Hohmann transfer orbit

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https://en.wikipedia.org/wiki/Hohmann_transfer_orbit

 

Hohmann transfer orbit, labelled 2, from an orbit (1) to a higher orbit (3).

 

An example of the Hohmann transfer orbit
   InSight ·   Earth ·   Mars

 

In orbital mechanics, the Hohmann transfer orbit (/ˈhoʊmən/) is an elliptical orbit used to transfer between two circular orbits of different radii around a central body in the same plane. The Hohmann transfer often uses the lowest possible amount of propellant in traveling between these orbits, but bi-elliptic transfers can beat it in some cases. 

The orbital maneuver to perform the Hohmann transfer uses two engine impulses, one to move a spacecraft onto the transfer orbit and a second to move off it. This maneuver was named after Walter Hohmann, the German scientist who published a description of it in his 1925 book Die Erreichbarkeit der Himmelskörper (The Attainability of Celestial Bodies). Hohmann was influenced in part by the German science fiction author Kurd Lasswitz and his 1897 book Two Planets. 

The elliptic transfer orbits between different bodies (planets, moons etc.) is often referred to as Hohmann transfer orbits. When used for travelling between celestial bodies, a Hohmann transfer orbit requires that the starting and destination points be at particular locations in their orbits relative to each other. Space missions using a Hohmann transfer must wait for this required alignment to occur, which opens a so-called launch window. For a space mission between Earth and Mars, for example, these launch windows occur every 26 months. A Hohmann transfer orbit also determines a fixed time required to travel between the starting and destination points; for an Earth-Mars journey this travel time is about 9 months. When transfer is performed between orbits close to cellestial bodies with significant gravitation, much less delta-v is usually required, as Oberth effect may be employed for the burns. 

They are also often used for these situations, but low-energy transfers which take into account the thrust limitations of real engines, and take advantage of the gravity wells of both planets can be more fuel efficient.

Explanation

The diagram shows a Hohmann transfer orbit to bring a spacecraft from a lower circular orbit into a higher one. It is one half of an elliptic orbit that touches both the lower circular orbit the spacecraft wishes to leave (blue and labeled 1 on diagram) and the higher circular orbit that it wishes to reach (red and labeled 3 on diagram). The transfer (yellow and labeled 2 on diagram) is initiated by firing the spacecraft's engine to accelerate it so that it will follow the elliptical orbit. This adds energy to the spacecraft's orbit. When the spacecraft has reached its destination orbit, its orbital speed (and hence its orbital energy) must be increased again to change the elliptic orbit to the larger circular one.

Due to the reversibility of orbits, Hohmann transfer orbits also work to bring a spacecraft from a higher orbit into a lower one; in this case, the spacecraft's engine is fired in the opposite direction to its current path, slowing the spacecraft and causing it to drop into the lower-energy elliptical transfer orbit. The engine is then fired again at the lower distance to slow the spacecraft into the lower circular orbit. 

The Hohmann transfer orbit is based on two instantaneous velocity changes. Extra fuel is required to compensate for the fact that the bursts take time; this is minimized by using high-thrust engines to minimize the duration of the bursts. For transfers in Earth orbit, the two burns are labelled the perigee burn and the apogee burn (or ''apogee kick); more generally, they are labelled periapsis and apoapsis burns. Alternately, the second burn to circularize the orbit may be referred to as a circularization burn. 

Type I and Type II

An ideal Hohmann transfer orbit transfers between two circular orbits in the same plane and traverses exactly 180° around the primary. In the real world, the destination orbit may not be circular, and may not be coplanar with the initial orbit. Real world transfer orbits may traverse slightly more, or slightly less, than 180° around the primary. An orbit which traverses less than 180° around the primary is called a "Type I" Hohmann transfer, while an orbit which traverses more than 180° is called a "Type II" Hohmann transfer.

Calculation

For a small body orbiting another much larger body, such as a satellite orbiting Earth, the total energy of the smaller body is the sum of its kinetic energy and potential energy, and this total energy also equals half the potential at the average distance ${\displaystyle a}$ (the semi-major axis):

${\displaystyle E={\frac {mv^{2}}{2}}-{\frac {GMm}{r}}={\frac {-GMm}{2a}}.}$

Solving this equation for velocity results in the vis-viva equation,

${\displaystyle v^{2}=\mu \left({\frac {2}{r}}-{\frac {1}{a}}\right),}$

where:

• ${\displaystyle v}$ is the speed of an orbiting body,
• ${\displaystyle \mu =GM}$ is the standard gravitational parameter of the primary body, assuming ${\displaystyle M+m}$ is not significantly bigger than ${\displaystyle M}$ (which makes ${\displaystyle v_{M}\ll v}$),
• ${\displaystyle r}$ is the distance of the orbiting body from the primary focus,
• ${\displaystyle a}$ is the semi-major axis of the body's orbit.

Therefore, the delta-v required for the Hohmann transfer can be computed as follows, under the assumption of instantaneous impulses:

${\displaystyle \Delta v_{1}={\sqrt {\frac {\mu }{r_{1}}}}\left({\sqrt {\frac {2r_{2}}{r_{1}+r_{2}}}}-1\right),}$

to enter the elliptical orbit at ${\displaystyle r=r_{1}}$ from the ${\displaystyle r_{1}}$ circular orbit

${\displaystyle \Delta v_{2}={\sqrt {\frac {\mu }{r_{2}}}}\left(1-{\sqrt {\frac {2r_{1}}{r_{1}+r_{2}}}}\right),}$

to leave the elliptical orbit at ${\displaystyle r=r_{2}}$ to the ${\displaystyle r_{2}}$ circular orbit, where ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ are respectively the radii of the departure and arrival circular orbits; the smaller (greater) of ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ corresponds to the periapsis distance (apoapsis distance) of the Hohmann elliptical transfer orbit. Typically, ${\displaystyle \mu }$ is given in units of m³/s², as such be sure to use meters, not kilometers, for ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$. The total ${\displaystyle \Delta v}$ is then:

${\displaystyle \Delta v_{\text{total}}=\Delta v_{1}+\Delta v_{2}.}$

Whether moving into a higher or lower orbit, by Kepler's third law, the time taken to transfer between the orbits is

${\displaystyle t_{\text{H}}={\sqrt {\frac {\pi ^{2}a_{\text{H}}^{3}}{\mu }}}=\pi {\sqrt {\frac {(r_{1}+r_{2})^{3}}{8\mu }}}}$

(one half of the orbital period for the whole ellipse), where ${\displaystyle a_{\text{H}}}$ is length of semi-major axis of the Hohmann transfer orbit. 

In application to traveling from one celestial body to another it is crucial to start maneuver at the time when the two bodies are properly aligned. Considering the target angular velocity being

${\displaystyle \omega _{2}={\sqrt {\frac {\mu }{r_{2}^{3}}}},}$

angular alignment α (in radians) at the time of start between the source object and the target object shall be

${\displaystyle \alpha =\pi -\omega _{2}t_{\text{H}}=\pi \left(1-{\frac {1}{2{\sqrt {2}}}}{\sqrt {\left({\frac {r_{1}}{r_{2}}}+1\right)^{3}}}\right).}$

Example

Total energy balance during a Hohmann transfer between two circular orbits with first radius ${\displaystyle r_{p}}$ and second radius ${\displaystyle r_{a}}$

Consider a geostationary transfer orbit, beginning at r₁ = 6,678 km (altitude 300 km) and ending in a geostationary orbit with r₂ = 42,164 km (altitude 35,786 km).

In the smaller circular orbit the speed is 7.73 km/s; in the larger one, 3.07 km/s. In the elliptical orbit in between the speed varies from 10.15 km/s at the perigee to 1.61 km/s at the apogee.

The Δv for the two burns are thus 10.15 − 7.73 = 2.42 and 3.07 − 1.61 = 1.46 km/s, together 3.88 km/s. 

This is greater than the Δv required for an escape orbit: 10.93 − 7.73 = 3.20 km/s. Applying a Δv at the Low Earth orbit (LEO) of only 0.78 km/s more (3.20−2.42) would give the rocket the escape speed, which is less than the Δv of 1.46 km/s required to circularize the geosynchronous orbit. This illustrates the Oberth effect that at large speeds the same Δv provides more specific orbital energy, and energy increase is maximized if one spends the Δv as quickly as possible, rather than spending some, being decelerated by gravity, and then spending some more to overcome the deceleration (of course, the objective of a Hohmann transfer orbit is different).

Worst case, maximum delta-v

As the example above demonstrates, the Δv required to perform a Hohmann transfer between two circular orbits is not the greatest when the destination radius is infinite. (Escape speed is √2 times orbital speed, so the Δv required to escape is √2 − 1 (41.4%) of the orbital speed.) The Δv required is greatest (53.0% of smaller orbital speed) when the radius of the larger orbit is 15.5817... times that of the smaller orbit. This number is the positive root of  x³ − 15 x² − 9 x − 1 = 0, which is  ${\displaystyle 5+4\,{\sqrt {7}}\cos \left({1 \over 3}\arctan {{\sqrt {3}} \over 37}\right)}$. For higher orbit ratios the Δv required for the second burn decreases faster than the first increases. 

Application to interplanetary travel

When used to move a spacecraft from orbiting one planet to orbiting another, the situation becomes somewhat more complex, but much less delta-v is required, due to the Oberth effect, than the sum of the delta-v required to escape the first planet plus the delta-v required for a Hohmann transfer to the second planet.

For example, consider a spacecraft travelling from the Earth to Mars. At the beginning of its journey, the spacecraft will already have a certain velocity and kinetic energy associated with its orbit around Earth. During the burn the rocket engine applies its delta-v, but the kinetic energy increases as a square law, until it is sufficient to escape the planet's gravitational potential, and then burns more so as to gain enough energy to get into the Hohmann transfer orbit (around the Sun). Because the rocket engine is able to make use of the initial kinetic energy of the propellant, far less delta-v is required over and above that needed to reach escape velocity, and the optimum situation is when the transfer burn is made at minimum altitude (low periapsis) above the planet. The delta-v needed is only 3.6 km/s, only about 0.4 km/s more than needed to escape Earth, even though this results in the spacecraft going 2.9 km/s faster than the Earth as it heads off for Mars (see table below). 

At the other end, the spacecraft will need a certain velocity to orbit Mars, which will actually be less than the velocity needed to continue orbiting the Sun in the transfer orbit, let alone attempting to orbit the Sun in a Mars-like orbit. Therefore, the spacecraft will have to decelerate in order for the gravity of Mars to capture it. This capture burn should optimally be done at low altitude to also make best use of Oberth effect. Therefore, relatively small amounts of thrust at either end of the trip are needed to arrange the transfer compared to the free space situation. 

However, with any Hohmann transfer, the alignment of the two planets in their orbits is crucial – the destination planet and the spacecraft must arrive at the same point in their respective orbits around the Sun at the same time. This requirement for alignment gives rise to the concept of launch windows. 

The term lunar transfer orbit (LTO) is used for the Moon. 

It is possible to apply the formula given above to calculate the Δv in km/s needed to enter a Hohmann transfer orbit to arrive at various destinations from Earth (assuming circular orbits for the planets). In this table, the column labeled "Δv to enter Hohmann orbit from Earth's orbit" gives the change from Earth's velocity to the velocity needed to get on a Hohmann ellipse whose other end will be at the desired distance from the Sun. The column labeled "v exiting LEO" gives the velocity needed (in a non-rotating frame of reference centered on the earth) when 300 km above the Earth's surface. This is obtained by adding to the specific kinetic energy the square of the speed (7.73 km/s) of this low Earth orbit (that is, the depth of Earth's gravity well at this LEO). The column "Δv from LEO" is simply the previous speed minus 7.73 km/s. 

Destination	Orbital radius (AU)	Δv (km/s)
		to enter Hohmann orbit from Earth's orbit	exiting LEO	from LEO
Sun	0	29.8	31.7	24.0
Mercury	0.39	7.5	13.3	5.5
Venus	0.72	2.5	11.2	3.5
Mars	1.52	2.9	11.3	3.6
Jupiter	5.2	8.8	14.0	6.3
Saturn	9.54	10.3	15.0	7.3
Uranus	19.19	11.3	15.7	8.0
Neptune	30.07	11.7	16.0	8.2
Pluto	39.48	11.8	16.1	8.4
Infinity	∞	12.3	16.5	8.8

Note that in most cases, Δv from LEO is less than the Δv to enter Hohmann orbit from Earth's orbit.

To get to the Sun, it is actually not necessary to use a Δv of 24 km/s. One can use 8.8 km/s to go very far away from the Sun, then use a negligible Δv to bring the angular momentum to zero, and then fall into the Sun. This can be considered a sequence of two Hohmann transfers, one up and one down. Also, the table does not give the values that would apply when using the Moon for a gravity assist. There are also possibilities of using one planet, like Venus which is the easiest to get to, to assist getting to other planets or the Sun. 

Comparison to other transfers

Bi-elliptic transfer

The bi-elliptic transfer consists of two half-elliptic orbits. From the initial orbit, a first burn expends delta-v to boost the spacecraft into the first transfer orbit with an apoapsis at some point ${\displaystyle r_{b}}$ away from the central body. At this point a second burn sends the spacecraft into the second elliptical orbit with periapsis at the radius of the final desired orbit, where a third burn is performed, injecting the spacecraft into the desired orbit.

While they require one more engine burn than a Hohmann transfer and generally require a greater travel time, some bi-elliptic transfers require a lower amount of total delta-v than a Hohmann transfer when the ratio of final to initial semi-major axis is 11.94 or greater, depending on the intermediate semi-major axis chosen.

The idea of the bi-elliptical transfer trajectory was first published by Ary Sternfeld in 1934.

Low-thrust transfer

Low-thrust engines can perform an approximation of a Hohmann transfer orbit, by creating a gradual enlargement of the initial circular orbit through carefully timed engine firings. This requires a change in velocity (delta-v) that is greater than the two-impulse transfer orbit and takes longer to complete.

Engines such as ion thrusters are more difficult to analyze with the delta-v model. These engines offer a very low thrust and at the same time, much higher delta-v budget, much higher specific impulse, lower mass of fuel and engine. A 2-burn Hohmann transfer maneuver would be impractical with such a low thrust; the maneuver mainly optimizes the use of fuel, but in this situation there is relatively plenty of it.

If only low-thrust maneuvers are planned on a mission, then continuously firing a low-thrust, but very high-efficiency engine might generate a higher delta-v and at the same time use less propellant than a conventional chemical rocket engine.

Going from one circular orbit to another by gradually changing the radius simply requires the same delta-v as the difference between the two speeds. Such maneuver requires more delta-v than a 2-burn Hohmann transfer maneuver, but does so with continuous low thrust rather than the short applications of high thrust. 

The amount of propellant mass used measures the efficiency of the maneuver plus the hardware employed for it. The total delta-v used measures the efficiency of the maneuver only. For electric propulsion systems, which tend to be low-thrust, the high efficiency of the propulsive system usually compensates for the higher delta-V compared to the more efficient Hohmann maneuver.

Transfer orbit using electrical propulsion or low-thrust engines optimize the transfer time to reach the final orbit and not the delta-v as in the Hohmann transfer orbit. For geostationary orbit, the initial orbit is set to be supersynchronous and by thrusting continuously in the direction of the velocity at apogee, the transfer orbit transforms to a circular geosynchronous one. This method however takes much longer to achieve due to the low thrust injected into the orbit.

Interplanetary Transport Network

In 1997, a set of orbits known as the Interplanetary Transport Network (ITN) was published, providing even lower propulsive delta-v (though much slower and longer) paths between different orbits than Hohmann transfer orbits. The Interplanetary Transport Network is different in nature than Hohmann transfers because Hohmann transfers assume only one large body whereas the Interplanetary Transport Network does not. The Interplanetary Transport Network is able to achieve the use of less propulsive delta-v by employing gravity assist from the planets.

Super heavy-lift launch vehicle

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Super_heavy-lift_launch_vehicle

 

Comparison of Energia, Falcon Heavy, Yenisei, Long March 9, SLS, N1, Saturn V, and Starship. Masses listed are the maximum payload to low Earth orbit in metric tons.

 

A super heavy-lift launch vehicle (SHLLV) is a launch vehicle capable of lifting more than 50 tonnes (110,000 lb) of payload into low Earth orbit (LEO).

Flown vehicles

Never made it to orbit

• N1, Soviet Moon rocket. Developed in late 1960s and early 1970s. Made 4 orbital launch attempts but did not reach orbit on any one of those flights. After the 4 failed launches, the project was cancelled in 1976.

 Retired

• Saturn V, with an Apollo program payload of a command module, service module, and Lunar Module. The three had a total mass of 45 t (99,000 lb). When the third stage and Earth-orbit departure fuel was included, Saturn V actually placed 140 t (310,000 lb) into low Earth orbit. The final launch of Saturn V placed Skylab, a 77,111 kg (170,001 lb) payload, into LEO.
• The Space Shuttle orbited a combined shuttle and cargo mass of 122,534 kg (270,142 lb) when launching the Chandra X-ray Observatory on STS-93. Chandra and its two-stage Inertial Upper Stage booster rocket weighed 22,753 kg (50,162 lb).
• The Energia system was designed to launch up to 105 t (231,000 lb) to low Earth orbit. Energia launched twice before the program was cancelled, but only one flight reached orbit. On the first flight, launching the Polyus weapons platform (approximately 80 t (180,000 lb)), the vehicle failed to enter orbit due to a software error on the kick-stage. The second flight successfully launched the Buran orbiter. 

The Space Shuttle and Buran differed from traditional rockets in that both launched what was essentially a reusable stage that carried cargo internally. 

Operational, but unproven as super heavy-lift

• Falcon Heavy is rated to launch 63.8 t (141,000 lb) to low Earth orbit (LEO) in a fully expendable configuration and an estimated 57 t (126,000 lb) in a partially reusable configuration, in which only two of its three boosters are recovered. Neither of these super-heavy lift configurations have been flown or are being planned to fly as of June 2019. The first test flight occurred on 6 February 2018, in a configuration in which recovery of all three boosters was attempted, with a small payload of 1,250 kg (2,760 lb) sent to an orbit beyond Mars. Since the vehicle is operational but has not yet been demonstrated to launch payloads over 50 tonnes (110,000 lb) to orbit, it is as yet unproven as a super heavy-lift capable launch vehicle.

Comparison

Rocket	Configuration	Organization	Nationality	LEO payload	Maiden flight	First >50t payload	Operational	Reusable
Saturn V	Apollo	NASA	United States	140 t (310,000 lb)A	1967	1967	Retired	No
N1	L3	Energia	Soviet Union	95 t (209,000 lb)	1969 (failed)	N/A	Cancelled	No
Space Shuttle		NASA	United States	122.5 t (270,142 lb)B	1981	1981	Retired	Partially
Energia	Buran	Energia	Soviet Union	100 t (220,000 lb)C	1987	1987	Retired	No
Falcon Heavy	ExpendedD	SpaceX	United States (private)	63.8 t (141,000 lb)	Not YetD	Not Yet	UnprovenD	No
	Recoverable side boostersE			57 t (126,000 lb)	Not YetD	Not Yet	UnprovenD	PartiallyE
Starship		SpaceX	United States (private)	150 t (330,000 lb)F	2020 (planned)	N/A	Development	Fully
SLS	Block 1	NASA	United States	95 t (209,000 lb)	2021 (planned)	N/A	Development	No
	Block 1B			105 t (231,000 lb)	2024 (planned)	N/A	Development	No
	Block 2			130 t (290,000 lb)	TBA	N/A	Development	No
Long March 9		CALT	China	140 t (310,000 lb)	2028 (planned)	N/A	Development	No
Yenisei	Yenisei	JSC SRC Progress	Russia	103 t (227,000 lb)	2028 (planned)	N/A	Development	No
	Don			130 t (290,000 lb)	2030 (planned)	N/A	Development	No
New Armstrong		Blue Origin	United States (private)	TBA	TBA	N/A	Proposed	Partially or Fully

^A Includes mass of Apollo command and service modules, Apollo Lunar Module, Spacecraft/LM Adapter, Saturn V Instrument Unit, S-IVB stage, and propellant for translunar injection; payload mass to LEO is about 122.4 t (270,000 lb)^[29] ^B Includes mass of orbiter and payload during STS-93; deployable payload is 27.5 t (61,000 lb) ^C Required upper stage or payload to perform final orbital insertion ^D Falcon Heavy has not yet flown in a configuration that would allow lifting 50 tonnes to LEO; to date it has only flown in the configuration that permits the possibility of recovery of the centre core (actually doing so is irrelevant) which is a configuration capable of lifting a maximum of 45 tonnes to LEO ^E Side booster cores recoverable and centre core intentionally expended. First re-use of the side boosters was demonstrated in 2019 when the ones used on the Arabsat-6A launch were reused on the STP-2 launch. ^F Does not include dry mass of spaceship

Proposed designs

The Space Launch System (SLS) is a US super heavy-lift expendable launch vehicle, which is under development as of August 2019. It is the primary launch vehicle of NASA's deep space exploration plans, including the planned crewed lunar flights of the Artemis program and a possible follow-on human mission to Mars.

The SpaceX Starship is both the second stage of a reusable launch vehicle and a spacecraft that is being developed by SpaceX, as a private spaceflight project.  It is being designed to be a long-duration cargo- and passenger-carrying spacecraft. While it is tested on its own initially, it will be used on orbital launches with an additional booster stage, the Super Heavy, where Starship would serve as the second stage on a two-stage-to-orbit launch vehicle. The combination of spacecraft and booster is called Starship as well.

New Armstrong is a super heavy-lift rocket proposed by Blue Origin. Payload and timeline are unknown.

Long March 9, a 140 t (310,000 lb) to LEO capable rocket has been proposed by China. It has a targeted capacity of 50 t (110,000 lb) to lunar transfer orbit and first flight by 2030.

Yenisei, a super heavy-lift launch vehicle using existing components instead of pushing the less-powerful Angara A5V project, has been proposed by Russia's RSC Energia in August 2016. This would allow Russia to launch missions towards establishing a permanent Moon base with simpler logistics, launching just one or two 80-to-160-tonne super-heavy rockets instead of four 40-tonne Angara A5Vs implying quick-sequence launches and multiple in-orbit rendezvous. In February 2018, the КРК СТК (space rocket complex of the super-heavy class) design was updated to lift at least 90 tonnes to LEO and 20 tonnes to lunar polar orbit, and to be launched from Vostochny Cosmodrome. The first flight is scheduled for 2028, with Moon landings starting in 2030.

ISRO is conducting preliminary research for the development of a super heavy-lift launch vehicle which is planned to have a lifting capacity of over 50-60 tonnes (presumably into LEO).

Cancelled designs

Comparison of Saturn V, Sea Dragon and Interplanetary Transport System

 

Numerous super-heavy lift vehicles have been proposed and received various levels of development prior to their cancellation.

As part of the Soviet Lunar Project four N1 rockets with a payload capacity of 95 t (209,000 lb), were launched but all failed shortly after lift-off (1969-1972). The program was suspended in May 1974 and formally cancelled in March 1976.

During project Aelita (1969-1972), the Soviets were developing a way to beat the Americans to Mars. They designed the UR-700m, a nuclear powered variant of the UR-700, to assemble the 1400 t (3,000,000 lb) MK-700 spacecraft in earth orbit in 2 launches. The rocket would have a payload capacity of 750 t (1,650,000 lb) and is the most capable rocket ever designed. It is often overlooked due too little information being known about the design. The only Universal Rocket to make it past the design phase was the UR-500 while the N1 was selected to be the Soviets HLV for lunar and Mars missions.

The U.S. Ares V for the Constellation program was intended to reuse many elements of the Space Shuttle program, both on the ground and flight hardware, to save costs. The Ares V was designed to carry 188 t (414,000 lb) and was cancelled in 2010, though much of the work has been carried forward into the Artemis program. 

A 1962 design proposal, Sea Dragon, called for an enormous 150 m (490 ft) tall, sea-launched rocket capable of lifting 550 t (1,210,000 lb) to low Earth orbit. Although preliminary engineering of the design was done by TRW, the project never moved forward due to the closing of NASA's Future Projects Branch.

SpaceX's first publicly released design of its Mars transportation infrastructure was the ITS launch vehicle unveiled in 2016. The payload capability was to be 550 t (1,210,000 lb) in an expendable configuration or 300 t (660,000 lb) in a reusable configuration. In 2017, it was succeeded by Starship.