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Determination of Design Meteoroid Mass for a Sporadic and Stream Meteoroid Environment

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Determination of Design Meteoroid Mass for a Sporadic and Stream Meteoroid Environment -- ,I DETERMINATION OF DESIGN METEOROID MASS FOR A SPORADIC AND STREAM METEOROID ENVIRONMENT by Donuld J. Kessler und Robert L. Patterson ', Munned Spucecruft Center I Houston, Texus , P NASA TN D-2828 TECH LIBRARY KAFB, NM DETERMINATION OF DESIGN METEOROID MASS FOR A SPORADIC AND STREAM METEOROID ENVIRONMENT By Donald J. Kessler and Robert L. Patterson Manned Spacecraft Center Houston, Texas NATIONAL AERONAUTICS AND SPACE ADMINLSTRATION For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - Price $2.00 DETERMTNATION OF DESIGN METEOROID MASS FOR A SPORADIC AND STRW METEOROID ENV1RO"T By Donald J. Kessler and Robert I;. Patterson Manned Spacecraft Center SUMMARY Meteor influx rates vary throughout the year, and, accordingly, the hazard to spacecraft changes. The total meteor activity can be represented as a fairly constant sporadic background with periods of increased activity associated with meteor streams. For a given probability of impact, the design meteoroid mass is a function of the meteoroid flux, the exposed area, and the time spent in the environment. A method is developed to compute the design mass for exposure to a specified meteoroid environment. INTRODUCTION Meteor activity during the course of a year varies considerably, with rea- sonably well defined, periodically recurring peaks. When the shower or stream meteors are separated from the total activity, the sporadic or background meteor activity is found to remain fairly constant. Thus, the sporadic activity can be presented as an annual average value. Individual stream activity is charac- terized by an increase to a maximum followed by a period of decay. When the periods of activity of a number of streams coincide, a curve of activity as a function of time becomes a complex curve. An example of the stream activity from October to December is illustrated in figure 1. The probability of encountering the design meteoroid mass or size is a function of the meteoroid flux, the exposed area, and the time spent in the environment. For the assumed time-invariant sporadic flux, it is a simple matter to compute the design mass, but for the time-variable stream flux, the calculation becomes more complicated. This paper presents a generalized method for determining the design meteoroid mass. zc -Individual stream flux ratio _-- - - Accumulative stream flux ratio 1 A"""'* 3 F 2 1 0 Figure 1. - Ratio of accumulative meteoroid stream flux to the sporadic meteoroid flux for masses 5 grams SYMBOLS A exposed area, (1 planetary shielding factor), sq ft *total - F stream-to-sporadic flux ratio m meteoroid mass, grams N cumulative flux, number/ft2-day of masses 2 m n number (for example, number of concurrent streams) probability of not encountering a particle of mass 2 m t time v average velocity of the meteoroid stream, km/sec a mete oroid f lux-ma s s constant 7 spacecraft exposure time, days 2 P defined in equation (13) Subs crip t s : SP sporadic meteoroids st meteoroid streams 0 streams and sporadics Superscripts: P, 8 exponents of variables in flux-mass relation THEORY The probability Po of not being hit by a particle of mass m, or larger, in time T on an exposed area A is given in reference 1 as -NAT Po = e The meteoroid stream flux equation of reference 2 can be written in gen- eral fom as or as -p -6 N=m V aF (2) Combining equations (1) and (2) and assuming the special case that F is a constant throughout the time interval T In general, however, F is a function of time t, and equation (3) in logarithmic form becomes c 3 If tl is 0, the elapsed time t2 - tl, or 7 as previously defined, is numerically equal to tg. Solving for mp and rewriting the limits of the integrat ion To illustrate the effect on the design mass m of various time dependent flux distributions F(t), consider then, mB = k17F(t)dt (58) The following examples are idealized flux distributions for a period of stream activity 7 0‘ The first case is for a constant flux C throughout the exposure time F =C 0 7 n~~=k/D’~Cdt 0 7 mp = kCp0) t 0 Case 1 For the second case of F varying linearly from 0 to C at t = 7 0’ C F =-t 7 0 7 mB=klo$-tdt0 0 7 0 t mp = kc(,--) Case 2 4 1- 7 In the third case, F increases linearly from 0 to C at t =- 0 2 and then decreases to 0 at t = T 0 2c F =-tr 0 = 2c(l-:) r 7 O I-0 0 O - I- 2 mp = klF 72Ct dt + kl 2C (1 - $-)dt Case 3 0 -0 2 mp = kc,(?) In the fourth case, F--2G (0 s 7 t ;) 0 Fc F =C 7 On --2ro 0 33 t F =3C(L-p) Case 4 I - 0 3 7 m = kl' dt + kl C dt +lo3C(l - t)dt 0 0 270 2T0 mB = kC - 3 The preceding integrations are based on the assumption that the exposure includes the entire stream period 7 However, if the time of exposure 7 is 0' less than To, the design mass could be expressed as a function of I-PO. This has been done for each of the preceding cases, and the result is sham in 5 figure 2, where a design mass of 1 is the value for case 1 and 7 = 70' Thus, the relative severity of various types of idealized distributions has been established. APPLICATION: STREAM METEOROIDS LO- The flux equation for stream meteor- a9 - oids from reference 2 is aa - 34 v-2. 68 m.-I. N= (6) P ar- c 2.92 x 106 a E (L6- r Substituting equation (6) into equa- -case3 tion s as - (1) f 5 44- 2 2.92 x io" loge p0 a3 - When the stream to sporadic flux ratio is time dependent, equation (7) must be written as 0 a2 a4 46 a8 LO 1.34 - -A V rh.0 m 6 2.92 x loge p0 Figure 2, - Design mass as a function drbfor various io stream distributions "I equation (8), the design mass may be determined for any stream as a function of time for a given A and Po. However, several streams my be active at the same time, and it is desirable to determine the design mass for the exposure to all active streams. If n different streams are active simul- taneously, the effect on the design mass in equation (8) would be a change in F(t) and V so that 6 Equation (9) can be simplified as follows: 1.34 m 1.34 1.34 + m = ml ... + n (10) Consider next the design mass for a single stream as being summed Over several time increments, that is or simply This equation may be used to determine a design mass using the design masses for shorter time increments. For example, if (m is the tl’t2 design mass to the 1.34th power for staying in the environmeni from the first day to the beginning of the second day, and 1.34 is the design mass (mt*9t3) for exposure from the second to the third day, then is the design mass to the 1.34th power for exposure to 2 consecutive days. Using the generalities expressed in equations (10) and (12), it is pos- sible to construct a table where m would be calculated for each day and each stream of reference 2. Then, in order to determine the design mass for. several days, m 1’34 for each stream and each day’s exposure could be added together and the 0.746th power taken of the sum. This is the basis for table I, with the parameter CL defined as follows: m = CL (lopCAPo) The equation for computing the values of P of table I is derived by solving equations (13) and (8) for p. 7 /” F(t)dt - %t - 2.92 x io6 v 2.68 When n meteoroid streams and the sporadic meteoroid background are com- bined, the design mass is defined as follows: 0.746 = (% l0geAPJ where - PLa - Psp + Pst Table I includes the values Pst and 1pst for the meteoroid streams throughout the year for an exposure time of 1 day, beginning on the date shown. The F values used from reference 2 are for masses grams. Plots of for any day of the year where = 3.63 X are presented in figure 3. PSP - I 1 1 1 1 --I I. 1- 1 1 L - -1 2- 2- 31 10 20 29 10 20 31 10 20 30 10 20 31 January February March April MY 12 x 10-11 (a) January to May I 2- 2- 1- ddI 10 20 30 10 20 31 10 20 31 10 20 30 10 20 31 June July August September October 01) June to October Figure 3. - Design mass area probability parameter a *u*.swm To illustrate the computations of 26 x 10-11 the design mass, consider the period @om November 25 to November 27 for an exposed spacecraft area of 500 sq ft 24 - and Po = 0.9999. The factor % for 22- the 3 days in question is, @om fig- ure 3(4, % = (22.8 + 24.4 + 23.9) x c1 sp 3.63 x lo-” - = 71.1 X and from equation (15) m 4= 71.1 x 10”’ m = x IO’* gram U 1.46 A computer program, more flexible a- than the method presented in the text, has been prepared to compute the critical 6- mass for various spacecram exposures.
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