Question

4. Later this year, a team from Technion University in Israel is planning to launch three small satellites as part of their ongoing SAMSON project. The team will test the use of radiation pressure from sunlight, incident on a satellite’s solar panels, to provide the energy needed to increase the radius of the satellite’s orbit. To confirm the theoretical viability of this idea, consider a nanosatellite of mass 2.10 kg in a circular orbit around Earth with a period of 96.0 minutes. The satellite’s solar panels have and albedo of 20.0% and area of 0.200 m2 , and they are maintained in an orientation perpendicular to the incident sunlight for the quarter of the satellite’s orbit in which the satellite achieves the greatest linear displacement away from the sun. Find the increase in the radius of the satellite’s orbit after one day. (This is a work-energy problem. You will need to find the original radius of the orbit, and then consider the work done by the radiation force exerted on the solar panels, then relate this to the initial and final energy of the circular orbit.) Assume the intensity of sunlight is 1400 W/m2 . ANSWER IS 19 METERS

Answer 1

4. consider a nanosatellite of mass m = 2.10 kg
   in a circular orbit around the earth
   time period of satellite, T = 96 m
   solar panel albedo, a = 0.2
   area of panels, A = 0.2 m^2

   original orbit = r
   speed of satellite = v
   now
   GMm/r^2 = mv^2/r
   GM/r = v^2 .. (1)

   also
   v = 2*pi*r/T
   hence
   GM/r = 4*pi^2*r^2/T^2
   r^3 = GMT^2/4*pi^2
   G = 6.67*10^-11
   M = 5.972*10^24 kg
   T = 96 min = 96*60 s
   hence
   r = 6.9458*10^6 m

   now intensity of sunlight = I = 1400 W/m^2
   hence force exerted on the satellite due to radiation = F
   F = IA/c = 9.3333*10^-7 N
   lets say this radiation pressure pushes the satellite by distance d
   then new orbit = r'
   r' - r = d

   -GMm/2r + F*a*(r' - r) = -GMm/2r'
   1.866666*10^-7*r'^2 - 60216106*r' + 4.1824902*10^13 = 0
   hence
   r' = 6945809.9865 m
   d = r' - r = 19.985 m
   hence the satellite is pushed by 19.985 m