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The contribution of grains to the activity of comets
Icarus 195 (2008) 340–347 www.elsevier.com/locate/icarus
The contribution of grains to the activity of comets II. The brightness of the coma E. Beer a , D. Prialnik b , M. Podolak b,∗ a NASA Ames Research Center, Moffett Field, CA 94035-1000, USA b Department of Geophysics and Planetary Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel 69978
Received 28 September 2006; revised 21 November 2007 Available online 31 December 2007
Abstract We use the model of grain behavior in the coma developed by Beer et al. [Beer, E.H., Podolak, M., Prialnik, P., 2006. Icarus 180, 473–486] to compute the contribution of ice grains to the brightness of the coma. The motion of an ice grain along the comet–Sun axis is computed, taking into account gas drag, the gravity of the nucleus, and radiation pressure of sunlight. The sublimation of the grains is also included. We assume that the maximum distance that a grain travels along this axis is indicative of the size of the coma, and we compute the resultant brightness as a function of heliocentric distance. The results are then compared to observations. © 2007 Elsevier Inc. All rights reserved. Keywords: Comets, coma; Comets, dust; Comets, composition
1. Introduction The brightness of a comet comes, to a large extent, from the reflection of sunlight by dust grains in the coma. These grains are ejected from the nucleus and move under the influence of gas drag, gravity, and radiation pressure. The problem of gas emission from a comet nucleus and its influence on grain motion in the coma has a long history. The motion of grains under the combined effects of solar gravity and solar radiation was analyzed nearly 100 years ago by Eddington (1910), while the effect of gas drag on the grains was examined in great detail by Finson and Probstein (1968). If the grains are composed of ice, there is the additional complication that these grains will sublimate in the solar radiation field. Evidence for the existence of such grains was recently presented by Kawakita et al. (2004), but the issue has been discussed in the literature for some time. An early discussion of the relevant physics was given by Schwehm and Kneissel (1981). They took the mass change due to sublimation into account in computing the mo* Corresponding author. Fax: +972 3 6409282.
E-mail address: [email protected] (M. Podolak). 0019-1035/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2007.12.004
tion of the grains, but computed the temperature of the grains only from the balance of absorbed and emitted radiation, and neglected the effect of sublimation in cooling the grain. Mukai et al. (1985) considered dirty ice grains, but computed their motion using an empirical formula based on the work of Wallis (1982). In the inner coma, near the surface of the nucleus, the gas density is sufficiently high so that vapor condensation onto the grains must be considered in addition to sublimation. A very detailed discussion of the complications that arise in this region has been presented in a series of papers by Crifo and his collaborators who have considered, among other things, non-uniform sublimation from the surface of the comet nucleus, non-spherical nuclei, and details of the gasdynamic flow (Crifo, 1995a, 1995b; Crifo and Rodionov, 1997; Rodionov et al., 2002). While these details are necessary for understanding the behavior of a particular comet, this can only be done if the exact shape, albedo, topography, and spin of the comet are known. For most comets this is not the case. In this paper we use an alternative approach: we employ a more crude approximation that has the advantage of containing the pertinent physical processes, while requiring much less
Coma bightness of comets
computation time than the more complex models. We use this model to compute comet behavior for an ensemble of comets. Although this approach will not give a terribly accurate estimate of the grain abundance in the coma, the uncertainties in the available parameters for most comets generally do not justify investing more effort. Furthermore, by comparing our results to the observations for a group of comets, we hope to effectively average over some of the unknown parameters. Our goal, therefore is to understand an observed trend, in this case the variation of brightness with heliocentric distance, in the context of the behavior of ice grains in the coma. Thus, for example, we neglect the effect of the coma gas on the energy balance of the grains. As noted above, near the nucleus the gas density can be quite high at small heliocentric distances. For example, at 1 AU the vapor pressure due to water ice sublimation should give a pressure of roughly 0.1 Pa and a gas temperature of nearly 200 K. Under such conditions, contact with the gas is the major factor in determining the grain temperature, and calculations including the effect of the gas show that for small (0.1 µm) grains of pure ice, vapor condensation onto the grain is possible. The rate of grain growth is very low, however, of the order of 10−3 µm s−1 , and is limited to a region about one cometary radius from the surface. By two cometary radii from the surface, the condensation rate has dropped by an order of magnitude, and sublimation of the grain begins shortly thereafter. All this is for pure ice grains (very high albedo) at 1 AU from the Sun. If dirt is mixed with the grain, or the grain is significantly larger than 0.1 µm, or the heliocentric distance is increased, the rate of vapor condensation onto the grain is even smaller, and often non-existent. We thus feel justified, for this calculation, to neglect the effect of the coma gas on the grain temperature. Further discussion of the behavior of ice grains can be found in Beer et al. (2006). Our approach will be to use the grain evolution model (GEM) that we have previously developed (Beer et al., 2006) to calculate the brightness of a cometary coma as a function of heliocentric distance. In particular, we will compare the slope of the theoretical light curve to those observed for different comets in order to infer something about the comet’s composition. As in Beer et al. (2006), we consider three types of grains: pure ice grains which absorb well in the IR, but are nearly transparent in the visible (where the solar radiation peaks), “dirty” ice grains, which we take to be a homogeneous mixture of water ice and some generic absorbing material (“dirt”), and refractory grains, which are taken to be composed completely of the absorbing material. The absorber is assumed to have a real refractive index of 1.45, and an imaginary index of 0.1. These values are representative of material produced by UV photolysis of simple carbon and nitrogen compounds (Khare et al., 1984) and should behave similarly to the dark material found in comets. This dirt will absorb in the visible, and will allow us to gauge the importance of the visible irradiation. The mass fraction of ice in the grain is denoted by X1 . In Section 2 we review the physical assumptions of our model, in Section 3 we present our theoretical light curves and compare them to observations, and in Section 4 we give our conclusions.
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2. The model 2.1. Grain velocity The brightness of the dust coma depends directly on the number and size of the reflectors, but much less so on their distance from the nucleus. As a result, the theoretical light curves will not be very sensitive to the actual grain velocity. The simplest assumption (Delsemme, 1982) is to take the grain velocity to be equal to the gas velocity. This approximation is obviously an overestimate, and also gives grain velocities independent of grain size. A more elaborate approximation is based on the equation of motion for an individual grain in a radially expanding cometary atmosphere (Wallis, 1982; Gombosi et al., 1986). This approximation assumes that the gas drag on the grain is characterized by Cd 2 πa ρgas (Vgas − Vgrain )2 , (1) 2 where Fd is the drag force, ρgas is the gas density, a is the grain radius, Cd is the drag coefficient, and Vgas and Vgrain are the gas and grain velocities, respectively. In this approximation both Cd and Vgas are assumed to be constant and ρgas is assumed to decrease as R −2 , where R is the distance from the nucleus. In such a case the equation of motion of a dust grain can be directly integrated to give Vgrain Vgrain Rn (2) + ln 1 − , = Xa 1 − Vgas − Vgrain Vgas R Fd =
where Rn is the radius of the nucleus, and Xa is a constant that depends on the parameters of the problem: Xa =
3Cd ZRn . 16aρgrain Vgas
(3)
Here ρgrain is the bulk density of a grain and Z is the rate of sublimation from the surface of the nucleus (mass per unit surface area per unit time). In this case the velocity of a grain increases with increasing distance from the nucleus. Clearly, there are additional effects that must be considered. In the first place, both the gas velocity and the gas temperature vary with distance from the nucleus. In addition, there is an exchange of energy and momentum between the gas and the grains. Much of the relevant physics has been discussed in an excellent review by Probstein (1968). Additional complications are dealt with by Crifo (1995a). As we pointed out above, the details of the grain velocity are of secondary importance in determining the light curve, so we have chosen an intermediate approximation: rather than do a detailed gas flow computation to determine the gas flow speed and temperature, we have assumed that gas speed is constant, and have computed the gas drag force, and the radiation pressure force. In addition, we have included the gravitational force exerted by the nucleus, since this will affect the larger grains. This allows us to compute the trajectories (in the frame of reference of the comet) of grains of different sizes.
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2.1.1. Gas drag The drag force on a grain depends on its size relative to the mean free path of the gas molecules. If mgas is the mass of a gas molecule, and T is its temperature, then the thermal velocity of the molecule is given by 8kT Vth = (4) , mgas π where k is Boltzmann’s constant. We assume Vgas = f Vth , where f is a constant. For simplicity we have chosen f = 1. We have also conducted a series of runs with f = 0.5 and in most cases the differences in the light curve were indistinguishable. The gas flux leaving the comet surface is given by mgas J = Pvap (5) , 2πkT where Pvap is the vapor pressure. If we set this flux equal to the product of gas density and velocity near the surface, we get mgas . ρ0 Vgas = Pvap (6) 2πkT Here ρ0 is the gas density near the surface. The mean free path, λ, is given by mgas λ= √ , 2ρ0 NA σgas
(7)
where NA is Avogadro’s number and σgas = 10−15 cm2 , is the collision cross section of a gas molecule. Even at a distance of 0.7 AU, the temperature is just above 199 K and the vapor pressure is approximately 1.7 dyn cm−2 (Fanale and Salvail, 1984). Under these conditions λ = 45.8 cm, while the maximum radius of a grain that can be lifted off from the surface of a 1 km radius nucleus is only 15.5 cm. At greater heliocentric distances the temperature is lower, the vapor pressure is lower, the mean free path is larger, and the maximum radius of a grain that can be lifted off is smaller. As a result, one may always use the relationship for Epstein drag (Fuchs, 1964). In this case the drag force is given by 4π 2 (8) a ρgas Vth (Vgas − Vgrain ). 3 Note that because of the linear dependence in the relative velocity of the gas and the grain for Epstein drag, the force can be quite different from that given in Eq. (1). For a constant gas flow speed 2 Rn ρgas (R) = ρ0 (9) . R Fdrag =
Substituting Vgas = Vth , and applying Newton’s law Fdrag =
mgas 2πkT
mgas 2 R 2πkT n
(11)
so that if only gas drag were included, the acceleration of the grain would be Vgas − Vgrain dVgrain . =β dt R2
(12)
2.1.2. Gravitational force The gravitational force on the grain due to the nucleus is GMn mgrain (13) , R2 where G is the universal gravitational constant, Mn is the nucleus mass and mgrain is the mass of a grain. The acceleration of the grain due to this force alone is
Fgrav = −
dVgrain GMn =− 2 . dt R For convenience we define ς ≡ GMn .
(14)
(15)
2.1.3. Radiation force The radiation force depends on the grain’s cross section for interaction with radiation. Since the grain radius is often comparable to the wavelength of the radiation, the true cross section will be some efficiency factor, Q, times the geometric cross section, where Q can be quite different from 1. For spherical grains, Mie theory can be used to compute efficiency factors for both scattering and absorption. Mie calculations for perfect spheres tend to produce various resonance peaks that do not appear if the grains are not completely spherical (see, e.g., Hansen and Travis, 1974). To avoid such spurious peaks we did our calculations for a narrow distribution of radii around the radius of interest. This averages out the effect of the resonances. The back reaction on the grain of scattered light will depend on the direction of scattering. For simplicity we use the asymmetry parameter, g, to approximate the relative amount of forward scattering; g too can be computed from Mie theory. The force on a grain due to radiation is then L πa 2 [Qabs + (1 − g)Qscat ] (16) , 4πd 2 c where L is the solar luminosity, d is the distance from the Sun, c is the speed of light, and Qabs and Qscat are the efficiency factors for absorption and scattering, respectively. Again, it will be useful to put all the constants together and define Frad = −
δ≡
3L [Qabs + (1 − g)Qscat ] . 16πd 2 acρgrain
(17)
dR = Vgrain dt we get
gives
Pvap β≡ aρgrain
Adding the three forces together, and noting that
dVgrain 4π 3 a ρgrain 3 dt
Pvap dVgrain = dt aρgrain
Let
Rn R
2 (Vgas − Vgrain ).
(10)
β(Vgrain − Vgas) + ς dVgrain dVgrain − δ. = Vgrain =− dt dR R2
(18)
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Fig. 1. Grain velocity as a function of distance from the nucleus for a 1 km nucleus at a distance of 1.09 AU from the Sun for three different models of the grain velocity. Further details are given in the text.
Fig. 2. Grain velocity as a function of time for a 1 km nucleus at a distance of 1.09 AU from the Sun for three different types of grains. Further details are given in the text.
This equation can be integrated, allowing for the sublimation of the ice grains, using a fourth-order Runge–Kutta scheme (Kunz, 1957). The grain density was taken to be 1.0 g cm−3 for pure and dirty ice grains, and 2.5 g cm−3 for refractory grains. In practice, to minimize computational time, the efficiency factors and the asymmetry parameter for the radiation force were computed for a wavelength of 0.6 µm, where the solar radiation has its peak. Note that the grain is allowed to sublimate as it moves along its trajectory, so that the parameters β and δ are allowed to vary with time. In addition, since the radiation force acts in a direction parallel to the Sun–comet axis, a proper calculation would treat the three forces as vectors and would give a parabolic coma (Eddington, 1910). For simplicity we consider motion of grains only in the sunward direction and treat the coma as an isotropic sphere. This should not have an important effect on the computed brightness of the coma, since that depends on the amount of material in the coma, and is not sensitive to its precise spatial distribution. The coma radius we calculate is meant to give an idea of the expected coma size, and of the amount of time that ice grains will remain in the coma.
methods can be quite different. In this, and in the next two figures, the trajectories for each grain were computed until the grain sublimated to a size smaller than 0.01 µm, which is the smallest grain size we considered. The figure illustrates the effect of including sublimation of the grain in computing its velocity. The 40 µm grain, for example, initially has a small acceleration because of its large mass. After moving a few kilometers from the surface, the gas density is low enough so that the grain can no longer be efficiently accelerated and its velocity becomes nearly constant. However, because the grain is sublimating, it eventually becomes small enough so that the gas can once again accelerate it efficiently, and the velocity increases once more. This seems to be contradicted by the behavior of the 0.6 µm grains which get further from the nucleus before being completely sublimated. This is only apparent, since the smaller grains are moving much faster than the larger ones, and therefore cover the same distance in a much shorter time. This can be seen in Fig. 2, which shows the velocity of ice grains as a function of time for grain radii of 0.6 and 40 µm. The curves are for pure ice grains (X1 = 1), dirty ice grains (X1 = 0.9) and refractory grains (X1 = 0). Note that the refractory grains are assumed to have a density of 2.5 g cm−3 . For this reason their speeds are noticeably lower than those of the pure and dirty ice grains, whose density is assumed to be 1 g cm−3 . The curves are all for a comet at 1.09 AU from the Sun, and again the trajectories are computed along a line joining the comet to the Sun. Typically, the largest grain in the distribution will have the longest lifetime against sublimation, and will be least affected by the radiation force, so it will reach the largest distance before evaporating or being turned back. Our computations show that this is true for refractory grains as well. In Fig. 2 we see that the 40 µm dirty ice grains do indeed have a lifetime considerably longer than the 0.6 µm grains. Fig. 3 shows the coma size. For our purposes we define this as the farthest a grain travels along a line from the comet to
2.2. Computed velocities Fig. 1 shows the difference in the grain velocity as computed by the three methods (Delsemme, 1982; Gombosi et al., 1986; and the present method) described above for the case of a 1 km nucleus at a distance of 1.09 AU from the Sun as a function of distance from the nucleus. In all cases the grains are assumed to travel on a line from the comet to the Sun. The velocities are normalized to the gas velocity, so that Delsemme’s velocity is the horizontal line at one. The other curves show the ratio of grain to gas velocities for dirty ice grains with radii of 40 and 0.6 µm as computed from the Gombosi et al. (1986) model and in the present paper. In both cases grain sublimation was included. As can be seen, the speeds computed by the three
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tering properties of the grains depend strongly on the ratio of the grain size to the wavelength of the light, we would expect that the brightness variation across the halo would be different in different wavelengths. Observations of brightness variation across the halo for different wavelengths, or the variation of the apparent coma size as a function of wavelength could then be used to help determine the size distribution and composition of the emitted grains. 2.3. Magnitude computation
Fig. 3. Coma size as a function of heliocentric distance. The coma size is taken to be the maximum distance from the nucleus that a grain can reach along the comet–Sun axis. The plots are for 0.6 and 40 µm grains composed of pure ice, dirty ice, and refractory material.
the Sun before either being turned back by radiation pressure or sublimating away completely. We show the coma size as a function of heliocentric distance for particles of 0.6 and 40 µm grains composed of pure ice (X1 = 1), dirty ice (X1 = 0.9), and refractory grains (X1 = 0). With the exception of the large refractory grains, the curves all exhibit the same general form: a small coma very near the Sun, a larger coma at intermediate distances, and a small coma far from the Sun. For the small refractory grains radiation pressure keeps the coma small near the Sun, while for the ice grains it is a combination of radiation pressure and sublimation. The large refractory grains are not affected by either process and form a large coma even very close to the Sun. As the comet recedes from the Sun, both the radiation pressure and the sublimation rate decrease, and the coma can be larger. As the comet gets still further from the Sun, the gas flux from the comet decreases enough so that the larger grains can no longer be accelerated efficiently. In addition, the sublimation rate of the grains decreases substantially. The resulting coma size depends on the combined effect of these processes. For 40 µm pure ice grains (circles in Fig. 3), sublimation is the major factor limiting the size of the coma. At larger heliocentric distances this sublimation drops quickly, and the coma can be substantially larger. Its size is then limited by radiation pressure. At still larger heliocentric distances the gas flow from the nucleus that accelerates the grains decreases more quickly than the radiation pressure and the coma radius decreases. 40 µm dirty ice grains (pluses in Fig. 3) show a similar behavior, except that they sublimate more quickly, and the coma they produce is always smaller than that of pure ice grains. Refractory grains (×’s in the figure) do not sublimate at all, but they are more dense, so that their acceleration by the gas flow from the nucleus is smaller. Near the Sun they produce a larger coma than the pure ice grains, but further from the Sun their coma is smaller. The coma size for the smaller grains in the figure can be understood using the same ideas. Since the scat-
As in Beer et al. (2006), we assume that the dust ejected from the nucleus follows a power law size distribution N (a) = Ca −3.5 . The largest grain in the distribution is always chosen to be the largest grain that can be lifted off the nuclear surface at that heliocentric distance. The mass of dust ejected at each heliocentric distance is 20% by mass of the gas sublimated from the nucleus. The constant, C is chosen at each heliocentric distance to give the appropriate total mass of ejected grains. The grains are allowed to sublimate as they travel and their scattering cross sections, albedos, and asymmetry parameters are computed from Mie theory as described in Beer et al. (2006). The magnitude of the coma is then computed by calculating the total amount of light returned by the dust. Let n(a) da be the number of grains with radii between a and a + da that leave the surface of the comet per unit area per unit time. The total scattering cross section of grains emitted per unit time is then amax
Qs (a)πa 2 n(a)(1 − g) da,
dA = amin
where Qs (a) is the efficiency factor for scattering, and g is the asymmetry parameter that estimates the fraction of the light that is scattered in the forward direction. In reality, the brightness of the coma should also be a function of the phase angle of the observation, but this formula should be sufficient for the purposes of our calculation. These factors are computed from Mie theory assuming the grains are spheres and that their real and imaginary indices are known. Note that this scattering efficiency already includes the effects of the grains’ single scattering albedo. At any given time, assuming that the rate of emission of grains is constant for the given interval, the coma will consist of grains that were emitted from some initial time τ0 till the present. The exact value of τ0 is difficult to determine. If the grains are particularly long lived, it may be given by the time it takes for the grains to travel far enough away from the nucleus so that their space density is low enough to cause their reflected light to be indistinguishable from the background. It may also be limited by the field of view of the telescope (Jewitt, 1991). If the grains have short lifetimes, then these lifetimes will determine τ0 . Coma observations indicate that the dust coma is rather small. Harmon et al. (1989) found that, for Comet IRAS– Araki–Alcock, large grains were confined to within ∼103 km of the nucleus. Hanner et al. (1985) report that the apex distance (the extent of the dust coma in the sunward direction) was
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345
Fig. 4. Coma magnitude as a function of heliocentric distance for pure ice grains, dirty ice grains and refractory grains. Also shown are the brightnesses of several comets as a function of heliocentric distance.
Fig. 5. Coma magnitude as a function of heliocentric distance for comets with nuclei of 1, 5, and 10 km, respectively. The dots are observations of the brightness of several comets.
only 6 × 103 km for Comet Crommelin. Hale–Bopp showed a coma extending past 4.5 × 103 km, but the grains also had a relatively high projected outflow velocity of 0.41 km s−1 . In all these cases it would seem that τ0 = 1 day is sufficient. We ran our code with values of τ0 between 1 and 4 days, and found that although the computed magnitude of the coma decreased somewhat with increased integration time, the slope of the light curve as a function of heliocentric distance was independent of the choice of τ0 . As we explain below, this is the diagnostic we wish to use to learn something of the composition of comets. Because the grains sublimate, their radii will vary with time so that if we assume that there is no mutual shadowing, the total scattering cross section of the grains plus the nucleus is
for pure ice grains. Since the refractory grains do not evaporate, the resulting coma is brighter, especially at small heliocentric distances where evaporation is a key factor in limiting the mass of ice grains in the coma. In Figs. 4–7 the grain trajectories are computed for a 48 h period. At small heliocentric distances the pure ice grain coma has a brightness that is very similar to one with dirty ice grains. This is because at small heliocentric distances both the pure and dirty ice grains sublimate significantly. Further from the Sun, however, the lifetimes of the pure ice grains increase and the coma reflectivity they produce is closer to that of the refractory grains. As the heliocentric distance increases, the gas and grain production rates decrease, and the coma brightness would be expected to decrease as a result. However there is an additional effect due to the fact that when the comet is far from the Sun, only grains that are small with respect to the wavelength of light can be lifted off the comet into the coma. For such grains, the interaction with sunlight is strongly dependent on their imaginary index of refraction. For dirty grains, the cross section for absorption of sunlight is much larger than the cross section for pure ice grains, and hence, the radiation pressure is larger for dirty ice grains. This results in a pure ice grain coma having a larger radius and a correspondingly higher brightness. The figure also shows the measured light curves of several comets. In this and the following figures, the light curves are taken from the work of Shanklin (1997, 1998a, 1998b, 1999). Comparison with observations shows that the light curve of many comets does indeed have a slope similar to the theoretical curves we have computed. Comet P/Kushida behaves like the flat part of the pure ice grain light curve, and would seem to have a coma composed largely of such grains. Comet Zanotta–Brewington also has a light curve with a shallow slope, although it is considerably brighter than our theoretical model. The simplest explanation for this is that the nucleus is larger than the 1 km we have assumed in our models. The dependence
τ0 dA(t) + πRn2 A,
A= 0
where A is the albedo of the nucleus. For a comet at a distance r from the Sun and Δ from the Earth the magnitude of the coma will be given by 2 a A m − m = −2.5 log (19) + 5 log Δ, r2 where m is the magnitude of the Sun. We follow the usual convention and take Δ = 1 AU, so that the coma magnitude reduces to 2 a A . m = m − 2.5 log (20) r2 3. Results Fig. 4 shows the magnitude variation as a function of heliocentric distance for the three types of grains we have studied. In this calculation we have assumed that the nucleus has a radius of 1 km. The lower curve is for dirty ice grains with X1 = 0.9, the upper curve is for refractory grains and the middle curve is
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Fig. 6. Same as Fig. 4 with some additional comets shown for comparison. Fig. 7. Brightness curves for comets with γ = 10 (see text).
of brightness on the radius of the comet nucleus is shown in Fig. 5, and we will discuss it below. Fig. 5 shows the theoretical light curves for dirty ice grains emitted from comets with radii of 1, 5, and 10 km. Also shown are the observed light curves for 5 comets. All of the observed curves have amplitudes and slopes that fall well within the range of the theoretical curves and would seem to have radii of between 1 and 5 km. These values are quite typical of comet radii. In fact, most comets have active areas of only a few percent of the total surface area. For a given nucleus radius, the dynamics of a particular grain will be unaffected, but the number of grains will decrease in proportion to the active area. However, in computing the mass of the nucleus we have assumed a bulk density of 1 g cm−3 while the actual value is probably at least a factor of two lower (Solem, 1994; Rickman, 2002). Furthermore, we have assumed that the grain production to gas production ratio is 0.2, and this is also uncertain by a factor of two. As a result, the brightness computed by the GEM should only be viewed as indicative of the expected value. Not all comets, however, fit this model. Fig. 6 shows three comets whose light curves are quite different from those predicted by the GEM. Thus, Comet P/Machholtz 2 not only has a flatter slope, but is also considerably brighter than the GEM predictions. One possibility is that this comet underwent an outburst which caused an increase in its brightness. A more likely possibility is that this comet has a large abundance of some more volatile ices such as CO2 . Since the sublimation rate of CO2 even from deeper within the nucleus can be much higher than that of surface H2 O at large heliocentric distances (Prialnik, 2002), a comet rich in CO2 would appear much brighter than the GEM prediction. The other two comets shown in the figure present a different problem. Their brightness is consistent with the GEM predictions in magnitude, but have a much steeper slope. We suggest that this might be due to a very different grain size distribution within the comet. Fig. 7 shows the GEM predictions for
a modified grain size distribution. The original GEM calculations were done for a grain size distribution where the number of grains with radii between a and a + da was given by N (a) da = Ca −γ da.
(21)
Observations of grains around some comets have led to the conclusion that γ is in the neighborhood of 3 or 4. Fig. 7 shows the prediction of the GEM for γ = 10. The lowermost curve is for a distribution of dirty ice grains where the grain size goes from the minimum value of 0.01 µm to a maximum value of 10 µm. The middle curve is for the same composition, but with a minimum size of 10 µm and a maximum size which is the largest size that can be lifted off the comet at that distance. The uppermost curve is for the same size distribution as the middle curve, but for pure ice grains. As can be seen, these curves have steeper slopes, although they are still not steep enough to match the observations. It would seem, however, that these comets have a coma composed of large grains with size distribution that is more nearly monodisperse than most comets. 4. Conclusions Although a comprehensive model for the variation of comet brightness with heliocentric distance has not yet been constructed, the GEM can be considered a first step in that direction. For several comets it succeeds in reproducing the slope of the brightness change with heliocentric distance. Clearly more physics must be added to the model in order that it can represent reality more faithfully. We have computed the grain motion only along a comet–Sun line. Grains leaving the comet at some angle to this line will follow parabolic trajectories, and the shape of the resultant coma will differ substantially from that of a sphere. Such trajectories will have to be followed so that we can better evaluate the area of the coma. In addition, the grains are ejected from the nucleus as a size distribution, and the different sizes not only evolve differently in time, but they also follow different trajectories. Any given
Coma bightness of comets
point in the coma will include grains that have traveled along different trajectories to reach that point. Furthermore, a given line of sight through the coma will intersect a number of such trajectories and a Haser-type model (Haser, 1957) will be required to compute the grain surface area encountered along the line of sight. Perhaps the most important addition for a successful comparison to observations is the coupling of the GEM to a more realistic model for comet activity. Here we have assumed that the sublimation rate as a function of distance depends only on balance between solar heating of the comet nucleus with cooling by ice sublimation and thermal radiation. Heat conduction into and out of the nucleus was not considered. This effect of thermal inertia, and the more important effects of outbursts (Prialnik and Bar-Nun, 1987) and release of more volatile species from subsurface layers (Prialnik, 2002), can cause significant variations in the brightness of a comet as a function of heliocentric distance. Finally, for the case of pure ice grains, the extremely long lifetime of these grains means that the coma is unrealistically large. In fact, when the space density of the grains becomes low enough the contrast with the background will become low enough so that the coma will no longer be detectable. This will effectively limit the size of the coma and must be included in future work. References Beer, E.H., Podolak, M., Prialnik, P., 2006. The contribution of icy grains to the activity of comets. I. Grain lifetime and distribution. Icarus 180, 473– 486. Crifo, J.F., 1995a. Condensing or sublimating objects in rotation and translation across a rarefied polyatomic vapor I. Astron. Astrophys. 300, 597– 604. Crifo, J.F., 1995b. A general physicochemical model of the inner coma of active comets. I. Implications of spatially distributed gas and dust production. Astrophys. J. 445, 470–488. Crifo, J.F., Rodionov, A.V., 1997. The dependence of the circumnuclear coma structure on the properties of the nucleus. II. First investigation of the coma surrounding a homogeneous aspherical nucleus. Icarus 129, 72–93. Delsemme, A.H., 1982. Chemical composition of nuclei. In: Wilkening, L.L. (Ed.), Comets. University of Arizona Press, Tucson, pp. 85–163. Eddington, A.S., 1910. The envelopes of Comet Morehouse (1908). Mon. Not. R. Astron. Soc. 70, 442–458. Fanale, F.P., Salvail, J.R., 1984. An idealized short-period comet model: Surface insolation, flux, dust flux, and mantle evolution. Icarus 60, 476–511.
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Finson, M.L., Probstein, R.F., 1968. A theory of dust comets. I. Model and equations. Astrophys. J. 154, 327–352. Fuchs, N.A., 1964. Mechanics of Aerosols. Pergammon, Oxford, UK. Gombosi, T.I., Nagy, A.F., Cravens, T.E., 1986. Dust and neutral gas modeling of the inner atmospheres of comets. Rev. Geophys. 24, 667–700. Hanner, M.S., Knacke, R., Sekanina, Z., Tokunaga, A.T., 1985. Dark grains in Comet Crommelin. Astron. Astrophys. 152, 177–181. Hansen, J.E., Travis, L., 1974. Light scattering in planetary atmospheres. Space Sci. Rev. 16, 527–610. Harmon, J.K., Campbell, D.B., Fine, A.A., Shapiro, I.I., Marsden, B.G., 1989. Radar observations of Comet IRAS–Araki–Alcock 1983d. Astrophys. J. 338, 1071–1093. Haser, L., 1957. Distribution d’intensite dans la tete d’une comete. Bull. Acad. R. Sci. Liege 43, 740–750. Jewitt, D.C., 1991. Cometary photometry. In: Newburn, R., Neugebauer, M., Rahe, J. (Eds.), Comets in the Post-Halley Era. Kluwer Academic Publishers, Amsterdam, pp. 19–65. Kawakita, H., Watanabe, J.-I., Ootsobu, T., Nakamura, R., Fuse, T., Takato, N., Sasaki, S., Sasaki, T., 2004. Evidence of icy grains in Comet C/2002 T7 (LINEAR) at 3.52 AU. Astrophys. J. 601, L191–L194. Khare, B.N., Sagan, C., Arakawa, E.T., Suits, F., Callcott, T.A., Williams, M.W., 1984. Optical constants of organic tholins produced in a simulated Titanian atmosphere: From soft X-ray to microwave frequencies. Icarus 60, 127–137. Kunz, S.K., 1957. Numerical Analysis. McGraw–Hill Book Company, Inc., New York. Mukai, T., Mukai, S., Fechtig, H., Grün, E., Giese, R.H., 1985. Dirty ice grains in comets. Adv. Space Res. 5, 339–342. Prialnik, D., 2002. Modeling the comet nucleus interior: Application to Comet C/1995 O1 Hale–Bopp. Earth Moon Planets 89, 27–52. Prialnik, D., Bar-Nun, A., 1987. On the evolution and activity of cometary nuclei. Astrophys. J. 313, 893–905. Probstein, R.F., 1968. The dusty gasdynamics of comet heads. In: Lavrent’ev, M.A. (Ed.), Problems of Hydrodynamics and Continuum Mechanics. SIAM, Philadelphia, pp. 569–583. Rickman, H., 2002. The nucleus of Comet Halley: Surface structure, mean density, gas and dust production. Adv. Space Res. 9, 59–71. Rodionov, A.V., Crifo, J.-F., Szego, K., Lagerros, J., Fulle, M., 2002. An advanced physical model of cometary activity. Planet. Space Sci. 50, 983–1024. Schwehm, G.H., Kneissel, B., 1981. Optical and physical properties and dynamics of dust grains released by Comet Halley. In: The Comet Halley Dust and Gas Environment. ESA SP-174. ESTEC, Noordwijk, pp. 77–84. Shanklin, J.D., 1997. The comets of 1991. J. Br. Astron. Assoc. 107, 186–198. Shanklin, J.D., 1998a. The comets of 1992. J. Br. Astron. Assoc. 108, 90–98. Shanklin, J.D., 1998b. The comets of 1993. J. Br. Astron. Assoc. 108, 305–319. Shanklin, J.D., 1999. The comets of 1994. J. Br. Astron. Assoc. 109, 191–209. Solem, J.C., 1994. Density and size of Comet Shoemaker–Levy 9 deduced from a tidal breakup model. Nature 370, 349–351. Wallis, K.M., 1982. Dusty gas dynamics in real comets. In: Wilkening, L.L. (Ed.), Comets. University of Arizona Press, Tucson, pp. 357–369.
The contribution of grains to the activity of comets II. The brightness of the coma E. Beer a , D. Prialnik b , M. Podolak b,∗ a NASA Ames Research Center, Moffett Field, CA 94035-1000, USA b Department of Geophysics and Planetary Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel 69978
Received 28 September 2006; revised 21 November 2007 Available online 31 December 2007
Abstract We use the model of grain behavior in the coma developed by Beer et al. [Beer, E.H., Podolak, M., Prialnik, P., 2006. Icarus 180, 473–486] to compute the contribution of ice grains to the brightness of the coma. The motion of an ice grain along the comet–Sun axis is computed, taking into account gas drag, the gravity of the nucleus, and radiation pressure of sunlight. The sublimation of the grains is also included. We assume that the maximum distance that a grain travels along this axis is indicative of the size of the coma, and we compute the resultant brightness as a function of heliocentric distance. The results are then compared to observations. © 2007 Elsevier Inc. All rights reserved. Keywords: Comets, coma; Comets, dust; Comets, composition
1. Introduction The brightness of a comet comes, to a large extent, from the reflection of sunlight by dust grains in the coma. These grains are ejected from the nucleus and move under the influence of gas drag, gravity, and radiation pressure. The problem of gas emission from a comet nucleus and its influence on grain motion in the coma has a long history. The motion of grains under the combined effects of solar gravity and solar radiation was analyzed nearly 100 years ago by Eddington (1910), while the effect of gas drag on the grains was examined in great detail by Finson and Probstein (1968). If the grains are composed of ice, there is the additional complication that these grains will sublimate in the solar radiation field. Evidence for the existence of such grains was recently presented by Kawakita et al. (2004), but the issue has been discussed in the literature for some time. An early discussion of the relevant physics was given by Schwehm and Kneissel (1981). They took the mass change due to sublimation into account in computing the mo* Corresponding author. Fax: +972 3 6409282.
E-mail address: [email protected] (M. Podolak). 0019-1035/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2007.12.004
tion of the grains, but computed the temperature of the grains only from the balance of absorbed and emitted radiation, and neglected the effect of sublimation in cooling the grain. Mukai et al. (1985) considered dirty ice grains, but computed their motion using an empirical formula based on the work of Wallis (1982). In the inner coma, near the surface of the nucleus, the gas density is sufficiently high so that vapor condensation onto the grains must be considered in addition to sublimation. A very detailed discussion of the complications that arise in this region has been presented in a series of papers by Crifo and his collaborators who have considered, among other things, non-uniform sublimation from the surface of the comet nucleus, non-spherical nuclei, and details of the gasdynamic flow (Crifo, 1995a, 1995b; Crifo and Rodionov, 1997; Rodionov et al., 2002). While these details are necessary for understanding the behavior of a particular comet, this can only be done if the exact shape, albedo, topography, and spin of the comet are known. For most comets this is not the case. In this paper we use an alternative approach: we employ a more crude approximation that has the advantage of containing the pertinent physical processes, while requiring much less
Coma bightness of comets
computation time than the more complex models. We use this model to compute comet behavior for an ensemble of comets. Although this approach will not give a terribly accurate estimate of the grain abundance in the coma, the uncertainties in the available parameters for most comets generally do not justify investing more effort. Furthermore, by comparing our results to the observations for a group of comets, we hope to effectively average over some of the unknown parameters. Our goal, therefore is to understand an observed trend, in this case the variation of brightness with heliocentric distance, in the context of the behavior of ice grains in the coma. Thus, for example, we neglect the effect of the coma gas on the energy balance of the grains. As noted above, near the nucleus the gas density can be quite high at small heliocentric distances. For example, at 1 AU the vapor pressure due to water ice sublimation should give a pressure of roughly 0.1 Pa and a gas temperature of nearly 200 K. Under such conditions, contact with the gas is the major factor in determining the grain temperature, and calculations including the effect of the gas show that for small (0.1 µm) grains of pure ice, vapor condensation onto the grain is possible. The rate of grain growth is very low, however, of the order of 10−3 µm s−1 , and is limited to a region about one cometary radius from the surface. By two cometary radii from the surface, the condensation rate has dropped by an order of magnitude, and sublimation of the grain begins shortly thereafter. All this is for pure ice grains (very high albedo) at 1 AU from the Sun. If dirt is mixed with the grain, or the grain is significantly larger than 0.1 µm, or the heliocentric distance is increased, the rate of vapor condensation onto the grain is even smaller, and often non-existent. We thus feel justified, for this calculation, to neglect the effect of the coma gas on the grain temperature. Further discussion of the behavior of ice grains can be found in Beer et al. (2006). Our approach will be to use the grain evolution model (GEM) that we have previously developed (Beer et al., 2006) to calculate the brightness of a cometary coma as a function of heliocentric distance. In particular, we will compare the slope of the theoretical light curve to those observed for different comets in order to infer something about the comet’s composition. As in Beer et al. (2006), we consider three types of grains: pure ice grains which absorb well in the IR, but are nearly transparent in the visible (where the solar radiation peaks), “dirty” ice grains, which we take to be a homogeneous mixture of water ice and some generic absorbing material (“dirt”), and refractory grains, which are taken to be composed completely of the absorbing material. The absorber is assumed to have a real refractive index of 1.45, and an imaginary index of 0.1. These values are representative of material produced by UV photolysis of simple carbon and nitrogen compounds (Khare et al., 1984) and should behave similarly to the dark material found in comets. This dirt will absorb in the visible, and will allow us to gauge the importance of the visible irradiation. The mass fraction of ice in the grain is denoted by X1 . In Section 2 we review the physical assumptions of our model, in Section 3 we present our theoretical light curves and compare them to observations, and in Section 4 we give our conclusions.
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2. The model 2.1. Grain velocity The brightness of the dust coma depends directly on the number and size of the reflectors, but much less so on their distance from the nucleus. As a result, the theoretical light curves will not be very sensitive to the actual grain velocity. The simplest assumption (Delsemme, 1982) is to take the grain velocity to be equal to the gas velocity. This approximation is obviously an overestimate, and also gives grain velocities independent of grain size. A more elaborate approximation is based on the equation of motion for an individual grain in a radially expanding cometary atmosphere (Wallis, 1982; Gombosi et al., 1986). This approximation assumes that the gas drag on the grain is characterized by Cd 2 πa ρgas (Vgas − Vgrain )2 , (1) 2 where Fd is the drag force, ρgas is the gas density, a is the grain radius, Cd is the drag coefficient, and Vgas and Vgrain are the gas and grain velocities, respectively. In this approximation both Cd and Vgas are assumed to be constant and ρgas is assumed to decrease as R −2 , where R is the distance from the nucleus. In such a case the equation of motion of a dust grain can be directly integrated to give Vgrain Vgrain Rn (2) + ln 1 − , = Xa 1 − Vgas − Vgrain Vgas R Fd =
where Rn is the radius of the nucleus, and Xa is a constant that depends on the parameters of the problem: Xa =
3Cd ZRn . 16aρgrain Vgas
(3)
Here ρgrain is the bulk density of a grain and Z is the rate of sublimation from the surface of the nucleus (mass per unit surface area per unit time). In this case the velocity of a grain increases with increasing distance from the nucleus. Clearly, there are additional effects that must be considered. In the first place, both the gas velocity and the gas temperature vary with distance from the nucleus. In addition, there is an exchange of energy and momentum between the gas and the grains. Much of the relevant physics has been discussed in an excellent review by Probstein (1968). Additional complications are dealt with by Crifo (1995a). As we pointed out above, the details of the grain velocity are of secondary importance in determining the light curve, so we have chosen an intermediate approximation: rather than do a detailed gas flow computation to determine the gas flow speed and temperature, we have assumed that gas speed is constant, and have computed the gas drag force, and the radiation pressure force. In addition, we have included the gravitational force exerted by the nucleus, since this will affect the larger grains. This allows us to compute the trajectories (in the frame of reference of the comet) of grains of different sizes.
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2.1.1. Gas drag The drag force on a grain depends on its size relative to the mean free path of the gas molecules. If mgas is the mass of a gas molecule, and T is its temperature, then the thermal velocity of the molecule is given by 8kT Vth = (4) , mgas π where k is Boltzmann’s constant. We assume Vgas = f Vth , where f is a constant. For simplicity we have chosen f = 1. We have also conducted a series of runs with f = 0.5 and in most cases the differences in the light curve were indistinguishable. The gas flux leaving the comet surface is given by mgas J = Pvap (5) , 2πkT where Pvap is the vapor pressure. If we set this flux equal to the product of gas density and velocity near the surface, we get mgas . ρ0 Vgas = Pvap (6) 2πkT Here ρ0 is the gas density near the surface. The mean free path, λ, is given by mgas λ= √ , 2ρ0 NA σgas
(7)
where NA is Avogadro’s number and σgas = 10−15 cm2 , is the collision cross section of a gas molecule. Even at a distance of 0.7 AU, the temperature is just above 199 K and the vapor pressure is approximately 1.7 dyn cm−2 (Fanale and Salvail, 1984). Under these conditions λ = 45.8 cm, while the maximum radius of a grain that can be lifted off from the surface of a 1 km radius nucleus is only 15.5 cm. At greater heliocentric distances the temperature is lower, the vapor pressure is lower, the mean free path is larger, and the maximum radius of a grain that can be lifted off is smaller. As a result, one may always use the relationship for Epstein drag (Fuchs, 1964). In this case the drag force is given by 4π 2 (8) a ρgas Vth (Vgas − Vgrain ). 3 Note that because of the linear dependence in the relative velocity of the gas and the grain for Epstein drag, the force can be quite different from that given in Eq. (1). For a constant gas flow speed 2 Rn ρgas (R) = ρ0 (9) . R Fdrag =
Substituting Vgas = Vth , and applying Newton’s law Fdrag =
mgas 2πkT
mgas 2 R 2πkT n
(11)
so that if only gas drag were included, the acceleration of the grain would be Vgas − Vgrain dVgrain . =β dt R2
(12)
2.1.2. Gravitational force The gravitational force on the grain due to the nucleus is GMn mgrain (13) , R2 where G is the universal gravitational constant, Mn is the nucleus mass and mgrain is the mass of a grain. The acceleration of the grain due to this force alone is
Fgrav = −
dVgrain GMn =− 2 . dt R For convenience we define ς ≡ GMn .
(14)
(15)
2.1.3. Radiation force The radiation force depends on the grain’s cross section for interaction with radiation. Since the grain radius is often comparable to the wavelength of the radiation, the true cross section will be some efficiency factor, Q, times the geometric cross section, where Q can be quite different from 1. For spherical grains, Mie theory can be used to compute efficiency factors for both scattering and absorption. Mie calculations for perfect spheres tend to produce various resonance peaks that do not appear if the grains are not completely spherical (see, e.g., Hansen and Travis, 1974). To avoid such spurious peaks we did our calculations for a narrow distribution of radii around the radius of interest. This averages out the effect of the resonances. The back reaction on the grain of scattered light will depend on the direction of scattering. For simplicity we use the asymmetry parameter, g, to approximate the relative amount of forward scattering; g too can be computed from Mie theory. The force on a grain due to radiation is then L πa 2 [Qabs + (1 − g)Qscat ] (16) , 4πd 2 c where L is the solar luminosity, d is the distance from the Sun, c is the speed of light, and Qabs and Qscat are the efficiency factors for absorption and scattering, respectively. Again, it will be useful to put all the constants together and define Frad = −
δ≡
3L [Qabs + (1 − g)Qscat ] . 16πd 2 acρgrain
(17)
dR = Vgrain dt we get
gives
Pvap β≡ aρgrain
Adding the three forces together, and noting that
dVgrain 4π 3 a ρgrain 3 dt
Pvap dVgrain = dt aρgrain
Let
Rn R
2 (Vgas − Vgrain ).
(10)
β(Vgrain − Vgas) + ς dVgrain dVgrain − δ. = Vgrain =− dt dR R2
(18)
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343
Fig. 1. Grain velocity as a function of distance from the nucleus for a 1 km nucleus at a distance of 1.09 AU from the Sun for three different models of the grain velocity. Further details are given in the text.
Fig. 2. Grain velocity as a function of time for a 1 km nucleus at a distance of 1.09 AU from the Sun for three different types of grains. Further details are given in the text.
This equation can be integrated, allowing for the sublimation of the ice grains, using a fourth-order Runge–Kutta scheme (Kunz, 1957). The grain density was taken to be 1.0 g cm−3 for pure and dirty ice grains, and 2.5 g cm−3 for refractory grains. In practice, to minimize computational time, the efficiency factors and the asymmetry parameter for the radiation force were computed for a wavelength of 0.6 µm, where the solar radiation has its peak. Note that the grain is allowed to sublimate as it moves along its trajectory, so that the parameters β and δ are allowed to vary with time. In addition, since the radiation force acts in a direction parallel to the Sun–comet axis, a proper calculation would treat the three forces as vectors and would give a parabolic coma (Eddington, 1910). For simplicity we consider motion of grains only in the sunward direction and treat the coma as an isotropic sphere. This should not have an important effect on the computed brightness of the coma, since that depends on the amount of material in the coma, and is not sensitive to its precise spatial distribution. The coma radius we calculate is meant to give an idea of the expected coma size, and of the amount of time that ice grains will remain in the coma.
methods can be quite different. In this, and in the next two figures, the trajectories for each grain were computed until the grain sublimated to a size smaller than 0.01 µm, which is the smallest grain size we considered. The figure illustrates the effect of including sublimation of the grain in computing its velocity. The 40 µm grain, for example, initially has a small acceleration because of its large mass. After moving a few kilometers from the surface, the gas density is low enough so that the grain can no longer be efficiently accelerated and its velocity becomes nearly constant. However, because the grain is sublimating, it eventually becomes small enough so that the gas can once again accelerate it efficiently, and the velocity increases once more. This seems to be contradicted by the behavior of the 0.6 µm grains which get further from the nucleus before being completely sublimated. This is only apparent, since the smaller grains are moving much faster than the larger ones, and therefore cover the same distance in a much shorter time. This can be seen in Fig. 2, which shows the velocity of ice grains as a function of time for grain radii of 0.6 and 40 µm. The curves are for pure ice grains (X1 = 1), dirty ice grains (X1 = 0.9) and refractory grains (X1 = 0). Note that the refractory grains are assumed to have a density of 2.5 g cm−3 . For this reason their speeds are noticeably lower than those of the pure and dirty ice grains, whose density is assumed to be 1 g cm−3 . The curves are all for a comet at 1.09 AU from the Sun, and again the trajectories are computed along a line joining the comet to the Sun. Typically, the largest grain in the distribution will have the longest lifetime against sublimation, and will be least affected by the radiation force, so it will reach the largest distance before evaporating or being turned back. Our computations show that this is true for refractory grains as well. In Fig. 2 we see that the 40 µm dirty ice grains do indeed have a lifetime considerably longer than the 0.6 µm grains. Fig. 3 shows the coma size. For our purposes we define this as the farthest a grain travels along a line from the comet to
2.2. Computed velocities Fig. 1 shows the difference in the grain velocity as computed by the three methods (Delsemme, 1982; Gombosi et al., 1986; and the present method) described above for the case of a 1 km nucleus at a distance of 1.09 AU from the Sun as a function of distance from the nucleus. In all cases the grains are assumed to travel on a line from the comet to the Sun. The velocities are normalized to the gas velocity, so that Delsemme’s velocity is the horizontal line at one. The other curves show the ratio of grain to gas velocities for dirty ice grains with radii of 40 and 0.6 µm as computed from the Gombosi et al. (1986) model and in the present paper. In both cases grain sublimation was included. As can be seen, the speeds computed by the three
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tering properties of the grains depend strongly on the ratio of the grain size to the wavelength of the light, we would expect that the brightness variation across the halo would be different in different wavelengths. Observations of brightness variation across the halo for different wavelengths, or the variation of the apparent coma size as a function of wavelength could then be used to help determine the size distribution and composition of the emitted grains. 2.3. Magnitude computation
Fig. 3. Coma size as a function of heliocentric distance. The coma size is taken to be the maximum distance from the nucleus that a grain can reach along the comet–Sun axis. The plots are for 0.6 and 40 µm grains composed of pure ice, dirty ice, and refractory material.
the Sun before either being turned back by radiation pressure or sublimating away completely. We show the coma size as a function of heliocentric distance for particles of 0.6 and 40 µm grains composed of pure ice (X1 = 1), dirty ice (X1 = 0.9), and refractory grains (X1 = 0). With the exception of the large refractory grains, the curves all exhibit the same general form: a small coma very near the Sun, a larger coma at intermediate distances, and a small coma far from the Sun. For the small refractory grains radiation pressure keeps the coma small near the Sun, while for the ice grains it is a combination of radiation pressure and sublimation. The large refractory grains are not affected by either process and form a large coma even very close to the Sun. As the comet recedes from the Sun, both the radiation pressure and the sublimation rate decrease, and the coma can be larger. As the comet gets still further from the Sun, the gas flux from the comet decreases enough so that the larger grains can no longer be accelerated efficiently. In addition, the sublimation rate of the grains decreases substantially. The resulting coma size depends on the combined effect of these processes. For 40 µm pure ice grains (circles in Fig. 3), sublimation is the major factor limiting the size of the coma. At larger heliocentric distances this sublimation drops quickly, and the coma can be substantially larger. Its size is then limited by radiation pressure. At still larger heliocentric distances the gas flow from the nucleus that accelerates the grains decreases more quickly than the radiation pressure and the coma radius decreases. 40 µm dirty ice grains (pluses in Fig. 3) show a similar behavior, except that they sublimate more quickly, and the coma they produce is always smaller than that of pure ice grains. Refractory grains (×’s in the figure) do not sublimate at all, but they are more dense, so that their acceleration by the gas flow from the nucleus is smaller. Near the Sun they produce a larger coma than the pure ice grains, but further from the Sun their coma is smaller. The coma size for the smaller grains in the figure can be understood using the same ideas. Since the scat-
As in Beer et al. (2006), we assume that the dust ejected from the nucleus follows a power law size distribution N (a) = Ca −3.5 . The largest grain in the distribution is always chosen to be the largest grain that can be lifted off the nuclear surface at that heliocentric distance. The mass of dust ejected at each heliocentric distance is 20% by mass of the gas sublimated from the nucleus. The constant, C is chosen at each heliocentric distance to give the appropriate total mass of ejected grains. The grains are allowed to sublimate as they travel and their scattering cross sections, albedos, and asymmetry parameters are computed from Mie theory as described in Beer et al. (2006). The magnitude of the coma is then computed by calculating the total amount of light returned by the dust. Let n(a) da be the number of grains with radii between a and a + da that leave the surface of the comet per unit area per unit time. The total scattering cross section of grains emitted per unit time is then amax
Qs (a)πa 2 n(a)(1 − g) da,
dA = amin
where Qs (a) is the efficiency factor for scattering, and g is the asymmetry parameter that estimates the fraction of the light that is scattered in the forward direction. In reality, the brightness of the coma should also be a function of the phase angle of the observation, but this formula should be sufficient for the purposes of our calculation. These factors are computed from Mie theory assuming the grains are spheres and that their real and imaginary indices are known. Note that this scattering efficiency already includes the effects of the grains’ single scattering albedo. At any given time, assuming that the rate of emission of grains is constant for the given interval, the coma will consist of grains that were emitted from some initial time τ0 till the present. The exact value of τ0 is difficult to determine. If the grains are particularly long lived, it may be given by the time it takes for the grains to travel far enough away from the nucleus so that their space density is low enough to cause their reflected light to be indistinguishable from the background. It may also be limited by the field of view of the telescope (Jewitt, 1991). If the grains have short lifetimes, then these lifetimes will determine τ0 . Coma observations indicate that the dust coma is rather small. Harmon et al. (1989) found that, for Comet IRAS– Araki–Alcock, large grains were confined to within ∼103 km of the nucleus. Hanner et al. (1985) report that the apex distance (the extent of the dust coma in the sunward direction) was
Coma bightness of comets
345
Fig. 4. Coma magnitude as a function of heliocentric distance for pure ice grains, dirty ice grains and refractory grains. Also shown are the brightnesses of several comets as a function of heliocentric distance.
Fig. 5. Coma magnitude as a function of heliocentric distance for comets with nuclei of 1, 5, and 10 km, respectively. The dots are observations of the brightness of several comets.
only 6 × 103 km for Comet Crommelin. Hale–Bopp showed a coma extending past 4.5 × 103 km, but the grains also had a relatively high projected outflow velocity of 0.41 km s−1 . In all these cases it would seem that τ0 = 1 day is sufficient. We ran our code with values of τ0 between 1 and 4 days, and found that although the computed magnitude of the coma decreased somewhat with increased integration time, the slope of the light curve as a function of heliocentric distance was independent of the choice of τ0 . As we explain below, this is the diagnostic we wish to use to learn something of the composition of comets. Because the grains sublimate, their radii will vary with time so that if we assume that there is no mutual shadowing, the total scattering cross section of the grains plus the nucleus is
for pure ice grains. Since the refractory grains do not evaporate, the resulting coma is brighter, especially at small heliocentric distances where evaporation is a key factor in limiting the mass of ice grains in the coma. In Figs. 4–7 the grain trajectories are computed for a 48 h period. At small heliocentric distances the pure ice grain coma has a brightness that is very similar to one with dirty ice grains. This is because at small heliocentric distances both the pure and dirty ice grains sublimate significantly. Further from the Sun, however, the lifetimes of the pure ice grains increase and the coma reflectivity they produce is closer to that of the refractory grains. As the heliocentric distance increases, the gas and grain production rates decrease, and the coma brightness would be expected to decrease as a result. However there is an additional effect due to the fact that when the comet is far from the Sun, only grains that are small with respect to the wavelength of light can be lifted off the comet into the coma. For such grains, the interaction with sunlight is strongly dependent on their imaginary index of refraction. For dirty grains, the cross section for absorption of sunlight is much larger than the cross section for pure ice grains, and hence, the radiation pressure is larger for dirty ice grains. This results in a pure ice grain coma having a larger radius and a correspondingly higher brightness. The figure also shows the measured light curves of several comets. In this and the following figures, the light curves are taken from the work of Shanklin (1997, 1998a, 1998b, 1999). Comparison with observations shows that the light curve of many comets does indeed have a slope similar to the theoretical curves we have computed. Comet P/Kushida behaves like the flat part of the pure ice grain light curve, and would seem to have a coma composed largely of such grains. Comet Zanotta–Brewington also has a light curve with a shallow slope, although it is considerably brighter than our theoretical model. The simplest explanation for this is that the nucleus is larger than the 1 km we have assumed in our models. The dependence
τ0 dA(t) + πRn2 A,
A= 0
where A is the albedo of the nucleus. For a comet at a distance r from the Sun and Δ from the Earth the magnitude of the coma will be given by 2 a A m − m = −2.5 log (19) + 5 log Δ, r2 where m is the magnitude of the Sun. We follow the usual convention and take Δ = 1 AU, so that the coma magnitude reduces to 2 a A . m = m − 2.5 log (20) r2 3. Results Fig. 4 shows the magnitude variation as a function of heliocentric distance for the three types of grains we have studied. In this calculation we have assumed that the nucleus has a radius of 1 km. The lower curve is for dirty ice grains with X1 = 0.9, the upper curve is for refractory grains and the middle curve is
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Fig. 6. Same as Fig. 4 with some additional comets shown for comparison. Fig. 7. Brightness curves for comets with γ = 10 (see text).
of brightness on the radius of the comet nucleus is shown in Fig. 5, and we will discuss it below. Fig. 5 shows the theoretical light curves for dirty ice grains emitted from comets with radii of 1, 5, and 10 km. Also shown are the observed light curves for 5 comets. All of the observed curves have amplitudes and slopes that fall well within the range of the theoretical curves and would seem to have radii of between 1 and 5 km. These values are quite typical of comet radii. In fact, most comets have active areas of only a few percent of the total surface area. For a given nucleus radius, the dynamics of a particular grain will be unaffected, but the number of grains will decrease in proportion to the active area. However, in computing the mass of the nucleus we have assumed a bulk density of 1 g cm−3 while the actual value is probably at least a factor of two lower (Solem, 1994; Rickman, 2002). Furthermore, we have assumed that the grain production to gas production ratio is 0.2, and this is also uncertain by a factor of two. As a result, the brightness computed by the GEM should only be viewed as indicative of the expected value. Not all comets, however, fit this model. Fig. 6 shows three comets whose light curves are quite different from those predicted by the GEM. Thus, Comet P/Machholtz 2 not only has a flatter slope, but is also considerably brighter than the GEM predictions. One possibility is that this comet underwent an outburst which caused an increase in its brightness. A more likely possibility is that this comet has a large abundance of some more volatile ices such as CO2 . Since the sublimation rate of CO2 even from deeper within the nucleus can be much higher than that of surface H2 O at large heliocentric distances (Prialnik, 2002), a comet rich in CO2 would appear much brighter than the GEM prediction. The other two comets shown in the figure present a different problem. Their brightness is consistent with the GEM predictions in magnitude, but have a much steeper slope. We suggest that this might be due to a very different grain size distribution within the comet. Fig. 7 shows the GEM predictions for
a modified grain size distribution. The original GEM calculations were done for a grain size distribution where the number of grains with radii between a and a + da was given by N (a) da = Ca −γ da.
(21)
Observations of grains around some comets have led to the conclusion that γ is in the neighborhood of 3 or 4. Fig. 7 shows the prediction of the GEM for γ = 10. The lowermost curve is for a distribution of dirty ice grains where the grain size goes from the minimum value of 0.01 µm to a maximum value of 10 µm. The middle curve is for the same composition, but with a minimum size of 10 µm and a maximum size which is the largest size that can be lifted off the comet at that distance. The uppermost curve is for the same size distribution as the middle curve, but for pure ice grains. As can be seen, these curves have steeper slopes, although they are still not steep enough to match the observations. It would seem, however, that these comets have a coma composed of large grains with size distribution that is more nearly monodisperse than most comets. 4. Conclusions Although a comprehensive model for the variation of comet brightness with heliocentric distance has not yet been constructed, the GEM can be considered a first step in that direction. For several comets it succeeds in reproducing the slope of the brightness change with heliocentric distance. Clearly more physics must be added to the model in order that it can represent reality more faithfully. We have computed the grain motion only along a comet–Sun line. Grains leaving the comet at some angle to this line will follow parabolic trajectories, and the shape of the resultant coma will differ substantially from that of a sphere. Such trajectories will have to be followed so that we can better evaluate the area of the coma. In addition, the grains are ejected from the nucleus as a size distribution, and the different sizes not only evolve differently in time, but they also follow different trajectories. Any given
Coma bightness of comets
point in the coma will include grains that have traveled along different trajectories to reach that point. Furthermore, a given line of sight through the coma will intersect a number of such trajectories and a Haser-type model (Haser, 1957) will be required to compute the grain surface area encountered along the line of sight. Perhaps the most important addition for a successful comparison to observations is the coupling of the GEM to a more realistic model for comet activity. Here we have assumed that the sublimation rate as a function of distance depends only on balance between solar heating of the comet nucleus with cooling by ice sublimation and thermal radiation. Heat conduction into and out of the nucleus was not considered. This effect of thermal inertia, and the more important effects of outbursts (Prialnik and Bar-Nun, 1987) and release of more volatile species from subsurface layers (Prialnik, 2002), can cause significant variations in the brightness of a comet as a function of heliocentric distance. Finally, for the case of pure ice grains, the extremely long lifetime of these grains means that the coma is unrealistically large. In fact, when the space density of the grains becomes low enough the contrast with the background will become low enough so that the coma will no longer be detectable. This will effectively limit the size of the coma and must be included in future work. References Beer, E.H., Podolak, M., Prialnik, P., 2006. The contribution of icy grains to the activity of comets. I. Grain lifetime and distribution. Icarus 180, 473– 486. Crifo, J.F., 1995a. Condensing or sublimating objects in rotation and translation across a rarefied polyatomic vapor I. Astron. Astrophys. 300, 597– 604. Crifo, J.F., 1995b. A general physicochemical model of the inner coma of active comets. I. Implications of spatially distributed gas and dust production. Astrophys. J. 445, 470–488. Crifo, J.F., Rodionov, A.V., 1997. The dependence of the circumnuclear coma structure on the properties of the nucleus. II. First investigation of the coma surrounding a homogeneous aspherical nucleus. Icarus 129, 72–93. Delsemme, A.H., 1982. Chemical composition of nuclei. In: Wilkening, L.L. (Ed.), Comets. University of Arizona Press, Tucson, pp. 85–163. Eddington, A.S., 1910. The envelopes of Comet Morehouse (1908). Mon. Not. R. Astron. Soc. 70, 442–458. Fanale, F.P., Salvail, J.R., 1984. An idealized short-period comet model: Surface insolation, flux, dust flux, and mantle evolution. Icarus 60, 476–511.
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