- Email: [email protected]
Tempel 1: Evolution of the surface
Icarus 245 (2015) 348–354
Contents lists available at ScienceDirect
Icarus journal homepage: www.elsevier.com/locate/icarus
Comet 9P/Tempel 1: Evolution of the surface Konrad J. Kossacki Institute of Geophysics of Warsaw University, Pasteura 7, 02-093 Warsaw, Poland
a r t i c l e
i n f o
Article history: Received 24 February 2014 Revised 24 September 2014 Accepted 25 September 2014 Available online 5 October 2014 Keywords: Comets Comets, nucleus Comets, composition Ices
a b s t r a c t Comet 9P/Tempel 1 was imaged during two consecutive perihelion passages. According to Thomas et al. (Thomas, et al. [2013]. Icarus 222, 453–466) one scarp located at 40°S receded up to 50 m. I attempted to use this observation to constrain local material properties of the nucleus. For this purpose I simulated recession of a model scarp due to sub-dust sublimation of ice. I have found, that the observed recession of the scarp can be reproduced when the dust mantle has high porosity about 0.8 and low thermal conductivity about 10 mW m1 K1, as found for artificial dust layers (Krause, M., Blum, J., Skorov, Yu.V., Trieloff, M. [2011]. Icarus 214, 286–296). Taking into account the temperature dependence of the sublimation coefficient of ice reduces the calculated recession rate of the scarp by about 40%. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction In 2011 Stardust-NExT (SN in the following) flew by Comet Tempel 1, imaged previously by the Deep Impact mission in 2005, significantly extending image coverage of the nucleus surface. Comparison of high resolution images gives unique opportunity to investigate changes of the surface of a comet nucleus between two consecutive perihelion passages. The Deep Impact mission (DI in the following) imaged approximately one third of the nucleus (Thomas et al., 2007), while the SN mission extended the coverage about two times. Hence, evolution of a one third of the surface can be investigated. According to Veverka et al. (2013) the only significant change in morphology occurred along one long scarp located at 40°S (Fig. 1). According to Thomas et al. (2013) ‘‘at least two crudely triangular salients evident in 2005 have disappeared by 2011’’. These structures had sizes about 50 m, and their lack in newer images suggests significant retreat of the scarp, locally up to 50 m. The scarp region located at ~30°N was identified as the source for most jets observed during SN mission (Veverka et al., 2013; Farnham et al., 2013). The scarp located at 40°S, where the erosional changes were identified was not the source of any jet traced back to the surface. However, this does not mean, that this region does not produce jets. As noted by Farnham et al. (2013) ‘‘the illumination is from low on the horizon as the Sun moves to the North, and so there is likely not enough energy to drive the activity’’. Correlation of the observed activity with topographic features suggests significant non-uniformity of the material composing nucleus. The most obvious reason for E-mail address: [email protected] http://dx.doi.org/10.1016/j.icarus.2014.09.044 0019-1035/Ó 2014 Elsevier Inc. All rights reserved.
locally enhanced activity seems presence of exposed ice. However, no spots of sufficiently high albedo have been found. Some bright spots were found in SN images, but they have albedo only about 25% higher than the average at their size scale as imaged by the Stardust NAVCAM (Li et al., 2013). For comparison, the bright patches in DI images, where ice was identified had albedo 2.5 time higher than average (Sunshine et al., 2006). Thus, it should be expected that the active scarp is covered by some dust. I attempted to derive properties of the material underlying the scarp from the observed recession rate. For this purpose I simulated recession of a model scarp under different conditions. 2. Model 2.1. Physical approach In this work I investigate local problem, evolution of one scarp on the surface of the comet nucleus. – The scarp is long and high, when compare to the thermal skin depth. Thus, is sufficient to calculate the heat and vapor diffusion in 1D, in the direction perpendicular to the local surface. – The scarp is long and rounded, so I performed simulations for different parts of the scarp, oriented in directions from North-East to North-West. – Albedo of the scarp is either the same as of the surrounding surface, or higher. – The comet nucleus has layered structure. The uppermost layer is a dust mantle of non-uniform thickness. On the horizontal surface, above and beneath the scarp the dust layer is thicker
349
K.J. Kossacki / Icarus 245 (2015) 348–354
than on the scarp itself. Beneath the dust is a layer composed of crystalline H2O grains with mineral cores. – Amorphous water ice and CO are present at large depth. – Geometry of the scarp is characterized by the inclination b relative to the horizontal plane and the local azimuth cs (for a vertical slope facing equator b = 90°, and cs = 0°).
Surface at t1 Surface at t2
Horizontal recession
Recession in the direction normal to the slope
Sketch of the scarp is shown in Fig. 2. 2.2. Mathematical formulation The applied model is an improved version of the program was used to simulate emission of water from Comet 9P/Tempel 1 due to sublimation beneath the dust mantle (Kossacki and Szutowicz, 2008). The most important feature of this model is including the sintering of ice grains, and the temperature dependent sublimation coefficient of ice. These effects are included neither in the ATPM model described in Rozitis and Green (2011), nor in the model Rosenberg and Prialnik (2010) which includes calculation of the heat transport within comet nucleus in 3D. In Kossacki and Szutowicz (2008) was shown, that the sublimation beneath the dust mantle of non-uniform thickness can match observed outgassing versus distance to perihelion. The method of simulating the outgassing rate described in Kossacki and Szutowicz (2008) was successfully applied in Finklenburg et al. (2014) to calculate the outgassing rate of the nucleus and simulate of the near nucleus coma of Comet 9P/Tempel 1. Recent versions of the model (Kossacki and Szutowicz, 2008) were described in. The model used in this work includes: – evolution of the nucleus cohesion due to sintering of ice grains (so called Kelvin effect which modifies grain-to-grain contact areas, but does not affect degree of compaction); – changes of porosity, due to sublimation/condensation in the medium; – crystallization of amorphous water ice when it is present; – sublimation of the CO ice (also explosive);
β Fig. 2. Geometry of the scarp.
– sublimation of H2O ice covered by the dust mantle, taking into account the temperature dependent sublimation coefficient (Kossacki et al., 1999; Gundlach et al., 2011; Kossacki and Markiewicz, 2013; Kossacki and Leliwa-Kopystynski, 2014). – diffusion of the vapor through the dust and the resulting recession of the surface; – illumination dependent on the local orientation of the surface. The basic equations are those for the surface temperature, the diffusion of heat, the sintering of ice grains, and the temperature dependence of the sublimation coefficient. The formulas, except description of the temperature dependence of the sublimation coefficient can be found in Kossacki and Szutowicz (2008). The sublimation coefficient was investigated by different authors, recently Gundlach et al. (2011) and Kossacki and LeliwaKopystynski (2014). Below is presented short description of the equations. The surface temperature of the dust layer can be calculated different ways. When the dust is cold enough to ignore the radiative heat transport, it is possible to consider the thermal conductivity of dust as constant. This is reasonable when the temperature remains below 200 K, or pores are of sub-micrometer size. However, the surface of a thick, very fluffy dust layer can warm-up to a temperature above 250 K. Thus, the model takes into account radiation within pores in the dust layer. In general case equations for the surface temperature, and the temperature beneath the dust layer are:
Sc R2h
ð1 AÞ maxðcos a; 0Þ rd T 4s þ kd rT ¼ 0;
ð1Þ
and
kd rTjd þ krTji H F s ¼ 0:
Fig. 1. Images of the scarp: Deep Impact (upper panel), and Stardust-NExT (lower panel). Marked (black arrow) is the most prominent of the structure visible on the scarp only in the DI image. Courtesy: NASA/JPL-Caltech/Cornell.
ð2Þ
The parameter Sc denotes the solar constant, A is the surface albedo, Rh is the actual heliocentric distance in AU, a is the illumination angle (between the direction to the Sun and the local normal to the surface), r is the Stefan–Boltzmann constant, d is the emissivity of the dust. The parameter kd denotes the thermal conductivity of dust, k is the thermal conductivity of granular ice-dust material beneath the dust layer, F s is the flux of molecules subliming and further diffusing into space through the dust mantle, and H is the latent heat of sublimation/condensation,. The terms in Eq. (2) are: heat flux conducted through the dust mantle, heat flux conducted within underlying ice-dust medium, and the energy losses due to sublimation and subsequent diffusion of water molecules to space. The temperature gradients are positive when the temperature increases versus depth. Both thermal conductivity, kd , and k, are temperature dependent. The respective equations are:
kd ¼ kd0 þ r p rT 3 ;
ð3Þ
350
K.J. Kossacki / Icarus 245 (2015) 348–354 1.2
1=2 32l dp k ¼ kg hð1 wÞ2=3 þ r p w H sat : 9pRT dT
1.1
ð4Þ
Here kg is the thermal conductivity of ice grains with mineral cores, and h is the Hertz factor. The terms in the right hand side of Eq. (4) describe: heat transport in the solid matrix of grains, and heat transport in the pores. For greater details see Kömle and Steiner (1992) and Kossacki et al. (2006). The Hertz factor is defined as the ratio between the contact area of adjacent grains and the cross section of a grain. The value of h evolves due to sintering of grains. The rate of sintering depends on the temperature, the radii of grains rg and on the actual value of the Hertz factor h. In the case of coarse-grained material sintering is slow and h is always close to its initial value, but the sintering of small grains can be very fast, depending on the temperature. There are known several sintering mechanisms. For H2O ice under cometary conditions most efficient is so called Kelvin effect: transport of molecules from adjacent grains onto the connecting neck. It is important, that the considered sintering mechanism does not lead to any changes of porosity. The radii of grains also remain unchanged. However, the grains change their shapes. The rate of increase of h is
dh 2rn X2 cpS 2 2 rg rn 1 ; ¼ 2 þ dt rn r g 2plR T 12 R T rg r 2n g g
h¼
0:5 rg rn
0:5 r d ð1 v d Þ 32l b1 expðb2 =TÞ; 2 Dd s 9pRg T
1 a1
þ
1 T a2 p ; tanh a3 tan p a1 273 a2 2
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 180
190
200
210
220
230
240
250
260
270
Temperature [K] Fig. 3. Sublimation coefficient of H2O ice versus temperature.
(see Fig. 2). Thus, Eq. (8) needs to be converted. When the horizontal surfaces above, and below the slope are covered by a thick dust prohibiting sublimation Eq. (8) becomes
dx Fs ¼ ; dt .i v i sinðbÞ
ð9Þ
2.3. Parameters
ð6Þ
ð7Þ
where a1 ¼ 2:33, a2 ¼ 138:72 and a3 ¼ 9:3. Equation for the recession rate of the scarp in the direction perpendicular to the slope is
dl 1 ¼ F s .1 i vi ; dt
0.9
where b is the inclination of a slope to the horizontal surface.
where as is the temperature dependent sublimation coefficient, and the remaining parameters are: b1 = 3.56 1012 Pa, and b2 = 6141.667 K (Fanale and Salvail, 1984). The sublimation coefficient was investigated by various authors. Recent measurements dealing with the sublimation coefficient of H2O ice were described in Gundlach et al. (2011) and Kossacki and Leliwa-Kopystynski (2014). According to both works as = 1 at temperatures lower than 195 K, and as = 0.15, at temperatures higher than 220 K. At middle range the temperature dependence of as determined by Kossacki and Leliwa-Kopystynski (2014) is somewhat steeper. The profiles of as versus temperature are shown in Fig. 3. In this work I calculate as as described by Kossacki and Leliwa-Kopystynski (2014). Thus,
as ¼ 1
Kossacki et al. 1999 Gundlach et al. 2011 Kossacki and Leliwa-Kopystynski, 2014
1
ð5Þ
(Swinkels and Ashby, 1980; Kossacki et al., 1994). The symbol rn denotes the radius of the grain-to-grain contact area (radius of the sinter neck), Rg is the universal gas constant, and l is the molar mass of water. The dimensionless variable S describes fraction of the surface of a grain from which the material is removed (source area) and transported onto the sinter neck (sink area). The parameters c and X are the surface energy and molar volume, respectively. For water ice they are X2 and c ¼ 4 1011 m4 mole2 J. The flux F s of molecules subliming beneath a dust layer thicker than ten layers of dust grains can be calculated as
F s ¼ as
Sublimation coefficient α (T)
and
ð8Þ
where .i is the bulk density of water ice, and v i is the volume fraction of ice in the material. The available observations are sufficient to determine only movement of the scarp in the horizontal direction
The model nucleus is orbiting like Comet 9P/Tempel 1. Thus, eccentricity e ¼ 0:51359, semi-major axis a = 3.13592 AU. Orientation of the rotation axis is also the same, so I = 16.4°, and U = 346.8°. The latter is solar longitude at perihelion. The considered scarp is located at 40°S. However, non-spherical shape of the nucleus makes the illumination of the scarp different than can be expected for the given latitude. Moreover, the scarp is significantly rounded and has some latitudinal extent. Thus, I considered two latitudes: 40°S, and 60°S. The height and inclination of the slope are not well determined. Vincent et al. (2013) considered vertical slope, about 100 m high. However, vertical slope is unlikely to be covered by a dust mantle, but an exposed ice was not detected. Thus, I consider a slope inclined by the angle b = 45–67.5°. Albedo of the surface is A = 0, and A ¼ 0:2. The lower value is close to the average bond albedo Ab ¼ 0:015 (Li et al., 2013). According to the authors the brightest spots identified in SN images had albedo only 25% higher than the average one. However, some very small patches may have higher albedo. The highest value of albedo considered in this work A ¼ 0:2 should bracket all possible values. The dust mantle is characterized by three parameters: the porosity wd , the thermal conductivity at low temperature kd0 and the thickness of dust layer Dd . Non of them is well known, but some estimates are available. According to Krause et al. (2011) experimentally created dust layers. Extremely porous layers had the thermal conductivity as low as kd = 2 mW m1 K1, while moderately porous dust layers, wd ¼ 0:71—0:84, had thermal conductivity kd ¼ 2—8 mW m1 K1. Thermal conductivity can be also derived from the observed oscillations of the surface temperature. According to Groussin et al. (2007) kd 6 36 mW m1 K1 at .d = 1000 kg m3. However, values of the thermal inertia found by Davidsson et al. (2009) indicate, that the uppermost can be much more conductive. The dust layer of low thermal conductivity prohibits intensive sub-dust sublimation of ice. Thus, a fast receding surface is probably not covered by a thick, low conducting dust. I consider kd0 = 10–100 mW m1 K1, and porosity wd ¼ 0:8. The largest considered value of kd0 is not supported by laboratory experiments and is probably an overestimate. Thickness of the dust
351
K.J. Kossacki / Icarus 245 (2015) 348–354
3. Results In this section is described, how recession of the scarp depends on albedo, geometry of the slope, and material parameters. In our work change of the position means horizontal movement of the middle of the considered slope. The nucleus of Comet 9P/Tempel 1 is not spherical. Thus, an area located at the geocentric latitude 40°S is illuminated at different angles than areas at the same latitude, but on a spherical object. To determine significance of this effect I simulated recession of a slope located at latitudes within the range 40–60°S. The exact value of the latitude appeared to be insignificant. This is because at perihelion, close to the local noon the scarp inclined by a high angle is illuminated almost perpendicularly (cosine of the illumination angle close to unity). In Fig. 4 is shown position of the scarp versus time measured from aphelion. The slope is inclined toward equator, i.e. cs ¼ 0 , by b ¼ 45—67:5 . The albedo is 0, and 0.2. The remaining parameters are: Dd ¼ 0:25 cm, kd = 100 mW m1 K1, r d ¼ rg ¼ 50 lm, h0 = 0.1, and T 0 = 40 K. In the case of low albedo, A ¼ 0, during one orbital period the scarp moves in the horizontal direction by 25–40 m depending on the inclination of the scarp. Increase of the albedo to 0.2 reduces the recession rate by about 40%.
0
Recession of the scarp [m]
layer may increase, decrease, or remain constant. Kossacki and Szutowicz (2008) have found, that reproducing the observed water production versus time and the surface temperature in the subsolar point was possible only when the dust layer had constant thickness. Thus, in this work thickness of the dust remains constant, surface erosion is compensated by the thickening from bottom. Beneath the dust mantle nucleus is composed of ice grains with mineral cores. The characteristic radius of grains is r g = 0.5–50 lm. The effective pore radius rp is in this work equal to r g . Initial value of the Hertz factor is within the range 0.001–0.1. Porosity of the material is determined by the volume fractions: of water ice, v i , and of mineral cores, v m . The volume v m is constant, but v i evolves due to sublimation/condensation of ice. However, the changes of v i are small. I consider two cases: v m ¼ v i;0 ¼ 0:15, and v m ¼ v i;0 ¼ 0:25. The corresponding porosity w is: 0.7, or 0.5. Mineral cores of the grains are agglomerates of dust and have density 1000 kg m3. The model parameters are summarized in Table 1.
o
β = 67.5 , A = 0
-10
β = 45 o ,
A=0
β = 45 o ,
A = 0.2
-20
β = 45 o
-30
λ d = 100 mW m-1 K -1 Δ d = 0.25 cm, ψ d = 0.8
-40
r d = rg = 50 μ m, h 0 = 0.1 ψ = 0.7
-50 0
0.2
0.4
0.6
0.8
1
Time [orbital periods] Fig. 4. Position of the scarp versus time measured from aphelion, when the slope is inclined toward equator by 45°, and 67.5°. The albedo A is: 0, and 0.2. Values of the remaining parameters are in the legend.
In Fig. 5 are shown results obtained, when the slope is inclined in different directions (different values of cs , corresponding to different parts of the rounded scarp). The inclination angle b is 45°. Material parameters are the same as in the case of Fig. 4. The recession rate is highest when the slope is inclined toward equator. However, for all orientations within the investigated range ±45° the differences are smaller than 15%. Fig. 6 is aimed to show significance of the dust thickness Dd and of the thermal conductivity kd . It should be emphasized, that kd is temperature dependent. Constant is only kd0 , the thermal conductivity at low temperature. It can be seen, that the dust thickness has significant influence on the recession of the scarp. When kd0 = 100 mW m1 K1, the recession rate is: 40 m (Dd = 0.25 cm), 34 m (Dd = 0.5 cm), and 27 m (Dd = 1 cm). When kd0 = 10 mW m1 K1 the recession rate is: 23 m (Dd = 0.25 cm), 11 m (Dd = 0.5 cm), and 4 m (Dd = 1 cm). Thus, the decrease of kd0 by an order of magnitude reduces the recession rate by about 60–80% depending on the dust thickness. The effect is strongest when the dust mantle is thick. The rate of sublimation is determined by the energy balance at the interface between the dust mantle and the underlying material (Eq. (2)). A decrease of the dust thermal conductivity by a factor ten should reduce energy available for sublimation by the factor
Table 1 Parameters. Parameter
Symbol
Albedo Emissivity Thermal conductivity of the dust mantle at low temperature Porosity of the dust mantle ðwd ¼ 1 v d Þ Pore radius in the dust mantle Density of the mineral cores of dust-ice grains (agglomerates very fine dust particles) Thermal conductivity of the mineral cores of dust-ice grains Initial volume fraction of H2O ice Volume fraction of the mineral component Tortuosity
A
Grain/pore radius in the ice-dust medium Initial temperature Initial Hertz factor
rg ¼ rp T0 ho
[lm] [K]
b
[deg.]
45–67.5
I
[deg.] [deg.] [AU]
16.4 346.8 3.135922 0.51359 5.52 40.6
Other parameters Inclination of the scarp The spin axis orientation Obliquity Solar longitude at perihelion Semi-major axis Eccentricity Orbital period Rotational period
Unit
kd0 wd rd
.m km
[mW m1 K1] [lm] [kg m3] [mW m1 K1]
v i0 vm s
U a e P p
[year] [h]
Value 0, 0.2 0.9 10, 100 0.8 0.5, 5, 50 1000 1000 0.15, 0.25 0.15, 0.25 pffiffiffi 2 0.5–50 40, 80 0.001, 0.1
352
K.J. Kossacki / Icarus 245 (2015) 348–354 320
0
β = 45
A = 0, ψ = 0.7
-10
Temperature at local noon [K]
Recession of the scarp [m]
β = 45 o r d = rg = 50 μ m, h 0 = 0.1 λ d = 100 mW m -1 K -1 Δ d = 0.25 cm
-20
-30
-40
γ = 315
o
γ = 180
o
γ = 45
o
-50 0
0.2
0.4
0.6
0.8
A=0
280
r d = r g = 50 μ m
Δ d = 0.25 cm
λ d0 = 10 mW m
-1
K
-1
T surface T sub-dust
260 240 220 200 180 0.3
1
λ d0 = 100 mW m -1 K -1 T surface T sub-dust
o
300
0.35
0.4
Time [orbital periods]
0.45
0.5
0.55
0.6
0.65
Time [orbital periods]
Fig. 5. Position of the scarp versus time measured from aphelion, when the slope is inclined by 45° toward equator and toward directions within range ±45°. Values of the remaining parameters are in the legend.
Fig. 7. Temperature at the surface and beneath the dust mantle versus time. The dust mantle has thickness Dd ¼ 0:25 and kd0 (constant term in Eq. (3)) is either 10, or 100 mW m1 K1. The surface is inclined by 45° toward equator. Values of the remaining parameters are in legend.
0
-10
o
A = 0, ψ = 0.7, h 0 = 0.7
Recession of the scarp [m]
Recession of the scarp [m]
0 β = 45
r d = r g = 50 μ m
-20
-30 λ d0 = 100 mW m -1 K -1 Δ d = 1.00 cm Δ d = 0.50 cm Δ d = 0.25 cm
-40
λ d0 = 10 mW m-1 K -1
-50 0
0.2
Δ d = 1.00 cm Δ d = 0.50 cm Δ d = 0.25 cm
0.4
0.6
0.8
ψ = 0.7, h 0 = 0.1
-20
-30 rd = 0.5 μ m, rg = 0.5 μ m rd = 0.5 μ m, rg = 5 μ m rd = 5 μ m, rd = 5 μ m,
-40
rg = 5 μ m r g = 50 μ m
-50
Time [orbital periods]
0
0.2
0.4
0.6
0.8
1
Time [orbital periods] Fig. 8. Position of the scarp versus time measured from aphelion. The initial temperature T 0 = 40 K. The surface is inclined by 45°. Values of the remaining parameters are in the legends.
0 β = 45
Recession of the scarp [m]
ten. However, slower sublimation indicates less efficient cooling due to the latent heat exchange during sublimation. This effect reduces decrease of the sub-dust temperature resulting from the decrease of the dust conductivity. The change of the sub-dust temperature should modify the temperature gradient in the dust layer and the flux of energy conducted downward from the warm dust surface. In addition, an increase of the temperature beneath the dust enhances outflow of heat downward to the cold interior of the comet. Finally, after decrease of the conductivity by a factor ten the sublimation flux should decrease, but less than ten times. However, the sublimation rate exponentially depends on the temperature and at low temperature becomes negligible. Changes of the dust thickness have similar influence on the sub-dust sublimation, as changes of the dust thermal conductivity. In Fig. 7 are shown profiles of the surface temperature and the sub-dust temperature versus time. Plotted are the values of the diurnal maximums of the temperature. The dust mantle has thickness Dd ¼ 0:25 and kd0 is either 10, or 100 mW m1 K1. The surface temperature significantly depends on kd0 , while the sub-dust temperature moderately. Fig. 8 is aimed to present, how granulation of the dust mantle and of the underlying material affect recession of the scarp. The dust is thin, Dd ¼ 0:25 mm, and the sizes of pores, rd and rg , are independent one from the other. Comparison of the profiles corresponding to a fixed rd , and different rg indicates that granulation of the material beneath the dust layer is significant only when r d has moderate value 5 lm. The considered change of the radii r d of pores
A = 0, Δ d = 0.25 cm
rd = 50 μ m, rg = 5 μ m rd = 50 μ m, rg = 50 μ m
1
Fig. 6. Position of the scarp versus time measured from aphelion. The dust mantle has thickness Dd ¼ 0:25—1 cm, and kd0 (constant term in Eq. (3)) is either 10, or 100 mW m1 K1. The surface is inclined by 45° toward equator. Values of the remaining parameters are in legend.
β = 45 o
-10
o
A = 0, Δ d = 0.25 cm ψ = 0.7
-10
T
0
= 80 K
-20
-30
-40
r d = r g = 50 μ m, h
0
= 0.1
r d = r g = 50 μ m, h
0
= 0.001
-50 0
0.2
0.4
0.6
0.8
1
Time [orbital periods] Fig. 9. Position of the scarp versus time measured from aphelion, when the initial temperature is 80 K. The surface is inclined by 45°. Values of the remaining parameters are in the legend.
in the dust mantle by two orders of magnitude from 0.5 lm to 50 lm results in a change of the recession rate by the factor four. I investigated also influence of the initial temperature. In Fig. 9 is plotted position of the surface versus time, when T 0 = 80 K. The remaining parameters are: Dd = 0.25 cm, r d ¼ rg ¼ 50 lm, h0 = 0.1, and b ¼ 45 . It can be seen, that the initial value of the Hertz factor, h0 , has minor importance. This is due to negligible influence of the granulation of the material beneath the dust mantle.
K.J. Kossacki / Icarus 245 (2015) 348–354 0
β = 45 o
Recession of the scarp [m]
A = 0, r d = 50 μ m λ d = 10 mW m -1 K -1
-10
Δ d = 0.25 cm ψ = 0.7, h 0 = 0.1
-20
-30
α s (T) (Gundlach et al, 2011)
-40
α s (T) (Kossacki and Leliwa-Kopystynski 2014) αs = 1
-50 0
0.2
0.4
0.6
0.8
1
Time [orbital periods] Fig. 10. Position of the scarp versus time measured from aphelion, when the sublimation coefficient of ice is either unity, or depends on the temperature. The surface is inclined by 45°. Values of the remaining parameters are in the legend.
Comparison of the profiles shown in Fig. 9 with the corresponding profile shown in Fig. 8 indicates, that the surface of a the warm nucleus recedes faster than predicted for a cold one. However, the difference is smaller than 10% despite the change of T 0 from 40 to 80 K. I investigated also, how the temperature dependence of the sublimation coefficient as affects the calculated recession rate of the scarp. In Fig. 10 are shown results obtained, when the sublimation coefficient is either unity, or is temperature dependent. It can be seen, that taking into account the temperature dependence of the sublimation coefficient can reduce the calculated recession rate by about 40%. 4. Summary and discussion I attempted to derive material properties of the nucleus beneath the scarp from the observed recession of one scarp. According to observations the scarp and the surrounding surface are entirely covered by dust (Thomas et al., 2013). Thus, I calculated recession of the model scarp due to sub-dust sublimation of ice. I investigated influence of: – local inclination and azimuth of the slope, – thickness and thermal conductivity of the dust layer, – granulation of the dust mantle and of the material beneath the dust, – initial temperature and cohesivity of the nucleus, expressed by the Hertz factor, well as – albedo of the surface. The scarp is rounded, almost half-circular. Thus, it could be expected that different parts of the scarp recede with different rates. The simulations show, that the local azimuth of the slope has very low significance. Thus, the scarp should retain the current shape. Granulation of the material composing the dust mantle can be the same, as of underlying ice-dust material, or different. I have found, that the recession rate of the considered scarp is sensitive only to the granulation of the dust mantle. When the slope is inclined by 45°, and the dust mantle has high thermal conductivity kd0 = 100 mW m1 K1 the scarp moves by: 14–20 m when Dd ¼ 0:25 cm, and rd ¼ 5 lm, or by 40–42 m when Dd ¼ 0:25 cm and r d ¼ 50 lm. When kd0 = 10 mW m1 K1, Dd ¼ 0:25 cm, and r d ¼ 50 lm the scarp can recede by 22 m. The lower value of kd0 is probably more realistic, because it is consistent with high porosity (Krause et al., 2011). Thus, taking into account approximate determination of the recession of the scarp based on the observa-
353
tions (up to 50 m according to Thomas et al. (2013)). I estimated that if the dust thickness Dd ¼ 2:5 mm, the characteristic radius of pores in the dust layer is about 50 lm (1/50 of the dust thickness). It should be noted, that low value of kd0 does not mean, that kd is always low. At high temperature kd is determined by the radiative heat transport. In this work thickness Dd of the dust covering the slope was within the range 2.5–10 mm. When the dust layer was 10 mm thick, the scarp receded less than 30 m in the case of high conducting dust, kd0 = 100 mW m1 K1, but in the case of more realistic kd0 = 10 mW m1 K1 recession was negligible. Very small dust thickness, on average about 1 mm, could be also possible. However, in such case ice should be locally exposed. Taking into account the temperature dependent sublimation coefficient significantly reduces calculated recession rate of the scarp. The difference depends on the considered formulation of the temperature dependence of the sublimation coefficient. It is about 40% when the formula Gundlach et al. (2011) is used, and 10% when the formula Kossacki and Leliwa-Kopystynski (2014) is applied. Finally, the performed simulations indicate, that the scarp is covered by a dust layer few millimeters thick, with characteristic radius of pores smaller than 100 lm. Acknowledgment Thank you the anonymous reviewers for their valuable suggestions. References Davidsson, B.R., Gutirrez, P.J., Rickman, H., 2009. Physical properties of morphological units on Comet 9P/Tempel 1 derived from near-IR deep impact spectra. Icarus 201, 335–357. Fanale, F.P., Salvail, J.R., 1984. An idealized short-period comet model: Surface insolation, H2O flux, and mantle evolution. Icarus 60, 476–511. Farnham, T.L., Bodewits, D., Li, J.-Y., Thomas, P., Belton, M.J.S., 2013. Connections between the jet activity and surface features on Comet 9P/Tempel 1. Icarus 222, 540–549. Finklenburg, S., Thomas, N., Su, C.C., Wu, J.-S., 2014. The spatial distribution of water in the inner coma of Comet 9P/Tempel 1: Comparison between models and observations. Icarus 236, 9–23. Groussin, O., A’Hearn, M.F., Li, J.Y., Thomas, P.C., Sunshine, J.M., Lisse, C.M., Meech, K.J., Farnham, T.L., Feaga, L.M., Delamere, W.A., 2007. Surface temperature of the nucleus of Comet 9P/Tempel 1. Icarus 187, 16–25. Gundlach, B., Skorov, Y.V., Blum, J., 2011. Outgassing of icy bodies in the Solar System – I. The sublimation of hexagonal water ice through dust layers. Icarus 213, 710–719. Kömle, N.I., Steiner, G., 1992. Temperature evolution of porous ice samples covered by a dust mantle. Icarus 96, 204–212. Kossacki, K.J., Leliwa-Kopystynski, J., 2014. Temperature dependence of the sublimation rate of water ice: Influence of impurities. Icarus 233, 101–105. Kossacki, K.J., Markiewicz, W.J., 2013. Comet 67P/CG: Influence of the sublimation coefficient on the temperature and outgassing. Icarus 224, 172–177. Kossacki, K.J., Szutowicz, S., 2008. Comet 9P/Tempel 1: Sublimation beneath the dust cover. Icarus 195, 705–724. Kossacki, K.J., Norbert, I., Kömle, N.I., Kargl, G., Steiner, G., 1994. The influence of grain sintering on the thermoconductivity of porous ice. Planet. Space Sci. 42, 383–389. Kossacki, K.J., Markiewicz, W.J., Skorov, Y., Koemle, N.I., 1999. Sublimation coefficient of water ice under simulated cometary-like conditions. Planet. Space Sci. 47, 1521–1530. Kossacki, K.J., Leliwa-Kopystynski, J., Szutowicz, S., 2006. Evolution of depressions on Comet 67P/Churyumov Gerasimenko: Role of ice metamorphism. Icarus 184, 221–238. Krause, M., Blum, J., Skorov, Yu.V., Trieloff, M., 2011. Thermal conductivity measurements of porous dust aggregates: I. Technique, model and first results. Icarus 214, 286–296. Li, J.-Y., A’Hearn, M.F., Belton, M.J.S., Farnham, T.L., Klaasen, K.P., Sunshine, J.M., Thomas, P.C., Veverka, J., 2013. Photometry of the nucleus of Comet 9P/Tempel 1 from Stardust-NExT flyby and the implications. Icarus 222, 467–476. Rosenberg, E.D., Prialnik, D., 2010. The effect of internal inhomogeneity on the activity of comet nuclei – Application to Comet 67P/Churyumov Gerasimenko. Icarus 209, 753–765. Rozitis, B., Green, S.F., 2011. Directional characteristics of thermal-infrared beaming from atmosphereless planetary surfaces – A new thermophysical model. MNRAS 415, 2042–2062.
354
K.J. Kossacki / Icarus 245 (2015) 348–354
Sunshine, J.M. et al., 2006. Exposed water ice deposits on the surface of Comet Tempel 1. Science 311, 1453–1455. Swinkels, F.B., Ashby, M.F., 1980. A second report on sintering diagrams. Acta Metall. 29, 259–281. Thomas, Peter C., Veverka, J., Belton, Michael J.S., Hidy, Alan, A’Hearn, Michael F., Farnham, T.L., Groussi, Olivier, Li, Jian-Yang, McFadden, Lucy A., Sunshine, Jessica, Wellnitz, Dennis, Lisse, Carey, Schultz, Peter, Meech, Karen J., Alan Delamere, W., 2007. The shape, topography, and Geology of Tempel 1 from deep impact observations. Icarus 187 (1), 4–15. Thomas, P., A’Hearn, M., Belton, M.J.S., Brownlee, D., Carcich, B., Hermalyn, B., Klaasen, K., Sackett, S., Schultz, P.H., Veverka, J., Bhaskaran, S., Bodewits, D.,
Chesley, S., Clark, B., Farnham, T., Groussin, O., Harris, A., Kissel, J., Li, J.-Y., Meech, K., Melosh, J., et al., 2013. The nucleus of Comet 9P/Tempel 1: Shape and geology from two flybys. Icarus 222, 453–466. Veverka, J., Klaasen, K., A’Hearn, M., Belton, M., Brownlee, D., Chesley, S., Clark, B., Economou, T., Farquhar, R., Green, S.F., Groussin, O., Harris, A., Kissel, J., Li, J.-Y., Meech, K., Melosh, J., Richardson, J., Schultz, P., Silen, J., Sunshine, J., Thomas, P., Bhaskaran, S., Bodewits, D., Carcich, B., Cheuvront, A., Farnham, T., Sackett, S., Wellnitz, D., Wolf, A., 2013. Return to Comet Tempel 1: Overview of StardustNExT results. Icarus 222, 424–435. Vincent, J.B., Sierks, H., Rose, M., 2013. Jet activity on the cliffs of Comet 9P/Tempel 1. EPSC Abstracts 8, EPSC2013-429-1.
Contents lists available at ScienceDirect
Icarus journal homepage: www.elsevier.com/locate/icarus
Comet 9P/Tempel 1: Evolution of the surface Konrad J. Kossacki Institute of Geophysics of Warsaw University, Pasteura 7, 02-093 Warsaw, Poland
a r t i c l e
i n f o
Article history: Received 24 February 2014 Revised 24 September 2014 Accepted 25 September 2014 Available online 5 October 2014 Keywords: Comets Comets, nucleus Comets, composition Ices
a b s t r a c t Comet 9P/Tempel 1 was imaged during two consecutive perihelion passages. According to Thomas et al. (Thomas, et al. [2013]. Icarus 222, 453–466) one scarp located at 40°S receded up to 50 m. I attempted to use this observation to constrain local material properties of the nucleus. For this purpose I simulated recession of a model scarp due to sub-dust sublimation of ice. I have found, that the observed recession of the scarp can be reproduced when the dust mantle has high porosity about 0.8 and low thermal conductivity about 10 mW m1 K1, as found for artificial dust layers (Krause, M., Blum, J., Skorov, Yu.V., Trieloff, M. [2011]. Icarus 214, 286–296). Taking into account the temperature dependence of the sublimation coefficient of ice reduces the calculated recession rate of the scarp by about 40%. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction In 2011 Stardust-NExT (SN in the following) flew by Comet Tempel 1, imaged previously by the Deep Impact mission in 2005, significantly extending image coverage of the nucleus surface. Comparison of high resolution images gives unique opportunity to investigate changes of the surface of a comet nucleus between two consecutive perihelion passages. The Deep Impact mission (DI in the following) imaged approximately one third of the nucleus (Thomas et al., 2007), while the SN mission extended the coverage about two times. Hence, evolution of a one third of the surface can be investigated. According to Veverka et al. (2013) the only significant change in morphology occurred along one long scarp located at 40°S (Fig. 1). According to Thomas et al. (2013) ‘‘at least two crudely triangular salients evident in 2005 have disappeared by 2011’’. These structures had sizes about 50 m, and their lack in newer images suggests significant retreat of the scarp, locally up to 50 m. The scarp region located at ~30°N was identified as the source for most jets observed during SN mission (Veverka et al., 2013; Farnham et al., 2013). The scarp located at 40°S, where the erosional changes were identified was not the source of any jet traced back to the surface. However, this does not mean, that this region does not produce jets. As noted by Farnham et al. (2013) ‘‘the illumination is from low on the horizon as the Sun moves to the North, and so there is likely not enough energy to drive the activity’’. Correlation of the observed activity with topographic features suggests significant non-uniformity of the material composing nucleus. The most obvious reason for E-mail address: [email protected] http://dx.doi.org/10.1016/j.icarus.2014.09.044 0019-1035/Ó 2014 Elsevier Inc. All rights reserved.
locally enhanced activity seems presence of exposed ice. However, no spots of sufficiently high albedo have been found. Some bright spots were found in SN images, but they have albedo only about 25% higher than the average at their size scale as imaged by the Stardust NAVCAM (Li et al., 2013). For comparison, the bright patches in DI images, where ice was identified had albedo 2.5 time higher than average (Sunshine et al., 2006). Thus, it should be expected that the active scarp is covered by some dust. I attempted to derive properties of the material underlying the scarp from the observed recession rate. For this purpose I simulated recession of a model scarp under different conditions. 2. Model 2.1. Physical approach In this work I investigate local problem, evolution of one scarp on the surface of the comet nucleus. – The scarp is long and high, when compare to the thermal skin depth. Thus, is sufficient to calculate the heat and vapor diffusion in 1D, in the direction perpendicular to the local surface. – The scarp is long and rounded, so I performed simulations for different parts of the scarp, oriented in directions from North-East to North-West. – Albedo of the scarp is either the same as of the surrounding surface, or higher. – The comet nucleus has layered structure. The uppermost layer is a dust mantle of non-uniform thickness. On the horizontal surface, above and beneath the scarp the dust layer is thicker
349
K.J. Kossacki / Icarus 245 (2015) 348–354
than on the scarp itself. Beneath the dust is a layer composed of crystalline H2O grains with mineral cores. – Amorphous water ice and CO are present at large depth. – Geometry of the scarp is characterized by the inclination b relative to the horizontal plane and the local azimuth cs (for a vertical slope facing equator b = 90°, and cs = 0°).
Surface at t1 Surface at t2
Horizontal recession
Recession in the direction normal to the slope
Sketch of the scarp is shown in Fig. 2. 2.2. Mathematical formulation The applied model is an improved version of the program was used to simulate emission of water from Comet 9P/Tempel 1 due to sublimation beneath the dust mantle (Kossacki and Szutowicz, 2008). The most important feature of this model is including the sintering of ice grains, and the temperature dependent sublimation coefficient of ice. These effects are included neither in the ATPM model described in Rozitis and Green (2011), nor in the model Rosenberg and Prialnik (2010) which includes calculation of the heat transport within comet nucleus in 3D. In Kossacki and Szutowicz (2008) was shown, that the sublimation beneath the dust mantle of non-uniform thickness can match observed outgassing versus distance to perihelion. The method of simulating the outgassing rate described in Kossacki and Szutowicz (2008) was successfully applied in Finklenburg et al. (2014) to calculate the outgassing rate of the nucleus and simulate of the near nucleus coma of Comet 9P/Tempel 1. Recent versions of the model (Kossacki and Szutowicz, 2008) were described in. The model used in this work includes: – evolution of the nucleus cohesion due to sintering of ice grains (so called Kelvin effect which modifies grain-to-grain contact areas, but does not affect degree of compaction); – changes of porosity, due to sublimation/condensation in the medium; – crystallization of amorphous water ice when it is present; – sublimation of the CO ice (also explosive);
β Fig. 2. Geometry of the scarp.
– sublimation of H2O ice covered by the dust mantle, taking into account the temperature dependent sublimation coefficient (Kossacki et al., 1999; Gundlach et al., 2011; Kossacki and Markiewicz, 2013; Kossacki and Leliwa-Kopystynski, 2014). – diffusion of the vapor through the dust and the resulting recession of the surface; – illumination dependent on the local orientation of the surface. The basic equations are those for the surface temperature, the diffusion of heat, the sintering of ice grains, and the temperature dependence of the sublimation coefficient. The formulas, except description of the temperature dependence of the sublimation coefficient can be found in Kossacki and Szutowicz (2008). The sublimation coefficient was investigated by different authors, recently Gundlach et al. (2011) and Kossacki and LeliwaKopystynski (2014). Below is presented short description of the equations. The surface temperature of the dust layer can be calculated different ways. When the dust is cold enough to ignore the radiative heat transport, it is possible to consider the thermal conductivity of dust as constant. This is reasonable when the temperature remains below 200 K, or pores are of sub-micrometer size. However, the surface of a thick, very fluffy dust layer can warm-up to a temperature above 250 K. Thus, the model takes into account radiation within pores in the dust layer. In general case equations for the surface temperature, and the temperature beneath the dust layer are:
Sc R2h
ð1 AÞ maxðcos a; 0Þ rd T 4s þ kd rT ¼ 0;
ð1Þ
and
kd rTjd þ krTji H F s ¼ 0:
Fig. 1. Images of the scarp: Deep Impact (upper panel), and Stardust-NExT (lower panel). Marked (black arrow) is the most prominent of the structure visible on the scarp only in the DI image. Courtesy: NASA/JPL-Caltech/Cornell.
ð2Þ
The parameter Sc denotes the solar constant, A is the surface albedo, Rh is the actual heliocentric distance in AU, a is the illumination angle (between the direction to the Sun and the local normal to the surface), r is the Stefan–Boltzmann constant, d is the emissivity of the dust. The parameter kd denotes the thermal conductivity of dust, k is the thermal conductivity of granular ice-dust material beneath the dust layer, F s is the flux of molecules subliming and further diffusing into space through the dust mantle, and H is the latent heat of sublimation/condensation,. The terms in Eq. (2) are: heat flux conducted through the dust mantle, heat flux conducted within underlying ice-dust medium, and the energy losses due to sublimation and subsequent diffusion of water molecules to space. The temperature gradients are positive when the temperature increases versus depth. Both thermal conductivity, kd , and k, are temperature dependent. The respective equations are:
kd ¼ kd0 þ r p rT 3 ;
ð3Þ
350
K.J. Kossacki / Icarus 245 (2015) 348–354 1.2
1=2 32l dp k ¼ kg hð1 wÞ2=3 þ r p w H sat : 9pRT dT
1.1
ð4Þ
Here kg is the thermal conductivity of ice grains with mineral cores, and h is the Hertz factor. The terms in the right hand side of Eq. (4) describe: heat transport in the solid matrix of grains, and heat transport in the pores. For greater details see Kömle and Steiner (1992) and Kossacki et al. (2006). The Hertz factor is defined as the ratio between the contact area of adjacent grains and the cross section of a grain. The value of h evolves due to sintering of grains. The rate of sintering depends on the temperature, the radii of grains rg and on the actual value of the Hertz factor h. In the case of coarse-grained material sintering is slow and h is always close to its initial value, but the sintering of small grains can be very fast, depending on the temperature. There are known several sintering mechanisms. For H2O ice under cometary conditions most efficient is so called Kelvin effect: transport of molecules from adjacent grains onto the connecting neck. It is important, that the considered sintering mechanism does not lead to any changes of porosity. The radii of grains also remain unchanged. However, the grains change their shapes. The rate of increase of h is
dh 2rn X2 cpS 2 2 rg rn 1 ; ¼ 2 þ dt rn r g 2plR T 12 R T rg r 2n g g
h¼
0:5 rg rn
0:5 r d ð1 v d Þ 32l b1 expðb2 =TÞ; 2 Dd s 9pRg T
1 a1
þ
1 T a2 p ; tanh a3 tan p a1 273 a2 2
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 180
190
200
210
220
230
240
250
260
270
Temperature [K] Fig. 3. Sublimation coefficient of H2O ice versus temperature.
(see Fig. 2). Thus, Eq. (8) needs to be converted. When the horizontal surfaces above, and below the slope are covered by a thick dust prohibiting sublimation Eq. (8) becomes
dx Fs ¼ ; dt .i v i sinðbÞ
ð9Þ
2.3. Parameters
ð6Þ
ð7Þ
where a1 ¼ 2:33, a2 ¼ 138:72 and a3 ¼ 9:3. Equation for the recession rate of the scarp in the direction perpendicular to the slope is
dl 1 ¼ F s .1 i vi ; dt
0.9
where b is the inclination of a slope to the horizontal surface.
where as is the temperature dependent sublimation coefficient, and the remaining parameters are: b1 = 3.56 1012 Pa, and b2 = 6141.667 K (Fanale and Salvail, 1984). The sublimation coefficient was investigated by various authors. Recent measurements dealing with the sublimation coefficient of H2O ice were described in Gundlach et al. (2011) and Kossacki and Leliwa-Kopystynski (2014). According to both works as = 1 at temperatures lower than 195 K, and as = 0.15, at temperatures higher than 220 K. At middle range the temperature dependence of as determined by Kossacki and Leliwa-Kopystynski (2014) is somewhat steeper. The profiles of as versus temperature are shown in Fig. 3. In this work I calculate as as described by Kossacki and Leliwa-Kopystynski (2014). Thus,
as ¼ 1
Kossacki et al. 1999 Gundlach et al. 2011 Kossacki and Leliwa-Kopystynski, 2014
1
ð5Þ
(Swinkels and Ashby, 1980; Kossacki et al., 1994). The symbol rn denotes the radius of the grain-to-grain contact area (radius of the sinter neck), Rg is the universal gas constant, and l is the molar mass of water. The dimensionless variable S describes fraction of the surface of a grain from which the material is removed (source area) and transported onto the sinter neck (sink area). The parameters c and X are the surface energy and molar volume, respectively. For water ice they are X2 and c ¼ 4 1011 m4 mole2 J. The flux F s of molecules subliming beneath a dust layer thicker than ten layers of dust grains can be calculated as
F s ¼ as
Sublimation coefficient α (T)
and
ð8Þ
where .i is the bulk density of water ice, and v i is the volume fraction of ice in the material. The available observations are sufficient to determine only movement of the scarp in the horizontal direction
The model nucleus is orbiting like Comet 9P/Tempel 1. Thus, eccentricity e ¼ 0:51359, semi-major axis a = 3.13592 AU. Orientation of the rotation axis is also the same, so I = 16.4°, and U = 346.8°. The latter is solar longitude at perihelion. The considered scarp is located at 40°S. However, non-spherical shape of the nucleus makes the illumination of the scarp different than can be expected for the given latitude. Moreover, the scarp is significantly rounded and has some latitudinal extent. Thus, I considered two latitudes: 40°S, and 60°S. The height and inclination of the slope are not well determined. Vincent et al. (2013) considered vertical slope, about 100 m high. However, vertical slope is unlikely to be covered by a dust mantle, but an exposed ice was not detected. Thus, I consider a slope inclined by the angle b = 45–67.5°. Albedo of the surface is A = 0, and A ¼ 0:2. The lower value is close to the average bond albedo Ab ¼ 0:015 (Li et al., 2013). According to the authors the brightest spots identified in SN images had albedo only 25% higher than the average one. However, some very small patches may have higher albedo. The highest value of albedo considered in this work A ¼ 0:2 should bracket all possible values. The dust mantle is characterized by three parameters: the porosity wd , the thermal conductivity at low temperature kd0 and the thickness of dust layer Dd . Non of them is well known, but some estimates are available. According to Krause et al. (2011) experimentally created dust layers. Extremely porous layers had the thermal conductivity as low as kd = 2 mW m1 K1, while moderately porous dust layers, wd ¼ 0:71—0:84, had thermal conductivity kd ¼ 2—8 mW m1 K1. Thermal conductivity can be also derived from the observed oscillations of the surface temperature. According to Groussin et al. (2007) kd 6 36 mW m1 K1 at .d = 1000 kg m3. However, values of the thermal inertia found by Davidsson et al. (2009) indicate, that the uppermost can be much more conductive. The dust layer of low thermal conductivity prohibits intensive sub-dust sublimation of ice. Thus, a fast receding surface is probably not covered by a thick, low conducting dust. I consider kd0 = 10–100 mW m1 K1, and porosity wd ¼ 0:8. The largest considered value of kd0 is not supported by laboratory experiments and is probably an overestimate. Thickness of the dust
351
K.J. Kossacki / Icarus 245 (2015) 348–354
3. Results In this section is described, how recession of the scarp depends on albedo, geometry of the slope, and material parameters. In our work change of the position means horizontal movement of the middle of the considered slope. The nucleus of Comet 9P/Tempel 1 is not spherical. Thus, an area located at the geocentric latitude 40°S is illuminated at different angles than areas at the same latitude, but on a spherical object. To determine significance of this effect I simulated recession of a slope located at latitudes within the range 40–60°S. The exact value of the latitude appeared to be insignificant. This is because at perihelion, close to the local noon the scarp inclined by a high angle is illuminated almost perpendicularly (cosine of the illumination angle close to unity). In Fig. 4 is shown position of the scarp versus time measured from aphelion. The slope is inclined toward equator, i.e. cs ¼ 0 , by b ¼ 45—67:5 . The albedo is 0, and 0.2. The remaining parameters are: Dd ¼ 0:25 cm, kd = 100 mW m1 K1, r d ¼ rg ¼ 50 lm, h0 = 0.1, and T 0 = 40 K. In the case of low albedo, A ¼ 0, during one orbital period the scarp moves in the horizontal direction by 25–40 m depending on the inclination of the scarp. Increase of the albedo to 0.2 reduces the recession rate by about 40%.
0
Recession of the scarp [m]
layer may increase, decrease, or remain constant. Kossacki and Szutowicz (2008) have found, that reproducing the observed water production versus time and the surface temperature in the subsolar point was possible only when the dust layer had constant thickness. Thus, in this work thickness of the dust remains constant, surface erosion is compensated by the thickening from bottom. Beneath the dust mantle nucleus is composed of ice grains with mineral cores. The characteristic radius of grains is r g = 0.5–50 lm. The effective pore radius rp is in this work equal to r g . Initial value of the Hertz factor is within the range 0.001–0.1. Porosity of the material is determined by the volume fractions: of water ice, v i , and of mineral cores, v m . The volume v m is constant, but v i evolves due to sublimation/condensation of ice. However, the changes of v i are small. I consider two cases: v m ¼ v i;0 ¼ 0:15, and v m ¼ v i;0 ¼ 0:25. The corresponding porosity w is: 0.7, or 0.5. Mineral cores of the grains are agglomerates of dust and have density 1000 kg m3. The model parameters are summarized in Table 1.
o
β = 67.5 , A = 0
-10
β = 45 o ,
A=0
β = 45 o ,
A = 0.2
-20
β = 45 o
-30
λ d = 100 mW m-1 K -1 Δ d = 0.25 cm, ψ d = 0.8
-40
r d = rg = 50 μ m, h 0 = 0.1 ψ = 0.7
-50 0
0.2
0.4
0.6
0.8
1
Time [orbital periods] Fig. 4. Position of the scarp versus time measured from aphelion, when the slope is inclined toward equator by 45°, and 67.5°. The albedo A is: 0, and 0.2. Values of the remaining parameters are in the legend.
In Fig. 5 are shown results obtained, when the slope is inclined in different directions (different values of cs , corresponding to different parts of the rounded scarp). The inclination angle b is 45°. Material parameters are the same as in the case of Fig. 4. The recession rate is highest when the slope is inclined toward equator. However, for all orientations within the investigated range ±45° the differences are smaller than 15%. Fig. 6 is aimed to show significance of the dust thickness Dd and of the thermal conductivity kd . It should be emphasized, that kd is temperature dependent. Constant is only kd0 , the thermal conductivity at low temperature. It can be seen, that the dust thickness has significant influence on the recession of the scarp. When kd0 = 100 mW m1 K1, the recession rate is: 40 m (Dd = 0.25 cm), 34 m (Dd = 0.5 cm), and 27 m (Dd = 1 cm). When kd0 = 10 mW m1 K1 the recession rate is: 23 m (Dd = 0.25 cm), 11 m (Dd = 0.5 cm), and 4 m (Dd = 1 cm). Thus, the decrease of kd0 by an order of magnitude reduces the recession rate by about 60–80% depending on the dust thickness. The effect is strongest when the dust mantle is thick. The rate of sublimation is determined by the energy balance at the interface between the dust mantle and the underlying material (Eq. (2)). A decrease of the dust thermal conductivity by a factor ten should reduce energy available for sublimation by the factor
Table 1 Parameters. Parameter
Symbol
Albedo Emissivity Thermal conductivity of the dust mantle at low temperature Porosity of the dust mantle ðwd ¼ 1 v d Þ Pore radius in the dust mantle Density of the mineral cores of dust-ice grains (agglomerates very fine dust particles) Thermal conductivity of the mineral cores of dust-ice grains Initial volume fraction of H2O ice Volume fraction of the mineral component Tortuosity
A
Grain/pore radius in the ice-dust medium Initial temperature Initial Hertz factor
rg ¼ rp T0 ho
[lm] [K]
b
[deg.]
45–67.5
I
[deg.] [deg.] [AU]
16.4 346.8 3.135922 0.51359 5.52 40.6
Other parameters Inclination of the scarp The spin axis orientation Obliquity Solar longitude at perihelion Semi-major axis Eccentricity Orbital period Rotational period
Unit
kd0 wd rd
.m km
[mW m1 K1] [lm] [kg m3] [mW m1 K1]
v i0 vm s
U a e P p
[year] [h]
Value 0, 0.2 0.9 10, 100 0.8 0.5, 5, 50 1000 1000 0.15, 0.25 0.15, 0.25 pffiffiffi 2 0.5–50 40, 80 0.001, 0.1
352
K.J. Kossacki / Icarus 245 (2015) 348–354 320
0
β = 45
A = 0, ψ = 0.7
-10
Temperature at local noon [K]
Recession of the scarp [m]
β = 45 o r d = rg = 50 μ m, h 0 = 0.1 λ d = 100 mW m -1 K -1 Δ d = 0.25 cm
-20
-30
-40
γ = 315
o
γ = 180
o
γ = 45
o
-50 0
0.2
0.4
0.6
0.8
A=0
280
r d = r g = 50 μ m
Δ d = 0.25 cm
λ d0 = 10 mW m
-1
K
-1
T surface T sub-dust
260 240 220 200 180 0.3
1
λ d0 = 100 mW m -1 K -1 T surface T sub-dust
o
300
0.35
0.4
Time [orbital periods]
0.45
0.5
0.55
0.6
0.65
Time [orbital periods]
Fig. 5. Position of the scarp versus time measured from aphelion, when the slope is inclined by 45° toward equator and toward directions within range ±45°. Values of the remaining parameters are in the legend.
Fig. 7. Temperature at the surface and beneath the dust mantle versus time. The dust mantle has thickness Dd ¼ 0:25 and kd0 (constant term in Eq. (3)) is either 10, or 100 mW m1 K1. The surface is inclined by 45° toward equator. Values of the remaining parameters are in legend.
0
-10
o
A = 0, ψ = 0.7, h 0 = 0.7
Recession of the scarp [m]
Recession of the scarp [m]
0 β = 45
r d = r g = 50 μ m
-20
-30 λ d0 = 100 mW m -1 K -1 Δ d = 1.00 cm Δ d = 0.50 cm Δ d = 0.25 cm
-40
λ d0 = 10 mW m-1 K -1
-50 0
0.2
Δ d = 1.00 cm Δ d = 0.50 cm Δ d = 0.25 cm
0.4
0.6
0.8
ψ = 0.7, h 0 = 0.1
-20
-30 rd = 0.5 μ m, rg = 0.5 μ m rd = 0.5 μ m, rg = 5 μ m rd = 5 μ m, rd = 5 μ m,
-40
rg = 5 μ m r g = 50 μ m
-50
Time [orbital periods]
0
0.2
0.4
0.6
0.8
1
Time [orbital periods] Fig. 8. Position of the scarp versus time measured from aphelion. The initial temperature T 0 = 40 K. The surface is inclined by 45°. Values of the remaining parameters are in the legends.
0 β = 45
Recession of the scarp [m]
ten. However, slower sublimation indicates less efficient cooling due to the latent heat exchange during sublimation. This effect reduces decrease of the sub-dust temperature resulting from the decrease of the dust conductivity. The change of the sub-dust temperature should modify the temperature gradient in the dust layer and the flux of energy conducted downward from the warm dust surface. In addition, an increase of the temperature beneath the dust enhances outflow of heat downward to the cold interior of the comet. Finally, after decrease of the conductivity by a factor ten the sublimation flux should decrease, but less than ten times. However, the sublimation rate exponentially depends on the temperature and at low temperature becomes negligible. Changes of the dust thickness have similar influence on the sub-dust sublimation, as changes of the dust thermal conductivity. In Fig. 7 are shown profiles of the surface temperature and the sub-dust temperature versus time. Plotted are the values of the diurnal maximums of the temperature. The dust mantle has thickness Dd ¼ 0:25 and kd0 is either 10, or 100 mW m1 K1. The surface temperature significantly depends on kd0 , while the sub-dust temperature moderately. Fig. 8 is aimed to present, how granulation of the dust mantle and of the underlying material affect recession of the scarp. The dust is thin, Dd ¼ 0:25 mm, and the sizes of pores, rd and rg , are independent one from the other. Comparison of the profiles corresponding to a fixed rd , and different rg indicates that granulation of the material beneath the dust layer is significant only when r d has moderate value 5 lm. The considered change of the radii r d of pores
A = 0, Δ d = 0.25 cm
rd = 50 μ m, rg = 5 μ m rd = 50 μ m, rg = 50 μ m
1
Fig. 6. Position of the scarp versus time measured from aphelion. The dust mantle has thickness Dd ¼ 0:25—1 cm, and kd0 (constant term in Eq. (3)) is either 10, or 100 mW m1 K1. The surface is inclined by 45° toward equator. Values of the remaining parameters are in legend.
β = 45 o
-10
o
A = 0, Δ d = 0.25 cm ψ = 0.7
-10
T
0
= 80 K
-20
-30
-40
r d = r g = 50 μ m, h
0
= 0.1
r d = r g = 50 μ m, h
0
= 0.001
-50 0
0.2
0.4
0.6
0.8
1
Time [orbital periods] Fig. 9. Position of the scarp versus time measured from aphelion, when the initial temperature is 80 K. The surface is inclined by 45°. Values of the remaining parameters are in the legend.
in the dust mantle by two orders of magnitude from 0.5 lm to 50 lm results in a change of the recession rate by the factor four. I investigated also influence of the initial temperature. In Fig. 9 is plotted position of the surface versus time, when T 0 = 80 K. The remaining parameters are: Dd = 0.25 cm, r d ¼ rg ¼ 50 lm, h0 = 0.1, and b ¼ 45 . It can be seen, that the initial value of the Hertz factor, h0 , has minor importance. This is due to negligible influence of the granulation of the material beneath the dust mantle.
K.J. Kossacki / Icarus 245 (2015) 348–354 0
β = 45 o
Recession of the scarp [m]
A = 0, r d = 50 μ m λ d = 10 mW m -1 K -1
-10
Δ d = 0.25 cm ψ = 0.7, h 0 = 0.1
-20
-30
α s (T) (Gundlach et al, 2011)
-40
α s (T) (Kossacki and Leliwa-Kopystynski 2014) αs = 1
-50 0
0.2
0.4
0.6
0.8
1
Time [orbital periods] Fig. 10. Position of the scarp versus time measured from aphelion, when the sublimation coefficient of ice is either unity, or depends on the temperature. The surface is inclined by 45°. Values of the remaining parameters are in the legend.
Comparison of the profiles shown in Fig. 9 with the corresponding profile shown in Fig. 8 indicates, that the surface of a the warm nucleus recedes faster than predicted for a cold one. However, the difference is smaller than 10% despite the change of T 0 from 40 to 80 K. I investigated also, how the temperature dependence of the sublimation coefficient as affects the calculated recession rate of the scarp. In Fig. 10 are shown results obtained, when the sublimation coefficient is either unity, or is temperature dependent. It can be seen, that taking into account the temperature dependence of the sublimation coefficient can reduce the calculated recession rate by about 40%. 4. Summary and discussion I attempted to derive material properties of the nucleus beneath the scarp from the observed recession of one scarp. According to observations the scarp and the surrounding surface are entirely covered by dust (Thomas et al., 2013). Thus, I calculated recession of the model scarp due to sub-dust sublimation of ice. I investigated influence of: – local inclination and azimuth of the slope, – thickness and thermal conductivity of the dust layer, – granulation of the dust mantle and of the material beneath the dust, – initial temperature and cohesivity of the nucleus, expressed by the Hertz factor, well as – albedo of the surface. The scarp is rounded, almost half-circular. Thus, it could be expected that different parts of the scarp recede with different rates. The simulations show, that the local azimuth of the slope has very low significance. Thus, the scarp should retain the current shape. Granulation of the material composing the dust mantle can be the same, as of underlying ice-dust material, or different. I have found, that the recession rate of the considered scarp is sensitive only to the granulation of the dust mantle. When the slope is inclined by 45°, and the dust mantle has high thermal conductivity kd0 = 100 mW m1 K1 the scarp moves by: 14–20 m when Dd ¼ 0:25 cm, and rd ¼ 5 lm, or by 40–42 m when Dd ¼ 0:25 cm and r d ¼ 50 lm. When kd0 = 10 mW m1 K1, Dd ¼ 0:25 cm, and r d ¼ 50 lm the scarp can recede by 22 m. The lower value of kd0 is probably more realistic, because it is consistent with high porosity (Krause et al., 2011). Thus, taking into account approximate determination of the recession of the scarp based on the observa-
353
tions (up to 50 m according to Thomas et al. (2013)). I estimated that if the dust thickness Dd ¼ 2:5 mm, the characteristic radius of pores in the dust layer is about 50 lm (1/50 of the dust thickness). It should be noted, that low value of kd0 does not mean, that kd is always low. At high temperature kd is determined by the radiative heat transport. In this work thickness Dd of the dust covering the slope was within the range 2.5–10 mm. When the dust layer was 10 mm thick, the scarp receded less than 30 m in the case of high conducting dust, kd0 = 100 mW m1 K1, but in the case of more realistic kd0 = 10 mW m1 K1 recession was negligible. Very small dust thickness, on average about 1 mm, could be also possible. However, in such case ice should be locally exposed. Taking into account the temperature dependent sublimation coefficient significantly reduces calculated recession rate of the scarp. The difference depends on the considered formulation of the temperature dependence of the sublimation coefficient. It is about 40% when the formula Gundlach et al. (2011) is used, and 10% when the formula Kossacki and Leliwa-Kopystynski (2014) is applied. Finally, the performed simulations indicate, that the scarp is covered by a dust layer few millimeters thick, with characteristic radius of pores smaller than 100 lm. Acknowledgment Thank you the anonymous reviewers for their valuable suggestions. References Davidsson, B.R., Gutirrez, P.J., Rickman, H., 2009. Physical properties of morphological units on Comet 9P/Tempel 1 derived from near-IR deep impact spectra. Icarus 201, 335–357. Fanale, F.P., Salvail, J.R., 1984. An idealized short-period comet model: Surface insolation, H2O flux, and mantle evolution. Icarus 60, 476–511. Farnham, T.L., Bodewits, D., Li, J.-Y., Thomas, P., Belton, M.J.S., 2013. Connections between the jet activity and surface features on Comet 9P/Tempel 1. Icarus 222, 540–549. Finklenburg, S., Thomas, N., Su, C.C., Wu, J.-S., 2014. The spatial distribution of water in the inner coma of Comet 9P/Tempel 1: Comparison between models and observations. Icarus 236, 9–23. Groussin, O., A’Hearn, M.F., Li, J.Y., Thomas, P.C., Sunshine, J.M., Lisse, C.M., Meech, K.J., Farnham, T.L., Feaga, L.M., Delamere, W.A., 2007. Surface temperature of the nucleus of Comet 9P/Tempel 1. Icarus 187, 16–25. Gundlach, B., Skorov, Y.V., Blum, J., 2011. Outgassing of icy bodies in the Solar System – I. The sublimation of hexagonal water ice through dust layers. Icarus 213, 710–719. Kömle, N.I., Steiner, G., 1992. Temperature evolution of porous ice samples covered by a dust mantle. Icarus 96, 204–212. Kossacki, K.J., Leliwa-Kopystynski, J., 2014. Temperature dependence of the sublimation rate of water ice: Influence of impurities. Icarus 233, 101–105. Kossacki, K.J., Markiewicz, W.J., 2013. Comet 67P/CG: Influence of the sublimation coefficient on the temperature and outgassing. Icarus 224, 172–177. Kossacki, K.J., Szutowicz, S., 2008. Comet 9P/Tempel 1: Sublimation beneath the dust cover. Icarus 195, 705–724. Kossacki, K.J., Norbert, I., Kömle, N.I., Kargl, G., Steiner, G., 1994. The influence of grain sintering on the thermoconductivity of porous ice. Planet. Space Sci. 42, 383–389. Kossacki, K.J., Markiewicz, W.J., Skorov, Y., Koemle, N.I., 1999. Sublimation coefficient of water ice under simulated cometary-like conditions. Planet. Space Sci. 47, 1521–1530. Kossacki, K.J., Leliwa-Kopystynski, J., Szutowicz, S., 2006. Evolution of depressions on Comet 67P/Churyumov Gerasimenko: Role of ice metamorphism. Icarus 184, 221–238. Krause, M., Blum, J., Skorov, Yu.V., Trieloff, M., 2011. Thermal conductivity measurements of porous dust aggregates: I. Technique, model and first results. Icarus 214, 286–296. Li, J.-Y., A’Hearn, M.F., Belton, M.J.S., Farnham, T.L., Klaasen, K.P., Sunshine, J.M., Thomas, P.C., Veverka, J., 2013. Photometry of the nucleus of Comet 9P/Tempel 1 from Stardust-NExT flyby and the implications. Icarus 222, 467–476. Rosenberg, E.D., Prialnik, D., 2010. The effect of internal inhomogeneity on the activity of comet nuclei – Application to Comet 67P/Churyumov Gerasimenko. Icarus 209, 753–765. Rozitis, B., Green, S.F., 2011. Directional characteristics of thermal-infrared beaming from atmosphereless planetary surfaces – A new thermophysical model. MNRAS 415, 2042–2062.
354
K.J. Kossacki / Icarus 245 (2015) 348–354
Sunshine, J.M. et al., 2006. Exposed water ice deposits on the surface of Comet Tempel 1. Science 311, 1453–1455. Swinkels, F.B., Ashby, M.F., 1980. A second report on sintering diagrams. Acta Metall. 29, 259–281. Thomas, Peter C., Veverka, J., Belton, Michael J.S., Hidy, Alan, A’Hearn, Michael F., Farnham, T.L., Groussi, Olivier, Li, Jian-Yang, McFadden, Lucy A., Sunshine, Jessica, Wellnitz, Dennis, Lisse, Carey, Schultz, Peter, Meech, Karen J., Alan Delamere, W., 2007. The shape, topography, and Geology of Tempel 1 from deep impact observations. Icarus 187 (1), 4–15. Thomas, P., A’Hearn, M., Belton, M.J.S., Brownlee, D., Carcich, B., Hermalyn, B., Klaasen, K., Sackett, S., Schultz, P.H., Veverka, J., Bhaskaran, S., Bodewits, D.,
Chesley, S., Clark, B., Farnham, T., Groussin, O., Harris, A., Kissel, J., Li, J.-Y., Meech, K., Melosh, J., et al., 2013. The nucleus of Comet 9P/Tempel 1: Shape and geology from two flybys. Icarus 222, 453–466. Veverka, J., Klaasen, K., A’Hearn, M., Belton, M., Brownlee, D., Chesley, S., Clark, B., Economou, T., Farquhar, R., Green, S.F., Groussin, O., Harris, A., Kissel, J., Li, J.-Y., Meech, K., Melosh, J., Richardson, J., Schultz, P., Silen, J., Sunshine, J., Thomas, P., Bhaskaran, S., Bodewits, D., Carcich, B., Cheuvront, A., Farnham, T., Sackett, S., Wellnitz, D., Wolf, A., 2013. Return to Comet Tempel 1: Overview of StardustNExT results. Icarus 222, 424–435. Vincent, J.B., Sierks, H., Rose, M., 2013. Jet activity on the cliffs of Comet 9P/Tempel 1. EPSC Abstracts 8, EPSC2013-429-1.