1. Solar sail
05/04
E-mail: iavorv@hotmail.com
Consider a cosmic sail-boat moving under the influence of the pressure of sun-light at a distance
from the Sun similar to that of Earth. What kind of accelerations you can expect? How do those
light-pressure forces compare with the gravitational pull of the Sun?
A sail gains momentum through either reflection or absorption of photons. The most efficient sail is a high reflector,
because each photon loses momentum p = 2hν/c in reflection compared to only hν/c in absorption.
The Sun is a blackbody emitter with radius RS = 6.96 × 108
m and effective temperature T ≈ 6000 K. Therefore,
at any distance r from the Sun, the total radiation intensity (in W/m2
) is:
I(r) = σT4 R2
S
r2
, (1)
where σ = 5.67 × 10−8
W/m2
/K4
is the Stefan-Boltzmann constant. From Planck’s formula, the spectral intensity at
particular frequency ν is:
Iν(r, ν) =
2πhν3
c2
1
exp( hν
kT ) − 1
R2
S
r2
, (2)
where h = 6.626 × 10−34
J s is the Planck’s constant and k = 1.38 × 10−23
J/K is the Boltzmann’s constant. From
Eq.(2) one can obtain the number of photons ∆N of frequency ν passing through area A during a time interval ∆t:
∆N
A∆t
=
Iν(r, ν)
hν
(3)
The force exerted by these photons (each losing momentum p = 2hν/c) is:
Fν(r, ν) =
∆N
∆t
p =
2A
c
Iν(r, ν) (4)
The total force is then obtained after integration over the whole spectrum of the Solar radiation:
FS(r) =
∞
0
Fν(r, ν)dν =
2A
c
∞
0
Iν(r, ν) =
2A
c
I(r)
FS(r) =
2σT4
R2
S
c
A
r2
≈ 2 × 1017 A
r2
(5)
As an example, let’s consider a large sail of area A = 1 km2
situated at Earth’s orbit r = 1.496 × 1011
m. The force
exerted by the solar radiation, according to Eq.(5) is FS = 9 N.
Let’s compare the radiation pressure force Eq.(5) with the gravitational force, given by Newton’s formula:
FG(r) = GMS
m
r2
= 1.33 × 1020 m
r2
, (6)
where G = 6.67 × 10−11
m3
kg−1
s−2
is Newton’s gravitational constant, MS = 1.99 × 1030
kg is the solar mass, and
m is the mass of the sail.
It is clear from Eqs.(5,6) that the radiation pressure on the sail can compensate for the gravitational pull at any
distance from the Sun only if the following relation between sail’s area and mass is satisfied:
A =
cGMS
2σT4R2
S
m ≈ 663.3m. (7)
A sail made of aluminum (with density ρAl = 2643 kg/m3
) can be in equilibrium with the gravitational force only if
it has thickness:
d =
2σT4
R2
S
cGMSρAl
≈
1
663.3ρAl
≈ 570 nm (8)