Editing Orbital decay (section)

=== 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.