PX282 - C11 - radiation pressure

photons carry momentum:

E_{γ} = p c

hence, they exert a pressure
examples of radiation pressure in action:
- protoplanetary pressure
- kepler's second light (KS) mission
- dust tail of comets
consider a beam of photons, with a momentum, $p_{γ}$ , hits a surface at an angle, $θ$ with the $z$ -axis, and gets reflected with a momentum, $p_{γ}^{'}$
the change in the $z$ -component of the momentum:

Δ p_{z} = p_{z}^{'} - p_{z} = \frac{E \cos θ}{c} - \frac{- E \cos θ}{c} = \frac{2 E \cos θ}{c}

the energy:

E = I \cos θ d t d A d Ω

therefore, a small change in momentum can be given by:

d p = \frac{2}{c} I d t d A \cos^{2} θ d Ω

force, $F = \frac{d p}{d t}$ , leads to:

P = \frac{d p}{d t d A}

the radiation pressure is given by:

P_{r a d} = \int_{H} \frac{2}{c} I \cos^{2} θ d Ω

- where, the integral is over a hemisphere ( $H$ )

in a gas, there is no surface to reflect, so the integral will be over the entire sphere ( $S$ ), and the factor of $2$ will be removed:

P_{r a d} = \frac{1}{c} \int_{S} I \cos^{2} θ d Ω = \frac{I}{c} \int_{0}^{2 π} d ϕ \int_{0}^{π} d θ \cos^{2} θ \sin θ

∴ P_{r a d} = \frac{4 π}{3 c} I

since it is integrated over a sphere, the average intensity can be taken:

P_{r a d} = \frac{4 π}{3 c} ⟨ I ⟩

for a blackbody emitter:

I = \frac{σ T^{4}}{π}

∴ P_{r a d, b b} = \frac{4 σ}{3 c} T^{4} = \frac{1}{3} a T^{4}

where, $a = \frac{4 σ}{c} = 7.57 \times 10^{- 16} J m^{- 3} K^{- 4}$

for the earth, $T \sim 300 K$ , $P_{r a d} \sim 2 \times 10^{- 6} N m^{- 2} \sim 10^{- 11} b a r$
for the sun:
- $T_{s u r f a c e} \sim 5800 K$ , $P_{r a d} \sim 2.9 \times 10^{- 1} N m^{- 2} \sim 10^{- 6} b a r$
- $T_{c o r e} \sim 1.5 \times 10^{7} K$ , $P_{r a d} \sim 1.3 \times 10^{13 N m^{- 2}} \sim 10^{8} b a r$