Solar Sail calculations
Listening to Michael Longcor’s song “Windward Passage” started me wondering how workable a solar sail would be as a way of moving things around in space and how much of the answer I could calculate on the basis of simple physics. I like figuring things out for myself, both because it is fun and because I do not entirely trust someone else’s conclusions if I have no way of checking them. It is an attitude reinforced by my experience in the climate debate, where I have found that expert conclusions quite often should not be trusted.1 One of my climate posts argued the superiority of an approximate analysis that the reader could check over more expert and precise analyses that he couldn’t — and offered one.
Centrifugal acceleration, acceleration away from the sun driven by the pressure of light from the sun, is proportional to the ratio between the area of the sail and the mass of the ship; without additional assumptions about the material the sail is made of I have no way of bounding it. Centripedal acceleration, acceleration towards the sun, cannot use light pressure since it pushes in the wrong direction; the only force available is the sun’s gravity. That makes it possible to calculate how long it would take a spaceship starting at rest to cover a given distance towards the sun, such as the distance from the orbit of Mars to that of Earth.
The Centripedal Calculation
At the Earth’s orbit solar gravity is about .006 m/sec2, so that is the maximum acceleration that a spaceship depending on gravity can get in that direction. Multiplying by the number of seconds in a day gives us about 500 m/sec/day. If we assume a ship accelerating towards the sun for a hundred days and ignore the problem of stopping when it gets where it is going — that is centrifugal acceleration which I don’t yet have a limit on — it will reach a top speed of 50,000 m/sec, have an average speed of 25,000 m/sec, and cover a distance of 32.85 million km. Since distance covered at a constant acceleration is proportional to the square of time spent, the time to cover a greater distance at the same acceleration is 100 days times the square root of the ratio of distances.
Mars is about 78 million km farther from the sun than Earth. At a constant acceleration of .006 m/sec2, falling from the orbit of Mars to that of Earth would take about 156 days. Solar gravity is weaker the farther out you go, only .43 as high at the orbit of Mars as at the orbit of Earth; at the corresponding acceleration the distance would take about 238 days so the actual time will be between those values; this is an approximate calculation so I am not bothering to find the exact time. I conclude that using a vessel powered only by the solar wind and solar gravity to transport things between Earth and Mars is, on that basis, not quite impossible, not even between Earth and the Asteroid belt, although the return trip for that might take a year or more. Further out it becomes increasingly unworkable.
The Centrifugal Calculation
What about the other direction? The pressure of solar light at the Earth’s orbit is 9.08 μN/m² (micNewtons per square meter) =9.08x10-6 Newtons/m2.2
Start with what it takes to keep the ship from falling into the sun — enough force to accelerate the ship at .006 m/sec2, just balancing the acceleration due to solar gravity. That requires a force of .006 Newtons/kg which requires .006/9.08x10-6 square meters of sail/kg =6000/9.08 = 660m2/kg.
Assume the sail is made of aluminum and all the weight is the sail. The density of aluminum is about 2,710 kg/m3. The thinnest aluminum foil is .oo6mm=6x10-6 meters thick. So a kg of aluminum has a volume of 1/2,710 = 3.69x10-5 m3 and, converted into thin foil, an area of 36.9/6 = 6.1 m2. It follows that in order to build a solar sail that, starting at Earth’s orbit, will not fall into the sun, you need a material with an area to weight ratio at least a hundred times that of the thinnest aluminum foil.
Actually, it doesn’t matter how far from the sun you start. Both light pressure and solar gravity are proportional to the inverse square of distance from the sun, so if light pressure is less than solar gravity somewhere it is less everywhere.
According to Wikipedia, Eric Drexler proposed panels of thin aluminum film 30 to 100 nanometres thick. 30 nanometers is 3x10-8m, so could in theory do it, although it has never been done. Actual solar sails are made of aluminized polymers, 2x10-6m, a little thinner and lighter than the thinnest aluminum foil but not by enough.
I Was Being Stupid
Reading the Wikipedia article on solar sails I realized my mistake. What I calculated, I think correctly, was what it would take for a solar sailboat to hang still in space, the solar light pressure just balancing solar gravity. But there is no reason why it has to be standing still. Earth, after all, is not falling into the sun, the pull of solar gravity being balanced not by light pressure but centrifugal force.3 The same would be true of a solar sailboat moving at the same speed in the same orbit, so it could use solar pressure to accelerate centrifugally. Alternatively it could orient its sail at an angle to the sun’s light, use light pressure to increase its orbital speed and use the resulting increase in centrifugal force to push itself outward. At this point the calculations of optimal orbits and how fast they can get the ship away from the sun gets well beyond what I am inclined to try.
My calculations were answering the wrong question but the answer is still interesting. A solar sail of currently workable materials produces less acceleration away from the sun than solar gravity produces towards the sun. It follows that solar pressure alone is less effective than solar gravity alone, which suggests that the times I calculated for the centripedal problem are probably shorter than the corresponding times for the centrifugal problem, that it probably will take longer to get from Earth’s orbit to Mars’ orbit than the other direction. That would not be true for a solar sail using Drexler’s proposed film, however. And I say “suggests” because I have not worked out the implications of using the solar sail to increase velocity around the sun and the increased velocity to push the boat outward.
So far I have been considering a sail powered only by sunlight. Some variants of the solar sail idea use a laser located on or near Earth to provide additional acceleration to an interstellar spaceship.
That still has the problem of only providing a push in one direction, away from Earth. To solve that you need lasers at both ends of your route.
My favorite solution was offered by Robert Forward.44 His ship has two solar sails, a circle inside a larger circle. When you approach the target system you cut loose the outer ring and angle everything so that the laser beam misses the sail still attached to the ship, hits the other, bounces off it, and is reflected back into the first sail. The detached sail accelerates into space, driven by the beam, while the spaceship is slowed by the reflected beam hitting the sail still attached.
Before the second ship arrives, the first builds a second laser cannon to provide brakes. Nobody could expect a maneuver that complicated to work twice. Once you have a laser at each end of things, traveling back and forth gets a lot easier.
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A Newton is the amount of force required to accelerate a mass of one kilogram at a rate of one meter per second squared.
“Centrifugal force” is a misleading term but the easiest way to think of the problem. Strictly speaking nothing is balancing the attraction of the sun on the Earth, with the result that Earth is accelerating towards the Sun but not moving towards it; the acceleration is at right angles to the Earth’s velocity, just what is needed to bend the Earth into a roughly circular orbit. One might say that the Earth is falling around the sun. The same would be true of a solar sailboat in the same orbit.
I am simplifying his solution by leaving out the early stage, when the ship is using sunlight and the sun’s gravity to pick up speed. And I am omitting an alternative solution to the braking problem also proposed by Forward in which the ship aims to slightly miss the target star and uses the magnetic field of the star to turn itself in a 180° curve, ending up approaching the star from the far side, which allows it to be slowed down by the light of the laser cannon.
If you want to really rabbit hole on this, look into plasma sails.
TL;DR: the idea is to use (massless) magnetic fields in lieu of material sails, thereby circumventing the thinness requirements.
It's a pretty interesting experience looking behind the curtains. I was really interested in fusion for a while and surprisingly there are some insufficiently hyped possibilities. Spin polarization of the fuel could increase reactivity to the point of increasing the reaction power by 70%. It's predicted that a high-powered tokamak plasma will have a more stable boundary effect for the "scrape off layer", which may double the output. If the effects work in harmony you have a three times more powerful reactor. Overall you can find more reasons to doubt, but interestingly these are often as hyped up as the positives. People prefer a binary prediction.
Climate change has been a very interesting one too. If you get into the weeds of the existing studies you may not notice that they are located in a brightly lit section of the possibility space which is almost exclusively focused on costs. It appears to me that the more complicated models even obscure to the people using them how a number of small biases in one direction can add up to a lot.