Help with calculating laser power for a lightsail starship

[Junghalli]

Related to this article posted in this thread. I was intrigued and did some Googling on the subject, and found some articles indicating that substantially better performance could be achieved with dielectric sail instead of an aluminum sail. I tried to do a calculation for a silicon carbide sail but I'm not sure it's right, the power estimate seems a little low to me. I wonder if anybody might care to double-check my calculations.

The original article describes a ship with an 80,000 ton aluminum lightsail, propelled to .5 c at 3 m/s^2 by a laser with a power of 4.3 X 10^16 watts.

According to this article silicon carbide is the best sail material, with a figure of merit of 60 km^2/kg. The figure of merit is the acceleration in G with a power level of 1.5 kW/m^2. So, for instance, IIRC a 1 m^2 sail would require a power level of 1.5 kW for an acceleration of 60 G.

I would assume to correct for payload one divides acceleration by the fraction of the ship's mass that is payload, for instance if 1/2 the ship's mass is payload it's 30 G instead of 60 G. To match the performance of Forward's aluminum sail ship we need 3000 tons of payload and an acceleration of 3 m/s^2.

Dividing 80,000 tons by the surface area of Forward's aluminum sail gives a mass of 102 kg/km^2. I'm using this figure for convenience.

For the sake of argument we assume a sail of 1 km^2 massing 102 kg. It has 1/29,703 the mass of the payload so the figure of merit is divided by that, giving us an acceleration of .02 m/s^2. I haven't been able to get a figure for 3 m/s^2 yet, but I calculated that 2 m/s^2 would require a sail with a surface area of 100 km^2. At a power level of 1.5 kW/m^2 this would translate into a 150 GW laser.

That just seems a little too good to be true, given that the original proposal called for a laser several hundred thousand times as powerful. Could somebody take a look at my calculations and see if I messed up.

[Kuroneko]

... silicon carbide is the best sail material, with a figure of merit of 60 km^2/kg. The figure of merit is the acceleration in G with a power level of 1.5 kW/m^2. So, for instance, IIRC a 1 m^2 sail would require a power level of 1.5 kW for an acceleration of 60 G.

For light, p = h/λ = E/c, so a laser of power P can provide a theoretical maximum force of F = 2P/c, which occurs if there is complete reflection. So, for P = 1.5e3W, we have F = 1.0e-5N. If this is incident on A = 1.0m², then at σ = 60km²/kg, the acceleration should be Fσ/A = 61 gees. In other words, the sail operates at near-optimum reflective efficiency.

The equation F = 2P/c, where P is the total laser power, is pretty much all you need to know for a back-of-the-envelope calculation. You can incorporate an infefficiency if you wish by simply introducing another factor, but if your article is accurate, this is close enough to unity; all you would need to do is to make sure your sail is large enough not to be destroyed by the laser.

For the sake of argument we assume a sail of 1 km^2 massing 102 kg. It has 1/29,703 the mass of the payload so the figure of merit is divided by that, giving us an acceleration of .02 m/s^2.

Alright, let's try this: 2[(1.5kW/m²)(1.0km²)/c]/[29704(102kg)] = 3.3e-6 m/s².

[Academia Nut]

This is probably what you were thinking of

Or, at least, that is an article I know of that discussed just such an idea.