3411:) 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:
6530:, 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
454:
45:
6776:, 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
8495:
7392:
6962:= 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
7046:, 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.
5635:
7029:= 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
6731:
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
6722:
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
5348:
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).
7179:
1591:
2248:
7379:
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.
5467:
1275:
5120:
4593:
6109:
4270:
661:
6397:
5181:
7054:
2883:
4046:
3535:
3881:
2615:
4723:
1464:
3190:
7375:
5937:
2119:
2114:
2041:
6771:
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
3804:
7280:
3744:
2951:
2740:
1154:
4977:
4469:
3025:
6005:
4105:
1079:
1024:
6208:
3093:
544:
7059:
6298:
6603:
714:
4394:
1697:
3654:
4643:
1391:
2416:
5384:
6310:
1873:
5429:
2349:
2313:
1941:
1763:
1430:
5690:
5458:
2461:
5980:
4952:
4873:
5630:{\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}}},}
7027:
6960:
6887:
4350:
2778:
1311:
1149:
6633:
2660:
747:
7208:
6662:
5176:
4323:
3939:
3427:
3333:
3306:
3271:
2769:
2496:
1968:
3376:
953:
4802:
3424:
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:
2378:
2277:
1902:
1726:
1623:
1459:
6983:
6916:
6836:
6809:
5729:
3587:
3405:
2519:
1786:
537:
6736:
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
3809:
6695:
5787:
5756:
4772:
4745:
4424:
3564:
3308:
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
3244:
3217:
913:
874:
841:
780:
2524:
808:
1833:
1649:
1337:
4648:
6000:
5807:
5142:
4972:
4893:
4822:
4464:
4444:
4297:
1807:
3098:
7655:
7296:
7049:
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
5816:
2046:
1973:
3748:
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,
6777:
3751:
6780:, more complicated analysis based on the propagation of the spacecraft's state vector and the integration of thrust are used to predict orbital motion.
5343:{\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}}}}
7215:
3661:
2890:
2669:
7174:{\displaystyle {\begin{aligned}\Delta v\ &=v_{\text{e}}\ln {100 \over 100-80}\\&=v_{\text{e}}\ln 5\\&=1.61v_{\text{e}}.\\\end{aligned}}}
8524:
440:
1078:
7567:
7474:
8372:
2956:
1054:
independently developed the equation in 1912 when he began his research to improve rocket engines for possible space flight. German engineer
1044:
498:
6889:= 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:
8432:
3030:
212:
281:
6555:
1065:
exercise, Tsiolkovsky is honored as being the first to apply it to the question of whether rockets could achieve speeds necessary for
1586:{\displaystyle {\vec {P}}_{\Delta t}=\left(m-\Delta m\right)\left({\vec {V}}+\Delta {\vec {V}}\right)+\Delta m{\vec {V}}_{\text{e}}}
4279:
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
8427:
8307:
7726:
3920:. The rocket is subject to a constant force, but its total mass is decreasing steadily because it is expelling gas. According to
3595:
1112:
7501:
6723:
only accounts for the reaction force from the rocket engine; it does not include other forces that may act on a rocket, such as
1342:
8529:
8392:
8146:
2243:{\displaystyle {\vec {P}}_{\Delta t}-{\vec {P}}_{0}=m\Delta {\vec {V}}+{\vec {v}}_{\text{e}}\Delta m-\Delta m\Delta {\vec {V}}}
8534:
8465:
8104:
8095:
7832:
7291:
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
3592:
and realising that the integral of a resultant force over time is total impulse, assuming thrust is the only force involved,
102:
5731:
again standing for the rocket's final velocity (after expelling all its reaction mass and being reduced to a rest mass of
7451:
6119:
8412:
7882:
6444:
6228:
5759:
433:
1270:{\displaystyle \sum _{i}{\vec {F}}_{i}=\lim _{\Delta t\to 0}{\frac {{\vec {P}}_{\Delta t}-{\vec {P}}_{0}}{\Delta t}}}
669:
497:, who independently derived it and published it in 1903, although it had been independently derived and published by
8539:
8337:
8164:
6543:
5115:{\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)}}}}
4355:
3921:
3891:
717:
462:
7665:
2772:
1655:
1101:
8475:
4588:{\displaystyle {\tfrac {1}{2}}m_{p}v_{\text{eff}}^{2}={\tfrac {1}{2}}m_{p}u^{2}+{\tfrac {1}{2}}m(\Delta v)^{2}.}
8460:
7985:
6521:
4600:
2383:
490:
361:
276:
7550:
A Treatise on the Motion of
Rockets: to which is added, an Essay on Naval Gunnery, in theory and practice, etc
6104:{\displaystyle {\frac {\Delta v}{c}}={\frac {R^{\frac {2v_{\text{e}}}{c}}-1}{R^{\frac {2v_{\text{e}}}{c}}+1}}}
4265:{\displaystyle ~\Delta v=v_{f}-v_{0}=-v_{\text{e}}\left=~v_{\text{e}}\ln \left({\frac {m_{0}}{m_{f}}}\right).}
1109:
In the following derivation, "the rocket" is taken to mean "the rocket and all of its unexpended propellant".
5353:
1839:
656:{\displaystyle \Delta v=v_{\text{e}}\ln {\frac {m_{0}}{m_{f}}}=I_{\text{sp}}g_{0}\ln {\frac {m_{0}}{m_{f}}},}
8470:
7778:
7435:
6451:
that is needed to perform a maneuver such as launching from, or landing on a planet or moon, or an in-space
232:
5389:
2318:
2282:
1910:
1732:
1396:
8519:
8332:
7934:
7854:
7842:
6737:
6114:
5640:
5437:
2421:
1037:
494:
426:
149:
4898:
880:
Given the effective exhaust velocity determined by the rocket motor's design, the desired delta-v (e.g.,
8455:
8397:
8367:
8155:
8032:
8000:
7970:
7929:
7914:
7793:
7446:
7288:
6842:
6527:
5944:
4827:
334:
169:
77:
7631:
477:
is a mathematical equation that describes the motion of vehicles that follow the basic principle of a
8480:
8302:
8086:
7975:
7944:
7872:
7847:
7822:
7783:
7764:
7719:
7597:
7564:
7471:
6456:
6211:
207:
164:
154:
82:
6988:
6921:
6848:
4328:
1280:
1118:
8342:
8137:
7877:
7699:
7430:
6611:
6304:
5705:
725:
249:
87:
7186:
6640:
5147:
4302:
3311:
3284:
3249:
2747:
2622:
2466:
1946:
1151:) to the change in linear momentum of the whole system (including rocket and exhaust) as follows:
8015:
7904:
7802:
7613:
7563:К. Ціолковскій, Изслѣдованіе мировыхъ пространствъ реактивными приборами, 1903 (available online
7470:К. Ціолковскій, Изслѣдованіе мировыхъ пространствъ реактивными приборами, 1903 (available online
7397:
7043:
6892:
5701:
3341:
918:
322:
197:
6668:, which is the same as the effective exhaust velocity measured in m/s (or ft/s if g is in ft/s),
6392:{\displaystyle \Delta v=c\tanh \left({\frac {v_{\text{e}}}{c}}\ln {\frac {m_{0}}{m_{1}}}\right)}
4777:
2354:
2253:
1878:
1702:
1599:
1435:
8382:
8280:
8210:
7965:
7919:
7837:
7575:
7482:
7440:
6965:
6898:
6818:
6791:
6757:
6724:
6487:
6468:
6448:
6218:
5711:
3569:
3387:
2775:), so it is not subject to integration, then the above equation may be integrated as follows:
2501:
1768:
811:
519:
453:
237:
174:
53:
7694:
1058:
independently derived the equation about 1920 as he studied the feasibility of space travel.
8362:
8294:
8058:
8020:
7894:
7864:
7817:
7605:
6773:
6768:
6733:
6698:
6665:
6549:
6531:
6452:
3274:
783:
750:
407:
356:
121:
65:
6673:
5765:
5734:
4750:
4730:
4402:
3542:
3222:
3195:
891:
852:
819:
758:
8402:
7995:
7899:
7889:
7788:
7712:
7571:
7478:
6812:
3894:). From the moment its engine is started (clock set to 0) the rocket expels gas mass at a
885:
793:
412:
317:
227:
202:
4466:, the amount of energy converted to increase the rocket's and pellet's kinetic energy is
1812:
1628:
1316:
17:
7601:
4090:
to burn all this fuel. Integrating both sides of the equation with respect to time from
8498:
8450:
8442:
8437:
8322:
8317:
8248:
8228:
8219:
7812:
7798:
7774:
7769:
7744:
7588:
Blanco, Philip (November 2019). "A discrete, energetic approach to rocket propulsion".
7405:
6749:
6706:
6479:
6425:
5985:
5810:
5792:
5127:
4957:
4878:
4807:
4449:
4429:
4282:
2878:{\displaystyle -\int _{V}^{V+\Delta V}\,dV={v_{e}}\int _{m_{0}}^{m_{f}}{\frac {dm}{m}}}
1792:
1055:
1051:
506:
502:
384:
300:
294:
217:
142:
136:
131:
7638:
as the ratio of initial to final mass and w as the exhaust velocity, corresponding to
1765:
is the velocity of the mass added to the exhaust (and lost by the rocket) during time
8513:
8352:
8347:
8266:
7909:
7827:
7617:
7420:
6505:
4041:{\displaystyle ~a={\frac {dv}{dt}}=-{\frac {F}{m(t)}}=-{\frac {Rv_{\text{e}}}{m(t)}}}
3530:{\displaystyle \Delta v=\int _{t_{0}}^{t_{f}}{\frac {|T|}{{m_{0}}-{t}\Delta {m}}}~dt}
881:
389:
222:
179:
7522:
1943:
in the observer frame is related to the velocity of the exhaust in the rocket frame
44:
8417:
8327:
8201:
8184:
8042:
7939:
7807:
7424:
7410:
3379:
107:
97:
92:
7548:
8422:
8257:
8027:
8007:
7924:
6753:
6741:
6728:
5461:
1066:
271:
7609:
6788:
Assume an exhaust velocity of 4,500 meters per second (15,000 ft/s) and a
3876:{\displaystyle \Delta v=V_{\text{exh}}~\ln \left({\frac {m_{0}}{m_{f}}}\right)}
7990:
7415:
7387:
6761:
844:
458:
351:
307:
266:
2610:{\displaystyle \sum _{i}F_{i}=m{\frac {dV}{dt}}+v_{\text{e}}{\frac {dm}{dt}}}
8407:
7759:
6745:
6222:
4718:{\displaystyle \Delta v=v_{\text{eff}}{\frac {m_{p}}{\sqrt {m(m+m_{p})}}}.}
5762:
where the rocket started at rest (with the rest mass including fuel being
4597:
Using momentum conservation in the rocket's frame just prior to ejection,
7660:
6429:
3185:{\displaystyle m_{0}-m_{f}=m_{f}\left(e^{\Delta V/v_{\text{e}}}-1\right)}
2663:
1062:
513:
486:
7370:{\displaystyle \Delta v\ =v_{\text{e}}\ln(100/11.2)\ =2.19v_{\text{e}}.}
4774:
the initial fueled-up mass of the rocket. Divide the total mass of fuel
8312:
7704:
6415:
5932:{\displaystyle {\frac {m_{0}}{m_{1}}}=\left^{\frac {c}{2v_{\text{e}}}}}
3408:
3278:
7689:
4352:
such that the mechanical energy gained per unit fuel mass is given by
2109:{\displaystyle {\vec {V}}_{\text{e}}={\vec {V}}+{\vec {v}}_{\text{e}}}
2036:{\displaystyle {\vec {v}}_{\text{e}}={\vec {V}}_{\text{e}}-{\vec {V}}}
7523:"On the Motion of Rockets both in Nonresisting and Resisting Mediums"
6483:
1028:
The necessary wet mass grows exponentially with the desired delta-v.
482:
478:
4050:
Now, the mass of fuel the rocket initially has on board is equal to
6486:
per unit mass and burn time, and is used to determine the mass of
5704:
is taken into account, the following equation can be derived for a
3799:{\displaystyle {\frac {J}{\Delta m}}={\frac {F}{p}}=V_{\text{exh}}}
3273:
the velocity of the rocket exhaust with respect to the rocket (the
1043:
The equation had been derived earlier by the
British mathematician
8117:
7736:
7700:
Tsiolkovsky's rocket equations plot and calculator in
WolframAlpha
6460:
1100:
452:
72:
1061:
While the derivation of the rocket equation is a straightforward
7275:{\displaystyle \Delta v\ =3v_{\text{e}}\ln 5\ =4.83v_{\text{e}}}
3890:
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.
7708:
7284:
and the payload is 10% × 10% × 10% = 0.1% of the initial mass.
3739:{\displaystyle J~{\frac {\ln({m_{0}})-\ln({m_{f}})}{\Delta m}}}
2946:{\displaystyle \Delta V=v_{\text{e}}\ln {\frac {m_{0}}{m_{f}}}}
2735:{\displaystyle -m{\frac {dV}{dt}}=v_{\text{e}}{\frac {dm}{dt}}}
7632:"A Transparent Derivation of the Relativistic Rocket Equation"
7183:
With three similar, subsequently smaller stages with the same
6490:
required for the given manoeuvre through the rocket equation.
915:, the equation can be solved for the required propellant mass
7695:
Relativity
Calculator – Learn Tsiolkovsky's rocket equations
876:
is the final total mass without propellant, a.k.a. dry mass.
6534:, which is the fraction of initial weight that is payload.
7634:(see the right side of equation 15 on the last page, with
4399:
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
1432:
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.
7495:
7493:
7491:
6811:
of 9,700 meters per second (32,000 ft/s) (Earth to
7527:
Journal of
Natural Philosophy, Chemistry & the Arts
6548:
The effective exhaust velocity is often specified as a
3020:{\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
6231:
6122:
5947:
5392:
5356:
4617:
4603:
4549:
4514:
4474:
4360:
4358:
2625:
7443:- added terms for gravity and drag in vertical flight
7299:
7218:
7189:
7057:
6991:
6968:
6924:
6901:
6851:
6821:
6794:
6676:
6643:
6614:
6558:
6313:
6008:
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5440:
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5150:
5130:
4980:
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4881:
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4651:
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4285:
4108:
3942:
3812:
3754:
3664:
3598:
3572:
3545:
3430:
3390:
3344:
3314:
3287:
3252:
3225:
3198:
3101:
3033:
2959:
2893:
2781:
2750:
2672:
2527:
2504:
2469:
2424:
2386:
2357:
2321:
2285:
2256:
2122:
2049:
1976:
1949:
1913:
1881:
1842:
1815:
1795:
1771:
1735:
1705:
1658:
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1602:
1467:
1438:
1399:
1345:
1319:
1283:
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:
6203:{\textstyle R^{\frac {2v_{\text{e}}}{c}}=\exp \left}
3088:{\displaystyle m_{0}=m_{f}e^{\Delta V/v_{\text{e}}}}
8381:
8293:
8237:
8173:
8126:
8066:
8057:
7953:
7863:
7752:
7743:
7369:
7274:
7202:
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5931:
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5684:
5629:
5452:
5423:
5378:
5342:
5170:
5136:
5114:
4966:
4946:
4887:
4875:. The remaining mass of the rocket after ejecting
4867:
4816:
4796:
4766:
4739:
4717:
4637:
4587:
4458:
4438:
4418:
4388:
4344:
4317:
4291:
4264:
4040:
3875:
3798:
3738:
3648:
3581:
3558:
3529:
3399:
3370:
3327:
3300:
3265:
3238:
3211:
3184:
3087:
3019:
2945:
2877:
2763:
2734:
2654:
2609:
2513:
2490:
2455:
2410:
2372:
2343:
2307:
2271:
2242:
2108:
2035:
1962:
1935:
1896:
1867:
1827:
1801:
1780:
1757:
1720:
1691:
1643:
1617:
1585:
1453:
1424:
1385:
1331:
1305:
1269:
1143:
1018:
947:
907:
868:
835:
802:
774:
741:
708:
655:
531:
6293:{\textstyle \tanh x={\frac {e^{2x}-1}{e^{2x}+1}}}
6895:: suppose that the first stage should provide a
6598:{\displaystyle v_{\text{e}}=g_{0}I_{\text{sp}},}
5637:since the final remaining mass of the rocket is
5472:
3589:is the initial mass minus the final (dry) mass,
3219:is the initial total mass including propellant,
1191:
6985:of 4,700 meters per second (15,000 ft/s);
6918:of 5,000 meters per second (16,000 ft/s);
4747:be the initial fuel mass fraction on board and
2521:results in a decrease in rocket mass in time),
709:{\displaystyle v_{\text{e}}=I_{\text{sp}}g_{0}}
4389:{\textstyle {\tfrac {1}{2}}v_{\text{eff}}^{2}}
4102:allows a substitution on the right) obtains:
3277:, or, if measured in time, that multiplied by
1036:The equation is named after Russian scientist
7720:
505:also developed it independently in 1912, and
434:
8:
1692:{\displaystyle {\vec {V}}+\Delta {\vec {V}}}
7516:
7514:
6838:to overcome gravity and aerodynamic drag).
3649:{\displaystyle \int _{t_{0}}^{t_{f}}F~dt=J}
8494:
8063:
7749:
7727:
7713:
7705:
6508:which displays the required mission delta-
4954:. The overall speed change after ejecting
4638:{\textstyle u=\Delta v{\tfrac {m}{m_{p}}}}
1386:{\displaystyle {\vec {P}}_{0}=m{\vec {V}}}
441:
427:
29:
7358:
7334:
7316:
7298:
7266:
7238:
7217:
7194:
7188:
7158:
7126:
7094:
7082:
7058:
7056:
7009:
7002:
6990:
6967:
6942:
6935:
6923:
6900:
6869:
6862:
6850:
6820:
6793:
6681:
6675:
6648:
6642:
6619:
6613:
6586:
6576:
6563:
6557:
6478:is produced by reaction engines, such as
6376:
6366:
6360:
6343:
6337:
6312:
6272:
6251:
6244:
6230:
6174:
6164:
6137:
6127:
6121:
6079:
6069:
6044:
6034:
6027:
6009:
6007:
6002:allows this equation to be rearranged as
5987:
5964:
5954:
5948:
5946:
5919:
5906:
5884:
5861:
5852:
5836:
5826:
5820:
5818:
5794:
5773:
5767:
5742:
5736:
5713:
5661:
5648:
5642:
5616:
5606:
5600:
5588:
5563:
5551:
5521:
5515:
5510:
5500:
5475:
5469:
5439:
5406:
5397:
5391:
5366:
5355:
5331:
5310:
5298:
5287:
5277:
5259:
5237:
5231:
5219:
5208:
5198:
5183:
5154:
5149:
5129:
5098:
5084:
5058:
5033:
5027:
5015:
5004:
4994:
4979:
4959:
4933:
4912:
4900:
4880:
4857:
4851:
4835:
4829:
4809:
4788:
4779:
4758:
4752:
4732:
4700:
4677:
4671:
4665:
4650:
4626:
4616:
4602:
4576:
4548:
4539:
4529:
4513:
4504:
4499:
4489:
4473:
4471:
4451:
4431:
4410:
4404:
4380:
4375:
4359:
4357:
4336:
4330:
4304:
4284:
4247:
4237:
4231:
4215:
4194:
4175:
4154:
4138:
4125:
4107:
4015:
4005:
3978:
3952:
3941:
3900:exhaust velocity relative to the rocket v
3861:
3851:
3845:
3826:
3811:
3790:
3773:
3755:
3753:
3715:
3710:
3688:
3683:
3671:
3663:
3620:
3615:
3608:
3603:
3597:
3571:
3550:
3544:
3510:
3502:
3492:
3487:
3480:
3472:
3469:
3461:
3456:
3449:
3444:
3429:
3389:
3362:
3349:
3343:
3319:
3313:
3292:
3286:
3257:
3251:
3230:
3224:
3203:
3197:
3163:
3154:
3147:
3132:
3119:
3106:
3100:
3077:
3068:
3061:
3051:
3038:
3032:
3009:
3000:
2987:
2977:
2964:
2958:
2935:
2925:
2919:
2907:
2892:
2860:
2852:
2847:
2840:
2835:
2824:
2819:
2809:
2794:
2789:
2780:
2755:
2749:
2712:
2706:
2679:
2671:
2640:
2630:
2624:
2587:
2581:
2554:
2542:
2532:
2526:
2503:
2468:
2436:
2435:
2431:
2423:
2411:{\displaystyle \Delta m\Delta {\vec {V}}}
2397:
2396:
2385:
2359:
2358:
2356:
2335:
2324:
2323:
2320:
2299:
2288:
2287:
2284:
2258:
2257:
2255:
2229:
2228:
2204:
2193:
2192:
2177:
2176:
2161:
2150:
2149:
2136:
2125:
2124:
2121:
2100:
2089:
2088:
2073:
2072:
2063:
2052:
2051:
2048:
2022:
2021:
2012:
2001:
2000:
1990:
1979:
1978:
1975:
1954:
1948:
1927:
1916:
1915:
1912:
1880:
1841:
1814:
1794:
1770:
1749:
1738:
1737:
1734:
1704:
1678:
1677:
1660:
1659:
1657:
1630:
1604:
1603:
1601:
1593:and where, with respect to the observer:
1577:
1566:
1565:
1539:
1538:
1521:
1520:
1481:
1470:
1469:
1466:
1437:
1413:
1402:
1401:
1398:
1372:
1371:
1359:
1348:
1347:
1344:
1318:
1297:
1286:
1285:
1282:
1250:
1239:
1238:
1225:
1214:
1213:
1209:
1194:
1181:
1170:
1169:
1162:
1156:
1135:
1124:
1123:
1120:
1005:
996:
989:
979:
966:
960:
939:
926:
920:
899:
893:
860:
854:
827:
821:
795:
766:
760:
733:
727:
700:
690:
677:
671:
642:
632:
626:
614:
604:
589:
579:
573:
561:
546:
521:
5379:{\textstyle \Delta x={\frac {\phi }{N}}}
4446:and the remaining mass of the rocket is
485:by expelling part of its mass with high
7542:
7540:
7463:
6552:and they are related to each other by:
4275:Limit of finite mass "pellet" expulsion
3909:propelling the rocket that is equal to
1868:{\displaystyle \left(m-\Delta m\right)}
332:
247:
51:
37:
8373:Transposition, docking, and extraction
7042:In the case of sequentially thrusting
6767:The rocket equation can be applied to
5424:{\textstyle x_{j}={\frac {j\phi }{N}}}
1699:is the velocity of the rocket at time
1625:is the velocity of the rocket at time
1313:is the momentum of the rocket at time
3905:(m/s). This creates a constant force
2619:If there are no external forces then
2344:{\displaystyle {\vec {F}}_{\text{i}}}
2308:{\displaystyle {\vec {v}}_{\text{e}}}
1936:{\displaystyle {\vec {V}}_{\text{e}}}
1758:{\displaystyle {\vec {V}}_{\text{e}}}
1425:{\displaystyle {\vec {P}}_{\Delta t}}
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.
7:
7656:"The Tyranny of the Rocket Equation"
6664:is the specific impulse measured in
5685:{\displaystyle m_{f}=m_{0}(1-\phi )}
5453:{\displaystyle N\rightarrow \infty }
2456:{\displaystyle dm\,d{\vec {v}}\to 0}
6635:is the specific impulse in seconds,
6221:as well as the "power" identity at
5975:{\textstyle {\frac {m_{0}}{m_{1}}}}
4947:{\displaystyle m=m_{0}(1-j\phi /N)}
7300:
7219:
7062:
6969:
6902:
6822:
6795:
6500:For interplanetary missions delta-
6314:
6012:
5887:
5864:
5715:
5487:
5482:
5447:
5357:
5313:
5185:
4981:
4868:{\displaystyle m_{p}=\phi m_{0}/N}
4652:
4610:
4566:
4325:, with an effective exhaust speed
4312:
4112:
4064:. For the constant mass flow rate
3813:
3761:
3727:
3573:
3507:
3431:
3391:
3148:
3062:
2991:
2894:
2801:
2505:
2482:
2393:
2387:
2225:
2219:
2210:
2173:
2137:
1888:
1875:is the mass of the rocket at time
1854:
1809:is the mass of the rocket at time
1772:
1712:
1674:
1558:
1535:
1504:
1482:
1445:
1414:
1258:
1226:
1195:
990:
548:
523:
25:
8433:Kepler's laws of planetary motion
7690:How to derive the rocket equation
6463:. As used in this context, it is
5144:the last term in the denominator
213:Kepler's laws of planetary motion
8493:
8428:Interplanetary Transport Network
8308:Collision avoidance (spacecraft)
7645:in the notation of this article)
7390:
7033:engines, the tanks, and payload.
1077:
489:and can thereby move due to the
43:
8393:Astronomical coordinate systems
8147:Longitude of the ascending node
6493:For multiple manoeuvres, delta-
3924:, its acceleration at any time
3806:the integral can be equated to
2664:conservation of linear momentum
1098:Consider the following system:
8525:Eponymous equations of physics
8466:Retrograde and prograde motion
7342:
7328:
7022:{\displaystyle 1-e^{-4.7/4.5}}
6955:{\displaystyle 1-e^{-5.0/4.5}}
6882:{\displaystyle 1-e^{-9.7/4.5}}
6512:as a function of launch date.
5679:
5667:
5479:
5464:becomes the definite integral
5444:
5106:
5069:
5066:
5043:
4941:
4918:
4824:discrete pellets each of mass
4706:
4687:
4573:
4563:
4345:{\displaystyle v_{\text{eff}}}
4309:
4068:it will therefore take a time
4032:
4026:
3993:
3987:
3722:
3707:
3695:
3680:
3566:is the initial (wet) mass and
3481:
3473:
2447:
2441:
2402:
2364:
2329:
2293:
2263:
2234:
2198:
2182:
2155:
2130:
2094:
2078:
2057:
2027:
2006:
1984:
1921:
1743:
1683:
1665:
1609:
1571:
1544:
1526:
1475:
1407:
1377:
1353:
1306:{\displaystyle {\vec {P}}_{0}}
1291:
1244:
1219:
1201:
1175:
1144:{\displaystyle {\vec {F}}_{i}}
1129:
1:
6628:{\displaystyle I_{\text{sp}}}
5178:and can be neglected to give
3922:Newton's Second Law of Motion
3658:The integral is found to be:
2655:{\textstyle \sum _{i}F_{i}=0}
1113:Newton's second law of motion
742:{\displaystyle I_{\text{sp}}}
8413:Equatorial coordinate system
7427:increases the final velocity
7203:{\displaystyle v_{\text{e}}}
6732:is a limit to the amount of
6657:{\displaystyle v_{\text{e}}}
5171:{\displaystyle \phi /N\ll 1}
4318:{\displaystyle N\to \infty }
3932:divided by its current mass
3892:Newton's First Law of Motion
3328:{\displaystyle v_{\text{e}}}
3301:{\displaystyle v_{\text{e}}}
3281:-on-Earth acceleration). If
3266:{\displaystyle v_{\text{e}}}
2764:{\displaystyle v_{\text{e}}}
2491:{\displaystyle dm=-\Delta m}
1963:{\displaystyle v_{\text{e}}}
1907:The velocity of the exhaust
6469:physical change in velocity
5760:inertial frame of reference
3371:{\displaystyle m_{0}-m_{f}}
2498:(since ejecting a positive
2351:have the same direction as
948:{\displaystyle m_{0}-m_{f}}
367:Tsiolkovsky rocket equation
8556:
8165:Longitude of the periapsis
7502:"Reactive Flying Machines"
6544:Effective exhaust velocity
6541:
6538:Effective exhaust velocity
6519:
6445:spacecraft flight dynamics
6413:
4797:{\displaystyle \phi m_{0}}
4299:pellets consecutively, as
2373:{\displaystyle {\vec {V}}}
2272:{\displaystyle {\vec {V}}}
1897:{\displaystyle t=\Delta t}
1721:{\displaystyle t=\Delta t}
1618:{\displaystyle {\vec {V}}}
1454:{\displaystyle t=\Delta t}
718:effective exhaust velocity
463:effective exhaust velocity
336:Engineering and efficiency
155:Bi-elliptic transfer orbit
8489:
8476:Specific angular momentum
6482:, is proportional to the
6307:), this is equivalent to
3896:constant mass flow rate R
1115:relates external forces (
471:classical rocket equation
18:Classical rocket equation
7610:10.1088/1361-6552/ab315b
7452:Stigler’s law of eponymy
7423:- applying delta-v in a
6978:{\displaystyle \Delta v}
6911:{\displaystyle \Delta v}
6831:{\displaystyle \Delta v}
6804:{\displaystyle \Delta v}
6522:Propellant mass fraction
6210:(here "exp" denotes the
5724:{\displaystyle \Delta v}
3928:is its propelling force
3582:{\displaystyle \Delta m}
3400:{\displaystyle \Delta V}
3382:of propellant expended.
2773:Tsiolkovsky's hypothesis
2514:{\displaystyle \Delta m}
1781:{\displaystyle \Delta t}
888:), and a given dry mass
532:{\displaystyle \Delta v}
491:conservation of momentum
362:Propellant mass fraction
261:Gravitational influences
8471:Specific orbital energy
7553:. G. & S. Robinson.
7547:Moore, William (1813).
7521:Moore, William (1810).
7436:Reversibility of orbits
7210:for each stage, gives:
1094:Most popular derivation
233:Specific orbital energy
8530:Konstantin Tsiolkovsky
7883:Geostationary transfer
7371:
7276:
7204:
7175:
7023:
6979:
6956:
6912:
6883:
6832:
6805:
6691:
6658:
6629:
6599:
6504:is often plotted on a
6459:that has the units of
6447:, is a measure of the
6393:
6294:
6223:Logarithmic identities
6204:
6105:
5996:
5976:
5933:
5803:
5783:
5752:
5725:
5686:
5631:
5454:
5425:
5380:
5344:
5309:
5230:
5172:
5138:
5124:Notice that for large
5116:
5026:
4968:
4948:
4889:
4869:
4818:
4798:
4768:
4741:
4719:
4645:, from which we find
4639:
4589:
4460:
4440:
4420:
4390:
4346:
4319:
4293:
4266:
4042:
3877:
3800:
3740:
3650:
3583:
3560:
3531:
3401:
3372:
3329:
3302:
3267:
3240:
3213:
3186:
3089:
3021:
2947:
2879:
2771:is constant (known as
2765:
2736:
2656:
2611:
2515:
2492:
2457:
2418:are negligible (since
2412:
2374:
2345:
2309:
2273:
2244:
2110:
2037:
1964:
1937:
1898:
1869:
1829:
1803:
1782:
1759:
1722:
1693:
1645:
1619:
1587:
1455:
1426:
1387:
1333:
1307:
1271:
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
8535:Single-stage-to-orbit
8456:Orbital state vectors
8398:Characteristic energy
8368:Trans-lunar injection
8156:Argument of periapsis
7833:Prograde / Retrograde
7794:Hyperbolic trajectory
7447:Spacecraft propulsion
7372:
7277:
7205:
7176:
7024:
6980:
6957:
6913:
6884:
6843:Single-stage-to-orbit
6833:
6806:
6692:
6690:{\displaystyle g_{0}}
6659:
6630:
6600:
6528:aerospace engineering
6402:Terms of the equation
6394:
6295:
6205:
6106:
5997:
5977:
5934:
5804:
5784:
5782:{\displaystyle m_{0}}
5753:
5751:{\displaystyle m_{1}}
5726:
5687:
5632:
5455:
5426:
5381:
5345:
5283:
5204:
5173:
5139:
5117:
5000:
4969:
4949:
4890:
4870:
4819:
4799:
4769:
4767:{\displaystyle m_{0}}
4742:
4740:{\displaystyle \phi }
4720:
4640:
4590:
4461:
4441:
4421:
4419:{\displaystyle m_{p}}
4391:
4347:
4320:
4294:
4267:
4043:
3878:
3801:
3741:
3651:
3584:
3561:
3559:{\displaystyle m_{0}}
3532:
3402:
3373:
3330:
3303:
3268:
3241:
3239:{\displaystyle m_{f}}
3214:
3212:{\displaystyle m_{0}}
3187:
3090:
3022:
2948:
2880:
2766:
2737:
2657:
2612:
2516:
2493:
2458:
2413:
2375:
2346:
2310:
2274:
2245:
2116:Solving this yields:
2111:
2038:
1965:
1938:
1899:
1870:
1830:
1804:
1783:
1760:
1723:
1694:
1646:
1620:
1588:
1456:
1427:
1388:
1334:
1308:
1272:
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
8303:Bi-elliptic transfer
7823:Parabolic trajectory
7297:
7216:
7187:
7055:
6989:
6966:
6922:
6899:
6849:
6819:
6792:
6674:
6641:
6612:
6556:
6311:
6229:
6212:exponential function
6120:
6006:
5986:
5945:
5817:
5793:
5766:
5735:
5712:
5641:
5468:
5438:
5390:
5354:
5182:
5148:
5128:
4978:
4958:
4899:
4879:
4828:
4808:
4778:
4751:
4731:
4649:
4601:
4470:
4450:
4430:
4426:is ejected at speed
4403:
4356:
4329:
4303:
4283:
4106:
3940:
3810:
3752:
3662:
3596:
3570:
3543:
3428:
3388:
3342:
3312:
3285:
3250:
3246:the final mass, and
3223:
3196:
3099:
3031:
2957:
2891:
2779:
2748:
2670:
2623:
2525:
2502:
2467:
2422:
2384:
2355:
2319:
2283:
2254:
2120:
2047:
1974:
1947:
1911:
1879:
1840:
1813:
1793:
1769:
1733:
1703:
1656:
1629:
1600:
1465:
1436:
1397:
1343:
1317:
1281:
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
8343:Low-energy transfer
7630:Forward, Robert L.
7602:2019PhyEd..54f5001B
7431:Relativistic rocket
6778:electric propulsion
6305:Hyperbolic function
6225:) and the identity
5706:relativistic rocket
5520:
4974:pellets is the sum
4509:
4385:
3627:
3539:where T is thrust,
3468:
2859:
2808:
1828:{\displaystyle t=0}
1644:{\displaystyle t=0}
1332:{\displaystyle t=0}
379:Efficiency measures
282:Sphere of influence
251:Celestial mechanics
33:Part of a series on
8338:Inclination change
7986:Distant retrograde
7570:2011-08-15 at the
7477:2011-08-15 at the
7398:Spaceflight portal
7367:
7272:
7200:
7171:
7169:
7019:
6975:
6952:
6908:
6893:Two-stage-to-orbit
6879:
6828:
6801:
6774:impulsive maneuver
6738:non-rocket systems
6687:
6654:
6625:
6595:
6432:"), symbolised as
6389:
6290:
6200:
6101:
5992:
5972:
5929:
5799:
5779:
5748:
5721:
5702:special relativity
5696:Special relativity
5682:
5627:
5506:
5486:
5450:
5421:
5376:
5340:
5168:
5134:
5112:
4964:
4944:
4885:
4865:
4814:
4794:
4764:
4737:
4715:
4635:
4633:
4585:
4558:
4523:
4495:
4483:
4456:
4436:
4416:
4386:
4371:
4369:
4342:
4315:
4289:
4262:
4038:
3886:Acceleration-based
3873:
3796:
3736:
3646:
3599:
3579:
3556:
3527:
3440:
3397:
3368:
3325:
3298:
3263:
3236:
3209:
3182:
3085:
3017:
2943:
2875:
2831:
2785:
2761:
2732:
2652:
2635:
2607:
2537:
2511:
2488:
2453:
2408:
2370:
2341:
2305:
2269:
2240:
2106:
2033:
1960:
1933:
1894:
1865:
1825:
1799:
1778:
1755:
1718:
1689:
1641:
1615:
1583:
1451:
1422:
1383:
1329:
1303:
1267:
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
8540:Rocket propulsion
8507:
8506:
8481:Two-line elements
8289:
8288:
8211:Eccentric anomaly
8053:
8052:
7920:Orbit of the Moon
7779:Highly elliptical
7590:Physics Education
7441:Robert H. Goddard
7361:
7347:
7319:
7308:
7269:
7255:
7241:
7227:
7197:
7161:
7129:
7110:
7085:
7070:
6769:orbital maneuvers
6758:tether propulsion
6651:
6622:
6589:
6566:
6382:
6352:
6346:
6288:
6219:Natural logarithm
6184:
6177:
6147:
6140:
6099:
6089:
6082:
6054:
6047:
6022:
5995:{\displaystyle R}
5970:
5926:
5922:
5900:
5897:
5874:
5842:
5809:standing for the
5802:{\displaystyle c}
5622:
5591:
5579:
5554:
5542:
5503:
5471:
5419:
5374:
5338:
5280:
5268:
5201:
5137:{\displaystyle N}
5110:
5109:
4997:
4967:{\displaystyle j}
4888:{\displaystyle j}
4817:{\displaystyle N}
4710:
4709:
4668:
4632:
4557:
4522:
4502:
4482:
4459:{\displaystyle m}
4439:{\displaystyle u}
4378:
4368:
4339:
4292:{\displaystyle N}
4253:
4218:
4210:
4157:
4111:
4098:(and noting that
4036:
4018:
3997:
3970:
3945:
3867:
3834:
3829:
3793:
3781:
3768:
3734:
3670:
3633:
3520:
3516:
3415:Other derivations
3322:
3295:
3260:
3166:
3080:
3012:
2999:
2941:
2910:
2887:This then yields
2873:
2758:
2730:
2709:
2697:
2626:
2605:
2584:
2572:
2528:
2444:
2405:
2367:
2338:
2332:
2302:
2296:
2266:
2237:
2207:
2201:
2185:
2158:
2133:
2103:
2097:
2081:
2066:
2060:
2030:
2015:
2009:
1993:
1987:
1957:
1930:
1924:
1802:{\displaystyle m}
1752:
1746:
1686:
1668:
1612:
1580:
1574:
1547:
1529:
1478:
1410:
1380:
1356:
1294:
1265:
1247:
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
8547:
8497:
8496:
8438:Lagrangian point
8333:Hohmann transfer
8278:
8264:
8255:
8246:
8226:
8217:
8208:
8199:
8195:
8191:
8182:
8162:
8153:
8144:
8135:
8115:
8111:
8102:
8093:
8084:
8064:
8033:Heliosynchronous
7982:Lagrange points
7935:Transatmospheric
7750:
7729:
7722:
7715:
7706:
7677:
7676:
7674:
7673:
7664:. Archived from
7652:
7646:
7628:
7622:
7621:
7585:
7579:
7561:
7555:
7554:
7544:
7535:
7534:
7518:
7509:
7508:
7506:
7500:Tsiolkovsky, K.
7497:
7486:
7468:
7400:
7395:
7394:
7393:
7376:
7374:
7373:
7368:
7363:
7362:
7359:
7345:
7338:
7321:
7320:
7317:
7306:
7281:
7279:
7278:
7273:
7271:
7270:
7267:
7253:
7243:
7242:
7239:
7225:
7209:
7207:
7206:
7201:
7199:
7198:
7195:
7180:
7178:
7177:
7172:
7170:
7163:
7162:
7159:
7144:
7131:
7130:
7127:
7115:
7111:
7109:
7095:
7087:
7086:
7083:
7068:
7028:
7026:
7025:
7020:
7018:
7017:
7013:
6984:
6982:
6981:
6976:
6961:
6959:
6958:
6953:
6951:
6950:
6946:
6917:
6915:
6914:
6909:
6888:
6886:
6885:
6880:
6878:
6877:
6873:
6837:
6835:
6834:
6829:
6810:
6808:
6807:
6802:
6712:
6704:
6699:standard gravity
6696:
6694:
6693:
6688:
6686:
6685:
6663:
6661:
6660:
6655:
6653:
6652:
6649:
6634:
6632:
6631:
6626:
6624:
6623:
6620:
6604:
6602:
6601:
6596:
6591:
6590:
6587:
6581:
6580:
6568:
6567:
6564:
6550:specific impulse
6532:payload fraction
6471:of the vehicle.
6467:the same as the
6453:orbital maneuver
6398:
6396:
6395:
6390:
6388:
6384:
6383:
6381:
6380:
6371:
6370:
6361:
6353:
6348:
6347:
6344:
6338:
6299:
6297:
6296:
6291:
6289:
6287:
6280:
6279:
6266:
6259:
6258:
6245:
6209:
6207:
6206:
6201:
6199:
6195:
6185:
6180:
6179:
6178:
6175:
6165:
6149:
6148:
6143:
6142:
6141:
6138:
6128:
6113:Then, using the
6110:
6108:
6107:
6102:
6100:
6098:
6091:
6090:
6085:
6084:
6083:
6080:
6070:
6063:
6056:
6055:
6050:
6049:
6048:
6045:
6035:
6028:
6023:
6018:
6010:
6001:
5999:
5998:
5993:
5981:
5979:
5978:
5973:
5971:
5969:
5968:
5959:
5958:
5949:
5938:
5936:
5935:
5930:
5928:
5927:
5925:
5924:
5923:
5920:
5907:
5905:
5901:
5899:
5898:
5893:
5885:
5876:
5875:
5870:
5862:
5853:
5843:
5841:
5840:
5831:
5830:
5821:
5808:
5806:
5805:
5800:
5789:initially), and
5788:
5786:
5785:
5780:
5778:
5777:
5757:
5755:
5754:
5749:
5747:
5746:
5730:
5728:
5727:
5722:
5691:
5689:
5688:
5683:
5666:
5665:
5653:
5652:
5636:
5634:
5633:
5628:
5623:
5621:
5620:
5611:
5610:
5601:
5593:
5592:
5589:
5580:
5578:
5564:
5556:
5555:
5552:
5543:
5541:
5530:
5522:
5519:
5514:
5505:
5504:
5501:
5485:
5459:
5457:
5456:
5451:
5430:
5428:
5427:
5422:
5420:
5415:
5407:
5402:
5401:
5385:
5383:
5382:
5377:
5375:
5367:
5349:
5347:
5346:
5341:
5339:
5337:
5336:
5335:
5319:
5311:
5308:
5297:
5282:
5281:
5278:
5269:
5267:
5263:
5245:
5241:
5232:
5229:
5218:
5203:
5202:
5199:
5177:
5175:
5174:
5169:
5158:
5143:
5141:
5140:
5135:
5121:
5119:
5118:
5113:
5111:
5102:
5088:
5062:
5042:
5041:
5037:
5028:
5025:
5014:
4999:
4998:
4995:
4973:
4971:
4970:
4965:
4953:
4951:
4950:
4945:
4937:
4917:
4916:
4895:pellets is then
4894:
4892:
4891:
4886:
4874:
4872:
4871:
4866:
4861:
4856:
4855:
4840:
4839:
4823:
4821:
4820:
4815:
4803:
4801:
4800:
4795:
4793:
4792:
4773:
4771:
4770:
4765:
4763:
4762:
4746:
4744:
4743:
4738:
4724:
4722:
4721:
4716:
4711:
4705:
4704:
4683:
4682:
4681:
4672:
4670:
4669:
4666:
4644:
4642:
4641:
4636:
4634:
4631:
4630:
4618:
4594:
4592:
4591:
4586:
4581:
4580:
4559:
4550:
4544:
4543:
4534:
4533:
4524:
4515:
4508:
4503:
4500:
4494:
4493:
4484:
4475:
4465:
4463:
4462:
4457:
4445:
4443:
4442:
4437:
4425:
4423:
4422:
4417:
4415:
4414:
4395:
4393:
4392:
4387:
4384:
4379:
4376:
4370:
4361:
4351:
4349:
4348:
4343:
4341:
4340:
4337:
4324:
4322:
4321:
4316:
4298:
4296:
4295:
4290:
4271:
4269:
4268:
4263:
4258:
4254:
4252:
4251:
4242:
4241:
4232:
4220:
4219:
4216:
4208:
4204:
4200:
4199:
4198:
4180:
4179:
4159:
4158:
4155:
4143:
4142:
4130:
4129:
4109:
4047:
4045:
4044:
4039:
4037:
4035:
4021:
4020:
4019:
4016:
4006:
3998:
3996:
3979:
3971:
3969:
3961:
3953:
3943:
3882:
3880:
3879:
3874:
3872:
3868:
3866:
3865:
3856:
3855:
3846:
3832:
3831:
3830:
3827:
3805:
3803:
3802:
3797:
3795:
3794:
3791:
3782:
3774:
3769:
3767:
3756:
3745:
3743:
3742:
3737:
3735:
3733:
3725:
3721:
3720:
3719:
3694:
3693:
3692:
3672:
3668:
3655:
3653:
3652:
3647:
3631:
3626:
3625:
3624:
3614:
3613:
3612:
3588:
3586:
3585:
3580:
3565:
3563:
3562:
3557:
3555:
3554:
3536:
3534:
3533:
3528:
3518:
3517:
3515:
3514:
3506:
3498:
3497:
3496:
3485:
3484:
3476:
3470:
3467:
3466:
3465:
3455:
3454:
3453:
3406:
3404:
3403:
3398:
3377:
3375:
3374:
3369:
3367:
3366:
3354:
3353:
3334:
3332:
3331:
3326:
3324:
3323:
3320:
3307:
3305:
3304:
3299:
3297:
3296:
3293:
3275:specific impulse
3272:
3270:
3269:
3264:
3262:
3261:
3258:
3245:
3243:
3242:
3237:
3235:
3234:
3218:
3216:
3215:
3210:
3208:
3207:
3191:
3189:
3188:
3183:
3181:
3177:
3170:
3169:
3168:
3167:
3164:
3158:
3137:
3136:
3124:
3123:
3111:
3110:
3094:
3092:
3091:
3086:
3084:
3083:
3082:
3081:
3078:
3072:
3056:
3055:
3043:
3042:
3026:
3024:
3023:
3018:
3016:
3015:
3014:
3013:
3010:
3004:
2997:
2982:
2981:
2969:
2968:
2953:or equivalently
2952:
2950:
2949:
2944:
2942:
2940:
2939:
2930:
2929:
2920:
2912:
2911:
2908:
2884:
2882:
2881:
2876:
2874:
2869:
2861:
2858:
2857:
2856:
2846:
2845:
2844:
2830:
2829:
2828:
2807:
2793:
2770:
2768:
2767:
2762:
2760:
2759:
2756:
2741:
2739:
2738:
2733:
2731:
2729:
2721:
2713:
2711:
2710:
2707:
2698:
2696:
2688:
2680:
2661:
2659:
2658:
2653:
2645:
2644:
2634:
2616:
2614:
2613:
2608:
2606:
2604:
2596:
2588:
2586:
2585:
2582:
2573:
2571:
2563:
2555:
2547:
2546:
2536:
2520:
2518:
2517:
2512:
2497:
2495:
2494:
2489:
2462:
2460:
2459:
2454:
2446:
2445:
2437:
2417:
2415:
2414:
2409:
2407:
2406:
2398:
2379:
2377:
2376:
2371:
2369:
2368:
2360:
2350:
2348:
2347:
2342:
2340:
2339:
2336:
2334:
2333:
2325:
2314:
2312:
2311:
2306:
2304:
2303:
2300:
2298:
2297:
2289:
2278:
2276:
2275:
2270:
2268:
2267:
2259:
2249:
2247:
2246:
2241:
2239:
2238:
2230:
2209:
2208:
2205:
2203:
2202:
2194:
2187:
2186:
2178:
2166:
2165:
2160:
2159:
2151:
2144:
2143:
2135:
2134:
2126:
2115:
2113:
2112:
2107:
2105:
2104:
2101:
2099:
2098:
2090:
2083:
2082:
2074:
2068:
2067:
2064:
2062:
2061:
2053:
2042:
2040:
2039:
2034:
2032:
2031:
2023:
2017:
2016:
2013:
2011:
2010:
2002:
1995:
1994:
1991:
1989:
1988:
1980:
1969:
1967:
1966:
1961:
1959:
1958:
1955:
1942:
1940:
1939:
1934:
1932:
1931:
1928:
1926:
1925:
1917:
1903:
1901:
1900:
1895:
1874:
1872:
1871:
1866:
1864:
1860:
1834:
1832:
1831:
1826:
1808:
1806:
1805:
1800:
1787:
1785:
1784:
1779:
1764:
1762:
1761:
1756:
1754:
1753:
1750:
1748:
1747:
1739:
1727:
1725:
1724:
1719:
1698:
1696:
1695:
1690:
1688:
1687:
1679:
1670:
1669:
1661:
1650:
1648:
1647:
1642:
1624:
1622:
1621:
1616:
1614:
1613:
1605:
1592:
1590:
1589:
1584:
1582:
1581:
1578:
1576:
1575:
1567:
1554:
1550:
1549:
1548:
1540:
1531:
1530:
1522:
1514:
1510:
1489:
1488:
1480:
1479:
1471:
1460:
1458:
1457:
1452:
1431:
1429:
1428:
1423:
1421:
1420:
1412:
1411:
1403:
1392:
1390:
1389:
1384:
1382:
1381:
1373:
1364:
1363:
1358:
1357:
1349:
1338:
1336:
1335:
1330:
1312:
1310:
1309:
1304:
1302:
1301:
1296:
1295:
1287:
1276:
1274:
1273:
1268:
1266:
1264:
1256:
1255:
1254:
1249:
1248:
1240:
1233:
1232:
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:
21:
8555:
8554:
8550:
8549:
8548:
8546:
8545:
8544:
8510:
8509:
8508:
8503:
8485:
8403:Escape velocity
8384:
8377:
8358:Rocket equation
8285:
8277:
8271:
8262:
8253:
8244:
8233:
8224:
8215:
8206:
8197:
8193:
8189:
8180:
8169:
8160:
8151:
8142:
8133:
8122:
8113:
8109:
8105:Semi-minor axis
8100:
8096:Semi-major axis
8091:
8082:
8076:
8049:
7971:Areosynchronous
7955:
7949:
7930:Sun-synchronous
7915:Near-equatorial
7859:
7739:
7733:
7686:
7681:
7680:
7671:
7669:
7654:
7653:
7649:
7644:
7629:
7625:
7587:
7586:
7582:
7572:Wayback Machine
7562:
7558:
7546:
7545:
7538:
7520:
7519:
7512:
7504:
7499:
7498:
7489:
7479:Wayback Machine
7469:
7465:
7460:
7396:
7391:
7389:
7386:
7354:
7312:
7295:
7294:
7262:
7234:
7214:
7213:
7190:
7185:
7184:
7168:
7167:
7154:
7142:
7141:
7122:
7113:
7112:
7099:
7078:
7071:
7053:
7052:
7040:
6998:
6987:
6986:
6964:
6963:
6931:
6920:
6919:
6897:
6896:
6858:
6847:
6846:
6817:
6816:
6790:
6789:
6786:
6750:space elevators
6720:
6710:
6702:
6677:
6672:
6671:
6644:
6639:
6638:
6615:
6610:
6609:
6582:
6572:
6559:
6554:
6553:
6546:
6540:
6524:
6518:
6497:sums linearly.
6439:and pronounced
6418:
6412:
6404:
6372:
6362:
6339:
6336:
6332:
6309:
6308:
6268:
6267:
6247:
6246:
6227:
6226:
6170:
6166:
6163:
6159:
6133:
6129:
6123:
6118:
6117:
6075:
6071:
6065:
6064:
6040:
6036:
6030:
6029:
6011:
6004:
6003:
5984:
5983:
5960:
5950:
5943:
5942:
5915:
5911:
5886:
5877:
5863:
5854:
5848:
5847:
5832:
5822:
5815:
5814:
5791:
5790:
5769:
5764:
5763:
5738:
5733:
5732:
5710:
5709:
5698:
5657:
5644:
5639:
5638:
5612:
5602:
5584:
5568:
5547:
5531:
5523:
5496:
5466:
5465:
5436:
5435:
5408:
5393:
5388:
5387:
5352:
5351:
5327:
5320:
5312:
5273:
5246:
5233:
5194:
5180:
5179:
5146:
5145:
5126:
5125:
5029:
4990:
4976:
4975:
4956:
4955:
4908:
4897:
4896:
4877:
4876:
4847:
4831:
4826:
4825:
4806:
4805:
4784:
4776:
4775:
4754:
4749:
4748:
4729:
4728:
4696:
4673:
4661:
4647:
4646:
4622:
4599:
4598:
4572:
4535:
4525:
4485:
4468:
4467:
4448:
4447:
4428:
4427:
4406:
4401:
4400:
4354:
4353:
4332:
4327:
4326:
4301:
4300:
4281:
4280:
4277:
4243:
4233:
4227:
4211:
4190:
4171:
4164:
4160:
4150:
4134:
4121:
4104:
4103:
4084:
4078:
4062:
4056:
4022:
4011:
4007:
3983:
3962:
3954:
3938:
3937:
3918:
3903:
3888:
3857:
3847:
3841:
3822:
3808:
3807:
3786:
3760:
3750:
3749:
3726:
3711:
3684:
3673:
3660:
3659:
3616:
3604:
3594:
3593:
3568:
3567:
3546:
3541:
3540:
3488:
3486:
3471:
3457:
3445:
3426:
3425:
3422:
3417:
3386:
3385:
3358:
3345:
3340:
3339:
3315:
3310:
3309:
3288:
3283:
3282:
3253:
3248:
3247:
3226:
3221:
3220:
3199:
3194:
3193:
3159:
3143:
3142:
3138:
3128:
3115:
3102:
3097:
3096:
3073:
3057:
3047:
3034:
3029:
3028:
3005:
2983:
2973:
2960:
2955:
2954:
2931:
2921:
2903:
2889:
2888:
2862:
2848:
2836:
2820:
2777:
2776:
2751:
2746:
2745:
2722:
2714:
2702:
2689:
2681:
2668:
2667:
2636:
2621:
2620:
2597:
2589:
2577:
2564:
2556:
2538:
2523:
2522:
2500:
2499:
2465:
2464:
2420:
2419:
2382:
2381:
2353:
2352:
2322:
2317:
2316:
2286:
2281:
2280:
2252:
2251:
2191:
2148:
2123:
2118:
2117:
2087:
2050:
2045:
2044:
1999:
1977:
1972:
1971:
1950:
1945:
1944:
1914:
1909:
1908:
1877:
1876:
1847:
1843:
1838:
1837:
1811:
1810:
1791:
1790:
1767:
1766:
1736:
1731:
1730:
1701:
1700:
1654:
1653:
1627:
1626:
1598:
1597:
1564:
1519:
1515:
1497:
1493:
1468:
1463:
1462:
1434:
1433:
1400:
1395:
1394:
1346:
1341:
1340:
1315:
1314:
1284:
1279:
1278:
1257:
1237:
1212:
1211:
1168:
1153:
1152:
1122:
1117:
1116:
1096:
1091:
1075:
1034:
1001:
985:
975:
962:
957:
956:
935:
922:
917:
916:
895:
890:
889:
886:escape velocity
856:
851:
850:
823:
818:
817:
792:
791:
762:
757:
756:
729:
724:
723:
696:
686:
673:
668:
667:
638:
628:
610:
600:
585:
575:
557:
543:
542:
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:
8553:
8551:
8543:
8542:
8537:
8532:
8527:
8522:
8512:
8511:
8505:
8504:
8502:
8501:
8499:List of orbits
8490:
8487:
8486:
8484:
8483:
8478:
8473:
8468:
8463:
8458:
8453:
8451:Orbit equation
8448:
8440:
8435:
8430:
8425:
8420:
8415:
8410:
8405:
8400:
8395:
8389:
8387:
8379:
8378:
8376:
8375:
8370:
8365:
8360:
8355:
8350:
8345:
8340:
8335:
8330:
8325:
8323:Gravity assist
8320:
8318:Delta-v budget
8315:
8310:
8305:
8299:
8297:
8291:
8290:
8287:
8286:
8284:
8283:
8275:
8269:
8260:
8251:
8249:Orbital period
8241:
8239:
8235:
8234:
8232:
8231:
8229:True longitude
8222:
8220:Mean longitude
8213:
8204:
8187:
8177:
8175:
8171:
8170:
8168:
8167:
8158:
8149:
8140:
8130:
8128:
8124:
8123:
8121:
8120:
8107:
8098:
8089:
8079:
8077:
8075:
8074:
8071:
8067:
8061:
8055:
8054:
8051:
8050:
8048:
8047:
8046:
8045:
8037:
8036:
8035:
8030:
8025:
8024:
8023:
8010:
8005:
8004:
8003:
7998:
7993:
7988:
7980:
7979:
7978:
7976:Areostationary
7973:
7968:
7959:
7957:
7951:
7950:
7948:
7947:
7945:Very low Earth
7942:
7937:
7932:
7927:
7922:
7917:
7912:
7907:
7902:
7897:
7892:
7887:
7886:
7885:
7880:
7873:Geosynchronous
7869:
7867:
7861:
7860:
7858:
7857:
7855:Transfer orbit
7852:
7851:
7850:
7845:
7835:
7830:
7825:
7820:
7815:
7813:Lagrange point
7810:
7805:
7796:
7791:
7786:
7781:
7772:
7767:
7762:
7756:
7754:
7747:
7741:
7740:
7735:Gravitational
7734:
7732:
7731:
7724:
7717:
7709:
7703:
7702:
7697:
7692:
7685:
7684:External links
7682:
7679:
7678:
7647:
7642:
7623:
7580:
7556:
7536:
7510:
7487:
7462:
7461:
7459:
7456:
7455:
7454:
7449:
7444:
7438:
7433:
7428:
7418:
7413:
7408:
7406:Delta-v budget
7402:
7401:
7385:
7382:
7366:
7357:
7353:
7350:
7344:
7341:
7337:
7333:
7330:
7327:
7324:
7315:
7311:
7305:
7302:
7265:
7261:
7258:
7252:
7249:
7246:
7237:
7233:
7230:
7224:
7221:
7193:
7166:
7157:
7153:
7150:
7147:
7145:
7143:
7140:
7137:
7134:
7125:
7121:
7118:
7116:
7114:
7108:
7105:
7102:
7098:
7093:
7090:
7081:
7077:
7074:
7072:
7067:
7064:
7061:
7060:
7039:
7036:
7035:
7034:
7016:
7012:
7008:
7005:
7001:
6997:
6994:
6974:
6971:
6949:
6945:
6941:
6938:
6934:
6930:
6927:
6907:
6904:
6890:
6876:
6872:
6868:
6865:
6861:
6857:
6854:
6827:
6824:
6800:
6797:
6785:
6782:
6719:
6716:
6715:
6714:
6707:Imperial units
6684:
6680:
6669:
6647:
6636:
6618:
6594:
6585:
6579:
6575:
6571:
6562:
6542:Main article:
6539:
6536:
6520:Main article:
6517:
6514:
6480:rocket engines
6414:Main article:
6411:
6405:
6403:
6400:
6387:
6379:
6375:
6369:
6365:
6359:
6356:
6351:
6342:
6335:
6331:
6328:
6325:
6322:
6319:
6316:
6286:
6283:
6278:
6275:
6271:
6265:
6262:
6257:
6254:
6250:
6243:
6240:
6237:
6234:
6198:
6194:
6191:
6188:
6183:
6173:
6169:
6162:
6158:
6155:
6152:
6146:
6136:
6132:
6126:
6097:
6094:
6088:
6078:
6074:
6068:
6062:
6059:
6053:
6043:
6039:
6033:
6026:
6021:
6017:
6014:
5991:
5967:
5963:
5957:
5953:
5918:
5914:
5910:
5904:
5896:
5892:
5889:
5883:
5880:
5873:
5869:
5866:
5860:
5857:
5851:
5846:
5839:
5835:
5829:
5825:
5811:speed of light
5798:
5776:
5772:
5745:
5741:
5720:
5717:
5697:
5694:
5681:
5678:
5675:
5672:
5669:
5664:
5660:
5656:
5651:
5647:
5626:
5619:
5615:
5609:
5605:
5599:
5596:
5587:
5583:
5577:
5574:
5571:
5567:
5562:
5559:
5550:
5546:
5540:
5537:
5534:
5529:
5526:
5518:
5513:
5509:
5499:
5495:
5492:
5489:
5484:
5481:
5478:
5474:
5449:
5446:
5443:
5418:
5414:
5411:
5405:
5400:
5396:
5373:
5370:
5365:
5362:
5359:
5334:
5330:
5326:
5323:
5318:
5315:
5307:
5304:
5301:
5296:
5293:
5290:
5286:
5276:
5272:
5266:
5262:
5258:
5255:
5252:
5249:
5244:
5240:
5236:
5228:
5225:
5222:
5217:
5214:
5211:
5207:
5197:
5193:
5190:
5187:
5167:
5164:
5161:
5157:
5153:
5133:
5108:
5105:
5101:
5097:
5094:
5091:
5087:
5083:
5080:
5077:
5074:
5071:
5068:
5065:
5061:
5057:
5054:
5051:
5048:
5045:
5040:
5036:
5032:
5024:
5021:
5018:
5013:
5010:
5007:
5003:
4993:
4989:
4986:
4983:
4963:
4943:
4940:
4936:
4932:
4929:
4926:
4923:
4920:
4915:
4911:
4907:
4904:
4884:
4864:
4860:
4854:
4850:
4846:
4843:
4838:
4834:
4813:
4791:
4787:
4783:
4761:
4757:
4736:
4714:
4708:
4703:
4699:
4695:
4692:
4689:
4686:
4680:
4676:
4664:
4660:
4657:
4654:
4629:
4625:
4621:
4615:
4612:
4609:
4606:
4584:
4579:
4575:
4571:
4568:
4565:
4562:
4556:
4553:
4547:
4542:
4538:
4532:
4528:
4521:
4518:
4512:
4507:
4498:
4492:
4488:
4481:
4478:
4455:
4435:
4413:
4409:
4383:
4374:
4367:
4364:
4335:
4314:
4311:
4308:
4288:
4276:
4273:
4261:
4257:
4250:
4246:
4240:
4236:
4230:
4226:
4223:
4214:
4207:
4203:
4197:
4193:
4189:
4186:
4183:
4178:
4174:
4170:
4167:
4163:
4153:
4149:
4146:
4141:
4137:
4133:
4128:
4124:
4120:
4117:
4114:
4082:
4076:
4060:
4054:
4034:
4031:
4028:
4025:
4014:
4010:
4004:
4001:
3995:
3992:
3989:
3986:
3982:
3977:
3974:
3968:
3965:
3960:
3957:
3951:
3948:
3916:
3901:
3898:(kg/s) and at
3887:
3884:
3871:
3864:
3860:
3854:
3850:
3844:
3840:
3837:
3825:
3821:
3818:
3815:
3789:
3785:
3780:
3777:
3772:
3766:
3763:
3759:
3732:
3729:
3724:
3718:
3714:
3709:
3706:
3703:
3700:
3697:
3691:
3687:
3682:
3679:
3676:
3667:
3645:
3642:
3639:
3636:
3630:
3623:
3619:
3611:
3607:
3602:
3578:
3575:
3553:
3549:
3526:
3523:
3513:
3509:
3505:
3501:
3495:
3491:
3483:
3479:
3475:
3464:
3460:
3452:
3448:
3443:
3439:
3436:
3433:
3421:
3418:
3416:
3413:
3396:
3393:
3365:
3361:
3357:
3352:
3348:
3318:
3291:
3256:
3233:
3229:
3206:
3202:
3180:
3176:
3173:
3162:
3157:
3153:
3150:
3146:
3141:
3135:
3131:
3127:
3122:
3118:
3114:
3109:
3105:
3076:
3071:
3067:
3064:
3060:
3054:
3050:
3046:
3041:
3037:
3008:
3003:
2996:
2993:
2990:
2986:
2980:
2976:
2972:
2967:
2963:
2938:
2934:
2928:
2924:
2918:
2915:
2906:
2902:
2899:
2896:
2872:
2868:
2865:
2855:
2851:
2843:
2839:
2834:
2827:
2823:
2818:
2815:
2812:
2806:
2803:
2800:
2797:
2792:
2788:
2784:
2754:
2744:Assuming that
2728:
2725:
2720:
2717:
2705:
2701:
2695:
2692:
2687:
2684:
2678:
2675:
2651:
2648:
2643:
2639:
2633:
2629:
2603:
2600:
2595:
2592:
2580:
2576:
2570:
2567:
2562:
2559:
2553:
2550:
2545:
2541:
2535:
2531:
2510:
2507:
2487:
2484:
2481:
2478:
2475:
2472:
2452:
2449:
2443:
2440:
2434:
2430:
2427:
2404:
2401:
2395:
2392:
2389:
2366:
2363:
2331:
2328:
2315:are opposite,
2295:
2292:
2265:
2262:
2236:
2233:
2227:
2224:
2221:
2218:
2215:
2212:
2200:
2197:
2190:
2184:
2181:
2175:
2172:
2169:
2164:
2157:
2154:
2147:
2142:
2139:
2132:
2129:
2096:
2093:
2086:
2080:
2077:
2071:
2059:
2056:
2029:
2026:
2020:
2008:
2005:
1998:
1986:
1983:
1953:
1923:
1920:
1905:
1904:
1893:
1890:
1887:
1884:
1863:
1859:
1856:
1853:
1850:
1846:
1835:
1824:
1821:
1818:
1798:
1788:
1777:
1774:
1745:
1742:
1728:
1717:
1714:
1711:
1708:
1685:
1682:
1676:
1673:
1667:
1664:
1651:
1640:
1637:
1634:
1611:
1608:
1573:
1570:
1563:
1560:
1557:
1553:
1546:
1543:
1537:
1534:
1528:
1525:
1518:
1513:
1509:
1506:
1503:
1500:
1496:
1492:
1487:
1484:
1477:
1474:
1450:
1447:
1444:
1441:
1419:
1416:
1409:
1406:
1379:
1376:
1370:
1367:
1362:
1355:
1352:
1328:
1325:
1322:
1300:
1293:
1290:
1263:
1260:
1253:
1246:
1243:
1236:
1231:
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:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8552:
8541:
8538:
8536:
8533:
8531:
8528:
8526:
8523:
8521:
8520:Astrodynamics
8518:
8517:
8515:
8500:
8492:
8491:
8488:
8482:
8479:
8477:
8474:
8472:
8469:
8467:
8464:
8462:
8459:
8457:
8454:
8452:
8449:
8447:
8446:-body problem
8445:
8441:
8439:
8436:
8434:
8431:
8429:
8426:
8424:
8421:
8419:
8416:
8414:
8411:
8409:
8406:
8404:
8401:
8399:
8396:
8394:
8391:
8390:
8388:
8386:
8380:
8374:
8371:
8369:
8366:
8364:
8361:
8359:
8356:
8354:
8351:
8349:
8348:Oberth effect
8346:
8344:
8341:
8339:
8336:
8334:
8331:
8329:
8326:
8324:
8321:
8319:
8316:
8314:
8311:
8309:
8306:
8304:
8301:
8300:
8298:
8296:
8292:
8282:
8274:
8270:
8268:
8267:Orbital speed
8261:
8259:
8252:
8250:
8243:
8242:
8240:
8236:
8230:
8223:
8221:
8214:
8212:
8205:
8203:
8188:
8186:
8179:
8178:
8176:
8172:
8166:
8159:
8157:
8150:
8148:
8141:
8139:
8132:
8131:
8129:
8125:
8119:
8108:
8106:
8099:
8097:
8090:
8088:
8081:
8080:
8078:
8072:
8069:
8068:
8065:
8062:
8060:
8056:
8044:
8041:
8040:
8038:
8034:
8031:
8029:
8026:
8022:
8021:Earth's orbit
8019:
8018:
8017:
8014:
8013:
8011:
8009:
8006:
8002:
7999:
7997:
7994:
7992:
7989:
7987:
7984:
7983:
7981:
7977:
7974:
7972:
7969:
7967:
7964:
7963:
7961:
7960:
7958:
7952:
7946:
7943:
7941:
7938:
7936:
7933:
7931:
7928:
7926:
7923:
7921:
7918:
7916:
7913:
7911:
7908:
7906:
7903:
7901:
7898:
7896:
7893:
7891:
7888:
7884:
7881:
7879:
7878:Geostationary
7876:
7875:
7874:
7871:
7870:
7868:
7866:
7862:
7856:
7853:
7849:
7846:
7844:
7841:
7840:
7839:
7836:
7834:
7831:
7829:
7826:
7824:
7821:
7819:
7816:
7814:
7811:
7809:
7806:
7804:
7800:
7797:
7795:
7792:
7790:
7787:
7785:
7782:
7780:
7776:
7773:
7771:
7768:
7766:
7763:
7761:
7758:
7757:
7755:
7751:
7748:
7746:
7742:
7738:
7730:
7725:
7723:
7718:
7716:
7711:
7710:
7707:
7701:
7698:
7696:
7693:
7691:
7688:
7687:
7683:
7668:on 2022-03-06
7667:
7663:
7662:
7657:
7651:
7648:
7641:
7637:
7633:
7627:
7624:
7619:
7615:
7611:
7607:
7603:
7599:
7596:(6): 065001.
7595:
7591:
7584:
7581:
7577:
7573:
7569:
7566:
7560:
7557:
7552:
7551:
7543:
7541:
7537:
7532:
7528:
7524:
7517:
7515:
7511:
7503:
7496:
7494:
7492:
7488:
7484:
7480:
7476:
7473:
7467:
7464:
7457:
7453:
7450:
7448:
7445:
7442:
7439:
7437:
7434:
7432:
7429:
7426:
7422:
7421:Oberth effect
7419:
7417:
7414:
7412:
7409:
7407:
7404:
7403:
7399:
7388:
7383:
7381:
7377:
7364:
7355:
7351:
7348:
7339:
7335:
7331:
7325:
7322:
7313:
7309:
7303:
7292:
7290:
7287:A comparable
7285:
7282:
7263:
7259:
7256:
7250:
7247:
7244:
7235:
7231:
7228:
7222:
7211:
7191:
7181:
7164:
7155:
7151:
7148:
7146:
7138:
7135:
7132:
7123:
7119:
7117:
7106:
7103:
7100:
7096:
7091:
7088:
7079:
7075:
7073:
7065:
7050:
7047:
7045:
7044:rocket stages
7037:
7032:
7014:
7010:
7006:
7003:
6999:
6995:
6992:
6972:
6947:
6943:
6939:
6936:
6932:
6928:
6925:
6905:
6894:
6891:
6874:
6870:
6866:
6863:
6859:
6855:
6852:
6844:
6841:
6840:
6839:
6825:
6814:
6798:
6783:
6781:
6779:
6775:
6770:
6765:
6763:
6759:
6755:
6751:
6747:
6743:
6739:
6735:
6730:
6729:gravitational
6726:
6718:Applicability
6717:
6708:
6700:
6682:
6678:
6670:
6667:
6645:
6637:
6616:
6608:
6607:
6606:
6592:
6583:
6577:
6573:
6569:
6560:
6551:
6545:
6537:
6535:
6533:
6529:
6523:
6516:Mass fraction
6515:
6513:
6511:
6507:
6506:porkchop plot
6503:
6498:
6496:
6491:
6489:
6485:
6481:
6477:
6472:
6470:
6466:
6462:
6458:
6454:
6450:
6446:
6443:, as used in
6442:
6438:
6437:
6431:
6427:
6423:
6417:
6410:
6406:
6401:
6399:
6385:
6377:
6373:
6367:
6363:
6357:
6354:
6349:
6340:
6333:
6329:
6326:
6323:
6320:
6317:
6306:
6303:
6284:
6281:
6276:
6273:
6269:
6263:
6260:
6255:
6252:
6248:
6241:
6238:
6235:
6232:
6224:
6220:
6217:
6213:
6196:
6192:
6189:
6186:
6181:
6171:
6167:
6160:
6156:
6153:
6150:
6144:
6134:
6130:
6124:
6116:
6111:
6095:
6092:
6086:
6076:
6072:
6066:
6060:
6057:
6051:
6041:
6037:
6031:
6024:
6019:
6015:
5989:
5965:
5961:
5955:
5951:
5939:
5916:
5912:
5908:
5902:
5894:
5890:
5881:
5878:
5871:
5867:
5858:
5855:
5849:
5844:
5837:
5833:
5827:
5823:
5812:
5796:
5774:
5770:
5761:
5743:
5739:
5718:
5707:
5703:
5695:
5693:
5676:
5673:
5670:
5662:
5658:
5654:
5649:
5645:
5624:
5617:
5613:
5607:
5603:
5597:
5594:
5585:
5581:
5575:
5572:
5569:
5565:
5560:
5557:
5548:
5544:
5538:
5535:
5532:
5527:
5524:
5516:
5511:
5507:
5497:
5493:
5490:
5476:
5463:
5441:
5432:
5416:
5412:
5409:
5403:
5398:
5394:
5371:
5368:
5363:
5360:
5332:
5328:
5324:
5321:
5316:
5305:
5302:
5299:
5294:
5291:
5288:
5284:
5274:
5270:
5264:
5260:
5256:
5253:
5250:
5247:
5242:
5238:
5234:
5226:
5223:
5220:
5215:
5212:
5209:
5205:
5195:
5191:
5188:
5165:
5162:
5159:
5155:
5151:
5131:
5122:
5103:
5099:
5095:
5092:
5089:
5085:
5081:
5078:
5075:
5072:
5063:
5059:
5055:
5052:
5049:
5046:
5038:
5034:
5030:
5022:
5019:
5016:
5011:
5008:
5005:
5001:
4991:
4987:
4984:
4961:
4938:
4934:
4930:
4927:
4924:
4921:
4913:
4909:
4905:
4902:
4882:
4862:
4858:
4852:
4848:
4844:
4841:
4836:
4832:
4811:
4789:
4785:
4781:
4759:
4755:
4734:
4725:
4712:
4701:
4697:
4693:
4690:
4684:
4678:
4674:
4662:
4658:
4655:
4627:
4623:
4619:
4613:
4607:
4604:
4595:
4582:
4577:
4569:
4560:
4554:
4551:
4545:
4540:
4536:
4530:
4526:
4519:
4516:
4510:
4505:
4496:
4490:
4486:
4479:
4476:
4453:
4433:
4411:
4407:
4397:
4381:
4372:
4365:
4362:
4333:
4306:
4286:
4274:
4272:
4259:
4255:
4248:
4244:
4238:
4234:
4228:
4224:
4221:
4212:
4205:
4201:
4195:
4191:
4187:
4184:
4181:
4176:
4172:
4168:
4165:
4161:
4151:
4147:
4144:
4139:
4135:
4131:
4126:
4122:
4118:
4115:
4101:
4097:
4093:
4089:
4085:
4075:
4071:
4067:
4063:
4053:
4048:
4029:
4023:
4012:
4008:
4002:
3999:
3990:
3984:
3980:
3975:
3972:
3966:
3963:
3958:
3955:
3949:
3946:
3935:
3931:
3927:
3923:
3919:
3912:
3908:
3904:
3897:
3893:
3885:
3883:
3869:
3862:
3858:
3852:
3848:
3842:
3838:
3835:
3823:
3819:
3816:
3787:
3783:
3778:
3775:
3770:
3764:
3757:
3746:
3730:
3716:
3712:
3704:
3701:
3698:
3689:
3685:
3677:
3674:
3665:
3656:
3643:
3640:
3637:
3634:
3628:
3621:
3617:
3609:
3605:
3600:
3590:
3576:
3551:
3547:
3537:
3524:
3521:
3511:
3503:
3499:
3493:
3489:
3477:
3462:
3458:
3450:
3446:
3441:
3437:
3434:
3420:Impulse-based
3419:
3414:
3412:
3410:
3394:
3383:
3381:
3378:is the total
3363:
3359:
3355:
3350:
3346:
3336:
3316:
3289:
3280:
3276:
3254:
3231:
3227:
3204:
3200:
3178:
3174:
3171:
3160:
3155:
3151:
3144:
3139:
3133:
3129:
3125:
3120:
3116:
3112:
3107:
3103:
3074:
3069:
3065:
3058:
3052:
3048:
3044:
3039:
3035:
3006:
3001:
2994:
2988:
2984:
2978:
2974:
2970:
2965:
2961:
2936:
2932:
2926:
2922:
2916:
2913:
2904:
2900:
2897:
2885:
2870:
2866:
2863:
2853:
2849:
2841:
2837:
2832:
2825:
2821:
2816:
2813:
2810:
2804:
2798:
2795:
2790:
2786:
2782:
2774:
2752:
2742:
2726:
2723:
2718:
2715:
2703:
2699:
2693:
2690:
2685:
2682:
2676:
2673:
2665:
2649:
2646:
2641:
2637:
2631:
2627:
2617:
2601:
2598:
2593:
2590:
2578:
2574:
2568:
2565:
2560:
2557:
2551:
2548:
2543:
2539:
2533:
2529:
2508:
2485:
2479:
2476:
2473:
2470:
2463:), and using
2450:
2438:
2432:
2428:
2425:
2399:
2390:
2361:
2326:
2290:
2260:
2231:
2222:
2216:
2213:
2195:
2188:
2179:
2170:
2167:
2162:
2152:
2145:
2140:
2127:
2091:
2084:
2075:
2069:
2054:
2024:
2018:
2003:
1996:
1981:
1951:
1918:
1891:
1885:
1882:
1861:
1857:
1851:
1848:
1844:
1836:
1822:
1819:
1816:
1796:
1789:
1775:
1740:
1729:
1715:
1709:
1706:
1680:
1671:
1662:
1652:
1638:
1635:
1632:
1606:
1596:
1595:
1594:
1568:
1561:
1555:
1551:
1541:
1532:
1523:
1516:
1511:
1507:
1501:
1498:
1494:
1490:
1485:
1472:
1448:
1442:
1439:
1417:
1404:
1374:
1368:
1365:
1360:
1350:
1326:
1323:
1320:
1298:
1288:
1261:
1251:
1241:
1234:
1229:
1216:
1204:
1198:
1187:
1182:
1172:
1163:
1159:
1136:
1126:
1114:
1110:
1103:
1099:
1093:
1088:
1086:
1082:
1080:
1072:
1070:
1068:
1064:
1059:
1057:
1053:
1048:
1046:
1045:William Moore
1041:
1039:
1031:
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:
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:
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:
50:
46:
42:
41:
38:Astrodynamics
36:
32:
31:
19:
8461:Perturbation
8443:
8418:Ground track
8357:
8328:Gravity turn
8279:
8272:
8265:
8256:
8247:
8227:
8218:
8209:
8202:True anomaly
8200:
8185:Mean anomaly
8183:
8163:
8154:
8145:
8136:
8116:
8103:
8094:
8087:Eccentricity
8085:
8043:Lunar cycler
8016:Heliocentric
7956:other points
7905:Medium Earth
7803:Non-inclined
7670:. Retrieved
7666:the original
7659:
7650:
7639:
7635:
7626:
7593:
7589:
7583:
7559:
7549:
7530:
7526:
7466:
7425:gravity well
7411:Jeep problem
7378:
7293:
7286:
7283:
7212:
7182:
7051:
7048:
7041:
7030:
6815:, including
6787:
6766:
6754:launch loops
6746:gun launches
6721:
6547:
6525:
6509:
6501:
6499:
6494:
6492:
6475:
6473:
6464:
6440:
6435:
6433:
6424:(literally "
6421:
6419:
6408:
6301:
6215:
6112:
5940:
5699:
5433:
5123:
4726:
4596:
4398:
4278:
4099:
4095:
4091:
4087:
4080:
4073:
4069:
4065:
4058:
4051:
4049:
3933:
3929:
3925:
3914:
3910:
3906:
3899:
3895:
3889:
3747:
3657:
3591:
3538:
3423:
3384:
3380:working mass
3337:
3335:hypothesis.
2886:
2743:
2618:
1906:
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
8423:Hill sphere
8258:Mean motion
8138:Inclination
8127:Orientation
8028:Mars cycler
7966:Areocentric
7838:Synchronous
6762:light sails
6742:aerobraking
6725:aerodynamic
5813:in vacuum:
5462:Riemann sum
308:Halo orbits
272:Hill sphere
88:Inclination
8514:Categories
8363:Rendezvous
8059:Parameters
7895:High Earth
7865:Geocentric
7818:Osculating
7775:Elliptical
7672:2016-04-18
7533:: 276–285.
7458:References
7416:Mass ratio
6488:propellant
6455:. It is a
3338:The value
1089:Derivation
845:propellant
459:mass ratio
352:Mass ratio
267:Barycenter
8408:Ephemeris
8385:mechanics
8295:Maneuvers
8238:Variation
8001:Libration
7996:Lissajous
7900:Low Earth
7890:Graveyard
7789:Horseshoe
7618:202130640
7326:
7301:Δ
7248:
7220:Δ
7136:
7104:−
7092:
7063:Δ
7004:−
6996:−
6970:Δ
6937:−
6929:−
6903:Δ
6864:−
6856:−
6823:Δ
6796:Δ
6701:, 9.80665
6441:delta-vee
6358:
6330:
6315:Δ
6261:−
6236:
6190:
6157:
6058:−
6013:Δ
5888:Δ
5882:−
5865:Δ
5758:) in the
5716:Δ
5677:ϕ
5674:−
5598:
5576:ϕ
5573:−
5561:
5536:−
5517:ϕ
5508:∫
5488:Δ
5483:∞
5480:→
5448:∞
5445:→
5413:ϕ
5369:ϕ
5358:Δ
5325:−
5314:Δ
5285:∑
5257:ϕ
5251:−
5235:ϕ
5206:∑
5192:≈
5186:Δ
5163:≪
5152:ϕ
5096:ϕ
5082:ϕ
5076:−
5056:ϕ
5050:−
5031:ϕ
5002:∑
4982:Δ
4931:ϕ
4925:−
4845:ϕ
4782:ϕ
4735:ϕ
4653:Δ
4611:Δ
4567:Δ
4313:∞
4310:→
4225:
4188:
4182:−
4169:
4148:−
4132:−
4113:Δ
4100:R = dm/dt
4003:−
3976:−
3839:
3814:Δ
3762:Δ
3728:Δ
3705:
3699:−
3678:
3601:∫
3574:Δ
3508:Δ
3500:−
3442:∫
3432:Δ
3392:Δ
3356:−
3172:−
3149:Δ
3113:−
3063:Δ
2992:Δ
2989:−
2917:
2895:Δ
2833:∫
2802:Δ
2787:∫
2783:−
2674:−
2628:∑
2530:∑
2506:Δ
2483:Δ
2480:−
2448:→
2442:→
2403:→
2394:Δ
2388:Δ
2365:→
2330:→
2294:→
2264:→
2235:→
2226:Δ
2220:Δ
2217:−
2211:Δ
2199:→
2183:→
2174:Δ
2156:→
2146:−
2138:Δ
2131:→
2095:→
2079:→
2058:→
2028:→
2019:−
2007:→
1985:→
1922:→
1889:Δ
1855:Δ
1852:−
1773:Δ
1744:→
1713:Δ
1684:→
1675:Δ
1666:→
1610:→
1572:→
1559:Δ
1545:→
1536:Δ
1527:→
1505:Δ
1502:−
1483:Δ
1476:→
1446:Δ
1415:Δ
1408:→
1378:→
1354:→
1292:→
1259:Δ
1245:→
1235:−
1227:Δ
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
8174:Position
7799:Inclined
7770:Circular
7661:NASA.gov
7568:Archived
7475:Archived
7384:See also
6845:rocket:
6784:Examples
6740:such as
6705:m/s (in
6430:velocity
6216:see also
6115:identity
5941:Writing
1063:calculus
514:velocity
487:velocity
8383:Orbital
8353:Phasing
8313:Delta-v
8118:Apsides
8112:,
7910:Molniya
7828:Parking
7765:Capture
7753:General
7598:Bibcode
6734:payload
6697:is the
6449:impulse
6416:Delta-v
5708:, with
3409:delta v
3279:gravity
1032:History
810:is the
749:is the
716:is the
663:where:
8039:Other
7940:Tundra
7808:Kepler
7784:Escape
7737:orbits
7616:
7346:
7307:
7254:
7226:
7069:
7038:Stages
6713:ft/s).
6711:
6709:32.174
6703:
6605:where
6484:thrust
6474:Delta-
6457:scalar
6426:change
6420:Delta-
6407:Delta-
5350:where
4209:
4110:
3944:
3833:
3669:
3632:
3519:
3192:where
2998:
2666:) and
2043:thus,
1277:where
483:thrust
479:rocket
8281:Epoch
8070:Shape
8008:Lunar
7962:Mars
7954:About
7925:Polar
7745:Types
7614:S2CID
7576:RARed
7574:in a
7505:(PDF)
7483:RARed
7481:in a
6461:speed
5460:this
4804:into
473:, or
465:ratio
73:Apsis
8073:Size
8012:Sun
7991:Halo
7843:semi
7578:PDF)
7565:here
7485:PDF)
7472:here
7352:2.19
7340:11.2
7289:SSTO
7260:4.83
7152:1.61
6327:tanh
6233:tanh
5386:and
4727:Let
2279:and
1970:by:
1393:and
469:The
7848:sub
7760:Box
7606:doi
7332:100
7101:100
7097:100
7031:all
7015:4.5
7007:4.7
6948:4.5
6940:5.0
6875:4.5
6867:9.7
6813:LEO
6760:or
6727:or
6666:m/s
6526:In
6465:not
6428:in
6302:see
6154:exp
5982:as
5700:If
5590:eff
5553:eff
5502:eff
5473:lim
5434:As
5279:eff
5200:eff
4996:eff
4667:eff
4501:eff
4377:eff
4338:eff
4094:to
4072:= (
3828:exh
3792:exh
3095:or
3027:or
2250:If
1192:lim
884:or
782:is
124:by
8516::
8196:,
8192:,
7801:/
7777:/
7658:.
7612:.
7604:.
7594:54
7592:.
7539:^
7531:27
7529:.
7525:.
7513:^
7490:^
7323:ln
7245:ln
7133:ln
7107:80
7089:ln
6764:.
6756:,
6752:,
6748:,
6744:,
6621:sp
6588:sp
6355:ln
6214:;
6187:ln
5692:.
5595:ln
5558:ln
5431:.
4396:.
4222:ln
4185:ln
4166:ln
4086:)/
4079:–
4057:–
3936::
3913:×
3836:ln
3702:ln
3675:ln
2914:ln
2380:,
1461::
1339::
1069:.
955::
798:ln
735:sp
720:;
692:sp
621:ln
606:sp
568:ln
8444:n
8276:0
8273:t
8263:v
8254:n
8245:T
8225:l
8216:L
8207:E
8198:f
8194:θ
8190:ν
8181:M
8161:ϖ
8152:ω
8143:Ω
8134:i
8114:q
8110:Q
8101:b
8092:a
8083:e
7728:e
7721:t
7714:v
7675:.
7643:e
7640:v
7636:R
7620:.
7608::
7600::
7507:.
7365:.
7360:e
7356:v
7349:=
7343:)
7336:/
7329:(
7318:e
7314:v
7310:=
7304:v
7268:e
7264:v
7257:=
7251:5
7240:e
7236:v
7232:3
7229:=
7223:v
7196:e
7192:v
7165:.
7160:e
7156:v
7149:=
7139:5
7128:e
7124:v
7120:=
7084:e
7080:v
7076:=
7066:v
7011:/
7000:e
6993:1
6973:v
6944:/
6933:e
6926:1
6906:v
6871:/
6860:e
6853:1
6826:v
6799:v
6683:0
6679:g
6650:e
6646:v
6617:I
6593:,
6584:I
6578:0
6574:g
6570:=
6565:e
6561:v
6510:v
6502:v
6495:v
6476:v
6436:v
6434:Δ
6422:v
6409:v
6386:)
6378:1
6374:m
6368:0
6364:m
6350:c
6345:e
6341:v
6334:(
6324:c
6321:=
6318:v
6300:(
6285:1
6282:+
6277:x
6274:2
6270:e
6264:1
6256:x
6253:2
6249:e
6242:=
6239:x
6197:]
6193:R
6182:c
6176:e
6172:v
6168:2
6161:[
6151:=
6145:c
6139:e
6135:v
6131:2
6125:R
6096:1
6093:+
6087:c
6081:e
6077:v
6073:2
6067:R
6061:1
6052:c
6046:e
6042:v
6038:2
6032:R
6025:=
6020:c
6016:v
5990:R
5966:1
5962:m
5956:0
5952:m
5921:e
5917:v
5913:2
5909:c
5903:]
5895:c
5891:v
5879:1
5872:c
5868:v
5859:+
5856:1
5850:[
5845:=
5838:1
5834:m
5828:0
5824:m
5797:c
5775:0
5771:m
5744:1
5740:m
5719:v
5680:)
5671:1
5668:(
5663:0
5659:m
5655:=
5650:f
5646:m
5625:,
5618:f
5614:m
5608:0
5604:m
5586:v
5582:=
5570:1
5566:1
5549:v
5545:=
5539:x
5533:1
5528:x
5525:d
5512:0
5498:v
5494:=
5491:v
5477:N
5442:N
5417:N
5410:j
5404:=
5399:j
5395:x
5372:N
5364:=
5361:x
5333:j
5329:x
5322:1
5317:x
5306:N
5303:=
5300:j
5295:1
5292:=
5289:j
5275:v
5271:=
5265:N
5261:/
5254:j
5248:1
5243:N
5239:/
5227:N
5224:=
5221:j
5216:1
5213:=
5210:j
5196:v
5189:v
5166:1
5160:N
5156:/
5132:N
5107:)
5104:N
5100:/
5093:+
5090:N
5086:/
5079:j
5073:1
5070:(
5067:)
5064:N
5060:/
5053:j
5047:1
5044:(
5039:N
5035:/
5023:N
5020:=
5017:j
5012:1
5009:=
5006:j
4992:v
4988:=
4985:v
4962:j
4942:)
4939:N
4935:/
4928:j
4922:1
4919:(
4914:0
4910:m
4906:=
4903:m
4883:j
4863:N
4859:/
4853:0
4849:m
4842:=
4837:p
4833:m
4812:N
4790:0
4786:m
4760:0
4756:m
4713:.
4707:)
4702:p
4698:m
4694:+
4691:m
4688:(
4685:m
4679:p
4675:m
4663:v
4659:=
4656:v
4628:p
4624:m
4620:m
4614:v
4608:=
4605:u
4583:.
4578:2
4574:)
4570:v
4564:(
4561:m
4555:2
4552:1
4546:+
4541:2
4537:u
4531:p
4527:m
4520:2
4517:1
4511:=
4506:2
4497:v
4491:p
4487:m
4480:2
4477:1
4454:m
4434:u
4412:p
4408:m
4382:2
4373:v
4366:2
4363:1
4334:v
4307:N
4287:N
4260:.
4256:)
4249:f
4245:m
4239:0
4235:m
4229:(
4217:e
4213:v
4206:=
4202:]
4196:0
4192:m
4177:f
4173:m
4162:[
4156:e
4152:v
4145:=
4140:0
4136:v
4127:f
4123:v
4119:=
4116:v
4096:T
4092:0
4088:R
4083:f
4081:m
4077:0
4074:m
4070:T
4066:R
4061:f
4059:m
4055:0
4052:m
4033:)
4030:t
4027:(
4024:m
4017:e
4013:v
4009:R
4000:=
3994:)
3991:t
3988:(
3985:m
3981:F
3973:=
3967:t
3964:d
3959:v
3956:d
3950:=
3947:a
3934:m
3930:F
3926:t
3917:e
3915:v
3911:R
3907:F
3902:e
3870:)
3863:f
3859:m
3853:0
3849:m
3843:(
3824:V
3820:=
3817:v
3788:V
3784:=
3779:p
3776:F
3771:=
3765:m
3758:J
3731:m
3723:)
3717:f
3713:m
3708:(
3696:)
3690:0
3686:m
3681:(
3666:J
3644:J
3641:=
3638:t
3635:d
3629:F
3622:f
3618:t
3610:0
3606:t
3577:m
3552:0
3548:m
3525:t
3522:d
3512:m
3504:t
3494:0
3490:m
3482:|
3478:T
3474:|
3463:f
3459:t
3451:0
3447:t
3438:=
3435:v
3407:(
3395:V
3364:f
3360:m
3351:0
3347:m
3321:e
3317:v
3294:e
3290:v
3259:e
3255:v
3232:f
3228:m
3205:0
3201:m
3179:)
3175:1
3165:e
3161:v
3156:/
3152:V
3145:e
3140:(
3134:f
3130:m
3126:=
3121:f
3117:m
3108:0
3104:m
3079:e
3075:v
3070:/
3066:V
3059:e
3053:f
3049:m
3045:=
3040:0
3036:m
3011:e
3007:v
3002:/
2995:V
2985:e
2979:0
2975:m
2971:=
2966:f
2962:m
2937:f
2933:m
2927:0
2923:m
2909:e
2905:v
2901:=
2898:V
2871:m
2867:m
2864:d
2854:f
2850:m
2842:0
2838:m
2826:e
2822:v
2817:=
2814:V
2811:d
2805:V
2799:+
2796:V
2791:V
2757:e
2753:v
2727:t
2724:d
2719:m
2716:d
2708:e
2704:v
2700:=
2694:t
2691:d
2686:V
2683:d
2677:m
2662:(
2650:0
2647:=
2642:i
2638:F
2632:i
2602:t
2599:d
2594:m
2591:d
2583:e
2579:v
2575:+
2569:t
2566:d
2561:V
2558:d
2552:m
2549:=
2544:i
2540:F
2534:i
2509:m
2486:m
2477:=
2474:m
2471:d
2451:0
2439:v
2433:d
2429:m
2426:d
2400:V
2391:m
2362:V
2337:i
2327:F
2301:e
2291:v
2261:V
2232:V
2223:m
2214:m
2206:e
2196:v
2189:+
2180:V
2171:m
2168:=
2163:0
2153:P
2141:t
2128:P
2102:e
2092:v
2085:+
2076:V
2070:=
2065:e
2055:V
2025:V
2014:e
2004:V
1997:=
1992:e
1982:v
1956:e
1952:v
1929:e
1919:V
1892:t
1886:=
1883:t
1862:)
1858:m
1849:m
1845:(
1823:0
1820:=
1817:t
1797:m
1776:t
1751:e
1741:V
1716:t
1710:=
1707:t
1681:V
1672:+
1663:V
1639:0
1636:=
1633:t
1607:V
1579:e
1569:V
1562:m
1556:+
1552:)
1542:V
1533:+
1524:V
1517:(
1512:)
1508:m
1499:m
1495:(
1491:=
1486:t
1473:P
1449:t
1443:=
1440:t
1418:t
1405:P
1375:V
1369:m
1366:=
1361:0
1351:P
1327:0
1324:=
1321:t
1299:0
1289:P
1262:t
1252:0
1242:P
1230:t
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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