Comet tails

The following websites of the tutorial Understanding Spectra from the Earth 
https://www.seos-project.uni-oldenburg.de/earthspectra/earthspectra-c00-p01.ar.html
are handled:
•	1 Light and radiation
- Elektromagnetic Waves 1 to 4
- Photons 1 to 4
- Supplement 1.4: Mass, energy and momentum of particles and photons
- Supplement 1.6: The radiation pressure
•	3. Climate and climate change
- the greenhouse effect

If you point a telescope into the sky from Earth, you can occasionally observe fast-moving objects: Comets. Close to the sun, comets are mechanically unstable due to the sun's strong radiation and therefore lose dust and molecules. The lost material becomes visible as a bright comet tail. However, comets do not drag their tails behind them on their trajectory, as the components are repelled by the radiation pressure of the sunlight. On the other hand, the gravitational force emanating from the sun has an attractive effect on the particles in the tail.

Source: United States Naval Observatory

Sometimes a double tail appears: the blue tail consists of molecules, the white tail of larger dust particles. The air molecules in our atmosphere also scatter blue, while water vapour particles and water droplets in the clouds scatter white.
The comet Hale-Bopp in the right-hand image, taken on 11 March 1997, showed both tails very clearly. Their different orientation could be caused by the role of radiation pressure and gravity on particles of different sizes: the radiation pressure of sunlight deflects the molecules away from the sun, while gravity prevails for the much larger and more massive dust particles! 
This hypothesis is investigated in more detail here.   

1.	The radiation pressure
Electromagnetic waves exert a radiation pressure that is equal to the sum of the electric and magnetic field energy:
 					    	   (see Supplement 1.4)
If  is multiplied by the speed of light, the irradiance  is obtained, i.e. the power per area present in the radiation field. This gives the radiation pressure of a particle,
 					      	   (see Supplement 1.6)
with the reflection coefficient r. The reflection coefficient is zero for completely absorbing particles. It is one for completely reflecting particles; the reflected radiation has the same radiation pressure as the incoming radiation, the particle therefore absorbs twice the radiation pressure. The force acting on an absorbing spherical particle with a diameter and cross-sectional area  in the radiation field is

In the case of an ideally reflecting particle, a factor  is added to the right-hand side of the equation. 
To allow for arbitrary distances of a comet from the sun, we assume the total radiation of the sun's surface. In chapter 3, section Climate and climate change, the total radiation output of the sun into space is calculated: 387.5⋅1024 W. At a distance  R from the sun, this radiation is distributed over an (imaginary) spherical surface . The irradiance is then at this distance: 

The force in Newtons (N) on a particle is thus:

2.	The force of gravity
The gravitational force acting on a particle from the sun is

with the mass  of the sun, the mass m of the particle, and the gravitational constant . For a spherical particle with diameter d and density  is:

The force of gravity thus becomes:

3.	Tasks
a)	Calculate the ratio of gravity and force due to the radiation pressure. What is the dependence of the quotient on the density, the diameter and the distance to the sun?
b)	Small ice crystals in the comet's tail have the density of water, i.e. approx. 1000 kg/m3. For what particle diameter do the force of radiation pressure and the force of gravity cancel each other out? Is the hypothesis correct that small particles are repelled by the radiation pressure and large particles are attracted towards the sun? 
c)	The particles in comet tails are obviously reflective rather than absorbent. How does the diameter at which the two forces are equal change for ideally reflecting particles?