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Redalyc.THE IMPORTANCE of NUCLEUS ROTATION in DETERMINING the LARGEST GRAINS EJECTED from COMETS Revista Mexicana de Astronomía y Astrofísica ISSN: 0185-1101 [email protected] Instituto de Astronomía México Molina, A. THE IMPORTANCE OF NUCLEUS ROTATION IN DETERMINING THE LARGEST GRAINS EJECTED FROM COMETS Revista Mexicana de Astronomía y Astrofísica, vol. 46, núm. 2, octubre, 2010, pp. 323-330 Instituto de Astronomía Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=57115522010 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative Revista Mexicana de Astronom´ıa y Astrof´ısica, 46, 323–330 (2010) THE IMPORTANCE OF NUCLEUS ROTATION IN DETERMINING THE LARGEST GRAINS EJECTED FROM COMETS A. Molina1,2 Received 2010 February 18; accepted 2010 June 16 RESUMEN Se determina el valor m´aximo del di´ametro de las part´ıculas eyectadas de los n´ucleos cometarios haciendo uso del modelo cl´asico de expansi´on radial de los gases de los cometas. Se destaca la importancia de las fuerzas inerciales, principalmente de las debidas a la rotaci´on del cometa. Se muestra una sencilla expresi´on indicando el valor m´ınimo que debe tener el periodo de rotaci´on del cometa para que la part´ıcula pueda levantarse de la superficie. Los resultados obtenidos son comparados con los valores determinados a partir de medidas de radar. Tambi´ense analiza el cometa Churyumov-Gerasimenko, que alcanzar´ala misi´on ROSETTA en 2014, proponiendo valores para los di´ametros de las mayores part´ıculas que pueden desprenderse del cometa. ABSTRACT The maximum diameter of large boulders ejected from the cometary nuclei is investigated using the classical model of dust grain as dragged out by radially expanding cometary gases. The importance of the inertial forces, and particularly those due to the rotation of the comet is shown. Although a larger dust grain can be lifted from the nucleus surface if the rotation is faster, the rotation period has to be larger than a certain value. A simple expression for this critical value of rotation period is given. Our results are applied to different comets and a comparison with the maximum radius values obtained by radar measurements is made. Finally, comet Churyumov-Gerasimenko, the target of the ROSETTA mission to arrive on 2014, is going to be analyzed and a range for the maximum diameter of the dust grains that can be lifted is proposed. Key Words: comets: general — comets: individual (Halley, Wirtanen, Hyakutake, C/2001 A2 LINEAR, IRAS-Araki-Alcock, Churyumov-Gerasimenko) — interplanetary medium — meteoroids 1. INTRODUCTION eter. Taking into account the conditions to produce electrophonic sounds, Beech (1998) obtained a lower As is known, the cometary nuclei release dust © Copyright 2010: Instituto de Astronomía, Universidad Nacional Autónoma México bound of 1.24 m for the diameter of the largest me- grains as the comets approach the Sun. Small grains teoroids within the Leonid stream. Cometary dust are dominated by solar radiation pressure and leave trails have been observed using space-based tele- the nucleus to form the dust tail of the comet. Large scopes, as in Sykes et al. (1986) who used four grains can produce a meteroid stream. The study broad-band filters at 12, 25, 60 and 100 µm on the of meteoroid streams can be used as a probe of Infrared Astronomical Satellite. Reach, Kelley, & cometary structure (Beech 1998). Thus, knowledge Sykes (2007) observed debris trails of 27 short period of the characteristics of meteoroids can provide us comets due to mm-sized or larger particles using the information on the composition of cometary nuclei. 24 µm camera on the Spitzer Space Telescope. Also, The large grains can be up to a few meter in diam- meteoroid strems have been observed by ground- 1Departamento de F´ısica Aplicada, Facultad de Ciencias, based telescopes. Thus, Ishiguro et al. (2002) ob- Universidad de Granada, Granada, Spain. tained first evidence of a cometary dust trail in opti- 2Instituto de Astrof´ısica de Andaluc´ıa, CSIC, Granada, Spain. cal wavelengths using the 2K CCD camera attached 323 324 MOLINA 2 to the 105 cm Schmidt telescope at the Kiso Obser- + Ω ~rd − (Ω~ · ~rd)Ω~ + 2~vd × Ω~ . (1) vatory. Aside from the direct use of telescopes, these large meteoroids can be studied by means of the ob- Here, ~rd is the nucleus to grain vector, CD is servations of the flashes that occur when they impact the drag coefficient,m ˙ g is the global gas mass loss on the surface of the Moon. Ortiz et al. (2000) ana- rate (also appearing as Q in the literature), vg is the 3 2 lyzed five impact flashes observed on the night side gas velocity, Cp = 1.19 · 10− kg m− , Qp is the of the Moon on 18 November 1999, which were asso- scattering efficiency of the grain, β is the ratio of ciated to the flux of meteoroids of the Leonid meteor radiation pressure to gravitational force of the Sun, shower. They derived a mass range of 1.3 kg to 9 kg G is the gravitational constant, Mc is the mass of for the meteoroid that produced the brightest im- the comet, MS is the mass of the Sun, ~rc is the Sun pact. Assuming that the Leonids are the meteoroids to comet vector, µ = 1 − β, Ω~ is the angular velocity 3 with the lowest density values, 400 kg m− , of the of the rotation of the comet and ~vd = d~rd/dt where all known meteoroid streams (Babadzhanov 2002), t is the time. then that meteoroid was 0.18–0.35 m in diameter. The drag coefficient depends on the shape of the Large ejected grains can also be found around the particle and the flow conditions. Henderson (1976) cometary nuclei in pseudostable orbits. This consti- presented, for spherical particles, accurate expres- tutes a hazard for the spacecrafts visiting the neigh- sions for CD varying over a wide range of flow con- bourhood of a comet (for physical risks of landing on ditions. In this paper, a spherical shape for the par- a comet see Kuhrt, Knollenberg, & Keller 1997) and ticle is assumed and a constant value of 2 for CD is therefore the estimations of the size of the particles employed for all cases. The constant Cp is related 1 that can be orbiting around a cometary nucleus are to β by means of the expression β = CpQp(ρdd)− , of the greatest interest. where ρd is the density and d the diameter of the dust In this paper, I show in § 2 how the diameter particle. Excluding very small grains, the radiation of the largest ejected particle can be obtained. I pressure force increases with decreasing size (Burns, 3 2 use a classical outgassing theory with special at- Lamy, & Soter 1979). The value of 1.19·10− kg m− 1 tention to the rotation of the comet. I apply is obtained from Cp = 3Es(8cGMS)− , where Es is our expressions to comet 1P/Halley, a comet with the mean total solar radiation and c the speed of a very low rotation period (§ 3), and to comet light (see Finson & Probstein 1968). I also assume 46P/Wirtanen, a very fast spinning comet (§ 4). § 5 that the nucleus obliquity is 0. In the above expres- shows a comparison with results on the size of the sion we have used the same nomenclature as that ejected particles from several cometary nuclei ob- employed by Fulle (1997) except for the last three tained by radar measurements. Due to the interest terms, which are due to the rotation of the cometary in Comet 67P/Churyumov-Gerasimenko, the target nucleus, and are not present in the work of Fulle. of the Rossetta mission, I estimate the diameter of However, they can be important as I show in this the largest ejected particle from the nucleus of the paper. comet in § 6. The conclusions are presented in § 7. From equation (1), I obtain an expression for the particle with largest diameter that can be lifted 2. SIZE OF THE LARGEST EJECTED GRAINS from the surface (see, for example Molina, Moreno, & Jim´enez-Fern´andez 2008): Cometary gas of the nucleus drags dust particles from the surface of the comet. Some of those par- 1 3CDm˙ gvg © Copyright 2010: Instituto de Astronomía, Universidad Nacional Autónoma México d = , (2) ticles can be orbiting around the comet describing max ρ g 16πR2 pseudostable orbits (Fulle 1997). Once the particles d eff have left the nucleus surface, they suffer the grav- where R is the radius of the cometary nucleus and 2 2 itational forces of the nucleus and of the Sun, the geff = g −Ω R cos φ (g is the gravitational accelera- solar radiation pressure force, the gas drag force and tion of the comet and φ is the latitude on the surface the inertial forces, which appear because a reference where the dust particle is located). In this paper, I frame attached to the nucleus is considered. The show the importance of the rotation terms in the equation of motion can be written as follows: equation of motion, and, particularly, of the values obtained for d , and I discuss the results obtained 2 max d ~rd 3CDm˙ gvg ~rd for several comets of different rotational periods. For = β − GMc 2 3 2 2 dt 16πCpQp rd comets rotating slowly, Ω R cos φ ≪ g and then µM G 3~r · ~r ~r geff ≈ g. As g = G4πρnR/3, it will be much larger + S d c ~r − ~r + βM G c 3 2 c d S 3 than Ω2R cos2(φ) for comets with τ 2 ≫ 3π/ρ G, rc rc rc n LARGEST PARTICLES EJECTED FROM ROTATING COMETS 325 Fig.
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