . . . . . . . . . . . . . . . . 164 Scatter plots for nongravitational parameters A , A , and A . A and A are re-ported in absolute value. Circles correspond to comets with a period larger than 60yr or an eccentricity larger than 0.9. Crosses are for all other comets. . . . . . . . . . 175 Magnitude of the position difference between the reference and ballistic solutions,and between the wide and ballistic solutions, as a function of heliocentric distance.The dashed line is the semimajor axis of the 1 σ uncertainty ellipsoid of the ballisticsolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Projection on the b -plane of C/2013 A1 3 σ uncertainty according to different sce-narios for nongravitational perturbations. The ballistic and reference solutions arealmost indistinguishable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 Expected evolution of the b -plane position uncertainty. The curves represent thesemimajor axis of the projection on the b -plane of the 3 σ uncertainty ellipse forthe three scenarios. The vertical bar corresponds to Jun 18, 2014 when the solarelongation of C/2013 A1 becomes larger than 60 ◦ . . . . . . . . . . . . . . . . . . . . 208 Expected evolution of the 3 σ uncertainty of the closest approach epoch. The verticalbar corresponds to Jun 18, 2014 when the solar elongation of C/2013 A1 becomeslarger than 60 ◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Expected evolution of the A uncertainty (1 σ ). The horizontal dashed lines arefor the nominal values of A in the reference and wide scenarios. The vertical barcorresponds to Jun 18, 2014 when the solar elongation of C/2013 A1 becomes largerthan 60 ◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2210 Projection on the b -plane of particles ejected with ∆ v = 10 m/s and for β = 0 . v . . . . . . . . . . . . 2311 For different values of β , required ∆ v to reach Mars as a function of the heliocentricdistance at which the ejection takes place. . . . . . . . . . . . . . . . . . . . . . . . . 24 13 –12 Required ∆ v to reach Mars multiplied by (1 /β ) . . The dashed line corresponds to∆ v = 418 m/s β . (1 au /r ) . . Impacts are possible only for particles ejected morethan 16 au from the Sun and with β ∼ × − or smaller. . . . . . . . . . . . . . . 2513 Required ∆ v to reach Mars multiplied by (5 . × − /β ) . . The lower dashed line cor-responds to ∆ v = 1 . β/ . × − ) . (5 au /r ) . In this case impacts are pos-sible for particles ejected more than 13 au from the Sun and β ∼ × − or smaller.The upper dashed line corresponds to ∆ v = 3 m/s ( β/ . × − ) . (5 au /r ) . Inthis case impacts are possible also for β = 0 .
001 and particles ejected as close as ∼ −3 −2 −1 0 1 2 3−3−2−10123 Residuals in RA cos(DEC) (arcsec) R e s i dua l s i n D E C ( a r cs e c ) Fig. 1.— Scatter plot of the astrometric residuals in Right Ascension and Declination with respectto JPL solution 46. Crosses correspond to rejected observations, while dots correspond to theobservations included in the fit. 15 – −5 0 5x 10 −15−10−50 x 10 ξ (km) ζ ( k m ) MarsC/2013 A1
Fig. 2.— Projection of the 3 σ uncertainty of JPL solution 46 on the October 2014 b -plane. Thenominal prediction for the b -plane coordinates is ( ξ, ζ ) = ( − , , − , b -plane. The minimum distancebetween the orbits of Mars and C/2013 A1 is ∼ −14 −12 −10 −8 −6 −4 Comet Total Magnitude A ( au / d ) −14 −12 −10 −8 −6 −4 Comet Total Magnitude A ( au / d ) Fig. 3.— Estimated nongravitational parameters A and A for the comets in the catalog. A isreported in absolute value. Circles correspond to comets with a period larger than 60 yr or aneccentricity larger than 0.9. Crosses are for all other comets. The dashed line corresponds to thetotal magnitude of C/2013 A1. 17 – −12 −10 −8 −6 −4 −12 −10 −8 −6 −4 A (au/d ) A ( au / d ) −12 −10 −8 −6 −4 −12 −10 −8 −6 −4 A (au/d ) A ( au / d ) Fig. 4.— Scatter plots for nongravitational parameters A , A , and A . A and A are reportedin absolute value. Circles correspond to comets with a period larger than 60 yr or an eccentricitylarger than 0.9. Crosses are for all other comets. 18 – Heliocentric distance (au) P o s i t i on d i ff e r en c e R SS ( k m ) Reference − BallisticWide − BallisticPosition uncertainty
Fig. 5.— Magnitude of the position difference between the reference and ballistic solutions, andbetween the wide and ballistic solutions, as a function of heliocentric distance. The dashed line isthe semimajor axis of the 1 σ uncertainty ellipsoid of the ballistic solution. 19 – −0.5 0 0.5 1 1.5 2 2.5 3x 10 −15−10−505 x 10 ξ (km) ζ ( k m ) Mars WideBallisticReference
Fig. 6.— Projection on the b -plane of C/2013 A1 3 σ uncertainty according to different scenarios fornongravitational perturbations. The ballistic and reference solutions are almost indistinguishable. 20 – Days to Mars encounter b − p l ane σ un c e r t a i n t y S M A ( k m ) Jun 18, 2014 ReferenceBallisticWide
Fig. 7.— Expected evolution of the b -plane position uncertainty. The curves represent the semima-jor axis of the projection on the b -plane of the 3 σ uncertainty ellipse for the three scenarios. Thevertical bar corresponds to Jun 18, 2014 when the solar elongation of C/2013 A1 becomes largerthan 60 ◦ . 21 – −1 Days to Mars encounter T C A un c e r t a i n t y ( m i n ) WideBallistic ReferenceJun 18, 2014
Fig. 8.— Expected evolution of the 3 σ uncertainty of the closest approach epoch. The vertical barcorresponds to Jun 18, 2014 when the solar elongation of C/2013 A1 becomes larger than 60 ◦ . 22 – −9 −8 −7 −6 −5 −4 −3 −2 Days to Mars encounter A un c e r t a i n t y ( au / d ) Jun 18, 2014Reference A Wide A Fig. 9.— Expected evolution of the A uncertainty (1 σ ). The horizontal dashed lines are for thenominal values of A in the reference and wide scenarios. The vertical bar corresponds to Jun 18,2014 when the solar elongation of C/2013 A1 becomes larger than 60 ◦ . 23 – −0.5 0 0.5 1 1.5 2 2.5 3 3.5 4x 10 −1.5−1−0.500.51 x 10 ξ (km) ζ ( k m ) Mars
Fig. 10.— Projection on the b -plane of particles ejected with ∆ v = 10 m/s and for β = 0 .
01. Thesolid line represents the position of the particles with no ∆ v . 24 – −3 −2 −1 Heliocentric distance (au) M i n i m u m ∆ v f o r i m pa c t on M a r s ( m / s ) β = 10 −5 β = 1.43 × −4 β = 2 × −4 β = 0.001 β = 0.01 β = 0.1 Fig. 11.— For different values of β , required ∆ v to reach Mars as a function of the heliocentricdistance at which the ejection takes place. 25 – −1 Heliocentric distance (au) N o r m a li z ed ∆ v ( m / s ) β = 10 −5 β = 1.43 × −4 β = 2 × −4 β = 0.001 β = 0.01 β = 0.1 Fig. 12.— Required ∆ v to reach Mars multiplied by (1 /β ) . . The dashed line corresponds to∆ v = 418 m/s β . (1 au /r ) . . Impacts are possible only for particles ejected more than 16 aufrom the Sun and with β ∼ × − or smaller. 26 – −2 −1 Heliocentric distance (au) N o r m a li z ed ∆ v ( m / s ) β = 10 −5 β = 0.1 β = 0.01 β = 0.001 β = 2 × −4 β = 1.43 × −4 Fig. 13.— Required ∆ v to reach Mars multiplied by (5 . × − /β ) . . The lower dashed linecorresponds to ∆ v = 1 . β/ . × − ) . (5 au /r ) . In this case impacts are possible forparticles ejected more than 13 au from the Sun and β ∼ × − or smaller. The upper dashedline corresponds to ∆ v = 3 m/s ( β/ . × − ) . (5 au /r ) . In this case impacts are possible alsofor β = 0 .
001 and particles ejected as close as ∼ List of Tables σ formal uncertainties of the corresponding (last two)digits in the parameter value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Close approach data for JPL orbit solution 46. . . . . . . . . . . . . . . . . . . . . . 293 A priori values and 3 σ uncertainties of nongravitational parameters for the threescenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 Close approach parameters and uncertainties for the three scenarions. The tableshows the b -plane coordinates, the semimajor axis of the 3 σ uncertainty projectedon the b -plane, and the time of closest approach. . . . . . . . . . . . . . . . . . . . . 31 28 –Table 1: J2000 heliocentric ecliptic orbital parameters of JPL orbit solution 46. Numbers in paren-theses indicate the 1 σ formal uncertainties of the corresponding (last two) digits in the parametervalue. Epoch (TDB) 2013 Aug 1.0Eccentricity 1.0006045(61)Perihelion distance (au) 1.3990370(73)Time of perihelion passage (TDB) 2014 Oct 25.3868(14)Longitude of node ( ◦ ) 300.974337(84)Argument of perihelion ( ◦ ) 2.43550(33)Inclination ( ◦ ) 129.026659(32) 29 –Table 2: Close approach data for JPL orbit solution 46.Close approach epoch ( ± σ ) 2014 Oct 19 18:30 (TDB) ± ± σ ) 134,680 km ± v ∞ ) 55.96 km/sMOID 27,414 kmNode crossing distance 27,563 kmMars’s arrival at line of nodes 2014 Oct 19 20:09 (TDB)C/2013 A1’s arrival at line of nodes 2014 Oct 19 17:21 (TDB)Mars’s arrival at MOID 2014 Oct 19 20:11 (TDB)C/2013 A1’s arrival at MOID 2014 Oct 19 17:20 (TDB) 30 –Table 3: A priori values and 3 σ uncertainties of nongravitational parameters for the three scenarios.Scenario A (au/d ) A (au/d ) A (au/d )Ballistic 0 ± ± ± ± × − (0 ± × − (0 ± × − Wide (1 ± × − (0 ± × − (0 ± × −
31 –Table 4: Close approach parameters and uncertainties for the three scenarions. The table showsthe b -plane coordinates, the semimajor axis of the 3 σ uncertainty projected on the b -plane, and thetime of closest approach.Scenario ξ (km) ζ (km) 3 σ SMA (km) TCA (TDB) ± σ Ballistic -27,445 -132,407 4789 2014 Oct 19 18:30 ± ± ±±