Orbital decay

Orbital decay is a gradual decrease of the distance between two orbiting 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. If left unchecked, the decay eventually results in termination of the orbit when the smaller object strikes the surface of the primary; or for objects where the primary has an atmosphere, the smaller object 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 comets). Collisions of stellar-mass objects are usually accompanied by effects such as gamma-ray bursts and detectable gravitational waves. Orbital decay is caused by one or more mechanisms which absorb energy from the orbital motion, such as fluid friction, gravitational anomalies, or 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 km (93 mi) while the lowest perigee of an elliptical revolution is approximately 90 km (56 mi).

Modeling

Simplified model 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.

d R d t = α o ( R ) ⋅ T ( R ) π {\displaystyle {\frac {dR}{dt}}={\frac {\alpha _{o}(R)\cdot T(R)}{\pi }}}

Where R is the distance of the spacecraft from the planet's origin, αo 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 αo 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 αo as a function of orbit radius R using the drag equation below:

α o = 1 2 ρ ( R ) v 2 c d A m {\displaystyle \alpha _{o}\,=\,{\tfrac {1}{2}}\,\rho (R)\,v^{2}\,c_{\rm {d}}\,{\frac {A}{m}}}

ρ ( R ) {\displaystyle \rho (R)} is the mass density of the atmosphere which is a function of the radius R from the origin,

v {\displaystyle v} is the orbital velocity,

A {\displaystyle A} is the drag reference area,

m {\displaystyle m} is the mass of the satellite, and

c d {\displaystyle c_{\rm {d}}} is the dimensionless drag coefficient related to the satellite geometry, and accounting for skin friction and form drag (~2.2 for cube satellites).

Proof of simplified model By the conservation of mechanical energy, the energy of the orbit is simply the sum of kinetic and gravitational potential energies, in an unperturbed two-body orbit. By substituting the vis-viva equation into the kinetic energy component, the orbital energy of a circular orbit is given by:

U = K E + G P E = − G M E m 2 R {\displaystyle U=KE+GPE=-{\frac {GM_{E}m}{2R}}}

Where G is the gravitational constant, ME 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.

d U d R = G M E m 2 R 2 {\displaystyle {\frac {dU}{dR}}={\frac {GM_{E}m}{2R^{2}}}}

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.

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