3400:) is the integration over time of the magnitude of the acceleration produced by using the rocket engine (what would be the actual acceleration if external forces were absent). In free space, for the case of acceleration in the direction of the velocity, this is the increase of the speed. In the case of an acceleration in opposite direction (deceleration) it is the decrease of the speed. Of course gravity and drag also accelerate the vehicle, and they can add or subtract to the change in velocity experienced by the vehicle. Hence delta-v may not always be the actual change in speed or velocity of the vehicle.
1091:
443:
6519:, the propellant mass fraction is the portion of a vehicle's mass which does not reach the destination, usually used as a measure of the vehicle's performance. In other words, the propellant mass fraction is the ratio between the propellant mass and the initial mass of the vehicle. In a spacecraft, the destination is usually an orbit, while for aircraft it is their landing location. A higher mass fraction represents less weight in a design. Another related measure is the
34:
8484:
7381:
6765:, in which the propellant is discharged and delta-v applied instantaneously. This assumption is relatively accurate for short-duration burns such as for mid-course corrections and orbital insertion maneuvers. As the burn duration increases, the result is less accurate due to the effect of gravity on the vehicle over the duration of the maneuver. For low-thrust, long duration propulsion, such as
6951:= 0.671, therefore 67.1% of the initial total mass has to be propellant to the first stage. The remaining mass is 32.9%. After disposing of the first stage, a mass remains equal to this 32.9%, minus the mass of the tank and engines of the first stage. Assume that this is 8% of the initial total mass, then 24.9% remains. The second stage should provide a
7035:, the equation applies for each stage, where for each stage the initial mass in the equation is the total mass of the rocket after discarding the previous stage, and the final mass in the equation is the total mass of the rocket just before discarding the stage concerned. For each stage the specific impulse may be different.
5624:
7018:= 0.648, therefore 64.8% of the remaining mass has to be propellant, which is 16.2% of the original total mass, and 8.7% remains for the tank and engines of the second stage, the payload, and in the case of a space shuttle, also the orbiter. Thus together 16.7% of the original launch mass is available for
6720:
forces. As such, when using it to calculate the propellant requirement for launch from (or powered descent to) a planet with an atmosphere, the effects of these forces must be included in the delta-V requirement (see
Examples below). In what has been called "the tyranny of the rocket equation", there
6711:
The rocket equation captures the essentials of rocket flight physics in a single short equation. It also holds true for rocket-like reaction vehicles whenever the effective exhaust velocity is constant, and can be summed or integrated when the effective exhaust velocity varies. The rocket equation
1073:
In order to understand the principle of rocket propulsion, Konstantin
Tsiolkovsky proposed the famous experiment of "the boat". A person is in a boat away from the shore without oars. They want to reach this shore. They notice that the boat is loaded with a certain quantity of stones and have the
5337:
1074:
idea of throwing, one by one and as quickly as possible, these stones in the opposite direction to the bank. Effectively, the quantity of movement of the stones thrown in one direction corresponds to an equal quantity of movement for the boat in the other direction (ignoring friction / drag).
7168:
1580:
2237:
7368:
If the motor of a new stage is ignited before the previous stage has been discarded and the simultaneously working motors have a different specific impulse (as is often the case with solid rocket boosters and a liquid-fuel stage), the situation is more complicated.
5456:
1264:
5109:
4582:
6098:
4259:
650:
1068:
6386:
5170:
7043:
2872:
4035:
3524:
3870:
2604:
4712:
1453:
3179:
7364:
5926:
2108:
2103:
2030:
6760:
in order to determine how much propellant is needed to change to a particular new orbit, or to find the new orbit as the result of a particular propellant burn. When applying to orbital maneuvers, one assumes an
3793:
7269:
3733:
2940:
2729:
1143:
4966:
4458:
3014:
5994:
4094:
1013:
6197:
3082:
533:
7048:
6287:
6592:
703:
4383:
1686:
3643:
4632:
1380:
2405:
5373:
6299:
1862:
5418:
2338:
2302:
1930:
1752:
1419:
5679:
5447:
2450:
5969:
4941:
4862:
5619:{\displaystyle \lim _{N\to \infty }\Delta v=v_{\text{eff}}\int _{0}^{\phi }{\frac {dx}{1-x}}=v_{\text{eff}}\ln {\frac {1}{1-\phi }}=v_{\text{eff}}\ln {\frac {m_{0}}{m_{f}}},}
7016:
6949:
6876:
4339:
2767:
2761:
1300:
1138:
6622:
2649:
736:
7197:
6651:
5165:
4312:
3928:
3416:
3322:
3295:
3260:
2758:
2485:
1957:
3365:
942:
4791:
3413:
The equation can also be derived from the basic integral of acceleration in the form of force (thrust) over mass. By representing the delta-v equation as the following:
2367:
2266:
1891:
1715:
1612:
1448:
6972:
6905:
6825:
6798:
5718:
3576:
3394:
2508:
1775:
526:
6725:
that the rocket can carry, as higher amounts of propellant increment the overall weight, and thus also increase the fuel consumption. The equation does not apply to
3798:
6684:
5776:
5745:
4761:
4734:
4413:
3553:
3297:
is NOT constant, we might not have rocket equations that are as simple as the above forms. Many rocket dynamics researches were based on the
Tsiolkovsky's constant
3233:
3206:
902:
863:
830:
769:
2513:
797:
1822:
1638:
1326:
4637:
5989:
5796:
5131:
4961:
4882:
4811:
4453:
4433:
4286:
1796:
3087:
7644:
7285:
7038:
For example, if 80% of the mass of a rocket is the fuel of the first stage, and 10% is the dry mass of the first stage, and 10% is the remaining rocket, then
5805:
2035:
1962:
3737:
Realising that impulse over the change in mass is equivalent to force over propellant mass flow rate (p), which is itself equivalent to exhaust velocity,
3740:
6769:, more complicated analysis based on the propagation of the spacecraft's state vector and the integration of thrust are used to predict orbital motion.
5332:{\displaystyle \Delta v\approx v_{\text{eff}}\sum _{j=1}^{j=N}{\frac {\phi /N}{1-j\phi /N}}=v_{\text{eff}}\sum _{j=1}^{j=N}{\frac {\Delta x}{1-x_{j}}}}
7204:
3650:
2879:
2658:
7163:{\displaystyle {\begin{aligned}\Delta v\ &=v_{\text{e}}\ln {100 \over 100-80}\\&=v_{\text{e}}\ln 5\\&=1.61v_{\text{e}}.\\\end{aligned}}}
7556:
7463:
2945:
1043:
independently developed the equation in 1912 when he began his research to improve rocket engines for possible space flight. German engineer
6878:= 0.884, therefore 88.4% of the initial total mass has to be propellant. The remaining 11.6% is for the engines, the tank, and the payload.
947:
3019:
6766:
6544:
1054:
exercise, Tsiolkovsky is honored as being the first to apply it to the question of whether rockets could achieve speeds necessary for
1575:{\displaystyle {\vec {P}}_{\Delta t}=\left(m-\Delta m\right)\left({\vec {V}}+\Delta {\vec {V}}\right)+\Delta m{\vec {V}}_{\text{e}}}
4268:
The rocket equation can also be derived as the limiting case of the speed change for a rocket that expels its fuel in the form of
8513:
3909:. The rocket is subject to a constant force, but its total mass is decreasing steadily because it is expelling gas. According to
3584:
429:
7490:
6712:
only accounts for the reaction force from the rocket engine; it does not include other forces that may act on a rocket, such as
1331:
1067:
8361:
2232:{\displaystyle {\vec {P}}_{\Delta t}-{\vec {P}}_{0}=m\Delta {\vec {V}}+{\vec {v}}_{\text{e}}\Delta m-\Delta m\Delta {\vec {V}}}
7280:
rocket, also with a 0.1% payload, could have a mass of 11.1% for fuel tanks and engines, and 88.8% for fuel. This would give
3581:
and realising that the integral of a resultant force over time is total impulse, assuming thrust is the only force involved,
1033:
487:
8421:
201:
5720:
again standing for the rocket's final velocity (after expelling all its reaction mass and being reduced to a rest mass of
6108:
270:
6217:
1259:{\displaystyle \sum _{i}{\vec {F}}_{i}=\lim _{\Delta t\to 0}{\frac {{\vec {P}}_{\Delta t}-{\vec {P}}_{0}}{\Delta t}}}
658:
486:, who independently derived it and published it in 1903, although it had been independently derived and published by
8416:
8296:
7715:
5104:{\displaystyle \Delta v=v_{\text{eff}}\sum _{j=1}^{j=N}{\frac {\phi /N}{\sqrt {(1-j\phi /N)(1-j\phi /N+\phi /N)}}}}
4344:
1101:
7654:
1644:
8518:
8381:
8135:
4577:{\displaystyle {\tfrac {1}{2}}m_{p}v_{\text{eff}}^{2}={\tfrac {1}{2}}m_{p}u^{2}+{\tfrac {1}{2}}m(\Delta v)^{2}.}
8523:
8454:
8093:
8084:
7821:
4589:
2372:
91:
7539:
A Treatise on the Motion of
Rockets: to which is added, an Essay on Naval Gunnery, in theory and practice, etc
6093:{\displaystyle {\frac {\Delta v}{c}}={\frac {R^{\frac {2v_{\text{e}}}{c}}-1}{R^{\frac {2v_{\text{e}}}{c}}+1}}}
4254:{\displaystyle ~\Delta v=v_{f}-v_{0}=-v_{\text{e}}\left=~v_{\text{e}}\ln \left({\frac {m_{0}}{m_{f}}}\right).}
1098:
In the following derivation, "the rocket" is taken to mean "the rocket and all of its unexpended propellant".
5342:
1828:
645:{\displaystyle \Delta v=v_{\text{e}}\ln {\frac {m_{0}}{m_{f}}}=I_{\text{sp}}g_{0}\ln {\frac {m_{0}}{m_{f}}},}
6440:
that is needed to perform a maneuver such as launching from, or landing on a planet or moon, or an in-space
8401:
7871:
7440:
6433:
5378:
2307:
2271:
1899:
1721:
1385:
5629:
5426:
2410:
5748:
4887:
869:
Given the effective exhaust velocity determined by the rocket motor's design, the desired delta-v (e.g.,
422:
6211:
5933:
4816:
8528:
8326:
8153:
7620:
6762:
6532:
3910:
3880:
706:
451:
466:
is a mathematical equation that describes the motion of vehicles that follow the basic principle of a
8464:
7553:
7460:
6977:
6910:
6837:
4317:
1269:
1107:
8449:
7974:
7688:
6600:
6510:
714:
479:
350:
265:
7175:
6629:
5136:
4291:
3300:
3273:
3238:
2736:
2611:
2455:
1935:
1140:) to the change in linear momentum of the whole system (including rocket and exhaust) as follows:
8459:
7767:
7552:К. Ціолковскій, Изслѣдованіе мировыхъ пространствъ реактивными приборами, 1903 (available online
7459:К. Ціолковскій, Изслѣдованіе мировыхъ пространствъ реактивными приборами, 1903 (available online
7424:
3330:
907:
221:
6657:, which is the same as the effective exhaust velocity measured in m/s (or ft/s if g is in ft/s),
6381:{\displaystyle \Delta v=c\tanh \left({\frac {v_{\text{e}}}{c}}\ln {\frac {m_{0}}{m_{1}}}\right)}
4766:
2343:
2242:
1867:
1691:
1588:
1424:
8508:
8321:
7923:
7843:
7831:
6726:
6103:
1026:
483:
415:
138:
6954:
6887:
6807:
6780:
5700:
3558:
3376:
2764:), so it is not subject to integration, then the above equation may be integrated as follows:
2490:
1757:
508:
8444:
8386:
8356:
8144:
8021:
7989:
7959:
7918:
7903:
7782:
7683:
7435:
7277:
6831:
6516:
1090:
1047:
independently derived the equation about 1920 as he studied the feasibility of space travel.
323:
158:
66:
8469:
8291:
8075:
7964:
7933:
7861:
7836:
7811:
7772:
7753:
7708:
7586:
6662:
6445:
6200:
5754:
5723:
4739:
4719:
4391:
3531:
3211:
3184:
880:
841:
808:
747:
196:
153:
143:
71:
6414:
3883:). From the moment its engine is started (clock set to 0) the rocket expels gas mass at a
782:
8:
8331:
8126:
7866:
7419:
6293:
5694:
4455:, the amount of energy converted to increase the rocket's and pellet's kinetic energy is
1801:
1617:
1305:
238:
76:
7590:
4079:
to burn all this fuel. Integrating both sides of the equation with respect to time from
8004:
7893:
7791:
7602:
7577:
Blanco, Philip (November 2019). "A discrete, energetic approach to rocket propulsion".
7386:
7032:
6881:
5974:
5781:
5690:
5116:
4946:
4867:
4796:
4438:
4418:
4271:
2867:{\displaystyle -\int _{V}^{V+\Delta V}\,dV={v_{e}}\int _{m_{0}}^{m_{f}}{\frac {dm}{m}}}
1781:
311:
186:
7627:
as the ratio of initial to final mass and w as the exhaust velocity, corresponding to
1754:
is the velocity of the mass added to the exhaust (and lost by the rocket) during time
8371:
8269:
8199:
7954:
7908:
7826:
7606:
7564:
7471:
7429:
6746:
6713:
6476:
6457:
6437:
6207:
4030:{\displaystyle ~a={\frac {dv}{dt}}=-{\frac {F}{m(t)}}=-{\frac {Rv_{\text{e}}}{m(t)}}}
3519:{\displaystyle \Delta v=\int _{t_{0}}^{t_{f}}{\frac {|T|}{{m_{0}}-{t}\Delta {m}}}~dt}
800:
226:
163:
42:
7511:
1932:
in the observer frame is related to the velocity of the exhaust in the rocket frame
8351:
8283:
8047:
8009:
7883:
7853:
7806:
7594:
6757:
6722:
6687:
6654:
6538:
6520:
6441:
3263:
772:
739:
396:
345:
110:
54:
8391:
7984:
7888:
7878:
7777:
7701:
7560:
7537:
7467:
6801:
874:
401:
306:
216:
191:
2652:
8487:
8439:
8431:
8426:
8311:
8306:
8237:
8217:
8208:
7801:
7787:
7763:
7758:
7733:
7598:
7394:
6777:
Assume an exhaust velocity of 4,500 meters per second (15,000 ft/s) and a
6738:
6695:
6468:
5799:
3865:{\displaystyle \Delta v=V_{\text{exh}}~\ln \left({\frac {m_{0}}{m_{f}}}\right)}
1044:
1040:
495:
491:
373:
289:
283:
206:
131:
125:
120:
442:
8502:
8341:
8336:
8255:
7898:
7816:
7409:
6494:
2599:{\displaystyle \sum _{i}F_{i}=m{\frac {dV}{dt}}+v_{\text{e}}{\frac {dm}{dt}}}
870:
378:
211:
168:
8406:
8316:
8190:
8173:
8031:
7928:
7796:
7413:
7399:
4707:{\displaystyle \Delta v=v_{\text{eff}}{\frac {m_{p}}{\sqrt {m(m+m_{p})}}}.}
3368:
96:
86:
81:
5751:
where the rocket started at rest (with the rest mass including fuel being
4586:
Using momentum conservation in the rocket's frame just prior to ejection,
8411:
8246:
8016:
7996:
7913:
6742:
6730:
6717:
5450:
3174:{\displaystyle m_{0}-m_{f}=m_{f}\left(e^{\Delta V/v_{\text{e}}}-1\right)}
1055:
260:
7359:{\displaystyle \Delta v\ =v_{\text{e}}\ln(100/11.2)\ =2.19v_{\text{e}}.}
4763:
the initial fueled-up mass of the rocket. Divide the total mass of fuel
7979:
7693:
7404:
6750:
5921:{\displaystyle {\frac {m_{0}}{m_{1}}}=\left^{\frac {c}{2v_{\text{e}}}}}
833:
447:
340:
296:
255:
33:
7678:
4341:
such that the mechanical energy gained per unit fuel mass is given by
2098:{\displaystyle {\vec {V}}_{\text{e}}={\vec {V}}+{\vec {v}}_{\text{e}}}
2025:{\displaystyle {\vec {v}}_{\text{e}}={\vec {V}}_{\text{e}}-{\vec {V}}}
8396:
7748:
7512:"On the Motion of Rockets both in Nonresisting and Resisting Mediums"
6734:
1017:
The necessary wet mass grows exponentially with the desired delta-v.
4263:
4039:
Now, the mass of fuel the rocket initially has on board is equal to
7649:
6475:
per unit mass and burn time, and is used to determine the mass of
6418:
5693:
is taken into account, the following equation can be derived for a
3788:{\displaystyle {\frac {J}{\Delta m}}={\frac {F}{p}}=V_{\text{exh}}}
3262:
the velocity of the rocket exhaust with respect to the rocket (the
1051:
1032:
The equation had been derived earlier by the
British mathematician
502:
475:
7689:
Tsiolkovsky's rocket equations plot and calculator in
WolframAlpha
8301:
6404:
3397:
3267:
1050:
While the derivation of the rocket equation is a straightforward
7264:{\displaystyle \Delta v\ =3v_{\text{e}}\ln 5\ =4.83v_{\text{e}}}
3879:
Imagine a rocket at rest in space with no forces exerted on it (
1029:
who independently derived it and published it in his 1903 work.
7273:
and the payload is 10% × 10% × 10% = 0.1% of the initial mass.
6472:
3728:{\displaystyle J~{\frac {\ln({m_{0}})-\ln({m_{f}})}{\Delta m}}}
2935:{\displaystyle \Delta V=v_{\text{e}}\ln {\frac {m_{0}}{m_{f}}}}
2724:{\displaystyle -m{\frac {dV}{dt}}=v_{\text{e}}{\frac {dm}{dt}}}
471:
467:
7621:"A Transparent Derivation of the Relativistic Rocket Equation"
7172:
With three similar, subsequently smaller stages with the same
6479:
required for the given manoeuvre through the rocket equation.
904:, the equation can be solved for the required propellant mass
8106:
7725:
6449:
61:
7684:
Relativity
Calculator – Learn Tsiolkovsky's rocket equations
865:
is the final total mass without propellant, a.k.a. dry mass.
6523:, which is the fraction of initial weight that is payload.
7623:(see the right side of equation 15 on the last page, with
4388:
In the rocket's center-of-mass frame, if a pellet of mass
1094:
Tsiolkovsky's theoretical rocket from t = 0 to t = delta_t
1421:
is the momentum of the rocket and exhausted mass at time
1036:
in 1810, and later published in a separate book in 1813.
490:
in 1810, and later published in a separate book in 1813.
7484:
7482:
7480:
6800:
of 9,700 meters per second (32,000 ft/s) (Earth to
1061:
7516:
Journal of
Natural Philosophy, Chemistry & the Arts
6537:
The effective exhaust velocity is often specified as a
3009:{\displaystyle m_{f}=m_{0}e^{-\Delta V\ /v_{\text{e}}}}
470:: a device that can apply acceleration to itself using
16:
Mathematical equation describing the motion of a rocket
6220:
6111:
5936:
5381:
5345:
4606:
4592:
4538:
4503:
4463:
4349:
4347:
2614:
7477:
7432:- added terms for gravity and drag in vertical flight
7288:
7207:
7178:
7046:
6980:
6957:
6913:
6890:
6840:
6810:
6783:
6665:
6632:
6603:
6547:
6302:
5997:
5977:
5808:
5784:
5757:
5726:
5703:
5632:
5459:
5429:
5173:
5139:
5119:
4969:
4949:
4890:
4870:
4819:
4799:
4769:
4742:
4722:
4640:
4461:
4441:
4421:
4394:
4320:
4294:
4274:
4097:
3931:
3801:
3743:
3653:
3587:
3561:
3534:
3419:
3379:
3333:
3303:
3276:
3241:
3214:
3187:
3090:
3022:
2948:
2882:
2770:
2739:
2661:
2516:
2493:
2458:
2413:
2375:
2346:
2310:
2274:
2245:
2111:
2038:
1965:
1938:
1902:
1870:
1831:
1804:
1784:
1760:
1724:
1694:
1647:
1620:
1591:
1456:
1427:
1388:
1334:
1308:
1272:
1146:
1110:
1008:{\displaystyle m_{0}=m_{f}e^{\Delta v/v_{\text{e}}}.}
950:
910:
883:
844:
811:
785:
750:
717:
661:
536:
511:
7376:
6192:{\textstyle R^{\frac {2v_{\text{e}}}{c}}=\exp \left}
3077:{\displaystyle m_{0}=m_{f}e^{\Delta V/v_{\text{e}}}}
7358:
7263:
7191:
7162:
7010:
6966:
6943:
6899:
6870:
6819:
6792:
6678:
6645:
6616:
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6092:
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5963:
5920:
5790:
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5712:
5673:
5618:
5441:
5412:
5367:
5331:
5159:
5125:
5103:
4955:
4935:
4876:
4864:. The remaining mass of the rocket after ejecting
4856:
4805:
4785:
4755:
4728:
4706:
4626:
4576:
4447:
4427:
4407:
4377:
4333:
4306:
4280:
4253:
4029:
3864:
3787:
3727:
3637:
3570:
3547:
3518:
3388:
3359:
3316:
3289:
3254:
3227:
3200:
3173:
3076:
3008:
2934:
2866:
2752:
2723:
2643:
2598:
2502:
2479:
2444:
2399:
2361:
2332:
2296:
2260:
2231:
2097:
2024:
1951:
1924:
1885:
1856:
1816:
1790:
1769:
1746:
1709:
1680:
1632:
1606:
1574:
1442:
1413:
1374:
1320:
1294:
1258:
1132:
1007:
936:
896:
857:
824:
791:
763:
730:
697:
644:
520:
6282:{\textstyle \tanh x={\frac {e^{2x}-1}{e^{2x}+1}}}
8500:
6884:: suppose that the first stage should provide a
6587:{\displaystyle v_{\text{e}}=g_{0}I_{\text{sp}},}
5626:since the final remaining mass of the rocket is
5461:
3578:is the initial mass minus the final (dry) mass,
3208:is the initial total mass including propellant,
1180:
6974:of 4,700 meters per second (15,000 ft/s);
6907:of 5,000 meters per second (16,000 ft/s);
4736:be the initial fuel mass fraction on board and
2510:results in a decrease in rocket mass in time),
698:{\displaystyle v_{\text{e}}=I_{\text{sp}}g_{0}}
4378:{\textstyle {\tfrac {1}{2}}v_{\text{eff}}^{2}}
4091:allows a substitution on the right) obtains:
3266:, or, if measured in time, that multiplied by
1025:The equation is named after Russian scientist
7709:
6526:
494:also developed it independently in 1912, and
423:
7570:
1681:{\displaystyle {\vec {V}}+\Delta {\vec {V}}}
7505:
7503:
6827:to overcome gravity and aerodynamic drag).
3638:{\displaystyle \int _{t_{0}}^{t_{f}}F~dt=J}
8483:
7716:
7702:
6497:which displays the required mission delta-
4943:. The overall speed change after ejecting
4627:{\textstyle u=\Delta v{\tfrac {m}{m_{p}}}}
1375:{\displaystyle {\vec {P}}_{0}=m{\vec {V}}}
1082:
430:
416:
6467:is produced by reaction engines, such as
5991:allows this equation to be rearranged as
3889:exhaust velocity relative to the rocket v
2798:
2420:
2400:{\displaystyle \Delta m\Delta {\vec {V}}}
1582:and where, with respect to the observer:
7723:
7500:
6390:
5368:{\textstyle \Delta x={\frac {\phi }{N}}}
4435:and the remaining mass of the rocket is
1089:
474:by expelling part of its mass with high
441:
7531:
7529:
7488:
6541:and they are related to each other by:
4264:Limit of finite mass "pellet" expulsion
3898:propelling the rocket that is equal to
1857:{\displaystyle \left(m-\Delta m\right)}
8501:
8362:Transposition, docking, and extraction
7576:
7031:In the case of sequentially thrusting
6756:The rocket equation can be applied to
5413:{\textstyle x_{j}={\frac {j\phi }{N}}}
1688:is the velocity of the rocket at time
1614:is the velocity of the rocket at time
1302:is the momentum of the rocket at time
7697:
7535:
7509:
5684:
3894:(m/s). This creates a constant force
3874:
2608:If there are no external forces then
2333:{\displaystyle {\vec {F}}_{\text{i}}}
2297:{\displaystyle {\vec {v}}_{\text{e}}}
1925:{\displaystyle {\vec {V}}_{\text{e}}}
1747:{\displaystyle {\vec {V}}_{\text{e}}}
1414:{\displaystyle {\vec {P}}_{\Delta t}}
1062:Experiment of the Boat by Tsiolkovsky
832:is the initial total mass, including
528:(with no external forces acting) is:
498:derived it independently about 1920.
7645:"The Tyranny of the Rocket Equation"
7526:
6653:is the specific impulse measured in
5674:{\displaystyle m_{f}=m_{0}(1-\phi )}
5442:{\displaystyle N\rightarrow \infty }
3403:
2445:{\displaystyle dm\,d{\vec {v}}\to 0}
6624:is the specific impulse in seconds,
6210:as well as the "power" identity at
5964:{\textstyle {\frac {m_{0}}{m_{1}}}}
4936:{\displaystyle m=m_{0}(1-j\phi /N)}
13:
7289:
7208:
7051:
6958:
6891:
6811:
6784:
6489:For interplanetary missions delta-
6303:
6001:
5876:
5853:
5704:
5476:
5471:
5436:
5346:
5302:
5174:
4970:
4857:{\displaystyle m_{p}=\phi m_{0}/N}
4641:
4599:
4555:
4314:, with an effective exhaust speed
4301:
4101:
4053:. For the constant mass flow rate
3802:
3750:
3716:
3562:
3496:
3420:
3380:
3137:
3051:
2980:
2883:
2790:
2494:
2471:
2382:
2376:
2214:
2208:
2199:
2162:
2126:
1877:
1864:is the mass of the rocket at time
1843:
1798:is the mass of the rocket at time
1761:
1701:
1663:
1547:
1524:
1493:
1471:
1434:
1403:
1247:
1215:
1184:
979:
537:
512:
14:
8540:
8422:Kepler's laws of planetary motion
7679:How to derive the rocket equation
7672:
6452:. As used in this context, it is
5133:the last term in the denominator
202:Kepler's laws of planetary motion
8482:
8417:Interplanetary Transport Network
8297:Collision avoidance (spacecraft)
7634:in the notation of this article)
7379:
7022:engines, the tanks, and payload.
6706:
6504:
3408:
1066:
478:and can thereby move due to the
32:
8382:Astronomical coordinate systems
8136:Longitude of the ascending node
6482:For multiple manoeuvres, delta-
3913:, its acceleration at any time
3795:the integral can be equated to
2653:conservation of linear momentum
1087:Consider the following system:
8514:Eponymous equations of physics
8455:Retrograde and prograde motion
7637:
7613:
7546:
7453:
7331:
7317:
7011:{\displaystyle 1-e^{-4.7/4.5}}
6944:{\displaystyle 1-e^{-5.0/4.5}}
6871:{\displaystyle 1-e^{-9.7/4.5}}
6501:as a function of launch date.
5668:
5656:
5468:
5453:becomes the definite integral
5433:
5095:
5058:
5055:
5032:
4930:
4907:
4813:discrete pellets each of mass
4695:
4676:
4562:
4552:
4334:{\displaystyle v_{\text{eff}}}
4298:
4057:it will therefore take a time
4021:
4015:
3982:
3976:
3711:
3696:
3684:
3669:
3555:is the initial (wet) mass and
3470:
3462:
2436:
2430:
2391:
2353:
2318:
2282:
2252:
2223:
2187:
2171:
2144:
2119:
2083:
2067:
2046:
2016:
1995:
1973:
1910:
1732:
1672:
1654:
1598:
1560:
1533:
1515:
1464:
1396:
1366:
1342:
1295:{\displaystyle {\vec {P}}_{0}}
1280:
1233:
1208:
1190:
1164:
1133:{\displaystyle {\vec {F}}_{i}}
1118:
1:
7446:
6617:{\displaystyle I_{\text{sp}}}
5167:and can be neglected to give
3911:Newton's Second Law of Motion
3647:The integral is found to be:
2644:{\textstyle \sum _{i}F_{i}=0}
1102:Newton's second law of motion
1077:
731:{\displaystyle I_{\text{sp}}}
8402:Equatorial coordinate system
7416:increases the final velocity
7192:{\displaystyle v_{\text{e}}}
6721:is a limit to the amount of
6646:{\displaystyle v_{\text{e}}}
5160:{\displaystyle \phi /N\ll 1}
4307:{\displaystyle N\to \infty }
3921:divided by its current mass
3881:Newton's First Law of Motion
3317:{\displaystyle v_{\text{e}}}
3290:{\displaystyle v_{\text{e}}}
3270:-on-Earth acceleration). If
3255:{\displaystyle v_{\text{e}}}
2753:{\displaystyle v_{\text{e}}}
2480:{\displaystyle dm=-\Delta m}
1952:{\displaystyle v_{\text{e}}}
1896:The velocity of the exhaust
7:
7372:
6772:
6458:physical change in velocity
5749:inertial frame of reference
3360:{\displaystyle m_{0}-m_{f}}
2487:(since ejecting a positive
2340:have the same direction as
937:{\displaystyle m_{0}-m_{f}}
356:Tsiolkovsky rocket equation
10:
8545:
8154:Longitude of the periapsis
7491:"Reactive Flying Machines"
6533:Effective exhaust velocity
6530:
6527:Effective exhaust velocity
6508:
6434:spacecraft flight dynamics
6402:
6395:
4786:{\displaystyle \phi m_{0}}
4288:pellets consecutively, as
2362:{\displaystyle {\vec {V}}}
2261:{\displaystyle {\vec {V}}}
1886:{\displaystyle t=\Delta t}
1710:{\displaystyle t=\Delta t}
1607:{\displaystyle {\vec {V}}}
1443:{\displaystyle t=\Delta t}
1020:
707:effective exhaust velocity
452:effective exhaust velocity
325:Engineering and efficiency
144:Bi-elliptic transfer orbit
8478:
8465:Specific angular momentum
8370:
8282:
8226:
8162:
8115:
8055:
8046:
7942:
7852:
7741:
7732:
7026:
6471:, is proportional to the
6296:), this is equivalent to
3885:constant mass flow rate R
1104:relates external forces (
460:classical rocket equation
7599:10.1088/1361-6552/ab315b
7441:Stigler’s law of eponymy
7412:- applying delta-v in a
6967:{\displaystyle \Delta v}
6900:{\displaystyle \Delta v}
6820:{\displaystyle \Delta v}
6793:{\displaystyle \Delta v}
6511:Propellant mass fraction
6199:(here "exp" denotes the
5713:{\displaystyle \Delta v}
3917:is its propelling force
3571:{\displaystyle \Delta m}
3389:{\displaystyle \Delta V}
3371:of propellant expended.
2762:Tsiolkovsky's hypothesis
2503:{\displaystyle \Delta m}
1770:{\displaystyle \Delta t}
877:), and a given dry mass
521:{\displaystyle \Delta v}
480:conservation of momentum
351:Propellant mass fraction
250:Gravitational influences
8460:Specific orbital energy
7542:. G. & S. Robinson.
7536:Moore, William (1813).
7510:Moore, William (1810).
7425:Reversibility of orbits
7199:for each stage, gives:
1083:Most popular derivation
222:Specific orbital energy
8519:Konstantin Tsiolkovsky
7872:Geostationary transfer
7360:
7265:
7193:
7164:
7012:
6968:
6945:
6901:
6872:
6821:
6794:
6680:
6647:
6618:
6588:
6493:is often plotted on a
6448:that has the units of
6436:, is a measure of the
6382:
6283:
6212:Logarithmic identities
6193:
6094:
5985:
5965:
5922:
5792:
5772:
5741:
5714:
5675:
5620:
5443:
5414:
5369:
5333:
5298:
5219:
5161:
5127:
5113:Notice that for large
5105:
5015:
4957:
4937:
4878:
4858:
4807:
4787:
4757:
4730:
4708:
4634:, from which we find
4628:
4578:
4449:
4429:
4409:
4379:
4335:
4308:
4282:
4255:
4031:
3866:
3789:
3729:
3639:
3572:
3549:
3520:
3390:
3361:
3318:
3291:
3256:
3229:
3202:
3175:
3078:
3010:
2936:
2868:
2760:is constant (known as
2754:
2725:
2645:
2600:
2504:
2481:
2446:
2407:are negligible (since
2401:
2363:
2334:
2298:
2262:
2233:
2099:
2026:
1953:
1926:
1887:
1858:
1818:
1792:
1771:
1748:
1711:
1682:
1634:
1608:
1576:
1444:
1415:
1376:
1322:
1296:
1260:
1134:
1095:
1027:Konstantin Tsiolkovsky
1009:
938:
898:
859:
826:
793:
765:
732:
699:
646:
522:
501:The maximum change of
484:Konstantin Tsiolkovsky
455:
139:Hohmann transfer orbit
8524:Single-stage-to-orbit
8445:Orbital state vectors
8387:Characteristic energy
8357:Trans-lunar injection
8145:Argument of periapsis
7822:Prograde / Retrograde
7783:Hyperbolic trajectory
7436:Spacecraft propulsion
7361:
7266:
7194:
7165:
7013:
6969:
6946:
6902:
6873:
6832:Single-stage-to-orbit
6822:
6795:
6681:
6679:{\displaystyle g_{0}}
6648:
6619:
6589:
6517:aerospace engineering
6391:Terms of the equation
6383:
6284:
6194:
6095:
5986:
5966:
5923:
5793:
5773:
5771:{\displaystyle m_{0}}
5742:
5740:{\displaystyle m_{1}}
5715:
5676:
5621:
5444:
5415:
5370:
5334:
5272:
5193:
5162:
5128:
5106:
4989:
4958:
4938:
4879:
4859:
4808:
4788:
4758:
4756:{\displaystyle m_{0}}
4731:
4729:{\displaystyle \phi }
4709:
4629:
4579:
4450:
4430:
4410:
4408:{\displaystyle m_{p}}
4380:
4336:
4309:
4283:
4256:
4032:
3867:
3790:
3730:
3640:
3573:
3550:
3548:{\displaystyle m_{0}}
3521:
3391:
3362:
3319:
3292:
3257:
3230:
3228:{\displaystyle m_{f}}
3203:
3201:{\displaystyle m_{0}}
3176:
3079:
3011:
2937:
2869:
2755:
2726:
2646:
2601:
2505:
2482:
2447:
2402:
2364:
2335:
2299:
2263:
2234:
2105:Solving this yields:
2100:
2027:
1954:
1927:
1888:
1859:
1819:
1793:
1772:
1749:
1712:
1683:
1635:
1609:
1577:
1445:
1416:
1377:
1323:
1297:
1261:
1135:
1093:
1010:
939:
899:
897:{\displaystyle m_{f}}
860:
858:{\displaystyle m_{f}}
827:
825:{\displaystyle m_{0}}
794:
766:
764:{\displaystyle g_{0}}
742:in dimension of time;
733:
700:
647:
523:
464:ideal rocket equation
445:
335:Preflight engineering
67:Argument of periapsis
8292:Bi-elliptic transfer
7812:Parabolic trajectory
7286:
7205:
7176:
7044:
6978:
6955:
6911:
6888:
6838:
6808:
6781:
6663:
6630:
6601:
6545:
6300:
6218:
6201:exponential function
6109:
5995:
5975:
5934:
5806:
5782:
5755:
5724:
5701:
5630:
5457:
5427:
5379:
5343:
5171:
5137:
5117:
4967:
4947:
4888:
4868:
4817:
4797:
4767:
4740:
4720:
4638:
4590:
4459:
4439:
4419:
4415:is ejected at speed
4392:
4345:
4318:
4292:
4272:
4095:
3929:
3799:
3741:
3651:
3585:
3559:
3532:
3417:
3377:
3331:
3301:
3274:
3239:
3235:the final mass, and
3212:
3185:
3088:
3020:
2946:
2880:
2768:
2737:
2659:
2612:
2514:
2491:
2456:
2411:
2373:
2344:
2308:
2272:
2243:
2109:
2036:
1963:
1936:
1900:
1868:
1829:
1802:
1782:
1758:
1722:
1692:
1645:
1618:
1589:
1454:
1425:
1386:
1332:
1306:
1270:
1144:
1108:
948:
908:
881:
842:
809:
792:{\displaystyle \ln }
783:
748:
715:
659:
534:
509:
482:. It is credited to
446:A rocket's required
391:Propulsive maneuvers
8332:Low-energy transfer
7619:Forward, Robert L.
7591:2019PhyEd..54f5001B
7420:Relativistic rocket
6767:electric propulsion
6294:Hyperbolic function
6214:) and the identity
5695:relativistic rocket
5509:
4963:pellets is the sum
4498:
4374:
3616:
3528:where T is thrust,
3457:
2848:
2797:
1817:{\displaystyle t=0}
1633:{\displaystyle t=0}
1321:{\displaystyle t=0}
368:Efficiency measures
271:Sphere of influence
240:Celestial mechanics
22:Part of a series on
8327:Inclination change
7975:Distant retrograde
7559:2011-08-15 at the
7466:2011-08-15 at the
7387:Spaceflight portal
7356:
7261:
7189:
7160:
7158:
7008:
6964:
6941:
6897:
6882:Two-stage-to-orbit
6868:
6817:
6790:
6763:impulsive maneuver
6727:non-rocket systems
6676:
6643:
6614:
6584:
6421:"), symbolised as
6378:
6279:
6189:
6090:
5981:
5961:
5918:
5788:
5768:
5737:
5710:
5691:special relativity
5685:Special relativity
5671:
5616:
5495:
5475:
5439:
5410:
5365:
5329:
5157:
5123:
5101:
4953:
4933:
4874:
4854:
4803:
4783:
4753:
4726:
4704:
4624:
4622:
4574:
4547:
4512:
4484:
4472:
4445:
4425:
4405:
4375:
4360:
4358:
4331:
4304:
4278:
4251:
4027:
3875:Acceleration-based
3862:
3785:
3725:
3635:
3588:
3568:
3545:
3516:
3429:
3386:
3357:
3314:
3287:
3252:
3225:
3198:
3171:
3074:
3006:
2932:
2864:
2820:
2774:
2750:
2721:
2641:
2624:
2596:
2526:
2500:
2477:
2442:
2397:
2359:
2330:
2294:
2258:
2229:
2095:
2022:
1949:
1922:
1883:
1854:
1814:
1788:
1767:
1744:
1707:
1678:
1630:
1604:
1572:
1440:
1411:
1372:
1318:
1292:
1256:
1197:
1156:
1130:
1096:
1005:
934:
894:
855:
836:, a.k.a. wet mass;
822:
789:
761:
728:
695:
642:
518:
456:
187:Dynamical friction
8529:Rocket propulsion
8496:
8495:
8470:Two-line elements
8278:
8277:
8200:Eccentric anomaly
8042:
8041:
7909:Orbit of the Moon
7768:Highly elliptical
7579:Physics Education
7430:Robert H. Goddard
7350:
7336:
7308:
7297:
7258:
7244:
7230:
7216:
7186:
7150:
7118:
7099:
7074:
7059:
6758:orbital maneuvers
6747:tether propulsion
6640:
6611:
6578:
6555:
6371:
6341:
6335:
6277:
6208:Natural logarithm
6173:
6166:
6136:
6129:
6088:
6078:
6071:
6043:
6036:
6011:
5984:{\displaystyle R}
5959:
5915:
5911:
5889:
5886:
5863:
5831:
5798:standing for the
5791:{\displaystyle c}
5611:
5580:
5568:
5543:
5531:
5492:
5460:
5408:
5363:
5327:
5269:
5257:
5190:
5126:{\displaystyle N}
5099:
5098:
4986:
4956:{\displaystyle j}
4877:{\displaystyle j}
4806:{\displaystyle N}
4699:
4698:
4657:
4621:
4546:
4511:
4491:
4471:
4448:{\displaystyle m}
4428:{\displaystyle u}
4367:
4357:
4328:
4281:{\displaystyle N}
4242:
4207:
4199:
4146:
4100:
4087:(and noting that
4025:
4007:
3986:
3959:
3934:
3856:
3823:
3818:
3782:
3770:
3757:
3723:
3659:
3622:
3509:
3505:
3404:Other derivations
3311:
3284:
3249:
3155:
3069:
3001:
2988:
2930:
2899:
2876:This then yields
2862:
2747:
2719:
2698:
2686:
2615:
2594:
2573:
2561:
2517:
2433:
2394:
2356:
2327:
2321:
2291:
2285:
2255:
2226:
2196:
2190:
2174:
2147:
2122:
2092:
2086:
2070:
2055:
2049:
2019:
2004:
1998:
1982:
1976:
1946:
1919:
1913:
1791:{\displaystyle m}
1741:
1735:
1675:
1657:
1601:
1569:
1563:
1536:
1518:
1467:
1399:
1369:
1345:
1283:
1254:
1236:
1211:
1179:
1167:
1147:
1121:
997:
801:natural logarithm
725:
682:
669:
637:
596:
584:
553:
450:as a function of
440:
439:
290:Lagrangian points
227:Vis-viva equation
197:Kepler's equation
44:Orbital mechanics
8536:
8486:
8485:
8427:Lagrangian point
8322:Hohmann transfer
8267:
8253:
8244:
8235:
8215:
8206:
8197:
8188:
8184:
8180:
8171:
8151:
8142:
8133:
8124:
8104:
8100:
8091:
8082:
8073:
8053:
8052:
8022:Heliosynchronous
7971:Lagrange points
7924:Transatmospheric
7739:
7738:
7718:
7711:
7704:
7695:
7694:
7666:
7665:
7663:
7662:
7653:. Archived from
7641:
7635:
7617:
7611:
7610:
7574:
7568:
7550:
7544:
7543:
7533:
7524:
7523:
7507:
7498:
7497:
7495:
7489:Tsiolkovsky, K.
7486:
7475:
7457:
7389:
7384:
7383:
7382:
7365:
7363:
7362:
7357:
7352:
7351:
7348:
7334:
7327:
7310:
7309:
7306:
7295:
7270:
7268:
7267:
7262:
7260:
7259:
7256:
7242:
7232:
7231:
7228:
7214:
7198:
7196:
7195:
7190:
7188:
7187:
7184:
7169:
7167:
7166:
7161:
7159:
7152:
7151:
7148:
7133:
7120:
7119:
7116:
7104:
7100:
7098:
7084:
7076:
7075:
7072:
7057:
7017:
7015:
7014:
7009:
7007:
7006:
7002:
6973:
6971:
6970:
6965:
6950:
6948:
6947:
6942:
6940:
6939:
6935:
6906:
6904:
6903:
6898:
6877:
6875:
6874:
6869:
6867:
6866:
6862:
6826:
6824:
6823:
6818:
6799:
6797:
6796:
6791:
6701:
6693:
6688:standard gravity
6685:
6683:
6682:
6677:
6675:
6674:
6652:
6650:
6649:
6644:
6642:
6641:
6638:
6623:
6621:
6620:
6615:
6613:
6612:
6609:
6593:
6591:
6590:
6585:
6580:
6579:
6576:
6570:
6569:
6557:
6556:
6553:
6539:specific impulse
6521:payload fraction
6460:of the vehicle.
6456:the same as the
6442:orbital maneuver
6387:
6385:
6384:
6379:
6377:
6373:
6372:
6370:
6369:
6360:
6359:
6350:
6342:
6337:
6336:
6333:
6327:
6288:
6286:
6285:
6280:
6278:
6276:
6269:
6268:
6255:
6248:
6247:
6234:
6198:
6196:
6195:
6190:
6188:
6184:
6174:
6169:
6168:
6167:
6164:
6154:
6138:
6137:
6132:
6131:
6130:
6127:
6117:
6102:Then, using the
6099:
6097:
6096:
6091:
6089:
6087:
6080:
6079:
6074:
6073:
6072:
6069:
6059:
6052:
6045:
6044:
6039:
6038:
6037:
6034:
6024:
6017:
6012:
6007:
5999:
5990:
5988:
5987:
5982:
5970:
5968:
5967:
5962:
5960:
5958:
5957:
5948:
5947:
5938:
5927:
5925:
5924:
5919:
5917:
5916:
5914:
5913:
5912:
5909:
5896:
5894:
5890:
5888:
5887:
5882:
5874:
5865:
5864:
5859:
5851:
5842:
5832:
5830:
5829:
5820:
5819:
5810:
5797:
5795:
5794:
5789:
5778:initially), and
5777:
5775:
5774:
5769:
5767:
5766:
5746:
5744:
5743:
5738:
5736:
5735:
5719:
5717:
5716:
5711:
5680:
5678:
5677:
5672:
5655:
5654:
5642:
5641:
5625:
5623:
5622:
5617:
5612:
5610:
5609:
5600:
5599:
5590:
5582:
5581:
5578:
5569:
5567:
5553:
5545:
5544:
5541:
5532:
5530:
5519:
5511:
5508:
5503:
5494:
5493:
5490:
5474:
5448:
5446:
5445:
5440:
5419:
5417:
5416:
5411:
5409:
5404:
5396:
5391:
5390:
5374:
5372:
5371:
5366:
5364:
5356:
5338:
5336:
5335:
5330:
5328:
5326:
5325:
5324:
5308:
5300:
5297:
5286:
5271:
5270:
5267:
5258:
5256:
5252:
5234:
5230:
5221:
5218:
5207:
5192:
5191:
5188:
5166:
5164:
5163:
5158:
5147:
5132:
5130:
5129:
5124:
5110:
5108:
5107:
5102:
5100:
5091:
5077:
5051:
5031:
5030:
5026:
5017:
5014:
5003:
4988:
4987:
4984:
4962:
4960:
4959:
4954:
4942:
4940:
4939:
4934:
4926:
4906:
4905:
4884:pellets is then
4883:
4881:
4880:
4875:
4863:
4861:
4860:
4855:
4850:
4845:
4844:
4829:
4828:
4812:
4810:
4809:
4804:
4792:
4790:
4789:
4784:
4782:
4781:
4762:
4760:
4759:
4754:
4752:
4751:
4735:
4733:
4732:
4727:
4713:
4711:
4710:
4705:
4700:
4694:
4693:
4672:
4671:
4670:
4661:
4659:
4658:
4655:
4633:
4631:
4630:
4625:
4623:
4620:
4619:
4607:
4583:
4581:
4580:
4575:
4570:
4569:
4548:
4539:
4533:
4532:
4523:
4522:
4513:
4504:
4497:
4492:
4489:
4483:
4482:
4473:
4464:
4454:
4452:
4451:
4446:
4434:
4432:
4431:
4426:
4414:
4412:
4411:
4406:
4404:
4403:
4384:
4382:
4381:
4376:
4373:
4368:
4365:
4359:
4350:
4340:
4338:
4337:
4332:
4330:
4329:
4326:
4313:
4311:
4310:
4305:
4287:
4285:
4284:
4279:
4260:
4258:
4257:
4252:
4247:
4243:
4241:
4240:
4231:
4230:
4221:
4209:
4208:
4205:
4197:
4193:
4189:
4188:
4187:
4169:
4168:
4148:
4147:
4144:
4132:
4131:
4119:
4118:
4098:
4036:
4034:
4033:
4028:
4026:
4024:
4010:
4009:
4008:
4005:
3995:
3987:
3985:
3968:
3960:
3958:
3950:
3942:
3932:
3871:
3869:
3868:
3863:
3861:
3857:
3855:
3854:
3845:
3844:
3835:
3821:
3820:
3819:
3816:
3794:
3792:
3791:
3786:
3784:
3783:
3780:
3771:
3763:
3758:
3756:
3745:
3734:
3732:
3731:
3726:
3724:
3722:
3714:
3710:
3709:
3708:
3683:
3682:
3681:
3661:
3657:
3644:
3642:
3641:
3636:
3620:
3615:
3614:
3613:
3603:
3602:
3601:
3577:
3575:
3574:
3569:
3554:
3552:
3551:
3546:
3544:
3543:
3525:
3523:
3522:
3517:
3507:
3506:
3504:
3503:
3495:
3487:
3486:
3485:
3474:
3473:
3465:
3459:
3456:
3455:
3454:
3444:
3443:
3442:
3395:
3393:
3392:
3387:
3366:
3364:
3363:
3358:
3356:
3355:
3343:
3342:
3323:
3321:
3320:
3315:
3313:
3312:
3309:
3296:
3294:
3293:
3288:
3286:
3285:
3282:
3264:specific impulse
3261:
3259:
3258:
3253:
3251:
3250:
3247:
3234:
3232:
3231:
3226:
3224:
3223:
3207:
3205:
3204:
3199:
3197:
3196:
3180:
3178:
3177:
3172:
3170:
3166:
3159:
3158:
3157:
3156:
3153:
3147:
3126:
3125:
3113:
3112:
3100:
3099:
3083:
3081:
3080:
3075:
3073:
3072:
3071:
3070:
3067:
3061:
3045:
3044:
3032:
3031:
3015:
3013:
3012:
3007:
3005:
3004:
3003:
3002:
2999:
2993:
2986:
2971:
2970:
2958:
2957:
2942:or equivalently
2941:
2939:
2938:
2933:
2931:
2929:
2928:
2919:
2918:
2909:
2901:
2900:
2897:
2873:
2871:
2870:
2865:
2863:
2858:
2850:
2847:
2846:
2845:
2835:
2834:
2833:
2819:
2818:
2817:
2796:
2782:
2759:
2757:
2756:
2751:
2749:
2748:
2745:
2730:
2728:
2727:
2722:
2720:
2718:
2710:
2702:
2700:
2699:
2696:
2687:
2685:
2677:
2669:
2650:
2648:
2647:
2642:
2634:
2633:
2623:
2605:
2603:
2602:
2597:
2595:
2593:
2585:
2577:
2575:
2574:
2571:
2562:
2560:
2552:
2544:
2536:
2535:
2525:
2509:
2507:
2506:
2501:
2486:
2484:
2483:
2478:
2451:
2449:
2448:
2443:
2435:
2434:
2426:
2406:
2404:
2403:
2398:
2396:
2395:
2387:
2368:
2366:
2365:
2360:
2358:
2357:
2349:
2339:
2337:
2336:
2331:
2329:
2328:
2325:
2323:
2322:
2314:
2303:
2301:
2300:
2295:
2293:
2292:
2289:
2287:
2286:
2278:
2267:
2265:
2264:
2259:
2257:
2256:
2248:
2238:
2236:
2235:
2230:
2228:
2227:
2219:
2198:
2197:
2194:
2192:
2191:
2183:
2176:
2175:
2167:
2155:
2154:
2149:
2148:
2140:
2133:
2132:
2124:
2123:
2115:
2104:
2102:
2101:
2096:
2094:
2093:
2090:
2088:
2087:
2079:
2072:
2071:
2063:
2057:
2056:
2053:
2051:
2050:
2042:
2031:
2029:
2028:
2023:
2021:
2020:
2012:
2006:
2005:
2002:
2000:
1999:
1991:
1984:
1983:
1980:
1978:
1977:
1969:
1958:
1956:
1955:
1950:
1948:
1947:
1944:
1931:
1929:
1928:
1923:
1921:
1920:
1917:
1915:
1914:
1906:
1892:
1890:
1889:
1884:
1863:
1861:
1860:
1855:
1853:
1849:
1823:
1821:
1820:
1815:
1797:
1795:
1794:
1789:
1776:
1774:
1773:
1768:
1753:
1751:
1750:
1745:
1743:
1742:
1739:
1737:
1736:
1728:
1716:
1714:
1713:
1708:
1687:
1685:
1684:
1679:
1677:
1676:
1668:
1659:
1658:
1650:
1639:
1637:
1636:
1631:
1613:
1611:
1610:
1605:
1603:
1602:
1594:
1581:
1579:
1578:
1573:
1571:
1570:
1567:
1565:
1564:
1556:
1543:
1539:
1538:
1537:
1529:
1520:
1519:
1511:
1503:
1499:
1478:
1477:
1469:
1468:
1460:
1449:
1447:
1446:
1441:
1420:
1418:
1417:
1412:
1410:
1409:
1401:
1400:
1392:
1381:
1379:
1378:
1373:
1371:
1370:
1362:
1353:
1352:
1347:
1346:
1338:
1327:
1325:
1324:
1319:
1301:
1299:
1298:
1293:
1291:
1290:
1285:
1284:
1276:
1265:
1263:
1262:
1257:
1255:
1253:
1245:
1244:
1243:
1238:
1237:
1229:
1222:
1221:
1213:
1212:
1204:
1199:
1196:
1175:
1174:
1169:
1168:
1160:
1155:
1139:
1137:
1136:
1131:
1129:
1128:
1123:
1122:
1114:
1070:
1014:
1012:
1011:
1006:
1001:
1000:
999:
998:
995:
989:
973:
972:
960:
959:
943:
941:
940:
935:
933:
932:
920:
919:
903:
901:
900:
895:
893:
892:
864:
862:
861:
856:
854:
853:
831:
829:
828:
823:
821:
820:
798:
796:
795:
790:
773:standard gravity
770:
768:
767:
762:
760:
759:
740:specific impulse
737:
735:
734:
729:
727:
726:
723:
704:
702:
701:
696:
694:
693:
684:
683:
680:
671:
670:
667:
651:
649:
648:
643:
638:
636:
635:
626:
625:
616:
608:
607:
598:
597:
594:
585:
583:
582:
573:
572:
563:
555:
554:
551:
527:
525:
524:
519:
505:of the vehicle,
432:
425:
418:
397:Orbital maneuver
346:Payload fraction
326:
307:Lissajous orbits
241:
212:Orbital velocity
159:Hyperbolic orbit
55:Orbital elements
45:
36:
19:
18:
8544:
8543:
8539:
8538:
8537:
8535:
8534:
8533:
8499:
8498:
8497:
8492:
8474:
8392:Escape velocity
8373:
8366:
8347:Rocket equation
8274:
8266:
8260:
8251:
8242:
8233:
8222:
8213:
8204:
8195:
8186:
8182:
8178:
8169:
8158:
8149:
8140:
8131:
8122:
8111:
8102:
8098:
8094:Semi-minor axis
8089:
8085:Semi-major axis
8080:
8071:
8065:
8038:
7960:Areosynchronous
7944:
7938:
7919:Sun-synchronous
7904:Near-equatorial
7848:
7728:
7722:
7675:
7670:
7669:
7660:
7658:
7643:
7642:
7638:
7633:
7618:
7614:
7575:
7571:
7561:Wayback Machine
7551:
7547:
7534:
7527:
7508:
7501:
7493:
7487:
7478:
7468:Wayback Machine
7458:
7454:
7449:
7385:
7380:
7378:
7375:
7347:
7343:
7323:
7305:
7301:
7287:
7284:
7283:
7255:
7251:
7227:
7223:
7206:
7203:
7202:
7183:
7179:
7177:
7174:
7173:
7157:
7156:
7147:
7143:
7131:
7130:
7115:
7111:
7102:
7101:
7088:
7083:
7071:
7067:
7060:
7047:
7045:
7042:
7041:
7029:
6998:
6991:
6987:
6979:
6976:
6975:
6956:
6953:
6952:
6931:
6924:
6920:
6912:
6909:
6908:
6889:
6886:
6885:
6858:
6851:
6847:
6839:
6836:
6835:
6809:
6806:
6805:
6782:
6779:
6778:
6775:
6739:space elevators
6709:
6699:
6691:
6670:
6666:
6664:
6661:
6660:
6637:
6633:
6631:
6628:
6627:
6608:
6604:
6602:
6599:
6598:
6575:
6571:
6565:
6561:
6552:
6548:
6546:
6543:
6542:
6535:
6529:
6513:
6507:
6486:sums linearly.
6428:and pronounced
6407:
6401:
6393:
6365:
6361:
6355:
6351:
6349:
6332:
6328:
6326:
6325:
6321:
6301:
6298:
6297:
6261:
6257:
6256:
6240:
6236:
6235:
6233:
6219:
6216:
6215:
6163:
6159:
6155:
6153:
6152:
6148:
6126:
6122:
6118:
6116:
6112:
6110:
6107:
6106:
6068:
6064:
6060:
6058:
6054:
6053:
6033:
6029:
6025:
6023:
6019:
6018:
6016:
6000:
5998:
5996:
5993:
5992:
5976:
5973:
5972:
5953:
5949:
5943:
5939:
5937:
5935:
5932:
5931:
5908:
5904:
5900:
5895:
5875:
5873:
5866:
5852:
5850:
5843:
5841:
5837:
5836:
5825:
5821:
5815:
5811:
5809:
5807:
5804:
5803:
5783:
5780:
5779:
5762:
5758:
5756:
5753:
5752:
5731:
5727:
5725:
5722:
5721:
5702:
5699:
5698:
5687:
5650:
5646:
5637:
5633:
5631:
5628:
5627:
5605:
5601:
5595:
5591:
5589:
5577:
5573:
5557:
5552:
5540:
5536:
5520:
5512:
5510:
5504:
5499:
5489:
5485:
5464:
5458:
5455:
5454:
5428:
5425:
5424:
5397:
5395:
5386:
5382:
5380:
5377:
5376:
5355:
5344:
5341:
5340:
5320:
5316:
5309:
5301:
5299:
5287:
5276:
5266:
5262:
5248:
5235:
5226:
5222:
5220:
5208:
5197:
5187:
5183:
5172:
5169:
5168:
5143:
5138:
5135:
5134:
5118:
5115:
5114:
5087:
5073:
5047:
5022:
5018:
5016:
5004:
4993:
4983:
4979:
4968:
4965:
4964:
4948:
4945:
4944:
4922:
4901:
4897:
4889:
4886:
4885:
4869:
4866:
4865:
4846:
4840:
4836:
4824:
4820:
4818:
4815:
4814:
4798:
4795:
4794:
4777:
4773:
4768:
4765:
4764:
4747:
4743:
4741:
4738:
4737:
4721:
4718:
4717:
4689:
4685:
4666:
4662:
4660:
4654:
4650:
4639:
4636:
4635:
4615:
4611:
4605:
4591:
4588:
4587:
4565:
4561:
4537:
4528:
4524:
4518:
4514:
4502:
4493:
4488:
4478:
4474:
4462:
4460:
4457:
4456:
4440:
4437:
4436:
4420:
4417:
4416:
4399:
4395:
4393:
4390:
4389:
4369:
4364:
4348:
4346:
4343:
4342:
4325:
4321:
4319:
4316:
4315:
4293:
4290:
4289:
4273:
4270:
4269:
4266:
4236:
4232:
4226:
4222:
4220:
4216:
4204:
4200:
4183:
4179:
4164:
4160:
4153:
4149:
4143:
4139:
4127:
4123:
4114:
4110:
4096:
4093:
4092:
4073:
4067:
4051:
4045:
4011:
4004:
4000:
3996:
3994:
3972:
3967:
3951:
3943:
3941:
3930:
3927:
3926:
3907:
3892:
3877:
3850:
3846:
3840:
3836:
3834:
3830:
3815:
3811:
3800:
3797:
3796:
3779:
3775:
3762:
3749:
3744:
3742:
3739:
3738:
3715:
3704:
3700:
3699:
3677:
3673:
3672:
3662:
3660:
3652:
3649:
3648:
3609:
3605:
3604:
3597:
3593:
3592:
3586:
3583:
3582:
3560:
3557:
3556:
3539:
3535:
3533:
3530:
3529:
3499:
3491:
3481:
3477:
3476:
3475:
3469:
3461:
3460:
3458:
3450:
3446:
3445:
3438:
3434:
3433:
3418:
3415:
3414:
3411:
3406:
3378:
3375:
3374:
3351:
3347:
3338:
3334:
3332:
3329:
3328:
3308:
3304:
3302:
3299:
3298:
3281:
3277:
3275:
3272:
3271:
3246:
3242:
3240:
3237:
3236:
3219:
3215:
3213:
3210:
3209:
3192:
3188:
3186:
3183:
3182:
3152:
3148:
3143:
3136:
3132:
3131:
3127:
3121:
3117:
3108:
3104:
3095:
3091:
3089:
3086:
3085:
3066:
3062:
3057:
3050:
3046:
3040:
3036:
3027:
3023:
3021:
3018:
3017:
2998:
2994:
2989:
2976:
2972:
2966:
2962:
2953:
2949:
2947:
2944:
2943:
2924:
2920:
2914:
2910:
2908:
2896:
2892:
2881:
2878:
2877:
2851:
2849:
2841:
2837:
2836:
2829:
2825:
2824:
2813:
2809:
2808:
2783:
2778:
2769:
2766:
2765:
2744:
2740:
2738:
2735:
2734:
2711:
2703:
2701:
2695:
2691:
2678:
2670:
2668:
2660:
2657:
2656:
2629:
2625:
2619:
2613:
2610:
2609:
2586:
2578:
2576:
2570:
2566:
2553:
2545:
2543:
2531:
2527:
2521:
2515:
2512:
2511:
2492:
2489:
2488:
2457:
2454:
2453:
2425:
2424:
2412:
2409:
2408:
2386:
2385:
2374:
2371:
2370:
2348:
2347:
2345:
2342:
2341:
2324:
2313:
2312:
2311:
2309:
2306:
2305:
2288:
2277:
2276:
2275:
2273:
2270:
2269:
2247:
2246:
2244:
2241:
2240:
2218:
2217:
2193:
2182:
2181:
2180:
2166:
2165:
2150:
2139:
2138:
2137:
2125:
2114:
2113:
2112:
2110:
2107:
2106:
2089:
2078:
2077:
2076:
2062:
2061:
2052:
2041:
2040:
2039:
2037:
2034:
2033:
2011:
2010:
2001:
1990:
1989:
1988:
1979:
1968:
1967:
1966:
1964:
1961:
1960:
1943:
1939:
1937:
1934:
1933:
1916:
1905:
1904:
1903:
1901:
1898:
1897:
1869:
1866:
1865:
1836:
1832:
1830:
1827:
1826:
1803:
1800:
1799:
1783:
1780:
1779:
1759:
1756:
1755:
1738:
1727:
1726:
1725:
1723:
1720:
1719:
1693:
1690:
1689:
1667:
1666:
1649:
1648:
1646:
1643:
1642:
1619:
1616:
1615:
1593:
1592:
1590:
1587:
1586:
1566:
1555:
1554:
1553:
1528:
1527:
1510:
1509:
1508:
1504:
1486:
1482:
1470:
1459:
1458:
1457:
1455:
1452:
1451:
1426:
1423:
1422:
1402:
1391:
1390:
1389:
1387:
1384:
1383:
1361:
1360:
1348:
1337:
1336:
1335:
1333:
1330:
1329:
1307:
1304:
1303:
1286:
1275:
1274:
1273:
1271:
1268:
1267:
1246:
1239:
1228:
1227:
1226:
1214:
1203:
1202:
1201:
1200:
1198:
1183:
1170:
1159:
1158:
1157:
1151:
1145:
1142:
1141:
1124:
1113:
1112:
1111:
1109:
1106:
1105:
1085:
1080:
1064:
1023:
994:
990:
985:
978:
974:
968:
964:
955:
951:
949:
946:
945:
928:
924:
915:
911:
909:
906:
905:
888:
884:
882:
879:
878:
875:escape velocity
849:
845:
843:
840:
839:
816:
812:
810:
807:
806:
784:
781:
780:
755:
751:
749:
746:
745:
722:
718:
716:
713:
712:
689:
685:
679:
675:
666:
662:
660:
657:
656:
631:
627:
621:
617:
615:
603:
599:
593:
589:
578:
574:
568:
564:
562:
550:
546:
535:
532:
531:
510:
507:
506:
436:
407:
406:
402:Orbit insertion
392:
384:
383:
369:
361:
360:
336:
328:
324:
317:
316:
312:Lyapunov orbits
303:
302:
286:
276:
275:
251:
243:
239:
232:
231:
217:Surface gravity
192:Escape velocity
182:
174:
173:
154:Parabolic orbit
150:
149:
116:
114:
111:two-body orbits
102:
101:
92:Semi-major axis
57:
47:
43:
17:
12:
11:
5:
8542:
8532:
8531:
8526:
8521:
8516:
8511:
8494:
8493:
8491:
8490:
8488:List of orbits
8479:
8476:
8475:
8473:
8472:
8467:
8462:
8457:
8452:
8447:
8442:
8440:Orbit equation
8437:
8429:
8424:
8419:
8414:
8409:
8404:
8399:
8394:
8389:
8384:
8378:
8376:
8368:
8367:
8365:
8364:
8359:
8354:
8349:
8344:
8339:
8334:
8329:
8324:
8319:
8314:
8312:Gravity assist
8309:
8307:Delta-v budget
8304:
8299:
8294:
8288:
8286:
8280:
8279:
8276:
8275:
8273:
8272:
8264:
8258:
8249:
8240:
8238:Orbital period
8230:
8228:
8224:
8223:
8221:
8220:
8218:True longitude
8211:
8209:Mean longitude
8202:
8193:
8176:
8166:
8164:
8160:
8159:
8157:
8156:
8147:
8138:
8129:
8119:
8117:
8113:
8112:
8110:
8109:
8096:
8087:
8078:
8068:
8066:
8064:
8063:
8060:
8056:
8050:
8044:
8043:
8040:
8039:
8037:
8036:
8035:
8034:
8026:
8025:
8024:
8019:
8014:
8013:
8012:
7999:
7994:
7993:
7992:
7987:
7982:
7977:
7969:
7968:
7967:
7965:Areostationary
7962:
7957:
7948:
7946:
7940:
7939:
7937:
7936:
7934:Very low Earth
7931:
7926:
7921:
7916:
7911:
7906:
7901:
7896:
7891:
7886:
7881:
7876:
7875:
7874:
7869:
7862:Geosynchronous
7858:
7856:
7850:
7849:
7847:
7846:
7844:Transfer orbit
7841:
7840:
7839:
7834:
7824:
7819:
7814:
7809:
7804:
7802:Lagrange point
7799:
7794:
7785:
7780:
7775:
7770:
7761:
7756:
7751:
7745:
7743:
7736:
7730:
7729:
7724:Gravitational
7721:
7720:
7713:
7706:
7698:
7692:
7691:
7686:
7681:
7674:
7673:External links
7671:
7668:
7667:
7636:
7631:
7612:
7569:
7545:
7525:
7499:
7476:
7451:
7450:
7448:
7445:
7444:
7443:
7438:
7433:
7427:
7422:
7417:
7407:
7402:
7397:
7395:Delta-v budget
7391:
7390:
7374:
7371:
7355:
7346:
7342:
7339:
7333:
7330:
7326:
7322:
7319:
7316:
7313:
7304:
7300:
7294:
7291:
7254:
7250:
7247:
7241:
7238:
7235:
7226:
7222:
7219:
7213:
7210:
7182:
7155:
7146:
7142:
7139:
7136:
7134:
7132:
7129:
7126:
7123:
7114:
7110:
7107:
7105:
7103:
7097:
7094:
7091:
7087:
7082:
7079:
7070:
7066:
7063:
7061:
7056:
7053:
7050:
7049:
7028:
7025:
7024:
7023:
7005:
7001:
6997:
6994:
6990:
6986:
6983:
6963:
6960:
6938:
6934:
6930:
6927:
6923:
6919:
6916:
6896:
6893:
6879:
6865:
6861:
6857:
6854:
6850:
6846:
6843:
6816:
6813:
6789:
6786:
6774:
6771:
6708:
6705:
6704:
6703:
6696:Imperial units
6673:
6669:
6658:
6636:
6625:
6607:
6583:
6574:
6568:
6564:
6560:
6551:
6531:Main article:
6528:
6525:
6509:Main article:
6506:
6503:
6469:rocket engines
6403:Main article:
6400:
6394:
6392:
6389:
6376:
6368:
6364:
6358:
6354:
6348:
6345:
6340:
6331:
6324:
6320:
6317:
6314:
6311:
6308:
6305:
6275:
6272:
6267:
6264:
6260:
6254:
6251:
6246:
6243:
6239:
6232:
6229:
6226:
6223:
6187:
6183:
6180:
6177:
6172:
6162:
6158:
6151:
6147:
6144:
6141:
6135:
6125:
6121:
6115:
6086:
6083:
6077:
6067:
6063:
6057:
6051:
6048:
6042:
6032:
6028:
6022:
6015:
6010:
6006:
6003:
5980:
5956:
5952:
5946:
5942:
5907:
5903:
5899:
5893:
5885:
5881:
5878:
5872:
5869:
5862:
5858:
5855:
5849:
5846:
5840:
5835:
5828:
5824:
5818:
5814:
5800:speed of light
5787:
5765:
5761:
5734:
5730:
5709:
5706:
5686:
5683:
5670:
5667:
5664:
5661:
5658:
5653:
5649:
5645:
5640:
5636:
5615:
5608:
5604:
5598:
5594:
5588:
5585:
5576:
5572:
5566:
5563:
5560:
5556:
5551:
5548:
5539:
5535:
5529:
5526:
5523:
5518:
5515:
5507:
5502:
5498:
5488:
5484:
5481:
5478:
5473:
5470:
5467:
5463:
5438:
5435:
5432:
5407:
5403:
5400:
5394:
5389:
5385:
5362:
5359:
5354:
5351:
5348:
5323:
5319:
5315:
5312:
5307:
5304:
5296:
5293:
5290:
5285:
5282:
5279:
5275:
5265:
5261:
5255:
5251:
5247:
5244:
5241:
5238:
5233:
5229:
5225:
5217:
5214:
5211:
5206:
5203:
5200:
5196:
5186:
5182:
5179:
5176:
5156:
5153:
5150:
5146:
5142:
5122:
5097:
5094:
5090:
5086:
5083:
5080:
5076:
5072:
5069:
5066:
5063:
5060:
5057:
5054:
5050:
5046:
5043:
5040:
5037:
5034:
5029:
5025:
5021:
5013:
5010:
5007:
5002:
4999:
4996:
4992:
4982:
4978:
4975:
4972:
4952:
4932:
4929:
4925:
4921:
4918:
4915:
4912:
4909:
4904:
4900:
4896:
4893:
4873:
4853:
4849:
4843:
4839:
4835:
4832:
4827:
4823:
4802:
4780:
4776:
4772:
4750:
4746:
4725:
4703:
4697:
4692:
4688:
4684:
4681:
4678:
4675:
4669:
4665:
4653:
4649:
4646:
4643:
4618:
4614:
4610:
4604:
4601:
4598:
4595:
4573:
4568:
4564:
4560:
4557:
4554:
4551:
4545:
4542:
4536:
4531:
4527:
4521:
4517:
4510:
4507:
4501:
4496:
4487:
4481:
4477:
4470:
4467:
4444:
4424:
4402:
4398:
4372:
4363:
4356:
4353:
4324:
4303:
4300:
4297:
4277:
4265:
4262:
4250:
4246:
4239:
4235:
4229:
4225:
4219:
4215:
4212:
4203:
4196:
4192:
4186:
4182:
4178:
4175:
4172:
4167:
4163:
4159:
4156:
4152:
4142:
4138:
4135:
4130:
4126:
4122:
4117:
4113:
4109:
4106:
4103:
4071:
4065:
4049:
4043:
4023:
4020:
4017:
4014:
4003:
3999:
3993:
3990:
3984:
3981:
3978:
3975:
3971:
3966:
3963:
3957:
3954:
3949:
3946:
3940:
3937:
3905:
3890:
3887:(kg/s) and at
3876:
3873:
3860:
3853:
3849:
3843:
3839:
3833:
3829:
3826:
3814:
3810:
3807:
3804:
3778:
3774:
3769:
3766:
3761:
3755:
3752:
3748:
3721:
3718:
3713:
3707:
3703:
3698:
3695:
3692:
3689:
3686:
3680:
3676:
3671:
3668:
3665:
3656:
3634:
3631:
3628:
3625:
3619:
3612:
3608:
3600:
3596:
3591:
3567:
3564:
3542:
3538:
3515:
3512:
3502:
3498:
3494:
3490:
3484:
3480:
3472:
3468:
3464:
3453:
3449:
3441:
3437:
3432:
3428:
3425:
3422:
3410:
3407:
3405:
3402:
3385:
3382:
3354:
3350:
3346:
3341:
3337:
3307:
3280:
3245:
3222:
3218:
3195:
3191:
3169:
3165:
3162:
3151:
3146:
3142:
3139:
3135:
3130:
3124:
3120:
3116:
3111:
3107:
3103:
3098:
3094:
3065:
3060:
3056:
3053:
3049:
3043:
3039:
3035:
3030:
3026:
2997:
2992:
2985:
2982:
2979:
2975:
2969:
2965:
2961:
2956:
2952:
2927:
2923:
2917:
2913:
2907:
2904:
2895:
2891:
2888:
2885:
2861:
2857:
2854:
2844:
2840:
2832:
2828:
2823:
2816:
2812:
2807:
2804:
2801:
2795:
2792:
2789:
2786:
2781:
2777:
2773:
2743:
2733:Assuming that
2717:
2714:
2709:
2706:
2694:
2690:
2684:
2681:
2676:
2673:
2667:
2664:
2640:
2637:
2632:
2628:
2622:
2618:
2592:
2589:
2584:
2581:
2569:
2565:
2559:
2556:
2551:
2548:
2542:
2539:
2534:
2530:
2524:
2520:
2499:
2496:
2476:
2473:
2470:
2467:
2464:
2461:
2441:
2438:
2432:
2429:
2423:
2419:
2416:
2393:
2390:
2384:
2381:
2378:
2355:
2352:
2320:
2317:
2304:are opposite,
2284:
2281:
2254:
2251:
2225:
2222:
2216:
2213:
2210:
2207:
2204:
2201:
2189:
2186:
2179:
2173:
2170:
2164:
2161:
2158:
2153:
2146:
2143:
2136:
2131:
2128:
2121:
2118:
2085:
2082:
2075:
2069:
2066:
2060:
2048:
2045:
2018:
2015:
2009:
1997:
1994:
1987:
1975:
1972:
1942:
1912:
1909:
1894:
1893:
1882:
1879:
1876:
1873:
1852:
1848:
1845:
1842:
1839:
1835:
1824:
1813:
1810:
1807:
1787:
1777:
1766:
1763:
1734:
1731:
1717:
1706:
1703:
1700:
1697:
1674:
1671:
1665:
1662:
1656:
1653:
1640:
1629:
1626:
1623:
1600:
1597:
1562:
1559:
1552:
1549:
1546:
1542:
1535:
1532:
1526:
1523:
1517:
1514:
1507:
1502:
1498:
1495:
1492:
1489:
1485:
1481:
1476:
1473:
1466:
1463:
1439:
1436:
1433:
1430:
1408:
1405:
1398:
1395:
1368:
1365:
1359:
1356:
1351:
1344:
1341:
1317:
1314:
1311:
1289:
1282:
1279:
1252:
1249:
1242:
1235:
1232:
1225:
1220:
1217:
1210:
1207:
1195:
1192:
1189:
1186:
1182:
1178:
1173:
1166:
1163:
1154:
1150:
1127:
1120:
1117:
1084:
1081:
1079:
1076:
1063:
1060:
1045:Hermann Oberth
1041:Robert Goddard
1022:
1019:
1004:
993:
988:
984:
981:
977:
971:
967:
963:
958:
954:
931:
927:
923:
918:
914:
891:
887:
867:
866:
852:
848:
837:
819:
815:
804:
788:
778:
777:
776:
758:
754:
743:
721:
692:
688:
678:
674:
665:
641:
634:
630:
624:
620:
614:
611:
606:
602:
592:
588:
581:
577:
571:
567:
561:
558:
549:
545:
542:
539:
517:
514:
496:Hermann Oberth
492:Robert Goddard
438:
437:
435:
434:
427:
420:
412:
409:
408:
405:
404:
399:
393:
390:
389:
386:
385:
382:
381:
376:
374:Gravity assist
370:
367:
366:
363:
362:
359:
358:
353:
348:
343:
337:
334:
333:
330:
329:
322:
319:
318:
315:
314:
309:
301:
300:
292:
288:
287:
282:
281:
278:
277:
274:
273:
268:
263:
258:
252:
249:
248:
245:
244:
237:
234:
233:
230:
229:
224:
219:
214:
209:
207:Orbital period
204:
199:
194:
189:
183:
180:
179:
176:
175:
172:
171:
169:Decaying orbit
166:
161:
156:
148:
147:
141:
134:
132:Transfer orbit
130:
129:
128:
126:Elliptic orbit
123:
121:Circular orbit
117:
108:
107:
104:
103:
100:
99:
94:
89:
84:
79:
74:
69:
64:
58:
53:
52:
49:
48:
41:
38:
37:
29:
28:
24:
23:
15:
9:
6:
4:
3:
2:
8541:
8530:
8527:
8525:
8522:
8520:
8517:
8515:
8512:
8510:
8509:Astrodynamics
8507:
8506:
8504:
8489:
8481:
8480:
8477:
8471:
8468:
8466:
8463:
8461:
8458:
8456:
8453:
8451:
8448:
8446:
8443:
8441:
8438:
8436:
8435:-body problem
8434:
8430:
8428:
8425:
8423:
8420:
8418:
8415:
8413:
8410:
8408:
8405:
8403:
8400:
8398:
8395:
8393:
8390:
8388:
8385:
8383:
8380:
8379:
8377:
8375:
8369:
8363:
8360:
8358:
8355:
8353:
8350:
8348:
8345:
8343:
8340:
8338:
8337:Oberth effect
8335:
8333:
8330:
8328:
8325:
8323:
8320:
8318:
8315:
8313:
8310:
8308:
8305:
8303:
8300:
8298:
8295:
8293:
8290:
8289:
8287:
8285:
8281:
8271:
8263:
8259:
8257:
8256:Orbital speed
8250:
8248:
8241:
8239:
8232:
8231:
8229:
8225:
8219:
8212:
8210:
8203:
8201:
8194:
8192:
8177:
8175:
8168:
8167:
8165:
8161:
8155:
8148:
8146:
8139:
8137:
8130:
8128:
8121:
8120:
8118:
8114:
8108:
8097:
8095:
8088:
8086:
8079:
8077:
8070:
8069:
8067:
8061:
8058:
8057:
8054:
8051:
8049:
8045:
8033:
8030:
8029:
8027:
8023:
8020:
8018:
8015:
8011:
8010:Earth's orbit
8008:
8007:
8006:
8003:
8002:
8000:
7998:
7995:
7991:
7988:
7986:
7983:
7981:
7978:
7976:
7973:
7972:
7970:
7966:
7963:
7961:
7958:
7956:
7953:
7952:
7950:
7949:
7947:
7941:
7935:
7932:
7930:
7927:
7925:
7922:
7920:
7917:
7915:
7912:
7910:
7907:
7905:
7902:
7900:
7897:
7895:
7892:
7890:
7887:
7885:
7882:
7880:
7877:
7873:
7870:
7868:
7867:Geostationary
7865:
7864:
7863:
7860:
7859:
7857:
7855:
7851:
7845:
7842:
7838:
7835:
7833:
7830:
7829:
7828:
7825:
7823:
7820:
7818:
7815:
7813:
7810:
7808:
7805:
7803:
7800:
7798:
7795:
7793:
7789:
7786:
7784:
7781:
7779:
7776:
7774:
7771:
7769:
7765:
7762:
7760:
7757:
7755:
7752:
7750:
7747:
7746:
7744:
7740:
7737:
7735:
7731:
7727:
7719:
7714:
7712:
7707:
7705:
7700:
7699:
7696:
7690:
7687:
7685:
7682:
7680:
7677:
7676:
7657:on 2022-03-06
7656:
7652:
7651:
7646:
7640:
7630:
7626:
7622:
7616:
7608:
7604:
7600:
7596:
7592:
7588:
7585:(6): 065001.
7584:
7580:
7573:
7566:
7562:
7558:
7555:
7549:
7541:
7540:
7532:
7530:
7521:
7517:
7513:
7506:
7504:
7492:
7485:
7483:
7481:
7473:
7469:
7465:
7462:
7456:
7452:
7442:
7439:
7437:
7434:
7431:
7428:
7426:
7423:
7421:
7418:
7415:
7411:
7410:Oberth effect
7408:
7406:
7403:
7401:
7398:
7396:
7393:
7392:
7388:
7377:
7370:
7366:
7353:
7344:
7340:
7337:
7328:
7324:
7320:
7314:
7311:
7302:
7298:
7292:
7281:
7279:
7276:A comparable
7274:
7271:
7252:
7248:
7245:
7239:
7236:
7233:
7224:
7220:
7217:
7211:
7200:
7180:
7170:
7153:
7144:
7140:
7137:
7135:
7127:
7124:
7121:
7112:
7108:
7106:
7095:
7092:
7089:
7085:
7080:
7077:
7068:
7064:
7062:
7054:
7039:
7036:
7034:
7033:rocket stages
7021:
7003:
6999:
6995:
6992:
6988:
6984:
6981:
6961:
6936:
6932:
6928:
6925:
6921:
6917:
6914:
6894:
6883:
6880:
6863:
6859:
6855:
6852:
6848:
6844:
6841:
6833:
6830:
6829:
6828:
6814:
6803:
6787:
6770:
6768:
6764:
6759:
6754:
6752:
6748:
6744:
6740:
6736:
6732:
6728:
6724:
6719:
6718:gravitational
6715:
6707:Applicability
6697:
6689:
6671:
6667:
6659:
6656:
6634:
6626:
6605:
6597:
6596:
6595:
6581:
6572:
6566:
6562:
6558:
6549:
6540:
6534:
6524:
6522:
6518:
6512:
6505:Mass fraction
6502:
6500:
6496:
6495:porkchop plot
6492:
6487:
6485:
6480:
6478:
6474:
6470:
6466:
6461:
6459:
6455:
6451:
6447:
6443:
6439:
6435:
6432:, as used in
6431:
6427:
6426:
6420:
6416:
6412:
6406:
6399:
6388:
6374:
6366:
6362:
6356:
6352:
6346:
6343:
6338:
6329:
6322:
6318:
6315:
6312:
6309:
6306:
6295:
6292:
6273:
6270:
6265:
6262:
6258:
6252:
6249:
6244:
6241:
6237:
6230:
6227:
6224:
6221:
6213:
6209:
6206:
6202:
6185:
6181:
6178:
6175:
6170:
6160:
6156:
6149:
6145:
6142:
6139:
6133:
6123:
6119:
6113:
6105:
6100:
6084:
6081:
6075:
6065:
6061:
6055:
6049:
6046:
6040:
6030:
6026:
6020:
6013:
6008:
6004:
5978:
5954:
5950:
5944:
5940:
5928:
5905:
5901:
5897:
5891:
5883:
5879:
5870:
5867:
5860:
5856:
5847:
5844:
5838:
5833:
5826:
5822:
5816:
5812:
5801:
5785:
5763:
5759:
5750:
5732:
5728:
5707:
5696:
5692:
5682:
5665:
5662:
5659:
5651:
5647:
5643:
5638:
5634:
5613:
5606:
5602:
5596:
5592:
5586:
5583:
5574:
5570:
5564:
5561:
5558:
5554:
5549:
5546:
5537:
5533:
5527:
5524:
5521:
5516:
5513:
5505:
5500:
5496:
5486:
5482:
5479:
5465:
5452:
5430:
5421:
5405:
5401:
5398:
5392:
5387:
5383:
5360:
5357:
5352:
5349:
5321:
5317:
5313:
5310:
5305:
5294:
5291:
5288:
5283:
5280:
5277:
5273:
5263:
5259:
5253:
5249:
5245:
5242:
5239:
5236:
5231:
5227:
5223:
5215:
5212:
5209:
5204:
5201:
5198:
5194:
5184:
5180:
5177:
5154:
5151:
5148:
5144:
5140:
5120:
5111:
5092:
5088:
5084:
5081:
5078:
5074:
5070:
5067:
5064:
5061:
5052:
5048:
5044:
5041:
5038:
5035:
5027:
5023:
5019:
5011:
5008:
5005:
5000:
4997:
4994:
4990:
4980:
4976:
4973:
4950:
4927:
4923:
4919:
4916:
4913:
4910:
4902:
4898:
4894:
4891:
4871:
4851:
4847:
4841:
4837:
4833:
4830:
4825:
4821:
4800:
4778:
4774:
4770:
4748:
4744:
4723:
4714:
4701:
4690:
4686:
4682:
4679:
4673:
4667:
4663:
4651:
4647:
4644:
4616:
4612:
4608:
4602:
4596:
4593:
4584:
4571:
4566:
4558:
4549:
4543:
4540:
4534:
4529:
4525:
4519:
4515:
4508:
4505:
4499:
4494:
4485:
4479:
4475:
4468:
4465:
4442:
4422:
4400:
4396:
4386:
4370:
4361:
4354:
4351:
4322:
4295:
4275:
4261:
4248:
4244:
4237:
4233:
4227:
4223:
4217:
4213:
4210:
4201:
4194:
4190:
4184:
4180:
4176:
4173:
4170:
4165:
4161:
4157:
4154:
4150:
4140:
4136:
4133:
4128:
4124:
4120:
4115:
4111:
4107:
4104:
4090:
4086:
4082:
4078:
4074:
4064:
4060:
4056:
4052:
4042:
4037:
4018:
4012:
4001:
3997:
3991:
3988:
3979:
3973:
3969:
3964:
3961:
3955:
3952:
3947:
3944:
3938:
3935:
3924:
3920:
3916:
3912:
3908:
3901:
3897:
3893:
3886:
3882:
3872:
3858:
3851:
3847:
3841:
3837:
3831:
3827:
3824:
3812:
3808:
3805:
3776:
3772:
3767:
3764:
3759:
3753:
3746:
3735:
3719:
3705:
3701:
3693:
3690:
3687:
3678:
3674:
3666:
3663:
3654:
3645:
3632:
3629:
3626:
3623:
3617:
3610:
3606:
3598:
3594:
3589:
3579:
3565:
3540:
3536:
3526:
3513:
3510:
3500:
3492:
3488:
3482:
3478:
3466:
3451:
3447:
3439:
3435:
3430:
3426:
3423:
3409:Impulse-based
3401:
3399:
3383:
3372:
3370:
3367:is the total
3352:
3348:
3344:
3339:
3335:
3325:
3305:
3278:
3269:
3265:
3243:
3220:
3216:
3193:
3189:
3167:
3163:
3160:
3149:
3144:
3140:
3133:
3128:
3122:
3118:
3114:
3109:
3105:
3101:
3096:
3092:
3063:
3058:
3054:
3047:
3041:
3037:
3033:
3028:
3024:
2995:
2990:
2983:
2977:
2973:
2967:
2963:
2959:
2954:
2950:
2925:
2921:
2915:
2911:
2905:
2902:
2893:
2889:
2886:
2874:
2859:
2855:
2852:
2842:
2838:
2830:
2826:
2821:
2814:
2810:
2805:
2802:
2799:
2793:
2787:
2784:
2779:
2775:
2771:
2763:
2741:
2731:
2715:
2712:
2707:
2704:
2692:
2688:
2682:
2679:
2674:
2671:
2665:
2662:
2654:
2638:
2635:
2630:
2626:
2620:
2616:
2606:
2590:
2587:
2582:
2579:
2567:
2563:
2557:
2554:
2549:
2546:
2540:
2537:
2532:
2528:
2522:
2518:
2497:
2474:
2468:
2465:
2462:
2459:
2452:), and using
2439:
2427:
2421:
2417:
2414:
2388:
2379:
2350:
2315:
2279:
2249:
2220:
2211:
2205:
2202:
2184:
2177:
2168:
2159:
2156:
2151:
2141:
2134:
2129:
2116:
2080:
2073:
2064:
2058:
2043:
2013:
2007:
1992:
1985:
1970:
1940:
1907:
1880:
1874:
1871:
1850:
1846:
1840:
1837:
1833:
1825:
1811:
1808:
1805:
1785:
1778:
1764:
1729:
1718:
1704:
1698:
1695:
1669:
1660:
1651:
1641:
1627:
1624:
1621:
1595:
1585:
1584:
1583:
1557:
1550:
1544:
1540:
1530:
1521:
1512:
1505:
1500:
1496:
1490:
1487:
1483:
1479:
1474:
1461:
1437:
1431:
1428:
1406:
1393:
1363:
1357:
1354:
1349:
1339:
1315:
1312:
1309:
1287:
1277:
1250:
1240:
1230:
1223:
1218:
1205:
1193:
1187:
1176:
1171:
1161:
1152:
1148:
1125:
1115:
1103:
1099:
1092:
1088:
1075:
1071:
1069:
1059:
1057:
1053:
1048:
1046:
1042:
1037:
1035:
1034:William Moore
1030:
1028:
1018:
1015:
1002:
991:
986:
982:
975:
969:
965:
961:
956:
952:
929:
925:
921:
916:
912:
889:
885:
876:
872:
871:orbital speed
850:
846:
838:
835:
817:
813:
805:
802:
786:
779:
774:
756:
752:
744:
741:
719:
711:
710:
708:
690:
686:
676:
672:
663:
655:
654:
653:
639:
632:
628:
622:
618:
612:
609:
604:
600:
590:
586:
579:
575:
569:
565:
559:
556:
547:
543:
540:
529:
515:
504:
499:
497:
493:
489:
488:William Moore
485:
481:
477:
473:
469:
465:
461:
453:
449:
444:
433:
428:
426:
421:
419:
414:
413:
411:
410:
403:
400:
398:
395:
394:
388:
387:
380:
379:Oberth effect
377:
375:
372:
371:
365:
364:
357:
354:
352:
349:
347:
344:
342:
339:
338:
332:
331:
327:
321:
320:
313:
310:
308:
305:
304:
298:
294:
293:
291:
285:
284:N-body orbits
280:
279:
272:
269:
267:
266:Perturbations
264:
262:
259:
257:
254:
253:
247:
246:
242:
236:
235:
228:
225:
223:
220:
218:
215:
213:
210:
208:
205:
203:
200:
198:
195:
193:
190:
188:
185:
184:
178:
177:
170:
167:
165:
162:
160:
157:
155:
152:
151:
145:
142:
140:
136:
135:
133:
127:
124:
122:
119:
118:
112:
106:
105:
98:
95:
93:
90:
88:
87:Orbital nodes
85:
83:
80:
78:
75:
73:
70:
68:
65:
63:
60:
59:
56:
51:
50:
46:
40:
39:
35:
31:
30:
27:Astrodynamics
26:
25:
21:
20:
8450:Perturbation
8432:
8407:Ground track
8346:
8317:Gravity turn
8268:
8261:
8254:
8245:
8236:
8216:
8207:
8198:
8191:True anomaly
8189:
8174:Mean anomaly
8172:
8152:
8143:
8134:
8125:
8105:
8092:
8083:
8076:Eccentricity
8074:
8032:Lunar cycler
8005:Heliocentric
7945:other points
7894:Medium Earth
7792:Non-inclined
7659:. Retrieved
7655:the original
7648:
7639:
7628:
7624:
7615:
7582:
7578:
7572:
7548:
7538:
7519:
7515:
7455:
7414:gravity well
7400:Jeep problem
7367:
7282:
7275:
7272:
7201:
7171:
7040:
7037:
7030:
7019:
6804:, including
6776:
6755:
6743:launch loops
6735:gun launches
6710:
6536:
6514:
6498:
6490:
6488:
6483:
6481:
6464:
6462:
6453:
6429:
6424:
6422:
6413:(literally "
6410:
6408:
6397:
6290:
6204:
6101:
5929:
5688:
5422:
5112:
4715:
4585:
4387:
4267:
4088:
4084:
4080:
4076:
4069:
4062:
4058:
4054:
4047:
4040:
4038:
3922:
3918:
3914:
3903:
3899:
3895:
3888:
3884:
3878:
3736:
3646:
3580:
3527:
3412:
3373:
3369:working mass
3326:
3324:hypothesis.
2875:
2732:
2607:
1895:
1100:
1097:
1086:
1072:
1065:
1056:space travel
1049:
1038:
1031:
1024:
1016:
868:
530:
500:
463:
459:
457:
355:
164:Radial orbit
115:eccentricity
97:True anomaly
82:Mean anomaly
72:Eccentricity
8412:Hill sphere
8247:Mean motion
8127:Inclination
8116:Orientation
8017:Mars cycler
7955:Areocentric
7827:Synchronous
6751:light sails
6731:aerobraking
6714:aerodynamic
5802:in vacuum:
5451:Riemann sum
297:Halo orbits
261:Hill sphere
77:Inclination
8503:Categories
8352:Rendezvous
8048:Parameters
7884:High Earth
7854:Geocentric
7807:Osculating
7764:Elliptical
7661:2016-04-18
7522:: 276–285.
7447:References
7405:Mass ratio
6477:propellant
6444:. It is a
3327:The value
1078:Derivation
834:propellant
448:mass ratio
341:Mass ratio
256:Barycenter
8397:Ephemeris
8374:mechanics
8284:Maneuvers
8227:Variation
7990:Libration
7985:Lissajous
7889:Low Earth
7879:Graveyard
7778:Horseshoe
7607:202130640
7315:
7290:Δ
7237:
7209:Δ
7125:
7093:−
7081:
7052:Δ
6993:−
6985:−
6959:Δ
6926:−
6918:−
6892:Δ
6853:−
6845:−
6812:Δ
6785:Δ
6690:, 9.80665
6430:delta-vee
6347:
6319:
6304:Δ
6250:−
6225:
6179:
6146:
6047:−
6002:Δ
5877:Δ
5871:−
5854:Δ
5747:) in the
5705:Δ
5666:ϕ
5663:−
5587:
5565:ϕ
5562:−
5550:
5525:−
5506:ϕ
5497:∫
5477:Δ
5472:∞
5469:→
5437:∞
5434:→
5402:ϕ
5358:ϕ
5347:Δ
5314:−
5303:Δ
5274:∑
5246:ϕ
5240:−
5224:ϕ
5195:∑
5181:≈
5175:Δ
5152:≪
5141:ϕ
5085:ϕ
5071:ϕ
5065:−
5045:ϕ
5039:−
5020:ϕ
4991:∑
4971:Δ
4920:ϕ
4914:−
4834:ϕ
4771:ϕ
4724:ϕ
4642:Δ
4600:Δ
4556:Δ
4302:∞
4299:→
4214:
4177:
4171:−
4158:
4137:−
4121:−
4102:Δ
4089:R = dm/dt
3992:−
3965:−
3828:
3803:Δ
3751:Δ
3717:Δ
3694:
3688:−
3667:
3590:∫
3563:Δ
3497:Δ
3489:−
3431:∫
3421:Δ
3381:Δ
3345:−
3161:−
3138:Δ
3102:−
3052:Δ
2981:Δ
2978:−
2906:
2884:Δ
2822:∫
2791:Δ
2776:∫
2772:−
2663:−
2617:∑
2519:∑
2495:Δ
2472:Δ
2469:−
2437:→
2431:→
2392:→
2383:Δ
2377:Δ
2354:→
2319:→
2283:→
2253:→
2224:→
2215:Δ
2209:Δ
2206:−
2200:Δ
2188:→
2172:→
2163:Δ
2145:→
2135:−
2127:Δ
2120:→
2084:→
2068:→
2047:→
2017:→
2008:−
1996:→
1974:→
1911:→
1878:Δ
1844:Δ
1841:−
1762:Δ
1733:→
1702:Δ
1673:→
1664:Δ
1655:→
1599:→
1561:→
1548:Δ
1534:→
1525:Δ
1516:→
1494:Δ
1491:−
1472:Δ
1465:→
1435:Δ
1404:Δ
1397:→
1367:→
1343:→
1281:→
1248:Δ
1234:→
1224:−
1216:Δ
1209:→
1191:→
1185:Δ
1165:→
1149:∑
1119:→
1039:American
980:Δ
922:−
803:function;
613:
560:
538:Δ
513:Δ
181:Equations
109:Types of
8163:Position
7788:Inclined
7759:Circular
7650:NASA.gov
7557:Archived
7464:Archived
7373:See also
6834:rocket:
6773:Examples
6729:such as
6694:m/s (in
6419:velocity
6205:see also
6104:identity
5930:Writing
1052:calculus
503:velocity
476:velocity
8372:Orbital
8342:Phasing
8302:Delta-v
8107:Apsides
8101:,
7899:Molniya
7817:Parking
7754:Capture
7742:General
7587:Bibcode
6723:payload
6686:is the
6438:impulse
6405:Delta-v
5697:, with
3398:delta v
3268:gravity
1021:History
799:is the
738:is the
705:is the
652:where:
8028:Other
7929:Tundra
7797:Kepler
7773:Escape
7726:orbits
7605:
7335:
7296:
7243:
7215:
7058:
7027:Stages
6702:ft/s).
6700:
6698:32.174
6692:
6594:where
6473:thrust
6463:Delta-
6446:scalar
6415:change
6409:Delta-
6396:Delta-
5339:where
4198:
4099:
3933:
3822:
3658:
3621:
3508:
3181:where
2987:
2655:) and
2032:thus,
1266:where
472:thrust
468:rocket
8270:Epoch
8059:Shape
7997:Lunar
7951:Mars
7943:About
7914:Polar
7734:Types
7603:S2CID
7565:RARed
7563:in a
7494:(PDF)
7472:RARed
7470:in a
6450:speed
5449:this
4793:into
462:, or
454:ratio
62:Apsis
8062:Size
8001:Sun
7980:Halo
7832:semi
7567:PDF)
7554:here
7474:PDF)
7461:here
7341:2.19
7329:11.2
7278:SSTO
7249:4.83
7141:1.61
6316:tanh
6222:tanh
5375:and
4716:Let
2268:and
1959:by:
1382:and
458:The
7837:sub
7749:Box
7595:doi
7321:100
7090:100
7086:100
7020:all
7004:4.5
6996:4.7
6937:4.5
6929:5.0
6864:4.5
6856:9.7
6802:LEO
6749:or
6716:or
6655:m/s
6515:In
6454:not
6417:in
6291:see
6143:exp
5971:as
5689:If
5579:eff
5542:eff
5491:eff
5462:lim
5423:As
5268:eff
5189:eff
4985:eff
4656:eff
4490:eff
4366:eff
4327:eff
4083:to
4061:= (
3817:exh
3781:exh
3084:or
3016:or
2239:If
1181:lim
873:or
771:is
113:by
8505::
8185:,
8181:,
7790:/
7766:/
7647:.
7601:.
7593:.
7583:54
7581:.
7528:^
7520:27
7518:.
7514:.
7502:^
7479:^
7312:ln
7234:ln
7122:ln
7096:80
7078:ln
6753:.
6745:,
6741:,
6737:,
6733:,
6610:sp
6577:sp
6344:ln
6203:;
6176:ln
5681:.
5584:ln
5547:ln
5420:.
4385:.
4211:ln
4174:ln
4155:ln
4075:)/
4068:–
4046:–
3925::
3902:×
3825:ln
3691:ln
3664:ln
2903:ln
2369:,
1450::
1328::
1058:.
944::
787:ln
724:sp
709:;
681:sp
610:ln
595:sp
557:ln
8433:n
8265:0
8262:t
8252:v
8243:n
8234:T
8214:l
8205:L
8196:E
8187:f
8183:θ
8179:ν
8170:M
8150:ϖ
8141:ω
8132:Ω
8123:i
8103:q
8099:Q
8090:b
8081:a
8072:e
7717:e
7710:t
7703:v
7664:.
7632:e
7629:v
7625:R
7609:.
7597::
7589::
7496:.
7354:.
7349:e
7345:v
7338:=
7332:)
7325:/
7318:(
7307:e
7303:v
7299:=
7293:v
7257:e
7253:v
7246:=
7240:5
7229:e
7225:v
7221:3
7218:=
7212:v
7185:e
7181:v
7154:.
7149:e
7145:v
7138:=
7128:5
7117:e
7113:v
7109:=
7073:e
7069:v
7065:=
7055:v
7000:/
6989:e
6982:1
6962:v
6933:/
6922:e
6915:1
6895:v
6860:/
6849:e
6842:1
6815:v
6788:v
6672:0
6668:g
6639:e
6635:v
6606:I
6582:,
6573:I
6567:0
6563:g
6559:=
6554:e
6550:v
6499:v
6491:v
6484:v
6465:v
6425:v
6423:Δ
6411:v
6398:v
6375:)
6367:1
6363:m
6357:0
6353:m
6339:c
6334:e
6330:v
6323:(
6313:c
6310:=
6307:v
6289:(
6274:1
6271:+
6266:x
6263:2
6259:e
6253:1
6245:x
6242:2
6238:e
6231:=
6228:x
6186:]
6182:R
6171:c
6165:e
6161:v
6157:2
6150:[
6140:=
6134:c
6128:e
6124:v
6120:2
6114:R
6085:1
6082:+
6076:c
6070:e
6066:v
6062:2
6056:R
6050:1
6041:c
6035:e
6031:v
6027:2
6021:R
6014:=
6009:c
6005:v
5979:R
5955:1
5951:m
5945:0
5941:m
5910:e
5906:v
5902:2
5898:c
5892:]
5884:c
5880:v
5868:1
5861:c
5857:v
5848:+
5845:1
5839:[
5834:=
5827:1
5823:m
5817:0
5813:m
5786:c
5764:0
5760:m
5733:1
5729:m
5708:v
5669:)
5660:1
5657:(
5652:0
5648:m
5644:=
5639:f
5635:m
5614:,
5607:f
5603:m
5597:0
5593:m
5575:v
5571:=
5559:1
5555:1
5538:v
5534:=
5528:x
5522:1
5517:x
5514:d
5501:0
5487:v
5483:=
5480:v
5466:N
5431:N
5406:N
5399:j
5393:=
5388:j
5384:x
5361:N
5353:=
5350:x
5322:j
5318:x
5311:1
5306:x
5295:N
5292:=
5289:j
5284:1
5281:=
5278:j
5264:v
5260:=
5254:N
5250:/
5243:j
5237:1
5232:N
5228:/
5216:N
5213:=
5210:j
5205:1
5202:=
5199:j
5185:v
5178:v
5155:1
5149:N
5145:/
5121:N
5096:)
5093:N
5089:/
5082:+
5079:N
5075:/
5068:j
5062:1
5059:(
5056:)
5053:N
5049:/
5042:j
5036:1
5033:(
5028:N
5024:/
5012:N
5009:=
5006:j
5001:1
4998:=
4995:j
4981:v
4977:=
4974:v
4951:j
4931:)
4928:N
4924:/
4917:j
4911:1
4908:(
4903:0
4899:m
4895:=
4892:m
4872:j
4852:N
4848:/
4842:0
4838:m
4831:=
4826:p
4822:m
4801:N
4779:0
4775:m
4749:0
4745:m
4702:.
4696:)
4691:p
4687:m
4683:+
4680:m
4677:(
4674:m
4668:p
4664:m
4652:v
4648:=
4645:v
4617:p
4613:m
4609:m
4603:v
4597:=
4594:u
4572:.
4567:2
4563:)
4559:v
4553:(
4550:m
4544:2
4541:1
4535:+
4530:2
4526:u
4520:p
4516:m
4509:2
4506:1
4500:=
4495:2
4486:v
4480:p
4476:m
4469:2
4466:1
4443:m
4423:u
4401:p
4397:m
4371:2
4362:v
4355:2
4352:1
4323:v
4296:N
4276:N
4249:.
4245:)
4238:f
4234:m
4228:0
4224:m
4218:(
4206:e
4202:v
4195:=
4191:]
4185:0
4181:m
4166:f
4162:m
4151:[
4145:e
4141:v
4134:=
4129:0
4125:v
4116:f
4112:v
4108:=
4105:v
4085:T
4081:0
4077:R
4072:f
4070:m
4066:0
4063:m
4059:T
4055:R
4050:f
4048:m
4044:0
4041:m
4022:)
4019:t
4016:(
4013:m
4006:e
4002:v
3998:R
3989:=
3983:)
3980:t
3977:(
3974:m
3970:F
3962:=
3956:t
3953:d
3948:v
3945:d
3939:=
3936:a
3923:m
3919:F
3915:t
3906:e
3904:v
3900:R
3896:F
3891:e
3859:)
3852:f
3848:m
3842:0
3838:m
3832:(
3813:V
3809:=
3806:v
3777:V
3773:=
3768:p
3765:F
3760:=
3754:m
3747:J
3720:m
3712:)
3706:f
3702:m
3697:(
3685:)
3679:0
3675:m
3670:(
3655:J
3633:J
3630:=
3627:t
3624:d
3618:F
3611:f
3607:t
3599:0
3595:t
3566:m
3541:0
3537:m
3514:t
3511:d
3501:m
3493:t
3483:0
3479:m
3471:|
3467:T
3463:|
3452:f
3448:t
3440:0
3436:t
3427:=
3424:v
3396:(
3384:V
3353:f
3349:m
3340:0
3336:m
3310:e
3306:v
3283:e
3279:v
3248:e
3244:v
3221:f
3217:m
3194:0
3190:m
3168:)
3164:1
3154:e
3150:v
3145:/
3141:V
3134:e
3129:(
3123:f
3119:m
3115:=
3110:f
3106:m
3097:0
3093:m
3068:e
3064:v
3059:/
3055:V
3048:e
3042:f
3038:m
3034:=
3029:0
3025:m
3000:e
2996:v
2991:/
2984:V
2974:e
2968:0
2964:m
2960:=
2955:f
2951:m
2926:f
2922:m
2916:0
2912:m
2898:e
2894:v
2890:=
2887:V
2860:m
2856:m
2853:d
2843:f
2839:m
2831:0
2827:m
2815:e
2811:v
2806:=
2803:V
2800:d
2794:V
2788:+
2785:V
2780:V
2746:e
2742:v
2716:t
2713:d
2708:m
2705:d
2697:e
2693:v
2689:=
2683:t
2680:d
2675:V
2672:d
2666:m
2651:(
2639:0
2636:=
2631:i
2627:F
2621:i
2591:t
2588:d
2583:m
2580:d
2572:e
2568:v
2564:+
2558:t
2555:d
2550:V
2547:d
2541:m
2538:=
2533:i
2529:F
2523:i
2498:m
2475:m
2466:=
2463:m
2460:d
2440:0
2428:v
2422:d
2418:m
2415:d
2389:V
2380:m
2351:V
2326:i
2316:F
2290:e
2280:v
2250:V
2221:V
2212:m
2203:m
2195:e
2185:v
2178:+
2169:V
2160:m
2157:=
2152:0
2142:P
2130:t
2117:P
2091:e
2081:v
2074:+
2065:V
2059:=
2054:e
2044:V
2014:V
2003:e
1993:V
1986:=
1981:e
1971:v
1945:e
1941:v
1918:e
1908:V
1881:t
1875:=
1872:t
1851:)
1847:m
1838:m
1834:(
1812:0
1809:=
1806:t
1786:m
1765:t
1740:e
1730:V
1705:t
1699:=
1696:t
1670:V
1661:+
1652:V
1628:0
1625:=
1622:t
1596:V
1568:e
1558:V
1551:m
1545:+
1541:)
1531:V
1522:+
1513:V
1506:(
1501:)
1497:m
1488:m
1484:(
1480:=
1475:t
1462:P
1438:t
1432:=
1429:t
1407:t
1394:P
1364:V
1358:m
1355:=
1350:0
1340:P
1316:0
1313:=
1310:t
1288:0
1278:P
1251:t
1241:0
1231:P
1219:t
1206:P
1194:0
1188:t
1177:=
1172:i
1162:F
1153:i
1126:i
1116:F
1003:.
996:e
992:v
987:/
983:v
976:e
970:f
966:m
962:=
957:0
953:m
930:f
926:m
917:0
913:m
890:f
886:m
851:f
847:m
818:0
814:m
775:;
757:0
753:g
720:I
691:0
687:g
677:I
673:=
668:e
664:v
640:,
633:f
629:m
623:0
619:m
605:0
601:g
591:I
587:=
580:f
576:m
570:0
566:m
552:e
548:v
544:=
541:v
516:v
431:e
424:t
417:v
299:)
295:(
146:)
137:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
