# Is radiation pressure wavelength dependent (does a blue photon move the solar sail more then a red photon)?

I have read this question:

Where John Rennie says:

Does the wavelength of the light matter?

No

Now in the same question, another answer, by DavePhd says:

Does the wavelength of the light matter?

Yes, shorter wavelength photons have higher momentum. p=h/λ

If I'm floating in space and I turn on a flashlight, will I accelerate?

Even these two have contradictory answers about the wavelength dependence of radiation pressure.

There is this question too, but none of the answers talk about my specific question:

How can a red light photon be different from a blue light photon?

Now there are many questions on this site about solar sails and radiation pressure, and the momentum of classical light and individual photons. The only thing these agree about is that classical light and even individual photons do have momentum, and they do exert pressure on the surface of the solar sail. Some of them agree on the fact that this happens via:

1. scattering (elastic or inelastic), this makes up most of the momentum transfer

2. absorption, this only relates to a small number of photons, but they do transfer momentum to the sail too

Radiation pressure is the mechanical pressure exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field. This includes the momentum of light or electromagnetic radiation of any wavelength that is absorbed, reflected, or otherwise emitted

https://en.wikipedia.org/wiki/Radiation_pressure#Pressures_of_absorption_and_reflection

Now what none of them specifically describe, or agree on, is whether this is wavelength dependent, and how for example a higher energy photon (blue) would push the sail more then a lower energy photon (red).

The confusion comes from the agreed fact that only part of the momentum of the photons is transferred, and whether a higher energy photon should transfer more (relatively to a lower energy photon) momentum is questionable because of the different answers.

Questions:

1. Is radiation pressure wavelength dependent

2. does a blue photon move the solar sail more then a red photon?

• It just depends on the wavelength dependence of the reflectivity. You should modify your final question to contain one question, since at present it contains two with different answers. Nov 22, 2021 at 23:03
• @ProfRob thank you I edited, I separated the two questions, since as you mentioned, they might have different answers. Nov 23, 2021 at 2:48

## 3 Answers

They are both right. On a "per photon" basis, a 100% reflected photon (with $$k = 2\pi/\lambda$$) imparts a momentum change of:

$$\Delta p = -(p_f - p_i) = -(\hbar(-k) - \hbar k)=2\hbar k$$

which clearly depends on wavelength.

Meanwhile, given an energy ($$E$$) per area per time, the momentum transfer is:

$$\Delta p = 2E/c$$

which doesn't depend on wavelength.

Questions:

1. Is radiation pressure wavelength dependent
1. does a blue photon move the solar sail more then a red photon?

For 1, one has to go through the derivations of the Poynting vector and the energy density and momentum density of the classical electromagnetic field.

To answer it I would go through the number of photons in the classical beam. I would say that for the same number of photons in the beam , the blue beam would transfer more momentum.

For 2. Photons do not have colors. A photon with an energy $$E=hν$$ where $$ν$$ is the frequency of the wave that a multitude of same energy photons would build up, is only characterized by the value in energy.

A single photon will have higher momentum for higher energy (setting m to zero here),

so on average a higher energy photon will transfer a higher momentum (allowing for the various interactions).

A photon belonging to a blue classical monochromatic beam would transfer more momentum than one coming from a red monochromatic beam.

• Thank you so much! Nov 23, 2021 at 16:11

Radiation pressure: (source)

$$P_{\text {incident }}=\frac{\langle S\rangle}{c}=\frac{I_{f}}{c}$$

Where where $${\displaystyle P}$$ is pressure (usually in Pascals), $${\displaystyle I_{f}}$$ is the incident irradiance (usually in W/$${m^2}$$) and $${\displaystyle c}$$ is the speed of light in vacuum.

Irradiance of perpendicular to the sail beam depends on intensity thus number n of photons per unit of time in the beam or else called radiation flux, the irradiated area on the sail which depends in the case of the beam to the thickness of the beam (both of these properties in combination, intensity and irradiated area also referred sometimes as luminosity of the beam) and the frequency of photons in the monochromatic beam $${E_{photon}}$$=hf.

Thus, higher frequency photons should generate more pressure.

Therefore for two identical luminosity perpendicular to the sail beams I expect the blue beam to generate more pressure on the solar sail than the red beam.