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Unit 7 · Lesson 7-3
Solar Radiation Pressure and Third-Body Perturbations
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Solar Radiation Pressure and Third-Body Perturbations

Radiation pressure, eclipses, lunisolar gravity and which perturbation dominates where

[Echo 1](https://en.wikipedia.org/wiki/Project_Echo), 1960
Echo 1 was the simplest satellite ever flown: a Mylar balloon, 30 metres across, inflated in orbit. It carried no instruments, no transmitter and no power system. It was a mirror. Radio signals were bounced off it from one continent to another. It was also, at 76 kilograms for 700 square metres of cross-section, the lightest thing per unit area ever put in space. And it did not stay where it was put. Over months, its orbit changed in ways that gravity alone could not explain. The eccentricity grew and shrank on a cycle of roughly a year. The perigee dropped hundreds of kilometres. Something was pushing it. That something was sunlight. Photons have no rest mass. They also carry momentum — p=E/cp = E/c — and when they strike a surface they deliver it. The pressure is about 4.54.5 micronewtons per square metre at Earth's distance: roughly the weight of a grain of sand spread over a dinner table. On Echo 1, that came to about 3 millinewtons of continuous force. Tiny. Applied for years, without pause, it reshaped the orbit of a 76 kg spacecraft so dramatically that Echo 1 became the first practical measurement of solar radiation pressure — and the reason every high-altitude mission since has had to model it.
Watch first · presented by Shoaib Aziz
The numbers in this film are the labs' own: sunlight presses 4.5 μN/m² at 1 AU; Echo 1 carried about 9.2 m² per kilogram; the Moon perturbs Earth satellites 2.18 times as hard as the Sun; at geostationary altitude central gravity is 0.224 m/s², J₂ about 8.3 × 10⁻⁶, the Moon 7.3 × 10⁻⁶, the Sun 3.3 × 10⁻⁶ and sunlight 1.2 × 10⁻⁷ m/s²; sunlight overtakes drag between 600 and 700 km whatever the area-to-mass ratio; and an untended GEO satellite's inclination grows 0.95° in a year in the lab's model (about 0.85° in reality).
Solar radiation pressure at 1 AU follows directly from the solar constant divided by the speed of light:
Psr=Sc=1361 W/m22.998×108 m/s=4.54×10−6 N/m2P_{sr} = \frac{S}{c} = \frac{1361\ \text{W/m}^2}{2.998\times10^8\ \text{m/s}} = 4.54\times10^{-6}\ \text{N/m}^2

Solar radiation pressure. Absorbed light delivers this; perfectly reflected light delivers twice as much, because the photon reverses direction.

a⃗srp=−Psr Cr Am u^⊙\vec{a}_{srp} = -P_{sr}\,C_{r}\,\frac{A}{m}\,\hat{u}_{\odot}

The acceleration. CrC_{r} is the reflectivity coefficient: 1 for a perfect absorber, 2 for a perfect mirror, typically 1.2–1.5 for real spacecraft. Note A/m again — the same area-to-mass ratio that governs drag.

Figure 7.9
sunlight — 1361 W/m²umbrapenumbraP·C_r·A/mP = S/c = 4.54 × 10⁻⁶ N/m²About the weight of a grain of sand spread over a dinner table —applied without pause, for years. Entering shadow switches it offinstantly, and that once-per-orbit chopping is what pumps eccentricity.
Radiation pressure geometry, and the Earth's shadow. The umbra is total darkness; the penumbra is the partial shadow where the Sun is only half-hidden. A satellite entering shadow loses the force instantly, which makes SRP a periodic on-off perturbation rather than a smooth one — and that periodicity is what pumps eccentricity.
💡Why area-to-mass ratio decides everything here
Every gravitational perturbation is mass-independent. Radiation pressure is not, and neither is drag — both scale with A/mA/m. A dense compact satellite, A/m≈0.01A/m \approx 0.01 m²/kg: SRP acceleration around 5×10−85\times10^{-8} m/s². A minor nuisance. A modern GEO satellite with large solar arrays, A/m≈0.02A/m \approx 0.02–0.050.05: enough to require active modelling for station-keeping. Echo 1, A/m≈9A/m \approx 9: SRP acceleration around 4×10−54\times10^{-5} m/s² — a thousand times higher, and completely dominant over every other perturbation at its altitude. A solar sail is designed for A/m>100A/m > 100, turning what is a nuisance for everyone else into the propulsion system. IKAROS, launched by JAXA in 2010, was the first spacecraft to fly on sunlight alone — a 14 by 14 metre sail, 20 kg, reaching Venus without propellant. LightSail 2 did it in Earth orbit in 2019, deliberately raising its apogee by using the sail exactly as this equation predicts. The same physics that ruined Echo 1's orbit is the only propulsion system that needs no propellant at all.
Given ships or sails adapted to the breezes of heaven, there will be those who will not shrink from even that vast expanse.
– Johannes Kepler, letter to Galileo, 1610
The second family of perturbations comes from other bodies. The Moon and the Sun both pull on an Earth satellite — but what matters is not the raw pull. It is the difference between the pull on the satellite and the pull on the Earth itself:
a⃗3b=μ3(r⃗3−r⃗∣r⃗3−r⃗∣3−r⃗3∣r⃗3∣3)\vec{a}_{3b} = \mu_3\left(\frac{\vec{r}_3 - \vec{r}}{|\vec{r}_3 - \vec{r}|^3} - \frac{\vec{r}_3}{|\vec{r}_3|^3}\right)

Third-body perturbation. The first term is the pull on the satellite; the second removes the pull on Earth, since we are working in an Earth-centred frame. What is left is a tidal difference.

⚠Why the Sun is weaker than the Moon, despite being vastly more massive
The Sun's gravitational parameter is 27 million times the Moon's. It is also 390 times further away. Because the perturbation is a difference of two nearly-equal pulls, it falls off as 1/d31/d^3 rather than 1/d21/d^2. So: aMoonaSun≈μMoonμSun(dSundMoon)3=12.7×107×3903≈2.2\frac{a_{\text{Moon}}}{a_{\text{Sun}}} \approx \frac{\mu_{\text{Moon}}}{\mu_{\text{Sun}}}\left(\frac{d_{\text{Sun}}}{d_{\text{Moon}}}\right)^3 = \frac{1}{2.7\times10^7} \times 390^3 \approx 2.2 The Moon perturbs Earth satellites roughly twice as strongly as the Sun does. The same cube law is why the Moon dominates ocean tides despite the Sun's overwhelming mass — a tidal force is a gravity gradient, and gradients punish distance far more harshly than gravity itself does.
Figure 7.10
Moonpulled harderpulled lesspull on EarthIn an Earth-centred frame only the *difference* survives:a₃ᵦ = μ₃ [ (r₃ − r)/|r₃ − r|³ − r₃/|r₃|³ ]A gradient, not a pull — so it falls off as 1/d³, and the nearby Moon beats the vastly more massive Sun by about 2.2×.
The tidal nature of a third-body perturbation. The Moon pulls on both Earth and satellite; only the difference survives in the Earth-centred frame. Near-side and far-side positions are pulled in opposite senses relative to Earth — the same geometry that raises two ocean tides rather than one.
Which brings us to the single most useful chart in the field: which perturbation dominates at which altitude? Every mission's force model is chosen from this picture.
Figure 7.11
10³10⁴10⁵10⁻¹²10⁻⁹10⁻⁶10⁻³10⁰10¹altitude (km)acceleration (m/s²)LEOGPSGEOcentral gravityJ₂drag (B = 100)MoonSunSRP (A/m = 0.02)Read it twice: once forwhich term is *largest*,and once for which is*secular*. Rarely the same.
Perturbing acceleration against altitude, all on one logarithmic plot. Drag plunges off the left. J₂ falls steadily. Third-body and SRP are nearly flat, because neither depends much on where you are around the Earth. The crossings are where mission force models change — and every one of them is a real engineering decision.
AltitudeLargest perturbing accelerationWhat actually limits the missionConsequence
200–500 kmJ₂DragLifetime measured in months; reboost or die
500–800 kmJ₂J₂ (with drag over decades)Sun-synchronous design; long but finite lifetime
800–2000 kmJ₂J₂Debris persists for centuries — nothing removes it
MEO (~20,000 km)J₂J₂ and lunisolar togetherGPS needs all three modelled for metre-level orbits
GEO (35,786 km)J₂ ≈ MoonCombined lunisolarInclination grows ~0.85°/year without station-keeping
Above GEOLunisolarLunisolarEarth is barely the dominant body any more
⚠Largest acceleration versus largest secular effect
Read the chart carefully and something looks wrong. At 300 km, J₂ produces about 1.3×10−21.3\times10^{-2} m/s² while drag produces about 10−510^{-5} — J₂ is a thousand times larger. So why does everyone say drag dominates low Earth orbit? Because size is not the same as consequence. J₂ is large but conservative: it rotates the orbit and returns everything it borrows. 7-1 showed that aa has no secular drift under J₂, so no matter how large the acceleration, the orbit's size and energy are untouched. It will never bring anything down. Drag is small and dissipative. Every pass, it removes a little energy and never gives any back. Small and one-way beats large and cyclic, given enough time. So when choosing a force model, ask two separate questions: which term is biggest (that governs your short-term position accuracy) and which term is secular (that governs your mission lifetime). They are frequently different terms, and confusing them is a classic way to build a propagator that is precise and wrong.
📐Worked Example: what actually moves a GEO satellite
A geostationary satellite: r=42,164r = 42{,}164 km, A/m=0.02A/m = 0.02 m²/kg, Cr=1.3C_r = 1.3. Central gravity. μ/r2=3.986×1014/(4.2164×107)2=0.224\mu/r^2 = 3.986\times10^{14}/(4.2164\times10^7)^2 = 0.224 m/s². This is the baseline. J₂. Roughly 32J2(R⊕/r)2×μ/r2=1.5(1.083×10−3)(0.1513)2(0.224)=8.3×10−6\tfrac{3}{2}J_2(R_\oplus/r)^2 \times \mu/r^2 = 1.5(1.083\times10^{-3})(0.1513)^2(0.224) = 8.3\times10^{-6} m/s². Moon. μMoon=4903\mu_{\text{Moon}} = 4903 km³/s², d≈384,400d \approx 384{,}400 km. The tidal term is about 2μ3r/d3=2(4.903×1012)(4.2164×107)/(3.844×108)3=7.3×10−62\mu_3 r/d^3 = 2(4.903\times10^{12})(4.2164\times10^7)/(3.844\times10^8)^3 = 7.3\times10^{-6} m/s². Sun. By the ratio above, roughly 3.3×10−63.3\times10^{-6} m/s². SRP. 4.54×10−6×1.3×0.02=1.2×10−74.54\times10^{-6} \times 1.3 \times 0.02 = 1.2\times10^{-7} m/s². Ranking: J₂ ≈ Moon > Sun ≫ SRP. At GEO the lunisolar terms have caught up with the bulge, and together they exceed it. And here is what they do. The combined lunisolar pull is not aligned with Earth's equator — the Moon's orbit is tilted about 5° from the ecliptic, which is itself tilted 23.4° from the equator. So the perturbation has a persistent out-of-plane component, and it drives the inclination of an unattended geostationary satellite upward at about 0.85° per year. After fifteen years without station-keeping, that is nearly 13° of inclination. The satellite is no longer a fixed dot; it traces a figure-eight 26° tall, and no fixed dish on the ground can see it. The satellite still works perfectly. It has simply stopped being where it was sold to be.
🛰Using solar radiation pressure
Solar radiation pressure is usually a nuisance. Occasionally it is free propulsion, and not only for solar sails. Attitude control without propellant. Several GEO satellites use asymmetric or steerable solar array "windmill" configurations so that SRP produces a controllable torque, dumping momentum from reaction wheels without firing a thruster. Free, for the life of the mission. Eccentricity management. SRP naturally pumps eccentricity at GEO on an annual cycle. Operators do not fight it — they let it run and phase their east–west station-keeping burns to work with it. The Yarkovsky effect. An asteroid absorbs sunlight on its day side and re-radiates it as infrared, slightly later, from its afternoon side. That asymmetric emission is a tiny thrust. Over millions of years it moves asteroids across the solar system, and it is the main mechanism delivering them from the main belt into Earth-crossing orbits. The Yarkovsky effect on Bennu was measured by OSIRIS-REx: about 284 metres of drift per year. That number is now the dominant term in the calculation of whether Bennu will strike Earth in 2182 — a planetary defence question resting on photon pressure.
↓But there is a problem
So the model is comprehensively broken. All four assumptions from Unit 2 are down. Earth is not a point mass. Space is not empty. Sunlight pushes. Other bodies pull. And every one of these acts continuously, for the entire life of the spacecraft. There is no equilibrium and no configuration that is left alone. A LEO satellite spirals down. A GEO satellite tilts out of its plane at nearly a degree a year and drifts along the equator toward one of two gravitational valleys. An eccentric orbit's perigee walks around the ellipse. Which means that keeping a satellite where it is supposed to be is not a solved problem, or a design decision made once. It is work, done continuously, paid for in propellant, until the propellant runs out. And what happens on the day it does — where the satellite goes to die, and who decides — is now a matter of international law, driven by a scenario a NASA scientist described in 1978 that gets closer every year.