Consider a solar sailcraft of total mass 10,000 kg , which uses a large, low-mas
ID: 1453880 • Letter: C
Question
Consider a solar sailcraft of total mass 10,000 kg, which uses a large, low-mass sail and the energy and momentum of sunlight for propulsion.
(a) Should the sail be reflective or absorbing, and why?
(b) The total power output of Sun is 3.9 × 1026W. How large a sail (express in km2) is needed to propel the craft against the gravitational force of Sun?
(c) Consider the total radiation power emitted by Sun as stated in the previous part. How much of this power is received by Earth?
rSunEarth= 1.50 × 1011m
REarth = 6.37 × 106m
Explanation / Answer
The light energy is given by E = h*f, where h = Planck's constant and f = light frequency. The energy impacting the sail Es is the area of the sail divided by the area of the sphere at the distance r from the sun, times the energy output of the sun:
Es = As/(4r^2) * W*t
W = power output of the sun, t is the duration of light impact on the sail
The momentum of the photons reaching the sail is given by p = h*f/c; since h*f = E, p = E/c.Therefore, the momentum of the light reaching the sail is
p = Es/c = As/(4r^2) * W*(t/c)
The momentum transferred to the sail is 2p because of the reflection, and force is change in momentum with time, so the force on the sail is
2p/t = As*W/(c*2r^2); this must equal the gravitational force, so
As*W/(c*2r^2) = GMm/r^2
The radius (distance from the sun) cancels out leaving
As = 2cGMm/W
Both the gravitational attraction and the radiation force decrease as 1/r^2, so the distance doesn't matter.
c = 3*10^8 m/sec
G = 6.67*10^-11 m^3/(kg*sec^2)
M = 2*10^30 kg
m = 10^4 kg
W = 3.9*10^26 joule/sec
A = 6.45*10^6 m^2
A = 6.45 km^2