# How can I calculate the relationship between power and acceleration for a beam driven photon sail?

I am writing a novel, and although I have a background in physics, I am unsure of the exact equations to use.

Specifically:

For a photon-sail ship, how powerful would a driving laser need to be in order for the ship to reach an acceleration of ~0.5g?

Assuming we are only talking within the solar system here, and assuming that the ship is heavy enough to carry passengers.

Would such a thing ever be feasible? I am trying to strike the right balance between fun and feasibility for some method of regular interplanetary transport (think the space equivalent of commercial jets).

• A can not answer the physics question, but regarding the novel: 1) Even LASERs are not perfect beams. Max acceleration might drop off after a while if the beam disperses to far. (though with a large sail as a target this might not be a problem). 2) At a continuous 0.5G you get up to high speeds pretty fast. Small particles will become dangerous. Depending on how hard core you write you might need to keep the speed down or have shields. 3) Interesting reading Jul 8, 2014 at 16:52
• If you haven't read Forward's Rocheworld you should do so. Just saying. And Niven and Pournelle's A Mote in God's Eye, but especially Forward and he was thinking about near future possibilities with real physics. Jul 8, 2014 at 21:41

Photons generate what we call Radiation Pressure. From wikipedia, http://en.wikipedia.org/wiki/Radiation_pressure, we get the equation: $$P_{absorb} = \frac{E_f} {c} cos\space\alpha\\ \text{and} \space P_{reflect}=\frac{2E_f} {c} cos^2\space\alpha$$ Where:$P_{absorb}$ is the Radiation Pressure on an absorptive surface (in Pascals).
$P_{reflect}$ is the Radiation Pressure on a reflective surface e.g. mirror (in Pascals).
$E_f$ is the energy flux/intensity (in $\frac{W} {m^2}$)
$c$ is the speec of light, and
$\alpha$ is the angle between the surface normal and the incident radiation.
Assuming that the ship has a reflective sail and the sail is orthogonal the the laser beam, we get $P=\frac{2E_f} {c}$. From $F=ma$ and $F=PA$, $$\frac{2E_fA}{c}=ma$$ I will stop here since there are parameters to be filled (namely, $m$ and $A$, which depends on your design of the ship.) After plugging in all the values, including $a=0.5g$, you shall obtain $E_f$, which is proportional to the power of the laser.