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Nodal precession
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<h2 id="description">Description</h2> <p>A non-rotating body of planetary scale or larger would be pulled by gravity into a spherical shape. Virtually all bodies rotate, however. The centrifugal force deforms the body so that it has an <a href="/facts/Equatorial_bulge/8CIPWl0I">equatorial bulge</a>. Because of the bulge of the central body, the gravitational force on a satellite is not directed toward the center of the central body, but is offset toward its equator. Whichever hemisphere of the central body the satellite lies over, it is preferentially pulled slightly toward the equator of the central body. This creates a torque on the satellite. This torque does not reduce the inclination; rather, it causes a torque-induced gyroscopic <a href="/facts/Precession/44Sk1PMD">precession</a>, which causes the <a href="/facts/Orbital_nodes/E5tUmCsY">orbital nodes</a> to drift with time. </p> <h2 id="equation">Equation</h2> <p>The rate of precession depends on the <a href="/facts/Inclination/TQCFWyfd">inclination</a> of the orbital plane to the equatorial plane, as well as the orbital eccentricity. </p><p>For a satellite in a <a href="/facts/Prograde_orbit/1HCbwQqy">prograde orbit</a> around Earth, the precession is westward (nodal regression), that is, the node and satellite move in opposite directions.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> A good approximation of the precession rate is </p> ω p = − 3 2 R E 2 ( a ( 1 − e 2 ) ) 2 J 2 ω cos i {\displaystyle \omega _{\mathrm {p} }=-{\frac {3}{2}}{\frac {{R_{\mathrm {E} }}^{2}}{\left(a\left(1-e^{2}\right)\right)^{2}}}J_{2}\omega \cos i} <p>where </p> <i>ω</i>p is the precession rate (in <a href="/facts/Radian/Srd38Sxa">rad</a>/s), <i>R</i>E is the body's equatorial radius (6378137 m for Earth), a is the <a href="/facts/Semi-major_axis/l7JES2wo">semi-major axis</a> of the satellite's orbit, e is the eccentricity of the satellite's orbit, ω is the angular velocity of the satellite's motion (2π radians divided by its period in seconds), i is its inclination, <i>J</i>2 is the body's <a href="/facts/Second_dynamic_form_factor/8CIPWl0I">second dynamic form factor</a> <p>The nodal progression of low Earth orbits is typically a few degrees per day to the west (negative). For a satellite in a circular (e = 0) 800 km altitude orbit at 56° inclination about Earth: </p> R E = 6.378 137 × 10 6 m J 2 = 1.082 626 68 × 10 − 3 {\displaystyle {\begin{aligned}R_{\mathrm {E} }&=6.378\,137\times 10^{6}{\text{ m}}\\J_{2}&=1.082\,626\,68\times 10^{-3}\end{aligned}}} <p>The orbital period is 6052.4 s, so the angular velocity is 0.001038 rad/s. The precession is therefore </p> ω p = − 3 2 ⋅ 6 378 137 2 ( 7 178 137 ( 1 − 0 2 ) ) 2 ⋅ ( 1.082 626 68 × 10 − 3 ) ⋅ 0.001 038 ⋅ cos 56 ∘ = − 7.44 × 10 − 7 rad/s {\displaystyle {\begin{aligned}\omega _{\mathrm {p} }&=-{\frac {3}{2}}\cdot {\frac {6\,378\,137^{2}}{\left(7\,178\,137\left(1-0^{2}\right)\right)^{2}}}\cdot \left(1.082\,626\,68\times 10^{-3}\right)\cdot 0.001\,038\cdot \cos 56^{\circ }\\&=-7.44\times 10^{-7}{\text{ rad/s}}\end{aligned}}} <p>This is equivalent to −3.683° per day, so the orbit plane will make one complete turn (in inertial space) in 98 days. </p><p>The apparent motion of the sun is approximately +1° per day (360° per year / 365.2422 days per tropical year ≈ 0.9856473° per day), so apparent motion of the sun relative to the orbit plane is about 2.8° per day, resulting in a complete cycle in about 127 days. For retrograde orbits ω is negative, so the precession becomes positive. (Alternatively, ω can be thought of as positive but the inclination is greater than 90°, so the cosine of the inclination is negative.) In this case it is possible to make the precession approximately match the apparent motion of the sun, resulting in a <a href="/facts/Sun-synchronous_orbit/3HonupUd">heliosynchronous orbit</a>. </p><p>The J 2 {\displaystyle J_{2}} used in this equation is the dimensionless coefficient J 2 ~ = − J 2 G M E R E 2 {\displaystyle {\tilde {J_{2}}}=-{\frac {J_{2}}{GM_{E}R_{E}^{2}}}} from the <a href="/facts/Geopotential_model/g735k6yT">geopotential model</a> or gravity field model for the body. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Axial_precession_(astronomy)/NQ79x3m3">Axial precession</a>, or "precession of the equinoxes" for Earth</li> <li><a href="/facts/Apsidal_precession/YUWDqXDp">Apsidal precession</a>, another kind of orbital precession (the change in the <a href="/facts/Argument_of_periapsis/cthncYDb">argument of periapsis</a>)</li> <li><a href="/facts/Lunar_standstill/ynvINcPL">Lunar standstill</a>, in which the Moon's <a href="/facts/Declination/5X4S4k9u">declination</a> on the lunistices depends on <a href="/facts/Lunar_precession/U1ISnDzx">the precession</a> of <a href="/facts/Lunar_node/Q3SstDoh">its orbital nodes</a></li> <li><a href="/facts/Lunar_node/Q3SstDoh">Lunar node</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://yarchive.net/space/orbits/nodal_regression.html">Nodal regression description from USENET</a></li> <li><a href="https://archive.today/20130410123143/http://www.agi.com/resources/academic-resources/astro-primer/primer91.htm">Discussion of nodal regression from Analytical Graphics</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Brown, Charles (2002). Elements of spacecraft design. p. 106. ISBN 9781600860515. <a href="9781600860515" target="_blank">9781600860515</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>