## Laser Sails. A relatively tough astrophysics problem...

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### Laser Sails. A relatively tough astrophysics problem...

My astrophysics prof gave example for homework.

Consider a spacecraft of mass m whose engine is a perfectly absorbing laser sail that is initially at rest in space. No gravity is being exerted on it. We aim a laser at the sail and cause it to accelerate into deep space.

Derive a formula for its kinetic energy, as a function of its mass and the total amount of energy Ei that it absorbs from the laser beam during some time interval t. Assume that the spacecrafts velocity remains NON-relativistic.

I.E. You're firing radiation and using the pressure of light to accelerate it.

My work:

You have the radiation pressure=[math]F_rad=((<S>A)/c)cos(theta)[/math]

With this you have a force. We know F=ma, which i'm guessing can be integrated over a time interval to give you the speed (i'm not great at integrating physics).

This would give us the V in [math]KE=1/2mv^2[/math].

I havnt used the Ei and therefore I can assume that my answer is wrong. Can anyone figure this one out?

Consider a spacecraft of mass m whose engine is a perfectly absorbing laser sail that is initially at rest in space. No gravity is being exerted on it. We aim a laser at the sail and cause it to accelerate into deep space.

Derive a formula for its kinetic energy, as a function of its mass and the total amount of energy Ei that it absorbs from the laser beam during some time interval t. Assume that the spacecrafts velocity remains NON-relativistic.

I.E. You're firing radiation and using the pressure of light to accelerate it.

My work:

You have the radiation pressure=[math]F_rad=((<S>A)/c)cos(theta)[/math]

With this you have a force. We know F=ma, which i'm guessing can be integrated over a time interval to give you the speed (i'm not great at integrating physics).

This would give us the V in [math]KE=1/2mv^2[/math].

I havnt used the Ei and therefore I can assume that my answer is wrong. Can anyone figure this one out?

### Re: Laser Sails. A relatively tough astrophysics problem...

In your formula for F_rad (which I assume you treat as given?) what are the inputs you need and where do you get them from?

Better yet would be to think kinematically, all the radiation gets absorbed. Can you use a coservation law to your benefit?

Better yet would be to think kinematically, all the radiation gets absorbed. Can you use a coservation law to your benefit?

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### Re: Laser Sails. A relatively tough astrophysics problem...

Maybe a note that light has momentum, and the spacecraft is absorbing all of it. So a certain amount of photons per second are giving their momentum to the ship. Since the v of the ship is not relativistic, I suppose it would be reasonable to say that a certain amount of momentum is contributed per second and that contribution would stay consistent with time.

### Re: Laser Sails. A relatively tough astrophysics problem...

This seems to be the case.

### Re: Laser Sails. A relatively tough astrophysics problem...

You have a laser with power [imath]P[/imath] and frequency [imath]f[/imath]. The momentum of a photon is [imath]p=\frac{hf}{c}[/imath], and the energy of a photon is [imath]E=hf[/imath]. Therefore the number of photons striking the sail per second (ignoring doppler shifts and relativistic effects) is [imath]n=\frac{P}{E}=\frac{P}{hf}[/imath]. Thus, the momentum transferred to the ship in time t is [imath]p_{ship}=npt=\frac{Pt}{c}[/imath]. The momentum of the ship is (neglecting relativity) is then [imath]p_{ship}=m_{ship}v_{ship}[/imath] and so, assuming it started from rest, the speed of the ship is given by [math]v_{ship}=\frac{Pt}{m_{ship}c}[/math].

I think. Of course this should be derivable using radiation pressure as well, unless I got it wrong. Hopefully it helps and is right, I really should not be posting right now.

I think. Of course this should be derivable using radiation pressure as well, unless I got it wrong. Hopefully it helps and is right, I really should not be posting right now.

### Re: Laser Sails. A relatively tough astrophysics problem...

That works out great. But I guess i'm not entirely familiar with some of your terms. What does \frac mean?

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### Re: Laser Sails. A relatively tough astrophysics problem...

It's TeX for a fraction. You need to change to prosilver or prosilver_left to see it properly, as subsilver isn't supported any more on the forums.

### Re: Laser Sails. A relatively tough astrophysics problem...

The only energy input into the system is Ei. If the sail is perfectly absorbing that means none of this incident energy is being reflected or transmitted out of the system. Wouldn't this imply all the energy was being converted into kinetic energy (since we have no information on anything else that the spacecraft would use the energy for, internally)?

### Re: Laser Sails. A relatively tough astrophysics problem...

Chen wrote:The only energy input into the system is Ei. If the sail is perfectly absorbing that means none of this incident energy is being reflected or transmitted out of the system. Wouldn't this imply all the energy was being converted into kinetic energy (since we have no information on anything else that the spacecraft would use the energy for, internally)?

No. If it all went to kinetic energy it would violate conservation of momentum.

### Re: Laser Sails. A relatively tough astrophysics problem...

Laser sails are cool but extremely ineffective.

Here is a follow-up question he offered:

The efficiency e of the laser sail is given by the ratio e=KE/Ei. What is the cumulative efficiency of the sail, after it has absorbed an amount of energy equal to 1/100th of the total rest mass energy of the spacecraft plus sail?

Here is a follow-up question he offered:

The efficiency e of the laser sail is given by the ratio e=KE/Ei. What is the cumulative efficiency of the sail, after it has absorbed an amount of energy equal to 1/100th of the total rest mass energy of the spacecraft plus sail?

### Re: Laser Sails. A relatively tough astrophysics problem...

At high speeds the doppler effect would start to matter wouldn't it? But maybe that's not supposed to be considered.

### Re: Laser Sails. A relatively tough astrophysics problem...

When you start getting appreciable fractions of the speed of light relative to the laser source, relativistic effects would come into play and the laser light would begin to redshift. Luckly, we are to assume v << c, so the following relativistic Doppler effect equation is ok to ignore because:

Since v << c, the fraction " v /c " is basically zero, so the fraction collapses to 1 /1 = 1. IE, its ok to assume the wavelength the ship is seeing is basically the same frequency the whole time.

Since v << c, the fraction " v /c " is basically zero, so the fraction collapses to 1 /1 = 1. IE, its ok to assume the wavelength the ship is seeing is basically the same frequency the whole time.

### Re: Laser Sails. A relatively tough astrophysics problem...

Tass wrote:No. If it all went to kinetic energy it would violate conservation of momentum.

You could consider this an inelastic collision problem--firing photons of momentum m at an object, and they all "stick." Which would explain why energy doesn't appear conserved; there is heat produced which is not accounted for.

### Re: Laser Sails. A relatively tough astrophysics problem...

Comic JK wrote:Tass wrote:No. If it all went to kinetic energy it would violate conservation of momentum.

You could consider this an inelastic collision problem--firing photons of momentum m at an object, and they all "stick." Which would explain why energy doesn't appear conserved; there is heat produced which is not accounted for.

Yup

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