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</li> </ul> </li> <li id="toc-Stellar_collision" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stellar_collision"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Stellar collision</span> </div> </a> <ul id="toc-Stellar_collision-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mass_concentration" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Mass_concentration"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Mass concentration</span> </div> </a> <ul id="toc-Mass_concentration-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>References</span> </div> </a> <ul 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href="https://bn.wikipedia.org/wiki/%E0%A6%95%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A6%AF%E0%A6%BC" title="কক্ষীয় ক্ষয় – Bangla" lang="bn" hreflang="bn" data-title="কক্ষীয় ক্ষয়" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Deteriorament_orbital" title="Deteriorament orbital – Catalan" lang="ca" hreflang="ca" data-title="Deteriorament orbital" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Deterioro_orbital" title="Deterioro orbital – Spanish" lang="es" hreflang="es" data-title="Deterioro orbital" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B1%D8%B3%D8%A7%DB%8C%D8%B4_%D9%85%D8%AF%D8%A7%D8%B1%DB%8C" title="فرسایش مداری – Persian" lang="fa" hreflang="fa" data-title="فرسایش مداری" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/D%C3%A9clin_d%27orbite" title="Déclin d'orbite – French" lang="fr" hreflang="fr" data-title="Déclin d'orbite" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Orbitalno_ispadanje" title="Orbitalno ispadanje – Croatian" lang="hr" hreflang="hr" data-title="Orbitalno ispadanje" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Peluruhan_orbit" title="Peluruhan orbit – Indonesian" lang="id" hreflang="id" data-title="Peluruhan orbit" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Decadimento_orbitale" title="Decadimento orbitale – Italian" lang="it" hreflang="it" data-title="Decadimento orbitale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%93%D7%A2%D7%99%D7%9B%D7%AA_%D7%9E%D7%A1%D7%9C%D7%95%D7%9C_%D7%9C%D7%95%D7%95%D7%99%D7%99%D7%A0%D7%99" title="דעיכת מסלול לווייני – Hebrew" lang="he" hreflang="he" data-title="דעיכת מסלול לווייני" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Decaimento_orbital" title="Decaimento orbital – Portuguese" lang="pt" hreflang="pt" data-title="Decaimento orbital" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%95%E0%B0%95%E0%B1%8D%E0%B0%B7%E0%B1%8D%E0%B0%AF%E0%B0%BE_%E0%B0%95%E0%B1%8D%E0%B0%B7%E0%B1%80%E0%B0%A3%E0%B0%A4" title="కక్ష్యా క్షీణత – Telugu" lang="te" hreflang="te" 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Not to be confused with <a href="/wiki/Decaying_Orbit_(film)" title="Decaying Orbit (film)">Decaying Orbit (film)</a>.</div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Inspiral" redirects here. Not to be confused with <a href="/wiki/Inspiral_(horse)" title="Inspiral (horse)">Inspiral (horse)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Altitude_of_Tiangong-1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Altitude_of_Tiangong-1.svg/290px-Altitude_of_Tiangong-1.svg.png" decoding="async" width="290" height="148" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Altitude_of_Tiangong-1.svg/435px-Altitude_of_Tiangong-1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Altitude_of_Tiangong-1.svg/580px-Altitude_of_Tiangong-1.svg.png 2x" data-file-width="988" data-file-height="504" /></a><figcaption>Altitude of <a href="/wiki/Tiangong-1" title="Tiangong-1">Tiangong-1</a> during its final year of uncontrolled reentry.<sup id="cite_ref-cmse_1-0" class="reference"><a href="#cite_note-cmse-1"><span class="cite-bracket">[</span>1<span 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a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle">Astrodynamics</th></tr><tr><td class="sidebar-image" style="padding-bottom:0.85em;"><span typeof="mw:File"><a href="/wiki/File:Orbit_mechanics_icon.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Orbit_mechanics_icon.svg/60px-Orbit_mechanics_icon.svg.png" decoding="async" width="60" height="60" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Orbit_mechanics_icon.svg/90px-Orbit_mechanics_icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cc/Orbit_mechanics_icon.svg/120px-Orbit_mechanics_icon.svg.png 2x" data-file-width="48" data-file-height="48" /></a></span></td></tr><tr><th class="sidebar-heading" style="padding-bottom:0.55em;"> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Orbital_mechanics" title="Orbital mechanics"><span style="font-size:110%;">Orbital mechanics</span></a></div></th></tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Orbital_elements" title="Orbital elements">Orbital elements</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Apsis" title="Apsis">Apsis</a></li> <li><a href="/wiki/Argument_of_periapsis" title="Argument of periapsis">Argument of periapsis</a></li> <li><a href="/wiki/Orbital_eccentricity" title="Orbital eccentricity">Eccentricity</a></li> <li><a href="/wiki/Orbital_inclination" title="Orbital inclination">Inclination</a></li> <li><a href="/wiki/Mean_anomaly" title="Mean anomaly">Mean anomaly</a></li> <li><a href="/wiki/Orbital_node" title="Orbital node">Orbital nodes</a></li> <li><a href="/wiki/Semi-major_and_semi-minor_axes" title="Semi-major and semi-minor axes">Semi-major axis</a></li> <li><a href="/wiki/True_anomaly" title="True anomaly">True anomaly</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Types of <a href="/wiki/Two-body_problem" title="Two-body problem">two-body orbits</a> by <br />eccentricity</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Circular_orbit" title="Circular orbit">Circular orbit</a></li> <li><a href="/wiki/Elliptic_orbit" title="Elliptic orbit">Elliptic orbit</a></li></ul> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Transfer_orbit" title="Transfer orbit">Transfer orbit</a> <div class="hlist" style="font-size:90%"><ul><li>(<a href="/wiki/Hohmann_transfer_orbit" title="Hohmann transfer orbit">Hohmann transfer orbit</a></li><li><a href="/wiki/Bi-elliptic_transfer" title="Bi-elliptic transfer">Bi-elliptic transfer orbit</a>)</li></ul></div></div> <ul><li><a href="/wiki/Parabolic_trajectory" title="Parabolic trajectory">Parabolic orbit</a></li> <li><a href="/wiki/Hyperbolic_trajectory" title="Hyperbolic trajectory">Hyperbolic orbit</a></li> <li><a href="/wiki/Radial_trajectory" title="Radial trajectory">Radial orbit</a></li> <li><a class="mw-selflink selflink">Decaying orbit</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Dynamical_friction" title="Dynamical friction">Dynamical friction</a></li> <li><a href="/wiki/Escape_velocity" title="Escape velocity">Escape velocity</a></li> <li><a href="/wiki/Kepler%27s_equation" title="Kepler's equation">Kepler's equation</a></li> <li><a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws of planetary motion</a></li> <li><a href="/wiki/Orbital_period" title="Orbital period">Orbital period</a></li> <li><a href="/wiki/Orbital_speed" title="Orbital speed">Orbital velocity</a></li> <li><a href="/wiki/Surface_gravity" title="Surface gravity">Surface gravity</a></li> <li><a href="/wiki/Specific_orbital_energy" title="Specific orbital energy">Specific orbital energy</a></li> <li><a href="/wiki/Vis-viva_equation" title="Vis-viva equation">Vis-viva equation</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0.55em;"> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Celestial_mechanics" title="Celestial mechanics"><span style="font-size:110%;">Celestial mechanics</span></a></div></th></tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Gravitational influences</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Barycenter" class="mw-redirect" title="Barycenter">Barycenter</a></li> <li><a href="/wiki/Hill_sphere" title="Hill sphere">Hill sphere</a></li> <li><a href="/wiki/Perturbation_(astronomy)" title="Perturbation (astronomy)">Perturbations</a></li> <li><a href="/wiki/Sphere_of_influence_(astrodynamics)" title="Sphere of influence (astrodynamics)">Sphere of influence</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/N-body_problem" title="N-body problem">N-body orbits</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Lagrange_point" title="Lagrange point">Lagrangian points</a> <div class="hlist" style="font-size:90%"><ul><li>(<a href="/wiki/Halo_orbit" title="Halo orbit">Halo orbits</a>)</li></ul></div></div> <ul><li><a href="/wiki/Lissajous_orbit" title="Lissajous orbit">Lissajous orbits</a></li> <li><a href="/wiki/Lyapunov_stability" title="Lyapunov stability">Lyapunov orbits</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0.55em;"> <div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Aerospace_engineering" title="Aerospace engineering"><span style="font-size:110%;">Engineering and efficiency</span></a></div></th></tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Preflight engineering</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Mass_ratio" title="Mass ratio">Mass ratio</a></li> <li><a href="/wiki/Payload_fraction" title="Payload fraction">Payload fraction</a></li> <li><a href="/wiki/Propellant_mass_fraction" title="Propellant mass fraction">Propellant mass fraction</a></li> <li><a href="/wiki/Tsiolkovsky_rocket_equation" title="Tsiolkovsky rocket equation">Tsiolkovsky rocket equation</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Efficiency measures</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Gravity_assist" title="Gravity assist">Gravity assist</a></li> <li><a href="/wiki/Oberth_effect" title="Oberth effect">Oberth effect</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Propulsive maneuvers</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"> <ul><li><a href="/wiki/Orbital_maneuver" title="Orbital maneuver">Orbital maneuver</a></li> <li><a href="/wiki/Orbit_insertion" title="Orbit insertion">Orbit insertion</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Astrodynamics" title="Template:Astrodynamics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Astrodynamics" title="Template talk:Astrodynamics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Astrodynamics" title="Special:EditPage/Template:Astrodynamics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Orbital decay</b> is a gradual decrease of the <a href="/wiki/Distance" title="Distance">distance</a> between two <a href="/wiki/Orbit" title="Orbit">orbiting</a> bodies at their closest approach (the <a href="/wiki/Periapsis" class="mw-redirect" title="Periapsis">periapsis</a>) over many orbital periods. These orbiting bodies can be a <a href="/wiki/Planet" title="Planet">planet</a> and its <a href="/wiki/Satellite" title="Satellite">satellite</a>, a <a href="/wiki/Star" title="Star">star</a> and any object orbiting it, or components of any <a href="/wiki/Binary_system_(astronomy)" class="mw-redirect" title="Binary system (astronomy)">binary system</a>. If left unchecked, the decay eventually results in termination of the orbit when the smaller object <a href="/wiki/Collision" title="Collision">strikes</a> the surface of the primary; or for objects where the primary has an atmosphere, the smaller object <a href="/wiki/Atmospheric_entry" title="Atmospheric entry">burns, explodes, or otherwise breaks up</a> in the larger object's <a href="/wiki/Atmosphere" title="Atmosphere">atmosphere</a>; or for objects where the primary is a star, ends with incineration by the star's radiation (such as for <a href="/wiki/Comet" title="Comet">comets</a>). <a href="/wiki/Stellar_collision" title="Stellar collision">Collisions of stellar-mass objects</a> are usually accompanied by effects such as <a href="/wiki/Gamma-ray_burst" title="Gamma-ray burst">gamma-ray bursts</a> and detectable <a href="/wiki/Gravitational_wave" title="Gravitational wave">gravitational waves</a>. </p><p>Orbital decay is caused by one or more mechanisms which absorb energy from the orbital motion, such as <a href="/wiki/Drag_(physics)" title="Drag (physics)">fluid friction</a>, <a href="/wiki/Mass_concentration_(astronomy)" title="Mass concentration (astronomy)">gravitational anomalies</a>, or <a href="/wiki/Electromagnetic_interaction" class="mw-redirect" title="Electromagnetic interaction">electromagnetic</a> effects. For bodies in <a href="/wiki/Low_Earth_orbit" title="Low Earth orbit">low Earth orbit</a>, the most significant effect is <a href="/wiki/Atmospheric_drag" class="mw-redirect" title="Atmospheric drag">atmospheric drag</a>. </p><p>Due to atmospheric drag, the lowest altitude above the <a href="/wiki/Earth" title="Earth">Earth</a> at which an object in a circular orbit can complete at least one full revolution without propulsion is approximately 150 km (93 mi) while the lowest <a href="/wiki/Perigee" class="mw-redirect" title="Perigee">perigee</a> of an elliptical revolution is approximately 90 km (56 mi). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Modeling">Modeling<span class="anchor" id="Modelling_Orbit_Decay"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbital_decay&action=edit&section=1" title="Edit section: Modeling"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Simplified_model">Simplified model<span class="anchor" id="A_Simplified_Orbit_Decay_Model"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbital_decay&action=edit&section=2" title="Edit section: Simplified model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A simplified decay model for a near-circular two-body orbit about a central body (or planet) with an atmosphere, in terms of the rate of change of the orbital altitude, is given below.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dR}{dt}}={\frac {\alpha _{o}(R)\cdot T(R)}{\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>R</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> <mi>π</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dR}{dt}}={\frac {\alpha _{o}(R)\cdot T(R)}{\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73c2ec64a58b1bed2d61d79dfbd7405fb1eec2a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.73ex; height:5.843ex;" alt="{\displaystyle {\frac {dR}{dt}}={\frac {\alpha _{o}(R)\cdot T(R)}{\pi }}}"></span></dd></dl> <p>Where <b>R</b> is the distance of the spacecraft from the planet's origin, <b>α<sub>o</sub></b> is the sum of all accelerations projected on the along-track direction of the spacecraft (or parallel to the spacecraft velocity vector), and <b>T</b> is the Keplerian period. Note that <b>α<sub>o</sub></b> is often a function of <b>R</b> due to variations in atmospheric density in the altitude, and <b>T</b> is a function of <b>R</b> by virtue of <a href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws of planetary motion</a>. </p><p>If only atmospheric drag is considered, one can approximate drag deceleration <b>α<sub>o</sub></b> as a function of orbit radius <b>R</b> using the <a href="/wiki/Drag_equation" title="Drag equation">drag equation</a> below: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{o}\,=\,{\tfrac {1}{2}}\,\rho (R)\,v^{2}\,c_{\rm {d}}\,{\frac {A}{m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>m</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{o}\,=\,{\tfrac {1}{2}}\,\rho (R)\,v^{2}\,c_{\rm {d}}\,{\frac {A}{m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da1306a622e7b6160ab1ab95c6dfa61e64e18f7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.583ex; height:5.343ex;" alt="{\displaystyle \alpha _{o}\,=\,{\tfrac {1}{2}}\,\rho (R)\,v^{2}\,c_{\rm {d}}\,{\frac {A}{m}}}"></span></dd></dl> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0761345b878fb29f4c2481c267ef2aa61c2a1ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle \rho (R)}"></span> is the <a href="/wiki/Mass_density" class="mw-redirect" title="Mass density">mass density</a> of the atmosphere which is a function of the radius R from the origin,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span> is the <a href="/wiki/Flow_velocity" title="Flow velocity">orbital velocity</a>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is the drag reference <a href="/wiki/Area" title="Area">area</a>,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> is the <a href="/wiki/Mass" title="Mass">mass</a> of the satellite, and</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{\rm {d}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{\rm {d}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0de912922b9cec97d52925bd589a54edda98990a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.153ex; height:2.009ex;" alt="{\displaystyle c_{\rm {d}}}"></span> is the <a href="/wiki/Dimensionless_number" class="mw-redirect" title="Dimensionless number">dimensionless</a> <a href="/wiki/Drag_coefficient" title="Drag coefficient">drag coefficient</a> related to the satellite geometry, and accounting for <a href="/wiki/Skin_friction" class="mw-redirect" title="Skin friction">skin friction</a> and <a href="/wiki/Form_drag" class="mw-redirect" title="Form drag">form drag</a> (~2.2 for cube satellites).</dd></dl></dd></dl> <p>The orbit decay model has been tested against ~1 year of actual GPS measurements of <a rel="nofollow" class="external text" href="https://directory.eoportal.org/web/eoportal/satellite-missions/v-w-x-y-z/velox-ci">VELOX-C1</a>, where the mean decay measured via GPS was 2.566 km across Dec 2015 to Nov 2016, and the orbit decay model predicted a decay of 2.444 km, which amounted to a 5% deviation. </p><p>An open-source <a href="/wiki/Python_(programming_language)" title="Python (programming language)">Python</a> based software, <a rel="nofollow" class="external text" href="https://github.com/sammmlow/ORBITM">ORBITM</a> (ORBIT Maintenance and Propulsion Sizing), is available freely on GitHub for Python users using the above model. </p> <div class="mw-heading mw-heading3"><h3 id="Proof_of_simplified_model">Proof of simplified model<span class="anchor" id="Proof_of_Simplified_Orbit_Decay_Model"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbital_decay&action=edit&section=3" title="Edit section: Proof of simplified model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By the <a href="/wiki/Mechanical_energy" title="Mechanical energy">conservation of mechanical energy</a>, the energy of the orbit is simply the sum of kinetic and gravitational potential energies, in an unperturbed <a href="/wiki/Two-body_problem" title="Two-body problem">two-body orbit</a>. By substituting the <a href="/wiki/Vis-viva_equation" title="Vis-viva equation">vis-viva equation</a> into the kinetic energy component, the orbital energy of a circular orbit is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=KE+GPE=-{\frac {GM_{E}m}{2R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>K</mi> <mi>E</mi> <mo>+</mo> <mi>G</mi> <mi>P</mi> <mi>E</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mi>m</mi> </mrow> <mrow> <mn>2</mn> <mi>R</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=KE+GPE=-{\frac {GM_{E}m}{2R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9fdbcda0a6630e000d9a22b191cf0fe25781af8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.262ex; height:5.509ex;" alt="{\displaystyle U=KE+GPE=-{\frac {GM_{E}m}{2R}}}"></span></dd></dl> <p>Where <b>G</b> is the gravitational constant, <b>M<sub>E</sub></b> is the mass of the central body and <b>m</b> is the mass of the orbiting satellite. We take the derivative of the orbital energy with respect to the radius. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dU}{dR}}={\frac {GM_{E}m}{2R^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>U</mi> </mrow> <mrow> <mi>d</mi> <mi>R</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>G</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mi>m</mi> </mrow> <mrow> <mn>2</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dU}{dR}}={\frac {GM_{E}m}{2R^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69823cead2a3e8d718e0b94e726c94fe4bf49a69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.378ex; height:5.676ex;" alt="{\displaystyle {\frac {dU}{dR}}={\frac {GM_{E}m}{2R^{2}}}}"></span></dd></dl> <p>The total decelerating force, which is usually atmospheric drag for low Earth orbits, exerted on a satellite of constant mass <b>m</b> is given by some force <b>F</b>. The rate of loss of orbital energy is simply the rate at the external force does negative work on the satellite as the satellite traverses an infinitesimal circular arc-length <b>ds</b>, spanned by some infinitesimal angle <b>dθ</b> and angular rate <b>ω</b>. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dU}{dt}}={\frac {F\cdot ds}{dt}}={\frac {F\cdot R\cdot d\theta }{dt}}=F\cdot R\cdot \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>U</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo>⋅</mo> <mi>d</mi> <mi>s</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo>⋅</mo> <mi>R</mi> <mo>⋅</mo> <mi>d</mi> <mi>θ</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>F</mi> <mo>⋅</mo> <mi>R</mi> <mo>⋅</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dU}{dt}}={\frac {F\cdot ds}{dt}}={\frac {F\cdot R\cdot d\theta }{dt}}=F\cdot R\cdot \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ea7d025a274ee5e47d22aeb79297ffa0acafb80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:38.007ex; height:5.509ex;" alt="{\displaystyle {\frac {dU}{dt}}={\frac {F\cdot ds}{dt}}={\frac {F\cdot R\cdot d\theta }{dt}}=F\cdot R\cdot \omega }"></span></dd></dl> <p>The angular rate <b>ω</b> is also known as the <a href="/wiki/Mean_motion" title="Mean motion">Mean motion</a>, where for a two-body circular orbit of radius <b>R</b>, it is expressed as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={\frac {2\pi }{T}}={\sqrt {\frac {GM_{E}}{R^{3}}}}={\frac {F\cdot R\cdot d\theta }{dt}}=F\cdot R\cdot \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>G</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>F</mi> <mo>⋅</mo> <mi>R</mi> <mo>⋅</mo> <mi>d</mi> <mi>θ</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>F</mi> <mo>⋅</mo> <mi>R</mi> <mo>⋅</mo> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\frac {2\pi }{T}}={\sqrt {\frac {GM_{E}}{R^{3}}}}={\frac {F\cdot R\cdot d\theta }{dt}}=F\cdot R\cdot \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0330c784592ec6cfbf21782aaa01e0ac34d0117b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:44.213ex; height:7.676ex;" alt="{\displaystyle \omega ={\frac {2\pi }{T}}={\sqrt {\frac {GM_{E}}{R^{3}}}}={\frac {F\cdot R\cdot d\theta }{dt}}=F\cdot R\cdot \omega }"></span></dd></dl> <p>and... </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=m\cdot \alpha _{o}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>m</mi> <mo>⋅</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=m\cdot \alpha _{o}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e396d2c64d3568b595e7276895523c09bc9fe4b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.076ex; height:2.509ex;" alt="{\displaystyle F=m\cdot \alpha _{o}}"></span></dd></dl> <p>Substituting <b>ω</b> into the rate of change of orbital energy above, and expressing the external drag or decay force in terms of the deceleration <b>α<sub>o</sub></b>, the orbital energy rate of change with respect to time can be expressed as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dU}{dt}}=m\cdot \alpha _{o}\cdot {\sqrt {\frac {GM_{E}}{R}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>U</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>m</mi> <mo>⋅</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>G</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mrow> <mi>R</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dU}{dt}}=m\cdot \alpha _{o}\cdot {\sqrt {\frac {GM_{E}}{R}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf4a8b9207a6a765a7da1b7d7579ec7af93e7636" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.577ex; height:6.176ex;" alt="{\displaystyle {\frac {dU}{dt}}=m\cdot \alpha _{o}\cdot {\sqrt {\frac {GM_{E}}{R}}}}"></span></dd></dl> <p>Having an equation for the rate of change of orbital energy with respect to both radial distance and time allows us to find the rate of change of the radial distance with respect to time as per below. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {dR}{dt}}=\left(\left({\frac {dU}{dR}}\right)^{-1}\cdot {\frac {dU}{dt}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>R</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>U</mi> </mrow> <mrow> <mi>d</mi> <mi>R</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>U</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {dR}{dt}}=\left(\left({\frac {dU}{dR}}\right)^{-1}\cdot {\frac {dU}{dt}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdfbe9d35203a222cf8a7d00b3b7c180af22c8ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:25.698ex; height:7.509ex;" alt="{\displaystyle {\frac {dR}{dt}}=\left(\left({\frac {dU}{dR}}\right)^{-1}\cdot {\frac {dU}{dt}}\right)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =2\alpha _{o}\cdot {\sqrt {\frac {R^{3}}{GM_{E}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>2</mn> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow> <mi>G</mi> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =2\alpha _{o}\cdot {\sqrt {\frac {R^{3}}{GM_{E}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cd4e9c446d077a230ba2a1aa56f1ff76f9546a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:16.541ex; height:7.676ex;" alt="{\displaystyle =2\alpha _{o}\cdot {\sqrt {\frac {R^{3}}{GM_{E}}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {\alpha _{o}\cdot T}{\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> </mrow> </msub> <mo>⋅</mo> <mi>T</mi> </mrow> <mi>π</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {\alpha _{o}\cdot T}{\pi }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28f7caa50ab351990364c4455481692f5b248686" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.122ex; height:5.176ex;" alt="{\displaystyle ={\frac {\alpha _{o}\cdot T}{\pi }}}"></span></dd></dl> <p>The assumptions used in this derivation above are that the orbit stays very nearly circular throughout the decay process, so that the equations for orbital energy are more or less that of a circular orbit's case. This is often true for orbits that begin as circular, as drag forces are considered "re-circularizing", since drag magnitudes at the <a href="/wiki/Apsis" title="Apsis">periapsis</a> (lower altitude) is expectedly greater than that of the <a href="/wiki/Apsis" title="Apsis">apoapsis</a>, which has the effect of reducing the mean eccentricity. </p> <div class="mw-heading mw-heading2"><h2 id="Sources_of_decay">Sources of decay<span class="anchor" id="Sources_of_Orbital_Decay"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbital_decay&action=edit&section=4" title="Edit section: Sources of decay"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Atmospheric_drag">Atmospheric drag</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbital_decay&action=edit&section=5" title="Edit section: Atmospheric drag"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Atmospheric_drag" class="mw-redirect" title="Atmospheric drag">Atmospheric drag</a></div> <p>Atmospheric drag at orbital altitude is caused by frequent collisions of gas <a href="/wiki/Molecule" title="Molecule">molecules</a> with the satellite. It is the major cause of orbital decay for satellites in <a href="/wiki/Low_Earth_orbit" title="Low Earth orbit">low Earth orbit</a>. It results in the reduction in the <a href="/wiki/Altitude" title="Altitude">altitude</a> of a satellite's orbit. For the case of Earth, atmospheric drag resulting in satellite re-entry can be described by the following sequence: </p> <dl><dd>lower altitude → denser atmosphere → increased drag → increased heat → usually burns on re-entry</dd></dl> <p>Orbital decay thus involves a <a href="/wiki/Positive_feedback" title="Positive feedback">positive feedback</a> effect, where the more the orbit decays, the lower its altitude drops, and the lower the altitude, the faster the decay. Decay is also particularly sensitive to external factors of the space environment such as solar activity, which are not very predictable. During <a href="/wiki/Solar_maximum" title="Solar maximum">solar maxima</a> the Earth's atmosphere causes significant drag up to altitudes much higher than during <a href="/wiki/Solar_minima" class="mw-redirect" title="Solar minima">solar minima</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Atmospheric drag exerts a significant effect at the altitudes of <a href="/wiki/Space_station" title="Space station">space stations</a>, <a href="/wiki/Space_Shuttle" title="Space Shuttle">Space Shuttles</a> and other crewed Earth-orbit spacecraft, and satellites with relatively high "low Earth orbits" such as the <a href="/wiki/Hubble_Space_Telescope" title="Hubble Space Telescope">Hubble Space Telescope</a>. Space stations typically require a regular altitude boost to counteract orbital decay (see also <a href="/wiki/Orbital_station-keeping" title="Orbital station-keeping">orbital station-keeping</a>). Uncontrolled orbital decay brought down the <a href="/wiki/Skylab" title="Skylab">Skylab</a> space station,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and (relatively) controlled orbital decay was used to de-orbit the <a href="/wiki/Mir" title="Mir">Mir</a> space station.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Reboost" title="Reboost">Reboosts</a> for the Hubble Space Telescope are less frequent due to its much higher altitude. However, orbital decay is also a limiting factor to the length of time the Hubble can go without a maintenance rendezvous, the most recent having been performed successfully by <a href="/wiki/STS-125" title="STS-125">STS-125</a>, with Space Shuttle <i>Atlantis</i> in 2009. Newer <a href="/wiki/Space_telescope" title="Space telescope">space telescopes</a> are in much higher orbits, or in some cases in solar orbit, so orbital boosting may not be needed.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Tidal_effects">Tidal effects</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbital_decay&action=edit&section=6" title="Edit section: Tidal effects"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Tidal_acceleration" title="Tidal acceleration">Tidal acceleration</a></div> <p>An orbit can also decay by negative <a href="/wiki/Tidal_acceleration" title="Tidal acceleration">tidal acceleration</a> when the orbiting body is large enough to raise a significant <a href="/wiki/Tidal_bulge" class="mw-redirect" title="Tidal bulge">tidal bulge</a> on the body it is orbiting and is either in a <a href="/wiki/Retrograde_orbit" class="mw-redirect" title="Retrograde orbit">retrograde orbit</a> or is below the <a href="/wiki/Synchronous_orbit" title="Synchronous orbit">synchronous orbit</a>. This saps angular momentum from the orbiting body and transfers it to the primary's rotation, lowering the orbit's altitude. </p><p>Examples of satellites undergoing tidal orbital decay are Mars' moon <a href="/wiki/Phobos_(moon)" title="Phobos (moon)">Phobos</a>, Neptune's moon <a href="/wiki/Triton_(moon)" title="Triton (moon)">Triton</a>, and the extrasolar planet <a href="/wiki/TrES-3b" title="TrES-3b">TrES-3b</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Light_and_thermal_radiation">Light and thermal radiation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbital_decay&action=edit&section=7" title="Edit section: Light and thermal radiation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Poynting%E2%80%93Robertson_effect" title="Poynting–Robertson effect">Poynting–Robertson effect</a> and <a href="/wiki/Yarkovsky_effect" title="Yarkovsky effect">Yarkovsky effect</a></div> <p>Small objects in the <a href="/wiki/Solar_System" title="Solar System">Solar System</a> also experience an orbital decay due to the forces applied by asymmetric radiation pressure. Ideally, energy absorbed would equal <a href="/wiki/Blackbody" class="mw-redirect" title="Blackbody">blackbody</a> energy emitted at any given point, resulting in no net force. However, the <a href="/wiki/Yarkovsky_effect" title="Yarkovsky effect">Yarkovsky effect</a> is the phenomenon that, because absorption and radiation of heat are not instantaneous, objects which are not terminally locked absorb sunlight energy on surfaces exposed to the Sun, but those surfaces do not re-emit much of that energy until after the object has rotated, so that the emission is parallel to the object's orbit. This results in a very small acceleration parallel to the orbital path, yet one which can be significant for small objects over millions of years. The Poynting-Robertson effect is a force opposing the object's velocity caused by asymmetric incidence of light, i.e., <a href="/wiki/Aberration_of_light" class="mw-redirect" title="Aberration of light">aberration of light</a>. For an object with prograde rotation, these two effects will apply opposing, but generally unequal, forces. </p> <div class="mw-heading mw-heading3"><h3 id="Gravitational_radiation">Gravitational radiation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbital_decay&action=edit&section=8" title="Edit section: Gravitational radiation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Two-body problem in general relativity</a></div> <p><a href="/wiki/Gravitational_radiation" class="mw-redirect" title="Gravitational radiation">Gravitational radiation</a> is another mechanism of orbital decay. It is negligible for orbits of planets and planetary satellites (when considering their orbital motion on time scales of centuries, decades, and less), but is noticeable for systems of <a href="/wiki/Compact_star" class="mw-redirect" title="Compact star">compact objects</a>, as seen in observations of neutron star orbits. All orbiting bodies radiate gravitational energy, hence no orbit is infinitely stable. </p> <div class="mw-heading mw-heading3"><h3 id="Electromagnetic_drag">Electromagnetic drag</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbital_decay&action=edit&section=9" title="Edit section: Electromagnetic drag"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Satellites using an <a href="/wiki/Electrodynamic_tether" title="Electrodynamic tether">electrodynamic tether</a>, moving through the Earth's magnetic field, create drag force that could eventually deorbit the satellite. </p> <div class="mw-heading mw-heading2"><h2 id="Stellar_collision">Stellar collision</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbital_decay&action=edit&section=10" title="Edit section: Stellar collision"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Stellar_collision" title="Stellar collision">Stellar collision</a></div> <p>A stellar collision is the coming together of two <a href="/wiki/Binary_stars" class="mw-redirect" title="Binary stars">binary stars</a> when they lose energy and approach each other. Several things can cause the loss of energy including <a href="/wiki/Tidal_force" title="Tidal force">tidal forces</a>, <a href="/wiki/Roche_lobe" title="Roche lobe">mass transfer</a>, and <a href="/wiki/Gravitational_radiation" class="mw-redirect" title="Gravitational radiation">gravitational radiation</a>. The stars describe the path of a <a href="/wiki/Spiral" title="Spiral">spiral</a> as they approach each other. This sometimes results in a merger of the two stars or the creation of a <a href="/wiki/Black_hole" title="Black hole">black hole</a>. In the latter case, the last several revolutions of the stars around each other take only a few seconds.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Mass_concentration">Mass concentration</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbital_decay&action=edit&section=11" title="Edit section: Mass concentration"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Mass_concentration_(astronomy)" title="Mass concentration (astronomy)">Mass concentration (astronomy)</a></div> <p>While not a direct cause of orbital decay, uneven mass distributions (known as <i>mascons</i>) of the body being orbited can perturb orbits over time, and extreme distributions can cause orbits to be highly unstable. The resulting unstable orbit can mutate into an orbit where one of the direct causes of orbital decay can take place. </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Orbital_decay&action=edit&section=12" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-cmse-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-cmse_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://en.cmse.gov.cn/col/col1763/index.html">"Tiangong-1 Orbital Status"</a>. <i>Official Website of China Manned Space</i>. 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