Coma dust scattering concepts applied to the Rosetta mission

Icarus 257 (2015) 9–22

Contents lists available at ScienceDirect

Icarus journal homepage: www.elsevier.com/locate/icarus

Coma dust scattering concepts applied to the Rosetta mission Uwe Fink a,⇑, Giovanna Rinaldi b a b

Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, United States IAPS-INAF, via del Fosso del Cavaliere, 100, 00133 Roma, Italy

a r t i c l e

i n f o

Article history: Received 19 August 2014 Revised 13 March 2015 Accepted 3 April 2015 Available online 20 April 2015 Keyword: Comets, coma

a b s t r a c t This paper describes basic concepts, as well as providing a framework, for the interpretation of the light scattered by the dust in a cometary coma as observed by instruments on a spacecraft such as Rosetta. It is shown that the expected optical depths are small enough that single scattering can be applied. Each of the quantities that contribute to the scattered intensity is discussed in detail. Using optical constants of the likely coma dust constituents, olivine, pyroxene and carbon, the scattering properties of the dust are calculated. For the resulting observable scattering intensities several particle size distributions are considered, a simple power law, power laws with a small particle cut off and a log-normal distributions with various parameters. Within the context of a simple outflow model, the standard definition of Afq for a circular observing aperture is expanded to an equivalent Afq for an annulus and specific line-of-sight observation. The resulting equivalence between the observed intensity and Afq is used to predict observable intensities for 67P/Churyumov–Gerasimenko at the spacecraft encounter near 3.3 AU and near perihelion at 1.3 AU. This is done by normalizing particle production rates of various size distributions to agree with observed ground based Afq values. Various geometries for the column densities in a cometary coma are considered. The calculations for a simple outflow model are compared with more elaborate Direct Simulation Monte Carlo Calculation (DSMC) models to define the limits of applicability of the simpler analytical approach. Thus our analytical approach can be applied to the majority of the Rosetta coma observations, particularly beyond several nuclear radii where the dust is no longer in a collisional environment, without recourse to computer intensive DSMC calculations for specific cases. In addition to a spherically symmetric 1-dimensional approach we investigate column densities for the 2-dimensional DSMC model on the day and night side of the comet. Our calculations are also applied to estimates of the dust particle densities and flux which are useful for the in-situ experiments on Rosetta. Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction In an earlier paper (Fink and Rubin, 2012), we developed and discussed relationships that connected the observable measure of the dust output, Afq (A’Hearn et al., 1984), to its dust production rate and mass loss rate. We extend and clarify our methods in this paper to calculate line-of sight column densities and scattered light intensities for use in the analysis and interpretation of the dust environment of Comet 67P/Churyumov–Gerasimenko, (abbreviated to 67P) the target of the Rosetta space mission (e.g. Glassmeier et al., 2007). Most of our work was carried out during a visit of Giovanna Rinaldi to Tucson, AZ. Our analysis is based on an analytical approach rather than using computer modeling e.g. the code SCATRD (Vasil’ev, 2006). While a computer model can take into account effects such as

⇑ Corresponding author. http://dx.doi.org/10.1016/j.icarus.2015.04.005 0019-1035/Ó 2015 Elsevier Inc. All rights reserved.

multiple scattering, or 3D atmospheres they often contain assumptions and approximations that are not well understood or documented. It is important that their results be compared to the output of analytical determinations which can be more easily calculated and verified. We will be closely guided by the Direct Simulation Monte Carlo Calculation (DSMC) for cometary gas (Tenishev et al., 2008) and for the dust outflow (Tenishev et al., 2011). These simulations provide the acceleration of the dust particles by the evaporating gas, their velocities and spatial densities. They can also yield the dust distribution for a spherical nucleus as a function of latitude for a two dimensional model where one hemisphere is illuminated by sunlight and the other hemisphere is in darkness. The DSMC model, however, does not provide the absolute amount of dust lifted off a nucleus surface i.e. its production rate, nor the size distribution of the dust. To overcome this shortcoming we develop relationships that connect the production rate of the dust, observable column densities and scattered light intensities to the commonly used

10

U. Fink, G. Rinaldi / Icarus 257 (2015) 9–22

measure of a comet’s dust output Afq, applying a variety of particle size distributions which, to a large extent determine the overall dust scattered light intensity. Additionally, we find that the majority of the Rosetta coma observations can be analyzed with sufficient accuracy using our analytical approach, without requiring to run specific DSMC cases, which are very computer intensive. We concentrate on the remote sensing imaging spectrometer VIRTIS (Coradini et al., 2007), and the imaging cameras OSIRIS (Keller et al., 2007). However, our analysis is also useful for the in-situ dust sensing instruments GIADA (Colangeli et al., 2007), MIDAS (Riedler et al., 2007), COSIMA (Kissel et al., 2007) and the remote sensing microwave instrument MIRO (Gulkis et al., 2007). The remote sensing instruments will measure the line of sight (LOS) intensity of scattered sunlight Iðk; q; gÞ, as a function of wavelength ðkÞ, impact parameter ðqÞ and phase angle ðgÞ for various geometries of the nucleus environment of the comet. It is a fairly complex process to use these data to gain an understanding of fundamental properties of cometary dust that we are interested in: (a) The column density of the dust and from this quantity the dust mass loss rate of the whole nucleus, and thus the dust/gas mass ratio. (b) Constraints on the composition of the dust via its wavelength dependence. (c) Constraints on the dust particle size distribution mostly by observing the phase dependence of the scattered radiation. (d) The correlation between active areas and dust jets and a better understanding of the mechanism of dust production and dust lift off. (e) The dust evolution as a function of the comet’s heliocentric distance. 2. Observable single scattering intensity IðkÞ for a column of particles Our objective is to obtain a dust column density and the properties of the dust from the observed intensity and thus we start with the observable light intensity, IðkÞ, for single scattering and confine ourselves, for the present, to a single particle size, a. The expression, given below contains five quantities, which are described in turn.

IðkÞ ¼ F i ðkÞncol ðqÞrgeom qsca ðkÞ

pðgÞ 4p

F i ðkÞ is the solar flux at the comet at a particular wavelength. It

discussion which is given in the next section. The column density can be calculated using a simple spherical outflow model or using the more complex Direct Simulation Monte Carlo code, or DSMC model (Tenishev et al., 2011). An additional complication is the particle size distribution since the observed intensity combines the contributions from all particles. This will be addressed in Section 5. For purposes of illustration, we pick a wavelength of 1.0 lm which is in the middle of the VIRTIS instrument wavelength range, and a phase angle of 40°. Our formulation can readily be extended to other wavelengths, phase angles and other non-uniform column densities. 2.1. Scattering properties of the dust particles The last three quantities of the above equation contain the scattering properties of the dust particles which can be collected into an expression Sd , which is a function of their size, their composition as expressed by their complex index of refraction, n, the wavelength, k, the phase function, and the shape of the particles.

Sd ða; n; k; gÞ ¼ rgeom qsca ðkÞpðgÞ At present we do not have a good idea what the most appropriate scattering properties for the dust particles should be since we only have a rough idea of their shapes, and their composition. For the calculations in this paper, we used Mie scattering for spherical particles, although we expect the dust particles to be fluffy aggregates made up of many small particles. Modeling of relatively compact and spherical but porous interstellar grains (Voshchinnikov et al., 2005) and porous grains in debris disks (Kirchschlager and Wolf, 2013) have shown that the curves for the scattering, absorption and extinction efficiencies as a function of particle size and wavelength are only marginally different from calculations for Mie particles. Similar results were found in comparisons with laboratory measurements by Pollack and Cuzzi (1980) for scattering of randomly oriented particles possessing a variety of shapes, sizes and refractive indices. It is likely however, that larger differences will be encountered for cometary fluffy open aggregates made up of many small particles. At present we do not have scattering efficiencies available for such particles. It appears that with improving computing power, such efficiencies may become available using techniques such as Discrete Dipole Approximation or the multi-sphere T-matrix approach (e.g. Kolokolova and Mackowski, 2012). When such data become available to us we plan to substitute these values for our present ‘‘approximate’’ Mie scattering values.

is given by F srðkÞ where r h is the heliocentric distance in AU of the 2

2.2. Estimate of an effective index of refraction

comet from the Sun and F s ðkÞ is the solar flux at 1 AU (e.g. Neckel and Labs, 1984). ncol ðqÞ is the column density observed by the spacecraft. Calculation of that is not simple and a detailed description of this is given below. rgeom is the geometric cross section of each particle. It is straightforward to calculate for spherical particles. qsca ðkÞ is the scattering efficiency of each particle. This is a function particle size and composition. We use Mie scattering and an index of refraction of n = 1.70 + 0.02i, and give the rationale for this below. pðgÞ is the phase function for a particular particle size. It is norR malized so that 4p pðgÞdX ¼ 4p. For isotropic scattering pðgÞ ¼ 1. The phase function is calculated for each particle size from Mie scattering using the above index.

Analysis of Cometary Interplanetary Dust Particles (IDP’s), the so called ‘‘Chondritic Porous IDP’s’’ collected by high altitude aircraft (Bradley, 2003; Keller and Messenger, 2011), show that they are mostly primitive silicates with roughly 10% carbonaceous material. This number is slightly lower than the value of 22% of the grain population having grains dominated by carbon and/or organic matter found for Comet 1P/Halley by Fomenkova et al. (1994). Both the pyroxenes and the olivines are of the Mg rich (or Fe poor) type. It is thought that pyroxene is more abundant than olivine, (Keller, private communication). The organics consist of graphite as well as complex organic molecules. Dynamical calculations for the zodiacal dust by Nesvorny´ et al. (2010) indicate that very likely 90% of these particles are of cometary origin so that the IDP’s should serve as a good proxy for the composition of dust particles in the coma of a comet. Samples of coma collected by the Stardust spacecraft to Comet 81P/Wild 2 (Zolensky et al., 2006) corroborate this, finding that their composition is quite similar to that of anhydrous chondritic IDP’s and composed mainly of Fe poor pyroxene

h

Determination of the column density for the highly non-uniform coma of a comet is non-trivial and requires a detailed

11

U. Fink, G. Rinaldi / Icarus 257 (2015) 9–22 Table 1 Optical constants of plausible dust materials. Wavelength (lm)

Reference

0.50

1.00

2.00

Mg rich pyroxene (Enstatite) Mg0.7Fe0.3SiO3

nreal nimag Abs. coeff. (cm1)

1.64 0.0057 1430

1.62 0.0027 338

1.59 0.0017 108

Dorschner et al. (1995)

Mg rich olivine (Forsterite; San Carlos) Mg1.96Fe0.16Si0.89O4

nreal nimag Abs. coeff. (cm1)

1.63 2.30E05 5.86

1.63 2.30E04 24.60

1.63 5.80E05 3.66

Zeidler et al. (2011)

Carbon (graphite; CDE theory) (rough av. || and \ Polr.)

nreal nimag Abs. coeff. (cm1)

1.97 0.48 12,000

2.18 0.57 72,000

2.67 0.66 42,000

Edoh (1983)

Tholin

nreal nimag Abs. coeff. (cm1)

1.71 0.045 11,400

1.65 0.0012 150

1.63 0.001 45

Khare et al. (1984)

and olivine. The optical constants of these major components of cometary dust are summarized in Table 1, where references to these constants are also listed. The real part of the refractive index for Mg rich pyroxene (Enstatite) and Mg rich olivine (Forsterite) is roughly 1.63. The addition of Fe will increase this index quite substantially. Organic molecules represented by ‘‘Tholin’’ have a real index of about 1.65. The index for graphite is considerably higher with values easily exceeding 2.0. We thus picked a real index of refraction of 1.70, slightly higher than Enstatite or Forsterite but taking into account the admixture of iron and graphite. The imaginary indices of pure Enstatite and Forsterite are quite low of the order of 0.001 or lower. However the addition of only small amounts of iron or any other impurities increases that index quickly by a factor of 10 or more (Zeidler et al., 2011). The imaginary part of the index of graphite is clearly very large. For our calculations in this paper we settled on an imaginary index of 0.02. We note that the observed backscattering of the cometary phase curve requires an imaginary part of the refractive index of 0.01–0.04. If the index is too small, the backscattering becomes too large, and if it is too large there is no backscattering, (see Fig. 8 in Fink and Rubin, 2012). The chosen indices of refraction are representative and we realize that they exhibit considerable wavelength dependence. We hope that the coma data produced by the Rosetta mission will allow us to deduce the wavelength dependence of these parameters and thus constrain the

composition of the dust, as was done for the surface in our recent paper (Fink, 2015). 2.3. Scattering efficiency and phase curve The scattering efficiencies for spherical Mie particles that we used are shown in Fig. 1. A more comprehensive set of Mie scattering efficiencies and phase curves is illustrated in our earlier paper (Fink and Rubin, 2012). The present figure is included for completeness and to compare the scattering efficiencies used in the SCATRD code (Vasil’ev, 2006) vs. scattering efficiencies for Mie particles using varying the smoothing parameter b as defined in Hansen and Travis (1974). The results of the two approaches are very close. A comparison of the phase functions between Mie scattering and the SCATRD code is presented in Fig. 2. The latter uses Henyey–Greenstein approximations with a single asymmetry parameter for each particle size (the first order of a Legendre polynomial expansion). Fig. 2 shows that such approximations can introduce quite serious errors. A single asymmetry parameter does not allow for backscattering, which is observed for cometary comae. It also gives values that are much too small for forward scattering and yields values that are about a factor of 2–4 too small for small phase angles (large scattering angles). Thus the advantage of a multiple scattering code can be muted if it incorporates such simplifying assumptions.

Comparison of Qsca for n= 1.70+0.02i

4.5 4

GR Uf Qsca b=0.01 Uf Qsca b=0.05 Uf Qsca b=0.10

3.5 3

Qsca

2.5 2 1.5 1 0.5 0 0.1

1

10

100

1000

x= 2πa/λ Fig. 1. Illustrations of typical Mie scattering efficiencies for n = 1.70 + 0.02i vs. parameter x = 2pa/k. A comparison is shown between two different scattering codes labeled GR, and UF to check for consistency. The agreement is very close. For illustrative purposes, the curves labeled UF have different smoothing parameters b (Hansen and Travis, 1974).

12

U. Fink, G. Rinaldi / Icarus 257 (2015) 9–22

1000.00

Phase function comparison Mie and Henyey Greenstein Approx. n= 1.70+0.02i x=0.628 x= 6.28 x= 62 x=628 HG x= 0.628 a= 0.10um HG x= 6.28 a= 1.0 HG x=62 a= 10.0um HG x= 628 a= 100um

Phase function

100.00

10.00

1.00

0.10

0.01 0

20

40

60

80

100

120

140

160

180

Scattering angle Fig. 2. Illustrations of selected phase functions for Mie scattering and Henyey–Greenstein approximations with a single asymmetry parameter for each particle size. It can be seen that using such a single asymmetry parameter does not allow for backscattering, gives values that are too small for forward scattering and yields values that are about a factor of 2–4 too small for small phase angles (large scattering angles). For the Henyey–Greenstein cases of varying parameter x we also list the corresponding particle radius for a wavelength of 1.0 lm.

3. Column density DSMC and analytical calculations

h ¼ ð90  uÞ  arcsin

The geometry of a spacecraft observing the coma of a comet is given in Fig. 3. The spacecraft is a distance rsp from the comet nucleus. The spacecraft instruments can either look at various locations on the nucleus (e.g. nadir pointing) or they can look past the limb of the comet along the direction f with an impact parameter q, measured from the nucleus center. This limb scan geometry is of most interest to the study of the dust since it is not adulterated by the signal from the nucleus. It can yield the spatial distribution of the dust and it provides the highest scattered light intensities. It is desired to find the column density along the line-of-sight (LOS), along the direction f in our figure. The angle / is defined in Fig. 3 and can be considered to extend only to the radius vector to point A, if we are mainly interested in a strong jet region or it can extend to the radius vector rsp of the spacecraft. The point f = 0 (and also / = 0) is at the right angle intersection of the impact parameter with the LOS. The spacecraft is considered to be at position fSc at the beginning of the line of sight. We also define:

r 2 ¼ q2 þ f2 ;

r 2sp ¼ q2 þ f2Sc ;

u ¼ arctan

  fSc

q

 ¼ arccos

q



rsp

and note that the miss angle h, between a tangent to the surface of the comet and the line of sight can be determined as:

  rn , where rn is the appropriate nucleus r sp

radius vector. The geometry of the 2-dimensional axisymmetric DSMC model has the axis of symmetry (x axis) pointing toward the Sun while the solar illumination and thus the gas and dust production along the y and z axes vary as the cosine of the Sun’s angle of incidence. The one dimensional DSMC model uses a radially symmetric gas and dust outflow. Neither of the models include rotation of the nucleus. The numerical results from the DSMC model used in this paper were obtained on the site ‘‘Inner Coma Environment Simulator’’ (http://ices.engin.umich.edu). A 3D model which takes into account the rotation axis and the varying illumination of the somewhat complex surface of 67P is presently under development. The geometry of observation is similar to that for Earth based telescopes except that for those observations the limited spatial resolution essentially restricts measurements to the whole coma using a photometric aperture and the LOS extends from 1 to þ1. To interpret these measurements, the concept of Afq was developed. This concept assumes a spherically symmetric 1-dimensional dust outflow model which we will use in the development below. To compare our analytical results to the DSMC model, we use the density and velocity profile along the +x axis (the sunward direction) as representative for all radius vectors. Our analysis serves several useful purposes: 1. It allows us to compare ground based observations to spacecraft measurements and thus predict observable light intensities. 2. It yields the limits

Fig. 3. Geometry for observation of the line-of-sight (LOS) particle column density along the LOS f (labeled z in the figure), with impact parameter q as observed from a typical spacecraft location. LOS calculations can be performed for an arbitrary coma extent, e.g. A to B with includes angle 2u, allowing for exploration of the inner coma dust jets, as illustrated. (See case 3 in Section 3 of text).

13

U. Fink, G. Rinaldi / Icarus 257 (2015) 9–22

and magnitude of errors of the simpler analytical tool compared to the much more complex DSMC calculations. 3. It provides a check on the quite elaborate DSMC calculations.

ncol ðqÞ ¼

3.1. Volume density

nv ol ðrÞ ¼

Q d 1 bd ðrrn Þ e 4pv d r 2

Here Q d is the equivalent spherical dust production rate at the nucleus (in particles/s) in the sector of extent 2/, which is sampled by the column density measurement. The radial distance, r, is measured from the center of the nucleus. The dust outflow velocity is v d and bd is a parameter which allows for the destruction or production of dust. The use of the parameter bd is particularly germane to sublimating ice grains as analyzed for 103P/Hartley 2 by Protopapa et al. (2014). The equation above is similar to that described some time ago by Haser (1957) for parent molecules. Until Rosetta data shows evidence to the contrary, we will use bd ¼ 0, i.e. no production or destruction of dust after it leaves the nucleus, however, it is probably useful to leave it in the equation as a placeholder. In Fig. 4 we compare the particle density distribution, nv ol ðrÞ; of our simple analytical calculation with the DSMC model for the case of 67P at 1.3 AU for the sunward facing side, or x axis, for a particle size of 10 lm. We used the outflow velocity of 33.07 m/s as given by the DSMC calculations for the terminal dust velocity beyond about 10 km from the nucleus center. However, we scaled the DSMC calculations to a production rate of 1.0  1010 particles/s of 10.0 lm size to yield an approximate observed Afq value for 67P of 200 cm, once a particle size distribution is applied as is discussed in Section 6 of this paper. The figure shows that for distances larger than about 4–5 km from the nucleus (2–4 km above the surface), when the dust particles enter the free flowing zone and have reached their terminal velocity, the DSMC and analytical calculations agree quite well. Inside this distance the accelerating particle velocities of the DSMC model have to be used. 3.2. Column density

Particle density (# of 10.0 μm particles/m3)

Using our simple outflow model the analytical column density can be calculated by the expression below. We give four sample cases of the LOS column density.

f2

nv ol ðrÞdf ¼

f1

¼

The expression for the volume density for the simple analytical dust outflow model is:

Z

Qd 4p v d

Z

f2

f1

df

q2 þ f2

  Qd 1 f f arctan 2  arctan 1 4pv d q q q

1. The column density looking from the spacecraft at distance r sp to þ1 then becomes:

"Z # Z 1 0 Qd df df ncol ðqÞ ¼ nv ol ðrÞdf ¼ þ 4pv d fSc q2 þ f2 q2 þ f2 0 f1    Qd 1 p f þ arctan Sc ¼ 4pv d q 2 q Z

f2

2. Looking from the Earth through the coma from 1 to þ1 we get:

np ðqÞ ¼

Q d 1 p p Q 1  ¼ d 4pv d q 2 2 4v d q

3. Should we be interested in a strong coma jet or outburst, dominating the intensity, the case below provides integration between points A (at f1) and B (at f2) having an extent 2/, to which can be added to the more symmetric background coma contribution. We get:

np ðqÞ ¼

Qd 1 2u ðradiansÞ 4pv d q

4. Looking directly down at the nucleus (nadir).

ncol ðrsp  r n Þ ¼

Qd 4p v d

Z

r sp

rn

  1 Qd 1 1 dr ¼  2 r 4pv d r n r sp

The resulting column densities are illustrated in Fig. 5. We show the case of the spacecraft 100 km from the nucleus center and 22 km from the nucleus center (20 km above the surface). As expected from the comparison of the volume densities, outside 4–5 km from the nucleus center the simpler analytical calculation can be used with very little error. However, at a distance of 0.1 km above the surface the DSMC calculations yield a column density about a factor of 3.6 higher than the simple analytical case which assumes a constant outflow velocity. Spatially resolved column density profiles above the nucleus surface should be able to test the results produced by the DSMC model. Several other cases for 67P at 3.3 AU were also tested but are not illustrated because they gave similar results. We point out that the column density along the LOS from the spacecraft (at 100 km) to +100 km, for an impact parameter of 4 km, includes 97.5% of the total column from 1 to þ1, missing only 2.5%. For the spacecraft at 22 km the

1000.00 10.0 μm dust particle density at 1.3 AU Analytical and scaled DSMC model CG_1.3_au_00 a= 10μm; v= 33.07 m/s; Q=1.0E10 particles/s estimated Afρ= 200 cm

100.00

DSMC scaled Analytical Calculation

10.00

1.00

0.10

0.01 0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

Distance from nucleus center (km) Fig. 4. Comparison of dust particle number density vs. distance from the nucleus between a simple analytical approach and a one dimensional DSMC model for which the x axis particle density on the sunward side was assumed to be representative of a radially symmetric distribution. The particle density is plotted for a single size of 10 lm in radius. The production rate has been scaled to 1.0  1010 particles/s to yield values of Afq around 1–2 m as observed for 67P/Churyumov–Gerasimenko approaching perihelion. The comet nucleus is 2 km in radius and the outflow velocity at 10 km from the center as given by the DSMC model is 33.07 m/s. It can be seen that outside 4 km from the nucleus, the densities given by the DSMC model and the analytical approach are virtually indistinguishable.

14

U. Fink, G. Rinaldi / Icarus 257 (2015) 9–22

Dust column density Analytical Calculation and DSMC model a= 10 μm; v= 33.07 m/s; Q= 1.0 E10 particles/s rsp= 100 & 22 km from nucleus center

140000

Coumn Denstity ( #/m2)

120000

DSMC Num Integr rsp= 100km DSMC Num Integr rsp= 22km Analyt Calc. rsp= 100km Analyt.Calc rsp=22km

100000 80000 60000 40000 20000 0 0

5

10

15

20

25

Impact parameter ρ (km) Fig. 5. Comparison of the column density vs. impact parameter between the DSMC model and a simple analytical approach using the parameters of Fig. 4 for 67P at 1.3 AU and the geometry of Fig. 3. Two spacecraft distances from the comet center are shown, 22 km and 100 km. If the impact parameter for the spacecraft LOS is around 4–5 km or higher, (2–3 km from the nucleus surface), the analytical representation and the DSMC model agree very well. For an impact parameter of 2.1 km the DSMC model gives a column density about a factor of 3.6 higher, because the particle speeds are much lower. The figure also shows that the observable column density changes very little for spacecraft distances of 22 km and 100 km from the nucleus center, since the contribution to the total column density farther from the nucleus is relatively small.

fraction missed is about 11.6%. The majority of the column density contribution comes from the vicinity of the nucleus. This integration required the analytical approach since the DSMC calculations do not extend to such large distances. 3.3. The 2-dimensional case For the 2D case of a hypothetical spherical comet with one hemisphere illuminated by the Sun and a dust particle production rate and outflow velocity varying with latitude, a simple analytical approach will not work and we have to use the DSMC calculations. We present several sample DSMC calculations for 67P at 1.3 AU. We show in Fig. 6 the 2D scaled volume particle distribution for the dayside and the nightside along the y axis (or equivalently the z axis for an axisymmetric model) for constant x axis values of +4 km (Sun-facing) and 4 km (anti-sunward). For comparison we have also added a density profile along the y axis if the +x axis values were radially symmetric. Interestingly the DSMC 2D case along the y axis gives larger volume densities than the equivalent 1D

4.5

case, with its x axis pointing toward the Sun. Also, the dayside values exhibit a dip at y = 0 while the nightside values show a peak. In Fig. 7 we show the corresponding column densities on the dayside and nightside as a function of impact parameter, with the spacecraft 100 km from the nucleus looking along the y axes. This is roughly the geometry for the planned Rosetta terminator orbit, looking past the limb on either the dayside or the nightside. As expected, the largest column densities are achieved by looking on the dayside at various impact distances above the nucleus surface. 4. Relationships between observed intensity and its equivalent Afq In this section we develop relationships that will connect the observed intensity along a line of sight to an equivalent Afq. A large number of ground-based measurements exist for Afq (e.g. A’Hearn et al., 1995; Fink, 2009). These measurements show that the Afq/gas ratio can vary by up to a factor of 100 from comet to comet,

DSMC CG_1.3_au_00 Density along LOS ρimp = 4.0 km

4

LOS 2D Dayside y axis; x= +4000m LOS 1D ( Density radially symmetric) LOS 2D Nightside y axis; x=-4000m

Number density (#/m3)

3.5 3 2.5 2 1.5 1 0.5 0 0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

Density along LOS (m) Fig. 6. Particle dust volume density of 10 lm particles from the two dimensional DSMC model case CG_1.3_au_00 for three LOS geometries. As in Fig. 4, their production rate has been scaled to 1.0  1010 particles/s. Densities along the y axis on the dayside (x = +4 km) and on the nightside (x = 4 km) are shown. Also shown for comparison is a calculated one dimensional case along the y axis for which the x axis particle density on the sunward side was assumed to be representative of a radially symmetric distribution (its density at LOS = 0 is the same as the density in Fig. 4 at x = 4 km, y = 0). Interestingly, the dayside profile shows a dip at x = 4, y = 0 km followed by an increase in the dust particle density for y values of ±2 km with values considerably higher than for the corresponding radially symmetric 1D case. The nightside value shows a peak at x = 4, y = 0 km.

15

U. Fink, G. Rinaldi / Icarus 257 (2015) 9–22

DSMC CG_1.3_au_00 Dust column density scaled a= 10 μm; Q~ 1.0 E10 parcles/s rsp= 100 km from nucleus center

180000

Coumn Densty ( #/m2)

160000

1D (+x density radially symmetric) 2D Dayside 2D Nightside

140000 120000 100000 80000 60000 40000 20000 0 0

2

4

6

8

10

12

14

16

18

20

Impact parameter ρ (km) Fig. 7. Dust column density along a LOS along the y axis vs. impact parameter (x axis values) for the DSMC model case CG_1.3_au_00. The column densities are scaled to roughly reproduce the observed Afq around 2 m. Again, somewhat surprisingly, but in accord with Fig. 6 the dayside column densities along the y axis are larger than a calculated 1D radially symmetric profile. The difference in column density between the dayside and nightside are roughly a factor of 4.

and this ratio is not constant during the passage of a comet to perihelion. As mentioned, the DSMC model does not have a mechanism that can predict the amount of dust lifted off the nucleus. It cannot relate the amount of dust lifted to the gas production rate. The DSMC model assumes a certain amount of dust that is lifted from the surface into the coma. The dust densities calculated by the DSMC model, together with its chosen particle size distribution must therefore be scaled to produce an observed Afq. It must be understood that Afq is an ‘‘all inclusive’’ observed quantity that combines all of the complex factors that go into the calculation of the scattered light intensity into one quantity. In particular it includes the particle size distribution and the complex scattering efficiencies of the particle, resulting from their composite composition and irregular shape. To disentangle and determine all of the physical parameters that make up an observed Afq is a difficult undertaking, yet this is our ultimate goal for a physically meaningful interpretation of Afq. It is one of the main objectives of this paper to lay the groundwork for this difficult undertaking. Ground based observations of Afq are exceedingly useful to give a global view of the dust environment of the comet which can then be compared to localized spacecraft observations, and can also be used to predict the expected line of sight scattered light intensity for a specific spacecraft observation geometry. This works well as long as the scattering parameters of the dust stay approximately the same. In the development of our analytical relationships below we confine ourselves to a single particle size, a, noting however that for a comparison with observable intensities, the quantities required to calculate Afq must be summed over all particle sizes using a particle size distribution. We also note that the analytical derivation assumes an equivalent production rate of dust at the surface of the comet which could be confined to an azimuthal region of extent 2u, or could be considered an average over a large surface region of the comet or an average over the whole comet, depending on the observation geometry. For regions close to the nucleus, a varying outflow velocity must be taken into account as discussed in Section 3. The quantity Afq or its equivalent value can be defined for three different geometries. (i) A line of sight observation at a projected distance q from the nucleus and for a specific azimuthal position. The Afq thus defined

can vary as a function of q and can vary with azimuthal position of the orbiting spacecraft’s geometric LOS, for example due to day– night variations or dust jets. (ii) An observation of intensity in an annulus. This Afq can vary for different annuli at different distances q from the comet. (iii) An observation of the dust scattering within a photometric aperture. The relationship for Afq in terms of a localized production rate of the dust and its scattering properties is carried out in Fink and Rubin (2012):

Af q ¼ Sd ðaÞ

Q d ða; uÞ 2v d ðaÞ

(i) A specific line of sight observation The intensity by a spacecraft along a line of sight is:

Iobs ða; kÞ ¼ F i ðkÞncol ða; qÞ

Sd ðaÞ 4p

The column density along the line of sight can be determined numerically using the DSMC model or can be calculated analytically for various spacecraft geometries as carried out in the previous section. For the equations below we use the analytical approach with the column density and equivalent dust production rate confined within a coma region of extent 2u (case 3 of the previous section). We get:

Iobs ða; kÞ ¼ F i ðkÞ

Q d ða; uÞ 1 Sd ðaÞ 2u ðradiansÞ 4pv d ðaÞ q 4p

Using our relationship for Afq above, we can express the intensity along a line of sight in terms of Afq, and obtain:

Iobs ða; kÞ ¼ F i ðkÞðAf qÞ

1 2u q 8p2

Conversely we can obtain an equivalent Afq value from an observed intensity a long a specific spacecraft line of sight.

ðAf qÞ ¼

Iobs ða; kÞ8p2 q F i ðkÞ2u

We note that this Afq value is not necessarily representative for the whole comet but applies to the area above the comet nucleus included in the angle 2u.

16

U. Fink, G. Rinaldi / Icarus 257 (2015) 9–22

(ii) Flux in an annulus

Af q ¼

If the particle scattering properties, their production rate Q d ðaÞ and the particle outflow velocity v d ðaÞ (or equivalently the Afq value) are spherically uniform around the nucleus we can sum all the line of sight paths to get the Afq within an annulus. Assuming we are looking from Earth, through the coma from 1 to +1, and at a distance D from the comet, the angle 2u becomes

p and the solid angle of the annulus is DX ¼

5. Particle size distributions All of the calculations so far presented were for a single particle size (mono-size). To get the total dust production and dust spatial distribution we need to sum over all the particle sizes in the cometary coma. The correct particle size distribution is probably the most important and most uncertain input for the calculation of the observed scattered light intensity, the determination of the dust mass loss rate and their relationship to Afq. A brief summary of the literature on cometary dust particle size distribution is given in Fink and Rubin (2012). We hope that by combining the results of the Rosetta suite of instruments looking at the dust, GIADA, MIDAS, COSIMA, MIRO, etc. we will be able to constrain the particle size distribution better than is known presently. We consider three particle size distributions: a simple power law distribution, a power law distribution with a small particle cut-off, and a log normal distribution. All three are illustrated in Fig. 8. All of the distributions were normalized at a particle radius of 10.0 lm. We felt it was not appropriate to normalize to the total number of particles for a particular distribution because most distributions use a large and small particle cut off thus making the total number of particles dependent on these parameters. The large particle cut-off for cometary dust is usually picked as the largest liftable particle size for a given gas production rate. An expression for this is given in Tenishev et al. (2011). The largest liftable particle size for the DSMC model CG_1.3_au_00 was 6.3 mm. The largest liftable size for the model at 3.3 AU was 6.81 lm. The calculation of the largest liftable size assumes spherical particles with a density of 1.0 g/cm3. Early results from the GIADA dust collection experiment on Rosetta (Rotundi et al., 2015) however indicate that considerably larger fluffy particles of the order of 100 lm have been lifted off the surface of Comet 67P at a distance of about 3.4 AU. The resulting Intensity, mass loss rate and Afq values can differ by quite large factors depending on which distribution and which parameters are used. We describe the results of various selected cases in the next section.

2pq d D2

q. The collected

flux in the aperture then becomes:

F obs ðannulusÞ ¼ Iobs DX ¼ Iobs ¼ F i ðkÞðAf qÞ

2pdq

D2

¼ F i ðkÞðAf qÞ

1 p 2pq dq q 8p2 D2

dq 4 D2

Afq can then be defined as:

Af q ¼

F obs 4D2 dq F i ðkÞ

The latter quantity is similar to the quantity ‘‘Summa Af’’ used by Tozzi et al. (2007, 2011). It does not presume a 1/q fall off for the dust column density. As is shown by the DSMC model in Figs. 4 and 5, inside about 5 radii from the nucleus the particles are still accelerating and thus we do not have a simple 1/q column density fall off. (iii) An observation within a photometric aperture If we assume that dust column density falls off as 1/q as is closely approximated for most Earth based observations we can integrate the intensity to get an observed flux within an aperture:

F obs ðapertureÞ ¼

Z

p

Iobs dX ¼ F i ðkÞðAf qÞ

0

¼ F i ðkÞðAf qÞ

1 8 p D2

Z 0

p

1

q

2pqdq

1 q 4 D2

Rel. Prod.Rtes Q (a) #/s

This reverts to the original expression for Afq (A’Hearn et al., 1984). It also serves as a proof of our expression for Afq, above, taken from Fink and Rubin (2012).

1.0E+18 1.0E+17 1.0E+16 1.0E+15 1.0E+14 1.0E+13 1.0E+12 1.0E+11 1.0E+10 1.0E+09 1.0E+08 1.0E+07 1.0E+06 1.0E+05 1.0E+04 1.0E+03 1.0E+02 1.0E+01 0.10

F obs 4D2 F i ðkÞq

Comparison of sample parcle size distribuons used for calculang Afρ and mass loss rate dQ/da ~ a-4 Q(a) ~a-3 UF Distr. ap= 0.20 um LogNorm rm= 0.07 sigma= 4.7

1.00

10.00

100.00

1000.00

10000.00

Radius μm Fig. 8. Particle production rates for three sample particle size distributions, all normalized for 10 lm radius particles. The distribution Q(a)  a3, was used by the DSMC code. It results in a differential particle size distribution dQ/da  a4, which we show for illustrative purposes as a dashed line. Next we show a particle size distribution which reduces the number of small particles and has a peak particle size of 0.20 lm. At present this is our preferred particle size distribution. Also shown is a log normal distribution with parameters am = 0.07 and r = 4.7. The latter distribution reduces the number of small and large particles. The number of large particles is down by a factor of 100 for 1 mm particles and by a factor of 2000! for 6.3 mm particles with respect to the plotted power law. It is possible that this distribution diminishes the number of large particles too much.

17

U. Fink, G. Rinaldi / Icarus 257 (2015) 9–22

a: Geometry for intensity calculation is shown in Fig. 3 (rsp = 100 km; pimp = 12 km; coma = 40 km) for coma of 100 km numbers will be (2.901/2.5320 higher). b: Uses n = 1.70 + 0.02i for optical constants and a phase angle of 40° particle size distributions are normalized for 10 lm particles using DSMC model CG_1.3_au_00 as a guide. c: For dust/gas ratio, a gas output of 153 kg/s (5.0E27 molecules/s) is used as given by DSMC model. d: Virtis NESR is given as 5  E04 W/m2 lm ster yielding a SNR of 4.6 for 1 pixel in 1 s. e: For an impact parameter of 2.1 km and using DSMC integration, intensities will be a factor of 21 higher.

17.27 7.64 6.42

Note Note Note Note Note

0.050 0.274 0.613 3.60E04 3.60E04 3.60E04 2.32E03 2.30E03 2.31E03 7.57 41.96 93.77 1.10E+11 2.48E+11 2.95E+11 2.00 2.00 2.00

0.062 0.108 0.300 3.60E04 3.60E04 3.60E04 2.31E03 2.30E03 2.31E03 9.53 16.50 45.91 5.46E+10 9.93E+10 2.73E+11 2.00 2.00 2.00

3.60E04 2.31E03

Log-normal rm = 0.07, r = 4.7 rm = 0.05, r = 7.0 rm = 0.20, r = 6.5

A brief discussion of Fig. 8 reveals the following: The power law distribution of dQ/da  a4.0 seems somewhat unphysical in the sense of extending a power law over such a large range of particle sizes. Generally there are mechanisms at work in nature which limit both the smallest possible sizes and the largest sizes. It thus requires both a small and large particle cut-off. The distribution with reduced small particles alleviates that problem for the small particles, but still needs a large particle cut-off, although this can be determined by the largest liftable particle size, as mentioned earlier. The two free parameters in the log normal distribution can yield quite a variety of particle size distributions. As a starting point we

34.7 19.1 6.94

0  12 a 1 @ln am A a ln NðaÞ ¼ ln r 2

4.10

The scattering program SCATRD (Vasil’ev, 2006) through an auxiliary program called ARS supplies several particle size distributions (Mono-size, Log-normal, Modified Gamma and Gaussian). The most likely of these particle size distributions for cometary dust is the log-normal distribution. Log-normal distributions have been used quite extensively for atmospheric aerosols. However, it is not clear to us that such distributions will also apply to cometary dust. The expression for the log-normal distribution is:

Table 2 Dust parameters for 67P/CG at 1.3 AU for various particle size distributions.

5.3. A log-normal distribution

2.01E+10

This particle size distribution reverts to a standard power law with exponent a for particle sizes several times larger than a0. It has two free parameters. The maximum of the distribution is given by ap ¼ aa0 .

2.00

Afq (m) Prod rate 10 lm Total mass loss (kg/s) Calculated intensity Optical depth, s (particles/s) (W/m2 lm ster at k of 1.0 lm)

a a a0 dQ 0 ¼ g 0 eð a Þ da a

Afq (m) Mass loss (kg/s)

Normalizing to an observed Afq value of 2 m

CG_1.3_au_00

For a simple power law distribution the exponentially increasing number of small particles can overwhelm the resulting calculation of the intensity, Afq and mass loss rate. Thus a small particle cut off is usually necessary. This is based on evidence of thermal infra-red measurements by Hanner (1983). We use the following formula developed in Fink and Rubin (2012) for the small particle cut off.

DSMC values

5.2. A power law distribution with a small particle cut-off

0.025

Dust/gas Notes

as an example in Fig. 8. The quantity at some point in the coma will have a different distribution because the velocity of the particles varies roughly as a0.5, and the particles can fragment or evaporate or be swept into the tail by the solar wind after they depart from the nucleus.

165.4 157.6 159.3

dQ da

ðdn Þ, da

192.1

the difference in slope between the latter two we have plotted

– Too many small particles

This distribution with an exponent, a, of 4 was used by, Tenishev et al. (2011), in their DSMC dust model simulations. Q ðaÞ is the production rate of particles of size a near the surface of the comet. The term ðdn Þ is often used somewhat loosely to describe the particle da size distribution without giving a good definition of this term. Thus in our paper we are careful to use Q ðaÞ and dQ . To illustrate da

– Cuts small particles only slightly – Probably good compromise – Cuts small particles quite severely

dQ  aa da

100.4

or

DSMC 1.3_au_00 (Q  a3) UF ap = 0.10 lm UF ap = 0.20 lm UF ap = 0.50 lm

Q ðaÞ ¼ g 0 aaþ1

65.4 160.3 301

The most common particle size distribution that has been used for cometary dust is a simple power law, usually with step function for small and large particle cut offs.

– Cuts large particles quite severely – Cuts small & intermediate particles some cuts for large particles – Cuts small & intermediate particles strongly large particles are not cut

5.1. A simple power law distribution

18

U. Fink, G. Rinaldi / Icarus 257 (2015) 9–22

picked the values am = 0.007 lm and r = 4.7 lm. This distribution has a natural small particle cut-off but it also appears that the numbers of large particles above say 100 lm, are depressed too much. 6. Model results for the observable intensity and Afq We now present results by adding the contributions to the intensity and Afq by all the particles using various particle size distributions. The refractive index used to calculate the scattering efficiencies was n = 1.70 + 0.02i, the rational for which is given in Section 2. We looked in some detail at two DSMC model cases: a case near perihelion at a distance of 1.3 AU (CG_1.3_au_00) and a case near the Rosetta encounter with the spacecraft at 3.3 AU (CG_3.3_au_00). The case CG_1.3_au_00 stands for 67P/CG at 1.3 AU from the Sun and uses a total gas production rate of 5  1027 molecules/s (95% H2O and 5% CO) equivalent to 153 kg/s, with a largest liftable particle size of 6.81 mm. The case CG_3.3_au_00 is at 3.3 AU and assumes a gas production rate of 1.0  1024 molecules/s (95% H2O and 5% CO) equivalent to 0.031 kg/s, with a largest liftable particle size of 6.81 lm, (ICES web site, (http://ices.engin.umich.edu). Early results from Rosetta however showed that the water production rate was roughly 0.3 kg/s at a distance of 3.9 AU, increasing to about 1.2 kg/s at a distance of 3.45 AU in late August 2014 (Gulkis et al., 2015). Since the amount of dust expelled into the coma cannot be predicted by the DSMC model calculations, we will adjust our dust production rate to produce values of Afq that are in accord with ground based observations for 67P. For the ground based observations we rely on the careful and extensive photometry observations of 67P by Schleicher (2006). A summary of ground based observations by Snodgrass et al. (2013) yields similar values and extends the data to larger heliocentric distances. The latter values are higher because Snodgrass corrects his values to 0 phase angle, an extrapolation that is quite uncertain and a phase that is essentially impossible to observe, while we use a phase angle of 40°. There is considerable scatter in the data for 67P, but there is clearly a higher post-perihelion dust production. We will use a value of 2.0 m at a phase angle of 40° as reasonably representative of the

1.00E-03

pre-perihelion dust environment near 1.3 AU. Spectra of 67P yielding continuum Afq measurements as well as emissions by CN, C2, NH2 and OI 1D are analyzed in the summary paper by Fink (2009). The results for the first case are summarized in Table 2. The first two values represent Afq and mass loss rates of the DSMC model as published by Tenishev et al. (2011) and downloaded from the ICES web site (http://ices.engin.umich.edu). The resulting Afq value of 100 m seems much too high when compared to observed values of around 2 m for 67P near perihelion. This was already pointed out and discussed in the paper by Fink and Rubin (2012). That paper also explored power laws with five different exponents and various small particle cut-offs which we will not repeat here. The remaining rows in columns 2 and 3 of Table 1 show the result of other particle size distributions which had their 10 lm particle production rate Qd normalized to the value of 9.5  1011 particles/s as given by the DSMC model. The table clearly shows the strong effect that the particle size distribution has on the resulting Afq and mass loss values. These distributions with both small and large particle reductions tend to give lower values of Afq, but are still not quite able to reach the lower observed value of 2 m for 67P near perihelion. To have agreement with observed Afq values, we normalized the Afq values calculated for each of the distribution to 2.0 m by adjusting the particle production rate at 10 lm. This number is listed in the 5th column of Table 2. Having thus normalized each particle distribution we can calculate the mass loss rate in column 6, the expected scattered light intensity in column 7, the optical depth in column 8 and the dust/gas ratio in column 9. The calculated intensity and optical depth is for a spacecraft 100 km from the nucleus center looking through the coma with an impact parameter of 12 km, a phase angle of 40° and assuming a symmetric radial dust outflow. Of particular note is the low optical depth of 3.6  104, which shows that the assumption of single scattering in Section 2 is clearly justified. The particle production rate and density would have to be a factor of more than a thousand higher before multiple scattering has to be taken into account. As shown in Section 4 we also note that since the column density is directly proportional to Afq once this quantity is fixed, the resulting intensities and optical depths are the same for the various particle size

Expected Intensity 67P at 1.3 AU var. Part. Size Distr. ρ= 12 km; phase= 40 o

dQ/da ~ a-4; Afr Sum = 2.0 m

Intensity (watts/m2 μm ster)

UF ap= 0.20 um; Afr Sum = 2.0 m

1.00E-04 Log-Normal; Afr Sum = 2.0 m

1.00E-05

1.00E-06

1.00E-07 0.100

1.000

10.000

100.000

1000.000

10000.000

Particle radius (μm) Fig. 9. Expected contributions to the scattered light intensity by particles of various sizes at a wavelength of 1 lm for three different particle size distributions. The figure illustrates the complex nature of the contributions to the intensity and Afq once all the factors that go into calculating these quantities are combined. Since the observable intensity is directly proportional to Afq, as demonstrated in the text, this figure is also a proxy for the contributions to Afq by particles of various sizes. The spacecraft is assumed to be 100 km from the nucleus and is looking at the comet which is at a distance of 1.3 AU from the Sun with an impact parameter of 12 km and a phase angle of 40°. The summed Afq values over all particle sizes have been normalized to a value of 2.0 m which results in a summed intensity of 2.31  103 W/m2 lm ster for all three cases. While the particle size distribution in Fig. 8 have all been normalized to equal particle density for 10.0 lm particles, in this figure, the contribution to the intensity and Afq for 10 lm particles is different for the three types of distributions because it is their sum over all particle sizes has been normalized to an Afq of 2.0 m.

Table 3 Example of the calculation of mass loss rate, Afq, scattered light intensity and extinction. Case of small particle cut off 2 3 4 x param., rgeom (m2) Qsca k = 1.0 lm

5 6 Phase Sd (m2) function

7 Qext.

vx

1.000 1.468 2.154 3.162 4.642 6.813 10.000

6.283 9.224 13.53 19.87 29.17 42.81 62.83

0.216 0.096 0.077 0.066 0.053 0.058 0.058

2.63 2.47 2.35 2.28 2.22 2.19 2.13

96.4 81.2 68.2 57.1 47.6 39.7 33.1

3.14E12 6.77E12 1.46E11 3.14E11 6.77E11 1.46E10 3.14E10

2.074 1.820 1.613 1.454 1.326 1.256 1.192

1.41E12 1.18E12 1.81E12 3.03E12 4.73E12 1.05E11 2.16E11

8

9 (m/s) Q(a) (#/s)

10 11 12 13 14 15 16 17 18 19 Mass loss Afq (m) Col density, Obs. int. Obs. int. Extinction Q(a) (#/s) Mass loss Afq (m) Obs. int. (kg/s) n(p) (#/m2) (W/m2 lm ster) (W/m2 lm ster) (W/m2 lm ster) (#/s) (kg/s)

2.01E+13 6.36E+12 2.01E+12 6.37E+11 2.01E+11 6.36E+10 2.01E+10

0.137 0.137 0.137 0.137 0.137 0.137 0.137

0.1471 0.0461 0.0267 0.0169 0.0100 0.0084 0.0066

Sum

0.957

0.262

4.02E+06 1.51E+06 5.68E+05 2.15E+05 8.13E+04 3.08E+04 1.17E+04

Notes: Column 1: Radius of particle in lm; size bin 10.0 lm extends to 14.68 lm. Column 2: x parameter 2pa/k for Mie scattering. Column 4: Scattering efficiency using n = 1.70 + 0.02i. Column 5: Phase function for 140° scattering angle (40° phase angle). Column 6: Combined scattering properties of the dust Sd = Qsca ⁄ rgeom ⁄ p(g). Column 7: Mie scattering extinction efficiency. Column 8: Particle velocity at 10 km from the nucleus (DSMC model CG_1.3_au_00). Column 9: Production rate of particles scaled to 2.01E10 particles/s at 10 lm (Q = 2.01E05 ⁄ a  3). Column 10: Mass loss rate dM/dt = Q ⁄ d ⁄ 4/3phai3 (d is 1000 kg/m3); hai = 1.175 ⁄ alower. Column 11: Afq = Sd ⁄ Q(a)/2vd. Column 12: Column density n(q) = Q(a) ⁄ 2//4pvdq for q = 12 km and spacecraft at 100 km (2/ = 2.901). Column 13: Scattered light intensity is I = Fi ⁄ n(q) ⁄ Sd/4p (Fi at 1 lm and 1.3 AU is 432 W/m2 lm). Column 14: Equivalently calculated intensity from Afq I = Fi ⁄ Afq ⁄ 2 ⁄ //(q ⁄ 8p2). Column 15: Extinction s = n(p) ⁄ rgeom ⁄ Qext. Column 16: Production rate of particles with small particle cut off ap = 0.20 lm (scaled to 9.93E10 particles/s at 10 lm).

1.95E04 6.10E05 3.53E05 2.23E05 1.32E05 1.12E05 8.68E06

1.95E04 6.10E05 3.53E05 2.23E05 1.32E05 1.12E05 8.68E06

3.31E05 2.52E05 1.95E05 1.54E05 1.22E05 9.83E06 7.84E06

3.46E04

3.46E04

1.23E04

4.83E+13 1.97E+13 7.42E+12 2.64E+12 9.06E+11 3.02E+11 9.93E+10

0.33 0.42 0.50 0.57 0.62 0.65 0.67

0.353 0.143 0.098 0.070 0.045 0.040 0.032

4.67E04 1.89E04 1.30E04 9.28E05 5.95E05 5.31E05 4.29E05

3.765

0.782

1.03E03

U. Fink, G. Rinaldi / Icarus 257 (2015) 9–22

1 Size a (lm)

19

20

U. Fink, G. Rinaldi / Icarus 257 (2015) 9–22

distributions. The predicted Intensities should readily be observable by the imaging cameras of OSIRIS and, given sufficient integration time, by the spectrally resolved data from VIRTIS. As already pointed out in Fink and Rubin (2012), the dust/gas mass ratios are all less than one, and a value of about 0.10 seems a reasonable possibility. A graphical representation of the contribution to the scattered light intensity by the various particle sizes is given in Fig. 9. To avoid clutter we picked only three particle size distributions. The figure illustrates the complex nature of the contributions to the intensity once all the factors that go into its calculation are combined. Since the observable intensity is directly proportional to Afq, this figure is also a proxy for the contributions to Afq by particles of various sizes. While the particle size distribution in Fig. 8 have all been normalized to equal particle density for 10.0 lm particles, in Fig. 9 the contribution to the intensity and Afq for 10 lm particles is quite different for the three types of distributions because it is the Afq sum over all particle sizes that has been normalized to the same value. To clarify the steps in our calculations, which are difficult to explain and have led to some confusion, we produced Table 3. The table illustrates how we calculated quantities such as mass loss rate, Afq, the scattered light intensity, and the extinction. We show examples for two particle size distributions a simple power law with Q ðaÞ  a3 as employed by the DSMC and a distribution with a small particle cut off. We pick seven particle size bins from 1.0 to 10.0 lm as an excerpt from a larger table. Each decade in radius is divided into six logarithmically spaced bins. The size points are always placed at the lower limit of the bin and give the integral over the whole bin, e.g. 1.0–1.468 lm. The average weighted particle size hai in each bin is 1.175 times the lower value. This factor depends somewhat on the particular particle size distribution, but this dependence is minor. The largest size bin in the table goes from 10.0 lm to 14.68 lm. Each individual column that is not self-explanatory is explained in the legend of Table 3, and so is not repeated here. The equations used to calculate the various quantities are given in the various sections in the text. The DSMC model provides the particle outflow velocity and volume density as a function of distance from the nucleus. As shown in Fig. 4, once a roughly constant outflow speed is reached, at a distance of 10 km or larger, the calculation of the particle densities from the DSMC model and the analytical calculation converge. Thus we used the particle velocities and their densities at 10 km from the nucleus from the DSMC model CG_1.3_au_00, to calculate

100000

Particle number density ( #/m3)

10000

a particle production rate, Q ðaÞ, via the simple expression Q ðaÞ ¼ nv ol ðrÞ4pv d ðaÞr 2 . We then scaled the DSMC production rate to 2.01  1013 particles/s for 10 lm particles which yields an Afq of 2 m when the contributions of all the particles are summed. The column density is for a specific spacecraft geometry with the spacecraft at 100 km from the nucleus and looking at the coma with an impact parameter q = 12 km. As shown in the text, in the analytical case, the observed scattered light intensity can either be calculated from the column density or from the equivalent Afq. Thus both calculations are shown in columns 14 and 15 respectively. If the analytical case is not appropriate, the column density has to be integrated directly using a numerical model such as the DSMC. Columns 17–20 illustrate the case for a particle size distribution with a small particle cut-off. The sums at the bottom clearly only refer to the size bins included in the table and thus are illustrative only. In Table 4 we present data similar to that for Table 2 but for the encounter of the Rosetta spacecraft with 67P near 3.3 AU. The largest liftable size for this case is 6.81 lm. There is presently poor agreement with the larger particles reported by GIADA (Rotundi et al., 2015); a conflict which will have to be resolved. There are few observations to guide us as to the expected Afq at these larger heliocentric distances and so we used a likely value of 5 cm (Snodgrass et al., 2013). Recent early estimates from Rosetta (Rotundi et al., 2015) give a value of 8 ± 4 cm, however, the particle size distribution and model assumptions to arrive at that value appears to be substantially different from ours. The optical depth and Intensity values are clearly much lower and considerably longer integration times will be necessary to detect the dust. We note again that the geometry is for a spacecraft 100 km from the nucleus center looking through the coma with an impact parameter of 12 km, a phase angle of 40°. Moving the spacecraft closer to the nucleus will not increase the estimated expected light intensity very much. However changing the impact parameter to 2.1 km (100 m above the surface for an assumed nucleus radius of 2.0 km) will increase the observable intensity by about a factor of 15–20. Although we concentrated our calculation on the interpretation of the remote sensing instrument OSIRIS and VIRTIS our calculations can also be used to calculate quantities that are of interest to the in-situ instruments. Thus we present in Fig. 10 the estimated particle density 10 km from the nucleus center of 67P at 1.3 AU for three particle size distributions for particles from 0.10 to 100 lm in size. All three particle size distributions were normalized to yield

Estimated particle # Density for 67P at 1.3AU and 10 km from the nucleus center var. particle size distributions (norm. to Afρ=200 cm) can be scaled by 1/r2 for other nuleus distances

1000

Q~a-3 UF ap=0.20

Log-Norm rm=0.05,sigm=7.0

100 10 1 0.1 0.01 0.001 0.100

1.000

10.000

100.000

Particle radius μm Fig. 10. Estimated particle density for 67P near perihelion at 1.3 AU for three different size distributions. The particle densities are at 10 km from the center of the comet and are normalized to yield an Afq of 2.0 m. Within limits discussed in the text, they can be scaled to other distances by 1/r2.

21

U. Fink, G. Rinaldi / Icarus 257 (2015) 9–22 Table 4 Dust parameters for 67P/CG at 3.3 AU for various particle size distributions. Normalizing to an observed Afq value of 5 cm

DSMC values CG_3.3_au_00

Dust/gas 2

Afq (m)

Prod. rate 6.81 lm (particles/s)

Mass loss (kg/s)

Calc. inten. (W/m lm ster at k of 1.0 lm)

Optical depth, s

Afq (m)

Mass loss (kg/s)

DSMC 3.3_au_00

3.32

0.044

0.05

2.27E+07

6.63E04

1.03E05

8.46E06

0.022

(Q  a3) UF ap = 0.10 lm UF ap = 0.20 lm UF ap = 0.50 lm

0.689 0.385 0.13

0.0224 0.0176 0.0121

0.05 0.05 0.05

1.10E+08 1.96E+08 5.81E+08

1.63E03 2.29E03 4.65E03

1.02E05 1.03E05 1.03E05

8.46E06 8.46E06 8.46E06

0.053 0.074 0.152

Log-normal rm = 0.07, r = 4.7 rm = 0.05, r = 7.0 rm = 0.20, r = 6.5

0.344 0.15 0.09

0.0163 0.0107 0.0088

0.05 0.05 0.05

2.19E+08 5.03E+08 8.39E+08

2.37E03 3.57E03 4.89E03

1.03E05 1.02E05 1.03E05

8.46E06 8.46E06 8.46E06

0.077 0.116 0.159

Note a: Geometry for intensity calculation is shown in Fig. 3 (rs = 100 km; pimp = 12 km; coma = 100 km). Note b: Uses n = 1.70 + 0.02i for optical constants and a phase angle of 40° particle size distributions are normalized to 6.81 lm particles using DSMC model CG_3.3_au_00 as a guide. Note c: For dust/gas ratio, a gas output of 0.0307 kg/s (1.0E24 molecules/s) is used as given by DSMC model (down by a factor of 5000 over the case at 1.3 AU). Note d: Virtis NESR is given as 5  E04 W/m2 lm ster which yields a SNR  0.02 for 1 pixel in 1 s. Note e: For an impact parameter of 2.1 km and using DSMC integration, intensities will be a factor of 15 higher.

Particle Flux ( #/m2 s)

1.00E+08 1.00E+07

Estimated particle Flux for 67P at 1.3AU and 10 km from the nucleus center var. particle size distributions (norm. to Afρ=200 cm)

1.00E+06

can be scaled by 1/r 2 for other nuleus distances Q~a-3

1.00E+05

UF ap=0.20 LogNorm rm=0.05,sigm=7.0

1.00E+04 1.00E+03 1.00E+02 1.00E+01 1.00E+00 1.00E-01 1.00E-02 0.100

1.000

10.000

100.000

Parcle radius μm Fig. 11. Estimated particle flux looking down at the nadir from the spacecraft to the comet. The assumptions for this figure are similar to Fig. 10 and, for reasonable distances closer or farther from the nucleus, can be scaled by 1/r2. The particle fluxes can provide useful estimates for the in-situ particle detector experiments GIADA, MIDAS, and COSIMA.

the observed Afq value of 2 m. These densities can readily be scaled by 1/r2 for larger distances as well as smaller nuclear distances. As the line of sight approaches the nucleus though, the simple analytical representation can deviate considerably from the more accurate DSMC model densities as exemplified in Fig. 4. In Fig. 11, we calculate the flux of particles assuming that one looks straight down at the nucleus in #/m2 s for size bins from 0.10 to 100 lm. Again like Fig. 10, these numbers can be scaled to other distances. An estimate of the particle flux for several in-situ instruments can be made but is necessarily very rough, within at best an order of magnitude, considering the uncertainties in the particle size distribution, the distance of the spacecraft from the comet and the expected Afq. GIADA has a collecting area of 10  10 cm (102 m2) and is sensitive to particles roughly from 15 to 300 lm in size. Thus at perihelion of 67P and a spacecraft distance of 100 km from the nucleus, this instrument should detect roughly 0.14 (100 lm) particles/day. COSIMA is an instrument mainly meant to analyze the dust composition using secondary ion mass analysis. However it also has a camera called COSISCOPE with a viewing area of 1  1 cm (104 m2) and can

see particles from 15 to 200 lm in size. It should see 1.3 (10 lm) particles/day. MIDAS uses an atomic force microscope to image particles of nm to lm in size. With a collecting area of 108 m2 it should see 0.14 (1.0 lm) particles/day.

7. Summary In the present paper we provide a framework to take the output of the DSMC cometary dust acceleration and spatial distribution model and translate this into observable quantities of the remote sensing instruments on the Rosetta mission, i.e. the scattered light intensity along a spacecraft line of sight. We also relate these LOS intensity measurements to an equivalent Afq, since this quantity is commonly used as a measure of the dust output of a comet for ground based observations. This connects the spacecraft observations to ground based results. Ground based observations can thus provide the dust production of a comet as input for the DSMC model, and can be compared to close-up spacecraft results. The process of going from a dust particle density to an observable

22

U. Fink, G. Rinaldi / Icarus 257 (2015) 9–22

LOS intensity is not simple since it involves estimating the scattering efficiencies of the dust particles of various sizes and folding these into various particle size distributions which presently are not well known. Inverting our process will allow us to interpret the light scattering observations of the remote sensing instruments OSIRIS and VIRTIS and by incorporating data from the in-situ Rosetta dust experiments should allow us to deduce physical properties of the dust, within the confines of the model assumptions, such as the particle sizes, their size distribution, the dust/gas mass ratio in the coma and constraints on the dust composition. Acknowledgments This work was supported by NASA Rosetta Grant JPL-1270067, and by the Istituto Astrofisica Planetologia Spaziali (IAPS). References A’Hearn, M.F. et al., 1984. Comet Bowell 1980b. Astron. J. 89, 579–591. A’Hearn, Michael F. et al., 1995. The ensemble properties of comets: Results from narrowband photometry of 85 comets, 1976–1992. Icarus 118, 223–270. Bradley, J.P., 2003. Interplanetary dust particles. In: Davies, A.M. (Ed.), Treatise on Geochemistry: Meteorites, Comets and Planets, vol. 1. Elsevier, pp. 689–711. Colangeli, L. et al., 2007. The grain impact analyser and dust accumulator (GIADA) experiment for the Rosetta mission: Design, performances and first results. Space Sci. Rev. 128, 803–821. Coradini, A. et al., 2007. VIRTIS: An imaging spectrometer for the Rosetta mission. Space Sci. Rev. 128, 803–821. Dorschner, J. et al., 1995. Steps toward interstellar silicate mineralogy II. Study of Mg–Fe-silicate glasses of variable composition. Astron. Astrophys. 300, 503– 520. Edoh, O., 1983. Optical Properties of Carbon from the Far Infrared to the Far Ultraviolet. Ph.D. Thesis, University of Arizona, Tucson, AZ. Fink, U., 2009. A taxonomic survey of comet composition 1985–2004 using CCD spectroscopy. Icarus 201, 311–334. Fink, U., 2015. Considerations regarding the colors and low surface albedo of comets using the Hapke methodology. Icarus 252, 32–38. Fink, U., Rubin, M., 2012. The calculation of Afq and mass loss rate for comets. Icarus 221, 721–734. Fomenkova, M.N., Chang, S., Mukhin, L.M., 1994. Carbonaceous components in the Comet Halley dust. Geochim. Cosmochim. Acta 58, 4503–4512. Glassmeier, K.H. et al., 2007. The Rosetta mission: Flying towards the origin of the Solar System. Space Sci. Rev. 128, 1–21. Gulkis, S. et al., 2007. MIRO: Microwave instrument for Rosetta orbiter. Space Sci. Rev. 128, 561–597. Gulkis, S. et al., 2015. Subsurface properties and early activity of Comet 67P/ Churyumov–Gerasimenko. Science 347, 709–714. Hanner, M.S., 1983. The nature of cometary dust from remote sensing. In: Gombosi, T.I. (Ed.), Cometary Exploration, vol. 2. Akademiai Kiado, Budapest, pp. 1–22, 83. Hansen, J.E., Travis, L.D., 1974. Light scattering in planetary atmospheres. Space Sci. Rev. 16, 527–610.

Haser, L., 1957. Distribution d’intensite dans la tete d’une comete. Bull. Acad. R. Belqique, Classes Sci., Ser. 5 43, 740–750. ICES, 2014. Inner Coma Environment Simulator. Univ. of Michigan. . Keller, L.P., Messenger, S., 2011. On the origins of GEMS grains. Geochim. Cosmochim. Acta 75, 5356–5365. Keller, H.U. et al., 2007. OSIRIS, the scientific camera system onboard Rosetta. Space Sci. Rev. 128, 433–506. Khare, B.N. et al., 1984. Optical constants of organic tholins produced in a simulated titanian atmosphere – From soft X-ray to microwave frequencies. Icarus 60, 127–137. Kirchschlager, F., Wolf, S., 2013. Porous dust grains in debris disks. Astron. Astrophys. 552, A54. Kissel, J. et al., 2007. COSIMA high resolution time-of-flight secondary ion mass spectrometer for the analysis of cometary dust particles onboard Rosetta. Space Sci. Rev. 128, 823–867. Kolokolova, L., Mackowski, D., 2012. Polarization of light scattered by large aggregates. J. Quant. Spectrosc. Radiat. Trans. 113, 2567–3572. Neckel, H., Labs, D., 1984. The solar radiation between 3300 Å and 12,500 Å. Solar Phys. 90, 205–258. Nesvorny´, D. et al., 2010. Cometary origin of the zodiacal cloud and carbonaceous micrometeorites. Implications for hot debris disks. Astrophys. J. 713, 816–836. Pollack, J.B., Cuzzi, J.N., 1980. Scattering by non-spherical particles of size comparable to wavelength – A new semi-empirical theory and its application to tropospheric aerosols. J. Atmos. Sci. 37, 868–881. Protopapa, S. et al., 2014. Water ice and dust in the innermost coma of Comet 103P/ Hartley 2. Icarus 238, 191–204. Riedler, W. et al., 2007. MIDAS the micro-imaging dust analysis system for the Rosetta mission. Space Sci. Rev. 128, 869–904. Rotundi, A. et al., 2015. Dust measurements in the coma of Comet 67P/Churyumov– Gerasimenko inbound to the Sun. Science 347, 3905–3911. Schleicher, D.G., 2006. Compositional and physical results for Rosetta’s new target Comet 67P/Churyumov Gerasimenko from narrowband photometry and imaging. Icarus 181, 442–457. Snodgrass, C. et al., 2013. Beginning of activity in 67P/Churyumov–Gerasimenko and predictions for 2014–2015. Astron. Astrophys. 557, A33. Tenishev, V.M., Combi, M.R., Davidsson, B.J.R., 2008. A global kinetic model for cometary comae: The evolution of the coma of the Rosetta target Comet Churyumov–Gerasimenko throughout the mission. Astrophys. J. 685, 659–677. Tenishev, V.M., Combi, M.R., Rubin, M., 2011. Numerical simulation of dust in a cometary coma: Application to Comet 67P/Churyumov–Gerasimenko. Astrophys. J. 732, 104–120. Tozzi, G.P. et al., 2007. Dust observations of Comet 9P/Tempel 1 at the time of the Deep Impact. Astron. Astrophys. 476, 979–988. Tozzi, G.P. et al., 2011. Evolution of the dust coma in Comet 67P/Churyumov– Gerasimenko before the 2009 perihelion. Astron. Astrophys. 531, A54. Vasil’ev, A.V., 2006. Chislennoe modelirovanie intensivnosti mnogokratno rasseyannogo solnechnogo izlucheniya i proizvodnykh ot nee s uchetom sfericheskoi geometrii atmosfery (komp’yuternyi kod SCATRD). Vestn. S.Peterburgskogo un-ta. Cer. 4. Fizika, khimiya. Vyp. 3, S3–S14. Voshchinnikov, N.V., Il’in, V.B., Henning, Th., 2005. Modelling the optical properties of composite and porous interstellar grains. Astron. Astrophys. 429, 379–381. Zeidler, S. et al., 2011. Near-infrared absorption properties of oxygen-rich stardust analogs. The influence of coloring metal ions. Astron. Astrophys. 526, A68. Zolensky, M.E. et al., 2006. Mineralogy and petrology of Comet 81P/Wild 2 nucleus samples. Science 314, 1735–1739.