هفت ماه تا مریخ: نگاهی دقیق به فرایند سفر کاوشگر HX-1 به سیاره سرخ
۲۰۲۰ سالی هیجان‌انگیز برای سفر به مریخ محسوب می‌شود. یکی از مسافران مریخ در این سال، کاوشگر سه‌گانه‌ی HX-1 است که توسط سازمان فضایی چین ...
هوهمان
براساس اصول مکانیک سماوی، بهترین زمان برای ارسال مأموریت‌های مریخ، اواخر ماه ژوئیه سال جاری میلادی است. این محدودیت، به دلیل نیاز به استفاده از دوره‌ی زمانی (پنجره‌ی پرتاب) انتقالی <anchor>هوهمان</anchor> است که براساس آن، هر ۲۶ ماه یکبار مریخ و زمین در موقعیتی نسبت به یکدیگر قرار می‌گیرند که می‌توان با کمترین میزان سوخت سفر به مریخ را به انجام رساند. به همین دلیل، در کنار سازمان فضایی چین، کاوشگر Perseverance ناسا و مدارگرد امارات متحده‌ی عربی موسوم به «امید» هم در همین بازه‌ی زمانی راهی سیاره‌ی سرخ خواهند شد.
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## Contents

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- (Top)

- [1Example](#Example)Toggle Example subsection  - [1.1Type I and Type II](#Type_I_and_Type_II)
  - [1.2Uses](#Uses)
- [2Calculation](#Calculation)
- [3Example](#Example_2)
- [4Worst case, maximum delta- _v_](#Worst_case,_maximum_delta-v)
- [5Application to interplanetary travel](#Application_to_interplanetary_travel)
- [6Comparison to other transfers](#Comparison_to_other_transfers)Toggle Comparison to other transfers subsection  - [6.1Bi-elliptic transfer](#Bi-elliptic_transfer)
  - [6.2Low-thrust transfer](#Low-thrust_transfer)
  - [6.3Interplanetary Transport Network](#Interplanetary_Transport_Network)
- [7See also](#See_also)
- [8Citations](#Citations)
- [9General and cited sources](#General_and_cited_sources)
- [10Further reading](#Further_reading)

Toggle the table of contents

# Hohmann transfer orbit

32 languages

- [العربية](https://ar.wikipedia.org/wiki/%D9%85%D8%AF%D8%A7%D8%B1_%D9%87%D9%88%D9%87%D9%85%D8%A7%D9%86_%D8%A7%D9%84%D8%A7%D9%86%D8%AA%D9%82%D8%A7%D9%84%D9%8A "مدار هوهمان الانتقالي – Arabic")
- [Aragonés](https://an.wikipedia.org/wiki/Orbita_de_transferencia_de_Hohmann "Orbita de transferencia de Hohmann – Aragonese")
- [Беларуская](https://be.wikipedia.org/wiki/%D0%93%D0%BE%D0%BC%D0%B0%D0%BD%D0%B0%D1%9E%D1%81%D0%BA%D0%B0%D1%8F_%D1%82%D1%80%D0%B0%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D1%8B%D1%8F "Гоманаўская траекторыя – Belarusian")
- [Български](https://bg.wikipedia.org/wiki/%D0%A5%D0%BE%D0%BC%D0%B0%D0%BD%D0%BE%D0%B2%D0%B0_%D1%82%D1%80%D0%B0%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D1%8F "Хоманова траектория – Bulgarian")
- [Català](https://ca.wikipedia.org/wiki/%C3%92rbita_de_Hohmann "Òrbita de Hohmann – Catalan")
- [Čeština](https://cs.wikipedia.org/wiki/Hohmannova_elipsa "Hohmannova elipsa – Czech")
- [Dansk](https://da.wikipedia.org/wiki/Hohmann-bane "Hohmann-bane – Danish")
- [Deutsch](https://de.wikipedia.org/wiki/Hohmann-Transfer "Hohmann-Transfer – German")
- [Español](https://es.wikipedia.org/wiki/%C3%93rbita_de_transferencia_de_Hohmann "Órbita de transferencia de Hohmann – Spanish")
- [فارسی](https://fa.wikipedia.org/wiki/%D9%85%D8%AF%D8%A7%D8%B1_%D9%87%D9%88%D9%87%D9%85%D8%A7%D9%86 "مدار هوهمان – Persian")
- [Français](https://fr.wikipedia.org/wiki/Orbite_de_transfert "Orbite de transfert – French")
- [한국어](https://ko.wikipedia.org/wiki/%ED%98%B8%EB%A7%8C_%EC%A0%84%EC%9D%B4_%EA%B6%A4%EB%8F%84 "호만 전이 궤도 – Korean")
- [हिन्दी](https://hi.wikipedia.org/wiki/%E0%A4%B9%E0%A5%8B%E0%A4%B9%E0%A4%AE%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A8_%E0%A4%B8%E0%A5%8D%E0%A4%A5%E0%A4%BE%E0%A4%A8%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A4%B0%E0%A4%A3_%E0%A4%95%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%BE "होहमान्न स्थानांतरण कक्षा – Hindi")
- [Bahasa Indonesia](https://id.wikipedia.org/wiki/Orbit_transfer_Hohmann "Orbit transfer Hohmann – Indonesian")
- [Italiano](https://it.wikipedia.org/wiki/Trasferimento_alla_Hohmann "Trasferimento alla Hohmann – Italian")
- [עברית](https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%9C%D7%95%D7%9C_%D7%94%D7%95%D7%94%D7%9E%D7%9F "מסלול הוהמן – Hebrew")
- [Magyar](https://hu.wikipedia.org/wiki/Hohmann-p%C3%A1lya "Hohmann-pálya – Hungarian")
- [日本語](https://ja.wikipedia.org/wiki/%E3%83%9B%E3%83%BC%E3%83%9E%E3%83%B3%E9%81%B7%E7%A7%BB%E8%BB%8C%E9%81%93 "ホーマン遷移軌道 – Japanese")
- [Norsk bokmål](https://no.wikipedia.org/wiki/Hohmann-bane "Hohmann-bane – Norwegian Bokmål")
- [Polski](https://pl.wikipedia.org/wiki/Manewr_transferowy_Hohmanna "Manewr transferowy Hohmanna – Polish")
- [Português](https://pt.wikipedia.org/wiki/%C3%93rbita_de_transfer%C3%AAncia_de_Hohmann "Órbita de transferência de Hohmann – Portuguese")
- [Română](https://ro.wikipedia.org/wiki/Transfer_orbital_Hohmann "Transfer orbital Hohmann – Romanian")
- [Русский](https://ru.wikipedia.org/wiki/%D0%93%D0%BE%D0%BC%D0%B0%D0%BD%D0%BE%D0%B2%D1%81%D0%BA%D0%B0%D1%8F_%D1%82%D1%80%D0%B0%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D1%8F "Гомановская траектория – Russian")
- [Simple English](https://simple.wikipedia.org/wiki/Hohmann_transfer_orbit "Hohmann transfer orbit – Simple English")
- [Slovenčina](https://sk.wikipedia.org/wiki/Hohmannova_prechodov%C3%A1_dr%C3%A1ha "Hohmannova prechodová dráha – Slovak")
- [Suomi](https://fi.wikipedia.org/wiki/Hohmannin_siirtorata "Hohmannin siirtorata – Finnish")
- [Svenska](https://sv.wikipedia.org/wiki/Hohmannbana "Hohmannbana – Swedish")
- [தமிழ்](https://ta.wikipedia.org/wiki/%E0%AE%93%E0%AE%95%E0%AF%81%E0%AE%AE%E0%AE%BE%E0%AE%A9%E0%AF%8D_%E0%AE%AA%E0%AF%86%E0%AE%AF%E0%AE%B0%E0%AF%8D%E0%AE%B5%E0%AF%81_%E0%AE%B5%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%A3%E0%AF%88 "ஓகுமான் பெயர்வு வட்டணை – Tamil")
- [Türkçe](https://tr.wikipedia.org/wiki/Hohmann_transfer_y%C3%B6r%C3%BCngesi "Hohmann transfer yörüngesi – Turkish")
- [Українська](https://uk.wikipedia.org/wiki/%D0%93%D0%BE%D0%BC%D0%B0%D0%BD%D1%96%D0%B2%D1%81%D1%8C%D0%BA%D0%B0_%D1%82%D1%80%D0%B0%D1%94%D0%BA%D1%82%D0%BE%D1%80%D1%96%D1%8F "Гоманівська траєкторія – Ukrainian")
- [Tiếng Việt](https://vi.wikipedia.org/wiki/Qu%E1%BB%B9_%C4%91%E1%BA%A1o_chuy%E1%BB%83n_ti%E1%BA%BFp_Hohmann "Quỹ đạo chuyển tiếp Hohmann – Vietnamese")
- [中文](https://zh.wikipedia.org/wiki/%E9%9C%8D%E6%9B%BC%E8%BD%89%E7%A7%BB%E8%BB%8C%E9%81%93 "霍曼轉移軌道 – Chinese")

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This article may **require [cleanup](/wiki/Wikipedia:Cleanup "Wikipedia:Cleanup")** to meet Wikipedia's [quality standards](/wiki/Wikipedia:Manual_of_Style "Wikipedia:Manual of Style"). The specific problem is: **The language is clumsy in several places.** Please help [improve this article](/wiki/Special:EditPage/Hohmann_transfer_orbit "Special:EditPage/Hohmann transfer orbit") if you can._(June 2022)__([Learn how and when to remove this message](/wiki/Help:Maintenance_template_removal "Help:Maintenance template removal"))_

Transfer manoeuvre between two orbits

[![](//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Hohmann_transfer_orbit.svg/220px-Hohmann_transfer_orbit.svg.png)](/wiki/File:Hohmann_transfer_orbit.svg) Hohmann transfer orbit, labelled 2, from an orbit (1) to a higher orbit (3)[![](//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Animation_of_InSight_trajectory.gif/220px-Animation_of_InSight_trajectory.gif)](/wiki/File:Animation_of_InSight_trajectory.gif) An example of a Hohmann transfer orbit between Earth and Mars, as used by the [NASA](/wiki/NASA "NASA") [InSight](/wiki/InSight "InSight") probe:

InSight **·**[Earth](/wiki/Earth "Earth") **·**[Mars](/wiki/Mars "Mars")Part of a series onAstrodynamics[![](//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Orbit_mechanics_icon.svg/60px-Orbit_mechanics_icon.svg.png)](/wiki/File:Orbit_mechanics_icon.svg)

[Orbital mechanics](/wiki/Orbital_mechanics "Orbital mechanics")

[Orbital elements](/wiki/Orbital_elements "Orbital elements")

- [Apsis](/wiki/Apsis "Apsis")
- [Argument of periapsis](/wiki/Argument_of_periapsis "Argument of periapsis")
- [Eccentricity](/wiki/Orbital_eccentricity "Orbital eccentricity")
- [Inclination](/wiki/Orbital_inclination "Orbital inclination")
- [Mean anomaly](/wiki/Mean_anomaly "Mean anomaly")
- [Orbital nodes](/wiki/Orbital_node "Orbital node")
- [Semi-major axis](/wiki/Semi-major_and_semi-minor_axes "Semi-major and semi-minor axes")
- [True anomaly](/wiki/True_anomaly "True anomaly")

Types of [two-body orbits](/wiki/Two-body_problem "Two-body problem") by

eccentricity

- [Circular orbit](/wiki/Circular_orbit "Circular orbit")
- [Elliptic orbit](/wiki/Elliptic_orbit "Elliptic orbit")

[Transfer orbit](/wiki/Transfer_orbit "Transfer orbit")

- (Hohmann transfer orbit
- [Bi-elliptic transfer orbit](/wiki/Bi-elliptic_transfer "Bi-elliptic transfer"))

- [Parabolic orbit](/wiki/Parabolic_trajectory "Parabolic trajectory")
- [Hyperbolic orbit](/wiki/Hyperbolic_trajectory "Hyperbolic trajectory")
- [Radial orbit](/wiki/Radial_trajectory "Radial trajectory")
- [Decaying orbit](/wiki/Orbital_decay "Orbital decay")

Equations

- [Dynamical friction](/wiki/Dynamical_friction "Dynamical friction")
- [Escape velocity](/wiki/Escape_velocity "Escape velocity")
- [Kepler's equation](/wiki/Kepler%27s_equation "Kepler's equation")
- [Kepler's laws of planetary motion](/wiki/Kepler%27s_laws_of_planetary_motion "Kepler's laws of planetary motion")
- [Orbital period](/wiki/Orbital_period "Orbital period")
- [Orbital velocity](/wiki/Orbital_speed "Orbital speed")
- [Surface gravity](/wiki/Surface_gravity "Surface gravity")
- [Specific orbital energy](/wiki/Specific_orbital_energy "Specific orbital energy")
- [Vis-viva equation](/wiki/Vis-viva_equation "Vis-viva equation")

[Celestial mechanics](/wiki/Celestial_mechanics "Celestial mechanics")

Gravitational influences

- [Barycenter](/wiki/Barycenter "Barycenter")
- [Hill sphere](/wiki/Hill_sphere "Hill sphere")
- [Perturbations](/wiki/Perturbation_(astronomy) "Perturbation (astronomy)")
- [Sphere of influence](/wiki/Sphere_of_influence_(astrodynamics) "Sphere of influence (astrodynamics)")

[N-body orbits](/wiki/N-body_problem "N-body problem")

[Lagrangian points](/wiki/Lagrange_point "Lagrange point")

- ( [Halo orbits](/wiki/Halo_orbit "Halo orbit"))

- [Lissajous orbits](/wiki/Lissajous_orbit "Lissajous orbit")
- [Lyapunov orbits](/wiki/Lyapunov_stability "Lyapunov stability")

[Engineering and efficiency](/wiki/Aerospace_engineering "Aerospace engineering")

Preflight engineering

- [Mass ratio](/wiki/Mass_ratio "Mass ratio")
- [Payload fraction](/wiki/Payload_fraction "Payload fraction")
- [Propellant mass fraction](/wiki/Propellant_mass_fraction "Propellant mass fraction")
- [Tsiolkovsky rocket equation](/wiki/Tsiolkovsky_rocket_equation "Tsiolkovsky rocket equation")

Efficiency measures

- [Gravity assist](/wiki/Gravity_assist "Gravity assist")
- [Oberth effect](/wiki/Oberth_effect "Oberth effect")

Propulsive maneuvers

- [Orbital maneuver](/wiki/Orbital_maneuver "Orbital maneuver")
- [Orbit insertion](/wiki/Orbit_insertion "Orbit insertion")

- [v](/wiki/Template:Astrodynamics "Template:Astrodynamics")
- [t](/wiki/Template_talk:Astrodynamics "Template talk:Astrodynamics")
- [e](/wiki/Special:EditPage/Template:Astrodynamics "Special:EditPage/Template:Astrodynamics")

In [astronautics](/wiki/Astronautics "Astronautics"), the **Hohmann transfer orbit** ([/ˈhoʊmən/](/wiki/Help:IPA/English "Help:IPA/English")) is an [orbital maneuver](/wiki/Orbital_maneuver "Orbital maneuver") used to transfer a [spacecraft](/wiki/Spacecraft "Spacecraft") between two orbits of different altitudes around a central body. For example, a Hohmann transfer could be used to raise a satellite's orbit from [low Earth orbit](/wiki/Low_Earth_orbit "Low Earth orbit") to [geostationary orbit](/wiki/Geostationary_orbit "Geostationary orbit"). In the idealized case, the initial and target orbits are both [circular](/wiki/Circular_orbit "Circular orbit") and [coplanar](/wiki/Orbital_plane "Orbital plane"). The maneuver is accomplished by placing the craft into an [elliptical transfer orbit](/wiki/Elliptical_orbit "Elliptical orbit") that is [tangential](/wiki/Tangent "Tangent") to both the initial and target orbits. The maneuver uses two [impulsive](/wiki/Impulse_(physics) "Impulse (physics)") engine burns: the first establishes the transfer orbit, and the second adjusts the orbit to match the target.

The Hohmann maneuver often uses the lowest possible amount of impulse which is also known as (MINESH)

```
(which consumes a proportional amount of delta-v, and hence propellant) to accomplish the transfer, but requires a relatively longer travel time than higher-impulse transfers. In some cases where one orbit is much larger than the other, a bi-elliptic transfer can use even less impulse, at the cost of even greater travel time.

```

The maneuver was named after [Walter Hohmann](/wiki/Walter_Hohmann "Walter Hohmann"), the German scientist who published a description of it in his 1925 book _Die Erreichbarkeit der Himmelskörper_ ( _The Attainability of Celestial Bodies_).[\[1\]](#cite_note-1) Hohmann was influenced in part by the German science fiction author [Kurd Lasswitz](/wiki/Kurd_Lasswitz "Kurd Lasswitz") and his 1897 book _[Two Planets](/wiki/Two_Planets "Two Planets")_.

When used for traveling 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 [launch window](/wiki/Launch_window "Launch window"). For a mission between [Earth](/wiki/Earth "Earth") and [Mars](/wiki/Mars "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 celestial bodies with significant gravitation, much less [delta-v](/wiki/Delta-v "Delta-v") is usually required, as the [Oberth effect](/wiki/Oberth_effect "Oberth effect") may be employed for the burns.

They are also often used for these situations, but [low-energy transfers](/wiki/Low-energy_transfer "Low-energy transfer") 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.[\[2\]](#cite_note-2)[\[3\]](#cite_note-3)[\[4\]](#cite_note-4)

## Example

\[ [edit](/w/index.php?title=Hohmann_transfer_orbit&action=edit&section=1 "Edit section: Example")\]

The diagram shows a Hohmann transfer orbit to bring a spacecraft from a lower circular orbit into a higher one. It is an [elliptic orbit](/wiki/Elliptic_orbit "Elliptic orbit") that is tangential both to the lower circular orbit the spacecraft is to leave (cyan, labeled _1_ on diagram) and the higher circular orbit that it is to reach (red, labeled _3_ on diagram). The transfer orbit (yellow, labeled _2_ on diagram) is initiated by firing the spacecraft's engine to add energy and raise the [apogee](/wiki/Apogee "Apogee"). When the spacecraft reaches apogee, a second engine firing adds energy to raise the perigee, putting the spacecraft in the larger circular orbit.

[![](//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Hohmann_transfer_orbit2.svg/220px-Hohmann_transfer_orbit2.svg.png)](/wiki/File:Hohmann_transfer_orbit2.svg)

Due to the [reversibility of orbits](/wiki/Reversibility_of_orbits "Reversibility of orbits"), a similar Hohmann transfer orbit can be used 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 lowering its perigee to that of the 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](/wiki/Impulse_function "Impulse function"). 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_](/wiki/Apogee_kick_motor "Apogee kick motor")[\[5\]](#cite_note-5)); 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

\[ [edit](/w/index.php?title=Hohmann_transfer_orbit&action=edit&section=2 "Edit section: Type I and Type II")\]

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.[\[6\]](#cite_note-NASA-trajectories-6)[\[7\]](#cite_note-7)

Transfer orbits can go more than 360° around the primary. These multiple-revolution transfers are sometimes referred to as Type III and Type IV, where a Type III is a Type I plus 360°, and a Type IV is a Type II plus 360°.[\[8\]](#cite_note-8)

### Uses

\[ [edit](/w/index.php?title=Hohmann_transfer_orbit&action=edit&section=3 "Edit section: Uses")\]

A Hohmann transfer orbit can be used to transfer an object's orbit toward another object, as long as they co-orbit a more massive body. In the context of Earth and the [Solar System](/wiki/Solar_System "Solar System"), this includes any object which orbits the [Sun](/wiki/Sun "Sun"). An example of where a Hohmann transfer orbit could be used is to bring an asteroid, orbiting the Sun, into contact with the Earth.[\[9\]](#cite_note-9)

## Calculation

\[ [edit](/w/index.php?title=Hohmann_transfer_orbit&action=edit&section=4 "Edit section: Calculation")\]

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](/wiki/Kinetic_energy "Kinetic energy") and [potential energy](/wiki/Potential_energy "Potential energy"), and this total energy also equals half the potential at the [average distance](/wiki/Semi-major_axis#Average_distance "Semi-major axis")a{\\displaystyle a}![{\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc) (the [semi-major axis](/wiki/Semi-major_axis "Semi-major axis")):
E=mv22−GMmr=−GMm2a.{\\displaystyle E={\\frac {mv^{2}}{2}}-{\\frac {GMm}{r}}={\\frac {-GMm}{2a}}.}![{\displaystyle E={\frac {mv^{2}}{2}}-{\frac {GMm}{r}}={\frac {-GMm}{2a}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8010dfa8282b04628e9e3105bc883a94049352e0)

Solving this equation for velocity results in the [vis-viva equation](/wiki/Vis-viva_equation "Vis-viva equation"),
v2=μ(2r−1a),{\\displaystyle v^{2}=\\mu \\left({\\frac {2}{r}}-{\\frac {1}{a}}\\right),}![{\displaystyle v^{2}=\mu \left({\frac {2}{r}}-{\frac {1}{a}}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f33e334eeb92d36e2f3e0aeb2d299bf6173a81e1)
where:

- v{\\displaystyle v}![{\displaystyle v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597) is the speed of an orbiting body,
- μ=GM{\\displaystyle \\mu =GM}![{\displaystyle \mu =GM}](https://wikimedia.org/api/rest_v1/media/math/render/svg/183d080fad6c5e9292cd08bc28897f334189b75f) is the [standard gravitational parameter](/wiki/Standard_gravitational_parameter "Standard gravitational parameter") of the primary body, assuming M+m{\\displaystyle M+m}![{\displaystyle M+m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af820258afa74dafb69f8551e365ffe5d3577c43) is not significantly bigger than M{\\displaystyle M}![{\displaystyle M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd) (which makes vM≪v{\\displaystyle v\_{M}\\ll v}![{\displaystyle v_{M}\ll v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76879795bec6cfc21213c96d40205fd326dd8b59)), (for Earth, this is _μ_ ~3.986E14 m3 s−2)
- r{\\displaystyle r}![{\displaystyle r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538) is the distance of the orbiting body from the primary focus,
- a{\\displaystyle a}![{\displaystyle a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc) is the [semi-major axis](/wiki/Semi-major_axis "Semi-major axis") of the body's orbit.

Therefore, the [delta- _v_](/wiki/Delta-v "Delta-v") (Δv) required for the Hohmann transfer can be computed as follows, under the assumption of instantaneous impulses:
Δv1=μr1(2r2r1+r2−1),{\\displaystyle \\Delta v\_{1}={\\sqrt {\\frac {\\mu }{r\_{1}}}}\\left({\\sqrt {\\frac {2r\_{2}}{r\_{1}+r\_{2}}}}-1\\right),}![{\displaystyle \Delta v_{1}={\sqrt {\frac {\mu }{r_{1}}}}\left({\sqrt {\frac {2r_{2}}{r_{1}+r_{2}}}}-1\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/578391c4886859814917efb3b1db442c4dc9a6d3)
to enter the elliptical orbit at r=r1{\\displaystyle r=r\_{1}}![{\displaystyle r=r_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb56c93930b9ce8c99555c481fad15c05fedad53) from the r1{\\displaystyle r\_{1}}![{\displaystyle r_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1) circular orbit, where r2{\\displaystyle r\_{2}}![{\displaystyle r_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f) is the aphelion of the resulting elliptical orbit, and
Δv2=μr2(1−2r1r1+r2),{\\displaystyle \\Delta v\_{2}={\\sqrt {\\frac {\\mu }{r\_{2}}}}\\left(1-{\\sqrt {\\frac {2r\_{1}}{r\_{1}+r\_{2}}}}\\right),}![{\displaystyle \Delta v_{2}={\sqrt {\frac {\mu }{r_{2}}}}\left(1-{\sqrt {\frac {2r_{1}}{r_{1}+r_{2}}}}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1e0d094141ccc7caee76461acdb18004451909a)
to leave the elliptical orbit at r=r2{\\displaystyle r=r\_{2}}![{\displaystyle r=r_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ce471e9ca4aaf21c36edec03c02e71c39b9bfe1) to the r2{\\displaystyle r\_{2}}![{\displaystyle r_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f) circular orbit,
where r1{\\displaystyle r\_{1}}![{\displaystyle r_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1) and r2{\\displaystyle r\_{2}}![{\displaystyle r_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f) are respectively the radii of the departure and arrival circular orbits;
the smaller (greater) of r1{\\displaystyle r\_{1}}![{\displaystyle r_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1) and r2{\\displaystyle r\_{2}}![{\displaystyle r_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f) corresponds to the [periapsis distance](/wiki/Periapsis_distance "Periapsis distance") ( [apoapsis distance](/wiki/Apoapsis_distance "Apoapsis distance")) of the Hohmann elliptical transfer orbit. Typically, μ{\\displaystyle \\mu }![{\displaystyle \mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161) is given in units of m3/s2, as such be sure to use meters, not kilometers, for r1{\\displaystyle r\_{1}}![{\displaystyle r_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1) and r2{\\displaystyle r\_{2}}![{\displaystyle r_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f). The total Δv{\\displaystyle \\Delta v}![{\displaystyle \Delta v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e18b43e4225eeaafeeb25aefc4ee90bd86f004dc) is then:

Δvtotal=Δv1+Δv2.{\\displaystyle \\Delta v\_{\\text{total}}=\\Delta v\_{1}+\\Delta v\_{2}.}![{\displaystyle \Delta v_{\text{total}}=\Delta v_{1}+\Delta v_{2}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7657c38a413a685bf91e648b5461ab45322f4891)

Whether moving into a higher or lower orbit, by [Kepler's third law](/wiki/Kepler%27s_laws_of_planetary_motion#Kepler's_understanding_of_the_laws "Kepler's laws of planetary motion")\[ _[broken anchor](/wiki/MOS:BROKENSECTIONLINKS "MOS:BROKENSECTIONLINKS")_\], the time taken to transfer between the orbits is

tH=124π2aH3μ=π(r1+r2)38μ{\\displaystyle t\_{\\text{H}}={\\frac {1}{2}}{\\sqrt {\\frac {4\\pi ^{2}a\_{\\text{H}}^{3}}{\\mu }}}=\\pi {\\sqrt {\\frac {(r\_{1}+r\_{2})^{3}}{8\\mu }}}}![{\displaystyle t_{\text{H}}={\frac {1}{2}}{\sqrt {\frac {4\pi ^{2}a_{\text{H}}^{3}}{\mu }}}=\pi {\sqrt {\frac {(r_{1}+r_{2})^{3}}{8\mu }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb419eb6c093cac387da31a4a50791ccd0c7352)
(one half of the [orbital period](/wiki/Orbital_period "Orbital period") for the whole ellipse), where aH{\\displaystyle a\_{\\text{H}}}![{\displaystyle a_{\text{H}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f49b6ef5efdd14bc8b865b3f7b7b7236c34b228) is length of [semi-major axis](/wiki/Semi-major_axis "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
ω2=μr23,{\\displaystyle \\omega \_{2}={\\sqrt {\\frac {\\mu }{r\_{2}^{3}}}},}![{\displaystyle \omega _{2}={\sqrt {\frac {\mu }{r_{2}^{3}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/717b8008d23b84858df208fdbab202817f547bbf)
angular alignment α (in [radians](/wiki/Radian "Radian")) at the time of start between the source object and the target object shall be
α=π−ω2tH=π(1−122(r1r2+1)3).{\\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).}![{\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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56bf8cc590b106e75d45256da5db1bb714727fb3)

## Example

\[ [edit](/w/index.php?title=Hohmann_transfer_orbit&action=edit&section=5 "Edit section: Example")\]

[![](//upload.wikimedia.org/wikipedia/commons/thumb/5/59/Total_energy_during_Hohmann_transfer.png/260px-Total_energy_during_Hohmann_transfer.png)](/wiki/File:Total_energy_during_Hohmann_transfer.png) Total energy balance during a Hohmann transfer between two circular orbits with first radius rp{\\displaystyle r\_{p}}![{\displaystyle r_{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d02ec163fac8837ac757005d783b375e52808b97) and second radius ra{\\displaystyle r\_{a}}![{\displaystyle r_{a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da5f0781a46ba20129a554237056b6ade78b956f)

Consider a [geostationary transfer orbit](/wiki/Geostationary_transfer_orbit "Geostationary transfer orbit"), beginning at _r_ 1 = 6,678 km (altitude 300 km) and ending in a [geostationary orbit](/wiki/Geostationary_orbit "Geostationary orbit") with _r_ 2 = 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.

Therefore the Δv for the first burn is 10.15 − 7.73 = 2.42 km/s, for the second burn 3.07 − 1.61 = 1.46 km/s, and for both together 3.88 km/s.

This is _greater_ than the Δv required for an [escape orbit](/wiki/Escape_orbit "Escape orbit"): 10.93 − 7.73 = 3.20 km/s. Applying a Δv at the [Low Earth orbit](/wiki/Low_Earth_orbit "Low Earth orbit") (LEO) of only 0.78 km/s more (3.20−2.42) would give the rocket the [escape velocity](/wiki/Escape_speed "Escape speed"), which is less than the Δv of 1.46 km/s required to circularize the geosynchronous orbit. This illustrates the [Oberth effect](/wiki/Oberth_effect "Oberth effect") that at large speeds the same Δv provides more [specific orbital energy](/wiki/Specific_orbital_energy "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_

\[ [edit](/w/index.php?title=Hohmann_transfer_orbit&action=edit&section=6 "Edit section: Worst case, maximum delta-v")\]

As the example above demonstrates, the Δ _v_ 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.[\[10\]](#cite_note-10) This number is the positive root of _x_ 3 − 15 _x_ 2 − 9 _x_ − 1 = 0, which is 5+47cos⁡(13arctan⁡337){\\textstyle 5+4\\,{\\sqrt {7}}\\cos \\left({1 \\over 3}\\arctan {{\\sqrt {3}} \\over 37}\\right)}![{\textstyle 5+4\,{\sqrt {7}}\cos \left({1 \over 3}\arctan {{\sqrt {3}} \over 37}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/318073e15207b553b2276f24f1437db17d63b002). For higher orbit ratios the Δ _v_ required for the second burn decreases faster than the first increases.

## Application to interplanetary travel

\[ [edit](/w/index.php?title=Hohmann_transfer_orbit&action=edit&section=7 "Edit section: Application to interplanetary travel")\]

When used to move a spacecraft from orbiting one planet to orbiting another, the [Oberth effect](/wiki/Oberth_effect "Oberth effect") allows to use less delta- _v_ than the sum of the delta- _v_ for separate manoeuvres to escape the first planet, followed by a Hohmann transfer to the second planet, followed by insertion into an orbit around the other planet.

For example, consider a spacecraft travelling from [Earth](/wiki/Earth "Earth") to [Mars](/wiki/Mars "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](/wiki/Escape_velocity "Escape velocity"), and then burns more so as to gain enough energy to get into the Hohmann transfer orbit (around the [Sun](/wiki/Sun "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](/wiki/Periapsis "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 must decelerate for the [gravity of Mars](/wiki/Gravity_of_Mars "Gravity of Mars") to capture it. This capture burn should optimally be done at low altitude to also make best use of the 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](/wiki/Launch_window "Launch window").

The term lunar transfer orbit (LTO) is used for the [Moon](/wiki/Moon "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 "LEO height" 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 [escape velocity](/wiki/Escape_velocity "Escape velocity") (10.9 km/s) from this height. The column "LEO" is simply the previous speed minus the LEO orbital speed of 7.73 km/s.

Destination
Orbital

radius

( [AU](/wiki/Astronomical_unit "Astronomical unit"))
Δ _v_ (km/s) to enter Hohmann orbit from
Earth's orbit
LEO height
LEO
[Sun](/wiki/Sun "Sun")0
29.8
31.7
24.0
[Mercury](/wiki/Mercury_(planet) "Mercury (planet)")0.39
7.5
13.3
5.5
[Venus](/wiki/Venus "Venus")0.72
2.5
11.2
3.5
[Mars](/wiki/Mars "Mars")1.52
2.9
11.3
3.6
[Jupiter](/wiki/Jupiter "Jupiter")5.2
8.8
14.0
6.3
[Saturn](/wiki/Saturn "Saturn")9.54
10.3
15.0
7.3
[Uranus](/wiki/Uranus "Uranus")19.19
11.3
15.7
8.0
[Neptune](/wiki/Neptune "Neptune")30.07
11.7
16.0
8.2
[Pluto](/wiki/Pluto "Pluto")39.48
11.8
16.1
8.4
[Infinity](/wiki/Infinity "Infinity")∞
12.3
16.5
8.8

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

To get to the Sun, it is actually not necessary to use a Δ _v_ of 24 km/s. One can use 8.8 km/s to go very far away from the Sun, then use a negligible Δ _v_ 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](/wiki/Gravity_assist "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

\[ [edit](/w/index.php?title=Hohmann_transfer_orbit&action=edit&section=8 "Edit section: Comparison to other transfers")\]

### Bi-elliptic transfer

\[ [edit](/w/index.php?title=Hohmann_transfer_orbit&action=edit&section=9 "Edit section: Bi-elliptic transfer")\]

Main article: [Bi-elliptic transfer](/wiki/Bi-elliptic_transfer "Bi-elliptic transfer")

The bi-elliptic transfer consists of two half- [elliptic orbits](/wiki/Elliptic_orbit "Elliptic orbit"). From the initial orbit, a first burn expends delta-v to boost the spacecraft into the first transfer orbit with an [apoapsis](/wiki/Apoapsis "Apoapsis") at some point rb{\\displaystyle r\_{b}}![{\displaystyle r_{b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c22eac9d11ab84da51b891d29ee2e5de75eab1a) away from the [central body](/wiki/Central_body "Central body"). At this point a second burn sends the spacecraft into the second elliptical orbit with [periapsis](/wiki/Periapsis "Periapsis") at the radius of the final desired orbit, where a third burn is performed, injecting the spacecraft into the desired orbit.[\[11\]](#cite_note-Curtis-11)

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](/wiki/Semi-major_axis "Semi-major axis") is 11.94 or greater, depending on the intermediate semi-major axis chosen.[\[12\]](#cite_note-Vallado-12)

The idea of the bi-elliptical transfer trajectory was first\[ _[citation needed](/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")_\] published by [Ary Sternfeld](/wiki/Ary_Sternfeld "Ary Sternfeld") in 1934.[\[13\]](#cite_note-sternfeld1934-13)

### Low-thrust transfer

\[ [edit](/w/index.php?title=Hohmann_transfer_orbit&action=edit&section=10 "Edit section: Low-thrust transfer")\]

Main article: [Low thrust relative orbital transfer](/wiki/Low_thrust_relative_orbital_transfer "Low thrust relative orbital 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_)](/wiki/Delta-v "Delta-v") that is greater than the two-impulse transfer orbit[\[14\]](#cite_note-MIT_16.522-14) and takes longer to complete.

Engines such as [ion thrusters](/wiki/Ion_thruster "Ion thruster") 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](/wiki/Specific_impulse "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.[\[14\]](#cite_note-MIT_16.522-14) 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](/wiki/Electrically_powered_spacecraft_propulsion "Electrically powered spacecraft 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 orbits 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.[\[15\]](#cite_note-15)

### Interplanetary Transport Network

\[ [edit](/w/index.php?title=Hohmann_transfer_orbit&action=edit&section=11 "Edit section: Interplanetary Transport Network")\]

Main article: [Interplanetary Transport Network](/wiki/Interplanetary_Transport_Network "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.[\[16\]](#cite_note-16) 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](/wiki/Gravity_assist "Gravity assist") from the planets.\[ _[citation needed](/wiki/Wikipedia:Citation_needed "Wikipedia:Citation needed")_\]

## See also

\[ [edit](/w/index.php?title=Hohmann_transfer_orbit&action=edit&section=12 "Edit section: See also")\]

- ![](//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/RocketSunIcon.svg/28px-RocketSunIcon.svg.png)[Spaceflight portal](/wiki/Portal:Spaceflight "Portal:Spaceflight")

- [Bi-elliptic transfer](/wiki/Bi-elliptic_transfer "Bi-elliptic transfer")
- [Delta-v budget](/wiki/Delta-v_budget "Delta-v budget")
- [Geostationary transfer orbit](/wiki/Geostationary_transfer_orbit "Geostationary transfer orbit")
- [Halo orbit](/wiki/Halo_orbit "Halo orbit")
- [Lissajous orbit](/wiki/Lissajous_orbit "Lissajous orbit")
- [List of orbits](/wiki/List_of_orbits "List of orbits")
- [Orbital mechanics](/wiki/Orbital_mechanics "Orbital mechanics")

## Citations

\[ [edit](/w/index.php?title=Hohmann_transfer_orbit&action=edit&section=13 "Edit section: Citations")\]

01. **[^](#cite_ref-1)**Walter Hohmann, _The Attainability of Heavenly Bodies_ (Washington: NASA Technical Translation F-44, 1960) [Internet Archive](https://archive.org/details/nasa_techdoc_19980230631).
02. **[^](#cite_ref-2)**Williams, Matt (2014-12-26). ["Making the Trip to Mars Cheaper and Easier: The Case for Ballistic Capture"](https://www.universetoday.com/117615/making-the-trip-to-mars-cheaper-and-easier-the-case-for-ballistic-capture/). _Universe Today_. Retrieved 2019-07-29.
03. **[^](#cite_ref-3)**Hadhazy, Adam. ["A New Way to Reach Mars Safely, Anytime and on the Cheap"](https://www.scientificamerican.com/article/a-new-way-to-reach-mars-safely-anytime-and-on-the-cheap/). _Scientific American_. Retrieved 2019-07-29.
04. **[^](#cite_ref-4)**["An Introduction to Beresheet and Its Trajectory to the Moon"](https://gereshes.com/2019/04/08/an-introduction-to-beresheet-and-its-trajectory-to-the-moon/). _Gereshes_. 2019-04-08. Retrieved 2019-07-29.
05. **[^](#cite_ref-5)**Jonathan McDowell, " [Kick In the Apogee: 40 years of upper stage applications for solid rocket motors, 1957-1997](https://arc.aiaa.org/doi/pdf/10.2514/6.1997-3133)", 33rd AIAA Joint Propulsion Conference, July 4, 1997. [abstract](https://arc.aiaa.org/doi/abs/10.2514/6.1997-3133). Retrieved 18 July 2017.
06. **[^](#cite_ref-NASA-trajectories_6-0)**NASA, _Basics of Space Flight_, Section 1, Chapter 4, " [Trajectories](https://solarsystem.nasa.gov/basics/chapter4-1)". Retrieved 26 July 2017. Also available [spaceodyssey.dmns.org](https://spaceodyssey.dmns.org/media/57432/hohmann_transfer_orbits.pdf) [Archived](https://web.archive.org/web/20170728080707/https://spaceodyssey.dmns.org/media/57432/hohmann_transfer_orbits.pdf) 2017-07-28 at the [Wayback Machine](/wiki/Wayback_Machine "Wayback Machine").
07. **[^](#cite_ref-7)**Tyson Sparks, [Trajectories to Mars](http://ccar.colorado.edu/asen5050/projects/projects_2015/Students/Alpert_Brian/interplanetary_transfer.html) [Archived](https://web.archive.org/web/20171028085842/http://ccar.colorado.edu/asen5050/projects/projects_2015/Students/Alpert_Brian/interplanetary_transfer.html) 2017-10-28 at the [Wayback Machine](/wiki/Wayback_Machine "Wayback Machine"), Colorado Center for Astrodynamics Research, 12/14/2012. Retrieved 25 July 2017.
08. **[^](#cite_ref-8)**Langevin, Y. (2005). "Design issues for Space Science Missions," _Payload and Mission Definition in Space Sciences_, V. Mártínez Pillet, A. Aparicio, and F. Sánchez, eds., Cambridge University Press, p. 30. [ISBN](/wiki/ISBN_(identifier) "ISBN (identifier)") [052185802X](/wiki/Special:BookSources/052185802X "Special:BookSources/052185802X"), 9780521858021
09. **[^](#cite_ref-9)**Calla, Pablo; Fries, Dan; Welch, Chris (2018). "Asteroid mining with small spacecraft and its economic feasibility". [arXiv](/wiki/ArXiv_(identifier) "ArXiv (identifier)"):[1808.05099](https://arxiv.org/abs/1808.05099) \[ [astro-ph.IM](https://arxiv.org/archive/astro-ph.IM)\].
10. **[^](#cite_ref-10)**Vallado, David Anthony (2001). [_Fundamentals of Astrodynamics and Applications_](https://books.google.com/books?id=PJLlWzMBKjkC). Springer. p. 317. [ISBN](/wiki/ISBN_(identifier) "ISBN (identifier)") [0-7923-6903-3](/wiki/Special:BookSources/0-7923-6903-3 "Special:BookSources/0-7923-6903-3").
11. **[^](#cite_ref-Curtis_11-0)**Curtis, Howard (2005). [_Orbital Mechanics for Engineering Students_](https://books.google.com/books?id=6aO9aGNBAgIC). [Elsevier](/wiki/Elsevier "Elsevier"). p. 264. [ISBN](/wiki/ISBN_(identifier) "ISBN (identifier)") [0-7506-6169-0](/wiki/Special:BookSources/0-7506-6169-0 "Special:BookSources/0-7506-6169-0").
12. **[^](#cite_ref-Vallado_12-0)**Vallado, David Anthony (2001). [_Fundamentals of Astrodynamics and Applications_](https://books.google.com/books?id=PJLlWzMBKjkC). Springer. p. 318. [ISBN](/wiki/ISBN_(identifier) "ISBN (identifier)") [0-7923-6903-3](/wiki/Special:BookSources/0-7923-6903-3 "Special:BookSources/0-7923-6903-3").
13. **[^](#cite_ref-sternfeld1934_13-0)**Sternfeld, Ary J. (1934-02-12), ["Sur les trajectoires permettant d'approcher d'un corps attractif central à partir d'une orbite keplérienne donnée"](http://gallica.bnf.fr/ark:/12148/bpt6k31506/f711.image.langEN) \[On the allowed trajectories for approaching a central attractive body from a given Keplerian orbit\], _Comptes rendus de l'Académie des sciences_ (in French), **198** (1), Paris: 711–713.
14. ^ [_**a**_](#cite_ref-MIT_16.522_14-0) [_**b**_](#cite_ref-MIT_16.522_14-1)MIT, _16.522: Space Propulsion_, Session 6, " [Analytical Approximations for Low Thrust Maneuvers](https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-522-space-propulsion-spring-2015/lecture-notes/MIT16_522S15_Lecture6.pdf)", Spring 2015 (retrieved 26 July 2017)
15. **[^](#cite_ref-15)**Spitzer, Arnon (1997). [_Optimal Transfer Orbit Trajectory using Electric Propulsion_](https://patents.google.com/patent/US5595360). [USPTO](/wiki/USPTO "USPTO").
16. **[^](#cite_ref-16)**[Lo, M. W.](/wiki/Martin_Lo "Martin Lo"); Ross, S. D. (1997). ["Surfing the Solar System: Invariant Manifolds and the Dynamics of the Solar System"](http://www.gg.caltech.edu/~mwl/publications/publications2.htm). _Technical Report_. IOM. [JPL](/wiki/JPL "JPL"). pp. 2–4. 312/97.

## General and cited sources

\[ [edit](/w/index.php?title=Hohmann_transfer_orbit&action=edit&section=14 "Edit section: General and cited sources")\]

- Bate, R. R.; Mueller, D.D.; White, J.E. (1971). [_Fundamentals of Astrodynamics_](https://archive.org/details/fundamentalsofas00bate). New York: Dover Publications. [ISBN](/wiki/ISBN_(identifier) "ISBN (identifier)") [978-0-486-60061-1](/wiki/Special:BookSources/978-0-486-60061-1 "Special:BookSources/978-0-486-60061-1").
- Battin, R .H. (1999). _An Introduction to the Mathematics and Methods of Astrodynamics_. Washington, DC: American Institute of Aeronautics & Ast. [ISBN](/wiki/ISBN_(identifier) "ISBN (identifier)") [978-1-56347-342-5](/wiki/Special:BookSources/978-1-56347-342-5 "Special:BookSources/978-1-56347-342-5").
- Hohmann, Walter (1925). _Die Erreichbarkeit der Himmelskörper_. Munich: R. Oldenbourg Verlag. [ISBN](/wiki/ISBN_(identifier) "ISBN (identifier)") [3-486-23106-5](/wiki/Special:BookSources/3-486-23106-5 "Special:BookSources/3-486-23106-5").
- Thornton, Stephen T.; Marion, Jerry B. (2003). _Classical Dynamics of Particles and Systems_ (5th ed.). [Brooks Cole](/wiki/Brooks_Cole "Brooks Cole"). [ISBN](/wiki/ISBN_(identifier) "ISBN (identifier)") [0-534-40896-6](/wiki/Special:BookSources/0-534-40896-6 "Special:BookSources/0-534-40896-6").
- Vallado, D. A. (2001). _Fundamentals of Astrodynamics and Applications_ (2nd ed.). Springer. [ISBN](/wiki/ISBN_(identifier) "ISBN (identifier)") [978-0-7923-6903-5](/wiki/Special:BookSources/978-0-7923-6903-5 "Special:BookSources/978-0-7923-6903-5").

## Further reading

\[ [edit](/w/index.php?title=Hohmann_transfer_orbit&action=edit&section=15 "Edit section: Further reading")\]

- ["Orbital Mechanics"](https://web.archive.org/web/20120204054322/http://www.braeunig.us/space/orbmech.htm). _Rocket and Space Technology_. Robert A. Braeunig. Archived from [the original](http://www.braeunig.us/space/orbmech.htm) on 2012-02-04. Retrieved 2005-08-17.
- ["4\. Interplanetary Trajectories"](https://solarsystem.nasa.gov/basics/chapter4-1). _Basics of Spaceflight_. JPL: [NASA](/wiki/NASA "NASA").

- [v](/wiki/Template:Orbits "Template:Orbits")
- [t](/wiki/Template_talk:Orbits "Template talk:Orbits")
- [e](/wiki/Special:EditPage/Template:Orbits "Special:EditPage/Template:Orbits")

Gravitational [orbits](/wiki/Orbit "Orbit")

[Types](/wiki/List_of_orbits "List of orbits")

General

- [Box](/wiki/Box_orbit "Box orbit")
- [Capture](/wiki/Parabolic_trajectory "Parabolic trajectory")
- [Circular](/wiki/Circular_orbit "Circular orbit")
- [Elliptical](/wiki/Elliptic_orbit "Elliptic orbit") / [Highly elliptical](/wiki/Highly_elliptical_orbit "Highly elliptical orbit")
- [Escape](/wiki/Parabolic_trajectory "Parabolic trajectory")
- [Horseshoe](/wiki/Horseshoe_orbit "Horseshoe orbit")
- [Hyperbolic trajectory](/wiki/Hyperbolic_trajectory "Hyperbolic trajectory")
- [Inclined](/wiki/Inclined_orbit "Inclined orbit") / [Non-inclined](/wiki/Non-inclined_orbit "Non-inclined orbit")
- [Kepler](/wiki/Kepler_orbit "Kepler orbit")
- [Lagrange point](/wiki/Lagrange_point "Lagrange point")
- [Osculating](/wiki/Osculating_orbit "Osculating orbit")
- [Parabolic trajectory](/wiki/Parabolic_trajectory "Parabolic trajectory")
- [Parking](/wiki/Parking_orbit "Parking orbit")
- [Prograde / Retrograde](/wiki/Retrograde_and_prograde_motion "Retrograde and prograde motion")
- [Synchronous](/wiki/Synchronous_orbit "Synchronous orbit")
  - [semi](/wiki/Semi-synchronous_orbit "Semi-synchronous orbit")
  - [sub](/wiki/Subsynchronous_orbit "Subsynchronous orbit")
- Transfer orbit

[Geocentric](/wiki/Geocentric_orbit "Geocentric orbit")

- [Geosynchronous](/wiki/Geosynchronous_orbit "Geosynchronous orbit")
  - [Geostationary](/wiki/Geostationary_orbit "Geostationary orbit")
  - [Geostationary transfer](/wiki/Geostationary_transfer_orbit "Geostationary transfer orbit")
- [Graveyard](/wiki/Graveyard_orbit "Graveyard orbit")
- [High Earth](/wiki/High_Earth_orbit "High Earth orbit")
- [Low Earth](/wiki/Low_Earth_orbit "Low Earth orbit")
- [Medium Earth](/wiki/Medium_Earth_orbit "Medium Earth orbit")
- [Molniya](/wiki/Molniya_orbit "Molniya orbit")
- [Near-equatorial](/wiki/Near-equatorial_orbit "Near-equatorial orbit")
- [Orbit of the Moon](/wiki/Orbit_of_the_Moon "Orbit of the Moon")
- [Polar](/wiki/Polar_orbit "Polar orbit")
- [Sun-synchronous](/wiki/Sun-synchronous_orbit "Sun-synchronous orbit")
- [Transatmospheric](/wiki/Transatmospheric_orbit "Transatmospheric orbit")
- [Tundra](/wiki/Tundra_orbit "Tundra orbit")
- [Very low Earth](/wiki/Very_low_Earth_orbit "Very low Earth orbit")

About

other points

- Mars
  - [Areocentric](/wiki/Areocentric_orbit "Areocentric orbit")
  - [Areosynchronous](/wiki/Areosynchronous_orbit "Areosynchronous orbit")
  - [Areostationary](/wiki/Areostationary_orbit "Areostationary orbit")
- Lagrange points
  - [Distant retrograde](/wiki/Distant_retrograde_orbit "Distant retrograde orbit")
  - [Halo](/wiki/Halo_orbit "Halo orbit")
  - [Lissajous](/wiki/Lissajous_orbit "Lissajous orbit")
  - [Libration](/wiki/Libration_point_orbit "Libration point orbit")
- [Lunar](/wiki/Lunar_orbit "Lunar orbit")
- Sun
  - [Heliocentric](/wiki/Heliocentric_orbit "Heliocentric orbit")
    - [Earth's orbit](/wiki/Earth%27s_orbit "Earth's orbit")
  - [Mars cycler](/wiki/Mars_cycler "Mars cycler")
  - [Heliosynchronous](/wiki/Sun-synchronous_orbit "Sun-synchronous orbit")
- Other
  - [Lunar cycler](/wiki/Lunar_cycler "Lunar cycler")

[Parameters](/wiki/Orbital_elements "Orbital elements")

- Shape
- Size

- e [Eccentricity](/wiki/Orbital_eccentricity "Orbital eccentricity")
- a [Semi-major axis](/wiki/Semi-major_and_semi-minor_axes "Semi-major and semi-minor axes")
- b [Semi-minor axis](/wiki/Semi-major_and_semi-minor_axes "Semi-major and semi-minor axes")
- Q, q [Apsides](/wiki/Apsis "Apsis")

Orientation

- i [Inclination](/wiki/Orbital_inclination "Orbital inclination")
- Ω [Longitude of the ascending node](/wiki/Longitude_of_the_ascending_node "Longitude of the ascending node")
- ω [Argument of periapsis](/wiki/Argument_of_periapsis "Argument of periapsis")
- ϖ [Longitude of the periapsis](/wiki/Longitude_of_the_periapsis "Longitude of the periapsis")

Position

- M [Mean anomaly](/wiki/Mean_anomaly "Mean anomaly")
- ν, θ, f [True anomaly](/wiki/True_anomaly "True anomaly")
- E [Eccentric anomaly](/wiki/Eccentric_anomaly "Eccentric anomaly")
- L [Mean longitude](/wiki/Mean_longitude "Mean longitude")
- l [True longitude](/wiki/True_longitude "True longitude")

Variation

- T [Orbital period](/wiki/Orbital_period "Orbital period")
- n [Mean motion](/wiki/Mean_motion "Mean motion")
- v [Orbital speed](/wiki/Orbital_speed "Orbital speed")
- _t_ 0 [Epoch](/wiki/Epoch_(astronomy) "Epoch (astronomy)")

[Maneuvers](/wiki/Orbital_maneuver "Orbital maneuver")

- [Bi-elliptic transfer](/wiki/Bi-elliptic_transfer "Bi-elliptic transfer")
- [Collision avoidance (spacecraft)](/wiki/Collision_avoidance_(spacecraft) "Collision avoidance (spacecraft)")
- [Delta-v](/wiki/Delta-v "Delta-v")
- [Delta-v budget](/wiki/Delta-v_budget "Delta-v budget")
- [Gravity assist](/wiki/Gravity_assist "Gravity assist")
- [Gravity turn](/wiki/Gravity_turn "Gravity turn")
- Hohmann transfer
- [Inclination change](/wiki/Orbital_inclination_change "Orbital inclination change")
- [Low-energy transfer](/wiki/Low-energy_transfer "Low-energy transfer")
- [Oberth effect](/wiki/Oberth_effect "Oberth effect")
- [Phasing](/wiki/Orbit_phasing "Orbit phasing")
- [Rocket equation](/wiki/Tsiolkovsky_rocket_equation "Tsiolkovsky rocket equation")
- [Rendezvous](/wiki/Space_rendezvous "Space rendezvous")
- [Trans-lunar injection](/wiki/Trans-lunar_injection "Trans-lunar injection")
- [Transposition, docking, and extraction](/wiki/Transposition,_docking,_and_extraction "Transposition, docking, and extraction")

[Orbital\
\
mechanics](/wiki/Orbital_mechanics "Orbital mechanics")

- [Astronomical coordinate systems](/wiki/Astronomical_coordinate_systems "Astronomical coordinate systems")
- [Characteristic energy](/wiki/Characteristic_energy "Characteristic energy")
- [Escape velocity](/wiki/Escape_velocity "Escape velocity")
- [Ephemeris](/wiki/Ephemeris "Ephemeris")
- [Equatorial coordinate system](/wiki/Equatorial_coordinate_system "Equatorial coordinate system")
- [Ground track](/wiki/Ground_track "Ground track")
- [Hill sphere](/wiki/Hill_sphere "Hill sphere")
- [Interplanetary Transport Network](/wiki/Interplanetary_Transport_Network "Interplanetary Transport Network")
- [Kepler's laws of planetary motion](/wiki/Kepler%27s_laws_of_planetary_motion "Kepler's laws of planetary motion")
- [Lagrangian point](/wiki/Lagrange_point "Lagrange point")
- [_n_-body problem](/wiki/N-body_problem "N-body problem")
- [Orbit equation](/wiki/Orbit_equation "Orbit equation")
- [Orbital state vectors](/wiki/Orbital_state_vectors "Orbital state vectors")
- [Perturbation](/wiki/Perturbation_(astronomy) "Perturbation (astronomy)")
- [Retrograde and prograde motion](/wiki/Retrograde_and_prograde_motion "Retrograde and prograde motion")
- [Specific orbital energy](/wiki/Specific_orbital_energy "Specific orbital energy")
- [Specific angular momentum](/wiki/Specific_angular_momentum "Specific angular momentum")
- [Two-line elements](/wiki/Two-line_element_set "Two-line element set")

- ![](//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png)[List of orbits](/wiki/List_of_orbits "List of orbits")

Retrieved from " [https://en.wikipedia.org/w/index.php?title=Hohmann\_transfer\_orbit&oldid=1245390313](https://en.wikipedia.org/w/index.php?title=Hohmann_transfer_orbit&oldid=1245390313)"

[Categories](/wiki/Help:Category "Help:Category"):

- [Astrodynamics](/wiki/Category:Astrodynamics "Category:Astrodynamics")
- [Spacecraft propulsion](/wiki/Category:Spacecraft_propulsion "Category:Spacecraft propulsion")
- [Orbital maneuvers](/wiki/Category:Orbital_maneuvers "Category:Orbital maneuvers")
- [Types of orbit](/wiki/Category:Types_of_orbit "Category:Types of orbit")

Hidden categories:

- [Webarchive template wayback links](/wiki/Category:Webarchive_template_wayback_links "Category:Webarchive template wayback links")
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