Question

It has been proposed to propel spacecraft through the Solar System with a large sail that is struck by photons from the Sun.

(a). Which would be more effective, a black sail that absorbs photons or a shiny sail that reflects photons back toward the Sun? Explain briefly

(b). Suppose thatphotons hit a shiny sail per second, perpendicular to the sail. Each photon has energy. What is the force on the sail? Explain briefly

Short answer

(a). A shiny sail that reflects back the photons towards the sun will be more effective.

(b). The force (F)on the sail is$2N\frac{E}{c}$.

Step-by-step solution

The concept of rate of change of momentum

The applied force equals the rate of change of momentum.

Step1:Explanation for why shining sail is more effective sail

(a)

A shining sail that reflects photons back toward the Sun would be more effective; the shiny sail absorbs the energy emitted by the Sun and reflects the photons back to the Sun's surface. In this instance, the change in photon momentum will be higher, and the net force will be bigger (the net force can be expressed as the rate of change of momentum).

Calculation for net force

(b)

If N photons per second strike a gleaming sail perpendicular to the sail, and each photon has energy E, the energy of each photon (a massless radiation, hence no rest mass) is:

$$\begin{gathered} E=pc \\ p=\frac{E}{c} \end{gathered}$$

On shiny sail photon is absorbed and re-emitted in the opposite direction, so the change in its momentum:

$\begin{array}{rcl}\Delta p& =& p_f-p_i\\ & =& -p-p\\ & =& -2p\\ & =& -2\frac{E}{c}\\ & & \end{array}$

The magnitude of change in momentum for N photons/second is

$\left|\Delta p_{N/s}\right|=2N\frac{E}{c}$

The net force for N photons/second is the rate of change of momentum:

$\begin{array}{rcl}F& =& \frac{\Delta p}{\Delta t}\\ & =& 2N\frac{E}{c}\\ & & \end{array}$

Thus, the force ()on the sail is$2N\frac{E}{c}$ .