AIAA/AAS Astodynamics Specialist Confeence and Exhibit 5-8 August 22, Monteey, Califonia AIAA 22-4887 Tue Satellite Ballistic Coefficient Detemination fo HASDM Buce R. Bowman Ai Foce Space Command, Space Analysis Cente Peteson AFB, Coloado 8914-465 Abstact. The High Accuacy Satellite Dag Model (HASDM) equies tue satellite ballistic coefficients (B) in ode to estimate accuate atmospheic tempeatue/density coections. Using satellite tacking data tue B values wee obtained fo ove 4 satellites that have been in obit since 197. Diffeential obit coections wee computed fom 197 to 21 evey 3 days thoughout the 31-yea peiod fo each satellite. The tue B values wee computed by aveaging the nealy 32 estimated B values obtained fo each satellite. These tue B values wee validated by compaing the tue B values of two sphees with theoetical values based on thei known physical dimensions, and by compaing the tue B values obtained fo pais of satellites having vey simila size, shape, and mass. The estimated B vaiations fo a numbe of satellites wee then aveaged ove each yea fom 197 to 21, and compaed with sola indices plotted fo the last thee sola cycles. Intoduction Fo HASDM satellite dag is used to compute atmospheic density vaiations. Since the dag equations have the ballistic coefficient, B, and atmospheic density value factoed togethe, the accuate computation of the density can be done only if the tue ballistic coefficients of the satellites ae know to within a small eo. Thus, it was necessay to detemine the tue B values of as many HASDM satellites as possible to enable the HASDM estimation of accuate atmospheic density vaiations. The tue ballistic coefficient is given by B = C D A/M, whee C D is the dimensionless dag coefficient, A is the coss-section aea of the silhouette of the satellite viewed in the diection of satellite motion, and M is the mass of the satellite. The ballistic coefficient estimated duing the obit detemination pocess is significantly diffeent fom the tue B, due to eos in the atmospheic model poviding neutal density. To diffeentiate this estimated value fom the tue ballistic coefficient B, we denote it as B which is given by: t t 3 3 B B ρv dt ρ v dt whee ρ is the tue density, ρ is the model density, v is the velocity of the satellite, and dt is a diffeential element of time which is used to integate ove the fit span t. Stictly speaking, v 3 epesents v ( v el el vsat ), whee v el is the velocity of the satellite elative to the atmosphee and v sat is the velocity of the satellite elative to the inetial coodinate fame. v el is the magnitude of. v el Diffeential Coection Obit Fits A batch least squaes algoithm was used to fit each satellite s obsevational data to obtain B values. The batch diffeential coection (DC) data spans vaied fom 4 to 12 days depending upon the enegy dissipation ate and the numbe of obsevations available pe span. DCs wee computed evey 3 days stating fom Jan 1, 197 until Dec 31, 2. Almost 1, ada and optical obsevations pe satellite wee pocessed ove the 31-yea peiod. Appoximately 3 obs/dc wee available fo the 197-199 peiod, and appoximately 6 obs/dc wee available fo the 199-2 peiod. Appoximately 32 estimated B values wee obtained pe satellite, and these values wee used to obtain an aveage value, the tue B value, ove the 31-yea time peiod. The petubation model employed fo the DC fits used a 48 48 tuncation of the EGM-96 geopotential, lunasola thid body gavitation, solid eath and ocean tides, and the Jacchia 7 atmospheic model fo atmospheic dag. The 48 degee and ode value was selected fo the geopotential because it esults in a significantly faste compute execution speed than the full 7 degee This mateial is declaed a wok of the U.S. Govenment and is not subject to copyight potection in the United States.
model, and the neglected highe degee and ode tems (up to 7 7) poduced less than a one pecent standad deviation eo in the ballistic coefficient due to aliasing. A vaiable dag coefficient (C D ) appoach was equied fo the pocessing. As peviously epoted (Bowman, 21) the dag coefficient will vay with altitude based on the changing molecula constituents of the atmosphee (Afonso, 1985). Nomally the C D value is consideed constant at a value of 2.2 fo oughly spheical objects. Howeve, when sola activity becomes low (F 1.7 < 8) the dominant atmospheic species changes fom atomic oxygen to helium at altitudes as low as 5 km. The dag coefficient with espect to atomic oxygen is appoximately 2.2, while with espect to helium it appoximates 2.8. Above 15 km, when hydogen becomes the dominant species, the C D value is geate than 4.. Theefoe, the B value, the poduct of the dag coefficient and aea-tomass atio, can vay by as much as 8% ove a wide ange of altitudes. If the C D value does not vay based on molecula constituents then the eo in C D will be eflected in an eo in B not attibuted to just neutal density eos. Figue 1 shows the diffeence in the B coefficient fom compaing B fom obit fits using a constant and then vaiable C D value. Seveal satellites wee used to compute B values evey 3 days fo the 31- yea peiod, using fist a constant C D value of 2.2 fo all the obit fits, and then using a vaiable C D value based on molecula species amounts. Figue 1 shows the diffeences (using vaiable C D using constant C D ) in the B value, fo 3 diffeent satellites with peigee heights vaying fom 51 km to 9 km. The diffeences in Figue 1 show that at altitudes of 9 km the B value can be in eo fom the C D change by as much as 18% duing low sola activity. Delta B % 1-1 -2-3 -4 65 km 9 63 km 59 km 51 km 9 km 84 km F1.7 Ave 56 km 77 km 7 75 8 85 9 95 1 Figue 1. Change in B (vaiable C D constant C D ) Figue 2 shows a histoy of B values ove the 31-yea peiod fo satellite 6, Exploe 8, which had a peigee height of appoximately 4 km duing the entie time peiod. Vaiations of as much as 5% with 55 45 35 25 15 5 F1B a standad deviation of 18% can be obseved. Since the satellite is almost spheical in shape the vaiations ae due, not to satellite aea vaiability, but to unmodeled atmospheic density vaiations at 4 km altitude. DB/BTue % 1 5-5 -1 18. % STD 7 75 8 85 9 95 1 Figue 2. B vaiation (pecent diffeence fom tue 31-yea aveage) fo satellite 6, Exploe 8, with a peigee height of 4 km. Validation of Tue B Values The method of computing the tue B value was validated by compaing the tue B values of two sphees with theoetical values computed fom thei known physical dimensions. Vanguad 2 and AE-B wee used in the analysis. Vanguad 2 was launched in 1959 with a peigee height of appoximately 56 km, while AE-B was launched in 1966 with a peigee height of appoximately 28 km. Vanguad 2 is still in obit, while AE-B decayed in 1985. Aveage B values wee computed, fist fo Vanguad 2 using 31 yeas of data, then fo AE-B using 15 yeas of data fom 197 to 1985. Fom analysis of a numbe of othe satellites tue B values, it was detemined that using the 197-1985 time peiod poduced an aveage B that was 2.5% highe than using the 197-2 peiod. Thus, the aveage B obtained fo AE-B was educed by 2.5% to obtain the tue B value. The tue B values fo both satellites wee then compaed to the values computed fom the C D A/M expession. Table 1 shows the compaisons fo a ange of C D values. It has been epoted (Cook, 1965; Moe, 1996; Padini, 1999) that the C D value fo sphees inceases slightly fom 2.2 to 2.4 fo altitudes inceasing fom 25 to 5 km. These C D values ae estimated to have a standad eo of 5%. Using the tue B values and A/M computed fom the dimensions, the C D value can be computed that best fits the 31-yea tue B value. Fo Vanguad 2 the value is appoximately 2.27, while fo the lowe altitude AE-B it is appoximately 2.14. These values ae in vey good ageement with the expected C D values plus o minus
5%. Thus, thee does not appea to be a bias (to within 5%) in the tue B values computed using the 197-2 time peiod. SAT Q Ht A/M CD CD*A/M B31Y/ B 31 Y Inc m2/kg CD*A/M 11 56 km.2158 2.2.4748 3.3 % Vanguad 2 32.9 deg 2.27.4899.1 %.494 2.4.5179-5.3 % 2183 28 km.276 2.1.58 2.1 % AE-B 64.7 deg 2.14.591.2 %.592 2.2.67-2.5 % Table 1. Compaisons of dag coefficients fo two sphees with diffeent peigee heights. Peigee height (Q Ht) and inclination (Inc) ae included. Anothe validation method consisted of compaing the tue B values obtained fo pais of satellites with vey simila size, shape, and mass. Thee pais of satellites wee found that could be consideed the same satellites in size, shape, and weight. Table 2 lists the pais and the esulting tue B values. The fist two pais consist of uppe stage ocket bodies. The fist pai was used to launch the Tios 3 and 4 satellites into nealy identical obits. Since both ae solid fuel ockets, which bun until all the fuel is used, a good assumption is that the empty weights ae the same. Also, since they wee launched within a yea of each othe, it is a good assumption that they wee poduced as the same ocket body model. The tue B values diffe by less than 1% fo this pai. The second pai of satellites ae also uppe stage solid ocket bodies, also used to launch satellites into nealy identical obits. The diffeence of tue B values is less then.1% fo this case. Finally, the last pai consideed is the Elekton 1 and 3 satellites, launched within 6 months of each othe. They wee both placed in vey simila obits with peigee and apogee heights of 4 km and 71 km espectively. Even though the shape of these satellites is complex (cylinde with 6 paddles) the tue B values compaed to within.2%. Thus, the consistency of the tue B values is eadily demonstated. Below in Table 3 is a list of 31-yea tue B values fo a numbe of selected satellites. The shape of the object plus additional obital paametes ae povided fo each satellite. Intenat. NORAD Object Qht Inc B 31Y Diff. Desig. km deg % 61-17B 165 Delta 1 R/B 74 47.9.5113 62-2B 229 Delta 1 R/B 7 48.2.5157.9 % 65-72D 1583 Tho Altai R/B 649 98.6.4513 66-26B 2129 Tho Altai R/B 634 98.6.4512. % 64-6A 746 Elekton 1 395 6.8.163 64-38A 829 Elekton 3 45 6.8.16 -.2 % Table 2. Compaison of tue B values fo pais of simila satellites. 1 Aveage B Values Figues 3 and 4 show yealy aveaged B vaiations fo satellite goups at 4 km peigee altitude in vaious inclination obits, and then in high inclination obits at vaious altitudes, espectively. The vaiations ae a esult of the unmodeled density vaiations using the Jacchia 197 model atmosphee (Jacchia, 197). The yealy B values wee obtained fom a yealy aveaging of the obit fitted B values. The plots show a emakable consistency. The consistency at 4 km fo diffeent inclinations, and at high inclinations fo diffeent altitudes suggests that a global coection should account fo the majoity of the obseved vaiations. Also of inteest is the coelation of the yealy B vaiations with the 9-day aveage sola flux F 1.7 index. The minimum yealy B vaiation occus duing sola minimum, although the minimum B vaiation does not level out as does the sola flux index. This indicates that the vaiation in the F 1.7 index does not account fo all of the long tem vaiation thoughout the sola cycle. 21 Atmospheic Density Vaiations Figue 5 shows the esults of the B vaiations of a numbe of low inclination satellites fo the fist half of 21 when the HASDM data was collected. The B vaiations ae plotted as a pecent deviation fom the tue B value fo each satellite. Also plotted ae the sola flux indices F 1.7 and a p. The peiod fom day 1 to day 75 shows vey little sola vaiability even though the F 1.7 value is still modeately high (~15). Duing this peiod of time, the obseved density inceases moe with height than is accounted fo in the Jacchia model, as seen fom the incease in the B values with height. The highe the B value ove the tue value, the highe the tue atmospheic density elative to the model density. When sola activity vaiations incease afte day 75, the atmospheic density appeas to be moe globally consistent with the pedicted model since the B vaiations fo all the satellites ae close togethe.
Howeve, thee still appeas to be a sepaation by height even duing this peiod of time. Also of inteest is the anti-coelation of the B vaiations with the F 1.7 vaiations. This indicates that the Jacchia model is consistently ovecoecting the density with espect to the sola flux index duing the maxima, and unde coecting duing the minima. The HASDM model must account fo, and emove, all these vaiations in ode to be used as a global density coection model. Conclusion Tue B values can be obtained fom aveaging diffeential obit coected B values ove the time peiod 197-2. These tue B values show selfconsistency when compaed to identical satellites, and compae extemely well with values computed fom dimensions of spheical satellites. The tue B values can be estimated to be accuate to within 2-3% based on the above analysis. Acknowledgment I would like to thank my colleague, M. Mak Stoz, as well as M. William Bake and M. Steve Casali of Omiton Inc. fo thei valuable contibutions and insight. Refeences Afonso, G., F. Balie, C. Bege, F. Mignad, and J. J. Walch, Reassessment of the Chage and Neutal Dag of LAGEOS and its Geophysical Implications, Jou. Geophys Res., 9, 9381, 1985. Bowman, B. R., W. N. Bake, and W. G. Schick, Obit Petubation Analysis of West Fod Needles Clustes, AIAA-2-4236, AAS/AIAA Astodynamics Specialist Confeence, Denve, Coloado, August, 2. Bowman, B. R., Atmospheic Density Vaiations at 15-4 km Height Detemined fom Long Tem Obit Petubation Analysis, AAS-21-132, AAS/AIAA Astodynamics Specialist Confeence, Santa Babaa, Califonia, Febuay, 21. Cook, G. E., Dag Coefficients of Spheical Satellites, Ann. Geophys., 22, 53, 1966. Jacchia, L. G., New Static Models of the Themosphee and Exosphee with Empiical Tempeatue Pofiles, Smithson. Astophys. Special Repot 313, 197. Moe, K., M. Moe, and S. Wallace, Dag Coefficients of Sphees in Fee-Molecula Flow, AAS/AIAA Space Flight Mechanics Meeting, Austin, Texas, Febuay, 1996. Padini, C., and L. Anselmo, Calibation of Semi- Empiical Atmosphee Models Though the Obital Decay of Spheical Satellites, AAS/AIAA Astodynamics Specialist Confeence, Gidwood, Alaska, August, 1999.
SAT Intenat. Name Object Tue_B Q Ht Ap Ht Inc NORAD Desig. m2/kg 21 21 Deg 11 1959-1A VANGUARD 2 Sphee.494 56 3 32.9 22 1959-9A EXPLORER 7 Dbl Cone.2237 515 815 5.3 6 196-14A EXPLORER 8 Dbl Cone.2237 386 1145 49.9 63 196-16A TIROS 2 Cylinde.1428 526 58 48.5 17 1961-13A EXPLORER 11 Cylinde.268 485 1425 28.8 165 1961-17B DELTA 1 R/B Cylinde.5113 62 65 47.9 229 1962-2D DELTA 1 R/B Cylinde.5157 555 6 48.3 694 1963-47A ATLAS 2 R/B Cylinde.1678 465 1425 3.4 746 1964-6A ELEKTRON 1 Cyl+paddles.163 395 7125 6.8 829 1964-38A ELEKTRON 3 Cyl+paddles.16 45 725 6.8 1583 1965-72D THOR ALTAIR R/B Cylinde.4513 614 89 98.3 1613 1965-78A OV1-2 Cylinde.1669 45 26 144.2 2129 1966-26B THOR ALTAIR R/B Cylinde.4512 57 65 98.2 2389 1966-7A OV3-3 Octogon.1753 345 288 81.4 2611 1966-111B OV1-1 Cyl+booms.2322 54 56 93.4 4221 1969-97A AZUR (GRS A) Cone+Cyl.2111 375 193 12.7 Table 3. Tue B values fo selected satellites, with peigee height (Q Ht), apogee height (Ap Ht), and inclination (Inc) fo the beginning of 21. 4 2 DB/BTue Values (1 Ave) Fo 4km Height 1613 -(36i)- 45km 6-5i - 385km 73-74i - 385km 4221 -(77i)- 375km 5328-81i - 41km F1.7 Ave 5 4 DB/Btue (%) -2 3 2 F1.7 Ave -4 1-6 65 7 75 8 85 9 95 1 15 Figue 3. One yea aveage B vaiations fo satellites with peigee heights nea 4 km. The inclination, i, and peigee height (km) ae listed fo each satellite. The paentheses aound inclination epesents a etogade obit.
4 2 DB/Btue Values (1 Ave) fo High Inclination Satellites 6212 -(81i)- 65km 2611 -(87i)- 5km 5328-81i - 41km 73-74i - 385km 4221 -(77i)- 375km 8745-83i - 36km 5281-74i - 34km F1B 5 4 DB/Btue (%) -2 3 2 F1.7 Ave -4 1-6 65 7 75 8 85 9 95 1 15 Figue 4. One yea aveage B vaiations fo satellites of high inclinations geate than 74 degees. 21 DB/BTue Values fo Low Inclination Satellites 8 11-33i - 56km 17-29i - 485km 8 6 56 km 482-15i - 475km 694-3i - 465km 1613 - (36)i- 4km 7 4 485 km 453-3i - 27km 1269-26i - 265km F1.7 6 DB/BTue % 2-2 465 km 475 km 27 km 4 km 265 km ap 5 4 3 Sola Flux -4 F1.7 2-6 ap 1-8 25 5 75 1 125 15 175 2 21 Day Figue 5. 21 vaiations of B fo low inclination satellites with peigee heights of 265 km to 56 km.