# Maximum velocity achieved by solar sail

What is the maximum (theoretical) velocity that can be achieved by solar-sail technology? Can we get close to the speed of light with that?

• In the solar system ? Using only the sun as source of light ? – Antzi Feb 28 '18 at 8:27
• @Antzi a good answer would derive a simple equation, then allow one to plug in specific assumptions afterward. That would be a better way to go than preconstraining the question. – uhoh Feb 28 '18 at 9:42
• Hi @Tom11, is there anything else I can add to my answer? – uhoh Nov 11 '18 at 12:19

tl;dr:

Can we get close to the speed of light with that?

No, at least not very easily. The terminal velocity $v_{\infty}$ is only about 0.2% the speed of light if you start at 1 AU using a 10 nanometer thick sail, and scales only as the inverse square root of the distance to the Sun where you start accelerating (as well as sail thickness), so you'd melt without getting much benefit from starting really close to it. Passing by a second stars helps little because you'd be going past it so quickly that you wouldn't get much of a second kick.

The relationship between the momentum and energy of a photon is $p = E/c$, force is $dp/dt$, and acceleration is $F/m$. For a reflecting sail there is up to a factor of 2 for perfect reflection of normal incidence, and the total incident power (energy per unit time) would be the intensity $I$ of sunlight (energy per unit area per unit time) times the area $A$. So

$$a=\frac{F}{m} = \frac{2AI}{mc}.$$

To give this a test run, a one square kilometer sail made from a futuristic reflecting material with a thickness of 10 nanometers would weigh of order 10 kg. At 1 AU from the Sun, $I$ is about 1361 W/m^2 (the Solar Constant), giving an acceleration of 0.9 m/s^2 which is surprisingly big until you remember that you're redirecting a gigawatt of sunlight in the process.

If you balk at said futuristic 10nm thick mirror, a sub-wavelength spaced wire grid is discussed in the 1999 NASA Institute for Advanced Concepts report Ultra-Thin Solar Sails for Interstellar Travel: Phase I Final Report where values from 0.5 to 4 m/s^2 are plotted in Figure 7.

To get the asymptotic terminal velocity, we can make that work for all distances by normalizing to 1 AU :

$$a(r) = \frac{2AI}{mc} \frac{1AU}{r^2},$$

and then integrate it to infinity but unfortunately I can't remember how to solve differential equations any more, so I will cheat and use the result at the top of page 14 of that report:

$$v_{\infty} = 548,000 (m/s) \sqrt{a_{1AU} \frac{1AU}{r_{start}}},$$

which is only about 0.2% the speed of light starting at 1 AU, and scales only as the inverse square root of the distance to the Sun where you start accelerating.

Table 1 shows some the missions that can be performed for an ultra-light solar sail. An ultra-thin sheet of aluminum only a few nanometers thick can achieve ac of ~0.3 m/s2, could reach Pluto in ~100 days, and reach the Oort cloud at ~10,000 AU within a century. In contrast, current propulsive methods would take over a decade to reach Pluto and are totally impractical to reach interstellar space. A perforated light sail made of aluminum could reach the Oort cloud in a half century if we can achieve ac >0.5 m/s2, and could do so in 12 years if ac ~5 m/s2 can be achieved. In the far term, a sail made of doped carbon nanotubes could probably approach the sun within 4 solar radii, and if it had an ac of 10 m/s2 , could reach α Centauri in a century. A sail made of doped carbon nanostructures could reach our nearest star in a few decades if ac >100 m/s2 as suggested by extensions of microwave theory.

It can thus be seen that the ultrathin solar sail holds the potential to revolutionize the prospects for interstellar travel. Uniquely, with this technology, such missions could not only become feasible, but potentially cheap, since all the motive power is provided in the form of crude sunlight – no giant lasers or other power systems are required.

• Let's stress out that (if i understand correctly) the results does NOT include any payload – Antzi Mar 1 '18 at 1:07
• @Antzi mass is of course total mass. It couldn't possibly be anything else. – uhoh Mar 1 '18 at 1:33