3399:) 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.
1102:
454:
6518:, 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
45:
8483:
7380:
6764:, 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
6950:= 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
7034:, 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.
5623:
7017:= 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
6719:
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
6710:
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
1084:
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
5336:
1085:
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).
7167:
1582:
7367:
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.
2236:
5455:
5108:
4581:
6097:
4258:
1272:
661:
1079:
6385:
5169:
7042:
2871:
4034:
3523:
3869:
2603:
4711:
3178:
7363:
5925:
2105:
2032:
6759:
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
1458:
3792:
7268:
3732:
2939:
2728:
2110:
4965:
4457:
3013:
5993:
4093:
1154:
1024:
6196:
3081:
544:
7047:
6286:
6591:
714:
4382:
1688:
3642:
4631:
1388:
2404:
5372:
6298:
1864:
5417:
2337:
2301:
1932:
1754:
5678:
5446:
2449:
5968:
4940:
4861:
5618:{\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}}},}
7015:
6948:
6875:
4338:
2766:
2760:
1424:
1308:
1149:
6621:
2648:
747:
7196:
6650:
5164:
4311:
3927:
3415:
3321:
3294:
3259:
2757:
2484:
1959:
3364:
953:
4790:
3412:
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:
2366:
2265:
1893:
1717:
1614:
1453:
6971:
6904:
6824:
6797:
5717:
3575:
3393:
2507:
1777:
537:
6724:
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
3797:
6683:
5775:
5744:
4760:
4733:
4412:
3552:
3296:
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
3232:
3205:
913:
874:
841:
780:
2512:
808:
1824:
1640:
1334:
4636:
5988:
5795:
5130:
4960:
4881:
4810:
4452:
4432:
4285:
1798:
3086:
7643:
7284:
7037:
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
5804:
2037:
1964:
3736:
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,
3739:
6768:, more complicated analysis based on the propagation of the spacecraft's state vector and the integration of thrust are used to predict orbital motion.
5331:{\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}}}}
7203:
3649:
2878:
2657:
7162:{\displaystyle {\begin{aligned}\Delta v\ &=v_{\text{e}}\ln {100 \over 100-80}\\&=v_{\text{e}}\ln 5\\&=1.61v_{\text{e}}.\\\end{aligned}}}
17:
7555:
7462:
2944:
1054:
independently developed the equation in 1912 when he began his research to improve rocket engines for possible space flight. German engineer
6877:= 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.
958:
3018:
6765:
6543:
1065:
exercise, Tsiolkovsky is honored as being the first to apply it to the question of whether rockets could achieve speeds necessary for
4267:
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
8512:
3908:. The rocket is subject to a constant force, but its total mass is decreasing steadily because it is expelling gas. According to
3583:
440:
7489:
6711:
only accounts for the reaction force from the rocket engine; it does not include other forces that may act on a rocket, such as
1339:
1078:
8360:
7279:
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
3580:
and realising that the integral of a resultant force over time is total impulse, assuming thrust is the only force involved,
1577:{\displaystyle {\vec {P}}_{2}=\left(m-\Delta m\right)\left({\vec {V}}+\Delta {\vec {V}}\right)+\Delta m{\vec {V}}_{\text{e}}}
1044:
498:
8420:
212:
5719:
again standing for the rocket's final velocity (after expelling all its reaction mass and being reduced to a rest mass of
6107:
281:
2231:{\displaystyle {\vec {P}}_{2}-{\vec {P}}_{1}=m\Delta {\vec {V}}+{\vec {v}}_{\text{e}}\Delta m-\Delta m\Delta {\vec {V}}}
6216:
669:
497:, who independently derived it and published it in 1903, although it had been independently derived and published by
8415:
8295:
7714:
5103:{\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)}}}}
4343:
1112:
7653:
1646:
8517:
8380:
8134:
4576:{\displaystyle {\tfrac {1}{2}}m_{p}v_{\text{eff}}^{2}={\tfrac {1}{2}}m_{p}u^{2}+{\tfrac {1}{2}}m(\Delta v)^{2}.}
8522:
8453:
8092:
8083:
7820:
4588:
2371:
102:
7538:
A Treatise on the Motion of
Rockets: to which is added, an Essay on Naval Gunnery, in theory and practice, etc
6092:{\displaystyle {\frac {\Delta v}{c}}={\frac {R^{\frac {2v_{\text{e}}}{c}}-1}{R^{\frac {2v_{\text{e}}}{c}}+1}}}
4253:{\displaystyle ~\Delta v=v_{f}-v_{0}=-v_{\text{e}}\left=~v_{\text{e}}\ln \left({\frac {m_{0}}{m_{f}}}\right).}
1267:{\displaystyle \sum _{i}{\vec {F}}_{i}=\lim _{\Delta t\to 0}{\frac {{\vec {P}}_{2}-{\vec {P}}_{1}}{\Delta t}}}
1109:
In the following derivation, "the rocket" is taken to mean "the rocket and all of its unexpended propellant".
5341:
1830:
656:{\displaystyle \Delta v=v_{\text{e}}\ln {\frac {m_{0}}{m_{f}}}=I_{\text{sp}}g_{0}\ln {\frac {m_{0}}{m_{f}}},}
6439:
that is needed to perform a maneuver such as launching from, or landing on a planet or moon, or an in-space
8400:
7870:
7439:
6432:
5377:
2306:
2270:
1901:
1723:
5628:
5425:
2409:
5747:
4886:
880:
Given the effective exhaust velocity determined by the rocket motor's design, the desired delta-v (e.g.,
433:
6210:
5932:
4815:
8527:
8325:
8152:
7619:
6761:
6531:
3909:
3879:
717:
462:
477:
is a mathematical equation that describes the motion of vehicles that follow the basic principle of a
8463:
7552:
7459:
6976:
6909:
6836:
4316:
1393:
1277:
1118:
8448:
7973:
7687:
6599:
6509:
725:
490:
361:
276:
7174:
6628:
5135:
4290:
3299:
3272:
3237:
2735:
2610:
2454:
1937:
1151:) to the change in linear momentum of the whole system (including rocket and exhaust) as follows:
8458:
7766:
7551:К. Ціолковскій, Изслѣдованіе мировыхъ пространствъ реактивными приборами, 1903 (available online
7458:К. Ціолковскій, Изслѣдованіе мировыхъ пространствъ реактивными приборами, 1903 (available online
7423:
3329:
918:
232:
6656:, which is the same as the effective exhaust velocity measured in m/s (or ft/s if g is in ft/s),
6380:{\displaystyle \Delta v=c\tanh \left({\frac {v_{\text{e}}}{c}}\ln {\frac {m_{0}}{m_{1}}}\right)}
4765:
2342:
2241:
1869:
1693:
1590:
1429:
8507:
8320:
7922:
7842:
7830:
6725:
6102:
1037:
494:
426:
149:
6953:
6886:
6806:
6779:
5699:
3557:
3375:
2763:), so it is not subject to integration, then the above equation may be integrated as follows:
2489:
1759:
519:
8443:
8385:
8355:
8143:
8020:
7988:
7958:
7917:
7902:
7781:
7682:
7434:
7276:
6830:
6515:
1101:
1058:
independently derived the equation about 1920 as he studied the feasibility of space travel.
334:
169:
77:
8468:
8290:
8074:
7963:
7932:
7860:
7835:
7810:
7771:
7752:
7707:
7585:
6661:
6444:
6199:
5753:
5722:
4738:
4718:
4390:
3530:
3210:
3183:
891:
852:
819:
758:
207:
164:
154:
82:
6413:
3882:). From the moment its engine is started (clock set to 0) the rocket expels gas mass at a
793:
8:
8330:
8125:
7865:
7418:
6292:
5693:
4454:, the amount of energy converted to increase the rocket's and pellet's kinetic energy is
1803:
1619:
1313:
249:
87:
7589:
4078:
to burn all this fuel. Integrating both sides of the equation with respect to time from
8003:
7892:
7790:
7601:
7576:
Blanco, Philip (November 2019). "A discrete, energetic approach to rocket propulsion".
7385:
7031:
6880:
5973:
5780:
5689:
5115:
4945:
4866:
4795:
4437:
4417:
4270:
2866:{\displaystyle -\int _{V}^{V+\Delta V}\,dV={v_{e}}\int _{m_{0}}^{m_{f}}{\frac {dm}{m}}}
1783:
322:
197:
7626:
as the ratio of initial to final mass and w as the exhaust velocity, corresponding to
1756:
is the velocity of the mass added to the exhaust (and lost by the rocket) during time
8370:
8268:
8198:
7953:
7907:
7825:
7605:
7563:
7470:
7428:
6745:
6712:
6475:
6456:
6436:
6206:
4029:{\displaystyle ~a={\frac {dv}{dt}}=-{\frac {F}{m(t)}}=-{\frac {Rv_{\text{e}}}{m(t)}}}
3518:{\displaystyle \Delta v=\int _{t_{0}}^{t_{f}}{\frac {|T|}{{m_{0}}-{t}\Delta {m}}}~dt}
811:
237:
174:
53:
7510:
1934:
in the observer frame is related to the velocity of the exhaust in the rocket frame
8350:
8282:
8046:
8008:
7882:
7852:
7805:
7593:
6756:
6721:
6686:
6653:
6537:
6519:
6440:
3262:
783:
750:
407:
356:
121:
65:
8390:
7983:
7887:
7877:
7776:
7700:
7559:
7536:
7466:
6800:
885:
412:
317:
227:
202:
2651:
8486:
8438:
8430:
8425:
8310:
8305:
8236:
8216:
8207:
7800:
7786:
7762:
7757:
7732:
7597:
7393:
6776:
Assume an exhaust velocity of 4,500 meters per second (15,000 ft/s) and a
6737:
6694:
6467:
5798:
3864:{\displaystyle \Delta v=V_{\text{exh}}~\ln \left({\frac {m_{0}}{m_{f}}}\right)}
1055:
1051:
506:
502:
384:
300:
294:
217:
142:
136:
131:
453:
8501:
8340:
8335:
8254:
7897:
7815:
7408:
6493:
2598:{\displaystyle \sum _{i}F_{i}=m{\frac {dV}{dt}}+v_{\text{e}}{\frac {dm}{dt}}}
881:
389:
222:
179:
8405:
8315:
8189:
8172:
8030:
7927:
7795:
7412:
7398:
4706:{\displaystyle \Delta v=v_{\text{eff}}{\frac {m_{p}}{\sqrt {m(m+m_{p})}}}.}
3367:
107:
97:
92:
5750:
where the rocket started at rest (with the rest mass including fuel being
4585:
Using momentum conservation in the rocket's frame just prior to ejection,
8410:
8245:
8015:
7995:
7912:
6741:
6729:
6716:
5449:
3173:{\displaystyle m_{0}-m_{f}=m_{f}\left(e^{\Delta V/v_{\text{e}}}-1\right)}
1066:
271:
7358:{\displaystyle \Delta v\ =v_{\text{e}}\ln(100/11.2)\ =2.19v_{\text{e}}.}
4762:
the initial fueled-up mass of the rocket. Divide the total mass of fuel
7978:
7692:
7403:
6749:
5920:{\displaystyle {\frac {m_{0}}{m_{1}}}=\left^{\frac {c}{2v_{\text{e}}}}}
844:
458:
351:
307:
266:
44:
7677:
4340:
such that the mechanical energy gained per unit fuel mass is given by
2100:{\displaystyle {\vec {V}}_{\text{e}}={\vec {V}}+{\vec {v}}_{\text{e}}}
2027:{\displaystyle {\vec {v}}_{\text{e}}={\vec {V}}_{\text{e}}-{\vec {V}}}
8395:
7747:
7511:"On the Motion of Rockets both in Nonresisting and Resisting Mediums"
6733:
1028:
The necessary wet mass grows exponentially with the desired delta-v.
4262:
4038:
Now, the mass of fuel the rocket initially has on board is equal to
7648:
6474:
per unit mass and burn time, and is used to determine the mass of
6417:
5692:
is taken into account, the following equation can be derived for a
3787:{\displaystyle {\frac {J}{\Delta m}}={\frac {F}{p}}=V_{\text{exh}}}
3261:
the velocity of the rocket exhaust with respect to the rocket (the
1062:
1043:
The equation had been derived earlier by the
British mathematician
513:
486:
7688:
Tsiolkovsky's rocket equations plot and calculator in
WolframAlpha
8300:
6403:
3396:
3266:
1061:
While the derivation of the rocket equation is a straightforward
7263:{\displaystyle \Delta v\ =3v_{\text{e}}\ln 5\ =4.83v_{\text{e}}}
3878:
Imagine a rocket at rest in space with no forces exerted on it (
1040:
who independently derived it and published it in his 1903 work.
7272:
and the payload is 10% × 10% × 10% = 0.1% of the initial mass.
6471:
3727:{\displaystyle J~{\frac {\ln({m_{0}})-\ln({m_{f}})}{\Delta m}}}
2934:{\displaystyle \Delta V=v_{\text{e}}\ln {\frac {m_{0}}{m_{f}}}}
2723:{\displaystyle -m{\frac {dV}{dt}}=v_{\text{e}}{\frac {dm}{dt}}}
482:
478:
7620:"A Transparent Derivation of the Relativistic Rocket Equation"
7171:
With three similar, subsequently smaller stages with the same
6478:
required for the given manoeuvre through the rocket equation.
915:, the equation can be solved for the required propellant mass
8105:
7724:
6448:
72:
7683:
Relativity
Calculator – Learn Tsiolkovsky's rocket equations
876:
is the final total mass without propellant, a.k.a. dry mass.
6522:, which is the fraction of initial weight that is payload.
7622:(see the right side of equation 15 on the last page, with
4387:
In the rocket's center-of-mass frame, if a pellet of mass
1105:
Tsiolkovsky's theoretical rocket from t = 0 to t = delta_t
1426:
is the momentum of the rocket and exhausted mass at time
1047:
in 1810, and later published in a separate book in 1813.
501:
in 1810, and later published in a separate book in 1813.
7483:
7481:
7479:
6799:
of 9,700 meters per second (32,000 ft/s) (Earth to
1072:
7515:
Journal of
Natural Philosophy, Chemistry & the Arts
6536:
The effective exhaust velocity is often specified as a
3008:{\displaystyle m_{f}=m_{0}e^{-\Delta V\ /v_{\text{e}}}}
481:: a device that can apply acceleration to itself using
27:
Mathematical equation describing the motion of a rocket
6219:
6110:
5935:
5380:
5344:
4605:
4591:
4537:
4502:
4462:
4348:
4346:
2613:
7476:
7431:- added terms for gravity and drag in vertical flight
7287:
7206:
7177:
7045:
6979:
6956:
6912:
6889:
6839:
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6782:
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6301:
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5172:
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4273:
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3652:
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3418:
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3240:
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3186:
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2769:
2738:
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2515:
2492:
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2412:
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2345:
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2273:
2244:
2113:
2040:
1967:
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1904:
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1806:
1786:
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1726:
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1649:
1622:
1593:
1461:
1432:
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1342:
1316:
1280:
1157:
1121:
1019:{\displaystyle m_{0}=m_{f}e^{\Delta v/v_{\text{e}}}.}
961:
921:
894:
855:
822:
796:
761:
728:
672:
547:
522:
7375:
6191:{\textstyle R^{\frac {2v_{\text{e}}}{c}}=\exp \left}
3076:{\displaystyle m_{0}=m_{f}e^{\Delta V/v_{\text{e}}}}
7357:
7262:
7190:
7161:
7009:
6965:
6942:
6898:
6869:
6818:
6791:
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5672:
5617:
5440:
5411:
5366:
5330:
5158:
5124:
5102:
4954:
4934:
4875:
4863:. The remaining mass of the rocket after ejecting
4855:
4804:
4784:
4754:
4727:
4705:
4625:
4575:
4446:
4426:
4406:
4376:
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4028:
3863:
3786:
3726:
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3387:
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3315:
3288:
3253:
3226:
3199:
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3075:
3007:
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2865:
2751:
2722:
2642:
2597:
2501:
2478:
2443:
2398:
2360:
2331:
2295:
2259:
2230:
2099:
2026:
1953:
1926:
1887:
1858:
1818:
1792:
1771:
1748:
1711:
1682:
1634:
1608:
1576:
1447:
1418:
1382:
1328:
1302:
1266:
1143:
1018:
947:
907:
868:
835:
802:
774:
741:
708:
655:
531:
6281:{\textstyle \tanh x={\frac {e^{2x}-1}{e^{2x}+1}}}
8499:
6883:: suppose that the first stage should provide a
6586:{\displaystyle v_{\text{e}}=g_{0}I_{\text{sp}},}
5625:since the final remaining mass of the rocket is
5460:
3577:is the initial mass minus the final (dry) mass,
3207:is the initial total mass including propellant,
1191:
6973:of 4,700 meters per second (15,000 ft/s);
6906:of 5,000 meters per second (16,000 ft/s);
4735:be the initial fuel mass fraction on board and
2509:results in a decrease in rocket mass in time),
709:{\displaystyle v_{\text{e}}=I_{\text{sp}}g_{0}}
4377:{\textstyle {\tfrac {1}{2}}v_{\text{eff}}^{2}}
4090:allows a substitution on the right) obtains:
3265:, or, if measured in time, that multiplied by
1036:The equation is named after Russian scientist
7708:
6525:
505:also developed it independently in 1912, and
434:
7569:
1683:{\displaystyle {\vec {V}}+\Delta {\vec {V}}}
7504:
7502:
6826:to overcome gravity and aerodynamic drag).
3637:{\displaystyle \int _{t_{0}}^{t_{f}}F~dt=J}
8482:
7715:
7701:
6496:which displays the required mission delta-
4942:. The overall speed change after ejecting
4626:{\textstyle u=\Delta v{\tfrac {m}{m_{p}}}}
1383:{\displaystyle {\vec {P}}_{1}=m{\vec {V}}}
1093:
441:
427:
6466:is produced by reaction engines, such as
5990:allows this equation to be rearranged as
3888:exhaust velocity relative to the rocket v
2797:
2419:
2399:{\displaystyle \Delta m\Delta {\vec {V}}}
1584:and where, with respect to the observer:
7722:
7499:
6389:
5367:{\textstyle \Delta x={\frac {\phi }{N}}}
4434:and the remaining mass of the rocket is
1100:
485:by expelling part of its mass with high
452:
7530:
7528:
7487:
6540:and they are related to each other by:
4263:Limit of finite mass "pellet" expulsion
3897:propelling the rocket that is equal to
1859:{\displaystyle \left(m-\Delta m\right)}
14:
8500:
8361:Transposition, docking, and extraction
7575:
7030:In the case of sequentially thrusting
6755:The rocket equation can be applied to
5412:{\textstyle x_{j}={\frac {j\phi }{N}}}
1690:is the velocity of the rocket at time
1616:is the velocity of the rocket at time
1310:is the momentum of the rocket at time
7696:
7534:
7508:
5683:
3893:(m/s). This creates a constant force
3873:
2607:If there are no external forces then
2332:{\displaystyle {\vec {F}}_{\text{i}}}
2296:{\displaystyle {\vec {v}}_{\text{e}}}
1927:{\displaystyle {\vec {V}}_{\text{e}}}
1749:{\displaystyle {\vec {V}}_{\text{e}}}
1073:Experiment of the Boat by Tsiolkovsky
843:is the initial total mass, including
539:(with no external forces acting) is:
509:derived it independently about 1920.
7644:"The Tyranny of the Rocket Equation"
7525:
6652:is the specific impulse measured in
5673:{\displaystyle m_{f}=m_{0}(1-\phi )}
5441:{\displaystyle N\rightarrow \infty }
3402:
2444:{\displaystyle dm\,d{\vec {v}}\to 0}
6623:is the specific impulse in seconds,
6209:as well as the "power" identity at
5963:{\textstyle {\frac {m_{0}}{m_{1}}}}
4935:{\displaystyle m=m_{0}(1-j\phi /N)}
24:
7288:
7207:
7050:
6957:
6890:
6810:
6783:
6488:For interplanetary missions delta-
6302:
6000:
5875:
5852:
5703:
5475:
5470:
5435:
5345:
5301:
5173:
4969:
4856:{\displaystyle m_{p}=\phi m_{0}/N}
4640:
4598:
4554:
4313:, with an effective exhaust speed
4300:
4100:
4052:. For the constant mass flow rate
3801:
3749:
3715:
3561:
3495:
3419:
3379:
3136:
3050:
2979:
2882:
2789:
2493:
2470:
2381:
2375:
2213:
2207:
2198:
2161:
1879:
1866:is the mass of the rocket at time
1845:
1800:is the mass of the rocket at time
1763:
1703:
1665:
1549:
1526:
1495:
1439:
1255:
1195:
990:
548:
523:
25:
18:Tsiolkovsky's rocket equation
8539:
8421:Kepler's laws of planetary motion
7678:How to derive the rocket equation
7671:
6451:. As used in this context, it is
5132:the last term in the denominator
213:Kepler's laws of planetary motion
8481:
8416:Interplanetary Transport Network
8296:Collision avoidance (spacecraft)
7633:in the notation of this article)
7378:
7021:engines, the tanks, and payload.
6705:
6503:
3407:
1077:
489:and can thereby move due to the
43:
8381:Astronomical coordinate systems
8135:Longitude of the ascending node
6481:For multiple manoeuvres, delta-
3912:, its acceleration at any time
3794:the integral can be equated to
2652:conservation of linear momentum
1098:Consider the following system:
8513:Eponymous equations of physics
8454:Retrograde and prograde motion
7636:
7612:
7545:
7452:
7330:
7316:
7010:{\displaystyle 1-e^{-4.7/4.5}}
6943:{\displaystyle 1-e^{-5.0/4.5}}
6870:{\displaystyle 1-e^{-9.7/4.5}}
6500:as a function of launch date.
5667:
5655:
5467:
5452:becomes the definite integral
5432:
5094:
5057:
5054:
5031:
4929:
4906:
4812:discrete pellets each of mass
4694:
4675:
4561:
4551:
4333:{\displaystyle v_{\text{eff}}}
4297:
4056:it will therefore take a time
4020:
4014:
3981:
3975:
3710:
3695:
3683:
3668:
3554:is the initial (wet) mass and
3469:
3461:
2435:
2429:
2390:
2352:
2317:
2281:
2251:
2222:
2186:
2170:
2143:
2121:
2085:
2069:
2048:
2018:
1997:
1975:
1912:
1734:
1674:
1656:
1600:
1562:
1535:
1517:
1469:
1419:{\displaystyle {\vec {P}}_{2}}
1404:
1374:
1350:
1303:{\displaystyle {\vec {P}}_{1}}
1288:
1241:
1219:
1201:
1175:
1144:{\displaystyle {\vec {F}}_{i}}
1129:
13:
1:
7445:
6616:{\displaystyle I_{\text{sp}}}
5166:and can be neglected to give
3910:Newton's Second Law of Motion
3646:The integral is found to be:
2643:{\textstyle \sum _{i}F_{i}=0}
1113:Newton's second law of motion
1088:
742:{\displaystyle I_{\text{sp}}}
8401:Equatorial coordinate system
7415:increases the final velocity
7191:{\displaystyle v_{\text{e}}}
6720:is a limit to the amount of
6645:{\displaystyle v_{\text{e}}}
5159:{\displaystyle \phi /N\ll 1}
4306:{\displaystyle N\to \infty }
3920:divided by its current mass
3880:Newton's First Law of Motion
3316:{\displaystyle v_{\text{e}}}
3289:{\displaystyle v_{\text{e}}}
3269:-on-Earth acceleration). If
3254:{\displaystyle v_{\text{e}}}
2752:{\displaystyle v_{\text{e}}}
2479:{\displaystyle dm=-\Delta m}
1954:{\displaystyle v_{\text{e}}}
1898:The velocity of the exhaust
7:
7371:
6771:
6457:physical change in velocity
5748:inertial frame of reference
3359:{\displaystyle m_{0}-m_{f}}
2486:(since ejecting a positive
2339:have the same direction as
948:{\displaystyle m_{0}-m_{f}}
367:Tsiolkovsky rocket equation
10:
8544:
8153:Longitude of the periapsis
7490:"Reactive Flying Machines"
6532:Effective exhaust velocity
6529:
6526:Effective exhaust velocity
6507:
6433:spacecraft flight dynamics
6401:
6394:
4785:{\displaystyle \phi m_{0}}
4287:pellets consecutively, as
2361:{\displaystyle {\vec {V}}}
2260:{\displaystyle {\vec {V}}}
1888:{\displaystyle t=\Delta t}
1712:{\displaystyle t=\Delta t}
1609:{\displaystyle {\vec {V}}}
1448:{\displaystyle t=\Delta t}
1031:
718:effective exhaust velocity
463:effective exhaust velocity
336:Engineering and efficiency
155:Bi-elliptic transfer orbit
8477:
8464:Specific angular momentum
8369:
8281:
8225:
8161:
8114:
8054:
8045:
7941:
7851:
7740:
7731:
7025:
6470:, is proportional to the
6295:), this is equivalent to
3884:constant mass flow rate R
1115:relates external forces (
471:classical rocket equation
7598:10.1088/1361-6552/ab315b
7440:Stigler’s law of eponymy
7411:- applying delta-v in a
6966:{\displaystyle \Delta v}
6899:{\displaystyle \Delta v}
6819:{\displaystyle \Delta v}
6792:{\displaystyle \Delta v}
6510:Propellant mass fraction
6198:(here "exp" denotes the
5712:{\displaystyle \Delta v}
3916:is its propelling force
3570:{\displaystyle \Delta m}
3388:{\displaystyle \Delta V}
3370:of propellant expended.
2761:Tsiolkovsky's hypothesis
2502:{\displaystyle \Delta m}
1772:{\displaystyle \Delta t}
888:), and a given dry mass
532:{\displaystyle \Delta v}
491:conservation of momentum
362:Propellant mass fraction
261:Gravitational influences
8459:Specific orbital energy
7541:. G. & S. Robinson.
7535:Moore, William (1813).
7509:Moore, William (1810).
7424:Reversibility of orbits
7198:for each stage, gives:
1094:Most popular derivation
233:Specific orbital energy
8518:Konstantin Tsiolkovsky
7871:Geostationary transfer
7359:
7264:
7192:
7163:
7011:
6967:
6944:
6900:
6871:
6820:
6793:
6679:
6646:
6617:
6587:
6492:is often plotted on a
6447:that has the units of
6435:, is a measure of the
6381:
6282:
6211:Logarithmic identities
6192:
6093:
5984:
5964:
5921:
5791:
5771:
5740:
5713:
5674:
5619:
5442:
5413:
5368:
5332:
5297:
5218:
5160:
5126:
5112:Notice that for large
5104:
5014:
4956:
4936:
4877:
4857:
4806:
4786:
4756:
4729:
4707:
4633:, from which we find
4627:
4577:
4448:
4428:
4408:
4378:
4334:
4307:
4281:
4254:
4030:
3865:
3788:
3728:
3638:
3571:
3548:
3519:
3389:
3360:
3317:
3290:
3255:
3228:
3201:
3174:
3077:
3009:
2935:
2867:
2759:is constant (known as
2753:
2724:
2644:
2599:
2503:
2480:
2445:
2406:are negligible (since
2400:
2362:
2333:
2297:
2261:
2232:
2101:
2028:
1955:
1928:
1889:
1860:
1820:
1794:
1773:
1750:
1713:
1684:
1636:
1610:
1578:
1449:
1420:
1384:
1330:
1304:
1268:
1145:
1106:
1038:Konstantin Tsiolkovsky
1020:
949:
909:
870:
837:
804:
776:
743:
710:
657:
533:
512:The maximum change of
495:Konstantin Tsiolkovsky
466:
150:Hohmann transfer orbit
8523:Single-stage-to-orbit
8444:Orbital state vectors
8386:Characteristic energy
8356:Trans-lunar injection
8144:Argument of periapsis
7821:Prograde / Retrograde
7782:Hyperbolic trajectory
7435:Spacecraft propulsion
7360:
7265:
7193:
7164:
7012:
6968:
6945:
6901:
6872:
6831:Single-stage-to-orbit
6821:
6794:
6680:
6678:{\displaystyle g_{0}}
6647:
6618:
6588:
6516:aerospace engineering
6390:Terms of the equation
6382:
6283:
6193:
6094:
5985:
5965:
5922:
5792:
5772:
5770:{\displaystyle m_{0}}
5741:
5739:{\displaystyle m_{1}}
5714:
5675:
5620:
5443:
5414:
5369:
5333:
5271:
5192:
5161:
5127:
5105:
4988:
4957:
4937:
4878:
4858:
4807:
4787:
4757:
4755:{\displaystyle m_{0}}
4730:
4728:{\displaystyle \phi }
4708:
4628:
4578:
4449:
4429:
4409:
4407:{\displaystyle m_{p}}
4379:
4335:
4308:
4282:
4255:
4031:
3866:
3789:
3729:
3639:
3572:
3549:
3547:{\displaystyle m_{0}}
3520:
3390:
3361:
3318:
3291:
3256:
3229:
3227:{\displaystyle m_{f}}
3202:
3200:{\displaystyle m_{0}}
3175:
3078:
3010:
2936:
2868:
2754:
2725:
2645:
2600:
2504:
2481:
2446:
2401:
2363:
2334:
2298:
2262:
2233:
2107:Solving this yields:
2102:
2029:
1956:
1929:
1890:
1861:
1821:
1795:
1774:
1751:
1714:
1685:
1637:
1611:
1579:
1450:
1421:
1385:
1331:
1305:
1269:
1146:
1104:
1021:
950:
910:
908:{\displaystyle m_{f}}
871:
869:{\displaystyle m_{f}}
838:
836:{\displaystyle m_{0}}
805:
777:
775:{\displaystyle g_{0}}
753:in dimension of time;
744:
711:
658:
534:
475:ideal rocket equation
456:
346:Preflight engineering
78:Argument of periapsis
8291:Bi-elliptic transfer
7811:Parabolic trajectory
7285:
7204:
7175:
7043:
6977:
6954:
6910:
6887:
6837:
6807:
6780:
6662:
6629:
6600:
6544:
6299:
6217:
6200:exponential function
6108:
5994:
5974:
5933:
5805:
5781:
5754:
5723:
5700:
5629:
5456:
5426:
5378:
5342:
5170:
5136:
5116:
4966:
4946:
4887:
4867:
4816:
4796:
4766:
4739:
4719:
4637:
4589:
4458:
4438:
4418:
4414:is ejected at speed
4391:
4344:
4317:
4291:
4271:
4094:
3928:
3798:
3740:
3650:
3584:
3558:
3531:
3416:
3376:
3330:
3300:
3273:
3238:
3234:the final mass, and
3211:
3184:
3087:
3019:
2945:
2879:
2767:
2736:
2658:
2611:
2513:
2490:
2455:
2410:
2372:
2343:
2307:
2271:
2242:
2111:
2038:
1965:
1938:
1902:
1870:
1831:
1804:
1784:
1760:
1724:
1694:
1647:
1620:
1591:
1459:
1430:
1394:
1340:
1314:
1278:
1155:
1119:
959:
919:
892:
853:
820:
803:{\displaystyle \ln }
794:
759:
726:
670:
545:
520:
493:. It is credited to
457:A rocket's required
402:Propulsive maneuvers
8331:Low-energy transfer
7618:Forward, Robert L.
7590:2019PhyEd..54f5001B
7419:Relativistic rocket
6766:electric propulsion
6293:Hyperbolic function
6213:) and the identity
5694:relativistic rocket
5508:
4962:pellets is the sum
4497:
4373:
3615:
3527:where T is thrust,
3456:
2847:
2796:
1819:{\displaystyle t=0}
1635:{\displaystyle t=0}
1329:{\displaystyle t=0}
379:Efficiency measures
282:Sphere of influence
251:Celestial mechanics
33:Part of a series on
8326:Inclination change
7974:Distant retrograde
7558:2011-08-15 at the
7465:2011-08-15 at the
7386:Spaceflight portal
7355:
7260:
7188:
7159:
7157:
7007:
6963:
6940:
6896:
6881:Two-stage-to-orbit
6867:
6816:
6789:
6762:impulsive maneuver
6726:non-rocket systems
6675:
6642:
6613:
6583:
6420:"), symbolised as
6377:
6278:
6188:
6089:
5980:
5960:
5917:
5787:
5767:
5736:
5709:
5690:special relativity
5684:Special relativity
5670:
5615:
5494:
5474:
5438:
5409:
5364:
5328:
5156:
5122:
5100:
4952:
4932:
4873:
4853:
4802:
4782:
4752:
4725:
4703:
4623:
4621:
4573:
4546:
4511:
4483:
4471:
4444:
4424:
4404:
4374:
4359:
4357:
4330:
4303:
4277:
4250:
4026:
3874:Acceleration-based
3861:
3784:
3724:
3634:
3587:
3567:
3544:
3515:
3428:
3385:
3356:
3313:
3286:
3251:
3224:
3197:
3170:
3073:
3005:
2931:
2863:
2819:
2773:
2749:
2720:
2640:
2623:
2595:
2525:
2499:
2476:
2441:
2396:
2358:
2329:
2293:
2257:
2228:
2097:
2024:
1951:
1924:
1885:
1856:
1816:
1790:
1769:
1746:
1709:
1680:
1632:
1606:
1574:
1445:
1416:
1380:
1326:
1300:
1264:
1208:
1167:
1141:
1107:
1016:
945:
905:
866:
847:, a.k.a. wet mass;
833:
800:
772:
739:
706:
653:
529:
467:
198:Dynamical friction
8528:Rocket propulsion
8495:
8494:
8469:Two-line elements
8277:
8276:
8199:Eccentric anomaly
8041:
8040:
7908:Orbit of the Moon
7767:Highly elliptical
7578:Physics Education
7429:Robert H. Goddard
7349:
7335:
7307:
7296:
7257:
7243:
7229:
7215:
7185:
7149:
7117:
7098:
7073:
7058:
6757:orbital maneuvers
6746:tether propulsion
6639:
6610:
6577:
6554:
6370:
6340:
6334:
6276:
6207:Natural logarithm
6172:
6165:
6135:
6128:
6087:
6077:
6070:
6042:
6035:
6010:
5983:{\displaystyle R}
5958:
5914:
5910:
5888:
5885:
5862:
5830:
5797:standing for the
5790:{\displaystyle c}
5610:
5579:
5567:
5542:
5530:
5491:
5459:
5407:
5362:
5326:
5268:
5256:
5189:
5125:{\displaystyle N}
5098:
5097:
4985:
4955:{\displaystyle j}
4876:{\displaystyle j}
4805:{\displaystyle N}
4698:
4697:
4656:
4620:
4545:
4510:
4490:
4470:
4447:{\displaystyle m}
4427:{\displaystyle u}
4366:
4356:
4327:
4280:{\displaystyle N}
4241:
4206:
4198:
4145:
4099:
4086:(and noting that
4024:
4006:
3985:
3958:
3933:
3855:
3822:
3817:
3781:
3769:
3756:
3722:
3658:
3621:
3508:
3504:
3403:Other derivations
3310:
3283:
3248:
3154:
3068:
3000:
2987:
2929:
2898:
2875:This then yields
2861:
2746:
2718:
2697:
2685:
2614:
2593:
2572:
2560:
2516:
2432:
2393:
2355:
2326:
2320:
2290:
2284:
2254:
2225:
2195:
2189:
2173:
2146:
2124:
2094:
2088:
2072:
2057:
2051:
2021:
2006:
2000:
1984:
1978:
1948:
1921:
1915:
1793:{\displaystyle m}
1743:
1737:
1677:
1659:
1603:
1571:
1565:
1538:
1520:
1472:
1407:
1377:
1353:
1291:
1262:
1244:
1222:
1190:
1178:
1158:
1132:
1008:
812:natural logarithm
736:
693:
680:
648:
607:
595:
564:
461:as a function of
451:
450:
301:Lagrangian points
238:Vis-viva equation
208:Kepler's equation
55:Orbital mechanics
16:(Redirected from
8535:
8485:
8484:
8426:Lagrangian point
8321:Hohmann transfer
8266:
8252:
8243:
8234:
8214:
8205:
8196:
8187:
8183:
8179:
8170:
8150:
8141:
8132:
8123:
8103:
8099:
8090:
8081:
8072:
8052:
8051:
8021:Heliosynchronous
7970:Lagrange points
7923:Transatmospheric
7738:
7737:
7717:
7710:
7703:
7694:
7693:
7665:
7664:
7662:
7661:
7652:. Archived from
7640:
7634:
7616:
7610:
7609:
7573:
7567:
7549:
7543:
7542:
7532:
7523:
7522:
7506:
7497:
7496:
7494:
7488:Tsiolkovsky, K.
7485:
7474:
7456:
7388:
7383:
7382:
7381:
7364:
7362:
7361:
7356:
7351:
7350:
7347:
7333:
7326:
7309:
7308:
7305:
7294:
7269:
7267:
7266:
7261:
7259:
7258:
7255:
7241:
7231:
7230:
7227:
7213:
7197:
7195:
7194:
7189:
7187:
7186:
7183:
7168:
7166:
7165:
7160:
7158:
7151:
7150:
7147:
7132:
7119:
7118:
7115:
7103:
7099:
7097:
7083:
7075:
7074:
7071:
7056:
7016:
7014:
7013:
7008:
7006:
7005:
7001:
6972:
6970:
6969:
6964:
6949:
6947:
6946:
6941:
6939:
6938:
6934:
6905:
6903:
6902:
6897:
6876:
6874:
6873:
6868:
6866:
6865:
6861:
6825:
6823:
6822:
6817:
6798:
6796:
6795:
6790:
6700:
6692:
6687:standard gravity
6684:
6682:
6681:
6676:
6674:
6673:
6651:
6649:
6648:
6643:
6641:
6640:
6637:
6622:
6620:
6619:
6614:
6612:
6611:
6608:
6592:
6590:
6589:
6584:
6579:
6578:
6575:
6569:
6568:
6556:
6555:
6552:
6538:specific impulse
6520:payload fraction
6459:of the vehicle.
6455:the same as the
6441:orbital maneuver
6386:
6384:
6383:
6378:
6376:
6372:
6371:
6369:
6368:
6359:
6358:
6349:
6341:
6336:
6335:
6332:
6326:
6287:
6285:
6284:
6279:
6277:
6275:
6268:
6267:
6254:
6247:
6246:
6233:
6197:
6195:
6194:
6189:
6187:
6183:
6173:
6168:
6167:
6166:
6163:
6153:
6137:
6136:
6131:
6130:
6129:
6126:
6116:
6101:Then, using the
6098:
6096:
6095:
6090:
6088:
6086:
6079:
6078:
6073:
6072:
6071:
6068:
6058:
6051:
6044:
6043:
6038:
6037:
6036:
6033:
6023:
6016:
6011:
6006:
5998:
5989:
5987:
5986:
5981:
5969:
5967:
5966:
5961:
5959:
5957:
5956:
5947:
5946:
5937:
5926:
5924:
5923:
5918:
5916:
5915:
5913:
5912:
5911:
5908:
5895:
5893:
5889:
5887:
5886:
5881:
5873:
5864:
5863:
5858:
5850:
5841:
5831:
5829:
5828:
5819:
5818:
5809:
5796:
5794:
5793:
5788:
5777:initially), and
5776:
5774:
5773:
5768:
5766:
5765:
5745:
5743:
5742:
5737:
5735:
5734:
5718:
5716:
5715:
5710:
5679:
5677:
5676:
5671:
5654:
5653:
5641:
5640:
5624:
5622:
5621:
5616:
5611:
5609:
5608:
5599:
5598:
5589:
5581:
5580:
5577:
5568:
5566:
5552:
5544:
5543:
5540:
5531:
5529:
5518:
5510:
5507:
5502:
5493:
5492:
5489:
5473:
5447:
5445:
5444:
5439:
5418:
5416:
5415:
5410:
5408:
5403:
5395:
5390:
5389:
5373:
5371:
5370:
5365:
5363:
5355:
5337:
5335:
5334:
5329:
5327:
5325:
5324:
5323:
5307:
5299:
5296:
5285:
5270:
5269:
5266:
5257:
5255:
5251:
5233:
5229:
5220:
5217:
5206:
5191:
5190:
5187:
5165:
5163:
5162:
5157:
5146:
5131:
5129:
5128:
5123:
5109:
5107:
5106:
5101:
5099:
5090:
5076:
5050:
5030:
5029:
5025:
5016:
5013:
5002:
4987:
4986:
4983:
4961:
4959:
4958:
4953:
4941:
4939:
4938:
4933:
4925:
4905:
4904:
4883:pellets is then
4882:
4880:
4879:
4874:
4862:
4860:
4859:
4854:
4849:
4844:
4843:
4828:
4827:
4811:
4809:
4808:
4803:
4791:
4789:
4788:
4783:
4781:
4780:
4761:
4759:
4758:
4753:
4751:
4750:
4734:
4732:
4731:
4726:
4712:
4710:
4709:
4704:
4699:
4693:
4692:
4671:
4670:
4669:
4660:
4658:
4657:
4654:
4632:
4630:
4629:
4624:
4622:
4619:
4618:
4606:
4582:
4580:
4579:
4574:
4569:
4568:
4547:
4538:
4532:
4531:
4522:
4521:
4512:
4503:
4496:
4491:
4488:
4482:
4481:
4472:
4463:
4453:
4451:
4450:
4445:
4433:
4431:
4430:
4425:
4413:
4411:
4410:
4405:
4403:
4402:
4383:
4381:
4380:
4375:
4372:
4367:
4364:
4358:
4349:
4339:
4337:
4336:
4331:
4329:
4328:
4325:
4312:
4310:
4309:
4304:
4286:
4284:
4283:
4278:
4259:
4257:
4256:
4251:
4246:
4242:
4240:
4239:
4230:
4229:
4220:
4208:
4207:
4204:
4196:
4192:
4188:
4187:
4186:
4168:
4167:
4147:
4146:
4143:
4131:
4130:
4118:
4117:
4097:
4035:
4033:
4032:
4027:
4025:
4023:
4009:
4008:
4007:
4004:
3994:
3986:
3984:
3967:
3959:
3957:
3949:
3941:
3931:
3870:
3868:
3867:
3862:
3860:
3856:
3854:
3853:
3844:
3843:
3834:
3820:
3819:
3818:
3815:
3793:
3791:
3790:
3785:
3783:
3782:
3779:
3770:
3762:
3757:
3755:
3744:
3733:
3731:
3730:
3725:
3723:
3721:
3713:
3709:
3708:
3707:
3682:
3681:
3680:
3660:
3656:
3643:
3641:
3640:
3635:
3619:
3614:
3613:
3612:
3602:
3601:
3600:
3576:
3574:
3573:
3568:
3553:
3551:
3550:
3545:
3543:
3542:
3524:
3522:
3521:
3516:
3506:
3505:
3503:
3502:
3494:
3486:
3485:
3484:
3473:
3472:
3464:
3458:
3455:
3454:
3453:
3443:
3442:
3441:
3394:
3392:
3391:
3386:
3365:
3363:
3362:
3357:
3355:
3354:
3342:
3341:
3322:
3320:
3319:
3314:
3312:
3311:
3308:
3295:
3293:
3292:
3287:
3285:
3284:
3281:
3263:specific impulse
3260:
3258:
3257:
3252:
3250:
3249:
3246:
3233:
3231:
3230:
3225:
3223:
3222:
3206:
3204:
3203:
3198:
3196:
3195:
3179:
3177:
3176:
3171:
3169:
3165:
3158:
3157:
3156:
3155:
3152:
3146:
3125:
3124:
3112:
3111:
3099:
3098:
3082:
3080:
3079:
3074:
3072:
3071:
3070:
3069:
3066:
3060:
3044:
3043:
3031:
3030:
3014:
3012:
3011:
3006:
3004:
3003:
3002:
3001:
2998:
2992:
2985:
2970:
2969:
2957:
2956:
2941:or equivalently
2940:
2938:
2937:
2932:
2930:
2928:
2927:
2918:
2917:
2908:
2900:
2899:
2896:
2872:
2870:
2869:
2864:
2862:
2857:
2849:
2846:
2845:
2844:
2834:
2833:
2832:
2818:
2817:
2816:
2795:
2781:
2758:
2756:
2755:
2750:
2748:
2747:
2744:
2729:
2727:
2726:
2721:
2719:
2717:
2709:
2701:
2699:
2698:
2695:
2686:
2684:
2676:
2668:
2649:
2647:
2646:
2641:
2633:
2632:
2622:
2604:
2602:
2601:
2596:
2594:
2592:
2584:
2576:
2574:
2573:
2570:
2561:
2559:
2551:
2543:
2535:
2534:
2524:
2508:
2506:
2505:
2500:
2485:
2483:
2482:
2477:
2450:
2448:
2447:
2442:
2434:
2433:
2425:
2405:
2403:
2402:
2397:
2395:
2394:
2386:
2367:
2365:
2364:
2359:
2357:
2356:
2348:
2338:
2336:
2335:
2330:
2328:
2327:
2324:
2322:
2321:
2313:
2302:
2300:
2299:
2294:
2292:
2291:
2288:
2286:
2285:
2277:
2266:
2264:
2263:
2258:
2256:
2255:
2247:
2237:
2235:
2234:
2229:
2227:
2226:
2218:
2197:
2196:
2193:
2191:
2190:
2182:
2175:
2174:
2166:
2154:
2153:
2148:
2147:
2139:
2132:
2131:
2126:
2125:
2117:
2106:
2104:
2103:
2098:
2096:
2095:
2092:
2090:
2089:
2081:
2074:
2073:
2065:
2059:
2058:
2055:
2053:
2052:
2044:
2033:
2031:
2030:
2025:
2023:
2022:
2014:
2008:
2007:
2004:
2002:
2001:
1993:
1986:
1985:
1982:
1980:
1979:
1971:
1960:
1958:
1957:
1952:
1950:
1949:
1946:
1933:
1931:
1930:
1925:
1923:
1922:
1919:
1917:
1916:
1908:
1894:
1892:
1891:
1886:
1865:
1863:
1862:
1857:
1855:
1851:
1825:
1823:
1822:
1817:
1799:
1797:
1796:
1791:
1778:
1776:
1775:
1770:
1755:
1753:
1752:
1747:
1745:
1744:
1741:
1739:
1738:
1730:
1718:
1716:
1715:
1710:
1689:
1687:
1686:
1681:
1679:
1678:
1670:
1661:
1660:
1652:
1641:
1639:
1638:
1633:
1615:
1613:
1612:
1607:
1605:
1604:
1596:
1583:
1581:
1580:
1575:
1573:
1572:
1569:
1567:
1566:
1558:
1545:
1541:
1540:
1539:
1531:
1522:
1521:
1513:
1505:
1501:
1480:
1479:
1474:
1473:
1465:
1454:
1452:
1451:
1446:
1425:
1423:
1422:
1417:
1415:
1414:
1409:
1408:
1400:
1389:
1387:
1386:
1381:
1379:
1378:
1370:
1361:
1360:
1355:
1354:
1346:
1335:
1333:
1332:
1327:
1309:
1307:
1306:
1301:
1299:
1298:
1293:
1292:
1284:
1273:
1271:
1270:
1265:
1263:
1261:
1253:
1252:
1251:
1246:
1245:
1237:
1230:
1229:
1224:
1223:
1215:
1210:
1207:
1186:
1185:
1180:
1179:
1171:
1166:
1150:
1148:
1147:
1142:
1140:
1139:
1134:
1133:
1125:
1081:
1025:
1023:
1022:
1017:
1012:
1011:
1010:
1009:
1006:
1000:
984:
983:
971:
970:
954:
952:
951:
946:
944:
943:
931:
930:
914:
912:
911:
906:
904:
903:
875:
873:
872:
867:
865:
864:
842:
840:
839:
834:
832:
831:
809:
807:
806:
801:
784:standard gravity
781:
779:
778:
773:
771:
770:
751:specific impulse
748:
746:
745:
740:
738:
737:
734:
715:
713:
712:
707:
705:
704:
695:
694:
691:
682:
681:
678:
662:
660:
659:
654:
649:
647:
646:
637:
636:
627:
619:
618:
609:
608:
605:
596:
594:
593:
584:
583:
574:
566:
565:
562:
538:
536:
535:
530:
516:of the vehicle,
443:
436:
429:
408:Orbital maneuver
357:Payload fraction
337:
318:Lissajous orbits
252:
223:Orbital velocity
170:Hyperbolic orbit
66:Orbital elements
56:
47:
30:
29:
21:
8543:
8542:
8538:
8537:
8536:
8534:
8533:
8532:
8498:
8497:
8496:
8491:
8473:
8391:Escape velocity
8372:
8365:
8346:Rocket equation
8273:
8265:
8259:
8250:
8241:
8232:
8221:
8212:
8203:
8194:
8185:
8181:
8177:
8168:
8157:
8148:
8139:
8130:
8121:
8110:
8101:
8097:
8093:Semi-minor axis
8088:
8084:Semi-major axis
8079:
8070:
8064:
8037:
7959:Areosynchronous
7943:
7937:
7918:Sun-synchronous
7903:Near-equatorial
7847:
7727:
7721:
7674:
7669:
7668:
7659:
7657:
7642:
7641:
7637:
7632:
7617:
7613:
7574:
7570:
7560:Wayback Machine
7550:
7546:
7533:
7526:
7507:
7500:
7492:
7486:
7477:
7467:Wayback Machine
7457:
7453:
7448:
7384:
7379:
7377:
7374:
7346:
7342:
7322:
7304:
7300:
7286:
7283:
7282:
7254:
7250:
7226:
7222:
7205:
7202:
7201:
7182:
7178:
7176:
7173:
7172:
7156:
7155:
7146:
7142:
7130:
7129:
7114:
7110:
7101:
7100:
7087:
7082:
7070:
7066:
7059:
7046:
7044:
7041:
7040:
7028:
6997:
6990:
6986:
6978:
6975:
6974:
6955:
6952:
6951:
6930:
6923:
6919:
6911:
6908:
6907:
6888:
6885:
6884:
6857:
6850:
6846:
6838:
6835:
6834:
6808:
6805:
6804:
6781:
6778:
6777:
6774:
6738:space elevators
6708:
6698:
6690:
6669:
6665:
6663:
6660:
6659:
6636:
6632:
6630:
6627:
6626:
6607:
6603:
6601:
6598:
6597:
6574:
6570:
6564:
6560:
6551:
6547:
6545:
6542:
6541:
6534:
6528:
6512:
6506:
6485:sums linearly.
6427:and pronounced
6406:
6400:
6392:
6364:
6360:
6354:
6350:
6348:
6331:
6327:
6325:
6324:
6320:
6300:
6297:
6296:
6260:
6256:
6255:
6239:
6235:
6234:
6232:
6218:
6215:
6214:
6162:
6158:
6154:
6152:
6151:
6147:
6125:
6121:
6117:
6115:
6111:
6109:
6106:
6105:
6067:
6063:
6059:
6057:
6053:
6052:
6032:
6028:
6024:
6022:
6018:
6017:
6015:
5999:
5997:
5995:
5992:
5991:
5975:
5972:
5971:
5952:
5948:
5942:
5938:
5936:
5934:
5931:
5930:
5907:
5903:
5899:
5894:
5874:
5872:
5865:
5851:
5849:
5842:
5840:
5836:
5835:
5824:
5820:
5814:
5810:
5808:
5806:
5803:
5802:
5782:
5779:
5778:
5761:
5757:
5755:
5752:
5751:
5730:
5726:
5724:
5721:
5720:
5701:
5698:
5697:
5686:
5649:
5645:
5636:
5632:
5630:
5627:
5626:
5604:
5600:
5594:
5590:
5588:
5576:
5572:
5556:
5551:
5539:
5535:
5519:
5511:
5509:
5503:
5498:
5488:
5484:
5463:
5457:
5454:
5453:
5427:
5424:
5423:
5396:
5394:
5385:
5381:
5379:
5376:
5375:
5354:
5343:
5340:
5339:
5319:
5315:
5308:
5300:
5298:
5286:
5275:
5265:
5261:
5247:
5234:
5225:
5221:
5219:
5207:
5196:
5186:
5182:
5171:
5168:
5167:
5142:
5137:
5134:
5133:
5117:
5114:
5113:
5086:
5072:
5046:
5021:
5017:
5015:
5003:
4992:
4982:
4978:
4967:
4964:
4963:
4947:
4944:
4943:
4921:
4900:
4896:
4888:
4885:
4884:
4868:
4865:
4864:
4845:
4839:
4835:
4823:
4819:
4817:
4814:
4813:
4797:
4794:
4793:
4776:
4772:
4767:
4764:
4763:
4746:
4742:
4740:
4737:
4736:
4720:
4717:
4716:
4688:
4684:
4665:
4661:
4659:
4653:
4649:
4638:
4635:
4634:
4614:
4610:
4604:
4590:
4587:
4586:
4564:
4560:
4536:
4527:
4523:
4517:
4513:
4501:
4492:
4487:
4477:
4473:
4461:
4459:
4456:
4455:
4439:
4436:
4435:
4419:
4416:
4415:
4398:
4394:
4392:
4389:
4388:
4368:
4363:
4347:
4345:
4342:
4341:
4324:
4320:
4318:
4315:
4314:
4292:
4289:
4288:
4272:
4269:
4268:
4265:
4235:
4231:
4225:
4221:
4219:
4215:
4203:
4199:
4182:
4178:
4163:
4159:
4152:
4148:
4142:
4138:
4126:
4122:
4113:
4109:
4095:
4092:
4091:
4072:
4066:
4050:
4044:
4010:
4003:
3999:
3995:
3993:
3971:
3966:
3950:
3942:
3940:
3929:
3926:
3925:
3906:
3891:
3876:
3849:
3845:
3839:
3835:
3833:
3829:
3814:
3810:
3799:
3796:
3795:
3778:
3774:
3761:
3748:
3743:
3741:
3738:
3737:
3714:
3703:
3699:
3698:
3676:
3672:
3671:
3661:
3659:
3651:
3648:
3647:
3608:
3604:
3603:
3596:
3592:
3591:
3585:
3582:
3581:
3559:
3556:
3555:
3538:
3534:
3532:
3529:
3528:
3498:
3490:
3480:
3476:
3475:
3474:
3468:
3460:
3459:
3457:
3449:
3445:
3444:
3437:
3433:
3432:
3417:
3414:
3413:
3410:
3405:
3377:
3374:
3373:
3350:
3346:
3337:
3333:
3331:
3328:
3327:
3307:
3303:
3301:
3298:
3297:
3280:
3276:
3274:
3271:
3270:
3245:
3241:
3239:
3236:
3235:
3218:
3214:
3212:
3209:
3208:
3191:
3187:
3185:
3182:
3181:
3151:
3147:
3142:
3135:
3131:
3130:
3126:
3120:
3116:
3107:
3103:
3094:
3090:
3088:
3085:
3084:
3065:
3061:
3056:
3049:
3045:
3039:
3035:
3026:
3022:
3020:
3017:
3016:
2997:
2993:
2988:
2975:
2971:
2965:
2961:
2952:
2948:
2946:
2943:
2942:
2923:
2919:
2913:
2909:
2907:
2895:
2891:
2880:
2877:
2876:
2850:
2848:
2840:
2836:
2835:
2828:
2824:
2823:
2812:
2808:
2807:
2782:
2777:
2768:
2765:
2764:
2743:
2739:
2737:
2734:
2733:
2710:
2702:
2700:
2694:
2690:
2677:
2669:
2667:
2659:
2656:
2655:
2628:
2624:
2618:
2612:
2609:
2608:
2585:
2577:
2575:
2569:
2565:
2552:
2544:
2542:
2530:
2526:
2520:
2514:
2511:
2510:
2491:
2488:
2487:
2456:
2453:
2452:
2424:
2423:
2411:
2408:
2407:
2385:
2384:
2373:
2370:
2369:
2347:
2346:
2344:
2341:
2340:
2323:
2312:
2311:
2310:
2308:
2305:
2304:
2287:
2276:
2275:
2274:
2272:
2269:
2268:
2246:
2245:
2243:
2240:
2239:
2217:
2216:
2192:
2181:
2180:
2179:
2165:
2164:
2149:
2138:
2137:
2136:
2127:
2116:
2115:
2114:
2112:
2109:
2108:
2091:
2080:
2079:
2078:
2064:
2063:
2054:
2043:
2042:
2041:
2039:
2036:
2035:
2013:
2012:
2003:
1992:
1991:
1990:
1981:
1970:
1969:
1968:
1966:
1963:
1962:
1945:
1941:
1939:
1936:
1935:
1918:
1907:
1906:
1905:
1903:
1900:
1899:
1871:
1868:
1867:
1838:
1834:
1832:
1829:
1828:
1805:
1802:
1801:
1785:
1782:
1781:
1761:
1758:
1757:
1740:
1729:
1728:
1727:
1725:
1722:
1721:
1695:
1692:
1691:
1669:
1668:
1651:
1650:
1648:
1645:
1644:
1621:
1618:
1617:
1595:
1594:
1592:
1589:
1588:
1568:
1557:
1556:
1555:
1530:
1529:
1512:
1511:
1510:
1506:
1488:
1484:
1475:
1464:
1463:
1462:
1460:
1457:
1456:
1431:
1428:
1427:
1410:
1399:
1398:
1397:
1395:
1392:
1391:
1369:
1368:
1356:
1345:
1344:
1343:
1341:
1338:
1337:
1315:
1312:
1311:
1294:
1283:
1282:
1281:
1279:
1276:
1275:
1254:
1247:
1236:
1235:
1234:
1225:
1214:
1213:
1212:
1211:
1209:
1194:
1181:
1170:
1169:
1168:
1162:
1156:
1153:
1152:
1135:
1124:
1123:
1122:
1120:
1117:
1116:
1096:
1091:
1075:
1034:
1005:
1001:
996:
989:
985:
979:
975:
966:
962:
960:
957:
956:
939:
935:
926:
922:
920:
917:
916:
899:
895:
893:
890:
889:
886:escape velocity
860:
856:
854:
851:
850:
827:
823:
821:
818:
817:
795:
792:
791:
766:
762:
760:
757:
756:
733:
729:
727:
724:
723:
700:
696:
690:
686:
677:
673:
671:
668:
667:
642:
638:
632:
628:
626:
614:
610:
604:
600:
589:
585:
579:
575:
573:
561:
557:
546:
543:
542:
521:
518:
517:
447:
418:
417:
413:Orbit insertion
403:
395:
394:
380:
372:
371:
347:
339:
335:
328:
327:
323:Lyapunov orbits
314:
313:
297:
287:
286:
262:
254:
250:
243:
242:
228:Surface gravity
203:Escape velocity
193:
185:
184:
165:Parabolic orbit
161:
160:
127:
125:
122:two-body orbits
113:
112:
103:Semi-major axis
68:
58:
54:
28:
23:
22:
15:
12:
11:
5:
8541:
8531:
8530:
8525:
8520:
8515:
8510:
8493:
8492:
8490:
8489:
8487:List of orbits
8478:
8475:
8474:
8472:
8471:
8466:
8461:
8456:
8451:
8446:
8441:
8439:Orbit equation
8436:
8428:
8423:
8418:
8413:
8408:
8403:
8398:
8393:
8388:
8383:
8377:
8375:
8367:
8366:
8364:
8363:
8358:
8353:
8348:
8343:
8338:
8333:
8328:
8323:
8318:
8313:
8311:Gravity assist
8308:
8306:Delta-v budget
8303:
8298:
8293:
8287:
8285:
8279:
8278:
8275:
8274:
8272:
8271:
8263:
8257:
8248:
8239:
8237:Orbital period
8229:
8227:
8223:
8222:
8220:
8219:
8217:True longitude
8210:
8208:Mean longitude
8201:
8192:
8175:
8165:
8163:
8159:
8158:
8156:
8155:
8146:
8137:
8128:
8118:
8116:
8112:
8111:
8109:
8108:
8095:
8086:
8077:
8067:
8065:
8063:
8062:
8059:
8055:
8049:
8043:
8042:
8039:
8038:
8036:
8035:
8034:
8033:
8025:
8024:
8023:
8018:
8013:
8012:
8011:
7998:
7993:
7992:
7991:
7986:
7981:
7976:
7968:
7967:
7966:
7964:Areostationary
7961:
7956:
7947:
7945:
7939:
7938:
7936:
7935:
7933:Very low Earth
7930:
7925:
7920:
7915:
7910:
7905:
7900:
7895:
7890:
7885:
7880:
7875:
7874:
7873:
7868:
7861:Geosynchronous
7857:
7855:
7849:
7848:
7846:
7845:
7843:Transfer orbit
7840:
7839:
7838:
7833:
7823:
7818:
7813:
7808:
7803:
7801:Lagrange point
7798:
7793:
7784:
7779:
7774:
7769:
7760:
7755:
7750:
7744:
7742:
7735:
7729:
7728:
7723:Gravitational
7720:
7719:
7712:
7705:
7697:
7691:
7690:
7685:
7680:
7673:
7672:External links
7670:
7667:
7666:
7635:
7630:
7611:
7568:
7544:
7524:
7498:
7475:
7450:
7449:
7447:
7444:
7443:
7442:
7437:
7432:
7426:
7421:
7416:
7406:
7401:
7396:
7394:Delta-v budget
7390:
7389:
7373:
7370:
7354:
7345:
7341:
7338:
7332:
7329:
7325:
7321:
7318:
7315:
7312:
7303:
7299:
7293:
7290:
7253:
7249:
7246:
7240:
7237:
7234:
7225:
7221:
7218:
7212:
7209:
7181:
7154:
7145:
7141:
7138:
7135:
7133:
7131:
7128:
7125:
7122:
7113:
7109:
7106:
7104:
7102:
7096:
7093:
7090:
7086:
7081:
7078:
7069:
7065:
7062:
7060:
7055:
7052:
7049:
7048:
7027:
7024:
7023:
7022:
7004:
7000:
6996:
6993:
6989:
6985:
6982:
6962:
6959:
6937:
6933:
6929:
6926:
6922:
6918:
6915:
6895:
6892:
6878:
6864:
6860:
6856:
6853:
6849:
6845:
6842:
6815:
6812:
6788:
6785:
6773:
6770:
6707:
6704:
6703:
6702:
6695:Imperial units
6672:
6668:
6657:
6635:
6624:
6606:
6582:
6573:
6567:
6563:
6559:
6550:
6530:Main article:
6527:
6524:
6508:Main article:
6505:
6502:
6468:rocket engines
6402:Main article:
6399:
6393:
6391:
6388:
6375:
6367:
6363:
6357:
6353:
6347:
6344:
6339:
6330:
6323:
6319:
6316:
6313:
6310:
6307:
6304:
6274:
6271:
6266:
6263:
6259:
6253:
6250:
6245:
6242:
6238:
6231:
6228:
6225:
6222:
6186:
6182:
6179:
6176:
6171:
6161:
6157:
6150:
6146:
6143:
6140:
6134:
6124:
6120:
6114:
6085:
6082:
6076:
6066:
6062:
6056:
6050:
6047:
6041:
6031:
6027:
6021:
6014:
6009:
6005:
6002:
5979:
5955:
5951:
5945:
5941:
5906:
5902:
5898:
5892:
5884:
5880:
5877:
5871:
5868:
5861:
5857:
5854:
5848:
5845:
5839:
5834:
5827:
5823:
5817:
5813:
5799:speed of light
5786:
5764:
5760:
5733:
5729:
5708:
5705:
5685:
5682:
5669:
5666:
5663:
5660:
5657:
5652:
5648:
5644:
5639:
5635:
5614:
5607:
5603:
5597:
5593:
5587:
5584:
5575:
5571:
5565:
5562:
5559:
5555:
5550:
5547:
5538:
5534:
5528:
5525:
5522:
5517:
5514:
5506:
5501:
5497:
5487:
5483:
5480:
5477:
5472:
5469:
5466:
5462:
5437:
5434:
5431:
5406:
5402:
5399:
5393:
5388:
5384:
5361:
5358:
5353:
5350:
5347:
5322:
5318:
5314:
5311:
5306:
5303:
5295:
5292:
5289:
5284:
5281:
5278:
5274:
5264:
5260:
5254:
5250:
5246:
5243:
5240:
5237:
5232:
5228:
5224:
5216:
5213:
5210:
5205:
5202:
5199:
5195:
5185:
5181:
5178:
5175:
5155:
5152:
5149:
5145:
5141:
5121:
5096:
5093:
5089:
5085:
5082:
5079:
5075:
5071:
5068:
5065:
5062:
5059:
5056:
5053:
5049:
5045:
5042:
5039:
5036:
5033:
5028:
5024:
5020:
5012:
5009:
5006:
5001:
4998:
4995:
4991:
4981:
4977:
4974:
4971:
4951:
4931:
4928:
4924:
4920:
4917:
4914:
4911:
4908:
4903:
4899:
4895:
4892:
4872:
4852:
4848:
4842:
4838:
4834:
4831:
4826:
4822:
4801:
4779:
4775:
4771:
4749:
4745:
4724:
4702:
4696:
4691:
4687:
4683:
4680:
4677:
4674:
4668:
4664:
4652:
4648:
4645:
4642:
4617:
4613:
4609:
4603:
4600:
4597:
4594:
4572:
4567:
4563:
4559:
4556:
4553:
4550:
4544:
4541:
4535:
4530:
4526:
4520:
4516:
4509:
4506:
4500:
4495:
4486:
4480:
4476:
4469:
4466:
4443:
4423:
4401:
4397:
4371:
4362:
4355:
4352:
4323:
4302:
4299:
4296:
4276:
4264:
4261:
4249:
4245:
4238:
4234:
4228:
4224:
4218:
4214:
4211:
4202:
4195:
4191:
4185:
4181:
4177:
4174:
4171:
4166:
4162:
4158:
4155:
4151:
4141:
4137:
4134:
4129:
4125:
4121:
4116:
4112:
4108:
4105:
4102:
4070:
4064:
4048:
4042:
4022:
4019:
4016:
4013:
4002:
3998:
3992:
3989:
3983:
3980:
3977:
3974:
3970:
3965:
3962:
3956:
3953:
3948:
3945:
3939:
3936:
3904:
3889:
3886:(kg/s) and at
3875:
3872:
3859:
3852:
3848:
3842:
3838:
3832:
3828:
3825:
3813:
3809:
3806:
3803:
3777:
3773:
3768:
3765:
3760:
3754:
3751:
3747:
3720:
3717:
3712:
3706:
3702:
3697:
3694:
3691:
3688:
3685:
3679:
3675:
3670:
3667:
3664:
3655:
3633:
3630:
3627:
3624:
3618:
3611:
3607:
3599:
3595:
3590:
3566:
3563:
3541:
3537:
3514:
3511:
3501:
3497:
3493:
3489:
3483:
3479:
3471:
3467:
3463:
3452:
3448:
3440:
3436:
3431:
3427:
3424:
3421:
3409:
3406:
3404:
3401:
3384:
3381:
3353:
3349:
3345:
3340:
3336:
3306:
3279:
3244:
3221:
3217:
3194:
3190:
3168:
3164:
3161:
3150:
3145:
3141:
3138:
3134:
3129:
3123:
3119:
3115:
3110:
3106:
3102:
3097:
3093:
3064:
3059:
3055:
3052:
3048:
3042:
3038:
3034:
3029:
3025:
2996:
2991:
2984:
2981:
2978:
2974:
2968:
2964:
2960:
2955:
2951:
2926:
2922:
2916:
2912:
2906:
2903:
2894:
2890:
2887:
2884:
2860:
2856:
2853:
2843:
2839:
2831:
2827:
2822:
2815:
2811:
2806:
2803:
2800:
2794:
2791:
2788:
2785:
2780:
2776:
2772:
2742:
2732:Assuming that
2716:
2713:
2708:
2705:
2693:
2689:
2683:
2680:
2675:
2672:
2666:
2663:
2639:
2636:
2631:
2627:
2621:
2617:
2591:
2588:
2583:
2580:
2568:
2564:
2558:
2555:
2550:
2547:
2541:
2538:
2533:
2529:
2523:
2519:
2498:
2495:
2475:
2472:
2469:
2466:
2463:
2460:
2440:
2437:
2431:
2428:
2422:
2418:
2415:
2392:
2389:
2383:
2380:
2377:
2354:
2351:
2319:
2316:
2303:are opposite,
2283:
2280:
2253:
2250:
2224:
2221:
2215:
2212:
2209:
2206:
2203:
2200:
2188:
2185:
2178:
2172:
2169:
2163:
2160:
2157:
2152:
2145:
2142:
2135:
2130:
2123:
2120:
2087:
2084:
2077:
2071:
2068:
2062:
2050:
2047:
2020:
2017:
2011:
1999:
1996:
1989:
1977:
1974:
1944:
1914:
1911:
1896:
1895:
1884:
1881:
1878:
1875:
1854:
1850:
1847:
1844:
1841:
1837:
1826:
1815:
1812:
1809:
1789:
1779:
1768:
1765:
1736:
1733:
1719:
1708:
1705:
1702:
1699:
1676:
1673:
1667:
1664:
1658:
1655:
1642:
1631:
1628:
1625:
1602:
1599:
1564:
1561:
1554:
1551:
1548:
1544:
1537:
1534:
1528:
1525:
1519:
1516:
1509:
1504:
1500:
1497:
1494:
1491:
1487:
1483:
1478:
1471:
1468:
1444:
1441:
1438:
1435:
1413:
1406:
1403:
1376:
1373:
1367:
1364:
1359:
1352:
1349:
1325:
1322:
1319:
1297:
1290:
1287:
1260:
1257:
1250:
1243:
1240:
1233:
1228:
1221:
1218:
1206:
1203:
1200:
1197:
1193:
1189:
1184:
1177:
1174:
1165:
1161:
1138:
1131:
1128:
1095:
1092:
1090:
1087:
1074:
1071:
1056:Hermann Oberth
1052:Robert Goddard
1033:
1030:
1015:
1004:
999:
995:
992:
988:
982:
978:
974:
969:
965:
942:
938:
934:
929:
925:
902:
898:
878:
877:
863:
859:
848:
830:
826:
815:
799:
789:
788:
787:
769:
765:
754:
732:
703:
699:
689:
685:
676:
652:
645:
641:
635:
631:
625:
622:
617:
613:
603:
599:
592:
588:
582:
578:
572:
569:
560:
556:
553:
550:
528:
525:
507:Hermann Oberth
503:Robert Goddard
449:
448:
446:
445:
438:
431:
423:
420:
419:
416:
415:
410:
404:
401:
400:
397:
396:
393:
392:
387:
385:Gravity assist
381:
378:
377:
374:
373:
370:
369:
364:
359:
354:
348:
345:
344:
341:
340:
333:
330:
329:
326:
325:
320:
312:
311:
303:
299:
298:
293:
292:
289:
288:
285:
284:
279:
274:
269:
263:
260:
259:
256:
255:
248:
245:
244:
241:
240:
235:
230:
225:
220:
218:Orbital period
215:
210:
205:
200:
194:
191:
190:
187:
186:
183:
182:
180:Decaying orbit
177:
172:
167:
159:
158:
152:
145:
143:Transfer orbit
141:
140:
139:
137:Elliptic orbit
134:
132:Circular orbit
128:
119:
118:
115:
114:
111:
110:
105:
100:
95:
90:
85:
80:
75:
69:
64:
63:
60:
59:
52:
49:
48:
40:
39:
35:
34:
26:
9:
6:
4:
3:
2:
8540:
8529:
8526:
8524:
8521:
8519:
8516:
8514:
8511:
8509:
8508:Astrodynamics
8506:
8505:
8503:
8488:
8480:
8479:
8476:
8470:
8467:
8465:
8462:
8460:
8457:
8455:
8452:
8450:
8447:
8445:
8442:
8440:
8437:
8435:
8434:-body problem
8433:
8429:
8427:
8424:
8422:
8419:
8417:
8414:
8412:
8409:
8407:
8404:
8402:
8399:
8397:
8394:
8392:
8389:
8387:
8384:
8382:
8379:
8378:
8376:
8374:
8368:
8362:
8359:
8357:
8354:
8352:
8349:
8347:
8344:
8342:
8339:
8337:
8336:Oberth effect
8334:
8332:
8329:
8327:
8324:
8322:
8319:
8317:
8314:
8312:
8309:
8307:
8304:
8302:
8299:
8297:
8294:
8292:
8289:
8288:
8286:
8284:
8280:
8270:
8262:
8258:
8256:
8255:Orbital speed
8249:
8247:
8240:
8238:
8231:
8230:
8228:
8224:
8218:
8211:
8209:
8202:
8200:
8193:
8191:
8176:
8174:
8167:
8166:
8164:
8160:
8154:
8147:
8145:
8138:
8136:
8129:
8127:
8120:
8119:
8117:
8113:
8107:
8096:
8094:
8087:
8085:
8078:
8076:
8069:
8068:
8066:
8060:
8057:
8056:
8053:
8050:
8048:
8044:
8032:
8029:
8028:
8026:
8022:
8019:
8017:
8014:
8010:
8009:Earth's orbit
8007:
8006:
8005:
8002:
8001:
7999:
7997:
7994:
7990:
7987:
7985:
7982:
7980:
7977:
7975:
7972:
7971:
7969:
7965:
7962:
7960:
7957:
7955:
7952:
7951:
7949:
7948:
7946:
7940:
7934:
7931:
7929:
7926:
7924:
7921:
7919:
7916:
7914:
7911:
7909:
7906:
7904:
7901:
7899:
7896:
7894:
7891:
7889:
7886:
7884:
7881:
7879:
7876:
7872:
7869:
7867:
7866:Geostationary
7864:
7863:
7862:
7859:
7858:
7856:
7854:
7850:
7844:
7841:
7837:
7834:
7832:
7829:
7828:
7827:
7824:
7822:
7819:
7817:
7814:
7812:
7809:
7807:
7804:
7802:
7799:
7797:
7794:
7792:
7788:
7785:
7783:
7780:
7778:
7775:
7773:
7770:
7768:
7764:
7761:
7759:
7756:
7754:
7751:
7749:
7746:
7745:
7743:
7739:
7736:
7734:
7730:
7726:
7718:
7713:
7711:
7706:
7704:
7699:
7698:
7695:
7689:
7686:
7684:
7681:
7679:
7676:
7675:
7656:on 2022-03-06
7655:
7651:
7650:
7645:
7639:
7629:
7625:
7621:
7615:
7607:
7603:
7599:
7595:
7591:
7587:
7584:(6): 065001.
7583:
7579:
7572:
7565:
7561:
7557:
7554:
7548:
7540:
7539:
7531:
7529:
7520:
7516:
7512:
7505:
7503:
7491:
7484:
7482:
7480:
7472:
7468:
7464:
7461:
7455:
7451:
7441:
7438:
7436:
7433:
7430:
7427:
7425:
7422:
7420:
7417:
7414:
7410:
7409:Oberth effect
7407:
7405:
7402:
7400:
7397:
7395:
7392:
7391:
7387:
7376:
7369:
7365:
7352:
7343:
7339:
7336:
7327:
7323:
7319:
7313:
7310:
7301:
7297:
7291:
7280:
7278:
7275:A comparable
7273:
7270:
7251:
7247:
7244:
7238:
7235:
7232:
7223:
7219:
7216:
7210:
7199:
7179:
7169:
7152:
7143:
7139:
7136:
7134:
7126:
7123:
7120:
7111:
7107:
7105:
7094:
7091:
7088:
7084:
7079:
7076:
7067:
7063:
7061:
7053:
7038:
7035:
7033:
7032:rocket stages
7020:
7002:
6998:
6994:
6991:
6987:
6983:
6980:
6960:
6935:
6931:
6927:
6924:
6920:
6916:
6913:
6893:
6882:
6879:
6862:
6858:
6854:
6851:
6847:
6843:
6840:
6832:
6829:
6828:
6827:
6813:
6802:
6786:
6769:
6767:
6763:
6758:
6753:
6751:
6747:
6743:
6739:
6735:
6731:
6727:
6723:
6718:
6717:gravitational
6714:
6706:Applicability
6696:
6688:
6670:
6666:
6658:
6655:
6633:
6625:
6604:
6596:
6595:
6594:
6580:
6571:
6565:
6561:
6557:
6548:
6539:
6533:
6523:
6521:
6517:
6511:
6504:Mass fraction
6501:
6499:
6495:
6494:porkchop plot
6491:
6486:
6484:
6479:
6477:
6473:
6469:
6465:
6460:
6458:
6454:
6450:
6446:
6442:
6438:
6434:
6431:, as used in
6430:
6426:
6425:
6419:
6415:
6411:
6405:
6398:
6387:
6373:
6365:
6361:
6355:
6351:
6345:
6342:
6337:
6328:
6321:
6317:
6314:
6311:
6308:
6305:
6294:
6291:
6272:
6269:
6264:
6261:
6257:
6251:
6248:
6243:
6240:
6236:
6229:
6226:
6223:
6220:
6212:
6208:
6205:
6201:
6184:
6180:
6177:
6174:
6169:
6159:
6155:
6148:
6144:
6141:
6138:
6132:
6122:
6118:
6112:
6104:
6099:
6083:
6080:
6074:
6064:
6060:
6054:
6048:
6045:
6039:
6029:
6025:
6019:
6012:
6007:
6003:
5977:
5953:
5949:
5943:
5939:
5927:
5904:
5900:
5896:
5890:
5882:
5878:
5869:
5866:
5859:
5855:
5846:
5843:
5837:
5832:
5825:
5821:
5815:
5811:
5800:
5784:
5762:
5758:
5749:
5731:
5727:
5706:
5695:
5691:
5681:
5664:
5661:
5658:
5650:
5646:
5642:
5637:
5633:
5612:
5605:
5601:
5595:
5591:
5585:
5582:
5573:
5569:
5563:
5560:
5557:
5553:
5548:
5545:
5536:
5532:
5526:
5523:
5520:
5515:
5512:
5504:
5499:
5495:
5485:
5481:
5478:
5464:
5451:
5429:
5420:
5404:
5400:
5397:
5391:
5386:
5382:
5359:
5356:
5351:
5348:
5320:
5316:
5312:
5309:
5304:
5293:
5290:
5287:
5282:
5279:
5276:
5272:
5262:
5258:
5252:
5248:
5244:
5241:
5238:
5235:
5230:
5226:
5222:
5214:
5211:
5208:
5203:
5200:
5197:
5193:
5183:
5179:
5176:
5153:
5150:
5147:
5143:
5139:
5119:
5110:
5091:
5087:
5083:
5080:
5077:
5073:
5069:
5066:
5063:
5060:
5051:
5047:
5043:
5040:
5037:
5034:
5026:
5022:
5018:
5010:
5007:
5004:
4999:
4996:
4993:
4989:
4979:
4975:
4972:
4949:
4926:
4922:
4918:
4915:
4912:
4909:
4901:
4897:
4893:
4890:
4870:
4850:
4846:
4840:
4836:
4832:
4829:
4824:
4820:
4799:
4777:
4773:
4769:
4747:
4743:
4722:
4713:
4700:
4689:
4685:
4681:
4678:
4672:
4666:
4662:
4650:
4646:
4643:
4615:
4611:
4607:
4601:
4595:
4592:
4583:
4570:
4565:
4557:
4548:
4542:
4539:
4533:
4528:
4524:
4518:
4514:
4507:
4504:
4498:
4493:
4484:
4478:
4474:
4467:
4464:
4441:
4421:
4399:
4395:
4385:
4369:
4360:
4353:
4350:
4321:
4294:
4274:
4260:
4247:
4243:
4236:
4232:
4226:
4222:
4216:
4212:
4209:
4200:
4193:
4189:
4183:
4179:
4175:
4172:
4169:
4164:
4160:
4156:
4153:
4149:
4139:
4135:
4132:
4127:
4123:
4119:
4114:
4110:
4106:
4103:
4089:
4085:
4081:
4077:
4073:
4063:
4059:
4055:
4051:
4041:
4036:
4017:
4011:
4000:
3996:
3990:
3987:
3978:
3972:
3968:
3963:
3960:
3954:
3951:
3946:
3943:
3937:
3934:
3923:
3919:
3915:
3911:
3907:
3900:
3896:
3892:
3885:
3881:
3871:
3857:
3850:
3846:
3840:
3836:
3830:
3826:
3823:
3811:
3807:
3804:
3775:
3771:
3766:
3763:
3758:
3752:
3745:
3734:
3718:
3704:
3700:
3692:
3689:
3686:
3677:
3673:
3665:
3662:
3653:
3644:
3631:
3628:
3625:
3622:
3616:
3609:
3605:
3597:
3593:
3588:
3578:
3564:
3539:
3535:
3525:
3512:
3509:
3499:
3491:
3487:
3481:
3477:
3465:
3450:
3446:
3438:
3434:
3429:
3425:
3422:
3408:Impulse-based
3400:
3398:
3382:
3371:
3369:
3366:is the total
3351:
3347:
3343:
3338:
3334:
3324:
3304:
3277:
3268:
3264:
3242:
3219:
3215:
3192:
3188:
3166:
3162:
3159:
3148:
3143:
3139:
3132:
3127:
3121:
3117:
3113:
3108:
3104:
3100:
3095:
3091:
3062:
3057:
3053:
3046:
3040:
3036:
3032:
3027:
3023:
2994:
2989:
2982:
2976:
2972:
2966:
2962:
2958:
2953:
2949:
2924:
2920:
2914:
2910:
2904:
2901:
2892:
2888:
2885:
2873:
2858:
2854:
2851:
2841:
2837:
2829:
2825:
2820:
2813:
2809:
2804:
2801:
2798:
2792:
2786:
2783:
2778:
2774:
2770:
2762:
2740:
2730:
2714:
2711:
2706:
2703:
2691:
2687:
2681:
2678:
2673:
2670:
2664:
2661:
2653:
2637:
2634:
2629:
2625:
2619:
2615:
2605:
2589:
2586:
2581:
2578:
2566:
2562:
2556:
2553:
2548:
2545:
2539:
2536:
2531:
2527:
2521:
2517:
2496:
2473:
2467:
2464:
2461:
2458:
2451:), and using
2438:
2426:
2420:
2416:
2413:
2387:
2378:
2349:
2314:
2278:
2248:
2219:
2210:
2204:
2201:
2183:
2176:
2167:
2158:
2155:
2150:
2140:
2133:
2128:
2118:
2082:
2075:
2066:
2060:
2045:
2015:
2009:
1994:
1987:
1972:
1942:
1909:
1882:
1876:
1873:
1852:
1848:
1842:
1839:
1835:
1827:
1813:
1810:
1807:
1787:
1780:
1766:
1731:
1720:
1706:
1700:
1697:
1671:
1662:
1653:
1643:
1629:
1626:
1623:
1597:
1587:
1586:
1585:
1559:
1552:
1546:
1542:
1532:
1523:
1514:
1507:
1502:
1498:
1492:
1489:
1485:
1481:
1476:
1466:
1442:
1436:
1433:
1411:
1401:
1371:
1365:
1362:
1357:
1347:
1323:
1320:
1317:
1295:
1285:
1258:
1248:
1238:
1231:
1226:
1216:
1204:
1198:
1187:
1182:
1172:
1163:
1159:
1136:
1126:
1114:
1110:
1103:
1099:
1086:
1082:
1080:
1070:
1068:
1064:
1059:
1057:
1053:
1048:
1046:
1045:William Moore
1041:
1039:
1029:
1026:
1013:
1002:
997:
993:
986:
980:
976:
972:
967:
963:
940:
936:
932:
927:
923:
900:
896:
887:
883:
882:orbital speed
861:
857:
849:
846:
828:
824:
816:
813:
797:
790:
785:
767:
763:
755:
752:
730:
722:
721:
719:
701:
697:
687:
683:
674:
666:
665:
664:
650:
643:
639:
633:
629:
623:
620:
615:
611:
601:
597:
590:
586:
580:
576:
570:
567:
558:
554:
551:
540:
526:
515:
510:
508:
504:
500:
499:William Moore
496:
492:
488:
484:
480:
476:
472:
464:
460:
455:
444:
439:
437:
432:
430:
425:
424:
422:
421:
414:
411:
409:
406:
405:
399:
398:
391:
390:Oberth effect
388:
386:
383:
382:
376:
375:
368:
365:
363:
360:
358:
355:
353:
350:
349:
343:
342:
338:
332:
331:
324:
321:
319:
316:
315:
309:
305:
304:
302:
296:
295:N-body orbits
291:
290:
283:
280:
278:
277:Perturbations
275:
273:
270:
268:
265:
264:
258:
257:
253:
247:
246:
239:
236:
234:
231:
229:
226:
224:
221:
219:
216:
214:
211:
209:
206:
204:
201:
199:
196:
195:
189:
188:
181:
178:
176:
173:
171:
168:
166:
163:
162:
156:
153:
151:
147:
146:
144:
138:
135:
133:
130:
129:
123:
117:
116:
109:
106:
104:
101:
99:
98:Orbital nodes
96:
94:
91:
89:
86:
84:
81:
79:
76:
74:
71:
70:
67:
62:
61:
57:
51:
50:
46:
42:
41:
38:Astrodynamics
37:
36:
32:
31:
19:
8449:Perturbation
8431:
8406:Ground track
8345:
8316:Gravity turn
8267:
8260:
8253:
8244:
8235:
8215:
8206:
8197:
8190:True anomaly
8188:
8173:Mean anomaly
8171:
8151:
8142:
8133:
8124:
8104:
8091:
8082:
8075:Eccentricity
8073:
8031:Lunar cycler
8004:Heliocentric
7944:other points
7893:Medium Earth
7791:Non-inclined
7658:. Retrieved
7654:the original
7647:
7638:
7627:
7623:
7614:
7581:
7577:
7571:
7547:
7537:
7518:
7514:
7454:
7413:gravity well
7399:Jeep problem
7366:
7281:
7274:
7271:
7200:
7170:
7039:
7036:
7029:
7018:
6803:, including
6775:
6754:
6742:launch loops
6734:gun launches
6709:
6535:
6513:
6497:
6489:
6487:
6482:
6480:
6463:
6461:
6452:
6428:
6423:
6421:
6412:(literally "
6409:
6407:
6396:
6289:
6203:
6100:
5928:
5687:
5421:
5111:
4714:
4584:
4386:
4266:
4087:
4083:
4079:
4075:
4068:
4061:
4057:
4053:
4046:
4039:
4037:
3921:
3917:
3913:
3902:
3898:
3894:
3887:
3883:
3877:
3735:
3645:
3579:
3526:
3411:
3372:
3368:working mass
3325:
3323:hypothesis.
2874:
2731:
2606:
1897:
1111:
1108:
1097:
1083:
1076:
1067:space travel
1060:
1049:
1042:
1035:
1027:
879:
541:
511:
474:
470:
468:
366:
175:Radial orbit
126:eccentricity
108:True anomaly
93:Mean anomaly
83:Eccentricity
8411:Hill sphere
8246:Mean motion
8126:Inclination
8115:Orientation
8016:Mars cycler
7954:Areocentric
7826:Synchronous
6750:light sails
6730:aerobraking
6713:aerodynamic
5801:in vacuum:
5450:Riemann sum
308:Halo orbits
272:Hill sphere
88:Inclination
8502:Categories
8351:Rendezvous
8047:Parameters
7883:High Earth
7853:Geocentric
7806:Osculating
7763:Elliptical
7660:2016-04-18
7521:: 276–285.
7446:References
7404:Mass ratio
6476:propellant
6443:. It is a
3326:The value
1089:Derivation
845:propellant
459:mass ratio
352:Mass ratio
267:Barycenter
8396:Ephemeris
8373:mechanics
8283:Maneuvers
8226:Variation
7989:Libration
7984:Lissajous
7888:Low Earth
7878:Graveyard
7777:Horseshoe
7606:202130640
7314:
7289:Δ
7236:
7208:Δ
7124:
7092:−
7080:
7051:Δ
6992:−
6984:−
6958:Δ
6925:−
6917:−
6891:Δ
6852:−
6844:−
6811:Δ
6784:Δ
6689:, 9.80665
6429:delta-vee
6346:
6318:
6303:Δ
6249:−
6224:
6178:
6145:
6046:−
6001:Δ
5876:Δ
5870:−
5853:Δ
5746:) in the
5704:Δ
5665:ϕ
5662:−
5586:
5564:ϕ
5561:−
5549:
5524:−
5505:ϕ
5496:∫
5476:Δ
5471:∞
5468:→
5436:∞
5433:→
5401:ϕ
5357:ϕ
5346:Δ
5313:−
5302:Δ
5273:∑
5245:ϕ
5239:−
5223:ϕ
5194:∑
5180:≈
5174:Δ
5151:≪
5140:ϕ
5084:ϕ
5070:ϕ
5064:−
5044:ϕ
5038:−
5019:ϕ
4990:∑
4970:Δ
4919:ϕ
4913:−
4833:ϕ
4770:ϕ
4723:ϕ
4641:Δ
4599:Δ
4555:Δ
4301:∞
4298:→
4213:
4176:
4170:−
4157:
4136:−
4120:−
4101:Δ
4088:R = dm/dt
3991:−
3964:−
3827:
3802:Δ
3750:Δ
3716:Δ
3693:
3687:−
3666:
3589:∫
3562:Δ
3496:Δ
3488:−
3430:∫
3420:Δ
3380:Δ
3344:−
3160:−
3137:Δ
3101:−
3051:Δ
2980:Δ
2977:−
2905:
2883:Δ
2821:∫
2790:Δ
2775:∫
2771:−
2662:−
2616:∑
2518:∑
2494:Δ
2471:Δ
2468:−
2436:→
2430:→
2391:→
2382:Δ
2376:Δ
2353:→
2318:→
2282:→
2252:→
2223:→
2214:Δ
2208:Δ
2205:−
2199:Δ
2187:→
2171:→
2162:Δ
2144:→
2134:−
2122:→
2086:→
2070:→
2049:→
2019:→
2010:−
1998:→
1976:→
1913:→
1880:Δ
1846:Δ
1843:−
1764:Δ
1735:→
1704:Δ
1675:→
1666:Δ
1657:→
1601:→
1563:→
1550:Δ
1536:→
1527:Δ
1518:→
1496:Δ
1493:−
1470:→
1440:Δ
1405:→
1375:→
1351:→
1289:→
1256:Δ
1242:→
1232:−
1220:→
1202:→
1196:Δ
1176:→
1160:∑
1130:→
1050:American
991:Δ
933:−
814:function;
624:
571:
549:Δ
524:Δ
192:Equations
120:Types of
8162:Position
7787:Inclined
7758:Circular
7649:NASA.gov
7556:Archived
7463:Archived
7372:See also
6833:rocket:
6772:Examples
6728:such as
6693:m/s (in
6418:velocity
6204:see also
6103:identity
5929:Writing
1063:calculus
514:velocity
487:velocity
8371:Orbital
8341:Phasing
8301:Delta-v
8106:Apsides
8100:,
7898:Molniya
7816:Parking
7753:Capture
7741:General
7586:Bibcode
6722:payload
6685:is the
6437:impulse
6404:Delta-v
5696:, with
3397:delta v
3267:gravity
1032:History
810:is the
749:is the
716:is the
663:where:
8027:Other
7928:Tundra
7796:Kepler
7772:Escape
7725:orbits
7604:
7334:
7295:
7242:
7214:
7057:
7026:Stages
6701:ft/s).
6699:
6697:32.174
6691:
6593:where
6472:thrust
6462:Delta-
6445:scalar
6414:change
6408:Delta-
6395:Delta-
5338:where
4197:
4098:
3932:
3821:
3657:
3620:
3507:
3180:where
2986:
2654:) and
2034:thus,
1274:where
483:thrust
479:rocket
8269:Epoch
8058:Shape
7996:Lunar
7950:Mars
7942:About
7913:Polar
7733:Types
7602:S2CID
7564:RARed
7562:in a
7493:(PDF)
7471:RARed
7469:in a
6449:speed
5448:this
4792:into
473:, or
465:ratio
73:Apsis
8061:Size
8000:Sun
7979:Halo
7831:semi
7566:PDF)
7553:here
7473:PDF)
7460:here
7340:2.19
7328:11.2
7277:SSTO
7248:4.83
7140:1.61
6315:tanh
6221:tanh
5374:and
4715:Let
2267:and
1961:by:
1390:and
469:The
7836:sub
7748:Box
7594:doi
7320:100
7089:100
7085:100
7019:all
7003:4.5
6995:4.7
6936:4.5
6928:5.0
6863:4.5
6855:9.7
6801:LEO
6748:or
6715:or
6654:m/s
6514:In
6453:not
6416:in
6290:see
6142:exp
5970:as
5688:If
5578:eff
5541:eff
5490:eff
5461:lim
5422:As
5267:eff
5188:eff
4984:eff
4655:eff
4489:eff
4365:eff
4326:eff
4082:to
4060:= (
3816:exh
3780:exh
3083:or
3015:or
2238:If
1192:lim
884:or
782:is
124:by
8504::
8184:,
8180:,
7789:/
7765:/
7646:.
7600:.
7592:.
7582:54
7580:.
7527:^
7519:27
7517:.
7513:.
7501:^
7478:^
7311:ln
7233:ln
7121:ln
7095:80
7077:ln
6752:.
6744:,
6740:,
6736:,
6732:,
6609:sp
6576:sp
6343:ln
6202:;
6175:ln
5680:.
5583:ln
5546:ln
5419:.
4384:.
4210:ln
4173:ln
4154:ln
4074:)/
4067:–
4045:–
3924::
3901:×
3824:ln
3690:ln
3663:ln
2902:ln
2368:,
1455::
1336::
1069:.
955::
798:ln
735:sp
720:;
692:sp
621:ln
606:sp
568:ln
8432:n
8264:0
8261:t
8251:v
8242:n
8233:T
8213:l
8204:L
8195:E
8186:f
8182:θ
8178:ν
8169:M
8149:ϖ
8140:ω
8131:Ω
8122:i
8102:q
8098:Q
8089:b
8080:a
8071:e
7716:e
7709:t
7702:v
7663:.
7631:e
7628:v
7624:R
7608:.
7596::
7588::
7495:.
7353:.
7348:e
7344:v
7337:=
7331:)
7324:/
7317:(
7306:e
7302:v
7298:=
7292:v
7256:e
7252:v
7245:=
7239:5
7228:e
7224:v
7220:3
7217:=
7211:v
7184:e
7180:v
7153:.
7148:e
7144:v
7137:=
7127:5
7116:e
7112:v
7108:=
7072:e
7068:v
7064:=
7054:v
6999:/
6988:e
6981:1
6961:v
6932:/
6921:e
6914:1
6894:v
6859:/
6848:e
6841:1
6814:v
6787:v
6671:0
6667:g
6638:e
6634:v
6605:I
6581:,
6572:I
6566:0
6562:g
6558:=
6553:e
6549:v
6498:v
6490:v
6483:v
6464:v
6424:v
6422:Δ
6410:v
6397:v
6374:)
6366:1
6362:m
6356:0
6352:m
6338:c
6333:e
6329:v
6322:(
6312:c
6309:=
6306:v
6288:(
6273:1
6270:+
6265:x
6262:2
6258:e
6252:1
6244:x
6241:2
6237:e
6230:=
6227:x
6185:]
6181:R
6170:c
6164:e
6160:v
6156:2
6149:[
6139:=
6133:c
6127:e
6123:v
6119:2
6113:R
6084:1
6081:+
6075:c
6069:e
6065:v
6061:2
6055:R
6049:1
6040:c
6034:e
6030:v
6026:2
6020:R
6013:=
6008:c
6004:v
5978:R
5954:1
5950:m
5944:0
5940:m
5909:e
5905:v
5901:2
5897:c
5891:]
5883:c
5879:v
5867:1
5860:c
5856:v
5847:+
5844:1
5838:[
5833:=
5826:1
5822:m
5816:0
5812:m
5785:c
5763:0
5759:m
5732:1
5728:m
5707:v
5668:)
5659:1
5656:(
5651:0
5647:m
5643:=
5638:f
5634:m
5613:,
5606:f
5602:m
5596:0
5592:m
5574:v
5570:=
5558:1
5554:1
5537:v
5533:=
5527:x
5521:1
5516:x
5513:d
5500:0
5486:v
5482:=
5479:v
5465:N
5430:N
5405:N
5398:j
5392:=
5387:j
5383:x
5360:N
5352:=
5349:x
5321:j
5317:x
5310:1
5305:x
5294:N
5291:=
5288:j
5283:1
5280:=
5277:j
5263:v
5259:=
5253:N
5249:/
5242:j
5236:1
5231:N
5227:/
5215:N
5212:=
5209:j
5204:1
5201:=
5198:j
5184:v
5177:v
5154:1
5148:N
5144:/
5120:N
5095:)
5092:N
5088:/
5081:+
5078:N
5074:/
5067:j
5061:1
5058:(
5055:)
5052:N
5048:/
5041:j
5035:1
5032:(
5027:N
5023:/
5011:N
5008:=
5005:j
5000:1
4997:=
4994:j
4980:v
4976:=
4973:v
4950:j
4930:)
4927:N
4923:/
4916:j
4910:1
4907:(
4902:0
4898:m
4894:=
4891:m
4871:j
4851:N
4847:/
4841:0
4837:m
4830:=
4825:p
4821:m
4800:N
4778:0
4774:m
4748:0
4744:m
4701:.
4695:)
4690:p
4686:m
4682:+
4679:m
4676:(
4673:m
4667:p
4663:m
4651:v
4647:=
4644:v
4616:p
4612:m
4608:m
4602:v
4596:=
4593:u
4571:.
4566:2
4562:)
4558:v
4552:(
4549:m
4543:2
4540:1
4534:+
4529:2
4525:u
4519:p
4515:m
4508:2
4505:1
4499:=
4494:2
4485:v
4479:p
4475:m
4468:2
4465:1
4442:m
4422:u
4400:p
4396:m
4370:2
4361:v
4354:2
4351:1
4322:v
4295:N
4275:N
4248:.
4244:)
4237:f
4233:m
4227:0
4223:m
4217:(
4205:e
4201:v
4194:=
4190:]
4184:0
4180:m
4165:f
4161:m
4150:[
4144:e
4140:v
4133:=
4128:0
4124:v
4115:f
4111:v
4107:=
4104:v
4084:T
4080:0
4076:R
4071:f
4069:m
4065:0
4062:m
4058:T
4054:R
4049:f
4047:m
4043:0
4040:m
4021:)
4018:t
4015:(
4012:m
4005:e
4001:v
3997:R
3988:=
3982:)
3979:t
3976:(
3973:m
3969:F
3961:=
3955:t
3952:d
3947:v
3944:d
3938:=
3935:a
3922:m
3918:F
3914:t
3905:e
3903:v
3899:R
3895:F
3890:e
3858:)
3851:f
3847:m
3841:0
3837:m
3831:(
3812:V
3808:=
3805:v
3776:V
3772:=
3767:p
3764:F
3759:=
3753:m
3746:J
3719:m
3711:)
3705:f
3701:m
3696:(
3684:)
3678:0
3674:m
3669:(
3654:J
3632:J
3629:=
3626:t
3623:d
3617:F
3610:f
3606:t
3598:0
3594:t
3565:m
3540:0
3536:m
3513:t
3510:d
3500:m
3492:t
3482:0
3478:m
3470:|
3466:T
3462:|
3451:f
3447:t
3439:0
3435:t
3426:=
3423:v
3395:(
3383:V
3352:f
3348:m
3339:0
3335:m
3309:e
3305:v
3282:e
3278:v
3247:e
3243:v
3220:f
3216:m
3193:0
3189:m
3167:)
3163:1
3153:e
3149:v
3144:/
3140:V
3133:e
3128:(
3122:f
3118:m
3114:=
3109:f
3105:m
3096:0
3092:m
3067:e
3063:v
3058:/
3054:V
3047:e
3041:f
3037:m
3033:=
3028:0
3024:m
2999:e
2995:v
2990:/
2983:V
2973:e
2967:0
2963:m
2959:=
2954:f
2950:m
2925:f
2921:m
2915:0
2911:m
2897:e
2893:v
2889:=
2886:V
2859:m
2855:m
2852:d
2842:f
2838:m
2830:0
2826:m
2814:e
2810:v
2805:=
2802:V
2799:d
2793:V
2787:+
2784:V
2779:V
2745:e
2741:v
2715:t
2712:d
2707:m
2704:d
2696:e
2692:v
2688:=
2682:t
2679:d
2674:V
2671:d
2665:m
2650:(
2638:0
2635:=
2630:i
2626:F
2620:i
2590:t
2587:d
2582:m
2579:d
2571:e
2567:v
2563:+
2557:t
2554:d
2549:V
2546:d
2540:m
2537:=
2532:i
2528:F
2522:i
2497:m
2474:m
2465:=
2462:m
2459:d
2439:0
2427:v
2421:d
2417:m
2414:d
2388:V
2379:m
2350:V
2325:i
2315:F
2289:e
2279:v
2249:V
2220:V
2211:m
2202:m
2194:e
2184:v
2177:+
2168:V
2159:m
2156:=
2151:1
2141:P
2129:2
2119:P
2093:e
2083:v
2076:+
2067:V
2061:=
2056:e
2046:V
2016:V
2005:e
1995:V
1988:=
1983:e
1973:v
1947:e
1943:v
1920:e
1910:V
1883:t
1877:=
1874:t
1853:)
1849:m
1840:m
1836:(
1814:0
1811:=
1808:t
1788:m
1767:t
1742:e
1732:V
1707:t
1701:=
1698:t
1672:V
1663:+
1654:V
1630:0
1627:=
1624:t
1598:V
1570:e
1560:V
1553:m
1547:+
1543:)
1533:V
1524:+
1515:V
1508:(
1503:)
1499:m
1490:m
1486:(
1482:=
1477:2
1467:P
1443:t
1437:=
1434:t
1412:2
1402:P
1372:V
1366:m
1363:=
1358:1
1348:P
1324:0
1321:=
1318:t
1296:1
1286:P
1259:t
1249:1
1239:P
1227:2
1217:P
1205:0
1199:t
1188:=
1183:i
1173:F
1164:i
1137:i
1127:F
1014:.
1007:e
1003:v
998:/
994:v
987:e
981:f
977:m
973:=
968:0
964:m
941:f
937:m
928:0
924:m
901:f
897:m
862:f
858:m
829:0
825:m
786:;
768:0
764:g
731:I
702:0
698:g
688:I
684:=
679:e
675:v
651:,
644:f
640:m
634:0
630:m
616:0
612:g
602:I
598:=
591:f
587:m
581:0
577:m
563:e
559:v
555:=
552:v
527:v
442:e
435:t
428:v
310:)
306:(
157:)
148:(
20:)
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