113 lines
4.6 KiB
Markdown
113 lines
4.6 KiB
Markdown
---
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title: Solar radiation pressure
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tags: radiation-pressure, srp, sun-synchronous, photon, area-to-mass
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summary: 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.
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---
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# Solar radiation pressure
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A satellite is a body held together, not a body falling. The force that keeps
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a small object from accelerating away, or from being pulled back in, is the
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momentum carried by the light it reflects and absorbs. It is usually ignored
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in a low Earth orbit model and dominant in a high one.
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## Definition
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**Solar radiation pressure** is the momentum flux carried by sunlight at a
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distance $r$ from the Sun. It is not a force on a satellite but on the light
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it intercepts; the acceleration the satellite feels is the flux multiplied
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by the satellite's area-to-mass ratio and by a reflectivity factor.
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The pressure at one astronomical unit is about $4.56 \times 10^{-6}\,\text{N}/\text{m}^2$. Divided by
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the area-to-mass ratio of a spacecraft, it gives an acceleration of the same
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order as $J_2$ on a large satellite at low altitude.
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## Mathematics
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### The flux
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The flux falls as the inverse square of the distance,
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$$
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P(r) = P_{1\,\text{au}} \left( \frac{1\,\text{au}}{r} \right)^2
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$$
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with $P_{1\,\text{au}} = 4.56 \times 10^{-6}\,\text{N}/\text{m}^2$ for total solar irradiance at 1 au.
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### The acceleration
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For a satellite with projected area $A$, mass $m$, and a coefficient
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$C_R$ that is 1 for absorption, 2 for perfect reflection and between for a
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typical surface, the acceleration is
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$$
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a_{\text{srp}} = C_R \frac{P_{1\,\text{au}} A}{m} \left( \frac{1\,\text{au}}{r} \right)^2
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$$
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The $A/m$ ratio is the whole story. It ranges from about $0.002\,\text{m}^2/\text{kg}$ for a
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large communications satellite to about `0.05` for a small cube of a
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CubeSat, a factor of 25 that is entirely attributable to structure.
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### Three components
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The acceleration along the Sun-satellite line has three distinguishable
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parts, because the force acts where the light strikes:
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- A **constant radial component**, which slightly lowers the effective
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gravitational parameter and is absorbed into it.
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- A **transverse component** along the orbit normal or the velocity
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direction, which produces a secular change in the mean motion, that is in
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the semi-major axis.
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- A **short-period variation** with the orbital period, because the geometry
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changes over an orbit when the satellite has a nonzero area ratio.
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Only the middle one changes the orbit over time; the first is a constant and
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the third averages out.
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### Eclipse
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In eclipse the satellite receives no radiation pressure. The effect is a
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periodic reduction of the transverse acceleration, which produces a
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cyclical variation in semi-major axis with the eclipse season rather than the
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orbit, and it is why a radiation pressure constant is calibrated from the
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mean over a long arc.
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## Unit conventions
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- Flux in watts per square metre; $P_{1\,\text{au}} = 1361\,\text{W}/\text{m}^2$ for total solar
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irradiance, which corresponds to the pressure above.
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- Area-to-mass ratio in square metres per kilogram.
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- Distances in kilometres; the `1 au` ratio is dimensionless so the unit
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does not matter there, but the acceleration that results must be in
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kilometres per second squared to be consistent with the orbit.
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## Applicability
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The effect is a century-scale concern in the semi-major axis. For a high
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Earth orbit with a large area-to-mass ratio, the semi-major axis drifts
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noticeably over a few years, which is why the planning of a constellation in
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the geosynchronous or high Earth orbit includes radiation pressure in its
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station-keeping budget. For a low Earth orbit it is a perturbation beside
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$J_2$ and atmospheric drag.
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The mean motion acceleration from the transverse component is what a model
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should capture. The short-period term is smaller than it looks because it
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averages, but for a sun-synchronous orbit it is not a perturbation at all,
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because it is the mechanism: the nodal rate is chosen by tuning it against
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$J_2$, and the two balance.
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The eclipse term matters only where the radiation pressure is large, and it
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matters because it is biased rather than symmetric.
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## Standards
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- Solar irradiance: IAU 2015 Resolution B3
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- The pressure and flux conversions follow that resolution
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- The effect as modelled in practice is described in Vallado,
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*Fundamentals of Astrodynamics and Applications*
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## See also
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- [[perturbations]] for the size of the effect beside $J_2$ and drag
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- [[atmospheric-drag]] for the other century-scale term
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- [[constellations]] for why a sun-synchronous plane is chosen by balance |