On the Coefficients in Meteor Physics Equations aria Yu. Khanukaeva epartment of Applied Mathematics, Moscow Institute of Physics and Technology (State University), Institutsky 9, 141700 olgoprudny, Moscow region, Russia Abstract. rag coefficients of meteoroids have been investigated in various flow regimes. Some new simple approximations for the drag coefficient and convective heat transfer coefficient of the meteoroid are proposed. Analytic solutions of the meteoroid drag equation were obtained using these approximations. Some numerical solutions with all variable coefficients have also been obtained. The comparison of solutions with constant and variable coefficients shows significant differences, depending on the size of meteoroid. INTROUCTION Basic equations of physics of meteoritic phenomena are the drag equation [1] dv 1 m = CρV A (1) dt and the ablation equation [1] dm 1 3 Q = CHρV A, () dt where m meteoroid mass, V its velocity, А middle square area, ρ density of the atmosphere, Q effective enthalpy of ablation, C and C H drag and heat transfer coefficients respectively, which are under consideration in the present work. Mechanical characteristics of the body: position, velocity, deceleration, mass, kinetic energy, strength etc. are the unknown quantities in meteoritic physics problems. The dependences of atmosphere density on height and crosssectional area on body mass and density are given, as well as coefficients in equations should be set. The problem of meteoroid motion with constant coefficients has been accurately discussed in literature. The solution of the problem, including ablation and mechanical fragmentation, was obtained even in the case of a non-isothermal atmosphere []. In fact, these coefficients are not constant over the meteoroid trajectory. The values of C / and C H are respectively the part of momentum and energy of the flow transferred to the body. They depend essentially on the flow regime realized. The range of flow regimes over meteoroids changes from free-molecule in upper atmosphere to continuum with thin shock wave at low altitudes. rag and heat transfer coefficients can be found as the solution of aerodynamic problem in the frame of some particular flow model. But this solution may be incorrect on some parts of the trajectory. An alternative variant is the search of some approximations of the coefficients, applicable for any regime. At first, we will concentrate on the drag coefficient, and then use the results in the analysis of the convective heat transfer coefficient. The value of the drag coefficient for simple geometric forms can be found analytically in free-molecule and continuum limits [3, 4]. Empirical dependences, based on experimental data, are used for the interpolation of these values in transitional regime. The review of the investigations on this method is given in Ref. [5]. The existing approximations are rather bulky expressions, not convenient for the application in meteoritic problems. The present work is devoted to the search of some simple analytical dependence of the drag coefficient on the dimensionless parameters of the problem, in particular on the Reynolds number, and to the solution of the problem accounting this dependence. Special attention is paid to the free-molecule regime. Some analytic solutions are obtained. Also some extension of the results has been done in order to consider the ablation of meteoroids more adequately.
THE RAG COEFFICIENT IN VARIOUS REGIMES It is sufficient to use the Newton theory in order to determine the pressure on the surface of meteoroid, flowing with the stable hypersonic continuum stream [1, 4]. In this case the drag coefficient will depend only on the body geometry. In free-molecule regime the drag coefficient is determined by the nature of the particle-surface interactions, rather then body geometry [3]. It was mentioned in Ref. [6], that incident particles of free-molecule stream can drive out sufficient amount of atoms or molecules from the crystal lattice of the meteoroid material to produce a sort of microexplosion on its surface. Though the speed of the crashed mass outbreak is less then the thermal velocity, corresponding to the evaporation temperature, the resulting jet efficiency is larger in comparison with the case of gas outbreak. Therefore the process leads to the growth of the deceleration and drag coefficient. Considering this effect, the expression for the drag coefficient in free-molecule flow regime C was shown in [6] for hypersonic velocities to reduce to Q Q M mat V 9 vap C = + + +, (3) V М a Qvap 4 Q where M a =9 g/mol air molecular weight, Q vap effective enthalpy of evaporation, Q * effective enthalpy of crashing of meteoroid lattice material, which molecular weight M mat. Equation (1) with the drag coefficient defined by (3) appears to have an analytic solution. It can be written in general form, but here for the sake of shortness and simplicity concrete values of parameters are used. For iron and stony meteoroids M mat М а, Q * 0.6 km /s, Q vap 8 km /s. Using these values, the solution of the problem of meteoroid ballistics in the isothermal atmosphere, which trajectory is a straight line, inclined to horizon with θ, may be presented as 40 V = 10, [ V ] = [ Ve ] = km/ s, (4) 40 0.675hA 1+ exp ρ 1 Ve 10 + msin θ where V e entrance velocity of the meteoroid, h scale height of the atmosphere (for the Earth h=7 km). In transitional flow regime the drag coefficient is often approximated as a function of the Reynolds number, recommended as the main parameter in [5] ρvr Re0 =, (5) µ ( T0 ) γ 1 where r the body size, µ the dynamical viscosity coefficient. T T (1 + ) the stagnation temperature, 0 = Μ where Т is the free stream temperature, γ ratio of specific heats, М Mach number. The model of gas-particles interactions involved defines the dependence of viscosity on the temperature. Here we use the hard spheres model, which gives the viscosity coefficient as the square root of the temperature. NEW APPROXIMATION Re Simple dependence С (Re) was offered by G.A. Tirskiy (oral report). It looks like С 1+ e. This formula qualitatively models the change of the drag coefficient for the spherical shape: in continuum flow Re and С 1, in free-molecule flow Re 0 and С, in agreement with known results [3, 4]. Function С (Re), as it is known from works [1, 3, 5] has a derivative close to zero in the vicinity of Re=0. But the function е -Re has a negative and very large derivative, that is the curve drops sharply in the vicinity of Re=0. Besides, the convexity of the graph of function е -Re is downcast (the second derivative is positive), but real curves have negative convexity. The function Re е has the demanded first and second derivatives. Therefore it seems reasonable to write it as а Re е, where а<1 is a free coefficient which can be determined to fit the experimental data. In order that the formula discussed can be applied to the calculation of the drag coefficients of not only spherical bodies, we will use a general notation of aerodynamical variables in transitional flow regime. Then we obtain
c ( C C ) e С The values of drag coefficient in continuum C c and free-molecule according to known formulas [3, 4]. Reynolds number is defined by formula (5). c a Re = C +. (6) C limits are calculated separately,.5 C 1.5 1 0.5 0 1 10 100 1000 Re 0 Circles М=4.6, Т W /T =4.5; Triangulars М=19, Т W /T =.5; Crosses М=15, Т W /T =1.6. FIGURE 1. Theoretical with а=0.001 (solid line) and experimental data for the drag coefficient of a sphere. The calculations were fulfilled using the spherical shape of the meteoroid. According to experimental data of work [7] the value of free coefficient was found to be equal to а=10-3. The experiments in [7] were fulfilled under different Mach numbers and ratios of body surface Т W and free stream temperatures. Fig. 1 contains these series of experimental points and theoretical curve provided that а=0.001. Free-molecule and continuum limits were taken c traditionally equal to C = and C = 0. 9. SOLUTIONS AN ANALYSIS It was found that the analytic solution of equation (1) with drag coefficient defined by (6), may be written in the following form haρ π erf ρ V = V exp c c ( ) e C + ( C C ) msin θ ρ, (7) where is the product of known constants and body size r = A/ π : r µ γrt ( γ 1) M = a a, γ=1.4, Т =84 К, µ =1.7 10-5 Pа s. If the value of C, defined by (3), is substituted in expression (6) for C, one can obtain a numerical solution of equation (1) over the whole range of heights including the specialties of free-molecule flow regime. Fig. represents the meteoroid velocity change with height under various drag coefficient definitions. Cases a) and b) correspond to different values of entrance mass of the meteoroid m e =3 10 3 kg and m e =3 10-6 kg respectively, other initial parameters being the same, namely entrance velocity V e =1 km/s and inclination angle counted from the horizon, θ=30. The classical solution of equation (1) with constant drag coefficient looks like
C ρha Vсl = Ve exp. (8) msin θ Plotting curve on Fig. a), a numerical calculation of equation (1) including gravity was used in order to avoid full coincidence of curves 1 and. This means that solution (7) barely differs from classical one for a large body. It is rather clear that the change of the drag coefficient gives the change of the deceleration in two times. For massive bodies the absolute value of the drag force acceleration is very small almost over the whole trajectory. Therefore classical solution (8) can be used, when the velocity change of massive meteoroids is discussed. In this case C =1 should be taken, as large bodies move predominantly in continuum flow regime. 1 solution (8) provided C =1.5 solution (7) 3 numerical solution with variable C and C, 4 solution (4) FIGURE. Change of velocity with height of large (a) and small (b) meteoroid. The classical solution gives the most essential divergence for small bodies at high altitudes. One can determine the critical size of meteoroid r c, for which taking into account dependence (6) gives the correction to the classical value V cl no less then 10% at the altitude z=100 km. Taking the ratio of velocities, defined by expressions (7) and (8), equal to 0.9, we obtain the equation for r c. With the meteoroid density being δ=1 g/sm 3, it takes form erf ( ρ) = 0.03. rc ρ Here coefficient also depends on r c. A numerical solution of this equation for the isothermal atmosphere gives the value of r=7 mkm at the altitude z=100 km. As the drag of micrometeoroids takes place at higher altitudes in free-molecule flow regime, the dependence (3) should be used. This will result in even larger difference. The drag will become more intensive as the entrance velocity becomes larger. The curves on Fig. b) shows the velocity change of a small meteoroid. According to its size, it belongs to the class of meteoroids, representing more than one half of 70 million cosmic bodies entering the atmosphere daily. As it was predicted, solution (4) has given much stronger deceleration, than the others. However, it is worth keeping in mind, that the meteoritic body s motion is accompanied by the intensive ablation. Also, what is essential is the mass loss due to the driving out of groups of meteoroid particles ought to be considered in free-molecule regime along with probable melting and evaporation. As Q * <<Q vap, the rate of the
ablation will be an order of magnitude higher than it is usually estimated in the classical theory of meteoritic phenomena. Consequently, the meteor track may be much shorter. The value of Q * was introduced into the general scheme of the effective enthalpy calculation, taken from [8]. According to equation (), the solution of the problem including meteoroid mass change gives a complete mass loss at the altitude of ~70 km for the small body, under consideration. It means, that the curves of velocity change terminate at this altitude and do not reach zero, as it follows from the solution without ablation. The mass change and velocity change (curve 5) of the considered small meteoroid are given on Fig. 3. Curve 1 for the classical solution (8) and curve 3 for the solution with the variable drag coefficient and constant mass are given on Fig. 3a), for the sake of comparison. Variable heat transfer coefficient, used in the calculations of equations (1-), is discussed below. FIGURE 3. Velocity and mass change of small body (r~1 mm). THE HEAT TRANSFER COEFFICIENT The heat transfer to a meteor body is realized by the convective and radiative mechanisms [1]. Following the traditional way, we calculate the heat transfer coefficient of the meteoroid as a simple sum of convective C Hcon and radiative C Hrad components. As possible nonequilibrium states of the gas in the shock layer give no more than 30% change in the heat transfer to the body [9], they are not considered here. In meteor physics the equilibrium heat flows are usually approximated as functions of the air density, velocity and size of the body [1]. We use the approximation of the radiative heat transfer coefficient, given in [8]. It is proposed to use analogous procedures for approximation of the drag and convective heat transfer coefficients. The mechanism of convective heat transfer is similar to the mechanism of momentum transfer. The value of C Hcon decreases along the trajectory [8], that means it changes with flow regime, which may be characterized by the Reynolds number. So, the coefficient may be written in the form analogous to (6). In freemolecule limit total heat transfer coefficient is equal to unit [1]. Therefore for the convective component we have to c write CHcon =1 CHrad and in continuum regime C Hcon ~ 1/ Re, according to [1,8]. Then for any regime, including transitional one, we write ( ) c Re 1 C / 0 Hrad b e СHc = b / Re0 + Re 0, (9) where unknown coefficients b and c should be determined to fit the numerical data.
The convective component, calculated accordingly expression (9) was added to with the radiative component to find total heat transfer coefficient, which was substituted into equation (). The change of components and full coefficient, found in the process of numerical solution of system (1,), is presented on Fig. 4(a-c). Unlike the drag coefficient the heat transfer coefficient substantially influences the solution not only for small, but also for massive bodies. Curve on Fig. 4d) gives the change of mass with height, obtained in the numerical calculations for massive body (R~1 m). Curves 1 and 3 are given for the sake of comparison. They correspond to the solutions with C H =1 and C H =0.01 respectively, which are rather common estimates, used in meteoritic problems. Thus, it can be seen from Fig. 4, that the heat transfer coefficient changes dramatically on the trajectory of the meteoroid. It is not recommended to use constant values of this coefficient in considerations of real meteoritic falls. FIGURE 4. Change of convective, radiative and full heat transfer coefficient with height of massive body (R~1 m) and its mass change, calculated with various values of C H. One more aspect of the problem should be mentioned. It is a shielding effect, produced by ablated vapors of the body and diminishing the heat flux to the surface. The convective heat transfer is predominantly subjected to this phenomenon. We used special method of the effective enthalpy calculation, which is partly suitable for this case. But its accurate discussion is beyond the scope of the present work. CONCLUSIONS The behavior of the meteoroid drag coefficient has been considered in various flow regimes. An effect of microexplosions on the surface of the body on meteoroids drag was taken into account in free-molecule flow. The analytic solution of the drag equation was found with this condition. Simple and convenient approximation of the drag coefficient for any flow regime has been offered. Analytic solutions of the drag equation were found provided various values of the drag coefficient and compared with the numerical solution, obtained using variable drag coefficient, with free-molecule limit value C being variable. It was found the critical size of a micrometeoroid, for which the variability of the drag coefficient is substantial. The present investigation may serve as the justification of constant drag coefficient usage for massive meteoroids and as the proof of impossibility of this simplification in case of small particles. In modeling the meteoroid entry it often comes to deal with middle sizes. In this case the dependences offered in this work are especially convenient. The altitudes, where the boundaries of flow regimes are situated, are not known beforehand, and monitoring the transitions between the regimes is a rather laborious problem. The formulae, given in the present work, fully exclude this problem, as they are universal for all regimes.
The idea of the drag coefficient calculation was extended to the calculation of the heat transfer coefficient. The offered approximation was used in the solution of the problem of the meteoroid motion in the atmosphere with variable mass. The importance of accurate calculation of the heat transfer coefficient was demonstrated in the calculation of the meteoroid mass change with height. The results of the present investigation may be used in modeling of meteoroids entry without restrictions on their sizes and also of artificial objects, moving in the atmosphere with cosmic velocities. ACKNOWLEGMENTS The work was supported by RFBR grant 00-01-06-5. REFERENCES 1. Bronshten, V. A., Physics of Meteoric Phenomena, Reidel, ordrecht, 1983, 356p.. Tirskiy, G. A., Khanukaeva,. Yu., The Model of Interaction of Cosmic Bodies With Nonisothermal Atmosphere, in Near-Earth Astronomy of the XXI Century, edited by M. A. Smirnov at al., INASAS Conference Proceedings, Geos, Moscow, 001, pp. 367-378. (In Russian) 3. Bird, G. А., Molecular Gas ynamics, Clarendon Press, Oxford, 1976, 30p. 4. Lunev, V. V., Hypersonic Aerodynamics, Mashinostroenie, Мoscow, 1975, 37p. (In Russian) 5. Khlopkov, Yu. I., Statistical Modeling in Physical Gasdynamics, MIPT Press, Moscow, 1998, 140p. (In Russian) 6. Stanukovich, K. P., News of AS USSR, Ser. Mech. and Mash 5, 3-8 (1960). (In Russian) 7. Kussoy, M. I., Hortsman, C. C., AIAA Journal 8, N, 315-30 (1970). 8. ReVelle,. O., Planetary Sciences SR-76-1, 1-90 (1976). 9. Tirskiy, G. A., Shcherbak V. G., Chemical and Thermodynamical Nonequilibrium Air Flows under Small and Moderate Reynolds Numbers, MSU Press, Moscow, 1988, 40p. (In Russian)