4. A solar sail allows a spacecraft to use radiation pressure for propulsion, si

Question

4. A solar sail allows a spacecraft to use radiation pressure for propulsion, similar to the way wind propels a sailboat. The sails of such spacecraft are made out of enormous reflecting panels. The area of the panels is maximized to catch the largest number of incident photons, thus maximizing the momentum transfer from the incident radiation. If such a spacecraft were to be simply pushed away from a star by the incident photons, the force of the radiation pressure would have to be greater than the gravitational attraction from the star emitting the photons. The critical parameter is the area density (mass per unit area) of the sail. Consider a perfectly reflecting mirror oriented so that solar radiation of intensity is incident perpendicular to the reflective surface of the mirror. (a) If the mirror has surface area A, what is the magnitude of the average force on the mirror due to the radiation pressure of the sunlight? (b) Suppose that the mirror described is initially at rest a distance R away from the sun. What is the critical value of areal density (i.e. mass per unit area) for the mirror at which the radiation pressure exactly cancels out the gravitational attraction from the sun? Express your answer numerically in units of kg/m2. (c) What is the diameter of the sail if it is circular and the combined mass of the sail and the spacecraft is 1000 kg? (d) What is the force on the sail if R is the mean sun-to-earth distance?

Explanation / Answer

a)

Favg=2*I*A/C

b)

use,

Favg=F_gravitational

2*I*A/C=G*m1*m2/R^2

here,

m1 is the mass of the sun,

m2 is the mass of the solar sail and spacecraft

2*I*A/C=G*m1*m2/R^2

===> m2/A=2*I*R^2/(G*m1*C)

m2/A=2*I*R^2/(G*m1*C)

=2*3.2*10^25/(6.67*10^-11*2*10^30*3*10^8) ( here, I=3.2*10^25/R^2 W/m^2)

=0.0016 kg/m^2

c)

if m2=1000kg

==> m2/A=0.0016

1000/A=0.0016

1000/(pi*r^2)=0.0016

===> radiud, r=446.03 m

diameter, d=2*r

d=2*446.03

d=892.06 m

d)

F=G*m1*m2/R^2

F=6.67*10^-11*1000*2*10^30/(1.496*10^11)

F=8.92*10^11 N

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