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Fig. 14. Schematic diagram indicating the various flow regimes and physical and chemical processes in the cometary coma for a Halley-type comet at 1AU heliocentric distance

Fig. 14. Schematic diagram indicating the various flow regimes and physical and chemical processes in the cometary coma for a Halley-type comet at 1AU heliocentric distance approaches still form a valuable complementary method to compute the gas distribution.

In summary, the choice of mathematical method depends on the coma region and conditions studied:

• Immediately above the surface: Gas-kinetic approach (Boltzmann equation) include upper surface layers, Monte-Carlo models; this region is often called "Knudsen-layer."

• Collisionally dominated coma region: Set of hydrodynamic equations for the gas and dust components.

• Transition region: Description of viscous flow, gas-kinetic and/or hydro-dynamic approach (check both).

• Outer coma region: Free molecular gas flow, Monte-Carlo models.

• Gas and dust tails: Free flow; molecules and dust particles move in the solar gravitational field under the influence of solar radiation pressure on Keplerian trajectories.

Hydrodynamic Description of the Gas Coma Expansion

The inner coma region is studied by space missions and therefore of special interest. Furthermore, within the inner few hundred kilometers above the cometary surface, the starting conditions for the flow observed on a large scale, e.g., from ground, are defined.

If K < 0.1, the flow can be described by the Euler equations for inviscid flow. The mass conservation equation, describing the variation of density, p, by volume changes of the expanding gas and by external gas sources, Q:

Here, the external sources Q are simply given by the comets gas production rate, if no chemical coma reactions are taken into account. When including the production and destruction of gas molecules by chemical processes, changing number densities of each species need to be taken into account in the source term by solving in addition a chemical reaction network (see Sect. 5).

The momentum conservation equation describing the acceleration of a fluid element by pressure gradients, Vp, or external forces, F, e.g., gravitation or gas-dust interaction forces, is given by:

And finally, the energy conservation equation is:

with the energy density:

Here, e is the inner energy and h the specific enthalpy. If gas molecules that move with distinctively different velocities compared to the dominant water molecules are considered (e.g., H atoms), a set of hydrodynamic equations for each of these species is needed, including the source terms for momentum exchange. For ions and electrons, a set of magneto-hydrodynamic equations must be used, taking into account the interaction with the solar wind (see Sect. 3.4).

In addition, there are analog equations for the dust particles, pd, in each size interval i:

The hydrodynamic equations for gas and dust are complemented by the equation of state for an ideal gas:

P(Y - 1) Y - 1 The surface conditions can be expressed as

To = T (1 + 7 _ ) ; u0 = Co = yt^RgTo] p0 = —; p0 = poRgTo (14) V 2 J uo when assuming a reservoir outflow model with the reservoir temperature assumed equal to the nucleus surface temperature. These equations then apply after a few free scale lengths above the surface. We note that the knowledge of the starting conditions at the nucleus surface is an important but difficult constraint for any coma model. For the first few collisions above the surface, there is the Knudsen layer, as described above. However, detailed calculations show that the results of the reservoir model are similar to the detailed Knudsen layer calculation.

As a first estimate for the gas velocity, we can use the equation for the limiting velocity of expansion from a gas reservoir into vacuum:

If we set Tr, the temperature of the reservoir, equal to the surface temperature of an active region of a comet near 1 AU and assume 7 = | for water vapor, we obtain a terminal gas velocity of about 0.86kms-1. This velocity is close to the values measured in the intermediate coma of comets near 1 AU. Figure 15 shows how the gas velocity and temperature change for different cometary gas production rates, Q.

In real comae, the gas flow depends also on heating by dust particles, mass loading by fragmenting dust and, of course, on the heliocentric distance of a comet. A detailed discussion on gas coma velocities is found in [47].

Acceleration of Dust Particles in the Inner Coma and Their Effect on the Gas Flow

The cometary dust particles are coupled to the gas flow for the first few kilometers above the surface. They are accelerated by collisions with gas molecules to velocities of a few 10-100 ms_1, depending on their size, shape, and density. The dynamics of dust particles on a larger scale is discussed in Sect. 3.3.

Fig. 15. Gas velocity and temperature in the coma for different gas production rates [33]. Top diagram: top curve: Q = 1027 s_1, until bottom curve: 1030 s_1. Bottom diagram: again from low to high gas production rates. In the dotted regions, fluid dynamics does not apply anymore

The main factors affecting the acceleration of dust in the cometary coma can be discussed already when looking at the simple scenario of acceleration of a spherical dust particle by the gas flow in the free molecular flow approximation [104]:

dt 2

with:

md 4 pda

The acceleration depends on dust particle mass, md, radius, a, relative gas velocity, u, and dust velocity, ud • Pg denotes the gas density. The main factor affecting the coupling of dust particles to the gas is the cross-sectional area to mass ratio, a. The drag coefficient, CD, depends only slightly on the shape and structure of the dust particles. So we find the following general behavior of dust particle acceleration:

• Light particles are accelerated more efficiently than massive particles.

• For a given density, small particles are accelerated more easily than large particles (Fig. 16). Therefore, small grains follow the gas flow longer than large grains.

• The gas density decreases rapidly with increasing nucleocentric distance. Furthermore, the drag coefficient, CD, is a strong function of temperature that also drops steeply in the inner coma. Therefore, CD decreases in the

Fig. 16. Dust particle acceleration depends on the radius of the particle. The figure shows the dust particle velocity for different radii. The solid line shows for comparison the gas velocity in the dust loaded flow [132]

Fig. 16. Dust particle acceleration depends on the radius of the particle. The figure shows the dust particle velocity for different radii. The solid line shows for comparison the gas velocity in the dust loaded flow [132]

first kilometers above the surface and dust acceleration is efficient only in the innermost coma. • In the intermediate coma, the dust particles decouple from the gas flow and move on trajectories according to their velocity just before decoupling. Further out, gravitation and solar radiation pressure determine their dynamics.

What happens to the gas flow when dust particles are added? The mass loading by dust reduces the initial gas outflow velocity. However, the hotter dust particles then heat the flow and lead to a faster acceleration of the gas. Finally, again gas velocities around 0.86kms-1 are reached for the gas flow in the intermediate coma.

The images obtained by the HMC camera on board the Giotto spacecraft visiting comet Halley provided indications for fragmentation of dust particles in the coma within a few kilometers above the surface. Such small and hot fragmenting dust particles heat the near-nucleus coma and can therefore also modify the near-nucleus dynamics.

Measured Coma Gas Velocities and Temperatures

Figure 17 shows the velocity of H2O gas measured in situ at comet Halley by the Giotto spacecraft [134,139]. The acceleration of the molecules can be seen. In the inner coma, velocities are near the value predicted from adiabatic gas expansion (0.86kms-1). The velocity reaches 1.1kms_1 within the first 40000 km above the nucleus.

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