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### Solar Sails

Posted: Thu May 03, 2007 2:54 am UTC
[edit2]In one of the books I came across while researching, it discussed this and came up with the result that's below. I'm leaving this up in case anyone else is interested- it should be correct now.

So, I'm writing a paper on solar sails, and as part of it I'm including the maximum mass density (by area, not volume, since its depth is unimportant) that a material can have to be a solar sail.

However, I've come up with a number that's really, really not correct, and I can't tell where my math (or science) is wrong. [edit- this is the revised version]

Fr=2LA/(c 4πr^2) (Force due to radiation pressure is 2 times the intensity of the source times the area of the object divided by the speed of light (this should be required to move from intensity to force density) divided by 4 Pi r2 (to reduce total intensity to the part shining on you))
Fg=GMm/r^2 (Force due to gravity is G times the mass of the sun M and mass of the sail m, divided by r2)

Setting these equal (to determine maximum density that'll give us equilibrium), we should get:
GMm/r^2=2LA/(c 4πr^2)
m/A=L/(GMc 2π)
Plugging in constants (obtained from wikipedia):
m/A=(3.827*10^26)/(2*π*6.674*10^-11*1.988*10^30*3*10^8)
m/A=1.53*10^-3

Cosmologicon is probably right.

Posted: Thu May 03, 2007 4:05 am UTC
Your first equation should have c in the denominator, not c^2.

As you know, for photons:

p = E / c

Differentiate both sides with respect to time to get:

F = L / c

Where L is the sun's luminosity. I think this is what you're calling the intesity, S, but both that word and that letter are often used for other things, so I suggest luminosity (or power radiated) L.

Posted: Thu May 03, 2007 4:27 am UTC
L is power density, which is W/m2, right?

The antiderivative of L will then be J/m2, which should be energy density. It looks like that's right, and so I think my mistake was using momentum density instead of force density.

[edit- Yay calculator error!]However, that changes the value to 6*10^-3 kg/m2, which I think is off by about a factor of 2. It's probably a flaw in one of the constants, and so I'll have to check those again.

Posted: Thu May 03, 2007 4:33 am UTC
No, L is power. If you want power per unit area, that's called the flux F. I see now that this is what you meant by S. If you do it like that, which makes sense, then you still need only a single factor of c below, but you also need to get rid of the 4pi r^2:

F = 2SA / c

because S = L / (4pi r^2). The value of the flux varies with distance from the sun, so it's not somethnig you'll be able to look up without a fixed distance. (The flux at 1AU is called the solar constant; that's easy to find, but I don't think it will do you as much good as L.)

Posted: Thu May 03, 2007 4:35 am UTC
Right.

(I hate doing dimensional analysis late at night)

Posted: Mon May 28, 2007 4:04 am UTC
I might not be following you at all.
But with a solar sail,
if you don't have enough power to overcome gravity,
you start out in orbit around the sun, and tack, using the solar wind pressure* at an angle to speed up, so that you are orbiting at further and further distances, so there's no weight problem. A heavier sail slows you down, but doesn't stop you.

* i wasn't sure i had the right terms here.

Posted: Mon May 28, 2007 5:38 am UTC
you start out in orbit around the sun, and tack, using the solar wind pressure at an angle to speed up, so that you are orbiting at further and further distances, so there's no weight problem. A heavier sail slows you down, but doesn't stop you.
Errr... that seems nearly totally wrong to me.

Posted: Mon May 28, 2007 5:52 am UTC
Why? It obeys conservation of energy and momentum. Just because you don't get enough force to thrust directly out of the gravity well doesn't mean you can't do it gradually; it's like a ramp. It's much easier to push a shopping cart up a ramp than it is to hoist it off the ground, even though the energy output is the same.

Posted: Mon May 28, 2007 5:59 am UTC
Why? It obeys conservation of energy and momentum. Just because you don't get enough force to thrust directly out of the gravity well doesn't mean you can't do it gradually; it's like a ramp. It's much easier to push a shopping cart up a ramp than it is to hoist it off the ground, even though the energy output is the same.
The phrase "solar wind pressure" strikes me as rather questionable; I was under the impression that the solar wind didn't convey much useful momentum, compared to radiation pressure.

I'm not questioning revolving around the sun to leave- that would work (although at 2 AM I don't feel particularly up to working it out). The primary application of solar sails that I was looking into were things that would stay "in place", and thus having them revolve around the sun would be counterproductive.

Posted: Mon May 28, 2007 7:34 am UTC
Ah. I didn't even notice "solar wind pressure" - I just thought "radiation pressure" instead. But yeah, using solar wind to ramp up an orbit rather than going straight out would be fine - I just misinterpreted your goal. Wouldn't you have problems with things like Jupiter if you were just trying to use radiation pressure to stay stationary?

Posted: Mon May 28, 2007 3:10 pm UTC
Vaniver wrote:The primary application of solar sails that I was looking into were things that would stay "in place", and thus having them revolve around the sun would be counterproductive.

"In Place" with respect to what? The Solar System is entirely composed of moving objects....
Ah, I see you're using the same r for your light source and your gravity well, so you must intend the Sun as your reference for both. Scratch my question. The equation you generated does solve the problem of "what's the maximum mass density for a sail, that will allow it to reverse its orbit without moving any closer to the Sun?" But I don't think that was the problem you stated...
As for the maximum mass density that a material can have to "be a solar sail," I'm afraid there isn't one. Every body with a non-zero albedo reacts to radiation pressure, and once it has achieved a stable orbit, any spacecraft that can change its response to radiation pressure can, eventually, sail anywhere in the Solar System. It's just a question of efficiency, as the time required for higher mass densities to acquire a lot of delta-V can quickly rise to millennia.

Posted: Mon May 28, 2007 5:34 pm UTC
As for the maximum mass density that a material can have to "be a solar sail," I'm afraid there isn't one. Every body with a non-zero albedo reacts to radiation pressure, and once it has achieved a stable orbit, any spacecraft that can change its response to radiation pressure can, eventually, sail anywhere in the Solar System. It's just a question of efficiency, as the time required for higher mass densities to acquire a lot of delta-V can quickly rise to millennia.

Yeah. But there is a maximum density if you want to make a statite (a solar sail that is stationary in the sun's frame, and stays that way by "sitting" on radiation pressure instead of orbiting). Such a sail would have to be MUCH finer that many current solar sail proposals, but may just be within theoretical limits of materials science. If it is possible, that would be a quick and easy way to build a Dyson sphere.

Interestingly, the maximum density for a statite is constant with distance from the sun. That's because luminosity and gravity both obey inverse-square laws, so the balance between the two is the same at any height (or radius).

The phrase "solar wind pressure" strikes me as rather questionable; I was under the impression that the solar wind didn't convey much useful momentum, compared to radiation pressure.

Yup, I've heard that too.