1. THERMAL EVOLUTION OF NON-SPHERICAL COMET NUCLEI
M.C. De Sanctis (1), J. Lasue (2), G. Magni (1), M.T.Capria (1), A. Coradini(3)
(1) IASF-INAF, Italy
(2) LPI, Houston, TX, USA
(3) IFSI-INAF, Italy

2. The Rome nucleus model
The nucleus is composed by ices of water, minor species (CO, CO2, and others) and dust particles. The dust grains are embedded in the ice matrix. Initially, the water ice can be considered in the amorphous phase.
The numerical code computes the heat diffusion in the porous cometary material, leading to water ice phase transition and the sublimation of the volatile ices. The gases diffuse inside the pore system, either re-condensing or escaping in space.
The dust grains are released by the sublimation of the ices and undergo the drag exerted by the escaping gas, so that they can be blown off by the flux and lost in space, or they can accumulate on the nucleus surface to form a dust crust.
The model follows the thermal evolution and differentiation of the nucleus along the orbit, giving gas and dust fluxes, internal composition and the temperature distribution.

3. Dust crust formation
The dust grains are seen as up to two populations of spherical particles distributed in different classes. They are initially embedded in the icy matrix.
The grains are liberated as the ices sublimate from the outer layers (free particles). Step by step, the critical radius and the amount of grains flying away from the nucleus (dust flux) or remaining on the surface (dust mantle) are computed.
The dust mantle formation is due to the combined effects of centrifugal forces, gas drag and gravitational force that act on the dust particles.

4. Current developments
A new model has been developed to take into account non-spherical shapes.
Influence of the topography on the thermal evolution and erosion of the nucleus and the coma features (jets)
Influence of shape and internal structure of the nucleus on its thermal evolution
Ref.: Lasue et al., 2008 PSS, De Sanctis et al.,2010 Icarus, De Sanctis et al, 2010, AJ.

5. Nucleus shape and basic assumptions
The shape of the comet can be described through a two-dimensional discreet grid defined with the angles θ and ϕ corresponding to the latitude and longitude of the points considered on the comet.
The local illumination is calculated through the cosine of the angle between the local normal to the surface and the direction to the Sun.
Shadowing effects are taken into account : for each shape or altered shape, the shadow of each point on the surface of the nucleus is calculated.
Thermal evolution are calculated locally, using as input the solar illumination and the different parameters of the cometary material beneath the surface. Lateral heat transfer is negligible with respect to the normal heat transfer.

6. 67P/C-G models We applied the model to comet 67P/C-G to study:
Influence of different shapes
Influence of obliquity
The results obtained are fully described in
De sanctis et al., Icarus 207 (2010), pp
De Sanctis et al, AJ, in press (June)

7. Application to 67P/C-G: assumptions
The Jupiter Family comet 67P/Churyumov-Gerasimenko has a perihelion distance of 1.29 AU and a period of 6.57 years.
Lamy et al.,
A&A 2006
Interaction with Jupiter in 1959 decreased its perihelion distance from q = 2.75 AU to q= 1.29 AU. (Beliaev et al. (1986) ; Carusi et al. (1985))
albedo
0.04
dust/ice 1
radius
2000 m
porosity
0.6
rotation period
12.55 h
obliquity
0° /-45°
initial temperature
30 K
dust grain thermal cond.
3 W /K/m
dust grain density
1000 kg/m3
CO²/H2O
0.01
CO/H2O
orbit a e
Multistage 1 50
0.5
Multistage 2 25
0.4
Multistage 3 8
Injection orbit A
3.84
0.43
Injection orbit B
4.3
0.36
Injection orbit C
3.51
0.63

8. Application to 67P/C-G: observations
Activity of 67P/C-G
 At perihelion, observations show a water flux ~ molec/s
 A peak of dust of 2000 kg/s in 1982 at 1.36 AU (Weiler et al., 2004)
 The gas and dust fluxes show a large pre/post perihelion asymmetry and dependence on perihelion distance.
(Rauer and Weiler)
67P light curve, Ferrin 2005

9. Results: spherical shape
Illumination, fluxes of H2O, fluxes (in molec/sec/m2) and stratification for CO2 and CO at perihelion.
The erosion plots show the difference in meters between the current local sublimation fronts and the local radius.
Very different water and volatile flux distribution
No stable dust mantle forms

10. Results: ellipsoidal shape
Mercator projection on the surface of the comet of water , CO2 and CO fluxes at perihelion.
Dust-covered regions (several spots appear in different locations of the comet surface) at the end of the evolutions on orbit b (far from the Sun). Not stable on the present orbit.

11. Results: spherical harmonic shapes
When the ice is on the surface, the water flux is coming mainly from the comet dayside, and it is concentrated where the illumination is maximum, in the sunward direction.
CO and CO2 are less influenced by comet shape.

12. Results: spherical harmonic shapes with pole coordinates lambda = 45°, beta = 7°
A nucleus similar to the observed one
(Lamy et al., 2007): irregular spherical harmonic shapes and obliquity -45°.
The simulations give us a scenario with an irregular distribution of water from the surface, coming mainly from the illuminated south sub-polar regions at perihelion.

13. Ellipsoidal shapes: effects of the obliquity
Asymmetry for CO and CO2 fluxes
North pole illuminated only near aphelion increases by ~100 the water flux at aphelion, implications for Rosetta.

14. Dust formation: ellipsoidal model with obliquity
Evolution of the dust covered regions during orbit b for a model with axial tilt
(a) Condition at the beginning of second revolution on orbit b;
(b) condition after three revolutions on orbit b and
(c) condition at the end of orbit b.
Effects of the obliquity:
Different location of the dust covered regions
Different thickness
Different stability

15. Dust formation: general results
In all the cases investigated, dust mantles form locally during the inner Solar System orbits at large heliocentric distance.
Dust-covered regions depend on comet obliquity. The comet axis inclination is a key parameter in the dust mantling.
One of the effects of a dust mantle is the quenching of the cometary activity: water sublimates from beneath the surface, diffusing through the crust. The gas diffusion through the dust mantle gives lower fluxes from the dust-covered regions.
This implies that, when the dust mantle is present on some areas of the comet surface, the water flux comes prevalently from those regions not covered by dust where the ice is exposed on the surface. These areas can be defined as ‘‘active regions” and during the nucleus rotation, several spots or areas can be activated when illuminated.

16. 67P C-G: comparison withobservations
Evolution of the global gas fluxes of the different models in days from perihelion in the inner Solar System orbit (from Lamy et al. (2007)).
The bold curves represent the simulated fluxes, the thin curves the simulated fluxes with a reduction factor of 20%.
Implication on the active surface fraction
We have seen that the comet water flux varies over the comet surface in relation with the shape and illumination, so it is not reasonable to assume a constant sublimation rate to evaluate the fraction of active surface, as this will underestimate the fraction of active area.

17. 67P C-G fluxes: comparison with observation
In our models, the maximum of the total water activity is about 5 x 1028 mol/s. This is larger than the observed water production rate (1– mol/s).
To match observations, we can think that a smaller area is covered by a thick dust mantle or that a thin dust mantle is covering the comet surface.
If the spin axis is inclined and water sublimates from a layer below a thin dust mantle, we can expect a steep rise of the activity before the perihelion, followed by smoother activity curves after the perihelion, similar to the behavior showed by the minor species in our models.
Model results and observations suggest that 67P/Churyumov–Gerasimenko could have water gas coming out from sub-surface sources, located in the southern hemisphere.

18. Discussion of the results in terms of Rosetta forthcoming measurements
The comet water activity is strongly localized and linked with the illumination condition. We foresee a large difference between night and day water fluxes and a strong dependence of water emission on the illumination, and, consequently, on spin axis inclinations.
If the pole orientation of 67P/Churyumov–Gerasimenko is as we have simulated and the comet is not covered by thick dust crust, the water gas distribution at perihelion is localized in the southern hemisphere and concentrated around the south polar regions, showing not only diurnal variations but also changes of intensity linked to the irregularity and shadows on the comet.
M model results implies a seasonal effect caused by active regions that rapidly come into illumination as the comet approaches the Sun.
When Rosetta will observe the comet at about 3 AU, the most illuminated and active regions will be the equatorial ones, but these regions will change in southern regions, approaching the perihelion.
The dust emission has a similar behavior: the dust particles are lifted off by gas drag and are strictly correlated with gas emission.

19. Comet nuclei shapes
The recent in situ measurements of cometary nuclei have shown typical non-spherical shape.
Crater-like depression and other topographic features are quite common on comet nuclei.

20. Backup slides

21. Current thermal evolution models
PDE solved numerically on a discreet grid: finite difference method
Numerical codes can be solved
1D (Espinasse et al., 1991, 1992)
1.5D diurnal effects (Capria et al. 1996, De Sanctis et al., 1999)
2.5 D with lateral heat conduction (Enzian et al., 1997, 1999)
Quasi-3D diurnal and latitudinal effects (Cohen et al., 2003, De Sanctis et al., 2005, 2007)
Figure adapted from Huebner et al., ‘Heat and gas diffusion in comet nuclei’, ISSI science Book series, 2006.

22. Relevant physics Heat diffusion in the nucleus 
Gas diffusion in the nucleus 
Sublimation/recondensation of volatiles in the nucleus 
Crystallization of amorphous H2O ice 
Release of gases trapped in amorphous H2O ice 
Gas fluxes computation 
Dust release, dust flux and dust mantle formation 
Radiogenic heating 
Nucleus rotation with diurnal effects 

23. The heat and gas diffusion equation
Where T is the temperature, t the time, K the heat conduction coefficient,  the density and c the specific heat of the comet material;
QH2O, QCO2 and QCO are the energies exchanged by the sublimation and recondensation of ices; Qtr is the heat released by the water ice phase transition from amorphous to crystalline form
The gas flow through the pore system is described by the mass conservation equation
 is the gas density,  is the flux and Q' is the gas source term