1. Recent results of Comet Thermal Modeling E. Kührt, N
Recent results of Comet Thermal Modeling E. Kührt, N. Gortsas, DLR Berlin

2. Outline
Activity of comets
Thermal modeling of comets
Conclusion

3. Activity of comets → to study activity is a main scientific objective of VIRTIS
Porosity
ACTIVITY
Composition
Strength
Heat + mass transport (K)
Internal energy sources
External energy (Sun)
Topography

4. Key observations to understand activity
Hale-Bopp ground based observations
activity of highly volatile ices (e.g. CO) scales nearly as the solar energy input (Biver et al. 2002), therefore one can conclude, that these volatiles are near the surface
activity is localized, strong CO jet near 20° n.l. (Bockelée-Morvan et al. 2009)
Lab
amorphous ice and trapping of gasses confirmed experimentally
however, amorphous ice was never identified in the solar system
KOSI (comet simulation): it is hard to keep activity alive

5. Deep Impact at Tempel-1
K < W/Km (Groussin et al. 2007)
K >1 W/mK (Davidsson 2009)
different source areas of H2O and CO2 (Feaga 2007)
below 1 m depth original composition
low density = 400 kg/m3
From IR spectroscopy: only 0.03 km2 of the surface is water ice, but: this is much too less to explain the observed activity (1.3 km2 ) (Sunshine 2006)
is this conclusive?

6. white: active, T1
black: inactive, T2

7. → Surface ice may be hidden in a mixture!
L. Moroz, private communication
→ Surface ice may be hidden in a mixture!

8. Main Puzzles with respect to activity
What is the structural/compositional difference between more and less active areas?
What is the degree of inhomogeneity?
How is the heat conductivity (4 orders of magnitude range)
Are there internal heat sources (phase transitions, chemical reactions?)
What is the trigger for outbursts and splits?
Is there amorphous ice in comets?
cons
has not been detected in interstellar clouds, star forming regions or other solar system objects
hard to form
phase transition with impurities is endothermic (Kouchi 2001)
DI: CO2 and water activities are separated
CO activity at HB beyond 20 AU, but no water
pros
CO (30 K) cannot condensate in nuclei but is present
trapping effect and energy release are needed may help to understand HB-activity
Lab results (Bar-Nun)
possible energy source for outbursts

9. 2. Thermal modeling of comets
Motivation
powerful tool to study activity and nucleus temperature
important to interpret space experiments (Mupus, Virtis, Osiris)
Our philosophy
as simple as possible since we know too less about comets
not too many free parameters
no fantasy results but include the available measurements
strict control of energy conservation and numerical stability

10. Status of Hale-Bopp modeling (best data base)
Prialnik (1998)
K=567/T
wrong spin axis
water curve failed
trapped CO > 10 m below surface

11. Enzian (1999)
K=0.1 W/mK
wrong spin axis
amorphous ice
trapped CO is released
water curve failed

12. Capria (2002)
K=3 W/mK
wrong spin axis
trapped CO is set free
extended source
water curve failed
CO > 10 m below surface

13. Our approach
we consider active areas without permanent dust mantles that mainly contribute to activity
water ice, dust and CO are incorporated but no amorphous ice phase
from observations we expect a low heat conductivity in the nucleus that requires an exact treatment as a Stefan problem (moving boundary problem)
heat conduction, heat advection, gas diffusion, sublimation, and condensation processes are taken into consideration, no extended source
obliquity of spin axis and argument of the ascending node (correlates the seasons with elliptical orbit) are taken from observations
observational evidence that activity is mainly from northern hemisphere and near equator (Lederer 2002, Bockelee-Morvan 2009) is considered.

14. Equations Heat conduction equ. Upper boundary cond.
(energy conservation)
Lower bound. cond.
Initial condition
Stefan equation
bulk sublimation and gas diffusion

15. Stefan problem (ablation)
velocity of erosion
velocity of heat wave
Surface x1(t)
Surface x1(t+Δt)
H2O + dust
Vp ~ 100 K=1
Vp ~ 3 K=0.001
Ve ~ 3 mm/h
Interface x2(t)
Z: sublimation rate
T: temperature
ρ: density
K: heat conductivity
τ: spin period
Interface x2(t + Δt)
H2O +
CO +
dust

16. for K < 0.01 erosion velocity exceeds penetration velocity of thermal wave:
→ no effective heating of the nucleus
→ Stefan problembecomes important

17. Workflow of numerical code
(Crank-Nicholson scheme)
T (x, t)
Δx (Δt)
common
Interpolation with
cubic splines
T(0,x)
t = t + Δt
T‘(x,t) T1 t
t + Δt
T1‘ Δx

18. Results HB

19. Water production rates CO production rates
K = 0.01 W/Km

20. Results CG

21. Water production rates CO production rates

22. Depth of CO T-profile at perihelion
k1 = W/mK
k2 = 0.01 W/Km
k3 = 0.1 W/Km

23. Importance of thermal transport by gas for activity
no SP, lat=0

24. Importance of thermal transport by gas for T-profile
no SP, lat=0

25. Importance of thermal transport by gas with moving boundary

27. 3. Conclusions
Cometary activity is still puzzling, Rosetta should help to understand it
Modeling the Hale-Bopp activity is the key to verify thermal models
Rigorous Stefan treatment is mandatory for low heat conductivity,
in combination with an inhomogeneous composition (as observed) but without amorphous ice it gives good results for HB
Exact Stefan solutions lead to important consequences:
heat penetration is obscured
temperature profiles are extremely steep near perihelion
volatiles as CO can be very close below the surface
Incorporation of gas diffusion does not remarkably change the modelled activity
Seasonal effects are important for activity