T k r
ð Þ ¼ T 0 1 À
2
π
1 þ
R N
r
2 !
À T 0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À
R N
r
2
r
ð3:82Þ
and
T ┴ r
ð Þ ¼
T 0
2
1 þ
R N
r
2
À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À
R N
r
2
r
"
#
ð3:83Þ
It should be noted that there is no description of temperature in the force-free
radial outflow case (Fig. 3.24).
As an aside, Knollenberg et al. (2016) noted that the stationary solution of the
expansion of a free axisymmetric supersonic gas jet into a vacuum can well be
approximated in the far field by a so-called virtual source flow, where all streamlines
are directed radially away from a common source point. This description applies in
the far field, at distances larger than ~10 times the source diameter (Koppenwallner
et al. 1986).
3.4.4 The Knudsen Layer and Low Density Flow
3.4.4.1 The Boltzmann Equation and the Direct Simulation Monte
Carlo Method
At the interface between the subliming ice surface and the vacuum of space, the VDF
cannot be Maxwellian because the surface itself prevents particle motion in one
direction. As was noted above, it is often assumed that the VDF at this interface on a
comet can be described by a half-Maxwellian (see Fig. 3.21). As the gas expands
Fig. 3.24 The ratio of T k to
T ┴ for the free expansion
model. Note the extremely
rapid change in the ratio
close to the nucleus
224
3 Gas Emissions Near the Nucleus
ð Þ ¼ T 0 1 À
2
π
1 þ
R N
r
2 !
À T 0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À
R N
r
2
r
ð3:82Þ
and
T ┴ r
ð Þ ¼
T 0
2
1 þ
R N
r
2
À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 À
R N
r
2
r
"
#
ð3:83Þ
It should be noted that there is no description of temperature in the force-free
radial outflow case (Fig. 3.24).
As an aside, Knollenberg et al. (2016) noted that the stationary solution of the
expansion of a free axisymmetric supersonic gas jet into a vacuum can well be
approximated in the far field by a so-called virtual source flow, where all streamlines
are directed radially away from a common source point. This description applies in
the far field, at distances larger than ~10 times the source diameter (Koppenwallner
et al. 1986).
3.4.4 The Knudsen Layer and Low Density Flow
3.4.4.1 The Boltzmann Equation and the Direct Simulation Monte
Carlo Method
At the interface between the subliming ice surface and the vacuum of space, the VDF
cannot be Maxwellian because the surface itself prevents particle motion in one
direction. As was noted above, it is often assumed that the VDF at this interface on a
comet can be described by a half-Maxwellian (see Fig. 3.21). As the gas expands
Fig. 3.24 The ratio of T k to
T ┴ for the free expansion
model. Note the extremely
rapid change in the ratio
close to the nucleus
224
3 Gas Emissions Near the Nucleus