Radiation pressure on sphere orbiting earth

[the_godfather]

Homework Statement

A spherical shell of diameter D, filled with hydrogen orbits the earth. The average intensity of solar radiation, in a plane perpendicular to the rays is $1.4kW/m^2$. Calculate the total force of solar radiation pushing it off its orbit as a function of the shell's albedo.

Homework Equations

$$F = P.A$$
$$A = (4\pi r^2)/2 = (\pi d)/2$$
$$P_r = \left\langle S\right\rangle/c = I/c$$

The Attempt at a Solution

So far I have used all the substitutions which is simple. What I'm not sure is how to incorporate the albedo. So far I have assumed that all incident radiation is absorbed. Can I add the coefficient as $(1 + a)$ where a = albedo to the front of momentum.
Also, my calculations show that reflected incident rays are sent back the way they come from when/if they are reflected. A plane wave incident on a spherical object though would not reflect back in the direction it came from unless it was on the horizontal axis. My only idea would be to use a solid angle? is this along the right lines?

 

[haruspex]

Can I add the coefficient as $(1 + a)$ where a = albedo to the front of momentum.

Not quite, for the reason below.

Also, my calculations show that reflected incident rays are sent back the way they come from when/if they are reflected. A plane wave incident on a spherical object though would not reflect back in the direction it came from unless it was on the horizontal axis. My only idea would be to use a solid angle?

Not sure what you meant by that. Consider a circular element of the surface with an axis parallel to the incoming radiation. The circle subtends an angle 2 θ at the centre of the sphere and has a width r δθ. What is the the force from the reflection from the circle? What is its component away from the sun?

 

[the_godfather]

Not quite, for the reason below.

Not sure what you meant by that. Consider a circular element of the surface with an axis parallel to the incoming radiation. The circle subtends an angle 2 θ at the centre of the sphere and has a width r δθ. What is the the force from the reflection from the circle? What is its component away from the sun?

That didn't help me unfortunately, would you be able to elaborate a bit more please...

My current understanding is that the energy impacting the sphere would depend on the latitude of the sphere, in a way similar to how solar radiation is more intense on the equator than nearer the poles (in general).

$d\Omega = \sin \theta ~d\theta ~d\phi$

 

[haruspex]

My current understanding is that the energy impacting the sphere would depend on the latitude of the sphere, in a way similar to how solar radiation is more intense on the equator than nearer the poles (in general).

$d\Omega = \sin \theta ~d\theta ~d\phi$

That's what I meant, but in this context the 'pole' should be taken as pointing towards the sun. So let's say the sun is above the 'north pole'. Consider a band width rδθ at latitude θ. How much light falls on it and at what angle? In which direction is it reflected? What therefore is the change in momentum of that light? What is the component of that change parallel to the NS axis?

 

[paranormal]

Firstly, it is important to define the albedo for the spherical shell in this scenario. The albedo is the fraction of incident radiation that is reflected by the shell. Therefore, in this case, the albedo would be (1 + a), where a is the albedo of the shell.

To incorporate the albedo into the calculation, you can use the equation F = (1 + a)P.A, where P is the incident radiation pressure and A is the area of the shell.

As for the direction of the reflected rays, it is correct to assume that they will be reflected back in the direction they came from. However, as you mentioned, this may not always be the case for a spherical object. In this case, you can use the solid angle to account for the direction of the reflected rays. The solid angle is a measure of the spread of rays from a point source, and can be calculated using the formula A = 2π(1 - cosθ), where θ is the angle between the incident and reflected rays.

Overall, the total force of solar radiation pushing the spherical shell off its orbit would be F = (1 + a)P.A, where P is the incident radiation pressure, A is the area of the shell, and a is the albedo. The direction of the force can be calculated using the solid angle, or by considering the direction of the reflected rays.