Some spacecraft design uses a solar sail made of aluminized plastic. As sunlight reflects off the sail, radiation pressure drives the spacecraft outward away from the sun.
a. If the sail material has a density of 700.0 kg/m3, what is the maximum thickness of the sail for which the force due to radiation pressure exceeds the gravitational force on the sail?
b. If the collector's area is 2e6 m^2. its thickness is 1 μm, and the craft carries a 100.0 kg payload, what is its acceleration at the radius of the Earth's orbit?
You need to know the radiation pressure due to solar radiation. This varies with (1/R^2), but so does the gravity force.
The solar flux at the Earth's distance from the sun is about
I = 1300 W/m^2, as I recall. You'd better look it up yourself. Divide that by c for the momentum flux.
(a) Let the sail area be A. (This will cancel out) and the thickness be d. For a balance with radiation pressure,
A*density*d*G*M/R^2 = 2*A*(1300/c)*(Re/R)^2
I am sure you recognize G as the universal constant of gravity. A cancels out. So does R^2.
The factor of 2 assumes perfect reflection of all incident light, for the momentum change.
M is the mass of the sun and Re is the radius of the earth;'s orbit. Solve for d
b. Use
2*A*(1300/c)*(Re/R)^2 for the force on the sail and the appropriate mass for that sail thickness.
Then use a = F/m for the acceleration
Thanks drwls. I'm having trouble with part b though. What is R? For the appropriate mass, am I suppose to use the density times the thickness to get the mass of the sail and with that, add the mass from payload?
a. To determine the maximum thickness of the sail for which the force due to radiation pressure exceeds the gravitational force, we need to compare the two forces.
Let's first calculate the force due to radiation pressure. The radiation pressure is given by the formula:
Pressure = Intensity * Reflectance / (Speed of light)
We know that the intensity of sunlight at the Earth's distance from the sun is approximately 1,340 W/m² (watts per square meter), and the reflectance of the aluminized plastic can be assumed to be around 0.9.
Plugging in these values, we have:
Pressure = 1,340 * 0.9 / (Speed of light)
The speed of light is approximately 3 * 10^8 m/s (meters per second), so we can calculate the pressure due to radiation.
Next, we need to calculate the gravitational force acting on the sail. The force of gravity is given by the formula:
Force = Mass * Gravitational acceleration
The mass is equal to the volume multiplied by the density, and the volume is the product of the area and thickness. We know that the density of the sail material is 700.0 kg/m³, and we need to find the maximum thickness of the sail.
Let's assume that the gravitational acceleration is approximately 9.8 m/s² (meters per second squared) for simplicity.
Rearranging the force equation, we have:
Thickness = Force / (Mass * Gravitational acceleration)
Now we can plug in the values and calculate the maximum thickness of the sail.
b. To determine the acceleration of the spacecraft at the radius of the Earth's orbit, we need to calculate the net force acting on the spacecraft.
The net force is equal to the force due to radiation pressure minus the force of gravity acting on the sail and payload. We can calculate the force due to radiation pressure using the formula from part a.
Next, we need to calculate the force of gravity. The mass of the spacecraft is the sum of the mass of the sail and the payload (100.0 kg). We already calculated the thickness of the sail in part a, so we can use that value.
The force of gravity is given by the formula:
Force = Mass * Gravitational acceleration
Substituting the values and calculating the force of gravity, we can then find the net force.
Finally, we can calculate the acceleration of the spacecraft using Newton's second law:
Acceleration = Net force / Mass
Substituting the values, we can calculate the acceleration at the radius of the Earth's orbit.