Dimensional analysis on Luminosity

[BERGXK]

I am having trouble with one of the problems on my hw. The question is:

Because light carries momentum, it creates pressure when it shines on something. This has led people to propse using "solar sails" instead of conventional rocket engines on interplanetary or interstellar space craft.

A. Use dimensional analysis to estimate the light pressure at a ditance d from a star with luminosity L

B.What is the force on a solar sail with the area of 1km^2 that is the same distance from the sun as earth?

C.Suppose the sail is pulling a ton spacecraft . How long would it take to reach jupiter's orbit, which is 5.2 au from the sun? For this estimate you may assume the acceleration remains constant, even though we found above that the pressure varies with teh distance from the star.

The only part of this problem that is difficult for me is the relationship between luminosity pressure and distance. I feel once i can find the relationship between them I can solve the rest of the problem easily. Luminosity is J/sec which is M L^2 T^-3 ~ so we have to find Pressure and distance units that is equal to this right? I am having a problem figuring out how to do this.

Ive been looking at the examples in my notes. I understand how to we find the relationship between p and kT by dividing their units and seeing L^-3 which is n. We then get the relationship p~nKT but I don't understand how we can intuitively assume some other equations likeP~h^i c^j (kT)^l for a relativistic gas then solve and get P~(kT)^4/(hc)^3 . I understand the algebra but i just don't get how we can intuitively assume h c and (kT) will be factors in the equation for P and not anything else. I think if i can understand this example in the notes i will also get the question for the hw.

Sorry for the long question and thanks in advance

 

[George Jones]

Luminosity is J/sec which is M L^2 T^-3

[pressure] = M L^-1 T^-2 = (M L^2 T^-3)/(L^2 L/T)

pressure = luminosity/(area*speed)

 

[BERGXK]

Thanks for the repy, I still have a question though Does A=L^2? The energy released by the star also relates to the distance could the A value be distance^2?

 

[tony873004]

[pressure] = M L^-1 T^-2 = (M L^2 T^-3)/(L^2 L/T)

pressure = luminosity/(area*speed)

How can you get a pressure without distance from the light source, as the force diminishes with distance?

 

[George Jones]

Thanks for the repy, I still have a question though Does A=L^2? The energy released by the star also relates to the distance could the A value be distance^2?

I was trying to steer you towards something like this. A = b*x^2, where b is a constant. What is b?

I'm not sure what your prof/text intends. If A is taken to be 1, then the result is an order of magnitude off from what I think is the correct result.

Also there is a factor of 2 difference between the cases of total absorption and total reflection.

 

[BERGXK]

Im very confused on this subject. In the class example we did we found P relation to KT by dividing their units and receiving L^-3 as the unit of difference. Therefore reaching the conclusion P~nKT.

For the second problem we did a relativistic Degenerate gas. We started this problem knowing what pressure in the degenerate relativistic gas depended on Paulis exclusion principle and the pressure from this gas was electron degeneracy pressure. The next part is the part i don't understand. We assumed that h(plancks constant) M(electron mass) and C(speed of light) would be related to pressure. How do we know this? Since he gas is degenerate I know KT is out of the picture but why is h(plancks constant) M and C in this relationship? MC is a momentum isn't Plancks constant the factor multiplied by frequency to get an energy? why is it in this equation? The rest of the problem was just algebra. We did p~h^i c^j n^k and solved for ijk in a systems of equations and received p~hcn^(4/3)

I feel if i can understand this example problem i will get this homework problem more. Can you explain the reasoning to be why hcn are used in the equation of a degenerate relativistic gas?

I feel I can't use the simple method of just dividing the units in this homework problem and need something like the 2nd example and solve a system of equations. since i am dealing with the speed of light shouldn't i use h somewhere?

I had the answer P~L/(A*V)
But this relationship does not answer the question how distance and luminosity relate since A is not neccesarly D^2.

 

[George Jones]

The next part is the part i don't understand. We assumed that h(plancks constant) M(electron mass) and C(speed of light) would be related to pressure. How do we know this? Since he gas is degenerate I know KT is out of the picture but why is h(plancks constant) M and C in this relationship?

I think dimensional analysis is important and useful, but I don't buy completely into the type of stuff you'er doing in this course. I'll have a go anyway. Take what I write with a grain of salt.

This is a quantum situation, so Planck's constant comes into play. It's an electron gas, so the mass of an electron is a characteristic mass. It's relativistic, and hence the speed of light.

I feel I can't use the simple method of just dividing the units in this homework problem and need something like the 2nd example and solve a system of equations. since i am dealing with the speed of light shouldn't i use h somewhere?

A simple, but less hand-wavy, analysis shows that h drops out.

I had the answer P~L/(A*V) But this relationship does not answer the question how distance and luminosity relate since A is not neccesarly D^2.

You have an area and a speed. What characteristic area and speed are relevant for this situation?

At distance D, the light from the star is distributed uniformly across the surface of an imaginary sphere that is centred on the star, and that has surface area 4*pi*D^2, which shows that pressure diminishes with distance, just as tony said. Light moves at the speed of light.

The simple, less handy-wavy analysis produces the same result in a much clearer way with only a little more effort, so I don't see the benefit in using the dimensional analysis argument.

Dimensional analysis is useful for making sure that results make sense.

The prof I had for a real analysis course that I took would devise "interesting" questions for the test, i.e., they often weren't based on assignment questions or examples from the lectures. I was pondering one such question on the theory of integration and getting nowhere, so I decided to see if giving plausible physical dimensions to each of the quantities in the question would help. This seemed to indicate that the question was impossible, so I told the prof that I thought there was a mistake in the question. After working on the question for a bit, he agreed, and gave the class a corrected version.

He asked me "How did you find this out?" When I replied "The units didn't match up," he just gave me a puzzled look. He was a pure mathematician, with no background in any of the physical sciences.