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Sun-Synchronous Circular Orbit, Inclination vs Altitude (LEO,J2 perturbed)
This code is a MATLAB script that can be used to design and analyze Sun-synchronous orbits. A Sun-synchronous orbit is a geocentric orbit which combines altitude and inclination in such a way that an object in this orbit has an a nodal regression rate which is equals to Earth’s orbital rotation speed around the Sun. The object in this orbit constantly illuminated by the Sun.
Output: Inclination vs Altitude Plot
clc; clear all; mu = 398600.440; % Earth’s gravitational parameter [km^3/s^2] Re = 6378; % Earth radius [km] J2 = 0.0010826269; % Second zonal gravity harmonic of the Earth we = 1.99106e-7; % Mean motion of the Earth in its orbit around the Sun [rad/s] % Input Alt = 250:5:1000; % Altitude,Low Earth orbit (LEO) a = Alt + Re; % Mean semimajor axis [km] e = 0.0; % Eccentricity
h = a*(1 - e^2); % [km] n = (mu./a.^3).^0.5; % Mean motion [s-1] tol = 1e-10; % Error tolerance % Initial guess for the orbital inclination i0 = 180/pi*acos(-2/3*(h/Re).^2*we./(n*J2)); err = 1e1; while(err >= tol ) % J2 perturbed mean motion np = n.*(1 + 1.5*J2*(Re./h).^2.*(1 - e^2)^0.5.*(1 - 3/2*sind(i0).^2)); i = 180/pi*acos(-2/3*(h/Re).^2*we./(np*J2)); err = abs(i - i0); i0 = i; end
plot(Alt,i,'.b'); grid on;hold on; xlabel('Altitude,Low Earth orbit (LEO)'); ylabel('Mean orbital inclination'); title('Sun-Synchronous Circular Orbit,Inclination vs Altitude(LEO,J2 perturbed)'); hold off;
Sun-Synchronous (Heliosynchronous) Orbit, Mean Orbital Inclination (J2 perturbed)
This code is a MATLAB script that can be used to design and analyze Sun-synchronous orbits. A Sun-synchronous orbit is a geocentric orbit which combines altitude and inclination in such a way that an object in this orbit has an a nodal regression rate which is equals to Earth’s orbital rotation speed around the Sun. The object in this orbit constantly illuminated by the Sun. Input: a = Mean semimajor axis, e = Eccentricity
Output: i = Mean orbital inclination
clc; clear all; mu = 398600.440; % Earth’s gravitational parameter [km^3/s^2] Re = 6378; % Earth radius [km] J2 = 0.0010826269; % Second zonal gravity harmonic of the Earth we = 1.99106e-7; % Mean motion of the Earth in its orbit around the Sun [rad/s] % Input a = 7378; % Mean semimajor axis [km] e = 0.05; % Eccentricity
h = a*(1 - e^2); % [km] n = (mu/a^3)^0.5; % Mean motion [s-1] tol = 1e-10; % Error tolerance % Initial guess for the orbital inclination i0 = 180/pi*acos(-2/3*(h/Re)^2*we/(n*J2)); err = 1e1; while(err >= tol ) % J2 perturbed mean motion np = n*(1 + 1.5*J2*(Re/h)^2*(1 - e^2)^0.5*(1 - 3/2*sind(i0)^2)); i = 180/pi*acos(-2/3*(h/Re)^2*we/(np*J2)); err = abs(i - i0); i0 = i; end fprintf('Mean orbital inclination %4.5f [deg] \n',i);
Mean orbital inclination 99.43667 [deg]
GPS Satellite Constellation
The Global Positioning System (GPS) is a space-based satellite navigation system that provides location and time information in all weather conditions, anywhere on or near the Earth where there is an unobstructed line of sight to four or more GPS satellites.
clc; clear all; fname = 'GPS.txt'; % TLE file name Earth3DPlot(1); % Open the TLE file and read TLE elements fid = fopen(fname, 'rb'); while ~feof(fid) [ h, i, RAAN, e,omega,theta ] = TLE2OE( fid ); [Rv, alfa,delta ] = J2PropagR( h, i, RAAN, e,omega,theta) ; scatter3(Rv(:,1),Rv(:,2),Rv(:,3),'.'); end fclose(fid); % Ground Track EarthTopographicMap(2,820,420); fid = fopen(fname, 'rb'); while ~feof(fid) [ h, i, RAAN, e,omega,theta ] = TLE2OE( fid ); [Rv, alfa,delta ] = J2PropagR( h, i, RAAN, e,omega,theta) ; scatter(alfa,delta,'.'); end fclose(fid); text(280,-80,'smallsats.org','Color',[1 1 1], 'VerticalAlignment','middle',... 'HorizontalAlignment','left','FontSize',14 ); title('GPS Satellite Constellation');
TLE Data Source. http://www.celestrak.com/ January 20, 2012
GPS BIIA-10 (PRN 32) 1 20959U 90103A 13018.84945698 .00000048 00000-0 10000-3 0 8427 2 20959 54.4303 229.9128 0117665 335.2539 24.2342 2.00550891162214 GPS BIIA-14 (PRN 26) 1 22014U 92039A 13019.14607454 -.00000071 00000-0 10000-3 0 6346 2 22014 56.1801 290.1218 0208290 69.5584 292.7075 2.00565875143955 GPS BIIA-15 (PRN 27) 1 22108U 92058A 13018.75434799 -.00000089 00000-0 10000-3 0 5581 2 22108 56.2534 343.0502 0222435 303.9410 53.9615 2.00569608149127 GPS BIIA-21 (PRN 09) 1 22700U 93042A 13018.74470584 -.00000088 00000-0 00000+0 0 8817 2 22700 56.4056 344.6226 0167916 93.9179 268.0033 2.00662093143282 GPS BIIA-22 (PRN 30) 1 22779U 93054A 13018.51983509 -.00000069 00000-0 10000-3 0 5571 2 22779 55.1533 39.1818 0106140 92.6721 268.5873 2.00571221142039 GPS BIIA-23 (PRN 04) 1 22877U 93068A 13018.41796082 .00000048 00000-0 10000-3 0 2373 2 22877 53.7295 164.7980 0102485 54.0859 306.8291 2.00556733140921 GPS BIIA-24 (PRN 06) 1 23027U 94016A 13018.12046811 -.00000048 00000-0 10000-3 0 1955 2 23027 53.8565 101.4536 0076290 335.9203 23.7554 2.00561802138223 GPS BIIA-25 (PRN 03) 1 23833U 96019A 13019.13913939 -.00000060 00000-0 10000-3 0 3935 2 23833 53.4736 96.9153 0157826 72.6851 289.0689 2.00564035123227 GPS BIIA-26 (PRN 10) 1 23953U 96041A 13018.50101030 .00000050 00000-0 10000-3 0 8803 2 23953 54.1034 225.9914 0120793 44.6889 316.3194 2.00554012121004 GPS BIIR-2 (PRN 13) 1 24876U 97035A 13019.00325574 -.00000072 00000-0 10000-3 0 3517 2 24876 56.3255 290.5219 0046517 120.2494 240.2582 2.00575732113546 GPS BIIA-28 (PRN 08) 1 25030U 97067A 13019.63536166 -.00000082 00000-0 10000-3 0 2975 2 25030 57.1991 351.0337 0128256 195.9414 163.6595 2.00570836111451 GPS BIIR-3 (PRN 11) 1 25933U 99055A 13019.25617045 .00000010 00000-0 10000-3 0 7874 2 25933 50.9146 148.4260 0136447 65.1740 296.2085 2.00561387 97361 GPS BIIR-4 (PRN 20) 1 26360U 00025A 13016.87981762 .00000037 00000-0 10000-3 0 8761 2 26360 53.2134 222.1388 0052448 73.0764 287.5347 2.00576559 93011 GPS BIIR-5 (PRN 28) 1 26407U 00040A 13019.21994987 -.00000062 00000-0 10000-3 0 8291 2 26407 56.2531 48.0068 0187923 257.4922 100.4450 2.00568287 91732 GPS BIIR-6 (PRN 14) 1 26605U 00071A 13018.81568033 -.00000070 00000-0 10000-3 0 8445 2 26605 55.9113 288.9048 0069360 245.5421 113.7790 2.00561684 89323 GPS BIIR-7 (PRN 18) 1 26690U 01004A 13018.69048158 .00000051 00000-0 10000-3 0 7237 2 26690 53.1752 225.1291 0138193 238.6253 120.0544 2.00553471 87732 GPS BIIR-8 (PRN 16) 1 27663U 03005A 13019.05927186 -.00000062 00000-0 10000-3 0 1540 2 27663 56.3455 47.6478 0068033 359.8082 0.2361 2.00576901 73096 GPS BIIR-9 (PRN 21) 1 27704U 03010A 13018.60667007 .00000047 00000-0 10000-3 0 377 2 27704 53.3697 165.0440 0198594 235.2103 122.8804 2.00553928 71879 GPS BIIR-10 (PRN 22) 1 28129U 03058A 13018.73549856 .00000051 00000-0 10000-3 0 6108 2 28129 53.0451 225.2919 0063340 244.9211 114.4621 2.00564098 66589 GPS BIIR-11 (PRN 19) 1 28190U 04009A 13019.17516771 -.00000046 00000-0 10000-3 0 6040 2 28190 55.1066 110.1249 0086959 14.6182 345.6476 2.00565996 64762 GPS BIIR-12 (PRN 23) 1 28361U 04023A 13018.96324076 -.00000066 00000-0 10000-3 0 4238 2 28361 54.8505 285.6669 0081990 194.9306 164.8791 2.00552385 62846 GPS BIIR-13 (PRN 02) 1 28474U 04045A 13018.96367060 .00000044 00000-0 10000-3 0 3107 2 28474 53.7850 163.8242 0119722 206.9062 152.4399 2.00573321 60202 GPS BIIRM-1 (PRN 17) 1 28874U 05038A 13018.32217387 -.00000042 00000-0 10000-3 0 5695 2 28874 55.2624 107.1585 0081479 231.4916 127.7975 2.00551100 53614 GPS BIIRM-2 (PRN 31) 1 29486U 06042A 13018.93123473 -.00000087 00000-0 10000-3 0 9604 2 29486 56.1978 346.2221 0082935 312.3064 46.9875 2.00549243 46326 GPS BIIRM-3 (PRN 12) 1 29601U 06052A 13019.37101994 -.00000060 00000-0 10000-3 0 9912 2 29601 56.3013 46.5841 0041062 4.7967 355.2950 2.00567097 45242 GPS BIIRM-4 (PRN 15) 1 32260U 07047A 13018.68384066 -.00000063 00000-0 00000+0 0 6980 2 32260 54.0003 283.4336 0051688 4.0264 356.0623 2.00561108 38627 GPS BIIRM-5 (PRN 29) 1 32384U 07062A 13018.50083400 -.00000043 00000-0 10000-3 0 7369 2 32384 55.2952 107.6507 0020298 293.2614 66.5488 2.00557842 37341 GPS BIIRM-6 (PRN 07) 1 32711U 08012A 13019.09384597 -.00000088 00000-0 10000-3 0 4945 2 32711 55.9093 345.9893 0061210 195.1431 164.6734 2.00559437 35567 GPS BIIRM-8 (PRN 05) 1 35752U 09043A 13018.56516204 .00000053 00000-0 10000-3 0 2489 2 35752 54.3995 224.8319 0030098 13.9303 346.1949 2.00578808 25132 GPS BIIF-1 (PRN 25) 1 36585U 10022A 13018.91284136 -.00000065 00000-0 10000-3 0 648 2 36585 55.6614 44.4964 0018446 28.7965 331.3534 2.00571492 19380 GPS BIIF-2 (PRN 01) 1 37753U 11036A 13019.28928872 .00000046 00000-0 00000+0 0 5261 2 37753 55.0147 164.8917 0014277 20.5806 339.4484 2.00559276 11077 GPS BIIF-3 (PRN 24) 1 38833U 12053A 13018.77758384 -.00000091 00000-0 00000+0 0 731 2 38833 54.9671 344.4909 0005362 333.1108 26.8609 2.00559339 2129
Iridium Satellite Constellation
The Iridium satellite constellation is a large group of satellites providing voice and data coverage to satellite phones, pagers and integrated transceivers over Earth’s entire surface. For Algorithms used in this post look for J2 Propagator example in previous posts. TLE Data Source. http://www.celestrak.com/ January 20, 2012
clc; clear all; fname = 'Iridium.txt'; % TLE file name Earth3DPlot(1); % Open the TLE file and read TLE elements fid = fopen(fname, 'rb'); while ~feof(fid) [ h, i, RAAN, e,omega,theta ] = TLE2OE( fid ); [Rv, alfa,delta ] = J2PropagR( h, i, RAAN, e,omega,theta) ; scatter3(Rv(:,1),Rv(:,2),Rv(:,3),'.'); %scatter(alfa,delta,'.'); end fclose(fid); % Ground Track EarthTopographicMap(2,820,420); fid = fopen(fname, 'rb'); while ~feof(fid) [ h, i, RAAN, e,omega,theta ] = TLE2OE( fid ); [Rv, alfa,delta ] = J2PropagR( h, i, RAAN, e,omega,theta) ; scatter(alfa,delta,'.'); end fclose(fid); text(280,-80,'smallsats.org','Color',[1 1 1], 'VerticalAlignment','middle',... 'HorizontalAlignment','left','FontSize',14 ); title('Iridium Satellite Constellation');
IRIDIUM 8 [+] 1 24792U 97020A 13019.25828200 .00000326 00000-0 10925-3 0 1963 2 24792 86.3934 271.5793 0002226 81.2735 278.8697 14.34220916822440 IRIDIUM 7 [+] 1 24793U 97020B 13019.25197876 -.00000503 00000-0 -18675-3 0 1983 2 24793 86.3945 271.7767 0002220 84.7209 275.4206 14.34215565822447 IRIDIUM 6 [+] 1 24794U 97020C 13020.15258204 -.00000555 00000-0 -20547-3 0 64 2 24794 86.3943 271.3667 0002055 91.6836 268.4734 14.34211576822570 IRIDIUM 5 [+] 1 24795U 97020D 13019.23925838 .00000329 00000-0 11041-3 0 3920 2 24795 86.3940 271.8568 0002884 94.1872 265.9660 14.34220500822523 IRIDIUM 4 [-] 1 24796U 97020E 13019.18979874 .00000271 00000-0 89606-4 0 2337 2 24796 86.3937 271.4666 0002262 84.3259 275.8183 14.34306383822451 IRIDIUM 914 [-] 1 24836U 97030A 13019.20345150 .00000212 00000-0 63561-4 0 7657 2 24836 86.3963 292.5492 0002155 58.6600 103.8688 14.37894932817517 IRIDIUM 12 [+] 1 24837U 97030B 13019.25535178 .00000381 00000-0 12913-3 0 2111 2 24837 86.3966 303.0298 0002355 78.3314 281.8152 14.34224536816196 IRIDIUM 10 [+] 1 24839U 97030D 13019.24267022 .00000398 00000-0 13513-3 0 2139 2 24839 86.3963 303.0252 0002313 79.6634 280.4838 14.34223673816207 IRIDIUM 13 [+] 1 24840U 97030E 13019.33146516 .00000486 00000-0 16659-3 0 1596 2 24840 86.3971 303.0933 0002297 80.2477 279.8974 14.34224246816208 IRIDIUM 16 [-] 1 24841U 97030F 13019.29063765 .00000373 00000-0 12216-3 0 1877 2 24841 86.4054 303.7554 0003596 109.2676 250.8904 14.35757128816500 IRIDIUM 911 [-] 1 24842U 97030G 13020.20773527 .00000234 00000-0 62641-4 0 8551 2 24842 86.4500 305.0286 0015683 38.2954 111.5834 14.43796646820680 IRIDIUM 15 [+] 1 24869U 97034A 13019.39200756 .00000441 00000-0 15046-3 0 2562 2 24869 86.3964 334.6667 0002522 80.2051 279.9444 14.34216310813184 IRIDIUM 17 [-] 1 24870U 97034B 13019.40804263 .00000374 00000-0 12479-3 0 1762 2 24870 86.3958 334.2377 0002456 71.7924 288.3538 14.34899576813250 IRIDIUM 920 [-] 1 24871U 97034C 13018.95829477 .00000217 00000-0 64458-4 0 8045 2 24871 86.4013 324.4734 0010511 225.2632 163.0156 14.38557564814770 IRIDIUM 18 [+] 1 24872U 97034D 13019.35396626 .00000423 00000-0 14393-3 0 1927 2 24872 86.3969 334.8716 0002418 86.6122 273.5367 14.34215618813176 IRIDIUM 921 [-] 1 24873U 97034E 13020.22218146 .00002090 00000-0 16212-3 0 2998 2 24873 86.3868 101.9576 0010785 117.9015 299.2404 15.00783240845815 IRIDIUM 26 [-] 1 24903U 97043A 13019.29134437 .00000171 00000-0 53633-4 0 3142 2 24903 86.3933 208.3646 0002274 83.3209 276.8239 14.34476984807115 IRIDIUM 25 [+] 1 24904U 97043B 13019.44172922 .00000050 00000-0 10965-4 0 2125 2 24904 86.3924 208.2554 0002148 81.2923 278.8522 14.34218816807095 IRIDIUM 46 [+] 1 24905U 97043C 13019.45441752 -.00000537 00000-0 -19870-3 0 2245 2 24905 86.3943 208.6532 0002113 86.0161 274.1278 14.34214094807109 IRIDIUM 23 [S] 1 24906U 97043D 13019.42961336 .00000077 00000-0 20290-4 0 2301 2 24906 86.3925 208.1768 0002105 83.6697 276.4740 14.34214706807103 IRIDIUM 22 [+] 1 24907U 97043E 13019.07385532 -.00000454 00000-0 -16920-3 0 1899 2 24907 86.3935 208.8899 0002169 79.5028 280.6412 14.34219764807055 DUMMY MASS 1 [-] 1 24925U 97048A 13019.70792417 .00000238 00000-0 25125-4 0 4000 2 24925 86.3327 65.0051 0005583 346.5268 149.0153 14.84265252832774 DUMMY MASS 2 [-] 1 24926U 97048B 13019.70170321 .00000226 00000-0 23822-4 0 3890 2 24926 86.3328 65.8623 0006587 351.2172 148.1866 14.84042008832651 IRIDIUM 29 [+] 1 24944U 97051A 13019.15337614 .00000137 00000-0 41734-4 0 1490 2 24944 86.3920 240.0018 0002175 82.5044 277.6386 14.34219380803630 IRIDIUM 32 [+] 1 24945U 97051B 13019.17240299 -.00001114 00000-0 -40484-3 0 1864 2 24945 86.3918 240.0169 0002170 87.6880 272.4570 14.34213936803655 IRIDIUM 33 [-] 1 24946U 97051C 13020.18831512 .00000175 00000-0 56862-4 0 2786 2 24946 86.3852 239.7658 0010014 87.9435 13.0341 14.32857018803575 IRIDIUM 28 [-] 1 24948U 97051E 13019.18549225 .00000209 00000-0 66975-4 0 392 2 24948 86.3943 240.0163 0002504 109.4543 39.7656 14.34732376803653 IRIDIUM 30 [+] 1 24949U 97051F 13019.16606417 .00000083 00000-0 22655-4 0 2285 2 24949 86.3914 240.0916 0002069 84.5363 275.6059 14.34219125803635 IRIDIUM 31 [+] 1 24950U 97051G 13019.15971742 .00000095 00000-0 26733-4 0 1881 2 24950 86.3912 240.0988 0002754 57.9016 302.2436 14.34223532803648 IRIDIUM 19 [+] 1 24965U 97056A 13019.21386943 -.00002494 00000-0 -89800-3 0 1758 2 24965 86.3939 271.5895 0002324 82.2007 277.9396 14.34210541801796 IRIDIUM 35 [+] 1 24966U 97056B 13019.36611351 .00000384 00000-0 13002-3 0 2272 2 24966 86.3946 271.5378 0002265 79.6507 280.4928 14.34221101801794 IRIDIUM 36 [-] 1 24967U 97056C 13020.21122838 .00000195 00000-0 61766-4 0 1524 2 24967 86.3974 271.1775 0002148 99.2787 47.2898 14.34747586802089 IRIDIUM 37 [+] 1 24968U 97056D 13019.20119965 .00000299 00000-0 99535-4 0 1822 2 24968 86.3939 271.7452 0002237 78.7252 281.4205 14.34220808801779 IRIDIUM 34 [+] 1 24969U 97056E 13019.29000192 .00000393 00000-0 13336-3 0 1697 2 24969 86.3946 271.6484 0002281 83.3693 276.7770 14.34221331801789 IRIDIUM 43 [+] 1 25039U 97069A 13019.34760118 .00000451 00000-0 15383-3 0 1764 2 25039 86.3975 334.8350 0002732 75.4578 284.6933 14.34216124795571 IRIDIUM 41 [+] 1 25040U 97069B 13019.34125956 .00000124 00000-0 37101-4 0 1693 2 25040 86.3975 334.7833 0002400 87.8563 272.2897 14.34219942795574 IRIDIUM 40 [+] 1 25041U 97069C 13019.29686359 -.00000498 00000-0 -18492-3 0 1975 2 25041 86.3971 334.7525 0002527 85.4810 274.6730 14.34212842795583 IRIDIUM 39 [+] 1 25042U 97069D 13019.30321589 .00000452 00000-0 15445-3 0 1781 2 25042 86.3990 335.0132 0002068 97.2505 262.8938 14.34216624795564 IRIDIUM 38 [-] 1 25043U 97069E 13018.93734748 .00000211 00000-0 66912-4 0 1650 2 25043 86.3999 333.7245 0003080 121.7814 299.2150 14.35164863795709 IRIDIUM 42 [+] 1 25077U 97077A 13019.29053708 .00000433 00000-0 14738-3 0 1818 2 25077 86.3969 334.8569 0002536 82.3130 277.8369 14.34215449791427 IRIDIUM 44 [-] 1 25078U 97077B 13020.18276462 .00000167 00000-0 49514-4 0 6521 2 25078 86.4006 327.7485 0003735 121.1671 9.7924 14.37114346792453 IRIDIUM 45 [+] 1 25104U 97082A 13019.23876329 -.00000896 00000-0 -32699-3 0 1362 2 25104 86.3930 208.3285 0002196 82.3576 277.7865 14.34214498789613 IRIDIUM 24 [-] 1 25105U 97082B 13020.21010638 .00000243 00000-0 72689-4 0 7090 2 25105 86.3936 194.3185 0010600 229.8673 255.8995 14.38833044791407 IRIDIUM 47 [+] 1 25106U 97082C 13019.25144752 -.00000094 00000-0 -40610-4 0 1143 2 25106 86.3919 208.2578 0001665 90.0600 270.0783 14.34217923789611 IRIDIUM 49 [+] 1 25108U 97082E 13019.47355223 .00000270 00000-0 89230-4 0 1422 2 25108 86.3928 208.2251 0002076 86.5174 273.6268 14.34222005789652 IRIDIUM 52 [+] 1 25169U 98010A 13019.29340342 .00000496 00000-0 16991-3 0 9024 2 25169 86.3966 303.2330 0002373 77.8507 282.2951 14.34224631781034 IRIDIUM 56 [+] 1 25170U 98010B 13019.28704736 -.00002350 00000-0 -84673-3 0 6938 2 25170 86.3978 303.3126 0002513 78.8408 281.3229 14.34207251781015 IRIDIUM 54 [+] 1 25171U 98010C 13019.24901286 .00000359 00000-0 12110-3 0 1403 2 25171 86.3978 303.2636 0002412 78.2931 281.8543 14.34223411781171 IRIDIUM 50 [+] 1 25172U 98010D 13019.35053981 .00000438 00000-0 14943-3 0 1770 2 25172 86.3973 303.1858 0002393 77.9142 282.2320 14.34231173781026 IRIDIUM 53 [+] 1 25173U 98010E 13019.36952395 .00000522 00000-0 17935-3 0 1553 2 25173 86.3965 303.1853 0002380 80.3006 279.8457 14.34224163781031 IRIDIUM 51 [S] 1 25262U 98018A 13019.25184820 .00000149 00000-0 46028-4 0 2932 2 25262 86.3949 271.8491 0002131 87.7931 272.3473 14.34226063780400 IRIDIUM 61 [+] 1 25263U 98018B 13020.11453228 -.00000705 00000-0 -25904-3 0 1030 2 25263 86.3937 271.3336 0002223 85.9093 274.2468 14.34210888776231 IRIDIUM 55 [+] 1 25272U 98019A 13019.28022689 -.00000320 00000-0 -12146-3 0 1071 2 25272 86.3913 239.8919 0002123 85.6132 274.5266 14.34216614775311 IRIDIUM 57 [+] 1 25273U 98019B 13019.18508789 -.00000818 00000-0 -29914-3 0 1412 2 25273 86.3918 240.1671 0002253 83.8519 276.2933 14.34214753775301 IRIDIUM 58 [+] 1 25274U 98019C 13019.12166106 .00000158 00000-0 49323-4 0 9577 2 25274 86.3922 240.3440 0002085 85.1053 275.0389 14.34223937775298 IRIDIUM 59 [+] 1 25275U 98019D 13019.47684150 -.00000017 00000-0 -13194-4 0 7403 2 25275 86.3971 240.1372 0002140 59.8338 300.3065 14.34221068775343 IRIDIUM 60 [+] 1 25276U 98019E 13019.13435185 .00000122 00000-0 36325-4 0 1836 2 25276 86.3910 239.9614 0002201 84.4079 275.7369 14.34219041775290 IRIDIUM 62 [+] 1 25285U 98021A 13019.37487346 .00000270 00000-0 89388-4 0 1607 2 25285 86.3964 176.7700 0001974 86.2148 273.9282 14.34218061774289 IRIDIUM 63 [+] 1 25286U 98021B 13019.38116229 -.00000555 00000-0 -20531-3 0 1745 2 25286 86.3965 176.9067 0002040 80.1666 279.9793 14.34210551774281 IRIDIUM 64 [+] 1 25287U 98021C 13019.38755872 .00000194 00000-0 62115-4 0 1077 2 25287 86.3966 176.7928 0002635 83.5815 276.5707 14.34217563774266 IRIDIUM 65 [+] 1 25288U 98021D 13019.39390155 .00000224 00000-0 72901-4 0 1481 2 25288 86.3960 176.7232 0002088 87.7553 272.3894 14.34218178774282 IRIDIUM 66 [+] 1 25289U 98021E 13019.40024514 .00000276 00000-0 91434-4 0 924 2 25289 86.3961 176.6380 0002040 86.6304 273.5143 14.34217879774284 IRIDIUM 67 [+] 1 25290U 98021F 13019.40658708 .00000208 00000-0 67066-4 0 9203 2 25290 86.3963 176.7468 0002122 83.6298 276.5154 14.34217387774296 IRIDIUM 68 [+] 1 25291U 98021G 13019.41293488 .00000170 00000-0 53509-4 0 1459 2 25291 86.3961 176.8325 0002003 84.0717 276.0724 14.34217307774264 IRIDIUM 69 [-] 1 25319U 98026A 13020.07089449 .00000234 00000-0 71292-4 0 9457 2 25319 86.3987 202.1317 0003488 127.2588 21.4608 14.37647589771757 IRIDIUM 71 [-] 1 25320U 98026B 13020.08413631 .00000227 00000-0 68229-4 0 6460 2 25320 86.3977 198.8183 0001951 167.4208 343.1557 14.38170592772112 IRIDIUM 70 [+] 1 25342U 98032A 13019.36852929 .00000267 00000-0 88145-4 0 1779 2 25342 86.3971 177.0171 0001906 90.0985 270.0444 14.34218090768366 IRIDIUM 72 [+] 1 25343U 98032B 13019.35584896 .00000236 00000-0 77164-4 0 1588 2 25343 86.3962 176.7399 0002115 84.7147 275.4303 14.34217456768366 IRIDIUM 73 [-] 1 25344U 98032C 13018.99395123 .00000351 00000-0 89616-4 0 325 2 25344 86.4467 162.8525 0002511 92.8910 46.4311 14.47776697774467 IRIDIUM 74 [S] 1 25345U 98032D 13019.39443214 .00000319 00000-0 89489-4 0 2230 2 25345 86.4473 176.9786 0002184 76.2355 283.9111 14.42954688770617 IRIDIUM 75 [+] 1 25346U 98032E 13019.36218660 .00000245 00000-0 80357-4 0 1820 2 25346 86.3963 176.9319 0002114 85.1603 274.9840 14.34218193768937 IRIDIUM 3 [+] 1 25431U 98048A 13019.48613037 .00000058 00000-0 13785-4 0 1049 2 25431 86.3927 208.4422 0001870 86.5887 273.5525 14.34218568754896 IRIDIUM 76 [+] 1 25432U 98048B 13019.43539516 .00000072 00000-0 18554-4 0 442 2 25432 86.3925 208.2691 0001892 73.6047 286.5363 14.34222014754873 IRIDIUM 82 [+] 1 25467U 98051A 13019.33493506 .00000093 00000-0 26174-4 0 2780 2 25467 86.3969 334.9495 0002785 91.1479 268.9988 14.34221851755870 IRIDIUM 81 [+] 1 25468U 98051B 13019.32859284 .00000439 00000-0 14956-3 0 998 2 25468 86.3971 334.6434 0002525 80.8422 279.3069 14.34216495752094 IRIDIUM 80 [+] 1 25469U 98051C 13019.30955266 .00000415 00000-0 14112-3 0 1027 2 25469 86.3970 334.8621 0002510 82.0657 278.0836 14.34215664752097 IRIDIUM 77 [+] 1 25471U 98051E 13019.19299270 .00000244 00000-0 80167-4 0 2703 2 25471 86.3991 334.6473 0001773 113.0566 332.8838 14.34217028757272 IRIDIUM 2 [-] 1 25527U 98066A 13019.19680509 .00003627 00000-0 26863-3 0 6142 2 25527 85.5597 251.7497 0009484 162.5069 345.4067 15.02787347774833 IRIDIUM 86 [+] 1 25528U 98066B 13019.27437687 .00000427 00000-0 14550-3 0 3297 2 25528 86.3960 302.8257 0002107 91.8263 268.3170 14.34224594748613 IRIDIUM 84 [+] 1 25530U 98066D 13019.30609118 .00000515 00000-0 17664-3 0 9815 2 25530 86.3974 302.9487 0002370 75.9862 284.1612 14.34224925746250 IRIDIUM 83 [+] 1 25531U 98066E 13019.26802892 .00000420 00000-0 14287-3 0 1180 2 25531 86.3962 303.0253 0002377 80.7990 279.3477 14.34223754743604 IRIDIUM 20 [+] 1 25577U 98074A 13019.46710345 .00000113 00000-0 33362-4 0 232 2 25577 86.3925 208.2006 0002124 81.4888 278.6554 14.34222908737712 IRIDIUM 11 [+] 1 25578U 98074B 13019.41001739 .00000102 00000-0 29296-4 0 2598 2 25578 86.3930 208.4419 0002261 87.6235 272.5223 14.34222606746671 IRIDIUM 14 [S] 1 25777U 99032A 13019.40982228 -.00000544 00000-0 -13171-3 0 2770 2 25777 86.5147 176.7905 0002385 89.9292 270.2159 14.54943661722722 IRIDIUM 21 [+] 1 25778U 99032B 13019.34950351 .00000174 00000-0 55150-4 0 955 2 25778 86.3965 176.8302 0001874 41.5150 318.6195 14.34217775717370 IRIDIUM 91 [+] 1 27372U 02005A 13019.10898298 -.00000869 00000-0 -31753-3 0 7026 2 27372 86.3931 240.5136 0005044 96.2705 263.9068 14.34214548580852 IRIDIUM 90 [S] 1 27373U 02005B 13019.31019289 .00000596 00000-0 17240-3 0 6888 2 27373 86.4473 302.9185 0003863 1.3957 358.7246 14.42958259579206 IRIDIUM 94 [S] 1 27374U 02005C 13019.42902793 .00000050 00000-0 10651-4 0 8654 2 27374 86.3933 208.5398 0002273 302.6836 57.4141 14.34222560585227 IRIDIUM 95 [+] 1 27375U 02005D 13019.28657011 -.00000247 00000-0 -95157-4 0 6651 2 27375 86.3927 240.2321 0002078 80.8445 279.2983 14.34216182580163 IRIDIUM 96 [S] 1 27376U 02005E 13019.15023926 .00000338 00000-0 95724-4 0 8549 2 27376 87.1864 245.4340 0002449 324.5213 35.5824 14.42843918585269 IRIDIUM 97 [+] 1 27450U 02031A 13019.37245032 -.00001841 00000-0 -66474-3 0 5682 2 27450 86.3939 271.7596 0004818 90.0074 270.1675 14.34210479559491 IRIDIUM 98 [S] 1 27451U 02031B 13019.33556017 .00000559 00000-0 16135-3 0 6567 2 27451 86.4457 335.2830 0005725 86.7150 273.4715 14.42953981560067
J2 Propagator
In this example we will implement the J2 Perturbation (first-order) propagator which accounts for secular variations in the orbit elements due to Earth oblateness. We do not consider atmospheric drag or solar or lunar gravitational forces.It uses two-line element (TLE) sets as an input. Additionally as an example we wil use the implemented propagator to plot orbits of RADUGA satellites.
% TLE Data Source. http://www.celestrak.com/ January 20, 2012 clc; clear all; Earth3DPlot(1); mu = 398600; % Earth’s gravitational parameter [km^3/s^2] R_earth = 6378; % Earth radius [km] J2 = 0.0010836; we = 360*(1 + 1/365.25)/(3600*24); % Earth's rotation [deg/s] fname = 'Raduga.txt'; % TLE file name % Open the TLE file and read TLE elements fid = fopen(fname, 'rb'); while ~feof(fid) % Reading TLE File L1 = fscanf(fid,'%23c%*s',1); L2 = fscanf(fid,'%d%6d%*c%5d%*3c%*2f%f%f%5d%*c%*d%5d%*c%*d%d%5d',[1,9]); L3 = fscanf(fid,'%d%6d%f%f%f%f%f%f%f%f',[1,9]); epoch = L2(1,4)*24*3600; % Epoch Date and Julian Date Fraction Db = L2(1,5); % Ballistic Coefficient i = L3(1,3); % Inclination [deg] RAAN = L3(1,4); % Right Ascension of the Ascending Node [deg] e = L3(1,5)/1e7; % Eccentricity omega = L3(1,6); % Argument of periapsis [deg] M = L3(1,7); % Mean anomaly [deg] n = L3(1,8); % Mean motion [Revs/day] % Orbital parametres a = (mu/(n*2*pi/(24*3600))^2)^(1/3); % Semi-major axis [km] T = 2*pi*sqrt(a^3/mu)/60; % Period in [min] rp = a*(1-e); h = (mu*rp*(1 + e))^0.5; % Angular momentum E = keplerEq(M,e,eps); theta = acos((cos(E) -e)/(1 - e*cos(E)))*180/pi; % [deg] True anomaly % J2 Propagator dRAAN = -(1.5*mu^0.5*J2*R_earth^2/((1-e^2)*a^3.5))*cosd(i)*180/pi; domega = dRAAN*(2.5*sind(i)^2 - 2)/cosd(i); eps = 1E-9; dt = 5; % time step [sec] ti = 0; j = 1; while(ti <= T); E = 2*atan(tand(theta/2)*((1-e)/(1+e))^0.5); M = E - e*sin(E); t0 = M/(2*pi)*T; t = t0 + dt; M = 2*pi*t/T; E = keplerEq(M,e,eps); theta = 2*atan(tan(E/2)*((1+e)/(1-e))^0.5)*180/pi; RAAN = RAAN + dRAAN*dt ; omega = omega + domega*dt; [R V] = Orbital2State( h, i, RAAN, e,omega,theta); % Considering Earth's rotation fi_earth = we*ti; Rot = [cosd(fi_earth), sind(fi_earth),0;... -sind(fi_earth),cosd(fi_earth),0;0,0,1]; R = Rot*R; ti = ti+dt; Rv(j,:) = R'; j = j+1; end scatter3(Rv(:,1),Rv(:,2),Rv(:,3),'.'); end fclose(fid); hold off; % Ground Track % Open the TLE file and read TLE elements EarthTopographicMap( 2,820,420); fid = fopen(fname, 'rb'); while ~feof(fid) [ h, i, RAAN, e,omega,theta ] = TLE2OE( fid ); [Rv, alfa,delta ] = J2PropagR( h, i, RAAN, e,omega,theta) ; scatter(alfa,delta,'.'); end fclose(fid); figure(2); legend('RADUGA 32', 'RADUGA-1 4', 'RADUGA-1 5', 'RADUGA-1 7', 'RADUGA-1M 1'); text(270,-80,'smallsats.org','Color',[1 1 1], 'VerticalAlignment','middle',... 'HorizontalAlignment','left','FontSize',14 ); title('RADUGA satellites ground track');
Input TLE File: Source www.celestrak.com
Raduga.txt RADUGA 32 1 23448U 94087A 13017.58278277 .00000000 00000-0 10000-3 0 5863 2 23448 12.7840 37.1920 0005679 124.1377 235.9487 1.00272544 83823 RADUGA-1 4 1 25642U 99010A 13017.71286665 .00000130 00000-0 10000-3 0 6727 2 25642 11.7705 50.8546 0001080 106.9153 253.1213 1.00296963 50866 RADUGA-1 5 1 26477U 00049A 13019.59753669 -.00000103 00000-0 10000-3 0 5404 2 26477 9.3910 51.0039 0002612 354.7657 5.2534 1.00203902 45393 RADUGA-1 7 1 28194U 04010A 13018.70907761 .00000086 00000-0 10000-3 0 6657 2 28194 6.8757 64.2408 0001484 24.4802 335.5290 1.00365778 32290 RADUGA-1M 1 1 32373U 07058A 13018.90229375 -.00000043 00000-0 00000+0 0 4653 2 32373 0.0263 70.5780 0002023 210.8419 231.8461 1.00272457 19621
Small Satellites 