Practical relations for assessments of the dusty-gas coma parameters

crossref(2020)

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摘要
<p>To ensure the safety of a spacecraft and efficiency of the instrument operations it is indispensable to have simple (i.e. with minimal number of parameters and which does not require long time simulations) models for the assessments of the dusty-gas coma parameters. The dusty-gas flow preserves typical general features regardless the particular coma model and the characteristics of the real cometary coma. Therefore, elementary models which account only for the main factors (due to the absence of reliable information on the surface and the interior) affecting the dusty-gas motion could be used for rough estimations of integral characteristics and asymptotic behavior of dusty-gas motion (e.g. terminal velocity, and distance and time when it is reached). At the same time, over-simplification of coma representation is undesirable also. For example, the assumption of spherical symmetry of the flow makes no difference between day and night sides of the coma.</p> <p>We propose a simple approximation of the gas coma parameters based on the numerical solutions of axisymmetric coma with different activity of the night side (see for example Crifo et al. 2002). This model is given by heliocentric distance <em>r<sub>h</sub></em>, total production rate <em>Q</em>, radius of the nucleus <em>R<sub>n</sub></em>, surface temperature <em>T<sub>n</sub></em>, mechanism of gas production (surface sublimation or diffusion from interior) and level of activity of the night side <em>a<sub>n</sub></em>. This model is able to reproduce typical anisotropy of gas density distribution in real coma and it better conform the physics of real coma than the frequently used model of spherical expansion. In the polar frame with axis directed to the Sun and polar angle <em>&#966;</em> (solar zenith angle), the spatial distribution of gas density is approximated by third order polynomial of cos(<em>&#966;</em>):</p> <p><img src="data:image/png;base64, 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" alt="" /></p> <p>where coefficients <em>a<sub>i</sub></em> are tabulated for a given mechanism of gas production (surface sublimation or diffusion from interior) and level of activity of the night side <em>a<sub>n</sub></em>. This approximation gives the relative deviation from numerical solution less than 10% in the most part of the coma.</p> <p>For the dust environment it is assumed that dust grains are spherical homogeneous isothermal particles, non-rotating, with invariable mass (i.e. non-sublimating and noncondensing). The grains released from the surface are submitted to the nucleus gravity <em>F<sub>G</sub></em>, the gas drag <em>F<sub>A</sub></em>, and the solar radiation pressure <em>F<sub>S</sub></em>. We do not allow for solar tidal effects, nor for mutual dust collisions, even though these effects are not always negligible. The theoretical basis of such approach is given in Crifo et al. 2005.</p> <p>In order to cover a broad range of physical conditions we use dimensionless description of dust dynamics proposed in Zakharov et al. 2018. In this case it is possible to limit the parameter space to three general dimensionless factors <em>Iv</em>, <em>Fu</em>, <em>Ro</em>. The factor <em>Iv</em> characterizes the efficiency of entrainment of the particle within the gas flow; <em>Fu</em> characterizes the efficiency of gravitational attraction; <em>Ro</em> characterizes the contribution of solar radiation pressure. In contrast to the gas environment, the structure of the dust environment can change drastically depending on the particular combination of <em>Iv</em>, <em>Fu</em>, <em>Ro</em>. Therefore, the spatial distribution of dust density can be approximated only for separate combinations of <em>Iv</em>, <em>Fu</em>, <em>Ro</em> within a certain range. The present study covers the range of 5&#183;10<sup>-6</sup> < <em>Iv</em> < 0.1, 10<sup>-7</sup> < <em>Fu</em> < 3&#183;10<sup>-7</sup>, 1.5&#183;10<sup>-10</sup> <<em>Ro</em> < 6&#183;10<sup>-6</sup> (i.e. 10<sup>-3</sup> < <em>Ro</em>/<em>Fu</em> < 40.0). For the dust density along sunward direction we propose the approximation:</p> <p><img src="data:image/png;base64, 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" alt="" /></p> <p>where coefficients <em>b<sub>i</sub></em> are tabulated for a given mechanism of gas production (surface sublimation or diffusion from interior) and level of activity of the night side <em>a<sub>n</sub></em>. This approximation gives the relative deviation from numerical solution less than 5% for <em>n</em>=4 and 17% for <em>n</em>=3.</p> <p><strong>Acknowledgements:</strong></p> <p>This research was also supported by the Italian Space Agency (ASI) within the ''Partecipazione italiana alla fase 0 della missione ESA Comet Interceptor'' (ASIINAF agreement n.Accordo Attuativo" numero 2020-4-HH.0 ).</p> <p><strong>References:</strong></p> <p>Crifo,J.F., Lukianov,G.A.,&#160; Rodionov,A.V.,&#160; Khanlarov,G.O., Zakharov, V.V., Comparison between Navier&#8211;Stokes and Direct Monte&#8211;Carlo Simulations of the Circumnuclear Coma. I. Homogeneous, Spherical Source, Icarus 156, 249&#8211;268, 2002.</p> <p>Crifo, J.-F., Loukianov, G.A., Rodionov, A.V., Zakharov, V.V., Direct Monte Carlo and multifluid modeling of the circumnuclear dust coma Spherical grain dynamics revisited. Icarus 176, 192&#8211;219, 2005.</p> <p>Zakharov, V.V., Ivanovski, S.L., Crifo, J.-F., Della Corte, V., Rotundi, A. , Fulle, M., Asymptotics for spherical particle motion in a spherically expanding flow, Icarus, Volume 312, p. 121-127, 2018.</p> <p>&#160;</p>
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