A question on electromagnetic waves (I think)

[runciblesp00n]

"Show that spherical dust grains within the Solar System which have a radius less than a critical radius r will be repelled by the sun, whilst those with a radius greater than r will be attracted towards it. Calculate r given that the mass of the sun is 2x10^30kg, its luminosity is 4x10^26W and the density of the dust grains is of order 10^3kgm^-3. (Neglect the pressure exerted by the Solar Wind).


If possible, I'd like an explanation on how to approach this problem, but not a full answer, as I'd rather do the working, etc myself (I can't really learn just by copying, I take information in a lot better by practicing applications). I just don't really know where to start with it.

Thanks!

 

[wildman]

Obtain the equations for 1) particles are attracted by the sun due to gravity and 2) replused by the energy in the photons (luminosity) and set them equal to each other.

 

[Moetasim]

Sure, I can provide some guidance on how to approach this problem. First, we need to understand the concept of radiation pressure, which is the pressure exerted by electromagnetic waves on a surface. In this case, the surface is the spherical dust grains within the Solar System.

The radiation pressure can be calculated using the formula: P = L/(4πr^2c), where P is the pressure, L is the luminosity of the source, r is the distance from the source, and c is the speed of light.

In this problem, we are given the luminosity of the sun (L = 4x10^26W) and the density of the dust grains (ρ = 10^3kgm^-3). We also know that the sun's mass (M = 2x10^30kg) and its luminosity are related through the equation L = GM^2/R^2, where G is the gravitational constant and R is the distance from the center of the sun.

Now, let's consider two scenarios: one where the dust grain has a radius less than the critical radius and one where it has a radius greater than the critical radius.

For the first scenario, the dust grain will be repelled by the sun if the radiation pressure is greater than the gravitational force exerted by the sun. This can be expressed as:

P > GMρ/r^2

Substituting the values we have, we get:

L/(4πr^2c) > GMρ/r^2

Simplifying and rearranging, we get:

r < (GMρ/cL)^1/2

This is the critical radius (r) at which the radiation pressure is equal to the gravitational force. For a dust grain with a radius less than this, the radiation pressure will be greater and it will be repelled by the sun.

For the second scenario, the dust grain will be attracted towards the sun if the gravitational force is greater than the radiation pressure. This can be expressed as:

GMρ/r^2 > L/(4πr^2c)

Simplifying and rearranging, we get:

r > (GMρ/cL)^1/2

This is the critical radius (r) at which the gravitational force is greater than the radiation pressure. For a dust grain with a radius greater than this, the gravitational force will be greater and it will be attracted towards the sun.

To calculate the value of