View source for Molniya orbit

{{good article}}
{{short description|Type of high-latitude satellite orbit}}
{{Other uses|Molniya (disambiguation)}}
[[File:NASA molniya oblique.png|thumb|Figure 1: The Molniya orbit. Usually the period from [[perigee]] +2 hours to perigee +10 hours is used to transmit to the northern hemisphere.]]
[[File:SDS3 const.png|thumb|upright=0.9|Figure 2: The [[Satellite Data System|SDS]] constellation, which uses satellites in a mixture of geostationary and Molniya orbits. The constellation of Molniya-orbiting satellites uses three satellites on different orbital planes, with apogees comparable to those of the geostationary satellites.]]

A '''Molniya orbit''' ({{lang-rus|Молния|p=ˈmolnʲɪjə|a=Ru-молния.ogg}}, "Lightning") is a type of satellite [[orbit]] designed to provide communications and [[remote sensing]] coverage over high [[latitude]]s. It is a [[highly elliptical orbit]] with an [[inclination]] of 63.4 [[Degree (angle)|degree]]s, an [[argument of perigee]] of 270 degrees, and an [[orbital period]] of approximately half a [[sidereal time|sidereal day]].<ref name="exam">{{cite conference |url=http://issfd.org/ISSFD_2009/CollisionRiskII/Kolyuka.pdf |title=Examination of the Lifetime, Evolution and Re-Entry Features for the "Molniya" Type Orbits |date=28 September 2009 |first1=Yu. F. |last1=Kolyuka |first2=N. M. |last2=Ivanov |first3=T. I. |last3=Afanasieva |first4=T. A. |last4=Gridchina |access-date=22 May 2018 |page=2 |publisher=Mission Control Center 4, Korolev, Moscow |location=Toulouse, France |conference=21st International Symposium of Space Flight Dynamics }}</ref> The name comes from the ''[[Molniya (satellite)|Molniya]]'' satellites, a series of [[Soviet Union|Soviet]]/[[Russia]]n civilian and military [[communications satellite]]s which have used this type of orbit since the mid-1960s. A variation on the Molniya orbit is the so-called Three Apogee (TAP) orbit, whose period is a third of a sidereal day.

The Molniya orbit has a long dwell time over the [[Hemispheres of Earth|hemisphere]] of interest, while moving very quickly over the other. In practice, this places it over either [[Russia]] or [[Canada]] for the majority of its orbit, providing a high [[angle of incidence (optics)|angle of view]] to communications and monitoring satellites covering these high-latitude areas. [[Geostationary orbit]]s, which are necessarily inclined over the [[equator]], can only view these regions from a low angle, hampering performance. In practice, a satellite in a Molniya orbit serves the same purpose for high latitudes as a geostationary satellite does for equatorial regions, except that multiple satellites are required for continuous coverage.<ref name="gsmo">{{cite book |url=https://books.google.com/books?id=K9Q5DwAAQBAJ&pg=PA57 |title=Global Satellite Meteorological Observation (GSMO) Theory |volume=1 |publisher=Springer International Publishing |isbn=978-3-319-67119-2 |first=Stojče Dimov |last=Ilčev |page=57 |date=2017 |access-date=16 April 2019 }}</ref>

Satellites placed in Molniya orbits have been used for television broadcasting, telecommunications, military communications, relaying, weather monitoring, [[Early warning satellite|early warning systems]] and some classified purposes.

==History==
The Molniya orbit was discovered by Soviet scientists in the 1960s as a high-[[latitude]] communications alternative to [[geostationary orbit]]s, which require large launch energies to achieve a high [[perigee]] and to [[Orbital inclination change|change inclination]] to orbit over the equator (especially when launched from Russian latitudes). As a result, [[Energia (corporation)|OKB-1]] sought a less energy-demanding orbit.<ref name=seh>{{cite book |page=416 |url=https://books.google.com/books?id=2ZNxDwAAQBAJ&q=molniya+orbit+OKB-1+history&pg=PA416 |isbn=978-1-85109-514-8 |title = Space Exploration and Humanity: A Historical Encyclopedia |volume=1 |author=History Committee of the American Astronautical Society |editor-first=Stephen B. |editor-last=Johnson |publisher=Greenwood Publishing Group |date=23 August 2010 |access-date=17 April 2019 }}</ref> Studies found that this could be achieved using a highly elliptical orbit with an [[apogee]] over Russian territory.<ref name="Martin2000">{{cite book |last=Martin |first=Donald H. |title=Communication Satellites |url=https://books.google.com/books?id=_azf94TByF8C&pg=PA218 |access-date=17 April 2019 |year=2000 |publisher=[[American Institute of Aeronautics and Astronautics]] |edition=4 |isbn=978-1-884989-09-4 |pages=215–232 }}</ref> The orbit's name refers to the "lightning" speed with which the satellite passes through the perigee.<ref>{{cite book |isbn=978-3-319-03416-4 |title=Handbook of Satellite Orbits: From Kepler to GPS |last1=Capderou |first1=Michel |date=23 April 2014 |page=393 |publisher=Springer Science & Business |url=https://books.google.com/books?id=1_68BAAAQBAJ&q=lightning |access-date=16 April 2019 |bibcode=2014hso..book.....C }}</ref>

The first use of the Molniya orbit was by the [[communications satellite]] series [[Molniya (satellite)|of the same name]]. After two launch failures, and one satellite failure in 1964, the first successful satellite to use this orbit, Molniya 1-1, launched on 23 April 1965.<ref name="Martin2000"/><ref>{{cite report|publisher=[[CIA]]: Office of Scientific Intelligence|date=12 December 2003|url=https://www.cia.gov/library/readingroom/docs/CIA-RDP78T04759A000500010016-5.pdf|archive-url=https://web.archive.org/web/20170123021401/https://www.cia.gov/library/readingroom/docs/CIA-RDP78T04759A000500010016-5.pdf|url-status=dead|archive-date=January 23, 2017|access-date=16 April 2016|title=Preliminary Analysis of the First Successful Soviet Communications Satellite|page=3}}</ref> The early Molniya-1 satellites were used for civilian television, telecommunication and long-range military communications, but they were also fitted with cameras used for weather monitoring, and possibly for assessing clear areas for [[Zenit (satellite)|Zenit spy satellites]].<ref name=seh/><ref>{{cite journal|title=A History of Soviet/Russian Meteorological Satellites|url=http://www.bis-space.com/belgium/wp-content/uploads/2015/05/meteor.pdf|first=Bart|last=Hendrickx|page=66|journal=[[Journal of the British Interplanetary Society]]|volume=57 (Suppl. 1)|year=2004|access-date=2018-03-27|archive-date=2018-03-27|archive-url=https://web.archive.org/web/20180327212247/http://www.bis-space.com/belgium/wp-content/uploads/2015/05/meteor.pdf|url-status=dead}}</ref> The original Molniya satellites had a lifespan of approximately 1.5 years, as their orbits were disrupted by [[Orbital perturbation analysis|perturbations]], and they had to be constantly replaced.<ref name="exam"/>

The succeeding series, the Molniya-2, provided both military and civilian broadcasting and was used to create the [[Orbita (TV system)|Orbita]] [[television network]], spanning the Soviet Union. These were in turn replaced by the Molniya-3 design.<ref name="Martin2000"/> A satellite called Mayak was designed to supplement and replace the Molniya satellites in 1997, but the project was cancelled,<ref name="tiros">{{cite report|publisher=Tiros Space Information News Bulletin|location=IAC 2017|volume=41|number=3|date=December 2015|editor-first=Jos|editor-last=Heyman|title=Cancelled projects: Russian comsats|first=Jos|last=Heyman|url=https://www.spaceindustry.com.au/Documents/nb2015-12.pdf|page=4|access-date=16 April 2019|archive-url=https://web.archive.org/web/20190305152203/https://www.spaceindustry.com.au/Documents/nb2015-12.pdf|archive-date=5 March 2019|url-status=dead}}</ref> and the Molniya-3 was replaced by the [[Meridian (satellite)|Meridian]] satellites, the first of which launched in 2006.<ref>{{cite web|title=Soyuz 2-1a launches with Russian Meridian 4 military satellite|publisher=NASASpaceflight.com|first=William |last=Graham|date=4 May 2011|access-date=16 April 2019 |url=https://www.nasaspaceflight.com/2011/05/soyuz-2-1a-russian-meridian-4-military-satellite/}}</ref> The Soviet [[US-K]] [[early warning satellite]]s, which watch for American rocket launches, were launched in Molniya orbits from 1967, as part of the [[Oko]] system.<ref name="cato">{{cite journal| title = Reducing a Common Danger: Improving Russia's Early-Warning System| journal = Cato Policy Analysis No. 399|page=5|date=May 3, 2001|first= Geoffrey|last= Forden|url = http://www.cato.org/pubs/pas/pa399.pdf|access-date=16 April 2019}}</ref><ref name="Podvig-History">{{Cite journal |first = Pavel |last = Podvig |year = 2002 |title = History and the Current Status of the Russian Early-Warning System |journal = Science and Global Security |volume = 10 |issue = 1 |pages = 21–60 |issn = 0892-9882 |doi = 10.1080/08929880212328 |bibcode = 2002S&GS...10...21P |url = http://iis-db.stanford.edu/pubs/20734/Podvig-S&GS.pdf |url-status = dead |archive-url = https://web.archive.org/web/20120315024323/http://iis-db.stanford.edu/pubs/20734/Podvig-S%26GS.pdf |archive-date = 2012-03-15 |citeseerx = 10.1.1.692.6127|s2cid = 122901563 }}</ref><ref>{{cite web|url=https://themoscowtimes.com/articles/russia-blinded-by-loss-of-missile-detection-satellite-36742 |title=Russia blinded by loss of missile detection satellite |publisher=Moscow Times| date=26 June 2014 |access-date=16 April 2019}}</ref>

From 1971, the American [[Jumpseat (satellite)|Jumpseat]] and [[Trumpet (satellite)|Trumpet]] military satellites were launched into Molniya orbits (and possibly used to intercept Soviet communications from the Molniya satellites). Detailed information about both projects remains classified {{as of|2019|lc=y}}.<ref>{{cite web|url=https://www.nasaspaceflight.com/2017/09/atlas-v-preparing-to-launch-nrol-42/ |publisher=NASASpaceflight.com |title=Atlas V launches NROL-42 spy satellite |first=William|last=Graham |date=23 September 2017|access-date=16 April 2019}}</ref> This was followed by the American [[Satellite Data System|SDS]] constellation, which operates with a mixture of Molniya and geostationary orbits. These satellites are used to relay signals from lower flying satellites back to ground stations in the United States and have been active in some capacity since 1976.<ref>{{cite book|first=Jeffrey T |last=Richelson|date=2002|title=The Wizards of Langley. Inside the CIA's Directorate of Science and Technology.|publisher=[[Westview Press]] |isbn=978-0-8133-4059-3 |location=Boulder|url=https://archive.org/details/wizardsoflangley00rich|url-access=registration |access-date=17 April 2019}}</ref> A Russian satellite constellation called Tyulpan was designed in 1994 to support communications at high latitudes, but it did not progress past the planning phase.<ref name="tiros"/>

In 2015 and 2017 Russia launched two ''Tundra'' satellites into a Molniya orbit, despite their name, as part of its [[EKS (satellite system)|EKS]] early warning system.<ref>{{Cite web|url=https://www.spaceflightinsider.com/missions/defense/russian-soyuz-2-1b-rocket-successfully-launches-tundra-satellite/|title=Russian Soyuz-2.1b rocket successfully launches Tundra satellite|date=November 17, 2015 |author=Tomasz Nowakowski |publisher=Spaceflight Insider}}</ref><ref>{{Cite web|url=https://www.spaceflightinsider.com/organizations/roscosmos/soyuz-rocket-successfully-delivers-eks-2-early-warning-satellite-rare-orbit/|title=Soyuz rocket successfully delivers EKS-2 early-warning satellite to rare orbit|date=May 25, 2017 |author=Curt Godwin |publisher=Spaceflight Insider}}</ref><ref>{{Cite web|url=https://spaceflightnow.com/2017/05/25/russia-sends-military-satellite-into-orbit-for-missile-warnings/|first=Stephen|last=Clark|date=25 May 2017|title=Russia sends military satellite into orbit for missile warnings – Spaceflight Now}}</ref>

{{multiple image
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| header = Animation of [[EKS (satellite system)|EKS]]
| image1 = Animation of EKS orbit around Earth.gif
| caption1 = Equatorial view
| image2 = Animation of EKS orbit around Earth - polar view.gif
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| image3 = Animation of EKS orbit - ECEF - front view.gif
| caption3 =  [[ECEF|Earth fixed frame]], front view
| image4 = Animation of EKS orbit - ECEF.gif
| caption4 =  [[ECEF|Earth fixed frame]], side view
| footer = {{legend2| OrangeRed | Kosmos 2510}}{{·}}{{legend2|Gold| Kosmos 2518}}{{·}}{{legend2|Cyan| Kosmos 2541}}{{·}}{{legend2|Lime| Kosmos 2546}}{{·}}{{legend2|RoyalBlue|Earth}}}}
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==Uses==
[[File:molniya.jpg|thumb|Figure 3: Groundtrack of a Molniya orbit. In the operational part of the orbit (four hours on each side of apogee), the satellite is north of 55.5° N (latitude of, for example, central Scotland, Moscow and southern part of Hudson Bay). A satellite in this orbits spends most of its time over the northern hemisphere and passes quickly over the southern hemisphere.]]
Much of the area of the former [[Soviet Union]], and [[Russia]] in particular, is located at high northern latitudes. To broadcast to these latitudes from a geostationary orbit (above the Earth's [[equator]]) requires considerable power due to the low [[angle of incidence (optics)|elevation angles]], and the extra distance and atmospheric attenuation that comes with it. Sites located above 81° latitude are unable to view geostationary satellites at all, and as a rule of thumb, elevation angles of less than 10° can cause problems, depending on the communications frequency.<ref name="gsmo"/>{{rp|499}}<ref>{{cite journal|url=http://www.ngs.noaa.gov/CORS/Articles/SolerEisemannJSE.pdf|page=123|title=Determination of Look Angles To Geostationary Communication Satellites|first1=Tomás |last1=Soler|first2= David W. |last2=Eisemann|journal=Journal of Surveying Engineering |volume=120|issue=3|date=August 1994|issn=0733-9453|access-date=16 April 2019|doi=10.1061/(ASCE)0733-9453(1994)120:3(115)}}</ref>

A satellite in a Molniya orbit is better suited to communications in these regions, because it looks more directly down on them during large portions of its orbit. With an apogee altitude as high as {{Convert|40000|km|mi}} and an apogee sub-satellite point of 63.4 degrees north, it spends a considerable portion of its orbit with excellent visibility in the northern hemisphere, from Russia as well as from northern Europe, Greenland and Canada.<ref name="gsmo"/>

While satellites in Molniya orbits require considerably less launch energy than those in geostationary orbits (especially launching from high latitudes),<ref name="Martin2000"/> their ground stations need steerable antennas to track the spacecraft, links must be switched between satellites in a constellation and range changes cause variations in signal amplitude. Additionally, there is a greater need for [[Orbital station-keeping|station-keeping]],<ref name="smad" /><ref name="art">{{cite journal|page=517 |date=18 August 1989 |journal=Journal of Atmospheric and Oceanic Technology|volume=7|issue=3 |title= On the Use of Satellites in Molniya Orbits of Meteorological Observation of Middle and High Latitudes |first1=Stanley Q. |last1=Kidder |first2=Thomas H. |last2=Vonder Haar|doi=10.1175/1520-0426(1990)007<0517:OTUOSI>2.0.CO;2|bibcode=1990JAtOT...7..517K |doi-access=free }}</ref><ref>{{cite journal|title=The Orbital Lifetime of Molniya Satellites|journal=Journal of the British Interplanetary Society |volume=28|url=https://www.researchgate.net/publication/252561825|first=D. G.|last=King-Hele|date=January 1975|pages=783–796|bibcode=1975JBIS...28..783K}}</ref> and the spacecraft will pass through the [[Van Allen radiation belt]] four times per day.<ref>{{cite book|page=67|title=Mission Design & Implementation of Satellite Constellations: Proceedings of an International Workshop held in Toulouse, France|date=November 1997|publisher=Springer-Science|url=https://books.google.com/books?id=rUDwCAAAQBAJ&q=van+allen+belts+molniya&pg=PA67|editor-first=Jozef C. |editor-last=van der Ha|isbn=9401061378|access-date=16 April 2019}}</ref>

===Southern hemisphere proposals===
Similar orbits with an argument of perigee of 90° could allow high-latitude coverage in the southern hemisphere. A proposed constellation, the ''Antarctic Broadband Program'', would have used satellites in an inverted Molniya orbit to provide [[Satellite internet access|broadband internet service]] to facilities in [[Antarctica]].<ref>{{Cite web|url=https://rsaa.anu.edu.au/research/established-projects/antarctic-broadband-program|title=Antarctic Broadband program|website=rsaa.anu.edu.au|publisher=Australian National University|access-date=12 April 2019}}</ref><ref name=aab>{{Cite conference|url=https://www.researchgate.net/publication/312119913|title=Antarctic Broadband: Fast Internet for the Bottom of the Earth |conference=IAC 2012 |date=October 2012 |first1= Grant|last1= Bonin|first2=Robert|last2=Zee|first3=Michael|last3=Brett|first4=Jan |last4=King|first5=Daniel |last5=Faber|access-date=12 April 2019}}</ref> Initially funded by the now defunct [[Australian Space Agency|Australian Space Research Programme]], it did not progress beyond initial development.<ref>{{cite report|title=Final evaluation of the Australian Space Research Program|publisher=Department of Industry, Innovation and Science|date=17 November 2015|editor-first=Cameron|editor-last=Bird|url=http://www.spaceindustry.com.au/Documents/Final_evaluation.pdf|access-date=12 April 2019|archive-date=5 March 2019|archive-url=https://web.archive.org/web/20190305154405/https://www.spaceindustry.com.au/Documents/Final_evaluation.pdf|url-status=dead}}</ref><ref>{{Cite web|url=http://theconversation.com/as-the-details-emerge-on-australias-new-space-agency-we-might-finally-have-lift-off-96542|title=As the details emerge on Australia's new space agency, we (might) finally have lift-off|first=Andrew|last=Dempster|website=The Conversation|date=15 May 2018 |access-date=12 April 2019}}</ref>

===Molniya constellations===
Permanent high-latitude coverage of a large area of Earth (like the whole of Russia, where the southern parts are about 45°{{nbsp}}N) requires a constellation of at least three spacecraft in Molniya orbits. If three spacecraft are used, then each spacecraft will be active for a period of eight hours per orbit, centered around apogee,<ref name="gsmo"/> as illustrated in figure 4. Figure 5 shows the satellite's field of view around the apogee. 

The Earth completes half a rotation in twelve hours, so the apogees of successive Molniya orbits will alternate between one half of the northern hemisphere and the other. For the original Molniya orbit, the apogees were placed over Russia and North America, but by changing the [[longitude of the ascending node|right ascension of the ascending node]] this can be varied.<ref name=smad/> The coverage from a satellite in a Molniya orbit over Russia is shown in figures 6 to 8, and over North America in figures 9 to 11.

The orbits of the three spacecraft should then have the same orbital parameters, but different right ascensions of the ascending nodes, with their passes over the apogees separated by 7.97 hours.<ref name="gsmo"/><ref>{{cite journal|page=519 |date=June 1990 |journal=Journal of Atmospheric and Oceanic Technology|volume=7|title= On the Use of Satellites in Molniya Orbits for Meteorological Observation of Middle and High Latitudes |first1=Stanley Q. |last1=Kidder |first2=Thomas H. |last2=Vonder Haar|issue=3 |doi=10.1175/1520-0426(1990)007<0517:OTUOSI>2.0.CO;2|bibcode=1990JAtOT...7..517K |doi-access=free }}</ref> Since each satellite has an operational period of approximately eight hours, when one spacecraft travels four hours after its apogee passage (see figure 8 or figure 11), then the next satellite will enter its operational period, with the view of the earth shown in figure 6 (or figure 9), and the switch-over can take place. Note that the two spacecraft at the time of switch-over are separated by about {{Convert|1500|km|mi}}, so that the ground stations only have to move their antennas a few degrees to acquire the new spacecraft.<ref>{{cite journal|page=9947 |volume=4 |year=2016 |title=Systems Engineering of a Terabit Elliptic Orbit Satellite and Phased Array Ground Station for IoT Connectivity and Consumer Internet Access |first1=R. L. |last1=Sturdivant|first2=E. K. P.|last2=Chon |journal=IEEE Access|doi=10.1109/ACCESS.2016.2608929|doi-access=free }}</ref>

===Diagrams===
{{Gallery|width=200
| File:Molniya 3 spacecraft configuration.svg | Figure 4: A constellation of three Molniya spacecraft providing service for the Northern hemisphere. '''P''' is the orbital period. A green line corresponds to service for Asia and Europe with the visibility of figures 6–8. A red line corresponds to service for North America with the visibility of figures 9–11.
| File:Mats.svg | Figure 5: Illumination zones (at least 10° elevation) from a Molniya orbit. At apogee, the green illumination zone applies. At three hours before or after apogee, the red zone applies. At four hours before or after apogee, the blue zone applies. The plane of the figure is the longitudinal plane of apogee rotating with the Earth. In the eight-hour period centered at the apogee passage, the longitudinal plane is almost fixed, the longitude of the satellite varies by only ±2.7°. The views of the Earth from these three points are displayed in figures 6–11.
| File:Molniya earth view Em4.svg | Figure 6: View of the Earth four hours before apogee from a Molniya orbit under the assumption that the longitude of the apogee is 90°{{nbsp}}E. The spacecraft is at an altitude of 24,043&nbsp;km over the point 92.65°{{nbsp}}E 47.04°{{nbsp}}N.
| File:Molniya earth view E.svg | Figure 7: View of the Earth from the apogee of a Molniya orbit under the assumption that the longitude of the apogee is 90°{{nbsp}}E. The spacecraft is at an altitude of 39,867&nbsp;km over the point 90°{{nbsp}}E 63.43°{{nbsp}}N.
| File:Molniya earth view Ep4.svg | Figure 8: View of the Earth four hours after apogee from a Molniya orbit under the assumption that the longitude of the apogee is 90° E. The spacecraft is at an altitude of 24,043&nbsp;km over the point 87.35°{{nbsp}}E 47.04°{{nbsp}}N
| File:Molniya earth view Wm4.svg | Figure 9: View of the Earth four hours before apogee from a Molniya orbit under the assumption that the longitude of the apogee is 90°{{nbsp}}W. The spacecraft is at an altitude of 24,043&nbsp;km over the point 87.35°{{nbsp}}W 47.04°{{nbsp}}N.
| File:Molniya earth view W.svg | Figure 10: View of the Earth from the apogee of a Molniya orbit under the assumption that the longitude of the apogee is 90°{{nbsp}}W. The spacecraft is at an altitude of 39,867&nbsp;km over the point 90°{{nbsp}}W 63.43°{{nbsp}}N.
| File:Molniya earth view Wp4.svg | Figure 11: View of the Earth 4 hours after apogee from a Molniya orbit under the assumption that the longitude of the apogee is 90°{{nbsp}}W. The spacecraft is at an altitude of 24,043&nbsp;km over the point 92.65°{{nbsp}}W 47.04°{{nbsp}}N.
}}
{{multiple image
| align = left
| direction = horizontal
| width = 240
| header = Comparison of [[Tundra orbit]], [[Quasi-Zenith Satellite System|QZSS orbit]] and Molniya orbit - equatorial view
| image1 = Animation of Tundra and QZSS orbit - front view.gif
| caption1 = Front view
| image2 = Animation of Tundra and QZSS orbit - side view.gif
| caption2 =  Side view
| image3 = Animation of Tundra and QZSS orbit - ECEF - front view.gif
| caption3 = [[ECEF|Earth fixed frame]], Front view
| image4 = Animation of Tundra and QZSS orbit - ECEF - side view.gif
| caption4 =  [[ECEF|Earth fixed frame]], Side view
| footer = {{legend2| OrangeRed | Tundra orbit}}{{·}}{{legend2|Lime| QZSS orbit}}{{·}}{{legend2|Cyan|Molniya orbit}}{{·}}{{legend2|RoyalBlue|Earth}}}}
{{clear}}

==Properties==
A typical Molniya orbit has the following properties:

* Argument of perigee: 270°
* Inclination: 63.4°<ref name="art"/>
* Period: 718 minutes<ref name="exam"/>
* Eccentricity: 0.74
* [[Semi-major and semi-minor axes#Astronomy|Semi-major axis]]: {{Convert|26600|km|mi|abbr=on}}

===Argument of perigee===
The argument of perigee is set at 270°, causing the satellite to experience apogee at the most northerly point of its orbit. For any future applications over the southern hemisphere, it would instead be set at 90°.<ref name=aab/>

===Orbital inclination===
In general, the [[oblate spheroid|oblateness]] of the Earth [[Orbital perturbation analysis|perturbs]] the [[argument of perigee]] (<math>\omega</math>), so that it gradually changes with time. If we only consider the first-order coefficient <math>J_2</math>, the perigee will change according to equation {{EquationNote|1}}, unless it is constantly corrected with station-keeping thruster burns.

{{NumBlk|:|<math>\dot{\omega} = \frac{3}{4}\cdot n J_2\cdot\left(\frac{R_E}{a}\right)^2 \frac{4-5\sin^2{i}}{(1-e^2)^2}</math>|{{EquationRef|1}}}}

where <math>i</math> is the orbital inclination, <math>e</math> is the eccentricity, <math>n</math> is mean motion in degrees per day, <math>J_2</math> is the perturbing factor, <math>R_E</math> is the radius of the earth, <math>a</math> is the semimajor axis, and <math>\dot{\omega}</math> is in degrees per day.

To avoid this expenditure of fuel, the Molniya orbit uses an inclination of 63.4°, for which the factor <math>4-5\sin^2{i}</math> is zero, so that there is no change in the position of perigee over time.<ref name="art"/><ref name="smad">{{cite book|title=Space Mission Analysis and Design|publisher=Microcosm Press and Kluwer Academic Publishers |editor1-first=Wiley J. |editor1-last=Larson |editor2-first=James R. |editor2-last=Wertz |bibcode=1999smad.book.....W |last1=Wertz |first1=James Richard |last2=Larson |first2=Wiley J. |year=1999|isbn=1-881883-10-8}}</ref>{{rp|143}} An orbit designed in this manner is called a [[frozen orbit]].

===Orbital period===
To ensure the geometry relative to the ground stations repeats every 24 hours, the [[Orbital period|period]] should be about half a [[sidereal day]], keeping the longitudes of the apogees constant.

However, the [[oblate spheroid|oblateness]] of the Earth also perturbs the right ascension of the [[Orbital node|ascending node]] (<math>\Omega</math>), changing the [[nodal period]] and causing the [[ground track]] to drift over time at the rate shown in equation {{EquationNote|2}}.

{{NumBlk|:| <math>\dot{\Omega} = -\frac{3}{2}\cdot n J_2\cdot\left(\frac{R_E}{a}\right)^2 \frac{\cos{i}} {(1-e^2)^2}</math> |{{EquationRef|2}}}}

where <math>\dot{\Omega}</math> is in degrees per day.<ref name=smad/>{{rp|143}}

Since the inclination of a Molniya orbit is fixed (as above), this perturbation is <math>\dot{\Omega} = -0.3</math> degrees per day. To compensate, the orbital period is adjusted so that the longitude of the apogee changes enough to cancel out this effect.<ref name="art"/>

===Eccentricity===
The eccentricity of the orbit is based on the differences in altitudes of its apogee and perigee. To maximise the amount of time that the satellite spends over the apogee, the eccentricity should be set [[Kepler's laws of planetary motion#Second law of Kepler|as high as possible]]{{Broken anchor|date=2024-09-06|bot=User:Cewbot/log/20201008/configuration|target_link=Kepler's laws of planetary motion#Second law of Kepler|reason= The anchor (Second law of Kepler) [[Special:Diff/968688731|has been deleted]].}}. However, the perigee needs to be high enough to keep the satellite substantially above the [[atmosphere]] to minimize drag (~600km), and the orbital period needs to be kept to approximately half a sidereal day (as above). These two factors constrain the eccentricity, which becomes approximately 0.737.<ref name="art"/>

===Semi-major axis===
The exact height of a satellite in a Molniya orbit varies between missions, but a typical orbit will have a perigee altitude of approximately {{Convert|600|km|mi}} and an apogee altitude of {{Convert|39700|km|mi}}, for a semi-major axis of {{Convert|26600|km|mi}}.<ref name="art"/>

==Modelling==
To track satellites using Molniya orbits, scientists use the SDP4 [[simplified perturbations models|simplified perturbations model]], which calculates the location of a satellite based on orbital shape, drag, radiation, gravitation effects from the sun and moon, and earth resonance terms.<ref name="spacetrackreport">{{cite report|last1= Hoots |first1=Felix R.|first2=Ronald L. |last2=Roehrich |title=Models for Propagation of NORAD Element Sets|publisher=United States Department of Defense Spacetrack Report|date=31 December 1988|issue=3|url=http://www.celestrak.com/NORAD/documentation/spacetrk.pdf|access-date=16 June 2010}}</ref>

==See also==
* [[List of orbits]]
* [[Tundra orbit]]

==References==
{{Reflist}}

==External links==
* [https://www.youtube.com/watch?v=O_Iykeouj3g Illustration of the communication geometry provided by satellites in 12-hour Molniya orbits] (video)

{{Orbits}}
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[[Category:Earth orbits]]
[[Category:Satellite broadcasting]]
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