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Researa REPORT NO. 17, January 1966, NASA Grantf24b-61 When One Encounters a Discussion of the Nature of Lunar Craters and the P

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Researa REPORT NO. 17, January 1966, NASA Grantf24b-61 When One Encounters a Discussion of the Nature of Lunar Craters and the P c - RESEARa REPORT NO. 17, January 1966, NASA GRANTF24b-61 8 fsG- William J. Abrams INTRODUCTION When one encounters a discussion of the nature of lunar t craters and the processes which resulted in their formation, one inevitably finds nlSchrkergsRule" mentioned as a factor to be considered in evaluating the merits of the impact theory . versus other proposed theories of formation. The generally accepted statement of Schr8ter's Rule is (Kopal('), Gilvarry(2)) that the volume of material in the . rim of a lunar crater is roughly equal to the volume of the depression formed below the surrounding ground level. Some authors (Bald~in'~),Khabakov(*) ) however prefer a statement of the rule which implies a strict one-to-one ratio between the rim volume and the crater volume. It is important to realize at this point that SchrSter's . statement was made as a result of direct observation of lunar craters (Schrker (5)) ; in discussions concerning the relative c merits of the various processes of crater formation it is introduced as empirical evidence in support of the meteoric . -a -2- impact origin of lunar cratcro, The rea6onin~behind the me of a statement of Schr8- t~r'sRule in any form as a substantiation of the impact theory of crater fmmation is obvious. If tnt only mechanism causing a crater Is one which displacee the orieinal terrain by digging out a ?it and piling up the original contents of' the pit around its edge, then there can not be more material piled around the pit tnan wa8 originally dug out from the ground. Implicit in this argument is tne assumation that the mechanism which operates on the terrain to produce the depression does not add appreciable material to the system by virtue of its own volume. That Is, the volume of the impacting body must be small compared to the volume of lunar surface which it displaces to form the crater. RIX - CRATLR DISTiEICTIOrJ: In order to determine the relationship between the volume of an impact crater and the volume of material in the rim around the crater it is first necessary to establish a criterion for determining which portions of' a lunar formation constitue the crater and which constitute the rim, h tMa study "crater volume" was taken as any depreaeion below the * - undiaturbed eround level, and "rim volume" as any elevation above the original level in the region of the crater. The problem of distinguishing between rim and crater then resolves to one of finding the original eurface level -3- of the area in wnich a given crater ha8 bcen formed. This la aCCOmpli6hed by determining the undisturbed surface level as near the crater as possible and then making tne assumption that the reelon in which the crater was fomed was originally at the same level as the surrounding regions, or constituted 8 level SU~~CPconnecting the addacent regions, This means of' determinine the original surface level can be used with available lunar surface contour maps; the original level is taken as tne first contour encountered, when proceEding radially from the center of the crater, which does not exhibit polar symmetry about the center of the crater and therefor6 probably existed prlor to the time of the crater's formation. Havins thus established a crlterion for distinguish- ing crater volum~ from rim volume as well as defining the outer limits of the rim extension from the center of the cra- ter, it is possible to test Schr8ter's Rule by using obser- vational data in tne form of lunar contour maps. CALCULATION OF VOKJMES Data for the evaluYtion of Schrtilter'e Eule were taken from multi2le digital crose-sections of selected lunar cra- ters pre2area frolg Aeronautical Chart and Information Service lunar contour maps at Pjoston University ('I. The geometry of' a typical crose-section as adaPted from the Catalogue of Lunar Craters Cross Sectians 2 Is illustrated below: I I I*. -a -4- I- f‘ ’P -2 Tne line 0 is the chosen _=alar axis of the crzter; d is tine height of original ground level DOVE an aroltrary zero reference level; z is the heignt of the post-impact surface above (OF below) €3; f is the radius. l- if z, and ++I are two adjacent points in the aigl- tal cross-section snd are connected by a straight line, f(p), the moment about 0 of the element of cross-aection between Zn and %+I is: -5- By the tkorem of Pappus, the momEnT; of an element of cross-sectional area rcultiplied by an angle of rotation 9 (in radians) about tne axis 0 represents a volum~eleaent of the solid If it is assumed to possess 2olrr spmetry tmough- out that m6le of rotation, Since aome error is introduced By thin assumption into the volume calculations, the resul- tant volumes and tnelr ratios are only meant to be approxi- mztions. In most cases the symmetry anale is only 45' so that the approximations should be better than order-of-magnitude , The probable error in the relative height of a given contour line for the data used in this study was estimated by the ACIS to be 190 meters. This error introduces an uncertainty into the determinat: on of the orleinal surface level of the impact area, and thus also into tne final rim volume and crater volume as shown In Fig. 2 below: Figure 2 + assumed surface level L 'vC (crater volume The ?robable error,6Vr, in the rim volume resulting fron the uncertainty *o(, in original surface level is 3 P .s vr = e (R~-- RIZjoc , iri and the probable error, 6Vc, in the crater volume le -6- dvc = e ~l~d. (4) R1 fs the average rrdius of‘ the crater(dep&ssion); R2 Is the everage outer radius of the rim; dls the uncertainty in height ( 100 iEErters in this case) of the original sur- face level. If a Scussian distribution Is assumed for tne prob- sble error o( the probable error in the ratlo of rlm-to- crater volume is (sce for exznple, Baird(7)) , where R Is the ratio, The mixed term Is retained In equation (7) because SVr and $Vc are not independent perturbations; Fig. 2 shows that when the error* Is such that5 Vr Is positive, then $Vc is negative. A Fortran I1 program (Aopenaix A, ProgrraP 1) wzs writ- ten to perform the volurne calculations, find the ratios of rim-to-crater volume, rna determine tile uncertainty in these ratios. Another program (Appendix I;, Frogram 2) was writte~tc ntit data cerds g, sre~rin~~sS~~L~TZ r-- in J (which produced NASA Report i’o. 16) into conveniat form for input to Proszm 1. -7- RESULTS Values of tni rim volume, tht crattr volu~;ic, tnE ratio of rim volme to crater volume, and the probable error in he ratio have been obtained for tnirtg-nine lunar craters from the output of ProEram 1; these results appear in Table 1, paEe 8. A glace at tk values of atlo and klta Ratio in tne Table, which represent the rim-to-crater volme ratios znd the protable error in those ratioa respectively, indicates that a large gro2ortion of the ratios are smaller then their corres3onding errors, while most of tae remaining values of katio navt errors comparable to their magnitudes. Since the Indicated errors iravc arisen only as a result of the uncer- tainty in thc oriE:inal lunar surface levcl, tm Inglication is that, given a ;robable Error of at lesst lG0 nctero in surface heizht, it ia not possible to distinguish rim volume from crater voluine wiLi sufficient accuracy to ffiake a valid statement concernins iiielr ratio. Furthermore, slnce the same ?robable error of 100 metera applies to every detcrain- atlon of surface altitude use5 in this study, the probable errors in the ratioa determined here represent minimum 9robeble errors since only the tffect of the error &xi the outermost contour heidht has been considered. Also, tne error intro- duced by assuming polar symmetry for a given crater through the anglg 0 has .lot been estimated. In Figure 3-a, tile values obtained for tse ratio of -8- TABLE 1 CRATER RIM VOL CRATER VOL RATIO DELTA RATIO (KM3) (KM3) ABULFEOA 1 10166E 02 50582E 03 2 089E-02 1 340E-0 1 ACDTPOA nunrr a n 2 50l76E C2 20247E 03 2 3 l. 3E-0 1 2 0 Si$€-0 1 ALFRACANUS 3 10514E 02 30386E 02 4 0 47 1E-0 1 40127E-01 ARCHIMEDES 4 40229E 03 10179E 03 3o588E 00 5o133E-02 AUTOLYCUS 5 10736E 03 80073E 02 20150E 00 3 5 36E-0 1 BULLIALOUS 6 10489E 03 30296E 03 4 5 19E-01 2 785E-0 1 CAPELLA 7 80564E 02 90618E 02 80904E-01 5 179E-0 1 CONON 8 5o891E 01 50203E 02 10132E-01 300396-01 COPERNICUS 9 1o389E 04 40763E 03 20916E 00 6 066E-0 I DAM01 SEAU 10 30881E 02 40596E 02 8 443E-0 1 lr031E 00 DOPELMAYER 11 10575E 03 20967E 02 50273E 00 60702E 00 ERATOSTHENES 12 2o950E 03 10710E 03 10725E 00 3 081E-0 1 EULER 13 30280E 02 10933E 02 10696E 00 30398E-01 GASSENDI 14 Lo517E 03 10862E 03 80149E-01 1.197E 00 GASSEMDI A 15 20479E 02 30087E 02 8 032E-01 70503E-01 GASENDI B 16 80112E 01 20465E 02 3 291 E-0 1 8o9SOE-01 GAUDIBERT 17 2o410E 02 10977E 02 10219E 00 10101E 00 GAY LUSSAC 18 30055E 02 10286E 02 20376E 00 30040E-01 GUTENBERG 19 1o912E 03 10833E 03 10043E 00 8 393E-0 1 HAMSTTEN 20 60888E 02 20034E 00 3o387E 02 60758E 04 !<ANT 21 90214E 02 60482E 02 10421E 00 3 547 E-0 1 LANSBERG 22 50994E 02 10057E 03 50668E-01 40504E-01 MAN1 LTUS 23 30383E 02 10048E 03 3 227E-0 1 4 496E-01 MERSENIUS P 24 20862E 02 20832E 02 10011E 00 10327E 00 MOST I NG 25 10237E 03 20538E 02 4o873E 00 10057E 01 PI TAT IUS 26 10785E 03 70887E 02 20263S 00 20040E 00 PLINIUS 27 80953E 02 10996E 03 4 48 2E-0 1 2 5 11 E-0 1 REINER 28 1o432E 03 20641E 02 50422E 00 10099E 01 REINHOLD 29 30413E 02 10598E 04 20136E-02 30687E-02 ROSS 30 2.2976 Q1 30752E 02 60123E-02 3 643E-0 1 SA8XNE 31 10185E 02 30387E 02 30499E-01 6.34lE-01 51RSAL f S 32 40440E 02 10566E 03 20836E-01 30300E-01 TARUNTIUS 33 40142E 02 20091E 02 10981E 00 6 040E-0 1 THEOPHILUS 34 30497E 03 loO16E 04 30437E-01 20298E-01 TIMOCHARIS 35 602336 02 5.5236 02 10129E 00 4 9 17E-0 1 TRI ESNECI'ER 36 1o581E 03 40424E 00 3.574E 02 4o254E 04 UKERT 37 1o47lE 03 20 195E-01 60702E 03 10003E 07 VI TELL0 38 502795 02 40211E 02 10254E 00 90045E-01 ZOLLNER 39 1o797E 03 70691E 02 20336E 00 4a262E-04 0 0 2.i W .
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