# How does light transfer momentum to a solar sail?

Last month the Planetary Society launched Solar Sail 2. After reading about it a bit I found that its a type of payload carrier that works on the principle of transfer of momentum from light.

Light is made of packets of energy called photons. While photons have no mass, they have momentum. Solar sails capture this momentum with sheets of large, reflective material such as Mylar. As photons bounce off the sail, most of their momentum is transferred, pushing the sail forward.

Source: PLanetry Society FAQ

This statement clearly contradicts the equation of momentum. The FAQ had an explanation for this but I wasn't able to understand it. The page quoted-

$$P=mv$$ only works for non-relativistic masses. For objects traveling near the speed of light, the universal equation for momentum is $$E^2=(pc)^2 + (mc^2)^2$$ This allows photons to have momentum, and that momentum can be transferred to another object like a solar sail.

First of all, I wasn't aware of the term "non relativistic mass", I looked for it and got a few articles for "relativistic mass". Though I wasn't convinced how can mass change with velocity but that is a whole another question.

Secondly, I found another paragraph on the same FAQ that states-

In one month of constant sunlight, LightSail's speed would increase by 549 kilometers per hour, roughly the speed of a jet airliner at cruising speed. In 16 months of constant sunlight, LightSail's speed would increase by 8,556 kilometers per hour, fast enough to escape the moon's gravity well.

The speeds mentioned here are far less than speed of light and thus the universal equation of momentum $$E^2=(pc)^2 + (mc^2)^2$$ should not be valid here.

So my question is "How is the solar sail then able to get momentum from light particicles?"

• I'm sure someone else will jump on with the equation for momentum for massless particles. But just in general, when you transfer momentum, the next object can be anywhere from slower to faster. And by non relativistic mass they just mean a mass moving at velocities less than like 0.1 times the speed of light. – user234190 Jul 25 '19 at 4:53
• So, you want to say that "$E^2=(pc)^2 + (mc^2)^2$" this equation can be used on any object irrespective of their speed? – Aditya Jain Jul 25 '19 at 5:00
• That equation is valid for all speeds. – G. Smith Jul 25 '19 at 5:10
• But, we have another equation $E^2=(mc)^2$. So, substituting $E^2$ we get $pc=0$ i.e. $p=0$ i.e $m=0$ ???? – Aditya Jain Jul 25 '19 at 5:15
• Yes the original equation is always valid, I just mean when mass is 0 then the momentum is given by energy, which is dependent on wavelength. Momentum is never really 0. – user234190 Jul 25 '19 at 5:20

Photons are elementary particles in the SM, massless, and they travel at speed c in vacuum, when measured locally.

Photons have energy and momentum.

The total equation for energy looks like this:

$$E^2 = m^2c^4 + p^2c^2$$

Now for massless particles, like photons:

$$E = pc$$,

Photons, though massless, they do have momentum, and can excerpt pressure on the surface they interact with.

When a photon interacts with an atom, three things can happen:

1. elastic scattering, the photon keeps its energy and changes angle

2. inelastic scattering, the photon keeps part of its energy and changes angle

3. absorption, the photon gives all its energy to the atom

Now with the Solar Sail, all three happen. It is the ratio between them, that is different for different materials.

For the Solar Sail, most of the photon get 1. elastically scattered, some get inelastically scattered and some absorbed.

Now since most of the photons get elastically scattered (like a mirror), the photons will give momentum to the sail itself, moving it relatively in space, thus giving it thrust.

Yes. Actually photons exert pressure on any surfaces exposed to them. For example, photons emitted by the Sun exert pressure of 9.08μN/m2 on the Earth.

Radiation pressure is the 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 which is absorbed, reflected, or otherwise emitted (e.g. black-body radiation) by matter on any scale (from macroscopic objects to dust particles to gas molecules). The reason they use mirrors, is because these objects are the ones with the highest ratio of elastic scattering.

It is basically because of conservation of momentum.

There are basically two types of reflection:

1. specular (like mirrors)

2. diffuse (like walls)

In the case of specular reflection, the photons keep their relative energy, phase, angle (this is the only way to build a mirror image).

Specular reflection is basically elastic scattering. In the case of elastic scattering, the sail will have the highest momentum transfer.

• I am tired of reading this: "When a photon interacts with an atom, three things can happen: elastic scattering, the photon keeps its energy and changes angle inelastic scattering, the photon keeps part of its energy and changes angle absorption, the photon gives all its energy to the atom " – my2cts Jul 25 '19 at 14:00
• @my2cts correct, will link it other times. Though, this time, the explanation is really needed here, since he is asking for the specific reason, photons transfer their momentum to the sail, and why it is made of mirrors. I am giving a really detailed answer to the question. – Árpád Szendrei Jul 25 '19 at 14:10
• In 2. you say the photon keeps part of its energy (momentum). That means, when part is transferred onto the sail, the photon now must have lower energy thus lowered its wavelength? – try-catch-finally Oct 3 '19 at 19:23
• @try-catch-finally 2. inelastic scattering is only a very small part of the photons. These usually heat up the material of the sail. Most of the photons get elastically scattered. physics.stackexchange.com/questions/503445/… – Árpád Szendrei Oct 3 '19 at 20:20

A photon has momentum $$\vec P=\hbar \vec k$$. The foil that the sail is made of is highly reflective, so each photon imparts $$\Delta \vec P=2 \hbar \vec k$$ momentum to the sail.