Chapter 7: Asteroids and CometsChapter 7: Asteroids and Comets

• composition, origin, fate• tail formation; the physics of sublimation

Review: asteroidsReview: 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 SourcesInternal Heat Sources•Many asteroids solidified from molten rock or

metal (igneous)•Gravitational energy?

RGMEg

2

53

kgJR

GMMEg /108.3

53 4•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 KkgJcV //840

KKkgJkgJ

McET

V

2.45//840/108.3 4

Impact HeatingImpact Heating• Two asteroids orbit the Sun in virtually identical

orbits. One has a radius of 40 km and a density of 3300 kg/m3. 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 sourcesHeat 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.

Isotope Half-life(109 yr)

Isotope fraction in

present-day chondrites

Element abundanc

e

Heating rate Isotope(J/yr/kg)

Chondrite (J/Gyr/kg)

40K 1.25 0.00011 5.6x10-4 920 5.7x104

87Rb 50.0 0.293 2.2x10-6 0.544 3.51x102

232Th 13.9 1.00 2.9x10-8 837 2.4x104

235U 0.71 0.0072 8.2x10-9 18000 1.1x103

238U 4.5 0.993 8.2x10-9 2970 2.4x104

Total: 1.1x105

ConductionConduction

Note that while the heat acquired in various ways is proportional to mass (and thus R3), the heat radiated away is proportional to surface area (thus R2). smaller bodies preferentially cool faster.

where A is the surface area, dx is the distance over which the heat is transmitted, kc is the thermal conductivity of the material.

The total thermal energy of a body is given by: 3

4~3 TcRTMcU v

v

dxdTAk

tQ

c

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:

c

v

kcR

tQU

3~)/(

~2

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:

coreccore

c TRkR

TRktQ 44 2

Cooling timesCooling 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 R2.

•For rocky material, the thermal diffusivity kc/ρcv=1x10-6m2/s so τyr~1x104Rkm

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.

c

v

kcR

tQU

3

~)/(

~2

HeatingHeating•Consider the equilibrium case where the internal heating rate

changes slowly, relative to the cooling rate.•Assuming a roughly linear temperature gradient:

RTT

dxdT surfacecore

• The rate of heat loss is given by:

RTT

RktQ surfacecore

c24

• If L is the heat production per mass (W/kg) then setting the total heating rate LM equal to the cooling rate gives:

RkMLTT

TTRktQLM

csurfacecore

surfacecorec

4

4

E.g. CeresE.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=9.5x1020 kgThe diameter is D=1000 km.For typical carbonaceous chondrites, the energy

produced by radioactivity is L~4x10-12 W/kg, and the thermal conductivity is kc=3W/K/m

BreakBreak

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

The nature of cometsThe 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 104-105km across and comet tails can extend to distances of 106-108km.

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?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 compositionComet composition•Comets become visible as such at a distance of about 2.5-3

AU. What temperature does this correspond to?

177

5.2280

/280

11

2/1

4/1

AUdK

AATIR

V

• At this temperature, ice can sublime to form water vapour (the solar wind pressure is ~10-20 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.

SublimationSublimation•The vapour pressure of a given substance at temperature T is

given by :

TRH

TRHpp

g

L

g

Lv

00 exp

where HL is the latent heat of vaporization, and p0 is the vapour pressure at some temperature T0.Rg =1.9871 cal/mole/K is the gas constant.

Gas P0

(Pa)

T0

(K)

HL

(cal/mole)

N2 133 47 1600CH4 133 67 2200CO2 133 139 5500NH3 133 164 7200H2O 133 256 11000

Calculate the vapour pressure of water at 273 K, and at 177 K.

SublimationSublimationIn 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:

kTvmH

23

2

2

The number of molecules per unit volume striking a surface, per unit time, is just nvZ

31

21

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 pV, we get:

TRH

TRH

mkTp

mkTp

mkT

kTpZ

g

L

g

L

H

H

v

H

v

0

0 exp12

123

61

TRH

TRHpp

g

L

g

Lv

00 expusing

Energy BalanceEnergy Balance1. Heating: radiation absorbed from the Sun, with

efficiency (1-Av)2. Cooling:

a) Reradiation in the thermal infrared, with efficiency (1-AIR)b) Sublimation carries off an energy 4R2ZHL

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.

SublimationSublimation

• 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

TRH

TRH

mkTpZ

g

L

g

L

H 0

0 exp12

For water:1225 2731276.20exp2731038.1

sm

TTKZ

Next lecture: More cometsNext lecture: More comets•Composition•Tail formation•Orbits