We have seen (Sect. 2.9.3.1) that if the thermal parameter, Θ, approaches
100, then the surface of an object becomes isothermal with rotation. For a thermal
inertia of 50 TIU this implies that rotation faster than 0.7 rad s
À1 will result in a
particle that is thermally a fast rotator and the net rocket effect will tend to zero. We
can look at the proportionalities for torque-induced rotation with a simple set of
assumptions.
The angular acceleration is
dΩ d
dt
¼
d l  F A
I
ð4:114Þ
where F A is the applied force, d l is the lever arm with respect to the rotation axis, and
I is the moment of inertia. The net imbalancing force can be written as the change in
momentum (as above) with sublimation from a surface area that is proportional to
the square of the particle radius. For this exercise we assume that a torque arises from
a sublimating area that is ¼ of the cross-sectional area and directed in the plane
orthogonal to the rotation axis. This gives
F A ¼ Z
πa
2
4
m H2O v H2O
ð4:115Þ
If we further assume that the lever arm, d l , is also a function of radius (we will use
2a/3) and if a moment of inertia for a filled sphere is used we can arrive at the
equation
dΩ d
dt
¼
5
16
Z
m H2O v H2O
ρ d a 2
ð4:116Þ
For decimetre-sized (0.1 m) particles, the resulting angular acceleration is of the
order of 10
À3 rad s
À2 assuming free sublimation and a water vapour ejection velocity
of 100 m s
À1 . This result implies that a decimetre particle would become isothermal
in around 1000 seconds. The numerical constant on the right-hand side depends on
the numerous assumptions that have been made (in particular the surface area of the
imbalancing force and the effective sublimation rate, both of which could an order of
magnitude smaller). However, the fundamental points are that the angular acceleration is proportional to 1/a
2 and that any net rocket effect from sub-decimetre
particles will be lost within, at most, a few hours as the surface temperature becomes
isothermal.
4.11.3 Neck-Lines and Dust Trails
Another phenomenon associated with the presence of slow moving large particles is
commonly referred to as a “neck-line structure”. Figure 4.59 shows this structure in a
370
4 Dust Emission from the Surface
100, then the surface of an object becomes isothermal with rotation. For a thermal
inertia of 50 TIU this implies that rotation faster than 0.7 rad s
À1 will result in a
particle that is thermally a fast rotator and the net rocket effect will tend to zero. We
can look at the proportionalities for torque-induced rotation with a simple set of
assumptions.
The angular acceleration is
dΩ d
dt
¼
d l  F A
I
ð4:114Þ
where F A is the applied force, d l is the lever arm with respect to the rotation axis, and
I is the moment of inertia. The net imbalancing force can be written as the change in
momentum (as above) with sublimation from a surface area that is proportional to
the square of the particle radius. For this exercise we assume that a torque arises from
a sublimating area that is ¼ of the cross-sectional area and directed in the plane
orthogonal to the rotation axis. This gives
F A ¼ Z
πa
2
4
m H2O v H2O
ð4:115Þ
If we further assume that the lever arm, d l , is also a function of radius (we will use
2a/3) and if a moment of inertia for a filled sphere is used we can arrive at the
equation
dΩ d
dt
¼
5
16
Z
m H2O v H2O
ρ d a 2
ð4:116Þ
For decimetre-sized (0.1 m) particles, the resulting angular acceleration is of the
order of 10
À3 rad s
À2 assuming free sublimation and a water vapour ejection velocity
of 100 m s
À1 . This result implies that a decimetre particle would become isothermal
in around 1000 seconds. The numerical constant on the right-hand side depends on
the numerous assumptions that have been made (in particular the surface area of the
imbalancing force and the effective sublimation rate, both of which could an order of
magnitude smaller). However, the fundamental points are that the angular acceleration is proportional to 1/a
2 and that any net rocket effect from sub-decimetre
particles will be lost within, at most, a few hours as the surface temperature becomes
isothermal.
4.11.3 Neck-Lines and Dust Trails
Another phenomenon associated with the presence of slow moving large particles is
commonly referred to as a “neck-line structure”. Figure 4.59 shows this structure in a
370
4 Dust Emission from the Surface