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title	tags	summary
Solar radiation pressure	radiation-pressure, srp, sun-synchronous, photon, area-to-mass	The acceleration a sunlit surface receives from photon momentum, why it is a century scale effect rather than a perturbation, and how it shapes a constellation.

Solar radiation pressure

A satellite is a body held together, not a body falling. The force that keeps a small object from accelerating away, or from being pulled back in, is the momentum carried by the light it reflects and absorbs. It is usually ignored in a low Earth orbit model and dominant in a high one.

Definition

Solar radiation pressure is the momentum flux carried by sunlight at a distance r from the Sun. It is not a force on a satellite but on the light it intercepts; the acceleration the satellite feels is the flux multiplied by the satellite's area-to-mass ratio and by a reflectivity factor.

The pressure at one astronomical unit is about 4.56e-6 N/m^2. Divided by the area-to-mass ratio of a spacecraft, it gives an acceleration of the same order as J_2 on a large satellite at low altitude.

Mathematics

The flux

The flux falls as the inverse square of the distance,


P(r) = P_{1\,\text{au}} \left( \frac{1\,\text{au}}{r} \right)^2

with P_1au = 4.56e-6 N/m^2 for total solar irradiance at 1 au.

The acceleration

For a satellite with projected area A, mass m, and a coefficient C_R that is 1 for absorption, 2 for perfect reflection and between for a typical surface, the acceleration is


a_{\text{srp}} = C_R \frac{P_{1\,\text{au}} A}{m} \left( \frac{1\,\text{au}}{r} \right)^2

The A/m ratio is the whole story. It ranges from about 0.002 m^2/kg for a large communications satellite to about 0.05 for a small cube of a CubeSat, a factor of 25 that is entirely attributable to structure.

Three components

The acceleration along the Sun-satellite line has three distinguishable parts, because the force acts where the light strikes:

A constant radial component, which slightly lowers the effective gravitational parameter and is absorbed into it.
A transverse component along the orbit normal or the velocity direction, which produces a secular change in the mean motion, that is in the semi-major axis.
A short-period variation with the orbital period, because the geometry changes over an orbit when the satellite has a nonzero area ratio.

Only the middle one changes the orbit over time; the first is a constant and the third averages out.

Eclipse

In eclipse the satellite receives no radiation pressure. The effect is a periodic reduction of the transverse acceleration, which produces a cyclical variation in semi-major axis with the eclipse season rather than the orbit, and it is why a radiation pressure constant is calibrated from the mean over a long arc.

Unit conventions

Flux in watts per square metre; P_1au = 1361 W/m^2 for total solar irradiance, which corresponds to the pressure above.
Area-to-mass ratio in square metres per kilogram.
Distances in kilometres; the 1 au ratio is dimensionless so the unit does not matter there, but the acceleration that results must be in kilometres per second squared to be consistent with the orbit.

Applicability

The effect is a century-scale concern in the semi-major axis. For a high Earth orbit with a large area-to-mass ratio, the semi-major axis drifts noticeably over a few years, which is why the planning of a constellation in the geosynchronous or high Earth orbit includes radiation pressure in its station-keeping budget. For a low Earth orbit it is a perturbation beside J_2 and atmospheric drag.

The mean motion acceleration from the transverse component is what a model should capture. The short-period term is smaller than it looks because it averages, but for a sun-synchronous orbit it is not a perturbation at all, because it is the mechanism: the nodal rate is chosen by tuning it against J_2, and the two balance.

The eclipse term matters only where the radiation pressure is large, and it matters because it is biased rather than symmetric.

Standards

Solar irradiance: IAU 2015 Resolution B3
The pressure and flux conversions follow that resolution
The effect as modelled in practice is described in Vallado, Fundamentals of Astrodynamics and Applications

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