Solar radiation pressure

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1. Where does the force come from? (Derive from scratch)

HOW we build the pressure, step by step.

Step 1 — Energy flux. The Sun emits power $L_\odot \approx 3.828\times10^{26}\,\text{W}$. At distance $r$, spread over a sphere of area $4\pi r^2$, the intensity (power per area) is $S(r) = \frac{L_\odot}{4\pi r^2}.$ Why this step? Energy is conserved, so the same power passes through every sphere centred on the Sun — bigger sphere, thinner flux.

At Earth ($r = 1\,\text{AU}$), $S \approx 1361\,\text{W/m}^2$ — the solar constant.

Step 2 — From energy flux to momentum flux. Each joule of light carries $1/c$ of momentum. So the momentum arriving per area per second is $\frac{S}{c}.$ Why this step? Momentum flux = force per area = pressure. We just converted "energy per second per area" into "momentum per second per area" using $p=E/c$.

Step 3 — The base pressure. $\boxed{P_{\text{rad}} = \frac{S}{c}}$ Why this step? This is the pressure on a perfectly absorbing surface facing the Sun. At 1 AU: $P_{\text{rad}} = \frac{1361}{3\times10^8} \approx 4.5\times10^{-6}\,\text{Pa}.$

Tiny — but constant and everywhere.

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2. Absorbing vs reflecting (the factor that trips people up)

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3. Acceleration of a spacecraft

WHY acceleration, not force? Orbits care about acceleration ($a=F/m$). A light spacecraft is pushed more.

For a spacecraft of mass $m$, cross-section $A$, absorbing (worst simple case): $a_{\text{SRP}} = \frac{P_{\text{rad}}(1+\eta)A}{m} = \frac{S(r)}{c}\,\frac{(1+\eta)A}{m}.$

The ratio $A/m$ is the area-to-mass ratio — the single most important design number for SRP sensitivity.

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4. Worked examples

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5. Common mistakes (Steel-man then fix)

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6. Active recall

Recall Test yourself (hide the answers)

• Why does a mirror feel double the force of a black surface? → recoil doubles the momentum change.
• What is the solar constant? → $\approx 1361\,\text{W/m}^2$.
• What single spacecraft number governs SRP sensitivity? → area-to-mass ratio $A/m$.
• Why doesn't SRP/gravity ratio change with distance? → both $\propto 1/r^2$.

Recall Feynman: explain to a 12-year-old

Sunlight is made of tiny bullets called photons. Each one is super light, but it still gives a little kick when it hits something — like millions of grains of sand hitting a sail. In space there's no air pushing back, so even a gentle, never-stopping push slowly speeds up a spacecraft. A shiny surface gets pushed harder because the photons bounce off (like a trampoline) instead of just splatting.

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7. Connections

• Orbital perturbations — SRP is a non-gravitational perturbing force.
• Solar sails — propulsion using SRP.
• Atmospheric drag — competing non-conservative force in LEO.
• Photon momentum and relativity — $E=pc$ origin.
• Inverse-square law — why intensity falls as $1/r^2$.
• Yarkovsky effect — thermal re-emission cousin of SRP on asteroids.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, sunlight sirf roshni nahi hai — usme momentum bhi hota hai. Photon ka mass zero hota hai, par relativity kehti hai $p = E/c$, matlab har photon thoda sa momentum carry karta hai. Jab ye photons kisi spacecraft pe girte hain, to ek chhota sa dhakka dete hain. Ek photon se kuch nahi hota, par Sun se har second mein arbon-kharbon photons aate hain, to total force real ban jaata hai. Isi ko bolte hain solar radiation pressure (SRP).

Formula banana simple hai. Sun ki power $L_\odot$ ko distance $r$ pe ek sphere $4\pi r^2$ pe faila do — to intensity $S = L_\odot/4\pi r^2$ milti hai (ye $1/r^2$ law hai). Fir har joule energy $1/c$ momentum laati hai, to pressure ban jaata hai $P = S/c$. Earth pe (1 AU) ye sirf $4.5\times10^{-6}$ Pascal hai — bahut chhota, par continuous.

Ek important baat: agar surface light ko absorb karta hai to $\Delta p = p$, par agar mirror ki tarah reflect karta hai to photon ulta bounce hota hai, isliye $\Delta p = 2p$ — double push! Isiliye solar sails shiny banaye jaate hain. Force $F = P_{rad}A(1+\eta)\cos\theta$ hota hai.

Ye matter kyun karta hai? Kyunki halke, badi-area wale spacecraft (jaise solar sails ya balloon satellites) ke liye SRP bahut asar dikhata hai — orbit ko months mein disturb kar deta hai. Yaad rakho: sabse important number hai area-to-mass ratio $A/m$. Jitna zyada $A/m$, utna zyada SRP feel hota hai. Dense probe ko farak nahi padta, par solar sail ke liye ye pura propulsion system ban jaata hai!