1357:
1069:
877:
1274:
1064:{\displaystyle z={\sqrt {\chi _{2}^{2}(p)}}{\begin{pmatrix}\cos \theta \\\sin \theta \end{pmatrix}}\Rightarrow x(\theta )=\mu +{\sqrt {\chi _{2}^{2}(p)}}\Sigma ^{1/2}{\begin{pmatrix}\cos \theta \\\sin \theta \end{pmatrix}}}
1448:
Golombek, M.; Kipp, D.; Warner, N.; Daubar, I. J.; Fergason, R.; Kirk, R. L.; Beyer, R.; Huertas, A.; Piqueux, S.; Putzig, N. E.; Campbell, B. A.; Morgan, G. A.; Charalambous, C.; Pike, W. T.; Gwinner, K. (2017-10-01).
292:
639:
65:
have no aerial factors.) By aggregating such numerous variables it is possible to model a spacecraft's landing zone to a certain degree of precision. By simulating entry under varying conditions an
765:
and is the definition of the chi-squared statistic used to construct the resulting distribution. So for the bivariate
Gaussian distribution, the boundary of the ellipse at a given percentile is
763:
869:
701:
820:
523:
1113:
336:
138:
1332:
461:
1151:
421:
1301:
1175:
386:
358:
1321:
551:
488:
57:, the landing point of a spacecraft will depend upon the degree of control (if any), entry angle, entry mass, atmospheric conditions, and drag. (Note that
17:
1183:
1432:
154:
559:
464:
145:
1504:
1154:
1494:
1323:
then define the landing footprint for a given level of confidence, which is expressed through the choice of percentile.
1280:
1499:
706:
829:
530:
89:
647:
768:
1116:
46:
493:
1077:
303:
16:
111:
526:
434:
1121:
97:
73:
1362:
85:
396:
1470:
1450:
1428:
1286:
389:
93:
54:
21:
1160:
371:
1462:
1418:
1410:
72:
can be calculated; the size of the ellipse represents the degree of uncertainty for a given
28:
343:
1306:
536:
473:
361:
1488:
144:. It is commonly assumed that the resulting distribution of landing sites follows a
84:
To create a landing footprint for a spacecraft, the standard approach is to use the
1403:"Stardust Entry: Landing and Population Hazards in Mission Planning and Operations"
1342:
1337:
364:
1269:{\displaystyle \Sigma =V\Lambda V^{T}\implies \Sigma ^{1/2}=V\Lambda ^{1/2}V^{T}}
424:
66:
1402:
1401:
Tooley, Jeff; Lyons, Daniel; Desai, Prasun; Wawrzyniak, Geoffrey (2006-08-21).
1466:
1352:
468:
50:
1474:
1423:
101:
1379:
105:
62:
58:
1414:
141:
69:
41:
823:
287:{\displaystyle f(x)={\frac {1}{2\pi {\sqrt {|\Sigma |}}}}\exp \left}
525:
with a multivariate
Gaussian joint distribution, the square of the
467:
from the numerical simulations, an ellipse can be calculated for a
634:{\displaystyle (x-\mu )^{T}\Sigma ^{-1}(x-\mu )\sim \chi _{n}^{2}}
15:
1407:
AIAA/AAS Astrodynamics
Specialist Conference and Exhibit
1409:. American Institute of Aeronautics and Astronautics.
1028:
920:
1309:
1289:
1186:
1163:
1124:
1080:
880:
832:
771:
709:
650:
562:
539:
496:
476:
437:
399:
374:
346:
306:
157:
114:
338:
is the vector containing the longitude/latitude pair
1333:
List of landing ellipses on extraterrestrial bodies
1315:
1295:
1268:
1169:
1145:
1107:
1063:
863:
814:
757:
695:
633:
545:
517:
482:
455:
415:
380:
352:
330:
286:
132:
53:'s landing zone on an astronomical body. After
96:and atmospheric parameters, solve the reentry
758:{\displaystyle Q=z_{1}^{2}+\cdots +z_{n}^{2}}
8:
490:. It is known that for a real-valued vector
1213:
1209:
864:{\displaystyle {\sqrt {\chi _{2}^{2}(p)}}}
1422:
1308:
1288:
1260:
1246:
1242:
1222:
1218:
1203:
1185:
1162:
1133:
1129:
1123:
1079:
1023:
1013:
1009:
988:
983:
977:
915:
898:
893:
887:
879:
844:
839:
833:
831:
797:
792:
776:
770:
749:
744:
725:
720:
708:
668:
661:
649:
625:
620:
589:
579:
561:
538:
509:
505:
504:
495:
475:
436:
408:
400:
398:
373:
345:
305:
255:
245:
219:
195:
187:
185:
173:
156:
113:
696:{\displaystyle z=\Sigma ^{-1/2}(x-\mu )}
644:This can be seen by defining the vector
1451:"Selection of the InSight Landing Site"
1370:
815:{\displaystyle z^{T}z=\chi _{2}^{2}(p)}
1157:of the covariance matrix, from which
518:{\displaystyle x\in \mathbb {R} ^{n}}
7:
1108:{\displaystyle \theta \in [0,2\pi )}
826:centered at the origin with radius
1290:
1239:
1215:
1196:
1187:
1164:
1126:
1006:
658:
586:
447:
405:
375:
331:{\displaystyle x=(\lambda ,\phi )}
252:
192:
14:
1378:Lakdawalla, Emily (13 May 2008).
1355:
133:{\displaystyle (\lambda ,\phi )}
146:bivariate Gaussian distribution
1210:
1102:
1087:
1000:
994:
965:
959:
953:
910:
904:
856:
850:
809:
803:
690:
678:
610:
598:
576:
563:
456:{\displaystyle (\mu ,\Sigma )}
450:
438:
409:
401:
325:
313:
276:
264:
242:
229:
196:
188:
167:
161:
127:
115:
1:
1146:{\displaystyle \Sigma ^{1/2}}
871:, leading to the equations:
1521:
1467:10.1007/s11214-016-0321-9
416:{\displaystyle |\Sigma |}
20:The landing footprint of
1296:{\displaystyle \Lambda }
1155:eigenvalue decomposition
531:chi-squared distribution
427:of the covariance matrix
100:, and catalog the final
94:initial entry conditions
80:Mathematical explanation
1382:. The Planetary Society
1283:lie on the diagonal of
1170:{\displaystyle \Sigma }
381:{\displaystyle \Sigma }
1317:
1297:
1270:
1171:
1153:can be found from the
1147:
1109:
1065:
865:
816:
759:
697:
635:
547:
519:
484:
457:
417:
382:
354:
332:
288:
134:
31:
29:Meridiani Planum, Mars
1505:Statistical intervals
1455:Space Science Reviews
1318:
1298:
1271:
1172:
1148:
1110:
1066:
866:
817:
760:
698:
636:
548:
520:
485:
458:
418:
383:
355:
333:
289:
135:
19:
1495:Spaceflight concepts
1307:
1287:
1184:
1161:
1122:
1078:
878:
830:
824:equation of a circle
769:
707:
648:
560:
553:degrees of freedom:
537:
527:Mahalanobis distance
494:
474:
435:
431:Once the parameters
397:
372:
353:{\displaystyle \mu }
344:
304:
155:
112:
1415:10.2514/6.2006-6412
1177:can be written as:
993:
903:
849:
802:
754:
730:
630:
98:equations of motion
74:confidence interval
47:area of uncertainty
1363:Spaceflight portal
1313:
1293:
1266:
1167:
1143:
1117:matrix square root
1115:is the angle. The
1105:
1061:
1055:
979:
947:
889:
861:
835:
812:
788:
755:
740:
716:
693:
631:
616:
543:
515:
480:
453:
413:
378:
350:
328:
284:
130:
86:Monte Carlo method
32:
1500:Atmospheric entry
1434:978-1-62410-048-2
1380:"Landing Ellipse"
1316:{\displaystyle x}
1003:
913:
859:
703:, which leads to
546:{\displaystyle n}
483:{\displaystyle p}
390:covariance matrix
227:
203:
200:
55:atmospheric entry
36:landing footprint
1512:
1479:
1478:
1445:
1439:
1438:
1426:
1424:2060/20070022844
1398:
1392:
1391:
1389:
1387:
1375:
1365:
1360:
1359:
1358:
1322:
1320:
1319:
1314:
1303:. The values of
1302:
1300:
1299:
1294:
1275:
1273:
1272:
1267:
1265:
1264:
1255:
1254:
1250:
1231:
1230:
1226:
1208:
1207:
1176:
1174:
1173:
1168:
1152:
1150:
1149:
1144:
1142:
1141:
1137:
1114:
1112:
1111:
1106:
1070:
1068:
1067:
1062:
1060:
1059:
1022:
1021:
1017:
1004:
992:
987:
978:
952:
951:
914:
902:
897:
888:
870:
868:
867:
862:
860:
848:
843:
834:
821:
819:
818:
813:
801:
796:
781:
780:
764:
762:
761:
756:
753:
748:
729:
724:
702:
700:
699:
694:
677:
676:
672:
640:
638:
637:
632:
629:
624:
597:
596:
584:
583:
552:
550:
549:
544:
524:
522:
521:
516:
514:
513:
508:
489:
487:
486:
481:
462:
460:
459:
454:
422:
420:
419:
414:
412:
404:
387:
385:
384:
379:
359:
357:
356:
351:
337:
335:
334:
329:
293:
291:
290:
285:
283:
279:
263:
262:
250:
249:
228:
220:
204:
202:
201:
199:
191:
186:
174:
139:
137:
136:
131:
38:, also called a
1520:
1519:
1515:
1514:
1513:
1511:
1510:
1509:
1485:
1484:
1483:
1482:
1447:
1446:
1442:
1435:
1400:
1399:
1395:
1385:
1383:
1377:
1376:
1372:
1361:
1356:
1354:
1351:
1329:
1305:
1304:
1285:
1284:
1256:
1238:
1214:
1199:
1182:
1181:
1159:
1158:
1125:
1120:
1119:
1076:
1075:
1054:
1053:
1041:
1040:
1024:
1005:
946:
945:
933:
932:
916:
876:
875:
828:
827:
772:
767:
766:
705:
704:
657:
646:
645:
585:
575:
558:
557:
535:
534:
503:
492:
491:
472:
471:
433:
432:
395:
394:
370:
369:
342:
341:
302:
301:
251:
241:
215:
211:
178:
153:
152:
110:
109:
82:
12:
11:
5:
1518:
1516:
1508:
1507:
1502:
1497:
1487:
1486:
1481:
1480:
1440:
1433:
1393:
1369:
1368:
1367:
1366:
1350:
1347:
1346:
1345:
1340:
1335:
1328:
1325:
1312:
1292:
1277:
1276:
1263:
1259:
1253:
1249:
1245:
1241:
1237:
1234:
1229:
1225:
1221:
1217:
1212:
1206:
1202:
1198:
1195:
1192:
1189:
1166:
1140:
1136:
1132:
1128:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1072:
1071:
1058:
1052:
1049:
1046:
1043:
1042:
1039:
1036:
1033:
1030:
1029:
1027:
1020:
1016:
1012:
1008:
1002:
999:
996:
991:
986:
982:
976:
973:
970:
967:
964:
961:
958:
955:
950:
944:
941:
938:
935:
934:
931:
928:
925:
922:
921:
919:
912:
909:
906:
901:
896:
892:
886:
883:
858:
855:
852:
847:
842:
838:
822:. This is the
811:
808:
805:
800:
795:
791:
787:
784:
779:
775:
752:
747:
743:
739:
736:
733:
728:
723:
719:
715:
712:
692:
689:
686:
683:
680:
675:
671:
667:
664:
660:
656:
653:
642:
641:
628:
623:
619:
615:
612:
609:
606:
603:
600:
595:
592:
588:
582:
578:
574:
571:
568:
565:
542:
512:
507:
502:
499:
479:
452:
449:
446:
443:
440:
429:
428:
411:
407:
403:
392:
377:
367:
362:expected value
349:
339:
327:
324:
321:
318:
315:
312:
309:
295:
294:
282:
278:
275:
272:
269:
266:
261:
258:
254:
248:
244:
240:
237:
234:
231:
226:
223:
218:
214:
210:
207:
198:
194:
190:
184:
181:
177:
172:
169:
166:
163:
160:
129:
126:
123:
120:
117:
81:
78:
13:
10:
9:
6:
4:
3:
2:
1517:
1506:
1503:
1501:
1498:
1496:
1493:
1492:
1490:
1476:
1472:
1468:
1464:
1460:
1456:
1452:
1444:
1441:
1436:
1430:
1425:
1420:
1416:
1412:
1408:
1404:
1397:
1394:
1381:
1374:
1371:
1364:
1353:
1348:
1344:
1341:
1339:
1336:
1334:
1331:
1330:
1326:
1324:
1310:
1282:
1261:
1257:
1251:
1247:
1243:
1235:
1232:
1227:
1223:
1219:
1204:
1200:
1193:
1190:
1180:
1179:
1178:
1156:
1138:
1134:
1130:
1118:
1099:
1096:
1093:
1090:
1084:
1081:
1056:
1050:
1047:
1044:
1037:
1034:
1031:
1025:
1018:
1014:
1010:
997:
989:
984:
980:
974:
971:
968:
962:
956:
948:
942:
939:
936:
929:
926:
923:
917:
907:
899:
894:
890:
884:
881:
874:
873:
872:
853:
845:
840:
836:
825:
806:
798:
793:
789:
785:
782:
777:
773:
750:
745:
741:
737:
734:
731:
726:
721:
717:
713:
710:
687:
684:
681:
673:
669:
665:
662:
654:
651:
626:
621:
617:
613:
607:
604:
601:
593:
590:
580:
572:
569:
566:
556:
555:
554:
540:
532:
528:
510:
500:
497:
477:
470:
466:
444:
441:
426:
393:
391:
368:
366:
363:
347:
340:
322:
319:
316:
310:
307:
300:
299:
298:
280:
273:
270:
267:
259:
256:
246:
238:
235:
232:
224:
221:
216:
212:
208:
205:
182:
179:
175:
170:
164:
158:
151:
150:
149:
147:
143:
124:
121:
118:
107:
103:
99:
95:
91:
90:distributions
87:
79:
77:
75:
71:
68:
64:
60:
56:
52:
48:
44:
43:
37:
30:
26:
24:
18:
1458:
1454:
1443:
1406:
1396:
1384:. Retrieved
1373:
1343:Moon landing
1338:Mars landing
1278:
1073:
643:
430:
423:denotes the
296:
88:to generate
83:
39:
35:
33:
22:
1461:(1): 5–95.
1281:eigenvalues
425:determinant
23:Opportunity
1489:Categories
1349:References
1279:where the
469:percentile
51:spacecraft
1475:1572-9672
1291:Λ
1240:Λ
1216:Σ
1211:⟹
1197:Λ
1188:Σ
1165:Σ
1127:Σ
1100:π
1085:∈
1082:θ
1051:θ
1048:
1038:θ
1035:
1007:Σ
981:χ
972:μ
963:θ
954:⇒
943:θ
940:
930:θ
927:
891:χ
837:χ
790:χ
735:⋯
688:μ
685:−
663:−
659:Σ
618:χ
614:∼
608:μ
605:−
591:−
587:Σ
573:μ
570:−
501:∈
465:estimated
448:Σ
442:μ
406:Σ
376:Σ
348:μ
323:ϕ
317:λ
274:μ
271:−
257:−
253:Σ
239:μ
236:−
217:−
209:
193:Σ
183:π
142:touchdown
125:ϕ
119:λ
102:longitude
63:asteroids
45:, is the
1327:See also
106:latitude
67:probable
61:and the
59:the Moon
40:landing
388:is the
360:is the
297:where:
70:ellipse
42:ellipse
1473:
1431:
1074:where
529:has a
365:vector
1386:7 May
533:with
108:pair
49:of a
25:rover
1471:ISSN
1429:ISBN
1388:2018
463:are
1463:doi
1459:211
1419:hdl
1411:doi
1045:sin
1032:cos
937:sin
924:cos
206:exp
140:at
92:of
27:on
1491::
1469:.
1457:.
1453:.
1427:.
1417:.
1405:.
148::
76:.
34:A
1477:.
1465::
1437:.
1421::
1413::
1390:.
1311:x
1262:T
1258:V
1252:2
1248:/
1244:1
1236:V
1233:=
1228:2
1224:/
1220:1
1205:T
1201:V
1194:V
1191:=
1139:2
1135:/
1131:1
1103:)
1097:2
1094:,
1091:0
1088:[
1057:)
1026:(
1019:2
1015:/
1011:1
1001:)
998:p
995:(
990:2
985:2
975:+
969:=
966:)
960:(
957:x
949:)
918:(
911:)
908:p
905:(
900:2
895:2
885:=
882:z
857:)
854:p
851:(
846:2
841:2
810:)
807:p
804:(
799:2
794:2
786:=
783:z
778:T
774:z
751:2
746:n
742:z
738:+
732:+
727:2
722:1
718:z
714:=
711:Q
691:)
682:x
679:(
674:2
670:/
666:1
655:=
652:z
627:2
622:n
611:)
602:x
599:(
594:1
581:T
577:)
567:x
564:(
541:n
511:n
506:R
498:x
478:p
451:)
445:,
439:(
410:|
402:|
326:)
320:,
314:(
311:=
308:x
281:]
277:)
268:x
265:(
260:1
247:T
243:)
233:x
230:(
225:2
222:1
213:[
197:|
189:|
180:2
176:1
171:=
168:)
165:x
162:(
159:f
128:)
122:,
116:(
104:/
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