Problem — Solar sail

Problem

A solar sail is a thin, reflective sheet of area $A$, exposed to sunlight outside the atmosphere ($I_\odot\approx 1{,}36$ kW/m²). (a) If it absorbs all the light the radiation pressure is $P_\text{rad}=I/c$; if it reflects the light it is $P_\text{rad}=2I/c$: why? (b) Calculate the force on a reflective square sail of side $100$ m. (c) Estimate the acceleration if the sail and payload have total mass $1$ kg.

Solution

(a) Absorption versus reflection. Light carries momentum: a beam of energy $E$ carries $p=E/c$. If the sail absorbs, it picks up all the incident momentum: change $\Delta p=p$. If it reflects, the light bounces back, reversing its own momentum: the change is double, $\Delta p=2p$. Since force is the momentum transferred per unit time, the pressure of a reflective sail is double:

$P_\text{rad}^\text{ass}=\frac{I}{c}\qquad P_\text{rad}^\text{rifl}=\frac{2I}{c}$

(b) Force on the reflective sail. Radiation pressure:

$P_\text{rad}=\frac{2I}{c}=\frac{2\cdot 1360}{3{,}00\cdot 10^8}\approx 9{,}1\cdot 10^{-6}\;\text{Pa}$

Area of the square sail:

$A=(100)^2=1\cdot 10^{4}\;\text{m}^2$

Force:

$F=P_\text{rad}\,A=9{,}1\cdot 10^{-6}\cdot 1\cdot 10^{4}\approx 9{,}1\cdot 10^{-2}\;\text{N}$

$\ev{F\approx 0{,}091\;\text{N}}$

(c) Acceleration.

$a=\frac{F}{m}=\frac{0{,}091}{1}\approx 0{,}091\;\text{m/s}^2$

$\ev{a\approx 0{,}091\;\text{m/s}^2}$

Small, but applied without pause for days it produces large velocities: this is the principle of solar-sail propulsion.

Textbook typo $F\approx 91$ N and $a\approx 91$ m/s²: it is wrong by a factor of $10^3$ (the $\mu$ prefix in the pressure was dropped, $9{,}1\;\mu$Pa read as $9{,}1$ mPa). The correct value is $F\approx 0{,}091$ N.

The textbook incorrectly gives

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Topics: Onde elettromagnetiche Concepts: Pressione di radiazione