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Hartley 2
Icarus 222 (2013) 653–661
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The sources of the unusual dust jets seen in Comet 103P/Hartley 2 Michael J.S. Belton ⇑ Belton Space Exploration Initiatives, LLC, 430 S. Randolph Way, Tucson, AZ 85716, USA
a r t i c l e
i n f o
Article history: Received 7 October 2012 Revised 27 November 2012 Accepted 10 December 2012 Available online 21 December 2012 Keywords: Comets Comets, Nucleus Rotational dynamics
a b s t r a c t We show that the unusual behavior of dust jets seen embedded in the sunward coma of 103P/Hartley 2 originate in active regions migrating over the two lobes of the nucleus following the Sun. The slowly changing orientation of the jets and their rapid changes in brightness is due to the shape and local topography of the nucleus coupled with the complex spin state. The intermittent appearance of a second jet is due to periodic deviations in the direction of the ejection of dust from the small lobe of the nucleus. The release of dust into the structures is likely due to the sublimation of H2O. The jets are characterized by injection speeds from the nucleus of 50–210 m/s, a radiation pressure parameter 0.08 < b < 1, and a particle life-time near 7 h. Within the jets, the average particle size decreases and the injection speed increases with distance from the nucleus. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction to the 103P/Hartley 2 dust jets Comet 103P/Hartley 2 (hereafter 103P) displays at least three different types of jet phenomena: a highly structured, CO2 driven, jet complex seen in EPOXI encounter images at the small end of the nucleus (A’Hearn et al., 2011), two large scale jets seen in CN, OH, and other molecular emissions (e.g., Knight and Schleicher, 2011; Samarasinha et al., 2011), and two fainter, approximately linear, dust jet or ‘‘cloud’’ structures seen embedded in the sunward coma (Lara et al., 2011; Mueller et al., 2012; Tozzi et al., 2012; Knight and Schleicher, 2012). The focus of this paper is the dust jets, in particular an elucidation of their sources. In Fig. 1 we show examples of the dust jets from the above observers. To summarize the observations taken near the time of the EPOXI encounter (November 4, 2010, 13:59:47), the most prominent dust jet was found to generally point sunward on the sky and was always present in data taken under good conditions. It was also slightly curved towards larger position angles (PA). It changed its projected orientation slowly from night-to-night but was never far away from the projected direction of the Sun (PA 104°). It had a projected length of 3000–4000 km, a width that grew with distance from the nucleus, and, according to Knight and Schleicher (2012), showed no strong evidence for the effects of radiation pressure. It was also shown to have variable brightness even over a time interval as short as a few hours, was ‘‘grey’’ in color, and reddens towards the nucleus (Lara et al., 2011; Knight and Schleicher, 2012). The brightness of the jet also fell-off with
⇑ Fax: +1 520 795 6286. E-mail address: [email protected] 0019-1035/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.icarus.2012.12.007
increasing distance q from the nucleus. Mueller et al. (2012) find that the brightness fall-off was faster than 1/q. Less is known about the second jet. It was not always present or very faint, being visible on 3 of the 17 nights recorded in the literature, possibly appearing intermittently. Its projected direction deviates from the direction of the Sun by some 30° towards PA 140° in the November, 2010, time frame. The limited data also seem to indicate that it may possibly repeat after 50 h, but this is uncertain. For example, in the Mueller et al. (2012) and Knight and Schleicher (2012) data it appears on two nights in images separated by 47 h. Knight and Schleicher (2012) found no evidence for a gas jet structure (i.e., in OH, CN, C2, C3, NH) associated with these jets in images in which the background coma was filtered out. While both jets were observed in November, only a single jet was seen in October and December, 2010 (Mueller et al., 2012). From this behavior all observers have suggested that the jets might originate from discrete active areas on the surface of the nucleus and both Knight and Schleicher (2012) and Mueller et al. (2012) have suggested that the source of the jets is near the root, at the nucleus, of the comet’s total rotational angular momentum vector (AMV) or its reverse. In an attempt to constrain the position of the sources on the nucleus, Mueller et al. (2012) posit that the source of the dominant jet may have been in sunlight the entire time and suggest spin in the short axis mode (SAM) as a possibility. There is disagreement between observers on the fluid driving the jet. Knight and Schleicher (2012) argue that the lack of gas features associated with the jet suggest an unobservable such as CO, or more likely, CO2, while Mueller et al. (2012) argue for water driven jets. The distribution of color in the dominant jet and its changing brightness suggests that the grains are relatively large when they leave the nucleus and probably sublimate or disintegrate with
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Fig. 1. Dust jets in 103P from different observers using different enhancement techniques around the time of the EPOXI encounter. In all cases North is up and East is to the left and the images are centered on the nucleus. The bright feature to the West of the nucleus is the comet tail. The features of interest are to the East of the nucleus. The position angle (PA) of the Sun varies systematically from image to image, but is approximately 104°. Note that at different times there are two dust jet features one near PA 104° and the other near PA 140°. The left two columns marked A are by Knight and Schleicher (2012) who enhance the images by division with an azimuthal median profile. These images are 8000 km on the side and taken through a blue continuum filter. The top row is for two times during November 2, 2010, the 2nd row is for November 3, 2010, the 3rd row is for November 4, 2010 and the last row for November 7, 2010. The images marked B are by Mueller et al. (2012) and are 8400 km on the side. The images, which are for November 2, 2010 (left) and November 4, 2010 (right), were enhanced by dividing the original by an azimuthally averaged radial profile. The images marked C, are from Lara et al. (2011) and are 5800 km on the side. These images, which were taken on October 27, 2010, are in filters centered at 443 nm (left) and 648 nm (right) and each is the quotient of two flux calibrated images taken a few hours apart. The images marked D, are from Tozzi et al. (2012) and are 2000 km on the side. Taken on November 5 and November 4, 2010 respectively, they were enhanced by subtracting an image of the comet at minimum activity. All of the images shown here are reprinted with permission from the various observers.
time, i.e., with distance from the nucleus. Mueller et al. (2012) and Tozzi et al. (2012) find that the minimum speed of the grains is 80 m/s and 15–20 m/s respectively, while Mueller et al. (2012) place constraints of 0.013, 0.029, and 0.007 (km/s)2 on V 2d =b in October, November, and December, 2010, respectively. Here Vd is the average speed of the dust grains as they leave the nucleus (the injection speed) and b is the average ratio of radiation pressure to solar gravity acting on the grains. In this paper we investigate the source of the dust jets using a shape model of the comet’s nucleus (Thomas et al., 2012) and a model of the comet’s complex spin state (Belton et al., 2012) based on EPOXI observations. We show that the dust jets likely arise from Type 1a active regions (see Belton (2010) for a proposed taxonomy of active areas) centered at the subsolar points on the two lobes of the nucleus as they migrate over the surface during the spin cycle, i.e., the jets are the result of sublimation, presumably mainly of H2O, in the vicinity of the subsolar regions on each lobe of 103P’s nucleus. In this picture, slow changes in the projected direction of the dominant jet are a reflection of the changing topography of the surface at the sub-solar region during the spin cycle. The behavior of the intermittently visible second jet is a result of an interaction between the peculiar shape of the nucleus (Thomas et al., 2012) and its complex spin (Belton et al., 2012) that causes the ejection of dust from the small lobe to briefly wander to larger PAs during the spin cycle. Thus we see the two jets as part of a
single outflow system arising from the variable extent of the sub-solar region on the nucleus that occasionally bifurcates because of the peculiar, bi-lobate, shape of the nucleus and its complex spin. In Section 2 we describe relevant properties of the nucleus needed for our simulations, and, in particular, discuss the release of particulates from its surface and their acceleration in the comet’s atmosphere. In Section 3 we present a simplified simulation of the axis of the dust jets and compare the results with observations. In Section 4 we present a discussion of the results.
2. Relevant properties of the nucleus and the release and acceleration of dust from the cometary surface To fully simulate the release of dust from the illuminated surface and the formation of a jet it is necessary to have: (i) a quantitative assessment of the shape and topography of the nucleus and the location of any active areas; (ii) a quantitative description of its spin, the magnitude and orientation of the AMV, and their rates of change; (iii) a practical theory of how dust particulates – ice crystals, silicates, organics, aggregates, etc., – are released from the surface; (iv) a theory of how the cometary atmosphere and, the solar gravity and radiation fields accelerates them; (v) a quantitative assessment of the properties of the atmosphere and its time
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dependence; and (vi) knowledge of the size distribution of the particles and how they might sublimate and fragment while in the jet. Few of these requirements can be met for most comets. However, in the case of 103P, items (i) and (ii) are now available is considerable detail. For item (iv) recent studies by Tenishev et al. (2011) and Combi et al. (2012) give a secure picture of how dust particles are accelerated by the atmosphere once released from the surface. When free of the atmosphere, it is already well known how to parameterize the motion of the grains in the competing radiation and gravitational fields of the Sun (e.g. Fulle, 2004). Items (iii), (v) and (vi) are largely terra incognito. With regard to (v), a considerable amount of information is latent in the spacecraft and Earth-based observations acquired at the EPOXI fly-by, but this has not, to our knowledge, as yet received any systematic quantitative treatment. However, enough is known to allow the somewhat simplified simulation treatment made in this study. Some information on items (iii) and (vi) is becoming available with the observations of Clark et al. (2004) at 81P/Wild 2 and Economou et al. (2012) at 9P/Tempel 1, which show that many, if not all, particles leave the surface as aggregates that subsequently break-up in the outer atmosphere. But the crucial elements of how sunlight triggers the break-off of such aggregates from the surface and what the initial particle size distribution might be, are still largely unexplored. Neither do we have a quantitative basis on how to properly parameterize fragmentation and sublimation of jet aggregates as they move in the jet. Most of the work done previously on this topic (e.g., Patashnick and Rupprecht, 1975; Hanner, 1981; Lien, 1990) involves single particles. In our simulations we will make simplifications that, of necessity, avoid these issues. 2.1. Shape, size, orientation and spin of the nucleus The shape and size (Table 1) of 103P is well defined by the EPOXI observations (Thomas et al., 2012) with the local topography specified in a shape model down to 2 2° elements. On the average, each element corresponds to an area of 400 m2 but varies over the nucleus. This is to be compared with a total surface area of 5.24 106 m2 (Thomas et al., 2012). In our simulations we calculate the normals to the surface relative to such small elements using the edges of the mesh elements. The nucleus is bi-lobate in shape, far from spherical, with one lobe having 1.5 times the surface area of the other. The small lobe has almost spherical curvature while the large lobe crudely approximates a cylindrical shape. In our simulations we shall treat each lobe separately as we expect that the curvature of the surface will play a role in defining the direction and cross-section of the jets. In order to provide a concrete visualization of the nucleus, we show, in Fig. 2, the shape model oriented to simulate the nucleus as it would have appeared to an observer on the Earth at the time of EPOXI encounter. To give an idea of the complexity of the spin, the Supplementary Online Material (SOM) contains Movie 1 that shows three rotations of the nucleus and includes a normal to an arbitrary point on the small end of the nucleus that illustrates the complexity of the motion. The orientation of the AMV is also shown. The spin state, including the orientation of the AMV, is well defined by the EPOXI observations (Belton et al., 2012). The characteristics of the spin state that we adopt for our simulations are also included in Table 1. Note that with a 18.40 h precession of the long axis around the AMV and a 26.72 h roll period around the long axis, a complete spin cycle, i.e., the time to return to essentially the same geometric configuration as seen by a remote observer, is 55 h. We recognize that there is presently some controversy over the direction of the AMV in space owing to divergent estimates published by Earth-based observers (Knight and Schleicher, 2011; Samarasinha et al., 2011; Waniak et al., 2012) who base their estimates on interpretations of the changing morphology of gas jets, particularly in
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CN emission. For the present, we justify our choice of spin state by the fact that it not only satisfies the totality of the EPOXI data but also provides, as pointed out by Belton et al. (2012), a satisfactory explanation of the radar observations of Harmon et al. (2011). The spin model used in our simulations also includes the effects of systematic changes in the precession and roll rates with time, although it is always assumed that the AMV direction is constant. To represent these changes a linear approximation is used, but we anticipate that it should provide reliable simulations for observations taken in October and November, 2010, but possibly not in December, 2010. As noted above, the SOM contains Movie 1 that shows the nature of the spin. 2.2. The release and acceleration of dust from the cometary surface It is not clear what physical processes are involved in releasing particulates from the surface of the nucleus. Sunlight obviously plays a dominant role on the illuminated part, but, to a much lesser extent, particulates are also observed being released on the darkside of the nucleus of 103P (Feaga et al., 2012). Several authors (e.g. Gombosi et al., 1985; Belton, 2010) have followed the assumption of Horanyi et al. (1984) that the flux of dust particles leaving the surface is proportional to the flux of subliming gases with the coefficient of proportionality being a function of particle size and comet. However, observations from the Stardust and Stardust-NExT missions indicate that a fraction, possibly all, of the dust is released as aggregates that disintegrate in the atmosphere (Clark et al., 2004; Economou et al., 2012). In their analysis Clark et al. (2004) point out that other physical processes could be involved such as thermal stress, internal gas pressure, and turbulence to name a few. They also quote experiments by Hartmann (1993) and Gruen et al. (1993) that show the liberation of large aggregates and particles in episodic events. It is not known how these processes respond to changing incidence angle of solar illumination, and so the commonly made assumption that dust production falls off smoothly with the cosine of the solar incident angle on a curved surface must be questioned. Regardless of how particulates, or aggregates, are released from the surface, our understanding of their acceleration in the cometary atmosphere appears to be far more secure and is described in two recent studies by Tenishev et al. (2011) and Combi et al. (2012), both made in the context of the forthcoming Rosetta mission to 67P/Churyumov–Gerasimenko. We expect that the results contained in these studies are, at least qualitatively, applicable to 103P even though 103P has strong sources of H2O vapor in its atmosphere and 67P may not. For example, Belton (2012) finds that the loss of H2O vapor from the surface of 103P lies between 6 1025 and 5 1026 molecules/s and since 67P has 13 times the surface area this would correspond to 8 1026 to 7 1027 molecules/s on 67P. In the Tenishev et al. (2011) and Combi et al. (2012) calculations for 1.29 AU, a total loss rate of 5 1027 molecules/s is assumed, which falls in this range. This gives some reason to believe that these calculations could have some applicability to 103P, at least qualitatively, in the context of accelerating dust particles to the speed at which they are injected into the dust jets. Tenishev et al. (2011) and Combi et al. (2012) present detailed calculations of the dynamics of dust particles released. Evidently, over a restricted area, the escaping solid material forms a ballistic (i.e., with essentially no collisions) system of particles moving on near parallel paths normal to the surface with a spread of terminal speeds that reflects the size distribution of the particles. In the Combi et al. (2012) study, which specifically concerns a small active area, the particle trajectories become detached from the gas motion a few km (1–10 nucleus radii) above the surface and form a highly collimated jet-like structure, while the gas rapidly expands both outwards and laterally to quickly fill the entire
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M.J.S. Belton / Icarus 222 (2013) 653–661 Table 1 Relevant properties of the shape model and spin state of the nucleus. The tabulated values are from Thomas et al. (2012) and Belton et al. (2012) and are collected here for convenience. The principal moments of inertia Il, Ii, and Is are related to the long, intermediate, and short axes respectively. The values for the moments of inertias are computed assuming that the interior mass distribution is homogeneous and are given per unit mass of the nucleus. Quantity
Value
Time of EPOXI closest approach (UTC)
2010-11-04, 13:59:47.7 JD 2455505.0831866 580 ± 18
Mean radius (m) Ratios of model principal moments of inertia Il (m2) Il/Is Ii/Is Body moment orientations (Latitude, E longitude) Il Ii Is Coordinates of the center of mass (latitude, longitude, and radius vector [m]) Direction of long axis (small end) at closest approach (RA, Dec; J2000) Direction of minimum moment at closest approach (RA, Dec; J2000) Direction of intermediate moment at closest approach (RA, Dec; J2000) Direction of the total rotational angular momentum vector Surface area (m2) Volume (m3) Length of the long axis (m) Estimated mean density (kg/m3) Estimated mass of nucleus (kg) Circulation period of long axis about M, the total rotational angular momentum vector (h) Roll period around the long axis (h) Sense of spin Mean tilt of the long axis to M Mean instantaneous period (h) dM/dt (m2/s2 per unit mass) dE/dt (m2/s3 per unit mass; E = rotational energy)
hemisphere without forming a gas-jet. In the Tenishev et al. (2011) study, the particles similarly detach from the gas flow a few nucleus radii above the surface, and, for the most part, move radially outward from the assumed spherical nucleus. This behavior forms the basis for our modeling of the dust jets on 103P. They are streams of particulates moving, at least initially, in parallel having been injected into the dust jets in a direction normal to the local surface. At any particular time there is a range of injection speeds, Vd(a), in the jet that reflect the initial particulate size (a) distribution and its response to the accelerative forces applied by the atmosphere close to the surface. An examination of the numerical results of the Tenishev et al. (2011) and Combi et al. (2012) calculations (their Figs. 10 and 7 respectively), shows that an interpolation formula
V d ðaÞ ¼ Constant1 aa
5.302 ± 0.4 104 0.166 ± 0.004 0.979 ± 0.002 89.73°, 207.56°E 0.05°, 106.14°E 0.25°, 16.14°E +19.57°, 240.39°E, 5.0 RA: 226.12 ± 0.9° Dec: 39.37 ± 0.7° RA: 225.92 ± 0.9° Dec: 39.60 ± 0.7° RA:301.8° Dec: 16.5° RA: 8° ± 4° Dec: 54° ± 1° 5.24 106 8.09 108 2330 230–300 1.9–2.4 1011 18.40 ± 0.13 26.72 ± 0.06 Prograde 81.2° ± 0.6° 14.1 ± 0.3 4.4 107 4.0 1011
3. Simulation of the dust jets The observational basis that we use to provide a foundation for our simulation of the jet complex is that the jets always seems to be present, are generally projected towards the Sun, and that they can vary considerably in brightness. These facts imply that a variable area around the sub-solar region may be involved in the production of the jet and that, since the nucleus spins as a LAM (long axis mode), the active area cannot be tied to the surface but that it must migrate following the Sun, i.e., we are considering a Type 1a active area as defined by Belton (2010). The secondary jet, apparently intermittent, could be associated with an active area fixed to the surface. However, as we shall find, the two jets are really part of a single, but complex outflow, from the bi-lobate nucleus.
ð1Þ 3.1. Simulation technique
provides a good description of the numerical results where a is the particle radius. The exponent a can be expected to change from comet to comet and with heliocentric distance. In the case of the Tenishev et al. (2011) calculations we find a = 0.44 and we will apply this value to 103P. Once in the jet, the particles are dynamically under control of solar gravity and radiation pressure characterized by the parameter b, where, according to Fulle (2004), b(a) is proportional to a1 for particles ranging in diameter from 1 lm to 1 cm, i.e. those most commonly found in comets, and so
V d ðaÞ ¼ Constant2 bðaÞa
ð2Þ
These numerical relationships provide a convenient way to parameterize the dynamics of particles in the jet. We choose specific values for Vd, b and a to characterize the jet and then for a range of speeds in the jet take Vd, (2/3)Vd and (1/3)Vd, etc. Finally, we use Eq. (2) to consistently calculate the relevant values of b.
There is not enough information about conditions and processes on 103P to perform a full-up simulation of the brightness distribution in the dust jets, i.e., to predict their 3-D structure, light scattering properties, particle size distribution, mode of release from the surface, etc. In this work we take an incremental approach where, as a first step, we attempt to simulate only the geometry of the dust jet and its projection on the sky. We do this in terms of only three parameters (Vd, b, a) as described above. With this philosophy we put aside the complications of fragmentation and sublimation in the jets that could lead to an expansion or contraction of the spatial domain of the jets. This approach, as we shall see, is enough to define the sources of the dust jets and explain much of the jet’s behavior. We calculate the jet axis using the following formalism: In vector notation (vectors are in bold font), the position rd (t) of a typical
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where ’ represents differentiation with respect to time and s = (t – t0). We define the jet axis at the time of observation, t, as the locus of particles at rd(t) that were injected into the dust jets with a range of velocities Vd at a sequence of times t0. The simulated jet can therefore be thought of as the summation of a sequence of ‘‘subjets’’ each defined by a separate value of injection speed, Vd, and, through Eq. (2), a different value of b. We ignore details of the release from the surface and acceleration of the dust particle in the atmosphere close to the nucleus. The calculations of Tenishev et al. (2011) and Combi et al. (2012) show that dust particles are accelerated in a region very close to the nucleus (we estimate 1– 10 nucleus radii or 0.6–6 km in the case of 103P) and, for the sub-solar region, the acceleration is essentially normal to the surface. Since the distance over which the acceleration takes place is miniscule as compared to the dimension of the dust jets, it is sufficient to parameterize the calculation with injection velocities parallel to the local surface normal N, but emanating from the center of mass of the nucleus. The coefficients in Eq. (3) are easily calculated from those appropriate for the nucleus itself presumed to be moving on a two-body orbit, e.g.,
prefer its use to the alternative of direct integration of the dust trajectories because it exposes the physics of what is going on in each of its separate terms. In performing the calculations we check at every step the convergence of the series by ensuring that the ratio of successive terms is 1. In addition, we take the calculation to the third order in s. This formalism is, of course, the basis of the old Bessel–Bredikhin theory of comet tail forms (e.g., Bobrovnikoff, 1951), although that theory uses a second Taylor expansion to bring the coefficients back to evaluation at time t of observation. It also generally ignores the motion of material perpendicular to the orbital plane. We do not employ either of these latter steps that can severely limit the application of Bessel–Bredikhin theory. The calculation proceeds as follows: We choose a sequence of values for t0 (20 values) starting a half a rotational cycle (9 h) back from the time of observation and for each time calculate the rotational state of the nucleus. This time interval was chosen after some experimentation and does not have a physical justification other than it produces acceptable simulations. The locations on each lobe, i = 1, 2, with the smallest solar zenith angle are determined as are their vector normals N1 and N2 to the surface. We take these ‘‘sub-solar’’ points as the injection points of dust particles that will form the axis of the jet. Vd,i = Vd Ni defines the injection velocity from lobe i where Vd is the injection speed parameter. In the simulations shown below we have found it sufficient to employ three sub-jets to satisfactorily emulate the effect of a distribution of injection speeds. These speeds are taken as Vd, (2/3)Vd, and (1/3)Vd. Note that since Vd 10–500 m/s (Tenishev et al., 2011; Combi et al., 2012), we can ignore the velocity imparted at the surface by rotation which is <0.1 m/s. While this approach is adequate for defining the basic geometry of the jets it would not be adequate for computing the brightness distribution where a full description of the injection velocity distribution would be needed. As a measure of the extent of the most active regions on the nucleus with respect to sublimation, we arbitrarily map a region on the surface within which the solar zenith angle is less the 24.5° and calculate its area in units of mesh elements. The coordinates of the dust particles at the time of observation, t, that were injected at t0 are then calculated in Equatorial coordinates relative to the nucleus for parameters Vd, b, and a using Eqs. (2)–(4) and where the positions and velocities of the nucleus are calculated using the 2-body orbital elements for 103P obtained from the JPL Horizons System. The final step is to transform the coordinates to the observer frame, again in Equatorial coordinates. In the SOM we include Movie 2 that follows the spin of the nucleus over a complete spin cycle (55 h) and shows how the sub-solar points move over the surfaces of the two lobes, how the sub-solar region, as defined above, waxes and wanes and how the directions of the normals to the surface at the sub-solar points are modulated. With respect to the latter, it is interesting that the projected normals from the sub-solar points retain approximate parallelism over most of the cycle except for brief periods when the normal to the small lobe wanders-off temporarily to considerably larger PAs. It is this behavior that is responsible for the apparently intermittent appearance of the secondary jet as we discuss below.
rd ðt0 Þ ¼ rn ðt 0 Þ
3.2. Application to the 103P/Hartley dust jets
Fig. 2. The appearance of the nucleus of 103P/Hartley 2 as seen from the Earth at the time of the EPOXI encounter. North (up) and East (left) are denoted by vertical and horizontal lines. The line at position angle 104° is the projected direction of the Sun. The terminator and dark limb are clearly seen. The dark grey areas on the illuminated part of the nucleus are the sub-solar regions where the solar zenith angle is less than 24.5°.
dust particle at time t that was injected from the vicinity of the nucleus at position rd (t0) at time t0 with injection velocity Vd and moves under the influence of the solar gravity and radiation fields characterized by b can be written as a Taylor expansion: 3 rd ðtÞ ¼ rd ðt 0 Þ þ r0d ðt0 Þ s þ ð1=2Þr00d ðt 0 Þ s2 þ ð1=6Þr000 d ðt 0 Þ s
þ
r0d ðt0 Þ
¼
r0n ðt 0 Þ
ð3Þ
þ Vd N
r00d ðt0 Þ ¼ k r00n ðt0 Þ ¼ kGMs rn ðt 0 Þ=r n ðt 0 Þ3
ð4Þ
3 5 0 0 r000 d ðt 0 Þ ¼ kGM s frn ðt 0 Þ=r n ðt 0 Þ 3rn ðt 0 Þ½rn ðt 0 Þrn ðt 0 Þ=r n ðt 0 Þ g
þ higher terms; Here the subscript n refers to the nucleus and k = (1 b). Ms is the mass of the Sun and G the gravitational constant. Even though convergence of this expansion is not assured for arbitrary times, s, we
3.2.1. The injection direction of the particles as a function of time In Fig. 3 we plot the PA’s of injection velocities from the sub-solar regions on the two lobes of the nucleus over an interval a little longer than a full spin cycle as seen from the Earth. Also included in the plot is the PA of the projected direction of the Sun. The PA of particles that were injected from the small lobe show strong periodic excursions from the sun-line for short intervals of time. The periodicity of these excursions is about that of the precessional
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Fig. 3. Plot of the PA’s of particles injected into the dust jets from the sub-solar regions on the small (full line) and large (dashed line) lobes of the nucleus over an interval slightly longer than a complete spin cycle (55 h) as seen from the Earth. The particles from both lobes show strong, but brief, excursions from the projected direction of the projected sun-line (nearly horizontal full line). The excursions from the two lobes are out-of-phase by about 9 h with those characterizing the large lobe having 1/3 the amplitude of those of the small lobe.
period of the long axis of the nucleus around the AMV. The PAs of particles injected from the large lobe also show excursions from the sun-line that are similarly short lived, but, in this case, the excursions have an amplitude only 1/3 that of the particles escaping from the small lobe. In addition the excursions from the large lobe are 9 h out-of-phase from those from the small lobe. For a majority of the time, the PAs of the injection velocity from both lobes hovers close to the sun-line, although those of the small lobe average 2–4° to the North. These results are compared to observations in Fig. 4. In the upper panel we plot the PAs noted by Knight and Schleicher (2011) and Mueller et al. (2012) in the week before the EPOXI encounter. In the case of the latter authors, a range of PAs was given and this is also shown. Presented in this way, there appears to be no obvious agreement between the observations and the model PAs. However, if the observed points are shifted by 7 h, as in the lower panel, a correlation appears between the model PAs of injection velocity from the small lobe and the PAs observed for the secondary jet. Away from the main peaks, the observed PAs and those of the injection velocity from the large lobe mainly deviate in the sense that observed PAs tend to be greater than those of the model injection velocity. Nevertheless, and with the exception of a single observation, the deviations are not large even though they are systematic. We briefly discuss the origin of this shift in Section 4. 3.2.2. The sub-solar dust jet complex In Figs. 5–7 we present simulations of the axis of the dust jet as observed on November 2–4, 2010, by Mueller et al. (2012). In each case the simulations are defined, as noted above, by a = 0.44 and three injection speeds (Vd, (2/3)Vd, and (1/3)Vd) with Vd = 210 m/s. Vd with b = 1 is taken to correspond to the smallest particles expected in the jets (cf. Table 1 in Fulle (2004)), while the bs for the slower (and larger) particulates are calculated from Eq. (2). In order to improve correspondence of the model to the observations we have found necessary to impose a lifetime to the particles in the jet of 6.8 h, In addition, for the case of November 4, the simulation can be improved by imposing a cut-off, or significant reduction, in the flow of particles about 4.5 h before the observation was made. This implies that the entire structure seen on that night was formed over a period of 3 h. In the case of the November 2 observations, we found a better correspondence between observation and simulation if Vd was reduced to 150 m/s.
Fig. 4. (Upper panel) Same as Fig. 3 but with observed PAs of dust jets superposed. Observations marked N are from Knight and Schleicher (2011) those marked are by Mueller et al. (2012). The latter authors give the spread of PAs for each observation; this is denoted by a vertical error bar. (Lower panel) Same as above but with the observed points shifted by 7 h, which brings them into better alignment with the model curves.
In evaluating these simulations the reader should recall that the spatial distribution of points does not represent the brightness distribution in the jet but is simply a locus of points of particles released at a sequence of different times. Nevertheless, the simulations show reasonable accord with the data in that the particles generally emerge from the nucleus, with scatter, on, or near, the sunline, but that, at certain times, material is ejected towards more southeasterly directions, intermittently forming what appears to be a secondary jet. Since the simulations are calculated in three dimensions it is easy to see what complexity there is in the structure of the jets as seen from other vantage points. In Fig. 8 we look at the simulation of November 4, 2010 from the direction of an observer on the Sun. The structure is seen to broadly spread out from the sun-line as a result of the changes in the local topography on the nucleus at the times when the individual particles were injected. The jets are also observed to vary rapidly with brightness. The simulations provide an explanation for this in the rapid changes of cross-section that the nucleus presents to the Sun during a spin cycle. This is seen more clearly in Fig. 9 where, in the upper panel, we plot the number of mesh element in the shape model that fall within the sub-solar region as defined above, as a function of time. We consider this as a surrogate for following the rise and fall of gas and dust production, The time interval covered is the last spin cycle (55 h) before the EPOXI encounter. As might be expected for the odd shape of 103P, this measure of production rate varies strongly during the spin cycle with peaks of variable strength that occur every 9 h and rapid changes are seen to take place over intervals as short as 3 h.
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Fig. 5. (Left panel) Image of dust jets for UT November 4, 2010, 11:28, from Mueller et al. (2012). The image is 8400 km on a side, North is up and East is to the left. (Right panel) Simulation of the dust jet axis. The vertical line points North and the horizontal line Eastwards. The scale of the simulations is the same as the image. The inclined line towards the PA = 104° is the projected sun-line. Markers , +, x, trace out the locus of particles with injection speeds Vd, (2/3)Vd and (1/3)Vd. In this simulation Vd = 210 m/s, b = 1, and a = 0.44 and the particles were injected every 27 min over a period of 9 h immediately preceding the time of observation. (Center panel) Same as the right panel except that a lifetime of 6.8 h has been imposed on the particles in the jet. In this case, we have also curtailed the injection of particles from the nucleus for 4.5 h before the time of observation. The bright material at the nucleus falls on the projected sun-line and was apparently emitted shortly before the time of observation.
Fig. 6. Same as Fig. 5 but for UT November 3, 08:28. In this case there is no curtailment of the ejection of particles before the time of observation. The panels are 8400 km on a side.
Fig. 7. Same as Fig. 5 but for UT November 2, 12:32 and two values of Vd. In this case there is no curtailment of the injection of particles before the time of observation. The simulation in the lower panel, where Vd = 150 m/s, provides a better correspondence with the observations. The panels are 8400 km on a side.
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lobes inject material toward the same PA, i.e., they are essentially parallel reinforcing each other, in a direction 5° North of the solar PA. However, three times in the spin cycle the PA of the injection velocity from the small lobe makes a large excursion from the solar direction to PA 140°, over an interval of 5 h, at times when the production rates are low and not changing rapidly. We expect, from this simulation that the observed rapid changes in brightness should occur in observations were the formation time of the jet spans an interval of rapidly changing active area. 4. Discussion
Fig. 8. The dust jet simulation for UT November 4, 2010, 11:28, as seen from the direction of an observer at the Sun. The simulations are 8400 km on the side. (Right panel) Same simulation as in the right panel of Fig. 5. (Left panel) The simulation to the right has been rotated to present a view along the sun-line. The threedimensional structure of the dust jet, which is due to the changing topography at the sub-solar point, becomes very clear in this comparison.
Fig. 9. (Upper panel) Plot of the number of mesh elements of the shape model that fall within the sub-solar region, i.e., the region where the solar zenith angle is <24.5°, with time. The plot covers a full spin cycle of 55 h ending at the time of the EPOXI encounter (November 4 13:59:47.310, 2010). We use this as a crude surrogate for the production of dust and gas as the nucleus spins. The peaks are, as expected, separated by 9 h. i.e., half the precession period of the long axis around the AMV. The rapid changes of the area of the sub-solar region with time are thought to be the cause of the observed rapid changes occasionally seen in the brightness of the jets. (Bottom panel) A repeat of Fig. 3, which covers the same time frame as in the upper panel and is presented for easy correlation with the upper panel. Note that for much of the time that the PAs of the injection velocity from the two lobes more or less coincide are times of relatively high gas and dust production. The times when the PA of the injection velocity from both lobes deviates strongly from the solar PA tend to occur at times of minimum gas and dust production.
The simulations in Figs. 5–7 and the rapid changes in the area active around the sub-solar point in Fig. 9 show that the model for the sources of the sunward dust jets proposed here provide an adequate basis for understanding the geometrical and dynamical properties of this phenomenon. The complex spin of 103P’s bi-lobate nucleus coupled with ejection of particulates from the sub-solar regions on the two lobes provides a good accounting of the PAs of the dust jets and how their geometry changes from night to night. In addition, the theoretical modeling of dust acceleration in cometary atmospheres by Tenishev et al. (2011) and Combi et al. (2012) appears to give a reasonably reliable picture of how particulates fill out the axis of the dust jets. In this picture the active areas are not fixed to the surface but follow the Sun and should be classified as Type 1a in the scheme devised by Belton (2010). Whatever the detailed processes by which particulates are released from the nucleus surface, insolation and, presumably, sublimation of H2O are involved. As shown in Fig. 9, which we see as a surrogate for instantaneous particulate production rate, the brightness of the dust jets should vary dramatically and rapidly throughout the spin cycle. The observations show such changes, but it is beyond the scope of this paper (or the tools we have available) to model the observed timing and extent of these brightness changes and this is clearly an area for future research. In our discussion of the relationship of observed PAs to model PAs (Fig. 4) we noted that a shift of 7 h was needed to bring them into reasonable accord. As a result of the simulations, we understand this shift as the characteristic time it takes to renew the jet structure into the observed configuration. It represents the balance between the lifetime of the particles and the time to refill the structure. The picture of the dust jets that we have developed here is that they are a collisionless system of particulates in which the individual particles have a short lifetime of 7 h. The particles move on ballistic trajectories under the influence of both solar gravity and radiation pressure with a great deal of parallelism near the axis of the jet. The structure of the jets, which apparently never reach a steady state, is such that the regions further from the nucleus are dominated by smaller, faster moving, particles and regions near the nucleus more heavily weighted towards larger, slower moving particles. While we have not modeled the color of scattered light from the jet, this seems a good basis for understanding the observed reddening of the jets closer to the nucleus. The focus of this paper has been to understand the sources of the dust jets in the context of the peculiar bi-lobate shape of 103P and its complex spin state. Future modeling of the brightness distribution and color of these jets holds the potential of learning new details about the structure of the inner coma of 103P and new insight into how particles are released from the surface as a result of solar irradiation. Acknowledgments
This behavior can be compared with the PAs of injection velocities from the both lobes of the nucleus as shown in the lower panel. During times when the production rates are peaking the two
This work was supported by NASA Grant NNX07AG24G under the PMDAP program and NNM07AA99C both to the University of
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Maryland and sub-contracted to Belton Space Exploration Initiatives, LLC, through a Memorandum of Agreement with the National Optical Astronomy Observatories. The principal author thanks the Principal Investigators of the above Grants, Jian-Yang Li and Michael F. A’Hearn, for their enthusiastic support. We thank Drs. B.E.A Mueller, L.M. Lara, M.M. Knight and G.-P. Tozzi for providing digital images and permission to use figures from their papers. In addition, we thank Drs. Mueller and N.H. Samarasinha for insightful discussions during the development of various aspects of these simulations. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.icarus.2012. 12.007. References A’Hearn, M.F. et al., 2011. EPOXI at Comet Hartley 2. Science 332, 1369–1400. Belton, M.J.S., 2010. Cometary activity, active areas, and a mechanism for collimated outflows. Icarus 210, 881–897. Belton, M.J.S., 2012. Cometary evolution and cryovolcanism. Can. J. Phys. 90, 807– 815. Belton, M.J.S. et al., 2012. The complex spin state of 103P/Hartley 2: Kinematics and orientation in space. Icarus 222, 595–609. Bobrovnikoff, N., 1951. Comets. In: Hynek, J.A. (Ed.), Astrophysics. McGraw-Hill, New York. Clark, B.C. et al., 2004. Release and fragmentation of aggregates to produce heterogeneous, lumpy coma streams. J. Geophys. Res. 109, E12S03. Combi, M.R., Tenishev, V.M., Rubin, M., Fougere, N., Gombosi, T.I., 2012. Narrow dust jets in a diffuse gas coma: A natural product of small active regions on comets. Astrophys. J. 749 (article 29). Economou, T.E., Green, S.F., Brownlee, D.E., Clark, B.C., 2012. Dust flux monitor instrument measurements during stardust-NExT flyby of Comet 9P/Tempel 1. Icarus 222, 526–539. Feaga, L.M. et al., 2012. Volatile distribution and heterogeneities in 103P/Hartley 2 as observed by the deep impact HRI-IR. Presented at Astroids, Comets, Meteors Conference, Niigata, Japan.
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Fulle, M., 2004. Motion of cometary dust. In: Festou, M., Keller, H.U., Weaver, H.A. (Eds.), Comets II. Univ. of Arizona Press, Tucson AZ, p. 565. Gombosi, T.I., Cravens, T.E., Nagy, A.F., 1985. Time-dependent dust gasdynamical flow near cometary nuclei. Astrophys. J. 293, 328–341. Gruen, E. et al., 1993. Development of a dust mantle on the surface of an isolated ice-dust mixture: Results from the KOSI-9 experiment. J. Geophys. Res. 98, 15091–15104. Hanner, M.S., 1981. On the detectability of icy grains in the comae of comets. Icarus 47, 342–350. Harmon, J.K., Nolan, M.C., Howell, E.S., Giorgini, J.D., Taylor, P.A., 2011. Radar observations of Comet 103P/Hartley 2. Astrophys. J. 734, L2, (4pp). Hartmann, W.K., 1993. Physical mechanism of comet outburst: An experimental result. Icarus 104, 226–233. Horanyi, M. et al., 1984. The friable sponge model of a cometary nucleus. Astrophys. J. 278, 449–455. Knight, M.M., Schleicher, D.G., 2011. CN morphology studies of Comet 103P/Hartley 2. Astron. J. 141, 183–197. Knight, M.M., Schleicher, D.G., 2012. The highly unusual outgassing of Comet 103P/ Hartley 2 from narrowband photometry and imaging of the coma. Icarus 222, 691–706. Lara, L.M., Lin, Z.-Y., Meech, K., 2011. Comet 103P/Hartley 2 at perihelion: Gas and dust activity. Astron. Astrophys. 532. A87, (11pp). Lien, D.J., 1990. Dust in comets. I. Thermal properties of homogeneous and heterogeneous grains. Astrophys. J. 355, 680–692. Mueller, B.E.A., Samarasinha, N.H., Farnham, T.L., A’Hearn, M.F., 2012. Analysis of the sunward continuum features of Comet 103P/Hartley 2 from ground-based images. Icarus 222, 799–807. Patashnick, H., Rupprecht, G., 1975. The size dependence of sublimation rates for interplanetary ice particles. Astrophys. J. 197, L79–L82. Samarasinha, N.H., Mueller, B.E.A., A’Hearn, M.F., Farnham, T.L., Gersch, A., 2011. Rotation of Comet 103P/Hartley 2 from structures in the coma. Astrophys. J. 734, L3, (6pp). Tenishev, V., Combi, M.R., Rubin, M., 2011. Numerical simulation of dust in a cometary coma: Application to Comet 67P/Churyumov–Gerasimenko. Astrophys. J. 732 (article 104). Thomas, P.C. et al., 2012. Shape, density, and geology of the nucleus of Comet 103P/ Hartley 2. Icarus 222, 550–558. Tozzi, G-P., et al. 2012. Activity of comet 103P/Hartley 2 at the time of the EPOXI mission flyby. Earth and Plan. Astrophys., arXiv:1206.1185 [astro-ph.EP]. Waniak, W., Borisov, G., Drahus, M., Bonev, T., 2012. Rotation-stimulated structures in the CN and C3 comae of Comet 103P/Hartley 2 around the EPOXi encounter. Astron. Astrophys. 543, A32–A39.
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The sources of the unusual dust jets seen in Comet 103P/Hartley 2 Michael J.S. Belton ⇑ Belton Space Exploration Initiatives, LLC, 430 S. Randolph Way, Tucson, AZ 85716, USA
a r t i c l e
i n f o
Article history: Received 7 October 2012 Revised 27 November 2012 Accepted 10 December 2012 Available online 21 December 2012 Keywords: Comets Comets, Nucleus Rotational dynamics
a b s t r a c t We show that the unusual behavior of dust jets seen embedded in the sunward coma of 103P/Hartley 2 originate in active regions migrating over the two lobes of the nucleus following the Sun. The slowly changing orientation of the jets and their rapid changes in brightness is due to the shape and local topography of the nucleus coupled with the complex spin state. The intermittent appearance of a second jet is due to periodic deviations in the direction of the ejection of dust from the small lobe of the nucleus. The release of dust into the structures is likely due to the sublimation of H2O. The jets are characterized by injection speeds from the nucleus of 50–210 m/s, a radiation pressure parameter 0.08 < b < 1, and a particle life-time near 7 h. Within the jets, the average particle size decreases and the injection speed increases with distance from the nucleus. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction to the 103P/Hartley 2 dust jets Comet 103P/Hartley 2 (hereafter 103P) displays at least three different types of jet phenomena: a highly structured, CO2 driven, jet complex seen in EPOXI encounter images at the small end of the nucleus (A’Hearn et al., 2011), two large scale jets seen in CN, OH, and other molecular emissions (e.g., Knight and Schleicher, 2011; Samarasinha et al., 2011), and two fainter, approximately linear, dust jet or ‘‘cloud’’ structures seen embedded in the sunward coma (Lara et al., 2011; Mueller et al., 2012; Tozzi et al., 2012; Knight and Schleicher, 2012). The focus of this paper is the dust jets, in particular an elucidation of their sources. In Fig. 1 we show examples of the dust jets from the above observers. To summarize the observations taken near the time of the EPOXI encounter (November 4, 2010, 13:59:47), the most prominent dust jet was found to generally point sunward on the sky and was always present in data taken under good conditions. It was also slightly curved towards larger position angles (PA). It changed its projected orientation slowly from night-to-night but was never far away from the projected direction of the Sun (PA 104°). It had a projected length of 3000–4000 km, a width that grew with distance from the nucleus, and, according to Knight and Schleicher (2012), showed no strong evidence for the effects of radiation pressure. It was also shown to have variable brightness even over a time interval as short as a few hours, was ‘‘grey’’ in color, and reddens towards the nucleus (Lara et al., 2011; Knight and Schleicher, 2012). The brightness of the jet also fell-off with
⇑ Fax: +1 520 795 6286. E-mail address: [email protected] 0019-1035/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.icarus.2012.12.007
increasing distance q from the nucleus. Mueller et al. (2012) find that the brightness fall-off was faster than 1/q. Less is known about the second jet. It was not always present or very faint, being visible on 3 of the 17 nights recorded in the literature, possibly appearing intermittently. Its projected direction deviates from the direction of the Sun by some 30° towards PA 140° in the November, 2010, time frame. The limited data also seem to indicate that it may possibly repeat after 50 h, but this is uncertain. For example, in the Mueller et al. (2012) and Knight and Schleicher (2012) data it appears on two nights in images separated by 47 h. Knight and Schleicher (2012) found no evidence for a gas jet structure (i.e., in OH, CN, C2, C3, NH) associated with these jets in images in which the background coma was filtered out. While both jets were observed in November, only a single jet was seen in October and December, 2010 (Mueller et al., 2012). From this behavior all observers have suggested that the jets might originate from discrete active areas on the surface of the nucleus and both Knight and Schleicher (2012) and Mueller et al. (2012) have suggested that the source of the jets is near the root, at the nucleus, of the comet’s total rotational angular momentum vector (AMV) or its reverse. In an attempt to constrain the position of the sources on the nucleus, Mueller et al. (2012) posit that the source of the dominant jet may have been in sunlight the entire time and suggest spin in the short axis mode (SAM) as a possibility. There is disagreement between observers on the fluid driving the jet. Knight and Schleicher (2012) argue that the lack of gas features associated with the jet suggest an unobservable such as CO, or more likely, CO2, while Mueller et al. (2012) argue for water driven jets. The distribution of color in the dominant jet and its changing brightness suggests that the grains are relatively large when they leave the nucleus and probably sublimate or disintegrate with
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Fig. 1. Dust jets in 103P from different observers using different enhancement techniques around the time of the EPOXI encounter. In all cases North is up and East is to the left and the images are centered on the nucleus. The bright feature to the West of the nucleus is the comet tail. The features of interest are to the East of the nucleus. The position angle (PA) of the Sun varies systematically from image to image, but is approximately 104°. Note that at different times there are two dust jet features one near PA 104° and the other near PA 140°. The left two columns marked A are by Knight and Schleicher (2012) who enhance the images by division with an azimuthal median profile. These images are 8000 km on the side and taken through a blue continuum filter. The top row is for two times during November 2, 2010, the 2nd row is for November 3, 2010, the 3rd row is for November 4, 2010 and the last row for November 7, 2010. The images marked B are by Mueller et al. (2012) and are 8400 km on the side. The images, which are for November 2, 2010 (left) and November 4, 2010 (right), were enhanced by dividing the original by an azimuthally averaged radial profile. The images marked C, are from Lara et al. (2011) and are 5800 km on the side. These images, which were taken on October 27, 2010, are in filters centered at 443 nm (left) and 648 nm (right) and each is the quotient of two flux calibrated images taken a few hours apart. The images marked D, are from Tozzi et al. (2012) and are 2000 km on the side. Taken on November 5 and November 4, 2010 respectively, they were enhanced by subtracting an image of the comet at minimum activity. All of the images shown here are reprinted with permission from the various observers.
time, i.e., with distance from the nucleus. Mueller et al. (2012) and Tozzi et al. (2012) find that the minimum speed of the grains is 80 m/s and 15–20 m/s respectively, while Mueller et al. (2012) place constraints of 0.013, 0.029, and 0.007 (km/s)2 on V 2d =b in October, November, and December, 2010, respectively. Here Vd is the average speed of the dust grains as they leave the nucleus (the injection speed) and b is the average ratio of radiation pressure to solar gravity acting on the grains. In this paper we investigate the source of the dust jets using a shape model of the comet’s nucleus (Thomas et al., 2012) and a model of the comet’s complex spin state (Belton et al., 2012) based on EPOXI observations. We show that the dust jets likely arise from Type 1a active regions (see Belton (2010) for a proposed taxonomy of active areas) centered at the subsolar points on the two lobes of the nucleus as they migrate over the surface during the spin cycle, i.e., the jets are the result of sublimation, presumably mainly of H2O, in the vicinity of the subsolar regions on each lobe of 103P’s nucleus. In this picture, slow changes in the projected direction of the dominant jet are a reflection of the changing topography of the surface at the sub-solar region during the spin cycle. The behavior of the intermittently visible second jet is a result of an interaction between the peculiar shape of the nucleus (Thomas et al., 2012) and its complex spin (Belton et al., 2012) that causes the ejection of dust from the small lobe to briefly wander to larger PAs during the spin cycle. Thus we see the two jets as part of a
single outflow system arising from the variable extent of the sub-solar region on the nucleus that occasionally bifurcates because of the peculiar, bi-lobate, shape of the nucleus and its complex spin. In Section 2 we describe relevant properties of the nucleus needed for our simulations, and, in particular, discuss the release of particulates from its surface and their acceleration in the comet’s atmosphere. In Section 3 we present a simplified simulation of the axis of the dust jets and compare the results with observations. In Section 4 we present a discussion of the results.
2. Relevant properties of the nucleus and the release and acceleration of dust from the cometary surface To fully simulate the release of dust from the illuminated surface and the formation of a jet it is necessary to have: (i) a quantitative assessment of the shape and topography of the nucleus and the location of any active areas; (ii) a quantitative description of its spin, the magnitude and orientation of the AMV, and their rates of change; (iii) a practical theory of how dust particulates – ice crystals, silicates, organics, aggregates, etc., – are released from the surface; (iv) a theory of how the cometary atmosphere and, the solar gravity and radiation fields accelerates them; (v) a quantitative assessment of the properties of the atmosphere and its time
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dependence; and (vi) knowledge of the size distribution of the particles and how they might sublimate and fragment while in the jet. Few of these requirements can be met for most comets. However, in the case of 103P, items (i) and (ii) are now available is considerable detail. For item (iv) recent studies by Tenishev et al. (2011) and Combi et al. (2012) give a secure picture of how dust particles are accelerated by the atmosphere once released from the surface. When free of the atmosphere, it is already well known how to parameterize the motion of the grains in the competing radiation and gravitational fields of the Sun (e.g. Fulle, 2004). Items (iii), (v) and (vi) are largely terra incognito. With regard to (v), a considerable amount of information is latent in the spacecraft and Earth-based observations acquired at the EPOXI fly-by, but this has not, to our knowledge, as yet received any systematic quantitative treatment. However, enough is known to allow the somewhat simplified simulation treatment made in this study. Some information on items (iii) and (vi) is becoming available with the observations of Clark et al. (2004) at 81P/Wild 2 and Economou et al. (2012) at 9P/Tempel 1, which show that many, if not all, particles leave the surface as aggregates that subsequently break-up in the outer atmosphere. But the crucial elements of how sunlight triggers the break-off of such aggregates from the surface and what the initial particle size distribution might be, are still largely unexplored. Neither do we have a quantitative basis on how to properly parameterize fragmentation and sublimation of jet aggregates as they move in the jet. Most of the work done previously on this topic (e.g., Patashnick and Rupprecht, 1975; Hanner, 1981; Lien, 1990) involves single particles. In our simulations we will make simplifications that, of necessity, avoid these issues. 2.1. Shape, size, orientation and spin of the nucleus The shape and size (Table 1) of 103P is well defined by the EPOXI observations (Thomas et al., 2012) with the local topography specified in a shape model down to 2 2° elements. On the average, each element corresponds to an area of 400 m2 but varies over the nucleus. This is to be compared with a total surface area of 5.24 106 m2 (Thomas et al., 2012). In our simulations we calculate the normals to the surface relative to such small elements using the edges of the mesh elements. The nucleus is bi-lobate in shape, far from spherical, with one lobe having 1.5 times the surface area of the other. The small lobe has almost spherical curvature while the large lobe crudely approximates a cylindrical shape. In our simulations we shall treat each lobe separately as we expect that the curvature of the surface will play a role in defining the direction and cross-section of the jets. In order to provide a concrete visualization of the nucleus, we show, in Fig. 2, the shape model oriented to simulate the nucleus as it would have appeared to an observer on the Earth at the time of EPOXI encounter. To give an idea of the complexity of the spin, the Supplementary Online Material (SOM) contains Movie 1 that shows three rotations of the nucleus and includes a normal to an arbitrary point on the small end of the nucleus that illustrates the complexity of the motion. The orientation of the AMV is also shown. The spin state, including the orientation of the AMV, is well defined by the EPOXI observations (Belton et al., 2012). The characteristics of the spin state that we adopt for our simulations are also included in Table 1. Note that with a 18.40 h precession of the long axis around the AMV and a 26.72 h roll period around the long axis, a complete spin cycle, i.e., the time to return to essentially the same geometric configuration as seen by a remote observer, is 55 h. We recognize that there is presently some controversy over the direction of the AMV in space owing to divergent estimates published by Earth-based observers (Knight and Schleicher, 2011; Samarasinha et al., 2011; Waniak et al., 2012) who base their estimates on interpretations of the changing morphology of gas jets, particularly in
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CN emission. For the present, we justify our choice of spin state by the fact that it not only satisfies the totality of the EPOXI data but also provides, as pointed out by Belton et al. (2012), a satisfactory explanation of the radar observations of Harmon et al. (2011). The spin model used in our simulations also includes the effects of systematic changes in the precession and roll rates with time, although it is always assumed that the AMV direction is constant. To represent these changes a linear approximation is used, but we anticipate that it should provide reliable simulations for observations taken in October and November, 2010, but possibly not in December, 2010. As noted above, the SOM contains Movie 1 that shows the nature of the spin. 2.2. The release and acceleration of dust from the cometary surface It is not clear what physical processes are involved in releasing particulates from the surface of the nucleus. Sunlight obviously plays a dominant role on the illuminated part, but, to a much lesser extent, particulates are also observed being released on the darkside of the nucleus of 103P (Feaga et al., 2012). Several authors (e.g. Gombosi et al., 1985; Belton, 2010) have followed the assumption of Horanyi et al. (1984) that the flux of dust particles leaving the surface is proportional to the flux of subliming gases with the coefficient of proportionality being a function of particle size and comet. However, observations from the Stardust and Stardust-NExT missions indicate that a fraction, possibly all, of the dust is released as aggregates that disintegrate in the atmosphere (Clark et al., 2004; Economou et al., 2012). In their analysis Clark et al. (2004) point out that other physical processes could be involved such as thermal stress, internal gas pressure, and turbulence to name a few. They also quote experiments by Hartmann (1993) and Gruen et al. (1993) that show the liberation of large aggregates and particles in episodic events. It is not known how these processes respond to changing incidence angle of solar illumination, and so the commonly made assumption that dust production falls off smoothly with the cosine of the solar incident angle on a curved surface must be questioned. Regardless of how particulates, or aggregates, are released from the surface, our understanding of their acceleration in the cometary atmosphere appears to be far more secure and is described in two recent studies by Tenishev et al. (2011) and Combi et al. (2012), both made in the context of the forthcoming Rosetta mission to 67P/Churyumov–Gerasimenko. We expect that the results contained in these studies are, at least qualitatively, applicable to 103P even though 103P has strong sources of H2O vapor in its atmosphere and 67P may not. For example, Belton (2012) finds that the loss of H2O vapor from the surface of 103P lies between 6 1025 and 5 1026 molecules/s and since 67P has 13 times the surface area this would correspond to 8 1026 to 7 1027 molecules/s on 67P. In the Tenishev et al. (2011) and Combi et al. (2012) calculations for 1.29 AU, a total loss rate of 5 1027 molecules/s is assumed, which falls in this range. This gives some reason to believe that these calculations could have some applicability to 103P, at least qualitatively, in the context of accelerating dust particles to the speed at which they are injected into the dust jets. Tenishev et al. (2011) and Combi et al. (2012) present detailed calculations of the dynamics of dust particles released. Evidently, over a restricted area, the escaping solid material forms a ballistic (i.e., with essentially no collisions) system of particles moving on near parallel paths normal to the surface with a spread of terminal speeds that reflects the size distribution of the particles. In the Combi et al. (2012) study, which specifically concerns a small active area, the particle trajectories become detached from the gas motion a few km (1–10 nucleus radii) above the surface and form a highly collimated jet-like structure, while the gas rapidly expands both outwards and laterally to quickly fill the entire
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M.J.S. Belton / Icarus 222 (2013) 653–661 Table 1 Relevant properties of the shape model and spin state of the nucleus. The tabulated values are from Thomas et al. (2012) and Belton et al. (2012) and are collected here for convenience. The principal moments of inertia Il, Ii, and Is are related to the long, intermediate, and short axes respectively. The values for the moments of inertias are computed assuming that the interior mass distribution is homogeneous and are given per unit mass of the nucleus. Quantity
Value
Time of EPOXI closest approach (UTC)
2010-11-04, 13:59:47.7 JD 2455505.0831866 580 ± 18
Mean radius (m) Ratios of model principal moments of inertia Il (m2) Il/Is Ii/Is Body moment orientations (Latitude, E longitude) Il Ii Is Coordinates of the center of mass (latitude, longitude, and radius vector [m]) Direction of long axis (small end) at closest approach (RA, Dec; J2000) Direction of minimum moment at closest approach (RA, Dec; J2000) Direction of intermediate moment at closest approach (RA, Dec; J2000) Direction of the total rotational angular momentum vector Surface area (m2) Volume (m3) Length of the long axis (m) Estimated mean density (kg/m3) Estimated mass of nucleus (kg) Circulation period of long axis about M, the total rotational angular momentum vector (h) Roll period around the long axis (h) Sense of spin Mean tilt of the long axis to M Mean instantaneous period (h) dM/dt (m2/s2 per unit mass) dE/dt (m2/s3 per unit mass; E = rotational energy)
hemisphere without forming a gas-jet. In the Tenishev et al. (2011) study, the particles similarly detach from the gas flow a few nucleus radii above the surface, and, for the most part, move radially outward from the assumed spherical nucleus. This behavior forms the basis for our modeling of the dust jets on 103P. They are streams of particulates moving, at least initially, in parallel having been injected into the dust jets in a direction normal to the local surface. At any particular time there is a range of injection speeds, Vd(a), in the jet that reflect the initial particulate size (a) distribution and its response to the accelerative forces applied by the atmosphere close to the surface. An examination of the numerical results of the Tenishev et al. (2011) and Combi et al. (2012) calculations (their Figs. 10 and 7 respectively), shows that an interpolation formula
V d ðaÞ ¼ Constant1 aa
5.302 ± 0.4 104 0.166 ± 0.004 0.979 ± 0.002 89.73°, 207.56°E 0.05°, 106.14°E 0.25°, 16.14°E +19.57°, 240.39°E, 5.0 RA: 226.12 ± 0.9° Dec: 39.37 ± 0.7° RA: 225.92 ± 0.9° Dec: 39.60 ± 0.7° RA:301.8° Dec: 16.5° RA: 8° ± 4° Dec: 54° ± 1° 5.24 106 8.09 108 2330 230–300 1.9–2.4 1011 18.40 ± 0.13 26.72 ± 0.06 Prograde 81.2° ± 0.6° 14.1 ± 0.3 4.4 107 4.0 1011
3. Simulation of the dust jets The observational basis that we use to provide a foundation for our simulation of the jet complex is that the jets always seems to be present, are generally projected towards the Sun, and that they can vary considerably in brightness. These facts imply that a variable area around the sub-solar region may be involved in the production of the jet and that, since the nucleus spins as a LAM (long axis mode), the active area cannot be tied to the surface but that it must migrate following the Sun, i.e., we are considering a Type 1a active area as defined by Belton (2010). The secondary jet, apparently intermittent, could be associated with an active area fixed to the surface. However, as we shall find, the two jets are really part of a single, but complex outflow, from the bi-lobate nucleus.
ð1Þ 3.1. Simulation technique
provides a good description of the numerical results where a is the particle radius. The exponent a can be expected to change from comet to comet and with heliocentric distance. In the case of the Tenishev et al. (2011) calculations we find a = 0.44 and we will apply this value to 103P. Once in the jet, the particles are dynamically under control of solar gravity and radiation pressure characterized by the parameter b, where, according to Fulle (2004), b(a) is proportional to a1 for particles ranging in diameter from 1 lm to 1 cm, i.e. those most commonly found in comets, and so
V d ðaÞ ¼ Constant2 bðaÞa
ð2Þ
These numerical relationships provide a convenient way to parameterize the dynamics of particles in the jet. We choose specific values for Vd, b and a to characterize the jet and then for a range of speeds in the jet take Vd, (2/3)Vd and (1/3)Vd, etc. Finally, we use Eq. (2) to consistently calculate the relevant values of b.
There is not enough information about conditions and processes on 103P to perform a full-up simulation of the brightness distribution in the dust jets, i.e., to predict their 3-D structure, light scattering properties, particle size distribution, mode of release from the surface, etc. In this work we take an incremental approach where, as a first step, we attempt to simulate only the geometry of the dust jet and its projection on the sky. We do this in terms of only three parameters (Vd, b, a) as described above. With this philosophy we put aside the complications of fragmentation and sublimation in the jets that could lead to an expansion or contraction of the spatial domain of the jets. This approach, as we shall see, is enough to define the sources of the dust jets and explain much of the jet’s behavior. We calculate the jet axis using the following formalism: In vector notation (vectors are in bold font), the position rd (t) of a typical
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where ’ represents differentiation with respect to time and s = (t – t0). We define the jet axis at the time of observation, t, as the locus of particles at rd(t) that were injected into the dust jets with a range of velocities Vd at a sequence of times t0. The simulated jet can therefore be thought of as the summation of a sequence of ‘‘subjets’’ each defined by a separate value of injection speed, Vd, and, through Eq. (2), a different value of b. We ignore details of the release from the surface and acceleration of the dust particle in the atmosphere close to the nucleus. The calculations of Tenishev et al. (2011) and Combi et al. (2012) show that dust particles are accelerated in a region very close to the nucleus (we estimate 1– 10 nucleus radii or 0.6–6 km in the case of 103P) and, for the sub-solar region, the acceleration is essentially normal to the surface. Since the distance over which the acceleration takes place is miniscule as compared to the dimension of the dust jets, it is sufficient to parameterize the calculation with injection velocities parallel to the local surface normal N, but emanating from the center of mass of the nucleus. The coefficients in Eq. (3) are easily calculated from those appropriate for the nucleus itself presumed to be moving on a two-body orbit, e.g.,
prefer its use to the alternative of direct integration of the dust trajectories because it exposes the physics of what is going on in each of its separate terms. In performing the calculations we check at every step the convergence of the series by ensuring that the ratio of successive terms is 1. In addition, we take the calculation to the third order in s. This formalism is, of course, the basis of the old Bessel–Bredikhin theory of comet tail forms (e.g., Bobrovnikoff, 1951), although that theory uses a second Taylor expansion to bring the coefficients back to evaluation at time t of observation. It also generally ignores the motion of material perpendicular to the orbital plane. We do not employ either of these latter steps that can severely limit the application of Bessel–Bredikhin theory. The calculation proceeds as follows: We choose a sequence of values for t0 (20 values) starting a half a rotational cycle (9 h) back from the time of observation and for each time calculate the rotational state of the nucleus. This time interval was chosen after some experimentation and does not have a physical justification other than it produces acceptable simulations. The locations on each lobe, i = 1, 2, with the smallest solar zenith angle are determined as are their vector normals N1 and N2 to the surface. We take these ‘‘sub-solar’’ points as the injection points of dust particles that will form the axis of the jet. Vd,i = Vd Ni defines the injection velocity from lobe i where Vd is the injection speed parameter. In the simulations shown below we have found it sufficient to employ three sub-jets to satisfactorily emulate the effect of a distribution of injection speeds. These speeds are taken as Vd, (2/3)Vd, and (1/3)Vd. Note that since Vd 10–500 m/s (Tenishev et al., 2011; Combi et al., 2012), we can ignore the velocity imparted at the surface by rotation which is <0.1 m/s. While this approach is adequate for defining the basic geometry of the jets it would not be adequate for computing the brightness distribution where a full description of the injection velocity distribution would be needed. As a measure of the extent of the most active regions on the nucleus with respect to sublimation, we arbitrarily map a region on the surface within which the solar zenith angle is less the 24.5° and calculate its area in units of mesh elements. The coordinates of the dust particles at the time of observation, t, that were injected at t0 are then calculated in Equatorial coordinates relative to the nucleus for parameters Vd, b, and a using Eqs. (2)–(4) and where the positions and velocities of the nucleus are calculated using the 2-body orbital elements for 103P obtained from the JPL Horizons System. The final step is to transform the coordinates to the observer frame, again in Equatorial coordinates. In the SOM we include Movie 2 that follows the spin of the nucleus over a complete spin cycle (55 h) and shows how the sub-solar points move over the surfaces of the two lobes, how the sub-solar region, as defined above, waxes and wanes and how the directions of the normals to the surface at the sub-solar points are modulated. With respect to the latter, it is interesting that the projected normals from the sub-solar points retain approximate parallelism over most of the cycle except for brief periods when the normal to the small lobe wanders-off temporarily to considerably larger PAs. It is this behavior that is responsible for the apparently intermittent appearance of the secondary jet as we discuss below.
rd ðt0 Þ ¼ rn ðt 0 Þ
3.2. Application to the 103P/Hartley dust jets
Fig. 2. The appearance of the nucleus of 103P/Hartley 2 as seen from the Earth at the time of the EPOXI encounter. North (up) and East (left) are denoted by vertical and horizontal lines. The line at position angle 104° is the projected direction of the Sun. The terminator and dark limb are clearly seen. The dark grey areas on the illuminated part of the nucleus are the sub-solar regions where the solar zenith angle is less than 24.5°.
dust particle at time t that was injected from the vicinity of the nucleus at position rd (t0) at time t0 with injection velocity Vd and moves under the influence of the solar gravity and radiation fields characterized by b can be written as a Taylor expansion: 3 rd ðtÞ ¼ rd ðt 0 Þ þ r0d ðt0 Þ s þ ð1=2Þr00d ðt 0 Þ s2 þ ð1=6Þr000 d ðt 0 Þ s
þ
r0d ðt0 Þ
¼
r0n ðt 0 Þ
ð3Þ
þ Vd N
r00d ðt0 Þ ¼ k r00n ðt0 Þ ¼ kGMs rn ðt 0 Þ=r n ðt 0 Þ3
ð4Þ
3 5 0 0 r000 d ðt 0 Þ ¼ kGM s frn ðt 0 Þ=r n ðt 0 Þ 3rn ðt 0 Þ½rn ðt 0 Þrn ðt 0 Þ=r n ðt 0 Þ g
þ higher terms; Here the subscript n refers to the nucleus and k = (1 b). Ms is the mass of the Sun and G the gravitational constant. Even though convergence of this expansion is not assured for arbitrary times, s, we
3.2.1. The injection direction of the particles as a function of time In Fig. 3 we plot the PA’s of injection velocities from the sub-solar regions on the two lobes of the nucleus over an interval a little longer than a full spin cycle as seen from the Earth. Also included in the plot is the PA of the projected direction of the Sun. The PA of particles that were injected from the small lobe show strong periodic excursions from the sun-line for short intervals of time. The periodicity of these excursions is about that of the precessional
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Fig. 3. Plot of the PA’s of particles injected into the dust jets from the sub-solar regions on the small (full line) and large (dashed line) lobes of the nucleus over an interval slightly longer than a complete spin cycle (55 h) as seen from the Earth. The particles from both lobes show strong, but brief, excursions from the projected direction of the projected sun-line (nearly horizontal full line). The excursions from the two lobes are out-of-phase by about 9 h with those characterizing the large lobe having 1/3 the amplitude of those of the small lobe.
period of the long axis of the nucleus around the AMV. The PAs of particles injected from the large lobe also show excursions from the sun-line that are similarly short lived, but, in this case, the excursions have an amplitude only 1/3 that of the particles escaping from the small lobe. In addition the excursions from the large lobe are 9 h out-of-phase from those from the small lobe. For a majority of the time, the PAs of the injection velocity from both lobes hovers close to the sun-line, although those of the small lobe average 2–4° to the North. These results are compared to observations in Fig. 4. In the upper panel we plot the PAs noted by Knight and Schleicher (2011) and Mueller et al. (2012) in the week before the EPOXI encounter. In the case of the latter authors, a range of PAs was given and this is also shown. Presented in this way, there appears to be no obvious agreement between the observations and the model PAs. However, if the observed points are shifted by 7 h, as in the lower panel, a correlation appears between the model PAs of injection velocity from the small lobe and the PAs observed for the secondary jet. Away from the main peaks, the observed PAs and those of the injection velocity from the large lobe mainly deviate in the sense that observed PAs tend to be greater than those of the model injection velocity. Nevertheless, and with the exception of a single observation, the deviations are not large even though they are systematic. We briefly discuss the origin of this shift in Section 4. 3.2.2. The sub-solar dust jet complex In Figs. 5–7 we present simulations of the axis of the dust jet as observed on November 2–4, 2010, by Mueller et al. (2012). In each case the simulations are defined, as noted above, by a = 0.44 and three injection speeds (Vd, (2/3)Vd, and (1/3)Vd) with Vd = 210 m/s. Vd with b = 1 is taken to correspond to the smallest particles expected in the jets (cf. Table 1 in Fulle (2004)), while the bs for the slower (and larger) particulates are calculated from Eq. (2). In order to improve correspondence of the model to the observations we have found necessary to impose a lifetime to the particles in the jet of 6.8 h, In addition, for the case of November 4, the simulation can be improved by imposing a cut-off, or significant reduction, in the flow of particles about 4.5 h before the observation was made. This implies that the entire structure seen on that night was formed over a period of 3 h. In the case of the November 2 observations, we found a better correspondence between observation and simulation if Vd was reduced to 150 m/s.
Fig. 4. (Upper panel) Same as Fig. 3 but with observed PAs of dust jets superposed. Observations marked N are from Knight and Schleicher (2011) those marked are by Mueller et al. (2012). The latter authors give the spread of PAs for each observation; this is denoted by a vertical error bar. (Lower panel) Same as above but with the observed points shifted by 7 h, which brings them into better alignment with the model curves.
In evaluating these simulations the reader should recall that the spatial distribution of points does not represent the brightness distribution in the jet but is simply a locus of points of particles released at a sequence of different times. Nevertheless, the simulations show reasonable accord with the data in that the particles generally emerge from the nucleus, with scatter, on, or near, the sunline, but that, at certain times, material is ejected towards more southeasterly directions, intermittently forming what appears to be a secondary jet. Since the simulations are calculated in three dimensions it is easy to see what complexity there is in the structure of the jets as seen from other vantage points. In Fig. 8 we look at the simulation of November 4, 2010 from the direction of an observer on the Sun. The structure is seen to broadly spread out from the sun-line as a result of the changes in the local topography on the nucleus at the times when the individual particles were injected. The jets are also observed to vary rapidly with brightness. The simulations provide an explanation for this in the rapid changes of cross-section that the nucleus presents to the Sun during a spin cycle. This is seen more clearly in Fig. 9 where, in the upper panel, we plot the number of mesh element in the shape model that fall within the sub-solar region as defined above, as a function of time. We consider this as a surrogate for following the rise and fall of gas and dust production, The time interval covered is the last spin cycle (55 h) before the EPOXI encounter. As might be expected for the odd shape of 103P, this measure of production rate varies strongly during the spin cycle with peaks of variable strength that occur every 9 h and rapid changes are seen to take place over intervals as short as 3 h.
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Fig. 5. (Left panel) Image of dust jets for UT November 4, 2010, 11:28, from Mueller et al. (2012). The image is 8400 km on a side, North is up and East is to the left. (Right panel) Simulation of the dust jet axis. The vertical line points North and the horizontal line Eastwards. The scale of the simulations is the same as the image. The inclined line towards the PA = 104° is the projected sun-line. Markers , +, x, trace out the locus of particles with injection speeds Vd, (2/3)Vd and (1/3)Vd. In this simulation Vd = 210 m/s, b = 1, and a = 0.44 and the particles were injected every 27 min over a period of 9 h immediately preceding the time of observation. (Center panel) Same as the right panel except that a lifetime of 6.8 h has been imposed on the particles in the jet. In this case, we have also curtailed the injection of particles from the nucleus for 4.5 h before the time of observation. The bright material at the nucleus falls on the projected sun-line and was apparently emitted shortly before the time of observation.
Fig. 6. Same as Fig. 5 but for UT November 3, 08:28. In this case there is no curtailment of the ejection of particles before the time of observation. The panels are 8400 km on a side.
Fig. 7. Same as Fig. 5 but for UT November 2, 12:32 and two values of Vd. In this case there is no curtailment of the injection of particles before the time of observation. The simulation in the lower panel, where Vd = 150 m/s, provides a better correspondence with the observations. The panels are 8400 km on a side.
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lobes inject material toward the same PA, i.e., they are essentially parallel reinforcing each other, in a direction 5° North of the solar PA. However, three times in the spin cycle the PA of the injection velocity from the small lobe makes a large excursion from the solar direction to PA 140°, over an interval of 5 h, at times when the production rates are low and not changing rapidly. We expect, from this simulation that the observed rapid changes in brightness should occur in observations were the formation time of the jet spans an interval of rapidly changing active area. 4. Discussion
Fig. 8. The dust jet simulation for UT November 4, 2010, 11:28, as seen from the direction of an observer at the Sun. The simulations are 8400 km on the side. (Right panel) Same simulation as in the right panel of Fig. 5. (Left panel) The simulation to the right has been rotated to present a view along the sun-line. The threedimensional structure of the dust jet, which is due to the changing topography at the sub-solar point, becomes very clear in this comparison.
Fig. 9. (Upper panel) Plot of the number of mesh elements of the shape model that fall within the sub-solar region, i.e., the region where the solar zenith angle is <24.5°, with time. The plot covers a full spin cycle of 55 h ending at the time of the EPOXI encounter (November 4 13:59:47.310, 2010). We use this as a crude surrogate for the production of dust and gas as the nucleus spins. The peaks are, as expected, separated by 9 h. i.e., half the precession period of the long axis around the AMV. The rapid changes of the area of the sub-solar region with time are thought to be the cause of the observed rapid changes occasionally seen in the brightness of the jets. (Bottom panel) A repeat of Fig. 3, which covers the same time frame as in the upper panel and is presented for easy correlation with the upper panel. Note that for much of the time that the PAs of the injection velocity from the two lobes more or less coincide are times of relatively high gas and dust production. The times when the PA of the injection velocity from both lobes deviates strongly from the solar PA tend to occur at times of minimum gas and dust production.
The simulations in Figs. 5–7 and the rapid changes in the area active around the sub-solar point in Fig. 9 show that the model for the sources of the sunward dust jets proposed here provide an adequate basis for understanding the geometrical and dynamical properties of this phenomenon. The complex spin of 103P’s bi-lobate nucleus coupled with ejection of particulates from the sub-solar regions on the two lobes provides a good accounting of the PAs of the dust jets and how their geometry changes from night to night. In addition, the theoretical modeling of dust acceleration in cometary atmospheres by Tenishev et al. (2011) and Combi et al. (2012) appears to give a reasonably reliable picture of how particulates fill out the axis of the dust jets. In this picture the active areas are not fixed to the surface but follow the Sun and should be classified as Type 1a in the scheme devised by Belton (2010). Whatever the detailed processes by which particulates are released from the nucleus surface, insolation and, presumably, sublimation of H2O are involved. As shown in Fig. 9, which we see as a surrogate for instantaneous particulate production rate, the brightness of the dust jets should vary dramatically and rapidly throughout the spin cycle. The observations show such changes, but it is beyond the scope of this paper (or the tools we have available) to model the observed timing and extent of these brightness changes and this is clearly an area for future research. In our discussion of the relationship of observed PAs to model PAs (Fig. 4) we noted that a shift of 7 h was needed to bring them into reasonable accord. As a result of the simulations, we understand this shift as the characteristic time it takes to renew the jet structure into the observed configuration. It represents the balance between the lifetime of the particles and the time to refill the structure. The picture of the dust jets that we have developed here is that they are a collisionless system of particulates in which the individual particles have a short lifetime of 7 h. The particles move on ballistic trajectories under the influence of both solar gravity and radiation pressure with a great deal of parallelism near the axis of the jet. The structure of the jets, which apparently never reach a steady state, is such that the regions further from the nucleus are dominated by smaller, faster moving, particles and regions near the nucleus more heavily weighted towards larger, slower moving particles. While we have not modeled the color of scattered light from the jet, this seems a good basis for understanding the observed reddening of the jets closer to the nucleus. The focus of this paper has been to understand the sources of the dust jets in the context of the peculiar bi-lobate shape of 103P and its complex spin state. Future modeling of the brightness distribution and color of these jets holds the potential of learning new details about the structure of the inner coma of 103P and new insight into how particles are released from the surface as a result of solar irradiation. Acknowledgments
This behavior can be compared with the PAs of injection velocities from the both lobes of the nucleus as shown in the lower panel. During times when the production rates are peaking the two
This work was supported by NASA Grant NNX07AG24G under the PMDAP program and NNM07AA99C both to the University of
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Maryland and sub-contracted to Belton Space Exploration Initiatives, LLC, through a Memorandum of Agreement with the National Optical Astronomy Observatories. The principal author thanks the Principal Investigators of the above Grants, Jian-Yang Li and Michael F. A’Hearn, for their enthusiastic support. We thank Drs. B.E.A Mueller, L.M. Lara, M.M. Knight and G.-P. Tozzi for providing digital images and permission to use figures from their papers. In addition, we thank Drs. Mueller and N.H. Samarasinha for insightful discussions during the development of various aspects of these simulations. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.icarus.2012. 12.007. References A’Hearn, M.F. et al., 2011. EPOXI at Comet Hartley 2. Science 332, 1369–1400. Belton, M.J.S., 2010. Cometary activity, active areas, and a mechanism for collimated outflows. Icarus 210, 881–897. Belton, M.J.S., 2012. Cometary evolution and cryovolcanism. Can. J. Phys. 90, 807– 815. Belton, M.J.S. et al., 2012. The complex spin state of 103P/Hartley 2: Kinematics and orientation in space. Icarus 222, 595–609. Bobrovnikoff, N., 1951. Comets. In: Hynek, J.A. (Ed.), Astrophysics. McGraw-Hill, New York. Clark, B.C. et al., 2004. Release and fragmentation of aggregates to produce heterogeneous, lumpy coma streams. J. Geophys. Res. 109, E12S03. Combi, M.R., Tenishev, V.M., Rubin, M., Fougere, N., Gombosi, T.I., 2012. Narrow dust jets in a diffuse gas coma: A natural product of small active regions on comets. Astrophys. J. 749 (article 29). Economou, T.E., Green, S.F., Brownlee, D.E., Clark, B.C., 2012. Dust flux monitor instrument measurements during stardust-NExT flyby of Comet 9P/Tempel 1. Icarus 222, 526–539. Feaga, L.M. et al., 2012. Volatile distribution and heterogeneities in 103P/Hartley 2 as observed by the deep impact HRI-IR. Presented at Astroids, Comets, Meteors Conference, Niigata, Japan.
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