Editing Orbital decay (section)

=== 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.