Editing Sun-synchronous orbit (section)

== Technical details ==
The angular [[Nodal precession|precession]] per orbit for an Earth orbiting satellite is approximately given by
: <math>\Delta \Omega = -3\pi \frac{J_2 R_\text{E}^2}{p^2} \cos i,</math> 
where
: {{math|''J''<sub>2</sub> {{=}} {{val|1.08263e-3}}}} is the [[geopotential model|coefficient for the second zonal term]] related to the [[oblateness]] of the Earth,
: {{math|''R''<sub>E</sub> ≈ 6378 km}} is the mean radius of the Earth,
: {{mvar|p}} is the [[Conic section#Conic parameters|semi-latus rectum]] of the orbit,
: {{mvar|i}} is the inclination of the orbit to the equator.

An orbit will be Sun-synchronous when the precession rate {{math|''ρ'' {{=}} {{sfrac|d''Ω''|d''t''}} }} equals the mean motion of the Earth about the Sun {{math|''n''<sub>E</sub>}}, which is 360° per [[sidereal year]] ({{val|1.99096871e-7|u=[[radian|rad]]/s}}), so we must set {{math| ''n''<sub>E</sub> {{=}} {{sfrac|Δ''Ω''<sub>E</sub>|''T''<sub>E</sub>}} {{=}} ''ρ'' {{=}} {{sfrac|Δ''Ω''|''T''}} }}, where {{mvar|T<sub>E</sub>}} is the Earth orbital period, while {{mvar|T}} is the period of the spacecraft around the Earth.

As the orbital period of a spacecraft is
: <math>T = 2\pi \sqrt{\frac{a^3}{\mu}},</math>

where {{mvar|a}} is the [[semi-major axis]] of the orbit, and {{mvar|μ}} is the [[standard gravitational parameter]] of the planet ({{val|398600.440|u=km<sup>3</sup>/s<sup>2</sup>}} for Earth); as {{math|''p'' ≈ ''a''}} for a circular or almost circular orbit, it follows that
: <math>\begin{align}
   \rho &\approx -\frac{3J_2 R_\text{E}^2 \sqrt{\mu}\cos i}{2a^{7/2}} \\
        &= -(360^\circ\text{ per year}) \times \left(\frac{a}{12\,352\text{ km}}\right)^{-7/2} \cos i \\
        &= -(360^\circ\text{ per year}) \times \left(\frac{T}{3.795\text{ h}}\right)^{-7/3} \cos i,
\end{align}</math>

or when {{mvar|ρ}} is 360° per year,
: <math>
   \cos i \approx -\frac{2\rho}{3 J_2 R_\text{E}^2 \sqrt{\mu}} a^{7/2} =
  -\left(\frac{a}{12\,352\text{ km}}\right)^{7/2} =
  -\left(\frac{T}{3.795\text{ h}}\right)^{7/3}.
</math>

As an example, with {{mvar|a}} = {{val|7200|u=km}}, i.e., for an altitude {{math|''a'' − ''R''<sub>E</sub>}} ≈ {{val|800|u=km}} of the spacecraft over Earth's surface, this formula gives a Sun-synchronous inclination of 98.7°.<!-- 98.696° -->

Note that according to this approximation {{math|cos ''i''}} equals −1 when the semi-major axis equals {{val|12352|u=km}}, which means that only lower orbits can be Sun-synchronous. The period can be in the range from 88&nbsp;minutes for a very low orbit ({{mvar|a}} = {{val|6554|u=km}}, {{mvar|i}} = 96°) to 3.8&nbsp;hours ({{mvar|a}} = {{val|12352|u=km}}, but this orbit would be equatorial, with {{mvar|i}} = 180°). A period longer than 3.8&nbsp;hours may be possible by using an eccentric orbit with {{mvar|p}} < {{val|12352|u=km}} but {{mvar|a}} > {{val|12352|u=km}}.

If one wants a satellite to fly over some given spot on Earth every day at the same hour, the satellite must complete a whole number of orbits per day. Assuming a circular orbit, this comes down to between 7 and 16 orbits per day, as doing less than 7 orbits would require an altitude above the maximum for a Sun-synchronous orbit, and doing more than 16 would require an orbit inside the Earth's atmosphere or surface. The resulting valid orbits are shown in the following table. (The table has been calculated assuming the periods given. The orbital period that should be used is actually slightly longer. For instance, a retrograde equatorial orbit that passes over the same spot after 24&nbsp;hours has a true period about {{sfrac|365|364}} ≈ 1.0027 times longer than the time between overpasses. For non-equatorial orbits the factor is closer to 1.)

: {| class="wikitable" style="text-align:right;"
! Orbits <br/>per day
! colspan=2 | Period ([[hour|h]])
! Altitude <br/>(km)
! Maximal <br/>latitude
! Inclin-<br/>ation
|-
| 16 || {{sfrac|1|1|2}}   || = 1:30 || 274 || 83.4° || 96.6°
|-
| 15 || {{sfrac|1|3|5}}   || = 1:36 || 567 || 82.3° || 97.7°
|-
| 14 || {{sfrac|1|5|7}}   || ≈ 1:43 ||  894 || 81.0° ||  99.0°
|-
| 13 || {{sfrac|1|11|13}} || ≈ 1:51 || 1262 || 79.3° || 100.7°
|-
| 12 || 2                 ||        || 1681 || 77.0° || 103.0°
|-
| 11 || {{sfrac|2|2|11}}  || ≈ 2:11 || 2162 || 74.0° || 106.0°
|-
| 10 || {{sfrac|2|2|5}}   || = 2:24 || 2722 || 69.9° || 110.1°
|-
| 9 || {{sfrac|2|2|3}}    || = 2:40 || 3385 || 64.0° || 116.0°
|-
| 8 || 3                  ||        || 4182 || 54.7° || 125.3°
|-
| 7 || {{sfrac|3|3|7}}    || ≈ 3:26 || 5165 || 37.9° || 142.1°
|}

When one says that a Sun-synchronous orbit goes over a spot on the Earth at the same ''local time'' each time, this refers to [[mean solar time]], not to [[apparent solar time]]. The Sun will not be in exactly the same position in the sky during the course of the year (see [[Equation of time]] and [[Analemma]]).

Sun-synchronous orbits are mostly selected for [[Earth observation satellite]]s, with an altitude typically between 600 and {{val|1000|u=km}} over the Earth surface. Even if an orbit remains Sun-synchronous, however, other orbital parameters such as [[argument of periapsis]] and the [[orbital eccentricity]] evolve, due to higher-order perturbations in the Earth's gravitational field, the pressure of sunlight, and other causes. Earth observation satellites, in particular, prefer orbits with constant altitude when passing over the same spot. Careful selection of eccentricity and location of perigee reveals specific combinations where the rate of change of perturbations are minimized, and hence the orbit is relatively stable{{snd}} a [[frozen orbit]], where the motion of position of the periapsis is stable.<ref>{{cite journal |last1=Low |first1=Samuel Y. W. |title=Designing a Reference Trajectory for Frozen Repeat Near-Equatorial Low Earth Orbits |journal=AIAA Journal of Spacecraft and Rockets |date=January 2022 |volume=59 |issue=1 |pages=84–93 |doi=10.2514/1.A34934 |bibcode=2022JSpRo..59...84L |s2cid=236275629 }}</ref> The [[European Remote-Sensing Satellite|ERS-1, ERS-2]] and [[Envisat]] of [[European Space Agency]], as well as the [[MetOp]] spacecraft of [[EUMETSAT]] and [[Radarsat-2|RADARSAT-2]] of the [[Canadian Space Agency]], are all operated in such Sun-synchronous frozen orbits.<ref>{{cite conference |bibcode=1989ommd.proc...49R |first=Mats |last=Rosengren |title=Improved technique for Passive Eccentricity Control (AAS 89-155) |volume=69 |publisher=AAS/NASA |book-title=Advances in the Astronautical Sciences |year=1989}}</ref>