Chapter 7: Asteroids and Comets composition, origin, fate tail formation; the physics of sublimation

Review: asteroids Mostly rocky bodies Found in the asteroid belt between Mars and Jupiter  Also the Trojans at the Jupiter Lagrangian points The source of most meteorites  Asteroids get perturbed from their orbits, into Earth-crossing trajectories

Internal Heat Sources Many asteroids solidified from molten rock or metal (igneous) Gravitational energy? For Ceres, Not all this energy will actually be available  If accretion rate is slow, energy will be radiated away rather than stored The heat capacity of typical rock is so this corresponds to a temperature rise of

Impact Heating Two asteroids orbit the Sun in virtually identical orbits. One has a radius of 40 km and a density of 3300 kg/m 3. The other has a radius of only 100 m, and the same density. Due to a small difference in initial velocity, the two asteroids approach each other and are attracted gravitationally. a)With what velocity does the smaller asteroid hit the larger one? b)Assume that all the energy released in the impact is retained in the immediate neighbourhood of the impact, in an amount of matter equal to twice the mass of the smaller asteroid. By how much will the temperature of the heated matter rise?

Heat sources Radioactive heating?  Unstable isotopes present in solar or carbonaceous chondrite mixture can be powerful sources of energy  Especially at earlier times when more of the radioactive isotope was available. IsotopeHalf-life (10 9 yr) Isotope fraction in present-day chondrites Element abundance Heating rate Isotope (J/yr/kg) Chondrite (J/Gyr/kg) 40 K x x Rb x x Th x x U x x U x x10 4 Total:1.1x10 5

Conduction Note that while the heat acquired in various ways is proportional to mass (and thus R 3 ), the heat radiated away is proportional to surface area (thus R 2 ).  smaller bodies preferentially cool faster. where A is the surface area, dx is the distance over which the heat is transmitted, k c is the thermal conductivity of the material. The total thermal energy of a body is given by: For a spherical asteroid, radius R, if we assume the surface temperature is T=0, and the T gradient is linear: Therefore, the timescale to conduct away all the thermal energy is: In solid rock, the main method of heat transport is conduction. The rate of heat loss (dQ/dt) is related to the conduction efficiency and temperature gradient:

Cooling times For two bodies of the same composition and different radii the rate of cooling is then just proportional to the surface areas – i.e R 2. For rocky material, the thermal diffusivity k c /ρc v =1x10 -6 m 2 /s so τ yr ~1x10 4 R km 2  More massive (larger) bodies will take longer to cool  A body the size of an asteroid (R~500 km) will lose any internal energy in less than ~2.5 Gyr.

Heating Consider the equilibrium case where the internal heating rate changes slowly, relative to the cooling rate. Assuming a roughly linear temperature gradient: The rate of heat loss is given by: If L is the heat production per mass (W/kg) then setting the total heating rate LM equal to the cooling rate gives:

E.g. Ceres Estimate the core temperature of Ceres, assuming it is in equilibrium with the flux of radiation from the Sun. You will need the following information: The mass is M=9.5x10 20 kg The diameter is D=1000 km. For typical carbonaceous chondrites, the energy produced by radioactivity is L~4x W/kg, and the thermal conductivity is k c =3W/K/m

Break

Comets Comet Hale-Bopp in 1997, photographed at Mono Lake, CA

The nature of comets Comets are distinguished from all other SS bodies by their appearance which includes a bright coma and long tails  hidden within these is the solid body “source” – the nucleus.  The coma is large, typically km across and comet tails can extend to distances of km.  The nucleus within is only ~10km across. The appearance of a comet changes as it moves through its orbit, becoming brighter as it approaches the Sun and fainter again as it moves away.

Asteroid or Comet? The only distinction between asteroids and comets is the presence of a Coma and tail  These features get dimmer as a comet moves farther away from the Sun.  E.g. comet Wilson-Harrington, discovered in 1949, was “rediscoverd” in 1979 as an asteroid.  Similarly, the asteroid 2060 Chiron moves in a cometlike orbit, and in 1988 it came closer to the Sun and became brighter and more cometlike. Coma and tail are caused by sublimation of ice. Thus distinction is simply one of ice content.

Comet composition Comets become visible as such at a distance of about AU. What temperature does this correspond to? At this temperature, ice can sublime to form water vapour (the solar wind pressure is ~ atm).

As expected, comets are warmer on their sun-facing side, as this temperature map from the Deep Impact mission shows (comet Tempel 1) Sublimation occurs more rapidly on one side than the other.

Sublimation The vapour pressure of a given substance at temperature T is given by : where H L is the latent heat of vaporization, and p 0 is the vapour pressure at some temperature T 0. R g = cal/mole/K is the gas constant. GasP 0 (Pa) T 0 (K) H L (cal/mole) N2N CH CO NH H2OH2O Calculate the vapour pressure of water at 273 K, and at 177 K.

Sublimation In a vapour in thermal equilibrium, the molecules of mass m are moving randomly, with velocities v, and the kinetic energy is equal to the thermal energy: The number of molecules per unit volume striking a surface, per unit time, is just where the factor 1/6 is due to the fact that we only count particles moving in the (say) +x direction Using the ideal gas law to relate n and vapour pressure p V, we get: using

Energy Balance 1.Heating: radiation absorbed from the Sun, with efficiency (1-A v ) 2.Cooling: a)Reradiation in the thermal infrared, with efficiency (1-A IR ) b)Sublimation carries off an energy 4  R 2 ZH L To calculate the temperature at radius r, and the sublimation rate Z, you have to solve the energy balance equation by setting the heating rate equal to the cooling rate.

Sublimation Calculations of the gas outflow rate as a function of heliocentric distance, for different ices. Water begins to sublime at about 3 AU. Sublimation requires a lot of energy, effectively cooling the surface of the comet For water:

Next lecture: More comets Composition Tail formation Orbits