Editing Orbital decay

{{Short description|Process that leads to gradual decrease of the distance between two orbiting bodies}}
{{Redirect-distinguish|Decaying Orbit|Decaying Orbit (film)}}{{Redirect-distinguish|Inspiral|Inspiral (horse)}}
[[File:Altitude of Tiangong-1.svg|thumb|upright=1.3|Altitude of [[Tiangong-1]] during its final year of uncontrolled reentry.<ref name="cmse">{{cite web|title=Tiangong-1 Orbital Status|url=http://en.cmse.gov.cn/col/col1763/index.html|website=Official Website of China Manned Space|publisher=China Manned Space Engineering Office|access-date=1 April 2018|date=1 April 2018}}</ref>]]
{{Astrodynamics}}

'''Orbital decay''' is a gradual decrease of the [[distance]] between two [[orbit]]ing bodies at their closest approach (the [[periapsis]]) over many orbital periods. These orbiting bodies can be a [[planet]] and its [[satellite]], a [[star]] and any object orbiting it, or components of any [[binary system (astronomy)|binary system]]. If left unchecked, the decay eventually results in termination of the orbit when the smaller object [[collision|strikes]] the surface of the primary; or for objects where the primary has an atmosphere, the smaller object [[Atmospheric entry|burns, explodes, or otherwise breaks up]] in the larger object's [[atmosphere]]; or for objects where the primary is a star, ends with incineration by the star's radiation (such as for [[comet]]s).  [[stellar collision|Collisions of stellar-mass objects]] are usually accompanied by effects such as [[gamma-ray burst]]s and detectable [[Gravitational wave|gravitational waves]].

Orbital decay is caused by one or more mechanisms which absorb energy from the orbital motion, such as [[drag (physics)|fluid friction]], [[mass concentration (astronomy)|gravitational anomalies]], or [[electromagnetic interaction|electromagnetic]] effects. For bodies in [[low Earth orbit]], the most significant effect is [[atmospheric drag]].

Due to atmospheric drag, the lowest altitude above the [[Earth]] at which an object in a circular orbit can complete at least one full revolution without propulsion is approximately 150&nbsp;km (93&nbsp;mi) while the lowest [[perigee]] of an elliptical revolution is approximately 90&nbsp;km (56&nbsp;mi).

== Modeling<span class="anchor" id="Modelling Orbit Decay"></span> ==

=== Simplified model<span class="anchor" id="A Simplified Orbit Decay Model"></span> ===
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.<ref>{{cite journal |last1=Low |first1=Samuel Y. W. |title=Assessment of Orbit Maintenance Strategies for Small Satellites |journal=AIAA/USU Conference on Small Satellites |date=August 2018 |volume=32 |doi=10.26077/bffw-p652}}</ref>

:<math> \frac{dR}{dt}=\frac{\alpha_o(R) \cdot T(R)}{\pi}  </math>

Where '''R''' is the distance of the spacecraft from the planet's origin, '''α<sub>o</sub>''' is the sum of all accelerations projected on the along-track direction of the spacecraft (or parallel to the spacecraft velocity vector), and '''T''' is the Keplerian period. Note that '''α<sub>o</sub>''' is often a function of '''R''' due to variations in atmospheric density in the altitude, and '''T''' is a function of '''R''' by virtue of [[Kepler's laws of planetary motion]].

If only atmospheric drag is considered, one can approximate drag deceleration '''α<sub>o</sub>''' as a function of orbit radius '''R''' using the [[drag equation]] below:

:<math>\alpha_o\, =\, \tfrac12\, \rho(R)\, v^2\, c_{\rm d}\, \frac{A}{m}</math>

::<math>\rho(R)</math> is the [[mass density]] of the atmosphere which is a function of the radius R from the origin,
::<math>v</math> is the [[flow velocity|orbital velocity]],
::<math>A</math> is the drag reference [[area]],
::<math>m</math> is the [[mass]] of the satellite, and
::<math>c_{\rm d}</math> is the [[dimensionless number|dimensionless]] [[drag coefficient]] related to the satellite geometry, and accounting for [[skin friction]] and [[form drag]] (~2.2 for cube satellites).

The orbit decay model has been tested against ~1 year of actual GPS measurements of [https://directory.eoportal.org/web/eoportal/satellite-missions/v-w-x-y-z/velox-ci VELOX-C1], where the mean decay measured via GPS was 2.566&nbsp;km across Dec 2015 to Nov 2016, and the orbit decay model predicted a decay of 2.444&nbsp;km, which amounted to a 5% deviation.

An open-source [[Python (programming language)|Python]] based software, [https://github.com/sammmlow/ORBITM ORBITM] (ORBIT Maintenance and Propulsion Sizing), is available freely on GitHub for Python users using the above model.

=== Proof of simplified model<span class="anchor" id="Proof of Simplified Orbit Decay Model"></span> ===

By the [[Mechanical energy|conservation of mechanical energy]], the energy of the orbit is simply the sum of kinetic and gravitational potential energies, in an unperturbed [[Two-body problem|two-body orbit]]. By substituting the [[vis-viva equation]] into the kinetic energy component, the orbital energy of a circular orbit is given by:

:<math> U = KE + GPE = -\frac{G M_E m}{2R} </math>

Where '''G''' is the gravitational constant, '''M<sub>E</sub>''' is the mass of the central body and '''m''' is the mass of the orbiting satellite.  We take the derivative of the orbital energy with respect to the radius.

:<math> \frac{dU}{dR} = \frac{G M_E m}{2R^2} </math>

The total decelerating force, which is usually atmospheric drag for low Earth orbits, exerted on a satellite of constant mass '''m''' is given by some force '''F'''. 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 '''ds''', spanned by some infinitesimal angle '''dθ''' and angular rate '''ω'''.

:<math> \frac{dU}{dt}=\frac{F \cdot ds}{dt}=\frac{F \cdot R \cdot d\theta}{dt}=F \cdot R \cdot \omega </math>

The angular rate '''ω''' is also known as the [[Mean motion]], where for a two-body circular orbit of radius '''R''', it is expressed as:

:<math> \omega = \frac{2\pi}{T} = \sqrt{\frac{G M_E}{R^3}} </math>

and...

:<math> F = m \cdot \alpha_o </math>

Substituting '''ω''' into the rate of change of orbital energy above, and expressing the external drag or decay force in terms of the deceleration '''α<sub>o</sub>''', the orbital energy rate of change with respect to time can be expressed as:

:<math> \frac{dU}{dt}= m \cdot \alpha_o \cdot \sqrt{\frac{G M_E}{R}}</math>

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.

:<math> \frac{dR}{dt} = \left( \left( \frac{dU}{dR} \right)^{-1} \cdot \frac{dU}{dt} \right) </math>
:<math> = 2\alpha_o \cdot \sqrt{\frac{R^3}{G M_E}} </math>
:<math> = \frac{\alpha_o \cdot T}{\pi} </math>

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 [[Apsis|periapsis]] (lower altitude) is expectedly greater than that of the [[Apsis|apoapsis]], which has the effect of reducing the mean eccentricity.

==Sources of decay<span class="anchor" id="Sources of Orbital Decay"></span>==

===Atmospheric drag===
{{further|Atmospheric drag}}
Atmospheric drag at orbital altitude is caused by frequent collisions of gas [[molecule]]s with the satellite.
It is the major cause of orbital decay for satellites in [[low Earth orbit]]. It results in the reduction in the [[altitude]] of a satellite's orbit. For the case of Earth, atmospheric drag resulting in satellite re-entry can be described by the following sequence:

: lower altitude → denser atmosphere → increased drag → increased heat → usually burns on re-entry

Orbital decay thus involves a [[positive feedback]] 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 [[Solar maximum|solar maxima]] the Earth's atmosphere causes significant drag up to altitudes much higher than during [[solar minima]].<ref>{{cite arXiv|last1=Nwankwo|first1=Victor U. J.|last2=Chakrabarti|first2=Sandip K.|title=Effects of Plasma Drag on Low Earth Orbiting Satellites due to Heating of Earth's Atmosphere by Coronal Mass Ejections|date=1 May 2013|class=physics.space-phn|eprint=1305.0233 <!-- unsupported parameter |url=https://arxiv.org/abs/1305.0233 -->}} </ref>

Atmospheric drag exerts a significant effect at the altitudes of [[space station]]s, [[Space Shuttle]]s and other crewed Earth-orbit spacecraft, and satellites with relatively high "low Earth orbits" such as the [[Hubble Space Telescope]]. Space stations typically require a regular altitude boost to counteract orbital decay (see also [[orbital station-keeping]]). Uncontrolled orbital decay brought down the [[Skylab]] space station,<ref>{{cite web|title=The Biggest Spacecraft Ever to Fall Uncontrolled From Space|author=Wall, Mike|date=May 5, 2021|url=https://www.space.com/13049-6-biggest-spacecraft-falls-space.html|publisher=space.com|access-date=April 29, 2023}}</ref> and (relatively) controlled orbital decay was used to de-orbit the [[Mir]] space station.<ref>{{cite web|title=20 Years Ago: Space Station Mir Reenters Earth's Atmosphere|date=March 23, 2021|url=https://www.nasa.gov/feature/20-years-ago-space-station-mir-reenters-earth-s-atmosphere|publisher=NASA|access-date=April 29, 2023}}</ref>

[[Reboost]]s 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 [[STS-125]], with Space Shuttle ''Atlantis'' in 2009.  Newer [[space telescope]]s are in much higher orbits, or in some cases in solar orbit, so orbital boosting may not be needed.<ref>[http://hubble.nasa.gov/missions/sm4.html The Hubble Program – Servicing Missions – SM4<!-- Bot generated title -->]</ref>

=== Tidal effects ===

{{further|Tidal acceleration}}
An orbit can also decay by negative [[tidal acceleration]] when the orbiting body is large enough to raise a significant [[tidal bulge]] on the body it is orbiting and is either in a [[retrograde orbit]] or is below the [[synchronous orbit]].  This saps angular momentum from the orbiting body and transfers it to the primary's rotation, lowering the orbit's altitude.

Examples of satellites undergoing tidal orbital decay are Mars' moon [[Phobos (moon)|Phobos]], Neptune's moon [[Triton (moon)|Triton]], and the extrasolar planet [[TrES-3b]].

===Light and thermal radiation===
{{main|Poynting–Robertson effect|Yarkovsky effect}}
Small objects in the [[Solar System]] also experience an orbital decay due to the forces applied by asymmetric radiation pressure. Ideally, energy absorbed would equal [[blackbody]] energy emitted at any given point, resulting in no net force. However, the [[Yarkovsky effect]] is the phenomenon that, because absorption and radiation of heat are not instantaneous, objects which are not [[tidal locking|tidally 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., [[aberration of light]]. For an object with prograde rotation, these two effects will apply opposing, but generally unequal, forces.

=== Gravitational radiation ===
{{main|Two-body problem in general relativity}}
[[Gravitational radiation]] 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 [[compact star|compact objects]], as seen in observations of neutron star orbits. All orbiting bodies radiate gravitational energy, hence no orbit is infinitely stable.

=== Electromagnetic drag ===

Satellites using an [[electrodynamic tether]], moving through the Earth's magnetic field, create drag force that could eventually deorbit the satellite.

==Stellar collision==
{{further|Stellar collision}}
A stellar collision is the coming together of two [[binary stars]] when they lose energy and approach each other. Several things can cause the loss of energy including [[tidal force]]s, [[Roche lobe|mass transfer]], and [[gravitational radiation]]. The stars describe the path of a [[spiral]] as they approach each other. This sometimes results in a merger of the two stars or the creation of a [[black hole]]. In the latter case, the last several revolutions of the stars around each other take only a few seconds.<ref>{{cite web|title=INSPIRAL GRAVITATIONAL WAVES|url=http://www.ligo.org/science/GW-Inspiral.php|website=LIGO|access-date=1 May 2015}}</ref>

==Mass concentration==
{{further|Mass concentration (astronomy)}}
While not a direct cause of orbital decay, uneven mass distributions (known as ''mascons'') 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.

== References ==
{{reflist}}

{{portal|Physics}}

{{DEFAULTSORT:Orbital decay}}
[[Category:Effects of gravity]]
[[Category:Orbits]]
[[Category:Black holes]]